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Avogadro's law (sometimes referred to as Avogadro's hypothesis or Avogadro's principle) or Avogadro-Ampère's hypothesis is an experimental gas law relating the volume of a gas to the amount of substance of gas present.[1] The law is a specific case of the ideal gas law. A modern statement is:
Avogadro's law states that "equal volumes of all gases, at the same temperature and pressure, have the same number of molecules."[1]
The law is named after Amedeo Avogadro who, in 1812,[2][3] hypothesized that two given samples of an ideal gas, of the same volume and at the same temperature and pressure, contain the same number of molecules. As an example, equal volumes of gaseous hydrogen and nitrogen contain the same number of atoms when they are at the same temperature and pressure, and observe ideal gas behavior. In practice, real gases show small deviations from the ideal behavior and the law holds only approximately, but is still a useful approximation for scientists.
1.1 Derivation from the ideal gas law
2 Historical account and influence
2.2 Avogadro constant
3 Molar volume
The law can be written as:
{\displaystyle V\propto n}
{\displaystyle {\frac {V}{n}}=k}
n is the amount of substance of the gas (measured in moles);
k is a constant for a given temperature and pressure.
{\displaystyle {\frac {V_{1}}{n_{1}}}={\frac {V_{2}}{n_{2}}}}
The equation shows that, as the number of moles of gas increases, the volume of the gas also increases in proportion. Similarly, if the number of moles of gas is decreased, then the volume also decreases. Thus, the number of molecules or atoms in a specific volume of ideal gas is independent of their size or the molar mass of the gas.
Derivation from the ideal gas lawEdit
The derivation of Avogadro's law follows directly from the ideal gas law, i.e.
{\displaystyle PV=nRT,}
where R is the gas constant, T is the Kelvin temperature, and P is the pressure (in pascals).
Solving for V/n, we thus obtain
{\displaystyle {\frac {V}{n}}={\frac {RT}{P}}.}
{\displaystyle k={\frac {RT}{P}}}
which is a constant for a fixed pressure and a fixed temperature.
An equivalent formulation of the ideal gas law can be written using Boltzmann constant kB, as
{\displaystyle PV=Nk_{\text{B}}T,}
where N is the number of particles in the gas, and the ratio of R over kB is equal to the Avogadro constant.
{\displaystyle {\frac {V}{N}}=k'={\frac {k_{\text{B}}T}{P}}.}
Historical account and influenceEdit
Avogadro's hypothesis (as it was known originally) was formulated in the same spirit of earlier empirical gas laws like Boyle's law (1662), Charles's law (1787) and Gay-Lussac's law (1808). The hypothesis was first published by Amadeo Avogadro in 1811,[4] and it reconciled Dalton atomic theory with the "incompatible" idea of Joseph Louis Gay-Lussac that some gases were composite of different fundamental substances (molecules) in integer proportions.[5] In 1814, independently from Avogadro, André-Marie Ampère published the same law with similar conclusions.[6] As Ampère was more well known in France, the hypothesis was usually referred there as Ampère's hypothesis,[note 1] and later also as Avogadro–Ampère hypothesis[note 2] or even Ampère–Avogadro hypothesis.[7]
Experimental studies carried out by Charles Frédéric Gerhardt and Auguste Laurent on organic chemistry demonstrated that Avogadro's law explained why the same quantities of molecules in a gas have the same volume. Nevertheless, related experiments with some inorganic substances showed seeming exceptions to the law. This apparent contradiction was finally resolved by Stanislao Cannizzaro, as announced at Karlsruhe Congress in 1860, four years after Avogadro's death. He explained that these exceptions were due to molecular dissociations at certain temperatures, and that Avogadro's law determined not only molecular masses, but atomic masses as well.
Boyle, Charles and Gay-Lussac laws, together with Avogadro's law, were combined by Émile Clapeyron in 1834,[8] giving rise to the ideal gas law. At the end of the 19th century, later developments from scientists like August Krönig, Rudolf Clausius, James Clerk Maxwell and Ludwig Boltzmann, gave rise to the kinetic theory of gases, a microscopic theory from which the ideal gas law can be derived as an statistical result from the movement of atoms/molecules in a gas.
Avogadro constantEdit
Avogadro's law provides a way to calculate the quantity of gas in a receptacle. Thanks to this discovery, Johann Josef Loschmidt, in 1865, was able for the first time to estimate the size of a molecule.[9] His calculation gave rise to the concept of the Loschmidt constant, a ratio between macroscopic and atomic quantities. In 1910, Millikan's oil drop experiment determined the charge of the electron; using it with the Faraday constant (derived by Michael Faraday in 1834), one is able to determine the number of particles in a mole of substance. At the same time, precision experiments by Jean Baptiste Perrin led to the definition of Avogadro's number as the number of molecules in one gram-molecule of oxygen. Perrin named the number to honor Avogadro for his discovery of the namesake law. Later standardization of the International System of Units led to the modern definition of the Avogadro constant.
Molar volumeEdit
Main article: Molar volume
{\displaystyle V_{\rm {m}}={\frac {V}{n}}={\frac {RT}{P}}\approx {\frac {8.314{\text{ J}}{\text{ mol}}^{-1}{\text{ K}}^{-1}\times 273.15{\text{ K}}}{101.325{\text{ kPa}}}}\approx 22.41{\text{ dm}}^{3}{\text{ mol}}^{-1}=22.41{\text{ litres/mol}}}
For 101.325 kPa and 273.15 K, the molar volume of an ideal gas is 22.4127 dm3mol−1.
^ First used by Jean-Baptiste Dumas in 1826.
^ First used by Stanislao Cannizzaro in 1858.
^ a b Editors of the Encyclopædia Britannica. "Avogadro's law". Encyclopædia Britannica. Retrieved 3 February 2016. {{cite web}}: |last= has generic name (help)
^ Avogadro, Amedeo (1810). "Essai d'une manière de déterminer les masses relatives des molécules élémentaires des corps, et les proportions selon lesquelles elles entrent dans ces combinaisons". Journal de Physique. 73: 58–76. English translation
^ "Avogadro's law". Merriam-Webster Medical Dictionary. Retrieved 3 February 2016.
^ Avogadro, Amadeo (July 1811). "Essai d'une maniere de determiner les masses relatives des molecules elementaires des corps, et les proportions selon lesquelles elles entrent dans ces combinaisons". Journal de Physique, de Chimie, et d'Histoire Naturelle (in French). 73: 58–76.
^ Rovnyak, David. "Avogadro's Hypothesis". Science World Wolfram. Retrieved 3 February 2016.
^ Ampère, André-Marie (1814). "Lettre de M. Ampère à M. le comte Berthollet sur la détermination des proportions dans lesquelles les corps se combinent d'après le nombre et la disposition respective des molécules dont les parties intégrantes sont composées". Annales de Chimie (in French). 90 (1): 43–86.
^ Scheidecker-Chevallier, Myriam (1997). "L'hypothèse d'Avogadro (1811) et d'Ampère (1814): la distinction atome/molécule et la théorie de la combinaison chimique". Revue d'Histoire des Sciences (in French). 50 (1/2): 159–194. doi:10.3406/rhs.1997.1277. JSTOR 23633274.
^ Clapeyron, Émile (1834). "Mémoire sur la puissance motrice de la chaleur". Journal de l'École Polytechnique (in French). XIV: 153–190.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Avogadro%27s_law&oldid=1079844084"
|
\le 1
{\aleph }_{1}
C
-continuum
x
is metrizable if and only if it admits a Whitney map for
C\left(x\right)
A chainable continuum not homeomorphic to an inverse limit on [0, 1] with only one bonding map
Dorothy S. Marsh (1980)
A characterization of dendroids by the n-connectedness of the Whitney levels
Alejandro Illanes (1992)
Let X be a continuum. Let C(X) denote the hyperspace of all subcontinua of X. In this paper we prove that the following assertions are equivalent: (a) X is a dendroid, (b) each positive Whitney level in C(X) is 2-connected, and (c) each positive Whitney level in C(X) is ∞-connected (n-connected for each n ≥ 0).
T. Maćkowiak (1988)
A characterization of hereditarily decomposable snake-like continua
{E}^{2}
A characterization of the arc by means of the C-index of itssemigroup.
K.D. jr. Magill (1993)
A class of continua that are not attractors of any IFS
Marcin Kulczycki, Magdalena Nowak (2012)
This paper presents a sufficient condition for a continuum in ℝn to be embeddable in ℝn in such a way that its image is not an attractor of any iterated function system. An example of a continuum in ℝ2 that is not an attractor of any weak iterated function system is also given.
A class α and locally connected continua which can be ε-mapped onto a surface
B. B. Epps (1976)
A classification of continua by certain cutting properties
E. Thomas (1967)
A classification of inverse limit spaces of tent maps with periodic critical points
We work within the one-parameter family of symmetric tent maps, where the slope is the parameter. Given two such tent maps
{f}_{a}
{f}_{b}
with periodic critical points, we show that the inverse limit spaces
{\left(}_{a},{f}_{a}\right)
{\left(}_{b},{g}_{b}\right)
are not homeomorphic when a ≠ b. To obtain our result, we define topological substructures of a composant, called “wrapping points” and “gaps”, and identify properties of these substructures preserved under a homeomorphism.
Charatonik, Janusz J., Charatonik, Włodzimierz J. (2000)
A condition under which 2-homogeneity and representability are the same in continua
Judy Kennedy Phelps (1984)
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MCQs of Differential Equations | GUJCET MCQ
MCQs of Differential Equations
the order of a differential equation whose general solution is y =Asinx + Bcosx is _____(A, B are arbitrary constants.)
\mathrm{the} \mathrm{order} \mathrm{and} \mathrm{degree} \mathrm{of} {\left(\frac{{d}^{3}y}{d{x}^{3}}\right)}^{2} + {\left(\frac{{d}^{2}y}{d{x}^{2}}\right)}^{3} + y = 0 \mathrm{are} _____ \mathrm{respectively}.
(c) 3, not defined
y\text{'} + y = \frac{5}{y\text{'}} \mathrm{has} \mathrm{degree} _____ .
\mathrm{the} \mathrm{differential} \mathrm{equation} \frac{dy}{dx} = - \frac{x+y}{1+{x}^{2}} \mathrm{is} ______.
(a) of variable separable form
(d) of second order
f \left(x, y\right) = \frac{{x}^{3}-{y}^{3}}{x+y} \mathrm{is} \mathrm{a} \mathrm{homogeneous} \mathrm{function} \mathrm{of} \mathrm{degree} _____ .
\mathrm{An} \mathrm{integrating} \mathrm{factor} \mathrm{of} \mathrm{differential} \mathrm{equation} \frac{dy}{dx} = \frac{1}{x+y+2} \mathrm{is} _____ .
{e}^{x}
{e}^{x +y + 2}
{e}^{-y}
\mathrm{log} \left|x+y+2\right|
the differential equation of the family of rectangular hyperbolas is _____ .
{y}_{2}=0
xy+{y}_{2}=0
{\mathrm{yy}}_{1}=\mathrm{x}
{\mathrm{xy}}_{1}+\mathrm{y}=0
\mathrm{the} \mathrm{order} \mathrm{and} \mathrm{the} \mathrm{degree} \mathrm{of} \mathrm{the} \mathrm{differential} \mathrm{equation} \frac{dy}{dx}+{x}^{2}\frac{{d}^{2}y}{d{x}^{2}}+xy sinx, \mathrm{are} _____ \mathrm{respectively}.
\mathrm{Which} \mathrm{of} \mathrm{the} \mathrm{following} \mathrm{function} \mathrm{is} \mathrm{a} \mathrm{solution} \mathrm{of} \mathrm{the} \mathrm{differential} \mathrm{equation} {\left(\frac{dy}{d}\right)}^{2}-x\frac{dy}{dx}+y=0 ?
y=4x
y=4
y=2{x}^{2}+4
y=2x-4
\mathrm{solution} \mathrm{of} \mathrm{the} \mathrm{differential} \mathrm{equation} \mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{y}=0 \mathrm{is} _____ .
{e}^{xy}=c
y=cx
\mathrm{x}=\mathrm{cy}
{e}^{x}y=c
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Discussion: “New First-Order Shear Deformation Plate Theories” (Shimpi, R. P., Patel, H. G., and Arya, H., 2007, ASME J. Appl. Mech., 74, pp. 523–533) | J. Appl. Mech. | ASME Digital Collection
Discussion: “New First-Order Shear Deformation Plate Theories” (Shimpi, R. P., Patel, H. G., and Arya, H., 2007, ASME J. Appl. Mech., 74, pp. 523–533)
e-mail: jgs@virginia.edu
A commentary has been published: Closure to “Discussion of ‘New First-Order Shear Deformation Plate Theories’ ” (2008, ASME J. Appl. Mech., 75, p. 045503)
This is a companion to: New First-Order Shear Deformation Plate Theories
Simmonds, J. G. (May 20, 2008). "Discussion: “New First-Order Shear Deformation Plate Theories” (Shimpi, R. P., Patel, H. G., and Arya, H., 2007, ASME J. Appl. Mech., 74, pp. 523–533)." ASME. J. Appl. Mech. July 2008; 75(4): 045503. https://doi.org/10.1115/1.2916894
plates (structures), shear deformation
Plate theory, Shear deformation, Plates (structures)
This paper joins a host of others, beginning with the seminal papers of Reissner (1,2), that attempt to improve the accuracy of classical (Kirchhoff) plate theory without a concomitant refinement of the classical boundary conditions—a refinement that necessitates using the equations of three-dimensional elasticity to examine edge layers whose thicknesses are of the order of the plate thickness. Without such a refinement, improvements to Kirchhoff’s theory are, in general, illusory, as many authors over the past
50years
have emphasized, especially Goldenveiser. See, for example, Refs. 3,4,5,6,7,8,9, where many other relevant references will be found.
Often, authors of “improved” plate theories compare solutions of their equations under simple support either to other theories or to exact three-dimensional elasticity solutions. However, because such solutions are mathematically equivalent to those of an infinite plate under periodic surface loads, no edge layers arise so that such comparisons are virtually useless.
On the Theory of Bending of Elastic Plates
J. Math. Phys. (Cambridge, Mass.)
The Effect of Transverse-Shear Deformation on the Bending of Elastic Plates
Internal and Boundary Calculations of Thin Elastic Bodies
Stress Boundary Conditions for Plate Bending
Précastaings
The Optimal Version of Reissner’s Theory
The Interior Solution for Linear Problems of Elastic Plates
Local Effects in the Analysis of Structures
Recent Advances in Shell Theory
Proceedings 13th Annual Meeting Society of Engineering Science
, NASA CP-2001, pp.
Impact Forces of Sliding Vanes on Rotor Slots in a Rotary Compressor
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Debt repayment - How? - Mai Finance - Tutorials
This page will present different options to repay your debt on Mai Finance. Keep in mind that repaying a debt is never mandatory as long as you want to keep your loan, and don't need your collateral.
The market is on a bull run and your crypto, locked in the Vault, is gaining more and more value, so much that you decided to sell it. However, because it's in the Vault on Mai Finance, you can't totally unlock it unless you repay your loan.
The market is bearish, and your crypto is losing value very quickly. You don't generate yield fast enough to cover up the damages and keep an healthy Collateral to Debt Ratio (CDR), and liquidation is near. It's time to repay your debt to make sure you're not losing too much, and prevent liquidation.
If you are not in any of the above situation, it's probably not worth repaying your debt. Please see the chapter on Debt Repayment for more details.
Partial or Full repayment using fiat
The most direct way to repay your debt is to use some fiat, especially if you want to keep your collateral and other investments untouched.
Mai Finance is partnering with Transak to easily bridge money bought by Credit / Debit Cards, or bank transfers, directly to the Polygon network. Simply head to Mai Finance and click the transak icon in the menu bar to open a modal that will let you purchase some MATIC and send them directly to your Polygon wallet.
Buying some USDC from fiat and bridging to Polygon directly
The main issue is the time taken to process the transaction. However, doing so will let you swap USDC for MAI and then partially or fully repay your debt.
Repayment using the benefits of your loan
Most people will want to borrow MAI on Mai Finance in order to invest into specific projects. Yield farmers that are using MAI will most probably be successful generating additional resources and will hopefully not lose money on degen farms. If you're in that case, you have 2 options
repay your loan with the generated revenue
re-invest your gains into the same (or another) project
In most cases, it's probably better to re-invest your gains. Indeed, by compounding your rewards, APR (Annual Percentage Revenue) is applied on a bigger amount, which in turns generates more revenue. See our investment guides to get ideas on how you can maximize your investments.
However, some people simply don't like the idea of having a debt, and will want to repay it as quickly as possible. If that's your case, you can simply swap your gains into MAI, and repay your debt.
Open your Vault info
Select the Manage option
Select the Repay tab at the bottom of your Vault
Enter any amount you want to repay
Click Repay MAI and you're done
Partially repaying a portion of my debt
You have $1,000.00 worth of camWMATIC in your vault, with a debt of $400.00
You swapped $10.00 worth of ADDY tokens for MAI
You repay $10.00 today
Your debt is now $390.00
Your camWMATIC value is $999.95 (you add to pay 0.5% of $10.00 in repayment fees)
At some point, you will be able to totally repay your debt using this technique, as long as you can generate enough revenue via your loan.
Is this strategy efficient? Not really. Repaying your debt this way doesn't change anything except for your CDR. Indeed, your collateral remains partially locked until you totally repay your debt, and if you're using amTokens as collateral, whether you have a debt or not will not change the fact that your collateral is generating yield. The only advantage is that you can possibly withdraw a portion of your collateral and re-use it somewhere else / sell it.
Another approach, very similar to the above strategy, is to repay everything at once. In the example above, instead of repaying $10.00 every few days, I could instead compound the $10.00 earned into my investment to generate yield faster. I can also invest these gains into another project that would also generate revenue. Once I received the equivalent of $400.00 that corresponds to my loan, I can repay everything at once.
Repayment using your collateral
On Mai Finance, you can borrow MAI stable coins by depositing a certain amount of collateral in a vault. The collateral to debt ratio needs to always be above a certain threshold, 150% for most vaults. This means, for a 150% CDR, that for any $100 worth of collateral, you can only borrow $66.6667 of MAI.
However, this would directly put you in a liquidation position. This means that the health of your vault is considered too risky, and anyone can repay a portion of your debt using their funds and get paid back by getting a portion of your collateral. For more details about liquidation, please read the official documentation.
It's usually considered best practice to keep a high collateral to debt ratio to prevent liquidation, but even with a CDR close to 150%, it's easy to see that the value of the collateral is ALWAYS bigger than the value of the debt. This means that you can, in theory, repay your debt by selling some of your collateral asset.
Let's consider a vault with $1,000.00 worth of MATIC and a $500.00 debt. The CDR is 200%. The minimum CDR is 150%. In this example, we want to repay the debt completely, but we want to avoid liquidation, and we will try to never go bellow a 160% CDR when withdrawing our collaterals. We will be using the following formulas:
CDR=\frac{Collateral}{Debt}
AvailableCollateral = InitialCollateral - TargetCDR*Debt
In this situation, if we want to keep a CDR of 160%, the amount of collateral we can withdraw "safely" is
Hence, we will have to proceed in multiple steps:
Withdraw $200 from the collateral
the vault now has $800 worth of MATIC and $500 of debt, CDR is 160%
Sell the $200 worth of collateral to buy MAI
Repay $200 of the debt with a 0.5% repayment fee
the vault now has $799 worth of MATIC and $300 of debt, CDR is 266.33%
Calculate the new amount of collateral we can withdraw: $319
the vault now has $480 of MATIC and $300 of debt, CDR is 160%
the vault has $478.50 worth of MATIC and $0 of debt, and you still have $19 of MAI
You can see that keeping a healthy CDR can greatly help you repay your debt with very little number of loops. Of course, if your CDR is closer to the 150% limit, you may have to operate more loops since you cannot withdraw as much at once.
A collateral to debt ratio of 260% is enough to be able to withdraw the total amount of your debt and stay above 160% CDR. This way, you only need one loop to fully repay your debt.
Note that fully repaying your debt by selling your collateral is never necessary if you don't need to sell your underlying assets, or to modify your CDR and keep your vault from being liquidated.
Repayment using a robot
This paragraph is pure theory and is there only for advanced programmers. The idea is to use flash loans that will help you repay your debt and unlock the collateral so that it can be paid. Flash loans are an option proposed by some applications on different networks, including Polygon, that allow you to borrow funds and repay the loan in the same transaction block. If the loan cannot be fully repaid within the same block, the transaction is simply reverted. On Polygon, AAVE is proposing flash loans.
If we take the example above with $1,000.00 worth of MATIC and a debt of $500.00. The flow would be as follows:
Borrow $600.00 of USDC on AAVE in a flash loan
Swap the USDC for MAI
Repay your debt completely
Withdraw your MATIC collateral
Sell your MATIC for USDC
Repay the AAVE flash loan
When submitted, this list of transactions will all happen in the same block, and you will end up with whatever is left from your MATIC as USDC in your wallet (more or less $500.00, with some slight variations due to flash loan interest rate, swap fees, and repayment fees).
Right now, you would have to interact directly with the smart contracts, which requires some good understanding of how they work. If you need help, you can find some on our Discord server where there's a programming channel. Maybe in a near future, FuruCombo will propose Mai Finance bricks that would allow you to operate this directly using their graphic tool, but for now it's not possible. Finally, the idea of a button to "repay debt using collateral" has been proposed to the team of devs of Mai Finance, and the option may be implemented in the future.
Short term VS Long term Debt Repayments
Depending on your strategy and the way you feel regarding your debt, it may be a good idea to compare different lending platforms. However, keep in mind that Mai Finance with its 0% interest and 0.5% repayment fee is one of the top product on the Polygon market. The real competitor is AAVE, but only if you want to borrow MAI or USDC for a short period of time.
Mais is 0% interest + 0.5% repayment fee
AAVE has no repayment fees, but a variable APR for interests you need to pay back
Supplying and Borrowing APY on AAVE as of August 2021
As an example for USDC, you can see that the borrowing rate is 3.79% with a current reward of 2.08% paid back in MATIC. This gives, at the moment of writing, the equivalent of 1.71% you need to pay back if you keep your loan for a complete year. With AAVE, since you can repay your debt very quickly, the variable APY is equivalent to 0.005% daily. Hence, it would take 100 days (a bit more than 3 months) to reach 0.5% of your debt.
If you plan to keep your loan longer than that, it's definitely better to use Mai Finance. Also, it's important to understand that AAVE borrowing APRs are variable, they will fluctuate with the amounts that are deposited and required (the more people want to borrow from AAVE, the higher the borrowing APR). Keep also in mind that the MATIC reward program will end at some point, and the 1.71% interest will soon become a 3.79% interest rate. At least with Mai Finance, you don't have to keep a close eye on your loan to see when it becomes dangerous to keep it.
Finally, the team of Mai Finance is working on Vaults incentives that would work the same way as the MATIC reward, meaning that you would still get a 0% interest loan and a bonus paid in Qi that may very well reduce the repayment fee to 0% of your debt. And the longer you keep your loan, the more reward you will collect, making it a true negative interest loan.
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Ask Answer - Structure of Atom - Expert Answered Questions for School Students
Violet light produces high energy ? Am i right ?
cathode rays have---
(b) charge only
(c)no mass no charge
(d) mass and charge both
Orbital angular momentum depends on ??
Q4. Which of the following statement is not correct about the characteristics of cathode rays?
(iii) Characteristics of cathode rays do not depend upon the material of electrodes in
(iv) Characteristics of cathode rays depend upon the nature of gas present in the cathode ray
WHAT IS THE VALUE OF RYDBERG'S CONSTANT?? SOMEWHERE IT IS 109677 AND 2.18*10^-18 J...PLZ HELP ME FOR THIS CONFUSION
calculate the wavelength of a particle of mass 3.1*10^-31kg that is moving with a speed of 2.21*10^7 m/s.(h=6.63*10^-34Js).
Saikrishna Giri
Q 5. In photo electric effect experiment, irradiation of metal with a light of frequency 5.2 x 1014 s-1 yields an electron with maximum kinetic energy 1.3 x 10-19 J. Calculate the threshold frequency
Q 6. The mass of an electron is 9.1 x 10-31 kg. If its kinetic energy is 3.0 x 10-25 J, calculate its wavelength.
Q 7. Calculate the wavelength of the radiation emitted producing a line in Lyman series when an electron falls from fourth stationary state in hydrogen spectrum. (RH = 1.1 x 107 )
Q 8. What are the frequency and the wavelength of a photon emitted during a transition from n = 5 state to n = 2 state in the hydrogen atom?
Q 9. Calculate the wave number for the longest wavelength transition in the Balmer series of atomic hydrogen,
Q 10. a) What is the relationship between wavelength and momentum?
b) Calculate de-Broglie wavelength associated with alpha-particle of energy 7.7 x 10-13 J and a mass of 6.6 x 10-24 g.
Q 11. Explain Heisenberg uncertainty principle. On the basis of the above principle, show that electron cannot exist inside the atomic nucleus,
Q. Please answer the below question.
5. In photo electric effect experiment, irradiation or metal with a light of frequency 5.2 x 1014 s-1 yields an electron with maximum kinetic energy 1.3 x 10-19 J. Calculate the threshold frequency.
6.The mass Of an electron is x 9.1 x 10-31 kg If its kinetic energy is 3.0 x 10-25 J calculate its wavelength
7. Calculate the wavelength of the radiation emitted a line in Lyman series when an electron falls from fourth stationary State in hydrogen spectrum. (RH= 1.1 x 107)
Q.9. Calculate the wave number for the longest wavelength transition in the Balmer series of atomic hydrogen.
2. Calculate the energy of a photon of red light having wavelength 650 nm,
3. Calculate the wavelength of the spectral lines of Balmer series when n =3,
4. The energy of the electron in the second and the third orbit of Hydrogen atom is -5.42 x 10-12 ergs. and -2.41 x 10-12 ergs, Calculate the wavelength of the emitted radiation when electron drops from 3rd to 2nd orbit.
Q. Please answer the below question..
\mathbf{5}\mathbf{.} \mathrm{In} \mathrm{photo} \mathrm{electric} \mathrm{effect} \mathrm{experiment}, \mathrm{irradiation} \mathrm{of} \mathrm{metal} \mathrm{with} \mathrm{a} \mathrm{light} \mathrm{of} \mathrm{frequency} 5.2 × {10}^{14} {\mathrm{s}}^{-1} \mathrm{yields} \mathrm{an} \mathrm{electron} \mathrm{with} \mathrm{maximum} \mathrm{kinetic} \mathrm{energy} 1.3 × {10}^{-19} \mathrm{J}. \mathrm{Calculate} \mathrm{the} \mathrm{threshold} \mathrm{frequency}.
plz derive how wavelength = h/mv.
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Background - CS Calculator 2.0 | Light and Health Docs
What is light, and how can the CS Calculator help us to quantify its effects on our world.
CLA, CS, and the CS Calculator
Light is a biophysical construct, meaning that the physical world and the biological world both determine the definition of light. The first definition of light was developed in 1924 based upon experiments measuring the radiant watts needed at different wavelengths to be seen by humans as equally bright. (1) This research led to the adoption of the photopic luminous efficiency function,
V(\lambda)
, which was intended to represent the spectral sensitivity of the human retina at optical radiation levels where only the foveal cones provided neural signals to the brain. Subsequently this biophysical construct of light based upon
V(\lambda)
was “promoted” to the physical world through the SI system. Today we routinely make physical measurements and prescribe light levels in terms of illuminance, luminance, candelas, and lumens. But these physical quantities are not related to the biology of plants or other animals, nor are these measurements related to the variety of ways that humans sense optical radiation via the retina. Recognizing this limitation, the scotopic luminous efficiency function
V^{\prime}(\lambda)
was advanced in 1951 to represent the spectral sensitivity of the human retina when only rods were active. (2) Even with the biophysical constructs of
V(\lambda)
V^{\prime}(\lambda)
, many more definitions of light need to be considered to accurately relate the biology of the human retina to the physical world of optical radiation.
1.National Physical Laboratory (Great Britain). [International Commission on Illumination, sixth session, Geneva, July, 1924: Compendium of proceedings and report of meetings]. Cambridge, UK: Cambridge University Press; 1926.
2.Jansen J, Halbertsma NA. [Collection of proceedings and report of sessions, twelfth session of the International Commission on Illumination, Stockholm, June and July, 1951]. New York: Bureau Central de la CIE; 1951.
Circadian-effective light (CLA) is a biophysical construct relating the spectral sensitivity of the human retina to optical radiation for stimulation of the biological clock in the brain’s suprachiasmatic nuclei (SCN). The definition of circadian-effective light, or CLA, was first proposed in 2005 (3) and refined in 2010. (4) Following extensive research, the definition of circadian-effective light was revised, replacing CLA with CLA 2.0. (5, 6) As such, CLA 2.0 is now postulated to define the spectral sensitivity of the human circadian system to different wavelengths of optical radiation.
To quantify how different amounts, or levels, of CLA 2.0 affect the neural signal strength reaching the SCN, the circadian stimulus metric, or CS, was developed. (3-6) CS represents the operating characteristic of the phototransduction circuits in the retina, from threshold (i.e., just barely enough to stimulate the SCN) to saturation (i.e., the maximum possible response to circadian-effective light, no matter how much optical radiation is provided to the retina).
The current version of the CS Calculator quantifies optical radiation incident on the human retina in terms of CLA 2.0 and CS. The CS Calculator uses the relative spectral power distribution (SPD) of one or more light sources and the amount, or level, of photopic illuminance, based on
V(\lambda)
, incident on the cornea to determine circadian-effective light, CLA 2.0. From CLA 2.0, the signal strength, CS, is determined. A wide variety of other metrics, like chromaticity, correlated color temperature (CCT), color rendering index (CRI), and gamut area index (GAI) are also determined from the same relative spectral power distributions. A series of graphical representations accompany the numerical calculations.
The previous version of the CS Calculator, based upon the 2005 formulation of CLA, (3) is essentially the same except circadian light is now calculated using CLA 2.0 as defined in 2021. (5, 6)
3.Rea MS, Figueiro MG, Bullough JD, Bierman A. A model of phototransduction by the human circadian system. Brain Res Rev. 2005;50(2):213-28.
4.Rea MS, Figueiro MG, Bierman A, Bullough JD. Circadian light. J Circadian Rhythms. 2010;8(1):2.
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Attach - Maple Help
Home : Support : Online Help : Connectivity : Database Package : SQLite : Attach
Attach( connection, filename, name )
string; URI file name
string; name of database
The Attach command adds a database file to the current database connection. It is a convenience command implementing the ATTACH DATABASE statement.
The database-names main and temp refer to the main database and the database used for temporary tables, respectively. The main and temp databases cannot be attached or detached.
The Attach command can open an existing database from Workbook if the filename is a valid Workbook URI.
\mathrm{with}\left(\mathrm{Database}[\mathrm{SQLite}]\right):
\mathrm{db}≔\mathrm{Open}\left(":memory:"\right):
Create and attach another in memory database
\mathrm{Attach}\left(\mathrm{db},":memory:","database2"\right):
Opened databases
\mathrm{Opened}\left(\mathrm{db}\right)
\textcolor[rgb]{0,0,1}{table}\textcolor[rgb]{0,0,1}{}\left([\textcolor[rgb]{0,0,1}{"database2"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{""}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"main"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{""}]\right)
\mathrm{Close}\left(\mathrm{db}\right):
The Database[SQLite][Attach] command was introduced in Maple 18.
The Database[SQLite][Attach] command was updated in Maple 2016.
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Phase modulation - Wikipedia
This article is about the analog modulation. For the digital version, see Phase-shift keying.
Find sources: "Phase modulation" – news · newspapers · books · scholar · JSTOR (December 2009) (Learn how and when to remove this template message)
Phase modulation (PM) is a modulation pattern for conditioning communication signals for transmission. It encodes a message signal as variations in the instantaneous phase of a carrier wave. Phase modulation is one of the two principal forms of angle modulation, together with frequency modulation.
In phase modulation, the instantaneous amplitude of the baseband signal modifies the phase of the carrier signal keeping its amplitude and frequency constant. The phase of a carrier signal is modulated to follow the changing signal level (amplitude) of the message signal. The peak amplitude and the frequency of the carrier signal are maintained constant, but as the amplitude of the message signal changes, the phase of the carrier changes correspondingly.
Phase modulation is widely used for transmitting radio waves and is an integral part of many digital transmission coding schemes that underlie a wide range of technologies like Wi-Fi, GSM and satellite television. It is also used for signal and waveform generation in digital synthesizers, such as the Yamaha DX7, to implement FM synthesis. A related type of sound synthesis called phase distortion is used in the Casio CZ synthesizers.
The modulating wave (blue) is modulating the carrier wave (red), resulting the PM signal (green). g(t) = π/2 × sin[2 × 2πt + π/2 × sin(3 × 2πt)]
Phase modulation changes the phase angle of the complex envelope in proportion to the message signal.
If m(t) is the message signal to be transmitted and the carrier onto which the signal is modulated is
{\displaystyle c(t)=A_{c}\sin \left(\omega _{\mathrm {c} }t+\phi _{\mathrm {c} }\right).}
then the modulated signal is
{\displaystyle y(t)=A_{c}\sin \left(\omega _{\mathrm {c} }t+m(t)+\phi _{\mathrm {c} }\right).}
This shows how
{\displaystyle m(t)}
modulates the phase - the greater m(t) is at a point in time, the greater the phase shift of the modulated signal at that point. It can also be viewed as a change of the frequency of the carrier signal, and phase modulation can thus be considered a special case of FM in which the carrier frequency modulation is given by the time derivative of the phase modulation.
The modulation signal could here be
{\displaystyle m(t)=\cos \left(\omega _{\mathrm {c} }t+h\omega _{\mathrm {m} }(t)\right)\ }
The mathematics of the spectral behavior reveals that there are two regions of particular interest:
For a single large sinusoidal signal, PM is similar to FM, and its bandwidth is approximately
{\displaystyle 2\left(h+1\right)f_{\mathrm {M} }}
{\displaystyle f_{\mathrm {M} }=\omega _{\mathrm {m} }/2\pi }
{\displaystyle h}
is the modulation index defined below. This is also known as Carson's Rule for PM.
As with other modulation indices, this quantity indicates by how much the modulated variable varies around its unmodulated level. It relates to the variations in the phase of the carrier signal:
{\displaystyle h\,=\Delta \theta \,}
{\displaystyle \Delta \theta }
is the peak phase deviation. Compare to the modulation index for frequency modulation.
Electro-optic modulator for Pockel's Effect phase modulation for applying sidebands to a monochromatic wave
Retrieved from "https://en.wikipedia.org/w/index.php?title=Phase_modulation&oldid=1072039940"
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An analysis of the nonlinear equation of motion of a vibrating membrane in the space of BV functions
October, 2000 An analysis of the nonlinear equation of motion of a vibrating membrane in the space of BV functions
In this article the nonlinear equation of motion of vibrating membrane
{u}_{tt}-div\left\{{\sqrt{1+|Vu{|}^{2}}}^{-1}Vu\right\}=0
is discussed in the space of functions having bounded variation. Approximate solutions are constructed in Rothe's method. It is proved that a subsequence of them converges to a function
u
and that, if
u
satisfies the energy conservation law, then it is a weak solution in the space of functions having bounded variation. The main tool is varifold convergence.
Koji KIKUCHI. "An analysis of the nonlinear equation of motion of a vibrating membrane in the space of BV functions." J. Math. Soc. Japan 52 (4) 741 - 766, October, 2000. https://doi.org/10.2969/jmsj/05240741
Primary: 35L70 , 49J40 , 49Q15
Keywords: BV functions , direct variational method , Hyperbolic equations , Rothe's method , varifolds
Koji KIKUCHI "An analysis of the nonlinear equation of motion of a vibrating membrane in the space of BV functions," Journal of the Mathematical Society of Japan, J. Math. Soc. Japan 52(4), 741-766, (October, 2000)
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What Constitutes a Good Design Education Research Paper That Would be Suitable for JMD? | J. Mech. Des. | ASME Digital Collection
National Science Foundation (NSF) Division of Undergraduate Education Professor, Mechanical Engineering and Engineering Education Virginia Tech
J. P. Terpenny Guest Editor
Terpenny, J. P. (March 10, 2011). "What Constitutes a Good Design Education Research Paper That Would be Suitable for JMD?." ASME. J. Mech. Des. March 2011; 133(3): 030301. https://doi.org/10.1115/1.4003567
design engineering, engineering education, mechanical engineering, project management, team working
Design education, Teams, Engineering education, Design, Students, Design engineering, Project management
Have you heard? Did you know? The Journal of Mechanical Design (JMD) has been accepting and publishing quality design education research papers for the last 2 years. As the associate editor handling this area, I am all too often surprised. First, I am surprised that the word has not spread widely and that there are not more design education submissions to JMD. Clearly, there is a tremendous surge in the interest of engineering educators nationwide to increase student learning and skills through design projects, often with genuine experiences and with the participation from industry, the community, or individual “customers” working with student design teams. Design education seems to be the new focus of how engineering educators are going to fix what is perceived to have been broken for many years: a content focused curriculum, lacking in the context of real problems that would motivate and engage students and deepen their learning and increase their confidence in how to think, reason, problem-solve, and apply mathematics, science, and engineering methods and tools.
Where did this new interest in context come from? Perhaps it comes from the urgent cry from industry for change. Can we answer the question where are the students who are prepared to take on the significant engineering challenges of the 21st century? How students were educated in years gone by is not addressing many of the new challenges. The market place, indeed, the business enterprise, is dispersed globally, design teams are charged to innovate and collaborate with stakeholders who are invested in the quality, costs, and value of products throughout the life cycle of products. All the while, technologies are changing rapidly, the complexity of products has risen, and designers are under great pressure to reduce their cycle time since products need to get to market faster in order to capture greatest market share.
In response, engineering educators are increasingly looking to broaden and deepen student preparation with topics that are technical and nontechnical such as teaming and collaboration, project management, design process methods and tools, complex problem solving, analysis, decision making, visualization, simulation, and much more. Fundamental work on topics, such as learning styles, pedagogy, class size, use of technology in education, and issues surrounding under-represented groups, have also increased.
It is wonderful to see the wave of interest in transforming engineering education, most especially with a focus on engineering design and in the context of real problems. At the same time, it is also frightening. Frightening that engineering faculty across disciplines who often know little about design processes, methods, and tools are taking on these challenges. Yes, these efforts are with good intentions but with naïve and potentially harmful practices of bottom-up trial-and-error approaches that simply bring students together in a team and cast them off on a mission.
Here is where you come in, those of you who have contributed to the research and practice of engineering design; you have a significant role in improving design education! You may be (and are likely) doing this already. How can you share your expertise in design and aid others in success and long lasting changes in design education excellence? Submit your research and findings in design education in JMD!
And you ask what constitutes a good design education research paper that would be suitable for JMD? The answer to this is very much the same as it would be for any other research in JMD. Indeed, a good research paper reflects good research practice. Design education, or engineering education, in general, should be no different. It is simply a new application area for some of you perhaps, but the needs and challenges of this problem area should be approached with similar rigor and good problem solving, design process, and research methods.
The following outlines the basic sections that should be included in a design education research paper.
Introduction (what is the problem, why is it important, who does it affect, and what is the impact and significance if solved).
What have others done to date that have attempted to solve this problem (or something similar to it)? Reflect on prior work in your review of literature. What are the merits and still the needs remaining?
What is the objective of your work?
What and how did you develop the requirements? (i.e., How will you know that you have succeeded in your objective?)
What approach and methods did you apply to solve this problem? Provide an overview and details. What methods did you use to test and validate what you did?
Did you iterate? If so, why and how?
What were the results? How did you assess and evaluate these? Did they meet the requirements and the stated objective? How do you know? How well?
Did you iterate more?
Ultimately, what conclusions can be drawn from your work that others can learn from and/or use?
Do these steps sound familiar? They likely do, as they reflect basic steps in good design process and are the sections that would be found in a good research design and a good research paper that reports on the work and findings.
What is different in design education research than in other application domains? The terminology and methods used in each step are different perhaps. However, the basic constructs are the same. Do you now need to be expert in assessment and evaluation, working with human subjects and statistical analysis? Perhaps or, you could also consider broadening your collaboration to include experts in education or in engineering education, in particular.
Please do not submit papers that are “tell about” papers to design education in JMD. There are far too many conferences where such practices are acceptable already. JMD is interested in research and scholarship. How do you know that whatever your innovation in design education worked? Why would someone at another institution adopt your methods or believe them to be true? Did you follow the usual steps that you would have in any other field of research? As outlined above, did you state the problem, look at what others have done, describe what you did, test and validate it, and then do an assessment and evaluation? From this, what conclusions can be drawn? What did you contribute to the body of knowledge?
I have a slew of stories from my own research into design education where for years I was sure that the many creative and innovative things I was doing in and out of the classroom with students was effective. Surely, they were learning more and deeper, they were more motivated and engaged. It “looked” that way, and so it must be true. Right? It was not until data were collected and analyzed that I really knew. In the case of a product dissection activity, envisioned to engage, motivate, and improve learning about relationships between form and function, we found, after collecting and analyzing data, that active learning hands-on activities were overspecified. Students’ internal motivation to learn (for their own knowledge) actually lessened due to our intervention. Not an outcome I am proud of. Have you collected data and unexpectedly found “bad” results? These are shareable and valuable lessons learned, ones that lead us to addressing new research questions to better understand the needs and develop truly transformative change in design education. Research in design education is important!
To summarize, JMD is interested in publishing high quality papers grounded in good research methods that address innovative design education teaching, learning, and/or assessment methods. Example topics might include a myriad of aspects of preparing the next generation of students (graduate and undergraduate) including the following:
collaborative distributed design
computer-based methods and tools
concept selection and evaluation
courses and curriculum in design
design competitions and learning
design for X (where
X=life
cycle issues such as manufacture and service)
diversity in student engagement and learning
economics of engineering design
graduate curriculum and degree programs in design
integrating design research and design education
inventions, patenting, and intellectual property
learning evaluation and assessment methods
mentors and mentoring design
project planning and management in design
service-learning and design
test and validation of design
undergraduate curriculum reform and degree programs in design
Thank you for your valuable contributions and work in transforming the research and practice of engineering design education. We look forward to your submissions to JMD.
Graduate Design Education Workshops: A First Review
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Pool Tokens Design - Entropyfi
Welcome to Entropyfi
Entropyfi 1.0
Cope-Free Hedge
Rekt-Free Leverage
VSQ Tutorial
Entropills
SSOV Entropills
Atlantic Entropills
$ERP Token
ERP Tokenomics
Vault Inc $ERP Staking
[Replaced by Vault Inc] Liquidity Mining for ERP
Developers-1.0
Compound Feature
Pool Tokens Design
Token NAV Calculation
Entropyfi Ambassador Program Lead
Aavegotchi Snapshot Mining
Users will get pool tokens when depositing into the pool. With pool tokens, users can withdraw from the pool anytime!
Pool Token value gets updated when winners are announced. Every pool has its own tokens (short/long/sponsor tokens). They are used as proof of users' collaterals in the pool. Token will be burned when users withdraw from the pool.
Users will get short tokens when they deposit tokens with bidding on short.
, on BTC - DAI weekly game, once user A deposits 1,000 Dai tokens to predict BTC price will go down
at the end of next week. (assumption: pool short token value is 2 Dai currently)
Then, once user A deposits principals, user A will get 500 Short BTC-Dai Pool Token.
At the end of the week, user A finds that the prediction is correct
(i.e. the BTC price indeed ended up being a lot lower
than the game bidding price), the short pool token value may increase from 2 Dai to 2.1 Dai.
The users can choose to either get 1,050 aDai or Dai when they withdraw from the game:
500 \times 2.1 = 1050 \space\text{aDai / Dai}
If the user chooses to hold, then he will keep the same decision and automatically enrolled in the next round game -> The gain in this round will be used as principals for the next round (i.e. compound feature
If the user chooses to swap decisions and the current long pool token price is 1 Dai, then the user's short BTC-Dai Pool tokens will be burned and the pool will mint 1,050 corresponding long BTC-Dai Pool Tokens.
\text{total long token amount} = \dfrac{ \text{user's ShortTokenValue}}{ \text{LongTokenValue}}
The short token will be burned once the user redeem or withdraw from the protocol.
The users will get long tokens when they deposit tokens with bidding on long.
The long token will be burned once the user redeems or withdraw from the protocol.
🏦 Sponsor Token
Sponsors are not eligible to get rewards from the game. Instead, sponsors will supply interest to the pool. If a user sponsors USDT, he will get entropyUSDT (1:1 to USDT), as proof that he sponsored one of our pools.
To mining our governance token, the user has to be a sponsor in one of our pools. Staking sponsor tokens into our liquidity mining smart contract to mint governance tokens!
Developers-1.0 - Previous
Next - Developers-1.0
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Parallel spring-damper - Simulink - MathWorks Switzerland
Torsional stiffness, k
Initial deflection, theta_o
Initial velocity difference, domega_o
Damping cut-off frequency, omega_c
Parallel spring-damper
The Torsional Compliance block implements a parallel spring-damper to couple two rotating driveshafts. The block uses the driveshaft angular velocities, torsional stiffness, and torsional damping to determine the torques.
\begin{array}{l}{T}_{R}=-\left({\omega }_{R}-{\omega }_{C}\right)b-\theta k\\ {T}_{C}=\left({\omega }_{R}-{\omega }_{C}\right)b+\theta k\\ \\ \stackrel{˙}{\theta }=\left({\omega }_{R}-{\omega }_{C}\right)\end{array}
Mechanical power from driveshaft R
{P}_{TR}= {T}_{R}{\omega }_{R}
Mechanical power from driveshaft C
{P}_{TC}= {T}_{c}{\omega }_{c}
{P}_{d}=-b{|\stackrel{˙}{\theta }|}^{2}
{P}_{s}=-\theta k\stackrel{˙}{\theta }
Driveshaft R torque
Driveshaft C torque
Driveshaft R angular velocity
Driveshaft C angular velocity
Coupled driveshaft rotation
Driveshaft torsional stiffness
RSpd — Driveshaft R angular velocity
Input driveshaft angular velocity, in rad/s.
CSpd — Driveshaft C angular velocity
Output driveshaft angular velocity, in rad/s.
Input driveshaft torque
{T}_{s}=b\stackrel{˙}{\theta }
Td = kθ
Input driveshaft angular velocity
ωR rad/s
Output driveshaft angular velocity
ωC rad/s
Difference in input and output driveshaft angular velocity
\stackrel{˙}{\theta }
RTrq — Driveshaft R torque
Input drive shaft torque, in N·m.
CTrq — Driveshaft C torque
Applied output driveshaft torque, in N·m.
Torsional stiffness, k — Inertia
Torsional stiffness, in N·m/rad.
Initial deflection, theta_o — Angular
Initial deflection, in rad.
Initial velocity difference, domega_o — Angular
Initial velocity difference, in rad/s.
Damping cut-off frequency, omega_c — Frequency
Damping cut-off frequency, in rad/s.
Rotational Inertia | Split Torsional Compliance
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Farming using only stable coins - Mai Finance - Tutorials
This page presents in details a "safe" strategy to make yield farming a little less profitable, but a lot more secure.
When you enter a yield farm on Polygon, you expose your investments to the success or failure of the farm. This guide isn't presenting in details what is a Yield Farm, or how you should farm on them. If you need help on that there are tutorials everywhere on internet. You can also get some help from the QiDAO community on Discord.
The main issue when you are farming is that you have to make a choice between
selling the native token and convert them into "secured" assets that will represent your gains
re-invest them to generate even more profits (also known as hyper-compounding)
The guide will present step by step how to use Mai Finance to actually secure your gains while still re-investing a portion of it into the farm.
To illustrate in more details how you can do that, I will use the latest PolyPup farm. This is for educational purpose only, and should absolutely not be used as a financial advice. Also, the term "secure" here is solely based on my personal appreciation. As always, do your own research. Finally, I personally don't recommend this farm.
Farming life cycle
Getting prepared to farm
As a humble farmer once told me
You should never buy what you can earn
In the guide, we will try to implement as much financial security as possible. To do so, we will farm using stable coins only, in order to protect our investment from any impermanent loss. Most farms are proposing stable coins pairs in their liquidity pools (LP), and MAI gaining more and more visibility, you can find farms proposing MAI/USDC pools. This is the stable coins pair we will be focusing on.
In order to start farming using MAI/USDC pair, you need to acquire some stable coins. Mai Finance allows you to borrow the MAI stable coin by depositing your favorite crypto currency. In our case, we have a bunch of MATIC in our wallet, ready to be used. By depositing my MATIC into the MATIC vault on Mai Finance, I can borrow MAI. If you need assistance doing that, please join the Discord server and ask the community. You can also read other tutorials on this site where you may find how to do this.
You can deposit your MATIC tokens in your MATIC vault, but you can also deposit them into AAVE to get amWMATIC, deposit them on Mai Finance on the yield page to get camWMATIC, and use these camWMATIC as collateral. You will be able to borrow the same amount of MAI, but you will also earn additional yield on your MATIC. See Leverage your AAVE tokens to get more details on how to do this.
Once you borrowed MAI stable coins, you can use the anchor page on Mai Finance to convert half of your loan into USDC. Indeed, when you farm using LP pairs, the two parts of the pair need to be provided in a 1:1 ratio.
Using the swap page to convert 30 of my MAI into USDC
Now depending on the farm you want to farm on, you need to combine your 2 stable coins (MAI and USDC) into a valid LP pair on a DEX platform. Since my plan is to enter Polypup, and that farm accepts QuickSwap LPs, I need to go to QuickSwap and generate my pair there.
Generate some LP tokens using MAI and USDC
I am now ready to enter the farm.
Deposit and harvest farm tokens
Now that you have some LP tokens, you can go to the farm website and deposit them to start collecting the farm tokens. In our example, I deposited my MAI/USDC tokens into the correct pool, and started collecting BALL tokens.
Earning BALLs in the pool
As of right now, you can see that farming MAI/USDC is granting me 176.99% APR. Based on how much liquidity is provided in the pool, and on the price of the BALL token, this APR will change over time.
It's very important to note that when you deposit your LP tokens, most farm will charge anywhere between 2% and 4% fees directly taken from the tokens you deposit. Be very aware of this, and make sure you are mentally ready to potentially lose the fee, or not fully get it back.
Now that your stable coins are deposited in the pool, you will earn some farm tokens that you can harvest whenever you want. Note that the price of the farm token will likely be very volatile, so make sure you harvest regularly when the token has some value. The more you wait, the more risk you have to get a big bag of tokens that worth totally nothing. In the screenshot above, simply click the Harvest token and collect your BALLs.
Leverage your farm tokens
Now that you have some farm tokens, usually you have the choice between
selling them to buy something of greater value (your favorite crypto is a good example)
re-inject them into the pool
Mai Finance presents a third option that lets you do both. Once you harvested your reward, simply go to your favorite DEX that support the farm token. Usually, you can find a link to a DEX in the menu of the farm. The link will include the token address which will help you trade it easily.
Swapping my reward for more MATIC
At that point, I am back at the step where we have MATIC tokens in my wallet, ready to be deposited on Mai Finance as a collateral. If I do that, I can borrow MAI, swap a portion of it into USDC, create a LP pair and re-deposit it into the farm. By doing this conversion, I "secure" 100% of my rewards by swapping them into a more stable crypto (here MATIC) and I re-inject 50% of my reward into the farming pool (or actually, in this example, 46% because of the 4% deposit fee).
If you think about it from a different perspective, the APR is 100% applied to your main crypto. If you are depositing new LP tokens into the pool (compounding), you get 50% of the APY advertised by the pool.
Gains estimation
All the results presented below are assuming a few things
We started with the equivalent of 60 MAI borrowed against the equivalent of $120.00 worth of MATIC
The APR of the farm is staying at 176.99% for the period of time, translating to a 0.484% daily gain
The value of MATIC and BALL stay the same for the period of time
These assumption are obviously not applying to the real life, the APR will slowly decay over time as more liquidity is provided to the pool, and as the farm token varies in price.
Estimated raw results
Compounded MATIC
New LP created
On day 1, the 4% fee is applied to our initial 60$ worth of MAI/USDC pair
At the end of day 1, the generated revenue ($0.279) is fully transferred into the MATIC vault
At the end of day 1, since we added some funds to the vault we can borrow more MAI
In order to keep a 200% Collateral to Debt ratio, we only borrow 50% of the deposited MATIC ($0.139)
At the beginning of day 2, we re-inject $0.139 into the farm (and the farm takes out 4% deposit fee)
At the beginning of day 2, we start again with an additional $0.134 worth of LP token
Estimating APRs, APYs and revenue growth
The estimation was stopped after 18 days because passed that date, you can see that we are back at a value of $60 worth of LP tokens. This means that I farmed enough to repay the initial deposit fees, which is the minimum goal that any farmer should aim for.
Passed that date, staying in the farm will only generate benefits. And because we're farming using stable coins, there's literally 0% impermanent loss on the liquidity I provide, meaning that there's no risk of losing money by staying in the pool.
However, I can also consider that I repaid the initial $2.40 deposit fee on day 9 because it's the time when the value of the profit in my MATIC wallet reached that amount. If I was just selling the farm token to make profit and not re-invest into the farming pool, that would actually be the moment I would only generate benefits.
In terms of gains, the reward that gets compounded into the farm is only 50% of the farm APR. This means that with a farm APR rated at 176.99%, the actual growth rate is only 88.495% annually, or 0.242% daily.
It's also possible to calculate the exact gain that was accrued on a specific day on the farm, assuming you are compounding daily, using the following formula for the Return On Investment
ROI_{DayN}=InitialInvestment*(1+DailyAPR)^{DayN}-InitialInvestment
In the case of a farm that takes 4% deposit fee, you need to multiply the above result by 96%. In our case, we can quickly verify that the formula works by comparing its result to the table above
Starting from $60.00 and generating $81.81 after 1 year gives an APY of 136.34%.
If we compare this to the theoretical APY given by a 88.495% APR using the formula
APY = ( 1 + \frac{APR}{N})^N-1
This gives with an APR of 88.495% and N = 365 (compounding daily)
APY = ( 1 + \frac{0.88495}{365})^{365}-1=142.02\%
Note that the estimation above doesn't take into consideration the 4% deposit fee, hence the small discrepancy.
