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A geometrical mass and its extremal properties for metrics on S 2
15 July 2005 A geometrical mass and its extremal properties for metrics on
{S}^{2}
Green's function for the Laplacian on surfaces is considered, and a mass-like quantity is derived from a regularization of Green's function. A heuristic argument, inspired by the role of the positive mass theorem in the solution to the Yamabe problem, gives rise to a geometrical mass that is a smooth function on a compact surface without boundary. The geometrical mass is shown to be independent of the point on the sphere, and it is also a spectral invariant. Moreover, a connection to a sharp Sobolev-type inequality reveals that it is actually minimized at the standard round metric. The behavior of the geometrical mass on the sphere is markedly different from that on other surfaces.
Jean Steiner. "A geometrical mass and its extremal properties for metrics on
{S}^{2}
." Duke Math. J. 129 (1) 63 - 86, 15 July 2005. https://doi.org/10.1215/S0012-7094-04-12913-6
Jean Steiner "A geometrical mass and its extremal properties for metrics on
{S}^{2}
," Duke Mathematical Journal, Duke Math. J. 129(1), 63-86, (15 July 2005) |
Astrophysics/Quiz - Wikiversity
< Astrophysics
The ALMA correlator is one of the most powerful supercomputers in the world. Credit: ALMA (ESO/NAOJ/NRAO), S. Argandoña.
Astrophysics is a lecture and an article about the application of laboratory physics to astronomical phenomena. It is part of the astronomy course on the principles of radiation astronomy.
You are free to take this quiz based on astrophysics at any time.
1 True or False, A calculation of energy is not possible unless a mass is involved.
2 Which of the following is not an electron volt?
3 Yes or No, The force of gravity is a major portion of the strong nuclear force.
5 True or False, The average value of the radius of the Earth's orbit around the Sun is a displacement.
6 The science of physical and logical laws is called
7 True or False, The temperature of the cores of stars may be determined by the balance between the gravitational attraction and the gas pressure.
9 True or False, A unit vector is a direction with a magnitude of one.
11 True or False, Any space in the real universe is completely empty of microwaves.
13 True or False, When the magnetic poles of the Sun reverse during the solar cycle, there is a short time in which the polar diameter is greater than the equatorial diameter.
17 True or False, The force of gravity is the first astronomical source of the strong nuclear force.
20 A first astronomical source has?
a temporal distribution with at least one datum
a spectral distribution
a spatial distribution
a position or location
a geognosy
scientific or observational investigations
{\displaystyle a={\sqrt {a_{x}^{2}+a_{y}^{2}+a_{z}^{2}}}}
More questions aimed at astrophysics theory and laboratory efforts may be better.
Retrieved from "https://en.wikiversity.org/w/index.php?title=Astrophysics/Quiz&oldid=2365218" |
Frank Cce Everyday Science for Class 8 Science Chapter 3 - Synthetic Fibres And Plastics
Frank Cce Everyday Science Solutions for Class 8 Science Chapter 3 Synthetic Fibres And Plastics are provided here with simple step-by-step explanations. These solutions for Synthetic Fibres And Plastics are extremely popular among Class 8 students for Science Synthetic Fibres And Plastics Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Frank Cce Everyday Science Book of Class 8 Science Chapter 3 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Frank Cce Everyday Science Solutions. All Frank Cce Everyday Science Solutions for class Class 8 Science are prepared by experts and are 100% accurate.
Teflon is used for making non-stick pans.
This is used as a substitute for glass because it is transparent and unbreakable.
Plastic is a poor conductor of electricity.
Jute is a natural fibre obtained from plants.
Polythene is highly inflammable.
Thermocol is obtained from polystyrene.
Polythene tetraphthalate (PET) is a very popular form of polyester.
Nylon is the first synthetic fibre produced. It was made by an American chemist, W.H. Carothers in the year, 1935.
Non-stick utensils are coated with teflon.
(a) Thermosetting polymer
Melamine is an example of a thermosetting polymer.
1. PVC is the short form of polyvinyl chloride.
2. Teflon is used for making non-stick cookware.
3. Plastics are synthetic material.
4. A material that can be decomposed by natural processes is called a natural material.
5. Bakelite is a thermosetting plastic.
1. Rayon (e) Bandages
2. Nylon (d) Tooth brush
3. Melamine (a) Unbreakable kitchen ware
4. Teflon (b) Non-stick pans
5. Bakelite (c) Handle of saucepan
Polyester does not shrink on washing.
Plastics are sensitive towards heat and they melt very fast on heating.
Acrilon fibre is obtained from the acrylonitrile monomer by the process of polymerisation.
PVC is an insulator and is used as a covering for electrical wiring.
Following are two synthetic fibres:
ii) Polyethylene
Rayon is called artificial silk because it has a similar appearance, shine and texture as that of silk. It is prepared from cellulose of wood pulp.
The following are two properties of a nylon fibre:
i) It is very strong and fairly elastic.
ii) It absorbs very little water; hence, it dries up rapidly.
Perspex can be used as a substitute for glass.
Teflon is used to make non-stick cookware.
W.H. Carothers (1935) made the first synthetic fibre (nylon).
The following are two natural fibres:
i) Cotton
ii) Jute
Acrylic resembles wool in its properties.
PVC (polyvinyl chloride) is used to make raincoats.
The 4 Rs that should be practised to minimise environmental pollution are reduce, reuse, recycle and recover.
Polymers are a long chains of small units called monomers that are joined together by the process of polymerisation.
Example: rayon, nylon etc.
Polymerisation is the process through which a large number of monomers are linked together to form a polymer.
n Monomers
\stackrel{\mathrm{Polymerisation} }{\to }
The following are two types of plastic:
i) Thermoplastics: Examples are polythene, polystyrene and PVC.
ii) Thermosetting plastics: Examples are bakelite and melamine.
Synthetic fibres are superior to natural fibres in the following aspects:
1. Synthetic fibres are strong and cheaper, whereas natural fibres are not.
2. Synthetic fibres are crease-resistant, whereas natural fibres are not.
3. Synthetic fibres are not attacked by moths and moulds, whereas natural fibres can be easily destroyed by them.
4. Synthetic fibres are light and durable but natural fibres are not durable.
The following are the two uses of PVC (polyvinyl chloride):
(i) It is used as an insulating covering for electrical wires.
(ii) It is used in making hand bags, raincoats, floor-covering materials and suitcase covers.
The following are the advantages of synthetic fibres over natural fibres:
i) They are strong and cheaper.
ii) They are crease-resistant.
iii) They are not attacked by moths and moulds.
iv) They are hydrophobic; hence, they dry quickly.
v) They are light and durable.
The following are three qualities of polyesters:
i) Polyesters are strong and wrinkle-resistant.
ii) They absorb very little water; hence, they dry quickly.
iii) They are resistant to the action of chemicals.
Bakelite and melamine are two thermosetting plastics.
Non-biodegradable materials are those materials that are not degraded naturally or by the actions of microorganisms. Example: plastic
Synthetic clothes are not suitable in summer because air does not pass freely through such clothes. Also, they do not absorb sweat as clothes made up of natural fibres like cotton do.
Synthetic fibres are those fibres that are made artificially, using chemicals. The general process used to obtain synthetic fibres is polymerisation. It is a process in which small units (monomers) are joined together to form a polymer.
n \mathrm{Monomer}\stackrel{\mathrm{Polymerisation} }{\to } \mathrm{Polymer}
Examples of synthetic fibres: nylon, rayon and polyester
The following are four advantages of synthetic fibres:
1. Synthetic fibres are strong and cheaper. Example: Polyester and nylon are strong and comparatively cheaper than natural fibres such as cotton.
2. Synthetic fibres do not retain creases. Example: Both nylon and polyester are resistant to wrinkles.
3. Synthetic fibres are resistant to the attacks of moths and moulds. Example: Nylon and acrylic resist the attack of moths, moulds and other insects.
4. Synthetic fibres are light, durable and easy to maintain. Examples: Nylon, acrylic and polyester are light, durable, washable and easy to maintain.
Thermoplastics are plastic substances that can be melted by heating. They can be moulded again and again by heating and can be given different shapes. Examples of thermoplastics: polythene and polyvinyl chloride
Uses of thermoplastics:
i) Polythene is used for making waterproof material.
ii) PVC is used as an insulating covering for electrical wiring.
Thermosetting plastics are a type of plastic, which can be moulded only once. They do not soften on heating. Softening and moulding are irreversible characteristics of thermosetting plastics. Examples of thermosetting plastics: bakelite and melamine
Uses of thermosetting plastics:
i) Bakelite is used for making plugs, switches, telephone cases and other electrical fittings.
ii) Melamine is used for making unbreakable dinner ware and decoration pieces.
Plastics are considered environmental hazards because they are usually non-biodegradable, which means that they do not get decomposed by microbes into simpler compounds. Some of the problems caused by the excessive use of plastics are as follows:
1. Plastic bags block the drains and cause the overflowing of waste water. Blocked drains are a breeding place for mosquitoes.
2. Sometimes animals eat discarded plastic bags along with garbage, which leads to their death.
3. Recycled plastic used as containers for food causes health hazards.
4. Burning of plastic causes air pollution.
Following are the steps to reduce the dangers of plastic pollution:
1. The use of plastic bags and other items made of plastic should be reduced.
2. For shopping, we can carry our own cloth bags or jute bags.
3. Anything made of plastic should not be burnt.
5. Practise and follow the 4-R principle of reduce, reuse, recycle and recover to minimise environmental pollution.
Biodegradable materials are decomposed naturally in the environment. Non-biodegradable materials are not decomposed naturally.
They are present only for a small time in the environment. They are present for a longer time in the environment.
They do not cause much environmental hazard . They cause considerable environmental hazards.
Example: waste paper and wood crumbs. Example: plastic bags and cans. |
Synthetic Asset Model - THORChain Docs
How THORChain enables synthetic assets with no IL and with single asset exposure.
THORChain synthetics are unique in that they are 50% backed by their own asset, with the other 50% backing being provided by RUNE. This is achieved by using pool ownership to collateralise the synth, which ensures always-on liquidity and pricing.
Virtual Depths were initially applied to Synth Swaps (Minting and Redeeming). VirtualMultSynths multiplies the pool depth (R and A) before the swap is calculated. This was intended to to implement less slip and thus users paying less fees, but was disabled (VirtualMultSynths set to 1) after it was discovered that this would allow front-running.
Synthetic Assets are created by adding Rune to a pool (or swapping from an asset into RUNE, then adding that) for a synthetic asset of that pool. This is known as Minting.
synthAmount = (r * R * A)/(r + R)^2
R = Rune Depth (before)
A = Asset Depth (before)
The total Synth Supply is updated;
synthSupply += synthAmount
SynthUnits represent the RUNE collateral value that needs to be kept in the pool. They are computationally derived at any point, this ensures there is only enough at any time to represent the outstanding supply.
The ratio of Synth Units to Liquidity Pool units should be the same as the ratio of synth assets to the total value of the pool without the synth assets (since LP units are all pool units without the synth units).
S = Synth Supply
A = Asset Depth
L = Liquidity Units
U = Synth Units
\frac{U}{L} = \frac{S}{(2A-S)}
U = L * \frac{S}{(2A-S)}
SynthUnits are issued to cover the new liquidity minted, but held by the pool (not owned by anyone). PoolUnits are therefore the sum of liquidityUnits + synthUnits.
Synthetic Assets Minting is capped to 33% of the assets in the pool or about 16.5% of the pool depth. Minting assets increases the total RUNE pooled amount, which cannot be greater than the total bonded.
Synthetic Assets are burned by swapping the Synth for Rune. This is known as Burning or Redeeming. A Synthetic Asset can be redeemed to Rune at any time (or swapped to Rune then to an asset).
Synth Assets hold the value to normal assets and can be redeemed 1:1. Thus swapping 1 BTC for Rune then minting Synthetic BTC will give 1 Synthetic BTC. This can later be redeemed to Rune and swapped for 1 BTC, excluding fees.
runeAmount = (s * A * R)/(s + A)^2
s = Synths to Redeem
Pool Units, Synth Supply and Rune Pool Depth are correspondingly decremented.
Synth Swaps can be done as follows:
Layer 1 to Synth: L1 in, Rune moved over to the pool, synth MINTED
Synth to Layer 1: Synth REDEEMED, RUNE moved to next pool, Layer 1 swapped out
Synth to Synth: Synth REDEEMED, RUNE moved over to the pool, synth MINTED
To specify the destination asset is synth, simply provide a THOR address. If it is Layer 1, then provide a layer 1 address, e.g. a BTC address.
Synth holders do not experience any gain or any loss caused by price changes when minting / redeeming a synth. They do, however, pay a slip-based fee on entry or exit and tx fees.
The dynamic synth unit accounting is to ensure that gain or loss caused by price divergence in the synths is quickly realised to LPs. As Liquidity Providers have Impermanent Loss Protection, as long as they stay for longer than 100 days, the Protocol Reserve is taking on the price risk.
Due to the collateralisation method, THORChain Synthetic Assets are impervious to impermanent loss and offer single asset exposure.
If an active pool that minted synths becomes staged, then swaps are disabled. However, synth holders can always redeem for RUNE, or the underlying asset, by specifying that on the way out.
Alternate Synth Derivation |
Price Elasticity of Demand Calculator | Dash Calculator
Use our simple price elasticity of demand calculator to determine the elasticity of demand given the initial and final quantities demanded and price.
Let's calculate the price elasticity of demand using the Midpoint Method.
\textrm{Price elasticity of demand} = \frac{\textrm{Percentage change in quantity}}{\textrm{Percentage change in price}}
\textrm{Percentage change in quantity} = \frac{\textrm{Final quantity} - \textrm{Initial quantity}}{ (\textrm{Final quantity} + \textrm{Initial quantity}) \div 2} \times 100
\textrm{Percentage change in quantity} = \frac{20 - 25}{ (20 + 25) \div 2} \times 100 = -22.22
\textrm{Percentage change in price} = \frac{\textrm{Final price} - \textrm{Initial price}}{ (\textrm{Final price} + \textrm{Initial price}) \div 2} \times 100
\textrm{Percentage change in price} = \frac{100 - 50}{ (100 + 50) \div 2} \times 100 = 66.67
\textrm{Price elasticity of demand} = \frac{\textrm{Percentage change in quantity}}{\textrm{Percentage change in price}}
\textrm{Price elasticity of demand} = \frac{-22.22}{66.67} = -0.33
How to calculate the price elasticity of demand
The formula for the price elasticity of demand is:
The price elasticity of demand measures how responsive is the quantity demanded of a good or service to a change in its price.
Price elasticity of demand : To what extent does the quantity demanded of a good changes in response to a change in its price?
For example, a coffee shop knows that if it raises the price of an ice coffee then the quantity of coffee sold will decrease. How much will it decrease? This depends on how much the quantity demanded responds to a change in price. The price elasticity of ice coffee determines this.
The price elasticity of demand formula compares the percentage change in quantity demanded to the percentage change in price:
You might notice that the calculation of percentage change does not use the initial price or quantity as the base, but rather the average of the initial and finance price or quantity. This is known as the Midpoint Method and its purpose is to adjust for rises and falls in prices.
We will learn more about the Midpoint Method below.
Let’s say that a coffee shop raises the price of an ice coffee from $4 to $6. The percentage change is calculated by the following formula.
In this case, the percentage change would be 50%.
Now, imagine that the coffee shop lowers the price of an ice coffee from $6 to $4. The percentage change would be:
The percentage change is −33%.
The price change in both scenarios is the same —$2, but the percentage change is different because the initial price is different in each scenario. We want a percentage change that is not dependent on the direction of the price change, which brings us to the Midpoint Method.
The Midpoint Method divides the change in price by the average price rather than the initial price. The average price is in the middle of the initial and final price, which is why it is called the Midpoint Method.
The formula for the percentage change in price based on the Midpoint Method is:
Using the Midpoint Method, the percentage change in price for a given dollar change in price will be the same regardless of direction.
In the previous examples where the ice coffee price change was $2 and the initial and final prices were $4 and $6, the percentage change using the midpoint method is:
The percentage change when the price of an ice coffee rises from $4 to $6 is:
The percentage change when the price of an ice coffee falls from $6 to $4 is:
The average price is the same whether the price rises or falls and so the magnitude of the percentage change for the same dollar change is the same. Here, it is 40%.
The Midpoint Method is used to calculate both the percentage change in price and the percentage change in quantity demanded.
Let’s say that when the price of an ice coffee falls from $6 to $4, the quantity demanded increases from 50 cups a day to 175 cups a day. The percentage change in quantity demanded using the Midpoint Method is:
When we look at demand elasticity, we use the absolute value, or the magnitude of the calculate price elasticity of demand. This allows us to look at how responsive quantity is to a change in price.
For example, the percentage change in quantity can be greater than, equal to, or less than the percentage change in price.
Elastic demand: The percentage change in quantity demanded is greater than the percentage change in price. What does this mean? This means that when there is a small change in price, there is a big change in the quantity demanded. Let’s say the item of a cupcake dropped by 5%. If demand for cupcakes is elastic, then the quantity demanded will increase by more than 5%. People will be buying a lot more cupcakes.
Unit elastic demand: The percentage change in quantity demanded is equal to the percentage change in price.
Inelastic demand: The percentage change in quantity demanded is less than the percentage change in price.
This means that when there is a big change in price, there is a small change in the quantity demanded. An example might be the cost of milk. When the price of milk doubles, you might decrease your consumption by a little, but not significantly since milk might be a staple in your home.
To determine if demand is elastic or inelastic, take the absolute value of the calculated price elasticity of demand and use the following table.
Price elasticity of demand (absolute value)
Greater than 1 Demand is elastic
Equal to 1 Demand is unit elastic
Less than 1 Demand is inelastic
What determines whether the demand for a good is elastic or inelastic? There are several factors that influence demand elasticity.
Demand for a good will be elastic if there is an abundance of substitutes available. For example, if the price of an ice coffee goes up at Starbucks, but does not at Dunkin’ Donuts, you might just go to Dunkin’ Donuts for your coffee fix.
If substitutes are very hard to find, then the demand for a good will be inelastic. Water, electricity, accessing the internet, and gasoline are inelastic goods because there are few alternatives.
The amount of time to find a substitute also affects the demand elasticity. If a substitute can be found easily and quickly, then demand will be elastic. If it takes a long time to find a substitute and you need the good immediately, then demand will be inelastic.
Soft drinks have elastic demand because there is an abundance of substitutes that can be found quickly — usually in the same aisle at the supermarket.
Necessities, such as food, water, and housing, usually have inelastic demand, whereas luxuries, items that you want but don’t need to have, tend to have more substitutes and generally has elastic demand.
Percentage of income spend on the good
Goods that cost a small fraction of your income tend to be inelastic. Examples are pens and salt. The reason is because if the price of salt doubles, the impact on your income is very little since the item cost very little to begin with.
Relatedly, high priced goods tend to have elastic demand and low priced goods generally have inelastic demand.
To summarize the factors influencing demand elasticity, here are the cases when demand is more elastic and when demand is less elastic.
Demand is elastic
Lots of substitutes available Few substitutes available
High priced goods Low priced goods
High percentage of income is spent on good Low percentage of income is spent on good
Luxury items Necessities
Examples of price elasticity in real life
Now that we know how price elasticity is calculated in theory, what is the price elasticity of some real life goods? A group of scholars from the Mackinac Center for Public Policy collected data to determine the elasticity of a number of goods.
Estimated elasticity of demand
Toothpicks 0.1
Airline travel (short-term) 0.1
Legal services 0.4
Doctor visits 0.6
Private school 1.1
Going out to a restaurant 2.3
International travel 4.0
Fresh tomatoes 4.6
Price Elasticity of Demand and Marketing: Mastering elasticity to market strategically by John Story Ph.D.
Price Elasticity of Demand by Patrick L. Anderson, Richard D. McLellan, Joseph P. Overton, and Dr. Gary L. Wolfram |
Hypostatic abstraction - Wikiversity
Hypostatic abstraction is a formal operation that takes an element of information, as expressed in a proposition
{\displaystyle X~{\text{is}}~Y,\!}
and conceives its information to consist in the relation between that subject and another subject, as expressed in the proposition
{\displaystyle X~{\text{has}}~Y\!{\text{-ness}}.\!}
The existence of the abstract subject
{\displaystyle Y\!{\text{-ness}}\!}
consists solely in the truth of those propositions that contain the concrete predicate
{\displaystyle Y.\!}
Hypostatic abstraction is known by many names, for example, hypostasis, objectification, reification, and subjectal abstraction. The object of discussion or thought thus introduced is termed a hypostatic object.
The above definition is adapted from the one given by introduced Charles Sanders Peirce (CP 4.235, “The Simplest Mathematics” (1902), in Collected Papers, CP 4.227–323).
The way that Peirce describes it, the main thing about the formal operation of hypostatic abstraction, insofar as it can be observed to operate on formal linguistic expressions, is that it converts an adjective or some part of a predicate into an extra subject, upping the arity, also called the adicity, of the main predicate in the process.
For example, a typical case of hypostatic abstraction occurs in the transformation from “honey is sweet” to “honey possesses sweetness”, which transformation can be viewed in the following variety of ways:
The grammatical trace of this hypostatic transformation tells of a process that abstracts the adjective “sweet” from the main predicate “is sweet”, thus arriving at a new, increased-arity predicate “possesses”, and as a by-product of the reaction, as it were, precipitating out the substantive “sweetness” as a new second subject of the new predicate, “possesses”.
Hypostatic Abstraction → ThoughtMesh
J. Jay Zeman, Peirce on Abstraction
Hypostatic Abstraction @ InterSciWiki
Hypostatic Abstraction @ MyWikiBiz
Hypostatic Abstraction @ Subject Wikis
Hypostatic Abstraction @ Wikiversity
Hypostatic Abstraction @ Wikiversity Beta
Hypostatic Abstraction, InterSciWiki
Hypostatic Abstraction, MyWikiBiz
Hypostatic Abstraction, PlanetMath
Hypostatic Abstraction, ThoughtMesh
Hypostatic Abstraction, Wikinfo
Hypostatic Abstraction, Wikiversity
Hypostatic Abstraction, Wikiversity Beta
Hypostatic Abstraction, Wikipedia
Retrieved from "https://beta.wikiversity.org/w/index.php?title=Hypostatic_abstraction&oldid=179365" |
Numerical Survey on Hyperbolicity of the Homogenized Equation for Van Der Waals Gas in Eulerian Coordinates | EMS Press
Numerical Survey on Hyperbolicity of the Homogenized Equation for Van Der Waals Gas in Eulerian Coordinates
University of Kansas, Lawrence, United States
We describe numerical methods for the location of the zero set of the periodic Evans function
D(\xi, \lambda)
\lambda, \xi
sufficiently small, or equivalently the spectrum of a linearized operator
L
with periodic coefficients through the homogenization. We demonstrate these methods for an example system, van der Waals gas in eulerian coordinates. We observe the hyperbolicity of the system. The hyperbolicity is necessary to show asymptotic behavior of a multi-dimensional single periodic wave of systems of conservation laws with viscosity under small perturbation.
Myunghyun Oh, Numerical Survey on Hyperbolicity of the Homogenized Equation for Van Der Waals Gas in Eulerian Coordinates. Z. Anal. Anwend. 27 (2008), no. 3, pp. 315–322 |
m (moved Stars Lecture 9 to Fusion: changed to topic)
{\displaystyle \left(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V\right)\psi =E\psi .\,\!}
{\displaystyle E={\frac {k^{2}\hbar ^{2}}{2m}}.\,\!}
{\displaystyle k}
{\displaystyle V=V_{0}}
{\displaystyle E-V_{0}={\frac {k^{2}\hbar ^{2}}{2m}}.\,\!}
{\displaystyle \sigma (E)={\frac {S(E)}{E}}e^{-(E_{G}/E)^{1/2}}.\,\!}
{\displaystyle S(E)}
{\displaystyle E}
{\displaystyle E_{G}}
{\displaystyle E_{G}=(1{\rm {\;MeV}})Z_{1}^{2}Z_{2}^{2}{\frac {m_{r}}{m_{p}}}.\,\!}
{\displaystyle n_{1}}
{\displaystyle n_{2}}
{\displaystyle \ell _{2}={\frac {1}{n_{1}\sigma }}\,\!}
{\displaystyle \tau _{2}={\frac {1}{n_{1}\sigma v}}.\,\!}
{\displaystyle r_{12}={\frac {n_{2}}{\tau _{2}}}=n_{1}n_{2}\sigma v.\,\!}
{\displaystyle r_{12}=n_{1}n_{2}<\sigma (E)v>.\,\!}
{\displaystyle <\sigma (E)v>}
{\displaystyle <\sigma (E)v>=\int d^{3}v\;prob(v)\sigma (E)v.\,\!}
{\displaystyle r_{12}=n_{1}n_{2}\int d^{3}v\sigma (E)v\left({\frac {m_{r}}{2\pi kT}}\right)^{3/2}e^{-{\frac {{\frac {1}{2}}m_{r}v^{2}}{kT}}}.\,\!}
{\displaystyle E={\frac {1}{2}}m_{r}v^{2},\,\!}
{\displaystyle dE=m_{r}vdv,\,\!}
{\displaystyle d^{3}v=4\pi v^{2}dv=4\pi {\frac {v^{2}}{v}}{\frac {dE}{m_{r}}},\,\!}
{\displaystyle vd^{3}v={\frac {8\pi E}{m_{r}}}{\frac {dE}{m_{r}}}.\,\!}
{\displaystyle <\sigma (E)v>=\left({\frac {2}{kT}}\right)^{3/2}{\frac {1}{\sqrt {\pi m_{r}}}}\int dEE\sigma (E)e^{-E/kT}.\,\!}
{\displaystyle \sigma (E)}
{\displaystyle <\sigma (E)v>=\left({\frac {2}{kT}}\right)^{3/2}{\frac {1}{\sqrt {\pi m_{r}}}}\int dES(E)e^{-(E_{G}/E)^{1/2}\;-\;E/kT}.\,\!}
{\displaystyle S(E)}
{\displaystyle <\sigma (E)v>=\left({\frac {2}{kT}}\right)^{3/2}{\frac {1}{\sqrt {\pi m_{r}}}}S(E)I.\,\!}
{\displaystyle I=\int _{0}^{\infty }e^{-(E_{G}/E)^{1/2}\;-\;E/kT}dE.\,\!}
{\displaystyle E_{0}}
{\displaystyle E_{0}}
{\displaystyle f(E)}
{\displaystyle {\frac {df}{dE}}=0={\frac {1}{kT}}-{\frac {E_{G}^{1/2}}{2E^{3/2}}}.\,\!}
{\displaystyle E_{0}=\left({\frac {1}{2}}E_{G}^{1/2}kT\right)^{2/3}.\,\!}
{\displaystyle E_{G}}
{\displaystyle E_{0}=(5.7\;{\rm {keV}})Z_{1}^{2/3}Z_{2}^{2/3}T_{7}^{2/3}\left({\frac {m_{r}}{m_{p}}}\right)^{1/3}.\,\!}
{\displaystyle E_{G}}
{\displaystyle kT}
{\displaystyle f(E)=f(E_{0})+{\frac {1}{2}}(E-E_{0})^{2}+f^{''}(E_{0}),\,\!}
{\displaystyle f^{''}(E_{0})={\frac {3E_{G}^{1/2}}{4E_{0}^{5/2}}}.\,\!}
{\displaystyle I}
{\displaystyle I={\frac {e^{-f(E_{0})}{\sqrt {2\pi }}}{\sqrt {f^{''}(E_{0}}}}.\,\!}
{\displaystyle <\sigma (E)v>=2.6S(E_{0}){\frac {E_{G}^{1/6}}{(kT)^{2/3}{\sqrt {m_{r}}}}}e^{-3(E_{G}/4kT)^{1/3}}.\,\!}
{\displaystyle \epsilon }
{\displaystyle L=\int \epsilon dM_{r}=\int \epsilon 4\pi r^{2}\rho dr.\,\!}
{\displaystyle {\frac {dL_{r}}{dr}}=4\pi r^{2}\rho \epsilon .\,\!}
{\displaystyle Q}
{\displaystyle r_{12}}
{\displaystyle \epsilon }
{\displaystyle \epsilon _{12}={\frac {r_{12}Q}{\rho }}.\,\!}
{\displaystyle n_{1}={\frac {X_{1}\rho }{m_{1}}}.\,\!}
{\displaystyle X_{1}}
{\displaystyle \epsilon _{12}={\frac {2.6QS(E_{0})X_{1}X_{2}}{m_{1}m_{2}{\sqrt {m_{r}}}(kT)^{2/3}}}\rho E_{G}^{1/6}e^{-3(E_{G}/4kT)^{1/3}}.\,\!}
{\displaystyle \epsilon \propto \rho ^{\alpha }T^{\beta }.\,\!}
{\displaystyle \alpha }
{\displaystyle \beta }
{\displaystyle \alpha =1}
{\displaystyle \beta }
{\displaystyle \epsilon }
{\displaystyle \beta ={\frac {d\ln \epsilon }{d\ln T}}.\,\!}
{\displaystyle \epsilon }
{\displaystyle \beta =-{\frac {2}{3}}+\left({\frac {E_{G}}{4kT}}\right)^{1/3}.\,\!}
{\displaystyle \beta \approx 4.3}
{\displaystyle \epsilon _{pp}\propto \rho T^{4.3}\,\!}
{\displaystyle 10^{7}}
{\displaystyle T_{c}\sim 10^{7}}
{\displaystyle \rho \sim 1}
{\displaystyle ^{-3}}
{\displaystyle S(E)}
{\displaystyle Q}
{\displaystyle \epsilon }
{\displaystyle \epsilon \sim 10^{20}{\rm {\;erg/s/g}}.\,\!}
{\displaystyle L=\int dM_{r}\epsilon \sim \epsilon M_{\odot }.\,\!}
{\displaystyle L\sim 10^{54}{\rm {\;erg/s}}\sim 10^{20}L_{\odot }.\,\!}
{\displaystyle 10^{20}}
{\displaystyle E_{G}}
{\displaystyle 4p\rightarrow {}^{4}{\rm {He}}+{\rm {energy}}.\,\!}
{\displaystyle p+p\rightarrow {}^{2}{\rm {H}}+e^{+}+\nu _{e}.\,\!}
{\displaystyle S(keV)\approx 3.78\times 10^{-22}}
{\displaystyle {}^{2}{\rm {H}}+p\rightarrow {}^{3}{\rm {He}}+\gamma ,\,\!}
{\displaystyle \times 10^{-4}}
{\displaystyle {}^{3}{\rm {He}}+{}^{3}{\rm {He}}\rightarrow {}^{4}{\rm {He}}+2p,\,\!}
{\displaystyle \epsilon _{cycle}=r_{p-p\;step}Q_{cycle}/\rho .\,\!}
{\displaystyle \epsilon _{pp}\propto \rho T^{-2/3}e^{-15.7T_{7}^{-1/3}}.\,\!}
{\displaystyle \epsilon _{pp}=(5\times 10^{5}){\frac {\rho X^{2}}{T^{2/3}}}e^{-15.7T_{7}^{-1/3}}{\rm {erg/s/g}}.\,\!}
{\displaystyle L=\int \epsilon dM\sim \epsilon (center)M_{\odot },\,\!}
{\displaystyle L_{\odot }\sim 10^{7}{\frac {M_{\odot }}{T_{7}^{2/3}}}e^{-15.7T_{7}^{-1/3}},\,\!}
{\displaystyle T_{c}\approx 10^{7}K.\,\!} |
Associated_bundle Knowpia
In mathematics, the theory of fiber bundles with a structure group
{\displaystyle G}
(a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from
{\displaystyle F_{1}}
{\displaystyle F_{2}}
, which are both topological spaces with a group action of
{\displaystyle G}
. For a fiber bundle F with structure group G, the transition functions of the fiber (i.e., the cocycle) in an overlap of two coordinate systems Uα and Uβ are given as a G-valued function gαβ on Uα∩Uβ. One may then construct a fiber bundle F′ as a new fiber bundle having the same transition functions, but possibly a different fiber.
A simple case comes with the Möbius strip, for which
{\displaystyle G}
is the cyclic group of order 2,
{\displaystyle \mathbb {Z} _{2}}
. We can take as
{\displaystyle F}
any of: the real number line
{\displaystyle \mathbb {R} }
{\displaystyle [-1,\ 1]}
, the real number line less the point 0, or the two-point set
{\displaystyle \{-1,\ 1\}}
{\displaystyle G}
on these (the non-identity element acting as
{\displaystyle x\ \rightarrow \ -x}
in each case) is comparable, in an intuitive sense. We could say that more formally in terms of gluing two rectangles
{\displaystyle [-1,\ 1]\times I}
{\displaystyle [-1,\ 1]\times J}
together: what we really need is the data to identify
{\displaystyle [-1,\ 1]}
to itself directly at one end, and with the twist over at the other end. This data can be written down as a patching function, with values in G. The associated bundle construction is just the observation that this data does just as well for
{\displaystyle \{-1,\ 1\}}
{\displaystyle [-1,\ 1]}
In general it is enough to explain the transition from a bundle with fiber
{\displaystyle F}
, on which
{\displaystyle G}
acts, to the associated principal bundle (namely the bundle where the fiber is
{\displaystyle G}
, considered to act by translation on itself). For then we can go from
{\displaystyle F_{1}}
{\displaystyle F_{2}}
, via the principal bundle. Details in terms of data for an open covering are given as a case of descent.
Associated bundles in generalEdit
φi : π−1(Ui) → Ui × F
ψij(u,f) := φi o φj−1(u,f) = (u,gij(u)f) for each (u,f) ∈ (Ui ∩ Uj) × F.
ψ′ij(u,f′) = (u, gij(u) f′) for (u,f′) ∈(Ui ∩ Uj) × F′
Principal bundle associated with a fibre bundleEdit
Fiber bundle associated with a principal bundleEdit
Define a right action of G on P × F via[3][4]
{\displaystyle (p,f)\cdot g=(p\cdot g,\rho (g^{-1})f)\,.}
{\displaystyle [p\cdot g,f]=[p,\rho (g)f]{\mbox{ for all }}g\in G.}
Reduction of the structure groupEdit
The companion concept to associated bundles is the reduction of the structure group of a
{\displaystyle G}
{\displaystyle B}
. We ask whether there is an
{\displaystyle H}
{\displaystyle C}
, such that the associated
{\displaystyle G}
-bundle is
{\displaystyle B}
, up to isomorphism. More concretely, this asks whether the transition data for
{\displaystyle B}
can consistently be written with values in
{\displaystyle H}
. In other words, we ask to identify the image of the associated bundle mapping (which is actually a functor).
Examples of reductionEdit
^ All of these constructions are due to Ehresmann (1941-3). Attributed by Steenrod (1951) page 36
^ Effectiveness is a common requirement for fibre bundles; see Steenrod (1951). In particular, this condition is necessary to ensure the existence and uniqueness of the principal bundle associated with E.
^ Husemoller, Dale (1994), p. 45.
^ Sharpe, R. W. (1997), p. 37.
Steenrod, Norman (1951). The Topology of Fibre Bundles. Princeton: Princeton University Press. ISBN 0-691-00548-6.
Husemoller, Dale (1994). Fibre Bundles (Third ed.). New York: Springer. ISBN 978-0-387-94087-8. |
To Asa Gray 12 October [1856]1
My dear Dr. Gray
I received yesterday your most kind letter of the 23d. 2 & your “Statistics” & two days previously another copy.3 I thank you cordially for them. Botanists write, of course, for Botanists; but as far as the opinion of an “outsider” goes, I think your paper admirable. I have read carefully a good many papers & works on Geograph. Distribution, & I know of only one Essay (viz Hooker’s N. Zealand)4 that makes any approach to the clearness with which your paper makes a non-Botanist appreciate the character of the Flora of a country. It is wonderfully condensed (what labour it must have required!): you ask whether such details are worth giving, in my opinion there is literally not one word too much.
I thank you sincerely for the information about “social” & “varying plants”; & likewise for giving me some idea about the proportion (ie
\frac{1}{4}
) of European plants, which you think do not range to the extreme north: this proportion is very much greater than I had anticipated from what I picked up in conversation &c.—5
To return to your Statistics: I daresay you will give how many genera (& orders) your 260 introduced plants belong to: I see they include 113 genera non indigenous: as you have probably a list of the introduced plants, would it be asking too great a favour to send me per Hooker or otherwise just the total number of genera & orders to which the introduced plants belong:6 I am much interested on this, & have found De Candolles remarks on this subject very instructive.7
Nothing has surprised me more than the greater generic & specific affinity with E. Asia than with W. America. Can you tell me (& I will promise to inflict no other question) whether climate explains this greater affinity? or it is one of the many utterly inexplicable problems in Bot. Geography? Is E. Asia nearly as well known as West America? so that does the state of knowledge allow a pretty fair comparison?8
I presume it would be impossible, but I think it would make in one point your tables of generic ranges more clear (admirably clear as they seem to me) if you could show, even roughly, what proportion of the genera in Common to Europe (ie nearly half) are very general or mundance rangers; as your results now stand at the first glance the affinity seems so very strong to Europe, owing, as I presume to the (nearly) half of the genera including very many genera common to the world or large portions of it. Europe is thus unfairly exalted.— Is this not so? If we had the number of genera strictly or nearly strictly European, one could compare better with Asia & southern America &c. But I daresay this is a Utopian wish owing to difficulty of saying what genera to call mundane. Nor have I my ideas at all clear on subject, & I have expressed them even less clearly than I have them.
I am so very glad that you intend to work out N. range of the 321 Europæan species; for it seems to me the by far most important element in their distribution.—
And I am equally glad that you intend to work out range of species in regard to size of genera ie number of species in genus.— I have been attempting to do this in a very few cases; but it is folly for any one but Botanist to attempt it: I must think that De Candolle has fallen into error in attempting to do this for Orders instead of for genera,—for reasons with which I will not trouble you.—9
In second column Heading p. 27 (or p. 229) there is misprint “and,” for “not”, which might seriously mislead an idle reader who only looked at general totals.
Many of our Societies always page their separate copies of papers with the proper pages for reference. Is not this good scheme & worth Prof. Silliman attending to? by a reference in body of your own paper I have corrected your paging.10
Hooker has lately returned from his continental trip & I am going to see him on Friday. Have you seen his Review on Decandolle: I cannot but think he is rather too severe on want of originality, & I hope much too severe on whole great & noble subject of Bot. Geograph.—11
With most sincere & hearty thanks for all your great kindness. Your’s very truly | C. Darwin
Dated by the reference to A. Gray 1856–7 and to the letter from Asa Gray, 23 September 1856.
Letter from Asa Gray, 23 September 1856.
A. Gray 1856–7. CD’s annotated copy is in DAR 135 (3).
J. D. Hooker 1853–5.
CD had written ‘extremely few’ in the manuscript of Natural selection (p. 539). Later he added the note: ‘Asa Gray thinks there are not a few plants common to U.S. & Europe, which do not range to Arctic regions.’ To this Hooker added, ‘Certainly J.D.H.’. See also letter from Asa Gray, 4 November 1856.
Gray did not give the desired figure in the second part of A. Gray 1856–7. In A. Gray 1856a, pp. xxv–xxviii, he had listed the number of introduced species (giving the total of 260 species, as mentioned by CD in the letter), but these had only been allocated to their taxonomic orders, not genera. The same list was repeated, with additional information but still excluding the number of genera, in A. Gray 1856–7, pp. 208–11. In CD’s copy of A. Gray 1856a there is a manuscript list, in the hand of an amanuensis, giving the names and genera of these introduced species. It is not clear whether CD had the list drawn up at Down House or whether it was sent to him by Gray at a later date. The information was eventually used in Natural selection, p. 232 n. 3.
A. de Candolle 1855. CD cited pages 745, 759, and 803 on the subject of naturalised plants. Alphonse de Candolle’s statistics are compared with Gray’s in Natural selection, p. 232.
See Correspondence vol. 5, letter to J. D. Hooker, 8 [November 1855], n. 3. CD thought the statistical relationships Candolle had discerned were probably due only to ‘parentage’ and common descent when applied to large groups like families and orders.
In CD’s copy of A. Gray 1856–7, he added the correct page numbers in pencil to to the pages of his independently paginated reprint.
See letter to J. D. Hooker, 9 October [1856].
Gray, Asa. 1856–7. Statistics of the flora of the northern United States. American Journal of Science and Arts 2d ser. 22: 204–32; 23: 62–84, 369–403.
Thanks AG for the first part of his "Statistics [of the flora of the northern U. S.", Am. J. Sci. 2d ser. 22 (1856): 204–32; 2d ser. 23 (1857): 62–84, 369–403]
and for information on social and varying plants.
Would like to know number of genera of introduced plants in U. S.
Is surprised at some affinities of northern U. S. flora and asks for any climatic explanations.
Asks what proportion of genera common to U. S. and Europe are mundane.
Is glad AG will work out the northern ranges of the European species and the ranges of species with regard to size of genera.
Archives of the Gray Herbarium, Harvard University (6) |
Difference between revisions of "Some APN functions CCZ-equivalent to x^3 + tr n(x^9) and CCZ-inequivalent to the Gold functions over GF(2^n)" - Boolean Functions
Difference between revisions of "Some APN functions CCZ-equivalent to x^3 + tr n(x^9) and CCZ-inequivalent to the Gold functions over GF(2^n)"
Some APN functions CCZ-equivalent to <math>x^3+tr_{n}(x^9)</math> and CCZ-inequivalent to the Gold functions over <math>\mathbb{F}_{2^n}</math> (constructed in <ref>Budaghyan L, Carlet C, Leander G. Constructing new APN functions from known ones. Finite Fields and Their Applications. 2009 Apr 1;15(2):150-9.</ref>)
Some APN functions CCZ-equivalent to <math>x^3+tr_{n}(x^9)</math> and CCZ-inequivalent to the Gold functions over <math>\mathbb{F}_{2^n}</math><ref>L. Budaghyan, C. Carlet, G. Leander. Constructing new APN functions from known ones. Finite Fields and Their Applications, v. 15, issue 2, pp. 150-159, April 2009. https://doi.org/10.1016/j.ffa.2008.10.001</ref>.
Some APN functions CCZ-equivalent to
{\displaystyle x^{3}+tr_{n}(x^{9})}
and CCZ-inequivalent to the Gold functions over
{\displaystyle \mathbb {F} _{2^{n}}}
{\displaystyle N^{\circ }}
{\displaystyle d^{\circ }}
{\displaystyle 1}
{\displaystyle x^{3}+tr_{n}(x^{9})+(x^{2}+x)tr_{n}(x^{3}+x^{9})}
{\displaystyle n\geqslant 5}
{\displaystyle \gcd(i,n)=1}
{\displaystyle 3}
{\displaystyle 2}
{\displaystyle x^{3}+tr_{n}(x^{9})+(x^{2}+x+1)tr_{n}(x^{3})}
{\displaystyle n\geqslant 4}
{\displaystyle \gcd(i,n)=1}
{\displaystyle 3}
{\displaystyle 3}
{\displaystyle {\Big (}x+tr_{n}^{3}(x^{6}+x^{12})+tr_{n}(x)tr_{n}^{3}(x^{3}+x^{12}){\Big )}^{3}+}
{\displaystyle tr_{n}{\Big (}\left(x+tr_{n}^{3}(x^{6}+x^{12})+tr_{n}(x)tr_{n}^{3}(x^{3}+x^{12})\right)^{9}{\Big )}}
{\displaystyle 6|n}
{\displaystyle \gcd(i,n)=1}
{\displaystyle 4}
{\displaystyle 4}
{\displaystyle \left(x^{\frac {1}{3}}+tr_{n}^{3}(x+x^{4})\right)^{-1}+tr_{n}\left(\left(\left(x^{\frac {1}{3}}+tr_{n}^{3}(x+x^{4})\right)^{-1}\right)^{9}\right)}
{\displaystyle 3|n}
{\displaystyle n}
{\displaystyle 4}
↑ L. Budaghyan, C. Carlet, G. Leander. Constructing new APN functions from known ones. Finite Fields and Their Applications, v. 15, issue 2, pp. 150-159, April 2009. https://doi.org/10.1016/j.ffa.2008.10.001
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An Efficient Random Algorithm for Box Constrained Weighted Maximin Dispersion Problem
An Efficient Random Algorithm for Box Constrained Weighted Maximin Dispersion Problem
Jinjin Huang
The box-constrained weighted maximin dispersion problem is to find a point in an n-dimensional box such that the minimum of the weighted Euclidean distance from given m points is maximized. In this paper, we first reformulate the maximin dispersion problem as a non-convex quadratically constrained quadratic programming (QCQP) problem. We adopt the successive convex approximation (SCA) algorithm to solve the problem. Numerical results show that the proposed algorithm is efficient.
Maximin Dispersion Problem, Successive Convex Approximation Algorithm, Quadratically Constrained Quadratic Programming (QCQP)
The weighted maximin problem model with box constraints is as follows:
{\mathrm{max}}_{x\in \chi }\left\{f\left(x\right):={\mathrm{min}}_{i=1,\cdots ,m}{\omega }_{i}{‖x-{x}^{i}‖}^{2}\right\}
\chi =\left\{y\in {R}^{n}|{\left({y}_{1}^{2},\cdots ,{y}_{n}^{2},1\right)}^{\text{T}}\in \kappa \right\}
\kappa
is a convex cone;
{x}_{1},\cdots ,{x}_{m}\in {R}^{n}
are m given points; these m points are equivalent to m locations;
{\omega }_{i}>0
i=1,\cdots ,m
‖\cdot ‖
denotes the Euclidean norm. In our numerical calculation,
{\omega }_{i}
is equal to 1. The goal is to find a point in a closed set
\chi ={\left[-1,1\right]}^{n}
such that the minimum of the weighted Euclidean distance from given set of points
{x}_{1},\cdots ,{x}_{m}
{R}^{n}
is maximized. The weighted maximin problem has been widely used in spatial management, facility location, and pattern recognition.
The weighted maximin dispersion problem with box constraints is known to be NP-hard in general [1] . For the low-dimensional cases (
n\le 3
\chi
being a polyhedral set), it is solvable in polynomial time [2] [3] . For
n>4
, a heuristic algorithm [2] [4] is used to solve this problem.
In paper [5] , they look for approximate solution through convex relaxation,
and prove that the approximation bound is
\frac{1-O\left(\sqrt{\mathrm{ln}\left(m\right){\gamma }^{\ast }}\right)}{2}
{\gamma }^{\ast }
\chi
\chi ={\left\{-1,1\right\}}^{n}
\chi ={\left[-1,1\right]}^{n}
{\gamma }^{*}=O\left(\frac{1}{n}\right)
. In paper [1] , they
use the linear programming relaxation method to give the approximate bounds of the ball problem, which is
\frac{1-O\left(\sqrt{\mathrm{ln}\left(m\right)/n}\right)}{2}
. In paper [5] , they consider the problem of finding a point in a unit n-dimensional
{\mathcal{l}}_{p}-\text{ball}
(p ≥ 2) such that the minimum of the weighted Euclidean distance from given m points is maximized. They show in paper [6] that the SDP-relaxation-based approximation algorithm provides the first theoretical approximation bound of
\frac{1-O\left(\sqrt{\mathrm{ln}\left(m\right)/n}\right)}{2}
In this paper, firstly, we model the maximin dispersion problem as a Quadratically constrained quadratic programming (QCQP), noting that (1) is a non-smooth, non-convex optimization problem, because the point-wise minimum of convex quadratics is non-differentiable and non-concave. We solve this problem with a general approximation framework, which is successive convex approximation (SCA), which can be summarized as follows: each quadratic component of (1) is locally linearized at the current iteration to construct its convex approximation function, so we obtain a convex subproblem. The solution of each subproblem is then used as the point about which we construct a convex surrogate function in the next iteration, repeat the steps, and then adopt the random block coordinate descent method (RBCDM) to obtain the solution of subproblem.
The remainder of the paper is organized as follows. In Section 2, we give technical preliminaries. In Section 3, we first reformulate maximin dispersion problem as a QCQP problem. Then, we describe the overall SCA approach and use the proposed methods (RBCDM) for solving each subproblem. In Section 4, we present some numerical results. Conclusions are made in Section 5.
The following concepts or definitions are adopted in our paper.
We use
{R}^{n}
to denote the space of n dimensional real valued vectors, and
x\in {R}^{n}
, we denote the ith component of x by
{x}_{i}
. Thus, each
x\in {R}^{n}
x=\left(\begin{array}{c}{x}_{1}\\ {x}_{2}\\ ⋮\\ {x}_{n}\end{array}\right)
Let
y\in {R}^{n}
\chi ={\left[-1,1\right]}^{n}
be a set, then the distance of the point y from the set
\chi
d\left(y,\chi \right)=\underset{x\in \chi }{\mathrm{inf}}{‖x-y‖}_{2}
3. Algorithm of Generation
We now reformulation (1) into the following equivalent form,
\underset{x\in \chi }{\mathrm{max}}\underset{i=1,\cdots ,m}{\mathrm{min}}{\omega }_{i}{‖x-{x}^{i}‖}^{2}⇔-\underset{x\in \chi }{\mathrm{min}}\underset{i=1,\cdots ,m}{\mathrm{max}}-{\omega }_{i}{‖x-{x}^{i}‖}^{2},
\underset{x\in \chi }{\mathrm{min}}\underset{i=1,\cdots ,m}{\mathrm{max}}-{\omega }_{i}{‖x-{x}^{i}‖}^{2},
and we will work with this formulation, note that the problem still remains non-convex.
We first introduce our algorithm ideas. First, we construct a convex optimization function of the non-convex objective function (3) by locally linearizing each quadratic component of (3) about the iterate point
{x}^{\left(r\right)}
, we obtain a n-dimensional convex subproblem. Second, we adopt random block coordinate descent method (RBCDM) to transform the n-dimensional convex subproblem into one-dimensional convex subproblem to reduce the computational complexity, here, the optimization variables be decomposed into n independent blocks. At each iteration of this method, random one of the components of variable is optimized, while the remaining variables are held fixed, until all components of a variable are updated, remember as a round, repeat the above steps until we achieve the effect we want. Such block structure can lead to low-complexity algorithms. Finally, to solve the one-dimensional subproblem.
f\left(x\right):=\underset{i=1,\cdots ,m}{\mathrm{max}}{u}_{i}\left(x\right)
{u}_{i}\left(x\right):=-{\omega }_{i}{‖x-{x}^{i}‖}^{2},\text{\hspace{0.17em}}i=1,\cdots ,m
{u}_{i}\left(x\right)
is concave for
i=1,\cdots ,m
, on locally linearizing
{u}_{i}\left(x\right)
about the current iterate point
x={x}^{\left(r\right)}
, we can obtain a global upper-bound of original objective
f\left(x\right)
. At the point
x={x}^{\left(r\right)}
, we construct a convex approximation function of
f\left(x\right)
x={x}^{\left(r\right)}
\begin{array}{c}{u}_{i}\left(x\right)\le {u}_{i}\left({x}^{\left(r\right)}\right)+\nabla {u}_{i}{\left({x}^{\left(r\right)}\right)}^{\text{T}}\left(x-{x}^{\left(r\right)}\right)\\ =2{\omega }_{i}{\left({x}^{i}-{x}^{\left(r\right)}\right)}^{\text{T}}x+{\omega }_{i}\left({\left({x}^{\left(r\right)}\right)}^{\text{T}}{x}^{\left(r\right)}-{\left({x}^{i}\right)}^{\text{T}}{x}^{i}\right)\\ ={\left({x}^{\left(r\right)}\right)}^{\text{T}}x+{d}_{i}^{\left(r\right)},\end{array}
{c}_{i}^{\left(r\right)}=2{\omega }_{i}\left({x}^{i}-{x}^{\left(r\right)}\right),\text{}{d}_{i}^{\left(r\right)}={\omega }_{i}\left({\left({x}^{\left(r\right)}\right)}^{\text{T}}{x}^{\left(r\right)}-{\left({x}^{i}\right)}^{T}{x}^{i}\right)
i=1,\cdots ,m
v\left(x,{x}^{\left(r\right)}\right):=\underset{i=1,\cdots ,m}{\mathrm{max}}{\left({c}_{i}^{\left(r\right)}\right)}^{\text{T}}x+{d}_{i}^{\left(r\right)}
, the piecewise linear function
v\left(x,{x}^{\left(r\right)}\right)
is an upper bound of the original function
f\left(x\right)
x={x}^{\left(r\right)}
, which is tight at
x={x}^{\left(r\right)}
[7] . We replace
f\left(x\right)
with its piecewise linear approximation about
{x}^{\left(r\right)}
to obtain the non-smooth, convex subproblem.
\underset{x\in \chi }{\mathrm{min}}\underset{i=1,\cdots ,m}{\mathrm{max}}{\left({c}_{i}^{\left(r\right)}\right)}^{\text{T}}x+{d}_{i}^{\left(r\right)}
This subproblem is computationally expensive, so we transform the high-dimensional problem into one-dimensional problem to reduce the complexity.
The concrete steps are as follows: We random update the jth component
{x}_{j}
of x at the current iterate point
{x}^{\left(r\right)}
and keep the other components unchanged，it must be noted that the
{x}_{j}
is a component of random selection, let
x={\left({x}_{1}^{\left(r\right)},\cdots ,{x}_{j-1}^{\left(r\right)},{x}_{j},{x}_{j+1}^{\left(r\right)},\cdots ,{x}_{n}^{\left(r\right)}\right)}^{\text{T}}
\begin{array}{l}{v}_{j}\left({x}_{j},{x}^{\left(r\right)}\right)\\ =\underset{i=1,\cdots ,m}{\mathrm{max}}2{\left({x}^{i}-{x}^{\left(r\right)}\right)}^{\text{T}}x+\left({\left({x}^{\left(r\right)}\right)}^{\text{T}}{x}^{\left(r\right)}-{\left({x}^{i}\right)}^{\text{T}}{x}^{i}\right)\\ =\underset{i=1,\cdots ,m}{\mathrm{max}}2\left(\left({x}_{1}^{i}-{x}_{1}^{\left(r\right)}\right),\cdots ,\left({x}_{j}^{i}-{x}_{j}^{\left(r\right)}\right),\cdots ,\left({x}_{n}^{i}-{x}_{n}^{\left(r\right)}\right)\right){\left({x}_{1}^{\left(r\right)},\cdots ,{x}_{j-1}^{\left(r\right)},{x}_{j},{x}_{j+1}^{\left(r\right)},\cdots ,{x}_{n}^{\left(r\right)}\right)}^{\text{T}}\\ \text{}+\left({\left({x}^{\left(r\right)}\right)}^{\text{T}}{x}^{\left(r\right)}-{\left({x}^{i}\right)}^{\text{T}}{x}^{i}\right)\end{array}
\begin{array}{l}=\underset{i=1,\cdots ,m}{\mathrm{max}}2\left({x}_{j}^{i}-{x}_{j}^{\left(r\right)}\right){x}_{j}+\left({\left({x}^{\left(r\right)}\right)}^{\text{T}}{x}^{\left(r\right)}-{\left({x}^{i}\right)}^{\text{T}}{x}^{i}\right)\\ \text{}+\underset{l=1}{\overset{n}{\sum }}2\left({x}_{l}^{i}-{x}_{l}^{\left(r\right)}\right){x}_{l}^{\left(r\right)}-2\left({x}_{j}^{i}-{x}_{j}^{\left(r\right)}\right){x}_{j}^{\left(r\right)}\\ =\underset{i=1,\cdots ,m}{\mathrm{max}}{a}_{i}^{\left(r\right)}{x}_{j}+{b}_{i}^{\left(r\right)}\end{array}
{a}_{i}^{\left(r\right)}=2\left({\left({x}^{i}\right)}_{j}-{\left({x}^{\left(r\right)}\right)}_{j}\right)
{b}_{i}^{\left(r\right)}=\left({\left({x}^{\left(r\right)}\right)}^{\text{T}}{x}^{\left(r\right)}-{\left({x}^{i}\right)}^{\text{T}}{x}^{i}\right)+\underset{i=1}{\overset{n}{\sum }}\left({\left({x}^{i}\right)}_{l}-{\left({x}^{\left(r\right)}\right)}_{l}\right){\left({x}^{\left(r\right)}\right)}_{l}-2\left({\left({x}^{i}\right)}_{j}-{\left({x}^{\left(r\right)}\right)}_{j}\right){\left({x}^{\left(r\right)}\right)}_{j}
obtain the one-dimensional convex subproblem
\underset{{x}_{j}\in {\chi }^{*}}{\mathrm{min}}\underset{i=1,\cdots ,m}{\mathrm{max}}{a}_{i}^{\left(r\right)}{x}_{j}+{b}_{i}^{\left(r\right)},
In order to solve the solution of one-dimensional piecewise linear function (5), we first arrange the
{a}_{i}^{\left(r\right)}
of the m lines from small to large, i.e.
