text stringlengths 256 16.4k |
|---|
Chesmistry
Mensuration is the branch of mathematics that studies the measurement of the geometric figures and their parameters like length, volume, shape, surface area, lateral surface area, etc.
Here, the concepts of mensuration are explained and all the important mensuration formulas provided.
A branch of mathematics that talks about the length, volume, or area of different geometric shapes is called Mensuration.
These shapes exist in 2 dimensions or 3 dimensions.
Here we can discuss about what is mensuration
Mensuration is a topic in Geometry which is a branch of mathematics.
Mensuration deals with length, area and volume of different kinds of shape- both 2D and 3D. So moving ahead in the introduction to Mensuration.
2D shape is a shape that is bounded by three or more straight lines or a closed circular line in a plane. These shapes have no depth or height. They have two dimensions length and breadth. Therefore called 2D figures or shapes. For 2D shapes, we measure Perimeter (P) and Area (A).
3D shape is a shape that is bounded by a number of surfaces or planes. These are also referred to as solid shapes. These shapes have height or depth unlike 2D shapes, they have three dimensions Length, Breadth and Height/Depth and therefore they are called 3D figures. 3D shapes are actually made up of a number of 2D shapes. Also, know as solid shapes, for 3D shapes we measure Volume (V), Curved Surface Area (CSA), Lateral Surface Area (LSA) and Total Surface Area (TSA).
Click here for more details about mensuration
CLOSED FIGURE : the plain figure which have same starting and ending points is called closed figure.
OPEN FIGURE : the plain figure which have different starting and ending points is called open figure.
CONFEX FIGURE : the plain which have all angles than 180o then is called confex figure.
In plain figure all diagnols lies inside the confex figure.
if you take any two points in a figure the line segment joining those two points is also inside that figure then it is called confex figure.
CONCAVE FIGURE : at least one angle is greater than 1800 then the figure is called concave figure.
if you take any two points in the figure the line segment joining those two points is not belongs to interior of the diagram then it is called concave figure.
POLYGON : a closed and confex figure which has 3 or more sides is called polygon.
TRIANGLE : a closed , confex figure which has 3 sides is called triangle.
PERIMETER : total length of boundary line of the plain figure is called perimeter.
AREA : the amount of surface of enclosed plain figure is called area.
NOTE: the perimeter of a triangle is /AB+/BC+//CA = A+B+C
AREA OF RIGHT ANGLED TRIANGLE :
=\frac{1}{2}×base×height\phantom{\rule{0ex}{0ex}}=\frac{1}{2}bh\phantom{\rule{0ex}{0ex}}
AREA EQUALITARAL TRIANGLE :
AREA\quad OF\quad EQUALITARAL\quad =\quad \frac{1}{2}\times B\times H\phantom{\rule{0ex}{0ex}}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad =\quad \frac{1}{2}\times a\times \frac{\sqrt{3}}{2}\times a\phantom{\rule{0ex}{0ex}}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad =\frac{\sqrt{3}}{4}{a}^{2}\phantom{\rule{0ex}{0ex}}
A{C}^{2}=A{D}^{2}+D{C}^{2}\phantom{\rule{0ex}{0ex}}{a}^{2}={h}^{2}+\frac{{a}^{2}}{4}\phantom{\rule{0ex}{0ex}}{a}^{2}–\frac{{a}^{2}}{4}={h}^{2}\phantom{\rule{0ex}{0ex}}{h}^{2\quad }=\frac{4{a}^{2}–{a}^{2}}{4}\phantom{\rule{0ex}{0ex}}{h}^{2}=\quad \frac{3{a}^{2}}{4}\phantom{\rule{0ex}{0ex}}h\quad =\frac{\sqrt{3}}{2}a\phantom{\rule{0ex}{0ex}}therefore\quad the\quad height\quad of\quad the\quad equilateral\quad triangle\quad is\quad h=\frac{\sqrt{3}}{2}a
HERONS FORMULA :
\sqrt{S\left(s–a\right)}\left(s–b\right)\left(s–c\right)\phantom{\rule{0ex}{0ex}}S\quad =\quad seri\quad perimeter\quad \phantom{\rule{0ex}{0ex}}S=\frac{a+b+c}{2}
AREA OF A RIGHT ANGLE ISOSCELES TRIANGLE :
=\frac{1}{2}×a×a\phantom{\rule{0ex}{0ex}}=\frac{1}{2}{a}^{2}
QUADRILATERAL : a closed convex figure which has 4sides is called quadrilateral.
area\quad of\quad quadrilateral\quad :\quad \left[\frac{1}{2}d\left({h}_{1}+{h}_{2}\right)\right]
TRAPEZIUM : In a quadrilateral one pair of opposite sides are parallel.
area\quad of\quad trapezium\quad :\quad \left[\frac{1}{2}h\left(a+b\right)\right]
PARALLELOGRAM : In a quadrilateral opposite sides are parallel the lengths of the diagonals are not equal is called parallelogram.
area\quad of\quad paralle\mathrm{log}ram\quad \left(l{l}^{gm}\right)=bh
RECTANGLE : In a parallelogram one angle is 90 degrees then it is called rectangle.
area\quad of\quad rec\mathrm{tan}gle\quad =\quad l\times b\phantom{\rule{0ex}{0ex}}perimeter\quad of\quad rec\mathrm{tan}gle\quad =2\left(l+b\right)
RHOMBUS : In a quadrilateral all sides are equal but lengths of the diagonals are not equal then is called rhombus.
{d}_{1\quad ,\quad }{d}_{2\quad }\quad are\quad lengths\quad of\quad the\quad diagonals\phantom{\rule{0ex}{0ex}}=\quad area\quad of\quad rhombus\quad =\quad \frac{{d}_{1}\quad {d}_{2}}{2}
SQUARE : In a quadrilateral all sides equal and diagonals are also equal then it is called square. (or)
in a rhombus one angle is 90 degrees then it is called square.
area\quad of\quad the\quad square\quad ={a}^{2}\phantom{\rule{0ex}{0ex}}perimeter\quad of\quad the\quad square\quad =\quad 4a
all squares are rhombus.
all squares are parallelograms.
all squares are trapeziums.
some rhombus are squares.
some rhombus are rectangles.
some rectangles are rhombus.
some rectangles are squares.
all rhombus are parallelogram.
all rhombus are trapeziums.
all rhombus are quadrilaterals.
all rectangles are parallelograms.
all rectangles are trapeziums.
all rectangles are quadrilaterals.
all parallelograms are trapeziums.
all parallelograms are quadrilaterals.
all trapeziums are quadrilaterals.
some quadrilaterals are trapezium.
some quadrilaterals are parallelograms.
some quadrilaterals are rectangles.
some quadrilaterals are rhombus.
some quadrilaterals are squares.
some trapeziums are parallelograms.
some trapeziums are rectangles.
some trapeziums are rhombus.
some trapeziums are squares.
some parallelograms are rectangles.
some parallelograms are rhombus.
some parallelograms are squares.
types of quadrilaterals no of measurements we need for construction
CIRCLE : The locus of the points which are equal distance from the fixed point is called a circle denoted by (o).
the fixed point is called centre of the circle.
RADIUS : The distance between the center of the circle and the point on the circle is called radius.
CHORD : A line segment joining any two points on a circle is called chord.
DIAMETER : The largest chord of the circle is called diameter.
NOTE : All diameters passes through the centre of the circle.
we can draw infinite diameter on the circle. it is denoted by (d).
ARC : The part of the circle is called arc.
SECTOR : The area between two radius OA,OB and ARC AB is called sector.
Area\quad of\quad circle\quad =\quad {\mathrm{\pi r}}^{2\quad }\quad \mathrm{or}\quad \frac{{\mathrm{\pi d}}^{2}}{4}\phantom{\rule{0ex}{0ex}}\mathrm{circumference}\quad \mathrm{of}\quad \mathrm{a}\quad \mathrm{circle}\quad =\quad 2\mathrm{\pi r}\quad \mathrm{or}\quad \mathrm{\pi d}\phantom{\rule{0ex}{0ex}}\mathrm{diameter}\quad \mathrm{of}\quad \mathrm{the}\quad \mathrm{circle}\quad =\quad \mathrm{d}–2\mathrm{r}\phantom{\rule{0ex}{0ex}}\mathrm{area}\quad \mathrm{of}\quad \mathrm{the}\quad \mathrm{semicircle}\quad =\quad \frac{{\mathrm{\pi r}}^{2}}{2}\phantom{\rule{0ex}{0ex}}\mathrm{circumference}\quad \mathrm{of}\quad \mathrm{semicircle}=\quad \mathrm{\pi r}\phantom{\rule{0ex}{0ex}}\mathrm{area}\quad \mathrm{of}\quad \mathrm{the}\quad \mathrm{sector}\quad =\quad \frac{\mathrm{lr}}{2}(\mathrm{here}\quad \mathrm{l}=\mathrm{lrength}\quad \mathrm{of}\quad \mathrm{the}\quad \mathrm{arc},\quad \mathrm{r}=\quad \mathrm{radius})\quad \mathrm{or}\quad \frac{{\mathrm{x}}^{0}}{{360}^{0}}\times {\mathrm{\pi r}}^{2}\phantom{\rule{0ex}{0ex}}\mathrm{length}\quad \mathrm{of}\quad \mathrm{the}\quad \mathrm{arc}\quad =\quad \mathrm{l}=\frac{{\mathrm{x}}^{0}}{{360}^{0}}\times 2\mathrm{\pi r}\phantom{\rule{0ex}{0ex}}\mathrm{area}\quad \mathrm{of}\quad \mathrm{the}\quad \mathrm{rectangular}\quad \mathrm{path}\quad =\quad \mathrm{area}\left(\mathrm{outerpath}\right)–\mathrm{area}\left(\mathrm{innerpath}\right)\phantom{\rule{0ex}{0ex}}\mathrm{path}\quad \mathrm{of}\quad \mathrm{the}\quad \mathrm{square}=\mathrm{area}(\mathrm{outer}\quad \mathrm{square})–\mathrm{area}(\mathrm{inner}\quad \mathrm{space})\phantom{\rule{0ex}{0ex}}\mathrm{path}\quad \mathrm{of}\quad \mathrm{the}\quad \mathrm{circle}=\mathrm{area}\left(\mathrm{outercircle}\right)–\mathrm{area}(\mathrm{inner}\quad \mathrm{circle})
3-d figures : the plane figures which have 3 measures (length,breadth,height) is called 3-d figures.
eg: cube,cuboid,cylinder,cone,sphere,hemisphere
\mathit{c}\mathit{u}\mathit{b}\mathit{e}\mathbf{\quad }\mathbf{:}\mathbf{\quad }\phantom{\rule{0ex}{0ex}}lateral\quad surface\quad area\quad of\quad cube\quad :\quad 4{a}^{2}\phantom{\rule{0ex}{0ex}}total\quad surface\quad area\quad of\quad cube\quad :\quad b{a}^{2}\phantom{\rule{0ex}{0ex}}volume\quad of\quad cube\quad :\quad {a}^{3}
\mathit{c}\mathit{u}\mathit{b}\mathit{o}\mathit{i}\mathit{d}\mathbf{\quad }\mathbf{:}\mathbf{\quad }\phantom{\rule{0ex}{0ex}}lateral\quad surface\quad area\quad of\quad cuboid\quad :\quad 2h\left(l+b\right)\phantom{\rule{0ex}{0ex}}total\quad surface\quad area\quad of\quad cuboid\quad :\quad 2\left(lb+bh+hl\right)\phantom{\rule{0ex}{0ex}}volume\quad of\quad cuboid\quad :\quad lbh\phantom{\rule{0ex}{0ex}}length\quad of\quad the\quad diagnoal\quad of\quad cuboid\quad :\quad \sqrt{{l}^{2}+{b}^{2}+{h}^{2}}\phantom{\rule{0ex}{0ex}}
\mathit{c}\mathit{y}\mathit{l}\mathit{i}\mathit{n}\mathit{d}\mathit{e}\mathit{r}\mathbf{\quad }\mathbf{:}\mathbf{\quad }\phantom{\rule{0ex}{0ex}}curved\quad surface\quad area\quad of\quad cylinder\quad :\quad 2\mathrm{\pi rh}\phantom{\rule{0ex}{0ex}}\mathrm{total}\quad \mathrm{surface}\quad \mathrm{area}\quad \mathrm{of}\quad \mathrm{cylinder}\quad :\quad 2\mathrm{\pi r}\left(\mathrm{r}+\mathrm{h}\right)\phantom{\rule{0ex}{0ex}}\mathrm{volume}\quad \mathrm{of}\quad \mathrm{cylinder}\quad :\quad {\mathrm{\pi r}}^{2}\mathrm{h}
\mathit{c}\mathit{o}\mathit{n}\mathit{e}\mathbf{\quad }\mathbf{:}\phantom{\rule{0ex}{0ex}}\mathit{r}\mathbf{\quad }\mathbf{:}\mathbf{\quad }radius\quad of\quad cone\phantom{\rule{0ex}{0ex}}h:\quad height\quad of\quad cone\quad \phantom{\rule{0ex}{0ex}}l\quad :\quad slant\quad heigjht\quad of\quad cone\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}curved\quad surface\quad area\quad of\quad cone\quad :\quad \mathrm{\pi rl}\phantom{\rule{0ex}{0ex}}\mathrm{total}\quad \mathrm{surface}\quad \mathrm{area}\quad \mathrm{of}\quad \mathrm{cone}\quad :\quad \mathrm{\pi r}\left(\mathrm{l}+\mathrm{r}\right)\phantom{\rule{0ex}{0ex}}\mathrm{volume}\quad \mathrm{of}\quad \mathrm{cone}\quad :\quad \frac{1}{2}{\mathrm{\pi r}}^{2}\mathrm{h}
\mathit{s}\mathit{p}\mathit{h}\mathit{e}\mathit{r}\mathit{e}\mathbf{\quad }\mathbf{:}\mathbf{\quad }\phantom{\rule{0ex}{0ex}}curved\quad surface\quad area\quad of\quad sphere\quad :\quad 4{\mathrm{\pi r}}^{2}\phantom{\rule{0ex}{0ex}}\mathrm{total}\quad \mathrm{surface}\quad \mathrm{area}\quad \mathrm{of}\quad \mathrm{sphere}\quad :\quad 4{\mathrm{\pi r}}^{2}\phantom{\rule{0ex}{0ex}}\mathrm{volume}\quad \mathrm{of}\quad \mathrm{sphere}\quad :\quad \frac{4}{3}{\mathrm{\pi r}}^{3}\phantom{\rule{0ex}{0ex}}
\mathit{h}\mathit{e}\mathit{m}\mathit{i}\mathit{s}\mathit{p}\mathit{h}\mathit{e}\mathit{r}\mathit{e}\mathbf{\quad }\mathbf{:}\mathbf{\quad }\phantom{\rule{0ex}{0ex}}curved\quad surface\quad area\quad of\quad hemisphere\quad :\quad 2{\mathrm{\pi r}}^{2}\phantom{\rule{0ex}{0ex}}\mathrm{total}\quad \mathrm{surface}\quad \mathrm{area}\quad \mathrm{of}\quad \mathrm{hemisphere}\quad :\quad 3{\mathrm{\pi r}}^{2}\phantom{\rule{0ex}{0ex}}\mathrm{volume}\quad \mathrm{of}\quad \mathrm{hemisphere}\quad :\quad \frac{2}{3}{\mathrm{\pi r}}^{3}
Mensuration in Mathematics
mensuration formula list
mcq questions for mensuration
what is mensuration in science
atomic structure class 11
environmental issues class 12
biodiversity and conservation class 12
what is a ecosystem
what is reproductive health
mensuration class 10
ncert solution class 8 maths chapter 5
square and square roots class 8
ncert 10 maths solution
AatmaNirbhar Bharat App
class 8 maths algebraic expressions
playing with numbers class 8
visualising solid shapes class 8
PM Cares Fund for Children Scheme |
Revision as of 17:51, 19 August 2019 by Nikolay (talk | contribs) (Created page with "The following tables list a lower bound on the Hamming distance between all known CCZ-inequivalent APN representatives up to dimension 11 using the methods described used in <...")
{\displaystyle l(F)=\lceil {\frac {m_{F}}{3}}\rceil +1}
{\displaystyle l(F)}
{\displaystyle (n,n)}
{\displaystyle F}
{\displaystyle m_{F}}
{\displaystyle m_{F}=\min _{b,\beta \in \mathbb {F} _{2^{n}}}|\{a\in \mathbb {F} _{2^{n}}:(\exists x\in \mathbb {F} _{2^{n}})(F(x)+F(a+x)+F(a+\beta )=b)\}|}
{\displaystyle m_{F}} |
Hodge cycles on abelian varieties associated to the complete binary trees
JANUARY, 2006 Hodge cycles on abelian varieties associated to the complete binary trees
Fumio HAZAWA
The structure of the ring of Hodge cycles on a certain family of abelian varieties of CM-type is investigated. This leads to an interesting combinatorial problem related to posets based on complete
p
-ary trees. A complete solution to the problem is given for the case
p=2
Fumio HAZAWA. "Hodge cycles on abelian varieties associated to the complete binary trees." J. Math. Soc. Japan 58 (1) 55 - 82, JANUARY, 2006. https://doi.org/10.2969/jmsj/1145287093
Primary: 06A11 , 11G10 , 14C30
Keywords: abelian variety , Binary tree , Hodge cycle
Fumio HAZAWA "Hodge cycles on abelian varieties associated to the complete binary trees," Journal of the Mathematical Society of Japan, J. Math. Soc. Japan 58(1), 55-82, (JANUARY, 2006) |
Spread Contracts Guide - Delta Exchange - User Guide
Spread Contracts: Motivation & Use Cases
Spread contracts are a special class of derivative where the underlying is a spread. The spread here could be the difference in prices of any two assets or even derivative contracts. Thus, a position in a spread contract is equivalent to offsetting (long and short respectively) positions in the assets/ contracts that form the spread.
Calendar spread contracts: These spread contracts are defined over the price difference of two futures contracts having the same underlying, but different maturities. To understand this better, let’s consider Bitcoin futures spread contract. This contract is defined as the difference in price of a longer-dated BTC futures (e.g. BTC December futures) and the price of a shorter-dated BTC futures (e.g. BTC September futures). Therefore, a position in the BTC futures spread contract represents a long and short position in the two underlying futures contracts.
Long BTC spread futures contract = Long longer-dated BTC futures + Short shorter-dated BTC futures
Short BTC spread futures contract = Short longer-dated BTC futures + Long shorter-dated BTC futures
It is evident from the above equations, calendar spread contracts should not be used for directional trades on the underlying asset (BTC in this case). Instead, these contracts should be used to express views on the relative pricing of the constituent futures.