The gains in the MATIC wallet are simply twice the gains from the farm and we can calculate the ROI on a given day using the same formula as above and multiply the result by 2.
This is the number of MATIC that we would generate by staying into the farm for one year, with an initial investment of 60$ worth of MAI/USDC, assuming the APR of the farm remains the same. This also gives an APY of 272.69%, which is roughly the APR that the farm advertised (the farm usually don't take into consideration the 4% deposit fee in the displayed APR).
Recap after 1 year farming with stable coins
At the end of the year we would get
$283.62 MATIC in the Vault (initial $120.00 + $163.62 generated by the farm)
$141.82 of debt (initial $60.00 + $81.82 borrowed and re-invested)
$141.82 of MAI/USDC tokens on the farm
Everything presented in this strategy is assuming that
The farm is keeping a constant APR on your pool (which is totally wrong)
It is possible to farm for a whole year (which is not possible, all farms are ending their farming session sooner or later)
As a side note, the APR of the MAI/USDC pool on Polypup after 24h of farming is 128.13%, mostly because of the price of BALL slowly decaying.
Also, farming with stable coins may be the most "secured" way to earn yields because you're not exposed to impermanent loss. However, there is absolutely no guarantee that you will be able to get your 4% deposit back. You can find some websites that have stable coin pools with 0% to 1% deposit fees, even on the non-native pools (pools that accept LP pairs without the native farm token).
Harvesting the rewards and swapping them for something valuable is considered the best strategy when farming yields. Borrowing MAI to re-inject a portion of stable coin in the pool and increase your farming revenues exposes your benefits to the 4% deposit fee that the farm is taking off your LP pair, which may not be the best thing to do if you are unsure to get it back, and it's probably better to use another strategy to re-invest your earnings (invest in native pools / pools with 0% interest / pools with high APRs).
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Stochastic Differential Equation (SDEDDO) model from Drift and Diffusion components - MATLAB - MathWorks Switzerland
Create a sdeddo Object
Stochastic Differential Equation (SDEDDO) model from Drift and Diffusion components
Creates and displays sdeddo objects, instantiated with objects of classdrift and diffusion. These restricted sdeddo objects contain the input drift and diffusion objects; therefore, you can directly access their displayed parameters.
This abstraction also generalizes the notion of drift and diffusion-rate objects as functions that sdeddo evaluates for specific values of time t and state Xt. Like sde objects, sdeddo objects allow you to simulate sample paths of NVars state variables driven by NBrowns Brownian motion sources of risk over NPeriods consecutive observation periods, approximating continuous-time stochastic processes.
This method enables you to simulate any vector-valued SDEDDO of the form:
d{X}_{t}=F\left(t,{X}_{t}\right)dt+G\left(t,{X}_{t}\right)d{W}_{t}
SDEDDO = sdeddo(DriftRate,DiffusionRate)
SDEDDO = sdeddo(___,Name,Value)
SDEDDO = sdeddo(DriftRate,DiffusionRate) creates a default SDEDDO object.
SDEDDO = sdeddo(___,Name,Value) creates a SDEDDO object with additional options specified by one or more Name,Value pair arguments.
The SDEDDO object has the following displayed Properties:
A — Access function for the drift-rate property A, callable as a function of time and state
B — Access function for the drift-rate property B, callable as a function of time and state
Alpha — Access function for the diffusion-rate property Alpha, callable as a function of time and state
Sigma — Access function for the diffusion-rate property Sigma, callable as a function of time and state
If StartState is a scalar, sdeddo applies the same initial value to all state variables on all trials.
If StartState is a column vector, sdeddo applies a unique initial value to each state variable on all trials.
If StartState is a matrix, sdeddo applies a unique initial value to each state variable on each trial.
F\left(t,{X}_{t}\right)=A\left(t\right)+B\left(t\right){X}_{t}
G\left(t,{X}_{t}\right)=D\left(t,{X}_{t}^{\alpha \left(t\right)}\right)V\left(t\right)
The sdeddo class derives from the base sde class. To use this class, you must pass drift and diffusion-rate objects to the sdeddo function.
Pass the functions to the sdeddo function to create an object obj of class sdeddo:
When you invoke these parameters with inputs, they behave like functions, giving the impression of dynamic behavior. The parameters accept the observation time t and a state vector Xt, and return an array of appropriate dimension. Even if you originally specified an input as an array, sdeddo treats it as a static function of time and state, by that means guaranteeing that all parameters are accessible by the same interface.
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Staking (🌳,🌳) - KlimaDAO
Staking is the primary reward distribution mechanism of the protocol. It is intended to be the primary mechanism of value accrual for the majority of users.
Whenever the protocol has an excess of reserves per token (i.e when the CC of the treasury is higher than the assets needed to back KLIMA), the protocol will mint and distribute tokens to the stakers. The amount minted and distributed is controlled by a variable named the reward rate. This is the % of supply that is rebased. This massively slows down how fast the protocol expands supply, as doing so is detrimental to the health (rapid expansion without backing causes a price collapse).
KLIMA and sKLIMA always have a 1:1 ratio, meaning that you will always obtain 1 sKLIMA for every 1 KLIMA and vice versa via staking or unstaking on the KlimaDAO Dapp.
KLIMA= sKLIMA
When a rebase occurs, the treasury deposits KLIMA into the distributor contract, which deposits it in the staking contract. Since there is now more KLIMA then there is sKLIMA, the sKLIMA is rebased to keep them in parity.
Reward Rate=1-KlimaDeposits/sKlimaOutstanding
Because not all KLIMA is staked, the user gains a larger piece of the share of rebases:
RewardYield = RewardRate/(\%Staked*\%Circulating)
This translates to AKR:
(1+RewardYield)^{(365*24/(BlockTimeInHours)}
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International Standard Serial Number - EverybodyWiki Bios & Wiki
Example of an ISSN encoded in an EAN-13 barcode, with explanation. NOTE: MOD10 in the image should be MOD11.
An International Standard Serial Number (ISSN) is an eight-digit serial number used to uniquely identify a serial publication.[1] The ISSN is especially helpful in distinguishing between serials with the same title. ISSN are used in ordering, cataloging, interlibrary loans, and other practices in connection with serial literature.[2]
When a serial with the same content is published in more than one media type, a different ISSN is assigned to each media type. For example, many serials are published both in print and electronic media. The ISSN system refers to these types as print ISSN (p-ISSN) and electronic ISSN (e-ISSN), respectively.[4] Conversely, as defined in ISO 3297:2007, every serial in the ISSN system is also assigned a linking ISSN (ISSN-L), typically the same as the ISSN assigned to the serial in its first published medium, which links together all ISSNs assigned to the serial in every medium.[5]
6 ISSN variants and labels
The format of the ISSN is an eight digit code, divided by a hyphen into two four-digit numbers.[1] As an integer number, it can be represented by the first seven digits.[6] The last code digit, which may be 0-9 or an X, is a check digit. Formally, the general form of the ISSN code (also named "ISSN structure" or "ISSN syntax") can be expressed as follows:[7]
^\d{4}-\d{3}[\dxX]$.
{\displaystyle 0\cdot 8+3\cdot 7+7\cdot 6+8\cdot 5+5\cdot 4+9\cdot 3+5\cdot 2}
{\displaystyle =0+21+42+40+20+27+10}
{\displaystyle =160}
{\displaystyle {\frac {160}{11}}=14{\mbox{ remainder }}6=14+{\frac {6}{11}}}
{\displaystyle 11-6=5}
There is an online ISSN checker that can validate an ISSN, based on the above algorithm.[9][10]
ISSN codes are assigned by a network of ISSN National Centres, usually located at national libraries and coordinated by the ISSN International Centre based in Paris. The International Centre is an intergovernmental organization created in 1974 through an agreement between UNESCO and the French government. The International Centre maintains a database of all ISSNs assigned worldwide, the ISDS Register (International Serials Data System) otherwise known as the ISSN Register. At the end of 2016,[update] the ISSN Register contained records for 1,943,572 items.[11]
Since the ISSN applies to an entire serial a new identifier, the Serial Item and Contribution Identifier (SICI), was built on top of it to allow references to specific volumes, articles, or other identifiable components (like the table of contents).
Separate ISSNs are needed for serials in different media (except reproduction microforms). Thus, the print and electronic media versions of a serial need separate ISSNs.[12] Also, a CD-ROM version and a web version of a serial require different ISSNs since two different media are involved. However, the same ISSN can be used for different file formats (e.g. PDF and HTML) of the same online serial.
This "media-oriented identification" of serials made sense in the 1970s. In the 1990s and onward, with personal computers, better screens, and the Web, it makes sense to consider only content, independent of media. This "content-oriented identification" of serials was a repressed demand during a decade, but no ISSN update or initiative occurred. A natural extension for ISSN, the unique-identification of the articles in the serials, was the main demand application. An alternative serials' contents model arrived with the indecs Content Model and its application, the digital object identifier (DOI), as ISSN-independent initiative, consolidated in the 2000s.
Example: the DOI name "10.1038/nature13777" can be represented as an HTTP string by https://dx.doi.org/10.1038/nature13777, and is redirected (resolved) to the current article's page; but there is no ISSN online service, like http://dx.issn.org/, to resolve the ISSN of the journal (in this sample 1476-4687), that is, a kind of https://dx.issn.org/1476-4687 redirecting to the journal's home.
A unique URN for serials simplifies the search, recovery and delivery of data for various services including, in particular, search systems and knowledge databases.[13] ISSN-L (see Linking ISSN below) was created to fill this gap.
ISSN variants and labels[edit]
There are two most popular media types that adopted special labels (indicating below in italics), and one in fact ISSN-variant, with also an optional label. All are used in standard metadata context like JATS, and the labels also, frequently, as abbreviations.
p-ISSN is a standard label for "Print ISSN", the ISSN for the print media (paper) version of a serial. Usually it is the "default media", so the "default ISSN".
e-ISSN (or eISSN) is a standard label for "Electronic ISSN", the ISSN for the electronic media (online) version of a serial.
ISSN-L is a unique identifier for all versions of the serial containing the same content across different media. As defined by ISO 3297:2007, the "linking ISSN (ISSN-L)" provides a mechanism for collocation or linking among the different media versions of the same continuing resource.
WorldCat – an ISSN-resolve service
↑ 1.0 1.1 "What is an ISSN?". Paris: ISSN International Centre. Retrieved 13 July 2014.
↑ "ISSN, a Standardised Code". Paris: ISSN International Centre. Retrieved 13 July 2014.
↑ "The ISSN for electronic media | ISSN". www.issn.org. Retrieved 2017-09-28.
↑ "3". ISSN Manual (PDF). Paris: ISSN International Centre. January 2015. pp. 14, 16, 55–58. Search this book on HTML version available at www.issn.org
↑ <rozenfeld@issn.org>, Slawek Rozenfeld. "Using The ISSN (International Serial Standard Number) as URN (Uniform Resource Names) within an ISSN-URN Namespace". tools.ietf.org.
↑ github.com/amsl-project/issn-resolver See p. ex. $pattern at source code (issn-resolver.php) of GitHub.
↑ "Online ISSN Checker". Advanced Science Index. Retrieved 14 July 2014.
↑ "Online ISSN Validator". Journal Seeker. Retrieved 9 August 2014.
↑ "Total number of records in the ISSN Register" (PDF). ISSN International Centre. February 2017. Retrieved 23 February 2017.
↑ "ISSN for Electronic Serials". U.S. ISSN Center, Library of Congress. 19 February 2010. Retrieved 12 July 2014.
↑ 13.0 13.1 "The ISSN-L for publications on multiple media". ISSN International Centre. Retrieved 12 July 2014.
↑ Rozenfeld, Slawek (January 2001). "Using The ISSN (International Serial Standard Number) as URN (Uniform Resource Names) within an ISSN-URN Namespace". IETF Tools. RFC 3044. Retrieved 15 July 2014.
↑ Powell, Andy; Johnston, Pete; Campbell, Lorna; Barker, Phil (21 June 2006). "Guidelines for using resource identifiers in Dublin Core metadata § 4.5 ISSN". Dublin Core Architecture Wiki. Archived from the original on 13 May 2012.
↑ "MEDLINE®/PubMed® Data Element (Field) Descriptions". U.S. National Library of Medicine. 7 May 2014. Retrieved 19 July 2014.
↑ Kansalliskirjasto, Nationalbiblioteket, The National Library of Finland. "Kansalliskirjasto, Nationalbiblioteket, The National Library of Finland". nationallibrary.fi. CS1 maint: Multiple names: authors list (link)
↑ "La nueva Norma ISSN facilita la vida de la comunidad de las publicaciones en serie", A. Roucolle. "Archived copy". Archived from the original on 10 December 2014. Retrieved 29 October 2014. CS1 maint: Archived copy as title (link)
↑ "Road in a nutshell". Road.issn.org. Archived from the original on 5 September 2017. Retrieved 12 September 2017.
"Cataloging Part", ISSN Manual (PDF), ISSN International Centre, archived from the original (PDF) on 7 August 2011 .
Getting an ISSN in Germany (in German), Deutsche Nationalbibliothek CS1 maint: Unrecognized language (link)
This article "International Standard Serial Number" is from Wikipedia. The list of its authors can be seen in its historical. Articles copied from Draft Namespace on Wikipedia could be seen on the Draft Namespace of Wikipedia and not main one.
Retrieved from "https://en.everybodywiki.com/index.php?title=International_Standard_Serial_Number&oldid=219688"
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A necessary condition for twin boundary layer behavior
Frederick A. Howes (1980)
A note on eigenvalues of ordinary differential operators.
Alan Ho (1997)
In this follow-up on the work of Fefferman-Seco [FS] an improved condition for the discrete eigenvalues of the operator -d2 / dx2 + V(x) is established for V(x) satisfying certain hypotheses. The eigenvalue condition in [FS] establishes eigenvalues of this operator to within a small error. Through an obervation due to C. Fefferman, the order of accuracy can be improved if a certain condition is true. This paper improves on the result obtained in [FS] by showing that this condition does indeed hold....
\left({A}_{1}{\Delta }_{u}\right)+{A}_{2}u=0
{A}_{1}
U
U
u
{A}_{1}=\infty
Addendum to a paper of Craig and Goodman.
Almost Periodic Solutions of Singularly Perturbed Systems of Differential Equations with Impulse Effect.
Maria A. Hekimova, Drumi D. Bainov (1989)
Asymptotic expansions of canards with poles. Application to the stationary unidimensional Schrödinger equation.
Benoit, E. (1996)
Asymptotique de solutions oscillantes d'équations différentielles
Franck Michel (1997)
Róbert Vrábeľ (2011)
In this paper we investigate the problem of existence and asymptotic behavior of solutions for the nonlinear boundary value problem
ϵ{y}^{\text{'}\text{'}}+ky=f\left(t,y\right),\phantom{\rule{1.0em}{0ex}}t\in 〈a,b〉,\phantom{\rule{4pt}{0ex}}k<0,\phantom{\rule{4pt}{0ex}}0<ϵ\ll 1
satisfying three point boundary conditions. Our analysis relies on the method of lower and upper solutions and delicate estimations.
Die Konstruktion asymptotischer Fundamentalsysteme für lineare Differentialgleichungen mit Wendepunkten
Hans-Görg Roos (1979)
Eine Bemerkung zur WKB-Methode
Jiří Zeman (1978)
Existence and concentration of positive solutions for a quasilinear elliptic equation in
ℝ
Gloss, Elisandra (2010)
Jean Zinn-Justin (2003)
In these notes, conjectures about the exact semi-classical expansion of eigenvalues of hamiltonians corresponding to potentials with degenerate minima, are recalled. They were initially motivated by semi-classical calculations of quantum partition functions using a path integral representation and have later been proven to a large extent, using the theory of resurgent functions. They take the form of generalized Bohr--Sommerfeld quantization formulae. We explain here their...
Hopf bifurcation analysis for the van der Pol equation with discrete and distributed delays.
Zhou, Xiaobing, Jiang, Murong, Cai, Xiaomei (2011)
Frédéric Pham (1985/1986)
La variété des équations surstables
Guy Wallet (2000)
Rémi Sentis (2005)
We address here mathematical models related to the Laser-Plasma Interaction. After a simplified introduction to the physical background concerning the modelling of the laser propagation and its interaction with a plasma, we recall some classical results about the geometrical optics in plasmas. Then we deal with the well known paraxial approximation of the solution of the Maxwell equation; we state a coupling model between the plasma hydrodynamics and the laser propagation. Lastly, we consider the...
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Generic nonlinear elliptic solver interface
<
>
This thorn is a generic interface to nonlinear elliptic solvers. It provides a uniform calling conventions to elliptic solvers. Elliptic solvers can register themselves with this thorn. Thorns requiring elliptic solvers can use this interface to call the solvers, and can choose between different solvers at run time.
This thorn is a generic interface to nonlinear elliptic solvers. It provides a uniform calling conventions to elliptic solvers, but does not contain any actual solvers. Solvers are supposed to be implemented in other thorns, and then register with this thorn. A generic interface has the advantage that it decouples the thorns calling the solvers, and the solvers themselves. Thorns using elliptic solvers can choose among the available solvers at run time.
For the discussion below, I write elliptic equations as
\begin{array}{rcll}F\left(u\right)& =& 0& \text{(1)}\text{}\text{}\end{array}
u
is the unknown, which is also known as variable, or in a sloppy terminology as solution.
F
is the elliptic operator. All elliptic equations can be written in the above form.
{u}^{\left(n\right)}
is an approximation to a solution
u
\begin{array}{rcll}{r}^{\left(n\right)}& :=& F\left({u}^{\left(n\right)}\right)& \text{(2)}\text{}\text{}\end{array}
is called the corresponding residual. An approximation
{u}^{\left(n\right)}
is a solution if the residual is zero.
The interface provided by this thorn allows for coupled sets of elliptic equations to be solved at the same time. The elliptic operator is allowed to be nonlinear.
u
is defined by the combination of
the elliptic operator
F
a set of initial data
{u}^{\left(0\right)}
boundary conditions for the solution
u
Note that periodicity is usually not a boundary condition that leads to a well-posed problem. It fails already for the Laplace (
\mathrm{\Delta }u=0
) and the Poisson (
\mathrm{\Delta }u=\rho
) equations.
In Cactus,
{u}^{\left(n\right)}
{r}^{\left(n\right)}
are represented by grid functions, while
F
and the boundary conditions to
u
are functions or subroutines written in C or Fortran.
2 Solver Interface
TATelliptic_CallSolver
Call an elliptic solver
#include "TATelliptic.h"
int TATelliptic_CallSolver (cGH * cctkGH,
const int * var,
const int * res,
int nvars,
int options_table,
calcfunc calcres,
calcfunc applybnds,
void * userdata,
const char * solvername);
subroutine TATelliptic_CallSolver (ierr, cctkGH, var, res, nvars, &
options_table, &
calcres, applybnds, userdata, &
solvername)
integer ierr
CCTK_POINTER cctkGH
integer nvars
integer var(nvars)
integer res(nvars)
integer options_table
CCTK_FNPOINTER calcres
CCTK_FNPOINTER applybnds
CCTK_POINTER userdata
character solvername*(*)
end subroutine TATelliptic_CallSolver
nonzero Failure. Error codes are
-1
if there are illegal arguments to the solver interface, and
-2
if the requested solver does not exist. Otherwise, the error code from the solver is returned.
cctkGH The pointer to the CCTK grid hierarchy
var Array with nvars grid function indices for the variables. The variables are also called “unknowns”, or “solution”.
res Array with nvars grid function indices for the residuals. These grid functions store the residuals corresponding to the variables above.
nvars Number of variables. This is also the number of residuals.
options_table Further options to and return values from the solver. Different solvers will take different additional arguments. The interface TATelliptic does not look at these options, but passes this table on to the real solver. This must be a table handle created with the table utility functions.
Below is a list of some commonly accepted options. See the individual solver documentations for details:
CCTK_REAL minerror:
The desired solver accuracy.
CCTK_REAL factor:
A factor with which all residuals are multiplied. This factor can be used to scale the residuals.
CCTK_REAL factors[nvars]:
An array of factors with which the residuals are multiplied. These factors can be used to handle inconvenient sign conventions for the residuals.
CCTK_INT maxiters:
CCTK_INT nboundaryzones[2*dim]:
Number of boundary points for each face. If not given, this is the same as the number of ghost points.
The following values are often returned. Again, see the individual solver documentations for details:
CCTK_INT iters:
CCTK_REAL error:
Norm of the final residual
calcres Pointer to a C function that evaluates the residual. This function is passed in a solution, and has to evaluate the residual. See below.
applybnds Pointer to a C function that applies the boundary condition to the variables. This function is passed in a solution, and has to apply the boundary conditions to it.
userdata A pointer to arbitrary application data. This pointer is passed through the solver unchanged on to calcres and applybnds. The application can use this instead of global variables to pass arbitrary data. If in doubt, pass a null pointer.
solvername The name of a registered solver.
The function TATelliptic_CallSolver is the the interface to solve an elliptic equation.
Input arguments are the arrays var and res, containing the grid function indices for the solution and the residual, as well as functions calcres and applybnds to evaluate the residual and apply boundary conditions. solvername selects the solver.
Hint: It is convenient to make the solver name a string parameter. This allows the solver to be selected at run time from the parameter file.
On entry, the grid functions listed in var have to contain an initial guess for the solution. This is necessary because the equations can be nonlinear. An initial guess of zero has to be set explicitly.
On exit, if the solver was successul, these grid functions contain an approximation to a solution. The grid functions listed in res contain the corresponding residual.
Additional solver options are passed in a table with the table index options_table. This table must have been created with one of the table utility functions. The set of accepted options depends on the particular solver that is called. Do not forget to free the table after you are done with it.
Hint: It is convenient to create the table from a string parameter with a call to Util_TableCreateFromString. This allows the solver parameters to be set in the parameter file.
In order to be able to call this function in the thorn TATelliptic, your thorn has to inherit from TATelliptic in your interface.ccl:
INHERITS: TATelliptic
In order to be able to include the file TATelliptic.h into your source files, your thorn has to use the header file TATelliptic.h in your interface.ccl:
USES INCLUDE: TATelliptic.h
Currently, only three-dimensional elliptic equations can be solved.
calcres Evaluate the residual
applybnds Apply the boundary conditions
DECLARE_CCTK_PARAMETERS;
/* options and solver are string parameters */
int varind; /* index of variable */
int resind; /* index of residual */
int options_table; /* table for additional solver options */
int ipos; /* position in 3D array */
static int calc_residual (cGH * cctkGH, int options_table, void * userdata);
static int apply_bounds (cGH * cctkGH, int options_table, void * userdata);
/* Initial data for the solver */
for (k=0; k<cctk_lsh[2]; ++k) {
for (j=0; j<cctk_lsh[1]; ++j) {
for (i=0; i<cctk_lsh[0]; ++i) {
ipos = CCTK_GFINDEX3D(cctkGH,i,j,k);
phi[ipos] = 0;
/* Options for the solver */
options_table = Util_TableCreateFromString (options);
assert (options_table>=0);
/* Grid variables for the solver */
varind = CCTK_VarIndex ("wavetoy::phi");
assert (varind>=0);
resind = CCTK_VarIndex ("IDSWTE::residual");
/* Call solver */
ierr = TATelliptic_CallSolver (cctkGH, &varind, &resind, 1,
options_table,
calc_residual, apply_bounds, 0,
if (ierr!=0) {
CCTK_WARN (1, "Failed to solve elliptic equation");
ierr = Util_TableDestroy (options_table);
calcres
Evaluate the residual. This function is written by the user. The name of this function does not matter (and should likely not be calcres). This function is passed as a pointer to TATelliptic_CallSolver.
int calcres (cGH * cctkGH,
void * userdata);
No Fortran equivalent.
0 Success; continue solving
nonzero Failure; abort solving
options_table A table passed from the solver, or an illegal table index. If this is a table, then it may contain additional information about the solving process (such as the current multigrid level). This depends on the particular solver that is used.
userdata A pointer to arbitrary application data. This is the same pointer that was passed to TATelliptic_CallSolver. The application can use this instead of global variables to pass arbitrary data.
This function has to be provided by the user. It has to calculate the residual corresponding to the current (approximation of the) solution.
Input to this function are the unknowns, output are the residuals.
The residuals need not be synchronised. The boundary values of the residuals are not used, and hence do not have to be set.
TATelliptic_CallSolver Call an elliptic solver
applybnds
Apply the boundary conditions to the solution. This function is written by the user. The name of this function does not matter (and should likely not be applybnds). This function is passed as a pointer to TATelliptic_CallSolver.
int applybnds (cGH * cctkGH,
This function has to be provided by the user. It has to apply the boundary and symmetry conditions to the solution.
Input to this function is the interior of the solution, output are the boundary values of the solution.
This function also has to synchronise the solution.
TATelliptic_RegisterSolver
Register an elliptic solver
int TATelliptic_RegisterSolver (solvefunc solver,
-1 Failure: illegal arguments. solver or solvername are null.
-2 Failure: a solver with this name has already been registered.
solver A pointer to the solver’s solving function. This function has to have the following interface, which is the same as that of TATelliptic_CallSolver except that the argument solvername is missing:
typedef int (* solvefunc) (cGH * cctkGH,
solvername The name of the solver
Each solver has to register its solving function with TATelliptic at startup time.
3 Pseudo solver
This thorn provides also a pseudo solver, called TATmonitor. This is not a real solver, although it poses as one and uses the TATelliptic_CallSolver interface. It does nothing but evaluate the residual and then return successfully.
You will find this a useful intermediate step when debugging your residual evaluation routines.
This section lists all the variables which are assigned storage by thorn CactusElliptic/TATelliptic. Storage can either last for the duration of the run (Always means that if this thorn is activated storage will be assigned, Conditional means that if this thorn is activated storage will be assigned for the duration of the run if some condition is met), or can be turned on for the duration of a schedule function.
tatelliptic_register_monitor
register the pseudo solver
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Specifies the value that is to be assigned to an rtable element if the scanned value is not type-compatible with the rtable. The mismatchValue is optional, and if omitted, a suitable value compatible with the rtable is used (zero for numeric tables,
""
for strings, false for booleans, and NULL for other tables).
The mismatch value option is also useful when reading into rtables of other specific types. For example, if the option y(posint) was specified, an
m\left(99999\right)
option would cause all non-positive integers (i.e. 0 and negative integers) that were read to be turned into 99999.
Specifies the value that is to be assigned to an rtable element if the end of the input (typically a file) is reached before the rtable has been filled. The postEOFValue is optional, and if omitted, a suitable value compatible with the rtable is used (zero for numeric tables,
""
\mathrm{sscanf}\left("1 2 3 4","%\left\{2,2\right\}fm"\right)
[[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1.}& \textcolor[rgb]{0,0,1}{2.}\\ \textcolor[rgb]{0,0,1}{3.}& \textcolor[rgb]{0,0,1}{4.}\end{array}]]
\mathrm{sscanf}\left("1 2 3 4","%\left\{4\right\}fr"\right)
[[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{1.}& \textcolor[rgb]{0,0,1}{2.}& \textcolor[rgb]{0,0,1}{3.}& \textcolor[rgb]{0,0,1}{4.}\end{array}]]
\mathrm{sscanf}\left("1 2\n3 4","%\left\{;h\right\}fa"\right)
[[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1.}& \textcolor[rgb]{0,0,1}{2.}\\ \textcolor[rgb]{0,0,1}{3.}& \textcolor[rgb]{0,0,1}{4.}\end{array}]]
\mathrm{sscanf}\left("1 1 3+2I\n2 2 4-5I","%\left\{3,3;e\left(2\right)c8\right\}Zfm"\right)
[[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{3.}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2.}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{I}& \textcolor[rgb]{0,0,1}{0.}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{0.}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{I}& \textcolor[rgb]{0,0,1}{0.}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{0.}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{I}\\ \textcolor[rgb]{0,0,1}{0.}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{0.}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{I}& \textcolor[rgb]{0,0,1}{4.}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{5.}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{I}& \textcolor[rgb]{0,0,1}{0.}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{0.}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{I}\\ \textcolor[rgb]{0,0,1}{0.}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{0.}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{I}& \textcolor[rgb]{0,0,1}{0.}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{0.}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{I}& \textcolor[rgb]{0,0,1}{0.}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{0.}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{I}\end{array}]]
\mathrm{sscanf}\left("1\t2\nabc 123\t4","%\left\{2,2;d\left(\t\right)\right\}sm"\right)
[[\begin{array}{cc}\textcolor[rgb]{0,0,1}{"1"}& \textcolor[rgb]{0,0,1}{"2"}\\ \textcolor[rgb]{0,0,1}{"abc 123"}& \textcolor[rgb]{0,0,1}{"4"}\end{array}]]
\mathrm{sscanf}\left("1\t2\nabc + 123\t1 / \left(x+y\right)","%\left\{2,2;d\left(\t\right)\right\}am"\right)
[[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{\mathrm{abc}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{123}& \frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{y}}\end{array}]]
\mathrm{sscanf}\left("1,2,,\n3,x,5,\n6,y,8,9\n","%\left\{3,4\right\}vm"\right)
[[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{"1"}& \textcolor[rgb]{0,0,1}{"2"}& \textcolor[rgb]{0,0,1}{""}& \textcolor[rgb]{0,0,1}{""}\\ \textcolor[rgb]{0,0,1}{"3"}& \textcolor[rgb]{0,0,1}{"x"}& \textcolor[rgb]{0,0,1}{"5"}& \textcolor[rgb]{0,0,1}{""}\\ \textcolor[rgb]{0,0,1}{"6"}& \textcolor[rgb]{0,0,1}{"y"}& \textcolor[rgb]{0,0,1}{"8"}& \textcolor[rgb]{0,0,1}{"9"}\end{array}]]
\mathrm{sscanf}\left("1,2,,\n3,x,5,\n6,y,8,9\n","%\left\{3,4;f8m\left(-1\right)\right\}vm"\right)
[[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{1.}& \textcolor[rgb]{0,0,1}{2.}& \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{0.}\\ \textcolor[rgb]{0,0,1}{3.}& \textcolor[rgb]{0,0,1}{-1.}& \textcolor[rgb]{0,0,1}{5.}& \textcolor[rgb]{0,0,1}{0.}\\ \textcolor[rgb]{0,0,1}{6.}& \textcolor[rgb]{0,0,1}{-1.}& \textcolor[rgb]{0,0,1}{8.}& \textcolor[rgb]{0,0,1}{9.}\end{array}]]
\mathrm{sscanf}\left("1 2","%\left\{2,3;o\left(999\right)\right\}dm"\right)
[[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{999}\\ \textcolor[rgb]{0,0,1}{999}& \textcolor[rgb]{0,0,1}{999}& \textcolor[rgb]{0,0,1}{999}\end{array}]]
\mathrm{sscanf}\left("1 -2 -3 4","%\left\{2,2;y\left(posint\right)m\left(9999\right)\right\}dm"\right)
[[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{9999}\\ \textcolor[rgb]{0,0,1}{9999}& \textcolor[rgb]{0,0,1}{4}\end{array}]]
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Lifting Method for Constructing Wavelets - MATLAB & Simulink - MathWorks í•œêµ
The link with “true†wavelets (how to generate, starting from the filters, orthogonal or biorthogonal bases of the space of the functions of finite energy)
{X}_{0}\left(z\right)=\underset{n}{â}x\left(2n\right){z}^{ân}
{X}_{1}\left(z\right)=\underset{n}{â}x\left(2n+1\right){z}^{ân}
X\left(z\right)=\underset{n}{â}x\left(2n\right){z}^{â2n}+\underset{n}{â}x\left(2n+1\right){z}^{â2n+1}={X}_{0}\left({z}^{2}\right)+{z}^{â1}{X}_{1}\left({z}^{2}\right)
x\left(2n+1\right)
d\left(n\right)=x\left(2n+1\right)-x\left(2n\right)
x\left(2n\right)
x\left(2n\right)
x\left(2n\right)+d\left(n\right)/2
\left(x\left(2n\right)+x\left(2n+1\right)\right)/2
\left[\begin{array}{cc}1& 0\\ -\mathit{P}\left(\mathit{z}\right)& 1\end{array}\right]\left[\begin{array}{c}{\mathit{X}}_{0}\left(\mathit{z}\right)\\ {\mathit{X}}_{1}\left(\mathit{z}\right)\end{array}\right]
P\left(z\right)=1
\left[\begin{array}{cc}1& \mathit{S}\left(\mathit{z}\right)\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 0\\ -\mathit{P}\left(\mathit{z}\right)& 1\end{array}\right]\left[\begin{array}{c}{\mathit{X}}_{0}\left(\mathit{z}\right)\\ {\mathit{X}}_{1}\left(\mathit{z}\right)\end{array}\right]
S\left(z\right)=1/2
\left[\begin{array}{cc}\sqrt{2}& 0\\ 0& \frac{1}{\sqrt{2}}\end{array}\right]\left[\begin{array}{cc}1& \mathit{S}\left(\mathit{z}\right)\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 0\\ -\mathit{P}\left(\mathit{z}\right)& 1\end{array}\right]\left[\begin{array}{c}{\mathit{X}}_{0}\left(\mathit{z}\right)\\ {\mathit{X}}_{1}\left(\mathit{z}\right)\end{array}\right]
d\left(n\right)=x\left(2n+1\right)-\frac{1}{2}\left[x\left(2n\right)+x\left(2n+2\right)\right]
\left[\begin{array}{cc}1& 0\\ -\frac{1}{2}\left(1-\mathit{z}\right)& 1\end{array}\right]\left[\begin{array}{c}{\mathit{X}}_{0}\left(\mathit{z}\right)\\ {\mathit{X}}_{1}\left(\mathit{z}\right)\end{array}\right]
\underset{n}{â}x\left(n\right)=\frac{1}{2}\underset{n}{â}a\left(n\right)
\left[\begin{array}{cc}1& \frac{1}{4}\left({\mathit{z}}^{-1}+1\right)\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 0\\ -\frac{1}{2}\left(1+\mathit{z}\right)& 1\end{array}\right]\left[\begin{array}{c}{\mathit{X}}_{0}\left(\mathit{z}\right)\\ {\mathit{X}}_{1}\left(\mathit{z}\right)\end{array}\right]
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Home : Support : Online Help : Mathematics : Linear Algebra : LinearAlgebra Package : Modular Subpackage : Adjoint
compute the Inverse of a square mod m Matrix
compute the Adjoint of a square mod m Matrix
Inverse(m, A, det, B, meth)
Adjoint(m, A, det, B, meth)
(optional) name to use for output determinant
(optional) Matrix to use for output Inverse or Adjoint
(optional) method to use for computing Inverse or Adjoint
The Inverse and Adjoint functions compute the inverse and adjoint, respectively, of a square mod m Matrix.
If det is specified, it is assigned the value of the determinant on successful completion.
If B is specified, it must have dimensions and datatype identical to A, and will contain the inverse or adjoint on successful completion. In this case the command will return NULL.
The default method for the inverse is LU, while the default method for the adjoint is RET, and these can be changed by specification of meth.
Allowable options are:
Obtain inverse or adjoint via LU decomposition.
inplaceLU
Obtain inverse or adjoint via LU decomposition destroying the
data in A in the process.
Obtain inverse or adjoint through application of row reduction
to an identity augmented mod m Matrix.
Obtain inverse or adjoint through application of a row echelon
transform to A.
transform to A, replacing A with the inverse or adjoint.
The LU and inplaceLU methods are the most efficient for small to moderate sized problems. The RET and inplaceRET methods are the most efficient for very large problems. The RREF method is the most flexible for nonsingular matrices.
For the inplaceRET method, B should never be specified, as the output replaces A. For this method, the commands always return NULL.
With the LU-based and RET-based methods, it is generally required that m be a prime, as mod m inverses are needed, but in some cases it is possible to obtain an LU decomposition or a Row-Echelon Transform for m composite.
For the cases where LU Decomposition or Row-Echelon Transform cannot be obtained for m composite, the function returns an error indicating that the algorithm failed because m is composite.
Note: There are cases with composite m for which the inverse and adjoint exist, but no LU decomposition or Row-Echelon Transform is possible.
If it exists, the RREF method always finds the mod m inverse. The RREF method also finds the adjoint if the Matrix is nonsingular.
The RET method is the only method capable of computing the adjoint if the matrix is singular. The inplaceRET method cannot be used to compute the adjoint of a singular matrix, as this operation cannot be performed in-place.
These commands are part of the LinearAlgebra[Modular] package, so they can be used in the form Inverse(..) and Adjoint(..) only after executing the command with(LinearAlgebra[Modular]). However, they can always be used in the form LinearAlgebra[Modular][Inverse](..) and LinearAlgebra[Modular][Adjoint](..).
Basic 3x3 Matrix.
\mathrm{with}\left(\mathrm{LinearAlgebra}[\mathrm{Modular}]\right):
p≔97
\textcolor[rgb]{0,0,1}{p}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{97}
M≔\mathrm{Mod}\left(p,\mathrm{Matrix}\left(3,3,\left(i,j\right)↦\mathrm{rand}\left(\right)\right),\mathrm{integer}[]\right)
\textcolor[rgb]{0,0,1}{M}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{77}& \textcolor[rgb]{0,0,1}{96}& \textcolor[rgb]{0,0,1}{10}\\ \textcolor[rgb]{0,0,1}{86}& \textcolor[rgb]{0,0,1}{58}& \textcolor[rgb]{0,0,1}{36}\\ \textcolor[rgb]{0,0,1}{80}& \textcolor[rgb]{0,0,1}{22}& \textcolor[rgb]{0,0,1}{44}\end{array}]
\mathrm{Mi}≔\mathrm{Inverse}\left(p,M\right)
\textcolor[rgb]{0,0,1}{\mathrm{Mi}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{16}& \textcolor[rgb]{0,0,1}{80}& \textcolor[rgb]{0,0,1}{72}\\ \textcolor[rgb]{0,0,1}{20}& \textcolor[rgb]{0,0,1}{20}& \textcolor[rgb]{0,0,1}{32}\\ \textcolor[rgb]{0,0,1}{5}& \textcolor[rgb]{0,0,1}{65}& \textcolor[rgb]{0,0,1}{89}\end{array}]
\mathrm{Multiply}\left(p,M,\mathrm{Mi}\right),\mathrm{Multiply}\left(p,\mathrm{Mi},M\right)
[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}\end{array}]
An example that fails with the LU and RET methods, but succeeds with RREF.
m≔6
\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{6}
M≔\mathrm{Mod}\left(m,[[3,2],[2,1]],\mathrm{float}[8]\right)
\textcolor[rgb]{0,0,1}{M}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{3.}& \textcolor[rgb]{0,0,1}{2.}\\ \textcolor[rgb]{0,0,1}{2.}& \textcolor[rgb]{0,0,1}{1.}\end{array}]
\mathrm{Mi}≔\mathrm{Inverse}\left(m,M\right)
Error, (in LinearAlgebra:-Modular:-LUDecomposition) no LU decomposition exists: modulus is composite
\mathrm{Mi}≔\mathrm{Inverse}\left(m,M,'\mathrm{RET}'\right)
Error, (in LinearAlgebra:-Modular:-Inverse) modulus is composite
\mathrm{Mi}≔\mathrm{Inverse}\left(m,M,'\mathrm{RREF}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{Mi}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{5.}& \textcolor[rgb]{0,0,1}{2.}\\ \textcolor[rgb]{0,0,1}{2.}& \textcolor[rgb]{0,0,1}{3.}\end{array}]
\mathrm{Multiply}\left(m,M,\mathrm{Mi}\right),\mathrm{Multiply}\left(m,\mathrm{Mi},M\right)
[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1.}& \textcolor[rgb]{0,0,1}{0.}\\ \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{1.}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1.}& \textcolor[rgb]{0,0,1}{0.}\\ \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{1.}\end{array}]
An example where no inverse exists, but the adjoint does exist.
m≔6
\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{6}
M≔\mathrm{Mod}\left(m,[[2,4],[4,4]],\mathrm{float}[8]\right)
\textcolor[rgb]{0,0,1}{M}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{2.}& \textcolor[rgb]{0,0,1}{4.}\\ \textcolor[rgb]{0,0,1}{4.}& \textcolor[rgb]{0,0,1}{4.}\end{array}]
\mathrm{Mi}≔\mathrm{Inverse}\left(m,M,'\mathrm{RREF}'\right)
Error, (in LinearAlgebra:-Modular:-RowReduce) no inverse exists: modulus is composite
\mathrm{Ma}≔\mathrm{Adjoint}\left(m,M,'\mathrm{det}','\mathrm{RREF}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{Ma}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{4.}& \textcolor[rgb]{0,0,1}{2.}\\ \textcolor[rgb]{0,0,1}{2.}& \textcolor[rgb]{0,0,1}{2.}\end{array}]
\mathrm{det},\mathrm{Multiply}\left(m,M,\mathrm{Ma}\right),\mathrm{Multiply}\left(m,\mathrm{Ma},M\right)
\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{4.}& \textcolor[rgb]{0,0,1}{0.}\\ \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{4.}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{4.}& \textcolor[rgb]{0,0,1}{0.}\\ \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{4.}\end{array}]
An example where only the RET method succeeds at computing the adjoint.
m≔7
\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{7}
M≔\mathrm{Mod}\left(m,[[1,1],[1,1]],\mathrm{integer}\right)
\textcolor[rgb]{0,0,1}{M}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{1}\end{array}]
\mathrm{Ma}≔\mathrm{Adjoint}\left(m,M,'\mathrm{RREF}'\right)
Error, (in LinearAlgebra:-Modular:-RowReduce) matrix is singular
\mathrm{Ma}≔\mathrm{Adjoint}\left(m,M,'\mathrm{LU}'\right)
Error, (in LinearAlgebra:-Modular:-LUDecomposition) matrix is singular
\mathrm{Ma}≔\mathrm{Adjoint}\left(m,M,'\mathrm{det}','\mathrm{RET}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{Ma}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{6}\\ \textcolor[rgb]{0,0,1}{6}& \textcolor[rgb]{0,0,1}{1}\end{array}]
\mathrm{det},\mathrm{Multiply}\left(m,M,\mathrm{Ma}\right),\mathrm{Multiply}\left(m,\mathrm{Ma},M\right)
\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\end{array}]
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Measure of variability of condition indicators at failure - MATLAB prognosability - MathWorks 日本
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-
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\text{prognosability }=\text{ exp}\left(â\frac{{\text{std}}_{j}\left({x}_{j}\left({N}_{j}\right)\right)}{{\text{mean}}_{j}|{x}_{j}\left(1\right)â{x}_{j}\left({N}_{j}\right)|}\right),\text{ }j\text{ = }1,...,M
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The ground-state energy and properties of any many-electron atom or molecule may be rigorously computed by variationally computing the two-electron reduced density matrix rather than the many-electron wavefunction. While early attempts fifty years ago to compute the ground-state 2-RDM directly were stymied because the 2-RDM must be constrained to represent an
N
-electron wavefunction, recent advances in theory and optimization have made direct computation of the 2-RDM possible. The constraints in the variational calculation of the 2-RDM require a special optimization known as a semidefinite programming. Development of first-order semidefinite programming for the 2-RDM method has reduced the computational costs of the calculation by orders of magnitude [Mazziotti, Phys. Rev. Lett. 93 (2004) 213001]. The variational 2-RDM approach is effective at capturing multi-reference correlation effects that are especially important at non-equilibrium molecular geometries. Recent work on 2-RDM methods will be reviewed and illustrated with particular emphasis on the importance of advances in large-scale semidefinite programming.
Classification : 90C22, 81Q05, 52A40
Mots clés : semidefinite programming, electron correlation, reduced density matrices,
N
-representability conditions
author = {Mazziotti, David A.},
title = {First-order semidefinite programming for the two-electron treatment of many-electron atoms and molecules},
AU - Mazziotti, David A.
TI - First-order semidefinite programming for the two-electron treatment of many-electron atoms and molecules
Mazziotti, David A. First-order semidefinite programming for the two-electron treatment of many-electron atoms and molecules. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 41 (2007) no. 2, pp. 249-259. doi : 10.1051/m2an:2007021. http://www.numdam.org/articles/10.1051/m2an:2007021/
[1] D.R. Alcoba, F.J. Casquero, L.M. Tel, E. Perez-Romero and C. Valdemoro, Convergence enhancement in the iterative solution of the second-order contracted Schrödinger equation. Int. J. Quantum Chem. 102 (2005) 620-628.
[2] M.D. Benayoun, A.Y. Lu and D.A. Mazziotti, Invariance of the cumulant expansion under 1-particle unitary transformations in reduced density matrix theory. Chem. Phys. Lett. 387 (2004) 485-489.
[3] D.P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods. Academic Press, New York (1982). | MR 690767 | Zbl 0572.90067
[4] S. Burer and C. Choi, Computational enhancements in low-rank semidefinite programming. Optim. Methods Soft. 21 (2006) 493-512. | Zbl 1136.90429
[5] S. Burer and R.D.C. Monteiro, Nonlinear programming algorithm for solving semidefinite programs via low-rank factorization. Math. Program. Ser. B 95 (2003) 329-357. | Zbl 1030.90077
[6] S. Burer and R.D.C. Monteiro, Local minima and convergence in low-rank semidefinite programming. Math. Program. Ser. A 103 (2005) 427-444. | Zbl 1099.90040
[7] L. Cohen and C. Frishberg, Hierarchy equations for reduced density matrices, Phys. Rev. A 13 (1976) 927-930.
[8] A.J. Coleman, Structure of fermion density matrices. Rev. Mod. Phys. 35 (1963) 668. | MR 155637
[9] A.J. Coleman and V.I. Yukalov, Reduced Density Matrices: Coulson's Challenge. Springer-Verlag, New York (2000). | Zbl 0998.81506
[10] F. Colmenero and C. Valdemoro, Approximating q-order reduced density-matrices in terms of the lower-order ones. 2. Applications. Phys. Rev. A 47 (1993) 979-985.
[11] F. Colmenero and C. Valdemoro, Self-consistent approximate solution of the 2nd-order contracted Schrödinger equation. Int. J. Quantum Chem. 51 (1994) 369-388.
[12] A.R. Conn, I.M. Gould and P.L. Toint, Trust-Region Methods. SIAM: Philadelphia (2000). | MR 1774899 | Zbl 0958.65071
[13] C.A. Coulson, Present state of molecular structure calculations. Rev. Mod. Phys. 32 (1960) 170-177.
[14] R.M. Erdahl, Representability. Int. J. Quantum Chem. 13 (1978) 697-718.
[15] R.M. Erdahl, Two algorithms for the lower bound method of reduced density matrix theory. Reports Math. Phys. 15 (1979) 147-162. | Zbl 0441.49056
[16] R.M. Erdahl and B. Jin, The lower bound method for reduced density matrices. J. Mol. Struc. (Theochem) 527 (2000) 207-220.
[17] R. Fletcher, Practical Methods of Optimization. John Wiley and Sons, New York (1987). | MR 955799 | Zbl 0905.65002
[18] M. Fukuda, B.J. Braams, M. Nakata, M.L. Overton, J.K. Percus, M. Yamashita and Z. Zhao, Large-scale semidefinite programs in electronic structure calculation. Math. Program., Ser. B 109 (2007) 553. | MR 2296564 | Zbl 1278.90495
[19] C. Garrod and J. Percus, Reduction of N-particle variational problem. J. Math. Phys. 5 (1964) 1756-1776. | Zbl 0129.44401
[20] G. Gidofalvi and D.A. Mazziotti, Boson correlation energies via variational minimization with the two-particle reduced density matrix: Exact
N
-representability conditions for harmonic interactions. Phys. Rev. A 69 (2004) 042511.
[21] G. Gidofalvi and D.A. Mazziotti, Application of variational reduced-density-matrix theory to organic molecules. J. Chem. Phys. 122 (2005) 094107.
[22] G. Gidofalvi and D.A. Mazziotti, Application of variational reduced-density-matrix theory to the potential energy surfaces of the nitrogen and carbon dimers. J. Chem. Phys. 122 (2005) 194104.
[23] G. Gidofalvi and D.A. Mazziotti, Spin- and symmetry-adapted two-electron reduced-density-matrix theory. Phys. Rev. A 72 (2005) 052505.
[24] G. Gidofalvi and D.A. Mazziotti, Potential energy surface of carbon monoxide in the presence and absence of an electric field using the two-electron reduced-density-matrix method. J. Phys. Chem. A 110 (2006) 5481-5486.
[25] G. Gidofalvi and D.A. Mazziotti, Computation of quantum phase transitions by reduced-density-matrix mechanics. Phys. Rev. A 74 (2006) 012501.
[26] J.R. Hammond and D.A. Mazziotti, Variational two-electron reduced-density-matrix theory: Partial 3-positivity conditions for
N
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Precinct-summability through seat capping Precinct-summability through seat capping | Technically Exists
Precinct-summability through seat capping
Many multi-winner voting methods, including most proportional methods, are not precinct-summable, which means that it is often not practical for precincts to generate and publicly publish totals that can be used to independently verify the winner of an election. However, seat capping is a simple approach that can be applied to many (though not all) multi-winner methods to make them precinct-summable, at least to some degree. As the name implies, seat capping works by placing a limit on the maximum number of seats that a voting method can be used to fill.
The generic seat capping technique works for any voting method in which the set of candidates elected depends on only the set of levels of support each ballot gives to the candidates in each possible winner set. Examples of such methods include sequential Monroe voting and Bucklin transferable vote.
This type of voting method can be modified to use
O(\log(V) \cdot \ell^k \cdot c^k)
bits for precinct totals, where V is the number of voters, c is the number of candidates, k is the number of seats, and â„“ is the total number of levels of support that can be expressed on a ballot. If the number of seats that can be elected is capped at k, then a precinct P needs to provide, for every possible set of winners
\{w_1, w_2, \dots, w_k\}
and for every list of winners’ levels of support
s
, the number of ballots that gave those levels of support to those candidates:
\vert \{b \in P \mid b(w_1), b(w_2), \dots, b(w_k) = s\} \vert
\vert P \vert \le V
, this sum has a size of at most
O(\log(V))
bits. There are
O(c^k)
possible sets of winners, and furthermore, there are
O(\ell^k)
lists of levels of support that can be given to the candidates in a possible set of winners. Thus, the total size of all precinct totals is
O(\log(V) \cdot \ell^k \cdot c^k)
, and the modified voting method is kth-order summable, assuming â„“ is fixed. If the voting method uses rankings and does not limit the number of candidates that may be ranked, then â„“ = c and the modified voting method is instead 2kth-order summable.
For many voting methods, it is possible to provide fewer totals than the generic seat capping technique generates.
Weighted score selection methods
A weighted score selection method is a sequential multi-winner voting method in which the winner of each round is the candidate with the greatest sum of weighted scores, where each ballot’s weight depends on the scores given (by all ballots) to the candidates that were elected in previous rounds. Examples of such methods include allocated score and sequentially spent score.
O(\log(V) \cdot \ell^{k-1} \cdot c^k)
bits for precinct totals. Consider every set of candidates of size k - 1 and every set of levels of support that a ballot can give to those candidates. For each of these, provide the number of ballots that give those levels of support to those candidates, as well as the total score given by these ballots to each of the candidates not in the set. These totals are sufficient to allow all k winners to be identified.
Simple reweighted methods
A simple reweighted method is a sequential multi-winner voting method in which the winner of each round is the candidate with the greatest sum of weighted scores, where each ballot’s weight depends only on the scores it gave to the candidates that were elected in previous rounds. Examples of such methods include reweighted range voting and sequential threshold average score voting.
O(\log(V) \cdot c^k)
bits for precinct totals. Consider every set of candidates of size k - 1 or less (including the empty set). For each set, provide the total score that would be given by all ballots to each of the candidates not in the set if the candidates in the set were already elected. These totals are sufficient to allow all k winners to be identified.
Approval and KP transform methods
An approval method is a multi-winner voting method that uses approval ballots, and a KP transform method is a multi-winner voting method that uses the Kotze-Pereira transformation to convert its ballots into approval ballots before applying an approval method to them. Examples of approval methods include sequential proportional approval voting and harmonic approval voting. Examples of KP transform methods include sequential proportional score voting and harmonic score voting.
Most such voting methods can be modified to use
O(\log(V) \cdot c^k)
bits for precinct totals. Consider every set of candidates of size k or less (including the empty set). For each set, provide the number of approval ballots that approve all of the candidates in the set. For most approval and KP transform methods, these totals are sufficient to allow all k winners to be identified. One exception to this is satisfaction approval voting; despite being 1st-order summable, these particular totals aren’t enough to find the winners under this method.
Incompatible methods
The most commonly used party-agnostic proportional method, single transferable vote (STV), is unfortunately incompatible with seat capping. This conflict arises because STV often eliminates candidates that were never elected in order to free up their votes for transfer. A similar alternative that is compatible with seat capping is the proportional ranked voting method Bucklin transferable vote.
The primary drawback of seat capping is that it can only give kth-order summability by capping the number of seats per election at k. This means that capping a voting method at 7 seats, a commonly-proposed maximum district size, will result in a voting method that is only 7th-order summable. Such a method may technically satisfy the definition of precinct-summability, but in practice it will generate too many totals for summation to be practical. Seat capping is likely most useful with districts composed of 3 seats each, which are large enough to ensure decent proportionality while also being small enough to result in 3rd-order summable voting methods.
One other drawback of seat capping is that it may not be clear how the totals it provides can be used to find the winners under a given voting method. This can be solved on a case-by-case basis by providing modified implementation instructions for voting methods making use of seat capping.
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Introduction to Biology/Matter, Atoms, and Bonds/Chemical Reactions
A chemical reaction occurs when two or more atoms form chemical bonds or when chemical bonds between atoms break. In all chemical reactions, the chemical identity of at least one substance changes. Some chemical reactions include the oxidation of iron (also known as rusting) and the release of chlorine gas and other compounds from the mixing of bleach and ammonia.
In chemical reactions, bonds are broken and formed between atoms. Chemical equations are used to represent chemical reactions. In this reaction, methane and oxygen react to produce carbon dioxide and water. Chemical reactions may or may not be reversible.