{a}_{1}^{\left(r\right)}\le {a}_{2}^{\left(r\right)}\le \cdots \le {a}_{m}^{\left(r\right)}
. For the convenience of description, we remember these m lines as
{y}_{i}={a}_{i}x+{b}_{i}\left(i=1,2,\cdots ,m\right)
x=\left[-1,1\right]
. The following is the algorithmic frameworks for solving one-dimensional subproblem.
In order to benchmark the performance of our proposed algorithms, we do some simple numerical comparisons. We do numerical experiments on 4 random instances when dimension n takes different values, respectively, such as n = 100, 500, 1000, 2000. The corresponding m we chose smaller than n, the same as n, and bigger than n. where all weights
{\omega }_{1},\cdots ,{\omega }_{m}
are equal to 1, all the numerical tests are implemented in MATLAB R2016a and run on a laptop with 2.50 GHz processor and 4 GB RAM.
All the input points
{x}^{i}
n\times 45000
Table 1. Algorithmic frameworks of subproblem.
Table 2. Numerical results.
rand\left(state,0\right);X=4\ast rand\left(n,450\right)-2;
We report the numerical results in Table 1. The columns
v{\left(CR\right)}^{\prime }
present the optimal objective function values of convex relaxation [1] of the 26 instances. In [1] , they first reformulate (1) as an equivalent smooth optimization problem as following
\begin{array}{l}\underset{x,\zeta }{\mathrm{max}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\zeta \\ s.t.\text{}{\omega }_{i}\left({‖x‖}^{2}-2{\left({x}^{i}\right)}^{\text{T}}x+{‖{x}^{i}‖}^{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\in \chi \end{array}
product the following convex relaxation (CR) when
\chi ={\left[-1,1\right]}^{n}
\begin{array}{l}\underset{x,\zeta }{\mathrm{max}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\zeta \\ s.t.\text{}{\omega }_{i}\left(n-2{\left({x}^{i}\right)}^{\text{T}}x+{‖{x}^{i}‖}^{2}\right)\\ \text{}x\in \chi \end{array}
we solved it with CVX solver [8] .
The next column present the statistical results over the 1000 runs of the general algorithm proposed in [1] , the subcolumns “max”, “min”, “ave” and “time 1” give the best, the worst, the average objective function values and running time found among 1000 tests, respectively. The last column is the result of our algorithm, where we choose 0 vector as the initial point. Finally, add a rounding (i.e., if
{x}_{h}^{\left(0\right)}\ge 0
{x}_{h}^{\left(0\right)}=1
{x}_{h}^{\left(0\right)}=-1
h=1,\cdots ,n
{x}^{\left(0\right)}
obtained by the iteration. The subcolumns “f(x1)”, “f(x2)” and “time 2” represents the numerical result corresponding to no add rounding, add rounding and running time of our algorithm, respectively. Numerical results show that the effect of “f(x2)” is the best. Table 2 shows that the qualities of the solutions returned by our algorithm are generally higher than those obtained by the general algorithm in [1] .
In this paper, we reformulate the maximin dispersion problem as QCQP problem and the original non-convex problem is approximated by a sequence of convex problems. Then, we adopt the random block coordinate descent method (RBCDM) to obtain the solution of subproblem. Numerical results show that the proposed algorithm is efficient.
Huang, J.J. (2019) An Efficient Random Algorithm for Box Constrained Weighted Maximin Dispersion Problem. Advances in Pure Mathematics, 9, 330-336. https://doi.org/10.4236/apm.2019.94015
1. Wang, S. and Xia, Y. (2016) On the Ball-Consterained Weighted Maximin Dispersion Problem. SIAM Journal on Optimization, 26, 1565-1588.
2. White, D.J. (1996) A Heuristic Approach to a Weighted Maxmin Disperation Problem. IMA Journal of Management Mathematics, 7, 219-231. https://doi.org/10.1093/imaman/7.3.219
3. Ravi, S.S., Rosenkrantz, D.J. and Tayi, G.K. (1994) Heuristic and Special Case Algorithms for Dispersion Problems. Operations Research, 42, 299-310. https://doi.org/10.1287/opre.42.2.299
4. Dasarthy, B. and White, L.J. (1980) A Maximin Location Problem. Operations Research, 28, 1385-1401. https://doi.org/10.1287/opre.28.6.1385
5. Wu, Z.P., Xia, Y. and Wang, S. (2017) Approximating the Weighted Maximin Dispersion Problem over an l_p-ball: SDP Relaxation Is Misleading. Optimization Letters, 12, 875-883.
6. Haines, S., Loeppky, J., Tseng, P. and Wang, X. (2013) Convex Relaxations of the Weighted Maxmin Dispersion Problem. SIAM Journal on Optimization, 23, 2264-2294. https://doi.org/10.1137/120888880
7. Konar, A. and Sidiropoulos, N.D. (2017) Fast Approximation Algorithms for a Class of Nonconvex QCQP Problems Using First-Order Methods. IEEE Transactions on Signal Processing, 65, 3494-3509.
8. Grant, M. and Boyd, S. (2010) CVX User’s Guide: For CVX Version 1.21. User’s Guide, 24-75. |
Calculating Power - CryptoBlades Wiki
Unaligned Character Power
Unaligned Character Power is the variable used to determine the range of enemy power rolls.
The required variables to calculate this are the following:
Character Power is the listed power displayed on the upper left hand side of the screen above the stamina bar when a character is selected.
Weapon Attribute Base
Weapon Attribute Base is the sum of all the weapon's attribute values without taking into consideration elemental matching.
Weapon Bonus Power
Weapon Bonus Power is the listed bonus power value if the weapon has been reforged.
The current formula to calculate for unaligned power is
unalignedPower = (((attributeTotal * 0.0025) + 1) * charPower) +bonusPower
After we calculate for unaligned power, we apply a ±10% to determine the range of values that the enemy power values might be.
Let's do a sample calculation assuming the following values below:
Character Power - 1000 (a level one character)
Attribute Total - 800 (a max attribute 4-star weapon)
Bonus Power - 1500 (a 100/100 LB, 0/25 4B, 0/10 5B weapon)
Unaligned Power comes out at 4500.
Minimum Enemy Power is 4500 * 0.9 rounded down to 4050.
Maximum Enemy Power is 4500 * 1.1 rounded down to 4950.
Aligned Character Power
Aligned Character Power is the variable used in determining the player's combat roll in conjunction with Trait Bonus.
The required variables to calculate this are similar to Unaligned Character Power above but instead of
Aligned Character Power uses
Weapon Attribute Multiplied
Weapon Attribute Multiplied is the sum of all the weapon's attribute values after applying a multiplier to each attribute based on elemental matching.
Instead of the formula simply summing up all the weapon's attributes, we instead evaluate each attribute separately and apply the following calculations to determine their value
if attributeElement != charElement (attributeValue * 0.0025)
if attributeElement == PWR (attributeValue * 0.002575)
if attributeElement == charElement (attributeValue * 0.002675)
Once each attribute has been evaluated, they get totaled and used in the same formula as unaligned power to get the aligned power.
alignedPower = ((evaluatedAttributeTotal + 1) * charPower) + bonusPower
Let's do another sample calculation assuming the following values below:
Character Element - Fire
Attribute One - STR 400
Attribute Two - CHA 400
Aligned Power comes out to 4570.
Aligned Power is used as is when calculating experience gain, or multiplied with Trait Bonus when calculating the player's combat roll.
Trait Bonus is a variable multiplied to Aligned Power and used to determine the player's combat roll.
The formula to determine Trait Bonus is outlined below.
TraitBonus = 1
if charElement == weaponElement (TraitBonus += 0.075)
if charElement > enemyElement (TraitBonus += 0.075)
if charElement < enemyElement (TraitBonus -= 0.075)
The elemental advantage in regards to character against enemy is as follows:
Earth beats Lightning
Lightning beats Water
Trait Bonus gets evaluated and then multiplied with Aligned Power to get the players final power value.
A ±10% is then applied to the final value to determine the player's combat roll.
Taking the Aligned Power calculated above, let's assume the following variables:
Weapon Element - Water
Enemy Element - Earth
Trait Bonus comes out to 1.075.
Final Power Value after applying Trait Bonus to the Aligned Power above is 4912.
Minimum Player Roll is 4912 * 0.9 rounded down to 4420.
Maximum Player Roll is 4912 * 1.1 rounded down to 5403.
Enemy Power is a simple ±10% calculation applied to the listed enemy power of whatever enemy the player chose.
The numerical value listed on the combat screen button is used to determine experience and SKILL payouts.
The calculated value with the ±10% applied is used to determine the enemy's rolls in combat
Taking the previous values into account, let's finalize the sample combat simulation
Minimum Enemy Power = 4050
Maximum Enemy Power = 4950
Let's assume the player chose an enemy with a listed power value of 4700.
Enemy Power = 4700
Minimum Enemy Roll is 4700 * 0.9 rounded down to 4230.
Maximum Enemy Roll is 4700 * 1.1 rounded down to 5170.
From this information, we know that the player can roll between 4420 - 5403 and the enemy can roll between 4230 - 5170. |
Measure of energy in a thermodynamic system
{\displaystyle c=}
{\displaystyle T}
{\displaystyle \partial S}
{\displaystyle N}
{\displaystyle \partial T}
{\displaystyle \beta =-}
{\displaystyle 1}
{\displaystyle \partial V}
{\displaystyle V}
{\displaystyle \partial p}
{\displaystyle \alpha =}
{\displaystyle 1}
{\displaystyle \partial V}
{\displaystyle V}
{\displaystyle \partial T}
{\displaystyle U(S,V)}
{\displaystyle H(S,p)=U+pV}
{\displaystyle A(T,V)=U-TS}
{\displaystyle G(T,p)=H-TS}
{\displaystyle H=\sum _{k}H_{k},}
{\displaystyle H=\int (\rho h)\,dV,}
{\displaystyle dU=\delta Q-\delta W,}
{\displaystyle dU=T\,dS-p\,dV.}
{\displaystyle dU+d(pV)=T\,dS-p\,dV+d(pV),}
{\displaystyle d(U+pV)=T\,dS+V\,dp.}
{\displaystyle dH(S,p)=T\,dS+V\,dp.}
Other expressions[edit]
{\displaystyle dH=C_{p}\,dT+V(1-\alpha T)\,dp.}
{\displaystyle \alpha ={\frac {1}{V}}\left({\frac {\partial V}{\partial T}}\right)_{p}.}
{\displaystyle dH=T\,dS+V\,dp}
{\displaystyle dH=C_{p}\,dT.}
{\displaystyle dH=T\,dS+V\,dp+\sum _{i}\mu _{i}\,dN_{i},}
Characteristic functions and natural state variables[edit]
Relationship to heat[edit]
{\displaystyle dU=\delta Q-p\,dV.}
{\displaystyle dH=dU+d(pV).}
{\displaystyle {\begin{aligned}dH&=\delta Q+V\,dp+p\,dV-p\,dV\\&=\delta Q+V\,dp.\end{aligned}}}
{\displaystyle dH=\delta Q.}
Heat of reaction[edit]
{\displaystyle \Delta H=H_{\mathrm {f} }-H_{\mathrm {i} },}
Specific enthalpy[edit]
Enthalpy changes[edit]
{\displaystyle dU=\delta Q+dU_{\text{in}}-dU_{\text{out}}-\delta W,}
{\displaystyle \delta W=d(p_{\text{out}}V_{\text{out}})-d(p_{\text{in}}V_{\text{in}})+\delta W_{\text{shaft}}.}
{\displaystyle dU_{\text{cv}}=\delta Q+dU_{\text{in}}+d(p_{\text{in}}V_{\text{in}})-dU_{\text{out}}-d(p_{\text{out}}V_{\text{out}})-\delta W_{\text{shaft}}.}
{\displaystyle dU_{\text{cv}}=\delta Q+dH_{\text{in}}-dH_{\text{out}}-\delta W_{\text{shaft}}.}
{\displaystyle {\frac {dU}{dt}}=\sum _{k}{\dot {Q}}_{k}+\sum _{k}{\dot {H}}_{k}-\sum _{k}p_{k}{\frac {dV_{k}}{dt}}-P,}
{\displaystyle {\dot {H}}_{k}=h_{k}{\dot {m}}_{k}=H_{\mathrm {m} }{\dot {n}}_{k},}
{\displaystyle P=\sum _{k}\left\langle {\dot {Q}}_{k}\right\rangle +\sum _{k}\left\langle {\dot {H}}_{k}\right\rangle -\sum _{k}\left\langle p_{k}{\frac {dV_{k}}{dt}}\right\rangle ,}
Some basic applications[edit]
Throttling[edit]
{\displaystyle 0={\dot {m}}h_{1}-{\dot {m}}h_{2}.}
{\displaystyle h_{1}=h_{2},}
{\displaystyle h_{\mathbf {f} }=x_{\mathbf {f} }h_{\mathbf {g} }+(1-x_{\mathbf {f} })h_{\mathbf {h} }.}
{\displaystyle 0=-{\dot {Q}}+{\dot {m}}h_{1}-{\dot {m}}h_{2}+P.}
{\displaystyle 0=-{\frac {\dot {Q}}{T_{\mathrm {a} }}}+{\dot {m}}s_{1}-{\dot {m}}s_{2}.}
{\displaystyle {\frac {P_{\text{min}}}{\dot {m}}}=h_{2}-h_{1}-T_{\mathrm {a} }(s_{2}-s_{1}).}
{\displaystyle {\frac {P_{\text{min}}}{\dot {m}}}=\int _{1}^{2}(dh-T_{\mathrm {a} }\,ds).}
{\displaystyle {\frac {P_{\text{min}}}{\dot {m}}}=\int _{1}^{2}v\,dp.}
The term expresses the obsolete concept of heat content,[21] as dH refers to the amount of heat gained in a process at constant pressure only,[22] but not in the general case when pressure is variable.[23] Josiah Willard Gibbs used the term "a heat function for constant pressure" for clarity.[note 2]
{\displaystyle \alpha T={\frac {T}{V}}\left({\frac {\partial ({\frac {nRT}{P}})}{\partial T}}\right)_{p}={\frac {nRT}{PV}}=1}
Retrieved from "https://en.wikipedia.org/w/index.php?title=Enthalpy&oldid=1088131582" |
For each quadratic function below, use the idea of completing the square to write it in graphing form. Then state the vertex of each parabola.
f(x)=x^2+4x+5
Draw tiles to help you make the expression into a square.
Put it into graphing form and from there figure out the vertex.
f(x)=(x+2)^2+1
(−2,1)
f(x)=x^2−6x
In both graphs, does the vertex represent the maximum or minimum value of the parabola?
Is the vertex the highest (maximum) or lowest (minimum) point on the graph? |
Sure a general grasp of data structures and algorithms helps to build software, but software development is not data structures and algorithms. Data structures and algorithms are variables (
x_1, x_2
) that contribute to the function of software development (
f(x_1..x_n)
), which is dependent on a bunch of other variables like verbal and critical thinking ability. Personally, I’d take working with someone who’s great at naming functions and variables over someone who can code a solution to the knapsack problem. |
Character Levels - CryptoBlades Wiki
Each character starts out at level one, requiring experience attained through successful combat to increase.
Character levels determine the amount of power they have during combat calculations, and in turn determines the SKILL payout on victories.
Level milestones are specific points where the character receives a large boost in power, and in turn increases their SKILL payout.
Currently milestones occur every ten levels, starting from 11 then 21, 31, 41, etc.
To calculate whether or not it is time to claim, you can make a copy of the following spreadsheet:
Link to the spreadsheet here: CryptoBlades Experience Calculator
An overview of the experience table can be found below:
Experience Table - July 3rd, 2021
Claiming Experience
Experience won through battles is stored in the Rewards bar, similar to SKILL.
As claiming experience costs a gas fee for the transaction, it is recommended to only claim your experience for a character if it will push them to the next milestone if the character in question is under level 41.
Past level 41, it becomes beneficial to claim their experience every level thereafter as the boost in power will result in more SKILL gained through fight payouts.
Often times due to varying win rates, character levels may become desynced.
It is always a good idea to claim experience before fighting with a character if that experience pushes them to the next milestone.
If other characters will not yet hit their respective milestones, it might be beneficial to stop fighting with that character and let others catch up assuming their stamina isn't full.
Power Per Level
To calculate the amount of power a character gets at a certain level, we refer to the formula below:
charPower = 1000 + ((charLevel - 1) * 10) * (Math.Floor((charLevel - 1) / 10) + 1
More information on how character power is used to determine combat calculations and payouts can be found here:
Characters - Previous
Next - Weapons |
Generalized Fuzzy Quasi-Ideals of an Intraregular Abel-Grassmann's Groupoid
2012 Generalized Fuzzy Quasi-Ideals of an Intraregular Abel-Grassmann's Groupoid
Bijan Davvaz, Madad Khan, Saima Anis, Shamsul Haq
We have introduced a new nonassociative class of Abel-Grassmann's groupoid, namely, intraregular and characterized it in terms of its
\left(\in ,\in {\vee }_{q}\right)
-fuzzy quasi-ideals.
Bijan Davvaz. Madad Khan. Saima Anis. Shamsul Haq. "Generalized Fuzzy Quasi-Ideals of an Intraregular Abel-Grassmann's Groupoid." J. Appl. Math. 2012 1 - 16, 2012. https://doi.org/10.1155/2012/627075
Bijan Davvaz, Madad Khan, Saima Anis, Shamsul Haq "Generalized Fuzzy Quasi-Ideals of an Intraregular Abel-Grassmann's Groupoid," Journal of Applied Mathematics, J. Appl. Math. 2012(none), 1-16, (2012) |
A circle passes through the origin anf has its centre on y = x If it cuts x2 + - Maths - Three Dimensional Geometry - 11580858 | Meritnation.com
A circle passes through the origin anf has its centre on y = x. If it cuts x2 + y2 - 4x - 6y + 10 = 0 orthogonally, the equation of the circle is
Let the equation of circle be {x}^{2}+{y}^{2}+2gx+2hy+f=0.\phantom{\rule{0ex}{0ex}}Therefore centre of circle is \left(-g,-h\right).\phantom{\rule{0ex}{0ex}}Since centre lies on line y=x, \therefore -h=-g\phantom{\rule{0ex}{0ex}}⇒h=g\phantom{\rule{0ex}{0ex}}Thus equation of circle becomes {x}^{2}+{y}^{2}+2gx+2gy+f=0.\phantom{\rule{0ex}{0ex}}Since the circle passes through origin,\phantom{\rule{0ex}{0ex}}⇒0+0+2g\left(0\right)+2g\left(0\right)+f=0\phantom{\rule{0ex}{0ex}}⇒f=0\phantom{\rule{0ex}{0ex}}Hence, equation of circle becomes {x}^{2}+{y}^{2}+2gx+2gy=0. \phantom{\rule{0ex}{0ex}}Since it cuts {x}^{2}+{y}^{2}-4x-6y+10=0 orthogonally,\phantom{\rule{0ex}{0ex}}\therefore 2g\left(-2\right)+2g\left(-3\right)=10\phantom{\rule{0ex}{0ex}}⇒-4g-6g=10\phantom{\rule{0ex}{0ex}}⇒-10g=10\phantom{\rule{0ex}{0ex}}⇒g=\frac{10}{-10}=-1\phantom{\rule{0ex}{0ex}}Thus required equation of circle is {x}^{2}+{y}^{2}-2x-2y=0. \phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\left[If circle {x}^{2}+{y}^{2}+2gx+2hy+f=0 intersects circle {x}^{2}+{y}^{2}+2{g}_{1}x+2{h}_{1}y+{f}_{1}=0, then 2g{g}_{1}+2h{h}_{1}=f+{f}_{1}.\right]\phantom{\rule{0ex}{0ex}}
If you have more doubts just ask here on the forum and our experts will try to help you out as soon as possible. |
Stability of multidimensional undercompressive shock waves | EMS Press
Stability of multidimensional undercompressive shock waves
This paper is devoted to the study of linear and nonlinear stability of undercompressive shock waves for first order systems of hyperbolic conservation laws in several space dimensions. We first recall the framework proposed by Freistühler to extend Majda's work on classical shock waves to undercompressive shock waves. Then we show how the so-called uniform stability condition yields a linear stability result in terms of a maximal
L^2
estimate. We follow Majda's strategy on shock waves with several improvements and modifications inspired from Métivier's work. The linearized problems are solved by duality and the nonlinear equations by mean of a Newton type iteration scheme. Finally, we show how this work applies to phase transitions in an isothermal van der Waals fluid.
Jean-François Coulombel, Stability of multidimensional undercompressive shock waves. Interfaces Free Bound. 5 (2003), no. 4, pp. 367–390 |
October, 2000 Asymptotic rigidity of Hadamard 2-spaces
We classify locally compact, geodesically complete, 2-dimensional Hadamard spaces whose Tits ideal boundaries have the minimal diameter
\pi
. Furthermore, we classify the universal covering spaces of certain 2-dimensional nonpositively curved spaces, which is an extension of the result obtained in the polyhedral case by W. Ballmann, M. Brin, and S. Barré.
Koichi NAGANO. "Asymptotic rigidity of Hadamard 2-spaces." J. Math. Soc. Japan 52 (4) 699 - 723, October, 2000. https://doi.org/10.2969/jmsj/05240699
Keywords: Hadamard space , ideal boundary , Tits metric
Koichi NAGANO "Asymptotic rigidity of Hadamard 2-spaces," Journal of the Mathematical Society of Japan, J. Math. Soc. Japan 52(4), 699-723, (October, 2000) |
Dictionary:Anisotropy (electrical) - SEG Wiki
Variation of an electrical property depending on the direction in which it is measured. The resistivity anisotropy coefficient is the square root of the ratio of the resistivity measured perpendicular to the bedding to that parallel to the bedding; also called pseudo-anisotropy. It usually has a value between 1 and 2. For a sequence of isotropic layers with thicknesses zi and resistivities
{\displaystyle \rho _{i}}
the unit resistance RT is
{\displaystyle R_{T}=\sum z_{i}\rho _{i}}
and the pseudo-anisotropy
{\displaystyle \lambda }
{\displaystyle \lambda ={\sqrt {\frac {\sum z_{i}}{\sum (z_{i}/\rho _{i})}}}}
See dar Zarrouk. The anisotropy of induced polarization in rocks is less than the anisotropy of resistivity. In layered rocks the resistivity parallel to the layering is less than that perpendicular to the layering. Anisotropy as measured in a borehole is caused by cyclic thin sequences of alternating sand and shale, sorting of sand grains, and fractures (healed or fluid-filled).
Retrieved from "https://wiki.seg.org/index.php?title=Dictionary:Anisotropy_(electrical)&oldid=117788" |
REcontent - Maple Help
Home : Support : Online Help : Mathematics : Discrete Mathematics : Summation and Difference Equations : LREtools : REcontent
content of a recurrence operator
primpart of a recurrence operator
REcontent(problem)
REprimpart(problem)
REcontent returns the content of the shift operator (that is, INFO[shifteqn]), thus returning the greatest common divisor of the coefficients of the operator.
Similarly, REprimpart returns INFO[shifteqn]/REcontent(problem). Note: Whereas the sign is removed from the content, it is not removed from the primitive part.
\mathrm{with}\left(\mathrm{LREtools}\right):
\mathrm{REcontent}\left(\left(n+1\right)u\left(n+2\right)+\left(n+1\right)u\left(n\right),u\left(n\right),\varnothing \right)
\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}
\mathrm{REprimpart}\left(\left(n+1\right)u\left(n+2\right)+\left(n+1\right)u\left(n\right),u\left(n\right),\varnothing \right)
{{\textcolor[rgb]{0,0,1}{\mathrm{&Shift}}}_{\textcolor[rgb]{0,0,1}{n}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{u}\right)}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}
\mathrm{REcontent}\left(\left(n+1\right)u\left(n+2\right)+\left(n+2\right)u\left(n\right),u\left(n\right),\varnothing \right)
\textcolor[rgb]{0,0,1}{1}
\mathrm{REprimpart}\left(\left(n+1\right)u\left(n+2\right)+\left(n+2\right)u\left(n\right),u\left(n\right),\varnothing \right)
{{\textcolor[rgb]{0,0,1}{\mathrm{&Shift}}}_{\textcolor[rgb]{0,0,1}{n}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{u}\right)}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{+}{{\textcolor[rgb]{0,0,1}{\mathrm{&Shift}}}_{\textcolor[rgb]{0,0,1}{n}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{u}\right)}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2} |
Tangible Book Value Per Share (TBVPS) Definition
Tangible book value per share (TBVPS) is a method by which a company’s value is determined on a per-share basis by measuring its equity without the inclusion of any intangible assets. Intangible assets are those that lack physical substance, thus making their valuation a more difficult undertaking than the valuation of tangible assets.
TBVPS is similar to price-to-tangible book value (PTBV).
Tangible book value per share (TBVPS) is the value of a company’s tangible assets divided by its current outstanding shares.
TBVPS determines the potential value per share of a company in the event that it must liquidate its assets.
Assets such as property and equipment are considered tangible assets. Intangible assets, such as goodwill, are not included in the calculation of TBVPS.
One of the criticisms of TBVPS’s validity is the lack of accuracy in the accounting of a company’s tangible assets.
\begin{aligned} &\text{TBVPS} = \frac { \text{Total Tangible Assets} }{ \text{Total Number of Shares Outstanding} } \\ &\textbf{where:} \\ &\text{TBVPS} = \text{tangible book value per share} \\ \end{aligned}
TBVPS=Total Number of Shares OutstandingTotal Tangible Assetswhere:TBVPS=tangible book value per share
Tangible book value (TBV) of a company is what common shareholders can expect to receive if a firm goes bankrupt—thereby forcing the liquidation of its assets at the book value price. Intangible assets, such as goodwill, are not included in tangible book value because they cannot be sold during liquidation. However, companies with high tangible book values tend to offer shareholders more downside protection in the case of bankruptcy.
Tangible book value per share thus focuses solely on the value of an organization's tangible assets, such as buildings and equipment. Once the value of the tangible assets is determined, that amount is divided by the number of the company’s current outstanding shares. The amount determined in this process is recognized as the company’s TBVPS.
TBV provides an estimate regarding the value of the company if it goes bankrupt and is forced to liquidate the entirety of its assets. Since certain intrinsic characteristics such as goodwill or employee knowledge cannot be liquidated for a price, TBV does not include intangible assets. The TBV applies only to physical items that can be handled and sold at an easily determined market value.
Certain online databases and websites allow potential investors to examine the progress of a company’s TBVPS over time.
Requirements for Tangible Book Value Per Share
An organization's tangible assets can include any physical products the company produces, as well as any materials used to produce them. Should an organization be in the business of producing bicycles, for instance, any completed bicycles, unused bicycle parts, or raw materials used during the process of fabricating bicycles would qualify as tangible assets. The value of these assets is determined based on what price they would draw should the company be forced to liquidate, most commonly in the event of a bankruptcy.
Aside from assets related to the production of a product, any equipment used to create the product can be included as well. This can include any tools or machinery required to complete production, as well as any real estate owned and used for the purposes of production. Additional business equipment, such as computers and filing cabinets, may also be considered tangible assets for the purpose of valuation.
Criticism of TBVPS
Book value refers to the ratio of stockholder equity to the number of shares outstanding. It takes into account only the accounting valuation, which is not always an accurate reflection of the current market valuation, or of what could be received during a sale. |
Configure Matrix Panels - xLights Manual
Facebook Support GroupWeb SiteApplication Bug ReportingForum
Configure Matrix Panels
Coming from LOR
Editing xLights_rgb.xml
Configure P10 Matrix Panels
In this example, the setup consists of 8 P10 panels (4H x 2 W) driven via a Beagle Black Bone (BBB) with an octoscroller running the Falcon Player (referred to as the FPBBB). This process should be the same for Raspberry Pi's running FPP with a Pi Hat or Colorlight Card.
First Add a New Ethernet Controller to the Controller Tab. It is recommended to use a DDP Protocol for FPP based P10/P5 panels. An E131 Protocol can also be used but for this guide DDP was selected.
Set the Name to FPPBB. Set the IP Address of the device IP(192.168.5.200 in this example). The Id is a unique controller ID for xLights to use, this Id must be different for each out. The default Id of 1 is okay for this example. The number of channels is the total number of channels for the P10 matrix. To calculate this number channels, multiple the number of channels per LED(3 in most cases) by the number of LEDs per panel(512 for P10's) and then multiple that result by the number of Panels(eight in this example). The final values is the total channel for the Matrix.
Channels Per LED * LED Per Panel * NumberOfPanel = Channel Count
For this example, the total number of channels is: 3 x 512 x 8 = 12288 channels.
Set the Description Field To 'P10Matrix' and set the Vendor to 'FPP' and the Controller Type to 'PiHat'. The Description . Enable the 'Auto Upload Configuration' setting.
Next, Define a model ‘P10Matrix’ for in the Layout Tab
First Select the 'Matrix' model icon in the Model Toolbar.
Click the Left Mouse Button Down then Drag the Pointer to the Right, and Release.
Next, define the model settings. Set the model name to ‘P10Matrix’ (or any name of your choice) set it as a horizontal matrix.
Set ‘# Strings’ = 64 (corresponds to number of rows), ‘Nodes/String’ as 64 (corresponds to the columns) and ‘Strands /String’ = 1. 'Starting Location' = Top Left.
A single 32 * 16 P10 panel has 16 rows (height) and 32 columns (width). Right click on the image in your Layout and select Node Layout to view the node definition.
To set the Start Channel Click the Ellipsis(three periods) button in the Start Channel Box.
Set the start Channel by Clicking the 'Controller' Option and selecting the Controller Name set earlier, from the drop-down list. Keep the 'Start Channel' set to 1.
The start channel should be set to '!P10Matrix:1' and xLights will automatically calculate the end channel.
Click the 'Save' Button when done.
Upload Config to FPP
Switch back to the Controller Tab. Highlight the 'FPPBB' controller and click 'Upload Output'. This will automatically set the Panel Start channel In FPP for you.
BBB FPP definition
Open The Beaglebone controller web page by entering the IP address in your web browser of choose or by right-clicking on the DDP output and selecting 'Open Controller'.
Set the FPP controller to bridge mode for testing for now. When running your show you can use bridge mode or switch to FPP Remote Mode, both will work with the DDP setup. No channel inputs need to be setup the DDP protocol handle that for you.
Click the 'Input/Output Setup' Menu Banner and Select the 'Channel Outputs' Option
On the LED Panel page, Check the Enable LED Panel Output Option. The panel layout is 2 x 4. The Start channel should automatically been updated by the Upload To Controller Option used earlier. Make sure this matches the model definition in xLights. 'Single Panel Size' sets the Panel Size and Scan Rate. For most indoor P10 panels this is '32x16 1/8 Scan'. The channel count is automatically is calculated. This should also match the model in xLights. The 'Model Start Corner' should be the same as the 'Starting Location' in xLights. 'Default Panel Color Order' should match the panel vendors recommendation. Different vendors use different Color Orders. You may have to do try all the options to find the correct one. 'Brightness' is the Panel brightness, 10 is 100%, 1 is 10%. All the other options keep to the defaults.
LED Panel Output Page
The vertical arrows correspond to the Up/Down physical setting of the arrows on the panel backside. The LED Panel Layout orientation is as if viewed from the front of the panels. 'O-1', 'O-2', 'O-3', etc are the outputs ports on the Octoscroller. 'P-1', 'P-2', 'P-3', etc. is the panel order for each output. The first panel connected to the Octoscroller output is 'P-1', the second panel is 'P-2' and so forth.
Octoscroller
Appendicies - Previous |
Página de pruebas 3 - Sinfronteras
2 Data Analytics courses
3 Possible sources of data
4.1 Qualitative vs quantitative data
4.1.1 Discrete and continuous data
4.3 Data Levels and Measurement
4.4 What is an example
4.5 What is a dataset
6 Some real-world examples of big data analysis
8 Descriptive Data Analysis
8.1.1.1 When not to use the mean
8.1.4 Skewed Distributions and the Mean and Median
8.1.5 Summary of when to use the mean, median and mode
8.2.2 Quartile
8.2.6 Z Score
8.3 Shape of Distribution
8.3.1 Probability distribution
8.3.1.1 The Normal Distribution
8.3.5 Visualization of measure of variations on a Normal distribution
9 Simple and Multiple regression
9.1.1 Measuring Correlation
9.1.1.1 Pearson correlation coefficient - Pearson s r
9.1.1.2 The coefficient of determination
{\displaystyle R^{2}}
9.1.2 Correlation
{\displaystyle \neq }
9.1.3 Testing the "generalizability" of the correlation
9.4 RapidMiner Linear Regression examples
10 K-Nearest Neighbour
11.1.1 Basic explanation of the algorithm
11.1.2 Algorithms addressed in Noel s Lecture
11.1.2.1 The ID3 algorithm
11.1.2.2 The C5.0 algorithm
11.2 Example in RapidMiner
12 Random Forests
13.3 Mutually exclusive and collectively exhaustive
13.4 Marginal probability
13.5 Joint Probability
13.6.1 Kolmogorov definition of Conditional probability
13.6.2 Bayes s theorem
13.6.2.1 Likelihood and Marginal Likelihood
13.6.2.2 Prior Probability
13.6.2.3 Posterior Probability
13.7 Applying Bayes' Theorem
13.7.1 Scenario 1 - A single feature
13.7.2 Scenario 2 - Class-conditional independence
13.7.3 Scenario 3 - Laplace Estimator
13.8 Naïve Bayes - Numeric Features
13.9 RapidMiner Examples
14 Perceptrons - Neural Networks and Support Vector Machines
15.1 Gradient boosting
16.1 Clustering class of the Noel course
16.1.1 RapidMiner example 1
17 Principal Component Analysis PCA
18 Association Rules - Market Basket Analysis
18.1 Association Rules example in RapidMiner
20 Text Analytics / Mining
21 Model Evaluation
21.1 Why evaluate models
21.2 Evaluation of regression models
21.3 Evaluation of classification models
22.1 NumPy and Pandas
22.2 Data Visualization with Python
22.3 Text Analytics in Python
22.4 Dash - Plotly
22.5 Scrapy
23.1 R tutorial
25.1 Diploma in Predictive Data Analytics assessment
Possible sources of data
Data Levels and Measurement
What is an example
Some real-world examples of big data analysis
When not to use the mean
Skewed Distributions and the Mean and Median
Summary of when to use the mean, median and mode
measures-central-tendency-mean-mode-median-faqs.php
Visualization of measure of variations on a Normal distribution
Pearson correlation coefficient - Pearson s r
The coefficient of determination
{\displaystyle R^{2}}
{\displaystyle \neq }
Testing the "generalizability" of the correlation
RapidMiner Linear Regression examples
Basic explanation of the algorithm
Algorithms addressed in Noel s Lecture
The C5.0 algorithm
Example in RapidMiner
https://www.youtube.com/watch?v=J4Wdy0Wc_xQ&t=4s
The marginal probability is the probability of a single event occurring, independent of other events. A conditional probability, on the other hand, is the probability that an event occurs given that another specific event has already occurred. https://en.wikipedia.org/wiki/Marginal_distribution
Kolmogorov definition of Conditional probability
Bayes s theorem
Likelihood and Marginal Likelihood
Scenario 1 - A single feature
Scenario 2 - Class-conditional independence
Scenario 3 - Laplace Estimator
Naïve Bayes - Numeric Features
RapidMiner Examples
Clustering class of the Noel course
RapidMiner example 1
Association Rules example in RapidMiner
Text Analytics / Mining
Why evaluate models
Landis JR, Koch GG. The measurement of observer agreement for categorical data. Biometrics. 1977 Mar;33(1):159-174. DOI: 10.2307/2529310.
Diploma in Predictive Data Analytics assessment
Retrieved from "http://wiki.sinfronteras.ws/index.php?title=Página_de_pruebas_3&oldid=21431" |
CoefficientsInParameters - Maple Help
Home : Support : Online Help : Mathematics : Factorization and Solving Equations : RegularChains : ParametricSystemTools Subpackage : CoefficientsInParameters
CoefficientsInParameters
return the coefficients of a polynomial with respect to parameters
CoefficientsInParameters(p, d, R)
polynomial in the ring
The command CoefficientsInParameters(p, d, R) returns a list, lp, of polynomials involving the last d variables only.
The integer d should be positive and less than the number of variables. The last d variables are regarded as parameters,
{U}_{1},...,{U}_{d}
, and the other variables,
{X}_{1},...,{X}_{n}
, are regarded as unknowns.
The common zeros of the polynomials in
\mathrm{lp}
form the variety of
{K}^{d}
where the polynomial
p
is identically zero when regarded as a polynomial in
{X}_{1},...,{X}_{n}
K[{U}_{1},...,{U}_{d}]
K
is the algebraic closure of the ground field of R.
More precisely, the function will extract from p the polynomials of
K[{U}_{1},...,{U}_{d}]
, which are the coefficients of p when regarded as a polynomial in
\left(K[{U}_{1},...,{U}_{d}]\right)[{X}_{1},...,{X}_{n}]
. Moreover, the extracted polynomials might be simplified while preserving the variety defined by them.
This command is part of the RegularChains[ParametricSystemTools] package, so it can be used in the form CoefficientsInParameters(..) only after executing the command with(RegularChains[ParametricSystemTools]). However, it can always be accessed through the long form of the command by using RegularChains[ParametricSystemTools][CoefficientsInParameters](..).
\mathrm{with}\left(\mathrm{RegularChains}\right):
\mathrm{with}\left(\mathrm{ParametricSystemTools}\right):
R≔\mathrm{PolynomialRing}\left([x,a,b,c]\right)
\textcolor[rgb]{0,0,1}{R}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{polynomial_ring}}
p≔a{x}^{2}+{a}^{2}x+bc
\textcolor[rgb]{0,0,1}{p}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{a}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{c}
\mathrm{CoefficientsInParameters}\left(p,3,R\right)
[\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{c}] |
Effects of Bleed Flow on Heat/Mass Transfer in a Rotating Rib-Roughened Channel | J. Turbomach. | ASME Digital Collection
Effects of Bleed Flow on Heat/Mass Transfer in a Rotating Rib-Roughened Channel
Yun Heung Jeon,
Yun Heung Jeon
Suk Hwan Park,
Suk Hwan Park
e-mail: hhcho@yonsei.ac.kr
Jeon, Y. H., Park, S. H., Kim, K. M., Lee, D. H., and Cho, H. H. (July 25, 2006). "Effects of Bleed Flow on Heat/Mass Transfer in a Rotating Rib-Roughened Channel." ASME. J. Turbomach. July 2007; 129(3): 636–642. https://doi.org/10.1115/1.2720495
The present study investigates the effects of bleed flow on heat/mass transfer and pressure drop in a rotating channel with transverse rib turbulators. The hydraulic diameter
(Dh)
of the square channel is
40.0mm
. 20 bleed holes are midway between the rib turburators on the leading surface and the hole diameter
(d)
4.5mm
. The square rib turbulators are installed on both leading and trailing surfaces. The rib-to-rib pitch
(p)
is 10.0 times of the rib height
(e)
and the rib height-to-hydraulic diameter ratio
(e∕Dh)
is 0.055. The tests were conducted at various rotation numbers (0, 0.2, 0.4), while the Reynolds number and the rate of bleed flow to main flow were fixed at 10,000 and 10%, respectively. A naphthalene sublimation method was employed to determine the detailed local heat transfer coefficients using the heat/mass transfer analogy. The results suggest that for a rotating ribbed passage with the bleed flow of
BR=0.1
, the heat/mass transfer on the leading surface is dominantly affected by rib turbulators and the secondary flow induced by rotation rather than bleed flow. The heat/mass transfer on the trailing surface decreases due to the diminution of main flow. The results also show that the friction factor decreases with bleed flow.
heat transfer, gas turbines, engines, sublimation, mass transfer, channel flow, turbulence
Flow (Dynamics), Heat, Mass transfer, Friction, Rotation
Local Heat/Mass Transfer Distribution in a Square Channel With Full and V-Shaped Ribs
Convective Heat Transfer Distributions Over Plates with Square Ribs From Infrared Thermography Measurements
Developing and Periodically Developed Flow, Temperature and Heat Transfer in a Ribbed Duct
Average Heat Transfer Measurements Near a Sharp 180Degree Turn Channel for Different Aspect Ratios
IMechE Conference Transaction: In Optical Methods and Data Processing in Heat and Fluid Flow
Heat Transfer in Triangular Channels With Angled Roughness Ribs on Two Walls
Experiments on Local Heat Transfer in a Rotating Square-Ended U-Bend
Effect of Cross-Sectional Aspect Ratio on Turbulent Heat Transfer in an Orthogonally Rotating Rectangular Smooth Duct
Poinastte
Detailed Heat Transfer Distributions in Two-Pass Square Channels With Rib Turbulators and Bleed Holes
Heat Transfer and Friction in Turbulent Pipe Flow With Various Physical Properties
Detailed Heat/Mass Transfer Distributions in a Rotating Smooth Channel With Bleed Flow |
On the Cauchy problem for a one-dimensional compressible viscous polytropic ideal gas | EMS Press
JournalspmVol. 64, No. 1pp. 87–126
On the Cauchy problem for a one-dimensional compressible viscous polytropic ideal gas
Baylor University, Waco, United States
Water Conservancy and Hydroelectric Power, Zhengzhou, China
In this paper, we first prove the regularity and continuous dependence on initial data for
H^i
-solutions
(i=1,2,4)
for large initial data and then show the large-time behavior of
H^i(i=2,4)
-global solutions for small initial data to the Cauchy problem for the compressible Navier--Stokes equations of a one-dimensional viscous polytropic ideal gas. Moreover, we also obtain the large-time behavior of ``small'' classical solutions {\em in the norm of classical solutions} for this model.
Yuming Qin, Yumei Wu, Fagui Liu, On the Cauchy problem for a one-dimensional compressible viscous polytropic ideal gas. Port. Math. 64 (2007), no. 1, pp. 87–126 |
Radiation astronomy/Greens/Quiz - Wikiversity
< Radiation astronomy/Greens(Redirected from Green astronomy/Quiz)
Green astronomy is a lecture from the radiation astronomy department for the course on the principles of radiation astronomy.
You are free to take this quiz based on green radiation astronomy.
To improve your scores, read and study the lecture, the links contained within, listed under See also, External links and in the {{principles of radiation astronomy}} template. This should give you adequate background to get 100 %.
3 Which of the following is not a characteristic of green astronomy?
6 Alpha Centauri A as a star has what green astronomy property?
9 When ionization cones are present, what green characteristics are usually readily observed?
49 What underwater events are the milky green swirls in the Atlantic Ocean off the coast of El Hierro due to?
52 True or False, The O I 557.7 nm line occurs in some meteor wake spectra generally confined to the height interval of 80 - 90 km.
53 Which of the following phenomena are associated with earthshine?
54 True or False, The zodiacal light has components in the green that are linearly polarized.
55 True or False, There are clear trends in the significant day-to-day and month-to-month changes in the zodiacal light.
{\displaystyle {\frac {c2}{\lambda T}}{\frac {1}{e^{\frac {c2}{\lambda T}}-1}}e^{\frac {c2}{\lambda T}}-5=0}
60 If Osiris in ancient Egyptian mythology/observational astronomy corresponds to Saturn, and Horus is the son of Osiris, what classical planet does Horus correspond to?
61 True or False, Telluric mercury lines are light polution lines occurring in the Martian atmosphere.
63 True or False, Green radiation bursts are the most luminous electromagnetic events known to occur in the universe.
64 Which of the following is not a phenomenon associated with optical green astronomy?
65 True or False, The rocky surface of the planet Venus can be detected when Venus is observed using green astronomy.
66 Which of the following is not a phenomenon associated with green astronomy?
67 True or False, The dark green filter improves imaging of cloud patterns on Venus.
68 Observations of comets have benefited greatly from what phenomena of green astronomy?
70 Which of the following is a phenomenon associated with nebulae?
71 True or False, The red shift cannot affect green stars.
73 True or False, The Earth's atmosphere does not transmit green radiation between 480 and 500 nm in wavelength because of water vapor.
74 Green astronomy may help to detect what type of astronomical object?
75 True or False, The position of the Sun can be determined directly with the use of green astronomy.
76 Green astronomy has helped to verify what famous theories?
77 True or False, A green metallic or stony object that is the remains of a meteor is called a meteoroid.
78 Various green radiation observatories occur at different altitudes and geographic locations due to what effect?
79 True or False, The V 2 rocket was first used as a sounding rocket for green astronomy before being converted to a weapon.
80 Moldavite is a mineral that may be associated with what green astronomy phenomenon?
81 True or False, The emission of synchrotron light from the Sun may more accurately fit the spectral radiance of the Sun than black body radiation.
82 True or False, Green Bremsstrahlung radiation is not detected from the Sun because the photosphere isn't ionized enough to produce it.
84 What green astronomy phenomena are associated with Mars?
85 True or False, The Mars Exploration Rover uses its green filter to take panoramic mosaics.
86 Which of the following is not a green characteristic of solar active regions?
88 True or False, Naked sunspots seen in Hβ which are devoid of plage are never associated with coronal holes.
89 The Sun as a star has what green geographical property?
91 True or False, The PC-1 F502N is centered at 501.85 nm with a band pass of 2.97 nm.
92 When close binary stars are present, what characteristics are readily observed?
96 Astronauts onboard the International Space Station used a Nikon digital camera to take pictures of what green astronomy phenomenon?
97 True or False, Chemiluminescence caused mainly by oxygen and nitrogen reacting with hydroxyl ions at heights of a few hundred kilometers contributes to airglow.
100 Yes or No, Oxygen emissions can be green or brownish-red depending on the amount of energy absorbed.
Green astronomy studies all the astronomical objects of the Solar System.
Retrieved from "https://en.wikiversity.org/w/index.php?title=Radiation_astronomy/Greens/Quiz&oldid=2042231" |
High-Energy Atmospheric Physics: Ball Lightning
This article proposes an explanation for High-Energy Atmospheric phenomena through the frames of Hypersphere World-Universe Model (WUM). In WUM, Terrestrial Gamma-Ray Flashes (TGFs) are, in fact, Gamma-Ray Bursts (GRBs). The spectra of TGFs at very high energies are explained by Dark Matter particles annihilation in Geocorona. Lightning initiation problem is solved by GRBs that slam into thunderclouds and carve a conductive path through a thunderstorm. We introduce Multiworld consisting of Macro-World, Large-World, Small-World, and Micro-World, characterized by suggested Gravitational, Extremely-Weak, Super-Weak, and Weak interaction respectively. We propose a new model of Ball Lightning formation based on the Dark Matter Core surrounded by electron-positron plasma in the Small-World.
Hypersphere World-Universe Model, High-Energy Atmospheric Physics, Ball Lightning, Geocorona, Lightning Initiation Problem, Terrestrial Gamma-Ray Flashes, Gamma-Ray Bursts, Dark Matter Core, Electron-Positron Plasma, Multiworld
2.5\times {10}^{-7}\text{kg}/{\text{m}}^{\text{3}}
{G}_{g}
{G}_{g}={G}_{0}\times {Q}^{-1}\propto {\tau }^{-1}
{G}_{0}=\frac{{a}^{2}{c}^{4}}{8\text{π}hc}
{G}_{g}
Q=1
a=\alpha {\lambda }_{e}
{\lambda }_{e}
\alpha
{m}_{e}
{m}_{0}
\alpha ={m}_{e}/{m}_{0}
{m}_{0}
{m}_{0}=h/ac
Q=\tau /{t}_{0}
\tau
{t}_{0}
{t}_{0}=a/c=5.9059674\times {10}^{-23}\text{\hspace{0.17em}}\text{s}
Q=0.759972\times {10}^{40}
R=aQ=1.34558\times {10}^{26}\text{\hspace{0.17em}}\text{m}
{M}_{tot}
{M}_{tot}=6{\text{π}}^{2}{m}_{0}\times {Q}^{2}=4.26943\times {10}^{53}\text{\hspace{0.17em}}\text{kg}
{G}_{W}={G}_{O}\times {Q}^{-1/4}\propto {\tau }^{-1/4}
{G}_{SW}={G}_{O}\times {Q}^{-1/2}\propto {\tau }^{-1/2}
{G}_{EW}={G}_{O}\times {Q}^{-3/4}\propto {\tau }^{-3/4}
{G}_{W}
{G}_{g}
{R}_{W}
{\rho }_{0}
{\rho }_{0}=h/c{a}^{4}
{R}_{W}=a\times {Q}^{1/4}=1.65314\times {10}^{-4}\text{\hspace{0.17em}}\text{m}
~{10}^{-16}\text{\hspace{0.17em}}\text{-}\text{\hspace{0.17em}}{10}^{-17}\text{\hspace{0.17em}}\text{m}
{M}_{W}
{M}_{W}=4\text{π}{\sigma }_{0}{R}_{W}^{2}/{c}^{2}=4\pi {m}_{0}\times {Q}^{1/2}=1.36752\times {10}^{-7}\text{\hspace{0.17em}}\text{kg}
{\sigma }_{0}
{\sigma }_{0}=hc/{a}^{3}
{\rho }_{W}
{\rho }_{W}=3{\rho }_{0}\times {Q}^{-1/4}=7.22621\times {10}^{3}\text{\hspace{0.17em}}\text{kg}/{\text{m}}^{\text{3}}
{G}_{EW}
{G}_{g}
{R}_{EW}
{R}_{EW}=a\times {Q}^{3/4}=1.44115\times {10}^{16}\text{\hspace{0.17em}}\text{m}
{\sigma }_{0}
{M}_{EW}=4\text{π}{\sigma }_{0}{R}_{EW}^{2}/{c}^{2}=4\text{π}{m}_{0}\times {Q}^{3/2}=1.03928\times {10}^{33}\text{\hspace{0.17em}}\text{kg}
{\rho }_{EW}
{\rho }_{EW}=3{\rho }_{0}\times {Q}^{-3/4}=8.28918\times {10}^{-17}\text{\hspace{0.17em}}\text{kg}/{\text{m}}^{\text{3}}
{G}_{SW}
{G}_{g}
{R}_{SW}
{R}_{SW}=a\times {Q}^{1/2}=1.54351\times {10}^{6}\text{\hspace{0.17em}}\text{m}
{M}_{SW}
{M}_{SW}=4\text{π}{m}_{0}\times Q=1.19215\times {10}^{13}\text{\hspace{0.17em}}\text{kg}
{\rho }_{SW}
{\rho }_{SW}=3{\rho }_{0}\times {Q}^{-1/2}=7.73947\times {10}^{-7}\text{\hspace{0.17em}}\text{kg}/{\text{m}}^{\text{3}}
{\rho }_{\text{DMF1}}
{\rho }_{\text{DMF1}}\cong 2.5\times {10}^{-7}\text{\hspace{0.17em}}\text{kg}/{\text{m}}^{\text{3}}
{m}_{\text{DMF}1}
{M}_{micro}
{m}_{\text{DMF}1}\times {M}_{micro}=2{m}_{0}^{2}\times {Q}^{1/2}=2.71692\times {10}^{-36}\text{\hspace{0.17em}}{\text{kg}}^{2}
{m}_{\text{DMF1}}
{m}_{\text{DMF}1}=2.34419\times {10}^{-24}\text{\hspace{0.17em}}\text{kg}
{M}_{micro}\approx 1.16\times {10}^{-12}\text{\hspace{0.17em}}\text{kg}
{r}_{micro}
{M}_{micro}
{r}_{micro}\cong {10}^{-2}\text{\hspace{0.17em}}\text{m}
{r}_{micro}
{M}_{micro}
{M}_{macro}
{m}_{e}\times {M}_{macro}=2{m}_{0}^{2}\times {Q}^{1/2}
{m}_{e}
{m}_{e}\approx 9.11\times {10}^{-31}\text{\hspace{0.17em}}\text{kg}
{M}_{macro}\cong 3\times {10}^{-6}\text{\hspace{0.17em}}\text{kg}
{\rho }_{atm}\cong 1.25\text{\hspace{0.17em}}\text{kg}/{\text{m}}^{\text{3}}
{r}_{\mathrm{min}}
{r}_{\mathrm{min}}\cong 0.83\times {10}^{-2}\text{\hspace{0.17em}}\text{m}
{\rho }_{atm}
Netchitailo, V.S. (2019) High-Energy Atmospheric Physics: Ball Lightning. Journal of High Energy Physics, Gravitation and Cosmology, 5, 360-374. https://doi.org/10.4236/jhepgc.2019.52020
1. Netchitailo, V.S. (2019) Solar System. Angular Momentum. New Physics. JHEPGC, 5, 112-139. https://doi.org/10.4236/jhepgc.2019.51005
2. Netchitailo, V.S. (2016) 5D World-Universe Model. Gravitation. Journal of High Energy Physics, Gravitation and Cosmology, 2, 328-343. https://doi.org/10.4236/jhepgc.2016.23031
3. Charman, N. (1972) The Enigma of Ball Lightning. New Scientist, 56, 632-635. https://books.google.com/books?id=TCTpu1UVFsYC&pg=PA633#v=onepage&q&f=false
4. Tesla, N. (1904) The Transmission of Electrical Energy without Wires. Electrical World and Engineer. http://web.archive.org/web/20051222121927/http://tfcbooks.com/tesla/wireless01.htm
5. (1988) Tesla and Ball Lightning. https://www.bibliotecapleyades.net/tesla/esp_tesla_20.htm
6. Kapitsa, P.L. (1955) The Nature of Ball Lightning. In: Ritchie, D.J., Ed., Ball Lightning: A Collection of Soviet Research in English Translation, 1961 Edition, Consultants Bureau, New York, 11-16.
