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What to do with Qi on Polygon - Mai Finance - Tutorials
This guide will explain in details how you can use your Qi, the native token from Mai Finance, on Polygon. We will also analyze if Qi is a good investment opportunity, and what drives its price.
Qi ([tʃ Í] or chee), is the native token of Mai Finance. Some use it to vote and QIP (QiDAO Improvement Proposals), some stake it to get even more of them, and some simply farm with it. You can find some very good info on how you can use Qi on the Mai Finance plateforme in the guide dedicated to passive income.
This guide, as most investment tutorials, will focus primarily on farming and harvesting yields, this time using the Qi token. We will see how you can generate a lot of Qi, and how you can use them on the different platforms on Polygon.
Humble farmers will sometimes tell you that you need to remain humble, sell what you harvest and take profits. But personally, I'll go with:
So grab your fish rods, and follow the guide.
What to do with your Qi on Polygon
I will not spend too much time on this part, there's already a complete guide on the subject. Just keep in mind that Mai Finance is collecting revenues, and redistributes a very large portion to Qi stakers. Staking your Qi on Mai finance is one of the best use for the token, and as of September 2021, 23% of all circulating Qi is locked on average for 2 years.
LP pair Farming
Once again, there are already a few other pages presenting how you can include the Qi token in your farming strategy to generate yields and never sell any of your farm tokens. See Stack DApps like Lego bricks or Farming or Staking ? guides for details exampled.
As a quick reminder, Qi is paired to different tokens to form LP (Liquidity Provider) pairs on QuickSwap so that you can farm
Qi-WMATIC on Mai Finance and get rewarded with Qi tokens
Qi-WETH on QuickSwap and get rewarded with QUICK tokens
Qi-QUICK on QuickSwap and get rewarded with QUICK tokens
LP pool on QuickSwap for the Qi-WETH pair
Single Qi farming
Qi can also be used solely on Impermax. Impermax is a platform where you will be able to leverage a specific LP pair several times in order to increase your gains from QuickSwap.
The way it works, is that you will borrow the 2 tokens forming the pair that you want to farm, combine them into more LP tokens, and farm with a much higher position. In most cases, borrowing rates are largely compensated by farming APR, giving you some net positive rewards.
Leveraged Qi-WETH position on Impermax
We can see here that the final reward APR on Impermax is 393.88% after leveraging 5 times, based on an APY of 239.68% on QuickSwap.
Note also that Impermax gives estimated APRs (Annual Percentage Rewards) while QuickSwap gives estimated APYs (Annual Percentage Yields), meaning that QuickSwap assumes you compound your rewards daily. The 239.68% APY on QuickSwap corresponds to a 122.49% APR.
But then, on Impermax, in order to borrow Qi and WETH to leverage your position, you need to get them from somewhere. This is possible only because some other users (or yourself) also supply both tokens separately. The more token is borrowed, the higher the borrowing rate becomes, and the lower the final APR, sometimes going in the negatives.
Qi and WETH statistics for the Qi-WETH market on Impermax
For our example, we will focus on Qi. You can see that the total supply of Qi is $427.21 and the total amount used in leveraged position is $321.44, giving the utilization rate of 75.24%. Impermax has some internal mechanism that automatically calculates the supply APR (APR that people lending Qi will get) and the borrowing APR (percentage of the farmed reward that will be deduced to pay the loan).
This means that you can provide Qi solely on Impermax and get, in our example, 43.73% APR, at the moment of writing. As supply and demand varies, the supply APR will also increase / decrease. When you supply single tokens on Impermax, you will get rewarded with the token you provide, meaning that this strategy will make you accumulate more Qi over time.
While you are on Impermax, you can also use the leverage option to get IMX rewards. Please read the Stacking guide to get more details on how you can include Impermax in your farming strategy.
Balancer is a really nice toolbox for any strategy, especially the ones that include Qi and/or MAI. Balancer proposes an equivalent of LP pair mining, but with more than 2 tokens provided in a 1:1 ratio. The pool can have 3, 4 or 5 tokens (sometimes even more) with different weight, and the algorithm in charge of the pool makes sure that the ratio for each token is always respected, selling some and buying others to keep it balanced.
The pool that we want to use here is a pool that contains Qi, WMATIC, BAL, USDC and MAI. This pool will reward you with both Qi and BAL tokens, and you can already see that you will be able to compound both into the pool. Because another amazing advantage of pools on Balancer is that you don't need to provide all tokens forming the pool in the proper ratio, the algorithm will do it for you. This means that you can indeed only deposit Qi in the pool and let the algorithm do the rest to rebalance everything.
Details of the pool as of September 2021
As a side note, BAL tokens can (or will soon be) usable as collateral on Mai Finance, which mean that you will have the option to store your BAL tokens on the BAL vault on Mai Finance, and borrow MAI against then. In addition to that, borrowing MAI against your BAL tokens will make you eligible for Qi rewards that will feed the pool on Balancer.
Closed loop using Mai Finance and Balancer
The APR of BAL vaults will highly dictates how interesting it is to have the vault in the loop, or if compounding your Qi into the Balancer's pool will be better.
Understanding the price of Qi
Getting a lot of Qi is one thing, however if the token loses value over time, is it really a good strategy to keep it? In this section, we will try to understand the different factors that impact directly the price of Qi so that, when you start investing your Qis, you will get a better idea of how its price may vary, and what impact you will get on the Qi ecosystem.
Qi emission
One of the main factor that will affect the price of Qi is the rate at which they are created. Indeed, price is always driven by demand and supply. If there's a lot of supply and very low demand, the price will naturally collapse. Hence understanding how Qi is generated is crucial to estimate its value over time.
There are currently 2 sources of Qi emission: farming rewards and vault rewards.
LP farms on Mai Finance in September 2021
If you are farming yields on MAI finance, you have the choice between the MAI/USDC pair and the Qi/WMATIC pair (as of September 2021).
The MAI/USDC pair is rewarded with 0.5 Qi / block
The Qi/WMATIC pair is rewarded with 1 Qi / block
On Polygon, the expected block time is 2 seconds, and since there are 86,400 seconds in a day, this means that the MAI/USDC pool is rewarded with 21,600 Qi every day, and the Qi/WMATIC with 43,200 Qi.
The pools of Mai Finance alone are responsible for 64,800 new Qi daily.
As for Vaults, each vault gets an emission of 0.05 Qi / block, or a daily emission of 2,160 Qi, and there are currently 10 Vaults, for a total of 21,600 Qi allocated as Vault rewards.
This means that each day, the equivalent of 86,400 new Qi are minted and distributed to users.
Yield optimizers are platforms that will automatically compound the rewards with some pre-defined strategies, and allocate additional rewards for you to choose their site. However, a big part of the reward harvested is sold directly, and re-used in another way on these platforms.
As an example, Adamant offers you to farm the Qi/WMATIC LP pair on their platform, with the following reward distribution
Qi-WMATIC pool on Adamant
Details of the 179.23% APR granted by Adamant
You will notice that if the overall APR is higher than on Mai Finance, it's solely because Adamant is allocating additional ADDY rewards to the farmers. The amount of Qi that is actually redistributed to the farmer is 98.45% compared to the 134.42% you can get on Mai Finance.
From these 98.45% Qi reward, half of it is directly sold to buy WMATIC and form additional LP tokens that is then given to the farmer.
With 100$ worth of LP token, assuming the APR and the token prices remain the same for a complete year, and assuming there's no compounding, you would get after one year of farming
$134.42 worth of new Qi on Mai Finance
$98.45 worth of new Qi/WMATIC token, or $49.23 worth of new Qi on Adamant
This means that, in the process, $85.20 worth of Qi is simply sold directly on the market, which is more than 60% of the total emission allocated to the pool on Adamant.
And Adamant isn't the only platform proposing the same type of service. Some other examples are Beefy Finance and Kogecoin. From the $4.9M TVL in the Qi/WMATIC farm on Mai Finance, $2.3M are coming directly from Adamant, $41k from Beefy and $12k from Kogecoin, representing for these 3 platforms more than 50% of the value locked on Mai Finance. A raw estimation is that more than 30% of the total daily Qi emission is dumped by these platforms, putting some very negative sell pressure on the token, decreasing its price, which partially explains why Qi has difficulties keeping a high price.
Understanding LP pairs
When you farm yields by providing LP pairs, the LP token is actually used to provide liquidity to users who are swapping one token for another. In our example of Qi/WETH, when someone is buying WETH, some of the token can be taken off the LP pool and sold to the user requiring it.
At that point, because some WETH has been taken off the pool, there is a balance mismatch: less WETH for the same amount of Qi. The algorithm in charge of maintaining the pool ratio to 1:1 will then sell some Qi from this pool to buy back some more WETH and recreate a perfect 1:1 ratio. The opposite things also happen when someone buys Qi, i.e. WETH is sold to buy back some Qi.
The same phenomenon occurs when one of the 2 tokens composing the pair gains or loses value. As an example, we will assume that Qi price is $1 and ETH price is $1,000, and that we have a pool that has $100 worth of Qi and $100 worth of WETH. It means that the pools contains 100 Qi and 0.1 WETH.
Now, if the price of ETH goes up to $2,000, if the pool keeps the same amount of token, we would have $100 worth of Qi but $200 worth of WETH, and we would have lost the balance. Hence, the algorithm in charge of the pool will sell a little bit of ETH to buy some Qi. In our easy example, $50 worth of ETH will be sold to buy $50 worth of Qi, and the final state would be
150 Qi with a value of $150$
0.075 ETH with a value of $150
This also means that when the price of one of the 2 token goes up, the pool creates some demande for the other, also driving its price up. The opposite is also true: if one token loses value, the other one will be sold to maintain a 1:1 ratio, driving the price down. This also partially explains the price fluctuation of Qi when compared to the price fluctuation of WEHT and WMATIC (the 2 main tokens to which Qi is paired).
Price of Qi (left) VS Price of WMATIC (right)
Lack of use case
Finally, the lack of use for Qi, or the lack of known use cases, can explain why the price of the token is going down. People collecting Qi from Vault rewards and/or farming on Mai Finance will just sell it while it still has "some" value in order to realize a profit, with no long-term vision, which is actually a pretty reasonable strategy. This guide tries actually to promote different ways to use your Qi tokens without selling them, but if the price doesn't go up, or if you can't generate benefits in other tokens (a.k.a profit leak), there's very little advantages stacking them.
How can we help price to go up?
If we want the price of Qi to go up (and who wouldn't want to), there are a few options.
Reduce emission: with 86,400 new Qi minted daily, the supply is very high. If we cut down this emission, the supply may be lower, and with less supply, price should in theory go up. However, the current emission for the farms need to remain the same because they're part of a current partnership with QuickSwap. Vault incentives have been launched, and are an amazing way to promote the lending platform, pushing people to take loans, so cutting these emissions is probably another bad idea.
Farm responsibly: I'm not saying that Adamant is the main culprit or is a bad product. Indeed, I include this platform in most of my strategies and use it on a daily basis. However, I try to farm in pools for which I have less concerns that the token will be dumped. If you want to farm yields on Adamant in a pool that is using the Mai Finance farm, keep in mind that you participate in the Qi depreciation. If you are ok with that, that's totally fine.
Understand how things work: prices of ALL crypto-currencies are highly volatile, and most of the time, the price of BTC and ETH dictate the price of all other tokens. This is not done magically. Also, please understand that when some people earn money, it's because some other are loosing some. There's no free money, and magical internet money is actually not that magical.
Find new ways to use your Qi tokens. This guide will always provide ideas that will help you increase the volume of your portfolio, and sell as few tokens as possible. Closed loops are the best because the output of one product feeds the input of the next one, creating a nice bubble that will only grow over time.
Keep also in mind that
Value = Quantity * Price
When the price go down but your quantity goes up, your value may increase, or at least it won't decrease as fast.
This guide is actually a result of many discussion about Qi price on the Discord channel. I realized that some people complaining about the price depreciation didn't fully understand the reasons about why the price of Qi was on a consistant down trend. This leads to frustration and delusion, which is very negative for Qi, the Mai Finance product, and the whole DeFi ecosystem in general. However, there's absolutely no guarantee that the price will ever go up, so if you want to keep your Qis and invest them, please do your own research and do it at your own risk. Plan your strategy properly and stick to it.
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Metals and Non-metals - Test Papers
Chapter 03 Metals and Non Metals
Metal always found in free state is? (1)
Roasting is a method of heating ore: (1)
In the absence of water.
In the absence of air.
In the presence of air.
Ayush is putting
{\mathrm{H}}_{2}\mathrm{S}{\mathrm{O}}_{4}
in the test tube containing water whereas Piyush is putting water in the lest tube containing
{\mathrm{H}}_{2}\mathrm{S}{\mathrm{O}}_{4}
. Which of the two is likely to face danger? (1)
Both Ayush and Piyush
The group of metals which do not react with oxygen: (1)
Na, Cs
Pt, Cu
Magnesium imparts: (1)
Dazzling white colour flame
Yellowish orange colour flame
Brick red colour flame
A non-metal X exists in two different forms Y and Z. Y is hardest natural substance, whereas Z is a good conductor of electricity. Identify X, Y arid Z. (1)
Why are metals reducing agents where as non metals are oxidizing agents? (1)
Name two metals which will displace hydrogen from dilute acids, and two metals which will not? (1)
Name two metals that do not react with water at all. (1)
State the property utilised in the following: (3)
Graphite in making electrodes.
Electrical wires are coated with Polyvinyl Chloride (PVC) or a rubber-like material.
Metal alloys are used for making bells and strings of musical instruments.
Choose from the following metals to answer the questions below: (3)
Aluminium, calcium, copper, iron, magnesium, potassium, nickel and zinc. Name a metal which
is manufactured by the electrolysis of its molten oxide.
is used to galvanise iron
is alloyed with zinc to make brass.
reacts with aqueous copper (II) sulphate to give a pink solid.
does not react with cold water.
How will you show that silver is less reactive than copper? (3)
Why metals replace hydrogen from dilute acids, whereas non-metals do not? (3)
Distinguish between ionic and covalent compounds under the following properties: (5)
Strength of forces between constituent elements
Electrical conduction in substances
Explain how the following metals are obtained from their compounds by the reduction process:
Metal M which is in the middle of the reactivity series.
Metal N which is high up in the reactivity series. Give one example of each type.
Show the formation of
N{a}_{2}O
MgO
by the transfer of electrons.
What are the ions present in these compounds? (5)
Explanation: Gold, platinum, copper, silver are few metals which can be found in free state, because they are unreactive in normal conditions with air, water and other chemicals.
Explanation: Roasting is heating of an ore in a regular supply of air in a furnace.
Explanation: Piyush is likely to face danger. Addition of water to concentrated sulphuric acid is an exothermic reaction and can lead to explosions. Water should not be poured into a container containing an acid.
Explanation: Gold and Silver do not react with oxygen. They are less reactive metals and lie at the bottom of the reactivity series.
Explanation: Magnesium burns in air with a dazzling white flame.
The non-metal X is carbon (C). Y and Z are the allotropes of carbon (different physical forms of carbon). Y is diamond because diamond is the hardest natural substance known and Z is graphite which is a good conductor of electricity.
Metals have a tendency to donate electrons and get oxidized. Thus, they are reducing agents.
Non- metals on the other hand have a tendency to gain electrons and get reduced. Therefore, they are oxidizing agents.
Sodium and calcium, being more reactive than hydrogen can displace hydrogen from dilute acids.
Copper and silver, being less reactive than hydrogen cannot displace hydrogen from dilute acids.
Lead and copper.
Graphite in an allotrope of carbon which is a good conductor of electricity because of presence of free electron and it is cheap, insoluble in water, do not react with acids and bases (non-corrosive). Due to these properties, it is used in making electrodes.
Polyvinyl Chloride (PVC) or a rubber-like material are insulators means they are bad conductors of electricity and hence do not allow electrons to flow. Hence, these are used in coating the electrical wires.
Metal alloys are used for making bells and strings of musical instruments because they are sonorous.
The metal which is manufactured by the electrolysis of its molten oxide is Aluminium.
Zinc is used to galvanise iron, because it is more reactive.
Copper is alloyed with zinc to make brass.
Iron reacts with aqueous copper (II) sulphate to give a pink solid.
Iron has no reaction with cold water but it reacts with steam.
In activity series silver is placed below the copper. hence copper is more reactive than silver.
The more reactive element can displace the less reactive element from its solution.
take two test tubes in test tube -1 take silver Sulphate solution(white in color) & in test tube-2 take copper Sulphate solution (blue in color). Add copper turnings to test tube -1 and add silver piece to test tube -2 keep the test tubes undisturbed for 15 minutes. After sometime observe the color changes in both test tubes. The color in test tube – 1 changed from white to blue and there is no color change in test tube-2.this indicates in test tube-1 copper displaces silver from its silver Sulphate solution forms copper Sulphate. And in test tube- 2 no displacement reaction takes place. Hence from this experiment we are proving that copper is more reactive than silver.
The reason why non-metals do not displace hydrogen from dilute acids is because unlike metals, non-metals do not have a tendency to lose electrons but to gain electrons. Metals have a tendency to lose electrons. These electrons, which are readily lost by reactive metals like sodium, potassium etc are accepted by hydrogen ions of the acids, reducing them to hydrogen gas (H2)
However, non-metals do not lose electrons readily, because of which they do not displace hydrogen from acids. Another important point to note is that not all metals will displace hydrogen from acids. Only those metals which are reactive than hydrogen will displace H2 from acid.
Ionic compounds have strong force of attraction between the oppositely charged ions (e.g.,
N{a}^{+}
C{l}^{-}
), so they are solids. Covalent compounds have weak force of attraction between their molecules, so they are usually liquids or gases.
Ionic compounds are soluble in water but covalent compounds are insoluble in water.
Ionic compounds conduct electricity when dissolved in water or when melted because they contain ions (charged particles). But, covalent compounds like glucose do not conduct electricity because they do not contain ions.
The metal M which is in the middle of the reactivity series (such as iron, zinc, lead, copper, etc.) is moderately reactive. So, for obtaining such metals from their compounds, their sulphides and carbonates (in which they are present in nature) are first converted into their oxides by the process of roasting and calcination respectively. For example,
\underset{\underset{\left(Sulphideore\right)}{Zinc\phantom{\rule{thinmathspace}{0ex}}Sulphide}}{2ZnS\left(s\right)}+3{O}_{2}\left(g\right)\stackrel{Heat}{⟶}2ZnO\left(s\right)+2S{O}_{2}\left(g\right)
\underset{\underset{\left(Carbonate\phantom{\rule{thinmathspace}{0ex}}are\right)}{Zinc\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}Carbonate}}{ZnC{O}_{3}\left(s\right)}\stackrel{Heat}{⟶}ZnO\left(s\right)+C{O}_{2}\left(g\right)
The metal oxide (MO) are then reduced to the corresponding metals by using suitable reducing agents such as carbon. For example, zinc metal from its oxide is obtained as follow:
\underset{Zinc\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}oxide}{ZnO\left(s\right)}\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}C\left(s\right)\phantom{\rule{thinmathspace}{0ex}}\to \underset{Zinc}{Zn\left(s\right)}\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}CO\left(g\right)
The metal N which is high up in the reactivity series (such as sodium, magnesium, calcium, aluminium, etc.), is very reactive and cannot be obtained from its compound by heating with carbon.
Therefore, such metals are obtained by electrolytic reduction of their molten salt. For example, sodium is obtained by the electrolysis of molten sodium chloride (
NaCl
Electronic configuration K, L, M, N
No. of outermost electrons
Sodium Na 11 2,8, 1 1 Na.
Oxygen O 8 2,6 6 :O::
Magnesium Mg 12 2,8, 2 2 Mg:
N{a}_{2}O
The atomic number of sodium is 11 and it has only one valence electron.
Hence, electronic configuration of
{}_{11}Na
is 2, 8, 1.
The atomic number of oxygen is 8 and it has 6 electrons in its valence shell.
{}_{8}O
is 2, 6.
Sodium has a tendency to lose the valence electron and oxygen has a tendency to gain the electron lost by sodium. Since, sodium can lose only one electron of the valence shell, and oxygen atom needs two electrons to complete its octet in the valence electron, two atoms of sodium combine with one atom of oxygen. By losing valence electron, sodium is changed into
N{a}^{+}
and by gaining two electrons lost by two sodium atoms, oxygen atom is changed into an oxide anion,
{O}_{2}
.In this process , both the atoms, sodium and oxygen, obtain the stable electronic configuration of the noble gas neon.
\underset{2,8,1}{Na}\phantom{\rule{thinmathspace}{0ex}}\to \phantom{\rule{thinmathspace}{0ex}}\underset{2,8}{N{a}^{+}}\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}{e}^{-}
{O}^{2-}
\underset{2,6}{O\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}}\phantom{\rule{thinmathspace}{0ex}}2{e}^{-}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\to \phantom{\rule{thinmathspace}{0ex}}\underset{2,8}{{O}^{2-}}
2N{a}^{+}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{O}^{2}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\to \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}2N{a}^{+}{O}^{2-}\phantom{\rule{thinmathspace}{0ex}}or\phantom{\rule{thinmathspace}{0ex}}N{a}_{2}O
The oppositely charged sodium ion,
N{a}^{+}{O}^{2-}
and oxide ion,
{O}^{2-}
are now held together by electrostatic force of attraction or by ionic or electrovalent bond.
N{a}_{2}O
is, therefore, an ionic or electrovalent compound.
MgO
The atomic number of magnesium = 12
Its electronic configuration is
\underset{2,}{K,}\phantom{\rule{thinmathspace}{0ex}}\underset{8,}{L,}\underset{2}{M}
It has two electronic in its outermost shell. So, the magnesium atom donates its 2 valence electrons and forms a stable magnesium ion,
{Mg}^{2+}
to attain the electronic arrangement of neon atom.
\underset{2,8,2}{Mg}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\to \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\underset{2,8}{\overset{2+}{Mg}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{2}^{e-}
The atomic number of oxygen = 8
\underset{2}{K,}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\underset{6}{L}
It has 6 electrons in its valence shell. Therefore, it requires 2 more electrons to attain the stable electronic arrangement of neon gas. Thus, oxygen accepts 2 electrons donated by magnesium atom and forms a stable oxide ion, O2-
\underset{2,6}{O}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}2{e}^{-}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\to \underset{2,8}{\overset{2-}{O}}
The oppositely charged magnesium ions,
{Mg}^{2+}
, and oxide ions, are held together by a strong force of electrostatic attraction to form magnesium oxide compound.
M{g}^{2+}\phantom{\rule{thinmathspace}{0ex}}{O}^{2-}\phantom{\rule{thinmathspace}{0ex}}or\phantom{\rule{thinmathspace}{0ex}}MgO.
MgO is ionic compound.
The ions present in Na2O are sodium ions (2Na+) and oxide ion O2-.
The ions present in MgO are magnesium ion (Mg2+) and oxide ion O2-.
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Physics - Quantum Superrotor
Quantum Superrotor
Institute for Molecular Science, National Institutes of Natural Sciences, Myodaiji, Okazaki 444-8585, Japan
Molecules that rotate at very high rates exhibit surprising quantum coherence properties.
APS/Alan Stonebraker; top panel adapted from [1].
Figure 1: (Top) Optical centrifuge pulse (red) for making molecules rotate to form superrotors. Its linear polarization rotates, pulling the molecule with it. The unidirectional molecular rotation thus created is observed directly by coherent Raman scattering induced by the probe pulse (blue). (Bottom) Milner and colleagues have found that superrotors have interesting quantum coherence properties. Imagine the molecule as a dancer—slow rotation allows decoherence (the green dancer in the left; a fuzzy image) while fast rotation maintains coherence (the green dancer in the right; sharper image) against environmental influences (other dancers).(Top) Optical centrifuge pulse (red) for making molecules rotate to form superrotors. Its linear polarization rotates, pulling the molecule with it. The unidirectional molecular rotation thus created is observed directly by coherent Raman scattering ... Show more
The rotational motion of an isolated molecule is among the simplest manifestations of angular momentum. More generally, angular momentum governs many important physical phenomena such as nuclear collisions, chemical reactions of atoms and molecules, and the emergence of superconductivity and magnetism in condensed matter. Molecular rotation can also be coupled easily to other internal degrees of freedom such as vibrational and electronic states. This efficient coupling offers opportunities for active control of quantum dynamics for fundamental studies and also for guiding chemical reactions toward desired outcomes.
As an example of this kind of control, a collaboration led by Valery Milner at the University of British Columbia, Canada, now reports in Physical Review Letters that they have used a laser pulse to turn diatomic molecules of oxygen and nitrogen into “superrotors”—molecules that rotate faster than [1]. The energy of this superrotation is comparable to that of the chemical bond between the two atoms within the molecule and would require a temperature of to be realized in thermal equilibrium.
To drive these high rotational speeds, the UBC researchers used a sophisticated custom-tailored laser pulse whose duration is several tens of picoseconds. The polarization of this laser pulse is rotated unidirectionally and accelerated gradually within the pulse duration, forcing the molecule to follow (see Fig. 1). Milner and colleagues find that superrotors created in this way have a quantum coherence that is far more robust against collisions compared to the molecules in thermal equilibrium at room temperature.
This kind of optical manipulation of the amplitudes and phases of wave functions of matter is referred to as “coherent control” [2]. The concept was developed in the 1980s for controlling the dynamics of isolated molecules in the gas phase in processes such as ionization and dissociation induced by coherent laser light [3,4]. Since then it has attracted the attention of researchers studying a variety of physical systems, ranging from an isolated atom to many-body systems such as bulk solids and photosynthetic molecules.
One motivation arises from possible applications of coherent control in the development of novel quantum technologies such as quantum information processing and bond-selective chemistry. The latter is particularly attractive as a way of precisely breaking or forming specific chemical bonds of a molecule. Another important motivation is to better understand the wave nature of matter, and this knowledge is available only when we can control quantum systems actively. Coherent control of an isolated molecule in the gas phase has been one of the major targets, as it is the simplest system equipped with both electronic and nuclear degrees of freedom. In their work, Milner and colleagues have successfully controlled the rotational state of an isolated molecule in the gas phase.
Interest in spinning up a molecule into a superrotor was first raised by a research group in Canada [5]. They proposed a scheme in which a superrotor could be generated with a short laser pulse whose linear polarization axis rotates. This short laser pulse polarizes the molecule, so that its induced dipole follows the rotating laser polarization, a scheme called an “optical centrifuge.” The same group used an optical centrifuge to study a diatomic molecule, , experimentally, and they indirectly showed the generation of superrotors by observing the atoms that dissociate while in high rotational states [6]. However, there could be other possible sources of those fragments such as dissociation due to the electronic repulsion between two atoms within the molecule, even without any molecular rotation. It has therefore been necessary to verify the generation of a superrotor by its direct observation and to investigate its physical properties.
Researchers have made several advances in their ability to control the coherence of molecular rotation. Two independent groups, one in Israel and one in Japan, have created and observed unidirectional molecular rotation [7,8]. Milner and co-workers have demonstrated another approach to this unidirectional rotation of a molecule, as they report in their paper. Moreover, they have utilized this new approach to create and observe superrotors directly.
In their work, Milner and colleagues use the optical centrifuge to create the superrotation of and molecules. They create an ultrashort frequency-dispersed laser pulse, with its higher and lower frequency halves spatially separated. Independent phase modulation of the two spectral halves ensures that the frequencies of the two components increase and decrease linearly in picoseconds. These spectral components are circularly polarized in opposite directions, and when they are recombined they interfere with each other to create a linearly polarized pulse. When this is done, the polarization of this recombined laser pulse rotates unidirectionally with a difference frequency between the spectral components. As the difference frequency increases in time, the speed of the rotation becomes faster. The or molecule is irradiated with this recombined “centrifuge” laser pulse, so that its bond axis follows this unidirectional rotation of the laser polarization.
One important trick the authors use is to cut the high- and low-frequency tails of the spectral components, so that the amplitude of the laser field terminates suddenly at a certain point in time. The molecule follows the rotation of the laser polarization until this point and is released suddenly from the laser field to rotate freely like a boomerang. The other important trick is that they have employed a method known as coherent Raman scattering, which is induced by another short laser pulse to probe the molecular rotation. This allows the authors to directly observe, in real time, quantum coherence among multiple rotational eigenstates superposed coherently by the centrifuge pulse, in other words, a rotational wave packet. In addition, the Raman probe reveals the direction of the molecular rotation.
This ultrafast Raman probe was essential in allowing the authors to track the temporal evolution of the quantum coherence of the rotational wave packets during and after the centrifuge pulse. Remarkably, the quantum coherence lasts longer for the rotational quantum number than for by a factor of about , surviving as long as nanosecond at room temperature and atmospheric pressure. Milner and co-workers have thus verified that the quantum coherence of a superrotor is stable against collisions.
Studying why the coherence of the superrotor is more robust than that of a slow rotor may provide guidelines for the development of methods to decouple a quantum system from its environment. One possible reason for the quantum robustness of the superrotor may be its larger level spacing than that of a slow rotor; if so, researchers should be able to find a correlation between level spacing and coherence time.
The superrotor may also be useful for investigating many-body interactions. Theorists have predicted that the angular momenta of unidirectional molecular rotations could be converted to the angular momentum of a macroscopic vortex gas flow of an ensemble of those molecules [9]. The ultrahigh and unidirectional angular momenta of superrotors could allow us to see these vortices clearly. Coherent control started in isolated systems almost years ago, and is now moving on to many-body interacting systems [10]. The superrotor could offer many opportunities for us to look into those fundamental physics more deeply.
Aleksey Korobenko, Alexander A. Milner, and Valery Milner, “Direct Observation, Study, and Control of Molecular Superrotors,” Phys. Rev. Lett. 112, 113004 (2014)
K. Ohmori, “Wave-Packet and Coherent Control Dynamics,” Annu. Rev. Phys. Chem. 60, 487 (2009)
M. Shapiro, J. W. Hepburn, and P. Brumer, “Simplified Laser Control of Unimolecular Reactions: Simultaneous (
{\omega }_{1},{\omega }_{3}
) Excitation,” Chem. Phys. Lett. 149, 451 (1988)
D. J. Tannor, R. Kosloff, and S. A. Rice, “Coherent Pulse Sequence Induced Control of Selectivity of Reactions: Exact Quantum Mechanical Calculations,” J. Chem. Phys. 85, 5805 (1986)
J. Karczmarek, J. Wright, P. Corkum, and M. Ivanov, “Optical Centrifuge for Molecules,” Phys. Rev. Lett. 82, 3420 (1999)
D. M. Villeneuve, S. A. Aseyev, P. Dietrich, M. Spanner, M. Y. Ivanov, and P. B. Corkum, “Forced Molecular Rotation in an Optical Centrifuge,” Phys. Rev. Lett. 85, 542 (2000)
K. Kitano, H. Hasegawa, and Y. Ohshima, “Ultrafast Angular Momentum Orientation by Linearly Polarized Laser Fields,” Phys. Rev. Lett. 103, 223002 (2009)
O. Korech, U. Steinitz, R. J. Gordon, I. S. Averbukh, and Y. Prior, “Observing Molecular Spinning via the Rotational Doppler Effect,” Nature Photon. 7, 711 (2013)
U. Steinitz, Y. Prior, and I. S. Averbukh, “Laser-Induced Gas Vortices,” Phys. Rev. Lett. 109, 033001 (2012)
H. Katsuki et al., “All-Optical Control and Visualization of Ultrafast Two-Dimensional Atomic Motions in a Single Crystal of Bismuth,” Nature Commun. 4, 2801 (2013)
Kenji Ohmori is a Professor at the Institute for Molecular Science (IMS) of the National Institutes of Natural Sciences in Okazaki, Japan, where he is also the Chairman of the Department of Photo-Molecular Science. After receiving his Ph.D. from The University of Tokyo in 1992, he was a Research Associate and an Associate Professor at Tohoku University. In 2003 he was appointed a Full Professor at IMS. His research interests focus on exploring the quantum-classical boundary and developing quantum technology based on the wave nature of matter. He is currently leading a JST-CREST project in which he combines an ensemble of ultracold Rydberg atoms and ultrafast coherent control to look into the dynamics in the transition from a delocalized wave function to a localized particle. He has been honored with the Japan Academy Medal (2007), JSPS Prize (2007), is a Fellow of the American Physical Society (2009), and has received the Humboldt Research Award (2012).
Direct Observation, Study, and Control of Molecular Superrotors
Aleksey Korobenko, Alexander A. Milner, and Valery Milner
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22A10 Analysis on general topological groups
22A20 Analysis on topological semigroups
A coarse convergence group need not be precompact
Petr Simon, Fabio Zanolin (1987)
A completion functor for Cauchy groups.
Fric, R., Kent, Darrell C. (1981)
A density property of the tori and duality
A Dowker group
Klaas Pieter Hart, Heikki J. K. Junnila, Jan van Mill (1985)
A noncommutative Orlicz-Pettis type theorem
L. Drewnowski, I. Labuda (1981)
A non-standard metric in the group of reals
A. Schinzel (1986)
A note on embeddings manifolds into topological groups preserving dimensions
Hisao Kato (1992)
A quest for nice kernels of neighbourhood assignments
Raushan Z. Buzyakova, Vladimir Vladimirovich Tkachuk, Richard Gordon Wilson (2007)
Given a topological property (or a class)
𝒫
, the class
{𝒫}^{*}
𝒫
(with respect to neighbourhood assignments) consists of spaces
X
such that for any neighbourhood assignment
\left\{{O}_{x}:x\in X\right\}
Y\subset X
Y\in 𝒫
\bigcup \left\{{O}_{x}:x\in Y\right\}=X
. The spaces from
{𝒫}^{*}
are called dually
𝒫
. We continue the study of this duality which constitutes a development of an idea of E. van Douwen used to define
D
-spaces. We prove a number of results on duals of some general classes of spaces establishing, in particular, that any generalized ordered space...
A relatively free topological group that is not varietal free
Vladimir Pestov, Dmitri Shakhmatov (1998)
Answering a 1982 question of Sidney A. Morris, we construct a topological group G and a subspace X such that (i) G is algebraically free over X, (ii) G is relatively free over X, that is, every continuous mapping from X to G extends to a unique continuous endomorphism of G, and (iii) G is not a varietal free topological group on X in any variety of topological groups.
A solenoidal and monothetic minimally almost periodic group
J. Nienhuys (1971)
A solution to Comfort's question on the countable compactness of powers of a topological group
Artur Hideyuki Tomita (2005)
In 1990, Comfort asked Question 477 in the survey book “Open Problems in Topology”: Is there, for every (not necessarily infinite) cardinal number
\alpha \le {2}^{}
, a topological group G such that
{G}^{\gamma }
is countably compact for all cardinals γ < α, but
{G}^{\alpha }
is not countably compact? Hart and van Mill showed in 1991 that α = 2 answers this question affirmatively under
M{A}_{countable}
. Recently, Tomita showed that every finite cardinal answers Comfort’s question in the affirmative, also from
M{A}_{countable}
. However, the question has remained...
A Structure On A Set Of Subsets
R. Dacić (1968)
A survey on just-non-
𝔛
Otera, Daniele Ettore, Russo, Francesco G. (2010)
A Vanishing Theorem for Group Compactifications.
Elisabetta Strickland (1987)
Abelian pro-countable groups and orbit equivalence relations
Maciej Malicki (2016)
We study a class of abelian groups that can be defined as Polish pro-countable groups, as non-archimedean groups with a compatible two-sided invariant metric or as quasi-countable groups, i.e., closed subdirect products of countable discrete groups, endowed with the product topology. We show that for every non-locally compact, abelian quasi-countable group G there exists a closed L ≤ G and a closed, non-locally compact K ≤ G/L which is a direct product of discrete countable groups....
Actions of a locally compact group with zero II.
T.H.McH. Hanson (1971)
Algebraic obstructions to sequential convergence in Hausdorff abelian groups.
Clark, Bradd, Cates, Sharon (1998)
Almost all submaximal groups are paracompact and σ-discrete
O. Alas, I. Protasov, M. Tkačenko, V. Tkachuk, R. Wilson, I. Yaschenko (1998)
We prove that any topological group of a non-measurable cardinality is hereditarily paracompact and strongly σ-discrete as soon as it is submaximal. Consequently, such a group is zero-dimensional. Examples of uncountable maximal separable spaces are constructed in ZFC.
An Example of a Q-minimal Precompact Topological Group Containing a Nonminimal Closed Normal Subgroup.
Ulrich Schwanengel (1979)
Analytic Baire spaces
A. J. Ostaszewski (2012)
We generalize to the non-separable context a theorem of Levi characterizing Baire analytic spaces. This allows us to prove a joint-continuity result for non-separable normed groups, previously known only in the separable context.
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Raytracer – ebvalaim.log
The Raytracer is a program that lets you distort an arbitrary image as if there was a black hole in front of it.
"Select background" - this button lets you select the image that will be used as the background.
"Resolution" - the resolution of the resulting image, max. 512x512.
"Angular momentum/mass" - the ratio of the angular momentum of the black hole to its mass. The maximal value is one (and it is also the maximal physically meaningful value).
"FoV" - the field of view in degrees.
"Observer position" - the position of the observer. The numbers are: the distance from the black hole in Schwarzschild radii and two spherical angles (from 0 to
\pi
and from 0 to
2\pi
) - please refer to the Wikipedia article.
"Observer velocity" - the three components of the velocity of the observer: front-back, up-down and left-right.
"Observer orientation" - allows you to rotate the camera in place. The numbers correspond to the angles of up-down and left-right rotations. The setting (0,0) causes the camera to look straight at the black hole.
"Run" - starts rendering.
The background is treated as a panoramic photo of the whole field of view in all directions. So, first, it should have 2:1 proportions (if it doesn't, it will be stretched accordingly). Second, by default you see the fragment of the picture in the center. The left and right edges correspond to what is behind the observer, and up and down - to what is above and below them. It can be changed with the settings of position and orientation of the observer.
After the rendering is started, the program calculates the light deflection in the gravitational field of the black hole pixel by pixel. It does that by simulating the trajectories of light rays going in different directions. For increased performance it is done in multiple threads - as many as there are CPU cores in the running machine.
Download the program and have fun for yourself :)
Download “Raytracer (Win32)” raytracer-20150616-win32.zip – Downloaded 492 times – 6 MB
Download “Raytracer (Linux amd64)” raytracer-20150613-linux64.zip – Downloaded 424 times – 50 KB
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30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)
30F50 Klein surfaces
A criterion for the existence of a flat connection on a parabolic vector bundle.
Biswas, Indranil (2002)
A foliation of Teichmüller space by twist invariant disks.
Albert Marden, Howard Masur (1975)
Addendum to: A foliation of Teichmüller space by twist invariant disks.
Amenable coverings of complex manifolds and holomorphic probalitiy measures.
Curt McMullen (1992)
An extremal problem for harmonic measure.
Friedrich Huckemann (1977)
Analytic Self-Mapping Reducing to the Identity Mapping
Takao Kato (1974)
Bases for qualitativ differentials.
Irwin Kra, Bernard Maskit (1982)
Billards en polygones et groupes fuchsiens
Eugène Gutkin (1995/1996)
Cannon-Thurston Maps, i-bounded Geometry and a Theorem of McMullen
Mahan Mj (2009/2010)
The notion of i-bounded geometry generalises simultaneously bounded geometry and the geometry of punctured torus Kleinian groups. We show that the limit set of a surface Kleinian group of i-bounded geometry is locally connected by constructing a natural Cannon-Thurston map.
Enrico Arbarello, Joseph Harris (1981)
Classification of complete minimal surfaces in R3 with total curvature 12... .
C.J. Costa (1991)
Connected components of the strata of the moduli spaces of quadratic differentials
Erwan Lanneau (2008)
In two fundamental classical papers, Masur [14] and Veech [21] have independently proved that the Teichmüller geodesic flow acts ergodically on each connected component of each stratum of the moduli space of quadratic differentials. It is therefore interesting to have a classification of the ergodic components. Veech has proved that these strata are not necessarily connected. In a recent work [8], Kontsevich and Zorich have completely classified the components in the particular case where the quadratic...
Détermination du nombre des intégrales abéliennes de première espèce
Die Zerlegungscharaktere abelscher total reeller Erweiterungen reeller Funktionenkörper einer Variablen.
G. Martens (1977)
Differentials of the second kind for families of Mumford curves
Lothar Gerritzen (1983/1984)
David Cimasoni (2012)
\sum
be a flat surface of genus
g
with cone type singularities. Given a bipartite graph
\Gamma
isoradially embedded in
\sum
, we define discrete analogs of the
{2}^{2g}
Dirac operators on
\sum
. These discrete objects are then shown to converge to the continuous ones, in some appropriate sense. Finally, we obtain necessary and sufficient conditions on the pair
\Gamma \subset \sum
for these discrete Dirac operators to be Kasteleyn matrices of the graph
\Gamma
. As a consequence, if these conditions are met, the partition function of the dimer...
Einige Bemerkungen zum Weierstraßschen Lückensatz.
Andrei Duma (1976)
Erratum: Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards. Invent. Math. 97, 553-583 (1989).
W.A. Veech (1991)
Étude des intégrales abéliennes de troisième espèce
D. Emmanuel (1879)
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\left(r,p\right)
-absolutely summing operators on the space
C\left(T,X\right)
and applications.
A description of Banach space-valued Orlicz hearts
Coenraad Labuschagne, Theresa Offwood (2010)
Let Y be a Banach space, (Ω, Σ; μ) a probability space and φ a finite Young function. It is shown that the Y-valued Orlicz heart H φ(μ, Y) is isometrically isomorphic to the l-completed tensor product
{H}_{\varphi }\left(\mu \right){\stackrel{˜}{\otimes }}_{l}Y
of the scalar-valued Orlicz heart Hφ(μ) and Y, in the sense of Chaney and Schaefer. As an application, a characterization is given of the equality of
\left({H}_{\varphi }\left(\mu \right){\stackrel{˜}{\otimes }}_{l}Y\right)*
{H}_{\varphi }\left(\mu \right)*{\stackrel{˜}{\otimes }}_{l}Y*
in terms of the Radon-Nikodým property on Y. Convergence of norm-bounded martingales in H φ(μ, Y) is characterized in terms of the Radon-Nikodým...
p
{DPP}_{p}
1<p<\infty
L\left(X,Y\right)
W\left(X,Y\right)
K\left(X,Y\right)
U\left(X,Y\right)
{C}_{p}\left(X,Y\right)
p
X
Y
W\left(X,Y\right)
K\left(X,Y\right)
{C}_{p}\left(X,Y\right)
{C}_{p}\left(X,Y\right)
U\left(X,Y\right)
L\left(X,Y\right)
M
-ideals in certain algebras of operators.
Cho, Chong-Man, Roh, Woo Suk (2000)
{c}_{0}
A note on Riesz spaces with property-
b
Ş. Alpay, B. Altin, C. Tonyali (2006)
A unified approach to the strong approximation property and the weak bounded approximation property of Banach spaces
Aleksei Lissitsin (2012)
We consider convex versions of the strong approximation property and the weak bounded approximation property and develop a unified approach to their treatment introducing the inner and outer Λ-bounded approximation properties for a pair consisting of an operator ideal and a space ideal. We characterize this type of properties in a general setting and, using the isometric DFJP-factorization of operator ideals, provide a range of examples for this characterization, eventually answering a question...
Addendum to the paper The Projecitve Tenso Product II: The Radon-Nikodym Property.
J. Diestel, Fourie, J., J. Swart (2009)
Almost Weakly Compact Operators
Ioana Ghenciu, Paul Lewis (2006)
Dunford-Pettis type properties are studied in individual Banach spaces as well as in spaces of operators. Bibasic sequences are used to characterize Banach spaces which fail to have the Dunford-Pettis property. The question of whether a space of operators has a Dunford-Pettis property when the dual of the domain and the codomain have the respective property is studied. The notion of an almost weakly compact operator plays a consistent and important role in this study.
Amenability for dual Banach algebras
V. Runde (2001)
We define a Banach algebra 𝔄 to be dual if 𝔄 = (𝔄⁎)* for a closed submodule 𝔄⁎ of 𝔄*. The class of dual Banach algebras includes all W*-algebras, but also all algebras M(G) for locally compact groups G, all algebras ℒ(E) for reflexive Banach spaces E, as well as all biduals of Arens regular Banach algebras. The general impression is that amenable, dual Banach algebras are rather the exception than the rule. We confirm this impression. We first show that under certain conditions an amenable...
An approximation property with respect to an operator ideal
Juan Manuel Delgado, Cándido Piñeiro (2013)
Given an operator ideal , we say that a Banach space X has the approximation property with respect to if T belongs to
{\overline{S\circ T:S\in ℱ\left(X\right)}}^{{\tau }_{c}}
for every Banach space Y and every T ∈ (Y,X),
{\tau }_{c}
being the topology of uniform convergence on compact sets. We present several characterizations of this type of approximation property. It is shown that some of the existing approximation properties in the literature may be included in this setting.
An ideal characterization of when a subspace of certain Banach spaces has the metric compact approximation property
J. Cabello, E. Nieto (1998)
C.-M. Cho and W. B. Johnson showed that if a subspace E of
{\ell }_{p}
, 1 < p < ∞, has the compact approximation property, then K(E) is an M-ideal in ℒ(E). We prove that for every r,s ∈ ]0,1] with
{r}^{2}+{s}^{2}<1
, the James space can be provided with an equivalent norm such that an arbitrary subspace E has the metric compact approximation property iff there is a norm one projection P on ℒ(E)* with Ker P = K(E)⊥ satisfying ∥⨍∥ ≥ r∥Pf∥ + s∥φ - Pf∥ ∀⨍ ∈ ℒ(E)*. A similar result is proved for subspaces of upper p-spaces...
An introduction to polynomials on Banach spaces.
Richard Aron (2002)
Answer to a question by M. Feder about K(X,Y).
G. Emmanuele (1993)
We show that a Banach space constructed by Bourgain-Delbaen in 1980 answers a question put by Feder in 1982 about spaces of compact operators.
A. Pełczyński (1971)
Mikael Lindström, R.A. Ryan (1992)
Approximation of holomorphic mappings on infinite dimensional spaces.
Erhan Çaliskan (2004)
In this article we examine necessary and sufficient conditions for the predual of the space of holomorphic mappings of bounded type, Gb(U), to have the approximation property and the compact approximation property and we consider when the predual of the space of holomorphic mappings, G(U), has the compact approximation property. We obtain also similar results for the preduals of spaces of m-homogeneous polynomials, Q(mE).
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G. E. Chatzarakis, H. Péics, I. P. Stavroulakis, "Oscillations in Difference Equations with Deviating Arguments and Variable Coefficients", Abstract and Applied Analysis, vol. 2014, Article ID 902616, 9 pages, 2014. https://doi.org/10.1155/2014/902616
G. E. Chatzarakis,1 H. Péics,2 and I. P. Stavroulakis3
1Department of Electrical and Electronic Engineering Educators, School of Pedagogical and Technological Education (ASPETE), 14121 N. Heraklio, Athens, Greece
2Faculty of Civil Engineering, University of Novi Sad, 24000 Subotica, Serbia
New sufficient conditions for the oscillation of all solutions of difference equations with several deviating arguments and variable coefficients are presented. Examples illustrating the results are also given.
In this paper we study the oscillation of all solutions of difference equation with several variable retarded arguments of the form and the (dual) difference equation with several variable advanced arguments of the form where , , , are sequences of positive real numbers and , , are sequences of integers such that and , , are sequences of integers such that Here, denotes the forward difference operator and denotes the backward difference operator .
Strong interest in is motivated by the fact that it represents a discrete analogue of the differential equation (see [1–3] and the references cited therein) where, for every , is a continuous real-valued function in the interval and is a continuous real-valued function on such that while represents a discrete analogue of the advanced differential equation (see [1, 2] and the references cited therein) where, for every , is a continuous real-valued function in the interval and is a continuous real-valued function on such that
By a solution of , we mean a sequence of real numbers which satisfies for all . Here It is clear that, for each choice of real numbers , , , , there exists a unique solution of which satisfies the initial conditions , , , .
By a solution of , we mean a sequence of real numbers which satisfies for all .
A solution (or ) of (or ) is called oscillatory, if the terms of the sequence are neither eventually positive nor eventually negative. Otherwise, the solution is said to be nonoscillatory.
In the last few decades, the oscillatory behavior of the solutions of difference and differential equations with several deviating arguments and variable coefficients has been studied. See, for example, [1–14] and the references cited therein.
In 2006, Berezansky and Braverman [5] proved that if where , for all , then all solutions of oscillate.
Recently, Chatzarakis et al. [7–9] established the following theorems.
Theorem 1 (see [9]). Assume that the sequences , , are increasing, (1) (2) holds, and where , for all , , for all , or then all solutions of [] oscillate.
Theorem 2 (see [7, 8]). Assume that the sequences , , are increasing and (1) (2) holds. Set If , and or then all solutions of [] oscillate.
The authors study further and and derive new sufficient oscillation conditions. These conditions are the improved and generalized discrete analogues of the oscillation conditions for the corresponding differential equations, which were studied in 1982 by Ladas and Stavroulakis [2]. Examples illustrating the results are also given.
2. Oscillation Criteria
2.1. Retarded Difference Equations
We present new sufficient conditions for the oscillation of all solutions of .
Theorem 3. Assume that , , are increasing sequences of integers such that (1) holds and , , are sequences of positive real numbers and define , , by (11). If , , and then all solutions of oscillate.
Proof. Assume, for the sake of contradiction, that is a nonoscillatory solution of . Then it is either eventually positive or eventually negative. As is also a solution of , we may restrict ourselves only to the case where for all large . Let be an integer such that for all . Then, there exists such that In view of this, becomes which means that the sequence is eventually decreasing.
Next choose a natural number such that Set It is obvious that Now we will show that for . Indeed, assume that for some , . For this , by , we have
At this point, we will establish the following claim.
Claim 1 (cf. [8]). For each , there exists an integer for each such that , and where is an arbitrary real number with .
To prove this claim, let us consider an arbitrary real number with . Then by (11) we can choose an integer such that Assume, first, that and choose . Then . Moreover, we have and, by (23), So, (21) and (22) are fulfilled. Next, we suppose that . It is not difficult to see that (23) guarantees that . In particular, it holds Thus, as , there always exists an integer so that and (21) holds. We assert that . Otherwise, . We also have . Hence, in view of (27), we get On the other hand, (23) gives We have arrived at a contradiction, which shows our assertion that . Furthermore, by using (23) (for the integer ) as well as (27), we obtain and consequently (22) holds true. Our claim has been proved.
Now, summing up (20) from to , we find or Summing up (20) from to , we find or Combining (32) and (34), we obtain or which means that is bounded. This contradicts our assumption that . Therefore for every .
Dividing both sides of by , for , we obtain or Summing up (38) from to for , we find But or Combining (39) and (41), we obtain or Taking limit inferiors on both sides of the above inequalities (43), we obtain and by adding we find Set Clearly Since for the function has a maximum at the critical point since the quadratic form Since , the maximum of at the critical point should be nonnegative. Thus, that is, Hence or which contradicts (14).
Taking into account the fact that for (see proof of Theorem 3), by using (44) and the fact that we obtain Adding these inequalities we have or which contradicts (56).
2.2. Advanced Difference Equations
Similar oscillation theorems for the (dual) advanced difference equation can be derived easily. The proofs of these theorems are omitted, since they follow a similar procedure as in Section 2.1.
In the case where , , are positive real constants and are constant retarded arguments of the form , [ are constant advanced arguments of the form ], , , equation [] takes the form For this equation, as a consequence of Theorems 3 [5] and 4 [6], we have the following corollary.
Corollary 7. Assume that or Then all solutions of oscillate.
Remark 8. A research question that arises is whether Theorems 3–6 are valid, even in the case where the coefficients oscillate (see [15, 16]). Then our results would be comparable to those in [15, 16]. This is a question that we currently study and expect to have some results soon.
The following two examples illustrate that the conditions for oscillations (65) and (66) are independent. They are chosen in such a way that only one of them is satisfied.
Example 1. Consider the retarded difference equation Here , , , , and It is easy to see that That is, condition (65) of Corollary 7 is satisfied and therefore all solutions of equation (67) oscillate.
However, That is, condition (66) of Corollary 7 is not satisfied.
Observe that Thus Also, and therefore none of the conditions (9), (12), (13), (8), and (10) are satisfied.
Example 2. Consider the advanced difference equation Here , , , and It is easy to see that That is, condition (66) of Corollary 7 is satisfied and therefore all solutions of (74) oscillate.
Observe that Thus Also, and therefore none of the conditions (9), (12), and (13) are satisfied.
At this point, we give an example with general retarded arguments illustrating the main result of Theorem 3. Similarly, one can construct examples to illustrate Theorems 4–6.
Example 3. Consider the delay difference equation with .
Here and denote the integer parts of and . Observe that the sequences and are increasing, , , and Observe that, for a positive decreasing function , the following inequality holds: Based on the above inequality, we will show that for any , , and any real number . Indeed, It is easy to see that From the above, it follows that Therefore Hence That is, condition (14) of Theorem 3 is satisfied and therefore all solutions of (81) oscillate.
Observe, however, that Thus Also, and therefore none of the conditions (8), (9), (12), and (13) are satisfied.
The second author was supported by the Serbian Ministry of Science, Technology and Development for Scientific Research Grant no. III44006.
