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This workshop, which continued the triennial series at Oberwolfach on Real and Harmonic Analysis that started in 1986, has brought together experts and young scientists working in harmonic analysis and its applications (such as to dispersive PDE's and ergodic theory) with the objective of furthering the important interactions between these fields. Three prominent experts, Elon Lindenstrauss (Princeton), Amos Nevo (Technion, Haifa), and Terence Tao (UCLA), gave survey respectively introductory lectures. Their topics included "Effective equidistribution on the torus", "Non-Euclidean lattice point counting problems, and the ergodic theory of lattice subgroups," and "The van der Corput lemma, equidistribution in nilmanifolds, and the primes." \medskip Major further areas and results represented at the workshop are: \begin{itemize} \item Application of Time Frequency analysis: this is an outgrowth of the method of "tile decomposition" which has been so successful in solving the problems of the bilinear Hilbert transform. Recent progress includes applications of these techniques and the theory of multilinear singular integral operators to ergodic theory and an extension of the celebrated Carleson-Hunt theorem to the "polynomial Carleson operator." \item Estimates for maximal functions: this includes recent progress on best weak
(1,1)
constants for the Hardy-Littlewood maximal function on metric measure spaces, estimates for maximal functions associated to monomial polyhedra, with applications to sharp estimates for the Bergman kernel on a general class of weakly pseudoconvex domains of finite type in
\C^n,
as well as estimates for maximal functions for the Schr\"odinger and the wave equation. \item Fourier and spectral multipliers: a breakthrough has been obtained on the characterization of radial Fourier multipliers. Contrary to a general belief that for
p\ne 1,2
\infty,
no "concrete" characterization of Fourier multipliers for
L^p(\R^d)
would be possible, radial Fourier multipliers have been characterized for the range
1<p<2d/(d+1),
at least when acting on radial functions, and in sufficiently high dimension even when acting on arbitrary
L^p
- functions, in terms of Fourier localized pieces of the convolution kernels. Moreover, improvements on Wolff's inequality for the cone multiplier have been achieved. \smallskip Also, a theory of Hardy spaces on metric measure spaces with exponential growth has been developed, which allows for instance to significantly improve on a spectral multiplier theorem for Riemannian manifolds with bounded geometry by M.Taylor. For instance, the new results apply to complex powers of the Laplacian, which could not be handled before. \item Oscillatory and Fourierintegral operators: this includes endpoint
L^p - L^q
and Sobolev inequalities for certain broad classes of highly degenerate Radon-like averaging operators. \item Applications to PDE's: this includes optimal global existence theorems by means of abstract Strichartz estimates for small amplitude nonlinear wave equations associated to certain linear wave equations involving compact perturbations of the standard Laplacian, global well-posedness and scattering in
H^1
for defocusing nonlinear Schr\"odinger equations on hyperbolic space, and a smoothing property for the
L^2
-critical nonlinear Schr\"odinger equation. \end{itemize} \medskip The meeting took place in a lively and active atmosphere, and greatly benefited from the ideal environment at Oberwolfach. It was attended by 43 participants. The program consisted of 3 survey lecture series and 25 lectures. The organisers made an effort to include young mathematicians, and greatly appreciate the support through the joint Oberwolfach/NSF program "US Junior Oberwolfach Fellows," which allowed to invite several outstanding young scientists from the United States.
Detlef Müller, Elias M. Stein, Real Analysis, Harmonic Analysis and Applications. Oberwolfach Rep. 5 (2008), no. 3, pp. 1771–1850
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From E. A. Darwin [after 31 March 1864?]1
I enclose 2 pods 2
\frac{1}{4}
gr each my little all which I hope will tide you over till you get some from Blunts.3
The date is conjectured from the relationship between this letter and the letter from E. A. Darwin to Emma Darwin, 30 [March 1864?].
Erasmus Alvey Darwin refers to the medicine podophyllin, which CD had started taking on 24 March 1864 (see letter from E. A. Darwin to Emma Darwin, 30 [March 1864?] and n. 4).
CD’s Classed account book (Down House MS) for 1864 included payments to Blunts, or Blunt and Salter, a Shrewsbury dispensing chemist. Blunts was ‘believed at Down to be the best chemist in the world’ (Emma Darwin (1915) 2: 118 n. 2; see also Correspondence vol. 2, letter to Catherine Darwin, [16 September 1842]).
Sends "2 pods ¼ gr each" to tide CD over.
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Possible world — Wikipedia Republished // WIKI 2
Concept of philosophy and logic used to express modal claims
"Possible worlds" redirects here. For other uses, see Possible Worlds (disambiguation).
A possible world is a complete and consistent way the world is or could have been. Possible worlds are widely used as a formal device in logic, philosophy, and linguistics in order to provide a semantics for intensional and modal logic. Their metaphysical status has been a subject of controversy in philosophy, with modal realists such as David Lewis arguing that they are literally existing alternate realities, and others such as Robert Stalnaker arguing that they are not.
Are Possible Worlds Real? Modal Realism Part 1 – Philosophy Tube
What are Possible Worlds? - Gentleman Thinker
Jean Houston: Part 1 Complete: Possible Human, Possible World - Thinking Allowed DVD
Ricardian model possible world trade prices
2 Argument from ways
3 Philosophical issues and applications
3.2 Explicating necessity and possibility
4 History of the concept
See also: Modal logic § Semantics, and Intensional logic
Possible worlds are one of the foundational concepts in modal and intensional logics. Formulas in these logics are used to represent statements about what might be true, what should be true, what one believes to be true and so forth. To give these statements a formal interpretation, logicians use structures containing possible worlds. For instance, in the relational semantics for classical propositional modal logic, the formula
{\displaystyle \Diamond P}
{\displaystyle P}
Possible worlds play a central role in the work of both linguists and philosophers working in formal semantics. Contemporary formal semantics is couched in formal systems rooted in Montague grammar, which is itself built on Richard Montague's intensional logic.[1] Contemporary research in semantics typically uses possible worlds as formal tools without committing to a particular theory of their metaphysical status. The term possible world is retained even by those who attach no metaphysical significance to them.
Possible worlds are often regarded with suspicion, which is why their proponents have struggled to find arguments in their favor.[2] An often-cited argument is called the argument from ways. It defines possible worlds as "ways how things could have been" and relies for its premises and inferences on assumptions from natural language,[3][4][5] for example:
(1) Hillary Clinton could have won the 2016 US election.
The central step of this argument happens at (2) where the plausible (1) is interpreted in a way that involves quantification over "ways". Many philosophers, following Willard Van Orman Quine,[6] hold that quantification entails ontological commitments, in this case, a commitment to the existence of possible worlds. Quine himself restricted his method to scientific theories, but others have applied it also to natural language, for example, Amie L. Thomasson in her easy approach to ontology.[7] The strength of the argument from ways depends on these assumptions and may be challenged by casting doubt on the quantifier-method of ontology or on the reliability of natural language as a guide to ontology.
Further information: Modal logic § The ontology of possibility, and Modal realism
The ontological status of possible worlds has provoked intense debate. David Lewis famously advocated for a position known as modal realism, which holds that possible worlds are real, concrete places which exist in the exact same sense that the actual world exists. On Lewis's account, the actual world is special only in that we live there. This doctrine is called the indexicality of actuality since it can be understood as claiming that the term "actual" is an indexical, like "now" and "here". Lewis gave a variety of arguments for this position. He argued that just as the reality of atoms is demonstrated by their explanatory power in physics, so too are possible worlds justified by their explanatory power in philosophy. He also argued that possible worlds must be real because they are simply "ways things could have been" and nobody doubts that such things exist. Finally, he argued that they could not be reduced to more "ontologically respectable" entities such as maximally consistent sets of propositions without rendering theories of modality circular. (He referred to these theories as "ersatz modal realism" which try to get the benefits of possible worlds semantics "on the cheap".)[8][9]
Modal realism is controversial. W.V. Quine rejected it as "metaphysically extravagant".[10] Stalnaker responded to Lewis's arguments by pointing out that a way things could have been is not itself a world, but rather a property that such a world can have. Since properties can exist without them applying to any existing objects, there's no reason to conclude that other worlds like ours exist. Another of Stalnaker's arguments attacks Lewis's indexicality theory of actuality. Stalnaker argues that even if the English word "actual" is an indexical, that doesn't mean that other worlds exist. For comparison, one can use the indexical "I" without believing that other people actually exist.[11] Some philosophers instead endorse the view of possible worlds as maximally consistent sets of propositions or descriptions, while others such as Saul Kripke treat them as purely formal (i.e. mathematical) devices.[12]
True propositions are those that are true in the actual world (for example: "Richard Nixon became president in 1969").
False propositions are those that are false in the actual world (for example: "Ronald Reagan became president in 1969").
Possible propositions are those that are true in at least one possible world (for example: "Hubert Humphrey became president in 1969"). (Humphrey did run for president in 1968, and thus could have been elected.) This includes propositions which are necessarily true, in the sense below.
Impossible propositions (or necessarily false propositions) are those that are true in no possible world (for example: "Melissa and Toby are taller than each other at the same time").
Necessarily true propositions (often simply called necessary propositions) are those that are true in all possible worlds (for example: "2 + 2 = 4"; "all bachelors are unmarried").[13]
Contingent propositions are those that are true in some possible worlds and false in others (for example: "Richard Nixon became president in 1969" is contingently true and "Hubert Humphrey became president in 1969" is contingently false).
Possible worlds play a central role in many other debates in philosophy. These include debates about the Zombie Argument, and physicalism and supervenience in the philosophy of mind. Many debates in the philosophy of religion have been reawakened by the use of possible worlds.
The idea of possible worlds is most commonly attributed to Gottfried Leibniz, who spoke of possible worlds as ideas in the mind of God and used the notion to argue that our actually created world must be "the best of all possible worlds". Arthur Schopenhauer argued that on the contrary our world must be the worst of all possible worlds, because if it were only a little worse it could not continue to exist.[14] Scholars have found implicit earlier traces of the idea of possible worlds in the works of René Descartes,[15] a major influence on Leibniz, Al-Ghazali (The Incoherence of the Philosophers), Averroes (The Incoherence of the Incoherence),[16] Fakhr al-Din al-Razi (Matalib al-'Aliya)[17] and John Duns Scotus.[16]
The modern philosophical use of the notion was pioneered by David Lewis and Saul Kripke.
Standard translation, an embedding of modal logics into first-order logic which captures their possible world semantics
Modal fictionalism
Impossible world
Modal realism
Extended modal realism
^ "Formal Semantics: Origins, Issues, Early Impact". Baltic International Yearbook of Cognition, Logic and Communication. This Proceeding of the Symposium for Cognition, Logic and Communication. Vol. 6. 2011.
^ Laan, David A. Vander (1997). "The Ontology of Impossible Worlds". Notre Dame Journal of Formal Logic. 38 (4): 597–620. doi:10.1305/ndjfl/1039540772.
^ Berto, Francesco; Jago, Mark (2018). "Impossible Worlds". The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. Retrieved 14 November 2020.
^ Menzel, Christopher (2017). "Possible Worlds". The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. Retrieved 14 November 2020.
^ Quine, Willard V. (1948). "On What There Is". Review of Metaphysics. 2 (1): 21–38.
^ Thomasson, Amie L. (2014). Ontology Made Easy. Oup Usa. p. 248.
^ W. V. O. Quine, "Proportional Objects" in Ontological Relativity and Other Essays', 1969, pp.140-147
^ See "A Priori and A Posteriori" (author: Jason S. Baehr), at Internet Encyclopedia of Philosophy: "A necessary proposition is one the truth value of which remains constant across all possible worlds. Thus a necessarily true proposition is one that is true in every possible world, and a necessarily false proposition is one that is false in every possible world. By contrast, the truth value of contingent propositions is not fixed across all possible worlds: for any contingent proposition, there is at least one possible world in which it is true and at least one possible world in which it is false." Accessed 7 July 2012.
^ Arthur Schopenhauer, "Die Welt als Wille und Vorstellung," supplement to the 4th book "Von der Nichtigkeit und dem Leiden des Lebens" p. 2222, see also R.B. Haldane and J. Kemp's translation "On the Vanity and Suffering of Life" pp 395-6
^ a b Taneli Kukkonen (2000), "Possible Worlds in the Tahâfut al-Falâsifa: Al-Ghazâlî on Creation and Contingency", Journal of the History of Philosophy, 38 (4): 479–502, doi:10.1353/hph.2005.0033, S2CID 170995877
^ Adi Setia (2004), "Fakhr Al-Din Al-Razi on Physics and the Nature of the Physical World: A Preliminary Survey", Islam & Science, 2, retrieved 2010-03-02
David Lewis, On the Plurality of Worlds (1986. Oxford & New York: Basil Blackwell) ISBN 0-631-13994-X
"Possible Worlds" entry in the Stanford Encyclopedia of Philosophy
"Possible worlds: what they are good for and what they are" — Alexander Pruss.
"Possible Objects" entry in the Stanford Encyclopedia of Philosophy
"Impossible Worlds" entry in the Stanford Encyclopedia of Philosophy
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2019 The Aronsson Equation, Lyapunov Functions, and Local Lipschitz Regularity of the Minimum Time Function
{C}^{1}
{C}^{1}
{C}^{1}
{C}^{1}
Pierpaolo Soravia. "The Aronsson Equation, Lyapunov Functions, and Local Lipschitz Regularity of the Minimum Time Function." Abstr. Appl. Anal. 2019 1 - 9, 2019. https://doi.org/10.1155/2019/6417074
Received: 17 July 2019; Accepted: 8 October 2019; Published: 2019
Pierpaolo Soravia "The Aronsson Equation, Lyapunov Functions, and Local Lipschitz Regularity of the Minimum Time Function," Abstract and Applied Analysis, Abstr. Appl. Anal. 2019(none), 1-9, (2019)
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(x − 2)^2 − 3 = 1
graphically. That is, graph
y = (x − 2)^2 − 3
y = 1
on the same set of axes and find the
x
-value(s) of any points of intersection. Then use algebraic strategies to solve the equation and verify that your graphical solutions are correct.
Look at the points of intersection. What is the value of
x
at each point of intersection?
Check to see if these values make the original equation true.
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1 October 2012 Heisenberg categorification and Hilbert schemes
Given a finite subgroup
\Gamma \subset {\mathrm{SL}}_{2}\left(\mathbb{C}\right)
we define an additive
2
{\mathcal{H}}_{\Gamma }
whose Grothendieck group is isomorphic to an integral form
{\mathfrak{h}}_{\Gamma }
of the Heisenberg algebra. We construct an action of
{\mathcal{H}}_{\Gamma }
on derived categories of coherent sheaves on Hilbert schemes of points on the minimal resolutions
\stackrel{^}{{ℂ}^{2}/\Gamma }
Sabin Cautis. Anthony Licata. "Heisenberg categorification and Hilbert schemes." Duke Math. J. 161 (13) 2469 - 2547, 1 October 2012. https://doi.org/10.1215/00127094-1812726
Sabin Cautis, Anthony Licata "Heisenberg categorification and Hilbert schemes," Duke Mathematical Journal, Duke Math. J. 161(13), 2469-2547, (1 October 2012)
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EUDML | Some results on -approximation of the -th derivate of Fourier series. EuDML | Some results on -approximation of the -th derivate of Fourier series.
{L}^{1}
-approximation of the
r
-th derivate of Fourier series.
Tomovski, Živorad
Volume: 3, Issue: 1, page Paper No. 10, 11p., electronic only-Paper No. 10, 11p., electronic only
Tomovski, Živorad. "Some results on -approximation of the -th derivate of Fourier series.." JIPAM. Journal of Inequalities in Pure & Applied Mathematics [electronic only] 3.1 (2002): Paper No. 10, 11p., electronic only-Paper No. 10, 11p., electronic only. <http://eudml.org/doc/122202>.
@article{Tomovski2002,
author = {Tomovski, Živorad},
keywords = {Fourier series; Sidon-Telyakovskij class; -convergence; cosine series; sine series; -convergence},
pages = {Paper No. 10, 11p., electronic only-Paper No. 10, 11p., electronic only},
title = {Some results on -approximation of the -th derivate of Fourier series.},
AU - Tomovski, Živorad
TI - Some results on -approximation of the -th derivate of Fourier series.
SP - Paper No. 10, 11p., electronic only
EP - Paper No. 10, 11p., electronic only
KW - Fourier series; Sidon-Telyakovskij class; -convergence; cosine series; sine series; -convergence
Fourier series, Sidon-Telyakovskij class,
{L}^{1}
-convergence, cosine series, sine series,
{L}^{1}
Articles by Tomovski
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Your note has followed us here, where we arrived 4 or 5 days ago.—1 As we wished to try 6 or 8 weeks of sea-side for my eldest girl we thought we would make this awfully long (to us) stretch & come here. We are charmed with the view from this crescent & with the walks all around & we have got a very good House. But is no joke bringing 16 souls &
\frac{3}{4}
tun of luggage so far.— My eldest girl is still a sad invalid; but certainly improves: she gets up twice every day now & can walk one or two hundred yards.2
I am glad for myself to have this outing & change for I have been a poor wretch for many months. You say not one word about yourself. Are you not a pretty sort of man? It seems indeed strange to hear of your having two daughters married.—3
Poor dear Henslow’s death has been a sad loss to many. He wrote me by dictation a most kind note from his death-bed.—4 L. Jenyns is going to write a biographical notice of him.—5
I shall not go to Manchester;6 but it would be a great temptation to get a sight of you.— How long it is since we met! Farewell my dear old friend | Yours affecty | C. Darwin
My eldest son is going to join as Partner in a Bank at Southampton: I had so good an offer it seemed a pity to reject it.—7
We hear that the Darwins of—(Elston) are coming here.—8
Fox’s letter has not been found. The Darwins left for Torquay on 1 July 1861, spending the night en route in Reading. See ‘Journal’ (Appendix II).
Henrietta Emma Darwin was recuperating slowly from what was thought to be typhus fever suffered in the spring of 1860.
Two of Fox’s daughters by his first marriage had recently married. Eliza Ann Fox married Henry Martyn Sanders, vicar of Skidby, Hull, and Harriet Emma Fox married Samuel Charlesworth Overton (Darwin pedigree).
The letter from John Stevens Henslow, apparently written shortly before his death on 16 May 1861, has not been found.
Jenyns ed. 1862 (see Correspondence vol. 9, Appendix X).
The British Association for the Advancement of Science was to hold its 1861 meeting in Manchester.
CD was arranging for his son William Erasmus Darwin to become a partner with the Southampton and Hampshire Bank. See letters to John Lubbock, 10 July [1861], 1 August [1861], and [2 August 1861].
Elston Hall, near Newark, Nottinghamshire, was the seat of the senior branch of the Darwin family. CD’s grandfather Erasmus Darwin was born at Elston Hall. The head of the family was Francis Rhodes Darwin, husband of Charlotte Maria Cooper Darwin, the granddaughter of Erasmus Darwin’s older brother, William Alvey Darwin. Francis Rhodes took the Darwin family name in 1850 under the provisions of the will of his brother-in-law Robert Alvey Darwin. See Darwin pedigree and Freeman 1978.
JY 8 61
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3.2 What are the goldilocks conditions for life? - Big History School
To work through '3.2 What are the goldilocks conditions for life?' you need to complete the Activities in order. So first complete ‘Learning Plan’ then move to Activity '3.2.1' followed by Activity '3.2.2' through to '3.2.5', and then finish with ‘Learning Summary’.
What are the Goldilocks Conditions for life?
In What are the goldilocks conditions for life? you will learn all about the Goldilocks conditions for life, how scientists believe life began on Earth and how natural selection and evolution have led to the diversity of plants and animals we have today.
To get started, read carefully through the What are the goldilocks conditions for life? learning goals below. Make sure you tick each of the check boxes to show that you have read all your What are the goldilocks conditions for life? learning goals.
As you read through the learning goals you may come across some words that you haven’t heard before. Please don’t worry. By the time you finish What are the goldilocks conditions for life? you will have become very familiar with them!
You will come back to these learning goals at the end of What are the goldilocks conditions for lief? to see if you have confidently achieved them.
Life and how it began
Use claim testers to evaluate claims about living things
Describe the Goldilocks Conditions for life on Earth
Begin to understand the scientific view of how life began
Understand what Biologists study
Identify key developments in the history of life on Earth
Describe an example of natural selection
Begin to understand the role of evolution in creating diversity
2Al
You may remember from your pre-mission critical thinking skills training that a ‘claim’ is information that someone presents as fact - not an opinion. They are asking you to trust that what they are saying is based on reliable information.
You’ve probably heard a lot of claims about life and living things. Well before you explore life on Earth any further, you’re going to play a game of Snap Judgement. In this game you will have to decide which claims about ‘life’ you trust and which ones you don’t trust. Once you watch Mission video 16: Life and how it began in the next activity, you will have an opportunity to come back to your ‘snap judgements’ and review your responses.
Do you remember what the four claim testers are? If you need a reminder, take a look at Infographic: the four claim testers in Helpful Resources.
If you’re playing this game with other students your teacher may have already placed some ‘claims’ on display for you. Your teacher will ask you to go to each one of the three claims on display and write on a post-it note if you trust or don’t trust the claim with a short sentence explaining why. You then need to decide which claim tester you are using e.g. intuition, logic, authority or evidence and place your post-it note under that claim tester heading before moving on to the next claim.
If your teacher has instructed you to complete this activity on your own or in a pair, you will use the Claims: snap judgement life worksheet. Read each one of the three claims on the worksheet carefully and write down whether you trust or don’t trust each of the claims. Write at least one sentence explaining why and circle which claim testers you used to come to your decision.
Infographic: the four claim testers
You will learn about life and how it began in the next activity. You will then have an opportunity to come back to the snap judgements that you’ve made in this activity, think about your responses and see if you’ve changed your mind.
2Al
In the last few activities you learned about how Earth evolved; its layered structure and the tectonic plates which are constantly reshaping the Earth’s crust. You are now ready to learn how all these changes helped to create the 'Goldilocks conditions' for life on Earth.
Mission video 16: Life and how it began explains what the Goldilocks conditions for life are and how scientists believe life on Earth began.
While you watch Mission video 16: Life and how it began look out for the answers to the following questions:
1. What are the Goldilocks conditions for life?
3. What is the scientific view of how life began on Earth?
4. When did life begin on Earth?
5. What were the first life forms?
6. What are the 6 features all living things have in common?
So where does ‘life begins’ appear on our History of the Universe Timeline?
So now that you have a clearer idea about how life began on Earth and the differences between living and non-living things, go back to the claims you responded to in the Snap Judgement game in the last activity.
If you are part of a class, your teacher will lead a class discussion where you and your classmates will share your responses to the claims, analyze which claim testers you used and decide whether, after watching Mission video 16: Life and how it began, any of you have since changed your minds about any of the claims and why.
2Al
When we talk about the ‘Goldilocks conditions’ for life what does it remind you of? Could it be the classic fairytale ‘Goldilocks and the Three Bears’?
The ‘Goldilocks zone’ for life is the zone around a star where the temperature is ‘just right’ – not too hot or not too cold – for liquid water to exist on a planet.
With temperatures as hot as 460oC and a very thick atmosphere full of carbon dioxide, the water on Venus eventually boiled away. With temperatures as cold as -153oC and a very thin atmosphere, the water on Mars either evaporated into space or turned into ice.
As we know from watching Mission video 16: Life and how it began, Earth is the ‘just right’ distance from the Sun for liquid water to exist and it also has an oxygen-rich atmosphere which protects us from extreme heat and cold.
In this activity you will create a comic based on the ‘Goldilocks and the Three Bears’ fairytale. In this version however, called ‘Commander Goldilocks and the three planets,’ the main character needs to find the ‘just right’ planet to have a well-deserved break from her intergalactic travels.
Use the Comic: Commander Goldilocks and the three planets worksheet to create your comic. Be as creative as possible as you draw your comic story about Commander Goldilocks’ intergalactic journey. Will she choose super-hot Venus, super-cold Mars or ‘just right’ Earth?
We’re guessing, if your Commander Goldilocks didn’t want to freeze to death or be cooked, that she chose to land on Earth. Would that be right?
Try to imagine that you were there when Commander Goldilocks decided to land on Earth. If she had never been to Earth before, how would you explain to her the difference between living and non-living things and what examples would you provide?
Then be sure to give her a tall refreshing glass of sparkling Earth water!
2Al
In the previous activities you learned how scientists believe life began on Earth. In this activity you will learn how life evolved from those first simple microscopic organisms into the amazing diversity of plant and animal life that surrounds us today.
Mission video 17: How life changes uses a timeline to trace the history of life on Earth. It also outlines the theories of natural selection and evolution which explain how life was able to change over time and become so interesting and diverse.
While you watch Mission video 17: How life changes look out for the answers to the following questions:
1. What were some key developments in the history of life on Earth?
2. Why was the development of the egg so important?
3. What is the scientific view of how dinosaurs became extinct?
4. What is a famous example of natural selection?
5. What does the theory of evolution explain?
To see an example of natural selection in action, play the ‘Peppermoths’ simulation game in Helpful Resources.
To begin the simulation, click on the circle with the heading, ‘A Bird’s Eye View of Natural Selection.’ Read through the instructions and conduct the experiment in the light forest and then the dark forest. How do the different results in the two different colored forests help to explain the process of natural selection?
http://peppermoths.weebly.com/
2Al
Charles Darwin was the first scientist to propose the theory of evolution. When he studied the finches on the Galapagos Islands he realized that finches which lived on different islands had different beaks which matched the particular foods available on the island they lived on.
Darwin came up with the idea that there would have originally been one type of finch but when some finches moved to other islands, the finches which had adaptations that best suited their new habitat were most successful. They survived and passed on those adaptations to the next generation.
Natural selection can be a bit of a tricky idea to get your head around. To help you have a better understanding of Darwin’s example of natural selection - the finches of the Galapagos and their different beaks - you’re going to have some fun with a hands-on demonstration.
In this demonstration you will see how some bird beaks are better adapted to eating certain foods than others. You will then reflect on how this explains the process of natural selection.
Your teacher will instruct you whether you will be conducting this demonstration as part of a group or individually.
If you are conducting the demonstration as part of a group you will follow the ‘group’ instructions on the Demo: bird beak adaptation instruction sheet:
Step 1: Form into groups of 4 and collect the tray of ‘food’ for your group as well as 4 different types of ‘beaks.’ Each member of the group should also have their own worksheet and cup.
Step 2: Refer to the Results Table. In the column under the heading ‘Food’ write down the 4 different types of ‘food’ in your tray. In the numbered row under the ‘Beaks’ heading write down the 4 different types of ‘beaks’ your group has.
Step 3: Each person in the group should select a different ‘beak’ and pick up their cup (stomach).
Step 4: Hold your ‘beak’ in one hand and your cup in the other hand. When your teacher says ‘go’ try to pick up as much food as you can with your ‘beak.’
Step 5: When your teacher says ‘stop,’ empty out your cup. Count the number of each type of ‘food’ you were able to pick up with your ‘beak.’ Record the amounts in that ‘Beak’ column.
Step 6: Place all the ‘food’ back into the tray and swap your ‘beak’ with someone else in your group and do the activity again, recording your results.
Step 7: Repeat step 6 two more times until you have tried all four of the ‘beaks.’
Step 8: Answer the reflection questions.
If you are conducting the demonstration individually you will follow the ‘individual’ instructions on the Demo: bird beak adaptation individual instruction sheet:
Step 1: Collect your worksheet, your tray of ‘food,’ your cup and the 4 different types of ‘beaks.’
Step 2: Refer to the Results Table. In the column under the heading ‘Food’ write down the 4 different types of ‘food’ in your tray. In the numbered row under the ‘Beaks’ heading write down the 4 different types of ‘beaks’ you have.
Step 3: Hold your first ‘beak’ in one hand and your cup (stomach) in your other hand.
Step 4: When your teacher says ‘go’ try to pick up as much food as you can with your ‘beak.’
Step 6: Place all the ‘food’ back into the tray and use your next ‘beak’ to do the activity again, recording your results in that ‘Beak’ column.
Step 7: Repeat step 6 two more times until you have tried all four of your ‘beaks.’
If you are part of a class, your teacher will lead a class discussion where you and your classmates will share your responses to the reflection questions on the Demo: bird beak adaptation instruction sheet:
Why were some beaks better than others at picking up particular types of food? Provide an example.
What would happen to a species of bird whose beak didn’t suit the type of food available in its habitat?
What would happen to a species of bird whose beak perfectly suited the type of food available in its habitat?
How is this an example of the process of natural selection?
2Al
In What are the goldilocks conditions for life? you learned all about the Goldilocks conditions for life, how scientists believe life began on Earth and how natural selection and evolution have led to the diversity of plants and animals we have today.
Now it’s time to revisit your What are the goldilocks conditions for life? learning goals and read through them again carefully.
Well done on completing your learning summary. Click here to go to 3.3 Mission Control Report (PBL Part B)
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Simulation Settings - MapleSim Help
Home : Support : Online Help : MapleSim : Using MapleSim : Simulating a Model : Specifying Simulation Settings : Simulation Settings
In the Simulation Settings tab ( ) you can specify the simulation duration time, the number of plot points, the solver, and other parameters for your model.
The following table describes the parameters available in the Simulation section. For information on the parameters available in the Advanced Simulation section, see Advanced Simulation Settings.
{t}_{d}
The duration time of the simulation. You can specify any positive value, including floating-point values.
Note: The duration time is not the same as the end time of your simulation. The end time for a simulation is given by td + ts, where ts is the start time for the simulation (see the Advanced Simulation Settings page).
The type of solver to use for the simulation.
Variable: use a variable time step to maintain error tolerances.
Fixed: use a fixed time step and disregard integration error.
Note: The fixed step solvers are identical to those used by MapleSim's exported code.
Variable: CK45 (non-stiff)
Fixed: Euler
DAE solver used during the simulation. The following choices are available when Solver Type is set to Variable.
CK45 (semi-stiff): use a semi-stiff DAE solver (ck45 method).
RKF45 (non-stiff): use a non-stiff DAE solver (rkf45 method).
Rosenbrock (stiff): use a stiff DAE solver (Rosenbrock method).
If your model is complex, you may want to use a stiff DAE solver to reduce the time required to simulate a model.
The following choices are available when Solver Type is set to Fixed.
Euler: use a forward Euler solver.
Implicit Euler: use an implicit Euler solver (suitable for stiff systems).
RK2: use a second-order Runge-Kutta solver.
RK3: use a third-order Runge-Kutta solver.
RK4: use a fourth-order Runge-Kutta solver.
{\mathrm{ε}}_{\mathrm{abs}}
1\cdot {10}^{-5}
The limit on the absolute error tolerance for a successful integration step if you are using a variable-step solver to run the simulation. You can specify a floating-point value for this option.
{\epsilon }_{\mathrm{rel}}
1\cdot {10}^{-5}
The limit on the relative error tolerance for a successful integration step if you are using a variable-step solver to run the simulation. You can specify a floating-point value for this option.
Uniform size of the sampling periods if you are using a fixed-step solver to run the simulation. You can specify a floating-point value for this option.
Minimum number of points to be plotted in a simulation. The data points are distributed evenly according to the simulation duration value. You can specify a positive integer. Additional points can be added for events (see Plot Events in the Advanced Simulation Settings page).
Note: This option allows you to specify the number of points for display purposes only. The actual number of points used during the simulation may differ from the number of points displayed in the simulation graph.
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Air pollutant concentrations/Citable Version - Citizendium
< Air pollutant concentrations
This version approved either by the Approvals Committee, or an Editor from the listed workgroup. The Engineering Workgroup is responsible for this citable version. While we have done conscientious work, we cannot guarantee that this version is wholly free of mistakes. See here (not History) for authorship.
Air pollutant concentrations, as measured or as calculated by air pollution dispersion modeling, must often be converted or corrected to be expressed as required by the regulations issued by various governmental agencies. Regulations that define and limit the concentration of pollutants in the ambient air or in gaseous emissions to the ambient air are issued by various national and state (or provincial) environmental protection and occupational health and safety agencies.
Such regulations involve a number of different expressions of concentration. Some express the concentrations as ppmv (parts per million by volume) and some express the concentrations as mg/m3 (milligrams per cubic meter), while others require adjusting or correcting the concentrations to reference conditions of moisture content, oxygen content or carbon dioxide content. This article presents methods for converting concentrations from ppmv to mg/m3 (and vice versa) and for correcting the concentrations to the required reference conditions.
All of the concentrations and concentration corrections in this article apply only to air and other gases. They are not applicable for liquids.
1 Converting air pollutant concentrations
2 Correcting concentrations for altitude
3 Correcting concentrations for reference conditions
3.1 Correcting to a dry basis
3.2 Correcting to a reference oxygen content
3.3 Correcting to a reference carbon dioxide content
Converting air pollutant concentrations
The conversion equations depend on the temperature at which the conversion is wanted (usually about 20 to 25 °C). At an ambient sea level atmospheric pressure of 1 atm (101.325 kPa or 1.01325 bar), the general equation is:
{\displaystyle \mathrm {ppmv} =\mathrm {mg} /\mathrm {m} ^{3}\cdot {\frac {(0.08205\cdot T)}{M}}}
and for the reverse conversion:
{\displaystyle \mathrm {mg} /\mathrm {m} ^{3}=\mathrm {ppmv} \cdot {\frac {M}{(0.08205\cdot T)}}}
mg/m3 = milligrams of pollutant per cubic meter of air at sea level atmospheric pressure and T
ppmv = air pollutant concentration, in parts per million by volume
T = ambient temperature in K = 273.15 + °C
0.08205 = Universal gas constant in atm·m3/(kmol·K)
M = molecular mass (or molecular weight) of the air pollutant
1 atm = absolute pressure of 101.325 kPa or 1.01325 bar
mol = gram mole and kmol = 1000 gram moles
Pollution regulations in the United States typically reference their pollutant limits to an ambient temperature of 20 to 25 °C as noted above. In most other nations, the reference ambient temperature for pollutant limits may be 0 °C or other values.
Although ppmv and mg/m3 have been used for the examples in all of the following sections, concentrations such as ppbv (i.e., parts per billion by volume), volume percent, mole percent and many others may also be used for gaseous pollutants.
Particulate matter (PM) in the atmospheric air or in any other gas cannot be expressed in terms of ppmv, ppbv, volume percent or mole percent. PM is most usually (but not always) expressed as mg/m3 of air or other gas at a specified temperature and pressure.
For gases, volume percent = mole percent
1 volume percent = 10,000 ppmv (i.e., parts per million by volume) with a million being defined as 106.
Care must be taken with the concentrations expressed as ppbv to differentiate between the British billion which is 1012 and the USA billion which is 109 (also referred to as the long scale and short scale billion, respectively).
Correcting concentrations for altitude
Air pollutant concentrations expressed as mass per unit volume of atmospheric air (e.g., mg/m3, µg/m3, etc.) at sea level will decrease with increasing altitude. The concentration decrease is directly proportional to the pressure decrease with increasing altitude. Some governmental regulatory jurisdictions require industrial sources of air pollution to comply with sea level standards corrected for altitude. In other words, industrial air pollution sources located at altitudes well above sea level must comply with significantly more stringent air quality standards than sources located at sea level (since it is more difficult to comply with lower standards). For example, New Mexico's Department of the Environment has a regulation with such a requirement.[1][2]
The change of atmospheric pressure with altitude can be obtained from this equation:[3]
{\displaystyle P_{\mathrm {h} }=P\,\cdot {\bigg (}{\frac {288-6.5h}{288}}{\bigg )}^{5.2558}}
Given an air pollutant concentration at sea-level atmospheric pressure, the concentration at higher altitudes can be obtained from this equation:
{\displaystyle C_{\mathrm {h} }=C\,\cdot {\bigg (}{\frac {288-6.5h}{288}}{\bigg )}^{5.2558}}
= altitude, in km
= atmospheric pressure at sea level
= atmospheric pressure at altitude h
= Air pollutant concentration, in mass per unit volume at sea level atmospheric pressure and specified temperature T
= Concentration, in mass per unit volume at altitude h and specified temperature T
As an example, given an air pollutant concentration of 260 mg/m3 at sea level, calculate the equivalent pollutant concentration at an altitude of 2800 meters:
Ch = 260 × [ { 288 - (6.5)(2.8) } / 288] 5.2558 = 260 × 0.71 = 185 mg/m3
The above equation for the decrease of air pollution concentrations with increasing altitude is applicable only for about the first 10 km of altitude in the troposphere (the lowest atmospheric layer) and is estimated to have a maximum error of about 3 percent. However, 10 km of altitude is sufficient for most purposes involving air pollutant concentrations.
Correcting concentrations for reference conditions
Many environmental protection agencies have issued regulations that limit the concentration of pollutants in gaseous emissions and define the reference conditions applicable to those concentration limits. For example, such a regulation might limit the concentration of NOx to 55 ppmv in a dry combustion exhaust gas (at a specified reference temperature and pressure) corrected to 3 volume percent O2 in the dry gas. As another example, a regulation might limit the concentration of total particulate matter to 200 mg/m3 of an emitted gas (at a specified reference temperature and pressure) corrected to a dry basis and further corrected to 12 volume percent CO2 in the dry gas.
Environmental agencies in the USA often use the terms "dscf" or "scfd" to denote a "standard" cubic foot of dry gas. Likewise, they often use the terms "dscm" or "scmd" to denote a "standard" cubic meter of gas. Since there is no universally accepted set of "standard" temperature and pressure, such usage can be and is very confusing. It is strongly recommended that the reference temperature and pressure always be clearly specified when stating gas volumes or gas flow rates. (See Reference conditions of gas temperature and pressure for more explanation)
Correcting to a dry basis
If a gaseous emission sample is analyzed and found to contain water vapor and a pollutant concentration of say 40 ppmv, then 40 ppmv should be designated as the "wet basis" pollutant concentration. The following equation can be used to correct the measured "wet basis" concentration to a "dry basis" concentration:
{\displaystyle C_{\mathrm {dry\,basis} }={\frac {C_{\mathrm {wet\,basis} }}{1-w}}}
= Concentration of the air pollutant in the emitted gas
= fraction, by volume, of the emitted gas
As an example, a wet basis concentration of 40 ppmv in a gas having 10 volume percent water vapor would have a:
Cdry basis = 40 ÷ ( 1 - 0.10 ) = 44.4 ppmv.
Correcting to a reference oxygen content
The following equation can be used to correct a measured pollutant concentration in a dry emitted gas with a measured O2 content to an equivalent pollutant concentration in a dry emitted gas with a specified reference amount of O2:[4]
{\displaystyle C_{\mathrm {r} }=C_{\mathrm {m} }\cdot {\frac {(20.9-\mathrm {reference\,volume\,\%\,O_{2}} )}{(20.9-\mathrm {measured\,volume\,\%\,O_{2}} )}}}
= corrected concentration of a dry gas with a specified reference volume % O2
= measured concentration in a dry gas having a measured volume % O2
As an example, a measured NOx concentration of 45 ppmv in a dry gas having 5 volume % O2 is:
45 × ( 20.9 - 3 ) ÷ ( 20.9 - 5 ) = 50.7 ppmv of NOx
when corrected to a dry gas having a specified reference O2 content of 3 volume %.
The measured gas concentration Cm must first be corrected to a dry basis before using the above equation.
Correcting to a reference carbon dioxide content
The following equation can be used to correct a measured pollutant concentration in an emitted gas (containing a measured CO2 content) to an equivalent pollutant concentration in an emitted gas containing a specified reference amount of CO2:[4]
{\displaystyle C_{\mathrm {r} }=C_{\mathrm {m} }\cdot {\frac {(\mathrm {reference\,volume\,\%\,CO_{2}} )}{(\mathrm {measured\,volume\,\%\,CO_{2}} )}}}
= corrected concentration of a dry gas having a specified reference volume % CO2
= measured concentration of a dry gas having a measured volume % CO2
As an example, a measured particulates concentration of 200 mg/m3 in a dry gas that has a measured 8 volume % CO2 is:
200 × ( 12 ÷ 8 ) = 300 mg/m3
when corrected to a dry gas having a specified reference CO2 content of 12 volume %.
↑ Draft Programmatic Environmental Impact Statement (EIS) for Stockpile Stewardship and Management(See section 03.05 of the EIS which involves the Los Alamos National Laboratory in New Mexico)
↑ Air Quality Impact Analysis (Developed for the United States Bureau of Land Management, Socorro Field Office, New Mexico)
↑ United States Department of Defense MIL-STD-810F, 30 August 2002 (See page 161 of 164 pdf pages)
↑ 4.0 4.1 David A. Lewandowski (1999). Design of Thermal Oxidation Systems for Volatile Organic Compounds, 1st Edition. CRC Press. ISBN 1-56670-410-3.
Retrieved from "https://citizendium.org/wiki/index.php?title=Air_pollutant_concentrations/Citable_Version&oldid=569394"
Emergency management Subgroup Citable Versions
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Trin. Coll. Camb
Please read the enclosed & return them.1 If you will let me know what answer to give to Zacharias I will write. I am much surprized to hear that any publisher will publish it.2 Would it be impertinent to tell him to pocket any profits,—it is so excessively improbable that there will be any.? Jevons’ letter is very pleasing to me & encourages me to believe that I perhaps may do something health notwithstanding.3 I have just finished an account of the Globes for the Philosoph. Mag., & I hope they will put it in, as it has never appeared anywhere yet.4 It took me rather longer than I expected to draw the figures & write it.
I have always been in the habit of going in very late to hall so as to escape a sit of 25 min. before I get anything to eat, but 3 of the very dullest men the world has seen have just taken their M.A’s & come to our table & always sit at the bottom so that I have been cut off from human intercourse for 4 or 5 days— So last night I thought it better to try the waiting dodge, & I shall pretty often in future as the other is very depressing.5
I was repaid last night by meeting a very pleasant American Prof. Gilman, who has come to Europe to get hints about Universities. He is to be president of a new Univ. at Baltimore to wh. some one has given
\frac{1}{4}
million dolls.6 He had met Leo. in S. Francisco, & knew the Nortons well tho’ not a Bostonian.7 How small the world is! Tell Horace that Rendal has got a fellowship.8 We have just reelected Cayley wh. is a good thing, as his professional stipend is not very high.9
I received a pamphlet from Germany this A.M10
Doctor G. H. D Esq.
Our titles seem an endless mystery to foreigners.
I had hardly written the other day when the sickness began again, tho’ not very bad; but today I am too unwell to do anything except write letters. It is the usual bilious business.11 I do not expect wine abstinence will do much for me. At times I feel an intense desire for something strong tasting & eat salt to satisfy it, but I suppose it is the wine I want; however I shall persevere for my month & certainly my average for the last 10 days has been very much higher.
I’m afraid my pitch experiments must wait again for a few days, as it requires making observations every 5 min. for a long time together.12 I have had one short turn at it & find it very difficult as one has to look at a watch & observe an index at same time. I almost think I shall have to get a chronometer & someone to help me, but I shall persevere without for a time & see whether it is likely to lead to any results. It will require hundreds of observations & each of them requires several hours preparation because I must get the pitch to a given temperature thro’out before I can begin.
The fear haunts me that it won’t be of any value when I do get my results. However after a month or two, if I can work, I shall begin to see my way
The enclosures have not been found, but were probably letters regarding the translation of an article written by George on marriages between cousins (G. H. Darwin 1875a; see n. 2, below).
Otto Zacharias had mentioned his desire to arrange for a German translation of George’s work on cousin marriage (G. H. Darwin 1875a) in his letter to CD of 19 August 1875. The translation, with an introduction by Zacharias, was published in 1876 (G. H. Darwin 1876), and a favourable review of it appeared in the Zeitschrift für Ethnologie (Lewkowitsch 1876).
The letter from William Stanley Jevons has not been found. George had written in support of Jevons’s Theory of political economy (Jevons 1871) in the Fortnightly Review in February 1875 (G. H. Darwin 1875d). Jevons suffered from chronic health problems but managed to continue to produce work in economics and logic (ODNB). George also suffered from periodic bouts of illness; see n. 11, below.
George’s paper, ‘Maps of the world’ (G. H. Darwin 1875c) appeared in the Philosophical Magazine in December 1875.
George, who was a fellow of Trinity College, Cambridge, refers to meals served in the dining hall. The men who had recently taken their MA degrees have not been identified.
Daniel Coit Gilman became the first president of Johns Hopkins University, Baltimore, in January 1875. In 1867, Johns Hopkins had endowed $3.5 million for the establishment of the university (ANB).
Gilman was educated in Connecticut and New York before spending a year at Harvard. Charles Eliot Norton and his family had spent four months living near Down in 1868, and had socialised frequently with the Darwins (see letter to C. E. Norton, 7 October 1875 and n. 5). Before becoming president of Johns Hopkins University, Gilman had been president of the University of California at Berkeley. He evidently met Leonard Darwin when Leonard was in San Francisco on his return from the transit of Venus expedition in New Zealand.
Gerald Henry Rendall, a contemporary of Horace Darwin, was elected a fellow of Trinity College, Cambridge, in 1875 (Alum. Cantab.).
Arthur Cayley had been a fellow of Trinity College, Cambridge, from 1842 until 1852; he was an honorary fellow from 1872 and was re-elected fellow in 1875. He was also the first Sadlerian (now spelled Sadleirian) Professor of mathematics at Cambridge (Alum. Cantab.).
The pamphlet has not been identified.
George suffered from chronic stomach problems, as did CD; both tried various diets and other treatments. Andrew Clark, their current physician, relied on a strict dietary regimen (see Correspondence vol. 21, letter from Andrew Clark, 3 September 1873, and Correspondence vol. 24, letter from Andrew Clark, 8 July 1876).
George’s experiments on pitch related to his study of tidal friction and the rigidity of the earth. He later published several papers on the motion of viscous and elastic spheroids that benefited from his observations on the flow of pitch (see G. H. Darwin 1907–16, vol. 2).
Darwin, George Howard. 1907–16. Scientific papers. 5 vols. Cambridge: Cambridge University Press.
Darwin, Horace. 1876. [Description of a dead-weight rotary dynamometer.] Institution of Mechanical Engineers. Proceedings (1876): 231–4.
Jevons, William Stanley. 1871. The theory of political economy. London and New York: Macmillan.
Lewkowitsch, H. 1876. Die Ehen zwischen Geschwisterkindern und ihre Folgen. [Review of G. H. Darwin 1876.] Zeitschrift für Ethnologie 8: 158–62
Sends an article for CD’s opinion.
Has finished an account of the globes for the Philosophical Magazine ["On maps of the world", 50 (1875): 431–44].
His poor health has interfered with his pitch experiments.
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EBITDA-To-Sales Ratio Definition
What Is the EBITDA-To-Sales Ratio?
The EBITDA-to-sales ratio, also known as EBITDA margin, is a financial metric used to assess a company's profitability by comparing its gross revenue with its earnings. More specifically, since EBITDA itself is derived in part from revenue, this metric indicates the percentage of a company's earnings remaining after operating expenses. A higher value indicates the company is able to produce earnings more efficiently by keeping costs low.
The EBITDA-to-sales ratio (EBITDA margin) shows how much cash a company generates for each dollar of sales revenue, before accounting for interest, taxes, and amortization & depreciation.
A low EBITDA-to-sales ratio suggests that a company may have problems with profitability as well as its cash flow, while a high result may indicate a solid business with stable earnings.
Because the ratio excludes the impact of debt interest, highly leveraged companies should cot be evaluated using this metric.
The Formula for the EBITDA-To-Sales Ratio
EBITDA\;\text{margin} = \frac{EBITDA}{\text{Net sales}}
EBITDAmargin=Net salesEBITDA
How to Calculate the EBITDA-To-Sales Ratio
EBITDA is an abbreviation for "earnings before interest, taxes, depreciation, and amortization." Thus, it is calculated adding back these line items to net income, and so does include operating expenses such as the cost of goods sold (COGS) and selling, general, and administrative (SG&A) expenses.
The EBITDA/sales ratio is therefore able to focus on the impact of direct operating costs while excluding the effects of the company's capital structure, tax exposure, and accounting quirks.
What Does the EBITDA-To-Sales Ratio Tell You?
The purpose of EBITDA is to report earnings while exlcluding certain expenses that are considered uncontrollable. EBITDA provides deeper insight into the operational efficiency of an organization based on only those costs management can control.
The EBITDA-to-sales ratio divides the EBITDA by a company's net sales. A ratio equal to 1 implies that a company has no interest, taxes, depreciation, or amortization. It is thus virtually guaranteed that the calculation of a company’s EBITDA-to-sales ratio will be less than 1 because of the deduction of those expenses in the numerator. As a result, the EBITDA-to-sales ratio should not return a value greater than 1. A value greater than 1 is an indicator of a miscalculation. Still, a good EBITDA-to-sales ratio is a number higher in comparison with its peers.
EBITDA-to-sales can be construed as a liquidity measurement, because a comparison is being made between the total revenue earned and the residual net income before certain expenses, showing the total amount a company can expect to receive after operating costs have been paid. Although this is not a true sense of the concept of liquidity, the calculation still reveals how easy it is for a business to cover and pay for certain costs.
Limitations of the EBITDA-To-Sales Ratio
The EBITDA-to-sales ratio for a given company is most useful when comparing to similar-sized companies within the same industry to one another. Because different companies have different cost structures across industries, the EBITDA-to-sales ratio calculations won't tell much during comparison if used to compare against industries with different cost structures.
For example, certain industries may experience more favorable taxation due to tax credits and deductions. These industries incur lower income tax figures and higher EBITDA-to-sales ratio calculations.
Another aspect related to the usefulness of the EBITDA-to-sales ratio concerns the use of depreciation and amortization methods. Because companies can select different depreciation methods, EBITDA-to-sales ratio calculations eliminate the depreciation expense from consideration to improve consistency between companies. Finally, the exclusion of debt interest has its drawbacks when measuring the performance of a company. Companies with high debt levels should not be measured using the EBITDA-to-sale ratio since large and regular interest payments should be included in the financial analysis of such companies.
What Adjusted EBITDA Tells Us
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Multiple solutions of superlinear elliptic equations | EMS Press
Utah State University, Logan, United States
Jiabao Su
In this paper we give some multiplicity results on existence of nontrivial solutions for superlinear elliptic equations with a saddle structure near
0
. We make use of a combination of bifurcation theory and minimax methods.
Paul H. Rabinowitz, Zhi-Qiang Wang, Jiabao Su, Multiple solutions of superlinear elliptic equations. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 18 (2007), no. 1, pp. 97–108
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EUDML | The forms and representations of the Lie algebra . EuDML | The forms and representations of the Lie algebra .
The forms and representations of the Lie algebra
{\text{sl}}_{2}\left(ℤ\right)
Yushchenko, A. V.. "The forms and representations of the Lie algebra .." Sibirskij Matematicheskij Zhurnal 43.5 (2002): 1197-1207 (2002); translation in Sib. Math. J. 43. <http://eudml.org/doc/50694>.
@article{Yushchenko2002,
author = {Yushchenko, A. V.},
keywords = {simple Lie algebra ; diagonal module; Lie algebra; irreducible module; diagonal algebra; -form; simple Lie algebra ; -form},
title = {The forms and representations of the Lie algebra .},
AU - Yushchenko, A. V.
TI - The forms and representations of the Lie algebra .
KW - simple Lie algebra ; diagonal module; Lie algebra; irreducible module; diagonal algebra; -form; simple Lie algebra ; -form
s{l}_{2}\left(ℤ\right)
, diagonal module, Lie algebra, irreducible module, diagonal algebra,
{ℤ}_{p}
-form, simple Lie algebra
s{l}_{2}\left(ℤ\right)
{ℤ}_{p}
Articles by Yushchenko
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Filter outliers using Hampel identifier - MATLAB - MathWorks í•œêµ
\frac{Lenâ1}{2}
Data input, specified as a vector or a matrix. The object accepts multichannel inputs, that is, m-by-n size inputs, where m ≥ 1, and n > 1. m is the number of samples in each frame (channel), and n is the number of channels. The object also accepts variable-size inputs. After the object is locked, you can change the size of each input channel, but you cannot change the number of channels.
Given a sequence x1, x2, x3, …, xn and a sliding window of length k, define point-to-point median and standard-deviation estimates using:
{m}_{i}=\mathrm{median}\left({x}_{iâk},{x}_{iâk+1},{x}_{iâk+2},â¦,{x}_{i},â¦,{x}_{i+kâ2},{x}_{i+kâ1},{x}_{i+k}\right)
{\mathrm{Ï}}_{i}=\mathrm{κ}\mathrm{median}\left(|{x}_{iâk}â{m}_{i}|,â¦,|{x}_{i+k}â{m}_{i}|\right),
\mathrm{κ}=\frac{1}{\sqrt{2}{\mathrm{erfc}}^{â1}\left(1/2\right)}â1.4826
The quantity σi /κ is known as the median absolute deviation (MAD).
|{x}_{i}â{m}_{i}|>{n}_{\mathrm{Ï}}{\mathrm{Ï}}_{i}
for a given threshold nσ, then the Hampel identifier declares xi an outlier and replaces it with mi. If nσ is 0, then the Hampel filter behaves as a regular median filter.
Computes the local median, mi, and standard deviation, σi, over the current window of data.
Compares the current sample with nσ × σi, where nσ is the threshold value. If
|{x}_{s}â{m}_{i}|>{n}_{\mathrm{Ï}}Ã{\mathrm{Ï}}_{i}
In this example, the Hampel filter slides a window of length 5 (Len) over the data. The filter has a threshold value of 2 (nσ). To have a complete window at the beginning of the frame, the filter algorithm prepends the frame with Len – 1 zeros. To compute the first sample of the output, the window centers on the
{\left[\frac{Lenâ1}{2}+1\right]}^{\text{th}}
ma{d}_{i}=\mathrm{median}\left(|{x}_{iâk}â{m}_{i}|,â¦,|{x}_{i+k}â{m}_{i}|\right)
mad=\mathrm{median}\left(|0â0|,â¦,|1â0|\right)=0
Standard deviation: σi = κ × madi = 0, where
\mathrm{κ}=\frac{1}{\sqrt{2}{\mathrm{erfc}}^{â1}\left(1/2\right)}â1.4826
\left[|{x}_{s}â{m}_{i}|=0\right]>\left[\left({n}_{\mathrm{Ï}}Ã{\mathrm{Ï}}_{i}\right)=0\right]
{\left[Endâ\frac{Lenâ1}{2}\right]}^{\text{th}}
\frac{Lenâ1}{2}
[2] Liu, Hancong, Sirish Shah, and Wei Jiang. “On-line outlier detection and data cleaning.†Computers and Chemical Engineering. Vol. 28, March 2004, pp. 1635–1647.
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Deep learning solution of nonstiff ordinary differential equation (ODE) - MATLAB dlode45 - MathWorks Italia
dlode45
Apply Neural ODE Operation
InitialStepSize
Deep learning solution of nonstiff ordinary differential equation (ODE)
Y = dlode45(odefun,tspan,Y0,theta)
Y = dlode45(odefun,tspan,Y0,theta,DataFormat=FMT)
Y = dlode45(odefun,tspan,Y0,theta,Name=Value)
The neural ordinary differential equation (ODE) operation returns the solution of a specified ODE.
The dlode45 function applies the neural ODE operation to dlarray data. Using dlarray objects makes working with high dimensional data easier by allowing you to label the dimensions. For example, you can label which dimensions correspond to spatial, time, channel, and batch dimensions using the "S", "T", "C", and "B" labels, respectively. For unspecified and other dimensions, use the "U" label. For dlarray object functions that operate over particular dimensions, you can specify the dimension labels by formatting the dlarray object directly, or by using the DataFormat option.
