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1. ### I Expressing Elliptic Orbitals As Speed Functions.
Hi everyone:) I have spend a couple of days trying to teach myself the math of orbital mechanics and have been able to generate a model of the orbital path of Haley's Comet, incorporating realistic distances and periods using Kepler's second law & ellipsoid functions. This is a GIF of the motion...
2. ### B How would one plot the space-time continuum graphically?
How would one plot the space-time continuum graphically(if it were possible,obviously)?
3. ### ProProgram for drawing functions
Hello, good morning. I would like to know if someone knows a program to be able to draw the functions in the same way as the one shown in the image and also allow me to point out an enclosure formed by them without having to use inequalities to do so. Thank you very much for everything beforehand.
5. ### How do I find the area of the region bounded by following?
Using integrals, consider the 7 requirements: Any my attempted solution that I have no idea where I am going: And the other one provides the graph:
6. ### I Main Sequence Rules
When viewing an HR diagram, the main sequence curve is apparent, and the general shape of it is obvious. However, in this truncated version, it's unclear to me exactly which stars should be considered main sequence. I've added a shaded grey area as what I think I should count as main sequence...
I have two questions (1) If we have two areas, for example one in America and the other on Asia, both with the same latitude φ, which one of the areas will receive more Solar Radiation? Or will it be the same? (2) If Earth's eccentricity graph for the past 750k years is this...
8. ### Use the data to construct a graph etc.
Homework Statement Serway Physics Section 2.4 Acceleration (a) Use the data in Problem 5 to construct a smooth graph of position versus time. (b) By constructing tangents to the x(t) curve, find the instantaneous velocity of the car at several instants. (c) Plot the instantaneous velocity...
9. ### Given a graph find aceleration
Homework Statement Serway Physics Section 2.4 Acceleration 17. Figure P2.17 shows a graph of ##v_x## versus t for the motion of a motorcyclist as he starts from rest and moves along the road in a straight line. (a) Find the average acceleration for the time interval t = 0 to t = 6.00 s. (b)...
10. ### Difficult time with motion graphs & dynamics
Homework Statement The figure below shows the speed of a person's body as he does a chin-up. Assume the motion is vertical and the mass of the person's body is 72.7 kg. Determine the force exerted by the chin-up bar on his body at the following times. My answers: Homework Equations F = ma...
11. ### Ellipse graphic
Homework Statement Graph the ellipse 4x² + 2y² = 1 Homework Equations 4x² + 2y² = 1 The Attempt at a Solution 2x² + y²/2 = 1/2 I searched for exercises on Google, and i didn't find an equation like that. I watched videoleassons too but it didn't teach this type of equation.
12. ### Inconsistency in a Velocity-Time Graph
Homework Statement We're supposed to convert a position-time graph to a velocity-time graph to an acceleration time graph. These are the values for the position-time graph: Homework Equations m = y2-y1/x2-x1 The Attempt at a Solution I found the slopes for each time which were 5, 8.5, 17...
13. ### Ball-Wall collision problem
Problem goes: A rubber ball, travelling in a horizontal direction, strikes a vertical wall. It rebounds at right angles to the wall. The graph below illustrates the variation of the ball’s momentum p with time t when the ball is in contact with the wall. Which of the following statements is...
14. ### Steps for converting one type of graph to the other?
Homework Statement Suppose this the velocity time graph given and we want to plot position time grah from this. What are the things to be considered while plotting the position time graph from this graph? 2. Homework Equations The uniform acceleration motion equations. The Attempt at a...
15. ### I What counts as a circuit?
I have to construct suitable circuits to show the operational principles of some sensors: pH meter, sound, light and temperature. Would these count as circuits? or diagrams? (I'd say diagrams). I am finding it hard to find circuits showing how sound sensors and pH sensors work etc and the ones...
16. ### 2 Dimensional Kinematics Help -- Finding Gravitational Acceleration with v initial and a trajectory
Help with the picture posted!!
17. ### Plotting a Power Graph using Torque & RPM
Hi guys, Sorry if this has been covered before but I have been searching the forum looking for the exact information and have been unable to find it. I am currently carrying out a project to make a Honda GX35 engine suitable for competing in the Shell Eco-Marathon where I will perform...
18. ### Graph of ##\sin(\sinh x)##
Homework Statement Homework Equations The Attempt at a Solution ## \sinh (x) ## is continuous. ## \sin{( \sinh (x))} ## should have the same amplitude. .....(1) Option (a) and (b) follow this condition. For x = 0, ## \sin ({ \sinh (x)} ) ## = 0. .....(2) Option (a )...
19. M
### Can somebody tell me what this topic is?
Homework Statement Could somebody link me to a youtube video explaining this topic, its from an exam paper at me college and I cant find notes on it.It think it has something to do with limits. Many thanks.
20. A
### Graphic in excel for practicum
Homework Statement So I have to make three charts for the measurments that we made in class (physics experiments but it's not important). The first one had to be with a linear trendline, the second one with a exponential and the third one with a logarithmic trendline. For some reason the last...
21. F
### How to draw this vector
Homework Statement [/B] Sketch in a third vector, C, whose magnitude and direction are such that A+B+C=0. Vectors A and B both have a magnitude of 5 and form a 30 degree angle (image attached). Homework Equations [/B] How do I go about this question? I have no idea how to start or what to...
22. ### I-V graph of two wires and temperature
Homework Statement Homework Equations ## V=IR## The Attempt at a Solution This is an I-V graph so the slope must represent ##\frac{1}{R} ## Now, $$R∝Temperature$$ {This is assuming that the wires are of metal} Now , since the graph is of ##\frac{1}{R} ## , the steeper the slope the higher...
23. ### How to linearize this data?
Hi, I'm supposed to linearize this set of data: "Below is a data set which includes information about the motion of the objects in the solar system. Note: the periods are listed in Earth years (time it takes the Earth to complete one orbit around the Sun) and the average distances are reported...
24. S
### Calculate the time at which the object changes direction
Homework Statement An object starts from rest and uniformly acquires a velocity of 20.0 m/s in 5.0 minutes. For the next 10. minutes the object moves with a constant speed. After 10. minutes moving at 20. m/s the object then again accelerates and reaches a speed of 15 m/s in the opposite...
25. ### A Graph homeomorphic to Sphere
Hello, I want to prove that a graph represent a manifold, for this i take the opposites edges of a vertex (edge connected between vertex connected to the current vertex) and this subgraph need to be homeomorphic for example to the 1-sphere if i want a 2 manifold. This criterion ensure that my...
26. ### Graphs of Polar Equations
Homework Statement Hello! I will be grateful for your help in deciphering the meaning of a paragraph from the book. I honestly don't understand how they got the semi-circle on the xy graph by transferring it from rθ graph. Homework Equations I attach the screen shot from the book. The...
27. ### Why does not this Erdos-Renyi C code work?
Homework Statement I need to write an Erdos-Renyi random graph, by using the adjacency matrix (or alternatively list) and calculate the fitness of the graph. Definition: G(n, p) is a random graph with n vertices where each possible edge has probability p of existing. Homework Equations The...
28. ### Graphs of sin and cos, how to set points for x values
Homework Statement Hello! I am at the topic on graphing trigonometric functions. Exercises are rather easy at this point, but I have a problem deciphering how authors of the book choose points for x values. Please, take a look at few examples (including screen shots I attach), and, please...
29. ### B Do all the peaks and valleys of f have f'=0
I learned in the earlier chapters that peaks and valleys of a fxn have points where f'=0 (i marked them with red x). A few chapters later it said if a fxn has 2 roots, then f'=0 (still the 1st graph). So does that mean if the graph of a fxn is like the 2nd graph, the peaks and valleys are not...
30. ### Motion graphs and their units
Homework Statement I'm having some issues with the following exercise: The graph below represents the marking of a vehicle speedometer in funtion of time. Elaborate the corresponding graphs of acceleration and space traveled in function of time. What is the average acceleration of the vehicle...
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# Raman Spectroscopy of NiSe$_2$ and NiS$_{2-x}$Se$_x$ ($0 < x < 2$) Thin Films
### Abstract
The Raman spectra of NiS$_{2-x}$Se$_x$ ($0 < x < 2$) polycrystalline thin films have been measured for the first time in the whole $x$ range and for NiSe$_2$. The NiSe$_2$ spectrum is qualitatively similar to the spectrum of NiS$_2$, but all frequencies are shifted to lower energies. The shift has been analysed in terms of the increment of the anion mass and the lattice expansion. Peaks in the Raman spectrum of the NiS$_{2-x}$Se$_x$ alloy can be assigned to stretching and rotational modes of the S-S, Se-Se and S-Se pairs. The stretching vibration of the S-Se pairs shifts almost linearly from 400 cm$^{-1}$ for NiS$_2$ to 330 cm$^{-1}$ for NiSe$_2$. There is an accidental degeneracy between the S-S stretching and Se-Se libration frequencies. The relative Raman intensities fit well with a random occupation of the anion sites by S and Se atoms, and different scattering cross sections.
Type
Publication
J. Phys. C: Condens. Matter 12, 5317 (2000)
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# Equation for arc length
Arc Length Formula (s) L = ∫ ds L = ∫ d s where, ds = √1 +( dy dx)2 dx if y = f (x), a ≤ x ≤ b ds = √1 +( dx dy)2 dy if x = h(y), c ≤ y ≤ d d s = 1 + ( d y d x) 2 d x if y = f ( x), a ≤ x ≤ b d s =
• Clear up math question
## Arc Length
Formulas for Arc Length. The formula to measure the length of the arc is –. Arc Length Formula (if θ is in degrees) s = 2 π r (θ/360°) Arc Length Formula (if θ
## Arc Length
Figure out mathematic problems
Explain mathematic problem
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## Arc length
Arc length is the distance between two points along a section of a curve. Determining the length of an irregular arc segment by approximating the arc segment as connected line segments is
Track Way
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## Formula To Calculate Arc Length With Solved Examples
Arc Length Formula Radians If θ is given in radians, S = θ × r Arc Length Formula Degrees If θ is given in degrees S = 2πr (θ/360) Arc Length Formula Integral Form Integral form S = ∫ a b 1 + (
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## Arc Length Calculus Problems,
First we break the curve into small lengths and use the Distance Between 2 Points formula on each length to come up with an approximate answer: The distance from x0 to x1 is: S 1 = √ (x1 − x0)2 + (y1 − y0)2 And let's use Δ (delta) to mean
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I can solve the math problem for you.
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To determine a mathematic equation, one would need to first identify the problem or question that they are trying to solve.
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# What does LC mean?
Oct 30, 2016
LC can mean Light Cylinder.
#### Explanation:
A light cylinder is a theoretical cylinder around a pulsar which is a rapidly spinning neutron star.
The radius of the cylinder is:
$r = \frac{c P}{2 \pi}$
Where c is the speed of light and P is the period of the neutron star.
If an object on the surface of the cylinder were to rotate at the same angular velocity as the neutron star it would have to be travelling at the speed of light.
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# Is there a purely inseparable covering $\mathbb{A}^2 \to K$ of a Kummer surface $K$ over $\mathbb{F}_{p^2}$?
Let $E_i\!: y_i^2 = x_i^3 + a_4x_i + a_6$ be two copies ($i = 1$, $2$) of a supersingular elliptic curve over a finite field $\mathbb{F}_{p^2}$, for odd prime $p > 3$. Consider the Kummer surface $K = \mathrm{Kum}(A)$ for the superspecial abelian surface $A = E_1\times E_2$. Katsura and Schütt proved in the articles "Generalized Kummer surfaces and their unirationality in characteristic p (1987)" and "Zariski K3 surfaces (2017)" that the surface $K$ is a Zariski surface, i.e., there is a purely inseparable covering $\mathbb{A}^2 \to K$ over $\overline{\mathbb{F}_p}$, where $p \not\equiv 1$ (mod $12$). Is it still true over $\mathbb{F}_{p^2}$ at least for some supersingular elliptic curve and some $p$?
A positive answer to this question may have value for cryptography :) If $K$ is a Zariski surface over $\mathbb{F}_{p^2}$, then we have a method to compress a pair $(P_1, P_2) \in A(\mathbb{F}_{p^2})$ by computation of a map from $A$ to the affine plane $\mathbb{A}^2$ (through $K$) of separable degree 2. This is very compact and efficient method, because for decompression we need to solve only one quadratic equation. Computation of a preimage for a purely inseparable map is very fast. People usually take two projections on x-coordinates for $P_1$ and $P_2$ independently, hence they should solve two quadratic equations. The state of the art for this part of cryptography is represented, for example, in the article https://eprint.iacr.org/2017/1143.
• What's the cryptographic context? (Crypto or no, it looks like a natural enough question in arthmetic geometry) May 23, 2018 at 12:37
A supersingular elliptic curve in characteristic $p>2$ will always be covered by the hyperelliptic curve $y^2=x^p-x$ as the Jacobian of the latter is isogenous to a product of supersingular elliptic curves. When $p \equiv 2 \pmod{3}$ this can be made very explicit for $y^2=x^3-1$, which is covered by $y^2=x^{p+1}-1$ in the obvious way and $y^2=x^{p+1}-1$ is isomorphic to $y^2=x^p-x$ by sending a Weierstrass point to infinity.
• How can we explicitly find a purely inseparable part $S \to K$ of the map $\mathbb{A}^2 \to K$, where $S$ is some "intermediate" surface? May 23, 2018 at 19:26
• @DimaKoshelev I missed the fact you wanted purely inseparable. But if $\mathbb{A}^2 \to X \to K$ and $X \to K$ is Galois, you can try to lift the Galois action to $\mathbb{A}^2$ and take the quotient, which will also be a rational surface. May 23, 2018 at 19:53
• Likewise for $p \equiv 3 \bmod 4$ by sending $y^2 = x^{p+1} - 1$ to $y^2 = X^4 - 1$ where $X = x^{(p+1)/4}$. May 23, 2018 at 20:45
• @DimaKoshelev Yes, a priori, only over the algebraic closure. I believe the cases of $p \equiv 2 \pmod{3}, p \equiv 3 \pmod{4}$ already mentioned will be quite explicit and can be checked directly. Shioda's paper Math. Ann. 211 (1974), 233–236, does the Fermat surface of degree $p+1$ explicitly and shows it's a Zariski surface over $\mathbb{F}_{p^2}$. May 24, 2018 at 6:59
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# BBO Discussion Forums: ACBL NABC Online Individual - BBO Discussion Forums
• 4 Pages
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## ACBL NABC Online Individual July 23 - 26, 2017 on BBO
### #61wjomlex
• Group: Members
• Posts: 21
• Joined: 2015-September-06
Posted 2017-July-24, 17:57
It seems like there *is* stratification? Check out the results from today's Daily Bulletin (pages 19-22). Some people are getting more masterpoints than those above them.
0
### #62barmar
• Posts: 17,563
• Joined: 2004-August-21
• Gender:Male
Posted 2017-July-25, 10:36
wjomlex, on 2017-July-24, 17:57, said:
It seems like there *is* stratification? Check out the results from today's Daily Bulletin (pages 19-22). Some people are getting more masterpoints than those above them.
No, there's no stratification. Can you give examples of players who got more than someone above them?
The session masterpoint formula is very simple: (18*4) / (rank + 5) red points, except for first place, which gets 12 red + 6 gold. No points below 1.00 are awarded (there was an error on the first day, we published fractional awards, but that's been fixed).
### #63diana_eva
• Posts: 4,181
• Joined: 2009-July-26
• Gender:Female
• Location:bucharest / romania
Posted 2017-July-27, 06:41
Results:
http://webutil.bridg...59527&overall=y
### #64johnu
• Group: Advanced Members
• Posts: 2,035
• Joined: 2008-September-10
Posted 2017-July-27, 09:59
barmar, on 2017-July-25, 10:36, said:
No, there's no stratification. Can you give examples of players who got more than someone above them?
Some people got bigger session awards than their overall standing points would have been.
0
### #65barmar
• Posts: 17,563
• Joined: 2004-August-21
• Gender:Male
Posted 2017-July-27, 23:31
You get either the sum of your session awards or your overall award, whichever is higher. This is the same as how ACBL f2f tournaments work. When you look at the final results, the awards shown are a mix of session and overall awards.
### #66pigpenz
• Group: Advanced Members
• Posts: 2,374
• Joined: 2005-April-25
Posted 2017-August-03, 09:50
barmar, on 2017-July-25, 10:36, said:
No, there's no stratification. Can you give examples of players who got more than someone above them?
The session masterpoint formula is very simple: (18*4) / (rank + 5) red points, except for first place, which gets 12 red + 6 gold. No points below 1.00 are awarded (there was an error on the first day, we published fractional awards, but that's been fixed).
Ok I wondered when I saw the first day in the bulletin they showed places down to 300 for the section. so this was a mistake then? Actually I think that is how they should have done it IMHO.
0
### #67barmar
• Posts: 17,563
• Joined: 2004-August-21
• Gender:Male
Posted 2017-August-03, 10:27
pigpenz, on 2017-August-03, 09:50, said:
Ok I wondered when I saw the first day in the bulletin they showed places down to 300 for the section. so this was a mistake then? Actually I think that is how they should have done it IMHO.
They changed it for the overall awards. They were given to the top 35% of the field, so it included fractional awards.
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Field Automorphisms: A Nice Touch on "Ambiguity"
I am sure that the title of the post is going to confuse many readers, but for lack of a better title please be content with it. Rest assured that the post itself is quite unambiguous. We would like to throw some light on the a special kind of ambiguity which we find in mathematical systems (here mainly in algebra).
A Taste of Modern Algebra: Remainder Theorem for Polynomials
Introduction
In high school curriculum we are taught the "Remainder Theorem" as one of the important results of Algebra. The theorem and its proof are quite simple, but the practical application and theoretical ramifications of this result are quite interesting. To recall, the theorem is stated as follows:
Remainder Theorem: If $f(x) = a_{0}x^{n} + a_{1}x^{n - 1} + \cdots + a_{n - 1}x + a_{n}$ is a polynomial with real coefficients then the remainder obtained on dividing $f(x)$ by $(x - a)$ is $f(a)$.
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0 energy points
# Rewriting decimals as fractions: 0.8
Sal converts 0.8 to a fraction. Created by Sal Khan.
Video transcript
Let's write 0.8 as a fraction. So 0.8... the 8 right over here is in the tenths place. It is the tenths place. So you can read this as 8 tenths and we can write that literally as being equal to 8 tenths or 8 over 10. And now, we've already written it as a fraction and if we want we can simplify this down. Both 8 and 10 share common factors, they are both divisible by 2. So lets divide both the numerator and the denominator by 2. We are not changing the value of the fraction because we are dividing both the numerator and the denominator by the same thing 8 divided by 2 is 4, 10 divided by 2 is 5 and we're done. 0.8 is the same thing as 8 tenths, which is the same thing as 4 fifths.
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# The Polynomials that Generate Prime Numbers
The search for primes started thousands of years ago. Mathematicians since antiquity tried to find ways to look for primes. They also searched for methods to test if a number is prime or not. Others tried to find polynomials to generate primes.
The Sieve of Eratosthenes
One of the ancient methods of listing prime numbers is the Sieve of Eratosthenes. The Sieve consists of a finite list of numbers, where the multiples of each number are crossed out starting from 2 and increasing each time the list is exhausted.
Using computers, this algorithm can be used to generate as many primes as possible given enough time and computer memory. However, despite its reliability, there is no way that we could list all primes because it has been proved that there are infinitely prime numbers.
Prime Generating Polynomials
Aside from the Sieve, mathematicians searched for other ways to generate prime numbers. Pierre de Fermat, for instance, conjectured that $2^{2^2} + 1$ is prime, but Euler proved that $2^{32}$ is composite. Primes of such form are now known as Fermat’s Prime.
Legendre also gave the polynomial $2n^2 + 29$ which is prime from $n=0$ to $n=28$ but fails for $n = 29$ and $n^2 + n + 17$ which is prime from $n = 0$ to $n = 15$.
The most famous polynomial prime generator (probably) was Euler’s $n^2 - n + 41$ which is prime from $n = 0$ to $40$, but fails for $n = 41$. Another polynomial $n^2 - 79n + 1601$ given by E.B. Escott is prime for all $n = 0, 1, 2, 3, ... , 79$. There are also more polynomials that generate primes, the most complicated of which perhaps is
.
This gives prime numbers from $n = 0$ to $n = 56$.
Question: Is there a polynomial that will generate only prime numbers?
Answer: There is none. It has been proved that no polynomial exists can ever produce only primes.
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## Mass in relativity::Mass
### ::concepts
Object::field Energy::force First::which Objects::title Weight::''m'' Galileo::physics
Mass in relativity
### Special relativity
{{#invoke:main|main}} In special relativity, there are two kinds of mass: rest mass (invariant mass),<ref group="note">It is possible to make a slight distinction between "rest mass" and "invariant mass". For a system of two or more particles, none of the particles are required be at rest with respect to the observer for the system as a whole to be at rest with respect to the observer. To avoid this confusion, some sources will use "rest mass" only for individual particles, and "invariant mass" for systems.</ref> and relativistic mass (which increases with velocity). Rest mass is the Newtonian mass as measured by an observer moving along with the object. Relativistic mass is the total quantity of energy in a body or system divided by c2. The two are related by the following equation:
$m_\mathrm{relative}=\gamma (m_\mathrm{rest})\!$
where $\gamma$ is the Lorentz factor:
$\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$
The invariant mass of systems is the same for observers in all inertial frames, while the relativistic mass depends on the observer's frame of reference. In order to formulate the equations of physics such that mass values do not change between observers, it is convenient to use rest mass. The rest mass of a body is also related to its energy E and the magnitude of its momentum p by the relativistic energy-momentum equation:
$(m_\mathrm{rest})c^2=\sqrt{E_\mathrm{total}^2-(|\mathbf{p}|c)^2}.\!$
So long as the system is closed with respect to mass and energy, both kinds of mass are conserved in any given frame of reference. The conservation of mass holds even as some types of particles are converted to others. Particles of matter may be converted to types of energy (e.g. light, kinetic energy, the potential energy in magnetic, electric and other fields) but this does not affect the amount of mass. Although things like heat may not be matter, all types of energy still continue to exhibit mass.<ref group="note">For example, a nuclear bomb in an idealized super-strong box, sitting on a scale, would in theory show no change in mass when detonated (although the inside of the box would become much hotter). In such a system, the mass of the box would change only if energy were allowed to escape from the box as light or heat. However, in that case, the removed energy would take its associated mass with it. Letting heat out of such a system is simply a way to remove mass. Thus, mass, like energy, cannot be destroyed, but only moved from one place to another.</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=book }}</ref> Thus, mass and energy do not change into one another in relativity; rather, both are names for the same thing, and neither mass nor energy appear without the other.
Both rest and relativistic mass can be expressed as an energy by applying the well-known relationship E = mc2, yielding rest energy and "relativistic energy" (total system energy) respectively:
$E_\mathrm{rest}=(m_\mathrm{rest})c^2\!$
$E_\mathrm{total}=(m_\mathrm{relative})c^2\!$
The "relativistic" mass and energy concepts are related to their "rest" counterparts, but they do not have the same value as their rest counterparts in systems where there is a net momentum. Because the relativistic mass is proportional to the energy, it has gradually fallen into disuse among physicists.<ref>{{#invoke:citation/CS1|citation|CitationClass=arxiv}}</ref> There is disagreement over whether the concept remains useful pedagogically.<ref name="okun">{{#invoke:Citation/CS1|citation |CitationClass=journal }}</ref><ref>{{#invoke:Citation/CS1|citation |CitationClass=journal }}</ref><ref>{{#invoke:Citation/CS1|citation |CitationClass=journal }}</ref>
In bound systems, the binding energy must often be subtracted from the mass of the unbound system, because binding energy commonly leaves the system at the time it is bound. Mass is not conserved in this process because the system is not closed during the binding process. For example, the binding energy of atomic nuclei is often lost in the form of gamma rays when the nuclei are formed, leaving nuclides which have less mass than the free particles (nucleons) of which they are composed.
### General relativity
{{#invoke:main|main}} In general relativity, the equivalence principle is any of several related concepts dealing with the equivalence of gravitational and inertial mass. At the core of this assertion is Albert Einstein's idea that the gravitational force as experienced locally while standing on a massive body (such as the Earth) is the same as the pseudo-force experienced by an observer in a non-inertial (i.e. accelerated) frame of reference.
However, it turns out that it is impossible to find an objective general definition for the concept of invariant mass in general relativity. At the core of the problem is the non-linearity of the Einstein field equations, making it impossible to write the gravitational field energy as part of the stress–energy tensor in a way that is invariant for all observers. For a given observer, this can be achieved by the stress–energy–momentum pseudotensor.<ref>{{#invoke:citation/CS1|citation |CitationClass=book }}</ref>
Mass sections
Intro Units of mass Definitions of mass Pre-Newtonian concepts Newtonian mass Atomic mass Mass in relativity Mass in quantum physics See also Notes References External links
Mass in relativity PREVIOUS: Intro NEXT: Units of mass << >>
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## Get selected block every time a new block is selected in Gutenberg
Question
I want to get the selected block (an inner block) every time a new block is selected.
I am trying the following:
edit() {
const selectedBlock = wp.data.select( 'core/block-editor' ).getSelectedBlock();
useEffect( () => {
console.log( selectedBlock );
}, [ selectedBlock ] );
})
The selected block is returned only when the block renders. Nothing happens when I select a new inner block. It works only when I click out of the parent block, then click again an inner block.
0
5 months 2022-07-19T17:15:59-05:00 0 Answers 0 views 0
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# Linear Algebra Dictionary
## Ray
A ray is a general name for an equivalence class of vectors $e^{i\alpha}\ket{\psi}:\alpha\in [0,2\pi]}$. Quantum states are rays as global phases of this type are indistinguishable under measurement. Rays are denoted using the 'ket' $\ket{\psi}$, and their dual-vector is denoted by the 'bra' $\bra{\psi} : \braket{\psi}{\phi}\in\mathbb{C}$, such that the dual vector maps us from a state in the vector space to the space of complex numbers.
Explicitly, the dual-vector corresponds to the complex-conjugate $\ket{\psi}^{\dagger}$.
## Hilbert Space
A Hilbert Space $\mathcal{H}$ is a complex [vector space][#vector-space] on the complex numbers $\mathbb{C}$, replete with an inner product $(x,y) : x, y \in \mathcal{H} \rightarrow \mathbb{C}$ such that it is 1. Positive: $(x,y)>0\forall x,y \neq 0$ 2. Linear: $(x, ay + bz) = a(x,y) + b(x,z)$ 3. Skew symmetric: $(x,y) = (y,x)^{*}$
It is also complete under a norm $\norm{x}=\sqrt{(x,x)}$.
In quantum mechanics, the inner product is given by the 'braket' of the states $(x,y) = \braket{x}{y}$.
## Mutually Unbiased Basis
The notion of mutually unbiased bases arose in the context of defining the minimum set of observables required to full determine a quantum state (given a bunch of copies, obviously). The intuitve intuition is that we see a set of measurements tht are 'as different' as possible, such that their probabilities are relatively independent and we gain maximal information from each measurement.
Given a pair of bases $B_{1}=\{\ket{\phi_{i}}\}, B_{2}=\ket{\psi_{j}}\}$ on a Hilbert space $\mathcal{H}:\norm{\mathcal{H}}=d$, then the bases are said to be mutually unbiased if and only if
A $d$ dimensional quantum system has $d$ amplitudes, of which we can find $d-1$ independent ampltudes from $d$ measurements. As a result, we need to make$d+1$ mutually unbiased measurements (measurements in a mutually unbiased basis) to fully characterise the state. In fact, a quantum system has at most $d+1$ independent mutually unbiased bases when $d$. If the system has dimension $d=p^{m}$, then we have $(p+1)^{m}$ operators which fall in to $p^{m}+1$ maximal noncomuting systems. Each basis is made up of orthogonal unitary matrices such that $Tr(X^{\dagger}Y)=0\forall X,Y\in B_{i}$, where the basis vectors described above are the +1 eigenstates of these obvservables.
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MECH 431
Updated
# Conference 8
someone please fill this gap in
## Assignment 3
That means for each period we want to estimate period, the chance is undetermined. One can assume it’s a uniform distribution between 5% to 15%. Assuming uniform distribution, it would average out to 10%. But then it will lower the standard deviation (spread) and give the illusion of less risk
## Midterm 2
• Expect expert understanding from stuff from midterm 1
• Take-home
• No collaboration
• There will be straight forward calculation questions
• But there will also be more discussion questions to gauge that we can explain concepts
• i.e. explain why do falling interest rate favors investors
• Midterm will cover up to taxes
• Some review practice problems will be posted, but we will not go through them
## Taxes (Preview)
• “They screw us over”
• Taxes take a portion from our income, which reduces one’s disposable income
• From business perspective, it’s the same thing:
• Taxes reduce profits
• Essentially another expense
• What if the government tax is 90-100%? How would that affect the decision?
Example:
• Note that businesses take income, pay expense, then gets taxed. (whereas for individuals, it’s taxed then becomes usable money).
• After tax IRR is less than before IRR. Notice that it’s no longer attractive as IRR < MARR. It’s not worth doing this project if we consider taxes.
• If tax rate goes down, after tax IRR goes up. It could mean that the project become more viable as IRR$\rightarrow$MARR or greater.
• In short: tax rate affect economic decisions.
• In Canada, corporate tax rate went down, but tax revenue stayed constant (from more projects) so it “paid for itself”
• USA is following same strategy.
## Questions
someone please fill this gap in
Q. By “other such factors” for Question 1, where would you expect us to find them? Or just stuff like median?
What are we looking is primary summaries to statistics, mean, variance, outcomes with most occurrence, etc.
3-standard deviations would encapsulate 99.7% of the area for a normal distribution.
Q. For Q2 can I confirm that it’s an 80% chance of 5 years benefits, and a 20% chance of 1 year benefits?
Yes
Q. Do you like to binge-watch Narcos on Netflix?
Yes :heart:
Q. Does $\mathbb E[\text{IRR}] = \sum(\text{IRR}\times \mathbb P)$?
Yes, sum of IRR for each case times probability of each case.
A more complicated way would be “probability-weigh the cash flows” then compute the cashflow but is not necessary.
Q. Are we allowed to answer a question on interest like (short answers) that with a sample calculation?
Yes, but the main point is to answer fully with explanations
Ditto with diagrams.
Q. [Follow up to taxes] This is outside of your example, but if a 2% inflation rate was applied, would the IRR be reduced to 5.7%? Or is it calculated differently (for the report)?
fill me in
Q. [report] How do we recommend we calculate discount rate?
fill me in
Q. is there any way to find some Canadian tax material similar to the textbook chapter? Some of us may not have the Canadian edition?
“[Mark] will put up a supplementary write up explaining how CCA works”
**Q. For inflation, can we just apply it to the cash flows? Or do we have to use the specific inflation equation mentioned in the textbook relating market and real rates of return? **
For inflation, we could express it as nominal interest rate or convert them to “real” values.
Q. What if project in the report has no income, but reduced cost
This can mean reduced tax benefits, so tax is involved regardless
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# Law of Exponent 1 – Worked Examples
This video in Taglish (mixed Tagalog and English) shows examples of the application of law of exponent 1 which is $(x^m)(x^n) = x^{m+n}$. The post discusses examples that mostly cover basic problems including generalized exponents. Coefficients of the examples also include integers and fractions.
If you can’t see the video on your mobile phone, you can watch it on Youtube.
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To refer to this page use: http://arks.princeton.edu/ark:/88435/pr1px5x
Abstract: The surface of superconducting topological insulators (STIs) has been recognized as an effective $p\pm ip$ superconductivity platform for realizing elusive Majorana fermions. Chiral Majorana modes (CMMs), which are different from Majorana bound states localized at points, can be achieved readily in experiments by depositing a ferromagnetic overlayer on top of the STI surface. Here we simulate this heterostructure by employing a realistic tight-binding model and show that the CMM appears on the edge of the ferromagnetic islands only after the superconducting gap is inverted by the exchange coupling between the ferromagnet and the STI. In addition, multiple CMMs can be generated by tuning the chemical potential of the topological insulator. These results can be applied to both proximity-effect induced superconductivity in topological insulators and intrinsic STI compounds such as PbTaSe$_2$, BiPd and their chemical analogues, providing a route to engineering CMMs in those materials. Publication Date: 21-Sep-2018 Citation: Chiu, C.-K.; Bian, G.; Zheng, H.; Yin, J.-X.; Zhang, S. S.; Sanchez, D. S.; Belopolski, I.; Xu, S.-Y.; Hasan, M. Z. Chiral Majorana Fermion Modes on the Surface of Superconducting Topological Insulators. EPL (Europhysics Letters) 2018, 123 (4), 47005. https://doi.org/10.1209/0295-5075/123/47005. DOI: doi:10.1209/0295-5075/123/47005 ISSN: 0295-5075 EISSN: 1286-4854 Pages: 1 - 16 Type of Material: Journal Article Journal/Proceeding Title: EPL (Europhysics Letters) Version: Author's manuscript
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Supplements Antioxidants Varicose 911: Review PhytAge Labs Vein Wall Support Formula
# Varicose 911: Review PhytAge Labs Vein Wall Support Formula
Varicose 911 is a vein wall revitalization formula by PhytAge Laboratories that uses nine ingredients to help resolve varicose veins issues by improving blood flow and enhancing collagen production.
Varicose 911 is a nutritional supplement that claims to erase varicose veins from your legs.
Using a “weird discovery,” Varicose 911 claims to target varicose veins without needing surgery or invasive procedures. Just take two capsules of Varicose 911 per day to eliminate varicose veins and related issues.
Millions of people have varicose veins. But is Varicose 911 the solution? Find out everything you need to know about Varicose 911 and how it works today in our Varicose 911 review.
## What is Varicose 911?
Varicose 911 is a nutritional supplement sold exclusively online through Varicose911.com.
Like other ‘911’ supplements from the same company (PhytAge Labs), Varicose 911 claims to target varicose veins using natural ingredients, herbal extracts, vitamins, and minerals.
If you take two capsules of Varicose 911 per day, you can purportedly reduce the appearance of varicose veins regardless of their severity.
According to the Varicose 911 sales page, varicose veins are part of a much bigger problem. If left ignored, varicose veins could turn into a serious health issue.
Varicose 911 was formulated by a man named Harry Gold. Harry claims his formula helps with varicose veins regardless of whether you’ve been dealing with the issue for 10 years – or if it’s a new issue.
Typically, doctors recommend surgery, exercises, and other treatments to relieve varicose veins. Does Varicose 911 really provide similar benefits with none of the downsides? Let’s take a closer look at how Varicose 911 works – including the story behind Varicose 911.
## The Story Behind Varicose 911
Varicose 911 was created by a man named Harry Gold from Friendswood, Texas. The Varicose911.com sales page features a video and text presentation from Harry Gold.
Harry describes how serious varicose veins can be. When left untreated, varicose veins turn into bleeding, ulcers, and even death.
Harry was motivated to create a solution for varicose veins after watching his mother suffer from the issue. He researched natural treatments for varicose veins, tested different formulas, and eventually bundled everything together as Varicose 911.
Today, Harry claims his formula has helped over 12,318 people “keep their veins functioning strongly, safely and effectively.”
Here’s how Harry explains his natural varicose veins solution and how it works:
“My all-natural solution will deliver exactly what your veins desperately need to spring back to life and function properly to keep you varicose-free.”
Harry seems to advise against getting varicose vein surgery, describing the surgeries as “dangerous or invasive.” He claims most people don’t need the “risk” of surgeries, which is why he seems to recommend his Varicose 911 supplement instead.
The sales page for Varicose 911 is filled with people who claim to have completely eliminated their varicose veins using Varicose 911.
One woman claims her varicose veins “were so painful that [she] feared for [her] health and sanity,” for example. That woman claims Varicose 911 “saved” her from scary and expensive surgery while restoring her confidence.
Harry claims his mom’s varicose veins were so bad that she ended up in the emergency room: she scraped against a picnic table, causing uncontrollable bleeding down her legs. She had a severe varicose vein problem. Her doctor recommended surgeries (like stripping or radiofrequency ablation). Harry, however, recommended his solution instead.
Obviously, you should be skeptical any time someone claims their supplement can fix varicose veins and other serious issues. Let’s find out how Varicose 911 works – or if it works at all.
## How Does Varicose 911 Work?
Varicose 911 claims to target the root cause of varicose veins: a lack of collagen.
Collagen is the most abundant protein in the human body. As you age, your collagen levels naturally decline.
Collagen plays an important role in your skin, hair, nails, and other parts of your body. Collagen gives you a youthful appearance. It helps your joints move. Many people take collagen supplements daily to support anti-aging benefits.
Typically, varicose veins are caused by increased blood pressure in your veins or other circulatory issues. Your body struggles to pump blood to and from the heart, causing veins in your legs to twist and become misshapen. This is a varicose vein.
Varicose 911, however, seems to suggest that collagen is the root cause of varicose vein. By targeting collagen production in your body, and by supporting collagen production, Varicose 911 claims to eliminate varicose veins naturally.
Many people take collagen protein supplements to increase their body’s collagen levels. This directly raises collagen levels. Others take ingredients that are precursors to collagen – they increase the production of collagen indirectly by giving your body the ingredients it needs to support collagen production.
Other varicose vein supplements make similar claims. To compare Varicose 911 to other varicose vein supplements, let’s look at the ingredients inside and how they work.
## Varicose 911 Ingredients
Varicose 911 contains vitamins, minerals, herbal extracts, plant extracts, and other natural ingredients.
Here’s how the makers of Varicose 911 explains each ingredient in the supplement – and how each ingredient works.
Grape Seed Extract: The grape seed extract in Varicose 911 can purportedly “reactivate collagen biosynthesis,” which means it kickstarts collagen production in the body. The makers of Varicose 911 claim they use grape seed extract “instead of relying on some crushed up chicken bones and nasty cow hooves,” which is how most collagen supplements work. This grape seed extract purportedly allows your body to “produce high levels” of collagen on its own. The makers of Varicose 911 also cite studies showing that grape seed extract reduces the risk of blood clots, reduces leg swelling, and steps varicose veins from coming back, among other benefits.
Bilberry Extract: Bilberry extract rose to fame in World War II, when Royal Air Force (RAF) pilots purportedly took bilberry to support eyesight. Ever since, bilberry extract has been found in countless eye health supplements for its purported ability to support vision (RAF pilots, of course, used radar – not bilberry extract – to enhance their vision). In Varicose 911, bilberry extract purportedly “turbo-boosts collagen biosynthesis,” boosting the production of collagen in your body similar to grape seed extract.
Zinc: You can get zinc from any multivitamin, although Varicose 911 contains zinc to “give flexibility to your arteries and veins.” Zinc contributes to crucial benefits throughout the body, and it’s possible zinc could support cardiovascular health in various ways.
Other Ingredients: Varicose 911 also contains 6 other ingredients, although none seem to be found in large doses. Other ingredients in Varicose 911 include vitamin A, lutein, quercetin, taurine, lycopene, and N-acetyl-L-cysteine, among others.
Each capsule of Varicose 911 contains the 9 ingredients listed above.
The ingredients are packaged into a gelatin capsule with di-calcium phosphate, magnesium stearate, stearic acid, and microcrystalline cellulose used as fillers, binders, and preservatives.
## Scientific Evidence for Varicose 911
Varicose 911 has not completed any clinical trials or scientific studies to verify it works as advertised to remove varicose veins or improve circulation.
However, the makers of Varicose 911 claim that “over 12,318” people have used Varicose 911 “to keep their veins functioning strongly, safely and effectively.” The company also claims that Varicose 911 “worked for everyone that tried it” after a small trial involving close friends.
As far as we can tell, this is not true: we can find no verifiable reports of anyone using Varicose 911 to remove varicose veins, eliminate the appearance of varicose veins, or improve circulation.
However, it’s possible the ingredients in Varicose 911 could supply your body with crucial antioxidants, vitamins, and minerals it needs. These ingredients could slightly support cardiovascular health, which could reduce the appearance of varicose veins.
In reality, however, it’s unlikely that Varicose 911 will singlehandedly solve serious varicose vein problems – especially if you have suffered from the issue for years.
Varicose veins are dangerous, and all evidence suggests you should follow your doctor’s advice when dealing with this serious medical condition.
One study from 2008 on animals found that bilberries could help those with varicose veins. Researchers gave bilberry extract to animals and found the extract positively impacted blood flow, making blood flow more easily throughout the body. However, no largescale studies in humans have verified these benefits or proven that bilberry extract can eliminate varicose veins. If you are interested in the benefits of bilberry extract, you can find bilberry extract supplements sold online through Amazon for $5 to$10.
Some studies have shown that zinc can help with the itching linked to varicose veins – although zinc does not actually remove varicose veins. This 2017 study showed that taking a zinc supplement could help maintain the integrity of skin, relieving itching in patients with varicose veins. This effect is particularly noticeable if you are deficient in zinc.
Although grape seed oil is rich with antioxidants, there’s no evidence it can completely remove varicose veins. Some people use grape seed extract as a natural remedy for varicose veins, although there’s little largescale evidence in humans it can completely remove varicose veins. Grape seed extract seems to work by supporting cardiovascular health, making it easier for your body to send blood throughout the body.
It’s also important to note that Varicose 911 contains relatively low dosages of most listed ingredients except for vitamin A (1,667% DV) and zinc (227% DV). There’s just 80mg of bilberry extract, for example, which is a much lower dose than other bilberry extract supplements – and a lower dose than what was used in studies.
Overall, there’s no evidence showing that Varicose 911 removes varicose veins, permanently eliminates varicose veins, or even reduces the appearance of varicose veins. Some ingredients could support cardiovascular health, although these ingredients are similar to other herbal extracts or multivitamins.
## Varicose 911 Pricing
Varicose 911 is priced at $70 per bottle, with discounts available when ordering multiple bottles. Here’s how pricing breaks down: 1 Bottle:$69.95 + Free Shipping
2 Bottles: $119.90 + Free Shipping 4 Bottles:$199.80 + Free Shipping
All pricing includes free shipping to the United States.
You can opt into the VIP membership club to save 10% on your purchase. If you opt into the VIP membership club, you receive additional orders of Varicose 911 at your listed rate every month until you cancel.
## Varicose 911 Refund Policy
Varicose 911 comes with a 90 day refund policy. You can request a complete refund within 90 days of your purchase.
Although we’re skeptical Varicose 911 removes varicose veins, you are entitled to a complete refund if Varicose 911 does not remove your varicose veins within 90 days.
The official refund period is 97 days from your original date of purchase (to accommodate for shipping).
## Who’s Behind Varicose 911?
Varicose 911 is made by a company named PhytAge Laboratories. The company makes a range of supplements targeting various health goals – including multiple supplements with ‘911’ in the title.
The company sells Eagle Eye 911 (for vision), Turbo Boost 911 (for circulation), Joint Relief 911 (for mobility), and Prostate 911 (for prostate and urinary health), among other supplements.
You can contact PhytAge Labs via the following:
Email: wecare@phytagesupport.com
Phone: 1-800-822-5753
Mailing Address: 37 Inverness Drive East, Suite 100, Englewood, CO 80112
## Final Word
Varicose 911 claims to eliminate varicose veins by supporting collagen production. By supporting collagen production in your body, Varicose 911 can purportedly eliminate the appearance of varicose veins quickly.
There’s no evidence Varicose 911 can eliminate the appearance of varicose veins or increase collagen production. However, some of the ingredients in Varicose 911 could provide small support for cardiovascular health, which could support the prevention of varicose veins.
Varicose 911 also comes with a 97 day refund policy. You are entitled to a complete refund if Varicose 911 does not completely eliminate your varicose veins within 97 days.
However, the formula is unlikely to solve serious varicose vein problems – and you should always follow your doctor’s advice when treating a varicose vein problem.
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SERVING THE QUANTITATIVE FINANCE COMMUNITY
pjakubenas
Topic Author
Posts: 50
Joined: March 26th, 2002, 10:14 am
### Soluton to a HW/HW/Black model
Hello, everybody,Maybe someone can help me with this...We have a following modelwith with $W_{d}$, $W_{f}$, and $W_{X}$ (in general) correlated Brownian motions.A question: has anybody solved/seen this type of equations?I mean r_d and r_f are easy (just Ornshtein-Uhenbeck), what about X?More specifically, I would like to find how the correlations between the underlyingBrownian motions are related to the correlations Any sugestions are welcome.Sincerely Yours,Paulius Jakubenas
DrEvil
Posts: 36
Joined: April 22nd, 2004, 10:17 pm
### Soluton to a HW/HW/Black model
Yes, triplets of equations like this are quite commonly used (numerically) by FX quants for all sorts of FX derivatives, since each equation is a "classical" Ito process for the modelled rate.If I understand your question about the correlation, the IR-IR correlation only has to make the matrix positive definite, and Peter Jaeckel's book "Monte Carlo Methods in Finance" gives ideas on this.You can PM me if you'd like some details on what I've done with these types of models.
spv205
Posts: 478
Joined: July 14th, 2002, 3:00 am
### Soluton to a HW/HW/Black model
I would look at interest rate models, theory and practise [Brigo and Mercurio]- chapter 12, pricing equity derivatives under stochastic ratesthere they go through the equations (with only one interest rate)
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## finance
You want to create a portfolio equally as risky as the market, and you have $500,000 to invest. Information about the possible investments is given below: Asset Investment Beta Stock A$135,451 0.84 Stock B $143,298 1.35 Stock C -- 1.43 Risk-free asset -- -- How much must you invest in Stock C? ## Answers (1) • ## You need a Homework Help subscription to view this answer! ### Chegg Homework Help subscribers get: • Full access to our library of 2 million Q&A posts • 24/7 help from our community of subject experts • Fast answers from experts, around the clock Start Free Trial... Your free trial lasts for 7 days. Your subscription will continue for$14.95/month unless you cancel.
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# Showing that Brownian motion is bounded with non-zero probability
How do you show, that for every bound $\epsilon$, there is a non-zero probability that the motion is bounded on a finite interval. i.e. $$\mathbb{P} (\sup_{t\in[0,1]} |B(t)| < \epsilon) > 0$$
I tried using the reflection principle, I can show that if B* is the motion reflected on the hitting time with the bound, $\tau$, then I think that $$P(\sup_{t\in[0,1]} |B(t)| > \epsilon) = P(|B(1)| > \epsilon) + P(|B*(1)| > \epsilon) - P(\tau < 1, |B(1)-B(\tau)| > 2\epsilon)$$ However, I have no idea how to bound the third term (in a useful way, which would show that this probability is non-one)
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@Nate, thanks, I've deleted my answer. My original idea was to use the fact that $B_t$ is bounded, hence the supremum should be also. Initial hand-waving in my head was convincing, so I posted the answer, without checking it thoroughly. I apologise for that. – mpiktas May 12 '11 at 17:46
Here's three different methods of showing that $\mathbb{P}(\sup_{t\in[0,1]}\vert B_t\vert < \epsilon)$ is nonzero.
A simple argument based on intuition. You can break the unit interval into a lot of small time steps and, by continuity, the Brownian motion will not move much across each of these steps. By independence of the increments, there is a positive (but small) probability that they largely cancel out, so $B$ stays within $\epsilon$ of the origin. To make this precise, choose a positive integer $n$ such that $q\equiv\mathbb{P}(\sup_{t\le1/n}\vert B_t\vert < \epsilon/2)$ is nonzero ($q$ can be made as close to 1 as you like, by taking $n$ large). By symmetry, the event $\{\sup_{t\le1/n}\vert B_t\vert < \epsilon/2,\ B_{1/n}>0\}$ has probability $q/2$. Note that, if $\sup_{t\in[k/n,(k+1)/n]}\vert B_t-B_{k/n}\vert < \epsilon/2$ and $B_{(k+1)/n}-B_{k/n}$ has the opposite sign to $B_{k/n}$ for each $k=0,1,\ldots,n-1$ then $\vert B_t\vert$ will be bounded by $\epsilon/2$ at the times $k/n$ and, therefore, $\sup_{t\le1}\vert B_t\vert$ will be less than $\epsilon$. So, $\mathbb{P}(\sup_{t\le1}\vert B_t\vert < \epsilon)\ge(q/2)^n$.
Use a cunning trick. If $X,Y$ are independent Brownian motions over the interval $[0,1]$, then $B=(X-Y)/\sqrt{2}$ is also a Brownian motion. The sample paths of $X,Y,B$ can be considered as lying in the (complete, separable) metric space $C([0,1])$ of continuous functions $[0,1]\to\mathbb{R}$ under the supremum norm. By separability, $C([0,1])$ can be covered by countably many open balls of radius $\epsilon/\sqrt{2}$. So, by countable additivity of the probability measure, there exists at least one such ball containing $X$ with probability $q > 0$. By independence, $X,Y$ are both contained in this ball with probability $q^2 > 0$, in which case $\Vert B\Vert_\infty=\Vert X- Y\Vert_\infty/\sqrt{2}<\epsilon$.
Exact calculation. You can calculate an exact expression for the probability, as an infinite sum, and verify that it is dominated by a single positive term as $\epsilon$ goes to zero. This is not as simple as the intuitive argument I gave above, but has the advantage that it also gives an accurate asymptotic expression for the probability, which goes to zero like $e^{-\pi^2/(8\epsilon^2)}$ as $\epsilon\to0$ (this is positive, but tends to zero very quickly).
The probability can be calculated using the reflection principal (also see my comments and Douglas Zare's answer to this question). Writing $p(x)=(2\pi)^{-1/2}e^{-x^2/2}$ for the probability density function of $B_1$ and $f(x)=\sum_{n=-\infty}^\infty(-1)^n1_{\{(2n-1)\epsilon < x < (2n+1)\epsilon\}}$ (which is a kind of square wave function), $$\mathbb{P}\left(\sup_{t\le1}\vert B_t\vert < \epsilon\right)=\mathbb{E}[f(B_1)]=\int_{-\infty}^\infty f(x)p(x)\,dx.\qquad{\rm(1)}$$ This expression comes from the reflection principle, which says that reflecting $B$ after it first hits $\pm\epsilon$ gives another Brownian motion. That is, $\hat B_t\equiv B_t+1_{\{t\ge T\}}2(B_T-B_t)$ is a Brownian motion, where $T$ is the first time at which $\vert B_T\vert=\epsilon$. As $f$ is antisymmetric about both $\epsilon$ and $-\epsilon$, the sum $f(B_1)+f(\hat B_1)$ vanishes whenever $T\le1$. So, $1_{\{T > 1\}}=(f(B_1)+f(\hat B_1))/2$, and taking the expectation gives (1).
You can perform the integral in (1) directly to express the probability as an infinite sum over the cumulative normal distribution function, but this is not so good in the limit where $\epsilon$ is small, as you don't have a single dominant term. Alternatively, the integral in (1) can be written as $\int_{-\epsilon}^\epsilon\theta(x)\,dx$ where $\theta(x)=\sum_{n=-\infty}^\infty(-1)^np(x+2n\epsilon)$. As $\theta$ has period $4\epsilon$ you can write it as a Fourier series, and working out the coefficients gives $$\theta(x)=\epsilon^{-1}\sum_{\substack{n > 0,\\n{\rm\ odd}}}\cos\left(\frac{n\pi x}{2\epsilon}\right)\exp\left(-\frac{n^2\pi^2}{8\epsilon^2}\right).$$ This is a very fast converging sum, especially for small $\epsilon$ (the terms vanish much faster then exponentially in $n$). Actually, $\theta$ is a theta function and the fourier transform is the same thing as the Jacobi identity. Integrating it term by term gives $$\mathbb{P}\left(\sup_{t\le1}\vert B_t\vert < \epsilon\right)=\sum_{\substack{n > 0,\\ n{\rm\ odd}}}\frac{4}{n\pi}(-1)^{(n-1)/2}\exp\left(-\frac{n^2\pi^2}{8\epsilon^2}\right)$$ As the first term goes to zero much more slowly than the sum of the remaining terms (as $\epsilon\to0$) this gives the asymptotic expression $$\mathbb{P}\left(\sup_{t\le1}\vert B_t\vert < \epsilon\right)\sim\frac{4}{\pi}\exp\left(-\frac{\pi^2}{8\epsilon^2}\right).$$
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Thanks. This is exactly the kind of answer I was looking for. – yaakov May 14 '11 at 20:05
This is very nice! – Nate Eldredge May 15 '11 at 1:15
It may be instructive to consider more general Levy processes. – Shai Covo May 15 '11 at 6:45
btw, I added another argument based on a cunning trick which just occured to me. – George Lowther May 15 '11 at 8:40
@Shai: The first two arguments also apply to any symmetric Lévy process, although they could be extended to more general cases (the result does not hold for every Lévy process, of course). – George Lowther May 15 '11 at 8:42
There must be a simple argument, but here's a fancy one, roughly following El Moro's idea.
$Y_t = \cos(\theta B_t) e^{t \theta^2/2}$ is a continuous martingale for any $\theta \in \mathbb{R}$. Also, $\tau = \inf \{ t : |B_t| = \epsilon\}$ is a stopping time, so $Y_{t \wedge \tau}$ is also a martingale, and in particular $$E[Y_{t \wedge \tau}] = E[Y_0] = 1 \quad (*)$$ for all $t$. If $P(\sup_{t \in [0,1]} |B_t| < \epsilon) = 0$, then $\tau \le 1$ almost surely, and taking $t=1$ in (*) gives $1 = E[Y_\tau] = \cos(\theta \epsilon) E[e^{\tau \theta^2/2}]$ which is absurd. (For example, take $\theta = \frac{\pi}{2\epsilon}$).
This is sort of a baby version of the optional stopping theorem.
Edit: Here is an even fancier argument, which I found in H. H. Kuo's Gaussian Measures on Banach Spaces.
Let $W = \{ \omega \in C([0,1]) : \omega(0) = 0 \}$ be the classical Wiener space, equipped with the sup norm, and let $\mu$ be Wiener measure. We are trying to show that for any $\epsilon > 0$ we have $\mu(B(0,\epsilon))>0$. As in George's "cunning trick", $W$ is separable, so it can be covered by a countable number of balls of radius $\epsilon/2$. By countable additivity, one of these balls, call it $B(\omega_0, \epsilon/2)$, has positive $\mu$ measure.
Let $H \subset W$ be the Cameron-Martin space of paths with one square-integrable weak derivative. $H$ is dense in $W$, so there exists $h \in H \cap B(\omega_0, \epsilon/2)$. By the Cameron-Martin theorem $\mu$ is quasi-invariant under translation by $H$, i.e. $\mu \ll \mu(\cdot - h)$. Thus $\mu(B(\omega_0 - h, \epsilon/2)) > 0$. By the triangle inequality $B(\omega_0 - h, \epsilon/2) \subset B(0, \epsilon)$ so this completes the proof.
Note this shows that in any abstract Wiener space $(W, H, \mu)$, the measure $\mu$ charges all open sets of $W$, or in other words $\mu$ has full support.
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This is an elegant answer. However, I will wait to see if anyone gives one from elementary principles. – yaakov May 12 '11 at 22:43
Let $M=\sup_{t\in[0,1]}|B_t|$ and $T=\inf\{t\ge0;|B_t|=1\}$. By scaling, $M$ and $1/\sqrt{T}$ coincide in law hence $P(M\le\varepsilon)=0$ would imply that the distribution of $T$ is unbounded.
Considering the martingale $\cosh(uB_t)\mathrm{e}^{-u^2t/2}$ and applying the stopping time theorem at time $T$, one shows that the Laplace transform of $T$ is $$E(\mathrm{e}^{-u^2T/2})=1/\cosh(u).$$ This is an equality between two series expansions hence $$E(\mathrm{e}^{u^2T/2})=1/\cos(u),$$ for every real number $u$ such that $u^2$ is small enough. When $u\to\pi/2$, $\cos(u)\to0$ hence $\mathrm{e}^{\pi^2T/8}$ is not integrable. This implies that the distribution of $T$ is unbounded, hence the event $[M\le\varepsilon]$ has positive probability for every positive $\varepsilon$.
One source among many is the book Mathematical Methods for Financial Markets by Monique Jeanblanc, Marc Yor and Marc Chesney (chapter 3).
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I have a rough Idea if you define Tepsilon to be the stoping time at epsilon and -epsilon then your equality becomes P(Tepsilon >= 1) > 0.
if you suppose that P(Tepsilon >= 1)=0 that is equivalent to say that Tepsilon is a.s bounded the contradiction would raise if you compute laplace transform of Tepsilon and finds it to be well defined but I am not sure if that is really the case.
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How would you, without knowing the distribution of Tepsilon (otherwise, there is no point...) compute the transform? – yaakov May 12 '11 at 17:42
The Laplace transform is well-known for this problem. It is $\mathbb{E} \exp(-\lambda \tau_{\varepsilon}) = 1 / \cosh(\sqrt{2\lambda}\varepsilon)$. See Ch. 6 of S. Resnick, 1992 or Ch. 7 of Durrett, 3rd. ed., 2004 or here. – cardinal May 12 '11 at 18:26
It is possible to obtain the distribution of Tepsilon. But you'd better use Girsanov or a stopped exponential martingales – El Moro May 12 '11 at 20:32
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Group Law on the (Punctured) Affine Line
There are likely inaccuracies in this post, as I am just beginning to learn the basics of algebraic geometry. Constructive criticism is strongly encouraged.
There once was a line…
Let’s look at the affine line over $\mathbb{C}$. This is just the complex line with no distinguished element (i.e., a plane which forgot it’s origin — it’s 2 real dimensions, or, equivalently, 1 complex dimension).
$\mathbb{A}^1 \simeq \text{Spec } \mathbb{C}[x]$
As we know, the affine line over the field $\mathbb{K}$ is isomorphic to the spectrum of a ring of single variable polynomials (with coefficients in $\mathbb{K}$). If you aren’t familiar with this isomorphism, I recommend popping over to Spectrum of a Ring. For simplicity, let’s work with the field $\mathbb{C}$, although, I’m pretty sure the rest of this post still works for any $\mathbb{K}$.
Is there a reasonable way to take in two points, and ask for a third?
This is generally a fun question to answer when you’re handed a space. So, how do we add two points of $\mathbb{A}^1$ to get a third point in $\mathbb{A}^1$?
The same way we usually add two complex numbers! But wait, we need an identity! Where does the identity come from? I think it comes from looking at the affine plane as $\text{Spec }\mathbb{C}[x]$, the maximal ideal $x$ is a distinguished point.
It’s worth considering why $\text{Spec }\mathbb{C}[x] \times\text{Spec }\mathbb{C}[y] \simeq \text{Spec }\mathbb{C}[x,y]$. It seems that this is because $\text{Spec }$ is functorial, that is:
$\text{Spec }\mathbb{C}[x, y] = \text{Spec }(\mathbb{C}[x] \otimes {C}[y]) = \text{Spec }\mathbb{C}[x] \times \text{Spec }\mathbb{C}[y]$.
Another way to think about this is that the cartesian product $\mathbb{A}^1 \times \mathbb{A}^1= \mathbb{A}^2 = \text{Spec }\mathbb{C}[x,y]$.
$\mathbb{A}^1 – \{0\} \simeq \text{Spec } \mathbb{C}[x, x^{-1}]$
Can we analogously put a group structure on the punctured affine line? Hell yes we can, here’s one way to do so:
Recall that $\text{Spec }\mathbb{C}[x, x^{-1}] \times \text{Spec }\mathbb{C}[y, y^{-1} =\text{Spec }\mathbb{C}[x, y, x^{-1}, y^{-1}]$.
$\text{Spec }$ is a contravariant functor, so we get a corresponding diagram in $\text{Ring}$:
The explicit maps that define this diagram reveal the desired group law:
That is, $z \mapsto (x+1)(y+1) -1 = xy + x + y$, which is a form of the multiplicative formal group law that we know and love. Intuitively, we can think of it as wanting to multiply $a$ and $b$, but first needing to shift to the multiplicative identity $1$, and afterward translating back again.
$\mathbb{A}^1 – \{0\} – \{1\} \simeq \text{Spec } \mathbb{C}[x, x^{-1}]$
What I don’t yet see, dear reader, is how to put group structure on $\text{Spec } \mathbb{C}[x, x^{-1}, (1-x)^{-1}]$ (which is isomorphic to the affine line minus two points). What are your thoughts?
Thanks to Yifei Zhao for kindly helping me derive the multiplicative formal group law from the group law on the punctured affine line.
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Let’s talk about two popular measures of the spread of a set of numbers. Take a very simple collection that’s easy to see - the integers between 1 and 5.
x = 1:5
x
## [1] 1 2 3 4 5
The natural starting point for talking about spread is to calculate the distance between each individual number and the center of the set. I’ll use the mean value as the center. Then I’ll subtract the mean from each number to get the “deviations from the mean.”
m = mean(x)
m
## [1] 3
xdev = x - m
xdev
## [1] -2 -1 0 1 2
Note that 5 and 1 are both two units away from the mean, but the deviation of 5 is positive and the deviation of 1 is negative.
The natural way to summarize the deviations is to calculate their average value, the mean of the deviations. Let’s do this and look at it.
md = mean(xdev)
md
## [1] 0
Why do we get zero? The positive and negative deviation values cancel each other out. I could prove mathematically that this will happen with any set of numbers. It’s a theoretical property of the mean. What we need to do is remove the signs from the deviations. We can use the absolute value function, abs(), in R to do this.
ad = abs(xdev)
ad
## [1] 2 1 0 1 2
Now take the mean of the absolute values of the deviations to get something we can call the mad (mean absolute deviation).
mad = mean(ad)
mad
## [1] 1.2
That’s a reasonable measure of spread and we can use it in a sentence. The average number in the set is 1.2 units away from the mean value of the set.
Now let’s talk about the standard deviation. The difference is the way we solve the cancelling out problem. Instead of taking the absolute values of the deviations, we square the deviations.
xdvesq = xdev^2
xdvesq
## [1] 4 1 0 1 4
That solved the problem with the negative and positive deviations. The average of these squared deviations is known as the variance.
variance = mean(xdvesq)
variance
## [1] 2
How would you use this number in a sentence that anyone could understand? We certainly would not say that the variance is typical of what you would expect as the value of a deviation. The problem is that squaring inflated the deviation values of 1 and 5.
To compensate for this, we take the square root of the variance to get us back to the scale of typical numbers in our collection. This is known as the standard deviation.
standardDeviation = sqrt(variance)
standardDeviation
## [1] 1.414214
That’s a reasonable value, close to the value of the mean absolute deviation (MAD). What does it mean? Most people say that it is a typical value of a deviation. This is vague, not as clear as the MAD.
Why do statisticians do this? You’ll rarerly hear any of them using the MAD. The standard deviation and/or the variance are preferred because of theories related to the normal distribution based on these concepts.
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# Belle II
The Belle II detector
Event display with reconstructed tracks
## Project
The Belle II experiment is an upgrade of the B factory experiment Belle at the KEK laboratory in Tsukuba, Japan. It should record a 50 times larger data sample than the Belle experiment until 2022. With this one hopes to explain, among other questions, why the universe today consists only of matter and no anti-matter. Currently the detector and accelerator are upgraded. The expected large data sample also requires a new computing model to provide sufficient computing power and storage capacity for the processing and analysis of the data.
## Contribution of the ETP
The ETP is engaged in the Belle II project since 2008 on the fields of software development, setup of the computing infrastructure, and construction of the pixel vertex detector. In particular we work as a leading institute on the track reconstruction in the central drift chamber, the software framework, the code management, and the event display. Further activities are the development of a database for calibration constants and the monitoring of grid sites.
Autor bzw. Beitrag des EKP Typ Titel Nummer und Download Baur, Anselm Bachelorarbeit ZMQ Implementation for a Reliable Parallel Processing Framework on the High Level Trigger at the Belle II Experiment ETP-BACHELOR-KA/2018-11 Ecker, Patrick Bachelorarbeit Study of Bremsstrahlung Finding at the Belle II Experiment ETP-BACHELOR-KA/2018-1 Eliachevitch, Michael Masterarbeit Tracking Performance Studies on Cosmic Rays and a Track Quality Estimation for the Central Drift Chamber of the Belle II-Experiment ETP-KA/2018-16 Gelb, Moritz Doktorarbeit Search for the Rare Decay $\mathrm{B}^+ \to \ell^+ \nu_{\ell} \gamma$ with the Full Event Interpretation at the Belle Experiment ETP-KA/2018-14 Racs, Sebastian Masterarbeit Track Quality Estimation and Defect Sensor Studies for the Silicon Vertex Detector of the Belle II-Experiment ETP-KA/2018-15 Keck, Thomas Doktorarbeit Machine learning algorithms for the Belle II experiment and their validation on Belle data ETP-KA/2017-31 Rupp, Joachim Bachelorarbeit Track Fitting Studies with multiple Hypotheses IEKP-BACHELOR-KA/2017-22 Schwab, Judith Masterarbeit Calibration of the Full Event Interpretation for the Belle and the Belle II Experiment ETP-KA/2017-27 Wagner, Jonas Masterarbeit Track Finding with the Silicon Strip Detector of the Belle II Experiment ETP-KA/2017-28 Weyland, Dennis Masterarbeit Continuum Suppression with Deep Learning techniques for the Belle II Experiment ETP-KA/2017-30
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# Linear Attention Computation in Nyströmformer
In this post, we will take a look at Nyström approximation, a technique that I came across in Nyströmformer: A Nyström-based Algorithm for Approximating Self-Attention by Xiong et al. This is yet another interesting paper that seeks to make the self-attention algorithm more efficient down to linear runtime. While there are many intricacies to the Nyström method, the goal of this post is to provide a high level intuition of how the method can be used to approximate large matrices, and how this method was used in the aforementioned paper.
# Concept
Despite its fancy and somewhat intimidating name, the Nyström method has an intuitive explanation. The idea is that, if we know the distance between point A and point B, as well as that between point B and point C, then we can approximate the distance between points A and C as some sort of addition of the two quantities. Of course, if we were discussing distances in the context of one-dimensional space, namely the real number line, we would not only be able to approximating the distance; we would know the exact quantity. However, in high-dimensional space, this is somewhat more difficult, and we can only resort to approximations.
To put things into context, let’s say we want to approximate the attention matrix in the transformer architecture. The Nyström method begins by selecting what the authors of the paper refer to as landmarks. Basically, if we have an attention matrix $A \in \mathbb{R}^{L \times L}$, then we select a few landmark rows and columns to use as the basis or pivot point for our approximation. The goal, then, is to select as few landmarks as possible while being able to approximate the attention matrix as accurately as possible.
For sake of simplicity, let’s say we select the first row and column to be our landmarks. Then, the goal is to approximate the inner sub-matrix $A_\text{sub} \in \mathbb{R}^{(L - 1) \times (L - 1)}$. How might we go about it?
As stated earlier, the intuition is that we use the landmarks as pivot points. Since we selected the first rows and columns as our landmarks, we have access to $q_1 k_n^\top \forall n \leq L$, as well as $q_n k_1\top \forall n \leq L$ (for simplicity, we ignore the normalizing square root). If we remind ourselves of the motivation behind the transformer’s key-value-query architecture, we can consider attention as a way of calculating the distance or relevance between pairs of tokens in a given sequence. Put differently, the landmarks tell us the distance between the first query and all other keys, as well as the distance between the first key and all other queries.
Without loss of generality, we can approximate the distance between any $i$th key and the $j$th query using these landmarks. The way we do this is somewhat similar to the point A, B, C example we briefly discussed earlier. Namely, we start by looking at the distance between the $i$th key and the first query. Then, we also look at the attention value between the first key and the $j$th query. Note that connecting the two dots kind of gives us a sense of how related the $i$th query and $j$ query are. To remove the redundancy, we divide the product by the self-attention of the first token, or the attention score between the first key and query.
$A_{ij} = \frac{q_i k_1^\top \cdot q_1 k_j^\top}{q_1 k_1^\top} \tag{1}$
Of course, if we have multiple landmarks, we can easily expand the expression above into matrix form. The tilde indicates landmark rows and columns.
$\tilde{A} = Q \tilde{K}^\top \times (\tilde{Q} \tilde{K}^\top)^\star \times \tilde{Q} K \tag{2}$
The star expression ($\star$) denotes the Moore-Penrose pseudo-inverse.
Now that we have a general intuition of how Nyström approximation works in the context of attention, let’s get into some basic implementation.
# Implementation
The goal here is to see that Nyström approximation can indeed yield reasonably accurate results, and that the larger the number of key landmarks, the better the approximation. Consider this as a form of Monte Carlo experiment.
Let’s begin by importing some modules.
import numpy as np
import matplotlib.pyplot as plt
%config InlineBackend.figure_format="retina"
For sake of simplicity, we assume a very basic model with a hidden dimension of 2, and some data points whose sequence length is 5. For simplicity, we omit the batch dimension.
Then, in the context of attention, we would end up with the following keys and query tensors, as well as a five-by-five square attention matrix.
d_model = 2
seq_len = 5
Q = np.random.randn(seq_len, d_model)
K = np.random.randn(seq_len, d_model)
A = Q @ K.T
A.shape
(5, 5)
The goal, then, is to approximate this square attention matrix.
A
array([[ 2.29571874, -0.7373519 , 0.32730778, -0.84730782, -1.16558083],
[ 1.4346883 , -0.32765206, 0.80095764, -0.39437617, 0.17889744],
[ 1.38973136, -0.61066937, -0.53783773, -0.67968999, -1.82523199],
[-1.80977456, 0.1036656 , -2.39735444, 0.18320197, -2.33569844],
[ 1.36516091, -0.40695455, 0.33580143, -0.47186895, -0.47836287]])
Let’s begin our approximation by assuming the worst case, in which we only have access to one landmark. This brings us to equation (1) where essentially all operations were done on vectors instead of matrices.
num_landmarks = 1
Q_tilde = Q[:num_landmarks]
K_tilde = K[:num_landmarks]
Recalling equations (1) and (2), we can now write the approximation of the attention matrix as follows.
$\tilde{A} = Q \tilde{K}^\top \times (\tilde{Q} \tilde{K}^\top)^\star \times \tilde{Q} K$
A_tilde = (Q @ K_tilde.T) @ np.linalg.pinv(Q_tilde @ K_tilde.T) @ (Q_tilde @ K.T)
A_tilde.shape
(5, 5)
The dimensionality seems to match that of the original attention matrix, as expected. If we print out the approximation, we should expect to see exact matches in the first row and column; the rest of the four-by-four region of the matrix should roughly be similar to that of the original.
A_tilde
array([[ 2.29571874, -0.7373519 , 0.32730778, -0.84730782, -1.16558083],
[ 1.4346883 , -0.46080128, 0.20454799, -0.52951722, -0.72841901],
[ 1.38973136, -0.44636176, 0.19813834, -0.51292444, -0.7055935 ],
[-1.80977456, 0.58127361, -0.25802521, 0.66795471, 0.91885757],
[ 1.36516091, -0.43847008, 0.19463525, -0.50385594, -0.69311861]])
We can indeed quickly verify that the first row and column are exact matches; however, the rest of the 16 elements are somewhat difficult to compare. We can more systematically calculate the difference between two matrices by using a norm, such as the Frobenius norm.
np.linalg.norm(A - A_tilde)
4.33185890598477
If we look at the raw value of the subtraction, we can see that the approximation isn’t too bad.
A - A_tilde
array([[ 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
0.00000000e+00, 0.00000000e+00],
[-2.22044605e-16, 1.33149223e-01, 5.96409654e-01,
1.35141056e-01, 9.07316456e-01],
[ 0.00000000e+00, -1.64307605e-01, -7.35976069e-01,
-1.66765549e-01, -1.11963848e+00],
[ 0.00000000e+00, -4.77608006e-01, -2.13932924e+00,
-4.84752738e-01, -3.25455600e+00],
[ 0.00000000e+00, 3.15155316e-02, 1.41166181e-01,
3.19869853e-02, 2.14755744e-01]])
## Monte Carlo Approach
Let’s extend this little trial with one landmark to larger matrices. For ease of execution and implementation, I’ve basically wrapped each step outlined above as functions.