Characteristics of calendar spread contracts
Market neutral: Because calendar spread contracts represent offsetting positions in two futures contracts of the same underlying, by definition, price of the spread contract should remain unimpacted by the change in price of the underlying. Let us go back to our Bitcoin futures spread contract example. If BTC prices goes up by 5%, the profits from the long position in one of the underlying BTC futures will be offset by short position in the other BTC futures.
This also means that a trade in a calendar spread contract is actually a call on the relative pricing of the two underlying futures. If the longer-dated futures is trading richer compared to the shorter-dated futures, you should short the spread contract. Alternatively, if the longer-dated futures is trading cheaper compared to the shorter-dated futures, you should long the spread contract.
Lower risk: Lower risk or price volatility of calendar spread contracts is a natural corollary of their market neutral nature. Moreover, spreads tend to be mean reverting. Lower volatility allows us to offer higher leverage on spread contracts. This means you can take bigger positions with smaller margin.
Cost efficient: For traders looking to gain from mis-pricing in futures with the same underlying with different expiry dates, taking a position in a spread contract is a lot more cost efficient than taking long/ short position in the two futures contracts. This is because:
(a) you need not provide margin for the long and short positions separately. Moreover, leverage allowed in the futures contracts tend to be lower than leverage for spread contracts. Therefore, an equivalent position in spread contracts require lesser margin, and
(b) you end up paying lower fees because you need to execute just one trade, as opposed to two, when you take a position in a spread contract.
Mechanics of Calendar Spread Contracts
Quotation & Settlement Currencies
Our BTC futures spread contracts are defined over the differences in prices of our BTC quarterly futures. These futures are inverse contracts, i.e. they are quoted in USD, but their margining and settlement happens in BTC.
Despite the constituent futures contract being inverse, the spread contracts are vanilla or linear. This enables quoting of USD value of spread for a selected bitcoin notional and thus, makes the mechanics of the contract more intuitive for traders.
Another detail worth noting here is that even though our BTC futures calendar spread contracts are quoted in USD, the margining currency is USDT. The USDT/ USD conversion rate is built into the contract and has been set at 1.
The order-book on calendar spread contracts on Delta represent the price at which traders are willing to buy or sell the difference between futures on two different maturities.
For the BTC futures calendar spread contract, the spread between the prices of the constituent futures is denominated in USD. Therefore, the price of the spread contract is quoted in USD.
Margining for calendar spread contracts
The margin requirement for a spread contract does not depend on the price of the spread contract. Instead, margin is computed using the notional size of the contract. For example, the notional size of our BTC futures spread contract is 0.1BTC. Therefore, the margin required for opening a position in the contract a fraction of the USD price of 0.1BTC.
\small Margin = Margin\% * Num\_of\_Contracts * Contract\_Size * Underlying's\_Spot\_Price
\small Margin\% = 1 / Leverage
That means, higher the leverage, lower Margin% and hence, lower the margin requirement.
A position can be opened at a Margin% that is higher than the Initial Margin% specified the contract’s specifications. Also, Just like in futures, margin requirement scales up with position size. Details of Margin Scaling are available here.
Open positions in calendar spread contracts are marked using Fair Price Marking. The fair price of a calendar spread contract is computed as the difference of the mark prices of the constituent futures contracts.
Profit/ Loss Equation
The profit/ loss (PnL) equation for a position in a spread contract is essentially the same as for vanilla futures.
\small PnL = Num\_of\_Contracts * Contract\_Size * (Current\_Contract\_Price - Entry\_Contract\_Price)
\small PNL = - Num\_of\_Contracts * Contract\_Size * (Current\_Contract\_Price - Entry\_Contract\_Price)
Please note that the profit/ loss of our BTC futures spread contract is denominated in USDT.
A position in a calendar spread contract goes into liquidation, when Position Margin after factoring in unrealised losses is less than Maintenance Margin, i.e.
Position\ Margin + Unrealised PNL = Maintenance\ Margin
where, (a) Position Margin is greater than or equal to Initial Margin, and (b) Unrealised PNL is computed at the prevailing Mark Rate.
The liquidation mechanism is exactly the same as for futures contracts. Any given position is liquidated in a step-wise manner to reduce the market impact of liquidations. Details of the liquidation process are available here.
Traders also have the option of enabling Auto Margin Top-up to prevent their positions from getting liquidated.
For the BTC futures calendar spread contract, both maker and taker fees are set at 0.05%. Trading fees are charged on the notional value of the .
The trading fees schedule for all the contracts listed on Delta Exchange is a available here.
Expired Calendar Spread Contracts
The Settlement Prices of expired spread contracts as well as other contracts (MOVE and futures) are available on this page.
Derivatives Guide - Previous
Next - Derivatives Guide |
THORNode Overview - THORChain Docs
THORNodes service the THORChain network. Each THORNode is comprised of several independent servers in a cluster. All THORNodes communicate and operate in cooperation to create a cross-chain swapping network.
Running a node is a serious undertaking. While Node Operators are well compensated for running a node, there are also risks, skills required and costs.
See the Node Operator 101 Video to learn more before running a node.
To set up a node, you have three choices:
Set up manually (not recommended unless you are an expert)
Set up via Kubernetes (recommended)
Set up via Provider (coming soon).
THORNode Stack
thornode - this is a daemon that runs the THORChain chain itself and a HTTP server, that gives a RESTful API to the chain.
gateway: THORNode gateway proxy to get a single IP address for multiple deployments
midgard - this daemotn is a layer 2 REST API that provides front-end consumers with semi real-time rolled up data and analytics of the THORChain network. Most requests to the network will come through Midgard. This daemon is here to keep the chain itself from fielding large quantities of requests. You can think of it as a “read-only slave” to the chain. This keeps the resources of the network focused on processing transactions.
As new nodes join/leave the network, this triggers a “churning event”. Which means the list of validators that can commit blocks to the chain changes, and also creates a new Asgard vault, while retiring an old one. All funds in this retiring vault are moved to the new Asgard vault.
Normally, a churning event happens every 3 days (50,000 blocks), although it is possible for it to happen more frequently (such as when a node optionally requests to leave the network using the LEAVE memo).
On every churn, the network selects one or more nodes to be churned out of the network (which can be typically churned back in later). In a given churning event, multiple nodes may be selected to be churned out, but never more than 1/3rd of the current validator set. The criterion the network will take into account is the following:
Requests to leave the network (self-removal)
Banned by other nodes (network-removal)
How long an active nodes has been committing blocks (oldest gets removed)
Bad behavior (accrued slash points for poor node operation)
Churning In
On every churn, the network may select one or more nodes to be churned into the network but never adds more than one to the total. Which nodes that are selected are purely by validator bond size. Larger bond nodes are selected over lower bond nodes.
There is a endpoint on Midgard that has deep analytics in mean and median active & standby bond sizes to drive efficient discovery of the best "bond" size. Chaosnet minimum bond is currently: 300K Rune (MIMIR Override) however competition has raised this higher. In the long term it is likely to stabilise between 2m and 2.5m RUNE.
The network is safe when it is over-bonded, but it shrewd Node Operators will probably actively manage their bond and pool part of it instead to maximise yield.
Risk of Running a Node
Deciding to run a node should be carefully considered and thought through. While the payoffs/rewards can be significant, there can also be an equally significant costs.
Risks to Bond
To run a node, you must obtain a significant amount of Rune, minimums apply. This RUNE is sent into the network as “bond” and held as leverage on each node to ensure they behave in the best interest of the network.
Running a malicious node or stealing from the network results in a slashing of this bond. Here are the ways in which a validator’s bond can get slashed.
Double Sign (5% of minimum bond) - if it is observed that a single validator node is committing blocks on multiple chains. To avoid this, never run two nodes with the same node account at the same time.
Unauthorised transaction (1.5x transaction value) - if a node sends funds without authorization, the bond is slashed 1.5x the value of the stolen funds. The slashed bond is dumped into the pool(s) where the funds were stolen and added to the reserve.
Bond slashing takes directly from the bond and does not affect rewards.
Risk to Income
When a node is active, it earns rewards from the network in RUNE. Sufficient rewards are required to be earned in order for a Validator to be profitable. Running an unreliable node results in rewards being slashed. Here are the ways in which a validator’s rewards can be slashed.
Not Observing (2 slash pts) - if a node does not observe transactions for all chains, while other nodes do, they get slash points added.
Not signing a transaction (600 slash pts) - if a node does not sign an outbound transaction, as requested by the network, they will get slash points added.
Fail to keygen (1 hr of revenue) - When the network attempts to churn, and attempts to create a new Asgard pubkey for the network, and fails to do so due to a specific node(s), they will lose 1 hr of revenue from their bond.
Slash points undo profits made on the network. For every 1 slash point a node receives, they lose 1 block of rewards. Rewards slashing reduces earned rewards and does not affect a validator’s bond.
Bond Rewards
Node Operators receive rewards if they are bonded and active on the network and are paid out in Rune. While revenue is generated every block (every 5 seconds) to each operator, those funds are not available to the operator until they churn out of the network. Each operator makes the same amount of income, no matter how much they bond to the network.They're claimed whenever a node leaves the network. See Keeping Track of Rewards below for more details.
Nodes receive the same amount of rewards regardless of how much RUNE they've bonded. This stabilises the amount that nodes need to bond. Over time, this stability increases the median bonded amount and the security of the network.
Rewards are affected by the Emission Schedule and the Incentive Pendulum. Over time, the Emission Schedule decreases the amount of RUNE allocated to nodes. The Incentive Pendulum increases and decreases the amount of RUNE allocated to nodes according to the security and capital efficiency of the network.
When a node joins the network the current block height is recorded. The system creates one block unit for every active node for every active block, and has a running total of the block units created. When a node leaves, it cashes in its block units for a portion of the bond rewards. The spent block units are destroyed.
For example, there are 10000 RUNE in bond rewards outstanding. Node A has been active for 30 blocks, and has 33 block units, but accrued 3 slash points. There are 1000 block units in total. Node A leaves the network and cashes in its 30 block units (33 - 3). It receives 300 RUNE ((30/1000) * 10000), leaving 9700 RUNE in node rewards. Its 33 block units are destroyed, leaving 967 block units left.
Income for one node can be estimated based on a few inputs:
Number of active nodes
Reward emission rate
% of rewards allocated to nodes, set by the Incentive Pendulum
Price of RUNE*
These inputs should be plugged into the following formula:
{{RewardAllocation * EmissionRate} \over {NumberOfNodes}}
An example with mainnet day 1 inputs:
3.06 million RUNE rewards emitted per month
67% of rewards allocated to nodes (stable Incentive Pendulum)
{{{0.67 * 3060000} \over {33}}} = 62,127
In this example, an individual operator would receive 62,127 RUNE over the month.
Depending on how the node was set up, it will likely cost between $1000 and $2000 per month, potentially more as the blockchain scales. The main driver of costs is resource allocation to hosting each THORNode service.
Running a THORNode is no simple task. As an operator, you will need to run/maintain multiple linux servers with extremely high uptime. It is encouraged that only professional systems engineers run nodes to maintain the highest quality reliability and security of the network. The simple question to know if you have the skillsets to run a THORNode is:
If the answer is no, it’s probably best that you do not run a node and participate in the network in other ways. The following skill sets are required to be an effective node operator.
Advanced knowledge of Linux server administration and security
Advanced knowledge of Kubernetes
Advanced experience running a host of nodes on a hosted platform such as AWS, Google Cloud, Digital Ocean, etc
Knowledge of running full nodes for other chains such as Bitcoin, Ethereum, and Binance.
Willingness to be “on call” at all times to respond to issues when/if your node becomes unavailable
THORNode Details
When you run a THORNode, each THORNode will have its own node account. An example node account looks like this:
"node_address": "thor19h62vypuelrj0pv4jhl26wf79yr5zhxcmd5w85",
"pub_key_set": {
"secp256k1": "thorpub1addwnpepq0ylvhrqepmsm3rqxr4ecuyx42l4y29g2d932zlse258432ccuy8spmddj5",
"ed25519": "thorpub1addwnpepq0ylvhrqepmsm3rqxr4ecuyx42l4y29g2d932zlse258432ccuy8spmddj5"
"validator_cons_pub_key": "thorcpub1zcjduepqzknjn39xtkdzr6a2zuzry7f02rn3cnqvy8ar7p2admk80j8xageqj9slpw",
"bond": "30240000000000",
"active_block_height": "180",
"bond_address": "tbnb1rqhrnvyex4p5zchhu0slgr76dc4cl5dnvzxx2h",
"status_since": "123",
"signer_membership": [
"thorpub1addwnpepqwm3arc5xqgjf8yt70psygcjasfj3c776cux4yxcaacyd9vm859lckczll7"
"requested_to_leave": false,
"forced_to_leave": false,
"leave_height": "0",
"slash_points": "17",
"jail": {
"release_height": "2393",
"reason": "failed to perform keysign"
Most importantly, this will tell you how many slash points the node account has accrued, their status, and the size of their bond (which is in 1e8 notation, 1 Rune == 100000000).
Types of node status:
Unknown - this should never be possible for a valid node account
Whitelisted - node has been bonded, but hasn’t set their keys yet
Standby - waiting to have minimum requirements verified to become “ready” status. This check happens on each churn event (3 days on average).
Ready - node has met minimum requirements to be churned and is ready to do so. Could be selected to churn into the network. Cannot unbond while in this status.
Active - node is an active participant of the network, by securing funds and committing new blocks to the THORChain blockchain. Cannot unbond while in this status.
Disabled - node has been disabled by use of LEAVE, and cannot re-join.
To get node account information, make an HTTP call to your thor-api port which will look like the following:
http://<host>:1317/thorchain/nodeaccount/<node address>
http://<host>:1317/thorchain/nodeaccounts
THORNodes have the ability to vote on Mimir settings.
Mimir settings have specific abilities. The process for voting from a Node is:
Make mimir
=> Enter THORNode Mimir key: <key>
=> Enter THORNode Mimir value: <value>
Mimir keys and values are listed in the Mimir endpoint.
A node can vote at any time on any key value.
A node's vote is valid as long as they are active (and removed if they are not).
2/3rds of active nodes need to agree for the change to happen
If 2/3rds consensus is not reached, Mimir admin takes priority, or a constant if present.
A node can change their vote anytime.
A node can delete their vote by using -1 value
Voting costs one native transaction fee, which is deducted from their bond. |
Radiation dosage - Wikiversity
Various types of radiation including ionizing radiation may cause harm to people, researchers, and students under different situations.
This problem set is designed to help you calculate how much radiation and of what type you may be exposed to and how much damage it might cause.
The idea is forewarned is forearmed so that should you find yourself performing research requiring the use of radiation you will use proper and effective precaution.
3 Dose equivalents
Radiation sickness[edit | edit source]
Main articles: Radiation/Sickness and Radiation sickness
Def. any "illness produced by ionizing radiation with symptoms ranging from nausea through to death"[1] is called radiation sickness.
Main articles: Radiation/Doses and Dosages
Def. an "addition of a small measured amount of a substance to something"[2] is called a dosage.
Def. a "quantity of an agent (not always active) substance or radiation administered at any one time"[3] is called a dose.
{\displaystyle H=W_{R}\cdot D}
The weighting factor (sometimes referred to as a quality factor) is determined by the radiation type and energy range.[4]
{\displaystyle H_{T}=\sum _{R}W_{R}\cdot D_{T,R}\ ,}
The effective dose of radiation (E), absorbed by a person is obtained by averaging over all irradiated tissues with weighting factors adding up to 1:[4][5]
{\displaystyle E=\sum _{T}W_{T}\cdot H_{T}=\sum _{T}W_{T}\sum _{R}W_{R}\cdot D_{T,R}}
Gray[edit | edit source]
The gray (symbol: Gy) is the SI derived unit of absorbed radiation dose of ionizing radiation (for example, X-rays), and is defined as the absorption of one joule of ionizing radiation by one kilogram of matter (usually human tissue).[6] The rad is equivalent to 0.01 Gy.
{\displaystyle 1\ \mathrm {Gy} =1\ {\frac {\mathrm {J} }{\mathrm {kg} }}=1\ \mathrm {m} ^{2}\cdot \mathrm {s} ^{-2}}
The worldwide average natural [effective radiation] dose to humans is about 2.4 millisievert (mSv) per year.[8]
The biggest source of natural background radiation is airborne radon, a radioactive gas that emanates from the ground. Radon and its isotopes, parent radionuclides, and decay products all contribute to an average inhaled dose of 1.26 mSv/a. Radon is unevenly distributed and variable with weather, such that much higher doses apply to many areas of the world, where it represents a significant health hazard. Concentrations over 500 times higher than the world average have been found inside buildings in Scandinavia, the United States, Iran, and the Czech Republic.[9]
An average human contains about 30 milligrams of potassium-40 (40K) and about 10 nanograms (10−8 g) of carbon-14 (14C), which has a decay half-life of 5,730 years. Excluding internal contamination by external radioactive material, the largest component of internal radiation exposure from biologically functional components of the human body is from potassium-40. The decay of about 4,000 nuclei of 40K per second[10] makes potassium the largest source of radiation in terms of number of decaying atoms. The energy of beta particles produced by 40K is also about 10 times more powerful than the beta particles from 14C decay. 14C is present in the human body at a level of 3700 Bq with a biological half-life of 40 days.[11] There are about 1,200 beta particles per second produced by the decay of 14C. However, a 14C atom is in the genetic information of about half the cells, while potassium is not a component of DNA. The decay of a 14C atom inside DNA in one person happens about 50 times per second, changing a carbon atom to one of nitrogen.[12] The global average internal dose from radionuclides other than radon and its decay products is 0.29 mSv/a, of which 0.17 mSv/a comes from 40K, 0.12 mSv/a comes from the uranium and thorium series, and 12 μSv/a comes from 14C.[8]
A gamma-ray burst has occurred somewhere nearby to Earth. The burst at maximum intensity lasted 100 s. While gamma-rays are absorbed by the Earth's atmosphere, conditions are such that where you are walking outside, you receive and absorb 10 % of the intensity over the 100 s.
The gamma-rays are at 1.8 MeV to the ground. The flux upon you is 4 x 103 photons · cm-2 · s-1 · MeV-1.
What is your absorbed dose and your dose equivalent? Calculate your various effective doses and your dose rate.
Using the table below, describe your likely symptoms if any.
Whole-body absorbed dose ([Gray] Gy)
Greater Than 30 Gy
Fever None Moderate increase (10-100%) Moderate to severe (100%) Severe (100%) Severe (100%)
[Central nervous system] CNS function No impairment Cognitive impairment 6–20 h Cognitive impairment > 24 h Rapid incapacitation Seizures, Tremor, Ataxia, Lethargy
Epilation after 3 Gy Severe leukopenia
Mortality Without care 0–5% 5–100% 95–100% 100% 100%
With care 0–5% 5–50% 50–100% 100% 100%
Death 6–8 weeks 4–6 weeks 2–4 weeks 2 days–2 weeks 1–2 days
A few minutes after you've calculated your condition, a special report comes through an emergency channel on your cell phone to tell you that the initial report mentioned above was in error. The actual flux received whole body is 106 higher. Recalculate your situation and answer the above questions and calculations again. How are you doing? Will you live to get to Problem 2?
