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On the optimal map in the $ 2 $-dimensional random matching problem
1.
Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy
2.
ETH, Rämistrasse 101, 8092 Zürich, Switzerland
3.
Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy
We show that, on a $ 2 $-dimensional compact manifold, the optimal transport map in the semi-discrete random matching problem is well-approximated in the $ L^2 $-norm by identity plus the gradient of the solution to the Poisson problem $ - {\Delta} f^{n, t} = \mu^{n, t}-1 $, where $ \mu^{n, t} $ is an appropriate regularization of the empirical measure associated to the random points. This shows that the ansatz of [
As part of our strategy, we prove a new stability result for the optimal transport map on a compact manifold.
Mathematics Subject Classification:Primary: 60D05; Secondary: 49J55, 58J35, 35F21. Citation:Luigi Ambrosio, Federico Glaudo, Dario Trevisan. On the optimal map in the $ 2 $-dimensional random matching problem. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 7291-7308. doi: 10.3934/dcds.2019304
References:
[1] [2] [3] [4] [5]
S. H. Benton,
The Hamilton-Jacobi Equation: A Global Approach, Academic Press, 1977.
Google Scholar
[6] [7] [8]
S. Caracciolo, C. Lucibello, G. Parisi and G. Sicuro, Scaling hypothesis for the Euclidean bipartite matching problem,
[9]
I. Chavel,
[10] [11]
L. C. Evans and R. F. Gariepy,
[12]
A. Fathi, Regularity of $C^1$ solutions of the Hamilton-Jacobi equation, in
[13]
N. Gigli,
On Hölder continuity-in-time of the optimal transport map towards measures along a curve,
[14] [15]
M. Goldman, M. Huesmann and F. Otto, A large-scale regularity theory for the Monge-Ampère equation with rough data and application to the optimal matching problem, preprint, arXiv: 1808.09250.Google Scholar
[16] [17] [18] [19]
M. Ledoux, On optimal matching of Gaussian samples ii, https://perso.math.univ-toulouse.fr/ledoux/files/2019/03/SudakovII.pdf.Google Scholar
[20]
M. Ledoux, A fluctuation result in dual Sobolev norm for the optimal matching problem, https://perso.math.univ-toulouse.fr/ledoux/files/2019/02/matchingclt.pdf.Google Scholar
[21] [22]
P.-L. Lions,
[23]
J. Lott and C. Villani, Hamilton–Jacobi semigroup on length spaces and applications,
[24] [25]
W. Rudin,
[26]
F. Santambrogio,
[27] [28] [29] [30] [31] [32]
N. G. Trillos and D. Slepčev,
On the rate of convergence of empirical measures in $\infty$-transportation distance,
[33]
show all references
References:
[1] [2] [3] [4] [5]
S. H. Benton,
The Hamilton-Jacobi Equation: A Global Approach, Academic Press, 1977.
Google Scholar
[6] [7] [8]
S. Caracciolo, C. Lucibello, G. Parisi and G. Sicuro, Scaling hypothesis for the Euclidean bipartite matching problem,
[9]
I. Chavel,
[10] [11]
L. C. Evans and R. F. Gariepy,
[12]
A. Fathi, Regularity of $C^1$ solutions of the Hamilton-Jacobi equation, in
[13]
N. Gigli,
On Hölder continuity-in-time of the optimal transport map towards measures along a curve,
[14] [15]
M. Goldman, M. Huesmann and F. Otto, A large-scale regularity theory for the Monge-Ampère equation with rough data and application to the optimal matching problem, preprint, arXiv: 1808.09250.Google Scholar
[16] [17] [18] [19]
M. Ledoux, On optimal matching of Gaussian samples ii, https://perso.math.univ-toulouse.fr/ledoux/files/2019/03/SudakovII.pdf.Google Scholar
[20]
M. Ledoux, A fluctuation result in dual Sobolev norm for the optimal matching problem, https://perso.math.univ-toulouse.fr/ledoux/files/2019/02/matchingclt.pdf.Google Scholar
[21] [22]
P.-L. Lions,
[23]
J. Lott and C. Villani, Hamilton–Jacobi semigroup on length spaces and applications,
[24] [25]
W. Rudin,
[26]
F. Santambrogio,
[27] [28] [29] [30] [31] [32]
N. G. Trillos and D. Slepčev,
On the rate of convergence of empirical measures in $\infty$-transportation distance,
[33]
[1] [2]
Brendan Pass.
Multi-marginal optimal transport and multi-agent matching problems: Uniqueness and structure of solutions.
[3]
J. M. Mazón, Julio D. Rossi, J. Toledo.
Optimal matching problems with costs given by Finsler distances.
[4] [5] [6] [7]
Nalini Anantharaman, Renato Iturriaga, Pablo Padilla, Héctor Sánchez-Morgado.
Physical solutions of the Hamilton-Jacobi equation.
[8] [9]
María Barbero-Liñán, Manuel de León, David Martín de Diego, Juan C. Marrero, Miguel C. Muñoz-Lecanda.
Kinematic reduction and the Hamilton-Jacobi equation.
[10]
Larry M. Bates, Francesco Fassò, Nicola Sansonetto.
The Hamilton-Jacobi equation,
integrability, and nonholonomic
systems.
[11] [12] [13]
Angel Angelov, Marcus Wagner.
Multimodal image registration by elastic matching of edge sketches via optimal control.
[14]
Yoshikazu Giga, Przemysław Górka, Piotr Rybka.
Nonlocal spatially inhomogeneous Hamilton-Jacobi equation
with unusual free boundary.
[15]
Laura Caravenna, Annalisa Cesaroni, Hung Vinh Tran.
Preface: Recent developments related to conservation laws and Hamilton-Jacobi equations.
[16] [17]
Fabio Camilli, Paola Loreti, Naoki Yamada.
Systems of convex Hamilton-Jacobi equations with implicit obstacles and the obstacle problem.
[18]
Alexander Quaas, Andrei Rodríguez.
Analysis of the attainment of boundary conditions for a nonlocal diffusive Hamilton-Jacobi equation.
[19]
Giuseppe Marmo, Giuseppe Morandi, Narasimhaiengar Mukunda.
The Hamilton-Jacobi theory and the analogy between classical and
quantum mechanics.
[20]
2018 Impact Factor: 1.143
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Total $\{k\}$-domination in special graphs
1.
University of Science and Technology of China (USTC), Hefei, China
2.
Facebook Seattle, 1101 Dexter Ave N, Seattle, WA 98109, USA
For a positive integer $k$ and a graph $G = (V,E)$, a function $f:V \to \{0,1,...,k\}$ is called a
total $\{k\}$- dominating function of $G$ if $\sum_{u∈ N_G(v)}f(u)≥ k$ for each $v∈ V$, where $N_G(v)$ is the neighborhood of $v$ in $G$. The total $\{k\}$- domination number of $G$, denoted by $\gamma _t^{\left\{ k \right\}}\left( G \right)$, is the minimum weight of a total $\{k\}$-dominating function $G$, where the weight of $f$ is $\sum_{v∈ V}f(v)$. In this paper, we determine the exact values of the total $\{k\}$-domination number for several commonly-encountered classes of graphs including cycles, paths, wheels, and pans. Mathematics Subject Classification:05C10. Citation:Haisheng Tan, Liuyan Liu, Hongyu Liang. Total $\{k\}$-domination in special graphs. Mathematical Foundations of Computing, 2018, 1 (3) : 255-263. doi: 10.3934/mfc.2018011
References:
[1] [2] [3]
B. Bresar, P. Dorbec, W. Goddard, B. Hartnell, M. Henning, S. Klavzar and D. F. Rall,
Vizing's conjecture: A survey and recent results,
[4]
G. Domke, S. Hedetniemi, R. Laskar and G. Fricke,
Relationships between integer and fractional parameters of graphs,
[5]
T. Haynes, S. H. ST and P. Slater,
[6]
T. Haynes, S. H. ST and P. Slater,
[7]
J. He and H. Liang, Complexity of total $\{k\}$-domination and related problems,
[8] [9]
C. Lee,
[10]
show all references
References:
[1] [2] [3]
B. Bresar, P. Dorbec, W. Goddard, B. Hartnell, M. Henning, S. Klavzar and D. F. Rall,
Vizing's conjecture: A survey and recent results,
[4]
G. Domke, S. Hedetniemi, R. Laskar and G. Fricke,
Relationships between integer and fractional parameters of graphs,
[5]
T. Haynes, S. H. ST and P. Slater,
[6]
T. Haynes, S. H. ST and P. Slater,
[7]
J. He and H. Liang, Complexity of total $\{k\}$-domination and related problems,
[8] [9]
C. Lee,
[10]
[1] [2]
M. D. König, Stefano Battiston, M. Napoletano, F. Schweitzer.
On algebraic graph theory and the dynamics of innovation networks.
[3] [4]
E. Muñoz Garcia, R. Pérez-Marco.
Diophantine conditions in small divisors and transcendental number theory.
[5] [6]
Navin Keswani.
Homotopy invariance of relative eta-invariants and $C^*$-algebra $K$-theory.
[7] [8]
A. C. Eberhard, J-P. Crouzeix.
Existence of closed graph, maximal, cyclic pseudo-monotone relations and revealed preference theory.
[9]
Nigel Higson and Gennadi Kasparov.
Operator K-theory for groups which act properly and isometrically on Hilbert space.
[10]
Harsh Pittie and Arun Ram.
A Pieri-Chevalley formula in the K-theory of aG/B-bundle.
[11] [12] [13]
Yunhai Xiao, Junfeng Yang, Xiaoming Yuan.
Alternating algorithms for total variation image reconstruction
from random projections.
[14] [15] [16] [17] [18] [19] [20]
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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 space and $Y$ a topological space. I am trying to show that if $f_n : X \to Y$ is measurable for each $n$, and the pointwise limit of $\{f_n\}$ exists, then $f(x) = \lim_{n \to \infty} f_n(x)$ is a measurable function. Let $V$ be some open set in $Y$. I was able to show th...
I was wondering If it is easier to factor in a non-ufd then it is to factor in a ufd.I can come up with arguments for that , but I also have arguments in the opposite direction.For instance : It should be easier to factor When there are more possibilities ( multiple factorizations in a non-ufd...
Consider a non-UFD that only has 2 units ( $-1,1$ ) and the min difference between 2 elements is $1$. Also there are only a finite amount of elements for any given fixed norm. ( Maybe that follows from the other 2 conditions ? )I wonder about counting the irreducible elements bounded by a lower...
How would you make a regex for this? L = {w $\in$ {0, 1}* : w is 0-alternating}, where 0-alternating is either all the symbols in odd positions within w are 0's, or all the symbols in even positions within w are 0's, or both.
I want to construct a nfa from this, but I'm struggling with the regex part
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I am looking for the asymptotics of the following integral
$\int_{\mathbb{R}} H_m^2(x) {\rm e}^{-2 \alpha^2 x^2} {\rm d} x = 2^{m-1/2} \alpha^{-2m -1} (1-2\alpha^2)^m \ \Gamma(m+1/2) ~ _2F_1\left(-m,m,1/2-m,\frac{\alpha^2}{2\alpha^2-1}\right)$
where $H_m$ is the $m^{\rm th}$ Hermite polynomial (orthogonal under the weight ${\rm e}^{-x^2}$), and $_2F_1$ is the hypergeometric function.
I found this formula from p. 803 of "Table of Integrals, Series, and Products" by Gradshteyn-Ryzhik. However, I have idea about the asymptotics of the $_2F_1$ term. Can anyone enlighten me on the asymptotics of
$_2F_1\left(-m,m,1/2-m,\beta\right)$
when $m$ is large? In fact I tried mathematica and it seems $_2F_1\left(-m,m,1/2-m,\beta\right) \sim |4 \beta|^m$. Any reference on this issue?
Now given the above asymptotics is true, observe that the norm of $H_m$ under the weight ${\rm e}^{-2 \alpha^2 x^2}$ has the same exponent for all $alpha$, including the original weight ($\alpha^2 = 1/2$). Is this a common phenomenon for orthogonal polynomials?
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High-energy neutrinos from individual blazar flares
Pre-published on: 2019 July 22
Published on: —
Abstract
Motivated by the recently reported evidence of an association between a high-energy neutrino and a $\gamma$-ray flare from the blazar TXS 0506+056, we calculate the expected high-energy neutrino signal from past, individual flares, from twelve blazars, selected in declinations favourable for detection with IceCube. To keep the number of free parameters to a minimum, we mainly focus on BL~Lac objects and assume the synchrotron self-Compton mechanism produces the bulk of the high-energy emission. We consider a broad range of the allowed parameter space for the efficiency of proton acceleration, the proton content of BL Lac jets, and the presence of external photon fields. To model the expected neutrino fluence we use simultaneous multi-wavelength observations. We find that in the absence of external photon fields and with jet proton luminosity normalised to match the observed production rate of ultra-high-energy cosmic rays, individual flaring sources produce a modest neutrino flux in IceCube, $ N^{\mathrm{IC,10 yr}}_{\nu_{\mu},{\mathrm{> 100~TeV}}} \lesssim 10^{-3}$ muon neutrinos with energy exceeding 100~TeV, stacking ten years of flare periods selected in the $>800~$MeV Fermi energy range, from each source. Under optimistic assumptions about the jet proton luminosity and in the presence of external photon fields, we find that the two most powerful sources in our sample, AO\,0235+164, and OJ\,287, would produce, in total, $N^{\mathrm{IC \times 10,10 yr}}_{\nu_{\mu}, \rm all~flares, > 100~TeV} \approx 3$ muon neutrinos during Fermi flaring periods, in future neutrino detectors with total instrumented volume $\sim$ ten times larger than IceCube, or otherwise, constrain the proton luminosity of blazar jets.
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Search
Now showing items 1-10 of 34
Lifetime of helium metastable spin-states in a helium discharge
(1965)
The lifetime of a helium 23s1 metastable atom electronic spin-state is measured in helium gas using optical pumping techniques. The metastable atoms are created by an RF electrical discharge. The spin-state lifetime is ...
Surface studies using spin-polarized electron energy loss spectroscopy
(1990)
Spin-polarized electron energy loss spectroscopy (SPEELS) has been used to investigate several paramagnetic surfaces. In this technique, a low energy beam of spin-polarized electrons from a GaAs source is directed at the ...
Reaction dynamics of excited states of helium and magneto-optical trapping of helium metastable atoms
(1993)
The temperature dependence of conversion of $\rm He(2\sp3S\sb1)$ metastable atoms to $\rm He\sb2(a\sp3\Sigma\sbsp{u}{+})$ metastable molecules in the three-body reaction $\rm He(2\sp3S\sb1)+ 2He(1\sp1S\sb0)\to He\sb2(a\s ...
Electron beam excitation studies of helium
(1967)
A comparison is made of relative number densities of excited states produced in Helium gas excited by an RF discharge and by the high energy (200kev, maximum) electron beam, it being found that the 33D states are relatively ...
FT-ICR studies of giant carbon fullerenes
(1992)
FT-ICR studies of high mass $\rm (C\sb{>150})$ carbon clusters have brought insight to the controversial structures of carbon fullerenes. Laser vaporization followed by supersonic beam technique produced carbon clusters ...
Probing depths of low energy electrons in metals
(1992)
Spin-polarized electron energy-loss spectroscopy has been used to investigate the probing depth of low energy ($\sim$30 eV) electrons in metals. A beam of spin-polarized electrons is directed at the surface of the sample ...
Optical pumping dynamics and spin relaxation in gaseous He_
(1966)
The first part of this investigation is concerned with the dynamics of the optical polarization process in He3 gas subjected to an electrical discharge. The characteristic time for the build-up of polarization under the ...
A method of polarization analysis of electrons from optically pumped He_
(1967)
An experiment in progress to extract a polarized beam of electrons from an optically pumped helium source gas is described. Possible ionization mechanisms in helium gas are outlined and the methods of optical pumping are ...
Spectra of dense helium excited by electron impact
(1969)
This is an account of the visible and near infrared spectrum of dense helium excited by 160keV electrons. It has been suspected that metastable helium atoms are involved in production of strong 02 and N2 spectra when ...
Laser annealing of Ni (001)
(1983)
Experimental aspects of laser cleaning and annealing of a Ni(1) surface with a pulsed ruby laser are reported. Effects of applying laser energy densities of .4 J/cm to 1.1 J/cm to an Ar-ion sputter-cleaned surface are ...
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Can I be a pedant and say that if the question states that $\langle \alpha \vert A \vert \alpha \rangle = 0$ for every vector $\lvert \alpha \rangle$, that means that $A$ is everywhere defined, so there are no domain issues?
Gravitational optics is very different from quantum optics, if by the latter you mean the quantum effects of interaction between light and matter. There are three crucial differences I can think of:We can always detect uniform motion with respect to a medium by a positive result to a Michelson...
Hmm, it seems we cannot just superimpose gravitational waves to create standing waves
The above search is inspired by last night dream, which took place in an alternate version of my 3rd year undergrad GR course. The lecturer talks about a weird equation in general relativity that has a huge summation symbol, and then talked about gravitational waves emitting from a body. After that lecture, I then asked the lecturer whether gravitational standing waves are possible, as a imagine the hypothetical scenario of placing a node at the end of the vertical white line
[The Cube] Regarding The Cube, I am thinking about an energy level diagram like this
where the infinitely degenerate level is the lowest energy level when the environment is also taken account of
The idea is that if the possible relaxations between energy levels is restricted so that to relax from an excited state, the bottleneck must be passed, then we have a very high entropy high energy system confined in a compact volume
Therefore, as energy is pumped into the system, the lack of direct relaxation pathways to the ground state plus the huge degeneracy at higher energy levels should result in a lot of possible configurations to give the same high energy, thus effectively create an entropy trap to minimise heat loss to surroundings
@Kaumudi.H there is also an addon that allows Office 2003 to read (but not save) files from later versions of Office, and you probably want this too. The installer for this should also be in \Stuff (but probably isn't if I forgot to include the SP3 installer).
Hi @EmilioPisanty, it's great that you want to help me clear out confusions. I think we have a misunderstanding here. When you say "if you really want to "understand"", I've thought you were mentioning at my questions directly to the close voter, not the question in meta. When you mention about my original post, you think that it's a hopeless mess of confusion? Why? Except being off-topic, it seems clear to understand, doesn't it?
Physics.stackexchange currently uses 2.7.1 with the config TeX-AMS_HTML-full which is affected by a visual glitch on both desktop and mobile version of Safari under latest OS, \vec{x} results in the arrow displayed too far to the right (issue #1737). This has been fixed in 2.7.2. Thanks.
I have never used the app for this site, but if you ask a question on a mobile phone, there is no homework guidance box, as there is on the full site, due to screen size limitations.I think it's a safe asssumption that many students are using their phone to place their homework questions, in wh...
@0ßelö7 I don't really care for the functional analytic technicalities in this case - of course this statement needs some additional assumption to hold rigorously in the infinite-dimensional case, but I'm 99% that that's not what the OP wants to know (and, judging from the comments and other failed attempts, the "simple" version of the statement seems to confuse enough people already :P)
Why were the SI unit prefixes, i.e.\begin{align}\mathrm{giga} && 10^9 \\\mathrm{mega} && 10^6 \\\mathrm{kilo} && 10^3 \\\mathrm{milli} && 10^{-3} \\\mathrm{micro} && 10^{-6} \\\mathrm{nano} && 10^{-9}\end{align}chosen to be a multiple power of 3?Edit: Although this questio...
the major challenge is how to restrict the possible relaxation pathways so that in order to relax back to the ground state, at least one lower rotational level has to be passed, thus creating the bottleneck shown above
If two vectors $\vec{A} =A_x\hat{i} + A_y \hat{j} + A_z \hat{k}$ and$\vec{B} =B_x\hat{i} + B_y \hat{j} + B_z \hat{k}$, have angle $\theta$ between them then the dot product (scalar product) of $\vec{A}$ and $\vec{B}$ is$$\vec{A}\cdot\vec{B} = |\vec{A}||\vec{B}|\cos \theta$$$$\vec{A}\cdot\...
@ACuriousMind I want to give a talk on my GR work first. That can be hand-wavey. But I also want to present my program for Sobolev spaces and elliptic regularity, which is reasonably original. But the devil is in the details there.
@CooperCape I'm afraid not, you're still just asking us to check whether or not what you wrote there is correct - such questions are not a good fit for the site, since the potentially correct answer "Yes, that's right" is too short to even submit as an answer
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For a typical hypothesis test with simple null hypothesis $H_0$ and simple alternative hypothesis $H_1$, and a continuous random variable $X$ as the observation, let the pdf of $X$ be denoted by $f_0(x)$ whenever the null hypothesis is true and by $f_1(x)$ whenever the alternative hypothesis is true. If $\Gamma_1$ denotes the region such that the null hypothesis is rejected whenever $X \in \Gamma_1$, then the Type I error probability -- the probability of
rejecting the null when in fact the null hypothesis is true -- is $$P(E_I) = P\{X \in \Gamma_1 \mid H_0 ~\text{is true}\} = \int_{\Gamma_1} f_0(x) \, \mathrm dx\tag{1}$$while the Type II error probability -- the probability of failing to reject the null when in fact the null is false -- is $$P(E_{II}) = P\{X \in \Gamma_1^c \mid H_0 ~\text{is false}\} = 1 - \int_{\Gamma_1} f_1(x) \, \mathrm dx.\tag{2}$$
Now, one way of reducing $P(E_I)$ is to
shrink the size of $\Gamma_1$ to $\Gamma_1^\prime \subset \Gamma_1$ where we assume that $f_0(x)$ is nonzero in the region $\Gamma_1 - \Gamma_1^\prime$ so that the integral in $(1)$ is over a smaller region and we can be sure that $P(E_I)^\prime < P(E_I)$. But then, the integral in $(2)$ is also over a smaller region and so $P(E_{II})^\prime \geq P(E_{II})$. It is tempting to insist that it must be the case that $P(E_{II})^\prime$ is strictly larger than $P(E_{II})$ but this does not hold if $f_1(x)$ is identically $0$ on $\Gamma_1 - \Gamma_1^\prime$ and so shrinking $\Gamma_1$ has no effect on the Type II error probability. Note that on $\Gamma_1 - \Gamma_1^\prime$, $f_0(x)$ is nonzero while $f_1(x) = 0$, that is, the likelihood ratio $\displaystyle\frac{f_1(x)}{f_0(x)}$ equals $0$ and so why the original test perversely chose to include these points in $\Gamma_1$ is beyond me: perhaps only to show that there are exceptions to the rule that
Reducing Type I error will
always result in increasing the Type II error
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19 0
I am currently studying the Massive Thirring Model (MTM) with the Lagrangian
$$ \mathcal{L} = \imath {\bar{\Psi}} (\gamma^\mu {\partial}_\mu - m_0 )\Psi - \frac{1}{2}g: \left( \bar{\Psi} \gamma_\mu \Psi \right)\left( \bar{\Psi} \gamma^\mu \Psi \right): . $$ and Hamiltonian $$ \int \mathrm{d}x \imath \Psi^\dagger \sigma_z \partial_x \Psi + m_0 \Psi^\dagger \Psi + 2g \Psi^\dagger_1 \Psi^\dagger_2 \Psi_2\Psi_1\\ $$ Due to the infinite set of conservation laws, particle production is said to be absent from this theory. However why isn't it sufficient to show that particle production is absent if the number operator $$N=\int \mathrm{d}x \Psi^\dagger \Psi$$ commutes the the Hamiltonian? Also, by particle production being absent, is that just a statement that all Feynman diagrams with self energy insertions evaluate to 0 but all other Feynman diagrams are possible?
$$
\mathcal{L} = \imath {\bar{\Psi}} (\gamma^\mu {\partial}_\mu - m_0 )\Psi - \frac{1}{2}g: \left( \bar{\Psi} \gamma_\mu \Psi \right)\left( \bar{\Psi} \gamma^\mu \Psi \right): .
$$
and Hamiltonian
$$
\int \mathrm{d}x \imath \Psi^\dagger \sigma_z \partial_x \Psi + m_0 \Psi^\dagger \Psi + 2g \Psi^\dagger_1 \Psi^\dagger_2 \Psi_2\Psi_1\\
$$
Due to the infinite set of conservation laws, particle production is said to be absent from this theory. However why isn't it sufficient to show that particle production is absent if the number operator $$N=\int \mathrm{d}x \Psi^\dagger \Psi$$ commutes the the Hamiltonian? Also, by particle production being absent, is that just a statement that all Feynman diagrams with self energy insertions evaluate to 0 but all other Feynman diagrams are possible?
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Often in topos theory, one starts with a geometric morphism $f: \mathcal Y \to \mathcal X$, but quickly passes to the Grothendieck fibration $U_f: \mathcal Y \downarrow f^\ast \to \mathcal X$, which is "$\mathcal Y$ regarded as an $\mathcal X$-topos". This maneuver takes one outside of the category of toposes and geometric morphisms, and thus tends to remove me from thinking of toposes geometrically, as generalized spaces.
It seems there should be some geometric interpretation lurking here -- after all, Grothendieck fibrations are "categories varying over a base", and a topos is "a base that can be varied over". But although $\mathcal Y \downarrow f^\ast$ is a topos, the fibration $U_f: \mathcal Y \downarrow f^\ast \to \mathcal X$ is not (the direct image of) a geometric morphism! In fact it has a right adjoint $F^\ast$ which has a further right adjoint $F_\ast: \mathcal Y \downarrow f^\ast \to \mathcal X$. We in fact have a totally connected geometric morphism $F: \mathcal Y \downarrow f^\ast \to \mathcal X$.
Moreover, the fibers of $U_f$ are toposes, but the reindexing functors are most naturally viewed as the
inverse images of étale geometric morphisms. So we don't naturally have a "topos fibered in toposes" it seems.
For one more perspective, $\mathcal Y \downarrow f^\ast$ is a cocomma object in the 2-category of toposes. As such, $\mathcal Y \downarrow f^\ast \leftarrow \mathcal Y$ is the free co-fibration on $f$ in the 2-category of toposes. I think this is the point of departure for Rosebrugh and Wood. But I have no geometric intuition for what a co-Grothendieck fibration is. And anyway, the functor $U_f$ doesn't even live in the category of toposes.
I'm not really sure what to make of this. So here are some
Questions: Let $f: \mathcal Y \to \mathcal X$ be a geometric morphism.
Is there a geometric interpretation of the fibration $U_f: \mathcal Y \downarrow f^\ast \to \mathcal X$?
Does the totally connected geometric morphism $F: \mathcal Y \downarrow f^\ast \to \mathcal X$ enjoy some sort of universal property with respect to $f$? (for example, does the inclusion of the category of toposes and totally connected geometric morphisms into toposes and all geometric morphisms have a left adjoint which is being applied here?)
Is there a fibration in the 2-category of toposes lurking around here somewhere (see the Elephant B4.4 for some discussion of these)?
Is there a general geometric interpretation of an opfibration over a topos whose fibers are toposes, with étale geometric morphisms for reindexing?
Addendum: Here are a few places where "indexed" notions come up where the viewpoint that "everything is indexed over the codomain" seems kind of inflexible to me:
A proper geometric morphism $f: \mathcal Y \to \mathcal X$ is one such that $f_\ast$ preserves $\mathcal X$-indexed filtered colimits of subterminal objects. It's a bit awkward, for example, to even discuss the composition of proper morphisms if you're tied to the view that $\mathcal Y$ is internal to $\mathcal X$.
A separated geometric morphism is one such that the diagonal is proper; in particular, a Hausdorff topos is one such that $\mathcal X \to \mathcal X \times \mathcal X$ is proper. It seems very unnatural to me in this context to think of $\mathcal X$ as being primarily an object "internal to $\mathcal X \times \mathcal X$".
Let me also point out there there is at least one place in
Sketches of an Elephant where $\mathcal Y \downarrow f^\ast$ plays a role qua topos -- in Ch C3.6 on local geometric morphisms, where it's referred to as a "scone". See in particular Example 3.6.3(f), Lemma 3.6.4, and Corollary 3.6.13 The scone is the dual construction $f^\ast \downarrow \mathcal Y$. But in the end of Ch C3.6, Johnstone does consider $\mathcal Y \downarrow f^\ast$ qua topos, and shows that it is related to totally connected geometric morphisms in the same way that the scone is related to local geometric morphisms.
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Some questions on the main site have formulae in titles, and it's difficult to discuss such things in the meta without mutilating the titles. It would be nice to have MathJax enabled on the Meta as well.
Can someone from SE please revisit this request now that we're out of beta. Signal processing is sufficiently math heavy that it's sometimes required in meta discussions as well. For example, see this recent question.
Done (after a short hiccup that hopefully didn't inconvenience anyone). We usually do this when we turn MathJax on for the main site, somehow it fell through the cracks - sorry about that :)
It's enabled now, enjoy.
let's see if it works for an answer:
$$\begin{align} X(f) &\triangleq \int\limits_{-\infty}^{+\infty} x(t) \ e^{-j 2 \pi f t} \ dt \\ \\ x(t) &= \int\limits_{-\infty}^{+\infty} X(f) \ e^{+j 2 \pi f t} \ df \\ \end{align} $$
that seems to work too.
Thank you Tim.
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Here's a geometric argument, but it isn't as slick as some of the Calculus-based ones.
Consider the unit circle about $O$, through $R$ and $S$, with $\theta = \angle ROS$. The perpendicular from $S$ to $\overline{OR}$ has length $\sin\theta$, while the perpendicular from $R$ up to $T$ on the extension of $\overline{OS}$ has length $\tan\theta$. Let $M$ be the midpoint of $\overline{ST}$.
Then $$2\;|\text{area of sector}\;ROS| = \theta \qquad\text{and}\qquad 2\;|\triangle ORM| = \frac{1}{2}\left(\sin\theta + \tan\theta\right)$$
"All we need to do" is show that the triangle has more area than the sector. This seems pretty clear; after all, the triangle contains almost-all of the sector, except for the circular segment defined by $\overline{KR}$, where $K$ is the intersection of $\overline{RM}$ and the circle. There is a concern, though, that the excess area in the triangular region $KSM$ could be less than that of the tiny sliver of a circular segment for small $\theta$; we need to dispel that concern.
There's probably a simpler route to this, but I coordinatized and, with the help of
Mathematica, found$$M = \left(\frac{1 + \cos\theta}{2}, \frac{\sin\theta (1 + \cos\theta)}{2 \cos\theta}\right)$$$$K = \left(\frac{1 + 3 \cos\theta + 2 \cos^2\theta + 2 \cos^3\theta}{1 + 3 \cos\theta + 4 \cos^2\theta}, \frac{2 \sin\theta \cos\theta ( 1 + \cos\theta)}{1 + 3 \cos\theta + 4 \cos^2\theta}\right)$$so that (after a bit more symbol-crunching)$$\frac{|\overline{MK}|}{|\overline{KR}|} = \frac{1 + 3 \cos\theta}{4 \cos^2\theta} = 1 + \frac{1 + 3 \cos\theta - 4 \cos^2\theta}{4 \cos^2\theta} = 1 + \frac{(1-\cos\theta)(1 + 4 \cos\theta)}{4 \cos^2\theta} > 1$$
for $0 < \theta < \pi/2$.
This says that $\overline{MK}$ is longer than $\overline{KR}$, so that we could reflect $R$ in $K$ to get $R^\prime$, and copy circular segment $KR$ as circular segment $KR^\prime$ inside $\triangle ORM$ yet tangent to the unit circle (and therefore outside of sector $ORS$).
Consequently, the triangle definitely has more area than the sector, so we're done. $\square$
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For each $n = 1, 2, ....$, suppose that $X_n$ is a continuous random variable with density $$\hspace{10mm}\mathrm{f}(x) = \begin{cases} \frac{1}{2}(1+x)e^{-x}, & \text{if $x \ge 0$ } \\[2ex] 0, & \text{if $x\le 0$} \end{cases}$$ Set $Y_n$=min{$X_1,X_2,....X_n$}. Does n$\cdot$$Y_n$ converges in distribution as n$\to \infty $ ? What will be the limiting distribution of n$\cdot$$Y_n$? Attept: I was tryng to find the distribution of n$\cdot$$Y_n$. But it became too complicated. How should I proceed here. Any help would be much appreciated.
Let's try to find a distribution function of $nY_n$.\
$P(nY_n \leq t) = 1 - P(min \{ X_1, ..., X_n \} > \frac{t}{n}) = 1 - P(X_1 > \frac{t}{n})\cdot ... \cdot P(X_n > \frac{t}{n})$
We can calculate the tail of the random variable $X_1$ (and all of the variables because I suppose that they are i.i.d).
$P(X_1 > \frac{t}{n}) = \int_{\frac{t}{n}}^{\infty} \frac{1}{2}(1+x) \exp (-x) dx$.
Now, try to do this integral by parts.
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Consider the finite sum
rs[x_, n_] := x/n Sum[n^2/(i + (n - i) x)^2, {i, 1, n}]
Is there a way to bring
Mathematica to calculate the limit for
n -> ∞?
I have tried
Limit[] as well as
NLimit[] without success.
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Consider the finite sum
rs[x_, n_] := x/n Sum[n^2/(i + (n - i) x)^2, {i, 1, n}]
Is there a way to bring
Mathematica to calculate the limit for
n -> ∞?
I have tried
Limit[] as well as
NLimit[] without success.
For a fixed numerical value of x the sum limit can be found using Shanks transformation or Richardson extrapolation. The Richardson extrapolation transformation of the sequence gives faster convergence. The results seem to be in agreement with the plots in the answer by Willinski -- see the attached image.
Here is the code for Shanks:
Clear[Shanks]Shanks[A_, n_] := (A[n + 2] A[n] - A[n + 1]^2)/( A[n + 2] + A[n] - 2 A[n + 1]);Shanks[A_, 1, n_] := Shanks[A, n];Shanks[A_, k_, n_] := ( Shanks[A, -1 + k, n] Shanks[A, -1 + k, 2 + n] - Shanks[A, -1 + k, 1 + n]^2)/( Shanks[A, -1 + k, n] + Shanks[A, -1 + k, 2 + n] - 2 Shanks[A, -1 + k, 1 + n]);
Here is the code for Richardson:
Clear[Richardson]Richardson[A_, n_, N_] := Total@Table[(A[n + k]*(n + k)^N *If[OddQ[k + N], -1, 1])/( k! (N - k)!), {k, 0, N}]
This code makes the table in the image:
rs[x_, n_] := x/n Sum[n^2/(i + (n - i) x)^2, {i, 1, n}]sf[n_] := rs[1/2, n]ns = {1, 2, 3, 4, 5, 6, 7, 15, 25, 50, 90};res = Outer[#1[#2] &, {sf, Shanks[sf, #1] &, Shanks[sf, 3, #1] &, Richardson[sf, #1, 1] &, Richardson[sf, #1, 6] &}, ns];TableForm[Transpose[Map[N[#1, 20] &, res, {-1}]], TableHeadings -> {ns, {"function value", "Shanks-1", "Shanks-3", "Richardson-1", "Richardson-6"}}]
The book by Bender and Orszag "Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory" has good explanations of the Shanks and Richardson transformations. (See chapter 8.)
See EDIT #3 for a valid answer.
First answer
This is not an answer to my question but just a mathematical derivation of the limit the existence of which was even doubted in some comments and answers. I hope this does not spoil the creativity. The task for Mathematica is still open.
I came to the question considering something very elementary in calculus: the definition of the Riemann integral.
With this hint you surely can do the derivation by yourself. Here is mine:
The limit is equal to 1, as Willinski first calculated numerically.
Proof: consider the intergal
fi = Integrate[1/t^2, {t, 1, x}, Assumptions -> x > 1](* Out[395]= (-1 + x)/x *)
And now consider the definition of the Riemann integral of a function g[x] between x=a and x=b as a limit of this sum
gs[a_, b_, n_] := (b - a)/n Sum[g[a + (b - a) k/n], {k, 0, n - 1}]
Letting
{a -> 1, b -> x, g[t] -> 1/t^2}
we have
gs1[x_, n_] := (x - 1)/n Sum[1/(1 + (x - 1) k/n)^2, {k, 0, n - 1}]
This sum is equal to our sum except for the factor (x-1)/x (and the summand for k=n which, however, is negligible in the limit). Hence our limit is equal to x/(x-1) times fi, that is equal to 1. QED.
EDIT #1
One attempt to help Mathematica calculate the limit could be replacing the sum over i by an integral over i. Not strict but at least a heuristic approach.
Here we go
x/n Integrate[n^2/(i + (n - i) x)^2, {i, 1, n}, Assumptions -> {n > 1, x > 1}](*Out[481]= ((-1 + n) x)/(1 + (-1 + n) x)*)Limit[%, n -> \[Infinity]](* Out[480]= 1 *)
We must then justifiy the replacement.
EDIT #2
Let us try to callculate symbolically the sum involved
s[n_, x_] := Sum[1/(i + (n - i) x)^2, {i, 0, n}]Assuming[x > 1, Timing[s[n, x]]](* Out[27]= $Aborted *)
Mathematica won't do this for real x. Hence we adopt the strategy "guess from integers" which I use frequently in such cases : look for results for some interger values of x, and guess what the general expression might be.
Table[{x, Sum[1/(i + (n - i) x)^2, {i, 0, n}]}, {x, 1, 7}] // Column
$$\left( \begin{array}{cc} 2 & \psi ^{(1)}(-2 n)-\psi ^{(1)}(1-n) \\ 3 & \frac{1}{4} \left(\psi ^{(1)}\left(-\frac{1}{2} (3 n)\right)-\psi ^{(1)}\left(1-\frac{n}{2}\right)\right) \\ 4 & \frac{1}{9} \left(\psi ^{(1)}\left(-\frac{1}{3} (4 n)\right)-\psi ^{(1)}\left(1-\frac{n}{3}\right)\right) \\ 5 & \frac{1}{16} \left(\psi ^{(1)}\left(-\frac{1}{4} (5 n)\right)-\psi ^{(1)}\left(1-\frac{n}{4}\right)\right) \\ 6 & \frac{1}{25} \left(\psi ^{(1)}\left(-\frac{1}{5} (6 n)\right)-\psi ^{(1)}\left(1-\frac{n}{5}\right)\right) \\ 7 & \frac{1}{36} \left(\psi ^{(1)}\left(-\frac{1}{6} (7 n)\right)-\psi ^{(1)}\left(1-\frac{n}{6}\right)\right) \\ \end{array} \right)$$
Where $\psi ^{(1)}(z)$ = PolyGamma[1, z]
Our guess is
sg[n_, x_] := 1/(x - 1)^2 (-PolyGamma[1, 1 - n/(x - 1)] + PolyGamma[1, -((x n)/(x - 1))])
But still: no results for the limit
For real x > 1
Timing[Limit[n x sg[n, x], {n -> \[Infinity]}]](*Out[26]= {211.21, {Limit[( n x (-PolyGamma[1, 1 - n/(-1 + x)] + PolyGamma[1, -((n x)/(-1 + x))]))/(-1 + x)^2, n -> \[Infinity]]}}*)
And even not for x = 2
Timing[Limit[n x sg[n, x] /. x -> 2, {n -> \[Infinity]}]](*Out[28]= {18.1741, {Limit[2 n (-PolyGamma[1, 1 - n] + PolyGamma[1, -2 n]), n -> \[Infinity]]}}*)
Tough stuff for Mathematica!
EDIT #3
Now I've found a direct way to an answer my original question, viz. how we can bring Mathematica to calculate the limit.
This method generalizes to similar expressions of which we have to take the limit.
Letting
z -> x - 1;
our sum becomes
s := x/n Sum[1/(1 + k/n z)^2, {k, 0, n}]
We have to calculate the limit of s for
n->\[Infinity].
Expanding the summand into a power series
1/(1 + k/n z)^2 -> Sum[Binomial[-2, m] k^m z^m/n^m, {m, 0, \[Infinity]}](*Out[50]= 1/(1 + (k z)/n)^2 -> n^2/(n + k z)^2*)
and exchanging the order of summation gives us
s1 := x/n Sum[Binomial[-2, m] z^m c[n, m], {m, 0, \[Infinity]}]
where
c[m_, n_] = Sum[k^m/n^m, {k, 0, n}](*Out[53]= n^-m (0^m + HarmonicNumber[n, -m])*)
Now we take the limit
n->\[Infinity], taking into account the factor
1/n
clim = Limit[n^-1 c[m, n], n -> \[Infinity], Assumptions -> {m \[Element] Integers, m >= 0}](* Out[58]= 1/(1 + m) *)
and the m-sum becomes
s2 = Sum[x z^m Binomial[-2, m] clim, {m, 0, \[Infinity]}](* Out[59]= x/(1 + z) *)
Finally, replacing z gives
slim = s2 /. z -> x - 1(* Out[61]= 1 *)
QED.
This post tackles the convergence acceleration of the Riemann integral in the same spirit as Anton's answer, except that I use a slight variation of one of the algorithms presented. In particular, I'm using this as an excuse to present the van den Broeck-Schwartz modification of the Wynn ε algorithm:
wgvs[seq_?VectorQ, h_: 1] := Module[{n = Length[seq], ep, v, w}, Table[ ep[k] = seq[[k]]; w = 0; Do[v = w; w = ep[j]; ep[j] = v If[OddQ[k - j], h, 1] + 1/(ep[j + 1] - w), {j, k - 1, 1, -1}]; ep[Mod[k, 2, 1]], {k, n}]]
The default setting of the second parameter corresponds to the classical Wynn algorithm.
For the OP's example:
rs[x_, n_] := x/n Sum[n^2/(i + (n - i) x)^2, {i, 1, n}, Method -> "Procedural"]tab = Table[rs[1/2, 2^k], {k, 12}] // N;res = wgvs[tab];-Log10[Abs[tab - 1]] {0.51491, 0.770798, 1.04959, 1.33974, 1.6354, 1.93377, 2.23347, 2.53384, 2.83454, 3.1354, 3.43635, 3.73734}-Log10[Abs[res - 1]] {0.51491, 0.770798, 1.57696, 2.25408, 3.74516, 4.92739, 6.84944, 8.60221, 10.9223, 13.2548, 15.6536, 14.9546}
where we see that Wynn ε achieved $\approx 14$ good digits with little additional effort.
For comparison, let's change the value of the second parameter of
wgvs[] to
0; this corresponds to applying the iterated Aitken $\Delta^2$ process:
res = wgvs[tab, 0];-Log10[Abs[res - 1]] {0.51491, 0.770798, 1.57696, 2.25408, 3.82643, 4.65844, 6.64954, 7.52831, 8.61013, 10.7914, 13.1377, 14.8085}
In addition to Bender and Orszag, a good reference on convergence acceleration methods is Brezinski and Redivo-Zaglia's
Extrapolation Methods: Theory and Practice. Weniger's survey paper is also a useful reference.
It's not the solution you are looking for and surely you have tried the same.
rs[x_, n_] := x/n Sum[n^2/(i + (n - i) x)^2, {i, 1, n}]rs[1., \[Infinity]]
Infinity::indet: Indeterminate expression 0 [Infinity] encountered.
Sum::div: Sum does not converge.