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M. K. Grammatikopoulos, R. Koplatadze, and I. P. Stavroulakis, “On the oscillation of solutions of first order differential equations with retarded arguments,” Georgian Mathematical Journal, vol. 10, no. 1, pp. 63–76, 2003. View at: Google Scholar | MathSciNet
B. R. Hunt and J. A. Yorke, “When all solutions of
{x}^{\text{'}}={\sum }_{j=1}^{n}{q}_{j}\left(t\right)x\left(t-{T}_{j}\left(t\right)\right)
oscillate,” Journal of Differential Equations, vol. 53, no. 2, pp. 139–145, 1984. View at: Publisher Site | Google Scholar | MathSciNet
J. Jaroš and I. P. Stavroulakis, “Necessary and sufficient conditions for oscillations of difference equations with several delays,” Utilitas Mathematica, vol. 45, pp. 187–195, 1994. View at: Google Scholar | MathSciNet
X. N. Luo, Y. Zhou, and C. F. Li, “Oscillation of a nonlinear difference equation with several delays,” Mathematica Bohemica, vol. 128, no. 3, pp. 309–317, 2003. View at: Google Scholar | MathSciNet
X. H. Tang and R. Y. Zhang, “New oscillation criteria for delay difference equations,” Computers & Mathematics with Applications, vol. 42, no. 10-11, pp. 1319–1330, 2001. View at: Publisher Site | Google Scholar | MathSciNet
L. Berezansky, G. E. Chatzarakis, A. Domoshnitsky, and I. P. Stavroulakis, “Oscillations of difference equations with several oscillating coefficients,” Abstract and Applied Analysis, vol. 2014, Article ID 392097, 9 pages, 2014. View at: Publisher Site | Google Scholar | MathSciNet
M. Bohner, G. E. Chatzarakis, and I. P. Stavroulakis, “Qualitative behavior of solutions of difference equations with several oscillating coefficients,” Arabian Journal of Mathematics, vol. 3, no. 1, pp. 1–13, 2014. View at: Publisher Site | Google Scholar | MathSciNet
Copyright © 2014 G. E. Chatzarakis et al. 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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46G15 Functional analytic lifting theory
46G25 (Spaces of) multilinear mappings, polynomials
A Borel extension approach to weakly compact operators on
{C}_{0}\left(T\right)
Thiruvaiyaru V. Panchapagesan (2002)
X
be a quasicomplete locally convex Hausdorff space. Let
T
be a locally compact Hausdorff space and let
{C}_{0}\left(T\right)=\left\{f\phantom{\rule{0.222222em}{0ex}}T\to I
f
is continuous and vanishes at infinity
\right\}
be endowed with the supremum norm. Starting with the Borel extension theorem for
X
\sigma
-additive Baire measures on
T
, an alternative proof is given to obtain all the characterizations given in [13] for a continuous linear map
u\phantom{\rule{0.222222em}{0ex}}{C}_{0}\left(T\right)\to X
to be weakly compact.
A Characterization of Weakly Sequentially Complete Banach Lattices.
V. Caselles (1985)
Márcia Federson (2002)
We slightly modify the definition of the Kurzweil integral and prove that it still gives the same integral.
A Criterion for an Operator with Real Spectrum to Be of Scalar-type.
Werner Ricker (1983)
A Ford-Fulkerson type theorem concerning vector-valued flows in infinite networks
Michael M. Neumann (1984)
A full characterization of multipliers for the strong
\rho
-integral in the euclidean space
Lee Tuo-Yeong (2004)
We study a generalization of the classical Henstock-Kurzweil integral, known as the strong
\rho
-integral, introduced by Jarník and Kurzweil. Let
\left({𝒮}_{\rho }\left(E\right),\parallel ·\parallel \right)
be the space of all strongly
\rho
-integrable functions on a multidimensional compact interval
E
, equipped with the Alexiewicz norm
\parallel ·\parallel
. We show that each element in the dual space of
\left({𝒮}_{\rho }\left(E\right),\parallel ·\parallel \right)
can be represented as a strong
\rho
-integral. Consequently, we prove that
fg
is strongly
\rho
-integrable on
E
for each strongly
\rho
-integrable function
f
g
is almost everywhere...
A Generalization Of Fubini's Theorem For Banach Algebra-Valued Measures.
T.V. Panchapagesan, Diómedes Barcenas (1984)
A Generalization of the Strict Topology.
J. Hoffmann-Jorgensen (1972)
A generalized Pettis measurability criterion and integration of vector functions
I. Dobrakov, T. V. Panchapagesan (2004)
For Banach-space-valued functions, the concepts of 𝒫-measurability, λ-measurability and m-measurability are defined, where 𝒫 is a δ-ring of subsets of a nonvoid set T, λ is a σ-subadditive submeasure on σ(𝒫) and m is an operator-valued measure on 𝒫. Various characterizations are given for 𝒫-measurable (resp. λ-measurable, m-measurable) vector functions on T. Using them and other auxiliary results proved here, the basic theorems of [6] are rigorously established.
A new relationship between decomposability and convexity
Bianca Satco (2006)
In the present work we prove that, in the space of Pettis integrable functions, any subset that is decomposable and closed with respect to the topology induced by the so-called Alexiewicz norm
\left|∥·∥\right|
\left|∥f∥\right|={sup}_{\left[a,b\right]\subset \left[0,1\right]}\parallel {\int }_{a}^{b}f\left(s\right)ds\parallel
) is convex. As a consequence, any such family of Pettis integrable functions is also weakly closed.
A Note on a Support of a Linear Mapping
Ljubomir Čukić (1993)
A note on copies of
{c}_{0}
in spaces of weak* measurable functions
Juan Carlos Ferrando (2000)
\left(\Omega ,\Sigma ,\mu \right)
is a finite measure space and
X
a Banach space, in this note we show that
{L}_{{w}^{*}}^{1}\left(\mu ,{X}^{*}\right)
, the Banach space of all classes of weak* equivalent
{X}^{*}
-valued weak* measurable functions
f
\Omega
\parallel f\left(\omega \right)\parallel \le g\left(\omega \right)
a.e. for some
g\in {L}_{1}\left(\mu \right)
equipped with its usual norm, contains a copy of
{c}_{0}
{X}^{*}
contains a copy of
{c}_{0}
A Note on Smoothability in Banach Spaces.
Daniel C. Kemp (1975)
A note on the construction of measures taking their values in a Banach space with basis.
María Congost Iglesias (1983)
If E is a Banach space with a basis {en}, n belonging to N, a vector measure m: a --> E determines a sequence {mn}, n belonging to N, of scalar measures on a named its components. We obtain necessary and sufficient conditions to ensure that when given a sequence of scalar measures it is possible to construct a vector valued measure whose components were those given. Furthermore we study some relations between the variation of the measure m and the variation of its components.
2
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Specify Conditional Mean Models - MATLAB & Simulink - MathWorks 日本
{\mathrm{Î}}^{D}{y}_{t}=c+{\mathrm{Ï}}_{1}{\mathrm{Î}}^{D}{y}_{tâ1}+â¦+{\mathrm{Ï}}_{p}{\mathrm{Î}}^{D}{y}_{tâp}+{\mathrm{θ}}_{1}{\mathrm{ε}}_{tâ1}+â¦+{\mathrm{θ}}_{q}{\mathrm{ε}}_{tâq}+{\mathrm{ε}}_{t}.
\mathrm{Ï}\left(L\right){\left(1âL\right)}^{D}{y}_{t}=c+\mathrm{θ}\left(L\right){\mathrm{ε}}_{t}
\left(1â{\mathrm{Ï}}_{1}Lâ{\mathrm{Ï}}_{2}{L}^{2}\right){\left(1âL\right)}^{1}{y}_{t}=c+\left(1+{\mathrm{θ}}_{1}L\right){\mathrm{ε}}_{t},
\mathrm{Ï}\left(L\right){\left(1âL\right)}^{D}{y}_{t}=c+\mathrm{θ}\left(L\right){\mathrm{ε}}_{t}.
\mathrm{Ï}\left(L\right){y}_{t}={c}^{â}+{x}_{t}^{â²}\mathrm{β}+{\mathrm{θ}}^{â}\left(L\right){\mathrm{ε}}_{t},
where c* = c/(1–L)D and θ*(L) = θ(L)/(1–L)D.
If you specify a nonzero D, then Econometrics Toolbox differences the response series yt before the predictors enter the model. You should preprocess the exogenous covariates xt by testing for stationarity and differencing if any are unit root nonstationary. If any nonstationary exogenous covariate enters the model, then the false negative rate for significance tests of β can increase.
For the distribution of the innovations, εt, there are two choices:
Independent and identically distributed (iid) Gaussian or Student’s t with a constant variance,
{\mathrm{Ï}}_{\mathrm{ε}}^{2}
Dependent Gaussian or Student’s t with a conditional variance process,
{\mathrm{Ï}}_{t}^{2}
{y}_{t}=c+{\mathrm{Ï}}_{1}{y}_{tâ1}+{\mathrm{ε}}_{t}
{\mathrm{ε}}_{t}={\mathrm{Ï}}_{\mathrm{ε}}{z}_{t}
{y}_{t}={\mathrm{ε}}_{t}+{\mathrm{θ}}_{1}{\mathrm{ε}}_{tâ1}+{\mathrm{θ}}_{2}{\mathrm{ε}}_{tâ2}
{\mathrm{ε}}_{t}={\mathrm{Ï}}_{\mathrm{ε}}{z}_{t}
\left(1â0.8L\right)\left(1âL\right){y}_{t}=0.2+\left(1+0.6L\right){\mathrm{ε}}_{t}
{\mathrm{ε}}_{t}=0.1{z}_{t}
\left(1+0.5L\right){\left(1âL\right)}^{1}\mathrm{Î}{y}_{t}={x}_{t}^{â²}\left[\begin{array}{c}â5\\ 2\end{array}\right]+{\mathrm{ε}}_{t}
{\mathrm{ε}}_{t}~N\left(0,1\right)
{\mathrm{Ï}}_{1},â¦,{\mathrm{Ï}}_{p}
{y}_{t}=0.8{y}_{tâ1}â0.2{y}_{tâ2}+{\mathrm{ε}}_{t},
{y}_{t}={\mathrm{Ï}}_{1}{y}_{tâ1}+{\mathrm{Ï}}_{12}{y}_{tâ12}+{\mathrm{ε}}_{t},
{\mathrm{Ï}}_{1}=0.6
{\mathrm{Ï}}_{12}=â0.3,
\mathrm{β}={\left[\begin{array}{ccc}0.5& 7& â2\end{array}\right]}^{â²}.
Distribution Distribution of the innovation process Use this argument to specify a Student’s t innovation distribution. By default, the innovation distribution is Gaussian.
{\mathrm{θ}}_{1},â¦,{\mathrm{θ}}_{q}
{y}_{t}={\mathrm{ε}}_{t}+0.5{\mathrm{ε}}_{tâ1}+0.2{\mathrm{ε}}_{tâ2},
{y}_{t}={\mathrm{ε}}_{t}+{\mathrm{θ}}_{1}{\mathrm{ε}}_{tâ1}+{\mathrm{θ}}_{4}{\mathrm{ε}}_{tâ4},
{\mathrm{θ}}_{1}=0.5
{\mathrm{θ}}_{4}=0.2,
{\mathrm{Ï}}_{\mathrm{ε}}^{2}
{\mathrm{Ï}}_{t}^{2}
{\mathrm{Ï}}_{\mathrm{ε}}^{2}
{\mathrm{Ï}}_{t}^{2}
\mathrm{Φ}\left(L\right)=\left(1â{\mathrm{Φ}}_{1}{L}^{{p}_{1}}ââ¦â{\mathrm{Φ}}_{{p}_{s}}{L}^{{p}_{s}}\right)
\mathrm{Î}\left(L\right)=\left(1+{\mathrm{Î}}_{1}{L}^{{q}_{1}}+â¦+{\mathrm{Î}}_{{q}_{s}}{L}^{{q}_{s}}\right)
\mathrm{Ï}\left(L\right)=\left(1â{\mathrm{Ï}}_{1}Lââ¦â{\mathrm{Ï}}_{p}{L}^{p}\right)
\mathrm{θ}\left(L\right)=\left(1+{\mathrm{θ}}_{1}L+â¦+{\mathrm{θ}}_{q}{L}^{q}\right).
\mathrm{Ï}\left(L\right)\mathrm{Φ}\left(L\right){\left(1âL\right)}^{D}\left(1â{L}^{s}\right){y}_{t}=c+\mathrm{θ}\left(L\right)\mathrm{Î}\left(L\right){\mathrm{ε}}_{t}.
The innovation series can be an independent or dependent Gaussian or Student’s t process. The arima default for the innovation distribution is an iid Gaussian process with constant (scalar) variance.
{\mathrm{Φ}}_{1},â¦,{\mathrm{Φ}}_{{p}_{s}}
\left(1â0.8L\right)\left(1â0.2{L}^{12}\right){y}_{t}={\mathrm{ε}}_{t},
\left(1â\mathrm{Ï}L\right)\left(1â{\mathrm{Φ}}_{12}{L}^{12}\right){y}_{t}={\mathrm{ε}}_{t},
{\mathrm{Î}}_{1},â¦,{\mathrm{Î}}_{{q}_{s}}
{y}_{t}=\left(1+0.6L\right)\left(1+0.2{L}^{12}\right){\mathrm{ε}}_{t},
{y}_{t}=\left(1+{\mathrm{θ}}_{1}L\right)\left(1+{\mathrm{Î}}_{4}{L}^{4}\right){\mathrm{ε}}_{t},
Seasonality Seasonal periodicity, s To specify the degree of seasonal integration s in the seasonal differencing polynomial Δs = 1 – Ls. For example, to specify the periodicity for seasonal integration of monthly data, specify 'Seasonality',12.
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Physics - Entangling Photons with Mismatched Colors
The University of Queensland, Brisbane, Queensland 4072, Australia
Figure 1: To entangle independent photons of different color, researchers use down-conversion to generate two pairs of polarization-entangled photons, labeled (
A
a
B
b
). The frequencies of each pair differ by either
40
80\phantom{\rule{0.333em}{0ex}}\text{MHz}
. Photons a and b enter a beam splitter, resulting in a “hypoentangled” state of polarization and frequency. Two detectors record the arrival times,
{t}_{1}
{t}_{2}
, of the two photons, and this information is fed forward to a device that applies a time-dependent phase shift to photon
A
. In the end, the entanglement of
and
b
is swapped to
A
B
.To entangle independent photons of different color, researchers use down-conversion to generate two pairs of polarization-entangled photons, labeled (
A
a
B
b
40
80\phantom{\rule{0.333em}{0ex}}\text{MHz}
. Photons a and b enter a... Show more
Entanglement has been capturing the imagination of scientists as well as the public ever since it first emerged from the newly minted quantum mechanics formalism eight decades ago. Two particles are considered entangled when their states cannot be described separately, thus allowing counterintuitive action-at-a-distance whenever one particle is measured. Entanglement powers certain quantum information technologies that ultimately promise to outperform classical devices. The prerequisite to engineering quantum processors, simulators, and sensors is our ability to generate entanglement between large numbers of quantum systems such as photons. Typically, this entanglement-generation requires that the photons be indistinguishable. However, a new experiment described in Physical Review Letters manages, for the first time, to entangle independently created photons that have different frequencies, or colors [1]. The results could be useful for entangling photons from arrays of nonidentical quantum dots, which often emit photons with slightly different frequencies.
Photons are often entangled at their creation. For example, pairs of identical photons can be generated by “down-converting” single pump photons in a nonlinear crystal. Unfortunately, this process is inefficient because the down-conversion has a low probability. One alternative is to entangle photons coming from different sources. This can be done by sending two photons into a beam splitter and taking advantage of the quantum phenomenon of two-photon interference: If an observer behind the beam splitter cannot tell which path each individual photon took, their probabilistic outcomes can add up or cancel out, leaving those photons that exit the beam splitter through different ports in an entangled state. This effect has been the workhorse of the quantum photonics community for decades [2].
To achieve perfect interference, the photons involved must be indistinguishable when they are eventually measured. However, contrary to common intuition, they don’t necessarily have to be indistinguishable when they hit the beam splitter; they may well have very different properties—different color, polarization, spatial mode, etc.,—and yet we can still recover interference if our detectors are insensitive to those properties, or if the respective information is “erased” before detection. Tiang-Ming Zhao and colleagues from the University of Science and Technology of China in Hefei have developed a clever method to erase frequency information, so that photons from distinct sources and of different color can become entangled through their interference [1].
The heart of their idea works as follows: two polarization-encoded photons with slightly different frequencies are sent into a polarizing beam splitter. If the photons were indistinguishable, the output would be entangled (assuming the photons leave from different ports). But the photons in Zhao et al.’s experiment can of course be distinguished by their color. In consequence, the beam splitter will not output the photons in a clean polarization-entangled state but rather in what has sometimes been called a “hypoentangled” state [3], in which polarization and color are intrinsically linked to each other. If one tried to measure either the color or the polarization independently, the remaining state would end up being mixed, i.e., not entangled. One can however try to carefully tease out 1 degree of freedom selectively and keep the entanglement alive in the other degree.
This technique has been applied before in a very similar scenario involving color and polarization by my colleagues and me [3]. In this previous case, we subjected our hypoentangled state to a joint polarization projection, and the output was a discretely color-entangled two-photon state. Here, in contrast, Zhao et al. erase the color information, leaving the eventual state entangled in polarization. They do this by making high-resolution time measurements at the outputs of their beam splitter. Time is related to frequency, and the frequency information in the incoming state can just as well be interpreted as timing information. If we simply averaged over the arrival times of the two photons behind the beam splitter indiscriminately, the remaining state would be mixed. However, the authors show that picking a certain timing combination allows the remaining state to stay intact and indeed be entangled in polarization. The researchers’ timing protocol effectively erases the frequency-distinguishing information about the photons and allows them to create a final state that is entangled up to a time-dependent phase factor that can be corrected for.
One problem is that the frequency eraser measures the photons, so they are no longer available for evaluation of their polarization entanglement. To get around this, the researchers included an entanglement-swapping protocol [4] that transfers the entanglement to two other photons, which can be subsequently evaluated. Specifically, their experiment starts with the down-conversion of two pump beams that each produce a pair of polarization-entangled photons with wavelengths around nanometers (see Fig. 1). The first pair, labeled and , differs from the second pair, labeled and , by a frequency of either or megahertz ( ), which is to times the linewidth of the photons. The team sends photons and into a beam splitter to produce a hypoentangled state of polarization and frequency. The frequency-distinguishing information is erased by two photon detectors on the far side of the beam splitter. The photon arrival times, and , are forwarded to an optical component that adds a time-dependent phase shift to photon . As a result, the polarization entanglement between and is transferred with high fidelity to and , as verified by the authors with polarization measurements of and .
The results could be of interest in efforts to build scalable devices from sophisticated photon sources such as quantum dots. Quantum dots can generate photons virtually deterministically, but they do suffer from the drawback that dots cannot be created entirely identical and hence create photons with slightly different spectral characteristics. It would be very exciting indeed if temporal quantum erasure could be used to create large-scale entangled clusters of photons emitted from dissimilar quantum dots.
It is still a long way to go from here, though, as the current scheme will face many challenges before it becomes practicable. The main concern is whether it will be able to entangle photons with an efficiency of at least —the rule-of-thumb threshold required for scalability of photonic architectures. Another issue is that the finite temporal resolution of the photon detectors constrains the maximal spectral separation that can be erased in the time-resolved two-photon interference measurement. For the best available conventional photon detectors, the allowed spectral separation is on the order of . Therefore, one would need quantum dots that emit photons within a few of each other. Current dot fabrication is starting to reach that level of precision, but engineers may well remove the remaining indistinguishability between sources by external tuning, via temperature or electric field controls [5]. What is perhaps a more interesting venue for the research presented here is its fundamental angle, and in particular the possibility of applying more sophisticated temporal filters in the detection, which might be useful for quantum state engineering.
Tian-Ming Zhao, Han Zhang, Jian Yang, Zi-Ru Sang, Xiao Jiang, Xiao-Hui Bao, and Jian-Wei Pan, “Entangling Different-Color Photons via Time-Resolved Measurement and Active Feed Forward,” Phys. Rev. Lett. 112, 103602 (2014)
C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of Subpicosecond Time Intervals Between Two Photons by Interference,” Phys. Rev. Lett. 59, 2044 (1987)
S. Ramelow, L. Ratschbacher, A. Fedrizzi, N. K. Langford, and A. Zeilinger, “Discrete Tunable Color Entanglement,” Phys. Rev. Lett. 103, 253601 (2009)
C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an Unknown Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels,” Phys. Rev. Lett. 70, 1895 (1993)
R. B. Patel, A. J. Bennett, I. Farrer, C. A. Nicoll, D. A. Ritchie, and A. J. Shields, “Two-Photon Interference of the Emission from Electrically Tunable Remote Quantum Dots,” Nature Photon. 4, 632 (2010)
Alessandro Fedrizzi is a postdoctoral research fellow at the University of Queensland, Australia. He applies photonic engineering to topics ranging from fundamental inquiries to implementations of quantum computing, quantum simulation, and quantum communication. Alessandro received his Ph.D. from the University of Vienna in 2008, upon which he joined the Quantum Technology Laboratory in Queensland. He currently holds an Australian Research Council (ARC) Discovery Early Career Research Award, and is associated with the ARC Centres of Excellence for Engineered Quantum Systems (EQuS) and for Quantum Computation and Communication Technology (CQC2T).
Entangling Different-Color Photons via Time-Resolved Measurement and Active Feed Forward
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16N20 Jacobson radical, quasimultiplication
16N40 Nil and nilpotent radicals, sets, ideals, rings
16N60 Prime and semiprime rings
16N80 General radicals and rings
A chain of Kurosh may have an arbitrary finite length
Konstantin Igorevich Beidar (1982)
A concrete analysis of the radical concept.
de la Rosa, B., van Niekerk, J.S., Wiegandt, R. (1992)
𝒜
𝒜
ℳ
A generalization of the prime radical of an ideal.
A generalized Picard group for prime rings
S. Montgomery (1990)
A new approach to orders in simple rings with minimal one-sided ideals.
M. Petrich, V. Gould (1990)
A non-semiprime associative algebra with zero weak radical.
Abdelfattah Haily (1997)
The weak radical, W-Rad(A) of a non-associative algebra A, has been introduced by A. Rodríguez Palacios in [3] in order to generalize the Johnson's uniqueness of norm theorem to general complete normed non-associative algebras (see also [2] for another application of this notion). In [4], he showed that if A is a semiprime non-associative algebra with DCC on ideals, then W-Rad(A) = 0. In the first part of this paper we give an example of a non-semiprime associative algebra A with DCC on ideals and...
A note on centralizers.
Bell, Howard E. (2000)
A note on derivations in semiprime rings.
Vukman, Joso, Kosi-Ulbl, Irena (2005)
A Note on Posner s Theorem with Generalized Derivations on Lie Ideals
Vincenzo De Filippis, M. S. Tammam El-Sayiad (2009)
A Note on Radicals and Polynomial Rings.
B.J. Gardner (1972)
A note on rings which are multiplicatively generated by idempotents and nilpotents.
Abu-Khuzam, Hazar (1988)
A note on rings with certain variable identities.
A Note on Semi-prime Rings.
Roger Yue Chi Ming (1986)
A note on semiprime rings with derivation.
Hongan, Motoshi (1997)
A note on the endomorphism ring of a module artinian with respect to a preradical
Josef Jirásko (1985)
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3
{E}^{2}
2
3
{E}^{3}
{S}^{3}
{E}^{4}
4
{E}^{n}
n
4<n<\infty
{S}^{n-1}\subset {E}^{n}
3
A complete description of normal surfaces for infinite series of 3-manifolds.
Fominykh, E.A. (2002)
A conjecture on Khovanov's invariants
Stavros Garoufalidis (2004)
We formulate a conjectural formula for Khovanov's invariants of alternating knots in terms of the Jones polynomial and the signature of the knot.
A criterion for homeomorphism between closed Haken manifolds.
Derbez, Pierre (2003)
A filtration of the set of integral homological 3-spheres.
Ohtsuki, Tomotada (1998)
A finite graphic calculus for 3-manifolds.
Riccardo Benedetti, Carlo Petronio (1995)
A gauge-field approach to 3- and 4-manifold invariants
Bogusław Broda (1997)
An approach to construction of topological invariants of the Reshetikhin-Turaev-Witten type of 3- and 4-dimensional manifolds in the framework of SU(2) Chern-Simons gauge theory and its hidden (quantum) gauge symmetry is presented.
H. Doll (1992)
A geometric invariant of discrete groups.
W.D. Neumann, R. Bieri, R. Strebel (1987)
A locally connected non-movable continuum that fails to separate
{E}_{3}
D. McMillan (1977)
A new invariant on hyperbolic Dehn surgery space.
Dowty, James G. (2002)
A Note on 2-fold Branched Covering Spaces of S3.
Taizo Kanenobu (1981)
A note on a 3-dimensional homogeneous space
J. H. Rubinstein, C. Gardiner (1979)
A note on irreducible Heegaard diagrams.
Cavicchioli, Alberto, Spaggiari, Fulvia (2006)
A polynomial invariant of rational homology 3-spheres.
Tomotada Ohtsuki (1996)
François Laudenbach (2014)
Heegaard splittings and Heegaard diagrams of a closed 3-manifold
M
are translated into the language of Morse functions with Morse-Smale pseudo-gradients defined on
M
. We make use in a very simple setting of techniques which Jean Cerf developed for solving a famous pseudo-isotopy problem. In passing, we show how to cancel the supernumerary local extrema in a generic path of functions when
dimM>2
. The main tool that we introduce is an elementary swallow tail lemma which could be useful elsewhere.
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Wikipedia:Manual of Style/Music
14 Names of organizations and institutions
Use either the {{music}} template flat {{music|flat}} (♭) and sharp {{music|sharp}} (♯) symbols or the words flat and sharp. According to The Unicode Standard 5.0, chapter 15.11, these are distinct from b (the lowercase letter b) or # (the number sign), hence b and # should not be used to indicate "flat" or "sharp". This template has the advantage of working in Microsoft Internet Explorer; see Template:Music for details. Examples:
The {{music}} template is recommended for the natural sign, {{music|natural}} produces ♮, and for double sharps and flats, {{music|doublesharp}} and {{music|doubleflat}} produce
. Either {{music|flat stroke}}
or {{music|halfflat}}
may be used instead of ♭ for a half flat, while {{music|halfsharp}}
Chords, progressions, and figured bassEdit
The degree symbol, "°", indicates a diminished chord. It can be copied and pasted or inserted from the menus above or below Wikipedia edit boxes on desktop web browsers. It can also be produced by typing °, or (on Windows PCs) Alt+0176 on the numeric pad / (Mac) Option+Shift+8. A superscript lower case "o" (<sup>o</sup>) may be used instead. The slashed o, "ø", which may not display correctly for all readers, is produced by superscripting the character produced by typing ø, Alt+0248 (Windows), or Option+o (Mac). For both of these there is an application of the {{music}} template: {{music|dim}} becomes o and {{music|dimslash}} becomes ø (e.g. Co and Cø).
Superscript and subscript may be combined, as in figured bass, in math markup, <math>\mathrm C_4^6</math> renders as
{\displaystyle \mathrm {C} _{4}^{6}}
, or <chem>C4^6</chem> renders as
{\displaystyle {\ce {C4^6}}}
{see: Help:Displaying a formula or m:Help:Displaying a formula; also: Template:SubSup).
Classical music titlesEdit
"Wenn ich in deine Augen seh'" from Dichterliebe—note that the trailing apostrophe and the ending quote are handled using the {{'"}} template, to insert some spacing between the characters without using an unsemantic space character: ... Augen seh{{'"}} ...
Generic movement titles (such as tempo markings or terms like minuet and trio) are capitalized with a single initial capital—that is, only the first word is capitalized—and are not italicized. Often, movements are described by multiple tempo markings. In this case, the tempo markings should be separated by en dashes set off by spaces (consider using the {{spaced en dash}} template), and the first letter of each tempo marking should be capitalized. True movement titles are enclosed in quotation marks. Once again, foreign language terms are not italicized.
Opus, work, and measure numbersEdit
Use a non-breaking space ( or {{nbsp}} between the abbreviation and number, instead of a regular space.
DiscographiesEdit
Equivalent terms in different varieties of EnglishEdit
Images and notationEdit
The sharp (♯) and flat (♭) signs are {{music|sharp}} and {{music|flat}}, respectively. A natural (♮) can be entered with {{music|natural}}. As noted above, these symbols should not be used in headings or articles titles.
Superscript and subscript may be combined, as in figured bass, in math markup: <math>C_6^4</math> =
{\displaystyle C_{6}^{4}}
. See Wikipedia:TeX markup and m:Help:Formula.
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Italian music termsEdit
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The names of works, and other terms, should be marked up with the {{lang}} template, using the appropriate two-letter language code. For example, to link to the article for the work "Deutschlandlied", use "{{lang|de|[[Deutschlandlied]]}}", which will appear as "Deutschlandlied".
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Air-Side Heat Transfer Enhancement Utilizing Design Optimization and an Additive Manufacturing Technique | J. Heat Transfer | ASME Digital Collection
Martinus A. Arie,
Martinus A. Arie
e-mail: martinus@umd.edu
Amir H. Shooshtari,
e-mail: amir@umd.edu
Veena V. Rao,
e-mail: vrao@umd.edu
Serguei V. Dessiatoun,
Serguei V. Dessiatoun
e-mail: ser@umd.edu
Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received January 18, 2016; final manuscript received October 21, 2016; published online December 28, 2016. Assoc. Editor: Danesh K. Tafti.
Arie, M. A., Shooshtari, A. H., Rao, V. V., Dessiatoun, S. V., and Ohadi, M. M. (December 28, 2016). "Air-Side Heat Transfer Enhancement Utilizing Design Optimization and an Additive Manufacturing Technique." ASME. J. Heat Transfer. March 2017; 139(3): 031901. https://doi.org/10.1115/1.4035068
This paper focuses on the study of an innovative manifold microchannel design for air-side heat transfer enhancement that uses additive manufacturing (AM) technology. A numerical-based multi-objective optimization was performed to maximize the coefficient of performance and gravimetric heat transfer density (
Q/MΔT
) of air–water heat exchanger designs that incorporate either manifold-microchannel or conventional surfaces for air-side heat transfer enhancement. Performance comparisons between the manifold-microchannel and conventional heat exchangers studied under the current work show that the design based on the manifold-microchannel in conjunction with additive manufacturing promises to push the performance substantially beyond that of conventional technologies. Different scenarios based on manufacturing constraints were considered to study the effect of such constraints on the heat exchanger performance. The results clearly demonstrate that the AM-enabled complex design of the fins and manifolds can significantly improve the overall performance, based on the criteria described in this paper. Based on the current manufacturing limit, up to nearly 60% increase in gravimetric heat transfer density is possible for the manifold-microchannel heat exchanger compared to a wavy-fin heat exchanger. If the manufacturing limit (fin thickness and manifold width) can be reduced even further, an even larger improvement is possible.
Extended surfaces, Heat exchangers, Heat transfer enhancement, Heat transfer , Two-phase flow , Manufacturing, Heat transfer
Heat exchangers, Heat transfer, Manifolds, Microchannels, Water, Optimization, Manufacturing, Additive manufacturing, Design
Numerical Investigation of Thermal Enhancement in a Micro Heat Sink With Fan-Shaped Reentrant Cavities and Internal Ribs
Thermal-Flow Characteristics of the New Wave-Finned Flat Tube Bundles in Air-Cooled Condensers
Heat Transfer Enhancement by Winglet-Type Vortex Generator Arrays in Compact Plain-Fin-and-Tube Heat Exchangers
Heat Transfer Enhancement of Backstep Flow by Means of EHD Conduction Pumping
Thermal-Hydraulic Performance of Metal Foam Heat Exchangers
, West Lafayette, IN, July 16–19.
Numerical Study of Fluid Flow and Heat Transfer in the Enhanced Microchannel With Oblique Fins
Micro-Channel Heat Exchanger Optimization
Semiconductor Thermal Measurement and Management Symposium
Numerical Analysis of Heat Transfer in a Manifold Microchannel Heat Sink With High Efficient Copper Heat Spreader
Experimental Investigation of Heat Transfer Performance of a Manifold Microchannel Heat Sink for Cooling of Concentrated Solar Cells
59th Electronic Components and Technology Conference, San Diego, CA, May 26–29, pp.
Force Fed Microchannel High Heat Flux Cooling Utilizing Microgrooved Surface
, University of Maryland, College Park, MD.
Numerical Investigation and Sensitivity Analysis of Manifold Microchannel Coolers
Next Generation Micro Channel Heat Exchangers
, 1st, ed.,
Simulation and Thermal Optimization of a Manifold Microchannel Flat Plate Heat Exchanger
Numerical Modeling and Optimization of Single Phase Manifold-Microchannel Plate Heat Exchanger
Andhare
Experimental Heat Transfer and Pressure Drop Characteristic of a Single Phase Manifold-Microchannel Plate Heat Exchanger
,” M.S. thesis, University of Maryland, College Park, MD.
Paper No. V003T23A003.
Thermal Optimization of an Air-Cooling Heat Exchanger Utilizing Manifold-Microchannels
Thermal and Thermomechanical Phenomena in Electronic Systems
), Orlando, FL, May 27-30, pp.
Numerical Modeling and Thermal Optimization of a Single-Phase Flow Manifold-Microchannel Plate Heat Exchanger
Experimental Characterization of a Nickel Alloy-Based Manifold-Microgroove Evaporator
Heat Transfer and Pressure Drop Characteristics of a Flat Plate Manifold Microchannel Heat Exchanger in Counter Fow Configuration
,” Enertron, Inc., The Linde Group, Gilbert, AZ, accessed Dec. 2015, http://www.enertron-inc.com/resources-folded-fins.aspx
,” Cooler Master, Tacherting, Germany, accessed Dec. 2015, http://odm.coolermaster.com/manufacture.php?page_id=6
Aluminium Plate-Fin Heat Exchangers
,” Tacherting, Germany, accessed Dec. 2015, http://www.lindeus-engineering.com/internet.le.le.usa/en/images/P_3_2_e_12_150dpi136_5772.pdf
Fabrication of High Aspect Ratio Microstructure Arrays by Micro Reverse Wire-EDM
Fabrication of Micro-Channel Arrays on Thin Metallic Sheet Using Internal Fluid Pressure: Investigations on Size Effects and Development of Design Guidelines
Fabrication, Assembly, and Testing of Cu-and Al-Based Microchannel Heat Exchangers
Fabrication of Monolithic Microchannels for IC Chip Cooling
Forced Air Cooling by Using Manifold Microchannel Heat Sinks
, Istanbul, Turkey, Sept. 24–25.
Acceleration Threshold Switches From an Additive Electroplating MEMS Process
Options for Additive Rapid Prototyping Methods (3D Printing) in MEMS Technology
Scaffold Design and In Vitro Study of Osteochondral Coculture in a Three-Dimensional Porous Polycaprolactone Scaffold Fabricated by Fused Deposition Modeling
An Electrochemical Fabrication Process for the Assembly of Anisotropically Oriented Collagen Bundles
Laser Sintering Ushers in New Route to PM Parts
Metal Powder Report (MPR)
, 3rd, ed.,
, Stratasys Ltd., private communication.
, Baltimore Aircoil Company, private communication.
NSF/EPRI Joint Solicitation-Informational Webcast
,” EPRI and NSF, accessed Dec. 2015, http://mydocs.epri.com/docs/PublicMeetingMaterials/1346/NSF_EPRI_Pres.pdf
|
Energetically Consistent Calculations for Oblique Impact in Unbalanced Systems With Friction | J. Appl. Mech. | ASME Digital Collection
Energetically Consistent Calculations for Oblique Impact in Unbalanced Systems With Friction
Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received February 24, 2015; final manuscript received April 25, 2015; published online June 9, 2015. Editor: Yonggang Huang.
Stronge, W. J. (August 1, 2015). "Energetically Consistent Calculations for Oblique Impact in Unbalanced Systems With Friction." ASME. J. Appl. Mech. August 2015; 82(8): 081003. https://doi.org/10.1115/1.4030459
Analytical mechanics is used to derive original 3D equations of motion that represent impact at a point in a system of rigid bodies. For oblique impact between rough bodies in an eccentric (unbalanced) configuration, these equations are used to compare the calculations of energy dissipation obtained using either the kinematic, the kinetic, or the energetic coefficient of restitution (COR);
eN,eP
e*
. Examples demonstrate that for equal energy dissipation by nonfrictional sources, either
eN≤e*≤eP
eP≤e*≤eN
depending on whether the unbalance of the impact configuration is positive or negative relative to the initial direction of slip. Consequently, when friction brings initial slip to rest during the contact period, calculations that show energy gains from impact can result from either the kinematic or the kinetic COR. On the other hand, the energetic COR always correctly accounts for energy dissipation due to both hysteresis of the normal contact force and friction, i.e., it is energetically consistent.
Dynamics, Impact
Compression, Friction, Impulse (Physics), Kinetic energy
Dynamics of a System of Rigid Bodies
Discontinuity-Induced Bifurcations in Systems With Impact and Friction: Discontinuities in the Impact Law
Rigid-Body Collisions With Friction
A Dynamics Puzzle
Stanford Mechanics Alumni Club Newsletter
Dynamics: Theory and Applications
Two Dimensional Rigid-Body Collisions With Friction
Impact Dynamics of Multibody Systems With Frictional Contact Using Joint Coordinates and Canonical Equations of Motion
Collisions of Constrained Rigid Body Systems With Friction
Frictional Impact Analysis in Open-Loop Multi-Body Mechanical Systems
Collision With Friction: Part A: Newton's Hypotheses
Energetic Consistency Conditions for Standard Impacts, Part II: Poisson-Type Inequality Impact Laws
Treatment of Impact With Friction in Planar Multibody Mechanical Systems
Collision With Friction: Part B: Poisson's and Stronge's Hypotheses
On Newton's and Poisson's Rules of Percussive Dynamics
Painlevé Paradox During Oblique Impact With Friction
The Painlevé Paradox Studied at a 3D Slender Rod
Multibody Impact Motion With Friction—Analysis, Simulation and Experimental Validation
Restitution and Friction Laws in Rigid Body Collisions
An Energy Based Coefficient of Restitution for Planar Impacts of Slender Bar With Massive External Surfaces
Numerical Simulation of the Dynamics of an Impacting Bar
Oblique Frictional Impact of a Bar: Analysis and Comparison of Different Impact Laws
The Bounds on the Coefficients of Restitution for Frictional Impact of a Rigid Pendulum Against a Fixed Surface
ASME J Appl. Mech.
Analysis and Computation of Two Body Impact in Three Dimensions
Impacts With Friction
Effect of Pipeline Thermal Expansion on Direct Electric Heating Cable
|
Order of approximation - Wikipedia
(Redirected from Orders of approximation)
Expressions for approximation accuracy
In science, engineering, and other quantitative disciplines, order of approximation refers to formal or informal expressions for how accurate an approximation is.
1 Usage in science and engineering
1.1 Zeroth-order
1.2 First-order
1.3 Second-order
Usage in science and engineering[edit]
In formal expressions, the ordinal number used before the word order refers to the highest power in the series expansion used in the approximation. The expressions: a zeroth-order approximation, a first-order approximation, a second-order approximation, and so forth are used as fixed phrases. The expression a zero order approximation is also common. Cardinal numerals are occasionally used in expressions like an order zero approximation, an order one approximation, etc.
The omission of the word order leads to phrases that have less formal meaning. Phrases like first approximation or to a first approximation may refer to a roughly approximate value of a quantity.[1][2] The phrase to a zeroth approximation indicates a wild guess.[3] The expression order of approximation is sometimes informally used to mean the number of significant figures, in increasing order of accuracy, or to the order of magnitude. However, this may be confusing as these formal expressions do not directly refer to the order of derivatives.
The choice of series expansion depends on the scientific method used to investigate a phenomenon. The expression order of approximation is expected to indicate progressively more refined approximations of a function in a specified interval. The choice of order of approximation depends on the research purpose. One may wish to simplify a known analytic expression to devise a new application or, on the contrary, try to fit a curve to data points. Higher order of approximation is not always more useful than the lower one. For example, if a quantity is constant within the whole interval, approximating it with a second-order Taylor series will not increase the accuracy.
In the case of a smooth function, the nth-order approximation is a polynomial of degree n, which is obtained by truncating the Taylor series to this degree. The formal usage of order of approximation corresponds to the omission of some terms of the series used in the expansion (usually the higher terms). This affects accuracy. The error usually varies within the interval. Thus the numbers zeroth, first, second etc. used formally in the above meaning do not directly give information about percent error or significant figures.
Zeroth-order[edit]
Zeroth-order approximation is the term scientists use for a first rough answer. Many simplifying assumptions are made, and when a number is needed, an order-of-magnitude answer (or zero significant figures) is often given. For example, you might say "the town has a few thousand residents", when it has 3,914 people in actuality. This is also sometimes referred to as an order-of-magnitude approximation. The zero of "zeroth-order" represents the fact that even the only number given, "a few", is itself loosely defined.
A zeroth-order approximation of a function (that is, mathematically determining a formula to fit multiple data points) will be constant, or a flat line with no slope: a polynomial of degree 0. For example,
{\displaystyle x=[0,1,2]\,}
{\displaystyle y=[3,3,5]\,}
{\displaystyle y\sim f(x)=3.67\,}
could be – if data point accuracy were reported – an approximate fit to the data, obtained by simply averaging the x-values and the y-values. However, data points represent results of measurements and they do differ from points in Euclidean geometry. Thus quoting an average value containing three significant digits in the output with just one significant digit in the input data could be recognized as an example of false precision. With the implied accuracy of the data points of ±0.5, the zeroth order approximation could at best yield the result for y of ~3.7±2.0 in the interval of x from -0.5 to 2.5, considering the standard deviation.
If the data points are reported as
{\displaystyle x=[0.00,1.00,2.00]\,}
{\displaystyle y=[3.00,3.00,5.00]\,}
the zeroth-order approximation results in
{\displaystyle y\sim f(x)=3.67\,}
The accuracy of the result justifies an attempt to derive a multiplicative function for that average, for example,
{\displaystyle y\sim \ x+2.67}
One should be careful though because the multiplicative function will be defined for the whole interval. If only three data points are available, one has no knowledge about the rest of the interval, which may be a large part of it. This means that y could have another component which equals 0 at the ends and in the middle of the interval. A number of functions having this property are known, for example y = sin πx. Taylor series is useful and helps predict an analytic solution but the approximation alone does not provide conclusive evidence.
First-order[edit]
[3]First-order approximation is the term scientists use for a slightly better answer. Some simplifying assumptions are made, and when a number is needed, an answer with only one significant figure is often given ("the town has 4×103 or four thousand residents"). In the case of a first-order approximation, at least one number given is exact. In the zeroth order example above, the quantity "a few" was given but in the first order example, the number "4" is given.
A first-order approximation of a function (that is, mathematically determining a formula to fit multiple data points) will be a linear approximation, straight line with a slope: a polynomial of degree 1. For example,
{\displaystyle x=[0.00,1.00,2.00]\,}
{\displaystyle y=[3.00,3.00,5.00]\,}
{\displaystyle y\sim f(x)=x+2.67\,}
is an approximate fit to the data. In this example there is a zeroth order approximation that is the same as the first order but the method of getting there is different; i.e. a wild stab in the dark at a relationship happened to be as good as an 'educated guess'.
Second-order[edit]
Second-order approximation is the term scientists use for a decent-quality answer. Few simplifying assumptions are made, and when a number is needed, an answer with two or more significant figures ("the town has 3.9×103 or thirty-nine hundred residents") is generally given. In mathematical finance, second-order approximations are known as convexity corrections. As in the examples above, the term "2nd order" refers to the number of exact numerals given for the imprecise quantity. In this case, "3" and "9" are given as the two successive levels of precision, instead of simply the "4" from the first order, or "a few" from the zeroth-order found in the examples above.
A second-order approximation of a function (that is, mathematically determining a formula to fit multiple data points) will be a quadratic polynomial, geometrically, a parabola: a polynomial of degree 2. For example,
{\displaystyle x=[0.00,1.00,2.00]\,}
{\displaystyle y=[3.00,3.00,5.00]\,}
{\displaystyle y\sim f(x)=x^{2}-x+3\,}
is an approximate fit to the data. In this case, with only three data points, a parabola is an exact fit based on the data provided. However, the data points for most of the interval are not available, which advises caution (see "zeroth order").
Higher-order[edit]
While higher-order approximations exist and are crucial to a better understanding and description of reality, they are not typically referred to by number.
Continuing the above, a third-order approximation would be required to perfectly fit four data points, and so on. See polynomial interpolation.
These terms are also used colloquially by scientists and engineers to describe phenomena that can be neglected as not significant (e.g. "Of course the rotation of the Earth affects our experiment, but it's such a high-order effect that we wouldn't be able to measure it" or "At these velocities, relativity is a fourth-order effect that we only worry about at the annual calibration.") In this usage, the ordinality of the approximation is not exact, but is used to emphasize its insignificance; the higher the number used, the less important the effect. The terminology, in this context, represents a high level of precision required to account for an effect which is inferred to be very small when compared to the overall subject matter. The higher the order, the more precision is required to measure the effect, and therefore the smallness of the effect in comparison to the overall measurement.
Chapman-Enskog method
^ first approximation in Webster's Third New International Dictionary, Könemann, ISBN 3-8290-5292-8
^ to a first approximation in Online Dictionary and Translations Webster-dictionary.org
^ a b to a zeroth approximation in Online Dictionary and Translations Webster-dictionary.org
Retrieved from "https://en.wikipedia.org/w/index.php?title=Order_of_approximation&oldid=1082482727"
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Calculation Inputs - AMSET Documentation
AMSET Documentation
Calculation Inputs Calculation Inputs Table of contents
Dense uniform band structure and wave function coefficients
Dielectric constants, piezoelectric constants and polar-phonon frequency
Scattering Rates
Calculation Inputs¶
Structural relaxation¶
In order to obtain accurate results, the crystal structure should first be relaxed using "tight" calculation settings including high force and energy convergence criteria. Note, that this can often be expensive for very large structures.
VASP settings for tight convergence
Dense uniform band structure and wave function coefficients¶
AMSET should be run on a vasprun.xml file from a "dense" uniform band structure calculation. Typically a k-point mesh density at least twice that needed to converge the total energy will be necessary to converge transport properties. Note this refers to the initial DFT mesh before Fourier interpolation. In order to obtain accurate band gaps often a hybrid DFT functional such as HSE06 is required.
Wave function coefficients are required to calculate wave function overlaps. This requires the WAVECAR file to be written by VASP (achieved by setting LWAVE = True). Wave function coefficients can then be extracted using the amset wave command. Coefficients are stored in the wavefunction.h5 file.
VASP settings for uniform calculations
LWAVE = True
Elastic constants¶
Elastic constants can be calculated using finite differences in VASP. It is very important to first relax the structure using tight convergence settings, as described in the structural relaxation section. Details on the finite difference approach in VASP can be found on the IBRION documentation page.
VASP settings for elastic constants
Deformation potentials¶
The absolute deformation potential describes the change in energy of the bands with change in volume and is calculated as
\mathbf{D}_{n\mathbf{k}} = \delta \varepsilon_{n\mathbf{k}} / \delta S_{\alpha\beta}
\mathbf{S}
is the uniform stress tensor. The deformation potential should be averaged over contraction (–0.5 %) and expansion (+0.5 %) of the lattice and calculated separately for each component of the strain tensor. To account for shifts in the average electrostatic potential between deformed cells, the eigenvalues are aligned to the average energy level of the core states.
AMSET includes a tool to assist with the calculation of the deformation potentials. The initial input is a "tight" optimised structure as described in the structural relaxation section. Deformed structures are generated using the amset deform create command, which will generate a list of POSCARs each corresponding to a component of the strain tensor. Note that symmetry is automatically used to reduce the number of calculations needed. A single point calculation (no relaxation, i.e., NSW = 0) should be performed for each deformed POSCAR as well as the undeformed structure.
VASP settings for deformation calculations
ICORELEVEL = 1 # needed to write the core levels to OUTCAR
The deformation potentials can be calculated using the amset deform read command. This requires the paths to the undeformed and deformation calculations as inputs. The undeformed folder should be specified first, followed by the deformation folders. For example,
amset deform read undeformed def-1 def-2 def-3
This will write the deformations potentials to a deformation.h5 file in the current directory. You can specify to use this file when calculating scattering rates by setting the deformation_potential option to "deformation.h5". See the settings page for more details.
Dielectric constants, piezoelectric constants and polar-phonon frequency¶
Static and high-frequency dielectric constants, piezoelectric constants, and the "effective polar phonon frequency" can be obtained using density functional perturbation theory (DFPT). It is very important to first relax the structure using tight convergence settings, as described in the structural relaxation section. Details on DFPT in VASP can be found on the IBRION and LEPSILON documentation pages.
VASP settings for dielectric constants and phonon frequency
LEPSILON = True
Note, DFPT cannot be used with hybrid exchange-correlation functionals. In these cases the LCALCEPS flag should be used in combination with IBRION = 6.
The dielectric constants and polar phonon frequency can be extracted from the VASP outputs using the command:
amset phonon-frequency
The command should be run in a folder containing the vasprun.xml file output from the DFPT calculation.
The effective phonon frequency is determined from the phonon frequencies
\omega_{\mathbf{q}\nu}
\nu
is a phonon branch and
\mathbf{q}
is a phonon wave vector) and eigenvectors
\mathbf{e}_{\kappa\nu}(\mathbf{q})
\kappa
is an atom in the unit cell). In order to capture scattering from the full phonon band structure in a single phonon frequency, each phonon mode is weighted by the dipole moment it produces according to
w_{\nu} = \sum_\kappa \left [ \frac{1}{M_\kappa \omega_{\mathbf{q}\nu}} \right]^{1/2} \times \left[ \mathbf{q} \cdot \mathbf{Z}_\kappa^* \cdot \mathbf{e}_{\kappa\nu}(\mathbf{q}) \right ]
\mathbf{Z}_\kappa^*
is the Born effective charge. This naturally suppresses the contributions from transverse-optical and acoustic modes in the same manner as the more general formalism for computing Frölich based electron-phonon coupling.
The weight is calculated only for
\Gamma
-point phonon frequencies and averaged over the full unit sphere to capture both the polar divergence at
\mathbf{q} \rightarrow 0
and any anisotropy in the dipole moments. The effective phonon frequency is calculated as the weighted sum over all
\Gamma
-point phonon modes according to
\omega_\mathrm{po} = \frac{\omega_{\Gamma\nu} w_{\nu}}{\sum_{\nu} w_\nu}.
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Cryptography | Special Issue : Privacy-Preserving Techniques in Cloud/Fog and Internet of Things
Privacy-Preserving Techniques in Cloud/Fog and Internet of Things
Submit to Special Issue Submit Abstract to Special Issue Review for Cryptography Edit a Special Issue
Special Issue "Privacy-Preserving Techniques in Cloud/Fog and Internet of Things"
Deadline for manuscript submissions: 1 August 2022 | Viewed by 3042
Department of Library and Information Science, Fu Jen Catholic University, New Taipei City 24205, Taiwan
Interests: data security; cryptography; network security; mobile communications and computing; wireless communications
Special Issue in Sensors: Security and Privacy in Wireless Sensor Networks: Advances and Challenges
Special Issue in Electronics: Methods, Challenges and Opportunities in IoT/Network Security for Post-quantum World
Department of Computer Science, Islamic Azad University, Tehran, Iran
Interests: privacy-preserving on the Internet of Things (IoT); wireless sensor networks; security; smart city
Dr. Mohammad Javad Shayegan
Computer Engineering Department, University of Science and Culture, Tehran, Iran
Interests: cryptography; BlockChain; Web; data science; distributed systems
Dr. Milad Taleby Ahvanooey
Faculty Member at IMShool, Nanjing University, Nanjing, China
Interests: applied cryptography; information hiding; text mining & retrieval; malware analysis; misinformation detection; authentication systems of smartphones; IoT security
Department of Computer Science and Technology, Harbin Institute of Technology, Shenzhen, China
Interests: security and privacy problems; the privacy issues related to mobile and IoT devices
Recently, wireless networks have been developed using cloud infrastructure and software-based networks. Their connections to the new generation Internet and the Internet of Things have reduced costs and improved reliability. It is critical for people, factories, vehicles, road and transportation environments, and much more to use IoT sensors and devices for daily tasks in these vast and complex networks. It is also important to leverage privacy-preservation patterns in large networks such as BigData, software-based networks. Several supporting technologies for IoT are Cloud Computing and Fog Computing. However, the possibility of privacy breaches in these three technologies is high. The main purpose of this leading series is to present and compile articles about this topic that can help us.
This Special Issue is looking for original articles that are not under consideration for publication elsewhere. The "Submit Online" button on the journal's submission page allows authors to follow the journal's formatting and submission instructions. Please mention that your article is for this Special Issue in your cover letter.
The following topics are possible, but not limited to:
Privacy-preserving models for big data networks
Privacy-preserving models for sensor networks and the Internet of Things
Privacy-preserving mathematical models for IoT networks and Cloud Computing
Privacy-preservation in Fog Computing
Machine learning methods for privacy-preservation
{C}_{ANNL}
\left({C}_{ANLU}\right)
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Direct Measurement of the Permeability of Human Cervical Tissue | J. Biomech Eng. | ASME Digital Collection
Deptarment of Mechanical Engineering,
e-mail: jyv2101@columbia.edu
e-mail: ky2218@columbia.edu
e-mail: rw2191@columbia.edu
Kristin M. Myers 1
Contributed by the Bioengineering Division of ASME for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received October 3, 2012; final manuscript received January 4, 2013; accepted manuscript posted January 18, 2013; published online February 7, 2013. Editor: Victor H. Barocas.