The dlode45 function best suits neural ODE and custom training loop workflows. To solve ODEs for other workflows, use ode45.
Y = dlode45(odefun,tspan,Y0,theta) integrates the system of ODEs given by odefun on the time interval defined by the first and last elements of tspan, with the initial conditions Y0 and parameters theta.
Y = dlode45(odefun,tspan,Y0,theta,DataFormat=FMT) specifies the data format for the unformatted initial conditions Y0. The format must contain "S" (spatial), "C" (channel), and "B" (batch) dimension labels only.
Y = dlode45(odefun,tspan,Y0,theta,Name=Value) specifies additional options using one or more name-value arguments. For example, Y = dlode45(odefun,tspan,Y0,theta,GradientsMode="adjoint") integrates the system of ODEs given by odefun and computes gradients by solving the associated adjoint ODE system.
For the initial conditions, create a formatted dlarray object containing a batch of 128 28-by-28 images with 64 channels. Specify the format "SSCB" (spatial, spatial, channel, batch).
Y0 = rand(inputSize(1),inputSize(2),numChannels,miniBatchSize);
dlY0 = dlarray(Y0,"SSCB");
View the size and format of the initial conditions.
size(dlY0)
dims(dlY0)
Specify the ODE function. Define the function odeModel, listed in the ODE Function section of the example, which applies a convolution operation followed by a hyperbolic tangent operation to the input data.
odefun = @odeModel;
Initialize the parameters for the convolution operation in the ODE function. The output size of the ODE function must match the size of the initial conditions, so specify the same number of filters as the number of input channels.
numFilters = numChannels;
parameters.Weights = dlarray(rand(filterSize(1),filterSize(2),numChannels,numFilters));
parameters.Bias = dlarray(zeros(1,numFilters));
Specify an interval of integration of [0 0.1].
Apply the neural ODE operation.
dlY = dlode45(odefun,tspan,dlY0,parameters);
The ODE function odeModel takes as input the function inputs t (unused) and y, and the ODE function parameters p containing the convolution weights and biases, and returns the output of the convolution-tanh block operation. The convolution operation applies padding such that the output size matches the input size.
Function to solve, specified as a function handle that defines the function to integrate.
Specify odefun as a function handle with syntax z = fcn(t,y,p), where t is a scalar, y is a dlarray, and p is a set of parameters. The function returns a dlarray with the same size and format as y. The function must accept all three input arguments t, y, and p, even if not all the arguments are used in the function. The size of the ODE function output z must match the size of the initial conditions.
For example, specify the ODE function that applies a convolution operation followed by a tanh operation.
function z = dlconvtanh(t,y,p)
Note here that the t argument is unused.
numeric vector | unformatted dlarray vector
Interval of integration, specified as a numeric vector or an unformatted dlarray vector with two or more elements. The elements in tspan must be all increasing or all decreasing.
The solver imposes the initial conditions given by Y0 at the initial time tspan(1), then integrates the ODE function from tspan(1) to tspan(end).
If tspan has two elements, [t0 tf], then the solver returns the solution evaluated at point tf.
If tspan has more than two elements, [t0 t1 t2 ... tf], then the solver returns the solution evaluated at the given points [t1 t2 ... tf]. The solver does not step precisely to each point specified in tspan. Instead, the solver uses its own internal steps to compute the solution, then evaluates the solution at the points specified in tspan. The solutions produced at the specified points are of the same order of accuracy as the solutions computed at each internal step.
Specifying several intermediate points has little effect on the efficiency of computation, but for large systems it can affect memory management.
The behavior of the dlode45 function differs from the ode45 function.
If InitialStep or MaxStep is [], then the software uses the values of tspan to initialize the values.
If InitialStep is [], then the software uses the elements of tspan as an indication of the scale of the task. When you specify tspan with different numbers of elements, the solution of the solver can change.
If MaxStep is [], then the software calculates the maximum step size using the first and last elements of tspan. When you change the initial or final values of tspan, the solution of the solver can change because the solver uses a different step sequence.
Initial conditions, specified as a formatted or unformatted dlarray object.
If Y0 is an unformatted dlarray, then you must specify the format using the DataFormat option.
For neural ODE operations, the data format must contain "S", "C", and "B" dimension labels only. The initial conditions must not have a "T" or "U" dimension.
theta — Parameters of ODE function
dlarray object | cell array of dlarray objects | structure of dlarray objects | table
Parameters of ODE function, specified as one of the following:
Cell array of dlarray objects
Structure of dlarray objects or nested structures of dlarray objects
Table with the variables Layer, Parameter, and Value, where Layer and Parameter contain the layer and parameter names, and Value contains the parameter value. Specify the variables as dlarray objects.
Example: Y = dlode45(odefun,tspan,Y0,theta,GradientsMode="adjoint") integrates the system of ODEs given by odefun and computes gradients by solving the associated adjoint ODE system.
You must specify DataFormat when the Y0 is not a formatted dlarray.
Example: DataFormat="SSCB"
GradientMode — Method to compute gradients
"direct" (default) | "adjoint"
Method to compute gradients with respect to the initial conditions and parameters when using the dlgradient function, specified as one of the following:
"direct" – Compute gradients by backpropagating through the operations undertaken by the numerical solver. This option best suits large mini-batch sizes or when tspan contains many values.
"adjoint" – Compute gradients by solving the associated adjoint ODE system. This option best suits small mini-batch sizes or when tspan contains a small number of values.
When GradientMode is "adjoint", odefun must support function acceleration. Otherwise, the function can return unexpected results.
When GradientMode is "adjoint", the software traces the ODE function input to determine the computation graph used for automatic differentiation. This tracing process can take some time and can end up recomputing the same trace. By optimizing, caching, and reusing the traces, the software can speed up the gradient computation.
For more information on deep learning function acceleration, see Deep Learning Function Acceleration for Custom Training Loops.
Initial step size, specified as a positive scalar or [].
If InitialStepSize is [], then the function automatically determines the initial step size based on the interval of integration and the output of the ODE function corresponding to the initial conditions.
MaxStepSize — Maximum step size
Maximum step size, specified as a positive scalar or [].
If MaxStepSize is [], then the function uses a tenth of the interval of integration size.
RelativeTolerance — Relative error tolerance
Relative error tolerance, specified as a positive scalar. The relative tolerance applies to all components of the solution.
AbsoluteTolerance — Absolute error tolerance
Absolute error tolerance, specified as a positive scalar. The relative tolerance applies to all components of the solution.
Y — Solution of neural ODE
Solution of the neural ODE at the times given by tspan(2:end), returned as a dlarray object with the same underlying data type as Y0.
If Y0 is a formatted dlarray and tspan contains exactly two elements, then Y has the same format as Y0. If Y0 is not a formatted dlarray and tspan contains exactly two elements, then Y is an unformatted dlarray with the same dimension order as Y0.
If Y0 is a formatted dlarray and tspan contains more than two elements, then Y has the same format as Y0 with an additional appended "T" (time) dimension. If Y0 is not a formatted dlarray and tspan contains more than two elements, then Y is an unformatted dlarray with the same dimension order as Y0 with an additional appended dimension corresponding to time.
The neural ordinary differential equation (ODE) operation returns the solution of a specified ODE. In particular, given an input, a neural ODE operation outputs the numerical solution of the ODE
y\prime =f\left(t,y,\theta \right)
for the time horizon (t0,t1) and with the initial condition y(t0) = y0, where t and y denote the ODE function inputs and θ is a set of learnable parameters. Typically, the initial condition y0 is either the network input or the output of another deep learning operation.
The dlode45 function uses the ode45 function, which is based on an explicit Runge-Kutta (4,5) formula, the Dormand-Prince pair. It is a single-step solver–in computing y(tn), it needs only the solution at the immediately preceding time point, y(tn-1) [2] [3].
[1] Chen, Ricky T. Q., Yulia Rubanova, Jesse Bettencourt, and David Duvenaud. “Neural Ordinary Differential Equations.” Preprint, submitted June 19, 2018. https://arxiv.org/abs/1806.07366.
[2] Dormand, J. R., and P. J. Prince. “A Family of Embedded Runge-Kutta Formulae.” Journal of Computational and Applied Mathematics 6, no. 1 (March 1980): 19–26. https://doi.org/10.1016/0771-050X(80)90013-3.
dlarray | dlgradient | dlfeval
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Material cache (Armadylean yellow) - The RuneScape Wiki
Material cache (Armadylean yellow)
Material caches (Armadylean yellow) are Archaeology material caches that can be found in the Stormguard Citadel Dig Site at either the keshik memorial or relay station excavation sites, or at the Empyrean Citadel excavation site.
Once an Armadylean yellow cache is depleted, it takes 120 seconds for it to respawn.
Armadylean yellow 1 Always[d 1] 7,251 150
^ These are only found at the Stormguard Citadel Dig Site.
{\displaystyle {\frac {L+E}{250{,}000}}}
{\displaystyle L}
{\displaystyle E}
{\displaystyle {\frac {1}{125{,}000}}}
{\displaystyle {\frac {1}{1{,}042}}}
Retrieved from ‘https://runescape.wiki/w/Material_cache_(Armadylean_yellow)?oldid=35694514’
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Outlier removal using Hampel identifier - MATLAB hampel - MathWorks India
Remove Spikes from Sinusoid
Hampel Filtering of Multichannel Signal
Find Outliers in Multichannel Signal
Signal Statistics Returned by Hampel Filter
xmedian
Outlier removal using Hampel identifier
y = hampel(x)
y = hampel(x,k)
y = hampel(x,k,nsigma)
[y,j] = hampel(___)
[y,j,xmedian,xsigma] = hampel(___)
hampel(___)
y = hampel(x) applies a Hampel filter to the input vector, x, to detect and remove outliers. For each sample of x, the function computes the median of a window composed of the sample and its six surrounding samples, three per side. It also estimates the standard deviation of each sample about its window median using the median absolute deviation. If a sample differs from the median by more than three standard deviations, it is replaced with the median. If x is a matrix, then hampel treats each column of x as an independent channel.
y = hampel(x,k) specifies the number of neighbors, k, on either side of each sample of x in the measurement window. k defaults to 3.
y = hampel(x,k,nsigma) specifies a number of standard deviations, nsigma, by which a sample of x must differ from the local median for it to be replaced with the median. nsigma defaults to 3.
[y,j] = hampel(___) also returns a logical matrix that is true at the locations of all points identified as outliers. This syntax accepts any of the input arguments from previous syntaxes.
[y,j,xmedian,xsigma] = hampel(___) also returns the local medians and the estimated standard deviations for each element of x.
hampel(___) with no output arguments plots the filtered signal and annotates the outliers that were removed.
Generate 100 samples of a sinusoidal signal. Replace the sixth and twentieth samples with spikes.
x = sin(2*pi*(0:99)/100);
x(20) = -2;
Use hampel to locate every sample that differs by more than three standard deviations from the local median. The measurement window is composed of the sample and its six surrounding samples, three per side.
[y,i,xmedian,xsigma] = hampel(x);
Plot the filtered signal and annotate the outliers.
n = 1:length(x);
plot(n,x)
plot(n,xmedian-3*xsigma,n,xmedian+3*xsigma)
plot(find(i),x(i),'sk')
legend('Original signal','Lower limit','Upper limit','Outliers')
Repeat the computation, but now take just one adjacent sample on each side when computing the median. The function considers the extrema as outliers.
hampel(x,1)
Filter the signal using hampel with the default settings.
y = hampel(x);
Increase the length of the moving window and decrease the threshold to treat a sample as an outlier.
y = hampel(x,4,2);
Output the running median for each channel. Overlay the medians on a plot of the signal.
[y,j,xmd,xsd] = hampel(x,4,2);
plot(xmd,'--')
Generate a multichannel signal that consists of two sinusoids of different frequencies embedded in white Gaussian noise of unit variance.
x = sin(pi./[10;2]*t)'+randn(numel(t),2);
Apply a Hampel filter to the signal. Take as outliers those points that differ by more than two standard deviations from the median of a surrounding nine-sample window. Output a logical matrix that is true at the locations of the outliers.
[y,h] = hampel(x,k,nsig);
Plot each channel of the signal in its own set of axes. Draw the original signal, the filtered signal, and the outliers. Annotate the outlier locations.
hk = h(:,k);
ax = subplot(2,1,k);
plot(t,x(:,k))
plot(t,y(:,k))
plot(t(hk),x(hk,k),'*')
ax.XTick = t(hk);
x = sin(2*pi*n/100);
Use hampel to compute the local median and estimated standard deviation for every sample. Use the default values of the input parameters:
The window size is
2×3+1=7
The points that differ from their window median by more than three standard deviations are considered outliers.
plot(n,[1;1]*xmedian+3*[-1;1]*xsigma)
legend('Signal','Lower','Upper','Outliers')
Repeat the calculation using a window size of
2×10+1=21
and two standard deviations as the criteria for identifying outliers.
sds = 2;
adj = 10;
[y,i,xmedian,xsigma] = hampel(x,adj,sds);
plot(n,[1;1]*xmedian+sds*[-1;1]*xsigma)
Input signal, specified as a vector or matrix. If x is a matrix, then hampel treats each column of x as an independent channel.
k — Number of neighbors on either side
Number of neighbors on either side of the sample xs, specified as an integer scalar. Samples close to the signal edges that have fewer than k samples on one side are compared to the median of a smaller window.
nsigma — Number of standard deviations
Number of standard deviations by which a sample of x must differ from its local median to be considered an outlier. Specify nsigma as a real scalar. The function estimates the standard deviation by scaling the local median absolute deviation (MAD) by a factor of
\kappa =\frac{1}{\sqrt{2}{\mathrm{erf}}^{-1}\left(1/2\right)}\approx 1.4826
Filtered signal, returned as a vector or matrix of the same size as x.
j — Outlier index
Outlier index, returned as a vector or matrix of the same size as x.
xmedian — Local medians
Local medians, returned as a vector or matrix of the same size as x.
xsigma — Estimated standard deviations
Estimated standard deviations, returned as a vector or matrix of the same size as x.
{m}_{i}=\mathrm{median}\left({x}_{i-k},{x}_{i-k+1},{x}_{i-k+2},\dots ,{x}_{i},\dots ,{x}_{i+k-2},{x}_{i+k-1},{x}_{i+k}\right)
{\sigma }_{i}=\kappa \mathrm{median}\left(|{x}_{i-k}-{m}_{i}|,\dots ,|{x}_{i+k}-{m}_{i}|\right),
\kappa =\frac{1}{\sqrt{2}{\mathrm{erf}}^{-1}\left(1/2\right)}\approx 1.4826
|{x}_{i}-{m}_{i}|>{n}_{\sigma }{\sigma }_{i}
for a given threshold nσ, then the Hampel identifier declares xi an outlier and replaces it with mi.
Near the sequence endpoints, the function truncates the window used to compute mi and σi:
i < k + 1
{m}_{i}=\mathrm{median}\left({x}_{1},{x}_{2},{x}_{3},\dots ,{x}_{i},\dots ,{x}_{i+k-2},{x}_{i+k-1},{x}_{i+k}\right)
{\sigma }_{i}=\kappa \mathrm{median}\left(|{x}_{1}-{m}_{1}|,\dots ,|{x}_{i+k}-{m}_{i}|\right)
i > n – k
{m}_{i}=\mathrm{median}\left({x}_{i-k},{x}_{i-k+1},{x}_{i-k+2},\dots ,{x}_{i},\dots ,{x}_{n-2},{x}_{n-1},{x}_{n}\right)
{\sigma }_{i}=\kappa \mathrm{median}\left(|{x}_{i-k}-{m}_{i}|,\dots ,|{x}_{n}-{m}_{n}|\right)
[2] Suomela, Jukka. “Median Filtering Is Equivalent to Sorting.” 2014.
medfilt1 | median | filloutliers | filter | isoutlier | mad (Statistics and Machine Learning Toolbox) | movmad | movmedian | sgolayfilt
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On the regularity of weak solutions to <em>H</em>-systems | EMS Press
In this paper we prove that every weak solution to the
H
-surface equation is locally bounded, provided the prescribed mean curvature
H
satisfies a suitable condition at infinity. No smoothness assumption is required on
H
. We consider also the Dirichlet problem for the
H
-surface equation on a bounded regular domain with
L^{\infty}
boundary data and the
H
-bubble problem. Under the same assumptions on
H
, we prove that every weak solution is globally bounded.
Roberta Musina, On the regularity of weak solutions to <em>H</em>-systems. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 18 (2007), no. 3, pp. 209–219
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Machine-Learning-Based Identification of Nonlinear Magneto-Rheological Fluid Damper - MATLAB & Simulink - MathWorks Australia
Explore Estimation Options
Compare Estimated Models with Other Nonlinearities
This example shows how to estimate a black-box dynamic model of a nonlinear magneto-rheological (MR) fluid damper. Magneto-rheological dampers are used to reduce structural vibrations by applying a controlled force, which depends on the input voltage and current of the device. A nonlinear ARX model with a regression tree ensemble mapping function is used to model the MR damper.
The data used in this example was provided by Dr. Akira Sano (Keio University, Japan) and Dr. Jiandong Wang (Peking University, China) who performed the experiments in a laboratory of Keio University. For a more detailed description of the experimental system and some related studies see [2].
In the experiment, one end of an MR damper fixed to the ground. The other end is connected to a shaker table that generates vibrations. The data set contains 3,499 input-output samples. The input is the velocity
\mathit{V}
of the damper measured in cm/sec and the output is the damping force
\mathit{F}
measured in Newtons. Load the experimental data.
load('mrdamper.mat')
Construct an iddata object from
\mathit{F}
\mathit{V}
data = iddata(F,V,Ts);
Split the data into estimation data and validation data. To estimate a nonlinear ARX model with a regression tree ensemble mapping function, use the first 3,000 samples of the input-output data. Use the last 499 samples for validation.
% Estimation data
ze = data(1:3000);
% Validation data
zv = data(3001:end);
Plot the estimation and validation data
plot(zv)
Estimate a nonlinear ARX model using the tree ensemble nonlinearity. Set the nonlinear ARX model orders to [16 16 0]. Doing so configures the model to use 16 input regressors, 16 output regressors, and no input-output delays.
mdlTreeEnsDef = nlarx(ze,[16 16 0],idTreeEnsemble);
Plot the simulated response of the estimated model along with the estimation data.
compare(ze,mdlTreeEnsDef)
Plot the simulated response of the estimated model along with the validation data.
compare(zv,mdlTreeEnsDef)
Use nondefault values for the estimation options. For this example, set the fit method to 'lsboost-resampled'. Estimate a model and plot its simulated response.
ens.EstimationOptions.FitMethod = 'lsboost-reweighted';
ens.EstimationOptions.NumLearningCycles = 150;
mdlTreeEnsBoostReweight = nlarx(ze,[16 16 0],ens);
compare(zv,mdlTreeEnsBoostReweight)
Reduce the estimation time and obtain and compact regression ensemble model by using the Shrink estimation option.
ens.EstimationOptions.Shrink = true;
mdlTreeEnsDefShrink = nlarx(ze,[16 16 0],ens);
compare(ze,mdlTreeEnsDefShrink)
compare(zv,mdlTreeEnsDefShrink)
Use the training data to estimate nonlinear ARX models using wavelet network and sigmoid network nonlinearities.
mdlWaveletNet = nlarx(ze,[16 16 0],idWaveletNetwork);
mdlSigmoidNet = nlarx(ze,[16 16 0],idSigmoidNetwork);
Compare the identified nonlinear ARX models obtained in this example using the validation data.
compare(zv,mdlTreeEnsDef,mdlTreeEnsBoostReweight,...
mdlTreeEnsDefShrink,mdlWaveletNet,mdlSigmoidNet)
[1] Breiman, Leo, ed. Classification and Regression Trees. CRC Press repr. Boca Raton, Fla.: Chapman & Hall/CRC, 1984.
[2] Wang, Jiandong, Akira Sano, Tongwen Chen, and Biao Huang. ‘Identification of Hammerstein Systems without Explicit Parameterisation of Non-Linearity’. International Journal of Control 82, no. 5 (May 2009): 937–52. https://doi.org/10.1080/00207170802382376.
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Differential equation - zxc.wiki
A differential equation (also differential equation , often DGL , DG , DGI. Or Dgl. Is abbreviated) a mathematical equation for a sought function of one or more variables, which also discharges occur this function. Many laws of nature can be formulated using differential equations. Differential equations are therefore an essential tool in mathematical modeling . A differential equation describes the behavior of these variables in relation to one another. Differential equations are an important subject of analysis in analysis , which investigates its solution theory. Not only because no explicit solution representation is possible for many differential equations, the approximate solution using numerical methods plays an essential role. A differential equation can be illustrated by a direction field .
1 Types of Differential Equations
1.2 Partial differential equation
5 Higher levels of abstraction
There are different types of differential equations. They are roughly divided into the following sub-areas. All of the following types can essentially co-exist independently and simultaneously.
Main article : Ordinary differential equation
If the function you are looking for depends only on one variable, it is called an ordinary differential equation. There are only ordinary derivatives according to the one variable.
{\ displaystyle y '(x) = - 2 \ cdot y (x) +5, \ qquad {\ ddot {z}} (t) +4 \ cdot z (t) = \ sin (3 \ cdot t)}
Writes the ordinary differential equation for the function you are looking for in the form
{\ displaystyle y (x)}
{\ displaystyle F \ left (x, y (x), y '(x), \ ldots, y ^ {(n)} (x) \ right) = 0,}
so the ordinary differential equation is called implicit .
Is the differential equation solved for the highest derivative, i. i.e., it applies
{\ displaystyle y ^ {(n)} (x) = f \ left (x, y (x), y '(x), \ ldots, y ^ {(n-1)} (x) \ right), }
so one calls the ordinary differential equation explicit . In the applications, explicit ordinary differential equations are mathematically easier to process. The highest order of derivation that occurs is called the order of the differential equation . For example, an explicit ordinary first order differential equation has the form
{\ displaystyle n}
{\ displaystyle y '(x) = f (x, y (x)) \ ,.}
There is a closed theory of solving explicit ordinary differential equations.
An ordinary differential equation is linear if it is linear in function and its derivatives:
{\ displaystyle y ^ {(n)} (x) + a_ {n-1} (x) \, y ^ {(n-1)} (x) + \ cdots + a_ {0} (x) \, y (x) = b (x)}
It is semilinear if it is linear in the derivatives and the function on the left, but the function can also depend on the function and its derivatives, except for the highest derivative:
{\ displaystyle b (x)}
{\ displaystyle y}
{\ displaystyle y ^ {(n)} (x) + a_ {n-1} (x) \, y ^ {(n-1)} (x) + \ cdots + a_ {0} (x) \, y (x) = b (x, y (x), y '(x), \ cdots, y ^ {(n-1)} (x))}
If the function you are looking for depends on several variables and if partial derivatives occur in the equation for more than one variable, then one speaks of a partial differential equation. Partial differential equations are a large field and the theory is not mathematically closed, but is the subject of current research in several areas.
One example is the so-called heat conduction equation for a function
{\ displaystyle u (t, x)}
{\ displaystyle {\ tfrac {\ partial} {\ partial t}} u (t, x) = a {\ tfrac {\ partial ^ {2}} {\ partial x ^ {2}}} u (t, x )}
A distinction is made between different types of partial differential equations. First there are linear partial differential equations . The function you are looking for and its derivatives are included linearly in the equation. The dependency with regard to the independent variables can definitely be non-linear. The theory of linear partial differential equations is the most advanced, but far from complete.
One speaks of a quasilinear equation if all derivatives of the highest order occur linearly, but this no longer applies to the function and derivatives of lower order. A quasi-linear equation is more difficult to deal with. A quasi-linear partial differential equation is semilinear if the coefficient function before the highest derivatives does not depend on lower derivatives and the unknown function. Most of the results are currently being achieved in the area of quasi-linear and semilinear equations.
Finally, if one cannot determine a linear dependency with regard to the highest derivatives, the equation is called a nonlinear partial differential equation or a completely nonlinear partial differential equation .
The equations of the second order are of particular interest in the field of partial differential equations. In these special cases there are further classification options.
With the type of stochastic differential equations, stochastic processes occur in the equation . Actually, stochastic differential equations are not differential equations in the above sense, but only certain differential relations that can be interpreted as differential equations.
The type of algebro differential equations is characterized by the fact that, in addition to the differential equation, there are also algebraic relations as secondary conditions .
There are also so-called retarded differential equations . In addition to a function and its derivatives , function values or derivatives from the past also occur at a point in time .
{\ displaystyle t}
An integro-differential equation is an equation in which not only the function and its derivatives, but also integrations of the function appear. An important example is the Schrödinger equation in the momentum representation ( Fredholm 's integral equation ).
Depending on the area of application and methodology, there are other types of differential equations.
One speaks of a system of differential equations when there is a vector-valued mapping and several equations
{\ displaystyle y = (y_ {1}, \ ldots, y_ {k})}
{\ displaystyle F_ {l} \ left (x, y, Dy, \ ldots, D ^ {n} y \ right) = 0, \ qquad l = 1, \ ldots, k.}
must be fulfilled at the same time. If this implicit differential equation system cannot be converted locally into an explicit system everywhere, then it is an algebro differential equation .
The solution set of a differential equation is generally not uniquely determined by the equation itself, but requires additional initial or boundary values . So-called initial boundary value problems can also occur in the area of partial differential equations.
In the case of initial or initial boundary value problems, one of the variables is generally interpreted as time. With these problems, certain dates are prescribed at a certain point in time, namely the start point in time.
In the case of boundary value or initial boundary value problems, a solution to the differential equation is sought in a restricted or unrestricted area and we provide so-called boundary values as data, which are given on the boundary of the area. Depending on the type of boundary conditions, a distinction is made between other types of differential equations, such as Dirichlet problems or Neumann problems .
Due to the diversity of both the actual differential equations and the problem definitions, it is not possible to provide a generally applicable solution method. Only explicit ordinary differential equations can be solved with a closed theory. A differential equation is called integrable if it is possible to solve it analytically, i.e. to specify a solution function (the integral ). Many mathematical problems, in particular nonlinear and partial differential equations, cannot be integrated, including those that appear quite simple, such as the three-body problem , the double pendulum or most of the top types .
A structured general approach to solving differential equations is pursued through symmetry and continuous group theory. In 1870, Sophus Lie put the theory of differential equations on a generally applicable basis with the Lie theory . He showed that the older mathematical theories for solving differential equations can be summarized by the introduction of so-called Lie groups . A general approach to solving differential equations takes advantage of the symmetry property of differential equations. Continuous infinitesimal transformations are used that map solutions to (other) solutions of the differential equation. Continuous group theory, Lie algebras and differential geometry are used to grasp the deeper structure of the linear and nonlinear (partial) differential equations and to map the relationships that ultimately lead to the exact analytical solutions of a differential equation. Symmetry methods are used to solve differential equations exactly.
The questions of existence, uniqueness, representation and numerical calculation of solutions are therefore completely or not at all solved, depending on the equation. Due to the importance of differential equations in practice, the application of numerical solution methods, especially with partial differential equations, is more advanced than their theoretical underpinning.
One of the Millennium Problems is the proof of the existence of a regular solution to the Navier-Stokes equations . These equations occur in fluid mechanics , for example .
As a solution, differential equations have functions that fulfill conditions for their derivatives . An approximation usually takes place by dividing space and time into a finite number of parts using a computational grid ( discretization ). The derivatives are then no longer represented by a limit value, but are approximated by differences. In numerical mathematics , the resulting error is analyzed and estimated as well as possible.
Depending on the type of equation, different discretization approaches are chosen, in the case of partial differential equations, for example, finite difference methods , finite volume methods or finite element methods .
The discretized differential equation no longer contains any derivatives, but only purely algebraic expressions. This results in either a direct solution rule or a linear or non-linear system of equations , which can then be solved using numerical methods.
Appearance and applications
A multitude of phenomena in nature and technology can be described by differential equations and mathematical models based on them. Some typical examples are:
Many physical theories are based on differential equations: equations of motion or vibrations in Newtonian mechanics , the load behavior of components , electrodynamics is dominated by the Maxwell equations , quantum mechanics by the Schrödinger equation .
in astronomy the orbits of the celestial bodies and the turbulence in the interior of the sun,
in biology, for example, processes in growth , in currents or in muscles , or in evolutionary theory .
in chemistry the kinetics of reactions ,
in electrical engineering the behavior of networks with energy-storing elements,
the behavior of surfaces in differential geometry ,
in fluid mechanics the behavior of these very currents,
in economics the analysis of economic growth processes ( growth theory ).
in computer science, for example, image inpainting (calculating fonts or logos from images)
The field of differential equations has given mathematics decisive impulses. Many parts of current mathematics research the existence, uniqueness and stability theory of various types of differential equations.
Differential equations or systems of differential equations require that a system can be described and quantified in algebraic form . Furthermore, that the descriptive functions can be differentiated at least in the areas of interest . In the scientific and technical environment, these requirements are often met, but in many cases they are not met. Then the structure of a system can only be described on a higher abstraction level. See in the order of increasing abstraction:
Systems theory (disambiguation)
Formal concept analysis (mathematics)
Order relation (mathematics)
GH Golub, JM Ortega: Scientific Computing and Differential Equations. An introduction to numerical mathematics . Heldermann Verlag, Lemgo 1995, ISBN 3-88538-106-0 .
G. Oberholz: Differential equations for technical professions. 4th edition. Verlag Anita Oberholz, Gelsenkirchen 1995, ISBN 3-9801902-4-2 .
PJ Olver: Equivalence, Invariants and Symmetry . Cambridge Press, 1995.
L. Papula: Mathematics for Engineers and Natural Scientists Volume 2 . Vieweg's reference books of technology, Wiesbaden 2001, ISBN 3-528-94237-1 .
H. Stephani: Differential Equations: Their Solution Using Symmetries. Edited by M. MacCallum. Cambridge University Press, 1989.
H. Benker: Differential equations with MATHCAD and MATLAB . Springer-Verlag, Berlin / Heidelberg / New York 2005.
Differential equations Matheplanet - Instructions for solving various differential equations with examples
Mathematics online course on differential equation at the University of Stuttgart
Dörte Haftendorn: Differential equations: numerics, examples, isoclines, ... University of Lüneburg
↑ Guido Walz (ed.), Lexikon der Mathematik, Springer-Spektrum Verlag, 2017, article linear differential equation, semilinear differential equation
↑ Peterson, Ivars: Filling in Blanks . In: Society for Science & # 38 (Ed.): Science News . 161, No. 19, pp. 299-300. doi : 10.2307 / 4013521 . Retrieved May 11, 2008.
This page is based on the copyrighted Wikipedia article "Differentialgleichung" (Authors); it is used under the Creative Commons Attribution-ShareAlike 3.0 Unported License. You may redistribute it, verbatim or modified, providing that you comply with the terms of the CC-BY-SA.
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Higher-Order Hermite-Fejér Interpolation for Stieltjes Polynomials
2013 Higher-Order Hermite-Fejér Interpolation for Stieltjes Polynomials
Hee Sun Jung, Ryozi Sakai
{w}_{\lambda }\left(x\right):=\left(1-{x}^{2}{\right)}^{\lambda -1/2}
{P}_{\lambda ,n}
be the ultraspherical polynomials with respect to
{w}_{\lambda }\left(x\right)
. Then, we denote the Stieltjes polynomials
{E}_{\lambda ,n+1}
{w}_{\lambda }\left(x\right)
{\int }_{-1}^{1}\mathrm{}{w}_{\lambda }\left(x\right){P}_{\lambda ,n}\left(x\right){E}_{\lambda ,n+1}\left(x\right){x}^{m}d\mathrm{x }
\left(=0, 0\le m<n+1; \ne 0, m=n+1\right)
. In this paper, we consider the higher-order Hermite-Fejér interpolation operator
{H}_{n+1,m}
based on the zeros of
{E}_{\lambda ,n+1}
and the higher order extended Hermite-Fejér interpolation operator
{ℋ}_{2n+1,m}
{E}_{\lambda ,n+1}{P}_{\lambda ,n}
m
is even, we show that Lebesgue constants of these interpolation operators are
O\left({n}^{\text{m}\text{a}\text{x}\left\{\left(1-\lambda \right)m-2,0\right\}}\right)\left(0<\lambda <1\right)
O\left({n}^{\text{m}\text{a}\text{x}\left\{\left(1-2\lambda \right)m-2,0\right\}}\right)\left(0<\lambda <1/2\right)
∥{ℋ}_{2n+1,m}∥=O\left({n}^{\text{m}\text{a}\text{x}\left\{\left(1-2\lambda \right)m-2,0\right\}}\right)\left(0<\lambda <1\right)
∥{H}_{n+1,m}∥=O\left({n}^{\text{m}\text{a}\text{x}\left\{\left(1-\lambda \right)m-2,0\right\}}\right)\left(0<\lambda <1/2\right)
. In the case of the Hermite-Fejér interpolation polynomials
{ℋ}_{2n+1,m}\left[·\right]
1/2\le \lambda <1
, we can prove the weighted uniform convergence. In addition, when
m
is odd, we will show that these interpolations diverge for a certain continuous function on
\left[-1,1\right]
, proving that Lebesgue constants of these interpolation operators are similar or greater than log
n
Hee Sun Jung. Ryozi Sakai. "Higher-Order Hermite-Fejér Interpolation for Stieltjes Polynomials." J. Appl. Math. 2013 1 - 15, 2013. https://doi.org/10.1155/2013/542653
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H∞ Fuzzy Tracking Control Design for Nonlinear Active Fault Tolerant Control Systems | J. Dyn. Sys., Meas., Control. | ASME Digital Collection
H∞
Fuzzy Tracking Control Design for Nonlinear Active Fault Tolerant Control Systems
School of Automation Science and Electrical Engineering,
, Beijing 100083, P.R.C.
Ming-Zhen Bai
Wu, H., and Bai, M. (June 9, 2008). "
H∞
Fuzzy Tracking Control Design for Nonlinear Active Fault Tolerant Control Systems." ASME. J. Dyn. Sys., Meas., Control. July 2008; 130(4): 041010. https://doi.org/10.1115/1.2936875
This paper studies the problem of
H∞
fuzzy tracking control design for nonlinear active fault tolerant control systems based on the Takagi and Sugeno fuzzy model. Two random processes with Markovian transition characteristics are introduced to model the system component fault process and the fault detection and isolation decision process used to reconfigure the control law, respectively. The random behavior of the FDI process is conditioned on the fault process state. The parallel distributed compensation scheme is employed for the control design. As a result, a closed-loop fuzzy system with two Markovian jump parameters is obtained. Based on a stochastic Lyapunov function, a sufficient condition for stochastic stability of the closed-loop fuzzy system with a guaranteed
H∞
model reference tracking performance is first derived. A linear matrix inequality approach to the control design is then developed to reduce the effect of the external disturbance and reference input on tracking error as small as possible. Finally, a simulation example is presented to illustrate the effectiveness of the proposed design method.
closed loop systems, control system synthesis, fault tolerance, fuzzy control, linear matrix inequalities, Markov processes, nonlinear control systems, stability, tracking, fault tolerant control, fuzzy control, linear matrix inequality, nonlinear systems, stochastic stability, tracking
Control systems, Design, Fuzzy control, Fuzzy logic, Linear matrix inequalities, Stability, Tracking control, Closed loop systems, Control equipment, Markov processes, Design methodology, Errors
Design of Reliable Control Systems
Reliable H∞ Control for Affine Nonlinear Systems
Reliable Mixed L2∕H∞ Fuzzy Static Output Feedback Control for Nonlinear Systems With Sensor Faults
Reliable H∞ Fuzzy Control for Continuous-Time Nonlinear Systems With Actuator Failures
Application of Precomputed Control Laws in a Reconfigurable Aircraft Flight Control System
Stability of the Pseudo-Inverse Method for Reconfigurable Control Systems
Design of Reconfigurable Control System Using Eigenstructure Assignment
Theilliol
Fault-Tolerant Control in Dynamic Systems: Application to a Winding Machine
Integrated Active Fault-Tolerant Control Using IMM Approach
Fault Identification and Fault-Tolerant Control for a Class of Networked Control Systems
Int. J. Innov. Comp. Inf. Control
Detection Delays, False Alarm Rates and Reconfiguration of Control Systems
Stochastic Stability Analysis for Continuous-Time Fault Tolerant Control Systems
H∞-Control for Markovian Jumping Linear Systems With Parametric Uncertainty
Stochastic Stability Analysis of Fault Tolerant Control Systems in the Presence of Noise
Stability and Performance of the Stochastic Fault Tolerant Control Systems
, Maui, HI, Dec., pp.
Stabilizing Controller Parametrization of Fault Tolerant Control Systems
,” Proceedings of the 44th IEEE Conference on Decision and Control and European Control Conference,
Design of Stochastic Fault Tolerant Control for H2 Performance
Fault Diagnosis in Dynamic Systems: Theory and Applications
Berdjag
Nonlinear Model Decomposition for Robust Fault Detection and Isolation Using Algebraic Tools
H∞ Fault Detection Filter Design for Networked Control Systems Modelled by Discrete Markovian Jump Systems
A Survey on Analysis and Design of Model-based Fuzzy Control Systems
Fuzzy Stabilization of Power Systems in a Co-Generation Scheme Subject to Random Abrupt Variations of Operating Conditions
Stabilization of Nonlinear Systems Under Variable Sampling: A Fuzzy Control Approach
Exact Fuzzy Modeling and Optimal Control of the Inverted Pendulum on Cart
H∞ Fuzzy Output Feedback Control Design for Nonlinear Systems: An LMI Approach
Robust Fuzzy Control of Nonlinear Systems With Parametric Uncertainties
H2 Guaranteed Cost Fuzzy Control for Uncertain Nonlinear Systems Via Linear Matrix Inequalities
Robust H∞ Nonlinear Modeling and Control Via Uncertain Fuzzy Systems
Robust H∞ Fuzzy Control Approach for a Class of Markovian Jump Nonlinear Systems
Mode-Independent Robust Stabilization for Uncertain Markovian Jump Nonlinear Systems Via Fuzzy Control
Fuzzy Tracking Control Design for Nonlinear Dynamic Systems Via T-S Fuzzy Model
Output Tracking Control for Fuzzy Systems Via Output Feedback Design
Math Works Inc.
Reliable Robust H ∞ Fuzzy Control for Uncertain Nonlinear Systems With Markovian Jumping Actuator Faults
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Maxima and minima - Wikipedia
(Redirected from Maximize)
Largest and smallest value taken by a function takes at a given point
For use in statistics, see Sample maximum and minimum.
"Extreme value" redirects here. For the concept in statistics, see Extreme value theory. For the concept in calculus, see Extreme value theorem.
"Maximum" and "Minimum" redirect here. For other uses, see Maximum (disambiguation) and Minimum (disambiguation).
Local and global maxima and minima for cos(3πx)/x, 0.1≤ x ≤1.1
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given range (the local or relative extrema), or on the entire domain (the global or absolute extrema).[1][2][3] Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, for finding the maxima and minima of functions.
As defined in set theory, the maximum and minimum of a set are the greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of real numbers, have no minimum or maximum.
4 Functions of more than one variable
5 Maxima or minima of a functional
6 In relation to sets
A real-valued function f defined on a domain X has a global (or absolute) maximum point at x∗, if f(x∗) ≥ f(x) for all x in X. Similarly, the function has a global (or absolute) minimum point at x∗, if f(x∗) ≤ f(x) for all x in X. The value of the function at a maximum point is called the maximum value of the function, denoted
{\displaystyle \max(f(x))}
, and the value of the function at a minimum point is called the minimum value of the function. Symbolically, this can be written as follows:
{\displaystyle x_{0}\in X}
is a global maximum point of function
{\displaystyle f:X\to \mathbb {R} ,}
{\displaystyle (\forall x\in X)\,f(x_{0})\geq f(x).}
The definition of global minimum point also proceeds similarly.
If the domain X is a metric space, then f is said to have a local (or relative) maximum point at the point x∗, if there exists some ε > 0 such that f(x∗) ≥ f(x) for all x in X within distance ε of x∗. Similarly, the function has a local minimum point at x∗, if f(x∗) ≤ f(x) for all x in X within distance ε of x∗. A similar definition can be used when X is a topological space, since the definition just given can be rephrased in terms of neighbourhoods. Mathematically, the given definition is written as follows:
{\displaystyle (X,d_{X})}
be a metric space and function
{\displaystyle f:X\to \mathbb {R} }
{\displaystyle x_{0}\in X}
is a local maximum point of function
{\displaystyle f}
{\displaystyle (\exists \varepsilon >0)}
{\displaystyle (\forall x\in X)\,d_{X}(x,x_{0})<\varepsilon \implies f(x_{0})\geq f(x).}
The definition of local minimum point can also proceed similarly.
In both the global and local cases, the concept of a strict extremum can be defined. For example, x∗ is a strict global maximum point if for all x in X with x ≠ x∗, we have f(x∗) > f(x), and x∗ is a strict local maximum point if there exists some ε > 0 such that, for all x in X within distance ε of x∗ with x ≠ x∗, we have f(x∗) > f(x). Note that a point is a strict global maximum point if and only if it is the unique global maximum point, and similarly for minimum points.
A continuous real-valued function with a compact domain always has a maximum point and a minimum point. An important example is a function whose domain is a closed and bounded interval of real numbers (see the graph above).
Finding global maxima and minima is the goal of mathematical optimization. If a function is continuous on a closed interval, then by the extreme value theorem, global maxima and minima exist. Furthermore, a global maximum (or minimum) either must be a local maximum (or minimum) in the interior of the domain, or must lie on the boundary of the domain. So a method of finding a global maximum (or minimum) is to look at all the local maxima (or minima) in the interior, and also look at the maxima (or minima) of the points on the boundary, and take the largest (or smallest) one.
For differentiable functions, Fermat's theorem states that local extrema in the interior of a domain must occur at critical points (or points where the derivative equals zero).[4] However, not all critical points are extrema. One can distinguish whether a critical point is a local maximum or local minimum by using the first derivative test, second derivative test, or higher-order derivative test, given sufficient differentiability.[5]
For any function that is defined piecewise, one finds a maximum (or minimum) by finding the maximum (or minimum) of each piece separately, and then seeing which one is largest (or smallest).
The global maximum of x√x occurs at x = e.
x2 Unique global minimum at x = 0.
x3 No global minima or maxima. Although the first derivative (3x2) is 0 at x = 0, this is an inflection point. (2nd derivative is 0 at that point.)
{\displaystyle {\sqrt[{x}]{x}}}
Unique global maximum at x = e. (See figure at right)
x−x Unique global maximum over the positive real numbers at x = 1/e.
x3/3 − x First derivative x2 − 1 and second derivative 2x. Setting the first derivative to 0 and solving for x gives stationary points at −1 and +1. From the sign of the second derivative, we can see that −1 is a local maximum and +1 is a local minimum. This function has no global maximum or minimum.
|x| Global minimum at x = 0 that cannot be found by taking derivatives, because the derivative does not exist at x = 0.
cos(x) Infinitely many global maxima at 0, ±2π, ±4π, ..., and infinitely many global minima at ±π, ±3π, ±5π, ....
2 cos(x) − x Infinitely many local maxima and minima, but no global maximum or minimum.
cos(3πx)/x with 0.1 ≤ x ≤ 1.1 Global maximum at x = 0.1 (a boundary), a global minimum near x = 0.3, a local maximum near x = 0.6, and a local minimum near x = 1.0. (See figure at top of page.)
x3 + 3x2 − 2x + 1 defined over the closed interval (segment) [−4,2] Local maximum at x = −1−√15/3, local minimum at x = −1+√15/3, global maximum at x = 2 and global minimum at x = −4.
For a practical example,[6] assume a situation where someone has
{\displaystyle 200}
feet of fencing and is trying to maximize the square footage of a rectangular enclosure, where
{\displaystyle x}
is the length,
{\displaystyle y}
is the width, and
{\displaystyle xy}
is the area:
{\displaystyle 2x+2y=200}
{\displaystyle 2y=200-2x}
{\displaystyle {\frac {2y}{2}}={\frac {200-2x}{2}}}
{\displaystyle y=100-x}
{\displaystyle xy=x(100-x)}
{\displaystyle x}
{\displaystyle {\begin{aligned}{\frac {d}{dx}}xy&={\frac {d}{dx}}x(100-x)\\&={\frac {d}{dx}}\left(100x-x^{2}\right)\\&=100-2x\end{aligned}}}
Setting this equal to
{\displaystyle 0}
{\displaystyle 0=100-2x}
{\displaystyle 2x=100}
{\displaystyle x=50}
{\displaystyle x=50}
is our only critical point. Now retrieve the endpoints by determining the interval to which
{\displaystyle x}
is restricted. Since width is positive, then
{\displaystyle x>0}
{\displaystyle x=100-y}
, that implies that
{\displaystyle x<100}
. Plug in critical point
{\displaystyle 50}
, as well as endpoints
{\displaystyle 0}
{\displaystyle 100}
{\displaystyle xy=x(100-x)}
{\displaystyle 2500,0,}
{\displaystyle 0}
Therefore, the greatest area attainable with a rectangle of
{\displaystyle 200}
feet of fencing is
{\displaystyle 50\times 50=2500}
Functions of more than one variable[edit]
Main article: Second partial derivative test
Peano surface, a counterexample to some criteria of local maxima of the 19th century
The global maximum is the point at the top
Counterexample: The red dot shows a local minimum that is not a global minimum
For functions of more than one variable, similar conditions apply. For example, in the (enlargeable) figure on the right, the necessary conditions for a local maximum are similar to those of a function with only one variable. The first partial derivatives as to z (the variable to be maximized) are zero at the maximum (the glowing dot on top in the figure). The second partial derivatives are negative. These are only necessary, not sufficient, conditions for a local maximum, because of the possibility of a saddle point. For use of these conditions to solve for a maximum, the function z must also be differentiable throughout. The second partial derivative test can help classify the point as a relative maximum or relative minimum. In contrast, there are substantial differences between functions of one variable and functions of more than one variable in the identification of global extrema. For example, if a bounded differentiable function f defined on a closed interval in the real line has a single critical point, which is a local minimum, then it is also a global minimum (use the intermediate value theorem and Rolle's theorem to prove this by contradiction). In two and more dimensions, this argument fails. This is illustrated by the function
{\displaystyle f(x,y)=x^{2}+y^{2}(1-x)^{3},\qquad x,y\in \mathbb {R} ,}
whose only critical point is at (0,0), which is a local minimum with f(0,0) = 0. However, it cannot be a global one, because f(2,3) = −5.
Maxima or minima of a functional[edit]
If the domain of a function for which an extremum is to be found consists itself of functions (i.e. if an extremum is to be found of a functional), then the extremum is found using the calculus of variations.
In relation to sets[edit]
Maxima and minima can also be defined for sets. In general, if an ordered set S has a greatest element m, then m is a maximal element of the set, also denoted as
{\displaystyle \max(S)}
. Furthermore, if S is a subset of an ordered set T and m is the greatest element of S with (respect to order induced by T), then m is a least upper bound of S in T. Similar results hold for least element, minimal element and greatest lower bound. The maximum and minimum function for sets are used in databases, and can be computed rapidly, since the maximum (or minimum) of a set can be computed from the maxima of a partition; formally, they are self-decomposable aggregation functions.
In the case of a general partial order, the least element (i.e., one that is smaller than all others) should not be confused with a minimal element (nothing is smaller). Likewise, a greatest element of a partially ordered set (poset) is an upper bound of the set which is contained within the set, whereas a maximal element m of a poset A is an element of A such that if m ≤ b (for any b in A), then m = b. Any least element or greatest element of a poset is unique, but a poset can have several minimal or maximal elements. If a poset has more than one maximal element, then these elements will not be mutually comparable.
In a totally ordered set, or chain, all elements are mutually comparable, so such a set can have at most one minimal element and at most one maximal element. Then, due to mutual comparability, the minimal element will also be the least element, and the maximal element will also be the greatest element. Thus in a totally ordered set, we can simply use the terms minimum and maximum.
If a chain is finite, then it will always have a maximum and a minimum. If a chain is infinite, then it need not have a maximum or a minimum. For example, the set of natural numbers has no maximum, though it has a minimum. If an infinite chain S is bounded, then the closure Cl(S) of the set occasionally has a minimum and a maximum, in which case they are called the greatest lower bound and the least upper bound of the set S, respectively.
Mex (mathematics)
^ Larson, Ron; Edwards, Bruce H. (2009). Calculus (9th ed.). Brooks/Cole. ISBN 978-0-547-16702-2.
^ Thomas, George B.; Weir, Maurice D.; Hass, Joel (2010). Thomas' Calculus: Early Transcendentals (12th ed.). Addison-Wesley. ISBN 978-0-321-58876-0.
^ Weisstein, Eric W. "Minimum". mathworld.wolfram.com. Retrieved 2020-08-30.
^ Weisstein, Eric W. "Maximum". mathworld.wolfram.com. Retrieved 2020-08-30.
^ a b Garrett, Paul. "Minimization and maximization refresher".
Wikimedia Commons has media related to Extrema (calculus).
Look up maxima, minima, or extremum in Wiktionary, the free dictionary.
Thomas Simpson's work on Maxima and Minima at Convergence
Application of Maxima and Minima with sub pages of solved problems
Jolliffe, Arthur Ernest (1911). "Maxima and Minima" . Encyclopædia Britannica. Vol. 17 (11th ed.). pp. 918–920.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Maxima_and_minima&oldid=1076973581"
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EUDML | Edge-coloring and -coloring for various classes of graphs. EuDML | Edge-coloring and -coloring for various classes of graphs.
Edge-coloring and
f
-coloring for various classes of graphs.
Zhou, Xiao; Nishizeki, Takao
Zhou, Xiao, and Nishizeki, Takao. "Edge-coloring and -coloring for various classes of graphs.." Journal of Graph Algorithms and Applications 3.1-3 (1999): Paper No. 1, 18 p.-Paper No. 1, 18 p.. <http://eudml.org/doc/230347>.
author = {Zhou, Xiao, Nishizeki, Takao},
keywords = {-coloring; edge-coloring; sequential and parallel algorithms; -coloring},
title = {Edge-coloring and -coloring for various classes of graphs.},
TI - Edge-coloring and -coloring for various classes of graphs.
KW - -coloring; edge-coloring; sequential and parallel algorithms; -coloring
f
-coloring, edge-coloring, sequential and parallel algorithms,
f
-coloring
Articles by Zhou
Articles by Nishizeki
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2Al
1. What are two examples of how our world is connected?
2. What are some positives of our connected world?
3. What are some negatives of our connected world?
4. What are some possible solutions to the challenges we face in the 21st century?
2Al
Once you have decided on a particular technological solution, draw a labelled diagram of it in the first box. In the final box write a paragraph explaining how your technological solution works.
2Al
1. How did people 100 years ago believe we would live today?
2. What are some ideas of how we might live 100 years in the future?
3. What are some positive changes we are already making for the future?
2Al
The Future vision: 100 years from now worksheet contains a giant thought cloud. Inside the thought cloud you will create your vision of the world 100 years in the future. You can use textas/pencils to draw your vision and/or cut out pictures and words to create a collage.
2Al
2Al
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Fermat's principle - Wikipedia
1.2 A ray as a signal path (line of sight)
1.3 A ray as an energy path (beam)
2 Equivalence to Huygens' construction
3.1 Isotropic media: Rays normal to wavefronts
3.2 Homogeneous media: Rectilinear propagation
4.1 Formulation in terms of refractive index
4.2 Relation to Hamilton's principle
5.1 Fermat vs. the Cartesians
5.2 Huygens's oversight
5.3 Laplace, Young, Fresnel, and Lorentz
{\displaystyle v_{\mathrm {r} }}
{\displaystyle T=\int _{A}^{B}\!dt=\int _{A}^{B}{\frac {ds}{\,v_{\mathrm {r} }\,}}}
{\displaystyle \delta T=\,\delta \int _{A}^{B}{\frac {ds}{\,v_{\mathrm {r} }\,}}\,=\,0\,.}
{\displaystyle S\,=\int _{A}^{B}\!dS\,=\int _{A}^{B}{\frac {\,c\,}{v_{\mathrm {r} }}}\,ds=\int _{A}^{B}\!n_{\mathrm {r} }\,ds\,,}
{\displaystyle \,n_{\mathrm {r} }=c/v_{\mathrm {r} }\,}
{\displaystyle dS=n_{\mathrm {r} \,}ds\,,\,}
{\displaystyle \delta S=\delta \int _{A}^{B}\!n_{\mathrm {r} }\,ds=0.}
{\displaystyle {\begin{aligned}\delta S&=\,\delta \int _{A}^{B}\!n_{\mathrm {r} }\,{\sqrt {dx^{2}+dy^{2}+dz^{2}}}\\&=\,\delta \int _{A}^{B}\!n_{\mathrm {r} }\,{\sqrt {{\dot {x}}^{2}+{\dot {y}}^{2}+{\dot {z}}^{2}}}~ds\,=\,0\,.\end{aligned}}}
{\displaystyle L(x,y,z,{\dot {x}},{\dot {y}},{\dot {z}})\,=\,n(x,y,z)\,{\sqrt {{\dot {x}}^{2}+{\dot {y}}^{2}+{\dot {z}}^{2}}}\,,}
{\displaystyle \delta S=\,\delta \int _{A}^{B}\!L(x,y,z,{\dot {x}},{\dot {y}},{\dot {z}})\,ds\,=\,0\,.}
{\displaystyle L{\big (}x(z),y(z),{\dot {x}}(z),{\dot {y}}(z),z{\big )}=n(x,y,z)\,{\sqrt {1+{\dot {x}}^{2}+{\dot {y}}^{2}}}\,,}
{\displaystyle \delta S=\,\delta \int _{A}^{B}\!L{\big (}x(z),y(z),{\dot {x}}(z),{\dot {y}}(z),z{\big )}\,dz\,=\,0.}
Retrieved from "https://en.wikipedia.org/w/index.php?title=Fermat%27s_principle&oldid=1088809416"
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Eccentricity (math) - zxc.wiki
Eccentricity (math)
Ellipse with labels
Hyperbola with labels
The term eccentricity has two related meanings in mathematics in connection with non-degenerate conic sections (ellipses, hyperbolas, parabolas):
Circle, ellipse, parabola and hyperbola with numerical eccentricity and the same half parameter (= radius of the circle)
The linear eccentricity of an ellipse or hyperbola is the distance from a focal point to the center point and is designated by (see figure). It has the dimension of a length. Since a circle is an ellipse with coincident focal points , applies to the circle .
{\ displaystyle e}
{\ displaystyle (F_ {1} = F_ {2} = M)}
{\ displaystyle e = 0}
The numerical eccentricity for ellipses and hyperbolas is the ratio of the linear eccentricity to the major semi-axis and thus a dimensionless number.
{\ displaystyle \ varepsilon}
{\ displaystyle e / a}
The following applies to an ellipse . In the case , the ellipse is a circle .
{\ displaystyle 0 \ leq \ varepsilon <1}
{\ displaystyle \ varepsilon = 0}
The numerical eccentricity describes here the increasing deviation of an ellipse from the circular shape.