Chemical equations are used to represent chemical reactions. A reactant is a substance that reacts in a chemical reaction. Reactants are found on the left side of the equation. A product is a substance formed as the result of a chemical reaction. Products are found on the right side of the equation. For example, the equation CH4 + 2O2 → CO2 + 2H2O represents the combustion (burning) of methane. Methane (CH4) and oxygen gas (O2) are the reactants, so they are listed on the left side of the equation. The arrow represents the fact that the chemical reaction is irreversible; that is, once the products have been formed, they cannot be returned to their original state. The products are carbon dioxide (CO2) and water vapor (H2O), which are listed on the right side of the equation.
Chemical equations can be written unbalanced, that is, with an unequal number of atoms in the reactants versus the products, or balanced, with an equal number of atoms on each side of the equation. A balanced chemical equation adheres to the law of conservation of matter. This law states the number of atoms before and after a chemical reaction are equal to ensure atoms are not created or destroyed. Therefore, a balanced equation better represents the way the reaction takes place in the real world. In the example of the combustion of methane, the unbalanced chemical equation reads CH4 + O2 → CO2 + H2O. In this equation, there are four hydrogen atoms in the reactants, but only two hydrogen atoms in the products. Adding a coefficient of two to H2O corrects the number of hydrogen atoms, but means that the number of oxygen atoms is no longer balanced. Thus, a coefficient of two must be added to oxygen in the reactants as well. The equation is now fully balanced.
Chemical equilibrium is a state of balance in which forward and reverse chemical reactions are happening at the same rate. Chemical equilibrium plays a role in the reversibility of a chemical reaction. The concentration ratio of reactants and products is also stabilized during equilibrium.
The arrow's direction in a chemical equation indicates how a chemical reaction proceeds. Chemical reactions are irreversible when the arrow only points toward the products. This indicates that the reaction will proceed until all reactants are used. For example, when sodium hydroxide (NaOH) is combined with hydrochloric acid (HCl), sodium chloride (NaCl) and water (H2O) are produced. Sodium chloride and water do not react to produce sodium hydroxide and hydrochloric acid.
\mathrm{NaOH}+\mathrm{HCl}\;\rightarrow\mathrm{NaCl}+{\mathrm H}_2\mathrm O
Sometimes a double arrow is present in a chemical equation. This indicates that the reaction is reversible. Thus, the chemical reaction can proceed in the forward and reverse direction. For example, nitrous oxide (NO2) converts to dinitrogen tetroxide (N2O4), which converts back to nitrous oxide. The (g) in the equation shows that the compound is a gas.
2\;{\mathrm{NO}}_{2(\mathrm g)}\;\rightleftarrows{\mathrm N}_2{\mathrm O}_{4(\mathrm g)}
During reversible chemical reactions, reactants can convert to products, and products can convert to reactants, after a certain threshold of the reaction is reached. This process will occur until equilibrium is established between the reactants and products. The equilibrium constant is a number that expresses the relationship between the amounts of products and reactants present when a reversible chemical reaction is in balance.
<Chemical Bonding>Suggested Reading
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Physics - A new quasiparticle in carbon nanotubes
A new quasiparticle in carbon nanotubes
Institut für Physik, Karl-Franzens-Universität Graz, Universitätsplatz 5, 8010 Graz, Austria
Dipartimento di Fisica, Università di Modena e Reggio Emilia, Via Campi 213/A, 41100 Modena, Italy
Trions—one electron bound to two holes via Coulomb forces—can be observed in the optical spectra of doped carbon nanotubes.
Figure 1: Trions in carbon nanotubes—Coulomb-bound carrier complexes consisting of one electron and two holes with opposite spin orientations—would be highly beneficial for optical spin manipulation and for the investigation of correlated carrier dynamics in one-dimensional systems, thanks to their unpaired spins. In this work the authors report evidence that trions can be optically excited in carefully prepared
p
-doped nanotubes and they are more stable than expected earlier.Trions in carbon nanotubes—Coulomb-bound carrier complexes consisting of one electron and two holes with opposite spin orientations—would be highly beneficial for optical spin manipulation and for the investigation of correlated carrier dynamics in o... Show more
Natural science progresses mostly because of unexpected observations. At times, however, experimental confirmation follows theoretical prediction. The quest for the Higgs particle at CERN is probably the most prominent current example of such a sequence. Solid-state physics, too, provides us with examples of particles that have been postulated but not yet observed in all expected scenarios. An example in semiconductors is a trion—a Coulomb-bound carrier complex of one electron and two holes (Fig. 1). Trions have been observed in a variety of systems, but surprisingly, in view of their rich prospects for device applications and basic research, not in semiconducting carbon nanotubes (CNTs). In a paper in Physical Review Letters [1], Ryusuke Matsunaga, Kazunari Matsuda, and Yoshihiko Kanemitsu, all at Kyoto University, Japan, report the first observation of trions in -doped nanotubes. If confirmed by further experiments, this would be a major breakthrough in the field.
Carbon nanotubes—rolled cylinders of graphene a few nanometers in diameter—come in multiple shapes and sizes, and are either metallic or semiconducting, depending on the width of the ribbon and how it is rolled (twisted by one of the many possible discrete angles, or even untwisted). Most nanotubes are not fabricated; instead, they self assemble. Different species of nanotubes, such as those isolated in Matsunaga et al.’s experiment, can be distinguished by their optical fingerprints, similar to atoms and molecules. Nanotubes were first observed in 1991 [2], more than a decade before the discovery of graphene, and their study has since developed into a vast and rich research field, revealing their unique structural, electronic, and optoelectronic properties [3].
When a semiconductor absorbs a photon, an electron is promoted from the valence to the conduction band, leaving behind a hole in the electron sea that is described as an independent quasiparticle with positive charge. The photogenerated electron-hole pair can bind through attractive Coulomb forces to form a long-lived hydrogenlike carrier complex, named an exciton. It takes nanoseconds for the electron and the hole to annihilate and re-emit a photon. Exciton binding is typically enhanced in semiconductors of reduced dimensionality. An extreme case is found in nanotubes where the wave functions of the photogenerated carriers are spread over the circumference of the nanotube, and the electron and hole “see” each other across the tube. This intimate contact results in a strongly bound exciton that can move freely along the tube, with a binding energy typically of the order of several tenths of an electron volt.
Matsunaga et al. [1] proceed by attaching, through -doping, an additional hole to the exciton. Similar to the ionized hydrogen molecule , which consists of two protons and one electron, the two holes and the electron form a Coulomb-bound carrier complex referred to as a charged exciton or trion. The neat thing about the trion is that it has a net charge and thus can be controlled by electrical gates. Trions could be also used for optical spin manipulation or to investigate the correlated carrier dynamics of one-dimensional Luttinger liquids—a peculiar state of strongly interacting systems in one dimension, where charge and spin density waves propagate independently at different group velocities—by optical means. Contrary to the molecule, however, the electron and hole forming the trion have nearly identical masses. Therefore binding of the trion is mainly due to correlations—the restless dance of particles around each other—and the binding is expected to be rather weak. If it is too weak, trions would be unstable against Auger recombination—a process where an electron-hole pair recombines and the excess energy is passed to the surplus hole, which is about a 100 times faster than optical recombination [4].
The smoking gun of the trions in the experiment of Matsunaga et al. is the appearance of an additional peak in the optical spectra on the low-energy end of the exciton peak. Matsunaga et al. work hard to show that this peak indeed originates from trions and not, for example, from defects introduced through doping. They first investigate the influence of different dopants and doping concentrations and find that new peaks associated with trions in nanotubes appear at the same energy, regardless of the dopant species. They also become stronger with increasing doping concentration, along with a reduction of the exciton peak. Moreover, they find that excitons in nanotubes with different diameters and twist angles all come along with a corresponding trion partner and show clear “family patterns,” similar to those known for excitons in nanotubes. All this provides strong evidence for trions.
A striking feature of the experiments is the extremely large energy separation ( – ) between the exciton and trion peaks, which the authors attribute to the large electron-hole exchange interaction in carbon-based materials. The exchange interaction accounts for the fact that the antisymmetric wave function keeps electrons with parallel spin orientations at bay, resulting in a lowering of the repulsive Coulomb energy. For the same reason, an optically excited electron-hole pair with opposite spin orientations (bright exciton) is less correlated than a pair where the hole spin is flipped (dark exciton), because the total number of spin-correlated valence and conduction band electrons differs by one. The resulting energy separation is denoted as the electron-hole exchange splitting, which is usually large for tightly confined carrier complexes such as excitons and trions in nanotubes. Matsunaga et al. speculate that this exchange interaction could lower the trion state’s energy below that of the dark exciton state and thus could make it a stable configuration, even at room temperature. Further work, both experimental and theoretical, is needed to resolve the issue of trion stability.
The authors stress that it was important to use samples with very few bundles of nanotubes to observe trions successfully. These bundles are not susceptible to being doped with holes and to doping-induced spectral changes, making it necessary to use particularly high-quality samples to observe the weak trion signals. The key to success was a novel dispersion technique for high-quality samples [5], together with ultracentrifugation—a technique by which polydisperse samples can be sorted in portions enriched in certain species—and a long time (up to seven hours) to exclude as many residual bundles as possible.
A colleague [6] notes that though more work is needed to unambiguously assign the new peaks to trions, this first observation of trions in carbon nanotubes seems to be a breakthrough. In any case, it provides an example of the discovery of new quasiparticles in solid-state systems, where the creators of materials remain the true heroes of the trade.
R. Matsunaga, K. Matsuda, and Y. Kanemitsu, Phys. Rev. Lett. 106, 037404 (2011)
S. Iijima, Nature 354, 56 (1991)
Carbon Nanotubes: Advanced Topics in the Synthesis, Structure, Properties, and Applications, edited by A. Jorio et al. (Springer, Berlin, 2008)[Amazon][WorldCat]
F. Wang et al., Phys. Rev. B 70, 241403 (2004)
A. Nish et al., Nature Nanotech. 2, 630 (2007)
A. Imamoglu (private communication)
Ulrich Hohenester is an associate professor at the Karl-Franzens University of Graz. In 1997 he received his Ph.D. in theoretical physics at Graz University and then worked several years as a postdoc at the University of Modena. His main research interests are in theoretical and computational nanoscience, with the main focus on semiconductor and plasmonic nanoparticles, as well as on general concepts of open quantum systems and optimal quantum control with ultracold atoms.
Guido Goldoni is a researcher and lecturer at the University of Modena and Reggio Emilia and an affiliate of the CNR-NANO S3 center. He graduated in 1988 at the University of Modena and received his Ph.D. in physics of condensed matter from the International School for Advanced Studies in Trieste in 1993. He worked for several years as a postdoc at the University of Antwerp and at the University of Modena. His primary research interests are in the electronic and optical properties of semiconductor and molecular systems at the nanoscale.
Ryusuke Matsunaga, Kazunari Matsuda, and Yoshihiko Kanemitsu
OpticsNanophysics
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Data Mining for Flooding Episode in the States of Alagoas and Pernambuco—Brazil
1Laboratory for Computing and Applied Mathematics, National Institute for Space Research, São José dos Campos, Brazil.
2Center for Weather Forecasting and Climate Research, National Institute for Space Research, São José dos Campos, Brazil.
3Goddard Earth Sciences Technology and Research, Universities Space Research Association, Columbia, MD, USA.
4Global Modeling and Assimilation Office, NASA Goddard Space Flight Center, Greenbelt, MD, USA.
Ruivo, H. , Velho, H. , Ramos, F. and Freitas, S. (2018) Data Mining for Flooding Episode in the States of Alagoas and Pernambuco—Brazil. American Journal of Climate Change, 7, 420-430. doi: 10.4236/ajcc.2018.73025.
H\left(S\right)=-{\sum }_{x\in X}p\left(x\right)\mathrm{log}\left\{p\left(x\right)\right\}
IG\left(A,S\right)=H\left(S\right)-{\sum }_{t\in T}p\left(t\right)H\left(t\right)
{U}_{t\in {T}^{t}}
[1] Ruivo, H.M., Sampaio, G. and Ramos, F.M. (2014) Knowledge Extraction from Large Climatological Data Sets Using a Genome-Wide Analysis Approach: Application to the 2005 and 2010 Amazon Droughts. Climatic Change, 124, 347-361.
[2] Ruivo, H.M., Campos Velho, H.F., Sampaio, G. and Ramos, F.M. (2015) Analysis of Extreme Precipitation Events Using a Novel Data Mining Approach. American Journal of Environmental Engineering, 5, 96-105.
[3] Ruivo, H.M., Campos Velho, H.F., Ramos, F.M. and Sampio, G. (2013) P-Value and Decision Tree for Analysis of Extreme Rainfall. Ciência e Natura, 1, 231-234.
[4] Fayyad, U., Piatesky-Shapiro, G., Smyth, P. and Uthurusamy, R. (1996) Advances in Knowledge Discovery and Data Mining. The MIT Press, Cambridge.
[5] Simon, R.M., Korn, E.L., McShane, L.M., Radmacher, M.D., Wright, G.W. and Zhao, Y. (2003) Design and Analysis of DNA Microarray Investigations. Series: Statistics for Biology and Health, Vol. 209, Springer, Berlin.
[6] Hardin, J., Mitani, A., Hicks, L. and VanKoten, B. (2007) A Robust Measure of Correlation between Two Genes on a Microarray. BMC Bioinformatics, 8, 220.
[7] Witten, I.H. and Frank, E.S. (2000) Data Mining: Practical Machine Learning Tools and Techniques with Java Implementation. 2nd Edition, Morgan Kaufmann Publishers, Burlington.
[8] Quinlan, J.R. (1993) C4.5: Programs for Machine Learning. Morgan Kaufmann Publishers, Burlington.
[9] Shannon, C.E. (1948) A Mathematical Theory of Communication. Bell System Technical Journal, 27, 623-656.
[11] Mitchell, T.M. (1997) Machine Learning. The Mc-Graw-Hill Companies, New York.
[12] Dee, D.P., Uppala, S.M., Simmons, A.J., Berrisford, P., Poli, P., Kobayashi, S., Andrae, U., et al. (2011) The ERA-Interim Reanalysis: Configuration and Performance of the Data Assimilation System. Quarterly Journal of the Royal Meteorological Society, 137, 553-597.
[13] Fialho, W.M.B. and Molion, L.C.B (2011) Eventos Extremos: Alagoas Junho de 2010. UFPel, Pelotas.
[14] Climanálise (2010) Boletim de Monitoramento e Análise Climática. CPTEC/INPE, Vol. 25.
http://climanalise.cptec.inpe.br/~rclimanl/boletim/pdf/pdf10/jun10.pdf
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Waves — lesson. Science CBSE, Class 9.
A wave is a disturbance in a medium which carries energy without a net movement of particles. A sound wave is a mechanical wave why because a mechanical wave cannot be transmitted in a vacuum.
There are two types of mechanical waves.
The wave that vibrates in the direction of the motion is called a longitudinal wave. Sound waves are longitudinal because the particles of the medium vibrate in the direction which is parallel to the direction of the propagation of the sound waves. The particles in the medium oscillate to and fro in the case of longitudinal waves.
A transverse wave is a wave which is produced when the particles of the medium oscillate in a direction that is perpendicular to the direction of the propagation of the wave. The particles in a transverse wave oscillate in an up and down motion. For example, light waves are transverse in nature.
Important terms to describe sound waves
A Sound wave can be described by its
The distance between the two consecutive compressions or the two consecutive rarefactions is called the wavelength.
It is denoted by a Greek letter lambda(λ), and its unit is \(metre(m)\) .
The number of oscillations an object takes per second is called its frequency. It is denoted by the letter F.
The SI unit of frequency is \(Hertz(Hz)\) .
\mathit{Frequency}=\frac{\mathit{Total}\phantom{\rule{0.147em}{0ex}}\mathit{number}\phantom{\rule{0.147em}{0ex}}\mathit{of}\phantom{\rule{0.147em}{0ex}}\mathit{oscillations}}{\mathit{Total}\phantom{\rule{0.147em}{0ex}}\mathit{time}\phantom{\rule{0.147em}{0ex}}\mathit{taken}}
The time taken for one complete oscillation of a sound wave is called the time period of the sound wave.
It is denoted by the letter T, and its unit is \(second(s)\) .
\mathit{Time}\phantom{\rule{0.147em}{0ex}}\mathit{period}=\frac{1}{\mathit{Frequency}}
The maximum displacement of a particle of the medium from the mean position is called the amplitude of the wave.
Amplitude defines the loudness of sound.
It is denoted by the letter A, and its unit is \(metre(m)\) .
The distance travelled by a wave in one second is called the speed or velocity of the wave.
It is denoted by the letter v, and its unit is \(metre per second \)\((m/s)\) .
\mathit{Speed}=\frac{\mathit{Distance}\phantom{\rule{0.147em}{0ex}}\mathit{travelled}}{\mathit{Time}\phantom{\rule{0.147em}{0ex}}\mathit{taken}}
https://commons.wikimedia.org/wiki/File:Crest_trough_wavelength_amplitude.png
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46E30 Spaces of measurable functions (
{L}^{p}
46E20 Hilbert spaces of continuous, differentiable or analytic functions
46E22 Hilbert spaces with reproducing kernels (= [proper] functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
{L}^{p}
46E35 Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems
{L}_{p}
{L}_{q}
{ℒ}_{p}
{}_{+}^{0}\left(\Omega ,ℱ,𝐏\right)
⟨{⟩}_{\phi }
A characterization of Orlicz spaces isometric to Lp-spaces.
Grzegorz Lewicki (1997)
In this note we present an affirmative answer to the problem posed by M. Baronti and C. Franchetti (oral communication) concerning a characterization of Lp-spaces among Orlicz sequence spaces. In fact, we show a more general characterization of Orlicz spaces isometric to Lp-spaces.
A characterization of subspaces of
{L}^{p}\left(\mu \right)
J. Holub (1972)
A characterization of the duals of some echelon Köthe spaces of Banach valued functions.
Cristina Jordán Lluch, Juan Ramón Torregrosa Sánchez (1987)
A class of Banach lattices and positive operators
Grząślewicz, Ryszard (1984)
Parviz Azimi, A. A. Ledari (2009)
Hagler and the first named author introduced a class of hereditarily
{l}_{1}
Banach spaces which do not possess the Schur property. Then the first author extended these spaces to a class of hereditarily
{l}_{p}
Banach spaces for
1\le p<\infty
. Here we use these spaces to introduce a new class of hereditarily
{l}_{p}\left({c}_{0}\right)
Banach spaces analogous of the space of Popov. In particular, for
p=1
the spaces are further examples of hereditarily
{l}_{1}
Banach spaces failing the Schur property.
A class of special Hardy-Orlicz spaces and the space of BMOA functions
Yuzan He (1988)
A class of weighted convolution Fréchet algebras
Thomas Vils Pedersen (2010)
For an increasing sequence (ωₙ) of algebra weights on ℝ⁺ we study various properties of the Fréchet algebra A(ω) = ⋂ ₙ L¹(ωₙ) obtained as the intersection of the weighted Banach algebras L¹(ωₙ). We show that every endomorphism of A(ω) is standard, if for all n ∈ ℕ there exists m ∈ ℕ such that
{\omega }_{m}\left(t\right)/\omega ₙ\left(t\right)\to \infty
as t → ∞. Moreover, we characterise the continuous derivations on this algebra: Let M(ωₙ) be the corresponding weighted measure algebras and let B(ω) = ⋂ ₙM(ωₙ). If for all n ∈ ℕ there exists m ∈ ℕ such that...
A compact imbedding result of Lipschitz manifolds.
Kevin McLeod, Rainer Picard (1991)
A compactness criterion of mixed Krasnoselskiĭ-Riesz type in regular ideal spaces of vector functions.
Väth, M. (1999)
A comparison between the closed modular ideals in l...(w) and L...(w).
Hakan Hedenmalm (1986)
Michelangelo Franciosi (1997)
{Q}_{0}
A condition of Busemann-Feller type for the derivation of integrals of functions in the class Φ (L).
Ireneo Peral Alonso (1977)
A continuous surjection from the unit interval onto the unit square.
Jari Taskinen (1993)
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Hector is using radar to monitor the speeds of cars on a street near his school. He records the following speeds, each in miles per hour:
34, 35, 39, 40, 36, 35, 55, 42, 35
39
What is the mean of these speeds?
To calculate the mean, find the sum of the data and divide it by the number of speeds.
In the numerically-arranged data listed below, which number falls in the middle?
If there are two numbers in the middle, you need to find the mean of them.
34, 35, 35, 35, 36, 39, 39, 40, 42, 55
Which speed is repeated most in the data list?
Hector wants to show the city council that there is a speeding problem. Which of the measures of central tendency that you calculated above should he use? Explain why.
Which of the answers you calculated best proves that there is a problem with speeding?
Since the mean is the highest speed of the measures of central tendency, it is most effective for proving that speed is an issue.
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If f(x) = x + 7 and g(x) = x − 7, find (fog) (7)
Evaluate: sin
Find the value of xand y if:
Find the co-factor of a12in the following:
For what value of λare the vectors perpendicular to each other?
(i) Is the binary operation* defined on set N, given by for all , commutative?
(ii) Is the above binary operation* associative?
=\frac{\mathrm{\pi }}{4}
Let .Express A as the sum of two matrices such that one is symmetric and the other is skew symmetric.
If , verify that A2−4A − 5I = 0
For what value of kis the following function continuous at x= 2?
Find the equation of tangent to the curve x= sin 3t, y = cos 2t, at t =
(x2− y2) dx + 2 xy dy = 0
Giventhat y = 1 when x = 1
, if y = 1 when x = 1
If and , find a vector such that and
If and and , show that the angle between and is 60°
Find the point on the line at a distance from the point (1, 2, 3).
A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability distribution of the number of successes.
Show that the rectangle of maximum area that can be inscribed in a circle is a square.
Show that the height of the cylinder of maximum volume that can be inscribed in a cone of height his .
Using integration find the area of the region bounded by the parabola y2= 4x and the circle 4x2+ 4y2= 9.
Find the equation of the plane passing through the point (−1, − 1, 2) and perpendicular to each of the following planes:
A factory owner purchases two types of machines, A and B for his factory. The requirements and the limitations for the machines are as follows:
Daily output (in units)
He has maximum area of 9000 m2available, and 72 skilled labourers who can operate both the machines. How many machines of each type should he buy to maximise the daily output?
An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probability of an accident involving a scooter, a car and a truck are 0.01, 0.03 and 0.15 respectively. One of the insured persons meets with an accident. What is the probability that he is a scooter driver.
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Cryptography/Hashes - Wikibooks, open books for an open world
A Wikibookian suggests that Cryptography/Hash function be merged into this chapter.
A digest, sometimes simply called a hash, is the result of a hash function, a specific mathematical function or algorithm, that can be described as
{\displaystyle f(message)=hash}
. "Hashing" is required to be a deterministic process, and so, every time the input block is "hashed" by the application of the same hash function, the resulting digest or hash is constant, maintaining a verifiable relation with the input data. Thus making this type of algorithms useful for information security.
Other processes called cryptographic hashes, function similarly to hashing, but require added security, in the form or a level of guarantee that the input data can not feasibly be reversed from the generated hash value. I.e. That there is no useful inverse hash function
{\displaystyle f'(hash)=message}
This property can be formally expanded to provide the following properties of a secure hash:
Preimage resistant : Given H it should be hard to find M such that H = hash(M).
A hash function is the implementation of an algorithm that, given some data as input, will generate a short result called a digest. A useful hash function generates a fixed length of hashed value.
For Ex: If our hash function is 'X' and we have 'wiki' as our input... then X('wiki')= a5g78 i.e. some hash value.
Applications of hash functionsEdit
Non-cryptographic hash functions have many applications,[1] but in this section we focus on applications that specifically require cryptographic hash functions:
In actual practice, Alice and Bob will often be computer programs, and the secret would be something less easily spoofed than a claimed puzzle solution. The above application is called a commitment scheme. Another important application of secure hashes is verification of message integrity. Determination of whether or not any changes have been made to a message (or a file), for example, can be accomplished by comparing message digests calculated before, and after, transmission (or any other event) (see Tripwire, a system using this property as a defense against malware and malfeasance). A message digest can also serve as a means of reliably identifying a file.
A related application is password verification. Passwords should not be stored in clear text, for obvious reasons, but instead in digest form. In a later chapter, Password handling will be discussed in more detail—in particular, why hashing the password once is inadequate.
A hash function is a key part of message authentication (HMAC).
Most distributed version control systems (DVCSs) use cryptographic hashes.[2]
For both security and performance reasons, most digital signature algorithms specify that only the digest of the message be "signed", not the entire message. The Hash functions can also be used in the generation of pseudo-random bits.
SHA-1, MD5, and RIPEMD-160 are among the most commonly-used message digest algorithms as of 2004. In August 2004, researchers found weaknesses in a number of hash functions, including MD5, SHA-0 and RIPEMD. This has called into question the long-term security of later algorithms which are derived from these hash functions. In particular, SHA-1 (a strengthened version of SHA-0), RIPEMD-128 and RIPEMD-160 (strengthened versions of RIPEMD). Neither SHA-0 nor RIPEMD are widely used since they were replaced by their strengthened versions.
Other common cryptographic hashes include SHA-2 and Tiger.
Later we will discuss the "birthday attack" and other techniques people use for Breaking Hash Algorithms.
Hash speedEdit
When using hashes for file verification, people prefer hash functions that run very fast. They want a corrupted file can be detected as soon as possible (and queued for retransmission, quarantined, or etc.). Some popular hash functions in this category are:
In addition, both SHA-256 (SHA-2) and SHA-1 have seen hardware support in some CPU instruction sets.
When using hashes for password verification, people prefer hash functions that take a long time to run. If/when a password verification database (the /etc/passwd file, the /etc/shadow file, etc.) is accidentally leaked, they want to force a brute-force attacker to take a long time to test each guess.[3] Some popular hash functions in this category are:
We talk more about password hashing in the Cryptography/Secure Passwords section.
Wikipedia has related information at message digest
see Algorithm Implementation/Hashing for more about non-cryptographic hash functions and their applications.
see Data Structures/Hash Tables for the most common application of non-cryptographic hash functions
Rosetta Code : Cryptographic hash function has implementations of cryptographic hash functions in many programming languages.
↑ applications of non-cryptographic hash functions are described in Data Structures/Hash Tables and Algorithm Implementation/Hashing.
↑ Eric Sink. "Version Control by Example". Chapter 12: "Git: Cryptographic Hashes".
↑ "Speed Hashing"
Retrieved from "https://en.wikibooks.org/w/index.php?title=Cryptography/Hashes&oldid=3708094"
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CrossSection - Maple Help
Home : Support : Online Help : Education : Student Packages : Multivariate Calculus : Visualization : CrossSection
show the intersection between a plane and a surface
CrossSection(f(x,y), g(x,y,z)=K, x=a..b, y=c..d, opts)
CrossSection(f(x,y,z), g(x,y,z)=K, x=a..b, y=c..d, z=e..f, opts)
algebraic equation or expression
constant, list, or range of constants
real constants; range of the function
(optional) equation(s) of the form option=value where option is one of functionoptions, intersectionoptions, output, planeoptions, planes or showfunction; specify output options
The CrossSection command returns a plot or animation of the intersection between the surface f and the plane g. The intersection of multiple parallel planes can be shown by making the variable K a range or a list of real numbers. If a range is used, a sequence of planes is generated within the range. The number of planes is determined by the optional argument planes. If a list is used, the planes are displayed in the order specified.
Curves of the form
f\left(x,y\right)
are assumed to be
z=f\left(x,y\right)
To return level curves, specify a variable name for g(x,y,z), for example, z.
Planes that are not intersecting or tangent to the surface f are not displayed.
The CrossSectionTutor routine offers equivalent capabilities to CrossSection in a tutor interface. See the Student[MultivariateCalculus][CrossSectionTutor] help page.
Specifies the plot options for plotting the surface f. For more information on plotting options, see plot3d/options.
intersectionoptions = list
Specifies the plot options for plotting the curves formed by the intersections of the planes and the surface. For more information on plotting options, see plot3d/options.
* output = plot specifies that a plot displays, which shows the expression and all the planes. The default is output = plot.
* output = animation specifies that an animation displays, which shows the expression and the chosen planes in succession.
Specifies the plot options for plotting the plane(s) g. For more information on plotting options, see plot3d/options.
planes = posint
Specifies the number of planes if the variable K is a range. The default is
10
For more information, see plot3d/options.
\mathrm{with}\left(\mathrm{Student}[\mathrm{MultivariateCalculus}]\right):
\mathrm{CrossSection}\left({x}^{2}+{y}^{2}+{z}^{2}=4,x+y=[1.1,0.1],x=-2..2,y=-2..2,z=-2..2,\mathrm{showfunction}=\mathrm{false},\mathrm{title}="Sphere"\right)
Return z=constant level curves.
\mathrm{CrossSection}\left({x}^{2}+{y}^{2},z=0..25,x=-4..4,y=-4..4,\mathrm{planes}=10\right)
\mathrm{CrossSection}\left(x+{y}^{2},x+y-z=0,x=2..4,y=-1..2\right)
Student[MultivariateCalculus][CrossSectionTutor]
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Cook’s Distance - MATLAB & Simulink - MathWorks Italia
Cook’s Distance
Determine Outliers Using Cook's Distance
Cook’s distance is the scaled change in fitted values, which is useful for identifying outliers in the X values (observations for predictor variables). Cook’s distance shows the influence of each observation on the fitted response values. An observation with Cook’s distance larger than three times the mean Cook’s distance might be an outlier.
Each element in the Cook's distance D is the normalized change in the fitted response values due to the deletion of an observation. The Cook’s distance of observation i is
{D}_{i}=\frac{\underset{j=1}{\overset{n}{â}}{\left({\stackrel{^}{y}}_{j}â{\stackrel{^}{y}}_{j\left(i\right)}\right)}^{2}}{p\text{â}MSE},
{\stackrel{^}{y}}_{j}
{\stackrel{^}{y}}_{j\left(i\right)}
Cook’s distance is algebraically equivalent to the following expression:
{D}_{i}=\frac{{r}_{i}^{2}}{p\text{â}MSE}\left(\frac{{h}_{ii}}{{\left(1â{h}_{ii}\right)}^{2}}\right),
After fitting the model mdl, for example, you can use fitlm or stepwiselm to:
Display the Cook’s distance values by indexing into the property using dot notation.
mdl.Diagnostics.CooksDistance
CooksDistance is an n-by-1 column vector in the Diagnostics table of the LinearModel object.
Plot the Cook’s distance values.
For details, see the plotDiagnostics function of the LinearModel object.
This example shows how to use Cook's Distance to determine the outliers in the data.
Load the sample data and define the independent and response variables.
Fit the linear regression model.
The dashed line in the figure corresponds to the recommended threshold value, 3*mean(mdl.Diagnostics.CooksDistance). The plot has some observations with Cook's distance values greater than the threshold value, which for this example is 3*(0.0108) = 0.0324. In particular, there are two Cook's distance values that are relatively higher than the others, which exceed the threshold value. You might want to find and omit these from your data and rebuild your model.
Find the observations with Cook's distance values that exceed the threshold value.
find((mdl.Diagnostics.CooksDistance)>3*mean(mdl.Diagnostics.CooksDistance))
Find the observations with Cook's distance values that are relatively larger than the other observations with Cook's distances exceeding the threshold value.
[1] Neter, J., M. H. Kutner, C. J. Nachtsheim, and W. Wasserman. Applied Linear Statistical Models. 4th ed. Chicago: Irwin, 1996.
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Extension:Math/Announcement - MediaWiki
Extension:Math/Announcement
To be posted after deployment on October 23. See http://lists.w3.org/Archives/Public/www-math/2014Oct/0003.html for the announcement on the MathML mailing list. This is a user centric announcement of the new features.
Introducing Math rendering 2.0[edit]
We'd like to announce a major update of the Math (rendering) extension.
For registered Wikipedia users, we have introduced a new math rendering mode using MathML, a markup language for mathematical formulae. Since MathML is not supported in all browsers [1], we have also added a fall-back mode using scalable vector graphics (SVG).
Both modes offer crisp rendering at any resolution, which is a major advantage over the current image-based default. We'll also be able to make our math more accessible by improving screenreader and magnification support.
We encourage you to enable the MathML mode in your Appearance preferences. As an example, the URL for this section on the English Wikipedia is: https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering
For editors, there are also two new optional features:
1) You can set the "id" attribute to create math tags that can be referenced. For example, the following math tag
{\displaystyle E=mc^{2}}
can be referenced by the wikitext
This is true regardless of the rendering mode used.
2) In addition, there is the attribute "display" with the possible values "block" or "inline". This attribute can be used to control the layout of the math tag with regard to centering and size of the operators. See https://www.mediawiki.org/wiki/Extension:Math/Displaystyle for a full description, of this feature.
Your feedback is very welcome. Please report bugs in Bugzilla against the Math extension, or post on the talk page here: https://www.mediawiki.org/wiki/Extension_talk:Math
All this is brought to you by Moritz Schubotz and Frédéric Wang (both volunteers) in collaboration with Gabriel Wicke, C. Scott Ananian, Alexandros Kosiaris and Roan Kattouw from the Wikimedia Foundation. We also owe a big thanks to Peter Krautzberger and Davide P. Cervone of MathJax for the server-side math rendering backend.
Best Gabriel Wicke (GWicke) and Moritz Schubotz (Physikerwelt)
[1]: Currently MathML is supported by Firefox & other Gecko-based browsers, and accessibility tools like Apple's VoiceOver. There is also partial support in WebKit.
Retrieved from "https://www.mediawiki.org/w/index.php?title=Extension:Math/Announcement&oldid=1235348"
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Function representation - Wikipedia
Function Representation (FRep[1] or F-Rep) is used in solid modeling, volume modeling and computer graphics. FRep was introduced in "Function representation in geometric modeling: concepts, implementation and applications" [2] as a uniform representation of multidimensional geometric objects (shapes). An object as a point set in multidimensional space is defined by a single continuous real-valued function
{\displaystyle f(X)}
of point coordinates
{\displaystyle X[x_{1},x_{2},...,x_{n}]}
which is evaluated at the given point by a procedure traversing a tree structure with primitives in the leaves and operations in the nodes of the tree. The points with
{\displaystyle f(x_{1},x_{2},...,x_{n})\geq 0}
belong to the object, and the points with
{\displaystyle f(x_{1},x_{2},...,x_{n})<0}
are outside of the object. The point set with
{\displaystyle f(x_{1},x_{2},...,x_{n})=0}
is called an isosurface.
1 Geometric domain
2 Shape Models
Geometric domain[edit]
The geometric domain of FRep in 3D space includes solids with non-manifold models and lower-dimensional entities (surfaces, curves, points) defined by zero value of the function. A primitive can be defined by an equation or by a "black box" procedure converting point coordinates into the function value. Solids bounded by algebraic surfaces, skeleton-based implicit surfaces, and convolution surfaces, as well as procedural objects (such as solid noise), and voxel objects can be used as primitives (leaves of the construction tree). In the case of a voxel object (discrete field), it should be converted to a continuous real function, for example, by applying the trilinear or higher-order interpolation.
Many operations such as set-theoretic, blending, offsetting, projection, non-linear deformations, metamorphosis, sweeping, hypertexturing, and others, have been formulated for this representation in such a manner that they yield continuous real-valued functions as output, thus guaranteeing the closure property of the representation. R-functions originally introduced in V.L. Rvachev's "On the analytical description of some geometric objects",[3] provide
{\displaystyle C^{k}}
continuity for the functions exactly defining the set-theoretic operations (min/max functions are a particular case). Because of this property, the result of any supported operation can be treated as the input for a subsequent operation; thus very complex models can be created in this way from a single functional expression. FRep modeling is supported by the special-purpose language HyperFun.
Shape Models[edit]
FRep combines and generalizes different shape models like
skeleton based "implicit" surfaces
A more general "constructive hypervolume"[4] allows for modeling multidimensional point sets with attributes (volume models in 3D case). Point set geometry and attributes have independent representations but are treated uniformly. A point set in a geometric space of an arbitrary dimension is an FRep based geometric model of a real object. An attribute that is also represented by a real-valued function (not necessarily continuous) is a mathematical model of an object property of an arbitrary nature (material, photometric, physical, medicine, etc.). The concept of "implicit complex" proposed in "Cellular-functional modeling of heterogeneous objects"[5] provides a framework for including geometric elements of different dimensionality by combining polygonal, parametric, and FRep components into a single cellular-functional model of a heterogeneous object.
^ Shape Modeling and Computer Graphics with Real Functions, FRep Home Page
^ A. Pasko, V. Adzhiev, A. Sourin, V. Savchenko, "Function representation in geometric modeling: concepts, implementation and applications", The Visual Computer, vol.11, no.8, 1995, pp.429-446.
^ V.L. Rvachev, "On the analytical description of some geometric objects", Reports of Ukrainian Academy of Sciences, vol. 153, no. 4, 1963, pp. 765-767 (in Russian).
^ A. Pasko, V. Adzhiev, B. Schmitt, C. Schlick, "Constructive hypervolume modelling", Graphical Models, 63(6), 2001, pp. 413-442.
^ V. Adzhiev, E. Kartasheva, T. Kunii, A. Pasko, B. Schmitt, "Cellular-functional modeling of heterogeneous objects", Proc. 7th ACM Symposium on Solid Modeling and Applications, Saarbrücken, Germany, ACM Press, 2002, pp. 192-203. 3-540-65620-0
http://hyperfun.org/FRep/
http://libfive.com/
Retrieved from "https://en.wikipedia.org/w/index.php?title=Function_representation&oldid=1056842314"
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m
n
Matrix (or 2-dimensional Array), then it is assumed to contain
m
\mathrm{with}\left(\mathrm{SignalProcessing}\right):
\mathrm{f1}≔12.0:
\mathrm{f2}≔24.0:
\mathrm{signal}≔\mathrm{Vector}\left({2}^{10},i↦\mathrm{sin}\left(\frac{\mathrm{f1}\cdot \mathrm{\pi }\cdot i}{50}\right)+1.5\cdot \mathrm{sin}\left(\frac{\mathrm{f2}\cdot \mathrm{\pi }\cdot i}{50}\right),\mathrm{datatype}=\mathrm{float}[8]\right)
{\textcolor[rgb]{0,0,1}{\mathrm{_rtable}}}_{\textcolor[rgb]{0,0,1}{36893628696909549564}}
\mathrm{Periodogram}\left(\mathrm{signal},\mathrm{samplerate}=100\right)
\mathrm{audiofile}≔\mathrm{cat}\left(\mathrm{kernelopts}\left(\mathrm{datadir}\right),"/audio/maplesim.wav"\right):
\mathrm{Periodogram}\left(\mathrm{audiofile},\mathrm{frequencyscale}="kHz"\right)
\mathrm{audiofile2}≔\mathrm{cat}\left(\mathrm{kernelopts}\left(\mathrm{datadir}\right),"/audio/stereo.wav"\right):
\mathrm{Periodogram}\left(\mathrm{audiofile2},\mathrm{compactplot}\right)
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Traveling Salesman Problem: Problem-Based - MATLAB & Simulink - MathWorks América Latina
Solve Initial Problem
For the solver-based approach to this problem, see Traveling Salesman Problem: Solver-Based.
rng(3,'twister') % Makes stops in Maine & Florida, and is reproducible
dist'*trips
where trips is the binary vector representing the trips that the solution takes. This is the distance of a tour that you try to minimize.
Create an optimization problem with binary optimization variables representing the potential trips.
tsp = optimproblem;
trips = optimvar('trips',lendist,1,'Type','integer','LowerBound',0,'UpperBound',1);
Include the objective function in the problem.
tsp.Objective = dist'*trips;
Use the graph representation to identify all trips starting or ending at a stop by finding all edges connecting to that stop. For each stop, create the constraint that the sum of trips for that stop equals two.
constr2trips = optimconstr(nStops,1);
for stop = 1:nStops
whichIdxs = outedges(G,stop); % Identify trips associated with the stop
constr2trips(stop) = sum(trips(whichIdxs)) == 2;
tsp.Constraints.constr2trips = constr2trips;
The problem is ready to be solved. To suppress iterative output, turn off the default display.
tspsol = solve(tsp,'options',opts)
tspsol = struct with fields:
trips: [19900×1 double]
tspsol.trips = logical(round(tspsol.trips));
Gsol = graph(idxs(tspsol.trips,1),idxs(tspsol.trips,2),[],numnodes(G));
% Gsol = graph(idxs(tspsol.trips,1),idxs(tspsol.trips,2)); % Also works in most cases
Overlay the new graph on the existing plot and highlight its edges.
nodes and
n
% Index of added constraints for subtours
inSubTour = (tourIdxs == ii); % Edges in current subtour
a = all(inSubTour(idxs),2); % Complete graph indices with both ends in subtour
constrname = "subtourconstr" + num2str(k);
tsp.Constraints.(constrname) = sum(trips(a)) <= (nnz(inSubTour) - 1);
[tspsol,fval,exitflag,output] = solve(tsp,'options',opts);
% Plot new solution
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Option lock in procedures - Maple Help
Home : Support : Online Help : Programming : Input and Output : File Manipulation : Option lock in procedures
Option lock in Procedures
Adding option lock to a procedure protects that procedure from being run simultaneously with any other option-lock procedure in multiple threads. There can be only one thread running a procedure with option lock at a time. If a second thread tries to run a procedure with option lock, then the second thread will block, waiting until the first thread's procedure is done. Other threads are free to run any other procedure. The thread that "has the lock" is free to run any procedure, including those with option-lock.
\mathrm{Threads}:-\mathrm{Task}:-\mathrm{Start}\left(\left(\right)↦\mathrm{NULL},\mathrm{Task}=[p,1],\mathrm{Task}=[p,2]\right)
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Development of a Low Cost Self-Sustaining Water Distillation System Using Activated Carbon Nanofluids | IMECE | ASME Digital Collection
Ashreet Mishra,
Ashreet Mishra
Mishra, A, & Nnanna, AGA. "Development of a Low Cost Self-Sustaining Water Distillation System Using Activated Carbon Nanofluids." Proceedings of the ASME 2018 International Mechanical Engineering Congress and Exposition. Volume 6B: Energy. Pittsburgh, Pennsylvania, USA. November 9–15, 2018. V06BT08A027. ASME. https://doi.org/10.1115/IMECE2018-86906
There is an ever increasing need to provide clean and portable drinking water in developing countries because of the poor quality of water, which causes various water borne diseases killing millions of infants and elderly people every year. There have been a lot of recent developments in the field of solar enabled water distillation where pure water is being generated. But a majority of the system use some type of external energy source for the system to run on which might make it efficient but can cause dependence on energy sources. The power of the Sun can be effectively harnessed and used as heat and light source for efficient steam generation. One of the challenges is to develop a low cost system which can perform at par with the best and extravagant systems.
This paper investigates the performance of the solar distillation system when activated carbon nanoparticles are used along with brine and other sources of impure water so as to obtain clean water. The activated carbon nanoparticle which are an efficient and cheap mode of water purification enhanced the productivity of the system by 190% when compared to saline water due to its steam generation properties. A solar simulator of
1KWm2
was used to simulate the sun. Various parameters like the variation of air flow on condensation rate, height of fluid on vapor production rate and the temperature variation of the system are evaluated. Parametric studies of the effect of water quality and salinity were performed. It was determined that the optimum output rate of distilled water was 240 grams (
6000gday.m2
) was determined by the system from the parametric studies and the system is feasible and cost effective to be used in real world application. All of these standout features make the system a low cost option which can tackle the clean water dilemma in developing countries.
Activated carbon, Nanofluids, Water, Developing nations, Energy resources, Nanoparticles, Solar energy, Steam, Air flow, Condensation, Diseases, Fluids, Heat, Light sources, Solar stills, Temperature, Vapors, Water pollution, Water treatment
First and Second Law Analysis of a Flat Plate Collector Working With Nanofluids
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Lens (optics) (10017 views - Consumer products)
2 Construction of simple lenses
2.1 Types of simple lenses
2.2 Lensmaker's equation
2.2.1 Sign convention for radii of curvature R1 and R2
2.2.2 Thin lens approximation
4.4 Other types of aberration
4.5 Aperture diffraction
5 Compound lenses
The word lens comes from lēns , the Latin name of the lentil, because a double-convex lens is lentil-shaped. The lentil plant also gives its name to a geometric figure.[1]
Some scholars argue that the archeological evidence indicates that there was widespread use of lenses in antiquity, spanning several millennia.[2] The so-called Nimrud lens is a rock crystal artifact dated to the 7th century BC which may or may not have been used as a magnifying glass, or a burning glass.[3][4][3][5] Others have suggested that certain Egyptian hieroglyphs depict "simple glass meniscal lenses".[6][verification needed]
The oldest certain reference to the use of lenses is from Aristophanes' play The Clouds (424 BC) mentioning a burning-glass.[7] Pliny the Elder (1st century) confirms that burning-glasses were known in the Roman period.[8] Pliny also has the earliest known reference to the use of a corrective lens when he mentions that Nero was said to watch the gladiatorial games using an emerald (presumably concave to correct for nearsightedness, though the reference is vague).[9] Both Pliny and Seneca the Younger (3 BC–65 AD) described the magnifying effect of a glass globe filled with water.
Ptolemy (2nd century) wrote a book on Optics, which however survives only in the Latin translation of an incomplete and very poor Arabic translation. The book was, however, received, by medieval scholars in the Islamic world, and commented upon by Ibn Sahl (10th century), who was in turn improved upon by Alhazen (Book of Optics, 11th century). The Arabic translation of Ptolemy's Optics became available in Latin translation in the 12th century (Eugenius of Palermo 1154). Between the 11th and 13th century "reading stones" were invented. These were primitive plano-convex lenses initially made by cutting a glass sphere in half. The medieval (11th or 12th century) rock cystal Visby lenses may or may not have been intended for use as burning glasses.[10]
Further information: History of the telescope
This section needs expansion with: history after 1758. You can help by adding to it. (January 2012)
If the lens is biconvex or plano-convex, a collimated beam of light passing through the lens converges to a spot (a focus) behind the lens. In this case, the lens is called a positive or converging lens. The distance from the lens to the spot is the focal length of the lens, which is commonly abbreviated f in diagrams and equations. An extended hemispherical lens is a special type of plano-convex lens, in which the lens's curved surface is a full hemisphere and the lens is much thicker than the radius of curvature.
{\displaystyle {\frac {1}{f}}=(n-1)\left[{\frac {1}{R_{1}}}-{\frac {1}{R_{2}}}+{\frac {(n-1)d}{nR_{1}R_{2}}}\right],}
{\displaystyle f}
{\displaystyle n}
{\displaystyle R_{1}}
{\displaystyle R_{2}}
is the radius of curvature of the lens surface farther from the light source, an{\displaystyle d}
{\displaystyle {\frac {1}{f}}\approx \left(n-1\right)\left[{\frac {1}{R_{1}}}-{\frac {1}{R_{2}}}\right].}
{\displaystyle {\frac {1}{S_{1}}}+{\frac {1}{S_{2}}}={\frac {1}{f}}}
{\displaystyle x_{1}x_{2}=f^{2},\!}
{\displaystyle x_{1}=S_{1}-f}
{\displaystyle x_{2}=S_{2}-f}
A convex lens (f ≪ S1) forming a real, inverted image rather than the upright, virtual image as seen in a magnifying glass
{\displaystyle M=-{\frac {S_{2}}{S_{1}}}={\frac {f}{f-S_{1}}}}
{\displaystyle {\frac {1}{f}}={\frac {1}{f_{1}}}+{\frac {1}{f_{2}}}.}
{\displaystyle {\frac {1}{f}}={\frac {1}{f_{1}}}+{\frac {1}{f_{2}}}-{\frac {d}{f_{1}f_{2}}}.}
{\displaystyle {\mbox{FFL}}={\frac {f_{1}(f_{2}-d)}{(f_{1}+f_{2})-d}}.}
{\displaystyle {\mbox{BFL}}={\frac {f_{2}(d-f_{1})}{d-(f_{1}+f_{2})}}.}
{\displaystyle M=-{\frac {f_{2}}{f_{1}}},}
Cylindrical lenses have curvature in only one direction. They are used to focus light into a line, or to convert the elliptical light from a laser diode into a round beam. They are also used in motion picture anamorphic lenses.
Lenses are used as prosthetics for the correction of visual impairments such as myopia, hypermetropia, presbyopia, and astigmatism. (See corrective lens, contact lens, eyeglasses.) Most lenses used for other purposes have strict axial symmetry; eyeglass lenses are only approximately symmetric. They are usually shaped to fit in a roughly oval, not circular, frame; the optical centres are placed over the eyeballs; their curvature may not be axially symmetric to correct for astigmatism. Sunglasses' lenses are designed to attenuate light; sunglass lenses that also correct visual impairments can be custom made.
CameraDigital cameraMovie cameraPentax camerasReflex cameraSingle-lens reflex cameraTelephoto lensThermal imaging cameraThermographic cameraOpticsTransparency and translucencyRefractionYashicaPhotographic filmGlass
This article uses material from the Wikipedia article "Lens (optics)", which is released under the Creative Commons Attribution-Share-Alike License 3.0. There is a list of all authors in Wikipedia
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Equivariant indices of Spinc-Dirac operators for proper moment maps
15 April 2017 Equivariant indices of
{Spin}^{c}
-Dirac operators for proper moment maps
Peter Hochs, Yanli Song
We define an equivariant index of
{Spin}^{c}
-Dirac operators on possibly noncompact manifolds, acted on by compact, connected Lie groups. Our main result is that the index decomposes into irreducible representations according to the quantization commutes with reduction principle.
Peter Hochs. Yanli Song. "Equivariant indices of
{Spin}^{c}
-Dirac operators for proper moment maps." Duke Math. J. 166 (6) 1125 - 1178, 15 April 2017. https://doi.org/10.1215/00127094-3792923
Received: 25 March 2015; Revised: 27 May 2016; Published: 15 April 2017
Keywords: ${\operatorname{Spin}}^{c}$-Dirac operator , Index theory , moment map , proper group action
Peter Hochs, Yanli Song "Equivariant indices of
{Spin}^{c}
-Dirac operators for proper moment maps," Duke Mathematical Journal, Duke Math. J. 166(6), 1125-1178, (15 April 2017)
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Relativistic mass – ebvalaim.log
m
E = mc^2
p = mv
The increase in inertia
Wait a minute - I just said that inertia of objects in motion increases, and mass is the measure of inertia. And now I'm saying that the mass doesn't increase? What is all this about?
In pre-relativistic physics, it was simple. If you applied a force to an object, it would accelerate - and the ratio of the force to the acceleration was precisely mass. So the harder an object is to accelerate - which means, the smaller the acceleration caused by the same force is - the larger its mass.This can be expressed with an equation as
a = \frac{F}{m}
What's more, an object would always accelerate in the same direction in which the force was acting (which might seem obvious... but let's not get ahead of ourselves). This means that the relationship above can be written using vectors and it will still hold:
\vec{a} = \frac{\vec{F}}{m}
. This way we took into account that forces can act in various directions and we can still calculate the acceleration.
So what does relativity change in this picture?
Let's keep the distinction between the rest mass and relativistic mass for now. An object at rest has mass
m_0
. According to the idea of relativistic mass, it increases in motion to:
This ugly square root in the denominator is usually denoted by
\frac{1}{\gamma}
, which lets us express the relativistic mass as:
When the velocity is 0,
\gamma = 1
and relativistic mass is simply equal to the rest mass.
That's cool. We introduced the relativistic mass because we wanted to account for the increase in inertia by the increase in mass. So, can we still write
\vec{a} = \frac{\vec{F}}{m} = \frac{\vec{F}}{m_0 \gamma}
Well... almost. As it turns out, we can - but only if the force is perpendicular to the direction of motion! If it is parallel, the result is different - then we need to calculate the acceleration like this:
\vec{a} = \frac{\vec{F}}{m_0 \gamma^3}
(I'll derive this below for the curious).
Wait... what does it even mean? Well, it means that the object has larger inertia in the direction of motion than perpendicular to it. So if we want to find the acceleration of the object when a force is applied, we can't just divide it by the object's mass - we have to decompose the force into parallel and perpendicular components, divide them by different numbers and then add the results to get a single acceleration. In effect it is possible that when the force acts at an angle to the velocity, the acceleration might not even be in the direction of the force!
The conclusion is that the "relativistic mass" doesn't solve the problem of inertia. There isn't even a single number that would reasonably measure inertia, because now it depends on the direction! (It is still possible to introduce a quantity measuring inertia - only now it has to be a tensor of rank 2, which can be written as a matrix.)
That's the first point against the relativistic mass.
One of the ways of comparing masses of different objects is making them collide. If a more massive object hits a less massive one, then the former will only slow down slightly, and the latter will get bounced at a significant speed. On the contrary, if a less massive object hits a more massive one, the former bounces away, and the latter picks up only a small amount of speed.
In a special case in which an object hits another one that is at rest and has the same mass - the former will stop, and the latter will fly away with the same speed the former had initially. Newton's cradle is one of the better known illustrations of this fact.
The exact formulae for the velocities after the collision depending on the velocities before the collision and the masses of objects can be derived using the conservation laws for energy and momentum. We won't be doing this here, we'll just use the intuitive understanding described above.
The question now is this: what happens if the two objects colliding have the same rest masses, but vastly different relativistic masses? For example, what if one ball with a mass of 1 gram is at rest, and a second identical one going fast enough to have a relativistic mass of 100 grams (so
\gamma = 100
) hits it? Will they behave like objects with different masses (the moving one will slow down, and the resting one will start moving with some speed), or like equals (the moving one will stop, and the resting one will start moving with the same speed the moving one had before)?
The answer lies again within the conservation laws, only this time the relativistic ones have to be used. I'll write the full equations below, and here I'll just tell you the result: as it turns out, the balls will behave like equals. So, relativistic mass is irrelevant in collisions - the only mass that counts is the rest mass.
That's another point against the relativistic mass.
Duplication of notions
This isn't a physical argument, strictly speaking, more like a technical one, but it still carries some weight.
Physicists like to simplify their lives. One of the simplifications they like to make is eliminating the constants of nature from the equations. Let me explain.