7. Handel, P.H. (1975) Maser Theory of Ball Lightning. Bulletin of the American Physical Society Series II, 20, 26.
8. Ashby, D.E.T.F. and Whitehead, C. (1971) Is Ball Lightning Caused by Antimatter Meteorites? Nature, 230, 180-182.
9. Are Antimatter Meteorites Optical Illusions? New Scientist, 49. https://books.google.com/books?id=yXq8b86qFLQC&pg=PA661&lpg=PA661&dq=Are+antimatter+meteorites+optical+illusions?%22&source=bl&ots=5aH0iPzKNj&sig=gVpclonPkexWBhhzoIKi6AVEEk8&hl=en&sa=X&ved=2ahUKEwje9vCt1avfAhWqJTQIHR3eCv8Q6AEwAHoECAgQAQ#v=onepage&q=Are%20antimatter%20meteorites%20optical%20illusions%3F%22&f=false
10. NASA’s Fermi Catches Thunderstorms Hurling Antimatter into Space. https://www.nasa.gov/mission_pages/GLAST/news/fermi-thunderstorms.html
11. Vuyk, L. (1996) Practical Containment, Development and Exploitation of Energy Contained in Ball Lightning Discharges or Black Holes. https://nl.espacenet.com/publicationDetails/biblio?II=0&ND=3&adjacent=true&locale=nl_NL&FT=D&date=19970912&CC=NL&NR=1002570C1&KC=C1
12. Rabinowitz, M. (2002) Little Black Holes: Dark Matter and Ball Lightning. arXiv:0212251.
13. Thornhill, W. (2006) The IEEE, Plasma Cosmology and Extreme Ball Lightning. https://www.holoscience.com/wp/the-ieee-plasma-cosmology-and-extreme-ball-lightning/
14. Van Devender, P. (2011) Extreme Ball Lightning: New Physics, New Energy Source, or Just Great Fun. (Conference)/OSTI.GOV.
15. Wu, H.-C. (2014) Theory of Ball Lightning. arXiv:1411.4784.
16. Cen, J., Yuan, P. and Xue, S. (2014) Observation of the Optical and Spectral Characteristics of Ball Lightning. Physical Review Letters, 112, Article ID: 035001. https://doi.org/10.1103/PhysRevLett.112.035001
17. Dwyer, J.R. (2012) The Mystery of Lightning. http://www.insightcruises.com/events/sa24/PDF/The_Mysteries_of_Lightning.pdf
18. Gurevich, A.V., Milikh, G.M. and Roussel-Dupre, R. (1992) Runaway Electron Mechanism of Air Breakdown and Preconditioning during a Thunderstorm. Physics Letters A, 165, 463-468. https://doi.org/10.1016/0375-9601(92)90348-P
19. Fishman, G.J., et al. (1994) Discovery of Intense Gamma-Ray Flashes of Atmospheric Origin. Science, 264, 1313-1316. https://doi.org/10.1126/science.264.5163.1313
20. Dwyer, J.R., Liu, N. and Rassoul, H.K. (2013) Properties of the Thundercloud Discharges Responsible for Terrestrial Gamma-Ray Flashes. Geophysical Research Letters, 40, 4067-4073. https://doi.org/10.1002/grl.50742
21. Fishman, G.J., et al. (2011) Temporal Properties of Terrestrial Gamma-Ray Flashes from the Gamma-Ray Burst Monitor on the Fermi Observatory. Journal of Geophysical Research, 116, A07304. https://doi.org/10.1029/2010JA016084
22. Briggs, M.S., et al. (2013) Terrestrial Gamma-Ray Flashes in the Fermi Era: Improved Observations and Analysis Methods. Journal of Geophysical Research Space Physics, 118, 3805-3830. https://doi.org/10.1002/jgra.50205
23. Dwyer, J.R. and Smith, D.M. (2005) A Comparison between Monte Carlo Simulations of Runaway Breakdown and Terrestrial Gamma-Ray Flash Observations. Geophysical Research Letters, 32, L22804. https://doi.org/10.1029/2005GL023848
24. Tavani, M., et al. (2011) Terrestrial Gamma-Ray Flashes as Powerful Particle Accelerators. Physical Review Letters, 106, Article ID: 018501.
25. Dwyer, J.R. (2012) The Relativistic Feedback Discharge Model of Terrestrial Gamma Ray Flashes. Journal of Geophysical Research, 117, A02308. https://doi.org/10.1029/2011JA017160
26. Østgaard, N., Gjesteland, T., Hansen, R.S., Collier, A.B. and Carlson, B. (2012) The True Fluence Distribution of Terrestrial Gamma Flashes at Satellite Altitude. Journal of Geophysical Research, 117, A03327. https://doi.org/10.1029/2011JA017365
27. Carlson, B., Lehtinen, N. and Inanm, U. (2007) Constraints on Terrestrial Gamma-Ray Flash Production Derived from Satellite Observations. Geophysical Research Letters, 34, L08809. https://doi.org/10.1029/2006GL029229
28. Gjesteland, T., Østgaard, N., Connell, P.H., Stadsnes, J. and Fishman, G.J. (2010) Effects of Dead Time Losses on Terrestrial Gamma Ray Flash Measurements with the Burst and Transient Source Experiment. Journal of Geophysical Research, 115, A00E21. https://doi.org/10.1029/2009JA014578
29. Xu, W., Celestin, S. and Pasko, V. (2012) Source Altitudes of Terrestrial Gamma-Ray Flashes Produced by Lightning Leaders. Geophysical Research Letters, 39, L08801. https://doi.org/10.1029/2012GL051351
30. Celestin, S., Xu, W. and Pasko, V. (2012) Terrestrial Gamma Ray Flashes with Energies up to 100 MeV Produced by Nonequilibrium Acceleration of Electrons in Lightning. Journal of Geophysical Research, 117, A05315. https://doi.org/10.1029/2012JA017535
31. Dwyer, J.R. (2008) The Source Mechanisms of Terrestrial Gamma-Ray Flashes (TGFs). Journal of Geophysical Research, 113, D10103.
32. NASA (2012) Solar System. http://chandra.harvard.edu/xray_sources/solar_system.html
33. Wargelin, B.J., et al. (2014) Observation and Modeling of Geocoronal Charge Exchange X-Ray Emission during Solar Wind Gusts. The Astrophysical Journal, 796, 1. https://doi.org/10.1088/0004-637X/796/1/28
34. Cravensa, T.E., et al. (2009) Solar Wind Charge Exchange Contributions to the Diffuse X-Ray Emission. AIP Conference Proceedings, 1156, 37. https://doi.org/10.1063/1.3211832
35. Atmospheric Windows. http://www.pas.rochester.edu/~blackman/ast104/windows.html
36. Netchitailo, V.S. (2017) Burst Astrophysics. Journal of High Energy Physics, Gravitation and Cosmology, 3, 157-166. https://doi.org/10.4236/jhepgc.2017.32016
37. Oreshko, A.G. (2012) Observation of Dark Spherical Area after Passage of Ball Lightning through Thick Absorbers. https://www.researchgate.net/profile/Alexander_Oreshko/publication/312218738_Observation_of_Dark_Spherical_Area_After_Passage_of_Ball_Lightning_Through_Thick_Absorbers/links/5877307808ae329d6226e786/Observation-of-Dark-Spherical-Area-After-Passage-of-Ball-Lightning-Through-Thick-Absorbers.pdf
38. Netchitailo, V.S. (2013) World-Universe Model. viXra:1303.0077
39. Netchitailo, V.S. (2016) Overview of Hypersphere World-Universe Model. Journal of High Energy Physics, Gravitation and Cosmology, 2, 593. https://doi.org/10.4236/jhepgc.2016.24052 |
Fair Price Marking - Delta Exchange - User Guide
Leverage that is inherent in derivative contracts, combined with the high volatility of cryptocurrencies, can lead to unwarranted liquidations. Lack of liquidity could further exacerbate this situation as price swings in a derivative contract relative to the underlying index could widen even further.
To avoid such issues and ensure a smooth trading experience, Delta Exchange marks all open position at Fair Mark Price instead of Last Traded Price. The Mark Price has lesser volatility and is more resilient against attempts to manipulate the market.
It is worth noting that Fair Price marking is relevant only for the computation of Unrealized PnL and Liquidation price. Realized PnL is based off actual traded prices and is thus not impacted by Mark Price.
Before going on to explaing how Mark Price is computed, we first need to introduce the notion of Impact Price. This price tries to estimate the price at which a typical long or short position (called Impact Position) in the futures contract can be entered at any given time.
Impact Size, in terms of number is contracts to be traded, is provided in the specifications for each contract. It is easy to see that Impact Price is a function of: (a) Impact Size and (b) current state of the order book.
Impact\ Bid\ Price = Average\ fill\ price\ to\ execute\ a\ typical\ short\ trade
Impact\ Ask\ Price = Average\ fill\ price\ to\ execute\ a\ typical\ long\ trade
Impact\ Mid\ Price = Average\ of\ Impact\ Bid\ Price\ and\ Impact\ Ask\ Price
Mark Price of Futures & Perpetual Contracts
The price of a futures contract converges to the underlying index price at the time of contract maturity, i.e.
Futures\ Price = Underlying\ Index\ Price
At all other times, the price of a futures contract broadly moves in tandem with the price of the Underlying Index, with the difference between the two referred to as basis, i.e.
Basis = Futures\ Price - Underlying\ Index\ Price
Since the Underlying Index is the foundation of the futures contract, it is logical to assume that
Futures\ Fair\ Price = Underlying\ Index\ Price + Fair\ Basis
Underlying\_Index\_Price
is obviously independent of the trading happening on Delta Exchange and is sourced in real-time from leading spot exchanges.
Fair Basis Calculation
We first compute an annualised fair value basis, %AnnualisedBasis:
\%AnnualisedBasis = (Impact\ Mid\ Price/ Underlying\ Index\ Price - 1) * (365*86400/ time\ to\ expiry\ in\ sec)
Note that For perpetual contracts
time\_to\_expiry
is always 8 hours.
%AnnualisedBasis is computed only once every 5 seconds. Further, if the market is illiquid, i.e.
(Impact\ Ask\ Price - Impact\ Bid\ Price) > Maintenance\ Margin
%AnnualisedBasis is not updated.
Next, %FairBasis is computed as the moving average of the 12 most recent values of %AnnualisedBasis. To avoid anomalous values, the %FairBasis is bounded by certain hard limits, which can vary from contract to contract.
Fair Basis is computed using %FairBasis as per the following equation:
Fair\ Basis = Underlying\ Index\ Price * \%Fair\ Basis * (time\ to\ expiry\ in\ sec/ (365* 86400))
Now that we have the Fair\ Basis, the Mark Price of the contract can be easily computed:
Futures\ Mark\ Price = Underlying\ Index\ Price + Fair\ Basis
It is worth noting that only live positions are marked using the Mark Price. Thus, while unrealised PnL may swing with the Mark Price, realised PnL is unimpacted by Mark Price and depends only on the actual traded prices.
Mark Price of Calendar Spread Contracts
The mark price of calendar spread contracts is computed as the difference of the fair prices of the two futures contracts that underlie the spread contract, with the fair prices computed as per the following formula:
Fair\ Price = Spot\ Price + moving\ average (annualised\ basis) * (time\ to\ expiry\ in\ sec/ (365* 86400))
Therefore, the mark price of a calendar spread contract can be written as:
Mark\ Price = Fair\ Price\ longer\ dated\ futures - Fair\ Price\ shorter\ dated\ futures
Mark Rate of BitMex Funding Rate Swap
Open positions in the BitMex Funding Rate swap product are marked at the annualised implied rate from the basis of the BTC futures of the same maturity as the swap. BitMex published fair basis of the relevant BTC futures. We use the annual rate implied by this fair basis as the mark rate.
The basis implied annual rate tends to become quite volatile as the futures contract heads into expiry. To keep the mark rate stable and avoid unnecessary liquidations, when the maturity of the swap contract is less tha 10 days away, we start using the basis implied annual rate of the next quarterly BTC futures.
Fair Price Marking Leading into Settlement
Contracts on Delta Exchange are settled on the 30 minute TWAP (time-weighted average price) of its Underlying Index. To ensure that at the time of settlement there are no under-margined positions, the transition from using Underlying Index Price to TWAP of the Underlying Index Price in the calculation of Mark Price is done gradually.
When the settlement time of a contract is one hour away, we switch to a weighted average of Underlying Index Price and TWAP of the Underlying Index Price in Mark Price computation. At the time of switch, weight of
Underlying\_Index\_Price
(and weight of
TWAP\_Underlying\_Index\_Price
0\%
. Over the next 30 minutes, these weights are changed every minute such that weights converge to
0\%
100\%
respectively. This means that in the last 30 minutes leading into contract settlement, it is
TWAP\_Underlying\_Index\_Price
that is used in the computation of the Mark Price. |
Ashley Sheridan - The Affects of Covid on Web Accessibility
accessibility translations
We've been a little over 2 years now with Covid in our lives, but there's still so much we don't know about it. One aspect of that is what the enduring effects of it are for some people, or Long-Covid as it is known. These effects vary in their symptoms and severity, but they can have an impact on our lives and can prevent us from doing all the things we are used to.
Make Designs Cleaner
Rely on Traditional Navigation
Consider How Your Content Can Be Translated
The Office of National Statistics has this to say:
An estimated 1.3 million people living in private households in the UK (2.1% of the population) were experiencing self-reported long COVID
Given that this is all self-reported statistics, the actual number is likely to be larger, although it's impossible to really say by how much.
Of those that reported having long lasting symptoms after having Covid, the main ones that might have an impact on how people are able to use the web are:
These symptoms can translate into the following main accessibility issues for people:
Problems with concentration, which makes long or ccomplex user journeys difficult
Problems remembering non-obvious things, such as entries on a previous form screen
Difficulty with designs that might trigger dizziness, causing people to completely avoid or drop out from some online tasks
Pain using a traditional mouse or keyboard to navigate
Innability to listen to any audio, such as people speaking on a video
While these cognition related problems have always been accessibility barriers, they tend to be the least considered when working to make websites more accessible. Even looking over the WCAG 2.2 Guidelines you can see that most of the focus is visual, auditory, and motor problems. Also, it's a common enough misconception that accessibility is only for blind users that it's made the list of top accessibility myths. This is only further impacted by automated accessibility tests for cognitive issues being few and far between. It's a lot more difficult for a computer to determine how easy something is to use, or how much a user journey relies on a user remembering things. Far easier to detect missing text for an image, or a missing label on a checkbox.
The recent and sudden influx of people with these problems is highlighting the need to focus more on the cognitive impact of websites on individuals. Personally, I've found post-covid brain fog to be incredibly difficult to work with, and I'm finding myself having to take more time on a given task than I would have previously. While I am able to fully continue in my work with this, there are many people who are in a far worse situation, people who have had to take a step back from their normal roles and responsibilities. According to a report by The Lancet, 88% of people who report Long Covid symptoms have listed brain fog as a symptom.
Up until now, cognitive issues have generally been addressed less than others, so the obvious thing to do would be to consider these types of issues more. Given that automated tests, like those found in Lighthouse or aXe, are less capable of finding these problems, it will require a little more in the way of manual testing to discover and resolve problems like these:
The large majority of forms online are already pretty simple, like a login, or a search. Sometimes though, you need to collect a lot of data. Consider a form for applying for a job. Instead of one giant form, break it into smaller logical steps, and show the user where they are within that series of steps. Use clear input labelling, and highlight errors and supply error messages close to their respective fields to allow people to more easily understand what parts may need fixing.
For your users who might be easily distracted, they can easily come back to this form and see what the current state is. They're reminded of what is required, and what they've already done, and anything that needs their immediate attention is made more obvious.
Besides the visuals, there are other things going on that benefit those users who also rely on their screen readers (or similar) to navigate:
The ribbon is a series of links with the current one indicated using aria-current="step", allowing navigation between the steps.
Input fields with errors use the actual invalid state, not just a CSS class. They're also associated with their visible error messages using aria-describedby. Without that, these error messages might easily be completely missed by a screen reader.
Input labels are always visible. This means someone can start writing, go away from their computer, and still complete it correctly when they return. This is not possible with placeholder text.
Labels are properly associated with their fields, it's not just visually nearby text.
The collection of form fields for each section are grouped within a <fieldset> and the heading text becomes its <legend>. This means that they can be more easily navigated using assistive technology.
Visually busy designs can make it more difficult for people to hone in on the specific things they need to complete a given task. Consider the often used Lings Cars as an example of exactly what not to do:
There's so much going on with this design, it's incredibly difficult to navigate even if you're not in a distracted or confused state. There are multiple designs of navigation, as well as elements which look like they should be links but aren't. The background is incredibly busy and the lines of it make it difficult to focus on content. The large clash of colours and fonts make reading more difficult, especially if you have eyesight issues or you're in bad lighting conditions, and the sheer number of animations are an assault on our focus.
A cleaner and simpler design makes it easier for all of your users to find the content they need more quickly, but it can be especially beneficial to anyone who is already struggling. A good, but limited, font selection can help separate headings from body text.
Navigation is one of the fundamentals of the web that Tim Berners Lee had in mind when he created the World Wide Web, the ability to link from one page to another. In the years since its inception navigation has become more complex, but at its heart it remains true to its core.
As the web has grown, it's allowed us to do more and more with how we present our navigation. We can build dynamic radial menus that break the typical box-like structure that plagues the Web, or even the website accompanying the most recent version of The Witcher TV series which uses scroll-jacking to navigate between sections.
But, this introduces a new set of problems for users who rely on the familiar. Now they have to learn how to use this new type of navigation as well as continue on with the task they wanted to complete on your site. While it might seem trivial to use a new form of website navigation, but for people affected by things like Covid brain fog.
There are many different studies and opinions about how long a person takes to come to a decision on whether to remain on your site or leave, with anything from a few seconds right up to 15. One fact remains in all of this though: people will form an opinion, and they will leave if your website is not easy enough for them to use and they feel they can get the same information elsewhere.
There's a content pattern referred to as the 'F-shape' which came from a study from the Nielson Normal Group. This shape typically covers the key content and navigation of the website if it follows a traditional approach. Because we're almost trained to expect this pattern, relying on it to build your navigation can greatly improve the time needed for anyone to find what they're looking for.
The average reading age in the UK is estimated to be about that of a 9 year old, which means that if you want to maximise the audience for your content, you need to make it readable for a 9 year old.
This can be difficult or impossible for some types of content, but we should aim to use the simplest language for our audience. This means avoiding complex words or phrasing if it's not completely necessary, and you should try to use language that's simpler to read. There is a test which can give you an estimate about the reading age of your content called a Flesch-Kinkaid test. The exact formula for this is:
206.835-1.015\left(\frac{\mathrm{total words}}{\mathrm{total sentences}}\right)-84.6\left(\frac{\mathrm{total syllables}}{\mathrm{total words}}\right)
This live test is specifically for English content, but there is ongoing work to test the readability of other languages:
Test your content readability:
I did a fuller write-up of this test, along with the Javascript source code I wrote the live test in on my blog.
The mothertongue language of your users might not always be the same language that your websites content is written in. Under normal circumstances this can be a problem; they will either be translating it themselves if they know the language, or they will be relying on automated translations and trying to fill the gaps (automated services are good, but not perfect.) If people are then further impacted by Covid brain fog, they might be struggling far more to understand your websites content.
You could look at your visitor tracking/logs to see where people are coming from, and use that to make an educated guess about what languages you should offer translations. For example, if you notice a lot of visitors coming from Switzerland, then you might opt to add translations in German and French to make your content more available to the majority of your Swiss audience.
However, getting your content translated like this is not cheap, and might not be a cost-effective option for you, so you may have to rely on a browser offering automatic translations in the visitors preferred locale. A lot of these tools struggle with translating content that is in certain attributes in the HMTL tags, meaning the content ends up being presented to them in different languages, unexpectedly.
Avoid using the placeholder attribute for form fields. Instead, use visible labels using the <label> tag.
Use aria-labelledby to refer to existing page elements rather than aria-label, which some translation tools don't properly detect.
Mark up the content language for your page, and any places where it changes using the lang attribute. This helps automatic translators get their translations correct. They might try to guess but get it wrong, which could make their translations incorrect.
The problems I've highlighted here are by no means new, they have always been issues to think about and address. I do believe that these issues tend to feature lower down on the list of efforts. The way that Covid has left some people should be cause to rethink on the priority of some of our accessibility efforts, and ensure we consider the whole sprectrum of barriers that people can encounter on the Web.
Accessibility Concerns Over Chrome Removing Alert, Prompt, And Confirm Modals |
Classical Bohr Atom - AstroBaki
Classical Bohr Atom
Radiation and Matter, to Order of Magnitude (Aaron Parsons)
Bohr Hydrogen Model 1: Radius (Quantum Chemistry)
Bohr Hydrogen Model 2: Energy (Quantum Chemistry)
Rotational Spectroscopy Example (Quantum Chemistry)
Bohr Model (Wikipedia)
Diatomic Molecule (Wikipedia)
Rotational Levels (Tuckerman, NYU)
Atomic and Molecular Quantum Numbers
Rovibrational Transitions
1 Order of Magnitude Interaction of Radiation and Matter
Order of Magnitude Interaction of Radiation and Matter
It turns out that you can tell a cute little story about the energy levels in hydrogen-like atoms using a simple Bohr model of the hydrogen atom and classical physics. You just need a couple small doses of quantum mechanics in the beginning to get you started on the right track.
Electronic Transitions:
Sketch of Rydberg-
{\displaystyle \alpha }
and Lyman-
{\displaystyle \alpha }
for hydrogen
We’ll start with the Bohr atom, and follow a classical derivation/estimation path. The only quantum mechanics that we need to inject is that angular momentum comes in units of
{\displaystyle \hbar }
. So let’s begin by quantizing the angular momentum associated with the electron’s orbit around the nucleus. We’ll define the radius of the electron’s orbit as
{\displaystyle a_{0}}
(the Bohr radius), and the velocity that the electron travels at will be
{\displaystyle v_{e}}
. This gives us the following expression for the electron’s angular momentum for
{\displaystyle n}
{\displaystyle \hbar }
{\displaystyle m_{e}v_{e}a_{0}=n\hbar \,\!}
Next, we balance the force required to keep the
{\displaystyle e^{-}}
in a circular orbit against the electric force:
{\displaystyle m_{e}{\frac {v_{e}^{2}}{a_{0}}}={\frac {Ze^{2}}{a_{0}^{2}}}\,\!}
This gives us the following solution for the Bohr radius:
{\displaystyle a_{0}={\frac {\hbar ^{2}}{m_{e}Ze^{2}}}\approx 0.52{\frac {\mathrm {\AA} }{Z}}\,\!}
A Rydberg is the energy required to ionize an H atom from the ground state. It is
{\displaystyle \sim 13.6eV}
. We can estimate it as half of the energy we get integrating the electric force from
{\displaystyle r=a_{0}\rightarrow \infty }
, (the other half is the kinetic energy of the orbiting electron):
{\displaystyle {\rm {Rydberg}}\equiv {\frac {Ze^{2}}{2a_{0}}}={\frac {Z^{2}e^{4}m_{e}}{\hbar ^{2}}}=13.6\cdot Z^{2}eV\,\!}
Fine Structure:
Fine Structure comes from the interaction of the magnetic moment of the
{\displaystyle e^{-}}
with the a
{\displaystyle {\vec {B}}}
caused by the Lorentz-transformed Coulomb field of the proton (generated by the
{\displaystyle e^{-}}
’s motion). The energy of a dipole interaction is
{\displaystyle E={\vec {\mu }}\cdot {\vec {B}}}
, so we’d expect that:
{\displaystyle \Delta E\sim \mu B\,\!}
We can estimate B by just applying a Lorentz boost to the electric field of the nucleus:
{\displaystyle B\sim {\frac {Ze}{a_{0}^{2}}}{\frac {v}{c}},\,\!}
{\displaystyle {\frac {v}{c}}\sim {\frac {Ze^{2}}{\hbar c}}}
. For reference,
{\displaystyle {\frac {v}{c}}}
{\displaystyle Z=1}
{\displaystyle {\frac {e^{2}}{\hbar c}}\equiv \alpha \approx {\frac {1}{137}}}
, which is the fine structure constant.
Next we need to estimate the magnetic dipole of an electron
{\displaystyle \mu _{e}}
{\displaystyle {\vec {B}}}
of a magnetic dipole goes as
{\displaystyle B_{di}\sim {\frac {\mu }{r^{3}}}}
, so we can estimate the magnetic dipole of an electron as something that produces the
{\displaystyle {\vec {B}}}
of a spinning electron at
{\displaystyle r_{e}}
, the classical electron radius:
{\displaystyle \mu \sim B_{e}{\big |}_{r_{e}}r_{e}^{3}.\,\!}
{\displaystyle r_{e}}
by setting the rest mass energy of the
{\displaystyle e^{-}}
to the electrostatic potential energy of the electron:
{\displaystyle {\begin{aligned}m_{e}c^{2}&\sim {\frac {e^{2}}{r_{e}}}\\r_{e}&\sim {\frac {e^{2}}{m_{e}c^{2}}}\\\end{aligned}}\,\!}
{\displaystyle B_{e}}
, we can use the Maxwell Equations:
{\displaystyle {\begin{aligned}{\vec {\nabla }}\times {\vec {B}}&={\frac {4\pi }{c}}J\\{\frac {B_{e}}{2\pi r_{e}}}&={\frac {4\pi }{c}}{\frac {I}{4\pi r_{e}^{2}}}\\\end{aligned}}\,\!}
We can estimate the current,
{\displaystyle I}
from the spin timescale of the electron,
{\displaystyle I\sim {\frac {e}{t_{spin}}}}
, and because the electron spin is quantized,
{\displaystyle \hbar =m_{e}r_{e}^{2}{\frac {2\pi }{t_{spin}}}}
. After some algebra, we get that:
{\displaystyle \mu _{e}={\frac {e\hbar }{2m_{e}c}}\,\!}
(Bohr magneton for
{\displaystyle e^{-}}
{\displaystyle \mu _{p}={\frac {Ze\hbar }{2m_{p}c}}\,\!}
(Bohr magneton for nucleus) Finally, we plug these in for our expression for the energy:
{\displaystyle {\begin{aligned}\Delta E&\sim {\frac {e\hbar }{2m_{e}c}}{\frac {Ze}{a_{0}^{2}}}Z\alpha \\&\sim Z^{4}\alpha \cdot {\rm {Ryd}}\\\end{aligned}}\,\!}
where we take the Rydberg to be 13.6 eV and have factored out its
{\displaystyle Z}
dependence explicitly.
Hyperfine Structure:
Instead of using the Bohr magneton, we use the intrinsic magnetic moment (spin) of the nucleus:
{\displaystyle B{\big |}_{p}\sim {\mu _{p} \over a_{0}^{3}}\sim B_{fine}{m_{e} \over m_{p}}\,\!}
{\displaystyle \Delta E\sim \Delta E_{fine}{m_{e} \over m_{p}}\,\!}
Note that Fine and Hyperfine are magnetic dipole transitions.
{\displaystyle e^{-}}
level transitions are electric dipole transitions. Magnetic dipole transitions are generally weaker.
Vibrational Transitions in Molecules:
Our general technique with vibrational transitions is to model them as harmonic oscillators. Thus, they should have the characteristic harmonic energy series:
{\displaystyle E_{n}=(n+{\frac {1}{2}})\hbar \omega _{0}\,\!}
For a harmonic oscillator,
{\displaystyle \omega _{0}={\sqrt {k \over m}}}
. We estimate that since the force for a spring is
{\displaystyle k\cdot x}
, and that force should be about the Coulomb force on
{\displaystyle e^{-}}
’s. If we say that atoms stretch with respect to each other about a Bohr radius:
{\displaystyle ka_{0}\sim {e^{2} \over a_{0}^{2}}\,\!}
{\displaystyle \Delta E{\big |}_{vib \atop trans}\sim Ryd\cdot {\sqrt {m_{e} \over A\cdot m_{p}}}\,\!}
where A is the atomic mass # of our atoms.
Rotational Transitions in Molecules:
The thing to remember is that angular momentum comes in units of
{\displaystyle \hbar }
Retrieved from "http:///astrobaki/index.php?title=Classical_Bohr_Atom&oldid=5138" |
Coulomb Focusing - AstroBaki
Coulomb Focusing
Revision as of 17:38, 5 December 2017 by C207 (talk | contribs)
Short Topical Videos[edit]
Coulomb Focusing (Aaron Parsons)
Need to Review?[edit]
Collisional Excitations
1 Coulomb Focusing
Imagine an incident electron has kinetic energy
{\displaystyle >h\nu _{21}}
{\displaystyle {\frac {1}{2}}m_{r}v^{2}\approx {\frac {1}{2}}m_{e}v^{2}>h\nu _{21}\,\!}
Coulomb focusing gives
{\displaystyle {1 \over v^{2}}}
We want to know how far away an electron with
{\displaystyle v}
can be aimed and still hit the
{\displaystyle a_{0}}
radius cloud around the ion. This is
{\displaystyle b}
, the impact parameter. Our collision cross-section
{\displaystyle =\pi b^{2}}
. Our angular momentum is conserved, so
{\displaystyle m_{e}v\,b=m_{e}v_{f}a_{0}\,\!}
{\displaystyle v_{f}^{2}=v^{2}+v_{\perp }^{2}}
{\displaystyle v_{\perp }}
is the velocity
{\displaystyle \perp }
to the original electron velocity. This is a result of it falling toward the ion. Then:
{\displaystyle {\frac {1}{2}}m_{e}v_{\perp }^{2}\sim {Ze^{2} \over a_{0}}\,\!}
{\displaystyle v_{f}^{2}=v^{2}+{Ze^{2} \over m_{e}a_{0}}\,\!}
{\displaystyle b={a_{0}v_{f} \over v}\,\!}
{\displaystyle \pi b^{2}={\pi a_{0}^{2} \over v^{2}}\left[v^{2}+{Ze^{2} \over m_{e}a_{0}}\right]\,\!}
{\displaystyle =\pi a_{0}^{2}\left[1+\underbrace {Ze^{2} \over m_{e}v^{2}a_{0}} _{Coulomb\ focusing \atop factor}\right]\,\!}
Generally, the Coulomb focusing factor
{\displaystyle >1}
because we want to excite, not ionize.
{\displaystyle a_{0}={\hbar ^{2} \over Ze^{2}m_{e}}}
{\displaystyle \pi b^{2}={\pi \hbar ^{2} \over m_{e}v^{2}}\,\!}
{\displaystyle \sigma _{12}={\pi \hbar ^{2} \over m_{e}^{2}v^{2}}\overbrace {\left({\Omega (1,2) \over g_{1}}\right)} ^{quantum\ mechanical \atop correction\ factor}\,\!}
{\displaystyle \Omega }
is the “collisional strength”, and generally is 0 below the
{\displaystyle v}
threshold, goes to 1 at the threshold, and decreases for increasing
{\displaystyle v}
, with some occasional spikes. Generally, it is of order 1, with some slight temperature dependency.
{\displaystyle q_{12}=\left\langle \sigma _{12}v\right\rangle \propto \left\langle {1 \over v}\right\rangle \propto {1 \over {\sqrt {T}}}\,\!}
2000 K gas.
{\displaystyle v_{term}\sim {\sqrt {\gamma kT \over m}}}
{\displaystyle v\sim {\sqrt {2000 \over 100}}42\cdot 1{km \over s}\approx 160{km \over s}}
{\displaystyle \sigma _{12}\sim 10^{-14}cm^{2}\left({\Omega (1,2) \over g_{1}}\right)\,\!}
{\displaystyle \sigma _{12}{\big |}_{osterbrock}\sim 10^{-15}cm^{2}\,\!}
Retrieved from "http:///astrobaki/index.php?title=Coulomb_Focusing&oldid=5149" |
NCERT Solutions for Class 9 Maths Chapter 1 Number Systems are created by the expert faculty at BYJU’S. These Solutions of NCERT Maths help the students in solving the problems adroitly and efficiently for the first term. They also focus on formulating the solutions of Maths in such a way that it is easy for the students to understand. The NCERT Solutions for Class 9 aim at equipping the students with detailed and step-wise explanations for all the answers to the questions given in the exercises of this Chapter.
In NCERT Solutions for Class 9 Maths Chapter 1, students are introduced to a lot of important topics that are considered to be very important for those who wish to pursue Mathematics as a subject in their higher classes. Based on these NCERT Solutions, students can practise and prepare for their upcoming first term exams, as well as endow themselves with the basics of Class 10 for the term wise exams then. These Maths Solutions of NCERT Class 9 are helpful as they are prepared with respect to the latest update on CBSE syllabus for 2021-22 and its guidelines.
Download PDF of NCERT Solutions for Class 9 Maths Chapter 1- Number Systems
Access Answers to NCERT Class 9 Maths Chapter 1 – Number Systems
1. Is zero a rational number? Can you write it in the form p/q where p and q are integers and q ≠ 0?
We know that, a number is said to be rational if it can be written in the form p/q , where p and q are integers and q ≠ 0.
Taking the case of ‘0’,
Zero can be written in the form 0/1, 0/2, 0/3 … as well as , 0/1, 0/2, 0/3 ..
Since it satisfies the necessary condition, we can conclude that 0 can be written in the p/q form, where q can either be positive or negative number.
As we have to find 6 rational numbers between 3 and 4, we will multiply both the numbers, 3 and 4, with 6+1 = 7 (or any number greater than 6)
i.e., 3 × (7/7) = 21/7
and, 4 × (7/7) = 28/7. The numbers in between 21/7 and 28/7 will be rational and will fall between 3 and 4.
Hence, 22/7, 23/7, 24/7, 25/7, 26/7, 27/7 are the 6 rational numbers between 3 and 4.
There are infinite rational numbers between 3/5 and 4/5.
To find out 5 rational numbers between 3/5 and 4/5, we will multiply both the numbers 3/5 and 4/5
with 5+1=6 (or any number greater than 5)
i.e., (3/5) × (6/6) = 18/30
and, (4/5) × (6/6) = 24/30
The numbers in between18/30 and 24/30 will be rational and will fall between 3/5 and 4/5.
Hence,19/30, 20/30, 21/30, 22/30, 23/30 are the 5 rational numbers between 3/5 and 4/5
Natural numbers- Numbers starting from 1 to infinity (without fractions or decimals)
i.e., Natural numbers= 1,2,3,4…
Whole numbers- Numbers starting from 0 to infinity (without fractions or decimals)
i.e., Whole numbers= 0,1,2,3…
Or, we can say that whole numbers have all the elements of natural numbers and zero.
Every natural number is a whole number; however, every whole number is not a natural number.
Integers- Integers are set of numbers that contain positive, negative and 0; excluding fractional and decimal numbers.
i.e., integers= {…-4,-3,-2,-1,0,1,2,3,4…}
i.e., Whole numbers= 0,1,2,3….
Hence, we can say that integers include whole numbers as well as negative numbers.
Every whole number is an integer; however, every integer is not a whole number.
Rational numbers- All numbers in the form p/q, where p and q are integers and q≠0.
i.e., Rational numbers = 0, 19/30 , 2, 9/-3, -12/7…
Hence, we can say that integers includes whole numbers as well as negative numbers.
Every whole numbers are rational, however, every rational numbers are not whole numbers.
Irrational Numbers – A number is said to be irrational, if it cannot be written in the p/q, where p and q are integers and q ≠ 0.
i.e., Irrational numbers = π, e, √3, 5+√2, 6.23146…. , 0.101001001000….
Real numbers – The collection of both rational and irrational numbers are known as real numbers.
i.e., Real numbers = √2, √5, , 0.102…
Every irrational number is a real number, however, every real numbers are not irrational numbers.
The statement is false since as per the rule, a negative number cannot be expressed as square roots.
E.g., √9 =3 is a natural number.
But √2 = 1.414 is not a natural number.
Similarly, we know that there are negative numbers on the number line but when we take the root of a negative number it becomes a complex number and not a natural number.
E.g., √-7 = 7i, where i = √-1
The statement that every point on the number line is of the form √m, where m is a natural number is false.
The statement is false, the real numbers include both irrational and rational numbers. Therefore, every real number cannot be an irrational number.
Every irrational number is a real number, however, every real number is not irrational.
Step 1: Let line AB be of 2 unit on a number line.
Step 2: At B, draw a perpendicular line BC of length 1 unit.
Step 3: Join CA
AB2+BC2 = CA2
22+12 = CA2 = 5
⇒ CA = √5 . Thus, CA is a line of length √5 unit.
Step 4: Taking CA as a radius and A as a center draw an arc touching
whose center was A.
4. Classroom activity (Constructing the ‘square root spiral’) : Take a large sheet of paper and construct the ‘square root spiral’ in the following fashion. Start with a point O and draw a line segment OP1 of unit length. Draw a line segment P1P2 perpendicular to OP1 of unit length (see Fig. 1.9). Now draw a line segment P2P3 perpendicular to OP2. Then draw a line segment P3P4 perpendicular to OP3. Continuing in Fig. 1.9 :
Constructing this manner, you can get the line segment Pn-1Pn by square root spiral drawing a line segment of unit length perpendicular to OPn-1. In this manner, you will have created the points P2, P3,….,Pn,… ., and joined them to create a beautiful spiral depicting √2, √3, √4, …
0.4\overline{7}
0.4\overline{7} = 0.4777..
1. Visualise 3.765 on the number line, using successive magnification.
We know that, √5 = 2.2360679…
Here, 2.2360679…is non-terminating and non-recurring.
Now, substituting the value of √5 in 2 –√5, we get,
2-√5 = 2-2.2360679… = -0.2360679
Since the number, – 0.2360679…, is non-terminating non-recurring, 2 –√5 is an irrational number.
Since the number 3/1 is in p/q form, (3 +√23)- √23 is rational.
2√7/7√7 = ( 2/7)× (√7/√7)
We know that (√7/√7) = 1
Hence, ( 2/7)× (√7/√7) = (2/7)×1 = 2/7
Since the number, 2/7 is in p/q form, 2√7/7√7 is rational.
(1/√2) ×(√2/√2)= √2/2 ( since √2×√2 = 2)
Then, √2/2 = 1.4142/2 = 0.7071..
Since the number , 0.7071..is non-terminating non-recurring, 1/√2 is an irrational number.
We know that, the value of = 3.1415
Hence, 2 = 2×3.1415.. = 6.2830…
Since the number, 6.2830…, is non-terminating non-recurring, 2 is an irrational number.
(3+√3)(2+√2 )
Opening the brackets, we get, (3×2)+(3×√2)+(√3×2)+(√3×√2)
= 6+3√2+2√3+√6
(ii) (3+√3)(3-√3 )
(3+√3)(3-√3 ) = 32-(√3)2 = 9-3
(√5+√2)2 = √52+(2×√5×√2)+ √22
= 5+2×√10+2 = 7+2√10
(√5-√2)(√5+√2) = (√52-√22) = 5-2 = 3
3. Recall, π is defined as the ratio of the circumference (say c) of a circle to its diameter, (say d). That is, π =c/d. This seems to contradict the fact that π is irrational. How will you resolve this contradiction?
There is no contradiction. When we measure a value with a scale, we only obtain an approximate value. We never obtain an exact value. Therefore, we may not realize whether c or d is irrational. The value of π is almost equal to 22/7 or 3.142857…
4. Represent (√9.3) on the number line.
Step 1: Draw a 9.3 units long line segment, AB. Extend AB to C such that BC=1 unit.
Step 2: Now, AC = 10.3 units. Let the centre of AC be O.
Step 3: Draw a semi-circle of radius OC with centre O.
Step 4: Draw a BD perpendicular to AC at point B intersecting the semicircle at D. Join OD.
Step 5: OBD, obtained, is a right angled triangle.
Here, OD 10.3/2 (radius of semi-circle), OC = 10.3/2 , BC = 1
OB = OC – BC
⟹ (10.3/2)-1 = 8.3/2
OD2=BD2+OB2
⟹ (10.3/2)2 = BD2+(8.3/2)2
⟹ BD2 = (10.3/2)2-(8.3/2)2
⟹ (BD)2 = (10.3/2)-(8.3/2)(10.3/2)+(8.3/2)
⟹ BD2 = 9.3
⟹ BD = √9.3
5. Rationalize the denominators of the following:
Multiply and divide 1/√7 by √7
(1×√7)/(√7×√7) = √7/7
Multiply and divide 1/(√7-√6) by (√7+√6)
[1/(√7-√6)]×(√7+√6)/(√7+√6) = (√7+√6)/(√7-√6)(√7+√6)
= (√7+√6)/√72-√62 [denominator is obtained by the property, (a+b)(a-b) = a2-b2]
= (√7+√6)/(7-6)
= (√7+√6)/1
= √7+√6
Multiply and divide 1/(√5+√2) by (√5-√2)
[1/(√5+√2)]×(√5-√2)/(√5-√2) = (√5-√2)/(√5+√2)(√5-√2)
= (√5-√2)/(√52-√22) [denominator is obtained by the property, (a+b)(a-b) = a2-b2]
= (√5-√2)/(5-2)
= (√5-√2)/3
(iv) 1/(√7-2)
Multiply and divide 1/(√7-2) by (√7+2)
1/(√7-2)×(√7+2)/(√7+2) = (√7+2)/(√7-2)(√7+2)
= (√7+2)/(√72-22) [denominator is obtained by the property, (a+b)(a-b) = a2-b2]
= (√7+2)/(7-4)
= (√7+2)/3
641/2 = (8×8)1/2
= (82)½
= 81 [⸪2×1/2 = 2/2 =1]
(ii)321/5
321/5 = (25)1/5
= (25)⅕
= 21 [⸪5×1/5 = 1]
(125)1/3 = (5×5×5)1/3
= (53)⅓
= 51 (3×1/3 = 3/3 = 1)
93/2 = (3×3)3/2
322/5 = (2×2×2×2×2)2/5
= (25)2⁄5
= 22 [⸪5×2/5= 2]
(iii)163/4
163/4 = (2×2×2×2)3/4
125-1/3 = (5×5×5)-1/3
= (53)-1⁄3
= 5-1 [⸪3×-1/3 = -1]
22/3×21/5 = 2(2/3)+(1/5) [⸪Since, am×an=am+n____ Laws of exponents]
= 213/15 [⸪2/3 + 1/5 = (2×5+3×1)/(3×5) = 13/15]
(1/33)7 = (3-3)7 [⸪Since,(am)n = am x n____ Laws of exponents]
111/2/111/4 = 11(1/2)-(1/4)
= 111/4 [⸪(1/2) – (1/4) = (1×4-2×1)/(2×4) = 4-2)/8 = 2/8 = ¼ ]
(iv) 71/2×81/2
71/2×81/2 = (7×8)1/2 [⸪Since, (am×bm = (a×b)m ____ Laws of exponents]
NCERT Solutions for Class 9 Maths Chapter 1 – Number Systems
As the Number System is one of the important topics in Maths, it has a weightage of 8 marks in Class 9 Maths CBSE Term I exams. On an average three questions are asked from this unit.
Introduction of Number Systems
Representing Real Numbers on the Number Line.
List of Exercises in NCERT Solutions for Class 9 Maths Chapter 1:
Exercise 1.1 Solutions 4 Questions ( 2 long, 2 short)
Exercise 1.3 Solutions 9 Questions ( 9 long)
Exercise 1.5 Solutions 5 Questions ( 4 long 1 short)
NCERT Solutions for Class 9 Maths Chapter 1 Number System is the first chapter of class 9 Maths. Number System is discussed in detail in this chapter. The chapter discusses the Number Systems and their applications. The introduction of the chapter includes whole numbers, integers and rational numbers.
The chapter starts with the introduction of Number Systems in section 1.1 followed by two very important topics in sections 1.2 and 1.3
Irrational Numbers – The numbers which can’t be written in the form of p/q.
Real Numbers and their Decimal Expansions – Here you study the decimal expansions of real numbers and see whether it can help in distinguishing between rational and irrationals.
Next, it discusses the following topics.
Representing Real Numbers on the Number Line – In this the solutions for 2 problems in Exercise 1.4.
Operations on Real Numbers – Here you explore some of the operations like addition, subtraction, multiplication and division on irrational numbers.
Explore more about Number Systems and learn how to solve various kinds of problems only on NCERT Solutions For Class 9 Maths. It is also one of the best academic resources to revise for your term wise exams.
Key advantages of NCERT Solutions for Class 9 Maths Chapter 1- Number Systems
These NCERT Solutions for Class 9 Maths helps you solve and revise the whole term wise syllabus of Class 9.
It helps in scoring well in Maths in first term exams.
The faculty have curated the solutions in a lucid manner to improve the problem-solving abilities among the students. For a more clear idea about Number Systems students can refer to the study materials available at BYJU’S.
RD Sharma Solutions for Class 9 Maths Number Systems
Why should we download NCERT Solutions for Class 9 Maths Chapter 1?
The presentation of each solution in the Chapter 1 of NCERT Solutions for Class 9 Maths is described in a unique way by the BYJU’S experts in Maths. The solutions are explained in understandable language which improves grasping abilities among students. To score good marks, practising NCERT Solutions for Class 9 Maths can help to a great extent. This chapter can be used as a model of reference by the students to improve their conceptual knowledge and understand the different ways used to solve the problems.
Is BYJU’S website providing answers to NCERT Solutions for Class 9 Maths Chapter 1 in a detailed way?
Yes, BYJU’S website provides answers to NCERT Solutions for Class 9 Maths Chapter 1 in step by step manner. This makes the students to learn all the concepts in detail and also they can clear their doubts as well. Regular practising makes them score high in Maths Term I exams.
Give an overview of concepts present in NCERT Solutions for Class 9 Maths Chapter 1.
NCERT Solutions for Class 9 Maths Chapter 1 has 3 exercises. The concepts of this chapter are listed below.
1.Irrational Numbers
2.Real Numbers and their Decimal Expansions
3.Representing Real Numbers on the Number Line
4.Operations on Real Numbers
5.Laws of Exponents for Real Numbers
Jasmitha October 15, 2020 at 8:56 pm
Byjus is the best for all subjects
Sakshi Singh October 27, 2020 at 12:11 pm
Sandeep sugithan April 27, 2022 at 12:31 pm |
Robust Stability and H∞ Control of Discrete-Time Jump Linear Systems With Time-Delay: An LMI Approach* | J. Dyn. Sys., Meas., Control. | ASME Digital Collection
Robust Stability and H∞ Control of Discrete-Time Jump Linear Systems With Time-Delay: An LMI Approach*
E. K. Boukas,
E. K. Boukas
Mechanical Engineering Department, E´cole Polytechnique de Montre´al, P.O. Box 6079, station “Centre-ville,” Montre´al, Que´bec, H3C 3A7 Canada
Contributed by the Dynamic Systems and Control Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS for publication in the ASME JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received by the ASME Dynamic Systems and Control Division, Aug. 2001; final revision, Dec. 2002. Associate Editor. N. Olgac.
J. Dyn. Sys., Meas., Control. Jun 2003, 125(2): 271-277
Boukas, E. K., and Liu , Z. K. (June 4, 2003). "Robust Stability and H∞ Control of Discrete-Time Jump Linear Systems With Time-Delay: An LMI Approach." ASME. J. Dyn. Sys., Meas., Control. June 2003; 125(2): 271–277. https://doi.org/10.1115/1.1570858
This paper considers the class of discrete-time jump linear systems with time-delay and polytopic uncertain parameters. The problems of delay-independent robust stability, stabilization and
H∞
control are cast into the framework of linear matrix inequality (LMI) and thus solved by LMI Toolbox of Matlab. By extending the system state, the system with time-delay is converted into a higher dimension Markov jump system without time-delay, and thus can be handled as a standard jump linear system with uncertain parameters. Numerical examples are provided to show the usefulness of the theoretical results.
robust control, optimal control, discrete time systems, linear systems, delays, linear matrix inequalities, uncertain systems, Markov processes, Jump Linear System, Polytopic Uncertainty, Time Delay, LMI
Delays, Linear systems, Stability, Theorems (Mathematics)
Stability Results for Discrete-Time Markovian Jump Linear Systems With Markovian Jumping Parameter Systems
Wonham, W. M., 1971, “Random Differential Equations in Control Theory,” Probabilistic Methods in Applied Mathematics, 2, A. T. Bharucha-reid, ed., Academic Press, New York.
Feedback Regulators for Jump Parameter Systems With State and Control Dependent Transition Rates
Controllability, Observability, and Discrete-Time Markovian Jump Linear Quadratic Control
Control Theory and Advanced Technology
Control-Theory and Advanced Technology
Mariton, M., 1990, Jump Linear Systems in Automatic Control, Marcel Dekker, New York.
Boyd, S., El Ghaoui, L., Feron, E., and Balakrishnan, V., 1994, Linear Matrix Inequalities in System and Control Theory, SIAM.
Control of System With Controlled Jump Markov Disturbances
Stability of Discrete-Time Linear Systems With Markovian Jumping Parameters
Necessary and Sufficient Condition for Robust Stability and Stabilizability of Continuous-Time Linear Systems With Markovian Jumps
H∞ Control for Markovian Jumping Linear Systems With Parametric Uncertainty
H∞-Output Feedback Controller Design for Linear Systems With Time-Varying Delayed State
Mean Square Stochastic Stability of Linear Time-Delay System with Markovian Jumping Parameters
Robust Stabilizability of Uncertain Linear Time-Delay Systems With Markovian Jumping Parameters
Robust H∞ of Discrete-Time Markovian Jump Linear Systems With Mode-Dependent Time-Delays
Al-Muthairi
Design of Robust Controllers for Time-Delay Systems
Stability Analysis for a Class of Uncertain Discrete Singularly Perturbed Systems With Multiple Time Delays
Optimal Feedback Design of Delayed Linear Systems With Experimental Validation |
Some APN functions CCZ-equivalent to x^3 + tr n(x^9) and CCZ-inequivalent to the Gold functions over GF(2^n) - Boolean Functions
Some APN functions CCZ-equivalent to x^3 + tr n(x^9) and CCZ-inequivalent to the Gold functions over GF(2^n)
Revision as of 19:21, 9 July 2020 by Nikolay (talk | contribs)
{\displaystyle x^{3}+tr_{n}(x^{9})}
{\displaystyle \mathbb {F} _{2^{n}}}
{\displaystyle N^{\circ }}
{\displaystyle d^{\circ }}
{\displaystyle 1}
{\displaystyle x^{3}+tr_{n}(x^{9})+(x^{2}+x)tr_{n}(x^{3}+x^{9})}
{\displaystyle n\geqslant 5}
{\displaystyle \gcd(i,n)=1}
{\displaystyle 3}
{\displaystyle 2}
{\displaystyle x^{3}+tr_{n}(x^{9})+(x^{2}+x+1)tr_{n}(x^{3})}
{\displaystyle n\geqslant 4}
{\displaystyle \gcd(i,n)=1}
{\displaystyle 3}
{\displaystyle 3}
{\displaystyle {\Big (}x+tr_{n}^{3}(x^{6}+x^{12})+tr_{n}(x)tr_{n}^{3}(x^{3}+x^{12}){\Big )}^{3}+}
{\displaystyle tr_{n}{\Big (}\left(x+tr_{n}^{3}(x^{6}+x^{12})+tr_{n}(x)tr_{n}^{3}(x^{3}+x^{12})\right)^{9}{\Big )}}
{\displaystyle 6|n}
{\displaystyle \gcd(i,n)=1}
{\displaystyle 4}
{\displaystyle 4}
{\displaystyle \left(x^{\frac {1}{3}}+tr_{n}^{3}(x+x^{4})\right)^{-1}+tr_{n}\left(\left(\left(x^{\frac {1}{3}}+tr_{n}^{3}(x+x^{4})\right)^{-1}\right)^{9}\right)}
{\displaystyle 3|n}
{\displaystyle n}
{\displaystyle 4} |
Difference between revisions of "Stellar spectra" - AstroBaki
Difference between revisions of "Stellar spectra"
(oops, created page with proper content)
{\bf Stellar Spectra} \\
\documentclass[preprint]{aastex}
\subsection*{Stellar Spectra} \\
The spectrum that we see from a star gives us information from the photosphere from the star, and contains all the information that we get from stars, aside from neutrinos. Thus, even though it is a very small fraction of the star, it is important observationally. \\ \\
The surface (or photosphere) is where the mean free path is comparable to the scale height of the atmosphere. This is given by
1 Stellar Spectra
\The spectrum that we see from a star gives us information from the photosphere from the star, and contains all the information that we get from stars, aside from neutrinos. Thus, even though it is a very small fraction of the star, it is important observationally. \\The surface (or photosphere) is where the mean free path is comparable to the scale height of the atmosphere. This is given by
{\displaystyle \ell \sim H\sim {\frac {P}{\rho g}}\,\!}
Plugging in gas pressure, which dominates the photospheres of stars,
{\displaystyle {\frac {1}{\kappa \rho }}\sim {\frac {kT}{mg}}.\,\!}
This gives us the pressure and density in the photosphere, with
{\displaystyle P\approx {\frac {g}{\kappa }},\,\!}
{\displaystyle \rho \approx {\frac {mg}{kT\kappa }}.\,\!}
For the Sun,
{\displaystyle T_{\rm {eff}}=5800}
K, giving a number density of
{\displaystyle n\approx 10^{17}}
{\displaystyle ^{-3}}
{\displaystyle 10^{-9}}
the central number density of the Sun. The pressure is
{\displaystyle P\approx 10^{5}}
erg cm
{\displaystyle ^{-3}\approx 0.1}
atm. The pressure and density, along with the effective temperature, set the observable properties of stars. To first order, we have assumed that the surface of a star is a perfect blackbody emitter. This is not accurate though, as it ignores the presence of spectral lines, the strength of which depends on the density and pressure as well. The spectra of stars are the basis of a classification system, labeling stars O, B, A, F, G, K, M, ranging from the hottest stars to the coolest stars. There are also L and T spectral types, which apply not to stars, but to brown dwarfs. The nice thing about this is that for main sequence stars, the mass maps to an effective temperature, which maps to a spectral type. Our goal is to understand the second fact (since we already understand the first fact). \\The matter on the surface of a star is in thermodynamic equilibrium. This is fundamentally the reason that the temperature determines the spectral lines in a star. The ratio of number of states is just set by the Boltzmann factor, with
{\displaystyle {\frac {N_{2}}{N_{1}}}={\frac {g_{2}}{g_{1}}}e^{-(E_{2}-E_{1})/kT}.\,\!}
For neutral atoms and molecules, this describes the number of electrons in each state. For determining the fraction of neutral and ionized atoms, we can generalize the Boltzmann factor to an expression known as the Saha equation. In thermal equilibrium, the temperature, pressure, and chemical potential (
{\displaystyle \mu }
) will all come in to equality. These three quantities are the macroscopic properties of a gas. The implication of the chemical potential equilibrium is that for a reaction where
{\displaystyle A+B\rightarrow C+D,\,\!}
{\displaystyle C+D\rightarrow A+B,\,\!}
{\displaystyle \mu (A)+\mu (B)=\mu (C)+\mu (D).\,\!}
The fact that the reaction be able to go both ways is important to remember. This fact is the reason that we cannot apply chemical potential equilibrium to describe fusion in the centers of star. It is so energetically difficult to break apart a helium atom that there are many more reactions producing helium then there are reactions breaking helium apart to provide hydrogen. So, instead, we’ll apply this to ionization balance, which is a problem chemical equilibrium can solve, and is what we want to understand to understand the surfaces of stars. We can start by looking at the ionization/recombination of hydrogen:
{\displaystyle e^{-}+p\leftrightarrow H+\gamma .\,\!}
We can solve for the balance of this reaction by writing down the chemical potential of each side,
{\displaystyle \mu (e^{-})+\mu (p)\leftrightarrow \mu (H)+\mu (\gamma ).\,\!}
The phase space distribution of particles is
{\displaystyle n(p)={\frac {g}{h^{3}}}{\frac {1}{e^{(E_{p}-\mu )/kT}\pm 1}}.\,\!}
Remember that the plus or minus depends on whether we are dealing with Fermi-Dirac or Bose-Einstein statistics. In the limit of a classical gas, this cannot matter, so we must have the exponential term dominate, leaving
{\displaystyle n(p)={\frac {g}{h^{3}}}e^{-(E_{p}-\mu )/kT}.\,\!}
The energy of particles is
{\displaystyle E_{p}=mc^{2}+{\frac {p^{2}}{2m}}.\,\!}
The number density of particles in real space is the integral over the momentum of the particle,
{\displaystyle n=\int 4\pi p^{2}dpn(p)=\int 4\pi p^{2}dp{\frac {g}{h^{3}}}e^{-(mc^{2}-\mu )/kT}e^{-p^{2}/2mkT}.\,\!}
Only the last term needs to stay in the integral, and it is a Gaussian integral with a nice solution. The result is
{\displaystyle n=ge^{-(mc^{2}-\mu )/kT}\left({\frac {2\pi mkT}{h^{2}}}\right)^{3/2}.\,\!}
Or, in terms of the quantum number density,
{\displaystyle n=gn_{Q}e^{-(mc^{2}-\mu )/kT}.\,\!}
This allows us to finally write down what
{\displaystyle \mu }
is, with
{\displaystyle \mu =mc^{2}-kT\ln \left({\frac {gn_{Q}}{n}}\right).\,\!}
Now, we have everything we need to solve for the ionization balance of hydrogen. We can write the chemical potential of the proton, electron, and hydrogen atom with the above expression, and the chemical potential of the photon is zero (we can create this for free, so they cannot have a chemical potential). The difference in the mass between the hydrogen atom and the sum of electon and proton masses is just the binding energy of the hydrogen atom, which we’ll call
{\displaystyle \chi }
{\displaystyle -{\frac {\chi }{kT}}-\ln \left({\frac {g_{H}n_{Q,H}}{n_{H}}}\right)=-\ln \left({\frac {g_{p}n_{Q,p}}{n_{p}}}\right)-\ln \left({\frac {g_{e}n_{Q,e}}{n_{e}}}\right)\,\!}
Some rearranging, taking advantage of the log rules for adding/subtraction, and then exponentiating both sides, gives the Saha equation,
{\displaystyle {\frac {n_{e}n_{p}}{n_{H}}}={\frac {g_{e}g_{p}}{g_{H}}}n_{Q,e}e^{-\chi /kT}.\,\!}
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Elementary_event Knowpia
In probability theory, an elementary event, also called an atomic event or sample point, is an event which contains only a single outcome in the sample space.[1] Using set theory terminology, an elementary event is a singleton. Elementary events and their corresponding outcomes are often written interchangeably for simplicity, as such an event corresponding to precisely one outcome.