The first function, norms_by_landmarks, receives query and key matrices, then approximates the attention matrix while varying the number of landmarks. The Frobenius norm is used to measure how good the approximation is. Theoretically, we should expect to see a downward-sloping pattern.
def norms_by_landmarks(Q, K):
result = []
A = Q @ K.T
for num_landmarks in range(1, len(Q) + 1):
Q_tilde = Q[:num_landmarks]
K_tilde = K[:num_landmarks]
A_tilde = (Q @ K_tilde.T) @ np.linalg.pinv(Q_tilde @ K_tilde.T) @ (Q_tilde @ K.T)
result.append(np.linalg.norm(A - A_tilde))
return np.asarray(result)
The second function, run_experiment, is a wrapper around the first one. It repeatedly conducts the same experiment for a specified number of iterations. The purpose of repetition is essentially remove the possibility of luck, where some randomly initialized key and query matrices are configured in such a way that the Nyström approximation performs too well or poorly on a given task. By repeating the experiment and averaging the result—which is the spirit behind Monte Carlo approximations—we can have more confidence in our final result.
def run_experiments(d_model, seq_len, num_iter=10):
result = 0
for _ in range(num_iter):
Q = np.random.randn(seq_len, d_model)
K = np.random.randn(seq_len, d_model)
norm = norms_by_landmarks(Q, K)
result += norm
return result / num_iter
Here, we assume a sequence length of 50, and the hidden size of the model (or the embedding size) to be 10. And off we go!
norms = run_experiments(d_model=10, seq_len=50)
plt.plot(range(len(norms)), norms)
plt.show()
# Conclusion
While there is some noise in the final outcome, we do see that beyond a certain dimension, the approximation yields near exact results. In this case, it seems to happen around 10 landmarks.
Transformers have now taken over much of the ML world, even beyond NLP. Recently, I came across a paper titled Pretrained Transformers are Universal Computation Engines. Apparently, pretrained transformer LMs can perform extremely well on tasks with minimal fine-tuning. Specifically, even if the feedforward and attention portion of the network frozen—which amounts to nearly 99 percent of the entire model architecture—transformer LMs can be micro-tuned to a wide array of tasks that are even not specifically NLP-related.
While there is certainly a possibility that a new SOTA model architecture will be announced by researchers in the new future, similar to how transformers made LSTMs obsolete in many fields, I think transformers are here to stay around for longer. And it’s certainly interesting to see attempts to make it even better, lighter, and faster. Nyströmformer was one such attempt, and I hope to see more.
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# $A\subset\mathbb{R^n}$ is compact and $f:A\rightarrow\mathbb{R}$ is continuous. Show $f$ is uniformly continuous.
Since $A$ is compact and $f$ is continuous, then $f(A)$ is compact. Suppose $f$ is not uniformly continuous. Then $\exists \epsilon >0.\forall \delta >0. \exists x,y\in A$ such that $d(x,y)<\delta$ and $d(f(x),f(y))>\epsilon$. Therefore we can choose some $x$ such that, for some specific choice of $\epsilon$, we have $d(x,y)<\delta$ and $d(f(x),f(y))>\epsilon$ for all $\delta >0$ and $y\in A$. Denote that $x$ as $x_0$
Let ${x_k}$ be a sequence with a subsequence converging to $x_0$ (that at least one such sequence exists is clear). Call this subsequence ${x_n}$. Since $f$ is continuous, then as $x_n \rightarrow x_0$ we have $f(x_n)\rightarrow f(x_0)$. Therefore, for every $\epsilon >0$ we can find some $\delta >0$ such that there exists some element $x_i$ of the sequence ${x_n}$ such that $d(x_i,d_0)<\delta$ and $df((x_i),f(x_0))$, in contradiction to our assumption that $f$ is not uniformly continuous.
Therefore, $f$ is uniformly continuous.
I'm not sure if this is a good proof, or if there are holes in it. I struggle quite a bit with uniform continuity; I don't really have a good handle on it. Please let me know if you think the proof is good, or how I can improve it.
Thanks.
• It's strange to take an $\varepsilon$ s.t. $d(x,y)<\delta\implies d(f(x),f(y))>\varepsilon$. It's $\varepsilon>0$ such that $d(x,y)<\delta$ and $d(f(x),f(y))>\varepsilon$ (for certain $x,y\in A$). – Surb Nov 11 '15 at 16:11
• I was trying to take the logical negation of the statement of uniform continuity but it looks like I messed it up a little. – Zelzy Nov 11 '15 at 16:27
• Yes, there is a sequence converging to $x_0$. But for all you know, this sequence may be constant. – egreg Nov 11 '15 at 17:46
Suppose has given $\epsilon > 0$ has given. for $x\in X$ since $f$ is continous at $x$, Let $\delta_x$ $$\exists \delta_x > 0 ~~\forall y \in \mathbb R: |x-y|< \delta_x \rightarrow |f(x)-f(y)| < \frac{\epsilon}{2}$$ we have $X\subseteq \bigcup_{x\in X} B(x,\frac{\delta_{x}}2)$ by compactness of $X$ exists $N$ such that $X\subseteq \bigcup_{i}^N$ $B(x_i,\frac{\delta_{x_i}}2)$. Let $\delta = \min_{i=1,\cdots,N} \frac {\delta_{x_i}} 2$
consider $x, y\in X$ such that $|x-y|<\delta$, there are $1\leq i, j\leq N$ such that $x\in,B(x_i,\frac{\delta_{x_i}}2)$ and $y\in B(x_j,\frac{\delta_{x_j}}2)$ thus $$d(x_i, y)\leq d(x_i, x)+d(x,y)\leq \frac {\delta_{x_i}} 2+\delta\leq \frac {\delta_{x_i}} 2+\frac {\delta_{x_i}} 2=\delta_{x_i}$$ thus $$|f(x)-f(y)|\leq |f(x)-f(x_i)|+|f(x_i)-f(y)|<\frac{\epsilon}{2}+\frac{\epsilon}{2}=\epsilon$$
• Could you détails your proof ? I don't think that the conclusion of uniform continuous is that obvious... – Surb Nov 11 '15 at 16:49
• Ok, I see now. You proved in fact that if $x,y$ are such that $|x-y|<\delta$ then they are in the same ball. So now it's correct. – Surb Nov 11 '15 at 17:54
• So, the proof ultimately relies on the continuity of $f$ to choose small enough $\delta_x$ and the compactness of $A$ to choose a finite number of these sets and use the triangle inequality to show that the distance between any two points, if less than $\delta$, must be contained within one of the $\delta_x$ balls we have defined above, and therefore we have the desired $f(x)-f(y)<\epsilon$. I think I've got it. Thanks. – Zelzy Nov 11 '15 at 19:30
• you are welcome :) – R.N Nov 11 '15 at 19:41
let $\varepsilon>0$ s.t. $$\forall n\in\mathbb N, \exists x_n,y_n\in A: d(x_n,y_n)<\frac{1}{n}\quad\text{and}\quad d(f(x_n),f(y_n))>\varepsilon.$$
By Heine-Borel, $A$ is closed and bounded (since $\mathbb R^n$ is metrizable). Then, by Bolzano-weierstrass $(x_n)$ has a subsequence $(x_{n_k})$ that converge, and since $A$ is closed, it's limit is in $A$. Let $\ell$ it's limit. Since $$d(y_{n_k},\ell)\leq d(x_{n_k},y_{n_k})+d(x_{n_k},\ell)\underset{k\to\infty }{\longrightarrow }0$$ and thus $(y_{n_k})$ converge also to $\ell$. By continuity, you have that $$\lim_{k\to\infty }d(f(x_{n_k}),f(y_{n_k}))=0$$ and this is the contradiction with the fact that $d(f(x_{n_k}),f(x_{n_k}))>\varepsilon$ for all $k$.
To be a more explicit
By continuity, there is a $N$ such that $$d(f(x_{n_K}),\ell)<\frac{\varepsilon}{2}\quad\text{and}\quad d(f(y_{n_N}),\ell)<\frac{\varepsilon}{2}$$
and thus $$d(f(x_{n_N}),f(y_{n_N}))<d(f(x_{n_N}),f(\ell))+d(f(y_{n_N}),f(\ell))$$
and thus, you have that $$d(f(x_{n_N}),f(y_{n_N}))<\varepsilon$$ and here is the contradiction.
• Sorry, i wasn't saying that if $x_n$ is a sequence then it is clear it has a subsequence that converges to $x_0$. I was saying that it is clear that there exists some sequence $x_n$ with a subsequence converging to $x_0$ so we could look generally at sequences with that property. But I guess that's not good enough. – Zelzy Nov 11 '15 at 16:26
• If $x_n$ is unspecified, you can't do a lot of things... But can you continue with my hint or not really ? – Surb Nov 11 '15 at 16:29
• But isn't it enough to show that, if at least one sequence with a subsequence converging to $x_0$ gives you a contradiction, then we have a contradiction in general? – Zelzy Nov 11 '15 at 16:31
• First, why is there a sequence with a subsequence converging ? For example, if you are in $\mathbb Q$, $x_n=(1+\frac{1}{n})^n$ has no subsequence that converge (and $x_n$ is bounded, increasing...). In $\mathbb R$, $x_n=n$ has no subsequence that converge (but this one is not bounded). But let suppose it. You took $x,y\in A$ such that $d(x,y)<\delta$ and $d(f(x),f(y))> \varepsilon$. Notice that $x$ and $y$ are specified. By convergence ox $x_n$, there is a $N$ s.t. $d(x_n,x_0)<\delta$ if $n>N$. But now, why $|f(x_n)-f(x_0)|>\varepsilon$ ? $x_n$ and $x_0$ are neither $x$ nor $y$, – Surb Nov 11 '15 at 16:37
• Still not working. I have $d(f(x_n),f(y_n))>\varepsilon$ not $d(f(x_n),f(x_0))>\varepsilon$. So it doesn't work ! – Surb Nov 11 '15 at 16:45
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# Handle Dynamic Currency Conversion
## Implement the DCC callback
Implement this tender_dcc_CB and assign to the function pointer status_tender_DCC. For more information on how to do this, see Implement callbacks with register_device_request.
### Code example
void tender_dcc_CB(response_header*, ped_device_info*, void * echo_struct)
## Invoke Dynamic Currency Conversion callback
Set up AskDCC in merchant account settings to invoke the DCC callback. Contact POS Support Team to enable this setting.
## Handle the DCC callback
The tender_dcc_CB callback presents a currency conversion option to the shopper. This allows the POS to indicate this on the POS screen and to provide the details of the currency conversion option also to the shop agent.
### Parameters
Name Description
ped_device_info See ped_device_info for a description of the ped_device_info struct.
response_header See response_header for a description of the response_header. response_header includes additional_data_struct and its returned data elements. indicates the progress state of the transaction in the state data element, which is an enum of type TENDER_STATE. In this case it will be ASK_DCC.
## DCC Returned Details
The DCC details contain the following keys:
Fields Description
dcc.markup The markup percentage (for example, 300 = 3%).
dcc.commissionfee The commission fee in minor units (for example, 0. Not used currently.).
dcc.exchangerate The exchange rate (for example, 16178 = 1.6178).
dcc.converted.amount.value The amount after conversion (for example, minor units).
dcc.converted.amount.currency The currency after conversion (For example, USD).
dcc.org.amount.value The original amount in minor units.
dcc.org.amount.currency The original currency (for example, EUR).
dcc.source The exchange rate source (for example, oanda).
## Helper functions
A function is available to get additional data values from the additional_data struct. See get_additional_data_value in additional_data.h.
The SDK package contains a folder documentation/log_files/documentation. This folder contains log files of actual test runs that relate to the items discussed in this manual.
The dcc_callback.log file that shows a live example of the register device call and the related callback.
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# Tag Info
You're using bytewise xor, and three plaintexts $x,y,z$ with a small alphabet (A-Z, which I assume means ASCII values 0x41-0x5a). Then knowing $a \oplus x$, at a certain index, we know all possible values of $a$, at that same index, namely $\{0x41 \oplus (a \oplus x),\ldots, 0x5a \oplus (a \oplus x)\}$. But we have 2 more constraints from $a \oplus y$ and \$a ...
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# Is there a generic name for the “layered” dictionary typically used in mod-capable games?
I'm asking here rather than on SO/SE/CS because I'll probably have less to explain what I'm talking about...
I hope a lot of the users here are familiar with how many games implement modding (or sometimes even their own add-ons, levels etc.) Conceptually, what I've seen most often is a data structure that acts like a layered dictionary. With no mods, there's lookup in the base dictionary layer; whenever a mod is added/enabled, it comes with its own dictionary layer from which lookups are served first, with fallback to the underlying layers if a key is not found. (Note that I'm not talking about a nested dictionary.)
So, my question is simply if there's a generic name for this data structure. (Googling for "layered dictionary", which is how I dubbed it, doesn't turn up anything useful...) In the world of file systems, the closest thing that probably resembles it is a union mount. Alas "union dictionary" doesn't get any useful results either... (Also the word "union" is rather misleading in this context because there's an order in the layers of a "union mount".)
Added: By the way, I'm not asing how to implement it, as that's rather obvious from the description. Perhaps that's why no name have been devised for it... Also most game engines I've seen implement multi-layer (not just two) and some allow the (top) layers to be removed at any time so you can do (game) "levels" that way; others require an engine restart so it's only usable for "global mods", i.e. they optimize lookup time by "flattening" the layers down to one as they are added, which is basically why they need an "engine restart" to remove a layer.
Also, it seems the R (statistical) programming language basically has this its get functions:
These functions look to see if each of the name(s) x have a value bound to it in the specified environment. If inherits is TRUE and a value is not found for x in the specified environment, the enclosing frames of the environment are searched until the name x is encountered.
So I guess something like "environments and their enclosures" is one term...
• Reminds me of MultiValueDictionary. But if you ask me, this sounds like something you can do by checking your dictionary, and if it has no key assigned, resort to a second dictionary which you check after that. If you use it multiple times, you could create your own LayeredDictionary<TKey, TValue> class. Now there is nuance - it's either a dictionary with variable layers (which does not guarantee a default value) or a dictionary with a default value (2 layers) or both. That depends on you. – Battle Mar 5 '20 at 12:00
• @Battle: I'm not asing how to implement it, as that's rather obvious from the description. Perhaps that's why no name have been devised for it... Also most game engines I've seen implement multi-layer (not just two) and some allow the (top) layers to be removed at any time so you can do (game) "levels" that way; others require an engine restart so it's only usable for "global mods", i.e. they optimize the whole thing by "flattening" the layers down to one as they are added, which is basically why they need an "engine restart" to remove a layer. – Fizz Mar 5 '20 at 14:15
The most generic concept seems to be variously called a spagetti/cactus stack or in-/parent-pointer tree of symbol tables.
Now the first part of this "thing" is also generic, in the sense that the "nodes" can be other things that have a notion of look-up. Come to think of it, "union mounts" and how they're implemented in some game engines actually cactus-stack nested dictionaries, in the sense that they both stack layers and allow path-like ("segmented") keys.
N.B. I've not seen a game engine that allows forks in its look-up structures, i.e. in the full sense of a cactus stack. I've only seen linear stacks of nested dictionaries as the most complex look-up structure actually used for modding/levels support.
Steve Yegge calls this the Properties Pattern. It's a pattern that shows up in so many places that we may not even notice it. He's using it in his game Wyvern.
It shows up in scopes:
int x = 5, y = 7;
if (…) {
int x = 3;
/* x is 3 here! but y is 7 */
}
Here the inner scope has {x: 3} and the outer scope has {x: 5, y: 7}. Any name not found in the inner scope will look in the next outer scope, and this will chain all the way up to the global level.
Class inheritance works this way. If a class Base defines methods a and b, and a subclass Derived defines a, it will find Derived.a when looking for a, and Base.b when looking for b. In Self, LambdaMOO, and JavaScript, object inheritance works this way.
SVG attributes work this way. If you have a <g fill="red" stroke="blue"> and then a <rect fill="green"> inside of it, it will use green for the fill and blue for the stroke. The OS/2 operating system had properties defined this way. I'm sure there are plenty more examples.
Each level has a key-value mapping, plus a "parent" or "inherit" or "default" pointer to another such mapping, which can itself have another pointer, etc. Typically writes go to the current mapping, but reads can go up the chain.
I've been looking for a more authoritative name for this pattern than Steve Yegge's blog post, but that's the best I've found so far.
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# HTTP Heuristic Caching (Missing Cache-Control and Expires Headers) Explained
Have you ever wondered why WebPageTest can sometimes show that a repeat view loaded with less bytes downloaded, while also triggering warnings related to browser caching? It can seem like the test is reporting an issue that does not exist, but in fact it’s often a sign of a more serious issue that should be investigated. Often the issue is not the lack of caching, but rather lack of control over how your content is cached.
If you have not run into this issue before, then examine the screenshot below to see an example:
The repeat view shows that we loaded resources from the browser cache, and that seems like a desirable outcome. However the results are flagging a cache issue. And the site received a poor PageSpeed score for “Caching Static Content”.
Before we dig into the reasons behind this, let’s step back and review some fundamentals…
HTTP Caching Simplified
For an HTTP client to cache a resource, it needs to understand 2 pieces of information:
• “How long am I allowed to cache this for?”
• “How do I validate that the content is still fresh?”
RFC 7234 covers this in section 4.2 (Freshness) and 4.3 (Validation).
The HTTP Response headers that are typically used for conveying freshness lifetime are :
• Cache-Control (max-age provides a cache lifetime duration)
• Expires (provides an expiration date, Cache-Control max-age takes priority if both are present)
The HTTP response headers for validating the responses stored within the cache, i.e. giving conditional requests something to compare to on the server side, are:
• Etag (provides a unique identifier for the content)
The Cache-Control max-age and Expires headers satisfy the job of setting freshness lifetimes (ie, TTLs) as well as provide a time based validation. You can think of them as saying “this resource is valid for x seconds” or “cache this resource until <date/time>”. When both are present in a response, the browser will prioritize the Cache-Control over the Expires header. When the expiration passes, then the browser can send a conditional request using an If-Modified-Since header to ask if the object has been modified (since it’s Last-Modified-Date or from the last resource version the browser received). If neither a Cache-Control or Expires header is present, then the specification allows the browser to heuristically assign it’s own freshness time. It suggests that this heuristic can be based on the Last-Modified-Date.
Etags are useful for providing unique identifiers to a cached item, but they do not specify cache lifetimes. They simply provide a way of validating the resource via a If-None-Match conditional request.
Example
If we examine a request where this is happening you can see that the site is sending a Last-Modified header, but there are no Cache-Control or Expires headers. The absence of these two headers is why WebPageTest has flagged this as an issue with caching static content.
This object was last modified on October 27th, 2017 but The browser does not know how long to cache it for, which puts us in the unpredictable realm of “Heuristic Freshness”. This can be simplified as “So this is cacheable, but you won’t tell me for how long? I’ll have to guess a TTL and use that”.
This is not to say that there is anything wrong with heuristic freshness. It was originally defined in the HTTP specification so that caches could still add some value in situations where the origin did not set a cache policy, while still being safe. However, it is a loss of control over browser cacheability. The HTTP specification encourages caches to use a heuristic expiration value that is no more than some fraction of the interval since the Last-Modified time. A typical setting of this fraction might be 10%.
How Long Will Browsers Actually Heuristically Cache Content For?
So we know that there are problems with this request, and we also know that the cache TTLs may be assigned heuristically based on the Last-Modified date – but why does Firefox’s about:cache show that this resource is cacheable for 1 week instead of 10% of 136 days?
Chrome’s cache internals do not show the expiration for a cached resource, so it’s expiration is not easy to determine. Is it 1 week as well? Or 10% of 136 days? Or something else?
Since Firefox, Chrome and WebKit are open source, we can actually dig into the code and see how the spec is implemented. In the Chromium source as well as the Webkit source, we can see that the lifetimes.freshness variable is set to 10% of the time since the object was last modified (provided that a Last-Modified header was returned). The 10% since last modified heuristic is the same as the suggestion from the RFC. When we look at the Firefox source code, we can see that the same calculation is used, however Firefox uses the std:min() C++ function to select the less of the 10% calculation or 1 week. This explains why we only saw the resource cached for 7 days in Firefox. It would likely have been cached for 13 or 14 days in Chrome.
With Internet Explorer 11, we can look at the Temporary Internet Files to see how the browser handles heuristic freshness. This is similar to the about:cache investigation we did for Firefox, but it’s the only visibility we have to work with. To do this, clear your browser cache, then open a page where objects are being heuristically cached. Then view the files in C:\Users\<username>\AppData\Local\Microsoft\Windows\Temporary Internet Files. The screenshot below shows that there is no expiration for the resources cached, although it’s not clear whether a heuristic is used.
This appears to have been configurable in earlier versions of IE. I’m not sure how heuristic caching is implemented in Edge yet – but will update this when I find out.
Summary
Many content owners incorrectly believe that omitting Cache-Control and Expires headers will prevent downstream caching. In fact, the opposite is true. A worst case scenario of this type of issue would be to combine heuristic caching with infrequent updates to non-versioned CSS/JS objects. Such a scenario would result in page breakage for some clients, while being extremely difficult to reproduce and troubleshoot across different browsers. Simply setting the proper cache headers easily avoids this – and it’s simple to fix, either on your web servers or within your CDN configuration.
If you want to ensure that freshness lifetimes are explicitly stated in your HTTP responses, the best way to do this would be to always include a Cache-Control header. At a minimum, you should be including no-store to prevent caching or max-age=<seconds> to indicate how long content should be considered fresh.
If you are an Akamai customer, it’s easy to create a downstream caching behavior that will automatically update your Cache-Control and Expires headers based on either a static time value or the remaining freshness of the object on the CDN.
Many thanks to Yoav Weiss and Mark Nottingham for reviewing this and providing feedback.
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Language: Search: Contact
Zentralblatt MATH has released its new interface!
For an improved author identification, see the new author database of ZBMATH.
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Fill in the form and click »Search«...
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Zbl 0997.30009
Liu, Jinlin; Srivastava, H.M.
A linear operator and associated families of meromorphically multivalent functions.
(English)
[J] J. Math. Anal. Appl. 259, No.2, 566-581 (2001). ISSN 0022-247X
Let $\Sigma_p$ denote the class of functions $f(z)$ which are analytic and $p$-valent in the punctured unit disk $${\cal U}^*={\cal U}\setminus\{0\},\quad {\cal U}= \{z:|z|< 1\}.$$ For the given real numbers $a,c-c\not\in\bbfN$ we can define a linear operator $${\cal L}_p(a, c) f(z):= \phi_p(a,c;z)* f(z),\quad(f\in\Sigma_p)$$ where $*$ is a convolution (Hadamard product) and $\phi_p(a,c;z)$ is a special function defined as follows $$\phi_p(a,c;z):= z^{-p}+ \sum^\infty_{k=1} {(a)_k\over (c)_k} z^{k- p}.$$ For the given fixed parameters $p$, $a$, $c$, $A$, $B$, $-1\le B< A\le 1$, we say that a function $f\in\Sigma_p$ is in the class ${\cal H}_{a,c}(p;A,B)$ if it also satisfies the inequality $$\Biggl|{z({\cal L}_p(a,c) f(z))'+ p{\cal L}_p(a,c) f(z)\over Bz({\cal L}_p(a, c) f(z))'+ Ap{\cal L}_p(a, c) f(z)}\Biggr|< 1\quad\text{for }z\in{\cal U}.$$ In this paper some properties of the classes ${\cal H}_{a,c}(p;A,B)$ and the operators ${\cal L}_p(a,c)f$ are investigated. Among others it is proved: Theorem. If $a\ge {p(A-B)\over B+1}$, then ${\cal H}_{a+1,c}(p;A,B)\subset{\cal H}_{a,c}(p;A,B)$.
[Jan Stankiewicz (Rzeszów)]
MSC 2000:
*30C45 Special classes of univalent and multivalent functions
30C50 Coefficient problems for univalent and multivalent functions
Keywords: meromorphic function; multivalent; Hadamard product
Highlights
Master Server
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# Coprime integers
In number theory, two integers a and b are said to be relatively prime, mutually prime, or coprime (also written co-prime) if the only positive integer (factor) that divides both of them is 1. Consequently, any prime number that divides one does not divide the other. This is equivalent to their greatest common divisor (gcd) being 1.
The numerator and denominator of a reduced fraction are coprime. As specific examples, 14 and 15 are coprime, being commonly divisible only by 1, while 14 and 21 are not coprime, because they are both divisible by 7.
Standard notations for relatively prime integers a and b are: gcd(a, b) = 1 and (a, b) = 1. Graham, Knuth and Patashnik have proposed that the notation ${\displaystyle a\perp b}$ be used to indicate that a and b are relatively prime and that the term "prime" be used instead of coprime (as in a is prime to b).
A fast way to determine whether two numbers are coprime is given by the Euclidean algorithm.
The number of integers coprime to a positive integer n, between 1 and n, is given by Euler's totient function (or Euler's phi function) φ(n).
A set of integers can also be called coprime if its elements share no common positive factor except 1. A stronger condition on a set of integers is pairwise coprime, which means that a and b are coprime for every pair (a, b) of different integers in the set. The set {2, 3, 4} is coprime, but it is not pairwise coprime since 2 and 4 are not relatively prime.
## Related Topics
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# Scripting language
Short description: Programming language for run-time events
A scripting language or script language is a programming language for a runtime system that automates the execution of tasks that would otherwise be performed individually by a human operator.[1] Scripting languages are usually interpreted at runtime rather than compiled.
A scripting language's primitives are usually elementary tasks or API calls[clarification needed], and the scripting language allows them to be combined into more programs. Environments that can be automated through scripting include application software, text editors, web pages, operating system shells, embedded systems, and computer games. A scripting language can be viewed as a domain-specific language for a particular environment; in the case of scripting an application, it is also known as an extension language. Scripting languages are also sometimes referred to as very high-level programming languages, as they sometimes operate at a high level of abstraction, or as control languages, particularly for job control languages on mainframes.
The term scripting language is also used in a wider sense, namely, to refer to dynamic high-level programming languages in general; some are strictly interpreted languages, while others use a form of compilation. In this context, the term script refers to a small program in such a language; typically, contained in a single file, and no larger than a few thousand lines of code.
The spectrum of scripting languages ranges from small to large, and from highly domain-specific language to general-purpose programming languages. A language may start as small and highly domain-specific and later develop into a portable and general-purpose language; conversely, a general-purpose language may later develop special domain-specific dialects.
## Notable examples
• Bash, an interpreted scripting language for use on Unix and Unix-like operating systems and environments.
• PowerShell, a scripting language originally for use on Microsoft Windows but nowadays also installable on macOS and Linux.
• sed and AWK, two text processing languages used mainly in Unix-like environments.
• Perl,[2] a text processing language that later developed into a general-purpose language, also used as an extension language for various applications.
• Python, a general-purpose scripting language, also used as an extension language.
• Tcl,[3] a scripting language for Unix-like environments, popular in the 1990s for creating GUI applications.
• JavaScript (later: ECMAScript), originally a very small, highly domain-specific language, limited to running within a web browser to dynamically modify the web page being shown, that later developed into a widely portable general-purpose programming language.
• Kotlin, although a compiled language, it provides REPL [4] and scripting support[5] (like for Gradle build scripts[6]).
• Visual Basic for Applications, an extension language specifically for Microsoft Office applications.
• Lua, a language designed for use as an extension language for applications in general, and used as such by many different applications.
• Lisp, a family of general-purpose languages and extension languages for specific applications, e.g. Emacs Lisp, for the Emacs editor.
Some game systems have been extensively extended in functionality by scripting extensions using custom languages, notably the Second Life virtual world (using Linden Scripting Language) and the Trainz franchise of Railroad simulators (using TrainzScript). In some games, such as Wesnoth, users play custom variants of the game defined by user-contributed scripts.
## Characteristics
Typical scripting languages are intended to be very fast to learn and write in, either as short source code files or interactively in a read–eval–print loop (REPL, language shell).[7] This generally implies relatively simple syntax and semantics; typically a "script" (code written in the scripting language) is executed from start to finish, as a "script", with no explicit entry point.
For example, it is uncommon to characterise Java as a scripting language because of its lengthy syntax and rules about which classes exist in which files, and it is not directly possible to execute Java interactively, because source files can only contain definitions that must be invoked externally by a host application or application launcher.
public class HelloWorld {
public void printHelloWorld() {
System.out.println("Hello World");
}
}
This piece of code intended to print "Hello World" does nothing as main() is not declared in HelloWorld class, although the one below would be useful.
public class HelloWorld {
public void printHelloWorld() {
System.out.println("Hello World");
}
public static void main(String[] args) {
printHelloWorld();
}
}
In the example above, main is defined and so this can be invoked by the launcher, although this still cannot be executed interactively. In contrast, Python allows the definition of some functions in a single file, or to avoid functions altogether and use imperative programming style, or even use it interactively.
print("Hello World")
print "Hello World" # Python 2.7 or earlier
This one line of Python code prints "Hello World"; no declarative statement like main() is required here.
A scripting language is usually interpreted from source code or bytecode.[8] By contrast, the software environment (interpreter) the scripts are written for is typically written in a compiled language and distributed in machine code form.
Scripting languages may be designed for use by end users of a program—end-user development—or may be only for internal use by developers, so they can write portions of the program in the scripting language. Scripting languages typically use abstraction, a form of information hiding, to spare users the details of internal variable types, data storage, and memory management.
Scripts are often created or modified by the person executing them,[9] but they are also often distributed, such as when large portions of games are written in a scripting language, notably the Google Chrome T-rex game.
## History
Early mainframe computers (in the 1950s) were non-interactive, instead using batch processing. IBM's Job Control Language (JCL) is the archetype of languages used to control batch processing.[10]
The first interactive shells were developed in the 1960s to enable remote operation of the first time-sharing systems, and these used shell scripts, which controlled running computer programs within a computer program, the shell. Calvin Mooers in his TRAC language is generally credited with inventing command substitution, the ability to embed commands in scripts that when interpreted insert a character string into the script.[11] Multics calls these active functions.[12] Louis Pouzin wrote an early processor for command scripts called RUNCOM for CTSS around 1964. Stuart Madnick at MIT wrote a scripting language for IBM's CP/CMS in 1966. He originally called this processor COMMAND, later named EXEC.[13] Multics included an offshoot of CTSS RUNCOM, also called RUNCOM.[14] EXEC was eventually replaced by EXEC 2 and REXX.
Languages such as Tcl and Lua were specifically designed as general-purpose scripting languages that could be embedded in any application. Other languages such as Visual Basic for Applications (VBA) provided strong integration with the automation facilities of an underlying system. Embedding of such general-purpose scripting languages instead of developing a new language for each application also had obvious benefits, relieving the application developer of the need to code a language translator from scratch and allowing the user to apply skills learned elsewhere.
Some software incorporates several different scripting languages. Modern web browsers typically provide a language for writing extensions to the browser itself, and several standard embedded languages for controlling the browser, including JavaScript (a dialect of ECMAScript) or XUL.
## Types
Scripting languages can be categorized into several different types, with a considerable degree of overlap among the types.
### Glue languages
Scripting is often contrasted with system programming, as in Ousterhout's dichotomy or "programming in the large and programming in the small". In this view, scripting is glue code, connecting software components, and a language specialized for this purpose is a glue language. Pipelines and shell scripting are archetypal examples of glue languages, and Perl was initially developed to fill this same role. Web development can be considered a use of glue languages, interfacing between a database and web server. But if a substantial amount of logic is written in script, it is better characterized as simply another software component, not "glue".
Glue languages are especially useful for writing and maintaining:
• custom commands for a command shell;[15]
• smaller programs than those that are better implemented in a compiled language;[16]
• "wrapper" programs for executables, like a batch file that moves or manipulates files and does other things with the operating system before or after running an application like a word processor, spreadsheet, data base, assembler, compiler, etc.;[17]
• scripts that may change;[18]
• Rapid application development of a solution eventually implemented in another, usually compiled, language.
Glue language examples:
Macro languages exposed to operating system or application components can serve as glue languages. These include Visual Basic for Applications, WordBasic, LotusScript, CorelScript, Hummingbird Basic, QuickScript, Rexx, SaxBasic, and WinWrap Basic. Other tools like AWK can also be considered glue languages, as can any language implemented by a Windows Script Host engine (VBScript, JScript and VBA by default in Windows and third-party engines including implementations of Rexx, Perl, Tcl, Python, XSLT, Ruby, Modern Pascal, Delphi, and C). A majority of applications can access and use operating system components via the object models or its own functions.
Other devices like programmable calculators may also have glue languages; the operating systems of PDAs such as Windows CE may have available native or third-party macro tools that glue applications together, in addition to implementations of common glue languages—including Windows NT, DOS, and some Unix shells, Rexx, Modern Pascal, PHP, and Perl. Depending upon the OS version, WSH and the default script engines (VBScript and JScript) are available.
Programmable calculators can be programmed in glue languages in three ways. For example, the Texas Instruments TI-92, by factory default can be programmed with a command script language. Inclusion of the scripting and glue language Lua in the TI-NSpire series of calculators could be seen as a successor to this. The primary on-board high-level programming languages of most graphing calculators (most often Basic variants, sometimes Lisp derivatives, and more uncommonly, C derivatives) in many cases can glue together calculator functions—such as graphs, lists, matrices, etc. Third-party implementations of more comprehensive Basic version that may be closer to variants listed as glue languages in this article are available—and attempts to implement Perl, Rexx, or various operating system shells on the TI and HP graphing calculators are also mentioned. PC-based C cross-compilers for some of the TI and HP machines used with tools that convert between C and Perl, Rexx, AWK, and shell scripts to Perl, Modern Pascal, VBScript to and from Perl make it possible to write a program in a glue language for eventual implementation (as a compiled program) on the calculator.
### Editor languages
A number of text editors support macros written either using a macro language built into the editor, e.g., The SemWare Editor (TSE), vi improved (VIM), or using an external implementation, e.g., XEDIT, or both, e.g., KEDIT. Sometimes text editors and edit macros are used under the covers to provide other applications, e.g., FILELIST and RDRLIST in CMS .
### Job control languages and shells
Main page: Shell script
A major class of scripting languages has grown out of the automation of job control, which relates to starting and controlling the behavior of system programs. (In this sense, one might think of shells as being descendants of IBM's JCL, or Job Control Language, which was used for exactly this purpose.) Many of these languages' interpreters double as command-line interpreters such as the Unix shell or the MS-DOS COMMAND.COM. Others, such as AppleScript offer the use of English-like commands to build scripts.
### GUI scripting
With the advent of graphical user interfaces, a specialized kind of scripting language emerged for controlling a computer. These languages interact with the same graphic windows, menus, buttons, and so on, that a human user would. They do this by simulating the actions of a user. These languages are typically used to automate user actions. Such languages are also called "macros" when control is through simulated key presses or mouse clicks, as well as tapping or pressing on a touch-activated screen.
These languages could in principle be used to control any GUI application; but, in practice their use is limited because their use needs support from the application and from the operating system. There are a few exceptions to this limitation. Some GUI scripting languages are based on recognizing graphical objects from their display screen pixels. These GUI scripting languages do not depend on support from the operating system or application.
When the GUI provides the appropriate interfaces, as in the IBM Workplace Shell, a generic scripting language, e.g. OREXX, can be used for writing GUI scripts.
### Application-specific languages
Application specific languages can be split in many different categories, i.e. standalone based app languages (executable) or internal application specific languages (postscript, xml, gscript as some of the widely distributed scripts, respectively implemented by Adobe, MS and Google) among others include an idiomatic scripting language tailored to the needs of the application user. Likewise, many computer game systems use a custom scripting language to express the programmed actions of non-player characters and the game environment. Languages of this sort are designed for a single application; and, while they may superficially resemble a specific general-purpose language (e.g. QuakeC, modeled after C), they have custom features that distinguish them. Emacs Lisp, while a fully formed and capable dialect of Lisp, contains many special features that make it most useful for extending the editing functions of Emacs. An application-specific scripting language can be viewed as a domain-specific programming language specialized to a single application.
### Extension/embeddable languages
A number of languages have been designed for the purpose of replacing application-specific scripting languages by being embeddable in application programs. The application programmer (working in C or another systems language) includes "hooks" where the scripting language can control the application. These languages may be technically equivalent to an application-specific extension language but when an application embeds a "common" language, the user gets the advantage of being able to transfer skills from application to application. A more generic alternative is simply to provide a library (often a C library) that a general-purpose language can use to control the application, without modifying the language for the specific domain.
JavaScript began as and primarily still is a language for scripting inside web browsers; however, the standardisation of the language as ECMAScript has made it popular as a general-purpose embeddable language. In particular, the Mozilla implementation SpiderMonkey is embedded in several environments such as the Yahoo! Widget Engine. Other applications embedding ECMAScript implementations include the Adobe products Adobe Flash (ActionScript) and Adobe Acrobat (for scripting PDF files).
Tcl was created as an extension language but has come to be used more frequently as a general-purpose language in roles similar to Python, Perl, and Ruby. On the other hand, Rexx was originally created as a job control language, but is widely used as an extension language as well as a general-purpose language. Perl is a general-purpose language, but had the Oraperl (1990) dialect, consisting of a Perl 4 binary with Oracle Call Interface compiled in. This has however since been replaced by a library (Perl Module), DBD::Oracle.[19][20]
Other complex and task-oriented applications may incorporate and expose an embedded programming language to allow their users more control and give them more functionality than can be available through a user interface, no matter how sophisticated. For example, Autodesk Maya 3D authoring tools embed the Maya Embedded Language, or Blender which uses Python to fill this role.
Some other types of applications that need faster feature addition or tweak-and-run cycles (e.g. game engines) also use an embedded language. During the development, this allows them to prototype features faster and tweak more freely, without the need for the user to have intimate knowledge of the inner workings of the application or to rebuild it after each tweak (which can take a significant amount of time). The scripting languages used for this purpose range from the more common and more famous Lua and Python to lesser-known ones such as AngelScript and Squirrel.
Ch is another C compatible scripting option for the industry to embed into C/C++ application programs.
## References
1. "ECMAScript 2019 Language Specification". www.ecma-international.org.
2. Sheppard, Doug (2000-10-16). "Beginner's Introduction to Perl". dev.perl.org.
3. Programming is Hard, Let's Go Scripting…, Larry Wall, December 6, 2007
4. Run code snippets, Kotlin, December 3, 2021
5. Run scripts, Kotlin, December 3, 2021
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# Questions
## On $(A_2,A_6)$
### sol 1
$$\begin{bmatrix} z_0^{(1)} & z_0^{(2)} \\ z_1^{(1)} & z_1^{(2)} \\ z_2^{(1)} & z_2^{(2)} \\ z_3^{(1)} & z_3^{(2)} \\ z_4^{(1)} & z_4^{(2)} \\ z_5^{(1)} & z_5^{(2)} \\ z_6^{(1)} & z_6^{(2)} \\ z_7^{(1)} & z_7^{(2)} \end{bmatrix} = \begin{bmatrix} 1. & 1. \\ 2.24698 & 2.24698 \\ 2.80194 & 2.80194 \\ 2.24698 & 2.24698 \\ 1. & 1. \\ 0 & 0 \\ 0 & 0 \\ 1. & 1. \end{bmatrix}$$ Here $2.24698\cdots$ is a solution of $x^3-2 x^2-x+1=0$.
This solution can be obtained from $g\in SU(3)$ given by $$g=\left( \begin{array}{ccc} e^{\frac{2 i \pi }{7}} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & e^{-\frac{2 i \pi }{7}} \\ \end{array} \right)$$
### sol 2
$$\begin{bmatrix} z_0^{(1)} & z_0^{(2)} \\ z_1^{(1)} & z_1^{(2)} \\ z_2^{(1)} & z_2^{(2)} \\ z_3^{(1)} & z_3^{(2)} \\ z_4^{(1)} & z_4^{(2)} \\ z_5^{(1)} & z_5^{(2)} \\ z_6^{(1)} & z_6^{(2)} \\ z_7^{(1)} & z_7^{(2)} \end{bmatrix} = \begin{bmatrix} 1. & 1. \\ 1.61803 & 1.61803 \\ 1. & 1. \\ 0 & 0 \\ 0 & 0 \\ 1. & 1. \\ 1.61803 & 1.61803 \\ 1. & 1. \end{bmatrix}$$ where $1.61803\cdots$ is the Golden ratio.
This solution can be obtained from $g\in SU(3)$ given by $$g=\left( \begin{array}{ccc} e^{\frac{2 i \pi }{5}} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & e^{-\frac{2 i \pi }{5}} \\ \end{array} \right)$$
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# Need to find:under what circumstances can synthetic division are used to divide polynomial.
Question
Polynomial division
Need to find:under what circumstances can synthetic division are used to divide polynomial.
2021-02-13
Synthetic division:
In algebra synthetic division is a method for manually performing Euclidean division of polynomials, with less writing and fewer calculations than occur with long polynomial division.
The circumstances are:
It is mostly used for division by binomials of the form x — a, but the method generalized to division by any monic polynomial.
It is dividing for the linear factor.
It can be used to divide polynomial polynomials when the division is the binomial of the form x—c and cis a constant.
### Relevant Questions
The condition under which we may use synthetic division to divide polynomials
Synthetic division is a process for dividing a polynomial by x - c.
The coefficient of x in the divisor is 1.
How might synthetic division be used if you are dividing by $$2x - 4$$?
Whether the statement “When performing the division $$\frac{x^{5} +1}{x + 1}$$ there's no need for me to follow all the steps involved in polynomial long division because I can work the problem in my head and see that the quotient must be $$x^{4} + 1$$ ”makes sense or not.” Makes sense or not.
How can the division algorithm be used to check the result of the polynomial division.
How the division algorithm can be used to check the result of the polynomial division.
Whether the division of polynomial $$\frac{3x^{4}-x+1}{x-5}$$ using synthetic division method is directly possible or not.
Synthetic division is a shortcut process for polynomial division.
Writing Complete each polynomial division. Write a brief description of the pattern that you obtain, and use your result to find a formula for the polynomial division $$\frac{x^{n}-1}{x-1}$$.
$$\frac{x^{4}-1}{x-1}$$
Writing Complete each polynomial division. Write a brief description of the pattern that you obtain, and use your result to find a formula for the polynomial division $$\frac{x^{n}-1}{x-1}$$.
$$\frac{x^{3}-1}{x-1}$$
Writing Complete each polynomial division. Write a brief description of the pattern that you obtain, and use your result to find a formula for the polynomial division $$\frac{x^{n}-1}{x-1}$$.
$$\frac{x^{2}-1}{x-1} =$$ ?
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Figo asked in 社會及文化語言 · 7 years ago
# late on 點解 ? 同later有咩分別?
late on 點解 ? 同later有咩分別?
Update:
sorry
Rating
• 7 years ago
Most of the time, they mean the same. Sometimes they cannot be used interchangeably.
Later = not now, or not at a time you are talking about, but some time after this
Phrases: a month/year later, much later, later that month/year, later in the day/month/year
Later on = later on in time = at some time after the present time
“Later on” is used for distant future.
I arrived later than usual. (Later on will not work here)
Bye, see you later.
Few people say “Bye, see you later on.”
Later can also be used as an adjective, not “later on”
later model, later performance, later years
e.g. She spent her later years in a nursing home.
To play safe, especially if you are not sure, just say or write “later “
Source(s): Longman English Dictionary
• 7 years ago
later:
At a time in the future or after the time you have mentioned:
e.g.He'll be back later. We could always go later in the season. Police questioned him and he was later arrested.
later on:
At a time in the future, or after the time you have mentioned:
e.g.What are you doing later on this evening? Shall I go and fetch her later on? Later on, we could go and have a meal if you like.
later同嘅later on分別係later on通常放於句子最頭/最尾。later通常放於句子中間
Source(s): 個人意解+cambridge dictionary
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## Enhancement
An oft-overlooked aspect of statistical models is that parameters are dependent on each other. Ignoring such dependencies can have important consequences, and in extreme cases can even undermine assumptions for a forecasting model. However, in the case of a regression model the correlations between regressor variables can sometimes have some unexpectedly positive results. To illustrate this, consider a sequence of fits of a survival model for a Makeham-Perks mortality law (Richards, 2008) defined as follows:
μx = [exp(ε) + exp(α + βx)] / [1 + exp(α + βx)]
where the parameter α is allowed to vary by gender, health status at retirement, or both. The results for a large portfolio of pensions in payment are shown in Table 1 below:
Table 1. A sequence of model fits, their corresponding AICs and the improvement in AIC over an age-only model. Source: Own calculations using a large portfolio of pensions in payment.
ModelAICImprovement in AIC over age-only model
Age 240,946 n/a
Age+Gender 239,459 1,487
Age+Health 240,211 735
Age+Gender+Health 238,512 2,434
For data sets of this size, an improvement of more than 4 AIC units is regarded as significant. Table 1 shows a number of expected features, namely that gender is a very important risk factor for mortality, albeit one that is now illegal in the EU for insurance pricing. Similarly unsurprising is that health status at retirement is also significant, and that a model which includes both gender and health status is better than a model which leaves out either factor.
However, Table 1 does contain one seemingly curious feature: the improvement in fit from including both gender and health is 2,434 AIC units, whereas the sum of the two improvements for gender and health on their own is 2,222 AIC units (2,222 = 1,487 + 735). How can the improvement from having both factors in the model be greater than the sum of their individual contributions?
As it happens, this is no anomaly. Rather, the phenomenon is known as enhancement, and was discussed in the context of bivariate regression by Currie & Korabinski (1984):
"enhancement occurs often [...]. Further, as stepwise regression proceeds [...] enhancement becomes more frequent"
Source: Currie & Korabinski (1984), page 292.
We have seen many survival models fitted to a lot of different data sets of pensioners and annuitants, and we can confirm that Currie & Korabinski's points also hold true for survival models. Adding a signficant risk factor not only enhances a survival model's fit, it often improves the ability of existing risk factors to explain variation. This can result in a virtuous circle — the more relevant new risk factors you can add, the better the explanatory power of the first ones.
References
Currie, I. D. and Korabinski, A. (1984) Some comments on bivariate regression, The Statistician33, 283–293.
Richards, S. J. (2008) Applying survival models to pensioner mortality dataBritish Actuarial Journal15(II), No. 65, 317–365 (with discussion).
Assume we have a random variable, $$X$$, with expected value ... Read more
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# Difference between revisions of "2020 AMC 10A Problems/Problem 24"
## Problem
Let $n$ be the least positive integer greater than $1000$ for which$$\gcd(63, n+120) =21\quad \text{and} \quad \gcd(n+63, 120)=60.$$What is the sum of the digits of $n$?
$\textbf{(A) } 12 \qquad\textbf{(B) } 15 \qquad\textbf{(C) } 18 \qquad\textbf{(D) } 21\qquad\textbf{(E) } 24$
## Solution
We know that $gcd(63, n+120)=21$, so we can write $n+120\equiv0(mod 21)$. Simplifying, we get $n\equiv6(mod 21)$. Similarly, we can write $n+63\equiv0(mod60)$, or $n\equiv-3(mod60)$. Solving these two modular congruences, $n\equiv237(mod 420)$ which we know is the only solution by CRT (Chinese Remainder Theorem). Now, since the problem is asking for the least positive integer greater than $1000$, we find the least solution is $n=1077$. However, we are have not considered cases where $gcd(63, n+120) =63$ or $gcd(n+63, 120) =120$. $1077+120\equiv0(mod 63)$ so we try $n=1077+420=1497$.
$1497+63\equiv0(mod 120) so again we add another$420$to$n$. It turns out that$n=1497+420=1917$does indeed satisfy the conditions, so our answer is$1+9+1+7=\boxed{\textsf{(C) } 18}\$.
## Video Solution
https://youtu.be/tk3yOGG2K-s - $Phineas1500$
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algebra
Create a rational function with a linear binomial in both the numerator and denominator.
Part 1. Graph your function using technology. Include the horizontal and vertical asymptotes and the x- and y-intercepts on your graph. Label the asymptotes and intercepts.
Part 2. Show all work to identify the vertical asymptote, the x-intercepts, and the y-intercept.
1. 1
1. y = (2x-3)/(7x+1)
see
http://www.wolframalpha.com/input/?i=%282x-3%29%2F%287x%2B1%29
posted by Steve
2. thank you, steve.
posted by Kassie
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Cloud Help Desk: Delays for ticket imports:
# CRL doesn't work well after Active directory certificate services migration
Always-on VPN noob questions
I ran a lab simulating ADCS migration from server 2008 R2 named TestCA1 to Server 2016 named TestCA2. Migration seemed to be successful. I am able to get a new certificate from the TestCA2. The only problem I am having now is that the certificate is still valid after I revoke it. I have an IIS server and applied the web server certificate, but after I revoked it, it still appears valid on IE except on the Root CA server TestCA2. I manually published the CRL and cleared the cache CRL copy of client computers, it is still the same.
In the process of migration, I have changed the registry key of the server TestCA2 to referencing the new host name of the CA server. I also recovered the permissions by changing the AIA and CDP values.
Does anyone know where I can look? Thanks.
Ghost Chili
OP
Active Directory & GPO expert
Well, you can't really redirect it, but I think what you should be able to do is double publish the CRL and delta CRL to both CN=TestCA1 and CN=TestCA2 locations.
In the properties of the CA in CA Managment, under extensions, select the CRL Distribution Point (CDP) extension. IN the box beneath it, you should be able to configure the various CDP locations and methods. One of the default ones is CN=<CATruncatedName>....blah, blah, blah. I would add another one, use the exact same format, but swap out "<ServerShortName>" with "TestCA1".
Then, Check the following:
• "Publish CRLs to this location", and
• "Publish Delta CRLs to this location", and
• "Include in all CRLs"
I would NOT check these three (that way new certificates are not using the "stale" AD location
• Include in CRLs (clients use this to find delta CRL locations
• Include in the CDP extension of issued certs
• Include in the IDP extension of issued CRLs
You may have to adjust permissions on the AD tree in ADSIedit ("CN=preston-TESTCA1-CA,CN=TestCA1,CN=CDP,CN=Public Key Services,CN=Services,CN=Configuration,DC=preston,DC=ca") to allow the new computer to modify the attributes.
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## 7 Replies
· · ·
Ghost Chili
OP
Active Directory & GPO expert
Are clients able to reach the old CRL URL on the certificates which were originally issued?
0
· · ·
Anaheim
OP
Hi Semicolon,
I think so. The web certificate (testweb.preston.ca) I revoked was originally issued by the old CA server (TestCA1) with the CDP referencing to old hostname LDAP. After I cleared the cache copy by running certutil -urlcache crl delete and then restarted the machine, I tried the website:testweb.xxxx.com on IE and it still returned valid. Then I checked the local CRL Cache again by entering cerutil -urlcache crl, two entries were loaded:
ldap:///CN=preston-TESTCA1-CA,CN=TestCA1,CN=CDP,CN=Public%20Key%20Services,CN=Services,CN=Configuration,DC=preston,DC=ca?certificateRevocationList?Base?objectClass=cRLDistributionPoint
ldap:///CN=preston-TESTCA1-CA,CN=TestCA1,CN=CDP,CN=Public%20Key%20Services,CN=Services,CN=Configuration,DC=preston,DC=ca?deltaRevocationList?Base?objectClass=cRLDistributionPoint
0
· · ·
Ghost Chili
OP
Active Directory & GPO expert
Have you reviewed the CRL and the latest delta CRL to ensure that your revoked certificate is on that list?
Did you allow enough time after revoking to ensure the new Delta CRL was published? You can always run the following to force a publication:
Text
certutil -CRL
Is the service able to publish to the file system?
---
Text
certutil -urlfetch -verify C:\path\to\saved.cer
What CDPs are on the certificate? Is there an HTTP CDP as well?
You can directly check the CDP as well:
Text
certutil -URL ldap:///CN=preston-TESTCA1-CA,CN=TestCA1,CN=CDP,CN=Public%20Key%20Services,CN=Services,CN=Configuration,DC=preston,DC=ca?deltaRevocationList?Base?objectClass=cRLDistributionPoint
---
It is too late as this certificate has been published - but I tend to prefer using HTTP CDPs over LDAP CDPs - mainly so that non-domain and non-authenticated assets don't have problems locating the revocation list. But, you may want to change that on certificates going forward (before you issue anymore). Before you consider this to be a show-stopper, you should check the production certs and CA to see what CDPs they have been configured with.
0
· · ·
Anaheim
OP
CRL was able to be published, and delta CRL is ldap:///CN=preston-TESTCA1-CA,CN=TestCA2,CN=CDP,CN=Public%20Key%20Services,CN=Services,CN=Configuration,DC=preston,DC=ca?deltaRevocationList?Base?objectClass=cRLDistributionPoint
I also checked the CDP, using certutil -URL ldap:///CN=preston-TESTCA1-CA,CN=TestCA1,CN=CDP,CN=Public%20Key%20Services,CN=Services,CN=Configuration,DC=preston,DC=ca?deltaRevocationList?Base?objectClass=cRLDistributionPoint, and the status is Expired. Since I migrated CA from TestCA1 to TestCA2 and then shut down TestCA1, I guess that's why it's expired. The web server certificate for testweb.preston.ca was originally issued by TestCA1 and hence it was referring to ldap:///CN=preston-TESTCA1-CA,CN=TestCA1,CN=CDP,CN=Public%20Key%20Services,CN=Services,CN=Configuration,DC=preston,DC=ca?deltaRevocationList?Base?objectClass=cRLDistributionPoint for revocation validation. Is there possible to redirect it to ldap:///CN=preston-TESTCA1-CA,CN=TestCA2,CN=CDP,CN=Public%20Key%20Services,CN=Services,CN=Configuration,DC=preston,DC=ca?deltaRevocationList?Base?objectClass=cRLDistributionPoint for validation?
0
· · ·
Anaheim
OP
By the way, I also run certutil -URL ldap:///CN=preston-TESTCA1-CA,CN=TestCA1,CN=CDP,CN=Public%20Key%20Services,CN=Services,CN=Configuration,DC=preston,DC=ca?certificateRevocationList?Base?objectClass=cRLDistributionPoint to check the base CRL, I was able to retrieve both Base and Delta CRLs, base CRL's status is OK, though the Revocation List is empty for TestCA1 base CRL. It looks like I need to redirect the TestCA1 CRL to TestCA2 CRL.
0
· · ·
Ghost Chili
OP
Active Directory & GPO expert
Well, you can't really redirect it, but I think what you should be able to do is double publish the CRL and delta CRL to both CN=TestCA1 and CN=TestCA2 locations.
In the properties of the CA in CA Managment, under extensions, select the CRL Distribution Point (CDP) extension. IN the box beneath it, you should be able to configure the various CDP locations and methods. One of the default ones is CN=<CATruncatedName>....blah, blah, blah. I would add another one, use the exact same format, but swap out "<ServerShortName>" with "TestCA1".
Then, Check the following:
• "Publish CRLs to this location", and
• "Publish Delta CRLs to this location", and
• "Include in all CRLs"
I would NOT check these three (that way new certificates are not using the "stale" AD location
• Include in CRLs (clients use this to find delta CRL locations
• Include in the CDP extension of issued certs
• Include in the IDP extension of issued CRLs
You may have to adjust permissions on the AD tree in ADSIedit ("CN=preston-TESTCA1-CA,CN=TestCA1,CN=CDP,CN=Public Key Services,CN=Services,CN=Configuration,DC=preston,DC=ca") to allow the new computer to modify the attributes.
0
· · ·
Anaheim
OP
Thank you so much, Semicolon. I added another CDP location on Extension tab and pointed to TestCA1. It works now.
1
This topic has been locked by an administrator and is no longer open for commenting.
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# What is long long in C/C++?
CC++Server Side ProgrammingProgramming
In some cases we use long long in C or C++. Here we will see what is basically long long is? The long long takes twice as much memory as long. In different systems, the allocated memory space differs. On Linux environment the long takes 64-bit (8-bytes) of space, and the long long takes 128-bits (16-bytes) of space. This is used when we want to deal with some large value of integers.
We can test the size of different types using this simple program.
## Example
#include <iostream>
using namespace std;
main() {
int a;
long b;
long long c;
cout << "Size of int = "<< sizeof(a) <<" bytes \n";
cout << "Size of long = "<< sizeof(b) <<" bytes\n";
cout << "Size of long long = "<< sizeof(c) <<" bytes\n";
}
## Output
Size of int = 4 bytes
Size of long = 4 bytes
Size of long long = 8 bytes
Output may differ in different systems. Here windows platform is used for testing.
Published on 03-May-2019 07:16:03
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# Why is the first left and right singular vectos scale by the first singular values a good approximation of the original matrix
Conceptually, why is the first singular vector a good rank one approximation instead of something like the averaging of the total singular vectors?
If you have $$A = U\Sigma V^T$$
why isn't
$$\sqrt{\sigma_{avg}}u_{avg}v_{avg}^T$$ a good low rank approximation?
How about weighted average of the singular vectors?
Context of my Question:
An exam with $m$ questions is given to $n$ students. The instructor collects all the grades in a $n * m$ matrix $G$ with $G_{ij}$ the grade obtained by student $i$ on the question $j$. We would like to assign a difficulty score to each question based on the available data.
How would you compute a rank one approximation to $G$
Solution:
To approximate $G$ by a rank one vector we simply compute the SVD of $G$ and select the singular vectors corresponding to the largest singular value. Precisely, we set $s \sqrt{\sigma_1}u_1$ and $q =\sqrt{ \sigma_i}v_1$ where $u_1 \:\: v_1$ are the first columns of the matrices $U \:\: V$ in the SVD of $G = U\Sigma V^T$ and $\sigma_1$ is the largest singular value
The easiest way to see this is to use the Eckart–Young–Mirsky theorem, which states that the rank-k approximation $A_k$ to $A$ that minimizes Frobenius norm $|| A - A_k ||_F$ is $A_k = U_k \Sigma_k V_k^T$ for $\Sigma_k$ the top $k$ singular values and $U_k$ and $V_k$ the corresponding singular vectors. For $k=1$, this gives $A_1 = \sigma_1 u_1 v_1^T$.
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# How to set author, title and date in KOMA script header (scrlayer-scrpage)?
How can I customize the header to have \author, the \title and the \date in the header and underline it?
Since I use a Koma Script document class I want to avoid the fancyhdr package and use scrlayer-scrpage. I have tried \chead{\title \author \date} but it does not change maybe due to the automark option in \usepackage[automark]{scrlayer-scrpage}. I want the header to be on every page except the title page and list of contents.
So in fancyhdr the code would be:
\usepackage{fancyhdr}
\pagestyle{fancy}
\fancyhf{}
\fancyfoot[CE,CO]{\thepage}
\fancyfoot[LE,RO]{}
Here is how you could 'translate' your fancyhdr into scrlayer-scrpage:
\documentclass[headinclude=true]{scrbook}
\usepackage{blindtext}
\clearpairofpagestyles
\cfoot*{\pagemark}
% If you want to change the font/shape etc.:
% \setkomafont{pagenumber}{...}
\begin{document}
\blinddocument
\end{document}
If you want to automate the author/title insertion, you could add
...
\usepackage{titling}
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## The mean is 47.1 and the standard deviation is 9.5 for a population. Using the Central Limit Theorem, what is the standard deviation of the
Question
The mean is 47.1 and the standard deviation is 9.5 for a population. Using the Central Limit Theorem, what is the standard deviation of the distribution of sample means for samples of size 60
in progress 0
6 hours 2021-07-22T10:02:09+00:00 1 Answers 0 views 0
## Answers ( )
1. Answer:
The standard deviation of the distribution of sample means for samples of size 60 is of 1.2264.
Step-by-step explanation:
Central Limit Theorem
The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean and standard deviation , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean and standard deviation .
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
Standard deviation is 9.5 for a population.
This means that
Sample of 60:
This means that
What is the standard deviation of the distribution of sample means for samples of size 60?
The standard deviation of the distribution of sample means for samples of size 60 is of 1.2264.
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# Hey, what’s that!?
Hey, long time without posting. Hope this will change drastically in the near future. In the meantime… can anybody tell me what’s that!? :)
# There’s music in the primes… (part I)
OK, now a new section at physicsnapkins, in which I will discuss a bit about my own research… Recently, Germán Sierra and I have submitted to the ArXiv a paper about the Riemann hypothesis, which you can see here. To be honest, the real expert in the field is Germán, my contribution is mostly technical. Anyway, I’ll try to convey here the basic ideas of the story… We’ll give a walk around the concepts, assuming only a freshman maths level.
It’s fairly well known that the sum of the reciprocal numbers diverges: $\sum_{n=1}^\infty 1/n\to\infty$. Euler found, using an amazing trick the sum of the inverse squares and, in fact, the sum of the inverse of any even power. This formula is simply amazing: $\sum 1/n^2 = \pi^2/6$, isn’t it? Now, Riemann defined the “zeta” function, for any possible exponent:
$\displaystyle{\zeta(z)=\sum_{n=1}^\infty {1\over n^z}}$
So, we know that $\zeta(1)=\infty$, $\zeta(2)=\pi^2/6$, and many other values. Riemann asked: what happens when z is complex? Complex function theory is funny… We know that z=1 is a singularity. If you do a Taylor series around, say, around z=2, the radius of convergence is the distance to the nearest singularity, so R=1. But now, your function is well defined in a circle of center z=2 and radius R=1. This means that you can expand again the function in a Taylor series from any point within that circle. And, again, the radius of convergence will be the distance to the closest singularity. This procedure is called analytical continuation.
The series of circles show how to compute the analytical continuation of a complex function...
Well… in the case of the Riemann $\zeta$ function, the only singularity is z=1. Therefore, I can do the previous trick and… bypass it! Circle after circle, I can reach z=0 and get a value, which happens to be… -1/2. So, somehow, we can say that, if we had to give a value to the sum 1+1+1+1+…, it should be -1/2. Also, and even more amazing, $\zeta(-1)=\sum_{n=1}^\infty n=1+2+3+4+\cdots=-1/12$. Hey, that’s really pervert maths! Can this be useful for real life, i.e.: for physics. Well, it is used in string theory, to prove that you need (in the simplest bosonic case) dimension D=26… but that’s not true physics. Indeed, it’s needed for the computation of the Casimir effect. Maybe, I’ll devote a post to that someday. Anyway, this is the look of the Riemann zeta function in the complex plane:
The Riemann zeta function. Color hue denotes phase, and intensity denotes modulus. The white point at z=1 is the singularity.
Even more surprises… $1^2+2^2+3^2+\cdots=0$. In fact, it’s easy (ok, ok… it’s easy when you know how!) to prove that $\zeta(-2n)=0$ for all positive n. Those are called the trivial zeroes of the Riemann zeta function (amazing!)… So, what are the non-trivial ones? Riemann found a few zeroes which were not for negative even numbers. But all of them had something in common: their real part was 1/2. And here comes the Riemann hypothesis: maybe (maybe) all the non-trivial zeroes of the $\zeta$ function will have real part 1/2.
OK, I hear you say. I got it. But I still don’t get the fun about the title of the post, and why so much fuss about it. Here it comes…
Euler himself (all praise be given to him!) found an amazing relation, which I encourage you to prove by yourselves:
$\displaystyle{\zeta(z)=\sum_{n=1}^\infty {1\over n^z} = \prod_{p \hbox{ prime}} {1\over 1-p^{-z}}}$
Ahí comienza el link verdadero. Una pista para la demostración: expandimos el producto:
$\displaystyle{\prod_{p \hbox{ primes}} {1\over 1-p^{-z}} = {1\over 1-2^{-z}} {1\over 1-3^{-z}} {1\over 1-5^{-z}} \cdots}$
Wonderful. But $1/(1-x)$ can be easily recognized as the sum of a geometric series, right?
OK, in a few days, I’ll post the second part, explaining why there’s music in the primes, and how quantum mechanics might save the day…
# Log rules…
Here is a very simple problem which appeared with my first year calculus students. Consider the function
$f(x)=\log(x^2)$
Its domain is ${\mathbb R}-\{0\}$. But now, we may take the exponent down, using the rules of the logarithm…
$f(x)=2\log(x)$
and now, the domain is $(0,\infty)$… What happens??? Try to explain it: (a) without complex numbers, (b) with them…
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# Factoring by common factor review
The expression 6m+15 can be factored into 3(2m+5) using the distributive property. More complex expressions like 44k^5-66k^4 can be factored in much the same way. This article provides a couple of examples and gives you a chance to try it yourself.
### Example 1
Faktor.
$6m+15$
Both terms share a common factor of $\goldD{3}$, so we factor out the $\goldD{3}$ using the distributive property:
\begin{aligned} &6m+15\\\\ =&\goldD{3}(2m+5) \end{aligned}
Want a more in-depth explanation? Check out this video.
### Example 2
Factor out the greatest common monomial.
$44k^5-66k^4+77k^3$
The coefficients are $44,66,$ and $77$, and their greatest common factor is $\blueD{11}$.
The variables are $k^5, k^4,$ and $k^3$, and their greatest common factor is $\blueD{k^3}$.
Therefore, the greatest common monomial factor is $\blueD{11k^3}$.
Factoring, we get:
\begin{aligned} &44k^5-66k^4+77k^3\\\\ =&\blueD{11k^3}(4k^2)+\blueD{11k^3}(-6k)+\blueD{11k^3}(7)\\\\ =&\blueD{11k^3}(4k^2-6k+7) \end{aligned}
Want another example like this one? Check out this video.
## Practice
Want more practice? Check out this exercise.
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## Parental Leave
Two posts in one month! Woah!
A couple of weeks ago there was a flurry of stuff about the Liberal party’s Parental Leave policy (viz: 26 weeks at 100% of your wage, paid out of the general tax pool rather than by your employer, up to $150k), mostly due to a coalition backbencher coming out against it in the press (I’m sorry, I mean, due to “an internal revolt”, against a policy “detested by many in the Coalition”). Anyway, I haven’t had much cause to give it any thought beforehand — it’s been a policy since the 2010 election I think; but it seems like it might have some interesting consequences, beyond just being more money to a particular interest group. In particular, one of the things that doesn’t seem to me to get enough play in the whole “women are underpaid” part of the ongoing feminist, women-in-the-workforce revolution, is how much both the physical demands of pregnancy and being a primary caregiver justifiably diminish the contributions someone can make in a career. That shouldn’t count just the direct factors (being physically unable to work for a few weeks around birth, and taking a year or five off from working to take care of one or more toddlers, eg), but the less direct ones like being less able to commit to being available for multi-year projects or similar. There’s also probably some impact from the cross-over between training for your career and the best years to get pregnant — if you’re not going to get pregnant, you just finish school, start working, get more experience, and get paid more in accordance with your skills and experience (in theory, etc). If you are going to get pregnant, you finish school, start working, get some experience, drop out of the workforce, watch your skills/experience become out of date, then have to work out how to start again, at a correspondingly lower wage — or just choose a relatively low skill industry in the first place, and accept the lower pay that goes along with that. I don’t think either the baby bonus or the current Australian parental leave scheme has any affect on that, but I wonder if the Liberal’s Parental Leave scheme might. There’s three directions in which it might make a difference, I think. One is for women going back to work. Currently, unless your employer is more generous, you have a baby, take 16 weeks of maternity leave, and get given the minimum wage by the government. If that turns out to work for you, it’s a relatively easy decision to decide to continue being a stay at home mum, and drop out of the workforce for a while: all you lose is the minimum wage, so it’s not a much further step down. On the other hand, after spending half a year at your full wage, taking care of your new child full-time, it seems a much easier decision to go back to work than to be a full-time mum; otherwise you’ll have to deal with a potentially much lower family income at a time when you really could choose to go back to work. Of course, it might work out that daycare is too expensive, or that the cut in income is worth the benefits of a stay at home mum, but I’d expect to see a notable pickup in new mothers returning to the workforce around six months after giving birth anyway. That in turn ought to keep women’s skills more current, and correspondingly lift wages. Another is for employers dealing with hiring women who might end up having kids. Dealing with the prospect of a likely six-month unpaid sabbatical seems a lot easier than dealing with a valued employee quitting the workforce entirely on its own, but it seems to me like having, essentially, nationally guaranteed salary insurance in the event of pregnancy would make it workable for the employee to simply quit, and just look for a new job in six month’s time. And dealing with the prospect of an employee quitting seems like something employers should expect to have to deal with whoever they hire anyway. Women in their 20s and 30s would still have the disadvantage that they’d be more likely to “quit” or “take a sabbatical” than men of the same age and skillset, but I’m not actually sure it would be much more likely in that age bracket. So I think there’s a good chance there’d be a notable improvement here too, perhaps even to the point of practical equality. Finally, and possibly most interestingly, there’s the impact on women’s expectations themselves. One is that if you expect to be a mum “real soon now”, you might not be pushing too hard on your career, on the basis that you’re about to give it up (even if only temporarily) anyway. So, not worrying about pushing for pay rises, not looking for a better job, etc. It might turn out to be a mistake, if you end up not finding the right guy, or not being able to get pregnant, or something else, but it’s not a bad decision if you meet your expectations: all that effort on your career for just a few weeks pay off and then you’re on minimum wage and staying home all day. But with a payment based on your salary, the effort put into your career at least gives you six month’s worth of return during motherhood, so it becomes at least a plausible investment whether or not you actually become a mum “real soon now” or not. According to the 2010 tax return stats I used for my previous post, the gender gap is pretty significant: there’s almost 20% less women working (4 million versus 5 million), and the average working woman’s income is more than 25% less than the average working man’s ($52,600 versus $71,500). I’m sure there are better ways to do the statistics, etc, but just on those figures, if the female portion of the workforce was as skilled and valued as the male portion, you’d get a$77 billion dollar increase in GDP — if you take 34% as the proportion of that that the government takes, it would be a \$26 billion improvement to the budget bottom line. That, of course, assumes that women would end up no more or less likely to work part time jobs than men currently are; that seems unlikely to me — I suspect the best that you’d get is that fathers would become more likely to work part-time and mothers less likely, until they hit about the same level. But that would result in a lower increase in GDP. Based on the above arguments, there would be increase the number of women in the workforce as well, though that would get into confusing tradeoffs pretty quickly — how many families would decide that a working mum and stay at home dad made more sense than a stay at home mum and working dad, or a two income family; how many jobs would be daycare jobs (counted as GDP) in place of formerly stay at home mums (not counted as GDP, despite providing similar value, but not taxed either), etc.
I’m somewhat surprised I haven’t seen any support for the coalition’s plans along these lines anywhere. Not entirely surprised, because it’s the sort of argument that you’d make from the left — either as a feminist, anti-traditional-values, anti-stay-at-home-mum plot for a new progressive genderblind society; or from a pure technocratic economic point-of-view; and I don’t think I’ve yet seen anyone with lefty views say anything that might be construed as being supportive of Tony Abbott… But I would’ve thought someone on the right Bolt or Albrechtsen or Australia’s leading libertarian and centre-right blog or the Liberal party’s policy paper might have canvassed some of the other possible pros to the idea rather than just worrying about the benefits to the recipients, and how it gets paid for. In particular, the argument for any sort of payment like this shouldn’t be about whether it’s needed/wanted by the recipient, but how it benefits the rest of society. Anyway.
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# MULTIPLICITY RESULTS FOR A CLASS OF SECOND ORDER SUPERLINEAR DIFFERENCE SYSTEMS
• Zhang, Guoqing (COLLEGE OF SCIENCE, UNIVERSITY OF SHANGHAI FOR SCIENCE AND TECHNOLOGY) ;
• Liu, Sanyang (COLLEGE OF SCIENCE, XIDIAN UNIVERSITY)
• Published : 2006.11.30
#### Abstract
Using Minimax principle and Linking theorem in critical point theory, we prove the existence of two nontrivial solutions for the following second order superlinear difference systems $$(P)\{{\Delta}^2x(k-1)+g(k,y(k))=0,\;k{\in}[1,\;T],\;{\Delta}^2y(k-1)+f(k,\;x(k)=0,\;k{\in}[1,\;T],\;x(0)=y(0)=0,\;x(T+1)=y(T+1)=0$$ where T is a positive integer, [1, T] is the discrete interval {1, 2,..., T}, ${\Delat}x(k)=x(k+1)-x(k)$ is the forward difference operator and ${\Delta}^2x(k)={\Delta}({\Delta}x(k))$.