Fernandez, M., Vink, J., Yoshida, K., Wapner, R., and Myers, K. M. (February 7, 2013). "Direct Measurement of the Permeability of Human Cervical Tissue." ASME. J Biomech Eng. February 2013; 135(2): 021024. https://doi.org/10.1115/1.4023380
The mechanical integrity of the uterine cervix is critical for a pregnancy to successfully reach full term. It must be strong to retain the fetus throughout gestation and then undergo a remodeling and softening process before labor for delivery of the fetus. It is believed that cervical insufficiency (CI), a condition in pregnancy resulting in preterm birth (PTB), is related to a cervix with compromised mechanical strength which cannot resist deformation caused by external forces generated by the growing fetus. Such PTBs are responsible for infant developmental problems and in severe cases infant mortality. To understand the etiologies of CI, our overall research goal is to investigate the mechanical behavior of the cervix. Permeability is a mechanical property of hydrated collagenous tissues that dictates the time-dependent response of the tissue to mechanical loading. The goal of this study was to design a novel soft tissue permeability testing device and to present direct hydraulic permeability measurements of excised nonpregnant (NP) and pregnant (PG) human cervical tissue from women with different obstetric histories. Results of hydraulic permeability testing indicate repeatability for specimens from single patients, with an order of magnitude separating the NP and PG group means (
2.1 ± 1.4×10-14
3.2 ± 4.8×10-13m4/N·s
, respectively), and large variability within the NP and PG sample groups. Differences were found between samples with similar obstetric histories, supporting the view that medical history may not be a good predictor of permeability (and therefore mechanical behavior) and highlighting the need for patient-specific measurements of cervical mechanical properties. The permeability measurements from this study will be used in future work to model the constitutive material behavior of cervical tissue and to develop in vivo diagnostic tools to stage the progression of labor.
Biological tissues, Permeability
March of Dimes, PMNCH, Save the Children, WHO. Born Too Soon: The Global Action Report on Preterm Birth
National Vital Statistics Reports: Births: Final Data for 2006, CDC/National Center for Health Statistics
. Available at: http://www.cdc.gov/nchs/data/nvsr/nvsr57/nvsr57_07.pdf.
Glucose Regulation in Young Adults With Very Low Birth Weight
N. Engl J. Med.
.10.1056/NEJMoa067187
Real-Time Sonoelastography of the Cervix: Tissue Elasticity of the Normal and Abnormal Cervix
White-Traut
Quantitative Ultrasound Assessment of the Rat Cervix
Quantitative Ultrasound Assessment of Cervical Microstructure
Estimate of the Attenuation Coefficient using a Clinical Array Transducer for the Detection of Cervical Ripening in Human Pregnancy
Beyond Cervical Length: Emerging Technologies for Assessing the Pregnant Cervix
.10.1016/j.ajog.2012.05.015
The Effect of Pregnancy and Labor on the Human Cervix: Changes in Collagen, Glycoproteins, and Glycosaminoglycans
Glycoconjugates (Glycosaminoglycans and Glycoproteins) and Glycogen in the Human Cervix Uteri
Connective Tissue Changes in the Cervix during Normal Pregnancy and Pregnancy Complicated by Cervical Incompetence
Tzeranis
Changes in the Biochemical Constituents and Morphologic Appearance of the Human Cervical Stroma during Pregnancy
.10.1016/j.ejogrb.2009.02.008
Elastin and Collagen in the Human Uterus and Cervix
The Extracellular Matrix of the Uterus, Cervix and Fetal Membranes: Synthesis, Degradation and Hormonal Regulation
Physical and Biomechanical Characteristics of Rat Cervical Ripening are not Consistent with Increased Collagenase Activity
Relationships Between Mechanical Properties and Extracellular Matrix Constituents of the Cervical Stroma During Pregnancy
Cervical Softening during Pregnancy-Regulated Changes in Collagen Cross-Linking and Composition of Matricellular Proteins in the Mouse
.10.1095/biolreprod.110.089599
Connective Tissue Changes Incident to Cervical Effacement
The Morphology of the Human Cervix
Cervical Softening, Effacement, and Dilatation: A Complex Biochemical Cascade
Anatomy and Physiology of Cervical Ripening
The Biochemistry and Physiology of the Uterine Cervix during Gestation and Parturition
Prenat. Neonat. Med.
.10.1016/j.tem.2010.01.011
.10.1210/en.2010-1105
Cervical Remodeling in Term and Preterm Birth: Insights from an Animal Model
Ex Vivo Assessment of Mouse Cervical Remodeling Through Pregnancy via 23Na MRS
The Anisotropic Hydraulic Permeability of Human Lumbar Anulus Fibrosus: Influence of Age, Degeneration, Direction, and Water Content
Proceedings of the ASME Summer Bioengineering Conference
Viscoelastic and Poroelastic Mechanical Characterization of Hydrated Gels
Al-Sharji
.10.1093/biomet/73.1.13
Estimating Equations for Parameters in Means and Covariances of Multivariate Discrete and Continuous Responses
Second Harmonic Generation Imaging as a Potential Tool for Staging Pregnancy and Predicting Preterm Birth
Kehtarnavaz
Three-Dimensional Fiber Architecture of the Nonpregnant Human Uterus Determined Ex Vivo Using Magnetic Resonance Diffusion Tensor Imaging
.10.1002/ar.a.20274
Collagen Organization in the Cervix and Its Relation to Mechanical Function
Coll. Relat. Res.
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ℋ
A Hadamard type theorem for the space-time.
Frattarolo, Walter (1994)
A model of gauge theory with invariant variables.
Oprisan, Cristian-Dan (2006)
An introduction to the Einstein-Vlasov system
Alan Rendall (1997)
Philippe Bechouche, Nicolas Besse (2010)
We consider the spherically symmetric Vlasov-Einstein system in the case of asymptotically flat spacetimes. From the physical point of view this system of equations can model the formation of a spherical black hole by gravitational collapse or describe the evolution of galaxies and globular clusters. We present high-order numerical schemes based on semi-Lagrangian techniques. The convergence of the solution of the discretized problem to the exact solution is proven and high-order error estimates...
Around the bounded
{L}^{2}
curvature conjecture in general relativity
Sergiu Klainerman, Igor Rodnianski, Jeremie Szeftel (2008)
We report on recent progress obtained on the construction and control of a parametrix to the homogeneous wave equation
{\square }_{\mathbf{g}}\phi =0
\gg
is a rough metric satisfying the Einstein vacuum equations. Controlling such a parametrix as well as its error term when one only assumes
{L}^{2}
bounds on the curvature tensor
\mathbf{R}
\gg
is a major step towards the proof of the bounded
{L}^{2}
curvature conjecture.
Blow-up for solutions of hyperbolic PDE and spacetime singularities
Alan D. Rendall (2000)
An important question in mathematical relativity theory is that of the nature of spacetime singularities. The equations of general relativity, the Einstein equations, are essentially hyperbolic in nature and the study of spacetime singularities is naturally related to blow-up phenomena for nonlinear hyperbolic systems. These connections are explained and recent progress in applying the theory of hyperbolic equations in this field is presented. A direction which has turned out to be fruitful is that...
Boundary conditions for the Einstein-Christoffel formulation of Einstein's equations.
Arnold, Douglas N., Tarfulea, Nicolae (2007)
Cauchy data on a manifold
Yvonne Choquet-Bruhat, Demetrios Christodoulou, Mauro Francaviglia (1978)
Characteristic evolution and matching.
Winicour, Jeffrey (2009)
Living Reviews in Relativity [electronic only]
Compact spacelike hypersurfaces with constant mean curvature in the anti de Sitter space.
de Lima, Henrique F., de Lima, Joseilson R. (2009)
Constraint-preserving boundary conditions for the linearized Baumgarte-Shapiro-Shibata-Nakamura formulation.
Alekseenko, Alexander M. (2008)
Duality-symmetric approach to general relativity and supergravity.
Nurmagambetov, Alexei J. (2006)
Einstein Metrics, Spinning Top Motions and Monopoles.
H. Pedersen (1986)
3+1
Thornburg, Jonathan (2007)
Formulazione intrinseca del problema di Cauchy in relatività generale
Giorgio Ferrarese (1988)
Viene stabilita una formulazione intrinseca del problema di Cauchy in Relatività generale, per uno spazio-tempo riemanniano descritto da un mezzo continuo globale e non-polare. In termini di variabili proprie: metrica, velocità angolare e di deformazione, densità di pura materia, flusso termico e temperatura. Vengono altresì precisate le condizioni iniziali per i dati di Cauchy su una assegnata superficie spaziale
{\sigma }_{3}
; condizioni in involuzione nel senso d'E. Cartan, le quali mettono in evidenza,...
Gauge symmetries of an extended phase space for Yang-Mills and Dirac fields
Günter Schwarz, Jędrzej Śniatycki (1997)
Global existence of solutions of the Einstein-Boltzmann equation in the spatially homogeneous case
Piotr Mucha (2000)
Hyperbolic methods for Einstein's equations.
Reula, Oscar A. (1998)
Initial data for numerical relativity.
Cook, Gregory B. (2000)
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Determine the speed of the CPU, as well as latencies and bandwidths of caches and main memory. These provides ideal, but real-world values against which the performance of other routines can be compared.
1 Measuring Maximum Speeds
This thorn measures the maximum practical speed that can be attained on a particular system. This speed will be somewhat lower than the theoretical peak performance listed in a system’s hardware description.
This thorn measures
CPU floating-point peformance (GFlop/s),
CPU integer peformance (GIop/s),
Cache/memory read latency (ns),
Cache/memory read bandwidth (GByte/s),
Cache/memory write latency (ns),
Cache/memory write bandwidth (GByte/s).
Theoretical performance values for memory access are often quoted in slightly different units. For example, bandwidth is often measured in GT/s (Giga-Transactions per second), where a transaction transfers a certain number of bytes, usually a cache line (e.g. 64 bytes).
A detailed understanding of the results requires some knowledge of how CPUs, caches, and memories operate. [1] provides a good introduction to this as well as to benchmark design. [2] is also a good read (by the same authors), and their (somewhat dated) software lmbench is available here [3].
We use the following algorithms to determine the maximum speeds. We believe these algorithms and their implementations are adequate for current architectures, but this may need to change in the future.
Each benchmark is run
N
N
is automatically chosen such that the total run time is larger than
1
second. (If a benchmark finishes too quickly, then
N
is increased and the benchmark is repeated.)
2.1 CPU floating-point peformance
CPUs for HPC systems are typically tuned for dot-product-like operations, where multiplications and additions alternate. We measure the floating-point peformance with the following calculation:
for (int i=0; i<N; ++i) {
s := c_1 * s + c_2
s
is suitably initialized and
{c}_{1}
{c}_{2}
are suitably chosed to avoid overflow, e.g.
s=1.0
{c}_{1}=1.1
{c}_{2}=-0.1
s
is a double precision variable. The loop over
is explicitly unrolled
8
times, is explicitly vectorized using LSUThorns/Vectors, and uses fma (fused multiply-add) instructions where available. This should ensure that the loops runs very close to the maximum possible speed. As usual, each (scalar) multiplication and addition is counted as one Flop (floating point operation).
2.2 CPU integer performance
Many modern CPUs can handle integers in two different ways, treating integers either as data, or as pointers and array indices. For example, integers may be stored in two different sets of registers depending on their use. We are here interested in the performance of pointers and array indices. Most modern CPUs cannot vectorize these operations (some GPUs can), and we therefore do not employ vectorization in this benchmark.
In general, array index calculations require addition and multiplication. For example, accessing the element
A\left(i,j\right)
of a two-dimensional array requires calculating
i+{n}_{i}\cdot j
{n}_{i}
is the number of elements allocated in the
i
However, general integer multiplcations are expensive, and are not necessary if the array is accessed in a loop, since a running index
p
can instead be kept. Accessing neighbouring elements (e.g. in stencil calculations) require only addition and multiplication with small constants. In the example above, assuming that
p
is the linear index corresponding to
A\left(i,j\right)
, accessing
A\left(i+1,j\right)
requires calculating
p+1
, and accessing
A\left(i,j+2\right)
p+2\cdot {n}_{i}
. We thus base our benchmark on integer additions and integer multiplications with small constants.
We measure the floating-point peformance with the following calculation:
s := b + c * s
b
is a constant defined at run time, and
c
is a small integer constant (
c=1\dots 8
) known at compile time.
s
is an integer variable of the same size as a pointer, i.e. 64 bit on a 64-bit system. The loop over
is explicitly unrolled
8
times, each time with a different value for
c
. Each addition and multiplication is counted as one Iop (integer operation).
2.3 Cache/memory read latency
Memory read access latency is measured by reading small amounts of data from random locations in memory. This random access pattern defeats caches, because caching does not work for random access patterns. To ensure that the read operations are executed sequentially, each read operation needs to depend on the previous. The idea for the algorithm below was taken from [1].
To implement this, we set up a large linked list where the elements are randomly orderd. Traversing this linked list then measures the memory read latency. This is done as in the following pseudo-code:
struct L { L* next; };
... set up large circular list ...
L* ptr = head;
To reduce the overhead of the for loop, we explicitly unroll the loop 100 times.
We use the hwloc library to determine the sizes of the available data caches, the NUMA-node-local, and the global amount of memory. We perform this benchmark once for each cache level, and once each for the local and global memory:
for a cache, the list occupies 3/4 of the cache;
for the local memory, the list occupies 1/2 of the memory;
for the global memory, the list skips the local memory, and occupies 1/4 of the remaining global memory.
To skip the local memory, we allocate an array of the size of the local memory. Assuming that the operating system prefers to allocate local memory, this will then ensure that all further allocations will then use non-local memory. We do not test this assumption.
2.4 Cache/memory read bandwidth
Memory read access bandwidth is measured by reading a large, contiguous amount of data from memory. This access pattern benefits from caches (if the amount is less than the cache size), and also benefits from prefetching (that may be performed either by the compiler or by the hardware). This presents thus an ideal case where memory is read as fast as possible.
To ensure that data are actually read from memory, it is necessary to consume the data, i.e. to perform some operations on them. We assume that a floating-point dot-product is among the fastest operations, and thus use the following algorithm:
for (int i=0; i<N; i+=2) {
s := m[i] * s + m[i+1]
As in section 2.1 above,
s
is a double precision variable.
m\left[i\right]
denotes the memory accesses. The loop over
is explicitly unrolled
8
times, is explicitly vectorized using LSUThorns/Vectors, and uses fma (fused multiply-add) instructions where available. This should ensure that the loops runs very close to the maximum possible speed.
To measure the bandwidth of each cache level as well as the local and global memory, the same array sizes as in section 2.3 are used.
2.5 Cache/memory write latency
The notion of a “write latency” does not really make sense, as write operations to different memory locations do not depend on each other. This benchmark thus rather measures the speed at which independent write requests can be handled. However, since writing partial cache lines also requires reading them, this benchmark is also influence by read performance.
To measure the write latency, we use the following algorithm, writing a single byte to random locations in memory:
char array[N];
char* ptr = ...;
ptr += ...;
In the loop, the pointer is increased by a pseudo-random amount, but ensuring that it stays within the bound of the array.
To measure the bandwidth of each cache level as well as the local and global memory, the same array sizes as in section 2.3 are used as starting point. For efficiency reasons, these sizes are then rounded down to the nearest power of two.
2.6 Cache/memory write bandwidth
Memory write access bandwidth is measured by writing a large, contiguous amount of data from memory, in a manner very similar to measuring read bandwidth. The major difference is that the written data do not need to be consumed by the CPU, which simplifies the implementation.
We use memset to write data into an array, assuming that the memset function is already heavily optimized.
This benchmark should work out of the box on all systems.
The only major caveat is that it does allocate more than half of the system’s memory for its benchmarks, and this can severely degrade system performance if run on an interactive system (laptop or workstation). If run with MPI, then only the root process will run the benchmark.
Typical memory bandwidth numbers are in the range of multiple GByte/s. Given today’s memory amounts of many GByte, this means that this benchmark will run for tens of seconds. In addition to bencharking memory access, the operating system also needs to allocate the memory, which is surprisingly slow. A typical total execution time is several minutes.
The XSEDE system Kraken at NICS reports the following performance numbers (measured on June 21, 2013):
INFO (MemSpeed): Measuring CPU, cache, and memory speeds:
CPU floating point performance: 10.396 Gflop/sec for each PU
CPU integer performance: 6.23736 Giop/sec for each PU
Read latency:
D1 cache read latency: 1.15434 nsec
L2 cache read latency: 5.82695 nsec
L3 cache read latency: 29.4962 nsec
local memory read latency: 135.264 nsec
global memory read latency: 154.1 nsec
Read bandwidth:
D1 cache read bandwidth: 72.3597 GByte/sec for 1 PUs
L2 cache read bandwidth: 20.7431 GByte/sec for 1 PUs
L3 cache read bandwidth: 9.51587 GByte/sec for 6 PUs
local memory read bandwidth: 5.19518 GByte/sec for 6 PUs
global memory read bandwidth: 4.03817 GByte/sec for 12 PUs
Write latency:
D1 cache write latency: 0.24048 nsec
L2 cache write latency: 2.8294 nsec
L3 cache write latency: 9.32924 nsec
local memory write latency: 47.5912 nsec
global memory write latency: 58.1591 nsec
Write bandwidth:
D1 cache write bandwidth: 39.3172 GByte/sec for 1 PUs
L2 cache write bandwidth: 12.9614 GByte/sec for 1 PUs
L3 cache write bandwidth: 5.5553 GByte/sec for 6 PUs
local memory write bandwidth: 4.48227 GByte/sec for 6 PUs
global memory write bandwidth: 3.16998 GByte/sec for 12 PUs
The XSEDE system Kraken at NICS also reports the following system configuration via thorn hwloc (reported on June 21, 2013):
INFO (hwloc): Extracting CPU/cache/memory properties:
There are 1 PUs per core (aka hardware SMT threads)
There are 1 threads per core (aka SMT threads used)
Cache (unknown name) has type "data" depth 1
size 65536 linesize 64 associativity 2 stride 32768, for 1 PUs
Cache (unknown name) has type "unified" depth 2
size 524288 linesize 64 associativity 16 stride 32768, for 1 PUs
size 6291456 linesize 64 associativity 48 stride 131072, for 6 PUs
Memory has type "local" depth 1
size 8589541376 pagesize 4096, for 6 PUs
Memory has type "global" depth 1
size 17179082752 pagesize 4096, for 12 PUs
Kraken’s CPUs identify themselves as 6-Core AMD Opteron(tm) Processor 23 (D0).
Let us examine and partially interpret these numbers. (While the particular results will be different for other systems, the general behaviour will often be similar.)
Kraken’s compute nodes run at 2.6 GHz and execute 4 Flop/cycle, leading to a theoretical peak performance of 10.4 GFlop/s. Our measured number of 10.396 GFlop/s is surprisingly close.
For integer performance, we expect half of the floating point performance since we cannot make use of vectorization, which yields a factor of two on this architecture. The reported number of 6.2 GIop/s is somewhat larger. We assume that the compiler found some way to optimize the code that we did not foresee, i.e. that this benchmark is not optimally designed. Still, the results are close.
The read latency for the D1 cache is here difficult to measure exactly, since it is so fast, and the cycle time and thus the natural uncertainty is about 0.38 ns. (We assume this could be measured accuratly with sufficient effort, but we do not completely trust our benchmark algorithm.) We thus conclude that the D1 cache has a latency of about 1 ns or less. A similar argument holds for the D1 read bandwidth – we conclude that the true bandwidth is 72 GB/s or higher.
The L2 cache has a higher read latency and a lower read bandwidth (it is also significantly larger than the D1 cache). We consider these performance numbers now to be trustworthy.
The L3 cache has again a slightly slower read performance than the L2 cache. The major difference is that the L3 cache is shared between six cores, so that the bandwidth will be shared if several cores access it simultaneously.
The local memory has a slightly lower read bandwith, and a significantly higher read latency than the L3 cache. The global memory is measurably slower than the local memory, but not by a large margin.
The write latencies are, as expected, lower than the read latencies (see section 2.5).
The write bandwidths are, surprisingly, only about half as large as the read bandwidths. This could either be a true property of the system architecture, or may be caused by write-allocating cache lines. The latter means that, as soon as a cache line is partially written, the cache fills it by reading from memory or the next higher cache level, although this is not actually necessary as the whole cache line will eventually be written. This additional read from memory effectively halves the observed write bandwidth. In principle, the memset function should use appropriate write instructions to avoid these unnecessary reads, but this may either not be the case, or the hardware may not be offering such write instructions.
[1] Larry McVoy, Carl Staelin, lmbench: Portable tools for performance analysis, 1996, Usenix, http://www.bitmover.com/lmbench/lmbench-usenix.pdf
[2] Carl Staelin, Larry McVoy, mhz: Anatomy of a micro-benchmark, 1998, Usenix, http://www.bitmover.com/lmbench/mhz-usenix.pdf
[3] LMbench – Tools for Performance Analysis, http://www.bitmover.com/lmbench
skip_largemem_benchmarks
Description: Skip benchmarks that require much memory
Description: Verbose output
This section lists all the variables which are assigned storage by thorn CactusUtils/MemSpeed. 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.
memspeed_measurespeed
measure cpu, memory, cache speeds
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What the Rule of 72 Reveals About the Future of an Investment
The Rule of 72 Defined
For example, the Rule of 72 states that $1 invested at an annual fixed interest rate of 10% would take 7.2 years ((72/10) = 7.2) to grow to $2. In reality, a 10% investment will take 7.3 years to double ((1.107.3 = 2).
The Rule of 72 is reasonably accurate for low rates of return. The chart below compares the numbers given by the Rule of 72 and the actual number of years it takes an investment to double.
Notice that although it gives an estimate, the Rule of 72 is less precise as rates of return increase.
The Rule of 72 and Natural Logs
The Rule of 72 can estimate compounding periods using natural logarithms. In mathematics, the logarithm is the opposite concept of a power; for example, the opposite of 10³ is log base 10 of 1,000.
\begin{aligned} &\text{Rule of 72} = ln(e) = 1\\ &\textbf{where:}\\ &e = 2.718281828\\ \end{aligned}
Rule of 72=ln(e)=1where:e=2.718281828
e is a famous irrational number similar to pi. The most important property of the number e is related to the slope of exponential and logarithm functions, and it's first few digits are 2.718281828.
The natural logarithm is the amount of time needed to reach a certain level of growth with continuous compounding.
The time value of money (TVM) formula is the following:
\begin{aligned} &\text{Future Value} = PV \times (1+r)^n\\ &\textbf{where:}\\ &PV = \text{Present Value}\\ &r = \text{Interest Rate}\\ &n = \text{Number of Time Periods}\\ \end{aligned}
Future Value=PV×(1+r)nwhere:PV=Present Valuer=Interest Raten=Number of Time Periods
To see how long it will take an investment to double, state the future value as 2 and the present value as 1.
2 = 1 \times (1 + r)^n
2=1×(1+r)n
Simplify, and you have the following:
2 = (1 + r)^n
2=(1+r)n
To remove the exponent on the right-hand side of the equation, take the natural log of each side:
ln(2) = n \times ln(1 + r)
ln(2)=n×ln(1+r)
This equation can be simplified again because the natural log of (1 + interest rate) equals the interest rate as the rate gets continuously closer to zero. In other words, you are left with:
ln(2) = r \times n
ln(2)=r×n
The natural log of 2 is equal to 0.693 and, after dividing both sides by the interest rate, you have:
0.693/r = n
0.693/r=n
By multiplying the numerator and denominator on the left-hand side by 100, you can express each as a percentage. This gives:
69.3/r\% = n
69.3/r%=n
The Rule of 72 is more accurate if it is adjusted to more closely resemble the compound interest formula—which effectively transforms the Rule of 72 into the Rule of 69.3.
Many investors prefer to use the Rule of 69.3 rather than the Rule of 72. For maximum accuracy—particularly for continuous compounding interest rate instruments—use the Rule of 69.3.
How to Calculate the Rule of 72 Using Matlab
The calculation of the Rule of 72 in Matlab requires running a simple command of "years = 72/return," where the variable "return" is the rate of return on investment and "years" is the result for the Rule of 72. The Rule of 72 is also used to determine how long it takes for money to halve in value for a given rate of inflation. For example, if the rate of inflation is 4%, a command "years = 72/inflation" where the variable inflation is defined as "inflation = 4" gives 18 years.
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Physics - Islands of insight in the nuclear chart
Islands of insight in the nuclear chart
A one of a kind radioactive beam experiment yields new insight for neutron-rich nuclei.
Figure 1: Nuclear chart showing the stable nuclei (black) and those predicted to lie inside the proton and neutron drip lines (open blue). The archipelago of islands where the shell-model magic numbers are broken is shown in red. The nuclei
{}^{30}\text{Mg}
{}^{32}\text{Mg}
involved in this experiment are shown in green, with their shell-model structure for neutrons in the upper left-hand corner.Nuclear chart showing the stable nuclei (black) and those predicted to lie inside the proton and neutron drip lines (open blue). The archipelago of islands where the shell-model magic numbers are broken is shown in red. The nuclei
{}^{30}\text{Mg}
{}^{32}\text{Mg}
invol... Show more
Like the shell model for atoms, which identifies which elements are volatile and which are inert, the nuclear shell model has guided our understanding of nuclear properties. Certain “magic number” landmarks associated with “inert” nuclei have dominated the nuclear landscape for over 50 years. As the more exotic neutron-rich nuclei are being studied, however, one finds that nuclear properties associated with the traditional landmarks can suddenly disappear. The reason is that both the relative importance of the average (“mean field”) and residual nuclear interactions, as well as the mean field itself, change.
One of the regions of the chart of nuclides where this breakdown occurs is at ( is the number of neutrons) and it is called the “island of inversion” (Fig. 1). The island of inversion is now known to be part of an archipelago of “islands of shell breaking” associated with the magic neutron numbers , , , , and . In order to accurately account for the effects of interactions and understand how the shell model breaks down, theorists need high-precision spectroscopic data for the isotopes in these regions. Now, in a paper appearing in Physical Review Letters, an international collaboration (Wimmer et al.) presents precise spectroscopic measurements of the neutron-rich element magnesium- , which lies in the much explored island of inversion at [1]. The measurements, which are the first of their kind and were performed at the REX-ISOLDE radioactive beam facility at CERN, address a long standing question about the low-lying energy spectrum of this nucleus and provide important information about the collective effects of nuclear interactions.
A Google search for “island of inversion” yields about results, almost all of which refer to a group of neutron-rich nuclei in a region of the nuclear chart centered around sodium- , which has protons and neutrons (stable sodium has neutrons). The term goes back to a paper by Warburton, Becker, and Brown [2] in which they studied the unusual features for nuclei in this mass region with the nuclear shell model. The magic numbers quoted above are derived from the nuclear shell model, the foundation of which is the appearance of gaps in the spectrum of single-particle energies that arise from the average (mean-field) interaction of one nucleon with all of the other nucleons. To draw an analogy with atomic physics, like the highly stable elements with closed electronic orbitals, the single-particle energy levels of “magic number” nuclei are filled with protons or neutrons (fermions that obey the Pauli principle) up to an energy just below the gaps. The energies of the first excited states in the nuclei with magic numbers are relatively high compared with those of neighboring nuclei since a proton or neutron must be excited to states across the gap. In contrast, inside the “islands of shell breaking” the energy of the first excited state drops and no longer shows any indication of the magic numbers found for the nuclei closer to stability. This archipelago holds the secret to many puzzling microscopic features of nuclear structure.
Current experiments with radioactive beams of isotopes are concentrated on studying the properties of neutron-rich nuclei out to the neutron drip line (the bottom-right edge of the chart of nuclei), where the islands of inversion lie. Inside the neutron drip line, the nuclei beta decay toward stability with lifetimes characteristic of the weak interaction, on the order of seconds. With this long lifetime, the masses, excited states, and decay properties of these nuclei can be studied in some detail when they can be produced. Beyond the neutron drip line, the nuclei are unstable to neutron decay and have a lifetime characteristic of the strong interaction, on the order of seconds. The experimental study of these nuclei is limited to observation of the neutrons from their decay.
With these general remarks in mind, one can look at the region of the nuclear chart probed by the REX-ISOLDE experiment. Two configurations for magnesium- are displayed for neutrons (Fig. 1, upper left); the closed-shell configuration A, and configuration B consisting of two neutrons excited from the and orbitals into the and orbitals across the shell gap, making a two-particle, two-hole state. Near the shell gaps, configurations of type B usually appear as excited states and are sometimes called pairing vibrations. The energy of these states is lower than twice the shell gap due to the pairing correlations between the particles and between the holes.
In this island of inversion, configurations of the type B become the ground state rather than the excited state. This change is sudden, with B forming an excited state in silicon- ( neutrons and protons) and then becoming the ground state for magnesium- ( neutrons and protons). Two factors contribute to this sudden change: a gradual reduction in the spherical shell gap as one approaches the neutron drip line at fluorine ( protons), and the configuration for the protons suddenly changing from “closed shell” in silicon- to “open shell” in magnesium- , which leads to stronger proton-neutron correlations and deformation.
Many experiments have studied the states in the island of inversion with configurations similar to B. The work at REX-ISOLDE provides the first confirmation that the predicted coexisting excited state A exists and shows some of its properties, providing a much sought after example of an explicit disappearance of shell closure. It is the first experiment of its type where a radioactive beam of magnesium-30 reacts with a target loaded with radioactive tritium. Two neutrons from the tritium target are transferred to magnesium- , leaving magnesium- and a proton that is detected. The shell-model configuration of magnesium- is shown in Fig 1. When two neutrons are added to make a final state with spin and parity , they can go into the orbitals, making state A, or into the orbitals, making state B. (The labels here are those associated with a wave function from a spherical potential with radial quantum number , orbital quantum number , and total angular momentum ). The REX-ISOLDE team observed two states: the ground state and an excited state at . The energy of the excited state, which is presumed to correspond to configuration A, is lower than any of the theoretical predictions discussed in this paper. The simplest estimate based on the theoretical extrapolation for the energy of A together with the measured energy of the ground state B is [3]. Understanding the reason for this disagreement with theory will be crucial for improving the many-body models as they are used to predict the properties of nuclei in even more neutron-rich nuclei.
The cross sections for the population of these two states give indirect information on the details of their structure. It is inferred that there is mixing between A and B, and that the component of B is larger than expected. This may be a signal that the single-particle energy of the orbital, which is about one above that of the orbital in , is dropping relative to as one approaches the neutron drip line. This is a feature of loosely bound orbitals with [4] and may be important for understanding why the neutron drip line suddenly changes from 6 neutrons for the oxygen isotopes ( protons) to greater than neutrons for the fluorine isotopes ( protons).
With the advent of experiments like the one reported by the REX-ISOLDE collaboration, the likes of which were inconceivable even a few years ago, we are beginning to gain a deeper understanding of the complexity of the collective effects that shape the nuclear landscape. The reasons for the existence of the islands of shell breaking are complex, and much needs to be explored. In addition, new magic numbers may appear near the neutron drip line [5] (for example, shows evidence that is a magic number). Apart from problems of intrinsic interest to nuclear physics, several questions in astrophysics also hinge crucially on our understanding of nuclear stability. Ultimately, an important goal is to be able to confidently predict the properties of neutron-rich nuclei and their role in the stellar nucleo-synthesis of elements. Such an objective will require a coherent program of theoretical, computational, and experimental advances.
K. Wimmer et al., Phys. Rev. Lett. 105, 252501 (2010)
E. K. Warburton, J. A. Becker, and B. A. Brown, Phys. Rev. C 41, 1147 (1990)
B. A. Brown and W. A. Richter, Phys. Rev. C 74, 034315 (2006)
I. Hamamoto, Phys. Rev. C 76, 054319 (2007)
T. Otsuka, R. Fujimoto, Y. Utsuno, B. A. Brown, M. Honma, and T. Mizusaki, Phys. Rev. Lett. 87, 082502 (2001)
Alex Brown is a Professor in the Department of Physics and Astronomy at Michigan State University and a member of the nuclear theory group at the National Superconducting Cyclotron Laboratory. His research focuses on theoretical models for nuclear structure and their applications to experiments with rare isotope beams, nuclear astrophysics, and fundamental symmetries in the nucleus. He received his Ph.D. in 1974 from Stony Brook University. In 2008 he was recognized as an Outstanding Referee by the American Physical Society.
Discovery of the Shape Coexisting
{0}^{+}
State in
{}^{32}\mathrm{Mg}
by a Two Neutron Transfer Reaction
{0}^{+}
{}^{32}\mathrm{Mg}
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Physics - Graphene Majoranas
Graphene Majoranas
December 15, 2015 • Physics 8, s142
Graphene could host Majorana quasiparticles if brought into contact with a conventional superconductor.
P. San-Jose et al., Phys. Rev. X (2015)
Majorana particles are rare birds in physics. Unlike other fermionic particles, they are their own antiparticles. In low-dimensional systems, they are also expected to emerge as zero-energy quasiparticles called Majorana zero modes, which have an unusual property: swapping two such particles changes their joint quantum state in a manner that depends on the order in which the swapping is done. This property could be exploited to build the elementary units of a fault-tolerant, topological quantum computer. Pablo San-Jose from the Materials Science Institute (CSIC) in Madrid, Spain, and co-workers have now put forward a proposal for observing these quasiparticles in graphene—the one-atom-thick form of carbon renowned for its high electrical conductivity.
Several teams have previously reported experimental evidence of Majorana zero modes in a variety of solid-state systems, including semiconducting nanowires and atomic chains. But the evidence is not airtight. Therefore, researchers are hunting for alternative host materials to detect unequivocal signatures of these modes. San-Jose and colleagues demonstrate how bringing a conventional superconductor into contact with a graphene sample could create Majorana zero modes in the sample’s interior, provided that graphene is in a magnetic phase called canted antiferromagnetism.
Such magnetism produces a strong effective coupling between the host material’s electron spin and its momentum—an effect that would be otherwise negligible in graphene and is at the heart of most other Majorana proposals. The team believes that their proposal could soon be tested in the laboratory by measuring characteristic Majorana features in the electrical current flowing from a metallic contact to a graphene sample.
P. San-Jose, J. L. Lado, R. Aguado, F. Guinea, and J. Fernández-Rossier
{\text{CaH}}_{6}
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Physics - <i>Landmarks</i>: The <i>Physical Review</i>’s Explosive Secret
Landmarks: The Physical Review’s Explosive Secret
The Physical Review delayed publishing the 1941 discovery of plutonium–which was used in an atomic bomb–until 1946 because of wartime security concerns.
Five flavors of plutonium. This element, used in the Nagasaki bomb at the end of World War II, was discovered in 1941, but the publication was delayed by the Physical Review until after the war because of security concerns.
APS has put the entire Physical Review archive online, back to 1893. Focus Landmarks feature important papers from the archive.
In the early days of World War II, physicists around the world were intently watching the pages of the Physical Review, waiting for updates on one of the century’s greatest revelations: fission, the splitting of an atom’s nucleus accompanied by a prodigious release of energy. But they waited in vain. Because of fears that Germany would use American research to pursue an atomic weapon, the Physical Review agreed to withhold reports of significant advances. It was not until several months after an atomic bomb exploded over Nagasaki, Japan, that Phys. Rev. published the paper announcing the discovery of plutonium, the material used in that bomb. Physicist Abraham Pais later called the journal’s silence on the subject “the most important nonevent in the history of the Physical Review.”
In 1940 the Physical Review published the discovery of element 93, the first element in the periodic table beyond the fissionable element uranium [1]. Although neptunium–named after Uranus’s neighbor–was not useful for a bomb, nearly all publication on fission-related work ceased shortly thereafter because of security concerns. The Physical Review and hundreds of other scientific journals agreed to submit all articles on the topic to a government committee.
Man of the elements. Glenn Seaborg adjusts equipment during his work on plutonium in 1941. He discovered nine other elements and hundreds of new isotopes in the following decades.
But scientific investigations continued. During the winter of 1940-41, Glenn Seaborg and his colleagues at the University of California used the Berkeley cyclotron to bombard uranium oxide with deuterons–bullets made of a proton and a neutron. The bombardment turned uranium-238 into neptunium-238 by replacing a neutron with a proton. Some of the neptunium then decayed to the new element 94 by releasing a beta particle (an electron emitted by the nucleus), turning another neutron into a proton.
But the researchers were not sure of all this at first. In their first letter to Phys. Rev., submitted in January 1941, they reported that their sample emitted alpha-particles–helium nuclei consisting of two protons and two neutrons. They hypothesized that element 94 had appeared in the sample and was alpha-decaying back to uranium (element 92). But it was not until they performed more chemical analyses that they could be sure. They submitted the evidence in a second letter in March. Seaborg and his colleagues named the new element plutonium, after the final planet beyond Neptune.
Then Seaborg’s team collaborated with Berkeley colleague Emilio Segrè to create the isotope plutonium-239 by hitting a uranium-238 compound with neutrons. Unlike plutonium-238, the new isotope would make an excellent bomb material, the researchers told the government in a secret report, because it could be split by neutrons, and at a rate 50 percent faster than uranium-235. It also could be produced from the abundant isotope uranium-238, whereas uranium-235 was rare. They reported the fission results to Phys. Rev. in May 1941.
A year later, the government put Seaborg in charge of a large effort to develop a means of mass-producing plutonium for the Manhattan Project. He succeeded, and a plutonium bomb hit Nagasaki on August 9, 1945, just three days after a uranium bomb devastated Hiroshima.
In 1946 Phys. Rev. finally published Seaborg’s letters, and others, along with their original dates of submission and footnotes explaining that they were “withheld from publication until the end of the war.” In 1951, Seaborg won the Nobel Prize in Chemistry, in large part for his role in the discovery of plutonium.
In the end, the absence of publications on fission in the Physical Review was too glaring to go unnoticed. A Soviet scientist deduced from the Americans’ silence on the topic that they were pursuing an atomic bomb. The Soviets soon followed suit. Maurice Goldhaber, who withheld his own research on fission until after the war, took a lesson from the experience. “When you tell people not to think about it,” he says, “they’ll just do it.”
E. McMillan and P. H. Abelson, Phys. Rev. 57, 1185 (1940)
1951 Nobel Prize in Chemistry–McMillan and Seaborg
Properties of 94(239)
J. W. Kennedy, G. T. Seaborg, E. Segrè, and A. C. Wahl
Radioactive Element 94 from Deuterons on Uranium
G. T. Seaborg, A. C. Wahl, and J. W. Kennedy
Search for Spontaneous Fission in
{94}^{239}
Joseph W. Kennedy and Arthur C. Wahl
{94}^{239}
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\text{Shareholders' Equity} = \text{Total Assets} - \text{Total Liabilities}
Shareholders’ Equity=Total Assets−Total Liabilities
Treasury shares or stock (not to be confused with U.S. Treasury bills) represent stock that the company has bought back from existing shareholders. Companies may do a repurchase when management cannot deploy all of the available equity capital in ways that might deliver the best returns. Shares bought back by companies become treasury shares, and the dollar value is noted in an account called treasury stock, a contra account to the accounts of investor capital and retained earnings. Companies can reissue treasury shares back to stockholders when companies need to raise money.
Many view stockholders' equity as representing a company's net assets—its net value, so to speak, would be the amount shareholders would receive if the company liquidated all of its assets and repaid all of its debts.
On a company's balance sheet, the amount of funds contributed by the owners or shareholders plus the retained earnings (or losses). One may also call this stockholders' equity or shareholders' equity.
U.S. Securities and Exchange Commission. "Form 4."
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Cisco and Janet spent an afternoon making burritos and then put them all on one plate. Six burritos have beef inside,
17
have chicken filling, and
2
have black-bean filling. Find the probability of randomly choosing the following burritos from the plate, and write your answer as a fraction, decimal, and percent.
A burrito with black bean filling.
Find the total number of burritos and the number of black bean burritos.
6 + 17 + 2 = 25
2 black bean burritos
State the probability as a fraction.
\frac{2}{25}
Convert the fraction to a decimal and a percent.
\left(\frac{2}{25}\right)\left(\frac{4}{4}\right)=\frac{8}{100}=0.08=8\%
\frac{2}{25}, \ 0.08, \ 8\%
A burrito that does not have beef filling.
Find the number of burritos that do not have beef filling and repeat the same steps as in part (a).
As a decimal, the probability of picking a burrito without beef filling is
0.76
Can you write this as a fraction and a percent?
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\mathrm{x0}
L\left(y\right)=0
L\left(y\right)=0
x
\mathrm{x0}=0
f
\mathrm{v_x0}\left(f\right)
the valuation at the point x0, that is, the lowest power in the series expansion of
\mathrm{x0}
f={f}_{0}{\left(x-\mathrm{x0}\right)}^{\mathrm{v_x0}\left(f\right)}+\mathrm{terms of higher order}
{f}_{0}
is called the leading coefficient of
\mathrm{x0}
Q\left(u,v\right)
RxR
{\left(x,y\right)|x\le u\mathrm{and}y\ge v}
{a}_{i}\frac{{d}^{i}}{{\mathrm{dx}}^{i}}y\left(x\right)
L
M\left(L\right)
Q\left(i,\mathrm{v_x0}\left({a}_{i}\right)-i\right)
i=0..\mathrm{order}\left(L\right)
M\left(L\right)
{P}_{k}\left(u\right)
l\left(k\right)
k
{P}_{k}
\frac{l\left(k\right)}{\mathrm{denom}\left(k\right)}
k
{a}_{i}'s
[k,{P}_{k}\left(u\right)]
k
{P}_{k}
{P}_{0}\left(u\right)
{P}_{0}\left(u\right)
\mathrm{with}\left(\mathrm{DEtools}\right):
\mathrm{ode}≔{x}^{7}\mathrm{diff}\left(y\left(x\right),x,x,x,x\right)-\left(x+{x}^{7}\right)\mathrm{diff}\left(y\left(x\right),x\right)-{x}^{9}y\left(x\right)
\textcolor[rgb]{0,0,1}{\mathrm{ode}}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{7}}\textcolor[rgb]{0,0,1}{}\left(\frac{{\textcolor[rgb]{0,0,1}{ⅆ}}^{\textcolor[rgb]{0,0,1}{4}}}{\textcolor[rgb]{0,0,1}{ⅆ}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{4}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\right)\textcolor[rgb]{0,0,1}{-}\left({\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{7}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0,0,1}{ⅆ}}{\textcolor[rgb]{0,0,1}{ⅆ}\textcolor[rgb]{0,0,1}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\right)\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{9}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)
\mathrm{newton_polygon}\left(\mathrm{ode},y\left(x\right),u\right)
[[\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{u}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{u}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}]]
\mathrm{newton_polygon}\left(\mathrm{ode},y\left(x\right),u,x=1\right)
[[\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{u}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{u}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{11}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{u}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}]]
\mathrm{newton_polygon}\left(\mathrm{ode},y\left(x\right),u,x=\mathrm{\infty }\right)
[[\frac{\textcolor[rgb]{0,0,1}{3}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{u}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}]]
\mathrm{ode}≔{\left({x}^{2}+1\right)}^{5}x\mathrm{diff}\left(y\left(x\right),x,x\right)-{\left({x}^{2}+1\right)}^{3}\mathrm{diff}\left(y\left(x\right),x\right)-{x}^{9}y\left(x\right)
\textcolor[rgb]{0,0,1}{\mathrm{ode}}\textcolor[rgb]{0,0,1}{≔}{\left({\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)}^{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\left(\frac{{\textcolor[rgb]{0,0,1}{ⅆ}}^{\textcolor[rgb]{0,0,1}{2}}}{\textcolor[rgb]{0,0,1}{ⅆ}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\right)\textcolor[rgb]{0,0,1}{-}{\left({\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0,0,1}{ⅆ}}{\textcolor[rgb]{0,0,1}{ⅆ}\textcolor[rgb]{0,0,1}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\right)\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{9}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)
\mathrm{newton_polygon}\left(\mathrm{ode},y\left(x\right),u,x=\mathrm{RootOf}\left({\mathrm{xx}}^{2}+1,\mathrm{xx}\right)\right)
[[\frac{\textcolor[rgb]{0,0,1}{3}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{32}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{RootOf}}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{\mathrm{_Z}}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)]]
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Characteristics of a sound wave — lesson. Science State Board, Class 9.
A sound wave can be explained completely by five characteristics; they are
Amplitude (\(A\)):
The wave's amplitude is the maximum displacement of medium particles from their original undisturbed positions when a wave passes through the medium. The sound will be loud if the vibration of a particle has a large amplitude and soft if the vibration has a small amplitude. 'A' stands for amplitude. The \(metre\) \(m\) is its SI unit.
Time period (\(T\)):
\mathit{Time}\phantom{\rule{0.147em}{0ex}}\mathit{period}=\frac{1}{\mathit{Frequency}}
Frequency (\(F\)):
The number of oscillations an object takes per second is called its frequency.
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}}
Velocity or Speed of the sound (\(v\)):
The speed of sound is defined as the distance that sound travels in one second. The letter ‘\(v\)' stands for it.
It is mathematically represented as,
where '\(n\)' is the frequency and '' is the wavelength.
Distance travelled by the sound wave is found by,
A sound has a frequency of \(70\) \(Hz\) and a wavelength of \(10\) \(m\). What is the speed of the sound?
Frequency (\(n\)) \(=\) \(70\) \(Hz\)
Wavelength (\(λ\)) \(=\) \(10\) \(m\)
To find: Speed of the sound (\(v\))
We know the formula,
By applying these values we get,
\(v\) \(=\)
70·10
\(v\) \(=\) 700
m/s
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Physics - Slipping through blood flow
Slipping through blood flow
Howard A. Stone, Alison M. Forsyth, and Jiandi Wan
Simulations provide insight into how viscous flow transforms the shapes of red blood cells, which may influence their physiological properties.
Illustration: Carin Cain
Figure 1: The motion and shape of red blood cells depends on the flow environment. (Top) The shape of a red blood cell at rest is symmetric, with both sides of the membrane concave. (Bottom left) The velocity profile of the fluid that carries the cells through a blood vessel is assumed to have a symmetric parabolic shape. (Bottom right) The simulated shapes of the cells as a function of the surrounding flow velocity and their reduced area
\nu
, which is the ratio of their area to the area of a circle with a circumference equal to the perimeter of the cell. Below a certain value of
\nu
, the cells are unstable to forming an asymmetric shape (from Ref. [1]).The motion and shape of red blood cells depends on the flow environment. (Top) The shape of a red blood cell at rest is symmetric, with both sides of the membrane concave. (Bottom left) The velocity profile of the fluid that carries the cells through... Show more
Coronary artery disease—the leading cause of death in the United States—results from the formation of plaque in our arteries, which blocks the transport of blood. The movement and deformation of red blood cells can also affect the flow of blood, and vice versa, but the mechanics of this relationship is still being explored. Writing in Physical Review Letters, Badr Kaoui and Chaouqi Misbah at Université Joseph Fourier in Grenoble, France, and George Biros at the Georgia Institute of Technology in the US explore, with simulations, how flow deforms red blood cells [1]. Their numerical simulations show that an experimentally observed transition in the shape of red blood cells, from symmetric to asymmetric, occurs even when the cells (which the authors model as vesicles) move in a fluid with a symmetric flow velocity distribution. They suggest that this shape transition, which arises because the symmetric shape is unstable, may be able to influence the flow efficiency for red blood cells. As recent research links the chemical responses of red blood cells to their mechanics, such models of individual shape transitions of cells could offer further understanding of physiological flows.
In the most elementary model for flow in the circulatory system, the heart acts as a pump, which drives the fluid containing red blood cells (blood) through circular tubes. In this case we expect the profile of the velocity to have a parabolic shape (Hagen-Poiseuille flow), as in the lower left part of Fig. 1. For a constant pressure drop, the volumetric flow rate of the fluid is proportional to the radius of the tube to the fourth power, which suggests that the diameter of blood vessels will play an important role in controlling blood flow. The natural expectation for a single red blood cell in a microvessel flow, which is assumed to have a parabolic velocity profile, is that it will be confined and should form a symmetrical shape in the center of the symmetrical flow. This result, however, is not necessarily the case, as pointed out by Kaoui et al.
A common model for red blood cells is a vesicle, which is a drop of liquid that is completely enclosed by a bilayer made from the same kind of phospholipid molecules found in cell membranes. Red blood cells are different from vesicles in that they have a cytoskeleton protein network underneath the lipid bilayer membrane, which gives the system a shear elasticity and supports a biconcave shape (about in diameter) under static conditions (top of Fig. 1). Under flow conditions, however, the fact that the cell can deform contributes to the viscous energy dissipation of the flow. For example, a red blood cell can tumble, as a rigid body, and tank-tread, where the cell maintains a constant orientation in a flow while the membrane rotates around the cell’s cytoplasm. There is also a symmetric “parachute” cell morphology where the cell deforms as a result of viscous forces, but keeps a symmetric shape and therefore cannot tank-tread.
In vivo studies have already demonstrated that red blood cells do in fact form asymmetric shapes in vessels that are less than , which is only a little larger than the cell itself [2]. In the presence of large viscous forces, the cell deforms asymmetrically into a “slipper” shape [3]. These asymmetric cells tank-tread because asymmetric viscous forces—produced by the cell’s asymmetric shape or the nonsymmetric position of the cell relative to the long axis of the vessel—act on the membrane. This “slipper” shape, which is a consequence of the confinement of the cell in a close-fitting channel, is believed to substantially reduce viscous dissipation [4].
With their simulations, Kaoui et al. have tried to address the question of how the cell shape changes as a consequence of the detailed flow structure, largely in the absence of direct interactions with a wall. For simplicity, they model the red blood cells as vesicles and assume they move in a plane in a symmetric parabolic flow. They find that the shape transition results from a loss in stability of the shape, which occurs when a dimensionless vesicle deflation number , defined as the ratio of the actual area to the area of a circle with a circumference equal to the perimeter of the cell, is below a certain value. ( is always less than , unless the cell is a circle.) Below the critical value of , the symmetric parachutelike shape develops an instability and the cell transforms into an asymmetric slipperlike shape (Fig. 1, right). Most importantly, this transition is not dependent on either the confinement of the surrounding blood vessel walls or membrane shear elasticity, which provides a new perspective for understanding the dynamics of red blood cells under flow conditions. Kaoui et al. also point out that this shape transition causes a decrease in the velocity difference between the cell and the flow, which could potentially enhance the efficiency of blood flow.
Kaoui et al.’s model is limited to two-dimensional vesicles and it is unclear if their results translate to more realistic three-dimensional models of red blood cells. Moreover, confinement and shear elasticity were shown to be unnecessary in this particular case, e.g., there was no direct wall effect, but such confinement influences may still play roles in shape transitions under different circumstances. Nevertheless, experiments on red blood cells have shown that viscous shear stresses in the flow control the transition from a symmetric parachutelike shape to an asymmetric slipper shape and that confinement is not necessary for the slipper shape [5], both conclusions that are consistent with the results of Kaoui et al.
The study of Kaoui et al. is an important contribution to the field of cell dynamics, but the role this shape transition plays in other physiological factors has yet to be examined theoretically or experimentally. For example, the effect of the density of red blood cells on this shape transition has not been explored and could result in cell clustering [6]. Adenosine triphosphate (ATP), which is released by red blood cells and can cause blood vessel dilation in vivo, has been correlated with cell deformation [7,8]. The role that the shape transitions discussed above play in the release of ATP or other chemicals thus will have interesting ramifications for both circulatory physiology and pathophysiology. In addition, as suggested by Kaoui et al., it has not been elucidated what role tank-treading or the lack thereof, plays in oxygen transport, which is the main function of red blood cells.
B. Kaoui, G. Biros, and C. Misbah, Phys. Rev. Lett. 103, 188101 (2009)
R. Skalak and P. I. Branemark, Science 164, 717 (1969)
P. Gaehtgens, C. Dührssen, and K. H. Albrecht, Blood Cells 6, 799 (1980)
P. Gaehtgens and H. Schmid-Schönbein, Naturwissenschaften 69, 294 (1982)
M. Abkarian, M. Faivre, R. Horton, K. Smistrup, C. A. Best-Popescu, and H. A. Stone, Biomedical Materials 3, 13 (2008)
J. L. McWhirtera, H. Noguchi, and G. Gompper Proc. Natl. Acad. Sci. U.S.A. 106, 6039 (2009)
A. K. Price, D. J. Fischer, R. S. Martin, and D. M. Spence, Anal. Chem. 76, 4849 (2004)
J. Wan, W. D. Ristenpart, and H. A. Stone, Proc. Natl. Acad. Sci. U.S.A. 105, 16432 (2008)
Alison M. Forsyth is a Ph.D. candidate at Harvard University in the School of Engineering and Applied Sciences and is currently a visiting scholar at Princeton University in the Department of Mechanical and Aerospace Engineering. She completed her B.S. in bioengineering at Syracuse University in 2006. Her research involves red blood cell deformation and dynamics with implications for physiological responses in the cardiovascular system.
Jiandi Wan is currently a Research Associate in the Department of Mechanical and Aerospace Engineering at Princeton University. His degrees are in chemistry from Wuhan University (B.S., 1998, M.S., 2001) and Boston University (Ph.D., 2006). Dr. Wan worked as a postdoctoral researcher in the School of Engineering and Applied Sciences at Harvard University from 2006 to 2009 and moved to Princeton University in 2009. Dr. Wan’s research includes microfluidic approaches for studying red blood cell dynamics and multiphase emulsions, photoinduced electron transfer dynamics, and surface chemistry. His recent work focuses on the material and biophysical applications of microfluidics.
Badr Kaoui, George Biros, and Chaouqi Misbah
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A bound sets technique for Dirichlet problem with an upper-Carathéodory right-hand side
Martina Pavlačková (2010)
In this paper, the existence and the localization result will be proven for vector Dirichlet problem with an upper-Carathéodory right-hand side. The result will be obtained by combining the continuation principle with bound sets technique.
A constructive method for solving stabilization problems
Vadim Azhmyakov (2000)
The problem of asymptotic stabilization for a class of differential inclusions is considered. The problem of choosing the Lyapunov functions from the parametric class of polynomials for differential inclusions is reduced to that of searching saddle points of a suitable function. A numerical algorithm is used for this purpose. All the results thus obtained can be extended to cover the discrete systems described by difference inclusions.
A continuous version of the Filippov-Gronwall inequality for differential inclusions
António Ornelas (1990)
We give an estimate for the distance between a given approximate solution for a Lipschitz differential inclusion and a true solution, both depending continuously on initial data.