{\ displaystyle \ varepsilon}
The following applies to a hyperbola . As it grows , the hyperbola becomes more and more open, i.e. i.e., the angle between the asymptotes increases . Equilateral hyperbolas, i.e. those with right-angled asymptotes, result for .
{\ displaystyle 1 <\ varepsilon}
{\ displaystyle \ varepsilon}
{\ displaystyle \ varepsilon = {\ sqrt {2}}}
For a parabola one defines (for motivation see below).
{\ displaystyle \ varepsilon = 1}
The significance of the numerical eccentricity results from the fact that every two ellipses or hyperbolas are exactly similar if they have the same numerical eccentricity. Two parabolas ( ) are always similar.
{\ displaystyle \ varepsilon \ equiv 1}
In the case of ellipses and hyperbolas, the distance between the focal points and the center is also called the focal length. In the case of a parabola, however, the distance between the focal point and the vertex is called the focal length.
{\ displaystyle e}
In astronomy , mostly only numerical eccentricity is used and is simply called eccentricity , but in contrast to the notation in mathematics, it is often referred to as.
{\ displaystyle e}
1 Mathematical treatment
Ellipse with guidelines
To define the numerical eccentricity on the cone
With eccentricity is first described the deviation of an ellipse from the circular shape. The distance from a focal point to the center point was used as a measure of this deviation (see 1st picture). For you get a circle. Since a hyperbola also has a center point and focal points, the term was extended to the hyperbolic case, although one cannot speak of the proximity of a hyperbola to a circle here. A parabola does not have a center point and therefore initially has no eccentricity.
{\ displaystyle e}
{\ displaystyle e = 0}
Another way of describing the deviation of an ellipse from a circle is the ratio . It is . Again, you get for a circle. In this case , the parameter is also the relationship between the distance of an ellipse point to the focal point and the distance to a guideline , which is used to define the guideline for an ellipse (see 4th picture). (A circle cannot be defined with the help of a guideline.) If the guideline definition also allows values equal to or greater than 1, the curve obtained is a parabola if the ratio is, and hyperbolas in the case . The parameter allows to describe ellipses, parabolas and hyperbolas with a common family parameter. For example describes the equation
{\ displaystyle \ varepsilon = e / a}
{\ displaystyle 0 \ leq \ varepsilon <1}
{\ displaystyle \ varepsilon = 0}
{\ displaystyle \ varepsilon> 0}
{\ displaystyle \ varepsilon}
{\ displaystyle \ varepsilon}
{\ displaystyle \ varepsilon = 1}
{\ displaystyle \ varepsilon> 1}
{\ displaystyle \ varepsilon}
{\ displaystyle x ^ {2} (\ varepsilon ^ {2} -1) + 2px-y ^ {2} = 0, \ quad \ varepsilon \ geq 0, \ p> 0}
(see 3rd picture)
all ellipses (incl. circle), the parabola and all hyperbolas that have the zero point as a common vertex, the x-axis as a common axis and the same half parameter (see 1st picture). ( is also the common radius of curvature in the common vertex, see ellipse, parabola, hyperbola).
{\ displaystyle p}
{\ displaystyle p}
The parameter only exists in the case of ellipses and hyperbolas and is called linear eccentricity . is a length.
{\ displaystyle e}
{\ displaystyle e}
For the ellipse is .
{\ displaystyle {\ tfrac {x ^ {2}} {a ^ {2}}} + {\ tfrac {y ^ {2}} {b ^ {2}}} = 1}
{\ displaystyle e = {\ sqrt {a ^ {2} -b ^ {2}}} <a}
For is and the ellipse a circle. Is only a little smaller than , i.e. H. is small, then the ellipse is very flat.
{\ displaystyle a = b}
{\ displaystyle e = 0}
{\ displaystyle e}
{\ displaystyle a}
{\ displaystyle b}
For the hyperbola is and therefore for every hyperbola .
{\ displaystyle {\ tfrac {x ^ {2}} {a ^ {2}}} - {\ tfrac {y ^ {2}} {b ^ {2}}} = 1}
{\ displaystyle e = {\ sqrt {a ^ {2} + b ^ {2}}}}
{\ displaystyle e> a}
The parameter exists for ellipses, hyperbolas and parabolas and is called numerical eccentricity . is the ratio of two lengths, so it is dimensionless .
{\ displaystyle \ varepsilon}
{\ displaystyle \ varepsilon}
For ellipses and hyperbolas applies , for parabolas .
{\ displaystyle \ varepsilon = e / a = {\ sqrt {1 \ mp {\ frac {b ^ {2}} {a ^ {2}}}}} \ neq 1}
{\ displaystyle \ varepsilon = 1}
If one understands an ellipse / parabola / hyperbola as a plane section of a vertical circular cone, the numerical eccentricity can be explained
{\ displaystyle \ varepsilon = {\ frac {\ sin \ beta} {\ sin \ alpha}}, \ \ 0 <\ alpha <90 ^ {\ circ}, \ 0 \ leq \ beta \ leq 90 ^ {\ circ }}
express. Here is the angle of inclination of a cone-generating plane and the angle of inclination of the intersecting plane (see picture). For there are circles and for parabolas. (The plane must not contain the apex of the cone.)
{\ displaystyle \ alpha}
{\ displaystyle \ beta}
{\ displaystyle \ beta = 0}
{\ displaystyle \ beta = \ alpha}
Small encyclopedia of mathematics . Verlag Harri Deutsch, 1977, ISBN 3-87144-323-9 , pp. 192, 195, 328, 330.
Ayoub B. Ayoub: The Eccentricity of a Conic Section. In: The College Mathematics Journal , Vol. 34, No. 2 (March 2003), pp. 116-121 ( JSTOR 3595784 ).
Ilka Agricola, Thomas Friedrich: Elementary Geometry. AMS, 2008, ISBN 978-0-8218-9067-7 , pp. 63-70 ( excerpt (Google) ).
Hans-Jochen Bartsch: Pocket book of mathematical formulas for engineers and natural scientists. Hanser, 2014, ISBN 978-3-446-43735-7 , pp. 287-289 ( excerpt (Google) ).
Eric W. Weisstein : Eccentricity . In: MathWorld (English).
^ Jacob Steiner's lectures on synthetic geometry. BG Teubner, Leipzig 1867 (at Google Books: books.google.de ).
^ Graf, Barner: Descriptive Geometry. Quelle & Meyer-Verlag, 1973, pp. 169-173.
This page is based on the copyrighted Wikipedia article "Exzentrizit%C3%A4t_%28Mathematik%29" (Authors); it is used under the Creative Commons Attribution-ShareAlike 3.0 Unported License. You may redistribute it, verbatim or modified, providing that you comply with the terms of the CC-BY-SA.
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Table 1 Priors for f = 0, f = 1, and 0 < f < 1
unfolded
\left\{\begin{array}{c}1-\mathrm{\theta \beta }-D\phantom{\rule{2.36043pt}{0ex}}\phantom{\rule{2.36043pt}{0ex}}\phantom{\rule{1em}{0ex}}f=0\\ D\phantom{\rule{2.36043pt}{0ex}}\phantom{\rule{2.36043pt}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2.36043pt}{0ex}}\phantom{\rule{2.36043pt}{0ex}}\phantom{\rule{2.36043pt}{0ex}}\phantom{\rule{0.3em}{0ex}}f=1\\ \frac{\theta }{100f}\phantom{\rule{2.36043pt}{0ex}}\phantom{\rule{2.36043pt}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2.36043pt}{0ex}}\phantom{\rule{0.3em}{0ex}}0<f<1\end{array}\right\
\left\{\begin{array}{c}1-\theta -D\phantom{\rule{2.36043pt}{0ex}}\phantom{\rule{2.36043pt}{0ex}}\phantom{\rule{1em}{0ex}}f=0\\ D\phantom{\rule{2.36043pt}{0ex}}\phantom{\rule{2.36043pt}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2.36043pt}{0ex}}\phantom{\rule{0.3em}{0ex}}f=1\\ \frac{\theta }{99}\phantom{\rule{2.36043pt}{0ex}}\phantom{\rule{2.36043pt}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2.36043pt}{0ex}}\phantom{\rule{2.36043pt}{0ex}}\phantom{\rule{0.3em}{0ex}}0<f<1\end{array}\right\
\left\{\begin{array}{c}\frac{\left(1-\mathrm{\theta \beta }\right)}{2}\phantom{\rule{2.36043pt}{0ex}}\phantom{\rule{2.36043pt}{0ex}}\phantom{\rule{2.36043pt}{0ex}}\phantom{\rule{2.36043pt}{0ex}}\phantom{\rule{2.36043pt}{0ex}}\phantom{\rule{2.36043pt}{0ex}}\phantom{\rule{1em}{0ex}}f=0\\ \frac{\left(1-\mathrm{\theta \beta }\right)}{2}\phantom{\rule{2.36043pt}{0ex}}\phantom{\rule{2.36043pt}{0ex}}\phantom{\rule{2.36043pt}{0ex}}\phantom{\rule{2.36043pt}{0ex}}\phantom{\rule{2.36043pt}{0ex}}\phantom{\rule{2.36043pt}{0ex}}\phantom{\rule{1em}{0ex}}f=1\\ \frac{\theta }{200f\left(1-f\right)}\phantom{\rule{2.36043pt}{0ex}}\phantom{\rule{2.36043pt}{0ex}}\phantom{\rule{1em}{0ex}}0<f<1\end{array}\right\
\left\{\begin{array}{c}\frac{1-\theta }{2}\phantom{\rule{2.36043pt}{0ex}}\phantom{\rule{2.36043pt}{0ex}}\phantom{\rule{1em}{0ex}}f=0\\ \frac{1-\theta }{2}\phantom{\rule{2.36043pt}{0ex}}\phantom{\rule{2.36043pt}{0ex}}\phantom{\rule{1em}{0ex}}f=1\\ \frac{\theta }{99}\phantom{\rule{2.36043pt}{0ex}}\phantom{\rule{2.36043pt}{0ex}}\phantom{\rule{2.36043pt}{0ex}}\phantom{\rule{2.36043pt}{0ex}}\phantom{\rule{2.36043pt}{0ex}}\phantom{\rule{2.36043pt}{0ex}}\phantom{\rule{1em}{0ex}}0<f<1\end{array}\right\
We discretize the interval [0,1] with N d = 100 breakpoints. The numbers 99, 100 and 200 appearing in the formulæ in the table are normalization factors (respectively N d − 1, N d and 2N d ). β is a normalization constant for the divergent function 1/f,
\beta =\sum _{i=1}^{{N}_{d}}\frac{1}{i}=ln\left({N}_{d}\right)+\gamma
where γ = 0.57721… is the Euler-Mascheroni constant.
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Frequency-wavenumber DMO correction - SEG Wiki
Revision as of 13:46, 8 October 2014 by Ageary (talk | contribs) (added links)
Refer to Figure 5.1-1 and recall that our objective with DMO correction is to transform the normal-moveout-corrected prestack data Pn(yn, tn; h) from yn − tn coordinates to y0 − τ0 coordinates so as to obtain the dip-moveout-corrected zero-offset data P0(y0, τ0; h). Note, however, the transformation equations ( 4a ) and ( 4b ) require knowledge of the reflector dip ϕ to perform the DMO correction. To circumvent this requirement, Hale [1] developed a method for DMO correction in the frequency-wavenumber domain. First, we use the relation from Section D.1
{\displaystyle y_{0}=y_{n}-{\frac {h^{2}}{t_{n}A}}\left({\frac {2{\rm {sin}}\phi }{v}}\right),}
{\displaystyle \tau _{0}={\frac {t_{n}}{A}},}
{\displaystyle {\rm {sin}}\phi ={\frac {vk_{y}}{2{\omega _{0}}}},}
which states that the reflector dip ϕ can be expressed in terms of wavenumber ky and frequency ω0, which are the Fourier duals of midpoint y0 and event time τ0, respectively. By way of equation ( 11 ), the transformation equations ( 4a ) and ( 4b ) are recast explicitly independent of reflector dip as
{\displaystyle y_{0}=y_{n}-{\frac {h^{2}k_{y}}{t_{n}A\omega _{0}}},}
{\displaystyle \tau _{0}={\frac {t_{n}}{A}},}
where A of equation ( 5 ) now is of the form
{\displaystyle A={\sqrt {1+{\frac {h^{2}}{t_{n}^{2}}}\left({\frac {2{\rm {sin}}\phi }{v}}\right)^{2}}}.}
{\displaystyle A={\sqrt {1+{\frac {h^{2}k_{y}^{2}}{t_{n}^{2}\omega _{0}^{2}}}}}.}
The frequency-wavenumber domain dip-moveout correction process that transforms the normal-moveout-corrected prestack data with a specific offset 2h from yn − tn domain to y0 − τ0 domain is achieved by the integral
{\displaystyle P_{0}\left(k_{y},\omega _{0};h\right)=\int {\frac {2A^{2}-1}{A^{3}}}\times P_{n}\left(k_{y},t_{n};h\right){\rm {exp}}\left(-i\omega _{0}t_{n}A\right)dt_{n}.}
Derivation of the integral transform of equation ( 14a ) is given in Section E.2.
Once dip-moveout correction is applied, the data are inverse Fourier transformed
{\displaystyle P_{0}\left(y_{0},\tau _{0};h\right)=\int \int P_{0}\left(k_{y},\omega _{0};h\right)\times {\rm {exp}}\left(-ik_{y}y_{0}+i\omega _{0}\tau _{0}\right)dk_{y}d\omega _{0}.}
The amplitude scaling (2A2 − 1)/A3 in equation ( 14a ) is by Black [2], and is represented by A−1 in the original derivation by Hale [1]. The difference is due to the fact that Hale [1] defined the output time variable for DMO correction as t0 of equation ( 6 ), whereas Black [2] correctly defined the output time variable as τ0 of equation ( 4b ). Fortunately, the phase term exp(−iω0tnA) as in equation ( 14a ) is identical in the case of both derivations. There is one other variation of the amplitude term by Liner (1989) and Bleistein [3] given by (2A2 − 1)/A. Nevertheless, within the context of a conventional processing sequence which includes geometric spreading correction prior to DMO correction, the amplitude scaling (2A2 − 1)/A3 described here preserves relative amplitudes.
{\displaystyle t_{0}=t_{n}A.}
We now outline the steps in dip-moveout correction in the frequency-wavenumber domain:
Start with prestack data in midpoint-offset y − h coordinates, P(y, h, t) and apply normal moveout correction using a dip-independent velocity v.
Sort the data from moveout-corrected CMP gathers Pn(yn, h, tn) to common-offset sections Pn(yn, tn; h).
Perform Fourier transform of each common-offset section in midpoint yn direction, Pn(ky, tn; h).
For each output frequency ω0, apply the phase-shift exp(−iω0tnA), scale by (2A2 − 1)/A3, and sum the resulting output over input time tn as described by equation ( 14a ).
Finally, perform 2-D inverse Fourier transform to obtain the dip-moveout corrected common-offset section P0(y0, τ0; h) (equation 14b ).
A flowchart of the dip-moveout correction described above is presented in Figure 5.1-2.
We shall now test the frequency-wavenumber DMO correction using modeled data for point scatterers and dipping events. Figure 5.1-3 depicts six point scatterers buried in a constant-velocity medium. A synthetic data set that comprises 32 common-offset sections, each with 63 midpoints, was created. The offsets range is from 0 to 1550 m with an increment of 50 m.
Figure 5.1-4 shows two constant-velocity stacks (CVS) of the CMP gathers from the synthetic data set associated with the velocity-depth model depicted in Figure 5.1-3. The offset range used in stacking is 50 − 1550 m. At the apex of the traveltime trajectory for each point scatterer, the event dip is zero. Therefore, stack response is best with moveout velocity equal to the medium velocity (3000 m/s). Along the flanks of the traveltime trajectories, optimum stack response varies as the event dip changes. The steeper the dip, the higher the moveout (or stacking) velocity.
Figure 5.1-2 A flowchart for frequency-wavenumber dip-moveout correction algorithm. The scalar A is given by equation ( 13 ) and B = (2A2 − 1)/A3 as in equation ( 14a ).
Selected common-offset sections associated with the subsurface model in Figure 5.1-3 are shown in Figure 5.1-5a. The well-known nonhyperbolic table-top trajectories are apparent at large offsets. Selected CMP gathers from the model of Figure 5.1-3 are shown in Figure 5.1-5b. Only selected gathers that span the right side of the center midpoint are displayed, since the common-offset sections are symmetric with respect to the center midpoint (CMP 32). Note that the travel-times at the center midpoint are perfectly hyperbolic, while the traveltimes at CMP gathers away from the center are increasingly nonhyperbolic.
The following DMO processing is applied to the data as in Figure 5.1-5a:
Figure 5.1-5c shows the NMO-corrected gathers, with stretch muting applied. The medium velocity (3000 m/s) was used for NMO correction (equation 2 ), an essential requirement for subsequent DMO correction. As a result, the events at and in the vicinity of the center midpoint (CMP 32) are flat after NMO correction, while the events at midpoints away from the center midpoint are increasingly overcorrected.
The stacked section derived from these gathers (Figure 5.1-5c) is shown in Figure 5.1-4b. Because medium velocity was used for NMO correction, the stack response is best for zero dip. Note the poor stack response along the steeply dipping flanks. The desired section is the zero-offset section in Figure 5.1-4a.
We sort the NMO-corrected gathers (Figure 5.1-5c) into common-offset sections for DMO processing. These are shown in Figure 5.1-6a.
Each common-offset section is individually corrected for dip moveout. The impulse responses of the dip-moveout operator for the corresponding offsets are shown in Figure 5.1-6b, and the resulting common-offset sections are shown in Figure 5.1-6c. Note the following effects of DMO:
DMO is a partial migration process. The flanks of the nonhyperbolic trajectories have been moved updip just enough to make them look like zero-offset trajectories, which are hyperbolic. As a result, each common-offset section after NMO and DMO corrections is approximately equivalent to the zero-offset section (Figure 5.1-4a).
This partial migration is subtly different from conventional migration in one respect. Unlike conventional migration, note from the impulse responses in Figure 5.1-6b that the dip-moveout correction becomes greater at increasingly shallow depths.
While it does nothing to the zero-offset section, dip-moveout correction also is greater at increasingly large offsets (Figure 5.1-6c).
Finally, as with conventional migration, the steeper the event, the greater partial migration takes place, with flat events remaining unaltered (Figure 5.1-6c).
Following the DMO correction, the data are sorted back to CMP gathers (Figure 5.1-6d). Compare the gathers in Figure 5.1-6d to the CMP gathers without DMO correction (Figure 5.1-5b). The DMO correction has left the zero-dip events unchanged (at and in the vicinity of CMP 32), while it substantially corrected steeply dipping events on the CMP gathers away from the center midpoint (CMP 32). The events on the CMP gathers now are flattened (Figure 5.1-6d). Also, since DMO correction is a migration-like process, it causes the energy to move from one CMP gather to neighboring gathers in the updip direction. Energy depletion at the CMP gathers in Figure 5.1-6d farther from the center midpoint occurred because there was no other CMP gather to contribute energy beyond CMP 63.
Stacking the NMO- and DMO-corrected gathers (Figure 5.1-6d) yields a section (Figure 5.1-7c) that more closely represents the zero-offset section (Figure 5.1-7a) than the stacked section without the DMO correction (Figure 5.1-7b). Note the enhanced stack response along the steeply dipping flanks in Figure 5.1-7c. (The sections all have the same display gain.)
{\displaystyle t^{2}=t_{n}^{2}+{\frac {4h^{2}}{v^{2}}},}
Figure 5.1-4 Stack response of six point scatterers buried in a constant-velocity earth model (3000 m/s) as depicted in Figure 5.1-3: (a) zero-offset section, (b) stack with NMO velocity of 3000 m/s, (c) stack with NMO velocity of 3600 m/s.
Figure 5.1-6 Intermediate results from DMO processing the nonzero-offset synthetic data derived from the depth model in Figure 5.1-3: (a) common-offset sections with offset range from 50 to 1550 m and an increment of 300 m sorted from the NMO-corrected gathers as in Figure 5.1-4c; (b) impulse responses of the DMO operators applied to the common-offset gathers; (c) common-offset sections as in (a) after DMO correction; (d) CMP gathers sorted from the common-offset sections as in (c) at midpoint locations from 32 to 63 as denoted in Figure 5.1-3 with an increment of 3.
Figure 5.1-7 (a) Zero-offset section associated with the depth model in Figure 5.1-3, (b) stack derived from the CMP gathers as in Figure 5.1-5c, (c) DMO stack derived from the CMP gathers as in Figure 5.1-6d.
Figure 5.1-8 DMO processing of dipping events: (a) zero-offset section with the medium velocity of 3500 m/s; (b) stack using optimum velocity picks from velocity spectra along the line, such as that shown in Figure 5.1-12a; (c) stack using the medium velocity of 3500 m/s; (d) DMO stack using velocity picks from velocity spectra along the line, such as that shown in Figure 5.1-12b. Location A refers to an example of events with conflicting dips.
We now examine results of DMO processing of a modeled data set for dipping events. Figure 5.1-8a shows a zero-offset section that consists of events with dips from 0 to 45 degrees with a 5-degree increment. Medium velocity is constant (3500 m/s). Several velocity analyses were performed along the line; an example is shown in Figure 5.1-9a. Note the dip-dependent semblance peaks. Selected CMP gathers are shown in Figure 5.1-10a. By using the optimum stacking velocities picked from the densely spaced velocity analyses, we apply NMO correction to the CMP gathers (Figure 5.1-10b), then stack them (Figure 5.1-8b). Aside from the conflicting dips at location A, stack response is close to the zero-offset section (Figure 5.1-8a). The DMO processing requires NMO correction using medium velocity (Figure 5.1-10c). Stack response using the medium velocity (Figure 5.1-8c) clearly degrades at steep dips. By applying DMO correction (Figure 5.1-10d) to the NMO-corrected gathers (Figure 5.1-10b), we get the improved stacked section in Figure 5.1-8d. The DMO stack is closest to the zero-offset section (Figure 5.1-8a).
Figure 5.1-9 Velocity analysis (a) without and (b) with DMO correction at analysis. The stacked sections without and with DMO correction are shown in Figures 5.1-8b and d.
DMO correction also yields dip-corrected velocity functions that can be used in subsequent migration. Refer to the velocity analysis in Figure 5.1-9b and note that all events have semblance peaks at 3500 m/s, which is the medium velocity for this model data set.
↑ 1.0 1.1 1.2 Hale (1984), Hale, D., 1984, Dip moveout by Fourier transform: Geophysics, 49, 741–757.
↑ 2.0 2.1 Black et al. (1993), Black, J., Schleicher, K. L. and Zhang, L., 1993, True-amplitude imaging and dip moveout: Geophysics, 58, 47–66.
↑ Bleistein (1990), Bleistein, N., 1990, Born DMO revisited: 60th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 1366–1369.
Retrieved from "https://wiki.seg.org/index.php?title=Frequency-wavenumber_DMO_correction&oldid=19782"
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EUDML | An Ascoli theorem for sequential spaces. EuDML | An Ascoli theorem for sequential spaces.
An Ascoli theorem for sequential spaces.
Sonck, Gert
Sonck, Gert. "An Ascoli theorem for sequential spaces.." International Journal of Mathematics and Mathematical Sciences 26.5 (2001): 303-315. <http://eudml.org/doc/49288>.
@article{Sonck2001,
author = {Sonck, Gert},
keywords = {continuous convergence; uniformizable; ; function spaces; compactness; precompactness; evenly continuous; separable; -algebra; ; -algebra},
title = {An Ascoli theorem for sequential spaces.},
AU - Sonck, Gert
TI - An Ascoli theorem for sequential spaces.
KW - continuous convergence; uniformizable; ; function spaces; compactness; precompactness; evenly continuous; separable; -algebra; ; -algebra
continuous convergence, uniformizable,
{R}_{0}
, function spaces, compactness, precompactness, evenly continuous, separable,
{C}^{*}
{R}_{0}
{C}^{*}
Sequential spaces
Articles by Sonck
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Simple gear of base and follower wheels with adjustable gear ratio, friction losses, and triggered faults - MATLAB - MathWorks í•œêµ
The Simple Gear block represents a gearbox that constrains the connected driveline axes of the base gear, B, and the follower gear, F, to corotate with a fixed ratio that you specify. You choose whether the follower axis rotates in the same or opposite direction as the base axis. If they rotate in the same direction, the angular velocity of the follower, ωF, and the angular velocity of the base, ωB, have the same sign. If they rotate in opposite directions, ωF and ωB have opposite signs. You can easily add and remove backlash, faults, and thermal effects.
{r}_{F}{\mathrm{Ï}}_{F}={r}_{B}{\mathrm{Ï}}_{B}
ωF is the angular velocity of the follower gear.
ωB is the angular velocity of the base gear.
{g}_{FB}=\frac{{r}_{F}}{{r}_{B}}=\frac{{N}_{F}}{{N}_{B}}
{g}_{FB}{\mathrm{Ï}}_{B}+{\mathrm{Ï}}_{F}â{\mathrm{Ï}}_{loss}=0
Ï„B is the input torque.
Ï„F is the output torque.
Ï„loss is the torque loss due to friction.
{\mathrm{Ï}}_{loss}=0
{\mathrm{Ï}}_{loss}â 0
Coulomb friction between teeth surfaces on gears B and F, characterized by efficiency, η
Viscous coupling of driveshafts with bearings, parametrized by viscous friction coefficients, μ
In the constant efficiency case, η is constant, independent of load or power transferred.
In the load-dependent efficiency case, η depends on the load or power transferred across the gears. For either power flow,
{\mathrm{Ï}}_{Coul}={g}_{FB}{\mathrm{Ï}}_{idle}+k{\mathrm{Ï}}_{F}
Ï„Coul is the Coulomb friction dependent torque.
Ï„idle is the net torque acting on the input shaft in idle mode.
Efficiency, η, is related to τCoul in the standard, preceding form but becomes dependent on load:
\mathrm{η}=\frac{{\mathrm{Ï}}_{F}}{{g}_{FB}{\mathrm{Ï}}_{idle}+\left(k+1\right){\mathrm{Ï}}_{F}}
{v}_{Tooth}={r}_{B}{\mathrm{Ï}}_{B}â\mathrm{β}{r}_{F}{\mathrm{Ï}}_{F},
rF is the follower gear radius, where rF = NF/NB·rB, and the Follower (F) to base (B) teeth ratio (NF/NB) parameter represents NF/NB.
ωB and ωF are the angular velocities of the base and follower gears, respectively.
β is the gear direction sign. When you set:
Output shaft rotates to In same direction as input shaft, β = 1.
Output shaft rotates to In opposite direction as input shaft, β = -1.
The block treats the meshing gear tooth as a position, xTooth, with respect to the linear backlash, Backlash, where -1/2·Backlash < xTooth < 1/2·Backlash. Backlash is equivalent to the Linear backlash parameter. The Initial offset parameter is equivalent to the initial position of xTooth.
e=\frac{{v}_{Backlash,tâ}}{{v}_{Backlash,t+}},
M = 1 Forwards engaged with xtooth = 1/2·Backlash
M = -1 Backwards engaged with xtooth = -1/2·Backlash
Constant efficiency — Reduce torque transfer by a constant efficiency factor. This factor falls in the range 0 < η ≤ 1 and is independent from load.
Load-dependent efficiency — Reduce torque transfer by a variable efficiency factor. This factor falls in the range 0 < η < 1 and varies with the torque load.
Temperature-dependent efficiency — Reduce torque transfer by a constant efficiency factor that is dependent on temperature but does not consider the gear load. This factor falls in the range 0 < η ≤ 1 and is independent from load. Torque transfer is determined from user-supplied data for gear efficiency and temperature.
Temperature and load-dependent efficiency — Reduce torque transfer by a variable efficiency factor that is dependent on temperature and load. This factor falls in the range 0 < η < 1 and varies with the torque load. Torque transfer efficiency is determined from user-supplied data for gear loading and temperature.
Efficiency — Torque transfer efficiency, η
0.8 (default) | 0≤ η < 1
Torque transfer efficiency, η, between base and follower shafts. Efficiency is inversely proportional to the meshing power losses.
Net torque,Ï„idle, acting on the input shaft in idle mode, that is, when torque transfer to the output shaft equals zero. For nonzero values, the power input in idle mode completely dissipates due to meshing losses.
Output torque, Ï„F, at which to normalize the load-dependent efficiency.
Efficiency at nominal output torque — Nominal efficiency, η
0.8 (default) | 0 ≤ η < 1 | scalar
Torque transfer efficiency, η, at the nominal output torque. Larger efficiency values correspond to greater torque transfer between the input and output shafts.
Absolute value of the follower shaft angular velocity above which the full efficiency factor is in effect, ωF. Below this value, a hyperbolic tangent function smooths the efficiency factor to one, lowering the efficiency losses to zero when at rest.
Efficiency — Efficiency, η
[.95, .9, .85] (default) | 0 ≤ η < 1 | array
Initial translational position of the gear tooth with respect to the amount of backlash you specify in the Linear backlash parameter. Specify a position, xTooth, that meets the condition -1/2·Backlash < xTooth < 1/2·Backlash, where Backlash is equivalent to the value that you specify for the Linear backlash parameter.
0.05 (default) | 0 ≤ η < 1 | scalar
Rotational angle range for the faulted efficiency. For a value or multiples of 2Ï€ rad, the faulted efficiency is applicable throughout rotation.
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The Relative Vigor Index (RVI) is a momentum indicator used in technical analysis that measures the strength of a trend by comparing a security's closing price to its trading range while smoothing the results using a simple moving average (SMA).
The RVI's usefulness is based on the observed tendency for prices to close higher than they open during uptrends, and to close lower than they open in downtrends.
The Relative Vigor Index (RVI) is a technical momentum indicator.
The RVI oscillates across a pre-determined center line rather than a banded trend.
Divergences between the RVI indicator and price suggest there will be a near-term change in the trend.
The Formula for the Relative Vigor Index (RVI)
The RVI formula may look complicated, but it is really fairly intuitive:
\begin{aligned} &\text{NUMERATOR}=\frac{a+(2\times b)+(2\times c)+d}{6}\\[7pt] &\text{DENOMINATOR}=\frac{e+(2\times f)+(2\times g)+h}{6}\\[7pt] &\text{RVI}=\frac{\text{SMA of NUMERATOR for $N$ periods}}{\text{SMA of DENOMINATOR for $N$ periods}}\\[7pt] &\qquad\text{\ Signal Line }=\frac{\text{RVI}+(2\times i)+(2\times j)+k}{6}\\[7pt] &\textbf{where:}\\ &a = \text{Close}-\text{Open}\\ &b=\text{Close}-\text{Open One Bar Prior to }a\\ &c =\text{Close}-\text{Open One Bar Prior to }b\\ &d =\text{Close}-\text{Open One Bar Prior to }c\\ &e =\text{High}-\text{Low of Bar }a\\ &f =\text{High}-\text{Low of Bar }b\\ &g =\text{High}-\text{Low of Bar }c\\ &h =\text{High}-\text{Low of Bar }d\\ &i = \text{RVI Value One Bar Prior}\\ &j = \text{RVI Value One Bar Prior to }i\\ &k = \text{RVI Value One Bar Prior to }j\\ &N = \text{Minutes/Hours/Days/Weeks/Months} \end{aligned}
NUMERATOR=6a+(2×b)+(2×c)+dDENOMINATOR=6e+(2×f)+(2×g)+hRVI=SMA of DENOMINATOR for N periodsSMA of NUMERATOR for N periods Signal Line =6RVI+(2×i)+(2×j)+kwhere:a=Close−Openb=Close−Open One Bar Prior to ac=Close−Open One Bar Prior to bd=Close−Open One Bar Prior to ce=High−Low of Bar af=High−Low of Bar bg=High−Low of Bar ch=High−Low of Bar di=RVI Value One Bar Priorj=RVI Value One Bar Prior to ik=RVI Value One Bar Prior to jN=Minutes/Hours/Days/Weeks/Months
How To Calculate the Relative Vigor Index (RVI)
Choose an N period to examine.
Identify the open, high, low, and close values for the current bar.
Identify the open, high, low, and close values for lookback periods prior to the current bar.
Calculate SMAs for NUMERATOR and DENOMINATOR over the N period.
Divide NUMERATOR value from DENOMINATOR value.
Place the result in the signal line equation and plot it on a graph.
What Does the Relative Vigor Index (RVI) Tell You?
The RVI indicator is calculated in a similar fashion to the stochastics oscillator but it compares the close relative to the open rather than comparing the close relative to the low. Traders expect the RVI value to rise as the bullish trend gains momentum because, in this positive setting, a security's closing price tends to be at the top of the range while the open is near the low of the range.
The RVI is interpreted in the same way as many other oscillators, such as moving average convergence-divergence (MACD) or the relative strength index (RSI). While oscillators tend to fluctuate between set levels, they may remain at extreme levels over a prolonged period of time so that interpretation must be undertaken in a broad context to be actionable.
The RVI is instead a centered oscillator and not a banded (trend-following) oscillator, which means that it's typically displayed above or below the price chart, moving around a center line rather than the actual price. It's a good idea to use the RVI indicator in conjunction with other forms of technical analysis in order to find the highest probability outcomes.
Example of How To Use the Relative Vigor Index (RVI)
A trader might examine potential changes in a trend with the RVI indicator by looking for divergences with the current price and then identifying specific entry and exit points with traditional trendlines and chart patterns.
The two most popular trading signals include:
RVI Divergences: Divergence between the RVI indicator and price suggests there will be a near-term change in the trend in the direction of the RVI's trend. So, if a stock price is rising and the RVI indicator is falling, it predicts the stock will reverse over the near term.
RVI Crossovers: Like many oscillators, the RVI has a signal line that's often calculated with price inputs. A crossover above the signal line is a bullish indicator, while a crossover below the signal line is a bearish indicator. These crossovers are designed to be leading indicators of future price direction.
Limitations of Using the Relative Vigor Index (RVI)
The RVI works best in trending markets and tends to generate false signals in rangebound markets. Results can be improved by setting longer-term lookback periods, which help to reduce the impact of whipsaws and short-term countertrends.
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Shareholder Equity Ratio Definition
What Is the Shareholder Equity Ratio?
The shareholder equity ratio indicates how much of a company's assets have been generated by issuing equity shares rather than by taking on debt. The lower the ratio result, the more debt a company has used to pay for its assets. It also shows how much shareholders might receive in the event that the company is forced into liquidation.
The shareholder equity ratio is expressed as a percentage and calculated by dividing total shareholders' equity by the total assets of the company. The result represents the amount of the assets on which shareholders have a residual claim. The figures used to calculate the ratio are recorded on the company balance sheet.
The shareholder equity ratio shows how much of a company's assets are funded by issuing stock rather than borrowing money.
The closer a firm's ratio result is to 100%, the more assets it has financed with stock rather than debt.
The ratio is an indicator of how financially stable the company may be in the long run.
The Formula for the Shareholder Equity Ratio Is
\text{Shareholder Equity Ratio} = \dfrac{\text{Total Shareholder Equity}}{\text{Total Assets}}
Shareholder Equity Ratio=Total AssetsTotal Shareholder Equity
Total shareholders' equity comes from the balance sheet, following the accounting equation:
\begin{aligned} &\text{SE} = \text{A} - \text{L}\\ &\textbf{where:}\\ &SE = \text{Shareholders' Equity}\\ &A = \text{Assets}\\ &L = \text{Liabilities} \end{aligned}
SE=A−Lwhere:SE=Shareholders’ EquityA=AssetsL=Liabilities
What Does the Shareholder Equity Ratio Tell You?
If a company sold all of its assets for cash and paid off all of its liabilities, any remaining cash equals the firm's equity. A company's shareholders' equity is the sum of its common stock value, additional paid-in capital, and retained earnings. The sum of these parts is considered to be the true value of a business.
When a company's shareholder equity ratio approaches 100%, it means that the company has financed almost all of its assets with equity capital instead of taking on debt. Equity capital, however, has some drawbacks in comparison with debt financing. It tends to be more expensive than debt, and it requires some dilution of ownership and giving voting rights to new shareholders.
The shareholder equity ratio is most meaningful in comparison with the company's peers or competitors in the same sector. Each industry has its own standard or normal level of shareholders' equity to assets.
Example of the Shareholder Equity Ratio
Say that you're considering investing in ABC Widgets, Inc. and want to understand its financial strength and overall debt situation. You start by calculating its shareholder equity ratio.
From the company's balance sheet, you see that it has total assets of $3.0 million, total liabilities of $750,000, and total shareholders' equity of $2.25 million. Calculate the ratio as follows:
Shareholders' equity ratio = $2,250,000 / 3,000,000 = .75, or 75%
This tells you that ABC Widgets has financed 75% of its assets with shareholder equity, meaning that only 25% is funded by debt.
In other words, if ABC Widgets liquidated all of its assets to pay off its debt, the shareholders would retain 75% of the company's financial resources.
When a Company Liquidates
If a business chooses to liquidate, all of the company assets are sold and its creditors and shareholders have claims on its assets. Secured creditors have the first priority because their debts were collateralized with assets that can now be sold in order to repay them.
Other creditors, including suppliers, bondholders, and preferred shareholders, are repaid before common shareholders.
A low level of debt means that shareholders are more likely to receive some repayment during a liquidation. However, there have been many cases in which the assets were exhausted before shareholders got a penny.
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Checking solutions - SEG Wiki
Incorrect solutions can often be identified using simple checks, such as verifying that the dimensions are the same on both sides of an equation. For example, suppose that we remember the basic form of the one-dimensional wave equation but are not sure on which side of the equation the factor
{\displaystyle V^{2}}
belongs—or even that it is
{\displaystyle V^{2}}
{\displaystyle V}
. Thus we may want to decide which form of the four equations below is correct:
{\displaystyle {\begin{aligned}\mathrm {(i)} {\frac {\partial ^{2}\psi }{\partial x^{2}}}=V^{2}{\frac {\partial ^{2}\psi }{\partial t^{2}}},&\quad \mathrm {(ii)} {\frac {\partial ^{2}\psi }{\partial x^{2}}}=V{\frac {\partial ^{2}\psi }{\partial t^{2}}},\\\mathrm {(iii)} V^{2}{\frac {\partial ^{2}\psi }{\partial x^{2}}}={\frac {\partial ^{2}\psi }{\partial t^{2}}},&\quad \mathrm {(iv)} V{\frac {\partial ^{2}\psi }{\partial x^{2}}}={\frac {\partial ^{2}\psi }{\partial t^{2}}}.\end{aligned}}}
Both sides of all equations have
{\displaystyle \partial ^{2}\psi }
so we ignore this factor. Denoting dimensions of length and time by
{\displaystyle L}
{\displaystyle T}
{\displaystyle V}
has dimensions
{\displaystyle L/T}
, so the first equation equates
{\displaystyle \mathrm {L} ^{-2}}
{\displaystyle \mathrm {L} ^{2}\mathrm {T} ^{-4}}
, hence cannot be correct. The second equation equates
{\displaystyle \mathrm {L} ^{-2}}
{\displaystyle \mathrm {LT} ^{-3}}
, while the fourth equates
{\displaystyle \mathrm {L} ^{-1}\mathrm {T} ^{-1}}
{\displaystyle \mathrm {T} ^{-2}}
, hence both are incorrect. The third equation has dimensions
{\displaystyle \mathrm {T} ^{-2}}
on both sides and so is at least dimensionally correct. Note that dimensional analysis cannot prove that an equation is correct even though it can prove that one is not correct. As another example: which of the following equations for the traveltime of a head wave from a horizontal refractor are incorrect?
{\displaystyle {\begin{aligned}\mathrm {(a)} \,t=t_{i}+Vx;\quad \mathrm {(b)} \,t=t_{i}+x^{2}/V;\quad \mathrm {(c)} \,t=t_{i}+x/V.\end{aligned}}}
Since all terms in a sum must have the same dimensions, we examine the dimensions of each term and readily find that (a) and (b) are incorrect while (c) is dimensionally correct. As an additional example, consider the following equation for the angle of approach:
{\displaystyle {\begin{aligned}\alpha ={\rm {sin}}^{-1}\left(\Delta t/\Delta x\right).\end{aligned}}}
Recalling that the arguments of trigonometric, exponential, logarithmic, and similar functions must be dimensionless (because they can be expanded in infinite series), we see that the equation must be incorrect because the argument has the dimensions
{\displaystyle T/L}
Another check is to see if varying the parameters produces reasonable changes in the calculated quantity. For example, which of the following equations for the reflection from a horizontal bed must be incorrect:
{\displaystyle {\begin{aligned}(t/t_{0})^{2}=1+(Vt_{0}/x)^{2}\quad \mathrm {or} \quad (t/t_{0})^{2}=1+(x/Vt_{0})^{2}?\end{aligned}}}
In the first equation the time
{\displaystyle t}
becomes smaller as the distance
{\displaystyle x}
increases, which is not reasonable, hence the equation must be incorrect. In the second equation,
{\displaystyle t}
{\displaystyle x}
increases, which is reasonable (but not a proof of correctness). A somewhat different example is to determine which of the following equations relating the critical distance
{\displaystyle x'}
to the depth of a refractor is incorrect:
{\displaystyle {\begin{aligned}x'=\left(2h/V_{1}\right)(V_{1}^{2}-V_{2}^{2})^{1/2}\quad \mathrm {or} \quad x'=\left(2h/V_{1}\right)(V_{2}^{2}-V_{1}^{2})^{1/2}.\end{aligned}}}
{\displaystyle V_{2}>V_{1}}
for a head wave to exist,
{\displaystyle (V_{1}^{2}-V_{2}^{2})^{1/2}}
is imaginary, so the first equation equates an imaginary quantity to a real quantity and therefore must be incorrect.
At times, equations exhibit varying degrees of symmetry, and this may be useful, not only in remembering them, but also in detecting errors. The following equations illustrate the value of symmetry:
{\displaystyle {\begin{aligned}&&\varepsilon _{xy}={\dfrac {\partial v}{\partial x}}+{\dfrac {\partial u}{\partial y}},\quad \varepsilon _{yz}={\dfrac {\partial w}{\partial y}}+{\dfrac {\partial v}{\partial z}},\quad \varepsilon _{zx}={\dfrac {\partial u}{\partial z}}+{\dfrac {\partial w}{\partial x}},\\&&\sigma _{xx}=\lambda \Delta +2\mu \varepsilon _{xx},\quad \sigma _{yy}=\lambda \Delta +2\mu \varepsilon _{yy},\quad \sigma _{zz}=\lambda \Delta +2\mu \varepsilon _{zz},\\&&{\dfrac {\sin \theta _{1}}{\alpha _{1}}}={\dfrac {\sin \delta _{1}}{\beta _{1}}}={\dfrac {\sin \theta _{2}}{\alpha _{2}}}={\dfrac {\sin \delta _{2}}{\beta _{2}}},\\&&R\left(\omega \right)=X\left(\omega \right)*\left(1/j\omega \right),\quad X\left(\omega \right)=-R\left(\omega \right)*\left(1/j\omega \right)\\\end{aligned}}}
(the last pair might be termed “antisymmetric” because of the minus sign). As the complexity of the equation increases, the value of symmetry generally decreases rapidly, for example, compare the third and fourth of Zoeppritz’s equations, equations (3.2h) and (3.2i); nevertheless symmetry may still be of value; for example, if in deriving equation (3.2f) we obtained the term
{\displaystyle -B_{1}{\rm {\;sin\;}}\delta _{1}}
, we should be suspicious because of the lack of symmetry with equation (3.2e).
We must be on the lookout for singularities [places where a function becomes infinite,
{\displaystyle 1/\left(1-2\sigma \right)}
{\displaystyle \sigma }
approaches 0.5]. What do singularities mean in a “physical sense”? What happens in the real world? Singularities cause computer programs to bomb, so programs must always be analyzed to make certain that they do not involve any potential singularities.
Most problems are deterministic, that is, they have a definite answer (or answers in some cases); this is so whenever the number of unknowns
{\displaystyle n}
equals the number of independent equations
{\displaystyle m}
. However, when the number of equations is less than the number of unknowns
{\displaystyle (n>m)}
, the unknowns are “underdetermined” and the best we can do is to find
{\displaystyle \left(n-m\right)}
relations between the unknowns. In the “overdetermined” case, where
{\displaystyle m>n}
, only approximate “best-fit” solutions are possible. As an example, when we try to find a velocity function that represents a set of time-depth data, we often seek a least-squares solutions (see also Sheriff and Geldart, 1995, Section 9.5.5 and in this book, problem 9.33).
Frequently the physics of a situation provides the equation. If we are asked to define the boundary conditions which govern the behavior of waves generated at the boundary between a fluid and a solid, we know that both P- and S-waves will exist in the solid but only a P-wave in the fluid. Therefore, a wave incident on the boundary will in general give rise to three waves involving three unknowns (the amplitudes of these three waves) and to fix these we will need exactly three boundary conditions obtained by applying physical principles, in this case the continuity of normal stresses and strains and the vanishing of shear stress at the boundary (see problem 2.10).
Merely substituting numerical values into an equation may produce ambiguity as to the dimensions of the answer. Including the dimensions when substituting solves this problem. Thus, suppose we wish to calculate the acoustic impedance,
{\displaystyle Z=\rho V_{P}}
{\displaystyle \rho =1.0\,\mathrm {g/cm} ^{3}}
{\displaystyle V_{P}=2.0\,\mathrm {km/s} }
{\displaystyle {\begin{aligned}Z&=\rho V_{P}={\frac {1.0\,\mathrm {g} }{\mathrm {cm} ^{3}}}\times {\frac {2.0\,\mathrm {km} }{\mathrm {s} }}={\frac {2.0\,\mathrm {g\,km} }{\mathrm {cm} ^{3}\,\mathrm {s} }}\\&={\frac {2.0\,\mathrm {g\,km} }{\mathrm {cm} ^{3}\,\mathrm {s} }}\times {\frac {10^{3}\,\mathrm {m} }{1\,\mathrm {km} }}\times \left({\frac {10^{2}\,\mathrm {cm} }{\mathrm {m} }}\right)^{3}\times {\frac {1\,\mathrm {kg} }{10^{3}\,\mathrm {g} }}={\frac {2.0\times 10^{6}\,\mathrm {kg} }{\mathrm {m} ^{3}\,\mathrm {s} }}.\end{aligned}}}
Because the numerators and denominators of the three multiplying factors are equal, each has the value of unity, and multiplying by one does not change a value. Multiplying by one also provides a means of changing from one measurement system to another. Thus if we are given that
{\displaystyle V_{P}=6000\,\mathrm {ft/s} }
{\displaystyle \rho =1\,\mathrm {g/cm} ^{3}}
{\displaystyle {\begin{aligned}Z&={\rho }V_{P}={\frac {1.0\,\mathrm {g} }{\mathrm {cm} ^{3}}}\times {\frac {6000\,\mathrm {ft} }{\mathrm {s} }}={\frac {6000\,\mathrm {g\,ft} }{\mathrm {cm} ^{3}\,\mathrm {s} }}\\&={\frac {6000\,\mathrm {g\,ft} }{\mathrm {cm} ^{3}\,\mathrm {s} }}\times {\frac {1\,\mathrm {km} }{3281\,\mathrm {ft} }}\times {\frac {10^{3}\,\mathrm {m} }{1\,\mathrm {km} }}\times \left({\frac {10^{2}\,\mathrm {cm} }{\mathrm {m} }}\right)^{3}\times {\frac {1\,\mathrm {kg} }{10^{3}\,\mathrm {g} }}\\&={\frac {6000}{3281}}\times {\frac {10^{6}\,\mathrm {kg} }{\mathrm {m} ^{3}\,\mathrm {s} }}={\frac {1.8\times 10^{6}\,\mathrm {kg} }{\mathrm {m} ^{3}\,\mathrm {s} }}.\end{aligned}}}
Appendix K of The Encyclopedic Dictionary of Applied Geophysics (Sheriff, 2002) lists conversion factors that often occur in geophysics.
Solving equations The basic elastic constants
Retrieved from "https://wiki.seg.org/index.php?title=Checking_solutions&oldid=119404"
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What Does the S&P 500 Index Measure? - Your Investments
The S&P 500 measures the value of the stocks of the 500 largest corporations by market capitalization listed on the New York Stock Exchange or Nasdaq Composite. The intention of Standard & Poor’s is to have a price that provides a quick look at the stock market and economy. Indeed, the S&P 500 index is the most popular measure used by financial media and professionals, while the mainstream media and the general public might be more familiar with the Dow Jones Industrial Average.
The S&P 500 index is a float-adjusted market-cap weighted index. It’s calculated by taking the sum of the adjusted market capitalization of all S&P 500 stocks and then dividing it with an index divisor, which is a proprietary figure developed by Standard & Poor’s.
<br /> ~S&P ~500 ~Index = dfrac{text{Market Cap for All S&P 500 Stocks}}{text{Index Divisor}}<br />
S&P 500 Index=Index DivisorMarket Cap for All S&P 500 Stocks
One result of this methodology is that the index is weighted toward larger-cap companies.
For example, on Oct. 29, 2021, the largest component was Microsoft at $2.49 trillion. Compare that to the likes of Adobe, which had a $309.44 billion market cap. The total market capitalization of all the companies in the index, as of Oct. 29, 2021, was $41.1 trillion.
The weighted average market capitalization of each individual component is then determined by dividing the market capitalization of the individual component by $41.1 trillion. Microsoft’s weighting is determined by taking its market capitalization and dividing it by the total index market cap.
The formula for determining this weighting is as follows:
<br /> text{Weighting} = dfrac{text{Market cap of individual component}}{text{The total market cap of all S&P 500 stocks}}<br />
Weighting=The total market cap of all S&P 500 stocksMarket cap of individual component
Therefore, using the same example, Microsoft has a 6.1% weighting, while a smaller company like Adobe has a 1.32% weighting in the index. This leads to the mega-cap stocks having an outsized impact on the index. Sometimes, this index structure can mask strength or weakness in smaller companies if larger-cap companies are diverging. In other ways, this index structure better represents the overall economy than indexes in which weighting is determined by an equal share or an index that is price-weighted.
S&P 500 Positives
The S&P 500 is considered an effective representation for the economy due to its inclusion of around 500 companies, which covers all areas of the United States and across all industries. In contrast, the Dow Jones Industrial Average (DJIA) is comprised of 30 companies, leading to a more narrow reflection. Further, the DJIA is a price-weighted index, so the largest weighted components are determined by its stock price rather than some fundamental measure.
The S&P 500 is a broader representation, having more stocks and covering every industry. The DJIA is limited and the movement of a stock in the DJIA can have a greater impact than that of the S&P 500. The largest weighted stock in the S&P 500 likely has a smaller weight than the largest weighted stock in the DJIA. The movement of a few companies can have a profound impact on the DJIA.
Lazy Traders are Smart – Inner Game Concept
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To Emma Darwin [5 April 1840]
You are a good old soul for having written to me so soon.— I, like another good old soul, will give you an account of my proceedings from the beginning.— At the station I met Sir F. Knowles,1 but was fortunate enough to get in a separate carrigae from that chatter-box. In my carriage, there was rather an elegant female, like a thin Lady Alderson, but so virtuous that I did not venture to open my mouth to her. She came with some female friend, also a lady & talked at the door of the carriage in so loud a voice, that we all listened with silent admiration. It was chiefly about family prayers, & how she always had them at
\frac{1}{2}
past ten not to keep the servants up. She then charged her friend to write to her either on Saturday night or Monday morning, Sunday being omitted in the most marked manner.— Our companion answered in the most pious tone, “Yes Eliza I will write either on Saturday night or on Monday morning.”— as soon as we started our virtuous female pulled out of her pocket a religious tract in a black cover, & a very thick pencil,—she then took off her gloves & commenced reading with great earnestness & marking the best passages with the aforesaid thick lead-pencil.— Her next neighbour was an old gentleman with a portentously purple nose, who was studying a number of the Christian Herald, & his next neighbour was the primmest she Quaker I have often seen.— Was not I in good company?— I never opened my mouth & therefore enjoyed my journey.
At Bermingham, I was kept standing in the office
\frac{3}{4}
of an hour in doubt, whether I could have a place, & I was so tired, that I regretted much that I took one,—however to my surprise the Journey rested me, & I arrived very brisk at Shrewsbury. In the office at Bermingham, I was aghast to see Mr. J. Hunt, an indomitable proser, taking his place.— He did not know me, as I found by his addressing a chance remark to me, I instantly resolved on the desperate attempt of travelling the whole way incognito.— My hopes were soon cut off by the appearance of Mrs. Hunt, whom I shook hands with vast surprise & interest, & opened my eyes with astonishment at Mr. Hunt, as if he had dropped from the skies.— Our fourth in the Coach was Mr Parr of Lyth,—an old miserly squire.2 Mr. Hunt opened his battery of conversation,—I stood fire well at first & then pretended to become very sleepy,—the proser became really so, so we had the most tranquil Journey.— Old Parr, the miser, was sadly misused at the Lion, for he had ordered a Fly to take him home, & there was only one, & Mark3 persuaded the man to take me up first, & gave a hint to the Porters to take a wonderful time in getting old Parr’s things off the Coach, so that the poor old gentleman must have thought the Porters & Fly men all gone mad together, so slowly no doubt they did everything, whilst I was driving up with the most surprising alacrity.—
My Father is appearing very well.— I have begun to extract wisdom from him, which I will not now write.—
He does not seem able to form any opinion about your case.—but strongly urges your going on suckling a little for some time, even at the expense of slight headachs.— He says you probably will be able to guess with better chance of truth later about your condition—but that it will be only a guess.— You will be pleased to hear, that he objects to the Baby having medicine for every trifle— He says, as long as the Baby keeps well, & the motions4 appear pretty healthy, he thinks it of little consequence whether it has a dozen or one or even less than one in 24 hours, although he says it is desirable it should be more than once.— He is very strong against gruel, but not against other food.— He thinks there is no occasion to go on with Asses’ milk.— But I will tell you all this when I come back.—
I enjoy my visit & have been surprisingly well & have not been sick once.— My Father says I may often take Calomel.— He has recommended me nothing in particular.— I find I am a good deal thinner than I was, weighing less than Erasmus now.—5 I suspect the Journey & change will do me good— I could not, however, sleep but very little the first night, & I verily believe it was from the lonesomeness of the big bed,—in which respect I have shown much more sentimentality than, it appears you did. I have begun, like a true old Arthur Gride6 making a small collection & have picked up several nice little things, & have got some receipt for puddings &c & laid some strong effectual hints about jams, & now you may send the empty jars when-ever you please.—
Chucky is very flourishing, & wishes ⟨m⟩uch to see the killcrop.— Be sure you give Mr Hoddy Doddy a kiss for me.—7 She wants you to send a list of Philharmonic,8 as she has lost the one you sent her.—
Good Bye my dear old Titty. I am often thinking about you. Give my best love to dear old Katty,9
& believe your affectionate old Husband | C. D
Francis Charles Knowles.
Thomas Parr of Lythwood Hall, 2 miles from Shrewsbury.
Mark Briggs, R. W. Darwin’s coachman.
CD weighed 10 stone 8 lb on 4 April, having lost 10 lb 6 oz since 13 September 1839 (‘Weighing Account’ book, Down House MS).
A mean old usurer in Dickens’ Nicholas Nickleby.
‘Chucky’ is Susan Darwin (Emma Darwin (1904) 2: 13) and the ‘killcrop’ is William Erasmus Darwin, as is ‘Mr Hoddy Doddy’.