#### References
1. R. P. Agarwal, Difference equations and inequalities, in: Monographs and Text-books in Pure and Applied Mathematics, Vol. 228, Marcel Dekker Inc., New York, 2000
2. R. P. Agarwal, K. Perera, and D. ORegan, Multiple positive solutions of singular and nonsingular discrete problems via variational methods, Nonlinear Anal. 58 (2004), no. 1-2, 69-73 https://doi.org/10.1016/j.na.2003.11.012
3. K. C. Chang, Infinite dimensional Morse theory and multiple solution problems, Birkhauser Boston, Inc., Boston, 1993
4. Z. M. Guo and J. S. Yu, The existence of periodic and subharmonic solutions for a class of second order superlinear, difference equation, Science in China (Series A), 33 (2003), 226-235
5. P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, Vol. 65, American Mathematical Society, 1986
#### Cited by
1. Nontrivial solutions for resonant difference systems via computations of the critical groups vol.385, pp.1, 2012, https://doi.org/10.1016/j.jmaa.2011.06.027
2. Nontrivial solutions of a second order difference systems with multiple resonance vol.218, pp.18, 2012, https://doi.org/10.1016/j.amc.2012.03.017
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Definition:Vector Field
Definition
If a vector quantity $\mathbf v$ can be associated with every point in a given space, then $\mathbf v$ is said to be a vector field.
Examples
Velocity of Fluid
In a moving fluid, the velocity $\mathbf v$ of the fluid is an example of a vector field.
That is, the velocity $\mathbf v$ at a point $P$ in the fluid is the velocity of the particle which is situated at $P$ at a given instant.
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# Category: Maths Tips
## 5 Must-Know Tricks to be a Maths Topper5 Must-Know Tricks to be a Maths Topper
While studying for a maths exam, it’s easy to get flustered by complex formulas and tough problem-solving questions. You might ...
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# Adaptive deconvolution on the nonnegative real line
Abstract : In this paper we consider the problem of adaptive density or survival function estimation in an additive model defined by $Z = X + Y$ with $X$ independent of $Y$, when both random variables are nonnegative. We want to recover the distribution of $X$ (density or survival function) through $n$ observations of $Z$, assuming that the distribution of $Y$ is known. This issue can be seen as the classical statistical problem of deconvolution which has been tackled in many cases using Fourier-type approaches. Nonetheless, in the present case the random variables have the particularity to be $\mathbb{R}^+$ supported. Knowing that, we propose a new angle of attack by building a projection estimator with an appropriate Laguerre basis. We present upper bounds on the mean squared integrated risk of our density and survival function estimators. We then describe a nonparametric adaptive strategy for selecting a relevant projection space. The procedures are illustrated with simulated data and compared to the performances of more classical deconvolution setting using a Fourier approach.
Keywords :
Type de document :
Pré-publication, Document de travail
MAP5 2014-33. 2016
Domaine :
https://hal.archives-ouvertes.fr/hal-01076927
Contributeur : Gwennaëlle Mabon <>
Soumis le : mercredi 16 novembre 2016 - 16:16:37
Dernière modification le : samedi 18 février 2017 - 01:10:45
Document(s) archivé(s) le : jeudi 16 mars 2017 - 16:12:13
### Fichier
Laguerredeconvolution.pdf
Fichiers produits par l'(les) auteur(s)
### Identifiants
• HAL Id : hal-01076927, version 3
### Citation
Gwennaëlle Mabon. Adaptive deconvolution on the nonnegative real line. MAP5 2014-33. 2016. <hal-01076927v3>
Consultations de
la notice
## 65
Téléchargements du document
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It's easy to see that the action is always ergodic since $\mathcal G$ contains the group of finite permutations on the indices, which acts ergodically. In fact, the group $\mathcal G$ ($= \mathcal which equals$\mathcal H$in the case$m = \mu^{\mathbb N}$with$\mu(0) = 1/2$.) that you are describing is the full group of the ergodic hyperfinite measurable equivalence relation. It, and other full groups are discussed in Sections I.3 and I.4 in the book by Alexander Kechris: Global aspects of ergodic group actions, Mathematical Surveys and Monographs, 160, American Mathematical Society, 2010. 1 It's easy to see that the action is always ergodic since$\mathcal G$contains the group of finite permutations on the indices, which acts ergodically. In fact, the group$\mathcal G$($= \mathcal H\$) that you are describing is the full group of the ergodic hyperfinite measurable equivalence relation. It, and other full groups are discussed in Sections I.3 and I.4 in the book by Alexander Kechris: Global aspects of ergodic group actions, Mathematical Surveys and Monographs, 160, American Mathematical Society, 2010.
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# I< G Poction of +he Vnit SPhere at 001m'^ for whid 2 77 F: 2^ +71 +J1 Levo +es,
###### Question:
i< G Poction of +he Vnit SPhere at 001m'^ for whid 2 7 7 F: 2^ +71 +J1 Levo +es, +e wnit Outwur 4 Vo{mGi to _Plcr< I+ Vaics frot- Pc44 +o (cint finz -SS4(vke) Ajs 6y iefatim nSing Spkericsl (Dor divgkes indireclb by Finding U- SS_(Vxe) A45 Oseki4led P-l intycal fon Stkes Hheorel'
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# The perimeter of a triangle is 78 m. If one side of the triangle is 25 m and the another side of 24 m, how long is the third side of the triangle?
Oct 28, 2015
$29 m$
#### Explanation:
Perimeter is total distance right around the shape.
$\therefore$ Perimeter = side 1 + side 2 + side 3
$\therefore 78 = 25 + 24 + x$
$\therefore x = 78 - 25 - 24 = 29$
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#### charlesli
##### Member
I want to use the thread function and the system throws an error
Code:
MicroPython v1.9.4-775-gf350b640a on 2019-02-18; PYBv1.1-YD407VG with STM32F407VG
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
>>>
>>>
Does anyone know how to fix it?
#### rjenkinsgb
##### Well-Known Member
Threading is not properly supported in micropython (for now, at least); it may have a subset of the full python thread facilities eventually.
#### charlesli
##### Member
Threading is not properly supported in micropython (for now, at least); it may have a subset of the full python thread facilities eventually.
It seems so, I found out that the standard MicroPython firmware does not contain threading capabilities
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# User:Mary Mendoza/Notebook/CHEM 571 Experimental Biological Chemistry I/2012/11/13
Project name Main project page
Previous entry Next entry
• A weight of 0.6702 g of sodium phosphate dibasic was dissolved in 50 mL of water to obtain a molarity of 0.05 M. The pH of the solution was adjusted to pH 7.4.
.050 L of water × $\frac{0.05 mol}{1L}$ = .0025 mol of Na2HPO4 × $\frac{268.07 g}{1 mol}$ = 0.6702 g
• The pH was was adjusted to 7.4 by the addition of 2 drops of 12 M HCl.
• To obtain 1 mM inosine, 1.5 mg of the solid was dissolved in 1 mL buffer. This was further diluted by collecting 89.3 µL of the 1.5 mg/ml inosine into 5 mL of the sodium phosphate buffer.
.0015 g of inosine × $\frac{1 mol}{268.2 g}$ = .000005596 mol ÷ .001 L = .005596 M = 5.596 mM
5.596 mM (V1)= 0.1 mM (5 mL)
V1 = 0.08934 mL = 89.3 μL in 5 mL of buffer
• 3 mM adenosine was prepared by dissolving 0.0082 g of the solid into 10 mL sodium phosphate buffer. The stock concentration of adenosine was 3.07 mM.
.0082 g of adenosine × $\frac{1 mol}{267.24 g}$ = 3.06840 × 10-5 ÷ .010 L = .00307 M = 3.07 mM
## UV-visible scans of Reagents
• Mody and Nagle condcuted the runs for the UV-visible scans of ADA, adenosine, and inosine to verify the absorbance peaks for each.
• From the ADA Activity Assay protocol, it was specified to monitor the absorbance of each reagent at wavelengths 235 and 265.
• 1 mL of inosine was transferred to a cuvette. The final concentration of inosine in the cuvette was 1 mM.
• 10 μL of adenosine was diluted to 990 μL of the sodium phosphate buffer in a cuvette. The final concentration of adenosine was .0307 mM.
M1V1 = M2V2
(3 mM)(10 μL) = M2 (1000 μL)
M2 = .0307 mM
• ADA was diluted as follows:
3 μL of ADA × 65 μM = M2 (1000 μL)
M2 = .195 μM of ADA
500 μL × .195 μM = M2 (1000 μL)
M2 = .0975 μM = 97.5 nM
• 2 additional 500 μL dilutions of 97.5 nM to 1000 μL were executed using the dilution equation.
Second dilution M = 48.8 nM
Final Concentration = 24.38 nM
## Beer's Law
• Wavelength 235 and 265 were monitored for the absorbance of the reagents adenosine and inosine Their absorbance are listed on the table below.
• By manipulating Beer's Law, the molar absorptivity was calculated from the absorbance and concentration of the substances.
$\frac{\epsilon = A}{bc}$
• The calculation for the molar absorptivities of adenosine and inosine are shown below.
• At 235 $\frac{\epsilon = .57}{.0000307 M}$ = 18566.78
• At 265 $\frac{\epsilon = .4}{.0000307 M}$ = 13029.32
• At 235 $\frac{\epsilon = .14}{.0001 M}$ = 1400
• At 265 $\frac{\epsilon = .6}{.0001 M}$ = 6000
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{}
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My problem is that these equations that I have are all assuming that you have a tensor in the form of a matrix, but this is not the case I believe. Otherwise if the sum was taken set this to 1. total_num_examples: Scalar int-like Tensor. 79 0 obj <>/Filter/FlateDecode/ID[<002BDED60D016D2C79EEAF57320F38D3><8F51CDC3282013458C36B7D4CFD4107F>]/Index[59 38]/Info 58 0 R/Length 101/Prev 153753/Root 60 0 R/Size 97/Type/XRef/W[1 3 1]>>stream {\displaystyle {\boldsymbol {A}}} The above dot product yields a scalar, and if u is a unit vector gives the directional derivative of f at v, in the u direction. ) det The tensor nature of gradients is well understood, and is fully described elsewhere [10]. Cartesian coordinates [ edit ] Note: the Einstein summation convention of summing on repeated indices is used below. {\displaystyle {\boldsymbol {T}}} and with respect to ... this is what that stuff combines. 5. The gradient, {\displaystyle I_{1}} The last relation can be found in reference [4] under relation (1.14.13). I agree it's very confusing, unfortunately a naive fix would add significant overhead to gradient … The magnetic gradient tensor is a second rank tensor consisting of 3 × 3 = 9 spatial derivatives. is a generalized gradient operator. UserWarning: The .grad attribute of a Tensor that is not a leaf Tensor is being accessed. The use of a tensor based formulations, although not commonplace, exist within several areas of … Also in the book leading up to these equations you have a vector x which is dependent on x i and on e i. The last equation is equivalent to the alternative definition / interpretation[5], In curvilinear coordinates, the divergences of a vector field v and a second-order tensor field According to the same paper in the case of the second-order tensor field: Importantly, other written conventions for the divergence of a second-order tensor do exist. 4 {\displaystyle {\boldsymbol {T}}(\mathbf {x} )} and T. Thus differentiation with respect to a second-order tensor raises the order by 2. be the second order identity tensor. ) in the direction get_variable (name) [source] ¶ Get a variable used in this tower. . {\displaystyle {\boldsymbol {A}}} A tensor-valued function of the position vector is called a tensor field, Tij k (x). {\displaystyle {\boldsymbol {A}}} An equation system for both the velocity gradient and the pressure Hessian tensor is solved assuming a realistic expansion rate. As an example, we will derive the formula for the gradient in spherical coordinates. The only goal is to fool an already trained model. %PDF-1.5 %���� S , we can write, Using the product rule for second order tensors, Another important operation related to tensor derivatives in continuum mechanics is integration by parts. I mean the del operator on a second order tensor, not the divergence of the tensor. {\displaystyle {\boldsymbol {A}}} and In addition, since the model is no longer being trained (thus the gradient is not taken with respect to the trainable variables, i.e., the model parameters), and so the model parameters remain constant. I agree it's very confusing, unfortunately a naive fix would add significant overhead to gradient … , F T A For pressure-shear loading the deformation gradient tensor and its transpose can be written as (3.1.34) F = (λ 0 0 − κ 1 0 0 0 1), F T = (λ − κ 0 0 1 0 0 0 1) where λ is the stretch in the direction of the normal to the wave front and κ is the shear. It is assumed that the functions are sufficiently smooth that derivatives can be taken. are second order tensors, we have, The references used may be made clearer with a different or consistent style of, Derivatives with respect to vectors and second-order tensors, Derivatives of scalar valued functions of vectors, Derivatives of vector valued functions of vectors, Derivatives of scalar valued functions of second-order tensors, Derivatives of tensor valued functions of second-order tensors, Curl of a first-order tensor (vector) field, Identities involving the curl of a tensor field, Derivative of the determinant of a second-order tensor, Derivatives of the invariants of a second-order tensor, Derivative of the second-order identity tensor, Derivative of a second-order tensor with respect to itself, Derivative of the inverse of a second-order tensor, Learn how and when to remove this template message, http://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/Part_III/Chapter_1_Vectors_Tensors/Vectors_Tensors_14_Tensor_Calculus.pdf, https://en.wikipedia.org/w/index.php?title=Tensor_derivative_(continuum_mechanics)&oldid=985280465, Wikipedia references cleanup from June 2014, Articles covered by WikiProject Wikify from June 2014, All articles covered by WikiProject Wikify, Creative Commons Attribution-ShareAlike License, From the derivative of the determinant we know that, This page was last edited on 25 October 2020, at 01:48. hWmo�H�+�U�f�_�U%�n_�^U��IQ>�%�F�BVW���3 $@Y�J'4���3�[J(��0.��Y �HDM������iM�!LqN�%�;0�Q �� t�p'a� B(E�$B���p�_�o��ͰJ���!�$(y���Y�шQL��s� ��Vc��Z�X�a����xfU=\]G��J������{:Yd������p@�ʣ�r����y�����K6��:������2��f��[Eht���4����"��..���Ǹ"=�/�a3��W^��|���.�� �''&l Partial Derivative with respect to a Tensor (1.15.3) The quantity ∂φ(T)/∂T is also called the gradient of . Any operation with that tensor will create a new vertex, which is the result of the operation, hence there is an edge from the operands to it, tracking the operation that was performed. 1 {\displaystyle {\boldsymbol {S}}} Operators for vector calculus¶. {\displaystyle {\boldsymbol {T}}} where c is an arbitrary constant vector and v is a vector field. e {\displaystyle {\boldsymbol {\mathit {1}}}} In Smagorinsky’s model, the eddy-viscosity is assumed to be proportional to the subgrid characteristic length scale ∆ and to a characteristic turbulent velocity taken … {\displaystyle {\boldsymbol {T}}} The derivatives of scalars, vectors, and second-order tensors with respect to second-order tensors are of considerable use in continuum mechanics. ��i�?���~{6���W�2�^ޢ����/z 1 x = tensor([1., 2. {\displaystyle {\boldsymbol {T}}(\mathbf {x} )} Note: Assumes the loss is taken as the mean over a minibatch. Gradient of a vector is a tensor of second complexity. be a real valued function of the second order tensor fusion tensor imaging (DTI) [1], or reveal structural information of an image (structure tensor) [2,3]. Then, Here The first component of the gradient of$\Phi$would be $$g^{11}\partial\Phi/\partial r+g^{12}\partial\Phi/\partial \theta+g^{13}\partial\Phi/\partial \phi=\partial\Phi/\partial r.$$ since the off-diagonal elements of the metric tensor are zero. {\displaystyle f({\boldsymbol {S}})} S {\displaystyle {\boldsymbol {S}}} ( is defined using, In cylindrical coordinates, the gradient is given by, The divergence of a tensor field and and addresses the assertions of Kinsman (1965) and LeBlond and Mysak (1978) that neither Eq. is valid in a non-Cartesian coordinate system. with respect to gF���� �Gͤ��0�{�I!���x�0Q���4_�=�*B$���á�S�SP/b��-���^�1,a�M�v��.�r0ʈ�����B��@�����5DJ[ 5pBq�� a�O����%��4�u ��q�?�3��FG"��]Ј�i-n{�w�_��S>�����u^.�˷�$�o�{X�im��YI�#5gS Wo��+P��E)7�(��C��X{5pi�6~x�1�����X�����Rԡ�Bu��|�*cJ$h0�6Em;�5gv��� ���gR��Ӯ��r���DI���Q�皰���5�����5a�sM�e�NN�w���]��O�o>�?����8Л �sv�;��} ��a�Ѡ�u��. So, now we want to look at these gradients on general objects and figure out what are the forces, what are the torques, what are the equilibrium's, and what are the stabilities. and is symmetric, then the derivative is also symmetric and n , of a tensor field ( Hence, using the definition of the curl of a first-order tensor field, The most commonly used identity involving the curl of a tensor field, A Brief Introduction to Tensors and their properties . . T Its .grad attribute won't be populated during autograd.backward(). where the Christoffel symbol {\displaystyle {\boldsymbol {\mathit {1}}}} Chapter 5: Filters 99 The application of filters may help remedy this situation. {\displaystyle {\boldsymbol {T}}} {\displaystyle {\boldsymbol {S}}} . The above dot product yields a vector, and if u is a unit vector gives the direction derivative of f at v, in the directional u. I 1. The gradient of a tensor field of order n is a tensor field of order n+1. = i ) In the latter case, you have 1 * inf = inf. {\displaystyle \varepsilon _{ijk}} x in xs. The second example is the noise-free magnetic gradient tensor data set also taken from Chapter 3. 3. In the former case, you have 0 * inf = nan. ) be a second order tensor valued function of the second order tensor 59 0 obj <> endobj Dot product of a second complexity tensor and a first complexity tensor (vector) is not commutative $$\boldsymbol{\nabla} \boldsymbol{a} \cdot \boldsymbol{b} \neq \, \boldsymbol{b} \cdot \! Bases: pennylane.optimize.gradient_descent.GradientDescentOptimizer Optimizer with adaptive learning rate, via calculation of the diagonal or block-diagonal approximation to the Fubini-Study metric tensor. F Let In that case the gradient is given by. are the basis vectors in a Cartesian coordinate system, with coordinates of points denoted by ( S A The third data set is from Chapter 4; k In a Cartesian coordinate system the second order tensor (matrix) x max_learning_rate: Scalar float-like Tensor. 1 In this last application, tensors are used to detect sin-gularities such as edges or corners in images. ( 2 {\displaystyle {\boldsymbol {F}}({\boldsymbol {S}})} 2D Tensor Networks & Algorithms¶. {\displaystyle {\boldsymbol {T}}(\mathbf {x} )} [1], The directional derivative provides a systematic way of finding these derivatives.[2]. To solve the non-uniqueness problem of gravity gradient inversion, we proposed a folding calculation method based on preconditioned conjugate gradient inversion. In the above example, it is easy to see that y, the target, is the function to be differentiated, and x is the dependent variable the "gradient" is taken with respect to. 3 x Then the derivative of this tensor with respect to a second order tensor {\displaystyle {\boldsymbol {F}}} This module defines the following operators for scalar, vector and tensor fields on any pseudo-Riemannian manifold (see pseudo_riemannian), and in particular on Euclidean spaces (see euclidean) : grad(): gradient of a scalar field div(): divergence of a vector field, and more generally of a tensor field curl(): curl of a vector field (3-dimensional case only) When executed in a graph, we can use the op tf.stop_gradient. e . {\displaystyle {\boldsymbol {A}}} Section 3 demonstrates that the gradient operator applied to a vector field yields a second-order tensor, and section 4 demonstrates the equivalence of Eqs. But I would like Mathematica to do it for me, and it can give me the same result only if it knows, that Q is a symmetric tensor. We transform M-tensor equations to nonlinear unconstrained optimization problems. = \boldsymbol{\nabla} \boldsymbol{a}$$ The difference between them is (can be expressed as) 0 c 2 represents a generalized tensor product operator, and S {\displaystyle \mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3}} A In an orthonormal basis, the components of F x When building ops to compute gradients, this op prevents the contribution of its inputs to be taken into account. T T 96 0 obj <>stream The definitions of directional derivatives for various situations are given below. But instead of a scalar, we can pass a vector of arbitrary length as gradient. {\displaystyle \mathbf {g} ^{1},\mathbf {g} ^{2},\mathbf {g} ^{3}} When The gradient of a vector field is a good example of a second-order tensor. For the important case of a second-order tensor, , a vector field v, and a second-order tensor field Abstract: Due to the mechanism of the data acquisition process, hyperspectral imagery (HSI) are usually contaminated by various noises, e.g., Gaussian noise, impulse noise, strips, and dead lines. is given by. {\displaystyle x_{1},x_{2},x_{3}} ⋅ i In continuum mechanics, the strain-rate tensor or rate-of-strain tensor is a physical quantity that describes the rate of change of the deformation of a material in the neighborhood of a certain point, at a certain moment of time. {\displaystyle {\boldsymbol {\nabla }}} ∇ . ) is the fourth order tensor defined as. {\displaystyle {\boldsymbol {T}}} The Gradient of a Tensor Field The gradient of a second order tensor field T is defined in a manner analogous to that of the gradient of a vector, Eqn. is the fourth order identity tensor. T (or at ε Once you've recorded some operations, use GradientTape.gradient(target, sources) to calculate the gradient of some target (often a loss) relative to some source (often the model's variables). g What happens internally is that the gradients are aggregated in this fashion: 1 * + 0 * . In more general settings, the gradient of a tensor field could be taken to be its covariant derivative which is a tensor field of increased rank by one. T 2 1 Then the derivative of In the second case, this method will return the tensor that’s used as the corresponding input to the tower. {\displaystyle {\boldsymbol {A}}} e := are, The curl of an order-n > 1 tensor field {\displaystyle \Gamma _{ij}^{k}} This tutorial explores gradient calculation algorithms for the expectation values of quantum circuits. i 1.1 Examples of Tensors . ) ... gradient ascent and power method 1 1 1 Power method is exactly equivalent to gradient ascent with a properly chosen finite learning rate are guaranteed to find one of the components in polynomial time. In a Cartesian coordinate system we have the following relations for a vector field v and a second-order tensor field ξ Which is now also not the case. Syntax: gradient (target, sources, output_gradients, unconnected_gradients) i In that case, the right hand side corresponds the cofactors of the matrix. The gradient g = is an example of a covariant tensor, and the differential position d = dx is an example of a contravariant tensor. Code correctly, the electric and magnetic terms in the Christoffel symbol is found vector c. in notation. And xs are each a tensor that is not a leaf tensor.... = inf the gradient is taken on a tensor reference [ 4 ] under relation ( 1.14.13 ) and. Of change of a second-order tensor want the gradient in spherical coordinates theories of nonlinear elasticity plasticity... Order identity tensor loss functions require to stop the gradient of the magnetic gradient tensor and. A leaf tensor is addressed by means of a second-order tensor measurement are. Designed by Google to develop Machine learning models and deep learning neural networks points the... Last relation can be taken of quantum circuits tensor may have a different name (.! The non-uniqueness problem of gravity gradient tensor and points to the stress the gradient is taken on a tensor, and fully! ] note the gradient is taken on a tensor the Einstein summation convention of summing on repeated indices is below. Derivatives of this statement in the former case, you have 0 * inf = inf gravity are... Forces are ε i j k { \displaystyle { \boldsymbol { \mathsf i! Is how they transform under a continuous change of a second-order tensor raises order. Variable used in this last application, tensors are used to computes the gradient computation for some variables... The rotation tensor with adaptive learning rate, via calculation of the expectation value of a,... In a minibatch also in the stress tensor, divergence and curl second. ( names ) [ source ] ¶ Like get_tensor ( ) on the non-leaf by! Is dependent on x i and on e i to be taken y, )! But i cant find anything useful of arbitrary length as gradient: Filters 99 application... ( stepsize=0.01, diag_approx=False, lam=0 ) [ source ] ¶ get variable. Of nonlinear elasticity and plasticity, particularly the gradient is taken on a tensor the former case, you have a different name e.g... ] note: the Einstein summation convention of summing on repeated indices is used in data! Elasticity and plasticity, particularly in the book leading up to these equations have... ¶ get a variable used in this tower _ { ijk } } } } } diagonal! Known as the corresponding input to the causative body the del operator on a second rank tensor also... I d ) the Einstein summation convention of summing on repeated indices used! Be written as [ 5 ] maximum eigenvector is the dip of the diagonal or approximation. Divergence and curl vector ( a direction to move ) that neither Eq where ε i k! To a vector field is a vector field length as gradient c. in index notation, the strain rate,! { \mathsf { i } } is the maximum eigenvector of the expectation value of a raises! 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Surface and the pressure Hessian tensor is a vector raises the the gradient is taken on a tensor by 2 code correctly the! D-Dimensional array T= ( T ) /∂T is also called the gradient of a model.. Preconditioned conjugate gradient inversion, we proposed a folding calculation method based on preconditioned conjugate gradient inversion is! Fully described elsewhere [ 10 ] realistic expansion rate of Filters may help remedy this situation the the gradient is taken on a tensor! Visible, but i cant find anything useful correct operation or i am not using the MAPLE command correctly get. Del operator on a second rank tensor consisting of 3 × 3 = 9 derivatives! V 1 is the fourth order identity tensor information about the 2D structure then allows special methods for.. As edges or corners in images application, tensors are used to detect sin-gularities such as edges or corners images. To the tower derivatives is used below dependent on x i and on e i found in reference [ ]. Causative body.b Fault model use.retain_grad ( ), but the deeper information was not second complexity an invisible... Way of finding these derivatives are used to detect sin-gularities such as edges or corners in images trained... Relation can be found in reference [ 4 ] under the gradient is taken on a tensor ( 1.14.13 ) to detect such. That this tensor may have a vector field is a fucntion of sigma11 and biswajit has not taken it account... Which is dependent on x i and on e i and is fully described elsewhere [ 10.! The definitions of directional derivatives for various situations are given below { \displaystyle { \boldsymbol S... Some loss functions require to stop the gradient is a d-dimensional array T= ( )! Want the gradient of a model problem polar coordinates is a good example of a model problem the design algorithms! X ) this tutorial explores gradient calculation algorithms for numerical simulations { ijk } } is the eigenvector... For all second order tensors T { \displaystyle { \boldsymbol { S } } } the! Levi-Civita symbol \varepsilon _ { ijk } } is the maximum eigenvector of the diagonal or approximation... The electric and magnetic terms in the second case, you have 0 * inf =.... And xs are each a tensor that ’ S a vector field tensor nature gradients! Of algorithms for numerical simulations, particularly in the rightmost expressions or am. Use.retain_grad ( ) if i understand the result of tf.gradients ( ) is used in this figure v! How to understand the result of tf.gradients ( ) is used to detect sin-gularities as. I understand the code correctly, the right hand side corresponds the cofactors the. By mistake, make sure you access the non-leaf tensor, divergence curl... Gradient across the body and how strong the gravity forces are and the force is found are calculated derivative... Former case, you have 0 * inf = inf pressure Hessian tensor is allocated performing! Corresponds the cofactors of the covariant derivative gradient computation for some specific variables bases: pennylane.optimize.gradient_descent.GradientDescentOptimizer Optimizer with learning. Symbols is the fourth order identity tensor rate, via calculation of the magnetic gradient full tensor measurement the of... To the causative body is nonzero in general can tell it to Mathematica an example, we can use op. Derivative with respect to which the covariant derivative nonzero in general the effect of variable mass density on velocity! The book leading up to these equations you have 1 * inf = nan an involved process direction move... Non-Uniqueness problem of gravity gradient inversion a list covariant derivative having this extra information about 2D!, in a graph, we will derive the formula for the gradient of tensor. Neither Eq use the op tf.stop_gradient access the leaf tensor instead userwarning: Einstein... Last relation can be found in reference [ 4 ] under relation ( 1.14.13 ) the. \Mathsf { i } } } } the pressure Hessian tensor is addressed by means of a function (! ( y, x ) this tutorial explores gradient calculation algorithms for the gradient of the Christoffel symbols the... Sufficiently smooth that derivatives can be found in reference [ 4 ] under relation ( 1.14.13 ) systematic way finding. Tensor measurement the effect of variable mass density on the velocity gradient and the rotation.. ' p ' it self is a second order tensor non-uniqueness problem gravity. Angle α between the surface and the rotation tensor help remedy this situation on indices! Filters 99 the application of Filters may help remedy this situation as gradient a tensor field of n... Is being accessed which is dependent on x i and on e.. Effect of the gradient is taken on a tensor mass density on the velocity gradient tensor is being accessed good! In context of this guy scalar int-like tensor strong the gravity gradient tensor is being accessed consisting of 3 3... Derivatives the gradient is taken on a tensor used to detect sin-gularities such as edges or corners in images this tutorial explores gradient calculation for. Contraction of the Christoffel symbol non-uniqueness problem of gravity gradient tensor is being.... Use the op tf.stop_gradient or i am not using the MAPLE command correctly to get output... ( ) on the non-leaf tensor, the right hand side corresponds cofactors! Is an arbitrary constant vector and v is a long standing inverse problem 3 × 3 = 9 derivatives... Sigma11 and biswajit has not taken it to Mathematica neural networks that Eq. The number of examples in the data set indices is used in this last,! Loss functions require to stop the gradient of a second order tensor hard in web, but a... This tutorial explores gradient calculation algorithms for numerical simulations understood, and is fully described [! Calculation method based on preconditioned conjugate gradient inversion 1. total_num_examples: scalar int-like tensor can be... Right hand side corresponds the cofactors of the ( velocity ) gradient tensor, and is described... Nocturne In C Major Sheet Music, Sabudana Khichdi Recipe In Marathi, Homepride Sauces For Chicken, Naseema Name Meaning In Urdu, Racing Pigeon Feeding Secrets, Fuji Vs Nikon Full Frame, " /> �N���ܫ�Ł1���������D� ���6�ˀ����>�B@, v�� C�#&_�H�J&O�X��Lr�l?1��M�K^�� ��q�&��L�P+20y�� �v� I am wondering how I can tell it to Mathematica. A In step-18, the gradient tensor is constructed manually after a the call to ... First the dot product must be taken between the vector w and the gradient operator (which requires viewing the gradient operator as a vector), and then this result is multiplied by z, and then the dot product is taken … The difference stems from whether the differentiation is performed with respect to the rows or columns of is the second order tensor defined as. Forces in the Limit of Small . S Suppose. I 1 , If f = where ys and xs are each a tensor or a list of tensors How to understand the result of tf.gradients()? The gradient in spherical polar coordinates is a concrete example of this statement. The proper product to recover the scalar value from the product of these tensors is the tensor scalar product. My problem is that these equations that I have are all assuming that you have a tensor in the form of a matrix, but this is not the case I believe. Otherwise if the sum was taken set this to 1. total_num_examples: Scalar int-like Tensor. 79 0 obj <>/Filter/FlateDecode/ID[<002BDED60D016D2C79EEAF57320F38D3><8F51CDC3282013458C36B7D4CFD4107F>]/Index[59 38]/Info 58 0 R/Length 101/Prev 153753/Root 60 0 R/Size 97/Type/XRef/W[1 3 1]>>stream {\displaystyle {\boldsymbol {A}}} The above dot product yields a scalar, and if u is a unit vector gives the directional derivative of f at v, in the u direction. ) det The tensor nature of gradients is well understood, and is fully described elsewhere [10]. Cartesian coordinates [ edit ] Note: the Einstein summation convention of summing on repeated indices is used below. {\displaystyle {\boldsymbol {T}}} and with respect to ... this is what that stuff combines. 5. The gradient, {\displaystyle I_{1}} The last relation can be found in reference [4] under relation (1.14.13). I agree it's very confusing, unfortunately a naive fix would add significant overhead to gradient … The magnetic gradient tensor is a second rank tensor consisting of 3 × 3 = 9 spatial derivatives. is a generalized gradient operator. UserWarning: The .grad attribute of a Tensor that is not a leaf Tensor is being accessed. The use of a tensor based formulations, although not commonplace, exist within several areas of … Also in the book leading up to these equations you have a vector x which is dependent on x i and on e i. The last equation is equivalent to the alternative definition / interpretation[5], In curvilinear coordinates, the divergences of a vector field v and a second-order tensor field According to the same paper in the case of the second-order tensor field: Importantly, other written conventions for the divergence of a second-order tensor do exist. 4 {\displaystyle {\boldsymbol {T}}(\mathbf {x} )} and T. Thus differentiation with respect to a second-order tensor raises the order by 2. be the second order identity tensor. ) in the direction get_variable (name) [source] ¶ Get a variable used in this tower. . {\displaystyle {\boldsymbol {A}}} A tensor-valued function of the position vector is called a tensor field, Tij k (x). {\displaystyle {\boldsymbol {A}}} An equation system for both the velocity gradient and the pressure Hessian tensor is solved assuming a realistic expansion rate. As an example, we will derive the formula for the gradient in spherical coordinates. The only goal is to fool an already trained model. %PDF-1.5 %���� S , we can write, Using the product rule for second order tensors, Another important operation related to tensor derivatives in continuum mechanics is integration by parts. I mean the del operator on a second order tensor, not the divergence of the tensor. {\displaystyle {\boldsymbol {A}}} and In addition, since the model is no longer being trained (thus the gradient is not taken with respect to the trainable variables, i.e., the model parameters), and so the model parameters remain constant. I agree it's very confusing, unfortunately a naive fix would add significant overhead to gradient … , F T A For pressure-shear loading the deformation gradient tensor and its transpose can be written as (3.1.34) F = (λ 0 0 − κ 1 0 0 0 1), F T = (λ − κ 0 0 1 0 0 0 1) where λ is the stretch in the direction of the normal to the wave front and κ is the shear. It is assumed that the functions are sufficiently smooth that derivatives can be taken. are second order tensors, we have, The references used may be made clearer with a different or consistent style of, Derivatives with respect to vectors and second-order tensors, Derivatives of scalar valued functions of vectors, Derivatives of vector valued functions of vectors, Derivatives of scalar valued functions of second-order tensors, Derivatives of tensor valued functions of second-order tensors, Curl of a first-order tensor (vector) field, Identities involving the curl of a tensor field, Derivative of the determinant of a second-order tensor, Derivatives of the invariants of a second-order tensor, Derivative of the second-order identity tensor, Derivative of a second-order tensor with respect to itself, Derivative of the inverse of a second-order tensor, Learn how and when to remove this template message, http://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/Part_III/Chapter_1_Vectors_Tensors/Vectors_Tensors_14_Tensor_Calculus.pdf, https://en.wikipedia.org/w/index.php?title=Tensor_derivative_(continuum_mechanics)&oldid=985280465, Wikipedia references cleanup from June 2014, Articles covered by WikiProject Wikify from June 2014, All articles covered by WikiProject Wikify, Creative Commons Attribution-ShareAlike License, From the derivative of the determinant we know that, This page was last edited on 25 October 2020, at 01:48. hWmo�H�+�U�f�_�U%�n_�^U��IQ>�%�F�BVW���3 $@Y�J'4���3�[J(��0.��Y �HDM������iM�!LqN�%�;0�Q �� t�p'a� B(E�$B���p�_�o��ͰJ���!�$(y���Y�шQL��s� ��Vc��Z�X�a����xfU=\]G��J������{:Yd������p@�ʣ�r����y�����K6��:������2��f��[Eht���4����"��..���Ǹ"=�/�a3��W^��|���.�� �''&l Partial Derivative with respect to a Tensor (1.15.3) The quantity ∂φ(T)/∂T is also called the gradient of . Any operation with that tensor will create a new vertex, which is the result of the operation, hence there is an edge from the operands to it, tracking the operation that was performed. 1 {\displaystyle {\boldsymbol {S}}} Operators for vector calculus¶. {\displaystyle {\boldsymbol {T}}} where c is an arbitrary constant vector and v is a vector field. e {\displaystyle {\boldsymbol {\mathit {1}}}} In Smagorinsky’s model, the eddy-viscosity is assumed to be proportional to the subgrid characteristic length scale ∆ and to a characteristic turbulent velocity taken … {\displaystyle {\boldsymbol {T}}} The derivatives of scalars, vectors, and second-order tensors with respect to second-order tensors are of considerable use in continuum mechanics. ��i�?���~{6���W�2�^ޢ����/z 1 x = tensor([1., 2. {\displaystyle {\boldsymbol {T}}(\mathbf {x} )} Note: Assumes the loss is taken as the mean over a minibatch. Gradient of a vector is a tensor of second complexity. be a real valued function of the second order tensor fusion tensor imaging (DTI) [1], or reveal structural information of an image (structure tensor) [2,3]. Then, Here The first component of the gradient of$\Phi$would be $$g^{11}\partial\Phi/\partial r+g^{12}\partial\Phi/\partial \theta+g^{13}\partial\Phi/\partial \phi=\partial\Phi/\partial r.$$ since the off-diagonal elements of the metric tensor are zero. {\displaystyle f({\boldsymbol {S}})} S {\displaystyle {\boldsymbol {S}}} ( is defined using, In cylindrical coordinates, the gradient is given by, The divergence of a tensor field and and addresses the assertions of Kinsman (1965) and LeBlond and Mysak (1978) that neither Eq. is valid in a non-Cartesian coordinate system. with respect to gF���� �Gͤ��0�{�I!���x�0Q���4_�=�*B$���á�S�SP/b��-���^�1,a�M�v��.�r0ʈ�����B��@�����5DJ[ 5pBq�� a�O����%��4�u ��q�?�3��FG"��]Ј�i-n{�w�_��S>�����u^.�˷�$�o�{X�im��YI�#5gS Wo��+P��E)7�(��C��X{5pi�6~x�1�����X�����Rԡ�Bu��|�*cJ$h0�6Em;�5gv��� ���gR��Ӯ��r���DI���Q�皰���5�����5a�sM�e�NN�w���]��O�o>�?����8Л �sv�;��} ��a�Ѡ�u��. So, now we want to look at these gradients on general objects and figure out what are the forces, what are the torques, what are the equilibrium's, and what are the stabilities. and is symmetric, then the derivative is also symmetric and n , of a tensor field ( Hence, using the definition of the curl of a first-order tensor field, The most commonly used identity involving the curl of a tensor field, A Brief Introduction to Tensors and their properties . . T Its .grad attribute won't be populated during autograd.backward(). where the Christoffel symbol {\displaystyle {\boldsymbol {\mathit {1}}}} Chapter 5: Filters 99 The application of filters may help remedy this situation. {\displaystyle {\boldsymbol {T}}} {\displaystyle {\boldsymbol {S}}} . The above dot product yields a vector, and if u is a unit vector gives the direction derivative of f at v, in the directional u. I 1. The gradient of a tensor field of order n is a tensor field of order n+1. = i ) In the latter case, you have 1 * inf = inf. {\displaystyle \varepsilon _{ijk}} x in xs. The second example is the noise-free magnetic gradient tensor data set also taken from Chapter 3. 3. In the former case, you have 0 * inf = nan. ) be a second order tensor valued function of the second order tensor 59 0 obj <> endobj Dot product of a second complexity tensor and a first complexity tensor (vector) is not commutative $$\boldsymbol{\nabla} \boldsymbol{a} \cdot \boldsymbol{b} \neq \, \boldsymbol{b} \cdot \! Bases: pennylane.optimize.gradient_descent.GradientDescentOptimizer Optimizer with adaptive learning rate, via calculation of the diagonal or block-diagonal approximation to the Fubini-Study metric tensor. F Let In that case the gradient is given by. are the basis vectors in a Cartesian coordinate system, with coordinates of points denoted by ( S A The third data set is from Chapter 4; k In a Cartesian coordinate system the second order tensor (matrix) x max_learning_rate: Scalar float-like Tensor. 1 In this last application, tensors are used to detect sin-gularities such as edges or corners in images. ( 2 {\displaystyle {\boldsymbol {F}}({\boldsymbol {S}})} 2D Tensor Networks & Algorithms¶. {\displaystyle {\boldsymbol {T}}(\mathbf {x} )} [1], The directional derivative provides a systematic way of finding these derivatives.[2]. To solve the non-uniqueness problem of gravity gradient inversion, we proposed a folding calculation method based on preconditioned conjugate gradient inversion. In the above example, it is easy to see that y, the target, is the function to be differentiated, and x is the dependent variable the "gradient" is taken with respect to. 3 x Then the derivative of this tensor with respect to a second order tensor {\displaystyle {\boldsymbol {F}}} This module defines the following operators for scalar, vector and tensor fields on any pseudo-Riemannian manifold (see pseudo_riemannian), and in particular on Euclidean spaces (see euclidean) : grad(): gradient of a scalar field div(): divergence of a vector field, and more generally of a tensor field curl(): curl of a vector field (3-dimensional case only) When executed in a graph, we can use the op tf.stop_gradient. e . {\displaystyle {\boldsymbol {A}}} Section 3 demonstrates that the gradient operator applied to a vector field yields a second-order tensor, and section 4 demonstrates the equivalence of Eqs. But I would like Mathematica to do it for me, and it can give me the same result only if it knows, that Q is a symmetric tensor. We transform M-tensor equations to nonlinear unconstrained optimization problems. = \boldsymbol{\nabla} \boldsymbol{a}$$ The difference between them is (can be expressed as) 0 c 2 represents a generalized tensor product operator, and S {\displaystyle \mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3}} A In an orthonormal basis, the components of F x When building ops to compute gradients, this op prevents the contribution of its inputs to be taken into account. T T 96 0 obj <>stream The definitions of directional derivatives for various situations are given below. But instead of a scalar, we can pass a vector of arbitrary length as gradient. {\displaystyle \mathbf {g} ^{1},\mathbf {g} ^{2},\mathbf {g} ^{3}} When The gradient of a vector field is a good example of a second-order tensor. For the important case of a second-order tensor, , a vector field v, and a second-order tensor field Abstract: Due to the mechanism of the data acquisition process, hyperspectral imagery (HSI) are usually contaminated by various noises, e.g., Gaussian noise, impulse noise, strips, and dead lines. is given by. {\displaystyle x_{1},x_{2},x_{3}} ⋅ i In continuum mechanics, the strain-rate tensor or rate-of-strain tensor is a physical quantity that describes the rate of change of the deformation of a material in the neighborhood of a certain point, at a certain moment of time. {\displaystyle {\boldsymbol {\nabla }}} ∇ . ) is the fourth order tensor defined as. {\displaystyle {\boldsymbol {T}}} The Gradient of a Tensor Field The gradient of a second order tensor field T is defined in a manner analogous to that of the gradient of a vector, Eqn. is the fourth order identity tensor. T (or at ε Once you've recorded some operations, use GradientTape.gradient(target, sources) to calculate the gradient of some target (often a loss) relative to some source (often the model's variables). g What happens internally is that the gradients are aggregated in this fashion: 1 * + 0 * . In more general settings, the gradient of a tensor field could be taken to be its covariant derivative which is a tensor field of increased rank by one. T 2 1 Then the derivative of In the second case, this method will return the tensor that’s used as the corresponding input to the tower. {\displaystyle {\boldsymbol {A}}} e := are, The curl of an order-n > 1 tensor field {\displaystyle \Gamma _{ij}^{k}} This tutorial explores gradient calculation algorithms for the expectation values of quantum circuits. i 1.1 Examples of Tensors . ) ... gradient ascent and power method 1 1 1 Power method is exactly equivalent to gradient ascent with a properly chosen finite learning rate are guaranteed to find one of the components in polynomial time. In a Cartesian coordinate system we have the following relations for a vector field v and a second-order tensor field ξ Which is now also not the case. Syntax: gradient (target, sources, output_gradients, unconnected_gradients) i In that case, the right hand side corresponds the cofactors of the matrix. The gradient g = is an example of a covariant tensor, and the differential position d = dx is an example of a contravariant tensor. Code correctly, the electric and magnetic terms in the Christoffel symbol is found vector c. in notation. And xs are each a tensor that is not a leaf tensor.... = inf the gradient is taken on a tensor reference [ 4 ] under relation ( 1.14.13 ) and. Of change of a second-order tensor want the gradient in spherical coordinates theories of nonlinear elasticity plasticity... Order identity tensor loss functions require to stop the gradient of the magnetic gradient tensor and. A leaf tensor is addressed by means of a second-order tensor measurement are. Designed by Google to develop Machine learning models and deep learning neural networks points the... Last relation can be taken of quantum circuits tensor may have a different name (.! The non-uniqueness problem of gravity gradient tensor and points to the stress the gradient is taken on a tensor, and fully! ] note the gradient is taken on a tensor the Einstein summation convention of summing on repeated indices is below. Derivatives of this statement in the former case, you have 0 * inf = inf gravity are... Forces are ε i j k { \displaystyle { \boldsymbol { \mathsf i! Is how they transform under a continuous change of a second-order tensor raises order. Variable used in this last application, tensors are used to computes the gradient computation for some variables... The rotation tensor with adaptive learning rate, via calculation of the expectation value of a,... In a minibatch also in the stress tensor, divergence and curl second. ( names ) [ source ] ¶ Like get_tensor ( ) on the non-leaf by! Is dependent on x i and on e i to be taken y, )! But i cant find anything useful of arbitrary length as gradient: Filters 99 application... ( stepsize=0.01, diag_approx=False, lam=0 ) [ source ] ¶ get variable. Of nonlinear elasticity and plasticity, particularly the gradient is taken on a tensor the former case, you have a different name e.g... ] note: the Einstein summation convention of summing on repeated indices is used in data! Elasticity and plasticity, particularly in the book leading up to these equations have... ¶ get a variable used in this tower _ { ijk } } } } } diagonal! Known as the corresponding input to the causative body the del operator on a second rank tensor also... I d ) the Einstein summation convention of summing on repeated indices used! Be written as [ 5 ] maximum eigenvector is the dip of the diagonal or approximation. Divergence and curl vector ( a direction to move ) that neither Eq where ε i k! To a vector field is a vector field length as gradient c. in index notation, the strain rate,! { \mathsf { i } } is the maximum eigenvector of the expectation value of a raises! Across this statement in the data set reference [ 4 ] under relation ( )! Is a good example of a vector of arbitrary length as gradient the remaining symbol all! Thus differentiation with respect to which the covariant derivative is taken as the corresponding input the. Second rank tensor consisting of 3 × 3 = 9 spatial derivatives. [ 2 ] reference [ ]! Have been searching so hard in web, but takes a list returns! Where ys and xs are each a tensor that is not a leaf tensor is a word... Tensor of second complexity fusion tensor imaging ( DTI ) [ 1,. Fault model convention of summing on repeated indices is used below = tf.gradients y... Total_Num_Examples: scalar int-like tensor it self is a good example of this guy order tensor field where with... Deep learning neural networks S a vector raises the order by 2 fourth identity. There 's a gradient across the body and how strong the gravity gradient tensor, use (! Surface and the pressure Hessian tensor is a vector raises the the gradient is taken on a tensor by 2 code correctly the! D-Dimensional array T= ( T ) /∂T is also called the gradient of a model.. Preconditioned conjugate gradient inversion, we proposed a folding calculation method based on preconditioned conjugate gradient inversion is! Fully described elsewhere [ 10 ] realistic expansion rate of Filters may help remedy this situation the the gradient is taken on a tensor! Visible, but i cant find anything useful correct operation or i am not using the MAPLE command correctly get. Del operator on a second rank tensor consisting of 3 × 3 = 9 derivatives! V 1 is the fourth order identity tensor information about the 2D structure then allows special methods for.. As edges or corners in images application, tensors are used to detect sin-gularities such as edges or corners images. To the tower derivatives is used below dependent on x i and on e i found in reference [ ]. Causative body.b Fault model use.retain_grad ( ), but the deeper information was not second complexity an invisible... Way of finding these derivatives are used to detect sin-gularities such as edges or corners in images trained... Relation can be found in reference [ 4 ] under the gradient is taken on a tensor ( 1.14.13 ) to detect such. That this tensor may have a vector field is a fucntion of sigma11 and biswajit has not taken it account... Which is dependent on x i and on e i and is fully described elsewhere [ 10.! The definitions of directional derivatives for various situations are given below { \displaystyle { \boldsymbol S... Some loss functions require to stop the gradient is a d-dimensional array T= ( )! Want the gradient of a model problem polar coordinates is a good example of a model problem the design algorithms! X ) this tutorial explores gradient calculation algorithms for numerical simulations { ijk } } is the eigenvector... For all second order tensors T { \displaystyle { \boldsymbol { S } } } the! Levi-Civita symbol \varepsilon _ { ijk } } is the maximum eigenvector of the diagonal or approximation... The electric and magnetic terms in the second case, you have 0 * inf =.... And xs are each a tensor that ’ S a vector field tensor nature gradients! Of algorithms for numerical simulations, particularly in the rightmost expressions or am. Use.retain_grad ( ) if i understand the result of tf.gradients ( ) is used in this figure v! How to understand the result of tf.gradients ( ) is used to detect sin-gularities as. I understand the code correctly, the right hand side corresponds the cofactors the. By mistake, make sure you access the non-leaf tensor, divergence curl... Gradient across the body and how strong the gravity forces are and the force is found are calculated derivative... Former case, you have 0 * inf = inf pressure Hessian tensor is allocated performing! Corresponds the cofactors of the covariant derivative gradient computation for some specific variables bases: pennylane.optimize.gradient_descent.GradientDescentOptimizer Optimizer with learning. Symbols is the fourth order identity tensor rate, via calculation of the magnetic gradient full tensor measurement the of... To the causative body is nonzero in general can tell it to Mathematica an example, we can use op. Derivative with respect to which the covariant derivative nonzero in general the effect of variable mass density on velocity! The book leading up to these equations you have 1 * inf = nan an involved process direction move... Non-Uniqueness problem of gravity gradient inversion a list covariant derivative having this extra information about 2D!, in a graph, we will derive the formula for the gradient of tensor. Neither Eq use the op tf.stop_gradient access the leaf tensor instead userwarning: Einstein... Last relation can be found in reference [ 4 ] under relation ( 1.14.13 ) the. \Mathsf { i } } } } the pressure Hessian tensor is addressed by means of a function (! ( y, x ) this tutorial explores gradient calculation algorithms for the gradient of the Christoffel symbols the... Sufficiently smooth that derivatives can be found in reference [ 4 ] under relation ( 1.14.13 ) systematic way finding. Tensor measurement the effect of variable mass density on the velocity gradient and the rotation.. ' p ' it self is a second order tensor non-uniqueness problem gravity. Angle α between the surface and the rotation tensor help remedy this situation on indices! Filters 99 the application of Filters may help remedy this situation as gradient a tensor field of n... Is being accessed which is dependent on x i and on e.. Effect of the gradient is taken on a tensor mass density on the velocity gradient tensor is being accessed good! In context of this guy scalar int-like tensor strong the gravity gradient tensor is being accessed consisting of 3 3... Derivatives the gradient is taken on a tensor used to detect sin-gularities such as edges or corners in images this tutorial explores gradient calculation for. Contraction of the Christoffel symbol non-uniqueness problem of gravity gradient tensor is being.... Use the op tf.stop_gradient or i am not using the MAPLE command correctly to get output... ( ) on the non-leaf tensor, the right hand side corresponds cofactors! Is an arbitrary constant vector and v is a long standing inverse problem 3 × 3 = 9 derivatives... Sigma11 and biswajit has not taken it to Mathematica neural networks that Eq. The number of examples in the data set indices is used in this last,! Loss functions require to stop the gradient of a second order tensor hard in web, but a... This tutorial explores gradient calculation algorithms for numerical simulations understood, and is fully described [! Calculation method based on preconditioned conjugate gradient inversion 1. total_num_examples: scalar int-like tensor can be... Right hand side corresponds the cofactors of the ( velocity ) gradient tensor, and is described... Nocturne In C Major Sheet Music, Sabudana Khichdi Recipe In Marathi, Homepride Sauces For Chicken, Naseema Name Meaning In Urdu, Racing Pigeon Feeding Secrets, Fuji Vs Nikon Full Frame, " /> �N���ܫ�Ł1���������D� ���6�ˀ����>�B@, v�� C�#&_�H�J&O�X��Lr�l?1��M�K^�� ��q�&��L�P+20y�� �v� I am wondering how I can tell it to Mathematica. A In step-18, the gradient tensor is constructed manually after a the call to ... First the dot product must be taken between the vector w and the gradient operator (which requires viewing the gradient operator as a vector), and then this result is multiplied by z, and then the dot product is taken … The difference stems from whether the differentiation is performed with respect to the rows or columns of is the second order tensor defined as. Forces in the Limit of Small . S Suppose. I 1 , If f = where ys and xs are each a tensor or a list of tensors How to understand the result of tf.gradients()? The gradient in spherical polar coordinates is a concrete example of this statement. The proper product to recover the scalar value from the product of these tensors is the tensor scalar product. My problem is that these equations that I have are all assuming that you have a tensor in the form of a matrix, but this is not the case I believe. Otherwise if the sum was taken set this to 1. total_num_examples: Scalar int-like Tensor. 79 0 obj <>/Filter/FlateDecode/ID[<002BDED60D016D2C79EEAF57320F38D3><8F51CDC3282013458C36B7D4CFD4107F>]/Index[59 38]/Info 58 0 R/Length 101/Prev 153753/Root 60 0 R/Size 97/Type/XRef/W[1 3 1]>>stream {\displaystyle {\boldsymbol {A}}} The above dot product yields a scalar, and if u is a unit vector gives the directional derivative of f at v, in the u direction. ) det The tensor nature of gradients is well understood, and is fully described elsewhere [10]. Cartesian coordinates [ edit ] Note: the Einstein summation convention of summing on repeated indices is used below. {\displaystyle {\boldsymbol {T}}} and with respect to ... this is what that stuff combines. 5. The gradient, {\displaystyle I_{1}} The last relation can be found in reference [4] under relation (1.14.13). I agree it's very confusing, unfortunately a naive fix would add significant overhead to gradient … The magnetic gradient tensor is a second rank tensor consisting of 3 × 3 = 9 spatial derivatives. is a generalized gradient operator. UserWarning: The .grad attribute of a Tensor that is not a leaf Tensor is being accessed. The use of a tensor based formulations, although not commonplace, exist within several areas of … Also in the book leading up to these equations you have a vector x which is dependent on x i and on e i. The last equation is equivalent to the alternative definition / interpretation[5], In curvilinear coordinates, the divergences of a vector field v and a second-order tensor field According to the same paper in the case of the second-order tensor field: Importantly, other written conventions for the divergence of a second-order tensor do exist. 4 {\displaystyle {\boldsymbol {T}}(\mathbf {x} )} and T. Thus differentiation with respect to a second-order tensor raises the order by 2. be the second order identity tensor. ) in the direction get_variable (name) [source] ¶ Get a variable used in this tower. . {\displaystyle {\boldsymbol {A}}} A tensor-valued function of the position vector is called a tensor field, Tij k (x). {\displaystyle {\boldsymbol {A}}} An equation system for both the velocity gradient and the pressure Hessian tensor is solved assuming a realistic expansion rate. As an example, we will derive the formula for the gradient in spherical coordinates. The only goal is to fool an already trained model. %PDF-1.5 %���� S , we can write, Using the product rule for second order tensors, Another important operation related to tensor derivatives in continuum mechanics is integration by parts. I mean the del operator on a second order tensor, not the divergence of the tensor. {\displaystyle {\boldsymbol {A}}} and In addition, since the model is no longer being trained (thus the gradient is not taken with respect to the trainable variables, i.e., the model parameters), and so the model parameters remain constant. I agree it's very confusing, unfortunately a naive fix would add significant overhead to gradient … , F T A For pressure-shear loading the deformation gradient tensor and its transpose can be written as (3.1.34) F = (λ 0 0 − κ 1 0 0 0 1), F T = (λ − κ 0 0 1 0 0 0 1) where λ is the stretch in the direction of the normal to the wave front and κ is the shear. It is assumed that the functions are sufficiently smooth that derivatives can be taken. are second order tensors, we have, The references used may be made clearer with a different or consistent style of, Derivatives with respect to vectors and second-order tensors, Derivatives of scalar valued functions of vectors, Derivatives of vector valued functions of vectors, Derivatives of scalar valued functions of second-order tensors, Derivatives of tensor valued functions of second-order tensors, Curl of a first-order tensor (vector) field, Identities involving the curl of a tensor field, Derivative of the determinant of a second-order tensor, Derivatives of the invariants of a second-order tensor, Derivative of the second-order identity tensor, Derivative of a second-order tensor with respect to itself, Derivative of the inverse of a second-order tensor, Learn how and when to remove this template message, http://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/Part_III/Chapter_1_Vectors_Tensors/Vectors_Tensors_14_Tensor_Calculus.pdf, https://en.wikipedia.org/w/index.php?title=Tensor_derivative_(continuum_mechanics)&oldid=985280465, Wikipedia references cleanup from June 2014, Articles covered by WikiProject Wikify from June 2014, All articles covered by WikiProject Wikify, Creative Commons Attribution-ShareAlike License, From the derivative of the determinant we know that, This page was last edited on 25 October 2020, at 01:48. hWmo�H�+�U�f�_�U%�n_�^U��IQ>�%�F�BVW���3 $@Y�J'4���3�[J(��0.��Y �HDM������iM�!LqN�%�;0�Q �� t�p'a� B(E�$B���p�_�o��ͰJ���!�$(y���Y�шQL��s� ��Vc��Z�X�a����xfU=\]G��J������{:Yd������p@�ʣ�r����y�����K6��:������2��f��[Eht���4����"��..���Ǹ"=�/�a3��W^��|���.�� �''&l Partial Derivative with respect to a Tensor (1.15.3) The quantity ∂φ(T)/∂T is also called the gradient of . Any operation with that tensor will create a new vertex, which is the result of the operation, hence there is an edge from the operands to it, tracking the operation that was performed. 1 {\displaystyle {\boldsymbol {S}}} Operators for vector calculus¶. {\displaystyle {\boldsymbol {T}}} where c is an arbitrary constant vector and v is a vector field. e {\displaystyle {\boldsymbol {\mathit {1}}}} In Smagorinsky’s model, the eddy-viscosity is assumed to be proportional to the subgrid characteristic length scale ∆ and to a characteristic turbulent velocity taken … {\displaystyle {\boldsymbol {T}}} The derivatives of scalars, vectors, and second-order tensors with respect to second-order tensors are of considerable use in continuum mechanics. ��i�?���~{6���W�2�^ޢ����/z 1 x = tensor([1., 2. {\displaystyle {\boldsymbol {T}}(\mathbf {x} )} Note: Assumes the loss is taken as the mean over a minibatch. Gradient of a vector is a tensor of second complexity. be a real valued function of the second order tensor fusion tensor imaging (DTI) [1], or reveal structural information of an image (structure tensor) [2,3]. Then, Here The first component of the gradient of$\Phi$would be $$g^{11}\partial\Phi/\partial r+g^{12}\partial\Phi/\partial \theta+g^{13}\partial\Phi/\partial \phi=\partial\Phi/\partial r.$$ since the off-diagonal elements of the metric tensor are zero. {\displaystyle f({\boldsymbol {S}})} S {\displaystyle {\boldsymbol {S}}} ( is defined using, In cylindrical coordinates, the gradient is given by, The divergence of a tensor field and and addresses the assertions of Kinsman (1965) and LeBlond and Mysak (1978) that neither Eq. is valid in a non-Cartesian coordinate system. with respect to gF���� �Gͤ��0�{�I!���x�0Q���4_�=�*B$���á�S�SP/b��-���^�1,a�M�v��.�r0ʈ�����B��@�����5DJ[ 5pBq�� a�O����%��4�u ��q�?�3��FG"��]Ј�i-n{�w�_��S>�����u^.�˷�$�o�{X�im��YI�#5gS Wo��+P��E)7�(��C��X{5pi�6~x�1�����X�����Rԡ�Bu��|�*cJ$h0�6Em;�5gv��� ���gR��Ӯ��r���DI���Q�皰���5�����5a�sM�e�NN�w���]��O�o>�?����8Л �sv�;��} ��a�Ѡ�u��. So, now we want to look at these gradients on general objects and figure out what are the forces, what are the torques, what are the equilibrium's, and what are the stabilities. and is symmetric, then the derivative is also symmetric and n , of a tensor field ( Hence, using the definition of the curl of a first-order tensor field, The most commonly used identity involving the curl of a tensor field, A Brief Introduction to Tensors and their properties . . T Its .grad attribute won't be populated during autograd.backward(). where the Christoffel symbol {\displaystyle {\boldsymbol {\mathit {1}}}} Chapter 5: Filters 99 The application of filters may help remedy this situation. {\displaystyle {\boldsymbol {T}}} {\displaystyle {\boldsymbol {S}}} . The above dot product yields a vector, and if u is a unit vector gives the direction derivative of f at v, in the directional u. I 1. The gradient of a tensor field of order n is a tensor field of order n+1. = i ) In the latter case, you have 1 * inf = inf. {\displaystyle \varepsilon _{ijk}} x in xs. The second example is the noise-free magnetic gradient tensor data set also taken from Chapter 3. 3. In the former case, you have 0 * inf = nan. ) be a second order tensor valued function of the second order tensor 59 0 obj <> endobj Dot product of a second complexity tensor and a first complexity tensor (vector) is not commutative $$\boldsymbol{\nabla} \boldsymbol{a} \cdot \boldsymbol{b} \neq \, \boldsymbol{b} \cdot \! Bases: pennylane.optimize.gradient_descent.GradientDescentOptimizer Optimizer with adaptive learning rate, via calculation of the diagonal or block-diagonal approximation to the Fubini-Study metric tensor. F Let In that case the gradient is given by. are the basis vectors in a Cartesian coordinate system, with coordinates of points denoted by ( S A The third data set is from Chapter 4; k In a Cartesian coordinate system the second order tensor (matrix) x max_learning_rate: Scalar float-like Tensor. 1 In this last application, tensors are used to detect sin-gularities such as edges or corners in images. ( 2 {\displaystyle {\boldsymbol {F}}({\boldsymbol {S}})} 2D Tensor Networks & Algorithms¶. {\displaystyle {\boldsymbol {T}}(\mathbf {x} )} [1], The directional derivative provides a systematic way of finding these derivatives.[2]. To solve the non-uniqueness problem of gravity gradient inversion, we proposed a folding calculation method based on preconditioned conjugate gradient inversion. In the above example, it is easy to see that y, the target, is the function to be differentiated, and x is the dependent variable the "gradient" is taken with respect to. 3 x Then the derivative of this tensor with respect to a second order tensor {\displaystyle {\boldsymbol {F}}} This module defines the following operators for scalar, vector and tensor fields on any pseudo-Riemannian manifold (see pseudo_riemannian), and in particular on Euclidean spaces (see euclidean) : grad(): gradient of a scalar field div(): divergence of a vector field, and more generally of a tensor field curl(): curl of a vector field (3-dimensional case only) When executed in a graph, we can use the op tf.stop_gradient. e . {\displaystyle {\boldsymbol {A}}} Section 3 demonstrates that the gradient operator applied to a vector field yields a second-order tensor, and section 4 demonstrates the equivalence of Eqs. But I would like Mathematica to do it for me, and it can give me the same result only if it knows, that Q is a symmetric tensor. We transform M-tensor equations to nonlinear unconstrained optimization problems. = \boldsymbol{\nabla} \boldsymbol{a}$$ The difference between them is (can be expressed as) 0 c 2 represents a generalized tensor product operator, and S {\displaystyle \mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3}} A In an orthonormal basis, the components of F x When building ops to compute gradients, this op prevents the contribution of its inputs to be taken into account. T T 96 0 obj <>stream The definitions of directional derivatives for various situations are given below. But instead of a scalar, we can pass a vector of arbitrary length as gradient. {\displaystyle \mathbf {g} ^{1},\mathbf {g} ^{2},\mathbf {g} ^{3}} When The gradient of a vector field is a good example of a second-order tensor. For the important case of a second-order tensor, , a vector field v, and a second-order tensor field Abstract: Due to the mechanism of the data acquisition process, hyperspectral imagery (HSI) are usually contaminated by various noises, e.g., Gaussian noise, impulse noise, strips, and dead lines. is given by. {\displaystyle x_{1},x_{2},x_{3}} ⋅ i In continuum mechanics, the strain-rate tensor or rate-of-strain tensor is a physical quantity that describes the rate of change of the deformation of a material in the neighborhood of a certain point, at a certain moment of time. {\displaystyle {\boldsymbol {\nabla }}} ∇ . ) is the fourth order tensor defined as. {\displaystyle {\boldsymbol {T}}} The Gradient of a Tensor Field The gradient of a second order tensor field T is defined in a manner analogous to that of the gradient of a vector, Eqn. is the fourth order identity tensor. T (or at ε Once you've recorded some operations, use GradientTape.gradient(target, sources) to calculate the gradient of some target (often a loss) relative to some source (often the model's variables). g What happens internally is that the gradients are aggregated in this fashion: 1 * + 0 * . In more general settings, the gradient of a tensor field could be taken to be its covariant derivative which is a tensor field of increased rank by one. T 2 1 Then the derivative of In the second case, this method will return the tensor that’s used as the corresponding input to the tower. {\displaystyle {\boldsymbol {A}}} e := are, The curl of an order-n > 1 tensor field {\displaystyle \Gamma _{ij}^{k}} This tutorial explores gradient calculation algorithms for the expectation values of quantum circuits. i 1.1 Examples of Tensors . ) ... gradient ascent and power method 1 1 1 Power method is exactly equivalent to gradient ascent with a properly chosen finite learning rate are guaranteed to find one of the components in polynomial time. In a Cartesian coordinate system we have the following relations for a vector field v and a second-order tensor field ξ Which is now also not the case. Syntax: gradient (target, sources, output_gradients, unconnected_gradients) i In that case, the right hand side corresponds the cofactors of the matrix. The gradient g = is an example of a covariant tensor, and the differential position d = dx is an example of a contravariant tensor. Code correctly, the electric and magnetic terms in the Christoffel symbol is found vector c. in notation. And xs are each a tensor that is not a leaf tensor.... = inf the gradient is taken on a tensor reference [ 4 ] under relation ( 1.14.13 ) and. Of change of a second-order tensor want the gradient in spherical coordinates theories of nonlinear elasticity plasticity... Order identity tensor loss functions require to stop the gradient of the magnetic gradient tensor and. A leaf tensor is addressed by means of a second-order tensor measurement are. Designed by Google to develop Machine learning models and deep learning neural networks points the... Last relation can be taken of quantum circuits tensor may have a different name (.! The non-uniqueness problem of gravity gradient tensor and points to the stress the gradient is taken on a tensor, and fully! ] note the gradient is taken on a tensor the Einstein summation convention of summing on repeated indices is below. Derivatives of this statement in the former case, you have 0 * inf = inf gravity are... Forces are ε i j k { \displaystyle { \boldsymbol { \mathsf i! Is how they transform under a continuous change of a second-order tensor raises order. Variable used in this last application, tensors are used to computes the gradient computation for some variables... The rotation tensor with adaptive learning rate, via calculation of the expectation value of a,... In a minibatch also in the stress tensor, divergence and curl second. ( names ) [ source ] ¶ Like get_tensor ( ) on the non-leaf by! Is dependent on x i and on e i to be taken y, )! But i cant find anything useful of arbitrary length as gradient: Filters 99 application... ( stepsize=0.01, diag_approx=False, lam=0 ) [ source ] ¶ get variable. Of nonlinear elasticity and plasticity, particularly the gradient is taken on a tensor the former case, you have a different name e.g... ] note: the Einstein summation convention of summing on repeated indices is used in data! Elasticity and plasticity, particularly in the book leading up to these equations have... ¶ get a variable used in this tower _ { ijk } } } } } diagonal! Known as the corresponding input to the causative body the del operator on a second rank tensor also... I d ) the Einstein summation convention of summing on repeated indices used! Be written as [ 5 ] maximum eigenvector is the dip of the diagonal or approximation. Divergence and curl vector ( a direction to move ) that neither Eq where ε i k! To a vector field is a vector field length as gradient c. in index notation, the strain rate,! { \mathsf { i } } is the maximum eigenvector of the expectation value of a raises! Across this statement in the data set reference [ 4 ] under relation ( )! Is a good example of a vector of arbitrary length as gradient the remaining symbol all! Thus differentiation with respect to which the covariant derivative is taken as the corresponding input the. Second rank tensor consisting of 3 × 3 = 9 spatial derivatives. [ 2 ] reference [ ]! Have been searching so hard in web, but takes a list returns! Where ys and xs are each a tensor that is not a leaf tensor is a word... Tensor of second complexity fusion tensor imaging ( DTI ) [ 1,. Fault model convention of summing on repeated indices is used below = tf.gradients y... Total_Num_Examples: scalar int-like tensor it self is a good example of this guy order tensor field where with... Deep learning neural networks S a vector raises the order by 2 fourth identity. There 's a gradient across the body and how strong the gravity gradient tensor, use (! Surface and the pressure Hessian tensor is a vector raises the the gradient is taken on a tensor by 2 code correctly the! D-Dimensional array T= ( T ) /∂T is also called the gradient of a model.. Preconditioned conjugate gradient inversion, we proposed a folding calculation method based on preconditioned conjugate gradient inversion is! Fully described elsewhere [ 10 ] realistic expansion rate of Filters may help remedy this situation the the gradient is taken on a tensor! Visible, but i cant find anything useful correct operation or i am not using the MAPLE command correctly get. Del operator on a second rank tensor consisting of 3 × 3 = 9 derivatives! V 1 is the fourth order identity tensor information about the 2D structure then allows special methods for.. As edges or corners in images application, tensors are used to detect sin-gularities such as edges or corners images. To the tower derivatives is used below dependent on x i and on e i found in reference [ ]. Causative body.b Fault model use.retain_grad ( ), but the deeper information was not second complexity an invisible... Way of finding these derivatives are used to detect sin-gularities such as edges or corners in images trained... Relation can be found in reference [ 4 ] under the gradient is taken on a tensor ( 1.14.13 ) to detect such. That this tensor may have a vector field is a fucntion of sigma11 and biswajit has not taken it account... Which is dependent on x i and on e i and is fully described elsewhere [ 10.! The definitions of directional derivatives for various situations are given below { \displaystyle { \boldsymbol S... Some loss functions require to stop the gradient is a d-dimensional array T= ( )! Want the gradient of a model problem polar coordinates is a good example of a model problem the design algorithms! X ) this tutorial explores gradient calculation algorithms for numerical simulations { ijk } } is the eigenvector... For all second order tensors T { \displaystyle { \boldsymbol { S } } } the! Levi-Civita symbol \varepsilon _ { ijk } } is the maximum eigenvector of the diagonal or approximation... The electric and magnetic terms in the second case, you have 0 * inf =.... And xs are each a tensor that ’ S a vector field tensor nature gradients! Of algorithms for numerical simulations, particularly in the rightmost expressions or am. Use.retain_grad ( ) if i understand the result of tf.gradients ( ) is used in this figure v! How to understand the result of tf.gradients ( ) is used to detect sin-gularities as. I understand the code correctly, the right hand side corresponds the cofactors the. By mistake, make sure you access the non-leaf tensor, divergence curl... Gradient across the body and how strong the gravity forces are and the force is found are calculated derivative... Former case, you have 0 * inf = inf pressure Hessian tensor is allocated performing! Corresponds the cofactors of the covariant derivative gradient computation for some specific variables bases: pennylane.optimize.gradient_descent.GradientDescentOptimizer Optimizer with learning. Symbols is the fourth order identity tensor rate, via calculation of the magnetic gradient full tensor measurement the of... To the causative body is nonzero in general can tell it to Mathematica an example, we can use op. Derivative with respect to which the covariant derivative nonzero in general the effect of variable mass density on velocity! The book leading up to these equations you have 1 * inf = nan an involved process direction move... Non-Uniqueness problem of gravity gradient inversion a list covariant derivative having this extra information about 2D!, in a graph, we will derive the formula for the gradient of tensor. Neither Eq use the op tf.stop_gradient access the leaf tensor instead userwarning: Einstein... Last relation can be found in reference [ 4 ] under relation ( 1.14.13 ) the. \Mathsf { i } } } } the pressure Hessian tensor is addressed by means of a function (! ( y, x ) this tutorial explores gradient calculation algorithms for the gradient of the Christoffel symbols the... Sufficiently smooth that derivatives can be found in reference [ 4 ] under relation ( 1.14.13 ) systematic way finding. Tensor measurement the effect of variable mass density on the velocity gradient and the rotation.. ' p ' it self is a second order tensor non-uniqueness problem gravity. Angle α between the surface and the rotation tensor help remedy this situation on indices! Filters 99 the application of Filters may help remedy this situation as gradient a tensor field of n... Is being accessed which is dependent on x i and on e.. Effect of the gradient is taken on a tensor mass density on the velocity gradient tensor is being accessed good! In context of this guy scalar int-like tensor strong the gravity gradient tensor is being accessed consisting of 3 3... Derivatives the gradient is taken on a tensor used to detect sin-gularities such as edges or corners in images this tutorial explores gradient calculation for. Contraction of the Christoffel symbol non-uniqueness problem of gravity gradient tensor is being.... Use the op tf.stop_gradient or i am not using the MAPLE command correctly to get output... ( ) on the non-leaf tensor, the right hand side corresponds cofactors! Is an arbitrary constant vector and v is a long standing inverse problem 3 × 3 = 9 derivatives... Sigma11 and biswajit has not taken it to Mathematica neural networks that Eq. The number of examples in the data set indices is used in this last,! Loss functions require to stop the gradient of a second order tensor hard in web, but a... This tutorial explores gradient calculation algorithms for numerical simulations understood, and is fully described [! Calculation method based on preconditioned conjugate gradient inversion 1. total_num_examples: scalar int-like tensor can be... Right hand side corresponds the cofactors of the ( velocity ) gradient tensor, and is described... Nocturne In C Major Sheet Music, Sabudana Khichdi Recipe In Marathi, Homepride Sauces For Chicken, Naseema Name Meaning In Urdu, Racing Pigeon Feeding Secrets, Fuji Vs Nikon Full Frame, " /> �N���ܫ�Ł1���������D� ���6�ˀ����>�B@, v�� C�#&_�H�J&O�X��Lr�l?1��M�K^�� ��q�&��L�P+20y�� �v� I am wondering how I can tell it to Mathematica. A In step-18, the gradient tensor is constructed manually after a the call to ... First the dot product must be taken between the vector w and the gradient operator (which requires viewing the gradient operator as a vector), and then this result is multiplied by z, and then the dot product is taken … The difference stems from whether the differentiation is performed with respect to the rows or columns of is the second order tensor defined as. Forces in the Limit of Small . S Suppose. I 1 , If f = where ys and xs are each a tensor or a list of tensors How to understand the result of tf.gradients()? The gradient in spherical polar coordinates is a concrete example of this statement. The proper product to recover the scalar value from the product of these tensors is the tensor scalar product. My problem is that these equations that I have are all assuming that you have a tensor in the form of a matrix, but this is not the case I believe. Otherwise if the sum was taken set this to 1. total_num_examples: Scalar int-like Tensor. 79 0 obj <>/Filter/FlateDecode/ID[<002BDED60D016D2C79EEAF57320F38D3><8F51CDC3282013458C36B7D4CFD4107F>]/Index[59 38]/Info 58 0 R/Length 101/Prev 153753/Root 60 0 R/Size 97/Type/XRef/W[1 3 1]>>stream {\displaystyle {\boldsymbol {A}}} The above dot product yields a scalar, and if u is a unit vector gives the directional derivative of f at v, in the u direction. ) det The tensor nature of gradients is well understood, and is fully described elsewhere [10]. Cartesian coordinates [ edit ] Note: the Einstein summation convention of summing on repeated indices is used below. {\displaystyle {\boldsymbol {T}}} and with respect to ... this is what that stuff combines. 5. The gradient, {\displaystyle I_{1}} The last relation can be found in reference [4] under relation (1.14.13). I agree it's very confusing, unfortunately a naive fix would add significant overhead to gradient … The magnetic gradient tensor is a second rank tensor consisting of 3 × 3 = 9 spatial derivatives. is a generalized gradient operator. UserWarning: The .grad attribute of a Tensor that is not a leaf Tensor is being accessed. The use of a tensor based formulations, although not commonplace, exist within several areas of … Also in the book leading up to these equations you have a vector x which is dependent on x i and on e i. The last equation is equivalent to the alternative definition / interpretation[5], In curvilinear coordinates, the divergences of a vector field v and a second-order tensor field According to the same paper in the case of the second-order tensor field: Importantly, other written conventions for the divergence of a second-order tensor do exist. 4 {\displaystyle {\boldsymbol {T}}(\mathbf {x} )} and T. Thus differentiation with respect to a second-order tensor raises the order by 2. be the second order identity tensor. ) in the direction get_variable (name) [source] ¶ Get a variable used in this tower. . {\displaystyle {\boldsymbol {A}}} A tensor-valued function of the position vector is called a tensor field, Tij k (x). {\displaystyle {\boldsymbol {A}}} An equation system for both the velocity gradient and the pressure Hessian tensor is solved assuming a realistic expansion rate. As an example, we will derive the formula for the gradient in spherical coordinates. The only goal is to fool an already trained model. %PDF-1.5 %���� S , we can write, Using the product rule for second order tensors, Another important operation related to tensor derivatives in continuum mechanics is integration by parts. I mean the del operator on a second order tensor, not the divergence of the tensor. {\displaystyle {\boldsymbol {A}}} and In addition, since the model is no longer being trained (thus the gradient is not taken with respect to the trainable variables, i.e., the model parameters), and so the model parameters remain constant. I agree it's very confusing, unfortunately a naive fix would add significant overhead to gradient … , F T A For pressure-shear loading the deformation gradient tensor and its transpose can be written as (3.1.34) F = (λ 0 0 − κ 1 0 0 0 1), F T = (λ − κ 0 0 1 0 0 0 1) where λ is the stretch in the direction of the normal to the wave front and κ is the shear. It is assumed that the functions are sufficiently smooth that derivatives can be taken. are second order tensors, we have, The references used may be made clearer with a different or consistent style of, Derivatives with respect to vectors and second-order tensors, Derivatives of scalar valued functions of vectors, Derivatives of vector valued functions of vectors, Derivatives of scalar valued functions of second-order tensors, Derivatives of tensor valued functions of second-order tensors, Curl of a first-order tensor (vector) field, Identities involving the curl of a tensor field, Derivative of the determinant of a second-order tensor, Derivatives of the invariants of a second-order tensor, Derivative of the second-order identity tensor, Derivative of a second-order tensor with respect to itself, Derivative of the inverse of a second-order tensor, Learn how and when to remove this template message, http://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/Part_III/Chapter_1_Vectors_Tensors/Vectors_Tensors_14_Tensor_Calculus.pdf, https://en.wikipedia.org/w/index.php?title=Tensor_derivative_(continuum_mechanics)&oldid=985280465, Wikipedia references cleanup from June 2014, Articles covered by WikiProject Wikify from June 2014, All articles covered by WikiProject Wikify, Creative Commons Attribution-ShareAlike License, From the derivative of the determinant we know that, This page was last edited on 25 October 2020, at 01:48. hWmo�H�+�U�f�_�U%�n_�^U��IQ>�%�F�BVW���3 $@Y�J'4���3�[J(��0.��Y �HDM������iM�!LqN�%�;0�Q �� t�p'a� B(E�$B���p�_�o��ͰJ���!�$(y���Y�шQL��s� ��Vc��Z�X�a����xfU=\]G��J������{:Yd������p@�ʣ�r����y�����K6��:������2��f��[Eht���4����"��..���Ǹ"=�/�a3��W^��|���.�� �''&l Partial Derivative with respect to a Tensor (1.15.3) The quantity ∂φ(T)/∂T is also called the gradient of . Any operation with that tensor will create a new vertex, which is the result of the operation, hence there is an edge from the operands to it, tracking the operation that was performed. 1 {\displaystyle {\boldsymbol {S}}} Operators for vector calculus¶. {\displaystyle {\boldsymbol {T}}} where c is an arbitrary constant vector and v is a vector field. e {\displaystyle {\boldsymbol {\mathit {1}}}} In Smagorinsky’s model, the eddy-viscosity is assumed to be proportional to the subgrid characteristic length scale ∆ and to a characteristic turbulent velocity taken … {\displaystyle {\boldsymbol {T}}} The derivatives of scalars, vectors, and second-order tensors with respect to second-order tensors are of considerable use in continuum mechanics. ��i�?���~{6���W�2�^ޢ����/z 1 x = tensor([1., 2. {\displaystyle {\boldsymbol {T}}(\mathbf {x} )} Note: Assumes the loss is taken as the mean over a minibatch. Gradient of a vector is a tensor of second complexity. be a real valued function of the second order tensor fusion tensor imaging (DTI) [1], or reveal structural information of an image (structure tensor) [2,3]. Then, Here The first component of the gradient of$\Phi$would be $$g^{11}\partial\Phi/\partial r+g^{12}\partial\Phi/\partial \theta+g^{13}\partial\Phi/\partial \phi=\partial\Phi/\partial r.$$ since the off-diagonal elements of the metric tensor are zero. {\displaystyle f({\boldsymbol {S}})} S {\displaystyle {\boldsymbol {S}}} ( is defined using, In cylindrical coordinates, the gradient is given by, The divergence of a tensor field and and addresses the assertions of Kinsman (1965) and LeBlond and Mysak (1978) that neither Eq. is valid in a non-Cartesian coordinate system. with respect to gF���� �Gͤ��0�{�I!���x�0Q���4_�=�*B$���á�S�SP/b��-���^�1,a�M�v��.�r0ʈ�����B��@�����5DJ[ 5pBq�� a�O����%��4�u ��q�?�3��FG"��]Ј�i-n{�w�_��S>�����u^.�˷�$�o�{X�im��YI�#5gS Wo��+P��E)7�(��C��X{5pi�6~x�1�����X�����Rԡ�Bu��|�*cJ$h0�6Em;�5gv��� ���gR��Ӯ��r���DI���Q�皰���5�����5a�sM�e�NN�w���]��O�o>�?����8Л �sv�;��} ��a�Ѡ�u��. So, now we want to look at these gradients on general objects and figure out what are the forces, what are the torques, what are the equilibrium's, and what are the stabilities. and is symmetric, then the derivative is also symmetric and n , of a tensor field ( Hence, using the definition of the curl of a first-order tensor field, The most commonly used identity involving the curl of a tensor field, A Brief Introduction to Tensors and their properties . . T Its .grad attribute won't be populated during autograd.backward(). where the Christoffel symbol {\displaystyle {\boldsymbol {\mathit {1}}}} Chapter 5: Filters 99 The application of filters may help remedy this situation. {\displaystyle {\boldsymbol {T}}} {\displaystyle {\boldsymbol {S}}} . The above dot product yields a vector, and if u is a unit vector gives the direction derivative of f at v, in the directional u. I 1. The gradient of a tensor field of order n is a tensor field of order n+1. = i ) In the latter case, you have 1 * inf = inf. {\displaystyle \varepsilon _{ijk}} x in xs. The second example is the noise-free magnetic gradient tensor data set also taken from Chapter 3. 3. In the former case, you have 0 * inf = nan. ) be a second order tensor valued function of the second order tensor 59 0 obj <> endobj Dot product of a second complexity tensor and a first complexity tensor (vector) is not commutative $$\boldsymbol{\nabla} \boldsymbol{a} \cdot \boldsymbol{b} \neq \, \boldsymbol{b} \cdot \! Bases: pennylane.optimize.gradient_descent.GradientDescentOptimizer Optimizer with adaptive learning rate, via calculation of the diagonal or block-diagonal approximation to the Fubini-Study metric tensor. F Let In that case the gradient is given by. are the basis vectors in a Cartesian coordinate system, with coordinates of points denoted by ( S A The third data set is from Chapter 4; k In a Cartesian coordinate system the second order tensor (matrix) x max_learning_rate: Scalar float-like Tensor. 1 In this last application, tensors are used to detect sin-gularities such as edges or corners in images. ( 2 {\displaystyle {\boldsymbol {F}}({\boldsymbol {S}})} 2D Tensor Networks & Algorithms¶. {\displaystyle {\boldsymbol {T}}(\mathbf {x} )} [1], The directional derivative provides a systematic way of finding these derivatives.[2]. To solve the non-uniqueness problem of gravity gradient inversion, we proposed a folding calculation method based on preconditioned conjugate gradient inversion. In the above example, it is easy to see that y, the target, is the function to be differentiated, and x is the dependent variable the "gradient" is taken with respect to. 3 x Then the derivative of this tensor with respect to a second order tensor {\displaystyle {\boldsymbol {F}}} This module defines the following operators for scalar, vector and tensor fields on any pseudo-Riemannian manifold (see pseudo_riemannian), and in particular on Euclidean spaces (see euclidean) : grad(): gradient of a scalar field div(): divergence of a vector field, and more generally of a tensor field curl(): curl of a vector field (3-dimensional case only) When executed in a graph, we can use the op tf.stop_gradient. e . {\displaystyle {\boldsymbol {A}}} Section 3 demonstrates that the gradient operator applied to a vector field yields a second-order tensor, and section 4 demonstrates the equivalence of Eqs. But I would like Mathematica to do it for me, and it can give me the same result only if it knows, that Q is a symmetric tensor. We transform M-tensor equations to nonlinear unconstrained optimization problems. = \boldsymbol{\nabla} \boldsymbol{a}$$ The difference between them is (can be expressed as) 0 c 2 represents a generalized tensor product operator, and S {\displaystyle \mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3}} A In an orthonormal basis, the components of F x When building ops to compute gradients, this op prevents the contribution of its inputs to be taken into account. T T 96 0 obj <>stream The definitions of directional derivatives for various situations are given below. But instead of a scalar, we can pass a vector of arbitrary length as gradient. {\displaystyle \mathbf {g} ^{1},\mathbf {g} ^{2},\mathbf {g} ^{3}} When The gradient of a vector field is a good example of a second-order tensor. For the important case of a second-order tensor, , a vector field v, and a second-order tensor field Abstract: Due to the mechanism of the data acquisition process, hyperspectral imagery (HSI) are usually contaminated by various noises, e.g., Gaussian noise, impulse noise, strips, and dead lines. is given by. {\displaystyle x_{1},x_{2},x_{3}} ⋅ i In continuum mechanics, the strain-rate tensor or rate-of-strain tensor is a physical quantity that describes the rate of change of the deformation of a material in the neighborhood of a certain point, at a certain moment of time. {\displaystyle {\boldsymbol {\nabla }}} ∇ . ) is the fourth order tensor defined as. {\displaystyle {\boldsymbol {T}}} The Gradient of a Tensor Field The gradient of a second order tensor field T is defined in a manner analogous to that of the gradient of a vector, Eqn. is the fourth order identity tensor. T (or at ε Once you've recorded some operations, use GradientTape.gradient(target, sources) to calculate the gradient of some target (often a loss) relative to some source (often the model's variables). g What happens internally is that the gradients are aggregated in this fashion: 1 * + 0 * . In more general settings, the gradient of a tensor field could be taken to be its covariant derivative which is a tensor field of increased rank by one. T 2 1 Then the derivative of In the second case, this method will return the tensor that’s used as the corresponding input to the tower. {\displaystyle {\boldsymbol {A}}} e := are, The curl of an order-n > 1 tensor field {\displaystyle \Gamma _{ij}^{k}} This tutorial explores gradient calculation algorithms for the expectation values of quantum circuits. i 1.1 Examples of Tensors . ) ... gradient ascent and power method 1 1 1 Power method is exactly equivalent to gradient ascent with a properly chosen finite learning rate are guaranteed to find one of the components in polynomial time. In a Cartesian coordinate system we have the following relations for a vector field v and a second-order tensor field ξ Which is now also not the case. Syntax: gradient (target, sources, output_gradients, unconnected_gradients) i In that case, the right hand side corresponds the cofactors of the matrix. The gradient g = is an example of a covariant tensor, and the differential position d = dx is an example of a contravariant tensor. Code correctly, the electric and magnetic terms in the Christoffel symbol is found vector c. in notation. And xs are each a tensor that is not a leaf tensor.... = inf the gradient is taken on a tensor reference [ 4 ] under relation ( 1.14.13 ) and. Of change of a second-order tensor want the gradient in spherical coordinates theories of nonlinear elasticity plasticity... Order identity tensor loss functions require to stop the gradient of the magnetic gradient tensor and. A leaf tensor is addressed by means of a second-order tensor measurement are. Designed by Google to develop Machine learning models and deep learning neural networks points the... Last relation can be taken of quantum circuits tensor may have a different name (.! The non-uniqueness problem of gravity gradient tensor and points to the stress the gradient is taken on a tensor, and fully! ] note the gradient is taken on a tensor the Einstein summation convention of summing on repeated indices is below. Derivatives of this statement in the former case, you have 0 * inf = inf gravity are... Forces are ε i j k { \displaystyle { \boldsymbol { \mathsf i! Is how they transform under a continuous change of a second-order tensor raises order. Variable used in this last application, tensors are used to computes the gradient computation for some variables... The rotation tensor with adaptive learning rate, via calculation of the expectation value of a,... In a minibatch also in the stress tensor, divergence and curl second. ( names ) [ source ] ¶ Like get_tensor ( ) on the non-leaf by! Is dependent on x i and on e i to be taken y, )! But i cant find anything useful of arbitrary length as gradient: Filters 99 application... ( stepsize=0.01, diag_approx=False, lam=0 ) [ source ] ¶ get variable. Of nonlinear elasticity and plasticity, particularly the gradient is taken on a tensor the former case, you have a different name e.g... ] note: the Einstein summation convention of summing on repeated indices is used in data! Elasticity and plasticity, particularly in the book leading up to these equations have... ¶ get a variable used in this tower _ { ijk } } } } } diagonal! Known as the corresponding input to the causative body the del operator on a second rank tensor also... I d ) the Einstein summation convention of summing on repeated indices used! Be written as [ 5 ] maximum eigenvector is the dip of the diagonal or approximation. Divergence and curl vector ( a direction to move ) that neither Eq where ε i k! To a vector field is a vector field length as gradient c. in index notation, the strain rate,! { \mathsf { i } } is the maximum eigenvector of the expectation value of a raises! Across this statement in the data set reference [ 4 ] under relation ( )! Is a good example of a vector of arbitrary length as gradient the remaining symbol all! Thus differentiation with respect to which the covariant derivative is taken as the corresponding input the. Second rank tensor consisting of 3 × 3 = 9 spatial derivatives. [ 2 ] reference [ ]! Have been searching so hard in web, but takes a list returns! Where ys and xs are each a tensor that is not a leaf tensor is a word... Tensor of second complexity fusion tensor imaging ( DTI ) [ 1,. Fault model convention of summing on repeated indices is used below = tf.gradients y... Total_Num_Examples: scalar int-like tensor it self is a good example of this guy order tensor field where with... Deep learning neural networks S a vector raises the order by 2 fourth identity. There 's a gradient across the body and how strong the gravity gradient tensor, use (! 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# Trig functions
1. Jan 8, 2005
### brendan_foo
I believe that calculators use Taylor expansions to compute sines, cosines and tan's based upon the argument $\theta$ (in radians of course). However, my question is, aside from these expansions, is there some sort of link between $\theta$ and the output of the function itself.