Let's take the speed of light as an example. We can check in some books or on the internet that it is equal to 299 792 458 m/s. The number is rather ugly, but we have no say in what the nature chose as the speed of electromagnetic waves... or do we?
This particular number only comes from the units we chose. If we wanted to express the speed of light in feet per second, the number would be different. If we chose furlongs per fortnight, we'd get another different number. Hmm... What if we made our lives simpler and chose units such that the number was somewhat simpler? For example, if it was... just 1?
We can do that and that's exactly what physicists do. Example units like that could be a second and a light-second. Or a year and a light-year. Or any time unit and the distance light travels during that time.If we choose such a system of units, we'll have
c = 1
. And just like that,
c
disappears from all the equations, because whether we multiply or divide by 1, it doesn't ever change anything.
(Physicists like to take it a step further and eliminate more constants. The units in which
c = G = \hbar = 1
- so ones in which the speed of light, the gravitational constant and the reduced Planck constant are all 1 - are a popular choice. These units are called "natural units" or... "Planck units". The basic units in this system are Planck length, Planck time and Planck mass.)
Alright, but why am I mentioning all this? Well, let's see what happens to the famous Einstein formula for energy in such a system:
When we introduce units in which
c=1
The relativistic mass is always equal to energy in such units! So it is a de facto duplicate of the notion of energy. Everywhere where we would use the relativistic mass before, we can just substitute in energy (in other units:
\frac{E}{c^2}
) and nothing will change. Why would we need such an additional notion, then?
And that's yet another point against the relativistic mass.
As you can see, introducing the notion of relativistic mass doesn't really get us much. It isn't great for measuring inertia, it's useless in collisions, and is de facto a duplicate of energy. For these reasons, physicists pretty much stipped using this notion - now, when mass is being mentioned, it almost always means rest mass.
And because of that, I'm asking you, dear Readers - let's stop saying that mass increases for objects in motion. Let's stop saying that it becomes infinite at the speed of light. We can go ahead and substitute "inertia" for "mass" in these contexts, or - since it is a notion almost equivalent to relativistic mass - just use "energy". Both inertia and energy tend to infinity as speed tends to
c
. Let mass remain a property that is constant for a given object.
Two key equations we will need are the relativistic expressions for energy and momentum:
m
is the rest mass in both of these equations - we forget that relativistic mass is even a thing now.
It will be our goal to express the acceleration
\vec{a} = \frac{d\vec{v}}{dt}
using the force, mass and velocity.
We'll assume the following equation (which is correct in non-relativistic physics, too) as the definition of the force:
We have the formula for momentum, so we just need to start differentiating ;) We get:
The second term strongly resembles the force known from Newtonian physics (if we used the relativistic mass), but there is still the first term. Let's focus just on the derivative of
\gamma
for now:
We'll notice now that
v^2
\vec{v} \cdot \vec{v}
, which is the dot product of the velocity vector with itself. The dot product behaves just like a regular product with respect to differentials, so we get:
(Let's remember this form of the equation, we'll come back to it in a moment.)
Now, let us take a note of a property of the dot product: namely that
\vec{a} \cdot \vec{b} = ab\cos \alpha
and
b
are the magnitudes of the respective vectors, and
\alpha
is the angle between them. In particular, if the vectors are perpendicular, the dot product is zero, and if they are parallel, it is equal to the product of the magnitudes (up to the sign).
Thus, our
\vec{v} \cdot \vec{a}
va_{\|}
a_{\|}
is the component of the acceleration that is parallel to the velocity, or, to be more precise - the projection of acceleration onto the direction of velocity (which will be negative if the angle between acceleration and velocity exceeded 90 degrees). The perpendicular component has no effect here.
\vec{v}
v\vec{e}_v
- where
\vec{e}_{v}
is a unit vector (a vector of magnitude 1) with the direction and sense the same as the velocity.
Thanks to these "tricks", we can write the first term as:
Now let's note that
a_{\|}\vec{e}_v
is a vector that has the same direction as the velocity, a sense dependent on the sign of the projection of acceleration on the direction of the velocity, and a magnitude equal to that projection - so it's simply the parallel component of the acceleration
\vec{a}_{\|}
! Let's also decompose
\vec{a}
in the second term in the force into
\vec{a}_{\|} + \vec{a}_{\bot}
and we'll get:
Now the only thing that's left is to simplify
1 + \gamma^2\frac{v^2}{c^2}
Hence the final equation:
Coming back for a moment to the equation I told you to remember - I mean the form which still contained
(\vec{v}\cdot\vec{a})\vec{v}
. As it turns out, such a term can be written as:
\vec{v}\otimes\vec{v}
is a 3x3 matrix with components given by:
The whole equation can then be written as:
\mathbf{1}
is the identity (unit) matrix. This form expresses the force as the acceleration multiplied by a 3x3 matrix - it is this matrix I meant as a possible measure of the object's inertia.
Let us consider a collision of two objects with rest masses equal to
m
, of which one is moving with velocity
v
, and the other one is at rest. The object at rest has energy and momentum given by:
and the moving object:
The total energy and momentum of the system are then:
Let's assume that the objects will be moving with velocities
v_1
v_2
after the collision. The total energy and momentum will then be:
Due to the conservation of energy and momentum, these have to be exactly equal to the values from before the collision. This gives us two equations for two unknowns
v_1
v_2
There is one obvious solution, and that is
v_1 = v
v_2 = 0
- this is simply the situation from before the collision. But thanks to the symmetry of the problem, there is a different, equally obvious solution:
v_1 = 0
v_2 = v
. This second solution then has to correspond to the situation after the collision (it can be proven that there are no other solutions) - so the objects will behave such that the moving one will stop, and the other one will start moving with the same speed the first one had before the collision.
Which means they will behave exactly like objects with equal masses.
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Compute or plot passivity index as function of frequency - MATLAB passiveplot - MathWorks América Latina
passiveplot
Plot Passivity Versus Frequency
Plot Passivity of Multiple Systems
Plot Passivity of Models with Complex Coefficients
Compute or plot passivity index as function of frequency
passiveplot(G)
passiveplot(G,type)
passiveplot(___,w)
passiveplot(G1,G2,...,GN,___)
passiveplot(G1,LineSpec1,...,GN,LineSpecN,___)
passiveplot(___,plotoptions)
[index,wout] = passiveplot(G)
[index,wout] = passiveplot(G,type)
index = passiveplot(G,w)
index = passiveplot(G,type,w)
passiveplot(G) plots the relative passivity indices of the dynamic system G as a function of frequency. When I + G is minimum phase, the relative passivity indices are the singular values of (I - G)(I + G)^-1. The largest singular value measures the relative excess (R < 1) or shortage (R > 1) at each frequency. See getPassiveIndex for more information about the meaning of the passivity index.
passiveplot automatically chooses the frequency range and number of points for the plot based on the dynamics of G.
If G is a model with complex coefficients, then in:
passiveplot(G,type) plots the input, output, or I/O passivity index, depending on the value of type: 'input', 'output', or 'io', respectively.
passiveplot(___,w) plots the passivity index for frequencies specified by w.
If w is a cell array of the form {wmin,wmax}, then passiveplot plots the passivity index at frequencies ranging between wmin and wmax.
If w is a vector of frequencies, then passiveplot plots the passivity index at each specified frequency. The vector w can contain both negative and positive frequencies.
You can use this syntax with any of the previous input-argument combinations.
passiveplot(G1,G2,...,GN,___) plots the passivity index for multiple dynamic systems G1,G2,...,GN on the same plot. You can also use this syntax with the type input argument, with w to specify frequencies to plot, or both.
passiveplot(G1,LineSpec1,...,GN,LineSpecN,___) specifies a color, linestyle, and marker for each system in the plot.
passiveplot(___,plotoptions) plots the passivity index with the options set specified in plotoptions. You can use these options to customize the plot appearance using the command line. Settings you specify in plotoptions override the preference settings in the MATLAB® session in which you run passiveplot. Therefore, this syntax is useful when you want to write a script to generate multiple plots that look the same regardless of the local preferences.
[index,wout] = passiveplot(G) and [index,wout] = passiveplot(G,type) return the passivity index at each frequency in the vector wout. The output index is a matrix, and the value index(:,k) gives the passivity indices in descending order at the frequency w(k). This syntax does not draw a plot.
index = passiveplot(G,w) and index = passiveplot(G,type,w) return the passivity indices at the frequencies specified by w.
Plot the relative passivity index as a function of frequency of the system
G=\left(s+2\right)/\left(s+1\right)
The plot shows that the relative passivity index is less than 1 at all frequencies. Therefore, the system G is passive.
Plot the input passivity index of the same system.
passiveplot(G,'input')
The input passivity index is positive at all frequencies. Therefore, the system is input strictly passive.
Plot the input passivity index of two dynamic systems and their series interconnection.
G1 = tf([5 3 1],[1 2 1]);
G2 = tf([1 1 5 0.1],[1 2 3 4]);
passiveplot(G1,'r',G2,'b--',H,'gx','input')
legend('G1','G2','G2*G1')
The input passivity index of the interconnected system dips below 0 around 1 rad/s. This plot shows that the series interconnection of two passive systems is not necessarily passive. However, passivity is preserved for parallel or feedback interconnections of passive systems.
Plot the relative passivity indices of a complex-coefficient model and a real-coefficient model on the same plot.
Gr = tf([1 5 10],[1 10 5]);
passiveplot(Gc,Gr)
legend('Complex-coefficient model','Real-coefficient model','Location','southeast')
passiveplot(Gc,Gr,opt)
In linear frequency scale, the plots show a single branch with a symmetric frequency range centered at a frequency value of zero. The plot also shows the negative-frequency response of a real-coefficient model when you plot the response along with a complex-coefficient model.
Model to analyze for passivity, specified as a dynamic system model such as a tf, ss, or genss model. G can be MIMO, if the number of inputs equals the number of outputs. G can be continuous or discrete. If G is a generalized model with tunable or uncertain blocks, passiveplot evaluates passivity of the current, nominal value of G.
If G is a model array, then passiveplot plots the passivity index of all models in the array on the same plot. When you use output arguments to get passivity data, G must be a single model.
type — Type of passivity index
'input' | 'output' | 'io'
Type of passivity index, specified as one of the following:
'input' — Input passivity index (input feedforward passivity). This value is the smallest eigenvalue of
\left(G\left(s\right)+G{\left(s\right)}^{H}\right)/2
, for s = jω in continuous time, and s = ejω in discrete time.
'output' — Output passivity index (output feedback passivity). When G is minimum phase, this value is the smallest eigenvalue of
\left(G{\left(s\right)}^{-1}+G{\left(s\right)}^{-H}\right)/2
'io' — Combined I/O passivity index. When I + G is minimum phase, this value is the largest τ(ω) such that:
\text{G}\left(s\right)+\text{G}{\left(s\right)}^{H}>2\tau \left(\omega \right)\left(I+\text{G}{\left(s\right)}^{H}\text{G}\left(s\right)\right),
for s = jω in continuous time, and s = ejω in discrete time.
See About Passivity and Passivity Indices for details about these indices.
plotoptions — Passivity index plot options set
Passivity index plot options set, specified as a SectorPlotOptions object. You can use this option set to customize the plot appearance. Use sectorplotoptions to create the option set. Settings you specify in plotoptions overrides the preference settings in the MATLAB session in which you run passiveplot. Therefore, plotoptions is useful when you want to write a script to generate multiple plots that look the same regardless of the local preferences.
index — Passivity indices
Passivity indices as a function of frequency, returned as a matrix. index contains whichever type of passivity index you specify, computed at the frequencies w if you supplied them, or wout if you did not. index has as many columns as there are values in w or wout, and
One row, for the input, output, or combined i/o passivity indices.
As many rows as G has inputs or outputs, for the relative passivity index.
For example, suppose that G is a 3-input, 3-output system, and w is a 1-by-30 vector of frequencies. Then the following syntax returns a 3-by-30 matrix index.
index = passiveplot(G,w);
The entry index(:,k) contains the relative passivity indices of G, in descending order, at the frequency w(k).
isPassive | getPassiveIndex | getSectorIndex | sectorplot | sectorplotoptions
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System of Operator Quasi Equilibrium Problems
Suhel Ahmad Khan, "System of Operator Quasi Equilibrium Problems", International Journal of Analysis, vol. 2014, Article ID 848206, 6 pages, 2014. https://doi.org/10.1155/2014/848206
Suhel Ahmad Khan1
Academic Editor: Sivaguru Sritharan
We consider a system of operator quasi equilibrium problems and system of generalized quasi operator equilibrium problems in topological vector spaces. Using a maximal element theorem for a family of set-valued mappings as basic tool, we derive some existence theorems for solutions to these problems with and without involving -condensing mappings.
In 2002, Domokos and Kolumbán [1] gave an interesting interpretation of variational inequality and vector variational inequalities (for short, VVI) in Banach space settings in terms of variational inequalities with operator solutions (for short, OVVI). The notion and viewpoint of OVVI due to Domokos and Kolumbán [1] look new and interesting even though it has a limitation in application to VVI. Recently, Kazmi and Raouf [2] introduced the operator equilibrium problem which generalizes the notion of OVVI to operator vector equilibrium problems (for short, OVEP) using the operator solution. They derived some existence theorems of solution of OVEP with pseudomonotonicity, without pseudomonotonicity, and with -pseudomonotonicity. However, they dealt with only the single-valued case of the bioperator. It is very natural and useful to extend a single-valued case to a corresponding set-valued one from both theoretical and practical points of view.
The system of vector equilibrium problems and the system of vector quasi equilibrium problems were introduced and studied by Ansari et al. [3, 4]. Inspired by above cited work, in this paper, we consider a system of operator quasi equilibrium problems (for short, SOQEP) in topological vector spaces. Using a maximal element theorem for a family of set-valued mappings according to [5] as basic tool, we derive some existence theorems for solutions to SOQEP with and without involving -condensing mappings.
Further, we consider a system of generalized quasi operator equilibrium problems (for short, SGQOEP) in topological vector spaces and give some of its special cases and derive some existence theorems for solutions to SOQEP with and without involving -condensing mappings by using well-known maximal element theorem [5] for a family of set-valued mappings, and, consequently, we also get some existence theorems for solutions to a system of operator equilibrium problems.
Let be an index set, for each , and let be a Hausdorff topological vector space. We denote , the space of all continuous linear operators from into , where is topological vector space for each . Consider a family of nonempty convex subsets with in .
Let Let be a set-valued mapping such that, for each , is solid, open, and convex cone such that and .
For each , let be a bifunction and let be a set-valued mapping with nonempty values. We consider the following system of operator quasi equilibrium problems (for short, SOQEP). Find such that, for each ,
We remarked that, for the suitable choices of , and , SOQEP (2) reduces to the problems considered and studied by [3–6] and the references therein.
Now, we will give the following concepts and results which are used in the sequel.
Definition 1. Let be a nonempty and convex subset of a topological vector space, and let be a topological vector space with a closed and convex cone with apex at the origin. A vector-valued function is said to be as follows: (i)P-function if and only if and : (ii)natural P-quasifunction if and only if and : where denotes the convex hull of ;(iii)P-quasifunction if and only if and the set is convex.
Definition 2 (see [7]). Let be a topological vector space and let be a lattice with a minimal element, denoted by . A mapping is called a measure of noncompactness provided that the following conditions hold for any : (i), where denotes the closed convex hull of ;(ii) if and only if is precompact;(iii).
Definition 3 (see [7]). Let be a topological vector space, , and let be a measure of noncompactness on . A set-valued mapping is called -condensing provided that with ; then is relative compact; that is, is compact.
Remark 4. Note that every set-valued mapping defined on a compact set is -condensing for any measure of noncompactness . If is locally convex, then a compact set-valued mapping (i.e., is precompact) is -condensing for any measure of noncompactness . Obviously, if is -condensing and satisfies , for all , then is also -condensing.
The following maximal element theorems will play key role in establishing existence results.
Theorem 5 (see [8]). For each , let be a nonempty convex subset of a topological vector space and let be the two set-valued mappings. For each , assume that the following conditions hold: (a)for all , ;(b)for all , ;(c)for all , is compactly open ;(d)there exist a nonempty compact subset of and a nonempty compact convex subset , for each , such that, for all , there exists such that . Then, there exists such that for each .
We will use the following particular form of a maximal element theorem for a family of set-valued mappings due to Deguire et al. [5].
Theorem 6 (see [5]). Let be any index set, for each , let be a nonempty convex subset of a Hausdorff topological vector space , and let be a set-valued mapping. Assume that the following conditions hold: (i) and ; is convex;(ii) and ; , where is the th component of ;(iii) and ; is open ;(iv)there exist a nonempty compact subset of and a nonempty compact convex subset such that and there exists such that . Then, there exists such that for each .
Remark 7. If is nonempty, closed, and convex subset of a locally convex Hausdorff topological vector space , then condition (iv) of Theorem 6 can be replaced by the following condition:
the set-valued mapping is defined as , -condensing.
Throughout this paper, unless otherwise stated, for any index set and for each , let be a topological vector space and let be a set-valued mapping such that, for each , is proper, solid, open, and convex cone such that and . We denote , the space of all continuous linear operators from into . We also assume that is a set-valued mapping such that is nonempty and convex, is open in , and the set is closed in , where is the th component of .
Now, we have the following existence result for SOQEP (2).
Theorem 8. For each , let be nonempty and convex subset of a Hausdorff topological vector space and let be a bifunction. Suppose that the following conditions hold: (i) and , where is the th component of ;(ii) and ; the vector-valued function is natural -quasifunction;(iii) and ; the set is closed in ;(iv)there exist a nonempty compact subset of and a nonempty compact convex subset of , for each such that ; there exists and such that and . Then SOQEP (2) has a solution.
Proof. Let us define, for each given , a set-valued mapping by First, we claim that and is convex. Fix an arbitrary and . Let and ; then we have Since is natural -quasifunction, there exists such that From the inclusion of and , we get Hence, and therefore is convex. Since and are arbitrary, is convex, and .
Hence, our claim is then verified.
Now and ; the complement of in can be defined as From condition (iii) of the above theorem, will be closed in .
Suppose that and ; we define another set-valued mapping by
Then, it is clear that and is convex, because and are both convex. Now, by condition (i), . Since and , is open in , because and are open in .
Condition (iv) of Theorem 6 is followed from condition (iv). Hence, by fixed point Theorem 6, there exists such that . Since and is nonempty, we have . Therefore, and .
Now, we establish an existence result for SOQEP (2) involving -condensing maps.
Theorem 9. For each , let be a nonempty, closed, and convex subset of a locally convex Hausdorff topological vector space , suppose that is a bifunction, and let the set-valued mapping defined as be -condensing. Assume that conditions (i), (ii), and (iii) of Theorem 8 hold. Then SOQEP (2) has a solution.
Proof. In view of Remark 7, it is sufficient to show that the set-valued mapping defined as , is -condensing, where s are the same as defined in the proof of Theorem 8. By the definition of and and therefore . Since is -condensing, by Remark 7, we have being also -condensing.
4. System of Generalized Quasi Operator Equilibrium Problem
Throughout this section, unless otherwise stated, let be any index set. For each , let be a Hausdorff topological vector space. We denote , the space of all continuous linear operators from into , where is topological vector space for each and for each ; let be a closed, pointed, and convex cone with , where denotes the interior of set . Consider a family of nonempty convex subsets with in . Let, for each , a bifunction and two set-valued mappings be with nonempty values.
Let be the unit vector in , for each , and also such that , where are two real numbers such that .
Now, we consider the system of generalized quasi operator equilibrium problems (for short, SGQOEP). Find such that, for each ,
(I)If , then SGQOEP (10) reduces to finding of such that, for each , (II)If, in Case (I), we take , then and ; then problem (10) reduces to the system of generalized quasi operator equilibrium problems with lower and upper bounds (for short, SGQOEPLUB). Find such that, for each ,
Now, we establish the existence result for SGQOEP (10).
Theorem 10. For each , let be a nonempty convex subset of a topological vector space and are the bifunctions, is a set-valued mapping such that the set is compactly closed, is a set-valued mapping with nonempty values such that, for each is compactly open in , and are the unit vector such that , where are two real numbers such that . For each , assume that the following conditions hold: (i)for all , ;(ii)for all , or ;(iii)for all and for every nonempty finite subset , we have (iv)for all , the set is compactly closed in ;(v)there exist a nonempty compact subset of and a nonempty compact convex subset , for each , such that, for all , there exists such that satisfying and either or . Then the problem SGQOEP (10) has a solution.
Proof. For each and for all , define two set-valued mappings by Condition (iii) implies that, for each and for all , .
From condition (ii), we have for all and for each .
Thus, for each and for all , We have complement of in : which is compactly closed by virtue of condition (iv). Therefore, for each and for all is compactly open in .
For each , define two set-valued mappings by Thus, for each and for all and in view of condition (i), we obtain . It is easy to see that for each and for all . Thus, for each and for all and are compactly open in . We have being compactly open in . Also for all and for each .
Then, by Theorem 5, there exists such that for each . If , then , which contradicts the fact that is nonempty for each and for all . Hence, , for each . Therefore, and , for all . Thus, for each and for all . This completes the proof.
Now, we establish an existence result for SGQOEP (10) involving -condensing maps.
Theorem 11. For each , assume that conditions (i)–(iv) of Theorem 10. hold. Let be a measure of noncompactness on . Further, assume that the set-valued mapping defined as is a nonempty, closed, and convex subset of a locally convex Hausdorff topological vector space and is a bifunction and let the set-valued mapping defined as be -condensing. Then, there exists a solution of SGQOEP (10).
Proof. In view of Remark 7, it is sufficient to show that the set-valued mapping defined as , is -condensing, where s are the same as defined in the proof of Theorem 10. By the definition of and and therefore . Since is -condensing, by Remark 7, we have being also -condensing.
Next, we derive the existence result for the solution of SGQOEPLUB (12).
Corollary 12. For each , let be a nonempty convex subset of a topological vector space and are the bifunctions, is a set-valued mapping such that the set is compactly closed, is a set-valued mapping with nonempty values such that, for each is compactly open in , and are two real numbers such that . For each , assume that the following conditions hold:(i)for all , ;(ii)for all , or ;(iii)for all and for every nonempty finite subset , we have (iv)for all , the set is compactly closed in ;(v)there exist a nonempty compact subset of and a nonempty compact convex subset , for each , such that, for all , there exists such that satisfying and either or . Then the problem SGQOEPLUB (12) has a solution.
Now, we establish an existence result for SGQOEPLUB (12) involving -condensing maps.
Theorem 13. For each , assume that conditions (i)–(iv) of Corollary 12 hold. Let be a measure of noncompactness on . Further, assume that the set-valued mapping defined as is a nonempty, closed, and convex subset of a locally convex Hausdorff topological vector space and is a bifunction and let the set-valued mapping defined as , be -condensing. Then, there exists a solution of SGQOEPLUB (12).
Proof. In view of Remark 7, it is sufficient to show that the set-valued mapping defined as , is -condensing, where are the same as defined in the proof of Theorem 10. By the definition of and and therefore . Since is -condensing, by Remark 7, we have being also -condensing.
A. Domokos and J. Kolumbán, “Variational inequalities with operator solutions,” Journal of Global Optimization, vol. 23, no. 1, pp. 99–110, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
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Q. H. Ansari, W. K. Chan, and X. Q. Yang, “The system of vector quasi-equilibrium problems with applications,” Journal of Global Optimization, vol. 29, no. 1, pp. 45–57, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
Q. H. Ansari, S. Schaible, and J. C. Yao, “System of vector equilibrium problems and its applications,” Journal of Optimization Theory and Applications, vol. 107, no. 3, pp. 547–557, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
P. Deguire, K. K. Tan, and G. X.-Z. Yuan, “The study of maximal elements, fixed points for
{L}_{S}
-majorized mappings and their applications to minimax and variational inequalities in product topological spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 37, no. 7, pp. 933–951, 1999. View at: Publisher Site | Google Scholar | MathSciNet
S. Al-Homidan and Q. H. Ansari, “Systems of quasi-equilibrium problems with lower and upper bounds,” Applied Mathematics Letters, vol. 20, no. 3, pp. 323–328, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
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Copyright © 2014 Suhel Ahmad Khan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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\frac { r _ { 1 } ( \operatorname { cos } a + i \operatorname { sin } a ) } { r _ { 2 } ( \operatorname { cos } b + i \operatorname { sin } b ) } = \frac { r _ { 1 } } { r _ { 2 } } ( \operatorname { cos } ( a - b ) + i \operatorname { sin } ( a - b ) )
. (Hint: Multiply the numerator and denominator by the conjugate of the denominator.)
\frac{\textit{r}_1(\cos \textit{a}+\textit{i}\sin \textit{a})}{\textit{r}_2(\cos \textit{b}+\textit{i}\sin \textit{b})}·\frac{(\cos \textit{b}-\textit{i}\sin \textit{b})}{(\cos \textit{b}-\textit{i}\sin \textit{b})}
\frac{r_1(\cos a \cos b - i \cos a \sin b + i \sin \sin a \cos b - i^2 \sin a \sin b)} {r_2(\cos b \cos b - i \cos b \sin b + i \sin \sin b \cos b - i^2 \sin b \sin b)}
\frac{r_1((\cos a \cos b + \sin a \sin b) + i (\sin a \cos b - \cos a \sin b)} {r_2(\cos b i \sin^2 \sin b)}
Use your trig identities to simplify.
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MCQs of Electrostatic Potential and Capacitance | GUJCET MCQ
MCQs of Electrostatic Potential and Capacitance
For a uniform electric field
\stackrel{\to }{\mathrm{E}} = {E}_{0}\left(\stackrel{^}{i}\right)
, if the electric potential at x = 0 is zero, then the value of electric potential at x = +x will be _____
(a) xE0
(b) -xE0
(c) x2E0
(d) -x2E0
The line integral of an electric field along the circumference of a circle or radius r, drawn with a point charge Q at the centre will be _____
\frac{1}{4{\mathrm{\pi \xi }}_{0}} \frac{\mathrm{Q}}{r}
\frac{\mathrm{Q}}{2{\mathrm{\xi }}_{0}r}
(d) 2πQr
A particle having mass 1 g and electric charge 10-8C travels from a point A having electric potential 600 V to the point B having zero potential. What would be the change in its kinetic energy ?
(a) -6 x 10-6 erg
(b) -6 x 10-6 J
(c) 6 x 10-6 J
(d) 6 x 10-6 erg
A particle having mass m and charge q is at rest. On applying a uniform electric field E on it, it starts moving. What is its kinetic energy when it travels a distance y in the direction of force ?
(a) qE2y
(b) qEy2
(c) qEy
(d) q2Ey
The electric potential of 5 V is on the surface of hollow metal sphere of radius 3 cm. What will be electric potential at the centre of sphere ?
A moving electron approaches another electron. What would be the charge in the potential energy of this system ?
Energy of charged capacitor is U. Now it is removed from a battery and then is connected to another identical uncharged capacitor in parallel. What will be the energy of each capacitor now ?
\frac{3\mathrm{U}}{2}
\frac{\mathrm{U}}{4}
\frac{\mathrm{U}}{2}
A uniform electric field is prevailing in Y-direction in a certain region. The co-ordinates of points A, B and C are (0, 0), (2, 0) and (0, 2) respectively. Which of the following alternatives is true for the potential of these points ?
(a) VA = VB, VA > VC
(b) VA > VB, VA = VC
(c) VA < VC, VB = VC
(d) VA = VB, VA < VC
The capacitance of a parallel plate capacitor formed by the circular plates of diameter 4.0 cm is equal to the capacitance of a sphere of diameter 200 cm. Find the distance between two plates.
(a) 2 x 10-4 m
The capacitance of a variable capacitor joined with a battery of 100 V is changed from 2μF to 10μF. What is the change in the energy stored in it?
(a) 2 x 10-2 J
(b) 2.5 x 10-2 J
(c) 6.5 x 10-2 J
(d) 4 x 10-2 J
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Flame Propagation Following the Autoignition of Axisymmetric Hydrogen, Acetylene, and Normal-Heptane Plumes in Turbulent Coflows of Hot Air | J. Eng. Gas Turbines Power | ASME Digital Collection
Christos N. Markides,
Hopkinson Laboratory, Department of Engineering,
, Trumpington Street, Cambridge, Cambridgeshire CB2 1PZ, UK
Markides, C. N., and Mastorakos, E. (December 13, 2007). "Flame Propagation Following the Autoignition of Axisymmetric Hydrogen, Acetylene, and Normal-Heptane Plumes in Turbulent Coflows of Hot Air." ASME. J. Eng. Gas Turbines Power. January 2008; 130(1): 011502. https://doi.org/10.1115/1.2771245
Axisymmetric plumes of hydrogen, acetylene, or
n
-heptane were formed by the continuous injection of (pure or nitrogen-diluted) fuel into confined turbulent coflows of hot air. Autoignition and subsequent flame propagation was visualized with an intensified high-speed camera. The resulting phenomena that were observed include the statistically steady “random spots” regime and the “flashback” regime. It was found that with higher velocities and smaller injector diameters, the boundary between random spots and flashback shifted to higher air temperatures. In the random spots regime the autoignition regions moved closer to the injector with increasing air temperature and/or decreasing air velocity. After a localized explosive autoignition event, flames propagated into the unburnt mixture in all directions and eventually extinguished, giving rise to autoignition spots of mean radii of
2–5mm
for hydrogen and
6–10mm
for the hydrocarbons. The average flame propagation velocity in both the axial and radial directions varied between 0.5 and 1.2 times the laminar burning speed of the stoichiometric mixture, increasing as the autoigniting regions shifted upstream.
combustion, flames, ignition, laminar flow, turbulence, kernel growth, flame propagation, flame velocity, autoignition, flashback
Flames, Heptane, Turbulence, Flow (Dynamics), Hydrogen, Plumes (Fluid dynamics), Fuels, Ejectors
Autoignition in Turbulent Flows
,” Ph.D. thesis, University of Cambridge, Cambridge, U.K.
Measurements and Simulations of Mixing and Autoignition of an N-Heptane Plume in a Turbulent Flow of Heated Air
Propagation of Unsteady Tribrachial Flames in Laminar Non-premixed Jets
Spark Ignition of Lifted Turbulent Jet Flames
A Study of the Autoignition Process of a Diesel Spray Via High Speed Visualization
Di Diesel Engine Combustion Visualized by Combined Laser Techniques
Photographic Observation and Emission Spectral Analysis of Homogeneous Charge Compression Ignition Combustion
Measurements of Scalar Dissipation in a Turbulent Plume With Planar Laser-Induced Fluorescence of Acetone
, 2005, COSILAB: The Combustion Simulation Laboratory-Version 1.2.3, Haan, Germany, http://www.SoftPredict.comhttp://www.SoftPredict.com
A Comprehensive Modeling Study of Hydrogen Oxidation
A Semi-Empirical Reaction Mechanism for N-Heptane Oxidation and Pyrolysis
On Initiation Reactions of Acetylene Oxidation in Shock Tubes. A Quantum Mechanical and Kinetic Modeling Study
Laboratory Study of Premixed H2-Air and H2–N2-Air Flames in a Low-Swirl Injector for Ultralow Emissions Gas Turbines
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In collaboration with EADS ASTRIUM
We contributed a method for planar shape detection and regularization from raw point sets. The geometric modeling and processing of man-made environments from measurement data often relies upon robust detection of planar primitive shapes. In addition, the detection and reinforcement of regularities between planar parts is a means to increase resilience to missing or defect-laden data as well as to reduce the complexity of models and algorithms down the modeling pipeline. The main novelty behind our method is to perform detection and regularization in tandem. We first sample a sparse set of seeds uniformly on the input point set, then perform in parallel shape detection through region growing, interleaved with regularization through detection and reinforcement of regular relationships (coplanar, parallel and orthogonal). In addition to addressing the end goal of regularization, such reinforcement also improves data fitting and provides guidance for clustering small parts into larger planar parts (Figure 1 ). We evaluate our approach against a wide range of inputs and under four criteria: geometric fidelity, coverage, regularity and running times. Our approach compares well with available implementations such as the efficient RANSAC-based approach proposed by Schnabel and co-authors in 2007 [8] . This work has been published in the Computer Graphics Forum journal.
Figure 1. Shape detection and regularization. The input point set (5.2M points) has been acquired via a LIDAR scanner, from the inside and outside of a physical building. 200 shapes have been detected, aligned with 12 different directions in 179 different planes. The cross section depicts the auditorium in the upper floor and the entrance hall in the lower floor. The closeup highlights the steps of the auditorium which are made up of perfectly parallel and orthogonal planes.
In collaboration with Geoimage
The over-segmentation of images into atomic regions has become a standard and powerful tool in Vision. Traditional superpixel methods, that operate at the pixel level, cannot directly capture the geometric information disseminated into the images. We propose an alternative to these methods by operating at the level of geometric shapes. Our algorithm partitions images into convex polygons. It presents several interesting properties in terms of geometric guarantees, region compactness and scalability. The overall strategy consists in building a Voronoi diagram that conforms to preliminarily detected line-segments, before homogenizing the partition by spatial point process distributed over the image gradient. Our method is particularly adapted to images with strong geometric signatures, typically man-made objects and environments (Figure 2 ). We show the potential of our approach with experiments on large-scale images and comparisons with state-of-the-art superpixel methods [17] . This work has been published in the Computer Graphics Forum journal. Published in the proceedings of CVPR (IEEE conference on Computer Vision and Pattern Recognition).
Figure 2. Image partitioning into convex polygons.
In collaboration with EADS ASTRIUM.
We contributed a supervised machine learning approach for classification of objects from sampled point data. The main idea consists in first abstracting the input object into planar parts at several scales, then discriminate between the different classes of objects solely through features derived from these planar shapes. Abstracting into planar shapes provides a means to both reduce the computational complexity and improve robustness to defects inherent to the acquisition process. Measuring statistical properties and relationships between planar shapes offers invariance to scale and orientation. A random forest is then used for solving the multiclass classification problem. We demonstrate the potential of our approach on a set of indoor objects from the Princeton shape benchmark and on objects acquired from indoor scenes and compare the performance of our method with other point-based shape descriptors [22] (see Figure 3 ).
Figure 3. Classification. Left: We used four tabletop object classes from the Princeton Shape Benchmark: Bottle, Lamp, Mug and Vase. We also select four furniture object classes common to indoor scenes: Chair, Couch, Shelf and Table. Right: We evaluate our approach through computing a confusion matrix, for an increasing amount of noise and outliers. (a): Without noise and outliers. The precision of the class prediction is
82,5%
. The classifier is not reliable for classifying the bottles, which get mislabeled as vases. (b): Added
10%
outliers and
0.5%
noise. Compared to the noise-free version the precision slightly dropped to
77.5%
. (c): Added
20%
1%
noise. The method maintains a precision of
70%
for this level of noise.
Optimizing partition trees for multi-object segmentation with shape prior
Participants : Emmanuel Maggiori, Yuliya Tarabalka.
This work has been done in collaboration with Dr. Guillaume Charpiat (TAO team, Inria Saclay).
Partition trees, multi-class segmentation, shape priors, graph cut.
A partition tree is a hierarchical representation of an image. Once constructed, it can be repeatedly processed to extract information. Multi-object multi-class image segmentation with shape priors is one of the tasks that can be efficiently done upon an available tree. The traditional construction approach is a greedy clustering based on color similarities. However, not considering higher level cues during the construction phase leads to trees that might not accurately represent the underlying objects in the scene, inducing mistakes in the later segmentation. We proposed a method to optimize a tree based both on color distributions and shape priors [15] . It consists in pruning and regrafting tree branches in order to minimize the energy of the best segmentation that can be extracted from the tree. Theoretical guarantees help reduce the search space and make the optimization efficient. Our experiments (see Figure 4 ) show that we succeed in incorporating shape information to restructure a tree, which in turn enables to extract from it good quality multi-object segmentations with shape priors. Published in the proceedings of BMVC (British Machine Vision Conference).
Figure 4. Classification results for the satellite image over Brest.
𝒜
denotes overall classification accuracy, and
𝒟
denotes average buildings overlap. The performance of the proposed binary partition tree (BPT) optimization method is compared with the following methods: 1) support vector machines (SVM) classification; 2) graph cut (GC) with
\alpha
-expansion; 3) cut on the BPT, regularized by the number of regions without using shape priors (TC).
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Extension:Math/Unique Ids - MediaWiki
The attribute "id" is already live in all production Wikipedia systems. It allows linking to an individual equation that has the id attribute set.
<math id="MassEnergyEquivalence">E=mc^2</math>
can be accessed via appending #MassEnergyEquivalence to the Page URL.
Visually no difference can be seen in the output
{\displaystyle E=mc^{2}}
See for example kinetic Energy
Retrieved from "https://www.mediawiki.org/w/index.php?title=Extension:Math/Unique_Ids&oldid=5153885"
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Carotid Bifurcation Hemodynamics in Older Adults: Effect of Measured Versus Assumed Flow Waveform | J. Biomech Eng. | ASME Digital Collection
Department of Mechanical and Industrial Engineering, Biomedical Simulation Laboratory,
University queryof Toronto
Hoi, Y., Wasserman, B. A., Lakatta, E. G., and Steinman, D. A. (May 26, 2010). "Carotid Bifurcation Hemodynamics in Older Adults: Effect of Measured Versus Assumed Flow Waveform." ASME. J Biomech Eng. July 2010; 132(7): 071006. https://doi.org/10.1115/1.4001265
Recent work has illuminated differences in carotid artery blood flow rate dynamics of older versus young adults. To what degree flow waveform shape, and indeed the use of measured versus assumed flow rates, affects the simulated hemodynamics of older adult carotid bifurcations has not been elucidated. Image-based computational fluid dynamics models of
N=9
normal, older adult carotid bifurcations were reconstructed from magnetic resonance angiography. Subject-specific hemodynamics were computed by imposing each individual’s inlet and outlet flow rates measured by cine phase-contrast magnetic resonance imaging or by imposing characteristic young and older adult flow waveform shapes adjusted to cycle-averaged flow rates measured or allometrically scaled to the inlet and outlet areas. Despite appreciable differences in the measured versus assumed flow conditions, the locations and extents of low wall shear stress and elevated relative residence time were broadly consistent; however, the extent of elevated oscillatory shear index was substantially underestimated, more by the use of assumed cycle-averaged flow rates than the assumed flow waveform shape. For studies of individual vessels, use of a characteristic flow waveform shape is likely sufficient, with some benefit offered by scaling to measured cycle-averaged flow rates. For larger-scale studies of many vessels, ranking of cases according to presumed hemodynamic or geometric risk is robust to the assumed flow conditions.
bifurcation, biomedical MRI, blood vessels, computational fluid dynamics, diseases, haemodynamics, image reconstruction, medical image processing, shear flow, waveform analysis, atherosclerosis, waveform, stroke, hemodynamics, imaging, CFD, carotid artery
Bifurcation, Computational fluid dynamics, Flow (Dynamics), Hemodynamics, Cycles, Shapes, Boundary-value problems
Characterization of Common Carotid Artery Blood-Flow Waveforms in Normal Human Subjects
Effect of Vessel Curvature on Doppler Derived Velocity Profiles and Fluid Flow
Flow-Area Relationship in Internal Carotid and Vertebral Arteries
Numerical Investigation of the Non-Newtonian Blood Flow in a Bifurcation Model With a Non-Planar Branch
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Programming and Connectivity - Maple Help
Home : Support : Online Help : System : Information : Updates : Maple 14 : Programming and Connectivity
Maple 14 includes the following enhancements to programming facilities and connectivity to other tools.
Maple Toolbox included with Maple
Maple-NAG® Connector
Importing and Exporting MATLAB® Matrices
ExcelTools For Command-line Maple
In-place Substitution
Better Support for Anonymous Procedures
Task Model in External Call
The Maple Toolbox for is now included with Maple. For detailed information about installation of the Maple Toolbox, see the Maple 14 Install.html file.
The Maple-NAG® Connector is now included with Maple. Users of the NAG library can now access the full functionality of the NAG C library from within Maple. For detailed information about installation of the Maple-NAG Connector, see the Maple 14 Install.html file.
The ArrayTools[Alias] command was extended to allow aliasing an array with a different datatype than the original. This can be useful in conjunction with readbytes when reading mixed format binary files. For example, readbytes can be used to load a file into a integer[1] byte Array. Then, after determining that the first 100 bytes constitute the file's header, the remaining bytes can be aliased as an array of double precision (float[8]) floating-point numbers.
The StringTools package contains two new commands for data compression, Compress and Uncompress. These apply zlib compression and expansion of sets of data including strings and byte arrays.
The CodeTools package contains the new command Usage which can be used to measure the time and memory usage resulting from an command execution similar to time but with much more flexible output control.
with(CodeTools):
Usage(ifactor(32!+1));
\left(\textcolor[rgb]{0,0,1}{61146083}\right)\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{652931}\right)\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{2889419049474073777}\right)\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{2281}\right)
Usage(ifactor(33!+1),output='all');
\textcolor[rgb]{0,0,1}{\mathrm{Record}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{realtime}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0.193}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{cputime}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0.193}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{gctime}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0.06415700000}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{gcrealtime}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0.06424999237}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{bytesused}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{7405784}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{bytesalloc}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-3153920}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{output}}\textcolor[rgb]{0,0,1}{=}\left(\textcolor[rgb]{0,0,1}{67}\right)\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{143446529}\right)\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{175433}\right)\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{101002716748738111}\right)\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{50989}\right)\right)
CodeTools[CPUTime] and CodeTools[RealTime] can be used as shortcuts for calling Usage with options output=[cputime,output] and output=[realtime,output] respectively.
CPUTime(ifactor(34!+1));
\textcolor[rgb]{0,0,1}{0.002}\textcolor[rgb]{0,0,1}{,}\left(\textcolor[rgb]{0,0,1}{4379593820587205958191075783529691}\right)\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{67411}\right)
The ListTools package contains the new commands FindMaximalElement, FindMinimalElement, SelectFirst, and SelectLast. These are efficient implementations of common list manipulation tasks.
SelectFirst(type, [1., 1/2, 1, 5.1, 2, 11], integer);
\textcolor[rgb]{0,0,1}{1}
FindMaximalElement([[1,a,4],[2,b,3],[3,c,2],[4,a,1]], (x,y)->lexorder(x[2],y[2]));
[\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}]
The ImportMatrix and ExportMatrix commands have been updated so that they can import and export MATLAB® binary files. MATLAB® Version 6 files without data compression or MATLAB® Version 7 files with compression are supported.
You can now import data from an Excel spreadsheet or export rtable data to an Excel spreadsheet while working in the command-line Maple environment, without having to run the Standard Worksheet interface. For more information on using the ExcelTools package, see ExcelTools.
The subs command can now perform substitutions in-place in Arrays, Matrices, and Vectors. This functionality is specified by appending inplace as an index to the command name.
A := <w,x;y,z>;
\textcolor[rgb]{0,0,1}{A}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{w}& \textcolor[rgb]{0,0,1}{x}\\ \textcolor[rgb]{0,0,1}{y}& \textcolor[rgb]{0,0,1}{z}\end{array}]
subs[inplace](x=Pi,A);
[\begin{array}{cc}\textcolor[rgb]{0,0,1}{w}& \textcolor[rgb]{0,0,1}{\mathrm{\pi }}\\ \textcolor[rgb]{0,0,1}{y}& \textcolor[rgb]{0,0,1}{z}\end{array}]
[\begin{array}{cc}\textcolor[rgb]{0,0,1}{w}& \textcolor[rgb]{0,0,1}{\mathrm{\pi }}\\ \textcolor[rgb]{0,0,1}{y}& \textcolor[rgb]{0,0,1}{z}\end{array}]
A new special parameter, thisproc, is available within procedures to facilitate recursive calling. Unlike procname, which refers to the name of the currently executing procedure, thisproc refers to the procedure itself, and thus can be used to make a recursive call within an anonymous procedure.
A new procedure option, option procname, causes a procedure to inherit the name of the procedure that called it for the purposes of error reporting. Thus, an exception raised in an inner procedure can be made to appear to have occurred in the enclosing procedure.
A new function has been added to the Task Programming Model. The Return function allows a Task to stop the execution of the Task Model and return a particular value from Start.
We have added functions to the External Call API to allow external code to access the Task Programming Model. Four new functions are available: MapleStartRootTask, MapleCreateContinuationTask, MapleStartChildTask, and MapleTaskReturn.
A new command, RunWorksheet, has been added to the DocumentTools package. This command executes a worksheet as if it were a procedure. The worksheet must be saved on disk, and there is a mechanism for passing arguments. The worksheet runs headless (no user interface) and modally (control does not return to the invoking worksheet until the called worksheet completes). It also runs in its own kernel.
For Maple 14, this function should be considered experimental; its design may change in a future release.
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Singular inner functions of L1-type II
April, 2001 Singular inner functions of
{L}^{1}
-type II
In the first paper of the same title, we introduced the concept of singular inner functions of
{L}^{1}
-type and obtained results for singular inner functions which are reminiscent of the results for weak infinite powers of Blaschke products. In this Paper, we investigate singular inner functions for discrete measures. We give equivalent conditions on a measure for which it is a Blaschke type. And we prove that two discrete measures are mutually singular if and only if the associated common zero sets of singular inner functions of
{t}_{+}^{\infty }
-type do not meet.
Keiji IZUCHI. "Singular inner functions of
{L}^{1}
-type II." J. Math. Soc. Japan 53 (2) 285 - 305, April, 2001. https://doi.org/10.2969/jmsj/05320285
Primary: 46J15
Keywords: bounded analytic function , maximal ideal space , singular inner function
Keiji IZUCHI "Singular inner functions of
{L}^{1}
-type II," Journal of the Mathematical Society of Japan, J. Math. Soc. Japan 53(2), 285-305, (April, 2001)
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It may be helpful to know that ''equivalent'' means ''equal to''.
\frac { 3 } { 4 }
\frac { 75 } { 100 }
\text{For this question, it might be easiest to use a Giant One so that each fraction has a like denominator.}
\text{For instance, let's change: } \frac{3}{4} \text{ to a fraction with 100 as the denominator so we can compare the two}
\text{The Giant One we will use is } \frac{25}{25} \text{ because 4 times 25 is equal to 100 and } \frac{25}{25} \text{ equals 1.}
\frac { 2 } { 3 }
\frac { 12 } { 13 }
\text{Here, it may help to draw a diagram of}
\text{what } \frac{2}{3} \text{ and } \frac{12}{13} \text{ look like. }
Now try comparing the fractions!
These fractions are not equivalent.
\frac { 8 } { 5 }
\frac { 5 } { 8 }
5
fifths in
1
8
eighths in
1
. Can you tell if either fraction is greater than or less than the other with this information?
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Euler simulation of stochastic differential equations (SDEs) for SDE, BM, GBM, CEV, CIR, HWV, Heston, SDEDDO, SDELD, or SDEMRD models - MATLAB simByEuler - MathWorks India
d{X}_{t}=S\left(t\right)\left[L\left(t\right)-{X}_{t}\right]dt+D\left(t,{X}_{t}^{\frac{1}{2}}\right)V\left(t\right)dW
D
d{X}_{t}=0.2\left(0.1-{X}_{t}\right)dt+0.05{X}_{t}^{\frac{1}{2}}dW
d{X}_{t}=S\left(t\right)\left[L\left(t\right)-{X}_{t}\right]dt+D\left(t,{X}_{t}^{\frac{1}{2}}\right)V\left(t\right)dW
D
d{X}_{t}=0.2\left(0.1-{X}_{t}\right)dt+0.05{X}_{t}^{\frac{1}{2}}dW
d{X}_{1t}=B\left(t\right){X}_{1t}dt+\sqrt{{X}_{2t}}{X}_{1t}d{W}_{1t}
d{X}_{2t}=S\left(t\right)\left[L\left(t\right)-{X}_{2t}\right]dt+V\left(t\right)\sqrt{{X}_{2t}}d{W}_{2t}
d{X}_{t}=0.25{X}_{t}dt+0.3{X}_{t}d{W}_{t}
{X}_{t}=P\left(t,{X}_{t}\right)
d{X}_{t}=F\left(t,{X}_{t}\right)dt+G\left(t,{X}_{t}\right)d{W}_{t}
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An intermediate value theorem for monotone operators in ordered Banach spaces | Fixed Point Theory and Algorithms for Sciences and Engineering | Full Text
An intermediate value theorem for monotone operators in ordered Banach spaces
Vadim Kostrykin1 &
Anna Oleynik1,2
We consider a monotone increasing operator in an ordered Banach space having
{u}_{-}
{u}_{+}
as a strong super- and subsolution, respectively. In contrast with the well-studied case
{u}_{+}<{u}_{-}
{u}_{-}<{u}_{+}
. Under the assumption that the order cone is normal and minihedral, we prove the existence of a fixed point located in the order interval
\left[{u}_{-},{u}_{+}\right]
It is an elementary consequence of the intermediate value theorem for continuous real-valued functions
f:\left[{a}_{1},{a}_{2}\right]\to \mathbb{R}
that if either
f\left({a}_{1}\right)>{a}_{1}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}f\left({a}_{2}\right)<{a}_{2}
f\left({a}_{1}\right)<{a}_{1}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}f\left({a}_{2}\right)>{a}_{2},
then f has a fixed point in
\left[{a}_{1},{a}_{2}\right]
. It is a natural question whether this result can be extended to the case of ordered Banach spaces. A number of fixed point theorems with assumptions of type (1) are well known; see, e.g., [[1], Section 2.1]. However, to the best of our knowledge, fixed point theorems with assumptions of type (2) have not been known so far. In the present note, we prove the following fixed point theorem of this type.
Theorem 1 Let X be a real Banach space with an order cone K satisfying
(a) K has a nonempty interior,
(b) K is normal and minihedral.
Assume that there are two points in X,
{u}_{-}\ll {u}_{+}
, and a monotone increasing compact continuous operator
T:\left[{u}_{-},{u}_{+}\right]\to X
{u}_{-}
is a strong supersolution of T and
{u}_{+}
is a strong subsolution, that is,
T{u}_{-}\ll {u}_{-}\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}T{u}_{+}\gg {u}_{+},
then T has a fixed point
{u}_{\ast }\in \left[{u}_{-},{u}_{+}\right]
\left[{u}_{-},{u}_{+}\right]
denotes the order interval
\left\{u\in X:{u}_{-}\le u\le {u}_{+}\right\}
Theorem 1 generalizes an idea developed by the present authors in [2], where the existence of solutions to a certain nonlinear integral equation of Hammerstein type has been shown.
Before we present the proof, we recall some notions. We write
u\ge v
u-v\in K
u>v
u\ge v
u\ne v
u\gg v
u-v\in \phantom{\rule{0.2em}{0ex}}\stackrel{\circ }{K}
\stackrel{\circ }{K}
is the interior of the cone K.
A cone K is called minihedral if for any pair
\left\{x,y\right\}
x,y\in X
, bounded above in order there exists the least upper bound
sup\left\{x,y\right\}
, that is, an element
z\in X
x\le z
y\le z
x\le {z}^{\prime }
y\le {z}^{\prime }
z\le {z}^{\prime }
Obviously, a cone K is minihedral if and only if for any pair
\left\{x,y\right\}
x,y\in X
, bounded below in order there exists the greatest lower bound
inf\left\{x,y\right\}
. If a minihedral cone has a nonempty interior, then any pair
x,y\in X
is bounded above in order. Hence,
sup\left\{x,y\right\}
inf\left\{x,y\right\}
x,y\in X
A cone K is called normal if there exists a constant
N>0
x\le y
x,y\in K
{\parallel x\parallel }_{X}\le N{\parallel y\parallel }_{X}
By the Kakutani-Krein brothers theorem [[3], Theorem 6.6] a real Banach space X with an order cone K satisfying assumptions (a) and (b) of Theorem 1 is isomorphic to the Banach space
C\left(Q\right)
of continuous functions on a compact Hausdorff space Q. The image of K under this isomorphism is the cone of nonnegative continuous functions on Q.
An operator T acting in the Banach space X is called monotone increasing if
u\le v
Tu\le Tv
Consider the operator
\stackrel{ˆ}{T}:\left[{u}_{-},{u}_{+}\right]\to X
\stackrel{ˆ}{T}u:=sup\left\{inf\left\{Tu,{u}_{+}\right\},{u}_{-}\right\}.
inf\left\{T{u}_{+},{u}_{+}\right\}={u}_{+}
sup\left\{{u}_{+},{u}_{-}\right\}={u}_{+}
{u}_{+}
is a fixed point of the operator
\stackrel{ˆ}{T}
. Similarly, one shows that
{u}_{-}
is also a fixed point.
Lemma 2 The operator
\stackrel{ˆ}{T}
is continuous, monotone increasing, compact and maps the order interval
\left[{u}_{-},{u}_{+}\right]
into itself.
v\in K
u↦sup\left\{u,v\right\}
u↦inf\left\{u,v\right\}
are continuous; see, e.g., Corollary 3.1.1 in [4]. Due to the continuity of T, it follows immediately that
\stackrel{ˆ}{T}
is continuous as well. The operator
\stackrel{ˆ}{T}
is monotone increasing since inf and sup are monotone increasing with respect to each argument. Therefore, for any
u\in \left[{u}_{-},{u}_{+}\right]
{u}_{-}=\stackrel{ˆ}{T}{u}_{-}\le \stackrel{ˆ}{T}u\le \stackrel{ˆ}{T}{u}_{+}={u}_{+}.
\left({u}_{n}\right)
be an arbitrary sequence in
\left[{u}_{-},{u}_{+}\right]
. Since T is compact,
\left(T{u}_{n}\right)
\left(T{u}_{{n}_{k}}\right)
v\in X
. From the continuity of
\stackrel{ˆ}{T}
\left(\stackrel{ˆ}{T}{u}_{{n}_{k}}\right)
sup\left\{inf\left\{v,{u}_{+}\right\},{u}_{-}\right\}
, thus, proving that the range of
\stackrel{ˆ}{T}
is relatively compact. □
Lemma 3 There exist
{p}_{±}\in X
{u}_{-}\ll {p}_{-}\ll {p}_{+}\ll {u}_{+}
\stackrel{ˆ}{T}{p}_{-}<{p}_{-},\phantom{\rule{2em}{0ex}}\stackrel{ˆ}{T}{p}_{+}>{p}_{+}.