Gamma-ray bursts are often followed by an X-ray afterglow. Even though you are at high altitude for some skiing when the gamma-ray burst occurred, there is probably more than enough atmosphere to prevent any further damage from the X-rays. The afterglow lasts for two days at only 5 % of the final flux of gamma rays at 120 keV. If the atmosphere had not been there, answer the questions and calculations of Problem 1 for the X-rays.
This just isn't your day. On a separate military channel given to you by one of your Army buddies you find out that during the gamma ray burst about the same flux of protons and pions was received by you. Recalculate again and assess your condition. Do you need to know where the nearest hospital is?
Oh, yes, this just keeps getting better and better. One of the people in your skiing group has a contact at the local hospital. She has just learned that alpha particles at a comparable flux were included behind the gamma-ray burst for a day and a half. Re-assess your situation again. Will you live long enough to try that new restaurant in town?
A first aid worker was wearing a neutron detector and happen to fall ill near you. You glance at the detector and notice it has maxed out at L = 100 keV/µm. Before any of your fingers fall off you reassess your situation one last time. Are you still going to live?
Additional approaches to radiation dosage can produce novel problems and problem sets.
↑ radiation sickness. San Francisco, California: Wikimedia Foundation, Inc. January 10, 2014. https://en.wiktionary.org/wiki/radiation_sickness. Retrieved 2014-04-10.
↑ dosage. San Francisco, California: Wikimedia Foundation, Inc. February 9, 2014. https://en.wiktionary.org/wiki/dosage. Retrieved 2014-04-10.
↑ dose. San Francisco, California: Wikimedia Foundation, Inc. March 27, 2014. https://en.wiktionary.org/wiki/dose. Retrieved 2014-04-10.
↑ 4.0 4.1 The 2007 Recommendations. International Commission on Radiological Protection. http://www.icrp.org/docs/ICRP_Publication_103-Annals_of_the_ICRP_37(2-4)-Free_extract.pdf. Retrieved 2011-04-15.
↑ A D Wrixon. "New ICRP recommendations". Journal on Radiological Protection 28 (2). http://iopscience.iop.org/0952-4746/28/2/R02/pdf/0952-4746_28_2_R02.pdf. Retrieved 2011-04-15.
↑ The International System of Units (SI). Bureau International des Poids et Mesures (BIPM). http://www.bipm.org/utils/common/pdf/si_brochure_8_en.pdf. Retrieved 2010-01-31.
↑ 8.0 8.1 United Nations Scientific Committee on the Effects of Atomic Radiation (2008). Sources and effects of ionizing radiation. New York: United Nations. p. 4. ISBN 978-92-1-142274-0. http://www.unscear.org/unscear/en/publications/2008_1.html. Retrieved 9 November 2012.
↑ Radiation Exposure and Contamination. http://www.merckmanuals.com/professional/injuries_poisoning/radiation_exposure_and_contamination/radiation_exposure_and_contamination.html. Retrieved 2 June 2013.
{{Anthropology resources}}{{Chemistry resources}}{{Gene project}}{{Humanities resources}}{{Medicine resources}}{{Phosphate biochemistry}}
Learn more about Radiation dosage
Retrieved from "https://en.wikiversity.org/w/index.php?title=Radiation_dosage&oldid=2373354"
Anthropology/Problems
Biochemistry/Problems
Genetics/Problems
Humanities/Problems |
Solid harmonics - Citizendium
In mathematics, solid harmonics are defined as solutions of the Laplace equation in spherical polar coordinates. There are two kinds of solid harmonic functions: the regular solid harmonics
{\displaystyle \scriptstyle R_{\ell }^{m}(\mathbf {r} )}
, which vanish at the origin, and the irregular solid harmonics
{\displaystyle \scriptstyle I_{\ell }^{m}(\mathbf {r} )}
, which have an
{\displaystyle r^{-(\ell +1)}}
singularity at the origin. Both sets of functions play an important role in potential theory. Regular solid harmonics appear in chemistry in the form of s, p, d, etc. atomic orbitals and in physics as multipoles. Irregular harmonics appear in the expansion of scalar fields in terms of multipoles.
Both kinds of solid harmonics are simply related to spherical harmonics
{\displaystyle \scriptstyle Y_{\ell }^{m}}
(normalized to unity),
{\displaystyle R_{\ell }^{m}(\mathbf {r} )\equiv {\sqrt {\frac {4\pi }{2\ell +1}}}\;r^{\ell }Y_{\ell }^{m}(\theta ,\varphi ),\qquad I_{\ell }^{m}(\mathbf {r} )\equiv {\sqrt {\frac {4\pi }{2\ell +1}}}\;{\frac {Y_{\ell }^{m}(\theta ,\varphi )}{r^{\ell +1}}}.}
1 Derivation, relation to spherical harmonics
2 Connection between regular and irregular solid harmonics
3 Addition theorems
4 Real form
4.1 Linear combination
4.2 z-dependent part
4.3 (x,y)-dependent part
4.5 List of lowest functions
Derivation, relation to spherical harmonics
The following vector operator plays a central role in this section
{\displaystyle \mathbf {L} \equiv \mathbf {r} \times \mathbf {\nabla } .}
Parenthetically, we remark that in quantum mechanics
{\displaystyle \scriptstyle -i\hbar \mathbf {L} }
is the orbital angular momentum operator, where
{\displaystyle \scriptstyle \hbar \,}
is Planck's constant divided by 2π. In quantum mechanics the momentum operator is proportional to the gradient,
{\displaystyle \scriptstyle \mathbf {p} =-i\hbar \mathbf {\nabla } }
, so that L is proportional to r×p, the orbital angular momentum operator.
{\displaystyle L^{2}\equiv \mathbf {L} \cdot \mathbf {L} =\sum _{i,j}[r_{i}\nabla _{j}r_{i}\nabla _{j}-r_{i}\nabla _{j}r_{j}\nabla _{i}]\quad {\hbox{and}}\quad \nabla _{j}r_{i}-r_{i}\nabla _{j}=\delta _{ji}}
one can derive that
{\displaystyle L^{2}=r^{2}\nabla ^{2}-(\mathbf {r} \cdot \mathbf {\nabla } )^{2}-\mathbf {r} \cdot \mathbf {\nabla } .}
Expression in spherical polar coordinates gives:
{\displaystyle \mathbf {r} \cdot \mathbf {\nabla } =r{\frac {\partial }{\partial r}}}
{\displaystyle (\mathbf {r} \cdot \mathbf {\nabla } )^{2}+\mathbf {r} \cdot \mathbf {\nabla } ={\frac {1}{r}}{\frac {\partial ^{2}}{\partial r^{2}}}r.}
It can be shown by expression of L in spherical polar coordinates that L² does not contain a derivative with respect to r. Hence upon division of L² by r² the position of 1/r² in the resulting expression is irrelevant. After these preliminaries we find that the Laplace equation ∇² Φ = 0 can be written as
{\displaystyle \nabla ^{2}\Phi (\mathbf {r} )=\left({\frac {1}{r}}{\frac {\partial ^{2}}{\partial r^{2}}}r+{\frac {L^{2}}{r^{2}}}\right)\Phi (\mathbf {r} )=0,\qquad \mathbf {r} \neq \mathbf {0} .}
It is known that spherical harmonics Yml are eigenfunctions of L²:
{\displaystyle L^{2}Y_{\ell }^{m}=-\ell (\ell +1)Y_{\ell }^{m}.}
Substitution of Φ(r) = F(r) Yml into the Laplace equation gives, after dividing out the spherical harmonic function, the following radial equation and its general solution,
{\displaystyle {\frac {1}{r}}{\frac {\partial ^{2}}{\partial r^{2}}}rF(r)={\frac {\ell (\ell +1)}{r^{2}}}F(r)\Longrightarrow F(r)=Ar^{\ell }+Br^{-\ell -1}.}
The particular solutions of the total Laplace equation are regular solid harmonics:
{\displaystyle R_{\ell }^{m}(\mathbf {r} )\equiv {\sqrt {\frac {4\pi }{2\ell +1}}}\;r^{\ell }Y_{\ell }^{m}(\theta ,\varphi ),}
and irregular solid harmonics:
{\displaystyle I_{\ell }^{m}(\mathbf {r} )\equiv {\sqrt {\frac {4\pi }{2\ell +1}}}\;{\frac {Y_{\ell }^{m}(\theta ,\varphi )}{r^{\ell +1}}}.}
Racah's normalization (also known as Schmidt's semi-normalization) is applied to both functions
{\displaystyle \int _{0}^{\pi }\sin \theta \,d\theta \int _{0}^{2\pi }d\varphi \;R_{\ell }^{m}(\mathbf {r} )^{*}\;R_{\ell }^{m}(\mathbf {r} )={\frac {4\pi }{2\ell +1}}r^{2\ell }}
(and analogously for the irregular solid harmonic) instead of normalization to unity. This is convenient because in many applications the Racah normalization factor appears unchanged throughout the derivations.
Connection between regular and irregular solid harmonics
From the definitions follows immediately that
{\displaystyle I_{\ell }^{m}(\mathbf {r} )={\frac {R_{\ell }^{m}(\mathbf {r} )}{r^{2\ell +1}}}}
A more interesting relationship follows from the observation that the regular solid harmonics are homogeneous polynomials in the components x, y, and z of r. We can replace these components by the corresponding components of the gradient operator ∇. Thus, the left hand side in the following equation is well-defined:
{\displaystyle R_{\ell }^{m}(\mathbf {\nabla } )\;{\frac {1}{r}}=(-1)^{\ell }{\frac {(2\ell )!}{2^{\ell }\ell !}}\;I_{\ell }^{m}(\mathbf {r} ),\qquad r\neq 0.}
For a proof see Biedenharn and Louck (1981), p. 312.
The translation of the regular solid harmonic gives a finite expansion,
{\displaystyle R_{\ell }^{m}(\mathbf {r} +\mathbf {a} )=\sum _{\lambda =0}^{\ell }{\binom {2\ell }{2\lambda }}^{1/2}\sum _{\mu =-\lambda }^{\lambda }R_{\lambda }^{\mu }(\mathbf {r} )R_{\ell -\lambda }^{m-\mu }(\mathbf {a} )\;\langle \lambda ,\mu ;\ell -\lambda ,m-\mu |\ell m\rangle ,}
where the Clebsch-Gordan coefficient is given by
{\displaystyle \langle \lambda ,\mu ;\ell -\lambda ,m-\mu |\ell m\rangle ={\ell +m \choose \lambda +\mu }^{1/2}{\ell -m \choose \lambda -\mu }^{1/2}{2\ell \choose 2\lambda }^{-1/2}.}
The similar expansion for irregular solid harmonics gives an infinite series,
{\displaystyle I_{\ell }^{m}(\mathbf {r} +\mathbf {a} )=\sum _{\lambda =0}^{\infty }{\binom {2\ell +2\lambda +1}{2\lambda }}^{1/2}\sum _{\mu =-\lambda }^{\lambda }R_{\lambda }^{\mu }(\mathbf {r} )I_{\ell +\lambda }^{m-\mu }(\mathbf {a} )\;\langle \lambda ,\mu ;\ell +\lambda ,m-\mu |\ell m\rangle }
{\displaystyle |r|\leq |a|\,}
. The quantity between pointed brackets is again a Clebsch-Gordan coefficient,
{\displaystyle \langle \lambda ,\mu ;\ell +\lambda ,m-\mu |\ell m\rangle =(-1)^{\lambda +\mu }{\ell +\lambda -m+\mu \choose \lambda +\mu }^{1/2}{\ell +\lambda +m-\mu \choose \lambda -\mu }^{1/2}{2\ell +2\lambda +1 \choose 2\lambda }^{-1/2}.}
By a simple linear combination of solid harmonics of ±m these functions are transformed into real functions. The real regular solid harmonics, expressed in Cartesian coordinates, are homogeneous polynomials of order l in x, y, z. The explicit form of these polynomials is of some importance. They appear, for example, in the form of spherical atomic orbitals and real multipole moments. The explicit Cartesian expression of the real regular harmonics will now be derived.
We write in agreement with the earlier definition
{\displaystyle R_{\ell }^{m}(r,\theta ,\varphi )=(-1)^{(m+|m|)/2}\;r^{\ell }\;\Theta _{\ell }^{|m|}(\cos \theta )e^{im\varphi },\qquad -\ell \leq m\leq \ell ,}
{\displaystyle \Theta _{\ell }^{m}(\cos \theta )\equiv \left[{\frac {(\ell -m)!}{(\ell +m)!}}\right]^{1/2}\,\sin ^{m}\theta \,{\frac {d^{m}P_{\ell }(\cos \theta )}{d\cos ^{m}\theta }},\qquad m\geq 0,}
{\displaystyle P_{\ell }(\cos \theta )}
is a Legendre polynomial of order l. The m dependent phase is known as the Condon-Shortley phase.
The following expression defines the real regular solid harmonics:
{\displaystyle {\begin{pmatrix}C_{\ell }^{m}\\S_{\ell }^{m}\end{pmatrix}}\equiv {\sqrt {2}}\;r^{\ell }\;\Theta _{\ell }^{m}{\begin{pmatrix}\cos m\varphi \\\sin m\varphi \end{pmatrix}}={\frac {1}{\sqrt {2}}}{\begin{pmatrix}(-1)^{m}&\quad 1\\-(-1)^{m}i&\quad i\end{pmatrix}}{\begin{pmatrix}R_{\ell }^{m}\\R_{\ell }^{-m}\end{pmatrix}},\qquad m>0.}
{\displaystyle C_{\ell }^{0}\equiv R_{\ell }^{0}.}
Since the transformation is by a unitary matrix the normalization of the real and the complex solid harmonics is the same.