Plot[rs[2., n], {n, 1, 100000}, PlotPoints -> 100]
Addition
<< NumericalCalculus`NLimit[rs[2, n], n -> 3]
Infinity::indet: Indeterminate expression ComplexInfinity+ComplexInfinity encountered. >>
NLimit::notnum: The expression Indeterminate is not numerical at the point n == 4.`. >>
NLimit[2 n (PolyGamma[1, 1 - 2 n] - PolyGamma[1, 1 - n]), n -> 3]
Plot[PolyGamma[1, 1 - 2 n] - PolyGamma[1, 1 - n], {n, 1, 3}]
NLimit[PolyGamma[1, 1 - 2 n] - PolyGamma[1, 1 - n], n -> 3]
Infinity::indet: Indeterminate expression ComplexInfinity+ComplexInfinity encountered. >>
NLimit::notnum: The expression Indeterminate is not numerical at the point n == 4.`. >>
I would be surprised if there exists a limit.
We know, with Limit[] and NLimit[] we cannot obtain a solution. With DiscretePlot we can get an idea, but that is not a prove.
Table[DiscretePlot[rs[x, k], {k, 1000}, PlotRange -> All, AxesOrigin -> {0, 0}], {x, {0.1, 1., 2., 5.}}]
I'm looking forward to the limiting procedure.
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Works by Sarig and Benovadia have built symbolic dynamics for arbitrary diffeomorphisms of compact manifolds. This shows thatthere can be at most countably many ergodic hyperbolic equilibriummeasures for any Holder continuous or geometric potentials. We will explain how this yields uniqueness inside each homoclinic class of measures, i.e., of ergodic and hyperbolic measures that are homoclinically related. In some cases, further topological or geometric arguments can show global uniqueness.
This is a joint work with Sylvain Crovisier and Omri Sarig
Works by Sarig and Benovadia have built symbolic dynamics for arbitrary diffeomorphisms of compact manifolds. This shows thatthere can be at most countably many ergodic hyperbolic equilibriummeasures for any Holder continuous or geometric potentials. We will explain how this yields uniqueness inside each homoclinic class of measures, i.e., of ergodic and hyperbolic measures that are homoclinically related. In some cases, further topological or ...
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Smooth parametrizations of semi-algebraic sets were introduced by Yomdin in order to bound the local volume growth in his proof of Shub’s entropy conjecture for C∞ maps. In this minicourse we will present some refinement of Yomdin’s theory which allows us to also control the distortion. We will give two new applications: - for any C∞ surface diffeomorphism f with positive entropy the saddle periodic points with Lyapunov exponents $\delta$-away from zero for $\delta \in]0,htop(f)[$ are equidistributed along measures of maximal entropy. - for C∞ maps the entropy is physically greater than or equal to the top Lyapunov exponents of the exterior powers.
Smooth parametrizations of semi-algebraic sets were introduced by Yomdin in order to bound the local volume growth in his proof of Shub’s entropy conjecture for C∞ maps. In this minicourse we will present some refinement of Yomdin’s theory which allows us to also control the distortion. We will give two new applications: - for any C∞ surface diffeomorphism f with positive entropy the saddle periodic points with Lyapunov exponents $\delta$-away ...
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Research schools
Smooth parametrizations of semi-algebraic sets were introduced by Yomdin in order to bound the local volume growth in his proof of Shub’s entropy conjecture for C∞ maps. In this minicourse we will present some refinement of Yomdin’s theory which allows us to also control the distortion. We will give two new applications: - for any C∞ surface diffeomorphism f with positive entropy the saddle periodic points with Lyapunov exponents $\delta$-away from zero for $\delta \in]0,htop(f)[$ are equidistributed along measures of maximal entropy. - for C∞ maps the entropy is physically greater than or equal to the top Lyapunov exponents of the exterior powers.
Smooth parametrizations of semi-algebraic sets were introduced by Yomdin in order to bound the local volume growth in his proof of Shub’s entropy conjecture for C∞ maps. In this minicourse we will present some refinement of Yomdin’s theory which allows us to also control the distortion. We will give two new applications: - for any C∞ surface diffeomorphism f with positive entropy the saddle periodic points with Lyapunov exponents $\delta$-away ...
37C05 ; 37C40 ; 37D25
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Research schools
Smooth parametrizations of semi-algebraic sets were introduced by Yomdin in order to bound the local volume growth in his proof of Shub’s entropy conjecture for C∞ maps. In this minicourse we will present some refinement of Yomdin’s theory which allows us to also control the distortion. We will give two new applications: - for any C∞ surface diffeomorphism f with positive entropy the saddle periodic points with Lyapunov exponents $\delta$-away from zero for $\delta \in]0,htop(f)[$ are equidistributed along measures of maximal entropy. - for C∞ maps the entropy is physically greater than or equal to the top Lyapunov exponents of the exterior powers.
Smooth parametrizations of semi-algebraic sets were introduced by Yomdin in order to bound the local volume growth in his proof of Shub’s entropy conjecture for C∞ maps. In this minicourse we will present some refinement of Yomdin’s theory which allows us to also control the distortion. We will give two new applications: - for any C∞ surface diffeomorphism f with positive entropy the saddle periodic points with Lyapunov exponents $\delta$-away ...
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These lectures are a mostly self-contained sequel to Vaughn Climenhaga’s talks in week 1. The focus of the week 2 lectures will be on uniqueness of equilibrium states for rank 1 geodesic flows, and their mixing properties. Burns, Climenhaga, Fisher and myself showed recently that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. I will discuss the proof of this result. With this result in hand, the question of when the “pressure gap” hypothesis can be verified becomes crucial. I will sketch our proof of the “entropy gap”, which is a new direct constructive proof of a result by Knieper. I will also describe new joint work with Ben Call, which shows that all the unique equilibrium states provided above have the Kolmogorov property. When the manifold has dimension at least 3, this is a new result even for the Knieper-Bowen-Margulis measure of maximal entropy. The common thread that links all of these arguments is that they rely on weak orbit specification properties in the spirit of Bowen.
These lectures are a mostly self-contained sequel to Vaughn Climenhaga’s talks in week 1. The focus of the week 2 lectures will be on uniqueness of equilibrium states for rank 1 geodesic flows, and their mixing properties. Burns, Climenhaga, Fisher and myself showed recently that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. I will discuss the proof of this result. With this result in hand, ...
37D35 ; 37D40 ; 37C40 ; 37D25
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Research schools
These lectures are a mostly self-contained sequel to Vaughn Climenhaga’s talks in week 1. The focus of the week 2 lectures will be on uniqueness of equilibrium states for rank 1 geodesic flows, and their mixing properties. Burns, Climenhaga, Fisher and myself showed recently that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. I will discuss the proof of this result. With this result in hand, the question of when the “pressure gap” hypothesis can be verified becomes crucial. I will sketch our proof of the “entropy gap”, which is a new direct constructive proof of a result by Knieper. I will also describe new joint work with Ben Call, which shows that all the unique equilibrium states provided above have the Kolmogorov property. When the manifold has dimension at least 3, this is a new result even for the Knieper-Bowen-Margulis measure of maximal entropy. The common thread that links all of these arguments is that they rely on weak orbit specification properties in the spirit of Bowen.
These lectures are a mostly self-contained sequel to Vaughn Climenhaga’s talks in week 1. The focus of the week 2 lectures will be on uniqueness of equilibrium states for rank 1 geodesic flows, and their mixing properties. Burns, Climenhaga, Fisher and myself showed recently that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. I will discuss the proof of this result. With this result in hand, ...
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These lectures are a mostly self-contained sequel to Vaughn Climenhaga’s talks in week 1. The focus of the week 2 lectures will be on uniqueness of equilibrium states for rank 1 geodesic flows, and their mixing properties. Burns, Climenhaga, Fisher and myself showed recently that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. I will discuss the proof of this result. With this result in hand, the question of when the “pressure gap” hypothesis can be verified becomes crucial. I will sketch our proof of the “entropy gap”, which is a new direct constructive proof of a result by Knieper. I will also describe new joint work with Ben Call, which shows that all the unique equilibrium states provided above have the Kolmogorov property. When the manifold has dimension at least 3, this is a new result even for the Knieper-Bowen-Margulis measure of maximal entropy. The common thread that links all of these arguments is that they rely on weak orbit specification properties in the spirit of Bowen.
These lectures are a mostly self-contained sequel to Vaughn Climenhaga’s talks in week 1. The focus of the week 2 lectures will be on uniqueness of equilibrium states for rank 1 geodesic flows, and their mixing properties. Burns, Climenhaga, Fisher and myself showed recently that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. I will discuss the proof of this result. With this result in hand, ...
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Research talks;Dynamical Systems and Ordinary Differential Equations
I will survey recent results on the generic properties of probability measures invariant by the geodesic flow defined on a nonpositively curved manifold. Such a flow is one of the early example of a non-uniformly hyperbolic system. I will talk about ergodicity and mixing both in the compact and noncompact setting, and ask some questions about the associated frame flow, which is partially hyperbolic.
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Research talks;Dynamical Systems and Ordinary Differential Equations
We prove a couple of general conditional convergence results on ergodic averages for horocycle andgeodesic subgroups of any continuous $SL(2,\mathbb{R})$- action on a locally compact space. These results are motivated by theorems of Eskin, Mirzakhani and Mohammadi on the $SL(2,\mathbb{R})$-action on the moduli space of Abelian differentials. By our argument we can derive from these theorems an improved version of the “weak convergence” of push-forwards of horocycle measures under the geodesic flow and a new short proof of a theorem of Chaika and Eskin on Birkhoff genericity in almost all directions for the Teichmüller geodesic flow.
We prove a couple of general conditional convergence results on ergodic averages for horocycle andgeodesic subgroups of any continuous $SL(2,\mathbb{R})$- action on a locally compact space. These results are motivated by theorems of Eskin, Mirzakhani and Mohammadi on the $SL(2,\mathbb{R})$-action on the moduli space of Abelian differentials. By our argument we can derive from these theorems an improved version of the “weak convergence” of ...
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- vii; 177 p.
ISBN 978-2-85629-904-3
Astérisque , 0410
Localisation : Périodique 1er étage
hyperbolicté non-uniforme # sélection de paramètres # application unimodale # attracteur Hénon # dynamiques chaotiques # dynamiques en petite dimension # pièce de puzzle
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- vii; 326 p.
ISBN 978-3-319-43058-4
Lecture notes in mathematics , 2164
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Chaire Jean-Morlet # CIRM # dynamique # théorie ergodique # géométrie différentielle
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Astérisque , 0382
Localisation : Périodique 1er étage
forme modulaire de Hilbert # forme modulaire $\rho$-adique # forme modulaire surconvergente # représentation galoisienne # modularité # conjecture d'Artin # conjecture de Fontaine-Mazur
37A20 ; 37D25 ; 37D30 ; 37A50 ; 37C40
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- ix; 165 p.
ISBN 978-2-85629-778-0
Astérisque , 0358
Localisation : Périodique 1er étage
Cocycle abélien # équation cohomologique # invariant d'holonomie # principe d'invariance # cocycle linéaire # théorie de Livsic # exposant de Liapounoff # hyperbolicité partielle # rigidité # cocycle lisse
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- ix; 277 p.
ISBN 978-0-8218-9853-6
Graduate studies in mathematics , 0148
Localisation : Collection 1er étage
système dynamique # théorie ergodique # exposant de Lyapunov # dynamique topologique # hyperbolicité non-uniforme # flot géodésique
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- 339 p.
ISBN 978-0-8218-4274-4
Fields institute communications , 0051
Localisation : Collection 1er étage
système dynamique # théorie ergodique # ergodicité lisse # système hyperbolique # flots sur surface # méthode quasiconforme # théorie de Teichmüller # foliation # groupe de Kleinian # surface modulaire de Riemann
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Springer monographs in mathematics
Localisation : Ouvrage RdC (MARG)
système dynamique # courbure négative # fonction zéta # opérateur de transfert # orbite périodique # système d'Anosov # flot hyperbolique
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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 all numbers $1\dots N$ can be built? $f(n)=N$)
@TheSimpliFire You mentioned base, is it true that using digits $\lt b$ means we can represent some number $N$ using $\le (b+1)\log_b N$ digits, if only $+,\times$ are allowed?
If $b=2$, $3\log_2 N$ bound is given: https://arxiv.org/pdf/1310.2894.pdf and explained: " The upper bound can be obtained by writing $N$ in binary and finding a representation using Horner’s algorithm."
So if we actually allow $\le b$ digits, we have $log_b N$ digits and that many bases, so the bound would be $2\log_b N$? https://en.wikipedia.org/wiki/Horner%27s_method @TheSimpliFire
The problem is inverting the bound which is not trivial if $b\ne 2$.
For example, we can build $1=2-1$ using $1,2$ digits but adding onto $5$ and having now a set $1,2,5$ does NOT allow to rebuild $1$ since all digits must be used.
So keeping consecutive integers from $n-1$ digit case is not guaranteed. This is the issue.
The $d$ is fixed at $n$ digits and all need to be used.
Thats why I took $d_i=2^{i-1}$ digit sets since we can divide two largest to get the $n-1$ case and this allows to obtain bound $f(n)\ge2^n-1$ eventually.
Inductively.
$i=1,\dots,n$
This is not the issue if all digits are $1$'s also, on which they give bound $3\log_2 N\ge a(N)$ which can be translated to $f(n)\ge 2^{N/3}$ since multiplying two $1$'s reduces the case to $n-1$ and allows induction.
We need to inductively build digits $d_i$ so next set can achieve at least what previous one did.
Otherwise, it is hard to prove the next step is better when adding more digits.
For example we can add $d_0,d_0/2,d_0/2$ where $d_0$ can be anything since $d_0-d_0/2-d_0/2$ reduces us to case $n-3$. The comments discuss setting better bounds using similar construction (on my last question)
I'm not sure if you have the full context of the question or if this makes sense so sorry for clogging up the chat :P
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Wythoff's game is as follows: there are two players $A$ and $B$ ( $A$ being the first player ) and there are $2$ piles of stones. When his turn a player can remove one or more stones from anyone pile or same number of stones from both the piles. A player unable to make a move loses. Also it is assumed that both the players play optimally.
As mentioned in the link the losing configurations $(n_k,m_k)$ ( $n_k$ stones in one pile and $m_k$ stones in another ) are given by $$ n_k = \lfloor( k*\phi) \rfloor$$ $$ m_k = \lfloor( k*\phi^2) \rfloor$$ where $k$ is any natural number ( and $n_k \le m_k$ ) and $\phi=\frac{1+\sqrt{5}}{2}$. For example $\{1,2\}$ ( two stones in one pile and one stone in another ) is a losing position.
Modification to Wythoff's game"
In this modified game instead of ${\it2}$ piles there are ${\it 3}$ piles. And when his turn a player can move one or more stones from anyone pile or same number of stones from any $2$ piles or same number of stones from all the $3$ piles.
How do I compute the losing positions for this modified game efficiently ? The inefficient way of course is to find the grundy number of each configuration $\{a,b,c\}$. This method is inefficient because I want to calculate number of losing positions given that number of stones in each pile can be between $1$ and $1000$. This is a project Euler question ( stone game ). I would appreciate hints only.
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I want all displayed mathematics in my document to be typeset in the style of inline mathematics. In his answer to this question, Matthew Leingang explains how to do the opposite (typeset all inline mathematics in the style of displayed mathematics) by carefully modifying the
\everymath token list. Reasoning by analogy I figured that the following test document ought to work:
\documentclass[a4paper,reqno]{amsart}\usepackage{amssymb,amsmath,amsthm,fullpage}\everydisplay=\expandafter{\the\everydisplay\textstyle}\begin{document}\begin{align*}x&=\sum_{i=1}^{10}i^2\\&=t^3+\int_3^9y^a\,\mathrm{d}y\end{align*}\end{document}
But it does not. When trying to compile this document, TeXWorks gives the following error:
! Improper \halign inside $$'s.<recently read> \halign
The code it displays, as though to draw my attention to a mistake, is the code in the vicinity of
\end{align*}. This error does not appear if the line in the preamble modifying
\everydisplay is commented out. Why doesn't this work?
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Let $X_n(\Bbb{Z})$ be the simplicial complex whose vertex set is $\Bbb{Z}$ and such that the vertices $v_0,...,v_k$ span a $k$-simplex if and only if $|v_i-v_j| \le n$ for every $i,j$. Prove that $X_n(\Bbb{Z})$ is $n$-dimensional...
no kidding, my maths is foundations (basic logic but not pedantic), calc 1 which I'm pretty used to work with, analytic geometry and basic linear algebra (by basic I mean matrices and systems of equations only
Anyway, I would assume if removing $ fixes it, then you probably have an open math expression somewhere before it, meaning you didn't close it with $ earlier. What's the full expression you're trying to get? If it's just the frac, then your code should be fine
This is my first time chatting here in Math Stack Exchange. So I am not sure if this is frowned upon but just a quick question: I am trying to prove that a proper subgroup of $\mathbb{Z}^n$ is isomorphic to $\mathbb{Z}^k$, where $k \le n$. So we must have $rank(A) = rank(\mathbb{Z}^k)$ , right?
For four proper fractions $a, b, c, d$ X writes $a+ b + c >3(abc)^{1/3}$. Y also added that $a + b + c> 3(abcd)^{1/3}$. Z says that the above inequalities hold only if a, b,c are positive. (a) Both X and Y are right but not Z. (b) Only Z is right (c) Only X is right (d) Neither of them is absolutely right.
Yes, @TedShifrin the order of $GL(2,p)$ is $p(p+1)(p-1)^2$. But I found this on a classification of groups of order $p^2qr$. There order of $H$ should be $qr$ and it is present as $G = C_{p}^2 \rtimes H$. I want to know that whether we can know the structure of $H$ that can be present?
Like can we think $H=C_q \times C_r$ or something like that from the given data?
When we say it embeds into $GL(2,p)$ does that mean we can say $H=C_q \times C_r$? or $H=C_q \rtimes C_r$? or should we consider all possibilities?
When considering finite groups $G$ of order, $|G|=p^2qr$, where $p,q,r$ are distinct primes, let $F$ be a Fitting subgroup of $G$. Then $F$ and $G/F$ are both non-trivial and $G/F$ acts faithfully on $\bar{F}:=F/ \phi(F)$ so that no non-trivial normal subgroup of $G/F$ stabilizes a series through $\bar{F}$.
And when $|F|=pr$. In this case $\phi(F)=1$ and $Aut(F)=C_{p-1} \times C_{r-1}$. Thus $G/F$ is abelian and $G/F \cong C_{p} \times C_{q}$.
In this case how can I write G using notations/symbols?
Is it like $G \cong (C_{p} \times C_{r}) \rtimes (C_{p} \times C_{q})$?
First question: Then it is, $G= F \rtimes (C_p \times C_q)$. But how do we write $F$ ? Do we have to think of all the possibilities of $F$ of order $pr$ and write as $G= (C_p \times C_r) \rtimes (C_p \times C_q)$ or $G= (C_p \rtimes C_r) \rtimes (C_p \times C_q)$ etc.?
As a second case we can consider the case where $C_q$ acts trivially on $C_p$. So then how to write $G$ using notations?
There it is also mentioned that we can distinguish among 2 cases. First, suppose that the sylow $q$-subgroup of $G/F$ acts non trivially on the sylow $p$-subgroup of $F$. Then $q|(p-1) and $G$ splits over $F$. Thus the group has the form $F \rtimes G/F$.
A presentation $\langle S\mid R\rangle$ is a Dehn presentation if for some $n\in\Bbb N$ there are words $u_1,\cdots,u_n$ and $v_1,\cdots, v_n$ such that $R=\{u_iv_i^{-1}\}$, $|u_i|>|v_i|$ and for all words $w$ in $(S\cup S^{-1})^\ast$ representing the trivial element of the group one of the $u_i$ is a subword of $w$
If you have such a presentation there's a trivial algorithm to solve the word problem: Take a word $w$, check if it has $u_i$ as a subword, in that case replace it by $v_i$, keep doing so until you hit the trivial word or find no $u_i$ as a subword
There is good motivation for such a definition here
So I don't know how to do it precisely for hyperbolic groups, but if $S$ is a surface of genus $g \geq 2$, to get a geodesic representative for a class $[\alpha] \in \pi_1(S)$ where $\alpha$ is an embedded loop, one lifts it to $\widetilde{\alpha}$ in $\Bbb H^2$ by the locally isometric universal covering, and then the deck transformation corresponding to $[\alpha]$ is an isometry of $\Bbb H^2$ which preserves the embedded arc $\widetilde{\alpha}$
It has to be an isometry fixing a geodesic $\gamma$ with endpoints at the boundary being the same as the endpoints of $\widetilde{\alpha}$.
Consider the homotopy of $\widetilde{\alpha}$ to $\gamma$ by straightline homotopy, but straightlines being the hyperbolic geodesics. This is $\pi_1(S)$-equivariant, so projects to a homotopy of $\alpha$ and the image of $\gamma$ (which is a geodesic in $S$) downstairs, and you have your desired representative
I don't know how to interpret this coarsely in $\pi_1(S)$
@anakhro Well, they print in bulk, and on really cheap paper, almost transparent and very thin, and offset machine is really cheaper per page than a printer, you know, but you should be printing in bulk, its all economy of scale.
@ParasKhosla Yes, I am Indian, and trying to get in some good masters progam in math.
Algebraic graph theory is a branch of mathematics in which algebraic methods are applied to problems about graphs. This is in contrast to geometric, combinatoric, or algorithmic approaches. There are three main branches of algebraic graph theory, involving the use of linear algebra, the use of group theory, and the study of graph invariants.== Branches of algebraic graph theory ===== Using linear algebra ===The first branch of algebraic graph theory involves the study of graphs in connection with linear algebra. Especially, it studies the spectrum of the adjacency matrix, or the Lap...
I can probably guess that they are using symmetries and permutation groups on graphs in this course.
For example, orbits and studying the automorphism groups of graphs.
@anakhro I have heard really good thing about Palka. Also, if you do not worry about little sacrifice of rigor (e.g. counterclockwise orientation based on your intuition, rather than, on winding numbers, etc.), Howie's Complex analysis is good. It is teeming with typos here and there, but you will be fine, i think. Also, thisbook contains all the solutions in appendix!
Got a simple question: I gotta find kernel of linear transformation $F(P)=xP^{''}(x) + (x+1)P^{'''}(x)$ where $F: \mathbb{R}_3[x] \to \mathbb{R}_3[x]$, so I think it would be just $\ker (F) = \{ ax+b : a,b \in \mathbb{R} \}$ since only polynomials of degree at most 1 would give zero polynomial in this case
@chandx you're looking for all the $G = P''$ such that $xG + (x+1)G' = 0$; if $G \neq 0$ you can solve the DE to get $G'/G = -x/(x+1) = -1 + 1/(x+1) \implies \ln G = -x + \ln(1+x) \implies G = (1+x)e^(-x) + C$ which is obviously not a polyonomial, so $G = 0$ and thus $P = ax + b$
could you suppose that $\operatorname{deg} P \geq 2$ and show that you wouldn't have nonzero polynomials? Sure.
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Part a) of this is fine, but I'm really stuck on part b) and I have a test on this in an hours time, does anyone have any hints?
closed as off-topic by Toby Mak, Dietrich Burde, Shogun, mrtaurho, Yanior Weg Aug 12 at 19:32
This question appears to be off-topic. The users who voted to close gave this specific reason:
" This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Toby Mak, Dietrich Burde, Shogun, mrtaurho, Yanior Weg
I think the best place to start is to first give things names, e.g.
$$(a,b,c) \ast (\lambda, \mu, \nu) =: (x,y,z) $$
And then simplify the RHS of the expression
$$ F(x,y,z) = F(a,b,c) \cdot F(\lambda, \mu, \nu)$$
to get an expression for $x$, $y$ and $z$. You'll use part a) for this.
EDIT: $(\alpha, \beta, \gamma)$ were not the best choice of letters!
Good luck!
$(a\alpha^2+b\alpha+c)(d\alpha^2+e\alpha+f)=ad(\alpha^4)+(bd+ae)\alpha^3+(af+be+cd)\alpha^2+(bf+ce)\alpha+cf$
Then use the relation $\alpha^3=-3\alpha+24/5$ to reduce this to something like
$ad(-3\alpha^2+24/5\alpha)+(bd+ae)(-3\alpha+24/5)+(af+be+cd)\alpha^2+(bf+ce)\alpha+cf =(-3ad+af+be+cd)\alpha^2+(24ad/5-3bd-3ae+bf+ce)\alpha+(24bd/5+24ae/5+cf)$
which will give us a way to compose $(a,b,c)*(d,e,f)=(-3ad+af+be+cd,24ad/5-3bd-3ae+bf+ce,24bd/5+24ae/5+cf)$
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As Sasho suggested, I am putting my comment as an answer.
The separations between
monotone versions of $\mathsf{NC}^1/\mathsf{poly}$ and $\mathsf{P/poly}$ versions of complexity are long known (Karchmer-Wigderson, Grigni-Sipser, etc), but in the non-monotone world almost nothing was known. Fortunately, Ben Rossman has recently found the first separation of formulas vs. circuits in the bounded depth setting.
Let $\mathrm{Circuit}(S,d)$ (resp., $\mathrm{Formula}(S,d)$) denote the set of all boolean functions computable by unbounded fanin circuits (resp. formulas) of depth $\leq d$ and size $\leq S$. It is clear that$$\mathrm{Circuit}(S,d) \subseteq \mathrm{Formula}(S^d,d).$$In particular,$$\mathrm{Circuit}(n^{O(1)},d) \subseteq \mathrm{Formula}(n^{O(d)},d).$$What Ben has shown is that, if $d=d(n)\leq \log\log\log n$, then$$\mathrm{Circuit}(n^{O(1)},d) \not\subseteq \mathrm{Formula}(n^{o(d)},d).$$Even more important is that he shows this separation on an
explicit and basic function $\mathrm{STCONN}(n,k)$: given an $n$-vertex graph, decide whether it has an $s$-$t$ path of length $\leq k$. This function is in $\mathrm{Circuit}(n^{O(1)},\log k)$.His main result is: if $dk^3\leq \log n/\log\log n$ then$$\mathrm{STCONN}(n,k)\in \mathrm{Formula}(S,d)\ \Longrightarrow\ S\geq n^{\Omega(\log k)}.$$This implies a tight depth lower bound: if $k\leq \log\log n$ then$$\mathrm{STCONN}(n,k)\in \mathrm{Circuit}(n^{O(1)},d)\ \Longrightarrow\ d=\Theta(\log k).$$The existing techniques for small-depth circuit -- namely switching lemmas and approximation bylow-degree polynomials-- do not distinguish between formulas and circuits due to their bottom-up nature. Top-down arguments, as Karchmer-Wigderson games, are difficult to realize in thenon-monotone case. What Ben uses is a combination of these arguments.
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Excitable media
Vladimir S. Zykov (2008), Scholarpedia, 3(5):1834. doi:10.4249/scholarpedia.1834 revision #91246 [link to/cite this article]
Contents Definition
An excitable medium is a dynamical system distributed continuously in space, each elementary segment of which possesses the property of excitability. Neighboring segments of an excitable medium interact with each other by diffusion-like local transport processes. In an excitable medium it is possible for excitation to be passed from one segment to another by means of local coupling. Thus, an excitable medium is able to support propagation of undamped solitary excitation waves, as well as wave trains.
Originally, the term excitability referred to the property of living organisms (or of their constituent cells) to respond strongly to the action of a relatively weak external stimulus. A typical example of excitability is the generation of a spike of transmembrane potential (action potential) by a cardiac cell, induced by a short depolarizing electrical perturbation of a resting state. Usually, the shape of the generated action potential does not depend on the perturbation strength, as long as the perturbation exceeds some threshold (all-or-nothing principle). After the generation of this strong response, the system returns to its initial resting state. A subsequent excitation can be generated after a suitable length of time, called the refractory period, has passed. This obviously non-linear, dynamical behavior is characteristic of a large class of systems in biology, chemistry and physics (Winfree, 2000; Krinsky, Swinney 1991; Kapral, Showalter 1995; Zykov, 1987; Mikhailov, 1990; Kaplan, Glass, 1995; Izhikevich, 2007).
Examples
The most prominent examples of excitable media are
propagation of electrical excitation in various biological tissues, including nerve fiber and myocardium concentration waves in the bromate-malonic acid reagent (the Belousov-Zhabotinsky reaction) propagating waves during the aggregation of social amoeba (Dictyostelium) waves of spreading depression in the retina of eye concentration waves in yeast extract doing glycolysis calcium waves within frog eggs Mathematical models
In many applications the mathematical description of the dynamical processes in an excitable medium can be represented in the form of a reaction-diffusion system\[ \frac{\partial E_i}{\partial t}=F_i(\nabla E_i,\vec{E}) + \nabla(D_i\nabla E_i) + I_i(\vec{r},t). \]
Here the vector \(\vec{E}\) determines the state of the system, the \(F_i\) are nonlinear functions of \(\vec{E}\) and, perhaps, \(\nabla E_i\ ,\) the \(D_i\) are diffusion coefficients. The \(I_i\) are external actions varying in space and time, which can be used for initiation of excitation waves. The most famous example of such type of descriptions are the Oregonator model, Hodgkin-Huxley model, Noble model, other models of cardiac cell, etc.
The basic features of the self-sustained dynamics in excitable media can be reproduced by the relatively simple and widely used two-component activator-inhibitor system
\( {\partial u \over \partial t} = \nabla^2 u + f(u,v), \)
\( {\partial v \over \partial t} = \sigma \nabla^2 v + \epsilon g(u,v). \)
Here \(u(\vec {r},t)\) and \(v(\vec {r},t)\ ,\) describe the state of the system, nonlinear functions \(f(u,v)\) and \(g(u,v)\) specify the local dynamics and \(\sigma\) determines the ratio between two diffusion constants. If parameter \(\epsilon << 1\) this reaction-diffusion system exhibit relaxational dynamics with intervals of fast and slow motion. Depending on the particular shape of the nonlinear functions \(f(u,v)\) and \(g(u,v)\) this system is refered to as the Brusselator, Fitzhugh-Nagumo model, Rinzel-Keller model, Barkley model, Morris-Lecar model, etc.
These generic systems of partial differential equations can be used to simulate the dynamics of one-, two- or three-dimensional media (Tyson, Keener, 1988). They can be adjusted to particular applications. For instance, the diffusion flows can be anisotropic or can include cross-diffusion terms. Some local and/or global feedback loops can be induced to reproduce naturally existing or artificially created stabilizing or destabilizing circuits.
An important example of possible modification represents the bidomain model describing the cardiac tissue as consisting of two colocated continuous media termed the intracellular and extracellular domain. The intracellular and extracellular potentials \(\phi_i\) and \(\phi_e\) are specified by the bidomain equations with conductivity tensors \(G_i\) and \(G_e\ :\)
\( \nabla \cdot (G_i\nabla \phi_i)= I_m, \)
\( \nabla \cdot (G_e\nabla \phi_e)= -I_m, \)
where the transmembrane current density \(I_m\) is determined as the following
\( I_m=\beta (C_m {\partial V_m \over \partial t} +I _{ion} + I_s). \)
Here \(\beta\) is the membrane surface-to-volume ratio, \(C_m\) is the membrane capacitance per unit area, \(I_{ion}\) is the ionic current density generated by the cell membrane and \(I_s\) is an imposed stimulation current density. The ionic current \(I_{ion}\) depends on the transmembrane potential \(V_m=\phi_i-\phi_e\) and on the vector \(\vec{m}\) of gate variables in accordance to one of the models of cardiac cell. Each gate variable obeys first order ordinary differential equation as was firstly specified by the Hodgkin-Huxley model\[ {d m_i \over d t}= \alpha_i(v_m)(1-m_i)-\beta_i(V_m)m_i. \]
In the limit \( G_e \rightarrow \infty \) the extracellular potential becomes to be uniform in space and the bidomain description is reduced to a multi-component reaction-diffusion system reproducing many important features of cardiac dynamics. However, the bidomain model allows for to consider the effects of, and the effects on, the surrounding extracellular electrical field (Keener, Sneyd, 1998).
Cellular automata and simplified descriptions of kinematics of excitation waves represent additional effective tools to describe different aspect of excitable-medium dynamics. They minimize computation time and under some approximations allow for analytical results.
Self-organized patterns
The waves in passive, linear media differ considerably from waves in excitable media, which are nonlinear and driven by a distributed energy source. One-dimensional excitable media are able to support a single traveling wave and also wave trains propagating without decrement. If the medium size is sufficiently large, the propagation velocity and the wave profile do not depend on the initial and/or boundary conditions. This distinguishes the waves in excitable media from solitons propagating in nonlinear conservative systems, where velocities and pulse shapes strongly depend on the initial conditions.
In two-dimensional excitable media, one can observe expanding target patterns and rotating spiral waves. In three dimensions, rotating scroll waves are possible. All these dynamic phenomena represent well-known examples of self-organization in complex systems resulting in pattern formation.
The structure and the properties of the self-organized patterns are established due to a balance between the energy influx from internal distributed sources and the energy dissipation. Thus, the spatio-temporal patterns in excitable media belong to self-organization processes, which occur far away from the thermodynamical equilibrium.
References A.T. Winfree, The Geometry of Biological Time (Springer, Berlin, Heidelberg, 2000). V. Krinsky and H. Swinney (eds), Wave and patterns in biological and chemical excitable media, (North-Holland, Amsterdam, 1991). R. Kapral and K. Showalter (eds.), Chemical Waves and Patterns, (Kluwer, Dordrecht, 1995). V.S. Zykov, Simulation of Wave Processes in Excitable Media, (Manchester Univ. Press, Manchester, 1987). A.S. Mikhailov, Foundation of Synergetics, (Springer, New York, 1990). D. Kaplan and L. Glass, Understanding Nonlinear Dynamics, (Springer, New York, 1995). E.M. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting, (MTI Press, 2007). J.J. Tyson and J.P. Keener, Singular perturbation theory of traveling waves in excitable media, Physica D 32:327-361 (1988). J.P. Keener and J. Sneyd, Mathematical Physiology, (Springer, New York, 1998). Internal references Gregoire Nicolis and Catherine Rouvas-Nicolis (2007) Complex systems. Scholarpedia, 2(11):1473. James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629. Eugene M. Izhikevich (2007) Equilibrium. Scholarpedia, 2(10):2014. Eugene M. Izhikevich and Richard FitzHugh (2006) FitzHugh-Nagumo model. Scholarpedia, 1(9):1349. Hans Meinhardt (2006) Gierer-Meinhardt model. Scholarpedia, 1(12):1418. C. Frank Starmer (2007) Initiation of excitation waves. Scholarpedia, 2(2):1848. James Sneyd (2007) Models of calcium dynamics. Scholarpedia, 2(3):1576. Harold Lecar (2007) Morris-Lecar model. Scholarpedia, 2(10):1333. Martin Fink and Denis Noble (2008) Noble model. Scholarpedia, 3(2):1803. Richard J. Field (2007) Oregonator. Scholarpedia, 2(5):1386. Gregoire Nicolis and Anne De Wit (2007) Reaction-diffusion systems. Scholarpedia, 2(9):1475. John Dowling (2007) Retina. Scholarpedia, 2(12):3487. See also
Population dynamics, Turing instability, Ginzburg-Landau equation, Fisher-Kolmogorov-Petrovski-Piskunov model, Models of calcium dynamics, Models of heart, Cardiac arrhythmia, Neuronal networks, Gierer-Meinhardt model, Morphogenesis
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Randomized learning of the second-moment matrix of a smooth function
1.
Institute of Electrical Engineering, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
2.
Department of Electrical Engineering, Colorado School of Mines, Denver, CO 80401, USA
3.
Departments of Statistics and Computer Science, Rutgers University, Piscataway, NJ 08854, USA
4.
Department of Computer Science, University of Colorado Boulder, Boulder, CO 80309, USA
Consider an open set $ \mathbb{D}\subseteq\mathbb{R}^n $, equipped with a probability measure $ \mu $. An important characteristic of a smooth function $ f:\mathbb{D}\rightarrow\mathbb{R} $ is its
second-moment matrix $ \Sigma_{\mu}: = \int \nabla f(x) \nabla f(x)^* \mu(dx) \in\mathbb{R}^{n\times n} $, where $ \nabla f(x)\in\mathbb{R}^n $ is the gradient of $ f(\cdot) $ at $ x\in\mathbb{D} $ and $ * $ stands for transpose. For instance, the span of the leading $ r $ eigenvectors of $ \Sigma_{\mu} $ forms an active subspace of $ f(\cdot) $, which contains the directions along which $ f(\cdot) $ changes the most and is of particular interest in ridge approximation. In this work, we propose a simple algorithm for estimating $ \Sigma_{\mu} $ from random point evaluations of $ f(\cdot) $ without imposing any structural assumptions on $ \Sigma_{\mu} $. Theoretical guarantees for this algorithm are established with the aid of the same technical tools that have proved valuable in the context of covariance matrix estimation from partial measurements. Keywords:Active subspace, second-moment matrix, covariance estimation, ridge approximation, approximation theory. Mathematics Subject Classification:Primary: 68W25; Secondary: 68W20. Citation:Armin Eftekhari, Michael B. Wakin, Ping Li, Paul G. Constantine. Randomized learning of the second-moment matrix of a smooth function. Foundations of Data Science, 2019, 1 (3) : 329-387. doi: 10.3934/fods.2019015
References:
[1]
R. Adamczak,
Logarithmic Sobolev inequalities and concentration of measure for convex functions and polynomial chaoses,
[2]
F. P. Anaraki and S. Hughes, Memory and computation efficient PCA via very sparse random projections, in
[3]
M. Azizyan, A. Krishnamurthy and A. Singh, Extreme compressive sampling for covariance estimation,
[4]
D. S. Bernstein,
[5]
I. Bogunovic, V. Cevher, J. Haupt and J. Scarlett, Active learning of self-concordant like multi-index functions, in
[6] [7]
E. J. Candes,
[8]
Y. Chen, Y. Chi and A. J. Goldsmith,
Exact and stable covariance estimation from quadratic sampling via convex programming,
[9]
A. Cohen, I. Daubechies, R. DeVore, G. Kerkyacharian and D. Picard,
Capturing ridge functions in high dimensions from point queries,
[10] [11] [12]
P. G. Constantine, A. Eftekhari and R. Ward, A near-stationary subspace for ridge approximation,
[13]
R. D. Cook, Using dimension-reduction subspaces to identify important inputs in models of physical systems, in
[14]
G. Dasarathy, P. Shah, B. N. Bhaskar and R. D. Nowak,
Sketching sparse matrices, covariances, and graphs via tensor products,
[15]
R. DeVore, G. Petrova and P. Wojtaszczyk,
Approximation of functions of few variables in high dimensions,
[16] [17]
A. Eftekhari, L. Balzano and M. B. Wakin, What to expect when you are expecting on the Grassmannian,
[18]
A. Eftekhari, M. B. Wakin and R. A. Ward, MC$^2$: A two-phase algorithm for leveraged matrix completion,
[19]
M. Fornasier, K. Schnass and J. Vybiral,
Learning functions of few arbitrary linear parameters in high dimensions,
[20] [21] [22] [23]
K. Fukumizu, F. R. Bach and M. I. Jordan,
Dimensionality reduction for supervised learning with reproducing kernel Hilbert spaces,
[24] [25] [26] [27]
A. Gonen, D. Rosenbaum, Y. Eldar and S. Shalev-Shwartz, The sample complexity of subspace learning with partial information,
[28] [29]
N. Halko, P. G. Martinsson and J. A. Tropp,
Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions,
[30]
T. J. Hastie and R. J. Tibshirani,
[31]
J. Haupt, R. M. Castro and R. Nowak,
Distilled sensing: Adaptive sampling for sparse detection and estimation,
[32] [33] [34] [35]
S. Keiper, Analysis of generalized ridge functions in high dimensions, in
[36]
M. Kolar and E. P. Xing, Consistent covariance selection from data with missing values, in
[37] [38] [39] [40]
E. Liberty, F. Woolfe, P. G. Martinsson, V. Rokhlin and M. Tygert,
Randomized algorithms for the low-rank approximation of matrices,
[41]
P. L. Loh and M. J. Wainwright,
High-dimensional regression with noisy and missing data: Provable guarantees with non-convexity,
[42] [43] [44]
F. W. J. Olver,
[45] [46] [47]
F. Pourkamali-Anaraki, Estimation of the sample covariance matrix from compressive measurements,
[48]
C. E. Rasmussen and C. K. I. Williams,
[49]
P. Ravikumar, M. J. Wainwright, G. Raskutti and B. Yu,
High-dimensional covariance estimation by minimizing $l_1$-penalized log-determinant divergence,
[50] [51]
B. Recht, M. Fazel and P. A. Parrilo,
Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization,
[52]
A. M. Samarov,
Exploring regression structure using nonparametric functional estimation,
[53]
T. Sarlos, Improved approximation algorithms for large matrices via random projections, in
[54] [55]
J. F. Traub and H. Wozniakowski,
[56]
R. Tripathy, I. Bilionis and M. Gonzalez,
Gaussian processes with built-in dimensionality reduction: Applications to high-dimensional uncertainty propagation,
[57]
H. Tyagi and V. Cevher,
Learning non-parametric basis independent models from point queries via low-rank methods,
[58] [59]
F. Vivarelli and C. K. I. Williams, Discovering hidden features with Gaussian processes regression,
[60] [61]
H. Wendland,
[62]
Y. Xia, H. Tong, W. K. Li and L. X. Zhu,
An adaptive estimation of dimension reduction space,
[63] [64]
show all references
References:
[1]
R. Adamczak,
Logarithmic Sobolev inequalities and concentration of measure for convex functions and polynomial chaoses,
[2]
F. P. Anaraki and S. Hughes, Memory and computation efficient PCA via very sparse random projections, in
[3]
M. Azizyan, A. Krishnamurthy and A. Singh, Extreme compressive sampling for covariance estimation,
[4]
D. S. Bernstein,
[5]
I. Bogunovic, V. Cevher, J. Haupt and J. Scarlett, Active learning of self-concordant like multi-index functions, in
[6] [7]
E. J. Candes,
[8]
Y. Chen, Y. Chi and A. J. Goldsmith,
Exact and stable covariance estimation from quadratic sampling via convex programming,
[9]
A. Cohen, I. Daubechies, R. DeVore, G. Kerkyacharian and D. Picard,
Capturing ridge functions in high dimensions from point queries,
[10] [11] [12]
P. G. Constantine, A. Eftekhari and R. Ward, A near-stationary subspace for ridge approximation,
[13]
R. D. Cook, Using dimension-reduction subspaces to identify important inputs in models of physical systems, in
[14]
G. Dasarathy, P. Shah, B. N. Bhaskar and R. D. Nowak,
Sketching sparse matrices, covariances, and graphs via tensor products,
[15]
R. DeVore, G. Petrova and P. Wojtaszczyk,
Approximation of functions of few variables in high dimensions,
[16] [17]
A. Eftekhari, L. Balzano and M. B. Wakin, What to expect when you are expecting on the Grassmannian,
[18]
A. Eftekhari, M. B. Wakin and R. A. Ward, MC$^2$: A two-phase algorithm for leveraged matrix completion,
[19]
M. Fornasier, K. Schnass and J. Vybiral,
Learning functions of few arbitrary linear parameters in high dimensions,
[20] [21] [22] [23]
K. Fukumizu, F. R. Bach and M. I. Jordan,
Dimensionality reduction for supervised learning with reproducing kernel Hilbert spaces,
[24] [25] [26] [27]
A. Gonen, D. Rosenbaum, Y. Eldar and S. Shalev-Shwartz, The sample complexity of subspace learning with partial information,
[28] [29]
N. Halko, P. G. Martinsson and J. A. Tropp,
Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions,
[30]
T. J. Hastie and R. J. Tibshirani,
[31]
J. Haupt, R. M. Castro and R. Nowak,
Distilled sensing: Adaptive sampling for sparse detection and estimation,
[32] [33] [34] [35]
S. Keiper, Analysis of generalized ridge functions in high dimensions, in
[36]
M. Kolar and E. P. Xing, Consistent covariance selection from data with missing values, in
[37] [38] [39] [40]
E. Liberty, F. Woolfe, P. G. Martinsson, V. Rokhlin and M. Tygert,
Randomized algorithms for the low-rank approximation of matrices,
[41]
P. L. Loh and M. J. Wainwright,
High-dimensional regression with noisy and missing data: Provable guarantees with non-convexity,
[42] [43] [44]
F. W. J. Olver,
[45] [46] [47]
F. Pourkamali-Anaraki, Estimation of the sample covariance matrix from compressive measurements,
[48]
C. E. Rasmussen and C. K. I. Williams,
[49]
P. Ravikumar, M. J. Wainwright, G. Raskutti and B. Yu,
High-dimensional covariance estimation by minimizing $l_1$-penalized log-determinant divergence,
[50] [51]
B. Recht, M. Fazel and P. A. Parrilo,
Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization,
[52]
A. M. Samarov,
Exploring regression structure using nonparametric functional estimation,
[53]
T. Sarlos, Improved approximation algorithms for large matrices via random projections, in
[54] [55]
J. F. Traub and H. Wozniakowski,
[56]
R. Tripathy, I. Bilionis and M. Gonzalez,
Gaussian processes with built-in dimensionality reduction: Applications to high-dimensional uncertainty propagation,
[57]
H. Tyagi and V. Cevher,
Learning non-parametric basis independent models from point queries via low-rank methods,
[58] [59]
F. Vivarelli and C. K. I. Williams, Discovering hidden features with Gaussian processes regression,
[60] [61]
H. Wendland,
[62]
Y. Xia, H. Tong, W. K. Li and L. X. Zhu,
An adaptive estimation of dimension reduction space,
[63] [64]
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We summarize the discussion of parameters to express numbers suitable to be factored::
$$ N = C_1 r^t + C_2 s^u $$
by the Special Number Field Sieve (SNFS) from the paper "An Implementation of the Number Field Sieve" by Marije Elkenbracht-Huizing (1996).