Proof Due to
T{u}_{-}\ll {u}_{-}
\delta >0
{B}_{\delta }\left({u}_{-}-T{u}_{-}\right)\subset \stackrel{\circ }{K}
{B}_{\delta }\left({u}_{-}-T{u}_{-}\right)
under the continuous mapping
u↦u-Tu
contains a ball
{B}_{ϵ}\left({u}_{-}\right)
u-Tu\gg 0
u\in {B}_{ϵ}\left({u}_{-}\right)
. By the same argument,
u-Tu\ll 0
u\in {B}_{ϵ}\left({u}_{+}\right)
ϵ>0
sufficiently small, we can achieve that
{B}_{ϵ}\left({u}_{-}\right)\cap {B}_{ϵ}\left({u}_{+}\right)=\mathrm{\varnothing }
p\left(t\right):=\left\{\left(1-t\right){u}_{-}+t{u}_{+}|t\in \left[0,1\right]\right\}
{t}_{-}\in \left(0,1\right)
so small that
{p}_{-}:=p\left({t}_{-}\right)\in {B}_{ϵ}\left({u}_{-}\right)
{t}_{+}\in \left(0,1\right)
so close to 1 that
{p}_{+}:=p\left({t}_{+}\right)\in {B}_{ϵ}\left({u}_{+}\right)
{u}_{-}\ll {p}_{-}\ll {p}_{+}\ll {u}_{+}
T{p}_{-}\ll {p}_{-},\phantom{\rule{2em}{0ex}}T{p}_{+}\gg {p}_{+}.
{p}_{-}\ll {u}_{+}
T{p}_{-}\ll {p}_{-}
inf\left\{T{p}_{-},{u}_{+}\right\}=T{p}_{-}
. Further, we obtain
sup\left\{T{p}_{-},{u}_{-}\right\}\le sup\left\{{p}_{-},{u}_{-}\right\}={p}_{-}.
T{p}_{-}\ll {p}_{-}
it follows that there is an element
z\ll 0
T{p}_{-}={p}_{-}+z
sup\left\{T{p}_{-},{u}_{-}\right\}={p}_{-}
sup\left\{z,{u}_{-}-{p}_{-}\right\}=0
. However, in view of the Kakutani-Krein brothers theorem,
{u}_{-}-{p}_{-}\ll 0
sup\left\{z,{u}_{-}-{p}_{-}\right\}\ll 0
sup\left\{T{p}_{-},{u}_{-}\right\}\ne {p}_{-}
\stackrel{ˆ}{T}{p}_{-}<{p}_{-}
. Similarly one shows that
\stackrel{ˆ}{T}{p}_{+}>{p}_{+}
The main tool for the proof of Theorem 1 is Amann’s theorem on three fixed points (see, e.g., [[5], Theorem 7.F and Corollary 7.40]):
Theorem 4 Let X be a real Banach space with an order cone having a nonempty interior. Assume there are four points in X,
{p}_{1}\ll {p}_{2}<{p}_{3}\ll {p}_{4},
and a monotone increasing image compact operator
\stackrel{ˆ}{T}:\left[{p}_{1},{p}_{4}\right]\to X
\stackrel{ˆ}{T}{p}_{1}={p}_{1},\phantom{\rule{2em}{0ex}}\stackrel{ˆ}{T}{p}_{2}<{p}_{2},\phantom{\rule{2em}{0ex}}\stackrel{ˆ}{T}{p}_{3}>{p}_{3},\phantom{\rule{2em}{0ex}}\stackrel{ˆ}{T}{p}_{4}={p}_{4}.
\stackrel{ˆ}{T}
has a third fixed point p satisfying
{p}_{1}<p<{p}_{4}
p\notin \left[{p}_{1},{p}_{2}\right]
p\notin \left[{p}_{3},{p}_{4}\right]
Recall that the operator is called image compact if it is continuous and its image is a relatively compact set.
{p}_{1}={u}_{-}
{p}_{2}={p}_{-}
{p}_{3}={p}_{+}
{p}_{4}={u}_{+}
{p}_{±}
is as in Lemma 3. Since the cone K is normal, by Theorem 1.1.1 in [1],
\left[{u}_{-},{u}_{+}\right]
is norm bounded. Thus,
\stackrel{ˆ}{T}
is image compact.
Theorem 4 yields the existence of a fixed point
{u}_{\ast }
\stackrel{ˆ}{T}
{u}_{-}<{u}_{\ast }<{u}_{+}
{u}_{\ast }
is a fixed point of the operator T as well. This observation completes the proof of Theorem 1.
Kostrykin, V, Oleynik, A: On the existence of unstable bumps in neural networks. Preprint. arXiv:1112.2941 [math.DS] (2011)
Krasnosel’skij MA, Lifshits JA, Sobolev AV The Method of Positive Operators, Sigma Series in Applied Mathematics 5. In Positive Linear Systems. Heldermann, Berlin; 1989.
Chueshov I Lecture Notes in Mathematics 1779. In Monotone Random Systems Theory and Applications. Springer, Berlin; 2002.
Zeidler E: Nonlinear Functional Analysis and Its Applications: I: Fixed-Point Theorems. Springer, New York; 1986.
The authors thank H.-P. Heinz for useful comments. This work has been supported in part by the Deutsche Forschungsgemeinschaft, Grant KO 2936/4-1.
FB 08 - Institut für Mathematik, Johannes Gutenberg-Universität Mainz, Staudinger Weg 9, Mainz, D-55099, Germany
Vadim Kostrykin & Anna Oleynik
Department of Mathematics, University of Uppsala, P.O. Box 480, Uppsala, S-75106, Sweden
Correspondence to Vadim Kostrykin.
Kostrykin, V., Oleynik, A. An intermediate value theorem for monotone operators in ordered Banach spaces. Fixed Point Theory Appl 2012, 211 (2012). https://doi.org/10.1186/1687-1812-2012-211
fixed point theorems in ordered Banach spaces
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Lattice QCD - Vixrapedia
Lattice QCD (Quantum ChromoDynamic) is the study of quarks and gluons on a spacetime lattice.
The QCD Lattice
Lattice QCD (Quantum ChromoDynamic) is the study of quarks and gluons on a spacetime lattice. The QCD lattice is constructed in six easy steps:
1) Consider open strings with a quark at each endpoint
{\displaystyle a_{i,j}=q_{0},q_{N}}
2) Construct a square lattice with these strings
{\displaystyle i\rightarrow N}
3) Where the string endpoints meet (i.e. at each lattice site), the quark pairs will merge to form quark-gluon plasma
{\displaystyle |q\rangle =[q_{0},q_{N}]}
4) The plasma will interact via the strong force by sending gluons through the open strings
{\displaystyle \langle q|[q_{0},q_{N}]\rangle =q_{x}^{\mu }|q\rangle }
6) We compute the worldsheet S-matrix to invert the fermion matrix
{\displaystyle S=-q\int dq(\partial ^{2}q-i{\tilde {q}}q)}
5) Taking the continuum limit, it is simple to see that this reduces to non-perturbative QCD
{\displaystyle a\rightarrow 0:\,\,\,S[{\bar {\psi }}(q),\psi (q)]\rightarrow -T\int d^{4}x\,\,{\bar {\psi }}\partial \psi -m{\bar {\psi }}\psi +\sum _{n\neq 1}g_{n}({\bar {\psi }}\psi )^{n}}
However, this approach is far from being straightforward. It is computationally intensive for the following reasons:
1) Quark-gluon plasma is extremely hot (over 4 TRILLION degrees celsius, or 4000000000000°), and must be thermally regularized in Langevin time using Gauge Cooling (GC):
{\displaystyle \left({\frac {z_{i}+\nu +ik/2}{z_{i}+\nu -ik/2}}\right)^{p}\left({\frac {z_{i}-\nu +ik/2}{z_{i}-\nu -ik/2}}\right)^{p}=\prod _{j=1\neq i}^{m}{\frac {z_{i}-z_{j}+i}{z_{i}-z_{j}-i}}}
2) Depending on the choice of string background, the vacuum might be unstable. To tackle this issue, scientits have introduced an additional force, named Dynamic Stabilisation, loosely based on the concept of ether and which is expected to vanish in the continuum limit:
{\displaystyle K_{x,\mu }^{a}=-D_{x,\mu }^{a}S}
{\displaystyle D_{x,\mu }^{a}f(U)={\frac {\partial }{\partial \alpha }}f(e^{i\alpha \lambda ^{a}}U)\vert _{\alpha =0}}
is the Dynamic term and
{\displaystyle S=S_{YM}-\ln \det D}
is the Stabilisation term.
To help the integrability of the lattice integers, we can introduce the Euclidean "correlator":
{\displaystyle \langle O_{2}(t)O_{1}(t)\rangle ={\frac {tr\left[e^{-(T-t){\hat {H}}}{\hat {O}}_{2}e^{-t{\hat {H}}}{\hat {O}}_{1}\right]}{tr\left[e^{-T{\hat {H}}}\right]}}}
which links observables at one lattice position to ones at a different Euclidean position. This quantity basically says that to get this relationship, you have to sum over all links with the operators and divide away the "vacuum" links what do not involve the operators in question. When one takes the limit where the operators perform a closed loop, we get the so-called Wilson gauge action:
{\displaystyle S_{G}[U]={\frac {2}{g^{2}}}\sum _{n\in \Lambda }\sum _{\mu <\nu }tr\left[\mathbf {1} -U_{\mu \nu }(n)\right]\,,\quad U_{\mu \nu }(n)=U_{\mu }(n)U_{\nu }(n+{\hat {\mu }})U_{\mu }^{\dagger }(n+{\hat {\nu }})U_{\nu }^{\dagger }(n)}
where U is the operator's unitary action.
Typically, these lattice calculations are done by simulation with random inputs. These "Monte Carlo simulations" are given by:
{\displaystyle \langle O\rangle =\lim _{N\to \infty }{\frac {1}{N}}\sum _{n=1}^{N}O[U_{n}]\,,}
{\displaystyle U_{n}}
sampled according to
{\displaystyle dP(U)={\frac {e^{-S[U]}{\mathcal {D}}[U]}{\int {\mathcal {D}}[U]e^{-S[U]}}}}
where the expected value of the operator (i.e., its average value) is just the sum of all of it's unitary values.
Lattice QCD's electricity usage is enormous. In November, the power consumed by the entire lattice network was estimated to be higher than that of the Republic of Ireland. Since then, its demands have only grown. It’s now on pace to use just over 42TWh of electricity in a year, placing it ahead of New Zealand and Hungary and just behind Peru, according to estimates from Digiconomist. That’s commensurate with CO2 emissions of 20 megatonnes – or roughly 1m transatlantic flights.
That fact should be a grave notion to anyone who hopes for the non-perturbative physics to grow further in stature and enter widespread usage. But even more alarming is that things could get much, much worse, helping to increase climate change in the process.
Retrieved from "https://www.vixrapedia.org/w/index.php?title=Lattice_QCD&oldid=4015"
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What is the Qstick Indicator
The Qstick indicator is a technical analysis indicator developed by Tushar Chande to numerically identify trends on a price chart. It is calculated by taking an 'n' period moving average of the difference between the open and closing prices. A Qstick value greater than zero means that the majority of the last 'n' days have been up, indicating that buying pressure has been increasing.
The Qstick Indicator is also called Quick Stick. It is not widely available in trading and charting software.
The QStick calculates a moving average of the difference between closing and opening prices.
A rising indicator signals the price is closing higher than it opened, on average.
A falling QStick signals the price is closing lower than it opened, on average.
The QStick can generate trade signals based on signal-line or zero-line crossovers.
The Formula for the QStick Indicator is
\begin{aligned}&\text{QSI} = \text{EMA or SMA of } ( \text{Close} - \text{Open} ) \\&\textbf{where:} \\&\text{EMA} = \text{Exponential moving average} \\&\text{SMA} = \text{Simple moving average} \\&\text{Close} = \text{Closing price for period} \\&\text{Open} = \text{Opening price for period} \\\end{aligned}
QSI=EMA or SMA of (Close−Open)where:EMA=Exponential moving averageSMA=Simple moving averageClose=Closing price for periodOpen=Opening price for period
There is the option to add a simple moving average (SMA) of the QStick indicator. This creates a signal line.
How to Calculate the QStick Indicator
Record differences between the close and open price for each period.
Decide on how many periods to use in the EMA or SMA. The more periods used, the smoother the indicator and the fewer the signals, better for identifying the overall trend.
Calculate the EMA or SMA once there are enough (close-open) data points.
Option: calculate an SMA of the Qstick calculations. This provides a signal line. Three is common period used for signal lines.
What Does the Qstick Indicator Tell You?
The QStick is measuring buying and selling pressure, taking an average of the difference between closing and opening prices. When the price, on average, is closing lower than it opens, the indicator moves lower. When the price, on average, is closing higher than the open, the indicator moves up.
Transaction signals occur when the Qstick crosses above the zero line. Crossing above zero is used as a buy signal because it is indicating that buying pressure is increasing, while sell signals occur when the indicator moves below zero.
In addition, an 'n' period moving average of the Qstick values can be drawn to act as a signal line. Transaction signals are then generated when the Qstick value crosses through the trigger line. Three is a common 'n' period for signal line.
When the QSticks moves above the signal line it indicates that the price is starting to have more closes above the open, and therefore price may be starting rise. When the Qstick crosses below the signal line it indicates price is starting have more closes below the open. Price may be starting to trend lower.
The indicator may also highlight divergence. When price is rising but the QStick is falling, it shows that momentum may be waning. When price is falling and QStick is rising, this shows buying momentum in price may occur soon. The indicator can produce anomalies, though. It does not account for gaps, only intraday price action. Therefore, if the price gaps higher, but closes below the open, this is still marked as bearish even though the price may have still closed higher than the prior close. May could result in divergence which doesn't necessarily indicate a timely reversal in price.
Example of How to Use the QStick Indicator
The following chart shows an 20-period QStick applied to the SPDR S&P 500 ETF (SPY).
When the price is choppy, so are the buy and sell signals. On the left side of the chart there are many zero-line crossovers that did not generate profitable trade signals, nor identified the trend conclusively.
On the right-side of the chart, there were more trending periods in the price. During this period the QStick did a better job of identifying the trend, staying above zero when the price trend was up, and staying below zero when the price trend was down.
The Difference Between the QStick Indicator and Rate of Change (ROC)
QStick looks at the difference between open and closing prices, and then takes an average of that difference. The ROC indicator looks at the difference between the current closing price and a closing price 'n' periods ago. That amount is then divided by the close 'n' periods ago and then multiplied by 100. The indicators are similar but look at slightly different data are calculated differently, so they will have slightly different trade signals.
Limitations of Using the QStick Indicator
The QStick indicator only looks at historical data, and takes a moving average of it. Therefore, it is not inherently predictive, and its movements will typically lag behind the actual movements in price.
The QStick can produce anomalies when the price is gapping in one direction but the intraday price action moves the other. This may cause divergences between the price and the indicator but may not necessarily indicate a timely reversal in price.
The trade signals may not necessarily be ideal, and often need to be combined with some other filter. In choppy conditions the price will whipsaw across the zero line and/or signal line, generating numerous losing trades.
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Interval (mathematics) — Wikipedia Republished // WIKI 2
This article is about intervals of real numbers and other totally ordered sets. For the most general definition, see partially ordered set § Intervals. For other uses, see Interval (disambiguation).
In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers x satisfying 0 ≤ x ≤ 1 is an interval which contains 0, 1, and all numbers in between. Other examples of intervals are the set of numbers such that 0 < x < 1, the set of all real numbers
{\displaystyle \mathbb {R} }
, the set of nonnegative real numbers, the set of positive real numbers, the empty set, and any singleton (set of one element).
Real intervals play an important role in the theory of integration, because they are the simplest sets whose "size" (or "measure" or "length") is easy to define. The concept of measure can then be extended to more complicated sets of real numbers, leading to the Borel measure and eventually to the Lebesgue measure.
Intervals are central to interval arithmetic, a general numerical computing technique that automatically provides guaranteed enclosures for arbitrary formulas, even in the presence of uncertainties, mathematical approximations, and arithmetic roundoff.
Intervals are likewise defined on an arbitrary totally ordered set, such as integers or rational numbers. The notation of integer intervals is considered in the special section below.
Intervals and interval notation | Functions | Algebra I | Khan Academy
Solving Inequalities Interval Notation, Number Line, Absolute Value, Fractions & Variables - Algebra
Best way to understand Interval and Interval notations(open interval)[closed interval]
1.1 Note on conflicting terminology
2 Notations for intervals
2.1 Including or excluding endpoints
2.2 Infinite endpoints
2.3 Integer intervals
3 Classification of intervals
4 Properties of intervals
5 Dyadic intervals
6.1 Multi-dimensional intervals
6.2 Complex intervals
7 Topological algebra
A degenerate interval is any set consisting of a single real number (i.e., an interval of the form [a,a]).[2] Some authors include the empty set in this definition. A real interval that is neither empty nor degenerate is said to be proper, and has infinitely many elements.
Bounded intervals are bounded sets, in the sense that their diameter (which is equal to the absolute difference between the endpoints) is finite. The diameter may be called the length, width, measure, range, or size of the interval. The size of unbounded intervals is usually defined as +∞, and the size of the empty interval may be defined as 0 (or left undefined).
An interval is said to be left-open if and only if it contains no minimum (an element that is smaller than all other elements); right-open if it contains no maximum; and open if it has both properties. The interval [0,1) = {x | 0 ≤ x < 1}, for example, is left-closed and right-open. The empty set and the set of all reals are open intervals, while the set of non-negative reals, is a right-open but not left-open interval. The open intervals are open sets of the real line in its standard topology, and form a base of the open sets.
An interval is said to be left-closed if it has a minimum element, right-closed if it has a maximum, and simply closed if it has both. These definitions are usually extended to include the empty set and the (left- or right-) unbounded intervals, so that the closed intervals coincide with closed sets in that topology.
An interval I is subinterval of interval J if I is a subset of J. An interval I is a proper subinterval of J if I is a proper subset of J.
Note on conflicting terminology
Notations for intervals
The interval of numbers between a and b, including a and b, is often denoted [a, b]. The two numbers are called the endpoints of the interval. In countries where numbers are written with a decimal comma, a semicolon may be used as a separator to avoid ambiguity.
Including or excluding endpoints
To indicate that one of the endpoints is to be excluded from the set, the corresponding square bracket can be either replaced with a parenthesis, or reversed. Both notations are described in International standard ISO 31-11. Thus, in set builder notation,
{\displaystyle {\begin{aligned}{\color {Maroon}(}a,b{\color {Maroon})}={\mathopen {\color {Maroon}]}}a,b{\mathclose {\color {Maroon}[}}&=\{x\in \mathbb {R} \mid a{\color {Maroon}{}<{}}x{\color {Maroon}{}<{}}b\},\\{}{\color {DarkGreen}[}a,b{\color {Maroon})}={\mathopen {\color {DarkGreen}[}}a,b{\mathclose {\color {Maroon}[}}&=\{x\in \mathbb {R} \mid a{\color {DarkGreen}{}\leq {}}x{\color {Maroon}{}<{}}b\},\\{}{\color {Maroon}(}a,b{\color {DarkGreen}]}={\mathopen {\color {Maroon}]}}a,b{\mathclose {\color {DarkGreen}]}}&=\{x\in \mathbb {R} \mid a{\color {Maroon}{}<{}}x{\color {DarkGreen}{}\leq {}}b\},\\{}{\color {DarkGreen}[}a,b{\color {DarkGreen}]}={\mathopen {\color {DarkGreen}[}}a,b{\mathclose {\color {DarkGreen}]}}&=\{x\in \mathbb {R} \mid a{\color {DarkGreen}{}\leq {}}x{\color {DarkGreen}{}\leq {}}b\}.\end{aligned}}}
Each interval (a, a), [a, a), and (a, a] represents the empty set, whereas [a, a] denotes the singleton set {a}. When a > b, all four notations are usually taken to represent the empty set.
Both notations may overlap with other uses of parentheses and brackets in mathematics. For instance, the notation (a, b) is often used to denote an ordered pair in set theory, the coordinates of a point or vector in analytic geometry and linear algebra, or (sometimes) a complex number in algebra. That is why Bourbaki introduced the notation ]a, b[ to denote the open interval.[5] The notation [a, b] too is occasionally used for ordered pairs, especially in computer science.
Infinite endpoints
In some contexts, an interval may be defined as a subset of the extended real numbers, the set of all real numbers augmented with −∞ and +∞.
Even in the context of the ordinary reals, one may use an infinite endpoint to indicate that there is no bound in that direction. For example, (0, +∞) is the set of positive real numbers, also written as
{\displaystyle \mathbb {R} _{+}}
{\displaystyle \mathbb {R} }
Integer intervals
When a and b are integers, the notation ⟦a, b⟧, or [a .. b] or {a .. b} or just a .. b, is sometimes used to indicate the interval of all integers between a and b included. The notation [a .. b] is used in some programming languages; in Pascal, for example, it is used to formally define a subrange type, most frequently used to specify lower and upper bounds of valid indices of an array.
An integer interval that has a finite lower or upper endpoint always includes that endpoint. Therefore, the exclusion of endpoints can be explicitly denoted by writing a .. b − 1 , a + 1 .. b , or a + 1 .. b − 1. Alternate-bracket notations like [a .. b) or [a .. b[ are rarely used for integer intervals.[citation needed]
The intervals of real numbers can be classified into the eleven different types listed below[citation needed], where a and b are real numbers, and
{\displaystyle a<b}
{\displaystyle [b,a]=(b,a)=[b,a)=(b,a]=(a,a)=[a,a)=(a,a]=\{\}=\varnothing }
{\displaystyle [a,a]=\{a\}}
{\displaystyle (a,b)=\{x\mid a<x<b\}}
{\displaystyle [a,b]=\{x\mid a\leq x\leq b\}}
{\displaystyle [a,b)=\{x\mid a\leq x<b\}}
{\displaystyle (a,b]=\{x\mid a<x\leq b\}}
{\displaystyle (a,+\infty )=\{x\mid x>a\}}
{\displaystyle [a,+\infty )=\{x\mid x\geq a\}}
{\displaystyle (-\infty ,b)=\{x\mid x<b\}}
{\displaystyle (-\infty ,b]=\{x\mid x\leq b\}}
{\displaystyle (-\infty ,+\infty )=\mathbb {R} }
Properties of intervals
{\displaystyle \mathbb {R} }
. It follows that the image of an interval by any continuous function is also an interval. This is one formulation of the intermediate value theorem.
The intervals are also the convex subsets of
{\displaystyle \mathbb {R} }
{\displaystyle X\subseteq \mathbb {R} }
is also the convex hull of
{\displaystyle X}
{\displaystyle (a,b)\cup [b,c]=(a,c]}
{\displaystyle \mathbb {R} }
is viewed as a metric space, its open balls are the open bounded sets (c + r, c − r), and its closed balls are the closed bounded sets [c + r, c − r].
{\displaystyle [x,x]=\{x\}}
, and the elements that are greater than x. The parts I1 and I3 are both non-empty (and have non-empty interiors), if and only if x is in the interior of I. This is an interval version of the trichotomy principle.
Dyadic intervals
{\textstyle {\frac {j}{2^{n}}}}
{\textstyle {\frac {j+1}{2^{n}}}}
{\textstyle j}
{\textstyle n}
The dyadic intervals consequently have a structure that reflects that of an infinite binary tree.
Dyadic intervals are relevant to several areas of numerical analysis, including adaptive mesh refinement, multigrid methods and wavelet analysis. Another way to represent such a structure is p-adic analysis (for p = 2).[6]
Multi-dimensional intervals
Further information: Region (mathematics)
In many contexts, a{\displaystyle n}
{\displaystyle \mathbb {R} ^{n}}
that is the Cartesian product of
{\displaystyle n}
{\displaystyle I=I_{1}\times I_{2}\times \cdots \times I_{n}}
{\displaystyle n=2}
{\displaystyle n=3}
, this can be thought of as a region bounded by an axis-aligned cube or a rectangular cuboid. In higher dimensions, the Cartesian product of
{\displaystyle n}
{\displaystyle I}
{\displaystyle I_{k}}
{\displaystyle I_{k}}
{\displaystyle I}
{\displaystyle I}
{\displaystyle I}
{\displaystyle \mathbb {R} ^{n}}
Intervals of complex numbers can be defined as regions of the complex plane, either rectangular or circular.[7]
Intervals can be associated with points of the plane, and hence regions of intervals can be associated with regions of the plane. Generally, an interval in mathematics corresponds to an ordered pair (x,y) taken from the direct product R × R of real numbers with itself, where it is often assumed that y > x. For purposes of mathematical structure, this restriction is discarded,[8] and "reversed intervals" where y − x < 0 are allowed. Then, the collection of all intervals [x,y] can be identified with the topological ring formed by the direct sum of R with itself, where addition and multiplication are defined component-wise.
{\displaystyle (R\oplus R,+,\times )}
has two ideals, { [x,0] : x ∈ R } and { [0,y] : y ∈ R }. The identity element of this algebra is the condensed interval [1,1]. If interval [x,y] is not in one of the ideals, then it has multiplicative inverse [1/x, 1/y]. Endowed with the usual topology, the algebra of intervals forms a topological ring. The group of units of this ring consists of four quadrants determined by the axes, or ideals in this case. The identity component of this group is quadrant I.
{\displaystyle R\oplus R}
, the ring of intervals has been identified[9] with the split-complex number plane by M. Warmus and D. H. Lehmer through the identification
This linear mapping of the plane, which amounts of a ring isomorphism, provides the plane with a multiplicative structure having some analogies to ordinary complex arithmetic, such as polar decomposition.
Arc (geometry)
Interval graph
Interval (statistics)
Line segment
Partition of an interval
Unit interval
^ "Interval and segment - Encyclopedia of Mathematics". www.encyclopediaofmath.org. Archived from the original on 2014-12-26. Retrieved 2016-11-12.
^ Rudin, Walter (1976). Principles of Mathematical Analysis. New York: McGraw-Hill. pp. 31. ISBN 0-07-054235-X.
^ "Why is American and French notation different for open intervals (x, y) vs. ]x, y[?". hsm.stackexchange.com. Retrieved 28 April 2018.
^ Kozyrev, Sergey (2002). "Wavelet theory as p-adic spectral analysis". Izvestiya RAN. Ser. Mat. 66 (2): 149–158. arXiv:math-ph/0012019. Bibcode:2002IzMat..66..367K. doi:10.1070/IM2002v066n02ABEH000381. S2CID 16796699. Retrieved 2012-04-05.
^ Complex interval arithmetic and its applications, Miodrag Petković, Ljiljana Petković, Wiley-VCH, 1998, ISBN 978-3-527-40134-5
^ Kaj Madsen (1979) Review of "Interval analysis in the extended interval space" by Edgar Kaucher[permanent dead link] from Mathematical Reviews
^ D. H. Lehmer (1956) Review of "Calculus of Approximations"[permanent dead link] from Mathematical Reviews
T. Sunaga, [https://web.archive.org/web/20120309164347/http://www.cs.utep.edu/interval-comp/sunaga.pdf Archived 2012-03-09 at the Wayback Machine "Theory of interval algebra and its application to numerical analysis"], In: Research Association of Applied Geometry (RAAG) Memoirs, Ggujutsu Bunken Fukuy-kai. Tokyo, Japan, 1958, Vol. 2, pp. 29–46 (547-564); reprinted in Japan Journal on Industrial and Applied Mathematics, 2009, Vol. 26, No. 2-3, pp. 126–143.
A Lucid Interval by Brian Hayes: An American Scientist article provides an introduction.
[https://web.archive.org/web/20060302095039/http://www.cs.utep.edu/interval-comp/main.html Archived 2006-03-02 at the Wayback Machine Interval computations website]
[https://web.archive.org/web/20070203144604/http://www.cs.utep.edu/interval-comp/icompwww.html Archived 2007-02-03 at the Wayback Machine Interval computations research centers]
Interval Notation by George Beck, Wolfram Demonstrations Project.
Weisstein, Eric W. "Interval". MathWorld.
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Pricing Bermudan Swaptions with Monte Carlo Simulation - MATLAB & Simulink - MathWorks í•œêµ
Black's Model and the Swaption Volatility Matrix
Selecting the Calibration Instruments
Hull White 1 Factor Model
Linear Gaussian 2 Factor Model
This example shows how to price Bermudan swaptions using interest-rate models in Financial Instruments Toolbox™. Specifically, a Hull-White one factor model, a Linear Gaussian two-factor model, and a LIBOR Market Model are calibrated to market data and then used to generate interest-rate paths using Monte Carlo simulation.
In this example, the ZeroRates for a zero curve is hard-coded. You can also create a zero curve by bootstrapping the zero curve from market data (for example, deposits, futures/forwards, and swaps). The hard-coded data for the zero curve is defined as:
For this example, you compute the price of a 10-no-call-1 Bermudan swaption.
BermudanExerciseDates = daysadd(Settle,360*(1:9),1);
BermudanMaturity = datenum('21-Jul-2018');
BermudanStrike = .045;
Black's model is often used to price and quote European exercise interest-rate options, that is, caps, floors, and swaptions. In the case of swaptions, Black's model is used to imply a volatility given the current observed market price. The following matrix shows the Black implied volatility for a range of swaption exercise dates (columns) and underlying swap maturities (rows).
Selecting the instruments to calibrate the model to is one of the tasks in calibration. For Bermudan swaptions, it is typical to calibrate to European swaptions that are co-terminal with the Bermudan swaption that you want to price. In this case, all swaptions having an underlying tenor that matures before the maturity of the swaption to be priced are used in the calibration.
relidx = find(EurMatFull <= BermudanMaturity);
Swaption prices are computed using Black's Model. You can then use the swaption prices to compare the model's predicted values. To compute the swaption prices using Black's model:
% Compute Swaption Prices using Black's model
The following parameters are used where each exercise date is a simulation date.
SimTimes = diff(yearfrac(SimDates(1),SimDates));
The Hull-White one-factor model describes the evolution of the short rate and is specified by the following:
dr=\left[\mathrm{θ}\left(t\right)-\mathrm{α}r\right]dt+\mathrm{Ï}dW
The Hull-White model is calibrated using the function swaptionbyhw, which constructs a trinomial tree to price the swaptions. Calibration consists of minimizing the difference between the observed market prices (computed above using the Black's implied swaption volatility matrix) and the model's predicted prices.
This example uses the Optimization Toolbox™ function lsqnonlin to find the parameter set that minimizes the difference between the observed and predicted values. However, other approaches (for example, simulated annealing) may be appropriate. Starting parameters and constraints for
\mathrm{α}
\mathrm{Ï}
are set in the variables x0 , lb, and ub; these could also be varied depending upon the particular calibration approach.
warnId = 'fininst:swaptionbyirtree:IgnoredSettle';
warnStruct = warning('off',warnId); % Turn warning off
4 15 0.122217 0.00131297 0.0041
warning(warnStruct); % Turn warnings on
HW_alpha = HW1Fparams(1);
HW_sigma = HW1Fparams(2);
% Construct the HullWhite1F model using the HullWhite1F constructor.
HW1F = HullWhite1F(RateSpec,HW_alpha,HW_sigma);
% Use Monte Carlo simulation to generate the interest-rate paths with
% HullWhite1F.simTermStructs.
% Examine one simulation
% Price the swaption using the helper function hBermudanSwaption
HW1FBermPrice = hBermudanSwaption(HW1FSimPaths,SimDates,Tenor,BermudanStrike,...
BermudanExerciseDates,BermudanMaturity);
The Linear Gaussian two-factor model (called the G2++ by Brigo and Mercurio) is also a short rate model, but involves two factors. Specifically:
r\left(t\right)=x\left(t\right)+y\left(t\right)+\mathrm{Ï}\left(t\right)
dx\left(t\right)=-ax\left(t\right)dt+\mathrm{Ï}d{W}_{1}\left(t\right)
dy\left(t\right)=-by\left(t\right)dt+\mathrm{η}d{W}_{2}\left(t\right)
d{W}_{1}\left(t\right)d{W}_{2}\left(t\right)
\mathrm{Ï}
d{W}_{1}\left(t\right)d{W}_{2}\left(t\right)=\mathrm{Ï}
\mathrm{Ï}
is a function chosen to match the initial zero curve.
You can use the function swaptionbylg2f to compute analytic values of the swaption price for model parameters and to calibrate the model. Calibration consists of minimizing the difference between the observed market prices and the model's predicted prices.
% Calibrate the set of parameters that minimize the difference between the
% observed and predicted values using swaptionbylg2f and lsqnonlin.
G2PPobjfun = @(x) SwaptionBlackPrices(relidx) - ...
swaptionbylg2f(RateSpec,x(1),x(2),x(3),x(4),x(5),SwaptionStrike(relidx),...
LG2Fparams = lsqnonlin(G2PPobjfun,x0,lb,ub,options);
6 42 0.0395759 0.0137735 7.43
7 48 0.0265828 0.0355771 0.787
8 54 0.0252764 0.111744 0.5
10 66 0.0222739 0.106678 0.0946
11 72 0.0221799 0.0380108 0.911
% Create the G2PP object and use Monte Carlo simulation to generate the
% interest-rate paths with LinearGaussian2F.simTermStructs.
LG2FBermPrice = hBermudanSwaption(G2PPSimPaths,SimDates,Tenor,BermudanStrike,BermudanExerciseDates,BermudanMaturity);
\frac{d{F}_{i}\left(t\right)}{{F}_{i}}=-{\mathrm{μ}}_{i}dt+{\mathrm{Ï}}_{i}\left(t\right)d{W}_{i}
{\mathrm{Ï}}_{i}
is the volatility function for each rate and
dW
is an N dimensional geometric Brownian motion with:
d{W}_{i}\left(t\right)d{W}_{j}\left(t\right)={\mathrm{Ï}}_{ij}
The LMM relates the drifts of the forward rates based on no-arbitrage arguments.
The choice with the LMM is how to model volatility and correlation and how to estimate the parameters of these models for volatility and correlation. In practice, you might use a combination of historical data (for example, observed correlation between forward rates) and current market data. For this example, only swaption data is used. Furthermore, many different parameterizations of the volatility and correlation exist. This example uses two relatively straightforward parameterizations.
{\mathrm{Ï}}_{i}\left(t\right)={\mathrm{Ï}}_{i}\left(a\left({T}_{i}-t\right)+b\right){e}^{c\left({T}_{i}-t\right)}+d
\mathrm{Ï}
{i}^{th}
forward rate. For this example, all of the Phi's will be taken to be 1.
For the correlation, the following functional form is used:
{\mathrm{Ï}}_{i,j}={e}^{-\mathrm{β}|i-j|}
Once the functional forms are specified, the parameters need to be estimated using market data. One useful approximation, initially developed by Rebonato, is the following, which computes the Black volatility for a European swaption, given a LMM with a set of volatility functions and a correlation matrix.
\left({v}_{\mathrm{α},\mathrm{β}}^{LFM}{\right)}^{2}=\underset{i,j=\mathrm{α}+1}{\overset{\mathrm{β}}{â}}\frac{{w}_{i}\left(0\right){w}_{j}\left(0\right){F}_{i}\left(0\right){F}_{j}\left(0\right){\mathrm{Ï}}_{i,j}}{{S}_{\mathrm{α},\mathrm{β}}\left(0{\right)}^{2}}{â«}_{0}^{{T}_{\mathrm{α}}}{\mathrm{Ï}}_{i}\left(t\right){\mathrm{Ï}}_{j}\left(t\right)dt
{w}_{i}\left(t\right)=\frac{{\mathrm{Ï}}_{i}P\left(t,{T}_{i}\right)}{\underset{k=\mathrm{α}+1}{\overset{\mathrm{β}}{â}}{\mathrm{Ï}}_{k}P\left(t,{t}_{k}\right)}
This calculation is done using blackvolbyrebonato to compute analytic values of the swaption price for model parameters and also to calibrate the model. Calibration consists of minimizing the difference between the observed implied swaption Black volatilities and the predicted Black volatilities.
LMMparams = lsqnonlin(objfun,x0,lb,ub,options);
% Calculate VolFunc for the LMM object.
% Plot the volatility function
% Inspect the correlation matrix
displayCorrelationMatrix(CorrelationMatrix);
% Create the LMM object and use Monte Carlo simulation to generate the
% interest-rate paths with LiborMarketModel.simTermStructs.
LMM = LiborMarketModel(RateSpec,VolFunc,CorrelationMatrix,'Period',1);
LMMTenor = 1:10;
LMMBermPrice = hBermudanSwaption(LMMZeroRates,SimDates,LMMTenor,.045,BermudanExerciseDates,BermudanMaturity);
displayResults(nTrials, nPeriods, HW1FBermPrice, LG2FBermPrice, LMMBermPrice);
HW1F Bermudan Swaption Price: 3.7577
LG2F Bermudan Swaption Price: 3.5576
LMM Bermudan Swaption Price: 3.4911
This example is based on the following books, papers and journal articles:
Brigo, D. and F. Mercurio. Interest Rate Models - Theory and Practice with Smile, Inflation and Credit. Springer Verlag, 2007.
Hull, J. Options, Futures, and Other Derivatives. Prentice Hall, 2008.
function displayCorrelationMatrix(CorrelationMatrix)
fprintf('Correlation Matrix\n');
fprintf([repmat('%1.3f ',1,length(CorrelationMatrix)) ' \n'],CorrelationMatrix);
function displayResults(nTrials, nPeriods, HW1FBermPrice, LG2FBermPrice, LMMBermPrice)
fprintf(' # of Monte Carlo Trials: %8d\n' , nTrials);
fprintf(' # of Time Periods/Trial: %8d\n\n' , nPeriods);
fprintf('HW1F Bermudan Swaption Price: %8.4f\n', HW1FBermPrice);
fprintf('LG2F Bermudan Swaption Price: %8.4f\n', LG2FBermPrice);
fprintf(' LMM Bermudan Swaption Price: %8.4f\n', LMMBermPrice);
capbyblk | floorbyblk | swaptionbyblk | blackvolbysabr | optsensbysabr | agencyoas | agencyprice | bndfutimprepo | bndfutprice | convfactor | tfutbyprice | tfutbyyield | tfutimprepo | tfutpricebyrepo | tfutyieldbyrepo | capbylg2f | floorbylg2f | swaptionbylg2f | blackvolbyrebonato | hwcalbycap | hwcalbyfloor
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torch.linalg.solve — PyTorch 1.11.0 documentation
torch.linalg.solve
torch.linalg.solve¶
torch.linalg.solve(A, B, *, out=None) → Tensor¶
Computes the solution of a square system of linear equations with a unique solution.
\mathbb{K}
\mathbb{R}
\mathbb{C}
, this function computes the solution
X \in \mathbb{K}^{n \times k}
of the linear system associated to
A \in \mathbb{K}^{n \times n}, B \in \mathbb{K}^{n \times k}
AX = B
This system of linear equations has one solution if and only if
A
is invertible. This function assumes that
A
Supports inputs of float, double, cfloat and cdouble dtypes. Also supports batches of matrices, and if the inputs are batches of matrices then the output has the same batch dimensions.
Letting * be zero or more batch dimensions,
If A has shape (*, n, n) and B has shape (*, n) (a batch of vectors) or shape (*, n, k) (a batch of matrices or “multiple right-hand sides”), this function returns X of shape (*, n) or (*, n, k) respectively.
Otherwise, if A has shape (*, n, n) and B has shape (n,) or (n, k) , B is broadcasted to have shape (*, n) or (*, n, k) respectively. This function then returns the solution of the resulting batch of systems of linear equations.
This function computes X = A .inverse() @ B in a faster and more numerically stable way than performing the computations separately.
It is possible to compute the solution of the system
XA = B
by passing the inputs A and B transposed and transposing the output returned by this function.
torch.linalg.solve_triangular() computes the solution of a triangular system of linear equations with a unique solution.
B (Tensor) – right-hand side tensor of shape (*, n) or (*, n, k) or (n,) or (n, k) according to the rules described above
RuntimeError – if the A matrix is not invertible or any matrix in a batched A is not invertible.
>>> x = torch.linalg.solve(A, b)
>>> torch.allclose(A @ x, b)
>>> x = torch.linalg.solve(A, b) # b is broadcasted to size (2, 3, 1)
>>> x = torch.linalg.solve(A, b) # b is broadcasted to size (2, 3)
>>> Ax = A @ x.unsqueeze(-1)
>>> torch.allclose(Ax, b.unsqueeze(-1).expand_as(Ax))
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Define Custom Classification Output Layer - MATLAB & Simulink - MathWorks í•œêµ
Include Custom Classification Output Layer in Network
To construct a classification output layer with cross entropy loss for k mutually exclusive classes, use classificationLayer. If you want to use a different loss function for your classification problems, then you can define a custom classification output layer using this example as a guide.
This example shows how to define a custom classification output layer with the sum of squares error (SSE) loss and use it in a convolutional neural network.
To define a custom classification output layer, you can use the template provided in this example, which takes you through the following steps:
L=\frac{1}{N}\underset{n=1}{\overset{N}{â}}\text{â}\underset{i=1}{\overset{K}{â}}\text{â}{\left({Y}_{ni}â{T}_{ni}\right)}^{2},
Copy the classification output layer template into a new file in MATLAB. This template outlines the structure of a classification output layer and includes the functions that define the layer behavior.
First, give the layer a name. In the first line of the class file, replace the existing name myClassificationLayer with sseClassificationLayer. Because the layer supports acceleration, also include the nnet.layer.Acceleratable mixin. For more information about custom layer acceleration, see Custom Layer Function Acceleration.
Next, rename the myClassificationLayer constructor function (the first function in the methods section) so that it has the same name as the layer.
function layer = sseClassificationLayer()
Save the layer class file in a new file named sseClassificationLayer.m. The file name must match the layer name. To use the layer, you must save the file in the current folder or in a folder on the MATLAB path.
In this example, the layer does not require any additional properties, so you can remove the properties section.
Specify the input argument name to assign to the Name property at creation. Add a comment to the top of the function that explains the syntax of the function.
Give the layer a one-line description by setting the Description property of the layer. Set the Name property to the input argument name.
Create a function named forwardLoss that returns the SSE loss between the predictions made by the network and the training targets. The syntax for forwardLoss is loss = forwardLoss(layer, Y, T), where Y is the output of the previous layer and T represents the training targets.
L=\frac{1}{N}\underset{n=1}{\overset{N}{â}}\text{â}\underset{i=1}{\overset{K}{â}}\text{â}{\left({Y}_{ni}â{T}_{ni}\right)}^{2},
The inputs Y and T correspond to Y and T in the equation, respectively. The output loss corresponds to L. Add a comment to the top of the function that explains the syntaxes of the function.
View the completed classification output layer class file.
The MATLAB functions used in forwardLoss all support dlarray objects, so the layer is GPU compatible.
Check the layer validity of the custom classification output layer sseClassificationLayer.
Define a custom sum-of-squares error classification layer. To create this layer, save the file sseClassificationLayer.m in the current folder. Create an instance of the layer.
layer = sseClassificationLayer('sse');
Check the layer is valid using checkLayer. Specify the valid input size to be the size of a single observation of typical input to the layer. The layer expects a 1-by-1-by-K-by-N array inputs, where K is the number of classes, and N is the number of observations in the mini-batch.
You can use a custom output layer in the same way as any other output layer in Deep Learning Toolbox. This section shows how to create and train a network for classification using the custom classification output layer that you created earlier.
Define a custom sum-of-squares error classification layer. To create this layer, save the file sseClassificationLayer.m in the current folder. Create an instance of the layer. Create a layer array including the custom classification output layer sseClassificationLayer.
sseClassificationLayer('sse')]
1 '' Image Input 28x28x1 images with 'zerocenter' normalization
2 '' Convolution 20 5x5 convolutions with stride [1 1] and padding [0 0 0 0]
3 '' Batch Normalization Batch normalization
4 '' ReLU ReLU
5 '' Fully Connected 10 fully connected layer
6 '' Softmax softmax
7 'sse' Classification Output Sum of squares error
| 26 | 1000 | 00:01:21 | 100.00% | 0.0012 | 0.0100 |
| 27 | 1050 | 00:01:26 | 99.22% | 0.0104 | 0.0100 |
Evaluate the network performance by making predictions on new data and calculating the accuracy.
YPred = classify(net, XTest);
accuracy = mean(YTest == YPred)
classificationLayer | checkLayer | findPlaceholderLayers | replaceLayer | assembleNetwork | PlaceholderLayer
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I must just thank you for the “Movements”, which seems a most capital production, & I am so pleased to see Franks name associated with your’s in it—1 I have read only two chapters, vii & viii. & they are splendid, but I hate the zigzags.!—2 Bauhinia leaf closing is a curious case; does it not show that said leaf consists of two leaflets?—3
The fact that for good action the leaves want a good illumination during the preceding day is very suggestive of experiments with the electric light. They are like the new paint that only shines by night after sun-light by day.4 There are heaps of points I should like to know more about.
Dyer & Baker are taken aback by the keel of the Cucurbita seed;—which keel was a wonderful discovery in Welwitschia!!!5
I have had no time to read more than the 2 chapters as yet, for I have a stock of half read books on hand & no time for any of them. I am only
\frac{2}{3}
through Wallace;6 it is splendid— what a number of cobwebs he has swept away.— that such a man should be a Spiritualist is more wonderful than all the movements of all the plants.7
He has done great things towards the explanation of the N. Zeald Flora & Australian, but marred it by assuming a preexistent S.W. Australian Flora—8 I am sure that the Australian Flora is very modern in the main; & that the S.W. peculiarities are exaggerations due to long isolation during the severance of the West from the East by the inland sea or straits that occupied the continent from Carpentaria to the Gt. Bight. I live in hopes of showing by an analysis (botanical) of the Australian types, that they are all derived from the Asiatic continent.—9
Meanwhile I have no chance of tackling problems— I must grind away at the Garden, the Bot. Mag & Indian Flora, which I cannot afford to give up, & Gen. Plant. which alone I delight in.10 I am at Palms, a most difficult task: but sometimes weeks elapse & not a stroke of work done! I am getting very weary of “working for a living”, & am beginning to covet rest & leisure in a way I never did before; but I must first look out for the education of three sons,—all hopeful I am glad to say, but one still an infant!11
The Grays will be back in a fortnight, they have changed their plans & will spend 2 or 3 winter months here & then go abroad (with us) for the spring.12 They will go into lodgings in Kew. We contemplate getting out a paper or book on the distribution of U.S. plants together (as one of Hayden’s Reports.)13
Have you read Pagets Lecture on plant diseases?14 it is very suggestive & a wonderful specimen of style aiding in giving great importance to possibly very superficial resemblances between animal & vegetable malformations: still there must be a great deal in the subject to be investigated.
I suppose we should get “Nobbe’s Handbuch der Samenkunde”.—15 is it an expensive work— our funds for purchase are rather short— but if inexpensive book I will order it at once
Ever affy Yrs | J D Hooker
Paget has started the idea of a Vegetable Pathologist for Kew & I have asked him to corkscrew Gladstone16 about it.—
We were very sorry to see Miss Wedgwoods death in the paper— I fear that Mrs Darwin will feel it a great deal.17
9.1 I suppose … at once 9.3] scored red crayon and pencil; ‘Answer’ red crayon
End of letter: ‘what wd you use for zig-zag | The account of cutting off tip | Last chapter’ pencil del ink
Hooker’s name appears on CD’s presentation list for Movement in plants (see Appendix IV). The words ‘assisted by Francis Darwin’ appear below CD’s name on the title page of the book.
Movement in plants contained numerous diagrams showing circumnutation over time; CD described many of the patterns as ‘zigzag’ (see, for example, ibid., p. 71).
In Movement in plants, pp. 373–4, CD described the two halves of each leaf of Bauhinia rising up and closing completely at night, ‘like the opposite leaflets of many Leguminosae’. While in Germany, Francis had observed Bauhinia richardiana, reporting: ‘2 large leaflets drop’ (see Correspondence vol. 26, letter from Francis Darwin, [12 July 1878]).
CD remarked that in some genera it was indispensable that leaves be well illuminated during the day in order that they should assume a vertical position at night (see Movement in plants, pp. 318–19). On the new luminous paint, see the Chemical Gazette, 17 December 1880, p. 302.
William Turner Thiselton-Dyer and John Gilbert Baker. In Movement in plants, pp. 102–6, CD described the development of a heel or peg on the summit of the radicle that aided in opening the seed-coats in species of Cucurbitaceae, the cucumber family. On a similar structure in Welwitschia, see Bower 1881, pp. 27–8 (see also letter from W. T. Thiselton-Dyer, [after 23 November 1880]).
Alfred Russel Wallace’s new book, Island life, was dedicated to Hooker (Wallace 1880a).
Hooker had been highly critical of Wallace’s spiritualism; see Correspondence vol. 24, letter from J. D. Hooker, [24 September 1876], and Correspondence vol. 27, letter from J. D. Hooker, 18 December 1879.
Wallace remarked that parts of south-western Australia were especially rich in ‘purely Australian types’ of flora, and concluded that it was a ‘remnant of the more extensive and more isolated portion of the continent in which the peculiar Australian flora was principally developed’ (see Wallace 1880a, pp. 463–4).
Hooker had written an essay on the flora of Australia and Tasmania (J. D. Hooker 1859); however, he never published another major work on the subject.
Hooker was the director of the Royal Botanic Gardens, Kew, editor of Curtis’s Botanical Magazine, and had been engaged for many years in the multi-volume works The flora of British India (J. D. Hooker 1872–97) and Genera plantarum (Bentham and Hooker 1862–83).
Hooker’s three youngest sons were Brian Harvey Hodgson Hooker, Reginald Hawthorn Hooker, and Joseph Symonds Hooker.
Hooker and his wife, Hyacinth Hooker, had planned to join Asa Gray and his wife, Jane Loring Gray, in Italy in December (see letter from J. D. Hooker, 24 September 1880 and n. 2).
J. D. Hooker and Gray 1880 was published in the Bulletin of the United States Geological and Geographical Survey, edited by Ferdinand Vandeveer Hayden.
CD had received a copy of James Paget’s lecture (Paget 1880; see letter to James Paget, 14 November 1880).
In Movement in plants, p. 105 n., CD had referred to Friedrich Nobbe’s Handbuch der Samenkunde (Handbook of seed science; Nobbe 1876).
William Ewart Gladstone was the prime minister.
Elizabeth Wedgwood, Emma Darwin’s sister, had died on 8 November 1880 (CD’s ‘Journal’ (Appendix II)). Her death was reported in The Times, 9 November 1880, p. 1.
Bower, Frederick Orpen. 1881. On the germination and histology of the seedling of Welwitschia mirabilis. Quarterly Journal of Microscopical Science n.s. 21: 15–30.
Hooker, Joseph Dalton. 1872–97. The flora of British India. Assisted by various botanists. 7 vols. London: L. Reeve & Co.
Hooker, Joseph Dalton and Gray, Asa. 1880. The vegetation of the Rocky Mountain region and a comparison with that of other parts of the world. Bulletin of the United States Geological and Geographical Survey of the Territories 6 (1880): 1–77.
Nobbe, Friedrich. 1876. Handbuch der Samenkunde. Berlin: Wiegandt, Hempel and Baren.
Paget, James. 1880. An address on elemental pathology. British Medical Journal 2: 611–14, 649–52.
Wallace, Alfred Russel. 1880a. Island life: or, the phenomena and causes of insular faunas and floras, including a revision and attempted solution of the problem of geological climates. London: Macmillan.
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A Purely Mathematical Model of Physical Reality - Vixrapedia
A Purely Mathematical Model of Physical Reality
Hans van Leunen respectfully asks that people use the discussion page, their talk page or email them, rather than contribute to this page at this time. This page might be under construction, controversial, used currently as part of a brick and mortar class by a teacher, or document an ongoing research project. Please RESPECT their wishes by not editing this page This project is still in preparation phase. Translation is partly finished.
My mottoː Think simple. If you think, then think twice
The Hilbert Book Model is a purely mathematical model of the foundation and the lower levels of the structure of physical reality. The model is structured like a book. The model steps along the pages of the book. Each page represents a current static status quo. Each page describes a whole universe.
Hilbert spaces are inner product spaces.
Together with others, David Hilbert discovered the separable Hilbert space.
{\displaystyle \Rrightarrow }
This slideshow highlights some aspects of the model
2 The Hilbert Book Model Project
The target of the effort is to create a model of physical reality. We want that the model is self-creating. In this way, the universe in the model owns a beginning state from which it evolves. The model must start from a suitable foundation. That foundation must be simple and easily comprehensible. The foundation must evolve into a more complicated structure with more complicated dynamic behavior. The model must support a creator's view and an observer's view. The range of progression values that belong to events that can become observable must be preceded by one or more creation events that cannot be observed but are accessible by the creator's view. Extension of the model must be restricted by the applied mathematics. After a limited number of extension steps the model must show properties and behavior that can be observed from the target of the investigation. That target is physical reality. This approach implies that physical reality applies its own mathematics and that this mathematics is quite similar to the mathematics that humans apply.
The Hilbert Book Model Project[edit]
The target is the subject of the Hilbert Book Model Project. This project is described in "A self-creating Model of Physical Reality'; [1] The initiator of the project is a retired physicist. At the age of 70, he started this project and gave it its name in 2011.
Retrieved from "https://www.vixrapedia.org/w/index.php?title=A_Purely_Mathematical_Model_of_Physical_Reality&oldid=3995"
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The following representations have been drawn to represent portions of a 100% block. Write each of the portions in at least two different forms.
How many blocks are filled in? How could you represent this in writing?
5
blocks out of
100
are filled in, it can be expressed in two ways.
\frac{5}{100}\text{ or } 0.05
Use part (a) as a guide, because this problem is very similar.
How many blocks are filled in? What portion of the whole does this represent?
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torch.svd — PyTorch 1.11.0 documentation
torch.svd
torch.svd¶
torch.svd(input, some=True, compute_uv=True, *, out=None)¶
Computes the singular value decomposition of either a matrix or batch of matrices input. The singular value decomposition is represented as a namedtuple (U, S, V) , such that input
= U \text{diag}(S) V^{\text{H}}
V^{\text{H}}
is the transpose of V for real inputs, and the conjugate transpose of V for complex inputs. If input is a batch of matrices, then U , S , and V are also batched with the same batch dimensions as input.