A differential inclusion : the case of an isotropic set
Gisella Croce (2005)
In this article we are interested in the following problem: to find a map
u:\Omega \to {ℝ}^{2}
\left\{\begin{array}{cc}Du\in E\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\hfill & \mathit{\text{a.e.}}\phantom{\rule{4.0pt}{0ex}}\text{in}\phantom{\rule{4.0pt}{0ex}}\Omega \hfill \\ u\left(x\right)=\varphi \left(x\right)\hfill & x\in \partial \Omega \phantom{\rule{85.35826pt}{0ex}}\hfill \end{array}\right\
\Omega
is an open set of
{ℝ}^{2}
E
is a compact isotropic set of
{ℝ}^{2×2}
. We will show an existence theorem under suitable hypotheses on
\varphi
u:\Omega \to {ℝ}^{2}
\left\{\begin{array}{cc}Du\in E\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\hfill & \mathit{\text{a.e.}}\phantom{\rule{4.0pt}{0ex}}\text{in}\phantom{\rule{4.0pt}{0ex}}\Omega \hfill \\ u\left(x\right)=\varphi \left(x\right)\hfill & x\in \partial \Omega \hfill \end{array}\right\
where Ω is an open set of
{ℝ}^{2}
and E is a compact isotropic set of
{ℝ}^{2×2}
. We will show an existence theorem under suitable hypotheses on φ.
Bertrand Maury, Juliette Venel (2011)
The aim of this paper is to develop a crowd motion model designed to handle highly packed situations. The model we propose rests on two principles: we first define a spontaneous velocity which corresponds to the velocity each individual would like to have in the absence of other people. The actual velocity is then computed as the projection of the spontaneous velocity onto the set of admissible velocities (i.e. velocities which do not violate the non-overlapping constraint). We describe here the...
The aim of this paper is to develop a crowd motion model designed to handle highly packed situations. The model we propose rests on two principles: we first define a spontaneous velocity which corresponds to the velocity each individual would like to have in the absence of other people. The actual velocity is then computed as the projection of the spontaneous velocity onto the set of admissible velocities (i.e. velocities which do not violate the non-overlapping constraint). We describe here...
A Filippov type existence theorem for a class of second-order differential inclusions.
Cernea, Aurelian (2008)
Tsalyuk, Vadim Z. (2003)
A new approach to the existence results for orientor fields with Nicoletti's boundary conditions
Stanisław Domachowski (2005)
Applying a global bifurcation theorem for convex-valued completely continuous mappings we prove some existence theorems for convex-valued differential inclusions of the form x'∈ F(t,x), where x satisfies the Nicoletti boundary conditions.
A new method of proof of Filippov’s theorem based on the viability theorem
Sławomir Plaskacz, Magdalena Wiśniewska (2012)
Filippov’s theorem implies that, given an absolutely continuous function y: [t 0; T] → ℝd and a set-valued map F(t, x) measurable in t and l(t)-Lipschitz in x, for any initial condition x 0, there exists a solution x(·) to the differential inclusion x′(t) ∈ F(t, x(t)) starting from x 0 at the time t 0 and satisfying the estimation
\left|x\left(t\right)-y\left(t\right)\right|⩽r\left(t\right)=\left|{x}_{0}-y\left({t}_{0}\right)\right|{e}^{{\int }_{{t}_{0}}^{t}l\left(s\right)ds}+{\int }_{{t}_{0}}^{t}\gamma \left(s\right){e}^{{\int }_{s}^{t}l\left(\tau \right)d\tau }ds,
where the function γ(·) is the estimation of dist(y′(t), F(t, y(t))) ≤ γ(t). Setting P(t) = x ∈ ℝn: |x −y(t)| ≤ r(t), we may formulate the conclusion in Filippov’s theorem...
A note on multivalued differential equations on proximate retracts.
O'Regan, Donal (1999)
A note on the solutions of a second-order evolution inclusion in non separable Banach spaces
Aurelian Cernea (2017)
We consider a Cauchy problem associated to a second-order evolution inclusion in non separable Banach spaces under Filippov type assumptions and we prove the existence of mild solutions.
A remark on Scorza-Dragoni theorem for differential inclusions
Józef Myjak (1989)
A smooth Lyapunov function from a class-
\mathrm{𝒦ℒ}
estimate involving two positive semidefinite functions
Andrew R. Teel, Laurent Praly (2000)
\mathrm{𝒦ℒ}
We consider differential inclusions where a positive semidefinite function of the solutions satisfies a class-
\mathrm{𝒦ℒ}
estimate in terms of time and a second positive semidefinite function of the initial condition. We show that a smooth converse Lyapunov function, i.e., one whose derivative along solutions can be used to establish the class-
\mathrm{𝒦ℒ}
estimate, exists if and only if the class-
\mathrm{𝒦ℒ}
estimate is robust, i.e., it holds for a larger, perturbed differential inclusion. It remains an open question whether...
A strong relaxation theorem for maximal monotone differential inclusions with memory
We consider maximal monotone differential inclusions with memory. We establish the existence of extremal strong and then we show that they are dense in the solution set of the original equation. As an application, we derive a “bang-bang” principle for nonlinear control systems monitored by maximal monotone differential equations.
A study of second order differential inclusions with four-point integral boundary conditions
Bashir Ahmad, Sotiris K. Ntouyas (2011)
In this paper, we discuss the existence of solutions for a four-point integral boundary value problem of second order differential inclusions involving convex and non-convex multivalued maps. The existence results are obtained by applying the nonlinear alternative of Leray Schauder type and some suitable theorems of fixed point theory.
A sufficient condition for the existence of multiple periodic solutions of differential inclusions
Ralf Bader (1996)
|
q
A generalization of Zeeman’s family
Michał Sierakowski (1999)
E. C. Zeeman [2] described the behaviour of the iterates of the difference equation
{x}_{n+1}=R\left({x}_{n},{x}_{n-1},...,{x}_{n-k}\right)/Q\left({x}_{n},{x}_{n-1},...,{x}_{n-k}\right)
, n ≥ k, R,Q polynomials in the case
k=1,Q={x}_{n-1}
R={x}_{n}+\alpha
{x}_{1},{x}_{2}
positive, α nonnegative. We generalize his results as well as those of Beukers and Cushman on the existence of an invariant measure in the case when R,Q are affine and k = 1. We prove that the totally invariant set remains residual when the coefficients vary.
A generalized sum-difference inequality and applications to partial difference equations.
Wang, Wu-Sheng (2008)
A global convergence result for a higher order difference equation.
Iričanin, Bratislav D. (2007)
A Harnack inequality for solutions of difference differential equations of elliptic-parabolic type.
Masashi Misawa (1993)
A Hille-Wintner type comparison theorem for second order difference equations.
Hooker, John W. (1983)
A Mathematical Analysis of Miller's Algorithm.
R.V.M. Zahar (1976/1977)
A model of cardiac tissue as an excitable medium with two interacting pacemakers having refractory time
Alexander Loskutov, Sergei Rybalko, Ekaterina Zhuchkova (2003)
A quite general model of the nonlinear interaction of two impulse systems describing some types of cardiac arrhythmias is developed. Taking into account a refractory time the phase locking phenomena are investigated. Effects of the tongue splitting and their interweaving in the parametric space are found. The results obtained allow us to predict the behavior of excitable systems with two pacemakers depending on the type and intensity of their interaction and the initial phase.
A necessary and sufficient condition for the existence of a unique solution of a discrete boundary value problem.
Abu-Saris, Raghib, Ahmad, Wajdi (2003)
A new generalization of Ostrowski type inequality on time scales.
Liu, Wenjun J., Ngô, Qúôc-Anh, Chen, Wenbing B. (2009)
A new view on one problem of asymptotic behavior of solutions of delay difference equations.
Shaikhet, L. (2006)
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Spache readability formula - Wikipedia
(Redirected from Spache Readability Formula)
The Spache readability formula is a readability test for writing in English, designed by George Spache. It works best on texts that are for children up to fourth grade. For older children, the Dale–Chall readability formula is more appropriate.
It was introduced in 1953 in Spache's "A new readability formula for primary-grade reading materials," (The Elementary School Journal, 53, 410–413), and has subsequently been revised.
The method compares words in a text to a set list of everyday words. The number of words per sentence and the percentage of unfamiliar words determine the reading age.
{\displaystyle {\mbox{Grade Level}}=\left(0.141\times {\mbox{Average sentence length}}\right)+\left(0.086\times {\mbox{Percentage of unique unfamiliar words}}\right)+0.839}
{\displaystyle {\mbox{Grade Level}}=\left(0.121\times {\mbox{Average sentence length}}\right)+\left(0.082\times {\mbox{Percentage of unique unfamiliar words}}\right)+0.659}
Spache, G. (1953). "A New Readability Formula for Primary-Grade Reading Materials". The Elementary School Journal. 53 (7): 410–13. doi:10.1086/458513. JSTOR 998915. S2CID 145135468.
Clarence R. Stone. "Measuring Difficulty of Primary Reading Material: A Constructive Criticism of Spache's Measure." The Elementary School Journal, Vol. 57, No. 1 (Oct. 1956), pp. 36–41
Retrieved from "https://en.wikipedia.org/w/index.php?title=Spache_readability_formula&oldid=1011150388"
|
p
{ℝ}^{3}
{P}_{\lambda }
{u}_{\mid \partial \Omega }=0
{P}_{\lambda }
{u}_{\lambda }\in {\bigcap }_{p\ge 2}{W}^{2,p}\left(\Omega \right)
li{m}_{\lambda \to 0⁺}||{u}_{\lambda }{||}_{{W}^{2,p}\left(\Omega \right)}=0
\lambda ↦{I}_{\lambda }\left({u}_{\lambda }\right)
{I}_{\lambda }
{P}_{\lambda }
A boundary integral equation for Calderón's inverse conductivity problem.
Kari Astala, Lassi Päivärinta (2006)
Towards a constructive method to determine an L∞-conductivity from the corresponding Dirichlet to Neumann operator, we establish a Fredholm integral equation of the second kind at the boundary of a two dimensional body. We show that this equation depends directly on the measured data and has always a unique solution. This way the geometric optics solutions for the L∞-conductivity problem can be determined in a stable manner at the boundary and outside of the body.
A class of weighted function spaces, and intermediate Caccioppoli-Schauder estimates
Giovanni M. Troianiello (1988)
Barbara I. Wohlmuth (2002)
Domain decomposition techniques provide a powerful tool for the numerical approximation of partial differential equations. We focus on mortar finite element methods on non-matching triangulations. In particular, we discuss and analyze dual Lagrange multiplier spaces for lowest order finite elements. These non standard Lagrange multiplier spaces yield optimal discretization schemes and a locally supported basis for the associated constrained mortar spaces. As a consequence, standard efficient iterative...
B. Lucquin-Desreux, S. Mas-Gallic (1992)
A convergent adaptive finite element method with optimal complexity.
Becker, Roland, Mao, Shipeng, Shi, Zhong-Ci (2008)
A counterexample to the
{L}^{p}
-Hodge decomposition
We construct a bounded domain
\Omega \subset {ℝ}^{2}
with the cone property and a harmonic function on Ω which belongs to
{W}_{0}^{1,p}\left(\Omega \right)
for all 1 ≤ p < 4/3. As a corollary we deduce that there is no
{L}^{p}
-Hodge decomposition in
{L}^{p}\left(\Omega ,{ℝ}^{2}\right)
for all p > 4 and that the Dirichlet problem for the Laplace equation cannot be in general solved with the boundary data in
{W}^{1,p}\left(\Omega \right)
for all p > 4.
A discrete maximum principle [Book]
Tadeusz Styś (1981)
A domain decomposition algorithm for elliptic problems in three dimensions.
Barry F. Smith (1991/1992)
Andreas Rieder (1998)
A double
S
-shaped bifurcation curve for a reaction-diffusion model with nonlinear boundary conditions.
Goddard, Jerome II, Lee, Eun Kyoung, Shivaji, R. (2010)
A FETI-DP preconditioner for mortar methods in three dimensions.
Kim, Hyea Hyun (2007)
A Finite Eement Scheme for Domains with Corners.
Ivo Babuska (1972/1973)
|
Erik Schnetter <eschnetter@perimeterinstitute.ca>, Federico Cipolletta
OpenCL is a programming standard for heterogeneous systems, i.e. for programming CPUs, GPUs, and other types of accelerators. OpenCL is implemented as a library, and OpenCL codes are compiled at run time by passing OpenCL routines, as strings, to the OpenCL library. This is different e.g. from CUDA, which is implemented as a language such as C or C++.
This thorn OpenCL provides the configuration bits that ensure that Cactus applications can use OpenCL libraries.
OpenCL describes itself as:
OpenCL is the first open, royalty-free standard for cross-platform, parallel programming of modern processors found in personal computers, servers and handheld/embedded devices. OpenCL (Open Computing Language) greatly improves speed and responsiveness for a wide spectrum of applications in numerous market categories from gaming and entertainment to scientific and medical software.
More information is available at http://www.khronos.org/opencl/.
There seem to be four OpenCL implementations available at this time. Unfortunately, they each have their drawbacks:
Available at http://developer.amd.com/zones/openclzone/pages/default.aspx. This supports both CPUs and ATI GPUs. Unfortunately, the OpenCL compiler seems to produce code with a low quality.
Included with the operating system, available by default. This supports both CPU and GPU. The compiler is based on LLVM. Unfortunately, there seem to be serious bugs – for example, I can’t get the
cos
function to provide correct results.
Available at http://software.intel.com/en-us/articles/opencl-sdk/. This supports only (Intel?) CPUs. The compiler is based on LLVM, and the implementation is also based on Intel’s TBB (Threading Building Blocks).
Available at http://developer.nvidia.com/opencl, included in their CUDA distribution. This supports only GPUs.
Open source, available at https://launchpad.net/pocl. This OpenCL implementation has not yet been released (current version is 0.6), and is based on LLVM.
In addition, Wikipedia http://en.wikipedia.org/wiki/OpenCL lists two IBM implementations for their Power processor and for Intel compatible CPUs, respectively. The latter may be identical with or similar to AMD’s implementation.
Since OpenCL can run on CPUs, good OpenCL implementation are available at no cost for virtually all platforms.
It is possible to install several OpenCL implementations (platforms) at the same time, to build against any one of them, and then to choose at run time which devices from which platforms to use. For example, it is possible to build an application using the Intel implementation, and then at run time use the Nvidia platform to access a GPU (assuming that both Intel and Nvidia implementations are installed). On Unix, this is implemented via a system-wide configuration directory /etc/OpenCL/vendors that lists all OpenCL platforms that will be available at run time.
3 OpenCL Programming
OpenCL is very similar to C. However, it differs from C in several key aspects:
much smaller run-time library, consisting mostly of mathematical functions (such as sqrt) and printf;
built-in support for fine-grained and coarse-grainded multi-threading;
built-in support for vectorisation.
Given this, it is not possible to write a whole application in OpenCL. Instead, only the expensive parts (so-called compute kernels) are written in OpenCL, and are launched e.g. from C or C++.
In addition, the hardware architecture of GPUs and other accelerators differs from CPUs in one key aspect:
memory is separate from the host (regular CPU) memory.
That means that one has to explicitly copy data between the host memory and the device memory before and/or after calling compute kernels.
4 OpenCL Programming in Cactus
Cactus supports OpenCL programming at several levels. At the lowest level, one can use this thorn OpenCL directly. While this works fine, it is somewhat tedious because one has to write a certain amount of boilerplate code to detect and initialise the device, to copy data between host and device, and to build and run compute kernels.
Since OpenCL is implemented as a library, the flesh knows only little about OpenCL. For example, there are no configuration options to spedify an OpenCL compiler, since code is compiled at run time via a library call to which the source code is passed as string. There is, however, one way in which the flesh supports OpenCL: Files with a .cl suffix are converted into a string and placed into the executable. These strings have the type char const * in C, and can be accessed at run time under a (globally visible) name OpenCL_source_THORN_FILE, where THORN and FILE and are the thorn name and file name, respectively. (This is also explained in the users’ guide.)
5 High-Level OpenCL Programming in Cactus
Cactus also offers a higher-level way of OpenCL programming, implemented in the thorns OpenCLRunTime and Accelerator.
Thorn OpenCLRunTime provides a convenient function for executing OpenCL code. This function expects, as input, a string containing the OpenCL kernel code, and then calls this code. Lower-level tasks such as identifying available compute devices, initialising them, compiling the kernel (once, and then remembering it), and handling arguments and parameters are taken care of automatically. Details are described in this thorn’s documentation.
Thorn Accelerator simplifies memory management for GPUs and other types of devices. One declares in the thorn’s schedule which routines read and write what variables, and Accelerator then keeps track which variables need to be copied at what time. It keeps track where (host and/or device) a variable has valid values, and copies data only when necessary, taking time level cycling, synchronisation, and I/O into account. Details are described in that thorn’s documentation.
This section lists all the variables which are assigned storage by thorn ExternalLibraries/OpenCL. 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.
opencl_printinfo
print opencl system information
|
57S30 Discontinuous groups of transformations
A comparison principle and extension of equivariant maps.
Alexander Kushkuley, Zalman Balanov (1994)
A Converse to the P.A. Smith Theorem for Nonunitary Homology Spheres.
Reinhard Schultz (1985)
A Counterexample to a Conjecture of Steenrod.
A decomposition into homeomorphic handlebodies with naturally equivalent involutions.
Nelson, Roger B. (1990)
A Homology Transfer for a Class of Simplicial Maps.
G. Brumfiel, R. Kubelka (1982)
A linearity theorem for group actions on spheres with applications to homotopy representations.
Stefan Bauer (1989)
A Narasimhan-Seshardri-Donaldson Correspondance over Non-orientable Surfaces.
Shuguang Wang (1996)
A note on the converse of the Lefschetz theorem for G-maps
M. Izydorek, A. Vidal (1993)
The purpose of this note is to prove the converse of the Lefschetz fixed point theorem (CLT) together with an equivariant version of the converse of the Lefschetz deformation theorem (CDT) in the category of finite G-simplicial complexes, where G is a finite group.
A split exact sequence of Mackey functors.
P.J. Webb (1991)
Marie Blanc (2001)
{S}^{3}
John W. Morgan (1980/1981)
Actions of
{Z}_{n}
on some surface-bundles over
{S}^{1}
Józef H. Przytycki (1982)
An Algebraic Model for G-Simple Homotopy Types.
Mel Rothenberg, Triantafillou Georgia (1984)
An Equivariant Lefschetz Formula for Finite Reductive Groups.
G. Ellingsrud, K. Lonsted (1980)
Äquivariante 2-Felder auf Mannigfaltigkeiten mit Involution.
Roland Schwänzl (1982)
Äquivariante Homotopie, äquivarianter Bordismus und freie differenzierbare Involutionen auf Sphären.
Peter Löffler (1980)
Balanced splittings of Semi-free actions of finite groups on homotopy spheres.
Douglas R. Anderson, I. Hambleton (1980)
Cancellation of hyperbolic forms and topological four-manifolds.
Ian Hambleton, Matthias Kreck (1993)
Centralizers of gap groups
Toshio Sumi (2014)
A finite group G is called a gap group if there exists an ℝG-module which has no large isotropy groups except at zero and satisfies the gap condition. The gap condition facilitates the process of equivariant surgery. Many groups are gap groups and also many groups are not. In this paper, we clarify the relation between a gap group and the structures of its centralizers. We show that a nonsolvable group which has a normal, odd prime power index proper subgroup is a gap group.
|
Note: '''Plot''' and '''Result''' may not be created with None inputs.
==Integrating Python with SCons==
"'sfspike n1="'+str(nx)+"'n2="'+str(nz))
sfspike n1=
Notice that the formatters now have the name of the variable inside parentheses: \
Flow('spike-'+count,None,'sfspike n1=100')
Python functions are incredibly useful because you can compartmentalize repeated uses of commonly used '''Flow''' s, and then use them in multiple SConstructs. For the best use of Python functions you should use the following conventions:
*always use keyword=value arguments to help document your code,
*list file names for input and output first, then other arguments,
nsp=1 mag=1.0 k1=
sfricker1 frequency=
Of course, functions can be compiled into groups and then placed into Python modules for widespread re-use throughout your SConstructs. Python modules are currently located in $RSFROOT/book/Recipes, which is where you should place your modules as well. As usual, you must use correct Python syntax to access functions contained within modules. For example, if you create a module called myutil.py, then you can access your functions in the following manner:
The least used, but most powerful part of Python that you can bring into your SConstructs are Python classes. For example, if you are writing a script to process multiple models in the exact same way, but that have different parameters you would have to write separate Flow statements to process each of them, OR you could write a Python class that takes the model parameters and uses those parameters to generate Flow statements automatically, similar to functions. However, a class can allow you to group functions together into a single coherent body and allow you to drastically reduce the amount of code that must be reused.
We refer the reader to the Python documentation for more information on creating and using classes.
The Authors tutorials demonstrate how one can create reproducible documents using the Madagascar processing package and L<sup>A</sup>TEX together. By the end of the Authors tutorials, you should be able to:
*build papers, including: SEG and EAGE abstracts, manuscripts for Geophysics, and handouts,
*build a CSM thesis,
*build a CWP report,
*build slides,
===Downloading L<sup>A</sup>TEX===
To begin, you need to download a full installation of L<sup>A</sup>TEX for your operating system. To do so, you may need to contact your system administrator. If you have administrative rights, then you can download a full install for your platform from the following locations:
*Linux - use your package management software to install a full texlive (you may need additional packages depending on your distribution).
or download a stable release from http://sourceforge.net/projects/segtex/files/ and unpack it into the <tt>texmf</tt> directory.
===Updating your L<sup>A</sup>TEX install===
Once the class files are successfully downloaded, you will need to run '''texhash''' or '''texconfig rehash''' to update L<sup>A</sup>TEX about the new class files. For reference, a successful run of '''texhash''' produces the following output:
jgodwin$ texhash
*<font color="#cd4b19">\sideplot</font>,
To add your own L<sup>A</sup>TEX class files, place them in this same directory, and then request SEG\TeX access to commit them to the main repository.
{\displaystyle N}
{\displaystyle N}
{\displaystyle 10s}
{\displaystyle \#}
{\displaystyle |}
{\displaystyle |}
{\displaystyle |}
|
A Periodic Flow with Infinite Epstein Hierarchy.
Elmar Vogt (1977)
A sufficient condition for a group of homeomorphisms to be affine.
Andrew Vogt (1988)
Actions localement libres du groupe affine.
E. Ghys (1985)
Algebraic Hulls and the Folner Property.
A. Iozzi, A. Nevo (1996)
An elementary proof of a Lima's theorem for surfaces.
Francisco Javier Turiel Sandín (1989)
An elementary proof of the following theorem is given:THEOREM. Let M be a compact connected surface without boundary. Consider a C∞ action of Rn on M. Then, if the Euler-Poincaré characteristic of M is non zero there exists a fixed point.
Asymptotic Properties of Foliations
J.F. Plante (1972)
Automorphism Groups of Second Order Differential Equations.
Ottmar Loos (1983)
Fiedler, Bernold, Sandstede, Björn, Scheel, Arnd, Wulff, Claudia (1996)
Bundles with Totally Disconnected Structure Group
John W. Wood (1971)
Caractérisation des opérations d'algèbres sur les modules différentiels
A. Legrand (1988)
Compact solvmanifolds of dimension at most 4.
Gorbatsevich, V.V. (2009)
Completing Lie algebra actions to Lie group actions.
Kamber, Franz W., Michor, Peter W. (2004)
G
Bernd Stratmann (2001)
Dimension des orbites d'une action de ... sur une variété compacte.
P. Molina, F.J. Turiel (1988)
Enden von Räumen mit eigentlichen Transformationsgruppen
Herbert Abels (1972)
Philippe Jouan (2010)
The aim of this paper is to prove that a control affine system on a manifold is equivalent by diffeomorphism to a linear system on a Lie group or a homogeneous space if and only if the vector fields of the system are complete and generate a finite dimensional Lie algebra. A vector field on a connected Lie group is linear if its flow is a one parameter group of automorphisms. An affine vector field is obtained by adding a left invariant one. Its projection on a homogeneous space, whenever it exists,...
Robert J. Zimmer (1982)
Examples of Manifolds which are Homogeneous Spaces de Lie Groups of Arbitrarily Large Dimension.
Floris Takens, P. de de Harpe (1975)
{M}^{3}
{ℝ}^{2}
Jose Luis Arraut, Marcos Craizer (1995)
In this paper we give a geometric characterization of the 2-dimensional foliations on compact orientable 3-manifolds defined by a locally free smooth action of
{ℝ}^{2}
|
Physics - Molecular Currents
October 30, 2009 • Phys. Rev. Focus 24, 17
Researchers measured how the electrical conductance between two
{\text{C}}_{60}
molecules depends on their distance and orientation.
Multiple Vision. Top image: A scanning tunneling microscope image made with a clean probe reveals a layer of
{\text{C}}_{60}
molecules (lower right) as well as four individual gold atoms and a small cluster of them. Lower image: After picking up one
{\text{C}}_{60}
molecule on the probe tip, scanning the individual atoms reveals the detailed orientation of the attached molecule.Multiple Vision. Top image: A scanning tunneling microscope image made with a clean probe reveals a layer of
{\text{C}}_{60}
{\text{C}}_{60}
molecule... Show more
European researchers have measured the electrical conductance between a single pair of precisely oriented molecules. In the 13 November Physical Review Letters they describe picking up one molecule on the tip of an ultrafine scanning-tunneling-microscope probe and monitoring the current when it is positioned over another molecule. Understanding and controlling electric current between neighboring molecules will be important if electronic circuits are to be built using individual molecules.
For the ultimate in miniaturization, researchers want to learn all they can about using molecules in electronic circuits. They have measured currents between molecules but not with a precise understanding of the configuration and positions of the molecular electrons.
To explore molecular conduction in greater detail, a team led by Richard Berndt of the University of Kiel, Germany, worked with spherical “buckyballs.” They first scattered the molecules onto a pristine gold or copper surface, which generated single-molecule-thick regions interspersed with voids of bare metal. They then picked up a single molecule on the tip of the needle-like probe of a scanning tunneling microscope (STM) and scanned the probe across the surface in the usual imaging mode. This procedure allowed stray, isolated metal atoms on the surface to act as “upside down” STM probes that provided detailed images of the suspended buckyball from underneath. In the images, the ordinarily spherical atoms show up in the shape of the buckyball. The team used these images to map the detailed orientation and electronic structure of the buckyball.
The researchers then measured the conductance (the inverse of resistance) for current flowing from the probe through the buckyball and into the metal surface. This measurement was tricky because the current and the conductance continue to increase as the probe moves closer to the surface, with no clear moment when the buckyball is touching but not “squishing” against the surface. For several molecular configurations, the team chose a representative point on the conductance curve using a method similar to previous experimenters and found that the largest value was about three times as big as the smallest value. It depended roughly on how many metal and atoms were in contact, and the team explained these variations by comparing them with detailed calculations. The conductance values were not far from the ideal value for a perfect quantum channel where e is the electron charge and h is Planck’s constant, or about 1/(13 kilo-ohm)).
To measure the current between molecules, they moved the suspended buckyball over other buckyballs. The resulting current had to flow from the metal probe through one molecule, then through the second molecule to the metal surface. For two molecules in this arrangement, the maximum current was 100 times smaller than for a single molecule. Team member Guillaume Schull, now at the Strasbourg Institute for Materials Physics and Chemistry (ICPMS) in France, traces this lower current to the poor electronic interface between buckyballs. “The bonding between these two molecules is really bad. There is no chemical bond when we approach one to the other,” he says. In fact, if the researchers try to push them any closer than their separation in crystals, the bottom molecule simply slides sideways, so the current doesn’t increase.
Nonetheless, the experiments give very detailed information about conduction between molecules. “[With our technique] we can play with the distance between the molecules or the molecular orientation,” Schull says. “All of these parameters are really vital for understanding how this would work at a bigger scale.” Paul Weiss, of the University of California, Los Angeles, agrees that measuring conduction between molecules is important. “If we’re ever going to make devices out of molecules,” he says, “we need to know how they couple,” both for desired connections and for undesired “cross-talk” between neighbors.
Guillaume Schull, Thomas Frederiksen, Mads Brandbyge, and Richard Berndt
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The shape of a black hole’s event horizon – ebvalaim.log
When I wrote my response, I had a hunch. In the case of a Schwarzschild black hole, r = const actually describes a sphere, as it is a spherically symmetric solution. In the case of a rotating black hole (called a Kerr black hole, from the physicist who first derived the metric describing them) it doesn't have to be so. r is just one of the coordinates, which can be assigned to the space arbitrarily and the points with the same r don't have to lie on a sphere. In order to be sure, we have to calculate it.
One of the ways of describing a surface is a so-called metric - it describes a way of calculating distances between points with given coordinates (I wrote a bit more about it in my series of articles about the mathematics of relativity). It's a fundamental tool in the General Theory of Relativity - the properties of space-time are described there exactly with a metric. Spinning black holes are described, as I already mentioned, by the Kerr metric:
M
- the black hole's mass
a
- the black hole's angular momentum (divided by mass - there is a limit
a \leq M
t
r
\vartheta
\varphi
- space-time coordinates
\Delta = r^2 - 2Mr + a^2
\Sigma = r^2 + a^2\cos^2 \vartheta
(everything is in units of
c = G = 1
, for simplicity; if we put
\frac{GM}{c^2}
M
ct
t
, we would get SI units)
The event horizon (actually, two of them!) appears in the place where
\Delta = 0
- so
r = M \pm \sqrt{M^2 - a^2}
. We are only interested in the outer horizon, as it is what decides the shape of the black hole.
Since we can give a concrete value for
, which corresponds to the horizon, it means that the horizon is described by r = const. We aren't interested in the evolution in time, either (and there is none - the Kerr metric is stationary), so we also assume t = const. This means that in order to limit the metric to the horizon itself, we only have to set dr = dt = 0. Such a metric (after additionally changing its sign - the spatial part of the metric is negative, but it doesn't matter when we don't mix space with time, and we are only concerned with space here) would look like this:
It's not very beautiful, but much more pleasant than the full metric.
Now we use the fact that we know the precise value of r:
r = M + \sqrt{M^2 - a^2}
. We then get:
r^2 + a^2 = 2Mr
\Delta = 0
= 2M (M + \sqrt{M^2 - a^2})
\Sigma = r^2 + a^2 \cos^2 \vartheta = r^2 + a^2 - a^2 \sin^2 \vartheta = 2M (M + \sqrt{M^2 - a^2}) - a^2 \sin^2 \vartheta
The metric can then be written as:
Well, that's worse, but we can simplify it a bit. First, we can drop the leading
2M^2
. It only controls the size of the horizon, but not its shape - so we don't care about it, and if we need it, we can always restore it. Second, we have
\frac{a}{M}
in many places - let's denote this by
\alpha
. As I already mentioned, there is a limit of
a \leq M
0 \leq \alpha \leq 1
(0 will get us back to the Schwarzschild metric, 1 is so-called extremal Kerr). This brings the metric to the following form:
We will describe our generated surface by a parametrization:
x(\vartheta, \varphi)
y(\vartheta, \varphi)
z(\vartheta, \varphi)
. Such a form means that every point on the surface is described by a pair of coordinates
(\vartheta, \varphi)
. For every such pair we can then calculate corresponding 3D coordinates. For example, we could parametrize a sphere like so:
Since we still have a rotational symmetry in the Kerr metric, fortunately, we can use a trick: we say that z is the axis of symmetry, and we transform x and y into the distance from the z axis (denoted by
\rho
) and an angle
\varphi
- so we basically introduce cyllindrical coordinates. This lets us write:
(In the case of a sphere, we would have
\rho(\vartheta) = r\sin\vartheta
z(\vartheta) = r\cos\vartheta
Now we need to calculate the metric of our surface. The metric of a flat 3D space in the x, y, z coordinates is very simple:
dx^2 + dy^2 + dz^2
. We just have to limit it to our surface and express it using
\vartheta
\varphi
We will do it by transforming the differential forms to the new coordinates:
After substituting this in the original 3D metric:
We would like this to be equal to the
h''
metric. For it to be like this, the following must hold:
The first condition gives us:
\rho(\vartheta) = \frac{\sin\vartheta}{\sqrt{\xi(\vartheta)}}
. We can calculate the derivative, which gives:
The second equation is an ordinary differential equation for
z(\vartheta)
- it can be solved numerically, and we will have the full description of our surface.
So now - what does such a surface look like? Let's note that when we calculate
(\rho(\vartheta), z(\vartheta))
\vartheta
0
\pi
, we get x and z coordinates of points on our surface for
\varphi = 0
- so it's something like a "zero meridian". We just rotate the resulting curve around the z axis and voila! We have our surface.
Let's try it. Shown below are the "zero meridians" for
\alpha = 0
(which should be half a circle - the Schwarzschild case, a sphere),
\alpha = 0.5
\alpha = 0.8
The shape of the horizon for alpha=0
The shape of the horizon for alpha=0.5
If you look closely, you can see some flattenning for
\alpha=0.5
, and quite a bit of it for 0.8.
And what happens for larger values of
\alpha
...? I tried to see - and the program calculating
z(\vartheta)
numerically started throwing complex numbers at me. What the...?
I started suspecting a mistake in the calculations, so I checked them multiple times, but everything looked OK. I concluded, then, that maybe this is the correct result - but what would be its interpretation?
The problem stems from the way of calculating
\frac{dz}{d\vartheta}
- namely, we take a coefficient of the surface's metric that multiplies
d\vartheta^2
, and we subtract
\left(\frac{d\rho}{d\vartheta}\right)^2
from it. Well, but for large enough values of
\alpha
, the second part turns out to be larger than the first one, and the square of the derivative of z turns out negative.
But wait... what is larger than what here? The first part is a coefficient of the metric, which is basically a square of the length of the line that we draw by moving along the surface by
d\vartheta
. The second one is the square of the change in our distance from the axis of rotation when we move like that. A negative result means that by moving along the surface a bit, we move away from the axis of symmetry by more than we move along the surface - WTF? Such things shouldn't happen, should they?
Well, it turns out that, although counter-intuitive, it's actually possible. The source of the problem is that we try to divide the horizon into circles - circles of latitude (constant
\vartheta
), and we try to "stack" them along the z axis so that the distances on the surface created that way were the same as on the horizon. However, if the geometry of space is sufficiently curved, the curvature of circles of latitude can vary in a way completely independent from the distance on the surface - and this is what happens here. Our distance from the axis of rotation (
\rho
) is just the radius of curvature of the circle of latitude, and it varies so fast that the distance on the surface "doesn't catch up". Such a situation is possible in a curved space, but not in a flat one - like the one we live in and try to picture the shape of the horizon.
So, simply put: the shape of the horizon is something that just doesn't exist in a flat space. Oh well. We won't see what an extremal horizon looks like (
\alpha = 1
; my master's thesis was actually about such horizons).
We won't be able to see the actual shape of the horizon - but an observer near such a black hole would surely see something, right? He could take a picture and print it on a 2-dimensional piece of paper - what would be the result?
A black hole strongly deflects rays of light that pass near it, and absorbs some of them - as a result, an observer sees a black spot on the sky, surrounded by a distorted image of the background. The shape of this spot - called a "shadow" of the black hole - is not the shape of the horizon, though. It is determined by how the black hole affects light. When the black hole is spinning, light passing it on the one side is deflected a bit differently, than on the other side, which causes the image to be a bit flattened, indeed - but only on one side, and in the direction perpendicular to the axis of rotation, not parallel.
The picture below is a result of simulating a black hole with
\alpha \approx 0.95
. The axis of rotation is vertical here. You can clearly see the flattened left side - caused by the fact that light on this side can pass a bit closer to the black hole without falling in, than on the other side.
A view of a black hole with alpha = approx. 0.95
In summary: I haven't expected that some horizons can't be pictured in 3D - which is kind of a consequence of my being sure that the horizon is always a sphere ;) A random person on the internet asked a simple question, and I actually learned a lot from it. This was a fascinating experience :)
Update 2017-07-26: I've learned that, not surprisingly, it is actually a well-known result that the surface of the horizon cannot be embedded in a flat 3D space for
\alpha > \frac{\sqrt{3}}{2}
(L. Smarr Surface Geometry of Charged Rotating Black Holes, Phys. Rev. D 7, 289 (1973)). It was fun to find it out by myself, nevertheless :)
|
Physics - Statistics and the single molecule
Statistics and the single molecule
Institut für Physik, Humboldt-Universität zu Berlin, Newtonstrasse 15, D-12489 Berlin, Germany
Current technology permits tracking single molecules with exquisite precision, but the results need to be interpreted with care. Long-duration measurement of the motion of a single particle yields information that is different and complementary to that obtained from an ensemble average of many particles.
Figure 1: In continuous-time random walks, the walker’s position is governed only by the number of preceding steps. This number of steps
n\left(t\right)
constitutes the operational time of the problem as recorded by the walker’s own clock, which ticks once each time
n
is incremented. Since
n\left(t\right)
grows slower than linearly with the physical (clock) time, this watch is always behind, leading to the overall subdiffusive behavior, as compared with the otherwise normal random walk. He et al. have used this model to study time-averaged single-molecule behavior in comparison with ensemble averages of many molecules.In continuous-time random walks, the walker’s position is governed only by the number of preceding steps. This number of steps
n\left(t\right)
n
is... Show more
One hundred years ago, the atomic-molecular theory of matter was having a hard time, and many physicists considered it merely a kind of convenient shorthand rather than a real description of nature; after all, nobody had really seen a molecule, let alone an atom. Today, developments in micromanipulation and in single-molecule tracking have not only made individual molecules visible, but have led to real breakthroughs in understanding of the molecular basis of life. This ability to follow and to manipulate single molecules has opened new perspectives in nanoscience and nanotechnology. Experts in single-molecule tracking often say that observation of individual trajectories gives more information about the system than only looking at ensemble averages, which is the approach taken in statistical thermodynamics. The idea is that the closer one looks, the more information one can get.
In a paper published in Physical Review Letters however, Yong He, Stanislav Burov, Ralf Metzler, and Eli Barkai (at Bar Ilan University in Israel and the Technical University of Munich) show that the information obtained in such single-particle experiments is different from that given by the ensemble-averaged cases, so one has to be careful about interpreting the results [1]. This is especially the case when the measured motion exhibits subdiffusion (a process that is slower than normal Fick’s law diffusion) that might be nonergodic (the time and ensemble averages give different answers). This situation is often encountered in both nonliving physical systems such as disordered semiconductors and groundwater motion in geophysical formations, and in the crowded interiors of living cells.
He et al. base their theoretical analysis and numerical simulations on the so-called continuous-time random walk (CTRW) model, first introduced by Montroll and Weiss in 1965 [2]. CTRW was developed to handle a variety of complex diffusion processes by considering the motion of particles on lattices (Fig. 1). The importance of the model became clear after Scher and Montroll [3] successfully used it in 1975 to explain dispersive charge carrier transport in strongly disordered semiconductors (the ubiquitous working media of copy machines and laser printers). In the CTRW model, a particle hardly moves most of the time, and only occasionally gets an opportunity to jump to a new location. The motion is therefore described as a sequence of jumps into different directions interrupted by periods during which the particle is just waiting for the next jump.
Simple random walks were first discussed by Rayleigh [4] who concentrated on the dependence of the quantities of interest on the number of jumps. The theory of continuous-time random walks instead concentrates on the temporal aspect of the problem. If there exists a well-defined mean waiting time, the overall displacement follows the normal diffusion, in which both the mean squared displacement in the absence of the external force (where is the diffusion coefficient), and the mean displacement (where is the mobility) under the action of the constant external force F are proportional to each other and both grow as the first power of the time . A venerable example is the one that captured Einstein’s attention: colloidal particles undergoing diffusive Brownian motion, while at the same time falling downward due to gravity.
This proportionality has deep roots in the behavior of physical systems close to thermodynamic equilibrium; the mobility and the diffusion coefficient are not independent, but are connected to each other by Einstein’s relation . In normal diffusion the “average” can be understood either as an ensemble average over a large ensemble of moving particles, or as a temporal moving average over a very long trajectory of motion of duration for a single particle. Normal Fick’s law diffusion is an ergodic process (that is, both averages give the same result).
Strange things happen when the calculated mean waiting time diverges, as was the case with the carrier transport investigated by Scher and Montroll where the probability density followed a power law proportional to . When , the system is said to exhibit subdiffusion, characterized by in the ensemble average. Apart from disordered semiconductors, the CTRW model with power-law waiting-time distribution adequately describes such different phenomena as the spread of pollutants in underground water (where the particles can be caught in stagnant regions of the flow), and many biological situations in the interior of living cells, where the motion is strongly hindered by a bulky cytoskeleton and by the existence of other huge molecules around the molecule we are interested in.
Because nothing happens between the jumps in the CTRW model, it is the number of jumps that is the appropriate internal time variable describing the process, its so-called operational time. If a well-defined mean waiting time exists, the diffusion is normal, since both and are proportional to the mean number of steps , which in turn goes as . In the case of anomalous diffusion, the mean squared displacement and the mean displacement under a constant force are still proportional to each other, but the number of steps shows a different time dependence going as .
In the case of disordered semiconductors, the ensemble average makes sense owing to the multiparticle nature of the physical quantity of interest, namely, the electric current in the form of simultaneous motion of many charge carriers. On the other hand, in single molecule experiments the time average is often used. In their paper, He et al. show that in some cases the results of experiments on mRNA molecules and lipid granules are well described by the CTRW model and thus an ensemble average will differ from the single-particle time average. Contrary to what might be expected, one observes in the time-averaged picture not anomalous diffusion, but normal diffusion, albeit with strongly fluctuating diffusion coefficient. The result is easy to grasp. The mean squared displacement during the time interval between the two instants and is governed by the number of steps that occur in between. This grows on the average as , i.e., approximately as for . This proportionality to also survives after temporal integration assumed by the moving time average, giving rise to the overall seemingly normal diffusion behavior (where is the measured mean squared displacement) as opposed to the ensemble-averaged . So, we can be fooled by a single-molecule measurement into thinking that the entire ensemble is undergoing normal diffusion.
My colleagues and I found this basic result recently [5], [6] and used it to show that the standard models of potential landscapes are unable to describe equilibrium fluctuations in peptides. However, He et al. have gone much further in their discussion. In particular they generalized Einstein’s relation for a given specific situation, and moreover discuss in detail the distribution of measured mean squared displacement , which can be considered as a proxy for the distribution of the diffusion coefficients measured in experiment.
I would like to stress the aspect of universal fluctuations connected with this distribution. Typically, in ergodic systems, the longer the averaging time, the narrower is the distribution of the result. For example, the mean squared displacement measured as the moving time average for given time-lag t in the normal diffusion approaches a deterministic value when the averaging time grows, . The width of the distribution of the relative result tends to zero. In the case of subdiffusion considered above, the width of the distribution of stagnates, and increasing measurement time does not improve the result. The overall distribution of tends to a universal form depending only on the exponent (which contains the specific details of the system).
Being unaware of this nonergodicity, one could come to the wrong conclusion that the system under investigation is inhomogeneous, i.e., that each of different random walkers tracked is physically different, or, mathematically speaking, their motions correspond to realizations of different random processes (normal diffusion with different diffusion coefficients). However, in reality what we see is different realizations of the same random process corresponding to subdiffusion. The overall behavior strongly resembles what has been found in biological experiments (e.g., when following the motion of single viruses in the cell [7]), although one has always to be extremely cautious when comparing the results of theories based on one mechanism or cause, when in fact experiments are influenced by many different factors.
There is another important and interesting result reported by He et al. [1]. Up to now, we have discussed the situation in an infinite system, when the walkers’ motion is not restricted by any boundaries. The cells, on the contrary, are not only finite but relatively small. As we have seen, the time moving average in the infinite system exhibits normal diffusion (although the underlying process is anomalous). In a finite system, we still see hints of the underlying anomalies, as the authors show by direct numerical simulation of the time-averaged CTRW on a relatively small one-dimensional lattice. The results of these simulations resemble strongly the observations of Golding and Cox [8] on the motion of mRNA molecules inside bacterial cells and can probably explain these findings (although here one has to be cautious, too).
Statistical thermodynamics typically deals with systems that rapidly relax to equilibrium or to a stationary state, implying the system is ergodic. Systems far from equilibrium or showing very slow relaxation may be nonergodic, and subdiffusion as modeled by CTRW may be one of the simplest theoretical examples. One has to be cautious when applying our intuition gained for the close-to-equilibrium cases to such processes: the information contained in the time-averaged and ensemble-averaged results is different and is pertinent to different aspects of the system’s behavior. Understanding this fact is necessary when interpreting the results of existing experiments and when planning future studies.
Y. He, S. Burov, R. Metzler and E. Barkai, Phys. Rev. Lett. 101, 058101 (2008)
E. W. Montroll and G. H. Weiss, J. Math. Phys. 6, 167 (1965)
H. Scher and E. W. Montroll, Phys. Rev. B 12, 2455 (1975)
Lord Rayleigh, Nature 72, 318 (1905)
A. Lubelski, I. M. Sokolov, and J. Klafter, Phys. Rev. Lett. 100, 250602 (2008)
T. Neusius, I. Daidone, I. M. Sokolov, and J. C. Smith, Phys. Rev. Lett. 100, 188103 (2008)
G. Seisenberger, M. U. Reid, T. Endreß, H. Büning, M. Hallek, and C. Bräuchle, Science 294, 1929 (2001)
I. Golding and E. C. Cox, Phys. Rev. Lett. 96, 098102 (2006)
Igor M. Sokolov received his Ph.D. from the Moscow State University in 1984 and worked until 1990 at the department of theoretical physics at Lebedev Physical Institute in Moscow. In 1990 he moved to the University of Bayreuth, Germany, on a fellowship of the Alexander von Humboldt Foundation. From 1991 to 2001 he held a position at the University of Freiburg, Germany, and left for Berlin in 2001. Since 2005 he has been a full professor of physics at Humboldt University in Berlin. His main scientific interests include nonequilibrium thermodynamics, transport phenomena in flows and in disordered systems, chemical kinetics, networks, and polymer dynamics. In 2008 Igor M. Sokolov was recognized as one of the Outstanding Referees of the American Physical Society.
Y. He, S. Burov, R. Metzler, and E. Barkai
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Probability, Popular Questions: CBSE Class 9 ENGLISH, English Grammar - Meritnation
1-\frac{1}{2\left(\sqrt{3}\right)}
1-\frac{\mathrm{\pi }}{3\left(\sqrt{3}\right)}
1-\frac{\mathrm{\pi }}{2\sqrt{3}}
1-\frac{2\mathrm{\pi }}{3\sqrt{3}}
4.97 5.05 5.08 5.03 5.00 5.06 5.08 4.98 5.04 5.07 5.00
\frac{4}{5}
(b) 1 (c) 0 (d)
\frac{5}{4}
In a class of 50 students 70% were passed. What is the probability of a failing child?
Gulshan Khanna asked a question
Marks:1.00
Rohan draws a card from a deck of 52 cards. Each timewhen he draws a card, he notes the result and places the card back in the deck. He does this experiment 100 times. The table given below shows the occurrence of each type of card.
What is the difference between the probabilities of drawing a black card and a red card?
Jeevesh Verma asked a question
what is probablit that a card selected from a pack of 52 cards is-niether a black nor a king.
Sabir asked a question
The probabilities of a student getting grade A,B,C AND D are0.2,0.3,0.15,0.35 respectively. then the probability that a student gets at least C grade is
Poornima Rote asked a question
Outcome 3 Head 2 Head 1 Head No Head Frequency 23 72 77 28
Three coins are tossed simultaneously, compute the probability of
2 Heads coming up
1 Head coming up
Ritwik Vannarakkal asked a question
what is the probability of an event that cannot happen?
Iffat Fatima asked a question
how is 5 considered as an even number
Cards are numbered 5 to 50, from these cards a card is drawn. Find the probability that the card drawn has number multiple of 7. The experts have already solved this question but can you answer according to this question?
A dice is thowrn once ' what is the probability of getting odd primes
5. Weekly wages of workers in a factory are as follows:-
Weekly wages (in Rs) 325 - 350 350 - 375 375 - 400 400 - 425 425 – 450 No of workers 0 45 75 60 40
Find the probability that a worker selected at random earns: - (i) Rs 400 or more (ii) Rs 375 or more but less than Rs 425 (iii) Upto Rs 400 (iv) at least Rs 375
answers please in steps
The following table shows the daily earnings of 25 shops.
Find the probability that a shop earns :
(i) Rs 100 and above
(ii) At least Rs 60 but less than Rs 80
(iii) Less than Rs 40.
Please solve Q No.7
Kindly solve 16 17 and 18 please
Basanth Pvvs & 1 other asked a question
the probability of guessing a correct answer is x/2. if the probability of nat guessing it correct is 2/3, then x is..........
1.WHEN 3 COINS ARE TOSSED ONCE, FIND THE PROBABILITY OF GETTING AT LEAST ONE HEAD.
2. WHEN 3 COINS ARE TOSSED SIMULTANEOUSLY,FIND THE PROBABILITY OF GETTING AT LEAST 2 TAILS.
The probability of a month of January having 5 Sundays is 3/7. Explain how?
In a cricket match, a batsman hits a boundary 8 times out of 40 balls he plays. What is the probability that he didn't hit boundary ?
Abdul Haque asked a question
in a class there are x boys and y girls,a student is selected at random, find the probability of selecting a girl?
1) A bag contains X white, Y red and Z blue . A ball is drawn at random , then find the probability of drawing a blue ball ?
On a busy road the following data was observed about cars passing through it and the number of occupants. no. of occupants 1 2 3 4 5 no. of cars 29 26 23 17 5 Find the probablity that it has (i) Exactly 5 occupants (ii) Lesss than 5 occupants (iii) More than 2 occupants
Muhammed Shahil asked a question
A company selected 2300 families at random and surveyed them to determine a relationship between income level and the number of vehicles in a home.The information gathered is listed in the table.
If a family is chosen at random,find the probability that the family is,(a)earning rs 7000-13000 per month and owning exactly 1 vehicle,(b)earning not more than one vehicles,(c)earning more than rs 13000 and owning 2 or more than 2 vehicles,(d)owning no vehicle
Vishesh Jindal asked a question
solve it quickly.....
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(Redirected from Wireless energy transfer)
Transmission of electrical energy without wires as a physical link
2 Field regions
3 Near-field (nonradiative) techniques
3.1.1 Resonant inductive coupling
3.2.1 Resonant capacitive coupling
3.3 Electrodynamic Wireless Power Transfer
3.4 Magnetodynamic coupling
3.5 Zenneck Wave Transmission
4 Far-field (radiative) techniques
5 Atmospheric plasma channel coupling
7.1 19th century developments and dead ends
7.3 Near-field and non-radiative technologies
7.4 Microwaves and lasers
Field regions[edit]
The oscillating electric and magnetic fields surrounding moving electric charges in an antenna device can be divided into two regions, depending on distance Drange from the antenna.[14][17][18][24][30][31] [32] The boundary between the regions is somewhat vaguely defined.[17] The fields have different characteristics in these regions, and different technologies are used for transferring power:
Near-field (nonradiative) techniques[edit]
Inductive coupling[edit]
{\displaystyle M}
{\displaystyle D_{\text{range}}}
{\displaystyle k\;=\;M/{\sqrt {L_{1}L_{2}}}}
{\displaystyle L1}
{\displaystyle L2}
{\displaystyle L1}
{\displaystyle L2}
{\displaystyle k=1}
{\displaystyle k}
{\displaystyle k^{2}}
{\displaystyle D_{\text{ant}}}
Resonant inductive coupling[edit]
Main article: Resonant inductive coupling
Further information: Tesla coil § Resonant transformer
Capacitive coupling[edit]
Resonant capacitive coupling[edit]
Electrodynamic Wireless Power Transfer[edit]
{\displaystyle k}
{\displaystyle k<k_{crit}}
{\displaystyle k=k_{crit}}
{\displaystyle k>k_{crit}}
Magnetodynamic coupling[edit]
Zenneck Wave Transmission[edit]
Far-field (radiative) techniques[edit]
Atmospheric plasma channel coupling[edit]
See also: Electrolaser
{\displaystyle p}
Energy harvesting[edit]
Main article: Energy harvesting
19th century developments and dead ends[edit]
Near-field and non-radiative technologies[edit]
Microwaves and lasers[edit]
Retrieved from "https://en.wikipedia.org/w/index.php?title=Wireless_power_transfer&oldid=1086012043"
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{S}^{1}
{𝐑}^{2}
Gilles Chatelet, Harold Rosenberg, Daniel Weil (1974)
A free group acting without fixed points on the rational unit sphere
Kenzi Satô (1995)
A note on smooth toral reductions of spheres.
Charles P. Boyer (1998)
A uniformly quasiconformal group not isomorphic to a Möbius group.
Dojnikov, P.V. (2004)
Stefan Friedl, Stefano Vidussi (2013)
In this paper we show that given any 3-manifold
N
and any non-fibered class in
{H}^{1}\left(N;Z\right)
there exists a representation such that the corresponding twisted Alexander polynomial is zero. We obtain this result by extending earlier work of ours and by combining this with recent results of Agol and Wise on separability of 3-manifold groups. This result allows us to completely classify symplectic 4-manifolds with a free circle action, and to determine their symplectic cones.
{S}^{3}
Actions of Groups with Periodic Cohomology on Poincaré Duality Spaces.
Andreas Stieglitz (1978)
Actions of the Homeotopy Group of an Orientable 3-Dimensional Handle Body.
E. Luft (1978)
{Z}_{n}
{S}^{1}
Affine Actions of Higher Rank Lattices.
R. Feres (1993)
Almost transitive actions on spaces with the rational homotopy of sphere products.
Bletz-Siebert, Oliver (2005)
Analysis on real affine
G
Ramacher, Pablo (2005)
Analytische periodische Strömungen auf kompakten komplexen Räumen.
Harald Holmann (1977)
Area preserving group actions on surfaces.
Franks, John, Handel, Michael (2003)
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60Fxx Limit theorems
60F15 Strong theorems
{L}^{p}
-limit theorems
A central limit theorem for independent summands
Z. Govindarajulu (1976)
A central limit theorem for non stationary mixing processes
Dalibor Volný (1989)
A central limit theorem for piecewise convex transformations of the unit interval
M. Jabłoński, J. Malczak (1984)
A central limit theorem for processes generated by a family of transformations [Book]
Marian Jabłoński (1991)
A central limit theorem for random fields.