The following are examples of elementary events:
{\displaystyle \{k\},}
{\displaystyle k\in \mathbb {N} }
if objects are being counted and the sample space is
{\displaystyle S=\{1,2,3,\ldots \}}
(the natural numbers).
{\displaystyle \{HH\},\{HT\},\{TH\},{\text{ and }}\{TT\}}
if a coin is tossed twice.
{\displaystyle S=\{HH,HT,TH,TT\}}
{\displaystyle H}
stands for heads and
{\displaystyle T}
{\displaystyle \{x\},}
{\displaystyle x}
is a real number. Here
{\displaystyle X}
is a random variable with a normal distribution and
{\displaystyle S=(-\infty ,+\infty ).}
This example shows that, because the probability of each elementary event is zero, the probabilities assigned to elementary events do not determine a continuous probability distribution.
Probability of an elementary eventEdit
Elementary events may occur with probabilities that are between zero and one (inclusively). In a discrete probability distribution whose sample space is finite, each elementary event is assigned a particular probability. In contrast, in a continuous distribution, individual elementary events must all have a probability of zero because there are infinitely many of them— then non-zero probabilities can only be assigned to non-elementary events.
Some "mixed" distributions contain both stretches of continuous elementary events and some discrete elementary events; the discrete elementary events in such distributions can be called atoms or atomic events and can have non-zero probabilities.[2]
Under the measure-theoretic definition of a probability space, the probability of an elementary event need not even be defined. In particular, the set of events on which probability is defined may be some σ-algebra on
{\displaystyle S}
and not necessarily the full power set.
^ Wackerly, Denniss; William Mendenhall; Richard Scheaffer. Mathematical Statistics with Applications. Duxbury. ISBN 0-534-37741-6.
^ Kallenberg, Olav (2002). Foundations of Modern Probability (2nd ed.). New York: Springer. p. 9. ISBN 0-387-94957-7.
Pfeiffer, Paul E. (1978). Concepts of Probability Theory. Dover. p. 18. ISBN 0-486-63677-1.
Ramanathan, Ramu (1993). Statistical Methods in Econometrics. San Diego: Academic Press. pp. 7–9. ISBN 0-12-576830-3. |
Use reference angles, the symmetry of a circle, and the knowledge that cos
(\frac { \pi } { 3 })=\frac { 1 } { 2 }
to write three other true statements using cosine and angles that are multiples of
\frac { \pi } { 3 }
Draw a unit circle diagram for the given angle. Unit circle, with right triangle in first quadrant, labeled as follows: hypotenuse, 1, horizontal leg, 1 half, angle opposite vertical leg, pi thirds.
Reflect the triangle in each of the
4
quadrants. Quadrant
2
is shown for you. Added to the unit circle, a right triangle in second quadrant, labeled as follows: hypotenuse, 1, horizontal leg, negative 1 half.
\text{cos}\left( \frac{2\pi}{3} \right)=-\frac{1}{2} |
ZeroDimensionalDecomposition - Maple Help
Home : Support : Online Help : Mathematics : Algebra : Polynomials : PolynomialIdeals : ZeroDimensionalDecomposition
decompose an ideal into zero-dimensional ideals
ZeroDimensionalDecomposition(J)
The ZeroDimensionalDecomposition command computes a sequence of zero-dimensional ideals, some of which may lie in extended polynomial rings. That is, to make the resulting ideals zero-dimensional, ring variables can be moved into the coefficient field. If the ideals in the resulting sequence are contracted back to the original ring and intersected, you get the original ideal. In general, this decomposition is not unique.
This command allows you to run algorithms for zero-dimensional ideals on ideals of positive Hilbert dimension. Be aware that some algorithms do not interact well with the extension and contraction process. In particular, you cannot use this process to directly test whether an ideal is radical because the decomposition of a radical ideal may contain non-radical components that vanish under contraction and intersection. Valid applications include solving, testing whether an ideal is prime or primary, and computing prime or primary decompositions or radical of an ideal.
\mathrm{with}\left(\mathrm{PolynomialIdeals}\right):
J≔〈xy-y〉
\textcolor[rgb]{0,0,1}{J}\textcolor[rgb]{0,0,1}{≔}〈\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{y}〉
\mathrm{zdd}≔[\mathrm{ZeroDimensionalDecomposition}\left(J\right)]
\textcolor[rgb]{0,0,1}{\mathrm{zdd}}\textcolor[rgb]{0,0,1}{≔}[〈\textcolor[rgb]{0,0,1}{y}〉\textcolor[rgb]{0,0,1}{,}〈\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}〉]
\mathrm{Intersect}\left(\mathrm{op}\left(\mathrm{map}\left(\mathrm{Contract},\mathrm{zdd},{x,y}\right)\right)\right)
〈\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{y}〉
K≔〈-2{y}^{3}+3{x}^{3}z,-{y}^{2}{z}^{2}〉
\textcolor[rgb]{0,0,1}{K}\textcolor[rgb]{0,0,1}{≔}〈\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{3}}〉
\mathrm{zdd}≔[\mathrm{ZeroDimensionalDecomposition}\left(K\right)]
\textcolor[rgb]{0,0,1}{\mathrm{zdd}}\textcolor[rgb]{0,0,1}{≔}[〈{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{2}}〉\textcolor[rgb]{0,0,1}{,}〈{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{8}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{3}}〉]
\mathrm{map}\left(\mathrm{Simplify}@\mathrm{Radical},\mathrm{zdd}\right)
[〈\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}〉\textcolor[rgb]{0,0,1}{,}〈\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{z}〉]
\mathrm{Intersect}\left(\mathrm{op}\left(\mathrm{map}\left(\mathrm{Contract},,{x,y,z}\right)\right)\right)
〈\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}〉
\mathrm{Radical}\left(K\right)
〈\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}〉
L≔〈x-y,{x}^{3}-yzw〉
\textcolor[rgb]{0,0,1}{L}\textcolor[rgb]{0,0,1}{≔}〈\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{w}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}〉
\mathrm{ZeroDimensionalDecomposition}\left(L\right)
〈\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}〉\textcolor[rgb]{0,0,1}{,}〈\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{w}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}〉\textcolor[rgb]{0,0,1}{,}〈\textcolor[rgb]{0,0,1}{w}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}〉
\mathrm{map}\left(\mathrm{IsRadical},[]\right)
[\textcolor[rgb]{0,0,1}{\mathrm{true}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{true}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{false}}]
\mathrm{IsRadical}\left(L\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{L2}≔〈{x}^{2}-y,{x}^{3}-yzw〉
\textcolor[rgb]{0,0,1}{\mathrm{L2}}\textcolor[rgb]{0,0,1}{≔}〈{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{w}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}〉
\mathrm{ZeroDimensionalDecomposition}\left(\mathrm{L2}\right)
〈\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}〉\textcolor[rgb]{0,0,1}{,}〈{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{w}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}〉\textcolor[rgb]{0,0,1}{,}〈\textcolor[rgb]{0,0,1}{w}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{y}〉
\mathrm{map}\left(\mathrm{IsRadical},[]\right)
[\textcolor[rgb]{0,0,1}{\mathrm{false}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{true}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{false}}]
\mathrm{IsRadical}\left(\mathrm{L2}\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}} |
Papers on Boolean Functions - Boolean Functions
On Almost Perfect Nonlinear Functions Over
{\displaystyle \mathbb {F} _{2}^{n}}
T.P. Berger , A. Canteaut , P. Charpin , Y. Laigle-Chapuy
We investigate some open problems on almost perfect nonlinear (APN) functions over a finite field of characteristic 2. We provide new characterizations of APN functions and of APN permutations by means of their component functions. We generalize some results of Nyberg (1994) and strengthen a conjecture on the upper bound of nonlinearity of APN functions. We also focus on the case of quadratic functions. We contribute to the current works on APN quadratic functions by proving that a large class of quadratic functions cannot be APN
C. Blondeau, K. Nyberg
Recently, a number of relations have been established among previously known statistical attacks on block ciphers. Leander showed in 2011 that statistical saturation distinguishers are on average equivalent to multidimensional linear distinguishers. Further relations between these two types of distinguishers and the integral and zero-correlation distinguishers were established by Bogdanov et al. [6]. Knowledge about such relations is useful for classification of statistical attacks in order to determine those that give essentially complementary information about the security of block ciphers. The purpose of the work presented in this paper is to explore relations between differential and linear attacks. The mathematical link between linear and differential attacks was discovered by Chabaud and Vaudenay already in 1994, but it has never been used in practice. We will show how to use it for computing accurate estimates of truncated differential probabilities from accurate estimates of correlations of linear approximations. We demonstrate this method in practice and give the first instantiation of multiple differential cryptanalysis using the LLR statistical test on PRESENT. On a more theoretical side, we establish equivalence between a multidimensional linear distinguisher and a truncated differential distinguisher, and show that certain zero-correlation linear distinguishers exist if and only if certain impossible differentials exist.
C. Carlet, P. Charpin, V. Zinoviev
Almost bent functions oppose an optimum resistance to linear and differential cryptanalysis. We
Open Questions on Nonlinearity and on APN Functions
In a first part of the paper, we recall some known open questions on the nonlinearity of Boolean and vectorial functions and on the APN-ness of vectorial functions. All of them have been extensively searched and seem quite difficult. We also indicate related less well-known open questions. In the second part of the paper, we introduce four new open problems (leading to several related sub-problems) and the results which lead to them. Addressing these problems may be less difficult since they have not been much worked on.
For every positive integers
{\displaystyle n,m}
and every even positive integer
{\displaystyle \delta }
, we derive inequalities satisfied by the Walsh transforms of all vectorial
{\displaystyle (n,m)}
-functions and prove that the case of equality characterizes differential
{\displaystyle \delta }
-uniformity. This provides a generalization to all differentially
{\displaystyle \delta }
-uniform functions of the characterization of APN
{\displaystyle (n,n)}
-functions due to Chabaud and Vaudenay, by means of the fourth moment of the Walsh transform. Such generalization has been missing since the introduction of the notion of differential uniformity by Nyberg in 1994 and since Chabaud-Vaudenay’s result the same year. For each even
{\displaystyle \delta \geq 2}
, we find several such characterizations. In particular, when
{\displaystyle \delta =2}
{\displaystyle \delta =4}
, we have that, for any
{\displaystyle (n,n)}
-function (resp. any
{\displaystyle (n,n-1)}
-function), the arithmetic mean of
{\displaystyle W_{F}^{2}(u_{1},v_{1})W^{2})F(u_{2},v_{2})W_{F}^{2}(u_{1}+u_{2},v_{1}+v_{2})}
{\displaystyle u_{1},u_{2}}
range independently over
{\displaystyle \mathbb {F} _{n}^{2}}
{\displaystyle v_{1},v_{2}}
are nonzero and distinct and range independently over
{\displaystyle \mathbb {F} _{2}^{m}}
, is at least
{\displaystyle 2^{3n}}
{\displaystyle F}
is APN (resp. is differentially 4-uniform) if and only if this arithmetic mean equals
{\displaystyle 2^{3n}}
(which is the value we would get with a bent function if such function could exist). These inequalities give more knowledge on the Walsh spectrum of
{\displaystyle (n,m)}
-functions. We deduce in particular a property of the Walsh support of highly nonlinear functions. We also consider the completely open question of knowing if the nonlinearity of APN functions is necessarily non-weak (as it is the case for known APN functions); we prove new lower bounds which cover all power APN functions (and hence a large part of known APN functions), which explain why their nonlinearities are rather good, and we discuss the question of the nonlinearity of APN quadratic functions (since almost all other known APN functions are quadratic).
C. Carlet, C. Ding
Functions with high nonlinearity have important applications in cryptography, sequences and coding theory. The purpose of this paper is to give a well-rounded treatment of non-Boolean functions with optimal nonlinearity. We summarize and generalize known results, and prove a number of new results. We also present open problems about functions with high nonlinearity.
F. Chabaud, S. Vaudenay
Linear cryptanalysis, introduced last year by Matsui, will most certainly open-up the way to new attack methods which may be made more efficient when compared or combined with differential cryptanalysis.
M. Brinkmann, G. Leander
We classify the almost perfect nonlinear (APN) functions in dimensions 4 and 5 up to affine and CCZ equivalence using backtrack programming and give a partial model for the complexity of such a search. In particular, we demonstrate that up to dimension 5 any APN function is CCZ equivalent to a power function, while it is well known that in dimensions 4 and 5 there exist APN functions which are not extended affine (EA) equivalent to any power function. We further calculate the total number of APN functions up to dimension 5 and present a new CCZ equivalence class of APN functions in dimension 6.
New families of quadratic almost perfect nonlinear trinomials and multinomials
C. Bracken, E. Byrne, N. Markin, G. McGuire
We introduce two new infinite families of APN functions, one on fields of order
{\displaystyle 2^{2k}}
{\displaystyle k}
{\displaystyle 2}
, and the other on fields of order
{\displaystyle 2^{3k}}
{\displaystyle k}
{\displaystyle 3}
. The polynomials in the first family have between three and
{\displaystyle k+2}
terms, the second family's polynomials have three terms.
An APN permutation in dimension six
KA Browning, JF Dillon, MT McQuistan, AJ Wolfe
{\displaystyle f:GF(2^{m})\rightarrow GF(2^{m})}
is almost perfect nonlinear, abbreviated APN, if
{\displaystyle x\mapsto f(x+a)-f(x)}
is 2-to-1 for all nonzero
{\displaystyle a}
{\displaystyle GF(2^{m})}
{\displaystyle f(0)=0}
, then this condition is equivalent to the condition that the binary code
{\displaystyle C_{f}}
{\displaystyle 2^{m}-1}
with parity-check matrix
{\displaystyle H_{f}=\left[{\begin{array}{ccc}\ldots &\omega ^{j}&\ldots \\\ldots &f(\omega ^{j})&\ldots \end{array}}\right]}
is double-error-correcting, where $\omega$ is primitive in $GF(2^m)$. A commonly held belief is that, if the dimension
{\displaystyle m}
is even, then an APN map on
{\displaystyle GF(2^{m})}
cannot be a permutation. We give a counterexample in dimension
{\displaystyle m=6}
An infinite class of quadratic APN functions which are not equivalent to power mappings
L. Budaghyan, C. Carlet, P. Felke, G. Leander
We exhibit an infinite class of almost perfect nonlinear quadratic polynomials from F2n to F2n (n ges 12, n divisible by 3 but not by 9). We prove that these functions are EA-inequivalent to any power function and that they are CCZ-inequivalent to any Gold function. In a forthcoming full paper, we shall also prove that at least some of these functions are CCZ-inequivalent to any Kasami function
L. Budaghyan, C. Carlet, G. Leander
This paper introduces the first found infinite classes of almost perfect nonlinear (APN) polynomials which are not Carlet-Charpin-Zinoviev (CCZ)-equivalent to power functions (at least for some values of the number of variables). These are two classes of APN binomials from F2n to F2n (for n divisible by 3, resp., 4). We prove that these functions are extended affine (EA)-inequivalent to any power function and that they are CCZ-inequivalent to the Gold, Kasami, inverse, and Dobbertin functions when n ges 12. This means that for n even they are CCZ-inequivalent to any known APN function. In particular, for n = 12,20,24, they are therefore CCZ-inequivalent to any power function.
Classes of Quadratic APN Trinomials and Hexanomials and Related Structures
L. Budaghyan, C. Carlet
A method for constructing differentially 4-uniform quadratic hexanomials has been recently introduced by J. Dillon. We give various generalizations of this method and we deduce the constructions of new infinite classes of almost perfect nonlinear quadratic trinomials and hexanomials from
{\displaystyle F_{2^{2m}}}
{\displaystyle F_{2^{2m}}}
. We check for
{\displaystyle m=3}
that some of these functions are CCZ-inequivalent to power functions.
L. Budaghyan, C. Carlet, A. Pott
New infinite classes of almost bent and almost perfect nonlinear polynomials are constructed. It is shown that they are affine inequivalent to any sum of a power function and an affine function
A new class of monomial bent functions
A. Canteaut, P. Charpin, G. Kyureghyan
We study the Boolean functions
{\displaystyle f_{\lambda }:\mathbb {F} _{2^{n}}\rightarrow \mathbb {F} _{2^{n}},n=6r}
{\displaystyle f(x)={\rm {{Tr}(\lambda x^{d})}}}
{\displaystyle d=2^{2r}+2^{r}+1}
{\displaystyle \lambda \in \mathbb {F} _{2^{n}}}
. Our main result is the characterization of those
{\displaystyle \lambda }
{\displaystyle f_{\lambda }}
are bent. We show also that the set of these cubic bent functions contains a subset, which with the constantly zero function forms a vector space of dimension
{\displaystyle 2^{r}}
{\displaystyle \mathbb {F} _{2}}
. Further we determine the Walsh spectra of some related quadratic functions, the derivatives of the functions
{\displaystyle f_{\lambda }}
We present a method for constructing new quadratic APN functions from known ones. Applying this method to the Gold power functions we construct an APN function
{\displaystyle x^{3}+{\rm {{Tr}(x^{9})}}}
{\displaystyle \mathbb {F} _{2^{n}}}
. It is proven that for
{\displaystyle n\geq 7}
this function is CCZ-inequivalent to the Gold functions, and in the case
{\displaystyle n=7}
it is CCZ-inequivalent to any power mapping (and, therefore, to any APN function belonging to one of the families of APN functions known so far).
Biham E, Shamir A.
The Data Encryption Standard (DES) is the best known and most widely usedc ryptosystem for civilian applications.It was developed at IBM and adopted by the National Buraeu of Standards in the mid 70's, and has successfully withstood all the attacks published so far in the open literature. In this paper we develop a new type of cryptanalytic attack which can break the reduced variant of DES with eight rounds in a few minutes on a PC and can break any reduced variant of DES (with up to 15 rounds) in less than 256 operations.The new attack can be applied to a variety of DES-like substitution/permutation cryptosystems, and demonstrates the crucial role of the (unpublished) design rules.
We introduce a new method for cryptanalysis of DES cipher, which is essentially a known-plaintext attack. As a result, it is possible to break 8-round DES cipher with
{\displaystyle 2^{21}}
known-plaintexts and 16-round DES cipher with
{\displaystyle 2^{47}}
known-plaintexts, respectively. Moreover, this method is applicable to an only-ciphertext attack in certain situations. For example, if plaintexts consist of natural English sentences represented by ASCII codes, 8-round DES cipher is breakable with
{\displaystyle 2^{29}}
ciphertexts only.
This work is motivated by the observation that in DES-like ciphers it is possible to choose the round functions in such a way that every non-trivial one-round characteristic has small probability. This gives rise to the following definition. A mapping is called differentially uniform if for every non-zero input difference and any output difference the number of possible inputs has a uniform upper bound. The examples of differentially uniform mappings provided in this paper have also other desirable cryptographic properties: large distance from affine functions, high nonlinear order and efficient computability.
Almost Perfect Nonlinear Power Functions on
{\displaystyle GF(2^{n})}
: A New Case for n Divisible by 5
{\displaystyle d=2^{4s}+2^{3s}+2^{2s}+2^{s}-1}
the power function
{\displaystyle x^{d}}
is almost perfect nonlinear (APN) on
{\displaystyle L=GF(2^{5s})}
, i.e. for each
{\displaystyle a\in L}
{\displaystyle (x+1)^{d}+x^{d}=a}
has either no or precisely two solutions in
{\displaystyle L}
. The proof of this result is based on a new “multi-variate” technique which was recently introduced by the author in order to confirm the conjectured APN property of Welch and Niho power functions.
{\displaystyle GF(2^{n})}
: The Niho Case
Almost perfect nonlinear (APN) mappings are of interest for applications in cryptography. We prove for odd
{\displaystyle n}
and the exponent
{\displaystyle d=2^{2r}+2^{r}-1}
{\displaystyle 4r+1\equiv 0modn}
, that the power functions
{\displaystyle x^{d}}
{\displaystyle GF(2^{n})}
is APN. The given proof is based on a new class of permutation polynomials which might be of independent interest. Our result supports a conjecture of Niho stating that the power function
{\displaystyle x^{d}}
is even maximally nonlinear or, in other terms, that the crosscorrelation function between a binary maximum-length linear shift register sequences of degree
{\displaystyle n}
and a decimation of that sequence by
{\displaystyle d}
takes on precisely the three values
{\displaystyle -1}
{\displaystyle -1\pm 2^{(n+1)/2}}
{\displaystyle GF(2^{n})}
: The Welch Case
We summarize the state of the classification of almost perfect nonlinear (APN) power functions
{\displaystyle x^{d}}
{\displaystyle GF(2^{n})}
and contribute two new cases. To prove these cases we derive new permutation polynomials. The first case supports a well-known conjecture of Welch stating that for odd
{\displaystyle n=2m+1}
, the power function
{\displaystyle x^{2m+3}}
is even maximally nonlinear or, in other terms, that the crosscorrelation function between a binary maximum-length linear shift register sequence of degree
{\displaystyle n}
{\displaystyle 2^{m}+3}
{\displaystyle -1}
{\displaystyle -1\pm 2^{m+1}}
Y. Edel, A. Pott
Following an example in [13], we show how to change one coordinate function of an almost perfect nonlinear (APN) function in order to obtain new examples. It turns out that this is a very powerful method to construct new APN functions. In particular, we show that the approach can be used to construct “non-quadratic” APN functions. This new example is in remarkable contrast to all recently constructed functions which have all been quadratic.
A new APN function which is not equivalent to a power mapping
Y. Edel, G. Kyureghyan, A. Pott
A new almost-perfect nonlinear function (APN) on
{\displaystyle \mathbb {F} _{2^{10}}}
which is not equivalent to any of the previously known APN mappings is constructed. This is the first example of an APN mapping which is not equivalent to a power mapping.
Retrieved from "https://boolean.h.uib.no/mediawiki/index.php?title=Papers_on_Boolean_Functions&oldid=110" |
Mathematical observations
Width and density
Growth of
w(\mathcal O_n)
w(On)
Programming exploration
Formalization in Lean
The other day my friend Leo presented the following puzzle to me. Given a partition
\left\{ \mathcal O_n \right\}_{n\in I}
{On}n∈I of positive integers with
\mathcal O_n
On all infinite, find a condition that is equivalent to
I finite. The motivation for the problem was juggling: each
\mathcal O_n
On is the "orbit" of the
n^{\text{th}}
nth ball, and we want to know when we are juggling finitely many balls.
After playing around with some examples we came up with the definition of width of an infinite subset of
\mathbb Z_+
Z+: the width of
\mathcal O
O denoted by
w(\mathcal O)
w(O) is the supremum of distances between consecutive elements. Intuitively, width is kind of the reciprocal of natural density. Recall that the natural density of
\mathcal O
O is the limit
d(\mathcal O) = \lim_{n\to \infty} a(n)/n
d(O)=limn→∞a(n)/n where
a(n) is the number of elements of
\mathcal O
\left\{ 1,...,n \right\}
{1,...,n}. Therefore
1/w(\mathcal O) \leq d(\mathcal O)
1/w(O)≤d(O). Let's see why that is.
\mathcal O = \left\{ x_i \right\}_1^\infty \subset \mathbb Z_+
O={xi}1∞⊂Z+, let
y_0 = x_1
y0=x1 and
y_i = x_{i+1} - x_i
yi=xi+1−xi so that
y_i
yi's are the "gaps" between consecutive terms of the sequence
\mathcal O
O. Then
w(\mathcal O) = \sup \left\{ y_i: i\geq 1 \right\}
w(O)=sup{yi:i≥1}. On the other hand, for
n = x_i
n=xi
\begin{aligned} a(n)/n &= i/x_i \text{\hspace{2cm} because $a(x_i) = i$}\\ &= \frac{i}{\sum_{j=1}^i y_j} \geq \frac{i}{\sum_{j=1}^i w(\mathcal O)} = 1/w(\mathcal O). \end{aligned}
a(n)/n=i/xibecause a(xi)=i=∑j=1iyji≥∑j=1iw(O)i=1/w(O).
\lim_i \to \infty
limi→∞ we get
d(\mathcal O) \geq 1/w(\mathcal O)
d(O)≥1/w(O). Additionally, if all but finitely many gaps
y_i
yi are equal to
w(\mathcal O)
w(O), then
d(\mathcal O)
d(O) exists and equals
1/w(\mathcal O)
1/w(O).
\liminf_{i\to\infty}
liminfi→∞ of (1) actually shows that even the lower density
\underline{d}(\mathcal O) = \liminf_n a(n)/n
d(O)=liminfna(n)/n is at least
1/w(\mathcal O)
1/w(O). The advantage of using the lower density is that it is defined for all subsets of natural numbers. Let's record this observation.
O_1
O1: for
\mathcal O \subset \mathbb Z_+
O⊂Z+,
1/w(\mathcal O) \leq \underline{d}(\mathcal O)
1/w(O)≤d(O), where
1/w(\mathcal O)=0
1/w(O)=0 if
w(\mathcal O) = \infty
w(O)=∞.
w(\mathcal O_n)
Here is another easy observation:
O_2
O2: If
\forall n \in I, w(\mathcal O_n) \leq M
∀n∈I,w(On)≤M for some fixed integer
M, then
I is finite.
I is infinite, consider
M+1 sets
\mathcal O_1,...,\mathcal O_{M+1}
O1,...,OM+1 and an interval
[a,a+M-1]
[a,a+M−1] where
a is large enough that the smallest element of each
\mathcal O_n, n\leq M+1
On,n≤M+1 is below
a. Then all
M+1 disjoint sets
\mathcal O_1,...,\mathcal O_{M+1}
O1,...,OM+1 have at least one element in
[a,a+M-1]
[a,a+M−1], which is impossible by pigeonhole principle.
w(\mathcal O_n) = O(1)
w(On)=O(1), then
I is finite – we are juggling finitely many balls. This leads to another conjecture: if
\forall n \in I, w(\mathcal O_n) \leq M_n
∀n∈I,w(On)≤Mn, then
I is finite (the only difference is that now
M_n
Mn depends on
n). This one is not true. Let the first set fill half of naturals, the second – half of the remaining half, etc:
\mathcal O_1 = 2\mathbb Z_+,\ \mathcal O_2 = 1 + 4\mathbb Z_+,\ \mathcal O_3 = 3+8\mathbb Z_+,\ ... ,\ \mathcal O_n = 2^{n-1}-1 + 2^n\mathbb Z_+.
O1=2Z+, O2=1+4Z+, O3=3+8Z+, ..., On=2n−1−1+2nZ+.
w(\mathcal O_n) = 2^n
w(On)=2n with
I infinite.
So far we know that if
w(\mathcal O_n) = O(1)
I is finite, but
w(\mathcal O_n) = O(2^n)
w(On)=O(2n) is not strong enough to guarantee that. A natural question is:
Question: is there an increasing function
f(n) such that
w(\mathcal O_n) = O(f(n))
w(On)=O(f(n)) if and only if
I is finite? Perhaps
f(n) = n^2
f(n)=n2 or another polynomial?
We already saw that
1/w(\mathcal O) \leq \underline{d}(\mathcal O)
1/w(O)≤d(O). So it might be fruitful to think about the (lower) densities of
\mathcal O_n
On. Intuitively, the sum of densities of a partition cannot be larger than 1 because each density is the "proportion" of the set in
\mathbb Z_+
Z+. Indeed, this is easy to formally state and prove:
O_3
O3: if sets
\left\{ \mathcal O_n \right\}_{n=1}^\infty
{On}n=1∞ are disjoint, then
\sum_{n=1}^\infty \underline{d}(\mathcal O_n) \leq 1
∑n=1∞d(On)≤1.
Proof: It suffices to prove that
\sum_{n=1}^N \underline{d}(\mathcal O_n) \leq 1
∑n=1Nd(On)≤1 for all
N so fix any
N. We have
\begin{aligned} \sum_{n=1}^N \underline{d}(\mathcal O_n) &= \liminf_{i\to\infty} \sum_{n=1}^N \frac{|\mathcal O_n\cap [i]|}{i} = \liminf_{i\to\infty} \frac{1}{i}\sum_{n=1}^N |\mathcal O_n\cap [i]|\\ &\leq \liminf_{i\to\infty} \frac{1}{i}\times i = 1 \text{ \hspace{1cm} because $\mathcal O_n$'s are disjoint}. \end{aligned}
n=1∑Nd(On)=i→∞liminfn=1∑Ni∣On∩[i]∣=i→∞liminfi1n=1∑N∣On∩[i]∣≤i→∞liminfi1×i=1 because On’s are disjoint.
Putting observations
O_1
O1 and
O_3
O3 together we obtain
O_4
I is infinite, then
\sum_{n=1}^\infty 1/w(\mathcal O_n) \leq 1
∑n=1∞1/w(On)≤1. In particular,
1/w(\mathcal O_n)
1/w(On) is asymptotically smaller than
1/n. Formally,
w(\mathcal O_n) = \omega(n)
w(On)=ω(n).
See wikipedia for the
\omega and
\Omega notation.
The contrapositive of the above is
w(\mathcal O_n) = O(n) \implies |I| < \infty
w(On)=O(n)⟹∣I∣<∞. Is the converse true? For example, can we find an infinite partition
\left\{ \mathcal O_n \right\}
{On} of
\mathbb Z_+
Z+ with
w(\mathcal O_n) \leq n^2?
w(On)≤n2? In order to answer this question I wrote a program that explores the "greedy" algorithm of making successive sets
\mathcal O_n
On.
We want to see if we can "fit" infinitely many disjoint sets
\left\{ \mathcal O_n \right\}
{On} in
\mathbb Z_+
w(\mathcal O_n) \leq n^2
w(On)≤n2 (starting at
n=2). The naive greedy algorithm for doing this is as follows. Let
\mathcal O_2 = 4,8,12,16,\text{etc}
O2=4,8,12,16,etc. We now want to let
\mathcal O_3 = 9,18,27,36,\text{etc.}
O3=9,18,27,36,etc. but there is a collision at
36. So change
36 to
35 and keep going at strides of
9. In general, place the next element of
\mathcal O_n
On, go to the current element plus the width
n^2
n2. If this spot is empty, put it there, if not, try the spot one below, etc. If we are ever unable to find the next spot because on our search down we returned back to the current element, then this strategy fails. If no such failure occurs, then we succeed.
The aforementioned program successfully uses this strategy for
10^8
108 numbers, i.e. we fill the set
\left\{ 1,\dots,10^8 \right\}
{1,…,108} with sets
\mathcal O_2,\dots,\mathcal O_{\sqrt{10^8}}
O2,…,O108 without any problems. Additionally letting the width of
\mathcal O_n
On be
c \cdot n^{1+\epsilon}
c⋅n1+ϵ for higher constants
c and smaller values of
\epsilon>0 suggests that this strategy is successful for
w(\mathcal O_n) = \Theta(n^{1+\epsilon})
w(On)=Θ(n1+ϵ) for
\epsilon>0 arbitrarily small. So we have the conjecture
C_1
C1: For any
\epsilon>0 the "greedy" algorithm produces infinitely many sets
\mathcal O_n
On with
w(\mathcal O_n) = \Theta(n^{1+\epsilon})
w(On)=Θ(n1+ϵ).
If this conjecture is true, then we have the following two facts:
|I| = \infty \implies w(\mathcal O_n) = \omega(n)
∣I∣=∞⟹w(On)=ω(n).
w(\mathcal O_n) = \Omega(n^{1+\epsilon}) \implies |I| = \infty
w(On)=Ω(n1+ϵ)⟹∣I∣=∞.
At that point it is still unclear what happens at, say,
w(\mathcal O_n) = \Theta(n\log^2 n)
w(On)=Θ(nlog2n), since
\sum \frac{1}{n\log^k n}
∑nlogkn1 converges for
k>1. But if
C_1
C1 is true, perhaps the following is true as well.
C_2
C2: For any increasing function
f such that
\sum_n 1/f(n)
∑n1/f(n) converges, the "greedy" algorithm produces infinitely many sets
\mathcal O_n
w(\mathcal O_n) = \Theta(f(n))
w(On)=Θ(f(n)).
This would finally give the equivalence
|I|=\infty \iff \sum_{n\in I} 1/w(\mathcal O_n) < \infty.
∣I∣=∞⟺n∈I∑1/w(On)<∞.
I is finite then any sum over
n\in I converges, so stating the statement above takes a bit more precision but we'll cross that bridge when we get there.
Lean is a proof assistant that allows people to "explain" mathematics to a computer. For example, here is my proof of the squeeze theorem. It would be cool to formalize the above observations in Lean. |
From Asa Gray 15 July [1862]1
On my way into Boston to mail the present envoi,2 I received yours of July 1..3 I open my envelope to acknowledge it.—
First, I am very sorry to hear that your health has suffered.4
2. Platanthera orbiculata has discs still wider apart than P. Hookeri, but no division into “2 bridal chambers”5
I am curious to know what you will say of my notes on P. hyperborea.6 I hope to be able to repeat the observations—in the field. Your son’s obs. on minute insects fertilising is to be noted.7 My pupil, Rothrock, catches Thrips, and only Thrips—in Houstonia.8
About “a third form in some genus—both stamens & pistil short”—9 I think you have a Mertensia in mind which you have referred to in your dimorphic paper;10 but it was noticed only on one or 2 specimens,—of a rare plant—and I do not think much of it. You want facts which can be verified and re-examined. I doubt you have made quite enough of it already,—unless it jibes in with some other better-observed facts.
See end of my over sheet.11
Mitchella repens, L
Two sorts of flowers,
I have had no time to look at them, fresh or dry.13
If young enough the differences would doubtless be as evident (between the pollen & stigmas of the two) as in Houstonia, of which I have sent, or will send, details.14
I have failed to get Rhexia,15
1.1 On my … suffered. 2.1] crossed ink
4.1 I am … A. G. 6.1] crossed ink
Verso: ‘Stamps | Review—Separate notice16 | Mitchella | Pogonia | Houstonia— your pupil experiment | Lythrum | Trübner’17 ink *S1 !alignleft!CD note:18 *S2 The [‘pistil’ del] stigma will project in one form & the anther hidden— Anther projects in other & stigma hidden— Pollen in both, nearly same size; perhaps little larger in short-styled. Pollen
\frac{11}{7000}
\frac{12}{7000}
in short-style long diameter— ink
The year is established by the relationship to the letter to Asa Gray, 1 July [1862].
The letter from Gray to which this is a postscript (see letter from Asa Gray, 21 July 1862) has not been found; it was apparently returned with the letter to Asa Gray, 28 July [1862]. Some indication of its content is conveyed by CD’s annotations, and by his reply (letter to Asa Gray, 28 July [1862]).
Letter to Asa Gray, 1 July [1862].
In the letter to Asa Gray, 1 July [1862], CD complained that he had felt ‘baddish for 2 or 3 weeks’.
In the letter to Asa Gray, 1 July [1862], CD acknowledged receipt of Gray’s notes on P. Hookeri, stating that the species was ‘really beautiful & quite a new case’, and that it was ‘almost laughable the viscid discs getting so far apart that the front of the flower has to be divided into two bridal chambers!’
Gray’s notes on P. hyperborea, which were sent with the letter from Asa Gray, 2–3 July 1862, have not been found; however, see CD’s response in the letter to Asa Gray, 23[–4] July [1862] and n. 23.
Gray refers to George Howard Darwin’s observations of insects pollinating Orchis maculata and Herminium monorchis (see letter to Asa Gray, 1 July [1862] and nn. 8–9).
In the letter from Asa Gray, 2–3 July 1862, Gray had promised to send Joseph Trimble Rothrock’s observations on the dimorphic plant, Houstonia, when completed. See also letter from Asa Gray, 4 August 1862.
See letter to Asa Gray, 1 July [1862] and n. 14.
‘Dimorphic condition in Primula’, pp. 95–6 (Collected papers 2: 62).
This portion of Gray’s package has not been found; it was possibly a sheet protecting Gray’s several enclosures. See also n. 12, below.
This enclosure has been identified by reference to CD’s annotation on the letter respecting Mitchella (see n. 13, below), and to his comments on the subject in the letter to Asa Gray, 28 July [1862]. The paper on which the enclosure is written is different from that of the rest of the letter, and may be part of the ‘over sheet’ referred to by Gray.
Gray had suggested that the dimorphic plant Mitchella repens would be a good case for CD to experiment upon (see Correspondence vol. 9, letter from Asa Gray, 11 October 1861). In the letter from Asa Gray, 2–3 July 1862, Gray expressed a hope that he would be able to collect specimens of the plant during his stay in Beverly, Massachusetts, beginning on 10 July 1862; he apparently enclosed such specimens with this letter (see CD’s annotations, and the letters to Asa Gray, 28 July [1862] and 9 August [1862]).
Gray sent CD observations on the pollen and stigmas of Houstonia in the letter from Asa Gray, [2 June 1862]. See also letters from Asa Gray, 2–3 July 1862 and 4 August 1862.
Gray had evidently enclosed a copy of his review of Orchids (A. Gray 1862a; see letter from Asa Gray, 2–3 July 1862, and letter to Asa Gray, 28 July [1862] and n. 3). Although the review is listed in the List of reviews (DAR 261 (DH/MS* 8: 6–18)) that served as CD’s index to his collection of reviews of his own books, it is absent from the Darwin Pamphlet Collection–CUL.
The annotation refers to Nicholas Trübner, Gray’s London agent (see also letter to Asa Gray, 28 July [1862] and n. 18).
The note is on the verso of the enclosure.
Observations on Platanthera.
Possibility of trimorphism in Mertensia.
DAR 110 (ser. 2): 116, DAR 165: 113
ALS 2pp † encl 1p |
Suppose you are offered the following peculiar gamble. You flip a symmetric coin until you get tails. Once you get tails, you get paid
2^n
2n dollars, where
n is the number of times you flipped the coin (remember that you must stop flipping once you get tails). For example, if you get tails on your first flip, you get 2 dollars. If you get tails on your second flip, you get $4, etc. How much would you be willing to pay to play this game? You would definitely be willing to pay $2, but what about $100, $1000 or a million?
It is probably not a good idea to pay more than $1000 to play this game, because the chance of getting 10 heads in a row is less than 0.1%. In other words, with probability 99.9% you would lose money. So where is the paradox? Well, you can expect to make infinite amount of money from playing this game just once. More precisely, if X is how much money you will be paid (a so-called random variable), the expected value of
X is
\frac{1}{2}\cdot 2 + \frac{1}{4}\cdot 4 + \frac{1}{8}\cdot 8 + … = 1 + 1 + 1 + … = \infty
21⋅2+41⋅4+81⋅8+…=1+1+1+…=∞. So in theory you should be willing to pay any amount, no matter how large, in order to play this game just once. It is interesting to observe that the expectation is infinite despite the fact that with probability 1 you will receive a finite amount of money.
Let's try to resolve this paradox. Suppose that instead of playing this game once, you can play it as many times as you want (let's disregard how long it would take to flip a coin a billion times) and receive the average of your winnings. Further suppose that it costs
S dollars (say,
S = 1 million) to play the game, and you get to play it as many times as you choose. But you must choose how many times you will play before you start playing; otherwise you could just keep playing until you make enough profit, which seems unfair. To summarize:
S = the cost of playing the game, e.g. one million
N = number of times you decide to play the game
X_i
Xi = the (random) amount you win at game i. Since
X_i
Xi are iid, we just use
X when there is no confusion
Now the question becomes: given
S, how many times
N will you play the game to come out positive? How will you go about figuring it out, at least approximately?
The relevant terms from probability theory are Law of Large Numbers and the Central Limit Theorem. But we must be careful using them in our setting. Not only is the second moment of our random variable X infinite, the first moment is as well! Instead we can analyze a truncated random variable
X'
X′, which agrees with
X on the first
2^n
2n values, but is equal to 0 for larger values. For example, if
n = 10
n=10,
X'(1) = 2, X'(2) = 4, X'(3) = 8, …, X'(10) = 2^{10}, X'(11) = 0 = X'(12) = \dots
X′(1)=2,X′(2)=4,X′(3)=8,…,X′(10)=210,X′(11)=0=X′(12)=…. So
X'
X′ is an underestimate of the amount of money you actually win. Intuitively,
X'
X′ makes a lot of sense: if the probability of flipping over 10 heads in a row is less than 0.1%, we may as well treat it as zero, and just pretend that it will never happen. It is convenient that the expected value of
X'
X′ is exactly n, instead of infinity. Now that we got our hands on a simpler random variable, let's see what will happen to the average of
X'_1, …., X'_N
X1′,….,XN′.
\bar X' = 1/M\cdot \sum_{i=1}^N X'_i
Xˉ′=1/M⋅∑i=1NXi′ be the average of our new random variables
X'_1, …., X'_N
X1′,….,XN′. Since
X'
X′ are all iid (independent and identically distributed), the expected value of
\bar X'
Xˉ′ is n as well. Also, the variance of
\bar X'
Xˉ′ is
\frac{1}{M} \cdot (\text{variance of }X')
M1⋅(variance of X′). The variance of X is
\mathbb{E}[X'^2] - \mathbb{E}[X']^2 = 2^{n+1} – 1 – n^2 \approx 2^{n+1}
E[X′2]−E[X′]2=2n+1–1–n2≈2n+1. So The variance of
\bar X'
Xˉ′ is approximately
2^{n+1}/M
2n+1/M.
Let's pause here to ponder what we just computed. Since we only use n to simplify our analysis, we are free to choose it any way we like. Increasing n increases the expected value for
X'
X′ but it also increases its variance exponentially! Recall that variance is a measure of how sure we can be that
X'
X′ lands close to its expected value. So there seems to be a trade-off between higher expected value and being confident that we are not too far off from the expected value.
Now we will play fast and loose with the Central Limit Theorem by treating
\bar X'
Xˉ′ as a normal random variable (it is not necessary but will simplify the analysis). Since it costs S dollars to play the game, we want the expected value of
\bar X'
Xˉ′ to be at least S. So let's take
n to be
2S. Now the variance of
\bar X'
2^{2S+1}/M
22S+1/M. Now taking
M to be
2^{2S+1} = 2 \cdot (2^S)^2
22S+1=2⋅(2S)2 ensures that the variance is 1. Since we are treating
\bar X'
Xˉ′ as a normal random variable, the probability that
\bar X'
Xˉ′ is below
S is very low. Since
X'
X′ is an underestimate of
X,
\bar X'
Xˉ′ is an underestimate of the average of
X_1, …, X_M
X1,…,XM. So with overwhelming probability, the average of your winnings will be above
S.
We have now answered the question: taking
M to be
2 \cdot (2^S)^2
2⋅(2S)2 ensures coming out positive when the cost is
S. The important bit here is that
M is exponential in
S. For example, if
S = 10
S=10, i.e. it costs $10 to play, then our computations suggest
M = 2 \cdot 2^{20} \approx 1,000,000
M=2⋅220≈1,000,000. This seems like a gross over-estimate. Perhaps using finer techniques such as tail bounds on subgaussian random variables will yield a more realistic estimate. |
Null structure and almost optimal local regularity for the Dirac-Klein-Gordon system | EMS Press
We prove almost optimal local well-posedness for the coupled Dirac-Klein-Gordon (DKG) system of equations in
1+3
dimensions. The proof relies on the null structure of the system, combined with bilinear spacetime estimates of Klainerman-Machedon type. It has been known for some time that the Klein-Gordon part of the system has a null structure; here we uncover an additional null structure in the Dirac equation, which cannot be seen directly, but appears after a duality argument.
Piero D'Ancona, Damiano Foschi, Sigmund Selberg, Null structure and almost optimal local regularity for the Dirac-Klein-Gordon system. J. Eur. Math. Soc. 9 (2007), no. 4, pp. 877–899 |
Gear train with sun, planet, and ring gears - MATLAB - MathWorks í•œêµ
{r}_{C}{\mathrm{Ï}}_{C}={r}_{S}{\mathrm{Ï}}_{S}+{r}_{P}{\mathrm{Ï}}_{P}
{r}_{R}{\mathrm{Ï}}_{R}={r}_{C}{\mathrm{Ï}}_{C}+{r}_{P}{\mathrm{Ï}}_{P}
{r}_{C}={r}_{S}+{r}_{P}
{r}_{R}={r}_{C}+{r}_{P}
ωC is the angular velocity of the carrier gear.
ωS is the angular velocity of the sun gear.
ωp is the angular velocity of the planet gears.
{g}_{RS}={r}_{R}/{r}_{S}={N}_{R}/{N}_{S},
\left(\text{1 }+ {g}_{RS}\right){\mathrm{Ï}}_{C} = {\mathrm{Ï}}_{S} + {g}_{RS}{\mathrm{Ï}}_{R}.
{g}_{RS}{\mathrm{Ï}}_{S} + {\mathrm{Ï}}_{R}– {\mathrm{Ï}}_{loss}=\text{ }0,
Ï„S is torque transfer for the sun gear.
Ï„R is torque transfer for the ring gear.
Ï„loss is torque transfer loss.
In the ideal case where there is no torque loss, Ï„loss = 0.
In the nonideal case, τloss ≠0. For more information, see Model Gears with Losses.
Constant efficiency — Transfer of torque between the gear wheel pairs is reduced by a constant efficiency, η, such that 0 < η ≤ 1.
Vector of torque transfer efficiencies, [ηSP ηRP], for sun-planet and ring-carrier gear wheel pair meshings, respectively.
Vector of output-to-input power ratios that describe the power flow from the sun gear to the planet gear, ηSP. The block uses the values to construct a 1-D temperature-efficiency lookup table.
Vector of output-to-input power ratios that describe the power flow from the ring gear to the planet gear, ηRP. The block uses the values to construct a 1-D temperature-efficiency lookup table.
Vector of viscous friction coefficients, [μS, μP], for the sun-carrier and planet-carrier gear motions, respectively. |
SolveFeedback - Maple Help
Home : Support : Online Help : Education : Grading : SolveFeedback
generate feedback for a step-by-step solution
SolveFeedback(solution,options)
solution steps {canvas-string,table,rtable,list}
(optional) equation(s) of the form option = value where option is one of variables, equations, or solutions
variables = list, the variable(s) to solve for
equations = list, the initial expression
solutions = list, the solutions that are relevant
The SolveFeedback command is the procedure behind the SolvePractice command. It analyzes a step-by-step solution to the given problem and provides feedback for each step of the solution. It can also be called directly, or used behind the scenes for other interactive applications of your own design.
The solution argument, when part of an interactive canvas-based application, is a XML-based string representing a canvas, or a table representing the embedded component name mapping. In this case, the returned result is a DocumentTools:-Canvas:-Script.
When called directly, solution can also be a list or array of expressions. In this case, the returned result is a list of feedback strings. The result has one extra string than the number of input steps, concluding with a (possibly empty) summary statement.
The variables = [x] option is useful to provide when the expression to solve for is multivariate. This clarifies exactly which variable is to be solved for. As of Maple 2020, the list must contain exactly one variable.
The solutions = [x=0] option allows you to specify which solutions are relevant to the given problem, in order to avoid feedback that prompts to solve for more solutions.
The equations = [...] option allows you to specify the initial expression, as opposed to showing it as the first expression in a canvas. This way it is up to the user to come up with the equation themselves.
\mathrm{with}\left(\mathrm{Grading}\right):
\mathrm{SolveFeedback}\left([2x+3=4x-5,2x-4x=-5-3,-2x=-8,x=4]\right)
[\textcolor[rgb]{0,0,1}{""}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"ok"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"ok"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"Good job, this is the correct solution"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{""}]
\mathrm{SolveFeedback}\left([2x+3=4x-5,2x-4x=-5+3,-2x=-2,x=1]\right)
[\textcolor[rgb]{0,0,1}{""}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"Check this step..."}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"consistent with previous step"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"consistent with previous step"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"no correct and fully simplified solution found; please try again"}]
\mathrm{SolveFeedback}\left([2x+y=4y,2x=3y,y=\frac{2}{3}x],\mathrm{variables}=[y]\right)
[\textcolor[rgb]{0,0,1}{""}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"ok"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"Good job, this is the correct solution"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{""}]
In this example we create an interactive canvas presenting a word problem. The input equation is not shown. Note that the t=-2 root is not part of the solutions list, as it does not make physical sense.
\mathrm{with}\left(\mathrm{Grading}\right):
\mathrm{with}\left(\mathrm{DocumentTools}:-\mathrm{Canvas}\right):
\mathrm{cv}≔\mathrm{NewCanvas}\left(["An object is launched at 19.6 meters per second \left(m/s\right) from a 58.8-meter tall platform. The equation for the object\text{'}s height s at time t seconds after launch is",s\left(t\right)=3.4220022349×{10}^{9}{t}^{2}+19.6t+58.8,"where s is in meters. When does the object strike the ground?",\mathrm{ScriptButton}\left("Check Page","Grading\left[SolveFeedback\right]\left(_canvas,equations=\left[t^2-4*t-12=0\right],variables=\left[t\right],solutions=\left[t=6\right]\right);",\mathrm{position}=[800,90],\mathrm{encode}=\mathrm{false}\right)]\right):
\mathrm{ShowCanvas}\left(\mathrm{cv},\mathrm{input}=["0 = t^2 -4*t -12","0 = \left(t-6\right)*\left(t+2\right)","t=6"]\right)
The feedback can be generated directly as follows:
\mathrm{SolveFeedback}\left(["0 = t^2 -4*t -12","0 = \left(t-6\right)*\left(t+2\right)","t=6"],\mathrm{equations}=[{t}^{2}-4t-12=0],\mathrm{variables}=[t],\mathrm{solutions}=[t=6]\right)
[\textcolor[rgb]{0,0,1}{""}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"ok"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"Good job, this is a correct solution"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"Good job, you found the correct solution"}]
The canvas can also be shared via Maple Learn:
\mathrm{ShareCanvas}\left(\mathrm{cv}\right)
The Grading:-SolveFeedback command was introduced in Maple 2021. |
Category:Submarines - Azur Lane Wiki
A submarine is a watercraft capable of independent operation underwater. It differs from a submersible, which has more limited underwater capability. Although experimental submarines had been built before, submarine design took off during the 19th century, and they were adopted by several navies. Submarines were first widely used during World War I (1914–1918), and now figure in many navies large and small. Military uses include attacking enemy surface ships (merchant and military), attacking other submarines, aircraft carrier protection, blockade running and reconnaissance.
The navy best known to use submarines during the 2nd world war is Germany, although other navies had submarines relatively in high amount such as the IJN whom possess midget submarines, medium-range submarines, purpose-built supply submarines and long-range fleet submarines. They also had submarines with the highest submerged speeds during World War II (I-201-class submarines) and submarines that could carry multiple aircraft (I-400-class submarines). They were also equipped with one of the most advanced torpedoes of the conflict, the oxygen-propelled Type 95. Nevertheless, despite their technical prowess, Japan chose to utilize its submarines for fleet warfare, and consequently were relatively unsuccessful, as warships were fast, maneuverable and well-defended compared to merchant ships.
The USN used submarines similar to the IJN in the Pacific which, though only about 2 percent of the U.S. Navy, destroyed over 30 percent of the Japanese Navy, including 8 aircraft carriers, 1 battleship and 11 cruisers. US submarines also destroyed over 60 percent of the Japanese merchant fleet, crippling Japan's ability to supply its military forces and industrial war effort.