She describes the Number Field Sieve this way:
Let $n$ be the odd number to be factored. It is easy to check whether $n$ is a prime number or a prime power ,and we assume that it is neither. Like [Multiple Polynomial Quadratic Sieve], the NFS tries to find a solution of the equation $v^2=w^2 \bmod{n}.$ For at least half of the pairs $(v \bmod n, w \bmod n)$ with $v^2\neq w^2 \bmod n$ and $v$ and $w$ relatively prime to $n$, the greatest common divisor of $n$ and $v-w$
gives a nontrivial factor of $n$.
To construct $v$ and $w$ we first choose two polynomials $$ f_1(x) = c_{1,d_1}x^{d_1} + c_{1,d_1-1}x^{d_1-1} + \ldots + c_{1,0} $$ $$ f_2(x) = c_{2,d_2}x^{d_2} + c_{2,d_2-1}x^{d_2-1} + \ldots + c_{2,0} $$ over $\mathbb{Z}$, with $f_1 \neq \pm f_2$, both irreducible over $\mathbb{Z}$ and having
content $\text{cont }f_i := \gcd(c_{i,d_i},\ldots,c_{i,0})$ equal to $1$; we also choose an integer $m$ that is a common root modulo $n$ of $f_1$ and $f_2$. In our implementation this is the only step in which the SNFS and the GNFS differ: in the SNFS we use the special form of $n$ to pick these polynomials by hand. One
polynomial will have very small coefficients compared to the coefficients of the polynomials we will use with the GNFS, where we search for a pair of polynomials with help of the computer. This makes SNFS faster than GNFS.
Tying what makes "SNFS faster than GNFS" to a manual process of picking a pair of (integer) polynomials (so that "one polynomial will have very small coefficients" compared to the polynomials "where we search... with help of the computer") makes the distinction contingent on the (advancing) state of the art in computer search for polynomials. Potentially the algorithms for polynomial selection will improve so far as to practically erase the distinction.
Analysis of what effect small coefficients have on the algorithm (in particular on the speed of the sieving phase) is beyond the limits of the present post. However useful perspective comes from examining the cases set out by Elkenbracht-Huizing's paper. Here are simplified tables of various factoring successes listed there:
Table 1: SNFS Example Polynomials and Timings$$ \begin{array}{|l|c|c|c|}\hlinen \text{ factor of } N & f_1(x) & f_2(x) & \begin{array}{c} \text{sieve} \\ \text{hours} \end{array} \\ \hline\begin{array}{c} \text{C98 from } \\ 7^{128} + 6^{128} \end{array} & x^4 + 1 & 6^{32}x - 7^{32} & 450 \\ \hline\begin{array}{c} \text{C106 from } \\ 2^{543} + 1 \end{array} & \begin{array}{c} 4x^4 \\+2x^2 + 1 \end{array} & x - 2^{90} & 250 \\ \hline\begin{array}{c} \text{C119 from } \\ 3^{319} - 1 \end{array} & \begin{array}{c} x^5 + x^4 \\ - 4x^3 -3x^2 \\+3x + 1\end{array} & \begin{array}{c} 3^{29}x \\- (3^{58} +1) \end{array} & 800 \\ \hline\begin{array}{c} \text{C123 from } \\ 2^{511} - 1 \end{array} & \begin{array}{c} x^6 -10x^4 \\+24x^2 -8\end{array} & \begin{array}{c} 2^{36}x \\- (2^{73} +1) \end{array} & 700 \\ \hline\begin{array}{c} \text{C135 from } \\ 73^{73} - 1 \end{array} & x^5 + 73^2 & x - 73^{15} & 2150 \\ \hline\begin{array}{c} \text{C165 from } \\ 12^{151} -1 \end{array} & 12x^5 - 1 & x - 12^{30} & \lt 1680 \\ \hline\end{array} $$
Table 2: GNFS Example Polynomials and Timings$$ \begin{array}{|l|c|c|c|}\hlinen \text{ factor of } N & f_1(x) & f_2(x) & \begin{array}{c} \text{sieve} \\ \text{hours} \end{array} \\ \hline\begin{array}{c} \text{C87 from } \\ 72^{99} + 1 \end{array} &\begin{array}{c} 1.5E20x^2 \\ - 1.4E21x \\- 8.3E21 \end{array} & \begin{array}{c} 1.0E21x^2 \\ + 7.0E21x \\- 7.4E22\end{array} & 2100 \\ \hline\begin{array}{c} \text{C87 from } \\ 72^{99} + 1 \end{array} &\begin{array}{c} 7.3E8x^2 \\ + 2.0E20x \\- 7.8E33\end{array} & \begin{array}{c} 2.2E9x^2 \\ - 7.9E21x \\- 2.3E34\end{array} & 1500 \\ \hline\begin{array}{c} \text{C97 from } \\ 12^{441} + 1 \end{array} &\begin{array}{c} -3.0E10x^2 \\+ 4.4E23x \\+ 3.6E36 \end{array} & \begin{array}{c} -5.4E11x^2 \\ - 4.8E24x \\+ 6.3E37 \end{array} & 3500 \\ \hline\begin{array}{c} \text{C105 from } \\ 3^{367} - 1 \end{array} &\begin{array}{c} 3.4E11x^2 \\ + 8.7E25x \\+ 5.4E38\end{array} & \begin{array}{c} 1.2E12x^2 \\ - 9.1E26x \\+ 1.3E41 \end{array} & \lt 7680 \\ \hline\begin{array}{c} \text{C106 from } \\ 12^{157} + 1 \end{array} &\begin{array}{c} 1.9E11x^2 \\ - 1.0E26x \\- 3.2E40\end{array} & \begin{array}{c} -7.9E11x^2 \\ - 4.0E26x \\+ 1.6E41 \end{array} & 11900 \\ \hline\begin{array}{c} \text{C107 from } \\ 6^{223} + 1 \end{array} &\begin{array}{c} -5.4E11x^2 \\ - 4.3E26x \\- 4.7E40\end{array} & \begin{array}{c} -2.4E11x^2 \\ + 7.6E26x \\- 3.1E41 \end{array} & 11200 \\\hline\end{array} $$
I'll append some additional remarks, but the first columns describe the number of digits of a composite factor of an expression, the second and third columns give the pair of polynomials used (in "scientific notation" for the large integer coefficients of the GNFS; see paper for exact values), and the last column gives a rough timing for the sieving (of relations) step.
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@JosephWright Well, we still need table notes etc. But just being able to selectably switch off parts of the parsing one does not need... For example, if a user specifies format 2.4, does the parser even need to look for e syntax, or ()'s?
@daleif What I am doing to speed things up is to store the data in a dedicated format rather than a property list. The latter makes sense for units (open ended) but not so much for numbers (rigid format).
@JosephWright I want to know about either the bibliography environment or \DeclareFieldFormat. From the documentation I see no reason not to treat these commands as usual, though they seem to behave in a slightly different way than I anticipated it. I have an example here which globally sets a box, which is typeset outside of the bibliography environment afterwards. This doesn't seem to typeset anything. :-( So I'm confused about the inner workings of biblatex (even though the source seems....
well, the source seems to reinforce my thought that biblatex simply doesn't do anything fancy). Judging from the source the package just has a lot of options, and that's about the only reason for the large amount of lines in biblatex1.sty...
Consider the following MWE to be previewed in the build in PDF previewer in Firefox\documentclass[handout]{beamer}\usepackage{pgfpages}\pgfpagesuselayout{8 on 1}[a4paper,border shrink=4mm]\begin{document}\begin{frame}\[\bigcup_n \sum_n\]\[\underbrace{aaaaaa}_{bbb}\]\end{frame}\end{d...
@Paulo Finally there's a good synth/keyboard that knows what organ stops are! youtube.com/watch?v=jv9JLTMsOCE Now I only need to see if I stay here or move elsewhere. If I move, I'll buy this there almost for sure.
@JosephWright most likely that I'm for a full str module ... but I need a little more reading and backlog clearing first ... and have my last day at HP tomorrow so need to clean out a lot of stuff today .. and that does have a deadline now
@yo' that's not the issue. with the laptop I lose access to the company network and anythign I need from there during the next two months, such as email address of payroll etc etc needs to be 100% collected first
@yo' I'm sorry I explain too bad in english :) I mean, if the rule was use \tl_use:N to retrieve the content's of a token list (so it's not optional, which is actually seen in many places). And then we wouldn't have to \noexpand them in such contexts.
@JosephWright \foo:V \l_some_tl or \exp_args:NV \foo \l_some_tl isn't that confusing.
@Manuel As I say, you'd still have a difference between say \exp_after:wN \foo \dim_use:N \l_my_dim and \exp_after:wN \foo \tl_use:N \l_my_tl: only the first case would work
@Manuel I've wondered if one would use registers at all if you were starting today: with \numexpr, etc., you could do everything with macros and avoid any need for \<thing>_new:N (i.e. soft typing). There are then performance questions, termination issues and primitive cases to worry about, but I suspect in principle it's doable.
@Manuel Like I say, one can speculate for a long time on these things. @FrankMittelbach and @DavidCarlisle can I am sure tell you lots of other good/interesting ideas that have been explored/mentioned/imagined over time.
@Manuel The big issue for me is delivery: we have to make some decisions and go forward even if we therefore cut off interesting other things
@Manuel Perhaps I should knock up a set of data structures using just macros, for a bit of fun [and a set that are all protected :-)]
@JosephWright I'm just exploring things myself “for fun”. I don't mean as serious suggestions, and as you say you already thought of everything. It's just that I'm getting at those points myself so I ask for opinions :)
@Manuel I guess I'd favour (slightly) the current set up even if starting today as it's normally \exp_not:V that applies in an expansion context when using tl data. That would be true whether they are protected or not. Certainly there is no big technical reason either way in my mind: it's primarily historical (expl3 pre-dates LaTeX2e and so e-TeX!)
@JosephWright tex being a macro language means macros expand without being prefixed by \tl_use. \protected would affect expansion contexts but not use "in the wild" I don't see any way of having a macro that by default doesn't expand.
@JosephWright it has series of footnotes for different types of footnotey thing, quick eye over the code I think by default it has 10 of them but duplicates for minipages as latex footnotes do the mpfoot... ones don't need to be real inserts but it probably simplifies the code if they are. So that's 20 inserts and more if the user declares a new footnote series
@JosephWright I was thinking while writing the mail so not tried it yet that given that the new \newinsert takes from the float list I could define \reserveinserts to add that number of "classic" insert registers to the float list where later \newinsert will find them, would need a few checks but should only be a line or two of code.
@PauloCereda But what about the for loop from the command line? I guess that's more what I was asking about. Say that I wanted to call arara from inside of a for loop on the command line and pass the index of the for loop to arara as the jobname. Is there a way of doing that?
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It is often said that gaussian process regression corresponds (GPR) to bayesian linear regression with a (possibly) infinite amount of basis functions. I am currently trying to understand this in detail to get an intuition for what kind of models I can express using GPR.
Do you think that this is a good approach to try to understand GPR?
In the book Gaussian Processes for Machine learning Rasmussen and Williams show that the set of gaussian processes described by the parameterised exponential squared kernel $$k(x,x';l)= \sigma_p^2\exp\left(-\frac{(x-x)^2}{2l^2}\right)$$ can be equivalently described as bayesian regression with prior belief $w \sim \mathcal{N}(0,\sigma_p^2 I)$ on the weights and an infinite amount of basis functions of the form $$\phi_c(x;l)=\exp\left(-\frac{(x-c)^2}{2l^2}\right) $$ Thus, the parameterisation of the kernel could by fully translated into a parameterisation of the basis functions.
Can the parameterisation of a differentiable kernel always be translated into parameterisation of the prior and the basis functions or are there differentiable kernels where e.g. the number of the basis functions depends on the configuration?
My understanding so far is that for a fixed kernel function k(x,x') Mercer's Theorem tells us that $k(x,x')$ can be expressed as $$k(x,x')=\sum_{i=1}^\infty \lambda_i\phi_i(x)\phi_i(x')$$ where $\phi_i$ is a function either into the reals or the complex numbers. Thus, for a given kernel the corresponding bayesian regression model has prior $w \sim \mathcal{N}(0,\text{diag}([\lambda_1^2,\ldots]))$ and basis functions $\phi_i$. Thus, every GP can even by formulated as bayesian linear regression model with diagonal prior. However, if we now use mercers theorem for every configuration of a parameterised kernel $k(x,x',\theta)$ that is differentiable at every $\theta$ the corresponding eigenvalues and eigenfunctions might by different for every configuration.
My next question is about the inverse of mercers theorem.
Which sets of basis functions lead to valid kernels?
And the extension
Which sets of parameterised basis functions lead to valid differentiable kernels?
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also, if you are in the US, the next time anything important publishing-related comes up, you can let your representatives know that you care about this and that you think the existing situation is appalling
@heather well, there's a spectrum
so, there's things like New Journal of Physics and Physical Review X
which are the open-access branch of existing academic-society publishers
As far as the intensity of a single-photon goes, the relevant quantity is calculated as usual from the energy density as $I=uc$, where $c$ is the speed of light, and the energy density$$u=\frac{\hbar\omega}{V}$$is given by the photon energy $\hbar \omega$ (normally no bigger than a few eV) di...
Minor terminology question. A physical state corresponds to an element of a projective Hilbert space: an equivalence class of vectors in a Hilbert space that differ by a constant multiple - in other words, a one-dimensional subspace of the Hilbert space. Wouldn't it be more natural to refer to these as "lines" in Hilbert space rather than "rays"? After all, gauging the global $U(1)$ symmetry results in the complex line bundle (not "ray bundle") of QED, and a projective space is often loosely referred to as "the set of lines [not rays] through the origin." — tparker3 mins ago
> A representative of RELX Group, the official name of Elsevier since 2015, told me that it and other publishers “serve the research community by doing things that they need that they either cannot, or do not do on their own, and charge a fair price for that service”
for example, I could (theoretically) argue economic duress because my job depends on getting published in certain journals, and those journals force the people that hold my job to basically force me to get published in certain journals (in other words, what you just told me is true in terms of publishing) @EmilioPisanty
> for example, I could (theoretically) argue economic duress because my job depends on getting published in certain journals, and those journals force the people that hold my job to basically force me to get published in certain journals (in other words, what you just told me is true in terms of publishing)
@0celo7 but the bosses are forced because they must continue purchasing journals to keep up the copyright, and they want their employees to publish in journals they own, and journals that are considered high-impact factor, which is a term basically created by the journals.
@BalarkaSen I think one can cheat a little. I'm trying to solve $\Delta u=f$. In coordinates that's $$\frac{1}{\sqrt g}\partial_i(g^{ij}\partial_j u)=f.$$ Buuuuut if I write that as $$\partial_i(g^{ij}\partial_j u)=\sqrt g f,$$ I think it can work...
@BalarkaSen Plan: 1. Use functional analytic techniques on global Sobolev spaces to get a weak solution. 2. Make sure the weak solution satisfies weak boundary conditions. 3. Cut up the function into local pieces that lie in local Sobolev spaces. 4. Make sure this cutting gives nice boundary conditions. 5. Show that the local Sobolev spaces can be taken to be Euclidean ones. 6. Apply Euclidean regularity theory. 7. Patch together solutions while maintaining the boundary conditions.
Alternative Plan: 1. Read Vol 1 of Hormander. 2. Read Vol 2 of Hormander. 3. Read Vol 3 of Hormander. 4. Read the classic papers by Atiyah, Grubb, and Seeley.
I am mostly joking. I don't actually believe in revolution as a plan of making the power dynamic between the various classes and economies better; I think of it as a want of a historical change. Personally I'm mostly opposed to the idea.
@EmilioPisanty I have absolutely no idea where the name comes from, and "Killing" doesn't mean anything in modern German, so really, no idea. Googling its etymology is impossible, all I get are "killing in the name", "Kill Bill" and similar English results...
Wilhelm Karl Joseph Killing (10 May 1847 – 11 February 1923) was a German mathematician who made important contributions to the theories of Lie algebras, Lie groups, and non-Euclidean geometry.Killing studied at the University of Münster and later wrote his dissertation under Karl Weierstrass and Ernst Kummer at Berlin in 1872. He taught in gymnasia (secondary schools) from 1868 to 1872. He became a professor at the seminary college Collegium Hosianum in Braunsberg (now Braniewo). He took holy orders in order to take his teaching position. He became rector of the college and chair of the town...
@EmilioPisanty Apparently, it's an evolution of ~ "Focko-ing(en)", where Focko was the name of the guy who founded the city, and -ing(en) is a common suffix for places. Which...explains nothing, I admit.
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The halting problem states there is no algorithm that will determine if a given program halts. As a consequence, there should be programs about which we can not tell whether they terminate or not. What are the simplest (smallest) known examples of such programs?
A pretty simple example could be a program testing the Collatz conjecture:
$$ f(n) = \begin{cases} \text{HALT}, &\text{if $n$ is 1} \\ f(n/2), & \text{if $n$ is even} \\ f(3n+1), & \text{if $n$ is odd} \end{cases} $$
It's known to halt for $n$ up to at least $5 × 2^{60} ≈ 5.764 × 10^{18}$, but in general it's an open problem.
The halting problem states there is no algorithm that will determine if a given program halts. As a consequence, there should be programs about which we can not tell whether they terminate or not.
"We" are not an algorithm =) There is no
general algorithm that could determine if a given program halts for every program.
What are the simplest (smallest) known examples of such programs?
Consider the following program:
n = 3while true: if is_perfect(n): halt() n = n + 2
Function is_perfect checks whether n is a perfect number. It is unknown whether there are any odd perfect numbers, so we don't know whether this program halts or not.
You write:
The halting problem states there is no algorithm that will determine if a given program halts. As a consequence, there should be programs about which we can not tell whether they terminate or not.
This is a non-sequitur, in both directions. You succumb to a common fallacy that is worth addressing.
Given any fixed program $P$, its halting problem ("Does $P$ always halt?") is
always decidable, because the answer is either "yes" or "no". Even if you can not tell which it is, you know that one of the two trivial algorithms that answer always "yes" resp. "no" solves the $P$-halting problem.
Only if you require that the algorithm should solve the Halting problem for all¹ programs can you show that no such algorithm can exist.
Now, knowing that the Halting problem is undecidable does not imply that there are any programs
nobody can not prove termination or looping of. Even if you are not more powerful than a Turing machine (which is only a hypothesis, not proven fact), all we know is that no single algorithm/person can provide such proof for all programs. There may be a different person being able to decide for each program.
Some more related reading:
How can it be decidable whether $\pi$ has some sequence of digits? Human computing power: Can humans decide the halting problem on Turing Machines? Algorithm to solve Turing's "Halting problem" Program synthesis, decidability and the halting problem Is it possible to solve the halting problem if you have a constrained or a predictable input? Why are the total functions not enumerable?
So you see that your actual question (as repeated below) has nothing to do with whether the halting problem is computable. At all.
What are the simplest (smallest) known examples of [programs we don't know to halt or loop]?
This in itself is a valid question; others have given good answers. Basically, you can transform every statement $S$ with unknown truth value into an example, provided it
does have a truth value:
$\qquad\displaystyle g(n) = \begin{cases}1, &S \text{ true},\\ g(n+1), &\text{else}.\end{cases}$
Granted, these are not very "natural".
Not necessarily all, but "many" in some sense. Infinitely many, at least.
Any open problem regarding the existence of a number with particular properties gives rise to such a program (the one which searches for such a number). For example, take the Collatz conjecture; since we don't know if it is true, we also don't know if the following program terminates:
n:=1; found:=false; while not found do s:={}; i:=n; while i not in s do add i to s; if i even then i:=i/2 else i:=3i+1 if 1 not in s then found:=true; n:=n+1
Given that the Busy Beaver problem is not solved for a 5-state-2-symbol Turing machine, there must be a Turing machine with only five states and only two symbols which has not been shown to halt or not when started for an empty tape. That is a very short, concise, and closed program.
the question is tricky because decidability (the CS equivalent formalization/ generalization of halting problem) is associated with
languages so it needs to be recast in that format. this seems to not be pointed out much, but many open problems in math/ CS can be readily converted to problems (languages) of unknown decidability. this is because of a tight correspondence between theorem proving and (un)decidability analysis. for example (somewhat like the other answer wrt odd perfect numbers), take the twin primes conjecture which dates to the Greeks (over 2 millenia ago) and is subject to major recent research advances eg by Zhang/ Tao. convert it to an algorithmic problem as follows:
Input:
n. Output: Y/N there exists at least ntwin primes.
the algorithm searches for twin primes and halts if it finds
n of them. it is not known if this language is decidable. resolution of the twin primes problem (which asks if there are a finite or infinite number) would also resolve the decidability of this language (if it is also proven/ discovered how many there are, if finite).
another example, take the Riemann hypothesis and consider this language:
Input:
n. Output: Y/N there exist at least nnontrivial zeroes of the Riemann zeta function.
the algorithm searches for nontrivial zeroes (the code is not especially complex, its similar to root finding, and there are other equivalent formulations that are relatively simple, which basically calculate sums of "parity" of all primes less than
x etc) and halts if it finds n of them and again, its not known if this language is decidable and resolution is "nearly" equivalent to solving the Riemann conjecture.
now, how about an even more spectacular example? (
caveat, probably more controversial as well)
Input: c: Output: Y/N there exists an O(n
c) algorithm for SAT.
similarly, resolution of decidability of this language is nearly equivalent to the P vs NP problem. however there is less obvious case for "simple" code for the problem in this case.
Write a simple program that checks whether for every n, $1 ≤ n ≤ 10^{50}$, the Collatz sequence starting with n will reach the number 1 in less than a billion iterations. When it has the answer, let the program stop if the answer is "Yes", and let it loop forever if the answer is "No".
We cannot tell whether this program terminates or not. (Who is we? Let's say "we" is anyone who could add a comment to my answer). However, someone with an incredibly powerful computer might tell. Some genius mathematician might be able to tell. There might be a rather small n, say n ≈ $10^{20}$ where a billion iterations are needed; that would be in reach of someone with a lot of determination, a lot of time, and a lot of money. But right now, we cannot tell.
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I have a formula $ \neg((q \implies \neg q) \vee p \vee (\neg q \implies (r \wedge p))) $.
As it contains 3 subformulas between the $\vee$'s, how can i put it into a parse tree, as a parse tree contains 2 branches from each node.
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I have a formula $ \neg((q \implies \neg q) \vee p \vee (\neg q \implies (r \wedge p))) $.
As it contains 3 subformulas between the $\vee$'s, how can i put it into a parse tree, as a parse tree contains 2 branches from each node.
As it is, your formula is ambiguous; it is not clear which pair of parentheses to insert in order to get a binary tree. Both
$\qquad \displaystyle \varphi_1 \lor (\varphi_2 \lor \varphi_3)$
and
$\qquad \displaystyle (\varphi_1 \lor \varphi_2) \lor \varphi_3$
are feasible. Luckily for you, $\lor$ is associative, so you can choose either one:
[source]
For the purposes of parsing, you would either require formulae to be completely parenthesised, or specify operators to be left- (or right-) associative. Operator precedences would be the next thing to look into, in order to disambiguate formulae like
$\qquad \displaystyle a \land b \lor c$.
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I have an encoding of an unambiguous grammar $G$. Can I apply a morphism $σ$ so that the grammar becomes ambiguous?
I have no clear concepts about this, only the idea: "grammar $G$ is ambiguous if there's a string $x \in L(G)$ having two different leftmost derivations in $G$". I would like a good explanation and if it's possible, an example.
Morphism:
I will consider two alphabets, $Σ_1$ and $Σ_2$, where $\Sigma_1$ is the terminal alphabet of $G$. A morphism is a function of the form $h\colon Σ_1^* \to Σ_2^*$ which satisfies: $h(xy)=h(x)h(y)$ for all $x,y \in \Sigma_1^*$. When applying the morphism to a grammar on $\Sigma_1$, the result is a grammar on $\Sigma_2$ (that is, the morphism is only applied to the terminals in $G$).
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Error analysis of discontinuous Galerkin method for the time fractional KdV equation with weak singularity solution
1.
School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, China
2.
School of Mathematic and Quantitative Economics, Shandong University of Finance and Economics, Jinan 250014, China
3.
Labroatory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing 100088, China
In this work, the time fractional KdV equation with Caputo time derivative of order $ \alpha \in (0,1) $ is considered. The solution of this problem has a weak singularity near the initial time $ t = 0 $. A fully discrete discontinuous Galerkin (DG) method combining the well-known L1 discretisation in time and DG method in space is proposed to approximate the time fractional KdV equation. The unconditional stability result and O$ (N^{-\min \{r\alpha,2-\alpha\}}+h^{k+1}) $ convergence result for $ P^k \; (k\geq 2) $ polynomials are obtained. Finally, numerical experiments are presented to illustrate the efficiency and the high order accuracy of the proposed scheme.
Keywords:Time fractional KdV equation, weak singularity, discontinuous Galerkin method, stability, error estimate. Mathematics Subject Classification:Primary: 35R11, 65M60; Secondary: 65M12. Citation:Na An, Chaobao Huang, Xijun Yu. Error analysis of discontinuous Galerkin method for the time fractional KdV equation with weak singularity solution. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 321-334. doi: 10.3934/dcdsb.2019185
References:
[1]
N. An, C. Huang and X. Yu,
Error analysis of direct discontinuous Galerkin method for two-dimensional fractional diffusion-wave equation,
[2]
W. Bu and A. Xiao,
An h-p version of the continuous Petrov-Galerkin finite element method for Riemann-Liouville fractional differential equation with novel test basis functions,
[3] [4]
Y. Cheng and C.-W. Shu,
A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives,
[5]
P. G. Ciarlet,
[6] [7]
P. A. Farrell, A. F. Hegarty, J. J. H. Miller, E. O'Riordan and G. I. Shishkin,
[8] [9]
D. Henry,
[10] [11]
C. Huang, N. An and X. Yu,
A fully discrete direct discontinuous Galerkin method for the fractional diffusion-wave equation,
[12]
C. Huang, M. Stynes and N. An,
Optimal ${L}^\infty ({L}^2)$ error analysis of a direct discontinuous Galerkin method for a time-fractional reaction-diffusion problem,
[13]
C. Huang, X. Yu, C. Wang, Z. Li and N. An,
A numerical method based on fully discrete direct discontinuous Galerkin method for the time fractional diffusion equation,
[14]
D. J. Korteweg and G. de Vries,
On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves,
[15] [16] [17]
S. Momani and A. Yıldı rım,
Analytical approximate solutions of the fractional convection-diffusion equation with nonlinear source term by He's homotopy perturbation method,
[18] [19]
K. Mustapha and W. McLean,
Discontinuous Galerkin method for an evolution equation with a memory term of positive type,
[20]
K. Mustapha and W. McLean,
Uniform convergence for a discontinuous Galerkin, time-stepping method applied to a fractional diffusion equation,
[21]
K. Mustapha, M. Nour and B. Cockburn,
Convergence and superconvergence analyses of HDG methods for time fractional diffusion problems,
[22]
I. Podlubny,
[23]
I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation,
[24]
J. Russell, Report of the committee on waves, Rep. Meet. Brit. Assoc. Adv. Sci., 7th Liverpool, 1837, London, John Murray.Google Scholar
[25]
M. Stynes, E. O'Riordan and J. Gracia,
Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation,
[26] [27]
L. Wei, Y. He, A. Yildirim and S. Kumar,
Numerical algorithm based on an implicit fully discrete local discontinuous Galerkin method for the time-fractional KdV-Burgers-Kuramoto equation,
[28]
G. H. Weiss, R. Klages, G. Radons and I. M. Sokolov (eds.), Anomalous transport: Foundations and applications [book review of WILEY-VCH Verlag GmbH & Co., Weinheim, 2008],
[29]
G. B. Witham,
[30]
N. Zabusky and M. Kruskal,
Interactions of solitons in a collisionless plasma and the recurrence of initial states,
[31]
Q. Zhang, J. Zhang, S. Jiang and Z. Zhang,
Numerical solution to a linearized time fractional KdV equation on unbounded domains,
show all references
References:
[1]
N. An, C. Huang and X. Yu,
Error analysis of direct discontinuous Galerkin method for two-dimensional fractional diffusion-wave equation,
[2]
W. Bu and A. Xiao,
An h-p version of the continuous Petrov-Galerkin finite element method for Riemann-Liouville fractional differential equation with novel test basis functions,
[3] [4]
Y. Cheng and C.-W. Shu,
A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives,
[5]
P. G. Ciarlet,
[6] [7]
P. A. Farrell, A. F. Hegarty, J. J. H. Miller, E. O'Riordan and G. I. Shishkin,
[8] [9]
D. Henry,
[10] [11]
C. Huang, N. An and X. Yu,
A fully discrete direct discontinuous Galerkin method for the fractional diffusion-wave equation,
[12]
C. Huang, M. Stynes and N. An,
Optimal ${L}^\infty ({L}^2)$ error analysis of a direct discontinuous Galerkin method for a time-fractional reaction-diffusion problem,
[13]
C. Huang, X. Yu, C. Wang, Z. Li and N. An,
A numerical method based on fully discrete direct discontinuous Galerkin method for the time fractional diffusion equation,
[14]
D. J. Korteweg and G. de Vries,
On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves,
[15] [16] [17]
S. Momani and A. Yıldı rım,
Analytical approximate solutions of the fractional convection-diffusion equation with nonlinear source term by He's homotopy perturbation method,
[18] [19]
K. Mustapha and W. McLean,
Discontinuous Galerkin method for an evolution equation with a memory term of positive type,
[20]
K. Mustapha and W. McLean,
Uniform convergence for a discontinuous Galerkin, time-stepping method applied to a fractional diffusion equation,
[21]
K. Mustapha, M. Nour and B. Cockburn,
Convergence and superconvergence analyses of HDG methods for time fractional diffusion problems,
[22]
I. Podlubny,
[23]
I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation,
[24]
J. Russell, Report of the committee on waves, Rep. Meet. Brit. Assoc. Adv. Sci., 7th Liverpool, 1837, London, John Murray.Google Scholar
[25]
M. Stynes, E. O'Riordan and J. Gracia,
Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation,
[26] [27]
L. Wei, Y. He, A. Yildirim and S. Kumar,
Numerical algorithm based on an implicit fully discrete local discontinuous Galerkin method for the time-fractional KdV-Burgers-Kuramoto equation,
[28]
G. H. Weiss, R. Klages, G. Radons and I. M. Sokolov (eds.), Anomalous transport: Foundations and applications [book review of WILEY-VCH Verlag GmbH & Co., Weinheim, 2008],
[29]
G. B. Witham,
[30]
N. Zabusky and M. Kruskal,
Interactions of solitons in a collisionless plasma and the recurrence of initial states,
[31]
Q. Zhang, J. Zhang, S. Jiang and Z. Zhang,
Numerical solution to a linearized time fractional KdV equation on unbounded domains,
N = 32 N = 64 N = 128 N = 256 N = 1024 3.0496E-2 1.1110E-2 3.9235E-3 1.3578E-3 4.6379E-4 1.5729E-4 1.4567 1.5016 1.5307 1.5498 1.5600 3.8341E-2 1.5127E-2 5.8825E-3 2.2665E-3 8.6831E-4 3.3157E-4 1.3417 1.3626 1.3759 1.3842 1.3888 5.9953E-2 2.6607E-2 1.1728E-2 5.1485E-3 2.2540E-3 9.8512E-4 1.1720 1.1817 1.1878 1.1916 1.1941
N = 32 N = 64 N = 128 N = 256 N = 1024 3.0496E-2 1.1110E-2 3.9235E-3 1.3578E-3 4.6379E-4 1.5729E-4 1.4567 1.5016 1.5307 1.5498 1.5600 3.8341E-2 1.5127E-2 5.8825E-3 2.2665E-3 8.6831E-4 3.3157E-4 1.3417 1.3626 1.3759 1.3842 1.3888 5.9953E-2 2.6607E-2 1.1728E-2 5.1485E-3 2.2540E-3 9.8512E-4 1.1720 1.1817 1.1878 1.1916 1.1941
Polynomial M Order Order 5 5.3831E-01 - 3.2328E-01 - 10 7.8579E-02 2.7762 4.7729E-02 2.7598 20 9.9319E-03 2.9840 6.2124E-03 2.9416 40 1.1426E-04 3.1196 7.5845E-04 3.0340 5 1.7236E-02 - 1.3819E-02 - 10 1.1399E-03 3.9184 8.7589E-04 3.9798 15 2.2712E-04 3.9406 1.7695E-04 3.9667 20 7.2979E-05 3.9418 6.1408E-04 3.9070
Polynomial M Order Order 5 5.3831E-01 - 3.2328E-01 - 10 7.8579E-02 2.7762 4.7729E-02 2.7598 20 9.9319E-03 2.9840 6.2124E-03 2.9416 40 1.1426E-04 3.1196 7.5845E-04 3.0340 5 1.7236E-02 - 1.3819E-02 - 10 1.1399E-03 3.9184 8.7589E-04 3.9798 15 2.2712E-04 3.9406 1.7695E-04 3.9667 20 7.2979E-05 3.9418 6.1408E-04 3.9070
N = 32 N = 64 N = 128 N = 256 N = 1024 2.6605E-2 9.8042E-3 3.4860E-3 1.2119E-3 4.1549E-4 1.4179E-4 1.4402 1.4918 1.5243 1.5444 1.5510 3.0086E-2 1.2002E-2 4.6980E-3 1.8177E-3 6.9850E-4 2.6770E-4 1.3258 1.3531 1.3699 1.3797 1.3836 4.1374E-2 1.8226E-2 7.9818E-3 3.4836E-3 1.5178E-3 6.6104E-4 1.1827 1.1912 1.1961 1.1985 1.1992
N = 32 N = 64 N = 128 N = 256 N = 1024 2.6605E-2 9.8042E-3 3.4860E-3 1.2119E-3 4.1549E-4 1.4179E-4 1.4402 1.4918 1.5243 1.5444 1.5510 3.0086E-2 1.2002E-2 4.6980E-3 1.8177E-3 6.9850E-4 2.6770E-4 1.3258 1.3531 1.3699 1.3797 1.3836 4.1374E-2 1.8226E-2 7.9818E-3 3.4836E-3 1.5178E-3 6.6104E-4 1.1827 1.1912 1.1961 1.1985 1.1992
Polynomial M Order Order 5 3.8931E-01 - 2.3483E-01 - 10 5.6563E-02 2.7829 3.4418E-02 2.7704 20 7.1139E-03 2.9911 4.4696E-03 2.9449 40 7.7979E-04 3.1894 5.4210E-04 3.0435 5 1.2812E-02 - 1.0564E-02 - 10 8.3809E-03 3.9342 6.7270E-04 3.9731 15 1.6615E-04 3.9552 1.2940E-04 4.0071 20 5.3162E-05 3.9564 4.3749E-05 3.9578
Polynomial M Order Order 5 3.8931E-01 - 2.3483E-01 - 10 5.6563E-02 2.7829 3.4418E-02 2.7704 20 7.1139E-03 2.9911 4.4696E-03 2.9449 40 7.7979E-04 3.1894 5.4210E-04 3.0435 5 1.2812E-02 - 1.0564E-02 - 10 8.3809E-03 3.9342 6.7270E-04 3.9731 15 1.6615E-04 3.9552 1.2940E-04 4.0071 20 5.3162E-05 3.9564 4.3749E-05 3.9578
N = 64 N = 128 N = 256 N = 512 N = 1024 2.4959E-3 8.4989E-4 2.8741E-4 9.6635E-5 3.2367E-5 1.5542 1.5641 1.5720 1.5784 6.0125E-3 2.3383E-3 8.9915E-4 3.4367E-4 1.3092E-4 1.3624 1.3788 1.3875 1.3923 1.0359E-2 4.6808E-3 2.0899E-3 1.2645E-4 4.0879E-4 1.1460 1.1633 1.1736 1.1803
N = 64 N = 128 N = 256 N = 512 N = 1024 2.4959E-3 8.4989E-4 2.8741E-4 9.6635E-5 3.2367E-5 1.5542 1.5641 1.5720 1.5784 6.0125E-3 2.3383E-3 8.9915E-4 3.4367E-4 1.3092E-4 1.3624 1.3788 1.3875 1.3923 1.0359E-2 4.6808E-3 2.0899E-3 1.2645E-4 4.0879E-4 1.1460 1.1633 1.1736 1.1803
N = 64 N = 128 N = 256 N = 512 N = 1024 6.0380E-3 2.2285E-3 7.2999E-4 2.4866E-4 8.4023E-5 1.5110 1.5371 1.5536 1.5653 1.1214E-2 4.4769E-3 1.7480E-3 6.7438E-4 2.5839E-4 1.3248 1.3567 1.3741 1.3839 1.5602E-2 7.0291E-3 3.1157E-3 1.3694E-3 5.9926E-4 1.1503 1.1737 1.1859 1.1923
N = 64 N = 128 N = 256 N = 512 N = 1024 6.0380E-3 2.2285E-3 7.2999E-4 2.4866E-4 8.4023E-5 1.5110 1.5371 1.5536 1.5653 1.1214E-2 4.4769E-3 1.7480E-3 6.7438E-4 2.5839E-4 1.3248 1.3567 1.3741 1.3839 1.5602E-2 7.0291E-3 3.1157E-3 1.3694E-3 5.9926E-4 1.1503 1.1737 1.1859 1.1923
[1]
Mahboub Baccouch.
Superconvergence of the semi-discrete local discontinuous Galerkin method for nonlinear KdV-type problems.
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Konstantinos Chrysafinos, Efthimios N. Karatzas.
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School, from our house as the crow flies, is 5.73 km. If we neglect air resistance and deal strictly with ballistic flight then we can materialize a wonderful fantasy. Starting in the backyard, extending over the top of the house, is a
launch-o-rocket, a rail-like launcher that accelerates the school-bound student until he or she can cruise over the city and arrive without bother of traffic. Our charter is to find the acceleration of the student from the launch-o-rocket. Finding the Initial Velocity
We rely on the well-known fact that the maximum distance in a throw occurs when the departure angle is 45°. The vertical speed and the horizontal speed are equal. We denote these two identical speeds as $s$. Since distance is time multiplied by speed, the distance from home to school $d$ is
$$ d = t\cdot s.$$
We know the distance $d = $ 5.73 km.
Turning to the vertical speed, the student departs the launch-o-rocket with vertical speed $s$, but is immediately subject to gravitational acceleration. Since the student’s upward flight is exactly matched by his or her downward flight. Because the flight is matched, the student spends $t/2$ time rising and $t/2$ time descending. Since the student has no vertical speed at the top, we know that his or her speed is
$$ s = g\frac{t}{2},$$
where $g$ is the gravitational acceleration 9.8 m/s
2.
Now, we have a system of equations
$$ d = t\cdot s $$
$$ s = g\frac{t}{2}.$$ The system looks like it has a many variables, but really there are only two, $s$ and $t$. We know $g$ and $d$. To solve the system we substitute for $s$ in the first equation with the second to get
$$ d = tg\frac{t}{2} = g\frac{t^2}{2}$$
Solve for $t$ $$ t = \sqrt{\frac{2d}{g}} = \sqrt{\frac{2\cdot 5.73\,\text{m}}{9.8\,\text{m/s}^2}} \approx 34.2\,\text{s}. $$ Not a bad commute, a little over half a minute.
With $t$ in hand, we can find the magnitude of the initial velocity. Remember that the initial velocity is $s$ in the horizontal direction and $s$ in the vertical direction, so the speed when leaving the launcher is
$$ \left| \mathbf{v}_0\right| = \sqrt{s^2 + s^2} = \sqrt{2s^2} = s\sqrt{2}. $$ The initial speed the student must attain is given by the very first equation, $d = s\cdot t$. Solving for $s$ with the value of $t$ we found, we get
$$ s = \frac{5.73\,\text{km}}{34.2\,\text{s}} = 168\,\text{m/s}. $$
Finding the Acceleration
The ramp lives on a footprint that is about 80 ft, or 24.4 m. It is also 24.4 m tall, so special zoning is surely required! The rail of the launch-o-rocket is the hypotenuse of a triangle, and that triangle has sides 24.4 m, and a total length of $\sqrt{2}\cdot 24.4\,\text{m} = 34.5\,\text{m}$.
The formula for position after a period of acceleration is
$$ p = \frac{1}{2}a\tau^2.$$ For our system, we also know that the acceleration is the change in speed divided by the change in time. Our speed goes from zero to 168 m/s in $\tau$. Again, we have a system of equations,
$$ 34.5\, \text{m} = \frac{1}{2}a\tau^2 $$
$$ a = \frac{168\,\text{m/s}}{\tau}.$$
Solve for $a$ by first solving the second equation for $\tau$, and then substituting that result into the first equation to get
$$ 34.5\, \text{m} = \frac{1}{2}a\left(\frac{168\,\text{m/s}}{a}\right)^2 $$
$$ a = \frac{\left( 168\, \text{m/s}\right)^2}{2 \cdot 34.5\,\text{m}}
= 407\, \text{m/s}^2 = 41.5\, g. $$
The typical onset of death occurs when acceleration exceeds about $10g$, so unfortunately,
the launch-o-rocket is a single try system.