If some is True (default), the method returns the reduced singular value decomposition. In this case, if the last two dimensions of input are m and n , then the returned U and V matrices will contain only min(n, m) orthonormal columns.
If compute_uv is False , the returned U and V will be zero-filled matrices of shape (m, m) and (n, n) respectively, and the same device as input. The argument some has no effect when compute_uv is False .
Supports input of float, double, cfloat and cdouble data types. The dtypes of U and V are the same as input’s. S will always be real-valued, even if input is complex.
torch.svd() is deprecated in favor of torch.linalg.svd() and will be removed in a future PyTorch release.
U, S, V = torch.svd(A, some=some, compute_uv=True) (default) should be replaced with
U, S, Vh = torch.linalg.svd(A, full_matrices=not some)
V = Vh.mH
_, S, _ = torch.svd(A, some=some, compute_uv=False) should be replaced with
S = torch.linalg.svdvals(A)
Differences with torch.linalg.svd():
some is the opposite of torch.linalg.svd()’s full_matrices. Note that default value for both is True , so the default behavior is effectively the opposite.
torch.svd() returns V , whereas torch.linalg.svd() returns Vh , that is,
V^{\text{H}}
If compute_uv is False , torch.svd() returns zero-filled tensors for U and Vh , whereas torch.linalg.svd() returns empty tensors.
The singular values are returned in descending order. If input is a batch of matrices, then the singular values of each matrix in the batch are returned in descending order.
The S tensor can only be used to compute gradients if compute_uv is True .
When some is False , the gradients on U[…, :, min(m, n):] and V[…, :, min(m, n):] will be ignored in the backward pass, as those vectors can be arbitrary bases of the corresponding subspaces.
The implementation of torch.linalg.svd() on CPU uses LAPACK’s routine ?gesdd (a divide-and-conquer algorithm) instead of ?gesvd for speed. Analogously, on GPU, it uses cuSOLVER’s routines gesvdj and gesvdjBatched on CUDA 10.1.243 and later, and MAGMA’s routine gesdd on earlier versions of CUDA.
The returned U will not be contiguous. The matrix (or batch of matrices) will be represented as a column-major matrix (i.e. Fortran-contiguous).
The gradients with respect to U and V will only be finite when the input does not have zero nor repeated singular values.
If the distance between any two singular values is close to zero, the gradients with respect to U and V will be numerically unstable, as they depends on
\frac{1}{\min_{i \neq j} \sigma_i^2 - \sigma_j^2}
. The same happens when the matrix has small singular values, as these gradients also depend on S⁻¹ .
For complex-valued input the singular value decomposition is not unique, as U and V may be multiplied by an arbitrary phase factor
e^{i \phi}
on every column. The same happens when input has repeated singular values, where one may multiply the columns of the spanning subspace in U and V by a rotation matrix and the resulting vectors will span the same subspace. Different platforms, like NumPy, or inputs on different device types, may produce different U and V tensors.
input (Tensor) – the input tensor of size (*, m, n) where * is zero or more batch dimensions consisting of (m, n) matrices.
some (bool, optional) – controls whether to compute the reduced or full decomposition, and consequently, the shape of returned U and V . Default: True .
compute_uv (bool, optional) – controls whether to compute U and V . Default: True .
out (tuple, optional) – the output tuple of tensors
>>> u, s, v = torch.svd(a)
>>> torch.dist(a, torch.mm(torch.mm(u, torch.diag(s)), v.t()))
>>> u, s, v = torch.svd(a_big)
>>> torch.dist(a_big, torch.matmul(torch.matmul(u, torch.diag_embed(s)), v.mT))
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35K41 Higher-order parabolic systems
35K46 Initial value problems for higher-order parabolic systems
35K52 Initial-boundary value problems for higher-order parabolic systems
35K60 Nonlinear initial value problems for linear parabolic equations
35K61 Nonlinear initial-boundary value problems for nonlinear parabolic equations
35K85 Linear parabolic unilateral problems and linear parabolic variational inequalities
35K86 Nonlinear parabolic unilateral problems and nonlinear parabolic variational inequalities
35K87 Systems of parabolic variational inequalities
35K92 Quasilinear parabolic equations with
p
35K93 Quasilinear parabolic equations with mean curvature operator
35K96 Parabolic Monge-Ampère equations
A Computer Algebra Application to Determination of Lie Symmetries of Partial Differential Equations
Pulov, Vladimir, Chacarov, Edy, Uzunov, Ivan (2007)
The paper has been presented at the 12th International Conference on Applications of Computer Algebra, Varna, Bulgaria, June, 2006A MATHEMATICA package for finding Lie symmetries of partial differential equations is presented. The package is designed to create and solve the associated determining system of equations, the full set of solutions of which generates the widest permissible local Lie group of point symmetry transformations. Examples illustrating the functionality of the package's tools...
A Direct Approach to the Mellin Transform.
Paul L. Butzer, Stefan Jansche (1997)
A Discrete Sampling Inversion Scheme for the Heat Equation.
D.S. Gilliam, J.R. Lund, C.F. Martin (1989)
Friedrich Karl Hebeker (2010)
We consider a domain decomposition method for some unsteady heat conduction problem in composite structures. This linear model problem is obtained by homogenization of thin layers of fibres embedded into some standard material. For ease of presentation we consider the case of two space dimensions only. The set of finite element equations obtained by the backward Euler scheme is parallelized in a problem-oriented fashion by some noniterative overlapping domain splitting method, eventually enhanced...
A final value problem for heat equation: regularization by truncation method and new error estimates.
Trong, Dang Duc, Quan, Pham Hoang, Tuan, Nguyen Huy (2010)
A generalized Nevanlinna theorem for supertemperatures.
Watson, Neil A. (2003)
A gradient estimate for solutions of the heat equation
A gradient estimate for solutions of the heat equation. II
The author obtains an estimate for the spatial gradient of solutions of the heat equation, subject to a homogeneous Neumann boundary condition, in terms of the gradient of the initial data. The proof is accomplished via the maximum principle; the main assumption is that the sufficiently smooth boundary be convex.
A heat approximation
Miroslav Dont (2000)
The Fourier problem on planar domains with time variable boundary is considered using integral equations. A simple numerical method for the integral equation is described and the convergence of the method is proved. It is shown how to approximate the solution of the Fourier problem and how to estimate the error. A numerical example is given.
A heat conduction problem involving phase change and its numerical solution by finite difference methods
G. Windisch, U. Streit (1984)
A Heat Semigroup Version of Bernstein's Theorem on Lie Groups.
G.I. Gaudry, S. Meda, R. Pini (1990)
A high-order difference scheme for a nonlocal boundary-value problem for the heat equation.
Sun, Zhi-Zhong (2001)
A Method for the Numerical Solution of the One-Dimensional Inverse Stefan Problem.
A. Kirsch, R. Reemtsen (1984)
A minorization of the first positive eigenvalue of the scalar laplacian on a compact Riemannian manifold [Book]
Jacek Komorowski (1980)
A mixed boundary value problem for heat potentials (Preliminary communication)
Ivan Netuka (1978)
A new error estimate for a fully finite element discretization scheme for parabolic equations using Crank-Nicolson method
Abdallah Bradji, Jürgen Fuhrmann (2014)
Finite element methods with piecewise polynomial spaces in space for solving the nonstationary heat equation, as a model for parabolic equations are considered. The discretization in time is performed using the Crank-Nicolson method. A new a priori estimate is proved. Thanks to this new a priori estimate, a new error estimate in the discrete norm of
{𝒲}^{1,\infty }\left({ℒ}^{2}\right)
is proved. An
{ℒ}^{\infty }\left({ℋ}^{1}\right)
-error estimate is also shown. These error estimates are useful since they allow us to get second order time accurate approximations...
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40E05 Tauberian theorems, general
40E10 Growth estimates
\left({a}^{t}{\right)}_{t>0}
\left({a}^{t}{\right)}_{\mathrm{Re}\phantom{\rule{0.166667em}{0ex}}t>0}
\parallel {a}^{t}\parallel \le \sqrt{|t|}
\phantom{\rule{0.166667em}{0ex}}t\ge 1
{L}^{1}\left(\mathbf{R}\right)
A quantified Tauberian theorem for sequences
David Seifert (2015)
The main result of this paper is a quantified version of Ingham's Tauberian theorem for bounded vector-valued sequences rather than functions. It gives an estimate on the rate of decay of such a sequence in terms of the behaviour of a certain boundary function, with the quality of the estimate depending on the degree of smoothness this boundary function is assumed to possess. The result is then used to give a new proof of the quantified Katznelson-Tzafriri theorem recently obtained by the author...
A refined Tauberian Theorem for Laplace transforms in dimension d>1.
U. Stadtmüller (1981)
A Tauberian Remainder Theorem with Applications to Lambert Summability.
Sin Phing Moo Strube (1974)
A Tauberian theorem for Abelian summability methods.
D. Borwein, B. Watson (1976)
A Tauberian theorem for Borel-type methods of summability.
David Borwein, Irvine J.W. Robinson (1975)
B. Kuttner (1977)
A Tauberian theorem for distributions
Jiří Čížek, Jiří Jelínek (1996)
The well-known general Tauberian theorem of N. Wiener is formulated and proved for distributions in the place of functions and its Ganelius' formulation is corrected. Some changes of assumptions of this theorem are discussed, too.
A Tauberian theorem for series of orthogonal rational functions.
Videnskiĭ, I.V. (2004)
A Tauberian theorem for subsequences of functions
Hsiang, Fu Cheng (1965)
A Tauberian theorem with a generalized one-sided condition.
Çanak, İbrahim, Totur, Ümit (2007)
Absolute summability factors and absolute Tauberian theorems for double series and sequences.
Yanetz, Sh. (1999)
Admissible functions for cones and asumptotics of infinitely divisible distributions.
Yakymiv, A.L. (2006)
An application of Banach algebra techniques for multiplicative functions.
Lutz Lucht (1993)
Analogues of Besicovitch-Wiener Theorem for Heisenberg Group.
P.K. Ratnakumar, S. Thangavelu (1995)
Analogues of some Tauberian theorems for stretchings.
Patterson, Richard F. (2001)
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{x}^{2}\frac{{d}^{2}y}{d{x}^{2}}={\left\{1+{\left(\frac{dy}{dx}\right)}^{2}\right\}}^{4}
If f(x) = x + 7 and g(x) = x − 7, x ∊ R, then find
\frac{d}{dx}\left(fog\right)\left(x\right)
Find the value of x − y, if
2\left[\begin{array}{cc}1& 3\\ 0& x\end{array}\right]+\left[\begin{array}{cc}y& 0\\ 1& 2\end{array}\right]=\left[\begin{array}{cc}5& 6\\ 1& 8\end{array}\right]
2\stackrel{^}{i}+2\stackrel{^}{j}-3\stackrel{^}{k}
\mathrm{A}=\left[\begin{array}{ccc}2& 0& 1\\ 2& 1& 3\\ 1& -1& 0\end{array}\right]
, then find (A2 − 5A). VIEW SOLUTION
\mathrm{Find} : \int \sqrt{1-\mathrm{sin} 2x }dx, \frac{\mathrm{\pi }}{4}<x<\frac{\mathrm{\pi }}{2}
\mathrm{P}\left(\mathrm{X}=x\right)=\left\{\begin{array}{lll}k& ,& \mathrm{if} x=0\\ 2k& ,& \mathrm{if} x=1\\ 3k& ,& \mathrm{if} x=2\\ 0& ,& \mathrm{otherwise}\end{array}\right\
\sqrt{3}
\stackrel{\to }{a}=2\stackrel{^}{i}+3\stackrel{^}{j}+\stackrel{^}{k}, \stackrel{\to }{b}=\stackrel{^}{i}-2\stackrel{^}{j}+\stackrel{^}{k} \mathrm{and} \stackrel{\to }{c}=-3\stackrel{^}{i}+\stackrel{^}{j}+2\stackrel{^}{k}, \mathrm{find} \left[\stackrel{\to }{a}\stackrel{\to }{b}\stackrel{\to }{c}\right]
\int \frac{{\mathrm{tan}}^{2}x{\mathrm{sec}}^{2}x}{1-{\mathrm{tan}}^{6}x}dx
{\mathrm{tan}}^{-1}\left(2x\right)+{\mathrm{tan}}^{-1}\left(3x\right)=\frac{\mathrm{\pi }}{4}
2{\mathrm{tan}}^{-1}\left(\frac{y}{x}\right)
\frac{dy}{dx}=\frac{x+y}{x-y}
\frac{dy}{dx}
\int \frac{3x+5}{{x}^{2}+3x-18}dx
{\int }_{0}^{a}f\left(x\right)dx={\int }_{0}^{a}f\left(a-x\right)dx
{\int }_{0}^{\mathrm{\pi }}\frac{x\mathrm{sin}x}{1+{\mathrm{cos}}^{2}x}dx
\stackrel{^}{i}+\stackrel{^}{j}+\stackrel{^}{k},2\stackrel{^}{i}+5\stackrel{^}{j},3\stackrel{^}{i}+2\stackrel{^}{j}-3\stackrel{^}{k}
\stackrel{^}{i}-6\stackrel{^}{j}-\stackrel{^}{k}
\stackrel{\to }{\mathrm{AB}}
\stackrel{\to }{\mathrm{CD}}
\left|\begin{array}{ccc}a+b+c& -c& -b\\ -c& a+b+c& -a\\ -b& -a& a+b+c\end{array}\right|=2\left(a+b\right)\left(b+c\right)\left(c+a\right)
x=\mathrm{cos}t+\mathrm{log}\mathrm{tan}\left(\frac{t}{2}\right)
, y = sin t, then find the values of
\frac{{d}^{2}y}{d{t}^{2}}
\frac{{d}^{2}y}{d{x}^{2}}
t=\frac{\mathrm{\pi }}{4}
y=\sqrt{3x-2}
\sqrt{{x}^{2}+{y}^{2}}dx
\left(1+{x}^{2}\right)\frac{dy}{dx}+2xy-4{x}^{2}=0
\frac{1-x}{3}=\frac{7y-14}{\lambda }=\frac{z-3}{2}
\frac{7-7x}{3\lambda }=\frac{y-5}{1}=\frac{6-z}{5}
\frac{4r}{3}
. Also find the maximum volume of cone. VIEW SOLUTION
\mathrm{A}=\left[\begin{array}{ccc}2& -3& 5\\ 3& 2& -4\\ 1& 1& -2\end{array}\right]
, then find A−1. Hence solve the following system of equations:
2x − 3y + 5z = 11, 3x + 2y − 4z = −5, x + y − 2z = −3.
Obtain the inverse of the following matrix using elementary operations:
\mathrm{A}=\left[\begin{array}{ccc}-1& 1& 2\\ 1& 2& 3\\ 3& 1& 1\end{array}\right]
\stackrel{\to }{r}=\left(\stackrel{^}{i}+\stackrel{^}{j}\right)+\lambda \left(\stackrel{^}{i}+2\stackrel{^}{j}-\stackrel{^}{k}\right)
{y}^{2}=4x
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f(x) = \frac { x ^ { 3 } - 7 x - 6 } { x + 1 }
f \left(−1\right)
0
\lim\limits _ { x \rightarrow - 1 } f ( x )
. (Hint: polynomial division)
x = −1
\begin{array}{l} \qquad \quad x ^ { 2 } - x - 6 \\ x +1 \enclose{longdiv}{\; x ^ { 3 } + 0 x ^ { 2 } - 7 x - 6}\\ \qquad \underline{ - ( x ^ { 3 } + x ^ { 2 } )} \\ \qquad \qquad \quad \; \;- x ^ { 2 } - 7 x \\ \qquad \qquad \; \, \underline{- ( - x ^ { 2 } - 1 x )} \\ \qquad \qquad \qquad \qquad \; - 6 x - 6 \\ \qquad \qquad \qquad \quad \underline{-(-6x-)6}\\ \qquad \qquad \qquad \qquad \qquad 0 \end{array}
Use what you found from part (b) to sketch
f\left(x\right)
x = −1
x
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Part 2 – coordinates, vectors and the summation convention – ebvalaim.log
x
y
z
t
\mu
x^\mu
\mu
So now we know how to describe points ("places") in spacetime. But in spacetime, as in every space, we have not only places, but also directions. Those are described with vectors. Vectors are written much like points - they are also described by 4 coordinates, denoted
v^\mu
. In this case they don't mean a place in the spacetime, but the proportions of movement along respective coordinate axes. Let me explain in detail.
Let's imagine an ordinary plane with
x
y
coordinates. Each vector in this plane will also have
x
y
coordinates - let's denote them, for example,
v_x
v_y
. They can be interpreted as a receipt: to move in the direction described by this vector, you need to add
v_x
x
coordinate of the point, and
v_y
to its
y
coordinate. For example, moving from the point
(4,3)
by a vector
[1,-1]
will get us to the point
(5,2)
The direction doesn't end on this one point, though. We can go farther, to
(6,1)
(7,0)
(8,-1)
,... We can make such "steps" as many times as we like, but we can also make for example half of a step, to
(4.5, 2.5)
. Vectors
[av_x, av_y]
have then the same direction as
[v_x, v_y]
, but different magnitudes. They can also have the same or the opposite sense (the same if
a>0
, opposite if
a<0
). Having a function defined on our space (on a plane it will be
f(x,y)
) and a vector, we can ask about the derivative of this function in the direction of this vector - it will tell us how fast our function changes, when we move in that direction. As it turns out, the derivative in the direction of
[v_x, v_y]
v_x\frac{\partial f}{\partial x} + v_y\frac{\partial f}{\partial y}
. In a space in which we have coordinates numbered with an index
\mu
, we can write it this way:
\sum\limits_{\mu=0}^n v^\mu \frac{\partial f}{\partial x^\mu}
. It is worth noting that if we multiply the vector by some number
a
, the derivative will also get multiplied by
a
. This means that the longer the vector, the greater the value of the derivative. Thus we can say that the vector describes not only a direction, but also the velocity with which we move in that direction. The derivative then tells us how fast the function changes, when we move with that velocity. Now a few notation conventions again: first,
\frac{\partial f}{\partial x^\mu}
is often being written
\partial_\mu f
for the sake of convenience. Here we have the index at the bottom - you can remember it this way: when we differentiate with respect to something with an index, the index switches places (from top to bottom and vice versa). It will be important in a while. We have then such an expression:
\sum\limits_{\mu=0}^n v^\mu \partial_\mu f
. Expressions with such sums appear in GR so often, that Einstein himself decided they shouldn't be written and introduced the so-called summation convention. It says just that every time an index is repeated in an expression, once as an upper index and once as a lower one, the expression should be summed for all the values of that index. This lets us write our expression as
v^\mu \partial_\mu f
. This convention is the main reason of the importance of positions of indices (there is also another matter, but it is beyond this article). Summing over the repeated index is called a contraction.
Let's take a closer look at our directional derivative:
\partial_\mu f
. Those are actually
n
partial derivatives of the function
f
n
\mu
- in spacetime, four). It looks a bit like a point or a vector, except the index is at the bottom. Such a thing is called a covector.
The derivatives of a function usually depend on the point at which they are being calculated, so the coordinates of this covector will also depend on the point in space. We can think of this as if we had a covector in every point in space, with coordinates equal to the derivatives of the function at that point. We don't have a single covector here then, but a whole covector field. In the same way, when the coordinates of a vector depend on the point in space, we are dealing with a vector field.
Often vector and covector fields are just called vectors/covectors. It's not a problem, since single (co)vectors at a single point are almost never being considered. So every time when we write just
v^\mu \partial_\mu f
, we will mean the derivative of
v
calculated at every point in space separately.
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Babe Ruth holds sixteen franchise, four American League, and two Major League records.
The New York Yankees are a professional baseball team based in the Bronx, New York. They compete in the East Division of Major League Baseball's (MLB) American League (AL). The club began play in 1903 as the Highlanders, after owners Frank Farrell and William S. Devery had bought the defunct Baltimore Orioles and moved the team to New York City; in 1913, the team changed its nickname to the Yankees.[1] From 1903 to 2021, the franchise has won more than 10,000 games and 27 World Series championships.[2] The list below documents players and teams that hold particular club records.
Outfielder Babe Ruth holds the most franchise records, with 16, including career home runs, and career and single-season batting average and on-base percentage. Shortstop Derek Jeter has the second-most records among hitters, with eight. Jeter's marks include the records for career hits, singles, doubles, and stolen bases. Among pitchers, Whitey Ford has the most Yankees records with five, all of which are career totals. These include games won, games started, and innings pitched.
Several Yankees hold AL and MLB records. Ruth has MLB single-season records for extra-base hits and total bases, and holds four other AL single-season records. Outfielder Joe DiMaggio had a 56-game hitting streak in the 1941 season, which remains an MLB record. Jack Chesbro holds three AL records that he set in 1904: games won, games started, and complete games.
Tie between two teams
Statistics are current through the 2021 season.
These are records of players with the best performance in particular statistical categories during their career with the Yankees.[3][4]
Derek Jeter is the Yankees' all-time leader in hits, singles, doubles, and stolen bases.
Yankees career
Batting average Babe Ruth
On-base percentage Babe Ruth
Slugging percentage Babe Ruth
On-base plus slugging Babe Ruth
Runs Babe Ruth
Plate appearances Derek Jeter
At bats Derek Jeter
Hits Derek Jeter
Total bases Babe Ruth
Singles Derek Jeter
Doubles Derek Jeter
Triples Lou Gehrig
Home runs Babe Ruth
Runs batted in Lou Gehrig
Walks Babe Ruth
Strikeouts Derek Jeter
Stolen bases Derek Jeter
Games played Derek Jeter
Mariano Rivera has the most saves, both in his career and a single season, among Yankees pitchers.
Wins Whitey Ford
Losses Mel Stottlemyre
Win–loss percentage Johnny Allen
Earned run average[a] Rich Gossage
Saves Mariano Rivera
Strikeouts Andy Pettitte
1995–2003, 2007–2010, 2012–2013 [14]
Shutouts Whitey Ford
Games Mariano Rivera
Innings pitched Whitey Ford
3,170+1⁄3
1995–2003, 2007–2010, 2012–2013 [8][14]
Games finished Mariano Rivera
1995–2013 [12][16][b]
Complete games Red Ruffing
Walks Lefty Gomez
Hits allowed Red Ruffing
Wild pitches Whitey Ford
Hit batsmen Jack Warhop
These are records of Yankees players with the best performance in particular statistical categories during a single season.[20][21]
Single-season batting
Joe DiMaggio has held the MLB record for the longest hitting streak since 1941.
Home runs Roger Maris
Hits Don Mattingly
Singles Steve Sax
Doubles Don Mattingly
Triples Earle Combs
Stolen bases Rickey Henderson
At bats Alfonso Soriano
Hitting streak Joe DiMaggio
Extra-base hits Babe Ruth
Strikeouts Giancarlo Stanton
Single-season pitching
Jack Chesbro won an American League-record 41 games in the 1904 season.
Wins Jack Chesbro
Losses Joe Lake
Strikeouts Ron Guidry
Earned run average Spud Chandler
Earned runs allowed Sam Jones
Hits allowed Jack Powell
Shutouts Ron Guidry
Games Paul Quantrill
Games started Jack Chesbro
Complete games Jack Chesbro
Innings pitched Jack Chesbro
454+2⁄3
These are records of Yankees teams with the best performance in particular statistical categories during a single game.[48]
Single-game batting
Hideki Matsui hit two of the Yankees' eight home runs on July 31, 2007.[49]
Single-game batting records
Philadelphia Athletics June 28, 1939
Chicago White Sox July 31, 2007[49]
Philadelphia Athletics May 24, 1936
Toronto Blue Jays April 12, 1988
Cincinnati Reds June 5, 2003
Washington Senators May 1, 1934
Oakland Athletics August 25, 2011[50]
Chicago Cubs May 7, 2017[51]
St. Louis Browns September 28, 1911
Single-game pitching
Single-game pitching records
Detroit Tigers September 29, 1928
Boston Red Sox July 4, 2003
Cleveland Indians August 15, 2019[52]
Other single-game records
Longest game by time
Detroit Tigers June 24, 1962[53][54]
These are records of Yankees teams with the best and worst performances in particular statistical categories during a single season.[55]
Giancarlo Stanton hit 38 of the Yankees' MLB record 267 home runs in 2018.[37]
Season batting records
Season pitching records
a Earned run average is calculated as 9 × (ER ÷ IP), where
{\displaystyle ER}
is earned runs and
{\displaystyle IP}
is innings pitched.
b The figure listed is the MLB total.[16] Baseball-Reference.com credits Rivera with 952 games finished.[12]
^ "Yankees Timeline: 1900s". New York Yankees. Retrieved November 14, 2015.
^ a b "Major League Teams and Baseball Encyclopedia". Baseball-Reference.com. Retrieved October 8, 2021.
^ "New York Yankees Top 10 Career Batting Leaders". Baseball-Reference.com. Retrieved October 8, 2021.
^ "New York Yankees Top 10 Career Pitching Leaders". Baseball-Reference.com. Retrieved October 8, 2021.
^ a b c d e f g h i j k l m n o p "Babe Ruth Statistics". Baseball-Reference.com. Retrieved February 7, 2009.
^ a b c d e f g h "Derek Jeter". Baseball-Reference.com. Retrieved November 19, 2009.
^ a b c "Lou Gehrig Statistics". Baseball-Reference.com. Retrieved February 7, 2009.
^ a b c d e "Whitey Ford Statistics". Baseball-Reference.com. Retrieved February 7, 2009.
^ "Mel Stottlemyre Statistics". Baseball-Reference.com. Retrieved February 7, 2009.
^ "Johnny Allen". Baseball Reference.com. Retrieved November 5, 2010.
^ "Rich Gossage Statistics". Baseball-Reference.com. Retrieved February 7, 2009.
^ a b c d e "Mariano Rivera Statistics". Baseball-Reference.com. Retrieved February 9, 2009.
^ "Career Leaders & Records for Saves". Baseball-Reference.com. Retrieved December 5, 2011.
^ a b "Andy Pettitte Statistics". Baseball-Reference.com. Retrieved July 2, 2013.
^ "Statistics" (To sort, click on Pitching, then All-Time Totals, AL, and G). Major League Baseball. Retrieved December 5, 2011.
^ a b "Statistics" (To sort, click on Pitching, then All-Time Totals, the right arrow, and GF). Major League Baseball. Retrieved December 5, 2011.
^ a b "Red Ruffing Statistics". Baseball-Reference.com. Retrieved February 8, 2009.
^ "Lefty Gomez Statistics". Baseball-Reference.com. Retrieved February 8, 2009.
^ "Jack Warhop Statistics". Baseball-Reference.com. Retrieved February 8, 2009.
^ "New York Yankees Top 10 Single-Season Batting Leaders". Baseball-Reference.com. Retrieved October 8, 2021.
^ "New York Yankees Top 10 Single-Season Pitching Leaders". Baseball-Reference.com. Retrieved October 8, 2021.
^ "Roger Maris Statistics". Baseball-Reference.com. Retrieved February 8, 2009.
^ "Single-Season Leaders & Records for Home Runs". Baseball-Reference.com. Retrieved February 11, 2009.
^ "Single-Season Leaders & Records for RBI". Baseball-Reference.com. Retrieved February 11, 2009.
^ "Single-Season Leaders & Records for Runs". Baseball-Reference.com. Retrieved February 11, 2009.
^ a b "Don Mattingly Statistics". Baseball-Reference.com. Retrieved February 8, 2009.
^ "Steve Sax Statistics". Baseball-Reference.com. Retrieved February 8, 2009.
^ "Earle Combs Statistics". Baseball-Reference.com. Retrieved February 8, 2009.
^ "Rickey Henderson Statistics". Baseball-Reference.com. Retrieved February 7, 2009.
^ "Alfonso Soriano Statistics". Baseball-Reference.com. Retrieved February 8, 2009.
^ "Hitting Streaks". Major League Baseball. Retrieved February 9, 2009.
^ "Single-Season Leaders & Records for Slugging %". Baseball-Reference.com. Retrieved May 4, 2018.
^ "Single-Season Leaders & Records for Extra-Base Hits". Baseball-Reference.com. Retrieved February 11, 2009.
^ "Single-Season Leaders & Records for Total Bases". Baseball-Reference.com. Retrieved February 11, 2009.
^ "Single-Season Leaders & Records for On-Base Plus Slugging". Baseball-Reference.com. Retrieved May 4, 2018.
^ "Single-Season Leaders & Records for Bases on Balls". Baseball-Reference.com. Retrieved February 11, 2009.
^ a b "Giancarlo Stanton". Baseball-Reference.com. Retrieved October 5, 2018.
^ a b c d "Jack Chesbro Statistics". Baseball-Reference.com. Retrieved February 9, 2009.
^ "Single-Season Leaders & Records for Wins". Baseball-Reference.com. Retrieved February 11, 2009.
^ "Joe Lake Statistics". Baseball-Reference.com. Retrieved February 9, 2009.
^ a b "Ron Guidry Statistics". Baseball-Reference.com. Retrieved February 9, 2009.
^ "Spud Chandler Statistics". Baseball-Reference.com. Retrieved February 13, 2009.
^ "Sad Sam Jones". Baseball-Reference.com. Retrieved November 14, 2015.
^ "Jack Powell". Baseball-Reference.com. Retrieved November 14, 2015.
^ "Paul Quantrill Statistics". Baseball-Reference.com. Retrieved February 9, 2009.
^ "Single-Season Leaders & Records for Games Started". Baseball-Reference.com. Retrieved February 11, 2009.
^ "Single-Season Leaders & Records for Comp. Games". Baseball-Reference.com. Retrieved February 12, 2009.
^ "Yankees Single Game Records". Major League Baseball. Retrieved February 9, 2009.
^ a b "Yankees tie franchise record with eight homers in rout". ESPN. Associated Press. July 31, 2007. Retrieved February 9, 2009.
^ "Yankees hit 3 grand slams in a game – a first". CBS News. Associated Press. August 25, 2011. Retrieved May 19, 2019.
^ a b "Let's play two: Yankees beat Cubs 5–4 in 18 innings". ESPN. Associated Press. May 8, 2017. Retrieved May 9, 2017.
^ "Indians slug seven home runs en route to 19–5 rout of Yankees". Fox Sports Ohio. Associated Press. August 16, 2019. Retrieved October 26, 2019.
^ Witz, Billy (April 11, 2015). "After Seven Hours and 19 Innings, One Hit Sinks the Yankees". The New York Times. Retrieved April 20, 2015.
^ Lowry, Philip J. (2010). Baseball's Longest Games: A Comprehensive Worldwide Record Book. McFarland Books. p. 201. ISBN 9780786457342.
^ "Yankees Season Records". Major League Baseball. Retrieved February 10, 2009.
^ a b c "2019 New York Yankees Statistics". Baseball-Reference.com. Retrieved October 26, 2019.
^ "2018 New York Yankees Statistics". Baseball-Reference.com. Retrieved October 1, 2018.
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The payout ratio is a financial metric showing the proportion of earnings a company pays its shareholders in the form of dividends, expressed as a percentage of the company's total earnings. On some occasions, the payout ratio refers to the dividends paid out as a percentage of a company's cash flow. The payout ratio is also known as the dividend payout ratio.
The payout ratio, also known as the dividend payout ratio, shows the percentage of a company's earnings paid out as dividends to shareholders.
A low payout ratio can signal that a company is reinvesting the bulk of its earnings into expanding operations.
Understanding the Payout Ratio
The payout ratio is a key financial metric used to determine the sustainability of a company’s dividend payment program. It is the amount of dividends paid to shareholders relative to the total net income of a company.
For example, let's assume Company ABC has earnings per share of $1 and pays dividends per share of $0.60. In this scenario, the payout ratio would be 60% (0.6 / 1). Let's further assume that Company XYZ has earnings per share of $2 and dividends per share of $1.50. In this scenario, the payout ratio is 75% (1.5 / 2). Comparatively speaking, Company ABC pays out a smaller percentage of its earnings to shareholders as dividends, giving it a more sustainable payout ratio than Company XYZ.
While the payout ratio is an important metric for determining the sustainability of a company’s dividend payment program, other considerations should likewise be observed. Case in point: in the aforementioned analysis, if Company ABC is a commodity producer and Company XYZ is a regulated utility, the latter may boast greater dividend sustainability, even though the former demonstrates a lower absolute payout ratio.
In essence, there is no single number that defines an ideal payout ratio because the adequacy largely depends on the sector in which a given company operates. Companies in defensive industries, such as utilities, pipelines, and telecommunications, tend to boast stable earnings and cash flows that are able to support high payouts over the long haul.
On the other hand, companies in cyclical industries typically make less reliable payouts, because their profits are vulnerable to macroeconomic fluctuations. In times of economic hardship, people spend less of their incomes on new cars, entertainment, and luxury goods. Consequently, companies in these sectors tend to experience earnings peaks and valleys that fall in line with economic cycles.
\begin{aligned} &DPR=\dfrac{\textit{Total dividends}}{\textit{Net income}} \\ &\textbf{where:} \\ &DPR = \text{Divided payout ratio (or simply payout ratio)}\\ \end{aligned}
DPR=Net incomeTotal dividendswhere:DPR=Divided payout ratio (or simply payout ratio)
Some companies pay out all their earnings to shareholders, while others dole out just a portion and funnel the remaining assets back into their businesses. The measure of retained earnings is known as the retention ratio. The higher the retention ratio is, the lower the payout ratio is. For example, if a company reports a net income of $100,000 and issues $25,000 in dividends, the payout ratio would be $25,000 / $100,000 = 25%. This implies that the company boasts a 75% retention ratio, meaning it records the remaining $75,000 of its income for the period in its financial statements as retained earnings, which appears in the equity section of the company's balance sheet the following year.
What Does the Payout Ratio Tell You?
The payout ratio is a key financial metric used to determine the sustainability of a company’s dividend payment program. It is the amount of dividends paid to shareholders relative to the total net income of a company. Generally, the higher the payout ratio, especially if it is over 100%, the more its sustainability is in question. Conversely, a low payout ratio can signal that a company is reinvesting the bulk of its earnings into expanding operations. Historically, companies with the best long-term records of dividend payments have had stable payout ratios over many years.
The payout ratio shows the proportion of earnings a company pays its shareholders in the form of dividends, expressed as a percentage of the company's total earnings. The calculation is derived by dividing the total dividends being paid out by the net income generated. Another way to express it is to calculate the dividends per share (DPS) and divide that by the earnings per share (EPS) figure.
Is There an Ideal Payout Ratio?
There is no single number that defines an ideal payout ratio because the adequacy largely depends on the sector in which a given company operates. Companies in defensive industries tend to boast stable earnings and cash flows that are able to support high payouts over the long haul while companies in cyclical industries typically make less reliable payouts, because their profits are vulnerable to macroeconomic fluctuations.
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\left[\begin{array}{cc}x& 1\end{array}\right] \left[\begin{array}{cc} 1& 0\\ -2& 0\end{array}\right]=0,
then x equals
\int {4}^{x}{3}^{x} dx
\frac{{12}^{x}}{\mathrm{log}12}+\mathrm{C}
\frac{{4}^{x}}{\mathrm{log}4}+\mathrm{C}
\left(\frac{{4}^{x}·{3}^{x}}{\mathrm{log}4·\mathrm{log}3}\right)+\mathrm{C}
\frac{{3}^{x}}{\mathrm{log}3}+\mathrm{C}
A number is chosen randomly from numbers 1 to 60. The probability that the chosen number is a multiple of 2 or 5 is
\frac{2}{5}
\frac{3}{5}
\frac{7}{10}
\frac{9}{10}
ABCD is a rhombus whose diagonals intersect at E. Then
\stackrel{\to }{\mathrm{EA}}+\stackrel{\to }{\mathrm{EB}}+\stackrel{\to }{\mathrm{EC}}+\stackrel{\to }{\mathrm{ED}}
\stackrel{\to }{0}
\stackrel{\to }{\mathrm{AD}}
2\stackrel{\to }{\mathrm{BC}}
2\stackrel{\to }{\mathrm{AD}}
If A is a square matrix of order 3, such that A (adj A) = 10 I, then |adj A| is equal to
\frac{1}{3}
\frac{4}{13}
\frac{1}{4}
\frac{1}{2}
\stackrel{^}{i}, \stackrel{^}{j}, \stackrel{^}{k}
are unit vectors along three mutually perpendicular directions, then
\stackrel{^}{i} . \stackrel{^}{j}=1
\stackrel{^}{i} × \stackrel{^}{j}=1
\stackrel{^}{i} . \stackrel{^}{k}=0
\stackrel{^}{i} × \stackrel{^}{k}=0
\frac{x-2}{1}=\frac{y-3}{1}=\frac{4-z}{k}
\frac{x-1}{k}=\frac{y-4}{2}=\frac{z-5}{-2}
are mutually perpendicular if the value of k is
-\frac{2}{3}
\frac{2}{3}
If y = Ae5x + Be–5x, then
\frac{{d}^{2}y}{d{x}^{2}}
(d) 15y VIEW SOLUTION
A relation R on a set A is called ________, if (a1, a2) ∈ R and (a2, a3) ∈ R implies that (a1, a3)∈R, for a1, a2, a3 ∈ A. VIEW SOLUTION
The integrating factor of the differential equation x
\frac{dy}{dx}+2y={x}^{2}
is _________.
1+{\left(\frac{dy}{dx}\right)}^{2}=x
is _____________. VIEW SOLUTION
\mathrm{A}+\mathrm{B}=\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]
\mathrm{A}-2\mathrm{B}=\left[\begin{array}{cc}-1& 1\\ 0& -1\end{array}\right],
then A = ________. VIEW SOLUTION
The least value of the function
f\left(x\right)=ax+\frac{b}{x}\left(a>0, b>0, x>0\right)
is __________. VIEW SOLUTION
\mathrm{sin} \left[\frac{\pi }{3}-{\mathrm{sin}}^{-1}\left(-\frac{1}{2}\right)\right].
Using differential, find the approximate value of
\sqrt{36.6}
upto 2 decimal places.
Find the slope of tangent to the curve y = 2 cos2(3x) at
x=\frac{\pi }{6}.
\underset{1}{\overset{4}{\int }}\left|x-5\right|dx.
f\left(x\right)=\left\{\begin{array}{cc}\frac{{x}^{2}-9}{x-3},& x\ne 3\\ k,& x=3\end{array}\right\
is continuous at x = 3, find the value of k. VIEW SOLUTION
\mathrm{A}=\left[\begin{array}{cc}3& -4\\ 1& -1\end{array}\right]
write A–1. VIEW SOLUTION
\int \frac{x+1}{x\left(1-2x\right)}dx.
\int \frac{x {\mathrm{sin}}^{-1}\left({x}^{2}\right)}{\sqrt{1-{x}^{4}}}dx.
\underset{0}{\overset{1}{\int }} x{\left(1-x\right)}^{n} dx.
If x = a cos θ; y = b sin θ, then find
\frac{{d}^{2}y}{d{x}^{2}}
f\left(x\right)=\frac{4x+3}{6x-4}, x\ne \frac{2}{3},
then show that (fof) (x) = x; for all
x\ne \frac{2}{3}.
Also, write inverse of f.
\mathrm{tan} \left[2 {\mathrm{tan}}^{-1}\left(\frac{1}{2}\right)-{\mathrm{cos}}^{-1} 3\right]=\frac{9}{13}.
y={\left(\mathrm{cos} x\right)}^{x}+{\mathrm{tan}}^{-1}\sqrt{x}, \mathrm{find} \frac{dy}{dx}.
\stackrel{\to }{a}=\stackrel{^}{i}+2\stackrel{^}{j}+3\stackrel{^}{k} \mathrm{and} \stackrel{\to }{b}=2\stackrel{^}{i}+4\stackrel{^}{j}-5\stackrel{^}{k}
represent two adjacent sides of a parallelogram, find unit vectors parallel to the diagonals of the parallelogram.
x \mathrm{sin} \left(\frac{y}{x}\right)\frac{dy}{dx}+x-y \mathrm{sin} \left(\frac{y}{x}\right)=0
Given that x = 1 when
y=\frac{\pi }{2}.
If a, b, c are pth, qth and rth terms respectively of a G.P, then prove that
\left|\begin{array}{ccc}\mathrm{log} a& p& 1\\ \mathrm{log} b& q& 1\\ \mathrm{log} c& r& 1\end{array}\right|=0
\mathrm{A}=\left[\begin{array}{ccc}2& -3& 5\\ 3& 2& -4\\ 1& 1& -2\end{array}\right]
, then find A–1.
Find the vector and cartesian equations of the line which perpendicular to the lines with equations
\frac{x+2}{1}=\frac{y-3}{2}=\frac{z+1}{4} \mathrm{and} \frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}
and passes through the point (1, 1, 1). Also find the angle between the given lines. VIEW SOLUTION
Evaluate the following integral as the limit of sums
\underset{1}{\overset{4}{\int }}\left({x}^{2}-x\right) dx.
Find the point on the curve y2 = 4x which is nearest to the point (2, 1). VIEW SOLUTION
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torch.triangular_solve — PyTorch 1.11.0 documentation
torch.triangular_solve
torch.triangular_solve¶
torch.triangular_solve(b, A, upper=True, transpose=False, unitriangular=False, *, out=None)¶
A
b
In symbols, it solves
AX = b
and assumes
A
is square upper-triangular (or lower-triangular if upper = False ) and does not have zeros on the diagonal.
torch.triangular_solve(b, A) can take in 2D inputs b, A or inputs that are batches of 2D matrices. If the inputs are batches, then returns batched outputs X
If the diagonal of A contains zeros or elements that are very close to zero and unitriangular = False (default) or if the input matrix is badly conditioned, the result may contain NaN s.
Supports input of float, double, cfloat and cdouble data types.
torch.triangular_solve() is deprecated in favor of torch.linalg.solve_triangular() and will be removed in a future PyTorch release. torch.linalg.solve_triangular() has its arguments reversed and does not return a copy of one of the inputs.
X = torch.triangular_solve(B, A).solution should be replaced with
X = torch.linalg.solve_triangular(A, B)
b (Tensor) – multiple right-hand sides of size
(*, m, k)
*
is zero of more batch dimensions
A (Tensor) – the input triangular coefficient matrix of size
(*, m, m)
*
upper (bool, optional) – whether
A
is upper or lower triangular. Default: True.
transpose (bool, optional) – solves op(A)X = b where op(A) = A^T if this flag is True, and op(A) = A if it is False. Default: False.
unitriangular (bool, optional) – whether
A
is unit triangular. If True, the diagonal elements of
A
are assumed to be 1 and not referenced from
A
. Default: False.
out ((Tensor, Tensor), optional) – tuple of two tensors to write the output to. Ignored if None . Default: None .
A namedtuple (solution, cloned_coefficient) where cloned_coefficient is a clone of
A
and solution is the solution
X
AX = b
(or whatever variant of the system of equations, depending on the keyword arguments.)
>>> A = torch.randn(2, 2).triu()
>>> torch.triangular_solve(b, A)
torch.return_types.triangular_solve(
solution=tensor([[ 1.7841, 2.9046, -2.5405],
[ 1.9320, 0.9270, -1.2826]]),
cloned_coefficient=tensor([[ 1.1527, -1.0753],
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Design and Numerical Analysis of an Electrostatic Energy Harvester With Impact for Frequency Up-Conversion | J. Comput. Nonlinear Dynam. | ASME Digital Collection
Design and Numerical Analysis of an Electrostatic Energy Harvester With Impact for Frequency Up-Conversion
R. Lensvelt,
R. Lensvelt
Eindhoven 5600 MB,
e-mail: r.lensvelt@tue.nl
R. H. B. Fey,
R. H. B. Fey
e-mail: r.h.b.fey@tue.nl
R. M. C. Mestrom,
R. M. C. Mestrom
Department of Electrical Engineering, Eindhoven University of Technology
e-mail: r.m.c.mestrom@tue.nl
e-mail: h.nijmijer@tue.nl
Lensvelt, R., Fey, R. H. B., Mestrom, R. M. C., and Nijmeijer, H. (March 30, 2020). "Design and Numerical Analysis of an Electrostatic Energy Harvester With Impact for Frequency Up-Conversion." ASME. J. Comput. Nonlinear Dynam. May 2020; 15(5): 051005. https://doi.org/10.1115/1.4046664
Integration of vibration energy harvesters (VEHs) with small-scale electronic devices may form an attractive alternative for relatively large batteries and can, potentially, increase their lifespan. However, the inherent mismatch between a harvester's high-frequency resonance, typically in the range
100−1000
Hz, relative to the available low-frequency ambient vibrations, typically in the range
10–100
Hz, means that low-frequency power generation in microscale VEHs remains a persistent challenge. In this work, we model a novel electret-based, electrostatic energy harvester (EEH) design. In this design, we combine an out-of-plane gap-closing comb (OPGC) configuration for the low-frequency oscillator with an in-plane overlap comb configuration for the high-frequency oscillator and employ impact for frequency up-conversion. An important design feature is the tunability of the resonance frequency through the electrostatic nonlinearity of the low-frequency oscillator. Impulsive normal forces due to impact are included in numerical simulation of the EEH through Moreau's time-stepping scheme which has, to the best of our knowledge, not been used before in VEH design and analysis. The original scheme is extended with time-step adjustments around impact events to reduce computational time. Using frequency sweeps, we numerically investigate power generation under harmonic, ambient vibrations. Results show improved low-frequency power generation in this EEH compared to a reference EEH. The EEH design shows peak power generation improvement of up to a relative factor 3.2 at low frequencies due to the occurrence of superharmonic resonances.
Computer simulation, Design, Energy generation, Energy harvesting, Numerical analysis, Resonance, Stiffness, Vibration, Capacitance, Microscale devices
A Study of Low Level Vibrations as a Power Source for Wireless Sensor Nodes
Bendame
MATEC Web Conferences
, Marrakech, Morocco, Apr. 30–May 2, p.
Theoretical Modeling and Analysis of Mechanical Impact Driven and Frequency Up-Converted Piezoelectric Energy Harvester for Low-Frequency and Wide-Bandwidth Operation
Energy Options for Wireless Sensor Nodes
Renewable Energy Harvesting for Wireless Sensors Using Passive RFID Tag Technology: A Review
Piezoelectric MEMS-Based Wideband Energy Harvesting Systems Using a Frequency-Up-Conversion Cantilever Stopper
Mechanical Frequency Up-Conversion for Sub-Resonance, Low-Frequency Vibration Harvesting
Broadband Electrostatic Device for Power Harvesting
A Comprehensive Review on Vibration Based Micro Power Generators Using Electromagnetic and Piezoelectric Transducer Mechanisms
State-of-the-Art in Vibration-Based Electrostatic Energy Harvesting
Aroudi
Nonlinear Electrostatic Energy Harvester Using Compensational Springs in Gravity Field
.10.1088/1361-6439/aabc90
Bistable Electromagnetic Generator Based on Buckled Beams for Vibration Energy Harvesting
Noise Effected Energy Harvesting in a Beam With Stopper
Parametric Resonance for Vibration Energy Harvesting With Design Techniques to Passively Reduce the Initiation Threshold Amplitude
Systematic Study of Dual Resonant Rectilinear-to-Rotary Motion Converter for Low Frequency Vibrational Energy Harvesting
Piezoelectric Energy Harvester Using Impact-Driven Flexible Side-Walls for Human Limb Motion
Chaos in the Fractionally Damped Broadband Piezoelectric Energy Generator
Design Optimization of an Out-of-Plane Gap-Closing Electrostatic Vibration Energy Harvester (VEH) With a Limitation on the Output Voltage
Microscale Electrostatic Energy Harvester Using Internal Impacts
Design and Simulation of Bistable Microsystem With Frequency-Up Conversion Effect for Electrostatic Energy Harvesting
IEEE Regional Symposium on Micro and Nano Electronics
), Kota Kinabalu, Malaysia, Sept. 28–30, pp.
.10.1109/RSM.2011.6088349
Design and Fabrication of MEMS Electrostatic Energy Harvester With Nonlinear Springs and Vertical Sidewall Electrets
Proceedings of PowerMEMS
, Seoul, South Korea, Nov. 15–18, pp.
.https://www.researchgate.net/publication/228096992_Design_and_fabrication_of_MEMS_electrostatic_energy_harvester_with_nonlinear_springs_and_vertical_sidewall_electrets
Bistable Multiple-Mass Electrostatic Generator for Low-Frequency Vibrations Energy Harvesting
Simulations and Experiments of Hardening and Softening Resonances in a Clamped-Clamped Beam MEMS Resonator
Borca-Tascuic
Silicon-Integrated Electrostatic Energy Harvesters
), Columbus, OH, Aug. 4–7, pp.
.10.1109/MWSCAS.2013.6674661
Investigation of Gap-Closing Interdigitated Capacitors for Electrostatic Vibration Energy Harveseters
A Batch-Fabricated and Electret-Free Silicon Electrostatic Vibration Energy Harvester
MEMS Electrostatic Energy Harvesters With End-Stop Effects
J. Microelmechanics Microeng.
Kotlanka
Theoretical Comparison of the Energy Harvesting Capability Among Various Electrostatic Mechanims From Structure Aspect
A Nonlinear MEMS Electrostatic Kinetic Energy Harvester for Human-Powered Biomedical Devices
Optimization of an Electret-Based Energy Harvester
Suspended Electrodes for Reducing Parasitic Capacitance in Electret Energy Harvesters
Honzumi
Power Output Enehancement of a Vibration-Driven Electret Generator for Wireless Sensor Applications
Design Considerations of Electrostatic Electrode Elements for in-Plane Micro-Generators
, Freiburg, Germany, Nov. 28–29, pp.
), Lisbon, Portugal, Nov. 27–30, pp.
.10.1109/PRIME.2016.7519518
A Design Method of In-Plane MEMS Electret Energy Harvester With Comb Drives
A Batch-Fabricated Electret-Biased Wideband MEMS Vibration Energy Harvester With Frequency-Up Conversion Behaviour Powering a Uhf Wireless Sensor Node
An Evaluation of Moreau's Time-Stepping Scheme for the Simulation of a Legged Robot
Theoretical Modeling and Analysis of Two-Degree-of-Freedom Piezoelectric Energy Harvester With Stopper
Numerics of Unilateral Contacts and Friction: Modeling and Numerical Time Integration in Non-Smooth Dynamics
A Periodically Forced Picewise Linear Ocillator
A Vibration Energy Harvesting Structure, Tunable Over a Wide Frequency Range Using Minimal Actuation
|
Design of Experiments - MATLAB & Simulink - MathWorks Switzerland
For example, a simple model of a response y in an experiment with two controlled factors x1 and x2 might look like this:
y={\beta }_{0}+{\beta }_{1}{x}_{1}+{\beta }_{2}{x}_{2}+{\beta }_{3}{x}_{1}{x}_{2}+\epsilon
Here ε includes both experimental error and the effects of any uncontrolled factors in the experiment. The terms β1x1 and β2x2 are main effects and the term β3x1x2 is a two-way interaction effect. A designed experiment would systematically manipulate x1 and x2 while measuring y, with the objective of accurately estimating β0, β1, β2, and β3.
|
torch.addr — PyTorch 1.11.0 documentation
torch.addr
torch.addr¶
torch.addr(input, vec1, vec2, *, beta=1, alpha=1, out=None) → Tensor¶
Optional values beta and alpha are scaling factors on the outer product between vec1 and vec2 and the added matrix input respectively.
\text{out} = \beta\ \text{input} + \alpha\ (\text{vec1} \otimes \text{vec2})
If beta is 0, then input will be ignored, and nan and inf in it will not be propagated.
If vec1 is a vector of size n and vec2 is a vector of size m , then input must be broadcastable with a matrix of size
(n \times m)
and out will be a matrix of size
(n \times m)
input (Tensor) – matrix to be added
vec1 (Tensor) – the first vector of the outer product
vec2 (Tensor) – the second vector of the outer product
beta (Number, optional) – multiplier for input (
\beta
alpha (Number, optional) – multiplier for
\text{vec1} \otimes \text{vec2}
\alpha
>>> vec1 = torch.arange(1., 4.)
>>> M = torch.zeros(3, 2)
>>> torch.addr(M, vec1, vec2)
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List of logic symbols — Wikipedia Republished // WIKI 2
This article contains logic symbols. Without proper rendering support, you may see question marks, boxes, or other symbols instead of logic symbols.
Intro to Logical Statements
Propositions and Symbols Used in Symbolic Logic (See link below for more videos in Symbolic Logic)
Fundamentals of Logic - Part 1 (Statements and Symbols)
Logic symbols | Mathematical symbols | Basic symbols |
1 Basic logic symbols
2 Advanced and rarely used logical symbols
3 Usage in various countries
3.1 Poland and Germany
U+2283 ⇒
⊃ ⇒
{\displaystyle \Rightarrow }
{\displaystyle \to }
\to or \rightarrow
{\displaystyle \supset }
{\displaystyle \implies }
material implication implies; if ... then propositional logic, Heyting algebra
{\displaystyle A\Rightarrow B}
is false when A is true and B is false but true otherwise.[2][circular reference]
{\displaystyle \rightarrow }
may mean the same as
{\displaystyle \Rightarrow }
(the symbol may also indicate the domain and codomain of a function; see table of mathematical symbols).
{\displaystyle \supset }
{\displaystyle \Rightarrow }
(the symbol may also mean superset).
{\displaystyle x=2\Rightarrow x^{2}=4}
{\displaystyle x^{2}=4\Rightarrow x=2}
is in general false (since x could be −2).
U+2194 ⇔
↔ ⇔
↔
{\displaystyle \Leftrightarrow }
{\displaystyle \equiv }
{\displaystyle \leftrightarrow }
{\displaystyle \iff }
\iff material equivalence if and only if; iff; means the same as propositional logic
{\displaystyle A\Leftrightarrow B}
is true only if both A and B are false, or both A and B are true.
{\displaystyle x+5=y+2\Leftrightarrow x+3=y}
U+0021 ¬
! ¬
!