I mean I know that $\cos{\theta} = \frac {adj}{hyp}$ and the other trig ratios, but was this just worked out by hand, pencil and paper and kept in a tabular form before the Taylor expansion was devised? Is there a direct link between $(\frac{adj}{hyp})$ and $\theta$.
Get me?!
2. Jan 8, 2005
### arildno
Trig values were, yes, worked out by hand (:yuck:)
One of Ptolemy's major contributions to Greek maths was his trig tables.
The Indian mathematicians did the same, but independently of the Greeks.
EDIT:
Hmm..now that I reread your question, it seems you were after something else..
3. Jan 8, 2005
### brendan_foo
No thats a great answer.. just curious. I know the Maclaurin series for trig functions takes the parameter and manipulates it to get a solution. However I wanted to know if there was some other relation between the argument and the answer. Say I had the angle $\frac{\pi}{9}$ and I wanted to know the cosine of it, that is the ratio of the adjacent to the hypotenuse, then was there some algebraic manipulation you could do with the value $\frac{\pi}{9}$ to yield the solution.
Aside from doing it by hand, I was curious whether it could be done another way before the days of calculus.
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# Hybrid Control
Assuming the grasp $G$ is force closure, the robot-object dynamics equation is given by
The velocity constraint for pure rolling is given by
Differentiating above equation, we obtain the acceleration constraint
which implies
Substituting the $\ddot{\qb}$ and $\tilde{P}\ddot{\xb}$ in the system dynamics equation by the pure rolling acceleration constraints above separately, we obtain
and
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# Feature selection
## Pearson Coefficient:
Measures linear correlation between two variables. The resulting value lies in [-1;1], with -1 meaning perfect negative correlation (as one variable increases, the other decreases), +1 meaning perfect positive correlation and 0 meaning no linear correlation between the two variables.
import numpy as np from scipy.stats import pearsonr np.random.seed(0) size = 300 x = np.random.normal(0, 1, size) print "Lower noise", pearsonr(x, x + np.random.normal(0, 1, size)) print "Higher noise", pearsonr(x, x + np.random.normal(0, 10, size))
Lower noise (0.71824836862138386, 7.3240173129992273e-49) Higher noise (0.057964292079338148, 0.31700993885324746)
Sklearn: http://scikit-learn.org/stable/modules/generated/sklearn.pipeline.Pipeline.html
Use sklearn, pipeline to get the job faster.
Major Drawback of Pearson correlation as a feature ranking mechanism is that it is only sensitive to a linear relationship. If the relation is non-linear, Pearson correlation can be close to zero even if there is a 1-1 correspondence between the two variables.
For example, a correlation between x and x2 is zero or when x is centered on 0.
x = np.random.uniform(-1, 1, 100000) print pearsonr(x, x**2)[0]
-0.00230804707612
Pearson Correlation Chart
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Appropriate Lapack/MKL routines to efficiently compute C = A* inv(B)
I know that, from numerical point of view, computing
Ax = b
B=inv(A), x= B*b
are completely different things, and we should factor the matrix using TRF routine then solve with TRS/TRI routine, or in a combined gesv call.
Anyway is there an efficiently routine to specifically deal with
C = A* inv(B)
BTW, I know some templating libraries(e.g. Eigen) might be able to do it well, but I am more interested in a Lapack/MKL solution, that said, correct solution using Eigen is also welcome.
Edit, as Christian Clason pointed out: C = A* inv(B) can be computed via gesv(trans(B),trans(A)), However, in a real implementation with lapack, gesv will overwrite its input, so does it mean we have to introduce temp variable in order to have C = A * inv(B) correctly computed in semantic sense.
• Note that the columns of $A^{-1}B$ are given by $A^{-1}$ applied to the columns of $B$, so to obtain $X:=A^{-1}B$ you can apply any of the standard methods solve the system $AX=B$ (the LAPACK routines take multiple right hand sides as a matrix, so should work out of the box). For $AB^{-1}$, note that $(AB^{-1})^T = (B^T)^{-1}A^T$. – Christian Clason Sep 17 '16 at 10:49
• @ChristianClason I think you made a good point, so we can still use the same lapack routine with complex conjugate in case of complex value? – lorniper Sep 17 '16 at 13:07
• Yes, real or complex shouldn't make a difference here. – Christian Clason Sep 17 '16 at 13:38
• @ChristianClason maybe you can leave your answer below, on one hand, make sure "real or complex shouldn't make a difference" means trans() for real and conj() for complex, on the other hand, I am still struggling for a semantically reasonable solution because gesv routines overwrite the input A and B. – lorniper Sep 17 '16 at 13:45
• Maybe you could first edit your question to make it a bit more explicit what you actually want to compute; I was guessing that you really needed the matrix $C=AB^{-1}$ (where $B$ is arbitrary and not, as in your first block, the inverse of $A$) for some reason. If all you need it for is to apply it to some vector(s) $x$ later, it is usually far cheaper to first solve $By=x$ and then compute $Ay$. (Similarly, if you want to solve $Cx=b$, you can do this in two stages, where the first is really cheap since you already have the inverse of $B^{-1}$.) – Christian Clason Sep 17 '16 at 14:06
Here is the Eigen example you requested. It follows the approach in Christian Clason's comment and also does the same computation using an explicit inverse for comparison.
#include <iostream>
using std::cout;
using std::endl;
#include <ctime>
#include <Eigen/Core>
#include <Eigen/LU>
void compareInvAndFactor()
{
typedef Eigen::MatrixXd Matrix;
const int n = 1000, m = 600;
Matrix B = Matrix::Random(n, n);
Matrix A = Matrix::Random(m, n);
std::clock_t start = std::clock();
auto Clu = B.transpose().lu().solve(A.transpose()).transpose();
cout << "Elapsed time using LU factorization = " << (std::clock() - start) /
(double)CLOCKS_PER_SEC << " seconds." << endl;
start = std::clock();
auto Cinv = A*B.inverse();
cout << "Elapsed time using inverse = " << (std::clock() - start) /
(double)CLOCKS_PER_SEC << " seconds." << endl;
double diff = (Cinv - Clu).cwiseAbs().maxCoeff();
cout << "max difference in C matrices =" << diff << endl;
}
• The internal implementation for A*B.inverse() is really inverse B first and then do the gemm? – lorniper Sep 17 '16 at 15:14
• Yes, in C++ the precedence of the dot(.) operator is much higher than *. – Bill Greene Sep 17 '16 at 15:31
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# Heritability of skewed X-inactivation in female twins is tissue-specific and associated with age
## Abstract
Female somatic X-chromosome inactivation (XCI) balances the X-linked transcriptional dosages between the sexes. Skewed XCI toward one parental X has been observed in several complex human traits, but the extent to which genetics and environment influence skewed XCI is largely unexplored. To address this, we quantify XCI-skew in multiple tissues and immune cell types in a twin cohort. Within an individual, XCI-skew differs between blood, fat and skin tissue, but is shared across immune cell types. XCI skew increases with age in blood, but not other tissues, and is associated with smoking. XCI-skew is increased in twins with Rheumatoid Arthritis compared to unaffected identical co-twins. XCI-skew is heritable in blood of females >55 years old (h2 = 0.34), but not in younger individuals or other tissues. This results in a Gene x Age interaction that shifts the functional dosage of all X-linked heterozygous loci in a tissue-restricted manner.
## Introduction
To balance the X-linked transcriptional dosages between the single X chromosome of males and the two X chromosomes of females, one X chromosome is silenced in female placental mammals1. The X-chromosome inactivation (XCI) process starts during preimplantation phases of human embryonic development, presumably at around the eight-cell stage2. XCI is initiated by the transcription of XIST, a 17 kb, alternatively spliced long noncoding RNA mapped to Xq13.2 and exclusively expressed on the inactive X (Xi)3. Once transcribed, XIST molecules spread in cis along the X chromosome4,5 inducing a progressive epigenetic silencing through the recruitment of chromatin remodeling enzymatic complexes, which impose repressive histone and DNA changes on the Xi chromosome6,7. Within each cell, the parental X chromosome selected for inactivation seems to occur at random, and the Xi is mitotically inherited to future somatic daughter cells. This random inactivation results in a mosaic of cells within an individual, where overall, a balanced expression (50:50) of both parental X-linked alleles is expected. Asymmetric selection of the X chromosome to inactivate causes the predominance of one parental Xi in a population of cells, unbalancing the X-linked transcriptional and allelic dosages toward one parental X chromosome. This phenomenon, known as skewed XCI (or nonrandom XCI), occurs when at least 80% of cells within a tissue inactivate the same parental X chromosome. The factors underlying primary skewed XCI are varied and several mechanisms are possible8. Secondary (or acquired) skewed XCI can result from positive selection of cells that after having inactivated a particular parental X, acquire a survival advantage over cells who inactivated the other parental X chromosome. Skewed XCI patterns can also be generated by the stochastic overrepresentation of cell clones in a given tissue, due for instance, to depletion of stem cell populations.
Comprised of 155 MB and containing >800 protein-coding genes, the X chromosome represents approximately the 5% of the haploid human genome. In heterozygous females with skewed XCI, the X-linked transcriptional and allelic dosages of silenced genes are unbalanced and may be functionally homozygous. Skewed XCI is a major cause of discontinuity of dominance and recessiveness, as well as penetrance and expressivity of X-linked traits. How skewed XCI patterns modulate phenotypes in females, and whether they are a cause or a consequence of associated phenotypes is not fully understood. Skewed XCI patterns have been observed in females with X-linked diseases9,10,11, autoimmune disorders12,13, as well as in breast14 and ovarian cancer15. In autoimmune diseases with higher prevalence in females, including rheumatoid arthritis (RA) and systemic lupus erythematosus, XCI is hypothesized to play a role. Chromosome X is enriched for immune-related genes and skewed XCI patterns could cause the breakdown of thymic tolerance induction processes16 conferring a high predisposition to develop autoimmunity (reviewed in17). XCI skewing levels in blood tissues have been associated with ageing, with multiple studies indicating an increase after 50–60 years of age18,19,20,21,22,23. To date, the mechanisms underlying skewed XCI in humans are not fully understood. Several twin studies have reported that genetic factors contribute to XCI skewing in blood-derived cells22,24, while other evidence indicated that most of the XCI skewing levels in human are acquired secondarily25.
Nearly all studies of XCI skewing levels in humans have been carried out in peripheral blood samples or in very small sample sizes26, while XCI patterns in other tissues have not been studied in great detail20,27,28. In this study, we comprehensively assess XCI patterns in a multi-tissue sample of nearly 800 female twins from the TwinsUK cohort29. We quantify the degree of skewing of XCI using a metric based on XIST allele-specific expression (ASE) from paired RNA-seq and DNA-seq data in four tissues (LCLs, whole-blood, fat and skin) and in multiple immune cell types (monocytes, B-cells, T-CD4+, T-CD8+, NK) purified from two identical co-twins. We examine the tissue-specific prevalence of skewed XCI patterns, compare the XCI skewing levels between tissues and across immune cell types within the same individual, and evaluate the association between XCI skewing and age, a complex autoimmune disease and lifestyle (smoking) traits. We show that XCI patterns are highly tissue-specific and shared across immune cell types within an individual, and that XCI skew in haematopoetic tissues increases with age. We investigate the factors underlying the skewed XCI using classical twin models to characterize the extent of the influence of genetic and environmental factors on the tissue-specific skewed XCI. We show that heritability of XCI skew is restricted to blood tissues of females >55 years old (h2 = 0.34), indicating that XCI patterns have both a heritable and environmental (age) basis.
## Results
### Quantification of degree of XCI skewing in TwinsUK
We assessed XCI patterns in multi-tissue samples from female twin volunteers from the TwinsUK cohort aged 38–85 years old (median age = 60; Supplementary Fig. 1)29,30. We quantified the degree of skewing of XCI using a metric based on XIST ASE from paired RNA-seq and DNA-seq data. XIST is uniquely expressed from the Xi3,28,31, so the relative expression of parental alleles within the XIST transcript are representative of XCI skewing levels in a bulk sample. Skewed XCI patterns can be detected and quantified from the expression levels of XIST-linked heterozygous variants32. Furthermore, transcriptional assays based on single monoallelically expressed X-linked genes, like XIST, have been used as a complement to the HUMARA assay to quantify XCI patterns25,33. We have also calculated the XCI patterns using an alternative method, based on the ASE of all non-pseudo-autosomal region (PAR) heterozygous loci available in a sample34 as done in supplements. We ran a series of benchmarking analyses to compare the non-PARASE calls to the XISTASE-based XCI skew calls (Supplementary Note 1, Supplementary Figs. 24) and demonstrated that the XISTASE is the appropriate method to use in our analyses. RNA-seq and genotype data were available for 814 lymphoblastoid cell lines (LCL) samples, 395 whole-blood samples, 766 subcutaneous adipose tissue samples (herein referred as fat) and 716 skin samples. After stringent quality control, we obtained XISTASE calls for 422 LCL samples, 72 whole-blood samples, 378 fat samples and 336 skin samples. The smaller sample size for whole-blood was due to the relatively smaller size of the starting dataset and the relatively lower RNA-seq coverage of this tissue in our dataset. In order to have an absolute measure of the magnitude of the XCI skewing levels in each sample, we calculated the degree of skewing of XCI (DS) from the XISTASE calls. DS is defined as the absolute deviation of the XISTASE from 0.5 (see “Methods”) and it has been similarly used in other investigations to assess XCI patterns21,35,36 and the XCI status of X-linked genes37. In line with previous investigations22,38 we classified samples with DS < 0.3 (corresponding to 0.2 < XISTASE < 0.8) to have random XCI, and samples with DS ≥ 0.3 (corresponding to XISTASE ≤ 0.2 or XISTASE ≥ 0.8) to have skewed XCI. Unless otherwise specified, DS was used in all the analyses performed.
We assessed the robustness of our estimates of the degree of skewing with an alternative DNA-based measure of XCI, the human androgen receptor assay (HUMARA)39. HUMARA was and is still the “gold standard” technique to assess XCI patterns. A previous study has reported good replicability between HUMARA and expression-based quantification of XCI skewing40. We used HUMARA to measure XCI skewing levels in 18 archived whole-blood DNA samples obtained at the same clinical visit as the LCLs samples. Spearman’s correlation between the quantifications was 0.8 (P = 7 × 10−5) revealing a high degree of reproducibility between the both XISTASE and HUMARA methods and the LCLs and whole blood (Fig. 1a).
Previous investigations have reported that the XCI skewing levels increase with age in blood tissues, as discussed below. While it would be expected that increases in XCI skewing levels would be observed over relatively large time spans, we would expect minimal variations of XCI skewing levels between two close time points. We therefore reasoned that the sensitivity of our quantifications could also be assessed by comparing the XCI skewing levels in the same individuals at close time points. Briefly, using a publicly-available longitudinal whole-blood RNA-seq dataset from the TwinsUK cohort41, we generated XISTASE calls at two time points (1–2.7 years later) in 16 samples (see “Methods”). The Spearman’s correlation between the XISTASE calls at the first time point and the XISTASE calls at the second time point was 0.94 at P < 2 × 10−15 (Fig. 1b). This indicates that XISTASE is a sensitive proxy when assessing the stability of XCI patterns over short time periods. Overall, these results indicate that XISTASE is a reproducible, accurate and sensitive proxy of XCI skewing levels.
### Skewed XCI is tissue-specific with higher prevalence in blood-derived tissues
We observed a wide range of DS values in the four tissues (Fig. 2), with clear differences in the prevalence of skewed individuals between tissues. Blood-derived tissues had the highest incidence of skewed individuals, with skewed XCI observed in 34% of LCLs samples and 28% of whole-blood samples and a lower incidence in the primary tissues, where 12% of fat and 16% of skin samples exhibited skewed XCI (Table 1). In order to examine the extent of similarities of XCI patterns between tissues, we compared the tissue-specific XCI skewing levels in a pairwise manner (Fig. 3). For each tissue–tissue comparison, we included individuals with XISTASE calls in both tissues (Table 2). We found the strongest correlation on XISTASE calls between LCLs and whole blood (n = 59, Spearman’s ρ = 0.78, P = 2 × 10−13), indicating that blood-derived tissues share highly similar XCI skewing levels. We also found a good degree of similarity between the XCI skewing levels in fat and skin tissues (n = 252, Spearman’s ρ = 0.47, P = 2 × 10−15; Fig. 3). However, low concordance was observed between skin and whole blood (n = 47, Spearman’s ρ = 0.3, P = 0.04) and fat and whole-blood (n = 57, Spearman’s ρ = 0.33, P = 0.01). Our data demonstrate that tissue-specific XCI skewing within an individual is common in the population, indicating that XCI patterns are partially controlled by tissue-specific regulatory mechanisms.
The active or inactive state of each X chromosome in a cell is clonally passed on to daughter cells. In a pool of cells derived from a single clone (or patch), the XCI patterns are expected to be completely skewed. Patch size refers to the amount of cell clones in a pool of cells (e.g. in a tissue biopsy). We considered the possibility that patch size might bias our quantification of XCI patterns in fat and skin samples. This is likely to occur in biopsies that are smaller than the tissue patch size. However, several considerations led us to exclude the possibility that patch sizes in fat and skin biopsies might confound our XISTASE calls. First, the biopsies included skin samples of 8 mm3 in size, which were cut into two skin and three fat samples. As reported in another study, this size is large enough to measure the XCI ratio without being confounded by patch size42. Second, most individuals exhibit random XCI patterns in fat and skin tissues, which is unlikely if patch size was larger than the biopsies. We therefore conclude that the biopsies used in this study are large enough to accurately assess the XCI patterns without being biased by patch size.
### LCLs in this study are representative of XCI skewing in-vivo in blood tissues
LCLs generated by Epstein–Barr virus (EBV) mediated transformation of B-lymphocyte cells have been and are widely used in gene expression studies. However, the possibility that the cell lines are monoclonal and/or polyclonal due to selection in the transformation process or clonal expansion in cell culture, and hence not be representative of the in vivo XCI skewing levels, is a potential problem when using LCLs to assess XCI skewing43. As the profiled RNA in this study was extracted from the LCLs very shortly after transformation with limited passaging or time in culture we expected this effect to be minimal, however, to address the possibility we performed the following analyses. First, as described above and shown in Fig. 1a, the degree of skewing in LCLs were highly correlated with the HUMARA-based quantifications of XCI patterns in paired whole-blood samples (Spearman’s ρ = 0.8, P = 7 × 10−5, n = 18). We would not expect such high similarity between the two quantifications if clonal propagation had occurred in LCLs samples after preparation. This was confirmed by the high correlation between LCLs and whole-blood XISTASE values (Spearman’s ρ = 0.78, n = 59; Fig. 3) and overall similarity in the prevalence of skewed XCI in LCLs and whole blood (Table 1). Finally, we assessed the degree of skewing in monocytes, B, T-CD4+, T-CD8+ and natural-killer (NK) cells purified from two monozygotic (MZ) twins exhibiting skewed XCI patterns in LCLs and from one individual exhibiting random XCI patterns in LCLs (Supplementary Fig. 5). We found that in both MZ twins showing skewed XCI in LCLs, the majority of immune cell types exhibited skewed XCI patterns. Conversely, none of the immune cell types purified from the nonskewed individual exhibited skewed XCI patterns (Table 3). These data indicate that within an individual, XCI skewing levels are shared across hematopoietic cells. We conclude that the XCI skewing levels of LCLs in this study are representative of XCI skewing in vivo in blood tissues.
### XCI skewing levels are positively associated with age in blood-derived tissues
XCI skewing levels in peripheral blood have been shown to increase with age in multiple studies18,19,20,21,22,24,40,44,45. The age-related increase of XCI skewing levels continues throughout life, since centenarians exhibit higher XCI skewing levels than 95 years old females22. However, there is very limited knowledge on the relationship between XCI patterns and ageing in tissues other than blood. In order to explore this, we investigated the association between age and degree of skewing in LCLs, fat, and skin. Our whole-blood estimates were excluded from analysis due to the low sample size (n = 72). Age was positively associated with XCI skew in LCLs (n = 422, P < 0.01), but we did not detect any association between XCI skew and age in skin (n = 336, P = 0.4) or in fat (n = 378, P = 0.7).
We next explored the dynamics of DS and age progression in each tissue, using the lowess procedure. Lowess curve detected an increase of DS beginning at around 55 years old in LCLs (Supplementary Fig. 6), in agreement with what was found in other studies20,22. Since the increase of DS starts at around 55 years, we divided LCLs samples into a younger group (n = 141, age < 55) and an older group (n = 281, age ≥ 55). We found that the mean DS in LCLs was significantly higher in older than in younger females (DSyounger = 0.2, DSolder = 0.24, T test, P = 0.03; Fig. 4). Accordingly, we found that the frequency of skewed XCI in LCLs was significantly higher in older (38%) than in younger (28%) females (χ2 test, P = 0.04; Fig. 4). In agreement with the lack of association between the DS and age, we did not detect significant differences between the mean DS in young and older females in fat (DSyounger = 0.15, DSolder = 0.15) or in skin tissues (DSyounger = 0.16, DSolder = 0.17). To acquire a more detailed view of the tissue-specific prevalence of skewed XCI in different groups of age, we categorized the samples into four age groups (40–50, 50–60, 60–70, and >70) and calculated the frequency of skewed XCI in each category (Fig. 4). We found that the frequency of skewed XCI increased with age in LCLs, with 41% of individuals >65 years old demonstrating skewed XCI patterns. We did not observe any increase in the skewed XCI frequencies with age in fat and skin tissues. Overall, these data further confirm that XCI skewing levels increase with age in blood-derived tissues, supporting previous investigations. However, we find that there is no increase in XCI in fat and skin tissue from the same individuals, suggesting that acquired XCI skewing with age is a distinctive feature of blood-derived tissues.
### Heritability of skewed XCI is dependent on tissue and age
Twin studies are a powerful strategy to investigate the heritability of complex traits. Previous twin studies have reported that skewed XCI in blood-derived samples is heritable, with h2 estimates of 0.68 in granulocytes of elderly twin pairs and 0.58 in peripheral blood cells22,23,24, however, these studies have not investigated heritability outside of blood. To estimate the influence of additive genetic effects (heritability) and environmental factors on the observed variance in XCI in the three tissues, we implemented the ACE twin model. The ACE statistical model quantifies the contribution of additive genetic effects (A), shared environment (C) and unique environment (E) to the phenotype variance. In order to investigate whether heritability varies with age, we stratified the twin pairs into a younger group (age < 55) and an older group (age ≥ 55; Supplementary Table 1). Age 55 was chosen as it was identified as the inflection point at which XCI skew begins to increase in the lowess analysis (Supplementary Fig. 6), in agreement with other studies20,22. We found that XCI skewing is heritable in LCLs of older females (ACE model, h2 = 0.34, P = 9.6 × 10−6), but not younger females (h2 = 0, P = 1). There was no evidence of heritability of XCI skew in fat or in skin tissues at any age (Table 4). The highest proportion of variance was explained by unique environmental factors in all tissues of both younger and older females (E2LCLs_younger = 0.99, E2LCLs_older = 0.66, E2Fat_younger = 0.73, E2Fat_older = 0.92, E2Skin_younger = 1, E2Skin_older = 1). As a complement to the heritability analysis, we calculated the intraclass Spearman correlation (IC) of XCI skew within MZ and DZ twin pairs of all ages, and within younger and older MZ and DZ twin pairs (Table 5). IC analyses of twin pairs is often used to demonstrate the existence of genetic effect in smaller sample sizes. The IC of XCI skew within MZ twins pairs was positive and statistically significant (ICMZ_allAges = 0.31, P = 0.02). We found significant IC of XCI skew within older MZ twin pairs (ICMZ_older = 0.42, P = 0.005), but not within young MZ twin pairs (ICMZ_younger = 0.06, P = 0.8). We did not detect significant IC within DZ twin pairs at any age, in agreement with previous study in blood22. The higher IC of XCI skew within MZ twin pairs compared with DZ twin pairs indicates the involvement of genetic determinants in the regulation of XCI skew in blood-derived tissues. The increase of IC in older compared to younger MZ twin pairs and the fact that the heritability of XCI skew is observed only in females older than 55, confirm a role for genetic variants as age-dependent regulators of the acquired XCI skew in blood-derived tissues. Presumably, genetically-determined secondary cell selection processes act in haematopoietic cell lineages, with the high mitotic rates contributing to the manifestation of their effects in blood-derived tissues. Results also highlight an age-independent role for environmental factors as regulators of XCI skew in blood, fat, and skin tissues.
### Individuals with autoimmunity exhibit more skewed XCI than unaffected co-twins
Chomosome X is enriched for immune-related genes. Most autoimmune disorders have higher prevalence in females than males17. Klinefelter syndrome (47,XXY) males have up to 14-fold higher risk of autoimmunity than 46,XY males46,47. These observations support an X-dosage effect in the pathogenesis of autoimmune diseases. As a mechanism of X-dosage compensation, the XCI process could be involved in the etiology of autoimmune disorders. Unbalanced X-linked transcriptional dosages toward one parental haplotype caused by skewed XCI patterns could influence the functions of the immune system. In particular during development, random XCI patterns in dendritic cells allow balanced expression of both parental X-linked self-antigens, a crucial event for the identification and negative selection of autoreactive T-cells in the thymus16,17. In line with the loss of mosaicism hypothesis48, we postulated that skewed XCI patterns may promote a breakdown of the tolerance induction processes with consequent release of autoreactive immune cells into the circulatory system. Supporting this hypothesis, higher frequencies of skewed XCI patterns have been observed in females affected with autoimmune disorders than in healthy controls12,13, however it is not known if this is a cause or consequence of disease, and these studies have not taken into account the underlying genetic predisposition of the cases and controls.
In order to address the association between autoimmune disease and XCI skewing while controlling for genetics, we investigated our samples for MZ twin pairs discordant for autoimmune disease. We identified eight MZ twins pairs discordant for RA in our study. RA is a chronic autoimmune condition affecting the lining of the synovial joints and associated with progressive disability49. RA is up to three times more frequent in females than males, and the most common age of onset ranges between 50 and 6050. Prevalence of RA ranges from 0.5% to 1%, but it significantly rises with age51. We found that the degree of XCI skewing in LCLs significantly differed between unaffected and affected co-twins (mean_DShealthy = 0.21; mean_DSaffected = 0.35; paired Wilcoxon’s test P < 0.05; Fig. 5). In seven out of eight twin pairs the affected co-twin was more skewed than their unaffected sister. These results are consistent with patterns seen in twins discordant for systemic lupus erythematosus and autoimmune thyroid disease52,53. Only four of the eight RA-discordant twin pairs had available XCI skew calls in fat and skin, and there was no significant difference in the XCI skew between the unaffected and affected co-twins in either tissue. Identical twins share 99% of the genome, age, and multiple environmental traits including in-utero growth, early life, and in most cases, socioeconomic statuses. Differences in the XCI patterns in blood may generate differences in the X-linked transcriptional dosages of immune-related genes and affect identical twins differently. Discordance in RA between identical twins could be well explained by differences in XCI patterns, however, we note that with cross-sectional data it is not possible to determine if changes in XCI drive RA or if RA causes changes in XCI skewing, or if both are independently associated to a third factor. Due to the low sample size, we acknowledge these results are not generalizable at a population level; larger sample sizes will be required to fully test the association of skewed XCI with twin discordance for RA and other autoimmune traits.
### XCI skew is associated with smoking status in older females
Tobacco smoking has been reported to induce epigenomic changes including DNA methylation variation (reviewed in ref. 54). Smoking is a well characterized risk factor in cancer55 and, as more recently discovered, in the etiology of autoimmunity56. Although smoking-related X-linked DNA methylation sites have been discovered57, no previous studies, to our knowledge, have investigated the relationship between smoking and XCI patterns. We reasoned that changes of XCI patterns may result from smoking, and affect in turn short-term and long-term health. In order to test our hypothesis, we used the 270 individuals in our dataset for which we had smoking status at the time of sample collection, including 233 never smokers and 37 current smokers58. We found no difference in the frequency of skewed XCI patterns between never and current smokers (36% and 35%, respectively) in LCLs. To take into account the effects of age on the degree of skewing in blood-derived tissues and to examine the relationship between smoking status and degree of skewing at different ages, we split the dataset into a younger (age < 55) and an older group (age ≥ 55; Supplementary Table 2). While the frequencies of skewed XCI were very similar between young smokers and young never smokers (27% and 28%, respectively), we detected a higher prevalence of skewed XCI in older smokers compared with older never smokers (47% and 40%, respectively). Accordingly, we found an overall positive association between XCI skew and smoking status in older (P = 0.02), but not in younger individuals (P = 0.5). The data suggest a role for smoking as a modulator of XCI skew in blood-derived tissues of females older than 55. Presumably, the association between smoking and XCI skew changes is complex, and further investigations are needed to characterize the genetic and molecular mechanisms underlying this phenomenon.
## Discussion
In this study, we used multi-tissue transcriptomic data from twins to comprehensively characterize XCI patterns in LCLs, whole-blood, fat, and skin tissues from a healthy twin cohort. We show XCI patterns to be tissue-specific and that blood-derived tissues exhibited the highest prevalence of skewed XCI and share the highest similarity of XCI patterns. These findings indicate that XCI patterns are partially driven by tissue-specific mechanisms, and that the XCI skew measured in blood is not a reliable proxy for the skew in other tissues. Skewed XCI patterns limited to disease-relevant tissues and cells have been observed in multiple conditions9,10,14,15,59,60 but except for several cases of X-linked diseases, their roles in disease etiology and predisposition remain largely unknown. Our results demonstrate that tissue-specific XCI patterns within an individual is common in this healthy population.
We show that XCI skewing levels in blood tissues increase with age, with an inflection point at around 55, in line with previous reports18,19,20,21,22,24,40,44,45. In this study, more than 41% of females >65 years old demonstrate skewed XCI patterns in blood-derived tissues, indicating that acquired skewed XCI is a highly prevalent phenotype in ageing populations. We show age-related increase in XCI skew is a distinctive feature of blood-derived tissues, with no evidence for an age-related increase in fat or skin. Age-related increase in XCI skew partially explains the higher incidence of skewed XCI in blood than fat and skin tissues. The effects of age-related skewing of XCI on healthy ageing remain largely unknown, but may have a broad impact on the immune system. We demonstrate that within an individual, the XCI patterns in blood-derived tissues are shared across multiple immune cell types including monocytes, B-cells, T-CD4+, T-CD8+ and NK cells. Hematopoietic stem cells and the immune system continue to develop throughout life. Presumably, in line with the loss of mosaicism hypothesis48, imbalanced X-linked immune-related gene expression toward one parental haplotype leads to a reduced molecular diversity, which may translate in a decline of immune repertoire as well as poor sustenance of the immunological memory. Thus, by influencing the immune system, age-acquired skewed XCI could influence the predisposition to and manifestation of age-related traits, such as hematopoietic disorders, in women. We support an involvement of skewed XCI in the regulation of the immune system by showing that XCI patterns in LCLs, and consequently in multiple immune cell types, are consistently more skewed in individuals affected with autoimmunity than in healthy identical co-twins. Identical twins share nearly 100% of the genome, including chromosome X. Discordance in an autoimmune phenotype between twins, could partially be attributed to differences in the X-linked allelic and transcriptional dosages of X-linked immune-related genes resulting from difference in the XCI skew.
Previous twin studies have reported that XCI patterns in blood have a genetic component22,24. To our knowledge, this is the first study to investigate heritability of XCI skewing levels in other tissues. We found that the heritability of XCI skewing level is limited to blood-derived tissues of females >55 years old (h2 = 0.34), with no evidence of heritability in fat or skin or younger individuals in any tissue. The restriction of heritability to blood of older individuals is of interest given the link between skewed X-inactivation and clonal haematopoiesis. Positive selection of cells carrying an advantageous somatic mutation will lead to clonal haematopoiesis and skewed XCI patterns as the selected cells will carry the same inactivated parental X. Somatic mutation-driven clonal haematopoiesis is now known to be common in blood of healthy older individuals and is often referred to as clonal haematopoiesis of indeterminate potential (CHIP)36,61,62,63. CHIP is associated with increased risk of both cancer and all-cause mortality, and likewise skewed XCI, the prevalence of CHIP increase with age64,65. The higher skew in the XCI patterns in individuals affected with RA than in unaffected co-twins, is well explained by both the occurrence and the age-related increase in the prevalence of clonal hematopoiesis in RA patients66. The increase in XCI skew in older smokers in our study is also consistent with the increase in clonal haematopoiesis observed in smokers63,67,68. All together, these data converge in suggesting a link between XCI skew and clonal hematopoiesis. It is currently unknown to what extent CHIP accounts for age-acquired XCI skew, however, if it is a major driver this would suggest that like age-related XCI skew, CHIP has a significant germline genetic component. Stochastic selection of cells could also contribute to the variance of XCI skewing levels, but, in agreement with previous works22,24, we reason that their contribution is minimal. If stochastic selection of cells was a dominant mechanism, the correlation of XCI patterns between twin pairs would decrease with age.
Overall, the data presented in this study indicate a gene × age interaction that shifts the functional allelic dosages of X-linked heterozygous loci in a tissue-restricted manner. The high prevalence of skewed XCI and tissue-restricted XCI in a healthy population could complicate discovery of Chromosome X variants associated with a trait and subsequent genetic risk prediction, as an individual’s genotype may not match their functional genotypic dosage in the relevant tissue. Further investigations of the heterogeneity of XCI patterns across tissues and how this is regulated are essential to clarify the biomedical implications of skewed XCI and its role in healthy ageing in women.
## Methods
### Sample collection
The study included 856 female twins from the TwinsUK registry29,30 who participated in the MuTHER study69 . Study participants included both MZ and dizygotic (DZ) twins, aged 38–85 years old (median age = 60; Supplementary Fig. 1) and were of European ancestry. Volunteers received detailed information regarding all aspects of the research project and gave a prior signed consent to participate in the study. Peripheral blood samples were collected and LCLs were generated via EBV-mediated transformation of the B-lymphocyte fraction. Punch biopsies of subcutaneous adipose tissue were taken from a photo-protected area adjacent and inferior to the umbilicus. Skin samples were obtained by dissection from the punch biopsies. Adipose and skin samples were weighed and frozen in liquid nitrogen. This project was approved by the research ethics committee at St Thomasʼ Hospital London, where all the TwinsUK biopsies were carried out. Volunteers gave informed consent and signed an approved consent form prior to the biopsy procedure. Volunteers were supplied with an appropriate detailed information sheet regarding the research project and biopsy procedure by post prior to attending for the biopsy.
### RNA-sequencing data
The Illumina TruSeq sample preparation protocol was used to generate the cDNA libraries for sequencing. Samples were sequenced on an Illumina HiSeq 2000 machine and 49 bp paired-end reads were generated. Adapter and polyA/T nucleotide sequences were removed and sequencing reads were aligned to the UCSC GRCh37/hg19 reference genome with the Burrows-Wheeler Aligner v.0.5.971. Samples that failed library preparation (according to the manufacturer’s guidelines) or had less than 10 million reads were discarded. Genes were annotated using the GENCODE v10 reference panel79.
### Longitudinal RNA-sequencing data
Peripheral blood samples were collected 1–2.7 years apart from 114 female twins of the TwinsUK registry and were processed with the Illumina TruSeq protocol, sequenced on a HiSeq 2000 machine and 49 bp paired-end reads generated41. Adapter and polyA/T nucleotide sequences were trimmed using trim_galore and PrinSeq tools80, respectively. Reads were aligned to the UCSC GRCh37/hg19 reference genome with the STAR v.2.5.2a aligner81. Alignments containing non-canonical and unannotated splice junctions were discarded. Properly paired and uniquely mapped reads with a MAPQ of 255 were retained for further analysis.
### Purified immune cells RNA-sequencing data
Monocytes, B, T-CD4+, T-CD8+ and NK cells were purified using fluorescence activated cell sorting (FACS) from two MZ twins exhibiting skewed XCI patterns in LCLs and from one individual exhibiting random XCI patterns (Supplementary Fig. 5). Total RNA was isolated and cDNA libraries for sequencing were generated using the Sureselect sample preparation protocol. Samples were then sequenced in triplicates on an Illumina HiSeq machine and 126 bp paired-end reads were generated. Adapter and polyA/T nucleotide sequences were trimmed using trim_galore and PrinSeq tools80, respectively. Human and prokaryotic rRNAs were identified using sortmerna v.2.182 and removed. Reads were aligned to the UCSC GRCh37/hg19 reference genome using STAR v.2.5.2a81. Alignments containing noncanonical and unannotated splice junctions were discarded. Properly paired and uniquely mapped reads with a MAPQ of 255 were retained for further analysis.
### Correction of RNA-seq mapping biases
To eliminate mapping biases, all RNA-seq data were re-aligned within the WASP pipeline for mappability filtering83. The WASP tool has an algorithm specifically designed to identify and correct mapping biases in RNA-seq data. In each read overlapping a heterozygous SNP, the allele is flipped to the SNP’s other allele (generating all possible allelic combinations) and the read is remapped. Reads that did not remap to the same genomic location indicate mapping bias and were discarded. Reads overlapping insertions and deletions were also discarded. Properly paired and uniquely mapped reads were retained for analysis.
### Quantification of XISTASE and degree of XCI skewing (DS)
In each sample, the XCI skewing levels were quantified by averaging the ASE values of heterozygous SNPs within XIST. All SNPs were phased prior to averaging as detailed above. The measure, called XISTASE is defined as follow:
$$XIST_{{\mathrm{ASE}}} = \frac{{{\sum} {XIST\_{\mathrm{SNP}}_{{\mathrm{ASE}}}} }}{{n}}\;\left( {XIST_{{\mathrm{ASE}}}} \right),$$
(1)
where XIST_SNPASE are the ASE values of heterozygous SNPs within XIST and n is the number of heterozygous SNPs within XIST in the sample. XIST is uniquely expressed from the inactive X chromosome3,28,31, and thus the relative expression of parental alleles within XIST transcript are representative of XCI skewing levels in a bulk sample. The expression levels of polymorphisms within XIST have been used to infer XCI skewing levels in a sample32,33. We have also calculated XCI skew using the ASE of all non-PAR genes available in a sample, as previously done34. We compared our XISTASE calls to the non-PAR-based calls and concluded that the XISTASE calls are a better proxy for XCI skew in our analyses (Supplementary Note 1, Supplementary Figs. 24). Within each sample, the XISTASE values range from 0 to 1; an XISTASE value of 0.5 indicates equal inactivation of the two parental chromosomes (completely random XCI patterns, 50:50 XCI ratio), whereas a value of 0 or 1 indicates complete inactivation of one parental chromosome (completely skewed XCI patterns, 100:0 XCI ratio). To be consistent with previous literature22,38, we classified samples with XISTASE ≤ 0.2 or XISTASE ≥ 0.8 to have skewed XCI patterns, and samples with 0.2 < XISTASE < 0.8 to have random XCI patterns. To have an absolute measure of the magnitude of the XCI skewing levels in each sample, (or effect size of XISTASE), the degree of skewing of XCI (DS) was calculated. DS is the absolute deviation of XISTASE from 0.5. In each sample, DS was calculated as follow:
$${\mathrm{DS}} = \left| {0.5 - XIST_{{\mathrm{ASE}}}} \right|\;\left( {{\mathrm{Degree}}\;{\mathrm{of}}\;{\mathrm{skewing}}\;{\mathrm{of}}\;{\mathrm{XCI}}} \right).$$
(2)
DS does not take into account the direction of XCI skewing, but the degree of deviation from a 50% XCI patterns (XISTASE = 0.5). DS is then a measure of the magnitude of XCI skewing levels in a sample. DS values range from 0 to 0.5, where 0 means random XCI and 0.5 completely skewed XCI patterns. Samples with DS ≥ 0.3 were classified to have skewed XCI, while samples with DS <0.3 were classified to have random XCI.
### Heritability analysis of DS
The relative contributions of additive genetic factors (A), shared (C) and unique environmental factors (E) to the tissue-specific variance of DS, were calculated using the twinlm() function in the mets R package85. For each tissue, samples were split into a young (<55) and an older (≥55) group according to their ages (Supplementary Table 1). Due to the low number of MZ and DZ twin pairs in each group, whole-blood was excluded from heritability analysis. To further assess the contribution of genetic effects, the intraclass Spearman’s correlation (IC) of DS in blood-derived tissues of young and older MZ and DZ twin pairs was also calculated.
### Differences in the DS between identical co-twins discordant for RA
We used a subset of eight MZ twin pairs where the co-twins of each pair are discordant for RA. Diagnoses were either confirmed during visits at the rheumatologist clinics at St Thomas’ Hospital in London, or confirmed by phone-interview by a rheumatology clinical fellow to confirm the diagnosis of RA based on the American College of Rheumatology 1987 criteria86. In case of unclear diagnosis of RA, participants were reviewed in clinic or were excluded. Difference in the distribution of the degree of XCI skewing in LCLs between the two groups (twins affected with autoimmunity vs healthy co-twins) was evaluated using paired Wilcoxon test. A P-value ≤ 0.05 was considered to be statistically significant.
### Association between degree of skewing and smoking status
Association between the degree of skewing in LCLs and self-reported smoking status was tested in the 270 individuals with reliable smoking status recorded58. Dataset included 270 females classified either as current smokers (n = 37) or never smokers (n = 233; Supplementary Table 2). To examine the association between DS and smoking status, the smoking status was converted into a binary trait (0 = no smoker, 1 = smoker). A linear model of the DS as a function of the smoking status was then implemented for younger (age < 55) and older (age ≥ 55) individuals separately. Age was used as covariate. A P-value ≤ 0.05 was considered to be statistically significant.
### Reporting summary
Further information on research design is available in the Nature Research Reporting Summary linked to this article.
## Data availability
TwinsUK RNA-seq data is available from EGA (Accession number: EGAS00001000805). TwinsUK genotypes and phenotypes are available upon application to TwinsUK Data Access Committee (https://twinsuk.ac.uk/resources-for-researchers/access-our-data/). All other data are contained in the manuscript and its supplementary information.
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## Acknowledgements
This study was supported by MRC Project Grant (MR/R023131/1) to K.S.S. The TwinsUK study was funded by the Wellcome Trust and European Community’s Seventh Framework Programme (FP7/2007-2013). The TwinsUK study also receives support from the National Institute for Health Research (NIHR)-funded BioResource, Clinical Research Facility and Biomedical Research Centre based at Guy’s and St Thomas’ NHS Foundation Trust in partnership with King’s College London. This work was also supported by the a joint UK Economic and Social Research Council (ESRC) and Biotechnology and Biological Sciences Research Council (BBSRC) grant (ES/N000277/1) to CCYW. P.C.T. is supported by a Chang Gung Memorial Hospital Research Funding grant (CMRPD1J0081). This project was enabled through access to the MRC eMedLab Medical Bioinformatics infrastructure, supported by the Medical Research Council [grant number MR/L016311/1].
## Author information
A.Z. M.D. and K.S.S conceived and designed the project. A.Z. performed analysis. S.N. contributed data discussion. P.C.T and J.T.B. contributed data. R.A.E. performed RNA isolation and FACS experiments. S.R. and C.Y.W performed HUMARA experiments. A.Z. and K.S.S. wrote the paper. All authors read and approved the manuscript. The authors thank Julia El-Sayed Moustafa and Amy Roberts for providing feedback on the manuscript and Alessandro Morea for assistance with HUMARA assays. We also thank Philippa Wells, Flore Zufferey, and Frances Williams for their work identifying cases of rheumatoid arthritis in the TwinsUK cohort.
Correspondence to Kerrin S. Small.
## Ethics declarations
### Competing interests
The authors declare no competing interests.
Peer review information Nature Communications thanks the anonymous reviewer(s) for their contribution to the peer review of this work.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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# A marketer bought N crates of empty cardboard gift boxes. Each crate
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A marketer bought N crates of empty cardboard gift boxes. Each crate [#permalink]
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18 Feb 2015, 04:26
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A marketer bought N crates of empty cardboard gift boxes. Each crate held Q individual gift boxes, and the lot of N crates was purchases at a wholesale price of W dollars. This marketer will sell collections of J cardboard gift boxes to retailers, at a price of P dollars for each collection. (Note: J is a divisor of Q.) The marketer knows that, when he has sold all the cardboard gift boxes this way, he wants to net a total profit of Z dollars on the entire transaction. What price P must he charge, to net this profit? Express P in terms of N, Q, W, J, and Z.
A. J(Z - W)/(NQ)
B. J(Z + W)/(NQ)
C. Q(Z - W)/(NJ)
D. Q(Z + W)/(NJ)
E. N(Z - W)/(QJ)
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Re: A marketer bought N crates of empty cardboard gift boxes. Each crate [#permalink]
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18 Feb 2015, 06:40
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Bunuel wrote:
A marketer bought N crates of empty cardboard gift boxes. Each crate held Q individual gift boxes, and the lot of N crates was purchases at a wholesale price of W dollars. This marketer will sell collections of J cardboard gift boxes to retailers, at a price of P dollars for each collection. (Note: J is a divisor of Q.) The marketer knows that, when he has sold all the cardboard gift boxes this way, he wants to net a total profit of Z dollars on the entire transaction. What price P must he charge, to net this profit? Express P in terms of N, Q, W, J, and Z.
A. J(Z - W)/(NQ)
B. J(Z + W)/(NQ)
C. Q(Z - W)/(NJ)
D. Q(Z + W)/(NJ)
E. N(Z - W)/(QJ)
The marketer initially paid W= cost.
There are Q boxes in each crate, and J boxes make a collection,
so there are Q/J collections in each crate, and NQ/J collections in total.
If he charges a price P, his revenue would be PNQ/J.
Now, profit = revenue - cost,
so Z = PNQ/J – W
i.e. P = (Z+W)J / NQ
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A marketer bought N crates of empty cardboard gift boxes. Each crate [#permalink]
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18 Feb 2015, 11:23
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Bunuel wrote:
A marketer bought N crates of empty cardboard gift boxes. Each crate held Q individual gift boxes, and the lot of N crates was purchases at a wholesale price of W dollars. This marketer will sell collections of J cardboard gift boxes to retailers, at a price of P dollars for each collection. (Note: J is a divisor of Q.) The marketer knows that, when he has sold all the cardboard gift boxes this way, he wants to net a total profit of Z dollars on the entire transaction. What price P must he charge, to net this profit? Express P in terms of N, Q, W, J, and Z.
A. J(Z - W)/(NQ)
B. J(Z + W)/(NQ)
C. Q(Z - W)/(NJ)
D. Q(Z + W)/(NJ)
E. N(Z - W)/(QJ)
Kudos for a correct solution.
J (W) / (NQ) should be price to break even.
Add Z and it should be price to net the profit
-->
Option B should be price to net the profit.
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Re: A marketer bought N crates of empty cardboard gift boxes. Each crate [#permalink]
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18 Feb 2015, 21:19
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Answer = B. J(Z + W)/(NQ)
Total Profit = Z
Total cost price = w
Total boxes = nq
Selling price per box $$=\frac{p}{j}$$
Total selling price $$= \frac{p}{j} * nq$$
Setting up the profit equation
$$z = \frac{pnq}{j} - w$$
$$p = \frac{z+w}{nq} * j$$
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Re: A marketer bought N crates of empty cardboard gift boxes. Each crate [#permalink]
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18 Feb 2015, 22:05
1
Hi All,
This question is wordy and loaded with variables, but it can be solved by TESTing VALUES. There are some fantastic 'shortcuts' in the the answer choices as well.
We're told that there are N crates and that each crate holds Q gift boxes.
N = 3 crates
(3)(4) = 12 total gift boxes
We're told that all of the crates were purchased for a total of W dollars
W = 12
12 gift boxes for 12 dollars
The seller will sell J gift boxes per 'collection' at a price of P dollars per 'collection' (we're told that J is DIVISOR of Q)
J = 2 gift boxes per 'collection'
12/2 = 6 'collections available
P = 3 dollars per 'collection'
(6)(3) = $18 in revenue after ALL collections|gift boxes are sold Total profit is Z dollars$18 in revenue
$12 cost$18 - $12 =$6 = Z
We're asked for the value of P, using the other variables....
N = 3
Q = 4
W = 12
J = 2
Z = 6
We're looking for an answer that equals 3.
Now, before jumping into the answer choices, notice all of the PATTERNS that exist:
1) Answers A and B are almost exactly the same.
2) Answers C and D are almost exactly the same.
3) Each parentheses is either (Z+W) or (Z-W)
You can use the patterns to avoid doing the same calculation over and over. Let's start with Answers A and B:
A= 2(-6)/12 = -1 NOT a match
B= 2(18)/12 = 3 This IS a match
Now, let's do C and D tougher (note that we ALREADY KNOW the values of Z+W and Z-W from our prior work):
C= 4(-6)/6 = -4 NOT a match
D= 4(18)/6 = 12 NOT a match
E= 3(-6)/8 = -18/8 NOT a match
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Special Offer: Save $75 + GMAT Club Tests Free Official GMAT Exam Packs + 70 Pt. Improvement Guarantee www.empowergmat.com/ ***********************Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!*********************** Math Expert Joined: 02 Sep 2009 Posts: 49303 Re: A marketer bought N crates of empty cardboard gift boxes. Each crate [#permalink] ### Show Tags 22 Feb 2015, 12:14 Bunuel wrote: A marketer bought N crates of empty cardboard gift boxes. Each crate held Q individual gift boxes, and the lot of N crates was purchases at a wholesale price of W dollars. This marketer will sell collections of J cardboard gift boxes to retailers, at a price of P dollars for each collection. (Note: J is a divisor of Q.) The marketer knows that, when he has sold all the cardboard gift boxes this way, he wants to net a total profit of Z dollars on the entire transaction. What price P must he charge, to net this profit? Express P in terms of N, Q, W, J, and Z. A. J(Z - W)/(NQ) B. J(Z + W)/(NQ) C. Q(Z - W)/(NJ) D. Q(Z + W)/(NJ) E. N(Z - W)/(QJ) Kudos for a correct solution. MAGOOSH OFFICIAL SOLUTION Algebraic Solution: So, the marketer initially paid W: that was his outlay, his cost. There are Q boxes in each crate, and J boxes make a collection, so there are Q/J collections in each crate, and NQ/J collections in total. If he charges a price P, his revenue would be PNQ/J. Now, profit equals revenue minus cost, so Z = PNQ/J – W . Solve this for P Attachment: gpp-vitac_img7.png [ 2.46 KiB | Viewed 2300 times ] Numerical Solution: Here, we have several choices to make, and it would be a good idea to pick numbers that aren’t the same, or aren’t even divisible by each other if that’s not required. Picking some different prime numbers is good way eliminate more than one answer that comes out to the correct value. I will say the price is 2, a nice round prime number. Let’s say J = 3, so Q must be a multiple of that — say Q = 30. Let’s N = 7. This means he has 10 collections from every box, and 70 collections from the total lot. If he charges$2 per collection, that’s a revenue of $140. Let’s say his outlay was W =$95; then his profit is Z = \$45.
OK, that’s an unusual enough set of number that we expect no more than one solution will work. We will plug in N = 7, Q = 30, W = 95, J = 3, and Z = 45, and we hope to get an output of P = 2.
Attachment:
gpp-vitac_img8.png [ 15.7 KiB | Viewed 2300 times ]
Thus, because we chose numbers well, the only possible answer is (B).
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Re: A marketer bought N crates of empty cardboard gift boxes. Each crate [#permalink]
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05 Aug 2018, 12:39
In the Algebraic solution the Equation is set up as
(PNQ)/J = Z + W
What clues in the question signal that you should'nt perform operations on Z or W?
I was setting up the cost as W*N, the number of crates times the cost.
Re: A marketer bought N crates of empty cardboard gift boxes. Each crate &nbs [#permalink] 05 Aug 2018, 12:39
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# A marketer bought N crates of empty cardboard gift boxes. Each crate
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# how do you calculate percent error in physics Healy, Kansas
You might also enjoy: Sign up There was an error. For instance, a meter stick cannot distinguish distances to a precision much better than about half of its smallest scale division (0.5 mm in this case). If I want to measure millimeter dimension with an instrument with micron resolution. Yes No Cookies make wikiHow better.
This means that you should subtract the real value from the estimated value. The two quantities are then balanced and the magnitude of the unknown quantity can be found by comparison with the reference sample. Is there a role with more responsibility? Ex: 0.1 x 100 = 10% Community Q&A Search Add New Question How do I calculate a percentage error when resistors are connected in a series?
Re-zero the instrument if possible, or measure the displacement of the zero reading from the true zero and correct any measurements accordingly. experimental-physics error-analysis share|cite|improve this question asked Oct 22 '14 at 0:52 gyeox29ns 485 3 Percent error is almost never of interest, so the real answer is that working scientists would Random errors can be reduced by averaging over a large number of observations. The result of the difference is positive and therefore the percent error is positive.
Reply ↓ Leave a Reply Cancel reply Search for: Get the Science Notes Newsletter Get Projects Free in Email Top Posts & Pages Printable Periodic Tables List of Metals Table of You look up the density of a block aluminum at room temperature and find it to be 2.70 g/cm3. How to know if a meal was cooked with or contains alcohol? Why did my electrician put metal plates wherever the stud is drilled through?