The Royal Navy Submarine Service was used primarily in the classic Axis blockade. Its major operating areas were around Norway, in the Mediterranean (against the Axis supply routes to North Africa), and in the Far East. In that war, British submarines sank 2 million tons of enemy shipping and 57 major warships, the latter including 35 submarines. Among these is the only documented instance of a submarine sinking another submarine while both were submerged.
Usage in Azur Lane
In Azur Lane, submarines function as support type ships, which unlike the other ship classes are not controllable by the player. Furthermore they have the following quirks:
Submarines can only be deployed on X-4 of each chapter (chapter 3 and onward), X-3 (chapter 8 and onward), X-2 (chapter 10 and onward), X-1 (chapter 11 and onward), SOS Rescue Missions, and event stages.
They have a limited hunting range, as shown by a red grid on the stage map.
They move 2 spaces for everytime you move your fleet regardless the distance your fleet does.
If an enemy fleet is present within their hunting range, they'll submerge and emerge at their target location and attack it after you move your fleet. Damage varies by your submarine's overall strength. This consumes their ammo supply.
{\displaystyle {\mbox{Node HP Reduction Percentage}}=0.15\times {\sqrt {\text{Submarine Fleet Power}}}+0.25\times ({\text{Average Submarine Fleet Level}}-{\text{Enemy Level}})}
They can support your fleet during combat should you battle within their hunting range, and can be summoned in battle with the submarine attack button. Summoning them consumes their ammo supply and fuel dictated by their fuel cost.
Once summoned they'll attack all present enemies until they run out of oxygen and are forced to resurface and eventually retreat.
Submarines have a certain amount of oxygen supply, which once runs out they'll resurface and become vulnerable to attacks. They can still fight back using the DD guns equipped to them.
Their automatic movement can be disabled on the formation menu.
You can adjust the position of subs, this will cost oil depending on the number of tiles and number of subs on the fleet. Subs will use the new tile as the center of their hunting range.
A theorized formula for the oil cost is
{\displaystyle 1+\left\lfloor 1.1\times {\text{number of subs}}\times {\text{distance moved}}\right\rfloor }
Submarine carriers
Pages in category ‘Submarines’
List of Submarines by Stats
Retrieved from ‘https://azurlane.koumakan.jp/w/index.php?title=Category:Submarines&oldid=113247’ |
Index Futures - DeFi Index - Delta Exchange - User Guide
DEFI Index is composed of the top ten DeFi tokens listed on Delta Exchange. DeFi Index tracks market performance of a basket of DeFi tokens. Index price is calculated using weighted average real time prices of component tokens on Delta Exchange. It is quoted in USDT.
Chainlink (LINK) , Aave (AAVE), Yearn.finance protocol (YFI) , Uniswap protocol token (UNI) , Ren (REN), Kyber network token (KNC), Balancer token (BAL), Band protocol (BAND) , Synthetix network token (SNX ), Compound (COMP)
DeFi Coins listed on Delta Exchange
DeFi Index is made of 10 components. These components are rebalanced at the end of every month. Weights of components depend on the market cap and 30 day volume of the respective components.
Weight of each token in the index is the average of capitalization and liquidity weight. Market cap weight is calculated as percentage share of the market cap of a taken compared to the market cap of index. Similarly, volume weight is also the percentage share of the volume of a taken compared to the total volume of all tokens in the index.
Weight = ( Capitalization\ Weight + Liquidity\ Weight)/2
Liquidity Weight = (30\ day\ volume\ of\ the\ token)/ (Sum\ of\ the\ 30\ day\ volume\ of\ all\ the\ tokens)
Liquidity Weight = (30\ day\ volume\ of\ the\ token)/ (Sum\ of\ the\ 30\ day\ volume\ of\ all\ the\ tokens)
If the capitalization or liquidity weight is greater than 30%, the weight is limited to 30%. Therefore, none of the component tokens can get weight greater than 30%.
Index is rebalanced at the end of every month. During rebalancing, new components can be added or existing components can be excluded according to the following criteria:
If a new DeFi token listed on Delta exchange gets higher ranking based on the capitalization and liquidity, it will be added to the index. Top 10 tokens in the rankings will be part of the index.
In special situation events like soft/hard fork, where token gets split, token may be excluded based on the capitalization and liquidity ranking.
In case any component token ceases to trade on Delta Exchange, it will be excluded from the DEFI Index.
Mkt Cap (M USD)
Volume (M 30D)
Cap Wt
Liquidity Wt
Adj Cap Wt
Adj Liquidity Wt |
(Redirected from Concrete syntax tree)
Parse tree to SAAB
A parse tree or parsing tree[1] or derivation tree or concrete syntax tree is an ordered, rooted tree that represents the syntactic structure of a string according to some context-free grammar. The term parse tree itself is used primarily in computational linguistics; in theoretical syntax, the term syntax tree is more common.
Concrete syntax trees reflect the syntax of the input language, making them distinct from the abstract syntax trees used in computer programming. Unlike Reed-Kellogg sentence diagrams used for teaching grammar, parse trees do not use distinct symbol shapes for different types of constituents.
A related concept is that of phrase marker or P-marker, as used in transformational generative grammar. A phrase marker is a linguistic expression marked as to its phrase structure. This may be presented in the form of a tree, or as a bracketed expression. Phrase markers are generated by applying phrase structure rules, and themselves are subject to further transformational rules.[2] A set of possible parse trees for a syntactically ambiguous sentence is called a "parse forest."[3]
A simple parse tree
A parse tree is made up of nodes and branches.[4] In the picture the parse tree is the entire structure, starting from S and ending in each of the leaf nodes (John, ball, the, hit). In a parse tree, each node is either a root node, a branch node, or a leaf node. In the above example, S is a root node, NP and VP are branch nodes, while John, ball, the, and hit are all leaf nodes.
Nodes can also be referred to as parent nodes and child nodes. A parent node is one which has at least one other node linked by a branch under it. In the example, S is a parent of both NP and VP. A child node is one which has at least one node directly above it to which it is linked by a branch of the tree. Again from our example, hit is a child node of V.
A nonterminal function is a function (node) which is either a root or a branch in that tree whereas a terminal function is a function (node) in a parse tree which is a leaf.
Constituency-based parse trees[edit]
The constituency-based parse trees of constituency grammars (phrase structure grammars) distinguish between terminal and non-terminal nodes. The interior nodes are labeled by non-terminal categories of the grammar, while the leaf nodes are labeled by terminal categories. The image below represents a constituency-based parse tree; it shows the syntactic structure of the English sentence John hit the ball:
Each node in the tree is either a root node, a branch node, or a leaf node.[5] A root node is a node that doesn't have any branches on top of it. Within a sentence, there is only ever one root node. A branch node is a parent node that connects to two or more child nodes. A leaf node, however, is a terminal node that does not dominate other nodes in the tree. S is the root node, NP and VP are branch nodes, and John (N), hit (V), the (D), and ball (N) are all leaf nodes. The leaves are the lexical tokens of the sentence.[6][page needed] A parent node is one that has at least one other node linked by a branch under it. In the example, S is a parent of both N and VP. A child node is one that has at least one node directly above it to which it is linked by a branch of a tree. From the example, hit is a child node of V. The terms mother and daughter are also sometimes used for this relationship.
Dependency-based parse trees[edit]
Phrase markers[edit]
Phrase markers, or P-markers, were introduced in early transformational generative grammar, as developed by Noam Chomsky and others. A phrase marker representing the deep structure of a sentence is generated by applying phrase structure rules. Then, this application may undergo further transformations.
Phrase markers may be presented in the form of trees (as in the above section on constituency-based parse trees), but are often given instead in the form of "bracketed expressions", which occupy less space in the memory. For example, a bracketed expression corresponding to the constituency-based tree given above may be something like :
{\displaystyle [_{S}\ [_{\mathit {N}}\ {\text{John}}]\ [_{\mathit {VP}}\ [_{V}\ {\text{hit}}]\ [_{\mathit {NP}}\ [_{\mathit {D}}\ {\text{the}}]\ [_{N}\ {\text{ball}}]]]]}
As with trees, the precise construction of such expressions and the amount of detail shown can depend on the theory being applied and on the points that the query author wishes to illustrate.
Parsing (syntax analysis)
^ See Chiswell and Hodges 2007: 34.
^ Noam Chomsky (26 December 2014). Aspects of the Theory of Syntax. MIT Press. ISBN 978-0-262-52740-8.
^ Billot, Sylvie, and Bernard Lang. "The structure of shared forests in ambiguous parsing."
^ "The parsetree Package for Drawing Trees in LaTeX". www1.essex.ac.uk.
^ See Carnie (2013:118ff.) for an introduction to the basic concepts of syntax trees (e.g. root node, terminal node, non-terminal node, etc.).
^ See Aho et al. 1986.
^ See for example Ágel et al. 2003/2006.
Aho, A. V., Sethi, R., and Ullman, J. D. 1986. Compilers: Principles, techniques, & tools. Reading, MA: Addison-Wesley.
Penn Treebank II Constituent Tags
Retrieved from "https://en.wikipedia.org/w/index.php?title=Parse_tree&oldid=1077659122" |
This paper is concerned with the global exact controllability of the semilinear heat equation (with nonlinear terms involving the state and the gradient) completed with boundary conditions of the form
\frac{\partial y}{\partial n}+f\left(y\right)=0
. We consider distributed controls, with support in a small set. The null controllability of similar linear systems has been analyzed in a previous first part of this work. In this second part we show that, when the nonlinear terms are locally Lipschitz-continuous and slightly superlinear, one has exact controllability to the trajectories.
Classification : 35K20, 93B05
Mots clés : controllability, heat equation, Fourier boundary conditions, semilinear
author = {Fern\'andez-Cara, Enrique and Gonz\'alez-Burgos, Manuel and Guerrero, Sergio and Puel, Jean-Pierre},
title = {Exact controllability to the trajectories of the heat equation with {Fourier} boundary conditions : the semilinear case},
AU - Fernández-Cara, Enrique
AU - González-Burgos, Manuel
AU - Guerrero, Sergio
AU - Puel, Jean-Pierre
TI - Exact controllability to the trajectories of the heat equation with Fourier boundary conditions : the semilinear case
Fernández-Cara, Enrique; González-Burgos, Manuel; Guerrero, Sergio; Puel, Jean-Pierre. Exact controllability to the trajectories of the heat equation with Fourier boundary conditions : the semilinear case. ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 3, pp. 466-483. doi : 10.1051/cocv:2006011. http://www.numdam.org/articles/10.1051/cocv:2006011/
[1] H. Amann, Parabolic evolution equations and nonlinear boundary conditions. J. Diff. Equ. 72 (1988) 201-269. | Zbl 0658.34011
[2] J. Arrieta, A. Carvalho and A. Rodríguez-Bernal, Parabolic problems with nonlinear boundary conditions and critical nonlinearities. J. Diff. Equ. 156 (1999) 376-406. | Zbl 0938.35077
[3] J.P. Aubin, L'analyse non linéaire et ses motivations économiques. Masson, Paris (1984). | Zbl 0551.90001
[4] O. Bodart, M. González-Burgos and R. Pŕez-García, Insensitizing controls for a semilinear heat equation with a superlinear nonlinearity. C. R. Math. Acad. Sci. Paris 335 (2002) 677-682. | Zbl 1021.35049
[5] A. Doubova, E. Fernández-Cara and M. González-Burgos, On the controllability of the heat equation with nonlinear boundary Fourier conditions. J. Diff. Equ. 196 (2004) 385-417. | Zbl 1049.35042
[6] A. Doubova, E. Fernández-Cara, M. González-Burgos and E. Zuazua, On the controllability of parabolic systems with a nonlinear term involving the state and the gradient. SIAM J. Control Optim. 41 (2002) 798-819. | Zbl 1038.93041
[7] L. Evans, Regularity properties of the heat equation subject to nonlinear boundary constraints. Nonlinear Anal. 1 (1997) 593-602. | Zbl 0369.35034
[8] C. Fabre, J.P. Puel and E. Zuazua, Approximate controllability of the semilinear heat equation. Proc. Roy. Soc. Edinburgh 125A (1995) 31-61. | Zbl 0818.93032
[9] L.A. Fernández and E. Zuazua, Approximate controllability for the semi-linear heat equation involving gradient terms. J. Optim. Theory Appl. 101 (1999) 307-328. | Zbl 0952.49003
[10] E. Fernández-Cara, M. González-Burgos, S. Guerrero and J.P. Puel, Null controllability of the heat equation with boundary Fourier conditions: The linear case. ESAIM: COCV 12 442-465. | Numdam | Zbl 1106.93009
[11] E. Fernández-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations. Ann. Inst. H. Poincaré, Anal. non Linéaire 17 (2000) 583-616. | Numdam | Zbl 0970.93023
[12] A. Fursikov and O.Yu. Imanuvilov, Controllability of Evolution Equations. Lecture Notes #34, Seoul National University, Korea (1996). | MR 1406566 | Zbl 0862.49004
[13] I. Lasiecka and R. Triggiani, Exact controllability of semilinear abstract systems with applications to waves and plates boundary control. Appl. Math. Optim. 23 (1991) 109-154. | Zbl 0729.93023
[14] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. Cambridge University Press, Cambridge (2000). | Zbl 0961.93003
[15] E. Zuazua, Exact boundary controllability for the semilinear wave equation, in Nonlinear Partial Differential Equations and their Applications, Vol. X, H. Brezis and J.L. Lions Eds. Pitman (1991) 357-391. | Zbl 0731.93011
[16] E. Zuazua, Exact controllability for the semilinear wave equation in one space dimension. Ann. I.H.P., Analyse non Linéaire 10 (1993) 109-129. | Numdam | Zbl 0769.93017 |
The spinor inner product of two spinors
S
T
of the same type is calculated by contracting each pair of corresponding spinor indices (one from
S
and one from
T
) with the appropriate epsilon spinor. For example, the inner product of two covariant rank 1 spinors with components
{S}_{A}
{T}_{B}
{\mathrm{ε}}^{\mathrm{AB}}{S}_{A}{T}_{B}.
The inner product of two contravariant rank 1 spinors
{S}_{}^{A}
{T}_{}^{B}
{\mathrm{ε}}_{\mathrm{AB}}^{}{S}_{}^{A}{T}_{}^{B}
. The inner product of two contravariant rank 2 spinors with components
{S}_{\mathrm{AB}}
{T}_{\mathrm{CD}}
{\mathrm{ε}}^{\mathrm{AC}}{\mathrm{ε}}^{\mathrm{BD}}
{S}_{\mathrm{AB}}
{T}_{\mathrm{CD}}
S
T
are odd rank spinors, then SpinorInnerProduct(S, T) = -SpinorInnerProduct(T, S) and therefore SpinorInnerProduct(S, S) = 0. (Strictly speaking, the spinor inner product is really just a bilinear pairing -- it is not a true inner product because it is not always symmetric in its arguments.)
S
T
are even rank spinors, then SpinorInnerProduct(S, T) = SpinorInnerProduct(T, S).
\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):
\mathrm{with}\left(\mathrm{Tensor}\right):
M
\left(x, y, z, t\right)
\left(\mathrm{z1}, \mathrm{z2}, \mathrm{w1}, \mathrm{w2}\right)
\mathrm{DGsetup}\left([x,y,z,t],[\mathrm{z1},\mathrm{z2},\mathrm{w1},\mathrm{w2}],M\right)
\textcolor[rgb]{0,0,1}{\mathrm{frame name: M}}
\mathrm{S1}
\mathrm{T1}
and calculate their inner product.
\mathrm{S1}≔\mathrm{evalDG}\left(a\mathrm{D_z1}+b\mathrm{D_z2}\right)
\textcolor[rgb]{0,0,1}{\mathrm{S1}}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"vector"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{M}\textcolor[rgb]{0,0,1}{,}[]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{5}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{a}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{6}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}]]]\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"vector"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{M}\textcolor[rgb]{0,0,1}{,}[]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{5}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{a}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{6}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}]]]\right)
\mathrm{T1}≔\mathrm{evalDG}\left(c\mathrm{D_z1}+d\mathrm{D_z2}\right)
\textcolor[rgb]{0,0,1}{\mathrm{T1}}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"vector"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{M}\textcolor[rgb]{0,0,1}{,}[]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{5}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{c}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{6}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{d}]]]\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"vector"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{M}\textcolor[rgb]{0,0,1}{,}[]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{5}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{c}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{6}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{d}]]]\right)
\mathrm{SpinorInnerProduct}\left(\mathrm{S1},\mathrm{T1}\right)
\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{c}
\mathrm{SpinorInnerProduct}\left(\mathrm{T1},\mathrm{S1}\right)
\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{c}
\mathrm{SpinorInnerProduct}\left(\mathrm{S1},\mathrm{S1}\right)
\textcolor[rgb]{0,0,1}{0}
Calculate the inner product of
\mathrm{S1}
\mathrm{T1}
from the definition.
\mathrm{\epsilon }≔\mathrm{EpsilonSpinor}\left("cov","spinor"\right)
\textcolor[rgb]{0,0,1}{\mathrm{ϵ}}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"tensor"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{M}\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{"cov_vrt"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"cov_vrt"}]\textcolor[rgb]{0,0,1}{,}[]]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-1}]]]\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"tensor"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{M}\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{"cov_vrt"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"cov_vrt"}]\textcolor[rgb]{0,0,1}{,}[]]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-1}]]]\right)
\mathrm{U1}≔\mathrm{ContractIndices}\left(\mathrm{\epsilon },\mathrm{S1},[[1,1]]\right)
\textcolor[rgb]{0,0,1}{\mathrm{U1}}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"tensor"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{M}\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{"cov_vrt"}]\textcolor[rgb]{0,0,1}{,}[]]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{5}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{b}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{6}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{a}]]]\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"tensor"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{M}\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{"cov_vrt"}]\textcolor[rgb]{0,0,1}{,}[]]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{5}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{b}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{6}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{a}]]]\right)
\mathrm{ContractIndices}\left(\mathrm{U1},\mathrm{T1},[[1,1]]\right)
\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{c}
Calculate the inner product of two rank 2 spinors
\mathrm{S2}
\mathrm{T2}
\mathrm{S2}≔\mathrm{evalDG}\left(a\mathrm{D_z1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dw2}+b\mathrm{D_z1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dw1}\right)
\textcolor[rgb]{0,0,1}{\mathrm{S2}}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"tensor"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{M}\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{"con_vrt"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"cov_vrt"}]\textcolor[rgb]{0,0,1}{,}[]]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{8}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{a}]]]\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"tensor"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{M}\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{"con_vrt"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"cov_vrt"}]\textcolor[rgb]{0,0,1}{,}[]]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{8}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{a}]]]\right)
\mathrm{T2}≔\mathrm{evalDG}\left(c\mathrm{D_z1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dw1}+d\mathrm{D_z2}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dw2}\right)
\textcolor[rgb]{0,0,1}{\mathrm{T2}}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"tensor"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{M}\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{"con_vrt"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"cov_vrt"}]\textcolor[rgb]{0,0,1}{,}[]]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{c}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{8}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{d}]]]\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"tensor"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{M}\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{"con_vrt"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"cov_vrt"}]\textcolor[rgb]{0,0,1}{,}[]]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{c}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{8}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{d}]]]\right)
\mathrm{SpinorInnerProduct}\left(\mathrm{S2},\mathrm{T2}\right)
\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{d}
Calculate the inner product of two rank 2 spinor-tensors
\mathrm{S3}
\mathrm{T3}
. Note that in this example the result is a rank 2 tensor.
\mathrm{S3}≔\mathrm{evalDG}\left(\mathrm{D_t}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dw1}+\mathrm{D_z}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dw2}\right)
\textcolor[rgb]{0,0,1}{\mathrm{S3}}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"tensor"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{M}\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{"con_bas"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"cov_vrt"}]\textcolor[rgb]{0,0,1}{,}[]]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{8}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}]]]\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"tensor"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{M}\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{"con_bas"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"cov_vrt"}]\textcolor[rgb]{0,0,1}{,}[]]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{8}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}]]]\right)
\mathrm{T3}≔\mathrm{evalDG}\left(\mathrm{D_y}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dw1}+\mathrm{D_x}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dw2}\right)
\textcolor[rgb]{0,0,1}{\mathrm{T3}}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"tensor"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{M}\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{"con_bas"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"cov_vrt"}]\textcolor[rgb]{0,0,1}{,}[]]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{8}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}]]]\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"tensor"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{M}\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{"con_bas"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"cov_vrt"}]\textcolor[rgb]{0,0,1}{,}[]]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{8}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}]]]\right)
\mathrm{SpinorInnerProduct}\left(\mathrm{S3},\mathrm{T3}\right)
\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"tensor"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{M}\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{"con_bas"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"con_bas"}]\textcolor[rgb]{0,0,1}{,}[]]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-1}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}]]]\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"tensor"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{M}\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{"con_bas"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"con_bas"}]\textcolor[rgb]{0,0,1}{,}[]]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-1}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}]]]\right) |
Warning, A new binding for the name `gamma` has been created. The global instance of this name is still accessible using the :- prefix, :-`gamma`. See ?protect for details. - Maple Help
Home : Support : Online Help : Warning, A new binding for the name `gamma` has been created. The global instance of this name is still accessible using the :- prefix, :-`gamma`. See ?protect for details.
\mathrm{start}≔1.3982
\textcolor[rgb]{0,0,1}{\mathrm{start}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{1.3982}
\mathrm{finish}≔12.2315
\textcolor[rgb]{0,0,1}{\mathrm{finish}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{12.2315}
\mathrm{diff}≔\mathrm{finish}-\mathrm{start}
\mathbf{local} \mathrm{diff}≔\mathrm{finish}-\mathrm{start}
\textcolor[rgb]{0,0,1}{\mathrm{diff}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{10.8333}
\mathrm{diff}
\textcolor[rgb]{0,0,1}{10.8333}
:-\mathrm{diff}\left(\mathrm{ln}\left(x\right),x\right)
\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{x}}
\mathrm{myname}≔"global version of myname"
\textcolor[rgb]{0,0,1}{\mathrm{myname}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{"global version of myname"}
\mathbf{local} \mathrm{myname}≔"local version of myname"
\textcolor[rgb]{0,0,1}{\mathrm{myname}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{"local version of myname"}
\mathrm{myname}
\textcolor[rgb]{0,0,1}{"local version of myname"}
:-\mathrm{myname}
\textcolor[rgb]{0,0,1}{"global version of myname"}
\mathrm{myProtectedName}≔19
\textcolor[rgb]{0,0,1}{\mathrm{myProtectedName}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{19}
\mathrm{protect}\left('\mathrm{myProtectedName}'\right)
\mathrm{myProtectedName}≔23
\mathrm{unprotect}\left('\mathrm{myProtectedName}'\right)
\mathrm{myProtectedName}≔23
\textcolor[rgb]{0,0,1}{\mathrm{myProtectedName}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{23}
\mathrm{eval}\left(\mathrm{arctan}\left(1\right)\right)
\frac{\textcolor[rgb]{0,0,1}{\mathrm{\pi }}}{\textcolor[rgb]{0,0,1}{4}}
\mathrm{unprotect}\left(\mathrm{\pi }\right)
\mathrm{\pi }≔4
\textcolor[rgb]{0,0,1}{\mathrm{\pi }}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{4}
\mathrm{eval}\left(\mathrm{arctan}\left(1\right)\right)
\textcolor[rgb]{0,0,1}{1} |
Choose Trajectories for Manipulator Paths - MATLAB & Simulink - MathWorks France
You can use the TimeScaling name-value argument of transformtraj as a workaround. This argument defines the trajectory time using an intermediate parameterization,
\mathit{s}
, such that transformtraj is defined using
\mathit{s}\left(\mathit{t}\right)
as time. In the default case used in this example, time scaling is uniform, so
\mathit{s}\left(\mathit{t}\right)=\mathit{t}
. The result is a linear motion between each pose. Instead, use time scaling defined by a minimum-jerk trajectory:
\mathit{s}\left(\mathit{t}\right)=
minjerkpolytraj
\left(\mathit{t}\right)
Time scaling is a discrete set of values,
\left[\mathit{s};\text{\hspace{0.17em}}\frac{d}{\mathrm{d}\mathit{t}}\mathit{s};\text{\hspace{0.17em}}\frac{{d}^{2}}{\mathrm{d}{\mathit{t}}^{2}}\mathit{s}\right]
, which sample the function
\mathit{s}\left(\mathit{t}\right)
, defined on the interval
\mathit{s}=\left[0,1\right] |
Profit Loss Math - Delta Exchange - User Guide
All open positions on Delta are marked at the fair price of the Futures contract. Thus, Unrealized PnL and Liquidation Prices are computed using Fair Prices, while Realized PnL is based on actual entry and exit prices.
PnL for a long/ short position in a Vanilla Futures
PnL = ± n*m*(Fut\_CurrentPrice - Fut\_EntryPrice)
PnL for a long/ short position in an Inverse Futures
PnL = ± n*m*(1/ Fut\_EntryPrice - 1/ Fut\_CurrentPrice)
m
is the multiplier and
n
is the position size (i.e. number of contracts).
If a position is acquired at multiple entry prices, an average entry price is computed and used for PnL computation. |
\epsilon \propto \rho T^\beta\ ,\ beta = -\frac{2}{3} + 23.6 T_&^{-1/3}
and therefore at $T \sim 10^7\ K$, the beta factor is $\sim 20$. \\
{\displaystyle \left(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V\right)\psi =E\psi .\,\!}
{\displaystyle E={\frac {k^{2}\hbar ^{2}}{2m}}.\,\!}
{\displaystyle k}
{\displaystyle V=V_{0}}
{\displaystyle E-V_{0}={\frac {k^{2}\hbar ^{2}}{2m}}.\,\!}
{\displaystyle \sigma (E)={\frac {S(E)}{E}}e^{-(E_{G}/E)^{1/2}}.\,\!}
{\displaystyle S(E)}
{\displaystyle E}
{\displaystyle E_{G}}
{\displaystyle E_{G}=(1{\rm {\;MeV}})Z_{1}^{2}Z_{2}^{2}{\frac {m_{r}}{m_{p}}}.\,\!}
{\displaystyle n_{1}}
{\displaystyle n_{2}}
{\displaystyle \ell _{2}={\frac {1}{n_{1}\sigma }}\,\!}
{\displaystyle \tau _{2}={\frac {1}{n_{1}\sigma v}}.\,\!}
{\displaystyle r_{12}={\frac {n_{2}}{\tau _{2}}}=n_{1}n_{2}\sigma v.\,\!}
{\displaystyle r_{12}=n_{1}n_{2}<\sigma (E)v>.\,\!}
{\displaystyle <\sigma (E)v>}
{\displaystyle <\sigma (E)v>=\int d^{3}v\;prob(v)\sigma (E)v.\,\!}
{\displaystyle r_{12}=n_{1}n_{2}\int d^{3}v\sigma (E)v\left({\frac {m_{r}}{2\pi kT}}\right)^{3/2}e^{-{\frac {{\frac {1}{2}}m_{r}v^{2}}{kT}}}.\,\!}
{\displaystyle E={\frac {1}{2}}m_{r}v^{2},\,\!}
{\displaystyle dE=m_{r}vdv,\,\!}
{\displaystyle d^{3}v=4\pi v^{2}dv=4\pi {\frac {v^{2}}{v}}{\frac {dE}{m_{r}}},\,\!}
{\displaystyle vd^{3}v={\frac {8\pi E}{m_{r}}}{\frac {dE}{m_{r}}}.\,\!}
{\displaystyle <\sigma (E)v>=\left({\frac {2}{kT}}\right)^{3/2}{\frac {1}{\sqrt {\pi m_{r}}}}\int dEE\sigma (E)e^{-E/kT}.\,\!}
{\displaystyle \sigma (E)}
{\displaystyle <\sigma (E)v>=\left({\frac {2}{kT}}\right)^{3/2}{\frac {1}{\sqrt {\pi m_{r}}}}\int dES(E)e^{-(E_{G}/E)^{1/2}\;-\;E/kT}.\,\!}
{\displaystyle S(E)}
{\displaystyle <\sigma (E)v>=\left({\frac {2}{kT}}\right)^{3/2}{\frac {1}{\sqrt {\pi m_{r}}}}S(E)I.\,\!}
{\displaystyle I=\int _{0}^{\infty }e^{-(E_{G}/E)^{1/2}\;-\;E/kT}dE.\,\!}
{\displaystyle E_{0}}
{\displaystyle E_{0}}
{\displaystyle f(E)}
{\displaystyle {\frac {df}{dE}}=0={\frac {1}{kT}}-{\frac {E_{G}^{1/2}}{2E^{3/2}}}.\,\!}
{\displaystyle E_{0}=\left({\frac {1}{2}}E_{G}^{1/2}kT\right)^{2/3}.\,\!}
{\displaystyle E_{G}}
{\displaystyle E_{0}=(5.7\;{\rm {keV}})Z_{1}^{2/3}Z_{2}^{2/3}T_{7}^{2/3}\left({\frac {m_{r}}{m_{p}}}\right)^{1/3}.\,\!}
{\displaystyle E_{G}}
{\displaystyle kT}
{\displaystyle f(E)=f(E_{0})+{\frac {1}{2}}(E-E_{0})^{2}f^{''}(E_{0}),\,\!}
{\displaystyle f^{''}(E_{0})={\frac {3E_{G}^{1/2}}{4E_{0}^{5/2}}}.\,\!}
{\displaystyle I}
{\displaystyle I={\frac {e^{-f(E_{0})}{\sqrt {2\pi }}}{\sqrt {f^{''}(E_{0}}}}.\,\!}
{\displaystyle <\sigma (E)v>=2.6S(E_{0}){\frac {E_{G}^{1/6}}{(kT)^{2/3}{\sqrt {m_{r}}}}}e^{-3(E_{G}/4kT)^{1/3}}.\,\!}
{\displaystyle \epsilon }
{\displaystyle L=\int \epsilon dM_{r}=\int \epsilon 4\pi r^{2}\rho dr.\,\!}
{\displaystyle {\frac {dL_{r}}{dr}}=4\pi r^{2}\rho \epsilon .\,\!}
{\displaystyle Q}
{\displaystyle r_{12}}
{\displaystyle \epsilon }
{\displaystyle \epsilon _{12}={\frac {r_{12}Q}{\rho }}.\,\!}
{\displaystyle n_{1}={\frac {X_{1}\rho }{m_{1}}}.\,\!}
{\displaystyle X_{1}}
{\displaystyle \epsilon _{12}={\frac {2.6QS(E_{0})X_{1}X_{2}}{m_{1}m_{2}{\sqrt {m_{r}}}(kT)^{2/3}}}\rho E_{G}^{1/6}e^{-3(E_{G}/4kT)^{1/3}}.\,\!}
{\displaystyle \epsilon \propto \rho ^{\alpha }T^{\beta }.\,\!}
{\displaystyle \alpha }
{\displaystyle \beta }
{\displaystyle \alpha =1}
{\displaystyle \beta }
{\displaystyle \epsilon }
{\displaystyle \beta ={\frac {d\ln \epsilon }{d\ln T}}.\,\!}
{\displaystyle \epsilon }
{\displaystyle \beta =-{\frac {2}{3}}+\left({\frac {E_{G}}{4kT}}\right)^{1/3}.\,\!}
{\displaystyle \beta \approx 4.3}
{\displaystyle \epsilon _{pp}\propto \rho T^{4.3}\,\!}
{\displaystyle 10^{7}}
{\displaystyle T_{c}\sim 10^{7}}
{\displaystyle \rho \sim 1}
{\displaystyle ^{-3}}
{\displaystyle S(E)}
{\displaystyle Q}
{\displaystyle \epsilon }
{\displaystyle \epsilon \sim 10^{20}{\rm {\;erg/s/g}}.\,\!}
{\displaystyle L=\int dM_{r}\epsilon \sim \epsilon M_{\odot }.\,\!}
{\displaystyle L\sim 10^{54}{\rm {\;erg/s}}\sim 10^{20}L_{\odot }.\,\!}
{\displaystyle 10^{20}}
{\displaystyle E_{G}}
{\displaystyle 4p\rightarrow {}^{4}{\rm {He}}+{\rm {energy}}.\,\!}
{\displaystyle p+p\rightarrow {}^{2}{\rm {H}}+e^{+}+\nu _{e}.\,\!}
{\displaystyle S(keV)\approx 3.78\times 10^{-22}}
{\displaystyle {}^{2}{\rm {H}}+p\rightarrow {}^{3}{\rm {He}}+\gamma ,\,\!}
{\displaystyle \times 10^{-4}}
{\displaystyle {}^{3}{\rm {He}}+{}^{3}{\rm {He}}\rightarrow {}^{4}{\rm {He}}+2p,\,\!}
{\displaystyle \epsilon _{cycle}=r_{p-p\;step}Q_{cycle}/\rho .\,\!}
{\displaystyle \epsilon _{pp}\propto \rho T^{-2/3}e^{-15.7T_{7}^{-1/3}}.\,\!}
{\displaystyle \epsilon _{pp}=(5\times 10^{5}){\frac {\rho X^{2}}{T^{2/3}}}e^{-15.7T_{7}^{-1/3}}{\rm {erg/s/g}}.\,\!}
{\displaystyle L=\int \epsilon dM\sim \epsilon (center)M_{\odot },\,\!}
{\displaystyle L_{\odot }\sim 10^{7}{\frac {M_{\odot }}{T_{7}^{2/3}}}e^{-15.7T_{7}^{-1/3}},\,\!}
{\displaystyle T_{c}\approx 10^{7}K.\,\!}
{\displaystyle p+p\rightarrow {}^{2}H+e^{+}+\nu _{e}\,\!}
{\displaystyle {}^{2}H+p\rightarrow {}^{3}He+\gamma \,\!}
{\displaystyle {}^{3}He+{}^{3}He\rightarrow {}^{4}He+2p\,\!}
{\displaystyle {}^{12}C+p\rightarrow {}^{13}N+\gamma \,\!}
{\displaystyle {}^{13}N\rightarrow {}^{13}C+e^{+}+\nu _{e}\,\!}
{\displaystyle {}^{13}C+p\rightarrow {}^{14}N+\gamma \,\!}
{\displaystyle {}^{14}N+p\rightarrow {}^{15}O+\gamma \,\!}
{\displaystyle {}^{15}O\rightarrow {}^{15}N+e^{+}+\nu _{e}\,\!}
{\displaystyle {}^{15}N+p\rightarrow {}^{12}C+{}^{4}He.\,\!}
{\displaystyle 10^{7}}
{\displaystyle 10^{-7}}
{\displaystyle 10^{-31}}
{\displaystyle 10^{24}}
{\displaystyle \epsilon _{CNO}\approx (4\times 10^{2}7){\frac {\rho }{T_{7}^{2/3}}}XZe^{-70.7T_{7}^{-1/3}}{\rm {\;erg/g/s}}.\,\!}
{\displaystyle \beta ={\frac {-2}{3}}+{\frac {23.6}{T_{7}^{1/3}}},\,\!}
{\displaystyle \epsilon \propto \rho T^{\beta }\,\!}
{\displaystyle \sigma \sim 10^{-44}\left({\frac {E_{\nu }}{m_{e}c^{2}}}\right)^{2}{\rm {\;cm^{2}}}.\,\!}
{\displaystyle \ell ={\frac {1}{n\sigma }}.\,\!}
{\displaystyle E_{\nu }\sim }
{\displaystyle \ell \sim 10^{9}R_{\odot }.\,\!}
{\displaystyle {}^{37}Cl+\nu _{e}\rightarrow {}^{37}Ar+e^{-}.\,\!}
{\displaystyle 10^{22}}
{\displaystyle \nu _{e}+D\rightarrow p+p+e^{-}.\,\!}
{\displaystyle \nu +D\rightarrow p+n+\nu .\,\!} |
Line (Euclidean geometry) - Citizendium
(Redirected from Line (geometry))
Line AB (in red) through points A and B (in blue). Of course, a thick and bounded image can only hint at an ideal line.
In Euclidean geometry, a line (sometimes called, more explicitly, a straight line) is an abstract concept that models the common notion of a curve that does not bend, has no thickness and extends infinitely in both directions. It is closely related to other basic concepts of geometry, especially, distance: it provides the shortest path between any two of its points. In space it can also be described as the intersection of two planes.
Assuming a common (intuitive, physical) idea of the geometry of a plane, "line" can be defined in terms of distances, orthogonality, coordinates etc. In a more abstract approach (vector spaces) lines are defined as one-dimensional affine subspaces. In an axiomatic approach, basic concepts of elementary geometry, such as "point" and "line", are undefined primitives.
1 Non-axiomatic approach
1.1.2 Definition via betweenness
1.1.3 Planar-geometric definitions
1.1.4 Definition via distances
1.1.5 Definition via right angles (orthogonality) in disguise
1.1.6 Definition via Cartesian coordinates
1.2 Some properties of lines
1.2.1 Most basic properties
1.2.2 Further properties
2 Axiomatic approach
2.1 What is wrong with the definitions given above?
3 Modern approach
4 Beyond mathematics
Non-axiomatic approach
By lines we mean straight lines.
Lines are treated both in plane geometry and in solid geometry. Plane geometry (called also "planar geometry") is a part of solid geometry that restricts itself to a single plane ("the plane") treated as a geometric universe. In other words, plane geometry is the theory of the two-dimensional Euclidean space, while solid geometry is the theory of the three-dimensional Euclidean space.
To define a line is more complicated than it may seem.
It is tempting to define a line as a curve of zero curvature, where a curve is defined as a geometric object having length but no breadth or depth. However, this is not a good idea; such definitions are useless in mathematics, since they cannot be used when proving theorems. Straight lines are treated by elementary geometry, but the notions of curves and curvature are not elementary, they need more advanced mathematics and more sophisticated definitions. Fortunately, it is possible to define a line via more elementary notions, and this way is preferred in mathematics. Still, the definitions given below are tentative. They are criticized afterwards, see axiomatic approach.
Three equivalent definitions of line are given below. Any other definition is equally acceptable provided that it is equivalent to these. Note that a part of a line is not a line. In particular, a line segment is not a line.
We could also define a line as the intersection of two planes (neither parallel nor coinciding). However, this definition does not work in plane geometry.
The first definition (via betweenness) works both in plane geometry and in solid geometry. The other two definitions apply in plane geometry only.
Definition via betweenness
First we define betweenness via distances. A point B is said to lie between points A and C if
{\displaystyle |AB|+|BC|=|AC|,}
that is, the distance from A to B plus the distance from B to C is the distance from A to C. Less formally, B lies on the shortest path from A to C.
Now we define a line as a set of points that contains more than one point and satisfies the two conditions:
Among three points of the given set there is always one that lies between the two others.
If one of three distinct points lies between the two others, and if any two of these three points belong to the given set, then the third point also belongs to the given set.
Remark 1. A line segment satisfies the first condition but violates the second. A plane satisfies the second condition but violates the first.
Remark 2. A set satisfying the second condition is either the empty set, or a single point, or a line, or a plane ("the whole plane" in planar geometry), or the whole space (in solid geometry). In the first two cases the set fails to contain more than one point. The last two cases violate the first condition.
Remark 3. The second condition should not be confused with the following weaker condition:
If B lies between A and C, and the points A, C belong to the given set, then B also belongs to the given set.
A set satisfying this weaker condition is called convex. A line is convex; also a line segment is convex; a triangle (including interior) is convex; also a disk is convex (but its boundary, a circle, is not).
Planar-geometric definitions
The definition of "line" given below may be compared with the definition of "circle" as consisting of those points in a plane that are a given distance (the radius) away from a given point (the center). A circle is a set of points chosen according to their relation to some given parameters (center and radius). Similarly, a line is a set of points chosen according to their relation to some given objects (points, numbers etc). However, a circle determines its center and radius uniquely; for a line, the situation is different.
Below, all points and lines are situated in the plane (assumed to be a two-dimensional Euclidean space).
Definition via distances
Let two different points A and B be given. The set of all points C that are equally far from A and B — that is,
{\displaystyle |AC|=|BC|}
— is a line.
This is the line orthogonal to the line AB through the middle point of the line segment AB.
Definition via right angles (orthogonality) in disguise
Let two different points A and B be given. The set of all points C such that
{\displaystyle |AB|^{2}+|AC|^{2}=|BC|^{2}}
This is the line orthogonal to the line AB through the point A.
{\displaystyle |AB|^{2}+|AC|^{2}=|BC|^{2}}
means that the lines AB and AC are orthogonal, that is, the angle BAC is right (unless C=A), see Pythagorean theorem (orthogonality is not only sufficient but also necessary for the equality).
Definition via Cartesian coordinates
In terms of Cartesian coordinates x, y ascribed to every point of the plane, a line is the set of points whose coordinates satisfy a linear equation
{\displaystyle ax+by=c}
Here real numbers a, b and c are parameters such that at least one of a, b does not vanish.
Some properties of lines
Most basic properties
For every two different points there exists one and only one line that contains these two points.
Every line contains at least two points.
There exist three points not lying on a line.
For every line and every point outside the line there exists one and only one line through the given point which does not intersect the given line.
Two lines either do not intersect (are parallel), or intersect in a single point, or coincide.
Two lines perpendicular to the same line are parallel to each other (or coincide).
What is wrong with the definitions given above?
The definitions given above assume implicitly that the Euclidean plane (or alternatively the 3-dimensional Euclidean space) is already defined, together with such notions as distances and/or Cartesian coordinates, while lines are not defined yet. However, this situation never appears in mathematical theory.
In the axiomatic approach points and lines are undefined primitives.
The modern approach (below) defines lines in a completely different way.
Axiomatic approach is similar to chess in the following aspect.
A chess piece, say a rook, cannot be defined before the whole chess game is defined, since such a phrase as "the rook moves horizontally or vertically, forward or back, through any number of unoccupied squares" makes no sense unless it is already known that "chess is played on a square board of eight rows and eight columns" etc. And conversely, the whole chess game cannot be defined before each piece is defined; the properties of the rook are an indispensable part of the rules of the game. No chess without rooks, no rooks outside chess! One must introduce the game, its pieces and their properties in a single combined definition.
Likewise, Euclidean space, its points, lines, planes and their properties are introduced simultaneously in a set of 20 assumptions known as Hilbert's axioms of Euclidean geometry (solid). It is possible to exclude plane-related axioms thus obtaining axioms of Euclidean plane geometry. The "most basic properties of lines" listed above are roughly the line-related assumptions (Hilbert's axioms), while "further properties" are first line-related consequences (theorems).
The modern approach defines the three-dimensional Euclidean space more algebraically, via linear spaces and quadratic forms, namely, as an affine space whose difference space is a three-dimensional inner product space. For further details see Affine space#Euclidean space and space (mathematics). The Euclidean plane, that is, the two-dimensional Euclidean space is defined similarly.
In this approach a line in an n-dimensional affine space (n ≥ 1) is defined as a (proper or improper) one-dimensional affine subspace.
In geodesy, theodolites use light rays as straight lines. They can measure angles with accuracy about 10–6 of the right angle.
General relativity theory predicts that the sum of the three angles of a triangle may differ from two right angles because gravitation influences the geometry of the space, making it non-Euclidean. Near the Earth, the angle deficit is about S/(1024m2), where S is the area of the triangle. Thus, for a triangle with all sides of 1000 km, the angle deficit is about 10–12 of the right angle, — for now, far beyond accuracy of measurements.
Retrieved from "https://citizendium.org/wiki/index.php?title=Line_(Euclidean_geometry)&oldid=19528" |
Release Notes - Maple Help
Home : Support : Online Help : Connectivity : Web Features : Network Communication : Sockets Package : Information on Sockets Package : Release Notes
Sockets Package Release Notes
These release notes describe the platform support for the Sockets package as well as known run-time issues and security considerations.
The Sockets package is supported on all platforms that are supported by Maple.
Maple is neither designed nor intended for network-secure applications. It is strongly recommended that you do not deploy server applications written using this package on generally accessible networks. Maple processes running servers should be started by using the
-z
command line option. (For more information, see maple.) Under no circumstances should a Maple server be run with a real or effective user ID that has privileged access to the host computer.
This section discusses known run-time issues with the Sockets package. The information is organized by platform and by operating system.
The procedure Serve does not cause the calling process to fork or spawn new threads or lightweight processes. This is not a bug, but a design restriction imposed by licensing requirements. |
The third cleavage in frog's development is-Turito
The third cleavage in frog's development is
Meroblastic and vertical
Holoblastic and unequatorial
Holoblastic and equatorial
Vertical and equatorial
Answer:The correct answer is: Holoblastic and unequatorial
Statement ‐ I : The equation
z\overline{z}+\overline{a}z+a\overline{z}+\lambda =0
where a is a complex number, represents a circle in Argand plane if
\lambda
is real. Statement ‐II : The radius of the circle
z\overline{z}+\overline{a}z+a\overline{z}+\lambda =0
\sqrt{\lambda -a\overline{a}}
z\overline{z}+\overline{a}z+a\overline{z}+\lambda =0
\lambda
z\overline{z}+\overline{a}z+a\overline{z}+\lambda =0
\sqrt{\lambda -a\overline{a}}
Observe the following statements and choose correct answer Statement-I:A particle p moves along a straight line, starting from a fixed point ‘O’, obeying s=16+48t-
{t}^{3}.
the direction of p when t=5 is along
\overline{OP}
Statement-II: For a particle moving on a line, if velocity v<0 then the body moves towards the initial point.
Statement -I:
{t}^{3}.
\overline{OP}
The reason why the right kidney is slightly lower than the left is:
Excretion involves the processes in which substances of no further use or those present in excess qualities are thrown out of the body.
The length of the chord joining the points
\left(4\mathrm{cos}\theta , 4\mathrm{sin}\theta \right)
\left(4\mathrm{cos}\left(\theta +60°\right), 4\mathrm{sin}\left(\theta +60°\right)
of the circle
{x}^{2}+{y}^{2}=16
\left(4\mathrm{cos}\theta , 4\mathrm{sin}\theta \right)
\left(4\mathrm{cos}\left(\theta +60°\right), 4\mathrm{sin}\left(\theta +60°\right)
{x}^{2}+{y}^{2}=16
Assertion (A):If
a>0,b>0
c>0
\left(a+b+c\right)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\ge 9
Reason (R): For positive numbers
a,
b,
c,
AM\ge GM
AM\ge GM
=\succ \frac{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}{3}\ge \left(\begin{array}{cc}111& \\ -& -\\ \overline{b}a.& c\end{array}\right)\to \left(1\right)
D:{x}^{2}-3x-4<0\supset \left(x+1\right)\left(x-4\right)<0⇒1<x<4
{x}^{2}-3x+2>0Rejectx-2\right)\left(x-1\right)>0=x<1{a}^{r}x>2
\frac{\left(a+b+c\right)}{3}\ge \left(abc{\right)}^{\frac{1}{3}}\to \left(2\right)
multiply (1) and (2)
a>0,b>0
c>0
\left(a+b+c\right)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\ge 9
a,
b,
c,
AM\ge GM
AM\ge GM
=\succ \frac{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}{3}\ge \left(\begin{array}{cc}111& \\ -& -\\ \overline{b}a.& c\end{array}\right)\to \left(1\right)
D:{x}^{2}-3x-4<0\supset \left(x+1\right)\left(x-4\right)<0⇒1<x<4
{x}^{2}-3x+2>0Rejectx-2\right)\left(x-1\right)>0=x<1{a}^{r}x>2
\frac{\left(a+b+c\right)}{3}\ge \left(abc{\right)}^{\frac{1}{3}}\to \left(2\right)
The number of codons that code different amino acids is -
Finding of Miller's experiment on origin of life has provided evidence for:
Darwin's finches represents:
Evolutionary convergence is the development of:
The material used in determining the age of a fossil is:
Hugo deVries'theory of mutation:
A definition of the term "Species" based on reproduction has distinct limitations, which include its
Select the correct statement(s) A Microbial experiment show the pre-existing advantageous mutations when selected will result in the observation of new phenotypes Over few generation this would result in speciation B Neanderthal fossils represent a human relative C In 1938, a fish caught in South Africa happened to be a coelacanth (lobe fins) which was thought to be extinct These animals evolved into the first amphibian living on both land and water D Lichens can be used as water pollution indicators E Alfred Wallace, a natur st, who worked in Malay Archepelago (present Indonesia) has also come to similar conclusion on natural selection as reached by Darwinism
\mu F
capacitor is connected to 45 V battery through a circuit whose resistance is 2000
\Omega .
What is the final charge on the capacitor?
We know that in steady state the capacitor behaves like as an open circuit
ie
, capacitor will not pass the current. So, the potential difference across the capacitor = 45 V Hence , the final charge on the capacitor is
q=CV
C=20\mu F, V=45 V
\therefore q=20×{10}^{-6}×45
q=900×{10}^{-6}
q=9×{10}^{-4}C
\mu F
\Omega .
ie
q=CV
C=20\mu F, V=45 V
\therefore q=20×{10}^{-6}×45
q=900×{10}^{-6}
q=9×{10}^{-4}C |
Test for Cointegration Using the Johansen Test - MATLAB & Simulink - MathWorks América Latina
This example shows how to assess whether a multivariate time series has multiple cointegrating relations using the Johansen test.
To illustrate the input and output structure of jcitest when conducting multiple tests, test for the cointegration rank using the default H1 model and two different lag structures.
[h,pValue,stat,cValue] = jcitest(Y,'Model','H1','Lags',1:2);
The default "trace" test assesses null hypotheses
H\left(r\right)
of cointegration rank less than or equal to r against the alternative
H\left(n\right)
, where n is the dimension of the data. The summaries show that the first test rejects a cointegration rank of 0 (no cointegration) and just barely rejects a cointegration rank of 1, but fails to reject a cointegration rank of 2. The inference is that the data exhibit 1 or 2 cointegrating relationships. With an additional lag in the model, the second test fails to reject any of the cointegration ranks, providing little by way of inference. It is important to determine a reasonable lag length for the VEC model (as well as the general form of the model) before testing for cointegration.
Because the Johansen method, by its nature, tests multiple rank specifications for each specification of the remaining model parameters, jcitest returns the results in the form of tabular arrays, and indexes by null rank and test number.
Display the test results, h.
t2 false false false {'H1'} {'trace'} 0.05
Column headers indicate tests r0, r1, and r2, respectively, of
H\left(0\right)
H\left(1\right)
H\left(2\right)
H\left(3\right)
. Row headers t1 and t2 indicate the two separate tests (two separate lag structures) specified by the input parameters.
Access the result for the second test at null rank
r=0
using tabular array indexing.
h20 = h.r0(2)
h20 = logical |
Liquidation - Delta Exchange - User Guide
Each position on Delta Exchange has two associated prices:
Liquidation Price: At Liquidation Price, the difference of Position Margin minus Unrealized PnL of the position is equal to the Maintenance Margin or Liquidation Margin.
Bankruptcy Price: At Bankruptcy Price, the Unrealized Loss of a position equal to the Position Margin.
A position goes into liquidation when Mark Price reaches the Liquidation Price of the position. It is worth noting that since Delta Exchange uses Isolated Margin, liquidation of position in a particular contract has no bearing on other existing positions and open orders on other derivative contracts.
Liquidation mechanism on Delta Exchange is comprised of the following steps:
When position size < Position Threshold
When the size of the position in liquidation is lower than the Position Threshold the entire position is liquidated in one shot. The steps involved in liquidated such a position are as follows:
All open orders on the contract are cancelled. This may or may not free up some margin blocked for these order.
An Immediate-or-Cancel (IOC) order is submitted to the market to close the entire position. The limit price of this order is set to the Bankruptcy Price. This ensures that the Realised Loss will always be lower than or equal to the Position Margin.
If the liquidation results in being filled at a price better than the Bankruptcy Price, the remaining Position Margin is used to offset the liquidation charge.
In case the position is not fully liquidated (for example due to lack of liquidity), an off market trade for the remaining position is executed at the Bankruptcy Price between the trader and the Liquidation Engine. This essentially results in the Liquidation Engine taking over the remaining position.
When position size > Position Threshold
When the size of the position in liquidation is greater than the Position Threshold Incremental Liquidation is used. Since reducing the position size lowers the margin requirement, it is possible to liquidate part of a position and ensure that the remainder of the position has sufficiently margined. Thus, Incremental Liquidation helps to avoid full liquidation of a trader’s position. The steps involved in this are listed below:
The liquidation Engine computes the minimum position size (the Liquidation Size) required to be liquidated such that the liquidation price of the remaining position is 1% away from the Mark Price. Please note that if the Liquidation Size turns our to be equal to the position size, then the entire position will be liquidated.
The Partial Position in Liquidation (PPL) is treated as a separate position and is assumed to have pro-rata share of the position margin. The Maintenance Margin percent (MM%) for the PPL is computed Using the Liquidation Size. The Implied Bankruptcy Price of the PPL is assumed to be MM% away from the current Mark Price. An Immediate-or-Cancel order of size equal to Liquidation Size with limit price equal to the Implied Bankruptcy Price is submitted to the market.
If the IOC order is filled at a price better than the Implied Bankruptcy Price, a liquidation charge (equaling
Maintenance\ Margin_{min}
) is deducted from the position margin assigned to the PPL. Any leftover margin after this deduction is added to the position margin of the still open position.
In case the ICO order is not fully filled, an off market trade at the Implied Bankruptcy Price of the PPL is executed between the trader and the Liquidation Engine. This essentially results in the Liquidation Engine taking over the part of the PPL that could not be liquidated.
As is evident from the discussion above, the Liquidation Engine always takes over a position at its bankruptcy price. The goal of the Liquidation Engine is to close positions it acquires at price which is equal to or better than the acquisition price. To this end, as soon as the Liquidation Engine takes over a position, it places a limit order to close the position. Generally, the price of this limit order will be away from the current Mark Price Liquidation Engine has acquired the position. Generally, this limit price would be away from Mark Price and would provide an opportunity to traders attractive entry point. However, if the Mark Price breaches the limit price, then the Liquidation Engine cancels the limit order and triggers auto-deleveraging.
For illustrative purposes, let’s assume that the maintenance margin requirement for the BTCUSD perpetual contract is given by the following equation:
When position size < 5 BTC, MM% = 0.5%
When position size > 5 BTC, MM% = 0.5% + 0.075% * (position size - 5)
A trader is long 20000 contracts at an entry price of USD 10000 at maximum allowed leverage
Position size = 2 BTC => MM% = 0.5%
liquidation price = 9950 & bankruptcy price = 9901
When Mark Price goes below 9950, this position will go into liquidation and it will be liquidated in a single shot by placing a limit IOC order to sell 20000 contracts with a limit price of 9901. In case, some part of the limit IOC order remains unfilled, the Liquidation Engine takes over the remaining position through an off market trade at 9901.