z-dependent part
Upon writing u = cos θ the mth derivative of the Legendre polynomial can be written as the following expansion in u
{\displaystyle {\frac {d^{m}P_{\ell }(u)}{du^{m}}}=\sum _{k=0}^{\left\lfloor (\ell -m)/2\right\rfloor }\gamma _{\ell k}^{(m)}\;u^{\ell -2k-m}}
{\displaystyle \gamma _{\ell k}^{(m)}=(-1)^{k}2^{-\ell }{\ell \choose k}{2\ell -2k \choose \ell }{\frac {(\ell -2k)!}{(\ell -2k-m)!}}.}
Since z = r cosθ it follows that this derivative, times an appropriate power of r, is a simple polynomial in z,
{\displaystyle \Pi _{\ell }^{m}(z)\equiv r^{\ell -m}{\frac {d^{m}P_{\ell }(u)}{du^{m}}}=\sum _{k=0}^{\left\lfloor (\ell -m)/2\right\rfloor }\gamma _{\ell k}^{(m)}\;r^{2k}\;z^{\ell -2k-m}.}
(x,y)-dependent part
Consider next, recalling that x = r sinθcosφ and y = r sinθsinφ,
{\displaystyle r^{m}\sin ^{m}\theta \cos m\varphi ={\frac {1}{2}}\left[(r\sin \theta e^{i\varphi })^{m}+(r\sin \theta e^{-i\varphi })^{m}\right]={\frac {1}{2}}\left[(x+iy)^{m}+(x-iy)^{m}\right]}
{\displaystyle r^{m}\sin ^{m}\theta \sin m\varphi ={\frac {1}{2i}}\left[(r\sin \theta e^{i\varphi })^{m}-(r\sin \theta e^{-i\varphi })^{m}\right]={\frac {1}{2i}}\left[(x+iy)^{m}-(x-iy)^{m}\right].}
{\displaystyle A_{m}(x,y)\equiv {\frac {1}{2}}\left[(x+iy)^{m}+(x-iy)^{m}\right]=\sum _{p=0}^{m}{m \choose p}x^{p}y^{m-p}\cos \left((m-p){\frac {\pi }{2}}\right)}
{\displaystyle B_{m}(x,y)\equiv {\frac {1}{2i}}\left[(x+iy)^{m}-(x-iy)^{m}\right]=\sum _{p=0}^{m}{m \choose p}x^{p}y^{m-p}\sin \left((m-p){\frac {\pi }{2}}\right).}
{\displaystyle C_{\ell }^{m}(x,y,z)=\left[{\frac {(2-\delta _{m0})(\ell -m)!}{(\ell +m)!}}\right]^{1/2}\Pi _{\ell }^{m}(z)\;A_{m}(x,y),\qquad m=0,1,\ldots ,\ell }
{\displaystyle S_{\ell }^{m}(x,y,z)=\left[{\frac {2(\ell -m)!}{(\ell +m)!}}\right]^{1/2}\Pi _{\ell }^{m}(z)\;B_{m}(x,y),\qquad m=1,2,\ldots ,\ell .}
List of lowest functions
We list explicitly the lowest functions up to and including l = 5 . Here
{\displaystyle {\bar {\Pi }}_{\ell }^{m}(z)\equiv \left[{\frac {(2-\delta _{m0})(\ell -m)!}{(\ell +m)!}}\right]^{1/2}\Pi _{\ell }^{m}(z).}
{\displaystyle {\begin{matrix}{\bar {\Pi }}_{0}^{0}=&1&{\bar {\Pi }}_{3}^{1}=&{\frac {1}{4}}{\sqrt {6}}(5z^{2}-r^{2})&{\bar {\Pi }}_{4}^{4}=&{\frac {1}{8}}{\sqrt {35}}\\{\bar {\Pi }}_{1}^{0}=&z&{\bar {\Pi }}_{3}^{2}=&{\frac {1}{2}}{\sqrt {15}}\;z&{\bar {\Pi }}_{5}^{0}=&{\frac {1}{8}}z(63z^{4}-70z^{2}r^{2}+15r^{4})\\{\bar {\Pi }}_{1}^{1}=&1&{\bar {\Pi }}_{3}^{3}=&{\frac {1}{4}}{\sqrt {10}}&{\bar {\Pi }}_{5}^{1}=&{\frac {1}{8}}{\sqrt {15}}(21z^{4}-14z^{2}r^{2}+r^{4})\\{\bar {\Pi }}_{2}^{0}=&{\frac {1}{2}}(3z^{2}-r^{2})&{\bar {\Pi }}_{4}^{0}=&{\frac {1}{8}}(35z^{4}-30r^{2}z^{2}+3r^{4})&{\bar {\Pi }}_{5}^{2}=&{\frac {1}{4}}{\sqrt {105}}(3z^{2}-r^{2})z\\{\bar {\Pi }}_{2}^{1}=&{\sqrt {3}}z&{\bar {\Pi }}_{4}^{1}=&{\frac {\sqrt {10}}{4}}z(7z^{2}-3r^{2})&{\bar {\Pi }}_{5}^{3}=&{\frac {1}{16}}{\sqrt {70}}(9z^{2}-r^{2})\\{\bar {\Pi }}_{2}^{2}=&{\frac {1}{2}}{\sqrt {3}}&{\bar {\Pi }}_{4}^{2}=&{\frac {1}{4}}{\sqrt {5}}(7z^{2}-r^{2})&{\bar {\Pi }}_{5}^{4}=&{\frac {3}{8}}{\sqrt {35}}z\\{\bar {\Pi }}_{3}^{0}=&{\frac {1}{2}}z(5z^{2}-3r^{2})&{\bar {\Pi }}_{4}^{3}=&{\frac {1}{4}}{\sqrt {70}}\;z&{\bar {\Pi }}_{5}^{5}=&{\frac {3}{16}}{\sqrt {14}}\\\end{matrix}}}
The lowest functions
{\displaystyle A_{m}(x,y)\,}
{\displaystyle B_{m}(x,y)\,}
{\displaystyle 1\,}
{\displaystyle 0\,}
{\displaystyle x\,}
{\displaystyle y\,}
{\displaystyle x^{2}-y^{2}\,}
{\displaystyle 2xy\,}
{\displaystyle x^{3}-3xy^{2}\,}
{\displaystyle 3x^{2}y-y^{3}\,}
{\displaystyle x^{4}-6x^{2}y^{2}+y^{4}\,}
{\displaystyle 4x^{3}y-4xy^{3}\,}
{\displaystyle x^{5}-10x^{3}y^{2}+5xy^{4}\,}
{\displaystyle 5x^{4}y-10x^{2}y^{3}+y^{5}\,}
Thus, for example, the angular part of one of the nine normalized spherical g atomic orbitals is:
{\displaystyle C_{4}^{2}(x,y,z)={\sqrt {\textstyle {\frac {9}{4\pi }}}}{\bar {\Pi }}_{4}^{2}A_{2}={\sqrt {\textstyle {\frac {9}{4\pi }}}}{\sqrt {\textstyle {\frac {5}{16}}}}(7z^{2}-r^{2})(x^{2}-y^{2}).}
One of the 7 components of a real multipole of order 3 (octupole) of a system of N charges q i is
{\displaystyle S_{3}^{1}(x,y,z)={\bar {\Pi }}_{3}^{1}B_{1}={\frac {1}{4}}{\sqrt {6}}\sum _{i=1}^{N}q_{i}(5z_{i}^{2}-r_{i}^{2})y_{i}.}
Spherical harmonics in Cartesian form
The following expresses normalized spherical harmonics in Cartesian coordinates (Condon-Shortley phase):
{\displaystyle r^{\ell }\,{\begin{pmatrix}Y_{\ell }^{m}\\Y_{\ell }^{-m}\end{pmatrix}}=\left[{\frac {2\ell +1}{4\pi }}\right]^{1/2}{\bar {\Pi }}_{\ell }^{m}(z){\begin{pmatrix}(-1)^{m}(A_{m}+iB_{m})/{\sqrt {2}}\\\qquad (A_{m}-iB_{m})/{\sqrt {2}}\\\end{pmatrix}},\qquad m>0.}
{\displaystyle r^{\ell }\,Y_{\ell }^{0}\equiv {\sqrt {\frac {2\ell +1}{4\pi }}}{\bar {\Pi }}_{\ell }^{0}(z).}
{\displaystyle A_{m}(x,y)=\sum _{p=0}^{m}{m \choose p}x^{p}y^{m-p}\cos \left((m-p){\frac {\pi }{2}}\right),}
{\displaystyle B_{m}(x,y)=\sum _{p=0}^{m}{m \choose p}x^{p}y^{m-p}\sin \left((m-p){\frac {\pi }{2}}\right),}
and for m > 0:
{\displaystyle {\bar {\Pi }}_{\ell }^{m}(z)=\left[{\frac {(\ell -m)!}{(\ell +m)!}}\right]^{1/2}\sum _{k=0}^{\left\lfloor (\ell -m)/2\right\rfloor }(-1)^{k}2^{-\ell }{\ell \choose k}{2\ell -2k \choose \ell }{\frac {(\ell -2k)!}{(\ell -2k-m)!}}\;r^{2k}\;z^{\ell -2k-m}.}
{\displaystyle {\bar {\Pi }}_{\ell }^{0}(z)=\sum _{k=0}^{\left\lfloor \ell /2\right\rfloor }(-1)^{k}2^{-\ell }{\ell \choose k}{2\ell -2k \choose \ell }\;r^{2k}\;z^{\ell -2k}.}
{\displaystyle {\bar {\Pi }}_{m}^{\ell }(z)}
{\displaystyle A_{m}(x,y)\,}
{\displaystyle B_{m}(x,y)\,}
{\displaystyle Y_{3}^{1}=-{\frac {1}{r^{3}}}\left[{\textstyle {\frac {7}{4\pi }}\cdot {\frac {3}{16}}}\right]^{1/2}(5z^{2}-r^{2})(x+iy)=-\left[{\textstyle {\frac {7}{4\pi }}\cdot {\frac {3}{16}}}\right]^{1/2}(5\cos ^{2}\theta -1)(\sin \theta e^{i\varphi })}
{\displaystyle Y_{4}^{-2}={\frac {1}{r^{4}}}\left[{\textstyle {\frac {9}{4\pi }}\cdot {\frac {5}{32}}}\right]^{1/2}(7z^{2}-r^{2})(x-iy)^{2}=\left[{\textstyle {\frac {9}{4\pi }}\cdot {\frac {5}{32}}}\right]^{1/2}(7\cos ^{2}\theta -1)(\sin ^{2}\theta e^{-2i\varphi })}
Most books on angular momenta discuss solid harmonics. See, for instance,
D. M. Brink and G. R. Satchler, Angular Momentum, 3rd edition ,Clarendon, Oxford, (1993)
L. C. Biedenharn and J. D. Louck, Angular Momentum in Quantum Physics, volume 8 of Encyclopedia of Mathematics, Addison-Wesley, Reading (1981)
The addition theorems for solid harmonics have been proved in different manners by many different workers. See for two different proofs for example:
R. J. A. Tough and A. J. Stone, J. Phys. A: Math. Gen. Vol. 10, p. 1261 (1977)
M. J. Caola, J. Phys. A: Math. Gen. Vol. 11, p. L23 (1978)
Retrieved from "https://citizendium.org/wiki/index.php?title=Solid_harmonics&oldid=32062" |
Marty is saving money to buy a new computer. He received
\$200
for his birthday and saves
\$150
of each week’s paycheck.
Does this situation represent a proportional relationship? If it does, identify the constant of proportionality. If it does not, explain why not.
4
weeks does Marty have twice as much money as he does after
2
No. Be sure you know why this is not a proportional relationship. |
To M. T. Masters 24 July [1862]1
My poor Boy rallied last night & is now out of danger. He had recurrent scarlet-fever with every sort of mischief in the glands; & this followed by dreadful erysipelas of head with typhoid symptoms.2 The Doctors never saw such a complication of illness. But thank God Port-wine every
\frac{3}{4}
of hour, night & day, seems to have saved him.
I thank you cordially for taking the trouble of writing at such length: your letter is in many ways of great value to me.3 The distinction of the two sorts of Peloria, though so excessively obvious when pointed out, never occurred to me.—4 I shall now know what flowers to look to. It is quite likely I may make nothing of these peloric gentlemen; but I am contented if I get any result once out of four or five sets of experiments.5
Pray give my compliments & best thanks to your Father for his kind information.6 The seeds are not ripe, but apparently I have got some few from a few of the peloric Pelargoniums; but perhaps the seed will prove bad.—7 Many thanks for references to Bull. Bot. Soc;8 as until within a few weeks I did not see that Peloric flowers would have any bearing on my subjects, I never attended to them. That is a curious case of hereditariness, which you mention: I think Prosper Lucas gives an analogous case.9
I am glad to hear that you are continuing your work on malconformations in Plants.10
With sincere thanks for your valuable aid. Believe me, my dear Sir | Yours sincerely | C. Darwin
See also Emma Darwin’s diary (DAR 242), letter to Asa Gray, 23[–4] July [1862], and letter to W. E. Darwin, [24 July 1862].
CD had written to Masters requesting information on peloric flowers (letter to M. T. Masters, 8 July [1862]); in the letter from M. T. Masters, 12 July 1862, Masters promised to send ‘a few memoranda’ on the subject when he had more time, but no such correspondence has been found.
Masters published an account of his observations on two forms of peloric flowers the following year (Masters 1863); he classified the apparent malformations as follows (Masters 1863, p. 260): As the word Peloria itself merely signifies something strange and out of the common way, there can be no objection, I think, to the introduction of the terms Regular and Irregular Peloria. “Regular or Congenital Peloria” would include those flowers which, contrary to their usual habit, retain throughout the whole of their growth their primordial regularity of form and equality of proportion. “Irregular or Acquired Peloria”, on the other hand, would include those flowers in which the irregularity of growth that ordinarily characterizes some portions of the corolla is manifested in all of them. CD cited Masters 1863 in his discussion of this point in Variation 2: 58; his annotated copy of the number of the Natural History Review in which the article appeared is in the Darwin Library–CUL.
In the letter to M. T. Masters, 8 July [1862], CD had asked for suggestions as to what plants he might grow for experimentation the following season, with an expectation of their producing peloric flowers. CD described crossing experiments on the peloric form of Antirrhinum majus in Variation 2: 70; his notes from these experiments, dated 1863–5, are in DAR 51 (ser. 2): 18–23.
In the letter from M. T. Masters, 12 July 1862, Masters told CD that he had written to his father, the nurseryman William Masters, to ask for information on the fertility of the peloric flowers in Gloxinia and other cultivated plants. In Variation 2: 167, CD noted William Masters’s observations on the sterility of peloric flowers in pelargoniums; CD had told M. T. Masters of his interest in this question in the letter to M. T. Masters, 8 July [1862].
CD refers to the crossing experiments with the normally sterile peloric flowers of pelargoniums that he had begun in May 1862 (see letter to Daniel Oliver, 8 June [1862], letter to Asa Gray, 1 July [1862], and letter to M. T. Masters, 8 July [1862]). CD’s notes from these experiments are in DAR 51 (ser. 2): 4–9, 12–13. In Variation 2: 167, CD reported that he had made ‘many vain attempts’ to set seed from these peloric flowers, but that he had ‘sometimes succeeded in fertilising them with pollen from a normal flower of another variety’ and, conversely, had ‘several times fertilised ordinary flowers with peloric pollen.’ Only once, he reported, had he ‘succeeded in raising a plant from a peloric flower fertilised by pollen from a peloric flower borne by another variety’.
The references have not been identified, but may have included an article on peloric flowers in the genus Zingiber by Arthur Gris, in the Bulletin de la Société Botanique de France (Gris 1859), cited by Masters in his article on peloric flowers (Masters 1863, p. 262).
The reference has not been traced. The work of the French physician Prosper Lucas on inheritance (Lucas 1847–50) is extensively cited in Variation; there is a heavily annotated copy of this work in the Darwin Library–CUL (see Marginalia 1: 513–23).
Masters was making a special study of plant morphology and teratology, and, in 1869, published Vegetable teratology (Masters 1869).
Gris, Arthur. 1859. Note sur quelques cas remarquables de pélorie dans le genre Zingiber. Bulletin de la Société Botanique de France 6: 346–8.
CD grateful to have had the distinction of the two sorts of peloria pointed out to him.
His very sick son rallied; is out of danger, thanks to port wine. |
2013 A Simplicial Branch and Bound Duality-Bounds Algorithm to Linear Multiplicative Programming
Xue-Gang Zhou, Bing-Yuan Cao
A simplicial branch and bound duality-bounds algorithm is presented to globally solving the linear multiplicative programming (LMP). We firstly convert the problem (LMP) into an equivalent programming one by introducing
p
auxiliary variables. During the branch and bound search, the required lower bounds are computed by solving ordinary linear programming problems derived by using a Lagrangian duality theory. The proposed algorithm proves that it is convergent to a global minimum through the solutions to a series of linear programming problems. Some examples are given to illustrate the feasibility of the present algorithm.
Xue-Gang Zhou. Bing-Yuan Cao. "A Simplicial Branch and Bound Duality-Bounds Algorithm to Linear Multiplicative Programming." J. Appl. Math. 2013 1 - 10, 2013. https://doi.org/10.1155/2013/984168
Xue-Gang Zhou, Bing-Yuan Cao "A Simplicial Branch and Bound Duality-Bounds Algorithm to Linear Multiplicative Programming," Journal of Applied Mathematics, J. Appl. Math. 2013(none), 1-10, (2013) |
Hydrokenomicrolite, (□,H2O)2Ta 2(O,OH)6(H2O), a new microlite-group mineral from Volta Grande pegmatite, Nazareno, Minas Gerais, Brazil | American Mineralogist | GeoScienceWorld
Hydrokenomicrolite, (□,H2O)2Ta 2(O,OH)6(H2O), a new microlite-group mineral from Volta Grande pegmatite, Nazareno, Minas Gerais, Brazil
Departamento de Física e Informática, Instituto de Física de São Carlos, Universidade de São Paulo, Caixa Postal 369, 13560-970 São Carlos, SP, Brazil
Present address: Department of Geosciences, University of Arizona, 1040 East 4th Street, Tucson, Arizona 85721, U.S.A. E-mail: mabadean@terra.com.br
Institute of Problems of Chemical Physics, Russian Academy of Sciences, Chernogolovka, Moscow Region 142432, Russia
Marcelo B. Andrade, Daniel Atencio, Nikita V. Chukanov, Javier Ellena; Hydrokenomicrolite, (□,H2O)2Ta 2(O,OH)6(H2O), a new microlite-group mineral from Volta Grande pegmatite, Nazareno, Minas Gerais, Brazil. American Mineralogist 2013;; 98 (2-3): 292–296. doi: https://doi.org/10.2138/am.2013.4186
Hydrokenomicrolite, (□,H2O)2Ta2(O,OH)6(H2O) or ideally □2Ta2[O4(OH)2](H2O), is a new microlite-group mineral approved by the CNMNC (IMA 2011-103). It occurs as an accessory mineral in the Volta Grande pegmatite, Nazareno, Minas Gerais, Brazil. Associated minerals are: microcline, albite, quartz, muscovite, spodumene, “lepidolite”, cassiterite, tantalite-(Mn), monazite-(Ce), fluorite, “apatite”, beryl, “garnet”, epidote, magnetite, gahnite, zircon, “tourmaline”, bityite, and other microlite-group minerals under study. Hydrokenomicrolite occurs as euhedral octahedral crystals, occasionally modified by rhombododecahedra, untwinned, from 0.2 to 1.5 mm in size. The crystals are pinkish brown and translucent; the streak is white, and the luster is adamantine to resinous. It is non-fluorescent under ultraviolet light. Mohs hardness is 4½–5, tenacity is brittle. Cleavage is not observed; fracture is conchoidal. The calculated density is 6.666 g/cm3. The mineral is isotropic, ncalc = 2.055. The infrared spectrum contains bands of O-H stretching vibrations and H-O-H bending vibrations of H2O molecules. The chemical composition (n = 3) is [by wavelength-dispersive spectroscopy (WDS), H2O calculated from crystal-structure analysis, wt%]: CaO 0.12, MnO 0.27, SrO 4.88, BaO 8.63, PbO 0.52, La2O3 0.52, Ce2O3 0.49, Nd2O3 0.55, Bi2O3 0.57, UO2 4.54, TiO2 0.18, SnO2 2.60, Nb2O5 2.18, Ta2O5 66.33, SiO2 0.46, Cs2O 0.67, H2O 4.84, total 98.35. The empirical formula, based on 2 cations at the B site, is [□0.71(H2O)0.48Ba0.33Sr0.27U0.10Mn0.02Nd0.02Ce0.02La0.02Ca0.01 Bi0.01Pb0.01]Σ2.00 (Ta1.75Nb0.10Sn0.10Si0.04Ti0.01)Σ2.00[(O5.77(OH)0.23]Σ6.00[(H2O)0.97Cs0.03]Σ1.00. The strongest eight X-ray powder-diffraction lines [d in Å(I)(hkl)] are: 6.112(86)(111), 3.191(52)(311), 3.052(100)(222), 2.642(28)(400), 2.035(11)(511)(333), 1.869(29)(440), 1.788(10)(531), and 1.594(24)(622). The crystal structure refinement (R1 = 0.0363) gave the following data: cubic, Fd
3¯
m, a = 10.454(1) Å, V = 1142.5(2) Å3, Z = 8. The Ta(O,OH)6 octahedra are linked through all vertices. The refinement results and the approximate empirical bond-valences sums for the positions A (1.0 v.u.) and Y′ (0.5 v.u.), compared to valence calculations from electron microprobe analysis (EMPA) and ranges expected for H2O molecules, confirm the presence of H2O at the A(16d) site and displaced from the Y(8b) to the Y′(32e) position. The mineral is characterized by H2O dominance at the Y site, vacancy dominance at the A site, and Ta dominance at the B site.
Nazareno Brazil
Volta Grande Brazil
hydrokenomicrolite
Fluorcalciomicrolite, (Ca,Na,□)2Ta2O6F, a new microlite-group mineral from Volta Grande pegmatite, Nazareno, Minas Gerais, Brazil
Oxynatromicrolite, (Na,Ca,U)2Ta2O6(O,F), a new member of the pyrochlore supergroup from Guanpo, Henan Province, China
Oxycalciomicrolite, (Ca,Na)2(Ta,Nb,Ti)2O6(O,F), a new member of the microlite group (pyrochlore supergroup) from the Paleoproterozoic São João del Rei Pegmatite Province, Minas Gerais state, Brazil
Waimirite-(Y), orthorhombic YF3, a new mineral from the Pitinga mine, Presidente Figueiredo, Amazonas, Brazil and from Jabal Tawlah, Saudi Arabia: description and crystal structure
Strontiohurlbutite, SrBe2(PO4)2, a new mineral from Nanping No. 31 pegmatite, Fujian Province, Southeastern China |
Compton Scattering - AstroBaki
Compton Scattering (Brightstorm, Youtube)
Deriving the Compton Scattering Formula Part 1 (That SingaporeanGuy, Youtube)
Compton Scattering (Wayne Hu, U. Chicago)
Lorentz transformations for relativistic energy and four-momentum
1.1 1 Mathematical Derivation of Compton Scattering
1.2 2 Compton Scattering for a Moving Electron
Compton scattering is the inelastic scattering of a photon off of an electron. This is the quantum mechanical or high-energy extension to Thomson Scattering.
If the photon loses energy, its wavelength will increase and this is called Compton Scattering. If the electron has sufficient initial kinetic energy, the photon the photon can gain energy, this is called Inverse Compton Scattering, or “Compton Up-Scattering”. An alteration of the photon spectrum (either up-scattering or down-scattering) due to interactions with electrons is called “comptonization”
Some examples of applications of Compton scattering are:
Compton exchange keeps electrons in thermal equilibrium with photons at redshifts
{\displaystyle z\geq 10^{3}}
The spectra of AGN and xray binaries are altered by Compton Scattering (e.g. radio emission to optical wavelengths).