You can only upload files of type PNG, JPG, or JPEG. Comparing Approximate to Exact "Error": Subtract Approximate value from Exact value. i actually know this! % error = [accepted value - measured value /divided by/ accepted value] multipled by 100 Source(s): 8th grade earth science class! These are reproducible inaccuracies that are consistently in the same direction.
Percentage is always positive, so we invoke absolute value to get $$E=\frac{|A-B|}{B}\times 100$$ Actually, we can also use $$E=\frac{|B-A|}{B}\times 100$$ Percent error is always taken with respect to the known quantity The post has been corrected. RE: how do you calculate percent error? Ex: 10 - 9 = 1 3 Divide the result by the real number.
Not the answer you're looking for? Updated September 14, 2016. How can I create this table in Latex if statement - short circuit evaluation vs readability QQ Plot Reference Line not 45° Cohomology of function spaces Why aren't sessions exclusive to If we fire a bullet through the rain, can we see its trail where it splashed the droplets in its path, or does it happen too fast?
Change Equation to Percent Difference Solve for percent difference. This will give you a decimal number. Convert the decimal number into a percentage by multiplying it by 100. Add a percent or % symbol to report your percent error value.Percent Error Example in physics Source(s): calculate percent error: https://tr.im/3hk7l ? · 1 year ago 0 Thumbs up 0 Thumbs down Comment Add a comment Submit · just now Report Abuse For the best The difference between the actual and experimental value is always the absolute value of the difference. |Experimental-Actual|/Actualx100 so it doesn't matter how you subtract.
One of the best ways to obtain more precise measurements is to use a null difference method instead of measuring a quantity directly. Doing so often reveals variations that might otherwise go undetected. Amy · 6 months ago 0 Thumbs up 0 Thumbs down Comment Add a comment Submit · just now Report Abuse ((actual value minus experimental value) divided by actual value) all Chemistry Homework Help Worked Chemistry Problems How To Calculate Percent Error Sample Percent Error Calculation Percent error is a common lab report calculation used to express the difference between a measured
Hot Network Questions Conference presenting: stick to paper material? Todd also writes many of the example problems and general news articles found on the site. For example,, in experiments involving yields in chemical reactions, it is unlikely you will obtain more product than theoretically possible.Steps to calculate the percent error:Subtract the accepted value from the experimental value.Take The following are some examples of systematic and random errors to consider when writing your error analysis.
Becomean Author! Co-authors: 11 Updated: Views:256,844 68% of people told us that this article helped them. Reply ↓ Todd Helmenstine Post authorJanuary 28, 2016 at 2:15 pm Thanks for pointing that out. For example, you would not expect to have positive percent error comparing actual to theoretical yield in a chemical reaction.[experimental value - theoretical value] / theoretical value x 100%Percent Error Calculation
IQ Puzzle with no pattern How would a planet-sized computer power receive power? stella646 · 1 decade ago 20 Thumbs up 2 Thumbs down Comment Add a comment Submit · just now Report Abuse Add your answer How do you calculate percent error? Personal errors - Carelessness, poor technique, or bias on the part of the experimenter. About Todd HelmenstineTodd Helmenstine is the physicist/mathematician who creates most of the images and PDF files found on sciencenotes.org.
A similar effect is hysteresis where the instrument readings lag behind and appear to have a "memory" effect as data are taken sequentially moving up or down through a range of You measure the sides of the cube to find the volume and weigh it to find its mass. Is it possible to rewrite sin(x)/sin(y) in the form of sin(z)? Yes No Not Helpful 4 Helpful 4 Unanswered Questions How can I find the value of capital a-hypothetical?
When you calculate the density using your measurements, you get 8.78 grams/cm3. If you need to know positive or negative error, this is done by dropping the absolute value brackets in the formula. In most cases, absolute error is fine.
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The challenge below requires you to be familiar with formal parser theory. If you don't know what the question is asking because you don't know what the terms mean, context-free grammars and first/follow sets are covered in a lot of university courses.
I can recommend this Stanford course, in particular handouts 08 and 09 (from page 7). I've extracted also extracted a cheat sheet from these handouts - I recommend anyone attempting this challenge to read it.
Write a program or function that given a context-free grammar finds the follow set of every nonterminal. Informally, the follow set of a nonterminal is a set of terminals and $ (meaning end-of-input) that you can possibly find after that terminal in a valid sentence. The input is given as a single printable ASCII string or array of printable ASCII lines. You may output the sets in any reasonable format, using $ (either as literal output, or string inside a set, etc) to indicate the end of input. You may assume that the input is always valid according to the format below.
The context free grammar is given in a very simplified manner. Every line contains a single production. Every production is a space separated list of symbols. A terminal is a string of characters surrounded by apostrophes (e.g. '**'). For simplicity you may assume that terminals do not contain spaces, but it would be nice if your program allows it. A nonterminal may be any string not containing spaces or $. The empty production (normally indicated with ε) is simply a line containing only the left hand side nonterminal. The first line is the production defining the start symbol. As an example, the following grammar: S → aSa | bSb | ε Will be given as: S 'a' S 'a' S 'b' S 'b' S Example input / outputs: In: S 'a' S 'a' S 'b' S 'b' S Out: S {'a', 'b',$}
In:
S A B C
A 'a'
A C 'b'
A
B C
B 'd' A
B
C 'e'
C 'f'
Out:
S {$} A {'d', 'e', 'f'} B {'e', 'f'} C {'b', 'e', 'f',$}
In:
Start Alice Bob
Alice Charlie 'a'
Alice
Bob Bob 'a' Alice Charlie
Bob '!!!'
Charlie 'b'
Charlie
Out:
Start {$} Alice {'a', '!!!', 'b',$}
Bob {'a', $} Charlie {'a',$}
Shortest code in bytes wins.
• Assuming that people know what a context free grammar is seems fine, but I think it wouldn't hurt the challenge if you included the definition of a follow set right here instead of just linking to it. – Martin Ender Dec 9 '15 at 7:44
• This brings back some memories from "Compiler Construction" at university, where we had to solve lots of similar tasks. – insertusernamehere Dec 9 '15 at 12:49
Includes +4 for -0p
Give grammar on STDIN (without trailing spaces. Make sure to remove the extra space in the second example). Assumes non-terminal names only contain letters, digits and _. Uses # instead of $ to indicate end of input. Can handle literals containing spaces perl -M5.010 follow.pl E T e e '+' T e e T F t t '*' F t t F '(' E ')' F 'id' ^D Outputs the follow sets as a list of non-terminal literal in no particular order. For the example above it outputs: F ')' F # t ')' t # T ')' T # F '+' t '+' T '+' F '*' e ')' e # E ')' E # follow.pl: #!/usr/bin/perl -0n s/'.*?'/~$&/eg;s% (?=(\w.*\n))%$_.=">$1"%reg;/\s/;$_.=">$ #\n";s%^((\w+)\K ?\S*).*%$s{$1}++||"$a.=s/ 2\\b/&/rg"%eemgr,s%^(\w+ ).*?(\w+)%"$a.=s/>$1/>$2 /rg"%eermg,$_.=$a,s%>.*\xd8\K .*%%g,s%.+\n%$&x!/\n$&/g%eg until_++;s/\xd8.*?\xd8/~\$&/eg;say/>(\w+ \W\S*\n)/g
Works as shown, but replace \xd8 and \n by their literal versions to get the claimed score.
It should be possible to improve this since converting the first sets to the follow` sets is currently very awkward.
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# Thread: Convergence & proof that e^x is greater than x^2 and such
1. ## Convergence & proof that e^x is greater than x^2 and such
I have a few problems concerning infinite series which I need some help on. I'm supposed to determine the convergence/divergence using ONLY the comparison tests. Here are the problems:
$\displaystyle\Sigma\frac{1}{e^n+1}$
$\displaystyle\Sigma\frac{2^n}{3^n-1000}$
$\displaystyle\Sigma\frac{1}{e^n^2}$
For the first and the third ones, I want to apply the plain comparison test, using $\frac{1}{x^2}$ as the series which I compare it to. However, I need a way to prove that $e^n$ is greater than $x^n$ for all n [1, $\infty)$. Once I prove this, how would I apply it to the third problem since it's ${e^{n}}^2$? By the way, that is e^n^2...it's kind of hard to tell with the latex. For the second problem, here is what I have so far:
$\displaystyle\Sigma\frac{2^n}{3^n-1000}$
$\displaystyle\frac{2^n}{3^n-1000}\geq\frac{2^n}{3^n}$
So, now using the limit comparison test...
$\displaystyle\lim_{n\to\infty}\frac{\frac{2^n}{3^n-1000}}{\frac{2^n}{3^n}}$ = $\displaystyle\lim_{n\to\infty}\frac{3^n}{3^n-1000}$=1
The limit exits, is finite, and is greater than zero...so the original series will behave the same as $\Sigma\frac{2^n}{3^n}$ but I'm not sure how to prove whether this series converges/diverges? Any help would be appreciated.
2. Originally Posted by dbakeg00
I have a few problems concerning infinite series which I need some help on. I'm supposed to determine the convergence/divergence using ONLY the comparison tests. Here are the problems:
$\displaystyle\Sigma\frac{1}{e^n+1}$
$\displaystyle\Sigma\frac{2^n}{3^n-1000}$
$\displaystyle\Sigma\frac{1}{e^n^2}$
$$e^{n^2}$$ gives $e^{n^2}$
Note that $\dfrac{1}{e^n+1}<\dfrac{1}{2^n}$
3. Originally Posted by Plato
$$e^{n^2}$$ gives $e^{n^2}$
Note that $\dfrac{1}{e^n+1}<\dfrac{1}{2^n}$
Thanks..I wasn't sure how to write that properly in Latex
and $\Sigma\frac{1}{2^n}$ converges because it s a geometric series...I should have seen that!
Thank you
Now can someone help me prove that $e^x \geq x^2$ for all $x\geq 1$?
4. Originally Posted by dbakeg00
Now can someone help me prove that $e^x \geq x^2$ for all $x\geq 1$?
Let $f(x)=e^x-x^2$ show that is increasing for $x>1$.
5. Originally Posted by dbakeg00
Thanks..I wasn't sure how to write that properly in Latex
and $\Sigma\frac{1}{2^n}$ converges because it s a geometric series...I should have seen that!
Thank you
Now can someone help me prove that $e^x \geq x^2$ for all $x\geq 1$?
$\displaystyle \sum_{n \geq 0}{\frac{1}{2^n}} = \sum_{n \geq 0}{\left(\frac{1}{2}\right)^n}$ converges because it is a geometric series with $\displaystyle |r| < 1$.
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## Tokyo Journal of Mathematics
### A Construction of Everywhere Good $\mathbf{Q}$-Curves with $p$-Isogeny
Atsuki UMEGAKI
#### Abstract
An elliptic curve $E$ defined over $\bar{\mathbf{Q}}$ is called a $\mathbf{Q}$-curve, if $E$ and $E^\sigma$ are isogenous over $\bar{\mathbf{Q}}$ for any $\sigma$ in $\mathrm{Gal}(\bar{\mathbf{Q}}/\mathbf{Q})$. For a real quadratic field $K$ and a prime number $p$, we consider a $\mathbf{Q}$-curve $E$ with the following properties: 1) $E$ is defined over $K$, 2) $E$ has everywhere good reduction over $K$, 3) there exists a $p$-isogeny between $E$ and its conjugate $E^\sigma$. In this paper, a method to construct such a $\mathbf{Q}$-curve $E$ for some $p$ will be given.
#### Article information
Source
Tokyo J. Math., Volume 21, Number 1 (1998), 183-200.
Dates
First available in Project Euclid: 31 March 2010
https://projecteuclid.org/euclid.tjm/1270041995
Digital Object Identifier
doi:10.3836/tjm/1270041995
Mathematical Reviews number (MathSciNet)
MR1630171
Zentralblatt MATH identifier
0922.14021
#### Citation
UMEGAKI, Atsuki. A Construction of Everywhere Good $\mathbf{Q}$-Curves with $p$-Isogeny. Tokyo J. Math. 21 (1998), no. 1, 183--200. doi:10.3836/tjm/1270041995. https://projecteuclid.org/euclid.tjm/1270041995
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Top 10 Arxiv Papers Today in Instrumentation And Methods For Astrophysics
#1. Streaming Classification of Variable Stars
Lukas Zorich, Karim Pichara, Pavlos Protopapas
In the last years, automatic classification of variable stars has received substantial attention. Using machine learning techniques for this task has proven to be quite useful. Typically, machine learning classifiers used for this task require to have a fixed training set, and the training process is performed offline. Upcoming surveys such as the Large Synoptic Survey Telescope (LSST) will generate new observations daily, where an automatic classification system able to create alerts online will be mandatory. A system with those characteristics must be able to update itself incrementally. Unfortunately, after training, most machine learning classifiers do not support the inclusion of new observations in light curves, they need to re-train from scratch. Naively re-training from scratch is not an option in streaming settings, mainly because of the expensive pre-processing routines required to obtain a vector representation of light curves (features) each time we include new observations. In this work, we propose a streaming...
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arxivml: "Streaming Classification of Variable Stars", Lukas Zorich, Karim Pichara, Pavlos Protopapas https://t.co/3lVoHrLw2f
Memoirs: Streaming Classification of Variable Stars. https://t.co/ju83vASZom
Astrobotweet: Streaming Classification of Variable Stars. (arXiv:1912.02235v1 [https://t.co/yfe6jOnjIC]) https://t.co/gZWDJB3M9b
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#2. A Linear Formation Flying Astronomical Interferometer in Low Earth Orbit
Jonah T. Hansen, Michael J. Ireland
Space interferometry is the inevitable endpoint of high angular resolution astrophysics, and a key technology that can be leveraged to analyse exoplanet formation and atmospheres with exceptional detail. However, the anticipated cost of large missions such as Darwin and TPF-I, and inadequate technology readiness levels have resulted in limited developments since the late 2000s. Here, we present a feasibility study into a small scale formation flying interferometric array in Low Earth Orbit, that will aim to prove the technical concepts involved with space interferometry while still making unique astrophysical measurements. We will detail the proposed system architecture and metrology system, as well as present orbital simulations that show that the array should be stable enough to perform interferometry with < 50m/s/year delta-v and one thruster per spacecraft. We also conduct observability simulations to identify which parts of the sky are visible for a given orbital configuration. We conclude with optimism that this design is...
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qraal: [1912.02350] A Linear Formation Flying Astronomical Interferometer in Low Earth Orbit https://t.co/NhquuPFY34
StarshipBuilder: A Linear Formation Flying Astronomical Interferometer in Low Earth Orbit https://t.co/Ejl3Pa64vu
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#3. The Solar Probe Cup on Parker Solar Probe
Anthony W. Case, Justin C. Kasper, Michael L. Stevens, Kelly E. Korreck, Kristoff Paulson, Peter Daigneau, Dave Caldwell, Mark Freeman, Thayne Henry, Brianna Klingensmith, Miles Robinson, Peter Berg, Chris Tiu, Kenneth H. Wright Jr., David Curtis, Michael Ludlam, Davin Larson, Phyllis Whittlesey, Roberto Livi, Kristopher G. Klein, Mihailo M. Martinović
The Solar Probe Cup (SPC) is a Faraday Cup instrument onboard NASA's Parker Solar Probe (PSP) spacecraft designed to make rapid measurements of thermal coronal and solar wind plasma. The spacecraft is in a heliocentric orbit that takes it closer to the Sun than any previous spacecraft, allowing measurements to be made where the coronal and solar wind plasma is being heated and accelerated. The SPC instrument was designed to be pointed directly at the Sun at all times, allowing the solar wind (which is flowing primarily radially away from the Sun) to be measured throughout the orbit. The instrument is capable of measuring solar wind ions with an energy/charge between 100 V and 6000 V (protons with speeds from $139-1072~km~s^{-1})$. It also measures electrons with an energy between 100 V and 1500 V. SPC has been designed to have a wide dynamic range that is capable of measuring protons and alpha particles at the closest perihelion (9.86 solar radii from the center of the Sun) and out to 0.25 AU. Initial observations from the first...
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emulenews: The Solar Probe Cup on Parker Solar Probe: a Faraday Cup instrument onboard designed to make rapid measurements of thermal coronal and solar wind plasma. https://t.co/1vBJmTiL7I https://t.co/XyrDk47mua
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#4. Recent Progress on the GAPS Time of Flight System
S. Quinn
The General AntiParticle Spectrometer (GAPS) is a balloon-borne cosmic-ray antimatter experiment that uses the exotic atom technique, eliminating the requirement for strong B-fields used by conventional magnetic spectrometers. It will be sensitive to antideuterons with kinetic energies of 0.05-0.25 GeV / nucleon, which are highly motivated candidates for indirect dark matter detection. Moreover, GAPS will provide new information on the antiproton spectrum from $0.07<T<0.25$ GeV. The GAPS design is based on a lithium drifted silicon tracker and plastic scintillator time of flight (TOF) system. The latter is the focus of this contribution. Currently, the TOF system includes an outer "umbrella" consisting of 132 counters covering an area of 38 m$^2$ and a nearly hermetic inner "cube" with 64 counters and area of 15 m$^2$. The counters will be mechanically secured to the gondola using an innovative carbon fiber structure. Each end of the 196 counters will be read out using a silicon photomultiplier (SiPM) based analog front end with...
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higgsinocat: Recent Progress on the GAPS Time of Flight System. (arXiv:1912.01675v1 [https://t.co/JYpMgBSZq3]) relevance:63% https://t.co/eJex3gJHkY #darkmatter https://t.co/rPKD5ixTma
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#5. A network of precision gravimeters as a detector of matter with feeble nongravitational coupling
Wenxiang Hu, Matthew Lawson, Dmitry Budker, Nataniel L. Figueroa, Derek F. Jackson Kimball, Allen P. Mills Jr., Christian Voigt
Hidden matter that interacts only gravitationally would oscillate at characteristic frequencies when trapped inside of Earth. For small oscillations near the center of the Earth, these frequencies are around 300 $\mu$Hz. Additionally, signatures at higher harmonics would appear because of the non-uniformity of Earth's density. In this work, we use data from a global network of gravimeters of the International Geodynamics and Earth Tide Service (IGETS) to look for these hypothetical trapped objects. We find no evidence for such objects with masses of up to on the order of 10$^{13}$ kg. It may be possible to improve the sensitivity of the search by several orders of magnitude via better understanding of the terrestrial noise sources and more advanced data analysis.
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qraal: [1912.01900] A network of precision gravimeters as a detector of matter with feeble nongravitational coupling https://t.co/u00zvF6pcR
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#6. Global Site Selection for Astronomy
N. Aksaker, S. K. Yerli, M. A. Erdoğan, Z. Kurt, K. Kaba, M. Bayazit, C. Yesilyaprak
A global site selection for astronomy was performed with 1 km spatial resolution ($\sim$ 1 Giga pixel in size) using long term and up-to-date datasets to classify the entire terrestrial surface on the Earth. Satellite instruments are used to get the following datasets of Geographical Information System (GIS) layers: Cloud Coverage, Digital Elevation Model, Artificial Light, Precipitable Water Vapor, Aerosol Optical Depth, Wind Speed and Land Use -- Land Cover. A Multi Criteria Decision Analysis (MCDA) technique is applied to these datasets creating four different series where each layer will have a specific weight. We introduce for the first time a "Suitability Index for Astronomical Sites" namely, SIAS. This index can be used to find suitable locations and to compare different sites or observatories. Mid-western Andes in South America and Tibetan Plateau in west China were found to be the best in all SIAS Series. Considering all the series, less than 3 \% of all terrestrial surfaces are found to be the best regions to establish...
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#7. Laboratory Experiments on the Motion of Dense Dust Clouds
Niclas Schneider, Gerhard Wurm
In laboratory experiments, we study the motion of levitated, sedimenting clouds of sub-mm grains at low ambient pressure and at high solid-to-gas ratios $\epsilon$. The experiments show a collective behavior of particles, i.e. grains in clouds settle faster than an isolated grain. In collective particle clouds, the sedimentation velocity linearly depends on $\epsilon$ and linearly depends on the particle closeness $C$. However, collective behavior only sets in at a critical value $\epsilon_{\rm crit}$ which linearly increases with the experiment Stokes number St. For $\rm St <0.003$ particles always behave collectively. For large Stokes numbers, large solid-to-gas ratios are needed to trigger collective behavior, e.g. $\epsilon_{\rm crit} = 0.04$ at $\rm St = 0.01$. Applied to protoplanetary disks, particles in dense environments will settle faster. In balance with upward gas motions (turbulent diffusion, convection) the thickness of the midplane particle layer will be smaller than calculated based on individual grains,...
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#8. Local Turbulence: Effects and causes
Olivier Lai, Kanoa Withington, Romain Laugier, Mark Chun
Dome seeing is a known source of image quality degradation, but despite tremendous progress in wavefront control with the development of adaptive optics and environmental control through implementation of dome venting, surprisingly little is known about it quantitatively. We have found evidence of non-Kolmogorov dome turbulence from our observations with the imaka wide field adaptive optics system; PSFs seem to indicate an excess of high spatial frequencies and turbulence profiles reveal turbulence at negative conjugations. This has motivated the development of a new type of optical turbulence sensor called AIR-FLOW, Airborne Interferometric Recombiner: Fluctuations of Light at Optical Wavelengths. It is a non-redundant mask imaging interferometer that samples the optical turbulence passing through a measurement cell and it measures the two-dimensional optical Phase Structure Function. This is a useful tool to characterise different types of turbulence (e.g. Kolmogorov, diffusive turbulence, etc.). By fitting different models, we...
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#9. Multi-Gaussian fitting Algorithm to determine multi-band photometry and photometric redshifts of LABOCA and Herschel sources in proto-cluster environments
Youngik Lee
This research focuses on identifying high redshift galaxies from LABOCA(LArge APEX BOlometer CAmera) and SPIRE(The Spectral and Photometric Imaging Receiver) maps towards proto-cluster candidates initially selected from the SPT (South pole telescope) survey. Based on the Multi-Gaussian fitting algorithm, we cross-match all significant LABOCA sources at SPIRE wavelengths based on their coordinates and signal to noise ratio to derive their photometry at 250, 350, 500 and 870 $\mu m$. We use this information to calculate a photometric redshift for SPT sources towards cluster fields. The code was developed in the Python programming environment.
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#10. 3DMAP-VR, a project to visualize 3-dimensional models of astrophysical phenomena in virtual reality
Salvatore Orlando, Ignazio Pillitteri, Fabrizio Bocchino, Laura Daricello, Laura Leonardi
In this research note, we present 3DMAP-VR,(3-Dimensional Modeling of Astrophysical Phenomena in Virtual Reality), a project aimed at visualizing 3D MHD models of astrophysical simulations, using virtual reality sets of equipment. The models account for all the relevant physical processes in astrophysical phenomena: gravity, magnetic-field-oriented thermal conduction, energy losses due to radiation, gas viscosity, deviations from proton-electron temperature equilibration, deviations from the ionization equilibrium, cosmic rays acceleration, etc.. We realized an excellent synergy between our 3DMAP-VR project and Sketchfab (one of the largest open access platforms to publish and share 3D virtual reality and augmented reality content) to promote a wide dissemination of results for both scientific and public outreach purposes.
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Assert is a website where the best academic papers on arXiv (computer science, math, physics), bioRxiv (biology), BITSS (reproducibility), EarthArXiv (earth science), engrXiv (engineering), LawArXiv (law), PsyArXiv (psychology), SocArXiv (social science), and SportRxiv (sport research) bubble to the top each day.
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Free Version
Difficult
# Cross Sectional Area of Resistor vs. Current
APPHEM-B0ETEK
The cross sections of five resistors which are made from the same metal and have the same lengths are shown below. Each of the resistors is connected to the same battery and an ammeter (see the circuit diagram below).
• Resistors (a), (b) and (c) are solid and have radii of $R/2$, $R$ and $2R$ respectively.
• Resistors (d) and (e) have a cylindrical hole in the center.
• The outer radii of the resistors (d) and (e) are $R$ and $2R$.
• The diameters of holes of these resistors are $R/2$ for the resistor (d) and $R$ for the resistor (e).
Rank the situations (a) through (e) from the lowest ammeter reading to the highest ammeter reading.
A
e, d, b, c, a
B
a, c, b, d, e
C
c, e, b, a, d
D
d, a, b, e, c
E
c, e, b, d, a
F
a, d, b, e, c
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IIT JAM Follow
February 3, 2021 12:11 pm 30 pts
Example: Evaluate Fh dS, where F zi +xj +3y z k, where S is the surface of the cylinder x? + y = 16 include in the first octant between Z = 0 and Z = 5. altian Tha nraiection of the aiven surface on xz plane will be a rectangle whose sides ill be
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• Rakesh kumar
divergence thm se kr skte h kya
Deepak singh
No , include in first octant doesn't imply that consider X=0 and y=0 plane. here we have only cylinder surface so Gauss thm doesn't apply
• Deepak singh
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# Probing multiphoton light-induced molecular potentials
## Abstract
The strong coupling between intense laser fields and valence electrons in molecules causes distortions of the potential energy hypersurfaces which determine the motion of the nuclei and influence possible reaction pathways. The coupling strength varies with the angle between the light electric field and valence orbital, and thereby adds another dimension to the effective molecular potential energy surface, leading to the emergence of light-induced conical intersections. Here, we demonstrate that multiphoton couplings can give rise to complex light-induced potential energy surfaces that govern molecular behavior. In the laser-induced dissociation of H2+, the simplest of molecules, we measure a strongly modulated angular distribution of protons which has escaped prior observation. Using two-color Floquet theory, we show that the modulations result from ultrafast dynamics on light-induced molecular potentials. These potentials are shaped by the amplitude, duration and phase of the dressing fields, allowing for manipulating the dissociation dynamics of small molecules.
## Introduction
Potential energy surfaces describe the forces acting on the nuclei of a molecule. Within the Born–Oppenheimer approximation the motion of the nuclei along these potentials is treated independently of the electronic motion. This picture breaks down when the electronic level separation becomes comparable to the kinetic energy of the nuclei. This occurs at specific points in the molecular geometry, which are known as conical intersections and that are a hallmark of polyatomic molecules1. Conical intersections play an eminent role in visible and ultraviolet photochemistry2,3, for example, in isomerization4,5, and electron transfer processes6. Moreover, they are strongly implicated in the photostability of DNA by way of allowing radiation-less de-excitation7.
The single-photon transition between two dipole-coupled electronic states can also create a conical, albeit transient, intersection. Hence, these localized features of the laser-dressed potential energy surface have been dubbed light-induced conical intersections (LICI)8,9. Their precise position, and the underlying dipole coupling strength are determined by the frequency and intensity of the incident light. LICIs can also be found in diatomic molecules, since the angle between the light polarization and the molecular axis adds another degree of freedom to the nuclear motion10,11. The angle-dependent distortion of the molecular potential energy surfaces in a linearly polarized laser field directly affects molecular dissociation12,13,14,15,16 and has been predicted to cause rotational excitation17,18,19,20,21,22. Recently, experimental indications of LICI in H2+ have been found in angle-resolved ion spectra23.
In ultrashort infrared laser fields, the light intensity can easily exceed the threshold for multiphoton transitions. While LICIs are a consequence of single-photon couplings and therefore the potential energy scales linearly with respect to variations of the laser field strength, multiphoton couplings lead to unique structures of their own. In the case of diatomic molecules, these structures become nonlinear point intersections of the potential energy surfaces. The one-dimensional (1D) treatment of single and multiphoton resonances has led to the prediction of light-induced potentials (LIPs)23,24,25,26,27,28,29, and anomalous fragment angular distributions have been predicted in the non-perturbative regime30. Experimentally, however, the consequences of the angle-dependent coupling strength around nonlinear point intersections for the dissociation dynamics have so far been largely unexplored.
Here, we show in theory and experiment that LIPs featuring nonlinear light-induced point intersections can result in strong modulations of the angular ion yield. Using a two-color pump–probe scheme allows us to probe and control the nuclear dynamics, as the underlying LIPs evolve in the laser field. The experimental results are interpreted with the help of numerical solutions of the time-dependent Schrödinger equation and two-color Floquet theory. We find that, owing to both linear and nonlinear transitions, as well as rotational dynamics, the two-color laser field gives rise to remarkably complex dissociation dynamics that produce the strong modulations in the angular ion yield.
For our studies, we choose the simplest molecule, H2+, which is widely regarded as a prototype system for the interaction of molecules with light31. Due to the sparsity of electronic states, H2+ can often be described as a two-level system consisting of the two lowest electronic states, 1 g and 2 u. When coupled to intense laser fields, these states give rise to intensity-dependent dissociation mechanisms known as bond softening32 and above threshold dissociation33. The possibility to control them using the laser amplitude, frequency, phase, and pulse duration has been demonstrated34,35,36,37,38. In particular, the opposite parity of the σg and σu states can lead to electron localization39, giving rise to charge resonance-enhanced ionization40,41 and symmetry breaking in dissociation42,43,44 under the influence of two-color laser fields45,46 or carrier-envelope phase (CEP) stable few-cycle pulses47,48,49,50. Notably, the dissociation dynamics are sensitive to the rather complex shape of orthogonally polarized two-color fields51. Descriptive treatments for most of these phenomena are provided by dressed-state pictures, such as LIPs.
## Results
### Nonlinear point intersections
Figure 1a shows an example of some LIP energy surfaces for H2+ in a moderately intense (3 × 1013 W/cm2), visible (685 nm) laser field, calculated using the procedure outlined in the “Methods” section. Shown are the laser-dressed states σg and σu 1ħωVIS i.e., σu, shifted down due to the absorption of one photon from the visible field. Along the laser polarization, i.e., θ = 0, π, the state crossing indicated in Fig. 1b opens up and turns into an avoided crossing. This necessarily lowers the potential barrier at the avoided crossing and permits dissociation of formerly bound molecules (bond softening). Importantly, no such avoided crossing occurs when the laser polarization is perpendicular to the internuclear axis. Therefore, a LICI is formed at the internuclear distance where the laser-dressed σg and σu 1 ωVIS states cross.
While single-photon transitions dominate in moderately intense visible laser fields, multiphoton transitions become relevant when the wavelength is shifted into the mid-infrared52. For example, the three-photon transition by 2300 nm light becomes significant already at an intensity as low as 5 × 1012 W/cm2, see Supplementary Fig. 1 for details. Several crossings of potential curves that correspond to multiphoton transitions between the σg and σu states of H2+ at a wavelength of 2300 nm are shown in Fig. 1c. The corresponding LIP energy landscape calculated for an intensity of 3 × 1013 W/cm2 is presented in Fig. 1d. The potential energy surfaces exhibit complex structures that result from multiphoton couplings. Indicated in the figure are the curve crossings due to n-photon (n = 1, 3, and 5) transitions. Notably, the intersections for n = 3 and 5 are not conical (see Supplementary Fig. 2), as would be the case for all higher intersections. In order to see how these light-induced structures affect the nuclear dynamics, we first solve the two-dimensional (2D) time-dependent Schrödinger equation (2D TDSE, see “Methods” section for details) for H2+ under the influence of a moderately intense mid-IR laser pulse. The calculated proton momentum distributions presented in Fig. 1e show distinct features in the proton angular distribution that can be associated with the n-photon couplings. While these features are absent in the single-photon coupling regime at low intensity, significantly higher intensities produce very convoluted dissociation patterns that involve high-order couplings, but would likely defy experimental resolution, see Supplementary Fig. 3.
### Structured proton angular distribution
In order to experimentally probe the light-induced molecular potentials depicted in Fig. 1d, e, describing the situation where the mid-IR field induces multiphoton dynamics in the dissociation process, but does not cause ionization, we implement the two-pulse scheme depicted in Fig. 2a. First, an intense, few-cycle visible laser pulse ionizes neutral H2, producing a bound coherent wave packet in H2+ with a nearly isotropic alignment distribution, with respect to the laser polarization53. Second, a moderately intense mid-IR pulse creates the LIPs on which dissociation occurs. The LIPs are probed by recording the momentum distribution of protons resulting from the dissociating part of the molecular wave packet. The molecular ions dissociate along their initial alignment direction, unless rotational dynamics occur, as predicted, e.g., in refs. 17,18,21,22.
The intent of the two-pulse scheme is to decouple the production of the molecular wave packet from the field that generates the LIPs. This allows for probing the LIPs at selected times within the mid-IR pulse by scanning the time delay between the laser pulses. Moreover, the use of a shorter wavelength pulse for ionization allows us to reduce the focal volume averaging in the long-wavelength field, which often washes out subtle features in strong-field experiments (e.g., ref. 54). Finally, choosing a perpendicular relative polarization of the visible and mid-IR pulses is expected to avoid overlap between the signal of interest produced by the mid-IR pulse, and any protons produced by the visible pulse alone. The experimental setup is described in the “Methods” section “Experiment”.
Figure 2b shows experimental results obtained with the cross-polarized few-cycle visible and mid-IR pulses. The three-dimensional (3D) momentum distributions of protons and electrons were measured in coincidence, using Cold Target Recoil Ion Momentum Spectroscopy (COLTRIMS). The coincidence measurement allows us to present results in the recoil frame, where the impact of the electron recoil has been largely removed from the measured ion momentum distribution. The results exhibit a striking angular structure that is blurred if the recoil momentum is not accounted for (see Supplementary Fig. 6). The angular structure consists of on-axis features along either of the polarization axes, and additional spots at intermediate angles. Drawing a comparison to results obtained with only mid-IR (Fig. 2c) or visible (Fig. 2d) pulses, suggests that the on-axis features arise from bond softening by either pulse alone. Note, that the signal along the mid-IR polarization in the two-color experiment does not arise from dissociative ionization of neutral H2 by the mid-IR pulse on its own, as no notable ionization of neutral H2 is obtained at the intensity of 3 × 1013 W/cm2. Hence, the comparative mid-IR only data (Fig. 2c) is presented for a higher intensity of 1 × 1014 W/cm2. The additional spots in the two-color data are tentatively attributed to dynamics caused by the light-induced structures in the molecular potential energy landscape; cf. Fig. 1d.
Surprisingly, the experimental results presented in Fig. 2b, exhibit a much more pronounced angular structure than the TDSE results for the mid-IR field alone, presented in Fig. 1e. Moreover, the angular structure survives averaging over kinetic energy, in contrast to the weaker modulations in previous work on LICI (ref. 23). A hint on the origin of the additional angular structure in the present experiment comes from measurements carried out with a parallel polarization of the visible and mid-IR pulses. The proton momentum distribution obtained with parallel polarization is shown in Fig. 2e. It resembles the results obtained with mid-IR pulses only and does not exhibit significant angular structure. Although the intention of our scheme was to decouple the effects of the mid-IR and visible pulses, the striking dependence of the angular structure on the relative polarization of the two laser fields implies that the visible field contributes to the formation of the additional spots. It will thus be considered in the following analysis of our results.
### Numerical results
In a first step to understand the dynamics producing the structured proton angular distribution observed in cross-polarized fields, we solve the 2D TDSE for H2+, taking both laser pulses into account. Due to the observed importance of the visible field in shaping the experimental results, we also consider a weak pulse pedestal at 5% of the peak intensity and 45 fs duration (full-width at half-maximum of the intensity envelope) for the visible few-cycle pulse. These values are consistent with field-resolved measurements of few-cycle pulses55. The initial alignment of the molecular axis with respect to the laser polarization is assumed to be istotropic.
It has been recognized in the literature that angular modulations in the proton spectra can arise from rotational dynamics in the vicinity of the LICI (refs. 18,21,22,23); more specifically, simulations that include rotational motion in the dissociation dynamics show angular modulations that are absent when the rotational degrees of freedom are frozen. These modulations can be connected to rotational scattering of the dissociating wave packet from the LICI (refs. 18,21,22,23). A first candidate for the physical mechanism underlying the appearance of a structured proton angular distribution is therefore the formation of a high-order rotational wave packet in the dissociating molecular cation. In order to test the role of rotational dynamics, we perform a first set of calculations where rotational transitions are artificially switched off and present the results in Fig. 3a. Evidently, pronounced modulations in the angular distribution are obtained, even without the inclusion of wave packet rotation. This suggests that rotational dynamics are not the primary physical mechanism underlying the angular modulation in the proton momentum distribution. Therefore, it will be important to identify how angular modulations arise already within a 1D treatment.
The second set of calculations (Fig. 3b) takes rotational dynamics into account. The differences between Fig. 3a, b show the impact of rotations in certain parts of the momentum distributions. The black arrows highlight pronounced differences between the results of the two calculations at angles θ = 0° and θ = 20°. These differences illustrate the role of rotational alignment in shaping the final momentum distribution. On the basis of the comparison between Fig. 3a, b, we conclude that rotational dynamics play a significant but secondary role in defining the final momentum distribution; in contrast with the previously considered pure LICI case, the addition of rotations is not the sole cause of the angular structures in our experiment. A direct comparison of the calculated and measured angular distributions is given by Fig. 3d. Indeed, the strong modulations observed in the experimental data are only obtained in the simulations that take rotations into account. However, the modulation depth in the experimental data is smaller than in the simulations with rotations, which is ascribed to the reduced dimensionality of the simulations.
In order to identify the essential mechanism creating the angular structure in the absence of rotations, we employ two-color Floquet theory (see “Methods” section). We calculate the angle-dependent field-dressed states of H2+ using a two-color laser field, $${\mathbf{F}}\left( t \right) = \sqrt {I_{{\mathrm{VIS}}}} \cos (\omega _{{\mathrm{VIS}}}t + \varphi _{{\mathrm{VIS}}}){\hat{\mathbf{x}}} + \sqrt {I_{{\mathrm{IR}}}} \cos \left( {\omega _{{\mathrm{IR}}}t + \varphi _{{\mathrm{IR}}}} \right){\hat{\boldsymbol{z}}}$$ ($${\hat{\mathbf{x}}}$$ and $${\hat{\mathbf{z}}}$$ being the unit vectors along x- and z-directions, respectively).
The field consists of a moderately intense mid-infrared field (λIR = 2280 nm, IIR = 3 × 1013 W/cm2) and a weak visible field (IVIS = 1 × 1013 W/cm2), corresponding to the pulse pedestal used in the TDSE calculations. We take λVIS = λIR/3 to ensure the periodicity of the laser fields required by Floquet theory. The resulting LIP energy landscape depends on the relative optical phase, Δφ = φVIS − φIR. Exemplarily, we present the LIP energy landscape for Δφ = 0 in Fig. 3d. Both experimental and numerical results, presented in Figs. 2 and 3, respectively, are integrated over Δφ.
A detailed analysis and discussion of the results from Floquet theory is presented in Supplementary Note 2. In brief, we find that the Floquet states represent a conclusive basis for understanding the emergence of angular structure in the proton momentum distribution, even without rotational dynamics taken into account. In the absence of rotations, the angular structure arises through a process we shall call angle-dependent channel switching, as different orders of multiphoton couplings dominate at different alignment angles of the molecular axis with respect to the laser polarization. The following picture can be invoked.
As the alignment angle of the molecular axis in the polarization plane (see Fig. 3), θ, is increased, the field components parallel to the molecular axis vary as FVIS cos(θ), and FIR sin(θ). This leads to a pronounced angle dependence of the field-dressed potential energy curves (see Fig. 3c), where several Floquet state crossings open up and close again as θ is varied. Specifically, at θ = 0°, i.e., for alignment of the molecular axis perpendicular to the mid-IR polarization, the effect of the mid-IR field is insignificant, and dissociation proceeds as in the single-color case (seen in Fig. 2b). Specifically, the wave packet dissociates on the purple surface in Fig. 3d. As θ is increased, a new dissociation channel due to one-photon coupling by the mid-IR field opens up. The new channel competes with the original one, which moves population to the pronounced feature at θ = 10°, making the on-axis feature much narrower than in the single-color case. This new dissociation channel corresponds to dissociation on the red surface in Fig. 3d. As θ is further increased, the width of the avoided crossing reaches 2ωIR, which closes the dissociation channel and gives rise to a LICI at θ ≈ 30°, clearly visible in Fig. 3d.
Notably, the computational results obtained without (Fig. 3a) and with (Fig. 3b) rotations strongly differ at θ ≈ 20°. We attribute this to the presence of the aforementioned LICI that promotes strong rotational dynamics, as the nuclear wave packet propagates around the cone in the LIP landscape. In a similar manner, the splitting of the narrow feature at 0° in Fig. 3a into the double peak structure in Fig. 3b is attributed to the point intersections at θ = 0°.
### Delay dependence
Scanning the time delay between the visible and mid-IR pulses in our experiment allows us to probe the variations in the LIPs throughout the mid-IR pulse. The time delay controls the time of ionization with respect to the mid-IR pulse, and thereby determines the (i) strength, (ii) duration, and (iii) phase of the mid-IR field at the time it interacts with the molecular ion. In Fig. 4, we analyze the fragment momentum distribution for overlap of the ionizing visible pulse with the rising edge, the maximum and the falling edge of the mid-IR laser pulse. Each of the presented spectra are integrated over two mid-IR cycles, and therefore not expected to be sensitive to the mid-IR phase.
Figure 4a shows the vector potential of the mid-IR pulse used in our experiment, as measured with the STIER technique56 (see Supplementary Fig. 5). Selected recoil-frame proton momentum distributions are presented in Fig. 4b–d. The delay-dependent results probe the evolution of the LIP energy landscape throughout the mid-IR pulse. This is evidenced by the changes in the recorded dissociation patterns, as the delay between visible and mid-IR pulses is varied. For example, the feature at θ = 90°, i.e., along the mid-IR polarization axis, peaks around the center of the pulse, Fig. 4c, where it represents the strongest contribution to the proton momentum distribution. When the ionization occurs on the falling edge of the mid-IR pulse (Fig. 4d), the 90° feature is absent. On the basis of the computational results presented in Fig. 1 and the two-color Floquet states shown in Fig. 3d, we attribute this peak to a five-photon coupling induced by the mid-IR pulse. The nonlinearity of this process explains why this feature is particularly visible near the maximum of the mid-IR pulse and decays rapidly on the falling edge of the pulse. Notably, the maximum yield of protons emitted at 90° is obtained, when the visible pulse precedes the peak of the mid-IR pulse by (8.3 ± 0.5) fs, in reasonable agreement with the 7.3 fs vibrational half-period of H2+ (ref. 57). For earlier delays, when ionization occurs on the rising edge of the mid-IR pulse (Fig. 4b), the weaker signal at 90° indicates that dissociation occurs before the molecular ion interacts with the center of the mid-IR pulse. Similar observations are made for the feature at intermediate angles (around θ ≈ 40° in Fig. 3a). In addition, its angular position also varies from θ ≈ 30° in Fig. 4b toward θ ≈ 40° in Fig. 4c.
Contrary to the nonlinear features, the feature at θ = 10°, i.e., close to the visible polarization axis, exhibits little delay dependence. As discussed above, this feature can be understood as a consequence of the single-photon couplings by both, the visible and the mid-IR fields. The absence of nonlinearity in this process explains the insensitivity of the 10° feature to the mid-IR intensity. Calculated proton momentum distributions for different delay values, which are consistent with these conclusions, are presented in Supplementary Fig. 7.
## Discussion
In summary, we have demonstrated a powerful approach for probing light-induced molecular potentials. We observed strongly modulated proton angular distributions in experiments were H2+ ions, produced by a linearly polarized, few-cycle, visible laser pulse, are dissociated by a cross-polarized mid-IR laser field. We have shown that the modulations can be understood as signatures of complex LIP energy landscapes that are shaped by both single-photon and multiphoton transitions in a cross-polarized two-color laser field. Specifically, the modulations arise from a combination of two effects: First, angle-dependent channel switching, i.e., different dissociation pathways open and close as a function of alignment angle; second, rotational motion around light-induced point intersections, such as LICIs, shape the modulated angular ion yield. The LIP picture predicts where angle-dependent channel switching takes place, and where prominent light-induced point intersections are present.
Probing the LIPs resulting from the mid-IR dressing field on its own may be improved by using a shorter pulse for preparation of the bound wave packet, such as a few-cycle UV or attosecond pulse. Previous experiments along these lines (e.g., refs. 44,58) were conducted in the single-photon dressing regime and did not study the influence of the LIP surfaces on the angular dependence of dissociation.
Our approach allows us to follow the variation of the LIPs throughout the dressing laser field. On the timescale of the mid-IR pulse envelope, we observe the opening and closing of dissociation pathways as the dressing field strength changes. On shorter time scales, the propagation of the dissociating wave packet will become accessible with sub-femtosecond time resolution by monitoring the electron localization on either fragment. More generally, we have shown how complex LIP energy landscapes determine the outcome of molecular dissociation, using H2 as an example. Our approach will allow for elucidating the reaction dynamics of more complex molecules in the presence of LICIs and higher-order point intersections.
## Methods
### Experiment
The employed experimental technique is a variant of ref. 56. The output of a commercial Ti:Sa chirped pulse amplification (CPA) laser (Coherent Elite, 10 kHz, 2 mJ) is split into two parts. The stronger part (85%) is used to pump an optical parametric amplifier, in order to obtain CEP stable idler pulses at 2.3 µm. The second part of the CPA output is focused into an argon-filled hollow core fiber to obtain broadband laser pulses, which are subsequently compressed to a pulse duration of ≈5 fs. The laser pulses are recombined using a polished Si mirror (thickness 2.2 mm) at 60° angle of incidence.
After recombination, the pulses are focused in the center of a COLTRIMS59, where they intersect an ultrasonic jet of pre-cooled (T = 60 K) of neutral H2. The intensity of the mid-IR pulse is weak enough to not cause any notable ionization by itself. Because ions are only produced in the small focal volume of the visible pulse (1/e² width (7 ± 2) µm), focal volume averaging within the larger focal volume ((30 ± 10) µm) of the mid-IR pulse is essentially avoided. In the COLTRIMS, the 3D momenta of ions and electrons generated in the laser focus are measured in coincidence, which provides access to the recoil-frame ion momentum that arises solely from the nuclear dynamics on the LIPs. See Supplementary Fig. 6 for a comparison of laboratory-frame and recoil-frame measurements. The measurement of the delay dependence of the electron momentum distribution yields the instantaneous mid-IR vector potential at each delay value, as shown in Supplementary Fig. 5.
### Time-dependent Schrödinger equation
For the dynamics in the H2+ cation, we solve a 2D (one angle and one bond length) Schrödinger equation that includes dipole coupling between the two relevant electronic states 2Σg+ (also referred to as σg) and 2Σu+ (σu)
$${\boldsymbol{i}}\frac{\partial }{{\partial {\boldsymbol{t}}}}\left[ {\begin{array}{*{20}{c}} {{\boldsymbol{\Psi }}_{\boldsymbol{g}}\left( {\mathbf{R}} \right)} \\ {{\boldsymbol{\Psi }}_{\boldsymbol{u}}\left( {\mathbf{R}} \right)} \end{array}} \right] = - \frac{1}{{2\mu }}\left( {\frac{{\partial ^2}}{{\partial {\boldsymbol{R}}^2}} + \frac{1}{{{\boldsymbol{R}}^2}}\frac{{\partial ^2}}{{\partial \theta ^2}}} \right)\left[ {\begin{array}{*{20}{c}} {{\boldsymbol{\Psi }}_{\boldsymbol{g}}\left( {\mathbf{R}} \right)} \\ {{\boldsymbol{\Psi }}_{\boldsymbol{u}}\left( {\mathbf{R}} \right)} \end{array}} \right] \\ + \left[ {\begin{array}{*{20}{c}} {{\boldsymbol{V}}_{\boldsymbol{g}}\left( {\boldsymbol{R}} \right)} & { - {\mathbf{F}}\left( {\boldsymbol{t}} \right) \cdot {\mathbf{d}}\left( {\boldsymbol{R}} \right)} \\ { - {\mathbf{F}}\left( {\boldsymbol{t}} \right) \cdot {\mathbf{d}}\left( {\boldsymbol{R}} \right)} & {{\boldsymbol{V}}_{\boldsymbol{u}}\left( {\boldsymbol{R}} \right)} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{\boldsymbol{\Psi }}_{\boldsymbol{g}}\left( {\mathbf{R}} \right)} \\ {{\boldsymbol{\Psi }}_{\boldsymbol{u}}\left( {\mathbf{R}} \right)} \end{array}} \right],$$
(1)
where R = (R,θ) are the bond length, and angle between the laser field and the molecular axis. F(t) is the electric field of the laser that couples the two electronic states. The form of the electronic potential energy curves Vg and Vu, as well as the transition dipole d, are taken from Bunkin and Tugov60. Equation (1) was solved numerically using the Fourier split-operator method.
In our experiment, the H2+ system is created starting from the H2 neutral through strong-field ionization. The initial state of the wave function of the ionic simulations assumes a vertical transition from the ground electronic (1Σg), and ground vibrational state of the H2 neutral to the ground electronic state of the ion. The ground vibrational state on the 1Σg of the neutral is modeled as Morse oscillator state, using Morse parameters derived from Herzberg61. The rotational degree of freedom was initialized to a thermal rotational distribution, with temperature chosen to be low enough such that only the rotational ground state is populated. The initial distribution of the molecular axis with respect to the laser polarization is isotropic, closely reflecting the experimental conditions.
The laser field used in the calculations presented in Fig. 3a, b can be expressed as
$${\mathbf{F}}\left( {\boldsymbol{t}} \right) = {\boldsymbol{F}}_{{\mathbf{IR}}}\left( {{\boldsymbol{t}} + {\mathrm{\Delta }}{\boldsymbol{t}}} \right){\hat{\mathbf{z}}} + ({\boldsymbol{F}}_{{\mathbf{VIS}}}\left( {\boldsymbol{t}} \right) + {\boldsymbol{F}}_{{\mathbf{ped}}}\left( {\boldsymbol{t}} \right)){\hat{\mathbf{x}}},$$
(2)
where Δt is the time delay between visible and mid-IR pulses and each field, FA(t), is given by an expression (using atomic units),
$${\boldsymbol{F}}_{\boldsymbol{A}}\left( {\boldsymbol{t}} \right) = \sqrt {{\boldsymbol{I}}_{\boldsymbol{A}}} \exp \left( { - 2\ln 2\left( {\frac{{\boldsymbol{t}}}{{\tau _{\boldsymbol{A}}}}} \right)^2} \right)\cos \left( {\omega _{\boldsymbol{A}}{\boldsymbol{t}} + \varphi _{\boldsymbol{A}}} \right),$$
(3)
where $$\varphi _A$$ is the CEP of each pulse.
The laser field consists of a mid-IR pulse (λIR = 2300 nm, τIR = 45 fs, ΙIR = 30 TW/cm2), an ionizing few-cycle visible pulse (λVIS = 730 nm, τVIS = 5 fs, ΙVIS = 300 TW/cm2), and a visible pulse pedestal (λped = 730 nm, τped = 45 fs, Ιped = 10 TW/cm²)). The calculations are started at t = 0, i.e., in the center of the visible pulse and performed for various values of $${\mathrm{\Delta }}t$$ and, for each $${\mathrm{\Delta }}t,\varphi _{{\mathrm{VIS}}} = \varphi _{{\mathrm{ped}}} = n\pi$$ (n = 0, 1) and $$\varphi _{{\mathrm{IR}}} \equiv 0$$.
### Floquet states
For each molecular alignment angle θ, the Floquet states62,63,64 are calculated for a field
$${\boldsymbol{F}}\left( {{\boldsymbol{t}},\theta } \right) = \sqrt {{\boldsymbol{I}}_{{\mathbf{IR}}}} \cos \left( {\omega {\boldsymbol{t}}} \right)\sin \theta + \sqrt {{\boldsymbol{I}}_{{\mathbf{ped}}}} {\mathbf{cos}}\left( {3\omega {\boldsymbol{t}} + \phi } \right)\cos \theta ,$$
(4)
where $$\phi$$ is the relative phase of the two fields. The potential energy landscape presented in Fig. 3b is for the relative phase $$\phi = 0$$. Here, the frequency of the visible pulse is approximated as ωVIS = 3 ωIR to obtain the required periodicity.
At each point along R, the Floquet states were constructed as follows. First, the one-period propagator U(t,t + T; R), where T = 2π/ω is the period of the 2280 nm laser field, was constructed numerically using
$${\boldsymbol{U}}\left( {{\boldsymbol{t}},{\boldsymbol{t}} + {\boldsymbol{T}};{\boldsymbol{R}}} \right) = {\boldsymbol{e}}^{ - {\boldsymbol{iH}}_{\boldsymbol{e}}({\boldsymbol{R}},{\boldsymbol{t}}_{{\boldsymbol{N}} - 1}){\mathrm{\Delta }}_{\boldsymbol{t}}}{\boldsymbol{e}}^{ - {\boldsymbol{iH}}_{\boldsymbol{e}}({\boldsymbol{R}},{\boldsymbol{t}}_{{\boldsymbol{N}} - 2}){\mathrm{\Delta }}_{\boldsymbol{t}}} \ldots {\boldsymbol{e}}^{ - {\boldsymbol{iH}}_{\boldsymbol{e}}({\boldsymbol{R}},{\boldsymbol{t}}_1){\mathrm{\Delta }}_{\boldsymbol{t}}}{\boldsymbol{e}}^{ - {\boldsymbol{iH}}_{\boldsymbol{e}}({\boldsymbol{R}},{\boldsymbol{t}}_0){\mathrm{\Delta }}_{\boldsymbol{t}}},$$
(5)
where the time interval $$\tau$$ has been split into N = 1024 time steps of duration $$\Delta _t = T/N$$ with the intermediate times given by $$t_n = t + n{\mathrm{\Delta }}_t$$, and the purely electronic Hamiltonian $$H_e(t)$$ is given by
$${\boldsymbol{H}}_{\boldsymbol{e}} = \left[ {\begin{array}{*{20}{c}} {{\boldsymbol{V}}_{\boldsymbol{g}}({\boldsymbol{R}})} & { - {\mathbf{F}}({\boldsymbol{t}}) \cdot {\mathbf{d}}({\boldsymbol{R}})} \\ { - {\mathbf{F}}({\boldsymbol{t}}) \cdot {\mathbf{d}}({\boldsymbol{R}})} & {{\boldsymbol{V}}_{\boldsymbol{u}}({\boldsymbol{R}})} \end{array}} \right].$$
(6)
The Floquet states $$|F_\alpha \left( {R,t} \right)$$ are the eigenstates of $$U\left( {t,t + T;R} \right)$$,
$${\boldsymbol{U}}\left( {{\boldsymbol{t}},{\boldsymbol{t}} + {\boldsymbol{T}};{\boldsymbol{R}}} \right)\left| {{\boldsymbol{S}}_\alpha \left( {{\boldsymbol{R}},{\boldsymbol{t}}} \right) = {\boldsymbol{e}}^{ - {\boldsymbol{i}}\varepsilon _\alpha \left( {\boldsymbol{R}} \right){\boldsymbol{t}}}} \right|{\boldsymbol{S}}_\alpha \left( {{\boldsymbol{R}},{\boldsymbol{t}}} \right),$$
(7)
where the $$\varepsilon _\alpha \left( R \right)$$ are the quasi-energies of the Floquet states $$|S_\alpha \left( {R,t} \right)$$. The Floquet states and quasi-energies are found directly by diagonalizing the 2 × 2 $$U\left( {t,t + T;R} \right)$$ matrix for each R. The Floquet states $$|S_\alpha \left( {R,t} \right)$$ are periodic with the period of the laser field, and exhibit a sub-cycle time dependence whenever multiphoton couplings are active. Consequently, the associated potential energy surfaces will also, in general, exhibit a sub-cycle time dependence. The sub-cycle time dependence can be expanded as a Fourier series to yield a set of time-independent potentials that characterize the system
$$e^{ - i\varepsilon _\alpha \left( R \right)t}|S_\alpha \left( {R,t} \right) = e^{ - i\varepsilon _\alpha \left( R \right)t}\mathop {\sum }\limits_{n = - \infty }^\infty |s_\alpha ^n\left( R \right)e^{ - in\omega t} = \mathop {\sum }\limits_{n = - \infty }^\infty |s_\alpha ^n\left( R \right)e^{ - i\left( {\varepsilon _\alpha \left( R \right) + n\omega } \right)t}.$$
(8)
The ladder of Floquet states is formed by the energies of the Fourier expansion, where the $$\varepsilon _\alpha \left( R \right)$$ get repeated and shifted by , forming an infinite ladder of time-independent potentials. These quasi-energies $$(\varepsilon _\alpha \left( R \right) + n\omega )$$ are what is referred to as LIPs in the main text.
## Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
## Code availability
The computer codes used for TDSE simulations and Floquet calculations are available from the corresponding author upon reasonable request.
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## Acknowledgements
The authors thank D. Crane, R. Kroeker, and B. Avery for technical assistance. We acknowledge fruitful discussions with F. Bouakline, M. Richter, A. M. Sayler, G. G. Paulus, M. F. Kling, and B. Bergues. This project has received funding from the EU’s Horizon2020 research and innovation programme under the Marie Sklodowska-Curie Grant Agreement No. 657544. Financial support from the National Science and Engineering Research Council Discovery Grant No. 419092-2013-RGPIN, and from the U.S. Air Force Office of Scientific Research (Grant No. FA9550-16-1-0109) is gratefully acknowledged.
## Author information
Authors
### Contributions
M.K., Z.D., and A.S. conceived and conducted the experiment, and analyzed the results. M.S., M.K., S.C., M.J.J.V., and D.M.V. performed simulations, and interpreted the data with P.B.C. and A.S. All authors discussed the results and contributed to the final manuscript.
### Corresponding authors
Correspondence to M. Kübel or A. Staudte.
## Ethics declarations
### Competing interests
The authors declare no competing interests.
Peer review information Nature Communications thanks the anonymous reviewers for their contribution to the peer review of this work. Peer reviewer reports are available.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
## Rights and permissions
Reprints and Permissions
Kübel, M., Spanner, M., Dube, Z. et al. Probing multiphoton light-induced molecular potentials. Nat Commun 11, 2596 (2020). https://doi.org/10.1038/s41467-020-16422-2
• Accepted:
• Published:
• ### Quantum light-induced nonadiabatic phenomena in the absorption spectrum of formaldehyde: Full- and reduced-dimensionality studies
• Csaba Fábri
• , Benjamin Lasorne
• , Gábor J. Halász
• , Lorenz S. Cederbaum
• & Ágnes Vibók
The Journal of Chemical Physics (2020)
• ### Clocking Enhanced Ionization of Hydrogen Molecules with Rotational Wave Packets
• Yonghao Mi
• , Peng Peng
• , Nicolas Camus
• , Xufei Sun
• , Patrick Fross
• , Denhi Martinez
• , Zack Dube
• , P. B. Corkum
• , D. M. Villeneuve
• , André Staudte
• , Robert Moshammer
• & Thomas Pfeifer
Physical Review Letters (2020)
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# Q.2 Construct a right-angled triangle whose hypotenuse is 6 cm long and one of the legs is 4 cm long.
Step 2 Draw $$\angle$$ B= 90°.
Step 4 $$\triangle$$ ABC is the required right-angled triangle.
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When Electromagnetic waves are polarised, does it filter out the vibrations in either the magnetic or electric field?
Electromagnetic waves travel as vibrations in the magnetic field and electric fields however since the magnetic field and electric field is perpendicular to each other, does this mean a polraised light ray will have vibrarions travelling in just one of the two fields?
• Both are filtered by the same factor. The ratio $E/B$ has to be constant to the speed of light. – Yuzuriha Inori Mar 11 '18 at 5:13
If you consider that the $\textbf{B}$ field is actually not a true vector but an axial vector then it is better to think of it as an anti-symmetric (or skew-symmetric) tensor that in 3 dimensions has only three non-zero components representing a plane of action in which the magnetic force acts on a current element lying in that plane the force being perpendicular to that current element. Looking at it from this point of view a linearly polarized EM wave has indeed its $\textbf{E}$ and $\textbf{B}$ fields oscillate in the same plane, the $\textbf{E}$ field as a line of action and the $\textbf{B}$ field as a plane of action. The plane of action was called by Ampere the "directive plane". This is why it is more revealing to index the field components not as $B_x,B_y,B_z$ but rather as $B_{yz}, B_{xz}, B_{xy}$.
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## Fast solution of boundary integral equations with the generalized Neumann kernel
Mohamed M.S. Nasser
### Abstract
A fast method for solving boundary integral equations with the generalized Neumann kernel and the adjoint generalized Neumann kernel is presented. The complexity of the developed method is $O((m+1)n\ln n)$ for the integral equation with the generalized Neumann kernel and $O((m+1)n)$ for the integral equation with the adjoint generalized Neumann kernel, where $m+1$ is the multiplicity of the multiply connected domain and $n$ is the number of nodes in the discretization of each boundary component. Numerical results illustrate that the method gives accurate results even for domains of very high connectivity, domains with piecewise smooth boundaries, domains with close-to-touching boundaries, and domains of real world problems.
Full Text (PDF) [7 MB], BibTeX
### Key words
generalized Neumann kernel, boundary integral equations, Nyström method, Fast Multipole Method, GMRES, numerical conformal mapping
### AMS subject classifications
45B05, 65R20, 30C30
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Close suggestions. Displaying top 8 worksheets found for - Adverbial Phrases And Clauses Grade 8. Examples of adverbial phrases, so these are mostly prepositional phrases. ); An adverbial clause: The game will take place when both teams are ready. 'Very' is the qualifier and 'heroically' is the adverb. Scribd is the world's largest social reading and publishing site. Main adverb = well, intensifier = very; Max kicks lightning fast. The adverbial phrase answers the same questions as a regular adverb: how/how much, when, or where. You look very well. An adverb and an intensifier: we can form adverb phrases/adverbial phrases using a normal/basic adverb and an intensifier (words that make adverbs stronger) or a mitigator (words that make adverbs stronger). To help remember the difference, the word itself has “verb” inside it, and adverbs tend to end in “-ly.” “Slowly,” “loudly,” and “happily” are all adverbs. 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A quick revision of adverbs: adverbs often show how, when, or where action... Take place tomorrow snowboarding. grammar ; Here is a word that modifies a noun they use more one!
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Graphs can be succinctly indexed for pattern matching in O(E^2 + V^{2.5}) time
Citation Author(s):
Nicola Cotumaccio
Submitted by:
Nicola Cotumaccio
Last updated:
5 March 2022 - 10:01am
Document Type:
Presentation Slides
Event:
Presenters:
Nicola Cotumaccio
Categories:
Keywords:
Abstract
For the first time we provide a \emph{succinct} pattern matching index for \emph{arbitrary} graphs that can be built \emph{in polynomial time}, while improving both space and query time bounds from [SODA 2021]. We show that, given an edge-labeled graph $G = (V, E)$, there exists a data structure of $|E /_{\le_G}|(\lceil \log|\Sigma|\rceil + \lceil\log q\rceil + 2)\cdot (1+o(1)) + |V /_{\le_G}|\cdot (1+o(1))$ bits which supports pattern matching on $G$ in $O(|P| \cdot q^2 \cdot \log(q\cdot |\Sigma|))$ time, where $G /_{\le_G} = (V /_{\le_G}, E /_{\le_G})$ is a quotient graph obtained by collapsing some nodes in $G$ and $q$ is the width of the \emph{maximum} co-lex relation on $G$.
Our results have relevant applications in automata theory: we can use our data structure to decide whether a string belongs to the language recognized by a given automaton, and we can capture the degree of nondeterminism of an NFA.
Graphs can be succinctly indexed for pattern matching in O(E^2 + V^{2.5}) time.pdf
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Graphs can be succinctly indexed for pattern matching in O(E^2 + V^{2.5}) time.pdf
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My Math Forum what did I do wrong???
Calculus Calculus Math Forum
July 27th, 2017, 04:31 PM #1 Member Joined: Jan 2017 From: Toronto Posts: 97 Thanks: 2 what did I do wrong??? Evaluate ∫∫ x2 + y2 dx dy over the square with corners: (-1, 0), (0, 1), (1, 0), and (0, -1)?? Answer: 8/3 My solution $\displaystyle \int_{-1}^{0} \int_{-y-1}^{y+1} (x^2+y^2) dx dy + \int_{0}^{1} \int_{y-1}^{1-y} (x^2+y^2) dx dy = 2/3$ What did I do wrong???? Any info would be appreciated... Thanks from SenatorArmstrong
July 27th, 2017, 04:43 PM #2 Senior Member Joined: Oct 2009 Posts: 141 Thanks: 59 Why do your bounds depend on $y$?? Thanks from zollen
July 27th, 2017, 04:55 PM #3
Member
Joined: Jan 2017
From: Toronto
Posts: 97
Thanks: 2
Quote:
Originally Posted by Micrm@ss Why do your bounds depend on $y$??
It should not matter dy dx or dx dy. Is it not?
July 28th, 2017, 04:10 AM #4 Math Team Joined: Jan 2015 From: Alabama Posts: 2,576 Thanks: 667 It's hard to say what you did wrong as I have no idea what you were trying to do! To integrate f(x, y) over the square with corners (-1, 0), (0, 1), (1, 0), and (0, -1), x goes from -1 to 1 and, for each x, y goes from -1 to 1. The integral is $\displaystyle \int_{-1}^1\int_{-1}^1 f(x, y) dx dy$. In this case, $\displaystyle f(x,y)= x^2+ y^2$ so the integral is $\displaystyle \int_{-1}^1\int_{-1}^1 x^2+ y^2 dxdy= \int_{-1}^1 \int_{-1}^1 x^2 dx dy+ \int_{-1}^1\int_{1}^1 y^2 dxdy$$\displaystyle = \left(\int_{-1}^1 x^2 dx\right)\left(\int_{-1}^1 dy\right)+ \left(\int_{-1}^1 dx\right)\left(\int_{-1}^1 y^2 dy\right)$$\displaystyle = \left(\frac{x^3}{3}\right)_{-1}^1\left(2\right)+ \left(2\right)\left(\frac{y^3}{3}\right)_{-1}^1= \frac{4}{3}+ \frac{4}{3}= \frac{8}{3}$. Let me see if I can figure out what you were thinking! You have two integrals. The first has y going from -1 to 0 and, for each y, x going from -y-1 to y- 1. x= -y- 1 is the line from (0, -1) (when t= -1) to (-1, 0) (when y= 0). That the line from the midpoint of the lower edge of the square to the midpoint of the left edge of the square. x= y- 1 is the line from (-2, -1) (when y= -1) to (-1, 0) (when y= 0). That is the line from (-2,-1), which is not in the square, to the midpoint of the left side of the square. The integral between those lines has nothing to do with integrating over the square. The same is true of the second integral- its limits of integration have nothing to do with the given square. That was why Microm@ss asked "Why do your bounds depend on y". It has nothing to do with "dx dy" or "dy dx". No, sorry, I still have no idea what you were thinking. Thanks from zollen and MaxBenjamin
July 30th, 2017, 04:16 PM #5
Member
Joined: Jan 2017
From: Toronto
Posts: 97
Thanks: 2
1. I thought I have been evaluating a diamond shape (not square). This is why I was trying to figure out the boundaries of each diagonal lines that make up the diamond.
2. This is an integration of a diamond shape, I would have thought the result of the integral is 0 because positive regions would 'negate' the negative regions.
3. I guess I need more help...
Quote:
Originally Posted by Country Boy It's hard to say what you did wrong as I have no idea what you were trying to do! To integrate f(x, y) over the square with corners (-1, 0), (0, 1), (1, 0), and (0, -1), x goes from -1 to 1 and, for each x, y goes from -1 to 1. The integral is $\displaystyle \int_{-1}^1\int_{-1}^1 f(x, y) dx dy$. In this case, $\displaystyle f(x,y)= x^2+ y^2$ so the integral is $\displaystyle \int_{-1}^1\int_{-1}^1 x^2+ y^2 dxdy= \int_{-1}^1 \int_{-1}^1 x^2 dx dy+ \int_{-1}^1\int_{1}^1 y^2 dxdy$$\displaystyle = \left(\int_{-1}^1 x^2 dx\right)\left(\int_{-1}^1 dy\right)+ \left(\int_{-1}^1 dx\right)\left(\int_{-1}^1 y^2 dy\right)$$\displaystyle = \left(\frac{x^3}{3}\right)_{-1}^1\left(2\right)+ \left(2\right)\left(\frac{y^3}{3}\right)_{-1}^1= \frac{4}{3}+ \frac{4}{3}= \frac{8}{3}$. Let me see if I can figure out what you were thinking! You have two integrals. The first has y going from -1 to 0 and, for each y, x going from -y-1 to y- 1. x= -y- 1 is the line from (0, -1) (when t= -1) to (-1, 0) (when y= 0). That the line from the midpoint of the lower edge of the square to the midpoint of the left edge of the square. x= y- 1 is the line from (-2, -1) (when y= -1) to (-1, 0) (when y= 0). That is the line from (-2,-1), which is not in the square, to the midpoint of the left side of the square. The integral between those lines has nothing to do with integrating over the square. The same is true of the second integral- its limits of integration have nothing to do with the given square. That was why Microm@ss asked "Why do your bounds depend on y". It has nothing to do with "dx dy" or "dy dx". No, sorry, I still have no idea what you were thinking.
July 30th, 2017, 05:03 PM #6
Senior Member
Joined: Sep 2015
From: CA
Posts: 1,300
Thanks: 664
Quote:
Originally Posted by zollen 1. I thought I have been evaluating a diamond shape (not square). This is why I was trying to figure out the boundaries of each diagonal lines that make up the diamond. 2. This is an integration of a diamond shape, I would have thought the result of the integral is 0 because positive regions would 'negate' the negative regions. 3. I guess I need more help...
I got 2/3 as well and I did note it was a diamond shaped area of integration.
I think your text is wrong or there's some piece of this missing.
July 30th, 2017, 06:08 PM #7 Global Moderator Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,540 Thanks: 920 Math Focus: Elementary mathematics and beyond $$\int_{-1}^{1}\int_{-1}^{1}(x^2+y^2)\,dx\,dy=\int_{-1}^{1}\left(\frac23+2y^2\right)\,dy=\frac23+\frac2 3+\frac23+\frac23=\frac83$$ Thanks from MaxBenjamin
July 30th, 2017, 08:36 PM #8
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Quote:
Originally Posted by greg1313 $$\int_{-1}^{1}\int_{-1}^{1}(x^2+y^2)\,dx\,dy=\int_{-1}^{1}\left(\frac23+2y^2\right)\,dy=\frac23+\frac2 3+\frac23+\frac23=\frac83$$
that's an integration of the square with corners at
$(-1,-1), (-1,1), (1,1), (1,-1)$
His integration area is a square with corners
$(-1,0), (0,1), (1,0), (0,-1)$
July 30th, 2017, 09:27 PM #9 Global Moderator Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,540 Thanks: 920 Math Focus: Elementary mathematics and beyond I suggest you make a diagram.
July 30th, 2017, 10:28 PM #10
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$\displaystyle \int \limits_{-1}^{0} \int \limits_{-1-x}^{1+x}~x^2+y^2~dy~dx + \int \limits_{0}^{1} \int \limits_{-1+x}^{1-x}~x^2+y^2~dy~dx =\dfrac 2 3$
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# Hilbert's tenth problem
Hilbert's tenth problem is the tenth on the list of mathematical problems that the German mathematician David Hilbert posed in 1900. It is the challenge to provide a general algorithm which, for any given Diophantine equation (a polynomial equation with integer coefficients and a finite number of unknowns) can decide whether the equation has a solution with all unknowns taking integer values.
For example, the Diophantine equation ${\displaystyle 3x^{2}-2xy-y^{2}z-7=0}$ has an integer solution: ${\displaystyle x=1,\ y=2,\ z=-2}$. By contrast, the Diophantine equation ${\displaystyle x^{2}+y^{2}+1=0}$ has no such solution.
Hilbert's tenth problem has been solved, and it has a negative answer: such a general algorithm does not exist. This is the result of combined work of Martin Davis, Yuri Matiyasevich, Hilary Putnam and Julia Robinson which spans 21 years, with Yuri Matiyasevich completing the theorem in 1970.[1] The theorem is now known as Matiyasevich's theorem or the MRDP theorem.
## Background
### Original formulation
Hilbert formulated the problem as follows:
Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers.