Fazekas, István, Chuprunov, Alexey (2004)
A central limit theorem for sums of a random number of independent random variables
Zdzisław Rychlik (1976)
Fraydoun Rezakhanlou (2002)
Michael H. Neumann (2013)
We derive a central limit theorem for triangular arrays of possibly nonstationary random variables satisfying a condition of weak dependence in the sense of Doukhan and Louhichi [Stoch. Proc. Appl. 84 (1999) 313–342]. The proof uses a new variant of the Lindeberg method: the behavior of the partial sums is compared to that of partial sums of dependent Gaussian random variables. We also discuss a few applications in statistics which show that our central limit theorem is tailor-made for statistics...
A central limit theorem for weighted averages of spins in the high temperature region of the Sherrington-Kirkpatrick model.
Panchenko, Dmitry (2005)
{𝒫}_{n}
{𝒫}_{n}
{𝒫}_{n}
A Characterisation of Chebyshev-Cramer Expansions.
R. Michel, C. Hipp (1976)
A connection between controlled Markov chains and martingales
Petr Mandl (1973)
A contractive property in finite state Markov chains
Petr Kratochvíl, Antonín Lešanovský (1985)
A functional approach for random walks in random sceneries.
Dombry, Clement, Guillotin-Plantard, Nadine (2009)
A functional central limit theorem for a class of interacting Markov chain Monte Carlo methods.
Bercu, Bernard, Del Moral, Pierre, Doucet, Arnaud (2009)
Nathalie Eisenbaum (2002)
A generalization of the global limit theorems of R. P. Agnew.
Rosalsky, Andrew (1988)
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(Redirected from Given any)
2.2 Other connectives
3 Universal closure
4 As adjoint
{\displaystyle \forall }
{\displaystyle \exists }
{\displaystyle \forall n\!\in \!\mathbb {N} \;P(n)}
{\displaystyle \forall n\!\in \!\mathbb {N} \;{\bigl (}Q(n)\rightarrow P(n){\bigr )}}
{\displaystyle \lnot \forall x\;P(x)\quad {\text{is equivalent to}}\quad \exists x\;\lnot P(x)}
{\displaystyle \lnot }
{\displaystyle \forall x\in X\,P(x)}
{\displaystyle \lnot \ \forall x\in X\,P(x)}
{\displaystyle \forall x\in X\,P(x)}
{\displaystyle \exists x\in X\,\lnot P(x)}
{\displaystyle \lnot \ \exists x\in X\,P(x)\equiv \ \forall x\in X\,\lnot P(x)\not \equiv \ \lnot \ \forall x\in X\,P(x)\equiv \ \exists x\in X\,\lnot P(x)}
{\displaystyle {\begin{aligned}P(x)\land (\exists {y}{\in }\mathbf {Y} \,Q(y))&\equiv \ \exists {y}{\in }\mathbf {Y} \,(P(x)\land Q(y))\\P(x)\lor (\exists {y}{\in }\mathbf {Y} \,Q(y))&\equiv \ \exists {y}{\in }\mathbf {Y} \,(P(x)\lor Q(y)),&{\text{provided that }}\mathbf {Y} \neq \emptyset \\P(x)\to (\exists {y}{\in }\mathbf {Y} \,Q(y))&\equiv \ \exists {y}{\in }\mathbf {Y} \,(P(x)\to Q(y)),&{\text{provided that }}\mathbf {Y} \neq \emptyset \\P(x)\nleftarrow (\exists {y}{\in }\mathbf {Y} \,Q(y))&\equiv \ \exists {y}{\in }\mathbf {Y} \,(P(x)\nleftarrow Q(y))\\P(x)\land (\forall {y}{\in }\mathbf {Y} \,Q(y))&\equiv \ \forall {y}{\in }\mathbf {Y} \,(P(x)\land Q(y)),&{\text{provided that }}\mathbf {Y} \neq \emptyset \\P(x)\lor (\forall {y}{\in }\mathbf {Y} \,Q(y))&\equiv \ \forall {y}{\in }\mathbf {Y} \,(P(x)\lor Q(y))\\P(x)\to (\forall {y}{\in }\mathbf {Y} \,Q(y))&\equiv \ \forall {y}{\in }\mathbf {Y} \,(P(x)\to Q(y))\\P(x)\nleftarrow (\forall {y}{\in }\mathbf {Y} \,Q(y))&\equiv \ \forall {y}{\in }\mathbf {Y} \,(P(x)\nleftarrow Q(y)),&{\text{provided that }}\mathbf {Y} \neq \emptyset \end{aligned}}}
{\displaystyle {\begin{aligned}P(x)\uparrow (\exists {y}{\in }\mathbf {Y} \,Q(y))&\equiv \ \forall {y}{\in }\mathbf {Y} \,(P(x)\uparrow Q(y))\\P(x)\downarrow (\exists {y}{\in }\mathbf {Y} \,Q(y))&\equiv \ \forall {y}{\in }\mathbf {Y} \,(P(x)\downarrow Q(y)),&{\text{provided that }}\mathbf {Y} \neq \emptyset \\P(x)\nrightarrow (\exists {y}{\in }\mathbf {Y} \,Q(y))&\equiv \ \forall {y}{\in }\mathbf {Y} \,(P(x)\nrightarrow Q(y)),&{\text{provided that }}\mathbf {Y} \neq \emptyset \\P(x)\gets (\exists {y}{\in }\mathbf {Y} \,Q(y))&\equiv \ \forall {y}{\in }\mathbf {Y} \,(P(x)\gets Q(y))\\P(x)\uparrow (\forall {y}{\in }\mathbf {Y} \,Q(y))&\equiv \ \exists {y}{\in }\mathbf {Y} \,(P(x)\uparrow Q(y)),&{\text{provided that }}\mathbf {Y} \neq \emptyset \\P(x)\downarrow (\forall {y}{\in }\mathbf {Y} \,Q(y))&\equiv \ \exists {y}{\in }\mathbf {Y} \,(P(x)\downarrow Q(y))\\P(x)\nrightarrow (\forall {y}{\in }\mathbf {Y} \,Q(y))&\equiv \ \exists {y}{\in }\mathbf {Y} \,(P(x)\nrightarrow Q(y))\\P(x)\gets (\forall {y}{\in }\mathbf {Y} \,Q(y))&\equiv \ \exists {y}{\in }\mathbf {Y} \,(P(x)\gets Q(y)),&{\text{provided that }}\mathbf {Y} \neq \emptyset \\\end{aligned}}}
{\displaystyle \forall {x}{\in }\mathbf {X} \,P(x)\to P(c)}
{\displaystyle P(c)\to \ \forall {x}{\in }\mathbf {X} \,P(x).}
{\displaystyle \forall {x}{\in }\emptyset \,P(x)}
{\displaystyle P(y)\land \exists xQ(x,z)}
{\displaystyle \forall y\forall z(P(y)\land \exists xQ(x,z))}
{\displaystyle X}
{\displaystyle {\mathcal {P}}X}
{\displaystyle f:X\to Y}
{\displaystyle X}
{\displaystyle Y}
{\displaystyle f^{*}:{\mathcal {P}}Y\to {\mathcal {P}}X}
{\displaystyle \exists _{f}}
{\displaystyle \forall _{f}}
{\displaystyle \exists _{f}\colon {\mathcal {P}}X\to {\mathcal {P}}Y}
{\displaystyle S\subset X}
{\displaystyle \exists _{f}S\subset Y}
{\displaystyle \exists _{f}S=\{y\in Y\;|\;\exists x\in X.\ f(x)=y\quad \land \quad x\in S\},}
{\displaystyle y}
{\displaystyle S}
{\displaystyle f}
{\displaystyle \forall _{f}\colon {\mathcal {P}}X\to {\mathcal {P}}Y}
{\displaystyle S\subset X}
{\displaystyle \forall _{f}S\subset Y}
{\displaystyle \forall _{f}S=\{y\in Y\;|\;\forall x\in X.\ f(x)=y\quad \implies \quad x\in S\},}
{\displaystyle y}
{\displaystyle f}
{\displaystyle S}
{\displaystyle !:X\to 1}
{\displaystyle {\mathcal {P}}(1)=\{T,F\}}
{\displaystyle S(x)}
{\displaystyle {\begin{array}{rl}{\mathcal {P}}(!)\colon {\mathcal {P}}(1)&\to {\mathcal {P}}(X)\\T&\mapsto X\\F&\mapsto \{\}\end{array}}}
{\displaystyle \exists _{!}S=\exists x.S(x),}
{\displaystyle S}
{\displaystyle \forall _{!}S=\forall x.S(x),}
Retrieved from "https://en.wikipedia.org/w/index.php?title=Universal_quantification&oldid=1070603168"
|
35A35 Theoretical approximation to solutions
35A01 Existence problems: global existence, local existence, non-existence
35A02 Uniqueness problems: global uniqueness, local uniqueness, non-uniqueness
35A08 Fundamental solutions
35A09 Classical solutions
35A15 Variational methods
35A16 Topological and monotonicity methods
35A20 Analytic methods, singularities
35A21 Propagation of singularities
35A22 Transform methods (e.g. integral transforms)
35A23 Inequalities involving derivatives and differential and integral operators, inequalities for integrals
35A24 Methods of ordinary differential equations
35A25 Other special methods
35A27 Microlocal methods; methods of sheaf theory and homological algebra in PDE
35A30 Geometric theory, characteristics, transformations
A Boundary Value Problem Connected with Response of Semi-space to a Short Laser Pulse
Gaetano Fichera (1997)
In this paper a mixed boundary value problem for the fourth order hyperbolic equation with constant coefficients which is connected with response of semi-space to a short laser pulse» and belongs to generalized Thermoelasticity is studied. This problem was considered by R. B. Hetnarski and J. Ignaczak, who established some important physical consequences. The present paper contains proof of the existence, uniqueness and continuous dependence of a solution on the datum, together with an effective...
A decomposition method for a semilinear boundary value problem with a quadratic nonlinearity.
Gordon, Michael S. (2005)
A Finite-Difference Approximation for the Eigenvalue of the Clamped Plate.
J.R. KUTTLER (1971)
A free boundary problem for the
p
-Laplacian: uniqueness, convexity, and successive approximation of solutions.
Acker, Andrew F., Meyer, R. (1995)
D. Chapelle, A. Gariah, P. Moireau, J. Sainte-Marie (2013)
We address the issue of parameter variations in POD approximations of time-dependent problems, without any specific restriction on the form of parameter dependence. Considering a parabolic model problem, we propose a POD construction strategy allowing us to obtain some a priori error estimates controlled by the POD remainder – in the construction procedure – and some parameter-wise interpolation errors for the model solutions. We provide a thorough numerical assessment of this strategy with the...
A hybrid semi-primitive shock capturing scheme for conservation laws.
Dubey, Ritesh Kumar (2010)
A mathematical model of suspension bridges
Gabriela Liţcanu (2004)
We prove the existence of weak T-periodic solutions for a nonlinear mathematical model associated with suspension bridges. Under further assumptions a regularity result is also given.
A mixed finite element method close to the equilibrium model applied to plane elastostatics
Jaroslav Haslinger, Ivan Hlaváček (1976)
George Avalos, Matthew Dvorak (2008)
We consider a coupled PDE model of various fluid-structure interactions seen in nature. It has recently been shown by the authors [Contemp. Math. 440, 2007] that this model admits of an explicit semigroup generator representation 𝓐:D(𝓐)⊂ H → H, where H is the associated space of fluid-structure initial data. However, the argument for the maximality criterion was indirect, and did not provide for an explicit solution Φ ∈ D(𝓐) of the equation (λI-𝓐)Φ =F for given F ∈ H and λ > 0. The present...
A new second order finite difference conservative scheme.
Guevara-Jordan, J.M., Rojas, S., Freites-Villegas, M., Castillo, J.E. (2005)
A nonlinear diffusion equation with nonlinear boundary conditions: Method of lines
Ján Filo (1988)
A note on hyperbolic partial differential equations. II.
A note on the unique solvability of an inverse problem with integral overdetermination.
Gözükızıl, Omer Faruk, Yaman, Metin (2008)
A parabolic - elliptic variational inequality.
Ulrich Hornung (1982)
A Picard-Maclaurin theorem for initial value PDEs.
Parker, G.Edgar, Sochacki, James S. (2000)
Karen Veroy, Dimitrios V. Rovas, Anthony T. Patera (2002)
We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic coercive partial differential equations with affine parameter dependence. The essential components are (i) (provably) rapidly convergent global reduced-basis approximations – Galerkin projection onto a space
{W}_{N}
spanned by solutions of the governing partial differential equation at
N
selected points in parameter space; (ii) a posteriori error estimation – relaxations of the error-residual equation...
A Posteriori Error Estimation for Reduced-Basis Approximation of Parametrized Elliptic Coercive Partial Differential Equations: “Convex Inverse” Bound Conditioners
We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic coercive partial differential equations with affine parameter dependence. The essential components are (i ) (provably) rapidly convergent global reduced-basis approximations – Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii ) a posteriori error estimation – relaxations of the error-residual equation...
A posteriori estimations of approximate solutions for some types of boundary value problems
Kodnár, Rudolf (1986)
A random walk for the solution sought: Remarks on the difference scheme approach to nonlinear semigroups and evolution operators.
M.A. Freedman (1987)
A review of some recent results for the approximate analytical solutions of nonlinear differential equations.
Pamuk, Serdal (2009)
|
35J05 Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation
35J08 Green's functions
35J46 First-order elliptic systems
35J48 Higher-order elliptic systems
35J58 Boundary value problems for higher-order elliptic systems
35J66 Nonlinear boundary value problems for nonlinear elliptic equations
35J67 Boundary values of solutions to elliptic equations
35J86 Linear elliptic unilateral problems and linear elliptic variational inequalities
35J87 Nonlinear elliptic unilateral problems and nonlinear elliptic variational inequalities
35J88 Systems of elliptic variational inequalities
35J92 Quasilinear elliptic equations with
p
35J93 Quasilinear elliptic equations with mean curvature operator
35J96 Elliptic Monge-Ampère equations
A calculus for a class of finitely degenerate pseudodifferential operators
Ingo Witt (2003)
For a class of degenerate pseudodifferential operators, local parametrices are constructed. This is done in the framework of a pseudodifferential calculus upon adding conditions of trace and potential type, respectively, along the boundary on which the operators degenerate.
A characterization of convex calibrable sets in
{R}^{N}
with respect to anisotropic norms
V. Caselles, A. Chambolle, S. Moll, M. Novaga (2008)
A counterexample to the regularity of the degenerate complex Monge-Ampère equation
Szymon Pliś (2005)
We modify an example due to X.-J. Wang and obtain some counterexamples to the regularity of the degenerate complex Monge-Ampère equation on a ball in ℂⁿ and on the projective space ℙⁿ.
A doubly degenerate elliptic system
González Montesinos, María Teresa, Francisco, Ortegón Gallego (2002)
Xinfu Chen, Avner Friedman (1992)
Rafael de la Llave, Enrico Valdinoci (2009)
M. Goldman (2013)
A global bifurcation result of a Neumann problem with indefinite weight.
El Khalil, Abdelouahed, Ouanan, Mohammed (2004)
A Liouville-type theorem for very weak solutions of nonlinear partial differential equations.
Alberto Fiorenza (1997)
\infty
Rosset, Edi (1996)
James Serrin (2006)
A multiplicity result for a class of quasilinear elliptic and parabolic problems.
do Rosário Grossinho, Maria, Omari, Pierpaolo (1997)
A nonlinear eigenvalue problem with indefinite weights related to the Sobolev trace embedding.
In this paper we study the Sobolev trace embedding W1,p(Ω) → LpV (∂Ω), where V is an indefinite weight. This embedding leads to a nonlinear eigenvalue problem where the eigenvalue appears at the (nonlinear) boundary condition. We prove that there exists a sequence of variational eigenvalues λk / +∞ and then show that the first eigenvalue is isolated, simple and monotone with respect to the weight. Then we prove a nonexistence result related to the first eigenvalue and we end this article with the...
Boumediene Abdellaoui, Ireneo Peral (2006)
The paper analyzes the influence on the meaning of natural growth in the gradient of a perturbation by a Hardy potential in some elliptic equations. Indeed, in the case of the Laplacian the natural problem becomes
-\Delta u-{\Lambda }_{N}\frac{u}{{|x|}^{2}}={\left|\nabla u+\frac{N-2}{2}\frac{u}{{|x|}^{2}}x\right|}^{2}{|x|}^{\left(N-2\right)/2}+\lambda f\left(x\right)
\Omega
u=0
\partial \Omega
{\Lambda }_{N}={\left(\left(N-2\right)/2\right)}^{2}
. This problem is a particular case of problem (2). Notice that
\left(N-2\right)/2
is optimal as coefficient and exponent on the right hand side.
A note on a degenerate elliptic equation with applications for lakes and seas.
Bresch, Didier, Lemoine, Jerome, Guillen-Gonzalez, Francisco (2004)
A note on non-negative singular infinity-harmonic functions in the half-space.
Tilak Bhattacharya (2005)
In this work we study non-negative singular infinity-harmonic functions in the half-space. We assume that solutions blow-up at the origin while vanishing at infinity and on a hyperplane. We show that blow-up rate is of the order |x|-1/3.
A note on the regularity of the degenerate complex Monge-Ampère equation
We prove the almost
{}^{1,1}
regularity of the degenerate complex Monge-Ampère equation in a special case.
A one-dimensional nonlinear degenerate elliptic equation.
Catrina, Florin, Wang, Zhi-Qiang (2001)
A penalty method for approximations of the stationary power-law Stokes problem.
Lefton, Lew, Wei, Dongming (2001)
A refinement of the radial Pohozaev identity
Florin Catrina (2010)
In this article we produce a refined version of the classical Pohozaev identity in the radial setting. The refined identity is then compared to the original, and possible applications are discussed.
|
60J22 Computational methods in Markov chains
60J75 Jump processes
A formula for densities of transition functions
A generalization of Ueno's inequality for n-step transition probabilities
Andrzej Nowak (1998)
We provide a generalization of Ueno's inequality for n-step transition probabilities of Markov chains in a general state space. Our result is relevant to the study of adaptive control problems and approximation problems in the theory of discrete-time Markov decision processes and stochastic games.
A Markov property for two parameter Gaussian processes.
David Nualart Rodón, M. Sanz (1979)
This paper deals with the relationship between two-dimensional parameter Gaussian random fields verifying a particular Markov property and the solutions of stochastic differential equations. In the non Gaussian case some diffusion conditions are introduced, obtaining a backward equation for the evolution of transition probability functions.
A non-Markovian process with unbounded
p
-variation.
Manstavičius, Martynas (2005)
A note on integral representation of Feller kernels
R. Rębowski (1991)
We consider integral representations of Feller probability kernels from a Tikhonov space X into a Hausdorff space Y by continuous functions from X into Y. From the existence of such a representation for every kernel it follows that the space X has to be 0-dimensional. Moreover, both types of representations coincide in the metrizable case when in addition X is compact and Y is complete. It is also proved that the representation of a single kernel is equivalent to the existence of some non-direct...
A note on Markov operators and transition systems
Bartosz Frej (2002)
On a compact metric space X one defines a transition system to be a lower semicontinuous map
X\to {2}^{X}
. It is known that every Markov operator on C(X) induces a transition system on X and that commuting of Markov operators implies commuting of the induced transition systems. We show that even in finite spaces a pair of commuting transition systems may not be induced by commuting Markov operators. The existence of trajectories for a pair of transition systems or Markov operators is also investigated.
Markus Kunze (2011)
Motivated by applications to transition semigroups, we introduce the notion of a norming dual pair and study a Pettis-type integral on such pairs. In particular, we establish a sufficient condition for integrability. We also introduce and study a class of semigroups on such dual pairs which are an abstract version of transition semigroups. Using our results, we give conditions ensuring that a semigroup consisting of kernel operators has a Laplace transform which also consists of kernel operators....
A semigroup charaterization of the multiparameter Wiener process.
Chang C.Y. Dorea (1983)
A simple proof of the
{L}^{p}
continuity of the higher order Riesz transforms with respect to the gaussian measure
{\gamma }_{d}
Liliana Forzani, Roberto Scotto, Wilfredo Urbina (2001)
A symbolic calculus and a parametrix construction for pseudodifferential operators with non-smooth negative definite symbols.
Alexander Potrykus (2009)
Additive functionals and excursions of Kuznetsov processes.
Boutabia, Hacène (2005)
An application of Markov operators in differential and integral equations
Jan Malczak (1992)
An extended generator and Schrödinger equations.
Getoor, R.K. (1999)
{L}^{2}\left(\mu \right)
{L}^{2}\left(\mu \right)
Asymptotic Properties of Stochastic Semilinear Equations by the Method of Lower Measures
B. Maslowski, I. Simão (1997)
Carl Graham (1985)
Branching Semigroups on Banach Lattices.
Egbert Dettweiler (1982)
Central limit theorem for an additive functional of a Markov process, stable in the Wesserstein metric
Anna Walczuk (2008)
We study the question of the law of large numbers and central limit theorem for an additive functional of a Markov processes taking values in a Polish space that has Feller property under the assumption that the process is asymptotically contractive in the Wasserstein metric.
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Number of nonzero linear coefficients - MATLAB - MathWorks í•œêµ
ncoeffs = 4×1
ncoeffs2 = 4×1
Regularization is the process of finding a small set of predictors that yield an effective predictive model. For linear discriminant analysis, there are two parameters, γ and δ, that control regularization as follows. cvshrink helps you select appropriate values of the parameters.
Let Σ represent the covariance matrix of the data X, and let
\stackrel{^}{X}
D=\text{diag}\left({\stackrel{^}{X}}^{T}*\stackrel{^}{X}\right).
\stackrel{Ë}{\mathrm{Σ}}
\stackrel{Ë}{\mathrm{Σ}}=\left(1â\mathrm{γ}\right)\mathrm{Σ}+\mathrm{γ}D.
Whenever γ ≥ MinGamma,
\stackrel{Ë}{\mathrm{Σ}}
Let μk be the mean vector for those elements of X in class k, and let μ0 be the global mean vector (the mean of the rows of X). Let C be the correlation matrix of the data X, and let
\stackrel{Ë}{C}
\stackrel{Ë}{C}=\left(1â\mathrm{γ}\right)C+\mathrm{γ}I,
{\left(xâ{\mathrm{μ}}_{0}\right)}^{T}{\stackrel{Ë}{\mathrm{Σ}}}^{â1}\left({\mathrm{μ}}_{k}â{\mathrm{μ}}_{0}\right)=\left[{\left(xâ{\mathrm{μ}}_{0}\right)}^{T}{D}^{â1/2}\right]\left[{\stackrel{Ë}{C}}^{â1}{D}^{â1/2}\left({\mathrm{μ}}_{k}â{\mathrm{μ}}_{0}\right)\right].
The parameter δ enters into this equation as a threshold on the final term in square brackets. Each component of the vector
\left[{\stackrel{Ë}{C}}^{â1}{D}^{â1/2}\left({\mathrm{μ}}_{k}â{\mathrm{μ}}_{0}\right)\right]
is set to zero if it is smaller in magnitude than the threshold δ. Therefore, for class k, if component j is thresholded to zero, component j of x does not enter into the evaluation of the posterior probability.
The DeltaPredictor property is a vector related to this threshold. When δ ≥ DeltaPredictor(i), all classes k have
|{\stackrel{Ë}{C}}^{â1}{D}^{â1/2}\left({\mathrm{μ}}_{k}â{\mathrm{μ}}_{0}\right)|â¤\mathrm{δ}.
Therefore, when δ ≥ DeltaPredictor(i), the regularized classifier does not use predictor i.
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Air pollution — lesson. Science CBSE, Class 9.
We know that air has different uses and is essential for all living things. We contaminate the atmosphere through a variety of environmental contributions, such as modernization.
Air pollution is the change in air's chemical composition due to the presence of pollutants that can be hazardous to humans and the environment.
Various sources pollute the air, such as carbon dioxide, nitrogen oxide, sulphur oxide, lead, smoke, soot, carbon monoxide, ozone, etc. These sources of pollutants are from the emission of industrialisation, the combustion process, and so on.
All gases except ozone are absorbed into the atmosphere, which pollutes the air. These sources of pollution and their effects can lead to various issues regarding health and the environment.
Carbon dioxide (
{\mathit{CO}}_{2}
) - The emission of this gas is from the complete combustion process, vehicles and industries etc. This lend smog (smoke + fog) formation in the environment. The health risk includes an increase in heartbeat, vision problems, etc.
Nitrogen oxides (
\mathit{NO},\phantom{\rule{0.147em}{0ex}}{\mathit{NO}}_{2}
) - This is released from industries, electricity generation, automobiles etc. Nitrogen oxides also lead to the formation of smog. This leads to heart, lung diseases and even problems with breathing.
Sulphur oxide (
{\mathit{SO}}_{2}
) - This gas is also released from industries, automobiles, combustion of fossil fuels etc. Sulphur oxide lends acid rain and damages foliage(the leaves of a tree or plant). This pollutant also damages the buildings and monuments, causes skin problems, asthma, heart diseases, etc.
Lead (
\mathit{Pb}
) - This is released from metal processing, waste incineration, combustion of fossil fuels and so on. Lead has an adverse effect on the environment, causing the loss in biodiversity, neurological problems, decreased reproduction etc. This also harms humans, especially children's; when exposed, it causes a learning disability affecting s the neurological system.
Ozone (
{O}_{3}
) - Ozone is released from nitrogen oxides, emission from industries, volatile organic compounds etc. Ozone interferes with plants' absorption patterns, which causes irritation and inflammation of breathing passages.
The pollutants mentioned above are released into the atmosphere at a very high rate except for lead.
2. Air Toxics:
Likewise, there are hundreds of compounds considered harmful, although they are present in minuscule concentrations in the air. These are called air toxins. The hazardous air pollutants are arsenic, asbestos, benzene, beryllium, coke oven emissions, cyanide, mercury compounds, radon, radium, uranium, vinyl chloride etc.
The sources of air toxics are from weathering of rocks (arsenic), natural deposits (asbestos), vehicle exhaust (benzene), papermaking (chlorine), automobile exhaust (cyanide), uranium mines(uranium) sewage sludge incineration (nickel compounds), battery manufacturing (lead compounds), etc.
These air toxics may be released in sudden and catastrophic accidents. For example, the Bhopal disaster of \(1984\) releasedmethyl isocyanate. This air toxic immediately killed \(3,000\) people and eventually took the life around \(15,000\) to \(25,000\) people after the incident. Hazardous air pollutants may be released when equipment leaks or when the material is transferred, or they may be emitted from smokestacks.
The toxic air pollution levels are generic in large cities, where particulates and gases from transportation, heating, and manufacturing accumulate and linger in the atmosphere.
Self-awareness is required to reduce air pollution. Cooperating with nature will preserve our lives and the earth we live on.
|
p
A boundary value problem in the hyperbolic space.
Amster, P., Keilhauer, G., Mariani, M.C. (1999)
A Class of Elliptic Partial Differential Equations with Exponential Nonlinearities (Corrigendum).
P.A. Vuillermot (1988)
A class of nonlinear conservative elliptic equations in cylinders
Jean René Licois, Laurent Véron (1998)
A comparison of multilevel methods for total variation regularization.
Vassilevski, P.S., Wade, J.G. (1997)
A difference scheme for an elliptic system of nonlinear differential-functional equations with Dirichlet type boundary conditions. The existence and uniqueness of solution
Bogusław Bożek, Ryszard Mosurski (1984)
A Dirichlet problem in the strip.
Montefusco, Eugenio (1996)
A discontinuous problem with quasilinear operator.
Calahorrano Recalde, Marco, Mayorga Zambrano, Juan (2001)
A fibering map approach to a semilinear elliptic boundary value problem.
Brown, Kenneth J., Wu, Tsung-Fang (2007)
A free boundary problem for quasi-linear elliptic equations
A generalization of Ekeland's variational principle with applications.
El Amrouss, Abdel R., Tsouli, Najib (2006)
A generalization of the saddle point method with applications
We show that one can drop an important hypothesis of the saddle point theorem without affecting the result. We then show how this leads to stronger results in applications.
A global solution curve for a class of semilinear equations.
Korman, Philip (1997)
Antonin Chambolle, Erik Lindgren, Régis Monneau (2012)
In this paper we study the limit as p → ∞ of minimizers of the fractional Ws,p-norms. In particular, we prove that the limit satisfies a non-local and non-linear equation. We also prove the existence and uniqueness of solutions of the equation. Furthermore, we prove the existence of solutions in general for the corresponding inhomogeneous equation. By making strong use of the barriers in this construction, we obtain some regularity results.
A mathematical analysis of thermal explosions.
Taira, Kazuaki (2001)
|
Julian was studying a pattern made with toothpicks, and he started the table shown at right.
Do you notice the pattern Julian made? In figures 1, 2, and 3, the number of toothpicks increase by 3. Can you complete the table?
The completed table should look like the one below.
Number of Toothpicks
1
7
2
10
3
13
4
\mathbf{16}
5
\mathbf{19}
Draw axes and plot Julian’s points.
It may help to label your
x
-axis (horizontal) ''Figure Number'' and your
y
-axis (vertical) ''Number of Toothpicks''.
How can you describe what all of these points have in common?
Try taking a look at your graph. Do your points form any kind of figure or line?
\begin{array}{c|c} \quad {\text{Figure} \\ \text{Number}} \quad & \; \; {\text{Number of} \\ \text{Toothpicks}} \\ \hline 1 & 7 \\ \hline 2 & 10 \\ \hline 3 & 13 \\ \hline 4 \\ \hline 5 \end{array}
|
C
{C}^{*}
C{\left(\text{Ω}\right)}_{\beta }
A characterization of Valdivia compact spaces.
Ondrej Kalenda (2000)
A classification of separable Rosenthal compacta and its applications [Book]
S. A. Argyros, P. Dodos, V. Kanellopoulos (2008)
A connection between multiplication in C(X) and the dimension of X
Andrzej Komisarski (2006)
Let X be a compact Hausdorff topological space. We show that multiplication in the algebra C(X) is open iff dim X < 1. On the other hand, the existence of non-empty open sets U,V ⊂ C(X) satisfying Int(U· V) = ∅ is equivalent to dim X > 1. The preimage of every set of the first category in C(X) under the multiplication map is of the first category in C(X) × C(X) iff dim X ≤ 1.
A contribution to the topological classification of the spaces Ср(X)
Robert Cauty, Tadeusz Dobrowolski, Witold Marciszewski (1993)
We prove that for each countably infinite, regular space X such that
{C}_{p}\left(X\right)
{Z}_{\sigma }
-space, the topology of
{C}_{p}\left(X\right)
is determined by the class
{F}_{0}\left({C}_{p}\left(X\right)\right)
of spaces embeddable onto closed subsets of
{C}_{p}\left(X\right)
{C}_{p}\left(X\right)
, whenever Borel, is of an exact multiplicative class; it is homeomorphic to the absorbing set
{\Omega }_{\alpha }
for the multiplicative Borel class
{M}_{\alpha }
{F}_{0}\left({C}_{p}\left(X\right)\right)={M}_{\alpha }
. For each ordinal α ≥ 2, we provide an example
{X}_{\alpha }
{C}_{p}\left({X}_{\alpha }\right)
{\Omega }_{\alpha }
A few remarks on the set of finite-to-one maps of the Cantor set
A function space C(K) not weakly homeomorphic to C(K)×C(K)
Witold Marciszewski (1988)
A function space Cp(X) not linearly homeomorphic to Cp(X) × ℝ
We construct two examples of infinite spaces X such that there is no continuous linear surjection from the space of continuous functions
{c}_{p}\left(X\right)
{c}_{p}\left(X\right)
× ℝ
.Inparticular,
cp(X)
isnotlinearlyhomeomorphicto
×ℝ
. One of these examples is compact. This answers some questions of Arkhangel’skiĭ.
A generalization of a theorem of Polak
Kasahara, Shouro (1967)
A generalization of boundedly compact metric spaces
Gerald Beer, Anna Di Concilio (1991)
〈X,d〉
UC
space provided each continuous function on
X
into a metric target space is uniformly continuous. We introduce a class of metric spaces that play, relative to the boundedly compact metric spaces, the same role that
UC
spaces play relative to the compact metric spaces.
A Hilbert cube compactification of the function space with the compact-open topology
Atsushi Kogasaka, Katsuro Sakai (2009)
Let X be an infinite, locally connected, locally compact separable metrizable space. The space C(X) of real-valued continuous functions defined on X with the compact-open topology is a separable Fréchet space, so it is homeomorphic to the psuedo-interior s = (−1, 1)ℕ of the Hilbert cube Q = [−1, 1]ℕ. In this paper, generalizing the Sakai-Uehara’s result to the non-compact case, we construct a natural compactification
\overline{C}
(X) of C(X) such that the pair (
\overline{C}
(X), C(X)) is homeomorphic to (Q, s). In case...
A new characterization of compact sets in function spaces
M.M. Drešević (1974)
A new hyperspace topology and the study of the function space
{\theta }^{*}
LC\left(X,Y\right)
S. Ganguly, Sandip Jana, Ritu Sen (2009)
A new way to find compact zero-dimensional first countable preimages of first countable compact spaces
Vladimir Vladimirovich Tkachuk (1988)
A nice subclass of functionally countable spaces
X
is functionally countable if
f\left(X\right)
is countable for any continuous function
f:X\to ℝ
. We will call a space
X
exponentially separable if for any countable family
ℱ
of closed subsets of
X
, there exists a countable set
A\subset X
A\cap \bigcap 𝒢\ne \varnothing
𝒢\subset ℱ
\bigcap 𝒢\ne \varnothing
. Every exponentially separable space is functionally countable; we will show that for some nice classes of spaces exponential separability coincides with functional countability. We will also establish that the class of exponentially separable spaces has...
Jerzy Kąkol, Albert Kubzdela, Wiesƚaw Śliwa (2013)
We prove a non-archimedean Dugundji extension theorem for the spaces
{C}^{*}\left(X,𝕂\right)
of continuous bounded functions on an ultranormal space
X
with values in a non-archimedean non-trivially valued complete field
𝕂
𝕂
is discretely valued and
Y
X
we show that there exists an isometric linear extender
T:{C}^{*}\left(Y,𝕂\right)\to {C}^{*}\left(X,𝕂\right)
X
is collectionwise normal or
Y
is Lindelöf or
𝕂
is separable. We provide also a self contained proof of the known fact that any metrizable compact subspace
Y
of an ultraregular...
A note on Ascol's theorem for spaces of multifunctions.
M.M. Marjanovic, M.M. Dresevic (1972)
|
Cecil the Acrobat walked
3\frac{1}{2}
feet on his tightrope, backed up
1
foot, then walked
6\frac{1}{2}
feet to get to the other side.
Write an expression to represent Cecil’s walk.
Imagine making a diagram of this description. What would it look like?
Try writing the expression yourself before checking the answer.
3\frac{1}{2}-1+6\frac{1}{2}
How far did Cecil walk?
Cecil's distance will be the value of the expression you just wrote, since you were describing his walk.
Cecil started at the beginning again and walked
6\frac{1}{2}
feet, backed up
1
3\frac{1}{2}
feet. How far did he walk this time? How does this compare to your answer from part (b)? Explain why these answers are the same or different in a sentence or with a drawing.
Try writing an expression for this walk. Does it look like your expression for part (a)? Are there any differences other than the order of the numbers? Remember that order does not matter if you are only doing addition.
|
p
{ℒ}^{2,\text{Φ}}
A bifurcation result for equations with anisotropic
p
-Laplace-like operators.
Le, Vy Khoi (2001)
A bifurcation theory for some nonlinear elliptic equations
Biagio Ricceri (2003)
We deal with the problem ⎧ -Δu = f(x,u) + λg(x,u), in Ω, ⎨ (
{P}_{\lambda }
) ⎩
{u}_{\mid \partial \Omega }=0
where Ω ⊂ ℝⁿ is a bounded domain, λ ∈ ℝ, and f,g: Ω×ℝ → ℝ are two Carathéodory functions with f(x,0) = g(x,0) = 0. Under suitable assumptions, we prove that there exists λ* > 0 such that, for each λ ∈ (0,λ*), problem (
{P}_{\lambda }
) admits a non-zero, non-negative strong solution
{u}_{\lambda }\in {\bigcap }_{p\ge 2}{W}^{2,p}\left(\Omega \right)
li{m}_{\lambda \to 0⁺}||{u}_{\lambda }{||}_{{W}^{2,p}\left(\Omega \right)}=0
for all p ≥ 2. Moreover, the function
\lambda ↦{I}_{\lambda }\left({u}_{\lambda }\right)
is negative and decreasing in ]0,λ*[, where
{I}_{\lambda }
is the energy functional related to (
{P}_{\lambda }
A boundary blow-up for sub-linear elliptic problems with a nonlinear gradient term.
Zhang, Zhijun (2006)
{C}^{0}
p
q
-Laplacian type equation with potentials eigenvalue problem in
{ℝ}^{N}
Wu, Mingzhu, Yang, Zuodong (2009)
A compactness result in thin-film micromagnetics and the optimality of the Néel wall
Radu Ignat, Felix Otto (2008)
In this paper, we study a model for the magnetization in thin ferromagnetic films. It comes as a variational problem for
{S}^{1}
-valued maps
{m}^{\text{'}}
(the magnetization) of two variables
{x}^{\text{'}}
{E}_{\epsilon }\left({m}^{\text{'}}\right)=\epsilon \int {|{\nabla }^{\text{'}}·{m}^{\text{'}}|}^{2}d{x}^{\text{'}}+\frac{1}{2}\int {\left||{\nabla }^{\text{'}}{|}^{-1/2}{\nabla }^{\text{'}}·{m}^{\text{'}}\right|}^{2}d{x}^{\text{'}}
. We are interested in the behavior of minimizers as
\epsilon \to 0
. They are expected to be
{S}^{1}
{m}^{\text{'}}
of vanishing distributional divergence
{\nabla }^{\text{'}}·{m}^{\text{'}}=0
, so that appropriate boundary conditions enforce line discontinuities. For finite
\epsilon >0
, these line discontinuities are approximated by smooth transition layers, the so-called Néel walls. Néel...
A compactness theorem of n-harmonic maps
Chang You Wang (2005)
A comparison of some efficient numerical methods for a nonlinear elliptic problem
Balázs Kovács (2012)
The aim of this paper is to compare and realize three efficient iterative methods, which have mesh independent convergence, and to propose some improvements for them. We look for the numerical solution of a nonlinear model problem using FEM discretization with gradient and Newton type methods. Three numerical methods have been carried out, namely, the gradient, Newton and quasi-Newton methods. We have solved the model problem with these methods, we have investigated the differences between them...
A Counterexample in Elliptic Regularity Theory.
Michael Struwe (1981)
A Counter-Example to the Boundary Regularity of Solutions to Elliptic Quasilinear Systems.
Mariano Giaquinta (1978)
A deformation theorem in the noncompact nonsmooth setting and its applications.
Arioli, Gianni (2001)
A Dicontinuous Solution of a Mildly Nonlinear Elliptic System.
A difference method for a nonlinear system of elliptic equations with mixed derivatives
Z. Kowalski (1980)
A diffused interface whose chemical potential lies in a Sobolev space
Yoshihiro Tonegawa (2005)
We study a singular perturbation problem arising in the scalar two-phase field model. Given a sequence of functions with a uniform bound on the surface energy, assume the Sobolev norms
{W}^{1,p}
of the associated chemical potential fields are bounded uniformly, where
p>\frac{n}{2}
n
is the dimension of the domain. We show that the limit interface as
\epsilon
tends to zero is an integral varifold with a sharp integrability condition on the mean curvature.
A direct method approach for the existence of convex hypersurfaces with prescribed Gauss-Kronecker curvature.
Kaising Tso (1992)
A Dirichlet problem with asymptotically linear and changing sign nonlinearity.
Marcello Lucia, Paola Magrone, Huan-Song Zhou (2003)
This paper deals with the problem of finding positive solutions to the equation -∆[u] = g(x,u) on a bounded domain 'Omega' with Dirichlet boundary conditions. The function g can change sign and has asymptotically linear behaviour. The solutions are found using the Mountain Pass Theorem.
|
Greta is trying to determine the portion of green candies in various bags of green and yellow candies. Using the information below, determine the portion of green candies in each bag.
Bag A: Two thirds of the candies are yellow. What portion of the candies is green?
Remember that the whole bag is a portion of
1
. What portion would go with the two thirds to complete one whole?
\frac{1}{3}
Bag B:
29\%
of the candies are yellow. What portion of the candies is green?
100\%
represents the whole bag.
71\%
Bag C:
4
9
candies are yellow. What portion of the candies is green?
This problem is similar to both part (a) and part (b). Refer to those if you need help.
9
candies, and
4
of them are yellow, how many does that leave? The remainder are all green candies.
|
Quantitative nonorientability of embedded cycles
15 January 2018 Quantitative nonorientability of embedded cycles
Duke Math. J. 167(1): 41-108 (15 January 2018). DOI: 10.1215/00127094-2017-0035
We introduce an invariant linked to some foundational questions in geometric measure theory and provide bounds on this invariant by decomposing an arbitrary cycle into uniformly rectifiable pieces. Our invariant measures the difficulty of cutting a nonorientable closed manifold or mod-
2
cycle in
{\mathbb{R}}^{n}
into orientable pieces, and we use it to answer some simple but long-open questions on filling volumes and mod-
\nu
currents.
Robert Young. "Quantitative nonorientability of embedded cycles." Duke Math. J. 167 (1) 41 - 108, 15 January 2018. https://doi.org/10.1215/00127094-2017-0035
Received: 28 January 2016; Revised: 9 May 2017; Published: 15 January 2018
Keywords: filling volume , integral currents , orientability , uniform rectifiability
Robert Young "Quantitative nonorientability of embedded cycles," Duke Mathematical Journal, Duke Math. J. 167(1), 41-108, (15 January 2018)
|
MAI loans and Vaults incentives - Mai Finance - Tutorials
This article is a detailed explanation of how you can use Mai Finance to borrow MAI at 0% interest, and get paid to do so, transforming your 0% interest loan into a negative interest loan.
The core business of Mai Finance is a lending platform. Instead of selling their crypto to buy other assets, people are able to lock their funds on Mai Finance and borrow against them. This presents the opportunity to keep high value assets (WBTC, WETH ...) while still being able to get other assets and farm yields. In that case, the loan is used to generate revenue, while the collateral is gaining value.
One of the other big advantage of using Mai Finance is that there's no repayment schedule. In other words, you borrow MAI stable coin against your crypto, you don't pay any interests, and you can repay your debt whenever you want. See the different articles on debt management for more details. The only fee that you would ever pay is a repayment fee corresponding to 0.5% of the money you borrowed that you pay when you repay your loan, and that is taken out of your collateral.
As an example, if you deposited $200 worth of WETH to borrow $100 worth of MAI, when you repay your loan you would have to pay a fee of $0.50 directly taken out of your WETH deposit.
If that wasn't already an amazing opportunity, the Mai Finance team introduced in September 2021 Vault incentives paid in Qi, the native token of Mai Finance. In other words, by depositing your assets on Mai Finance in a vault to borrow MAI, you will also get paid to do it. This articles presents in details how this functionality works.
Vaults - What they are and how they work
On Mai Finance, vaults are special storages where one can deposit their assets. Currently, there are 10 types of vaults:
The different vault types you can create on Mai Finance
There are 2 different types of vaults:
The first 6 vaults in the list are for simple assets while the 4 last ones are for camTokens. camTokens are compounding AAVE market tokens, a representation of a deposit that you could have done on AAVE and then deposited on the yield pools of Mai Finance. While you assets is generating yields on AAVE (and while the rewards are automatically compounded by the yield pool), you can still borrow MAI stable coins against these tokens.
As a side note, you can see on the screenshot above that the creation page shows some very important informations:
MAI available: this corresponds to the maximum debt ceiling, the maximum number of MAI that can be minted from vault deposits.
Min Coll. ratio: this is the minimum Collateral to Debt ratio (CDR) for that vault
Vault incentives APR
Understanding Debt Ceiling
The maximum number of MAI that one can mint on a specific vault depends on how much assets is deposited on that vault. Debt ceiling are implemented in order to make sure that the market isn't flooded with MAI in a very short time, which may affect the price of the stable coin.
As an example, if a big institution would deposit 5,000 WBTC at once and was able to borrow $100,000,000 worth of MAI, swapping the totality for more WBTC, this could drive the price of MAI down so much that the price would deviate too much from its peg, putting the whole platform at risk. Debt ceiling is the mechanism that prevents this from happening: there's a maximum amount of MAI that can be minted for a given vault type.
When the debt ceiling is reached, the time at which there aren't any more available MAI to mint is recorded, and the system automatically increases the debt ceiling after 48 hours. This is considered enough time for the MAI price to stabilize (in case of high sell pressure following a big sell off of MAI).
This means that for 48h, nobody will be able to borrow more MAI from a vault that reached its debt ceiling, unless a debt is repaid.
As a side note, the more MAI on the market, the more stable the price is. Indeed, a massive sell of MAI is less invasive if there are more MAI in circulation.
If someone sells 1,000 MAI while there are only 10,000 MAI in circulation, the sell corresponds to 10%
If someone sells 1,000 MAI while there are 10,000,000 MAI in circulation, the sell corresponds to 0.01%
Hence, the debt ceiling isn't increased incrementally, but exponentially: the more MAI in circulation, the less impact a big sell would have, so the debt ceiling can be increased by a lot more.
When you borrow MAI, it can happen that the maximum amount of MAI that you can borrow is capped by the debt ceiling, regardless of the current value of your collateral and the current amount of MAI you already borrowed. When that's the case, you may wait up for 48h before you can actually borrow more MAI.
Understanding Collateral to Debt Ratio
The CDR, or Collateral to Debt Ratio is the ratio between the value of the deposited assets in your vault compared to the amount of MAI you borrowed.
As an example, if you deposited $200 worth of WETH to borrow $100 worth of MAI, your CDR would be
CDR=\frac{CollateralValue}{DebtValue}=\frac{200}{100}=200\%
Maintaining a CDR above 100% means that, at any point, there are more collateral than debt. This is mandatory to ensure that the MAI stable coin is over-collateralized, and is one of the foundations of the Mai Finance tokenomics. You can get more details from the official Mai Finance documentation.
Each vault type has a minimum CDR ratio accepted, a threshold under which the vault is considered at risk because the borrowed amount may not be backed by enough collateral. At this point, anyone can liquidate the vault, meaning a part of the debt is repaid by the liquidator that can then get a portion of the deposited collateral in repayment. Once again, you can find more details about liquidation process in the official documentation.
When you borrow MAI against a given collateral, you will get some hints on what's the maximum amount of MAI you can borrow, and what would be the impact on your health ratio depending on the amount borrowed, as you can see in the screenshot bellow:
Health mitigation depending on borrowed amount
It's very important to keep an eye on your CDR and keep a healthy ratio to
prevent liquidation
increase the health of the whole Mai Finance platform by ensuring the MAI volume in circulation is properly backed
The "healthy" CDR, as defined by the Mai Finance team, is between 25% and 270% above the minimum CDR value. As a side note, you can also check our strategy guides to see how you can use conservative/aggressive CDRs to invest in other projects, or repay your debt using your debt.
Understanding Vaults incentives APRs
In September 2021, Mai Finance introduced vault incentives. This is a reward allocated by the Mai Finance platform to anyone borrowing MAI and participating in the growth of the platform.
Each Vault type (among the 10 different types) receives 0.05 Qi per block, that is then distributed between all the users who have a healthy Collateral to Debt Ratio. The APR of the vault is defined by the current amount of MAI borrowed.
As an example, Ben and Kila are 2 friends who deposited their ETH in the WETH vaults on Mai Finance.
Ben deposited the equivalent of $2,000 worth of ETH and borrowed 1,000 MAI
Kila deposited the equivalent of $10,000 worth of ETH and borrowed 6,000 MAI
The current amount of MAI borrowed by users who deposited WETH in the vault is 1,000,000 MAI.
Both Ben and Kila qualify for the vault incentives because Ben has a CDR of 200% and Kila a CDR of 166.67%. Ben, with his loan, owns 0.1% of the total amount borrowed, while Kila owns 0.6%.
The total amount of Qi allocated to the WETH vault (or any vault) is
Qi=0.05*\frac{86400}{2}=2160
86,400 is the number of seconds in a day, and on Polygon, the block time is 2 seconds, meaning that the expected number of blocks every day is 86,400 / 2 = 43,200. Hence, the emission for each Vault is 2,160 Qi / day.
Note: Block time has increased lately and is around 2.6 seconds. However, all APRs and APYs displayed on all apps are assuming a block time of 2 seconds. Please DYOR and check the current block time on PolygonScan.
Hence, if the state of the Vault remains the same, Ben will get 0.1% of the 2,160 Qi distributed, while Kila will get 0.6% of the granted reward.
Ben will get 2.16 Qi every day, which is a daily reward of 0.216%, or an APR of 78.84%
Kila will get 12.96 Qi every day, which is also a daily reward of 0.216%, or an APR of 78.84%
On a side note, 2,160 Qi for 1,000,000 MAI is a daily reward of 0.216%, or 78.84%, which is the Vault's APR.
It's easy to see that the Vault's APR is directly linked to the amount of MAI borrowed. The more MAI is borrowed, the lower the APR. As a side note, the amount of MAI that can be borrowed is also capped by the debt ceiling, which is increased with the demand for MAI.
As a verification, we can calculate the theoretical APR for the MATIC vault based on numbers published on the analytics page on Mai Finance. The amount of MAI borrowed from the MATIC vault is 799,328. The reward is 216 Qi per day for that vault. That corresponds to a APR of
APR=\frac{QiReward*Qi_{Price}}{MAI_{borrowed}}*365=\frac{2160*0.441}{785008}*365=44.29\%
This corresponds more or less to the APR of the MATIC Vault, as displayed in the following screenshot:
APR of a MATIC vault on Mai Finance after the launch of Vault rewards
Calculating starting vaults' APRs
With the same data as the example above, it's possible to calculate the starting APRs for all vaults
As you can see, some vaults will generate more rewards than others. Also, you can see that it's super important to deposit your assets as soon as possible to benefit from high APRs before the debt ceiling is increased and more loan is taken (lowering the APR).
You can also see that if you keep your loan for one year of more, the 0.5% repayment fee will easily be compensated by the reward program.
Rewards allocated by the vault incentives will be distributed the same way as for staked Qi. Every Wednesday, the Qi allocated by the Vaults incentives program will be airdropped / claimed for the week prior to the pay day.
Vaults incentives FAQs
If you want to know more about the way Vault incentives are working, here's an official FAQ from the Discord server.
What vaults are receiving rewards?
Right now all the vault types have been allocated Qi rewards
How much rewards are given out for the borrowing incentives?
0.05 Qi/block for each vault type
How much MAI do I need to borrow to earn rewards?
For Vault Borrow Incentives, stay between 25% and 270% above the liquidation ratio to receive QI token airdrop. This means:
Matic - Liquidation ratio 150% - Eligible for Incentives between 175% and 420%
Tokens: - Liquidation ratio 130% - Eligible for Incentives between 155% and 400%
CamTokens: - Liquidation ratio 135% - Eligible for Incentives between 160% and 405%
How can I see if my vault is earning rewards?
If you see the fire emoji on your vault overview page that means that vault is earning rewards
Your percent of the reward pool is based on the percentage of MAI you borrowed compared to the total amount of MAI borrowed from that vault type.
How long will the incentives program last?
The scheduled length of the borrowing incentives program is to last 3 months. The DAO can vote to stop incentives before the 3 months is over or vote to extend the program.
How will we receive rewards?
Qi will be airdropped to eligible vault holders.
How is eligibility for rewards gathered?
Eligibility for rewards is calculated per block. You will earn rewards for the blocks you were eligible during the week.
When do tracking rewards for the week start?
We will follow the same schedule as eQi. You can find the block numbers on the boost page.
This guide has been written prior to the launch of Vault incentives, meaning that the APRs promoted in this document (as well as this document) are subject to modifications, and/or may not be accurate. The amount of MAI borrowed, the debt ceiling and the value of the Qi token will highly impact the final APR of each vault type. Please, make sure that you invest responsibly.
|
Positive Solutions for Three-Point Boundary Value Problem of Fractional Differential Equation with -Laplacian Operator
Shang-lin Yao, Guo-hui Wang, Zhi-ping Li, Li-jun Yu, "Positive Solutions for Three-Point Boundary Value Problem of Fractional Differential Equation with -Laplacian Operator", Discrete Dynamics in Nature and Society, vol. 2013, Article ID 376938, 7 pages, 2013. https://doi.org/10.1155/2013/376938
Shang-lin Yao,1,2 Guo-hui Wang,2 Zhi-ping Li,1 and Li-jun Yu2
1School of Energy Resources, China University of Geosciences (Beijing), Beijing 100083, China
2Research Institute of Petroleum Exploration and Development, PetroChina, Beijing 100083, China
We investigate the existence of multiple positive solutions for three-point boundary value problem of fractional differential equation with -Laplacian operator , where are the standard Riemann-Liouville derivatives with , and the constant is a positive number satisfying ; -Laplacian operator is defined as . By applying monotone iterative technique, some sufficient conditions for the existence of multiple positive solutions are established; moreover iterative schemes for approximating these solutions are also obtained, which start off a known simple linear function. In the end, an example is worked out to illustrate our main results.
In this paper, we study the existence of multiple positive solutions for the following three-point boundary value problem of fractional differential equation with -Laplacian operator where are the standard Riemann-Liouville derivatives with and the constant is a positive number satisfying ; -Laplacian operator is defined as .
Fractional differential equations have gained considerable importance due to their application in various sciences, such as physics, mechanics, chemistry, and engineering. In the recent years, there has been a significant development in ordinary and partial differential equations involving fractional derivatives; see the monographs of Kilbas et al. [1], Miller and Ross [2], Podlubny [3], and the papers [4–14] and the references therein.