CMB photons get upscattered by galaxy cluster plasma. This is called the Sunyaev-Zeldovich effect.
Inverse Compton scattering has been proposed as a likely emission mechanism for gamma ray bursts.
The Compton effect was first observed in the 1920’s and Arthur Holly Comton was awarded the 1927 Nobel prize for the discovery. The effect is often regarded as some of the first concrete experimental evidence for quantum mechanics.
1 Mathematical Derivation of Compton Scattering
To find the final energy of a scattered photon and the Compton shift (the change in the photon wavelength), we use the conservation of energy and momentum.
In the standard derivation of Compton scattering, the electron is assumed to be free and at rest. This is a good approximation considering the photon energies for which this process is significant are much larger than relevant electron binding energies.
Compton scattering geometry.
A photon with initial energy
{\displaystyle E_{\gamma _{i}}=h\nu _{i}}
travelling in the
{\displaystyle {\hat {x}}}
direction scatters of an electron at rest (
{\displaystyle E_{e_{i}}=m_{e}c^{2}}
). After the scatter, the photon has energy
{\displaystyle E_{\gamma _{f}}=h\nu _{f}}
and is travelling at an angle
{\displaystyle \theta }
relative to the original
{\displaystyle {\hat {x}}}
direction. The electron has energy given by
{\displaystyle E_{e_{f}}={\sqrt {p_{e}^{2}c^{2}+m_{e}^{2}c^{4}}}}
and has scattered at a different angle. See figure below for a depiction of the relevant variables.
The conservation of energy tells us
{\displaystyle {\begin{aligned}E_{\gamma _{i}}+E_{e_{i}}&=E_{\gamma _{f}}+E_{e_{f}}\\E_{\gamma _{i}}+m_{e}c^{2}&={\sqrt {p_{e}^{2}c^{2}+m_{e}^{2}c^{4}}}+E_{\gamma _{f}}\end{aligned}}\,\!}
Rearranging and squaring both sides gives
{\displaystyle (E_{\gamma _{i}}-E_{\gamma _{f}}+m_{e}c^{2})^{2}=p_{e}^{2}c^{2}+m_{e}^{2}c^{4}.\,\!}
We will use the conservation of momentum to write the
{\displaystyle p_{e}^{2}c^{2}}
factor in terms of the photon energies, so we will come back to this equation.
The conservation of momentum tells us
{\displaystyle {\vec {p_{\gamma _{i}}}}={\vec {p_{\gamma _{f}}}}+{\vec {p_{e_{f}}}}\,\!}
{\displaystyle ({\vec {p_{\gamma _{i}}}}-{\vec {p_{\gamma _{f}}}})^{2}={\vec {p_{e_{f}}}}^{2}\,\!}
{\displaystyle p_{\gamma _{i}}^{2}+p_{\gamma _{f}}^{2}-2p_{\gamma _{i}}p_{\gamma _{f}}\cos \theta =p_{e_{f}}^{2}\,\!}
{\displaystyle \theta }
is the angle between the initial photon direction
{\displaystyle {\hat {x}}}
and the final photon direction.
Multiplying the above equation by
{\displaystyle c^{2}}
and using the relation
{\displaystyle E_{\gamma }=p_{\gamma }c}
{\displaystyle E_{\gamma _{i}}^{2}+E_{\gamma _{f}}^{2}-2E_{\gamma _{i}}E_{\gamma _{f}}\cos \theta =p_{e}^{2}c^{2}.\,\!}
Now, we can plug this into our final expression for the conservation of energy above and expand the squared brackets on the left side at the same time:
{\displaystyle {E_{\gamma _{i}}^{2}}+{E_{\gamma _{f}}^{2}}+{m_{e}^{2}c^{4}}+{2}E_{\gamma _{i}}m_{e}c^{2}-{2}E_{\gamma _{f}}m_{e}c^{2}-{2}E_{\gamma _{i}}E_{\gamma _{f}}={E_{\gamma _{i}}^{2}}+{E_{\gamma _{f}}^{2}}-{2}E_{\gamma _{i}}E_{\gamma _{f}}\cos \theta +{m_{e}^{2}c^{4}}.\,\!}
Cancelling off similar terms on both sides gives us
{\displaystyle E_{\gamma _{i}}m_{e}c^{2}-E_{\gamma _{f}}m_{e}c^{2}-E_{\gamma _{i}}E_{\gamma _{f}}=-E_{\gamma _{i}}E_{\gamma _{f}}\cos \theta .\,\!}
Solving for the final photon energy gives
{\displaystyle E_{\gamma _{f}}={\frac {E_{\gamma _{i}}}{1+{\frac {E_{\gamma _{i}}}{m_{e}c^{2}}}(1-\cos \theta )}},\,\!}
and the change in the photon wavelength, referred to as the Compton shift, is found to be
{\displaystyle {\frac {1}{E_{\gamma _{f}}}}-{\frac {1}{E_{\gamma _{i}}}}={\frac {1}{m_{e}c^{2}}}(1-\cos \theta )\,\!}
{\displaystyle \lambda _{f}-\lambda _{i}={\frac {h}{m_{e}c}}(1-\cos \theta )\,\!}
The prefactor
{\displaystyle \lambda ={\frac {h}{m_{e}c}}}
is called the Compton wavelength.
{\displaystyle \lambda _{c}=0.02}
A gives the order of change in wavelength during a Compton scatter interaction; therefore, Compton scattering is only relevant for high energy (low wavelength) photons.
{\displaystyle \lambda _{1}-\lambda >0}
: the shift here is tiny. The maximum possible value is
{\displaystyle \lambda _{1}-\lambda =2hm_{e}c=0.04}
A. The momentum tends to be shared between the electron and photon, but no so much the energy.
Remember this is scattering, not absorption. Photon # is conserved.
{\displaystyle \lambda _{i}\gg \lambda _{c}}
{\displaystyle h\nu \ll m_{e}c^{2}}
), the scattering can be approximated as elastic (Thomson scattering) and
{\displaystyle \Delta E_{\gamma }=0}
The derivation for the cross section of Compton scattering, which is given by the Klein-Nishina formula, is “outside of the scope of [R & L]”. An important thing to know is that the scattering angle is anisotropic and depends on energy. In the limit that
{\displaystyle h\nu \gg m_{e}c^{2}}
{\displaystyle \sigma \sim \sigma _{T}\left({\frac {m_{e}c^{2}}{h\nu }}\right)}
, and this additional term is called the Klein-Nishina correction.
2 Compton Scattering for a Moving Electron
What we’ve done so far was for a stationary electron. For a moving electron, we need to consider the dependence on the angle at which the photon is coming in with respect to the direction of velocity (
{\displaystyle \theta }
). To make this situation similar to the one we just considered, we need to be in the frame of the electron. In this frame, the photons has a new energy as a result of time dilation:
{\displaystyle E^{\prime }=\gamma E(1-{v \over c}\cos \theta )\,\!}
{\displaystyle v \over c}
term is just the classic Doppler shift of the photon. Now suppose that the photon (which entered at angle
{\displaystyle \theta ^{\prime }}
in the electron frame) rebounds at an angle
{\displaystyle \theta _{1}^{\prime }}
. To relate this to our previous derivation, we want to find
{\displaystyle \phi }
. So note that
{\displaystyle \theta _{1}^{\prime }-\phi ^{\prime }=\theta ^{\prime }}
. Thus, in the electron’s frame:
{\displaystyle E_{1}^{\prime }={E^{\prime } \over 1+{E^{\prime } \over m_{e}c^{2}}(1-\cos \phi ^{\prime })}\,\!}
Transforming this back into the lab frame:
{\displaystyle E_{1}=E_{1}^{\prime }\gamma (1+{v \over c}\cos \theta _{1}^{\prime })\,\!}
This generally follows Rybicki & Lightman. However, it might help to know that in R&L, 7.7b follows from 7.8a, which follows from 7.7a.
{\displaystyle E^{\prime }\ll m_{e}c^{2}}
{\displaystyle {\begin{aligned}E_{1}&\approx E^{\prime }\gamma (1+{v \over c}\cos \theta _{1}^{\prime })\\&\approx \gamma ^{2}E(1+{v \over c}\cos \theta _{1}^{\prime })(1-{v \over c}\cos \theta )\\\end{aligned}}\,\!}
In a “typical” collision,
{\displaystyle \theta \sim \theta _{1}^{\prime }\sim {\pi \over 2}}
{\displaystyle E_{1}\sim \gamma ^{2}E}
{\displaystyle E^{\prime }\gg m_{e}c^{2}}
{\displaystyle {\begin{aligned}E_{1}^{\prime }&={E^{\prime } \over 1+{E^{\prime } \over m_{e}c^{2}}(1-\cos \phi ^{\prime })}\approx m_{e}c^{2}\\E_{1}&=E_{1}^{\prime }\gamma (1+{v \over c}\cos \theta _{1}^{\prime })\\&\approx m_{e}c^{2}\gamma (1+{v \over c}\cos \theta _{1}^{\prime })\approx \gamma m_{e}c^{2}\\\end{aligned}}\,\!}
This final term defines the maximum rebound of the photon.
Retrieved from "http:///astrobaki/index.php?title=Compton_Scattering&oldid=5840" |
Research on the Impact of Internet Finance on the Efficiency of Chinese Commercial Banks
\mathrm{ln}{y}_{it}={\beta }_{0}+{\beta }_{t}t+\beta {x}_{it}+\mathrm{ln}{v}_{it}+\mathrm{ln}{u}_{it},i=1,\cdots ,N;t=1,\cdots ,T
{m}_{it}={\delta }_{0}+{\delta }_{t}t+\delta {z}_{it}+{W}_{it}
C{E}_{it}=\mathrm{exp}\mathrm{exp}\left({u}_{it}\right)=\mathrm{exp}\mathrm{exp}\left({z}_{it}\delta +{W}_{it}\right)
T{E}_{it}=\mathrm{exp}\mathrm{exp}\left(-{u}_{it}\right)=\mathrm{exp}\mathrm{exp}\left(-{z}_{it}\delta -{W}_{it}\right)
{\sigma }_{v}^{2}
{\sigma }_{v}^{2}
{\sigma }^{2}
\begin{array}{l}\mathrm{ln}PTA={\alpha }_{0}+\sum _{i=1}^{3}{\alpha }_{i}\mathrm{ln}{y}_{i}+{\sum _{i=1}^{3}{\beta }_{i}\mathrm{ln}w}_{i}+{t}_{1}T+{\tau }_{1}\mathrm{ln}R\\ \text{}+\frac{1}{2}\left[\sum _{i=1}^{3}\sum _{j=1}^{3}{\delta }_{ij}\mathrm{ln}{y}_{i}\mathrm{ln}{y}_{j}+\sum _{i=1}^{3}\sum _{j=1}^{3}{\gamma }_{ij}\mathrm{ln}{w}_{i}\mathrm{ln}{w}_{j}+{t}_{11}{T}^{2}+{\tau }_{11}\mathrm{ln}{R}^{2}\right]\\ \text{}+\sum _{i=1}^{3}\sum _{j=1}^{3}{\rho }_{ij}\mathrm{ln}{y}_{i}\mathrm{ln}{w}_{j}+\sum _{i=1}^{3}\sum _{j=1}^{3}{\phi }_{i}\mathrm{ln}{y}_{i}T+\sum _{i=1}^{3}\sum _{j=1}^{3}{\vartheta }_{i}\mathrm{ln}{w}_{i}T\\ \text{}+\sum _{i=1}^{3}\sum _{j=1}^{3}{\theta }_{ij}\mathrm{ln}{y}_{i}\mathrm{ln}R+\sum _{i=1}^{3}\sum _{j=1}^{3}{\mu }_{ij}\mathrm{ln}{w}_{i}\mathrm{ln}R+\sum _{i=1}^{3}\sum _{j=1}^{3}{\pi }_{ij}T\mathrm{ln}R-\mathrm{ln}{u}_{it}+\mathrm{ln}{v}_{it}\end{array}
{\sigma }_{v}^{2}
{\sigma }_{v}^{2}
{\delta }_{ij}={\delta }_{ji}
{\gamma }_{ij}={\gamma }_{ji}
{\rho }_{ij}={\rho }_{ji}
\begin{array}{l}\mathrm{ln}\frac{PTA}{{w}_{3}\ast A}={\alpha }_{0}+\sum _{i=1}^{3}{\alpha }_{i}\mathrm{ln}\frac{{y}_{i}}{A}+\sum _{i=1}^{3}{\beta }_{i}\mathrm{ln}\frac{{w}_{i}}{{w}_{3}}+{t}_{1}T+{\tau }_{1}\mathrm{ln}R\\ \text{}+\frac{1}{2}\left[\sum _{i=1}^{3}\sum _{j=1}^{3}\mathrm{ln}\frac{{y}_{i}}{A}\mathrm{ln}\frac{{y}_{j}}{A}+\sum _{i=1}^{3}\sum _{j=1}^{3}{\gamma }_{ij}\mathrm{ln}\frac{{w}_{i}}{{w}_{3}}\mathrm{ln}\frac{{w}_{j}}{{w}_{3}}+{t}_{11}{T}^{2}+{\tau }_{11}\mathrm{ln}{R}^{2}\right]\\ \text{}+\sum _{i=1}^{3}\sum _{j=1}^{3}{\rho }_{ij}\mathrm{ln}\frac{{y}_{i}}{A}\mathrm{ln}\frac{{w}_{j}}{{w}_{3}}+\sum _{i=1}^{3}\sum _{j=1}^{3}{\phi }_{i}\mathrm{ln}\frac{{y}_{i}}{A}T+\sum _{i=1}^{3}\sum _{j=1}^{3}{\vartheta }_{i}\mathrm{ln}\frac{{w}_{i}}{{w}_{3}}T\\ \text{}+\sum _{i=1}^{3}\sum _{j=1}^{3}{\theta }_{ij}\mathrm{ln}\frac{{y}_{i}}{A}\mathrm{ln}R+\sum _{i=1}^{3}\sum _{j=1}^{3}{\mu }_{ij}\mathrm{ln}\frac{{w}_{i}}{{w}_{3}}\mathrm{ln}R+\sum _{i=1}^{3}\sum _{j=1}^{3}{\pi }_{ij}T\mathrm{ln}R\\ \text{}-\mathrm{ln}{u}_{it}+\mathrm{ln}{v}_{it}\end{array}
\begin{array}{l}EF{F}_{it}={\beta }_{0}+{\beta }_{1}\ast NLR+{\beta }_{2}\ast BS+{\beta }_{3}\ast PRS+{\beta }_{4}\ast MS\\ \text{}+{\beta }_{5}\ast TPP+{\beta }_{6}\ast GOP+{\epsilon }_{it}\end{array}
Zhao, S. (2018) Research on the Impact of Internet Finance on the Efficiency of Chinese Commercial Banks. American Journal of Industrial and Business Management, 8, 898-911. https://doi.org/10.4236/ajibm.2018.84062
1. Repková, I. (2015) Banking Efficiency Determinants in the Czech Banking Sector. Procedia Economics and Finance, 23, 191-196.
2. Tan, Y. and Floros, C. (2013) Risk, Capital and Efficiency in Chinese Banking. Journal of International Financial Markets, Institutions & Money, 26, 378-393.
3. Mosko, A. and Bozdo, A. (2016) Modeling the Relationship between Bank Efficiency, Capital and Risk in Albanian Banking System. Procedia Economics and Finance, 39, 319-327.
4. Fiordelisi, F., Marques-Ibanez, D. and Molyneux, P. (2011) Efficiency and Risk in European Banking. Journal of Banking & Finance, 35, 1315-1326.https://doi.org/10.1016/j.jbankfin.2010.10.005
5. Delis, M., Iosifidi, M. and Tsionas, M.G. (2017) Endogenous Bank Risk and Efficiency. European Journal of Operational Research, 260, 376-387.
6. Wang, C. and Zou, P.F. (2006) Study on X-Efficiency of Chinese Commercial Banks Based on Capital Structure and Risk. Management World, 11, 6-12.
7. Cheng, M.Y. and Zhao, H. (2011) Analysis of Influence of Market Power on Bank Efficiency. The Journal of Quantitative & Technical Economics, 10, 78-91.
8. Liu, R.X., Lv, X. and Luo, Y. (2016) The Measurement of China’s Commercial Banks’ Efficiency under the Restriction of Bad Loans—Based on the Two-Phase DEA Model. Journal of Nanjing Audit University, 6, 41-50.
9. He, Y. (2014) Analysis of Cost Efficiency and Influencing Factors of China’s Banking Industry: A Study Based on Stochastic Frontier Approach. Central China Normal University Journal of Postgraduates, 21, No. 12.
10. Zhu, J.C. (2013) Research on the Background, Current situation and the Trend of Internet Finance. Rural Finance Research, 10, 5-8.
11. Qiu, F. (2013) Internet Financial Shock and Commercial Bank Response. Financial Accounting, 11, 45-49.
12. Zhang, M. (2014) Internet Finance and Traditional Finance. China Policy Review, 2, 46-49.
13. Battese, G.E. and Coelli, T.J. (1995) A Model for Technical Inefficiency Effects in a Stochastic Frontier Production Function for Panel Data. Empirical Economics, 20, 325-332. https://doi.org/10.1007/BF01205442
14. Yao, S.J., Jiang, C.X. and Feng, G.F. (2011) The Reform and Efficiency of Chinese Banking: 1995-2008. Economic Research Journal, 8, 4-14. |
* [[Classification of Quadratic APN Trinomials, Quadrinomials, Pentanomials, Hexanomials (CCZ-inequivalent with infinite monomial families) in Small Dimensions with all Coefficients equal to 1]]
* [[Differentially 4-uniform permutations ]]
* [[CCZ-invariants for all known APN functions in dimension 7]]
{\displaystyle \mathbb {F} _{2^{n}}} |
A Modified Average Reynolds Equation for Rough Bearings With Anisotropic Slip | J. Tribol. | ASME Digital Collection
Hsiang-Chin Jao,
Hsiang-Chin Jao
No. 1 University Road,
e-mail: q28991067@mail.ncku.edu.tw
Kuo-Ming Chang,
Kuo-Ming Chang
No. 415 Chien Kung Road,
e-mail: koming@cc.kuas.edu.tw
Li-Ming Chu,
No. 1 Nantai Street,
e-mail: lmchu@mail.stust.edu.tw
e-mail: wlli@mail.ncku.edu.tw
Contributed by the Tribology Division of ASME for publication in the JOURNAL OF TRIBOLOGY. Manuscript received February 14, 2015; final manuscript received June 14, 2015; published online August 6, 2015. Assoc. Editor: Min Zou.