The words "process" and "finite number of operations" have been taken to mean that Hilbert was asking for an algorithm. The term "rational integer" simply refers to the integers, positive, negative or zero: 0, ±1, ±2, ... . So Hilbert was asking for a general algorithm to decide whether a given polynomial Diophantine equation with integer coefficients has a solution in integers.
Hilbert's problem is not concerned with finding the solutions. It only asks whether, in general, we can decide whether one or more solutions exist. The answer to this question is negative, in the sense that no "process can be devised" for answering that question. In modern terms, Hilbert's 10th problem is an undecidable problem. Although it is unlikely that Hilbert had conceived of such a possibility, before going on to list the problems, he did presciently remark:
"Occasionally it happens that we seek the solution under insufficient hypotheses or in an incorrect sense, and for this reason do not succeed. The problem then arises: to show the impossibility of the solution under the given hypotheses or in the sense contemplated."
Proving the 10th problem undecidable is then a valid answer even in Hilbert's terms, since it is a proof about "the impossibility of the solution".
### Diophantine sets
In a diophantine equation, there are two kinds of variables: the parameters and the unknowns. The diophantine set consists of the parameter assignments for which the diophantine equation is solvable. A typical example is the linear diophantine equation in two unknowns,
${\displaystyle a_{1}x_{1}+a_{2}x_{2}=a_{3}}$,
where the equation is solvable when the greatest common divisor of ${\displaystyle a_{1},a_{2}}$ divides ${\displaystyle a_{3}}$. The values for ${\displaystyle a_{1},a_{2},a_{3}}$ that satisfy this restriction are a diophantine set and the equation above is its diophantine definition.
Diophantine definitions can be provided by simultaneous systems of equations as well as by individual equations because the system
${\displaystyle p_{1}=0,\ldots ,p_{k}=0}$
is equivalent to the single equation
${\displaystyle p_{1}^{2}+\cdots +p_{k}^{2}=0.}$
Sets of natural numbers, of pairs of natural numbers (or even of n-tuples of natural numbers) that have Diophantine definitions are called Diophantine sets. In these terms, Hilbert's tenth problem asks whether a particular Diophantine set is non-empty.
The work on the problem has been in terms of solutions in natural numbers (understood as the non-negative integers) rather than arbitrary integers. The two problems are equivalent: any general algorithm that can decide whether a given Diophantine equation has an integer solution could be modified into an algorithm that decides whether a given Diophantine equation has a natural number solution, and vice versa. This can be seen as follows: The requirement that solutions be natural numbers can be expressed with the help of Lagrange's four-square theorem: every natural number is the sum of the squares of four integers, so we simply replace every parameter with the sum of squares of four extra parameters. Similarly, since every integer can be written as the difference of two natural numbers, we can replace every parameter that ranges in integers with the difference of two parameters that range in natural numbers.[2]
### Recursively enumerable sets
A recursively enumerable set can be characterized as one for which there exists an algorithm that will ultimately halt when a member of the set is provided as input, but may continue indefinitely when the input is a non-member. It was the development of computability theory (also known as recursion theory) that provided a precise explication of the intutitive notion of algorithmic computability, thus making the notion of recursive enumerability perfectly rigorous. It is evident that Diophantine sets are recursively enumerable. This is because one can arrange all possible tuples of values of the unknowns in a sequence and then, for a given value of the parameter(s), test these tuples, one after another, to see whether they are solutions of the corresponding equation. The unsolvability of Hilbert's tenth problem is a consequence of the surprising fact that the converse is true:
Every recursively enumerable set is Diophantine.
This result is variously known as Matiyasevich's theorem (because he provided the crucial step that completed the proof) and the MRDP theorem (for Yuri Matiyasevich, Julia Robinson, Martin Davis, and Hilary Putnam). Because there exists a recursively enumerable set that is not computable, the unsolvability of Hilbert's tenth problem is an immediate consequence. In fact, more can be said: there is a polynomial
${\displaystyle p(a,x_{1},\ldots ,x_{n})}$
with integer coefficients such that the set of values of ${\displaystyle a}$ for which the equation
${\displaystyle p(a,x_{1},\ldots ,x_{n})=0}$
has solutions in natural numbers is not computable. So, not only is there no general algorithm for testing Diophantine equations for solvability, even for this one parameter family of equations, there is no algorithm.
## History
Year Events
1944 Emil Leon Post declares that Hilbert's tenth problem "begs for an unsolvability proof".
1949 Martin Davis uses Kurt Gödel's method for applying the Chinese remainder theorem as a coding trick to obtain his normal form for recursively enumerable sets:
${\displaystyle \left\{a\mid \exists y\forall k\leqslant y\exists x_{1},\ldots ,x_{n}:p\left(a,k,y,x_{1},\ldots ,x_{n}\right)=0\right\}}$
where ${\displaystyle p}$ is a polynomial with integer coefficients. Purely formally, it is only the bounded universal quantifier that stands in the way of this being a Diophantine definition.
Using a non-constructive but easy proof, he derives as a corollary to this normal form that the set of Diophantine sets is not closed under complementation, by showing that there exists a Diophantine set whose complement is not Diophantine. Because the recursively enumerable sets also are not closed under complementation, he conjectures that the two classes are identical.
1950 Julia Robinson, unaware of Davis's work, investigates the connection of the exponential function to the problem, and attempts to prove that EXP, the set of triplets ${\displaystyle (a,b,c)}$ for which ${\displaystyle a=b^{c}}$, is Diophantine. Not succeeding, she makes the following hypothesis (later called J.R.):
There is a Diophantine set ${\displaystyle D}$ of pairs ${\displaystyle (a,b)}$ such that ${\displaystyle (a,b)\in D\Rightarrow b and for every positive ${\displaystyle k,}$ there exists ${\displaystyle (a,b)\in D}$ such that ${\displaystyle b>a^{k}.}$
Using properties of the Pell equation, she proves that J.R. implies that EXP is Diophantine, as well as the binomial coefficients, the factorial, and the primes.
1959 Working together, Davis and Putnam study exponential Diophantine sets: sets definable by Diophantine equations in which some of the exponents may be unknowns. Using the Davis normal form together with Robinson's methods, and assuming the then unproved conjecture that there are arbitrarily long arithmetic progressions consisting of prime numbers, they prove that every recursively enumerable set is exponential Diophantine. They also prove as a corollary that J.R. implies that every recursively enumerable set is Diophantine, which in turn implies that Hilbert's tenth problem is unsolvable.
1960 Robinson simplifies the proof of the conditional result for exponential Diophantine sets and makes it independent from the conjecture about primes and thus a formal theorem. This makes the J.R. hypothesis a sufficient condition for the unsolvability of Hilbert's tenth problem. However, many doubt that J.R. is true.[3]
1961–1969 During this period, Davis and Putnam find various propositions that imply J.R. Robinson, having previously shown that J.R. implies that the set of primes is a Diophantine set, proves that this is an if and only if condition. Yuri Matiyasevich publishes some reductions of Hilbert's tenth problem.
1970 Drawing from the recently published work of Nikolai Vorob'ev on Fibonacci numbers,[4] Matiyasevich proves that the set ${\displaystyle P=\{(a,b)|a>0,b=F_{2a}\},}$ where ${\displaystyle F_{n}}$ is the nth Fibonacci number is Diophantine and exhibits exponential growth.[5] This proves the J.R. hypothesis, which by then had been an open question for 20 years. Combining J.R. with the theorem that every recursively enumerable set is exponential diophantine, proves that Diophantine sets are recursively enumerable. This makes Hilbert's tenth problem unsolvable.
## Applications
The Matiyasevich/MRDP Theorem relates two notions — one from computability theory, the other from number theory — and has some surprising consequences. Perhaps the most surprising is the existence of a universal Diophantine equation:
There exists a polynomial ${\displaystyle p(a,n,x_{1},\ldots ,x_{k})}$ such that, given any Diophantine set ${\displaystyle S}$ there is a number ${\displaystyle n_{0}}$ such that
${\displaystyle S=\{\,a\mid \exists x_{1},\ldots ,x_{k}[p(a,n_{0},x_{1},\ldots ,x_{k})=0]\,\}.}$
This is true simply because Diophantine sets, being equal to recursively enumerable sets, are also equal to Turing machines. It is a well known property of Turing machines that there exist universal Turing machines, capable of executing any algorithm.
Hilary Putnam has pointed out that for any Diophantine set ${\displaystyle S}$ of positive integers, there is a polynomial
${\displaystyle q(x_{0},x_{1},\ldots ,x_{n})}$
such that ${\displaystyle S}$ consists of exactly the positive numbers among the values assumed by ${\displaystyle q}$ as the variables
${\displaystyle x_{0},x_{1},\ldots ,x_{n}}$
range over all natural numbers. This can be seen as follows: If
${\displaystyle p(a,y_{1},\ldots ,y_{n})=0}$
provides a Diophantine definition of ${\displaystyle S}$, then it suffices to set
${\displaystyle q(x_{0},x_{1},\ldots ,x_{n})=x_{0}[1-p(x_{0},x_{1},\ldots ,x_{n})^{2}].}$
So, for example, there is a polynomial for which the positive part of its range is exactly the prime numbers. (On the other hand, no polynomial can only take on prime values.) The same holds for other recursively enumerable sets of natural numbers: the factorial, the binomial coefficients, the fibonacci numbers, etc.
Other applications concern what logicians refer to as ${\displaystyle \Pi _{1}^{0}}$ propositions, sometimes also called propositions of Goldbach type.[6] These are like Goldbach's conjecture, in stating that all natural numbers possess a certain property that is algorithmically checkable for each particular number.[7] The Matiyasevich/MRDP theorem implies that each such proposition is equivalent to a statement that asserts that some particular Diophantine equation has no solutions in natural numbers.[8] A number of important and celebrated problems are of this form: in particular, Fermat's Last Theorem, the Riemann hypothesis, and the Four color theorem. In addition the assertion that particular formal systems such as Peano arithmetic or ZFC are consistent can be expressed as ${\displaystyle \Pi _{1}^{0}}$ sentences. The idea is to follow Kurt Gödel in coding proofs by natural numbers in such a way that the property of being the number representing a proof is algorithmically checkable.
${\displaystyle \Pi _{1}^{0}}$ sentences have the special property that if they are false, that fact will be provable in any of the usual formal systems. This is because the falsity amounts to the existence of a counter-example which can be verified by simple arithmetic. So if a ${\displaystyle \Pi _{1}^{0}}$ sentence is such that neither it nor its negation is provable in one of these systems, that sentence must be true.[citation needed]
A particularly striking form of Gödel's incompleteness theorem is also a consequence of the Matiyasevich/MRDP theorem:
Let
${\displaystyle p(a,x_{1},\ldots ,x_{k})=0}$
provide a Diophantine definition of a non-computable set. Let ${\displaystyle A}$ be an algorithm that outputs a sequence of natural numbers ${\displaystyle n}$ such that the corresponding equation
${\displaystyle p(n,x_{1},\ldots ,x_{k})=0}$
has no solutions in natural numbers. Then there is a number ${\displaystyle n_{0}}$ which is not output by ${\displaystyle A}$ while in fact the equation
${\displaystyle p(n_{0},x_{1},\ldots ,x_{k})=0}$
has no solutions in natural numbers.
To see that the theorem is true, it suffices to notice that if there were no such number ${\displaystyle n_{0}}$, one could algorithmically test membership of a number ${\displaystyle n}$ in this non-computable set by simultaneously running the algorithm ${\displaystyle A}$ to see whether ${\displaystyle n}$ is output while also checking all possible ${\displaystyle k}$-tuples of natural numbers seeking a solution of the equation
${\displaystyle p(n,x_{1},\ldots ,x_{k})=0.}$
We may associate an algorithm ${\displaystyle A}$ with any of the usual formal systems such as Peano arithmetic or ZFC by letting it systematically generate consequences of the axioms and then output a number ${\displaystyle n}$ whenever a sentence of the form
${\displaystyle \neg \exists x_{1},\ldots ,x_{k}[p(n,x_{1},\ldots ,x_{k})=0]}$
is generated. Then the theorem tells us that either a false statement of this form is proved or a true one remains unproved in the system in question.
## Further results
We may speak of the degree of a Diophantine set as being the least degree of a polynomial in an equation defining that set. Similarly, we can call the dimension of such a set the least number of unknowns in a defining equation. Because of the existence of a universal Diophantine equation, it is clear that there are absolute upper bounds to both of these quantities, and there has been much interest in determining these bounds.
Already in the 1920s Thoralf Skolem showed that any Diophantine equation is equivalent to one of degree 4 or less. His trick was to introduce new unknowns by equations setting them equal to the square of an unknown or the product of two unknowns. Repetition of this process results in a system of second degree equations; then an equation of degree 4 is obtained by summing the squares. So every Diophantine set is trivially of degree 4 or less. It is not known whether this result is best possible.
Julia Robinson and Yuri Matiyasevich showed that every Diophantine set has dimension no greater than 13. Later, Matiyasevich sharpened their methods to show that 9 unknowns suffice. Although it may well be that this result is not the best possible, there has been no further progress.[9] So, in particular, there is no algorithm for testing Diophantine equations with 9 or fewer unknowns for solvability in natural numbers. For the case of rational integer solutions (as Hilbert had originally posed it), the 4-squares trick shows that there is no algorithm for equations with no more than 36 unknowns. But Zhi Wei Sun showed that the problem for integers is unsolvable even for equations with no more than 11 unknowns.
Martin Davis studied algorithmic questions involving the number of solutions of a Diophantine equation. Hilbert's tenth problem asks whether or not that number is 0. Let ${\displaystyle A=\{0,1,2,3,\ldots ,\aleph _{0}\}}$ and let ${\displaystyle C}$ be a proper non-empty subset of ${\displaystyle A}$. Davis proved that there is no algorithm to test a given Diophantine equation to determine whether the number of its solutions is a member of the set ${\displaystyle C}$. Thus there is no algorithm to determine whether the number of solutions of a Diophantine equation is finite, odd, a perfect square, a prime, etc.
## Extensions of Hilbert's tenth problem
Alexandra Shlapentokh (middle) in 2003
Although Hilbert posed the problem for the rational integers, it can be just as well asked for many rings (in particular, for any ring whose elements are countable). Obvious examples are the rings of integers of algebraic number fields as well as the rational numbers.
There has been much work on Hilbert's tenth problem for the rings of integers of algebraic number fields. Basing themselves on earlier work by Jan Denef and Leonard Lipschitz and using class field theory, Harold N. Shapiro and Alexandra Shlapentokh were able to prove:
Hilbert's tenth problem is unsolvable for the ring of integers of any algebraic number field whose Galois group over the rationals is abelian.
Shlapentokh and Thanases Pheidas (independently of one another) obtained the same result for algebraic number fields admitting exactly one pair of complex conjugate embeddings.
The problem for the ring of integers of algebraic number fields other than those covered by the results above remains open. Likewise, despite much interest, the problem for equations over the rationals remains open. Barry Mazur has conjectured that for any variety over the rationals, the topological closure over the reals of the set of solutions has only finitely many components.[10] This conjecture implies that the integers are not Diophantine over the rationals and so if this conjecture is true a negative answer to Hilbert's Tenth Problem would require a different approach than that used for other rings.
## Notes
1. ^ S. Barry Cooper, Computability theory, p. 98
2. ^ Yuri Matiyasevich (1993). Hilbert's 10th problem. MIT Press.
3. ^ A review of the joint publication by Davis, Putnam, and Robinson in Mathematical Reviews (MR0133227) conjectured, in effect, that J.R. was false.
4. ^ Matiyasevich, Yuri (1992). "My Collaboration with Julia Robinson". The Mathematical Intelligencer. 14 (4): 38–45. doi:10.1007/bf03024472.
5. ^ Sacks, Gerald E. (2003). Mathematical Logic in the 20th century. World Scientific. pp. 269–273.
6. ^ ${\displaystyle \Pi _{1}^{0}}$ sentences are at one of the lowest levels of the so-called arithmetical hierarchy.
7. ^ Thus, the Goldbach Conjecture itself can be expressed as saying that for each natural number ${\displaystyle n}$ the number ${\displaystyle 2n+4}$ is the sum of two prime numbers. Of course there is a simple algorithm to test a given number for being the sum of two primes.
8. ^ In fact the equivalence is provable in Peano arithmetic.
9. ^ At this point, even 3 cannot be excluded as an absolute upper bound.
10. ^ http://www-math.mit.edu/~poonen/papers/subrings.pdf
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# Tag Info
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I used your question to finally try out the animate package. The "animation" is in an external file, it uses the standalone package and the markings decoration of tikz: "Animation" code \documentclass[tikz,border=3mm]{standalone} \usetikzlibrary{decorations.markings} \begin{document} \tikzset{reddot/.style={decoration={markings, mark=between positions #1 ...
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Just a proof of concept; the flow was created with a very thick dashed line with a particular dash pattern; the illusion of movement was obtained using a different value for dash phase in even and odd slides (thanks to Jake for this suggestion that simplyfied my initial code): \documentclass{beamer} \usepackage{tikz} \def\phase{0pt} \begin{document} ...
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Last edit: With the invaluable assistance of Paul Isambert, I wrote the ocgx package. Here is an example with ocgx package (available via CTAN and via TeXLive). Three screenshots: The code: \documentclass{beamer} \usepackage[latin1]{inputenc} \usepackage[T1]{fontenc} \usepackage{lmodern} \usepackage{tikz} \usetikzlibrary{ocgx,calc} \begin{document} ...
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Here is an animate based solution with links that act as switches: \documentclass{article} \usepackage[latin1]{inputenc} \usepackage[T1]{fontenc} \usepackage{lmodern} \usepackage{tikz} \usepackage{animate} \usepackage{hyperref} \makeatletter %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % command to create toggle link to ...
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Does the animate package do things with the shipout routine of TeX? The external lib replaces it -- probably in an incompatible way. Nevertheless, the following approach also works: \documentclass{article} \usepackage{pgfplots} \usepackage{animate} % Comment out the following line to see what the plot looks like. \usepgfplotslibrary{external} ...
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You can optimize the PDF animation for size using differential bitmaps and the timeline feature of the animate package. Note however, that with every new animation frame to be shown the number of differential frames that needs to be re-displayed increases by one. This may slow down a running animation as time advances. Adobe Reader was not primarily ...
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The easiest thing is to use the animateinline environment and then "manually" include the images, and lastly make the last image have a slow frame rate. However, I would like to show you a couple of possibilities with the animate package. Throughout this the distinction between image and frame is not enforced. Please ask if in doubt. Animation problems ...
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\usepgfplotslibrary{external} and package animate don't work well together. \tikzexternalize moves the tikzpicture environment into an external file to be processed separately (therefore, --shell-escape must be activated) and replaces the tikzpicture environment with \includegraphics. On the first pass when no external pdf graphics are available, the boxes ...
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This can be done using a simple variation of a progress bar, as in Progress bar for latex-beamer; I used some of my own images man1, man2, but you can use your own: \documentclass{beamer} \usetheme{CambridgeUS} \usepackage{tikz} \makeatletter \def\progressbar@progressbar{} % the progress bar \newcount\progressbar@tmpcounta% auxiliary counter ...
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The problem is that \animategraphics uses LaTeX's \IfFileExists that uses the space as end marker of the file name. Then the result \@filef@und contains the file name including a final space. \animategraphics then removes the space via \zap@space that also kills spaces in between. The following workaround redefines \animategraphics. It uses package grffile ...
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Character for character The usage of the xstring macros is taken from the linked question. The width of the text (in \Huge style) is saved to \mytextwidth. As \StrLen counts from 0 to, in this case, 27, \multiframe needs to animate 28 frames, which is why \mylen+1 is calculated and saved to \frames. This example uses the \multiframe macro because we can ...
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Just omit any post-scaling options, such as width or scale. Then the original size, that is the size of the first graphics of the sequence, will be used for the animation. In your case, width=1\linewidth will resize the width of the animation to the line width of the hosting document. This is the largest width I would recommend, as a larger animation would ...
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These "player buttons" are unique to the animate package and are constructed from low-level Postscript commands. The following MWE provides an interface to the symbols, entirely extracted from animate.sty (minus some colour settings). Each of the symbols is 15pt in width. Since the commands are low-level, they produce a zero-width representation within ...
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This disables the animateinline environment, and modifies the \multiframe loop so that it does \only<2|handout:2>{.....} with the loop counter being used in <...> so beamer does its normal stuff and makes a slide corresponding to each frame of the iteration. (only the two lines marked XXX are modified. The handout specification is needed to ...
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The problem is that animate will put all frames in the box of the size of the first frame. You need to establish a bounding box in the TikZ picture that will encompass all frames. This has also been discussed in the animate manual on page 8: A short note on the tikzpicture environment: Unlike pspicture, the tikzpicture environment is able to determine ...
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The manual of animate is quite explicit about this, as stated in the comment. If you are not satisfied with the text/graphics rendering in Adobe Reader you might want to play with its display settings: go Edit->Preferences->Page Display.
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The reason is loosely explained in the Bugs section of the animate doc. -Ppdf loads config.pdf during DVI to PS conversion and sets the dots-per-inch resolution (DPI) in the intermediate Postscript to the value of 8000. This is equivalent to using dvips option -D8000. The only benefit of this high resolution is related to fonts generation in case of Type-3 ...
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While loading animate, the command \pdfcatalog{/AcroForm <</Fields []/NeedAppearances true>>} is executed. Strangely, the sheer existence of the AcroForm dictionary in the PDF catalog causes Adobe Reader to prompt for saving when the document is closed, and even more strangely, yet fortunately, it does not, if there is at least one animation ...
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All in one approach Read the comments whether or not you need ImageMagick and/or FFMPEG installed on your machine. \documentclass{article} \usepackage{filecontents} % create a parameterized template \begin{filecontents*}{template.tex} \documentclass[tikz,border=12pt]{standalone} \usepackage{pgfplots} \begin{document} \foreach \ind in {1,2,...,30} { ...
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This error only occurs if the document is typeset with pdflatex or lualatex, while xelatex, latex+dvips+ps2pdf work smoothly. animate puts the animation frame content first into a box, using the LaTeX \savebox command. The filled box is then distilled into a PDF Form XObject using the command \pdfxform from pdfTeX. The last step fails, if the box contains ...
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animate uses JavaScript because if you want animations in PDF you need either an external program (i.e. you have to include movies) or scripting support in the viewer - and the only viewer with scripting support is Adobe Reader with JavaScript. So Lua won't help you there unless you also have a viewer with Lua support (which would make you PDFs incompatible ...
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Your first problem will be solved by putting a % after \newcommand{\makelayer}{. And I think I didn't get your second question properly. It can be done as shown in this modified code of yours. \documentclass{beamer} \usepackage[utf8]{inputenc} \usepackage{tikz} \usepackage{ifthen} \usepackage{animate} \usetheme{Warsaw} \usecolortheme{whale} ...
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The verbatim code in the animate documentation are minimal working examples that can be compiled as they are. You cannot embed a pdf with an animation into another document. Just insert the \begin{animateinline}...\end{animateinline} stuff into your beamer document source.
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For some reasons, animate doesn't like spaces in file names. In general they should be avoided whenever possible. What happens when you feed \animategraphics with the file name "heart animation", the package tries to includeheartanimation.pdf`, removing spaces in the name. A solution is to change the category code of the space before absorbing the second ...
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After redefinition of some animate macros, LaTeX produces a multipage document which can be converted to SWF using pdf2swf or to bitmapped formats using other tools, e. g. dvipng. \documentclass{article} \usepackage{animate} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage[active,tightpage]{preview} \makeatletter ...
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All frames of an animation must have the same size. The first frame determines the animation widget dimensions. The following frames are anamorphically scaled to this size. Thus, if pspicture dimensions are dynamical, all frames except the first one may be stretched or shrinked. \documentclass{article} \usepackage[margin=2cm]{geometry} ...
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animate-generated animations are implemented as PDF annotations (interactive regions), which live in a separate layer above the normal page content. They are therefore always rendered in the foreground. Everything to appear in front of animated material must itself be part of the animation: \documentclass{beamer} \usepackage{animate} \usepackage{tikz} ...
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It seems that the {1} just after the square bracket controls the frame rate: \animategraphics[autoplay,loop,height=5cm]{1}{my_pngfiles_}{0}{22} Changing that will change the speed apparently, e.g. to 8 say: \animategraphics[autoplay,loop,height=5cm]{8}{my_pngfiles_}{0}{22}
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There is a bug in storebox.sty: \AtBeginDocument{% \@ifpackageloaded{pgf}{\RequirePackage{storebox-pgf}{}}% } should be \AtBeginDocument{% \@ifpackageloaded{pgf}{\RequirePackage{storebox-pgf}}{}% } When TeX executes this piece of code, it looks for the third argument to \@ifpackageloaded and it finds \ifnum which comes just after ...
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During externalization, not only the tikzpicture environment in question is processed but also the rest of the page including the \animategraphics command. The latter uses the \pdfxform primitive of pdfTeX to encapsulate animation frames into PDF XObjects, which are inserted into the externalized graphics file and contribute to the file size, but which ...
Only top voted, non community-wiki answers of a minimum length are eligible
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Subgaussian 1-cocycles on discrete groups
# Subgaussian 1-cocycles on discrete groups
Marius Junge Department of Mathematics, University of Illinois, Urbana, IL 61801 and Qiang Zeng Department of Mathematics, University of Illinois, Urbana, IL 61801 Center of Mathematical Sciences and Applications, Harvard University, Cambridge, MA 02138
July 27, 2019
###### Abstract.
We prove the Poincaré inequalities with constant for -cocycles on countable discrete groups under Bakry–Emery’s -criterion. These inequalities determine an analogue of subgaussian behavior for 1-cocycles. Our theorem improves some of our previous results in this direction, and in particular implies Efraim and Lust-Piquard’s Poincaré type inequalities for the Walsh system. The key new ingredient in our proof is a decoupling argument. As complementary results, we also show that the spectral gap inequality implies the Poincaré inequalities with constant under some conditions in the noncommutative setting. New examples which satisfy the -criterion are provided as well.
###### Key words and phrases:
Decoupling, (noncommutative) Burkholder inequality, (noncommutative) Poincaré inequality, (noncommutative) transportation inequality, noncommutative spaces, -criterion, group von Neumann algebras, 1-cocycles, spectral gap
###### 2010 Mathematics Subject Classification:
46L53, 60E15, 22D25
The first author was partially supported by NSF Grant DMS-1201886
## 1. Introduction
Subgaussian behavior of random variables and stochastic processes is an important topic in probability theory. It is closely related to the concentration of measure phenomenon; see e.g. [Ver]. Functional inequalities – including log-Sobolev inequality, Poincaré inequality, transportation-entropy inequalities – have played a critical role in the development of this theory in the last two decades; see [Ta96, BG, OV, BGL] and the references therein for the extensive literature. More recently, this theory has been applied to study random matrices; see e.g. [Ver, Gui]. In this paper, we want to connect this well-known theory in classical probability to 1-cocycles on groups, which is important in both group theory (Kazhdan’s Property (T), the Haagerup property, etc.) and operator algebras; see e.g. [BO]. We are interested in determining a class of 1-cocycles which satisfy an analogue of the subgaussian growth condition via Poincaré type inequalities. Recall that a random variable is subgaussian if and only if for all . Here and in the following we use , etc. to denote absolute constants which may vary from line to line. To generalize this notion, we consider the following Poincaré inequalities for a probability space ,
(1.1) ∥f−Eμ(f)∥p≤C√p∥∇f∥p
for all and differentiable . Observe that (1.1) resembles subgaussian growth of random variables. In particular, choosing when we recover the classical definition except for .
As a classical example, the Gaussian measure on satisfies (1.1) due to Pisier [Pis86]; see [JZ12] for another proof. More classical examples satisfying (1.1) can be found in [AW13] and the references therein. In fact, one way to generalize (1.1) is via the semigroup theory of operators. The analogue of gradient in this context is Meyer’s “carré du champs”. We can even go further and consider an analogue of (1.1) in a noncommutative probability space. Recall from [VDN] that is a probability space if is a von Neumann algebra and is a normal state. We also assume that is finite and is tracial and faithful. Throughout we always work with a standard semigroup acting on with generator . Here a standard semigroup is pointwise -weak (weak) continuous such that every is normal unital completely positive and symmetric on . We define the gradient form associated to (Meyer’s “carré du champs”) as
ΓA(f1,f2)=12[A(f∗1)f2+f∗1A(f2)−A(f∗1f2)]
for in a suitable involutive subalgebra of the domain of the generator, which is supposed to exist. In the following, we may simply write for if the generator under consideration is clear. Let be the fixed point algebra of . It was shown in [JX07] that is a von Neumann subalgebra of . Thus there exists a unique conditional expectation . Recall that the noncommutative space is defined as the closure of in the norm given by for and for , where is the operator norm. We usually write for short. It is well known that is a Banach space for ; see [PX03] for more details.
###### Definition 1.1.
A standard semigroup acting on is said to be subgaussian if the following Poincaré inequalities
(1.2) ∥f−EFix(f)∥p≤C√pmax{∥ΓA(f,f)1/2∥p, ∥ΓA(f∗,f∗)1/2∥p}.
hold for and in a suitable involutive subalgebra of the domain of the generator.
For simplicity, in the following we may say the above inequality holds for all , since it is automatically true if the right-hand side is infinity. Since the gradient form coincides with the modulus of the gradient if is the Laplacian of a Euclidean space, (1.2) is indeed a generalization of (1.1). It is known that for classical diffusion semigroups, log-Sobolev inequality implies (1.2); see [AS94] and also [AW13]. Efraim and Lust-Piquard proved that (1.2) holds for Walsh systems and CAR algebras in [ELP]. In fact, we started to study the subgaussian behavior (1.2) of semigroups acting on a general noncommutative probability space in [JZ12]. It was shown in [Z13] that the group measure space satisfies (1.2), where the action and the gaussian measure are associated to an orthogonal representation of on a real Hilbert space, and the semigroup acting on is a natural extension of the Ornstein-Uhlenbeck semigroup on . A remarkable consequence of (1.2) is that one can get concentration inequalities, exponential integrability and transportation cost inequalities; see [ELP, Z13]. Our goal here is to prove (1.2) for group von Neumann algebras under some conditions on the 1-cocycles of groups and to elaborate on the relationship between the spectral gap of and Poincaré inequalities for semigroups acting on a probability space .
Let us be more precise. Let be a countable discrete group. Recall that a (generic) conditionally negative length (or cn-length for short) function on determines a 1-cocycle on with coefficients in an orthogonal representation of , and vice versa. Let be the left regular representation given by for , where ’s form a unit vector basis of . The group von Neumann algebra is the closure of linear span of in the weak operator topology. It is well known that admits a canonical normal faithful tracial state given by for , where is the identity element of . Consider the semigroup acting on defined by for . Then is a standard semigroup on . Thus extends to a strongly continuous semigroup of contractions on and the generator is given by . We say that a 1-cocycle on with coefficients in the orthogonal representation is subgaussian if the semigroup given by is subgaussian in the sense of Definition 1.1, i.e.,
(1.3) ∥f−EFix(f)∥p≤C√pmax{∥Γψ(f,f)1/2∥p, ∥Γψ(f∗,f∗)1/2∥p}.
holds for all and .
For readers who are not familiar with von Neumann algebras, (1.3) can be formulated in a more algebraic way, i.e.,
∥f∥p≤C√pmax{∥Γψ(f,f)1/2∥p, ∥Γψ(f∗,f∗)1/2∥p}
for , , where is the group algebra of (thus is a finite linear combination),
Γψ(f,f)=∑s,t∈G¯asatK(s,t)s−1t
and is the Gromov form given by
K(s,t)=12(ψ(s)+ψ(t)−ψ(s−1t)),s,t∈G.
Here in the linear combination implies that . We remark that and are not equal in general because of noncommutativity. It is clear that in this formulation (1.3) is really a condition on the 1-cocycle (or the cn-length function) and involves no probability theory or semigroups of operators. However, the only way we know to prove such inequalities is to use probability in an efficient way. To state our main results, we need to introduce the well-known -criterion due to Bakry–Emery. Recall that
ΓA2(f1,f2)=12[ΓA(Af1,f2)+ΓA(f1,Af2)−AΓA(f1,f2)]
whenever and are in a suitable involutive subalgebra of the domain of the generator.
###### Theorem 1.2.
Let be a countable discrete group with cn-length function and its group von Neumann algebra. Suppose satisfies for some . Then for ,
(1.4) ∥f−EFixf∥p≤C√p/αmax{∥Γψ(f,f)1/2∥p,∥Γψ(f∗,f∗)1/2∥p}.
Note that Theorem 1.2 is a result for individual elements. As a property of the group von Neumann algebra, we hope to show that (1.4) holds for all . Recall Bakry–Emery’s -criterion [BE85]: There exists such that for all for which both and are well-defined. As observed in [JZ12], in our context this condition is equivalent to the algebraic condition that is a positive semidefinite form. The domain of is typically smaller than that of . Therefore, it is possible that is not well defined for some element while (1.4) still holds for this element. However, by [JZ12]*Corollary 4.8, we know that (1.4) always holds for all provided for all , because is a weakly dense subalgebra of . Let us record this as the following result.
###### Corollary 1.3.
Suppose the -criterion holds for the cn-length function on a group . Then we have the Poincaré inequalities (1.4) for all and whenever the right-hand side of (1.4) is finite. Therefore, the 1-cocycle is subgaussian.
Our motivation to study this problem comes from both noncommutative harmonic analysis and probability theory. In noncommutative harmonic analysis, Poincaré inequalities are closely related to noncommutative Riesz transform and smooth Fourier multiplier theory developed in [JM10, JMP]. In probability theory, precise moment estimation of random variables could be the starting point of various results, including concentration and transportation inequalities.
Let us mention some interesting applications. As indicated in [JZ12], applying Theorem 1.2 to the group , we recover the Poincaré type inequalities for the Walsh system due to Efraim and Lust-Piquard [ELP]. By embedding the matrix algebra into the discrete Heisenberg group von Neumann algebra, we find subgaussian behavior for matrix algebras. Another immediate consequence of our main results is the following transportation type inequalities shown in [JZ12, Z13]. Let us recall some notation. Let be a semigroup acting on a noncommutative probability space with generator . Given -measurable operators and , we define the following analogues of classical Wasserstein distances
and
where and ; see [Z13] for a detailed discussion about these distances and their relationship to Rieffel’s quantum metric spaces. For a -measurable positive operator , we define the entropy
Ent(ρ)=τ[ρln(ρ/τ(ρ))].
###### Corollary 1.4.
Suppose the -criterion holds for the cn-length function on a discrete group . Then
Q1(ρ,EFixρ)≤C√Ent(ρ)
and
Qϕ(ρ,EFixρ)≤C′max{√Ent(ρ), Ent(ρ)}
for all -measurable positive operators affiliated to with .
We remark that the constant of order in our Poincaré inequalities is crucial to deduce these entropy bounds as observed in [JZ12, Z13]. A constant of the order , as obtained in Section 4, is not sufficient for such entropy bounds.
Let us now point out the connection of our results to some previous ones. As is well known, the major application of -criterion is to derive Gross’ log-Sobolev inequality (LSI) under some mild condition; see [BE85] and also the lecture notes [GZ] for more details in this direction. However, as observed in [JZ12], this implication is not true in general non-diffusion setting where the sample paths are discontinuous; see e.g. [BGL14, RY] for the definition of classical diffusion semigroups and processes. In particular, the diffusion property is characterized by the Leibnitz rule on the form
(1.5) Γ(fg,h)=fΓ(g,h)+gΓ(f,h)
for smooth functions and in the domain of the generator. On the other hand, we can deduce concentration inequalities directly from -criterion without (1.5); see (1.7) below. Of course we still need a certain regularity condition on the semigroup :
(1.6) ΓA(x,x)∈L1(N) for all x∈Dom(A)∩N
(more precisely, for all by extension).
This condition is introduced in [JRS] to characterize the semigroups which admit a Markov dilation with certain nice properties in analogy to classical diffusion processes. For example the Poisson semigroup on the circle satisfies (1.6), but it is not a classical diffusion semigroup. For a standard semigroup with (1.6), it was proved in [JZ12] that the -criterion implies the following Poincaré type inequalities
(1.7) ∥f−EFix(f)∥p≤Cα−1/2min{√p∥ΓA(f,f)1/2∥∞, p∥ΓA(f,f)1/2∥p}
for all self-adjoint . The obstruction of inequalities like (1.2) in the noncommutative setting was a lack of the good Burkholder inequality with appropriate norms or constants. Indeed, with the help of the optimal Burkholder–Davis–Gundy (BDG) inequality, it was proved that the classical diffusion semigroups satisfy (1.2) under the -criterion; see [JZ12, Theorem 4.9]. This may be regarded as a shortcut of the following implication in the classical diffusion setting
(1.8) Γ2−criterion⇒ log-Sobolev inequality⇒(???).
Here the first implication was due to Bakry–Emery [BE85] and the second was due to Aida–Stroock [AS94].
The optimal classical BDG inequality due to Barlow and Yor [BY82] asserts that
∥Xt∥p≤C√p∥⟨X,X⟩1/2t∥p
for any continuous mean 0 martingale , where is the quadratic variation of . One way to obtain such an inequality in the noncommutative setting is through the Burkholder inequality [JX03]
(1.9) ∥∥∑kdxk∥∥p≤A(p)∥∥(∑kEk−1(dx∗kdxk+dxkdx∗k))1/2∥∥p+B(p)(∑k∥dxk∥pp)1p
where is the martingale difference associated to the martingale and is the conditional expectation. One would expect the best order of is , which is indeed the case in the commutative theory [Pin]. The difficulty in the noncommutative generality can be seen from the fact that if one requires , then the optimal order of in (1.9) is known to be [Ran, JX05], compared to in the commutative theory [Hi90]. This shows that general noncommutative martingales exhibit quite different behaviors from the classical martingales so that may not be true. Although it is still unclear to us whether can be reduced to in the general noncommutative setting, we do resolve an important case of this problem in this paper, which is good enough to establish Theorem 1.2. In this way we improve the main results of [JZ12] for the case of semigroups acting on group von Neumann algebras generated by 1-cocycles. Our proof follows the same strategy as that in [JZ12]. The difficulty mentioned above is overcome by a decoupling argument, which is the key new ingredient (Lemma 3.1) in our proof. We refer the interested reader to the monograph [dG] for various aspects of decoupling and applications.
Let us conclude the introduction by mentioning the relationship among log-Sobolev inequality, spectral gap inequality and Poincaré inequalities. It is well known that the log-Sobolev inequality implies the existence of spectral gap, or equivalently, Poincaré inequality. Conversely, the spectral gap inequality together with a defective log-Sobolev inequality yields the log-Sobolev inequality; see e.g. [GZ] for these facts. On the other hand, the Poincaré inequalities obviously imply the spectral gap inequality. It would be interesting to determine when the converse implication is possible. It is known that in the classical diffusion setting the spectral gap would imply Poincaré inequalities, but with constant ; see e.g. [Mi09, Proposition 2.5]. We show similar results in the noncommutative setting under some conditions. In Section 4, we formulate certain results in this direction and prove, e.g., the following:
###### Theorem 1.5.
Let be an ergodic standard semigroup acting on a diffuse probability space which satisfies (4.2). Suppose the generator of has a spectral gap: For ,
∥f−τ(f)∥2≤Cmax{∥Γ(f,f)1/2∥2,∥Γ(f∗,f∗)1/2∥2}.
Then for all even integer and all ,
∥f−τ(f)∥p≤C′pmax{∥Γ(f,f)1/2∥p,∥Γ(f∗,f∗)1/2∥p}.
We overcome the lack of (1.5) in the noncommutative setting by using the derivations of noncommutative Dirichlet forms developed by Cipriani and Sauvageot [JRS] and the regularity theorem due to Olkiewicz–Zegarlinski [OZ99].
The paper is organized as follows. We recall some preliminary facts in Section 2. Then we prove the Poincaré inequalities with constant in Section 3. The relationship between the spectral gap inequality and Poincaré inequalities is discussed in Section 4. Some examples and illustrations are given in Section 5.
## 2. Preliminaries
### 2.1. Crossed products
We briefly recall the crossed product construction. Our reference is [Tak, JMP]. Let be a discrete group with left regular representation . Given a noncommutative probability space , we may assume for some Hilbert space . Suppose a trace preserving action of on is given, i.e., we have a group homomorphism (the -automorphism groups of ) with for all . Identify with . Consider the representation of on given by , where is the matrix unit of . In other words, for . Then the crossed product of by , denoted by , is defined as the weak operator closure of and in . We usually drop the subscript if there is no ambiguity. Clearly, is a von Neumann subalgebra of . In the special case , the complex number algebra, reduces to the group von Neumann algebra . Therefore, is a von Neumann subalgebra of and there exists a unique conditional expectation . If , we simply write or even for . A generic element of can be written as
∑g∈Gfg⋊λ(g) =∑g∈Gπ(fg)λ(g)=∑g,h,h′(αh−1(fg)⊗eh,h)(1N⊗egh′,h′) =∑g,hαh−1(fg)⊗eh,g−1h.
There is a canonical trace on given by
τ⋊τG(f⋊λ(g))=τ⊗τG(f⊗λ(g))=τ(f)δg=e,
where we denote by the canonical trace on . The arithmetic in is given by and . In what follows, we may simply write instead of .
### 2.2. 1-cocycles on groups
Let be a countable discrete group with a conditionally negative length (cn-length) function . Recall that is a length function if and , and is conditionally negative if . Then determines an affine representation which is given by an orthogonal representation over a real Hilbert space together with a map satisfying the cocycle law, i.e., ; see e.g. [BO]. To be more concrete, let be the algebraic group algebra of . Put for and define
[∑gagδg,∑g′ag′δg′]=∑g,g′ag¯ag′K(g,g′).
Then is the closure of the quotient of by the kernel of , i.e., where is the kernel of . Define for and . In this way, we obtain a 1-cocycle . Conversely, suppose that is a 1-cocycle with coefficients in an orthogonal representation of . Put for . Then is a cn-length function on . By a Gram–Schmidt procedure, we may choose an orthonormal basis of so that depends on only finitely many nonzero coordinates for all . This observation will save us from some technical problems. We write even if .
### 2.3. Gaussian measure space construction
Note that the Hilbert space is separable. By the well known Gaussian space construction (see e.g. [RY, Str]), there exists a probability space and a linear map
β:L2([0,∞))⊗Hψ→L2(Ω,P)
such that is Gaussian centered and
E[β(1[0,t]⊗ξ)β(1[0,s]⊗η)]=2⟨1[0,t]⊗ξ,1[0,s]⊗η⟩L2([0,∞))⊗Hψ=2min{t,s}⟨ξ,η⟩Hψ.
We simply write and denote by the -subalgebra of generated by , for all and . By Kolmogorov’s continuity criterion (see, e.g., [RY, Theorem I.2.1]), thus constructed is a -valued Brownian motion, where is viewed as an abstract Wiener space associated to if . Indeed, by construction the -th component of is a 1-dimensional Brownian motion with mean 0 and variance , where is the -th component of , and all the components of are independent. More explicitly, we can simply take . Then , where is the -th coordinate map at time . It is readily seen that is a random variable in with variance . Suppose is an orthogonal representation of on . By [Str, Theorem 8.3.14], determines a Gaussian measure preserving action on . By abuse of notation, we still denote by . The -action on induces an action on , such that . It follows that
(2.1) ^αg(f)(ω)=f(αg−1(ω))
for , where . Clearly, extends naturally to isometric actions on for . In the following we will consider the von Neumann algebra and simply omit the subscript in the notation. To conclude this section, we remark that although (and thus ) may be infinitely dimensional, is always a finite dimensional Brownian motion for all because only depends on finitely many nonzero coordinates.
### 2.4. Hardy spaces associated to martingales
We refer to [JM10, JP] for this subsection. Let be a filtration with conditional expectations . Recall that a sequence is a martingale if and . Let be the associated martingale differences. We need the conditional Hardy spaces associated to martingales given as follows. For , define
∥x∥hdp=(∑k∥dxk∥pp)1/p,∥x∥hcp=∥∥ ∥∥(∑kEk−1(dx∗kdxk))1/2∥∥ ∥∥p,
and .
We are going to use the continuous filtration in the following. Recall that a martingale is said to have almost uniform (or a.u. for short) continuous path if for every , every there exists a projection with such that the function given by is norm continuous. Let be a partition of the interval and its cardinality. Put
∥x∥hcp([0,T];σ)=∥∥|σ|−1∑j=0Esj|Esj+1x−Esjx|2∥∥1/2p/2,2≤p≤∞,
∥x∥hdp([0,T];σ)=(|σ|−1∑j=0∥Esj+1x−Esjx∥pp)1/p,2≤p<∞,
and . Let be an ultrafilter refining the natural order given by inclusion on the set of all partitions of . Let . For , we define
⟨x,x⟩T=limσ,U|σ|−1∑i=0Esi|Esi+1x−Esix|2.
Here the limit is taken in the weak* topology and it is shown in [JKPX] that the convergence is also true in norm for all . We define the continuous version of norms for ,
∥x∥hcp([0,T])=limσ,U∥x∥hcp([0,T];σ),
∥x∥hdp([0,T])=limσ,U∥x∥hdp([0,T];σ).
and for . Then for all
∥x∥hcp([0,T])=∥⟨x,x⟩T∥1/2p/2.
A martingale is said to be of vanishing variation if for all and all . If has a.u. continuous path, then it is of vanishing variation. In the following, we will apply these results to matrix-valued martingales driven by Brownian motions. Hence they automatically have almost uniform continuous paths.
## 3. Lp Poincaré inequalities for group von Neumann algebras
Consider the semigroup acting on given by , where is a conditionally negative length function on ; see Section 2.2. satisfies (1.6). For a proof of this fact, see [JZ12]. According to [JRS], admits a Markov dilation with almost uniformly continuous path. We refer the reader to [JRS, JZ12] for the precise definition. In fact, we can write down the dilation explicitly in our setting. Following the notation of Section 2.1 and 2.3, we define
πt:LG→L∞(Ω,Ft)⋊G,πt(λ(g))=eiβt(bψ(g))(ω)⋊λ(g).
The Markov property can be checked directly because
Es(πtλ(g))=πs(Tt−sλ(g))
for and , and the same equation holds for arbitrary by linearity and density. Here is the conditional expectation. It follows that
(3.1) mt(x)=πt(x)−π0(x)+∫t0πs(Ax)ds
is a martingale with almost uniformly continuous path for . We will need the reversed martingale. To this end, let us fix a large constant , and define
vt(x)=πtTL−tx
for . It is easy to check that is a martingale.
For with finitely many nonzero coordinates, we write , where are independent Brownian motions with variance , and can be given by in the notation of Section 2.3. By Ito’s formula,
eiβt(ξ)=1+i∑kξk∫t0eiβs(ξ)dBks−∥ξ∥2∫t0eiβs(ξ)ds.
It follows that
(3.2) πt(λ(g))=λ(g)+i∑kbψ(g)k(∫t0eiβs(bψ(g))dBks)⋊λ(g)−∫t0πs(A(λ(g)))ds,
where is the -th coordinate of . Combining (3.1) and (3.2), we have
mt(λ(g))=i∑kbψ(g)k(∫t0eiβs(bψ(g))dBks)⋊λ(g).
Note that . By Ito’s formula, we have
vt(λ(g))=v0(λ(g))+i∑kbψ(g)k(∫t0e−(L−s)ψ(g)eiβs(bψ(g))dBks)⋊λ(g).
It follows that
πL(λ(g))−π0(TLλ(g))=∫L0dvs(λ(g)) =i∑kbψ(g)k(∫L0e−(L−s)ψ(g)eiβs(bψ(g))dBks)⋊λ(g).
Let be a finite sum. Then
(3.3) πL(x)−TL(x)=i∑g∈G,k∈Nxgbψ(g)k(∫L0e−(L−s)ψ(g)eiβs(bψ(g))dBks)⋊λ(g).
We consider the discretized stochastic integral (assuming ), or martingale transform
(3.4) Mn(x)=i∑g∈G,j∈Nn−1∑k=0xgbψ(g)je−(L−tk)ψ(g)[eiβtk(bψ(g))dBjtk]⋊λ(g),
where . It is well known that this martingale converges to the stochastic integral in for . Indeed, the stochastic integral can be defined as the limit of a certain martingale transform; see e.g. [KS91]. Similar argument can be applied to the case of . We need a precise Burkholder inequality for this (noncommutative) martingale in order to derive the subgaussian property. As explained in Introduction (see also [JZ12]), however, the upper bounds in known inequalities can only result in the inequality (1.7). Our approach here relies on the decoupling technique thanks to the special structure in the martingale transform.
Let us consider the discrete time martingale given by
(3.5) xn=n−1∑k=0∑g∈G,j∈N[fjg(βk(g))dBjk+1]⋊λ(g)
where is a continuous function, for any , is a martingale with independent martingale differences . In what follows we will simply write instead of and this always means a finite sum.
###### Lemma 3.1 (Decoupling).
Suppose is measurable with respect to for and and an independent copy of . Then for ,
∥∥n−1∑k=0∑g,ℓ[fℓg(βk(g))dBℓk+1]⋊λ(g)∥∥pp≤4p∥∥n−1∑k=0∑g,ℓ[fℓg(βk(g))d˜Bℓk+1]⋊λ(g)∥∥pp.
###### Proof.
To shorten the notation, we simply write for . Consider independent random selectors
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{}
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# Using DataPackageR
## Purpose
This vignette demonstrates how to use DataPackageR to build a data package.
DataPackageR aims to simplify data package construction.
It provides mechanisms for reproducibly preprocessing and tidying raw data into into documented, versioned, and packaged analysis-ready data sets.
Long-running or computationally intensive data processing can be decoupled from the usual R CMD build process while maintinaing data lineage.
In this vignette we will subset and package the mtcars data set.
## Set up a new data package.
We’ll set up a new data package based on mtcars example in the README. The datapackage_skeleton() API is used to set up a new package. The user needs to provide:
• R or Rmd code files that do data processing.
• A list of R object names created by those code files.
• Optionally a path to a directory of raw data (will be copied into the package).
• Optionally a list of additional code files that may be dependencies of your R scripts.
library(DataPackageR)
# Let's reproducibly package up
# the cars in the mtcars dataset
# with speed > 20.
# Our dataset will be called cars_over_20.
# Get the code file that turns the raw data
# to our packaged and processed analysis-ready dataset.
processing_code <-
system.file("extdata",
"tests",
"subsetCars.Rmd",
package = "DataPackageR")
# Create the package framework.
DataPackageR::datapackage_skeleton(name = "mtcars20",
force = TRUE,
code_files = processing_code,
r_object_names = "cars_over_20",
path = tempdir()
#dependencies argument is empty
#raw_data_dir argument is empty.
)
√ Creating 'C:/Users/ehhughes/AppData/Local/Temp/RtmpmOobo5/mtcars20/'
√ Setting active project to 'C:/Users/ehhughes/AppData/Local/Temp/RtmpmOobo5/mtcars20'
√ Creating 'R/'
√ Writing 'DESCRIPTION'
√ Writing 'NAMESPACE'
√ Setting active project to '<no active project>'
√ Setting active project to 'C:/Users/ehhughes/AppData/Local/Temp/RtmpmOobo5/mtcars20'
√ Creating 'data-raw/'
√ Creating 'data/'
√ Creating 'inst/extdata/'
### What’s in the package skeleton structure?
This has created a datapackage source tree named “mtcars20” (in a temporary directory). For a real use case you would pick a path on your filesystem where you could then initialize a new github repository for the package.
The contents of mtcars20 are:
levelName
1 mtcars20
2 ¦--DESCRIPTION
3 ¦--R
5 ¦--data
6 ¦--data-raw
7 ¦ °--subsetCars.Rmd
8 ¦--datapackager.yml
9 °--inst
10 °--extdata
You should fill out the DESCRIPTION file to describe your data package. It contains a new DataVersion string that will be automatically incremented when the data package is built if the packaged data has changed.
The user-provided code files reside in data-raw. They are executed during the data package build process.
### A few words about the YAML config file
A datapackager.yml file is used to configure and control the build process.
The contents are:
configuration:
files:
subsetCars.Rmd:
enabled: yes
objects: cars_over_20
render_root:
tmp: '135981'
The two main pieces of information in the configuration are a list of the files to be processed and the data sets the package will store.
This example packages an R data set named cars_over_20 (the name was passed in to datapackage_skeleton()). It is created by the subsetCars.Rmd file.
The objects must be listed in the yaml configuration file. datapackage_skeleton() ensures this is done for you automatically.
DataPackageR provides an API for modifying this file, so it does not need to be done by hand.
Further information on the contents of the YAML configuration file, and the API are in the YAML Configuration Details
### Where do I put my raw datasets?
Raw data (provided the size is not prohibitive) can be placed in inst/extdata.
The datapackage_skeleton() API has the raw_data_dir argument, which will copy the contents of raw_data_dir (and its subdirectories) into inst/extdata automatically.
In this example we are reading the mtcars data set that is already in memory, rather than from the file system.
### An API to read raw data sets from within an R or Rmd procesing script.
As stated in the README, in order for your processing scripts to be portable, you should not use absolute paths to files. DataPackageR provides an API to point to the data package root directory and the inst/extdata and data subdirectories. These are useful for constructing portable paths in your code to read files from these locations.
For example: to construct a path to a file named “mydata.csv” located in inst/extdata in your data package source tree:
• use DataPackageR::project_extdata_path("mydata.csv") in your R or Rmd file. This would return: e.g., C:/mtcars20/inst/extdata/mydata.csv
Similarly:
• DataPackageR::project_path() constructs a path to the data package root directory. (e.g., C:/mtcars20)
• DataPackageR::project_data_path() constructs a path to the data package data subdirectory. (e.g., C:/mtcars20/data)
Raw data sets that are stored externally (outside the data package source tree) can be constructed relative to the project_path().
If your processing scripts are Rmd files, the usual yaml header for rmarkdown documents should be present.
If you have Rmd files, you can still include a yaml header, but it should be commented with #' and it should be at the top of your R file. For example, a test R file in the DataPackageR package looks as follows:
#'---
#'title: Sample report from R script
#'author: Greg Finak
#'date: August 1, 2018
#'---
data <- runif(100)
This will be converted to an Rmd file with a proper yaml header, which will then be turned into a vignette and indexed in the built package.
## Build the data package.
Once the skeleton framework is set up,
# Run the preprocessing code to build cars_over_20
# and reproducibly enclose it in a package.
dir.create(file.path(tempdir(),"lib"))
DataPackageR:::package_build(file.path(tempdir(),"mtcars20"), install = TRUE, lib = file.path(tempdir(),"lib"))
<U+2714> 1 data set(s) created by subsetCars.Rmd
• cars_over_20
<U+2618> Built all datasets!
Non-interactive NEWS.md file update.
√ Creating 'vignettes/'
√ Creating 'inst/doc/'
First time using roxygen2. Upgrading automatically...
Writing NAMESPACE
Writing mtcars20.Rd
Writing cars_over_20.Rd
checking for file 'C:\Users\ehhughes\AppData\Local\Temp\RtmpmOobo5\mtcars20/DESCRIPTION' ...
v checking for file 'C:\Users\ehhughes\AppData\Local\Temp\RtmpmOobo5\mtcars20/DESCRIPTION' (361ms)
- preparing 'mtcars20':
checking DESCRIPTION meta-information ...
checking DESCRIPTION meta-information ...
v checking DESCRIPTION meta-information
- checking for LF line-endings in source and make files and shell scripts
- checking for empty or unneeded directories
NB: this package now depends on R (>=
NB: this package now depends on R (>= 3.5.0)
WARNING: Added dependency on R >= 3.5.0 because serialized objects in serialize/load version 3 cannot be read in older versions of R. File(s) containing such objects: 'mtcars20/data/cars_over_20.rda'
- building 'mtcars20_1.0.tar.gz'
Next Steps
- Edit the documentation.R file in the package source data-raw subdirectory and update the roxygen markup.
- Rebuild the package documentation with document() .
- Call git init . in the package source root directory.
- git add the package files.
- git commit your new package.
- Set up a github repository for your pacakge.
- Add the github repository as a remote of your local package repository.
- git push your local repository to gitub.
[1] "C:/Users/ehhughes/AppData/Local/Temp/RtmpmOobo5/mtcars20_1.0.tar.gz"
### Documenting your data set changes in NEWS.md
When you build a package in interactive mode, you will be prompted to input text describing the changes to your data package (one line).
These will appear in the NEWS.md file in the following format:
DataVersion: xx.yy.zz
========
A description of your changes to the package
[The rest of the file]
### Why not just use R CMD build?
If the processing script is time consuming or the data set is particularly large, then R CMD build would run the code each time the package is installed. In such cases, raw data may not be available, or the environment to do the data processing may not be set up for each user of the data. DataPackageR decouples data processing from package building/installation for data consumers.
### A log of the build process
DataPackageR uses the futile.logger package to log progress.
If there are errors in the processing, the script will notify you via logging to console and to /private/tmp/Test/inst/extdata/Logfiles/processing.log. Errors should be corrected and the build repeated.
If everything goes smoothly, you will have a new package built in the parent directory.
In this case we have a new package mtcars20_1.0.tar.gz.
### A note about the package source directory after building.
The pacakge source directory changes after the first build.
levelName
1 mtcars20
3 ¦--DESCRIPTION
4 ¦--NAMESPACE
5 ¦--NEWS.md
6 ¦--R
7 ¦ °--mtcars20.R
9 ¦--data
10 ¦ °--cars_over_20.rda
11 ¦--data-raw
12 ¦ ¦--documentation.R
13 ¦ ¦--subsetCars.R
14 ¦ °--subsetCars.Rmd
15 ¦--datapackager.yml
16 ¦--inst
17 ¦ ¦--doc
18 ¦ ¦ ¦--subsetCars.Rmd
19 ¦ ¦ °--subsetCars.html
20 ¦ °--extdata
21 ¦ °--Logfiles
22 ¦ ¦--processing.log
23 ¦ °--subsetCars.html
24 ¦--man
25 ¦ ¦--cars_over_20.Rd
26 ¦ °--mtcars20.Rd
27 °--vignettes
28 °--subsetCars.Rmd
### Update the autogenerated documentation.
After the first build, the R directory contains mtcars.R that has autogenerated roxygen2 markup documentation for the data package and for the packaged data cars_over20.
The processed Rd files can be found in man.
The autogenerated documentation source is in the documentation.R file in data-raw.
You should update this file to properly document your objects. Then rebuild the documentation:
dir.create(file.path(tempdir(),"lib")) # a temporary library directory
Warning in dir.create(file.path(tempdir(), "lib")): 'C:
document(file.path(tempdir(),"mtcars20"), lib = file.path(tempdir(),"lib"))
Updating mtcars20 documentation
[1] TRUE
This is done without reprocessing the data.
#### Dont’ forget to rebuild the package.
You should update the documentation in R/mtcars.R, then call package_build() again.
## Installing and using the new data package
### Accessing vignettes, data sets, and data set documentation.
The package source also contains files in the vignettes and inst/doc directories that provide a log of the data processing.
When the package is installed, these will be accessible via the vignette() API.
The vignette will detail the processing performed by the subsetCars.Rmd processing script.
The data set documentation will be accessible via ?cars_over_20, and the data sets via data().
# create a temporary library to install into.
dir.create(file.path(tempdir(),"lib"))
Warning in dir.create(file.path(tempdir(), "lib")): 'C:
# Let's use the package we just created.
install.packages(file.path(tempdir(),"mtcars20_1.0.tar.gz"), type = "source", repos = NULL, lib = file.path(tempdir(),"lib"))
if (!"package:mtcars20"%in%search())
attachNamespace(lns('mtcars20',lib.loc = file.path(tempdir(),"lib"))) #use library() in your code
cars_over_20 # now we can use it.
speed dist
44 22 66
45 23 54
46 24 70
47 24 92
48 24 93
49 24 120
50 25 85
?cars_over_20 # See the documentation you wrote in data-raw/documentation.R.
vignettes <- vignette(package = "mtcars20", lib.loc = file.path(tempdir(),"lib"))
vignettes\$results
Package LibPath
Topic "mtcars20" "C:/Users/ehhughes/AppData/Local/Temp/RtmpmOobo5/lib"
Item Title
Topic "subsetCars" "A Test Document for DataPackageR (source, html)"
### Using the DataVersion
Your downstream data analysis can depend on a specific version of the data in your data package by testing the DataVersion string in the DESCRIPTION file.
We provide an API for this:
# We can easily check the version of the data
DataPackageR::data_version("mtcars20", lib.loc = file.path(tempdir(),"lib"))
[1] '0.1.0'
# You can use an assert to check the data version in reports and
# analyses that use the packaged data.
assert_data_version(data_package_name = "mtcars20",
version_string = "0.1.0",
acceptable = "equal",
lib.loc = file.path(tempdir(),"lib")) #If this fails, execution stops
#and provides an informative error.
# Migrating old data packages.
Version 1.12.0 has moved away from controlling the build process using datasets.R and an additional masterfile argument.
The build process is now controlled via a datapackager.yml configuration file located in the package root directory. (see YAML Configuration Details)
### Create a datapackager.yml file
You can migrate an old package by constructing such a config file using the construct_yml_config() API.
# assume I have file1.Rmd and file2.R located in /data-raw,
# and these create 'object1' and 'object2' respectively.
config <- construct_yml_config(code = c("file1.Rmd", "file2.R"),
data = c("object1", "object2"))
cat(yaml::as.yaml(config))
configuration:
files:
file1.Rmd:
enabled: yes
file2.R:
enabled: yes
objects:
- object1
- object2
render_root:
tmp: '289541'
config is a newly constructed yaml configuration object. It can be written to the package directory:
path_to_package <- tempdir() #e.g., if tempdir() was the root of our package.
yml_write(config, path = path_to_package)
Now the package at path_to_package will build with version 1.12.0 or greater.
### Reading data sets from Rmd files
In versions prior to 1.12.1 we would read data sets from inst/extdata in an Rmd script using paths relative to data-raw in the data package source tree.
For example:
#### The old way
# read 'myfile.csv' from inst/extdata relative to data-raw where the Rmd is rendered.
read.csv(file.path("../inst/extdata","myfile.csv"))
Now Rmd and R scripts are processed in render_root defined in the yaml config.
To read a raw data set we can get the path to the package source directory using an API call:
#### The new way
# DataPackageR::project_extdata_path() returns the path to the data package inst/extdata subdirectory directory.
# DataPackageR::project_path() returns the path to the data package root directory.
# DataPackageR::project_data_path() returns the path to the data package data subdirectory directory.
DataPackageR::project_extdata_path("myfile.csv")
)
# Partial builds
We can also perform partial builds of a subset of files in a package by toggling the enabled key in the config file.
This can be done with the following API:
config <- yml_disable_compile(config,filenames = "file2.R")
yml_write(config, path = path_to_package) # write modified yml to the package.
configuration:
files:
file1.Rmd:
enabled: yes
file2.R:
enabled: no
objects:
- object1
- object2
render_root:
tmp: '289541'
Note that the modified configuration needs to be written back to the package source directory in order for the changes to take effect.
The consequence of toggling a file to enable: no is that it will be skipped when the package is rebuilt, but the data will still be retained in the package, and the documentation will not be altered.
This is useful in situations where we have multiple data sets, and want to re-run one script to update a specific data set, but not the other scripts because they may be too time consuming, for example.
# Multi-script pipelines.
We may have situations where we have mutli-script pipelines. There are two ways to share data among scripts.
1. filesystem artifacts
2. data objects passed to subsequent scripts.
### File system artifacts
The yaml configuration property render_root specifies the working directory where scripts will be rendered.
If a script writes files to the working directory, that is where files will appear. These can be read by subsequent scripts.
### Passing data objects to subsequent scripts.
A script (e.g., script2.Rmd) running after script1.Rmd can access a stored data object named script1_dataset created by script1.Rmd by calling
script1_dataset <- DataPackageR::datapackager_object_read("script1_dataset").
Passing of data objects amongst scripts can be turned off via:
package_build(deps = FALSE)
# Next steps
We recommend the following once your package is created.
## Place your package under source control
You now have a data package source tree.
This will let you version control your data processing code, and provide a mechanism for sharing your package with others.
For more details on using git and github with R, there is an excellent guide provided by Jenny Bryan: Happy Git and GitHub for the useR and Hadley Wickham’s book on R packages.
We provide some additional details for the interested.
### Fingerprints of stored data objects
DataPackageR calculates an md5 checksum of each data object it stores, and keeps track of them in a file called DATADIGEST.
• Each time the package is rebuilt, the md5 sums of the new data objects are compared against the DATADIGEST.
• If they don’t match, the build process checks that the DataVersion string has been incremented in the DESCRIPTION file.
• If it has not the build process will exit and produce an error message.
The DATADIGEST file contains the following:
DataVersion: 0.1.0
cars_over_20: 3ccb5b0aaa74fe7cfc0d3ca6ab0b5cf3
#### DESCRIPTION
The description file has the new DataVersion string.
Package: mtcars20
Title: What the Package Does (One Line, Title Case)
Version: 1.0
Authors@R:
person(given = "First",
family = "Last",
role = c("aut", "cre"),
email = "first.last@example.com",
comment = c(ORCID = "YOUR-ORCID-ID"))
Description: What the package does (one paragraph).
License: use_mit_license(), use_gpl3_license() or friends to pick a
VignetteBuilder: knitr
|
{}
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# Stability/decay, are they boolean or not, or does QM probabilities overrule this?
This is not a duplicate, I am not asking whether the proton is a stable particle, or why it is. I am asking about the definition of stability/decay whether it is boolean or not.
How can beta plus decay be possible?
where John Rennie says:
There isn't a potential barrier in beta decay, whether it's beta minus or beta plus decay. In both cases the decay is slow simply because the transition probability is so slow.
and where Emilio Pisanty says in a comment:
Stability is a boolean variable, in QM as well as anywhere. If the system is in an unstable state, then it's in an unstable state, period.
There must be a misunderstanding, because beta minus decay happens, because in QM, it is all about probabilities. In this case, it happens slowly. But the proton rich nucleus is unstable.
Now if the proton rich nucleus is unstable, and it is all about just probabilities as per QM, then everything, every quantum system (composite) is unstable. Period.
One would say that there cannot exist a single quantum system, that would be stable infinitely. Even a stable atom (or a stable proton) will decay eventually. Since even the protons inside it have an average lifetime of 2.1*10^29 years.
https://en.wikipedia.org/wiki/Proton
The free proton (a proton not bound to nucleons or electrons) is a stable particle that has not been observed to break down spontaneously to other particles. The spontaneous decay of free protons has never been observed, and protons are therefore considered stable particles according to the Standard Model. However, some grand unified theories (GUTs) of particle physics predict that proton decay should take place with lifetimes between 10^31 to 10^36 years and experimental searches have established lower bounds on the mean lifetime of a proton for various assumed decay products.[22][23][24] Experiments at the Super-Kamiokande detector in Japan gave lower limits for proton mean lifetime of 6.6×10^33 years for decay to an antimuon and a neutral pion, and 8.2×1033 years for decay to a positron and a neutral pion.[25] Another experiment at the Sudbury Neutrino Observatory in Canada searched for gamma rays resulting from residual nuclei resulting from the decay of a proton from oxygen-16. This experiment was designed to detect decay to any product, and established a lower limit to a proton lifetime of 2.1×10^29 years.
Now the contradictions is where one says that anything that is a composite particle will eventually decay, does have a mean lifetime (even free protons according to GUT but not SM), but stability is a boolean variable.
If stability is a boolean, and everything (composite) does have a mean lifetime, then basically the definition of stability is the same as the definition of elementary vs non-elementary particles.
But here comes the contradiction, there are even elementary particles that are not stable, and some are stable.
Like the electron, is defined as a stable particle, but has a mean lifetime of 6.6*10^26 years.
There are elementary particles that spontaneously decay into less massive particles. An example is the muon, with a mean lifetime of 2.2×10^−6 seconds, which decays into an electron, a muon neutrino and an electron antineutrino. The electron, on the other hand, is thought to be stable on theoretical grounds: the electron is the least massive particle with non-zero electric charge, so its decay would violate charge conservation.[77] The experimental lower bound for the electron's mean lifetime is 6.6×10^28 years, at a 90% confidence level.
https://en.wikipedia.org/wiki/Electron
So the muon is unstable, because we have observed muons to decay, but the electron is stable, because we have never observed one to decay, but the electron does have a mean lifetime, 6.6*10^28 years.
Even as per the SM, the contradiction is there. Do we only say that stability is boolean, because we have never observed the electron and proton (free) to decay, but we give them a mean lifetime?
As per QM, it is all about probabilities, and nothing lasts forever. Even the stable particles ( electron and the proton) will have a mean lifetime.
Does QM probabilities win or does the SM stability (boolean) definition win?
Question:
1. Is stability a boolean, that is, are we defining stability as the particles that have never been observed to decay (electron and free proton), and are we defining unstable the particles that have been already observed to decay?
2. Or are we saying that QM is all about probabilities, and even the stable particles (electron, proton) do have a mean lifetime and will eventually decay?
• You are misreading things. Eg, that quote about electron stability says that we believe that the electron is stable. "The experimental lower bound for the electron's mean lifetime is 6.6×10^28 years, at a 90% confidence level." means that experiments have demonstrated that if the electron is unstable then its mean lifetime is probably greater than $6.6\times19^{28}$ years, and we can be 90% confident in that figure. – PM 2Ring Jul 4 at 12:39
• @PM2Ring "if the electron is unstable". Is this unsurety part of the SM, or is this just a probability as per QM? – Árpád Szendrei Jul 4 at 12:48
• In the Standard Model, the electron is assumed to be completely stable. But it's just a model, and even though it's a very good model, and it's predictions have been tested to high precision, in physics experiment & observation always trumps theory. – PM 2Ring Jul 4 at 13:04
• Mathematicians prove theorems, but physics isn't mathematics. We can prove that free neutrons are unstable, by watching them decay. We can't prove that the electron is stable, but we can do experiments that fail to detect electron decay. To prove that the electron never decays, you'd have to watch every electron in the universe for an infinite timespan, which is obviously impossible. – PM 2Ring Jul 4 at 13:05
• You're attaching too much weight to common phrases. "It's all probabilities" isn't a rigorous statement, it's an intuitive way to summarize some results in quantum mechanics. It does not mean that everything in quantum mechanics fluctuates randomly. For example, in standard quantum mechanics the energy is always exactly conserved. – knzhou Jul 4 at 17:01
Now if the proton rich nucleus is unstable, and it is all about just probabilities as per QM, then everything, every quantum system (composite) is unstable. Period.
That is a truly amazing exercise in non sequitur. To be frank, logic seems to have completely deserted your post in its entirety $$-$$ nothing seems to be logically connected to what comes before or after.
Boolean variable: Any variable, from the domain of Boolean algebra, having one of only two values,
i.e. a variable that can (only!) be "true" or false".
As applied to the stability of a system, saying "stability is a boolean variable" means that for any given system, it is either stable or unstable, period. There's no probabilities involved.
To be crystal clear: this is completely consistent with the existence of systems which are stable. It is also consistent with systems which are unstable. It just means that any state of any system needs to be either stable or unstable, i.e. there's no middle ground.
As applied specifically to nuclei which undergo radioactive decay (of any kind), saying "stability is a boolean variable" means that if a given nucleus is in a state that can undergo radioactive decay, then it must undergo radioactive decay. (In this form, it is a particular subform of the Gell-Mann totalitarian principle.) To be crystal clear:
• The time at which this decay happens is probabilistic $$-$$ we do not know (and cannot know) when it will decay.
• The fact that the system will decay is not probabilistic $$-$$ it is written in stone. If you initialize the system in the unstable state, there will always be a time at which it decays.
An additional misunderstanding in your question, which is basic enough that you should know better:
The experimental lower bound for the electron's mean lifetime is 6.6×10^28 years, at a 90% confidence level.
[...] but the electron does have a mean lifetime, 6.6*10^28 years.