A list of Philharmonic Society concerts. On 22 February 1840 CD noted in his Account Book (Down House MS) ‘Mr Cramer Philharmonic tickets [£]12/12’.
Catherine Darwin.
An amusing description of his railway journey to Shrewsbury.
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Chain rule - Simple English Wikipedia, the free encyclopedia
In differential calculus, the chain rule is a way of finding the derivative of a function. It is used where the function is within another function. This is called a composite function.
{\displaystyle F(x)}
equals the composite function of the form:
{\displaystyle F(x)=f(g(x))}
where g is a function differentiable at x and f is a function differentiable at g(x), then the derivative of
{\displaystyle F(x)}
{\displaystyle F'(x)}
, exists, and is equal to
{\displaystyle F'(x)=f'(g(x))g'(x)}
1. Find the derivative of the outside function (all of it at once).
2. Find the derivative of the inside function (the bit between the brackets).
3. Multiply the answer from the first step by the answer from the second step. This is basically the last step in solving for the derivative of a function.
{\displaystyle F(x)=(x^{2}+5)^{3}\rightarrow 3(x^{2}+5)^{2}}
{\displaystyle F'(x)=3(x^{2}+5)^{2}(2x)=6x(x^{2}+5)^{2}}
In this example, the cubed sign (3) is the outside function and
{\displaystyle x^{2}+5}
is the inside function. The derivative of the outside function would be
{\displaystyle 3x^{2}}
, where x is replaced by the inside function. The derivative of the inside function would be 2x, which is multiplied by
{\displaystyle 3(x^{2}+5)^{2}}
{\displaystyle 6x(x^{2}+5)^{2}}
↑ "Chain Rule for Derivative — The Theory". Math Vault. 2016-06-05. Retrieved 2020-09-19.
↑ "Chain rule (article)". Khan Academy. Retrieved 2020-09-19.
↑ Kouba, Duane (May 6, 1997). "Differentiation Using the Chain Rule". math.ucdavis.edu. Retrieved September 19, 2020. {{cite web}}: CS1 maint: url-status (link)
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Chain_rule&oldid=7116138"
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Charge (physics) - zxc.wiki
A charge , a common symbol or , in physics, is a quantity that is defined and interpreted differently depending on the theoretical structure of physics . What all definitions have in common is that in the borderline case of a "subordinate" theory they agree with the definition there.
{\ displaystyle Q}
{\ displaystyle q}
The only charge that plays a role in everyday practical life is the electrical charge . If the term cargo is therefore used without further specification, this cargo is usually meant.
3 Non-relativistic quantum mechanics
5 mass as charge
The only charge that occurs in classical physics is the electric charge. It describes the strength with which a particle (physics) interacts with the electric field . The electrical charge is classically identical to a coupling constant between force fields and matter. In classical physics, a moving charge is called (electrical) current ; the current determines how strongly a particle interacts with the magnetic field . Conversely, every charge emits an electric field and every current a magnetic field. When describing the equations of motion for the electric and magnetic field, the Maxwell equations , charge density and current density are included as parameters. The continuity equation follows from Maxwell's equations , which states that charge is a conservation quantity.
{\ displaystyle I}
{\ displaystyle \ rho = \ partial Q / \ partial V}
{\ displaystyle {\ vec {j}} = \ partial I / \ partial {\ vec {A}}}
At the time when the special theory of relativity was developed in 1905, only the electric charge was known. Since the Maxwell equations are also relativistically valid equations, the continuity equation can be represented in a relativistically covariant formulation, whereby the charge density is understood as the zeroth component of the four-fold current density:
{\ displaystyle j ^ {\ mu} = {\ begin {pmatrix} c \ rho & {\ vec {j}} \ end {pmatrix}}}
In non-relativistic quantum mechanics , the charge, as it is the property of a particle in classical physics, is the coupling constant with which a wave function couples to the electrical potential and the vector potential . The formalism used for this is called minimal coupling . Since in quantum mechanics the wave function itself is subject to a continuity equation , according to which the absolute probability of encountering a particle is obtained, the size of the charge associated with the particle is also preserved. This equation of continuity cannot be represented in a relativistically covariant manner.
The quantum field theory describes the union of quantum mechanics with the special theory of relativity. In it no longer only force fields are viewed as fields, but also matter. In quantum field theory, the term charge is used twice, on the one hand as a charge operator and on the other hand as its eigenvalue . The charge operator is defined using Noether's theorem . The Noether theorem is also valid in classical mechanics and states that every continuous symmetry of a system has a conservation quantity. For fields it yields a relativistic covariant continuity equation with the charge density
{\ displaystyle {\ hat {Q}}}
{\ displaystyle q}
{\ displaystyle {\ hat {\ rho}} ^ {a} = g \ psi _ {i} ^ {\ dagger} T_ {ij} ^ {a} \ psi _ {j} -f ^ {abc} A ^ {\ nu b} F_ {0 \ nu} ^ {c}}
{\ displaystyle \ psi}
the fermionic field operators,
{\ displaystyle A}
the vector bosonic field operators,
{\ displaystyle F}
the field strength tensor
{\ displaystyle T}
the generators of the symmetry group
{\ displaystyle f}
the structural constants of the symmetry group
{\ displaystyle g}
a coupling constant
The charge operator exchanges with the Hamilton operator , so a common eigenbase can be chosen so that the observable particles always have a well-defined charge as eigenvalue to the operator. In particular, the definition of the charge operator contains the coupling constant, but charge and coupling constant are different objects.
In unbroken theories, the charge operator annihilates the vacuum; denotes the quantum vacuum, is . This is not the case in theories with spontaneously broken symmetry; it grips the Fabri-Picasso's theorem , according to which the operator norm of the charge operator is infinite: . The definition of a charge as an eigenvalue of such an operator is not possible. Therefore, in the Standard Model, there is both a strong charge, called a color charge , which can be assigned to the, and an electrical charge which results from the refraction of the , but not a “weak charge”.
{\ displaystyle | \ Omega \ rangle}
{\ displaystyle {\ hat {Q}} | \ Omega \ rangle = 0}
{\ displaystyle \ | {\ hat {Q}} \ | = \ infty}
{\ displaystyle SU (3) _ {C}}
{\ displaystyle SU (2) _ {L} \ otimes U (1) _ {Y}}
{\ displaystyle U (1) _ {Q}}
In the case of non-Abelian symmetry groups like that , these Noether charges are still retained, but corrections of higher orders to their eigenvalues are no longer gauge-invariant. As in the classic case, the charge is determined from the potential with the aid of the coupling constant for these corrections. The violation of the gauge invariance is a direct consequence of the Weinberg-Witten theorem .
{\ displaystyle SU (3)}
Mass as charge
The mass is separated from the perspective of classical physics semantically from the concept of charge, but may charge as the gravitational be construed. A quantity that obeys a continuity equation according to Noether's theorem is the energy-momentum tensor , whose component in the classical Limes corresponds to the mass density.
{\ displaystyle T ^ {\ mu \ nu}}
{\ displaystyle T ^ {00}}
Charge carrier (physics)
Ian JR Aitchison and Anthony JG Hey: Gauge Theories in Particle Physics, Volume 2: Non-Abelian Gauge Theories: QCD and the Electroweak Theory . 3. Edition. Institute of Physics Publishing, Bristol, Philadelphia 2004, ISBN 0-7503-0950-4 (English).
Lewis H. Ryder: Quantum Field Theory . 2nd Edition. Cambridge University Press, Cambridge 1996, ISBN 0-521-47242-3 (English).
Mattew D. Schwartz: Quantum Field Theory and the Standard Model . 1st edition. Cambridge University Press, Cambridge 2014, ISBN 978-1-107-03473-0 (English).
This page is based on the copyrighted Wikipedia article "Ladung_%28Physik%29" (Authors); it is used under the Creative Commons Attribution-ShareAlike 3.0 Unported License. You may redistribute it, verbatim or modified, providing that you comply with the terms of the CC-BY-SA.
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Wikizero - Circumference
For the circumference of a graph, see Circumference (graph theory).
Circumference (C in black) of a circle with diameter (D in blue), radius (R in red), and centre (O in green). Circumference = π × diameter = 2π × radius.
In geometry, the circumference (from Latin circumferens, meaning "carrying around") is the perimeter of a circle or ellipse.[1] That is, the circumference would be the arc length of the circle, as if it were opened up and straightened out to a line segment.[2] More generally, the perimeter is the curve length around any closed figure. Circumference may also refer to the circle itself, that is, the locus corresponding to the edge of a disk. The circumference of a sphere is the circumference, or length, of any one of its great circles.
1.1 Relationship with π
The circumference of a circle is the distance around it, but if, as in many elementary treatments, distance is defined in terms of straight lines, this cannot be used as a definition. Under these circumstances, the circumference of a circle may be defined as the limit of the perimeters of inscribed regular polygons as the number of sides increases without bound.[3] The term circumference is used when measuring physical objects, as well as when considering abstract geometric forms.
When a circle's diameter is 1, its circumference is
{\displaystyle \pi .}
When a circle's radius is 1—called a unit circle—its circumference is
{\displaystyle 2\pi .}
Relationship with π
The circumference of a circle is related to one of the most important mathematical constants. This constant, pi, is represented by the Greek letter
{\displaystyle \pi .}
The first few decimal digits of the numerical value of
{\displaystyle \pi }
are 3.141592653589793 ...[4] Pi is defined as the ratio of a circle's circumference
{\displaystyle C}
to its diameter
{\displaystyle d:}
{\displaystyle \pi ={\frac {C}{d}}.}
{\displaystyle {C}=\pi \cdot {d}=2\pi \cdot {r}.\!}
The use of the mathematical constant π is ubiquitous in mathematics, engineering, and science.
In Measurement of a Circle written circa 250 BCE, Archimedes showed that this ratio (
{\displaystyle C/d,}
since he did not use the name π) was greater than 310/71 but less than 31/7 by calculating the perimeters of an inscribed and a circumscribed regular polygon of 96 sides.[5] This method for approximating π was used for centuries, obtaining more accuracy by using polygons of larger and larger number of sides. The last such calculation was performed in 1630 by Christoph Grienberger who used polygons with 1040 sides.
Main article: Ellipse § Circumference
Circumference is used by some authors to denote the perimeter of an ellipse. There is no general formula for the circumference of an ellipse in terms of the semi-major and semi-minor axes of the ellipse that uses only elementary functions. However, there are approximate formulas in terms of these parameters. One such approximation, due to Euler (1773), for the canonical ellipse,
{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1,}
{\displaystyle C_{\rm {ellipse}}\sim \pi {\sqrt {2\left(a^{2}+b^{2}\right)}}.}
Some lower and upper bounds on the circumference of the canonical ellipse with
{\displaystyle a\geq b}
are:[6]
{\displaystyle 2\pi b\leq C\leq 2\pi a,}
{\displaystyle \pi (a+b)\leq C\leq 4(a+b),}
{\displaystyle 4{\sqrt {a^{2}+b^{2}}}\leq C\leq \pi {\sqrt {2\left(a^{2}+b^{2}\right)}}.}
{\displaystyle 2\pi a}
{\displaystyle 4{\sqrt {a^{2}+b^{2}}}}
is the perimeter of an inscribed rhombus with vertices at the endpoints of the major and minor axes.
The circumference of an ellipse can be expressed exactly in terms of the complete elliptic integral of the second kind.[7] More precisely,
{\displaystyle C_{\rm {ellipse}}=4a\int _{0}^{\pi /2}{\sqrt {1-e^{2}\sin ^{2}\theta }}\ d\theta ,}
{\displaystyle a}
{\displaystyle e}
is the eccentricity
{\displaystyle {\sqrt {1-b^{2}/a^{2}}}.}
Arc length – Distance along a curve
Isoperimetric inequality – Geometric inequality which sets a lower bound on the surface area of a set given its volume
^ San Diego State University (2004). "Perimeter, Area and Circumference" (PDF). Addison-Wesley. Archived from the original (PDF) on 6 October 2014.
^ Bennett, Jeffrey; Briggs, William (2005), Using and Understanding Mathematics / A Quantitative Reasoning Approach (3rd ed.), Addison-Wesley, p. 580, ISBN 978-0-321-22773-7
^ Katz, Victor J. (1998), A History of Mathematics / An Introduction (2nd ed.), Addison-Wesley Longman, p. 109, ISBN 978-0-321-01618-8
^ Almkvist, Gert; Berndt, Bruce (1988), "Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, π, and the Ladies Diary", American Mathematical Monthly, 95 (7): 585–608, doi:10.2307/2323302, JSTOR 2323302, MR 0966232, S2CID 119810884
Look up circumference in Wiktionary, the free dictionary.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Circumference&oldid=1066896111"
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Triakis octahedron - Wikipedia
Conway notation kO
Vertices by type 8{3}+6{8}
Symmetry group Oh, B3, [4,3], (*432)
In geometry, a triakis octahedron (or trigonal trisoctahedron[1] or kisoctahedron[2]) is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated cube.
It can be seen as an octahedron with triangular pyramids added to each face; that is, it is the Kleetope of the octahedron. It is also sometimes called a trisoctahedron, or, more fully, trigonal trisoctahedron. Both names reflect that it has three triangular faces for every face of an octahedron. The tetragonal trisoctahedron is another name for the deltoidal icositetrahedron, a different polyhedron with three quadrilateral faces for every face of an octahedron.
This convex polyhedron is topologically similar to the concave stellated octahedron. They have the same face connectivity, but the vertices are in different relative distances from the center.
If its shorter edges have length 1, its surface area and volume are:
{\displaystyle {\begin{aligned}A&=3{\sqrt {7+4{\sqrt {2}}}}\\V&={\frac {3+2{\sqrt {2}}}{2}}\end{aligned}}}
{\displaystyle \alpha ={\sqrt {2}}-1}
, then the 14 points
{\displaystyle (\pm \alpha ,\pm \alpha ,\pm \alpha )}
{\displaystyle (\pm 1,0,0)}
{\displaystyle (0,\pm 1,0)}
{\displaystyle (0,0,\pm 1)}
are the vertices of a triakis octahedron centered at the origin.
The length of the long edges equals
{\displaystyle {\sqrt {2}}}
, and that of the short edges
{\displaystyle 2{\sqrt {2}}-2}
The faces are isosceles triangles with one obtuse and two acute angles. The obtuse angle equals
{\displaystyle \arccos({\frac {1}{4}}-{\frac {1}{2}}{\sqrt {2}})\approx 117.200\,570\,380\,16^{\circ }}
and the acute ones equal
{\displaystyle \arccos({\frac {1}{2}}+{\frac {1}{4}}{\sqrt {2}})\approx 31.399\,714\,809\,92^{\circ }}
The triakis octahedron has three symmetry positions, two located on vertices, and one mid-edge:
A triakis octahedron is a vital element in the plot of cult author Hugh Cook's novel The Wishstone and the Wonderworkers.
The triakis octahedron is one of a family of duals to the uniform polyhedra related to the cube and regular octahedron.
The triakis octahedron is a part of a sequence of polyhedra and tilings, extending into the hyperbolic plane. These face-transitive figures have (*n32) reflectional symmetry.
3D model of a triakis octahedron
Animation of triakis octahedron and other related polyhedra
Spherical triakis octahedron
*n32 symmetry mutation of truncated tilings: t{n,3}
t{12i,3}
t{9i,3}
The triakis octahedron is also a part of a sequence of polyhedra and tilings, extending into the hyperbolic plane. These face-transitive figures have (*n42) reflectional symmetry.
^ "Clipart tagged: 'forms'". etc.usf.edu.
^ Conway, Symmetries of things, p. 284
Wenninger, Magnus (1983), Dual Models, Cambridge University Press, doi:10.1017/CBO9780511569371, ISBN 978-0-521-54325-5, MR 0730208 (The thirteen semiregular convex polyhedra and their duals, Page 17, Triakisoctahedron)
Eric W. Weisstein, Triakis octahedron (Catalan solid) at MathWorld.
Conway Notation for Polyhedra Try: "dtC"
Retrieved from "https://en.wikipedia.org/w/index.php?title=Triakis_octahedron&oldid=1047997749"
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EUDML | C4-Extensions of Sn as Galois Groups. EuDML | C4-Extensions of Sn as Galois Groups.
C4-Extensions of Sn as Galois Groups.
Crespo, Teresa. "C4-Extensions of Sn as Galois Groups.." Mathematica Scandinavica 76.2 (1995): 214-220. <http://eudml.org/doc/167338>.
@article{Crespo1995,
author = {Crespo, Teresa},
keywords = {explicit computations; Galois embedding problems; central extension of ; criterium for the solvability of the embedding problem; spinor norm; Clifford algebra},
title = {C4-Extensions of Sn as Galois Groups.},
AU - Crespo, Teresa
TI - C4-Extensions of Sn as Galois Groups.
KW - explicit computations; Galois embedding problems; central extension of ; criterium for the solvability of the embedding problem; spinor norm; Clifford algebra
Teresa Crespo, Galois realization of central extensions of the symmetric group with kernel a cyclic 2-group
explicit computations, Galois embedding problems, central extension of
{S}_{n}
, criterium for the solvability of the embedding problem, spinor norm, Clifford algebra
Articles by Teresa Crespo
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Staking your $CRV - Curve Finance
Starting on the 19th of September 2020, 50% of all trading fees are distributed to veCRV holders. This is the result of a community-led proposal to align incentives between liquidity providers and governance participants (veCRV holders).
Collected fees will be used to buy 3CRV (LP token for 3Pool) and distribute them to veCRV holders. This currently represents over $15M in trading fees per year. veCRV stands for vote escrowed $CRV, they are $CRV vote locked in the Curve DAO.
You can also lock $CRV to obtain a boost on your provided liquidity.
Locking your $CRV
You can extend a lock and add $CRV to it at any point but you cannot have $CRV with different expiry dates.
Claiming your trading fees
How to calculate the APY for staking CRV?
The formula below can help you calculate the daily APY:
DailyTradingVolume * 0.0002 * 365 / (TotalveCRV * CRVPrice) * 100%
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A probabilistic approach to convex (ϕ)-entropy decay for Markov chains
April 2022 A probabilistic approach to convex
\left(\mathit{\varphi }\right)
-entropy decay for Markov chains
Giovanni Conforti1
1Département de Mathématiques Appliquées, École Polytechnique
We study the exponential dissipation of entropic functionals along the semigroup generated by a continuous time Markov chain and the associated convex Sobolev inequalities, including MLSI and Beckner inequalities. We propose a method that combines the Bakry–Émery approach and coupling arguments, which we use as a probabilistic alternative to the discrete Bochner identities. In particular, the validity of the method is not limited to the perturbative setting and we establish convex entropy decay for interacting random walks beyond the high temperature/weak interaction regime. In this framework, we show that the exponential contraction of the Wasserstein distance implies MLSI. We also revisit classical examples often obtaining new inequalities and sometimes improving on the best known constants. In particular, we analyse the zero range dynamics, hardcore and Bernoulli–Laplace models and the Glauber dynamics for the Curie–Weiss and Ising model.
The author wishes to thank Paolo Dai Pra and Matthias Erbar for providing insightful comments at an early stage of this work.
Giovanni Conforti. "A probabilistic approach to convex
\left(\mathit{\varphi }\right)
-entropy decay for Markov chains." Ann. Appl. Probab. 32 (2) 932 - 973, April 2022. https://doi.org/10.1214/21-AAP1700
Received: 1 May 2020; Revised: 1 April 2021; Published: April 2022
Primary: 39B62 , 60J80 , 60K35
Keywords: Beckner inequalities , convexity of the entropy , coupling , Glauber dynamics , hardcore models , Markov chains , Modified logarithmic Sobolev inequality
Giovanni Conforti "A probabilistic approach to convex
\left(\mathit{\varphi }\right)
-entropy decay for Markov chains," The Annals of Applied Probability, Ann. Appl. Probab. 32(2), 932-973, (April 2022)
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Trin: Coll:
Sat. Ap. 18.’74
I enclose a paper of questions; will you write your answers across them— I don’t think you will want the slips for this—as reference to the book will suffice—even if that is necessary.—1
I continue very much the same— tho’ I got thro’ the journey well eno’, I had a turn of sickness afterwds.—2 I’ve had none since except a little yesterday. I’ve played tennis for
\frac{1}{2}
an hour each morning, but I’m so feeble & play so badly that its only just better than a constitutional. The place is v. desolate, but men return today. We were only 2 in hall yesty.—Aldis Wright3 & I. That Waring parchment is very curious & I’m delighted at having got it—4 I had a long talk with Wright (who’s a great antiquarian) about it last night in my rooms after hall & he thinks it will be worth printing in Notes & Queries or somewhere else.5 There are a great many v. curious words. It is a remarkable thing that none of the bedrooms have either washing stands or mirrors in them & there are no pictures at all altho’ the houses were clearly well furnished—every article is mentioned down to the minutest. Frank comes up today & I’m going to have the Ruck family to lunch tomorr.— I daresay Mrs. Ruck will bring the fiend this time6
I’ve answered Mr Forster of Exeter, but told him to take no further trouble, as I almost think the whole cousin business will collapse—for I’m less satisfied than I was with my own statistics—7
However I suppose must buckle to & write out what I can on the subject soon—but its awfully disheartening.
I dont think I shall be up to laboratory work yet.
I think I shall have don the book except index in 3 days!!
Yours affly | George Darwin
Mill’s Logic vol II p 18 footnote
“Mr Darwin’s remarkable speculation on the Origin of Species is another unimpeachable example of a legitimate hypothesis. What he terms “natural selection” is not only a vera causa, but one proved to be capable of producing effects of the same kind with those which the hypothesis ascribes to it: the question of possibility is entirely one of degree. It is unreasonable to accuse Mr. Darwin (as has been done) of violating the rules of Induction. The rules of Induction are concerned with the conditions of Proof. Mr Darwin has never pretended that his doctrine was proved. He was not bound by the rules of Induction, but by those of Hypothesis. And these last have seldom been more completely fulfilled. He has opened a path of inquiry full of promise, the results of which none can foresee. And is it not a wonderful feat of scientific knowledge & ingenuity to have tendered so bold a suggestion, which the first impulse of everyone was to reject at once, admissable & discussable, even as a conjecture”8
(Accurate copy GHD)
Verso of last page: ‘Cousins’ pencil
The list of questions has not been found. George had helped to prepare Descent 2d ed. and was checking the proofs of the book.
George had returned to Cambridge on 15 April 1874 (Emma Darwin’s diary (DAR 242)).
William Aldis Wright was senior bursar of Trinity College, Cambridge (ODNB).
The parchment has not been identified. The name ‘Waring’ in the Darwin family originated from Robert Waring, CD’s great-great-great-grandfather (Freeman 1978).
Notes and Queries is a literary journal founded in 1849 and published by Oxford University Press.
Francis Darwin was engaged to Amy Ruck. The other members of the Ruck family were Lawrence and Mary Anne Ruck and their children, Arthur Ashley, Laurence Ithel, Oliver Edwal, and Richard Matthews, and a married sister, Mary Elizabeth Atkin. The ‘fiend’ was Laurence Ithel Ruck (see letter from G. H. Darwin, 20 April 1874).
For George’s findings on cousin marriage, see the letter from G. H. Darwin, 6 February 1874. George probably refers to William Robson Scott. He acknowledged ‘Dr Scott of Exeter’ for information on children born deaf and dumb from first-cousin marriages, and for offering to put him in communication with superintendents of other institutions for the deaf and dumb in his article on cousin marriage; he also received information from Samuel Strong Forster of Worcester College for the Blind (G. H. Darwin 1875a, pp. 169–170).
The quotation is from the fifth edition of John Stuart Mill’s System of logic (Mill 1862, 2: 18 n.). For a previous discussion of the passage, see Correspondence vol. 11, letter from E. A. Darwin, 9 November 1863.
Sends queries [on proofs of Descent, 2d ed.]. Will be finished, except for the index, in two days.
Is now less satisfied than formerly with his statistics on cousin marriage.
[Enclosure is a copy by GHD of J. S. Mill’s statement about Origin (Logic 2: 18 n.).]
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Home : Support : Online Help : Programming : Data Types : Type Checking : Types : intersect
check for an object of type union
check for an object of type minus
check for an object of type intersect
check for an object of type subset
type(expr, `union`)
type(expr, `minus`)
type(expr, `intersect`)
type(expr, `subset`)
These procedures check to see if expr is a function of type union, minus, intersect, or subset respectively. See union for more information on these functions.
It is important to put quotes around the words union, minus, intersect, and subset as they are keywords. Not using quotes will cause a syntax error.
Note that in the case of the third example, the call to the minus function is made before the call to the type function, and therefore the return is false. The set difference,
b
, which is clearly not a function of type minus, is passed to the type function.
\mathrm{type}\left(a\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{union}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}b,\mathrm{`union`}\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{type}\left(\mathrm{`intersect`}\left(a,b,c\right),\mathrm{`intersect`}\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{type}\left({a,b,c}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{minus}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{a,c},\mathrm{`minus`}\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{type}\left(a\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{subset}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}b,\mathrm{`subset`}\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
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Ask Answer - Number Systems - Expert Answered Questions for School Students
Shreeja Dixit
\sqrt{3-2\sqrt{2}}
\sqrt{2}=1.4142
\sqrt{\frac{\sqrt{2}-1}{\sqrt{2}+1}}
Q.12. If x = 2 -
\sqrt{3}
{\left(x-\frac{1}{x}\right)}^{3}
Kanishq Bhardwaj
Q). Show that
1.414141......=1.\overline{)41}
\frac{p}{q}
, where p and q integers and q
\ne
Find two rational numbers between3/7 and 4/5
(2 root 2 + 5 root 3)(2 root 5 - 3 root 3)
(2?2+5?3)(2?5-3?3)
Explain root 13 spiral
Root 13 spiral
Q.4. Find the value of
{\left[\frac{{x}^{-4}}{{x}^{-10}}\right]}^{5/4}
Tuheen Talukdar
express 0.6 + 0.7tab + 0.47tab in the form p by q
Simplify\phantom{\rule{0ex}{0ex}}{\left(\frac{{5}^{-1}×{7}^{2}}{{5}^{2}×{7}^{-4}}\right)}^{7/2}×{\left(\frac{{5}^{-2}×{7}^{3}}{{5}^{3}×{7}^{-5}}\right)}^{-5/2}
Jyotir Aditya Srivastava
\frac{12}{3.6}, \frac{23}{3.1}, \left(\frac{23}{3.1} on number line\right)\phantom{\rule{0ex}{0ex}}\left(\sqrt{3} on number line\right)\phantom{\rule{0ex}{0ex}}\left(\frac{8}{3} on number line\right)
Nishtha Sinha
please help me to solve question number 6
Q6. If x = (2 +
\sqrt{5}
) , find the value of x2 + 1/x2
Determine rational number p and q
\mathbf{1}\mathbf{\right)}\mathbf{ }\mathbf{ }Determine rational number p and q\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\frac{7 - \sqrt{5}}{7 - \sqrt{5}} - \frac{7 - \sqrt{5}}{7 + \sqrt{5}} = P - 7\sqrt{59}
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Googlewhack - Wikipedia
A Googlewhack is a contest to find a Google Search query that returns a single result. A Googlewhack must consist of two words found in a dictionary and is only considered legitimate if both of the search terms appear in the result.The term googlewhack, coined by Gary Stock, first appeared on the web at Blinking on 8 January 2002
Published googlewhacks are short-lived since when published to a website, the new number of hits will become at least two: one to the original hit found, and one to the publishing site.[1]
The term googlewhack, coined by Gary Stock, first appeared on the web at Blinking on 8 January 2002.[2] Subsequently, Stock created The Whack Stack, at googlewhack.com, to allow the verification and collection of user-submitted Googlewhacks.
Googlewhacks were the basis of British comedian Dave Gorman's comedy tour Dave Gorman's Googlewhack Adventure and book of the same name.[3] In these Gorman tells the true story of how, while attempting to write a novel for his publisher, he became obsessed with Googlewhacks and traveled across the world finding people who had authored them. Although he never completed his original novel, Dave Gorman's Googlewhack Adventure went on to be a Sunday Times #1 best seller in the UK.
Participants at Googlewhack.com discovered the sporadic "cleaner girl" bug in Google's search algorithm where "results 1–1 of thousands" were returned for two relatively common words[4] such as Anxiousness Scheduler[5] or Italianate Tablesides.[6]
Googlewhack went offline in November 2009 after Google stopped providing definition links.[definition needed] Gary Stock stated on the game's web page soon afterward that he was pursuing solutions for Googlewhack to remain viable.[citation needed]
Some people propose the googlewhack "score", which is the product of the hits of the individual words.[7] Thus a googlewhack score is highest when the individual words produce a large number of hits.
comparative unicyclist[8]
maladroit wheezer[8]
New Scientist has discussed the idea of a Googlewhackblatt, which is similar to a Googlewhack except that it involves finding a single word that produces only one Google result. Lists of these have become available, but as with Googlewhacks they result in the Googlewhackblatt status of the word being destroyed—unless it is blocked by robots.txt or the word does not produce any Google results before it is added to the list, thus forming the Googlewhackblatt Paradox. Those words that do not produce any Google search results at all are known as Antegooglewhackblatts before they are listed—and subsequently elevated to Googlewhackblatt status if it is not blocked by robots.txt.
Feedback stories are also available on the New Scientist website, thus resulting in the destruction of any existing Googlewhackblatts that is ever printed in the magazine. Antegooglewhackblatts that are posted on the Feedback website become known as Feedbackgooglewhackblatts as their Googlewhackblatt status is created. In addition, New Scientist has more recently discovered another way to obtain a Googlewhackblatt without falling into the Googlewhackblatt Paradox. One can write the Googlewhackblatt on a website, but backward, and then search on elgooG to view the list properly while still keeping the Googlewhackblatt's status as a Googlewhackblatt.
In contrast to Googlewhacks, many Googlewhackblatts and Antegooglewhackblatts are nonsense words or uncommon misspellings that are not in dictionaries and probably never will be.
Practical use of specially constructed Googlewhackblatts was proposed by Leslie Lamport (although he did not use the term).[9]
Research applicationsEdit
The probabilities of internet search result values for multi-word queries was studied in 2008 with the help of Googlewhacks.[10][11][12] Based on data from 351 Googlewhacks from the "WhackStack" a list of previously documented Googlewhacks,[13] the Heaps' law
{\displaystyle \beta }
coefficient for the indexed World Wide Web (about 8 billion pages in 2008) was measured to be
{\displaystyle \beta =0.52}
. This result is in line with previous studies which used under 20,000 pages.[14] The googlewhacks were a key in calibrating the model so that it could be extended automatically to analyse the relatedness of word pairs.
Statistically improbable phrase – finds phrases in Amazon books unlikely to appear in any other book indexed
^ "Googlewhack official rules". Googlewhack.com. Archived from the original on 18 August 2017. Retrieved 28 March 2014.
^ "Googlewhacking: The Search for The One True Googlewhack". Unblinking.com. Retrieved 28 March 2014.
^ Gorman, Dave (2005). Dave Gorman's Googlewhack! adventure. London: Ebury. ISBN 0091897424.
^ "Googlewhack NACK!". Googlewhack.com. Archived from the original on 6 August 2017. Retrieved 28 March 2014.
^ Essex, Mike (13 February 2012). "Anxiousness Scheduler". Blog.blagman.co.uk. Archived from the original on 29 March 2014. Retrieved 28 March 2014.
^ "italianate tablesides". Googlewhack.com. Archived from the original on 21 January 2013. Retrieved 28 March 2014.
^ googlewhack scoring is discussed numerous places, e.g.,: [1] [2] [3]
^ a b "What is Googlewhacking? - Definition from WhatIs.com". WhatIs.com. Retrieved 17 December 2021.
^ Archival References to Web Pages, Ninth International World Wide Web Conference: Poster Proceedings (May 2000)
^ Lansey JC, Bukiet B (January 2009). "Internet Search Result Probabilities, Heaps' Law and Word Associativity". Journal of Quantitative Linguistics. 16 (1): 40–66. doi:10.1080/09296170802514153.
^ Googlewhacks for Fun and Profit on YouTube Google Tech Talk 2008
^ "Poster Presentation" (PDF). Retrieved 28 March 2014.
^ "The Whack Stack". Googlewhack. 13 February 2010. Archived from the original on 21 January 2013. Retrieved 17 December 2021.
^ Ricardo Baeza-Yates and Berthier Ribeiro-Neto, Modern Information Retrieval, ACM Press, 1999.
Bowman, Lisa M (29 January 2002). "Have you Googlewhacked?". CNET News. Retrieved 31 December 2012.
"'Googlewhacking' a new activity for the searchers". London: Reuters. 6 February 2002.
"A New Word on the Internet". New York: The New Yorker. 1 July 2015.
GoogleWhack.com
UnBlinking.com
List of Googlewhacks found at GoogleWhack.com
Retrieved from "https://en.wikipedia.org/w/index.php?title=Googlewhack&oldid=1078771699"
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11.3 Practical applications | Summary statistics | Siyavula
\dfrac{\text{8 + 8}}{\text{2}} = \text{8}
\text{6; 10; 8; 6; 11; 5; 10; 11; 6; 7; 5}
\text{5; 5; 6; 6; 6; 7; 8; 10; 10; 11; 11}
\text{6; 6; 14; 14; 13; 14; 15; 13; 18; 6}
\text{6; 6; 6; 13; 13; 14; 14; 14; 15; 18}
\dfrac{\text{13 + 14}}{\text{2}} = \dfrac{\text{27}}{\text{2}} = \text{13.5}
\text{18; 4; 4; 6; 4; 11; 4; 8; 11; 4; 4}
\text{4; 4; 4; 4; 4; 4; 6; 8; 11; 11; 18}
\text{5; 4; 4; 4; 4; 4; 3; 5; 4; 15}
\text{3; 4; 4; 4; 4; 4; 4; 5; 5; 15}
\text{7; 6; 10; 7; 8; 6; 9; 11; 10; 11; 9}
\text{6; 6; 7; 7; 8; 9; 9; 10; 10; 11}
\text{15; 15; 19; 20; 18; 20; 17; 15; 21}
20
\text{15 + 15 + 19 + 20 + 18 + 20 + 17 + 15 + 21 + 20}
\text{180}
\dfrac{\text{sum of the number of nuts}}{\text{number of packets}} = \dfrac{\text{180}}{\text{10}} = \text{18}
\text{7; 11; 7; 11; 5; 10; 7; 10}
\text{7}
\text{11}
\text{7}
\text{11}
\text{5}
\text{10}
\text{7}
\text{10}
\text{68}
\dfrac{\text{sum of the values}}{\text{number of values}} = \dfrac{\text{68}}{\text{8}} = \text{8.5}
\text{6}
\text{8}
\text{7}
\text{10}
\text{8}
\text{7}
\text{10}
\text{9}
\text{7}
\text{72}
\dfrac{\text{sum of the values}}{\text{number of values}} = \dfrac{\text{72}}{\text{9}} = \text{8}
\text{11; 6; 8; 7; 9; 9; 10}
\text{11}
\text{6}
\text{8}
\text{7}
\text{9}
\text{9}
\text{10}
\text{60}
\dfrac{\text{sum of the values}}{\text{number of values}} = \dfrac{\text{60}}{\text{7}} = \text{8.57142...} \approx 8.57
\text{0.9}
\text{1.2}
\text{0.8}
\text{1.3}
\text{1.1}
\text{1.7}
\text{1.1}
\text{8.1}
\dfrac{\text{sum of the rainfall (in mm)}}{\text{number of days}} = \dfrac{\text{8.1}}{\text{7}} = \text{1.15714...mm} \approx \text{1.16 mm}
\text{Mean}
\dfrac{\text{sum of the values}}{\text{number of values}}
\dfrac{\text{original number of playing cards}}{\text{group size}}
\therefore \text{Original number of playing cards}
\text{mean}\times \text{group size}
\text{6}\times\text{7}
\text{42}
\text{Mean}
\dfrac{\text{final number of playing cards}}{\text{group size}}
\therefore \text{Final number of playing cards}
\text{mean}\times \text{group size}
\text{7.6}\times\text{5}
\text{38}
\text{Mean}
\dfrac{\text{original number of sweets}}{\text{group size}}
\therefore \text{Original number of sweets}
\text{mean}\times \text{group size}
\text{6}\times\text{6}
\text{36}
\text{Mean}
\dfrac{\text{number of sweets after friends eat their sweets}}{\text{group size}}
\therefore \text{Number of sweets after friends eat their sweets}
\text{mean}\times \text{group size}
\text{8}\times\text{4}
\text{32}
\text{Original number of marbles}
\text{mean}\times \text{group size}
\text{1}\times\text{7}
\text{7}
\text{Final number of marbles}
\text{mean}\times \text{group size}
\text{0.6}\times\text{5}
\text{3}
\text{7; 6; 10; 11; 9; 6; 5}
\text{7 + 6 + 10 + 11 + 9 + 6 + 5 = 54 }
\text{Mean}
\dfrac{\text{sum of all the values}}{\text{number of values}}
\frac{54}{7}
\text{7.71428...}
\text{5; 6; 6; 7; 9; 10; 11}
\text{7 + 5 + 9 + 6 + 7 + 5 + 11 + 6 + 8 + 11 + 7 + 9 = 91}
39 + 28 + 55 + 55 + 11 + 18 + 11 = 217
11;\ 11;\ 18;\ 28;\ 39;\ 55;\ 55
13;\ 13;\ 14;\ 15;\ 15;\ 15;\ 16;\ 17
4;\ 6;\ 9;\ 9;\ 13;\ 18;\ 24;\ 27;\ 27
\text{Range} = 17 - 13 = 4
\text{Range} = 27 - 4 = 23
6;\ 7;\ 8;\ 10;\ 5;\ 7;\ 9;\ 8
5;\ 5;\ 6;\ 7;\ 7;\ 9;\ 9;\ 10;\ 11
16;\ 30;\ 16;\ 30;\ 8;\ 16;\ 27;\ 5
5;\ 8;\ 16;\ 16;\ 16;\ 27;\ 30;\ 30
105;\ 106;\ 106;\ 107;\ 107;\ 108;\ 108;\ 109;\ 110;\ 110;\ 111;\ 111
\text{ 20; 16; 10; 3; 12; 10; 11; 14; 5; 19 }
\text{ 13; 12; 11; 13; 13; 11; 12; 12; 11; 12 }
\text{ 3; 5; 10; 10; 11; 12; 14; 16; 19; 20 }
\text{ 11; 11; 11; 12; 12; 12; 12; 13; 13; 13 }
\text{Range of marks} = 20 - 3 = 17
\text{Range of marks} = 13 - 11 = 2
\text{28; 29; 29; 30; 32; 36; 36; 41 }
41 - 28 = 13
\text{11; 15; 16; 18; 21; 27; 48; 49 }
49 - 11 = 38
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Dynamical System Modeling Using Neural ODE - MATLAB & Simulink - MathWorks Italia
Synthesize Data of Target Dynamics
Train Model Using Custom Training Loop
This example shows how to train a neural network with neural ordinary differential equations (ODEs) to learn the dynamics of a physical system.
Neural ODEs [1] are deep learning operations defined by the solution of an ODE. More specifically, neural ODE is an operation that can be used in any architecture and, given an input, defines its output as the numerical solution of the ODE
{y}^{\prime }=f\left(t,y,\theta \right)
\left({t}_{0},{t}_{1}\right)
y\left({t}_{0}\right)={y}_{0}
. The right-hand side
f\left(t,y,\theta \right)
of the ODE depends on a set of trainable parameters
\theta
, which the model learns during the training process. In this example,
f\left(t,y,\theta \right)
is modeled with a model function containing fully connected operations and nonlinear activations. The initial condition
{y}_{0}
is either the input of the entire architecture, as in the case of this example, or is the output of a previous operation.
This example shows how to train a neural network with neural ODEs to learn the dynamics
x
of a given physical system, described by the following ODE:
{x}^{\prime }=Ax
A
is a 2-by-2 matrix.
The neural network of this example takes as input an initial condition and computes the ODE solution through the learned neural ODE model.
The neural ODE operation, given an initial condition, outputs the solution of an ODE model. In this example, specify a block with a fully connected layer, a tanh layer, and another fully connected layer as the ODE model.
In this example, the ODE that defines the model is solved numerically with the explicit Runge-Kutta (4,5) pair of Dormand and Prince [2]. The backward pass uses automatic differentiation to learn the trainable parameters
\theta
by backpropagating through each operation of the ODE solver.
The learned function
f\left(t,y,\theta \right)
is used as the right-hand side for computing the solution of the same model for additional initial conditions.
Define the target dynamics as a linear ODE model
{x}^{\prime }=Ax
, with x0 as its initial condition, and compute its numerical solution xTrain with ode45 in the time interval [0 15]. To compute an accurate ground truth data, set the relative tolerance of the ode45 numerical solver to
{10}^{-7}
. Later, you use the value of xTrain as ground truth data for learning an approximated dynamics with a neural ODE model.
A = [-0.1 -1; 1 -0.1];
trueModel = @(t,y) A*y;
numTimeSteps = 2000;
odeOptions = odeset(RelTol=1.e-7);
t = linspace(0, T, numTimeSteps);
[~, xTrain] = ode45(trueModel, t, x0, odeOptions);
xTrain = xTrain';
Visualize the training data in a plot.
plot(xTrain(1,:),xTrain(2,:))
title("Ground Truth Dynamics")
The model function consists of a single call to dlode45 to solve the ODE defined by the approximated dynamics
f\left(t,y,\theta \right)
for 40 time steps.
neuralOdeTimesteps = 40;
dt = t(2);
timesteps = (0:neuralOdeTimesteps)*dt;
Define the learnable parameters to use in the call to dlode45 and collect them in the variable neuralOdeParameters. The function initializeGlorot takes as input the size of the learnable parameters sz and the number of outputs and number of inputs of the fully connected operations, and returns a dlarray object with underlying type single with values set using Glorot initialization. The function initializeZeros takes as input the size of the learnable parameters, and returns the parameters as a dlarray object with underlying type single. The initialization example functions are attached to this example as supporting files. To access these functions, open this example as a live script. For more information about initializing learnable parameters for model functions, see Initialize Learnable Parameters for Model Function.
neuralOdeParameters = struct;
Initialize the parameters for the fully connected operations in the ODE model. The first fully connected operation takes as input a vector of size stateSize and increases its length to hiddenSize. Conversely, the second fully connected operation takes as input a vector of length hiddenSize and decreases its length to stateSize.
stateSize = size(xTrain,1);
neuralOdeParameters.fc1 = struct;
sz = [hiddenSize stateSize];
neuralOdeParameters.fc1.Weights = initializeGlorot(sz, hiddenSize, stateSize);
neuralOdeParameters.fc1.Bias = initializeZeros([hiddenSize 1]);
sz = [stateSize hiddenSize];
neuralOdeParameters.fc2.Weights = initializeGlorot(sz, stateSize, hiddenSize);
neuralOdeParameters.fc2.Bias = initializeZeros([stateSize 1]);
Display the learnable parameters of the model.
neuralOdeParameters.fc1
Bias: [2×1 dlarray]
Define Neural ODE Model
Create the function odeModel, listed in the ODE Model section of the example, which takes as input the time input (unused), the corresponding solution, and the ODE function parameters. The function applies a fully connected operation, a tanh operation, and another fully connected operation to the input data using the weights and biases given by the parameters.
Create the function model, listed in the Model Function section of the example, which computes the outputs of the deep learning model. The function model takes as input the model parameters and the input data. The function outputs the solution of the neural ODE.
Create the function modelLoss, listed in the Model Loss Function section of the example, which takes as input the model parameters, a mini-batch of input data with corresponding targets, and returns the loss and the gradients of the loss with respect to the learnable parameters.
Specify options for Adam optimization.
Train for 1200 iterations with a mini-batch-size of 200.
Every 50 iterations, solve the learned dynamics and display them against the ground truth in a phase diagram to show the training path.
Initialize the averageGrad and averageSqGrad parameters for the Adam solver.
Construct a mini-batch of data from the synthesized data with the createMiniBatch function, listed in the Create Mini-Batches Function section of the example.
Evaluate the model loss and gradients and loss using the dlfeval function and the modelLoss function, listed in the Model Loss Function section of the example.
Update the model parameters using the adamupdate function.
numTrainingTimesteps = numTimeSteps;
trainingTimesteps = 1:numTrainingTimesteps;
plottingTimesteps = 2:numTimeSteps;
% Create batch
[X, targets] = createMiniBatch(numTrainingTimesteps, neuralOdeTimesteps, miniBatchSize, xTrain);
% Evaluate network and compute loss and gradients
[loss,gradients] = dlfeval(@modelLoss,timesteps,X,neuralOdeParameters,targets);
% Update network
[neuralOdeParameters,averageGrad,averageSqGrad] = adamupdate(neuralOdeParameters,gradients,averageGrad,averageSqGrad,iter,...
learnRate,gradDecay,sqGradDecay);
% Plot loss
currentLoss = double(loss);
addpoints(lineLossTrain,iter,currentLoss);
% Plot predicted vs. real dynamics
if mod(iter,plotFrequency) == 0 || iter == 1
% Use ode45 to compute the solution
y = dlode45(@odeModel,t,dlarray(x0),neuralOdeParameters,DataFormat="CB");
plot(xTrain(1,plottingTimesteps),xTrain(2,plottingTimesteps),"r--")
plot(y(1,:),y(2,:),"b-")
title("Predicted vs. Real Dynamics")
legend("Training Ground Truth", "Predicted")
Use the model to compute approximated solutions with different initial conditions.
Define four new initial conditions different from the one used for training the model.
tPred = t;
x0Pred1 = sqrt([2;2]);
x0Pred2 = [-1;-1.5];
x0Pred3 = [0;2];
x0Pred4 = [-2;0];
Numerically solve the ODE true dynamics with ode45 for the four new initial conditions.
[~, xTrue1] = ode45(trueModel, tPred, x0Pred1, odeOptions);
Numerically solve the ODE with the learned neural ODE dynamics.
xPred1 = dlode45(@odeModel,tPred,dlarray(x0Pred1),neuralOdeParameters,DataFormat="CB");
Visualize the predicted solutions for different initial conditions against the ground truth solutions with the function plotTrueAndPredictedSolutions, listed in the Plot True and Predicted Solutions Function section of the example.
plotTrueAndPredictedSolutions(xTrue1, xPred1);
The model function, which defines the neural network used to make predictions, is composed of a single neural ODE call. For each observation, this function takes a vector of length stateSize, which is used as initial condition for solving numerically the ODE with the function odeModel, which represents the learnable right-hand side
f\left(t,y,\theta \right)
of the ODE to be solved, as right hand side and a vector of time points tspan defining the time at which the numerical solution is output. The function uses the vector tspan for each observation, regardless of the initial condition, since the learned system is autonomous. That is, the odeModel function does not explicitly depend on time.
function X = model(tspan,X0,neuralOdeParameters)
X = dlode45(@odeModel,tspan,X0,neuralOdeParameters,DataFormat="CB");
The odeModel function is the learnable right-hand side used in the call to dlode45. It takes as input a vector of size stateSize, enlarges it so that it has length hiddenSize, and applies a nonlinearity function tanh. Then the function compresses the vector again to have length stateSize.
function y = odeModel(~,y,theta)
y = tanh(theta.fc1.Weights*y + theta.fc1.Bias);
y = theta.fc2.Weights*y + theta.fc2.Bias;
This function takes as inputs a vector tspan, a set of initial conditions X0, the learnable parameters neuralOdeParameters, and target sequences targets. It computes the predictions with the model function, and compares them with the given targets sequences. Finally, it computes the loss and the gradient of the loss with respect to the learnable parameters of the neural ODE.
function [loss,gradients] = modelLoss(tspan,X0,neuralOdeParameters,targets)
% Compute predictions.
X = model(tspan,X0,neuralOdeParameters);
% Compute L1 loss.
loss = l1loss(X,targets,NormalizationFactor="all-elements",DataFormat="CBT");
gradients = dlgradient(loss,neuralOdeParameters);
Create Mini-Batches Function
The createMiniBatch function creates a batch of observations of the target dynamics. It takes as input the total number of time steps of the ground truth data numTimesteps, the number of consecutive time steps to be returned for each observation numTimesPerObs, the number of observations miniBatchSize, and the ground truth data X.
function [x0, targets] = createMiniBatch(numTimesteps,numTimesPerObs,miniBatchSize,X)
% Create batches of trajectories.
s = randperm(numTimesteps - numTimesPerObs, miniBatchSize);
x0 = dlarray(X(:, s));
targets = zeros([size(X,1) miniBatchSize numTimesPerObs]);
targets(:, i, 1:numTimesPerObs) = X(:, s(i) + 1:(s(i) + numTimesPerObs));
Plot True and Predicted Solutions Function
The plotTrueAndPredictedSolutions function takes as input the true solution xTrue, the approximated solution xPred computed with the learned neural ODE model, and the corresponding initial condition x0Str. It computes the error between the true and predicted solutions and plots it in a phase diagram.
function plotTrueAndPredictedSolutions(xTrue,xPred)
xPred = squeeze(xPred)';
err = mean(abs(xTrue(2:end,:) - xPred), "all");
plot(xTrue(:,1),xTrue(:,2),"r--",xPred(:,1),xPred(:,2),"b-",LineWidth=1)
title("Absolute Error = " + num2str(err,"%.4f"))
legend("Ground Truth","Predicted")
[1] Chen, Ricky T. Q., Yulia Rubanova, Jesse Bettencourt, and David Duvenaud. “Neural Ordinary Differential Equations.” Preprint, submitted December 13, 2019. https://arxiv.org/abs/1806.07366.
[2] Shampine, Lawrence F., and Mark W. Reichelt. “The MATLAB ODE Suite.” SIAM Journal on Scientific Computing 18, no. 1 (January 1997): 1–22. https://doi.org/10.1137/S1064827594276424.
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Compute RLC parameters of radial copper cables with single screen - MATLAB - MathWorks Nordic
Cable Parameters Tool
Open the Cable Parameters Tool App
Geometric mean distance between cables
Phase-Screen insulator
Outer Screen insulator
RLC matrices
Send RLC matrices to block
Send to workspace
Load Typical parameters
Load User parameters
Compute RLC parameters of radial copper cables with single screen
The Cable Parameters Tool app computes the RLC parameters of radial cooper cables that have a single screen. The app computes the parameters based on conductor and insulator characteristics.
For a set of N cables, the Cable Parameters Tool computes the self- and mutual impedances, the phase-to-screen, and screen to ground capacitances of radial cables with screen.
The Cable Parameters Tool assumes that a cable consists of an inner copper phase conductor with an outer screen conductor, using cross-linked polyethylene (XLPE) insulator material.
The Cable and Insulator Parameters
The following figure shows a typical high-voltage cable.
The variables used in the equations are:
N: The number of cables
n: the number of strands contained in the phase conductor.
d: the diameter of one strand (m)
f: the nominal frequency of the cable application
r: the radius of the phase conductor
µr: the relative permeability of phase conductor
rint, rext: the internal and external radius of the screen conductor
GMD: Geometric mean distance between the phase conductors.
ρ: Resistivity of the screen conductor
ɛrax: Relative permittivity of the phase-screen insulator
ɛrxe: Relative permittivity of the outer screen insulator
dax,Dax: the internal and external diameter of phase-screen insulator
dxe,Dxe: the internal and external diameter of the outer screen insulator
Self-Impedance of Phase Conductor(s)
The self-impedance of the copper phase conductor is calculated as:
\begin{array}{cc}{Z}_{aa}={R}_{\varphi }+{R}_{e}+j{k}_{1}\mathrm{log}\left(\frac{{D}_{e}}{GM{R}_{\varphi }}\right)& \Omega /\text{km}\end{array}
The DC resistance of phase conductor is given by:
\begin{array}{cc}{R}_{\varphi }={\rho }_{Cu}\frac{1000}{{S}_{Cu}}=\left(17.8e-9\right)\frac{1000}{n\pi {\left(d/2\right)}^{2}}& \Omega /\text{km}\end{array}
The resistance of earth return is given by:
\begin{array}{cc}{R}_{e}={\pi }^{2}\cdot {10}^{-4}\cdot f& \Omega /\text{km}\end{array}
The frequency factor is given by:
\begin{array}{cc}{k}_{1}=0.0529\cdot \frac{f}{0.3048\cdot 60}& units\text{ }\left(\Omega /\text{km}\right)\end{array}
The distance to equivalent earth return path is given by:
\begin{array}{cc}{D}_{e}=1650\sqrt{{\rho }_{e}/\left(2\pi f\right)}& m\\ {\rho }_{Cu}=17.8e-9& \Omega /m\end{array}
The geometric mean radius of phase conductor is given by:
GM{R}_{\varphi }=r\cdot \mathrm{exp}\left(-\frac{{\mu }_{r}}{4}\right)
Self Impedance of Screen Conductor(s)
The self-impedance of the screen conductor is calculated as:
\begin{array}{cc}{Z}_{xx}={R}_{N}+{R}_{e}+j{k}_{1}\mathrm{log}\left(\frac{{D}_{e}}{GM{R}_{N}}\right)& \Omega /\text{km}\end{array}
The DC resistance of the screen conductor is given by:
\begin{array}{cc}{R}_{N}=\rho \frac{1000}{S}& \Omega /\text{km}\end{array}
The geometric mean radius of the screen conductor is given by:
GM{R}_{N}={r}_{\mathrm{int}}+\frac{\left({r}_{ext}-{r}_{\mathrm{int}}\right)}{2}
Mutual Impedance Between the Phase and Screen Conductors
The mutual impedance between the phase conductor and its corresponding screen conductor is calculated as:
\begin{array}{cc}{Z}_{ax}={R}_{e}+j{k}_{1}\mathrm{log}\left(\frac{{D}_{e}}{{D}_{n}}\right)& \Omega /\text{km}\end{array}
Dn corresponds to the distance between the phase conductor and the mean radius of the screen conductor.
Mutual Impedance Between the Phase Conductors
If more than one cable is modeled (N>1), the mutual impedance between the N phase conductors is calculated as:
\begin{array}{cc}{Z}_{ab}={R}_{e}+j{k}_{1}\mathrm{log}\left(\frac{{D}_{e}}{GMD}\right)& \Omega /\text{km}\end{array}
In general, the geometric mean distance (GMD) between the phase conductors of a given set of cables can be calculated as
GMD=\sqrt[n]{\prod _{1}^{n}{d}_{xy}}
where n is the total number of distances between the conductors. However, the GMD value is not calculated by the app and needs to be specified directly as an input parameter.
Capacitance Between the Phase and Screen Conductors
The capacitance between the phase conductor and its corresponding screen conductor is calculated as:
\begin{array}{cc}{C}_{ax}=\frac{1}{0.3048}\left(\frac{0.00736{\epsilon }_{rax}}{\text{log }\left({D}_{ax}/{d}_{ax}\right)}\right)& \mu F/\text{km}\end{array}
The cross-linked polyethylene (XLPE) insulator material is assumed in this equation.
Capacitance Between the Screen Conductor and the Ground
The same equation is used to calculate the capacitance between the screen conductor and the ground:
\begin{array}{cc}{C}_{xe}=\frac{1}{0.3048}\left(\frac{0.00736{\epsilon }_{rxe}}{\text{log }\left({D}_{xe}/{d}_{xe}\right)}\right)& \mu F/\text{km}\end{array}
Capacitance Between the Phase Conductors
The capacitive effect between the phase conductors is negligible and therefore not computed by the power_cableparam function.