Jao, H., Chang, K., Chu, L., and Li, W. (August 6, 2015). "A Modified Average Reynolds Equation for Rough Bearings With Anisotropic Slip." ASME. J. Tribol. January 2016; 138(1): 011702. https://doi.org/10.1115/1.4030901
A lubrication theory that includes the coupled effects of surface roughness and anisotropic slips is proposed. The anisotropic-slip phenomena originate from the microscale roughness at the atomic scale (microtexture) and surface properties of the lubricating surfaces. The lubricant flow between rough surfaces (texture) is defined as the flow in nominal film thickness multiplied by the flow factors. A modified average Reynolds equation (modified ARE) as well as the related factors (pressure and shear flow factors, and shear stress factors) is then derived. The present model can be applied to squeeze film problems for anisotropic-slip conditions and to sliding lubrication problems with restrictions to symmetric anisotropic-slip conditions (the two lubricating surfaces have the same principal slip lengths, i.e.,
b1x=b2x
b1y=b2y
). The performance of journal bearings is discussed by solving the modified ARE numerically. Different slenderness ratios 5, 1, and 0.2 are considered to discuss the coupled effects of anisotropic slip and surface roughness. The results show that the existence of boundary slip can dilute the effects of surface roughness. The boundary slip tends to “smoothen” the bearings, i.e., the derived flow factors with slip effects deviate lesser from the values at smooth cases (pressure flow factors
φxxp,φyyp=1
; shear flow factors
φxxs=0
; and shear stress factors
φf,φfp=1
φfs=0
) than no-slip one. The load ratio increases as the dimensionless slip length (
B
) decreases exception case is also discussed or the slenderness ratio (
b/d
) increases. By controlling the surface texture and properties, a bearing with desired performance can be designed.
Fluid film lubrication, Hydrodynamic lubrication, Journal bearings, Surface roughness, Asperities
Anisotropy, Bearings, Flow (Dynamics), Surface roughness, Journal bearings, Pressure, Shear stress, Stress
Hydrodynamic Force Measurements: Boundary Slip of Water on Hydrophilic Surfaces and Electrokinetic Effects
V. S. J.
Shear-Dependent Boundary Slip in an Aqueous Newtonian Liquid
Limits of Hydrodynamic No-Slip Boundary Condition
Apparent Slip of Newtonian Fluids Past Adsorbed Polymer Layers
The No Slip Boundary Condition Switches to Partial Slip When the Fluid Contains Surfactant
Direct Experimental Evidence of Slip in Hexadecane: Solid Interfaces
Slippery Questions About Complex Fluids Flowing Past Solids
Slippage of Water Over Hydrophobic Surfaces
Int. J. Miner. Process.
C. L. M. H.
,” Mémoires de l'Académie Royale des Sciences de l'Institut de France, 6, pp.
The Half-Wetted Bearing, Part 1: Extended Reynolds Equation
The Half-Wetted Bearing, Part 2: Potential Application in Low Load Contacts
Equation for Slip of Simple Liquids at Smooth Solid Surfaces
The Partially Wetted Bearing—Extended Reynolds Equation
T. V. V. L. N.
Wall Slip Effects in (Elasto) Hydrodynamic Journal Bearings
Microstructure and Lubricating Property of Ultrafast Laser Pulse Textured Silicon Carbide Seals
Low-Friction Characteristics of Nanostructured Surfaces on Silicon Carbide for Water-Lubricated Seals
Direct Velocity Measurements of the Flow Past Drag Reducing Ultrahydrophobic Surfaces
Cottin-Bizonne
Slippage of Water Past Superhydrophobic Carbon Nanotube Forests in Microchannels
Ulmanella
Effective Slip and Friction Reduction in Nanograted Superhydrophobic Microchannels
Anisotropic Wetting on Microstrips Surface Fabricated by Femtosecond Laser
A Low Friction Bearing Based on Liquid Slip at the Wall
Patterning Flows Using Grooved Surfaces
Tensorial Hydrodynamic Slip
Effective Slip in Pressure-Driven Flow Past Superhydrophobic Stripes
Characteristics of Journal Bearings With Anisotropic Slip
Surface Roughness Effects in Journal Bearings With Non-Newtonian Lubricants
An Average Reynolds Equation for Non-Newtonian Fluid, With Application to the Lubrication of the Magnetic Head-Disk Interface
Surface Roughness Effects in Hydrodynamic Lubrication Involving the Mixture of Two Fluids
Some Discussions on the Flow Factor Tensor-Considerations of Roughness Orientation and Flow Rheology
An Average Flow Model for Couple Stress Fluids
, Chap. 10, Sec. 3.
Modeling of Head Disk Interface an Average Flow Model
Cavitation in Bearings
A Penalty Formulation and Numerical Approximation of the Reynolds–Hertz Problem of Elastohydrodynamic Lubrication
Review of Fluid Slip Over Superhydrophobic Surfaces and Its Dependence on the Contact Angle
Effects of Roughness Orientations on Thin Film Lubrication of Magnetic Recording System
Improvement and Analysis for a Gaseous Cavitation Model Applied in a Tilting-Pad Journal Bearing |
Spectrographs - Wikiversity
The image shows a cap-off view of the insides of a spectrograph. Credit: Buil/AstroSurf.
This problem set is designed for astronomy to help the student, teacher, and researcher understand the inner workings of a spectrograph.
The image shows a plastic prism. Credit: .
{\displaystyle n_{0}}
{\displaystyle n_{1}}
{\displaystyle n_{2}}
{\displaystyle \theta '}
indicate the ray angles after refraction. Credit: NathanHagen.
A prism is a transparent optical element with flat, polished surfaces that refract light [over a range of wavelengths]. At least two of the flat surfaces must have an angle [α] between them. The exact angles between the surfaces depend on the application. The traditional geometrical shape is that of a triangular prism with a triangular base and rectangular sides, and in colloquial use "prism" usually refers to this type.
{\displaystyle {\begin{aligned}\theta '_{0}&=\,{\text{arcsin}}{\Big (}{\frac {n_{0}}{n_{1}}}\,\sin \theta _{0}{\Big )}\\\theta _{1}&=\alpha -\theta '_{0}\\\theta '_{1}&=\,{\text{arcsin}}{\Big (}{\frac {n_{1}}{n_{2}}}\,\sin \theta _{1}{\Big )}\\\theta _{2}&=\theta '_{1}-\alpha \end{aligned}}}
For a prism in air
{\displaystyle n_{0}=n_{2}\simeq 1}
{\displaystyle n=n_{1}}
{\displaystyle \delta }
{\displaystyle \delta =\theta _{0}+\theta _{2}=\theta _{0}+{\text{arcsin}}{\Big (}n\,\sin {\Big [}\alpha -{\text{arcsin}}{\Big (}{\frac {1}{n}}\,\sin \theta _{0}{\Big )}{\Big ]}{\Big )}-\alpha }
{\displaystyle \theta _{0}}
{\displaystyle \alpha }
{\displaystyle \sin \theta \approx \theta }
{\displaystyle {\text{arcsin}}x\approx x}
{\displaystyle \delta }
{\displaystyle \delta \approx \theta _{0}-\alpha +{\Big (}n\,{\Big [}{\Big (}\alpha -{\frac {1}{n}}\,\theta _{0}{\Big )}{\Big ]}{\Big )}=\theta _{0}-\alpha +n\alpha -\theta _{0}=(n-1)\alpha \ .}
{\displaystyle \delta (\lambda )\approx [n(\lambda )-1]\alpha }
The image at the top of this resource appears to have a deviation angle of 45°. The detector may be about 4 cm from the prism. Using a refractive index n = 1.732 and a representative wavelength for the optical colors calculate the width of each wavelength channel and the total detector width to capture the incoming photons. Use an apex angle of 35°.
Let the deviation angle be 30°, the detector distance be half a meter with the same refractive index and apex angle of 37.5°.
For the optical colors calculate the width of each wavelength channel and the total detector width to capture the incoming photons.
Using the configuration of Problem 2 and assuming a prism for X-rays and gamma rays existed, calculate the width of each wavelength channel, for five representative wavelengths of each, and the total detector width to capture the incoming photons.
Using the configuration of Problem 1 and representative wavelengths for each of the infrared bands described in infrared astronomy, calculate the width of each wavelength channel and the total detector width to capture the incoming photons.
Using each configuration of the problems above and representative wavelengths for submillimeter, microwave, and radio waves, calculate the width of each wavelength channel and the total detector width to capture the incoming photons.
Amateur astronomers may be able to build or buy a spectrograph.
Problems/Astronomy
Retrieved from "https://en.wikiversity.org/w/index.php?title=Spectrographs&oldid=2370199" |
Prediction of Ingress Through Turbine Rim Seals—Part II: Combined Ingress | J. Turbomach. | ASME Digital Collection
, Bath BA2 7AY, U.K.
A companion article has been published: Prediction of Ingress Through Turbine Rim Seals—Part I: Externally Induced Ingress
Owen, J. M., Pountney, O., and Lock, G. (July 15, 2011). "Prediction of Ingress Through Turbine Rim Seals—Part II: Combined Ingress." ASME. J. Turbomach. May 2012; 134(3): 031013. https://doi.org/10.1115/1.4003071
In Part I of this two-part paper, the orifice equations were solved for the case of externally induced (EI) ingress, where the effects of rotational speed are negligible. In Part II, the equations are solved, analytically and numerically, for combined ingress (CI), where the effects of both rotational speed and external flow are significant. For the CI case, the orifice model requires the calculation of three empirical constants, including
Cd,e,RI
Cd,e,EI
, the discharge coefficients for rotationally induced (RI) and EI ingress. For the analytical solutions, the external distribution of pressure is approximated by a linear saw-tooth model; for the numerical solutions, a fit to the measured pressures is used. It is shown that although the values of the empirical constants depend on the shape of the pressure distribution used in the model, the theoretical variation of
Cw,min
(the minimum nondimensional sealing flow rate needed to prevent ingress) depends principally on the magnitude of the peak-to-trough pressure difference in the external annulus. The solutions of the orifice model for
Cw,min
are compared with published measurements, which were made over a wide range of rotational speeds and external flow rates. As predicted by the model, the experimental values of
Cw,min
could be collapsed onto a single curve, which connects the asymptotes for RI and EI ingress at the respective smaller and larger external flow rates. At the smaller flow rates, the experimental data exhibit a minimum value of
Cw,min
, which undershoots the RI asymptote. Using an empirical correlation for
Cd,e
, the model is able to predict this undershoot, albeit smaller in magnitude than the one exhibited by the experimental data. The limit of the EI asymptote is quantified, and it is suggested how the orifice model could be used to extrapolate the effectiveness data obtained from an experimental rig to engine-operating conditions.
engines, orifices (mechanical), swirling flow, turbines
Annulus, Discharge coefficient, Engines, Flow (Dynamics), Pressure, Turbines, Sealing (Process), Clearances (Engineering), Fluids, Wheels
Theoretical Prediction of Ingress Through Turbine Rim Seals—Part I: Externally Induced Ingress
A Numerical Investigation Into the Effect of an External Flow Field on the Sealing of a Rotor-Stator Cavity
,” Ph.D. thesis, University of Sussex, UK.
Flow and Heat Transfer in Rotating Disc Systems, Volume 1—Rotor-Stator Systems
Flow and Heat Transfer in Rotating Disc Systems, Volume 2—Rotating Cavities |
Find intersection of line and circle in Cartesian coordinates - MATLAB linecirc - MathWorks Benelux
linecirc
Find Intersection of Line and Circle
Find Intersection of Vertical Line and Circle
Find intersection of line and circle in Cartesian coordinates
[xout,yout] = linecirc(slope,intercpt,centerx,centery,radius)
[xout,yout] = linecirc(slope,intercpt,centerx,centery,radius) finds the intersection of a line with the specified slope and intercept and a circle with the specified center and radius, in Cartesian coordinates.
Find the intersection of the line
\mathit{y}=2\mathit{x}-1
and a circle with its center at (3, 4) and a radius of 5.
[xout,yout] = linecirc(2,-1,3,4,5)
xout = 1×2
yout = 1×2
\mathit{x}=-1
[xout,yout] = linecirc(Inf,-1,3,4,5)
slope — Slope of line
numeric scalar | Inf
Slope of the line, specified as a numeric scalar or Inf. Specify Inf when the line is vertical.
intercpt — Intercept of line
Intercept of the line, specified as a numeric scalar.
When slope is a numeric scalar, this argument is the y-intercept of the line.
When slope is Inf, this argument is the x-intercept of the line.
centerx — x-coordinate of center of circle
x-coordinate of the center of the circle, specified as a numeric scalar.
centery — y-coordinate of center of circle
y-coordinate of the center of the circle, specified as a numeric scalar.
Radius of the circle, specified as a positive scalar.
xout — x-coordinates of intersections
x-coordinates of the intersections, returned as a two-element vector.
When the line is tangent to the circle, the elements of the vector are equal.
When the line does not intersect the circle, both elements are NaN.
yout — y-coordinates of intersections
y-coordinates of the intersections, returned as a two-element vector.
circcirc |
2011 Slip Effects on Fractional Viscoelastic Fluids
Muhammad Jamil, Najeeb Alam Khan
Unsteady flow of an incompressible Maxwell fluid with fractional derivative induced by a sudden moved plate has been studied, where the no-slip assumption between the wall and the fluid is no longer valid. The solutions obtained for the velocity field and shear stress, written in terms of Wright generalized hypergeometric functions
{}_{p}{\Psi }_{q}
, by using discrete Laplace transform of the sequential fractional derivatives, satisfy all imposed initial and boundary conditions. The no-slip contributions, that appeared in the general solutions, as expected, tend to zero when slip parameter is
\theta \to 0
. Furthermore, the solutions for ordinary Maxwell and Newtonian fluids, performing the same motion, are obtained as special cases of general solutions. The solutions for fractional and ordinary Maxwell fluid for no-slip condition also obtained as limiting cases, and they are equivalent to the previously known results. Finally, the influence of the material, slip, and the fractional parameters on the fluid motion as well as a comparison among fractional Maxwell, ordinary Maxwell, and Newtonian fluids is also discussed by graphical illustrations.
Muhammad Jamil. Najeeb Alam Khan. "Slip Effects on Fractional Viscoelastic Fluids." Int. J. Differ. Equ. 2011 (SI1) 1 - 19, 2011. https://doi.org/10.1155/2011/193813
Received: 23 May 2011; Accepted: 7 September 2011; Published: 2011
Muhammad Jamil, Najeeb Alam Khan "Slip Effects on Fractional Viscoelastic Fluids," International Journal of Differential Equations, Int. J. Differ. Equ. 2011(SI1), 1-19, (2011) |
Classification of all Jacobian elliptic fibrations on certain K3 surfaces
July, 2006 Classification of all Jacobian elliptic fibrations on certain K3 surfaces
In this paper we classify all configurations of singular fibers of elliptic fibrations on the double cover of
{\mathbf{P}}^{2}
ramified along six lines in general position.
Remke KLOOSTERMAN. "Classification of all Jacobian elliptic fibrations on certain K3 surfaces." J. Math. Soc. Japan 58 (3) 665 - 680, July, 2006. https://doi.org/10.2969/jmsj/1156342032
Keywords: elliptic fibrations , singular fiber configuration
Remke KLOOSTERMAN "Classification of all Jacobian elliptic fibrations on certain K3 surfaces," Journal of the Mathematical Society of Japan, J. Math. Soc. Japan 58(3), 665-680, (July, 2006) |
Solve the problem below by writing and solving an equation. Be sure to define your variable.
122
countries in Africa, Europe, and North America (as of 2012). Europe has twice as many countries as North America, and Africa has seven more than Europe. How many countries are in each of these three continents? Write an equation and solve it to answer this question.
n=
the number of countries in North America.
2n=
number of countries in Europe
2n+7=
the number of countries in Africa
The total number of countries in all three continents is
122
n+(2n)+(2n+7)=122
n=23
23
countries in North America.
Remember to find the number of countries in Europe and Africa. |
Compare Approaches to Cointegration Analysis - MATLAB & Simulink - MathWorks América Latina
Comparing inferences and estimates from the Johansen and Engle-Granger approaches can be challenging, for a variety of reasons. First of all, the two methods are essentially different, and may disagree on inferences from the same data. The Engle-Granger two-step method for estimating the VEC model, first estimating the cointegrating relation and then estimating the remaining model coefficients, differs from Johansen's maximum likelihood approach. Secondly, the cointegrating relations estimated by the Engle-Granger approach may not correspond to the cointegrating relations estimated by the Johansen approach, especially in the presence of multiple cointegrating relations. It is important, in this context, to remember that cointegrating relations are not uniquely defined, but depend on the decomposition
C=A{B}^{\prime }
of the impact matrix.
Nevertheless, the two approaches should provide generally comparable results, if both begin with the same data and seek out the same underlying relationships. Properly normalized, cointegrating relations discovered by either method should reflect the mechanics of the data-generating process, and VEC models built from the relations should have comparable forecasting abilities.
As the following shows in the case of the Canadian interest rate data, Johansen's H1* model, which is the closest to the default settings of egcitest, discovers the same cointegrating relation as the Engle-Granger test, assuming a cointegration rank of 2:
beta = [1; -b];
[~,~,~,~,mles] = jcitest(Y,'model','H1*');
Model: H1*
BJ2 = mles.r2.paramVals.B;
c0J2 = mles.r2.paramVals.c0;
% Normalize the 2nd cointegrating relation with respect to
% the 1st variable, to make it comparable to Engle-Granger:
BJ2n = BJ2(:,2)/BJ2(1,2);
c0J2n = c0J2(2)/BJ2(1,2);
% Plot the normalized Johansen cointegrating relation together
% with the original Engle-Granger cointegrating relation:
plot(dates,Y*beta-c0,'LineWidth',2,'Color',COrd(4,:))
plot(dates,Y*BJ2n+c0J2n,'--','LineWidth',2,'Color',COrd(5,:))
legend('Engle-Granger OLS','Johansen MLE','Location','NW')
title('{\bf Cointegrating Relation}') |
Telescopes and cameras - Wikiversity
{\displaystyle f}
{\displaystyle D}
{\displaystyle {\frac {1}{f}}=(n-1)\left[{\frac {1}{R_{1}}}-{\frac {1}{R_{2}}}+{\frac {(n-1)d}{nR_{1}R_{2}}}\right],}
What is the formula for a plano-convex lens?
Calculate the focal length of a double convex lens in which R1 = 10 m and R2 = 200 m. Let n = 1.732 and d = 2 cm.
What is the focal length of a hemispherical plano-convex lens of radius 10 m? Let n= 1.324 and d = 1 dm.
What is a likely focal length for a plano-convex lens where the convex surface is parabolic rather than spherical?
Diagram of decreasing apertures, that is, increasing f-numbers, in one-stop increments; each aperture has half the light gathering area of the previous one. Credit: Cbuckley.
For a focal length of 100 mm, what are the aperture or entrance pupil diameters for each of the apertures shown if no optical components are between the entrance pupil and the aperture?