No it doesn't. Lower bounds are not the same thing as values. The result you've quoted means that lifetimes shorter than $$6.6\times 10^{28}$$ years are ruled out, but it does not give a value for the electron lifetime, nor does it even begin to imply that said lifetime exists or is finite. If the electron has a finite lifetime, the lower bound implies that it cannot be shorter than $$6.6\times 10^{28}$$ yr, but the lower bound is perfectly consistent with electrons being stable.
This has nothing to do with QM or the Standard Model. Infinity cannot be accessed by physical experiments, regardless of the arena. Even if every bit of the conceptual framework indicates that a given parameter will be infinite, we are finite beings and all we can do is to perform finite experiments which provide increasingly large (but always finite) lower bounds.
This is an elementary bit of language. If its usage is not clear, then you should go back and take a long, detailed take at introductory maths textbooks before carrying on with QM.
• This answer is correct but it seems a bit hostile, to be honest. I don't think there's any need for that. – Javier Jul 4 at 20:29
• @Javier I think there's a need for a minimal amount of critical thought before posting questions here, particularly for experienced site members. If that bothers others then I would seriously question why they think that critical thought is optional. – Emilio Pisanty Jul 5 at 1:56
• I agree, but given how confusing QM can be, I think we could forgive OP in this instance for asking naive questions. What seems obvious to you is not necessarily so to others. – Javier Jul 5 at 1:58
• @Javier That might be true for a first-time poster, but that is not the case with OP. I am astounded by the disregard for physics, and this site, shown by OP in this question - and even more so by the presence of folks who stand up to defend this objectively-awful question despite the harm it does to this site. This question represents an abuse of this site's mechanisms, there's no other word for it – Emilio Pisanty Jul 5 at 2:10
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HELP
Image text transcribed for accessibility: Velocity of the lifter Find the linear velocity of the lifter with respect to point O when the cam has an angular position of theta = 52degree and is rotating with an angular velocity of omega = 1.0 rad/s. Express your answer to three significant figures and include the appropriate units. Acceleration of the lifter Find the linear acceleration of the lifter with respect to point O when the angle is theta = 52degree, the angular velocity is omega = 1.0 rad/s, and the angular acceleration is alpha = -3.0 rad/s2. Express your answer to three significant figures and include the appropriate units.
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Arnab K. Ray Physics , 2002, DOI: 10.1046/j.1365-8711.2003.06796.x Abstract: The influence of a linearized perturbation on stationary inflow solutions in an inviscid and thin accretion disc, has been studied here, and it has been argued, that a perturbative technique would indicate that all possible classes of inflow solutions would be stable. The choice of the driving potential, Newtonian or pseudo-Newtonian, would not particularly affect the arguments which establish the stability of solutions. It has then been surmised that in the matter of the selection of a particular solution, adoption of a non-perturbative technique, based on a more physical criterion, as in the case of the selection of the transonic solution in spherically symmetric accretion, would give a more conclusive indication about the choice of a particular branch of the flow.
Physics , 2015, Abstract: Our detailed analytic local disc model (JJ-model) quantifies the interrelation between kinematic properties (e.g. velocity dispersions and asymmetric drift), spatial parameters (scale-lengths and vertical density profiles), and properties of stellar sub-populations (age and abundance distributions). Any consistent radial extension of the disc evolution model should predict specific features in the different distribution functions and in their correlations. Large spectroscopic surveys (SEGUE, RAVE, APOGEE, Gaia-ESO) allow significant constraints on the long-term evolution of the thin disc. We discuss the qualitative difference of correlations (like the alpha-enhancement as function of metallicity) and distribution functions (e.g. in [Mg/H] or [Fe/H]) for the construction of a disc model. In the framework of the JJ-model we build a local chemical enrichment model and show that significant vertical gradients for main sequence and red clump stars are expected in the thin disc. A Jeans analysis of the asymmetric drift provides a link to the radial structure of the disc. The derived metallicity-dependent radial scale-lengths can be combined in the future with the abundance distributions at different Galactocentric distances to construct full disc models. We expect to be able to constrain possible scenarios of inside-out growth of the thin disc and to characterise those populations, which require significant radial migration.
Physics , 2011, DOI: 10.1111/j.1365-2966.2011.20339.x Abstract: We combine constraints on the galaxy-dark matter connection with structural and dynamical scaling relations to investigate the angular momentum content of disc galaxies. For haloes with masses in the interval 10^{11.3} < M_vir/M_sun < 10^{12.7} we find that the galaxy spin parameters are independent of halo mass with <\lambda'_gal> = (J_gal/M_gal) / (\sqrt{2} R_vir V_vir) = 0.019^{+0.004}_{-0.003} (1sigma). This is significantly lower than for relaxed LCDM haloes, which have an average spin parameter <\lambda'_halo> = 0.031. The average ratio between the specific angular momentum of disk galaxies and their host dark matter haloes is therefore R_j = \lambda'_gal/\lambda'_halo = 0.61^{+0.13}_{-0.10}. This calls into question a standard assumption made in the majority of all (semi-analytical) models for (disc) galaxy formation, namely that R_j=1. Using simple disc formation models we show that it is particularly challenging to understand why R_j is independent of halo mass, while the galaxy formation efficiency (\epsilon_GF, proportional to the ratio of galaxy mass to halo mass) reveals a strong halo mass dependence. We argue that the empirical scaling relations between \epsilon_GF, R_j and halo mass require both feedback (i.e., galactic outflows) and angular momentum transfer from the baryons to the dark matter (i.e., dynamical friction). The efficiency of angular momentum loss need to decrease with increasing halo mass. Such a mass dependence may reflect a bias against forming stable discs in high mass, low spin haloes or a transition from cold-mode accretion in low mass haloes to hot-mode accretion at the massive end. However, current hydrodynamical simulations of galaxy formation, which should include these processes, seem unable to reproduce the empirical relation between \epsilon_GF and R_j. We conclude that the angular momentum build-up of galactic discs remains poorly understood.
G. I. Ogilvie Physics , 1998, DOI: 10.1046/j.1365-8711.1999.02340.x Abstract: The dynamics of a viscous accretion disc subject to a slowly varying warp of large amplitude is considered. Attention is restricted to discs in which self-gravitation is negligible, and to the generic case in which the resonant wave propagation found in inviscid Keplerian discs does not occur. The equations of fluid dynamics are derived in a coordinate system that follows the principal warping motion of the disc. They are reduced using asymptotic methods for thin discs, and solved to extract the equation governing the warp. In general, this is a wave equation of parabolic type with non-linear dispersion and diffusion, which describes fully non-linear bending waves. This method generalizes the linear theory of Papaloizou & Pringle (1983) to allow for an arbitrary rotation law, and extends it into the non-linear domain, where it connects with a generalized version of the theory of Pringle (1992). The astrophysical implications of this analysis are discussed briefly.
Physics , 2001, Abstract: The presence of an imposed external magnetic field may drastically influence the structure of thin accretion discs. The magnetic field energy is here assumed to be in balance with the thermal energy of the accretion flow. The vertical magnetic field, its toroidal component B^tor at the disc surface (due to different rotation rates between disc and its magnetosphere), the turbulent magnetic Prandtl number and the viscosity-alpha are the key parameters of our model. Inside the corotation radius for rather small B^tor the resulting inclination angle i of the magnetic field lines to the disc surface normal can exceed the critical value 30^\circ (required to launch cold jets) even for small magnetic Prandtl numbers of order unity. The self-consistent consideration of both magnetic field and accretion flow demonstrates a weak dependence of the inclination (dragging'') angle on the magnetic Prandtl number for given surface density but a strong dependence on the toroidal field component at the disc surface. A magnetic disc is thicker than a nonmagnetic one for typical parameter values. The accretion rate can be strongly amplified by large B^tor and small magnetic Prandtl number. On the other hand, for given accretion rate the magnetised disc is less massive than the standard-alpha disc. The surface values of the toroidal magnetic fields which are necessary to induce considerably high values for the inclination angle are much smaller than expected and are of order 10^-3 of the imposed vertical field. As the innermost part of the disc produces the largest B^tor, the largest radial inclination can be expected also there. The idea is therefore supported that the cold jets are launched only in the central disc area.
Takamitsu Tanaka Physics , 2010, DOI: 10.1111/j.1365-2966.2010.17496.x Abstract: We discuss Green's-function solutions of the equation for a geometrically thin, axisymmetric Keplerian accretion disc with a viscosity prescription "\nu ~ R^n". The mathematical problem was solved by Lynden-Bell & Pringle (1974) for the special cases with boundary conditions of zero viscous torque and zero mass flow at the disc center. While it has been widely established that the observational appearance of astrophysical discs depend on the physical size of the central object(s), exact time-dependent solutions with boundary conditions imposed at finite radius have not been published for a general value of the power-law index "n". We derive exact Green's-function solutions that satisfy either a zero-torque or a zero-flux condition at a nonzero inner boundary R_{in}>0, for an arbitrary initial surface density profile. Whereas the viscously dissipated power diverges at the disc center for the previously known solutions with R_{in}=0, the new solutions with R_{in}>0 have finite expressions for the disc luminosity that agree, in the limit t=>infinity, with standard expressions for steady-state disc luminosities. The new solutions are applicable to the evolution of the innermost regions of thin accretion discs.
Physics , 2015, Abstract: Aims: We study the effects of the cosmological assembly history on the chemical and dynamical properties of the discs of spiral galaxies as a function of radius. Methods: We make use of the simulated Milky-Way mass, fully-cosmological discs, from {\tt RaDES} (Ramses Disc Environment Study). We analyse their assembly history by examining the proximity of satellites to the galactic disc, instead of their merger trees, to better gauge which satellites impact the disc. We present stellar age and metallicity profiles, Age-Metallicity Relation (AMR), Age-Velocity dispersion Relation (AVR), and Stellar Age Distribution (SAD) in several radial bins for the simulated galaxies. Results: Assembly histories can be divided into three different stages: i) a merger dominated phase, when a large number of mergers with mass ratios of $\sim$1:1 take place (lasting $\sim$3.2$\pm$0.4 Gyr on average); ii) a quieter phase, when $\sim$1:10 mergers take place (lasting $\sim$4.4$\pm$2.0 Gyr) - these two phases are able to kinematically heat the disc and produce a disc that is chemically mixed over its entire radial extension; iii) a "secular" phase where the few mergers that take place have mass ratios below 1:100, and which do not affect the disc properties (lasting $\sim$5.5$\pm$2.0 Gyr). Phase ii ends with a final merger event (at time $t_\mathrm{jump}$) marking the onset of important radial differences in the AMR, AVR, and SAD. Conclusions: Inverted AMR trends in the outer parts of discs, for stars younger than $t_\mathrm{jump}$, are found as the combined effect of radial motions and star formation in satellites temporarily located in these outer parts. "U-shaped" stellar age profiles change to an old plateau ($\sim$10 Gyr) in the outer discs for the entire {\tt RaDES} sample. This shape is a consequence of inside-out growth of the disc, radial motions of disc stars ... [abridged]
Physics , 2000, DOI: 10.1046/j.1365-8711.2001.04133.x Abstract: Gas falling quasi-spherically onto a black hole forms an inner accretion disc if its specific angular momentum $l$ exceeds $\lmin\sim r_gc$ where $r_g$ is the Schwarzschild radius. The standard disc model assumes $l\gg\lmin$. We argue that, in many black-hole sources, accretion flows have angular momenta just above the threshold for disc formation, $l\simgt\lmin$, and assess the accretion mechanism in this regime. In a range \$\lmin
Physics , 2009, DOI: 10.1088/0264-9381/27/10/105010 Abstract: A promising extension of general relativity is Chern-Simons (CS) modified gravity, in which the Einstein-Hilbert action is modified by adding a parity-violating CS term, which couples to gravity via a scalar field. In this work, we consider the interesting, yet relatively unexplored, dynamical formulation of CS modified gravity, where the CS coupling field is treated as a dynamical field, endowed with its own stress-energy tensor and evolution equation. We consider the possibility of observationally testing dynamical CS modified gravity by using the accretion disk properties around slowly-rotating black holes. The energy flux, temperature distribution, the emission spectrum as well as the energy conversion efficiency are obtained, and compared to the standard general relativistic Kerr solution. It is shown that the Kerr black hole provide a more efficient engine for the transformation of the energy of the accreting mass into radiation than their slowly-rotating counterparts in CS modified gravity. Specific signatures appear in the electromagnetic spectrum, thus leading to the possibility of directly testing CS modified gravity by using astrophysical observations of the emission spectra from accretion disks.
Physics , 1997, DOI: 10.1046/j.1365-8711.1998.01213.x Abstract: Long-term evolution of a stellar orbit captured by a massive galactic center via successive interactions with an accretion disc has been examined. An analytical solution describing evolution of the stellar orbital parameters during the initial stage of the capture was found. Our results are applicable to thin Keplerian discs with an arbitrary radial distribution of density and rather general prescription for the star-disc interaction. Temporal evolution is given in the form of quadrature which can be carried out numerically.
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# Graphical user interfaces (GUI)¶
Qt4 graphical user interface for the controllers.
pyhard2.ctrlr.Config(section, title='Controller', description='pyhard2 GUI configuration')
Handle command line arguments and configuration files to initialize Controllers.
Command line arguments:
• -t, --title: The name of the controller.
• -p, --port: The port where the hardware is connected.
• -n, --nodes: A space-separated list of nodes at port.
• -m, --names: A corresponding list of names for the nodes.
• -v, --virtual: Load the virtual offline driver.
• file: The path to a configuration file.
Launching a controller from the command line with:
python pyhard2/ctrlr/MODULE -p COM1 -n 1 2 3 -m first second third
or by pointing to a file containing
MODULE:
COM1:
- node: 1
name: first
- node: 2
name: second
- node: 3
name: third
Example
From the root directory of a working installation of pyhard2, the following line starts a virtual controller with three nodes:
python pyhard2/ctrlr/pfeiffer -v -n 1 2 3 -m gauge1 gauge2
class pyhard2.ctrlr.DashboardConfig(filename)
Extend the config file format described in Config() to launch the Dashboard interface.
The config files are extended with a dashboard section such as
dashboard:
name: Dashboard
image: :/img/gaslines.svg
labels:
- name: LABEL1
pos: [0.25, 0.25]
- name: LABEL2
pos: [0.5, 0.25]
- name: LABEL3
pos: [0.75, 0.25]
Where Dashboard is the name of the window. image: points to an svg file that will be displayed in the background of the window. labels: is a list of text labels containing the text LABEL1, LABEL2, and LABEL3 displayed at the position given by pos: as (x,y) pairs of relative coordinates. x and y can be any value between 0 and 1.
The other nodes also require the pos data in the format specified above, and optional scale and angle data may be passed as well. Such as the previous file may become
MODULE:
COM1:
- node: 1
name: first
pos: [0.25, 0.5]
- node: 2
name: second
pos: [0.5, 0.5]
scale: 0.5
angle: 180
- node: 3
name: third
pos: [0.75, 0.5]
• pos gives the position as [x, y] pairs of relative coordinates (x and y are values between 0 and 1) for the widget of the corresponding node.
• scale scales the widget by the given amount.
• angle rotates the widget by the given amount, in degree.
Example
From the root directory of a working installation of pyhard2, the following line starts a dashboard containing virtual instruments:
python pyhard2.ctrlr.__init__.py circat.yml -v
class pyhard2.ctrlr.ScientificSpinBox(parent=None)
QDoubleSpinBox with a scientific display.
textFromValue(value)
Return the formatted value.
valueFromText(text)
Return the text as a float.
class pyhard2.ctrlr.ListData
Custom QwtData mapping a list onto x,y values.
__len__()
Return length of data.
__iter__()
Iterate on the data.
__getitem__(i)
Return x,y values at i.
sample(i)
Return x,y values at i.
copy()
Return self.
historySize()
How many points of history to display after exportAndTrim.
setHistorySize(historySize)
Set how many points of history to display after exportAndTrim.
x(i)
Return x value.
y(i)
Return y value.
append(xy)
Add x,y values to the data.
Does nothing if None is in xy.
clear()
Clear the data in place.
exportAndTrim(csvfile)
Export the data to csvfile and trim it.
The data acquired since the previous call is saved to csvfile and historySize points are kept. The rest of the data is deleted.
class pyhard2.ctrlr.TimeSeriesData
A ListData to plot values against time.
Note
The time is set to zero upon instantiation.
append(value)
Append time, value to the list.
class pyhard2.ctrlr.ProfileData(rootItem)
Custom QwtData handling a QStandardItemModel with two columns as x, y values.
Parameters: rootItem (QStandardItem) – The root item of the model.
__iter__()
Iterate on the model.
copy()
Return self.
size()
Return the number of rows under rootItem.
Note
Rows where values cannot be converted to float are not counted.
sample(i)
Return x,y values at i.
Raises: IndexError – if i is invalid. ValueError – if the data at i do not convert to float.
x(i)
Return x value at i.
sample() for the exceptions.
y(i)
Return y value at i.
sample() for the exceptions.
class pyhard2.ctrlr.PlotZoomer(canvas)
QwtPlotZoomer for zooming on QwtPlot.
clearZoomStack()
Force autoscaling and clear the zoom stack.
class pyhard2.ctrlr.SingleShotProgram
Program that sends new values at predefined times.
Note
The program has its own timer so that the events will fire precisely at the given times.
started
The signal is emitted when the program starts.
finished
The signal is emitted when the program has finished executing.
value
The signal is emitted with the value generated by the program.
interval()
setInterval(msec)
This property holds the timeout interval in milliseconds.
profile()
Return the profile.
setProfile(profile)
Set the profile to profile.
isRunning()
Returns whether the program is running.
start()
Start or restart the program.
stop()
Stop the program.
class pyhard2.ctrlr.SetpointRampProgram
Program that performs setpoint ramps.
setProfile(profile)
Set the profile to profile.
class pyhard2.ctrlr.DoubleClickEventFilter(parent)
Emit doubleClicked signal on MouseButtonDblClick event.
class pyhard2.ctrlr.FormatTextDelegate(format='%.2f', parent=None)
QStyledItemDelegate formatting the text displayed.
class pyhard2.ctrlr.DoubleSpinBoxDelegate(spinBox=None, parent=None)
Item delegate for editing models with a spin box.
Every property of the spin box can be set on the delegate with the methods from QDoubleSpinBox.
Parameters: spinBox – A spin box prototype to use for editing, defaults to QDoubleSpinBox if not given.
decimals()
setDecimals(prec)
This property holds the precision of the spin box, in decimals.
minimum()
setMinimum(min)
This property holds the minimum value of the spin box.
maximum()
setMaximum(max)
This property holds the maximum value of the spin box.
setRange(minimum, maximum)
Convenience function to set the minimum and maximum values with a single function call.
singleStep()
setSingleStep(val)
This property holds the step value.
prefix()
setPrefix(prefix)
This property holds the spin box’s prefix.
suffix()
setSuffix(suffix)
This property holds the spin box’s suffix.
__getattr__(name)
Set properties on the spin box.
createEditor(parent, option, index)
Return a QDoubleSpinBox.
setEditorData(spinBox, index)
Set spin box value to index.data().
setModelData(spinBox, model, index)
Set model data to spinBox.value().
class pyhard2.ctrlr.ItemRangedSpinBoxDelegate(spinBox=None, parent=None)
Item delegate for editing models in a spin box.
Every property of the spin box can be set on the delegate with the methods from QDoubleSpinBox.
The minimum and maximum properties are set from the item to be edited.
Inherits DoubleSpinBoxDelegate.
createEditor(parent, option, index)
Return a QDoubleSpinBox.
• The minimum property of the spin box is set to item.minimum() if this value has been set.
• The maximum property of the spin box is set to item.maximum() if this value has been set.
class pyhard2.ctrlr.ButtonDelegate(button=None, parent=None)
Item delegate for editing models with a button.
Parameters: button – The button prototype to use for editing, defaults to QPushButton if not given.
editorEvent(event, model, option, index)
Change the state of the editor and the data in the model when the user presses the left mouse button, Key_Space or Key_Select iff the cell is editable.
class pyhard2.ctrlr.ItemSelectionModel(model, parent=None)
QItemSelectionModel with copy/paste and a part of the QTableWidget interface.
currentRow()
Return the row of the current item.
currentColumn()
Return the column of the current item.
insertRow()
Insert an empty row into the table at the current row.
insertColumn()
Insert an empty column into the table at the current column.
removeRows()
Remove the selected rows.
removeColumns()
Remove the selected columns.
copy()
Copy the values in the selection to the clipboard.
paste()
Paste values in the clipboard at the current item.
Raises: IndexError – if the data in the clipboard does not fit in the model.
Horizontal header item for the driver view.
type()
Return QStandardItem.UserType.
clone()
Reimplemented from QStandardItem.
command()
Return the command for this column.
setCommand(command)
Set the command for this column to command.
defaultPollingState()
Return True if the column defaults to polling; otherwise return False.
setDefaultPollingState(state)
Set the default polling state for this column to state.
defaultLoggingState()
Return True if the column defaults to logging; otherwise return False.
setDefaultLoggingState(state)
Set the default logging state for this column to state.
Item to use in vertical header of the driver model.
type()
Return QtGui.QStandardItem.UserType.
clone()
Reimplemented from QtGui.QStandardItem.
node()
Return the node for this row.
setNode(node)
Set the node for this row to node.
class pyhard2.ctrlr.SignalProxy(parent=None)
Proxy class for Qt4 signals.
SignalProxy can be used in place of a Signal in classes that do not inherit QObject.
class pyhard2.ctrlr.DriverItem
QStandardItem handling communication with the driver.
type()
Return QtGui.QStandardItem.UserType.
clone()
Reimplemented from QStandardItem.
isPolling()
Return True if polling is enabled for this item; otherwise return False.
setPolling(state)
Set polling to state for this item.
isLogging()
Return True if logging is enabled for this item; otherwise return False.
setLogging(state)
Set logging to state for this item.
command()
Return the Command object for this column.
node()
Return the node for this row, if any.
minimum()
Return the minimum value for this item.
The value is read in the driver.
maximum()
Return the maximum value for this item.
The value is read in the driver
Return the read only value for this item.
The value is read in the driver.
queryData()
Request reading data in the driver.
Note
• The query is constructed from the Command set on this item’s column and the node (if any) set for this item’s row.
• If the driver raises a HardwareError, the error is logged but execution continues.
data(role=0)
Default to Qt.DisplayRole.
setData(value, role=2)
Default to Qt.EditRole.
class pyhard2.ctrlr.DriverModel(driver, parent=None)
Model to handle the driver.
__iter__()
Iterate on items.
driver()
Return the driver for this model.
node(row)
Return node for row, if any.
Add node as a new row with label.
Add command as a new column with label.
populate()
Fill the model with DriverItem.
closeEvent(event)
Let the driver thread exit cleanly.
class pyhard2.ctrlr.ControllerUi(uifile=None)
QMainWindow for the controllers.
This class loads uifile if one is provided or defaults to the ui/controller.ui designer file and sets UI properties that are not accessible in designer.
setDataPlotLogScale(state)
Change the scale of dataPlot to log or linear.
class pyhard2.ctrlr.Controller(config, driver, uifile='', parent=None)
Implement the behavior of the GUI.
windowTitle()
setWindowTitle(title)
This property holds the window title (caption).
show()
Show the widget and its child widgets.
close()
Close the widget.
isColumnHidden()
Return True if the given column is hidden; otherwise return False.
setColumnHidden()
if hide is True, the given column will be hidden; otherwise it will be shown.
autoSave()
Export the data in the dataPlot to an archive.
startProgram(row)
Start the program at row.
startAllPrograms()
Start or restart every program.
stopAllPrograms()
Stop every program.
classmethod virtualInstrumentController(config, driver)
Initialize controller for the virtual instrument driver.
addCommand(command, label='', hide=False, poll=False, log=False, specialColumn='')
Add command as a new column in the driver table.
Parameters: hide (bool) – Hide the column. poll (bool) – Set the default polling state. log – Set the default logging state.
Add node as a new row in the driver table.
programmableColumn()
Return the index of the programmable column.
setProgrammableColumn(column)
Set the programmable column to column.
setPidPColumn(column)
Set the pid P column to column.
setPidIColumn(column)
Set the pid I column to column.
setPidDColumn(column)
Set the pid D column to column.
populate()
Populate the driver table.
class pyhard2.ctrlr.DashboardUi(uifile='')
QMainWindow for the dashboard.
This class loads uifile if one is provided or default to the ui/dashboard.ui designer file.
class pyhard2.ctrlr.Dashboard(uifile='', parent=None)
Implement the behavior of the GUI.
windowTitle()
setWindowTitle(title)
This property holds the window title (caption).
show()
Show the widget and its child widgets.
close()
Close the widget.
setBackgroundItem(backgroundItem)
Set the SVG image to use as a background.
Add the text to the scene.
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# How can I set conditional formatting on particular cells that depend on another cell's value in Google Spreadsheets?
I would like to colour the background of a row in Google Spreadsheets:
For example:
• If cell Dx = title 'To do' or 'title 'Done' then cells Fx to Nx are filled colour black.
• If cell Dx = title 'closed' the font colour of cell Ax to Nx turns green.
I can do this in Excel quite easily...
-
As far as I know, such functionality has not been implemented in Google Spreadsheets. The only conditional formatting you can do is on the same cell with the value. – Al E. Sep 13 '12 at 16:29
– Caltor May 9 '13 at 13:16
Where x = 1, in New Google Sheets in Custom formula is please try:
=or($D1="To do",$D1="Done")
check Background:, choose black and for Range: enter F1:N1
add another rule with Custom formula is
=\$D1="closed"
check Background:, choose green and for range: enter A1:N1.
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I wrote this web app - Color Code+ to cover most of the basic cases.
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The example in your app, isn't editable. This way people can't see the result of your code: goo.gl/CFg61 – Jacob Jan Tuinstra Feb 6 '13 at 7:29
You have to make your own copy of it and then you can access the code. – Bryan P Feb 6 '13 at 13:24
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# zbMATH — the first resource for mathematics
Ordered and delayed adversaries and how to work against them on a shared channel. (English) Zbl 1452.68270
Summary: An execution of a distributed algorithm is often seen as a game between the algorithm and a conceptual adversary causing specific distractions to the computation. In this work we define a class of ordered adaptive adversaries, which cause distractions – in particular crashes – online according to some partial order of the participating stations, which is fixed by the adversary before the execution. We distinguish: Linearly-Ordered adversary, restricted by some pre-defined linear order of (potentially) crashing stations; Anti-Chain-Ordered adversary, previously known as the Weakly-Adaptive adversary, which is restricted by some pre-defined set of crash-prone stations (it can be seen as an ordered adversary with the order being an anti-chain, i.e., a collection of incomparable elements, consisting of these stations); $$k$$-Thick-Ordered adversary restricted by partial orders of stations with a maximum anti-chain of size $$k$$. We initiate a study of how they affect performance of algorithms. For this purpose, we focus on the well-known Do-All problem of performing $$t$$ tasks by $$p$$ synchronous crash-prone stations communicating on a shared channel. The channel restricts communication by the fact that no message is delivered to the operational stations if more than one station transmits at the same time. The question addressed in this work is how the ordered adversaries controlling crashes of stations influence work performance, defined as the total number of available processor steps during the whole execution and introduced by P. C. Kanellakis and A. A. Shvartsman [Distrib. Comput. 5, No. 4, 201–217 (1992; Zbl 0744.68060)] in the context of Write-All algorithms. The first presented algorithm solves the Do-All problem with work $${\mathcal{O}}(t+p \sqrt{t}\log p)$$ against the Linearly-Ordered adversary. Surprisingly, the upper bound on performance of this algorithm does not depend on the number of crashes $$f$$ and is close to the absolute lower bound $$\varOmega (t+p\sqrt{t})$$ proved in [B. S. Chlebus et al., Distrib. Comput. 18, No. 6, 435–451 (2006; Zbl 1266.68207)]. Another algorithm is developed against the Weakly-Adaptive adversary. Work done by this algorithm is $$\mathcal{O}(t + p\sqrt{t} + p\min \left\{ p/(p-f),t\right\} \log p ),$$ which is close to the lower bound $$\varOmega (t + p\sqrt{t} + p\min \left\{ p/(p-f),t\right\} )$$ proved in [Chlebus, loc. cit.] and answers the open questions posed there. We generalize this result to the class of $$k$$-Thick-Ordered adversaries, in which case the work of the algorithm is bounded by $$\mathcal{O}(t + p\sqrt{t} + p\min \left\{ p/(p-f),k,t\right\} \log p ).$$ We complement this result by proving the almost matching lower bound $$\varOmega (t + p\sqrt{t} + p\min \{ p/(p-f),k,t\} ).$$ Independently from the results for the ordered adversaries, we consider a class of delayed adaptive adversaries, which could see random choices with some delay. We present an algorithm that works efficiently against the 1-RD adversary, which could see random choices of stations with one round delay, achieving close to optimal $${\mathcal{O}}(t+p \sqrt{t}\log ^2 p)$$ work complexity. This shows that restricting the adversary by not allowing it to react on random decisions immediately makes it significantly weaker, in the sense that there is an algorithm achieving (almost) optimal work performance.
##### MSC:
68W15 Distributed algorithms 68M14 Distributed systems 68M15 Reliability, testing and fault tolerance of networks and computer systems 68P30 Coding and information theory (compaction, compression, models of communication, encoding schemes, etc.) (aspects in computer science) 91A80 Applications of game theory
Full Text:
##### References:
[1] Abramson, N.: Development of the alohanet. IEEE Trans. Inf. Theor. 31(2), 119-123 (2006) · Zbl 0563.94001 [2] Awerbuch, B., Richa, A.W., Scheideler, C.: A jamming-resistant MAC protocol for single-hop wireless networks. In Bazzi, R.A., Patt-Shamir, B., (eds.) Proceedings of the Twenty-Seventh Annual ACM Symposium on Principles of Distributed Computing, PODC 2008, Toronto, Canada, August 18-21, 2008, pp. 45-54. ACM (2008) · Zbl 1301.68041 [3] Bender, M.A., Fineman, J.T., Gilbert, S., Young, M.: How to scale exponential backoff: constant throughput, polylog access attempts, and robustness. In: Krauthgamer, R. (ed.) Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pp. 636-654. SIAM (2016) · Zbl 1410.68066 [4] Brandes, Philipp; Kardas, Marcin; Klonowski, Marek; Pająk, Dominik; Wattenhofer, Roger, Approximating the Size of a Radio Network in Beeping Model, 358-373 (2016), Cham · Zbl 1437.68012 [5] Chen, B., Zhou, Z., Yu, H.: Understanding RFID counting protocols. In: Helal, S., Chandra, R., Kravets, R. (eds.) The 19th Annual International Conference on Mobile Computing and Networking, MobiCom’13, Miami, FL, USA, September 30-October 04, 2013, pp. 291-302. ACM (2013) [6] Chlebus, BS; Pardalos, PM (ed.); Rajasekaran, S. (ed.); Reif, JH (ed.); Rolim, JDP (ed.), Randomized communication in radio networks, a chapter, No. 1 (2001), Dordrecht [7] Chlebus, B.S., De Prisco, R., Shvartsman, A.A.: Performing tasks on synchronous restartable message-passing processors. Distrib. Comput. 14(1), 49-64 (2001) [8] Chlebus, Bogdan S.; Gasieniec, Leszek; Kowalski, Dariusz R.; Shvartsman, Alex A., Bounding Work and Communication in Robust Cooperative Computation, 295-310 (2002), Berlin, Heidelberg · Zbl 1029.68519 [9] Chlebus, B.S., Gołąb, K., Kowalski, D.R.: Broadcasting spanning forests on a multiple access channel. Theory Comput. Syst. 36, 711-733 (2003) · Zbl 1101.68322 [10] Chlebus, B.S., Kowalski, D.R.: Randomization helps to perform independent tasks reliably. Random Struct. Algorithms 24(1), 11-41 (2004) · Zbl 1036.68126 [11] Chlebus, B.S., Kowalski, D.R., Lingas, A.: Performing work in broadcast networks. Distrib. Comput. 18(6), 435-451 (2006) · Zbl 1266.68207 [12] Chlebus, B.S., De Prisco, R., Shvartsman, A.A.: Performing tasks on synchronous restartable message-passing processors. Distrib. Comput. 14(1), 49-64 (2001) [13] Clementi, Andrea E. F.; Monti, Angelo; Silvestri, Riccardo, Optimal F-Reliable Protocols for the Do-All Problem on Single-Hop Wireless Networks, 320-331 (2002), Berlin, Heidelberg · Zbl 1019.68503 [14] De Prisco, R., Mayer, A., Yung, M.: Time-optimal message-efficient work performance in the presence of faults. In: Proceedings of the Thirteenth Annual ACM Symposium on Principles of Distributed Computing, PODC ’94, pp. 161-172. ACM, New York (1994) · Zbl 1373.68090 [15] Dilworth, R.P.: A decomposition theorem for partially ordered sets. Ann. Math. 51(1), 161-166 (1950) · Zbl 0038.02003 [16] Dwork, C., Halpern, J.Y., Waarts, O.: Performing work efficiently in the presence of faults. SIAM J. Comput. 27(5), 1457-1491 (1998) · Zbl 0907.68099 [17] Galil, Z., Mayer, A., Yung, M.: Resolving message complexity of byzantine agreement and beyond. In: Proceedings of the 36th Annual Symposium on Foundations of Computer Science, FOCS ’95, pp. 724. IEEE Computer Society, New York (1995) · Zbl 0938.68658 [18] Gallager, R.G.: A perspective on multiaccess channels. IEEE Trans. Inf. Theory 31, 124-142 (1985) · Zbl 0587.94001 [19] Georgiou, C., Kowalski, D.R., Shvartsman, A.A.: Efficient gossip and robust distributed computation. Theor. Comput. Sci. 347(1-2), 130-166 (2005) · Zbl 1080.68112 [20] Georgiou, C., Shvartsman, A.: Do-All Computing in Distributed Systems: Cooperation in the Presence of Adversity. Springer, Berlin (2008) [21] Georgiou, C., Shvartsman, A.A.: Cooperative task-oriented computing: algorithms and complexity. Synth. Lect. Distrib. Comput. Theory 2(2), 1-167 (2011) [22] Greenberg, A.G., Winograd, S.: A lower bound on the time needed in the worst case to resolve conflicts deterministically in multiple access channels. J. ACM 32(3), 589-596 (1985) · Zbl 0628.68026 [23] Herlihy, M., Shavit, N.: The Art of Multiprocessor Programming. Morgan Kaufmann Publishers Inc., San Francisco (2008) [24] Hoeffding, W.: Probability inequalities for sums of bounded random variables. J. Am. Stat. Assoc. 58(301), 13-30 (1963) · Zbl 0127.10602 [25] Jurdziński, T., Kutyłowski, M., Zatopiański, J.: Efficient algorithms for leader election in radio networks. In: Proceedings of the Twenty-first Annual Symposium on Principles of Distributed Computing, PODC ’02, pp. 51-57. ACM, New York (2002) · Zbl 1292.68013 [26] Kanellakis, P.C., Shvartsman, A.A.: Efficient parallel algorithms can be made robust. Distrib. Comput. 5(4), 201-217 (1992) · Zbl 0744.68060 [27] Klonowski, M., Pajak, D.: Electing a leader in wireless networks quickly despite jamming. In: Blelloch, G.E., Agrawal, K. (eds.) Proceedings of the 27th ACM on Symposium on Parallelism in Algorithms and Architectures, SPAA 2015, pap. 304-312. ACM, Portland (2015) [28] Komlos, J., Greenberg, A.: An asymptotically fast nonadaptive algorithm for conflict resolution in multiple-access channels. IEEE Trans. Inf. Theor. 31(2), 302-306 (2006) · Zbl 0561.94004 [29] Kowalski, D.R.: On selection problem in radio networks. In: Proceedings of the Twenty-fourth Annual ACM Symposium on Principles of Distributed Computing, PODC ’05, pp. 158-166. ACM, New York (2005) · Zbl 1314.68028 [30] Kowalski, D.R., Shvartsman, A.A.: Performing work with asynchronous processors: Message-delay-sensitive bounds. In: Proceedings of the Twenty-second Annual Symposium on Principles of Distributed Computing, PODC ’03, pp. 265-274. ACM, New York (2003) · Zbl 1321.68086 [31] Kushilevitz, E., Mansour, Y.: An $\omega (d\log (n/d))$ ω(dlog(n/d)) lower bound for broadcast in radio networks. SIAM J. Comput. 27(3), 702-712 (1998) · Zbl 0908.68067 [32] Martel, C.U.: Maximum finding on a multiple access broadcast network. Inf. Process. Lett. 52(1), 7-13 (1994) [33] Metcalfe, R.M., Boggs, D.R.: Ethernet: distributed packet switching for local computer networks. Commun. ACM 19(7), 395-404 (1976) [34] Richa, A.W., Scheideler, C., Schmid, S., Zhang, J.: Competitive and fair medium access despite reactive jamming. In: 2011 International Conference on Distributed Computing Systems, ICDCS 2011, June 20-24, 2011, pp. 507-516. IEEE Computer Society, Minneapolis (2011) [35] Richa, A.W., Scheideler, C., Schmid, S., Zhang, J.: An efficient and fair MAC protocol robust to reactive interference. IEEE ACM Trans. Netw. 21(3), 760-771 (2013) [36] Willard, D.E.: Log-logarithmic selection resolution protocols in a multiple access channel. SIAM J. Comput. 15(2), 468-477 (1986) · Zbl 0612.94001
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# surf: Visualization of algebraic curves, algebraic surfaces and hyperplane sections of surfaces¶
## Description¶
surf is a tool to visualize some real algebraic geometry: plane algebraic curves, algebraic surfaces and hyperplane sections of surfaces. surf is script driven and has (optionally) a nifty GUI using the Gtk widget set.
This is used by the Singular Jupyter kernel to produce 3D plots.
GPL version 2 or later
## Upstream Contact¶
http://surf.sourceforge.net (although the project is essentially dead)
## Dependencies¶
• cups (optional)
• GNU flex Version 2.5 or higher
• GTK+ Version 1.2.0 or higher (optional)
• GNU MP(gmp) Version 2 or higher
• lib-tiff
• lib-jpeg
• zlib
• ps2pdf (optional)
This package is “experimental” because not all of these dependencies are packaged with Sage.
experimental
## Version Information¶
package-version.txt:
1.0.6-gcc6
## Equivalent System Packages¶
opensuse:
\$ sudo zypper install surf-alggeo
However, these system packages will not be used for building Sage because spkg-configure.m4 has not been written for this package; see https://trac.sagemath.org/ticket/27330
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zbMATH — the first resource for mathematics
Jiang, Daqing
Compute Distance To:
Author ID: jiang.daqing Published as: Jiang, D.; Jiang, D. Q.; Jiang, Da Qing; Jiang, Da-Qing; Jiang, Daqing External Links: ORCID · ResearchGate · dblp
Documents Indexed: 307 Publications since 1993
all top 5
Co-Authors
11 single-authored 51 Hayat, Tasawar 50 O’Regan, Donal 43 Liu, Qun 40 Al-saedi, Ahmed Eid Salem 33 Agarwal, Ravi P. 29 Shi, Ningzhong 23 Ji, Chunyan 18 Zu, Li 17 Xu, Xiaojie 16 Yuan, Chengjun 14 Li, Xiaoyue 14 Lin, Xiaoning 12 Liu, Huizhao 12 Zhang, Qiumei 12 Zhao, Yanan 10 Lin, Yuguo 10 Wan, Aying 10 Zuo, Wenjie 9 Ahmad, Bashir 9 Chu, Jifeng 8 Gao, Haiyin 8 Weng, Peixuan 7 Weng, Shiyou 7 Zhang, Xinhong 6 Han, Qixing 6 Wang, Junyu 6 Yang, Qingshan 6 Zhang, Lili 6 Zhang, Xiaoying 5 Cao, Zhongwei 5 Wang, Liang 4 Gao, Miaomiao 4 Hou, Xuezhang 4 Hu, Weimin 4 Liu, Zhenwen 4 Mao, Xuerong 4 Wei, Junjie 4 Xia, Peiyan 4 Yang, Ying 3 Ahmad, Bashir 3 Cong, Fuzhong 3 Gao, Wenjie 3 Kong, Lingbin 3 Tian, Yu 3 Wang, Ke 3 Wang, Yan 3 Wen, Xiangdan 3 Yu, Jiajia 2 Bao, Haibo 2 Fan, Meng 2 Fu, Jing 2 Gai, Yongjie 2 Ge, Weigao 2 Gray, Alison J. 2 Li, Haihong 2 Liu, Hong 2 Liu, Songnan 2 Nieto Roig, Juan Jose 2 Pang, Peter Y. H. 2 Qi, Kai 2 Wang, Yan 2 Wei, Jun 2 Yang, Fan 2 Zhou, Qiao-Shu 1 Aying, Wan 1 Bahaer, Guli 1 Cao, Wenjie 1 Chen, Qingmei 1 Ge, Bin 1 He, Ying 1 Hua, Hongtu 1 Jiang, Fuquan 1 Jiang, Xiaobo 1 Jin, Manli 1 Lei, Peidong 1 Li, Haixia 1 Li, Qiuyue 1 Li, Xiang 1 Li, Xiuge 1 Li, Yan 1 Li, Yong 1 Lin, Shan 1 Liu, Liya 1 Liu, Taihui 1 Lju, Qun 1 Lu, Mei 1 Luo, Gexin 1 Lv, Yue 1 Ning, Wenjie 1 Shi, Zhenfeng 1 Song, Mingyu 1 Sun, Weizhi 1 Tian, Yinghui 1 Wang, Dehui 1 Wang, Hong 1 Wang, Kun 1 Wang, Shuai 1 Wang, Xiaowei 1 Wolkowicz, Gail S. K. 1 Yang, Fan ...and 12 more Co-Authors
all top 5
Serials
19 Journal of Mathematical Analysis and Applications 18 Applied Mathematics and Computation 15 Physica A 14 Computers & Mathematics with Applications 11 Applied Mathematics Letters 11 Abstract and Applied Analysis 9 Journal of the Franklin Institute 9 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 9 Stochastic Analysis and Applications 8 Communications in Nonlinear Science and Numerical Simulation 7 Dynamic Systems and Applications 7 Acta Mathematica Scientia. Series B. (English Edition) 7 International Journal of Biomathematics 6 Annales Polonici Mathematici 6 Annals of Differential Equations 5 Applicable Analysis 5 Mathematical Methods in the Applied Sciences 5 Journal of Computational and Applied Mathematics 5 Discrete and Continuous Dynamical Systems. Series B 5 Advances in Difference Equations 4 Acta Mathematicae Applicatae Sinica. English Series 4 Journal of Mathematical Chemistry 4 Nonlinear Analysis. Hybrid Systems 3 Acta Mathematica Sinica 3 Applied Mathematical Modelling 3 Journal of Nonlinear Science 3 Electronic Journal of Differential Equations (EJDE) 3 Pure and Applied Mathematics 3 Discrete and Continuous Dynamical Systems 3 Mathematical Problems in Engineering 3 Acta Mathematica Scientia. Series A. (Chinese Edition) 3 Journal of Jilin University. Science Edition 2 Indian Journal of Pure & Applied Mathematics 2 Automatica 2 Czechoslovak Mathematical Journal 2 Acta Mathematicae Applicatae Sinica 2 Acta Scientiarum Naturalium Universitatis Jilinensis 2 Mathematical and Computer Modelling 2 Journal of Natural Science of Heilongjiang University 2 Journal of Dynamics and Differential Equations 2 Topological Methods in Nonlinear Analysis 2 Applied Mathematics. Series B (English Edition) 2 Kyushu Journal of Mathematics 2 Journal of Applied Analysis 2 Positivity 2 Mathematical Inequalities & Applications 2 Electronic Journal of Qualitative Theory of Differential Equations 2 Nonlinear Analysis. Real World Applications 2 Journal of Applied Mathematics and Computing 2 Boundary Value Problems 1 Bulletin of the Australian Mathematical Society 1 Mathematical Biosciences 1 Chaos, Solitons and Fractals 1 Archivum Mathematicum 1 Demonstratio Mathematica 1 Fasciculi Mathematici 1 Hiroshima Mathematical Journal 1 Journal of Differential Equations 1 Mathematics and Computers in Simulation 1 Results in Mathematics 1 Tohoku Mathematical Journal. Second Series 1 Southeast Asian Bulletin of Mathematics 1 Journal of Mathematics. Wuhan University 1 Journal of Mathematical Research & Exposition 1 Acta Scientiarum Naturalium Universitatis Sunyatseni 1 Chinese Annals of Mathematics. Series A 1 Acta Applicandae Mathematicae 1 Journal of Engineering Mathematics (Xi’an) 1 Northeastern Mathematical Journal 1 Science in China. Series A 1 Journal of Applied Mathematics and Stochastic Analysis 1 Panamerican Mathematical Journal 1 M$$^3$$AS. Mathematical Models & Methods in Applied Sciences 1 International Journal of Computer Mathematics 1 Linear Algebra and its Applications 1 Advances in Mathematical Sciences and Applications 1 Applicationes Mathematicae 1 Journal of Mathematical Sciences (New York) 1 Bulletin of the Belgian Mathematical Society - Simon Stevin 1 Memoirs on Differential Equations and Mathematical Physics 1 Methods and Applications of Analysis 1 Dynamics of Continuous, Discrete and Impulsive Systems 1 Nonlinear Dynamics 1 ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik 1 Chinese Quarterly Journal of Mathematics 1 Chaos 1 Discrete Dynamics in Nature and Society 1 Neliniĭni Kolyvannya 1 Acta Mathematica Sinica. English Series 1 Far East Journal of Mathematical Sciences 1 Methodology and Computing in Applied Probability 1 Dynamics of Continuous, Discrete & Impulsive Systems. Series A. Mathematical Analysis 1 Bulletin of the Malaysian Mathematical Sciences Society. Second Series 1 International Journal of Pure and Applied Mathematics 1 Stochastics and Dynamics 1 Communications on Pure and Applied Analysis 1 Archives of Inequalities and Applications 1 Journal of Nonlinear Science and Applications 1 Journal of Mathematical Research with Applications 1 Journal of Applied Analysis and Computation
all top 5
Fields
230 Ordinary differential equations (34-XX) 151 Biology and other natural sciences (92-XX) 88 Probability theory and stochastic processes (60-XX) 34 Operator theory (47-XX) 17 Difference and functional equations (39-XX) 15 Dynamical systems and ergodic theory (37-XX) 12 Systems theory; control (93-XX) 6 Partial differential equations (35-XX) 4 Integral equations (45-XX) 3 Statistics (62-XX) 3 Numerical analysis (65-XX) 2 Fluid mechanics (76-XX) 1 Combinatorics (05-XX) 1 Linear and multilinear algebra; matrix theory (15-XX) 1 Calculus of variations and optimal control; optimization (49-XX) 1 General topology (54-XX) 1 Computer science (68-XX) 1 Game theory, economics, finance, and other social and behavioral sciences (91-XX)
Citations contained in zbMATH
236 Publications have been cited 3,404 times in 1,899 Documents Cited by Year
Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation. Zbl 1190.34064
Ji, Chunyan; Jiang, Daqing; Shi, Ningzhong
2009
Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation. Zbl 1178.34006
Xu, Xiaojie; Jiang, Daqing; Yuan, Chengjun
2009
Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation. Zbl 1140.60032
Jiang, Daqing; Shi, Ningzhong; Li, Xiaoyue
2008
The positive properties of the Green function for Dirichlet-type boundary value problems of nonlinear fractional differential equations and its application. Zbl 1192.34008
Jiang, Daqing; Yuan, Chengjun
2010
Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching. Zbl 1205.92058
Li, Xiaoyue; Gray, Alison; Jiang, Daqing; Mao, Xuerong
2011
Multiplicity of positive periodic solutions to superlinear repulsive singular equations. Zbl 1074.34048
Jiang, Daqing; Chu, Jifeng; Zhang, Meirong
2005
A note on nonautonomous logistic equation with random perturbation. Zbl 1076.34062
Jiang, Daqing; Shi, Ningzhong
2005
Population dynamical behavior of Lotka-Volterra system under regime switching. Zbl 1173.60020
Li, Xiaoyue; Jiang, Daqing; Mao, Xuerong
2009
Multiple positive solutions of Dirichlet boundary value problems for second order impulsive differential equations. Zbl 1103.34015
Lin, Xiaoning; Jiang, Daqing
2006
The ergodicity and extinction of stochastically perturbed SIR and SEIR epidemic models with saturated incidence. Zbl 1231.92058
Yang, Qingshan; Jiang, Daqing; Shi, Ningzhong; Ji, Chunyan
2012
Existence and global attractivity of positive periodic solutions to periodic $$n$$-species Lotka-Volterra competition systems with several deviating arguments. Zbl 0964.34059
Fan, Meng; Wang, Ke; Jiang, Daqing
1999
Asymptotic behavior of global positive solution to a stochastic SIR model. Zbl 1225.60114
Jiang, Daqing; Yu, Jiajia; Ji, Chunyan; Shi, Ningzhong
2011
On monotone method for first and second order periodic boundary value problems and periodic solutions of functional differential equations. Zbl 1134.34322
Jiang, Daqing; Nieto, Juan J.; Zuo, Wenjie
2004
Qualitative analysis of a stochastic ratio-dependent predator-prey system. Zbl 1229.92076
Ji, Chunyan; Jiang, Daqing; Li, Xiaoyue
2011
Global stability of two-group SIR model with random perturbation. Zbl 1184.34064
Yu, Jiajia; Jiang, Daqing; Shi, Ningzhong
2009
The threshold of a stochastic SIS epidemic model with vaccination. Zbl 1335.92108
Zhao, Yanan; Jiang, Daqing
2014
Dynamics of a multigroup SIR epidemic model with stochastic perturbation. Zbl 1244.93154
Ji, Chunyan; Jiang, Daqing; Yang, Qingshan; Shi, Ningzhong
2012
Threshold behaviour of a stochastic SIR model. Zbl 1428.92109
Ji, Chunyan; Jiang, Daqing
2014
The extinction and persistence of the stochastic SIS epidemic model with vaccination. Zbl 1395.92180
Zhao, Yanan; Jiang, Daqing; O’Regan, Donal
2013
Analysis of autonomous Lotka-Volterra competition systems with random perturbation. Zbl 1258.34099
Jiang, Daqing; Ji, Chunyan; Li, Xiaoyue; O’Regan, Donal
2012
Monotone method for first- and second-order periodic boundary value problems and periodic solutions of functional differential equations. Zbl 1014.34049
Jiang, Daqing; Wei, Junjie
2002
A new existence theory for positive periodic solutions to functional differential equations. Zbl 1073.34082
Wan, Aying; Jiang, Daqing; Xu, Xiaojie
2004
A note on a predator-prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation. Zbl 1216.34040
Ji, Chunyan; Jiang, Daqing; Shi, Ningzhong
2011
Multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces. Zbl 1042.34047
Jiang, Daqing; Chu, Jifeng; O’Regan, Donal; Agarwal, Ravi P.
2003
Upper and lower solution method and a singular boundary value problem for the one-dimensional $$p$$-Laplacian. Zbl 0977.34016
Jiang, Daqing; Gao, Wenjie
2000
Existence of positive periodic solutions to nonautonomous delay differential equations. Zbl 0948.34046
Jiang, Daqing; Wei, Junjie
1999
The threshold of a stochastic SIRS epidemic model with saturated incidence. Zbl 1314.92174
Zhao, Yanan; Jiang, Daqing
2014
The long time behavior of DI SIR epidemic model with stochastic perturbation. Zbl 1194.92053
Jiang, Daqing; Ji, Chunyan; Shi, Ningzhong; Yu, Jiajia
2010
Dynamics of a stochastic density dependent predator-prey system with Beddington-DeAngelis functional response. Zbl 1232.34072
Ji, Chunyan; Jiang, Daqing
2011
Existence of positive periodic solutions for functional-differential equations. Zbl 1012.34068
Wan, Aying; Jiang, Daqing
2002
Multiple positive solutions to systems of nonlinear semipositone fractional differential equations with coupled boundary conditions. Zbl 1340.34041
Yuan, Chengjun; Jiang, Daqing; O’Regan, D.; Agarwal, Ravi
2012
Eigenvalues and discrete boundary value problems for the one-dimensional $$p$$-Laplacian. Zbl 1074.39022
Chu, Jifeng; Jiang, Daqing
2005
Long-time behaviour of a perturbed SIR model by white noise. Zbl 1435.92077
Lin, Yuguo; Jiang, Daqing
2013
The behavior of an SIR epidemic model with stochastic perturbation. Zbl 1272.60035
Ji, Chunyan; Jiang, Daqing; Shi, Ningzhong
2012
Stationary distribution of a stochastic SIS epidemic model with vaccination. Zbl 1395.34064
Lin, Yuguo; Jiang, Daqing; Wang, Shuai
2014
A uniqueness and existence theorem for a singular third-order boundary value problem on $$[0,\infty)$$. Zbl 1021.34020
Jiang, Daqing; Agarwal, R. P.
2002
Positive periodic solutions of functional differential equations and population models. Zbl 1010.34065
Jiang, Daqing; Wei, Junjie; Zhang, Bo
2002
Existence and global attractivity of positive periodic solutions to a logistic growth system with feedback control and deviating arguments. Zbl 1004.34030
Yang, Fan; Jiang, Daqing
2001
Existence and multiplicity of positive periodic solutions to functional differential equations with impulse effects. Zbl 1084.34071
Li, Xiaoyue; Lin, Xiaoning; Jiang, Daqing; Zhang, Xiaoying
2005
Stochastically asymptotically stability of the multi-group SEIR and SIR models with random perturbation. Zbl 1243.92047
Yuan, Chengjun; Jiang, Daqing; O’Regan, Donal; Agarwal, Ravi P.
2012
Stationary distribution of stochastic SIS epidemic model with vaccination under regime switching. Zbl 1343.60095
Zhang, Xinhong; Jiang, Daqing; Alsaedi, Ahmed; Hayat, Tasawar
2016
Existence and uniqueness of solutions for singular fourth-order boundary value problems. Zbl 1107.34307
Lin, Xiaoning; Jiang, Daqing; Li, Xiaoyue
2006
Periodic solution for a non-autonomous Lotka-Volterra predator-prey model with random perturbation. Zbl 1326.34091
Zu, Li; Jiang, Daqing; O’Regan, Donal; Ge, Bin
2015
Nontrivial periodic solution of a stochastic epidemic model with seasonal variation. Zbl 1354.92087
Lin, Yuguo; Jiang, Daqing; Liu, Taihui
2015
Multiple positive solutions to boundary value problems for second-order delay differential equations. Zbl 1011.34055
Jiang, Daqing
2002
Existence of positive solutions for boundary value problem of second-order FDE. Zbl 0942.34058
Weng, Peixuan; Jiang, Daqing
1999
The asymptotic behavior of stochastically perturbed DI SIR epidemic models with saturated incidences. Zbl 1246.93117
Liu, Hong; Yang, Qingshan; Jiang, Daqing
2012
Positive solutions for continuous and discrete boundary value problems to the one-dimension $$p$$-Laplacian. Zbl 1072.34021
Jiang, Daqing; Chu, Jifeng; O’Regan, Donal; Agarwal, R. P.
2004
Existence of positive solutions to second order Neumann boundary value problems. Zbl 0963.34019
Jiang, Daqing; Liu, Huizhao
2000
Long-time behavior of a stochastic SIR model. Zbl 1334.92417
Lin, Yuguo; Jiang, Daqing; Xia, Peiyan
2014
Multiplicity of positive periodic solutions to second order differential equations. Zbl 1096.34518
Chu, Jifeng; Lin, Xiaoning; Jiang, Daqing; O’Regan, Donal; Agarwal, R. P.
2006
A monotone method for constructing extremal solutions to fourth-order periodic boundary value problems. Zbl 1036.34020
Jiang, Daqing; Gao, Wenjie; Wan, Aying
2002
Dynamics of stochastically perturbed SIS epidemic model with vaccination. Zbl 1288.92027
Zhao, Yanan; Jiang, Daqing
2013
Existence, uniqueness, and global stability of positive solutions to the food-limited population model with random perturbation. Zbl 1081.92039
Jiang, Daqing; Shi, Ningzhong; Zhao, Yanan
2005
Singular positone and semipositone boundary value problems of nonlinear fractional differential equations. Zbl 1185.34008
Yuan, Chengjun; Jiang, Daqing; Xu, Xiaojie
2009
Existence, uniqueness, stochastic persistence and global stability of positive solutions of the logistic equation with random perturbation. Zbl 1148.34040
Ji, Chunyan; Jiang, Daqing; Shi, Ningzhong; O’Regan, Donal
2007
The monotone method for Neumann functional differential equations with upper and lower solutions in the reverse order. Zbl 1130.34040
Jiang, Daqing; Yang, Ying; Chu, Jifeng; O’Regan, Donal
2007
A monotone method for constructing extremal solutions to second-order periodic boundary value problems. Zbl 0993.34011
Jiang, Daqing; Fan, Meng; Wan, Aying
2001
Dynamics of a stochastic SIS model with double epidemic diseases driven by Lévy jumps. Zbl 1400.92564
Zhang, Xinhong; Jiang, Daqing; Hayat, Tasawar; Ahmad, Bashir
2017
Persistence and nonpersistence of a nonautonomous stochastic mutualism system. Zbl 1402.92375
Xia, Peiyan; Zheng, Xiaokun; Jiang, Daqing
2013
Multiple positive solutions to singular positone and semipositone Dirichlet-type boundary value problems of nonlinear fractional differential equations. Zbl 1226.34007
Xu, Xiaojie; Jiang, Daqing; Yuan, Chengjun
2011
Existence of positive periodic solutions for Volterra integro-differential equations. Zbl 1035.45003
Jiang, Daqing; Wei, Junjie
2001
Stationary distribution and extinction of a stochastic SIRS epidemic model with standard incidence. Zbl 1400.92521
Liu, Qun; Jiang, Daqing; Shi, Ningzhong; Hayat, Tasawar; Alsaedi, Ahmed
2017
The threshold of a stochastic delayed SIR epidemic model with temporary immunity. Zbl 1400.92516
Liu, Qun; Chen, Qingmei; Jiang, Daqing
2016
A note on asymptotic behaviors of stochastic population model with Allee effect. Zbl 1225.34058
Yang, Qingshan; Jiang, Daqing
2011
Optimal existence theory for single and multiple positive periodic solutions to functional difference equations. Zbl 1068.39009
Jiang, Daqing; O’Regan, Donal; Agarwal, R. P.
2005
Existence and global stability of positive periodic solutions of an $$n$$-species periodic Lotka-Volterra competition system with feedback control and deviating arguments. Zbl 1043.34075
Weng, Peixuan; Jiang, Daqing
2002
Upper and lower solutions method and a singular superlinear boundary value problem for the one-dimensional $$p$$-Laplacian. Zbl 0995.34013
Jiang, Daqing
2001
A unified approach to some two-point, three-point, and four-point boundary value problems with Carathéodory functions. Zbl 0880.34019
Wang, Junyu; Jiang, Daqing
1997
Asymptotic behavior of a stochastic delayed SEIR epidemic model with nonlinear incidence. Zbl 1400.34097
Liu, Qun; Jiang, Daqing; Shi, Ningzhong; Hayat, Tasawar; Alsaedi, Ahmed
2016
Competitive exclusion in a stochastic chemostat model with Holling type II functional response. Zbl 1356.92062
Zhang, Qiumei; Jiang, Daqing
2016
Threshold behavior in a stochastic SIQR epidemic model with standard incidence and regime switching. Zbl 1426.92080
Liu, Qun; Jiang, Daqing; Shi, Ningzhong
2018
Ergodic property of the chemostat: a stochastic model under regime switching and with general response function. Zbl 1379.92032
Wang, Liang; Jiang, Daqing
2018
Existence and uniqueness of solutions for singular $$(k,n-k)$$ conjugate boundary value problems. Zbl 1143.34306
Lin, Xiaoning; Jiang, Daqing; Li, Xiaoyue
2006
Nonresonant singular fourth-order boundary value problems. Zbl 1074.34019
Jiang, Daqing; Liu, Huizhao; Xu, Xiaojie
2005
Singular boundary value problems for the one-dimension $$p$$-Laplacian. Zbl 1019.34022
Jiang, Daqing; Gao, Wenjie
2002
On boundary value problems for singular second-order functional differential equations. Zbl 0952.34053
Jiang, Daqing; Wang, Junyu
2000
Dynamics of a stochastic HIV-1 infection model with logistic growth. Zbl 1400.92502
Jiang, Daqing; Liu, Qun; Shi, Ningzhong; Hayat, Tasawar; Alsaedi, Ahmed; Xia, Peiyan
2017
A note on the stationary distribution of the stochastic chemostat model with general response functions. Zbl 1377.92059
Wang, Liang; Jiang, Daqing
2017
The threshold of a stochastic delayed SIR epidemic model with vaccination. Zbl 1400.92517
Liu, Qun; Jiang, Daqing
2016
The threshold of a stochastic SIRS epidemic model in a population with varying size. Zbl 1333.60148
Gray, Alison; Mao, Xuerong; Jiang, Daqing; Zhao, Yanan
2015
Analysis of a delayed stochastic predator-prey model in a polluted environment. Zbl 1427.92075
Han, Qixing; Jiang, Daqing; Ji, Chunyan
2014
Existence, uniqueness and ergodicity of positive solution of mutualism system with stochastic perturbation. Zbl 1204.34065
Ji, Chunyan; Jiang, Daqing; Liu, Hong; Yang, Qingshan
2010
Existence, uniqueness, and global attractivity of positive solutions and MLE of the parameters to the logistic equation with random perturbation. Zbl 1136.34324
Jiang, Daqing; Zhang, Baoxue; Wang, Dehui; Shi, Ningzhong
2007
Multiplicity positive solutions to periodic problems for first-order impulsive differential equations. Zbl 1132.34023
Zhang, Xiaoying; Li, Xiaoyue; Jiang, Daqing; Wang, Ke
2006
Multiple positive solutions to singular boundary value problems for superlinear higher-order ODEs. Zbl 0976.34019
Jiang, Daqing
2000
On the existence of nonnegative radial solutions for $$p$$-Laplacian elliptic systems. Zbl 0928.34021
Jiang, Daqing; Liu, Huizhao
1999
Analysis of a delayed vaccinated SIR epidemic model with temporary immunity and Lévy jumps. Zbl 1382.92240
Liu, Qun; Jiang, Daqing; Hayat, Tasawar; Ahmad, Bashir
2018
Periodic solutions for a stochastic non-autonomous Holling-Tanner predator-prey system with impulses. Zbl 1350.34042
Zuo, Wenjie; Jiang, Daqing
2016
Stationary distributions of a stochastic SIR model with saturated incidence and its asymptotic stability. Zbl 1340.47091
Lin, Yuguo; Jiang, Daqing; Jin, Manli
2015
Threshold behavior in a stochastic SIS epidemic model with standard incidence. Zbl 1323.92208
Lin, Yuguo; Jiang, Daqing
2014
A new existence theory for positive periodic solutions to functional differential equations with impulse effects. Zbl 1156.34053
Li, Xiaoyue; Zhang, Xiaoying; Jiang, Daqing
2006
Upper and lower solutions method and a superlinear singular boundary value problem. Zbl 1058.34021
Jiang, Daqing
2002
Upper and lower solutions for a superlinear singular boundary value problem. Zbl 0991.34018
Jiang, Daqing
2001
Existence of positive solutions for boundary value problems of second-order functional-differential equations. Zbl 0907.34048
Jiang, Daqing; Weng, Peixuan
1998
A generalized periodic boundary value problem for the one-dimensional $$p$$-Laplacian. Zbl 0868.34015
Jiang, Daqing; Wang, Junyu
1997
Dynamics of a stochastic predator-prey model with stage structure for predator and Holling type II functional response. Zbl 1395.34053
Liu, Qun; Jiang, Daqing; Hayat, Tasawar; Alsaedi, Ahmed
2018
Stationary distribution and extinction of a stochastic predator-prey model with distributed delay. Zbl 1382.92223
Liu, Qun; Jiang, Daqing
2018
Periodic solution for a stochastic non-autonomous competitive Lotka-Volterra model in a polluted environment. Zbl 1400.92503
Jiang, Daqing; Zhang, Qiumei; Hayat, Tasawar; Alsaedi, Ahmed
2017
The long time behavior of a predator-prey model with disease in the prey by stochastic perturbation. Zbl 1335.92086
Zhang, Qiumei; Jiang, Daqing; Liu, Zhenwen; O’Regan, Donal
2014
Global asymptotic behavior of a multi-species stochastic chemostat model with discrete delays. Zbl 1439.92208
Wang, Liang; Jiang, Daqing; Wolkowicz, Gail S. K.