A trade is long 200000 contracts at an entry price of USD 10000 at maximum allowed leverage
Position size = 2 BTC => MM% = 1.63% & IM% = 3.25%
liquidation price = 9940.5 & bankruptcy price = 9685.5
Lets assume current Mark Price is 9840. In this situation, incremental liquidation will apply. Because of adverse price movement, the remaining position margin is sufficient to support a position of only 62611 contracts. The system will thus attempt to liquidate a long position of 137389. This is the Partial Position in Liquidation or PPL. The maintenance margin required for a position of PPL’s size is 1.16%. Therefore, the liquidation price of PPL is set to 9728, i.e. 1% away from the current Mark Price of 9840. This means a sell limit IOC order with price of 9728 is sent to the order book to close PPL.
Let’s further assume that the order book currently does not have sufficient depth to fill this limit IOC order and 50000 contracts remain unfilled. In this case, the Liquidation Engine would take over the unfilled part and would acquire a long position of 50000 contracts at 9728. Recall that the Mark Price is still at 9840. Now, the Liquidation Engine will place a sell limit order at 9728 to close this recently acquired position.
With his position partially liquidated, the trader is now left with a long position of 62611 contracts. Because the position size is smaller, so is the margin requirement for it. The new liquidation price and bankruptcy price for the remaining position are 9648 and 9593 respectively. |
Cerebral venous thrombosis due to vaccine-induced immune thrombotic thrombocytopenia after a second ChAdOx1 nCoV-19 dose | Blood | American Society of Hematology
Katarzyna Krzywicka,
1Department of Neurology, Amsterdam University Medical Centers, University of Amsterdam, Amsterdam, The Netherlands;
Anita van de Munckhof,
Anita van de Munckhof
Julian Zimmermann,
2Department of Neurology, Universitätsklinikum Bonn, Bonn, Germany;
Felix J. Bode,
Giovanni Frisullo,
Giovanni Frisullo
3Department of Neurology, Fondazione Policlinico Universitario Agostino Gemelli Istituto di Ricovero e Cura a Carattere Scientifico, Rome, Italy;
Theodoros Karapanayiotides,
4Second Department of Neurology, School of Medicine and Faculty of Health Sciences, Aristotle University of Thessaloniki, American Hellenic Educational Progressive Association University Hospital, Thessaloniki, Greece;
Bernd Pötzsch,
5Institute of Experimental Hematology and Transfusion Medicine, Universitätsklinikum Bonn, Bonn, Germany;
Mayte Sánchez van Kammen,
Mirjam R. Heldner,
Marcel Arnold,
7Department of Hematology, Inselspital, Bern University Hospital, University of Bern, Bern, Switzerland; and
8Instituto de Medicina Molecular João Lobo Antunes, Faculdade de Medicina, and
Diana Aguiar de Sousa,
9Department of Neurosciences and Mental Health, Hospital de Santa Maria, Centro Hospitalar Universitario Lisboa Norte, Universidade de Lisboa, Lisbon, Portugal
for the Cerebral Venous Sinus Thrombosis With Thrombocytopenia Syndrome Study Group
This is a related article to: Second-dose VITT: rare but real
Katarzyna Krzywicka, Anita van de Munckhof, Julian Zimmermann, Felix J. Bode, Giovanni Frisullo, Theodoros Karapanayiotides, Bernd Pötzsch, Mayte Sánchez van Kammen, Mirjam R. Heldner, Marcel Arnold, Johanna A. Kremer Hovinga, José M. Ferro, Diana Aguiar de Sousa, Jonathan M. Coutinho; for the Cerebral Venous Sinus Thrombosis With Thrombocytopenia Syndrome Study Group, Cerebral venous thrombosis due to vaccine-induced immune thrombotic thrombocytopenia after a second ChAdOx1 nCoV-19 dose. Blood 2022; 139 (17): 2720–2724. doi: https://doi.org/10.1182/blood.2021015329
Cerebral venous thrombosis (CVT) is the most common and severe manifestation of vaccine-induced immune thrombotic thrombocytopenia (VITT), which is a rare side effect of the SARS-CoV-2 vaccine ChAdOx1 nCoV-19 (Vaxzevria, AstraZeneca/Oxford).1-4 The absolute risk of VITT and VITT-related CVT is estimated at 20 and 8 per million first doses of ChAdOx1 nCoV-19, respectively.5,6
So far, no definite VITT cases occurring after a second ChAdOx1 nCoV-19 vaccine dose have been reported, raising the question of whether VITT only occurs after a first dose. Two pharmacovigilance studies reported cases of thrombosis with thrombocytopenia after a second ChAdOx1 nCoV-19 dose, but because of lack of clinical data, none of these could be classified as VITT.7-9 Knowledge on whether VITT can occur after a second ChAdOx1 nCoV-19 dose is relevant for clinicians and policymakers, especially in low- and middle-income countries, which are currently the main users of adenovirus-based vaccines.10
We used data from the “CVT after SARS-CoV-2 vaccination” registry4,11 to identify VITT-related CVT cases occurring after a second ChAdOx1 nCoV-19 dose.
Details of this registry have been published.4 Briefly, this ongoing study collects data on patients with CVT with symptom onset
≤
28 days from SARS-CoV-2 vaccination, regardless of the type and dose of vaccine. The study is endorsed by the European Academy of Neurology and the European Stroke Organization. Investigators are instructed to report consecutive cases from their hospitals. The ethical review board of the Academic Medical Centre issued a waiver of formal approval for this observational study. Each center obtained local permission to carry out the study and acquired informed consent for the use of pseudonymized care data according to national law.
We used the case definition criteria of the United Kingdom expert hematology panel to classify cases as definite, probable, possible, or unlikely VITT after ChAdOx1 nCoV-19 administration among CVT cases reported until 1 December 2021.9
Within the study period, 202 CVT cases after SARS-CoV-2 vaccination were reported from 24 countries (Figure 1). Of the 124 patients with CVT following ChAdOx1 nCoV-19 vaccination, 120 were after a first dose, and 4 were after a second dose. There were 61 definite, 20 probable, 10 possible, and 29 unlikely VITT cases after a first ChAdOx1 nCoV-19 dose. Of the 4 cases after the second dose, 1 was definite, 1 was probable, 1 was possible, and 1 was an unlikely VITT. There were no possible, probable, or definite VITT cases after the second dose of any of the other vaccines.
Flowchart of patient selection. Out of 202 reported patients with CVT after SARS-CoV-2 vaccination, we excluded 13, 5, and 8 cases with symptom onset outside of the 0-28 day interval, with no radiological confirmation, and duplicate and/or incomplete cases, respectively. Out of the remaining 176 cases, 124 cases developed CVT after ChAdOx1 nCoV-19 vaccination. Of these, 120 developed CVT after a first dose (61 definite, 20 probable, 10 possible, and 29 unlikely VITT), and 4 after a second dose (1 definite, 1 probable, 1 possible, and 1 unlikely).
Details of the 4 cases after a second ChAdOx1 nCoV-19 dose are provided in Table 1. A timeline of the clinical course of each of the cases is provided in supplemental Figures 1-4, available on the Blood Web site. None of the patients reported any symptoms after the first dose of ChAdOx1 nCoV-19. The patients (3 men, 1 woman) were between their forties and sixties. None had preexistent comorbidities. The interval between receiving the second vaccine dose and symptom onset varied between 1 and 6 days. The 2 patients who met the criteria for probable and definite VITT (patients 1 and 2) both died of brain herniation.
Clinical details of CVT cases after a second ChAdOx1 nCoV-19 dose
Patient 1
VITT classification* Probable Definite Possible Unlikely
Age† 60s 50s 40s 60s
Medical history Unremarkable Thrombophilia Unremarkable Unremarkable
Prior COVID-19 infection at any time No No No No
Interval between first and second vaccination (d)‡ 90 44 62 77
Interval between second vaccination and symptom onset (d) 5 6 1 4
Interval between symptom onset and diagnosis (d) 0 1 0 0
Headache No Yes Yes No
Focal neurologic deficits Yes Yes Yes Yes
Coma Yes Yes No No
Seizure No No Yes Yes
Intracerebral hemorrhage Yes Yes No No
Location of CVT Superior sagittal sinus Superior sagittal sinus, left transverse and sigmoid sinus, straight sinus, left jugular vein Right transverse and sigmoid sinuses Superior sagittal sinus, right transverse and sigmoid sinus, right jugular vein
Platelet count at admission, ×109/L 188 40 109 175
Platelet count nadir, ×109/L 55 14 55 124
Anti-PF4 antibody ELISA Negative Positive Negative Negative
Type ELISA test Lifecodes PF4 IgG from Immucor PF4 IgG from Immucor Lifecodes PF4 IgG from Immucor ZYMUTEST HIA IgG, HYPHEN BIOMED
Optical density ELISA 0.06 2.12§ 0.12 0.03
Optical density test threshold ≥0.4 ≥0.4 ≥0.4 ≥0.3
Functional assay to detect platelet- activating PF4 antibodies Positiveǁ Not performed Positiveǁ Negative¶
Type of functional assay Modified HIPA NA Modified HIPA Multiplate HIMEA
D-dimer, ug/L FEU 35 200 29 100 2400 513
Fibrinogen, g/L 4.17 2.63 3.34 4.14
ref <3.50 ref <4.00 ref <3.50 ref <4.50
Anticoagulation Argatroban None# Argatroban followed by dabigatran Fondaparinux followed by dabigatran
IVIG Yes No Yes No
Decompressive hemicraniectomy Yes No No No
Major bleeding during admission Yes** No No No
New VTE during admission No Yes, pelvic veins No No
Outcome at hospital discharge Dead Dead No disability No disability
Days between symptom onset and death 2 3 NA NA
Cause of death Brain herniation Brain herniation NA NA
ELISA, enzyme-linked immunosorbent assay; FEU, fibrinogen equivalent units; HIMEA, heparin-induced multiple electrode aggregometry; HIPA, heparin-induced platelet aggregation; IVIG, intravenous immune globulin; NA, not applicable; VTE, venous thromboembolism.
According to the United Kingdom expert hematology panel.9
To avoid the possibility of patient identification, exact age has been removed.
In all cases, the first vaccination was ChAdOx1 nCoV-19.
Blood was drawn from the patient at admission, stored at 4°C for 1 wk, then stored at −20°C for 327 d before it was tested.
Modified HIPA assay was performed as previously described.15
HIMEA assay was performed as previously described.16
Reason: multiple intracerebral hemorrhages and diffuse subarachnoid hemorrhage.
Worsening of intracerebral hemorrhages.
In patient 3 with symptom onset on day 1, the rapid onset could be explained if circulating anti-PF4 antibodies were present after the first vaccination, suggesting immunological preconditioning similar to that described in heparin-induced thrombocytopenia.12
Of note, no specific events were observed after the first dose of this vaccine, suggesting that the development of VITT after the second dose of ChAdOx1 nCoV-19 cannot be predicted on clinical grounds. Although the numbers are small, the clinical severity appears comparable to CVT-VITT after a first ChAdOx1 nCoV-19 dose, as 2 patients had an intracerebral hemorrhage, 1 had a concurrent venous thrombosis, and 2 patients died during admission.4,5,13
Based on reported CVT cases to the registry, VITT appears to be much less common after a second ChAdOx1 nCoV-19 dose than after a first. However, since many countries, especially in Europe, restricted the use of the ChAdOx1 nCoV-19 vaccine after the emergence of VITT, the lower frequency of reported VITT after a second dose could partly be explained by the fact that fewer people received a second dose of ChAdOx1 nCoV-19 than a first dose. Even so, data from the European Centre for Disease Prevention and Control show that, until week 33 of 2021, 39 million first doses and 29 million second doses were administered in the European Economic Area.14 Therefore, this imbalance cannot fully explain the difference in the incidence of VITT. Still, due to the risk of reporting bias, data from our registry must be interpreted cautiously when concluding that VITT is much less common after a second than after a first dose.
In conclusion, CVT-VITT can occur after the second dose of the ChAdOx1 nCoV-19 vaccine but was reported less often than after a first vaccine dose. Symptom onset of VITT may be more rapid after a second than after a first dose, but the low number of cases precludes firm conclusions.
This work was supported by The Netherlands Organisation for Health Research and Development (ZonMw, grant number 10430072110005) (J.M.C.) and the Dr. C. J. Vaillant Foundation (J.M.C.). The sponsors of this study are public or nonprofit organizations that support science in general. They had no role in gathering, analyzing, or interpreting the data.
K.K., A.v.d.M., and M.S.v.K. are PhD candidates at the University of Amsterdam. This work is submitted in partial fulfillment of the requirement for the doctoral degree.
Contribution: D.A.d.S. and J.M.C. conceptualized the study; K.K., A.v.d.M., and M.S.v.K. provided data curation; K.K. and A.v.d.M. provided formal analysis; K.K., A.v.d.M., J.Z., F.J.B., G.F., T.K., B.P., J.A.K.H., and M.S.v.K. provided investigation for the study; M.A., J.M.F., D.A.d.S., and J.M.C. are responsible for methodization; K.K., A.v.d.M., M.S.v.K., M.R.H., D.A.d.S., and J.M.C. provided project administration; M.A., J.M.F., D.A.d.S., and J.M.C. provided resources; D.A.d.S. and J.M.C. supervised the study; K.K. and A.v.d.M. contributed validation and visualization; K.K., D.A.d.S., and J.M.C. wrote the original draft of the manuscript; K.K., A.v.d.M., J.Z., F.J.B., G.F., T.K., B.P., M.S.v.K., J.A.K.H., M.A., M.R.H., J.M.F., D.A.d.S., and J.M.C. wrote, reviewed, and edited the manuscript; and K.K., A.v.d,M., and J.M.C. had full access to all the data in the study and take responsibility for the integrity of the data and the accuracy of the data analysis.
Conflict-of-interest disclosure: M.R.H. has received grants from the Swiss Heart Foundation and Bangerter Foundation; travel support from Bayer; personal fees for data safety monitoring board or advisory board participation from Amgen; and is a member of the European Stroke Organisation Board of Directors and European Stroke Organisation Education Committee. M.A. has received personal fees from AstraZeneca, Bayer, Bristol Myers Squibb, Covidien, Daiichi Sankyo, Medtronic, Novartis, Sanofi, Pfizer, and Amgen; and grants from the Swiss National Science Foundation and Swiss Heart Foundation. J.A.K.H. has received grants from Baxalta as well as personal fees paid to her institution from Shire, Ablynx, Roche, Sobi, and the Swiss Federal Office of Public Health. J.M.F. has received personal fees from Boehringer Ingelheim, Bayer, and Daiichi Sankyo as well as grants from Bayer. D.A.d.S. has received travel support from Boehringer Ingelheim; speaker fees from Bayer; and personal fees for advisory board participation from AstraZeneca. J.M.C. has received grants paid to his institution from Boehringer Ingelheim and Bayer and payments paid to his institution for data safety monitoring board participation by Bayer. The remaining authors declare no competing financial interests.
A complete list of the members of the Cerebral Venous Sinus Thrombosis With Thrombocytopenia Syndrome Study Group appears in the supplemental appendix.
Correspondence: Jonathan M. Coutinho, Department of Neurology, Amsterdam University Medical Centers, Location AMC, Meibergdreef 9, 1105 AZ Amsterdam, The Netherlands; e-mail: j.coutinho@amsterdamumc.nl.
For original data, please contact j.coutinho@amsterdamumc.nl.
Sánchez van Kammen
Cerebral Venous Sinus Thrombosis With Thrombocytopenia Syndrome Study Group
Guidance from the Expert Haematology Panel (EHP) on Covid-19 vaccine-induced immune thrombocytopenia and thrombosis.
. Available at: https://b-s-h.org.uk/media/20499/guidance-version-22-20210903.pdf. Accessed 9 December 2021.
Officially reported COVID-19 vaccination data - dashboard
. Updated 09-12-21. https://app.powerbi.com/view?r=eyJrIjoiMWNjNzZkNjctZTNiNy00YmMzLTkxZjQtNmJiZDM2MTYxNzEwIiwidCI6ImY2MTBjMGI3LWJkMjQtNGIzOS04MTBiLTNkYzI4MGFmYjU5MCIsImMiOjh9. Accessed 9 December 2021.
. COVID-19 vaccination and CVT study. Available at: https://cerebralvenousthrombosis.com/professionals/covid-cvt/. Accessed 9 December 2021.
Tamborska
CVT After Immunisation Against COVID-19 (CAIAC) collaborators
European Centre for Disease Prevention and Control COVID-19 Vaccine Tracker
. Available at: https://qap.ecdc.europa.eu/public/extensions/COVID-19/vaccine-tracker.html#uptake-tab. Accessed 9 December 2021.
Diagnosis and management of vaccine-related thrombosis following AstraZeneca COVID-19 vaccination: guidance statement from the GTH
Hamostaseologie.
Heparin-induced multiple electrode aggregometry is a promising and useful functional tool for heparin-induced thrombocytopenia diagnosis: confirmation in a prospective study
D.A.d.S. and J.M.C. contributed equally to this study. |
H. G. Nagaraja, G. Somashekhara, Savithri Shashidhar, "On Generalized Sasakian-Space-Forms", International Scholarly Research Notices, vol. 2012, Article ID 309486, 12 pages, 2012. https://doi.org/10.5402/2012/309486
H. G. Nagaraja,1 G. Somashekhara,1 and Savithri Shashidhar1
1Department of Mathematics, Bangalore University, Central College Campus, Bangalore, Karnataka 560 001, India
Academic Editor: E. H. Saidi
The purpose of the present paper is to characterize pseudoprojectively flat and pseudoprojective semisymmetric generalized Sasakian-space-forms.
Alegre et al. [1] introduced and studied the generalized Sasakian-space-forms. The authors Alegre and Carriazo [2], Somashekhara and Nagaraja [3, 4], and De and Sarkar [5, 6] studied the generalized Sasakian-space-forms. An almost contact metric manifold is said to be a generalized Sasakian-space-form if there exist differentiable functions such that curvature tensor of is given by for any vector fields , , on , where In this paper, we study the curvature properties like flatness, symmetry, and semisymmetry properties in a generalized Sasakian-space-form by considering a pseudoprojective curvature tensor.
The paper is organized as follows. Section 2 of this paper contains some preliminary results on the generalized Sasakian-space-forms. In Section 3, we study the pseudoprojectively flat generalized Sasakian-space-form and obtain necessary and sufficient conditions for a generalized Sasakian-space-form to be pseudoprojectively flat. In the next section, we deal with pseudoprojectively semisymmetric generalized Sasakian-space-forms, and it is proved that a generalized Sasakian-space-form is pseudoprojectively semisymmetric if and only if the space form is pseudoprojectively flat and . The last section is devoted to the study of -flat and -semi symmetric generalized Sasakian-space-forms. In this section, we prove that the associated functions are linearly dependent.
In a -dimensional almost contact metric manifold, the pseudoprojective curvature tensor [7] is defined by where and are constants and , , and are the Riemannian curvature tensor of type , the Ricci tensor, and the scalar curvature of the manifold, respectively. If , , then (1.3) takes the form where is the projective curvature tensor. A manifold shall be called pseudoprojectively flat if the pseudoprojective curvature tensor . It is known that the pseudoprojectively flat manifold is either projectively flat (if ) or Einstein (if and ).
A -dimensional -differentiable manifold is said to admit an almost contact metric structure if it satisfies the following relations: where is a tensor field of type , is a vector field, is a 1-form, and is a Riemannian metric on . A manifold equipped with an almost contact metric structure is called an almost contact metric manifold. An almost contact metric manifold is called a contact metric manifold if it satisfies for all vector fields and .
In a generalized Sasakian-space-form, the following hold:
3. Pseudoprojectively Flat Generalized Sasakian-Space-Forms
If the generalized Sasakian-space-form under consideration is pseudoprojectively flat, then from (1.3) we have where and are constants and .
Now taking in (3.1) and using (2.1), (2.2), (2.7), and (2.9), we get Again putting in (3.2), we get The aforementioned equation implies That is, either or If , and , then, from (1.3), it follows that . Thus in this case pseudoprojective flatness and projective flatness are equivalent.
If , and , then comparing (2.10) and (3.6), we get Using (3.7) in (2.9), we get Let be an orthonormal basis of the tangent space at each point of the manifold. Taking and summing over , we obtain This shows that is Einstein with a scalar curvature . Thus we state the following.
Theorem 3.1. A pseudoprojectively flat generalized Sasakian-space-form is either projectively flat or an Einstein manifold with a scalar curvature .
Suppose that (3.7) holds. Then in view of (2.7) and (2.9), we can write (1.3) as where Replacing by and by , we get Let be an orthonormal basis of the tangent space at each point of the manifold.
Taking and summation over , , we get Again putting = = and taking summation over , , we get with . In view of (3.7), we get .
Now (2.7) reduces to the form from which we have , and consequently By using (3.14) and (3.15) in (1.3), we get . This leads to the following.
Theorem 3.2. A -dimensional generalized Sasakian-space-form is pseudoprojectively flat if and only if , , and .
Alegre and Carriazo [2] proved that any contact metric generalized Sasakian-space-form with a dimension greater than or equal to five is a Sasakian manifold and , , and must be constants.
Thus from (3.14), we have the following theorem.
Theorem 3.3. A -dimensional generalized Sasakian-space-form with a dimension greater than or equal to 5 is of constant curvature if and only if , , , and .
4. Pseudoprojective Semisymmetric Generalized Sasakian-Space-Form
Definition 4.1. If a generalized Sasakian-space-form satisfies then the manifold is said to be pseudoprojectively semisymmetric manifold.
By using (1.3), (2.1), (2.2), (2.7), and (2.9), we have Taking in (4.2), we get Again putting in (4.2), we get From (4.1), we have Taking and contracting the above with respect to , we get Putting in (4.6) and with the help of (4.2) and (4.3), we get either or Let be an orthonormal basis of the tangent space at each point of the manifold of the manifold. Putting and taking summation over , , and using (4.2) and (4.4), we obtain where Now contracting (4.9), we obtain Using (4.10) in (4.11), we get In view of (2.10), (4.12) yields From (2.9) and (4.13), we have Now using (4.12) and (4.14) in (4.2), we get Plugging (4.15) in (4.6), we obtain Therefore by taking into account (4.7) and (4.16), we have either or is pseudoprojectively flat.
Conversely, suppose that . Then, from (2.1), (2.2) and (2.7), we have . Hence . If the space-form is pseudoprojectively flat then clearly it is pseudoprojectively semisymmetric. Hence we can state the following.
Theorem 4.2. A -dimensional generalized Sasakian-space-form is pseudoprojectively semisymmetric if and only if the space form is either pseudoprojectively flat or .
By combining Theorems 3.2 and 4.2, we have the following.
Corollary 4.3. A -dimensional generalized Sasakian-space-form is pseudoprojectively flat if and only if or and .
5. -Curvature Tensor in a Generalized Sasakian-Space-Form
In a -dimensional Riemannian manifold , the -curvature tensor is given by [8] where are some smooth functions on . For different values of , the -curvature tensor reduces to the curvature tensor, quasiconformal curvature tensor, conformal curvature tensor, conharmonic curvature tensor, concircular curvature tensor, pseudoprojective curvature tensor, projective curvature tensor, -projective curvature tensor, -curvature tensors , and -curvature tensors .
Suppose that is -flat. Then from (5.1), we have In view of (2.7), (2.8), and (2.9) in (5.2), we have Putting in (5.3), we get If we choose a unit vector orthogonal to and taking , then making use of (2.1) and (2.3) in (5.4), we obtain Putting in (5.5), we have where Thus we have the following.
Theorem 5.1. If a -dimensional generalized Sasakian-space-form is -flat, then (5.6) holds.
From the above theorem, we discuss the following cases.
Case . If is quasiconformally flat, then , , . Putting these in (5.7), we obtain .
If is conharmonically flat, then , , , . Putting these in (5.7), we get .
Similarly for -flat, -flat, -flat, -flat spaces, (5.7) gives .
Case . If is conformally flat, then , , , .
Putting these in (5.7), we obtain . Hence .
Case . If is pseudoprojectively flat, then ,.
By putting these values in (5.7), we have .
If is projectively flat, then .
Making use of the above functional values in (5.7), we get .
Similarly for concircularly flat, -projectively flat, -flat, -flat, -flat, -flat, and -flat spaces, (5.7) gives .
Case . If is -flat, then .
Putting these in (5.7), we obtain that .
If is -flat, then . Putting these in (5.7), we have .
Similarly, for a -flat space, (5.7) gives .
Summarizing the above cases, we have the following corollaries.
Corollary 5.2. If a -dimensional generalized Sasakian-space-form is either quasiconformally flat, conharmonically flat, -flat, -flat, -flat, or -flat, then ,, and are linearly dependent.
Corollary 5.3. If a -dimensional generalized Sasakian-space-form is conformally flat, then .
The above corollary was already proved by Kim [9] and Sarkar and De [10].
Corollary 5.4. If a -dimensional generalized Sasakian-space-form is either pseudoprojectively flat, projectively flat, concircularly flat, -projectively flat, -flat, -flat, -flat, -flat, or -flat, then and are linearly dependent.
Corollary 5.5. If a -dimensional generalized Sasakian-space-form is either -flat, -flat, or -flat, then and are linearly dependent.
5.1. -Semisymmetric Generalized Sasakian-Space-Form
Definition 5.6. is -semisymmetric if holds in .
We know that From (5.8) and (5.9), we have By using (5.1) in (5.10), we have Let be an orthonormal basis of the tangent space at each point of the manifold. Contracting (5.11) with respect to and putting , also taking summation over , , and making use of (2.1), (2.4), (2.7), (2.9), and (2.8), we have where Changing to in (5.12) and also in view of (2.1) and (2.2), (2.9) yields Thus we can state the following.
Theorem 5.7. A -semisymmetric generalized Sasakian-space-form is -Einstein provided .
P. Alegre, D. E. Blair, and A. Carriazo, “Generalized Sasakian-space-forms,” Israel Journal of Mathematics, vol. 141, no. 1, pp. 157–183, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
P. Alegre and A. Carriazo, “Structures on generalized Sasakian-space-forms,” Differential Geometry and its Applications, vol. 26, no. 6, pp. 656–666, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
G. Somashekhara and H. G. Nagaraja, “Generalized Sasakian-space-forms and trans-Sasakian mani-folds,” Acta Mathematical Paedagogicae Academiae Nyiregyhazienis, vol. 28, no. 2, 2012. View at: Google Scholar
G. Somashekhara and H. G. Nagaraja, “On K-torseforming vector field in a trans-Sasakian generalized Sasakian-space-form,” International Journal of Mathematical Archive, vol. 3, no. 7, pp. 2583–2588, 2012. View at: Google Scholar
U. C. De and A. Sarkar, “Some results on generalized Sasakian-space-forms,” Thai Journal of Mathematics, vol. 8, no. 1, pp. 1–10, 2010. View at: Google Scholar | Zentralblatt MATH | MathSciNet
U. C. De and A. Sarkar, “On the projective curvature tensor of generalized Sasakian-space-forms,” Quaestiones Mathematicae, vol. 33, no. 2, pp. 245–252, 2010. View at: Publisher Site | Google Scholar | MathSciNet
B. Prasad, “A pseudo projective curvature tensor on a Riemannian manifold,” Bulletin of the Calcutta Mathematical Society, vol. 94, no. 3, pp. 163–166, 2002. View at: Google Scholar | Zentralblatt MATH | MathSciNet
M. M. Tripathi and P. Gupta, “
\tau
-curvature tensor on a semi-Riemannian manifold,” Journal of Advanced Mathematical Studies, vol. 4, no. 1, pp. 117–129, 2011. View at: Google Scholar | Zentralblatt MATH | MathSciNet
U. K. Kim, “Conformally flat generalized Sasakian-space-forms and locally symmetric generalized Sasakian-space-forms,” Note di Matematica, vol. 26, no. 1, pp. 55–67, 2006. View at: Google Scholar | Zentralblatt MATH | MathSciNet
A. Sarkar and U. C. De, “Some curvature properties of generalized Sasakian-space-forms,” Lobachevskii Journal of Mathematics, vol. 33, no. 1, pp. 22–27, 2012. View at: Publisher Site | Google Scholar | MathSciNet
Copyright © 2012 H. G. Nagaraja 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. |
Compute fundamental value of signal - Simulink - MathWorks Nordic
Fundamental (PLL-Driven)
Compute fundamental value of signal
The Fundamental (PLL-Driven) block computes the fundamental value of the input 3 over a running window of one cycle of fundamental frequency given by input 1. The reference frame required for the computation is given by the input 2.
Based on the Fourier analysis of a periodic signal, the fundamental value of a signal f(t) can be expressed as
\begin{array}{l}Fundamental\left(f\left(t\right)\right)=a\mathrm{cos}\left({\omega }_{0}t\right)+b\mathrm{sin}\left({\omega }_{0}t\right)\\ a=\frac{2}{T}\underset{\left(t-T\right)}{\overset{t}{\int }}f\left(t\right)\mathrm{cos}\left({\omega }_{0}t\right)dt\\ b=\frac{2}{T}\underset{\left(t-T\right)}{\overset{t}{\int }}f\left(t\right)\mathrm{sin}\left({\omega }_{0}t\right)dt\\ T=\frac{1}{{f}_{0}}\text{ }{f}_{0}:\text{Fundamental frequency}\\ {\omega }_{0}=2×\Pi ×{f}_{0}\end{array}
The magnitude and phase of the fundamental is calculated by
Magnitude=\sqrt{{a}^{2}+{b}^{2}}\text{ }Phase=a\mathrm{tan}\left(b/a\right)
To resolve these equations, the block uses the input 1 (Freq) for f0 and input 2 (wt) for ω0t. These two input signals are normally connected to the outputs of a PLL block.
Specify the minimum frequency value that sets the buffer size of the Variable Time Delay block used inside the block to compute the fundamental value. Default is 45.
Initial input [ Mag, Phase-relative-to-PLL (degrees) ]
Specify the initial magnitude and phase in degrees of the input signal. Default is [1, 0].
Fundamental frequency (Hz) required by the computation. This input is normally connected to the output Freq of a PLL block.
Angle of the reference frame (rad/s) required for the computation. This input is normally connected to the output wt of a PLL block.
Connects to the signal to be analyzed. Typical input signals are voltages or currents measured by the Current Measurement or Voltage Measurement block.
Returns the magnitude of the fundamental in the same unit as the input signal.
\angle \text{u}
Returns the phase of the fundamental, in degrees, relative to the reference frame wt (input 2).
Dimensionalized Yes, for Input 3 (In)
The power_FundamentalPLLDriven model compares the Fourier block (with a specified fundamental frequency of 60 Hz) output to the Fundamental (PLL-Driven) block output. The PLL-Driven block outputs accurate magnitude and phase even if the fundamental frequency of the input signal varies during the simulation. |
Zu Chongzhi - New World Encyclopedia
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Zhu Chung Zih bronze staute. Partly damaged.
Zu Chongzhi (Traditional Chinese: 祖沖之; Simplified Chinese: 祖冲之; Hanyu Pinyin: Zǔ Chōngzhī; Wade-Giles: Tsu Ch'ung-chih, 429–500), courtesy name Wenyuan (文遠), was a prominent Chinese mathematician and astronomer during the Liu Song and Southern Qi Dynasties.
China is one of the countries which had the most advanced mathematics before fourteenth century. Zu Chongzhi is known for his accurate approximation for π for the following 900 years. His best approximation was between 3.1415926 and 3.1415927 (355/113). Zu also calculated one year as
{\displaystyle \textstyle {365{9589 \over 39491}}}
(≒365.24281481) days, which is close to today's 365.24219878 days. Zu also developed the Daming calendar (大明曆) in 465, and his son completed his work. It became the official calender of Ming Dynasty.
2 Zhui Shu
4 The South Pointing Chariot
Chinese mechanical engineer Ma Jun (c. 200-265 C.E.) originally invented the South Pointing Chariot, a two-wheeled vehicle which was designed to constantly point south by the use of differential gears without magnetic compass. Zu Chongzhi made a major improvement to it including the adoption of new bronze gears.
Zu Chongzhi's ancestry was from modern Baoding, Hebei. To flee from the ravages of war, Zu's grandfather Zu Chang moved to the Yangtze, as part of the massive population movement during the Eastern Jin. Zu Chang (祖昌) at one point held the position of "Minister of Great Works" (大匠卿) within the Liu Song and was in charge of government construction projects. Zu's father, Zu Shuo (祖朔) also served the court and was greatly respected for his erudition.
Zu was born in Jiankang. His family had historically been involved in astronomy research, and from childhood Zu was exposed to both astronomy and mathematics. When he was only a youth his talent earned him much repute. When Emperor Xiaowu of Liu Song heard of him, he was sent to an Academy, the Hualin Xuesheng (華林學省), and later at the Imperial Nanjing University (Zongmingguan) to perform research. In 461 in Nanxu (today Zhenjiang, Jiangsu), he was engaged in work at the office of the local governor.
Zu Chongzhi, along with his son Zu Gengzhi, wrote a mathematical text entitled Zhui Shu (Method of Interpolation). It is said that the treatise contains formulas for the volume of the sphere, cubic equations and the accurate value of pi. Sadly, this book didn't survive to the present day, since it has been lost since the Song Dynasty.
His mathematical achievements included:
the Daming calendar (大明曆) introduced by him in 465. His son continued his work and completed the calender. The Daming calender became official calender of Liang Dynasty (梁朝; Pinyin: Liáng cháo) (502-557).
distinguishing the Sidereal Year and the Tropical Year, and he measured 45 years and 11 months per degree between those two, and today we know the difference is 70.7 years per degree.
calculating one year as
{\displaystyle \textstyle {365{9589 \over 39491}}}
(≒365.24281481) days, which is very close to 365.24219878 days as we know today.
calculating the number of overlaps between sun and moon as 27.21223, which is very close to 27.21222 as we know today; using this number he successfully predicted an eclipse four times during 23 years (from 436 to 459).
calculating the Jupiter year as about 11.858 Earth years, which is very close to 11.862 as we know of today.
deriving two approximations of pi, which held as the most accurate approximation for π for over 900 years. His best approximation was between 3.1415926 and 3.1415927, with 355⁄113 (密率, Milu, detailed approximation) and 22⁄7 (約率, Yuelu, rough approximation) being the other notable approximations. He obtained the result by approximating a circle with a 12,288 (= 212 × 3) sided polygon. This was an impressive feat for the time, especially considering that the device Counting rods he used for recording intermediate results were merely a pile of wooden sticks laid out in certain patterns. Japanese mathematician Yoshio Mikami pointed out, "
{\displaystyle {\tfrac {22}{7}}}
was nothing more than the π value obtain several-hundred years earlier by the Greek mathematician Archimedes,however Milu
{\displaystyle \pi ={\tfrac {355}{113}}}
could not be found in any Greek, Indian or Arabian manuscripts, not until 1585 Dutch mathematician Adriaan Anthoniszoom obtained this fraction; the Chinese possessed this most extraodinary fraction over a whole millennium earlier than Europe." Hence Mikami strongly urged that the fraction
{\displaystyle {\tfrac {355}{113}}}
be named after Zu Chongzhi as Zu Chongzhi fraction.[1] In Chinese literature, this fraction is known as "Zu rate." Zu rate is a best rational approximation to π, and is the closest rational approximation to π from all fractions with denominator less than 16,600.[2]
finding the volume of a sphere as πD3/6 where D is diameter (equavilent to 4πr3/3).
discovering the Cavalieri's principle, 1,000 years before Bonaventura Cavalieri in the West.
Most of Zu's great mathematical works, are recorded in his lost text Zhui Shu. Most scholars argue about his complexity. Since traditionally, the Chinese developed mathematics as algebraic, and equational. Logically, scholars assume that his work, Zhui Shu yields methods of cubic equations. His works on the accurate value of pi describes the lengthy calculations. Zu used the method of exhaustion, inscribing a 12,288-gon. Interestingly, Zu's value of pi is precise to eight decimal places. No mathematician since his time, computed a value this precise until another 900 years. Zu also worked on deducing the formula for the volume of the sphere.
The South Pointing Chariot
Traditional Chinese: 指南車
Simplified Chinese: 指南车
- Hanyu Pinyin: zhi3 nan2 che1
- Jyutping: zi2 naam4 ce1
Reconstruction of a South Pointing Chariot, 2005
Model in the Science Museum in London
The South Pointing Chariot device was invented by a number of engineers since antiquity in China, including Zhang Heng (CE 78–139), and Ma Jun (c. 200-265 C.E.). It was a two-wheeled vehicle that incorporated an early use of differential gears to operate a fixed figurine that would constantly point south, hence enabling one to accurately measure their directional bearings. It is a non-magnetic compass vehicle.
Although the chariot can technologically be made to point to any direction, the south was selected based upon ancient Chinese thought that the "Son-of-heaven" (天子) faces the south. In ancient Chinese thought, geographical direction is not value neutral but highly value loaded. The idea was incorporated into Feng shui, a general geographical-astronomical theory of fortune.
The literal translation of this chariot in Chinese character, "指南車," is a combination of two characters, "vehicle" (車) and "instruction" or "teaching." The character of "teaching" (指南) consists of two Characters, "pointing" (指) and "south" (南). Hence, "teaching" is expressed as "pointing to the sought." Thus, the chariot is a vehicle for a teacher or a master or Xian, Toaist immortal saint.
This effect was achieved not by magnetics (like in a compass), but through intricate mechanics, the same design that allows equal amounts of torque applied to wheels rotating at different speeds for the modern automobile. After the Three Kingdoms period, the device fell out of use temporarily. However, it was Zu Chongzhi who successfully re-invented it in 478 C.E., as described in the texts of the Song Shu (c. 500 C.E.) and the Nan Chi Shu, with a passage from the latter below:
When Emperor Wu of Liu Song subdued Guanzhong he obtained the south-pointing carriage of Yao Xing, but it was only the shell with no machinery inside. Whenever it moved it had to have a man inside to turn (the figure). In the Sheng-Ming reign period, Gao Di commissioned Zi Zu Chongzhi to reconstruct it according to the ancient rules. He accordingly made new machinery of bronze, which would turn round about without a hitch and indicate the direction with uniformity. Since Ma Jun's time such a thing had not been.[3]
Zu Chongzhi made a new improved vehicle with bronze gears for Emperor Shun of Liu Song. The first true differential gear used in the Western world was by Joseph Williamson in 1720.[4] Joseph Williamson used a differential for correcting the equation of time for a clock that displayed both mean and solar time.[4] Even then, the differential was not fully appreciated in Europe until James White emphasized its importance and provided details for it in his Century of Inventions (1822).[4]
Named for him
{\displaystyle \pi ={\tfrac {355}{113}}}
as Zu Chongzhi
{\displaystyle \pi }
rate. Zu Chongzhi computed π to be between 3.1415926 and 3.1415927 and gave two approximations of π, 22⁄7 and 355⁄113 in the fifth century.
{\displaystyle \pi ={\tfrac {355}{113}}}
{\displaystyle \pi }
rate.
The lunar crater Tsu Chung-Chi
1888 Zu Chong-Zhi is the name of asteroid 1964 VO1.
↑ Yoshio Mikami, The Development of Mathematics in China and Japan. (New York: Chelsea Pub. Co, 1913), 50
↑ The next "best rational approximation" to π is
{\displaystyle {\frac {52163}{16604}}=3.1415923874}
↑ 4.0 4.1 4.2 Needham, Volume 4, Part 2, 298.
Du, Shiran and Shaogeng He. "Zu Chongzhi". Encyclopedia of China (Mathematics Edition), 1st ed. Retrieved February 24, 2009.
Lloyd, G. E. R. Principles and Practices in Ancient Greek and Chinese Science. Aldershot, Hampshire, Great Britain: Ashgate/Variorum, 2006. ISBN 9780860789932
Mikami, Yoshio. The Development of Mathematics in China and Japan. New York: Chelsea Pub. Co, 1913.
Needham, Joseph. 1986. Science and Civilization in China: Volume 4, Part 2. Taipei: Caves Books, Ltd.
Needham, Joseph, Shigeru Nakayama, and Nathan Sivin. Chinese Science; Explorations of an Ancient Tradition. Cambridge: MIT Press, 1973. ISBN 9780262140119
Needham, Joseph. The Grand Titration: Science and Society in East and West. London: Allen & Unwin, 1969. ISBN 9780049310056
Temple, Robert K. G., and Joseph Needham. The Genius of China: 3,000 Years of Science, Discovery, and Invention. New York: Simon and Schuster, 1986. ISBN 9780671620288
Volkov, Alexei. 1997. Zhao Youqin and his calculation of (Pi). Historia Mathematica. 24(3): 301.
Zhong, Shizu. Ancient China's Scientists. Hong Kong: Commercial Press, 1984. ISBN 9789620710537
Zu Chongzhi at the University of Maine
Zu Chongzhi at Encyclopedia Britannica
Zu_Chongzhi history
Milü history
South_Pointing_Chariot history
Liang_Dynasty history
History of "Zu Chongzhi"
Retrieved from https://www.newworldencyclopedia.org/p/index.php?title=Zu_Chongzhi&oldid=1044042 |
ChiSquare - Maple Help
Home : Support : Online Help : Statistics and Data Analysis : Statistics Package : Distributions : ChiSquare
ChiSquare(nu)
ChiSquareDistribution(nu)
The chi-square distribution is a continuous probability distribution with probability density function given by:
f\left(t\right)={\begin{array}{cc}0& t<0\\ \frac{{t}^{\frac{\mathrm{\nu }}{2}-1}{ⅇ}^{-\frac{t}{2}}}{{2}^{\frac{\mathrm{\nu }}{2}}\mathrm{\Gamma }\left(\frac{\mathrm{\nu }}{2}\right)}& \mathrm{otherwise}\end{array}
0<\mathrm{\nu }
The ChiSquare variate with nu degrees of freedom is equivalent to the Gamma variate with scale
2
and shape nu/2: ChiSquare(nu) ~ Gamma(2,nu/2).
The ChiSquare variate is related to the FRatio variate by the formula FRatio(nu,omega) ~ (ChiSquare(nu)*omega)/(ChiSquare(omega)*nu)
The ChiSquare variate is related to the Normal variate and the StudentT variate by the formula StudentT(nu) ~ Normal(0,1)/sqrt(ChiSquare(nu)/nu)
Note that the ChiSquare command is inert and should be used in combination with the RandomVariable command.
\mathrm{with}\left(\mathrm{Statistics}\right):
X≔\mathrm{RandomVariable}\left(\mathrm{ChiSquare}\left(\mathrm{\nu }\right)\right):
\mathrm{PDF}\left(X,u\right)
{\begin{array}{cc}\textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{<}\textcolor[rgb]{0,0,1}{0}\\ \frac{{\textcolor[rgb]{0,0,1}{u}}^{\frac{\textcolor[rgb]{0,0,1}{\mathrm{\nu }}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{u}}{\textcolor[rgb]{0,0,1}{2}}}}{{\textcolor[rgb]{0,0,1}{2}}^{\frac{\textcolor[rgb]{0,0,1}{\mathrm{\nu }}}{\textcolor[rgb]{0,0,1}{2}}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{\Gamma }}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0,0,1}{\mathrm{\nu }}}{\textcolor[rgb]{0,0,1}{2}}\right)}& \textcolor[rgb]{0,0,1}{\mathrm{otherwise}}\end{array}
\mathrm{PDF}\left(X,0.5\right)
\frac{\textcolor[rgb]{0,0,1}{0.7788007831}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{0.5}}^{\textcolor[rgb]{0,0,1}{0.5000000000}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{\nu }}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1.}}}{{\textcolor[rgb]{0,0,1}{2.}}^{\textcolor[rgb]{0,0,1}{0.5000000000}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{\nu }}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{\Gamma }}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{0.5000000000}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{\nu }}\right)}
\mathrm{Mean}\left(X\right)
\textcolor[rgb]{0,0,1}{\mathrm{\nu }}
\mathrm{Variance}\left(X\right)
\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{\nu }}
Stuart, Alan, and Ord, Keith. Kendall's Advanced Theory of Statistics. 6th ed. London: Edward Arnold, 1998. Vol 1: Distribution Theory. |
Miniworkshop: L2-Spectral Invariants and the Integrated Density of States | EMS Press
Miniworkshop: L2-Spectral Invariants and the Integrated Density of States
H. Daniel Lenz
Jozef Dodziuk
The CUNY Graduate Center, New York, United States
Both the study of
L^2
-spectral invariants in geometry and the investigation of the integrated density of states in mathematical physics have attracted much attention in recent years. While the two topics are strongly related, the corresponding communities are rather unaware of each others work and methods. The main aim of this mini-workshop was to bring together people from both fields and provide a basis for interaction.
Accordingly, the first two days of the conference were spent with survey talks solicited by the organizers to highlight concepts and methods. There were 9 such talks with durations between 60 and 90 minutes. The second half of the conference was devoted to more detailed investigations. Most participants used the opportunity to present their current research in the area of the meeting. There were 13 such talks.
The results presented in those talks contained significant contributions e.g.~to the Atiyah conjecture about integrality of
L^2
-Betti numbers for a completely new class of groups by Peter Linnell, a mathematically rigorous derivation using von Neumann traces of the asymptotics of the specific heat near absolute zero by Mikhael Shubin, and approximation results for the integrated density of states in various new contexts.
Altogether the conference was attended by 17 participants.
H. Daniel Lenz, Thomas Schick, Jozef Dodziuk, Ivan Veselić, Miniworkshop: L2-Spectral Invariants and the Integrated Density of States. Oberwolfach Rep. 3 (2006), no. 1, pp. 511–552 |
Science In Everyday Life for Class 6 Science Chapter 11 - Measurement And Motion
Science In Everyday Life Solutions for Class 6 Science Chapter 11 Measurement And Motion are provided here with simple step-by-step explanations. These solutions for Measurement And Motion are extremely popular among Class 6 students for Science Measurement And Motion Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Science In Everyday Life Book of Class 6 Science Chapter 11 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Science In Everyday Life Solutions. All Science In Everyday Life Solutions for class Class 6 Science are prepared by experts and are 100% accurate.
1. Rectilinear motion ...........................................................................................................
2. Curvilinear motion ..........................................................................................................
3. Rotational motion ...........................................................................................................
4. Periodic motion ..............................................................................................................
5. Non-periodic motion .....................................................................................................
(i) A train moving on a straight track.
(ii) A car moving on a straight road.
(i) A car taking a turn.
(ii) A child going down a slide.
(i) Motion of blades of a rotating fan.
(ii) Motion of blades of a windmill.
(i) To and fro motion of a pendulum.
(ii) Motion of hands of a clock.
5. Non-periodic motion
(i) A car moving on a road.
(ii) Birds gliding across the sky.
An example of a standard unit of measurement is
(a) hand span
(b) pace
Because it does not vary from person to person and place to place.
Metre is the SI unit of
2500 m equals
Since 1000 m =1 km
∴ 2500 m =
\frac{1}{1000}×2500=2.5 \mathrm{km}
An object is said to be in motion if its position (with respect to another object)
(a) changes with time
(b) does not change with time
(d) is the same as before
A body is said to be in motion if its position with respect to the another object changes with time.
A car moving on a straight road is in
(a) curvilinear motion
It is because the movement of an object in a straight line is rectilinear.
1. ..................... (Kilometer/Cubit) is a non-standard unit of length.
2. The distance of an object from one end to the other is called ..................... (length/kilometre).
3. When a body is in translational motion, all its parts move ..................... (equal/unequal) distances in a given time.
4. The rotation of Earth on its axis is ..................... (translational/rotational) motion.
5. A type of motion that repeats itself after equal intervals of time is called ..................... (curvilinear/periodic) motion.
1. Cubit is a non-standard unit of length.
2. The distance of an object from one end to the other is called length.
3. When a body is in translational motion, all its parts move equal distances in a given time.
4. The rotation of Earth on its axis is rotational motion.
5. A type of motion that repeats itself after equal intervals of time is called periodic motion.
Why do we need standard units of measurements?
The units, in which our different body parts are measured, are not reliable because the length of our body parts varies from person to person. Therefore, we need standard units of measurement.
Give one advantage of using Sl units.
The International System of Units or SI units are the modern form of the metric system.
Advantage: Scientists of different countries can communicate their results easily to one another.
Write a short note on the relevance of estimation.
The 'idea of how much' is estimation. It is not necessary to make accurate measurements in all cases.
Example: A rough idea of the quantity of each ingredient is enough to cook a meal. Estimation skills are very useful in our life. In such type of calculations, we can find a number that is close enough to the right answer.
Let us assume we want to plant a row of flowers.
The length of the row = 58.3 cm = 60 cm (approx)
The distance between two adjacent flowers = 6 cm
∴ Total number of flowers planted in a row = 60/6 = 10 (approx)
Therefore, estimation is relevant for our everyday life.
Give two differences between rectilinear and curvilinear motions.
Rectilinear Motion Curvilinear Motion
When an object in translational motion moves in a straight line, it is said to be in rectilinear motion.
Example: Motion of a car on a straight road. When an object in translational motion moves along a curved path, it is said to be in curvilinear motion.
Example: Motion of a car taking a turn.
Give one similarity between rectilinear and curvilinear motions.
Both the rectilinear motion and curvilinear motion are types of translational motion. In this type of motion, all parts of an object move the same distance in a given time.
Usha wants to buy a dupatta that is 250 cm long. The shopkeeper wants her to give the measurement in metres. Can you help her?
or 1 cm =
\frac{1}{100}\mathrm{m}
∴ 250 cm =
\frac{1}{100}×250=2.50 \mathrm{m}
Thus, Usha should tell the shopkeeper that she needs a dupatta of 2.50 metre in length.
Maya's Physics text book is 11 mm thick and her Maths text book is 1 cm thick. Which text book is thicker?
Thickness of Physics textbook = 11 mm
Thickness of Math textbook = 1 cm
To compare the thickness of books, the measurements should be in the same unit.
∴ Thickness of Math textbook = 10 mm
Thus, Physics textbook is thicker.
Amar lives in a hostel which is 20,00,000 m away from his house. Calculate the distance in km.
Distance between Amar's hostel and his house = 20,00,000 m
or 1 m =
\frac{1}{1000}\mathrm{km}
∴ 20,00,000 m =
\frac{1}{1000}× 20,00,000=2000 \mathrm{km}
Thus, Amar lives in a hostel that is 2,000 km away from his house.
Discuss with examples how people measured length in ancient times.
In ancient times, people used different body parts to measure length.
Hand span: It is the distance between the tip of the thumb and the tip of the little finger of a fully stretched hand.
Cubit: It is the distance between the tip of the middle finger and the elbow.
Fathom: It is the length of the outstretched arms.
However, these units were not reliable, so people realised the need for standard units.
Discuss the precautions that one should take while measuring length with a ruler.
The following precautions should be taken while measuring length with a ruler:
1. The ruler should be kept along the length of the object carefully.
2. Measurement should only be started from a mark that is fully clear. Example: If you measure from 3-cm mark to the 7-cm mark, the length is 4 cm (7 − 3 = 4 cm).
3. Eyes should be exactly above the point where the measurement is to be taken, otherwise you might get faulty or inaccurate readings.
Explain how one can measure length of a curved line using a string.
One can measure the length of a curved line using a string and a ruler.
Take a non-stretchable string and tie a knot at one of its ends.
Hold the string steadily with your fingers and stretch it along the curved line until you reach the other end.
Make a mark on the thread where it reaches the other end.
Place the string along a ruler and measure the length between the knot and the marked point.
Thus, the length of a curved line is measured using a string.
Describe the different types of motion with examples.
The different types of motion are explained with examples as given below:
1. Translational motion
A motion in which all parts of an object move equal distance in a given time is called translational motion.
Types of translational motion:
(i) Rectilinear motion
(ii) Curvilinear motion
Example: Motion of a car on a straight road.
When an object in translational motion moves along a curved path, it is said to be in curvilinear motion.
When an object moves about an axis and different parts of it move by unequal distances in a given interval of time, it is said to be in rotational motion.
Example: Motion of blades of a windmill.
A motion that repeats itself after equal intervals of time is called periodic motion.
Example: The to and fro motion of a pendulum.
A motion that does not repeat itself at regular intervals of time is called non-periodic motion.
Example: Birds gliding across the sky.
The distance of something from one end to the other is called length.
An object is said to be in motion if its position (with respect to another object) changes with time.
When an object moves about an axis and different parts of it move by different distances in a given interval of time, it is said to be in rotational motion. |
Mini-Workshop: Thick Subcategories - Classifications and Applications | EMS Press
Mini-Workshop: Thick Subcategories - Classifications and Applications
University of Toronto at Scarborough, Canada
Thick subcategories of triangulated categories have been the main topic of this workshop. Triangulated categories arise in many areas of modern mathematics, for instance in algebraic geometry, in representation theory of groups and algebras, or in stable homotopy theory. We give three typical examples of such triangulated categories: \begin{itemize} \item the category of perfect complexes of
\mathcal O_X
-modules over a scheme
X
, \item the stable category of finite dimensional representations of a finite group, \item the stable homotopy category of finite spectra. \end{itemize} In each case, there is a classification of thick subcategories under some appropriate conditions. Recall that a subcategory of a triangulated category is {\em thick}, if it is a triangulated subcategory and closed under taking direct factors. Historically, the first classification has been established by Hopkins and Smith for the stable homotopy category, using the nilpotence theorem. A similar idea was then applied by Hopkins and Neeman to categories of perfect complexes over commutative noetherian rings. Later, Thomason extended this classification to schemes. For stable categories of finite group representations, the classification of thick subcategories is due to Benson, Carlson, and Rickard.
The format of the workshop has been a combination of introductory survey lectures and more specialized talks on recent progress and open problems. The mix of participants from different mathematical areas and the relatively small size of the workshop provided an ideal atmosphere for fruitful interaction and exchange of ideas. It is a pleasure to thank the administration and the staff of the Oberwolfach Institute for their efficient support and hospitality.
Henning Krause, Stefan Schwede, Ragnar-Olaf Buchweitz, Mini-Workshop: Thick Subcategories - Classifications and Applications. Oberwolfach Rep. 3 (2006), no. 1, pp. 461–510 |
Symbolic cumulative sum - MATLAB cumsum - MathWorks Italia
Cumulative Sum of Symbolic Vector
Cumulative Sum of Each Column and Row in Symbolic Matrix
Reverse Cumulative Sum of 3-D Symbolic Array
Symbolic Vector with NaN Values
Symbolic cumulative sum
B = cumsum(A) returns the cumulative sum of A starting at the beginning of the first array dimension in A whose size does not equal 1. The output B has the same size as A.
If A is a matrix, then cumsum(A) returns a matrix containing the cumulative sums of each column of A.
B = cumsum(A,dim) returns the cumulative sum along dimension dim. For example, if A is a matrix, then cumsum(A,2) returns the cumulative sum of each row.
B = cumsum(___,direction) specifies the direction using any of the previous syntaxes. For instance, cumsum(A,2,'reverse') returns the cumulative sum within the rows of A by working from end to beginning of the second dimension.
Create a symbolic vector. Find the cumulative sum of its elements.
A = (1:5)*x
\left(\begin{array}{ccccc}x& 2 x& 3 x& 4 x& 5 x\end{array}\right)
In the vector of cumulative sums, element B(2) is the sum of A(1) and A(2), while B(5) is the sum of elements A(1) through A(5).