In [15], Li et al. were concerned with the nonlinear differential equation of fractional order subject to the boundary conditions By using some fixed point theorems, the existence and multiplicity results of positive solutions were established.
On the other hand, the differential equations with -Laplacian have also been widely studied owing to the fact that -Laplacian boundary value problems have important application in theory and application of mathematics and physics. For example, in [16], by using the fixed point index, Yang and Yan investigated the existence of positive solution for the third-order Sturm-Liouville boundary value problems with -Laplacian operator: However, there are few articles dealing with the existence of solutions to boundary value problems for fractional differential equation with -Laplacian operator. In [17], the authors investigated the nonlinear nonlocal problem where . By using Krasnoselskii’s fixed point theorem and Leggett-Williams theorem, some sufficient conditions for the existence of positive solutions to the above BVP are obtained. In [18], by using upper and lower solutions method, under suitable monotone conditions, Wang et al. investigated the existence of positive solutions to the following nonlocal problem: where . Recently, Chai [19] investigated the two-point boundary value problem of fractional differential equation with -Laplacian operator: By means of the fixed point theorem on cones, some existence and multiplicity results of positive solutions are obtained.
Motivated by the above mentioned works, in this paper, we consider the multiplicity results of positive solutions for the three point boundary value problem of fractional differential equation with -Laplacian operator. Difference to [15–19], by using monotone iterative technique, we not only establish the existence of multiple positive solutions but also obtain the iterative sequences of these positive solutions.
Definition 2 (see [1–3]). The Riemann-Liouville fractional derivative of order of a function is given by where , denotes the integer part of number , provided that the right-hand side is pointwise defined on .
Proposition 3 (see [1–3]). (1) If , then
Proposition 4 (see [1–3]). Let , and is integrable, then where , is the smallest integer greater than or equal to .
Definition 5. A function is called a nonnegative solution of BVP (1), if on and satisfies (1). Moreover, if , then is said to be a positive solution of BVP (1).
For forthcoming analysis, we first consider the following fractional differential equation:
Lemma 6 (see [15]). If and , then the boundary value problem (13) has the unique solution where where .
Lemma 7 (see [15]). The Green function in Lemma 6 has the following properties:(i) is continuous on ,(ii) for any . And if , the Green function also satisfies(iii) for any ,(iv)there exists a positive function such that where
Let be the set of positive integers, let be the set of real numbers, and let be the set of nonnegative real numbers. Let . Denote by the Banach space of all continuous functions from into with the norm
Define the cone in as Let satisfy the relation , where is given by (1).
To study BVP (1), we first consider the associated linear BVP: for and .
Let . By Proposition 4, the solution of initial value problem is given by . From the relations , it follows that , and so Noting that , from (22), we know that the solution of (20) satisfies By Lemma 6, the solution of (23) can be written as Since , , we have , , and so which implies that the solution of (23) is given by For the convenience, we make the following assumptions.(H1) is continuous and nondecreasing, and there exists a constant such that, for any , (H2) is nonnegative on (0, 1), and
Remark 8. By (27), for any , clearly, Now, for any , define one operator as follows: Then by (20) and (23), the BVP (1) is equivalent to the fixed point problems of the operators .
Lemma 9. Assume that (H1) and (H2) hold. Then are continuous, compact, and nondecreasing.
Proof. In fact, for any , On the other hand, by Lemma 7, So .
Next, supposing is a bounded set, then for any , there exists a constant such that . Thus for any , we have which implies is bounded. On the other hand, according to the Arzela-Ascoli theorem and Lebesgue dominated convergence theorem, we easily see is completely continuous. In the end, noticing the monotonicity of on and the definition of , we also have that the operator is nondecreasing.
Define two constants
Theorem 10. Suppose conditions (H1) and (H2) hold. If there exists a positive constant such that where and are defined by (34), then the BVP (1) has the maximal and minimal solutions and , which are positive, and there exist two positive constants such that Moreover for initial values , define the iterative sequences by Then for uniformly, respectively.
Proof. Let ; we firstly prove . In fact, for any , we have By the assumption , we have It follows from Lemma 9 that is completely continuous operator; thus by (35) and (40), we have which implies that .
Let ; then . Leting , we have . Denote It follows from that . Since is compact, we obtain that is a sequentially compact set.
Since , we have By the induction, we get Consequently, there exists such that . Letting , from the continuity of and , we obtain , which implies that is a nonnegative solution of boundary value problem (1). Since , we know the zero function is not the solution of boundary value problem (1), thus ; by , we have that is, is a positive solution of boundary value problem (1).
On the other hand, let ; then . Leting , from the previous expressions, we have . Thus let us denote It follows from that Since is compact by Lemma 9, we can assert that is a sequentially compact set.
Now, since , we have It follows from Lemma 9 that is nondecreasing, so By the induction, we have Consequently, there exists such that . Letting , from the continuity of and , we obtain , which implies that is a nonnegative solution of boundary value problem (1).
Next, noting , thus it follows from monotonicity of that ; by the induction, we have , which implies that . Thus by (45) we have This means that is also a positive solution of boundary value problem (1).
In the end, let be any fixed point of in , then and then By induction, we have Taking the limit, we have This implies that and are maximal and minimal solutions of the BVP (1).Let , then we have The proof is completed.
Remark 11. If , then holds naturally, and in this case we take Thus we have the following Corollary 12.
Corollary 12. Suppose condition (H1) holds and . If there exists a positive constant such that where is defined by (34), then the BVP (1) has the maximal and minimal solutions and , which are positive, and there exist two positive constants such that Moreover for initial values , define the iterative sequences by Then for uniformly, respectively.
Corollary 13. Suppose conditions hold. If Then there exists a constant such that the BVP (1) has the maximal and minimal solutions and , which are positive, and there exist two positive constants such that Moreover for initial values , define the iterative sequences by Then for uniformly, respectively.
Proof . It follows from that which implies that there exists large enough such that Notice that ; (67) is equivalent to By Theorem 10, the conclusion of Corollary 13 holds.
Remark 14. In Corollary 13, we obtain that the BVP (1) has the maximal and minimal solutions and only by comparing to . But note that and are irrelative, so (62) is easy to be satisfied; this implies that Corollary 13 is very interesting.
Let , and , then For any and , we have Taking , then which implies that (62) holds. By Corollary 13, we know the BVP (69) has at least two positive solutions.
A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, “Theory and applications of fractional differential equations,” in North-Holland Mathematics Studies, vol. 204, Elsevier, Amsterdam, The Netherlands, 2006. View at: Google Scholar
I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, AcademicPress, New York, NY, USA, 1999.
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46M10 Projective and injective objects
46M15 Categories, functors
46M18 Homological methods (exact sequences, right inverses, lifting, etc.)
46M40 Inductive and projective limits
𝒞
In this article we give some properties of the tensor product, with the
ϵ
\pi
topologies, of two locally convex spaces. As a consequence we prove that the theory of M. de Wilde of the closed graph theorem does not contain the closed graph theorem of L. Schwartz.
A construction of a base for the m fold tensor product of a Banach space.
Hemasinha, Rohan (1989)
A note on reflexivity of projective tensor products
Díaz, J.C., López Molina, J.A. (1991)
A Note on the Approximation of Elements in the Riesz Tensor Product.
J.J. Grobler, C.C.A. Labuschagne (1989)
A Problem of Mixed Interpolation.
Robin E. Harte (1975)
A Representation of the Space ... .
A version of the Bartle--Graves theorem for temperate functions.
Aqzzouz, Belmesnaoui, Nouira, Redouane (2004)
About certain isomorphic properties of Banach spaces in projective tensor products.
Giovanni Emmanuele (1990)
This note is an announcement of results contained in the papers [4], [5], [6] concerning isomorphic properties of Banach spaces in projective tensor products (for this definition and some property we refer to [1]). At the end, some new result is obtained too.
Absolute bases, tensor products and a theorem of Bohr
Séan Dineen, Richard Timoney (1989)
Absolutely p-summing operators and Banach spaces containing all
{l}_{p}^{n}
uniformly complemented
Absolutely summing operators between Banach spaces of finite cotype.
E. A. Sánchez Pérez (2000)
Addendum to: "Linear operations, tensor products, and contractive projections in function spaces" (Studia Math. 38 (1970), pp. 131-186)
M. Rao (1973)
Algebraic quantum field theory and noncommutative moment problems. II
J. Yngvason (1988)
{L}^{1}
p
Jean Fresnel, Bernard De Mathan (1978)
Amenability for discrete convolution semigroup algebras.
J. Duncan, A.L.T. Paterson (1990)
An additivity formula for the strict global dimension of C(Ω)
Seytek Tabaldyev (2014)
Let A be a unital strict Banach algebra, and let K + be the one-point compactification of a discrete topological space K. Denote by the weak tensor product of the algebra A and C(K +), the algebra of continuous functions on K +. We prove that if K has sufficiently large cardinality (depending on A), then the strict global dimension is equal to .
An associative law for infinite topological A-tensor product A-algebras
Kyriazis, Athanasios (1989)
|
Exclusive disjunction - Simple English Wikipedia, the free encyclopedia
{\displaystyle \scriptstyle A\oplus B}
{\displaystyle \scriptstyle A\oplus B\oplus C}
{\displaystyle ~\oplus ~}
{\displaystyle ~\Leftrightarrow ~}
Exclusive disjunction (also called exclusive or, XOR) is a logic operation on two values. It is often represented by the symbol
{\displaystyle \veebar }
{\displaystyle \oplus }
).[1] It will be true, if exactly one of the two values is true. Otherwise, it will be false. This also means that the result of 'XOR' will be true precisely both the values are different. Same values will result in a false.[2]
The best way to remember a XOR operation is: "One or the other, but not both".
Because of that, this is different from inclusive disjunction.[3]
Truth table[change | change source]
{\displaystyle p\,\mathrm {XOR} \,q}
{\displaystyle p\oplus q}
{\displaystyle p\veebar q}
,[1] or
{\displaystyle p\neq q}
) is as follows:[2]
↑ 1.0 1.1 "Comprehensive List of Logic Symbols". Math Vault. 2020-04-06. Retrieved 2020-09-03.
↑ 2.0 2.1 Weisstein, Eric W. "XOR". mathworld.wolfram.com. Retrieved 2020-09-03.
↑ "Disjunction | logic". Encyclopedia Britannica. Retrieved 2020-09-03.
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Exclusive_disjunction&oldid=7597321"
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26A09 Elementary functions
26A12 Rate of growth of functions, orders of infinity, slowly varying functions
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.)
26A18 Iteration
26A24 Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems
26A36 Antidifferentiation
26A46 Absolutely continuous functions
26A51 Convexity, generalizations
A class of commutators for multilinear fractional integrals in nonhomogeneous spaces.
Lian, Jiali, Wu, Huoxiong (2008)
A direct extension of Meller's calculus.
Koh, E.L. (1982)
A fixed point approach to the Mittag-Leffler-Hyers-Ulam stability of a fractional integral equation
Nasrin Eghbali, Vida Kalvandi, John M. Rassias (2016)
In this paper, we have presented and studied two types of the Mittag-Leffler-Hyers-Ulam stability of a fractional integral equation. We prove that the fractional order delay integral equation is Mittag-Leffler-Hyers-Ulam stable on a compact interval with respect to the Chebyshev and Bielecki norms by two notions.
A Fractional Analog of the Duhamel Principle
Umarov, Sabir, Saydamatov, Erkin (2006)
Mathematics Subject Classification: 35CXX, 26A33, 35S10The well known Duhamel principle allows to reduce the Cauchy problem for linear inhomogeneous partial differential equations to the Cauchy problem for corresponding homogeneous equations. In the paper one of the possible generalizations of the classical Duhamel principle to the time-fractional pseudo-differential equations is established.* This work partially supported by NIH grant P20 GMO67594.
A fractional calculus approach to the mechanics of fractal media.
Carpinteri, A., Chiaia, B., Cornetti, P. (2000)
A Fractional LC − RC Circuit
Ayoub, N., Alzoubi, F., Khateeb, H., Al-Qadi, M., Hasan (Qaseer), M., Albiss, B., Rousan, A. (2006)
Mathematics Subject Classification: 26A33, 30B10, 33B15, 44A10, 47N70, 94C05We suggest a fractional differential equation that combines the simple harmonic oscillations of an LC circuit with the discharging of an RC circuit. A series solution is obtained for the suggested fractional differential equation. When the fractional order α = 0, we get the solution for the RC circuit, and when α = 1, we get the solution for the LC circuit. For arbitrary α we get a general solution which shows how the...
A general method to solve fractional differential equations on
R
B. Stanković (2010)
A global uniqueness result for fractional order implicit differential equations
Said Abbas, Mouffak Benchohra (2012)
In this paper we investigate the global existence and uniqueness of solutions for the initial value problems (IVP for short), for a class of implicit hyperbolic fractional order differential equations by using a nonlinear alternative of Leray-Schauder type for contraction maps on Fréchet spaces.
A new application of the homotopy analysis method in solving the fractional Volterra's population system
Mehdi Ghasemi, Mojtaba Fardi, Reza Khoshsiar Ghaziani (2014)
This paper considers a Volterra's population system of fractional order and describes a bi-parametric homotopy analysis method for solving this system. The homotopy method offers a possibility to increase the convergence region of the series solution. Two examples are presented to illustrate the convergence and accuracy of the method to the solution. Further, we define the averaged residual error to show that the obtained results have reasonable accuracy.
A new class of analytic functions involving certain fractional derivative operators.
Bhatt, S., Raina, R.K. (1999)
Branislav Martić (1973)
A note on fractional integration.
B. Martic (1973)
A note on fractional Sumudu transform.
Gupta, V.G., Shrama, Bhavna, Kiliçman, Adem (2010)
A numerical solution using an adaptively preconditioned Lanczos method for a class of linear systems related with the fractional Poisson equation.
Ilić, M., Turner, I.W., Anh, V. (2008)
A Poster about the Old History of Fractional Calculus
Tenreiro Machado, J., Kiryakova, Virginia, Mainardi, Francesco (2010)
MSC 2010: 26A33, 05C72, 33E12, 34A08, 34K37, 35R11, 60G22The fractional calculus (FC) is an area of intensive research and development. In a previous paper and poster we tried to exhibit its recent state, surveying the period of 1966-2010. The poster accompanying the present note illustrates the major contributions during the period 1695-1970, the "old history" of FC.
A Poster about the Recent History of Fractional Calculus
Machado, Tenreiro, Kiryakova, Virginia, Mainardi, Francesco (2010)
MSC 2010: 26A33, 05C72, 33E12, 34A08, 34K37, 35R11, 60G22In the last decades fractional calculus became an area of intense re-search and development. The accompanying poster illustrates the major contributions during the period 1966-2010.
A Proof of the Hardy-Littlewood Theorem on Fractional Integration and a Generalization
Miroslav Pavlović (1996)
A remark on local fractional calculus and ordinary derivatives
Ricardo Almeida, Małgorzata Guzowska, Tatiana Odzijewicz (2016)
In this short note we present a new general definition of local fractional derivative, that depends on an unknown kernel. For some appropriate choices of the kernel we obtain some known cases. We establish a relation between this new concept and ordinary differentiation. Using such formula, most of the fundamental properties of the fractional derivative can be derived directly.
|
p
A conjecture concerning the exponential diophantine equation
{a}^{x}+{b}^{y}={c}^{z}
A note on a conjecture of Jeśmanowicz
Moujie Deng, G. Cohen (2000)
Let a, b, c be relatively prime positive integers such that
{a}^{2}+{b}^{2}={c}^{2}
. Jeśmanowicz conjectured in 1956 that for any given positive integer n the only solution of
{\left(an\right)}^{x}+{\left(bn\right)}^{y}={\left(cn\right)}^{z}
in positive integers is x=y=z=2. If n=1, then, equivalently, the equation
{\left({u}^{2}-{v}^{2}\right)}^{x}+{\left(2uv\right)}^{y}={\left({u}^{2}+{v}^{2}\right)}^{z}
, for integers u>v>0, has only the solution x=y=z=2. We prove that this is the case when one of u, v has no prime factor of the form 4l+1 and certain congruence and inequality conditions on u, v are satisfied.
A note on Catalan's equation
A note on Jeśmanowicz' conjecture
A note on Jeśmanowicz' conjecture concerning primitive Pythagorean triplets
A note on perfect powers of the form
{x}^{m-1}+...+x+1
A note on ternary purely exponential diophantine equations
Yongzhong Hu, Maohua Le (2015)
Let a,b,c be fixed coprime positive integers with mina,b,c > 1, and let m = maxa,b,c. Using the Gel’fond-Baker method, we prove that all positive integer solutions (x,y,z) of the equation
{a}^{x}+{b}^{y}={c}^{z}
satisfy maxx,y,z < 155000(log m)³. Moreover, using that result, we prove that if a,b,c satisfy certain divisibility conditions and m is large enough, then the equation has at most one solution (x,y,z) with minx,y,z > 1.
\left({a}^{n}{x}^{m}±1\right)/\left({a}^{n}x±1\right)={y}^{n}+1
Jiagui Luo (2001)
{a}^{x}+{b}^{y}={c}^{z}
Zhenfu Cao (1999)
\left(\genfrac{}{}{0pt}{}{k}{2}\right)-1={q}^{n}+1
In this note we prove that the equation
\left(\genfrac{}{}{0pt}{}{k}{2}\right)-1={q}^{n}+1
q\ge 2,n\ge 3
, has only finitely many positive integer solutions
\left(k,q,n\right)
. Moreover, all solutions
\left(k,q,n\right)
k{10}^{{10}^{182}}
q{10}^{{10}^{165}}
n2·{10}^{17}
x²+{b}^{y}={c}^{z}
{x}^{2}+{b}^{Y}={c}^{z}
a
b
c
r
{a}^{2}+{b}^{2}={c}^{r}
min\left(a,b,c,r\right)>1
gcd\left(a,b\right)=1,a
r
is odd. In this paper we prove that if
b\equiv 3\phantom{\rule{4.44443pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}4\right)
b
c
is an odd prime power, then the equation
{x}^{2}+{b}^{y}={c}^{z}
has only the positive integer solution
\left(x,y,z\right)=\left(a,2,r\right)
min\left(y,z\right)>1
{x}^{2}+{q}^{m}={y}^{3}
\left({x}^{m}-1\right)/\left(x-1\right)={y}^{n}
A Note on the Diophantine Equation x2 + 4D = yp.
Le Maohua (1993)
A note on the equation
a{x}^{n}-b{y}^{n}=c
Maurice Mignotte (1996)
A note on the exponential Diophantine equation
{\left(4m²+1\right)}^{x}+{\left(5m²-1\right)}^{y}={\left(3m\right)}^{z}
Jianping Wang, Tingting Wang, Wenpeng Zhang (2015)
Let m be a positive integer. Using an upper bound for the solutions of generalized Ramanujan-Nagell equations given by Y. Bugeaud and T. N. Shorey, we prove that if 3 ∤ m, then the equation
{\left(4m²+1\right)}^{x}+{\left(5m²-1\right)}^{y}={\left(3m\right)}^{z}
has only the positive integer solution (x,y,z) = (1,1,2).
A note on the number of solutions of the generalized Ramanujan-Nagell equation
x²-D={k}^{n}
{x}^{2}-D={p}^{n}
Yuan-e Zhao, Tingting Wang (2012)
D
be a positive integer, and let
p
be an odd prime with
p\nmid D
. In this paper we use a result on the rational approximation of quadratic irrationals due to M. Bauer, M. A. Bennett: Applications of the hypergeometric method to the generalized Ramanujan-Nagell equation. Ramanujan J. 6 (2002), 209–270, give a better upper bound for
N\left(D,p\right)
, and also prove that if the equation
{U}^{2}-D{V}^{2}=-1
has integer solutions
\left(U,V\right)
, the least solution
\left({u}_{1},{v}_{1}\right)
{u}^{2}-p{v}^{2}=1
p\nmid {v}_{1}
D>C\left(p\right)
C\left(p\right)
is an effectively computable constant...
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MCQs of Moving Charges & Magnetism | GUJCET MCQ
Moving Charges & Magnetism MCQs
MCQs of Moving Charges & Magnetism
At a place an electric field and a magnetic field are in the downward direction. There is an electron moves in the downward direction. Hence this electron _____ .
(a) will bend towards left
(b) will bend towards right
(c) will gain velocity
(d) will lose velocity.
When a charged particle moves in a magnetic field its kinetic energy _____ .
(b) can increase
(c) can decrease
(d) can increase or decrease
There are 100 turns per cm length in a very long solenoid. It carries a current of 5 A. The magnetic field at its centre on the axis is _____ T.
(d) 12.56 × 10-2
A conducting wire of 1 m length is used to form a circular loop. If it carries a current of 1 ampere, its magnetic moment will be _____ Am2.
2\mathrm{\pi }
\frac{\mathrm{\pi }}{2}
\frac{\mathrm{\pi }}{4}
\frac{1}{4\mathrm{\pi }}
A long wire carries a steady current. When it is bent in a circular form, the magnetic field at its centre is B. Now if this wire is bent in circular loop of n turns, what is the magnetic field at its centre ?
{\mathrm{n}}^{2}\mathrm{B}
2{\mathrm{n}}^{2}\mathrm{B}
Two parallel thin wires, each carrying current I are kept at a separation r from each other. Hence the magnitude of force per unit length of one wire due to the other wire is _____
\frac{{\mu }_{0}{I}^{2}}{{r}^{2}}
\frac{{\mu }_{0}{I}^{2}}{2\mathrm{\pi }r}
\frac{{\mu }_{0}I}{2\mathrm{\pi }r}
\frac{{\mu }_{0}I}{2\mathrm{\pi }{r}^{2}}
A particle of charge q and mass m moves on a circular path of radius r in a plane inside and normal to a uniform magnetic field B. The time taken by this particle to complete one revolution is _____ .
\frac{2\mathrm{\pi mq}}{\mathrm{B}}
\frac{2{\mathrm{\pi q}}^{2}B}{\mathrm{m}}
\frac{2\mathrm{\pi qB}}{\mathrm{m}}
\frac{2\mathrm{\pi m}}{\mathrm{Bq}}
At each of the two ends of a rod of length 2r, a particle of mass m and charge q is attached. If this rod is rotated about its centre with angular speed ω, the ratio of its magnetic dipole moment to the total angular momentum of this particle is _____ .
\frac{q}{2\mathrm{m}}
\frac{q}{\mathrm{m}}
\frac{2q}{\mathrm{m}}
\frac{q}{\mathrm{\pi m}}
A very long solenoid of length L has n layers. There are N turns in each layer. Diameter of the solenoid is D and it carries current I. The magnetic field at the centre of the solenoid is _____ .
(a) directly proportional to D
(b) inversely proportional to D.
(c) independent of D
(d) directly proportional to L.
The angular speed of the charged particle is independent of _____ .
(b) its linear speed
(c) charge of particle
(d) magnetic field.
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11D41 Higher degree equations; Fermat's equation
11D04 Linear equations
11D09 Quadratic and bilinear equations
11D25 Cubic and quartic equations
11D61 Exponential equations
11D68 Rational numbers as sums of fractions
11D72 Equations in many variables
p
-adic and power series fields
A “class group” obstruction for the equation
C{y}^{d}=F\left(x,z\right)
Denis Simon (2008)
In this paper, we study equations of the form
C{y}^{d}=F\left(x,z\right)
F\in ℤ\left[x,z\right]
is a binary form, homogeneous of degree
n
, which is supposed to be primitive and irreducible, and
is any fixed integer. Using classical tools in algebraic number theory, we prove that the existence of a proper solution for this equation implies the existence of an integral ideal of given norm in some order in a number field, and also the existence of a specific relation in the class group involving this ideal. In some cases, this result...
A class number criterion for the equation
\left({x}^{p}-1\right)/\left(x-1\right)=p{y}^{q}
Benjamin Dupuy (2007)
A hyperelliptic diophantine equation related to imaginary quadratic number fields with class number 2.
B.M.M. de de Weger (1992)
B.M.M. Weger (1993)
A note on a cyclotomic diophantine equation
Veikko Ennola (1975)
A note on Format's conjecture
K. Inkeri (1976)
{x}^{2}+{q}^{m}={y}^{3}
Hui Lin Zhu (2011)
A note on the Fermat equation
R. Tijdeman (1987)
A note on the integer solutions ofhyperelliptic equations
Maohua Le (1995)
A quantitative version of Runge's theorem on diophantine equations
P. G. Walsh (1992)
A quantitative version of Siegel's theorem: integral points on elliptic curves and Catalan curves.
Joseph H. Silverman (1987)
A rationality condition for the existence of odd perfect numbers.
Davis, Simon (2003)
A result for case I of the Fermat problem.
W.P. Coleman (1979)
A special case of Vinogradov's mean value theorem
R. C. Vaughan, T. D. Wooley (1997)
A symmetric diophantine system concerning fifth powers
Ajai Choudhry, Jarosław Wróblewski (2012)
Algorithms for finding good examples for the
abc
and Szpiro conjectures.
Nitaj, Abderrahmane (1993)
Almost powers in the Lucas sequence
Yann Bugeaud, Florian Luca, Maurice Mignotte, Samir Siksek (2008)
The famous problem of determining all perfect powers in the Fibonacci sequence
{\left({F}_{n}\right)}_{n\ge 0}
and in the Lucas sequence
{\left({L}_{n}\right)}_{n\ge 0}
has recently been resolved [10]. The proofs of those results combine modular techniques from Wiles’ proof of Fermat’s Last Theorem with classical techniques from Baker’s theory and Diophantine approximation. In this paper, we solve the Diophantine equations
{L}_{n}={q}^{a}{y}^{p}
a>0
p\ge 2
, for all primes
q<1087
and indeed for all but
13
q<{10}^{6}
. Here the strategy of [10] is not sufficient due to the sizes of...
An application of Hilbert's irreducibility theorem to diophantine equations
Andrzej Schinzel (1982)
An effective p-adic analogue of a theorem of Thue
J. Coates (1969)
An effective p-adic analogue of a theorem of Thue II. The greatest prime factor of a binary form
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Geometrical and Spectral Properties of the Orthogonal Projections of the Identity
2013 Geometrical and Spectral Properties of the Orthogonal Projections of the Identity
Luis González, Antonio Suárez, Dolores García
We analyze the best approximation
AN
(in the Frobenius sense) to the identity matrix in an arbitrary matrix subspace
AS
A\in {ℝ}^{n×n}
nonsingular,
S
being any fixed subspace of
{ℝ}^{n×n}
). Some new geometrical and spectral properties of the orthogonal projection
AN
are derived. In particular, new inequalities for the trace and for the eigenvalues of matrix
AN
are presented for the special case that
AN
is symmetric and positive definite.
Luis González. Antonio Suárez. Dolores García. "Geometrical and Spectral Properties of the Orthogonal Projections of the Identity." J. Appl. Math. 2013 (SI03) 1 - 8, 2013. https://doi.org/10.1155/2013/435730
Luis González, Antonio Suárez, Dolores García "Geometrical and Spectral Properties of the Orthogonal Projections of the Identity," Journal of Applied Mathematics, J. Appl. Math. 2013(SI03), 1-8, (2013)
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Physics - An Efficient Way to Predict Water’s Phases
An Efficient Way to Predict Water’s Phases
June 9, 2021 • Physics 14, s67
A machine-learning technique maps water’s phase space as reliably as gold standard ab initio calculations but at a much smaller computational cost.
L. Zhang et al. [1]
Given its familiarity, water is a surprisingly tricky substance to simulate. In addition to having liquid and gas phases, water adopts at least 14 distinct solid configurations—in both molecular and ionic forms—depending on the temperature and pressure. This complex phase space makes water an ideal testbed for theoretical models that predict material behaviors. Now, researchers have tested an efficient new model and found that it can predict all of water’s phases with an accuracy approaching that of much more computationally demanding techniques [1].
The first step in mapping a material’s phase space is to construct its “potential energy surface” (PES) under various conditions. The PES describes the probability of finding a specific configuration at a given temperature and pressure. The gold standard way to approach this task is to use ab initio calculations, which capture the quantum-mechanical behavior of the system’s electrons. But the computational cost of this method is so high that simulating more than a narrow range of conditions is impractical. So far, the majority of water’s phase space has been mapped using quicker, less reliable approximations.
Linfeng Zhang at Princeton University and colleagues achieve both accuracy and speed by using carefully selected ab initio calculations to train a machine-learning algorithm. Beginning with a rough phase diagram based on experimental data, the algorithm identifies points on the map where ab initio calculations are most needed. After performing these calculations, the process repeats until the error in the approximation falls below an acceptable threshold.
The researchers plan to use the technique to map parts of water’s phase space that are uncharted by experiment. They also say that future applications could include nuclear quantum effects on water’s various properties.
L. Zhang et al., “Phase diagram of a deep potential water model,” Phys. Rev. Lett. 126, 236001 (2021).
Phase Diagram of a Deep Potential Water Model
Linfeng Zhang, Han Wang, Roberto Car, and Weinan E
{\text{CaH}}_{6}
|
Discrete-time or continuous-time two-degree-of-freedom PID controller - Simulink - MathWorks Australia
D\left[\frac{N}{1+N\alpha \left(z\right)}\right],
\alpha \left(z\right)=\frac{{T}_{s}}{z-1}.
\alpha \left(z\right)=\frac{{T}_{s}z}{z-1}.
\alpha \left(z\right)=\frac{{T}_{s}}{2}\frac{z+1}{z-1}.
u=P\left(br-y\right)+I\frac{1}{s}\left(r-y\right)+D\frac{N}{1+N\frac{1}{s}}\left(cr-y\right),
u=P\left(br-y\right)+I\alpha \left(z\right)\left(r-y\right)+D\frac{N}{1+N\beta \left(z\right)}\left(cr-y\right),
u=P\left[\left(br-y\right)+I\frac{1}{s}\left(r-y\right)+D\frac{N}{1+N\frac{1}{s}}\left(cr-y\right)\right].
u=P\left[\left(br-y\right)+I\alpha \left(z\right)\left(r-y\right)+D\frac{N}{1+N\beta \left(z\right)}\left(cr-y\right)\right],
\alpha \left(z\right)=\frac{{T}_{s}}{z-1}.
\alpha \left(z\right)=\frac{{T}_{s}z}{z-1}.
\alpha \left(z\right)=\frac{{T}_{s}}{2}\frac{z+1}{z-1}.
D\left[\frac{N}{1+N\alpha \left(z\right)}\right],
\alpha \left(z\right)=\frac{{T}_{s}}{z-1}.
\alpha \left(z\right)=\frac{{T}_{s}z}{z-1}.
\alpha \left(z\right)=\frac{{T}_{s}}{2}\frac{z+1}{z-1}.
{u}_{i}=\int \left(r-y\right)I\text{\hspace{0.17em}}dt.
u=P\left(br-y\right)+I\frac{1}{s}\left(r-y\right)+D\frac{N}{1+N\frac{1}{s}}\left(cr-y\right),
u=P\left(br-y\right)+I\alpha \left(z\right)\left(r-y\right)+D\frac{N}{1+N\beta \left(z\right)}\left(cr-y\right),
u=P\left[\left(br-y\right)+I\frac{1}{s}\left(r-y\right)+D\frac{N}{1+N\frac{1}{s}}\left(cr-y\right)\right].
u=P\left[\left(br-y\right)+I\alpha \left(z\right)\left(r-y\right)+D\frac{N}{1+N\beta \left(z\right)}\left(cr-y\right)\right],
D\frac{z-1}{z{T}_{s}}\left(cr-y\right).
{z}_{pole}=1-N{T}_{s}
{z}_{pole}=\frac{1}{1+N{T}_{s}}
{z}_{pole}=\frac{1-N{T}_{s}/2}{1+N{T}_{s}/2}
\begin{array}{l}{F}_{par}\left(s\right)=\frac{\left(bP+cDN\right){s}^{2}+\left(bPN+I\right)s+IN}{\left(P+DN\right){s}^{2}+\left(PN+I\right)s+IN},\\ {C}_{par}\left(s\right)=\frac{\left(P+DN\right){s}^{2}+\left(PN+I\right)s+IN}{s\left(s+N\right)},\end{array}
\begin{array}{l}{F}_{id}\left(s\right)=\frac{\left(b+cDN\right){s}^{2}+\left(bN+I\right)s+IN}{\left(1+DN\right){s}^{2}+\left(N+I\right)s+IN},\\ {C}_{id}\left(s\right)=P\frac{\left(1+DN\right){s}^{2}+\left(N+I\right)s+IN}{s\left(s+N\right)}.\end{array}
{Q}_{par}\left(s\right)=\frac{\left(\left(b-1\right)P+\left(c-1\right)DN\right)s+\left(b-1\right)PN}{s+N}.
{Q}_{id}\left(s\right)=P\frac{\left(\left(b-1\right)+\left(c-1\right)DN\right)s+\left(b-1\right)N}{s+N}.
|
Cooperative bargaining - Wikipedia
Cooperative bargaining is a process in which two people decide how to share a surplus that they can jointly generate. In many cases, the surplus created by the two players can be shared in many ways, forcing the players to negotiate which division of payoffs to choose. Such surplus-sharing problems (also called bargaining problem) are faced by management and labor in the division of a firm's profit, by trade partners in the specification of the terms of trade, and more.
The present article focuses on the normative approach to bargaining. It studies how the surplus should be shared, by formulating appealing axioms that the solution to a bargaining problem should satisfy. It is useful when both parties are willing to cooperate in implementing the fair solution. The five axioms, any Nash Bargaining Solution should satisfy are Pareto Optimality (PAR), Individual Rationality (IR), Independent of Expected Utility Representations (INV), Independence of Irrelevant Alternatives (IIA) and Symmetry (SYM). While SYM and PAR restrict the behavior of the solution to only a specific bargaining problem, INV and IIA require consistency of solution across bargaining problems in Game Theory. Such solutions, particularly the Nash solution, were used to solve concrete economic problems, such as management–labor conflicts, on numerous occasions.[1]
An alternative approach to bargaining is the positive approach. It studies how the surplus is actually shared. Under the positive approach, the bargaining procedure is modeled as a non-cooperative game. The most common form of such game is called sequential bargaining.
1.1 Feasibility set
1.2 Disagreement point
2 Nash bargaining game
4 Bargaining solutions
4.1 Nash bargaining solution
4.2 Kalai–Smorodinsky bargaining solution
4.3 Egalitarian bargaining solution
5 Experimental solutions
7 Bargaining solutions and risk-aversion
A two-person bargain problem consists of:
A feasibility set
{\displaystyle F}
, a closed subset of
{\displaystyle \mathbb {R} ^{2}}
that is often assumed to be convex, the elements of which are interpreted as agreements.
{\displaystyle F}
is often assumed to be convex because, for any two feasible outcomes, a convex combination (a weighted average) of them is typically also feasible.
A disagreement, or threat, point
{\displaystyle d=(d_{1},d_{2})}
{\displaystyle d_{1}}
{\displaystyle d_{2}}
are the respective payoffs to player 1 and player 2, which they are guaranteed to receive if they cannot come to a mutual agreement.
The problem is nontrivial if agreements in
{\displaystyle F}
are better for both parties than the disagreement point. A solution to the bargaining problem selects an agreement
{\displaystyle \phi }
{\displaystyle F}
Feasibility set[edit]
The feasible agreements typically include all possible joint actions, leading to a feasibility set that includes all possible payoffs. Often, the feasible set is restricted to include only payoffs that have a possibility of being better than the disagreement point for both agents.[2]
Disagreement point[edit]
The disagreement point
{\displaystyle d}
is the value the players can expect to receive if negotiations break down. This could be some focal equilibrium that both players could expect to play. This point directly affects the bargaining solution, however, so it stands to reason that each player should attempt to choose his disagreement point in order to maximize his bargaining position. Towards this objective, it is often advantageous to increase one's own disagreement payoff while harming the opponent's disagreement payoff (hence the interpretation of the disagreement as a threat). If threats are viewed as actions, then one can construct a separate game wherein each player chooses a threat and receives a payoff according to the outcome of bargaining. It is known as Nash's variable threat game.
Nash bargaining game[edit]
John Forbes Nash was the first to study cooperative bargaining. His solution is called the Nash bargaining solution. It is the unique solution to a two-person bargaining problem that satisfies the axioms of scale invariance, symmetry, efficiency, and independence of irrelevant alternatives. According to Walker,[3] Nash's bargaining solution was shown by John Harsanyi to be the same as Zeuthen's solution[4] of the bargaining problem.
The Nash bargaining game is a simple two-player game used to model bargaining interactions. In the Nash bargaining game, two players demand a portion of some good (usually some amount of money). If the total amount requested by the players is less than that available, both players get their request. If their total request is greater than that available, neither player gets their request.
Nash (1953) presents a non-cooperative demand game with two players who are uncertain about which payoff pairs are feasible. In the limit as the uncertainty vanishes, equilibrium payoffs converge to those predicted by the Nash bargaining solution.[2]
Strategies are represented in the Nash demand game by a pair (x, y). x and y are selected from the interval [d, z], where d is the disagreement outcome and z is the total amount of good. If x + y is equal to or less than z, the first player receives x and the second y. Otherwise both get d; often
{\displaystyle d=0}
There are many Nash equilibria in the Nash demand game. Any x and y such that x + y = z is a Nash equilibrium. If either player increases their demand, both players receive nothing. If either reduces their demand they will receive less than if they had demanded x or y. There is also a Nash equilibrium where both players demand the entire good. Here both players receive nothing, but neither player can increase their return by unilaterally changing their strategy.
In Rubinstein's alternating offers bargaining game,[5] players take turns acting as the proposer for splitting some surplus. The division of the surplus in the unique subgame perfect equilibrium depends upon how strongly players prefer current over future payoffs. In particular, let d be the discount factor, which refers to the rate at which players discount future earnings. That is, after each step the surplus is worth d times what it was worth previously. Rubinstein showed that if the surplus is normalized to 1, the payoff for player 1 in equilibrium is 1/(1+d), while the payoff for player 2 is d/(1+d). In the limit as players become perfectly patient, the equilibrium division converges to the Nash bargaining solution.
Bargaining solutions[edit]
Various solutions have been proposed based on slightly different assumptions about what properties are desired for the final agreement point.
Nash bargaining solution[edit]
John Forbes Nash Jr. proposed[6] that a solution should satisfy certain axioms:
Invariant to affine transformations or Invariant to equivalent utility representations
Nash proved that the solutions satisfying these axioms are exactly the points
{\displaystyle (x,y)}
{\displaystyle F}
which maximize the following expression:
{\displaystyle (u(x)-u(d))(v(y)-v(d))}
where u and v are the utility functions of Player 1 and Player 2, respectively, and d is a disagreement outcome. That is, players act as if they seek to maximize
{\displaystyle (u(x)-u(d))(v(y)-v(d))}
{\displaystyle u(d)}
{\displaystyle v(d)}
, are the status quo utilities (the utility obtained if one decides not to bargain with the other player). The product of the two excess utilities is generally referred to as the Nash product. Intuitively, the solution consists of each player getting their status quo payoff (i.e., noncooperative payoff) in addition to a share of the benefits occurring from cooperation.[7]: 15–16
Kalai–Smorodinsky bargaining solution[edit]
Main article: Kalai–Smorodinsky bargaining solution
Independence of Irrelevant Alternatives can be substituted with a Resource monotonicity axiom. This was demonstrated by Ehud Kalai and Meir Smorodinsky.[8] This leads to the so-called Kalai–Smorodinsky bargaining solution: it is the point which maintains the ratios of maximal gains. In other words, if we normalize the disagreement point to (0,0) and player 1 can receive a maximum of
{\displaystyle g_{1}}
with player 2's help (and vice versa for
{\displaystyle g_{2}}
), then the Kalai–Smorodinsky bargaining solution would yield the point
{\displaystyle \phi }
on the Pareto frontier such that
{\displaystyle \phi _{1}/\phi _{2}=g_{1}/g_{2}}
Egalitarian bargaining solution[edit]
The egalitarian bargaining solution, introduced by Ehud Kalai,[9] is a third solution which drops the condition of scale invariance while including both the axiom of Independence of irrelevant alternatives, and the axiom of resource monotonicity. It is the solution which attempts to grant equal gain to both parties. In other words, it is the point which maximizes the minimum payoff among players. Kalai notes that this solution is closely related to the egalitarian ideas of John Rawls.
Irrelevant-independence
Resource-monotonicity
Nash (1950) Maximizing the product of surplus utilities
Kalai-Smorodinsky (1975) Equalizing the ratios of maximal gains
Kalai (1977) Maximizing the minimum of surplus utilities
Experimental solutions[edit]
A series of experimental studies[10] found no consistent support for any of the bargaining models. Although some participants reached results similar to those of the models, others did not, focusing instead on conceptually easy solutions beneficial to both parties. The Nash equilibrium was the most common agreement (mode), but the average (mean) agreement was closer to a point based on expected utility.[11] In real-world negotiations, participants often first search for a general bargaining formula, and then only work out the details of such an arrangement, thus precluding the disagreement point and instead moving the focal point to the worst possible agreement.
Kenneth Binmore has used the Nash bargaining game to explain the emergence of human attitudes toward distributive justice.[12][13] He primarily uses evolutionary game theory to explain how individuals come to believe that proposing a 50–50 split is the only just solution to the Nash bargaining game. Herbert Gintis supports a similar theory, holding that humans have evolved to a predisposition for strong reciprocity but do not necessarily make decisions based on direct consideration of utility.[14]
Bargaining solutions and risk-aversion[edit]
Some economists have studied the effects of risk aversion on the bargaining solution. Compare two similar bargaining problems A and B, where the feasible space and the utility of player 1 remain fixed, but the utility of player 2 is different: player 2 is more risk-averse in A than in B. Then, the payoff of player 2 in the Nash bargaining solution is smaller in A than in B.[15]: 303–304 However, this is true only if the outcome itself is certain; if the outcome is risky, then a risk-averse player may get a better deal as proved by Alvin E. Roth and Uriel Rothblum[16]
For a comprehensive discussion of the Nash bargaining solution and the huge literature on the theory and application of bargaining - including a discussion of the classic Rubinstein bargaining model - see Abhinay Muthoo's book Bargaining Theory and Application.[17]
Rubinstein bargaining model
^ Thomson, William (1994-01-01), "Chapter 35 Cooperative models of bargaining", Handbook of Game Theory with Economic Applications, Elsevier, vol. 2, pp. 1237–1284, retrieved 2021-02-05
^ a b Nash, John (1953-01-01). "Two-Person Cooperative Games". Econometrica. 21 (1): 128–140. doi:10.2307/1906951. JSTOR 1906951.
^ Walker, Paul (2005). "History of Game Theory". Archived from the original on 2000-08-15. Retrieved 2008-05-03.
^ Zeuthen, Frederik (1930). Problems of Monopoly and Economic Warfare.
^ Rubinstein, Ariel (1982-01-01). "Perfect Equilibrium in a Bargaining Model". Econometrica. 50 (1): 97–109. CiteSeerX 10.1.1.295.1434. doi:10.2307/1912531. JSTOR 1912531.
^ Nash, John (1950). "The Bargaining Problem". Econometrica. 18 (2): 155–162. doi:10.2307/1907266. JSTOR 1907266.
^ Muthoo, Abhinay (1999). Bargaining theory with applications. Cambridge University Press.
^ Kalai, Ehud & Smorodinsky, Meir (1975). "Other solutions to Nash's bargaining problem". Econometrica. 43 (3): 513–518. doi:10.2307/1914280. JSTOR 1914280.
^ Kalai, Ehud (1977). "Proportional solutions to bargaining situations: Intertemporal utility comparisons" (PDF). Econometrica. 45 (7): 1623–1630. doi:10.2307/1913954. JSTOR 1913954.
^ Schellenberg, James A. (1 January 1990). "'Solving' the Bargaining Problem" (PDF). Mid-American Review of Sociology. 14 (1/2): 77–88. Retrieved 28 January 2017.
^ Felsenthal, D. S.; Diskin, A. (1982). "The Bargaining Problem Revisited: Minimum Utility Point, Restricted Monotonicity Axiom, and the Mean as an Estimate of Expected Utility". Journal of Conflict Resolution. 26 (4): 664–691. doi:10.1177/0022002782026004005.
^ Binmore, Kenneth (1998). Game Theory and the Social Contract Volume 2: Just Playing. Cambridge: MIT Press. ISBN 978-0-262-02444-0.
^ Binmore, Kenneth (2005). Natural Justice. New York: Oxford University Press. ISBN 978-0-19-517811-1.
^ Gintis, H. (11 August 2016). "Behavioral ethics meets natural justice". Politics, Philosophy & Economics. 5 (1): 5–32. doi:10.1177/1470594x06060617.
^ Osborne, Martin (1994). A Course in Game Theory. MIT Press. ISBN 978-0-262-15041-5.
^ Roth, Alvin E.; Rothblum, Uriel G. (1982). "Risk Aversion and Nash's Solution for Bargaining Games with Risky Outcomes". Econometrica. 50 (3): 639. doi:10.2307/1912605. JSTOR 1912605.
^ Abhinay Muthoo "Bargaining Theory with Applications", Cambridge University Press, 1999.
Binmore, K.; Rubinstein, A.; Wolinsky, A. (1986). "The Nash Bargaining Solution in Economic Modelling". RAND Journal of Economics. 17 (2): 176–188. doi:10.2307/2555382. JSTOR 2555382.
Nash bargaining solutions
Retrieved from "https://en.wikipedia.org/w/index.php?title=Cooperative_bargaining&oldid=1056609571"
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A Class of C*-algebras and Topological Markov Chains II: Reducible Chains and the Ext-functor for C*-algebras.
J. Cuntz (1981)
A natural localization of Hardy spaces in several complex variables
Mihai Putinar, Roland Wolff (1997)
Let H²(bΩ) be the Hardy space of a bounded weakly pseudoconvex domain in
{ℂ}^{n}
. The natural resolution of this space, provided by the tangential Cauchy-Riemann complex, is used to show that H²(bΩ) has the important localization property known as Bishop’s property (β). The paper is accompanied by some applications, previously known only for Bergman spaces.
All nuclear C*-algebras are amenable.
U. Haagerup (1983)
Ruy Exel (1997)
An implicit function theorem in Banach spaces
Jürgen Leiterer (1985)
An index formula for chains
Robin Harte, Woo Lee (1995)
We derive a formula for the index of Fredholm chains on normed spaces.
Analytic Structure in the Spectra of Certain uF-Algebras.
R.M. Brooks (1979)
Applications of the ‘Ham Sandwich Theorem’ to Eigenvalues of the Laplacian
Kei Funano (2016)
We apply Gromov’s ham sandwich method to get: (1) domain monotonicity (up to a multiplicative constant factor); (2) reverse domain monotonicity (up to a multiplicative constant factor); and (3) universal inequalities for Neumann eigenvalues of the Laplacian on bounded convex domains in Euclidean space.
BDF neboli věta o hlavních osách v nekonečné dimenzi
P. R. Halmos (1984)
Bundles of Banach algebras.
Kitchen, J.W., Robbins, D.A. (1994)
Classification of dimension groups and iterating systems.
Norbert Riedel (1981)
Cohomologie différentiable des algèbres de polynômes de leurs localisées ou de leurs complétées, et des variétés
Cohomology of operator algebras. III : reduction to normal cohomology
B.E. Johnson, Richard V. Kadison, John R. Ringrose (1972)
Cohomology of some non-selfadjoint operator algebras.
Jens P. Nielsen (1980)
Cohomology theory on non commutative algebras of continuous functions
G. A. Stavrakas (1989)
Completely bounded multilinear maps and C*-algebraic cohomology.
E. Christensen, E.G. Effros (1987)
Computing and estimating the global dimension in certain classes af Banach algebras.
Yu. Selivanov (1993)
Connes' Analogue of the Thom Isomorphism for the Kasparov Groups.
Georges Skandalis, Thierry Fack (1981)
|
hyad.es | A Triple Product Integral Identity for Vector Spherical Harmonics
A Triple Product Integral Identity for Vector Spherical Harmonics
As everyone knows, the product of three spherical harmonics can be related to the Wigner [3j]
3j
symbols (and therefore Clebsch-Gordan coefficients) in the form
[\int \mathrm d \Omega\ Y_{l_1}^{m_1} Y_{l_2}^{m_2} Y_{l_3}^{m_3} = \sqrt{(2 l_1 + 1)(2 l_2 + 1)(2 l_3 + 1) \over 4\pi}\left(\begin{array}{ccc} l_1 & l_2 & l_3 \\ m_1 & m_2 & m_3\end{array}\right) \left(\begin{array}{ccc} l_1 & l_2 & l_3 \\ 0 & 0 & 0\end{array}\right).\tag{1}]
Lately, however, I have been extensively using vector spherical harmonics, for which I have not been able to find equivalent identities to this scalar version. In particular I have been confronted with situations where I have needed to perform integrals of the form
[\int \mathrm d \Omega\ Y_1 (\nabla Y_2) \cdot (\nabla Y_3), \tag{2}]
where here I am schematically using 1, 2, 3 to stand in for the relevant multi-indices.
To proceed, let us first consider how one recovers the orthogonality relation of the poloidal vector spherical harmonics [\boldsymbol{\Psi}_l^m = r \nabla Y_l^m]
\boldsymbol{\Psi}_l^m = r \nabla Y_l^m
. Limiting our attention to the unit sphere, we have [\nabla_3 = \partial_r + {1 \over r} \nabla_2]
\nabla_3 = \partial_r + {1 \over r} \nabla_2
(although [Y_l^m]
Y_l^m
has no [r]
r
dependence in any case). Therefore
[\int \mathrm d \Omega\ \boldsymbol{\Psi}_l^m \cdot \boldsymbol{\Psi}_{l'}^{m'} = \int \mathrm d \Omega \ \nabla Y_l^m \cdot \nabla Y_{l'}^{m'},]
where the derivative here is understood to be purely horizontal (acting only on angles). We now integrate by parts to obtain
[\int \mathrm d \Omega\ \nabla \cdot \left(Y_{l'}^{m'} \nabla Y_l^m\right) - \int \mathrm d \Omega\ Y_{l'}^{m'} \nabla^2 Y_l^m.]
The first term vanishes by Stokes' Theorem as it is the integral of a 2-divergence over the sphere (which has no boundary). For the second term, we recall the definition of the spherical harmonics is that they are the eigenfunctions of the Laplacian: [\left(\nabla^2 + l(l+1)\right)Y_{l}^{m} = 0]
\left(\nabla^2 + l(l+1)\right)Y_{l}^{m} = 0
[\int \mathrm d \Omega \ \nabla Y_l^m \cdot \nabla Y_{l'}^{m'} = l(l+1) \int \mathrm d \Omega \ Y_l^m Y_{l'}^{m'} = l(l+1) \delta_{l,l'}\delta_{m,m'} \equiv L^2 \delta_{l,l'}\delta_{m,m'},]
Let us now turn our attention to the triple product integral, eq. 2. Again integrating by parts, we obtain
[\begin{aligned}I_{123} \equiv \int \mathrm d \Omega\ &Y_1 (\nabla Y_2) \cdot (\nabla Y_3) \\&= \int \mathrm d \Omega\ \nabla \cdot \left(Y_1 Y_2 (\nabla Y_3)\right) - \int \mathrm d \Omega\ Y_2 (\nabla Y_1) \cdot (\nabla Y_3) - \int \mathrm d \Omega\ Y_2 Y_1 \nabla^2 Y_3\\&= - \int \mathrm d \Omega\ \left[Y_2 (\nabla Y_3) \cdot (\nabla Y_1)\right] + l_3(l_3 + 1) \int \mathrm d \Omega\ Y_2 Y_1 Y_3 \\ &= -I_{231} + L_3^2 F_{123}\end{aligned}]
Again, the integral of the 2-divergence vanishes, and we are left with 2 terms. The first is an integral of similar form to eq. 2, but with the multi-indices having undergone a cyclic permutation (and with a sign change). The second is the scalar triple integral in eq. 1, which does not depend on the ordering of the multi-indices. Effecting this cyclic permutation twice more, we arrive at
[I_{123} = (L_3^2 - L_1^2 + L_2^2) F_{123} - I_{123} \implies I_{123} = \left(L_3^2 - L_1^2 + L_2^2 \over 2\right) F_{123}.]
[\int \mathrm d \Omega\ Y_{l_1}^{m_1} (\nabla Y_{l_2}^{m_2}) \cdot (\nabla Y_{l_3}^{m_3}) = \left(l_3(l_3 + 1) - l_1(l_1 + 1) + l_2(l_2 + 1) \over 2\right) \int \mathrm d \Omega\ Y_{l_1}^{m_1} Y_{l_2}^{m_2} Y_{l_3}^{m_3}.]
|
20N10 Ternary systems (heaps, semiheaps, heapoids, etc.)
n
\left(n\ge 3\right)
20N20 Hypergroups
3
-configurations with simple edge basis and their corresponding quasigroup identities
Václav J. Havel (1993)
A Class of Balanced Laws on Quasigroups (I)
Branka P. Alimpić (1976)
A Class of Balanced Laws on Quasigroups (Ii)
A class of Bol loops with a subgroup of index two
G
{C}_{2}
the cyclic group of order
2
. Consider the
8
\left(x,y\right)↦{\left({x}^{i}{y}^{j}\right)}^{k}
i
j
k\in \left\{-1,\phantom{\rule{0.166667em}{0ex}}1\right\}
. Define a new multiplication on
G×{C}_{2}
by assigning one of the above
8
multiplications to each quarter
\left(G×\left\{i\right\}\right)×\left(G×\left\{j\right\}\right)
i,j\in {C}_{2}
. We describe all situations in which the resulting quasigroup is a Bol loop. This paper also corrects an error in P. Vojtěchovsk’y: On the uniqueness of loops
M\left(G,2\right)
A class of commutative loops with metacyclic inner mapping groups
Aleš Drápal (2008)
We investigate loops defined upon the product
{ℤ}_{m}×{ℤ}_{k}
\left(a,i\right)\left(b,j\right)=\left(\left(a+b\right)/\left(1+t{f}^{i}\left(0\right){f}^{j}\left(0\right)\right),i+j\right)
f\left(x\right)=\left(sx+1\right)/\left(tx+1\right)
, for appropriate parameters
s,t\in {ℤ}_{m}^{*}
. Each such loop is coupled to a 2-cocycle (in the group-theoretical sense) and this connection makes it possible to prove that the loop possesses a metacyclic inner mapping group. If
s=1
, then the loop is an A-loop. Questions of isotopism and isomorphism are considered in detail.