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The travelling salesman problem (TSP) asks the following question: Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and returns to the origin city? It is an NP-hard problem in combinatorial optimization, important in operations research and theoretical computer science.
TSP can be formulated as an integer linear program. Label the cities with the numbers $0, \ldots, n$ and define: $$ x_{ij} = \begin{cases} 1 & \text{the path goes from city } i \text{ to city } j \\ 0 & \text{otherwise} \end{cases} $$ For $i = 1, \ldots, n$, let $u_i$ be an artificial variable, and finally take $c_{ij}$ to be the distance from city $i$ to city $j$. Then TSP can be written as the following integer linear programming problem:
\begin{align} \min &\sum_{i=0}^n \sum_{j\ne i,j=0}^nc_{ij}x_{ij} && \\ & 0 \le x_{ij} \le 1 && i,j=0, \cdots, n \\ & u_{i} \in \mathbf{Z} && i=0, \cdots, n \\ & \sum_{i=0,i\ne j}^n x_{ij} = 1 && j=0, \cdots, n \\ & \sum_{j=0,j\ne i}^n x_{ij} = 1 && i=0, \cdots, n \\ &u_i-u_j +nx_{ij} \le n-1 && 1 \le i \ne j \le n \end{align}
The first set of equalities requires that each city be arrived at from exactly one other city, and the second set of equalities requires that from each city there is a departure to exactly one other city. The last constraints enforce that there is only a single tour covering all cities, and not two or more disjointed tours that only collectively cover all cities. To prove this, it is shown below (1) that every feasible solution contains only one closed sequence of cities, and (2) that for every single tour covering all cities, there are values for the dummy variables $u_i$ that satisfy the constraints.
To prove that every feasible solution contains only one closed sequence of cities, it suffices to show that every subtour in a feasible solution passes through city 0 (noting that the equalities ensure there can only be one such tour). For if we sum all the inequalities corresponding to $x_{ij}=1$ for any subtour of $k$ steps not passing through city 0, we obtain: $$ nk \leq (n-1)k, $$ which is a contradiction.
It now must be shown that for every single tour covering all cities, there are values for the dummy variables $u_i$ that satisfy the constraints. Without loss of generality, define the tour as originating (and ending) at city 0. Choose $u_{i}=t$ if city $i$ is visited in step $t$ $(i, t = 1, 2, \ldots, n)$. Then $$ u_i-u_j\le n-1, $$ since $u_i$ can be no greater than $n$ and $u_j$ can be no less than 1; hence the constraints are satisfied whenever $x_{ij}=0$. For $x_{ij}=1$, we have $$u_{i} - u_{j} + nx_{ij} = (t) - (t+1) + n = n-1,$$ satisfying the constraint.
The problem has been shown to be NP-hard (more precisely, it is complete for the complexity class FPNP; see function problem), and the decision problem version (“given the costs and a number x, decide whether there is a round-trip route cheaper than x”) is NP-complete.
In the general case, finding a shortest travelling salesman tour is NPO-complete. If the distance measure is a metric and symmetric, the problem becomes APX-complete and Christofides’s algorithm approximates it within 1.5.
If the distances are restricted to 1 and 2 (but still are a metric) the approximation ratio becomes 8/7. In the asymmetric, metric case, only logarithmic performance guarantees are known, the best current algorithm achieves performance ratio $0.814 log(n)$; it is an open question if a constant factor approximation exists.
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Revisiting constraints on sterile neutrino mixing parameters using IceCube atmospheric neutrino data
Pre-published on: 2019 July 22
Published on: —
Abstract
Sterile neutrinos with $\sim 1$ eV mass scale can resolve anomalies reported in short-baseline neutrino oscillation experiments. The same sterile neutrinos can affect the atmospheric neutrino fluxes at the detectors such as IceCube in the TeV energy range. Recent IceCube search results for a sterile neutrino resulted in stringent constraints on its mass and mixing with active neutrinos, essentially excluding the parameter space favored by the short-baseline experiments. In our recent reanalysis of IceCube data we show that sterile-active neutrino mixing schemes affect interpretation of the excluded parameter regions. We present exclusion regions in the mixing angle range $0.01 \le \sin^2\theta_{24} \le 0.1$ and mass-squared difference range $0.1~{\rm eV}^2 \le \Delta m_{42}^2 \le 10~{\rm eV}^2$ for the mass-mixing and flavor-mixing schemes. We exclude the parameter space favored by recent MiniBooNE analysis at $\gtrsim 2\sigma$~CL for both schemes. Furthermore, we find that a prompt atmospheric neutrino flux contribution can relax the constraints on the sterile neutrino mass-squared difference for $\Delta m_{42}^2 \gtrsim 1~{\rm eV}^2$.
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Search
Now showing items 1-10 of 108
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 with the ALICE experiment at the CERN LHC in ...
Event-by-event mean pT fluctuations in pp and Pb–Pb collisions at the LHC
(Springer, 2014-10)
Event-by-event fluctuations of the mean transverse momentum of charged particles produced in pp collisions at s√ = 0.9, 2.76 and 7 TeV, and Pb–Pb collisions at √sNN = 2.76 TeV are studied as a function of the ...
Anomalous evolution of the near-side jet peak shape in Pb-Pb collisions at $\sqrt{s_{\mathrm{NN}}}$ = 2.76 TeV
(American Physical Society, 2017-09-08)
The measurement of two-particle angular correlations is a powerful tool to study jet quenching in a $p_{\mathrm{T}}$ region inaccessible by direct jet identification. In these measurements pseudorapidity ($\Delta\eta$) and ...
Enhanced production of multi-strange hadrons in high-multiplicity proton-proton collisions
(Nature Publishing Group, 2017)
At sufficiently high temperature and energy density, nuclear matter undergoes a transition to a phase in which quarks and gluons are not confined: the quark–gluon plasma (QGP)1. Such an exotic state of strongly interacting ...
Multiplicity dependence of the average transverse momentum in pp, p-Pb, and Pb-Pb collisions at the LHC
(Elsevier, 2013-12)
The average transverse momentum <$p_T$> versus the charged-particle multiplicity $N_{ch}$ was measured in p-Pb collisions at a collision energy per nucleon-nucleon pair $\sqrt{s_{NN}}$ = 5.02 TeV and in pp collisions at ...
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 been studied using the ALICE detector at the LHC. The spectra exhibit a significant hardening with ...
Forward-central two-particle correlations in p-Pb collisions at $\sqrt{s_{\rm NN}}$ = 5.02 TeV
(Elsevier, 2016-02)
Two-particle angular correlations between trigger particles in the forward pseudorapidity range ($2.5 < |\eta| < 4.0$) and associated particles in the central range ($|\eta| < 1.0$) are measured with the ALICE detector in ...
K$^{*}(892)^{0}$ and $\phi(1020)$ meson production at high transverse momentum in pp and Pb-Pb collisions at $\sqrt{s_\mathrm{NN}}$ = 2.76 TeV
(American Physical Society, 2017-06)
The production of K$^{*}(892)^{0}$ and $\phi(1020)$ mesons in proton-proton (pp) and lead-lead (Pb-Pb) collisions at $\sqrt{s_\mathrm{NN}} =$ 2.76 TeV has been analyzed using a high luminosity data sample accumulated in ...
Directed flow of charged particles at mid-rapidity relative to the spectator plane in Pb-Pb collisions at $\sqrt{s_{NN}}$=2.76 TeV
(American Physical Society, 2013-12)
The directed flow of charged particles at midrapidity is measured in Pb-Pb collisions at $\sqrt{s_{NN}}$=2.76 TeV relative to the collision plane defined by the spectator nucleons. Both, the rapidity odd ($v_1^{odd}$) and ...
Measurement of D-meson production versus multiplicity in p-Pb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV
(Springer, 2016-08)
The measurement of prompt D-meson production as a function of multiplicity in p–Pb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV with the ALICE detector at the LHC is reported. D$^0$, D$^+$ and D$^{*+}$ mesons are reconstructed ...
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I tried evaluating the following limit with WA, but I got a wrong answer. The limit is: $$\lim_{n\to\infty}\left| \frac{n\sin^2(n+1)}{(n+1)\sin^2(n)}\right|$$ The result that I obtained from WA was $0$, however it's wrong. Any thoughts? How do I report this?
You can check this by:
WolframAlpha[ "Limit Abs[n Csc(n)^2 Sin(n+1)^2/(n+1)] as n goes to infinity"]
0
$$$$ **Edit: Mathematica is not able evaluate the limit, instead other computing programs like Maple and Matlab seem to do so. Why so?
Why is the equivalent expression
Limit[ Abs[ n Csc[n]^2 Sin[n + 1]^2/(n + 1) ], n -> ∞]
returned unevaluated?
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Dear Uncle Colin Do you have any tips for sketching three-dimensional vectors? Every time we have an A-level question, my teacher says "draw a diagram!" but I don't know how to draw in 3D. - Got A Useless Sketching Situation Hi, GAUSS, and thank you for your message! Three-dimensional vectorsRead More →
That @solvemymaths is an excellent source of puzzles and whathaveyou: Meanwhile, back in 1940 when everything was basically shit... pic.twitter.com/A5eKXOunFC — Ed Southall (@solvemymaths) October 7, 2017 How would you find $\sqrt[3]{\frac{1-x^2}{x}}$ when $x=0.962$, using log tables or otherwise? I would start by trying to make the numbers nicer: IRead More →
Dear Uncle Colin, I'm sitting my GCSE Maths starting tomorrow. What will the grade boundaries be? - First Exams Are Redoubtable Hi, FEAR, I'm writing this, believe it or not, in early February1. I'm not even sure this summer's GCSE papers have been written yet, but I am going toRead More →
One of the puzzles in the MathsJam Shout looked impossible, so obviously I sat down with Mr Miller and had a go at it. I don't have it in front of me, but it went something like: A couple hosts a party to which five other couples are invited. AtRead More →
Dear Uncle Colin, I had a question in an exam that gave a cubic, $f(x) = x^3 - 8x^2 + cx + d$, with roots $\alpha$, $\beta$ and $\gamma$. When plotted on an Argand diagram, the triangle formed by the three roots has area 8. Given that $\alpha=2$, find $c$Read More →
In Episode 56 of Wrong, But Useful, we're joined by @zoelgriffiths (Zoe Griffiths), maths communicator from Think Maths. Zoe had her poem e, to thee, x in @chalkdustmagazine recently, and did a set about misleading statistics at @aeoud (An Evening Of Unnecessary Detail) Bad polls and fake stats, including thisRead More →
It turns out, I made an error in Cracking Mathematics. Not (in this case) a mathematical or historical error, although there are plenty of those1 but an error of etiquette: my potted biography of John Horton Conway emphasised the Game of Life above the rest of his work; I imagineRead More →
Dear Uncle Colin, I'm practicing for the Oxford PAT and have been asked how many terms of the binomial expansion would be needed to determine $(3.12)^5$ to one decimal place? I don't really know where to start. - Knows Expansions (Binomial); Lacks Explanations Hi, KEBLE, and thanks for your message!Read More →
A professor - according to Reddit - asked their class how many people you'd need to have in a room to be absolutely certain two of them would have Social Security numbers1 ending in the same four digits (in the same order). 10001, obviously. How about a probability of 99.9%?Read More →
In this post, I swap liberally between d-notation and '-notation for derivatives. Deal with it. Dear Uncle Colin, Why do we have to treat second-order ODEs differently when the auxiliary equation has a repeated root? Something Or Other Defies Expectations Hi, SOODE, and thanks for your message! First, some backgroundRead More →
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This question arose while addressing Comments on a previous blog post about exponentially distributed delays. One of my ongoing complaints is that many, if not most, popular load-test generation tools do not provide exponential variates as part of a library of time delays or think-time distributions. Not only is this situation bizarre, given that all load tests are actually performance models (and who doesn’t love an exponential distribution in their performance models?), but without the exponential distribution you are less likely to observe such things as buffer overflow conditons due to larger than normal (or uniform) queueing fluctuations. Exponential delays are both simple and useful for that purpose, but we are often left to roll our own code and then debug it.
Listing 2.2 on p. 35 of my Perl::PDQ book shows you how to generate exponential variates in
Perl. Here, however, I want to use R to compare exponential delays with both the uniform distribution (the default distribution available in all load-test simulators) and the normal distribution (the familiar “ bell curve“).
The R function that generates exponential variates directly is
rexp(n, rate = 1) where, for example, the parameter called
rate might correspond to the arrival rate of requests going into your test rig or
system under test (SUT). The R programming language uses the same notation as p. 57 of my Perl::PDQ book. In my books and classes, I usually write that rate as $\lambda$ to match conventional queueing theory symbology. The probability density function (PDF), or
dexp() in R, is usually written as:
\begin{equation} f(t) = \lambda e^{-\lambda t} \end{equation}
and shown in Figure 1.
The rate is $\lambda$, but the average or statistical mean of (1) is given by the inverse rate or $1/\lambda$. Since $\lambda$ is the average arrival
rate, $1/\lambda$ is the average interarrival time as would be seen by the SUT. The view from the load-test client corresponds to a think-time delay of $Z = 1/\lambda$ in your script. In either case, the delay is the time interval between requests, whether departing the client or arriving at the SUT. If you could apply the R function
rexp() directly to produce 10 exponentially distributed delays with a mean time of $Z=30$ seconds, you would write
rexp(10,1/30) with the result:
> rexp(10,1/30) [1] 126.841780 78.137417 12.989666 31.544992 9.598437 89.299508 2.063628 4.597275 [9] 24.792890 11.950121
Note that some delays are much smaller than the mean while other delays are much greater. Unfortunately, this R function is not available to you in load-test scripts so, you have to code your own. Here’s how that works.
Inverse Transformation In eqn. (1), we have the output $f(t)$ on the left and the corresponding delay $(t)$ on the right side (in the exponent). However, we would really prefer to have things the other way around: flip a coin to get an input on the right and find out what delay that corresponds to as an output on the left.
We can use the
inverse transform to do precisely that. The inverse function does not necessarily exist for an arbitrary probability distribution but, thankfully, the exponential distribution has a very simple form which allows it. The inverse of the exponential function is the natural logarithm function. The PDF in (1) lies in the range $0 \le f < \lambda$ on the $y$-axis, but we need to work with probabilities. The function which does this is the cumulative distribution function $F(t)$ in Figure 2:
\begin{equation} F(t) = 1 – e^{-\lambda t} \end{equation}
which is strictly bounded by the range $0 \le F < 1$. $F(t)$ is the corresponding
area under $f(t)$ and corresponds to
pexp(q, rate = 1) in R.
Typically, we would look along the $t$-axis (horizontal) for a particular time $(t)$ and then look up (to the curve) and across to the y-axis $(F)$ to find out the probability of that time occurring. Here, instead, we pick a random point on y-axis interval corresponding to $F$ (e.g., by flipping a coin). The corresponding delay is read off from the t-axis by following the dashed arrow in Figure 2, which shows this
inversion process for probability values $0.90$, $0.80$ and $0.30$. BTW, those probability values also correspond respectively to $90$th, $80$th, and $30$th percentiles, if you prefer to think of them that way. We can simulate the coin flip by using a variate $u \sim U(0,1)$ chosen from a uniform distribution $0 \le u < 1$.
Based on Figure 2, how can we calculate the corresponding interarrival delay $(t)$ that the load generator should use? Letting $u$ represent $F$ in (2) and transposing produces:
\begin{equation} e^{-\lambda t} = 1 – u \end{equation}
Next, we solve (3) for $t$ by taking natural logs of both sides—the
inverse function:
\begin{equation} \lambda t = – \ln(1 – u) \end{equation}
But the value of $u$ lies in the same interval as $(1-u)$, since they have the same uniform distribution. Hence, we can use the slightly simpler form:
\begin{equation} t = – \frac{\ln(u)}{\lambda} \end{equation}
Finally, we have arrived at the place where we wanted to be: flip a coin to get a
random input on the right hand side of (5) and find out what delay the client script should use as an output on the left. For load testing, the random delay $(t)$ is associated with a mean think time $Z = 1/\lambda$ and is therefore computed using:
\begin{equation} t = -Z \ln(u) \end{equation}
Equation (6) is what
rexp() uses under the covers, and it’s what you need to code in your client test scripts. A rather simple formula which, again, underscores the lunacy of not having it integrated into the load-test simulator.
Examples in R Using R, we first generate $10$ random variates (coin tosses) from a uniform distribution:
> uVariates <- runif(10) # create $10$ uniform variates > uVariates [1] 0.13737362 0.23143301 0.59315715 0.56576222 0.48420383 0.66065404 0.77418521 0.07673259 [9] 0.39985820 0.99247847
Next, we use these values as inputs into eqn. (6), with a mean time of 30 seconds, to calculate 10
exponentially distributed delays:
> thinkZ <- 30 # specify the average desired delay > tDelays <- -thinkZ*log(uVariates) # see eqn. (6) > tDelays [1] 59.5515274 43.9039444 15.6688773 17.0874419 21.7574800 12.4357489 7.6783242 77.0228628 [9] 27.4993587 0.2264988
Note the spread of delay times, which would also create significant fluctuations in queue depth as seen by buffers on the SUT side.
For comparison, here are $10$ delay samples produced by a
uniform distribution with the same mean as used for the exponential samples, i.e., the arithmetic mean $\frac{0+60}{2}=30$ seconds:
> runif(10,0,60) [1] 25.90247 51.94069 30.43872 10.11337 29.17908 25.90095 31.91967 42.46816 49.94629 34.62311
Similarly, here are $10$ delay samples produced by a
normal distribution with a mean of $30$ seconds:
> rnorm(10,30) [1] 30.57623 30.63393 30.45814 29.40551 29.59376 30.38025 29.91399 30.12652 29.67296 29.00941
Clearly, the exponential distribution produces a greater spread of delay times.
Related Posts
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I think that the matter of the paradox is that it causes explosion and trivializes the formal system, and if this explosion can be prevented, there is no problem.
I'm an unprofessional person but come up with an idea.
At first,
1.discard the general law of excluded middle and disjunctive syllogism.
Next, there are two kind of propositions. One is which must not contradict itself (namely, $P\wedge \lnot P$ can not be admitted), the other is which can contradict itself. An example of former is rigorous scientific claim and examples of latter are purely formal artificial statements, or sentences in ordinary contexts such as "I like him and I dislike him.".
Then,
2.admit law of excluded middle and disjunctive syllogism only for proposition of former kind.
3.the propositions "$X\in X, X\not\in X$" for the Russell set $X$ are of latter kind( namely, are artificial ones and don't appear in natural mathematical context), so doesn't cause explosion.
Now, I think that
4.in spite of both $X\in X\Rightarrow X\notin X$ and $X\notin X\Rightarrow X\in X$ are true, explosion doesn't occur because none of $X\in X$, $X\notin X$ and $X\in X \lor X\notin X$ is true (unless there are some extra axioms which prove them).
5.we can do usual mathematics because of 2.
Does this idea work?
Repply
How do you rigorously identify "artificial" statements?
->I define natural statements individually and identify artificial ones to the others. For example, one can assume statements which only relevant to sets in Gödel's constructible universe to be natural ones.
you may be trying to (re?)invent some form of paraconsistent logic.
->I don't know paraconsistent logic technically but I think that if we try to paste more than one scientific theories together or treat natural language, it is natural that there will be some contradictions and we have to treat them in some way. For this purpose, I discarded the full-strength explosion law.
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Relating to a question I posted here, I have formulated the following question:
Notation: $\max\{ x_1, \cdots, x_n \}$ denotes the maximal number among $x_1, \cdots, x_n$.
How to check the following system of equations is feasible? Assumption: $x_i, b_k$ are all in $[0,1]$
$\max\{x_i\mid i\in J_k\}=b_k$ where $1\leq k\leq m$ and $J_k\subseteq \{1, \cdots, n\}$
for example:
$\max\{x_1, x_4\}=0.7$ and $\max\{x_1, x_2, x_3\}=0.5$
What's the complexity of this problem? Obviously it is in NP, but is it NP-hard? It would be a surprise if such an easy problem does not have a polynomial algorithm. Does linear programming help here?
Many thanks.
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I'm a grad student in psychology, and as I pursue more and more independent studies in statistics, I am increasingly amazed by the inadequacy of my formal training. Both personal and second hand experience suggests that the paucity of statistical rigor in undergraduate and graduate training is rather ubiquitous within psychology. As such, I thought it would be useful for independent learners like myself to create a list of "Statistical Sins", tabulating statistical practices taught to grad students as standard practice that are in fact either superseded by superior (more powerful, or flexible, or robust, etc.) modern methods or shown to be frankly invalid. Anticipating that other fields might also experience a similar state of affairs, I propose a community wiki where we can collect a list of statistical sins across disciplines. Please, submit one "sin" per answer.
Failing to look at (plot) the data.
Most interpretations of p-values are sinful! The conventional usage of p-values is badly flawed; a fact that, in my opinion, calls into question the standard approaches to the teaching of hypothesis tests and tests of significance.
Haller and Krause have found that statistical instructors are almost as likely as students to misinterpret p-values. (Take the test in their paper and see how you do.) Steve Goodman makes a good case for discarding the conventional (mis-)use of the p -value in favor of likelihoods. The Hubbard paper is also worth a look.
Haller and Krauss. Misinterpretations of significance: A problem students share with their teachers. Methods of Psychological Research (2002) vol. 7 (1) pp. 1-20 (PDF)
Hubbard and Bayarri. Confusion over Measures of Evidence (p's) versus Errors (α's) in Classical Statistical Testing. The American Statistician (2003) vol. 57 (3)
Goodman. Toward evidence-based medical statistics. 1: The P value fallacy. Ann Intern Med (1999) vol. 130 (12) pp. 995-1004 (PDF)
Also see:
Wagenmakers, E-J. A practical solution to the pervasive problems of p values. Psychonomic Bulletin & Review, 14(5), 779-804.
for some clear cut cases where even the nominally "correct" interpretation of a p-value has been made incorrect due to the choices made by the experimenter.
Update (2016): In 2016, American Statistical Association issued a statement on p-values, see here. This was, in a way, a response to the "ban on p-values" issued by a psychology journal about a year earlier.
The most dangerous trap I encountered when working on a predictive model is not to reserve a test dataset early on so as to dedicate it to the "final" performance evaluation.
It's really easy to overestimate the predictive accuracy of your model if you have a chance to somehow use the testing data when tweaking the parameters, selecting the prior, selecting the learning algorithm stopping criterion...
To avoid this issue, before starting your work on a new dataset you should split your data as:
development set evaluation set
Then split your development set as a "training development set" and "testing development set" where you use the training development set to train various models with different parameters and select the bests according to there performance on the testing development set. You can also do grid search with cross validation but only on the development set. Never use the evaluation set while model selection is not 100% done.
Once you are confident with the model selection and parameters, perform a 10 folds cross-validation on the evaluation set to have an idea of the "real" predictive accuracy of the selected model.
Also if your data is temporal, it is best to choose the development / evaluation split on a time code: "It's hard to make predictions - especially about the future."
Reporting p-values when you did data-mining (hypothesis discovery) instead of statistics (hypothesis testing).
Testing the hypotheses $H_0: \mu=0$ versus $H_1: \mu\neq 0$ (for example in a Gaussian setting)
to justify that $\mu=0$ in a model (i.e mix "$H_0$ is not rejected" and "$H_0$ is true").
A very good example of that type of (very bad) reasoning is when you test whether the variances of two Gaussians are equal (or not) before testing if their mean are equal or not with the assumption of equal variance.
Another example occurs when you test normality (versus non normality) to justify normality. Every statistician has done that in is life ? it is baaad :) (and should push people to check robustness to non Gaussianity)
A few mistakes that bother me:
Assuming unbiased estimators are always better than biased estimators.
Assuming that a high $R^2$ implies a good model, low $R^2$ implies a bad model.
Incorrectly interpreting/applying correlation.
Reporting point estimates without standard error.
Using methods which assume some sort of Multivariate Normality (such as Linear Discriminant Analysis) when more robust, better performing, non/semiparametric methods are available.
Using p-value as a measure of strength between a predictor and the response, rather than as a measure of how much evidence there is of
somerelationship.
Dichotomization of a continuous predictor variable to either "simplify" analysis or to solve for the "problem" of non-linearity in the effect of the continuous predictor.
Not really answering the question, but there's an entire book on this subject:
Phillip I. Good, James William Hardin (2003).
Common errors in statistics (and how to avoid them). Wiley. ISBN 9780471460688
interpreting
Probability(data | hypothesis) as
Probability(hypothesis | data) without the application of Bayes' theorem.
Ritualized Statistics.
This "sin" is when you apply whatever thing you were taught, regardless of its appropriateness, because it's how things are done. It's statistics by rote, one level above letting the machine choose your statistics for you.
Examples are Intro to Statistics-level students trying to make everything fit into their modest t-test and ANOVA toolkit, or any time one finds oneself going "Oh, I have categorical data, I should use X" without ever stopping to look at the data, or consider the question being asked.
A variation on this sin involves using code you don't understand to produce output you only kind of understand, but know "the fifth column, about 8 rows down" or whatever is the answer you're supposed to be looking for.
Maybe stepwise regression and other forms of testing after model selection.
Selecting independent variables for modelling without having any
a priori hypothesis behind the existing relationships can lead to logical fallacies or spurious correlations, among other mistakes.
Useful references (from a biological/biostatistical perspective):
Kozak, M., & Azevedo, R. (2011). Does using stepwise variable selection to build sequential path analysis models make sense? Physiologia plantarum, 141(3), 197–200. doi:10.1111/j.1399-3054.2010.01431.x
Whittingham, M. J., Stephens, P., Bradbury, R. B., & Freckleton, R. P. (2006). Why do we still use stepwise modelling in ecology and behaviour? The Journal of animal ecology, 75(5), 1182–9. doi:10.1111/j.1365-2656.2006.01141.x
Frank Harrell,
Regression Modeling Strategies, Springer 2001.
Something I see a surprising amount in conference papers and even journals is making multiple comparisons (e.g. of bivariate correlations) and then reporting all the p<.05s as "significant" (ignoring the rightness or wrongness of that for the moment).
I know what you mean about psychology graduates, as well- I've finished a PhD in psychology and I'm still only just learning really. It's quite bad, I think psychology needs to take quantitative data analysis more seriously if we're going to use it (which, clearly, we should)
Being exploratory but pretending to be confirmatory. This
can happen when one is modifying the analysis strategy (i.e. model fitting, variable selection and so on) data driven or result driven but not stating this openly and then only reporting the "best" (i.e. with smallest p-values) results as if it had been the only analysis. This also pertains to the point if multiple testing that Chris Beeley made and results in a high false positive rate in scientific reports.
The one that I see quite often and always grinds my gears is the assumption that a statistically significant main effect in one group and a non-statistically significant main effect in another group implies a significant effect x group interaction.
Especially in epidemiology and public health - using arithmetic instead of logarithmic scale when reporting graphs of relative measures of association (hazard ratio, odds ratio or risk ratio).
More information here.
Correlation implies causation, which is not as bad as accepting the Null Hypothesis.
Analysis of rate data (accuracy, etc) using ANOVA, thereby assuming that rate data has Gaussian distributed error when it's actually binomially distributed. Dixon (2008) provides a discussion of the consequences of this sin and exploration of more appropriate analysis approaches.
A current popular one is plotting 95% confidence intervals around the raw performance values in repeated measures designs when they only relate to the variance of an effect. For example, a plot of reaction times in a repeated measures design with confidence intervals where the error term is derived from the MSE of a repeated measures ANOVA. These confidence intervals don't represent anything sensible. They certainly don't represent anything about the absolute reaction time. You could use the error term to generate confidence intervals around the effect but that is rarely done.
While I can relate to much of what Michael Lew says, abandoning p-values in favor of likelihood ratios still misses a more general problem--that of overemphasizing probabilistic results over effect sizes, which are required to give a result substantive meaning. This type of error comes in all shapes and sizes and I find it to be the most insidious statistical mistake. Drawing on J. Cohen and M. Oakes and others, I've written a piece on this at http://integrativestatistics.com/insidious.htm .
Failing to test the assumption that error is normally distributed and has constant variance between treatments. These assumptions aren't always tested, thus least-squares model fitting is probably often used when it is actually inappropriate.
My intro psychometrics course in undergrad spent at least two weeks teaching how to perform a stepwise regression. Is there any situation where stepwise regression is a good idea?
My old stats prof had a "rule of thumb" for dealing with outliers: If you see an outlier on your scatterplot, cover it up with your thumb :)
This may be more of a pop-stats answer than what you're looking for, but:
Using the mean as an indicator of location when data is highly skewed.
This isn't
necessarily a problem, if you and your audience knows what you're talking about, but this generally isn't the case, and the median is often likely to give a better idea of what's going on.
My favourite example is mean wages, which are usually reported as "average wages". Depending on the income/wealth inequality in a country, this can be vastly different from the median wage, which gives a much better indicator for where people are at in real life. For example, in Australia, where we have relatively low inequality, the median is 10-15% lower than the mean. In the US the difference is much starker, the median is less than 70% of the mean, and the gap is increasing.
Reporting on the "average" (mean) wage results in a rosier picture than is warranted, and could also give a large number of people the false impression that they aren't earning as much as "normal" people.
That the p-value is the probability that the null hypothesis is true and (1-p) is the probability that the alternative hypothesis is true, of that failing to reject the null hypothesis means the alternative hypothesis is false etc.
In similar vein to @dirkan - The use of p-values as a formal measure of evidence of the null hypothesis being true. It does have some good heuristic and intuitively good features, but is essentially an incomplete measure of evidence because it makes no reference to the alternative hypothesis. While the data may be unlikely under the null (leading to a small p-value), the data may be
even more unlikely under the alternative hypothesis.
The other problem with p-values, which also relates to some styles of hypothesis testing, is there is no principle telling you which statistic you should choose, apart from the very vague "large value" $\rightarrow$ "unlikely if null hypothesis is true". Once again, you can see the incompleteness showing up, for you should also have "large value" $\rightarrow$ "likely if
alternative hypothesis is true" as an additional heuristic feature of the test statistic.
Using pie charts to illustrate relative frequencies. More here.
Using statistics/probability in hypothesis testing to measure the "absolute truth". Statistics simply cannot do this, they can only be of use in deciding between
alternatives, which must be specified from "outside" the statistical paradigm. Statements such as "the null hypothesis is proved true by the statistics" are just incorrect; statistics can only tell you "the null hypothesis is favoured by the data, compared to the alternative hypothesis". If you then assume that either the null hypothesis or the alternative must be true, you can say "the null proved true", but this is only a trivial consequence of your assumption, not anything demonstrated by the data.
Repeating the same or similar experiment over 20 times on the same data and then reporting a statistically significant result with $\alpha = 0.05$. Incidentally there is a comic about this one.
And similarly to (or almost the same as) @ogrisel's answer, performing a Grid search and reporting only the best result.
(With a bit of luck this will be controversial.)
Using a Neyman-Pearson approach to statistical analysis of scientific experiments. Or, worse, using an ill-defined hybrid of Neyman-Pearson and Fisher.
Requesting, and perhaps obtaining
The Flow Chart: That graphical thing where you say what the level of your variables are and what sort of relationship you're looking for, and you follow the arrows down to get a Brand Name Test or a Brand Name Statistic. Sometimes offered with mysterious 'parametric' and 'non-parametric' paths.
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Let $E\to M$ be a real vector bundle of finite rank over a closed differentiable manifold $M$. Let $C^{\infty}(E)$ denote the space of smooth sections of $E$ and let $e\in C^{\infty}(E)$ be a section. I often see statements of the type:
"The tangent space to $C^{\infty}(E)$ at any given section $e\in C^{\infty}(E)$ is isomorphic to $C^{\infty}(E)$ itself, namely $T_{e}C^{\infty}(E)\simeq C^{\infty}(E)$."
I wonder what is the precise formulation of the statement above.
I know that if we only take smooth sections, $C^{\infty}(E)$ admits the structure of an infinite dimensional Frechet manifold with respect to the appropriate topology. The statement should be then that the isomorphism $T_{e}C^{\infty}(E)\simeq C^{\infty}(E)$ holds understanding $C^{\infty}(E)$ as a Frechet manifold and taking $T_{e}C^{\infty}(E)$ to be the tangent space of $C^{\infty}(E)$ as a Frechet manifold? Or rather, should it be understood by assuming that we have implicitly Sobolev-completed $C^{\infty}(E)$ into $H_{s}(E)$ using some appropriate Sobolev norm as to make $H_{s}(E)$ into a smooth Hilbert manifold and then the isomorphism that actually holds is $T_{e}H_{s}(E)\simeq H_{s}(E)$?
My second question is: can one make sense of an isomorphism of the type $T_{e}C^{\infty}(E)\simeq C^{\infty}(E)$ if the base is non-compact?
And lastly, let $Q\to M$ be a smooth fiber bundle over a closed manifold $M$ with typical fiber $F$ given by a smooth manifold. How should be understood the tangent space at a point of the space of sections of $Q$?
References are welcome.
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According to page 53 of Modern Computer Arithmetic (pdf), all of the steps in the Schönhage–Strassen Algorithm cost $O(N \cdot \lg(N))$ except for the recursion step which ends up costing $O(N\cdot \lg(N) \cdot \lg(\lg(N)))$.
I don't understand why the same inductive argument used to show the cost of the recursive step doesn't work for $O(N\cdot \lg(N))$.
Assume that, for all $X < N$, the time $F(X)$ is less than $c \cdot X \cdot \lg(X)$ for some $c$. So the recursive step costs $d \cdot \sqrt{N} F(\sqrt{N})$, and we know this is less than $d \cdot \sqrt{N} c \cdot \sqrt{N} \lg(\sqrt{N}) = \frac{c \cdot d}{2} \cdot N \cdot \lg(N)$ by the inductive hypothesis. If we can show that $d < 2$, then we're done because we've satisfied the inductive step. I'm pretty sure recursion overhead is negligible, so $d \approx 1$ and we have $\frac{c}{2} N \cdot \lg(N)$ left to do the rest. Easy: everything else is $O(N \cdot \lg(N))$ so we can pick a $c$ big enough for it to fit in our remaining time.
Basically, without digging into the details that will contradict this somehow, it
looks like things would work out if we assumed the algorithm costs $O(N \cdot \log(N))$. The same thing seems to happen if I expand the recursive invocations then sum it all up... so where is the penalty coming from?
My best guess is that it has to do with the $\lg(\lg(N))$ levels of recursion, since that's how many times you must apply a square root to get to a constant size. But how do we know each recursive pass is not getting cheaper, like in quickselect?
For example, if we group our $N$ initial items into words of size $O(\lg(N))$, meaning we have $O(N/\lg(N))$ items of size $O(\lg(N))$ to multiply when recursing, shouldn't that only take $O(N/\lg(N) \cdot \lg(N) \cdot \lg(\lg(N)) \cdot \lg(\lg(\lg(N)))) = O(N \cdot \lg(\lg(N)) \cdot \lg(\lg(\lg(N))))$ time to do. Not only is that well within the $N \cdot \lg(N)$ limit, it worked even though I used the larger $N\cdot \lg(N)\cdot \lg(\lg(N))$ cost for the recursive steps (for "I probably made a mistake" values of "worked").
My second guess is that there's some blowup at each level that I'm not accounting for. There are constraints on the sizes of things that might work together to slow down how quickly things get smaller, or to multiply how many multiplications have to be done.
Here's the recursive expansion.
Assume we get $N$ bits and split them into $\sqrt{N}$ groups of size $\sqrt{N}$. Everything except the recursion costs $O(N \lg N)$. The recursive multiplications can be done with $3 \cdot \sqrt{N}$ bits. So we get the relationship:
$M(N) = N \cdot \lg(N) + \sqrt{N} \cdot M(3 \cdot \sqrt{N})$
Expanding once:
$M(N) = N \cdot \lg(N) + \sqrt{N} \cdot (3 \cdot \sqrt{N} \cdot lg(3 \cdot \sqrt{N}) + \sqrt{3 \cdot \sqrt{N}} \cdot M(3 \cdot \sqrt{3 \cdot \sqrt{N}})$
Simplifying:
$M(N) = N \cdot \lg(N) + 3 \cdot N \cdot lg(3 \cdot \sqrt{N}) + \cdot \sqrt{3} \cdot N^{2-\frac{1}{2}} \cdot M(3^{2-\frac{1}{2}} \cdot \sqrt{\sqrt{N}})$
See the pattern? Each term will end up in the form $3^{2-2^{-i}} \cdot N \cdot lg(N^{2^{-i}} 3^{2-2^{-i}})$. So the overall sum is:
$\sum_{i=0}^{\lg(\lg(N))} 3^{2-2^{-i}} \cdot N \cdot \lg(N^{2^{-i}} 3^{2-2^{-i}})$
We can upper bound this by increasing the powers of 3 to just 3^2, since that can only increase the value in both cases:
$\sum_{i=0}^{\lg(\lg(N))} 9 \cdot N \cdot \lg(N^{2^{-i}} 9)$
Which is asymptotically the same as:
$\sum_{i=0}^{\lg(\lg(N))} N \cdot \lg(N^{2^{-i}})$
Moving the power out of the logarithm:
$\sum_{i=0}^{\lg(\lg(N))} N \cdot \lg(N) \cdot 2^{-i}$
Moving variables not dependent on $i$ out:
$N \cdot \lg(N) \sum_{i=0}^{\lg(\lg(N))} 2^{-i}$
The series is upper bounded by 2, so we're upper bounded by:
$N \cdot \lg(N)$
Not sure where the $\lg(\lg(N))$ went. All the twiddly factors and offsets (because many recurrence relations "solutions" are broken by those) I throw in seem to get killed off by the $\lg$ creating that exponentially decreasing term, or they end up not multiplied by $N$ and are asymptotically insignificant.
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@JosephWright Well, we still need table notes etc. But just being able to selectably switch off parts of the parsing one does not need... For example, if a user specifies format 2.4, does the parser even need to look for e syntax, or ()'s?
@daleif What I am doing to speed things up is to store the data in a dedicated format rather than a property list. The latter makes sense for units (open ended) but not so much for numbers (rigid format).
@JosephWright I want to know about either the bibliography environment or \DeclareFieldFormat. From the documentation I see no reason not to treat these commands as usual, though they seem to behave in a slightly different way than I anticipated it. I have an example here which globally sets a box, which is typeset outside of the bibliography environment afterwards. This doesn't seem to typeset anything. :-( So I'm confused about the inner workings of biblatex (even though the source seems....
well, the source seems to reinforce my thought that biblatex simply doesn't do anything fancy). Judging from the source the package just has a lot of options, and that's about the only reason for the large amount of lines in biblatex1.sty...
Consider the following MWE to be previewed in the build in PDF previewer in Firefox\documentclass[handout]{beamer}\usepackage{pgfpages}\pgfpagesuselayout{8 on 1}[a4paper,border shrink=4mm]\begin{document}\begin{frame}\[\bigcup_n \sum_n\]\[\underbrace{aaaaaa}_{bbb}\]\end{frame}\end{d...
@Paulo Finally there's a good synth/keyboard that knows what organ stops are! youtube.com/watch?v=jv9JLTMsOCE Now I only need to see if I stay here or move elsewhere. If I move, I'll buy this there almost for sure.
@JosephWright most likely that I'm for a full str module ... but I need a little more reading and backlog clearing first ... and have my last day at HP tomorrow so need to clean out a lot of stuff today .. and that does have a deadline now
@yo' that's not the issue. with the laptop I lose access to the company network and anythign I need from there during the next two months, such as email address of payroll etc etc needs to be 100% collected first
@yo' I'm sorry I explain too bad in english :) I mean, if the rule was use \tl_use:N to retrieve the content's of a token list (so it's not optional, which is actually seen in many places). And then we wouldn't have to \noexpand them in such contexts.
@JosephWright \foo:V \l_some_tl or \exp_args:NV \foo \l_some_tl isn't that confusing.
@Manuel As I say, you'd still have a difference between say \exp_after:wN \foo \dim_use:N \l_my_dim and \exp_after:wN \foo \tl_use:N \l_my_tl: only the first case would work
@Manuel I've wondered if one would use registers at all if you were starting today: with \numexpr, etc., you could do everything with macros and avoid any need for \<thing>_new:N (i.e. soft typing). There are then performance questions, termination issues and primitive cases to worry about, but I suspect in principle it's doable.
@Manuel Like I say, one can speculate for a long time on these things. @FrankMittelbach and @DavidCarlisle can I am sure tell you lots of other good/interesting ideas that have been explored/mentioned/imagined over time.
@Manuel The big issue for me is delivery: we have to make some decisions and go forward even if we therefore cut off interesting other things
@Manuel Perhaps I should knock up a set of data structures using just macros, for a bit of fun [and a set that are all protected :-)]
@JosephWright I'm just exploring things myself “for fun”. I don't mean as serious suggestions, and as you say you already thought of everything. It's just that I'm getting at those points myself so I ask for opinions :)
@Manuel I guess I'd favour (slightly) the current set up even if starting today as it's normally \exp_not:V that applies in an expansion context when using tl data. That would be true whether they are protected or not. Certainly there is no big technical reason either way in my mind: it's primarily historical (expl3 pre-dates LaTeX2e and so e-TeX!)
@JosephWright tex being a macro language means macros expand without being prefixed by \tl_use. \protected would affect expansion contexts but not use "in the wild" I don't see any way of having a macro that by default doesn't expand.
@JosephWright it has series of footnotes for different types of footnotey thing, quick eye over the code I think by default it has 10 of them but duplicates for minipages as latex footnotes do the mpfoot... ones don't need to be real inserts but it probably simplifies the code if they are. So that's 20 inserts and more if the user declares a new footnote series
@JosephWright I was thinking while writing the mail so not tried it yet that given that the new \newinsert takes from the float list I could define \reserveinserts to add that number of "classic" insert registers to the float list where later \newinsert will find them, would need a few checks but should only be a line or two of code.
@PauloCereda But what about the for loop from the command line? I guess that's more what I was asking about. Say that I wanted to call arara from inside of a for loop on the command line and pass the index of the for loop to arara as the jobname. Is there a way of doing that?
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Hey guys! I built the voltage multiplier with alternating square wave from a 555 timer as a source (which is measured 4.5V by my multimeter) but the voltage multiplier doesn't seem to work. I tried first making a voltage doubler and it showed 9V (which is correct I suppose) but when I try a quadrupler for example and the voltage starts from like 6V and starts to go down around 0.1V per second.
Oh! I found a mistake in my wiring and fixed it. Now it seems to show 12V and instantly starts to go down by 0.1V per sec.
But you really should ask the people in Electrical Engineering. I just had a quick peek, and there was a recent conversation about voltage multipliers. I assume there are people there who've made high voltage stuff, like rail guns, which need a lot of current, so a low current circuit like yours should be simple for them.
So what did the guys in the EE chat say...