Ground resistivity — Ground resistivity
Ground resistivity, in Ohm-meters.
Frequency — Cable frequency
Frequency, in hertz, that is used to evaluate RLC parameters.
Number of cables — Number of cables
Number of cables. A cable consists of an inner phase conductor, an outer screen conductor, and insulator. This parameter determines the dimension of the R, L, and C matrices as 2N-by-2N, where N is the number of cables.
Geometric mean distance between cables — Geometric mean distance between cables
Geometric mean distance between the cables.
To enable this parameter, set Number of cables to 2 or higher.
Number of strands — Phase conductor strands
Number of strands contained in the phase conductor.
Strand diameter — Phase conductor strand diameter
Diameter of one strand, in mm, cm, or m.
Resistivity — Phase conductor resistivity
1.78e-08 (default) | positive scalar
DC resistivity of conductor, in ohm*m.
Relative permeability — Phase conductor relative permeability
Relative permeability of the conductor material.
External diameter — Phase conductor external diameter
20.9 mm (default) | positive scalar
Phase conductor outside diameter, in mm, cm, or m.
Resistivity — Screen conductor resistivity
Total section — Screen conductor total section
0.000169 m^2 (default) | positive scalar
Total section of screen conductor, in mm^2, cm^2, or m^2.
The screen total section value is sometimes provided in datasheets. If you do not know this value, you can compute it as follows:
Total section = pi*r_out^2 – pi*r_in^2 where:
r_out is the external radius of screen conductor
r_in is the internal radius of screen conductor
Internal diameter — Screen conductor internal diameter
External diameter — Screen conductor external diameter
Relative permittivity — Phase-screen insulator relative permittivity
Relative permittivity of the phase-screen material.
Internal diameter — Phase-screen insulator internal diameter
External diameter — Phase-screen insulator external diameter
Relative permittivity — Outer screen insulator relative permittivity
Relative permittivity of the outer screen material.
Internal diameter — Outer screen insulator internal diameter
External diameter — Outer screen insulator external diameter
RLC matrices — Compute RLC matrices
Computes the RLC matrices for a given cable. After computing the parameters, the app displays the results in the Computed Parameters section. The obtained results are of the form of 2N-by-2N RLC matrices that can be directly used in the block you selected to model your cable. For an example, see the 4 Cables with screen (PI model) block in the power_cable example.
Block — Block selector
Confirms the block selection. The name of the selected block appears in the field to the right of the Block button.
Send RLC matrices to block — Send RLC matrices to selected block
Sends the RLC parameters to the Distributed Parameter Line or Pi Section Line blocks specified in the Block parameter.
Send to workspace — Block selector
Sends the R, L, and C matrices to the MATLAB workspace. The app creates the variables R_matrix, L_matrix, and C_matrix in your workspace.
Load Typical parameters — Load typical parameters
Load the default cable parameters provided with Simscape™ Electrical™ Specialized Power Systems software. This command opens a browser window where you can select the DefaultCableParameters.mat file, which represents the four-cable configuration used in the power_cable example..
Load User parameters — Load user parameters
Save — Save cable data
Saves your cable data by generating a MAT-file that contains the GUI information and cable data.
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Necessary and Sufficient Condition for Mann Iteration Converges to a Fixed Point of Lipschitzian Mappings
2012 Necessary and Sufficient Condition for Mann Iteration Converges to a Fixed Point of Lipschitzian Mappings
Chang-He Xiang, Jiang-Hua Zhang, Zhe Chen
E
is a real normed linear space,
C
is a nonempty convex subset of
E
T:C\to C
is a Lipschitzian mapping, and
{x}^{*}\in C
T
{x}_{0}\in C
, suppose that the sequence
\left\{{x}_{n}\right\}\subset C
is the Mann iterative sequence defined by
{x}_{n+1}=\left(1-{\alpha }_{n}\right){x}_{n}+{\alpha }_{n}T{x}_{n},n\ge 0
\left\{{\alpha }_{n}\right\}
is a sequence in [0, 1],
{\sum }_{n=0}^{\infty }{\alpha }_{n}^{2}<\infty
{\sum }_{n=0}^{\infty }{\alpha }_{n}=\infty
. We prove that the sequence
\left\{{x}_{n}\right\}
strongly converges to
{x}^{*}
if and only if there exists a strictly increasing function
\mathrm{\Phi }:\left[0,\infty \right)\to \left[0,\infty \right)
\mathrm{\Phi }\left(0\right)=0
{\mathrm{limsup} }_{n\to \infty }{\mathrm{inf} }_{j\left({x}_{n}-{x}^{*}\right)\in J\left({x}_{n}-{x}^{*}\right)}\left\{〈T{x}_{n}-{x}^{*},j\left({x}_{n}-{x}^{*}\right)〉-\parallel {x}_{n}-{x}^{*}{\parallel }^{2}+\mathrm{\Phi }\left(\parallel {x}_{n}-{x}^{*}\parallel \right)\right\}\le 0
Chang-He Xiang. Jiang-Hua Zhang. Zhe Chen. "Necessary and Sufficient Condition for Mann Iteration Converges to a Fixed Point of Lipschitzian Mappings." J. Appl. Math. 2012 (SI15) 1 - 9, 2012. https://doi.org/10.1155/2012/327878
Chang-He Xiang, Jiang-Hua Zhang, Zhe Chen "Necessary and Sufficient Condition for Mann Iteration Converges to a Fixed Point of Lipschitzian Mappings," Journal of Applied Mathematics, J. Appl. Math. 2012(SI15), 1-9, (2012)
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Homogenization of Scalar Problem for a Combined Structure with Singular or Thin Reinforcement | EMS Press
Homogenization of Scalar Problem for a Combined Structure with Singular or Thin Reinforcement
S.E. Pastukhova
Technical University, Moscow, Russian Federation
The homogenization of quadratic integral functionals for combined structures with singular or asymptotically singular reinforcement is studied in a model case in dimension
N=2.
Generalizations to more general cases in dimension
N=2
or to some model cases in dimension
N>2
are discussed. Such results are obtained in the frame of homogenization of problems depending on two parameters developed by V.~V.~Zhikov in %%% here alteration: citations have been expanded. [\textit{Funct. Anal. Appl.} 33 (1999)(1)], [\textit{Sb.~Math.} 191 (2000)(7-8)], and [{\it Izv.~Math.} 66 (2002)(2)]. In particular, an essential tool is the notion of two-scale convergence of sequences of functions belonging to Sobolev spaces with respect to variable measures.
Giuseppe Cardone, A. Corbo Esposito, S.E. Pastukhova, Homogenization of Scalar Problem for a Combined Structure with Singular or Thin Reinforcement. Z. Anal. Anwend. 26 (2007), no. 3, pp. 277–301
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Carrier wave - Wikipedia
(Redirected from Carrier frequency)
In telecommunications, a carrier wave, carrier signal, or just carrier, is a waveform (usually sinusoidal) that is modulated (modified) with an information-bearing signal for the purpose of conveying information.[1] This carrier wave usually has a much higher frequency than the input signal does. The purpose of the carrier is usually either to transmit the information through space as an electromagnetic wave (as in radio communication), or to allow several carriers at different frequencies to share a common physical transmission medium by frequency division multiplexing (as in a cable television system). The term originated in radio communication, where the carrier wave creates the waves which carry the information (modulation) through the air from the transmitter to the receiver. The term is also used for an unmodulated emission in the absence of any modulating signal.[2]
The frequency spectrum of a typical radio signal from an AM or FM radio transmitter. The horizontal axis is frequency; the vertical axis is signal amplitude or power. It consists of a signal (C) at the carrier wave frequency fC, with the modulation contained in narrow frequency bands called sidebands (SB) just above and below the carrier.
In music production, carrier signals can be controlled by a modulating signal to change the sound property of an audio recording and add a sense of depth and movement.[3]
2 Carrierless modulation systems
3 Carrier leakage
The term carrier wave originated with radio. In a radio communication system, such as radio or television broadcasting, information is transmitted across space by radio waves. At the sending end, the information, in the form of a modulation signal, is applied to an electronic device called a transmitter. In the transmitter, an electronic oscillator generates a sinusoidal alternating current of radio frequency; this is the carrier wave. The information signal is used to modulate the carrier wave, altering some aspects of the carrier, to impress the information on the wave. The alternating current is amplified and applied to the transmitter's antenna, radiating radio waves that carry the information to the receiver's location. At the receiver, the radio waves strike the receiver's antenna, inducing a tiny oscillating current in it, which is applied to the receiver. In the receiver, the modulation signal is extracted from the modulated carrier wave, a process called demodulation.
Most radio systems in the 20th century used frequency modulation (FM) or amplitude modulation (AM) to add information to the carrier. The frequency spectrum of a modulated AM or FM signal from a radio transmitter is shown above. It consists of a strong component (C) at the carrier frequency
{\displaystyle f_{C}}
with the modulation contained in narrow sidebands (SB) above and below the carrier frequency. The frequency of a radio or television station is considered to be the carrier frequency. However the carrier itself is not useful in transmitting the information, so the energy in the carrier component is a waste of transmitter power. Therefore, in many modern modulation methods, the carrier is not transmitted. For example, in single-sideband modulation (SSB), the carrier is suppressed (and in some forms of SSB, eliminated). The carrier must be reintroduced at the receiver by a beat frequency oscillator (BFO).
Carriers are also widely used to transmit multiple information channels through a single cable or other communication medium using the technique of frequency division multiplexing (FDM). For example, in a cable television system, hundreds of television channels are distributed to consumers through a single coaxial cable, by modulating each television channel on a carrier wave of a different frequency, then sending all the carriers through the cable. At the receiver, the individual channels can be separated by bandpass filters using tuned circuits so the television channel desired can be displayed. A similar technique called wavelength division multiplexing is used to transmit multiple channels of data through an optical fiber by modulating them on separate light carriers; light beams of different wavelengths.
Carrierless modulation systemsEdit
The information in a modulated radio signal is contained in the sidebands while the power in the carrier frequency component does not transmit information itself, so newer forms of radio communication (such as spread spectrum and ultra-wideband), and OFDM which is widely used in Wi-Fi networks, digital television, and digital audio broadcasting (DAB) do not use a conventional sinusoidal carrier wave.
Carrier leakageEdit
Carrier leakage is interference caused by cross-talk or a DC offset. It is present as an unmodulated sine wave within the signal's bandwidth, whose amplitude is independent of the signal's amplitude. See frequency mixers.
^ "Carrier wave with no modulation transports no information". University Of Texas. Archived from the original on 2008-04-14. Retrieved 2008-05-30.
^ Federal Standard 1037C and MIL-STD-188
The dictionary definition of carrier wave at Wiktionary
Retrieved from "https://en.wikipedia.org/w/index.php?title=Carrier_wave&oldid=1033885063"
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On η -Upper Sign Property and Upper Sign Continuity and Their Applications in Equilibrium-Like Problems
\eta
-Upper Sign Property and Upper Sign Continuity and Their Applications in Equilibrium-Like Problems
Ali Farajzadeh, Somaye Jafari, Chin-Tzong Pang
We first introduce the notion of
\eta
-upper sign property which is an extension of the upper sign property introduced in Castellani and Giuli, 2013, by relaxing convexity on the set. Afterwards, we establish a link between the solution sets of local dual equilibrium problem (Minty local equilibrium problem) and equilibrium problem for mappings whose domains are not necessarily convex by relaxing the upper sign continuity on the map, as it is assumed in the literature (Bianchi and Pini, 2005; Castellani and Giuli, 2013; Farajzadeh and Zafarani, 2010). Accordingly, it allows us to extend and obtain some existence results for equilibrium-like problems.
Ali Farajzadeh. Somaye Jafari. Chin-Tzong Pang. "On
\eta
-Upper Sign Property and Upper Sign Continuity and Their Applications in Equilibrium-Like Problems." Abstr. Appl. Anal. 2014 (SI71) 1 - 6, 2014. https://doi.org/10.1155/2014/207502
Ali Farajzadeh, Somaye Jafari, Chin-Tzong Pang "On
\eta
-Upper Sign Property and Upper Sign Continuity and Their Applications in Equilibrium-Like Problems," Abstract and Applied Analysis, Abstr. Appl. Anal. 2014(SI71), 1-6, (2014)
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No. 1 (not no 4) Carlton Place.
I was rather afraid the brokerage would turn out a failure, we shall have to choose one out of the ones Mr A. knows;2 I am very sorry to hear that poor Aunt Charlotte is so unwell, I suppose winter coming on is very bad for asthma.3
If I had known the boys had been coming home last Sunday, I would have come too; I think I shall come this day fortnight or three weeks.4 You will be astonished to hear that I am up to snuff enough to manage by myself for a short time, that is with the help of the head Clerk; Mrs A. went up to London yesterday, and Mr A. took her up, so that I had to wind up all the business yesterday and it was a very full day; he is not coming back till Tuesday, so that I shall be the cock of the walk on Monday; the head Clerk is a very nice old man, who has been there 45 years, so that in any difficulty I can always go to him. I like Mr A. better than ever and we get on perfectly together. I went up and had a quiet dinner with them on Friday, no body there but Capt. Forrest who is a very nice man, and I had a very pleasant evening.5 I have come round to the opinion that Mrs A. is a much greater beauty than I had thought her; I can see they despise Southampton society, and know very of the town’s people; she shuddered when I mentioned Capt Vignole’s dinner, and said she had been there once and never means to go again.6
I don’t think I have ever given you an account of my day here, and one day is as good as a hundred; 7.30. a.m. a very mild rap at the door; from 7.45. till 8 I generally manage to get up, I then dress up to decency pitch and go down and put the tea in the pot, and have a soliloquy before the fire what to have for breakfast. The general result is an egg, or some times a rasher, or perhaps yesterday’s cold chop warmed up, but this morning a lovely kidney came up quite unexpected like. I have breakfast over by about 8.45., I then stand before the fire or have a quiet strum, till Mrs. Pratt appears to have a solemn Confab. about dinner, and to praise yesterdays dinner or this mornings kidney, or anything to do with the establishment, of which she is mighty proud.7 After we have made up our minds about dinner, it is about time to be off, as I have to be there by 10 minutes to 10, and it is quarter of an hour’s walk; Mr A. does not come in till 10.30. I always find Mr Fall there with the keys, ready for me to open the save and strong room.8 when it is opened and the Clerks have taken out all the books, I get out the money, and the notes, and drafts and bills &c, and then I have to look among the bills to see if any are coming due, and enter the things, drafts and bills notes powers of attorney and all sorts of things that come down from Lubbock; then theres the daily letter to write to Lubbock’s to say that we have theirs, and to tell them whats what to do for us;9 then after that there is the Ledger to be culled over, that is to see if all the business done the day before, has been put down to each separate person concerned; by this time it gets to be about 11.30, or 12. And then the times comes in, and one takes it easy, till some odds & ends want doing, of which there are generally plenty; if there is nothing to do, I finish the times or read a book, till about 1—when I generally try to do a little mathematics, but they come rather stiff after a bustling morning, and I am getting in rather a fright about my degree10
at about 2 or
\frac{1}{2}
past I go out and get a turn, sometimes
\frac{1}{2}
an hour’s some times less sometimes more; before I go in I get a Sandwich or butter and roll or a bun or two, at a pastry cook’s just opposite; it is by this time from 3 to 3.30. so that there is only
\frac{1}{2}
hour more before we wind up, which is rather a long job if there is anything wrong, but we generally get away from 4.30 to 4.45: at about 5 wet or dry I go out for about a 6 mile walk, and get back for dinner at 7; I find mathematics an awful try after dinner, I sit before the fire reading a book or something, till I begin to feel sleepy, then comes the tug, am I to go to sleep or not, my general resource is to go the pianoforte and have a frantic strum, to annoyance I should guess of the widow below me; the widow and I are just Box and Cox,11 just as I go out she comes down to breakfast, and soon after I go out my walk she comes in from hers so that I have never seen her except her back once; if the piano wakes me I do some mathematics and then read a book and so to bed; and so on; one after the other without much change.
They are going to get up Volunteer Engineers here under patronage of Sir H. James, and Mr A. gave me a hint as if he knew for a fact they were going to come and ask me to belong; but I have had enough volunteering.12 Would it be the correct thing to go and call again on Sir H. James as he was not in; I should think not—13 it has been a lovely day but rather cold; on Sunday I have dinner at two to suit Mrs P. after dinner I went a good long walk;
I hope you will be up to your primula paper I suppose it will be rather longer than the generality.14
am I ever to have “Orley Farm”15
Dated by the references to CD’s Primula paper and to Francis and George Howard Darwin being home from school (see nn. 4 and 14, below). In 1861, 17 November fell on a Sunday.
See letter to W. E. Darwin, 15 November [1861]. ‘Mr A’ was William’s partner in the Southampton and Hampshire Bank, George Atherley.
Emma Darwin’s elder sister, Charlotte Langton, had been in poor health for some time and had gone to St Leonard’s-on-Sea in Sussex to recuperate (see letter to W. E. Darwin, 4 November [1861]).
Francis and George Howard Darwin, who attended Clapham Grammar School, came home for the weekend on Saturday, 9 November 1861. William returned to Down House for Christmas, arriving on 24 December (Emma Darwin’s diary).
John Henry Forrest, who lived in Winchester, was chief constable of Hampshire (Post Office directory of Hampshire, Wiltshire, and Dorsetshire 1859), and Atherley’s brother-in-law.
Possibly John Vignoles, a naval commander who had retired in 1855 (Navy list 1861).
William had taken rooms in the lodging house of Mary Pratt, 1 Carlton Place, Southampton (Post Office directory of Dorsetshire, Wiltshire, and Hampshire 1867).
William perhaps refers to Phillip Carteret Fall, who had been a partner in the Southampton and Hampshire Bank before his retirement in the summer of 1861 (Banking almanac 1861; see also letter to W. E. Darwin, [25 May 1861]).
The bank of Robarts, Lubbock & Co was the London agent for the Southampton and Hampshire Bank. It was through the intermediation of John Lubbock and his father John William Lubbock that William had learned about the vacancy in the Southampton bank. The relationship between provincial banks and their London agents was strengthened following the institution of the country clearing system in 1858. The object of the clearing system was to collect drafts payable between bankers and to expedite settlements. Country banks dispatched to London in one parcel all the cheques drawn on other country banks received in a day’s trading and received in return the bulk of the cheques payable by themselves (see Matthews 1921).
William left the University of Cambridge without taking a degree in order to take up the offer of a banking partnership with the Southampton and Hampshire Bank. Having kept the required number of terms (three terms for three years), he planned to enter for the mathematical tripos examination of 1862. As Francis Darwin later recalled: ‘He must have gone in for the Tripos with his mathematics in a rusty condition, otherwise he might perhaps have been higher than bracketed top of the “Apostles” where his name appears.’ (F. Darwin 1914, pp. 20–1).
In John Maddison Morton’s play Box and Cox, John Box and James Cox are two characters who share the same apartment, one using it by day and the other by night so that they scarcely ever meet.
Henry James was the director of the Ordnance Survey based in Southampton. William, who was a captain in the Farnborough Rifle Volunteer Corps, was trying to resign his commission (see letters to W. E. Darwin, 22 October [1861], [27 October 1861], and 15 November [1861]).
CD had written a letter of introduction for William to James (see letters to W. E. Darwin, 12 October [1861] and [27 October 1861]).
CD read a paper on the two forms of Primula to the Linnean Society of London on Thursday, 21 November 1861 (see Collected papers 2: 45–63).
Orley Farm, by Anthony Trollope, was published in parts in Harper’s Monthly Magazine, between May 1861 and December 1862 (Irwin 1926). It was published in book form in 1862.
Irwin, Mary Leslie. 1926. Anthony Trollope: a bibliography. New York: Burt Franklin.
Matthews, Philip W. 1921. The bankers’ clearing house: what it is and what it does. London: Sir Isaac Pitman & Sons.
Describes in detail his day at home and at the bank in Southampton.
ALS 11pp inc ?
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Plot conformal array directivity or pattern versus azimuth - MATLAB - MathWorks Nordic
Plot Azimuth Pattern of 5-Element Cross Sonar Array
patternAzimuth(sArray,FREQ)
patternAzimuth(sArray,FREQ,EL)
patternAzimuth(sArray,FREQ,EL,Name,Value)
patternAzimuth(sArray,FREQ) plots the 2-D array directivity pattern versus azimuth (in dBi) for the array sArray at zero degrees elevation angle. The argument FREQ specifies the operating frequency.
patternAzimuth(sArray,FREQ,EL), in addition, plots the 2-D array directivity pattern versus azimuth (in dBi) for the array sArray at the elevation angle specified by EL. When EL is a vector, multiple overlaid plots are created.
patternAzimuth(sArray,FREQ,EL,Name,Value) plots the array pattern with additional options specified by one or more Name,Value pair arguments.
PAT = patternAzimuth(___) returns the array pattern. PAT is a matrix whose entries represent the pattern at corresponding sampling points specified by the 'Azimuth' parameter and the EL input argument.
sArray — Conformal array
Conformal array, specified as a phased.ConformalArray System object.
Example: sArray= phased.ConformalArray;
M-by-1 complex-valued column vector
Array weights, specified as the comma-separated pair consisting of 'Weights' and an M-by-1 complex-valued column vector. Array weights are applied to the elements of the array to produce array steering, tapering, or both. The dimension M is the number of elements in the array.
Example: 'Weights',ones(10,1)
Parent — Handle to axis
Handle to the axes along which the array geometry is displayed specified as a scalar.
PAT — Array directivity or pattern
L-by-N real-valued matrix
Array directivity or pattern, returned as an L-by-N real-valued matrix. The dimension L is the number of azimuth values determined by the 'Azimuth' name-value pair argument. The dimension N is the number of elevation angles, as determined by the EL input argument.
Construct a 5-element acoustic cross array (UCA) using the ConformalArray System object™. Assume the operating frequency is 4 kHz. A typical value for the speed of sound in seawater is 1500.0 m/s. Plot the array patterns at two different elevation angles.
Construct and view array
y = [-1,0,1,0,0]*lam/2;
z = [0,0,0,-1,1]*lam/2;
'ElementNormal',[zeros(1,N);zeros(1,N)]);
viewArray(sArray)
Plot azimuth pattern for magnitude
patternAzimuth(sArray,fc,[0,20],...
Plot azimuth pattern for directivity
D=4\pi \frac{{U}_{\text{rad}}\left(\theta ,\phi \right)}{{P}_{\text{total}}}
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I send very imperfect answer to question, & which I have written on foreign paper, to save you copying & you can send when you write to Thomson in Calcutta.—1 Hereafter I shall be able to answer better your question about qualities induced in individual being inherited:2 gout in man,—loss of wool in sheep (which begins in 1st generation & takes 2 or 3 to complete) probably obesity (for it is rare with poor); probably obesity & early maturity in Short-horn Cattle, &c.—
I am very glad you like Huxley’s Lectures;3 I have been very much struck with them; especially with the philosophy of induction.—4 I have quarrelled with him with overdoing sterility & ignoring cases from Gärtner & Kölreuter about sterile varieties.5 His geology is obscure;6 & I rather doubt about man’s mind & language.—7 But it seems to me admirably done, & as you say “oh my” about the praise of the Origin:8 I can’t help liking it, which makes me rather ashamed of myself.—
I enclose Asa Gray;9 only last page &
\frac{1}{2}
will interest you; but look at red (?) & rewrite names.10 Do not allude to Gray that you have seen this letter, as he might not like it, as he speaks of your being wrong (& converted, alas not so!) about Crossing.11 The sentence about Strawberries made me look at Bentham, & I have enclosed remark for him;12 I can assure him his remark would make any good horticulturist’s hair stand on end.13 It is marvellous to see Asa Gray so cock-sure about the doom of Slavery.—14
You wrote me a famous long letter a few days ago: Emma is going to read De TocVille & so was glad to hear your remarks.—15 I am glad to hear that you are going to do some work which will bring a little grist to the mill; but good Heavens how do you find time with Genera Plantarum, official work, friends, & Heaven knows what!16
Many thanks about Poison for Plants.—17 I know nothing about leaf-insects, except that they are carnivorous.— Andrew Murray knows.—18
You ask what I think about Falconer;19 of course I am much pleased at the very kind way he refers to me;20 but, as I look at it, the great gain is for any good man to give up immutability of species: the road is then open for progress; it is comparatively immaterial whether he believes in N. Selection; but how any man can persuade himself that species change unless he sees how they become adapted to their conditions is to me incomprehensible.—21 I do not see force of Falconer’s remarks about spire of shells, Phyllotaxis, &c:22 I suppose he did not look at my chapter on what I call laws of variation.—23
How very well Falconer writes: by the way in one of your letters you insisted on importance of style;24 I have just been struck with excellent instance in Alex. Braun on Rejuvenescence in Ray Soc 1853; I have tried & literally I cannot read it.25 Have you read it?
I have just received long pamphet by Alph. De Candolle on Oaks & allies,26 in which he has worked out in very complete & curious manner individual variability of species, & has wildish speculations on their migrations & duration &c.27 It is really curious to see how blind he is to the conditions or struggle for life; he attributes the presence of all species of all genera of trees to dryness or dampness! At end he has discussion on “Origin”;28 I have not yet come to this, but suppose it will be dead against it. Should you like to see this pamphlet?
My hot-house will begin building in a week or so,29 & I am looking with much pleasure at catalogues to see what plants to get: I shall keep to curious & experimental plants. I see I can buy Pitcher plants for only 10s .6!30 But the job is whether we shall be able to manage them. I shall get Sarracenia Dichœa your Hedysarum, Mimosa & all such funny things,, as far as I can without great expence.31 I daresay I shall beg for loan of some few orchids; especially for Acropera Loddigesii.32 I fancy orchids cost awful sums; but I must get priced catalogue. I can see hardly any Melastomas in catalogues.—33
I had a whole Box of small Wedgwood medallions; but drat the children everything in this house gets lost & wasted; I can find only about a dozen little things as big as shillings, & I presume worth nothing; but you shall look at them when here & take them if worth pocketing.34
You sent us a gratuitous insult about the “chimney-pots” in dining room, for you shan’t have them; nor are they Wedgwood ware.—35
Remember Naudin36
When you return you must remember my list of experimental seeds.—37 I hope you will enjoy yourself38
Goodnight my dear old friend | C. Darwin
You have not lately mentioned Mrs. Hooker: remember us most kindly to her.—39
The reference is to the surgeon and botanist Thomas Thomson; Thomson lived in Calcutta only until 1860 or 1861 (DNB). The enclosure has not been found. See also letter from J. D. Hooker, [12 January 1863] and n. 2.
See letter from J. D. Hooker, [12 January 1863]. On 23 January 1863, CD began writing up his ‘Chapter on Inheritance’ for Variation, eventually published as chapters 12–14 (Variation 2: 1–84; see ‘Journal’ (Correspondence vol. 11, Appendix II)).
Thomas Henry Huxley presented an evening lecture series for working men at the Museum of Practical Geology in London during November and December 1862; the lectures were published as T. H. Huxley 1863a. See letter to T. H. Huxley, 10 [January 1863], and letter from J. D. Hooker, [12 January 1862].
T. H. Huxley 1863a, pp. 55–67. Huxley’s discussion of induction formed part of the third lecture, delivered on 24 November 1862 (‘The method by which the causes of the present and past conditions of organic nature are to be discovered.— The origination of living beings’). There is a lightly annotated copy of T. H. Huxley 1863a in the Darwin Library–CUL (see Marginalia 1: 425).
Huxley argued that the origin of species through natural selection could not be proven until artificial selection produced from a common stock varieties that were sterile with one another (T. H. Huxley 1863a, pp. 146–50). CD, by contrast, was impressed by the plant hybridisation experiments conducted by Karl Friedrich von Gärtner and Joseph Gottlieb Kölreuter (Origin, pp. 246–9, 257–9, 270–4; Gärtner 1844 and 1849; Kölreuter 1761–6). See letter to T. H. Huxley, 10 [January 1863], and Correspondence vol. 10, Appendix VI.
T. H. Huxley 1863a, pp. 29–52. CD refers particularly to pages 39–41, and to figure 5 on page 40, which he thought would be confusing to a non-geologist. See letters to T. H. Huxley, 7 December [1862] and n. 7, and 18 December [1862] (Correspondence vol. 10).
T. H. Huxley 1863a, pp. 153–6. While arguing that ‘man differs to no greater extent from the animals which are immediately below him than these do from other members of the same order’, Huxley wrote that it was largely the power of language that distinguished man ‘from the whole of the brute world’ (ibid., pp. 154–5).
See letter from J. D. Hooker, [12 January 1863], and letter to T. H. Huxley, 10 [January 1863] and n. 4.
In Asa Gray’s letter, CD marked some of the plant names with marginal crosses in red crayon, and Hooker clearly printed the names ‘Abronia’, ‘Nyctaginia’, ‘Pavonia’ for Pavonia hastata, and ‘Ruellia’. These were plants in which the plants flowering earlier in the season were pollinated in the bud (see Correspondence vol. 10, letter from Asa Gray, 29 December 1862).
In November and December 1862, CD and Hooker debated the effects of crossing on variation, with Hooker maintaining that self-fertilisation did not favour variation, ‘whereas crossing tends to variation by adding differences’ (see Correspondence vol. 10, letter from J. D. Hooker, 26 November 1862). CD agreed with Gray (A. Gray 1862d, p. 420) that: free cross-breeding of incipient varieties inter se and with their original types is just the way to blend all together, to repress all salient characteristics as fast as the mysterious process of variation originates them, and fuse the whole into a homogeneous form. See Correspondence vol. 10, letter to Asa Gray, 26[–7] November [1862].
The letter from Asa Gray, 29 December 1862 (Correspondence vol. 10), is incomplete; Gray’s statement concerning strawberries was made in a postscript that has not been located. However, in his account of strawberries in Variation 1: 351–4, CD considered it unlikely that hybrids of European and American strawberries were fertile enough to be worth cultivation. This fact was surprising to him ‘as these forms structurally are not widely distinct, and are sometimes connected in the districts where they grow wild, as I hear from Professor Asa Gray, by puzzling intermediate forms’ (Variation 1: 352). CD probably consulted George Bentham’s Handbook of the British flora (Bentham 1858; see n. 13, below). The enclosure for Bentham has not been found. See also letter to Asa Gray, 2 January [1863] and n. 17.
In his Handbook of the British flora, Bentham wrote that while several wild and cultivated strawberries had been proposed as species, ‘the great facility with which fertile hybrids are produced, gives reason to suspect that the whole genus … may prove to consist but of one species’ (Bentham 1858, pp. 191–2). CD’s annotated copy of Bentham 1858 is in the Rare Books Room–CUL (see Marginalia 1: 51).
The letter from Asa Gray, 29 December 1862 (Correspondence vol. 10), is incomplete; the portion containing Gray’s statement regarding events in the United States has not been found. Gray may have commented on Abraham Lincoln’s emancipation proclamation, which was to come into effect on 1 January 1863; from that time all slaves in territories still in rebellion were to be freed (see Denney 1992, pp. 248, 251).
Emma Darwin. CD refers to Alexis Henri Charles Maurice Clérel, comte de Tocqueville’s Democracy in America (H. Reeve trans. 1862). See Correspondence vol. 10, letter from J. D. Hooker, [21 December 1862], and this volume, letter from J. D. Hooker, 6 January 1863. CD had read Tocqueville’s De la démocratie en Amérique (Tocqueville 1836) in February 1849 (see Correspondence vol. 10, letter to J. D. Hooker, 24 December [1862], and Correspondence vol. 4, Appendix IV, 119: 22b).
Hooker had been commissioned to write a flora of New Zealand (J. D. Hooker 1864–7; see letter from J. D. Hooker, 6 January 1863). At the same time, Hooker was at work on Genera plantarum (Bentham and Hooker 1862–83), and also had official duties in his capacity as assistant director of the Royal Botanic Gardens, Kew.
In his letter to Hooker of 3 January [1863], CD asked for advice about how to prevent mould from growing on his children’s dried flower collections; for Hooker’s reply, see his letter of 6 January 1863. In his Account book–cash accounts (Down House MS), on 16 January 1863, CD recorded a payment of 9s. for ‘Poison for plants’ to the London importers and makers of chemical and photographic apparatus, Bolton & Barnitt of Holborn Bars, London.
Hooker had asked CD what he should feed newly hatched leaf insects from Java (see letter from J. D. Hooker, 6 January 1863). Andrew Murray was a botanist and entomologist with expertise in Coleoptera and insects harmful to crops (DNB).
Hooker had asked CD’s opinion of Falconer 1863a (see letter from J. D. Hooker, 6 January 1863).
Falconer 1863a, pp. 77–81 (see n. 21, below). See also letter to Hugh Falconer, 5 [and 6] January [1863] and n. 7.
In his article on fossil and recent elephants, Hugh Falconer praised CD and his theory of modified descent (Falconer 1863a, pp. 77, 80). At the same time, he argued that natural selection was an inadequate explanation for the origin of species since some species subject to variable conditions over time, such as the mammoths, had remained unchanged (Falconer 1863a, p. 80).
While Falconer conceded that forms like the mammoth and other extinct elephants were ‘modified descendants of earlier progenitors’ (Falconer 1863a, p. 80), he continued to argue against the adequacy of natural selection to explain this modification: The law of Phyllotaxis, which governs the evolution of leaves around the axis of a plant, is nearly as constant in its manifestation, as any of the physical laws connected with the material world. Each instance, however different from another, can be shown to be a term of some series of continued fractions. When this is coupled with the geometrical law governing the evolution of form, so manifest in some departments of the animal kingdom, e. g. the spiral shells of the Mollusca, it is difficult to believe, that there is not in nature, a deeper seated and innate principle, to the operation of which ‘Natural Selection’ is merely an adjunct.
Origin, pp. 131–70.
The reference has not been identified.
Braun 1851. The English title of the article was ‘Reflections on the phenomena of rejuvenescence in nature, especially in the life and development of plants’ (Henfrey trans. 1853). There is an annotated copy of Arthur Henfrey’s translation of Braun 1851 in the Darwin Library–CUL (see Marginalia 1: 366–7).
Alphonse de Candolle sent CD copies of A. de Candolle 1862a and 1862b. See Correspondence vol. 10, letter from Alphonse de Candolle, 18 September 1862; see also following letter. CD’s annotated copies of A. de Candolle 1862a and 1862b are in the Darwin Pamphlet Collection–CUL.
A. de Candolle 1862a, pp. 326–53. See following letter and n. 6.
A. de Candolle 1862a, pp. 354–61, 363. See Intellectual Observer 3 (1863): 81–6, for a translation of the last portion of A. de Candolle 1862b. See also following letter and n. 7.
See Correspondence vol. 10, letter to J. D. Hooker, 24 December [1862]), and this volume, letter to Asa Gray, 2 January [1863] and n. 24, and Appendix VI.
See DAR 157.1: 111 and 112 for CD’s botanical notes on experiments with Nepenthes (pitcher plants).
CD had experimented on the power of movement in Hedysarum and Mimosa in 1862 (see Correspondence vol. 10).
CD was keen to obtain fresh flowers of Acropera; for CD’s continuing investigation of this orchid genus, see Correspondence vol. 10, letter from John Scott, 11 November 1862, and letter to John Scott, 12 November [1862], and this volume, letter from John Scott, 6 January 1863 and nn. 3 and 4.
See letter to Hugh Falconer, 5 [and 6] January [1863] and n. 22.
Hooker had started to collect Wedgwood ware and was particularly interested in medallions. See Correspondence vol. 10, letter from J. D. Hooker, [27 or 28 December 1862], and this volume, letter to J. D. Hooker, 3 January [1863], and letter from J. D. Hooker, 6 January 1863.
See letter from J. D. Hooker, 6 January 1863.
With his letter to Hooker of 24 December [1862] (Correspondence vol. 10), CD enclosed a ‘memorandum of enquiry’ for Charles Victor Naudin, whom Hooker hoped to meet during his forthcoming visit to Paris (see n. 38, below).
In his letter to Hooker of 3 November [1862] (Correspondence vol. 10), CD enclosed a list of the seeds he wanted for experiments on sensitivity in plants. See also ibid., letter to J. D. Hooker, [10–]12 November [1862].
Hooker and Bentham departed for Paris on 17 January 1863 (Jackson 1906, p. 193).
Braun, Alexander Carl Heinrich. 1851. Betrachtungen über die Erscheinung der Verjüngung in der Natur, insbesondere in der Lebens- und Bildungsgeschichte der Pflanze. Leipzig: Wilhelm Engelmann.
Tocqueville, Charles Alexis Henri Maurice Clérel de. 1836. De la démocratie en Amérique. 4th edition. 2 vols. in 1. Paris: Charles Gosselin.
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EUDML | An -regularity result for generalized harmonic maps into spheres. EuDML | An -regularity result for generalized harmonic maps into spheres.
\epsilon
-regularity result for generalized harmonic maps into spheres.
Moser, Roger. "An -regularity result for generalized harmonic maps into spheres.." Electronic Journal of Differential Equations (EJDE) [electronic only] 2003 (2003): Paper No. 01, 7 p., electronic only-Paper No. 01, 7 p., electronic only. <http://eudml.org/doc/122819>.
author = {Moser, Roger},
keywords = {generalized harmonic maps; Sobolev spaces; spheres; regularity},
title = {An -regularity result for generalized harmonic maps into spheres.},
TI - An -regularity result for generalized harmonic maps into spheres.
KW - generalized harmonic maps; Sobolev spaces; spheres; regularity
generalized harmonic maps, Sobolev spaces, spheres, regularity
Articles by Moser
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Computer communications hash algorithm
In cryptography, an HMAC (sometimes expanded as either keyed-hash message authentication code or hash-based message authentication code) is a specific type of message authentication code (MAC) involving a cryptographic hash function and a secret cryptographic key. As with any MAC, it may be used to simultaneously verify both the data integrity and authenticity of a message.
HMAC can provide authentication using a shared secret instead of using digital signatures with asymmetric cryptography. It trades off the need for a complex public key infrastructure by delegating the key exchange to the communicating parties, who are responsible for establishing and using a trusted channel to agree on the key prior to communication.
An iterative hash function breaks up a message into blocks of a fixed size and iterates over them with a compression function. For example, SHA-256 operates on 512-bit blocks. The size of the output of HMAC is the same as that of the underlying hash function (e.g., 256 and 512 bits in the case of SHA-256 and SHA3-512, respectively), although it can be truncated if desired.
{\displaystyle {\begin{aligned}\operatorname {HMAC} (K,m)&=\operatorname {H} {\Bigl (}{\bigl (}K'\oplus opad{\bigr )}\parallel \operatorname {H} {\bigl (}\left(K'\oplus ipad\right)\parallel m{\bigr )}{\Bigr )}\\K'&={\begin{cases}\operatorname {H} \left(K\right)&K{\text{ is larger than block size}}\\K&{\text{otherwise}}\end{cases}}\end{aligned}}}
‖ denotes concatenation
Hash function H
b, bytes
L, bytes
SHA-1 64 20
SHA-224 64 28
SHA-512/224 128 28
SHA-384 128 48
SHA3-512 72 64
out = H( in )
L = length( out )
b = H's internal block length
The following pseudocode demonstrates how HMAC may be implemented. Block size is 512 bits (64 bytes) when using one of the following hash functions: SHA-1, MD5, RIPEMD-128.[2]
function hmac is
key: Bytes // Array of bytes
message: Bytes // Array of bytes to be hashed
hash: Function // The hash function to use (e.g. SHA-1)
blockSize: Integer // The block size of the hash function (e.g. 64 bytes for SHA-1)
outputSize: Integer // The output size of the hash function (e.g. 20 bytes for SHA-1)
// Compute the block sized key
block_sized_key = computeBlockSizedKey(key, hash, blockSize)
o_key_pad ← block_sized_key xor [0x5c blockSize] // Outer padded key
i_key_pad ← block_sized_key xor [0x36 blockSize] // Inner padded key
return hash(o_key_pad ∥ hash(i_key_pad ∥ message))
function computeBlockSizedKey is
// Keys longer than blockSize are shortened by hashing them
key = hash(key)
// Keys shorter than blockSize are padded to blockSize by padding with zeros on the right
return Pad(key, blockSize) // Pad key with zeros to make it blockSize bytes long
return key
The design of the HMAC specification was motivated by the existence of attacks on more trivial mechanisms for combining a key with a hash function. For example, one might assume the same security that HMAC provides could be achieved with MAC = H(key ∥ message). However, this method suffers from a serious flaw: with most hash functions, it is easy to append data to the message without knowing the key and obtain another valid MAC ("length-extension attack"). The alternative, appending the key using MAC = H(message ∥ key), suffers from the problem that an attacker who can find a collision in the (unkeyed) hash function has a collision in the MAC (as two messages m1 and m2 yielding the same hash will provide the same start condition to the hash function before the appended key is hashed, hence the final hash will be the same). Using MAC = H(key ∥ message ∥ key) is better, but various security papers have suggested vulnerabilities with this approach, even when two different keys are used.[1][3][4]
No known extension attacks have been found against the current HMAC specification which is defined as H(key ∥ H(key ∥ message)) because the outer application of the hash function masks the intermediate result of the internal hash. The values of ipad and opad are not critical to the security of the algorithm, but were defined in such a way to have a large Hamming distance from each other and so the inner and outer keys will have fewer bits in common. The security reduction of HMAC does require them to be different in at least one bit.[citation needed]
The cryptographic strength of the HMAC depends upon the size of the secret key that is used. The most common attack against HMACs is brute force to uncover the secret key. HMACs are substantially less affected by collisions than their underlying hashing algorithms alone.[6][7] In particular, Mihir Bellare proved that HMAC is a PRF under the sole assumption that the compression function is a PRF.[8] Therefore, HMAC-MD5 does not suffer from the same weaknesses that have been found in MD5.
RFC 2104 requires that "keys longer than B bytes are first hashed using H" which leads to a confusing pseudo-collision: if the key is longer than the hash block size (e.g. 64 bytes for SHA-1), then HMAC(k, m) is computed as HMAC(H(k), m).This property is sometimes raised as a possible weakness of HMAC in password-hashing scenarios: it has been demonstrated that it's possible to find a long ASCII string and a random value whose hash will be also an ASCII string, and both values will produce the same HMAC output.[9][10]
In 2011 an informational RFC 6151[13] was published to summarize security considerations in MD5 and HMAC-MD5. For HMAC-MD5 the RFC summarizes that – although the security of the MD5 hash function itself is severely compromised – the currently known "attacks on HMAC-MD5 do not seem to indicate a practical vulnerability when used as a message authentication code", but it also adds that "for a new protocol design, a ciphersuite with HMAC-MD5 should not be included".
HMAC_SHA256("The quick brown fox jumps over the lazy dogThe quick brown fox jumps over the lazy dog", "message") = 5597b93a2843078cbb0c920ae41dfe20f1685e10c67e423c11ab91adfc319d12
^ a b Bellare, Mihir; Canetti, Ran; Krawczyk, Hugo (1996). "Keying Hash Functions for Message Authentication": 1–15. CiteSeerX 10.1.1.134.8430. {{cite journal}}: Cite journal requires |journal= (help)
^ "Definition of HMAC". HMAC: Keyed-Hashing for Message Authentication. sec. 2. doi:10.17487/RFC2104. RFC 2104.
^ Preneel, Bart; van Oorschot, Paul C. (1995). "MDx-MAC and Building Fast MACs from Hash Functions". CiteSeerX 10.1.1.34.3855. {{cite journal}}: Cite journal requires |journal= (help)
^ Preneel, Bart; van Oorschot, Paul C. (1995). "On the Security of Two MAC Algorithms". CiteSeerX 10.1.1.42.8908. {{cite journal}}: Cite journal requires |journal= (help)
^ Keccak team. "Keccak Team – Design and security". Retrieved 31 October 2019. Unlike SHA-1 and SHA-2, Keccak does not have the length-extension weakness, hence does not need the HMAC nested construction. Instead, MAC computation can be performed by simply prepending the message with the key.
^ Bruce Schneier (August 2005). "SHA-1 Broken". Retrieved 9 January 2009. although it doesn't affect applications such as HMAC where collisions aren't important
^ IETF (February 1997). "Security". HMAC: Keyed-Hashing for Message Authentication. sec. 6. doi:10.17487/RFC2104. RFC 2104. Retrieved 3 December 2009. The strongest attack known against HMAC is based on the frequency of collisions for the hash function H ("birthday attack") [PV,BCK2], and is totally impractical for minimally reasonable hash functions.
^ Bellare, Mihir. "New Proofs for NMAC and HMAC: Security without Collision-Resistance" (PDF). Journal of Cryptology. Retrieved 15 December 2021. This paper proves that HMAC is a PRF under the sole assumption that the compression function is a PRF. This recovers a proof based guarantee since no known attacks compromise the pseudorandomness of the compression function, and it also helps explain the resistance-to-attack that HMAC has shown even when implemented with hash functions whose (weak) collision resistance is compromised.
^ "PBKDF2+HMAC hash collisions explained · Mathias Bynens". mathiasbynens.be. Retrieved 7 August 2019.
^ "Aaron Toponce : Breaking HMAC". Retrieved 7 August 2019.
^ Jongsung, Kim; Biryukov, Alex; Preneel, Bart; Hong, Seokhie (2006). "On the Security of HMAC and NMAC Based on HAVAL, MD4, MD5, SHA-0 and SHA-1" (PDF). {{cite journal}}: Cite journal requires |journal= (help)
^ Wang, Xiaoyun; Yu, Hongbo; Wang, Wei; Zhang, Haina; Zhan, Tao (2009). "Cryptanalysis on HMAC/NMAC-MD5 and MD5-MAC" (PDF). Retrieved 15 June 2015. {{cite journal}}: Cite journal requires |journal= (help)
^ "RFC 6151 – Updated Security Considerations for the MD5 Message-Digest and the HMAC-MD5 Algorithms". Internet Engineering Task Force. March 2011. Retrieved 15 June 2015.
Rust HMAC implementation
Retrieved from "https://en.wikipedia.org/w/index.php?title=HMAC&oldid=1085852149"
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March. 7th.
It is indeed an age since we have had any communication, & very glad I was to receive your note. Our long silence occurred to me a few weeks since, & I had then thought of writing but was idle. I congratulate & condole with you on your tenth child;1 but please to observe when I have a 10th, send only condolences to me. We have now seven children, all well Thank God, as well as their mother; of these 7, five are Boys; & my Father used to say that it was certain, that a Boy gave as much trouble as three girls, so that bonâ fide we have 17 children.
It makes me sick whenever I think of professions; all seem hopelessly bad, & as yet I cannot see a ray of light.— I should very much like to talk over this (By the way my three Bug-bears are Californian & Australian Gold, beggaring me by making my money on mortgage worth nothing2 —The French coming by the Westerham & Sevenoaks roads, & therefore enclosing Down3 —and thirdly Professions for my Boys.) & I shd like to talk about Education, on which you ask me what we are doing. No one can more truly despise the old stereotyped stupid classical education than I do, but yet I have not had courage to break through the trammels. After many doubts we have just sent our eldest Boy to Rugby, where for his age he has been very well placed. By the way, I may mention for chance of hereafter your wishing for such a thing for any friends, that Mr. Wharton Vicar of Mitcham, appear to us a really excellent preparatory tutor or small school keeper.—4 I honour, admire & envy you for educating your Boys at home.5 What on earth shall you do with your Boys?
Towards the end of this month, we go to see Willy at Rugby, & thence for 5 or 6 days to Susan at Shrewsbury;6 I then return home to look after the Babies; & Emma goes to the F. Wedgwoods of Etruria for a week.7 Very many thanks for your most kind & large invitation to Delamere;8 but I fear we can hardly compass it. I dread going anywhere, on account of my stomach so easily failing under any excitement. I rarely even now go to London; not that I am at all worse, perhaps rather better & lead a very comfortable life with my 3 hours of daily work, but it is the life of a hermit. My nights are always bad, & that stops my becoming vigorous.— You ask about water cure: I take at intervals of 2 or 3 month, 5 or 6 weeks of moderately severe treatment, & always with good effect.9
Do you come here, I pray & beg whenever you can find time: you cannot tell how much pleasure it would give me & Emma.
I have finished 1st. vol. for Ray Soc. of Pedunculated cirripedes, which, as I think you are a member, you will soon get. Read what I describe on sexes of Ibla & Scalpellum.— I am now at work on the Sessile cirripedes, & am wonderfully tired of my job: a man to be a systematic naturalist ought to work at least 8 hours per day.— You saw through me, when you said that I must have wished to have seen effects of Holmfirth Debacle,10 for I was saying a week ago to Emma, that had I been, as I was in old days, I would have been certainly off that hour—
You ask after Erasmus; he is much as usual, & constantly more or less ⟨unw⟩ell. Susan is much better, & very flourishing & happy. Catherine is at Rome & has enjoyed it in a degree that is quite astonishing to my old dry bones.—
And now I think I have told you enough & more than enough about the house of Darwin; so my dear old Friend Farewell. What pleasant times we had in drinking Coffee in your rooms at Christ Coll. And think of the glories of Crux Major.11 Ah in those days there were no professions for sons, no ill-health to fear for them, no Californian gold—no French invasions. How paramount the future is to the present, when one is surrounded by children. My dread is hereditary ill-health. Even death is better for them.
My dear Fox your sincere friend | C. Darwin.
Remember do if you ever can, come here.
You can at any time send Athenæum Newspaper addressed to me at the Athenæum Club, Pall Mall which is my House of call for Parcels of all kinds—
P.S. Susan has lately been working in a way, which I think truly heroic about the scandalous violation of the act against children climbing chimneys.12 We have set up a little Society in Shrewsbury to prosecute those who break the Law.13 It is all Susan’s doing. She has had very nice letters from Ld. Shaftesbury & the D. of Sutherland, but the brutal Shropshire Squires are as hard as stone to move. The act out of London seems most commonly violated. It makes one shudder to fancy one of one’s own children at 7 years old being forced up a chimney—to say nothing of the consequent loathsome disease, & ulcerlated limbs, & utter moral degradation.14 If you think strongly on this subject, do make some enquiries— add to your many good works—this other one, & try to stir up the magistrates. There are several people making a stir in different parts of England on this subject.— It is not very likely that you would wish for such but I could send you some essays & information if you so liked, either for yourself or to give away.—
Emma desires me to give her very kind remembrances to Mrs Fox, in which I beg to join.—
Ellen Elizabeth Fox, born 26 February 1852, Fox’s fifth child by his second wife.
CD refers to the 1849 Californian gold-rush (see also Correspondence vol. 4, letter to Syms Covington, 23 November 1850) and the 1851 gold-rush in New South Wales and Victoria, Australia.
During 1851, Louis Napoleon, president of the French republic, challenged the Republicans and made clear his wish to re-establish the Empire. His attempts to achieve this culminated in the coup d’état of 2 December 1851, which was widely considered in England to be the first stage in the restoration of the monarchy and prompted fears of Napoleonic aggression. Shortly after the coup, Bartholomew James Sulivan held forth at a dinner party at Down House on the subject of ‘how easily a small invading force might overrun our south-eastern counties … Those present urged him to write to the papers on the subject.’ This he did in letters to the Naval and Military Gazette (10 and 31 January 1852), proposing the establishment of a volunteer corps (Sulivan ed. 1896, p. 426).
Henry James Wharton had been William Darwin’s tutor from autumn 1850 until he entered Rugby School in February 1852.
Fox had three sons at this time: Samuel William Darwin, 10
\frac{1}{2}
years old; Charles Woodd, 5 years old; and Robert Gerard, 2
\frac{1}{2}
According to Emma Darwin’s diary, she and CD, with Henrietta and George, were in Rugby on 24 March when Ernest Hensleigh Wedgwood and William joined them for dinner. They then travelled to Shrewsbury to stay with CD’s sister, Susan Elizabeth Darwin. She and Catherine Darwin continued to live at the Mount, the family residence in Shrewsbury. CD returned home on 1 April (‘Journal’; Correspondence vol. 5, Appendix I).
On 2 April, Emma and Susan Darwin travelled to Barlaston to visit Francis (Frank) and Fanny Mosley Wedgwood, Emma’s brother and sister-in-law. Emma returned home on 10 April (Emma Darwin’s diary).
Fox was rector of Delamere, Cheshire.
Fox had introduced the idea of hydropathy to CD in 1849 (see Correspondence vol. 4, letters to W. D. Fox, 6 February [1849] and 7 [July 1849]). After his initial visit to James Manby Gully’s hydropathic establishment in March 1849, CD twice returned to Malvern for therapy. He also continued the treatment at home. He may have consulted Gully about his own health in March 1851 when he took his daughter Anne to Malvern for treatment, but his Health diary (Down House MS) shows no treatments during the week he was away from Down on that visit.
The village of Holmfirth, in the West Riding, Yorkshire, had been destroyed when the dam of a reservoir burst on 5 February 1852. A full account of the disaster is in the Annual Register (1852): 478–81.
Panagæus crux major. In his letters to Fox, CD frequently recalled their capturing this beetle. See, among other letters, Correspondence vol. 1, letters to W. D. Fox, May 1832, [7–11] March 1835, and 15 February 1836; and Correspondence vol. 2, [25 March 1843]. See also Autobiography, p. 63.
The Parliamentary Acts of 1834 and 1840 prohibiting the use of boys under the age of sixteen as apprentices to chimney-sweeps failed to provide for enforcement. Lord Shaftesbury introduced bills in the House of Lords in 1851 and 1852 to strengthen the laws regulating chimney-sweeps, but an Act was passed only in 1864. See Strange 1982, p. xiv, and Hansard’s Parliamentary Debates, 3d ser. 176 (1864): index.
CD’s Account book (Down House MS) shows a contribution of £5 to the ‘Chimney Sweep Society per Catherine’ on 26 June 1852.
CD had read Henry Mayhew’s London labour and the London poor soon after publication in April 1851 (Correspondence vol. 4, Appendix IV, 119: 23b), in which a detailed account of the chimney-sweeps’ climbing boys is given. See Strange 1982 for a full description of the legislation brought in to prevent the use of climbing boys in chimneys.
Strange, Kathleen H. 1982. Climbing boys; a study of sweeps’ apprentices, 1773–1875. London and New York: Allison & Busby.
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Convert real Schur form to complex Schur form - MATLAB rsf2csf - MathWorks Italia
Transform Real Schur Form to Complex Schur Form
[Unew,Tnew] = rsf2csf(U,T)
[Unew,Tnew] = rsf2csf(U,T) transforms the outputs of [U,T] = schur(X) for real matrices X from real Schur form to complex Schur form. This operation transforms how the eigenvalues of X are expressed in T, and transforms U such that X = Unew*Tnew*Unew' and Unew'*Unew = eye(size(X)).