\left(\begin{array}{ccccc}x& 3 x& 6 x& 10 x& 15 x\end{array}\right)
Create a 3-by-3 symbolic matrix A whose elements are all equal to 1.
A = sym(ones(3))
\left(\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right)
Compute the cumulative sum of elements of A. By default, cumsum returns the cumulative sum of each column.
\left(\begin{array}{ccc}1& 1& 1\\ 2& 2& 2\\ 3& 3& 3\end{array}\right)
To compute the cumulative sum of each row, set the value of the dim option to 2.
\left(\begin{array}{ccc}1& 2& 3\\ 1& 2& 3\\ 1& 2& 3\end{array}\right)
Create a 3-by-3-by-2 symbolic array.
A(:,:,1) = [x y 3; 3 x y; y 2 x];
A(:,:,2) = [x y 1/3; 1 y x; 1/3 x 2];
\left(\begin{array}{ccc}x& y& 3\\ 3& x& y\\ y& 2& x\end{array}\right)
\left(\begin{array}{ccc}x& y& \frac{1}{3}\\ 1& y& x\\ \frac{1}{3}& x& 2\end{array}\right)
Compute the cumulative sum along the rows by specifying dim as 2. Specify the 'reverse' option to work from right to left in each row. The result is the same size as A.
\left(\begin{array}{ccc}x+y+3& y+3& 3\\ x+y+3& x+y& y\\ x+y+2& x+2& x\end{array}\right)
\left(\begin{array}{ccc}x+y+\frac{1}{3}& y+\frac{1}{3}& \frac{1}{3}\\ x+y+1& x+y& x\\ x+\frac{7}{3}& x+2& 2\end{array}\right)
To compute the cumulative sum along the third (page) dimension, specify dim as 3. Specify the 'reverse' option to work from largest page index to smallest page index.
\left(\begin{array}{ccc}2 x& 2 y& \frac{10}{3}\\ 4& x+y& x+y\\ y+\frac{1}{3}& x+2& x+2\end{array}\right)
\left(\begin{array}{ccc}x& y& \frac{1}{3}\\ 1& y& x\\ \frac{1}{3}& x& 2\end{array}\right)
Create a symbolic vector containing NaN values. Compute the cumulative sums.
A = [sym('a') sym('b') 1 NaN 2]
\left(\begin{array}{ccccc}a& b& 1& \mathrm{NaN}& 2\end{array}\right)
\left(\begin{array}{ccccc}a& a+b& a+b+1& \mathrm{NaN}& \mathrm{NaN}\end{array}\right)
\left(\begin{array}{ccccc}a& a+b& a+b+1& a+b+1& a+b+3\end{array}\right)
Input array, specified as a symbolic vector, matrix, or multidimensional array.
cumsum(A,1) works on successive elements in the columns of A and returns the cumulative sum of each column.
cumsum(A,2) works on successive elements in the rows of A and returns the cumulative sum of each row.
NaN condition, specified as:
Cumulative sum array, returned as a vector, matrix, or multidimensional array of the same size as the input A.
cumprod | fold | int | symprod | symsum |
F. J. Duarte - Wikiquote
All the indistinguishable photons illuminate the array of N slits, or grating, simultaneously. If only one photon propagates, at any given time, then that individual photon illuminates the whole array of N slits simultaneously.
Francisco Javier "Frank" Duarte (born 1 September 1954) is a Chilean-American laser physicist and author who has contributed widely to dye lasers, tunable lasers, multiple-prism optics, and quantum interferometry.
in Introduction to Lasers, F. J. Duarte (2003). Tunable Laser Optics. Elsevier Academic. p. 3. ISBN 0-12-222696-8.
Feynman uses Dirac's notation to describe the quantum mechanics of stimulated emission... he applies that physics to... dye molecules... In this regard, Feynman could have predicted the existence of the tunable laser.
in Introduction to Lasers, F. J. Duarte (2003). Tunable Laser Optics. Elsevier Academic. p. 3. ISBN 0-12-222696-8. (while discussing The Feynman Lectures on Physics).
The Dirac notation, though originally applied to the propagation of single particles, also applies to describing the propagation of ensembles of coherent, or indistinguishable, photons.
in Dirac Optics, F. J. Duarte (2003). Tunable Laser Optics. Elsevier Academic. p. 25. ISBN 0-12-222696-8.
The intimate relation between interference and diffraction has its origin in the interference equation itself.
Dirac wrote the first chapter in laser optics.
I find the concept of a final theory, or a theory of everything, rather limiting.
Multiple-prism arrays were first introduced by Newton (1704) in his book Opticks. In that visionary volume Newton reported on arrays of nearly isosceles prisms in additive and compensating configurations to control the propagation path and the dispersion of light. Further, he also illustrated slight beam expansion in a single isosceles prism.
in The Physics of Multiple-Prism Optics, F. J. Duarte (2003). Tunable Laser Optics. Elsevier Academic. p. 57. ISBN 0-12-222696-8.
The longer the cavity and the narrower the beam waist, the better the beam quality of the laser emission, or
{\displaystyle |\langle x|s\rangle |^{2}\ =\sum _{j=1}^{\mathbb {N} }\,\Psi (r_{j})^{2}\ +2\sum _{j=1}^{\mathbb {N} }\,\Psi (r_{j}){\bigg (}\sum _{m=j+1}^{\mathbb {N} }\,\Psi (r_{m})cos(\Omega _{m}-\Omega _{j}){\bigg )}}
in Pulsed Narrow-Linewidth Tunable Laser Oscillators, F. J. Duarte (2003). Tunable Laser Optics. Elsevier Academic. p. 147. ISBN 0-12-222696-8.
Personally, I find the concept of a "final theory," or a "theory of everything," rather limiting. The fun of discovery will most likely last as long as the human race continues.
in F. J. Duarte (2012). Laser Physicist. Optics Journal. p. 154. ISBN 978-0-9760383-1-3.
The EPR claim of an "all values" spread in the coordinate depends on an idealized absolute and exact measurement of
{\displaystyle p}
{\displaystyle \Delta p\ =0}
. Since this is physically impossible, the claim of "no physical reality" can be negated.
in F. J. Duarte (2013). Quantum Optics for Engineers. New York: CRC. ISBN 978-1439888537.
The most efficient and practical interpretation of quantum mechanics is... no interpretation at all.
Quotes about Duarte[edit]
The sciences revolted under the guidance of several student activists, Frank Duarte in particular... we were fortunate that Duarte somehow established close links to the Federal Government, which was now the source of all funds.
J. C. Ward, Memoirs of a Theoretical Physicist (Optics Journal, New York, 2004) p. 26.
In 1994, Duarte first reported on solid-state dye laser oscillators.
R. G. Driggers, Encyclopedia of Optical Engineering (CRC, New York, 2003) p. 2853.
After some algebra, a successive formula can be derived from Duarte's original equation.
K. Osvay et al., Measurements of non-compensated angular dispersion and the subsequent temporal lengthening of femtosecond pulses in a CPA laser, Optics Communications 248, 201-209 (2005).
Another connection with Newton's pioneering work on spectra has been found by Duarte in modern quantum optics.
P. Rowlands (2018). Newton and Modern Physics. World Scientific. p. 176. ISBN 978-1786343291.
Tunable Laser Optics at Google
Duarte's biography at the Optical Society
Duarte's home page
Retrieved from "https://en.wikiquote.org/w/index.php?title=F._J._Duarte&oldid=2818624" |
\cos x = -\frac{4}{5}
x ∈ [π, \frac { 3 \pi } { 2 }]
, find exact values for
\sin 2x
\cos 2x
\sin\left(\frac{x}{2}\right)
\cos\left(\frac{x}{2}\right)
. Start by drawing a triangle that where
\cos x = \frac{4}{5}
, find the missing sides and then determine if the value of the function you are using in the formulas is positive or negative based on the location of the angle.
\sin\left(2α\right) = 2\sin\left(α\right)\cos\left(α\right)
\cos\left(2α\right) = \cos^{2}\left(α\right) − \sin^{2}\left(α\right)
\sin\left ( \frac{\theta }{2} \right )=\pm \sqrt{\frac{1-\cos(\theta)}{2}}
\cos\left(\frac{\theta }{2} \right )=\pm \sqrt{\frac{1+\cos(\theta)}{2}} |
<td><math>i + 1</math></td>
<td><ref>Heeralal Janwa and Richard M Wilson, ''Hyperplane sections of fermat varieties in <math>P^3</math> in char. 2 and some applications to cyclic codes'', International Symposium on Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes, pp. 180-194, Springer, 1993</ref><ref>Tadao Kasami, ''The weight enumerators for several classes of subcodes of the 2nd order binary Reed-Muller codes'', Information and Control, 18(4):369-394, 1971</ref></td>
<td><math>n = 2t + 1</math></td>
<td><math>n-1</math></td>
<td><ref>Thomas Beth and Cunsheng Ding, ''On almost perfect nonlinear permutations'', Workshop on the Theory and Application of Cryptographic Techniques, pp. 65-76, Springer, 1993</ref><ref name="kaisa_ref" />
<td><ref name="kaisa_ref" /><ref>Thomas Beth and Cunsheng Ding, ''On almost perfect nonlinear permutations'', Workshop on the Theory and Application of Cryptographic Techniques, pp. 65-76, Springer, 1993</ref>
{\displaystyle F(x)=x^{d}}
{\displaystyle \deg(x^{d})}
{\displaystyle 2^{i}+1}
{\displaystyle \gcd(i,n)=1}
2 [1][2]
{\displaystyle 2^{2i}-2^{i}+1}
{\displaystyle \gcd(i,n)=1}
{\displaystyle i+1}
{\displaystyle 2^{t}+3}
{\displaystyle n=2t+1}
{\displaystyle 3}
{\displaystyle 2^{t}+2^{t/2}-1,t}
{\displaystyle n=2t+1}
{\displaystyle (t+2)/2}
{\displaystyle 2^{t}+2^{(3t+1)/2}-1,t}
{\displaystyle t+1}
{\displaystyle 2^{2t}-1}
{\displaystyle n=2t+1}
{\displaystyle n-1}
{\displaystyle 2^{4i}+2^{3i}+2^{2i}+2^{i}-1}
{\displaystyle n=5i}
{\displaystyle i+3}
↑ Robert Gold, Maximal recursive sequences with 3-valued recursive cross-correlation functions (corresp.), IEEE transactions on Information Theory, 14(1):154-156, 1968
↑ 2.0 2.1 Kaisa Nyberg, Differentially uniform mappings for cryptography, Workshop on the Theory and Application of Cryptographic Techniques, pp. 55-64, Springer, 1993
↑ Heeralal Janwa and Richard M Wilson, Hyperplane sections of Fermat varieties in
{\displaystyle P^{3}}
in char. 2 and some applications to cyclic codes, International Symposium on Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes, pp. 180-194, Springer, 1993
↑ Tadao Kasami, The weight enumerators for several classes of subcodes of the 2nd order binary Reed-Muller codes, Information and Control, 18(4):369-394, 1971
↑ Hans Dobbertin, Almost perfect nonlinear power functions on
{\displaystyle GF(2^{n})}
: the Welch case, IEEE Transactions on Information Theory, 45(4):1271-1275, 1999
{\displaystyle GF(2^{n})}
: the Niho case, Information and Computation, 151(1-2):57-72, 1999
↑ Thomas Beth and Cunsheng Ding, On almost perfect nonlinear permutations, Workshop on the Theory and Application of Cryptographic Techniques, pp. 65-76, Springer, 1993
↑ Hans Dobbertin, Almost perfect nonlinear power functions over
{\displaystyle GF(2^{n})}
: a new case for
{\displaystyle n}
divisible by 5, Proceedings of the fifth conference on Finite Fields and Applications FQ5, pp.113-121 |
To Asa Gray 23[–4] July [1862]1
I received several days ago two large packets, but have as yet read only your letter;2 for we have been in fearful distress & I could attend to nothing. Our poor Boy had the rare case of second rash & sore throat, besides mischief in kidneys; & as if this was not enough a most serious attack of erysipelas with typhoid symptoms.3 I despaired of his life; but this evening he has eaten one mouthful & I think has passed the crisis. He has lived on Port-wine every
\frac{3}{4}
of an hour day & night. This evening to our astonishment he asked whether his stamps were safe & I told him of the one sent by you,4 & that he shd. see it tomorrow. He answered “I should awfully like to see it now”; so with difficulty he opened his eyelids & glanced at it & with a sigh of satisfaction said “all right”.— Children are one’s gretest happiness, but often & often a still greter misery. A man of science ought to have none,—perhaps not a wife; for then there would be nothing in this wide world worth caring for & a man might (whether he would is another question) work away like a Trojan.— I hope in a few days to get my Brains in order & then I will pick out all your orchid letters (& read by & bye your last)5 & return them in hopes of your making use of them—6 Planthanthera would be eminently well worth giving & as much as feel safe about Cypripedium;7 in part I am not sure that I understand the passages by which insects crawl in & out. Could you give a diagram?8 I have such an arrear of letters & such a number of experiments,9 all going to the dogs, that I have not time to make abstract of your letters. Will you return me such, as you do not use: but I hope you will be led to use all some time or another.—10 I shall be very glad to hear of Rosmacks *? observations on Houstonia; you only just alluded to them.—11 You did formerly tell me about Specularia:12 in viola & oxalis the case seems to me to be much too remarkable to be called “precocious flowering”.13
*I hope he will publish note; I hear the French say that my paper on Primula is all pure imagination; but I cannot hear that this is grounded on any observations—14
You will never read my horrid writing, if I write on both pages, of thin paper which I have taken in obedience to orders.—15 Of all the carpenters for knocking the right nail on the head, you are the very best: no one else has perceived that my chief interest in my orchid book, has been that it was a “flank movement” on the enemy.16 I live in such solitude that I hear nothing, & have no idea to what you allude about Bentham & the orchids & Species.17 But I must enquire.—
By the way one of my chief enemies (the sole one who has annoyed me) namely Owen, I hear has been lecturing on Birds, & admits that all have descended from one, & advances as his own idea that the oceanic wingless Birds have lost their wings by gradual disuse.18 He never alludes to me or only with bitter sneers & coupled with Buffon, & the Vestiges.—19
Well it has been an amusement to me this first evening scribbling as egotistically as usual about myself & my doings; so you must forgive me, as I know well your kind heart will do.— I have managed to skim the news-paper, but had not heart to read all the bloody details. Good God what will the end be; perhaps we are too despondent here; but I must think you are too hopeful on your side of the water. I never believed the “canard” of the army of the Potomac having capitulated.20 My good dear wife & self are come to wish for Peace at any price.
Good Night my good friend. I will scribble no no more— C. D.
One more word. I shd like to hear what you think about what I say in last Ch. of Orchid Book on the meaning & cause of the endless diversity of means for same general purpose.— It bears on design—that endless question—21
Good Night Good Night.
P.S. Last night after writing the above, I read the great bundle of notes.22 Little did I think what I had to read. What admirable observations! You have distanced me on my own hobby-horse! I have not had for weeks such a glow of pleasure as your observations gave me.— Plat. hyperborea is indeed a most curious case & especially interesting to me. How like the Bee ophrys.23 Does it live in arctic regions where insects may be scarce? It would be very good to ascertain whether there actually is any occasional crossing, or removal of pollinia in this species.24 How curious about the nectary. See my note p. 324 about Aceras.25 Aceras, I now find, leads, also, most closely into the rare O. hircina.26 How organic beings are connected! How excellently you have worked Cyp. spectabilis. I daresay I may be altogether wrong, & fertilisation may always be by small insects bodily crawling in: I wish you could get some 2 youths to watch on warm day for 2 or 3 hours a fine plant of some Cypripedium.—27 What diversity in Platanthera— Your observations seem to me much too good to be sunk in any review of my Book; they won’t be noticed.—28 But I am so very sorry I did not return your M.S. earlier: I shall be so grieved if I thus cause you inconvenience; but in truth it was physically impossible for me before last night to read or attend to anything.
Farewell my good Friend | C. Darwin
The year is established by the reference to Leonard Darwin’s illness (see n. 3, below).
Gray had sent a ‘great bundle of notes’ with the letter from Asa Gray, 2–3 July 1862 (see n. 22, below); the envelope to that letter bears a London postmark: ‘JY 18 | 62’.
Leonard Darwin had been sent home from school on 12 June 1862 suffering from scarlet fever; his recovery was interrupted by a recurrence of symptoms in July (see Emma Darwin’s diary (DAR 242), and the letters to W. E. Darwin, 4 [July 1862], 9 July [1862], and [after 14 July 1862]).
Gray had sent a three-cent postage stamp in the letter from Asa Gray, 2–3 July 1862.
Letter from Asa Gray, 2–3 July 1862 (see n. 2, above).
Since starting to read Orchids in May, Gray had sent CD a number of notes on American species of orchids (see letters from Asa Gray, 18 May 1862, [2 June 1862], [late June 1862], and 2–3 July 1862). CD had urged Gray to publish his observations, either in reviewing Orchids, or separately, and had offered to return the notes to Gray to enable him to do so (see letters to Asa Gray, 10–20 June [1862], 1 July [1862], and 14 July [1862]). In the letter from Asa Gray, 2–3 July 1862, Gray asked CD to indicate those observations that seemed to him ‘worth touching on’, and to send back the appropriate notes.
Gray had expressed a wish to examine most of the species of Cypripedium further before publishing on the subject (letter from Asa Gray, 2–3 July 1862). He incorporated an account of American species of Cypripedium and Platanthera in the follow-up article to his review of Orchids (A. Gray 1862b), stating in regard to Cypripedium that it was a subject on which he would ‘hazard a few remarks’ (p. 427).
In Orchids, pp. 274–5, CD had suggested that Cypripedium must be pollinated by an insect inserting its proboscis into one of the two lateral entrances at the base of the labellum, directly over one of the two lateral anthers, and thus either placing the pollen onto the flower’s own stigma, or carrying it away to another flower. In ‘Fertilization of orchids’, pp. 155–6 (Collected papers 2: 152), CD stated: Prof. Asa Gray, after examining several American species of Cypripedium, wrote to me … that he was convinced that I was in error, and that the flowers are fertilized by small insects entering the labellum through the large opening on the upper surface, and crawling out by one of the two small orifices close to either anther and the stigma. Gray detailed his observations in A. Gray 1862b, but did not provide any illustrations, concluding: ‘The beauty of these adaptations can be appreciated only by actual inspection of the parts or of a series of figures.’
On CD’s experiments, see, for instance, the letters to J. D. Hooker, 30 May [1862] and n. 7, and 23 June [1862] and n. 4, the letter to Alphonse de Candolle, 17 June [1862] and n. 2, and the letter to M. T. Masters, 8 July [1862] and n. 3.
Gray’s notes to CD on American species of orchids have not been found in the Darwin Archive–CUL, or in the Gray Herbarium Archives.
‘Rosmack’ is a misspelling; Joseph Trimble Rothrock was a student of Gray’s (see letter from Asa Gray, 2–3 July 1862). Gray forwarded Rothrock’s observations on Houstonia in the letter from Asa Gray, 4 August 1862.
See letter from Asa Gray, 2–3 July 1862, and Correspondence vol. 9, letter from Asa Gray, 11 October 1861.
See letter from Asa Gray, 2–3 July 1862.
Gray advised CD to use thinner writing paper, thereby reducing postage charges, in the letter from Asa Gray, 2–3 July 1862.
See letter from Asa Gray, 2–3 July 1862 and n. 16. The reference is to George Bentham’s presidential address to the anniversary meeting of the Linnean Society of London on 24 May 1862 (Bentham 1862).
CD probably refers to Richard Owen’s lectures at the Museum of Practical Geology on the ‘Characters, Organisation, Geographical Distribution, and Geological Relations of Birds’. The series of six lectures ran from 14 to 30 May 1862 (Athenæum, 10 May 1862, p. 613). Neither the printed accounts of these lectures, nor the manuscript text which survives for some of them, report the details cited by CD (see letter to Armand de Quatrefages, 11 July [1862], n. 7). However, CD apparently learned of the content of the lectures from one of his acquaintances (see letter to Charles Lyell, 22 August [1862] and n. 7).
CD refers to the evolutionary views of George Louis Leclerc, comte de Buffon (see letter from Armand de Quatrefages, [after 11 July 1862], n. 5) and to the anonymous evolutionary work, Vestiges of the natural history of creation ([Chambers] 1844). Owen discussed CD’s theory in common with the views of Buffon and Robert Chambers in R. Owen 1861a, pp. 442–3. See also R. Owen 1862a. In his fifth lecture, on the geographical distribution of birds (Natural History Museum, London, OC38.3/318; see also Medical Times and Gazette (1862), pt 1: 617), Owen stated: we must remember that by the word ‘creation’ we mean “a process we know not what.” We have not yet ascertained the secondary causes which operated when “the earth brought forth grass & herb yielding seed after its kind” and when “the waters brought forth abundantly the moving creature that hath life.” And if the ‘spontaneous generation’ of a fruit-bearing tree, or of a fish, were conceivable, & the whole process demonstrable, we should still retain as strongly the idea which is the chief of the ‘mode’ or group of ideas called ‘creation’; viz., that the process was ordained by and had originated from an … all-wise First Cause of all things. On Owen’s views concerning evolution, see Rupke 1994, pp. 220–58.
The reference is to the failed attempt made by the Union army of the Potomac in June 1862 to seize the Confederate capital, Richmond, Virginia. Between 25 June and 1 July 1862 the Confederates drove the Union forces away from Richmond; the ensuing battles, resulting in 30,000 casualties, were the bloodiest yet seen in the conflict (McPherson 1988, pp. 461–71).
In chapter 7 of Orchids, pp. 348–9, CD stated: In my examination of Orchids, hardly any fact has so much struck me as the endless diversity of structure,—the prodigality of resources,—for gaining the very same end, namely, the fertilisation of one flower by the pollen of another. The fact to a certain extent is intelligible on the principle of natural selection. Gray and CD had corresponded at length on the question of design in nature (see Correspondence vols. 8 and 9). For Gray’s response to CD’s question, see the letter from Asa Gray, 22 September 1862 and n. 11.
CD refers to the notes on orchids sent with the letter from Asa Gray, 2–3 July 1862 (see n. 2, above, and the letter from Asa Gray, 15 July [1862] and n. 6); the notes have not been found.
Gray’s notes on this subject have not been found; however, in the portion of A. Gray 1862c that was published in the September number of the American Journal of Science and Arts, Gray disputed Joseph Dalton Hooker’s claim that Platanthera hyperborea and P. dilatata constituted a single species, reporting that he had recently observed that ‘while P. dilatata … can rarely if ever self-fertilise, P. hyperborea readily does so, much in the manner of Ophrys apifera as recently illustrated by Darwin’ (p. 259). (Gray gave further details of pollination in P. hyperborea in the follow-up article to his review of Orchids (A. Gray 1862b, p. 426).) In Orchids, pp. 72–3, CD similarly distinguished Ophrys arachnites as a species separate from O. apifera (the bee ophrys) on the basis that the former was adapted for cross-fertilisation, while the latter was adapted for self-fertilisation. One of CD’s objectives in Orchids was to demonstrate that cross-fertilisation was the ‘main object’ of the contrivances by which orchids were pollinated (p. 1), and in the conclusion, he noted that the bee ophrys was anomalous in being adapted for self-fertilisation (p. 359). He had subsequently been speculating about possible explanations for this apparent anomaly (see letters from G. C. Oxenden, [before 30 May 1862], 21 June 1862, and 8 July 1862, and letter to A. G. More, 7 June [1862]).
In A. Gray 1862b, p. 426, Gray described how it would be possible for an insect to cross-pollinate P. hyperborea, continuing: If the rule holds here as elsewhere, that a stigma is more sensitive to the pollen of another flower than to that of its own, there will be no lack of sufficient crossing in this species, wherever proper insects abound; where they do not, it will be prolific without them.
Gray did not mention the nectary of P. hyperborea in either of his published accounts (A. Gray 1862b and 1862c). CD’s account of the monstrous flowers of Aceras in Orchids, p. 324 n., referred to the pollinia not having viscid discs, and to the two anther cells being widely separated; this resembles Gray’s statement that P. hyperborea had smaller viscid discs than P. dilatata, with the anther-cells more divergent, and the stalks of the pollinia being ‘very attenuated and weak’ (A. Gray 1862c, p. 260).
CD had recently received specimens of Orchis hircina from George Chichester Oxenden (see letter from G. C. Oxenden, 4 June [1862] and n. 2). See also Orchids 2d ed., pp. 25–6.
See n. 8, above. In his account of insect pollination in Cypripedium (A. Gray 1862b, p. 428), Gray stated that CD’s theory might account for fertilisation in the genus, ‘but hardly in C. spectabile’. He went on to note that the ‘rigid, sharp-pointed papillæ, all directed forwards’, that were particularly striking on the stigma of C. spectabile, offered ‘no slight confirmation’ of his own hypothesis, since they would act like a ‘wool-card’ in removing pollen from any insect ‘working its way upwards to the base of the labellum’. However, Gray noted with respect to his hypothesis that he had ‘not been able to detect insects actually at work’.
Gray’s observations on American species of Platanthera and Cypripedium were incorporated into the follow-up article to his review of Orchids (A. Gray 1862b). |
(Created page with "We consider the 490 APN functions in dimension 7 constructed by the matrix method <ref name="yu">Yu, Yuyin, Mingsheng Wang, and Yongqiang Li. "A Matrix Approach for Constructi...")
<th><math>\Gamma</math>-rank</th>
<th><math>\Gamma-rank</math></th>
<th>Indices</th>
{\displaystyle \Gamma }
{\displaystyle \Gamma }
{\displaystyle \Gamma -rank} |
Solve the following quadratic equations by factoring and using the Zero Product Property. Be sure to check your solutions.
x^2-13x+42=0
Draw a generic rectangle and diamond. Draw a rectangle and divide it into four equal parts. Blank Diamond Problem.
Factor. Added to the rectangle: Interior upper right box is negative 42. Interior lower left box is X squared. Added to the diamond problem: Left blank, Right blank, Top 42, x squared, Bottom negative 13, x.
(x−6)(x−7)=0
Added to the rectangle: Interior upper left box is negative 6, x. Interior lower right box is negative 7, X. Exterior left edge top is negative 6. Exterior left edge bottom is X. Exterior bottom edge left is X. Exterior bottom edge right is negative 7. Added to the diamond problem: Left negative 6, x, Right negative 7, x.
x=6
Make sure you find the other solution. Added to the diamond problem: Left negative 6, x, Right negative 7, x.
0=3x^2+10x-8
0=\left(3x-2\right)\left(?\ +\ ?\right)
2x^2-10x=0
There are only
2
terms, so factor using the GCF.
4x^2+8x-60=0
4
x=3
x=-5 |
Event_(probability_theory) Knowpia
In probability theory, an event is a set of outcomes of an experiment (a subset of the sample space) to which a probability is assigned.[1] A single outcome may be an element of many different events,[2] and different events in an experiment are usually not equally likely, since they may include very different groups of outcomes.[3] An event consisting of only a single outcome is called an elementary event or an atomic event; that is, it is a singleton set. An event
{\displaystyle S}
is said to occur if
{\displaystyle S}
contains the outcome
{\displaystyle x}
of the experiment (or trial) (that is, if
{\displaystyle x\in S}
). The probability (with respect to some probability measure) that an event
{\displaystyle S}
occurs is the probability that
{\displaystyle S}
{\displaystyle x}
of an experiment (that is, it is the probability that
{\displaystyle x\in S}
). An event defines a complementary event, namely the complementary set (the event not occurring), and together these define a Bernoulli trial: did the event occur or not?
Typically, when the sample space is finite, any subset of the sample space is an event (that is, all elements of the power set of the sample space are defined as events). However, this approach does not work well in cases where the sample space is uncountably infinite. So, when defining a probability space it is possible, and often necessary, to exclude certain subsets of the sample space from being events (see Events in probability spaces, below).
An Euler diagram of an event.
{\displaystyle B}
{\displaystyle A}
By the ratio of their areas, the probability of
{\displaystyle A}
is approximately 0.4.
Since all events are sets, they are usually written as sets (for example, {1, 2, 3}), and represented graphically using Venn diagrams. In the situation where each outcome in the sample space Ω is equally likely, the probability
{\displaystyle P}
{\displaystyle A}
is the following formula:
{\displaystyle \mathrm {P} (A)={\frac {|A|}{|\Omega |}}\,\ \left({\text{alternatively:}}\ \Pr(A)={\frac {|A|}{|\Omega |}}\right)}
Events in probability spacesEdit
Defining all subsets of the sample space as events works well when there are only finitely many outcomes, but gives rise to problems when the sample space is infinite. For many standard probability distributions, such as the normal distribution, the sample space is the set of real numbers or some subset of the real numbers. Attempts to define probabilities for all subsets of the real numbers run into difficulties when one considers 'badly behaved' sets, such as those that are nonmeasurable. Hence, it is necessary to restrict attention to a more limited family of subsets. For the standard tools of probability theory, such as joint and conditional probabilities, to work, it is necessary to use a σ-algebra, that is, a family closed under complementation and countable unions of its members. The most natural choice of σ-algebra is the Borel measurable set derived from unions and intersections of intervals. However, the larger class of Lebesgue measurable sets proves more useful in practice.
In the general measure-theoretic description of probability spaces, an event may be defined as an element of a selected 𝜎-algebra of subsets of the sample space. Under this definition, any subset of the sample space that is not an element of the 𝜎-algebra is not an event, and does not have a probability. With a reasonable specification of the probability space, however, all events of interest are elements of the 𝜎-algebra.
A note on notationEdit
Even though events are subsets of some sample space
{\displaystyle \Omega ,}
they are often written as predicates or indicators involving random variables. For example, if
{\displaystyle X}
is a real-valued random variable defined on the sample space
{\displaystyle \Omega ,}
{\displaystyle \{\omega \in \Omega \mid u<X(\omega )\leq v\}\,}
{\displaystyle u<X\leq v\,.}
{\displaystyle \Pr(u<X\leq v)=F(v)-F(u)\,.}
{\displaystyle u<X\leq v}
is an example of an inverse image under the mapping
{\displaystyle X}
{\displaystyle \omega \in X^{-1}((u,v])}
{\displaystyle u<X(\omega )\leq v.}
Complementary event – Opposite of a probability event
^ Leon-Garcia, Alberto (2008). Probability, statistics and random processes for electrical engineering. Upper Saddle River, NJ: Pearson.
^ Pfeiffer, Paul E. (1978). Concepts of probability theory. Dover Publications. p. 18. ISBN 978-0-486-63677-1.
^ Foerster, Paul A. (2006). Algebra and trigonometry: Functions and applications, Teacher's edition (Classics ed.). Upper Saddle River, NJ: Prentice Hall. p. 634. ISBN 0-13-165711-9.
Wikimedia Commons has media related to Event (probability theory).
"Random event", Encyclopedia of Mathematics, EMS Press, 2001 [1994] |
Inequalities - Maple Help
Home : Support : Online Help : Education : Grading : Inequalities
Inequalities(ineq)
Inequalities(curves, str, pt)
list of inequalities in variables x and y
list of LinearFunction objects
list of strings "strict" or "nonstrict", indicating type of inequality
a GridPoint object or rtable/list representing a point in the feasible region
The Inequalities constructor generates and returns an object representing a set of inequalities. Currently, only linear inequalities are supported.
The first calling sequence requires a list of inequalities in the variables x and y to be provided.
The second calling sequence allows a feasible region to be defined indirectly, through a list of curves. The str parameter indicates whether the inequality associated with each curve is strict or not; this list must have the same number of elements as curves.
The pt parameter can be any point in the feasible region. If the feasible region is empty, then an empty list should be given as pt.
\mathrm{with}\left(\mathrm{Grading}\right)
[\textcolor[rgb]{0,0,1}{\mathrm{AbsoluteValueFunction}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Draw}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{ExponentialFunction}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{GetData}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{GetDomain}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{GetExpression}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{GradePlot}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{GridPoint}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Inequalities}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{IsQuadraticFormula}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{LinearFunction}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{LogarithmicFunction}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{QuadraticFunction}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Quiz}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Segment}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{SolveFeedback}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{SolvePractice}}]
\mathrm{I1}≔\mathrm{Inequalities}\left([x+y\le 1,2<2x-y]\right)
\textcolor[rgb]{0,0,1}{\mathrm{I1}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{<< Inequalities: \left[x+y <= 1, 2 < 2*x-y\right]>>}}
\mathrm{I2}≔\mathrm{Inequalities}\left([\mathrm{LinearFunction}\left([0,1],[-2,-2]\right),\mathrm{LinearFunction}\left([1,0],[-2,0]\right)],["nonstrict","strict"],[-2,1]\right)
\textcolor[rgb]{0,0,1}{\mathrm{I2}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{<< Inequalities: \left[3/2*x+1 <= y, 0 < y\right]>>}}
The Grading:-Inequalities command was introduced in Maple 18. |
statplots(deprecated)/histogram - Maple Help
Home : Support : Online Help : statplots(deprecated)/histogram
stats[statplots, histogram]
Histograms of statistical data
stats[statplots, histogram](data,
\mathrm{arg}=\mathrm{value}
statplots[histogram](data,
\mathrm{arg}=\mathrm{value}
histogram(data,
\mathrm{arg}=\mathrm{value}
statistical list(s)
area=x
where x is a number or a descriptive statistics item
numbars=y
where y is an integer greater than zero
The function histogram of the subpackage stats[statplots] plots a bar graph for one or more statistical lists.
The command histogram(data1) will tally the data in data1 into bars of equal area. Classes (ranges) contained in data1 are plotted as boxes proportional to their weight.
When more than one statistical list is specified, as in histogram(data1, data2, ...), a 3-D plot is created. One histogram is created for each data set, and they are plotted on the same graph. The data1 plot is closest to the viewer, and each subsequent data set is behind it. The i^th data set, is plotted along the plane z=i, with the z-axis pointed away from the viewer.
By default, a probability histogram will be produced: the total area of the bars will be 1. This can be changed by using the area parameter; area=10 will give a histogram with the bars having a total area of 10. To make the total area of the bars equal to the total weight of the data, use area=count.
The parameter numbars= allows the user to specify how many divisions the data should be broken up into. If the data is spread uniformly, then numbars=y should produce a histogram with y columns. Less uniform data may result in empty columns. When this parameter is not specified, a representative default value is chosen.
By default, bars will be of equal area and not necessarily equal width. When using the area=... parameter, the default changes so that the bars are always of equal width.
The function stats[transform,split] can be used to generate histograms in which each bar has the same area.
Another choice is to use classes that have the same width. The function stats[transform, tallyinto] can be used to collect point data into class data. It is usually a good idea to group data into between 5 and 20 classes. The boundaries of the classes should not include actual data points.
One disadvantage of using histograms for the display of data is that their appearance can change drastically depending on the positions of the boundaries of the classes. This is somewhat lessened if the class marks (mid-points of the classes) are chosen to be actual data items. For example, the class 1..3 would be acceptable if the data contains the item 2, but would be rejected if the data does not contain the item 2.
Missing data are ignored.
\mathrm{with}\left(\mathrm{stats}\right):
\mathrm{with}\left(\mathrm{stats}[\mathrm{statplots}]\right):
\mathrm{data1}≔[\mathrm{Weight}\left(1..3,5\right),\mathrm{Weight}\left(3..5,10\right),\mathrm{Weight}\left(5..7,8\right)]:
\mathrm{histogram}\left(\mathrm{data1},\mathrm{color}=\mathrm{cyan}\right)
Some randomly generated numbers:
\mathrm{histogram}\left([\mathrm{random}[\mathrm{normald}]\left(250\right)]\right)
Here is a more advanced example. Data2 was generated via a standard normal random number generator. We want to plot a histogram of this and also have the theoretical distribution.
\mathrm{data2}≔[-1.96,-0.814,1.86,1.96,0.519,0.739,-0.0540,0.702,0.663,0.591,0.580,0.475,0.589,-1.33,0.0420,-0.460,-0.482,1.58,0.778,0.530,-0.507,-0.233,-0.195,0.193,-0.136]:
\mathrm{histogram}\left(\mathrm{data2},\mathrm{area}=\mathrm{count}\right)
default: probability histogram, equal area regions
p≔\mathrm{histogram}\left(\mathrm{data2},\mathrm{color}=\mathrm{yellow}\right):
q≔\mathrm{plot}\left(\mathrm{stats}[\mathrm{statevalf},\mathrm{pdf},\mathrm{normald}],-3..3,\mathrm{color}=\mathrm{red}\right):
\mathrm{plots}[\mathrm{display}]\left({p,q}\right)
probability histogram, with equal width regions, and overlay theoretical
p≔\mathrm{histogram}\left(\mathrm{data2},\mathrm{color}=\mathrm{gray},\mathrm{area}=1\right):
q≔\mathrm{plot}\left(\mathrm{stats}[\mathrm{statevalf},\mathrm{pdf},\mathrm{normald}],-3..3,\mathrm{color}=\mathrm{red}\right):
\mathrm{plots}[\mathrm{display}]\left({p,q}\right)
\mathrm{data2classified}≔\mathrm{stats}[\mathrm{transform},\mathrm{scaleweight}[\frac{1}{\mathrm{nops}\left(\mathrm{data2}\right)}]]\left(\mathrm{stats}[\mathrm{transform},\mathrm{tallyinto}]\left(\mathrm{data2},[-3..-1,-1..1,1..3]\right)\right):
\mathrm{plots}[\mathrm{display}]\left({\mathrm{plot}\left(\mathrm{stats}[\mathrm{statevalf},\mathrm{pdf},\mathrm{normald}],-3..3,\mathrm{color}=\mathrm{red}\right),\mathrm{statplots}[\mathrm{histogram}]\left(\mathrm{data2classified}\right)}\right)
describe(deprecated)[median]
describe(deprecated)[quartile] |
Group delay response of discrete-time filter - MATLAB grpdelay - MathWorks Benelux
Compute Group Delay of RRC Filter
Group delay response of discrete-time filter
[gd,w] = grpdelay(rcfilter)
[gd,w] = grpdelay(rcfilter,n)
[gd,w] = grpdelay(___,'Arithmetic',arithType)
grpdelay(rcfilter)
[gd,w] = grpdelay(rcfilter) returns gd, the group delay of the specified filter based on the filter coefficients. The output w contains the frequencies (in radians per sample) at which the function evaluates the group delay. The group delay is defined as
-\frac{d}{dw}\left(angle\left(w\right)\right)
[gd,w] = grpdelay(rcfilter,n) returns the group delay of the specified filter and the corresponding frequencies at n points that are equally spaced around the upper-half of the unit circle (from 0 to π).
[gd,w] = grpdelay(___,'Arithmetic',arithType) computes the group delay of the filter System object™, specifies the type of arithmetic that the function uses to compute the group delay. You can use any input combination from the previous syntaxes.
grpdelay(rcfilter) plots the group delay of the specified filter by using the fvtool object function.
For more input options, see the grpdelay function.
Compute the group delay of an RRC filter.
rcfilter = comm.RaisedCosineTransmitFilter;
gd = grpdelay(rcfilter,32);
gd(1:5)'
rcfilter — Filter
Filter, specified as one of these System objects.
n — Number of points over which the group delay is computed
Number of points over which the group delay is computed, specified as a positive integer. For faster computations (performed using FFTs) specify n as a power of two.
Group delay, returned as a column vector of length n.
w — Frequencies used for group delay evaluation
Frequencies in radians/sample used for group delay evaluation, returned as a column vector of length n. Unit are in radians per sample. The frequencies are equally spaced around the upper-half of the unit circle (from 0 to π).
For faster computations (performed using FFTs), specify n, the number of points over which the function computes the group delay, as a power of two.
info | coeffs | order | freqz | fvtool | filter | impz | grpdelay |
Emission Schedule - THORChain Docs
Describes the Emission Schedule from the Reserve to Nodes and Liquidity providers.
There are a maximum of 500m RUNE. All 100% was created at genesis and distributed via different mechanisms:
In return for capital: 5% (SEED) and 16% (IDO) was sold for capital to start the network and give it value. They took on risk to support the network.
In return for time and effort: 10% was allocated to a group of devs who worked since 2018. They took on risk to deliver the network.
In return for bootstrap participation: 24% was given to users who participated in the bootstrapping of the network.
In return for through-life participation: 44% has been placed in the Protocol to pay out to Nodes and LPs for the next 10+ years
The Reserve also backstops Impermanent Loss Protection, and is used to underwrite debt. The Reserve is depleted by block rewards and continually topped up by system income.
Block rewards are calculated as such:
blockReward = \frac{ \frac{reserve}{emissionCurve}}{blocksPerYear} = \frac{ \frac{180,000,000}{8}}{5256000} = 4.28
So if the reserve has 180m rune, a single block will emit ~4.28 Rune from the reserve, which means 2/3rds of that is awarded to the node operators. The rest is paid to Liquidity providers.
The emission curve is designed to start at around 30% APR and target 2% after 10 years. At that point, the majority of the revenue will come from fees.
System Income
The Reserve is continually topped up by income, such as transfer fees and outbound fees. Other sources of revenue include THORName fees and excess liquidation fees on collateral. |
\begin{array}{l} y \ge \frac{3}{4}x - 2 \\ y < -\frac{1}{2}x + 3 \end{array}
For both lines, test a point by inputting the coordinates into the equation. If the point makes the equation true, shade the side of the line containing that point. If the point makes the equation false, shade the opposite side. The answer to the system of inequalities is the region of overlap.
Use the eTool below to complete the graph for the problem.
Click the link at right for the full eTool version: 9-106 HW eTool (Desmos) |
Allowed Trading Bands - Delta Exchange - User Guide
To prevent market manipulation as well as idiosyncratic price moves trading on Delta Exchange can take place only within the allowed trading band. Generally, speaking the allowed trading band is defined around the prevailing Mark Price. However, the exact methodology for creating the trading band differs from different product types.
Please do note that trading bands are not applicable to liquidation orders.
Futures & Perpetual Contracts
Band1: Price Volatility Band
Band1 is defined as a +/-2 standard deviation band around the current Mark Price. The standard deviation of the Mark Price in the last 15 minutes.
UpperBand1=Mark\ Price + 2 * Standard\ Deviation (Mark\ Price)
LowerBand1 = Mark\ Price - 2 * Standard\ Deviation (Mark\ Price)
Thus, Band1 expands when market is volatile and contracts when there’s little volatility in the market.
Band2: Price Range Band
Band2 is defined around the current Mark Price in terms of percentage of Mark Price.
UpperBand2=Mark\ Price + Mark\ Price * Range/ 100
LowerBand2 = Mark\ Price - Mark\ Price * Range/ 100
where value of Range is provided as Price Band in a contract’s specification.
The allowed trading band for futures and perpetual contracts is computed by combining Band1 and Band2 in such a way that on either side the wider of the two bands is selected.
UpperBand = max (UpperBand1, UpperBand2)
LowerBand = min (LowerBand1, LowerBand2)
Calendar Spread Contracts
The allowed trading band for calendar spread contracts is created by combining two different bands which are described below:
UpperBand1=Mark\ Price + 2 * Standard\ Deviation (Mark\ Price)
LowerBand1 = Mark\ Price - 2 * Standard\ Deviation (Mark\ Price)
Band2 is defined around the current Mark Price in terms of percentage of Spot Price.
UpperBand2=Mark\ Price + Spot\ Price * Range/100
LowerBand2 = Mark\ Price - Spot\ Price * Range/100
Band1 and Band2 are combined in such a way that on either side the wider of the two bands is selected.
UpperBand = max (UpperBand1, UpperBand2)
LowerBand = min (LowerBand1, LowerBand2)
The allowed trading band for options (call, put and MOVE) contracts is created by combining three different bands which are described below:
UpperBand1=Mark\ Price + 2 * Standard\ Deviation (Mark\ Price)
LowerBand1 = Mark\ Price - 2 * Standard\ Deviation (Mark\ Price)
Band2: Implied Volatility Band
This band is created by computing the theoretical prices of the options contract at mid implied volatility +/- IV Range. The mid implied volatility is average of implied volatility for impact bids and impact offers.
UpperBand2 = Black\ Scholes\ Price (Mid\ Implied\ Volatility + IV\ Range, Spot\ Price, Time\ to\ Expiry)
LowerBand2 = Black\ Scholes\ Price (Mid\ Implied\ Volatility - IV\ Range, Spot\ Price, Time\ to\ Expiry)
UpperBand3=Mark\ Price + Spot\ Price * Range/100
LowerBand3 = Mark\ Price - Spot\ Price * Range/100
The allowed trading band for options is computed by combining Band1, Band2 and Band3 in such a way that on either side the wider of the two bands is selected.
UpperBand = max (UpperBand1, UpperBand2, UpperBand3)
LowerBand = min (LowerBand1, LowerBand2, LowerBand3)
The allowed trading band for interest rate swap (IRS) contracts is created by combining three different bands which are described below:
Band1: Rate Volatility Band
Band1 is defined as a +/-2 standard deviation band around the current Mark Rate. The standard deviation of the Mark Rate in the last 15 minutes.
UpperBand1=Mark\ Rate + 2 * Standard\ Deviation (Mark\ Rate)
LowerBand1 = Mark\ Rate - 2 * Standard\ Deviation (Mark\ Rate)
Band2: Rate Range Band
Band3 is defined around the current Mark Rate in terms of percentage of maximum (Vmax) and minimum (Vmin) values that the interest rate over which the swap contract is defined is allowed to take.
UpperBand2=Mark\ Price + max (Vmax, abs(Vmin)) * Range/100
LowerBand2=Mark\ Price - max (Vmax, abs(Vmin)) * Range/100
Band3: Rate Limit Band
Since the interest rate over which the IRS contract is defined is bounded between (
Vmin, Vmax
), no trading should happen outside this band. Therefore,
UpperBand3 = Vmax
LowerBand3=Vmin
Vmax
Vmin
are part of the contract’s specification.
The allowed trading band for IRS contracts is derived by combining Band1, Band2 and Band3 as per the following equation:
UpperBand = min (max (UpperBand1, UpperBand2), UpperBand3)
LowerBand = max (min (LowerBand1, LowerBand2), LowerBand3)
Impact of the Trading Band on various types of orders
Market orders: Any part of your market order that will breach the trading band on execution will be converted into a limit order with price equal to the max (for buy orders) or min (for sell orders) allowed price.
Limit orders: If the price of your buy limit order exceeds the Max Price, the price of the order will be automatically set to the max buy price. Conversely, if the price of your sell limit order is below the Min Price, the price of the order will be automatically set to the min sell price.
Stop market orders: Once triggered, a stop market order will behave in the exact same manner as a market order. This means that part or all of the order could be converted into a limit order if its execution will breach the trading band in force at the time the stop market order is triggered.
Stop limit orders: Once triggered, a stop limit order will behave in the exact same manner as a limit order. This means that the limit price of the order may be changed in accordance with the trading band in force at the time the stop limit order is triggered.
Bracket orders: Bracket orders are akin to two stop market orders (a take-profit order and a stoploss order) wherein triggering of one order cancels the another. Once one of the orders in a bracket order is triggered, it behaves in the exact same manner as a stop market order.
Changing behaviour of market orders using order validity attributes
Order validity attributes are also known as time in force flags. The default time in force flag for market orders as well as stop market orders is GTC (Good till cancelled). Traders using the website do not have the option to changing this flag. However, if you are trading using APIs, then you have the option to place IOC (immediate or cancel) market order and stop market orders. When you use an IOC flag, any part of the market (or stop market) order that will breach the trading band will be cancelled. |
CAPM Calculator | Dash Calculator
This CAPM calculator will tell you the expected return on asset, risk premium, and market risk premium given a risk-free rate, expected return on the market, and beta coefficient.
What is the expected return on the market?
What is the beta of the asset?
Asset moves in same direction as market.
The expected return on the asset is 6.90%.
6.90% Expected return on the asset
4.40% Risk premium of the asset
8.00% Expected return of the market
5.50% Risk premium of the market
Let's calculate the expected return on the asset using the CAPM formula. Continue reading below to learn more about the formula.
E(r_i) = r_f + \beta_i \times (E(r_m)-r_f)
E(r_i) = 2.5 + 0.8 \times(8-2.5)
E(r_i) = 0.07
E(r_i) = 6.90 \%
The risk-free rate is usually the rate of a government bond, such as a 30-year Treasury Bill.
This is usually the historical return of a market benchmarket such as the S&P 500.
Time period of return
S&P 500 7.96% 1957 to 2018
Dow Jones Industrial Average 5.42% 1896 to 2018
Russell 2000 7.70% 1999 to 2019
MSCI EAFE 4.00% 1999 to 2019
Beta is the level of the asset return's sensitivity compared to the market. For example:
Beta <= −1 Asset moves in opposite direction as the market. Movement is greater than market.
−1 < Beta < 0 Asset moves in opposite direction as the market.
Beta = 0 No correlation between asset and market.
0 < Beta < 1 Asset moves in same direction as market.
Beta = 1 Asset and market are perfectly correlated. Both both in the same direction by the same amount.
Beta > 1 Asset moves in same direction as market. Movement is greater than market.
The capital asset pricing model, or CAPM, calculates the exposure of an asset to market risk. The model was developed in the mid-1960s by William Sharpe, John Lintner, and Jan Mossin.
CAPM is a model that describes the relationship between the risk premium of an individual asset and the risk premium of the market. The risk premium of an individual asset is equal to the risk premium of the market, adjusted by the beta of the asset.
The purpose of CAPM is to understand whether an asset is fairly priced relative to its beta and the market premium.
is the expected return on asset
is the risk-free rate
is the beta coefficient
is the expected return on the market
CAPM can also be represented as:
is considered to be the risk premium of the asset
is considered to be the market premium
Let’s go through each of the inputs into the CAPM equation.
Expected return on asset
The expected return on the asset is what CAPM calculates. This is what an investor expects to earn on the asset over time.
The risk-free rate is the rate you would receive on an asset that has no risk. This is typically the yield on a long-term government bond, where the asset is based. For a U.S. stock, this could be the yield on the 10-year Treasury Bill.
The expected return on the market is the return of a market benchmark, such as the S&P 500, Russell 2000, Dow Jones Industrial Average, or another benchmark that encompasses most of the market.
Investors generally use the historical rate of return for the S&P 500, which was 8% between 1957 and 2018.
An asset’s beta measures the risk involved with investing in the asset relative to the market risk and the risk-free rate.
Beta reflects the sensitivity of the asset to the market risk. A beta of 1 signifies that the asset has the same risk as the market. When the market goes up a little, the asset goes up a little. When the market goes down a lot, the asset goes down a lot. The two are perfectly correlated.
A beta of 0 means the asset and the market are not at all correlated. The two move independently of each other.
A positive beta means the asset and the market move in the same direction, while a negative beta means the two move in opposite directions.
The risk premium of the asset is the difference between its expected return and the risk-free rate.
The market premium is the difference between the expected return of the market and the risk-free rate.
Introduction to Finance: Markets, Investments, and Financial Management by Ronald W. Melicher and Edgar A. Norton
Financial Algebra: Advanced Algebra with Financial Applications by Robert Gerver and Richard J. Sgroi
"The Capital Asset Pricing Model: Some Empirical Tests" by Michael C. Jensen, Fischer Black, and Myron S. Scholes
"A New Look at the Capital Asset Pricing Model" by Marshall E. Blume and Irwin Friend |
fit(deprecated)/leastsqu - Maple Help
Home : Support : Online Help : fit(deprecated)/leastsqu
stats[fit, leastsquare]
Fit a curve to data using the least square method
stats[fit, leastsquare[vars, eqn, parms]]( data)
fit[leastsquare[vars, eqn, parms]]( data)
list of statistical lists
list of variables, corresponding, in order, to the lists in data
The equation to fit (optional, default = a linear equation with the last variable in vars as dependent variable and with a constant term).
Set of parameters that will be replaced (optional, default = indets(eqn) minus op(vars)).
The function leastsquare of the subpackage stats[fit, ...] fits a curve to the given data using the method of least squares.
The equation to fit must be linear in the unknown parameters. The equation itself need not be linear. For example,
y=a{x}^{2}+bx+c
with the parameters a, b, c is accepted. Note that some equations which have their parameters appearing nonlinearly can be transformed to linear ones.
y=a{ⅇ}^{bx}
\mathrm{ln}\left(y\right)=A+bx,
A=\mathrm{log}\left(a\right)
. The leastsquares command does not apply this transformation automatically, but the Statistics[ExponentialFit] command can be used instead.
Missing data and ranges cannot be handled.
Weighted data are handled in the following fashion. The weight associated with the dependent variable is the weight given to the corresponding point. The weight specifications corresponding to the independent variables are ignored.
Data fitting routines are also available in the Statistics package. For more information, see the Statistics[Regression] help page.
The command with(stats[fit],leastsquare) allows the use of the abbreviated form of this command.
\mathrm{with}\left(\mathrm{stats}\right):
\mathrm{fit}[\mathrm{leastsquare}[[x,y,z]]]\left([[1,2,3,5],[2,4,6,8],[3,5,7,10]]\right)
\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{y}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}
Here's an example using Weight
\mathrm{fit}[\mathrm{leastsquare}[[x,y,z]]]\left([[1,2,3,5,5,5],[2,4,6,8,8,8],[3,5,7,10,15,15]]\right)
\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{=}\frac{\textcolor[rgb]{0,0,1}{13}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}}{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{7}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}}{\textcolor[rgb]{0,0,1}{6}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}
\mathrm{fit}[\mathrm{leastsquare}[[x,y,z]]]\left([[1,2,3,5,5],[2,4,6,8,8],[3,5,7,10,\mathrm{Weight}\left(15,2\right)]]\right)
\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{=}\frac{\textcolor[rgb]{0,0,1}{13}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}}{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{7}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}}{\textcolor[rgb]{0,0,1}{6}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}
A fit to a quadratic:
\mathrm{Xvalues}≔[1,2,3,4]
\textcolor[rgb]{0,0,1}{\mathrm{Xvalues}}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}]
\mathrm{Yvalues}≔[0,6,14,24]
\textcolor[rgb]{0,0,1}{\mathrm{Yvalues}}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{14}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{24}]
\mathrm{eq_fit}≔\mathrm{fit}[\mathrm{leastsquare}[[x,y],y=a{x}^{2}+bx+c,{a,b,c}]]\left([\mathrm{Xvalues},\mathrm{Yvalues}]\right)
\textcolor[rgb]{0,0,1}{\mathrm{eq_fit}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{=}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{4}
The {a,b,c} parameter is optional in this case, since there are no extra Maple variables in the equation (compare with y=a*x^2+b*x+c+Pi*x^3, where Pi is definitely not a parameter.)