The voltage multiplier should be ok on a capacitive load. It will drop the voltage on a resistive load, as mentioned in various Electrical Engineering links on the topic. I assume you have thoroughly explored the links I have been posting for you...
A multimeter is basically an ammeter. To measure voltage, it puts a stable resistor into the circuit and measures the current running through it.
Hi all! There is theorem that links the imaginary and the real part in a time dependent analytic function. I forgot its name. Its named after some dutch(?) scientist and is used in solid state physics, who can help?
The Kramers–Kronig relations are bidirectional mathematical relations, connecting the real and imaginary parts of any complex function that is analytic in the upper half-plane. These relations are often used to calculate the real part from the imaginary part (or vice versa) of response functions in physical systems, because for stable systems, causality implies the analyticity condition, and conversely, analyticity implies causality of the corresponding stable physical system. The relation is named in honor of Ralph Kronig and Hans Kramers. In mathematics these relations are known under the names...
I have a weird question: The output on an astable multivibrator will be shown on a multimeter as half the input voltage (for example we have 9V-0V-9V-0V...and the multimeter averages it out and displays 4.5V). But then if I put that output to a voltage doubler, the voltage should be 18V, not 9V right? Since the voltage doubler will output in DC.
I've tried hooking up a transformer (9V to 230V, 0.5A) to an astable multivibrator (which operates at 671Hz) but something starts to smell burnt and the components of the astable multivibrator get hot. How do I fix this? I check it after that and the astable multivibrator works.
I searched the whole god damn internet, asked every god damn forum and I can't find a single schematic that converts 9V DC to 1500V DC without using giant transformers and power stage devices that weight 1 billion tons....
something so "simple" turns out to be hard as duck
In peskin book of QFT the sum over zero point energy modes is an infinite c-number, fortunately, it's experimental evidence doesn't appear, since experimentalists measure the difference in energy from the ground state. According to my understanding the zro pt energy is the same as the ground state, isn't it?
If so, always it is possible to substract a finite number (higher exited state for e.g.) from this zero point enrgy (which is infinte), it follows that, experimentally we always obtain infinte spectrum.
@AaronStevens Yeah, I had a good laugh to myself when he responded back with "Yeah, maybe they considered it and it was just too complicated". I can't even be mad at people like that. They are clearly fairly new to physics and don't quite grasp yet that most "novel" ideas have been thought of to death by someone; likely 100+ years ago if it's classical physics
I have recently come up with a design of a conceptual electromagntic field propulsion system which should not violate any conservation laws, particularly the Law of Conservation of Momentum and the Law of Conservation of Energy. In fact, this system should work in conjunction with these two laws ...
I rememeber that Gordon Freeman's thesis was "Observation of Einstein-Podolsky-Rosen Entanglement on Supraquantum Structures by Induction Through Nonlinear Transuranic Crystal of Extremely Long Wavelength (ELW) Pulse from Mode-Locked Source Array "
In peskin book of QFT the sum over zero point energy modes is an infinite c-number, fortunately, it's experimental evidence doesn't appear, since experimentalists measure the difference in energy from the ground state. According to my understanding the zro pt energy is the same as the ground state, isn't it? If so, always it is possible to substract a finite number (higher exited state for e.g.) from this zero point enrgy (which is infinte), it follows that, experimentally we always obtain infinte spectrum.
@ACuriousMind What confuses me is the interpretation of Peskin to this infinite c-number and the experimental fact
He said, the second term is the sum over zero point energy modes which is infnite as you mentioned. He added," fortunately, this energy cannot be detected experm., since the experiments measure only the difference between from the ground state of H".
@ACuriousMind Thank you, I understood your explanations clearly. However, regarding what Peskin mentioned in his book, there is a contradiction between what he said about the infinity of the zero point energy/ground state energy, and the fact that this energy is not detectable experimentally because the measurable quantity is the difference in energy between the ground state (which is infinite and this is the confusion) and a higher level.
It's just the first encounter with something that needs to be renormalized. Renormalizable theories are not "incomplete", even though you can take the Wilsonian standpoint that renormalized QFTs are effective theories cut off at a scale.
according to the author, the energy differenc is always infinite according to two fact. the first is, the ground state energy is infnite, secondly, the energy differenc is defined by substituting a higher level energy from the ground state one.
@enumaris That is an unfairly pithy way of putting it. There are finite, rigorous frameworks for renormalized perturbation theories following the work of Epstein and Glaser (buzzword: Causal perturbation theory). Just like in many other areas, the physicist's math sweeps a lot of subtlety under the rug, but that is far from unique to QFT or renormalization
The classical electrostatics formula $H = \int \frac{\mathbf{E}^2}{8 \pi} dV = \frac{1}{2} \sum_a e_a \phi(\mathbf{r}_a)$ with $\phi_a = \sum_b \frac{e_b}{R_{ab}}$ allows for $R_{aa} = 0$ terms i.e. dividing by zero to get infinities also, the problem stems from the fact that $R_{aa}$ can be zero due to using point particles, overall it's an infinite constant added to the particle that we throw away just as in QFT
@bolbteppa I understand the idea that we need to drop such terms to be in consistency with experiments. But i cannot understand why the experiment didn't predict such infinities that arose in the theory?
These $e_a/R_{aa}$ terms in the big sum are called self-energy terms, and are infinite, which means a relativistic electron would also have to have infinite mass if taken seriously, and relativity forbids the notion of a rigid body so we have to model them as point particles and can't avoid these $R_{aa} = 0$ values.
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Does anybody know an efficient mechanism to prove the possession of a digital signature (e.g. RSA) on a certain attribute (message) in zero-knowledge? That is, without revealing the actual signature (against tracking, for privacy) prove that you have one? Thanks a lot in advance!
Guillou and Quisquater (link) present a zero-knowledge proof of an RSA signature. Basically, the scheme is as follows:
Public knowledge: RSA modulus $n$, public RSA exponent $v$, preimage $X$.
Secret knowledge for prover: $A$, such that $A^v = X \mod n$.
$$ \begin{matrix} \mathcal{P} & & \mathcal{V} \\ r \xleftarrow{\$} \mathbb{Z}_n^* \phantom{\mod n} & & \\ T \leftarrow r^v \mod n & & \\ & \xrightarrow{\quad{}T\quad{}} & \\ & & d \xleftarrow{\$} \{0,1,\ldots,v-1\} \\ & \xleftarrow{d} & \\ t \leftarrow A^dr \mod n & & \\ & \xrightarrow{\quad{}t\quad} & \\ & & t^v \stackrel{?}{=} X^{d}T \mod n \end{matrix} $$
In this diagram, $\leftarrow$ denotes assignment of a value to a variable and $\xleftarrow{\$}$ denotes uniformly random selection from a finite set.
One-Way Aggregate Signatures (OWAS) (also called composite signatures) can be used to do this. They are based on BLS signatures. I will skip the notation except mention that they are based on bilinear pairings. The above links will give the details.
Let $H$ be a hash function and $x_1$ be the private key. The public key is $y_1=g^{x_1}$. For any message $m_1$, the signature is $\sigma_1={h_1}^{x_1}$, where $h_1 = H(m_1)$. To verify the signature, test that: $\hat{e}(\sigma_1, g) \stackrel{?}{=} \hat{e}(h_1, y_1)$ holds.
Let $x_2$ be another private key. The public key is $y_2=g^{x_2}$. As before, for any message $m_2$, the signature is $\sigma_2={h_2}^{x_2}$, where $h_2 = H(m_2)$. To verify the signature, test that: $\hat{e}(\sigma_2, g) \stackrel{?}{=} \hat{e}(h_2, y_2)$ holds.
We can also combine $\sigma_1, \sigma_2$ into an aggregate signature $\sigma$ as follows: $\sigma=\sigma_{1}\cdot \sigma_{2}$. To verify, we check that: $$\hat{e}(\sigma, g) \stackrel{?}{=} \hat{e}(h_1, y_1)\cdot\hat{e}(h_2, y_2)$$
The signatures satisfy the standard security as shown in the Aggregate Signatures Paper. That is, assuming that the Diffie-Hellman problem is hard, presentation of $\sigma$ proves that $y_1$ signed $m_1$ and $y_2$ signed $m_2$ (even if the original signatures are not provided).
The security of OWAS relies on the fact that given just $\sigma$, it is hard to compute either $\sigma_1$ or $\sigma_2$ (as hard as solving Diffle-Hellman problem). In fact, for two combined signatures, the resulting signature is a
zero-knowledge proof of knowledge of each individual signature. This can be shown as follows:
Suppose I can control the output of the hash function (i.e., we are using the "random oracle model"), then instead of doing it the correct way as described above, given $(g, h_1, y_1)$, I can compute a fake tuple $(\sigma, h_2, y_2)$ such that the last verification equation above holds. First generate random $a, b$. The compute:
$$y_2=g^a y_1$$ $$h_2=\frac{g^b}{h_1}$$ $$\sigma=\frac{{y_1}^b\cdot g^{ab}}{{h_1}^a}$$
It can be checked that the above values satisfy the aggregate signature verification equation. Yet, I did this without knowing $\sigma_1$. Hence this is zero-knowledge.
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By "surface bundle over a surface" I mean a compact, oriented 4-manifold $X$ which is the total space of an oriented fiber bundle $X\to B $ over an oriented 2-manifold $B$. Assume that the signature of the 4-manifold $X$ is non-trivial.
Conjecture 1: $X$ and $B$ can be given complex structures such that the map $X \to B$ is holomorphic.
Conjecture 2: $X$ and $B$ can be given the structure of complex algebraic varieties such that the map $X \to B$ is algebraic.
Question: Are the above conjectures true? If not, can you provide a counter example?
Background: The bundle $X\to B$ induces a map $f:B \to M_g$ where $M_g$ is the moduli space of curves and $g$ is the genus of the fiber. The signature of $X$ is given by
$$\sigma(X) = 4\int _B f^*(\lambda)$$
where $\lambda$ is the first Chern class of the Hodge bundle. Since the signature of $X$ is non-zero, the map $f$ is non-trivial in homology and one might hope that an argument along the following lines is true: Find a representative $\tilde{f}$ in the same homotopy class as $f$ which has minimal energy. Using the fact that $M_g$ has a hyperbolic metric ($g>2$ since $\sigma(X)\neq 0$), prove this minimal energy map is holomorphic or even algebraic. $\tilde{f}$ provides $X$ with the desired structure.
This question is a variant on this recent question: A four-dimensional counterexample? the counter example given there has zero signature.
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Dear Uncle Colin, I have the simultaneous equations $3x^2 - 3y = 0$ and $3y^2 - 3x = 0$. I've worked out that $x^2 = y$ and $y^2 = x$, but then I'm stuck! - My Expertise Relatedto1 Simultaneous Equations? Not Nearly Enough! Hi, MERSENNE, and thanks for your message!Read More →
Because I'm insufferably vain, I have a search running in my Twitter client for the words "The Maths Behind", in case someone mentions my book (which is, of course, available wherever good books are sold). On the minus side, it rarely is; on the plus side, the search occasionally throwsRead More →
Dear Uncle Colin, I have been given the series $\frac{1}{2} + \frac{1}{3} + \frac{1}{8} + \frac{1}{30} + \frac{1}{144} + ...$, which appears to have a general term of $\frac{1}{k! + (k+1)!}$ - but I can't see how to sum that! Any ideas? - Series Underpin Maths! Hi, SUM, and thanksRead More →
Since it's Christmas (more or less), let's treat ourselves to a colourful @solvemymaths puzzle: Have a go, if you'd like to! Below the line will be spoilers. Consistency The first and most obvious thing to ask is, is Ed's claim reasonable? At a glance, yes, it makes sense: there's aRead More →
Dear Uncle Colin, How do I verify the identity $\frac{\cos(\theta)}{1 - \sin(\theta)} \equiv \tan(\theta) + \sec(\theta)$ for $\cos(\theta) \ne 0$? - Struggles Expressing Cosines As Nice Tangents Hi, SECANT, and thanks for your message! The key questions for just about any trigonometry proof are "what's ugly?" and "how can IRead More →
Midway through the second half of You Can’t Polish A Nerd, Steve Mould neatly encapsulates the show in one line: “It creates images on your oscilloscope. It’s so cool!” Because of course you have an oscilloscope. And of course you would use it - or failing that, a balloon andRead More →
Dear Uncle Colin, How would you factorise $63x^2 + 32x - 63$? I tried the method where you multiply $a$ and $c$ (it gives you -3969) - but I'm not sure how to find factors of that that sum to 32! Factors Are Troublesomely Oversized, Urgh Hi, FATOU, and thanksRead More →
In this month’s episode of Wrong, But Useful, we’re joined by @ch_nira, who is Dr Nira Chamberlain in real life - and the World’s Most Interesting Mathematician. Nira is a professional mathematical modeller, president-designate of the IMA, and a visiting fellow at Loughborough university. We discuss Nira’s entry in theRead More →
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I answer the question in two steps: in the first one I prove that the class functions $f$ in question are Riemann integrable even if not bounded. To see this let's recall the following definitions
Definition 1. The measure of an interval $[a,b]$ is its length $\mu ([a,b])=\ell ([a-b])=a-b$.
Definition 2. A set $S\subset\mathbb{R}$ is said to be a set of measure zero if, given any $\varepsilon>0$ there exists a countable (i.e finite or denumerable) family of intervals $\{[a_n,b_n]\}_{n\in\mathbb{N}}$ which covers $S$ and whose total length is less than $\varepsilon$.
According to definition 2, the set $D$ of discontinuity points of $f$ is a set of measure zero, since as already noted by Arian,$$D\subseteq\bigcup_{0\leq n\leq N(\varepsilon)}[a_n,b_n]=D_\varepsilon$$with$$\mu(D_\varepsilon)=\sum_{n=0}^{N(\varepsilon)} \mu ([a_n,b_n]) =\sum_{n=0}^{N(\varepsilon)}\ell ([a_n-b_n])<\varepsilon\quad \forall \varepsilon>0$$where $N(\varepsilon)$ is the (finite) cardinality of the covering family. It necessary to consider it as a function $N:\mathbb{R}_+\to \mathbb{N}$ since, from the hypothesis given in the question, we only know that the discontinuity set of $f$ is covered by a finite number of intervals: this is true even when $D$ it is a denumerable set with a finite number of accumulation points, but in this case $N$ increases as $\varepsilon$ goes to $0$. Now the crucial point: for every $0<\varepsilon<b-a$,$$[a,b]\setminus D_\varepsilon\text{ is the union of a finite number of (half open or open) intervals.}$$This fact can be easily checked if the intervals in $\{[a_n,b_n]\}_{0\leq n\leq N(\varepsilon)}$ are pairwise disjoint or if they, on the contrary, overlap each other: to see that this fact is true even in the intermediate cases, it suffices to consider the connected components of $\bigcup_{0\leq n\leq N(\varepsilon)}[a_n,b_n]$. Therefore $f$ is Riemann integrable in every set $[a,b]\cap D_\varepsilon$ since it is continuous on each of its finite number of subintervals: passing to the limit we get$$\int\limits_{[a,b]\setminus D_\varepsilon} f(x)\mathrm{d}x\underset{\varepsilon\to 0}\longrightarrow\int\limits_{[a,b]} f(x)\mathrm{d}x. \tag{1} \label{1}$$Thus, under the hypothesis of the question,
$f$ is Riemann integrable even without assuming its boundedness.
The second step is to note that, while $f$ is R-integrable, the limit in \eqref{1} can possibly be $\infty$: however, if $f$ is also bounded, this possibility is ruled out, since if $M\geq0$ is the supremum of $|f|$, then$$\left\vert\:\int\limits_{[a,b]\setminus D_\varepsilon} f(x)\mathrm{d}x\right|\leq M\!\!\!\!\!\!\int\limits_{[a,b]\setminus D_\varepsilon}\!\!\!\!\mathrm{d}x < M (b-a),$$and thus$$\int\limits_{[a,b]} f(x)\mathrm{d}x<M(b-a) \tag{2} \label{2}$$
Notes
Lebesgue's criterion for Riemann integrability ([1], §1.7 p.20) states that if $f:[a,b]\to \mathbb{R}$ a real function of one variable, continuous outside a set $D\subset[a,b]$$$f\text{ is Riemann integrable }\iff \mu(D)=0$$
The result is a consequence of the general properties of Riemann's integral and is true even without the hypothesis of boundedness of $f$: it implies that the R-integrability of a function doe not depend on the cardinality of its discontinuity set nor on other topological properties it may have.
[1] Shilov, G. E. and Gurevich, B. L. (1977),
Integral, Measure, and Derivative: A Unified Approach, revised edition, Dover books on advanced mathematics, New York: Dover Publications, pp. xiv+233, ISBN 0-486-63519-8, MR 0466463, Zbl 0391.28007.
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Let $ X_n $ be a family of random variables an $X$ a rv with $X_n \overset{D}\rightarrow X$. Let $ a_n$ a sequence of real numbers with $a_n \overset{n \rightarrow \infty} \rightarrow a $. We say that the Distribution function of $X$ is continuous.
a) Proof that for all $x \in \mathbb{R}$ an every $\epsilon >0$ there exist a $N_{\epsilon,x} \in \mathbb{N}$, thus for all $n \geq N_{\epsilon,x}:$
$P(X+a \leq x - \epsilon) - \epsilon \leq P(X_n + a_n \leq x) \leq P(X+a \leq x + \epsilon) + \epsilon$
b) Show that
$ X_n + a_n \overset{D}\rightarrow X+a$ for $n \rightarrow \infty$
Can anybody give me a hint? I have no idea how to start with the excersice.
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First, a quick reminder regarding Fourier series. It's well-known that in the Hilbert space of measurable functions $f:[0,T]\to\mathbb{C}$ such that $\int_0^T|f(t)|^2dt<\infty$ (with inner product $\langle f,g\rangle = \int_0^T\overline{f(t)}g(t)dt$), henceforth denoted $L^2([0,T])$, the trigonometric system forms a closed orthonormal system. Therefore, every such function can be represented as a series of trigonometric functions (which converges, I should clarify, in
norm, and not necessarily pointwise). That's the Fourier series for such functions.
However, in our case $t\mapsto 8e^tu(t-6)$, defined on $\mathbb{R}$, isn't periodic. As such, it has no Fourier series. We therefore might turn to the notion of the Fourier transform: if a measurable function has the property $\int_\mathbb{R}|f(t)|dt<\infty$ (we say, $f\in L^1(\mathbb{R})$) we can define $F(\omega) = \int_\mathbb{R}f(t)e^{-i\omega t}dt$ (i.e., the integral converges, so $F(\omega)$ is well-defined). This is the Fourier transform. Another well-known fact is that in the Hilbert space of measurable functions $f:\mathbb{R}\to\mathbb{C}$ such that $\int_\mathbb{R}|f(t)|^2dt<\infty$ (with inner product $\langle f,g\rangle = \int_\mathbb{R}\overline{f(t)}g(t)dt$), henceforth denoted $L^2(\mathbb{R})$, functions which are members of $L^1(\mathbb{R})\cap L^2(\mathbb{R})$ are dense; this--along with the fact that the Fourier transform is an isometry for such functions (see Parseval's equality)--allows us to extend the transform to functions which are $L^2$ but not $L^1$.
However, once more, $f(t)=8e^tu(t-6)$ is neither $L^1$ nor $L^2$. In such cases, sometimes we can still define a Laplace transform, which helps 'dampen' growing rates which are exponential or sub-exponential. This is done by defining $F(z) = \int_\mathbb{R}f(t)e^{-zt}dt$, which usually converges on a band or half-plane of $\mathbb{C}$. The transform of the function you mentioned could probably be found in transformation tables on Wiki, but for $z\in\mathbb{C}$ the function $e^{-zt}8e^tu(t-6)\in L^1(\mathbb{R})$ iff $\operatorname{Re}(z)>1$, and there we have$$\int_{\mathbb{R}}e^{-zt}8e^tu(t-6)dt = 8\int_6^\infty e^{(1-z)t} = \frac{8}{z-1}e^{6(1-z)}.$$This is therefore the Laplace transform (and its area of convergence) of $f$.
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An In-Depth Look at Spherical Aberration Compensation Plates
Optical aberrations are deviations from a perfect, mathematical model. It is important to note that they are not caused by any physical, optical, or mechanical flaws. Rather, they can be caused by the lens shape itself, or placement of optical elements within a system, due to the wave nature of light. Optical aberrations are named and characterized in several different ways. For simplicity, consider aberrations divided into two groups: chromatic aberrations (present when using more than one wavelength of light) and monochromatic aberrations (present with a single wavelength of light). For more information on aberrations, view Chromatic and Monochromatic Optical Aberrations and Comparison of Optical Aberrations.
Figure 1: Illustration of Spherical Aberration
One of the most common types of monochromatic aberrations is spherical aberration. Spherical aberration is the result of light focusing at different locations based on its radial distance from the lens center resulting in poor system performance (Figure 1). Though spherical aberration is present in all spherical optics, an innovative way to correct for it is by employing spherical aberration compensation plates to reduce or remove known quantities of spherical aberration in a system.
WHAT ARE SPHERICAL ABERRATION COMPENSATION PLATES?
By compensating and correcting for a known amount of spherical aberration, spherical aberration compensation plates are single-element optical components that can be easily inserted into a system, reducing spot size and drastically improving image quality (Figures 2a – 2b). These corrector plates signify a change in the way aberration correction can be handled. By correcting for known amounts of spherical aberration, they save in design time, reduction of system weight as well as manufacturing costs.
Spherical aberration compensation plates are designed to be used in collimated space near a pupil. They should be used for systems that have small fields of view such as laser systems or applications imaging point-like objects. These corrector plates can be combined to induce the desired amount of compensatory spherical aberration. Negative sign plates create over corrected spherical aberration, while positive plates create under corrected spherical aberration.
Spherical aberration compensation plates are optically flat windows with low wavefront distortion that have been magnetorheologically polished to impart a mild aspheric surface. Aspheric surfaces are traditionally defined with the following surface profile (sag):
(1)$$ Z\left(s\right)= \frac {Cs^2}{1+\sqrt{1-\left(1+k\right)C^2s^2}}+A_4 s^4+A_6 s^6+A_8 s^8+... $$
Where Z is the sag of the surface parallel to the optical axis; s is the radial distance from the optical axis; C is the lens curvature, inverse of radius; k is the conic constant; and A4, A6, A8 are 4th, 6th, 8th… order aspheric terms.
However, in the case of spherical aberration compensation plates, there is no optical power (curvature) to the surface (i.e. C = 0). The corrector plates achieve their known amounts of spherical aberration by having at least one non-zero aspheric coefficient. Equation 1 then reduces to:
(2)$$ Z\left(s\right)= A_4 s^4+A_6 s^6+A_8 s^8+... $$ Figure 2a: Spot Diagram of an Optical System with Uncorrected Spherical Aberrations Figure 2b: Spot Diagram of an Optical System with Spherical Aberration Compensation Plate What are the Benefits of Spherical Aberration Compensation Plates?
Spherical aberration compensation plates represent a shift in the paradigm for how optical designers and industrial end-users compensate and overcome spherical aberrations. They generate a new level of flexibility allowing for aberration correction to take place at the design phase, prototyping phase, or post production phase. Additionally, these corrector plates allow users to passively correct for known amounts of aberrations without a complete system redesign and without the inclusion of software and adaptive optics controls, saving time and money.
Historically, options for correcting spherical aberration have been expensive and cumbersome. These options include the use of adaptive optical systems, liquid lenses, or magnetorheological finishing of the final element in an assembly. In each of these cases, the process for reducing the spherical aberrations can be costly and extremely time intensive; making these solutions not well suited for OEM applications. Fortunately, the implementation of a single spherical aberration compensation plate is two orders of magnitude less expensive than most readily available adaptive optical systems.
Depending on their implementation, spherical aberration compensation plates can be used to improve system performance while reducing the total number of optical elements and thus reducing system weight, assembly time, and cost. As a component-level optic, the applications and benefits of these corrector plates are only limited by the creativity of their end-users.
Spherical aberration compensation plates represent the beginning of a new concept for complete aberration correction. As a result, it now seems possible to solve other aberrations by simply implementing a single optical component into the system level design without requiring a complete system redesign. These corrector plates represent a change in how aberration correction is done and pave the way for additional aberration correction plates (i.e., astigmatism, coma, Petzval, etc.).
How are Spherical Aberration Compensation Plate Innovative?
Optical designers have dealt with trying to compensate for spherical aberrations in their systems for centuries and will continue to do so as improvements in emerging technologies push the limits of optical components. Additionally, optical designers and manufacturers continue to look for designs that reduce aberrations as well as new components that eliminate spherical aberrations all together. With the advent of doublet lenses and aspheric lenses, many optical designers are able to compensate for spherical aberrations within their system during the early stages of design. However, industrial end-users and laboratory researchers often do not have a simple or cost effective solution which easily compensates for these errors.
Current aberration correction methods include the use of adaptive optics: more specifically, deformable mirrors or liquid lenses. The current state of these technologies require the end-user to have an in-depth knowledge of electro-optics and computer programming to successfully integrate a closed loop adaptive optics system. This makes it extremely difficult to quickly and easily improve system performance. Spherical aberration compensation plates represent a truly unique, passive solution for correcting spherical aberrations.
The component nature of the corrector plates drastically increases the product line's lifetime and overall usability. They can be integrated into a system at any point during the production or use-cycles. As a passive optical component, software and electronic advancements do not reduce or restrict the lifetime and usefulness of these products. As long as diverging or converging light is sent through glass (even as a window) spherical aberrations will continue to be a concern for optical designers and industrial end-users. These characteristics make spherical aberration plates a mainstay in the optics industry for years to come.
Real-World Application Examples
To truly understand and appreciate the benefits of incorporating spherical aberration compensation plates into an existing application setup, consider two real-world examples with accompanying equations, illustrations, and Zemax simulations.
Application 1: Spherical Aberration as a Function of Beam Diameter and Wavelength
Spherical aberration compensation plates specify the total amount of spherical aberration imparted on a collimated beam of light covering its entire clear aperture. However, it is often necessary to know the amount of spherical aberration generated by the corrector plate at beam diameters smaller than its clear aperture. If the beam diameter is smaller than the clear aperture, then consider the question, "How much spherical aberration is generated?"
For a collimated beam, the total wavefront error, W(λ,ρ), generated by the corrector plate is a function of both the wavelength and incident beam diameter:
(3)$$ W \left (\lambda, \rho \right)= W_{040} \rho^4 $$
where W(λ,ρ) is the transmitted wavefront error (WFE) due to spherical aberration in units of waves, or λ; ρ is the incident beam diameter divided by the clear aperture of the plate; and W
040, which depends on wavelength, is the wavefront aberration coefficient based on the individual plate in units of λ. Figure 3: Spherical Aberration vs. Incident Beam Diameter for #66-749 12.5mm Diameter +0.25λ Spherical Aberration Compensation Plate
At 587.6nm, W
040 is equal to the wavefront error indicated in the specifications of the individual corrector plate. For instance, for #66-749 12.5mm Diameter +0.25λ plate, W 040 equals +0.25 at 587.6nm and the clear aperture (CA) is 11.25mm. It is important to note that Equation 3 is valid only when a collimated beam is incident on the corrector plate; it is not valid if the incident beam is converging or diverging. To illustrate the amount of spherical aberration generated as a function of incident beam diameter, consider #66-749 at 587.6nm (Figure 3).
As mentioned earlier, the amount of spherical aberration generated by the spherical aberration compensation plate is also affected by the wavelength of the light source. Figure 4 shows that #66-749 produces more spherical aberration at shorter wavelengths than at longer ones. Also, this is true regardless of the sign of W
040 (Figures 4 – 5). In summary, the magnitude of the spherical aberration introduced by the corrector plate increases with aperture and decreases with wavelength. Figure 4: W 040 as a Function of Wavelength for #66-749 12.5mm Diameter +0.25λ Spherical Aberration Compensation Plate Figure 5: W 040 as a Function of Wavelength for #66-750 12.5mm Diameter -0.25λ Spherical Aberration Compensation Plate Application 2: Correcting Aberration from a Positive Focal Length Optical Lens
A positive focal length optical lens always introduces positive spherical aberration. This is evident from its transmitted wavefront error (WFE) profile and optical path difference (OPD) graph (Figure 6). To correct the positive lens' induced spherical aberration, use a negative spherical aberration compensation plate.
Consider the specific example of using #66-760 25mm Diameter -1.00λ Spherical Aberration Compensation Plate with #32-891 25mm Diameter 200m EFL Plano-Convex (PCX) Lens operating at f/8.89. Figure 6 illustrates the WFE profile and OPD graph of the PCX lens alone, whereas Figure 7 the corrector plate placed on the collimated side of the lens. Without the plate, the PCX lens produces +0.9162λ of spherical aberration; however, with the plate, the resulting transmitted WFE is +0.9162λ – 1λ ≈ -0.0836λ —less than λ/10! Though the numerical difference may be small, the optical difference is significant to anyone trying to correct for spherical aberration.
Figure 6: WFE (Left) and OPD at ƒ/8.89 (Right) for #32-891 25mm Diameter 200mm EFL Plano-Convex Lens Figure 8: Wavefront Error (WFE) Generated by #66-749 12.5mm Diameter +0.25λ Spherical Aberration Compensation Plate vs. f/#
It is important to note that because the spherical aberration compensation plate was placed in collimated space, the transmitted WFE is independent of the orientation of the aspheric surface on the plate. If the plate is added on the side of the lens where light is converging, then the amount of spherical aberration added by the plate is equal to the amount of spherical aberration produced by a plane parallel plate of the same thickness as the corrector plate plus the amount added if it were used in collimated space. To further understand this concept, consider the amount of induced spherical aberration for a plate placed in a convergent/divergent wavefront:
(4)$$ W\left( \lambda, \rho ,t,n, f/\# \right)= -\frac {t}{\left(f/\# \right)^4} \cdot {\frac {n^2-1}{128 \lambda n^3}}+W_{040} \rho^4 $$
where W(λ, ρ, t, n, f/#) is the transmitted WFE due to spherical aberration in units of waves, or λ; ρ is the incident beam diameter divided by the clear aperture of the plate; W
040, which depends on wavelength, is the wavefront aberration coefficient based on the individual plate in units of λ; t is the thickness of the plate; n is the index of refraction of the plate at wavelength λ; and f/# is the f-number of the convergent/divergent beam.
For an f/# ≥ 10, the wavefront error approaches +0.25 at 587.6nm when using #66-749 12.5mm Diameter 0.25λ Spherical Aberration Compensation Plate (Figure 8).
Optical aberrations exist in all optical, imaging, and photonics systems. The key to achieving the best systems is to understand and correct for these aberrations with the best methods and components. Spherical aberration compensation plates are one of the tools with which optical designers can save in design time, reduction of system weight and manufacturing costs. Spherical aberration compensation plates correct for known amounts of spherical aberration within a system, thereby allowing the implementation a single optical component without requiring a complete system redesign.
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Update:Since, it seems there is no progress regarding this question, any idea, conjecture, hunch, or advice is welcome. For example, are there any partial or incomplete results? What are the main challenges? Which techniques may be amenable to make any progress?Any observation (irrespective of whether it is insightful or not; trivial or not) is also welcome. Background:
Decision tree complexity or query complexity is a simple model of computation defined as follows. Let $f:\{0,1\}^n\to \{0,1\}$ be a Boolean function. The deterministic query complexity of $f$, denoted $D(f)$, is the minimum number of bits of the input $x\in\{0,1\}^n$ that need to be read (in the worse case) by a deterministic algorithm that computes $f(x)$. Note that the measure of complexity is the number of bits of the input that are read; all other computation is free.
We define the zero error or Las Vegas randomized query complexity of $f$, denoted $R_0(f)$, as the minimum number of input bits that need to be read in expectation by a zero-error randomized algorithm that computes $f(x)$. A zero-error algorithm always outputs the correct answer, but the number of input bits read by it depends on the internal randomness of the algorithm. (This is why we measure the expected number of input bits read.)
We define the bounded error or Monte Carlo randomized query complexity of $f$, denoted $R_2(f)$, to be the minimum number of input bits that need to be read by a bounded-error randomized algorithm that computes $f(x)$. A bounded-error algorithm always outputs an answer at the end, but it only needs to be correct with probability $\geq$ $1 - \delta$ ($2/3$, say).
Work on Recursive Boolean Functions:
There has been a line of work on the decision tree complexity of recursive boolean functions as mentioned below. The techniques focus on applying Yao's Lemma and using the distributional perspective guaranteed by it. This means we define a probability distribution on the inputs and the cost incurred by the best algorithm for this distribution gives a lower bound on the randomized decision tree complexity of the function. The worst possible distribution will give the actual randomized decision tree complexity.
The techniques in these works focus on giving a lower bound on the cost incurred by reading the "minority" bits (or vertices in the function tree) of the input via some form of induction. Another direction of attack could be to find the most "hard" distribution.
Some Notions
We define: The distribution $D^*$ on an input set $I$ is hard for a given function $f$, if $\forall D$ on $I$, $C(A,D) \leq C(A^*, D^*) $, where $C(A,D)$ is the expected cost (i.e. number of input bits read on expectation) incurred by the deterministic decision tree $A$ when the input follows the probability distribution is $D$. where $A^* = \operatorname{argmin}_A C(A, D^*), A = \operatorname{argmin}_A C(A, D)$
A distribution $D_1 < D_2$, if $C_m(D_1) < C_m(D_2)$, where $C_m(D_i) = C(A_i, D_i)$, and $A_i = \operatorname{argmin}_A C(A, D_i)$ . In other words $D_2$ is harder than $D_1$ means the best possible algorithm for $D_2$ does worse than the best possible algorithm for $D_1$. Note: The algorithm must be correct in the whole domain, and not just in the support of the distributions. For the base case of a recursive boolean function like say 2 bits or 4 bits, it is often easy to show a certain distribution to be hard. Often it is an easy observation or an obvious fact. In many cases, it may seem natural that the "hard" distribution is the recursive extension of the hard distribution. However, this may not be true in general, especially if the function is not symmetric over the input bits and rather skewed i.e. not all input bits are equally important to infer the value of the function on certain inputs (or a subset thereof).
Question:
Is there any work on how to approach the problem of finding the "hard" distribution of the recursive function, from that of the base case function?
Is there any interesting connection of this problem with any other problems? Any comments are welcome.
References:
[1] M. Saks, A. Wigderson Probabilistic Boolean Decision Trees and the Complexity of Evaluating Game Trees Proceedings of the 27th Foundations of Computer Science, pp. 29-38, October 1986.
[2] M. Santha. On the Monte Carlo boolean decision tree complexity of read-once formulae. Random Structures and Algorithms, 6(1):75{87, 1995.
[3] Frédéric Magniez, Ashwin Nayak, Miklos Santha, and David Xiao. Improved bounds for the randomized decision tree complexity of recursive majority. In Luca Aceto, Monika Henzinger, and Jiri Sgall, editors, ICALP (1), volume 6755 of Lecture Notes in Computer Science, pages 317–329. Springer, 2011.
[3a]Frederic Magniez, Ashwin Nayak, Miklos Santha, Jonah Sherman, Gabor Tardos, David Xiao. Improved bounds for the randomized decision tree complexity of recursive majority. http://arxiv.org/abs/1309.7565 (Submitted on 29 Sep 2013)
[4] Nikos Leonardos. An improved lower bound for the randomized decision tree complexity of recursive majority. In Fedor V. Fomin, Rusins Freivalds, Marta Z. Kwiatkowska, and David Peleg, editors, Proceedings of 40th International Colloquium on Automata, Languages and Programming, volume 7965 of Lecture Notes in Computer Science, pages 696{708. Springer, 2013.
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Thank you to Lazarus from Colyton Grammar School and Andrei fromTudor Vianu National College, Bucharest for your solutions to thisproblem. We see that in all three cases the ratio $XZ$ to $XY$ isthe Golden Ratio.
In the first diagram, taking the circle to have unit radius, $AO =BO = CO = ZO$.
Triangle $BXO$ is similar to triangle $XPO$ (angles $30^\circ,60^\circ, 90^\circ$) so $BX = BO\cos30 = {\sqrt 3\over 2}$ and $XO= BO\sin 30 = {1\over 2}$, $OP = XO\sin 30 = {1\over 4}$ and $XP =PY = XO \cos 30 ={\sqrt 3\over 4}$ so $XY = {\sqrt 3 \over2}$.
Now consider triangle $OPZ$. Angle $OPZ = 90^\circ$. Thereforeusing Pythagoras' Theorem, $PZ= \sqrt{1 - {1\over 16}}={\sqrt{15}\over 4}$. So $XZ= XP + PZ = {\sqrt{3}\over 4}\left(1 +\sqrt{5}\right)$ and $${XZ\over XY} = {1\over 2}\left(1 +\sqrt{5}\right)$$ which is the Golden Ratio.
In the second diagram, taking the side of the square $XY$ as $1$unit and using Pythagoras Theorem for triangle $OSY$ gives$OS={\sqrt 5 \over 2}$ which is the radius of the circle. So $OZ={\sqrt 5 \over 2}$. Hence $XZ = {1\over 2}\left(1 +\sqrt{5}\right)$ and $${XZ\over XY} = {1\over 2}\left(1 +\sqrt{5}\right)$$ again the Golden Ratio.
In the third diagram take the sides of the pentagon to be $1$ unitlong and the chords $XZ$ and $WZ$ etc. to be $x$ units.
Triangles $UXY$, $YVZ$ and $XZW$ are similar isosceles triangles(angles of $36^\circ$, $72^\circ$ and $72^\circ$) with $VY =VZ = XY= 1$ and $YZ = x-1$. We see that the ratio we want to find $XZ/XY =x$.
By similar triangles, taking the ratio of the long to the shortsides, we get $${x\over 1 } = {1 \over{x-1}}$$ which gives thequadratic equation $$x^2 - x - 1 = 0$$ with solutions ${1\over2}\left(1 + \sqrt{5}\right)$ or ${1\over 2}\left(1 -\sqrt{5}\right)$. The latter is negative so does not make sensegeometrically so the ratio must be the Golden Ratio: $${1 \over2}\left(1 + \sqrt{5}\right)$$
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Resonance Contents Classical Derivations Series Resonance
The classical circuit to demonstrate series resonance is the RLC circuit shown in the figure right, which shows a voltage source connected to R, L and C impedances in series. Given a fixed ac voltage source U operating at angular frequency [math] \omega [/math], the current in the circuit is given by the following:
[math]I = \frac{U}{Z} = \frac{U}{R + j \left(\omega L - \frac{1}{\omega C} \right)} [/math] [math]= \frac{U}{R + j \left(\frac{\omega^{2} LC - 1}{\omega C} \right)} [/math]
The current is at a maximum when the impedance is at a minimum. So given constant R, L and C, the minimum impedance occurs when:
[math]\omega^{2} LC - 1 = 0 \, [/math]
or
[math]\omega = \frac{1}{\sqrt{LC}} [/math]
This angular frequency is called the
resonant frequency of the circuit. At this frequency, the current in the series circuit is at a maximum and this is referred to as a point of series resonance. The significance of this in practice is when harmonic voltages at the resonant frequency cause high levels of current distortion. Parallel Resonance
The classical circuit to demonstrate series resonance is the RLC circuit shown in the figure right, which shows a current source connected to R, L and C impedances in parallel. Given a fixed ac current source I operating at angular frequency [math] \omega [/math], the voltage across the impedances is given by the following:
[math]V = IZ = \frac{I}{\frac{1}{R} + j \left(\omega C - \frac{1}{\omega L} \right)} [/math] [math]= \frac{I}{\frac{1}{R} + j \left(\frac{\omega^{2} LC - 1}{\omega L} \right)} [/math]
The voltage is at a maximum when the impedance is also at a maximum. So given constant R, L and C, the maximum impedance occurs when:
[math]\omega^{2} LC - 1 = 0 \, [/math]
or
[math]\omega = \frac{1}{\sqrt{LC}} [/math]
Notice that the resonant frequency is the same as that in the series resonance case. At this resonant frequency, the voltage in the parallel circuit is at a maximum and this is referred to as a point of parallel resonance. The significance of this in practice is when harmonic currents at the resonant frequency cause high levels of voltage distortion.
Resonance in Practical Circuits Series Resonance
Here a distorted voltage at the input of the transformer can cause high harmonic current distortion ([math]I_{h}[/math]) at the resonant frequency of the RLC circuit.
Parallel Resonance
In this more common scenario, a harmonic current source ([math]I_{h}[/math]) can cause high harmonic voltage distortion on the busbar at the resonant frequency of the RLC circuit. The harmonic current source could be any non-linear load, e.g. power electronics interfaces such as converters, switch-mode power supplies, etc.
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We let $\mathcal{F}_\infty=\bigvee_n\mathcal{F}_n$. (What does $\bigvee$ usually denote? Is $\bigvee$ different from $\cup$? Can't find it in the text.)
The theorem tells us when we have $\lim_n E(Y|\mathcal{F}_n)=E(Y|\mathcal{F}_\infty)$.
I'd like to know:
Is $\mathcal{F}_\infty$ a $\sigma$-algebra? Is it true that $A\in\mathcal{F}_\infty$ implies $A\in\mathcal{F}_n$ for some $n$?
It seems the answer to the former (assuming the latter is true) is no: If $A\in \mathcal{F}_1$ and $B\in\mathcal{F}_2$, then $A\cup B$ need not be an element of $\mathcal{F_n}$ for any $n$.
But then $E(Y|\mathcal{F}_\infty)$ is nonsensical, since conditional expectation (as a random variable) is defined using $\sigma$-algebras.
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The problem is:
Define a sequence by $a_{1} = 1, a_{2} = 1, a_{n+2} = \sqrt{a_{n+1} + a_{n}}, \forall n \geq 1.$
(a) Prove that $a_{n} < 2$, for all positive integer $n$.
(b) Prove that for all positive integer $n$ such that $n \geq 2$, we have $a_{n+1} > a_{n}$.
I have trouble with part (b). What I did is as follows:
For $n = 2$, \begin{equation} a_{n+1} = a_{3} = \sqrt{a_{2} + a_{1}} = \sqrt{1+1} = \sqrt{2} > a_{2} = 1; \tag{1} \end{equation} For $n = k$, assume $a_{k+1} > a_{k}$;
For $n = k+1$, \begin{equation} a_{n+1} = a_{k+2} = \sqrt{a_{k+1} + a_{k}} > \sqrt{a_{k} + a_{k}} = \sqrt{2a_{k}} . \tag{2} \end{equation}
I need to prove $a_{k+2} > a_{k+1}$, but (2) does not lead to it. How should I solve this problem?
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There are reasons that any modern example is likely to resemble the status of Legendre's constant. Most (but not all) interesting numbers admit a polynomial-time algorithm to compute their digits. In fact, there is an interesting semi-review by Borwein and Borwein that shows that most of the usual numbers in calculus (for example, $\exp(\sqrt{2}+\pi)$) have a
quasilinear time algorithm on a RAM machine, meaning $\tilde{O}(n) = O(n(\log n)^\alpha)$ time to compute $n$ digits. Once you have $n$ digits, you can use the continued fraction algorithm to find the best rational approximation with at most $n/2-O(1)$ digits in the denominator. The continued fraction algorithm is equivalent to the Euclidean algorithm, which also has a quasilinear time version according to Wikipedia.
Euler's constant has been to computed almost 30 billion digits, using a quasilinear time algorithm due to Brent and McMillan.