In real Schur form, T has real eigenvalues on the diagonal, and complex eigenvalues are expressed as 2-by-2 real blocks along the main diagonal:
\left[\begin{array}{cccccc}{\lambda }_{1}& {t}_{12}& {t}_{13}& {t}_{14}& {t}_{15}& \cdots \\ & a& e& {t}_{24}& {t}_{25}& \cdots \\ & f& a& {t}_{34}& {t}_{35}& \cdots \\ & & & c& g& \cdots \\ & & & h& c& \cdots \\ & & & & & \ddots \end{array}\right]
The eigenvalues represented by these blocks are
a±i\sqrt{-fe}
c±i\sqrt{-hg}
In complex Schur form, Tnew is upper triangular with all eigenvalues, real or complex, on the main diagonal:
\left[\begin{array}{cccccc}{\lambda }_{1}& tne{w}_{12}& tne{w}_{13}& tne{w}_{14}& tne{w}_{15}& \cdots \\ & a+bi& tne{w}_{23}& tne{w}_{24}& tne{w}_{25}& \cdots \\ & & a-bi& tne{w}_{34}& tne{w}_{35}& \cdots \\ & & & c+di& tne{w}_{45}& \cdots \\ & & & & c-di& \cdots \\ & & & & & \ddots \end{array}\right]
Apply Schur decomposition to a real matrix, and then transform the matrix factors so that the eigenvalues are directly on the main diagonal.
Create a real matrix and calculate the Schur decomposition. The U factor is unitary so that
{\mathit{U}}^{\mathit{T}}\mathit{U}={\mathit{I}}_{\mathit{N}}
, and the T factor is in real Schur form with complex conjugate eigenvalue pairs expressed as 2-by-2 blocks on the diagonal.
-2 1 1 4];
[U,T] = schur(X)
T has two real eigenvalues on the diagonal and one 2-by-2 block representing a complex conjugate pair of eigenvalues.
Transform U and T so that Tnew is upper triangular with the eigenvalues on the diagonal, and Unew satisfies X = Unew*Tnew*Unew'.
Unew = 4×4 complex
-0.4980 + 0.0000i -0.1012 + 0.2163i -0.1046 + 0.2093i 0.8001 + 0.0000i
-0.6751 + 0.0000i 0.1842 + 0.3860i -0.1867 - 0.3808i -0.4260 + 0.0000i
-0.2337 + 0.0000i 0.2635 - 0.6481i 0.3134 - 0.5448i 0.2466 + 0.0000i
Tnew = 4×4 complex
4.8121 + 0.0000i -0.9697 + 1.0778i -0.5212 + 2.0051i -1.0067 + 0.0000i
U — Unitary matrix
Unitary matrix, specified as the matrix returned by [U,T] = schur(X). The matrix U satisfies U'*U = eye(size(X)).
T — Schur form
Schur form, specified as the matrix returned by [U,T] = schur(X). The matrix T satisfies X = U*T*U'. The Schur form has real eigenvalues on the diagonal, and complex eigenvalues are expressed as 2-by-2 real blocks along the main diagonal.
Unew — Transformed unitary matrix
Transformed unitary matrix, returned as a matrix. The matrix Unew satisfies Unew'*Unew = eye(size(X)).
Tnew — Transformed Schur form
Transformed Schur form, returned as a matrix. Tnew is upper triangular with the eigenvalues of X on the diagonal, and it satisfies X = Unew*Tnew*Unew'.
You can use ordeig to obtain the same eigenvalue ordering as rsf2csf from the results of a Schur decomposition. However, rsf2csf also returns the remainder of the Schur matrix T and Schur vector matrix U, transformed to complex representation.
schur | cdf2rdf | ordeig
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Multisignal 1-D inverse wavelet packet transform - MATLAB idwpt - MathWorks 日本
Inverse Wavelet Packet Transform
Change Boundary Extension Mode
PR Biorthogonal Filters
Multisignal 1-D inverse wavelet packet transform
xrec = idwpt(wpt,l)
xrec = idwpt(wpt,l,wname)
xrec = idwpt(wpt,l,LoR,HiR)
xrec = idwpt(___, 'Boundary',ExtensionMode)
xrec = idwpt(wpt,l) inverts the discrete wavelet packet transform (DWPT) of the terminal node wavelet packet tree wpt using the bookkeeping vector l. The idwpt function assumes that you obtained wpt and l using dwpt with the fk18 wavelet and default settings.
If the input to dwpt was one signal, xrec is a column vector. If the input was a multichannel signal, xrec is a matrix, where each matrix column corresponds to a channel.
xrec = idwpt(wpt,l,wname) uses the wavelet specified by wname to invert the DWPT. wname must be recognized by wavemngr. The specified wavelet must be the same wavelet used to obtain the DWPT.
xrec = idwpt(wpt,l,LoR,HiR) uses the scaling (lowpass) filter, LoR, and wavelet (highpass) filter, HiR. The synthesis filter pair LoR and HiR must be associated with the same wavelet used in the DWPT.
xrec = idwpt(___, 'Boundary',ExtensionMode) specifies the mode to extend the signal. ExtensionMode can be either 'reflection' (default) and 'periodic'. By setting ExtensionMode to 'periodic' or 'reflection', the wavelet packet coefficients at each level are extended based on the modes 'per' or 'sym' in dwtmode, respectively. ExtensionMode must be the same mode used in the DWPT.
This example shows how to perform the inverse wavelet packet transform using synthesis filters.
Obtain the DWPT of an ECG signal using dwpt with default settings.
[wpt,l] = dwpt(wecg);
By default, dwpt uses the fk18 wavelet. Obtain the synthesis (reconstruction) filters associated with the wavelet.
[~,~,lor,hir] = wfilters('fk18');
Invert the DWPT using the synthesis filters and demonstrate perfect reconstruction.
xrec = idwpt(wpt,l,lor,hir);
norm(wecg-xrec,'inf')
Obtain the DWPT of an ECG signal using dwpt and periodic extension.
[wpt,l] = dwpt(wecg,'Boundary','periodic');
By default, idwpt uses symmetric extension. Invert the DWPT using periodic and symmetric extension modes.
xrecA = idwpt(wpt,l,'Boundary','periodic');
xrecB = idwpt(wpt,l);
Demonstrate perfect reconstruction only when the extension modes of the forward and inverse DWPT agree.
fprintf('Periodic/Periodic : %f\n',norm(wecg-xrecA,'inf'))
Periodic/Periodic : 0.000000
fprintf('Periodic/Symmetric: %f\n',norm(wecg-xrecB,'inf'))
Periodic/Symmetric: 1.477907
This example shows how to take an expression of a biorthogonal filter pair and construct lowpass and highpass filters to produce a perfect reconstruction (PR) pair in Wavelet Toolbox™.
The LeGall 5/3 filter is the wavelet used in JPEG2000 for lossless image compression. The lowpass (scaling) filters for the LeGall 5/3 wavelet have five and three nonzero coefficients respectively. The expressions for these two filters are:
{H}_{0}\left(z\right)=1/8\left(-{z}^{2}+2z+6+2{z}^{-1}-{z}^{-2}\right)
{H}_{1}\left(z\right)=1/2\left(z+2+{z}^{-1}\right)
Create these filters.
H0 = 1/8*[-1 2 6 2 -1];
H1 = 1/2*[1 2 1];
Many of the discrete wavelet and wavelet packet transforms in Wavelet Toolbox rely on the filters being both even-length and equal in length in order to produce the perfect reconstruction filter bank associated with these transforms. These transforms also require a specific normalization of the coefficients in the filters for the algorithms to produce a PR filter bank. Use the biorfilt function on the lowpass prototype functions to produce the PR wavelet filter bank.
[LoD,HiD,LoR,HiR] = biorfilt(H0,H1);
The sum of the lowpass analysis and synthesis filters is now equal to
\sqrt{2}
sum(LoD)
sum(LoR)
The wavelet filters sum, as required, to zero. The L2-norms of the lowpass analysis and highpass synthesis filters are equal. The same holds for the lowpass synthesis and highpass analysis filters.
Now you can use these filters in discrete wavelet and wavelet packet transforms and achieve a PR wavelet packet filter bank. To demonstrate this, load and plot an ECG signal.
Obtain the discrete wavelet packet transform of the ECG signal using the LeGall 5/3 filter set.
[wpt,L] = dwpt(wecg,LoD,HiD);
Now use the reconstruction (synthesis) filters to reconstruct the signal and demonstrate perfect reconstruction.
plot([wecg xrec])
axis tight, grid on;
You can also use this filter bank in the 1-D and 2-D discrete wavelet transforms. Read and plot an image.
im = imread('woodsculp256.jpg');
image(im); axis off;
Obtain the 2-D wavelet transform using the LeGall 5/3 analysis filters.
[C,S] = wavedec2(im,3,LoD,HiD);
Reconstruct the image using the synthesis filters.
imrec = waverec2(C,S,LoR,HiR);
image(uint8(imrec)); axis off;
The LeGall 5/3 filter is equivalent to the built-in 'bior2.2' wavelet in Wavelet Toolbox. Use the 'bior2.2' filters and compare with the LeGall 5/3 filters.
[LD,HD,LR,HR] = wfilters('bior2.2');
hl = stem([LD' LoD']);
hl(1).MarkerFaceColor = [0 0 1];
hl(1).Marker = 'o';
hl(2).Marker = '^';
title('Lowpass Analysis')
hl = stem([HD' HiD']);
title('Highpass Analysis')
hl = stem([LR' LoR']);
title('Lowpass Synthesis')
hl = stem([HR' HiR']);
title('Highpass Synthesis')
wpt — Terminal node wavelet packet tree
Terminal node wavelet packet tree, specified as a cell array. wpt is the output of dwpt with the 'FullTree' value set to false.
Example: [wpt,l] = dwpt(X,'Level',3,'FullTree',false) returns the terminal node wavelet packet tree of the three-level wavelet packet decomposition of X.
Bookkeeping vector, specified as a vector of positive integers. The vector l is the output of dwpt. The bookkeeping vector contains the length of the input signal and the number of coefficients by level, and is required for perfect reconstruction.
'fk18' (default) | character vector | string scalar
Wavelet to use in the inverse DWPT, specified as a character vector or string scalar. wname must be recognized by wavemngr. The specified wavelet must be the same wavelet used to obtain the DWPT.
You cannot specify both wname and a filter pair, LoD and HiD.
Example: xrec = idwpt(wpt,l,"sym4") specifies the sym4 wavelet.
LoR,HiR — Wavelet synthesis filters
Wavelet synthesis (reconstruction) filters to use in the inverse DWPT, specified as a pair of real-valued vectors. LoR is the scaling (lowpass) synthesis filter, and HiR is the wavelet (highpass) synthesis filter. The synthesis filter pair must be associated with the same wavelet as used in the DWPT. You cannot specify both wname and a filter pair, LoR and HiR. See wfilters for additional information.
idwpt does not check that LoR and HiR satisfy the requirements for a perfect reconstruction wavelet packet filter bank. See PR Biorthogonal Filters for an example of how to take a published biorthogonal filter and ensure that the analysis and synthesis filters produce a perfect reconstruction wavelet packet filter bank using idwpt.
ExtensionMode — Wavelet packet transform boundary handling
'reflection' (default) | 'periodic'
Wavelet packet transform boundary handling, specified as 'reflection' or 'periodic'. When set to 'reflection' or 'periodic', the wavelet packet coefficients are extended at each level based on the 'sym' or 'per' mode in dwtmode, respectively. ExtensionMode must be the same mode used in the DWPT. If unspecified, ExtensionMode defaults to 'reflection'.
[3] Mesa, Hector. “Adapted Wavelets for Pattern Detection.†In Progress in Pattern Recognition, Image Analysis and Applications, edited by Alberto Sanfeliu and Manuel Lazo Cortés, 3773:933–44. Berlin, Heidelberg: Springer Berlin Heidelberg, 2005. https://doi.org/10.1007/11578079_96.
dwpt | imodwpt
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By: Zhiyong Zhang
Keywords:distribution; kurtosis; skewness
Moments are quantitative measures of a distribution function. Formally, the nth moment about a value c of a distribution f(x) is defined as
\begin{array}{l}{\mu }_{n}=E\left[\left(x-c{\right)}^{n}\right]=\\ \text{ }\text{ }\left\{\begin{array}{c}\sum {\left(x-c\right)}^{n}f\left(x\right)\text{ }\text{Discrete distibution}\\ \int {\left(x-c\right)}^{n}f\left(x\right)dx\text{ }\text{Continuous distribution}\end{array}\right\}.\end{array}
[Page 1084]When c = 0, they are called the raw moments, and when c is set at the mean of the distributions, they are called central moments. The first raw moment is the mean and the first central moment is 0. For the second and higher moments, the central moments are often used. For some distributions, their moments can be flexibly obtained through their moment-generating functions. Certain distributions can be uniquely determined by a few moments. For example, a normal distribution can be determined by its first two moments. Although higher moments of a distribution can be available, the first four moments are ...
Zhang, Z. (2018). Moments of a distribution. In B. Frey (Ed.), The SAGE encyclopedia of educational research, measurement, and evaluation (Vol. 1, pp. 1084-1085). SAGE Publications, Inc., https://dx.doi.org/10.4135/9781506326139.n441
Zhang, Zhiyong. "Moments of a Distribution." In The SAGE Encyclopedia of Educational Research, Measurement, and Evaluation, edited by Frey, Bruce B., 1084-85. Thousand Oaks,, CA: SAGE Publications, Inc., 2018. https://dx.doi.org/10.4135/9781506326139.n441.
Zhang, Z. 2018. Moments of a Distribution. In: Bruce B. Frey Editor, 2018. The SAGE Encyclopedia of Educational Research, Measurement, and Evaluation, Thousand Oaks,, CA: SAGE Publications, Inc. pp. 1084-1085 Available at: <https://dx.doi.org/10.4135/9781506326139.n441> [Accessed 23 May 2022].
Zhang, Zhiyong. "Moments of a Distribution." The SAGE Encyclopedia of Educational Research, Measurement, and Evaluation. Edited by Bruce B. Frey. Vol. 1. Thousand Oaks,: SAGE Publications, Inc., 2018, pp. 1084-85. SAGE Knowledge. 23 May 2022, doi: https://dx.doi.org/10.4135/9781506326139.n441.
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eqn - Maple Help
Home : Support : Online Help : Programming : codegen Package : eqn
produce output suitable for troff/eqn printing
eqn(expr, filename)
(optional) output to a file
The eqn function produces output which is suitable for printing the input expression expr with a troff/eqn processor. It knows how to format integrals, limits, sums, products and matrices.
The mathematical format is taken, in general, from the CRC handbook or the Handbook of Mathematical functions.
The functions sum, product, int, diff, limit, and log are aliased by Sum, Product, Int, Diff, Limit, and Log, so that these can be used to prevent evaluation by Maple.
User-defined function printing can be interfaced by including a procedure `eqn/function-name`.
Note eqn will not produce .EQ or .EN lines in the output file.
The function eqn produces output as a side-effect, and returns NULL as the function value. Therefore the ditto commands, % and %%, will not recall the output of an eqn command.
\mathrm{with}\left(\mathrm{codegen}\right):
\mathrm{eqn}\left({x}^{2}+{y}^{2}={z}^{2}\right)
{{{ "x" sup 2 }^+^{ "y" sup 2 }}~~=~~{ "z" sup 2 }}
Put this output in the file EqnFile
\mathrm{eqn}\left({x}^{2}+{y}^{2}={z}^{2},\mathrm{EqnFile}\right)
\mathrm{eqn}\left(\mathrm{Int}\left(\frac{1}{{x}^{2}+1},x\right)=\mathrm{int}\left(\frac{1}{{x}^{2}+1},x\right)\right)
{{int { {( {{ "x" sup 2 }^+^1 } )} sup -1 }~d "x" }~~=~~{arctan (
"x" )}}
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Brightness & Efficiency calculations - FPbase Help
Creating a microscope page
Brightness & Efficiency calculations
An explanation on how FPbase calculates excitation and emission efficiency on microscope page
The brightness/efficiency readout on an FPbase microscope page
Caution! These calculations cannot take many critical parameters into account (such as FP expression level, photobleaching, illumination power, filter delamination, etc... )! As such, they are merely intended as theoretical predictions, and may not reflect the actual comparative brightness of two fluorophores on your system.
The first number in this field (11.71 below) provides a rough estimate of the apparent brightness for a given fluorophore/filter-set combination (when extinction coefficient and quantum yield are available for the fluorophore). The numbers in parentheses give the excitation efficiency and collection efficiency with the current filter combination.
Brightness is calculated as the product of the excitation and collection efficiencies (described below) and the extinction coefficient and quantum yield of the selected fluorophore, all divided by 1000. If the EC and QY are not available for a given probe, then only excitation and collection efficiencies will be shown. The absolute value of this number is not particularly meaningful, but it can be used to compare the relative brightness of different fluorophore/filter arrangements.
"Standard" Mode:
In the normal mode, excitation efficiency is interpreted as the percentage of light incident upon the sample that can be absorbed by the fluorophore. It is calculated as the area under the curve for the combined light + excitation filters (and dichroics) + fluorophore excitation spectra, divided by the area under the curve of the light + excitation filters alone:
\frac{ \int ( \epsilon_{ex} \times fluor_{ex})}{\int \epsilon_{ex} }
\epsilon_{ex}
is the effective excitation spectrum:
\epsilon_{ex} = light \times filter_{ex} \times filter_{dichroic}
For example, a (narrow band) laser at the peak absorption wavelength of a fluorophore would have near 100% efficiency; but a very broadband excitation spectrum, even if it overlaps the peak absorption wavelength, can have relatively poor excitation efficiency if it contains excess off-peak energy. Even though the 460/80x filter shown in the first image below covers much of the EGFP excitation spectrum, it has lower excitation efficiency (58%, represented by the area with diagonal lines) than a 488nm laser right at the peak excitation wavelength of EGFP (99.8% efficiency):
"lower" excitation efficiency in standard mode
"higher" excitation efficiency in standard mode
This mode really favors light sources that efficiently excite the fluorophore, without a lot of off-peak excitation (which would be particularly useful in a live-cell setting, where you want to minimize the energy impinging on the sample). One downside of this mode is that broader-band excitation filters tend to render a lower efficiency (and therefore "brightness") score than narrow-band filters centered on the peak excitation wavelength... which can be a bit counter-intuitive. Which is why "broadband mode" is also available, below.
Another potential confusion here is caused by the fact that we cannot make any assumptions about the power of the light source (it is assumed to be "sufficiently high"). As an admittedly "strange" example: a laser at peak absorption wavelength (here, 488nm) will still have excellent excitation efficiency even if a poorly matched excitation filter that effectively blocks the laser (eg. 515/20) is added to the light path (since one could theoretically just turn up the laser infinitely).
"Broadband" mode
this setting is available in the settings (gear icon)
Because the standard mode of excitation efficiency penalizes broadband "off-peak" excitation, it can lead to unexpected results when comparing the expected brightness of two excitation filters. In "broadband" mode, the light source is assumed to be of constant power, and excitation efficiency is interpreted as coverage of the excitation spectrum. Here, efficiency is calculated as the area under the curve for the combined light + excitation filters (and dichroics) + fluorophore excitation spectra, divided by the area under the curve of fluorophore excitation spectra alone:
\frac{ \int ( \epsilon_{ex} \times fluor_{ex})}{\int fluor_{ex}}
\epsilon_{ex}
\epsilon_{ex} = light \times filter_{ex} \times filter_{dichroic}
If you've got a broadband light-source (such as a metal-halide or multi-LED light source) and you are trying to determine the expected brightness of a fluorophore given different excitation filters, this mode is more likely to behave as you would expect.
The problem with this mode is that it makes laser illumination look terrible: as a monochromatic light-source, will never cover much of the fluorophore excitation spectrum, but rather puts a lot of power into (hopefully) the most effective wavelengths for excitation. For this reason, the "standard mode" above is default.
Emission (Collection) Efficiency
Excitation efficiency is the percentage of emission photons that can be collected given the emission path. It is calculated as the area under the curve for the combined emission filters (and dichroics) + camera QE + fluorophore emission spectra, divided by the area under the full fluorophore emission spectrum.
\frac{ \int ( \epsilon_{em} \times fluor_{em})}{\int fluor_{em}}
\epsilon_{em}
is the combined emission path spectrum:
\epsilon_{em} = filter_{em} \times filter_{dichroic} \times camera_{QE}
In the image below, the EGFP emission spectrum is relatively well matched to the 525/50m filter, and the collection efficiency, represented by the area with diagonal lines, is about 58% the area of the full fluorophore emission spectrum.
As a side-note: don't forget that many additional photons are lost as a result of not being collected by the objective lens in the first place, or by scattering or absorption somewhere in the emission path. So "100%" collection efficiency here by no means that you collected every photon emitted from the fluorophore.
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Riemannian manifold - formulasearchengine
{{#invoke:Hatnote|hatnote}} Template:Distinguish2
In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space (M,g) is a real smooth manifold M equipped with an inner product
{\displaystyle g_{p}}
{\displaystyle T_{p}M}
{\displaystyle p}
that varies smoothly from point to point in the sense that if X and Y are vector fields on M, then
{\displaystyle p\mapsto g_{p}(X(p),Y(p))}
is a smooth function. The family
{\displaystyle g_{p}}
of inner products is called a Riemannian metric (tensor). These terms are named after the German mathematician Bernhard Riemann. The study of Riemannian manifolds constitutes the subject called Riemannian geometry.
2.1 Riemannian manifolds as metric spaces
3 Riemannian metrics
3.2 The pullback metric
3.3 Existence of a metric
4 Riemannian manifolds as metric spaces
4.2 Geodesic completeness
In 1828, Carl Friedrich Gauss proved his Theorema Egregium (remarkable theorem in Latin), establishing an important property of surfaces. Informally, the theorem says that the curvature of a surface can be determined entirely by measuring distances along paths on the surface. That is, curvature does not depend on how the surface might be embedded in 3-dimensional space. See differential geometry of surfaces. Bernhard Riemann extended Gauss's theory to higher-dimensional spaces called manifolds in a way that also allows distances and angles to be measured and the notion of curvature to be defined, again in a way that was intrinsic to the manifold and not dependent upon its embedding in higher-dimensional spaces. Albert Einstein used the theory of Riemannian manifolds to develop his general theory of relativity. In particular, his equations for gravitation are constraints on the curvature of space.
The tangent bundle of a smooth manifold M assigns to each fixed point of M a vector space called the tangent space, and each tangent space can be equipped with an inner product. If such a collection of inner products on the tangent bundle of a manifold varies smoothly as one traverses the manifold, then concepts that were defined only pointwise at each tangent space can be extended to yield analogous notions over finite regions of the manifold. For example, a smooth curve α(t): [0, 1] → M has tangent vector α′(t0) in the tangent space TM(α(t0)) at any point t0 ∈ (0, 1), and each such vector has length ‖α′(t0)‖, where ‖·‖ denotes the norm induced by the inner product on TM(α(t0)). The integral of these lengths gives the length of the curve α:
{\displaystyle L(\alpha )=\int _{0}^{1}{\|\alpha '(t)\|\,\mathrm {d} t}.}
Smoothness of α(t) for t in [0, 1] guarantees that the integral L(α) exists and the length of this curve is defined.
In many instances, in order to pass from a linear-algebraic concept to a differential-geometric one, the smoothness requirement is very important.
Every smooth submanifold of Rn has an induced Riemannian metric g: the inner product on each tangent space is the restriction of the inner product on Rn. In fact, as follows from the Nash embedding theorem, all Riemannian manifolds can be realized this way. In particular one could define Riemannian manifold as a metric space which is isometric to a smooth submanifold of Rn with the induced intrinsic metric, where isometry here is meant in the sense of preserving the length of curves. This definition might theoretically not be flexible enough, but it is quite useful to build the first geometric intuitions in Riemannian geometry.
Riemannian manifolds as metric spaces
Usually a Riemannian manifold is defined as a smooth manifold with a smooth section of the positive-definite quadratic forms on the tangent bundle. Then one has to work to show that it can be turned to a metric space:
If γ: [a, b] → M is a continuously differentiable curve in the Riemannian manifold M, then we define its length L(γ) in analogy with the example above by
{\displaystyle L(\gamma )=\int _{a}^{b}\|\gamma '(t)\|\,\mathrm {d} t.}
With this definition of length, every connected Riemannian manifold M becomes a metric space (and even a length metric space) in a natural fashion: the distance d(x, y) between the points x and y of M is defined as
d(x,y) = inf{ L(γ) : γ is a continuously differentiable curve joining x and y}.
Even though Riemannian manifolds are usually "curved," there is still a notion of "straight line" on them: the geodesics. These are curves which locally join their points along shortest paths.
Assuming the manifold is compact, any two points x and y can be connected with a geodesic whose length is d(x,y). Without compactness, this need not be true. For example, in the punctured plane R2 \ {0}, the distance between the points (−1, 0) and (1, 0) is 2, but there is no geodesic realizing this distance.
In Riemannian manifolds, the notions of geodesic completeness, topological completeness and metric completeness are the same: that each implies the other is the content of the Hopf–Rinow theorem.
{{#invoke:main|main}} Let M be a differentiable manifold of dimension n. A Riemannian metric on M is a family of (positive definite) inner products
{\displaystyle g_{p}\colon T_{p}M\times T_{p}M\longrightarrow \mathbf {R} ,\qquad p\in M}
such that, for all differentiable vector fields X,Y on M,
{\displaystyle p\mapsto g_{p}(X(p),Y(p))}
defines a smooth function M → R.
In other words, a Riemannian metric g is a symmetric (0,2)-tensor that is positive definite (i.e. g(X, X) > 0 for all tangent vectors X ≠ 0).
In a system of local coordinates on the manifold M given by n real-valued functions x1,x2, …, xn, the vector fields
{\displaystyle \left\{{\frac {\partial }{\partial x^{1}}},\dotsc ,{\frac {\partial }{\partial x^{n}}}\right\}}
give a basis of tangent vectors at each point of M. Relative to this coordinate system, the components of the metric tensor are, at each point p,
{\displaystyle g_{ij}(p):=g_{p}{\Biggl (}\left({\frac {\partial }{\partial x^{i}}}\right)_{p},\left({\frac {\partial }{\partial x^{j}}}\right)_{p}{\Biggr )}.}
Equivalently, the metric tensor can be written in terms of the dual basis {dx1, …, dxn} of the cotangent bundle as
{\displaystyle g=\sum _{i,j}g_{ij}\mathrm {d} x^{i}\otimes \mathrm {d} x^{j}.}
Endowed with this metric, the differentiable manifold (M, g) is a Riemannian manifold.
{\displaystyle {\frac {\partial }{\partial x^{i}}}}
identified with ei = (0, …, 1, …, 0), the standard metric over an open subset U ⊂ Rn is defined by
{\displaystyle g_{p}^{\mathrm {can} }\colon T_{p}U\times T_{p}U\longrightarrow \mathbf {R} ,\qquad \left(\sum _{i}a_{i}{\frac {\partial }{\partial x^{i}}},\sum _{j}b_{j}{\frac {\partial }{\partial x^{j}}}\right)\longmapsto \sum _{i}a_{i}b_{i}.}
Then g is a Riemannian metric, and
{\displaystyle g_{ij}^{\mathrm {can} }=\langle e_{i},e_{j}\rangle =\delta _{ij}.}
Equipped with this metric, Rn is called Euclidean space of dimension n and gijcan is called the (canonical) Euclidean metric.
Let (M,g) be a Riemannian manifold and N ⊂ M be a submanifold of M. Then the restriction of g to vectors tangent along N defines a Riemannian metric over N.
More generally, let f: Mn→Nn+k be an immersion. Then, if N has a Riemannian metric, f induces a Riemannian metric on M via pullback:
{\displaystyle g_{p}^{M}\colon T_{p}M\times T_{p}M\longrightarrow \mathbf {R} ,}
{\displaystyle (u,v)\longmapsto g_{p}^{M}(u,v):=g_{f(p)}^{N}(T_{p}f(u),T_{p}f(v)).}
This is then a metric; the positive definiteness follows on the injectivity of the differential of an immersion.
Let (M, gM) be a Riemannian manifold, h:Mn+k→Nk be a differentiable map and q∈N be a regular value of h (the differential dh(p) is surjective for all p∈h−1(q)). Then h−1(q)⊂M is a submanifold of M of dimension n. Thus h−1(q) carries the Riemannian metric induced by inclusion.
In particular, consider the following map :
{\displaystyle h\colon \mathbf {R} ^{n}\longrightarrow \mathbf {R} ,\qquad (x^{1},\dotsc ,x^{n})\longmapsto \sum _{i=1}^{n}(x^{i})^{2}-1.}
Then, 0 is a regular value of h and
{\displaystyle h^{-1}(0)=\left\{x\in \mathbf {R} ^{n}\vert \sum _{i=1}^{n}(x^{i})^{2}=1\right\}=\mathbf {S} ^{n-1}}
is the unit sphere Sn − 1 ⊂ Rn. The metric induced from Rn on Sn − 1 is called the canonical metric of Sn − 1.
Let M1 and M2 be two Riemannian manifolds and consider the cartesian product M1 × M2 with the product structure. Furthermore, let π1: M1 × M2 → M1 and π2: M1 × M2 → M2 be the natural projections. For (p,q) ∈ M1 × M2, a Riemannian metric on M1 × M2 can be introduced as follows :
{\displaystyle g_{(p,q)}^{M_{1}\times M_{2}}\colon T_{(p,q)}(M_{1}\times M_{2})\times T_{(p,q)}(M_{1}\times M_{2})\longrightarrow \mathbf {R} ,}
{\displaystyle (u,v)\longmapsto g_{p}^{M_{1}}(T_{(p,q)}\pi _{1}(u),T_{(p,q)}\pi _{1}(v))+g_{q}^{M_{2}}(T_{(p,q)}\pi _{2}(u),T_{(p,q)}\pi _{2}(v)).}
{\displaystyle T_{(p,q)}(M_{1}\times M_{2})\cong T_{p}M_{1}\oplus T_{q}M_{2}}
allows us to conclude that this defines a metric on the product space.
The torus S1 × … × S1 = Tn possesses for example a Riemannian structure obtained by choosing the induced Riemannian metric from R2 on the circle S1 ⊂ R2 and then taking the product metric. The torus Tn endowed with this metric is called the flat torus.
Let g0, g1 be two metrics on M. Then,
{\displaystyle {\tilde {g}}:=\lambda g_{0}+(1-\lambda )g_{1},\qquad \lambda \in [0,1],}
is also a metric on M.
The pullback metric
If f:M→N is a differentiable map and (N,gN) a Riemannian manifold, then the pullback of gN along f is a quadratic form on the tangent space of M. The pullback is the quadratic form f*gN on TM defined for v, w ∈ TpM by
{\displaystyle (f^{*}g^{N})(v,w)=g^{N}(df(v),df(w))\,.}
where df(v) is the pushforward of v by f.
The quadratic form f*gN is in general only a semi definite form because df can have a kernel. If f is a diffeomorphism, or more generally an immersion, then it defines a Riemannian metric on M, the pullback metric. In particular, every embedded smooth submanifold inherits a metric from being embedded in a Riemannian manifold, and every covering space inherits a metric from covering a Riemannian manifold.
Existence of a metric
Every paracompact differentiable manifold admits a Riemannian metric. To prove this result, let M be a manifold and {(Uα, φ(Uα))|α ∈ I} a locally finite atlas of open subsets U of M and diffeomorphisms onto open subsets of Rn
{\displaystyle \phi \colon U_{\alpha }\to \phi (U_{\alpha })\subseteq \mathbf {R} ^{n}.}
Let τα be a differentiable partition of unity subordinate to the given atlas. Then define the metric g on M by
{\displaystyle g:=\sum _{\beta }\tau _{\beta }\cdot {\tilde {g}}_{\beta },\qquad {\text{with}}\qquad {\tilde {g}}_{\beta }:={\tilde {\phi }}_{\beta }^{*}g^{\mathrm {can} }.}
where gcan is the Euclidean metric. This is readily seen to be a metric on M.
Let (M, gM) and (N, gN) be two Riemannian manifolds, and f: M → N be a diffeomorphism. Then, f is called an isometry, if
{\displaystyle g^{M}=f^{*}g^{N}\,,}
or pointwise
{\displaystyle g_{p}^{M}(u,v)=g_{f(p)}^{N}(df(u),df(v))\qquad \forall p\in M,\forall u,v\in T_{p}M.}
Moreover, a differentiable mapping f: M → N is called a local isometry at p ∈ M if there is a neighbourhood U ⊂ M, p ∈ U, such that f: U → f(U) is a diffeomorphism satisfying the previous relation.
A connected Riemannian manifold carries the structure of a metric space whose distance function is the arclength of a minimizing geodesic.
Specifically, let (M,g) be a connected Riemannian manifold. Let c: [a,b] → M be a parametrized curve in M, which is differentiable with velocity vector c′. The length of c is defined as
{\displaystyle L_{a}^{b}(c):=\int _{a}^{b}{\sqrt {g(c'(t),c'(t))}}\,\mathrm {d} t=\int _{a}^{b}\|c'(t)\|\,\mathrm {d} t.}
By change of variables, the arclength is independent of the chosen parametrization. In particular, a curve [a,b] → M can be parametrized by its arc length. A curve is parametrized by arclength if and only if
{\displaystyle \|c'(t)\|=1}
{\displaystyle t\in [a,b]}
The distance function d : M×M → [0,∞) is defined by
{\displaystyle d(p,q)=\inf L(\gamma )}
where the infimum extends over all differentiable curves γ beginning at p ∈ M and ending at q ∈ M.
This function d satisfies the properties of a distance function for a metric space. The only property which is not completely straightforward is to show that d(p,q) = 0 implies that p = q. For this property, one can use a normal coordinate system, which also allows one to show that the topology induced by d is the same as the original topology on M.
The diameter of a Riemannian manifold M is defined by
{\displaystyle \mathrm {diam} (M):=\sup _{p,q\in M}d(p,q)\in \mathbf {R} _{\geq 0}\cup \{+\infty \}.}
The diameter is invariant under global isometries. Furthermore, the Heine–Borel property holds for (finite-dimensional) Riemannian manifolds: M is compact if and only if it is complete and has finite diameter.
A Riemannian manifold M is geodesically complete if for all p ∈ M, the exponential map
{\displaystyle \exp _{p}}
{\displaystyle v\in T_{p}M}
, i.e. if any geodesic
{\displaystyle \gamma (t)}
starting from p is defined for all values of the parameter t ∈ R. The Hopf-Rinow theorem asserts that M is geodesically complete if and only if it is complete as a metric space.
If M is complete, then M is non-extendable in the sense that it is not isometric to an open proper submanifold of any other Riemannian manifold. The converse is not true, however: there exist non-extendable manifolds which are not complete.
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Elliptic_function Knowpia
In the mathematical field of complex analysis, elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those integrals occurred at the calculation of the arc length of an ellipse.
Important elliptic functions are Jacobi elliptic functions and the Weierstrass
{\displaystyle \wp }
Further development of this theory led to hyperelliptic functions and modular forms.
A meromorphic function is called an elliptic function, if there are two
{\displaystyle \mathbb {R} }
-linear independent complex numbers
{\displaystyle \omega _{1},\omega _{2}\in \mathbb {C} }
{\displaystyle f(z+\omega _{1})=f(z)}
{\displaystyle f(z+\omega _{2})=f(z),\quad \forall z\in \mathbb {C} }
So elliptic functions have two periods and are therefore also called doubly periodic.
Period lattice and fundamental domainEdit
Parallelogram where opposite sides are identified
I{\displaystyle f}
is an elliptic function with periods
{\displaystyle \omega _{1},\omega _{2}}
{\displaystyle f(z+\gamma )=f(z)}
for every linear combination
{\displaystyle \gamma =m\omega _{1}+n\omega _{2}}
{\displaystyle m,n\in \mathbb {Z} }
{\displaystyle \Lambda :=\langle \omega _{1},\omega _{2}\rangle _{\mathbb {Z} }:=\mathbb {Z} \omega _{1}+\mathbb {Z} \omega _{2}:=\{m\omega _{1}+n\omega _{2}\mid m,n\in \mathbb {Z} \}}
is called the period lattice.
The parallelogram generated by
{\displaystyle \omega _{1}}
{\displaystyle \omega _{2}}
{\displaystyle \{\mu \omega _{1}+\nu \omega _{2}\mid 0\leq \mu ,\nu \leq 1\}}
is called fundamental domain.
Geometrically the complex plane is tiled with parallelograms. Everything that happens in the fundamental domain repeats in all the others. For that reason we can view elliptic function as functions with the quotient group
{\displaystyle \mathbb {C} /\Lambda }
as their domain. This quotient group, called an elliptic curve, can be visualised as a parallelogram where opposite sides are identified, which topologically is a torus.[1]
Liouville's theoremsEdit
The following three theorems are known as Liouville's theorems (1847).
1st theoremEdit
A holomorphic elliptic function is constant.[2]
This is the original form of Liouville's theorem and can be derived from it.[3] A holomorphic elliptic function is bounded since it takes on all of its values on the fundamental domain which is compact. So it is constant by Liouville's theorem.
2nd theoremEdit
Every elliptic function has finitely many poles in
{\displaystyle \mathbb {C} /\Lambda }
and the sum of its residues is zero.[4]
This theorem implies that there is no elliptic function not equal to zero with exactly one pole of order one or exactly one zero of order one in the fundamental domain.
3rd theoremEdit
A non-constant elliptic function takes on every value the same number of times in
{\displaystyle \mathbb {C} /\Lambda }
counted with multiplicity.[5]
Weierstrass ℘-functionEdit
One of the most important elliptic functions is the Weierstrass
{\displaystyle \wp }
-function. For a given period lattice
{\displaystyle \Lambda }
{\displaystyle \wp (z)={\frac {1}{z^{2}}}+\sum _{\lambda \in \Lambda \setminus \{0\}}\left({\frac {1}{(z-\lambda )^{2}}}-{\frac {1}{\lambda ^{2}}}\right).}
It is constructed in such a way that it has a pole of order two at every lattice point. The term
{\displaystyle -{\frac {1}{\lambda ^{2}}}}
is there to make the series convergent.
{\displaystyle \wp }
is an even elliptic function, that means
{\displaystyle \wp (-z)=\wp (z)}
{\displaystyle \wp '(z)=-2\sum _{\lambda \in \Lambda }{\frac {1}{(z-\lambda )^{3}}}}
{\displaystyle \wp '(-z)=-\wp '(z).}
One of the main results of the theory of elliptic functions is the following: Every elliptic function with respect to a given period lattice
{\displaystyle \Lambda }
can be expressed as a rational function in terms of
{\displaystyle \wp }
{\displaystyle \wp '}
{\displaystyle \wp }
-function satisfies the differential equation
{\displaystyle \wp '^{2}(z)=4\wp (z)^{3}-g_{2}\wp (z)-g_{3}.}
{\displaystyle g_{2}}
{\displaystyle g_{3}}
are constants that depend on
{\displaystyle \Lambda }
{\displaystyle g_{2}(\omega _{1},\omega _{2})=60G_{4}(\omega _{1},\omega _{2})}
{\displaystyle g_{3}(\omega _{1},\omega _{2})=140G_{6}(\omega _{1},\omega _{2})}
{\displaystyle G_{4}}
{\displaystyle G_{6}}
are so called Eisenstein series.[8]
In algebraic language: The field of elliptic functions is isomorphic to the field
{\displaystyle \mathbb {C} (X)[Y]/(Y^{2}-4X^{3}+g_{2}X+g_{3})}
where the isomorphism maps
{\displaystyle \wp }
{\displaystyle X}
{\displaystyle \wp '}
{\displaystyle Y}
{\displaystyle \wp }
-function with period lattice
{\displaystyle \Lambda =\mathbb {Z} +e^{2\pi i/6}\mathbb {Z} }
Derivative of the
{\displaystyle \wp }
Relation to elliptic IntegralsEdit
The relation to elliptic Integrals has mainly a historical background. Elliptic Integrals had been studied by Legendre, whose work was taken on by Niels Henrik Abel and Carl Gustav Jacobi.
Abel discovered elliptic functions by taking the inverse function
{\displaystyle \varphi }
of the elliptic integral function
{\displaystyle \alpha (x)=\int _{0}^{x}{\frac {dt}{\sqrt {(1-c^{2}t^{2})(1+e^{2}t^{2})}}}}
{\displaystyle x=\varphi (\alpha )}
Additionally he defined the functions[10]
{\displaystyle f(\alpha )={\sqrt {1-c^{2}\varphi ^{2}(\alpha )}}}
{\displaystyle F(\alpha )={\sqrt {1+e^{2}\varphi ^{2}(\alpha )}}}
After continuation to the complex plane they turned out to be doubly periodic and are known as Abel elliptic functions.
Jacobi elliptic functions are similarly obtained as inverse functions of elliptic integrals.
Jacobi considered the integral function
{\displaystyle \xi (x)=\int _{0}^{x}{\frac {dt}{\sqrt {(1-t^{2})(1-k^{2}t^{2})}}}}
and inverted it:
{\displaystyle x=\operatorname {sn} (\xi )}
{\displaystyle \operatorname {sn} }
stands for sinus amplitudinis and is the name of the new function.[11] He then introduced the functions cosinus amplitudinis and delta amplitudinis, which are defined as follows:
{\displaystyle \operatorname {cn} (\xi ):={\sqrt {1-x^{2}}}}
{\displaystyle \operatorname {dn} (\xi ):={\sqrt {1-k^{2}x^{2}}}}
Only by taking this step, Jacobi could prove his general transformation formula of elliptic integrals in 1827.[12]
Shortly after the development of infinitesimal calculus the theory of elliptic functions was started by the Italian mathematician Giulio di Fagnano and the Swiss mathematician Leonhard Euler. When they tried to calculate the arc length of a lemniscate they encountered problems involving integrals that contained the square root of polynomials of degree 3 and 4.[13] It was clear that those so called elliptic integrals could not be solved using elementary functions. Fagnano observed an algebraic relation between elliptic integrals, what he published in 1750.[13] Euler immediately generalized Fagnano's results and posed his algebraic addition theorem for elliptic integrals.[13]
Except for a comment by Landen[14] his ideas were not pursued until 1786, when Legendre published his paper Mémoires sur les intégrations par arcs d’ellipse.[15] Legendre subsequently studied elliptic integrals and called them elliptic functions. Legendre introduced a three-fold classification –three kinds– which was a crucial simplification of the rather complicated theory at that time. Other important works of Legendre are: Mémoire sur les transcendantes elliptiques (1792),[16] Exercices de calcul intégral (1811–1817),[17] Traité des fonctions elliptiques (1825–1832).[18] Legendre's work was mostly left untouched by mathematicians until 1826.
Subsequently, Niels Henrik Abel and Carl Gustav Jacobi resumed the investigations and quickly discovered new results. At first they inverted the elliptic integral function. Following a suggestion of Jacobi in 1829 these inverse functions are now called elliptic functions. One of Jacobi's most important works is Fundamenta nova theoriae functionum ellipticarum which was published 1829.[19] The addition theorem Euler found was posed and proved in its general form by Abel in 1829. Note that in those days the theory of elliptic functions and the theory of doubly periodic functions were considered to be different theories. They were brought together by Briout and Bouquet in 1856.[20] Gauss discovered many of the properties of elliptic functions 30 years earlier but never published anything on the subject.[21]
^ Rolf Busam (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 259, ISBN 978-3-540-32058-6
^ Jeremy Gray (2015), Real and the complex : a history of analysis in the 19th century (in German), Cham, pp. 118f, ISBN 978-3-319-23715-2
^ a b K. Chandrasekharan (1985), Elliptic functions (in German), Berlin: Springer-Verlag, p. 28, ISBN 0-387-15295-4
^ Gray, Jeremy (14 October 2015), Real and the complex : a history of analysis in the 19th century (in German), Cham, p. 74, ISBN 978-3-319-23715-2
^ a b c Gray, Jeremy (2015). Real and the complex : a history of analysis in the 19th century. Cham. pp. 23f. ISBN 978-3-319-23715-2. OCLC 932002663.
^ John Landen: An Investigation of a general Theorem for finding the Length of any Arc of any Conic Hyperbola, by Means of Two Elliptic Arcs, with some other new and useful Theorems deduced therefrom. In: The Philosophical Transactions of the Royal Society of London 65 (1775), Nr. XXVI, S. 283–289, JSTOR 106197.
^ Adrien-Marie Legendre: Mémoire sur les intégrations par arcs d’ellipse. In: Histoire de l’Académie royale des sciences Paris (1788), S. 616–643. – Ders.: Second mémoire sur les intégrations par arcs d’ellipse, et sur la comparaison de ces arcs. In: Histoire de l’Académie royale des sciences Paris (1788), S. 644–683.
^ Adrien-Marie Legendre: Mémoire sur les transcendantes elliptiques, où l’on donne des méthodes faciles pour comparer et évaluer ces trancendantes, qui comprennent les arcs d’ellipse, et qui se rencontrent frèquemment dans les applications du calcul intégral. Du Pont & Firmin-Didot, Paris 1792. Englische Übersetzung A Memoire on Elliptic Transcendentals. In: Thomas Leybourn: New Series of the Mathematical Repository. Band 2. Glendinning, London 1809, Teil 3, S. 1–34.
^ Adrien-Marie Legendre: Exercices de calcul integral sur divers ordres de transcendantes et sur les quadratures. 3 Bände. (Band 1, Band 2, Band 3). Paris 1811–1817.
^ Adrien-Marie Legendre: Traité des fonctions elliptiques et des intégrales eulériennes, avec des tables pour en faciliter le calcul numérique. 3 Bde. (Band 1, Band 2, Band 3/1, Band 3/2, Band 3/3). Huzard-Courcier, Paris 1825–1832.
^ Carl Gustav Jacob Jacobi: Fundamenta nova theoriae functionum ellipticarum. Königsberg 1829.
^ Gray, Jeremy (2015). Real and the complex : a history of analysis in the 19th century. Cham. p. 122. ISBN 978-3-319-23715-2. OCLC 932002663.
^ Gray, Jeremy (2015). Real and the complex : a history of analysis in the 19th century. Cham. p. 96. ISBN 978-3-319-23715-2. OCLC 932002663.
Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 16". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. pp. 567, 627. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253. See also chapter 18. (only considers the case of real invariants).
N. I. Akhiezer, Elements of the Theory of Elliptic Functions, (1970) Moscow, translated into English as AMS Translations of Mathematical Monographs Volume 79 (1990) AMS, Rhode Island ISBN 0-8218-4532-2
Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, New York, 1976. ISBN 0-387-97127-0 (See Chapter 1.)
E. T. Whittaker and G. N. Watson. A course of modern analysis, Cambridge University Press, 1952
Wikimedia Commons has media related to Elliptic functions.
"Elliptic function", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
MAA, Translation of Abel's paper on elliptic functions.
Elliptic Functions and Elliptic Integrals on YouTube, lecture by William A. Schwalm (4 hours)
Johansson, Fredrik (2018). "Numerical Evaluation of Elliptic Functions, Elliptic Integrals and Modular Forms". arXiv:1806.06725 [cs.NA].
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Use Distributed Arrays to Solve Systems of Linear Equations with Iterative Methods - MATLAB & Simulink - MathWorks India
For large-scale mathematical computations, iterative methods can be more efficient than direct methods. This example shows how you can solve systems of linear equations of the form
Ax=b
in parallel using distributed arrays with iterative methods.
When you use the distributed function, MATLAB automatically starts a parallel pool using your default cluster settings. This example uses the Wathen matrix from the MATLAB gallery function. This matrix is a sparse, symmetric, and random matrix with overall dimension
N=3{n}^{2}+4n+1
You can now define the right hand vector
b
b
is defined as the row sum of
A
, which leads to an exact solution to
Ax=b
{x}_{\text{exact}}=\left[1,...,1{\right]}^{T}
The MATLAB function pcg provides the conjugate gradient (CG) method, which iteratively generates a series of approximate solutions for
\mathit{x}
, improving the solution with each step.
The iterative computation ends when the series of approximate solutions converges to a specific tolerance or after the maximum number of iteration steps. For both distributed and on-client arrays,
pcg
uses the same default settings:
The default maximum tolerance is
{10}^{-6}
You can improve the efficiency of solving your system using the preconditioned conjugate gradient (PCG) method. First, precondition your system of linear equations using a preconditioner matrix
\mathit{M}
. Next, solve your preconditioned system using the CG method. The PCG method can take much fewer iterations than the CG method.
The MATLAB function pcg is also used for the PCG method. You can supply a suitable preconditioner matrix
\mathit{M}
as an additional input.
An ideal preconditioner matrix is a matrix whose inverse
{\mathit{M}}^{-1}
is a close approximation to the inverse of the coefficient matrix,
{\mathit{A}}^{-1}
, but is easier to compute. This example uses the diagonal of
A
to precondition the system of linear equations.
The Wathen matrix used in this example is a good demonstration of how a good preconditioner can dramatically improve the efficiency of the solution. The Wathen matrix has relatively small off-diagonal components, so choosing
M=\text{diag}\left(A\right)
gives a suitable preconditioner. For an arbitrary matrix
\mathit{A}
, finding a preconditioner might not be so straightforward.
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Plant Models for Gain-Scheduled Controller Tuning - MATLAB & Simulink - MathWorks Benelux
Obtaining the Family of Linear Models
Set Up for Gain Scheduling by Linearizing at Design Points
Trim Plant at Design Points
Linearize at Design Points
Create slTuner Interface with Block Substitution
Sample System at Simulation Snapshots
Sample System at Varying Parameter Values
Eliminate Samples at Unneeded Design Points
LPV Plants in MATLAB
Gain scheduling is a control approach for controlling a nonlinear plant. To tune a gain-scheduled control system, you need a collection of linear models that approximate the nonlinear dynamics near selected design points. Generally, the dynamics of the plant are described by nonlinear differential equations of the form:
\begin{array}{l}\stackrel{˙}{x}=f\left(x,u,\sigma \right)\\ y=g\left(x,u,\sigma \right).\end{array}
Here, x is the state vector, u is the plant input, and y is the plant output. These nonlinear differential equations can be known explicitly for a particular system. More commonly, they are specified implicitly, such as by a Simulink® model.
You can convert these nonlinear dynamics into a family of linear models that describe the local behavior of the plant around a family of operating points (x(σ),u(σ)), parameterized by the scheduling variables, σ. Deviations from the nominal operating condition are defined as:
\delta x=x-x\left(\sigma \right),\text{ }\delta u=u-u\left(\sigma \right).
These deviations are governed, to first order, by linear parameter-varying dynamics:
\begin{array}{cc}\stackrel{˙}{\delta }x=A\left(\sigma \right)\delta x+B\left(\sigma \right)\delta u,& \delta y=C\left(\sigma \right)\delta x+D\left(\sigma \right)\delta u,\\ & \\ A\left(\sigma \right)=\frac{\partial f}{\partial x}\left(x\left(\sigma \right),u\left(\sigma \right)\right)& B\left(\sigma \right)=\frac{\partial f}{\partial u}\left(x\left(\sigma \right),u\left(\sigma \right)\right)\\ C\left(\sigma \right)=\frac{\partial g}{\partial x}\left(x\left(\sigma \right),u\left(\sigma \right)\right)& D\left(\sigma \right)=\frac{\partial g}{\partial u}\left(x\left(\sigma \right),u\left(\sigma \right)\right).\end{array}
This continuum of linear approximations to the nonlinear dynamics is called a linear parameter-varying (LPV) model:
\begin{array}{l}\frac{dx}{dt}=A\left(\sigma \right)x+B\left(\sigma \right)u\\ y=C\left(\sigma \right)x+D\left(\sigma \right)u.\end{array}
The LPV model describes how the linearized plant dynamics vary with time, operating condition, or any other scheduling variable. For example, the pitch axis dynamics of an aircraft can be approximated by an LPV model that depends on incidence angle, α, air speed, V, and altitude, h.
In practice, you replace this continuum of plant models by a finite set of linear models obtained for a suitable grid of σ values This replacement amounts to sampling the LPV dynamics over the operating range and selecting a representative set of σ values, your design points.
Gain-scheduled controllers yield best results when the plant dynamics vary smoothly between design points.
If you do not have this family of linear models, there are several approaches to obtaining it, including:
If you have a Simulink model, trim and linearize the model at the design points.
Linearize the Simulink model using parameter variation.
If the scheduling variable is time, linearize the model at a series of simulation snapshots.
If you have nonlinear differential equations that describe the plant, linearize them at the design points.
For tuning gain schedules, after you obtain the family of linear models, you must associate it with an slTuner interface to build a family of tunable closed-loop models. To do so, use block substitution, as described in Multiple Design Points in slTuner Interface.
This example shows how to linearize a plant model at a set of design points for tuning of a gain-scheduled controller. The example then uses the resulting linearized models to configure an slTuner interface for tuning the gain schedule.
Open the rct_CSTR model.
mdl = 'rct_CSTR';
In this model, the Concentration controller and Temperature controller both depend on the output concentration Cr. To set up this gain-scheduled system for tuning, you linearize the plant at a set of steady-state operating points that correspond to different values of the scheduling parameter Cr. Sometimes, it is convenient to use a separate model of the plant for trimming and linearization under various operating conditions. For example, in this case, the most straightforward way to obtain these linearizations is to use a separate open-loop model of the plant, rct_CSTR_OL.
mdl_OL = 'rct_CSTR_OL';
open_system(mdl_OL)
Suppose that you want to control this plant at a range of Cr values from 4 to 8. Trim the model to find steady-state operating points for a set of values in this range. These values are the design points for tuning.
Cr = (4:8)'; % concentrations
for k=1:length(Cr)
opspec = operspec(mdl_OL);
opspec.Outputs(1).y = Cr(k);
% Compute equilibrium condition
[op(k),report(k)] = findop(mdl_OL,opspec,findopOptions('DisplayReport','off'));
op is an array of steady-state operating points. For more information about steady-state operating points, see About Operating Points.
Linearizing the plant model using op returns an array of LTI models, each linearized at the corresponding design point.
G = linearize(mdl_OL,'rct_CSTR_OL/CSTR',op);
To tune the control system rct_CSTR, create an slTuner interface that linearizes the system at those design points. Use block substitution to replace the plant in rct_CSTR with the linearized plant-model array G.
blocksub.Name = 'rct_CSTR/CSTR';
blocksub.Value = G;
tunedblocks = {'Kp','Ki'};
ST0 = slTuner(mdl,tunedblocks,blocksub);
For this example, only the PI coefficients in the Concentration controller are designated as tuned blocks. In general, however, tunedblocks lists all the blocks to tune.
For more information about using block substitution to configure an slTuner interface for gain-scheduled controller tuning, see Multiple Design Points in slTuner Interface.
For another example that illustrates using trimming and linearization to generate a family of linear models for gain-scheduled controller tuning, see Trimming and Linearization of the HL-20 Airframe.
If you are controlling the system around a reference trajectory (x(σ),u(σ)), use snapshot linearization to sample the system at various points along the σ trajectory. Use this approach for time-varying systems where the scheduling variable is time.
To linearize a system at a set of simulation snapshots, use a vector of positive scalars as the op input argument of linearize, slLinearizer, or slTuner. These scalars are the simulation times at which to linearize the model. Use the same set of time values as the design points in tunable surfaces for the system.
If the scheduling variable is a parameter in the Simulink model, you can use parameter variation to sample the control system over a parameter grid. For example, suppose that you want to tune a model named suspension_gs that contains two parameters, Ks and Bs. These parameters each can vary over some known range, and a controller gain in the model varies as a function of both parameters.
To set up such a model for tuning, create a grid of parameter values. For this example, let Ks vary from 1 – 5, and let Bs vary from 0.6 – 0.9.