\mathrm{eq_fit}≔\mathrm{fit}[\mathrm{leastsquare}[[x,y],y=a{x}^{2}+bx+c]]\left([\mathrm{Xvalues},\mathrm{Yvalues}]\right)
\textcolor[rgb]{0,0,1}{\mathrm{eq_fit}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{=}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{4}
Transform this into a procedure
\mathrm{eq_function}≔\mathrm{unapply}\left(\mathrm{rhs}\left(\mathrm{eq_fit}\right),x\right)
\textcolor[rgb]{0,0,1}{\mathrm{eq_function}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{↦}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{4}
Then give the predicted values (we could have used map() in this case, since the data does not involve classes or weights)
\mathrm{Yvalues_predicted}≔\mathrm{transform}[\mathrm{apply}[\mathrm{eq_function}]]\left(\mathrm{Xvalues}\right)
\textcolor[rgb]{0,0,1}{\mathrm{Yvalues_predicted}}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{14}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{24}]
Find the residuals:
\mathrm{Residuals}≔\mathrm{transform}[\mathrm{multiapply}[\left(x,y\right)↦x-y]]\left([\mathrm{Yvalues},\mathrm{Yvalues_predicted}]\right)
\textcolor[rgb]{0,0,1}{\mathrm{Residuals}}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}]
The residuals are all zero since all the points fall on the quadratic.
describe(deprecated)[linearcorrelation]
Statistics[Regression] |
Existence of Solutions for some Quasiaffine PDEs | EMS Press
Existence of Solutions for some Quasiaffine PDEs
Giovanni Pisante
Università degli Studi Federico II, Caserta, Italy
We prove the existence of solutions of problems of the type
\left\{ \begin{aligned} \Phi(Du(x)) &= f(x) &\quad &\text{\rm in } \, \Omega\\ u(x) &=\xi_0 x &\quad &\text{\rm on } \, \partial \Omega \end{aligned} \right.
\Phi:\mathbb{R}^{n\times n} \to \mathbb{R}
quasiaffine function and
\xi_0\in \mathbb{R}^{n\times n}
Flavia Giannetti, Giovanni Pisante, Existence of Solutions for some Quasiaffine PDEs. Z. Anal. Anwend. 28 (2009), no. 1, pp. 47–56 |
Chapter 2 Solutions – Statistical Methods in Bioinformatics | R-bloggers
Chapter 2 Solutions – Statistical Methods in Bioinformatics
As I have mentioned previously, I have begun reading Statistical Methods in Bioinformatics by Ewens and Grant and working selected problems for each chapter. In this post, I will give my solution to two problems. The first problem is pretty straightforward.
Suppose that a parent of genetic type Mm has three children. Then the parent transmits the M gene to each child with probability 1/2, and the genes that are transmitted to each of the three children are independent. Let
I_1 = 1
if children 1 and 2 had the same gene transmitted, and
I_1 = 0
otherwise. Similarly, let
I_2 = 1
if children 1 and 3 had the same gene transmitted,
I_2 = 0
otherwhise, and let
I_3 = 1
I_3 = 0
The question first asks us to how that the three random variables are pairwise independent but not independent. The pairwise independence comes directly from the bolded phrase in the problem statement. Now, to show that the three random variables are not independent, denote by
p_j
I_j = 1
j = 1, 2, 3
. If we had independence, then the following statement would be true:
However, notice that the event in the lefthand side can never happen because if
I_1 = 1
I_2 = 1
I_3
must be 1. Hence, the lefthand side must equal 0, while the righthand side equals 1/8. Therefore, the three random variables are not independent.
The question also asks us to discuss why the variance of
I_1 + I_2 + I_3
is equal to the sum of the individual variances. Often, this is only the case of the random variables are independent. But because the random variables here are pairwise independent, the covariances must be 0. Thus, the equality must hold.
Problems 2.23 – 2.27
While I worked the above problem because of its emphasis on genetics, the following set of problems is much more fun in terms of the mathematics because of its usage of approximations.
i = 1, \ldots, n
X_i
i
th lifetime of certain cellular proteins until degradation. We assume that
X_1, \ldots, X_n
are iid random variables, each of which is exponentially distributed with rate parameter
\lambda > 0
n = 2m + 1
be an odd integer.
This set of questions is concerned with the mean and variance of the sample median,
X_{(m + 1)}
X_{(i)}
i
th order statistic. First, note that the mean and variance of the minimum value
X_{(1)}
1/(n\lambda)
1/(n\lambda)^2
, respectively. From the memoryless property of the exponential distribution, the mean value of the time until the next protein degrades is independent of the previous. However, there are now
n - 1
proteins remaining. Thus, the mean and variance of
X_{(2)}
1/(n\lambda) + 1/((n-1)\lambda)
1/(n\lambda)^2 + 1/((n-1)\lambda)^2
, respectively. Continuining in this manner, we have
E[X_{(m + 1)}]
Now, we wish to approximate the mean with a much simpler formula. First, from (B.7) in Appendix B, we have
\gamma
is Euler’s constant. Then, we can write the expected sample median as
Hence, as
n \rightarrow \infty
, this approximation goes to
\frac{\log 2}{\lambda}
, which is the median of an exponentially distributed random variable. Specifically, the median is the solution to
F_X(x) = 1/2
F_X
denotes the cumulative distribution function of the random variable
X
Improved Approximation of
E[X_{(m + 1)}]
It turns out that we can improve this approximation with the following two results:
Following the derivation of our above approximation, we have that
Var[X_{(m + 1)}]
We can also approximate
Var[X_{(m + 1)}]
using the approximation
a = m+1
b = 2m + 1 |
Combat Rewards - CryptoBlades Wiki
Stamina per Fight
Players may decide how much stamina they wish to spend on a single fight, and can spend up to 200 stamina in one go.
Gas offset is paid once per transaction, so if the user decides to spend 200 stamina they would receive 5x the evaluated baseline rewards, but gas offset only once.
The expected dollar value gains are equivalent for all tiers of stamina spending when accounting for gas, however earnings across an average period of time may differ if the player loses multiple high stamina fights.
Experience gained in fights are also multiplied proportionally depending on whatever stamina value the player chooses to spend.
Skill Payout
The formula to determine SKILL payout is as follows:
payout = gasOffset + (baseline * √(enemyPower/1000))
Gas Offset is shown in the earnings calculator as follows:
Baseline is also shown in the earnings calculator as follows:
These numbers are dynamically adjusted by the Oracle, taking into account SKILL dollar value.
Note that the "power" variable indicated in the formula is the listed power value of whatever enemy the player chooses to fight.
Next - Combat |
Compound interest is interest that's calculated both on the initial principal of a deposit or loan, and on all previously accumulated interest.
For example, let's say you have a deposit of $100 that earns a 10% compounded interest rate. The $100 grows into $110 after the first year, then $121 after the second year. Each year the base increases by 10%. The reason the second year's gain is $11 instead of $10 is as a result of the same rate (10% in this example) being applied to a larger base ($110 compared to $100, our starting point).
Or let's say, $100 is the principal of a loan, and the compound interest rate is 10%. After one year you have $100 in principal and $10 in interest, for a total base of $110. In year two, the interest rate (10%) is applied to the principal ($100, resulting in $10 of interest) and the accumulated interest ($10, resulting in $1 of interest), for a total of $11 in interest gained that year, and $21 for both years.
It's similar to the Compounded Annual Growth Rate (CAGR). For CAGR, you are computing a rate that links the return over a number of periods. For compound interest, you most likely know the rate already; you are just calculating what the future value of the return might be.
For the formula for compound interest, just algebraically rearrange the formula for CAGR. You need the beginning value, interest rate, and number of periods in years. The interest rate and number of periods need to be expressed in annual terms, since the length is presumed to be in years. From there you can solve for the future value. The equation reads:
\begin{aligned}&\text{Beginning Value}\\&\times\left(1+\left(\frac{\text{interest rate}}{\text{NCPPY}}\right)\right)^{(\text{years}\ \times\ \text{NCPPY)}\ =\ \text{Future Value}}\\&\textbf{where:}\\&NCPPY=\text{number of compounding periods per year}\end{aligned}
Beginning Value×(1+(NCPPYinterest rate))(years × NCPPY) = Future Valuewhere:
This formula looks more complex than it really is, because of the requirement to express it in annual terms. Keep in mind, if it's an annual rate, then the number of compounding periods per year is one, which means you're dividing the interest rate by one and multiplying the years by one. If compounding occurs quarterly, you would divide the rate by four, and multiply the years by four.
Financial modeling best practices require calculations to be transparent and easily auditable. The trouble with piling all of the calculations into a formula is that you can't easily see what numbers go where, or what numbers are user inputs or hard-coded.
There are two ways to set this up in Excel. The most easy to audit and understand is to have all the data in one table, then break out the calculations line by line. Conversely, you could calculate the whole equation in one cell to arrive at just the final value figure. Both are detailed below:
How Can I Calculate Compounding Interest on a Loan in Excel? |
Preliminary Observation and Significance of Changes on Rash of Skin Prick Test during SLIT
Preliminary Observation and Significance of Changes on Rash of Skin Prick Test during SLIT
Weinian Lin*, Jinchao Lin, Jun Liao, Runfeng Chen, Zhiwei Huang, Xiaodong Zhang, Wen Lin
Department of Otorhinolaryngology, Quanzhou First Affiliated Hospital of Fujian Medical University, Quanzhou, China
Objective: To observe the changes on skin wheal and erythema of skin prick test for the patients with allergic rhinitis during SLIT. Methods: Since March 2010 the 103 cases of SLIT attacked by allergic rhinitis patients, divided into four age groups, respectively measured the diameter of skin wheal and erythema before treatment, six months, one year and 2 years after SLIT. The data were analyzed by analysis of variance method; P < 0.01 showed the difference was statistically significant. Results: The results showed that the most changes of skin erythema diameter were statistically significant in N1, N2, N3 age group during test observation compare with the data before SLIT (p < 0.01); and the most changes of allergen wheal diameter were not statistically significant, but the N4 group had no significant change of wheal and erythema diameter. Conclusion: Although most of the skin test wheal did not change significantly during the treatment of SLIT, the erythema reaction decreased to a certain extent, indicating that the intensity of histamine release may be reduced during the treatment.
Rhinitis, Allergy, Immunotherapy, Skin Tests, Therapeutic Evaluation
Skin prick test (SPT) and detection of serum specific IgE are the main methods for the diagnosis of allergic rhinitis, combined with medical history, clinical symptoms and nasal signs [1] [2] [3] [4] . Specific immunotherapy (SIT) is increasingly considered as a main first-line treatment for allergic rhinitis [1] [2] [3] [4] . Sublingual immunotherapy (SLIT) is widely used because of its convenience, operability and good patient compliance. In this study, we investigated the changes and significance of test results of wheal and erythema in allergen SPT in the treatment of SLIT.
A total of 103 patients (72 males and 31 females), with complete data, underwent SLIT at our hospital from March 2010 to February 2016, and were recruited in this study. All case accorded with the guidelines for the diagnosis and treatment of allergic rhinitis (2015, Tianjin, China) were on the indication of with moderate to severe AR classification, and the duration of the disease was 1 - 16 years, with an average of 1years and 8 months, The subjects were divided into four groups, with 22 patients (17 males and 5 females) in N1 group (4 - 8 years old), 38 patients (27 males and 11 females) in N2 group (8 years and 1 month to 14 years old), 35 patients (24 males and 11 females) in N3 group (14 years and 1 month to 40 years old), and eight patients (4 males and 4 females) in N4 group (>40 years old).
The standard allergen SPT was performed on the skin inside forearm of patients with suspected allergic rhinitis. Common inhaled allergen provided by ALK-Abello was used, comprised include common ragweed, mugwort, compound mite (house dust and dust), Penicillium notatum, Humulus scandens pollen, tree compound II (willow, poplar, elm), cockroach, cocoon silk, cat hair and dog hair allergen liquid and cross reaction control substance. The normal skin inside forearm was selected and wiped with cotton ball dipped in saline water, the standard allergen solution was dripped onto the skin at 2 cm intervals, then the skin was pricked by an independent lancet, the forearm was exposed for 20 minutes, and the test results were analyzed.
According to the guidelines for the diagnosis and treatment of allergic rhinitis (2015, Tianjin, China) [1] , the skin index SI is used as one of the diagnostic indicators of allergic rhinitis, so the SI was adopted, and the maximum and minimum diameters (the vertical line at the midpoint of maximum diameter was taken) of wheal of mite allergen and histamine prick point were measured respectively, to calculate their average wheal diameters of mite allergen and histamine prick point (the maximum and minimum diameters addition are equal to 2 equal to the average diameter), and the ratio of the two was SI, which was classified into four grades: 0.3 ≤ SI < 0.5: +, 0.5 ≤ SI < 1.0: ++, 1.0 ≤ SI < 2.0: +++, SI ≥ 2.0: ++++.
2.3. Test Data Acquisition
In clinical practice, the patients with allergic rhintis were selected with the SI for skin test to compound mite allergen liquid was +++ to ++++. After the patients with positive reactions to other allergens were excluded, the specific SLIT with Dermatophagoides farinae drops was conducted. The results of SPT to mite allergens before treatment, as well as at 6 months, 1 year and 2 years after treatment were recorded. The maximum diameters of wheal and erythema of the mite allergen prick point were also recorded along with the clinical symptoms.
All patients and their families were given medication notes at the first visit. Follow-up files were established, contact information was noted, regular follow-up interviews were conducted to guide and resolve patients’ care needs, the results of allergen SPT were detected, medication for home care was prescribed, environmental health was advised, dose adjustment and adverse drug reaction were noted, all of which improved the treatment compliance. The patients were asked to discontinue oral or local antihistamine and hormone drugs for one week before next SPT examination.
In this study, SPSS 13.0 statistical software was used for all analyses. The quantitative data were expressed as mean ± standard deviation (
\stackrel{¯}{x}
± s). The data were approximately normally distributed, and repeated measures analysis of variance was used. p < 0.01 was considered as significant difference.
In this study, after eliminating antihistamines in the case of pricked erythema, as Table 1 shows that the diameter of erythema at almost every observation time in the N1, N2 and N3 groups was significantly changed (p < 0.01) as compared to the data before treatment, except for the date in N2 group at 2 years after treatment and the date in N3 group at 1 years after treatment. However, most changes on the diameter of allergen wheal were not statistically significant, and the diameters of erythema in the N4 group did not significantly change during SLIT at each time point.
Table 1. Changes in the diameter of wheal blush of SPT in the four groups at different time points (unit: cm).
SIT is one of first-line treatments for AR [1] [2] [3] [4] . In 2013, the World Allergy Organization (WAO) stated in the sublingual specific immunotherapy position paper that SLIT can be used as a therapeutic strategy for initial and early respiratory diseases [3] . Dermatophagoides farinae Drops (Chang di), as the main product of SLIT for AR, has been used for more than ten years in China, and its biological stability and clinical efficacy are generally recognized. In this study, Dermatophagoides farinae Drops (Chang di) treatment of patients with AR sensitive to dust mite achieved an ideal effect by alleviating clinical symptoms, reducing the occurrence of AR complications, and effectively reducing the use of antihistamines and other drugs [4] [5] [6] [7] .
There is no simple and acceptable indicator to determine the clinical efficacy of specific immunity in patients with AR. The possible tests that may help to evaluate the efficacy include sIgE, sIgG4, cytokines, antigen specific T-cell detection, nasal provocation test, allergen exposed compartment, histamine release test, etc. The allergen SPT only included histamine release test. In clinical practice, skin sensitivity test included the adjacent reactions of wheal and erythema. The positive result of skin sensitive test with erythema reaction is generally believed to be stronger than simple wheal reaction. We found that during SLIT with mite allergen, most patients showed reduction of clinical symptoms and nasal mucosa pale edema, which became ruddy. However, most of the results of prick test in clinical practice were not significantly improved. The erythema around wheal improved in some patients. However, this improvement was not found in all patients throughout the treatment period. Whether the intensity of histamine release improved in some patients during treatment remains unknown. Because SPT is affected by numerous conditions, more extensive clinical data for comparative analysis are needed.
In this study, after eliminating antihistamines in the case of pricked erythema, the data at almost every observation time point in the N1, N2 and N3 groups were significantly changed as compared to the data before treatment. However, most changes in allergen wheal were not statistically significant, and the diameters of erythema in the N4 group did not significantly change during SLIT at each time point, and this may be related to too little sample. With the progress of SLIT, clinical symptoms improved but the wheal reaction of allergen SPT did not significantly change, indicating that the results of erythema reaction of histamine release intensity significantly reduced. Moreover, and there was no significant change in the N4 group (>40 years old).
A recent study showed that in a group of 4 - 60 years old patients with AR, who underwent SLIT (Dermatophagoides pteronyssinus and Dermatophagoides farinae Drops) for six months, the clinical nasal symptoms were significantly improved, the onset was rapid, with an average onset time of 14 weeks, the levels of dust mite and house dust mite specific IgG4 significantly increased, but the levels of serum specific IgE did not significantly change as compared to the control group. Specific IgG4, a tolerogenic antibody, competed with serum specific IgE for the same allergen binding site, which prevented the release of inflammatory mediators [2] [6] [7] . One study [8] have shown that these IgG antibodies not only have the function of “blocking antibodies” but also have the ability to prevent SIgE mediated allergen histamine release, which seems to provide a clue as to answer why the reduction in the erythema diameter of some age groups found in our study.
SIT is the process of inducing immune tolerance by giving increasing doses of allergenic allergen vaccine to patients with AR [2] [6] . When the patient is re-exposed to the corresponding allergen, the clinical symptoms are absent or reduced. However, the sensitization of allergens was unchanged, perhaps because immune tolerance and minor histamine release played clinical therapeutic effects. Clinical trials showed that the younger the age, the more pronounced were the erythema changes of histamine release intensity. These preliminary results need to be confirmed in a large sample size clinical study.
Lin, W.N., Lin, J.C., Liao, J., Chen, R.F., Huang, Z.W., Zhang, X.D. and Lin, W. (2018) Preliminary Observation and Significance of Changes on Rash of Skin Prick Test during SLIT. International Journal of Otolaryngology and Head & Neck Surgery, 7, 209-213. https://doi.org/10.4236/ijohns.2018.74022
1. Group of Experts on Rhinology (2015) Chinese Guidelines for Diagnosis and Treatment of Allergic Rhinitis. Chinese Journal of Otorhinolaryngol Head and Neck Surgery, 51, 6-18.
2. Cheng, L. (2016) Allergen-Specific Immunotherapy as the First-Line Treatment for Allergic Rhinitis. Journal of Otolaryngology and Ophthalmology of Shandong University, 30, 1-2.
3. Cheng, L. (2017) Some Issues Concerning the Prevention and Treatment of Allergic Rhinitis. Journal of Clinical Otorhinolaryngology Head and Neck Surgery, 31, 1-2.
4. Canonica, G.W., Cox, L., Pawankar, R., Baena-Cagnani, C.E., Blaiss, M., Bonini, S., et al. (2014) Sublingual Immunotherapy: World Allergy Organization Position Paper 2013 Update. World Allergy Organization Journal, 7, 6. https://doi.org/10.1186/1939-4551-7-6
5. Wang, D.H., Chen, L., Cheng, L., Li, K.N., Yuan, H., Lu, J.H., et al. (2013) Fast Onset of Action of Sublingual Immunotherapy in House Dust Mite-Induced Allergic Rhinitis: A Multicenter, Randomized, Double-Blind, Placebo-Controlled Trial. Laryngoscope, 123, 1334-1340. https://doi.org/10.1002/lary.23935
6. Canonica, G.W., Bousquet, J., Casale, T., et al. (2009) Sub-Lingual Immunotherapy: World Allergy Organization Position Paper 2009. Allergy, 64, 1-59. https://doi.org/10.1111/j.1398-9995.2009.02309.x
7. Jutel, M., Agache, I., Bonini, S., Burks, A.W., Calderon, M., Canonica, W., et al. (2016) International Consensus on Allergen Immunotherapy II: Mechanisms, Standardization, and Pharmacoeconomics. The Journal of Allergy and Clinical Immunology, 137, 358-368. https://doi.org/10.1016/j.jaci.2015.12.1300
8. Dearon, M. (1997) Negative Regulation Regulation of Mast Cell Activation by Receptors for IgG. International Archives of Allergy and Immunology, 113, 138-141. https://doi.org/10.1159/000237528 |
Hom-stacks and restriction of scalars
15 July 2006 Hom-stacks and restriction of scalars
Martin C. Olsson
Martin C. Olsson1
1Department of Mathematics, University of Texas at Austin
Fix an algebraic space
S
\mathcal{X}
\mathcal{Y}
be separated Artin stacks of finite presentation over
S
with finite diagonals (over
S
). We define a stack
{\underline{\mathrm{Hom}}}_{S}\left(\mathcal{X},\mathcal{Y}\right)
classifying morphisms between
\mathcal{X}
\mathcal{Y}
\mathcal{X}
is proper and flat over
S
, and assume fppf locally on
S
that there exists a finite finitely presented flat cover
Z\to \mathcal{X}
Z
an algebraic space. Then we show that
{\underline{\mathrm{Hom}}}_{S}\left(\mathcal{X},\mathcal{Y}\right)
is an Artin stack with quasi-compact and separated diagonal
Martin C. Olsson. "Hom-stacks and restriction of scalars." Duke Math. J. 134 (1) 139 - 164, 15 July 2006. https://doi.org/10.1215/S0012-7094-06-13414-2
Martin C. Olsson "Hom-stacks and restriction of scalars," Duke Mathematical Journal, Duke Math. J. 134(1), 139-164, (15 July 2006) |
Density - Simple English Wikipedia, the free encyclopedia
Density is a measurement that compares the amount of matter an object has to its volume. An object with much matter in a certain volume has high density. An object with little matter in the same amount of volume has a low density. Density is found by dividing the mass of an object by its volume:
{\displaystyle \rho ={\frac {m}{V}}}
where ρ is the density, m is the mass, and V is the volume.[1]
Changes of densityEdit
In general, density can be changed by changing either the pressure or the temperature. Increasing the pressure always increases the density of a material. Increasing the temperature usually lowers the density, but there are exceptions. For example, the density of water increases slightly between its melting point at 0 °C and 4 °C. When water freezes, it expands by about 9% in volume, making ice that is less dense than liquid water. Water expands as it drops below 4 °C.
Fresh water is often used as a standard of relative density. This is called specific gravity.
The most common SI units for density are g/cm3 and kg/m3. When the numerator is much larger than the denominator, that means the substance has a higher density. When the denominator is much larger than the numerator, the substance has a lower density.
"Density" sometimes means the ratio between other properties instead of mass and volume. Then it means how much of a property can be found in a specific piece of what they are looking at. For example, population density is how many people live within the same amount of land area. The population density in the city is higher than the country side because people live closer to each other in the city. In computers, storage density is how much data can fit on a data storage device in relation to its physical size. A Blu-ray disc has a higher storage density than a DVD which has a higher storage density than a CD, even though they all have almost exactly the same volume.
↑ "Density definition in Oil Gas Glossary".
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Density&oldid=8148932" |
Difference between revisions of "Walsh spectra of all known APN functions over GF(2^8)" - Boolean Functions
Difference between revisions of "Walsh spectra of all known APN functions over GF(2^8)"
The tables below contain the Walsh spectra for all quadratic APN functions of dimension 8 as given in the appendix of [https://eprint.iacr.org/2013/007.pdf "A Matrix Approach for Constructing Quadratic APN Functions"]. All 8180 functions listed in the appendix have one of the following three Walsh spectra:
The tables below contain the Walsh spectra for all [[Known instances of APN functions over GF(2^8)]]. All of these 8180 functions have one of the following three Walsh spectra:
* <math>\{ -32^{2380}, -16^{20400}, 0^{16320}, 16^{23120}, 32^{3060} \}</math> (same as the Gold functions)
* <math>\{ -64^6, -32^{2240}, -16^{20880}, 0^{15600}, 16^{23664}, 32^{2880}, 64^{10} \}</math> (type 1)
* <math>\{ -64^{12}, -32^{2100}, -16^{21360}, 0^{14880}, 16^{24208}, 32^{2700}, 64^{20} \}</math> (type 2)
There are 12 functions with a Walsh spectrum of type 2 (given in the table below), 487 functions with a Walsh spectrum of type 1, and 7681 functions with a Gold-like Walsh spectrum (not listed below due to space limitations). Magma code listing all functions with a [[:File:Gold spectra.txt|Gold-like Walsh spectrum]], a [[:File:T1 spectra.txt|Walsh spectrum of type 1]] and a [[:File:T2 spectra.txt|Walsh spectrum of type 2]] are available.
There are 12 functions with a Walsh spectrum of type 2 (given in the table below), 487 functions with a Walsh spectrum of type 1, and 7681 functions with a Gold-like Walsh spectrum (not listed below due to space limitations). Magma code listing all functions with a [[:File:Gold spectra.txt|Gold-like Walsh spectrum]], a [[:File:T1 spectra.txt|Walsh spectrum of type 1]] and a [[:File:T2 spectra.txt|Walsh spectrum of type 2]] is available.
{\displaystyle \{-32^{2380},-16^{20400},0^{16320},16^{23120},32^{3060}\}}
{\displaystyle \{-64^{6},-32^{2240},-16^{20880},0^{15600},16^{23664},32^{2880},64^{10}\}}
{\displaystyle \{-64^{12},-32^{2100},-16^{21360},0^{14880},16^{24208},32^{2700},64^{20}\}}
{\displaystyle \alpha ^{130}\cdot x^{192}+\alpha ^{160}\cdot x^{160}+\alpha ^{117}\cdot x^{144}+\alpha ^{230}\cdot x^{136}+\alpha ^{228}\cdot x^{132}+\alpha ^{162}\cdot x^{130}+\alpha ^{25}\cdot x^{129}+\alpha ^{79}\cdot x^{96}+\alpha ^{204}\cdot x^{80}+\alpha ^{83}\cdot x^{72}+\alpha ^{159}\cdot x^{68}+\alpha ^{234}\cdot x^{66}+\alpha ^{36}\cdot x^{65}+\alpha ^{67}\cdot x^{48}+\alpha ^{151}\cdot x^{40}+\alpha ^{17}\cdot x^{36}+\alpha ^{81}\cdot x^{34}+\alpha ^{52}\cdot x^{33}+\alpha ^{9}\cdot x^{24}+\alpha ^{116}\cdot x^{20}+\alpha ^{102}\cdot x^{18}+\alpha ^{97}\cdot x^{17}+\alpha ^{74}\cdot x^{12}+\alpha ^{48}\cdot x^{10}+\alpha ^{144}\cdot x^{9}+\alpha ^{58}\cdot x^{6}+\alpha ^{146}\cdot x^{5}+\alpha ^{123}\cdot x^{3}}
{\displaystyle \alpha ^{154}\cdot x^{192}+\alpha ^{36}\cdot x^{160}+\alpha ^{83}\cdot x^{144}+\alpha ^{160}\cdot x^{136}+\alpha ^{253}\cdot x^{132}+\alpha ^{215}\cdot x^{130}+\alpha ^{221}\cdot x^{129}+\alpha ^{76}\cdot x^{96}+\alpha ^{137}\cdot x^{80}+\alpha ^{206}\cdot x^{72}+\alpha ^{185}\cdot x^{68}+\alpha ^{165}\cdot x^{66}+\alpha ^{201}\cdot x^{65}+\alpha ^{226}\cdot x^{48}+\alpha ^{25}\cdot x^{40}+\alpha ^{65}\cdot x^{36}+\alpha ^{11}\cdot x^{33}+\alpha ^{170}\cdot x^{24}+\alpha ^{247}\cdot x^{20}+\alpha ^{155}\cdot x^{18}+\alpha \cdot x^{17}+\alpha ^{146}\cdot x^{12}+\alpha ^{204}\cdot x^{10}+\alpha ^{121}\cdot x^{9}+\alpha ^{202}\cdot x^{6}+\alpha ^{246}\cdot x^{5}+\alpha ^{170}\cdot x^{3}}
{\displaystyle \alpha ^{183}\cdot x^{192}+\alpha ^{178}\cdot x^{160}+\alpha ^{103}\cdot x^{144}+\alpha ^{97}\cdot x^{136}+\alpha ^{37}\cdot x^{132}+\alpha ^{172}\cdot x^{130}+\alpha ^{102}\cdot x^{129}+\alpha ^{62}\cdot x^{96}+\alpha ^{145}\cdot x^{80}+\alpha ^{96}\cdot x^{72}+\alpha ^{132}\cdot x^{68}+\alpha ^{210}\cdot x^{66}+\alpha ^{69}\cdot x^{65}+\alpha ^{69}\cdot x^{48}+\alpha ^{11}\cdot x^{40}+x^{36}+\alpha ^{4}\cdot x^{34}+\alpha ^{76}\cdot x^{33}+\alpha ^{122}\cdot x^{24}+\alpha ^{6}\cdot x^{20}+\alpha ^{145}\cdot x^{18}+\alpha ^{155}\cdot x^{17}+\alpha ^{41}\cdot x^{12}+\alpha ^{40}\cdot x^{10}+\alpha ^{106}\cdot x^{9}+\alpha ^{144}\cdot x^{6}+\alpha ^{102}\cdot x^{5}+\alpha ^{246}\cdot x^{3}}
{\displaystyle \alpha ^{22}\cdot x^{192}+\alpha ^{167}\cdot x^{160}+\alpha ^{178}\cdot x^{144}+\alpha ^{84}\cdot x^{136}+\alpha ^{219}\cdot x^{132}+\alpha ^{248}\cdot x^{130}+\alpha ^{130}\cdot x^{129}+\alpha ^{221}\cdot x^{96}+\alpha ^{84}\cdot x^{80}+\alpha ^{123}\cdot x^{72}+\alpha ^{140}\cdot x^{68}+\alpha ^{26}\cdot x^{66}+\alpha ^{108}\cdot x^{65}+\alpha ^{50}\cdot x^{48}+\alpha ^{15}\cdot x^{40}+\alpha ^{211}\cdot x^{36}+\alpha ^{116}\cdot x^{34}+\alpha ^{19}\cdot x^{33}+\alpha ^{228}\cdot x^{24}+\alpha ^{176}\cdot x^{20}+\alpha ^{42}\cdot x^{18}+\alpha ^{80}\cdot x^{17}+\alpha ^{180}\cdot x^{12}+\alpha ^{203}\cdot x^{10}+\alpha ^{104}\cdot x^{9}+\alpha ^{72}\cdot x^{6}+\alpha ^{151}\cdot x^{5}+\alpha ^{247}\cdot x^{3}}
{\displaystyle \alpha ^{156}\cdot x^{192}+\alpha ^{25}\cdot x^{160}+\alpha ^{158}\cdot x^{144}+\alpha ^{20}\cdot x^{136}+\alpha ^{50}\cdot x^{132}+\alpha ^{140}\cdot x^{130}+\alpha ^{203}\cdot x^{129}+\alpha ^{184}\cdot x^{96}+\alpha ^{152}\cdot x^{80}+\alpha ^{228}\cdot x^{72}+\alpha ^{194}\cdot x^{68}+\alpha ^{203}\cdot x^{66}+\alpha ^{131}\cdot x^{65}+\alpha ^{25}\cdot x^{48}+\alpha ^{192}\cdot x^{40}+\alpha ^{191}\cdot x^{36}+\alpha ^{125}\cdot x^{34}+\alpha ^{136}\cdot x^{33}+\alpha ^{132}\cdot x^{24}+\alpha ^{85}\cdot x^{20}+\alpha ^{191}\cdot x^{18}+\alpha ^{120}\cdot x^{17}+\alpha ^{212}\cdot x^{12}+\alpha ^{244}\cdot x^{10}+\alpha ^{133}\cdot x^{9}+\alpha ^{78}\cdot x^{6}+\alpha ^{161}\cdot x^{5}+\alpha \cdot x^{3}}
{\displaystyle \alpha ^{193}\cdot x^{192}+\alpha ^{33}\cdot x^{160}+\alpha ^{22}\cdot x^{144}+\alpha ^{204}\cdot x^{136}+\alpha ^{173}\cdot x^{132}+\alpha ^{50}\cdot x^{130}+\alpha ^{66}\cdot x^{129}+\alpha ^{42}\cdot x^{96}+\alpha ^{69}\cdot x^{80}+\alpha ^{175}\cdot x^{72}+\alpha ^{230}\cdot x^{68}+\alpha ^{253}\cdot x^{66}+\alpha ^{16}\cdot x^{65}+\alpha ^{52}\cdot x^{48}+\alpha ^{54}\cdot x^{40}+\alpha ^{9}\cdot x^{36}+\alpha ^{177}\cdot x^{34}+\alpha ^{99}\cdot x^{33}+\alpha ^{12}\cdot x^{24}+\alpha ^{37}\cdot x^{20}+\alpha ^{83}\cdot x^{18}+\alpha ^{230}\cdot x^{17}+\alpha ^{78}\cdot x^{12}+\alpha \cdot x^{10}+\alpha ^{64}\cdot x^{9}+\alpha ^{225}\cdot x^{6}+\alpha ^{68}\cdot x^{5}+\alpha ^{204}\cdot x^{3}}
{\displaystyle \alpha ^{88}\cdot x^{192}+\alpha ^{8}\cdot x^{160}+\alpha ^{11}\cdot x^{144}+\alpha ^{121}\cdot x^{136}+\alpha ^{205}\cdot x^{132}+\alpha ^{165}\cdot x^{130}+\alpha ^{206}\cdot x^{129}+\alpha ^{164}\cdot x^{96}+\alpha ^{235}\cdot x^{80}+\alpha ^{94}\cdot x^{72}+\alpha ^{173}\cdot x^{68}+\alpha ^{142}\cdot x^{66}+\alpha ^{238}\cdot x^{65}+\alpha ^{102}\cdot x^{48}+\alpha ^{113}\cdot x^{40}+\alpha ^{183}\cdot x^{36}+\alpha ^{187}\cdot x^{34}+\alpha ^{157}\cdot x^{33}+\alpha ^{2}\cdot x^{24}+\alpha ^{23}\cdot x^{20}+\alpha ^{122}\cdot x^{18}+\alpha ^{21}\cdot x^{17}+\alpha ^{154}\cdot x^{12}+\alpha ^{78}\cdot x^{10}+\alpha ^{117}\cdot x^{9}+\alpha ^{177}\cdot x^{6}+\alpha ^{111}\cdot x^{5}+\alpha ^{60}\cdot x^{3}}
{\displaystyle \alpha ^{212}\cdot x^{192}+\alpha ^{198}\cdot x^{160}+\alpha ^{175}\cdot x^{144}+\alpha ^{80}\cdot x^{136}+\alpha ^{196}\cdot x^{132}+\alpha ^{167}\cdot x^{130}+\alpha ^{2}\cdot x^{129}+\alpha ^{65}\cdot x^{96}+\alpha ^{243}\cdot x^{80}+\alpha ^{91}\cdot x^{72}+\alpha ^{171}\cdot x^{68}+\alpha ^{211}\cdot x^{66}+\alpha ^{182}\cdot x^{65}+\alpha ^{247}\cdot x^{48}+\alpha ^{86}\cdot x^{40}+\alpha ^{89}\cdot x^{36}+\alpha ^{87}\cdot x^{34}+\alpha ^{83}\cdot x^{33}+\alpha ^{138}\cdot x^{24}+\alpha ^{45}\cdot x^{20}+\alpha ^{149}\cdot x^{18}+\alpha ^{100}\cdot x^{17}+\alpha ^{188}\cdot x^{12}+\alpha ^{17}\cdot x^{10}+\alpha ^{243}\cdot x^{9}+\alpha ^{237}\cdot x^{6}+\alpha ^{112}\cdot x^{5}+\alpha ^{137}\cdot x^{3}}
{\displaystyle \alpha ^{117}\cdot x^{192}+\alpha ^{61}\cdot x^{160}+\alpha ^{230}\cdot x^{144}+\alpha ^{105}\cdot x^{136}+\alpha ^{191}\cdot x^{132}+\alpha ^{113}\cdot x^{130}+\alpha ^{245}\cdot x^{129}+\alpha ^{139}\cdot x^{96}+\alpha ^{166}\cdot x^{80}+\alpha ^{210}\cdot x^{72}+\alpha ^{221}\cdot x^{68}+\alpha ^{138}\cdot x^{66}+\alpha ^{146}\cdot x^{65}+\alpha ^{120}\cdot x^{48}+\alpha ^{124}\cdot x^{40}+\alpha ^{252}\cdot x^{36}+\alpha ^{182}\cdot x^{34}+\alpha ^{5}\cdot x^{33}+\alpha ^{8}\cdot x^{24}+\alpha ^{136}\cdot x^{20}+\alpha ^{235}\cdot x^{18}+\alpha ^{61}\cdot x^{17}+\alpha ^{45}\cdot x^{12}+\alpha ^{149}\cdot x^{10}+\alpha ^{158}\cdot x^{9}+\alpha ^{13}\cdot x^{6}+\alpha ^{169}\cdot x^{5}+\alpha ^{121}\cdot x^{3}}
{\displaystyle \alpha ^{34}\cdot x^{192}+\alpha ^{57}\cdot x^{160}+\alpha ^{187}\cdot x^{144}+\alpha ^{36}\cdot x^{136}+\alpha ^{137}\cdot x^{132}+\alpha ^{63}\cdot x^{130}+\alpha ^{98}\cdot x^{129}+\alpha ^{236}\cdot x^{96}+\alpha ^{161}\cdot x^{80}+\alpha ^{66}\cdot x^{72}+\alpha ^{191}\cdot x^{68}+\alpha ^{117}\cdot x^{66}+\alpha ^{241}\cdot x^{65}+\alpha ^{7}\cdot x^{48}+\alpha ^{9}\cdot x^{40}+\alpha ^{153}\cdot x^{36}+\alpha ^{118}\cdot x^{34}+\alpha ^{154}\cdot x^{33}+\alpha ^{194}\cdot x^{24}+\alpha ^{157}\cdot x^{20}+\alpha ^{14}\cdot x^{18}+\alpha ^{116}\cdot x^{17}+\alpha ^{119}\cdot x^{12}+\alpha ^{113}\cdot x^{10}+\alpha ^{13}\cdot x^{9}+\alpha ^{138}\cdot x^{6}+\alpha ^{143}\cdot x^{5}+\alpha ^{35}\cdot x^{3}}
{\displaystyle \alpha ^{140}\cdot x^{192}+\alpha ^{233}\cdot x^{160}+\alpha ^{150}\cdot x^{144}+\alpha ^{146}\cdot x^{136}+\alpha ^{99}\cdot x^{132}+\alpha ^{249}\cdot x^{130}+\alpha ^{211}\cdot x^{129}+\alpha ^{66}\cdot x^{96}+\alpha ^{37}\cdot x^{80}+\alpha ^{35}\cdot x^{72}+\alpha ^{199}\cdot x^{68}+\alpha ^{170}\cdot x^{66}+\alpha ^{2}\cdot x^{65}+\alpha ^{217}\cdot x^{48}+\alpha ^{2}\cdot x^{40}+\alpha ^{192}\cdot x^{36}+\alpha ^{32}\cdot x^{34}+\alpha ^{229}\cdot x^{33}+\alpha ^{241}\cdot x^{24}+\alpha ^{200}\cdot x^{20}+\alpha ^{63}\cdot x^{18}+\alpha ^{17}\cdot x^{17}+\alpha ^{251}\cdot x^{12}+\alpha ^{44}\cdot x^{10}+\alpha ^{106}\cdot x^{9}+\alpha ^{25}\cdot x^{6}+\alpha ^{174}\cdot x^{5}+\alpha ^{127}\cdot x^{3}}
{\displaystyle \alpha ^{237}\cdot x^{192}+\alpha ^{133}\cdot x^{160}+\alpha ^{204}\cdot x^{144}+\alpha ^{169}\cdot x^{136}+\alpha ^{30}\cdot x^{132}+\alpha ^{127}\cdot x^{130}+\alpha ^{41}\cdot x^{129}+\alpha ^{12}\cdot x^{96}+\alpha ^{198}\cdot x^{80}+\alpha ^{151}\cdot x^{72}+\alpha ^{252}\cdot x^{68}+\alpha ^{29}\cdot x^{66}+\alpha ^{144}\cdot x^{65}+\alpha ^{120}\cdot x^{48}+\alpha ^{72}\cdot x^{40}+\alpha ^{123}\cdot x^{36}+\alpha ^{170}\cdot x^{34}+\alpha ^{159}\cdot x^{33}+\alpha ^{77}\cdot x^{24}+\alpha ^{227}\cdot x^{20}+\alpha ^{161}\cdot x^{18}+\alpha ^{231}\cdot x^{17}+\alpha ^{159}\cdot x^{12}+\alpha ^{253}\cdot x^{10}+\alpha ^{56}\cdot x^{9}+\alpha ^{35}\cdot x^{6}+\alpha ^{251}\cdot x^{5}+\alpha ^{99}\cdot x^{3}} |
Online Water Wash Tests of GE J85-13 | J. Turbomach. | ASME Digital Collection
Online Water Wash Tests of GE J85-13
Elisabet Syverud,
Department of Energy and Process Engineering, NTNU, Norwegian University of Science and Technology, Trondheim, Norway
Syverud, E., and Bakken, L. E. (January 1, 2007). "Online Water Wash Tests of GE J85-13." ASME. J. Turbomach. January 2007; 129(1): 136–142. https://doi.org/10.1115/1.2372768
This paper reports the results of a series of online water wash tests of a GE J85-13 jet engine at the test facilities of the Royal Norwegian Air Force. The engine performance was deteriorated by injecting atomized saltwater at the engine inlet. The engine was then online washed with water injected at three different droplet sizes (25, 75, and
200μm
) and at water-to-air ratios ranging from 0.4% to 3% by mass. Engine performance was measured using standard on-engine instrumentation. Extra temperature and pressure sensors in the compressor section provided additional information of the propagation of deposits in the aft stages. The measurements were supported by visual observations. The overall engine performance improved rapidly with online wash. The buildup of deposits in the aft stages was influenced both by the droplet size and the water-to-air ratio. The water-to-air ratio was the most important parameter to achieve effective online washing.
jet engines, aerospace testing, aircraft maintenance, aerospace test facilities, cleaning, compressor cleaning, axial compressor, stage characteristics, GE J85-13
Compressors, Jet engines, Test facilities, Water, Engines, Aerospace industry, Drops, Air Force, Aircraft, Instrumentation, Maintenance, Pressure sensors, Temperature, Testing
, “A Review of Online Washing Systems,” ASME Paper No. GT2004–53224.
,” ASME Paper No. 97-AA-135.
A Method for Cleaning a Stationary Gas Turbine Unit During Operation
,” World Patent No. WO2004/055334.
Effective Compressor Cleaning as a Result of Scientific Testing and Field Experience
,” FP Turbomachinery Consultants GmbH, Spinzalaan 177-C, 2273 XG Voorburg, The Netherlands.
Motion of Condensed Phase in the Blade Passages of an Axial Gas Turbine Stage
,” Teploenergetika, pp.
Water Ingestion Into Jet Engine Axial Compressors
, “Compressor Performance With Water Injection,” Proceedings of ASME Turbo Expo 2001, June 4–7, New Orleans, LA.
The Fundamental Equations of Gas-Droplet Multiphase Flow
, “The Effect of Water Injection on Multispool Gas Turbine Behavior,” ASME Paper No. GT2004–53320.
,” AGARD Advisory Report AGARD AR-332.
Axial Compressor Deterioration Caused by Saltwater Ingestion
Blade Row Dynamic Digital Compressor Program Volume 1 J85 Clean Inlet Flow and Parallel Compressor Models
,” NASA CR-134978.
Performance of a J85–13 Compressor With Clean and Distorted Inlet Flow
,” NASA TM X-3304.
Recommended Practices for Measurement of Gas Path Pressures and Temperatures for Performance Assessment of Aircraft Turbine Engines and Components
, ASME PTC-22, ASME, New York.
,” ASME PTC-19.1, ASME, New York.
Caguiat
, “Compressor Fouling Testing on Rolls Royce/Allison 501-K17 and General Electric LM 2500 Gas Turbine Engines,” ASME Paper No. GT2002–30262.
ASTM Standard E 799–92,
Standard Practice for Determining Data Criteria and Processing for Liquid Drop Size Analysis |
Radiative Processes in Astrophysics - AstroBaki
Revision as of 00:02, 9 February 2010 by WikiSysop (talk | contribs) (Created page with '== The Black Body == A blackbody is the simplest source: it absorbs and reemits radiation with 100% efficiency. The frequency content of blackbody radiation is given by the ''P…')
A blackbody is the simplest source: it absorbs and reemits radiation with 100% efficiency. The frequency content of blackbody radiation is given by the Planck Function:
{\displaystyle B_{\nu }={\frac {h\nu }{\lambda ^{2}}}{2 \over (e^{\frac {h\nu }{kT}}-1)}}
{\displaystyle B_{\nu }={\frac {2h\nu ^{3}}{c^{2}(e^{\frac {h\nu }{kT}}-1)}}\neq B_{\lambda }}
The # density of photons having frequency between
{\displaystyle \nu }
{\displaystyle \nu +d\nu }
has to equal the # density of phase-space cells in that region, multiplied by the occupation # per cell. Thus:
{\displaystyle n_{\nu }d\nu ={\frac {4\pi \nu ^{2}d\nu }{c^{3}}}{\frac {2}{e^{\frac {h\nu }{kT}}-1}}}
{\displaystyle h\nu {\frac {n_{\nu }c}{4\pi }}=I_{\nu }=B_{\nu }}
so we have it. In the limit that
{\displaystyle h\nu \gg kT}
{\displaystyle B_{\nu }\approx {\frac {2h\nu ^{3}}{c^{2}}}e^{-{\frac {h\nu }{kT}}}}
{\displaystyle h\nu \ll kT}
{\displaystyle B_{\nu }\approx {\frac {2kT}{\lambda ^{2}}}}
Note that this tail peaks at
{\displaystyle \sim {\tfrac {3kT}{h}}}
{\displaystyle \nu B_{\nu }=\lambda B_{\lambda }}
Retrieved from "http:///astrobaki/index.php?title=Radiative_Processes_in_Astrophysics&oldid=4" |
Plot custom microphone element directivity or pattern versus elevation - MATLAB - MathWorks América Latina
Plot custom microphone element directivity or pattern versus elevation
patternElevation(sElem,FREQ)
patternElevation(sElem,FREQ,AZ)
patternElevation(sElem,FREQ,AZ,Name,Value)
PAT = patternElevation(___)
patternElevation(sElem,FREQ) plots the 2-D element directivity pattern versus elevation (in dBi) for the element sElem at zero degrees azimuth angle. The argument FREQ specifies the operating frequency.
patternElevation(sElem,FREQ,AZ), in addition, plots the 2-D element directivity pattern versus elevation (in dBi) at the azimuth angle specified by AZ. When AZ is a vector, multiple overlaid plots are created.
patternElevation(sElem,FREQ,AZ,Name,Value) plots the element pattern with additional options specified by one or more Name,Value pair arguments.
PAT = patternElevation(___) returns the element pattern. PAT is a matrix whose entries represent the pattern at corresponding sampling points specified by the 'Elevation' parameter and the AZ input argument.
AZ — Azimuth angles for computing directivity and pattern
Azimuth angles for computing sensor or array directivities and patterns, specified as a 1-by-N real-valued row vector where N is the number of desired azimuth directions. Angle units are in degrees. The azimuth angle must lie between –180° and 180°.
The azimuth angle is the angle between the x-axis and the projection of the direction vector onto the xy plane. This angle is positive when measured from the x-axis toward the y-axis.
[-90:90] (default) | 1-by-P real-valued row vector
Elevation angles, specified as the comma-separated pair consisting of 'Elevation' and a 1-by-P real-valued row vector. Elevation angles define where the array pattern is calculated.
Example: 'Elevation',[-90:2:90]
Element directivity or pattern, returned as an P-by-N real-valued matrix. The dimension P is the number of elevation angles determined by the 'Elevation' name-value pair argument. The dimension N is the number of azimuth angles determined by the AZ argument.
D=4\pi \frac{{U}_{\text{rad}}\left(\theta ,\phi \right)}{{P}_{\text{total}}}
pattern | patternAzimuth |
A remark on the stability of saturated generic graphs
October, 2005 A remark on the stability of saturated generic graphs
We show that any saturated generic graph satisfying some property (*) is strictly stable or
\omega
-stable. As a corollary, we obtain that any saturated generic pseudoplane is strictly stable or
\omega
Koichiro IKEDA. "A remark on the stability of saturated generic graphs." J. Math. Soc. Japan 57 (4) 1229 - 1234, October, 2005. https://doi.org/10.2969/jmsj/1150287312
Keywords: generic graph , pseudoplane , stability
Koichiro IKEDA "A remark on the stability of saturated generic graphs," Journal of the Mathematical Society of Japan, J. Math. Soc. Japan 57(4), 1229-1234, (October, 2005) |
Perpetual Contracts Guide - Delta Exchange - User Guide
Perpetual Contracts: Motivation & Use Cases
Perpetual contracts are a type of derivatives that are similar to a futures contract, but with some key differentiating properties:
Unlike futures, perpetual contracts do not have an expiry date
The price of a futures contract and its underlying can be quite different, with these two prices being guaranteed to converge at the contract expiry. Perpetual contracts by design trade close to the price of the underlying (spot). The closeness of perpetual price and spot price is achieved through funding which is explained below.
The above property makes trading perpetual contracts akin to trading spot markets on leverage, i.e. margin trading.
Benefits of a perpetual contracts vs. futures
Since a futures contract has an expiry date, a trader looking to maintain his position will need to periodically roll to a next contract as the previous one expires. Perpetual contracts obviate the need to roll positions.
The difference between price of a futures and its underlying (i.e. the basis) can vary quite a bit. This exposes futures traders to basis risk. Since perpetual contract always trades close to spot market, the basis risk is minimal and bounded.
Funding Rate Explanation
Funding is the primary mechanism which tethers price of a perpetual contract to spot. Funding is a series of continuous payments that are exchanged between longs and shorts in a perpetual contract. Let’s understand how funding helps keep price of the perpetual contract close to the spot price.
Perpetual contract price > Spot price
When a perpetual contract trades at a premium to spot, funding tends to be positive, i.e. longs pay funding to shorts. This creates disincentive to stay long or enter into a new long position. Conversely, it creates incentive to stay short or enter into a new short position. These dynamics will serve to push the price of the perpetual contract down towards the spot price.
Perpetual contract price < Spot price
When a perpetual contract trades at a discount to spot, funding tends to be negative, i.e. shorts pay funding to longs. This creates disincentive to stay short or enter into a new short position. Conversely, it creates incentive to stay long or enter into a new long position. These dynamics will serve to push the price of the perpetual contract up towards the spot price.
Funding Rate is considered to be an 8-hourly rate and is the sum of two terms: (a) Premium and (b) Interest Rate.
P r e m i u m = ( M a r k P r i c e − U n d e r l y i n g \_ I n d e x \_ P r i c e ) / U n d e r l y i n g \_ I n d e x \_ P r i c e
The details on how the Mark Price is calculated are available here.
Premium is measured every minute and its 8-hour TWAP (Avg. Premium) is used in the computation Funding Rate.
The Interest Rate term in Funding calculation is a function of the differential of borrow rates of quote currency and base currency of the perpetual contract. The Interest Rate thus is a proxy for cost of holding a position in a perpetual contract.
Since borrow rates for different currencies can vary widely, the Interest Rate used in Funding calculations can vary from contract to contract. However, as of now, Interest Rate of 0.01%/ 8 hours is being used for contracts.
Funding Rate is computed using the following formula:
Funding\ Rate = Avg.\ Premium + clamp (Interest\ Rate - Avg.\ Premium, 0.05\%, -0.05\%)
The clamp function limits the value of (Interest Rate - Avg. Premium) to a band of (-0.05%, 0.05%).
This means that if (Interest Rate - Avg. Premium) is within +/-0.05%, Funding Rate is equal to:
Funding\ Rate = Avg.\ Premium + (Interest\ Rate - Avg.\ Premium) = Interest\ Rate
When (Interest Rate - Avg. Premium) < -0.05%, then Funding Rate is equal to:
Funding\ Rate = Avg.\ Premium - 0.05\%
And, when (Interest Rate - Avg. Premium) > 0.05%, then Funding Rate is equal to:
Funding\ Rate = Avg.\ Premium + 0.05\%
Funding Rate is computed 3 times in a 24 hour period at: 8am UTC, 4pm UTC and 12am UTC. At these times, the TWAP of Premium in the preceding 8 hours is used to compute the Funding Rate. This Funding Rate thus obtained remains applicable for the next 8 hours.
At any instant, there are two Funding Rates available: (a) the Funding Rate that is currently applicable and (b) the estimate of the Funding Rate that will be applicable in the next 8 hour interval. Both these rates are available on the price ticker on the top of the trading terminal.
Funding is exchanged between longs and shorts every minute. It is important to note that while accruals for funding paid or received happen every minute, entries for funding payments are made in the transaction log once every 2 hours or when a position is closed. Funding payments are completely peer-to-peer and Delta Exchange does not charge any fees on funding. Funding paid or received is computed as:
Funding\ Payment = Current\_Position\_Value * Funding\ Rate * 1/ (8 * 60)
where Current Position Value is value of a the position at the current Underlying Index Price.
A perpetual contract can be thought as an 8-hour futures contract that is being rolled into the next 8-hour futures every minute. Thus, at any time, the fair basis of a perpetual contract should be similar to a futures contract which will expire in 8 hours. With this in mind, we enforce pretty tight caps on the fair basis for perpetual contracts. As of now, most perpetual contracts have funding capped at 0.5% or 0.15% (for alt-btc pairs). But these caps are subject to change and are available in the contract specifications.
Lets say you have a long position of 10000 contracts in the BTCUSD Perpetual contract on Delta Exchange. Recall that 1 BTCUSD contract is 1 USD.
Between 8am UTC and 4pm UTC, the TWAP of Premium was 0.04%. Assuming Interest Rate is 0.01%, for the next 8 hours, i.e. between 4pm UTC and 12am UTC, the applicable Funding Rate will be:
Funding\ Rate = 0.04\% + clamp(0.01\% - 0.04\%, 0.05\%, -0.05\%) = 0.01\%
Since you are long and Funding Rate is positive, you’d be paying funding.
Let’s assume that you hold this long position from 5pm UTC through to 5:30pm UTC. During this time, both the Mark Price and Underlying Index Price stay flat at $4015 and $4000 respectively.
Funding\ Paid\ per\ min = 10000 * (1/ 4000) * 0.01\% * 1/ (8*60) = 0.00000052 BTC
And, the funding paid by you in 30 mins can be computed as:
Funding\ Paid\ for\ 30\ mins = 30 * 0.00001693 = 0.00001562 BTC
Funding for OTC Contracts
For OTC contracts, funding rate is not computed using the order book. Instead, funding rate is charged by the party providing liquidity to the party that demands liquidity. While the funding rate generally stays constant, it can change with the liquidity situation in the market or with sharp price moves. All other dynamics of funding remain unchanged, i.e. in OTC contracts too: (a) funding is peer to peer, (b) if funding rate is positive: longs pay shorts and if funding rate is negative: shorts pay longs, and (c) funding is exchanged between longs and shorts every minute. |
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