As a result, for any such number it's difficult to be surprised. You would need a mathematical coincidence that the number is rational, but with a denominator that is out of reach for modern computers. (This was Brent and MacMillian's stated motivation in the case of Euler's constant.) I think that it would be fairly newsworthy if it happened. On the other hand, if you can only compute the digits very slowly, then your situation resembles Legendre's.
I got e-mail asking for a reference to the paper of Borwein and Borwein. The paper is On the complexity of familiar functions and numbers. To summarize the relevant part of this survey paper, any value or inverse value of an elementary function in the sense of calculus, including also hypergeometric functions as primitives, can be computed in quasilinear time. So can the gamma or zeta function evaluated at a rational number.
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But if you don't want to have a Google account: Chrome is really good. Much faster than FF (I can't run FF on either of the laptops here) and more reliable (it restores your previous session if it crashes with 100% certainty).
And Chrome has a Personal Blocklist extension which does what you want.
: )
Of course you already have a Google account but Chrome is cool : )
Guys, I feel a little defeated in trying to understand infinitesimals. I'm sure you all think this is hilarious. But if I can't understand this, then I'm yet again stalled. How did you guys come to terms with them, later in your studies?
do you know the history? Calculus was invented based on the notion of infinitesimals. There were serious logical difficulties found in it, and a new theory developed based on limits. In modern times using some quite deep ideas from logic a new rigorous theory of infinitesimals was created.
@QED No. This is my question as best as I can put it: I understand that lim_{x->a} f(x) = f(a), but then to say that the gradient of the tangent curve is some value, is like saying that when x=a, then f(x) = f(a). The whole point of the limit, I thought, was to say, instead, that we don't know what f(a) is, but we can say that it approaches some value.
I have problem with showing that the limit of the following function$$\frac{\sqrt{\frac{3 \pi}{2n}} -\int_0^{\sqrt 6}(1-\frac{x^2}{6}+\frac{x^4}{120})^ndx}{\frac{3}{20}\frac 1n \sqrt{\frac{3 \pi}{2n}}}$$equal to $1$, with $n \to \infty$.
@QED When I said, "So if I'm working with function f, and f is continuous, my derivative dy/dx is by definition not continuous, since it is undefined at dx=0." I guess what I'm saying is that (f(x+h)-f(x))/h is not continuous since it's not defined at h=0.
@KorganRivera There are lots of things wrong with that: dx=0 is wrong. dy/dx - what/s y? "dy/dx is by definition not continuous" it's not a function how can you ask whether or not it's continous, ... etc.
In general this stuff with 'dy/dx' is supposed to help as some kind of memory aid, but since there's no rigorous mathematics behind it - all it's going to do is confuse people
in fact there was a big controversy about it since using it in obvious ways suggested by the notation leads to wrong results
@QED I'll work on trying to understand that the gradient of the tangent is the limit, rather than the gradient of the tangent approaches the limit. I'll read your proof. Thanks for your help. I think I just need some sleep. O_O
@NikhilBellarykar Either way, don't highlight everyone and ask them to check out some link. If you have a specific user which you think can say something in particular feel free to highlight them; you may also address "to all", but don't highlight several people like that.
@NikhilBellarykar No. I know what the link is. I have no idea why I am looking at it, what should I do about it, and frankly I have enough as it is. I use this chat to vent, not to exercise my better judgment.
@QED So now it makes sense to me that the derivative is the limit. What I think I was doing in my head was saying to myself that g(x) isn't continuous at x=h so how can I evaluate g(h)? But that's not what's happening. The derivative is the limit, not g(h).
@KorganRivera, in that case you'll need to be proving $\forall \varepsilon > 0,\,\,\,\, \exists \delta,\,\,\,\, \forall x,\,\,\,\, 0 < |x - a| < \delta \implies |f(x) - L| < \varepsilon.$ by picking some correct L (somehow)
Hey guys, I have a short question a friend of mine asked me which I cannot answer because I have not learnt about measure theory (or whatever is needed to answer the question) yet. He asks what is wrong with \int_0^{2 \pi} \frac{d}{dn} e^{inx} dx when he applies Lesbegue's dominated convergence theorem, because apparently, if he first integrates and then derives, the result is 0 but if he first derives and then integrates it's not 0. Does anyone know?
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Real-time computation (or estimation) of the “probability to win” is difficult. We’ve seem that in soccer games, in elections… but actually, as a professor, I see that frequently when I grade my students.
Consider a classical multiple choice exam. After each question, imagine that you try to compute the probability that the student will pass. Consider here the case where we have 50 questions. Students pass when they have 25 correct answers, or more. Just for simulations, I will assume that students just flip a coin at each question… I have n students, and 50 questions
1 2 3 set.seed(1) n=10 M=matrix(sample(0:1,size=n*50,replace=TRUE),50,n)
Let X_{i,j} denote the score of student i at question j. Let S_{i,j} denote the cumulated score, i.e. S_{i,j}=X_{i,1}+\cdots+X_{i,j}. At step j, I can get some sort of prediction of the final score, using \hat{T}_{i,j}=50\times S_{i,j}/j. Here is the code
1 2 SM=apply(M,2,cumsum) NB=SM*50/(1:50)
We can actually plot it
1 2 3 4 plot(NB[,1],type="s",ylim=c(0,50)) abline(h=25,col="blue") for(i in 2:n) lines(NB[,i],type="s",col="light blue") lines(NB[,3],type="s",col="red") But that’s simply the prediction of the final score, at each step. That’s not the computation of the probability to pass !
Let’s try to see how we can do it… If after j questions, the students has 25 correct answer, the probability should be 1 – i.e. if S_{i,j}\geq 25 – since he cannot fail. Another simple case is the following : if after j questions, the number of points he can get with all correct answers until the end is not sufficient, he will fail. That means if S_{i,j}+(50-i+1)< 25[/latex] the probability should be 0. Otherwise, to compute the probability to sucess, it is quite straightforward. It is the probability to obtain at least [latex]25-S_{i,j}[/latex] correct answers, out of [latex]50-j[/latex] questions, when the probability of success is actually [latex]S_{i,j}/j[/latex]. We recognize the survival probability of a binomial distribution. The code is then simply
1 2 3 4 5 6 7 PB=NB*NA for(i in 1:50){ for(j in 1:n){ if(SM[i,j]>=25) PB[i,j]=1 if(SM[i,j]+(50-i+1)<25) PB[i,j]=0 if((SM[i,j]<25)&(SM[i,j]+(50-i+1)>=25)) PB[i,j]=1-pbinom(25-SM[i,j],size=(50-i),prob=SM[i,j]/i) }}
So if we plot it, we get
1 2 3 4 plot(PB[,1],type="s",ylim=c(0,1)) abline(h=25,col="red") for(i in 2:n) lines(PB[,i],type="s",col="light blue") lines(PB[,3],type="s",col="red")
which is much more volatile than the previous curves we obtained ! So yes, computing the “probability to win” is a complicated exercice ! Don’t blame those who try to find it hard to do !
Of course, things are slightly different if my students don’t flip a coin… this is what we obtain if half of the students are good (2/3 probability to get a question correct) and half is not good (1/3 chance),
If we look at the probability to pass, we usually do not have to wait until the end (the 50 questions) to know who passed and who failed
PS : I guess a less volatile solution can be obtained with a Bayesian approach… if I find some spare time this week, I will try to code it…
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School, from our house as the crow flies, is 5.73 km. If we neglect air resistance and deal strictly with ballistic flight then we can materialize a wonderful fantasy. Starting in the backyard, extending over the top of the house, is a
launch-o-rocket, a rail-like launcher that accelerates the school-bound student until he or she can cruise over the city and arrive without bother of traffic. Our charter is to find the acceleration of the student from the launch-o-rocket. Finding the Initial Velocity
We rely on the well-known fact that the maximum distance in a throw occurs when the departure angle is 45°. The vertical speed and the horizontal speed are equal. We denote these two identical speeds as $s$. Since distance is time multiplied by speed, the distance from home to school $d$ is
$$ d = t\cdot s.$$
We know the distance $d = $ 5.73 km.
Turning to the vertical speed, the student departs the launch-o-rocket with vertical speed $s$, but is immediately subject to gravitational acceleration. Since the student’s upward flight is exactly matched by his or her downward flight. Because the flight is matched, the student spends $t/2$ time rising and $t/2$ time descending. Since the student has no vertical speed at the top, we know that his or her speed is
$$ s = g\frac{t}{2},$$
where $g$ is the gravitational acceleration 9.8 m/s
2.
Now, we have a system of equations
$$ d = t\cdot s $$
$$ s = g\frac{t}{2}.$$ The system looks like it has a many variables, but really there are only two, $s$ and $t$. We know $g$ and $d$. To solve the system we substitute for $s$ in the first equation with the second to get
$$ d = tg\frac{t}{2} = g\frac{t^2}{2}$$
Solve for $t$ $$ t = \sqrt{\frac{2d}{g}} = \sqrt{\frac{2\cdot 5.73\,\text{m}}{9.8\,\text{m/s}^2}} \approx 34.2\,\text{s}. $$ Not a bad commute, a little over half a minute.
With $t$ in hand, we can find the magnitude of the initial velocity. Remember that the initial velocity is $s$ in the horizontal direction and $s$ in the vertical direction, so the speed when leaving the launcher is
$$ \left| \mathbf{v}_0\right| = \sqrt{s^2 + s^2} = \sqrt{2s^2} = s\sqrt{2}. $$ The initial speed the student must attain is given by the very first equation, $d = s\cdot t$. Solving for $s$ with the value of $t$ we found, we get
$$ s = \frac{5.73\,\text{km}}{34.2\,\text{s}} = 168\,\text{m/s}. $$
Finding the Acceleration
The ramp lives on a footprint that is about 80 ft, or 24.4 m. It is also 24.4 m tall, so special zoning is surely required! The rail of the launch-o-rocket is the hypotenuse of a triangle, and that triangle has sides 24.4 m, and a total length of $\sqrt{2}\cdot 24.4\,\text{m} = 34.5\,\text{m}$.
The formula for position after a period of acceleration is
$$ p = \frac{1}{2}a\tau^2.$$ For our system, we also know that the acceleration is the change in speed divided by the change in time. Our speed goes from zero to 168 m/s in $\tau$. Again, we have a system of equations,
$$ 34.5\, \text{m} = \frac{1}{2}a\tau^2 $$
$$ a = \frac{168\,\text{m/s}}{\tau}.$$
Solve for $a$ by first solving the second equation for $\tau$, and then substituting that result into the first equation to get
$$ 34.5\, \text{m} = \frac{1}{2}a\left(\frac{168\,\text{m/s}}{a}\right)^2 $$
$$ a = \frac{\left( 168\, \text{m/s}\right)^2}{2 \cdot 34.5\,\text{m}}
= 407\, \text{m/s}^2 = 41.5\, g. $$
The typical onset of death occurs when acceleration exceeds about $10g$, so unfortunately,
the launch-o-rocket is a single try system.
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I am to calculate the rate of collision of gas molecules with the
walls of a container with dimensions $3m\times4m\times2m$ at room temperature ( 20°C) and pressure 1atm.For reasons of simplicity I may assume the air in the container to be an ideal gas with an atomic mass of 29.
There are a couple of intuitions on my mind but I can't quite connect the dots. For on I know that the average speed of a molecule is $$v_{avg}=\sqrt{\frac{8k_BT}{\pi m}}$$ pressure is defined as $$p=\frac{F}{A}$$ the force F can be calculated using the impulse $$I=\Delta p=F*\Delta t \rightarrow F=\frac{\Delta p}{\Delta t}=\frac{2mv}{\Delta t}$$ solving for $\Delta t$ using $$d=vt \rightarrow t=\frac{d}{v}=\frac{2L}{v}$$ (where L is the distance between opposite walls) results in a frequency of collision.
As I said I just can't connect the dots here and any help on how to tackle this problem would be greatly appreciated.
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It is not possible at an information-theoretic level to do what you want to do.
Let us suppose we have two pure states: $|\phi\rangle$ and $|\psi\rangle$, where$$|\phi\rangle = \frac{1}{\sqrt{N}}\sum_{i=1}^N |x_i\rangle$$and $|\psi\rangle$ is similar to $|\phi\rangle$ but with a few of the coefficients tweaked in some way. Notice that we're fixing just two states, and even if you're promised that you're given one of these two states it will not be possible to determine which one you're given with high probability, under the assumption that $|\phi\rangle$ and $|\psi\rangle$ are close together. (As DaftWullie has suggested, a variant of this problem where $|\phi\rangle$ is fixed and $|\psi\rangle$ is not known ahead of time is certainly no easier than the case in which $|\psi\rangle$ is known ahead of time.)
To keep things simple, let us suppose that we're given $|\phi\rangle$ with probability $1/2$ and $|\psi\rangle$ with probability $1/2$, and we're aiming to maximize the probability of correctly determining which of the two states we were given. A theorem sometimes called the Holevo-Helstrom theorem tells us exactly what the optimal probability of a correct guess is:$$\frac{1}{2} + \frac{1}{4} \bigl\| |\phi\rangle\langle \phi| - |\psi\rangle\langle \psi| \bigr\|_1,$$where the norm is the trace norm. Because we're working with pure states, this expression can be simplified to$$\frac{1}{2} + \frac{1}{2}\sqrt{1 - |\langle \psi | \phi \rangle|^2}.$$This is for the
optimal measurement; you cannot do any better than this, no matter what you try to do, assuming you start with one of the two states selected at random and by honest means do your best to determine which state you were given.
Now, the question suggests that the "unbalancedness," or variance among the coefficients of the states, is small. With that in mind we could define$$\varepsilon = 1 - |\langle \psi | \phi \rangle|^2$$and regard $\varepsilon$ as a small (but nonnegative) real number. This means that our probability of a correct guess is$$\frac{1}{2} + \frac{\sqrt{\varepsilon}}{2},$$which is not much better than just randomly guessing (which yields a correct answer with probability 1/2).
If you assume that you are given $k$ copies of $|\phi\rangle$ or $k$ copies of $|\psi\rangle$, the calculation is exactly the same, except replacing the states with $|\phi\rangle^{\otimes k}$ and $|\psi\rangle^{\otimes k}$, also as DaftWullie has suggested. We can see how this will affect the optimal correctness probability: it now becomes$$\frac{1}{2} + \frac{1}{2}\sqrt{1 - |\langle \psi | \phi \rangle|^{2k}}.$$Notice that we're placing no constraints on the measurements for distinguishing the two cases -- they could be arbitrarily correlated across the $k$ copies of the states.
We can compare this optimal correctness probability with the one-shot case by using the inequality$$1 - |\langle \psi | \phi \rangle|^{2k}\leq k \bigl(1 - |\langle \psi | \phi \rangle|^2\bigr).$$To prove this inequality, notice that it is trivial if $|\langle \psi | \phi \rangle| = 1$, and otherwise we can use the crude estimate$$\frac{1 - |\langle \psi | \phi \rangle|^{2k}}{1 - |\langle \psi | \phi \rangle|^2} = 1 + |\langle \psi | \phi \rangle|^2 + \cdots + |\langle \psi | \phi \rangle|^{2(k-1)} < k$$for the case $|\langle \psi | \phi \rangle| < 1$. We find that the optimal correctness probability is upper-bounded by$$\frac{1}{2} + \frac{\sqrt{k\varepsilon}}{2}.$$So, if $k$ is considered to be much smaller than $N$, and $\varepsilon$ is on the order of $1/N$, then we're still not doing much better than randomly guessing.
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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 Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV have been measured using the ALICE detector ...
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The first result is an immediate consequence of the fact that if $p$ is an odd prime, it has exactly as many QR as NR in the interval $[1,p-1]$.
For the second, I take it we want to find $\sum_1^{p-2}\left(\frac{k(k+1)}{p}\right)$.
Let $k^\ast$ be (the least positive residue of) the inverse of $k$ modulo $p$. Then$$\left(\frac{k(k+1)}{p}\right)=\left(\frac{k(k+kk\ast)}{p}\right)=\left(\frac{k^2(1+k^\ast)}{p}\right)=\left(\frac{1+k^\ast}{p}\right).$$
As $k$ travels through the numbers $1$ to $p-2$, the numbers $1+k^\ast$ travel through the integers from $2$ to $p-1$. So our sum is the sum of all the Legendre symbols $\left(\frac{z}{p}\right)$ except $\left(\frac{1}{p}\right)$. This Legendre symbol is $1$. By the first result, the sum of
all the Legendre symbols is $0$. This yields the second result.
Remark: We leave to you to show that from the second result that any prime $\gt 3$ has $2$ consecutive QR or two consecutive NR. (There are simpler ways to show that.)
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I'm attempting to prepare data in the same manner as section 2 of this paper.
I'm finding it a bit of a struggle. Could someone check (/improve upon) my interpretation regarding the 2 sections I have highlighted (below)?
In the first section (highlighted in yellow):
We note that price momentum is a cross-sectional phenomenon with winners having high past returns and losers having low past returns relative to other stocks. Thus we normalize each of the cumulative returns by calculating the z-score relative to the cross-section of all stocks for each month or day.
... I'm struggling to understand exactly what is being described.
As far as I can see, the process would be:
For each day ... for each stock ... Assemble a trailing length-33 vector of past prices Use this to compute stock's mean & standard deviation Use today's price $x$ to compute stock's z-value: $z = \frac{x-\mu}{\sigma}$ (If I understand correctly, $z$ is a basic indication of momentum). Now we have a z-vector over all stocks for this day. Normalize it! (?) Now we have a vector indicating relative momentum for each stock for this day. for each stock ...
And the section (highlighted in green):
Finally we use returns over the subsequent month, t + 1, to label the examples with returns below the median as belonging to class 1 and those with returns above the median to class 2.
... I think translates as:
get monthly returns for months t-13 through t+1 & compute median class = 1 if return for month t+1 < median else 2
So it looks as though class 2 stocks follow their normalised $z$ momentum-indicator, whereas class 1 don't.
Does this look correct?
PS No tag for 'data-processing'
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Understanding Spatial Filters
Spatial Filters are designed to be used with lasers to "clean up" the beam. Often times a laser system does not produce a beam with a smooth intensity profile. In order to produce a clean Gaussian beam, a spatial filter is used to remove the unwanted multiple-order energy peaks and pass only the central maximum of the diffraction pattern (see illustration). Also, when a laser beam passes through a system, dust in the air or on optical components can disrupt the beam and create scattered light. This scattered light can leave unwanted ring patterns in the beam profile. The spatial filter removes this additional spatial noise from the system. The spatial filter assembly consists of a microscope objective, a pinhole aperture, and a positioning mechanism. The positioning mechanism has precision X-Y movements that center the pinhole at the focal point of the objective lens. Our TECHSPEC® Laser Objectives are designed for HeNe lasers (632.8nm) and provide the smallest spot sizes possible. Choosing the correct pinhole and objective combination will yield optimal results. The following equations were used to determine the values for the Aperture Selection Chart.
(1)$$ \text{Beam Spot Diameter} \left[ \text{μm} \right] = 1.27 \frac{\lambda \, f}{D} $$
Where:
λ = wavelength of laser (microns) f = focal length of objective lens (mm) D = input beam diameter (mm)
Pinhole size is then determined for the table (see note):
(2)$$ \text{Pinhole Diameter} \left[ \text{μm} \right] = 1.5 \times \text{Beam Spot Size Diameter} \left[ \text{μm} \right] $$
Note: The factor of 1.5 in Equation 2.0 is determined as the optimal factor in order to pass the maximum amount of energy, while eliminating as much spatial noise as possible.
Aperture Selection Chart EFL of Microscope Objective Lens (mm) 5.5mm 8.0mm 8.5mm 9.0mm 1.84mm Input Beam Dia. 8μm 10μm 12.5μm 12.5μm 1.56mm Input Beam Dia. 8μm 12.5μm 12.5μm 15μm 1.44mm Input Beam Dia. 8μm 12.5μm 15μm 15μm 1.34mm Input Beam Dia. 10μm 15μm 15μm 15μm 1.31mm Input Beam Dia. 10μm 15μm 15μm 15μm 1.21mm Input Beam Dia. 10μm 15μm 15μm 15μm 0.92mm Input Beam Dia. 12.5μm 20μm 20μm 25μm
If you would like to see other topics covered or more detailed information, let us know. We invite you to discuss any suggestions or specific application requirements with our engineering department at 800-363-1992 or contacting us online.
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EDIT: I moved my original question down to the bottom. The question here at the top is related, at least I suppose that the same phenomenon is behind both of them.
In the article "Valeurs de fonctions L et périods d'intégrales" (p. 333/334), Deligne sketches how to define the motive $M(f)$ attached to a newform $f=\sum a_nq^n$. The construction involves taking the eigenspace (inside a certain cohomology) where the Hecke operators act with the eigenvalues given by the Fourier coefficients of $f$, i.e. the common kernel of all $T_p-a_p$. He then uses the differential form $\omega_f:=f(z)\,2\pi i\mathrm dz$ to calculate periods (I'm assuming that the weight is $2$ here, for simplicity).
Deligne claims that the
transposes ${}^tT_n$ of the Hecke operators act on $\omega_f$ by the eigenvalues $a_n$, so that $\omega_f$ represents a class in the de Rham realization of the motive $M(f^*)$ attached to the dual cusp form.
Why is this true? Let ${}^N_kW$ be the motive defined by Scholl having $M(f)$ as a submotive. If we view $\omega_f$ as a class in $\operatorname{fil}^1{}^N_kW_{\mathrm{dR}}\otimes\mathbb C\cong S_k(X(N),\mathbb C)$, then the complex comparison isomorphism to ${}^N_kW_{\mathrm B}\otimes\mathbb C$ (which is the Eichler-Shimura isomorphism) sends $\omega_f$ to the usual cocycle given by integrating $\omega_f$ along geodesics on the upper half plane, which corresponds to $f$ (and not $f^*$!). Since the Eichler-Shimura isomorphism is Hecke equivariant, $\omega_f$ should lie in the eigenspace for $f$ and not $f^*$, i.e. it should be just $f$ under the identification $\operatorname{fil}^1{}^N_kW_{\mathrm{dR}}\otimes\mathbb C\cong S_k(X(N),\mathbb C)$.
EDIT: This was my original question.
I have a question concerning a detail in Tony Scholl's construction of motives attached to modular forms.
To my understanding, the construction of Deligne's Galois representation attached to modular forms works as follows. Let $Y_1(N)$ be the modular curve over $\mathbb Q$ classifying elliptic curves with a point of exact order $N$, and let $f\colon E_1(N)\rightarrow Y_1(N)$ be the universal elliptic curve over it. Then one looks at the étale cohomology group $$ {}_k^NW_\ell:=H^1_{\mathrm p}(Y_1(N),\operatorname{Sym}^{k-2}R^1f_*\mathbb Q_\ell).$$ From the Hecke correspondences on $Y_1(N)$ one gets a Hecke action on this $\mathbb Q_\ell$-vector space, and using the (Eichler-Shimura) congruence relation and the "corrected scalar product", or "twisted Poincaré duality" (involving the Atkin-Lehner involution) one can show that $$ \det(1-F_pX,{}_k^NW_\ell)=1-T_p+p^{k-1}\langle p \rangle X^2 $$ where $F_p$ is a geometric Frobenius at $p$. Hence to get Deligne's Galois representation attached to a fixed newform $f$ of weight $k$ and level $N$ with Fourier coefficients in a number field $K$, one just tensors ${}_k^NW_\ell$ over the Hecke algebra $T$ with the morphism $T\rightarrow K_\lambda$ induced by $f$, where $\lambda$ is a prime of $K$ lying above $\ell$ (or equivalently, one projects to the eigenspace of ${}_k^nW_\ell\otimes_{\mathbb Q_\ell}K_\lambda$ where the Hecke operators act by the eigenvalues given by the Fourier coefficients of $f$).
Now in the article "Motives for modular forms" Scholl defines a motive over $\mathbb Q$ called ${}_k^NW$ whose $\ell$-adic étale realization is precisely ${}^N_kW_\ell$. He then shows that the Hecke correspondences induce endomorphisms of this motive and that on the Betti realization these endomorphisms are the same as the ones defined by Deligne (Prop. 4.1.1). Hence also on the $\ell$-adic realization these are the same as Deligne's.
But then Scholl projects to the eigenspace where the
transposes ${}^tT_p$ of the Hecke operators act by the eigenvalues given by $f$ to get his motive attached to $f$. He claims (4.2.2) that on the $\ell$-adic realization we get precisely Deligne's Galois representation attached to $f$. To me this seems to contradict the above, it should rather give the representation attached to the "dual" modular form $f^*$. Also Scholl writes that on the space of cusp forms (which is the intermediate step $\operatorname{fil}^1{}^N_kW_{\mathrm{dR}}=\operatorname{fil}^{k-1}{}^N_kW_{\mathrm{dR}}$ in the Hodge filtration on the de Rham realization of ${}^N_kW$) the endomorphisms of ${}^N_kW$ given by the Hecke correspondences induce the tranposes of the usual Hecke operators. But I do not see why this statement is true. If on the Betti realization they are the same as Deligne's, then by the Eichler-Shimura isomorphism one should see that the are the usual ones on the space of cusp forms... (?)
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We consider perturbations to the rest state of heat conductive compressible fluid in a three-dimensional exterior domain. In the regularity class $ {\cal V}_s $ , we show stability of L 2 -norms of these perturbations, while in the more restrictive class $ {\cal V}_d $ we show convergence to zero of these perturbations along a time sequence converging to infinity.
We consider the zero-velocity stationary problem of the Navier—Stokes equations of compressible isentropic flow describing the distribution of the density $ \varrho $ of a fluid in a spatial domain $ \Omega \subset {\rm R}^N $ driven by a time-independent potential external force $ \vec f = \triangledown F $ . We study the structure of the set of all solutions to the stationary problem having a prescribed mass m > 0 and a prescribed energy. Cardinality of the solution set depends on m and it is either continuum or at most two. Conditions on m for distinguishing these cases have been found. Uniqueness for the stationary system is also studied.
We consider local mesh adaptation for stationary flow control problems. The state equation is given by the incompressible Navier-Stokes equations with Neumann boundary control. As an example, the functional to be minimized is the drag coefficient of an immersed body. We use finite elements on locally refined meshes to discretize the first order necessary conditions based on the Lagrangian. An a posteriori error estimate is derived which is directly related to the control problem. It is used to successively enrich the finite element space until the computed solution satisfies prescribed accuracy requirements.
We prove the existence of globally defined weak solutions to the Navier—Stokes equations of compressible isentropic flows in three space dimensions on condition that the adiabatic constant satisfies $ \gamma > 3/2 $ .
We prove an optimal relationship between the regularity of a function and the asymptotic behavior of its Fourier transform. As an application of this result we show L p -estimates for the Stokes semigroup in $ \Bbb R^n $ and $ {\Bbb R}_+^n $ when $ 1\leqq p\leqq \infty $ .
For 2D Navier—Stokes equations defined in a bounded domain $ \Omega $ we study stabilization of solution near a given steady-state flow $ \hat v(x) $ by means of feedback control defined on a part $ \Gamma $ of boundary $ \partial\Omega $ . New mathematical formalization of feedback notion is proposed. With its help for a prescribed number $ \sigma > 0 $ and for an initial condition v 0(x) placed in a small neighbourhood of $ \hat v(x) $ a control u(t,x'), $ x' \in \Gamma $ , is constructed such that solution v(t,x) of obtained boundary value problem for 2D Navier—Stokes equations satisfies the inequality: $ \|v(t,\cdot)-\hat v\|_{H^1}\leqslant ce^{-\sigma t}\quad {\rm for}\; t \geqslant 0 $ . To prove this result we firstly obtain analogous result on stabilization for 2D Oseen equations.
A global-in-time unique smooth solution is constructed for the Cauchy problem of the Navier—Stokes equations in the plane when initial velocity field is merely bounded not necessary square-integrable. The proof is based on a uniform bound for the vorticity which is only valid for planar flows. The uniform bound for the vorticity yields a coarse globally-in-time a priori estimate for the maximum norm of the velocity which is enough to extend a local solution. A global existence of solution for a q-th integrable initial velocity field is also established when $ q > 2 $ .
We investigate the steady motion of a liquid in a lake, modeled as a thin domain. We assume the motion is governed by Navier—Stokes equations, while a Robin-type traction condition, and a friction condition is prescribed at the surface and at the bottom, respectively. We also take into account Coriolis forces. We derive an asymptotic model as the aspect ratio $ \delta $ = depth/width of the domain goes to 0. When the Reynolds number is not too large, this is mathematically justified and the three-dimensional limit velocity is given in terms of wind, bathymetry, depth and of a two-dimensional potential. Numerical simulation is carried out and the influence of traction condition reading is experienced.
Some conceptual ambiguities in the derivation of the equations of capillarity on the basis of the principle of virtual work are addressed, and hypotheses are proposed toward obtaining a physically correct characterization in general circumstances. It is shown that under the hypotheses, the classical equations of capillarity for an interface of an incompressible fluid with a fluid of negligible density can be obtained on the basis of global phenomenological reasoning, without recourse to consideration of intermolecular attractions. More generally, the procedure is applied to derive the specific equations arising from a compressible fluid configuration with idealized pressure-density relationship in a capillary tube, and a general necessary condition for existence of a solution is established. It is shown that for symmetric domains, the condition is also sufficient for existence of a unique symmetric solution.
Let $ \cal B $ ; be a homogeneous body of revolution around an axis a, with fore-and-aft symmetry. Typical examples are bodies of constant density having the shape of cylinders of circular cross-section, of prolate and oblate spheroids, etc. In this paper we prove that, provided a certain geometric condition is satisfied, the only possible orientations that $ \cal B $ ; can eventually achieve when dropped in a Navier-Stokes fluid under the action of the acceleration of gravity g and at a small and nonzero Reynolds number, is with a either parallel or perpendicular to g. This result is obtained by a rigorous calculation of the torque exerted by the fluid on the body. We also show that the above geometric condition is certainly satisfied if $ \cal B $ ; is a prolate spheroid. Moreover, in this case, we prove, by a "quasi-steady" argument, that, at first order in λ, the configuration with a perpendicular to g is stable to small disorientation, while the other is unstable, in accordance with experiments.
This paper is concerned with the Navier-Stokes flows in the homogeneous spaces of degree -1, the critical homogeneous spaces in the study of the existence of regular solutions for the Navier-Stokes equations by means of linearization. In order to narrow the gap for the existence of small regular solutions in $ \dot B^{-1}_{\infty,\infty}(R^n)^n $ , the biggest critical homogeneous space among those embedded in the space of tempered distributions, we study small solutions in the homogeneous Besov space $ \dot B^{-1+n/p}_{p,\infty}(R^n)^n $ and a homogeneous space defined by $ \hat M_n(R^n)^n $ , which contains the Morrey-type space of measures $ \tilde M_n(R^n)^n $ appeared in Giga and Miyakawa [20]. The earlier investigations on the existence of small regular solutions in homogeneous Morrey spaces, Morrey-type spaces of finite measures, and homogeneous Besov spaces are strengthened. These results also imply the existence of small forward self-similar solutions to the Navier-Stokes equations. Finally, we show alternatively the uniqueness of solutions to the Navier-Stokes equations in the critical homogeneous space $ C([0,\infty);L_n(R^n)^n) $ by applying Giga-Sohr's $ L_p(L_q) $ estimates on the Stokes problem.
In a previous paper, the present authors studied the asymptotic behaviour of solutions to steady compressible Navier-Stokes equations in barotropic and isothermal regime with sufficiently small external data, in particular, in the whole plane. Here, the same problem is investigated in a two dimensional exterior domain with the prescribed velocity at infinity $ v_{\infty}\ne 0 $ . Similar results as in Dutto and Novotny [DuNo] are found; in particular, it is proved that there exists a unique solution which possesses the similar pointwise decay and the wake structure as the fundamental Oseen tensor.
The boundary-value problems for the stationary Boussinesq heat transfer equations with general non-standard boundary conditions for the velocity and mixed boundary conditions for the temperature are considered. The local and global existence theorems are proved. The precise a priori estimates for the solution are derived.
An example of stratified shear flow is presented in which an explicit construction is given for unstable eigenvalues with smooth eigenfunctions for the Taylor—Goldstein equation. It is proved for any stratified, plane parallel shear flow that the unstable spectrum of the linear operator is purely discrete. A general theorem is then invoked to prove that the specific example is nonlinearly unstable. A sufficient condition for nonlinear stability for stratified shear flow is discussed.
The time-dependent Navier—Stokes problem on an interior or exterior smooth domain, with nonhomogeneous Dirichlet boundary condition, is treated in anisotropic L p Sobolev spaces (1 {\frac 1p} - 1 $ ; the present work extends the solvability to spaces with $ s > {\frac 1p} - 2 $ for zero initial data ( $ s > - 2 $ if f = 0), $ s > {\frac 2p} - 2 $ for nonzero initial data, with s,p subject to other conditions stemming from the nonlinearity.
The paper is concerned with the modelling of viscous incompressible flow in an unbounded exterior domain with the aid of the coupling of the nonlinear Navier—Stokes equations considered in a bounded domain with the linear Oseen system in an exterior domain. These systems are coupled on an artificial interface via suitable transmission conditions. The present paper is a continuation of the work [8], where the coupling of the Navier—Stokes problem with the Stokes problem is treated. However, the coupling "Navier—Stokes — Oseen" is physically more relevant. We give the formulation of this coupled problem and prove the existence of its weak solution for large data.
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Answer
The work done by the weightlifter is zero.
Work Step by Step
By definition, the work is only done when an object changes its position after a force acts on it. The weightlifter is just holding a barbell above his head. Due to this, the weightlifter’s body is just shaking but not moving from his position. So, there is no displacement done by the weightlifter. Since the displacement of the object is zero, therefore, the work done by the weightlifter is also zero. Example: Assume that a weightlifter is holding a barbell of $100\text{Kg}$ on his head. The force acting on the barbell is ${{\mathbf{F}}_{\mathbf{b}}}$. So, $\begin{align} & {{\mathbf{F}}_{\mathbf{b}}}=\text{mg} \\ & \text{=100 Kg}\times \text{9}\text{.8 m}{{\text{s}}^{\text{-2}}} \\ & =980\text{ N} \end{align}$ Since there is no displacement, thus the displacement is zero. Now work done by the weightlifter is: $\begin{align} & \text{W}=\mathbf{F}\cdot \mathbf{S} \\ & =980\text{ N}\times 0 \\ & =0 \end{align}$
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I'm trying to design a data structure that supports the following operations in $O(1)$ time:
$\operatorname{query}(\ell)$: Yield some auxiliary datum associated with list $\ell$ (just a single integer for my use case). $\operatorname{update}(\ell,v)$: Set the auxiliary datum of list $\ell$ to $v$. $\operatorname{list}(n)$: Get the list associated with node $n$. $\operatorname{ends}(\ell)$: Get the end nodes of list $\ell$. $\operatorname{concat}(n_0,n_1)$: Concatenate $\operatorname{list}(n_0)$ and $\operatorname{list}(n_1)$ so that nodes $n_0$ and $n_1$ are adjacent. Preconditions are that $n_0$ and $n_1$ are end nodes in two distinct lists and those two lists have the same auxiliary data (which becomes the auxiliary datum of the resulting list). $\operatorname{split}(n)$: Split $\operatorname{list}(n)$ after node $n$ (according to some arbitrary assignment of direction to the lists). The resulting two lists have the same auxiliary data as the original list.
Use a model as powerful as you want, eg, word RAM.
Basically these are just linked lists with some additional data. The tricky part is to ensure the datum for a list can be updated in $O(1)$ but also accessed from any node in $O(1)$.
I was unable to do this with only elementary data structures like arrays and linked lists. The naive strategies I can think of run in $O(N)$ time (where $N$ is the maximum total number of nodes over the lifetime of the data structure). I think any balanced BST can yield a working solution$^1$, but that's $O(\log N)$ and the approach I'm thinking of does not lend itself well to using standard library BST containers, which is a pain. I'm hoping that this can be done in $O(1)$ in a simple-to-implement way. Something like $O(\log^\ast N)$ or $O(\alpha(N))$ would be good as well, and I would even be interested in a simple $O(\log N)$ strategy that doesn't require implementing a balanced BST from scratch.
$^1$: The idea for implementing these operations with BSTs is to let the key of a node in the tree be its rank in the current list ordering. The $\operatorname{split}$ and $\operatorname{concat}$ operations can change that ordering so we just have to let the structure of the trees represent the ordering implicitly. In all BST implementations that I'm aware of, the balancing, $\operatorname{split}$, and $\operatorname{concat}$ operations don't compare keys explicitly but just look at the tree structure so this works.
For the particular application that originally inspired this, the nodes come from a constant universe $U=\{0,1,\dotsc,N-1\}$ and represent vertices in a rooted tree such that, at any given point, all lists will be descending paths in the tree. This makes the BST strategy particularly simple to implement: the key ordering is the (fixed) ordering by depth.
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I have a problem with what seems a very simple functional maximization. Let's define:
$$ J[z]=\int \left( u(z)-\frac{\dot z^2}{2} \right) dt $$
Where $u(z)=-z^2+5$. The problem is to find
$$ \arg\max_z J[z]$$
Said in a colloquial way, to maximize the function $u(z)$ without varying too much $z$ with time. The second variation of the functional for an arbitrary variation $h(t)$ is:
$$ \frac{\delta^2}{\delta z^2}J[z]=\int \left(h^2 u''(z)-\dot h^2 \right) dt = -\int \left(2 h^2+ \dot{h}^2 \right) dt \le 0 \quad \forall h $$
So then the functional is convex and any stationary point satisfying the Euler-Lagrange equations would be a global maximizer of $J$. The Euler-Lagrange equations for this functional reduce to the following differential equation:
$$ \ddot z=-u'(z)=2z $$
But this doesn't make any sense, since the maximum of $u(z)$ is at $z=0$, and all the trajectories starting in $z\ne0$ diverge from that point in an exponential way. Where did I go wrong?
Thank you
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1. Measurement with the ATLAS detector of multi-particle azimuthal correlations in p+Pb collisions at sNN=5.02 TeV
Physics Letters B, ISSN 0370-2693, 08/2013, Volume 725, Issue 1-3, pp. 60 - 78
In order to study further the long-range correlations (“ridge”) observed recently in collisions at , the second-order azimuthal anisotropy parameter of charged...
PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
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Physical Review Letters, ISSN 0031-9007, 06/2015, Volume 115, Issue 1, p. 012301
The second-order azimuthal anisotropy Fourier harmonics, nu(2), are obtained in p-Pb and PbPb collisions over a wide pseudorapidity (.) range based on...
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4. Measurement of the differential cross section and charge asymmetry for inclusive $$\mathrm {p}\mathrm {p}\rightarrow \mathrm {W}^{\pm }+X$$ p p → W ± + X production at $${\sqrt{s}} = 8$$ s = 8 TeV
The European Physical Journal C, ISSN 1434-6044, 8/2016, Volume 76, Issue 8, pp. 1 - 27
The differential cross section and charge asymmetry for inclusive $$\mathrm {p}\mathrm {p}\rightarrow \mathrm {W}^{\pm }+X \rightarrow \mu ^{\pm }\nu +X$$ p p...
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Physical Review C - Nuclear Physics, ISSN 0556-2813, 09/2015, Volume 92, Issue 3
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Nuclear Inst. and Methods in Physics Research, A, ISSN 0168-9002, 12/2015, Volume 803, Issue C, pp. 100 - 112
The response of silicon strip sensors to electrons from a Sr source was measured using a multi-channel read-out system with 25 ns sampling time. The...
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7. Search for heavy Majorana neutrinos in mu(+/-)mu(+/-) + jets events inproton-proton collisions at root s=8TeV
PHYSICS LETTERS B, ISSN 0370-2693, 09/2015, Volume 748, pp. 144 - 166
A search is performed for heavy Majorana neutrinos (N) using an event signature defined by two muons of the same charge and two jets (mu(+/-)mu(+/-)jj). The...
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8. Measurement of Prompt psi(2S) to J/psi. Yield Ratios in Pb-Pb and p-p Collisions at root sNN=2.76 TeV
PHYSICAL REVIEW LETTERS, ISSN 0031-9007, 12/2014, Volume 113, Issue 26
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9. Observation of Charge-Dependent Azimuthal Correlations in p-Pb Collisions and Its Implication for the Search for the Chiral Magnetic Effect
Physical Review Letters, ISSN 0031-9007, 03/2017, Volume 118, Issue 12
Charge-dependent azimuthal particle correlations with respect to the second-order event plane in p-Pb and PbPb collisions at a nucleon-nucleon center-of-mass...
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10. Search for heavy Majorana neutrinos in μ±μ± + jets events in proton–proton collisions at √s = 8 TeV
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11. Measurement of prompt ψ (2S) to J /ψ yield ratios in Pb-Pb and p - p collisions at sNN =2.76 TeV
Physical Review Letters, ISSN 0031-9007, 12/2014, Volume 113, Issue 26
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12. Searches for supersymmetry based on events with b jets and four W bosons in pp collisions at 8 TeV
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13. Measurement of the cross section ratio $\sigma_\mathrm{t \bar{t} b \bar{b}} / \sigma_\mathrm{t \bar{t} jj }$ in pp collisions at $\sqrt{s}$ = 8 TeV
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14. Distributions of topological observables in inclusive three- and four-jet events in pp collisions at √s = 7 TeV
ISSN 1434-6044, 2015
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The very first thing one usually does in a logic course is to defines a formal language for the propositional logic.
I don't want to bother what a "logic" is and why we need a "language" which we define know for a "logic".
A language consists of an ALPHABET and a GRAMMAR.
An alphabet $\mathcal{A}$ is a union of three different sets, we will call the element of an alphabet symbols. The first set are the symbols for the propositional variables like $A,B,C....$, the second set is the set of logical symbols $T,F,\wedge,\vee,\implies,\iff$ and the thirs set are non-logical symbols like $(,)$
Now here comes the first thing I don't understand yet we say a
propositional formula is a sequence of symbols that satisfies certain rules. What I don't understand is how exactly I can write the set I take an element from when I say: Assume $\phi$ is a propositional formula. Maybe somebody can help me.
The condition that has to be satisfied if $\phi$ wants to be an element of the set of all statements (that I don't know how to define in logical notation) are: $\phi$ is a propositional variable($\iff \phi$ is a function with domain $\{1\}$ and range $\mathcal{A}$ and $\phi(1)$ is a propositional variable) or $\phi$ is $F$( $\iff \phi$ is a function with domain $\{1\}$ and range $\mathcal{A}$ and $\phi(1)=F$) or $\phi$ is $T$ ( $\iff \phi$ is a function with domain $\{1\}$ and range $\mathcal{A}$ and $\phi(1)=T$) or there is a $ n\in\mathbb{N}$ such that $n\geq 2$ and $\phi$ is a function with domain $\{1,...,n\}$ and range $\mathcal{A}$ and $\phi(1)=\neg$ and $\psi:\{1,...,n-1\}\rightarrow \mathcal{A}$ with $\forall k\in\{1,...,n-1\}\psi(k)=\phi(k+1)$ is a propositional formula or there is a $n\in\mathbb{N}$ and a $k\in\mathbb{N}$ such that $n\geq 5$ and $k\in\{2,...,n-1\}$ and $\phi$ is a function with domain $\{1,...,n\}$ and range $\mathcal{A}$ and $\phi(1)=($ and $\phi(n)=)$ and $\phi(k)=\wedge\backslash\vee\backslash\implies\backslash\iff$ and $\psi_1:\{1,....,k-2\}\rightarrow \mathcal\{A\}$ with $\forall j\in{1,...,k-2}\psi_1(j)=\phi(j+1)$ and $\psi_2:\{1,...,(n-1)-k\}$ with $\forall z\in \{1,...,(n-1)-k\}\psi_2(z)=\phi(z+k)$ are propositional formulas.