Ks = 1:5;
Bs = [0.6:0.1:0.9];
[Ksgrid,Bsgrid] = ndgrid(Ks,Bs);
These values are the design points at which to sample and tune the system. For example, create an slTuner interface to the model, assuming one tunable block, a Lookup Table block named K that models the parameter-dependent gain.
params(1) = struct('Name','Ks','Value',Ksgrid);
params(2) = struct('Name','Bs','Value',Bsgrid);
STO = slTuner('suspension_gs','K',params);
slTuner samples the model at all (Ksgrid,Bsgrid) values specified in params.
Next, use the same design points to create a tunable gain surface for parameterizing K.
design = struct('Ks',Ksgrid,'Bs',Bsgrid);
shapefcn = @(Ks,Bs)[Ks,Bs,Ks*Bs];
K = tunableSurface('K',1,design,shapefcn);
setBlockParam(ST0,'K',K);
After you parameterize all the scheduled gains, you can create your tuning goals and tune the system with systune.
Sometimes, your sampling grid includes points that represent irrelevant or unphysical design points. You can eliminate such design points from the model grid entirely, so that they do not contribute to any stage of tuning or analysis. To do so, use voidModel, which replaces specified models in a model array with NaN. voidModel replaces specified models in a model array with NaN. Using voidModel lets your design over a grid of design points that is almost regular.
There are other tools for controlling which models contribute to design and analysis. For instance, you might want to:
Keep a model in the grid for analysis, but exclude it from tuning.
Keep a model in the grid for tuning, but exclude it from a particular design goal.
For more information, see Change Requirements with Operating Condition.
In MATLAB®, you can use an array of LTI plant models to represent an LPV system sampled at varying values of σ. To associate each linear model in the set with the underlying design points, use the SamplingGrid property of the LTI model array σ. One way to obtain such an array is to create a parametric generalized state-space (genss) model of the system and sample the model with parameter variation to generate the array. For an example, see Study Parameter Variation by Sampling Tunable Model.
slTuner | findop | voidModel
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Home : Support : Online Help : Science and Engineering : Dynamic Systems : System Object : Verify
Verify(sys)
The Verify command checks the validity of sys, a linear system object.
If the system object is verified, the procedure returns NULL. Otherwise an error message describing the problem is displayed. For some problems, a warning rather than an error is generated.
For a system to be verified, it must meet the following criteria.
It must be a module.
It must have the exports common to all the system types.
It must have the Matrix exports specific to its type.
The Matrix exports must have compatible dimensions and correspond to the number of input, output, and, possibly, state variables.
The Matrix elements must be of the appropriate type and correspond to a valid linear system.
\mathrm{with}\left(\mathrm{DynamicSystems}\right):
\mathrm{sys}≔\mathrm{NewSystem}\left(\frac{a}{s+b}\right):
\mathrm{PrintSystem}\left(\mathrm{sys}\right)
[\begin{array}{l}\textcolor[rgb]{0,0,1}{\mathbf{Transfer Function}}\\ \textcolor[rgb]{0,0,1}{\mathrm{continuous}}\\ \textcolor[rgb]{0,0,1}{\mathrm{1 output\left(s\right); 1 input\left(s\right)}}\\ \textcolor[rgb]{0,0,1}{\mathrm{inputvariable}}\textcolor[rgb]{0,0,1}{=}[\textcolor[rgb]{0,0,1}{\mathrm{u1}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{s}\right)]\\ \textcolor[rgb]{0,0,1}{\mathrm{outputvariable}}\textcolor[rgb]{0,0,1}{=}[\textcolor[rgb]{0,0,1}{\mathrm{y1}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{s}\right)]\\ {\textcolor[rgb]{0,0,1}{\mathrm{tf}}}_{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{=}\frac{\textcolor[rgb]{0,0,1}{a}}{\textcolor[rgb]{0,0,1}{s}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{b}}\end{array}
\mathrm{Verify}\left(\mathrm{sys}\right)
Make the transfer function an expression that is not a rational polynomial in s.
\mathrm{sys}:-\mathrm{tf}[[1,1]]≔\frac{\mathrm{exp}\left(s\right)}{{s}^{3}+5{s}^{2}+7s+6}
{\textcolor[rgb]{0,0,1}{\mathrm{tf}}}_{[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}]}\textcolor[rgb]{0,0,1}{≔}\frac{{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{s}}}{{\textcolor[rgb]{0,0,1}{s}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{s}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{7}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{s}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{6}}
\mathrm{Verify}\left(\mathrm{sys}\right)
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Linked lists - Padheye.com : Discover Excellence
Linear Data Structures Linked lists
by Guest on June 29, 2020 in Data Structure, Linear Data Structures
1. Grab the last letter, which is "G". Our tail pointer lets us do this in
O(1)
That's
O(1)
Why is it
O(1)
time? Because the runtime doesn't get bigger if the string gets bigger. No matter how many characters are in our string, we still just have to tweak a couple pointers for any append.
So with a dynamic array, our append would have a worst-case time cost of
O(n)
Linked lists have worst-case
O(1)
-time appends, which is better than the worst-case
O(n)
time of dynamic arrays.
That worst-case part is important. The average case runtime for appends to linked lists and dynamic arrays is the same:
O(1)
Bam.
O(1)
time again.
It's all in the step where we made room for the first letter. We had to move all
n
characters in our string. One at a time. That's
O(n)
So linked lists have faster prepends (
O(1)
time) than dynamic arrays (
O(n)
No "worst case" caveat this time—prepends for dynamic arrays are always
O(n)
time. And prepends for linked lists are always
O(1)
So if linked lists are so great, why do we usually store strings in an array? Because arrays have
O(1)
-time lookups. And those constant-time lookups come from the fact that all the array elements are lined up next to each other in memory.
Lookups with a linked list are more of a process, because we have no way of knowing where the
i
th node is in memory. So we have to walk through the linked list node by node, counting as we go, until we hit the
i
th item.
LinkedListNode * getIthItemInLinkedList(LinkedListNode* head, size_t i)
LinkedListNode* currentNode = head;
size_t currentPosition = 0;
while (currentNode != nullptr) {
if (currentPosition == i) {
currentNode = currentNode->next_;
ostringstream error;
error << "List has fewer than i + 1 (" << (i + 1) << ") nodes";
throw invalid_argument(error.str());
CC#C++JavaJavaScriptObjective-CPHPPython 2.7Python 3.6RubySwift
i + 1
steps down our linked list to get to the
i
th node (we made our function zero-based to match indices in arrays). So linked lists have
O(i)
-time lookups. Much slower than the
O(1)
-time lookups for arrays and dynamic arrays.
Labels: Data Structure, Linear Data Structures
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Table 2 Comparison between OPG and RANKL expression in different groups (
\overline{\mathbit{x}}
± s, n = 5)
RANKL (pmol/l)
Normal 16.3092 ± 0.8425▲▲△ 3.0708 ± 1.2391▲▲
Normal induction 23.8044 ± 1.7818 5.0082 ± 1.0355
GIOP 14.1432 ± 2.6551▲▲△△ 1.6027 ± 0.1123◆◆▲▲
GIOP induction 20.1376 ± 3.0025▲ 2.0739 ± 0.2375◆◆▲▲
◆ P < 0.05, ◆◆ P < 0.01, vs. normal group; ▲ P < 0.05, ▲▲ P < 0.01, vs. normal induction group; △ P < 0.05, △△ P < 0.01, vs. GIOP induction group.
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Negative binomial cumulative distribution function - MATLAB nbincdf - MathWorks India
Compute Negative Binomial Distribution CDF
Negative binomial cumulative distribution function
y = nbincdf(x,R,p)
y = nbincdf(x,R,p,'upper')
y = nbincdf(x,R,p) computes the negative binomial cdf at each of the values in x using the corresponding number of successes, R and probability of success in a single trial, p. x, R, and p can be vectors, matrices, or multidimensional arrays that all have the same size, which is also the size of y. A scalar input for x, R, or p is expanded to a constant array with the same dimensions as the other inputs.
y = nbincdf(x,R,p,'upper') returns the complement of the negative binomial cdf at each value in x, using an algorithm that more accurately computes the extreme upper tail probabilities.
The negative binomial cdf is
y=F\left(x|r,p\right)=\sum _{i=0}^{x}\left(\begin{array}{c}r+i-1\\ i\end{array}\right){p}^{r}{q}^{i}{I}_{\left(0,1,...\right)}\left(i\right)
The simplest motivation for the negative binomial is the case of successive random trials, each having a constant probability p of success. The number of extra trials you must perform in order to observe a given number R of successes has a negative binomial distribution. However, consistent with a more general interpretation of the negative binomial, nbincdf allows R to be any positive value, including nonintegers. When R is noninteger, the binomial coefficient in the definition of the cdf is replaced by the equivalent expression
\frac{\Gamma \left(r+i\right)}{\Gamma \left(r\right)\Gamma \left(i+1\right)}
p = nbincdf(x,3,0.5);
stairs(x,p)
cdf | nbinpdf | nbininv | nbinstat | nbinfit | nbinrnd
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2Al
Commander Ripley outlines some of the most important roles which would need to be fulfilled in a Mars community. Can you think of the names of different types of experts who could fulfil those roles?
As stated on your checklist, now is a good time to start collecting the materials you think you may need to create your 3D model. You may like to begin collecting things like recyclables, packaging, aluminium foil, boxes, LEGO.
You will need to organise tasks and set deadlines to be able to complete your Mars human habitat on time.
2Al
2Al
In this planning step you will organise how you will complete your Mars human habitat model and presentation by breaking the process down into individual design challenges. For each design challenge you will list the materials required and set deadlines. If you are working in Mission Teams you will also assign tasks to different team members.
2Al
Once you are all organised, it’s time to start building!
2Al
2Al
2Al
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Probability 5 Ways - Donnacha Oisín Kidney
Tags: Probability, Haskell
Ever since the famous pearl by Erwig and Kollmansberger (2006), probabilistic programming with monads has been an interesting and diverse area in functional programming, with many different approaches.
I’m going to present five here, some of which I have not seen before.
As presented in the paper, a simple and elegant formulation of probability distributions looks like this:
newtype Prob a
= Prob
{ runProb :: [(a, Rational)]
It’s a list of possible events, each tagged with their probability of happening. Here’s the probability distribution representing a die roll, for instance:
die :: Prob Integer
die = [ (x, 1/6) | x <- [1..6] ]
The semantics can afford to be a little fuzzy: it doesn’t hugely matter if the probabilities don’t add up to 1 (you can still extract meaningful answers when they don’t). However, I can’t see a way in which either negative probabilities or an empty list would make sense. It would be nice if those states were unrepresentable.
Its monadic structure multiplies conditional events:
instance Functor Prob where
fmap f xs = Prob [ (f x, p) | (x,p) <- runProb xs ]
instance Applicative Prob where
pure x = Prob [(x,1)]
fs <*> xs
[ (f x,fp*xp)
| (f,fp) <- runProb fs
, (x,xp) <- runProb xs ]
instance Monad Prob where
xs >>= f
[ (y,xp*yp)
| (x,xp) <- runProb xs
, (y,yp) <- runProb (f x) ]
In most of the examples, we’ll need a few extra functions in order for the types to be useful. First is support:
support :: Prob a -> [a]
support = fmap fst . runProb
And second is expectation:
expect :: (a -> Rational) -> Prob a -> Rational
expect p xs = sum [ p x * xp | (x,xp) <- runProb xs ]
probOf :: (a -> Bool) -> Prob a -> Rational
probOf p = expect (bool 0 1 . p)
It’s useful to be able to construct uniform distributions:
uniform xs = Prob [ (x,n) | x <- xs ]
n = 1 % toEnum (length xs)
die = uniform [1..6]
>>> probOf (7==) $ do
x <- die
y <- die
pure (x+y)
As elegant as the above approach is, it leaves something to be desired when it comes to efficiency. In particular, you’ll see a combinatorial explosion at every step. To demonstrate, let’s take the example above, using three-sided dice instead so it doesn’t take up too much space.
The probability table looks like this:
But the internal representation looks like this:
States are duplicated, because the implementation has no way of knowing that two outcomes are the same. We could collapse equivalent outcomes if we used a Map, but then we can’t implement Functor, Applicative, or Monad. The types:
class Applicative f => Monad f where
(>>=) :: f a -> (a -> f b) -> f b
Don’t allow an Ord constraint, which is what we’d need to remove duplicates. We can instead make our own classes which do allow constraints:
import Prelude hiding (Functor(..),Applicative(..),Monad(..))
type Domain f a :: Constraint
type Domain f a = ()
fmap :: Domain f b => (a -> b) -> f a -> f b
{-# MINIMAL pure, liftA2 #-}
pure :: Domain f a => a -> f a
liftA2 :: Domain f c => (a -> b -> c) -> f a -> f b -> f c
(<*>) :: Domain f b => f (a -> b) -> f a -> f b
(>>=) :: Domain f b => f a -> (a -> f b) -> f b
fail :: String -> a
return :: (Applicative f, Domain f a) => a -> f a
This setup gets over a couple common annoyances in Haskell, like making Data.Set a Monad:
type Domain Set a = Ord a
instance Applicative Set where
pure = Set.singleton
liftA2 f xs ys = do
pure (f x y)
instance Monad Set where
(>>=) = flip foldMap
And, of course, the probability monad:
newtype Prob a = Prob
{ runProb :: Map a Rational
type Domain Prob a = Ord a
fmap f = Prob . Map.mapKeysWith (+) f . runProb
pure x = Prob (Map.singleton x 1)
instance Ord a => Monoid (Prob a) where
mempty = Prob Map.empty
mappend (Prob xs) (Prob ys) = Prob (Map.unionWith (+) xs ys)
Prob xs >>= f
= Map.foldMapWithKey ((Prob .) . flip (Map.map . (*)) . runProb . f) xs
support = Map.keys . runProb
expect p = getSum . Map.foldMapWithKey (\k v -> Sum (p k * v)) . runProb
uniform xs = Prob (Map.fromList [ (x,n) | x <- xs ])
ifThenElse True t _ = t
Coming up with the right implementation all at once is quite difficult: luckily, there are more general techniques for designing DSLs that break the problem into smaller parts, which also give us some insight into the underlying composition of the probability monad.
The technique relies on an algebraic concept called “free objects”. A free object for some class is a minimal implementation of that class. The classic example is lists: they’re the free monoid. Monoid requires that you have an additive operation, an empty element, and that the additive operation be associative. Lists have all of these things: what makes them free, though, is that they have nothing else. For instance, the additive operation on lists (concatenation) isn’t commutative: if it was, they wouldn’t be the free monoid any more, because they satisfy an extra law that’s not in monoid.
For our case, we can use the free monad: this takes a functor and gives it a monad instance, in a way we know will satisfy all the laws. This encoding is used in several papers (Ścibior, Ghahramani, and Gordon 2015; Larsen 2011).
The idea is to first figure out what primitive operation you need. We’ll use weighted choice:
choose :: Prob a -> Rational -> Prob a -> Prob a
choose = ...
Then you encode it as a functor:
data Choose a
= Choose Rational a a
deriving (Functor,Foldable)
We’ll say the left-hand-choice has chance
p
, and the right-hand
1-p
. Then, you just wrap it in the free monad:
type Prob = Free Choose
And you already have a monad instance. Support comes from the Foldable instance:
support = toList
Expectation is an “interpreter” for the DSL:
expect p = iter f . fmap p
f (Choose c l r) = l * c + r * (1-c)
For building up the tree, we can use Huffman’s algorithm:
fromList :: (a -> Rational) -> [a] -> Prob a
fromList p = go . foldMap (\x -> singleton (p x) (Pure x))
go xs = case minView xs of
Nothing -> error "empty list"
Just ((xp,x),ys) -> case minView ys of
Just ((yp,y),zs) ->
go (insertHeap (xp+yp) (Free (Choose (xp/(xp+yp)) x y)) zs)
And finally, it gets the same notation as before:
uniform = fromList (const 1)
One of the advantages of the free approach is that it’s easy to define multiple interpreters. We could, for instance, write an interpreter that constructs a diagram:
>>> drawTree ((,) <$> uniform "abc" <*> uniform "de")
┌('c','d')
┌1 % 2┤
│ └('c','e')
1 % 3┤
│ ┌('a','d')
│ ┌1 % 2┤
│ │ └('a','e')
└1 % 2┤
│ ┌('b','d')
└('b','e')
There’s a lot to be said about free objects in category theory, also. Specifically, they’re related to initial and terminal (also called final) objects. The encoding above is initial, the final encoding is simply Cont:
type Prob = Cont Rational
Here, also, we get the monad instance for free. In contrast to previously, expect is free:
expect = flip runCont
Support, though, isn’t possible.
This version is also called the Giry monad: there’s a deep and fascinating theory behind it, which I probably won’t be able to do justice to here. Check out Jared Tobin’s post (2017) for a good deep dive on it.
The branching structure of the tree captures the semantics of the probability monad well, but it doesn’t give us much insight into the original implementation. The question is, how can we deconstruct this:
Eric Kidd (2007) pointed out that the monad is the composition of the writer and list monads:
type Prob = WriterT (Product Rational) []
but that seems unsatisfying: in contrast to the tree-based version, we don’t encode any branching structure, we’re able to have empty distributions, and it has the combinatorial explosion problem.
Adding a weighting to nondeterminism is encapsulated more concretely by the ListT transformer. It looks like this:
newtype ListT m a
= ListT
{ runListT :: m (Maybe (a, ListT m a))
It’s a cons-list, with an effect before every layer1.
While this can be used to give us the monad we need, I’ve found that something more like this fits the abstraction better:
data ListT m a
= ListT a (m (Maybe (ListT m a)))
It’s a nonempty list, with the first element exposed. Turns out this is very similar to the cofree comonad:
Just like the initial free encoding, we can start with a primitive operation:
data Perhaps a
= Impossible
| WithChance Rational a
And we get all of our instances as well:
{ runProb :: Cofree Perhaps a
} deriving (Functor,Foldable)
instance Comonad Prob where
extract (Prob xs) = extract xs
duplicate (Prob xs) = Prob (fmap Prob (duplicate xs))
foldProb :: (a -> Rational -> b -> b) -> (a -> b) -> Prob a -> b
foldProb f b = r . runProb
r (x :< Impossible) = b x
r (x :< WithChance p xs) = f x p (r xs)
uniform :: [a] -> Prob a
uniform (x:xs) = Prob (coiterW f (EnvT (length xs) (x :| xs)))
f (EnvT 0 (_ :| [])) = Impossible
f (EnvT n (_ :| (y:ys)))
= WithChance (1 % fromIntegral n) (EnvT (n - 1) (y:|ys))
expect p = foldProb f p
f x n xs = (p x * n + xs) / (n + 1)
probOf p = expect (\x -> if p x then 1 else 0)
pure x = Prob (x :< Impossible)
append :: Prob a -> Rational -> Prob a -> Prob a
append = foldProb f (\x y -> Prob . (x :<) . WithChance y . runProb)
f e r a p = Prob . (e :<) . WithChance ip . runProb . a op
ip = p * r / (p + r + 1)
op = p / (r + 1)
xs >>= f = foldProb (append . f) f xs
We see here that we’re talking about gambling-style odds, rather than probability. I wonder if the two representations are dual somehow?
The application of comonads to streams (ListT) has been explored before (Uustalu and Vene 2005); I wonder if there are any insights to be gleaned from this particular probability comonad.
Erwig, Martin, and Steve Kollmansberger. 2006. “Functional Pearls: Probabilistic Functional Programming in Haskell.” Journal of Functional Programming 16 (1): 21–34. doi:10.1017/S0956796805005721.
Kidd, Eric. 2007. “Build Your Own Probability Monads.”
Larsen, Ken Friis. 2011. “Memory Efficient Implementation of Probability Monads.”
Ścibior, Adam, Zoubin Ghahramani, and Andrew D. Gordon. 2015. “Practical Probabilistic Programming with Monads.” In Proceedings of the 2015 ACM SIGPLAN Symposium on Haskell, 50:165–176. Haskell ’15. New York, NY, USA: ACM. doi:10.1145/2804302.2804317.
Tobin, Jared. 2017. “Implementing the Giry Monad.” jtobin.io.
Uustalu, Tarmo, and Varmo Vene. 2005. “The Essence of Dataflow Programming.” In Proceedings of the Third Asian Conference on Programming Languages and Systems, 2–18. APLAS’05. Berlin, Heidelberg: Springer-Verlag. doi:10.1007/11575467_2.
Note this is not the same as the ListT in transformers; instead it’s a “ListT done right”.↩︎
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Rd Sharma 2017 for Class 7 Math Chapter 25 - Data Handling Iv Probability
Rd Sharma 2017 Solutions for Class 7 Math Chapter 25 Data Handling Iv Probability are provided here with simple step-by-step explanations. These solutions for Data Handling Iv Probability are extremely popular among Class 7 students for Math Data Handling Iv Probability Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Rd Sharma 2017 Book of Class 7 Math Chapter 25 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Rd Sharma 2017 Solutions. All Rd Sharma 2017 Solutions for class Class 7 Math are prepared by experts and are 100% accurate.
When a coin is tossed at random, what is the probability of getting (i) a head? (ii) a tail?
Total number of times a coin is tossed = 1000
Number of times a head comes up = 445
Number of times a tail comes up = 555
(i) Probability of getting a head =
\frac{\mathrm{Number} \mathrm{of} \mathrm{heads}}{\mathrm{Total} \mathrm{no}. \mathrm{of} \mathrm{trails}}=\frac{445}{1000}=0.445
\frac{\mathrm{Number} \mathrm{of} \mathrm{tails}}{\mathrm{Total} \mathrm{no}. \mathrm{of} \mathrm{trails}}=\frac{555}{1000}=0.555
A die is thrown 100 times and outcomes are noted as given below:
Outcome: 1 2 3 4 5 6
Frequency: 21 9 14 23 18 15
If a die is thrown at random, find the probability of getting a/an.
(iv) Even number
(v) Odd number
(vi) Number less than 3.
Number of times "1" comes up = 21
Number of times "2" comes up = 9
(i) Probability of getting 3
\frac{\mathrm{frequency} \mathrm{of} 3}{\mathrm{Total} \mathrm{no}. \mathrm{of} \mathrm{trails}}=\frac{14}{100}=0.14
(ii) Probability of getting 5
\frac{\mathrm{frequency} \mathrm{of} 5}{\mathrm{Total} \mathrm{no}. \mathrm{of} \mathrm{trails}}=\frac{18}{100}=0.18
(iii) Probability of getting 4
\frac{\mathrm{frequency} \mathrm{of} 4}{\mathrm{Total} \mathrm{no}. \mathrm{of} \mathrm{trails}}=\frac{23}{100}=0.23
(iv) Frequency of getting an even no. = Frequency of 2 + Frequency of 4 + Frequency of 6 = 9+ 23 + 15 = 47
Probability of getting an even no. =
\frac{\mathrm{frequency} \mathrm{of} \mathrm{even} \mathrm{no}.}{\mathrm{Total} \mathrm{no}. \mathrm{of} \mathrm{trails}}=\frac{47}{100}=0.47
(v) Frequency of getting an odd no. = Frequency of 1 + Frequency of 3 + Frequency of 5 = 21+ 14 + 18 = 53
Probability of getting an odd no. =
\frac{\mathrm{frequency} \mathrm{of} \mathrm{odd} \mathrm{no}.}{\mathrm{Total} \mathrm{no}. \mathrm{of} \mathrm{trails}}=\frac{53}{100}=0.53
(vi) Frequency of getting a no. less than 3 = Frequency of 1 + Frequency of 2= 21 + 9 = 30
Probability of getting a no. less than 3
\frac{\mathrm{frequency} \mathrm{of} \mathrm{no}. \mathrm{less} \mathrm{than} 3}{\mathrm{Total} \mathrm{no}. \mathrm{of} \mathrm{trails}}=\frac{30}{100}=0.30
No. of socks in the box = 4
S = {B,B,W,W}
No. of white socks left = 2-1 =1
Probability of getting a white sock
\frac{\mathrm{Number} \mathrm{of} \mathrm{white} \mathrm{socks} \mathrm{left} \mathrm{in} \mathrm{the} \mathrm{box}}{\mathrm{Total} \mathrm{no}. \mathrm{of} \mathrm{socks} \mathrm{left} \mathrm{in} \mathrm{the} \mathrm{box}}=\frac{1}{3}
Two coins are tossed simultaneously 500 times and the outcomes are noted as given below:
Outcome: Two heads
(HH) One head
(HT or TH) No head
If same pair of coins is tossed at random, find the probability of getting (i) Two heads (ii) One head (iii) No head.
Number of outcomes of two heads (HH) = 105
Number of outcomes of one head (HT or TH) = 275
Number of outcomes of no head (TT) = 120
(i) Probability of getting two heads =
\frac{\mathrm{frequency} \mathrm{of} \mathrm{getting} 2 \mathrm{heads}}{\mathrm{Total} \mathrm{number} \mathrm{of} \mathrm{trails}}=\frac{105}{500}=\frac{21}{100}
(ii) Probability of getting one head =
\frac{\mathrm{frequency} \mathrm{of} \mathrm{getting} 1 \mathrm{head}}{\mathrm{Total} \mathrm{number} \mathrm{of} \mathrm{trails}}=\frac{255}{500}=\frac{11}{20}
(iii) Probability of getting no head =
\frac{\mathrm{frequency} \mathrm{of} \mathrm{getting} \mathrm{no} \mathrm{head}}{\mathrm{Total} \mathrm{number} \mathrm{of} \mathrm{trails}}=\frac{120}{500}=\frac{6}{25}
An unbiased coin is tossed once, the probability of getting head is
\frac{1}{2}
\frac{1}{3}
\frac{1}{4}
Tossing a coin, either we get a head (H) or a tail (T). So, the probability of getting a head is
\frac{1}{2}
There are 10 cards numbered from 1 to 10. A card is drawn randomly. The probability of getting
an even numbered card is
\frac{1}{10}
\frac{1}{5}
\frac{1}{2}
\frac{2}{5}
The number on the cards are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
The even numbers on the cards are 2, 4, 6, 8, 10.
∴ Probability of getting an even numbered card =
\frac{\mathrm{Number} \mathrm{of} \mathrm{even} \mathrm{numbered} \mathrm{card}}{\mathrm{Number} \mathrm{of} \mathrm{cards} \mathrm{with} \mathrm{numbers} \mathrm{from} 1 \mathrm{to} 10}
=\frac{5}{10}=\frac{1}{2}
A dice is rolled. The probability of getting an even prime is
\frac{1}{6}
\frac{1}{3}
\frac{1}{2}
\frac{5}{6}
The possible numbers on a dice are 1, 2, 3, 4, 5, 6.
There is only one even prime number which is 2.
∴ Probability of getting an even prime =
\frac{\mathrm{Number} \mathrm{of} \mathrm{even} \mathrm{prime} \mathrm{numbers}}{\mathrm{Number} \mathrm{of} \mathrm{all} \mathrm{possible} \mathrm{outcomes} \mathrm{on} \mathrm{the} \mathrm{dice}}
=\frac{1}{6}
There are 100 cards numbered from 1 to 100 in a box. If a card is drawn from the box and
the probability of an event is
\frac{1}{2}
, then the number of favourable cases to the event is
\frac{50}{100}=\frac{1}{2}
So, if the the probability of an event is
\frac{1}{2}
, then the number of favourable cases has to be 50.
When a dice is thrown, the total number of possible outcomes is
The number on the faces of a dice are 1, 2, 3, 4, 5, and 6.
There are 10 marbles in a box which are marked with the distinct numbers from 1 to 10.
A marble is drawn randomly. The probability of getting prime numbered marble is
\frac{1}{2}
\frac{2}{5}
\frac{9}{3}
\frac{3}{10}
The numbers marked on the marbles are 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10.
Here, the prime numbers (favourable outcomes) are 2, 3, 5, and 7.
Probability of getting prime numbered marble =
\frac{4}{10}=\frac{2}{5}
The probability of getting a red card from a well shuffled pack of cards is
\frac{1}{4}
\frac{1}{2}
\frac{3}{4}
\frac{1}{3}
There are 52 cards in a standard deck. There are four different suits Diamonds (red), Clubs (black)
, Hearts (red), and Spades (black) each containing 13 cards.
∴ Number of red cards (favourable outcomes) = 13 + 13 = 26
Probability of getting a red card =
\frac{26}{52}=\frac{1}{2}
A coin is tossed 100 times and head is obtained 59 times. The probability of getting a tail is
\frac{59}{100}
\frac{41}{100}
\frac{29}{100}
\frac{43}{100}
Number of head obtained = 59
Number of tail obtained (favourable outcomes) = 100 − 59 = 41
Probability of getting a tail =
\frac{41}{100}
A dice is tossed 80 times and number 5 is obtained 14 times. The probability of not getting the number 5 is
\frac{7}{40}
\frac{7}{80}
\frac{33}{40}
Probability of getting 5 =
\frac{14}{80}=\frac{7}{40}
Probability of not getting 5 =
1-\frac{7}{40}=\frac{33}{40}
A bag contains 4 green balls, 4 red balls and 2 blue balls. If a ball is drawn from the bag, the
probability of getting neither green nor red ball is
\frac{2}{5}
\frac{1}{2}
\frac{4}{5}
\frac{1}{5}
The probability of getting neither green nor red ball is equal to the probability of getting blue balls.
Probability of getting neither green nor red ball =
\frac{2}{10}=\frac{1}{5}
|
Relators - Maple Help
Home : Support : Online Help : Mathematics : Group Theory : Relators
Relators( G )
The Relators( G ) command returns the relators of a finitely presented group.
\mathrm{with}\left(\mathrm{GroupTheory}\right):
G≔〈〈a,b〉|〈{a}^{2},{b}^{3},{\left(a·b\right)}^{5}=1〉〉
\textcolor[rgb]{0,0,1}{G}\textcolor[rgb]{0,0,1}{≔}⟨\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{∣}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{a}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{b}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}⟩
\mathrm{Relators}\left(G\right)
{[\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{a}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}]}
The GroupTheory[Relators] command was introduced in Maple 17.
|
An open tank is filled with hg upto a height of 76cm. find the -Turito
An open tank is filled with Hg upto a height of 76cm. Find the pressure at the bottom of the tank and middle of the tank. (If atmospheric pressure is 1 atm) respectively are
1.5 atm, 2 atm
2 atm, 1.5 atm
Answer:The correct answer is: 2 atm, 1.5 atm
For one mole of a van der Waals gas when b=0 and T = 300 K, the PV vs 1/V plot is shown below. The value of ‘a’ (atmliter2 .mol–2) is
Let If AB is a scalar multiple of B, then
The value of λ such that the system
The number of values of k for which the system of equations has infinite number of solutions is
In the relation P is pressure, Z is distance, k is Boltzmann constant and is the temperature. The dimensional formula of β b will be:
In the equation The value of x is :
A rocket is fired upwards. Its engine explodes fully is 12s. The height reached by the rocket as calculated from its velocity-time graph is
Height reached m=66 m
Figure given shows the distance–time graph of the motion of a car. It follows from the graph that the car is
Since which is in form
Thus, the motion is uniformly accelerated
Three persons and of same mass travel with same speed such that each one faces the other always. After how much time will they meet each other?
If and are acute angles such that and then
If then the value(s) of is are
\sqrt{3}\mathrm{cos} \theta -\mathrm{sin} \theta =1
\theta =
\sqrt{3}\mathrm{cos} \theta -\mathrm{sin} \theta =1
\theta =
From a high tower at time , one stone is dropped from rest and simultaneously another stone is projected vertically up with an initial velocity. The graph of the distance between the two stones, before either his the ground, plotted against time will be as
, Clearly,
If then principal value of
Let M and N be two 3×3 non- singular symmetric matrices such that MN NM .If denotes the transpose of the matrix P, then is equal to
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EUDML | On the Milnor -groups of complete discrete valuation fields. EuDML | On the Milnor -groups of complete discrete valuation fields.
On the Milnor
K
-groups of complete discrete valuation fields.
Nakamura, Jinya. "On the Milnor -groups of complete discrete valuation fields.." Documenta Mathematica 5 (2000): 151-200. <http://eudml.org/doc/120601>.
author = {Nakamura, Jinya},
keywords = {Milnor -theory; syntomic complex; Cartier operator; Milnor -theory},
title = {On the Milnor -groups of complete discrete valuation fields.},
AU - Nakamura, Jinya
TI - On the Milnor -groups of complete discrete valuation fields.
KW - Milnor -theory; syntomic complex; Cartier operator; Milnor -theory
Masato Kurihara, On the structure of Milnor
K
Milnor
K
-theory, syntomic complex, Cartier operator, Milnor
K
K
K
p
p
K
Articles by Nakamura
|
EUDML | Algebraic groups associated with abelian varieties. EuDML | Algebraic groups associated with abelian varieties.
Algebraic groups associated with abelian varieties.
Ichikawa, Takashi. "Algebraic groups associated with abelian varieties.." Mathematische Annalen 289.1 (1991): 133-142. <http://eudml.org/doc/164773>.
@article{Ichikawa1991,
author = {Ichikawa, Takashi},
keywords = {Hodge groups; Hodge cycles; non-simple abelian varieties; Tate modules of abelian varieties over -adic local fields},
title = {Algebraic groups associated with abelian varieties.},
TI - Algebraic groups associated with abelian varieties.
KW - Hodge groups; Hodge cycles; non-simple abelian varieties; Tate modules of abelian varieties over -adic local fields
Davide Lombardo, [unknown]
Hodge groups, Hodge cycles, non-simple abelian varieties,
\ell -adic
Tate modules of abelian varieties over
\ell
-adic local fields
Classical groups (geometric aspects)
Articles by Takashi Ichikawa
|
Kozodaev M.G., Chernikova, A.G., Korostylev E.V., Park M.H., Khakimov R.R., Hwang C.S., Markeev A.M. "Mitigating wakeup effect and improving endurance of ferroelectric HfO 2 -ZrO 2 thin films by careful La-doping" Journal of Applied Physics, Volume 125, Issue 3, 21 January 2019, 034101
R. I. Romanov, M. G. Kozodaev, D. I. Myakota, A. G. Chernikova, S. M. Novikov, V. S. Volkov, A. S. Slavich, S. S. Zarubin, P. S. Chizhov, R. R. Khakimov, A. A. Chouprik, C. S. Hwang, A. M. Markeev "Synthesis of Large Area Two-Dimensional MoS2 Films by Sulfurization of Atomic Layer Deposited MoO3Thin Film for Nanoelectronic Applications" ACS Applied Nano Materials 2019, 2, 12, 7521-75
M. G. Kozodaev, Y. Y. Lebedinskii, A. G. Chernikova, E. V. Korostylev, A. A. Chouprik, R. R. Khakimov, Andrey M. Markeev, C. S. Hwang "Temperature controlled Ru and RuO2 growth via
{\mathbf{O}}^{*}
radical-enhanced atomic layer deposition with Ru(EtCp)2" J. Chem. Phys. 151, 204701 (2019)
Vitalii Mikheev, Anastasia Chouprik, Yury Lebedinskii, Sergei Zarubin, Yury Matveyev, Ekaterina Kondratyuk, Maxim G. Kozodaev, Andrey M. Markeev, Andrei Zenkevich, Dmitrii Negrov "Ferroelectric Second-Order Memristor" ACS Appl. Mater. Interfaces 2019, 11, 35, 32108-32114
R. I. Romanov, D. I. Myakota, A. A. Chuprik, S. M. Novikov, Yu. Yu. Lebedinskii, A. G. Chernikova, and A. M. Markeev "Two-Dimensional and Screw Growth of MoS2 Films in the Process of Chemical Deposition from the Gas Phase" Russian Journal of Applied Chemistry Volume 92, Issue 5, 1 May 2019, Pages 596-601
Damir R. Islamov, Vladimir A. Gritsenko, Timofey V. Perevalov, Vladimir A. Pustovarov, Oleg M. Orlov, Anna G. Chernikova, Andrey M. Markeev, Stefan Slesazeck, Uwe Schroeder, Thomas Mikolajick, Gennadiy Ya Krasnikov "Identification of the nature of traps involved in the field cycling of Hf0.5Zr0.5O2-based ferroelectric thin films" , Acta Materialia Volume 166, March 2019, Pages 47-55
Roman I. Romanov, Aleksandr S. Slavich, Maxim G. Kozodaev, Denis I. Myakota,Yuri Y. Lebedinskii, Sergey M. Novikov, and Andrey M. Markeev "Band Alignment in As-Transferred and AnnealedGraphene/MoS2Heterostructures" Physica Status Solidi - Rapid Research Letters 2019,1900406
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2013 Applications of Soft Union Sets in the Ring Theory
Yongwei Yang, Xiaolong Xin, Pengfei He
The aim of the paper is to lay a foundation for providing a soft algebraic tool in considering many problems that contain uncertainties. In order to provide these soft algebraic structures, the notion of
\left(\lambda ,\mu \right)
-soft union rings which is a generalization of that of soft union rings is proposed. By introducing the notion of soft cosets, soft quotient rings based on
\left(\lambda ,\mu \right)
-soft union ideals are established. Moreover, through discussing quotient soft subsets, an approach for constructing quotient soft union rings is made. Finally, isomorphism theorems of
\left(\lambda ,\mu \right)
-soft union rings related to invariant soft sets are discussed.
Yongwei Yang. Xiaolong Xin. Pengfei He. "Applications of Soft Union Sets in the Ring Theory." J. Appl. Math. 2013 1 - 9, 2013. https://doi.org/10.1155/2013/474890
Yongwei Yang, Xiaolong Xin, Pengfei He "Applications of Soft Union Sets in the Ring Theory," Journal of Applied Mathematics, J. Appl. Math. 2013(none), 1-9, (2013)
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The bifurcation set of a complex polynomial function of two variables and the Newton polygons of singularities at infinity
January, 2002 The bifurcation set of a complex polynomial function of two variables and the Newton polygons of singularities at infinity
A. Némethi and A. Zaharia have defined the explicit set for a complex polynomial function
f
{C}^{n}\to C
and conjectured that the bifurcation set of the global fibration of
is given by the union of the set of critical values and the explicit set of
. They have proved only the case
n=2
f
is Newton non-degenerate. In the present paper we will prove this for the case
n=2
, containing the Newton degenerate case, by using toric modifications and give an expression of the bifurcation set of
in the words of Newton polygons.
Masaharu ISHIKAWA. "The bifurcation set of a complex polynomial function of two variables and the Newton polygons of singularities at infinity." J. Math. Soc. Japan 54 (1) 161 - 196, January, 2002. https://doi.org/10.2969/jmsj/1191593959
Keywords: bifurcation set , complex polynomial functions , Newton polygons , singularities at infinity , toric modifications
Masaharu ISHIKAWA "The bifurcation set of a complex polynomial function of two variables and the Newton polygons of singularities at infinity," Journal of the Mathematical Society of Japan, J. Math. Soc. Japan 54(1), 161-196, (January, 2002)
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Metabolism - Citizendium
Metabolism (from Greek μεταβολισμός "metabolismos") is the biochemical modification of chemical compounds by living organisms and cells. In common usage, the word is often used to refer to the basal metabolic rate, the "set point" that each person has in breaking down food energy and building up their own body. In multicellular creatures like humans, its meaning encompasses the overall ingestion of food and excretion of wastes, as well as the building up of muscles and the growth of the body. In terms of the whole organism, metabolism includes the chemical conversion of ingested items other than food, like drugs and poisons (see Drug metabolism). This article describes the actual biology of metabolism at a cellular level, which explains just how those processes are carried out.
Metabolism includes: (1) anabolism, in which a cell uses chemical energy and reducing power to construct complex molecules, and perform life functions such as creating cellular structure; and (2) catabolism, in which a cell breaks down complex molecules to yield the chemical energy and reducing power. Cell metabolism involves complex sequences of controlled chemical reactions called metabolic pathways. Just as the word metabolism can be used to describe processes in a whole organism, the terms "anabolism" and "catabolism" can similarly be used in this way. For example, anabolic processes can also refer to building up muscle and adding body weight, while catabolic processes can refer to the loss of muscle mass and body fat.
With proper training and nutrition, weight lifting promotes the anabolic process of bodybuilding. Natural hormones, produced in both men and women, aid muscle development in response to weight bearing exercise.
2 Overview: Harnessing energy and making chemical bonds
2.1 ATP: the energy currency of cells
2.1.1 Phototrophic
2.1.2 Chemotrophic
2.2 Reducing Power: obtaining electrons for chemical bonds
2.2.1 Organotrophic
2.2.2 Lithotrophic
3 Regulation of metabolism in animals
Santorio Santorio (1561-1636) in his steelyard balance, from Ars de statica medecina, first published in 1614.
The first controlled experiments on human metabolism were published by Santorio Santorio in 1614 in his book Ars de statica medecina, in which he described experiments in which he weighed himself in a chair suspended from a steelyard balance (see image), before and after eating, sleeping, working, sex, fasting, depriving from drinking, and excreting. He found that by far the greatest part of the food he took in was lost from the body through perspiratio insensibilis (insensible perspiration). In medicine and the health sciences, the term "insensible losses" is still used to refer to fluids that escape the body without leaving easily-measurable traces behind.
At about the same time, Jan Baptist van Helmont made the first observations regarding photosynthesis, when he discovered that growing plants drew almost no matter from the surrounding soil. The physical source of the plant's growth was not obvious until later experiments, which delved into the process now known as photosynthesis.
In the 18th century, Joseph Priestley discovered that green plants released a substance (later found to be oxygen) that could sustain the life of a mouse in an enclosed chamber. Jan Ingenhousz extended Priestley's experiments to show that oxygen was produced when light was cast on the plant, while Jean Senebier showed that carbon dioxide was absorbed by plants during photosynthesis. In 1804, Nicolas de Saussure discovered that plant growth was the result of both the fixation of atmospheric carbon dioxide (
{\displaystyle CO_{2}}
) into the plant, and the incorporation of water.
Between 1854 and 1864, Louis Pasteur discovered that glucose fermentation is due to microorganisms, and, in 1897, Eduard Buchner proved that cell-free yeast extracts could also perform these reactions, and so the ability to ferment was not limited to entire living creatures (cells)- but included certain portions of their physical contents. Subsequent investigations showed that living organisms, with few exceptions, metabolize glucose using the same mechanism, namely, by a biochemical pathway that breaks down sugar.
Overview: Harnessing energy and making chemical bonds
A few of the catabolic pathways in a cell. Proteins are broken down into amino acids, and fats into glycerol and fatty acids. Carbohydrates (mostly sugars and starch) are hydrolyzed into monosacharides like glucose. The mitochondrion (in green) contains the enzymes that catalyze the citric acid cycle and beta-oxidation, as well as the electron transport chain (where respiration occurs). ATP is a high-energy molecule. See text for details
Living things, like all things, obey the laws of thermodynamics. That means that energy and matter cannot be created from nothing; cool things always get colder rather than warmer, and each fragment of a whole are smaller than the whole itself. But, unlike inanimate things, cells and tissues are able to harness energy and matter to change in ways that give the illusion of defying those laws. A baby does grow. A walrus' body is warmer than its icy surroundings. An amoeba can divide and shortly be two amoebas, each one the same size of the original cell that split. The metabolism of the baby, the walrus, and the amoeba is responsible for all these processes. Of course, rather than defy the laws of thermodynamics, the chemical reactions that make up metabolic processes always obey them.
Enzymes present in cells can catalyze a large variety of chemical reactions with exquisite specificity. Generally, enzymes are protein molecules that make reactions go faster by bringing the reactant molecules close together in just the right orientation for a chemical change to occur. Sometimes these enzymes are floating free in the cytoplasm of the cell, other times they are corralled together within a compartment of the cell, a special organelle. For example, the mitochondrion of cells contains enzymes for oxidative phosphorylation (a catabolic process). The Golgi apparatus of cells contains many of the enzymes used for protein posttranslational modification (an anabolic process).
Often, the chemical reactions needed to synthesize useful cell components require energy. Chemists describe these reactions as involving a positive change in free energy. Such chemical transformations are not spontaneous, but "uphill", requiring more than just the mixing of the substrates. In these cases, specific enzymes may couple each "uphill" (non-spontaneous or energy requiring) reaction to a second, steep "downhill" (very spontaneous or energy releasing) reaction. Thus, thermodynamically favorable reactions can be used to "drive" each thermodynamically unfavorable one - such that the the overall process goes on its own, as a spontaneous series of reactions.
There is one particular energetically favourable reaction that is repeatedly used to drive "uphill" reactions in metabolism:
Adenosine triphosphate + water → Adenosine diphosphate + phosphate ion + hydrogen ion
This reaction, the hydrolysis of Adenosine triphosphate (ATP) into Adenosine diphosphate and two ions, occurs often in metabolic pathways. ATP is sometimes called the "energy currency" of cells because it is used to "finance" these uphill reactions. To regenerate it for further reactions a large amount of energy is needed to recombine the products shown in the equation above. This is done by coupling the uphill synthesis of ATP to other energy-releasing reactions and is so ubiquitous that organisms can be classified according to how they derive energy for the process. Organisms can be classified as either Phototrophic or Chemotrophic.
Phototrophic organisms can obtain energy from light. In these reactions, excitation of a photosynthetic reaction centre is caused by the absorption of a light photon. During the process, the reaction center loses an electron that excites (reduces) an electron acceptor, such as pheophytin, initiating a flow of electrons down an electron transport chain present in the thylakoid membrane. The energy released in the electron transfer steps serves to create a proton gradient across the membrane; its dissipation is used by ATP synthase as the energy to synthesise ATP from ADP and a phosphate anion by photophosphorylation (see Chemiosmotic hypothesis). Depending on the organism, the reaction center regains the lost electron by either recycling the excited electrons or taking one from an electron donor. In plants, a water molecule serves as the electron donor through a process called photolysis, that releases oxygen gas as a waste product.
A few of the anabolic pathways in a cell. Glucose can be stored as a glycogen polymer, or synthesized from lower molecular weight precursors. Excess acetyl-CoA can be stored as fatty acids, or converted into ketone bodies.
Chemotrophic organisms obtain energy from chemical reactions. For example, glucose can be oxidized to pyruvate through glycolysis. This yields two molecules of ATP for each molecule of glucose, by substrate-level phosphorylation, and four electrons, which reduce two NAD+ molecules to NADH.
For glycolysis to continue, the NADH must be recycled to NAD+ by donating the electrons to an electron acceptor. Respiration is said to occur if this electron acceptor is external to the metabolism, and may be either anaerobic or aerobic. Fermentation, on the other hand, does not use an external electron acceptor: in this case, the electron acceptor is a product of glycolysis, usually pyruvate or a pyruvate derivative.
Acetyl-CoA is a pivotal molecule during aerobic respiration. Acetyl-CoA is derived from pyruvate, but can also be formed through β-oxidation of fatty acids or through the catabolism of amino acids, and is oxidized to CO2 through the Krebs cycle. The Krebs cycle releases eight electrons from each acetyl-CoA molecule, which are eventually used in aerobic organisms to reduce oxygen (terminal electron acceptor) via an electron transport chain. This is part of the process to synthesis more ATP, known as oxidative phosphorylation, and is very similar to photophosphorylation in phototrophs.
Reducing Power: obtaining electrons for chemical bonds
Reducing power is an important input into many anabolic pathways, including the Calvin cycle of photosynthesis, the biosynthesis of amino acids, and the biosynthesis of fatty acids. Reducing power is usually supplied as hydrogen equivalents carried by NADPH. Organisms can be classified according to the primary source of this reducing power as:
These organisms use organic compounds (e.g. glucose) as the primary electron source.
These organisms use inorganic compounds (e.g. Fe2+, (iron ions)) as primary electron source.
Regulation of metabolism in animals
In animals, metabolism is controlled by the endocrine system through the secretion of hormones. Some hormones have anabolic actions on the body, others have mainly catabolic actions. For example, testosterone is an anabolic hormone, and synthetic steroids that produce the anabolic actions are known as anabolic steroids. Cortisol on the other hand, which is a steroid hormone produced by the adrenal gland, is a catabolic hormone.
Two hormones synthesized by the pancreas, insulin and glucagon, are particularly important. Insulin is secreted when blood glucose levels are high, and it stimulates glucose uptake by muscle, glycogen synthesis, and triacylglyceride synthesis by adipose tissue (fat). It also inhibits gluconeogenesis and glycogen degradation. Glucagon is secreted when blood glucose levels are low, and its effects are opposite to those of insulin. In the liver, glucagon stimulates glycogen degradation and the absorption of gluconeogenic aminoacids, and it inhibits glycogen synthesis and promotes the release of fatty acids by adipose tissue.
In mammals and other warm blooded animals, many metabolic process are ultimately controlled by the central nervous system, which regulates the endocrine system. The central nervous system and endocrine system are influenced by the balance between the energy demands of the organism, and its energy stores (see also Hunger). For example, fat stores secrete a hormone called leptin that acts at the hypothalamus to regulate hormone secretion. The hypothalamus is also sensitive to circulating concentrations of glucose and insulin, and to body temperature. When the ambient temperature is low, the metabolic rate of an endothermic animal will increase in order to generate more body heat (thermogenesis). In animals that hibernate, the body temperature drops down enough that the basal metabolic rate is quite low, conserving energy over a winter period of inactivity.
Some ectothermic animals, like reptiles, regulate their body temperature by their behavior. These "cold blooded" creatures, including lizards, snakes, and turtles, keep at an optimum body temperature by heating up in the sun (basking) and cooling down in the shade or the cool earth of a burrow. The metabolism of these animals also changes with body temperature, and explains the sluggish movements of an ectotherm in colder seasons or times of day.
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Number - Citizendium
The concept of number is one of the most elementary, or fundamental notions of mathematics. Such elementary concepts cannot be defined in terms of other concepts (trivially, if an elementary concept could be defined in terms of other concepts, then it would not in fact be fundamental). Rather, a fundamental concept such as number can only be explained by demonstration. Such an approach relies for its efficacy on the intuitive properties of the human mind and its ability to abstract and generalize.
There are philosophical problems bound up with the concept of number. First, there is the ontological problem of the various types of numbers — do they exist, or are they "mental concepts". Then there is the epistemological problem which is concerned with how we know anything about numbers.
In mathematics, a number is formally a member of a given set (possibly an ordered set). It conveys the ideas of :
counting (e.g., there are 26 simple latin letters),
ordering (e.g., e is smaller than pi in the real number set), and
measurement (e.g., the weight of 50 lbs in the Imperial system is approximately equal to 22.7 kg in the metric system).
However, due to the expressiveness of positional number systems, the usefulness of geometric objects, and the advances in different scientific fields, it can convey more properties.
A word written only with digits is called a numeral, and may represent a number. Numerals are often used for labeling (like telephone numbers), for ordering (like serial numbers), and for encoding (like ISBNs).
The writing of a number depends on the numeral system in use. For instance, the number 12 is written "1100" in base 2, "C" in base 16, and "XII" as a roman numeral. We can geometrically represent a number with unitless vectors in a cartesian system or by drawing simple shapes (e.g., squares and circles). There are other means to express a number.
Abstract algebra studies abstract number systems such as groups, rings and fields.
This section presents different number sets, but this list is not exhaustive.
The natural numbers (
{\displaystyle \scriptstyle \mathbb {N} }
) are used to count things (e.g., there are 52 weeks in a Julian year). This set contains many remarkable subsets : prime numbers, Fibonacci numbers, perfect numbers, catalan numbers, etc.
The integers (
{\displaystyle \scriptstyle \mathbb {Z} }
) also include negative numbers, that can be used to represent debits and credits, etc. (e.g., a company owes 60 millions US dollars to a bank). This set includes the natural numbers.
{\displaystyle \scriptstyle \mathbb {Q} }
) are any number that can be represented as a fraction (e.g., someone received half of her pay yesterday). This set includes the integers.
The irrational numbers (
{\displaystyle \scriptstyle \mathbb {J} }
) find application in many abstract mathematical fields, such as algebra and number theory. An irrational number can not be written as a fraction, and can indeed not be written out fully at all. The numbers
{\displaystyle \pi }
{\displaystyle {\sqrt {2}}}
are both irrational. This set does not share any member with the rational number set.
The real numbers (
{\displaystyle \scriptstyle \mathbb {R} }
) find applications in measurements and advanced mathematics. They are usually best written as decimal numbers (e.g., the value of e is approximately equal to 2.718281828). This set includes the rational numbers and the irrational numbers.
The complex numbers (
{\displaystyle \scriptstyle \mathbb {C} }
) have two parts, where one is real and the other is some number multiplied by the imaginary number
{\displaystyle i\!}
{\displaystyle i={\sqrt {-1}}}
. The complex numbers were discovered while searching solutions to some polynomials (e.g., the polynomial
{\displaystyle \scriptstyle x^{2}+1=0}
has two solutions, one being
{\displaystyle \scriptstyle {\sqrt {-1}}=(0,1)=i}
). Because the complex number set is algebraically closed, it finds applications in many scientific fields, such as engineering and applied mathematics. This set includes the real numbers.
A complex number that is solution to a polynomial in integer coefficients is an algebraic number. This set includes all rational numbers and a subset of the irrational numbers. Any other complex number is a transcendental number.
In order to meet their needs, scientists created other number sets. To ease the study of quadratic forms, Carl Friedrich Gauss introduced from 1829 to 1831 what is known today as the gaussian integers. While studying 3D mechanics, William Rowan Hamilton introduced the quaternions in 1843 (today, they are largely superseded by vectors). Octonions were discovered in 1843. Georg Cantor, through its naive set theory, formally defined the notion of infinity in 1895. Kurt Hensel first described the p-adic numbers in 1897, looking for a way to bring the ideas and the techniques of power series within number theory.
We can consider unitless vectors and unitless matrices as number sets, since they mathematically abstract phenomenas in a unique way and we can apply operations upon them.
The notation plays a central role in the perception of what a number is and what we can do with it. A good notation saves lots of work when operating on numbers (and more generally on any mathematical abstract objects). For instance, it is possible to add numbers written in roman numerals (e.g., MCMXCVIII plus CCXVII). However, it is faster to add numbers written in base 10 (e.g., 1998 plus 217). The gain is higher when multiplying numbers.
In the Western world, the positional number system in base ten is the most used number notation. In this system, a numeral is constructed by putting digits side by side, each position in the numeral having a different numerical weight (a power of 10). In some knowledge fields, other numeral systems allow better handling of information. For instance, electronic engineers use binary numbers when dealing with electronic circuits.
To convey more information and to ease reading, different symbols are added to the digits :
Integer numerals are prepended with the minus or the plus symbol ("-" and "+"). This applies to any numeral, as long as it does not represent a natural number.
Numerals may come with a radix point, the decimal separator in base 10 (the period "." in some systems, the comma "," in others).
In long numerals, digits are grouped and may contain a thousand separator (e.g., the speed of light in vacuum is written as 1,079,252,849 km/h in some systems, while it is written as 1 079 252 849 km/h in some others).
Percentages ("%") allow to write a numeral as a fraction with the denominator 100 (e.g., 14.5% =
{\displaystyle \scriptstyle {\frac {14.5}{100}}}
Per mills ("‰") allow to write a numeral as a fraction with the denominator 1,000 (e.g., 22.3‰ =
{\displaystyle \scriptstyle {\frac {22.3}{1000}}}
Per cent mille (pcm) allow to write a numeral as a fraction with the denominator 100,000 (e.g., 78.7 pcm =
{\displaystyle \scriptstyle {\frac {78.7}{100000}}}
Parts per million (ppm), parts per billion (ppb) and parts per trillion (ppt) are others way to write a numeral as a fraction with the denominators 1 million, 1 billion, and 1 trillion.