For a simple camera or telescope with an aperture lens diameter of 5 m, what is the focal length for each of the apertures or entrance pupil diameters in the diagram if no optical components are between the entrance pupil and the aperture?
For the following pairs of focal lengths and lens diameters, what would be the f-numbers? (a) 5 m, 3 m (b) 10 m, 3 mm (c) 50 cm, 2 m (d) 15 km, 25 m (e) 5,000 km, 16 cm.
{\displaystyle f/11}
The lens in the image at right has an aperture range of
{\displaystyle f/2.0}
{\displaystyle f/22.}
The image lens at right uses a standard f-stop scale, which is an approximately geometric sequence of numbers that corresponds to the sequence of the powers of the square root of 2:
{\displaystyle f/1,}
{\displaystyle f/1.4}
{\displaystyle f/2,}
{\displaystyle f/2.8,}
{\displaystyle f/4,}
{\displaystyle f/5.6,}
{\displaystyle f/8,}
{\displaystyle f/11,}
{\displaystyle f/16,}
{\displaystyle f/22.}
{\displaystyle f/1={\frac {f/1}{({\sqrt {2}})^{0}}},}
{\displaystyle f/1.4={\frac {f/1}{({\sqrt {2}})^{1}}},}
{\displaystyle f/2={\frac {f/1}{({\sqrt {2}})^{2}}},}
{\displaystyle f/2.8={\frac {f/1}{({\sqrt {2}})^{3}}},}
{\displaystyle f/4=,}
{\displaystyle f/5.6=,}
{\displaystyle f/8=,}
{\displaystyle f/11=,}
{\displaystyle f/16=,}
{\displaystyle f/22=.}
What are the missing terms?
Each "stop" is marked with its corresponding f-number, and represents a halving of the light intensity from the previous stop. This corresponds to a decrease of the pupil and aperture diameters by a factor of 1/
{\displaystyle \scriptstyle {\sqrt {2}}}
or about 0.7071, and hence a halving of the area of the pupil.
For a lens diameter of 35 mm, what is the focal length for each f-stop?
Shutter speeds are arranged so that each setting differs in duration by a factor of approximately two from its neighbour. Opening up a lens by one stop allows twice as much light to fall on the film in a given period of time. Therefore to have the same exposure at this larger aperture as at the previous aperture, the shutter would be opened for half as long (i.e., twice the speed). The film will respond equally to these equal amounts of light, since it has the property of reciprocity. This is less true for extremely long or short exposures, where we have reciprocity failure. Aperture, shutter speed, and film sensitivity are linked: for constant scene brightness, doubling the aperture area (one stop), halving the shutter speed (doubling the time open), or using a film twice as sensitive, has the same effect on the exposed image.
Shutter speed or exposure time is the length of time a camera's shutter is open when taking a photograph.[1] The amount of light that reaches the film or image sensor is proportional to the exposure time.
Exposure value (EV) is a single quantity that accounts for the shutter speed and the f-number.
Multiple combinations of shutter speed and f-number can give the same exposure value.
Doubling the exposure time doubles the amount of light (subtracts 1 EV). Making the f-number one stop brighter (reducing the f-number by a factor of
{\displaystyle \scriptstyle {\sqrt {2}}}
) also doubles the amount of light. A shutter speed of 1/50 s with an
{\displaystyle f/4.0}
lens gives the same exposure value as a 1/100 s shutter with an {
{\displaystyle f/2.8}
lens, and also the same exposure value as a 1/200 s shutter with an
{\displaystyle f/2.0}
lens.
A standardized 2:1 scale for shutter speed is such that opening one aperture stop and reducing the shutter speed by one step resulted in the identical exposure. The standards for shutter speeds are:[2]
What are the other f-numbers that match the shutter speeds?
Exposure value (EV) is a number that represents a combination of a camera's shutter speed and f-number, such that all combinations that yield the same exposure have the same EV value (for any fixed scene luminance). Exposure value also is used to indicate an interval on the photographic exposure scale, with 1 EV corresponding to a standard power-of-2 exposure step, commonly referred to as a stop.
Exposure value is a base-2 logarithmic scale:
{\displaystyle \mathrm {EV} =\log _{2}{\frac {N^{2}}{t}}\,,}
N is the relative aperture (f-number)
t is the exposure time (“shutter speed”) in seconds.
In a mathematical expression involving physical quantities, it is common practice to require that the argument to a transcendental function (such as the logarithm) be dimensionless. The definition of EV ignores the units in the denominator and uses only the numerical value of the exposure time in seconds; EV is not the expression of a physical law, but simply a number for encoding combinations of camera settings.
Match up the shutter speeds with f-numbers to give the same EV.
Alternate title could be Radiation telescopes/Problem set.
↑ Sidney F. Ray (2000). "Camera Features". In Ralph Eric Jacobson. Manual of Photography: A Textbook of Photographic and Digital Imaging (Ninth ed.). Focal Press. pp. 131–2. ISBN 0-240-51574-9. http://books.google.com/books?id=HHX4xB94vcMC&pg=PA132&ots=7Gq_Az_-zl&sig=bQ5bvKIS-y1_Q4km6Pm-yCZDcGo.
↑ Cub Kahn (1999). Essential Skills for Nature Photography. Amherst Media. ISBN 1-58428-009-3. http://books.google.com/books?id=EZhNY--TZjIC&pg=PT21&ei=h0MHSejTFI_gswOh3eDzDQ.
{{Radiation astronomy resources}}{{Reasoning resources}}{{Technology resources}}
Retrieved from "https://en.wikiversity.org/w/index.php?title=Telescopes_and_cameras&oldid=2246485"
Instruments/Problems
Photography/Problems
Semantics/Problems |
Robbie builds model rockets. One day he sets up a rocket, backs away from the launch pad, and then shoots the rocket off into the air. The rocket’s path is represented by the equation
y=-10x^2+130x-400
y
is the height in meters off the ground and
x
is the horizontal distance in meters from Robbie.
Use either the Zero Product Property or the Quadratic Formula to find the
x
-intercepts of the path of Robbie’s rocket. What do the
x
-intercepts tell you?
x=8
x=5
(5,0)
(8,0)
Robbie backed up
5
m from the launch pad.
The rocket landed
8
m away from him.perty or the Quadratic Formula to find the
x
-intercepts of the path of Robbie's rocket. What do the
x
When Robbie's rocket lands, how far is it from the launch pad?
The launch pad was 5m from Robbie, and the rocket landed 8m from him in the same direction.
Use the graph in the eTool below for this problem. |
On two problems studied by A. Ambrosetti | EMS Press
We study the Ambrosetti-Prodi and Ambrosetti-Rabinowitz problems. We prove for the first one the existence of a continuum of solutions with shape of a reflected C (
\supset
-shape). Next, we show that there is a relationship between these two problems.
David Arcoya, José Carmona, On two problems studied by A. Ambrosetti. J. Eur. Math. Soc. 8 (2006), no. 2, pp. 181–188 |
Trilinear forms and the central values of triple product L-functions
1 November 2008 Trilinear forms and the central values of triple product
L
Atsushi Ichino1
1Department of Mathematics, Graduate School of Science, Osaka City University; current: School of Mathematics, Institute for Advanced Study
We give an explicit formula for certain global trilinear forms that appear in Jacquet's conjecture in terms of local trilinear forms and the central values of triple product
L
Atsushi Ichino. "Trilinear forms and the central values of triple product
L
-functions." Duke Math. J. 145 (2) 281 - 307, 1 November 2008. https://doi.org/10.1215/00127094-2008-052
Atsushi Ichino "Trilinear forms and the central values of triple product
L
-functions," Duke Mathematical Journal, Duke Math. J. 145(2), 281-307, (1 November 2008) |
Numerical Analysis of Slag Carry-Over during Molten Steel Draining
Open Journal of Applied Sciences > Vol.7 No.11, November 2017
Numerical Analysis of Slag Carry-Over during Molten Steel Draining ()
Departmento De Materiales, Universidad Autonoma Metropolitana, Mexico City, Mexico.
DOI: 10.4236/ojapps.2017.711044 PDF HTML XML 815 Downloads 1,423 Views Citations
Slag carry-over during the draining of molten steel from a teeming ladle is numerically studied here. Two-phase isothermal transient 3D Computational Fluid Dynamics simulations were employed to simulate the draining process. Two nozzle diameters, two nozzle positions and three slag heights were considered. From mass balances, the slag carry-over in terms of mass flow rate was obtained for each of the above variables. Besides, the draining times of the teeming ladle were estimated from theoretical considerations and CDF simulations, and compared.
CFD Simulations, Draining Time, Multiphase Flow, Slag Carry-Over, Teeming Ladle
Flores-Sanchez, D. and Barron, M. (2017) Numerical Analysis of Slag Carry-Over during Molten Steel Draining. Open Journal of Applied Sciences, 7, 611-616. doi: 10.4236/ojapps.2017.711044.
Slag carry-over during draining of molten steel from teeming ladles is an important industrial issue given it affects the quality of the solid steel. Main problems of slag carry-over are [1] : 1) hindering of addition of alloys and conditioners; 2) high levels of FeO and MnO, which result in high oxygen content of steel; 3) increased processing time and treatment costs; 4) high inclusion formation, which causes steel cleanliness problems and increased risk of nozzle clogging during casting; 5) phosphorous reversion in the ladle; 6) poor removal of sulfur in the ladle; and 7) increased ladle refractory wear.
Tapping of molten steel without slag carry-over is a difficult task due to the formation of a draining vortex [2] . In [3] the mechanism of slag carry-over during drainage of metallurgical vessel is studied using a physical model. Vortex and drain sink formation are found to be the main mechanism of carry-over of slag to the underlying vessel. The mechanisms of vortex formation are studied in [4] using water modeling and computer simulations. The authors report that the critical bath height for vortex development increases with steel throughputs and nozzle opening.
In this work, the slag carry-over during molten steel draining from a teeming ladle (see Figure 1) is numerically studied using 3D transient isothermal Computational Fluid Dynamics (CFD) simulations. A circular nozzle is located at two positions of the bottom of the ladle: centered and off-centered (0.5 m from low border). Two diameters of the nozzles, and three heights of the slag layer are considered. The slag carry-over in terms of mass flow rate of slag is quantified through mass balances as function of nozzle diameter, nozzle position and slag height.
The flow of an isothermal incompressible Newtonian fluid and the mass conservation are represented by the Navier-Stokes equations and the continuity equation,
Figure 1. The teeming ladle. In this figure hms is the height of molten steel and hs is the height of slag.
respectively [5] . Turbulence in the mold is simulated by means of the classical two equations K-ε model [6] . The multiphase nature of the ladle flow is simulated by means of the Volume of Fluid (VOF) model [7] , which considers that all the present phases share the same flow field. The mass conservation principle forces that the whole of the phase volume fractions sums the unity.
A mass balance in the teeming ladle yields the following expression, which tracks the time evolution of the molten steel height:
\sqrt{{h}_{ms}}\left(t\right)=\sqrt{{h}_{ms0}}-\left(\frac{1}{2}{\left(\frac{{D}_{1}}{{D}_{2}}\right)}^{2}{C}_{D}\sqrt{2g}\right)t
where hms0 is the initial height of molten steel, CD is the nozzle discharge coefficient, g is gravity, and t is time.
On the other hand, the teeming ladle becomes empty when the molten steel height and the slag height become null. In this case, the draining time from Equation (1) is given by:
{t}_{d}=\frac{2\sqrt{{h}_{ms0}+{h}_{s}}}{{\left(\frac{{D}_{1}}{{D}_{2}}\right)}^{2}{C}_{D}\sqrt{2g}}
A cylindrical teeming ladle in which D2 = D3 (see Figure 1) is considered in the computer simulations. The mesh consists in 250,000 tetrahedral cells. The ladle model is solved using commercial CFD software. Table 1 shows the main parameters of the ladle.
Figure 2 and Figure 3 show the slag carry-over corresponding to the 0.05 m diameter nozzle, in centered and off-centered position, respectively, as function of the slag height. The molten metal height is kept constant at 0.75 m. These Figures show that for both nozzle positions, slag carry-over starts at 1230, 1270
Table 1. Main parameters of the teeming ladle.
and 1295 s of elapsed time for slag heights of 0.2, 0.15 and 0.10 m, respectively. Besides, for the centered position of the nozzle (Figure 2), the average carry-over for 0.15 and 0.20 slag heights is 4.5 kg/s, whereas for 0.1 m of slag height the average carry-over is 4.2 kg/s. Figure 3 shows that for the 0.05 m diameter off-centered nozzle the average slag carry-over for 0.1 and 0.5 m of slag height is around 4.2 kg/s.
On the other hand, Figure 4 and Figure 5 show the slag carry-over corresponding to the 0.1 m diameter nozzle, in centered and off-centered position, respectively, as function of the slag height. As in Figure 2 and Figure 3, the molten metal height is maintained constant at 0.75 m. Figure 4 shows that for the centered nozzle position and 0.1 m of nozzle diameter, slag carry-over starts at 307, 312 and 327 s of elapsed time for slag heights of 0.2, 0.15 and 0.10 m, respectively. For the off-centered position and 0.1 m nozzle diameter, Figure 5 shows that slag carry-over starts at around 302, 310 and 318 s of elapsed time for slag heights of 0.2, 0.15 and 0.10 m, respectively. That is, for 0.1 m of nozzle diameter, slag carry-over starts first for the off-centered nozzle position.
Related to the mass flow rate of slag from the ladle for the 0.1 m diameter centered nozzle, Figure 4 shows and average of 17.5 and 17.0 kg/s for slag heights
Figure 2. Time evolution of the slag carry-over for three different slag heights: 0.1 m (black), 0.15 m (red), 0.2 m (blue). Nozzle diameter, 0.05 m. Position of the nozzle: centered.
Figure 3. Time evolution of the slag carry-over for three different slag heights: 0.1 m (black), 0.15 m (red), 0.2 m (blue). Nozzle diameter, 0.05 m. Position of the nozzle: off-centered.
Figure 4. Time evolution of the slag carry-over for three different slag heights: 0.1 m (black), 0.15 m (red), 0.2 m (blue). Nozzle diameter, 0.1 m. Position of the nozzle: centered.
Figure 5. Time evolution of the slag carry-over for three different slag heights: 0.1 m (black), 0.15 m (red), 0.2 m (blue). Nozzle diameter, 0.1 m. Position of the nozzle: off-centered.
Table 2. Draining time for a 0.05 m diameter nozzle.
of 0.15 - 0.20 and 0.1 m, respectively. For the 0.1 m diameter off-centered nozzle, Figure 5 shows and average of 17 and 17.0 kg/s for slag heights of 0.15 - 0.20 and 0.1 m, respectively.
Finally, the draining times were determined from Equation (2) and from CFD computer simulations, considering a molten steel height of 0.75 m and a nozzle discharge coefficient of 1.0. These draining times are shown in Table 2 and Table 3. It can be observed that the draining times calculated from Equation (2) are slightly larger than those estimated through CFD simulations. This is due to the fact that in CFD simulations some slag is retained in the ladle, whereas Equation (2) considers that the molten steel and the slag are fully drained. Besides, in accordance to CFD simulations, draining time are slightly larger for the off-centered position than that of the centered position.
The slag carry-over from a teeming ladle was numerically studied. Two nozzle diameters, two nozzle positions and three slag heights were considered in the 3D transient isothermal CFD computer simulations. The following conclusions arise:
1) Slag carry-over in terms of mass flow rate is significantly increased as the nozzle diameter is increased.
2) Starting time of slag carry-over increases as the slag height decreases.
3) Mass flow rate of slag is slightly larger for the nozzle centered position than that corresponding to the off-centered position.
4) Draining time depends inversely on the nozzle diameter. As the nozzle diameter is increased, the draining time is decreased.
[1] Ametek Land (2017) https://www.landinst.com
[2] Mazumdar, S., Pradhan, N., Bhor, P.K. and Jagannathan, K.P. (1995) Entrainment during Tapping of a Model Converter Using Two Liquid Phases. ISIJ International, 35, 92-94.
[3] Koria, S.C. and Kanth, U. (1994) Model Studies of Slag Carry-Over during Drainage of Metallurgical Vessels. Steel Research, 65, 8-14. https://doi.org/10.1002/srin.199400919
[4] Morales, R.D., Davila-Maldonado, O., Calderon, I. and Morales-Higa, K. (2013) Physical and Mathematical Models of Vortex Flows during the Last Stages of Steel Draining Operations from a Ladle. ISIJ International, 53, 782-791.
[5] Bird, R.B., Stewart, W.E. and Lightfoot, E.N. (2002) Transport Phenomena. 2nd Edition, Wiley, New York.
[6] Thomas, B.G., Yuan, Q., Sivaramakrishnan, S., Shi, T., Vanka, S.P. and Assar, M.B. (2001) Comparison of Four Methods to Evaluate Fluid Velocities in a Continuous Slab Casting Mold. ISIJ International, 41, 1262-1271.
[7] Hirt, C.W. and Nichols, B.D. (1981) Volume of Fluid (VOF) Method for the Dynamics of Free Boundaries. Journal of Computational Physics, 39, 201-225. |
Interest Rate Swaps Guide - Delta Exchange - User Guide
BitMex Funding Rate Swap
DAI Savings Rate Swap
Interest Rate Swaps: Motivation & Use Cases
Interest Rate Swaps (IRS) are a class of derivative contracts in which two parties agree to exchange one stream of interest rate payments for another, over a set period of time. Typically, one of the interest rate streams varies with time and is known as as the ‘floating rate’. And, the other interest rate stream stays constant and is referred to as the ‘fixed rate’. Thus, a typical IRS entails swap of floating interest rate payments with fixed interest payments.
Floating rate: is typically an easily observable/ benchmark rate that changes periodically and is of interest to a lot of people.
From the traditional finance world, a good example of such a rate is LIBOR. LIBOR is short for London InterBank Offer Rate. This is the rate at which banks are willing to offer uncollateralised loans to each other. LIBOR is published everyday and is used as the floating leg on a multitude of interest rate swaps and other interest rates based derivatives. In the domain of cryptocurrencies, widely accepted benchmark interest rates like LIBOR are yet to emerge. However, interest rates on de-fi platforms (that typically change with every block on the ETH blockchain) and funding rates on perpetual contracts (that typically change every 8 hours).
Fixed rate: or the swap rate, is the rate which a party demands in exchange of assuming the uncertainty of paying floating rate over the duration of the IRS. It is logical to deduce that the fixed rate in an IRS will reflect the expected values of the floating rate in the future.
In traditional financial markets, the market-implied future values of benchmark rates like LIBOR can be imputed from a variety of interest rate derivatives. These ‘forward rates’ help traders in pricing LIBOR-based interest rate swaps. Unfortunately, market-implied forward rates are not available in crypto yet. Thus, traders will need to come up with their own estimates of future values of the floating rate of an IRS.