My question to you how can I prove this claim:
If a property holds for every constant (propositional variable,T,F) and the implications ( $a$ is a propositional formula and $a$ has property $\Rightarrow \neg a$ has property,$a$ and $b$ are propostioanl formulas and $a$ and $b$ have property then $a\wedge b$ has property (and so on for the other logical operationsyymbols)) are true then every propositional formula has the property.
I am trying to find a way to prove this by induction over the natural numbers $n\in\mathbb{N}$, however I didn't succeed in building this bridge. I was reading a book (H.-D Ebbinghaus,J.Flum,W.Thomas) and they have proposed to define the notion of a derivation with a lenght $n\in\mathbb{N}$ and then show that the claim above implies that every derivation fullfils the property. Which then in turn means every element fullfills the property because every element has (or is) a derivation.
An example of what a derivation is
$((A\wedge B)\vee C) $ is a propositional formula because there exists a derivation, namely
$C$ is a propositional constant
$A$ is a propositional constant
$B$ is a propositional constant
$(A\wedge B)$ is a propositional formula because of $3.$ and $2.$
$((A\wedge B)\vee C)$ is a propositional formula because of $4.$ and $1.$ This derivation has a length of $5$
I have tried to define what a
derivation is and to prove that every propositional formula has (or is[because it depends on how we define derivation]) a derivation.
I came to an unsatisfying result. Because I didn't see another way to prove that every propositional formula has a derivation other than changing the definition of what a propositional formula itself is. Thus I have first defined what a derivation is and then said what a propositional formula is, namely objects that we get from certain derivations (the derivation itself depends on "rules" thats why I said certain here). This however prompted a new problem. Namely that if I go the other way around and already have an element I cannot use the prior convenient definition anymore and because the definition that I came up with didn't match the definition which the author intended, I couldn't understand the following proofs in the book anymore. I hope somebody can help me to find a a way to define what a derivation is and then to prove that every propositional has a derivation and eventually to prove the claim above by using this definition of a derivation. My only goal is to find a proof for the claim which preserves the definition of a propositional formula. The above paragraph was just a proposal and described my efforts and thinking-process so far.
If you want to know more details about the work I have done so far please tell me I will edit then, and thank you for reading this long text.
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Why Mathematica 11.1.1 seems to hang on
DSolve[y'[x] == Log[1 + y[x]^2], y[x], x]
Since this ODE is separable, it becomes just an integration problem.
Mathematica knows that
Integrate[1/Log[y^2 + 1], y] have no closed form solution as it returns
instantly unevaluated the integration of the left side
The question is: What else could possibly make it hang on this?
I know CAS does not necessarily follow the same steps to solve problems as humans do by hand, so it must be trying something else and I am just curious what is Mathematica trying to do with this to make it seem to hang (I gave up waiting after few minutes).
Hand solution follows.
Solve
$$ \boxed{y'=\log \left(1+y^2\right)}$$
Solution
This is separable ODE of the form $y'(x) = p(x) f(y) $. Moving $\left(\log \left(y^2+1\right)\right)$ to the left side gives
$$ \frac{y'(x)}{\log \left(y^2+1\right)} = 1 $$ Integrating both sides
\begin{align*} \frac{1}{\log \left(y^2+1\right)} \frac{ \mathop{\mathrm{d}y}}{\mathop{\mathrm{d}x}} &= 1 \\ \int \frac{1}{\log \left(y^2+1\right)} \mathop{\mathrm{d}y} &= \int 1\mathop{\mathrm{d}x} \\ \int \frac{1}{\log \left(y^2+1\right)} \mathop{\mathrm{d}y} &= x + C_0 \end{align*}
The left side has no closed form solution, therefore solution can be written as $$ \boxed{\int^{y(x)} \frac{1}{\log \left(\tau ^2+1\right)} \,d\tau = x + C_0} $$
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I'm a Calculus 2 Student. But don't worry, this isn't a "do my HW" problem. I have a question about improper integrals.
I just realized something that's got me quite curious about the behaviors of infinite limits and I was hoping someone could explain what I'm observing.
If we're given the integral:
$$ \int_a^\infty [f(x)] dx $$
We're to take the limit of some number $M$ and solve the integral that way.
$$ \lim_{M\rightarrow\infty}\int_a^M [f(x)] dx $$ However, recalling what the definition of an integral is...
$$ \lim_{M\rightarrow\infty}[\lim_{N\rightarrow\infty} \sum_{k=1}^N[f(c_k) \Delta x] ] $$
What intrigues me is that we have two limits approaching infinity at the same time. One limit slicing the summation infinitely small, another expanding the scope of the summation infinitely long.
If something is becoming infinitely small and infinitely large at the same time, why doesn't it remain the same size? Does this imply $M$ is approaching infinity faster than $N$? Or does this mean the slices of $N$ (rather $\Delta x$ ) are "growing" with $M$ as it approaches infinity?
Thanks!
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Suppose we color the edges $\{1,\ldots, {n \choose 2}\}$ of the complete graph on $n$ vertices with $m$ colors each edge being assigned a color picked uniformly at random from $\{1,\ldots, m\}.$
I would like to estimate the probability that the obtained random coloring is proper, that is all edges incident with a vertex $v$ are colored with different colors.
One way to do this is as follows. Suppose $E_i$ is the event that the edge $e_i$ of $K_n$ is colored with a color that already occurs at its adjacent edges. Then $$Pr[E_i] = \frac{1}{m} (1 - (1-\frac{1}{m})^{2n-4})$$ and thus the probability that $G$ is not colored properly is estimated as $$Pr[E_1\cup \ldots E_{n \choose 2}] \leq \frac{n(n-1)}{2m} (1 - (1-\frac{1}{m})^{2n-4} )$$
Now this is not a good estimate since the right hand size is larger then one for any $m$ that is not "large enough".
Another way to estimate this is to define the event $V_i$ to be that the $i$th vertex of our graph is incident with edges of distinct colors and use a similar estimate. But again the given bound is not very sharp.
Hence I am wondering is there a more delicate way to estimate the mentioned probability? Is this known? Is there a different way to model this ?
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My reference is "A Course on Mathematical Logic" by S.M. Srivastava. I have trouble understanding some points in this remarkable book.
Substituting a term in place of a variable in a formula. Why is substitution defined as it is?
This is in the context of first order logic. Take a language $L$ (variables, constant symbols, function symbols, relation symbols (among which $=$), logical connectives ($ \neg,\vee)$ and the existential quantifier ($\exists$)). We are given a term $t=t[v_1,\dots,v_n]$ and a formula $\varphi=\varphi[x_1,\dots,x_m]$. This notation means that the variables that figure in $t$ are among $\lbrace v_1,\dots,v_n \rbrace$, and the free variables in $\varphi$ are among $\lbrace x_1,\dots,x_m \rbrace$.
The author defines that $t$ is substitutiable in place of a variable $x$ in $\varphi$, if for every variable $v$ effectively occurring in $t$ (thus $v\in\lbrace v_1,\dots,v_n \rbrace$), and every subformula $\exists v \psi$ of $\varphi$, if $x$ occurs in this subformula, it is not free (considered as an occurrence inside $\varphi$). If $t$ is substitutiable for $x$ in $\varphi$, you define $\varphi_x[t]$ to be the formula in which every free occurrence of $x$ in $\varphi$ has been replaced by the term $t$.
For instance, take $\varphi=\varphi[x,y]$ to be $\neg (x=0) \vee \exists x (x=y)$ in a language with a constant symbol $0$. Suppose there is a $1$-ary function symbol $f$ and define $t=f(x)$. By definition, $t$ is substitutiable for $x$ in $\varphi$. Indeed, there is only one subformula of the form $\exists x\psi$ in $\varphi$, and in it, the occurrences of $x$ are all bounded. If we go to replace $t$ in $\varphi$ everywhere $x$ is free, we get $$ \varphi_x[t]= \neg (t=0) \vee \exists x (x=y)=\neg (f(x)=0) \vee \exists x (x=y)$$ On the other hand, $t$ is not substitutiable for $y$ because $y$ occurs in the term $\exists x(x=y)$ but is free as an occurrence in $\varphi$. The $\neg(x=0)$ part of the forumla I only included so as to have a free occurrence of $x$ where actual substitution could be seen.
For simplicity, change $\varphi$ to be $\exists x(x=y)$, again, $t$ may not be substituted in place of $y$ inside $\varphi$. I intuitively understand that simply replacing $y$ with $t$ would give a formula that is
qualitatively different from $\varphi$ : $\exists x (x=f(x))$ is a very different matter than $\exists x(x=y)$ or $\exists x(x=f(z))=\varphi_y[f(z)]$
Can someone explain that 'intuition' and justify why Srivastava's definition is indeed the correct definition?
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What you want to keep in mind is that when we say a quantum state is represented by a state $\mid \psi \rangle$ in a Hilbert space we haven't yet committed ourselves to a particular Hilbert space.
When a system has a classical analogue we introduce hermitian operators $X$ and $P$ which obey a canonical commutation relation $[X,P]=i\hbar $. These operators have eigenvectors $ \mid x \rangle $ and $ \mid p \rangle $ which form a complete orthonormal basis in our space. They also generally have a continuous spectrum of eigenvalues $x$ and $p$.
So we have, $$X \mid x \rangle = x \mid x \rangle, \qquad P \mid p \rangle = p \mid p \rangle,$$
and any state $ \mid \psi \rangle $ can be written as a linear combination of these eigenvectors,
$$ \mid \psi \rangle = \sum_x \psi(x) \mid x \rangle \quad (\text{discrete spectrum }) $$
$$ \mid \psi \rangle = \int dx \quad \psi(x) \mid x \rangle \quad (\text{continuous spectrum }) $$
Where $\psi(x)$ are the coefficients of the $\mid x \rangle$'s in the expansion of the state $\mid \psi \rangle$. Using the orthonormality of the basis vectors we can conclude that $\psi(x) = \langle x \mid \psi \rangle$ this function of the eigenvalues is usually what we call the wavefunction. Since there is a $1-1$ correspondence between wave functions and the state vectors they represent we often become sloppy and refer to them as if they are the same thing.
Now a reasonable question to ask is what is the wave functions that correspond to the eigenstates of $P$? I'm not going to derive it here but it is possible to show (starting from the canonical commutation relation) that,
$$ \langle x \mid P \psi \rangle = \frac{\hbar }{i} \frac{\partial}{\partial x} \langle x \mid \psi \rangle = \frac{\hbar}{i} \frac{\partial \psi(x)}{\partial x}$$
Using this it is easy to show that the wave function for $\mid p \rangle$ is, $$ p(x) = \langle x \mid p \rangle = \frac{1}{\sqrt{2\pi \hbar}} e^{ixp/\hbar}$$
This gives us a way of converting wave functions in the $x$-basis to wave functions in the $p$ basis. Starting with the projection of $\mid \psi \rangle$ onto the $p$-basis we expand $\mid \psi \rangle$ in the $x$-basis and perform the integration,
$$ \psi(p) = \langle p \mid \psi \rangle = \int dx \quad \langle p \mid x \rangle \langle x \mid \psi \rangle = \frac{1}{\sqrt{2\pi\hbar}} \int dx \quad e^{-ixp/\hbar} \psi(x) $$
The last line above is obviously the fourier transform of $\psi(x)$.
To answer a question in the comments,
The form of the states $\mid x \rangle $ depends on the basis you represent them in. If I were to represent these states in the $p$-basis they would look like,
$$ x(p) = \langle p \mid x \rangle = \frac{1}{\sqrt{2\pi\hbar}} e^{-ixp/\hbar} $$
If I were to represent them in their own basis I would get dirac delta functions,
$$ x'(x) = \langle x \mid x' \rangle = \delta(x-x') $$
Similarly if I expand the $p$'s in their own basis I would get a delta function,
$$ p'(p) = \langle p \mid p' \rangle = \delta(p-p')$$
Think of the wave functions as the components of a good old arrow vector. You can get different components by using different basis vectors but at the end of the day the vector itself is unchanged. Similarly we can get different looking wave functions by looking at their expansions in different basis sets but at the end of the day they all correspond to the same state vector.
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I don't really understand the difference. Shouldn't roll down and pull to par be the same technically? If a bond is trading as a discount it "increases" in value because everyday gets closer to par, and it rolls into another issue which has gotten closer? I feel that pull to par is incorporated within roll down because of this.
Pull-to-par says that the bond's price will gradually converge toward par (100% of face value)
when yield is unchanged. This process is also known as accretion for a bond trading at a discount (since its price gradually goes higher toward par) and amortization for a bond trading at a premium (since its price gradually declines toward par). Pull-to-par says nothing about the shape of the yield curve.
Rolldown is all about the shape of the yield curve. If the yield curve is upward sloping, you "roll down" the yield curve (i.e., yield goes down) as time passes, resulting in capital gains. If yield curve is downward sloping, you "roll up" the yield curve with the passage of time (i.e., yield goes up).
Consider a 10-year zero coupon bond trading at a yield of 10%. Its initial price is $100 / (1 + 10\%)^{10} = 38.55$ (assuming annual compounding). After a year, its price, assuming the same yield, becomes $100 / (1 + 10\%)^9 = 42.41$. This increase is price is pull-to-par at work.
Now assume that the yield curve is upward sloping, such that the 9-year yield is 9%. Further, let's assume that the yield curve did not change over the year. Even though the yield curve is unchanged, because our original 10-year zero coupon bond has rolled down to the 9-year point, its yield is now 9% instead of 10%. The price is therefore $100 / (1 + 9\%)^9 = 46.04$. This is rolldown at work.
Pull-to-par just says that a bond's (clean) price will converge towards its face value as the bonds approaches maturity. There is nothing really interesting about pull-to-par - a bond's (clean) price has to converge to its face value, because a bond with just a few days to maturity is essentially a short-term cash deposit.
Look at it this way - the price of an $n$-year zero coupon bond is
$$ p_n = 100/(1+y)^n\approx 100 \times (1 - ny) $$
The approximation is okay for small $n$, i.e. when we are close to maturity. This clearly converges to 100 as $n$ approaches zero, no matter what the yield is. This is pull-to-par.
Roll-down is a statement about the capital appreciation or depreciation on a bond, assuming that the shape of the yield curve doesn't change. For example, say that you have the yield curve below (for simplicity, say it is the yield curve for zero coupon bonds).
The yield of the five-year bond is 4.08%, and so its price is $100 / (1.048)^5 = \$79.10$. In a years time, assuming that the yield curve is unchanged, it will be a four-year bond, with a yield of 3.7%, so its price will be $100 / (1.037)^4 = \$86.47$.
Therefore the return from holding the bond, assuming that its yield doesn't change, is
$$ R = \frac{86.47}{79.10} - 1 = 9.32\% $$
Note that this is much higher than the bond's yield, which is $4.08\%$! The difference between the bond's yield, and the expected return assuming no change in the yield curve, is the
roll-down. In this case the roll-down is
$$ 9.32\% - 4.08\% = 5.25\% $$
so the roll-down can be a very significant contributor to the return on a bond, especially in environments with steep yield curves and low yield volatility.
One way to understand carry, yield and roll-down is to look at the return on zero-coupon bonds. If the yield curve for a bond maturiting in $n$ years, at time $t$ is $y_{n,t}$ then the prices of zero coupon bonds are
$$ p_{n,t} = \frac{1}{(1 + y_{n,t})^n} $$
One year later, at $t+1$, that bond is priced using the yield $y_{n-1,t+1}$ (because after one year has passed, the bond has a tenor one year shorter) so its price is
$$ p_{n-1,t+1} = \frac{1}{(1 + y_{n-1,t+1})^{n-1}} $$
The return from holding the bond over the year is
$$ \begin{align} R & = \frac{\frac{1}{(1 + y_{n-1,t+1})^{n-1}}}{\frac{1}{(1 + y_{n,t})^n}} - 1 \\ & = \frac{(1 + y_{n,t})^n}{(1 + y_{n-1,t+1})^{n-1}} - 1 \\ & \approx ny_{n,t} - (n-1)y_{n-1,t+1} \\ & = y_{n,t} + (n-1)(y_{n,t} - y_{n-1,t}) - (n-1)(y_{n-1,t+1}-y_{n-1,t}) \\ & = y_{n,t} + (n-1)(y_{n,t} - y_{n-1,t}) - (n-1)\Delta y_{n-1,t} \end{align} $$
The first term is the
yield return, the second is the roll-down return and the final term is the duration return, i.e. the return due to changes in yield between $t$ and $t+1$.
Looking at the roll-down return, you can see that roll-down is larger for bonds that have higher duration (i.e. larger $n$) and larger for bonds that are on a steep part of the yield curve (i.e. $y_{n,t} - y_{n-1,t}$ is large).
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253 23 Homework Statement Two identical uniform triangular metal played held together by light rods. Caluclate the x coordinate of centre of mass of the two mass object. Give that mass per unit area of plate is 1.4g/cm square and total mass = 25.2g Homework Equations -
Not sure what I went wrong here, anyone can help me out on this? Thanks.
EDIT: Reformatted my request. Diagram: So as far as I know to calculate the center of mass for x, I have to use the following equation: COM(x): ##\frac{1}{M}\int x dm## And I also figured that to find center of mass, I will have to sum the mass of the 2 plates by 'cutting' them into stripes, giving me the following formula: ##dm = \mu * dx * y## where ##\mu## is the mass per unit area. So subbing in the above equation into the first, I get: ##\frac{1}{M}\int x (\mu * dx *y) ## ##\frac{\mu}{M}\int xy dx## Since the 2 triangles are identical, I can assume triangle on the left has equation ##y = 1/4x +4## This is the part where I'm not sure. Do I calculate each of the triangle's center of moment, sum them and divide by 2? Or am I suppose to use another method? Regardless of what, supposed I am correct: COM for right triangle: ##\frac{\mu}{M}\int_{4}^{16}x(\frac{1}{4}x+4) dx## = 8 (expected) COM for left triangle: ##\frac{\mu}{M}\int_{-11}^{1}x(-\frac{1}{4}x+4) dx## = 5.63... Total COM = ##8+5.63/2## which is wrong :( Thanks
EDIT: Reformatted my request.
Diagram:
So as far as I know to calculate the center of mass for x, I have to use the following equation:
COM(x):
##\frac{1}{M}\int x dm##
And I also figured that to find center of mass, I will have to sum the mass of the 2 plates by 'cutting' them into stripes, giving me the following formula:
##dm = \mu * dx * y## where ##\mu## is the mass per unit area.
So subbing in the above equation into the first, I get:
##\frac{1}{M}\int x (\mu * dx *y) ##
##\frac{\mu}{M}\int xy dx##
Since the 2 triangles are identical, I can assume triangle on the left has equation ##y = 1/4x +4##
This is the part where I'm not sure. Do I calculate each of the triangle's center of moment, sum them and divide by 2? Or am I suppose to use another method?
Regardless of what, supposed I am correct:
COM for right triangle:
##\frac{\mu}{M}\int_{4}^{16}x(\frac{1}{4}x+4) dx## = 8 (expected)
COM for left triangle:
##\frac{\mu}{M}\int_{-11}^{1}x(-\frac{1}{4}x+4) dx## = 5.63...
Total COM = ##8+5.63/2## which is wrong :(
Thanks
Last edited:
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How to create a sharp mesh from a function without even trying
In part 1 and part 2 of the series, we looked at the Marching Cubes algorithm, and how it can turn any function into a grid based mesh. In this article we’ll look at some of the shortcomings and how we can do better.
Marching Cubes is easy to implement, and therefore ubiquitous. But it has a number of problems:
Complexity
Even though you only need process one cube at a time, Marching Cubes ends up pretty complicated as there are a lot of different possible cases to consider.
Ambiguity
Some cases in Marching Cubes cannot be obviously resolved one way or another. In 2d, if you have two opposing corners, it’s impossible to say if they are meant to be joined or not.
In 3d, the problem is even worse, as inconsistent choices can leave you with a leaky mesh. We had to write extra code to deal with that in part 2.
Marching Cubes cannot do sharp edges and corners
Here’s a square approximated with Marching Cubes.
The corners have been sliced off. Adaptivity cannot help here – Marching Squares always creates straight lines on the interior of any cell, which is where the target square corner happens to lie.
So, what do do next?
Enter Dual Contouring Brevity Alert:This tutorial covers concepts and ideas more than methods and code. If you are more interested in the implementation, make sure you check out the python implementation (2d , 3d ), which contains commented example code with all the nitty gritty.
Dual Contouring solves these problems, and is more extensible to boot. The trade off is we need to know even more information about \( f(x) \), the function determining what is solid and what is not. Not only do we need to know the value of \( f(x) \), we also need to know the gradient \( f'(x) \). That extra information will improve the adaptivity from marching cubes.
Dual Contouring works by placing a single vertex in each cell, and then “joining the dots” to form the full mesh. Dots are joined across every edge that has a sign change, just like in marching cubes.
Aside:
The “dual” in the name comes from the fact that cells in the grid becomes vertices in the mesh, which relates to the
dual graph
Unlike Marching Cubes, we cannot evaluate cells independently. We must consider adjacent ones to “join the dots” and find the full mesh. But in fact, it’s a lot simpler algorithm than Marching Cubes because there are not a lot of distinct cases. You simply find every edge with a sign change, and connect the vertices for the cells adjacent to that edge.
Getting the gradient
In our simple example of a 2d circle of radius 2.5, \( f \) is defined as:
\[ f(x, y) = 2.5 – \sqrt{x^2 + y^2} \]
(in other words, 2.5 minus the distance from the origin)
With a bit of calculus, we can compute the gradient:
\[ f'(x, y) = \text{Vec2}\left(\frac{-x}{\sqrt{x^2 + y^2}}, \frac{-y}{\sqrt{x^2 + y^2}} \right) \]
The gradient is pair of numbers for every point, indicating how much the function changes when moving along the x-axis or y-axis.
But you don’t need complicated maths to get the gradient function. You can just measure the change in \( f \) when \( x \) and \( y \) are perturbed by a small amount \( d \).
\[ f'(x, y) \approx \text{Vec2}\left(\frac{f(x+d, y) – f(x-d, y)}{2d}, \frac{f(x, y+d) – f(x, y-d)}{2d} \right) \]
This works for any sufficiently smooth \( f \), providing you pick \( d \) small enough. In practice, even functions with sharp points are sufficiently smooth, as you don’t need evaluate the gradient near the sharp parts for it to work. Link to code .
Adaptivity
So far, we’ve just got the same sort of stepped look that Marching Cubes had. We need to add some adaptivity. For Marching Cubes, we chose where along the edge put a vertex. Now we have a free choice of anywhere in the interior of the cell.
We want to choose the point that is most consistent with the information provided to us, i.e. the evaluation of \( f(x) \) and its gradient. Note that we’ve sampled the gradient along the edges, not at the corners.
By picking the illustrated point, we ensure that the output faces from this cell conform as well as possible to the normals:
In practice, not all the normals around the cell are going to agree. We need to pick the point of best fit. I discuss how to handle picking that point in a later section.
Going 3d
The 2d and 3d cases aren’t really that different. A cell is now a cube, not a square. And we are outputting faces, not edges. But that’s it. The routine for picking a single point per cell is the same. And we still find edges with a sign change, and then connect the points of adjacent cells, but now that is 4 cells, giving us a 4-sided polygon:
Results
Dual contouring has a more natural look and flow to it than marching cubes, as you can see in this sphere constructed with it:
In 3d, this procedure is robust enough to pick points running along the edge of a sharp feature, and to pick out corners where they occur.
Choosing a vertex location
One major problem I glossed over before is how to choose the point location when the normals don’t point to a consistent location.
The problem is even worse in 3d when there are likely to be a lot of normals.
The way to solve this is to pick the point that is mutually the best for all the normals.
First, for each normal, we assign a penalty to locations further away from ideal. Then we sum up all the penalty functions, which will give an ellipse shaped penalty. Finally, we pick the point with the lowest penalty.
Mathematically, the individual penalty functions are the square of the distance from the ideal line for that normal. The sum of all those squared terms is a quadratic function, so the total penalty function is called the
QEF (quadratic error function). Finding the minimal point of a quadratic function is a standard routine available in most matrix libraries. In my python implementation , I use numpy’s lstsq function. Problems Colinear normals
Most tutorials would stop here but it’s a dirty secret that solving the QEF as described in the original Dual Contouring paper doesn’t actually work very well.
If you solve the QEF, you can find the point that is most consistent with the normals of the function. But there’s
no actual guarantee that the resulting point is inside the cell.
In fact, it’s quite common for it not to be if you have large flat surfaces. In that case all the sampled normals are the same or very close, as in this diagram.
I’ve seen a lot of advice dealing with this particular problem. Several people have given up on using the gradient information, instead taking the center of the cell, or average of the boundary positions. This is called Surface Nets, and it at least has simplicity going for it.
But what I’d recommend based on my own experiments is the combination of two techniques.
Technique 1: Constrained QEF solving
Recall we were finding the cell point by finding the point that minimized the value of a given function, called the QEF. With some small changes, we can find the minimizing point
within the cell. Technique 2: Biasing the QEF
We can add any quadratic function we like to the QEF and we get another quadratic function which is still solvable. So I add a quadratic which has a minimal point in the center of the cell.
This has the effect of pulling the solution of the overall QEF towards the center. In fact, it has a stronger effect when the normals are colinear and likely to give screwy results, while it barely affects the positions for the good case.
Using both techniques is somewhat redundant, but I think it gives the best visual results.
Better details on both techniques are demonstrated in the code .
Self interesctions
Another dual contouring annoyance is it can sometimes generate a 3d surface that intersects itself. For most uses, this is completely ignorable, so I have not addressed it.
There is a paper discussing how to deal with it: “Intersection-free Contouring on An Octree Grid” Ju and Udeshi 2006
Manifolds
While a dual contouring mesh is always water tight, it’s not always a well defined surface. As there’s only one point per cell, if two surfaces pass through a cell, they will share it. This is called a “non-manifold” mesh, and it can mess with some texturing algorithms. This issue is common if you have solids that are thinner than the cell size, or multiple objects nearly touching each other.
Extensions
Because of the relative simplicity of the meshing process, Dual Contouring is much easier to extend to cell layouts other than the standard grid used above. Most commonly, you can run it on an octtree to have varying sizes of cells precisely where you need the detail. The rough idea is identical – pick a point per cell using sampled normals, then for each edge showing a sign change find the adjacent 4 cells and combine their vertices into a face. With an octree, you can recurse to find those edges, and the adjacent cells. Matt Keeter has a detailed tutorial on what to do.
Another neat extension is that all you need for Dual Contouring is a measure of what is inside / outside, and corresponding normals. Though I’ve assumed you have a function for this, you can also derive the same information from another mesh. This allows you to “remesh”, i.e. generate a clean set of vertices and faces that cleans up the original mesh. For example, Blender’s remesh modifier.
Well, that’s all I’ve got time for. Let me know in the comments if you found it useful, or if there’s anything else you’d like me to explain.
Further Reading
Dual Contouring is only one of a handful of similar techniques. See SwiftCoder’s list for a number of other approaches, each with pros and cons.
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Preprints (rote Reihe) des Fachbereich Mathematik Refine Year of publication 1996 (22) (remove)
284
A polynomial function \(f : L \to L\) of a lattice \(\mathcal{L}\) = \((L; \land, \lor)\) is generated by the identity function id \(id(x)=x\) and the constant functions \(c_a (x) = a\) (for every \(x \in L\)), \(a \in L\) by applying the operations \(\land, \lor\) finitely often. Every polynomial function in one or also in several variables is a monotone function of \(\mathcal{L}\). If every monotone function of \(\mathcal{L}\)is a polynomial function then \(\mathcal{L}\) is called orderpolynomially complete. In this paper we give a new characterization of finite order-polynomially lattices. We consider doubly irreducible monotone functions and point out their relation to tolerances, especially to central relations. We introduce chain-compatible lattices and show that they have a non-trivial congruence if they contain a finite interval and an infinite chain. The consequences are two new results. A modular lattice \(\mathcal{L}\) with a finite interval is order-polynomially complete if and only if \(\mathcal{L}\) is finite projective geometry. If \(\mathcal{L}\) is simple modular lattice of infinite length then every nontrivial interval is of infinite length and has the same cardinality as any other nontrivial interval of \(\mathcal{L}\). In the last sections we show the descriptive power of polynomial functions of lattices and present several applications in geometry.
285
On derived varieties (1996)
Derived varieties play an essential role in the theory of hyperidentities. In [11] we have shown that derivation diagrams are a useful tool in the analysis of derived algebras and varieties. In this paper this tool is developed further in order to use it for algebraic constructions of derived algebras. Especially the operator \(S\) of subalgebras, \(H\) of homomorphic irnages and \(P\) of direct products are studied. Derived groupoids from the groupoid \(N or (x,y)\) = \(x'\wedge y'\) and from abelian groups are considered. The latter class serves as an example for fluid algebras and varieties. A fluid variety \(V\) has no derived variety as a subvariety and is introduced as a counterpart for solid varieties. Finally we use a property of the commutator of derived algebras in order to show that solvability and nilpotency are preserved under derivation.
279
It is shown that Tikhonov regularization for ill- posed operator equation \(Kx = y\) using a possibly unbounded regularizing operator \(L\) yields an orderoptimal algorithm with respect to certain stability set when the regularization parameter is chosen according to the Morozov's discrepancy principle. A more realistic error estimate is derived when the operators \(K\) and \(L\) are related to a Hilbert scale in a suitable manner. The result includes known error estimates for ordininary Tikhonov regularization and also the estimates available under the Hilbert scale approach.
293
Tangent measure distributions were introduced by Bandt and Graf as a means to describe the local geometry of self-similar sets generated by iteration of contractive similitudes. In this paper we study the tangent measure distributions of hyperbolic Cantor sets generated by contractive mappings, which are not similitudes. We show that the tangent measure distributions of these sets equipped with either Hausdorff or Gibbs measure are unique almost everywhere and give an explicit formula describing them as probability distributions on the set of limit models of Bedford and Fisher.
276
Let \(a_1,\dots,a_n\) be independent random points in \(\mathbb{R}^d\) spherically symmetrically but not necessarily identically distributed. Let \(X\) be the random polytope generated as the convex hull of \(a_1,\dots,a_n\) and for any \(k\)-dimensional subspace \(L\subseteq \mathbb{R}^d\) let \(Vol_L(X) :=\lambda_k(L\cap X)\) be the volume of \(X\cap L\) with respect to the \(k\)-dimensional Lebesgue measure \(\lambda_k, k=1,\dots,d\). Furthermore, let \(F^{(i)}\)(t):= \(\bf{Pr}\) \(\)(\(\Vert a_i \|_2\leq t\)), \(t \in \mathbb{R}^+_0\) , be the radial distribution function of \(a_i\). We prove that the expectation functional \(\Phi_L\)(\(F^{(1)}, F^{(2)},\dots, F^{(n)})\) := \(E(Vol_L(X)\)) is strictly decreasing in each argument, i.e. if \(F^{(i)}(t) \le G^{(i)}(t)t\), \(t \in {R}^+_0\), but \(F^{(i)} \not\equiv G^{(i)}\), we show \(\Phi\) \((\dots, F^{(i)}, \dots\)) > \(\Phi(\dots,G^{(i)},\dots\)). The proof is clone in the more general framework of continuous and \(f\)- additive polytope functionals.
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Let \(a_1,\dots,a_m\) be independent random points in \(\mathbb{R}^n\) that are independent and identically distributed spherically symmetrical in \(\mathbb{R}^n\). Moreover, let \(X\) be the random polytope generated as the convex hull of \(a_1,\dots,a_m\) and let \(L_k\) be an arbitrary \(k\)-dimensional subspace of \(\mathbb{R}^n\) with \(2\le k\le n-1\). Let \(X_k\) be the orthogonal projection image of \(X\) in \(L_k\). We call those vertices of \(X\), whose projection images in \(L_k\) are vertices of \(X_k\)as well shadow vertices of \(X\) with respect to the subspace \(L_k\) . We derive a distribution independent sharp upper bound for the expected number of shadow vertices of \(X\) in \(L_k\).
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A convergence rate is established for nonstationary iterated Tikhonov regularization, applied to ill-posed problems involving closed, densely defined linear operators, under general conditions on the iteration parameters. lt is also shown that an order-optimal accuracy is attained when a certain a posteriori stopping rule is used to determine the iteration number.
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This paper investigates the convergence of the Lanczos method for computing the smallest eigenpair of a selfadjoint elliptic differential operator via inverse iteration (without shifts). Superlinear convergence rates are established, and their sharpness is investigated for a simple model problem. These results are illustrated numerically for a more difficult problem.
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A regularization Levenberg-Marquardt scheme, with applications to inverse groundwater filtration problems (1996)
The first part of this paper studies a Levenberg-Marquardt scheme for nonlinear inverse problems where the corresponding Lagrange (or regularization) parameter is chosen from an inexact Newton strategy. While the convergence analysis of standard implementations based on trust region strategies always requires the invertibility of the Fréchet derivative of the nonlinear operator at the exact solution, the new Levenberg-Marquardt scheme is suitable for ill-posed problems as long as the Taylor remainder is of second order in the interpolating metric between the range and dornain topologies. Estimates of this type are established in the second part of the paper for ill-posed parameter identification problems arising in inverse groundwater hydrology. Both, transient and steady state data are investigated. Finally, the numerical performance of the new Levenberg-Marquardt scheme is studied and compared to a usual implementation on a realistic but synthetic 2D model problem from the engineering literature.
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This paper develops truncated Newton methods as an appropriate tool for nonlinear inverse problems which are ill-posed in the sense of Hadamard. In each Newton step an approximate solution for the linearized problem is computed with the conjugate gradient method as an inner iteration. The conjugate gradient iteration is terminated when the residual has been reduced to a prescribed percentage. Under certain assumptions on the nonlinear operator it is shown that the algorithm converges and is stable if the discrepancy principle is used to terminate the outer iteration. These assumptions are fulfilled , e.g., for the inverse problem of identifying the diffusion coefficient in a parabolic differential equation from distributed data.
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We extend the methods of geometric invariant theory to actions of non reductive groups in the case of homomorphisms between decomposable sheaves whose automorphism groups are non recutive. Given a linearization of the natural actionof the group Aut(E)xAut(F) on Hom(E,F), a homomorphism iscalled stable if its orbit with respect to the unipotentradical is contained in the stable locus with respect to thenatural reductive subgroup of the automorphism group. Weencounter effective numerical conditions for a linearizationsuch that the corresponding open set of semi-stable homomorphismsadmits a good and projective quotient in the sense of geometricinvariant theory, and that this quotient is in additiona geometric quotient on the set of stable homomorphisms.
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Functiones et Approximatio Commentarii Mathematici Funct. Approx. Comment. Math. Volume 60, Number 1 (2019), 87-96. A note on the Diophantine equation $2^{n-1}(2^{n}-1)=x^3+y^3+z^3$ Abstract
Motivated by a recent result of Farhi we show that for each $n\equiv \pm 1\pmod{6}$ the title Diophantine equation has at least two solutions in integers. As a consequence, we get that each (even) perfect number is a sum of three cubes of integers. Moreover, we present some computational results concerning the considered equation and state some questions and conjectures.
Article information Source Funct. Approx. Comment. Math., Volume 60, Number 1 (2019), 87-96. Dates First available in Project Euclid: 28 March 2018 Permanent link to this document https://projecteuclid.org/euclid.facm/1522202458 Digital Object Identifier doi:10.7169/facm/1700 Mathematical Reviews number (MathSciNet) MR3932606 Zentralblatt MATH identifier 07055566 Citation
Ulas, Maciej. A note on the Diophantine equation $2^{n-1}(2^{n}-1)=x^3+y^3+z^3$. Funct. Approx. Comment. Math. 60 (2019), no. 1, 87--96. doi:10.7169/facm/1700. https://projecteuclid.org/euclid.facm/1522202458
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This article provides answers to the following questions, among others:
How does a siphon work? What is the maximum height to the apex of the flexible tube that can be covered? Why does a siphon not contradict the law of conservation of energy? Introduction
If, for example, you want to empty a pool by a garden hose, you only have to place one end of the hose over the edge into the pool and the other end outside. The outside end of the hose only has to be lower than the water level. Once the water is sucked in, it runs out of the hose permanently. As long as the lower end is always held lower than the water level, the water is also carried over larger heights, such as over the edge of the pool (to the maximum height to be overcome later more). Such an arrangement is also called a
siphon or siphon spillway. Explanation
This paradoxical behavior is due to hydrostatic pressure. In order to better understand this, a plastic bottle filled with water is first considered. A flexible tube is mounted on the screw cap of the bottle. If the bottle is now turned upside down, the water will begin to flow out through the tube. The outflow of the water leads to an increase in the volume of air inside the bottle. Since no air can inflow through the relatively small hose, a negative pressure is created inside the bottle.
The resulting negative pressure in the bottle can then be used to suck water from another vessel. Only a second flexible tube has to be attached to the bottle from above. This tube is now placed in the vessel from which the water is to be sucked off. The water is sucked in so to speak by the negative pressure in the bottle, which is produced by the outflow of the water.
In principle, the air volume in the bottle at the beginning can be chosen arbitrarily small. And finally, even an initial air volume can be completely dispensed with. The pressure gradient required to pump the water will then form directly inside the water. The last step is to use another flexible tube instead of the bottle. Finally one obtains in this way a single tube, which sucks the water from a higher level over the apex and then drains it into a lower level.
The illustrated example with the bottle also clearly shows that the maximum height be overcome between the higher water level and the apex of the hose is limited by the maximum suction lift of about 10 meters. After all, the maximum negative pressure that can be generated is a vacuum. With an atmospheric air pressure of 1 bar, this ambient pressure can thus push a maximum water column of 10 meters upwards. The ambient pressure is therefore not sufficient for higher heights.
Mathematical derivation
If a flexible tube is used to pump water from a higher reservoir to a lower reservoir, then only the difference in height \(h\) in the water levels is relevant for driving the water flow. To show this, a stopcock is inserted into the highest point of the water-filled tube. The stopcock is first closed. To the left and right of the valve different pressures \(p_1\) and \(p_1\) will be formed. The negative pressure at these points causes the greater ambient pressure \(p_0\), which acts on the water surface, to push water upwards in the respective tube sections according to the drinking straw principle. In this state, the respective water columns remain in position in the tube sections.
The immersion depth of the tube ends in the respective reservoirs plays no role for the equilibrium, since the hydrostatic pressure of the surrounding water would in any case press the water in the tube to the same level (principle of communicating vessels). The same ambient pressure \(p_0\) acts not only outside but also inside the hose at the water level. This
upward acting pressure is obviously in equilibrium with the downward acting hydrostatic pressure of the water column in the tube (\(\rho g h\)) and the downward acting pressure \(p_1\) or (\p_2\). (p_2) above the water column:
\begin{align}
&p_0 \overset{!}{=} p_1 + \rho \cdot g \cdot h_1 \\[5px] &p_0 \overset{!}{=} p_2 + \rho \cdot g \cdot h_2 \\[5px] \end{align}
For the pressures to the left and right of the stopcock applies:
\begin{align}
&p_1 = p_0 – \rho \cdot g \cdot h_1 \\[5px] &p_2 = p_0 – \rho \cdot g \cdot h_2 \\[5px] \end{align}
Since the height \(h_2\) is obviously greater than the height \(h_1\), there is less pressure on the right side of the valve than on the left side. If the stopcock is now opened in thought, then the water obviously flows from the higher pressure in the direction of the lower pressure. The bigger the pressure difference, the stronger the water is pushed from the left side towards the right side. The pressure difference \(\delta p\) is thus a measure of the driving force of the water flow. This driving force is in turn only dependent on the height difference \(h\) of the two water levels:
\begin{align}
\require{cancel} \Delta p &= p_1 – p_2 \\[5px] &= p_0 – \rho \cdot g \cdot h_1 – \left(p_0 – \rho \cdot g \cdot h_2\right) \\[5px] &= \bcancel{p_0} – \rho \cdot g \cdot h_1 – \bcancel{p_0} + \rho \cdot g \cdot h_2 \\[5px] \label{g} &= \rho \cdot g \cdot h_2 – \rho \cdot g \cdot h_1 \\[5px] &= \rho \cdot g \cdot \left(h_2 – h_1\right) \\[5px] &= \rho \cdot g \cdot h \\[5px] \end{align}
As equation (\ref{g}) states, the water flow is driven by the difference between the hydrostatic pressures in the hose sections to the left and right of the apex. This in turn is only due to the height difference \(h\) in the water levels. So as long as there is a difference in height between the upper water level and the lower water level, water can be pumped from the upper to the lower reservoir (provided that the apex of the hose is not higher than the maximum geodetic suction lift). If the water is not discharged into a reservoir but is discharged through the end of the hose directly into the open, then the height difference \(h\) refers to the end of the hose.
Law of conservation of energy
The siphon principle does not contradict the law of energy conservation, since water is transported from points of higher potential energy to points of lower potential energy, even if water flows upwards against gravity in the suction hose section.
One can illustrate this situation with a chain placed over a pulley. If the chain section to the right of the idler pulley is longer and thus heavier, the greater weight will cause the chain to unroll. Only because chain links are moved upwards on the left side, this does not contradict the law of conservation of energy, since the centre of gravity of the chain moves downwards overall.
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Consider a $C^k$, $k\ge 2$, Lorentzian manifold $(M,g)$ and let $\Box$ be the usual wave operator $\nabla^a\nabla_a$. Given $p\in M$, $s\in\Bbb R,$ and $v\in T_pM$, can we find a neighborhood $U$ of $p$ and $u\in C^k(U)$ such that $\Box u=0$, $u(p)=s$ and $\mathrm{grad}\, u(p)=v$?
The tog is a measure of thermal resistance of a unit area, also known as thermal insulance. It is commonly used in the textile industry and often seen quoted on, for example, duvets and carpet underlay.The Shirley Institute in Manchester, England developed the tog as an easy-to-follow alternative to the SI unit of m2K/W. The name comes from the informal word "togs" for clothing which itself was probably derived from the word toga, a Roman garment.The basic unit of insulation coefficient is the RSI, (1 m2K/W). 1 tog = 0.1 RSI. There is also a clo clothing unit equivalent to 0.155 RSI or 1.55 tog...
The stone or stone weight (abbreviation: st.) is an English and imperial unit of mass now equal to 14 pounds (6.35029318 kg).England and other Germanic-speaking countries of northern Europe formerly used various standardised "stones" for trade, with their values ranging from about 5 to 40 local pounds (roughly 3 to 15 kg) depending on the location and objects weighed. The United Kingdom's imperial system adopted the wool stone of 14 pounds in 1835. With the advent of metrication, Europe's various "stones" were superseded by or adapted to the kilogram from the mid-19th century on. The stone continues...