A class of non-Moufang Bol loops isomorphic to all their loop isotopes.
V.S. Ramamurthi, B.L. Sharma (1985)
A class of quasigroups solving a problem of ergodic theory
Jonathan D. H. Smith (2000)
A pointed quasigroup is said to be semicentral if it is principally isotopic to a group via a permutation on one side and a group automorphism on the other. Convex combinations of permutation matrices given by the one-sided multiplications in a semicentral quasigroup then yield doubly stochastic transition matrices of finite Markov chains in which the entropic behaviour at any time is independent of the initial state.
A closure condition which is equivalent to the Thomsen condition in quasigroups.
M. A. Taylor (1983)
In this note it is shown that the closure condition, X1Y2 = X2Y1, X1Y4 = X2Y3, X3Y3 = X4Y1 --> X4Y2 = X3Y4, (and its dual) is equivalent to the Thomsen condition in quasigroups but not in general. Conditions are also given under which groupoids satisfying it are principal homotopes of cancellative, abelian semigroups, or abelian groups.
A construction of Bruck loops
A construction of commutative Moufang loops and quasimodules
A geometric approach to universal quasigroup identities
In the present paper we construct the accompanying identity
\stackrel{^}{I}
of a given quasigroup identity
I
. After that we deduce the main result:
I
is isotopically invariant (i.e., for every guasigroup
Q
it holds that if
I
is satisfied in
Q
I
is satisfied in every quasigroup isotopic to
Q
) if and only if it is equivalent to
\stackrel{^}{I}
(i.e., for every quasigroup
Q
it holds that in
Q
I,\stackrel{^}{I}
are both satisfied or both not).
A New Proof Of Belousov's Theorem For A Special Law Of Quasigroup Operations
Svetozar Milić (1971)
A new proof of Belousov's theorem for a special law of quasigroup operations.
S. Milic (1971)
A note on associative triples of elements in cancellation groupoids
A note on Bol loops of order 2nk.
Karl Robinson (1981)
A note on compatible reflexive relations on quasigroups
A note on loops of square-free order
Emma Leppälä, Markku Niemenmaa (2013)
Q
be a loop such that
|Q|
is square-free and the inner mapping group
I\left(Q\right)
is nilpotent. We show that
Q
is centrally nilpotent of class at most two.
STC
-groupoids
A note on simple medial quasigroups
K. K. Ščukin (1993)
A solvable primitive group with finitely generated abelian stabilizers is finite.
|
Linear optimization: Phases of the Simplex method 1 – Operations-Research-Wiki
Linear optimization: Phases of the Simplex method 1
A linear optimization tableau can be systemized in different Phases that have to be dealt with before the actual optimization step can be used, which is explained by the method of using the simplex algorithm.
Phase 0 Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \Rightarrow
Phase 0' Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \Rightarrow
1 Phase 0: equations
2 Phase 0': Free variables
3 Phase 1: Minimizing the infeasibility
4 Phase 2: Optimization
5 Excourse
Phase 0: equations
The locked variables are identifyable where equations are written as constraints. This means that the structure variable (slack variable) has to become 0 to solve the equation (locked variable). This structure variable must not stand in the basis, therefore it is urgent for it to leave the basis. The row where this blocked variable is findable becomes the pivot-row . Those coulumns that do not contain blocked variables as a non basis variable become the pivot coulumn. After the pivot element is found, a simplex iteration follows by the simple rules of simplex algorithm.
old constraints:
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \sum_{j}a_{ij}x_j \le b_i \Rightarrow y_i+\sum_{j}a_{ij}x_j =b_i ;~~y_i \ge 0
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \sum_{j} a_{ij}x_j = b_i
In the new constraints there are no
{\displaystyle y_{i}}
. This means that this structure variable (slack variable) has become 0
Phase 0': Free variables
Phase 0'
Free variables are variables for which the not negativity conditions are not valid respectivly are not defined. They have to be transfered to the basis as well. As long as free variables are located in the non-basis, the coulumn they stand in becomes a pivot-coulumn. If a free variable is relocated to the basis, it is blocked for the rest of the optimization procedure. This means the basis-row it stands in after the relocation cannot become a pivot-row again. Again a simplex iteration follows the search of the pivot element in this phase.
{\displaystyle x_{1}}
{\displaystyle x_{2}}
are the variables which have to be optimized. In the last line, that represents the non negativity conditions, in the adjoining tableau only
{\displaystyle x_{1}}
is mentioned and therefore restricted. Hence
{\displaystyle x_{2}}
is a free variable.
Phase 1: Minimizing the infeasibility
Constraints that contain a ≥ operator in it form an infeasible initial solution. In the simplex tableau this fact is dealt with a multiplication by -1. This means also that a negative right side is generated (infeasible). This often appears in constraints whereas the profit should be higher than a special value for example. The pivot row becomes the row where the negative right side is located. If possible the pivot-coulumn should be a coulumn where the pivot element has a negative sign. A simplex iteration is again required after finding the pivot element.
In the given example the third constraint is representing the actual phase. The restriction
{\displaystyle y_{3}}
has a negative sign for each value, because this line was multiplicated by -1 due to the infeasible initial solution.
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): y_3: ~30x_1+30x_2 \ge 600 \Rightarrow -30x_1-30x_2\le -600
If an optimal solution is not found by correcting the three Phases above, more simplex iterations are probably required. On the one hand an optimal solution is reached, where only positive values stand in the objective function coefficients. Moreover all elements on the right hand side have to be positive as well. On the other hand it can be shown that no feasible solution exists, if the algorithm is not able to replace a negative value in the objective function coefficients.
Short reminder of how to do the simplex optimization process
Selection of the Pivot element
Exchange basic variables (BV) and corresponding non basic variables (NBV)
Calculate elements of the Pivot row and the Pivot column
Calculate remaining columns
Check, whether another iteration is necessary
Primal degeneracy
The pivot row is free to choose, because there is more than one valid option. Thereby you have to decide whether you select the first or the second row as a pivot row in the given example.
Dual degeneracy
The pivot column is free to choose, because you have to decide between two or more different column, which contain the same negative value. In tableau above you can choose between column one or two, by the greatest change method or just randomly decide.
The following example illustrates the different phases of the simplex.
Linear Program:
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): 10x_1 + 20x_2 + 40000x_3 \rightarrow min!
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): x_1+ 2x_2-1760x_3 \le 8800
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): x_1+x_2 \ge 12000 \Rightarrow -x_1-x_2 \le -12000
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): x_1-2x_2=0
not negativity condition
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): x_1 \ge 0
By using the LP we create a first tableau
Eliminate Phase 0
The equation in the last row, that contains
{\displaystyle y_{3}}
, has to be dealt with. The slack variable
{\displaystyle y_{3}}
has to leave the basis. It's row becomes the pivot row. The pivot column has to be the column where
{\displaystyle x_{1}}
is located, because it's the only possible value. "-2" or "0" are not allowed to become the pivot element, because dividing by "0" is prohibited and "-2" would generate a negative right hand side. Therefore "1" (yellow marked) is left and becomes the pivot element.
Simplex iteration
After the simplex iteration we obtain the tableau shown above. When the slack variable
{\displaystyle y_{3}}
has moved into the non-basis, it is blocked until the end of the optimization process. Thus the blocked column cannot be a pivot column again.
Eliminate Phase 0'
By choosing the coulumn of
{\displaystyle x_{3}}
as a pivot column you start eliminating the free variables and transporting them to the basis. The only possible pivot element is "-1760" in the first row, where
{\displaystyle y_{1}}
is located. The zeroes in the lines below the corresponding column are strictly prohibited to use as pivot element. Normally this pivot element should have a positive sign, but in this special case there is a negative one. A simplex iteration is still possible though. It only creates a negative right hand side, which is to be dealt with in phase 1.
After the iteration the variable
{\displaystyle x_{3}}
is now in the basis. It's row is also blocked and therefore cannot be used again as a pivot row in the following optimization process. As mentioned above another negative right hand side is generated, which can be dealt with in the step of eliminating phase 1.
Eliminating Phase 1
The row in that the negative right hand side is located, becomes the new pivot row. Thereby we pick the row of
{\displaystyle y_{2}}
as a pivot row, because the first one is blocked for pivotization throughout the steps above, where
{\displaystyle x_{3}}
was set to a blocked status. It is useful to pick a pivot element with a negative sign to get rid of the negative sign of the right hand side. Therefore the pivot coulumn is the coulumn of
{\displaystyle x_{2}}
and the pivot element is "-3".
After the iteration, one can recognize that the right hand side in the column of
{\displaystyle x_{3}}
has a positive sign again. Therefore the solution is already optimal, because the objective function coefficients only consists of positive values. This makes the optimization step (Phase 2) obsolete and we are finished with the optimization process.
OR-script
OR-wiki : https://bisor.wiwi.uni-kl.de/orwiki/Phasen_des_Simplex-Algorithmus
OR-tutorium
Lucas Bertram, Niklas Drawert, Simon Liesenfeld
Von „https://www.wiwi.uni-kl.de/bisor-orwiki/index.php?title=Linear_optimization:_Phases_of_the_Simplex_method_1&oldid=4318“
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Physics - A New Phase of Solid Oxygen
A New Phase of Solid Oxygen
Institute for Integrated cell-Material Sciences, Kyoto University, Yoshida, Sakyo-ku, 606-8501, Japan
A large applied magnetic field can bring solid oxygen into a novel phase—a consequence of the strong spin-lattice coupling in the molecular crystal.
T. Nomura et al., Phys. Rev. Lett. (2014)
Figure 1: (a) Possible geometrical arrangements of the
{\text{O}}_{2}
{\text{O}}_{2}
dimer [2]: H geometry (parallel), S geometry (canted), or X geometry (crossed). (b) Evidence of a field-induced magnetic phase transition in solid oxygen. While previous studies explored a range of applied fields up to
\sim 50
T, an abrupt increase in magnetization is observed by Matsuda and co-workers as the field reaches
123
T [1]. The hysteresis loop suggests a first-order phase transition.(a) Possible geometrical arrangements of the
{\text{O}}_{2}
{\text{O}}_{2}
dimer [2]: H geometry (parallel), S geometry (canted), or X geometry (crossed). (b) Evidence of a field-induced magnetic phase transition in solid oxygen. While previous studies explored a range of a... Show more
Molecular crystals, like sugar, ice, or solid chalcogens (the elements of group of the periodic table, such as oxygen) are made of molecules held together by van der Waals forces. As they crystallize, they form structures that depend on the characteristics of the intermolecular interactions. Among various simple molecules, solid molecular oxygen ( ) is unique because it carries a magnetic moment with a spin quantum number . This results in a complex relationship between the magnetic interaction and crystal structure, which has long attracted the attention of scientists—solid oxygen is regarded as a unique example of a spin-controlled system. In Physical Review Letters [1], a team led by Yasuhiro Matsuda at the University of Tokyo, Japan, has reported the discovery of a novel phase of solid that can be reached by applying an extraordinarily strong magnetic field [ – tesla (T)]. The experiments indicate this is the result of a first-order transition that is both a magnetic and structural transition: the antiferromagnetic phase collapses and the crystal symmetry changes. The finding adds a new dimension to the phase diagram of oxygen and is a key demonstration of how strongly spin and lattice are coupled in the solid.
Molecular crystals exhibit various kinds of interesting structures because of the delicate balance of intermolecular interactions. In the case of ice, the hydrogen bonds determine the low-density structure that is unique among molecular crystals. In dry ice—the solid form of carbon dioxide ( )—an electrostatic quadrupole moment causes the characteristic cubic structure (also seen, e.g., in the phase of solid ). The case of solid molecular oxygen is particularly interesting. Faraday discovered the magnetism of in 1848, and since then, oxygen has been actively studied as a ubiquitous yet exotic molecular magnet. This magnetism plays an important role via the exchange interaction, leading to fascinating phenomenology in solid phases. At ambient pressure, three different phases ( , , and ) appear with decreasing temperature. Such phases differ in crystal structure as well as in magnetic properties. The application of pressure induces even more phases: [at gigapascal (GPa)], ( GPa), and ( GPa). The latter is a metallic phase, which even becomes superconducting below kelvin (K). A seventh phase ( ) has been observed in a high-pressure ( – GPa), high-temperature ( – K) range. But despite intense research, it is not yet very well established whether and how the magnetic interaction is essential for the molecular arrangement in the different phases of solid .
Why can magnetic properties influence structural properties in solid oxygen? First, the exchange interaction (as estimated from experiments) is comparable in strength to the van der Waals forces. Second, the exchange interaction between the magnetic moments of , including its sign, depends on the geometry of the molecular arrangement. Ab initio calculations have tackled the simpler case of a - dimer [2]. They predict that the stable arrangement of the two molecules depends on whether the magnetic moments align antiferromagnetically or ferromagnetically. Under normal conditions, an “H geometry” should occur [see Fig. 1(a)]: the two molecules are parallel, the exchange interaction is antiferromagnetic, and the singlet state is favored (that is, the two spins cancel each other, and the total spin is zero). The H geometry is indeed observed in the three phases of solid oxygen. But when the dimer is magnetized, two other geometries are stable: S (canted) and X (crossed) [Fig. 1(a)]. In both, the exchange interaction is ferromagnetic. This leads to the expectation of both magnetic and structural phase transitions resulting from the application of large magnetic fields.
Such theoretical predictions could be tested when, ten years ago, the - dimer was successfully synthesized by using nanoporous coordination polymers [3]. X-ray diffraction measurements at low temperatures revealed that the H geometry, as in the molecular solid, is robust in the dimer system. However, the magnetic measurements could not be interpreted based on Heisenberg models and H geometry only. To explain the experimental findings, researchers invoked a scenario in which there are excited states of other geometries, e.g., the S or X type [4]. This scenario indicates that a molecular rearrangement may occur in a magnetic field. It was also found that the thermally excited states lead to deviations from the H geometry at higher temperatures. These combined results, obtained on dimers, led to the anticipation of a field-induced structural phase transition in solid .
Matsuda’s team explored the possibility of this kind of phase transition starting from the phase of solid , using state-of-the-art techniques for generating microsecond pulses of ultrahigh magnetic fields (of up to T). Working at low temperatures ( K), they directly measured the magnetization using a pickup coil. Because oxygen has an antiferromagnetic ground state, the magnetization should linearly increase with increasing magnetic fields. This is true at low magnetic fields. However, a distinct, sudden increase of the magnetization occurs at around T [see Fig. 1(b)]. This is the clear signature of a magnetic phase transformation. The magnetization curve is also found to have a significant hysteresis between the field-increasing and field-decreasing processes. Hence the phase transition is suggested to be a first-order transition, probably associated with a structural change.
The authors used optical spectroscopy to probe structural changes, measuring transmission changes at visible wavelengths (around nanometers) as a function of the applied field. The existence of a bimolecular absorption resonance in this wavelength region is well established—it causes the blue color in solid . The key effect allowing optical detection of structural changes is the change of light scattering at the domain boundaries in polycrystalline oxygen. oxygen is structurally anisotropic (monoclinic). The interface between differently oriented crystal domains, each with anisotropic refractive indices, generates strong scattering, thus making polycrystalline oxygen opaque for visible light. But if the domains suddenly became isotropic, such scattering would be reduced. What the authors found is that the transmitted light intensity considerably increased at fields where the magnetic transition takes place. The effect is dramatic: The crystal becomes nearly transparent at ultrahigh magnetic fields. Such field-induced transparency clearly demonstrates that the crystal symmetry changes from anisotropic to isotropic. A similar phenomenon has been previously observed in the -to- phase transition, driven by increasing temperature at zero field, in which the crystal structure transforms from the rhombohedral ( ) to cubic ( ).
Magnetization measurements and magnetotransmission optical spectroscopy thus perfectly complement each other, providing compelling evidence of the novel phase of solid in ultrahigh magnetic fields. Recalling the magnetic field-induced rearrangement of the - dimer, the authors attribute the observed structural phase transition to the rearrangement of molecules. Dimer results suggest the antiferromagnetic coupling could become unstable in the new molecular arrangement, and ferromagnetic exchange interaction might become favored. This makes the result a noteworthy discovery in the long history of solid studies: the eighth phase of solid is completely different from the known seven phases at zero field where the exchange interaction between molecules is either paramagnetic or antiferromagnetic.
While optical spectroscopy clearly shows a structural transition, an outstanding question is the determination of the crystal structure of the novel phase. This is very challenging since the applied magnetic field is pulsed, with a duration of less than ten microseconds, which makes it difficult to probe the new structure with conventional, static diffraction measurements. In absence of reliable experiments, the puzzle may first be tackled by theoretical predictions of crystal structures. Finally, further studies at higher temperatures will be needed to gather a complete picture of the underlying physics that establishes the entire field-temperature phase diagram of solid oxygen.
T. Nomura, Y. H. Matsuda, S. Takeyama, A. Matsuo, K. Kindo, J. L. Her, and T. C. Kobayashi, “Novel Phase of Solid Oxygen Induced by Ultrahigh Magnetic Fields,” Phys. Rev. Lett. 112, 247201 (2014)
B. Bussery and P. E. S. Wormer, “A van der Waals Intermolecular Potential for (O
{}_{2}
{}_{2}
,” J. Chem. Phys. 99, 1230 (1993)
R. Kitaura et al., “Formation of a One-Dimensional Array of Oxygen in a Microporous Metal-Organic Solid,” Science 298, 2358 (2002)
A. Hori et al., ”Spin-Dependent Molecular Orientation of O
{}_{2}
{}_{2}
Dimer Formed in the Nanoporous Coordination Polymer” J. Phys. Soc. Jpn. 82, 084703 (2013)
Susumu Kitagawa is the director of the Institute for Integrated Cell-Material Sciences, Kyoto University and professor of functional chemistry, faculty of Engineering, Kyoto University. His research background is in inorganic chemistry, focusing on coordination network materials, which are now known as porous coordination polymers (PCPs) or metal-organic frameworks (MOFs). The focus of his group is to design and synthesize functional porous materials for science and technology of storage, separation, and conversion of gas substances. See more at http://www.sbchem.kyoto-u.ac.jp/kitagawa-lab/index-e.html
T. Nomura, Y. H. Matsuda, S. Takeyama, A. Matsuo, K. Kindo, J. L. Her, and T. C. Kobayashi
MagnetismChemical Physics
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How to do arbitrage with liquidity pools - LP-Swap Academy
How to do arbitrage with liquidity pools
When you add liquidity to a liquidity pool, for example in BUSD-WBNB one, you must provide quantities of both tokens by respecting the ratio
x^{\mathrm{BUSD}}/ x^{\mathrm{WBNB}}
of tokens already in the pool. Why does this correspond to providing equal values (expressed in dollars for instance) of BUSD and WBNB? This is because otherwise, arbitrage between the prices of the tokens and the ratio of the pool is made possible. We make this arbitrage opportunity precise in this article, but first let us reformulate the problematic with numbers.
p^{\mathrm{BUSD}}
p^{\mathrm{WBNB}}
be the respective prices of BUSD and WBNB expressed in dollars. If you have
Q
dollars to invest in this liquidity and you split these dollars equally in BUSD and WBNB, then you obtain
\frac{Q}{2}\times \frac{1}{p^{\mathrm{BUSD}}}
tokens BUSD
\frac{Q}{2}\times \frac{1}{p^{\mathrm{WBNB}}}
tokens WBNB. The ratio of tokens you obtain is then
p^{\mathrm{WBNB}}/{p^{\mathrm{BUSD}}}
. The arbitrage-free formula then states that this ratio matches the ratio of the pool:
\frac{p^{\mathrm{WBNB}}}{{p^{\mathrm{BUSD}}}}= \frac{x^{\mathrm{BUSD}}}{{x^{\mathrm{WBNB}}}}.
The arbitrage-free formula is also equivalent to the following equation:
p^{\mathrm{BUSD}} \times x^{\mathrm{BUSD}}= p^{\mathrm{WBNB}} \times x^{\mathrm{WBNB}}
which means that inside the BUSD-WBNB pool, the total value of BUSD tokens equal the total value of WBNB tokens.
Recall the swap formula that expresses the output quantity
q_{output}
of WBNB tokens obtained from swapping an input quantity
q_{input}
tokens BUSD:
q_{output}=\frac{(1-r)\times q_{input} \times x^{\mathrm{WBNB}} }{x^{\mathrm{BUSD}}+(1-r)\times q_{input}}
In the absence of transaction fees (
r = 0
) and of slippage (
x^{\text{BUSD}} + q_{input} \approx x^{\text{BUSD}}
), the output quantity of WBNB would be:
q_{output} = \frac{x^{\mathrm{WBNB}}}{x^{\mathrm{BUSD}}} q_{input}
In other words, in the absence of slippage and transaction fees, the ratio
x^{\mathrm{WBNB}} /x^{\mathrm{BUSD}}
gives the amount of WBNB you would get per BUSD. This ratio is called the pool price of BUSD. Similarly,
x^{\mathrm{BUSD}}/ x^{\mathrm{WBNB}}
is called the pool price of WBNB.
So the arbitrage-free formula states that the pool price equals the ratio of the actual prices, expressed in dollars or any currency:
\frac{x^{\mathrm{BUSD}}}{{x^{\mathrm{WBNB}}}} = \frac{p^{\mathrm{WBNB}}}{{p^{\mathrm{BUSD}}}}.
When the arbitrage-free formula does not hold, there is a possibility of arbitrage.
Let's say for instance that
x^{\mathrm{BUSD}}/x^{\mathrm{WBNB}}>p^{\mathrm{WBNB}}/ p^{\mathrm{BUSD}}.
p^{\mathrm{WBNB}}
dollars you can buy 1 WBNB token. Using the price pool (without transaction fees and slippage), you can then swap the WBNB token in the pool to obtain
x^{\mathrm{BUSD}}/x^{\mathrm{WBNB}}
tokens BUSD. Selling the BUSD tokens you get back
p^{\mathrm{BUSD}}\times x^{\mathrm{BUSD}}/x^{\mathrm{WBNB}}
dollars, which is bigger than your initial investment
p^{\mathrm{WBNB}}
x^{\mathrm{BUSD}}/x^{\mathrm{WBNB}}<p^{\mathrm{WBNB}}/ p^{\mathrm{BUSD}}
is symmetric: You can buy BUSD with dollars, then swap BUSD against WBNB using the pool and finally sell the resulting WBNB to obtain more dollars than you started with.
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Physics - Supersolidity or quantum plasticity?
Supersolidity or quantum plasticity?
Department of Physics, University of Alberta, Edmonton, Alberta T6G 2G7, Canada
New torsional oscillator experiments with plastically deformed helium show that what was thought to be defect-controlled supersolidity at low temperature may in fact be high-temperature softening from nonsuperfluid defect motion in the crystalline structure.
Illustration: (Left) Alan Stonebraker
Figure 1: (Left) Andronikashvili torsional oscillator. Viscous fluid is dragged by the moving disks, increasing the moment of inertia and reducing the oscillation frequency. In the superfluid phase, there is a frictionless component that does not move with the disks. The decoupling of this mass raises the oscillator frequency, giving a direct measurement of the superfluid density. (Right) What happens when solid helium is plastically deformed? Black curve: torsional oscillator frequency before deforming the helium. The increase at low temperatures suggests decoupling of some of the solid helium, i.e., supersolidity. Blue curve: the expected behavior if the defects produced by deformation enhance the supersolid fraction. Red curve: the behavior if the effect of the new defects is to elastically soften the crystal.(Left) Andronikashvili torsional oscillator. Viscous fluid is dragged by the moving disks, increasing the moment of inertia and reducing the oscillation frequency. In the superfluid phase, there is a frictionless component that does not move with the... Show more
Superfluidity—the frictionless flow of a liquid analogous to lossless current flow in a superconductor—is well known in liquid , but “supersolidity”—coexistence of crystalline order and superflow—seems counterintuitive. However, quantum mechanics allows atomic exchange even in a solid, especially if the atoms are light and the interatomic potential is weak. This makes helium the most quantum of solids and the possibility of supersolidity in helium was suggested more than 40 years ago [1]. Despite earlier searches, it was not until 2004 that Kim and Chan’s torsional oscillator experiments [2,3] provided convincing evidence of supersolidity. The torsional oscillator technique has been a cornerstone of our understanding of superfluidity since Andronikashvili’s classic experiment [4] in which he attached immersed disks in liquid and set them into torsional oscillation (Fig. 1, left) . Liquid moving with the discs contributed to the oscillator’s moment of inertia and so reduced its frequency (increased its period). Since a frictionless fluid is not dragged along with the disks, the oscillator frequency provided a direct measurement of the superfluid density in liquid helium. During the 1970s, John Reppy and co-workers at Cornell University in the US refined the “high torsional oscillator” into an exquisitely sensitive tool for studying superfluids. Now, Reppy reports new experiments in Physical Review Letters [5] that throw into question our current understanding of supersolidity.
Reppy describes the first experiments in which solid helium is plastically deformed inside a torsional oscillator. Shearing the crystal parallel to the walls of the oscillator changes its frequency, but only at high temperatures, casting doubt on current ideas that associate defects with supersolidity. Reppy proposes that the observed behavior might be due to elastic softening of crystals by defect motion at high temperatures, rather than to decoupling of a supersolid component at low temperatures. This interpretation is intriguing, but leaves a number of puzzles. As Reppy points out, it does not explain the behavior of helium confined in small pores [2] nor the dependence on quantum statistics [6] ( vs ). Most importantly, the “blocked annulus” experiment [3], which provides the strongest evidence of long range flow in solid helium, seems to require either superflow or extraordinary elastic properties.
In 2004, Kim and Chan [2,3] repeated the Andronikashvili experiment with solid helium. Using a torsion cell filled with at high pressures, they observed an increase in the torsional oscillator frequency below about , which they interpreted as a superfluidlike decoupling of about of the solid—the “non-classical rotational inertia” (NCRI) that characterizes a supersolid. Two other observations provided important support for this interpretation. First, the apparent decoupling decreased at high oscillation amplitudes, suggesting a critical velocity for superflow, like that seen in superfluid helium. Second, blocking the annular channel eliminated the decoupling, confirming that long range flow was involved. It soon became clear that such supersolidity must involve defects. Rittner and Reppy showed [7] that annealing (which removes defects) dramatically reduced the fraction of the helium that decoupled. Their later experiments [8] showed that the NCRI fraction could be enhanced by growing crystals in narrow channels and by rapid thermal quenching to increase defect densities. Dislocations are created during crystal growth and by plastic deformation and much recent work has focused on the possibility of superflow associated with a dislocation network.
A supersolid might be expected to exhibit other phenomena that characterize superfluids: superleaks, persistent currents, new sound modes, and quantized vortices. Despite much effort, experiments have not yet found such a “smoking gun” and the essential evidence for supersolidity still comes from torsional oscillators. However, elastic measurements [9,10] also show unusual effects that are clearly related to the oscillator behavior. The shear modulus decreases by about upon warming to , with the same dependence on temperature, amplitude, frequency, and impurity concentration as the NCRI. These elastic effects have a natural interpretation in terms of dislocation motion and pinning, but this does not directly involve supersolidity.
An obvious question is raised: are the two phenomena properties of a supersolid state or could changes in the helium’s shear modulus produce the observed torsional oscillator effects without invoking supersolidity? Numerical modelling of several realistic oscillator geometries [11] has shown that the effects of the observed shear modulus changes are much too small—extraordinary softening would be required to account for the apparent NCRI. Nonetheless, the new experiments by John Reppy suggest a nonsupersolid interpretation of the torsional oscillator behavior (Fig. 1, right). Plastic deformation creates dislocations and the higher defect density would be expected to produce a larger supersolid fraction in the helium. The additional mass decoupling would raise the oscillator frequency at low temperature, but would have no effect at high temperatures where there is no supersolid. Reppy’s experiments do not show this behavior. Deformation leaves the frequency unchanged at low temperature but reduces it at high temperatures. Annealing, which removes some defects, has the opposite effect, but again, it is only the high-temperature behavior which changes. Similar behavior is seen in the shear modulus—large strains and annealing both change the shear modulus at high temperatures, but not at low temperature [10]. This is expected for the modulus, since the effect of dislocations is to elastically weaken the crystal when they are mobile (at high temperatures), but not at low temperatures (where they are pinned).
Reppy is the first to plastically deform helium inside a torsional oscillator, but annealing has also been studied by others [12]. Why were the effects of annealing in those experiments interpreted as changes in a supersolid fraction? It is important to realize that the helium makes only a small contribution to the oscillator’s moment of inertia and that the apparent decoupling involves only a small fraction of the helium. The frequency changes associated with the NCRI are typically a few parts per million and must be extracted from a background that depends on temperature. Growing crystals at constant volume (as in most torsional oscillator experiments, including Reppy’s) introduces pressure and density gradients, as well as defects, into the solid helium. Annealing reduces these gradients by redistributing mass within the cell, with corresponding changes in the moment of inertia and oscillator frequency. Annealing also reduces the pressure in the oscillator (by as much as bar in some of Rittner and Reppy’s earlier work [8]). Depending on how rigid its walls are, these pressure changes can deform the oscillator and affect its frequency. It is not surprising that annealing produces frequency shifts and these have been noted by other groups. Given the focus on supersolid decoupling at low temperatures, it was natural for researchers to compare data by shifting their oscillator frequencies to make the background coincide at high temperatures. Reppy’s measurements suggest that instead it is the low-temperature frequencies that should coincide. However, this does not appear to be the case in all torsional oscillator measurements. In other experiments [13–15], annealing did not change the apparent NCRI but rather shifted the frequency uniformly at all temperatures (up in some experiments, down in others). In other cases, annealing lowered the frequency and also reduced the NCRI.
So what is special about Reppy’s measurements? Is it the design of his cell? His torsional oscillator results differ from measurements by other groups in some other ways. He has seen much larger NCRI fractions ( in this cell, up to in other experiments) and annealing produced more dramatic NCRI and pressure changes than other groups have observed [7,8]. In his current measurements, the narrowness of the annulus ( gap) may ensure that pressure and density gradients are primarily in the vertical direction. Reppy’s experiments also involve a particularly simple plastic deformation—pure vertical shear within the annular gap. Perhaps the deformation and annealing produce very little of the radial mass redistribution that would change the moment of inertia (and shift the overall torsional oscillator frequency), allowing the intrinsic behavior to be seen. This paper will certainly spur new torsional oscillator measurements to see how the effects it describes depend on details of cell design.
By plastically deforming helium within a torsional oscillator, Reppy has introduced a new tool to study the unusual properties of solid helium. If the behavior he sees proves to be universal, it could shift the focus of research to the unusual effects of defects on the elastic behavior of solid helium. We still don’t understand this quantum solid, but this paper poses a crucial question: are we dealing with supersolidity at low temperatures or some remarkable form of quantum plasticity that leads to extraordinary softening at high temperatures?
A. F. Andreev and I. M. Lifshitz, Sov. Phys. JETP 29, 1107 (1969); D. J. Thouless, Ann. Phys. 52, 403 (1969); G. V. Chester, Phys. Rev. A 2, 256 (1970); A. J. Leggett, Phys. Rev. Lett. 25, 1543 (1970)
E. L. Andronikashvili, Sov. Phys JETP 18, 424 (1948)
J. D. Reppy, Phys. Rev. Lett. 104, 255301 (2010)
J. T. West, O. Syshchenko, J. Beamish, and M. H. W. Chan, Nature Phys. 5, 598 (2009)
A. S. C. Rittner and J. D. Reppy, Phys. Rev. Lett. 97, 165301 (2006)
J. Day and J. Beamish, Nature 450, 853 (2007)
J. Day, O. Syshchenko, and J. Beamish, Phys. Rev. B 79, 214524 (2009)
A. C. Clark and M. H. W. Chan, Phys. Rev. B 77, 184513 (2008)
A. C. Clark, J. T. West, and M. H. W. Chan, Phys. Rev. Lett. 99, 135302 (2007)
M. H. W. Chan (private communication)
K. Shirahama (private communication)
H. Kojima (private communication)
Nonsuperfluid Origin of the Nonclassical Rotational Inertia in a Bulk Sample of Solid
{}^{4}\mathrm{He}
{}^{4}\mathrm{He}
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Using the Noise thorn
The Noise thorn can be used to place random values on Cactus grid functions at initial data and at the boundaries during evolution. This can be used to carry out “robust stability” tests, such as those proposed by Jeff Winicour.
To apply a random perturbation to initial data, set noise::apply_id_noise="yes". Then each grid function listed in the parameter noise::id_vars will be adjusted by a random factor. The maximum size of the random perturbation is controlled by the parameter noise::amplitude. The perturbations are applied during the CCTK_POSTINITIAL time bin.
A random number will be added to each point on the boundary of grid functions listed in the noise::bc_vars parameter if the flag noise::apply_bc_noise="yes" is set. As with the initial data, the maximum size of the perturbation is given by the noise::amplitude parameter. The adjustments are applied at each CCTK_POSTSTEP.
The following parameters can be used to apply a random adjustment of size
A=±0.0005
to the initial data and boundaries of the metric variables.
ActiveThorns = "... Noise ..."
noise::apply_id_noise = "yes"
noise::id_vars = "admbase::metric"
noise::apply_bc_noise = "yes"
noise::bc_vars = "admbase::metric"
noise::amplitude = 0.001
Description: Maximum absolute value of random data
Range Default: 0.000001
apply_bc_noise
Description: Add random noise to initial data
apply_id_noise
bc_vars
Description: Variables to modify with noise at boundary
Description: Initial data variables to modify with noise
noise_boundaries
Description: At which boundaries to apply noise
noise_stencil
Description: Number of boundary points
This section lists all the variables which are assigned storage by thorn CactusNumerical/Noise. 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.
id_noise
add noise to initial data
bc_noise
add noise to boundary condition
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How swap output is calculated - LP-Swap Academy
How swap output is calculated
When swapping on a decentralised exchange like PancakeSwap, have you ever wondered how the quantity of tokens you get after the swap is determined? Let us walk you through an example and explain you the maths behind it.
So let us say that you want to swap
q_{input}
BUSD for WBNB tokens, you are going to do the swap using a BUSD-WBNB pool that contains
x^{\mathrm{BUSD}}
x^{\mathrm{WBNB}}
tokens BUSD and WBNB.
In general there may be several pools joining two tokens, or there may not be any in which case you would need to find a route: a sequence of several pools joining your input token to the desired output token. Choosing the right pools and finding routes is the role of decentralised exchanges and AMMs like PancakeSwap.
Recall the constant product rule that relates the quantities of BUSD and WBNB in the pool before and after the swap:
x_{post}^{\mathrm{BUSD}}\times x_{post}^{\mathrm{WBNB}} = x_{pre}^{\mathrm{BUSD}} \times x_{pre}^{\mathrm{WBNB}}.
Another key ingredient is the fee of the pool. This fee is determined by a fee rate
0\leq r \leq 1
: Among your
q_{input}
BUSD tokens, you are charged
r\times q_{input}
tokens that are mainly used to pay the liquidity providers and are locked into the pool. Your remaining
(1-r)\times q_{input}
BUSD tokens are put into the pool to do the swap. So we have
x_{post}^{\mathrm{BUSD}}=x_{pre}^{\mathrm{BUSD}}+ (1-r)\times q_{input}.
But what is the quantity
q_{output}
of WBNB tokens that you get after the swap? Well, this is the quantity that is withdrawn from the pool:
x_{post}^{\mathrm{WBNB}}=x_{pre}^{\mathrm{WBNB}} - q_{output}.
So the constant product rule rewrites
(x_{pre}^{\mathrm{BUSD}}+(1-r)\times q_{input})\times (x_{pre}^{\mathrm{WBNB}}-q_{output})= x_{pre}^{\mathrm{BUSD}} \times x_{pre}^{\mathrm{WBNB}},
and so isolating your output
q_{output}
of WBNB tokens yields the swap formula:
q_{output}=\frac{(1-r)\times q_{input} \times x_{pre}^{\mathrm{WBNB}} }{x_{pre}^{\mathrm{BUSD}}+(1-r)\times q_{input}}.
Note that the ratio
\frac{q_{output}}{q_{input}}=\frac{(1-r)\times x_{pre}^{\mathrm{WBNB}} }{x_{pre}^{\mathrm{BUSD}}+(1-r)\times q_{input}}
gets smaller as the input quantity
q_{input}
increases. This phenomenon is known as slippage.
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Physics - Quantized topological surface states promise a quantum Hall state in topological insulators
Quantized topological surface states promise a quantum Hall state in topological insulators
Jacob W. Linder
Department of Physics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway
August 9, 2010 • Physics 3, 66
The first experimental observation of Landau levels in the surface states of a three-dimensional topological insulator confirms the levels obey unconventional quantization rules.
Illustration: (a),(b) Carin Cain, adapted from Novoselov et al., Nature Phys. 2, 177 (2006); (c) Cheng et al. [6]; (d) Hanaguri et al. [7]
Figure 1: (a) Illustration of the conventional integer quantum Hall effect found, e.g., in 2D semiconductor systems. (b) Illustration of the unconventional half-integer quantum Hall effect found, e.g., in a graphene layer. (c) The experimentally observed Landau quantization of the topological surface states in
{\text{Bi}}_{2}{\text{Se}}_{3}
for several magnetic field strengths. Curves are offset vertically for clarity. Note in particular the peak occurring at the Fermi level (around
-200\phantom{\rule{0.333em}{0ex}}\text{mV}
sample bias), which is independent of the field strength. This serves as a hallmark of the unconventional quantization form shown in (b). (d) STM picture of a
{\text{Bi}}_{2}{\text{Se}}_{3}
surface in the presence of triangular-shaped defects (bright white spots). By selecting one point in such a map and plotting the
d\phantom{\rule{0}{0ex}}I/d\phantom{\rule{0}{0ex}}V
curves as a function of bias voltage, one is able to probe the density of states similarly to (c).(a) Illustration of the conventional integer quantum Hall effect found, e.g., in 2D semiconductor systems. (b) Illustration of the unconventional half-integer quantum Hall effect found, e.g., in a graphene layer. (c) The experimentally observed Landa... Show more
The prediction [1] and experimental discovery [2,3] of a class of materials known as topological insulators is a major recent event in the condensed matter physics community. Why do two- and three-dimensional topological insulators (such as [2] and [3], respectively) attract so much interest? Thinking practically, these materials open a rich vista of possible applications and devices based on the unique interplay between spin and charge. More fundamentally, there is much to enjoy from a physics point of view, including the aesthetic spin-resolved Fermi surface topology [3], the possibility of hosting Majorana fermions (a fermion that is its own antiparticle) in a solid-state system [4], and the intrinsic quantum spin Hall effect, which can be thought of as two copies of the quantum Hall effect for spin-up and spin-down electrons [5]. Now, an exciting new addition to the above list comes from two teams that are reporting the first experimental observation of quantized topological surface states forming Landau levels in the presence of a magnetic field. The two papers - one appearing in Physical Review Letters by Peng Cheng and colleagues at Tsinghua University in China, and collaborators in the US, the other, appearing as a Rapid Communication in Physical Review B, by Tetsuo Hanaguri at Japan’s RIKEN Advanced Science Institute in Wako and scientists at the Tokyo Institute of Technology - pave the way for seeing a quantum Hall effect in topological insulators.
An ideal topological insulator is electrically insulating in its bulk, but has conducting electronic states that are formed on its surface (or edges in a two-dimensional material). These electronic states reside on a massless Dirac cone, a relativistic energy-momentum dispersion in reciprocal space. Dirac cones also appear in graphene, but there are essential differences between topological insulators and graphene. In a topological insulator, as opposed to the case for graphene, the surface states display a remarkable robustness against any type of disorder or perturbation that is invariant under time-reversal symmetry, even though the surface states in both of these systems are Dirac fermions. The reason for this is the seemingly trivial fact that the Brillouin zone in topological insulators features a single Dirac cone, thus an odd number, whereas in graphene the number of Dirac cones is even. This can be probed experimentally, e.g., via photoemission spectroscopy [3] to reveal how many Dirac cones exist in the Brillouin zone. In an environment with a single Dirac cone, Kramers’ theorem of degeneracy guarantees that the energy bands of the surface states in topological insulators must cross at the time-reversal invariant point at . As a result, no disorder or perturbation that respects this time-reversal invariance can induce a gap in the surface-state spectrum.
Let us extend this reasoning to a three-dimensional topological insulator, in which the surface can be thought of as a two-dimensional metal, with carriers that will feature different spins depending on their direction of propagation. In such a two-dimensional electron system, the much-celebrated phenomenon known as the quantum Hall effect (QHE) is known to occur under appropriate circumstances. The QHE [8] was originally discovered in the context of two-dimensional electron gases at low temperatures, and under the application of a strong magnetic field perpendicular to the sample. In such a scenario, quantized orbits of motion are formed and the electrons in the sample can only occupy energy states belonging to quantized energy eigenvalues. More recently, the QHE was experimentally demonstrated to occur in graphene [9], but with a twist compared to the two-dimensional electron gas case. The special properties of the Dirac fermions in graphene cause the quantized conductance to form its characteristic plateaus at half-integer values of [10], in contrast to the integer values of observed in conventional two-dimensional electron gases. To understand this, it is instructive to consider the analytical expression for the Landau energy levels for the linear dispersion,
Here, is the Fermi level, is an integer, is the Fermi velocity, is the unit electric charge, is the field strength, and is Planck’s reduced constant. The most remarkable aspect of the above equation is the fact that there exists a Landau level that is completely independent of the magnetic field strength, namely, at , . Compare this with the case of a conventional two-dimensional electron gas system with a parabolic dispersion (nonzero mass of the charge carriers), where such a zeroth Landau level is not permitted. How, then, is this related to the half-integer quantum Hall effect? The key observation here is that the conductance plateaus, serving as the hallmark of the QHE, occur whenever the chemical potential falls between two Landau levels. For a conventional QHE system, there is no zeroth Landau level, and thus a plateau appears at . In contrast, if a zeroth Landau level exists, a plateau should exist right before and right after the Fermi level. When particle-hole symmetry holds in the system, the conductance is an odd function of energy, measured relative to the Fermi level. It therefore follows that the first conductance quantization plateaus must occur precisely at values , thus being separated by as they should. Both of the above scenarios are illustrated in Fig. 1(a) and (b).
Since the surface states in topological insulators are Dirac fermions, it follows from the above discussion that 3D topological insulators should feature such a unique zero-energy Landau level, just like graphene, and thus display the unconventional quantum Hall effect. But do they? The papers by Cheng et al. [6] and Hanaguri et al. [7] answer this question in the affirmative. Both research teams utilized a scanning tunneling microscopy/spectroscopy (STM/STS) technique to probe the current-voltage characteristics on the surface of high-quality samples. The STM provides information about the density of states of the surface electrons, as shown in Fig. 1(d), where the density of states and the triangular defects in are shown.
In their work, Cheng et al. [6] demonstrate an indisputable signature of Landau levels forming at the surface of the topological insulator under the influence of a strong external magnetic field, namely, a well-defined series of peaks in the conductance spectrum as a function of the bias between the STM/STS tip and the surface of the sample [see Fig. 1(c)]. Most interestingly, a peak is consistently observed right at the Fermi level and does not change with increasing field strength. This is precisely the famous zeroth Landau level. Cheng et al. then proceed to confirm further the two-dimensional nature of the Landau states in three-dimensional topological insulators by adding impurity silver atoms directly on the surface by means of evaporation. With increasing concentration of impurities, the Dirac point shifts in energy due to the electron transfer between silver atoms and the substrate and, more importantly, the Landau quantization eventually disappears. This does not mean that the surface states disappear, only the quantization.
Hanaguri et al. [7] also verify the presence of Landau levels using similar experimental techniques to those used in Ref. [6]. But in addition to this, they also develop of a clever strategy to probe explicitly the energy-momentum dispersion of the topological surface states. As quasiparticle interference modulations are absent in the Dirac cone on a topological insulator, due to the above-mentioned robustness of these states, this conventional tool for probing the -space structure of electronic states is rendered ineffective. Instead, Hanaguri et al. derive the − dispersion directly from the Landau-level spectroscopy by making use of the fact that the Landau orbits are quantized in space, and also employing a Bohr-Sommerfeld quantization condition which results in the formula . In this way, a set of energy levels and belonging momenta can be obtained once and are specified, since scales with .
These experiments have established that the surfaces of topological insulators support Landau levels with an unconventional (half-integer) quantization. The next step to actually seeing the resulting quantum Hall effect will be transport measurements. However, important questions remain to be answered. For instance, what is the role of a possible Zeeman splitting with regard to these unconventional Landau levels? It is known that an out-of-plane exchange field will induce a mass gap in the surface-state spectrum of the topological insulators, thus destroying the surface states in a limited energy regime. If the Zeeman coupling to the external field is weak, an appreciable mass gap will only occur at high field strengths. But the profoundly different response of the topological surface states with respect to a Zeeman-field, because of the single Dirac cone compared to a conventional parabolic dispersion, suggests that more exciting effects than a mere spin-splitting of the Landau levels should occur. Such questions and more remain to be explored in this developing field of research, in addition to a direct experimental observation of the quantized conductance plateaus pertaining to the quantum Hall effect.
B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Science 314, 1757 (2006)
M. König et al., Science 318, 766 (2007)
D. Hsieh et al., Nature 452, 970 (2008); Y. Xia et al., Nature Phys. 5, 398 (2009)
N. Read and D. Green, Phys. Rev. B 61,10267 (2000); D. A. Ivanov, Phys. Rev. Lett. 86, 268 (2001); S. Das Sarma et al., Phys. Rev. B 73, 220502 (2006)
B. A. Bernevig and S.-C. Zhang, Phys. Rev. Lett. 96, 106802 (2006)
P. Cheng et al., Phys. Rev. Lett. 105, 076801 (2010)
T. Hanaguri, K. Igarashi, M. Kawamura, H. Takagi, and T. Sasagawa, Phys. Rev. B 82, 081305 (2010)
T. Ando, Y. Matsumoto, and Y. Uemura, J. Phys. Soc. Jpn 39, 279 (1975); K. Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45, 494 (1980)
Y. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim, Nature 438, 201 (2005)
V. P. Gusynin and S. G. Sharapov, Phys. Rev. Lett. 95, 146801 (2005)
Jacob Linder obtained his Ph.D. in physics from the Norwegian University of Science and Technology (NTNU) in 2009. In 2010, he received awards from the Royal Norwegian Society of Sciences and Letters and from ExxonMobil for his work on quantum transport and proximity effects in unconventional superconducting hybrid systems. He is now an Associate Professor at NTNU, Norway. His research addresses a variety of topics related to proximity effects and quantum transport in hybrid structures, magnetization dynamics and spin-transfer torque, and properties of unconventional superconductivity.
Landau Quantization of Topological Surface States in
{\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}
Peng Cheng, Canli Song, Tong Zhang, Yanyi Zhang, Yilin Wang, Jin-Feng Jia, Jing Wang, Yayu Wang, Bang-Fen Zhu, Xi Chen, Xucun Ma, Ke He, Lili Wang, Xi Dai, Zhong Fang, Xincheng Xie, Xiao-Liang Qi, Chao-Xing Liu, Shou-Cheng Zhang, and Qi-Kun Xue
Momentum-resolved Landau-level spectroscopy of Dirac surface state in
{\text{Bi}}_{2}{\text{Se}}_{3}
T. Hanaguri, K. Igarashi, M. Kawamura, H. Takagi, and T. Sasagawa
{\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}
{\text{Bi}}_{2}{\text{Se}}_{3}
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2021.10 - SIMBA Documentation
2021.10 2021.10 Table of contents
Improved Solver
Stop At Steady-State
New PWM Generator and Switch with Threshold models
New Dual Active Bridge Example
SIMBA 2021.10
Welcome to SIMBA 2021.10! This major release brings major changes including improved solver, AC Sweep and the new Stop at Steady State option.
If you haven’t yet, be sure to download it.
The solver is now about 10x more accurate with even faster simulation.
The maximum relative tolerance of the integration error is now set to 10E-7 instead of 10E-6 previously and this computational overhead is more than compensated by improvements and optimizations of the solver engine.
A new test bench is available: AC Sweep. It uses the steady-state algorithm to calculate the transfer function of a periodic system at different user-defined frequencies. More information here.
When Stop at Steady-State is enabled, the End Time parameter is not used, and the simulation stops when SIMBA detects the steady state.
This feature is particularly useful during a parametric analysis because each run can have different time constants and the Stop at Steady-State feature ensures that each run will stop at its steady state. To determine if the steady state is reached, SIMBA analyzes the RMS value and the highest non-DC harmonics (if any) of all simulated state variables.
A new search option is added to the Library tab to quickly search and find your models.
Two new models are added to the library:
Control>Sources>Controlled PWM Generator
Electrical>Switches>Controlled Switch with Threshold
This example shows a Dual Active Bridge converter with:
an input voltage of 95 V,
an output voltage of 380 V,
an output power of 1 kW.
Each bridge with a duty cycle of 50%. The phase-shift between the two bridges is set by the discrete controller (PI regulator) in order to regulate the output voltage.
Mosfets of this example are set with conduction parameters: a first R{on}
R{on}
resistance for the channel conduction mechanism (forward and reverse conduction when the transistor's gate id driven high) and a second R_{on}
R_{on}
resistance and a V_f
drop voltage for the body diode.
Note : with the command used in this example, mosfets are always used in synchronous rectification mode which means that the controlled conduction mechanism (channel) is the major contributor for conducting the reverse current compared to the body diode conduction.
The transformer ratio can be chosen according required values of primary and secondary voltages. Here a secondary voltage of 380V is required and a primary voltage of 95V is considered to be the lowest possible value (worst case), which leads to a transformer ratio of 4.
The expression below gives the maximum power which can be transferred for a phase-shift angle of \pi / 2
\pi / 2
P_{max} = \frac{V_1 V_2}{m 8 L_{ac} f_{sw}}
with m = \frac{V_2}{V_1}
m = \frac{V_2}{V_1}
, the transformer ratio.
A maximum power of 2 kW and a margin of 10 % si considered. At a switching frequency of 250 kHz, with V_1 = 95V
and V_2 = 380V
, this leads to a maximun inductor value of 2.051 \mu H
2.051 \mu H
This value is compatible with typical values of leakage inductors of high-frequency transformers with these levels of power, voltages and ratio.
[1] M. Blanc, Y. Lembeye, J.P. Ferrieux, Dual Active Bridge (DAB) pour la conversion continu-continu, Techniques de l'ingénieur E3975, 2019.
In 2021.10, several bugs are corrected and the general stability of SIMBA is improved. The complete list of changes is available here.
If you are interested, you can check our roadmap and a public GitHub project is available to share ideas, report bugs, and suggest new features.
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LoadConnection - Maple Help
Home : Support : Online Help : Connectivity : Database Package : LoadConnection
load a saved Connection
LoadConnection( name, opts )
(optional) string; the identifier for the connection to load
(optional) equation(s) of the form option=value where option is one of filename or password
LoadConnection re-establishes a connection that was saved using the Save command or Database Connection Maplet application.
Each connection that is saved using the Save command has an associated identifier. This identifier is used to distinguish between the connections saved in a file. If specified, name is used as the identifier. Otherwise, "default" is used as the identifier.
The filename option specifies the file from which this connection is loaded. When no filename is given, connections are loaded from $HOME/maple/toolbox/Database/data/default.con, where $HOME is the value returned by kernelopts( homedir ).
password = string or one of the literal symbols none or hidden
The password option is used to specify the password to use when re-establishing the connection to the database. If a password option is given, it overrides a password that is saved in the file. If no password is saved in the file and a password is required to connect to the database, then the password option must be given. If a string is given, that string is used as the password. If hidden is given, then a Maplet application opens to allow you to enter a password without it appearing in the worksheet. The default value is none.
Along with saving the parameters necessary to re-establish the connection, the Save command also stores the connection's current settings, as returned by the GetOptions command. When a connection is loaded using the LoadConnection command, these settings are also restored.
LoadConnection returns an expression sequence of two modules. The first module is the Database[Driver] module you used to open the connection. The second is the Database[Connection] module representing the opened connection.
\mathrm{driver}≔\mathrm{Database}[\mathrm{LoadDriver}]\left(\right):
\mathrm{conn}≔\mathrm{driver}:-\mathrm{OpenConnection}\left(\mathrm{url},\mathrm{name},\mathrm{pass}\right):
\mathrm{conn}:-\mathrm{Save}\left(\right):
\mathrm{conn}:-\mathrm{Close}\left(\right);
\mathrm{driver}:-\mathrm{Close}\left(\right)
Re-establish the connection saved as default.
\mathrm{driver},\mathrm{conn}≔\mathrm{Database}[\mathrm{LoadConnection}]\left(\mathrm{password}='\mathrm{hidden}'\right):
Modify this connection's options.
\mathrm{conn}:-\mathrm{SetOptions}\left('\mathrm{autocommit}'=\mathrm{true}\right)
Save using a different identifier and specify the password.
\mathrm{conn}:-\mathrm{Save}\left("ac1",'\mathrm{password}'=\mathrm{pass}\right):
\mathrm{conn}:-\mathrm{Close}\left(\right);
\mathrm{driver}:-\mathrm{Close}\left(\right)
Re-establish the "ac1" connection.
\mathrm{driver},\mathrm{conn}≔\mathrm{Database}[\mathrm{LoadConnection}]\left("ac1"\right):
Check this connection's options.