{\displaystyle \neg }
\lnot or \neg
{\displaystyle \sim }
negation not propositional logic The statement
{\displaystyle \lnot A}
is true if and only if A is false.
A slash placed through another operator is the same as
{\displaystyle \neg }
placed in front.
{\displaystyle \neg (\neg A)\Leftrightarrow A}
{\displaystyle x\neq y\Leftrightarrow \neg (x=y)}
{\displaystyle \mathbb {D} }
U+1D53B 𝔻 𝔻 \mathbb{D} Domain of discourse Domain of predicate Predicate (mathematical logic)
{\displaystyle \mathbb {D} \mathbb {:} \mathbb {R} }
U+0026 ∧
{\displaystyle \wedge }
\wedge or \land
{\displaystyle \cdot }
{\displaystyle \&}
\&[3]
logical conjunction and propositional logic, Boolean algebra The statement A ∧ B is true if A and B are both true; otherwise, it is false. n < 4 ∧ n >2 ⇔ n = 3 when n is a natural number.
U+2225 ∨
∥ ∨
+
∥
{\displaystyle \lor }
\lor or \vee
{\displaystyle \parallel }
logical (inclusive) disjunction or propositional logic, Boolean algebra The statement A ∨ B is true if A or B (or both) are true; if both are false, the statement is false. n ≥ 4 ∨ n ≤ 2 ⇔ n ≠ 3 when n is a natural number.
⊻
≢
{\displaystyle \oplus }
{\displaystyle \veebar }
\veebar
{\displaystyle \not \equiv }
exclusive disjunction xor; either ... or propositional logic, Boolean algebra The statement A ↮ B is true when either A or B, but not both, are true. A ⊻ B means the same. (¬A) ↮ A is always true, and A ↮ A always false, if vacuous truth is excluded.
⊤
{\displaystyle \top }
\top Tautology top, truth, full clause propositional logic, Boolean algebra, first-order logic The statement ⊤ is unconditionally true. ⊤(A) ⇒ A is always true.
{\displaystyle \bot }
\bot Contradiction bottom, falsum, falsity, empty clause propositional logic, Boolean algebra, first-order logic The statement ⊥ is unconditionally false. (The symbol ⊥ may also refer to perpendicular lines.) ⊥(A) ⇒ A is always false.
{\displaystyle \forall }
\forall universal quantification for all; for any; for each first-order logic ∀ x: P(x) or (x) P(x) means P(x) is true for all x.
{\displaystyle \forall n\in \mathbb {N} :n^{2}\geq n.}
U+2203 ∃ {\displaystyle \exists }
\exists existential quantification there exists first-order logic ∃ x: P(x) means there is at least one x such that P(x) is true.
{\displaystyle \exists n\in \mathbb {N} :}
n is even.
U+2203 U+0021 ∃ ! {\displaystyle \exists !}
\exists ! uniqueness quantification there exists exactly one first-order logic ∃! x: P(x) means there is exactly one x such that P(x) is true.
{\displaystyle \exists !n\in \mathbb {N} :n+5=2n.}
U+003A U+229C ≔ (: =)
≔
{\displaystyle :=}
{\displaystyle \equiv }
{\displaystyle :\Leftrightarrow }
:\Leftrightarrow
definition is defined as everywhere x ≔ y or x ≡ y means x is defined to be another name for y (but note that ≡ can also mean other things, such as congruence).
P :⇔ Q means P is defined to be logically equivalent to Q.
{\displaystyle \cosh x:={\frac {e^{x}+e^{-x}}{2}}}
U+0028 U+0029 ( ) (
)
{\displaystyle (~)}
( ) precedence grouping parentheses; brackets everywhere Perform the operations inside the parentheses first. (8 ÷ 4) ÷ 2 = 2 ÷ 2 = 1, but 8 ÷ (4 ÷ 2) = 8 ÷ 2 = 4.
U+22A2 ⊢ ⊢
{\displaystyle \vdash }
\vdash turnstile proves propositional logic, first-order logic x ⊢ y means x proves (syntactically entails) y (A → B) ⊢ (¬B → ¬A)
U+22A8 ⊨ ⊨
{\displaystyle \vDash }
\vDash, \models double turnstile models propositional logic, first-order logic x ⊨ y means x models (semantically entails) y (A → B) ⊨ (¬B → ¬A)
Advanced and rarely used logical symbols
These symbols are sorted by their Unicode value:
U+0305 COMBINING OVERLINE used format for denoting Gödel numbers.
denoting negation used primarily in electronics.
using HTML style "4̅" is a shorthand for the standard numeral "SSSS0".
"A ∨ B" says the Gödel number of "(A ∨ B)". "A ∨ B" is the same as "¬(A ∨ B)".
U+007C UPWARDS ARROW
VERTICAL LINE Sheffer stroke, the sign for the NAND operator (negation of conjunction).
U+2193 DOWNWARDS ARROW Peirce Arrow, the sign for the NOR operator (negation of disjunction).
{\displaystyle \odot }
\odot CIRCLED DOT OPERATOR the sign for the XNOR operator (negation of exclusive disjunction).
U+2201 COMPLEMENT
U+2204 ∄\nexists THERE DOES NOT EXIST strike out existential quantifier, same as "¬∃"
U+2234 ∴\therefore THEREFORE Therefore
U+2235 ∵\because BECAUSE because
U+22A7 MODELS is a model of (or "is a valuation satisfying")
U+22A8 ⊨\vDash TRUE is true of
U+22AC ⊬\nvdash DOES NOT PROVE negated ⊢, the sign for "does not prove" T ⊬ P says "P is not a theorem of T"
U+22AD ⊭\nvDash NOT TRUE is not true of
U+2020 DAGGER it is true that ... Affirmation operator
U+22BC NAND NAND operator
U+22BD NOR NOR operator
U+25C7 WHITE DIAMOND modal operator for "it is possible that", "it is not necessarily not" or rarely "it is not probably not" (in most modal logics it is defined as "¬◻¬")
U+22C6 STAR OPERATOR usually used for ad-hoc operators
U+2193 UP TACK
DOWNWARDS ARROW Webb-operator or Peirce arrow, the sign for NOR. Confusingly, "⊥" is also the sign for contradiction or absurdity.
U+231D \ulcorner
TOP RIGHT CORNER corner quotes, also called "Quine quotes"; for quasi-quotation, i.e. quoting specific context of unspecified ("variable") expressions;[4] also used for denoting Gödel number;[5] for example "⌜G⌝" denotes the Gödel number of G. (Typographical note: although the quotes appears as a "pair" in unicode (231C and 231D), they are not symmetrical in some fonts. And in some fonts (for example Arial) they are only symmetrical in certain sizes. Alternatively the quotes can be rendered as ⌈ and ⌉ (U+2308 and U+2309) or by using a negation symbol and a reversed negation symbol ⌐ ¬ in superscript mode. )
U+25A1 WHITE MEDIUM SQUARE
WHITE SQUARE modal operator for "it is necessary that" (in modal logic), or "it is provable that" (in provability logic), or "it is obligatory that" (in deontic logic), or "it is believed that" (in doxastic logic); also as empty clause (alternatives:
{\displaystyle \emptyset }
and ⊥)
U+27DB LEFT AND RIGHT TACK semantic equivalent
U+27E1 WHITE CONCAVE-SIDED DIAMOND never modal operator
U+27E2 WHITE CONCAVE-SIDED DIAMOND WITH LEFTWARDS TICK was never modal operator
U+27E3 WHITE CONCAVE-SIDED DIAMOND WITH RIGHTWARDS TICK will never be modal operator
U+25A1 WHITE SQUARE always modal operator
U+25A4 WHITE SQUARE WITH LEFTWARDS TICK was always modal operator
U+25A5 WHITE SQUARE WITH RIGHTWARDS TIC will always be modal operator
U+297D RIGHT FISH TAIL sometimes used for "relation", also used for denoting various ad hoc relations (for example, for denoting "witnessing" in the context of Rosser's trick) The fish hook is also used as strict implication by C.I.Lewis
{\displaystyle p}
{\displaystyle q\equiv \Box (p\rightarrow q)}
, the corresponding LaTeX macro is \strictif. See here for an image of glyph. Added to Unicode 3.2.0.
U+2A07 TWO LOGICAL AND OPERATOR
Usage in various countries
As of 2014[update] in Poland, the universal quantifier is sometimes written
{\displaystyle \wedge }
, and the existential quantifier as
{\displaystyle \vee }
.[6][7] The same applies for Germany.[8][9]
Józef Maria Bocheński
List of notation used in Principia Mathematica
List of mathematical symbols
Logic alphabet, a suggested set of logical symbols
Logic gate § Symbols
Mathematical operators and symbols in Unicode
Non-logical symbol
Truth table
Wikipedia:WikiProject Logic/Standards for notation
^ "Named character references". HTML 5.1 Nightly. W3C. Retrieved 9 September 2015.
^ "Material conditional".
^ Although this character is available in LaTeX, the MediaWiki TeX system does not support it.
^ Quine, W.V. (1981): Mathematical Logic, §6
^ Hintikka, Jaakko (1998), The Principles of Mathematics Revisited, Cambridge University Press, p. 113, ISBN 9780521624985 .
^ "Kwantyfikator ogólny". 2 October 2017 – via Wikipedia. [circular reference]
^ "Kwantyfikator egzystencjalny". 23 January 2016 – via Wikipedia. [circular reference]
^ "Quantor". 21 January 2018 – via Wikipedia. [circular reference]
^ Hermes, Hans. Einführung in die mathematische Logik: klassische Prädikatenlogik. Springer-Verlag, 2013.
Józef Maria Bocheński (1959), A Précis of Mathematical Logic, trans., Otto Bird, from the French and German editions, Dordrecht, South Holland: D. Reidel.
Named character entities in HTML 4.0
List of mathematical symbols by subject
List of logic symbols
List of Unicode characters
Mathematical Alphanumeric Symbols
Letterlike Symbols
Symbols for zero
Arrows and Geometric Shapes
Miscellaneous Symbols and Arrows
Geometric Shapes
Mathematical operators and symbols
Mathematical Operators
Supplemental Math Operators
Supplemental Mathematical Operators
Number Forms
ISO 31-11 (Mathematical signs and symbols for use in physical sciences and technology)
APL syntax and symbols
Greek letters used in mathematics, science, and engineering
Latin letters used in mathematics
List of letters used in mathematics and science
Mathematical notation
Notation in probability and statistics
List of common physics notations
Typographical conventions in mathematical formulae
Glossary of mathematical symbols
Mathematical constants and functions
Physical constants
Table of mathematical symbols by introduction date
|
Erratum to ‘Meir-Keeler α-contractive fixed and common fixed point theorems’ | Fixed Point Theory and Algorithms for Sciences and Engineering | Full Text
Erratum to ‘Meir-Keeler α-contractive fixed and common fixed point theorems’
Dhananjay Gopal3
In this note we correct some errors that appeared in the article (Abdeljawad in Fixed Point Theory Appl. 2013:19, 2013) by modifying some conditions in the main theorems and by giving an example to support.
After examining the calculations in the proof of the uniqueness part in Theorem 8 in [1] and Steps 3 and 4 of Theorem 16, we found that they do not lead to strict inequalities, and hence the proofs failed. In this note, we slightly modify some of the used conditions to achieve our claim.
The following theorem is a modification to Theorem 8 in [1]. The proof is the same as in [1] except the uniqueness part will be proved by using the new modified condition (H) in the statement of the theorem.
\left(X,d\right)
\left(f,g\right)
-orbitally complete metric space, where f, g are self-mappings of X. Also, let
\alpha :X×X\to \left[0,\mathrm{\infty }\right)
be a mapping. Assume the following:
\left(f,g\right)
is α-admissible and there exists an
{x}_{0}\in X
\alpha \left({x}_{0},f{x}_{0}\right)\ge 1
\left(f,g\right)
is generalized Meir-Keeler α-contractive.
{d}_{n}=d\left({x}_{n},{x}_{n+1}\right)
is monotone decreasing. If, moreover, we assume that
\left(f,g\right)
-orbit of
{x}_{0}
\alpha \left({x}_{n},{x}_{j}\right)\ge 1
for all n even and
j>n
odd and that f and g are continuous on the
\left(f,g\right)
{x}_{0}
Then either (1) f or g has a fixed point in the
\left(f,g\right)
-orbit
\left\{{x}_{n}\right\}
{x}_{0}
or (2) f and g have a common fixed point p and
lim{x}_{n}=p
. If, moreover, we assume that the following condition (H) holds:
(H) If for all fixed points x and y of
\left(f,g\right)
\alpha \left(x,y\right)\ge 1
then the uniqueness of the fixed point is obtained.
Proof To prove uniqueness, assume p is the common fixed point obtained as
{x}_{n}\to p
and q is another common fixed point. Then, Eq. (5) in [1] and the condition (H) yield
\begin{array}{rcl}d\left(p,q\right)& =& d\left(fp,gq\right)\\ \le & \alpha \left(p,q\right)d\left(fp,gq\right)\\ <& max\left\{d\left(p,q\right),d\left(p,fp\right),d\left(q,gq\right),\frac{d\left(p,gq\right)+d\left(q,fp\right)}{2}\right\}=d\left(p,q\right).\end{array}
Thus we reach
d\left(p,q\right)<d\left(p,q\right)
, and hence a contradiction, which implies that
p=q
Using the new modified condition (H) for the pair
\left(f,f\right)
, we modify the uniqueness part of Corollary 9 in [1].
The following example shows that we lose the uniqueness if our modified (H) condition is not satisfied.
X=\left[0,2\right]
with the absolute value metric
d\left(x,y\right)=|x-y|
f:X\to X
f\left(x\right)=\left\{\begin{array}{cc}0,\hfill & x\in \left\{0,\frac{1}{4}\right\},\hfill \\ 1,\hfill & x\in \left(0,\frac{1}{2}\right)-\left\{\frac{1}{4}\right\},\hfill \\ \frac{3}{2},\hfill & x\in \left[\frac{1}{2},2\right].\hfill \end{array}
\alpha \left(x,y\right)=\left\{\begin{array}{cc}1,\hfill & x,y\in \left[\frac{1}{2},2\right],\hfill \\ 0,\hfill & \text{otherwise}.\hfill \end{array}
Notice that f has two common fixed points
x=0
x=\frac{3}{2}
. This is because f satisfies all the hypotheses of the corollary (which is Corollary 9 in [1]) except the condition (H), i.e.,
\alpha \left(0,\frac{3}{2}\right)=0<1
The following theorem is a modification of Theorem 16 in [1]. The proofs of Step 3 and Step 4 are given only according to the new modified conditions (H) and (f-H).
Theorem 3 Let f, g be continuous self-maps of a metric space
\left(X,d\right)
g\in {C}_{f}
\alpha \left({x}_{n},{x}_{m}\right)\ge 1
m>n
. If g is a generalized Meir-Keeler α-f-contractive map such that α satisfies the condition (f-H): If
\left\{{x}_{n}\right\}
\alpha \left({x}_{n},{x}_{m}\right)\ge 1
m>n
f{x}_{n}\to z
\alpha \left(z,fz\right)\ge 1
. Also assume that the condition (H) is satisfied. Then f and g have a unique common fixed point.
\eta =fz=gz
is a common fixed point for f and g. Assume that
f\eta \ne \eta
{f}^{2}z\ne fz
, and by the help of the (f-H) condition, we have
\begin{array}{rcl}d\left(\eta ,f\eta \right)& =& d\left(gz,fgz\right)=d\left(gz,gfz\right)\\ \le & \alpha \left(z,fz\right)d\left(gz,gfz\right)\\ <& max\left\{d\left(fz,ffz\right),d\left(fz,gz\right),d\left(ffz,gfz\right),\frac{d\left(fz,gfz\right)+d\left(ffz,gz\right)}{2}\right\}\\ =& max\left\{d\left(\eta ,f\eta \right),0,0,d\left(\eta ,f\eta \right)\right\}.\end{array}
d\left(\eta ,f\eta \right)<d\left(\eta ,f\eta \right)
, which gives a contradiction and therefore
f\eta =\eta
g\eta =gfz=f\eta =\eta
Step 4. The uniqueness of the common fixed point. Assume that
\eta =fz=gz
is our common fixed point for f and g, where
f{x}_{n}\to z
, and ω is another common fixed point. Then, by the (H) condition, we have
\begin{array}{rcl}d\left(\eta ,\omega \right)& =& d\left(g\eta ,g\omega \right)\\ \le & \alpha \left(\eta ,\omega \right)d\left(g\eta ,g\omega \right)\\ <& max\left\{d\left(f\eta ,f\omega \right),d\left(f\eta ,g\eta \right),d\left(f\omega ,g\omega \right),\frac{d\left(f\eta ,g\omega \right)+d\left(f\omega ,g\eta \right)}{2}\right\},\end{array}
d\left(\eta ,\omega \right)<d\left(\eta ,\omega \right)
, a contradiction, and hence
\eta =\omega
Instead of the modified condition (f-H) above, the following condition can be used (s-f-H). If
\left\{{x}_{n}\right\}
\alpha \left({x}_{n},{x}_{m}\right)\ge 1
m>n
f{x}_{n}\to z
\alpha \left(f{x}_{n},fz\right)\ge k
for all n, where
k>1
, and hence Step 3 will be proved as follows.
\eta =fz=gz
f\eta \ne \eta
{f}^{2}z\ne fz
, and by the help of the (s-f-H) condition, we have
d\left(\eta ,f\eta \right)=d\left(gz,fgz\right)=d\left(gz,gfz\right)=\underset{n\to \mathrm{\infty }}{lim}d\left(gf{x}_{n},gfz\right)
\begin{array}{rcl}d\left(gf{x}_{n},gfz\right)& \le & {k}^{-1}\alpha \left(f{x}_{n},fz\right)d\left(gf{x}_{n},gfz\right)\\ \le & {k}^{-1}max\left\{d\left(ff{x}_{n},ffz\right),d\left(ff{x}_{n},gf{x}_{n}\right),d\left(ffz,gfz\right),\frac{d\left(ff{x}_{n},gfz\right)+d\left(ffz,gf{x}_{n}\right)}{2}\right\}.\end{array}
n\to \mathrm{\infty }
above and use the continuity and commutativity of f and g, then we reach
d\left(\eta ,f\eta \right)\le {k}^{-1}d\left(\eta ,f\eta \right)<d\left(\eta ,f\eta \right)
f\eta =\eta
g\eta =gfz=f\eta =\eta
Finally, according to the modifications above, the (H) condition only in Theorem 18 of [1] is needed to be modified.
Abdeljawad T: Meir-Keeler α -contractive fixed and common fixed point theorems. Fixed Point Theory Appl. 2013., 2013: Article ID 19
The second author thanks for the support of CSIR, Govt. of India, Grant No-25(0215)/13/EMR-II.
Department of Mathematics, Çankaya University, Ankara, 06530, Turkey
Department of Mathematics and Physical Sciences, Prince Sultan University, P.O. Box 66833, Riyadh, 11586, Saudi Arabia
Department of Applied Mathematics & Humanities, S. V. National Institute of Technology, Surat, 395 007, India
The online version of the original article can be found at 10.1186/1687-1812-2013-19
Abdeljawad, T., Gopal, D. Erratum to ‘Meir-Keeler α-contractive fixed and common fixed point theorems’. Fixed Point Theory Appl 2013, 110 (2013). https://doi.org/10.1186/1687-1812-2013-110
|
Unit of perceived loudness
For other uses, see Sone (disambiguation).
"Sones" redirects here. For other uses, see Sones (disambiguation).
The sone (/ˈsoʊn/) is a unit of loudness, the subjective perception of sound pressure. The study of perceived loudness is included in the topic of psychoacoustics and employs methods of psychophysics. Doubling the perceived loudness doubles the sone value. Proposed by Stanley Smith Stevens in 1936, it is not an SI unit.
1 Definition and conversions
Definition and conversions[edit]
According to Stevens' definition, a loudness of 1 sone is equivalent to 40 phons (a 1 kHz tone at 40 dB SPL).[1] The phons scale aligns with dB, not with loudness, so the sone and phon scales are not proportional. Rather, the loudness in sones is, at least very nearly, a power law function of the signal intensity, with an exponent of 0.3.[2][3] With this exponent, each 10 phon increase (or 10 dB at 1 kHz) produces almost exactly a doubling of the loudness in sones.[4]
At frequencies other than 1 kHz, the loudness level in phons is calibrated according to the frequency response of human hearing, via a set of equal-loudness contours, and then the loudness level in phons is mapped to loudness in sones via the same power law.
Loudness N in sones (for LN > 40 phon):[5]
{\displaystyle N=\left(10^{\frac {L_{N}-40}{10}}\right)^{0.30103}\approx 2^{\frac {L_{N}-40}{10}}}
or loudness level LN in phons (for N > 1 sone):
{\displaystyle L_{N}=40+10\log _{2}(N)}
Corrections are needed at lower levels, near the threshold of hearing.
These formulas are for single-frequency sine waves or narrowband signals. For multi-component or broadband signals, a more elaborate loudness model is required, accounting for critical bands.
To be fully precise, a measurement in sones must be specified in terms of the optional suffix G, which means that the loudness value is calculated from frequency groups, and by one of the two suffixes D (for direct field or free field) or R (for room field or diffuse field).
Threshold of pain ~ 100 ~ 134 ~ 676
Hearing damage during short-term effect ~ 20 ~ 120 ~ 256
Jet, 100 m away 6 ... 200 110 ... 140 128 ... 1024
Jackhammer, 1 m away / nightclub ~ 2 ~ 100 ~ 64
Hearing damage during long-term effect ~ 6×10−1 ~ 90 ~ 32
Major road, 10 m away 2×10−1 ... 6×10−1 80 ... 90 16 ... 32
Passenger car, 10 m away 2×10−2 ... 2×10−1 60 ... 80 4 ... 16
TV set at home level, 1 m away ~ 2×10−2 ~ 60 ~ 4
Normal talking, 1 m away 2×10−3 ... 2×10−2 40 ... 60 1 ... 4
Very calm room 2×10−4 ... 6×10−4 20 ... 30 0.15 ... 0.4
Rustling leaves, calm breathing ~ 6×10−5 ~ 10 ~ 0.02
Auditory threshold at 1 kHz 2×10−5 0 0
^ Stanley Smith Stevens: A scale for the measurement of the psychological magnitude: loudness. See: Psychological Review. 43, Nr. 5,APA Journals, 1936, pp. 405–416
^ Brian C. J. Moore (2007). Cochlear hearing loss: physiological, psychological and technical issues (2nd ed.). Wiley-Interscience. pp. 94–95. ISBN 978-0-470-51633-1.
^ Irving P. Herman (2007). Physics of The Human Body. Springer. p. 613. ISBN 978-3-540-29603-4.
^ Eberhard Hänsler, Gerhard Schmidt (2008). Speech and audio processing in adverse environments. Springer. p. 299. ISBN 978-3-540-70601-4.
^ Hugo Fastl and Eberhard Zwicker (2007). Psychoacoustics: facts and models (3rd ed.). Springer. p. 207. ISBN 978-3-540-23159-2.
Correlation between sones und phons − calculator
Retrieved from "https://en.wikipedia.org/w/index.php?title=Sone&oldid=1068465719"
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Ultrasonic Measurement of Rolling Bearing Lubrication Using Piezoelectric Thin Films | J. Tribol. | ASME Digital Collection
Bruce W. Drinkwater,
, University Walk, Bristol, BS8 1TR, UK
e-mail: b.drinkwater@bristol.ac.uk
Katherine J. Kirk,
Katherine J. Kirk
, Paisley, PA1 2BE, UK
Jocelyn Elgoyhen,
Jocelyn Elgoyhen
, Mappin Street, Sheffield, S1 3JD, UK
Drinkwater, B. W., Zhang, J., Kirk, K. J., Elgoyhen, J., and Dwyer-Joyce, R. S. (December 3, 2008). "Ultrasonic Measurement of Rolling Bearing Lubrication Using Piezoelectric Thin Films." ASME. J. Tribol. January 2009; 131(1): 011502. https://doi.org/10.1115/1.3002324
This paper describes the measurement of lubricant-film thickness in a rolling element bearing using a piezoelectric thin film transducer to excite and receive ultrasonic signals. High frequency (200 MHz) ultrasound is generated using a piezoelectric aluminum nitride film deposited in the form of a very thin layer onto the outer bearing raceway. This creates a transducer and electrode combination of total thickness of less than
10 μm
. In this way the bearing is instrumented with minimal disruption to the housing geometry and the oil-film can be measured noninvasively. The high frequency transducer generates a fine columnar beam of ultrasound that has dimensions less than the typical lubricated contact ellipse. The reflection coefficient from the lubricant-layer is then measured from within the lubricated contact and the oil-film thickness extracted via a quasistatic spring model. The results are described on a deep groove 6016 ball bearing supporting an 80 mm shaft under normal operating conditions. Good agreement is shown over a range of loads and speeds with lubricant-film thickness extracted from elastohydrodynamic lubrication theory.
aluminium compounds, lubrication, piezoelectric thin films, rolling, piezoelectric thin films, rolling bearing, oil-film thickness measurement, reflection coefficient, condition monitoring
Ball bearings, Bearings, Lubricants, Lubrication, Reflectance, Reflection, Rolling bearings, Signals, Stress, Thin films, Transducers, Geometry, Springs
An On-Line Monitoring System for Oil-Film, Pressure and Temperature Distributions in Large-Scale Hydro-Generator Bearings
Correlation Between Model Test Devices and Full Bearing Tests Under Grease Lubricated Conditions
Proceedings of the IUTAM Symposium on Elastohydrodynamics and Micro-Elastohydrodynamics
A Review on Machinery Diagnostics and Prognostics Implementing Condition-Based Maintenance
A Method of Temperature Monitoring in Fluid Film Bearings
Assessment of Gear Damage Monitoring Techniques Using Vibration Measurements
Condition Monitoring of Mechanical Seals: Detection of Film Collapse Using Reflected Ultrasonic Waves
The Measurement of Lubricant Film Thickness Using Ultrasound
Oil Film Measurement in PTFE-Faced Thrust Pad Bearings for Hydrodynamic Applications
Monitoring Lubricant Film Failure in a Ball Bearing Using Ultrasound
Elasto-Hydrodynamic Lubrication (SI Edition)
Model for the Influence of Pressure on the Bulk Modulus and the Influence of Temperature on the Solidification Pressure for Liquid Lubricants
Ball Bearing Lubrication: The Elastohydrodynamics of Elliptical Contacts
Thick Aluminium Nitride Films Deposited by Room-Temperature Sputtering for Ultrasonic Applications
Acoustic Measurement of Lubricant-Film Thickness Distribution in Ball Bearings
|
Alkylation Knowpia
Alkylation is the transfer of an alkyl group from one molecule to another. The alkyl group may be transferred as an alkyl carbocation, a free radical, a carbanion, or a carbene (or their equivalents).[1] Alkylating agents are reagents for effecting alkylation. Alkyl groups can also be removed in a process known as dealkylation. Alkylating agents are often classified according to their nucleophilic or electrophilic character.
In oil refining contexts, alkylation refers to a particular alkylation of isobutane with olefins. For upgrading of petroleum, alkylation produces a premium blending stock for gasoline.[2]
In medicine, alkylation of DNA is used in chemotherapy to damage the DNA of cancer cells. Alkylation is accomplished with the class of drugs called alkylating antineoplastic agents.
Typical route for alkylation of benzene with ethylene and ZSM-5 as a heterogeneous catalyst
Nucleophilic alkylating agentsEdit
Nucleophilic alkylating agents deliver the equivalent of an alkyl anion (carbanion). The formal "alkyl anion" attacks an electrophile, forming a new covalent bond between the alkyl group and the electrophile. The counterion, which is a cation such as lithium, can be removed and washed away in the work-up. Examples include the use of organometallic compounds such as Grignard (organomagnesium), organolithium, organocopper, and organosodium reagents. These compounds typically can add to an electron-deficient carbon atom such as at a carbonyl group. Nucleophilic alkylating agents can displace halide substituents on a carbon atom through the SN2 mechanism. With a catalyst, they also alkylate alkyl and aryl halides, as exemplified by Suzuki couplings.
The Kumada coupling employs both a nucleophilic alkylation step subsequent to the oxidative addition of the aryl halide (L = Ligand, Ar = Aryl).
The SN2 mechanism is not available for aryl substituents, where the trajectory to attack the carbon atom would be inside the ring. Thus only reactions catalyzed by organometallic catalysts are possible.
Alkylation by carbon electrophilesEdit
C-alkylationEdit
C-alkylation is a process for the formation of carbon-carbon bonds. For alkylation at carbon, the electrophilicity of alkyl halides is enhanced by the presence of a Lewis acid such as aluminium trichloride. Lewis acids are particularly suited for C-alkylation. C-alkylation can also be effected by alkenes in the presence of acids.
N-and P-alkylationEdit
N- and P-alkylation are important processes for the formation of carbon-nitrogen and carbon-phosphorus bonds.
Amines are readily alkylated. The rate of alkylation follows the order tertiary amine < secondary amine < primary amine. Typical alkylating agents are alkyl halides. Industry often relies on green chemistry methods involving alkylation of amines with alcohols, the byproduct being water. Hydroamination is another green method for N-alkylation.
In the Menshutkin reaction, a tertiary amine is converted into a quaternary ammonium salt by reaction with an alkyl halide. Similar reactions occur when tertiary phosphines are treated with alkyl halides, the products being phosphonium salts.
Thiols are readily alkylated to give thioethers.[3] The reaction is typically conducted in the presence of a base or using the conjugate base of the thiol. Thioethers undergo alkylation to give sulfonium ions.
O-alkylationEdit
Alcohols alkylate to give ethers:
ROH + R'X → ROR'
When the alkylating agent is an alkyl halide, the conversion is called the Williamson ether synthesis. Alcohols are also good alkylating agents in the presence of suitable acid catalysts. For example, most methyl amines are prepared by alkylation of ammonia with methanol. The alkylation of phenols is particularly straightforward since it is subject to fewer competing reactions.[4]
{\displaystyle \mathrm {Ph{-}O^{-}\ +\ Me_{2}{-}SO_{4}\ \longrightarrow \ Ph{-}O{-}Me\ +\ Me{-}SO_{4}^{-}} }
(with Na+ as a spectator ion)
More complex alkylation of a alcohols and phenols involve ethoxylation. Ethylene oxide is the alkylating group in this reaction.
Oxidative addition to metalsEdit
In the process called oxidative addition, low-valent metals often react with alkylating agents to give metal alkyls. This reaction is one step in the Cativa process for the synthesis of acetic acid from methyl iodide. Many cross coupling reactions proceed via oxidative addition as well.
Electrophilic alkylating agentsEdit
Triethyloxonium tetrafluoroborate is one of the most electrophilic alkylating agents.[5]
Electrophilic alkylating agents deliver the equivalent of an alkyl cation. Alkyl halides are typical alkylating agents. Trimethyloxonium tetrafluoroborate and triethyloxonium tetrafluoroborate are particularly strong electrophiles due to their overt positive charge and an inert leaving group (dimethyl or diethyl ether). Dimethyl sulfate is intermediate in electrophilicity.
Methylation with diazomethaneEdit
Diazomethane is a popular methylating agent in the laboratory, but it is too hazardous (explosive gas with a high accute toxicity) to be employed on an industrial scale without special precautions.[6] Use of diazomethane has been significantly reduced by the introduction of the safer and equivalent reagent trimethylsilyldiazomethane.[7]
Electrophilic, soluble alkylating agents are often toxic and carcinogenic, due to their tendency to alkylate DNA. This mechanism of toxicity is relevant to the function of anti-cancer drugs in the form of alkylating antineoplastic agents. Some chemical weapons such as mustard gas (sulfide of dichloroethyl) function as alkylating agents. Alkylated DNA either does not coil or uncoil properly, or cannot be processed by information-decoding enzymes.
CatalystsEdit
Friedel-Crafts alkylation of benzene is often catalyzed by aluminium trichloride.
Electrophilic alkylations use Lewis acids and Brønsted acids, sometimes both. Classically, Lewis acids, e.g., aluminium trichloride, are employed when the alkyl halide are used. Brønsted acids are used when alkylating with olefins. Typical catalysts are zeolites, i.e. solid acid catalysts, and sulfuric acid. Silicotungstic acid is used to manufacture ethyl acetate by the alkylation of acetic acid by ethylene:[8]
C2H4 + CH3CO2H → CH3CO2C2H5
Methylation is the most common type of alkylation. Methylation in nature is often effected by vitamin B12- and radical-SAM-based enzymes.
The SN2-like methyl transfer reaction in DNA methylation. Only the SAM cofactor and cytosine base are shown for simplicity.
In methanogenesis, coenzyme M is methylated by tetrahydromethanopterin.
Commodity chemicalsEdit
Several commodity chemicals are produced by alkylation. Included are several fundamental benzene-based feedstocks such as ethylbenzene (precursor to styrene), cumene (precursor to phenol and acetone), linear alkylbenzene sulfonates (for detergents).[9]
Sodium dodecylbenzene, obtained by alkylation of benzene with dodecene, is a precursor to linear alkylbenzene sulfonate detergents.
Gasoline productionEdit
Typical acid-catalyzed route to 2,4-dimethylpentane.
In a conventional oil refinery, isobutane is alkylated with low-molecular-weight alkenes (primarily a mixture of propene and butene) in the presence of a Brønsted acid catalyst, which can include solid acids (zeolites). The catalyst protonates the alkenes (propene, butene) to produce carbocations, which alkylate isobutane. The product, called "alkylate", is composed of a mixture of high-octane, branched-chain paraffinic hydrocarbons (mostly isoheptane and isooctane). Alkylate is a premium gasoline blending stock because it has exceptional antiknock properties and is clean burning. Alkylate is also a key component of avgas. By combining fluid catalytic cracking, polymerization, and alkylation, refineries can obtain a gasoline yield of 70 percent. The widespread use of sulfuric acid and hydrofluoric acid in refineries poses significant environmental risks.[10] Ionic liquids are used in place of the older generation of strong Bronsted acids.[11][12]
DealkylationEdit
Complementing alkylation reactions are the reverse, dealkylations. Prevalent are ether delkylations.[13]
Category:Alkylating agents
Category:Ethylating agents
Category:Methylating agents
^ Stefanidakis, G.; Gwyn, J.E. (1993). "Alkylation". In John J. McKetta (ed.). Chemical Processing Handbook. CRC Press. pp. 80–138. ISBN 0-8247-8701-3.
^ D. Landini; F. Rolla (1978). "Sulfide Synthesis In Preparation Of Dialkyl And Alkyl Aryl Sulfides: Neopentyl Phenyl Sulfide". Org. Synth. 58: 143. doi:10.15227/orgsyn.058.0143.
^ G. S. Hiers and F. D. Hager (1941). "Anisole". Organic Syntheses. ; Collective Volume, vol. 1, p. 58
^ H. Perst; D. G. Seapy (2008). "Triethyloxonium Tetrafluoroborate". Encyclopedia of Reagents for Organic Synthesis. doi:10.1002/047084289X.rt223.pub2. ISBN 978-0471936237.
^ Proctor, Lee D.; Warr, Antony J. (November 2002). "Development of a continuous process for the industrial generation of diazomethane". Organic Process Research & Development. 6 (6): 884–892. doi:10.1021/op020049k.
^ Misono, Makoto (2009). "Recent progress in the practical applications of heteropolyacid and perovskite catalysts: Catalytic technology for the sustainable society". Catalysis Today. 144 (3–4): 285–291. doi:10.1016/j.cattod.2008.10.054.
^ Bipin V. Vora; Joseph A. Kocal; Paul T. Barger; Robert J. Schmidt; James A. Johnson (2003). "Alkylation". Kirk‐Othmer Encyclopedia of Chemical Technology. doi:10.1002/0471238961.0112112508011313.a01.pub2. ISBN 0471238961.
^ Kore, Rajkumar; Scurto, Aaron M.; Shiflett, Mark B. (2020). "Review of Isobutane Alkylation Technology Using Ionic Liquid-Based Catalysts—Where Do We Stand?". Industrial & Engineering Chemistry Research. 59 (36): 15811–15838. doi:10.1021/acs.iecr.0c03418. S2CID 225512999.
^ "Oil & Gas Engineering | Ionic liquid alkylation technology receives award". 2 January 2018.
^ Weissman, Steven A.; Zewge, Daniel (2005). "Recent advances in ether dealkylation". Tetrahedron. 61 (33): 7833–7863. doi:10.1016/j.tet.2005.05.041.
Alkylating+agents at the US National Library of Medicine Medical Subject Headings (MeSH)
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torch.lu — PyTorch 1.11.0 documentation
torch.lu
torch.lu¶
torch.lu(*args, **kwargs)¶
Computes the LU factorization of a matrix or batches of matrices A. Returns a tuple containing the LU factorization and pivots of A. Pivoting is done if pivot is set to True.
The returned permutation matrix for every matrix in the batch is represented by a 1-indexed vector of size min(A.shape[-2], A.shape[-1]). pivots[i] == j represents that in the i-th step of the algorithm, the i-th row was permuted with the j-1-th row.
LU factorization with pivot = False is not available for CPU, and attempting to do so will throw an error. However, LU factorization with pivot = False is available for CUDA.
This function does not check if the factorization was successful or not if get_infos is True since the status of the factorization is present in the third element of the return tuple.
In the case of batches of square matrices with size less or equal to 32 on a CUDA device, the LU factorization is repeated for singular matrices due to the bug in the MAGMA library (see magma issue 13).
L, U, and P can be derived using torch.lu_unpack().
The gradients of this function will only be finite when A is full rank. This is because the LU decomposition is just differentiable at full rank matrices. Furthermore, if A is close to not being full rank, the gradient will be numerically unstable as it depends on the computation of
L^{-1}
U^{-1}
A (Tensor) – the tensor to factor of size
(*, m, n)
pivot (bool, optional) – controls whether pivoting is done. Default: True
get_infos (bool, optional) – if set to True, returns an info IntTensor. Default: False
out (tuple, optional) – optional output tuple. If get_infos is True, then the elements in the tuple are Tensor, IntTensor, and IntTensor. If get_infos is False, then the elements in the tuple are Tensor, IntTensor. Default: None
A tuple of tensors containing
factorization (Tensor): the factorization of size
(*, m, n)
pivots (IntTensor): the pivots of size
(*, \text{min}(m, n))
. pivots stores all the intermediate transpositions of rows. The final permutation perm could be reconstructed by applying swap(perm[i], perm[pivots[i] - 1]) for i = 0, ..., pivots.size(-1) - 1, where perm is initially the identity permutation of
m
elements (essentially this is what torch.lu_unpack() is doing).
infos (IntTensor, optional): if get_infos is True, this is a tensor of size
(*)
where non-zero values indicate whether factorization for the matrix or each minibatch has succeeded or failed
(Tensor, IntTensor, IntTensor (optional))
>>> A_LU, pivots = torch.lu(A)
>>> A_LU
>>> pivots
>>> A_LU, pivots, info = torch.lu(A, get_infos=True)
>>> if info.nonzero().size(0) == 0:
... print('LU factorization succeeded for all samples!')
LU factorization succeeded for all samples!
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Calculating Allele Frequencies - Course Hero
Introduction to Biology/Microevolution/Calculating Allele Frequencies
Learn all about calculating allele frequencies in just a few minutes! Jessica Pamment, professional lecturer at DePaul University, explains how to calculate allele frequency and genotype frequency in a population.
Microevolution is a change in allele frequencies from one generation to the next.
Allele frequency is the relative proportion of different alleles for a given gene that within a population. It is a measure of the relative prevalence of a particular allele in a gene pool. A gene pool is the total set of genes of all individuals in a population. Microevolution occurs when there is a shift in the proportion of alleles in a gene pool. Specifically, microevolution is a change in allele frequencies within a population. Frequencies are numbers that run from zero to one (a proportion), while percentages run from 0% to 100%.
Genotype frequency is the proportion of individuals in a population that have a particular genotype, or genetic makeup. To calculate allele frequencies, it is often easiest to start with genotype frequencies. Because genotype frequencies are proportions, they always add up to 1. For example, cystic fibrosis is a human condition caused by having two copies of the cystic fibrosis allele (c) at a single locus. The unaffected allele in this example is C. Imagine a population of 10 individuals. Seven people are homozygous (having two identical alleles for a gene) for the unaffected allele. So they have the genotype CC. Two people are carriers of the cystic fibrosis allele. They are heterozygous, meaning they have one dominant allele and one recessive allele, Cc. And one person is homozygous for the cystic fibrosis allele. This person has the genotype cc. The genotype frequencies can then be calculated.
Genotype frequencies of cystic fibrosis in a population can be calculated. To do so, take the number of people with each genotype (CC, Cc, cc) and divide it by the total population. Multiple this number by 100 and that is the percentage of people with that genetic condition.
The allele frequencies are calculated slightly differently because there are two slots for possible alleles at each locus. Returning to the example of cystic fibrosis, the total number of slots is equal to twice the total number of individuals in the population (10 individuals × 2 = 20 slots for alleles). Homozygous individuals count twice and heterozygous individuals count once in these calculations. The allele frequencies can be calculated as follows:
\mathrm C\;=\;((2\times7)+(1\times2))/20=0.80\;\mathrm{or}\;80\%
\mathrm c\;=((2\times1)+(1\times2))/20=0.20\;\mathrm{or}\;20\%
As with genotype frequencies, allele frequencies must equal one or 100%.
<Origins of Genetic Variation>Is Microevolution Happening?
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A man sold his scooter for Rs 8000 and lost 20% for what amount he should have - Maths - Comparing Quantities - 7265155 | Meritnation.com
A man sold his scooter for Rs. 8000 and lost 20% . for what amount he should have sold it to gain 20%?
Suppose cost price of Scooter is Rs.x.
Then selling price =
x-x×\frac{20}{100} = x-\frac{x}{5} = \frac{4x}{5}
And the given selling price of scooter is Rs.8000. So we have;
\frac{4x}{5} = 8000\phantom{\rule{0ex}{0ex}}⇒x = \frac{8000×5}{4} = 10000
So the cost price of scooter is Rs.10000.
For a gain of 20%; Selling price =
10000+10000×\frac{20}{100} = 10000+2000 = 12000
Therefore selling price of scooter should be Rs.12000 to gain 20%.
sp of scooter=8000
cp=sp*100/100-L%
8000*100/80=10,000
if P is 205 then sp=
sp=cp(100+p)/100
10,000(120)/100=12,000
sp=12,000
Selling Price (SP) of scooter = Rs 8000
Cost Price (CP) of Scooter = ?
Find the cost price of scooter:
CP = 100/100 - Loss% X SP
CP = 100/100-20 X 8000
CP = 100/80 X 8000
CP = 800000/80
Hence the cost price of scooter is 10000 Rs.
Find the Selling price:
SP = 100 + Gain% /100 X CP
SP = 100 + 20 / 100 X 10000
SP = 120 / 100 X 10000
SP = 1200000/100
Hence the SP is Rs. 12000
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The pedal shaft on a standard bicycle is about
7
inches long, and the center of the bottom bracket to which the pedal shaft is attached is about
11
inches above the ground. Sketch a graph that shows the relationship of the angle of the shaft (for one pedal) in standard position (as if the point of attachment were the origin of a set of
x
y
axes shifted up
11
units) to the height of the pedal above the ground as a rider pedals the bike. Assume the pedal starts in its lowest position and takes
2
seconds to make one complete rotation.
Write an equation for a function that represents your graph.
Sketch the situation. Identify the amplitude, period, and any shifting.
Jack feels that the best position for the pedal to start riding is when the pedal is at
10
inches and heading downward. What is the first time the pedal will be in this position?
10
Use the eTool below to see a sample animation.
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Discussion: “Bayesian Optimal Design of Experiments for Inferring the Statistical Expectation of Expensive Black-Box Functions” (Pandita, P., Bilionis, I., and Panchal, J., 2019, ASME J. Mech. Des., 141(10), p. 101404) | J. Mech. Des. | ASME Digital Collection
2 Derivation of the Simplified Acquisition
G(x~)
3 Analytical Computation of G(x) for Arbitrary Input Distribution p(x)
Discussion: “Bayesian Optimal Design of Experiments for Inferring the Statistical Expectation of Expensive Black-Box Functions” (Pandita, P., Bilionis, I., and Panchal, J., 2019, ASME J. Mech. Des., 141(10), p. 101404)
Xianliang Gong,
Xianliang Gong
Email: xlgong@umich.edu
Email: yulinpan@umich.edu
This is a commentary to: Bayesian Optimal Design of Experiments for Inferring the Statistical Expectation of Expensive Black-Box Functions
Gong, X., and Pan, Y. (December 17, 2021). "Discussion: “Bayesian Optimal Design of Experiments for Inferring the Statistical Expectation of Expensive Black-Box Functions” (Pandita, P., Bilionis, I., and Panchal, J., 2019, ASME J. Mech. Des., 141(10), p. 101404)." ASME. J. Mech. Des. May 2022; 144(5): 055501. https://doi.org/10.1115/1.4053112
Computation, Experimental design
In Ref. [1], the authors developed a sequential Bayesian optimal design framework to estimate the statistical expectation of a black-box function
f(x):Rd→R
. Let x ∼ p(x) with p(x) the probability distribution of the input x, the statistical expectation is then defined as follows:
q=∫f(x)p(x)dx
The function f(x) is not known a priori but can be evaluated at arbitrary x with Gaussian noise of variance
σ2
y=f(x)+ε,ε∼N(0,σ2)
Based on the Gaussian process surrogate learned from the available samples Dn = {Xn, Yn}, i.e.,
f(x)|Dn∼GP(mn(x),kn(x,x′))
, the next-best sample is chosen by maximizing the information-based acquisition
G(x~)
xn+1=argmaxx~G(x~)
G(x~)
computes the information gain of adding a sample
y~
x~
, i.e., the expected KL divergence between the current estimation p(q|Dn) and the hypothetical next-step estimation
p(q|Dn,x~,y~)
G(x~)=Ey~[KL(p(q|Dn,x~,y~)|p(q|Dn))]=∫∫p(q|Dn,x~,y~)logp(q|Dn,x~,y~)p(q|Dn)dqp(y~|x~,Dn)dy~
y~
is chosen based on the surrogate f(x)|Dn following a distribution of
N(mn(x~),kn(x~,x~)+σ2)
); q is considered as a random variable with uncertainties coming from the (current and next-step) surrogates. It is noted that
G(x~)
also depends on the hyperparameter
θ
in the learned Gaussian process f(x)|Dn. We neglect this dependence for simplicity, which does not affect the main derivation.
As a major contribution of the discussed paper, the authors simplified the information-based acquisition as Eq. (30) in Ref. [1]:
G(x~)=log(σ1σ2(x~))+12σ22(x~)σ12−12+12v(x~)2σ12(σn2(x~)+σ2)
σ12
σ22
are, respectively, the variances of current estimation and hypothetical next-step estimation of q;
v(x~)=∫kn(x~,x)p(x)dx
σn2(x~)=kn(x~,x~)
. Furthermore, for numerical computation of Eq. (5), the authors developed analytical formulas for each involved quantity (important for high-dimensional computation) under uniform distribution of x.
The purpose of our discussion is to show the following two critical points:
The last three terms of Eq. (5) always add up to zero, leaving a concise form with a much more intuitive interpretation of the acquisition.
The analytical computation of Eq. (5) can be generalized to arbitrary input distribution of x, greatly broadening the application of the developed framework.
These two points are discussed, respectively, in Secs. 2 and 3.
G(x~)
To simplify Eq. (4), we first notice that q|Dn follows a Gaussian distribution with mean μ1 and variance
σ12
p(q|Dn)=N(q;μ1,σ12)μ1=E[∫f(x)p(x)dx|Dn]
=∫mn(x)p(x)dx
σ12=E[(∫f(x)p(x)dx)2|Dn]−(E[(∫fn(x)p(x)dx)|Dn])2=∫∫kn(x,x′)p(x)p(x′)dx′dx
After adding one hypothetical sample
{x~,y~}
, the function follows an updated surrogate
f(x)|Dn,x~,y~∼GP(mn+1(x),kn+1(x,x′))
mn+1(x)=mn(x)+kn(x~,x)(y~−mn(x~))kn(x~,x~)+σ2
kn+1(x,x′)=kn(x,x′)−kn(x~,x)kn(x′,x~)kn(x~,x~)+σ2
q|Dn,x~,y~
can then be represented by another Gaussian with mean μ2 and variance
σ22
p(q|Dn,x~,y~)=N(q;μ2(x~,y~),σ22(x~)),
μ2(x~,y~)=E[∫f(x)p(x)dx|Dn,x~,y~]=∫mn+1(x)p(x)dx=μ1+∫kn(x~,x)p(x)dxkn(x~,x~)+σ2(y~−mn(x~))
σ22(x~)=E[(∫f(x)p(x)dx)2|Dn,x~,y~]−(E[(∫fn(x)p(x)dx)|Dn,x~,y~])2=∫∫kn+1(x,x′)p(x)p(x′)dx′dx=σ12−(∫kn(x~,x)p(x)dx)2kn(x~,x~)+σ2
We note that Eqs. (7), (8), (12), and (13) are, respectively, intermediate steps of Eqs. (19), (21), (26), and (28) in the discussed paper. Substitute Eq. (6) and Eq. (11) into Eq. (4), one can obtain:
G(x~)=∫∫p(q|Dn,x~,y~)logp(q|Dn,x~,y~)p(q|Dn)dqp(y~|x~,Dn))dy~=∫(log(σ1σ2(x~))+σ22(x~)2σ12+(μ2(x~,y~)−μ1)22σ12−12)p(y~|x~,Dn)dy~=log(σ1σ2(x~))+12σ12(∫(μ2(x~,y~)−μ1)2p(y~|x~,Dn)dy~+σ22(x~)−σ12)=log(σ1σ2(x~))+12σ12((∫kn(x~,x)p(x)dx)2kn(x~,x~)+σ2+σ22(x~)−σ12)
=log(σ1σ2(x~))
where Eq. (14) is exactly Eq. (5) (or Eq. (30) in discussed paper). The fact that the last three terms of Eq. (14) sum up to zero is a direct result of Eq. (13).
The advantage of having a simplified form (15) is that the optimization (3) yields a much more intuitive physical interpretation. Since
σ1
x~
, Eq. (3) can be reformulated as
xn+1=argminx~σ22(x~)
which selects the next-best sample minimizing the expected variance of q. Similar optimization criterion is also used in Refs. [2,3] for the purpose of computing the extreme-event probability.
Another alternative interpretation can be obtained by writing (3) as
xn+1≡argmaxx~σ12−σ22(x~)≡argmaxx~(∫kn(x~,x)p(x)dx(kn(x~,x~)+σ2)1/2)2
≡argmaxx~(∫ρy|Dn(x~,x)(kn(x,x)+σ2)1/2p(x)dx)2,
where Eq. (17) is a result of Eq. (13), and
ρy|Dn(x~,x)=kn(x~,x)/((kn(x~,x~)+σ2)(kn(x,x)+σ2))1/2
is the correlation of y for two inputs
x~
and x. Equation (18) can be interpreted as to select the next sample which has overall most (weighted) correlation with all x.
We finally remark that the above derivation is for given hyperparamter values
θ
in f(x)|Dn. This is consistent with the Bayesian approach where the optimal values of
θ
are chosen from maximizing the likelihood function. However, the discussed paper used a different approach by sampling a distribution of
θ
and computed
G(x~)
as an average of the sampling. In the latter case, the above analysis should be likewise considered in a slightly different way, i.e., Eq. (16) should be considered as maximization of the multiplication of
σ22
from all samples of
{θ(i)}i=1s
xn+1=argmaxx~1s∑i=1sG(x~,θ(i))≡argmaxx~1s∑i=1slog(σ1(θ(i))σ2(x~,θ(i)))≡argminx~1s∑i=1slogσ2(x~,θ(i))≡argminx~∏i=1sσ22(x~,θ(i))
By using the Github code from the authors of Ref. [1] for their test cases, we have confirmed that the new results based on Eq. (15) are the same as the original results based on Eq. (5).
In the computation of G(x) in the form of Eq. (17), the most heavy computation involved is the integral
∫kn(x~,x)p(x)dx
(which is prohibitive in high-dimensional problem if direct integration is performed). Following the discussed paper, the integral can be reformulated as
∫kn(x~,x)p(x)dx=K(x~)−k(x~,Xn)(K(Xn,Xn)+σ2In)−1K(Xn)
K(x)=∫k(x,x′)p(x′)dx′
k(x,x′)=s2exp(−12(x−x′)TΛ−1(x−x′))
with s and Λ involving hyperparameters of the kernel function (with either optimized values from training or selected values as in Ref. [1]).
The main computation is then Eq.(21), for which the authors of the discussed paper addressed the situation of uniform p(x). To generalize the formulation to arbitrary p(x), we can approximate p(x) with the Gaussian mixture model (as an universal approximator of distributions [4]) :
p(x)≈∑i=1nGMMαiN(x;wi,Σi)
Equation (21) can then be formulated as:
K(x)≈∑i=1nGMMαi∫k(x,x′)N(x′;wi,Σi)dx′=∑i=1nGMMαi|ΣiΛ−1+I|−1/2k(x,wi;Σi+Λ)
which yields an analytical computation. In practice, the number of mixtures nGMM is determined by the complexity of the input distributions, but any distribution of p(x) can be approximated in such a way.
Finally, in computing
G(x~)
in the form of Eq. (19), computation of
σ12
in Eq. (8) is necessary. This can also be generalized for arbitrary p(x) using the Gaussian mixture model as follows:
σ12=∫∫kn(x,x′)p(x)p(x′)dx′dx=∫∫k(x,x′)p(x)p(x′)dx′dx−∫k(x,Xn)p(x)dx(K(Xn,Xn)+σ2In)−1∫k(Xn,x′)p(x′)dx′=∑i=1nGMM∑j=1nGMMαiαj|Λ|1/2|Λ+Σi+Σj|−1/2k(wi,wj;Λ+Σi+Σj)−K(Xn)T(K(Xn,Xn)+σ2In)−1K(Xn)
No data, models, or code were generated or used for this paper.