Very small and very large numbers are usually expressed in scientific notation. Their numeral uses the product symbol × or E (e.g., the speed of light in vacuum is approximately
{\displaystyle \scriptstyle 3.0\times 10^{8}m/s=3.0E+8\,m/s}
There are other ways to represent a number.
Fractions contain a slash or a vinculum (e.g.,
{\displaystyle \scriptstyle a/b={\frac {a}{b}}}
). Ratios use the colon (e.g., 1.5 : 5).
To shorten some numerals (or to show some properties), numbers are represented using an exponentation (e.g., 34), a radical symbol (e.g.,
{\displaystyle \scriptstyle {\sqrt {2}}}
), a repeated pattern (e.g.,
{\displaystyle \scriptstyle {\frac {1}{3}}=0.{\overline {3}}}
). Almost any expression having only functions and numerals may represent a number (e.g.,
{\displaystyle \scriptstyle \sin({\frac {\pi }{3}})}
{\displaystyle \scriptstyle |-3.4|}
{\displaystyle \scriptstyle \zeta (3)}
Complex numbers are represented either by
{\displaystyle \scriptstyle (a,b)}
{\displaystyle \scriptstyle a+bi}
{\displaystyle \scriptstyle a,b\in \mathbb {R} }
In physics, vectors are usually represented as the sum of unit vectors :
{\displaystyle {\vec {\imath }},{\vec {\jmath }},\ldots }
. In mathematics, we may encounter the "hat notation" :
{\displaystyle {\hat {\imath }},{\hat {\jmath }},\ldots }
Named constants are another way to represent numbers :
{\displaystyle \pi }
, e,
{\displaystyle \gamma }
In geometry, a number can be represented in different ways. For instance, the length between two points in a cartesian coordinate system may represent a number. Fractions are sometimes represented by a rectangular grid. We could represent
{\displaystyle \scriptstyle {\frac {7}{12}}}
by a grid.
In statistics, numbers are represented by areas in histogram or by height in bar charts. In pie charts, values are proportional to the central angles. There are many other ways to represent numbers in statistics.
Many other scientific fields have their own notations.
There are cases where it is difficult to say if a symbol represents a number. Take for instance the units in International System of Units. When we write 30 cm, it means 30 ÷ 100 × meter. Officially, we should see "cm" as a centimeter, a unit of measure. However, 30 cm is the same as 0.3 m. For this reason, "30 c" represents a number: 0.3.
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Chemical symbol - Citizendium
Chemical symbols are the international standard way to denote chemical elements. In particular, they are used in chemical formulas to describe the composition and structure of molecules, and in reaction formulas. A chemical symbol consists of one or two letters: the initial letter of its scientific name in uppercase which, in most cases, is followed by a suitable lowercase letter from the name. (Sometimes, on a temporary basis, for new artificial elements initially 3-letter symbols are used.)
Well-known examples of one-letter symbols are H for hydrogen and O for oxygen; Ca is the two-letter symbol for calcium. For most elements the symbol fits its English name because this name is also derived from the scientific name (which usually is of Greek or Latin origin). However, in a few cases there is no relation between the English word and its symbol. For instance, Fe (derived from Latin "ferrum") is the symbol for iron.
Element number 112 has the symbol Uub, for its temporary name Ununbium (which is derived from Latin for one-one-two).
For a complete list of all chemical symbols see the alphabetical list.
In addition, subscripts and superscripts attached to the basic symbol are used to carry additional information.
The number of atoms in a molecule is indicated by a subscript (on the right). For example, O2 is oxygen, and O3 is ozone.
An ion of an element is indicated by a superscript, where + and - stand for positive and negative charge, respectively. For example, H+ means a hydrogen ion, and Ca2+ a calcium ion (with two electrons missing).
The ion H+ of hydrogen is a proton which—in atomic reactions—is also indicated as p. (Similarly, a neutron is indicated by n.)
In atom physics, isotopes of an element are distinguished by adding the atomic mass as a superscript (usually, but not always, attached to the left of the symbol).
The isotopes of hydrogen have symbols of their own: 1H is hydrogen H, 2H is deuterium D, and 3H is tritium T.
For convenience, sometimes the atomic number—which is already implied by the chemical symbol—is added as a subscript on the left.
A simple example for a chemical reaction formula is
{\displaystyle {\rm {2H_{2}+O_{2}\to 2H_{2}O}}}
which states that hydrogen and oxygen can react and produce water.
A hydrogen ion and a OH-group combine to water:
{\displaystyle {\rm {H^{+}+OH^{-}\to H_{2}O}}}
In an atomic fusion reaction, two deuterium atoms can either combine to a tritium atom and emit a proton:
{\displaystyle {\rm {{}^{2}D+{}^{2}D\to {}^{3}T+p}}}
or to a helium isotope and emit a neutron:
{\displaystyle {\rm {{}_{1}^{2}D+{}_{1}^{2}D\to {}_{2}^{3}He+n}}}
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A block of mass 300 kg is set into motion on a frictionless hor-Turito
A block of mass 300 kg is set into motion on a frictionless horizontal surface with the help of frictionless pulley and a rope system as shown in figure. What horizontal force F should be applied to produce in the block an acceleration of 1 ms-1
If is a polynomial function such that
A body of mass m rests on horizontal surface. The coefficient of friction between the body and the surface is if the body is pulled by a force P as shown in figure the limiting function between body and surface will be……..
A body of mass. 5 kg starts motion form the origin with an initial velocity If a constant force acts an the body, than the time in which the Y-component of the velocity becomes zero is……
Three blocks A,B, and C of equal mass m are placed one over the other, one on a smooth horizontal ground as shown in figure. Coefficient of friction between any two blocks of A,B and C is 0.5 What would he the maximum value of mass of block so that the blocks A,B and C move without slipping over each other
A train is moving along a horizontal track. A pendulum suspended from the roof makes an angle of 40 with the vertical. The acceleration of the train is-ms2(g=10ms2)
A bag of sand of mass m is suspended hy rope. A bullet of mass is fired at it with a velocity v and gets embedded into it . The velocity of the bag finally is…….
An L -shaped tube with a small office is held in a water stream as shown in fig. The upper end of the tube is? 10.6cm above the surface of water. What will be the. height of the set of water coming from the office velocity of water stream is 2.45 m/s2
Three blocks having equal mass of 2 kg are hanging on a string passing over a pulley as shown in figure. what will be the tension produced in a string connecting the blocks B and .C
f\left(x\right)=\left\{\begin{array}{l}\left[x\right] \text{ if }-3<x\le -1\\ |x| \text{ if }-1<x<1\\ |\left[-x\right]| \text{ if }1\le x\le 3\end{array}\text{ then }\left\{x:f\left(x\right)\ge 0\right\}=\right\
f\left(x\right)=\left\{\begin{array}{l}\left[x\right] \text{ if }-3<x\le -1\\ |x| \text{ if }-1<x<1\\ |\left[-x\right]| \text{ if }1\le x\le 3\end{array}\text{ then }\left\{x:f\left(x\right)\ge 0\right\}=\right\
A partly hanging uniform chain of length L is resting on a rough horizontal table. l is the maximum possible length that can hang in equilibrium The coefficient of friction between the chain and table is…….
Same forces act an two bodies of different mass 2 kg and 5 kg initially at rest. The ratio of times. required to acquire same final velocity is…..
An object of mass 3 kg is moving with a velocity of 5 m/s along a stright path. If a force of 12 N is applied for 3 sec on the object in a perpendicular to its direction of motion. The magnitude of velocity of the particle at the end of 3 sec is……..m/s.
A rope which can withstand a maximum tension of 400 N hangs from a tree. If a monkey of mass 30 kg climbs on the rope in which of the following cases-will the rope break? (take g=10 ms2 and neglect the mass of rope)
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Hodrick-Prescott filter for trend and cyclical components - MATLAB hpfilter - MathWorks Australia
Apply Hodrick-Prescott Filter to Matrix of Data
Apply Hodrick-Prescott Filter to Table Variables
Visually Compare Results from Various Smoothing Parameter Values
hpfilter supports name-value argument syntax for all optional inputs
Hodrick-Prescott filter for trend and cyclical components
[Trend,Cyclical] = hpfilter(Y)
[TTbl,CTbl] = hpfilter(Tbl)
[___] = hpfilter(___,Name=Value)
hpfilter(___)
hpfilter(ax,___)
[___,h] = hpfilter(___)
Separate one or more time series into trend and cyclical components by applying the Hodrick-Prescott filter [1]. hpfilter optionally plots the series and trend component, with cycles removed. The plot helps you select a smoothing parameter.
[Trend,Cyclical] = hpfilter(Y) returns the trend Trend and cyclical Cycilcal components from applying the Hodrick-Prescott Filter to each variable (column) of the input matrix of time series data Y. The smoothing parameter default is 1600, suggested in [1] for quarterly data.
[TTbl,CTbl] = hpfilter(Tbl) returns the tables or timetables TTbl and CTbl containing variables for the trend and cyclical components, respectively, from applying the Hodrick-Prescott Filter to each variable in the input table or timetable Tbl. To select different variables in Tbl to filter, use the DataVariables name-value argument.
[___] = hpfilter(___,Name=Value) uses additional options specified by one or more name-value arguments, using any input-argument combination in the previous syntaxes. hpfilter returns the output-argument combination for the corresponding input arguments. For example, hpfilter(Tbl,Smoothing=100,DataVariables=1:5) applies the Hodrick-Prescott filter to the first five variables in the input table Tbl and sets the smoothing parameter to 100.
hpfilter(___) plots time series variables in the input data and their respective trend components, computed by the Hodrick-Prescott filter, on the same axes.
hpfilter(ax,___) plots on the axes specified by ax instead of the current axes (gca). ax can precede any of the input argument combinations in the previous syntaxes.
[___,h] = hpfilter(___) plots the specified series and their trend components, and additionally returns handles to plotted graphics objects. Use elements of h to modify properties of the plot after you create it.
Plot the cyclical component of the US post-WWII, seasonally adjusted, quarterly, real gross national product (GNPR).
GNPR = DataTable.GNPR;
[trend,cyclical] = hpfilter(GNPR);
T = numel(trend)
trend and cyclical are 235-by-1 vectors containing the trend and cyclical components, respectively, resulting from applying the Hodrick-Prescott filter to the series with default smoothing parameter 1600.
plot(dates,cyclical)
ylabel("Real GNP Cyclical Component")
Apply the Hodrick-Prescott filter to all variables in input tabular variables.
Load the US equity indices data set, which contains the table DataTable of daily closing prices of the NYSE and NASDAQ composite indices from 1990 through 2011.
Create a timetable from the table.
Aggregate the daily data in the timetable to quarterly.
TTQ = convert2quarterly(TT);
TTQ contains the closing prices in TT at the end of each quarter.
Apply the Hodrick-Prescott filter to all variables in the quarterly timetable. The default smoothing parameter value is 1600. Display the last few observed components.
[TQTbl,CQTbl] = hpfilter(TTQ);
size(TQTbl)
tail(TQTbl)
Time NYSE NASDAQ
31-Mar-2000 605.09 2610.6
30-Jun-2000 614.21 2653.5
30-Sep-2000 622.64 2686.6
31-Dec-2000 630.51 2711.6
tail(CQTbl)
Time NYSE NASDAQ
31-Mar-2000 42.608 1962.3
30-Jun-2000 28.724 1312.6
30-Sep-2000 40.398 986.2
31-Dec-2000 26.359 -241.11
31-Mar-2001 -42.288 -890.66
30-Jun-2001 -23.346 -585.48
30-Sep-2001 -108.27 -1261.9
31-Dec-2001 -69.269 -823.78
TQTbl and CQTbl are 48-by-2 timetables containing the trend and cyclical components, respectively, of the series in TTQ. Variables in the input and output timetables correspond.
By default, hpfilter filters all variables in the input table or timetable. To select a subset of variables, set the DataVariables option.
To compare outputs, apply the Hodrick-Prescott filter to all variables in the table DataTable and the timetable TT.
% Table input of daily data
[DTTTbl,DTCTbl] = hpfilter(DataTable);
size(DTTTbl)
tail(DTTTbl)
NYSE NASDAQ
20-Dec-2001 584.4 1976.4
27-Dec-2001 587.04 1972
tail(DTCTbl)
NYSE NASDAQ
19-Dec-2001 1.2948 5.5523
20-Dec-2001 -1.1413 -57.834
21-Dec-2001 0.67903 -29.492
26-Dec-2001 2.2792 -12.408
28-Dec-2001 6.6419 16.425
% Timetable input of daily data
[TTbl,CTbl] = hpfilter(TT);
size(TTbl)
tail(TTbl)
tail(CTbl)
Because the data is unaggregated, the outputs of the daily data have more rows than from the quarterly data. The filter results of the daily inputs are equal among the corresponding outputs, but hpfilter returns tables of results, instead of timetables, when you supply data in a table.
Load the Nelson-Plosser macroeconomic data set Data_NelsonPlosser.mat, which contains series measured yearly in the table DataTable.
Filter the real and nominal GNP series, GNPR and GNPN, respectively. Plot the trend component with each series by additionally returning the vector of graphics objects. Set the smoothing parameter to 2000.
[TTbl,CTbl,h] = hpfilter(DataTable,Smoothing=2000, ...
DataVariables=["GNPR" "GNPN"]);
xticklabels(dates(g.XTick))
Experiment with the smoothing parameter value by filtering the series several more times and setting the smoothing parameter to 0, 10, 100, 1000, 10,000, and Inf. Plot each set of results by not returning any outputs.
smoothing = [10.^(0:4) Inf];
for j = 1:numel(smoothing)
hpfilter(DataTable,Smoothing=smoothing(j), ...
title("\lambda = " + string(smoothing(j)));
legend("off")
By default, hpfilter plots to the current axes (gca).
hpfilter removes, from the specified data, all rows containing at least one missing observation, represented by a NaN value.
Example: hpfilter(Tbl,Smoothing=100,DataVariables=1:5) applies the Hodrick-Prescott filter to the first five variables in the input table Tbl and sets the smoothing parameter to 100.
Smoothing — Trend component smoothing parameter
1600 (default) | Inf | nonnegative numeric scalar | nonnegative numeric vector
Trend component smoothing parameter, specified as a nonnegative numeric scalar or vector of length numVars.
For a scalar, hpfilter applies Smoothing to all specified input series. For a vector, hpfilter applies Smoothing(k) to specified input series k in the data.
If Smoothing(k) = 0, hpfilter does not smooth the corresponding trend components. In this case, these conditions apply:
Trend(:,k) = yk, where yk is the input data of specified variable k.
Cyclical(:,k) = zeros(numObs,1).
If Smoothing(k) = Inf, hpfilter applies maximum smoothing. In this case, these conditions apply:
Trend(:,k) is the linear time trend computed by least squares.
Cyclical(:,k) is the detrended series.
As the magnitude of the smoothing parameter increases, Trend approaches the linear time trend.
1] suggests values for the smoothing parameter that depend upon
the periodicity of the data:
The smoothing parameter value depends on the periodicity of the data. Although a best practice is to experiment with smoothing values for your data, these values are recommended in [1]:
14400 for monthly data
1600 for quarterly data
100 for yearly data
Example: Smoothing=100
Variables in Tbl that hpfilter filters, specified as a string vector or cell vector of character vectors containing variable names in Tbl.Properties.VariableNames, or an integer or logical vector representing the indices of names. The selected variables must be numeric.
Trend — Trend component τt
Trend component τt of each series in the data, returned as a numObs-by-numVars numeric matrix. hpfilter returns Trend when you supply the input Y.
Cyclical — Cyclical component ct
Cyclical component ct of each series in the data, returned as a numObs-by-numVars numeric matrix. hpfilter returns Cyclical when you supply the input Y.
TTbl — Trend component τt
Trend component τt of each specified series, returned as a numObs-by-numVars table or timetable, the same data type as Tbl. hpfilter returns TTbl when you supply the input Tbl.
CTbl — Cyclical component ct
Cyclical component ct of each specified series, returned as a numObs-by-numVars table or timetable, the same data type as Tbl. hpfilter returns CTbl when you supply the input Tbl.
Handles to plotted graphics objects, returned as a vector of graphics objects. hpfilter plots the data and trend only when you return no outputs or you return h.
The Hodrick-Prescott filter decomposes an observed time series yt (Y) into a trend component τt (Trend) and a cyclical component ct (Cyclical) such that yt = τt + ct.
The objective function of the filter is
f\left({\tau }_{t}\right)=\sum _{t=1}^{T}{\left({y}_{t}-{\tau }_{t}\right)}^{2}+\lambda \sum _{t=2}^{T-1}{\left[\left({\tau }_{t+1}-{\tau }_{t}\right)-\left({\tau }_{t}-{\tau }_{t-1}\right)\right]}^{2},
λ is the smoothing parameter (smoothing).
yt – τt = ct.
The programming problem is to minimize the objective function over τ1,…,τT. The objective penalizes the sum of squares for the cyclical component with the sum of squares of second-order differences for the trend component (trend acceleration penalty). If λ = 0, the minimum of the objective is 0 with τt = yt for all t. As λ increases, the penalty for a flexible trend increases, resulting in an increasingly smoother trend. When λ is arbitrarily large, the trend acceleration approaches 0, resulting in a linear trend.
This figure shows the effects of increasing the smoothing parameter on the trend component for a simulated series.
The filter is equivalent to a cubic spline smoother, where the smoothed component is τt.
For high-frequency series, the Hodrick-Prescott filter can produce anomalous endpoint effects. In this case, do not extrapolate the series using the results of the filter.
[1] Hodrick, Robert J., and Edward C. Prescott. "Postwar U.S. Business Cycles: An Empirical Investigation." Journal of Money, Credit and Banking 29, no. 1 (February 1997): 1–16. https://doi.org/10.2307/2953682.
R2022a: hpfilter supports name-value argument syntax for all optional inputs
hpfilter accepts the smoothing parameter as the name-value argument Smoothing. However, the function will continue to accept the previous syntax.
hpfilter(Data,smoothing)
smoothing is the value of the smoothing parameter.
hpfilter(Data,Smoothing=smoothing)
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Multiple Choice: If the probability of getting a particular result in an experiment is
75.3\%
, what is the probability of not getting that result? Explain your choice.
75.3\%+100\%
75.3\%−100\%
100\%−75.3\%
\frac{1}{75.3\%}
Probability is defined as ''a number between zero and one that states the likelihood of an event occurring.'' Use this information to help you decide which answer is the best.
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MATH Matters for Math Honours | Science Student Success Centre
Are you an incoming Math Honours student? Start your degree off on the right foot by attending MATH Matters for Math Honours, a subsection of the MATH Matters week-long program. While all Math Honours students will be automatically able to see the lecture content, the week-long program will include extra support including a fully graded post-test with comments and specific advice on tackling math proofs. The course is completely separate from MATH Matters for non-honours students.
During the one-week long program, you will review and learn the skills and concepts you need to be successful in your program. Topics covered in MATH Matters for Math Honours will include:
Exponent and Log Laws
(Note that all topics above will focus on proving why ideas work rather than just using the math.)
What will you get out of MATH Matters for Math Honours?
Most secondary schools do not focus enough on mathematical proof, so many students who take honours mathematics courses tend to have difficulty on understanding and producing proofs. MATH Matters for Math Honours will focus on going through details covered in high school (sets, inequalities, limits, quadratic formula, sequences, etc…) but focuses on proving why the ideas work rather than learning how to do the math.
Study strategies will be provided for students on how they should be tackling an honours math class. Some examples include: creating definitions sheets, understanding and using mathematical theorems, the difference between understanding and memorizing, and the different help options you will have available to you once classes start.
Some first year topics may be discussed as well to help give you a head start in your courses (induction, symbolic logic, etc…)
The lectures will be taught by an instructor who teaches first-year Math courses for Math Honours majors, and the study groups will be led by seasoned and successful Math Honours students. This will give you a solid head start in understanding the problems and concepts you will see as a Math Honours student.
Through MATH Matters for Math Honours, you will also meet and interact with other Math Honours students, find study partners, and strengthen your math foundation – all before classes begin!
How Prepared Are You For Honours Math?
If you have trouble answering these questions, then it would be a good idea to highly consider taking MATH Matters for Math Honours to help prior to taking honours math classes.
1. Can you prove why
{\left(-1\right)}^{2}=1
2. Can you prove why the quadratic formula works starting from
a{x}^{2 }+bx+c=0
1+2+3+4+...+n=\frac{n\left(n+1\right)}{2}
{\mathrm{log}}_{a}\left(x\right)+{\mathrm{log}}_{a}\left(y\right)={\mathrm{log}}_{a}\left(xy\right)
5. Can you prove why the product rule works
\left(f\left(x\right)g\left(x\right)\right)\text{'}=f\text{'}\left(x\right)g\left(x\right)+f\left(x\right)g\text{'}\left(x\right)
To register for the MATH Matters week-long program for Math Honours, follow the instructions at Registering for MATH Matters and the MATH Matters administrator will assign you to the lectures and study groups for Math Honours students.
Math Honours students will follow the same schedule as other MATH Matters week-long participants. Visit Information for Attendees for more information.
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The residual finiteness of positive one-relator groups | EMS Press
It is proven that every positive one-relator group which satisfies the
{\rm C}'({1\over6})
condition has a finite index subgroup which splits as a free product of two free groups amalgamating a finitely generated malnormal subgroup. As a consequence, it is shown that every
{\rm C}'({1\over6})
positive one-relator group is residually finite. It is shown that positive one-relator groups are generically
{\rm C}'({1\over6})
and hence generically residually finite. A new method is given for recognizing malnormal subgroups of free groups. This method employs a 'small cancellation theory' for maps between graphs.
Daniel T. Wise, The residual finiteness of positive one-relator groups. Comment. Math. Helv. 76 (2001), no. 2, pp. 314–338
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Do two lines always have only one intersection point? Consider this as you answer the questions below.
Write a system of linear equations that has an infinite number of solutions. Write your equations in
y = mx + b
form and graph your system on graph paper. Explain why it has an infinite number of solutions.
Write a system of equations using two lines that have the same slope and the same
y
y=2x+1
y=\frac{6}{3}x+1
Since the two lines have the same slope and the same
y
-intercept, the lines coincide (overlap).
So all of the points are solutions for both equations.
How can you algebraically determine that a system of linear equations has an infinite number of solutions? Solve your system of equations from part (a) algebraically and demonstrate how you know that the system has an infinite number of solutions
2x + 1 = 2x + 1
2x
on both sides results in
1 = 1
which is a true statement. Therefore, there are an infinite number of solutions to this equation.
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Indijska matematika - Wikipedia
Indijska matematika se razvila na Indijskom potkontinentu[1] od 1200. pne. [2] sve do kraja 18. vijeka. U klasičnom periodu indijske matematike (400 - 1200) su zabilježena značajna postignuća zahvaljujući učenjacima kao što su Aryabhata, Brahmagupta i Bhaskara II. Decimalni sistem brojeva koji se koristi danas[3] se prvi put koristio u indijskoj matematici.[4] Indijski matematičari su dali rani doprinos proučavanju koncepta nule kao broja,[5] negativnih brojeva,[6] aritmetike i algebre.[7] Uz to se u Indiji razvila i trigonometrija[8], uključujući suvremene defincije sinusa i kosinusa.[9] Ti matematički koncepti su se kasnije prenijeli na Bliski Istok, Kinu i Evropu[7] te bitno pridonijeli razvoju koncepata koji danas tvore temelj mnogih područja matematike.
Drevna i srednjovjekovni matematički tekstovi, svi napisani na sanskritu, najčešće su se sastojali od sutra u kojima su načela ili problemi izneseni u ekonomičnim stihovima kako bi ih učenik mogao što lakše upamtiti. Njih je slijedila druga sekscija koja se sastojala od komentara u prozi (ponekad nekoliko komentara od različitih učenjaka) koji su detaljnije obrazložili problem ili izložili njegovo rješenja. U proznom dijelu forma (i njena memorizacija) nisu bili tako važnio kao same ideje.[1][10] Sva matematička djela su se prenosila usmenom predajom sve do oko 500. pne. a nakon čega su se prenosili i usmeno i preko rukopisa. Najstariji sačuvani matematički dokument na Indijskom kontinentu je rukopis iz Bakhshalija, otkriven godine 1881. u Bakhshaliju kraj Peshawara (moderni Pakistan) a koji datira iz 7. vijeka.[11][12]
Važno poglavlje u historiji indijske matematike bio je razvoj ekspanzije nizova za trigonometrijske funkcije (sinuse, kosinuse i obrnute trigonometrijske funkcije) od strane Keralske škole u 15. vijeku. Njihovo djelo, napravljeno dva vijeka prije otkrića infinitezimalnog računa u Evropi, je predstavljao prvi primjer stepenog reda.[13] Međutim, ona nije razvila koncepte diferencijala i integracije, niti ima neposrednih dokaza da su se ta dostignuća proširila izvan Kerale.[14][15][16][17]
↑ 1,0 1,1 Encyclopaedia Britannica (Kim Plofker) 2007, str. 1
↑ (Hayashi 2005, pp. 360–361)
↑ Ifrah 2000, str. 346: "The measure of the genius of Indian civilisation, to which we owe our modern (number) system, is all the greater in that it was the only one in all history to have achieved this triumph. Some cultures succeeded, earlier than the Indian, in discovering one or at best two of the characteristics of this intellectual feat. But none of them managed to bring together into a complete and coherent system the necessary and sufficient conditions for a number-system with the same potential as our own."
↑ Plofker 2009, str. 44–47
↑ Bourbaki 1998, str. 46: "...our decimal system, which (by the agency of the Arabs) is derived from Hindu mathematics, where its use is attested already from the first centuries of our era. It must be noted moreover that the conception of zero as a number and not as a simple symbol of separation) and its introduction into calculations, also count amongst the original contribution of the Hindus."
↑ Bourbaki 1998, str. 49: Modern arithmetic was known during medieval times as "Modus Indorum" or method of the Indians. Leonardo of Pisa wrote that compared to method of the Indians all other methods is a mistake. This method of the Indians is none other than our very simple arithmetic of addition, subtraction, multiplication and division. Rules for these four simple procedures was first written down by Brahmagupta during 7th century AD. "On this point, the Hindus are already conscious of the interpretation that negative numbers must have in certain cases (a debt in a commercial problem, for instance). In the following centuries, as there is a diffusion into the West (by intermediary of the Arabs) of the methods and results of Greek and Hindu mathematics, one becomes more used to the handling of these numbers, and one begins to have other "representation" for them which are geometric or dynamic."
↑ 7,0 7,1 "algebra" 2007. Britannica Concise Encyclopedia. Encyclopædia Britannica Online. 16 May 2007. Quote: "A full-fledged decimal, positional system certainly existed in India by the 9th century (AD), yet many of its central ideas had been transmitted well before that time to China and the Islamic world. Indian arithmetic, moreover, developed consistent and correct rules for operating with positive and negative numbers and for treating zero like any other number, even in problematic contexts such as division. Several hundred years passed before European mathematicians fully integrated such ideas into the developing discipline of algebra."
↑ (Pingree 2003, p. 45) Quote: "Geometry, and its branch trigonometry, was the mathematics Indian astronomers used most frequently. Greek mathematicians used the full chord and never imagined the half chord that we use today. Half chord was first used by Aryabhata which made trigonometry much more simple. In fact, the Indian astronomers in the third or fourth century, using a pre-Ptolemaic Greek table of chords, produced tables of sines and versines, from which it was trivial to derive cosines. This new system of trigonometry, produced in India, was transmitted to the Arabs in the late eighth century and by them, in an expanded form, to the Latin West and the Byzantine East in the twelfth century."
↑ (Bourbaki 1998, p. 126): "As for trigonometry, it is disdained by geometers and abandoned to surveyors and astronomers; it is these latter (Aristarchus, Hipparchus, Ptolemy) who establish the fundamental relations between the sides and angles of a right angled triangle (plane or spherical) and draw up the first tables (they consist of tables giving the chord of the arc cut out by an angle
{\displaystyle \theta <\pi }
{\displaystyle 2r\sin \left(\theta /2\right)}
↑ Filliozat 2004, str. 140–143
↑ Hayashi 1995
↑ Encyclopaedia Britannica (Kim Plofker) 2007, str. 6
↑ Stillwell 2004, str. 173
↑ Bressoud 2002, str. 12 Quote: "There is no evidence that the Indian work on series was known beyond India, or even outside Kerala, until the nineteenth century. Gold and Pingree assert [4] that by the time these series were rediscovered in Europe, they had, for all practical purposes, been lost to India. The expansions of the sine, cosine, and arc tangent had been passed down through several generations of disciples, but they remained sterile observations for which no one could find much use."
↑ Plofker 2001, str. 293 Quote: "It is not unusual to encounter in discussions of Indian mathematics such assertions as that “the concept of differentiation was understood [in India] from the time of Manjula (... in the 10th century)” [Joseph 1991, 300], or that “we may consider Madhava to have been the founder of mathematical analysis” (Joseph 1991, 293), or that Bhaskara II may claim to be “the precursor of Newton and Leibniz in the discovery of the principle of the differential calculus” (Bag 1979, 294). ... The points of resemblance, particularly between early European calculus and the Keralese work on power series, have even inspired suggestions of a possible transmission of mathematical ideas from the Malabar coast in or after the 15th century to the Latin scholarly world (e.g., in (Bag 1979, 285)). ... It should be borne in mind, however, that such an emphasis on the similarity of Sanskrit (or Malayalam) and Latin mathematics risks diminishing our ability fully to see and comprehend the former. To speak of the Indian “discovery of the principle of the differential calculus” somewhat obscures the fact that Indian techniques for expressing changes in the Sine by means of the Cosine or vice versa, as in the examples we have seen, remained within that specific trigonometric context. The differential “principle” was not generalized to arbitrary functions—in fact, the explicit notion of an arbitrary function, not to mention that of its derivative or an algorithm for taking the derivative, is irrelevant here"
↑ Pingree 1992, str. 562 Quote:"One example I can give you relates to the Indian Mādhava's demonstration, in about 1400 A.D., of the infinite power series of trigonometrical functions using geometrical and algebraic arguments. When this was first described in English by Charles Matthew Whish, in the 1830s, it was heralded as the Indians' discovery of the calculus. This claim and Mādhava's achievements were ignored by Western historians, presumably at first because they could not admit that an Indian discovered the calculus, but later because no one read anymore the Transactions of the Royal Asiatic Society, in which Whish's article was published. The matter resurfaced in the 1950s, and now we have the Sanskrit texts properly edited, and we understand the clever way that Mādhava derived the series without the calculus; but many historians still find it impossible to conceive of the problem and its solution in terms of anything other than the calculus and proclaim that the calculus is what Mādhava found. In this case the elegance and brilliance of Mādhava's mathematics are being distorted as they are buried under the current mathematical solution to a problem to which he discovered an alternate and powerful solution."
↑ Katz 1995, str. 173–174 Quote:"How close did Islamic and Indian scholars come to inventing the calculus? Islamic scholars nearly developed a general formula for finding integrals of polynomials by A.D. 1000—and evidently could find such a formula for any polynomial in which they were interested. But, it appears, they were not interested in any polynomial of degree higher than four, at least in any of the material that has come down to us. Indian scholars, on the other hand, were by 1600 able to use ibn al-Haytham's sum formula for arbitrary integral powers in calculating power series for the functions in which they were interested. By the same time, they also knew how to calculate the differentials of these functions. So some of the basic ideas of calculus were known in Egypt and India many centuries before Newton. It does not appear, however, that either Islamic or Indian mathematicians saw the necessity of connecting some of the disparate ideas that we include under the name calculus. They were apparently only interested in specific cases in which these ideas were needed. ... There is no danger, therefore, that we will have to rewrite the history texts to remove the statement that Newton and Leibniz invented calculus. Thy were certainly the ones who were able to combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between them, and turn the calculus into the great problem-solving tool we have today."
'Index of Ancient Indian mathematics', MacTutor History of Mathematics Archive, St Andrews University, 2004.
Online course material for InSIGHT Arhivirano 2009-08-22 na Wayback Machine-u, a workshop on traditional Indian sciences for school children conducted by the Computer Science department of Anna University, Chennai, India.
Izvor: https://sh.wikipedia.org/w/index.php?title=Indijska_matematika&oldid=41135065
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This article is about mathematical spaces having five dimensions. For the musical group, see The 5th Dimension. For alternate planes of existence in fiction, see fourth dimension in literature.
A five-dimensional space is a space with five dimensions. In mathematics, a sequence of N numbers can represent a location in an N-dimensional space. If interpreted physically, that is one more than the usual three spatial dimensions and the fourth dimension of time used in relativistic physics.[1] Whether or not the universe is five-dimensional is a topic of debate.
A 2D orthogonal projection of a 5-cube
2 Five-dimensional geometry
Much of the early work on five-dimensional space was in an attempt to develop a theory that unifies the four fundamental interactions in nature: strong and weak nuclear forces, gravity and electromagnetism. German mathematician Theodor Kaluza and Swedish physicist Oskar Klein independently developed the Kaluza–Klein theory in 1921, which used the fifth dimension to unify gravity with electromagnetic force. Although their approaches were later found to be at least partially inaccurate, the concept provided a basis for further research over the past century.[1]
To explain why this dimension would not be directly observable, Klein suggested that the fifth dimension would be rolled up into a tiny, compact loop on the order of 10-33 centimeters.[1] Under his reasoning, he envisioned light as a disturbance caused by rippling in the higher dimension just beyond human perception, similar to how fish in a pond can only see shadows of ripples across the surface of the water caused by raindrops.[2] While not detectable, it would indirectly imply a connection between seemingly unrelated forces. The Kaluza–Klein theory experienced a revival in the 1970s due to the emergence of superstring theory and supergravity: the concept that reality is composed of vibrating strands of energy, a postulate only mathematically viable in ten dimensions or more. Superstring theory then evolved into a more generalized approach known as M-theory. M-theory suggested a potentially observable extra dimension in addition to the ten essential dimensions which would allow for the existence of superstrings. The other 10 dimensions are compacted, or "rolled up", to a size below the subatomic level.[1][2] The Kaluza–Klein theory today is seen as essentially a gauge theory, with the gauge being the circle group.[citation needed]
The fifth dimension is difficult to directly observe, though the Large Hadron Collider provides an opportunity to record indirect evidence of its existence.[1] Physicists theorize that collisions of subatomic particles in turn produce new particles as a result of the collision, including a graviton that escapes from the fourth dimension, or brane, leaking off into a five-dimensional bulk.[3] M-theory would explain the weakness of gravity relative to the other fundamental forces of nature, as can be seen, for example, when using a magnet to lift a pin off a table — the magnet is able to overcome the gravitational pull of the entire earth with ease.[1]
Mathematical approaches were developed in the early 20th century that viewed the fifth dimension as a theoretical construct. These theories make reference to Hilbert space, a concept that postulates an infinite number of mathematical dimensions to allow for a limitless number of quantum states. Einstein, Bergmann and Bargmann later tried to extend the four-dimensional spacetime of general relativity into an extra physical dimension to incorporate electromagnetism, though they were unsuccessful.[1] In their 1938 paper, Einstein and Bergmann were among the first to introduce the modern viewpoint that a four-dimensional theory, which coincides with Einstein-Maxwell theory at long distances, is derived from a five-dimensional theory with complete symmetry in all five dimensions. They suggested that electromagnetism resulted from a gravitational field that is “polarized” in the fifth dimension.[4]
The main novelty of Einstein and Bergmann was to seriously consider the fifth dimension as a physical entity, rather than an excuse to combine the metric tensor and electromagnetic potential. But they then reneged, modifying the theory to break its five-dimensional symmetry. Their reasoning, as suggested by Edward Witten, was that the more symmetric version of the theory predicted the existence of a new long range field, one that was both massless and scalar, which would have required a fundamental modification to Einstein's theory of general relativity.[5] Minkowski space and Maxwell's equations in vacuum can be embedded in a five-dimensional Riemann curvature tensor.[citation needed]
In 1993, the physicist Gerard 't Hooft put forward the holographic principle, which explains that the information about an extra dimension is visible as a curvature in a spacetime with one fewer dimension. For example, holograms are three-dimensional pictures placed on a two-dimensional surface, which gives the image a curvature when the observer moves. Similarly, in general relativity, the fourth dimension is manifested in observable three dimensions as the curvature path of a moving infinitesimal (test) particle. 'T Hooft has speculated that the fifth dimension is really the spacetime fabric.[citation needed]
Five-dimensional geometryEdit
According to Klein's definition, "a geometry is the study of the invariant properties of a spacetime, under transformations within itself." Therefore, the geometry of the 5th dimension studies the invariant properties of such space-time, as we move within it, expressed in formal equations.[6]
Main article: 5-polytope
An important uniform 5-polytope is the 5-demicube, h{4,3,3,3} has half the vertices of the 5-cube (16), bounded by alternating 5-cell and 16-cell hypercells. The expanded or stericated 5-simplex is the vertex figure of the A5 lattice,
. It and has a doubled symmetry from its symmetric Coxeter diagram. The kissing number of the lattice, 30, is represented in its vertices.[7] The rectified 5-orthoplex is the vertex figure of the D5 lattice,
. Its 40 vertices represent the kissing number of the lattice and the highest for dimension 5.[8]
Regular and semiregular polytopes in five dimensions
Aut(A5)
r{3,3,3,4}
A hypersphere in 5-space (also called a 4-sphere due to its surface being 4-dimensional) consists of the set of all points in 5-space at a fixed distance r from a central point P. The hypervolume enclosed by this hypersurface is:
{\displaystyle V={\frac {8\pi ^{2}r^{5}}{15}}}
List of regular 5-polytopes
^ a b c d e f g Paul Halpern (April 3, 2014). "How Many Dimensions Does the Universe Really Have". Public Broadcasting Service. Retrieved September 12, 2015.
^ a b Oulette, Jennifer (March 6, 2011). "Black Holes on a String in the Fifth Dimension". Discovery News. Archived from the original on November 1, 2015. Retrieved September 12, 2015.
^ Boyle, Alan (June 6, 2006). "Physicists probe fifth dimension". NBC news. Retrieved September 12, 2015.
^ Einstein, Albert; Bergmann, Peter (1938). "On A Generalization Of Kaluza's Theory Of Electricity". Annals of Mathematics. 39 (3): 683–701. doi:10.2307/1968642. JSTOR 1968642.
^ Witten, Edward (January 31, 2014). "A Note On Einstein, Bergmann, and the Fifth Dimension". arXiv:1401.8048 [physics.hist-ph].
^ Sancho, Luis (October 4, 2011). Absolute Relativity: The 5th dimension (abridged). p. 442.
^ Sphere packings, lattices, and groups, by John Horton Conway, Neil James Alexander Sloane, Eiichi Bannai [1]
Wesson, Paul S. (1999). Space-Time-Matter, Modern Kaluza-Klein Theory. Singapore: World Scientific. ISBN 981-02-3588-7.
Wesson, Paul S. (2006). Five-Dimensional Physics: Classical and Quantum Consequences of Kaluza-Klein Cosmology. Singapore: World Scientific. ISBN 981-256-661-9.
Weyl, Hermann, Raum, Zeit, Materie, 1918. 5 edns. to 1922 ed. with notes by Jūrgen Ehlers, 1980. trans. 4th edn. Henry Brose, 1922 Space Time Matter, Methuen, rept. 1952 Dover. ISBN 0-486-60267-2.
Anaglyph of a five dimensional hypercube in hyper perspective
Retrieved from "https://en.wikipedia.org/w/index.php?title=Five-dimensional_space&oldid=1086102681"
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Amplitude phase shift keying (APSK) modulation - MATLAB apskmod - MathWorks í•œêµ
Apply APSK Modulation
Apply APSK Modulation with Phase Offset
Apply APSK Modulation Modifying Symbol Ordering
Apply APSK Modulation to Input Bits
Amplitude phase shift keying (APSK) modulation
y = apskmod(x,M,radii)
y = apskmod(x,M,radii,phaseoffset)
y = apskmod(___,Name,Value)
y = apskmod(x,M,radii) performs APSK modulation on the input data, x, based on the specified number of constellation points per PSK ring, M, and the radius of each PSK ring, radii. For a description of APSK modulation, see Algorithms.
apskmod specifically applies to multiple ring PSK constellations. For a single ring PSK constellation, use pskmod.
y = apskmod(x,M,radii,phaseoffset) specifies an initial phase offset for each PSK ring of the APSK modulated signal.
y = apskmod(___,Name,Value) specifies options using one or more name-value pair arguments using any of the previous syntaxes. For example, 'OutputDataType','double' specifies the desired output data type as double. Specify name-value pair arguments after all other input arguments.
Modulate data using APSK with an unequal number of constellation points on each circle.
Define vectors for modulation order and PSK ring radii. Generate data for constellation points.
M = [4 8 20];
radii = [0.3 0.7 1.2];
modOrder = sum(M);
x = 0:modOrder-1;
Apply APSK modulation to the data.
y = apskmod(x,M,radii);
Plot the resulting constellation using a scatter plot.
Modulate a random data sequence using APSK with zero phase offset for the inner circle and pi/6 phase offset for the outer circle.
Define vectors for modulation order, PSK ring radii, and PSK ring phase offset. Generate random data.
M = [8 8];
radii = [0.5 1];
phOff = [0 pi/6];
x = randi([0 modOrder-1],100,1);
y = apskmod(x,M,radii,phOff);
Plot the resulting constellation using a scatter plot and observe the phase offset between the constellation circles.
Plot APSK constellations for gray and custom symbol mappings.
Define vectors for modulation order and PSK ring radii. Generate bit data for constellation points.
radii = [0.5 1.5];
The apskmod function assumes the single channel binary input is left-msb aligned and specified column-wise. Reshape the bit matrix to express the single channel signal in a single column.
xBit = int2bit(x,log2(modOrder));
Apply APSK modulation to the data using the default phase offset. Since element values for M are equal and element values for phase offset are equal, the symbol mapping defaults to 'gray'. Binary input is used to highlight the Gray nature of the constellation mapping. Plot the constellation.
y = apskmod(xBit(:),M,radii,'PlotConstellation',true,'InputType','bit');
Create a custom symbol mapping vector. This custom mapping happens to be another Gray mapping.
cmap = [0;1;9;8;12;13;5;4;2;3;11;10;14;15;7;6];
Apply APSK modulation with a custom symbol mapping. Plot the constellation. Binary input is used to highlight that the custom mapping defines different Gray symbol mapping.
z = apskmod(xBit(:),M,radii,'SymbolMapping',cmap,'PlotConstellation',true,'InputType','bit');
Modulate a random bit sequence using APSK and output data type single. Pass the signal through a noisy channel and display the constellation diagram.
Define vectors for modulation order and PSK ring radii. Generate random binary data.
M = [8 12 20 24];
radii = [0.8 1.2 2 2.5];
bitsPerSym = log2(sum(M));
x = randi([0 1],2000*bitsPerSym,1);
Apply APSK modulation to the data and use a name-value pair to output as data type single.
y = apskmod(x,M,radii,'InputType','bit','OutputDataType','single');
Pass through an AWGN channel with a 25 dB SNR.
yrec = awgn(y,25,'measured');
Plot the received constellation as a scatter plot.
scatterplot(yrec)
Input signal, specified as a scalar, vector, or matrix. The elements of x must be binary values or integers in the range [0, (sum(M)-1)].
To process the input signal as binary elements, set the 'InputType' name-value pair to 'bit'. For binary inputs, the number of rows must be an integer multiple of log2(sum(M)). Groups of log2(sum(M)) bits in a column are mapped onto a symbol, with the first bit representing the MSB and the last bit representing the LSB.
M — Constellation points per PSK ring
Constellation points per PSK ring, specified as a vector with more than one element. Each vector element indicates the number of constellation points in its corresponding PSK ring. The first element corresponds to the innermost circle, and so on, until the last element, which corresponds to the outermost circle. Element values must be multiples of four and sum(M) must be a power of two. The modulation order is the total number of points in the signal constellation and equals the sum of the vector elements, sum(M).
Example: [4 12 16] specifies a three PSK ring constellation with a modulation order of sum(M) = 32.
radii — Radius per PSK ring
Radius per PSK ring, specified as a vector with the same length as M. The first element corresponds to the innermost circle, and so on, until the last element, which corresponds to the outermost circle. The elements must be positive and arranged in increasing order.
Example: [0.5 1 2] defines radii for three constellation PSK rings. The inner ring has a radius of 0.5, the second ring has a radius of 1.0, and the outer ring has a radius of 2.0.
phaseoffset — Phase offset per PSK ring
[pi/M(1) pi/M(2) … pi/M(end)] (default) | scalar | vector
Phase offset per PSK ring in radians, specified as a scalar or vector with the same length as M. The first element corresponds to the innermost circle, and so on, until the last element, which corresponds to the outermost circle. The phaseoffset can be a scalar only if all the elements of M are the same value.
Example: [pi/4 pi/12 pi/16] defines three constellation PSK ring phase offsets. The inner ring has a phase offset of pi/4, the second ring has a phase offset of pi/12, and the outer ring has a phase offset of pi/16.
Example: y = apskmod(x,M,radii,'InputType','bit','OutputDataType','single');
SymbolMapping — Symbol mapping
'gray' | 'contourwise-gray' | integer vector
Symbol mapping, specified as the comma-separated pair consisting of 'SymbolMapping' and one of the following:
'contourwise-gray' — Uses Gray mapping along the contour in the phase dimension for each PSK ring.
'gray' — Uses Gray mapping along the contour in both the amplitude and phase dimensions. For Gray symbol mapping, all the values for M must be equal and all the values for phaseoffset must be equal. For a description of the Gray mapping used, see [2].
integer vector — Use custom symbol mapping. Vector must consist of sum(M) unique elements with values in the range [0, (sum(M)-1]. The first element corresponds to the constellation point in the first quadrant of the innermost circle, with subsequent elements positioned counterclockwise around the PSK rings.
The default symbol mapping depends on M and phaseOffset. When all the elements of M are equal and all the elements of phaseOffset are equal, the default is 'gray'. For all other cases, the default is 'contourwise-gray'.
Input type, specified as the comma-separated pair consisting of 'InputType' and either of these options:
'integer' –– The input signal must consist of integers in the range [0, (sum(M) – 1)].
'bit' –– The input signal must contain binary values, and the number of rows must be an integer multiple of log2(sum(M)). Binary input signals are assumed to be left-MSB aligned and specified column-wise. Groups of log2(sum(M)) bits in a column are mapped onto a symbol, with the first bit representing the MSB and the last bit representing the LSB.
Output data type, specified as the comma-separated pair consisting of 'OutputDataType' and either 'double' or 'single'.
PlotConstellation — Plot reference constellation
Plot reference constellation, specified as the comma-separated pair consisting of 'PlotConstellation' and a logical scalar. To plot the reference constellation, set PlotConstellation to true.
y — APSK modulated signal
APSK modulated signal, returned as a complex scalar, vector, or matrix. The dimensions of y depend on the specified 'InputType' value.
Dimensions of y
'integer' y has the same dimensions as input x.
'bit' The number of rows in y equals the number of rows in x divided by log2(sum(M)).
The function implements a pure APSK constellation.
\mathrm{Ï}=\left\{\begin{array}{cc}{R}_{1}\mathrm{exp}\left(j\left(\frac{2\mathrm{Ï}}{{M}_{1}}i+{\mathrm{θ}}_{1}\right)\right),& i=0,â¦,{M}_{1}â1,\\ {R}_{2}\mathrm{exp}\left(j\left(\frac{2\mathrm{Ï}}{{M}_{2}}i+{\mathrm{θ}}_{2}\right)\right),& i=0,â¦,{M}_{2}â1,\\ ⋮& ⋮\\ {R}_{{N}_{\text{C}}}\mathrm{exp}\left(j\left(\frac{2\mathrm{Ï}}{{M}_{{N}_{\text{C}}}}i+{\mathrm{θ}}_{\text{Nc}}\right)\right),& i=0,â¦,{M}_{{N}_{\text{C}}}â1,\end{array}
NC is the number of concentric rings. NC ≥ 2.
θl is the phase offset of the lth ring.
j=\sqrt{â1}
apskdemod | dvbsapskmod | mil188qammod | pskmod | qammod
comm.GeneralQAMModulator | comm.PSKModulator
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Decay estimates of solutions for dissipative wave equations in RN with lower power nonlinearities
April, 2004 Decay estimates of solutions for dissipative wave equations in
{R}^{N}
with lower power nonlinearities
Ryo IKEHATA, Yasuaki MIYAOKA, Takashi NAKATAKE
Optimal energy decay estimates will be derived for weak solutions to the Cauchy problem in
{R}^{N}\left(N=1,2,3\right)
of dissipative wave equations, which have lower power nonlinearities
|u{|}^{p-1}u
1+2/N<p\le N/\left[N-2{\right]}^{+}
Ryo IKEHATA. Yasuaki MIYAOKA. Takashi NAKATAKE. "Decay estimates of solutions for dissipative wave equations in
{R}^{N}
with lower power nonlinearities." J. Math. Soc. Japan 56 (2) 365 - 373, April, 2004. https://doi.org/10.2969/jmsj/1191418635
Keywords: Cauchy problem , Critical exponent , dissipation , optimal decay estimates , semilinear wave equation
Ryo IKEHATA, Yasuaki MIYAOKA, Takashi NAKATAKE "Decay estimates of solutions for dissipative wave equations in
{R}^{N}
with lower power nonlinearities," Journal of the Mathematical Society of Japan, J. Math. Soc. Japan 56(2), 365-373, (April, 2004)
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Reflection and transmission coefficients - SEG Wiki
Revision as of 17:08, 7 November 2019 by Ageary (talk | contribs) (remove)
1 Problem 3.7a
2 Problem 3.7b
Calculate the reflection and transmission coefficients,
{\displaystyle R}
{\displaystyle T}
{\displaystyle V_{ss}=2.43}
{\displaystyle V_{sh}=2.02}
{\displaystyle \rho _{ss}=2.08}
{\displaystyle ^{3}}
{\displaystyle \rho _{sh}=2.23}
{\displaystyle ^{3}}
{\displaystyle V_{ss}=3.35}
{\displaystyle V_{sh}=3.14}
{\displaystyle \rho _{ss}=2.21}
{\displaystyle ^{3}}
{\displaystyle \rho _{sh}=2.52}
{\displaystyle ^{3}}
{\displaystyle \psi =A\sin \omega t}
{\displaystyle \partial \psi /\partial t=\omega A\cos \omega t}
{\displaystyle {\frac {1}{2}}\rho \omega ^{2}A^{2}\cos ^{2}\omega t}
{\displaystyle \rho }
{\displaystyle \cos ^{2}\omega t=+1}
{\displaystyle {}=E={\frac {1}{2}}\rho \omega ^{2}A^{2}}
{\displaystyle E}
{\displaystyle R}
{\displaystyle T}
{\displaystyle E_{R}}
{\displaystyle E_{T}}
{\displaystyle \left({\frac {1}{2}}\rho \omega ^{2}A^{2}\right)\alpha }
{\displaystyle {\begin{aligned}E_{R}=R^{2},\qquad E_{T}={\frac {\rho _{2}\alpha _{2}\omega ^{2}A_{2}^{2}}{\rho _{1}\alpha _{1}\omega ^{2}A_{0}^{2}}}=\left({\frac {Z_{2}}{Z_{1}}}\right)T^{2}={\frac {4Z_{1}Z_{2}}{(Z_{1}+Z_{2})^{2}}}.\end{aligned}}}
{\displaystyle E_{R}}
{\displaystyle E_{T}}
{\displaystyle R}
{\displaystyle T}
{\displaystyle E_{R}}
{\displaystyle E_{T}}
{\displaystyle {\begin{aligned}Z_{ss}=2.08\times 2.43=5.05,\qquad Z_{sh}=2.23\times 2.02=4.50\end{aligned}}}
{\displaystyle {\begin{aligned}R=\left(4.50-5.05\right)/\left(4.50+5.05\right)=-0.55/9.55=-0.058.\end{aligned}}}
{\displaystyle {\begin{aligned}T=2\times 5.05/9.55=1.06.\end{aligned}}}
{\displaystyle R}
{\displaystyle Z_{ss}=2.21\times 3.35=7.40,\ Z_{sh}=2.52\times 3.14=7.91}
{\displaystyle {\begin{aligned}R&=\left(7.91-7.40\right)/\left(7.91+7.40\right)=0.51/15.3=0.033,\\T&=2\times 7.40/15.3=0.967.\end{aligned}}}
{\displaystyle {}=8.686}
{\displaystyle {\begin{aligned}R&=\ln \left(0.058\right)=-2.8\ \mathrm {nepers} =-24\ \mathrm {dB} ,\\T&=\ln \left(1.06\right)=0.058\ \mathrm {nepers} \ =0.51\ \mathrm {dB} .\end{aligned}}}
{\displaystyle {\begin{aligned}R&=\ln \left(0.033\right)=-3.4\ \mathrm {nepers} \ =-30\ \mathrm {dB} ,\\T&=\ln \left(0.967\right)=-0.034\ \mathrm {nepers} \ =-0.29\ \mathrm {dB} .\end{aligned}}}
{\displaystyle dB}
Calculate the energy coefficients
{\displaystyle E_{R}}
{\displaystyle E_{T}}
{\displaystyle {\begin{aligned}E_{R}&=R^{2}=(0.058)^{2}=0.0034,\\E_{T}&=\left(Z_{2}/Z_{1}\right)T^{2}=\left(4.50/5.05\right)\times 1.06^{2}=1.00.\end{aligned}}}
{\displaystyle {\begin{aligned}E_{R}&=0.033^{2}=0.001,\\E_{T}&=(\left(7.91/7.40\right)\times 0.967^{2}=1.00.\end{aligned}}}
{\displaystyle E_{R}+E_{T}=1}
Complex coefficient of reflection Amplitude/energy of reflections and multiples
Theory of Seismic Waves Geometry of seismic waves
Reflection/refraction at a solid/solid interface and displacement of a free surface
Complex coefficient of reflection
Reflection/transmission coefficients at small angles and magnitude
Variation of reflectivity with angle (AVA)
Retrieved from "https://wiki.seg.org/index.php?title=Reflection_and_transmission_coefficients&oldid=141071"
Partitioning at an interface
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EUDML | -matrices and Lyapunov scalar stability. EuDML | -matrices and Lyapunov scalar stability.
{P}^{\alpha }
-matrices and Lyapunov scalar stability.
Hershkowitz, Daniel; Mashal, Nira
Hershkowitz, Daniel, and Mashal, Nira. "-matrices and Lyapunov scalar stability.." ELA. The Electronic Journal of Linear Algebra [electronic only] 4 (1998): 39-47. <http://eudml.org/doc/119734>.
@article{Hershkowitz1998,
author = {Hershkowitz, Daniel, Mashal, Nira},
keywords = {inequalities involving eigenvalues; positive definite matrices; -stability; -matrices; Lyapunov diagonal stability; -stability; -matrices},
title = {-matrices and Lyapunov scalar stability.},
AU - Hershkowitz, Daniel
TI - -matrices and Lyapunov scalar stability.
KW - inequalities involving eigenvalues; positive definite matrices; -stability; -matrices; Lyapunov diagonal stability; -stability; -matrices
inequalities involving eigenvalues, positive definite matrices,
H
-stability,
P
-matrices, Lyapunov diagonal stability,
H
P
Articles by Hershkowitz
Articles by Mashal
|
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