Can you tell me why this question deserves to be negative?I tried to find faults and I couldn't: I did some research, I did all the calculations I could, and I think it is clear enough . I had deleted it and was going to abandon the site but then I decided to learn what is wrong and see if I ca...
I am a bit confused in classical physics's angular momentum. For a orbital motion of a point mass: if we pick a new coordinate (that doesn't move w.r.t. the old coordinate), angular momentum should be still conserved, right? (I calculated a quite absurd result - it is no longer conserved (an additional term that varies with time )
in new coordinnate: $\vec {L'}=\vec{r'} \times \vec{p'}$
$=(\vec{R}+\vec{r}) \times \vec{p}$
$=\vec{R} \times \vec{p} + \vec L$
where the 1st term varies with time. (where R is the shift of coordinate, since R is constant, and p sort of rotating.)
would anyone kind enough to shed some light on this for me?
From what we discussed, your literary taste seems to be classical/conventional in nature. That book is inherently unconventional in nature; it's not supposed to be read as a novel, it's supposed to be read as an encyclopedia
@BalarkaSen Dare I say it, my literary taste continues to change as I have kept on reading :-)
One book that I finished reading today, The Sense of An Ending (different from the movie with the same title) is far from anything I would've been able to read, even, two years ago, but I absolutely loved it.
I've just started watching the Fall - it seems good so far (after 1 episode)... I'm with @JohnRennie on the Sherlock Holmes books and would add that the most recent TV episodes were appalling. I've been told to read Agatha Christy but haven't got round to it yet
?Is it possible to make a time machine ever? Please give an easy answer,a simple one A simple answer, but a possibly wrong one, is to say that a time machine is not possible. Currently, we don't have either the technology to build one, nor a definite, proven (or generally accepted) idea of how we could build one. — Countto1047 secs ago
@vzn if it's a romantic novel, which it looks like, it's probably not for me - I'm getting to be more and more fussy about books and have a ridiculously long list to read as it is. I'm going to counter that one by suggesting Ann Leckie's Ancillary Justice series
Although if you like epic fantasy, Malazan book of the Fallen is fantastic
@Mithrandir24601 lol it has some love story but its written by a guy so cant be a romantic novel... besides what decent stories dont involve love interests anyway :P ... was just reading his blog, they are gonna do a movie of one of his books with kate winslet, cant beat that right? :P variety.com/2016/film/news/…
@vzn "he falls in love with Daley Cross, an angelic voice in need of a song." I think that counts :P It's not that I don't like it, it's just that authors very rarely do anywhere near a decent job of it. If it's a major part of the plot, it's often either eyeroll worthy and cringy or boring and predictable with OK writing. A notable exception is Stephen Erikson
@vzn depends exactly what you mean by 'love story component', but often yeah... It's not always so bad in sci-fi and fantasy where it's not in the focus so much and just evolves in a reasonable, if predictable way with the storyline, although it depends on what you read (e.g. Brent Weeks, Brandon Sanderson). Of course Patrick Rothfuss completely inverts this trope :) and Lev Grossman is a study on how to do character development and totally destroys typical romance plots
@Slereah The idea is to pick some spacelike hypersurface $\Sigma$ containing $p$. Now specifying $u(p)$ is trivial because the wave equation is invariant under constant perturbations. So that's whatever. But I can specify $\nabla u(p)|\Sigma$ by specifying $u(\cdot, 0)$ and differentiate along the surface. For the Cauchy theorems I can also specify $u_t(\cdot,0)$.
Now take the neigborhood to be $\approx (-\epsilon,\epsilon)\times\Sigma$ and then split the metric like $-dt^2+h$
Do forwards and backwards Cauchy solutions, then check that the derivatives match on the interface $\{0\}\times\Sigma$
Why is it that you can only cool down a substance so far before the energy goes into changing it's state? I assume it has something to do with the distance between molecules meaning that intermolecular interactions have less energy in them than making the distance between them even smaller, but why does it create these bonds instead of making the distance smaller / just reducing the temperature more?
Thanks @CooperCape but this leads me another question I forgot ages ago
If you have an electron cloud, is the electric field from that electron just some sort of averaged field from some centre of amplitude or is it a superposition of fields each coming from some point in the cloud?
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For a pair of positive integers $m$ and $n$, put
$$\displaystyle \Psi(m,n) = 2^{-\omega(m)} \sum_{a | m} \left(\frac{n}{a}\right),$$ where $\left(\frac{\cdot}{\cdot}\right)$ is the Jacobi symbol. Let $\sum^\ast$ denote a sum over square-free numbers. Let $X$ be a positive number. Can one evalute the sum
(1)
$$\displaystyle \sideset{}{^\ast}\sum_{\substack{D \leq X \\ d_1 d_2 d_3 = D \\ d_1 - d_2 + d_3 = 0}} \Psi(d_1, d_2) \Psi(d_2,d_3) \Psi(d_3,d_1)?$$
A similar, but more straightforward question, was addressed by Fouvry and Kluners in this paper, where they evaluated the sum
$$\displaystyle \sideset{}{^\ast} \sum_{\substack{D \leq X \\ d_1 d_2 = D}} \Psi(d_1, d_2)\Psi(d_2,d_1).$$
The cyclic nature of the sum (1) leads me to believe that a similar argument should work, but the presence of the linear relation between the divisors $d_1, d_2, d_3$ complicates things.
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The authors set to study the “averaging” properties of dropout in a quantitative manner in the context of fully connected, feed forward networks understood as DAGs. In particular, architectures other than sequential are included, cf. Figure 1. In the linear case with no activations, the output of some layer $h$ (no dropout yet) is:
$$ S^h_i = \sum_{l < h} \sum_j w^{h l}_{i j} S^l_j . $$
And if activations are included:
\begin{equation} O^h_i = A (S_i^h) = A \left( \sum_{l < h} \sum_j w^{h l}_{i j} O^l_j \right), \label{dag-network} \tag{1} \end{equation}
with $O^0_i = I_i$. The authors consider only sigmoid and exponential activation functions $A$.
Recall that dropout consists of randomly disabling (i.e. setting to 0)some fraction of the outputs at each layer.
1 This means that forsome fixed input, randomness is introduced in the model by the dropoutscheme. The authors only explicitly consider i.i.d. “Bernoulli gatingvariables” $\delta^l_j$ (at layer $l$, output $j$) which disableoutputs with probability $p^l_j$ (but mention that the results extendto other distributions):
\begin{equation} O^h_i = \sigma (S_i^h) = \sigma \left( \sum_{l < h} \sum_j w^{h l}_{i j} \delta^l_j O^l_j \right) . \label{eq:dropout-bernoulli} \tag{2} \end{equation}
Note that probabilities and expectations are therefore always over the set of all possible subnetworks, not over the input data.
The
key result is the following estimate on the expected value ofan output, using the Normalized Weighted Geometric Mean:
\begin{equation} \mathbb{E} (O^h_i) \overset{(\dagger)}{\approx} \text{NWGM} (O^h_i) \overset{(\ast)}{=} A_i^h (\mathbb{E} [S_i^h]) \overset{(\triangle)}{=} A_i^h (\sum_{l < h} \sum_j w^{h l}_{i j} p^l_j \mathbb{E} (O^l_j)), \label{eq:main-estimate} \tag{3} \end{equation}
where the NWGM is defined as the quotient $\text{NWGM} (x) = G (x) / (G (x) - G’ (x))$ where $G (x) = \prod_i x_i^{p_i}$ is the weighted geometric mean of the $x_i$ with weights $p_i$, and $G’ (x) = \prod_i (1 - x_i)^{p_i}$ the weighted geometric mean of their complements.
In (3), $(\ast)$ holds exactly only for sigmoid and constant functions (p. 2) and $(\triangle)$ follows from independence. The approximation $(\dagger)$ (Section 4) is shown to be exact for linear layers and to hold to first order in general. An interesting observation is that the Ky Fan inequality tells us:
$$ G \leqslant \frac{G}{G + G’} \leqslant E \text{, if } 0 < O_i \leqslant 0.5 \text{ for all } i, $$
and empirical tests show that:
This seems to indicate that the NWGM is in practice a good approximation when using sigmoidal units. Note however that the bound in eq. (22) $| \mathbb{E}- \text{NWGM} | \leqslant 2\mathbb{E} (1 -\mathbb{E}) | 1 - 2\mathbb{E} |$ seems rather rough, as Figure 3 shows:
Analysis of gradient descent: Using dropout means optimizingsimultaneously over the training set and the whole set of possiblenetworks. Therefore, two quantities of interest are the ensembleerror
$$ E_{\text{ENS}} = \frac{1}{2} \sum_i (t_i - O^i_{\text{ENS}})^2$$
and the
dropout error
$$ E_D = \frac{1}{2} \sum_i (t_i - O^i_D)^2 . $$
In the case of a single linear unit (!) it is show that:
$$ \mathbb{E} (\nabla E_D) = \nabla (E_{\text{ENS}} + R_{\text{ENS}}) $$
with the usual $l^2$ regularizer (here for just one training sample $I$)
$$ R_{\text{ENS}} = \frac{1}{2} \sum_j w^2_j I_j^2 \text{Var} (\delta_j) . $$
So
in expectation, the gradient of the dropout network is thegradient of a regularized ensemble. Observe that:
Dropout provides immediately the magnitude of the regularization term which is adaptively scaled by the inputs and by the variance of the dropout variables. Note that $p_i=0.5$ is the value that provides the highest level of regularization.
Analogously,
for a single sigmoid unit: the expected value of thegradient of the dropout network is approximately the gradient of aregularized ensemble network:
$$ \mathbb{E} (\nabla E_D) \approx \nabla E_{\text{ENS}} + \lambda \sigma’ (S_{\text{ENS}}) w_j I_j^2 \text{Var} (\delta_j) . $$
These results are extended to deeper networks in: The Dropout Learning Algorithm, (2014)
Simulations: the validity of the bounds is tested using MonteCarlo approximations to the ensemble distribution. It is shown inseveral examples how dropout favours the sparsity of activations and“increases the consistency of layers” after dropout layers. See Improving neural networks by preventing co-adaptation of feature detectors for the introduction of dropout. ⇧
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In this great question by Nathaniel Johnston, and in its answers, we can learn the following remarkable inequality: For all $v,w \in \mathbb{R}^n$ we have \begin{align*} \|v^2\| \, \|w^2\| - \langle v^2, w^2 \rangle \le \|v\|^2 \|w\|^2 - \langle v,w \rangle^2; \quad (*) \end{align*} here, $\langle\cdot,\cdot\rangle$ denotes the standard inner product on $\mathbb{R}^n$, $\|\cdot\|$ denotes the Euclidean norm and $v^2,w^2 \in \mathbb{R}^n$ denote the elementwise squares of $v$ and $w$. Both sides of $(*)$ are nonnegative by the Cauchy-Schwarz inequality, and the LHS gives a non-zero bound for the right RHS, in general.
What strikes me is that the RHS and the LHS of $(*)$ have different (linear) symmetry groups: the RHS does not change if we apply any orthogonal matrix $U \in \mathbb{R}^{n \times n}$ to both $v$ and $w$, while this is not true for the LHS. Hence, we can strengthen $(*)$ to \begin{align*} \sup_{U^*U = I}\Big(\|(Uv)^2\| \, \|(Uw)^2\| - \langle (Uv)^2, (Uw)^2 \rangle\Big) \le \|v\|^2 \|w\|^2 - \langle v,w \rangle^2. \quad (**) \end{align*} Unfortunately, I have no idea how to evaluate the LHS of $(**)$.
Question. Can we explicitely evaluate the LHS of $(**)$? Or, more generally, is there a version of $(*)$ for which both sides are invariant under multiplying both $v$ and $w$ by (identical) orthogonal matrices?
Admittedly, this question is a bit vague since it might depend on one's perspective which expressions one considers to be "explicit" and which inequalities one considers to be a "version" of $(*)$. Nevertheless, I'm wondering whether some people share my intuition that there should be a more symmetric version of $(*)$.
Edit. Maybe it is worthwhile to add the following motivating example: If we choose $n = 2$ and $v = (1,1)/\sqrt{2}$, $w = (1,-1)/\sqrt{2}$, then those vectors are orthogonal and the RHS of $(*)$ equals $1$, while the LHS of $(*)$ vanishes since $v^2$ and $w^2$ are linearly dependent. However, the LHS of $(**)$ is also equal to $1$; to see this, choose $U = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & -1 \\ 1 & 1\end{pmatrix}$and observe that $Uv = (0,1)$, $Uw = (1,0)$.
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Let $K$ compact in a metric space $M$, $N$ is a metric space, $\mathcal{C}(K,N)$ the space of continuous functions $f: K \longrightarrow N$ and $E \subset \mathcal{C}(K,N)$ a family of functions such that $\overline{E}$ is compact in $\mathcal{C}(K,N)$ with respect to the topology generated by the metric of uniform convergence (see the the post scriptum for the definition). Fix $a \in K$ and define $\varphi: \overline{E} \times K \longrightarrow \mathbb{R}$, $\varphi(f,x) := d(f(x),f(a))$. Show that $\varphi$ is continuous.
Initially, I was stuck because I don't know what is the metric defined on $\overline{E} \times K$ (observe that I can't take any metric since that I don't know if this space is a finite-dimensional normed space), but I tried prove the continuity of $\varphi$ via sequence and I would like to know if what I did is right.
$\textbf{My attempt:}$
Let be $(f_n,x_n)$ a sequence in $\overline{E} \times K$ such that $\lim\limits_{n \rightarrow \infty} (f_n,x_n) = (f,x)$. By continuity of $f$ and the metric $d$,
$$\lim\limits_{n \rightarrow \infty} \varphi(f_n,x_n) = \lim\limits_{n \rightarrow \infty} d(f_n(x_n),f_n(a)) = \lim\limits_{n \rightarrow \infty} d(f_n(x_n),f_n(a)) = d(\lim\limits_{n \rightarrow \infty} f_n(x_n), \lim\limits_{n \rightarrow \infty} f_n(a)) = d(\lim\limits_{n \rightarrow \infty} f_n(x_n), f(a)),$$
so the continuity of $\varphi$ will follow if I'm be able to show that $\lim\limits_{n \rightarrow \infty} f_n(x_n) = f(x)$. By the convergence of $(f_n)$ to $f$ and by the continuity of $f$ (observe that $f$ is continuous because $\overline{E}$ is compact by hypothesis, therefore is closed), we know that, given $\varepsilon > 0$, exist $N \in \mathbb{N}$ such that
$$n \geq N \Longrightarrow d(f_n(x),f(x)) < \frac{\varepsilon}{3}$$
and
$$n \geq N \Longrightarrow d(f(x_n),f(x_n)) < \frac{\varepsilon}{3}$$
Thus,
$$d(f_n(x_n),f(x)) \leq d(f_n(x_n),f_N(x_n)) + d(f_N(x_n),f_N(x)) + d(f_N(x),f(x)) < \frac{\varepsilon}{3} + \frac{\varepsilon}{3} + \frac{\varepsilon}{3} = \varepsilon,$$
whenever $n \geq N$, therefore $\lim\limits_{n \rightarrow \infty} f_n(x_n) = f(x)$ and the continuity of $\varphi$ is proved. $\square$
Thanks in advance!
$\textbf{EDIT:}$
$\textbf{Definition:}$ if $X \subset M$ is arbitrary and $f,g: X \longrightarrow M$ are bounded functions, then the metric of uniform convergence is defined by
$$d(f,g) := \sup_{x \in X} d(f(x),g(x)).$$
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Help:Editing
The MediaWiki software is extremely easy to use. Viewing pages is self-explanatory, but adding new pages and making edits to existing content is extremely easy and intuitive as well. No damage can be done that can't be easily fixed. Although there are some differences, editing SDIY wiki is much the same as editing on Wikipedia.
Contents 1 Editing the wiki 2 Use a sandbox page 3 Creating links and adding pages 4 Headings 5 Lists 6 Inserting files 7 Schematics 8 Tables 9 Formatting 10 Categories 11 Standard appendices 12 Templates 13 Talk pages 14 See also 15 Further reading 16 References Editing the wiki
By default the enhanced editing toolbar is disabled. To enable it go to Preferences:Editing and tick
Enable enhanced editing toolbar.
At the top of any wiki the page, you will see some tabs titled Page, Discussion, Edit, History, Move And Watch. Clicking the edit tab opens the editor, a large text entry box in the middle of the page. This is where to enter plain text. Very little formatting code (known as "wiki markup") is required, unlike regular websites using HTML and CSS. At the top of this text entry box is a row of buttons with small icons on them. Holding the mouse cursor over an icon displays a tool-tip telling you its function. These buttons make it very simple to use the formatting features of the wiki software. You can achieve the same effect by typing the correct wiki code, however using the buttons makes it very simple and also eases the process of learning the correct code syntax. Please do your best to always fill in the edit summary field. An enhanced editor can be enabled in user preferences.
Use a sandbox page
Use the sandbox page to play around and experiment with editing. It isn't for formal wiki info, just a place to play and explore. Any content here won't be preserved. You can create your own sandbox area by appending "/sandbox" to the URL of your user page, or click the Sandbox link in the personal toolbar area, if enabled in your preferences. Your own sandbox is where to rough out articles until they're ready for posting. Don't do articles in rough in the main wiki. Sandboxes will be indexed by search engines like any other page, unless the first line is
__NOINDEX__ or uses the template
{{User sandbox}}.
The third and fourth buttons are create "Internal link" and "External link". The third button creates an internal link (aka a wikilink) which, in the editor, has the format
[[Eurocard]] ie. surrounded by double square brackets. Use a vertical bar "|" (the "pipe" symbol) to create a link with a different name to original article eg.
[[Printed circuit board|PCB]], only PCB appears on the page.
Only the first occurence of a link on the page needs to be the link, any further uses of the word/phrase can be in plain text. If the page doesn't exist already the link will be in shown with red text. Following a redlink opens up the editor window for creating that page within the wiki structure. Linking articles in a structured way is the preferred method of adding new pages to the wiki. Except for names, use ordinary sentence case for article titles.
Using the fourth button will make an external link to a page elsewhere on the Internet. This has the form
[http://www.google.com Google], ie. the URL, followed by a space, followed by linking text in single square brackets.
Every article is part of a network of connected topics. Establishing such connections via internal links is a good way to establish context. Each article should be linked from more general subjects, and contain links where readers might want to use them to help clarify a detail. Only create relevant links. When you write a new article, make sure that one or more other pages link to it. There should always be an unbroken chain of links leading from the Main Page to every article in the wiki.
Always preview your edits before saving them and also check any links you have made to confirm that they do link to where you expect.
See also Wikipedia:Manual of Style/Linking
Headings
Headings help clarify articles and create a structure shown in the table of contents.
Headings are hierarchical. The article's title uses a level 1 heading, so you should start with level 2 heading (
==Some heading==) and follow it with a level 3 (
===A sub-heading===, and just use
'''Text made bold''' after that). Whether extensive subtopics should be kept on one page or moved to individual pages is a matter of personal judgment.
Headings should not be links. This is because headings in themselves introduce information and let the reader know what subtopics will be presented; links should be incorporated in the text of the section.
Except for names, use ordinary sentence case for headings, also don't have all words with initial caps.
Lists
In an article, significant items should normally be mentioned naturally within the text rather than merely listed. Where a "bulleted list" is required each item/line of the list is preceded by an asterix (
*) or for indenting a sublist use two asterixes
**). For numbered lists use a hash sign (
#) and further hash signs for subsections. Lists of links are usually bulleted, giving each link on a new line.
Definition lists
Are useful for more than just terms and definitions. Use semi-colons and colons:
Some term - this line starts with ; And then a definition - this line starts with a : Inserting files
The sixth button will enable to insert an image (or other media type) into the text. Relevant images add interest to the article. Currently there are limitations on the allowable size of uploads.
<mp3>Synth_filter_sweep.mp3</mp3> An example of a classic analog synthesizer
sound - a sawtooth bass filter sweep with
gradually increasing resonance.
You can also insert MP3 clips by using the tag <mp3>, but it needs to be put in an inline styled table to format it neatly. Use <br> tags to format any caption.
File names should be clear and descriptive, without being excessively long. It is helpful to have descriptive names for editors. Very generic filenames should not be used when uploading, as sooner or later someone else will use same name and this will overwrite the first file.
For a large selection of freely usable media see Wikimedia Commons.
Hotlinking from from Wikimedia Commons is allowed. You can first upload your file there, but be sure to use a long descriptive or unique file name. This is to avoid name clash. When files have the same name, some other other file might be displayed locally instead of the one expected.
Hotlinking is not recommended because anyone could change, vandalise, rename or delete a hotlinked image. There is no control over what is served locally. If you do hotlink, then it is still necessary to follow any licensing conditions.
Generally hotlinking is wrong because it exploits another servers bandwidth to supply the files. For files on sites other than Wikimedia, don't link directly to those files without permission. Either download a copy from the other site and then upload it to the wiki, or link to the other site's page on which the file can be found.
Schematics
For quickly illustrating articles with simple schematics and sketches. There's some suggestions listed at Wikipedia:Project Electronics/Programs and at StackExchange EE:Good Tools for Drawing Schematics.
Tables
Use wiki markup not HTML nor images. The easiest way to work with tables is to use Microsoft Excel. Paste and edit or create a table in Excel, then copy the table into the tab-delimited string to wiki markup converter. Other methods are described at Commons:Convert tables and charts to wiki code.
Enabling the enhanced editor in user preferences, gives an
Insert a table button. Clicking this produces the following. {| class="wikitable" border="1" |- !header1!!header 2!!header 3 |- | row 1, cell 1|| row 1, cell 2|| row 1, cell 3 |- | row 2, cell 1|| row 2, cell 2|| row 2, cell 3 |}
Which displays as
header1 header 2 header 3 row 1, cell 1 row 1, cell 2 row 1, cell 3 row 2, cell 1 row 2, cell 2 row 2, cell 3
For more in depth information on table markup see Wikipedia:Help:Table.
Formatting
Be sure to keep your content meaningful. Relying on styling to indicate meaning is a bad practice (e.g. for machine readability such as by search engines, screen readers using text-to-speech, and text browsers).
Inline styling
Some HTML tags and inline styling is allowed, for example
<code>,
<div>,
<span> and
<font>. These apply anywhere you insert them - depending upon which fonts are installed. Here is an example using <span style="font-family:Courier;font-size:100%;color:#0000ff;background-color:#dddddd"></span>. For further information see Mediawiki:Help:Formatting.
Indenting text
Use a colon
: to indent text.
Subscript and superscript
Foo<sub>Bar</sub> gives Foo
Bar and
Bar<sup>Baz</sup> gives Bar
Baz. Inserting symbols
Symbols and other special characters can be inserted through HTML entities. For example Ω will show Ω and > will show >. These are case-sensitive. For a list of HTML entities see Wikipedia:List of HTML entities
Text boxes
For preformatted text (in a dashed box) simply indent it by one space. Inline styling allows more options e.g.
<div style="background: #eeffff; border: 1px solid #999999; padding: 1em; width:88%;">
LaTeX formulae
SDIY wiki supports embedding mathematical formulas using TeX syntax. Use
<m> tags.
For example (use edit to see the source): <m>\operatorname{erfc}(x) = \frac{2}{\sqrt{\pi}} \int_x^{\infty} e^{-t^2}\,dt = \frac{e^{-x^2}}{x\sqrt{\pi}}\sum_{n=0}^\infty (-1)^n \frac{(2n)!}{n!(2x)^{2n}}</m>
Categories
Add one or more categories to pages or uploaded file, by simply adding eg.
[[Category:Whatever]]. Categories themselves need to be categorised to create a hierarchy for navigating through the wiki.
Standard appendices
Information that can follow after the body of the article should follow in this order:
A list of works created by the subject of the article See also, a list of internal links to related articles Notes and references Further reading, a list of recommended relevant books, articles, or other publications that have not been used as sources External links, a list of recommended relevant websites that have not been used as sources Templates
Template:Main article A template is a page that gets included in another page, (this is called transclusion). This is useful for text that is often repeated. For example, create a page called "Template:Main article" with the text
''The main article for this is
[[{{{1}}}]].''
and then to use the template insert "{{Main article|Whatever}}" where you want that text to appear.
Talk pages
Don't leave visible notes and comments in the article. At the top of every article, the second tab entitled Discussion, opens the articles "Talk page". This is where to dicuss the article or leave notes for other editors. Remember to sign you posts on talk pages, (second from last button). In articles to leave notes or explanations use HTML commenting. These will be hidden except from other editors. An HTML comment, which has the form:
<!--This is a comment.-->, will work fine in Mediawiki.
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The Annals of Statistics Ann. Statist. Volume 17, Number 1 (1989), 107-124. Uniformly Powerful Goodness of Fit Tests Abstract
The simple hypothesis is tested that the distribution of independent random variables $X_1, X_2, \cdots, X_n$ is a given probability measure $P_0$. Let $\pi_n$ be any sequence of partitions. The alternative hypothesis is the set of probability measures $P$ with $\sum_{A\in \pi_n}|P(A) - P_0(A)| \geq \delta$, where $\delta > 0$. Note the dependence of this set of alternatives on the sample size. It is shown that if the effective cardinality of the partitions is of the same order as the sample size, then sequences of tests exist with uniformly exponentially small probabilities of error. Conversely, if the effective cardinality is of larger order than the sample size, then no such sequence of tests exists. The effective cardinality is the number of sets in the partition which exhaust all but a negligible portion of the probability under the null hypothesis.
Article information Source Ann. Statist., Volume 17, Number 1 (1989), 107-124. Dates First available in Project Euclid: 12 April 2007 Permanent link to this document https://projecteuclid.org/euclid.aos/1176347005 Digital Object Identifier doi:10.1214/aos/1176347005 Mathematical Reviews number (MathSciNet) MR981439 Zentralblatt MATH identifier 0674.62032 JSTOR links.jstor.org Citation
Barron, Andrew R. Uniformly Powerful Goodness of Fit Tests. Ann. Statist. 17 (1989), no. 1, 107--124. doi:10.1214/aos/1176347005. https://projecteuclid.org/euclid.aos/1176347005
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This is an elementary question, but a little subtle so I hope it is suitable for MO.
Let $T$ be an $n \times n$ square matrix over $\mathbb{C}$.
The characteristic polynomial $T - \lambda I$ splits into linear factors like $T - \lambda_iI$, and we have the Jordan canonical form:
$$ J = \begin{bmatrix} J_1 \\\ & J_2 \\\ & & \ddots \\\ & & & J_n \end{bmatrix}$$
where each block $J_i$ corresponds to the eigenvalue $\lambda_i$ and is of the form
$$ J_i = \begin{bmatrix} \lambda_i & 1 \\\ & \lambda_i & \ddots \\\ & & \ddots & 1 \\\ & & & \lambda_i \end{bmatrix}$$
and each $J_i$ has the property that $J_i - \lambda_i I$ is nilpotent, and in fact has kernel strictly smaller than $(J_i - \lambda_i I)^2$, which shows that none of these Jordan blocks fix any proper subspace of the subspace which they fix. Thus, Jordan canonical form gives the closest possible to a diagonal matrix. The elements in the superdiagonals of the Jordan blocks are the obstruction to diagonalization.
So far, so good. What I want to prove is the assertion that "Almost all square matrices over $\mathbb{C}$ is diagonalizable". The measure on the space of matrices is obvious, since it can be identified with $\mathbb{C}^{n^2}$. How to prove, perhaps using the above Jordan canonical form explanation, that almost all matrices are like this?
I am able to reason out the algebra part as above, but is finding difficulty in the analytic part. All I am able to manage is the following. The characteristic equation is of the form
$$(x - \lambda_1)(x - \lambda_2) \cdots (x - \lambda_n)$$
and in the space generated by the $\lambda_i$'s, the measure of the set in which it can happen that $\lambda_i = \lambda_j$ when $i \neq j$, is $0$: this set is a union of hyperplanes, each of measure $0$.
But here I have cheated, I used only the characteristic equation instead of using the full matrix. How do I prove it rigorously?
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Let consider a steady state CSTR in which occurs the following reaction:
$$\ce{aA + bB \leftrightarrow cC}$$
Notations:
$F_i$ is the molar flow of $i$, $r$ is the rate of reaction, $V$ is the volume, $X$ is the conversion at the end of the reactor, $\nu_i$ is the stoichiometric coefficient of $i$ and $\xi$ is the extent of reaction.
Mass balance:
$$\ce{IN + PROD - CONS = OUT + VAR}$$
Therefore,
$$ \begin{cases} F_{A,in} + 0 - arV = F_{A,out} \\ F_{B,in} + 0 - brV = F_{B,out} \\ F_{C,in} + crV - 0 = F_{C,out} \\ \end{cases} $$
As $$\xi = -rV = -X\frac{F_{i,in}}{\nu_i}$$ we have for $\ce{A}$:
$$\begin{alignat}{} F_{A,out} &= F_{A,in} - arV \\ & = F_{A,in} - a(-\xi) \\ & = F_{A,in} - a\left(X\frac{F_{A,in}}{a}\right) \\ & = \left(1 - X \right)F_{A,in} \end{alignat}$$
For $B$ with can find as well:
$$F_{B,out} = \left(1 - X \right)F_{B,in} = \frac{b}{a}\left(1 - X \right)F_{A,in}$$
My problem with C:
Now if I follow my calculus for $\ce{C}$, here is what I get:
$$\begin{alignat}{} F_{C,out} &= F_{C,in} + crV \\ & = F_{C,in} + c(-\xi) \\ & = F_{C,in} + c\left(X\frac{F_{C,in}}{c}\right) \\ & = \left(1 + X \right)F_{C,in} \end{alignat}$$
Which states that if there is no $C$ at the beginning, there is no $C$ produced at all! Which is wrong. Can you tell me where I am wrong?
NB: I have no clue, which tags fit best for this question.
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This question is motivated by this one.
Suppose $l$ is the minimum measurable unit of length. What is entropy of a spinless particle contained in this interval?
We know that entropy of a two-level system depends on the probabilities of the respective levels, if the probability of the state 0 is $p_0$, then the entropy (in natural units) is:
$$S= -\sum_{i=0}^1 p_i \ln p_i = -p_0 \ln p_0 - (1 - p_0) \ln (1 - p_0)$$
So if $p_0=1/2$ then $S=\ln 2\: \mathrm{nat}$, equal to $1\: \mathrm{bit}$. A particle which has the maximum in the middle has entropy of $1\: \mathrm{bit}$ (it is equally likely to be measured to the right and to the left of the middle).
Since we cannot measure intervals smaller than $l$, we cannot make guesses about where the maximum of the probability for the particle is located. As such, if we assume that the particle is equally likely have the maximum of the probability in any point on the interval $x\in[0,l]$, the total entropy becomes $$S=\int_0^l \frac{-(1-\frac xl) \ln (1- \frac xl)-\frac xl \ln (\frac xl)}{l} \, dx=\int_0^1 -(1-x) \ln (1-x)-x \ln (x) \, dx=\frac12$$
An entropy of a similar particle contained in a square area with side $l$ will be twice more, that is $1\: \mathrm{nat}$.
Now if we assume that $l=2l_p$ where $l_p$ is the Planck length, we arrive that such spinless particle has entropy of $1\: \mathrm{nat}$ per 4 square Planck length or $1/4\: \mathrm{nat}$ for one square Planck length.
Thus from the only assumption that double Planck length is the minimum measurable interval, and double Planck length squared is expected to contain 1 particle on average we arrive at the standard value of the Black Hole entropy in nats:
$$S=\frac{A}{4l_p^2}=\frac14 A_p$$
Where $A_p$ is the area in Planck units.
Sometimes I encountered a claim that the fundamental unit of information is 1 bit. From the above considerations it follows that possibly the fundamental unit is 1/2 (or 1 or 1/4) nat.
UPDATE
Note that the distance of $2l_p$ between two particles is natural if we assume that the particles are planckons, whose radius is Planck length $l_p$. As such, the Black Hole can be viewed as a spherical shell consisting of one layer of planckons.
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Today, we will discuss about the Daily Magic Spell problem given on January 2, 2017:
Problem: Find the value of $\displaystyle \frac{20!-19!}{18!}$. Solution: Note that
$\displaystyle \frac{20!-19!}{18!} = \frac{20(19!) - 19!}{18!} = \frac{19(19!)}{18!} = 19\times 19 = 361$.
Some students may attempt to evaluate $20!$, $19!$, and $18!$ naively. Although this could yield the correct answer, the approach itself requires too much effort to solve the problem.
If you have any other approaches that you would like to share with us, feel free to post and let us know how you would tackle this problem! We would like to hear your thoughts!
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Now showing items 1-9 of 9
Measurement of $J/\psi$ production as a function of event multiplicity in pp collisions at $\sqrt{s} = 13\,\mathrm{TeV}$ with ALICE
(Elsevier, 2017-11)
The availability at the LHC of the largest collision energy in pp collisions allows a significant advance in the measurement of $J/\psi$ production as function of event multiplicity. The interesting relative increase ...
Multiplicity dependence of jet-like two-particle correlations in pp collisions at $\sqrt s$ =7 and 13 TeV with ALICE
(Elsevier, 2017-11)
Two-particle correlations in relative azimuthal angle (Δ ϕ ) and pseudorapidity (Δ η ) have been used to study heavy-ion collision dynamics, including medium-induced jet modification. Further investigations also showed the ...
The new Inner Tracking System of the ALICE experiment
(Elsevier, 2017-11)
The ALICE experiment will undergo a major upgrade during the next LHC Long Shutdown scheduled in 2019–20 that will enable a detailed study of the properties of the QGP, exploiting the increased Pb-Pb luminosity ...
Azimuthally differential pion femtoscopy relative to the second and thrid harmonic in Pb-Pb 2.76 TeV collisions from ALICE
(Elsevier, 2017-11)
Azimuthally differential femtoscopic measurements, being sensitive to spatio-temporal characteristics of the source as well as to the collective velocity fields at freeze-out, provide very important information on the ...
Charmonium production in Pb–Pb and p–Pb collisions at forward rapidity measured with ALICE
(Elsevier, 2017-11)
The ALICE collaboration has measured the inclusive charmonium production at forward rapidity in Pb–Pb and p–Pb collisions at sNN=5.02TeV and sNN=8.16TeV , respectively. In Pb–Pb collisions, the J/ ψ and ψ (2S) nuclear ...
Investigations of anisotropic collectivity using multi-particle correlations in pp, p-Pb and Pb-Pb collisions
(Elsevier, 2017-11)
Two- and multi-particle azimuthal correlations have proven to be an excellent tool to probe the properties of the strongly interacting matter created in heavy-ion collisions. Recently, the results obtained for multi-particle ...
Jet-hadron correlations relative to the event plane at the LHC with ALICE
(Elsevier, 2017-11)
In ultra relativistic heavy-ion collisions at the Large Hadron Collider (LHC), conditions are met to produce a hot, dense and strongly interacting medium known as the Quark Gluon Plasma (QGP). Quarks and gluons from incoming ...
Measurements of the nuclear modification factor and elliptic flow of leptons from heavy-flavour hadron decays in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 and 5.02 TeV with ALICE
(Elsevier, 2017-11)
We present the ALICE results on the nuclear modification factor and elliptic flow of electrons and muons from open heavy-flavour hadron decays at mid-rapidity and forward rapidity in Pb--Pb collisions at $\sqrt{s_{\rm NN}}$ ...
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 Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV have been measured using the ALICE detector ...
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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 been studied using the ALICE detector at the LHC. The spectra exhibit a significant hardening with ...
Forward-central two-particle correlations in p-Pb collisions at $\sqrt{s_{\rm NN}}$ = 5.02 TeV
(Elsevier, 2016-02)
Two-particle angular correlations between trigger particles in the forward pseudorapidity range ($2.5 < |\eta| < 4.0$) and associated particles in the central range ($|\eta| < 1.0$) are measured with the ALICE detector in ...
Measurement of D-meson production versus multiplicity in p-Pb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV
(Springer, 2016-08)
The measurement of prompt D-meson production as a function of multiplicity in p–Pb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV with the ALICE detector at the LHC is reported. D$^0$, D$^+$ and D$^{*+}$ mesons are reconstructed ...
Measurement of electrons from heavy-flavour hadron decays in p–Pb collisions at $\sqrt{s_{\rm NN}} = 5.02$ TeV
(Elsevier, 2016-03)
The production of electrons from heavy-flavour hadron decays was measured as a function of transverse momentum ($p_{\rm T}$) in minimum-bias p–Pb collisions at $\sqrt{s_{\rm NN}} = 5.02$ TeV with ALICE at the LHC for $0.5 ...
Direct photon production in Pb-Pb collisions at $\sqrt{s_{NN}}$=2.76 TeV
(Elsevier, 2016-03)
Direct photon production at mid-rapidity in Pb-Pb collisions at $\sqrt{s_{\rm NN}} = 2.76$ TeV was studied in the transverse momentum range $0.9 < p_{\rm T} < 14$ GeV/$c$. Photons were detected via conversions in the ALICE ...
Multi-strange baryon production in p-Pb collisions at $\sqrt{s_\mathbf{NN}}=5.02$ TeV
(Elsevier, 2016-07)
The multi-strange baryon yields in Pb--Pb collisions have been shown to exhibit an enhancement relative to pp reactions. In this work, $\Xi$ and $\Omega$ production rates have been measured with the ALICE experiment as a ...
$^{3}_{\Lambda}\mathrm H$ and $^{3}_{\bar{\Lambda}} \overline{\mathrm H}$ production in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV
(Elsevier, 2016-03)
The production of the hypertriton nuclei $^{3}_{\Lambda}\mathrm H$ and $^{3}_{\bar{\Lambda}} \overline{\mathrm H}$ has been measured for the first time in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV with the ALICE ...
Multiplicity dependence of charged pion, kaon, and (anti)proton production at large transverse momentum in p-Pb collisions at $\sqrt{s_{\rm NN}}$= 5.02 TeV
(Elsevier, 2016-09)
The production of charged pions, kaons and (anti)protons has been measured at mid-rapidity ($-0.5 < y < 0$) in p-Pb collisions at $\sqrt{s_{\rm NN}} = 5.02$ TeV using the ALICE detector at the LHC. Exploiting particle ...
Jet-like correlations with neutral pion triggers in pp and central Pb–Pb collisions at 2.76 TeV
(Elsevier, 2016-12)
We present measurements of two-particle correlations with neutral pion trigger particles of transverse momenta $8 < p_{\mathrm{T}}^{\rm trig} < 16 \mathrm{GeV}/c$ and associated charged particles of $0.5 < p_{\mathrm{T}}^{\rm ...
Centrality dependence of charged jet production in p-Pb collisions at $\sqrt{s_\mathrm{NN}}$ = 5.02 TeV
(Springer, 2016-05)
Measurements of charged jet production as a function of centrality are presented for p-Pb collisions recorded at $\sqrt{s_{\rm NN}} = 5.02$ TeV with the ALICE detector. Centrality classes are determined via the energy ...
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I will like to animate a list of parametric spirals over a list of packed spheres. The effect I am after is illustrated in this gif by Étienne Jacob :
By looking at the gif above, we will divide our implementation into two parts:
Sphere Packing, where we try to find a list of locations and radii for the sphere such that they are non intersecting and look packed. Generating Spirals, this part is relatively easy, all we have to do is find a parametric equation for the spiral over the spheres and replicate it over the list of locations and radii we have got. Sphere Packing
There are multiple algorithms we can use to generate point locations and radii. Some are iterative, others are generative (but expensive), and others are "approximative." We don't really care about execution time because the output is readily cached. Perfect packing is not needed either since smaller circles will usually go unnoticed.
Our algorithms will be as follows:
Initialize an empty vector and float lists for locations and radii. Define a function that generates a random location, preferably uniform in density. For some arbitrary number: Generate a random point $p$. Loop over the point and radii lists and compute the smallest signed distance between the point $p$ and the nearest sphere surface point. If the signed distance is positive (meaning the point is outside all of the existing spheres), append that point $p$ and radii (which is defined as any scalar that is smaller that or equal the positive distance).
The smallest signed distance between the point $p$ and the nearest surface point is equal to the distance between $p$ and the nearest sphere minus its radius.
If the distance is smaller than the radius, then the signed distance will be negative, telling us that the point $p$ is inside of the spheres, and that's why we don't append it. Moreover, the radius of the sphere at point $p$ has to be smaller than the positive distance so as not to intersect the other circle.
To find the smallest signed distance, we loop over the point and radii lists and reassign an initially infinity float parameter if the current signed distance is smaller than the value of the parameter. This is is the equivalent to a KD Tree search except we are subtracting some value defined as the radius.
Next, we create a loop that has an arbitrary number of iterations controlling the maximum number of spheres. However, it does not strictly define the number of spheres itself! The loop has two main parameters: (1) a vector list that will include the spheres locations and (2) a float list that will contain the radii locations. Both inputs are undefined, meaning that they are initially empty lists that we will append to later. In this loop, we generate a random point (we will talk about that algorithm later on), compute the smallest signed distance using the loop we created before and append the sphere location if it is positive. The append condition is implemented as a form of reassign condition for both lists. It is important that we copy the lists at each iteration, ensuring the append doesn't overwrite the original list rendering our reassign condition useless. I chose to define the radius as the positive distance bounded by some arbitrary constant called the
Max Radius, though any other positive value less than the distance will work.
Again, the random point generation is a choice and it may vary based on our needs. I can see that in the reference gif, the random points are uniformly distributed inside a circle, so our random generator can be created as define in the document about Disk Point Picking, that is, the random point is defined by the parametric equation:
$$ \begin{aligned} x &= \sqrt{r} \cos t\\ y &= \sqrt{r} \sin t \end{aligned} $$
Where $r$ and $t$ are uniform random variables, $t$ ranging from zero to two $\pi$ and $r$ ranging from zero to some max radius. Th result would be something like this:
Had I used a normalized random vector, it would have been like this:
Spiral
All we have to do now is generate a spiral on those spheres. A possible parametric equation would be the ones described in this question and answers, that is:
$$ \begin{aligned} x &= \sqrt{1-t^2}\cos (a \pi t)\\ y &= \sqrt{1-t^2}\sin (a \pi t)\\ z &= t \end{aligned} $$
Where $t$ ranges between negative one and one and $a$ controls the number of revolutions. Which results:
But since sin and cos and periodic, we can animate them by adding a phase shift:
We can now transform this by some transformation matrix defined by the sphere locations and radii. Create a spline from it. And we are going to put that in a group to be able to use it easily.
Final Implementation
By looping over spheres, randomizing some of the parameters and transforming, as follows:
We get:
You can now render this as you wish. Maybe through in some random colors or make it particles like the reference gif.
Blend File:
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The Leibniz rule is as follows:
$$\frac{d}{d\alpha} \int_{a(\alpha)}^{b(\alpha)} f(x, \alpha) dx = \frac{db(\alpha)}{d\alpha} f(b(\alpha), \alpha) - \frac{da(\alpha)}{d\alpha} f(a(\alpha), \alpha) + \int^{b(\alpha)}_{a(\alpha)} \frac{\partial}{\partial\alpha} f(x, \alpha) dx$$
What I would like to know is how to apply the above formula for the case of the partial derivative:
$$\frac{\partial}{\partial\alpha} \int_{a(\alpha)}^{b(\beta)} f(x, \alpha) dx.$$
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