\mathrm{conn}:-\mathrm{GetOptions}\left('\mathrm{autocommit}'\right)
[\textcolor[rgb]{0,0,1}{\mathrm{autocommit}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{true}}]
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Create pix2pixHD global generator network - MATLAB pix2pixHDGlobalGenerator - MathWorks India
Create Pix2PixHD Generator
Create Pix2PixHD Generator with Batch Normalization
pix2pixHD Generator Network
Create pix2pixHD global generator network
net = pix2pixHDGlobalGenerator(inputSize)
net = pix2pixHDGlobalGenerator(inputSize,Name,Value)
net = pix2pixHDGlobalGenerator(inputSize) creates a pix2pixHD generator network for input of size inputSize. For more information about the network architecture, see pix2pixHD Generator Network.
net = pix2pixHDGlobalGenerator(inputSize,Name,Value) modifies properties of the pix2pixHD network using name-value arguments.
Specify the network input size for 32-channel data of size 512-by-1024 pixels.
Create a pix2pixHD generator network that performs batch normalization after each convolution.
net = pix2pixHDGlobalGenerator(inputSize,"Normalization","batch")
Example: 'NumFiltersInFirstBlock',32 creates a network with 32 filters in the first convolution layer
Number of downsampling blocks in the network encoder module, specified as a positive integer. In total, the network downsamples the input by a factor of 2^NumDownsamplingBlocks. The decoder module consists of the same number of upsampling blocks.
Number of output channels, specified as a positive integer.
Filter size in intermediate convolution layers, specified as a positive odd integer or 2-element vector of positive odd integers of the form [height width]. The intermediate convolution layers are the convolution layers excluding the first and last convolution layer. When you specify the filter size as a scalar, the filter has identical height and width. Typical values are between 3 and 7.
Number of residual blocks, specified as a positive integer.
\left[\begin{array}{ccc}3& 1& 4\\ 1& 5& 9\\ 2& 6& 5\end{array}\right]\to \left[\begin{array}{ccccccc}2& 2& 2& 2& 2& 2& 2\\ 2& 2& 2& 2& 2& 2& 2\\ 2& 2& 3& 1& 4& 2& 2\\ 2& 2& 1& 5& 9& 2& 2\\ 2& 2& 2& 6& 5& 2& 2\\ 2& 2& 2& 2& 2& 2& 2\\ 2& 2& 2& 2& 2& 2& 2\end{array}\right]
\left[\begin{array}{ccc}3& 1& 4\\ 1& 5& 9\\ 2& 6& 5\end{array}\right]\to \left[\begin{array}{ccccccc}5& 1& 1& 5& 9& 9& 5\\ 1& 3& 3& 1& 4& 4& 1\\ 1& 3& 3& 1& 4& 4& 1\\ 5& 1& 1& 5& 9& 9& 5\\ 6& 2& 2& 6& 5& 5& 6\\ 6& 2& 2& 6& 5& 5& 6\\ 5& 1& 1& 5& 9& 9& 5\end{array}\right]
\left[\begin{array}{ccc}3& 1& 4\\ 1& 5& 9\\ 2& 6& 5\end{array}\right]\to \left[\begin{array}{ccccccc}5& 6& 2& 6& 5& 6& 2\\ 9& 5& 1& 5& 9& 5& 1\\ 4& 1& 3& 1& 4& 1& 3\\ 9& 5& 1& 5& 9& 5& 1\\ 5& 6& 2& 6& 5& 6& 2\\ 9& 5& 1& 5& 9& 5& 1\\ 4& 1& 3& 1& 4& 1& 3\end{array}\right]
\left[\begin{array}{ccc}3& 1& 4\\ 1& 5& 9\\ 2& 6& 5\end{array}\right]\to \left[\begin{array}{ccccccc}3& 3& 3& 1& 4& 4& 4\\ 3& 3& 3& 1& 4& 4& 4\\ 3& 3& 3& 1& 4& 4& 4\\ 1& 1& 1& 5& 9& 9& 9\\ 2& 2& 2& 6& 5& 5& 5\\ 2& 2& 2& 6& 5& 5& 5\\ 2& 2& 2& 6& 5& 5& 5\end{array}\right]
Probability of dropout, specified as a number in the range [0, 1]. If you specify a value of 0, then the network does not include dropout layers. If you specify a value greater than 0, then the network includes a dropoutLayer (Deep Learning Toolbox) in each residual block.
"GlobalGenerator_" (default) | string | character vector
Pix2pixHD generator network, returned as a dlnetwork (Deep Learning Toolbox) object.
A pix2pixHD generator network consists of an encoder module followed by a decoder module. The default network follows the architecture proposed by Wang et. al. [1].
The encoder module downsamples the input by a factor of 2^NumDownsamplingBlocks. The encoder module consists of an initial block of layers, NumDownsamplingBlocks downsampling blocks, and NumResidualBlocks residual blocks. The decoder module upsamples the input by a factor of 2^NumDownsamplingBlocks. The decoder module consists of NumDownsamplingBlocks upsampling blocks and a final block.
The table describes the blocks of layers that comprise the encoder and decoder modules.
An optional activation layer specified by the FinalActivationLayer name-value argument.
You can create the discriminator network for pix2pixHD by using the patchGANDiscriminator function.
Train the pix2pixHD GAN network using a custom training loop.
addPix2PixHDLocalEnhancer | encoderDecoderNetwork | blockedNetwork | cycleGANGenerator | unitGenerator
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Download - Maple Help
Home : Support : Online Help : Connectivity : Web Features : Web Page Access : URL package : Download
download data specified by a URL to a file
Download( url, destination, options )
string specifying the URL from which to request data
(optional) file or directory to which the data should be written
mode=one of text or binary
Specifies whether the reply data should be treated as a string (mode=text) or as an Array with data type integer[1] (mode=binary). This is only supported by the http and https schemes. FTP transfers are always treated as binary.
output=list or one of filename, headers, or size
A list of one or more of the symbols filename, headers, and size, or one of the symbols filename, headers, or size. The output will be an expression sequence with values corresponding to the specified keywords. The meaning of the symbols is as follows:
filename is the file name to which the downloaded data was written.
size refers to the size (in bytes) of the downloaded data. It is returned as an integer.
This option is only valid with http and https schemes. The default is filename.
The Download command fetches data from the address specified in url and saves it to a file.
If file is specified and a file with this name does not exist in the current directory, the data is saved to this filename.
If file is specified and corresponds to an existing file, the data is saved to a temporary filename in the same directory.
If file is specified and corresponds to an existing directory, the data is saved to a temporary filename in that directory.
If file is not specified, the data is saved to a temporary filename in the current directory.
The Download command uses a GET request to fetch the remote URL. To use other HTTP schemes or provide additional configuration details such as login credentials or proxy settings, see the URL package.
The current implementation of Download is not secure. Between generating the unique filename and creating the file, another process may be able to create a file with the same name. By doing so, the other process can read information from or write information into the file.
\mathrm{URL}:-\mathrm{Download}\left("http://www.maplesoft.com/"\right):
\mathrm{URL}:-\mathrm{Download}\left("https://www.maplesoft.com/images2015/resources/Maple/Maple_2015_logo.jpg","MapleLogo.jpg"\right)
The URL[Download] command was introduced in Maple 2019.
The URL[Download] command was updated in Maple 2020.
The cafile, password, proxy, proxypassword, proxyuser, redirections, requestheaders, responseheaders, timeout, user and verify options were introduced in Maple 2020.
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Filippo Tolli (2000)
A Central Limit Theorem for Approximate Martingale Arrays with Values in a Locally Compact Abelian Group.
Michael S. Bingham (1986)
Piotr Graczyk (1992)
{𝒫}_{n}
{𝒫}_{n}
{𝒫}_{n}
A Decomposition Theorem for Additive Set-Functions, with Applications to Pettis Integrals and Ergodic Means.
Michel Talagrand, David H. Fremlin (1979)
A Gaussian bound for convolutions of functions on locally compact groups
Nick Dungey (2006)
We give new and general sufficient conditions for a Gaussian upper bound on the convolutions
{K}_{m+n}\ast {K}_{m+n-1}\ast \cdots \ast {K}_{m+1}
of a suitable sequence K₁, K₂, K₃, ... of complex-valued functions on a unimodular, compactly generated locally compact group. As applications, we obtain Gaussian bounds for convolutions of suitable probability densities, and for convolutions of small perturbations of densities.
A Green's Function for Non-Homogegeous Random Walks on Free Products.
Alice R. Trenholme (1988)
A Law of the Iterated Logarithm for a Class of Polynomial Hypergroups.
Michael Voit (1990)
A Littlewood-Paley-Stein estimate on graphs and groups
We establish the boundedness in
{L}^{q}
spaces, 1 < q ≤ 2, of a “vertical” Littlewood-Paley-Stein operator associated with a reversible random walk on a graph. This result extends to certain non-reversible random walks, including centered random walks on any finitely generated discrete group.
{ℝ}^{*+}
{ℝ}^{d}
Emile Le Page, Marc Peigne (1994)
A New Proof of the Generalized Continuity Theorem of Paul Levy.
Eberhard Siebert (1978)
A remark on the norm of a random walk on surface groups
Andrzej Żuk (1997)
We show that the norm of the random walk operator on the Cayley graph of the surface group in the standard presentation is bounded by 1/√g where g is the genus of the surface.
A survey of results on random random walks on finite groups.
Hildebrand, Martin (2005)
A topological version of some ergodic theorems
G. Taylor (1978)
Absolut-Stetigkeit und Träger von Gauß-Verteilungen auf lokalkompakten Gruppen.
Actions of large semigroups and random walks on isometric extensions of boundaries
Yves Guivarc'h, Albert Raugi (2007)
Marek Kanter (1976)
Admissible transformations of measures.
P. L. Brockett (1976)
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Refer to the diagram at right to answer the questions below.
\array{++++\\ \qquad------}
Copy the diagram and add two pieces so that the result is a value of
-2
. You can add positive and/or negative pieces. Show the pieces you added to get this answer
This problem is very similar to problem 3-34. Last time, you were told how many pieces to add or subtract. This time, you will need to use your own understanding to determine how best to reach
-2
by adding two pieces.
Since the current value of this diagram is
-2
, you will need to add one negative piece
and one positive piece to keep the value of
-2
Copy the diagram again, but this time remove four pieces (again positives and/or negatives) so that the result is a value of
2
. Show the pieces you removed to get this answer.
You already know the current value is
-2
. By only removing pieces from the picture,
-2
2
? Think about this question, then proceed with the problem.
Use the following eTool to answer the questions.
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IMU simulation model - MATLAB - MathWorks 日本
MagneticField — Magnetic field vector in local navigation coordinate system (μT)
magReadings — Magnetometer measurement of IMU in sensor body coordinate system (μT)
NoiseDensity: [0 0 0] (rad/s)/√Hz
RandomWalk: [0 0 0] (rad/s)*√Hz
TemperatureBias: [0 0 0] (rad/s)/°C
TemperatureScaleFactor: [0 0 0] %/°C
AccelerationBias: [0 0 0] (rad/s)/(m/s²)
MeasurementRange: 19.62 m/s²
Resolution: 0.00059875 (m/s²)/LSB
ConstantBias: [0.4905 0.4905 0.4905] m/s²
NoiseDensity: [0.003924 0.003924 0.003924] (m/s²)/√Hz
BiasInstability: [0 0 0] m/s²
RandomWalk: [0 0 0] (m/s²)*√Hz
TemperatureBias: [0.34335 0.34335 0.5886] (m/s²)/°C
TemperatureScaleFactor: [0.02 0.02 0.02] %/°C
totalAcc=âacceleration+g
a=\left(orientation\right){\left(totalAcc\right)}^{T}
b={\left(\left[\begin{array}{ccc}1& \frac{{\mathrm{α}}_{2}}{100}& \frac{{\mathrm{α}}_{3}}{100}\\ \frac{{\mathrm{α}}_{1}}{100}& 1& \frac{{\mathrm{α}}_{3}}{100}\\ \frac{{\mathrm{α}}_{1}}{100}& \frac{{\mathrm{α}}_{2}}{100}& 1\end{array}\right]\left({a}^{T}\right)\right)}^{T}+\text{ConstantBias}
{\mathrm{β}}_{1}={h}_{1}*\left(w\right)\left(\text{BiasInstability}\right)
{H}_{1}\left(z\right)=\frac{1}{1â\frac{1}{2}{z}^{â1}}
{\mathrm{β}}_{2}=\left(w\right)\left(\sqrt{\frac{\text{SampleRate}}{2}}\right)\left(\text{NoiseDensity}\right)
{\mathrm{β}}_{3}={h}_{2}*\left(w\right)\left(\frac{\text{RandomWalk}}{\sqrt{\frac{\text{SampleRate}}{2}}}\right)
{H}_{2}\left(z\right)=\frac{1}{1â{z}^{â1}}
{\mathrm{Î}}_{e}=\left(\text{Temperature}â25\right)\left(\text{TemperatureBias}\right)
scaleFactorError=1+\left(\frac{\text{Temperature}â25}{100}\right)\left(\text{TemperatureScaleFactor}\right)
e=\left\{\begin{array}{c}\begin{array}{c}\text{MeasurementRange}\\ â\text{MeasurementRange}\end{array}\\ d\end{array}\begin{array}{c}\\ \\ \end{array}\begin{array}{c}\text{if}\\ \text{if}\\ \text{else}\end{array}\begin{array}{c}\\ \\ \end{array}\begin{array}{c}d>\text{MeasurementRange}\\ âd>\text{MeasurementRange}\\ \end{array}
accelReadings=\left(\text{Resolution}\right)\left(\mathrm{round}\left(\frac{e}{\text{Resolution}}\right)\right)
a=\left(orientation\right){\left(angularVelocity\right)}^{T}
b={\left(\left[\begin{array}{ccc}1& \frac{{\mathrm{α}}_{2}}{100}& \frac{{\mathrm{α}}_{3}}{100}\\ \frac{{\mathrm{α}}_{1}}{100}& 1& \frac{{\mathrm{α}}_{3}}{100}\\ \frac{{\mathrm{α}}_{1}}{100}& \frac{{\mathrm{α}}_{2}}{100}& 1\end{array}\right]\left({a}^{T}\right)\right)}^{T}+\text{ConstantBias}
{\mathrm{β}}_{1}={h}_{1}*\left(w\right)\left(\text{BiasInstability}\right)
{H}_{1}\left(z\right)=\frac{1}{1â\frac{1}{2}{z}^{â1}}
{\mathrm{β}}_{2}=\left(w\right)\left(\sqrt{\frac{\text{SampleRate}}{2}}\right)\left(\text{NoiseDensity}\right)
{\mathrm{β}}_{3}={h}_{2}*\left(w\right)\left(\frac{\text{RandomWalk}}{\sqrt{\frac{\text{SampleRate}}{2}}}\right)
{H}_{2}\left(z\right)=\frac{1}{1â{z}^{â1}}
{\mathrm{Î}}_{e}=\left(\text{Temperature}â25\right)\left(\text{TemperatureBias}\right)
scaleFactorError=1+\left(\frac{\text{Temperature}â25}{100}\right)\left(\text{TemperatureScaleFactor}\right)
e=\left\{\begin{array}{c}\begin{array}{c}\text{MeasurementRange}\\ â\text{MeasurementRange}\end{array}\\ d\end{array}\begin{array}{c}\\ \\ \end{array}\begin{array}{c}\text{if}\\ \text{if}\\ \text{else}\end{array}\begin{array}{c}\\ \\ \end{array}\begin{array}{c}d>\text{MeasurementRange}\\ âd>\text{MeasurementRange}\\ \end{array}
gyroReadings=\left(\text{Resolution}\right)\left(\mathrm{round}\left(\frac{e}{\text{Resolution}}\right)\right)
a=\left(orientation\right){\left(totalAcc\right)}^{T}
b={\left(\left[\begin{array}{ccc}1& \frac{{\mathrm{α}}_{2}}{100}& \frac{{\mathrm{α}}_{3}}{100}\\ \frac{{\mathrm{α}}_{1}}{100}& 1& \frac{{\mathrm{α}}_{3}}{100}\\ \frac{{\mathrm{α}}_{1}}{100}& \frac{{\mathrm{α}}_{2}}{100}& 1\end{array}\right]\left({a}^{T}\right)\right)}^{T}+\text{ConstantBias}
{\mathrm{β}}_{1}={h}_{1}*\left(w\right)\left(\text{BiasInstability}\right)
{H}_{1}\left(z\right)=\frac{1}{1â\frac{1}{2}{z}^{â1}}
{\mathrm{β}}_{2}=\left(w\right)\left(\sqrt{\frac{\text{SampleRate}}{2}}\right)\left(\text{NoiseDensity}\right)
{\mathrm{β}}_{3}={h}_{2}*\left(w\right)\left(\frac{\text{RandomWalk}}{\sqrt{\frac{\text{SampleRate}}{2}}}\right)
{H}_{2}\left(z\right)=\frac{1}{1â{z}^{â1}}
{\mathrm{Î}}_{e}=\left(\text{Temperature}â25\right)\left(\text{TemperatureBias}\right)
scaleFactorError=1+\left(\frac{\text{Temperature}â25}{100}\right)\left(\text{TemperatureScaleFactor}\right)
e=\left\{\begin{array}{c}\begin{array}{c}\text{MeasurementRange}\\ â\text{MeasurementRange}\end{array}\\ d\end{array}\begin{array}{c}\\ \\ \end{array}\begin{array}{c}\text{if}\\ \text{if}\\ \text{else}\end{array}\begin{array}{c}\\ \\ \end{array}\begin{array}{c}d>\text{MeasurementRange}\\ âd>\text{MeasurementRange}\\ \end{array}
magReadings=\left(\text{Resolution}\right)\left(\mathrm{round}\left(\frac{e}{\text{Resolution}}\right)\right)
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Constructive Analysis for Least Squares Regression with Generalized K-Norm Regularization
Cheng Wang, Weilin Nie, "Constructive Analysis for Least Squares Regression with Generalized K-Norm Regularization", Abstract and Applied Analysis, vol. 2014, Article ID 458459, 7 pages, 2014. https://doi.org/10.1155/2014/458459
Cheng Wang1 and Weilin Nie1
We introduce a constructive approach for the least squares algorithms with generalized K-norm regularization. Different from the previous studies, a stepping-stone function is constructed with some adjustable parameters in error decomposition. It makes the analysis flexible and may be extended to other algorithms. Based on projection technique for sample error and spectral theorem for integral operator in regularization error, we finally derive a learning rate.
In learning theory, we are always given a sample set , which is drawn from a joint distribution on the sample space . Here, the input space is a compact metric space and for a regression problem. For a function obtained via some algorithm, a loss functional is defined to measure its performance on a sample point . In regression problem, least square loss is most widely used. Then, we can use the generalization error to evaluate over the whole sample space:
From [1], we know the goal function is , which is called the regression function, minimizing the generalization error. Since is always unknown in practice, we have to find another function close to based on the sample. The famous empirical risk minimization (ERM) algorithm is introduced in [2, 3]. To avoid overfitting, a penalty term related to is added into this algorithm, which is usually called regularization. While the squared -norm regularization term is extensively studied in [4], and so forth, in this paper, we consider a more general model: with some . In this algorithm, minimization is restricted to a hypothesis space which is a reproducing kernel Hilbert space (RKHS) on . The RKHS [5] is defined as with , associated with a Mercer Kernel which is continuous, symmetric, and positive definite. Since is a compact metric space, Kernel is bounded and we denote in the following.
Though uniform bounded assumption was abandoned in previous work [6], we still assume almost surely for some constant throughout this paper for simplicity, since our analysis can be extended to the unbounded situation by choosing some different probability inequality.
For the hypothesis space, a polynomial decay condition is given to control the capacity. To state this condition, we have to firstly recall covering number.
Definition 1. Let be a pseudometric space and . For , the covering number of the set with respect to is defined to be the minimal number of balls of radius whose union covers . That is, where .
When metric is chosen to be , that is, , it is the classical uniform covering number. It is widely used in [4, 6–8], and so forth, and more detailed analysis can be found in [9, 10]. More recent references [11–14] use -empirical covering number to obtain a sharper upper bound for the excess generalization error .
Definition 2. Denote for some . For a set of functions on and , with notation and , the -empirical covering number of is given by
Now, we can describe the capacity condition of the hypothesis space .
Definition 3. We say that has empirical polynomial complexity with exponent , , if there exists a constant such that where is the ball with radius in .
The integral operator defined by is also important in learning theory and has been studied in [15]. In [1], the authors claim that, for a Mercer Kernel , the associated is a compact operator with nonincreasing positive eigenvalue sequence . And the induced fractional operator is well defined, for any with orthogonal basis of . In the following, we will make use of this notion in our construction analysis.
We additionally introduce the projection operator on the space of measurable function :
The main result is stated as follows which will be proved in Section 6.
Theorem 4. Assume (3), (7) hold for sample distribution and hypothesis space . The regression function satisfies for some . is obtained from (2). Then, by choosing appropriate (explicit expression can be found in the proof) with confidence for any , we have for some constant not depending on or and
3. Error Decomposition
Various error decomposition methods motivate our research, especially [7, 12, 14, 16, 17]. A general idea of error decomposition is to transform the excess generalization error (see [1] for details) to two parts, which can be bounded by some concentration inequality and approximation analysis. In our setting, let be a function to be determined in ; it can be expressed as where The first and second terms and are called sample error which will be studied in Section 5, while the third term is regularization error (or approximation error) which is our main work in this paper.
It is known that can be freely chosen in which is close to in some sense and in previous work is always naturally chosen to be the one minimizing . However, we will encounter difficulties if the minimizer does not exist or the expression of the minimizer is not explicit. In this paper, we construct a special function in the form with some to handle this problem.
4. Regularization Error
It is the main contribution of this section to conduct error analysis for the regularization error. Regularization error, also called approximation error, has already been studied in [18]. However, we will analyze this part of error in a different viewpoint. From [1], we know is a compact self-adjoint and positive operator. By applying the spectral theorem for compact operators, we can bound a compact positive operator with its eigenvalues. Firstly, we have to introduce a useful lemma.
Lemma 5. Let and ; one has
Proof. By simply taking derivative of the right-hand side with respect to , we can find that it reaches its maximum when ; that is, Since , where , the lemma is proved.
Proposition 6. Assume and (15); there holds with some constant depending on and by choosing appropriate and .
Proof. Since , we will analyze the two terms, respectively. Noting that and , we have Recall that , combining with Lemma 5; there holds
For the term , we have the following inequality as : This means
To minimize the sum of upper bounds (22) and (24) is the same to maximize the power of . We can choose Then, This proves the result with
Remark 7. Another choice can also lead to the same result except for the constants.
Remark 8. In the case , our result turns to which is consistent with the classical one [4]. In fact, for a general of interest, the bound is better than since , while .
In [7], the authors construct a function based on the generalized Fourier expansion of and derive that with some constant for any . The rate is always much less than and cannot achieve when . On the other hand, our result is better than , while .
Compared with [19], we get the same rate of upper bound. There, the authors find a connection between and with different . However, their analysis needs an existent result, while our method does not.
5. Sample Error
There are a vast number of literatures studying the sample error. Here, we will follow the analysis of [11]. Firstly, we should introduce the Bernstein inequality [20]. Denote , , and for an integral function on .
Lemma 9. Assume for some constant almost surely. Then, for any .
Now, we can obtain the sample error bound involving .
Proposition 10. Assume (3), for any , with confidence ; there holds
Proof. Let we have . Note that and It is easy to see that Then, and By Bernstein inequality, holds with . Set the right-hand side to be ; we can solve and the following bound This proves the proposition.
For the sample error term , it is more difficult since it involves the function which varies, while the sample size is different. So, we need a concentration inequality for a set of functions as in [21]. By setting , the inequality becomes as follows.
Lemma 11. Let be a set of measurable functions on , and is constant such that each function satisfies and . If for some and , then there exists a constant depending only on such that, for any , with probability at least , there holds where .
The result will be used to estimate . We apply this lemma to the function set and have the following proposition.
Proposition 12. Let be defined as above with some satisfying , whose expression will be given in the next section. Assume (3) and (7) hold. Then, we have for some constant depending only on with confidence .
Proof. From definition, we know that where is an element of .
In the following, we verify the conditions for in Lemma 11. For any function , it holds On the other hand, for any depending, respectively, on , This means and
Now, we can see from Lemma 11 that, with confidence , there holds This proves the proposition.
6. Total Error
Combining the regularization and sample error bounds, we can prove the main result as follows.
Proof of Theorem 4. By substituting the regularization error and sample error (in the error decomposition formula) with obtained bounds in the above two sections, we have Note that and radius is always larger than ; the bound becomes where From , we have the bound for the radius: , and the above inequality is now To balance the two terms, we choose and the result is proved with constant
Remark 13. In [11], the authors also use -empirical covering number and derive an optimal rate . Compared with their classical rate for squared -norm regularization, our result also can achieve the best one , while tends to . Though when , that is, , our rate is worse than , we will get a better rate than when . Moreover, by the iteration technique [4], we can expect that the radius for is close to the upper bound of , which leads to a sharper learning rate . This is always better than for any .
This work is supported by NSF of China (Grant no. 11326096), Foundation for Distinguished Young Talents in Higher Education of Guangdong, China (no. 2013LYM 0089), Doctor Grants of Huizhou University (Grant no. C511.0206), Major Project of Chinese National Statistics Bureau (no. 2013LZ52), NSF of Guangdong Province in China (no. S2013010014601), “12.5” Planning Project of Common Construction Subject for Philosophical and Social Sciences in Guangdong (no. GD12XYJ18), Project of Science and Technology Innovation in Guangdong Education Department (no. 2013KJCX0175), and Planning Fund Project of Humanities and Social Science Research in Chinese Ministry of Education (no. 14YJAZH040).
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47A06 Linear relations (multivalued linear operators)
47A07 Forms (bilinear, sesquilinear, multilinear)
47A11 Local spectral properties
47A13 Several-variable operator theory (spectral, Fredholm, etc.)
47A16 Cyclic vectors, hypercyclic and chaotic operators
47A20 Dilations, extensions, compressions
47A25 Spectral sets
47A30 Norms (inequalities, more than one norm, etc.)
47A45 Canonical models for contractions and nonselfadjoint operators
47A48 Operator colligations (= nodes), vessels, linear systems, characteristic functions, realizations, etc.
47A52 Ill-posed problems, regularization
47A55 Perturbation theory
47A56 Functions whose values are linear operators (operator and matrix valued functions, etc., including analytic and meromorphic ones)
47A57 Operator methods in interpolation, moment and extension problems
47A58 Operator approximation theory
47A63 Operator inequalities
47A64 Operator means, shorted operators, etc.
47A67 Representation theory
47A68 Factorization theory (including Wiener-Hopf and spectral factorizations)
47A80 Tensor products of operators
A characterization for the spectral capacity of a finite system of operators
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A constructive proof of the composition rule for Taylor's functional calculus
Mats Andersson, Sebastian Sandberg (2000)
We give a new constructive proof of the composition rule for Taylor's functional calculus for commuting operators on a Banach space.
A functional calculus description of real interpolation spaces for sectorial operators
Markus Haase (2005)
For a holomorphic function ψ defined on a sector we give a condition implying the identity
{\left(X,\left({A}^{\alpha }\right)\right)}_{\theta ,p}=x\in X|{t}^{-\theta Re\alpha }\psi \left(tA\right)\in L{⁎}^{p}\left(\left(0,\infty \right);X\right)
where A is a sectorial operator on a Banach space X. This yields all common descriptions of the real interpolation spaces for sectorial operators and allows easy proofs of the moment inequalities and reiteration results for fractional powers.
A functional calculus for quotient bounded operators.
Stoian, Sorin Mirel (2006)
A generalization of parallel addition.
S.-L. Eriksson-Bique, H. Leutwiler (1989)
A maximum problem for operators
Vlastimil Pták (1984)
A multi-dimensional spectral theory in C*-algebras
F. Vasilescu (1982)
A note on a formula for the fractional powers of infinitesimal generators of semigroups
Celso Martinez, Miguel Sanz (1996)
If -A is the generator of an equibounded
{C}_{0}
-semigroup and 0 < Re α < m (m integer), its fractional power
{A}^{\alpha }
can be described in terms of the semigroup, through a formula that is only valid if a certain function
{K}_{\alpha ,m}
is nonzero. This paper is devoted to the study of the zeros of
{K}_{\alpha ,m}
A note on the spectral mapping theorem
Carlos Hernández-Garciadiego (1986)
A notion of analytic generator for groups of unbounded operators
We introduce a notion of analytic generator for groups of unbounded operators, on Banach modules, arising from Esterle’s quasimultiplier theory. Characterizations of analytic generators are given in terms of the existence of certain functional calculi. This extends recent results about C₀ groups of bounded operators. The theory is applicable to sectorial operators, representations of
{H}^{\infty }
, and integrated groups.
A product integral representation of the generalized inverse
Charles W. Groetsch (1975)
A spectral mapping theorem for functions with finite Dirichlet integral.
H. Bercovici, C. Foias, C. Pearcy (1986)
A Spectral Mapping Theorem for Local Multipliers.
Erich Marschall (1982)
John J. Buoni (1975)
A useful algebra for functional calculus
Mohammed Hemdaoui (2019)
We show that some unital complex commutative LF-algebra of
{𝒞}^{\left(\infty \right)}
ℕ
-tempered functions on
{ℝ}^{+}
(M. Hemdaoui, 2017) equipped with its natural convex vector bornology is useful for functional calculus.
Evgenii Pustylnik (2001)
The Fourier expansion in eigenfunctions of a positive operator is studied with the help of abstract functions of this operator. The rate of convergence is estimated in terms of its eigenvalues, especially for uniform and absolute convergence. Some particular results are obtained for elliptic operators and hyperbolic equations.
W. Mlak (1974)
An axiomatic approach to joint spectra I
{M}_{q}\left(\right)
-functional calculus for power-bounded operators on certain UMD spaces
Earl Berkson, T. A. Gillespie (2005)
For 1 ≤ q < ∞, let
{}_{q}\left(\right)
denote the Banach algebra consisting of the bounded complex-valued functions on the unit circle having uniformly bounded q-variation on the dyadic arcs. We describe a broad class ℐ of UMD spaces such that whenever X ∈ ℐ, the sequence space ℓ²(ℤ,X) admits the classes
{}_{q}\left(\right)
as Fourier multipliers, for an appropriate range of values of q > 1 (the range of q depending on X). This multiplier result expands the vector-valued Marcinkiewicz Multiplier Theorem in the direction q >...
An operator-theoretic approach to theorems of the Pick-Nevanlinna and carathéodory types.
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How to use MAI in the real life - Mai Finance - Tutorials
Mai Finance proposes loans with 0% interest rates and a single 0.5% repayment fee. This is particularly useful to leverage investments on DeFi on Polygon, but you can use it in the real world too.
Case study for MAI
DogeOfWallStreet is very bullish on Ethereum. With all the craze revolving around the crypto, partially driven by the NFT hype and bullish case for Eth2.0, he wants to make sure he can get as much ETH as possible before the price really goes to the moon.
In order to do so, our friend already invested in a small mining rig with 6 GPUs. He is mining ETH 24/7/365 and makes on average $40.00 a day with his 6 RTX 3080.
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after EIP1559, his profit decreased, but only by 15% which isn't too bad
a lot of ETH miners are selling their cards because they think mining is no longer profitable
price of GPUs slightly dropped, and it's now possible to buy some on specialized websites and even in store
"The Merge" that would move Ethereum from P-o-W (Proof of Work) to P-o-S (Proof of Stake) will not occur before Q1 2022 and the Ethereum developers are famous for adding delays to what they are supposed to deliver
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After 1 month of mining, the clever miner checks his portfolio on the ETH mainnet after the pool paid him. He has the equivalent of $1,340.00 worth of ETH (mining profit and ETH gained almost 12% in 4 weeks) , which should be way enough to buy the card ... but then, dilemma:
should he keep his ETH because the price is on the rise?
should he sell his ETH because the sooner he buys his GPU, the sooner he will mine more ETH?
That's about when he discovered Mai Finance and the power of 0% interest loans.
Getting the 0% interest / 0.5% repayment fee loan
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DogeOfWallStreet deposits his ETH on AAVE and receives amWETH. He goes now to Mai Finance and deposits the amWETH to get camWETH tokens that will be used as collateral in the vaults. At that stage, he still owns all his ETH, and Mai Finance allows him to gain 1.39% APY on his asset. Not much, but still appreciated.
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At this point, DogeOfWallStreet kept almost 100% of his ETH that is now generating 1.39% annually, AND he has 562.00 USD in his bank account. It's time to move to the next step!
DogeOfWallStreet buys the 1660 super he wanted, paid some taxes and shipping fees for a grand total of $554.35. After a few days, he receives his GPU and installs it in his rig.
The extra GPU is increasing his mining revenue by $2.14 daily. It will take him 270 days to fully repay the card, which is an acceptable ROI according to him. Indeed, his $554.35 new investment will possibly generate a revenue of $781.10 ($2.14 * 365) annually, which is an APR of 140.90%. This needs to be added to the $18.34 worth of camToken that the vault on Mai Finance is generating annually.
Seeing those numbers, DogeOfWallStreet will certainly transfer all the ETH he is mining to Polygon and repeat the loop every month to acquire more cards, and mine more ETH.
At some point, if mining ETH becomes unprofitable, he can always switch to another coin (ETC, RVN, ERG...), but his mining rig is currently an amazing investment.
Initial investment and real life debt repayment strategies
Thinking about it, DogeOfWallStreet is now sad that he didn't know Mai Finance existed before he bought his rig. Indeed, this is what he could have done
Get a bank loan of $20,000.00 at 5% rate and a 2 years amortization ($877.43 repayment / mo)
Convert the USD to ETH, bridge it to Polygon, use AAVE and Mai Finance to borrow 10,000 MAI
Convert the MAI to USD and buy his rig + initial GPUs
Generate an average of $1,300.00 every month to repay his bank loan and keep the rest as ETH that can be used to repay his debt on Mai Finance, or increase his debt on Mai and invest into something else
Alternatively, he could also split the $1,300.00 worth of ETH into
A portion of ETH sold directly, but less than the amount owed every month to the bank
A portion of ETH bridged to Polygon and Mai Finance where he can borrow MAI and repay the rest of the bank loan
Assuming you get Earnings and need to repay a Debt with Earnings > Debt. Assuming that Debt is also bigger than half of your Earnings. Let's assume Fiat to be the portion of your Earnings you need to use to repay your Debt, and Crypto the portion that will be sent to Mai Finance to borrow MAI. We get the following equations
Earnings = Fiat + Crypto
Debt = Fiat+\frac{Crypto}{CDR}
Which leads to the following results
Crypto=Earnings - \frac{Debt*CDR-Earnings}{CDR-1}
Fiat = \frac{Debt*CDR-Earnings}{CDR-1}
In our example with $1,300.00 earned that need to pay a $878.00 debt, the math is
Fiat=\frac{878*2-1300}{2-1}=\$456.00
Crypto=1300-\frac{878*2-1300}{2-1}=\$844.00
Hence we need to sell $456.00 worth of ETH and use Mai Finance to borrow $422.00 of MAI from $844.00 worth of ETH deposited in the vault (assuming a collateral to debt ratio of 200%). It's clear that, by doing so, we're keeping more of our earnings and have to sell less crypto. It's a way to increase your savings while still repaying your debt outside of the crypto world.
Of course, this is moving the debt that you need to repay to the bank to a debt you need to repay to Mai Finance. However, the fact that the bank has a high interest fee model, and that you need to repay monthly explains why it's better to move it to Mai Finance (0% interest, no time limit to repay).
If you want to use a more agressive Collateral to Debt ratio, you would be able to sell less crypto to fiat, and deposit/borrow more from Mai vaults. As an example, with a CDR of 175%, you would sell the equivalent of $316.00 of ETH and keep $984.00 in Mai Finance to borrow $562.00 of MAI, still repaying your $878.00 debt to the bank every month.
Note: All the math here is not including transaction fees.
This tutorial is absolutely not financial advice and should absolutely not be taken for granted. Mining Ethereum is a risky business. Hardware is still pretty expensive and there's no guarantee that mining will still be profitable in the near future. Investing in mining should be done at your own risk, please understand how mining works, and make sure the ROI is acceptable for you. Mining revenues are highly depending on the demand for transactions, and may become unprofitable before you could fully repay your initial investment.
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Inductor (Coupled) - SIMBA Documentation
Case of two coupled inductors
Case of three coupled inductors
Inductor (Coupled)
This component provides two or more mutually coupled inductors. For two or three coupled inductors, it is electrically equivalent with a non-ideal transfomer (see details in section below).
The size n
of the matrix corresponds to the number of windings. L_i
is the self inductance of the internal inductor and M_{i,j}
M_{i,j}
is the mutual inductance (see below).
The inductance matrix should not be singular in order to be invertible. In case of ideally coupled inductors, the inductance matrix can become singular and an ideal transformer component should be preferred to model these ones. The mutual inductances M_{ i,j}
M_{ i,j}
can also be linked with a coupling factor K_{i,j}
K_{i,j}
such as: M_{ i,j} = k_{ i,j} \sqrt{ L_i L_j}
M_{ i,j} = k_{ i,j} \sqrt{ L_i L_j}
The electric model of two coupled inductors is equivalent to a non-ideal transformer model:
L_m
: the magnetizing inductance at the primary side,
L_p
: the leakage inductance at the primary side,
L_s
: the leakage inductance at the secondary side,
m
: the transformer ratio.
The electric model of two inductors and its parameters are then:
L_m = \frac{M}{m}
L_m = \frac{M}{m}
L_p = L_1 - \frac{M}{m}
L_p = L_1 - \frac{M}{m}
L_s = L_2 - m.M
The electric model of three coupled inductors is also equivalent to a non-ideal transformer model:
L_m = \frac{M_{ 13} M_{ 12}}{ M_{23}}
L_m = \frac{M_{ 13} M_{ 12}}{ M_{23}}
l_1 = L_1 - \frac{M_{13} M_{12}}{M_{23}}
l_1 = L_1 - \frac{M_{13} M_{12}}{M_{23}}
l_2 = L_2 - \frac{M_{23} M_{12}}{M_{13}}
l_3 = L_3 - \frac{M_{23} M_{13}}{M_{12}}
NumberOfWindings The number of ideal inductors.
Inductance The inductance matrix (H). In order to model a magnetic coupling between the windings, a square matrix must be entered.
Iinit Vector of Initial inductor currents (A). The vector size must be equal to the number of windings.
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n
\left(n\ge 3\right)
n
\left(n\ge 3\right)
A note on
-groups for
n\ge 3
Ušan, Janez (1999)
Addition and correction to the paper "Diagonal algebras"
Algebras of multiplace functions.
B.M. Schein, V.S. Trohimenko (1979)
Algorithm for the complement of orthogonal operations
Iryna V. Fryz (2018)
G. B. Belyavskaya and G. L. Mullen showed the existence of a complement for a
k
-tuple of orthogonal
n
-ary operations, where
k<n
, to an
n
-ary operations. But they proposed no method for complementing. In this article, we give an algorithm for complementing a
k
n
-ary operations to an
n
-ary operations and an algorithm for complementing a
k
k
-ary operations to an
n
-ary operations. Also we find some...
Asymptotics for the number of
n
-quasigroups of order 4.
Potapov, V.N., Krotov, D.S. (2006)
Aus der Theorie der Polyquasigruppen
Frank Walther (1984)
Cancellative relations and matrices
Ladislav Bican, Aleš Drápal, Tomáš Kepka (1987)
Domenico Boccioni (1965)
Commutation of operations and its relationship with Menger and Mann superpositions
Fedir M. Sokhatsky (2004)
The article considers a problem from Trokhimenko paper [13] concerning the study of abstract properties of commutations of operations and their connection with the Menger and Mann superpositions. Namely, abstract characterizations of some classes of operation algebras, whose signature consists of arbitrary families of commutations of operations, Menger and Mann superpositions and their various connections are found. Some unsolved problems are given at the end of the article.
Congruences of
n
-group and of associated Hosszú-Gluskin algebras.
Construction of all homomorphisms of groupoids
k
n
Jacek Michalski (1981)
C-S-maximal superassociative systems
H. Länger (1978)
C-S-maximal superassociative systems. II.
Helmut Länger (1984)
Die Darstellung von partiellen
K
-stelligen Operationen
Jan Kastl, Tomáš Tichý (1976)
K
-abgeschlossenen Teilmengen der Halbgruppen
Jan Kastl (1976)
Free commutative idempotent abelian groupoids and quasigroups
Jaroslav Ježek, Tomáš Kepka (1976)
Free Objects in Primitive Varieties of N-groupoids
Georgi Čupona, Smile Markovski (1995)
Frobenius n-group algebras
Biljana Zeković (2002)
Frobenius algebras play an important role in the representation theory of finite groups. In the present work, we investigate the (quasi) Frobenius property of n-group algebras. Using the (quasi-) Frobenius property of ring, we can obtain some information about constructions of module category over this ring ([2], p. 66-67).
|
Angela had a rectangular piece of paper and then cut a rectangle out of a corner as shown at right. Find the area and perimeter of the resulting shape.
Perimeter is the distance around the exterior of a figure.
Area is the number of 'squares' needed to cover the interior of a figure.
Divide this shape into smaller rectangles to find the area.
Answer for Perimeter:
6+7+15+9+21+16=74
|
Bezout Matrix - Maple Help
Home : Support : Online Help : Programming : Maplets : Examples : Advanced : Bezout Matrix
display a graphical interface to the BezoutMatrix function
BezoutMatrix()
The BezoutMatrix() calling sequence displays a Maplet application that returns a Bezout matrix.
A definition of the Bezout matrix is given in the Maplet application.
The text inputs in the Maplet application are two polynomials, which determine the matrix.
By using the Evaluate result check box, control whether the Maplet application returns the Bezout matrix or the calling sequence required to calculate the Bezout matrix in the worksheet. The default behavior is to evaluate the result, that is, return the matrix.
If the polynomials entered have more than one variable, when OK is clicked, a second window is displayed. From the drop-down menu, select the variable with respect to which the Bezout matrix is calculated.
The BezoutMatrix sample Maplet worksheet demonstrates how to write a Maplet application that functions similarly to the Maplet application displayed by this routine.
\mathrm{with}\left(\mathrm{Maplets}[\mathrm{Examples}][\mathrm{LinearAlgebra}]\right):
\mathrm{BezoutMatrix}\left(\right)
BezoutMatrix Sample Maplet
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2012 Viscous Flow over Nonlinearly Stretching Sheet with Effects of Viscous Dissipation
Javad Alinejad, Sina Samarbakhsh
The flow and heat transfer characteristics of incompressible viscous flow over a nonlinearly stretching sheet with the presence of viscous dissipation is investigated numerically. The similarity transformation reduces the time-independent boundary layer equations for momentum and thermal energy into a set of coupled ordinary differential equations. The obtained equations, including nonlinear equation for the velocity field
f
and differential equation by variable coefficient for the temperature field
\theta
, are solved numerically by using the fourth order of Runge-Kutta integration scheme accompanied by shooting technique with Newton-Raphson iteration method. The effect of various values of Prandtl number, Eckert number and nonlinear stretching parameter are studied. The results presented graphically show some behaviors such as decrease in dimensionless temperature
\theta
due to increase in Pr number, and curve relocations are observed when heat dissipation is considered.
Javad Alinejad. Sina Samarbakhsh. "Viscous Flow over Nonlinearly Stretching Sheet with Effects of Viscous Dissipation." J. Appl. Math. 2012 (SI07) 1 - 10, 2012. https://doi.org/10.1155/2012/587834
Javad Alinejad, Sina Samarbakhsh "Viscous Flow over Nonlinearly Stretching Sheet with Effects of Viscous Dissipation," Journal of Applied Mathematics, J. Appl. Math. 2012(SI07), 1-10, (2012)
|
{C}^{*}
{W}^{*}
K
{C}^{*}
{W}^{*}
{C}^{*}
{C}^{*}
{C}^{*}
{C}^{*}
{C}^{*}
{C}^{*}
{C}^{*}
K
{C}^{*}
A classification theorem for nuclear purely infinite simple
{C}^{*}
A Note on a Chern character on the circle.
Nicolae Anghel (1994)
K
-theory of Fredholm modules and
KK
Kandelaki, Tamaz (2006)
Almost commuting unitary elements in purely infinite simple C*-algebras.
Huaxin Lin (1995)
Approximate unitary equivalence of homomorphisms from ... .
Huaxin Lin, N. Chr. Philips (1995)
Atiyahova-Singerova věta o indexu
Martin Markl (2013)
Bilipschitz invariance of the first transverse characteristic map
Michel Hilsum (2012)
Given a smooth S¹-foliated bundle, A. Connes has shown the existence of an additive morphism ϕ from the K-theory group of the foliation C*-algebra to the scalar field, which factorizes, via the assembly map, the Godbillon-Vey class, which is the first secondary characteristic class, of the classifying space. We prove the invariance of this map under a bilipschitz homeomorphism, extending a previous result for maps of class C¹ by H. Natsume.
{C}^{*}
-algebras associated to coverings of
k
-graphs.
Kumjian, Alex, Pask, David, Sims, Aidan (2008)
Carla Farsi, Neil Watling (1994)
Centrally ergodic one-parameter automorphism groups on semifinite injective von Neumann algebras.
Yasuyuki Kawahigashi (1989)
Classification des C*-algèbres purement infinies nucléaires
Claire Anantharaman-Delaroche (1995/1996)
Classification of certain infinite simple C*-algebras, II.
George A. Elliott, Mikael Rordam (1995)
Classifying C*-algebras via ordered, mod-p K-theory.
Marius Dadarlat, Terry A. Loring (1996)
Complex structure on the smooth dual of
GL\left(n\right)
Brodzki, Jacek, Plymen, Roger (2002)
Cyclic cohomology of crossed products by algebraic.
Victor Nistor (1993)
Cyclic Cohomology, Supersymmetry and KMS States The KMS States as Generalized Elliptic Operators
Daniel Kastler (1992)
In this article, we present two possible extensions of the classical theory of equivariant cohomology. The first, due to P. Baum, R. MacPherson and the author, is called the “delocalized theory". We attempt to present it in very concrete form for a circle action on a smooth manifold. The second is the cyclic homology of the crossed- product algebra of the algebra of smooth functions on a manifold, by the convolution algebra of smooth functions on a Lie group, when such Lie group act on the manifold....
Diffeotopy functors of ind-algebras and local cyclic cohomology.
Puschnigg, Michael (2003)
|
47B06 Riesz operators; eigenvalue distributions; approximation numbers,
s
47B10 Operators belonging to operator ideals (nuclear,
p
{}^{\infty }
-vectors and boundedness
Jan Stochel, F. H. Szafraniec (1997)
The following two questions as well as their relationship are studied: (i) Is a closed linear operator in a Banach space bounded if its
{}^{\infty }
-vectors coincide with analytic (or semianalytic) ones? (ii) When are the domains of two successive powers of the operator in question equal? The affirmative answer to the first question is established in case of paranormal operators. All these investigations are illustrated in the context of weighted shifts.
A characterization of formally symmetric unbounded operators.
Jocić, Danko (1989)
A Convergence Theorem for Selfadjoint Operators Applicable to Dirac Operators with Cutoff Potentials.
Rainer Wüst (1973)
A generalization related to Schrödinger operatorswith a singular potential.
A Geometrie Proof of the Spectral Theorem for Unbounded Self-Adjoint Operators.
Herbert Leinfelder (1979)
Anne-Sophie Bonnet-Bendhia, Karim Ramdani (2002)
This paper is devoted to the spectral analysis of a non elliptic operator
A
, deriving from the study of superconducting micro-strip lines. Once a sufficient condition for the self-adjointness of operator
A
has been derived, we determine its continuous spectrum. Then, we show that
A
is unbounded from below and that it has a sequence of negative eigenvalues tending to
-\infty
. Using the Min-Max principle, a characterization of its positive eigenvalues is given. Thanks to this characterization, some conditions...
This paper is devoted to the spectral analysis of a non elliptic operator A , deriving from the study of superconducting micro-strip lines. Once a sufficient condition for the self-adjointness of operator A has been derived, we determine its continuous spectrum. Then, we show that A is unbounded from below and that it has a sequence of negative eigenvalues tending to -∞. Using the Min-Max principle, a characterization of its positive eigenvalues is given. Thanks to this characterization, some...
C*
Palle E. T. Jørgensen (1984)
A study of an operator arising in the theory of circular plates
Leopold Herrmann (1988)
{L}_{0}:{D}_{{L}_{0}}\subset H\to H
{L}_{0}u=\frac{1}{r}\frac{d}{dr}\left\{r\frac{d}{dr}\left[\frac{1}{r}\frac{d}{dr}\left(r\frac{du}{dr}\right)\right]\right\}
{D}_{{L}_{0}}=\left\{u\in {C}^{4}\left(\left[0,R\right]\right),{u}^{\text{'}}\left(0\right)={u}^{\text{'}\text{'}\text{'}\text{'}}\left(0\right)=0,u\left(R\right)={u}^{\text{'}}\left(R\right)=0\right\}
H={L}_{2,r}\left(0,R\right)
is shown to be essentially self-adjoint, positive definite with a compact resolvent. The conditions on
{L}_{0}
(in fact, on a general symmetric operator) are given so as to justify the application of the Fourier method for solving the problems of the types
{L}_{0}u=g
{u}_{tt}+{L}_{0}u=g
A theorem on "localized" self-adjointness of Schrödinger operators with
{L}^{1}-
potentials.
Cycon, Hans L. (1982)
A topological characterization of the product of two closed operators
Bekkai Messirdi, Mohammed Hichem Mortad, Abdelhalim Azzouz, Ghouti Djellouli (2008)
The purpose of this work is to give a topological condition for the usual product of two closed operators acting in a Hilbert space to be closed.
A unitary invariant for pairs of self-adjoint operators.
Richard W. Carey (1976)
Absence of the singular continuous component in the spectrum of analytic direct integrals.
Filonov, N., Sobolev, A.V. (2004)
Algebraic sum of unbounded normal operators and the square root problem of Kato
Toka Diagana (2003)
Almost commuting self-adjoint matrices - a short proof of Huaxin Lin's theorem.
Mikael Rordam, Peter Friis (1996)
Sokolov, Maksim S. (2003)
An approximate method for determination of eigenvalues and eigenvectors of self-adjoint operators
An Example of Quantum Exponential Process.
Andreas Boukas (1991)
|
Comments on tag 00CV—Kerodon
Subsection 2.1.4: Nonunital Monoidal Functors (cite)
Comment #164 by DnlGrgk on March 27, 2019 at 12:11
In Definition 2.1.4.3, there seems to be a tilde above the morphism
\mu_{X,Y}
that is probably not supposed to be there. Also, at the end of the definition, it should be 'morphisms' instead of 'isomorphisms': "In this case, we will refer to the isomorphisms
\{\mu_{X,Y}\}_{X,Y} \in \mathcal{C}
as the tensor constraints of
F
\mu_{X,Y}
\{\mu_{X,Y}\}_{X,Y \in \mathcal{C}}
F
Comment #168 by Kerodon on March 28, 2019 at 00:06
Comment #240 by Peng DU on June 19, 2019 at 19:08
2 lines before Definition 2.1.4.3, “are suitable compatible with” should be "are suitably compatible with".
Comment #247 by Kerodon on June 22, 2019 at 16:23
In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 00CV. The letter 'O' is never used.
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Simulate Propagation Channels - MATLAB & Simulink - MathWorks Australia
Resource Grid and Transmission Waveform
Plot Signal Before and After Fading Channel
This example shows how to simulate propagation channels. It demonstrates how to generate cell-specific reference signals, map them onto a resource grid, perform OFDM modulation, and pass the result through a fading channel.
Specify the cell-wide settings as fields in the structure enb. Many of the functions used in this example require a subset of these fields.
antennaPort = 0;
Generate a subframe resource grid. To create the resource grid, call the lteDLResourceGrid function. This function creates an empty resource grid for one subframe.
Generate cell-specific reference symbols (CellRS) and map them onto the resource elements (REs) of a resource grid using linear indices.
cellRSsymbols = lteCellRS(enb,antennaPort);
cellRSindices = lteCellRSIndices(enb,antennaPort,{'1based'});
subframe(cellRSindices) = cellRSsymbols;
Perform OFDM modulation of the complex symbols in a subframe, subframe, using cell-wide settings structure enb.
The first output argument, txWaveform, contains the transmitted OFDM modulated symbols. The second output argument, info, is a structure that contains details about the modulation process. The field info.SamplingRate provides the sampling rate,
{R}_{sampling}
, of the time domain waveform:
{R}_{sampling}\phantom{\rule{0.5em}{0ex}}=\frac{30.72\phantom{\rule{0.2777777777777778em}{0ex}}MHz}{2048×{N}_{FFT}},
{N}_{FFT}
is the size of the OFDM inverse Fourier transform (IFT).
Construct the LTE multipath fading channel. First, set up the channel parameters by creating a structure, channel.
The sampling rate in the channel model, channel.SamplingRate, must be set to the info field of the SamplingRate returned by the lteOFDMModulate function.
Pass data through the LTE fading channel by calling the lteFadingChannel function. This function generates an LTE multipath fading channel, as specified in TS 36.101 (See reference [1]). The first input argument, txWaveform, is an array of LTE transmitted samples. Each row contains the waveform samples for each of the transmit antennas. These waveforms are filtered with the delay profiles as specified in the parameter structure, channel.
The output argument, rxWaveform, is the channel output signal matrix. Each row corresponds to the waveform at each of the receive antennas. Since you have defined one receive antenna, the number of rows in the rxWaveform matrix is one.
size(rxWaveform)
Display a spectrum analyzer with before-channel and after-channel waveforms. Use SpectralAverages = 10 to reduce noise in the plotted signals
title = 'Waveform Before and After Fading Channel';
saScope = dsp.SpectrumAnalyzer('SampleRate',info.SamplingRate,'ShowLegend',true,...
'SpectralAverages',10,'Title',title,'ChannelNames',{'Before','After'});
saScope([txWaveform,rxWaveform]);
|
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