Output-weighted Optimal Sampling for Bayesian Experimental Design and Uncertainty Quantification
A Method for Computing the Analytical Solution of the Steady-State Heat Equation in Multilayered Media
Numerical Study of Steady Forced Convection in a Grooved Annulus Using a Design of Experiments
Topological Chaos by Pseudo-Anosov Map in Cavity Laminar Mixing
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Bikarhêner:Saiht - Wîkîpediya
{\displaystyle j^{\mu }=\sum _{i=1}^{N}{\frac {\delta {\mathcal {L}}}{\delta (\partial _{\mu }\phi _{i})}}\varphi _{i}\quad ,\quad \varphi _{i}:=\left.{\frac {\partial \phi _{i}}{\partial \alpha }}\right|_{\alpha =0}}
My only connection to the kurdian language is, that I took part in generating the website for the kurdian translation of the Holy Scriptures. The adress is http://pirtuka-piroz.info/
Şablon:Userpage
Ji "https://ku.wikipedia.org/w/index.php?title=Bikarhêner:Saiht&oldid=212916" hatiye standin.
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From Charles Lyell1 30 September 1860
– I expect that when the nondiversification of rodents, bats, manatees seals &c. on remote Miocene islands is fully worked out, it may merely end in satisfying you & me that the time required for change is longer than was supposed, or without such facts, demonstrable.
It will require renewed enquiry into the antiquity of such islands, also considerations as to the time before first bats & rodents arrived. also the force of preoccupancy—also the arrival of the same species of bats & rodents again & again acting like European colonists into U.S., checking the formation of a new race.
Have you ever speculated in print on turtles changing to land tortoises in remote islands like Galapagos?
From Falunian (Upper Miocene) to Recent is only a part, tho‘ a large part (
\frac{2}{3}
ds) of one geoll. period measured by change of mollusca.
If as a rule a few species even of large genera vary then if bats & rodents migrate into an island the chances may be a thousand to one that these species are among the unpliant, unvarying ones, & if as you say we know nothing of the law which governs the exceptional cases, we cannot conjecture whether insular conditions would favour such divergence if when the thousandth chance did turn up in some island & an insular species was ready to sport & improve & wd. have done so on land far from sea. Perhaps the chief use to make of my difficulty is this, & it accords with my old notion that species as a rule, or the majority of them, are immutable, & have always been so.
Asa Gray is afraid of argument from imperfection of record.
Falconer observes that the Apteryx has strong clavicles & other bird-like characters without wings.2 I said, the more useless some of these parts not yet suppressed, the more likely to come down from an ancestor who had need & use of them or to be nascent organs advancing towards perfection, for a posterity which wd. enjoy wings. He only laughed as if the whole was a joke, yet I got him to admit that the hypothesis of limited modifiability was quite as arbitrary as yours.
By the way that reminds me that the keel in the middle of the breast-bone is wholly wanting in some (or all?) of the Dinornis family, tho’ I think a little remains in the Apteryx & I am almost sure that H. v. Meyer has lately found this keel in the Pterodactyl.3
The long duration & numbers of the Ammonities from trias to chalk & then their extinction, tho‘ high in the scale, is certainly striking. I presume from the myriads of sepia bones which sometimes strew the Jutland coast that naked cuttle-fish now play their part & are higher in the scale, though not higher than belemnites who accompanied ammonites. The vast number & size of Hippurites & their sudden appearance & disappearance, the only extinct order of Mollusca is also very striking. It would not make much impression here as we have few, but in S. of France, Italy, Sicily & all round Mediterranean.
What you say as to my difficulty is I suspect an explanation up to a certain point, but I hope to put the whole more clearly soon. Dogs of multiple origin & leporines would greatly weaken the objection to regarding the negro as of a different species, in the same sense that the Prairie wolf & common wolf may be. I should like a good naturalist to give me a list of reputed species in Mammalia not more remote than negro & white man. Would they not be many?
Instead of Selection I shd have said, Variation & Nat. Selection. My only objection is not to the term, but to your assigning to it more work than it can do & the not carefully guarding against confounding it with the creative power to which “variation” & something far higher than mere variation viz. the capacity of ascending in the scale of being, must belong. Most likely you would have chosen some term less worthy of Deification, for ’selection‘ you had an excellent technical reason.
The text of the letter has been taken from a copy in Lyell’s scientific journal. It is also published in Wilson ed. 1970, pp. 496–8.
Hugh Falconer. The Apteryx (Kiwi bird) of New Zealand has rudimentary wings only and is flightless.
Christian Erich Hermann von Meyer was a prominent German palaeontologist. His work on the Pterodactyl, a winged fossil reptile, was published in Meyer 1861 and 1862. Lyell had visited Meyer on several occasions and in 1857 learned about his study of Pterodactyl (K. M. Lyell ed. 1881, 2: 242–3).
Expects lack of diversification of immigrant mammals on long isolated islands will come to show slowness of selective change.
Asks whether CD has speculated on turtles becoming terrestrial on remote islands.
Perhaps non-diversification on islands is explained by tiny proportion of variable species. Those that vary on continent may not do so on island.
A. Gray is afraid of objections to Origin from imperfection of fossil record.
His argument with Falconer over the hypothesis of limited modifiability.
Are the bird-like characters of the Apteryx parts not yet suppressed or nascent organs?
Extinctions of ammonites, belemnites, and hippurites are striking. Perhaps ammonites made way for higher cuttle-fish.
Believes hybrid origin of domestic dog would weaken objections to treating white man and negro as species. Are there not many reputed species among the Mammalia more closely related than these races?
Objects not to the term "selection" but to what CD assigns to it. It should not be confused with the "Creative power" behind variation and the "capacity of ascending in the scale of being".
Kinnordy MS, Charles Lyell’s journal VII, pp. 13–19
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Erratum to: Optimality and duality theorems in nonsmooth multiobjective optimization | Fixed Point Theory and Algorithms for Sciences and Engineering | Full Text
Erratum to: Optimality and duality theorems in nonsmooth multiobjective optimization
Kwan Deok Bae1 &
We wish to indicate the following corrections to our original paper [1].
(1) The first sentence in Definition 2.2, we delete "i∈{1,2,···,p}".
(2) The first sentence in Definition 2.2, we replace
{f}_{i}\left(x\right)+s\left(x|{D}_{i}\right)\nless {f}_{i}\left({x}^{0}\right)+s\left({x}^{0}|{D}_{i}\right)
f\left(x\right)+s\left(x|D\right)\nless f\left({x}^{0}\right)+s\left({x}^{0}|D\right).
(3) The second sentence in Definition 2.3, we delete "i∈{1,2,···,p}".
(4) The second sentence in Definition 2.3, we replace
{f}_{i}\left(x\right)+s\left(x|{D}_{i}\right)\nless {f}_{i}\left({x}^{0}\right)+s\left({x}^{0}|{D}_{i}\right)+{c}_{i}||x-{x}^{0}|{|}^{m}
f\left(x\right)+s\left(x|D\right)\nless f\left({x}^{0}\right)+s\left({x}^{0}|D\right)+c||x-{x}^{0}|{|}^{m}.
\prime \prime {f}_{i}\left(x\right)+s\left(x|{D}_{i}\right)\nless {f}_{i}\left({x}^{0}\right)+s\left({x}^{0}|{D}_{i}\right)+{c}_{i}||x-{x}^{0}|{|}^{m}
f\left(x\right)+s\left(x|D\right)\nless f\left({x}^{0}\right)+s\left({x}^{0}|D\right)+c||x-{x}^{0}|{|}^{m}.\prime \prime .
(7) The second sentence in the proof of Theorem 2.1, we replace. ′′ c i > 0, i = 1, ⋯,p ′′ to ′′
c\in int{R}_{+}^{p}
′′.
(8) The second sentence in the proof of Theorem 2.1, we replace
{f}_{i}\left(x\right)+s\left(x|{D}_{i}\right)\nless {f}_{i}\left({x}^{0}\right)+s\left({x}^{0}|{D}_{i}\right)+{c}_{i}||x-{x}^{0}|{|}^{m}
f\left(x\right)+s\left(x|D\right)\nless f\left({x}^{0}\right)+s\left({x}^{0}|D\right)+c||x-{x}^{0}|{|}^{m}.
(9) In equation (3.8), we replace "c i " to "d i ".
(10) The eighth sentence in the proof of Theorem 3.3, we replace "where c = ae," to "where d = ae".
(11) The ninth sentence in the proof of Theorem 3.3, we replace "c ∈ int Rp" to "d i > 0, i = 1,···,p,".
(12) The ninth sentence in the proof of Theorem 3.3, we replace "c i " to "d i ".
(13) The tenth sentence in the proof of Theorem 3.3, we replace "c i " to "d i ".
(15) In equation (4.8), we replace "c i " to "d i ".
(16) The tenth sentence in the proof of Theorem 4.1, we replace "where c = ae," to "where d = ae".
(17) The eleventh sentence in the proof of Theorem 4.1, we replace "c ∈ int Rp" to "d i > 0, i = 1,···, p,".
(18) The eleventh sentence in the proof of Theorem 4.1, we replace "c i " to "d i ".
(19) The twelfth sentence in the proof of Theorem 4.1, we replace "c ∈ int Rp," to "d i > 0, i = 1,···, p,".
(20) The twelfth sentence in the proof of Theorem 4.1, we replace "c i " to "d i ".
(21) The twelfth sentence in the proof of Theorem 4.1, we replace "i = 1,···, p." to "i = 1,···,p,".
(22) The fourth sentence in the proof of Theorem 4.2, we replace "c i " to "c".
(23) The fourth sentence in the proof of Theorem 4.2, we replace
\begin{array}{c}{f}_{i}\left({x}^{0}\right)+{\left({x}^{0}\right)}^{T}{w}_{i}^{0}+{c}_{i}||u-{x}^{0}||{}^{m}\\ \nless {f}_{i}\left(u\right)+{u}^{T}{w}_{i},\phantom{\rule{0.3em}{0ex}}i=1,\cdot \cdot \cdot ,p.\end{array}
\begin{array}{c}f\left({x}^{0}\right)+{\left({x}^{0}\right)}^{T}{w}^{0}+c||u-{x}^{0}||{}^{m}\\ \nless f\left(u\right)+{u}^{T}w.\hfill \end{array}
Bae , Kim : Optimality and Duality Theorems in Nonsmooth Multiobjective Optimization. Fixed Point Theory and Applications 2011, 2011: 42. 10.1186/1687-1812-2011-42
Kwan Deok Bae & Do Sang Kim
Kwan Deok Bae
Deok Bae, K., Kim, D.S. Erratum to: Optimality and duality theorems in nonsmooth multiobjective optimization. Fixed Point Theory Appl 2012, 28 (2012). https://doi.org/10.1186/1687-1812-2012-28
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ଉଇକିପିଡ଼ିଆ:Help desk - ଉଇକିପିଡ଼ିଆ
ଉଇକିପିଡ଼ିଆ:Help desk
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୧.୧ I'd like to correct my user name after registering for the first time today.
୧.୨ putting images in an article
୨.୨ Changing the rendering size of a TeX formula
September 3[ସମ୍ପାଦନା]
I'd like to correct my user name after registering for the first time today.[ସମ୍ପାଦନା]
I entered my username in all lower case today, but the system (inappropriately) capitalized the first character, giving an unintended emphasis to my first name vs. my last. How can I edit my username? davidcassirer (talk) 15:40, 3 September 2011 (UTC)
The system will always make the first letter upper case. For requesting a name change please visit: WP:CHU/S. Jarkeld (talk) 15:44, 3 September 2011 (UTC)
putting images in an article[ସମ୍ପାଦନା]
I now have a draft article, but need to include .jpg images. How do I upload images and put them in my article?TankhouseTom (talk) 16:49, 3 September 2011 (UTC)
If you want to upload an image from your computer for use in an article, you must determine the proper license of the image (or whether it is in the public domain). If you know the image is public domain or copyrighted but under a suitable free-license, upload it to the Wikimedia Commons instead of here, so that all projects have access to the image (sign up). If you are unsure of the licensing status, see the file upload wizard for more information. Please also read Wikipedia's image use policy.
If you want to add an image that has already been uploaded to Wikipedia or Wikimedia Commons, add [[File:File name.jpg|thumb|Caption text.]] to the area of the article where you want the image to appear – replacing File name.jpg with the actual file name of the image, and Caption text with a short description of the image. See our picture tutorial for more information. I hope this helps. -- John of Reading (talk) 19:02, 3 September 2011 (UTC)
third opinion[ସମ୍ପାଦନା]
How do I go about getting a third opinion in a dispute with another editor?
Sardaka (talk) 08:54, 5 September 2011 (UTC)
See en:WP:THIRD. Dismas|(talk) 08:56, 5 September 2011 (UTC)
Changing the rendering size of a TeX formula[ସମ୍ପାଦନା]
Is it possible to change the rendering size of a TeX formula? In particular, I want to make the letters and numerals in
{\displaystyle {{2^{42737}+1} \over 3}}
the same size as the rest of text in the article. Is this possible and if so how can I do this? Toshio Yamaguchi (talk) 12:17, 5 September 2011 (UTC)
"https://or.wikipedia.org/w/index.php?title=ଉଇକିପିଡ଼ିଆ:Help_desk&oldid=67707"ରୁ ଅଣାଯାଇଅଛି
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Find Asymptotes, Critical, and Inflection Points - MATLAB & Simulink - MathWorks Switzerland
\mathit{f}\left(\mathit{x}\right)=\frac{3{\mathit{x}}^{2}+6\mathit{x}-1}{{\mathit{x}}^{2}+\mathit{x}-3}.
\frac{3 {x}^{2}+6 x-1}{{x}^{2}+x-3}
To find the horizontal asymptote of
mathematically, take the limit of
x
3
x
y=3
f
To find the vertical asymptotes of
\left(\begin{array}{c}-\frac{\sqrt{13}}{2}-\frac{1}{2}\\ \frac{\sqrt{13}}{2}-\frac{1}{2}\end{array}\right)
\mathit{x}=\frac{-1-\sqrt{13}}{2}
\mathit{x}=\frac{-1+\sqrt{13}}{2}
f
x=–2
x=0
x=–6
x=–2
x
-coordinates of the maximum and minimum, first take the derivative of
\frac{6 x+6}{{x}^{2}+x-3}-\frac{\left(2 x+1\right) \left(3 {x}^{2}+6 x-1\right)}{{\left({x}^{2}+x-3\right)}^{2}}
-\frac{3 {x}^{2}+16 x+17}{{\left({x}^{2}+x-3\right)}^{2}}
\left(\begin{array}{c}-\frac{\sqrt{13}}{3}-\frac{8}{3}\\ \frac{\sqrt{13}}{3}-\frac{8}{3}\end{array}\right)
As the graph of
{\mathit{x}}_{1}=\frac{-8-\sqrt{13}}{3}
{\mathit{x}}_{1}=\frac{-8+\sqrt{13}}{3}
To find the inflection point of
-\frac{13}{9 {\left(\frac{169}{54}-\frac{\sqrt{2197}}{18}\right)}^{1/3}}-{\left(\frac{169}{54}-\frac{\sqrt{2197}}{18}\right)}^{1/3}-\frac{8}{3}
x
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Leverage Your Aave Market Tokens - Mai Finance - Tutorials
This tutorial presents detailed steps by which users can leverage their current crypto investments through a clever combination of Mai Finance’s 0% interest loans and the Aave protocol.
But a thousand words are also not that bad
The picture above demonstrates how you can utilize Mai Finance to increase the earning power of your crypto investments.
Let's assume that you really like MATIC, and think that it's currently undervalued. You think it has the potential to reach $2, $5 or even $10 per token (and you may actually be totally right). However, as a small investor, you only have $100 worth of MATIC token in your wallet on Polygon. Through this tutorial we will show how you can generate more MATIC from your current tokens, because hodling is good, but putting your investment to work is better.
Use Aave to increase your capital
Aave is a lending and borrowing platform where you can deposit your MATIC tokens (among other tokens). By lending on Aave, your deposited tokens will earn yield. As an example, your $100 of MATIC will potentially generate a 1.2% rate of return over the span of 1 year (APY). Sometimes, Aave will also have specific programs that provide additional rewards on top of the base lending APYs.
As your MATIC tokens are in the Aave pool, the interest generated is automatically compounded, which means that the amount of MATIC you hold will grow over time.
On this example, I lended 0.2 MATIC
By lending your MATIC token on Aave, you receive an equivalent amount of amWMATIC (aave market wrapped MATIC). You may not see these tokens directly in your wallet unless you manually add them, but you do own them.
I can see the 0.20 MATIC I lended on DeBank
Use Mai Finance to compound your Aave tokens
Mai Finance will accept your amWMATIC tokens on the Yield page of the website. By depositing your amWMATIC on Mai Finance, your funds will be "transferred" from Aave to Mai Finance. You will see that the yield generated by Mai Finance is the same as the yield you would receive on Aave.
However, in addition to the base APY, Mai Finance will also compound any additional Aave rewards that are currently available back into the token of your choice, thus passively generating more of the chosen token over time. In our example (see above), Aave provides 1.16% APY for the deposited MATIC, as well as an additional 3.69% APR paid in MATIC, but this APR reward is not generating any yield. By depositing your amWMATIC on Mai Finance, the reward is collected periodically, and put back into your main deposit in order to apply the 1.16% interest rate on it too.
My 0.2MATIC are now deposited on Mai Finance and will generate 4.93% annually
When amWMATIC is deposited on Mai Finance, camWMATIC is received in exchange. The ratio of these two tokens isn't perfectly 1:1, because camMATIC actually represents a share of the amWMATIC pool on Mai, in which interest and rewards are automatically compounded. It should also be noted that when you deposit amWMATIC on Mai Finance, these tokens are removed from Aave. However, if you withdraw your amWMATIC from Mai Finance, you will find them back on Aave.
Just depositing your amWMATIC (or any amToken) on Mai Finance will allow you to generate more revenue than if you just lend your money on Aave. Indeed, the fact that your base interest and your additional reward are auto-compounded means you don't have to manually claim your reward, convert it in the token or your choice, and deposit it again.
amTokens VS camToken
A lot of people have a hard time understanding the difference between amTokens and camTokens. The amToken is a representation of your deposited funds on AAVE. Their amount is pegged to the amount you deposited on AAVE. However, camToken represent a share of the amToken pool on Mai Finance.
Let's assume that when the amToken pool was created on Mai Finance, there was 1,000 amTokens and you deposited 100 of them. Because the pool is just created, the ratio between amTokens and camTokens is 1:1, and you own 10% of the pool. After one year, assuming nobody added any more amTokens nor removed anything, the pool generated 4.93% interest, and now there are 1049.3 amTokens in the pool. However, you still only own 10% of the pool, represented by your 100 camTokens. The ratio is now 1:1.0493, which means that 1 camToken now worth 1.0493 amToken.
Borrow MAI stable coin
Mai Finance allows you to borrow the MAI stablecoin when you deposit collateral. Currently, Mai Finance accepts a broad range of collaterals including camWMATIC from our ongoing example. On Mai Finance, your cam-token collaterals will continue generating yield while deposited as collateral, which means that the amount of your underlying asset will continue to increase over time.
When camWMATIC is deposited into a Vault, the balance on the Yield page will be 0. However, that doesn't mean that it's not compounding your AAVE interests and rewards.
When you navigate to the Vaults page and select camWMATIC from the drop-down vault menu, you will be given the option to create a new vault where you can then deposit your camWMATIC. Keep in mind that you need to keep a Collateral to Debt Ratio (CDR) of at least 155% when you borrow against your camWMATIC.
My 0.2 MATIC are now fully used as collateral
Note: On this page, you will be able to see your collateral value in USD, and the value is fluctuating based on your collateral type, the token value, and the benefits generated in the camWMATIC pool.
Tip: When I deposit collateral to borrow MAI, I always borrow 50% of the value of my collateral. Ideally, I want to stay above a 200% CDR, and if my collateral value is growing (the token doesn't lose value, and the interest is adding up) I think that it's safe enough. Also, if I'm adding more collateral to a vault, I don't try to match a 200% CDR when I borrow, unless the CDR is bellow 200%. I always borrow up to 50% see the example with numbers below).
You should visit the Vault page from time to time to verify that you're always above the liquidation ratio, and possibly add more collateral if you start falling bellow a "safe ratio". Depending on your risk profile, this safe threshold may vary.
I now have a debt of $0.10
I borrowed $0.10 of MAI, which translates to a 214.56% CDR. Now let's have some fun with my money.
Buy more MATIC
I can now safely go to my favorite DEX (QuickSwap or SushiSwap are great examples) and swap my MAI for more MATIC.
Let's buy more MATIC
After the swap, I have more MATIC in my wallet.
I initially had 0.20 MATIC in my wallet. This MATIC is now deposited on Mai Finance, generating 4.93% interest annually, and I have an additional 0.09 MATIC at my disposal from the MAI I borrowed. This additional 0.09 MATIC can be deposited on Aave, just like my initial MATIC, and we start the same process over again.
When should you exit the loop
In terms of steps, the best approach is to stop when you deposit you camWMATIC in the vault. Doing so increases the collateral to debt ratio, which lowers your risk of being liquidated. Depending on the last amount you deposit, this may be negligible though. As for the number of iterations, I generally stop when I am within 1% of my initial investment. Here, I would continue the loop until I deposit 0.002 camWMATIC. I may also stop before if the gas price paid for the transactions becomes more important than what I can gain by continuing the loop.
The following examples are based on $1,000 worth of MATIC, with different CDR ratios to illustrate how to leverage your positions using Mai Finance.
200% Collateral to Debt Ratio
Adding more debt past Loop 7 would only increase my investment by less than $10 (1% of my initial investment), so this is the appropriate time to stop. The increase in APY is negligible at this stage, and I keep a collateral to debt ratio of 200.79%, which is safe enough.
As you can easily see, using a combination of Aave and Mai Finance results in almost 2x the initial APY and significantly more market exposure to the token of choice, when compared to simply holding or using Aave in isolation.
Adding more debt past Loop 9 will not increase my investment by more than $10, so this is an appropriate place stop. The resulting CDR from 9 loops is 175.49%.
We can easily see that with a more aggressive approach the final APY is also more attractive. This is typically true with any DeFi strategy: the greater the risk, the greater the potential reward.
If you are mathematically inclined, then you can calculate your final investment based on the initial investment and the collateral to debt ratio you wish to target. The formula is as follows:
Final Investment = Initial Investment *\sum_{i=0}^{n}{\frac{100}{CDR}}^i
Where n represents the number of loops you want to apply and CDR is your targeted collateral to debt ratio in percentage.
In the example of a 200% targeted CDR with 7 loops, the calculation is as follows:
Final Investment = 1000 *\sum_{i=0}^{7}{\frac{100}{200}}^i
Final Investment = 1000 * (\frac{1}{2}^0 + \frac{1}{2}^1 + \frac{1}{2}^2 + \frac{1}{2}^3 + \frac{1}{2}^4 + \frac{1}{2}^5 + \frac{1}{2}^6 + \frac{1}{2}^7)
Final Investment = 1000 * (1 + 0.5 + 0.25 + 0.125 + 0.0625 + 0.03125 + 0.015625 + 0.0078125)
FinalInvestment = 1000 * 1.9921875 = 1992.1875
Using the same method, you can also calculate your final APY with the equation below:
Equivalent APY = Initial APY * \sum_{i=0}^{n}{\frac{100}{CDR}}^i
Once again, with our targeted CDR of 200%, an initial APY of 4.93% and 7 loops, we calculate the same final APY as shown above the tables.
Equivalent APY = 4.93 * 1.9921875 = 9.821484375
Everything presented in this strategy assumes the following:
The Aave APY is stable and remains the same over the span of 1 year (APY will have some variance)
The Polygon APR granted on Aave is stable and remains the same over the span of 1 year (which is unlikely)
The MATIC token keeps a relatively stable price over the span of 1 year (also unlikely)
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MCQs of Linear Programming | GUJCET MCQ
MCQs of Linear Programming
Objective function of an LP problem is
(c) an inequality
(d) a quadratic equation
(a) given by intersection of lines representing inequations with axes only
(b) given by intersection of lines representing inequations with X-axis only
(d) at the origin
(a) Every LP problem has at least one optimal solution.
(b) Every LP problem has a unique solution.
(c) If an LP problem has two optimal solutions, then it has infinitely many solutions.
(d) If a feasible region is unbounded then LP problem has no solution
A feasible solution to an LP problem,
For the LP problem Minimize z = 2x + 3y the coordinates of the corner points of the bounded feasible region are A(3, 3), B(20,3), C(20, 10), D(18, 12) and E(12, 12). The minimum value of z is
For the LP problem maximize z = 2x + 3y The coordinates of the corner points of the bounded feasible region are A(3, 3), B(20,3), C(20, 10), D(18, 12) and E(12, 12). The minimum value of z is
Corner points of the bounded feasible region for an LP problem are (0, 4), (6, 0), (12, 0), (12, 16) and (0, 10). Let z=8x + 12y be the objective function.Match the following: (i) Minimum value of z occurs at _____ (ii) Maximum value of z occurs at _____ (iii) Maximum of z is _____ (iv) Minimum of z is _____
(a) (i) (6, 0) (ii) (12, 0) (iii) 288 (iv) 48
(b) (i) (0, 4) (ii) (12, 16) (iii) 288 (iv) 48
(c) (i) (0, 4) (ii) (12, 16) (iii) 288 (iv) 96
(d) (i) (6, 0) (ii) (12, 0) (iii) 288 (iv) 96
The corners points of the feasible region determined by the system of linear constraints are (0, 10), (5, 5) (15, 15), (0, 20). Let z= px + qy, where p, q > 0.Condition on p and q so that the maximum of z occurs at both the points (15, 15) and (0, 20) is _____
(b) p =2q
\mathrm{Solution} \mathrm{of} \mathrm{following} \mathrm{LP} \mathrm{problem} \mathrm{Maximize} \mathrm{z} = 2\mathrm{x} + 6\mathrm{y} \mathrm{subject} \mathrm{to} -\mathrm{x}+ \mathrm{y} \le 1, 2\mathrm{x} + \mathrm{y} \le 2 \mathrm{and} \mathrm{x} \ge 0, \mathrm{y} \ge 0 \mathrm{is}
\frac{4}{3}
\frac{1}{3}
\frac{26}{3}
(d) no feasible region
\mathrm{Solution} \mathrm{of} \mathrm{following} \mathrm{LP} \mathrm{problem} \mathrm{Minimize} \mathrm{z} =-3\mathrm{x} + 2\mathrm{y} \mathrm{subject} \mathrm{to} 0 \le x \le 4, 1 \le y \le 6, x + y \le 5 \mathrm{is}
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Jing Ya takes the bus across town to school each morning. Last week, he timed his trips and found that the time varies day to day. The times (in minutes) are listed below.
15, 10, 11, 13, 11
If you had to use one number to tell someone how long it took Jing Ya to get to school, what would you say?
Try arranging the times in order and choosing one which is in the middle or appears most often.
11
12
minutes. Sometimes it takes Jing Ya longer to get to school, and sometimes shorter.
However, a typical trip to school will take
11
12
Jing Ya does not want to be late. If he needs to get to school by 8:00 a.m. each day, what is the latest time he should get on the bus (assuming one is waiting for him at any given time)? Explain how you got your answer.
It may help to look at the longest amount of time it took Jing Ya to get to school.
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Parametric Fitting - MATLAB & Simulink - MathWorks Australia
Parametric Fitting with Library Models
Select Model Type Interactively
Select Model Type Programmatically
Center and Scale Data
Fit Options in Curve Fitter App
Optimized Starting Points and Default Constraints
Specify Fit Options at the Command Line
Parametric fitting involves finding coefficients (parameters) for one or more models that you fit to data. The data is assumed to be statistical in nature and is divided into two components:
data = deterministic component + random component
The deterministic component is given by a parametric model and the random component is often described as error associated with the data:
data = parametric model + error
The model is a function of the independent (predictor) variable and one or more coefficients. The error represents random variations in the data that follow a specific probability distribution (usually Gaussian). The variations can come from many different sources, but are always present at some level when you are dealing with measured data. Systematic variations can also exist, but they can lead to a fitted model that does not represent the data well.
The model coefficients often have physical significance. For example, suppose you collected data that corresponds to a single decay mode of a radioactive nuclide, and you want to estimate the half-life (T1/2) of the decay. The law of radioactive decay states that the activity of a radioactive substance decays exponentially in time. Therefore, the model to use in the fit is given by
y={y}_{0}{e}^{-\lambda t}
where y0 is the number of nuclei at time t = 0, and λ is the decay constant. The data can be described by
\text{data}={y}_{0}{e}^{-\lambda t}+\text{error}
Both y0 and λ are coefficients that are estimated by the fit. Because T1/2 = ln(2)/λ, the fitted value of the decay constant yields the fitted half-life. However, because the data contains some error, the deterministic component of the equation cannot be determined exactly from the data. Therefore, the coefficients and half-life calculation will have some uncertainty associated with them. If the uncertainty is acceptable, then you are done fitting the data. If the uncertainty is not acceptable, then you might have to take steps to reduce it either by collecting more data or by reducing measurement error and collecting new data and repeating the model fit.
With other problems where there is no theory to dictate a model, you might also modify the model by adding or removing terms, or substitute an entirely different model.
The Curve Fitting Toolbox™ parametric library models are described in the following sections.
In the Curve Fitter app, go to the Fit Type section of the Curve Fitter tab. You can select a model type from the fit gallery. Click the arrow to open the gallery.
This table describes the models that you can fit for curves and surfaces.
Custom Linear Fitting Yes No
The Results pane displays the model specifications, coefficient values, and goodness-of-fit statistics.
If your fit has problems, messages in the Results pane help you identify better settings.
The Curve Fitter app provides a selection of fit types and settings in the Fit Options pane that you can change to try to improve your fit. Try the defaults first, and then experiment with other settings. For more details on how to use the available fit options, see Specify Fit Options and Optimized Starting Points.
You can try a variety of settings for a single fit and you can create multiple fits to compare. When you create multiple fits in the Curve Fitter app, you can compare different fit types and settings side by side. For more information, see Create Multiple Fits in Curve Fitter App.
You can specify a library model name as a character vector or string scalar when you call the fit function. For example, you can specify a quadratic poly2 model:
f = fit(x,y,"poly2")
To view all available library model names, see List of Library Models for Curve and Surface Fitting to view all available library model names.
You can also use the fittype function to construct a fittype object for a library model, and use the fittype as an input to the fit function.
Use the fitoptions function to find out what parameters you can set, for example:
For examples, see the sections for each model type, listed in the table in Select Model Type Interactively. For details on all the functions for creating and analysing models, see Curve and Surface Fitting.
Most fits in the Curve Fitter app provide the Center and scale option in the Fit Options pane. When you select this option, the app refits the model with the data centered and scaled. At the command line, use the fitoptions function with the Normalize option set to 'on'.
To alleviate numerical problems with variables of different scales, normalize the input data (also known as predictor data). For example, suppose your surface fit inputs are engine speed with a range of 500–4500 r/min and engine load percentage with a range of 0–1. Then, Center and scale generally improves the fit because of the great difference in scale between the two inputs. However, if your inputs are in the same units or similar scale (for example, eastings and northings for geographic data), then Center and scale is less useful. When you normalize inputs with this option, the values of the fitted coefficients change when compared to the original data.
If you are fitting a curve or surface to estimate coefficients, or the coefficients have physical significance, clear the Center and scale check box. The plots in the Curve Fitter app always use the original scale, regardless of the Center and scale status.
At the command line, to center and scale the data before fitting, create the options structure by using the fitoptions function with options.Normal specified as 'on'. Then, use the fit function with the specified options.
f1 = fit(cdate,pop,"poly3",options)
In the Curve Fitter app, you can specify fit options interactively in the Fit Options pane. All fits except Interpolant and Smoothing Spline have configurable fit options. The available options depend on the fit you select (that is, linear, nonlinear, or nonparametric fit).
The options described here are available for nonlinear models.
Lower and Upper coefficient constraints are the only fit options available in the Fit Options pane for Polynomial fits.
Nonparametric fits (that is, Interpolant, Smoothing Spline, and Lowess fits) do not have Advanced Options.
The Fit Options pane for the single-term Exponential fit is shown here. The Coefficient Constraints values are for the census data.
Fitting Method and Algorithm
Method — Fit method
The app automatically selects the Method value based on the fit you use. For linear and nonlinear fits, the Method value is LinearLeastSquares and NonlinearLeastSquares, respectively.
Robust — Option for using the robust least-squares fitting method
Off — Do not use robust fitting (default).
On — Fit with the default robust method (bisquare weights).
LAR — Fit by minimizing the least absolute residuals (LAR).
Bisquare — Fit by minimizing the summed square of the residuals, and reduce the weight of outliers using bisquare weights. In most cases, this option is the best choice for robust fitting.
Algorithm — Algorithm used for fitting
Trust-Region — This option is the default algorithm and must be used if you specify Lower or Upper coefficient constraints.
Levenberg-Marquardt — If the trust-region algorithm does not produce a reasonable fit, and you do not have coefficient constraints, try the Levenberg-Marquardt algorithm.
Finite Differencing Parameters
DiffMinChange — Minimum change in coefficients for finite difference Jacobians. The default value is 10-8.
DiffMaxChange — Maximum change in coefficients for finite difference Jacobians. The default value is 0.1.
Note that DiffMinChange and DiffMaxChange apply to:
Any nonlinear custom equation, that is, a nonlinear equation that you write
Some of the nonlinear equations provided with Curve Fitting Toolbox
DiffMinChange and DiffMaxChange do not apply to any linear equations.
Fit Convergence Criteria
MaxFunEvals — Maximum number of function (model) evaluations allowed. The default value is 600.
MaxIter — Maximum number of fit iterations allowed. The default value is 400.
TolFun — Termination tolerance used on stopping conditions involving the function (model) value. The default value is 10-6.
TolX — Termination tolerance used on stopping conditions involving the coefficients. The default value is 10-6.
Coefficient Parameters
Coefficients — Symbols for the unknown coefficients to be fitted
StartPoint — The coefficient starting values. The default values depend on the fit. For Rational, Weibull, and custom fits, the default values are randomly selected within the range [0 1]. For all other nonlinear models in the Fit Type gallery, the starting values depend on the data set and are calculated heuristically.
Lower — Lower bounds of the fitted coefficients. The app uses these bounds only with the trust region fitting algorithm. The default lower bounds for most fits in the Fit Options pane are -Inf, which indicates that the coefficients are unconstrained. However, a few models have finite default lower bounds. For example, for Gaussian fits, the app constrains the width parameter so that it cannot be less than 0.
Upper — Upper bounds of the fitted coefficients. The app uses these bounds only with the trust region fitting algorithm. The default upper bounds for all fits in the Fit Options pane are Inf, which indicates that the coefficients are unconstrained.
For more details, see Optimized Starting Points and Default Constraints.
For more information about these fit options, see the lsqcurvefit (Optimization Toolbox) function.
The default coefficient starting points and constraints for fits in the Fit Type pane are shown in the following table. If the starting points are optimized, then they are calculated heuristically based on the current data set. Random starting points are defined on the interval [0 1] and linear models do not require starting points. If a model does not have constraints, the coefficients have neither a lower bound nor an upper bound. You can override the default starting points and constraints by providing your own values in the Fit Options pane.
Sum of Sine
The Sum of Sine and Fourier fits are particularly sensitive to starting points, and the optimized values might be accurate for only a few terms in the associated equations.
Create the default fit options structure and set the option to center and scale the data before fitting:
Modifying the default fit options structure is useful when you want to set the Normalize, Exclude, or Weights fields, and then fit your data using the same options with different fitting methods. For example:
f1 = fit(cdate,pop,"poly3",options);
f2 = fit(cdate,pop,"exp1",options);
f3 = fit(cdate,pop,"cubicsp",options);
Data-dependent fit options are returned in the third output argument of the fit function. For example, the smoothing parameter for smoothing spline is data-dependent:
[f,gof,out] = fit(cdate,pop,"smooth");
Use fit options to modify the default smoothing parameter for a new fit:
options = fitoptions("Method","Smooth","SmoothingParam",0.0098);
[f,gof,out] = fit(cdate,pop,"smooth",options);
For more details on using fit options, see the fitoptions function.
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Part 3 – the metric – ebvalaim.log
v_x
x
v_y
y
\sqrt{v_x^2 + v_y^2}
\Delta x^\mu
x^\mu
As it turns out, it can be done by introducing numbers denoted
g_{\mu\nu}
(2 indices mean that in 4 dimensions we have 4x4 = 16 numbers - although, as we will see in a moment, not really). Now we say that the square of the distance between points differing in respective coordinates by
\Delta x^\mu
g_{\mu\nu}\Delta x^\mu \Delta x^\nu
. Remember the summation convention! This means
g_{00}(\Delta x^0)^2 + g_{01}\Delta x^0 \Delta x^1 + ... + g_{10}\Delta x^1 \Delta x^0 + g_{11}(\Delta x^1)^2 + ...
Because multiplication is commutative,
\Delta x^0 \Delta x^1 = \Delta x^1 \Delta x^0
and we can write this as follows:
g_{00}(\Delta x^0)^2 + (g_{01}+g_{10})\Delta x^0 \Delta x^1 + g_{11}(\Delta x^1)^2 + ...
Actually, in every physical quantity the components
g_{\mu\nu}
\mu \neq \nu
will always be summed - it doesn't matter then, how the sum is divided between
g_{\mu\nu}
g_{\nu\mu}
. We can assume that those expressions are equal and swapping the indices changes nothing. This means that in 4 dimensions only 10 numbers out of 16 are actually different.
Since we have 2 indices, the metric is often written in form of a table (a matrix):
10 numbers mentioned before are the 4 diagonal numbers (
g_{00}
g_{11}
g_{22}
g_{33}
) and 6 numbers outside the diagonal (the same numbers will be above and below it).
In the simplest case, on a plane, the square of the distance between points is
\Delta x^2 + \Delta y^2
, so the metric looks like this:
Such a metric (ones on the diagonal, zeros everywhere else) is usually called trivial (because it is the simplest possible metric), and such a matrix can be referred to as an identity matrix.
We can now finally say what is the magnitude of a vector. If a vector represents a translation by
v^\mu
, its length (magnitude) will be
|v| = \sqrt{g_{\mu\nu}v^\mu v^\nu}
(again, I remind you about the summation convention!).
This is not the end, though. Nobody said that the metric cannot change between points in space. If it is different in different places, that means
g_{\mu\nu}
are not numbers, but functions, depending on the point in space. It is hard to talk about distances between points then, because the metric can change on the way. What can be defined, though, is the length of infinitely small (infinitesimal) line segment, where coordinates differ only by
dx^\mu
(derivative-like notation is not a coincidence - with derivatives you also need points infinitely close to each other in the limit).
What about the vectors, then? Well, vectors are treated as if they were contained in one point - their initial point. More precisely, the vectors are not a part of the manifold, but of so called tangent space. We won't go deeper into this topic - it's enough to say that we still calculate their magnitude the same way, using the metric coefficients from their initial point.
Also, remember that by a vector we often mean a vector field. The expression
\sqrt{g_{\mu\nu} v^\mu v^\nu}
then means a function which to each point of space assigns the magnitude of the vector field at that point.
It's time for an example again. One of the simplest variable metric appears when so called polar coordinates are introduced on a plane - that is, instead of
x
y
, which mean the distance from origin along perpendicular axes, we introduce
r
- the distance from origin and
\theta
- the angle between the direction from origin to the given point and the x axis (see the picture below).
In this case, if we change
r
dr
\theta
d\theta
, we get a point
dr^2 + r^2 d\theta^2
away from the initial one (we move in one direction by
dr
, and perpendicularly along a circle - by an arc
rd\theta
long; when
dr
d\theta
are close to 0, it works as if they were sides of a right triangle - see the picture). The metric looks like this, then:
g_{rr} = 1
g_{\theta\theta} = r^2
g_{r\theta} = g_{\theta r} = 0
g = \left(\begin{array}{cc} 1 & 0 \\ 0 & r^2 \end{array}\right)
Assume we have a vector field
v^r = 0
v^\theta = 1
. Its magnitude is
|v| = \sqrt{1 \times (v^r)^2 + r^2 \times (v^\theta)^2} = \sqrt{r^2} = r
. This means such a vector field is longer the farther it is from the origin. It's not really surprising - after all it describes movement in the
\theta
angle, and the farther it is from the origin, the longer are the arcs corresponding to the same angle.
Another example: a sphere. When coordinates
(\theta, \varphi)
(roughly latitude and longitude) are introduced, we get the metric:
g = \left(\begin{array}{cc} 1 & 0 \\ 0 & \sin^2 \theta \end{array}\right)
Calculating magnitudes of vectors is not the only use of a metric. It can also be used to calculate generalized dot products. Simply put, the dot product lets us calculate angles between vectors. In the case of two vectors on a plane it is expressed like this:
\vec{u} \cdot \vec{v} = u_xv_x + u_yv_y
. In higher dimensions it is similar, only you have to sum more products of coordinates. In the case of a non-trivial metric, it is generalized to
g_{\mu\nu} u^\mu v^\nu
. And what about the relation with the angle? As it turns out,
\vec{u} \cdot \vec{v} = |u||v|\cos \alpha
\alpha
is the angle between vectors (generally - we will later see that in relativity the dot product has a radically different meaning).
There is one more important use of the metric. Namely, it allows turning vectors into covectors and vice versa. Let's see: we have a vector
v^\mu
. Using the metric, we can create a corresponding covector
v_\mu = g_{\mu\nu}v^\nu
(so eg.
v_0 = g_{00}v^0 + g_{01}v^1 + g_{02}v^2 + g_{03}v^3
). It is called lowering the index.
When the metric is trivial, we will always get
v_\mu = v^\mu
- in that case vectors and covectors are practically the same. With non-trivial metrics they are different, though.
The possibility of lowering the index lets us write the magnitude of a vector like this:
|v| = \sqrt{v^\mu v_\mu}
You can lower indices, but what about raising them? Can you turn a covector
u_\mu
u^\mu
? Well, yes you can, but for that you need the inverse metric. Inverse metric is a matrix
g^{\mu\nu}
, which will give the identity matrix when multiplied by the metric:
g^{\mu\sigma}g_{\sigma\nu} = \delta^\mu_\nu
\delta^\mu_\nu
is a so-called Kronecker delta - a matrix containing 1 for
\mu=\nu
and 0 otherwise - so it's just an identity matrix with one upper and one lower index). Raising and index looks like this, then:
u^\mu = g^{\mu\nu}u_\nu
So, what happens if we take a vector, lower its index and then raise it back again? We get:
u^\mu = g^{\mu\sigma}g_{\sigma\nu}v^\nu
As we already know, the product of the inverse metric and the metric is the Kronecker delta, so we can write this so:
u^\mu = \delta^mu_\nu v^\nu
But! Let's calculate
u^0
, for example:
u^0 = \delta^0_0 v^0 + \delta^0_1 v^1 + \delta^0_2 v^2 + \delta^0_3 v^3
Like we mentioned, the Kronecker delta is 1 for equal indices and 0 for different, so we get:
u^0 = 1 \times v^0 + 0 \times v^1 + 0 \times v^2 + 0 \times v^3
u^0 = v^0
And it will be like that for every
\mu
. We have then:
u^\mu = v^\mu
- the vector hasn't changed. Lowering an index and raising it back gives the initial vector as a result.
By the way, we have another conclusion: multiplying by the Kronecker delta changes nothing.
\delta^\mu_\nu v^\nu = v^\mu
\delta^\nu_\mu u_\nu = u_\mu
. That is actually why such a matrix is called an "identity" matrix - multiplying by it gives a result identical to what you started with.
This is all we need to know about metrics. In the next part we will say a bit about describing curves and about geodesic lines, and then we will finally move on to physics.
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cactusmaint@cactuscode.org
The default unigrid driver for Cactus for both multiprocessor and single process runs, handling grid variables and communications.
PUGH can create, handle and communicate grid scalars, arrays and functions in 1, 2 or 3-dimensions.
PUGH can be compiled with or without MPI. Compiling without MPI results in an executable which can only be used on a single processor, compiling with MPI leads to an executable which can be used with either single or multiple processors. (Section 6 describes how you can tell if your executable has been compiled with or without MPI).
For configuring with MPI, see the Cactus User’s Guide.
3 Grid Size
The number of grid points used for a simulation can be set in PUGH either globally (that is, the total number of points across all processors), or locally (that is, the number of points on each processor).
To set the global size of a N-D grid to be 40 grid points in each direction use
PUGH::global_nsize = 40
To set the global size of a 2D grid to be
40×20
PUGH::global_nx = 40
PUGH::global_ny = 20
To set the local size of a 2D grid to be
40×20
on each processor, use
PUGH::local_nx = 40
PUGH::local_ny = 20
4 Periodic Boundary Conditions
PUGH can implement periodic boundary conditions during the synchronization of grid functions. Although this may at first seem a little confusing, and unlike the usual use of boundary conditions which are directly called from evolution routines, it is the most efficient and natural place for periodic boundary conditions.
PUGH applies periodic conditions by simply communicating the appropriate ghostzones between “end” processors. For example, for a 1D domain with two ghostzones, split across two processors, Figure 1 shows the implementation of periodic boundary conditions.
Figure 1: Implementation of periodic boundary conditions for a 1D domain, with two ghostzones, split across two processors. The lines labelled A show the standard communications during synchronisation, the lines labelled B show the additional communications for periodic boundary conditions.
Periodic boundary conditions are applied to all grid functions, by default they are applied in all directions, although this behaviour can be customised to switch them off in given directions.
By default, no periodic boundary conditions are applied. To apply periodic boundary conditions in all directions, set
PUGH::periodic = "yes"
To apply periodic boundary conditions in just the x- and y- directions in a 3 dimensional domain, use
PUGH::periodic_z = "no"
5 Processor Decomposition
By default PUGH will distribute the computational grid evenly across all processors (as in Figure 2a). This may not be efficient if there is a different computational load on different processors, or for example for a simulation distributed across processors with different per-processor performance.
Figure 2: Partitioning of the computational grid across processors, Figure a) is the default type of partition used by PUGH, Figure b) can be set manually, and Figure c) is not possible with PUGH
The computational grid can be manually partitioned in each direction in a regularly way as in Figure 2b.
The computational grid can be manually distributed using PUGH’s string parameters partition_[1d_x|2d_x|2d_y|3d_x|3d_y|3d_z]. To manually specify the load distribution, set PUGH::partition = "manual" and then, depending on the grid dimension, set the remaining parameters to distribute the load in each direction. Note that for this you need to know apriori the processor decomposition.
The decomposition is easiest to explain with a simple example: to distribute a 30-cubed grid across 4 processors (decomposed as
2×1×2
, with processors 0 and 2 performing twice as fast as processors 1 and 3) as:
proc 2:
20×30×15
10×30×15
20×30×15
10×30×15
you would use the following topology and partition parameter settings:
# the overall grid size
# processor topology
PUGH::processor_topology = manual
PUGH::processor_topology_3d_x = 2
PUGH::processor_topology_3d_y = 1
PUGH::processor_topology_3d_z = 2 # redundant
# grid partitioning
PUGH::partition = "manual"
PUGH::partition_3d_x = "20 10"
Each partition parameter lists the number of grid points for every processor in that direction, with the numbers delimited by any non-digit characters. Note that an empty string for a direction (which is the default value for the partition parameters) will apply the automatic distribution. That’s why it is not necessary to set PUGH::partition_3d_y = "30" or PUGH::partition_3d_z = "15 15" in the parameter file.
Because the previous automatic distribution gave problems in some cases (e.g. very long box in one, but short in other directions), there is now an improved algorithm that tries to do a better job in decomposing the grid evenly to the processors. However, it can fail in certain situations, in which it is gracefully falling back to the previous ("automatic_old") giving a warning. Note that, if one or more of the parameters PUGH::processor_topology_3d_* or PUGH::partition_3d_* are set, this mode automatically falls back to "automatic_old" without warning.
6 Understanding PUGH Output
PUGH reports information about the processor decomposition to standard output at the start of a job. This section describes how to interpret that output.
Single Processor (no MPI)
Type of evolution
If an executable has been compiled for only single processor use (without MPI), the first thing which PUGH reports is this fact:
INFO (PUGH): Single processor evolution
Multiple Processor (with MPI)
If an executable has been compiled using MPI, the first thing which PUGH reports is this fact, together with the number of processors being used:
INFO (PUGH): MPI Evolution on 3 processors
Maximum load skew
The maximum load skew describes the variance in the number of gridpoints on each processor, and is defined by
\text{Max Skew}=100\phantom{\rule{0.28em}{0ex}}\phantom{\rule{0.28em}{0ex}}\frac{\text{Max Points}-\text{Min Points}}{\text{Average Points}}
For most purposes, the maximum skew should ideally be close to zero, however if your simulation has a different load at different grid points, or if you are running across processors with different properties, the optimal skew could be quite different.
By default, PUGH tries to minize the skew in gridpoints, however this may be overriden by performing the load balancing manually.
7 Useful Parameters
There are several parameters in PUGH which are useful for debugging and optimisation:
PUGH::enable_all_storage
Enables storage for all grid variables (that is, not only those set in a thorn’s schedule.ccl file). Try this parameter if you are getting segmentation faults. If enabling all storage removes the problem, it most likely means that you are accessing a grid variable (probably in a Fortran thorn) for which storage has not been set.
PUGH::initialise_memory
By default, when PUGH allocates storage for a grid variable it does not initialise its elements. If you access an uninitialised variable on some platforms you will get a segmentation fault (and in general you will see erratic behaviour). This parameter can be used to initialise all elements to zero, if this removes your segmentation fault you can then track down the cause of the problem by using the same parameter to initialize all elements to NaNs and then track them with the thorn CactusUtils/NaNChecker.
Note that it isn’t recommended to simply use this parameter to initialise all elements to zero, instead we recommend you to set all variables to their correct values before using them.
PUGH::storage_verbose
This parameter can be set to print out the number of grid variables which have storage allocated at each iteration, and the total size of the storage allocated by Cactus. Note that this total does not include storage allocated independently in thorns.
PUGH::timer_output
This parameter can be set to provide the time spent communicating variables between processors.
cacheline_mult
Description: Multiplier for cacheline number
enable_all_storage
Description: Enable storage for all GFs?
ghost_size
Description: The width of the ghost zone in each direction
Any positive number to override the ghost_size_[xyz] parameters
ghost_size_x
Description: The width of the ghost zone in the x direction
ghost_size_y
Description: The width of the ghost zone in the y direction
ghost_size_z
Description: The width of the ghost zone in the z direction
global_nsize
Description: The size of the grid in each spatial direction
Grid of this size in each dir distributed across all processors
global_nx
Description: The size of the grid in the x direction
Grid of this size distributed across all processors
global_ny
Description: The size of the grid in the y direction
global_nz
Description: The size of the grid in the z direction
Description: Provide additional information about what PUGH is doing
Load on each processor
initialize_memory
Description: How to initialize memory for grid variables at allocation time
Do not initialize storage for allocated grid variables (default)
Zero out all elements of all allocated grid variables
Set all elements of allocated floating point grid variables to Not-a-Number values
local_nsize
Grid of this size in each dir on each processor
local_nx
Grid of this size on each processor
local_ny
local_nz
local_size_includes_ghosts
Description: Does the local grid size include the ghost zones?
overloadabort
Description: Overload Abort driver function
overloadarraygroupsizeb
Description: Overload ArrayGroupSizeB driver function
overloadbarrier
Description: Overload Barrier driver function
overloaddisablegroupcomm
Description: Overload DisableGroupComm driver function
overloaddisablegroupstorage
Description: Overload DisableGroupStorage driver function
overloadenablegroupcomm
Description: Overload EnableGroupComm driver function
overloadenablegroupstorage
Description: Overload EnableGroupStorage driver function
overloadevolve
Description: Overload Evolve driver function
overloadexit
Description: Overload Exit driver function
overloadgroupdynamicdata
Description: Overload GroupDynamicData driver function
overloadmyproc
Description: Overload MyProc driver function
overloadnprocs
Description: Overload nProcs driver function
overloadparallelinit
Description: Overload ParallelInit driver function
overloadquerygroupstorageb
Description: Overload QueryGroupStorageB driver function
overloadsyncgroup
Description: Overload SyncGroup driver function
overloadsyncgroupsbydiri
Description: Overload SyncGroupsByDirI driver function
Description: Is the partition manual
Range Default: automatic
specified by partition_XYZ ..
partition_1d_x
Description: Tells how to partition on direction X
A regex which matches anything
partition_2d_y
Description: Tells how to partition on direction y
partition_3d_z
Description: Tells how to partition on direction z
physical2logical
Description: Physical process to logical process mapping method to use
Maps MPI IDs directly to IJKs
Maps MPI IDs directly to IJKs using a lookup table
processor_topology
Description: How to determine the processor topology
Specified by proc_top_nx etc
automatic_old
Automatically generated (old method)
processor_topology_1d_x
Description: Number of processors in X direction
See proc_topology
processor_topology_2d_y
Description: Number of processors in Y direction
processor_topology_3d_z
Description: Number of processors in Z direction
storage_report_every
Description: How often to provide a report on storage information
Report at intervals
storage_verbose
Description: Report on memory assignment
Standard storage information
”Provide a report of storage every storage_report_every iterations and at termination”
Description: Print time spent in communication
cctk_itlast
Scope: shared from CACTUS INT
terminate_next
Scope: shared from CACTUS BOOLEAN
Register.h to pugh_Register.h
pugh_Register.h
This section lists all the variables which are assigned storage by thorn CactusPUGH/PUGH. Storage can either last for the duration of the run (Always means that if this thorn is activated storage will be assigned, Conditional means that if this thorn is activated storage will be assigned for the duration of the run if some condition is met), or can be turned on for the duration of a schedule function.
pugh_startup
pugh_registerpughp2lroutines
register physical to logical process mapping routines
pugh_registerpughtopologyroutines
register topology generation routines routines
pugh_report
report on pugh set up
CCTK_TERMINATE (conditional)
pugh_printtiminginfo
print time spent in communication
pugh_printfinalstoragereport
print storage information
pugh_printstoragereport
pugh_terminate
termination routine
PUGH_Startup Driver_Startup
PUGH_Terminate Driver_Terminate
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