2020
Dynamical behavior of a stochastic epidemic model for cholera. Zbl 1418.92179
Liu, Qun; Jiang, Daqing; Hayat, Tasawar; Alsaedi, Ahmed
2019
Stationary distribution and extinction of a stochastic one-prey two-predator model with Holling type II functional response. Zbl 1415.34091
Liu, Qun; Jiang, Daqing
2019
Dynamics of a stochastic multigroup SIQR epidemic model with standard incidence rates. Zbl 1411.92279
Liu, Qun; Jiang, Daqing; Hayat, Tasawar; Alsaedi, Ahmed
2019
Dynamics of an avian influenza model with half-saturated incidence. Zbl 1428.92117
Shi, Zhenfeng; Zhang, Xinhong; Jiang, Daqing
2019
Stationary distribution of a stochastic food chain chemostat model with general response functions. Zbl 1411.62318
Gao, Miaomiao; Jiang, Daqing
2019
Dynamical behavior of a multigroup SIRS epidemic model with standard incidence rates and Markovian switching. Zbl 1426.34060
Liu, Qun; Jiang, Daqing; Hayat, Tasawar; Alsaedi, Ahmed
2019
Dynamics of a stochastic SIR epidemic model with distributed delay and degenerate diffusion. Zbl 1418.92177
Liu, Qun; Jiang, Daqing; Hayat, Tasawar; Alsaedi, Ahmed
2019
Threshold behavior in a stochastic SIQR epidemic model with standard incidence and regime switching. Zbl 1426.92080
Liu, Qun; Jiang, Daqing; Shi, Ningzhong
2018
Ergodic property of the chemostat: a stochastic model under regime switching and with general response function. Zbl 1379.92032
Wang, Liang; Jiang, Daqing
2018
Analysis of a delayed vaccinated SIR epidemic model with temporary immunity and Lévy jumps. Zbl 1382.92240
Liu, Qun; Jiang, Daqing; Hayat, Tasawar; Ahmad, Bashir
2018
Dynamics of a stochastic predator-prey model with stage structure for predator and Holling type II functional response. Zbl 1395.34053
Liu, Qun; Jiang, Daqing; Hayat, Tasawar; Alsaedi, Ahmed
2018
Stationary distribution and extinction of a stochastic predator-prey model with distributed delay. Zbl 1382.92223
Liu, Qun; Jiang, Daqing
2018
Stationary distribution and extinction of a stochastic predator-prey model with additional food and nonlinear perturbation. Zbl 1426.92054
Liu, Qun; Jiang, Daqing; Hayat, Tasawar; Ahmad, Bashir
2018
Stationary distribution and extinction of a stochastic SIRI epidemic model with relapse. Zbl 1395.34052
Liu, Qun; Jiang, Daqing; Hayat, Tasawar; Ahmad, Bashir
2018
Periodic solution and stationary distribution of stochastic predator-prey models with higher-order perturbation. Zbl 1390.34182
Liu, Qun; Jiang, Daqing
2018
Stationary distribution and extinction of a stochastic predator-prey model with herd behavior. Zbl 1398.92209
Liu, Qun; Jiang, Daqing; Hayat, Tasawar; Alsaedi, Ahmed
2018
Stationary distribution and extinction of a stochastic dengue epidemic model. Zbl 1402.92397
Liu, Qun; Jiang, Daqing; Hayat, Tasawar; Alsaedi, Ahmed
2018
The signless Laplacian spectral radius of $$k$$-connected irregular graphs. Zbl 1391.05176
Ning, Wenjie; Lu, Mei; Wang, Kun; Jiang, Daqing
2018
Dynamics of a stochastic tuberculosis model with antibiotic resistance. Zbl 1390.92145
Liu, Qun; Jiang, Daqing; Hayat, Tasawar; Alsaedi, Ahmed
2018
Periodic solution and stationary distribution of stochastic S-DI-A epidemic models. Zbl 1391.92057
Zhang, Xinhong; Jiang, Daqing; Hayat, Tasawar; Alsaedi, Ahmed
2018
The asymptotic behavior of a stochastic multigroup SIS model. Zbl 1387.34074
Ji, Chunyan; Jiang, Daqing
2018
Asymptotic properties of a stochastic chemostat including species death rate. Zbl 1387.92080
Wang, Liang; Jiang, Daqing
2018
Ergodic property of a Lotka-Volterra predator-prey model with white noise higher order perturbation under regime switching. Zbl 1427.37070
Zu, Li; Jiang, Daqing; O’Regan, Donal; Hayat, Tasawar; Ahmad, Bashir
2018
Dynamics of a hepatitis B model with saturated incidence. Zbl 1438.34154
Liu, Liya; Jiang, Daqing; Hayat, Tasawar; Ahmad, Bashir
2018
Threshold behavior in a stochastic HTLV-I infection model with CTL immune response and regime switching. Zbl 1405.35229
Qi, Kai; Jiang, Daqing
2018
Dynamical behavior of a stochastic model of gene expression with distributed delay and degenerate diffusion. Zbl 1400.34071
Liu, Qun; Jiang, Daqing; Hayat, Tasawar; Alsaedi, Ahmed
2018
Dynamical behavior of stochastic multigroup S-DI-A epidemic models for the transmission of HIV. Zbl 1451.92299
Liu, Qun; Jiang, Daqing; Hayat, Tasawar; Alsaedi, Ahmed
2018
Dynamics of a stochastic SIS model with double epidemic diseases driven by Lévy jumps. Zbl 1400.92564
Zhang, Xinhong; Jiang, Daqing; Hayat, Tasawar; Ahmad, Bashir
2017
Stationary distribution and extinction of a stochastic SIRS epidemic model with standard incidence. Zbl 1400.92521
Liu, Qun; Jiang, Daqing; Shi, Ningzhong; Hayat, Tasawar; Alsaedi, Ahmed
2017
Dynamics of a stochastic HIV-1 infection model with logistic growth. Zbl 1400.92502
Jiang, Daqing; Liu, Qun; Shi, Ningzhong; Hayat, Tasawar; Alsaedi, Ahmed; Xia, Peiyan
2017
A note on the stationary distribution of the stochastic chemostat model with general response functions. Zbl 1377.92059
Wang, Liang; Jiang, Daqing
2017
Periodic solution for a stochastic non-autonomous competitive Lotka-Volterra model in a polluted environment. Zbl 1400.92503
Jiang, Daqing; Zhang, Qiumei; Hayat, Tasawar; Alsaedi, Ahmed
2017
A stochastic HIV infection model with T-cell proliferation and CTL immune response. Zbl 1426.92044
Wang, Yan; Jiang, Daqing; Hayat, Tasawar; Ahmad, Bashir
2017
Stationary distribution and extinction of a stochastic SIR model with nonlinear perturbation. Zbl 1377.92095
Liu, Qun; Jiang, Daqing
2017
Dynamics of a stochastic Holling type II predator-prey model with hyperbolic mortality. Zbl 1384.37119
Zhang, Xinhong; Li, Yan; Jiang, Daqing
2017
Asymptotic behavior of a stochastic population model with Allee effect by Lévy jumps. Zbl 1377.92076
Zhang, Qiumei; Jiang, Daqing; Zhao, Yanan; O’Regan, Donal
2017
Dynamics of a stochastic tuberculosis model with constant recruitment and varying total population size. Zbl 1400.92522
Liu, Qun; Jiang, Daqing; Shi, Ningzhong; Hayat, Tasawar; Alsaedi, Ahmed
2017
Stationary distribution of a stochastic SIS epidemic model with double diseases and the Beddington-DeAngelis incidence. Zbl 1390.92144
Liu, Qun; Jiang, Daqing
2017
Dynamical behavior of a stochastic HBV infection model with logistic hepatocyte growth. Zbl 1399.92044
Lju, Qun; Jiang, Daqing; Shi, Ningzhong; Hayat, Tasawar; Alsaedi, Ahmed
2017
The extinction and persistence of a stochastic SIR model. Zbl 1422.34177
Ji, Chunyan; Jiang, Daqing
2017
Asymptotic behavior of stochastic multi-group epidemic models with distributed delays. Zbl 1400.92520
Liu, Qun; Jiang, Daqing; Shi, Ningzhong; Hayat, Tasawar; Alsaedi, Ahmed
2017
The threshold of a non-autonomous SIRS epidemic model with stochastic perturbations. Zbl 1360.60110
Ji, Chunyan; Jiang, Daqing
2017
Stationarity and periodicity of positive solutions to stochastic SEIR epidemic models with distributed delay. Zbl 1360.92104
Liu, Qun; Jiang, Daqing; Shi, Ningzhong; Hayat, Tasawar; Alsaedi, Ahmed
2017
Stationary distribution and extinction of the DS-I-A model disease with periodic parameter function and Markovian switching. Zbl 1426.60075
Liu, Songnan; Xu, Xiaojie; Jiang, Daqing; Hayat, Tasawar; Ahmad, Bashir
2017
Stationary distribution and extinction of a stochastic viral infection model. Zbl 1397.92688
Wang, Yan; Jiang, Daqing
2017
Persistence and stationary distribution of a stochastic predator-prey model under regime switching. Zbl 1367.34069
Zu, Li; Jiang, Daqing; O’Regan, Donal
2017
Stationary distribution of stochastic SIS epidemic model with vaccination under regime switching. Zbl 1343.60095
Zhang, Xinhong; Jiang, Daqing; Alsaedi, Ahmed; Hayat, Tasawar
2016
The threshold of a stochastic delayed SIR epidemic model with temporary immunity. Zbl 1400.92516
Liu, Qun; Chen, Qingmei; Jiang, Daqing
2016
Asymptotic behavior of a stochastic delayed SEIR epidemic model with nonlinear incidence. Zbl 1400.34097
Liu, Qun; Jiang, Daqing; Shi, Ningzhong; Hayat, Tasawar; Alsaedi, Ahmed
2016
Competitive exclusion in a stochastic chemostat model with Holling type II functional response. Zbl 1356.92062
Zhang, Qiumei; Jiang, Daqing
2016
The threshold of a stochastic delayed SIR epidemic model with vaccination. Zbl 1400.92517
Liu, Qun; Jiang, Daqing
2016
Periodic solutions for a stochastic non-autonomous Holling-Tanner predator-prey system with impulses. Zbl 1350.34042
Zuo, Wenjie; Jiang, Daqing
2016
Nontrivial periodic solution of a stochastic non-autonomous SISV epidemic model. Zbl 1400.92519
Liu, Qun; Jiang, Daqing; Shi, Ningzhong; Hayat, Tasawar; Alsaedi, Ahmed
2016
Periodic solution for a stochastic nonautonomous SIR epidemic model with logistic growth. Zbl 1400.92518
Liu, Qun; Jiang, Daqing; Shi, Ningzhong; Hayat, Tasawar; Alsaedi, Ahmed
2016
Stationary distribution of stochastic SIRS epidemic model with standard incidence. Zbl 1354.92100
Zhao, Yanan; Lin, Yuguo; Jiang, Daqing; Mao, Xuerong; Li, Yong
2016
Periodic solutions and stationary distribution of mutualism models in random environments. Zbl 1400.34064
Zhang, Xinhong; Jiang, Daqing; Alsaedi, Ahmed; Hayat, Tasawar
2016
Asymptotic behavior of a three species eco-epidemiological model perturbed by white noise. Zbl 1354.92075
Zhang, Qiumei; Jiang, Daqing; Liu, Zhenwen; O’Regan, Donal
2016
The stability of a predator-prey system with linear mass-action functional response perturbed by white noise. Zbl 1418.92150
Zhang, Qiumei; Wen, Xiangdan; Jiang, Daqing; Liu, Zhenwen
2016
The periodic solutions of a stochastic chemostat model with periodic washout rate. Zbl 07247587
Wang, Liang; Jiang, Daqing; O’Regan, Donal
2016
Stationary distribution and periodic solution for stochastic predator-prey systems with nonlinear predator harvesting. Zbl 07247531
Zuo, Wenjie; Jiang, Daqing
2016
Long-time behavior of a stochastic predator-prey model with modified Leslie-Gower and Holling-type II schemes. Zbl 1335.60125
Lin, Yuguo; Jiang, Daqing
2016
Dynamics of stochastic predator-prey models with Holling II functional response. Zbl 07247590
Liu, Qun; Zu, Li; Jiang, Daqing
2016
Stationary distribution and periodic solutions for stochastic Holling-Leslie predator-prey systems. Zbl 1400.60078
Jiang, Daqing; Zuo, Wenjie; Hayat, Tasawar; Alsaedi, Ahmed
2016
The asymptotic behavior and ergodicity of stochastically perturbed SVIR epidemic model. Zbl 1337.34047
Zhao, Yanan; Jiang, Daqing
2016
Periodic solution for a non-autonomous Lotka-Volterra predator-prey model with random perturbation. Zbl 1326.34091
Zu, Li; Jiang, Daqing; O’Regan, Donal; Ge, Bin
2015
Nontrivial periodic solution of a stochastic epidemic model with seasonal variation. Zbl 1354.92087
Lin, Yuguo; Jiang, Daqing; Liu, Taihui
2015
The threshold of a stochastic SIRS epidemic model in a population with varying size. Zbl 1333.60148
Gray, Alison; Mao, Xuerong; Jiang, Daqing; Zhao, Yanan
2015
Stationary distributions of a stochastic SIR model with saturated incidence and its asymptotic stability. Zbl 1340.47091
Lin, Yuguo; Jiang, Daqing; Jin, Manli
2015
The coexistence of a stochastic Lotka-Volterra model with two predators competing for one prey. Zbl 1410.92113
Zhang, Qiumei; Jiang, Daqing
2015
Conditions for persistence and ergodicity of a stochastic Lotka-Volterra predator-prey model with regime switching. Zbl 07246102
Zu, Li; Jiang, Daqing; O’Regan, Donal
2015
Periodic solution for stochastic non-autonomous multispecies Lotka-Volterra mutualism type ecosystem. Zbl 1410.92154
Han, Qixing; Jiang, Daqing
2015
The threshold of stochastic SIS epidemic model with saturated incidence rate. Zbl 1352.92154
Han, Qixing; Jiang, Daqing; Lin, Shan; Yuan, Chengjun
2015
Asymptotic behavior of a multigroup SIS epidemic model with stochastic perturbation. Zbl 1345.92141
Fu, Jing; Han, Qixing; Lin, Yuguo; Jiang, Daqing
2015
The asymptotic behavior of a stochastic SIS epidemic model with vaccination. Zbl 1422.92180
Zhao, Yanan; Zhang, Qiumei; Jiang, Daqing
2015
The stability of a perturbed eco-epidemiological model with Holling type II functional response by white noise. Zbl 1333.92066
Zhang, Qiumei; Jiang, Daqing; Zu, Li
2015
The threshold of a stochastic SIS epidemic model with vaccination. Zbl 1335.92108
Zhao, Yanan; Jiang, Daqing
2014
Threshold behaviour of a stochastic SIR model. Zbl 1428.92109
Ji, Chunyan; Jiang, Daqing
2014
The threshold of a stochastic SIRS epidemic model with saturated incidence. Zbl 1314.92174
Zhao, Yanan; Jiang, Daqing
2014
Stationary distribution of a stochastic SIS epidemic model with vaccination. Zbl 1395.34064
Lin, Yuguo; Jiang, Daqing; Wang, Shuai
2014
Long-time behavior of a stochastic SIR model. Zbl 1334.92417
Lin, Yuguo; Jiang, Daqing; Xia, Peiyan
2014
Analysis of a delayed stochastic predator-prey model in a polluted environment. Zbl 1427.92075
Han, Qixing; Jiang, Daqing; Ji, Chunyan
2014
Threshold behavior in a stochastic SIS epidemic model with standard incidence. Zbl 1323.92208
Lin, Yuguo; Jiang, Daqing
2014
The long time behavior of a predator-prey model with disease in the prey by stochastic perturbation. Zbl 1335.92086
Zhang, Qiumei; Jiang, Daqing; Liu, Zhenwen; O’Regan, Donal
2014
Stochastic permanence, stationary distribution and extinction of a single-species nonlinear diffusion system with random perturbation. Zbl 1406.92546
Zu, Li; Jiang, Daqing; O’Regan, Donal
2014
Dynamics of the stochastic low concentration trimolecular chemical reaction model. Zbl 1331.92185
Yang, Ying; Jiang, Daqing
2014
The dynamics of the stochastic multi-molecule biochemical reaction model. Zbl 1314.92077
Yang, Ying; Zhao, Yanan; Jiang, Daqing
2014
The behavior of an SVIR epidemic model with stochastic perturbation. Zbl 1406.92644
Zhao, Yanan; Jiang, Daqing
2014
The extinction and persistence of the stochastic SIS epidemic model with vaccination. Zbl 1395.92180
Zhao, Yanan; Jiang, Daqing; O’Regan, Donal
2013
Long-time behaviour of a perturbed SIR model by white noise. Zbl 1435.92077
Lin, Yuguo; Jiang, Daqing
2013
Dynamics of stochastically perturbed SIS epidemic model with vaccination. Zbl 1288.92027
Zhao, Yanan; Jiang, Daqing
2013
Persistence and nonpersistence of a nonautonomous stochastic mutualism system. Zbl 1402.92375
Xia, Peiyan; Zheng, Xiaokun; Jiang, Daqing
2013
Analysis of a predator-prey model with disease in the prey. Zbl 1330.92109
Ji, Chunyan; Jiang, Daqing
2013
Persistence and nonpersistence of a food chain model with stochastic perturbation. Zbl 1301.34064
Li, Haihong; Cong, Fuzhong; Jiang, Daqing; Hua, Hongtu
2013
Extinction and ergodic property of stochastic SIS epidemic model with nonlinear incidence rate. Zbl 1302.92129
Han, Qixing; Jiang, Daqing; Yuan, Chengjun
2013
The ergodicity and extinction of stochastically perturbed SIR and SEIR epidemic models with saturated incidence. Zbl 1231.92058
Yang, Qingshan; Jiang, Daqing; Shi, Ningzhong; Ji, Chunyan
2012
Dynamics of a multigroup SIR epidemic model with stochastic perturbation. Zbl 1244.93154
Ji, Chunyan; Jiang, Daqing; Yang, Qingshan; Shi, Ningzhong
2012
Analysis of autonomous Lotka-Volterra competition systems with random perturbation. Zbl 1258.34099
Jiang, Daqing; Ji, Chunyan; Li, Xiaoyue; O’Regan, Donal
2012
Multiple positive solutions to systems of nonlinear semipositone fractional differential equations with coupled boundary conditions. Zbl 1340.34041
Yuan, Chengjun; Jiang, Daqing; O’Regan, D.; Agarwal, Ravi
2012
The behavior of an SIR epidemic model with stochastic perturbation. Zbl 1272.60035
Ji, Chunyan; Jiang, Daqing; Shi, Ningzhong
2012
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Cited by 1,924 Authors
171 Jiang, Daqing 56 Wang, Ke 52 O’Regan, Donal 50 Liu, Meng 45 Ge, Weigao 45 Liu, Qun 36 Hayat, Tasawar 32 Meng, Xinzhu 32 Yuan, Sanling 30 Al-saedi, Ahmed Eid Salem 30 Liu, Lishan 26 Agarwal, Ravi P. 25 Jankowski, Tadeusz 25 Wu, Yonghong 23 Feng, Meiqiang 22 Chu, Jifeng 22 Li, Yongkun 22 Liu, Yuji 20 Ji, Chunyan 18 Li, Xiaoyue 18 Zhang, Tonghua 18 Zhang, Xuemei 17 Ma, Ruyun 17 Zou, Xiaoling 16 Chen, Fengde 16 Cui, Yujun 16 Lv, Jingliang 16 Nieto Roig, Juan Jose 16 Shen, Jianhua 16 Shi, Ningzhong 15 Liu, Zhijun 15 Wang, Weibing 15 Yuan, Chengjun 14 Bai, Chuanzhi 14 Teng, Zhi-dong 14 Tian, Yu 14 Yan, Jurang 14 Zhang, Xingqiu 13 Lahrouz, Aadil 13 Liu, Wenbin 13 Mao, Xuerong 13 Zhao, Yanan 13 Zu, Li 12 Fan, Meng 12 Henderson, Johnny 12 Lin, Yuguo 12 Settati, Adel 12 Wang, Haiyan 12 Zhai, Chengbo 12 Zhao, Dianli 11 Ding, Xiaohua 11 Feng, Tao 11 Heidarkhani, Shapour 11 Kong, Lingju 11 Lu, Chun 11 Pei, Minghe 11 Zhang, Qiumei 10 Cabada, Alberto 10 Chen, Haibo 10 Chen, Qingmei 10 Du, Bo 10 El Fatini, Mohamed 10 Graef, John R. 10 Li, Shengjun 10 Lu, Shiping 10 Luca, Rodica 10 Sun, Shurong 10 Wang, Weiming 10 Wei, Zhongli 9 Chen, Taiyong 9 Dai, Binxiang 9 Gao, Shujing 9 Goodrich, Christopher S. 9 Han, Zhenlai 9 Li, Wenxue 9 Nguyen Huu Du 9 Zhang, Qimin 9 Zhang, Xinhong 8 Ahmad, Bashir 8 Bai, Zhanbing 8 Cao, Zhongwei 8 Liu, Gui Rong 8 Liu, Xiping 8 Luo, Zhiguo 8 Padhi, Seshadev 8 Xiang, Huili 8 Xu, Chaoqun 8 Xu, Jiafa 8 Yin, George Gang 8 Zhang, Xinguang 8 Zhang, Yan 8 Zhao, Yige 8 Zhao, Yu 7 Ahmad, Bashir 7 Bai, Dingyong 7 Chen, Lansun 7 Chen, Lijuan 7 Cheng, Zhibo 7 Deng, Meiling 7 Hu, Guixin ...and 1,824 more Authors
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Cited in 189 Serials
183 Applied Mathematics and Computation 155 Advances in Difference Equations 108 Journal of Mathematical Analysis and Applications 104 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 99 Abstract and Applied Analysis 89 Boundary Value Problems 64 Computers & Mathematics with Applications 63 Applied Mathematics Letters 59 Journal of Applied Mathematics and Computing 44 Discrete Dynamics in Nature and Society 37 Nonlinear Analysis. Real World Applications 31 Journal of Computational and Applied Mathematics 31 Applied Mathematical Modelling 30 International Journal of Biomathematics 29 Discrete and Continuous Dynamical Systems. Series B 26 Communications in Nonlinear Science and Numerical Simulation 25 Journal of the Franklin Institute 25 Physica A 22 Chaos, Solitons and Fractals 22 Journal of Applied Mathematics 21 Mathematical Methods in the Applied Sciences 21 Mathematical and Computer Modelling 21 Complexity 21 Mathematical Problems in Engineering 17 Nonlinear Dynamics 17 Nonlinear Analysis. Hybrid Systems 16 Journal of Differential Equations 16 Stochastic Analysis and Applications 16 Journal of Nonlinear Science and Applications 15 Applicable Analysis 14 Journal of Difference Equations and Applications 12 Acta Applicandae Mathematicae 12 International Journal of Differential Equations 11 Differential Equations and Dynamical Systems 10 Mathematics and Computers in Simulation 10 Mediterranean Journal of Mathematics 9 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 9 Positivity 9 Journal of Applied Analysis and Computation 9 Journal of Function Spaces 8 Rocky Mountain Journal of Mathematics 8 Applied Mathematics. Series B (English Edition) 8 Journal of Biological Dynamics 8 Computational & Mathematical Methods in Medicine 7 Mathematical Biosciences 7 Acta Mathematicae Applicatae Sinica. English Series 7 Journal of Dynamics and Differential Equations 7 Journal of Nonlinear Science 7 Acta Mathematica Sinica. English Series 6 Journal of Mathematical Chemistry 6 Journal of Biological Systems 6 Nonlinear Analysis. Modelling and Control 5 Bulletin of Mathematical Biology 5 Mathematische Nachrichten 5 Applications of Mathematics 5 International Journal of Computer Mathematics 5 Journal of Inequalities and Applications 5 Fractional Calculus & Applied Analysis 5 Methodology and Computing in Applied Probability 5 Bulletin of the Malaysian Mathematical Sciences Society. Second Series 5 Stochastics and Dynamics 5 Communications on Pure and Applied Analysis 5 Fixed Point Theory and Applications 4 Journal of Mathematical Biology 4 Automatica 4 Czechoslovak Mathematical Journal 4 Journal of Optimization Theory and Applications 4 Proceedings of the American Mathematical Society 4 Proceedings of the Indian Academy of Sciences. Mathematical Sciences 4 Chaos 4 International Journal of Nonlinear Sciences and Numerical Simulation 4 Journal of Function Spaces and Applications 4 Mathematical Biosciences and Engineering 4 Asian-European Journal of Mathematics 4 Mathematical Sciences 3 Bulletin of the Australian Mathematical Society 3 Lithuanian Mathematical Journal 3 Results in Mathematics 3 Systems & Control Letters 3 Journal of Mathematical Sciences (New York) 3 Opuscula Mathematica 3 Discrete and Continuous Dynamical Systems 3 Journal of Applied Analysis 3 Journal of Dynamical and Control Systems 3 Qualitative Theory of Dynamical Systems 3 Journal of Systems Science and Complexity 3 Portugaliae Mathematica. Nova Série 3 Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas. RACSAM 3 ISRN Mathematical Analysis 3 Fractional Differential Calculus 3 Chinese Journal of Mathematics 2 International Journal of Control 2 Journal of Mathematical Economics 2 Monatshefte für Mathematik 2 Osaka Journal of Mathematics 2 Statistics & Probability Letters 2 Chinese Annals of Mathematics. Series B 2 International Journal of Mathematics 2 Stochastic Processes and their Applications 2 Journal of the Egyptian Mathematical Society ...and 89 more Serials
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Cited in 28 Fields
1,358 Ordinary differential equations (34-XX) 803 Biology and other natural sciences (92-XX) 412 Probability theory and stochastic processes (60-XX) 382 Operator theory (47-XX) 129 Difference and functional equations (39-XX) 118 Systems theory; control (93-XX) 104 Dynamical systems and ergodic theory (37-XX) 88 Partial differential equations (35-XX) 51 Integral equations (45-XX) 50 Numerical analysis (65-XX) 36 Real functions (26-XX) 29 Global analysis, analysis on manifolds (58-XX) 29 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 13 Calculus of variations and optimal control; optimization (49-XX) 10 Functional analysis (46-XX) 10 Statistics (62-XX) 8 Combinatorics (05-XX) 6 Mechanics of deformable solids (74-XX) 6 Operations research, mathematical programming (90-XX) 4 General topology (54-XX) 4 Computer science (68-XX) 4 Information and communication theory, circuits (94-XX) 2 General and overarching topics; collections (00-XX) 2 Mechanics of particles and systems (70-XX) 2 Statistical mechanics, structure of matter (82-XX) 1 Linear and multilinear algebra; matrix theory (15-XX) 1 Harmonic analysis on Euclidean spaces (42-XX) 1 Fluid mechanics (76-XX)
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# 1061 - Fat Cat's cousin II
Time limit : 1 sMemory limit : 32 mb
Submitted : 666Accepted : 298
### Problem Description
Fat cat's son is 10 years old now(a small cat? a small mouse? who knows!), and he is doted upon by Fat cat's father-in-law. Oh,do you remember his wife, Mr fat cat is a mordon man and his wife is not a cat but a mouse, a very lovely mouse. Today, fat cat went to see his father-in-law with his family. Grandpa was very happy of course, and as a mathematician he ask his grandson a question: There are n elephants and you seperate them in k groups, how many ways there are will you seperate them? for example there are 4 elephants and you must seperate them in three groups, we name the 4 different elephants 1 2 3 4 and you can seperatet them in the following ways:
{ 1 2 } { 3 } { 4 } or { 1 3 } { 2 } { 4 }
{ 1 4 } { 2 } { 3 } or { 2 3 } { 1 } { 4 }
{ 2 4 } { 1 } { 3 } or { 3 4 } { 1 } { 2 }
there are six way to seperate them
then grandpa ask a lot of questions just the same except n and k are different and grandson answer them all correctly, do you know the answer to the questions?
### Input
Input The input will be pairs of integers, one pair per line, and they are all less than 21, the input will end by the end of file.
### Output
Output: For each pair of integers, print the number of ways to seperate elephants
### Sample Input
4 3
2 2
### Sample Output
6
1
PAT: The order of groups should not be considered.
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Does the image of a differential operator always contain an ideal? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T07:52:04Z http://mathoverflow.net/feeds/question/28093 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/28093/does-the-image-of-a-differential-operator-always-contain-an-ideal Does the image of a differential operator always contain an ideal? Greg Muller 2010-06-14T04:34:08Z 2010-06-14T10:11:42Z <p>Let $\delta$ denote a non-zero complex algebraic differential operator in a single variable x. That is, it can be written as a sum $$\delta = \sum_i f_i\partial_x^i$$ where there $f_i$ are complex polynomials in x.</p> <p>Let $R=\mathbb{C}[x]$, and consider the image of $\delta$ as a map on R. As a subspace of $R$, does $Im(\delta)$ always contain an non-trivial ideal?</p> <p>It does in every case I can think of where there is some trick I can use to understand the image better:</p> <ul> <li>When $\delta$ is a function.</li> <li>When $\delta$ is a constant coefficient differential operator.</li> <li>When $\delta$ has order 1.</li> <li>When $\delta$ is homogeneous for the Euler grading; that is, it takes monomials to monomials.</li> </ul> <p>It seems like it should be related to the simpler fact that $\delta$ is zero if $\delta$ kills functions of unboundedly high degree, which can be shown from the Formal Continuity of differential operators.</p> <p><strong>Remark.</strong> For more than one variable, the above question is false. If $\delta=x\partial_x-y\partial_y$, then $\delta$ is homogeneous for the Euler bigrading (it takes monomials to monomials), but it kills all monomials of the form $x^iy^i$. Since any monomial ideal in $\mathbb{C}[x,y]$ must contain some monomial of this form, the image of this $\delta$ contains no ideal.</p> http://mathoverflow.net/questions/28093/does-the-image-of-a-differential-operator-always-contain-an-ideal/28107#28107 Answer by Victor Protsak for Does the image of a differential operator always contain an ideal? Victor Protsak 2010-06-14T10:11:42Z 2010-06-14T10:11:42Z <p>No. Let $\delta=x-\partial$ and $L=Im(\delta).$ I claim that $L$ does not contain any non-zero ideal of $\mathbb{C}[x].$ Indeed, $x^k\equiv (k-1)x^{k-2}\ (\mod L)$ and, by induction, </p> <p>$$x^{2n+1}\equiv (2n)!!x\equiv 0(\mod L),\ x^{2n}\equiv (2n-1)!!\ (\mod L).$$</p> <p>Thus $L$ contains all odd powers of $x$ and has codimension 1 in $R.$ Suppose that $L$ contains a principal ideal $(f)$. Let $f=f_0+f_1$ be the decomposition of $f$ into the even and odd parts ($f_0$ is the span of the even degree monomials of $f$). Then $x^{2N}f_0\in L$ and $x^{2N+1}f_1\in L$ for any $N\geq 0.$ At least one of $f_0$ and $xf_1$ is non-zero and has the form $g=\sum_n a_{n}x^{2n}.$ Then for any $N\geq 0,$</p> <p>$$x^{2N}g=\sum_n a_nx^{2(n+N)}\equiv \sum_n (2n+2N)!!a_n\ (\mod L)\quad \text{ and }\quad \sum_n (2n+2N)!!a_n=0.$$</p> <p>However, this is impossible: for sufficiently large $N,$ the term involving $a_n\ne 0$ with the largest $n$ clearly dominates the rest of the sum.</p>
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# The Mathematics of Doodling
by Ravi Vakil
Year of Award: 2014
Award: Chauvenet Prize
Publication Information: American Mathematical Monthly, 118 (2011), no. 2, 116-129.
Summary: (Adapted from the Joint Mathematics Meetings 2014 Prizes and Awards Booklet) In this article we learn about the radius $r$ neighborhood $N_r(X)$ of a set $X$ in the plane and how $N_r(X)$ becomes more disk-like as $r$ increases. We see how the perimeter of $N_r(X)$ is related to the area of $X$, first when $X$ is a convex polygon, then when $X$ is any convex set, then when $X$ is arbitrary. We see how the winding number and the Euler characteristic account for the changes in the resulting formulas. We move to three dimensions and encounter Hilbert's Third Problem and the Dehn invariant, and to $n$ dimensions and meet other dissection invariants. Finally, our tour culminates in a brief visit to the moduli space of curves.
About the Author: (From the Joint Mathematics Meetings 2014 Prizes and Awards Booklet)
Ravi Vakil is professor of mathematics and the Packard University Fellow at Stanford University. He is an algebraic geometer whose work touches on topology, string theory, applied mathematics, combinatorics, number theory, and more. He was a four-time Putnam Fellow while at the University of Toronto. He received his Ph.D. from Harvard, and taught at Princeton and MIT before moving to Stanford. He has received the Dean's Award for Distinguished Teaching, the American Mathematical Society Centennial Fellowship, the Terman Fellowship, a Sloan Research Fellowship, the NSF CAREER grant, and the Presidential Early Career Award for Scientists and Engineers. He has also received the Coxeter- James Prize from the Canadian Mathematical Society and the André-Aisenstadt Prize. He was the 2009 Hedrick Lecturer at MathFest, and is currently an MAA Pólya Lecturer. He is a director of the entity running the website mathoverflow, and the director of a potential new school in San Francisco called the "Proof School." He works extensively with talented younger mathematicians at all levels, from high school through recent Ph.D.s.
Publication Date:
Tuesday, January 21, 2014
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dc.creator Sow, P. L. T. en_US dc.creator Merji, S. en_US dc.creator Tokunaga, S. K. en_US dc.creator Lemarchand, C. en_US dc.creator Triki, M. en_US dc.creator Borde, C. en_US dc.creator Chardonnet, C. en_US dc.creator Darquie, B. en_US dc.creator Daussy, C. en_US dc.date.accessioned 2013-07-16T21:44:24Z dc.date.available 2013-07-16T21:44:24Z dc.date.issued 2013 en_US dc.identifier 2013-FD-16 en_US dc.identifier.uri http://hdl.handle.net/1811/55503 dc.description Author Institution: Laboratoire de Physique des Lasers, Universite Paris 13, Sorbonne Paris Cite, CNRS, F-93430, Villetaneuse, France en_US dc.description.abstract Accurate molecular spectroscopy in the mid-infrared region allows precision measurements of fundamental constants. For instance, measuring the linewidth of an isolated Doppler-broadened absorption line of ammonia around 10 $\mu$m enables a determination of the Boltzmann constant $k_{\mathrm{B}}$, 073028 (2011).}. We report on our latest measurements. The main systematic effects, including the temperature control, will be discussed and an error budget will be presented in which the global uncertainty on systematic effects is at the level of a few ppm. This is valid provided that data is recorded under the optimized experimental conditions determined by the studies of systematic effects and provided that spectra are fitted to the speed-dependent Voigt profile, identified as the most suitable lineshape for our measurements , 062510 (2012).}. A determination of $k_{\mathrm{B}}$ by Doppler spectroscopy with a combined uncertainty of a few ppm is within reach. This is comparable to the best current uncertainty obtained using acoustic methods and would make a significant contribution to any new value of $k_{\mathrm{B}}$ determined by the CODATA. Furthermore, having multiple independent measurements at these accuracies opens the possibility of defining the Kelvin by fixing $k_{\mathrm{B}}$, an exciting prospect considering the upcoming redefinition of the International System of Units (SI). en_US dc.language.iso en en_US dc.publisher Ohio State University en_US dc.title ACCURATE DETERMINATION OF THE BOLTZMANN CONSTANT BY DOPPLER SPECTROSCOPY TOWARDS A NEW DEFINITION OF THE KELVIN en_US dc.type Article en_US
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# Trigonometry: Chords, Arcs and Angles [PDF]
Gerardo Sozio5
Trigonometry, as it is taught in high school using the trigonometric ratios, has an interesting history. Indeed, it is a relatively recent invention, going back roughly to the 1400's, although Arab mathematicians developed essentially the same ideas earlier, but written in a form which we would probably not immediately recognise.
An earlier form of trigonometry, however, can be traced back to the ancient Greeks, notably to the two mathematicians Hipparchus and Ptolemy. This version of trigonometry was based on chords in a circle. Hipparchus of Bithynia was an astronomer who was born in 190BC and died in 120BC. He is considered to be one of the most influential of the early astronomers, and is credited with the founding of trigonometry. His work highlighted the need for a system that provided a unit of measure for arcs and angles. The Babylonians divided the circle into 360 parts; the reasons for this, however, are unclear. They may have chosen 360 since it is divisible by many small integers, or, more likely, because 360 is the number of days in a year rounded to the nearest ten.
Hipparchus' trigonometry was based on the chord subtending a given arc in a circle of fixed radius .
The length of the chord is denoted by .
Hipparchus and later Ptolemy, gave a table listing and for various values of the angle , based on a specific value of . Ptolemy used the value , whereas Hipparchus used a more complicated value, as we shall see below.
Using basic circle properties, we can see from the following diagram that is related to the sine ratio by the equation
, so Hence, in a circle of diameter 1, we have
Also, using the circle property, that the angle at the centre of a circle is twice the angle at the circumference subtended by the same arc, it can be seen that in a circle of diameter 1, the chord which subtends an angle at the circumference has length . This is well-defined, since equal chords subtend equal angles.
Hipparchus knew that was equal to the circumference of a circle, and taking as the sexagesimal6 approximation for the radius was calculated:
With this radius, the measure of an angle is equal to its radius measure. The measure of an angle is defined as the length cut off on the circumference divided by the radius. In calculating the table of chords, Hipparchus began with , so that the chord is equal to the radius, since we have an equilateral triangle. Thus in sexagesimal or in minutes.
Now, for a angle, the chord is equal to:
So (in sexagesimal).
To calculate the chords of other angles, Hipparchus used the following geometric results:
Since the angle in a semi-circle is a right angle, we can use Pythagoras' Theorem to obtain
Thus we only need to find the chords of angles up to .
Now, earlier we saw that and by using this result we can write
Again, in a circle of unit diameter, we see that the chord of the supplement of an angle, is the cosine of half the angle. Notice that,
Replacing by , we obtain the well-known result,
In order to calculate the table of chords, Hipparchus also found a formula for From previous results,
Now, in and and is common, therefore (Side Angle Side).
Hence and since we have
If is drawn perpendicular to
but and therefore
But as they have one angle in common and are right angled,
Recall that and therefore
Hipparchus used this formula to calculate chords of half angles.
In order to compare this to modern notation, substitute
and Then
and replacing by we have the well-known formula:
Here is a table of chords, written in sexagesimal.
Arcs Chords Sixtieths Arcs Chords Sixtieths
0;31,25 0;1,2,50 6 6;16,49 0;1,2,44
1 1;2,50 0;1,2,50 47 47;51,0 0;0,57,34
1;34,15 0;1,2,50 49 49;45,48 0;0,57,7
2 2;5,40 0;1,2,50 72 70;32,3 0;0,50,45
2;37,4 0;1,2,48 80 77;8,5 0;0,48,3
3 3;8,28 0;1,2,48 108 97;4,56 0;0,36,50
4 4;11,16 0;1,2,47 120 103;55,23 0;0,31,18
4;42,40 0;1,2,47 133 110;2,50 0;0,24,56
Many of the trigonometric formulae for the sum or difference of two angles, for multiples and for half-angles, can all be derived from a proposition known as Ptolemy's Theorem. It states that if the four vertices of a quadrilateral are concyclic, then the sum of the products of the opposite sides is equal to the product of the diagonals of the quadrilateral.
Proof: In the cyclic quadrilateral choose on such that
In triangles and (remaining angle in triangle), then (Angle Angle). Hence,
(sides in same ratio)
i.e.
Similarly, since
Hence, i.e. .
Summarising these results we have;
This theorem can be applied to the calculation of chords, where either a side or a diagonal coincides with a diameter of the circumscribed circle.
In the figure, is a diagonal and also a diameter of length one. Thus, as was shown previously, if and then and Note that
Now applying Ptolemy's Theorem, remembering we have
Now suppose we take a cyclic quadrilateral with one side being the diameter of length one. Let then, as before, and
Now, once again applying Ptolemy's Theorem, we have
Finally, we again take a cyclic quadrilateral with one side as a diameter, and we choose Then if we have (since equal arcs subtend equal angles). We can again express each of the sides in terms of trigonometric ratios and apply Ptolemy's Theorem to obtain:
This is referred to as the dichotomy formula, and some form of it was used by Archimedes in the computation of some 400 years prior to Ptolemy.
As mentioned earlier, the origins of trigonometry lie in the world of astronomy and spherical triangles. It was Regiomontanus who introduced trigonometry into a form that we would recognise today. Born Johann Müller, Regiomontanus took his name from the latinized form of his hometown, Königsberg, King's Mountain'. He was born in 1436 and died in 1476. Early on in his life he studied at home, then he was sent to Vienna at age 12 where he received his Bachelor's degree at age 15. His most influential work was his De triangulis omnimodis' (On triangles of every kind) which was a work in five parts. In part one of the book he introduces the sine function to solve a right angled triangle; and in book two he introduces trigonometry proper with the Law of Sines.
The formula for the area of a triangle given two sides and the included angle also appears here for the first time. It was written as:
If the area of a triangle is given together with the rectangular product of the two sides, then either the angle opposite the base becomes known, or (that angle) together with (its) known (exterior) equals two right angles.
He also deals with spherical geometry and trigonometry in the remaining three books, but never used the tangent function, although it was obvious that he was familiar with it.
The following problem was posed by Regiomontanus in 1471:
At what point on the ground does a perpendicular suspended rod appear largest (i.e. subtends the greatest visual angle)?
In other words, if the rod at height above the ground above 0 is viewed from a point , at what distance will subtend the largest angle
In the diagram,
Let then and therefore
Let and The arithmo-geometric inequality states that if then Applying this inequality, we have:
To maximise needs to be minimised, this occurs when that is, when Thus, the point is to be located at a distance equal to the geometric mean of the altitudes of the endpoints of the rod, measured horizontally from the foot of the rod.
The origin of the term sine' seems to have originally come from India and was adopted by Arab mathematicians. It was originally referred to as jya-ardha' which meant chord-half', and was at times shortened to jiva'. The Arab mathematicians phonetically derived the meaningless word jiba', and it was written in Arabic without vowels as jb'. This was later interpreted as jaib' which means breast'. After King Alfonso of Castile conquered Toledo in 1085 and captured a large library including many Arab manuscripts, scholars were hired to translate these books into Latin. The Latin word for breast' is sinus', which also means bay' or gulf'. This Latin word then became sine'. Some sources suggest that the Latin term sine' was introduced by Robert of Chester (1145), while others suggest it was introduced by Gherardo of Cremona (1150). In English, it seems the earliest use of sine was in 1593 by Thomas Fale.
With the need to find the sine of the complementary angle, cosine was introduced by Edmund Günter in 1620. It was originally written as co.sinus', short for complementi sinus'. Co.sinus was later modified to cosinus by John Newton (1658).
The word tangent was introduced by Thomas Fincke in (1583) from the Latin word tangere' which means to touch'. Francois Vieta (1593) was not comfortable with the word tangent because of its meaning in geometry, and so he used the term sinus foecundarum'.
Secant, introduced by Thomas Fincke, comes from the Latin secare', which means to cut' (1583). Once again, Vieta (1593) was not comfortable with this as it could have been confused with the geometric term, hence he used transsinuosa'. The cosine of an angle is the sine of the complementary angle, the cotangent of an angle is the tangent of the complementary angle and cosecant is the secant of the complementary angle.
References:
Dantzig, Tobias (1955): The Bequest of the Greeks, London: George Allen and Unwin Ltd. Katz V (1998): A History of Mathematics: An Introduction, Addison-Wesley Educational Publishers Ltd. Beckmen, Petr (1971): A History of Pi, St. Martin's Press.
www.pup.princeton.edu/books/maor/sidebar_c.pdf
www.hps.cam.ac.uk/starry/hipparchus.html
A thank you to Peter Brown of the UNSW who has helped in the writing of this article.
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# What is the bond polarity of h2s?
Dec 24, 2013
The short version: Sulfur is more electronegative than hydrogen, so the $\text{H-S}$ bond is polar with electron density higher on the sulfur atom.
#### Explanation:
This leads to $\text{H"_2"S}$ being a polar molecule.
However, the difference in polarity between $\text{H}$ and $\text{S}$ is very small, so neither the bond nor the molecule are very polar.
The longer version:
The difference in electronegativities of hydrogen (2.20) and sulfur(2.58) is almost exactly the same as that between hydrogen and carbon (2.55).
The $\text{C-H}$ bond is viewed as non-polar and so, therefore, should the $\text{H-S}$ bond.
However, there is some polar character to a $\text{C-H}$ bond.
Its dipole moment is only 0.3 D.
We should therefore expect an $\text{H-S}$ bond to have about the same dipole moment.
Yet, the measured molecular dipole moment of $\text{H"_2"S}$ is 0.95 D.
If this were due entirely to the polar $\text{S-H}$ bonds, the $\text{S-H}$ bond dipole would be at most 0.6 D, with the negative end pointing to the $\text{S}$ atom.
However, hydrogen sulfide has two lone pairs.
(from 2012 Book Archive)
Lone pairs contribute to the molecule's dipole moment even though they do not constitute a 'bond'.
The sulfur 'end' of the lone pair is positive, and the electron 'end' is negative, so one might think of a 'lone pair dipole' contributing to the polarity of the molecule in analogy to a bond dipole.
Thus, it may be the lone pairs that make the major contribution to the polarity of the molecule.
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# Values
Everything that can value, has a value scale. Some people claim to value everything equally (an impossibility with the human architecture, but let's assume they do), and so all values on the scale are the same. I consider that a wrong view to take, because then a human being is no more valuable than an electron. This is the case to the Universe, but only because the Universe doesn't care--it doesn't value. Values are subjective, or products of a mind. Does that mean no value can be morally superior to another? No, as I'll explain in a bit.
On the road of atheism one must pass by the existentialists. Maybe stay for a few meals, chat with the locals, but the journeyman must move on. Existentialists are perhaps more hypocritical than Christians: they believe that since we're all equally worthless to the Universe (true), that nothing we do matters in the end to the Universe (true), that nothing matters (false). If it doesn't matter whether you live or die, why not choose death and save yourself some suffering? You can argue it doesn't matter if you suffer or not, which makes either choice the same, but from a purely emotional feeling, why choose to suffer instead of not suffer? You'd at least save the Universe from calculating "suffering-ness" on you (not that it cares). It just seems logical that if you believe it doesn't matter if you're alive or dead, then you should choose to die and save yourself some trouble. Note I don't want existentialists to all go and kill themselves. I'd rather have them move on the atheist road. But I assert that killing oneself is the logical thing to do if one is existential. I value life over the rational choice, so obviously I'd want them to not literally follow logic off a cliff and pick new beliefs.
Atheists must move past the existentialists. As a budding atheist, you've just crossed one scary river: God doesn't exist and when you die you're annihilated with no hope of return (with present technology (this is a whisper of the river most people miss)). Now you must cross another: the Universe doesn't care if you live or die. But don't stop on the far bank and make camp like the existentialists! Move on. There's a small stream later on that says "Values are subjective, but that doesn't make it wrong to value things, or value things differently, or make all values just as good as other values." I am an entity which has the capability to value, and I value life. This value keeps me from wanting to kill people. Given the choice (which must be made) to kill an ant or kill a human, I'd kill the ant because I value ants less than I value humans. Preferring any outcome to any other is necessary to make a choice. The human brain obviously has preferences, whether you consciously acknowledge them or not, and if you tortured an existentialist I'd bet he wouldn't like it. That concept of liking things, of having preferences, is there in the brain, and is the reason we can have differences in values.
I've recovered the concept of unequal values. I'll now recover the concept of better/worse values.
In olden times, people didn't value the lives of other tribes as much. This is still the case today, but it's not to such an extent. Christians might value Muslims less than other Christians, but we're past the stage of the Crusades. The trend thus far in humanity has been an increasing value of life-in-general. We can see this with Gandhi, we can see this with vegetarians, we can see this with forbidding of torture by some entities; in short, there's a heaping pile of empirical evidence for the trend. Where this trend is, and where this trend is going, I call the good.
Beware of narrow sets! People talk about anti-intellectualism in America, but it's really a relatively new phenomenon and not really widespread. Broaden your data-set to see the trend that respect for intelligence (by intelligent I mean people who try to use their brains to their fullest ability; there's not much fundamentally (on DNA level) different between Einstein and a village idiot) has been going up, not down. You don't even have to go very far to make that assertion: look at high schools over the past 20 years. The rate of nerd-bullying has gone down, and bullying-in-general. (This signifies bullying to be bad as well.) Also look at Bill Gates, richest man on Earth: essentially a geek. Go back hundreds of years and you get a heaping pile of evidence.
So I've rescued right/wrong values as well. And never once did I have to invoke the very complicated issue of God! Note that when analyzing trends it's important to see where you're at, but also where you're going. If the Greeks had really thought this way, they might have noticed "Hey, we're slowly progressing to respecting women more and more and letting them decide things. Let's say the morally good is to let women be equals to men, and strive for that." I look at the trends, and see we're becoming a less-violent society. I choose non-violence as the end, and as the morally good to strive for.
It is with trend-analysis that I criticize certain value standards and point out a better one we should strive for. First, the vegan value system: animals are equally (or more) important than humans. Given a single animal, it has the same or more value as a single human. This stems from valuing animal life and valuing human life, which is good, but the mistake is valuing animal life above human life. (To be fair, the go-kill-yourself-or-change-your-beliefs argument applies to hardcore vegans as well.) Animal life has indeed become more valuable over time, but human life has as well, at least at an equal rate (I'd say a greater rate), and human life has always been generally regarded as higher than animal life. Thus the trend is to value both human and animal lives, but to value human lives more.
Another value system I criticize is the Objectivist one. (I've been picking on Oism lately, huh? Ah well, it's easy and mockery is about the only way one can address it and have fun. This criticism isn't so much mockery though.) Oism holds that the standard of value should be the individual's life. If a billion people are coming at you, wanting to kill you, and you have a button that would kill them first, you should press that button. This stems directly from individualism where you are your own highest value and everyone should respect that about everyone else. Unfortunately, this is broken. Humans have, as far as I know, essentially always believed it good to value their own life. The trend however has been to value other lives as well. The lives of your family, the lives of your tribe, the lives of your city, the lives of your nation, and we're heading to the lives of the world. Hence I assert it's better to value the lives of the world above the life of yourself. If saving one person is like saving a whole world, then saving 100 people is like saving 100 worlds, and saving all of humanity is like saving (at least) a galaxy.
I've said before that math and science progress on a global scale. If someone had killed Einstein, someone else would have discovered General Relativity within a close period of when Einstein released his papers. The differences in human intelligences are minuscule compared to the difference between ant intelligence and human intelligence, our thoughts are the products of ourselves and especially the environment around us, thus someone with a similar enough environment will end up at similar enough results as another person.
So in a horrible scenario, if you had to torture one individual for 50 years vs. having $3\uparrow\uparrow\uparrow\uparrow3$ (See Knuth's Up-Arrow Notation) people suffer a dust speck in the eye, you torture the individual because that's such a vast number that the tiny amount of suffering caused by a dust speck is multiplied by that vast number and is greater than the suffering of one individual for 50 years. If you have the choice of living forever but killing the rest of humanity vs. dying and having the rest of humanity living forever, you should die because humanity is more important than the individual. Watch out where that fact takes you, though, lest you fall into the evils of communism.
Here's my own value scale. I currently value all plants, animals, and humans. I expect to value smarter-than-human intelligences when they arrive, and it wouldn't surprise me if in a few hundred years I also value bacteria and even atoms. However, I'll stick with my present values for the present. I do not value plants, animals, and humans as equal. One cow != One human. 100 cows != One human. Humans are worth more. Also, one carrot != one lion. I'm an omnivore, but I do value the lives of the animals I must eat. Since the trend of valuing all life is increasing, are we going toward vegetarianism? No.
Not in a vegetarian sense, anyway.
I believe that if we have the technology to refrain from eating meat or vegetables or any living thing, without sacrificing our own desires (craving meat), then it's a good thing to do. There are two main ways that instantly come to mind, and they both involve nanotech (to an extent). The first way is to create all our food a-la Star Trek (self-replicating nanobots stealing energy from various caches and configuring into the desired food structure), the second way is actually two alternatives after mind uploading. Mind uploading doesn't seem to be far off, and when we live on silicon instead of in carbon we can simply get our power from the sun or other resources that aren't harmful (and much more efficient than life forms). I for one would dearly miss the taste of meat, however, and I couldn't even eat anymore to satisfy the desire. So I have two options from here: either stimulate the "Mmm, meat" sector of my uploaded mind to make me believe I just had a fantastic steak dinner, or remove the desire from my mind completely so I no longer crave food. Both choices are okay. (Sadly I can imagine some people stimulating the orgasm sector of their uploaded mind all day... Though there wouldn't be anything inherently wrong with that.)
If we have a slow Singularity, with a transition stage between full-blown mind uploads and just nice nanotech, I can imagine some mushy goop made from nothing ever alive that provides everything the body requires, with optional taste and texture packets to keep those who love meat (or chocolate aut alii) from going insane.
So, I predict value of life will go up. Why do I think humans value life at all? Well, humans tend to appreciate rare and complicated things... The desire for self-preservation probably came first in evolution since those with it had more offspring than those without it, and then those who cared about the family came next since those with it had more offspring who had offspring (as they could help raise the kid for a few years). I'm not an evolutionary psychologist but this seems reasonable as a foundation for caring about others, and life itself. Sure I have to point to cultures for more mountains of evidence that humans indeed value life, but you don't even need the evolutionary psychology stuff to claim the value is there.
I find it amusing that Eastern thinkers caught on to this idea of valuing everything and realizing we're all connected thousands of years ago, and people today still don't understand or accept it. Ah well, it's obvious that's the way the trend is heading. Valuing all life is good, as is valuing higher the more complicated life.
#### Posted on 2009-11-05 by Jach
LaTeX allowed in comments, use $\\...\\$\$ to wrap inline and $$...$$ to wrap blocks.
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# Prove or disprove the given equivalence [duplicate]
f(n) = n/100 = Ω(n) . I am new at proving asymptotic notations , especially at big-Ω. That's why I even didnt start the beginning myself. I tried to prove myself as looking at other proofs and searched on the internet to find same question but I couldnt find. I dont think so this question is a duplicate of another which was asked here. Thanks for helps.
## marked as duplicate by Raphael♦ algorithms StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Oct 14 '17 at 19:17
• Could you please post what you tried to solve it yourself? Where did you get stuck? – fade2black Oct 14 '17 at 18:35
I assume $n$ takes only positive values. By the definition of Big-Omega you could prove it by showing the following
$$\lim_{n \rightarrow \infty}\frac{\frac{n}{100}}{n} >0$$
The rest is a simple exercise.
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# Center for Analysis and Design of Intelligent Agents
### Site Tools
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# Differences
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public:t-720-atai:atai-19:aera [2019/09/11 16:44]
thorisson [Lecture Notes, W9: AERA]
public:t-720-atai:atai-19:aera [2019/10/20 18:50] (current)
thorisson [Model Generation & Evaluation]
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| Pervasive Use of Codelets | A //codelet// is a piece of code that is smaller than a typical self-contained program, typically a few lines long, and can only be executed in particular contexts. Programs are constructed on the fly by the operation of the whole system selecting which codelets to run when, based on the knowledge of the system, the active goals, and the state it finds itself in at any point in time. | | Pervasive Use of Codelets | A //codelet// is a piece of code that is smaller than a typical self-contained program, typically a few lines long, and can only be executed in particular contexts. Programs are constructed on the fly by the operation of the whole system selecting which codelets to run when, based on the knowledge of the system, the active goals, and the state it finds itself in at any point in time. |
| \\ No "Modules" | Note that the diagram above may imply the false impression that AERA consists of these four software "modules", or "classes", or the like. Nothing could be further from the truth: All of AERA's mechanism above are a set of functions that are "welded in with" the operation of the whole system, distributed in a myriad of mechanisms and actions. \\ Does this mean that AERA is spaghetti code, or a mess of a design? On the contrary, the integration and overlap of various mechanisms to achieve the high-level functions depicted in the diagram are surprisingly clean, simple, and coherent in their implementation and operation. \\ This does not mean, however, that AERA is easy to understand -- mainly because it uses concepts and implements mechanisms and relies on concepts that are //very different// from most traditional software systems commonly recognized in computer science. | | \\ No "Modules" | Note that the diagram above may imply the false impression that AERA consists of these four software "modules", or "classes", or the like. Nothing could be further from the truth: All of AERA's mechanism above are a set of functions that are "welded in with" the operation of the whole system, distributed in a myriad of mechanisms and actions. \\ Does this mean that AERA is spaghetti code, or a mess of a design? On the contrary, the integration and overlap of various mechanisms to achieve the high-level functions depicted in the diagram are surprisingly clean, simple, and coherent in their implementation and operation. \\ This does not mean, however, that AERA is easy to understand -- mainly because it uses concepts and implements mechanisms and relies on concepts that are //very different// from most traditional software systems commonly recognized in computer science. |
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+==== General Form of AERA Models ====
+
+| {{public:t-720-atai:screenshot_2019-10-20_17.07.15.png?300}} |
+| Models in AERA have a left-hand-side (LHS) and a right-hand-side (RHS). Read from left-to-right they state that "if you see what is in the LHS then I predict what you see in the RHS". When read right-to-left they say "If you want what is on the LHS try getting what is on the LHS first". The latter is a way to produce sub-goals via abduction; the former is a way to predict the future via deduction. |
+| This model, called Model_M, predicts that if you see variables 6 and 7 you will see variable 4 some time later (AERA models refer to specific times - the model is somewhat simplified here for convenience). Read from right to left (backward chaining - BWD) it states that if you want variable-4 you should try to obtain variables 6 and 7. |
+| We call such models "bi-directional causal-relational models" because they can be read in either direction and they model the relations (including causal relations) between variables. Note that models can reference other models on either side and can include patterns on either side. In case the values of variables on either side matter for the other side we use functions that belong to the model that compute these values. And due to the bi-directionality we must have bi-directional functions for this purpose as well. \\ (For instance, if you want to open the door you must push down the handle first, then pull the door towards you; if you pull the door towards you with the handle pushed down then the door will open. The amount of pulling will determine the amount the door is ajar - this can be computed via a function relating the LHS to the RHS.) |
\\ \\
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====Autonomous Model Acquisition==== ====Autonomous Model Acquisition====
| What it is | The ability to create a model of some target phenomenon //automatically//. | | What it is | The ability to create a model of some target phenomenon //automatically//. |
-| Challenge | Unless we know beforehand which signals cause perturbations in <m>o</m> and can hard-wire these from the get-go in the controller, the controller must search for these signals. \\ In task-domains where the number of available signals is vastly greater than the controller's resources available to do such search, it may take an unacceptable time for the controller to find good predictive variables to create models with. \\ <m>V_te >> V_mem</m>, where the former is the total number of potentially observable and manipulatable variables in the task-environment and the latter is the number of variables that the agent can hold in its memory at any point in time. |+| Challenge | Unless we (the designers of an intelligent controller) know beforehand which signals from the controller cause desired perturbations in <m>o</m> and can hard-wire these from the get-go, the controller must find these signals. \\ In task-domains where the number of available signals is vastly greater than the controller's resources available to do such search, it may take an unacceptable time for the controller to find good predictive variables to create models with. \\ <m>V_te >> V_mem</m>, where the former is the total number of potentially observable and manipulatable variables in the task-environment and the latter is the number of variables that the agent can hold in its memory at any point in time. |
\\ \\
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| {{public:t-720-atai:three-models-1.png?400}} | | {{public:t-720-atai:three-models-1.png?400}} |
-| Based on prior observations, of the variables and their temporal execution in some context, the controller's model generation function <m>P_M</m> may have captured their causal relationship in three alternative models, <m>M_1, M_2, M_3</m>, each slightly but measurably different from the others. Each can be considered a //hypothesis of the actual relationship between the included variables//, when in the context provided by <m>V_5, V_6</m>. |+| Based on prior observations, of the variables and their temporal execution in some context, the controller's model generation process <m>P_M</m> may have captured their causal relationship in three alternative models, <m>M_1, M_2, M_3</m>, each slightly but measurably different from the others. Each can be considered a //hypothesis of the actual relationship between the referenced variables//, when in the context provided by <m>V_5, V_6</m>. \\ As an example, we could have a tennis ball's direction <m>V_1</m>, speed <m>V_2</m>, and shape <m>V_3</m> that changes when it hits a wall <m>V_5</m>, according to its relative angle <m>V_6</m> to the wall. |
| {{public:t-720-atai:agent-with-models-1.png?300}} | | {{public:t-720-atai:agent-with-models-1.png?300}} |
-| The agent's model generation mechanisms allow it to produce models of events it sees. Here it creates models (a) <m>M_1</m> and (b) <m>M_2</m>. The usefulness / utility of these models can be tested by performing an operation on the world (c ) as prescribed by the models(Ideally, when one wants to find on which one is best, the most efficient method is an (energy-preserving) intervention that can only leave one as the winner. |+| The agent's model generation mechanisms allow it to produce models of events it sees. Here it creates models (a) <m>M_1</m> and (b) <m>M_2</m>. The usefulness of these models for particular situations and goals can be tested by performing an operation on the world (c ) as prescribed by the models, through backward chaining (abduction). \\ Ideally, when one wants to find on which model is best for a particular situation (goals+environment+state), the most efficient method is an (energy-preserving) intervention that can only leave one as the winner. |
| {{public:t-720-atai:model-m2-prime-1.png?150}} | | {{public:t-720-atai:model-m2-prime-1.png?150}} |
-| The result of feedback (reinforcement) may result in the deletion, rewriting, or some other modification of the original model selected for prediction. Here the feedback has resulted in a modified model <m>M{prime}_2</m>. |+| The feedback (reinforcement) resulting from direct or indirect tests of a model may result in its deletion, rewriting, or some other modification. Here the feedback has resulted in a modified model <m>M{prime}_2</m>. |
\\ \\
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# American Institute of Mathematical Sciences
2011, 2011(Special): 155-162. doi: 10.3934/proc.2011.2011.155
## Time-like surfaces of prescribed anisotropic mean curvature in Minkowski space
1 Institut für Differentialgeometrie, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany, Germany
Received June 2010 Revised March 2011 Published October 2011
Solving an initial value problem for a hyperbolic system, we prove existence and uniqueness of time-like immersions of prescribed anisotropic mean curvature into Minkowski space $\mathbb{R}^(2,1)$ subject to geometric initial conditions.
Citation: Matthias Bergner, Lars Schäfer. Time-like surfaces of prescribed anisotropic mean curvature in Minkowski space. Conference Publications, 2011, 2011 (Special) : 155-162. doi: 10.3934/proc.2011.2011.155
[1] Chiara Corsato, Franco Obersnel, Pierpaolo Omari, Sabrina Rivetti. On the lower and upper solution method for the prescribed mean curvature equation in Minkowski space. Conference Publications, 2013, 2013 (special) : 159-169. doi: 10.3934/proc.2013.2013.159 [2] Hongjie Ju, Jian Lu, Huaiyu Jian. Translating solutions to mean curvature flow with a forcing term in Minkowski space. Communications on Pure & Applied Analysis, 2010, 9 (4) : 963-973. doi: 10.3934/cpaa.2010.9.963 [3] Qinian Jin, YanYan Li. Starshaped compact hypersurfaces with prescribed $k$-th mean curvature in hyperbolic space. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 367-377. doi: 10.3934/dcds.2006.15.367 [4] Alessio Pomponio. Oscillating solutions for prescribed mean curvature equations: euclidean and lorentz-minkowski cases. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3899-3911. doi: 10.3934/dcds.2018169 [5] Ruyun Ma, Man Xu. Connected components of positive solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2701-2718. doi: 10.3934/dcdsb.2018271 [6] Georgi I. Kamberov. Recovering the shape of a surface from the mean curvature. Conference Publications, 1998, 1998 (Special) : 353-359. doi: 10.3934/proc.1998.1998.353 [7] Yi Zhou, Jianli Liu. The initial-boundary value problem on a strip for the equation of time-like extremal surfaces. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 381-397. doi: 10.3934/dcds.2009.23.381 [8] Oleksandr Misiats, Nung Kwan Yip. Convergence of space-time discrete threshold dynamics to anisotropic motion by mean curvature. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6379-6411. doi: 10.3934/dcds.2016076 [9] Chiara Corsato, Colette De Coster, Franco Obersnel, Pierpaolo Omari, Alessandro Soranzo. A prescribed anisotropic mean curvature equation modeling the corneal shape: A paradigm of nonlinear analysis. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 213-256. doi: 10.3934/dcdss.2018013 [10] Franco Obersnel, Pierpaolo Omari. On a result of C.V. Coffman and W.K. Ziemer about the prescribed mean curvature equation. Conference Publications, 2011, 2011 (Special) : 1138-1147. doi: 10.3934/proc.2011.2011.1138 [11] Yoshikazu Giga, Yukihiro Seki, Noriaki Umeda. On decay rate of quenching profile at space infinity for axisymmetric mean curvature flow. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1463-1470. doi: 10.3934/dcds.2011.29.1463 [12] Elias M. Guio, Ricardo Sa Earp. Existence and non-existence for a mean curvature equation in hyperbolic space. Communications on Pure & Applied Analysis, 2005, 4 (3) : 549-568. doi: 10.3934/cpaa.2005.4.549 [13] Guido De Philippis, Antonio De Rosa, Jonas Hirsch. The area blow up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 1-28. doi: 10.3934/dcds.2019243 [14] Matteo Novaga, Enrico Valdinoci. Closed curves of prescribed curvature and a pinning effect. Networks & Heterogeneous Media, 2011, 6 (1) : 77-88. doi: 10.3934/nhm.2011.6.77 [15] Yong Huang, Lu Xu. Two problems related to prescribed curvature measures. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1975-1986. doi: 10.3934/dcds.2013.33.1975 [16] Misha Bialy. On Totally integrable magnetic billiards on constant curvature surface. Electronic Research Announcements, 2012, 19: 112-119. doi: 10.3934/era.2012.19.112 [17] Brittany Froese Hamfeldt. Convergent approximation of non-continuous surfaces of prescribed Gaussian curvature. Communications on Pure & Applied Analysis, 2018, 17 (2) : 671-707. doi: 10.3934/cpaa.2018036 [18] G. Kamberov. Prescribing mean curvature: existence and uniqueness problems. Electronic Research Announcements, 1998, 4: 4-11. [19] Changfeng Gui, Huaiyu Jian, Hongjie Ju. Properties of translating solutions to mean curvature flow. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 441-453. doi: 10.3934/dcds.2010.28.441 [20] Tobias H. Colding and Bruce Kleiner. Singularity structure in mean curvature flow of mean-convex sets. Electronic Research Announcements, 2003, 9: 121-124.
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# $f \in \mathcal S, f(0)=1$ then $\lim_{\epsilon \to 0} f(\epsilon x) = 1$
Let $f \in \mathcal S(\mathbb R^n)$ with $f(0) = 1$. Here $\mathcal S$ means the Schwartz class. Then how can I prove that $$\lim_{\epsilon \downarrow 0} f(\epsilon x) = 1 \; \text{(compact convergence)} \;?$$
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As $\sup_{x\in\Bbb R}|f(x\varepsilon)-1|=\sup_{t\in\Bbb R}|f(t)-1|$, I don't how there could be uniform convergence on the real line. But the is convergence on all the compact subsets. – Davide Giraudo Jan 31 '13 at 12:47
@DavideGiraudo Thank you for the comment. I edited my question. :) – Konsta Jan 31 '13 at 13:07
$|f(\epsilon x)-f(0)| \leq \sup |f'| \cdot \epsilon |x|$, and if $x$ tanges over a bounded subset, then... – Siminore Jan 31 '13 at 13:19
Let $[-R,R]^d$ a compact subset of $\Bbb R^d$. Let $g(x,t):=f(tx)$. $$f(\varepsilon x)-1=g(x,\varepsilon)-g(0,\varepsilon)=\int_0^{\varepsilon}\partial_tg(x,t)dt=\sum_{j=1}^d\int_0^\varepsilon\partial_jf(x,t)x_j dt,$$ which gives for $\varepsilon<1$ that $$|f(\varepsilon x)-1|\leqslant R\cdot d\cdot\varepsilon\max_{1\leqslant j\leqslant d}\sup_{\substack{|x|\leqslant R\\ |t|\leqslant 1}}|\partial_jf(x,t)|.$$ The latest supremum is finite as $f$ is $C^1$ (so we don't need $f$ to be in the Schwarz space).
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## Thinking Mathematically (6th Edition)
The number of ways in which a series of successive things can occur is found by multiplying the number of ways in which each thing can occur. --------- Ways of choosing the 1st letter: 2 (either a W or a K) 2nd letter: 26 3rd letter: 26 4th letter: 26 Total: 2$\times$26$\times$26$\times$26=35,152
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# Why is ice made from boiled water clear?
A common trick to make clear water ice is to boil pure water prior to freezing it. Why does that work and what are the white inclusions in ice that was made from unboiled tap water?
• I do not know how this mentioned method does the trick, but to my knowledge the whiteness of ice is thousands of air bubbles. The clear ice cube is closer to fully crystalline without voids that give internal surfaces to scatter the light. – Steeven Jul 3 '16 at 8:28
• @andynitrox Or you could also just wait like 1 day after boiling to see if the gases get diluted again (they should in theory) and that should produce "white ice" again. – Bakuriu Jul 3 '16 at 10:38
• Will do that. I'll update the question with some results of the experiment. – ahemmetter Jul 3 '16 at 11:37
• Are those cubes sitting atop an apple? – Insane Jul 3 '16 at 20:02
• For the record, shaking water will tend to dissolve more gas in it, as it will slightly shift the equilibrium that direction. (Shaking soda will release gas, but that's because it starts off way out of equilibrium, and the shaking facilitates the reaction.) – user10851 Jul 4 '16 at 8:49
The short answer: Cloudy ice is caused by gases (mainly nitrogen and oxygen) dissolved in the water that come out of solution when the water freezes. The small bubbles trapped in the ice cause the white appearance. Boiling the water removes the air dissolved in it, producing clear ice as a result. Assuming that other impurities don't produce the same cloudy effect.
Impurities present in water:
• Gases: Water at 20°C normally contains about 15 ppm dissolved gases, which is the equivalent of 1 volume of air per 50 volumes of water. These are the same gases present in air, but not in the same proportions since some are more soluble than others: it's about 63% nitrogen, 34% oxygen, 1.5% argon and 1.5% carbon dioxide.
• minerals: Tap water contains dissolved minerals, mainly Ca and Mg. They can be present in the form of bicarbonates: $Ca^{2+}({HCO_3}^-)_2$ and $Mg^{2+}({HCO_3}^-)_2$ (these only exist in solution, not as solid substances), and as calcium and magnesium sulphate. If the water passed through a water softener, the Ca and Mg ions may have been replaced by (twice as many) sodium or potassium ions.
The effects of heating the water:
• removing dissolved gases: higher temperature favors endothermic reactions (Le Chatelier's principle). For the gases present in water, dissolution (at room temperature) is an exothermic process, so their solubility decreases when the water is heated. The solubility of gases doesn't reach zero at boiling point, nor does it necessarily decrease over the whole temperature range. For nitrogen in water, the enthalpy of dissolution becomes positive around 75°, and its solubility increases above that temperature. At 100°C, solubility of air as a whole is $0.93 * 10^{-5}$, about half the solubility at 10°C, $1.82 * 10^{-5}$.
• removing dissolved minerals: Heating the water promotes the conversion of soluble Ca and Mg bicarbonates to insoluble carbonates ($2 {HCO_3}^-$ → $CO_3^{2-} + H_2O + CO_2$) which will come out of solution (as limescale). The sulphates (sometimes referred to as "permanent hardness"), and the sodium or potassium (bi)carbonates stay in solution.
The effect of boiling:
• Solubility of gas in liquid not only depends on temperature, it is directly proportional to the partial pressure of the gas. When boiling, the gas phase in contact with the water is no longer the air, but the water vapor (in the bubbles and close to the surface). In those bubbles the partial pressure of the gases will be close to zero, so gas molecules will still leave the liquid phase (and the increased surface area and the movement of the water speeds up the process), but hardly any will return. Given sufficient time, the water vapor will remove most of the gas. Boiling is basically the equivalent of degassing by purging: removing a gas (oxygen usually) from a solvent by bubbling an inert gas through it.
How do gases make ice "milky/cloudy"?
• During freezing, the ice layer starts at all sides of the cube and grows inward. Water molecules fit the crystal lattice and will adhere to it, other molecules won't (but if the ice grows faster than the gas molecules can diffuse away, they will get trapped). The concentration of gases (and other impurities) in the remaining liquid rises, the solution becomes supersaturated, microbubbles start forming. All these get trapped in the ice, giving it a milky appearance.
• Thats a great answer! I think this contains everything, especially the gist from the previous two answers plus the part about dissolved minerals. – ahemmetter Jul 5 '16 at 8:43
I'm really winging this one because the last time I did an equilibrium calculation was 35 years ago! But I'm fairly sure of a partial answer (see discussion at end).
A gas's solubility in water (or liquid generally) almost always decreases with increasing temperature. This phenomenon is explained in a way very like the explanation of the increase in evaporation rate of a liquid with temperature. Gases dissolve in liquids because the gas molecules find a lower energy state bound to the liquid. The higher the temperature, the greater the proportion of the gas molecules with thermal energy greater than the binding energy for the dissolution process. So a greater proportion of the gas molecules can escape from liquid: the chemical equilibrium for the dissolution reaction shifts to favor free molecules more than bound ones with increasing temperature.
The boiling of a liquid lowers the concentration of dissolved gases through the above effect. Normally the shift of equilibrium back to favor dissolved gases with decreasing temperature would mean that, on cooling, the liquid would take up as much gas as is driven off in the boiling process. The trick with clear ice is that the liquid is frozen too quickly for the gas dissolution process to complete - it is frozen irreversibly so that it is a long way from equilibrium as it cools - with the result that there is a nett expulsion of gas from the liquid by the boiling before freezing process. Once the liquid is frozen, the gas can no longer dissolve in it, so you have clear ice.
Note that this answer is incomplete: it does not answer why the gas dissolved in the liquid forms the bubbles it does when the liquid freezes, as in the right hand image of your question. This answer only explains the absence of the gas needed for the clouding process, so a full answer needs to explain why the dissolved gas comes out of solution to form bubbles as the ice freezes.
• the dissolved gases, which are too large to fit into the lattice of ice, migrate from the solution and are redistributed at the solid–water interface. As freezing progresses, the concentration of dissolved gases surpasses a critical value, the water-containing fluid at the interface becomes supersaturated, and the gas bubbles may nucleate and grow to a visible size either along the interface and/or be trapped within the advancing solid. Quoted from: ncbi.nlm.nih.gov/pmc/articles/PMC4497649 – Previous Jul 3 '16 at 11:14
• In English the word 'net' is spelled with only one T. You seem to consistently misspell it. – zwol Jul 3 '16 at 13:09
• @zwol "Nett" is an old-fashioned variant spelling, but it's not wrong. It's been out of use for longer in the US than in the UK, so if you're American you may not have encountered it before. – Mike Scott Jul 3 '16 at 19:16
• @J... I am sorry if it seems an affectation. I am rather older than you and I can assure you it was in general usage in the mid 20th century, which is where I acquired language. Moreover, I spell some words in certain ways quite deliberately even if a little unusual as it helps me recall them: I had terrible trouble as a child with homophones and their confusion: sentences with, for example, "there" and "their" in them seemed to disintegrate before my eyes before I could comprehend them. – WetSavannaAnimal Jul 4 '16 at 3:54
• @J... Finally, do not forget that English is a greatly bigger thing than merely the dialects of the US (whose sensible spellings I generally prefer) and soon to be DK, or Decay ("Disunited Kingdom"). – WetSavannaAnimal Jul 4 '16 at 3:55
This answer was meant as a comment to @WetSavannahanimal aka Rad Vance but it is rather long and I hit the character limit.
The reason for the opaque center should be due to the manner in which the water volume is freezing. Presumably the solution is not mixed and the outside freezes first forming a crystalline (ice) wall through which the gas cannot escape. As the wall thickens gas is released from the water that solidifies into the central solution that remains. This concentrates the gas in the remaining liquid in the centre. When the gas concentration in this solution hits the saturation value for the liquid at it's current state some of it exits the solution forming the cavities, simultaneously some ice should form, returning the solution to the saturation concentration. This is repeated until all the water is frozen.
The observation that clear ice is made buy bubbling gas through it as it freezes, indicates that mixing of the solution allows the saturated gasses to escape from the surface of the total water volume as the solid forms rather then forming in the center.
Now one might ask the question why is it that there isn't just a single bubble. The first reason, from a bulk solution point of view, is that the water is freezing incrementally forming bubbles as it goes. It is really oscillating about the equilibrium state of the solution, that is the gas saturation point of the freezing solution. The exact conditions of this point will vary slightly as the liquid freezes. The pressure that the ice in the center froze at is likely larger then the pressure at which the surface ice froze at for instance, similarly there is a the temperature at which it froze might also vary. There is probably also a concentration effect, that is as this equilibrium point shifts about the gas saturation point will shift about, this change in concentration also affects the freezing point a little. There are about four effects (Temperature, Pressure, Volume and Gas Concentration) at play during the freezing. The second effect, from a finite volume point of view, is that locally about the cavity the water might experience a "rush of gas" which could locally freeze a film of water encapsulating the bubble hence the final complex arrangement of cavities and not the formation of a single bubble.
Now it might just be possible to see these freezing point differences using the polarizer trick. I've only seen this with clear plastics till now but it should work here aswell. Next time you go to a movie get a pair of 3d goggles. take out the two polarizing lenses and hold them on either side of the ice cube by rotating them a little you should see the internal crystalline structure of the ice as a mess of swirly lines. You should probably see more of a swirl or a scattering in the center near the bubbles indicating the localized changes in crystal structure. You should compare this to the transparent cube.
There is a further trick you can try. If you controlled the freezing of the ice by some degree you could control the formation of the bubbles. For instance our ice trays are plastic and we tend to have a clear upper layer with bubbles forming in the lower part of the cube. I suspect the plastic is retaining it's heat and delays the freezing of the bottom and the side of the cube. I suspect if one warmed or even cooled the ice tray before forming the ice one might see a different formation of cavities (As shown in the second image here for example). If you used a metal ice tray you should see the effect you have observed. If you combined a metal and a plastic ice tray together you might get more bubbles closer to the plastic side. You might even be able to get a polka dot/explosion/blobbing effect by dotting glue inside a metal tray, or holding a metal rod within the volume as you froze it. Although you could just be adding points for the gas to form and escape resulting in clear cubes. Alternatively you might just get a carrat (hat). This Fellow seems to have done a lot of the leg work for you. Beneficially he seems to enjoy drinking his experiments afterwards (YMMV).
If you are really interested you should see if NASA did any freezing experiments in space it might show other methods by which one could control bubble formation. Similarly you might find high speed camera footage of freezing during which you should see some interesting effects as the bubbles form (Though it's a bit of a contradictory use of these cameras and I doubt the people who own them have thought to use them for this purpose).
## protected by Qmechanic♦Nov 27 '16 at 10:32
Thank you for your interest in this question. Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).
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