Notional Amount: can be thought of as the ‘Principal Amount’ for which fixed or floating interest payments are being exchanged in an IRS. Notional Amount is plugged into the simple interest formula to compute interest rate payments that two parties in an IRS have to make to each other, i.e.
Interest\ Rate\ Payment = Notional\ Amount * Rate * Time
Using IRS for hedging
You can get into an IRS trade to convert floating-rate exposure/ liability into a fixed rate one. By buying floating-for-fixed, you can eliminate the uncertainty of the floating-rate liability.
Using IRS to speculate
You can use IRS to speculate on the future values of the floating rate.
Buy floating-for-fixed: Pay fixed rate, Receive floating rate
If you believe that the floating rate is likely to go up, you can choose to receive the floating rate and pay the fixed rate. This trade will be profitable if the realised value of the floating rate over the is higher than the expected value baked into the fixed rate.
Sell floating-for-fixed: Pay floating rate, Receive fixed rate
If you believe that floating rate is likely to do gown, you can choose to pay the floating rate and receive the fixed rate. Since you have locked in a higher fixed rate, your trade will be profitable if your expectation about floating rate going lower comes true.
Understanding the Math Behind IRS
In an IRS,
Value\ of\ Floating\ Payments\ = Notional\ Amount* \sum(floating\ rate_i * T_i)
floating\ rate_i
is the value of the floating rate in the i-th time interval. Obviously, these refer to future values of the floating rate and thus need to be estimated by the traders.
Value\ of\ Fixed\ Payments = Notional\ Amount* fixed\ rate * \sum T_i
For a fairly priced IRS, we have must have:
Value\ of\ Fixed\ Payments\ = Value\ of\ Floating\ Payments
Using the above equation, we can find the fair value of the fixed rate as:
fixed\ rate = \sum (floating\ rate_i * T_i) / \sum T_i
In the above equations, we have ignored time value of money, i.e. the present value of futures payments is not getting impacted by their timing. This approximation helps keep the calculations simple and does not have material impact on IRS of short tenures.
We have assumed that fixed and floating payment exchanges are synchronized. That need not be the case. In fact, in the IRS contracts listed on Delta Exchange, fixed payment is done in a single shot at the inception of the swap.
Profit/ loss Calculations
To be able to understand the profit/ loss calculations for an IRS, it is important for you to appreciate the fact that the profit/ loss from an IRS trade is path dependent. What this means is that it is not just your entry and exit levels but also the path the floating rate takes during the trade that determines your profit/ loss. To put this in context, Profit/ loss for futures is path independent as is evident from the futures PnL equation:
PnL = P_{exit} - P_{entry}
The path dependency of profit/ loss is due to the fixed-floating payment exchanges that happen while you are in an IRS trade. In fact, thinking in terms of cashflows is a good starting point for computing profit/ loss from an IRS trade. Cashflows can occur: (a) at the trade inception, (b) while the trade is open and fixed/ floating payments are being exchanged periodically and (c) at the time of exit from the trade. You make profit in a trade when the cumulative net cashflow (i.e. total incoming cashflow - total outgoing cashflow) is positive.
Recall that an IRS is a derivative contract in which two parties agree to exchange fixed/ floating payments for a pre-determined duration of time. When a party is looking to exit from an IRS position before the contract’s tenure is complete, it does so by taking opposite position in the same IRS for the remainder of tenure of the IRS. So, if you are receiving fixed/ paying floating, you do a trade in which you pay fixed/ receive floating to close your position.
As the above diagrams illustrate, on closing an IRS position, you end up paying or receiving the difference of the fixed rates in the two opposing swaps for the remaining duration of the IRS contract.
Therefore, the profit/ loss from an IRS contract trade can be written as:
PnL = Cashflow\ at\ inception + Net\ fixed/ floating\ payments + Cashflow\ at\ Exit
Interest Rate Swaps Listed on Delta Exchange
BitMex Funding Rate Swap This contracts enables you to swap the funding rate of BitMex’s XBTUSD perpetual contract that changes every 8 hours with a rate that stays fixed through the duration of the swap. You can use the BitMex funding rate swap either to hedge the risk of variability of funding you are paying on an open position in BitMex’s XBTUSD or to speculate on the movement in funding rate. Complete details of this contract are available here.
Dai Savings Rate Swap This contract enables you to swap variable Dai Savings Rate with a rate that stays fixed through the duration of the swap. You can use the Dai Savings Rate swap to both hedge the risk of change in Dai Savings Rate as well as to speculate on Dai and the broader DeFi ecosystem. Complete details of this contract are available here.
BitMex Funding Rate Swap
DAI Savings Rate Swap |
Rotation around a fixed axis - 3D CAD Models & 2D Drawings
Rotation around a fixed axis (8774 views - Mechanical Engineering)
{\displaystyle {\vec {F}}=m{\vec {a}}}
{\displaystyle F_{\mathrm {net} }=Ma_{\mathrm {cm} }\;\!}
{\displaystyle r}
{\displaystyle s}
{\displaystyle \theta }
{\displaystyle \theta ={\frac {s}{r}}}
{\displaystyle {\begin{aligned}1{\text{ revolution }}&=360^{\circ }=2\pi {\text{ radians, and}}\\1{\text{ rad}}&={\frac {180^{\circ }}{\pi }}\approx 57.27^{\circ }.\end{aligned}}}
{\displaystyle \Delta \theta =\theta _{2}-\theta _{1},\!}
{\displaystyle \Delta \theta }
{\displaystyle \theta _{1}}
{\displaystyle \theta _{2}}
{\displaystyle \omega }
{\displaystyle {\overline {\omega }}={\frac {\Delta \theta }{\Delta t}}={\frac {\theta _{2}-\theta _{1}}{t_{2}-t_{1}}}.}
{\displaystyle \omega (t)={\frac {d\theta }{dt}}.}
{\displaystyle v={\frac {ds}{dt}}}
{\displaystyle \omega ={\frac {d\theta }{dt}}={\frac {v}{r}},}
{\displaystyle v}
{\displaystyle \omega ={2\pi f}\!}
{\displaystyle {\overline {\alpha }}}
{\displaystyle {\overline {\alpha }}={\frac {\Delta \omega }{\Delta t}}={\frac {\omega _{2}-\omega _{1}}{t_{2}-t_{1}}}.}
{\displaystyle \alpha (t)={\frac {d\omega }{dt}}={\frac {d^{2}\theta }{dt^{2}}}.}
{\displaystyle a=r\alpha ,\!}
{\displaystyle a_{\mathrm {R} }={\frac {v^{2}}{r}}=\omega ^{2}r\!}
{\displaystyle T=I\alpha }
{\displaystyle \theta }
{\displaystyle \omega _{i}}
{\displaystyle \omega _{f}}
{\displaystyle \alpha }
{\displaystyle t}
{\displaystyle {\begin{aligned}\omega _{f}&=\omega _{i}+\alpha t\\\theta &=\omega _{i}t+{\frac {1}{2}}\alpha t^{2}\\\omega _{f}^{2}&=\omega _{i}^{2}+2\alpha \theta \\\theta &={\frac {1}{2}}\left(\omega _{f}+\omega _{i}\right)t\end{aligned}}}
{\displaystyle m}
{\displaystyle r}
{\displaystyle I=mr^{2}.}
{\displaystyle {\boldsymbol {\tau }}}
{\displaystyle {\boldsymbol {\tau }}=\mathbf {r} \times \mathbf {F} ,}
{\displaystyle {\boldsymbol {\tau }}=I{\boldsymbol {\alpha }},}
{\displaystyle W=\tau \theta .\!}
{\displaystyle P=\tau \omega .\!}
{\displaystyle \mathbf {L} }
{\displaystyle \mathbf {L} =\sum \mathbf {r} \times \mathbf {p} }
{\displaystyle \mathbf {L} =I{\boldsymbol {\omega }},}
{\displaystyle {\boldsymbol {\tau }}={\frac {d\mathbf {L} }{dt}},}
{\displaystyle K_{\text{rot}}={\frac {1}{2}}I\omega ^{2},}
{\displaystyle \Delta \theta }
Control panel (engineering)Crank (mechanism)CrankpinCrankshaftCrankshaft position sensorDifference engineEngineEngine tuningFour-stroke engineInternal combustion engineLinear motionLinear-motion bearingMechanical engineeringMid-engine designMotion controlPerpetual motionRadial engineReciprocating engineReciprocating motionShaft (mechanical engineering)Steam engineStirling engineTwo-stroke engineV12 engineWheel and axleAxle
This article uses material from the Wikipedia article "Rotation around a fixed axis", which is released under the Creative Commons Attribution-Share-Alike License 3.0. There is a list of all authors in Wikipedia |
According to the school secretary, the size of the graduating class of Gauss High has steadily increased since the school has opened. She claims that the number of graduates each year forms the sequence
t(n) = 42 + 12(n - 1)
, where n is the number of years the high school has had students graduate. At the end of the year, the administration mails out invitations to all previous graduates to attend the upcoming graduation ceremony.
In this situation, what does a sequence represent? What does a series represent?
Remember that a sequence is a set of discrete numbers while a series is the sum of those numbers.
10
th graduation ceremony, how many total graduates should there be? Explain how you found your answer.
S_{10}=\frac{10}{2}\left(t\left(1\right) + t\left(10\right)\right)
The total number of Gauss High graduates after
n
years can be represented by the graph below. If the alumni relations committee wants to invite all graduates to a party to celebrate the
n
th anniversary of the school, how may invitations will they need to send? |
Odds ratio - Citizendium
The odds ratio is a technical term often used in statistics, and especially in medical statistics and epidemiology. The odds ratio is the ratio of the relative incidence (odds) of a target disorder in the experimental group relative to the relative incidence in a control group. It reflects how the risk of having a particular disorder is influenced by the treatment. An odds ratio of 1 means that there is no benefit of treatment compared to the control group.[1]
The odds ratio is a difficult concept for non mathematicians, and recommendations for how to teach its use are available.[2]
For a mathematician, the relative incidence of a disorder would be called the 'odds on having the disorder'. An odds ratio is the ratio between the odds on some event under two circumstances.
Effect of t'ment on disorder
Experimental treatment A B
Control treatment C D
If we have A individuals with the disorder and B without in the experimental group, and C individuals with the disorder and D without in the control group, the empirical odds ratio is
{\displaystyle Odds{\text{ }}ratio={\frac {A*D}{B*C}}}
This example is from the Titanic (example from Power[3]):
Fate on the Titanic
Females 308 154
142 survived, 709 died
Odds of survival = 142/709 = 0.20
Probability (risk or chance) of survival = 142/(142+709) = 17%
Odds ratio (OR) for survival = 0.20/2.00 = 0.10
Relative risk (RR) for survival = 17%/67% = 0.25
The odds on a male passenger surviving the Titanic disaster was 1 to 5 against surviving. The odds on a female passenger surviving the Titanic disaster was 2 to 1 in favour of surviving. The odds ratio for survival between men and women is (1/5) / (2/1) = 1/10
Rather than classifying the passengers on the Titanic as male or female, and comparing their odds on survival, one could classify passengers as survivors or not survivors, and compare the odds on being male or female. The odds ratio for sex (male versus female) between survivors and non-survivors is also 0.10. This follows from the algebraic identity
{\displaystyle (AD/BC)=(A/B)/(C/D)=(A/C)/(B/D)}
The identity between the odds ratios is much used in epidemiology, especially in retrospective case-control studies. Consider for example the relation between lung cancer and exposure to asbestos. Instead of sampling from the two populations of people who are exposed to asbestos and people who have not been exposed to asbestos, and studying the odds ratio for lung cancer between the two populations defined by exposure status, one could just as well sample from the two populations of people who got lung cancer and people who did not get lung cancer, and study the odds ratio for asbestos-exposure between the two populations defined by health status. This is extremely valuable when studying illnesses which are rather rare, even in the population exposed to the risk factor of interest.
The odds ratio is generally used to measure the association between a risk factor and disease. However, using the odds ratio to measure the ability of a risk factor to diagnose disease is problematic.[4] The odds ratio should be at least 16 to have reasonable diagnostic ability.[5]
The odds ratio is similar to the relative risk ratio. The two ratios will be numerically similar if the rates in the two groups being compared are both similar and both less than 20% to 30%.[6][7][8]
The odds ratio may be the most stable ratio across different prevalences.[9]
The odds ratio may be used to derive the number needed to treat:[9][10]
For odds ratios less than 1:[10]
{\displaystyle NNT={\frac {1-CER*(1-OR)}{CER*(1-OR)*(1-CER)}}{\mbox{, where CER is control event rate and OR is odds ratio}}}
For odds ratios greater than 1:[10]
{\displaystyle NNT={\frac {1+CER*(OR-1)}{CER*(OR-1)*(1-CER)}}{\mbox{, where CER is control event rate and OR is odds ratio}}}
↑ Anonymous. Odds and odds ratio. Bandolier.
↑ Prasad K et al. (2008). "Tips for teachers of evidence-based medicine: understanding odds ratios and their relationship to risk ratios". J Gen Intern Med 23: 635–40. DOI:10.1007/s11606-007-0453-4. PMID 18181004. Research Blogging.
↑ Power M (2008). "Resource reviews". Evidence-based Medicine 13: 92. PMID 18515638. [e]
↑ Boyko EJ, Alderman BW (1990). "The use of risk factors in medical diagnosis: opportunities and cautions". J Clin Epidemiol 43: 851–8. PMID 2213074. [e]
↑ Pepe MS et al. (2004). "Limitations of the odds ratio in gauging the performance of a diagnostic, prognostic, or screening marker". Am J Epidemiol 159: 882–90. PMID 15105181.
↑ Sinclair JC, Bracken MB (1994). "Clinically useful measures of effect in binary analyses of randomized trials". J Clin Epidemiol 47: 881–9. PMID 7730891.
↑ Altman DG, Deeks JJ, Sackett DL (November 1998). "Odds ratios should be avoided when events are common". BMJ 317 (7168): 1318. PMID 9804732. PMC 1114216. [e]
↑ Page J, Attia J (2003). "Using Bayes' nomogram to help interpret odds ratios". ACP J Club 139: A11–2. PMID 12954046. Helpful chart
↑ 9.0 9.1 Furukawa TA et al. (2002). "Can we individualize the 'number needed to treat'? An empirical study of summary effect measures in meta-analyses". Int J Epidemiol 31: 72–6. PMID 11914297. [e] Cite error: Invalid <ref> tag; name "pmid11914297" defined multiple times with different content
↑ 10.0 10.1 10.2 McQuay HJ, Moore RA (1997). "Using numerical results from systematic reviews in clinical practice". Ann Intern Med 126: 712–20. PMID 9139558.
Retrieved from "https://citizendium.org/wiki/index.php?title=Odds_ratio&oldid=339040" |
Lagrangian point - Marspedia
2 Positions of the Lagrangian points
2.1 Calculating the position of
{\displaystyle L_{1}}
{\displaystyle L_{2}}
3 Current missions using the Lagrangian points of the Earth-Sun system
4 Objects observed in the Mars-Sun
{\displaystyle L_{4}}
{\displaystyle L_{5}}
5 Uses of the Mars
{\displaystyle L_{1}}
{\displaystyle L_{2}}
A Lagrangian point is a point of interplanetary gravitational stability in a two body orbital configuration. Devised by the Italian/French mathematician and astronomer, Joseph Louis Lagrange (1736-1813), Lagrangian points are currently used to describe the influence of the planets over local space, provides a location for Trojan asteroids and may possibly be used by future space missions as a "stepping stone" to Mars and the Moon.
Positions of the Lagrangian points
Example: The position of the L1, L2, L3 and L4,5 points in the Mars-Sun system.
The simplest Lagrangian point to understand is the "first Lagrangian" (or
{\displaystyle L_{1}}
) point between the Earth and the Sun. The point at which the gravitational pull of the Sun and the gravitational pull of the Earth cancels out (
{\displaystyle g_{Earth}+g_{Sun}=0}
) creates an island of gravitational stability where space observatories, or indeed space stations, can be positioned.
{\displaystyle L_{2}}
is located on the opposite side of the smallest orbital body (in this case, the Earth) to
{\displaystyle L_{1}}
. This very stable region is also useful to space observatories observing the cosmos. The Earth in this case will be constantly eclipsing the Sun, allowing sensitive optics to operate free of noise emitted from the Sun.
{\displaystyle L_{3}}
is a less-stable Lagrangian point on the far side of the Sun. In this case, the Earth's gravitational force is negligible, allowing other planet's orbits to interfere with the gravitational stability of the region.
{\displaystyle L_{4}}
{\displaystyle L_{5}}
are points leading and trailing the orbiting body at an angle of approximately 60° from the Earth-Sun line. These points are also known as "Trojan points" where asteroids (known as Trojan asteroids) become captured by the relative gravitational stability and orbit with the orbital body.
Calculating the position of
{\displaystyle L_{1}}
{\displaystyle L_{2}}
If the mass of the larger body is massively greater than that of the orbiting body (as is the case in the Earth, or Mars, around the larger Sun), the following equation[1] can be applied:
{\displaystyle r\approx R{\sqrt[{3}]{\frac {M_{2}}{3M_{1}}}}}
{\displaystyle r}
{\displaystyle L_{1}}
{\displaystyle L_{2}}
from the orbiting body,
{\displaystyle R}
is the distance between the bodies and
{\displaystyle M_{2}}
{\displaystyle M_{1}}
are the masses of the smaller and larger bodies respectively.
Current missions using the Lagrangian points of the Earth-Sun system
{\displaystyle L_{1}}
The Solar and Heliospheric Observatory (SoHO) - A multi-instrument observatory contantly observing the Sun.
Advanced Composition Explorer (ACE) - An in-situ observatory measuring the properties of the solar wind and solar radiation.
{\displaystyle L_{2}}
Wilkinson Microwave Anisotropy Probe (WMAP) - Surveying the sky, observing the microwave radiation left over by the Big Bang.
Objects observed in the Mars-Sun
{\displaystyle L_{4}}
{\displaystyle L_{5}}
Asteroids in the
{\displaystyle L_{4}}
{\displaystyle L_{5}}
Mars-Sun Lagrangian points are often called Mars Trojan asteroids. A handful of asteroids are in stable solar orbits, leading (
{\displaystyle L_{4}}
) and following (
{\displaystyle L_{5}}
) the path of Mars including an asteroid named "1999 UJ7" (at
{\displaystyle L_{4}}
) and "5261 Eureka" (at
{\displaystyle L_{5}}
Uses of the Mars
{\displaystyle L_{1}}
{\displaystyle L_{2}}
An "early warning system" to notify settlers about the onset of solar storms.
Communications satellites inserted in stable orbits around the Lagrangian points - long-term nodes of communication between Earth and Mars. Could be extended to include the
{\displaystyle L_{4}}
{\displaystyle L_{5}}
points to allow communication even when Mars is in conjunction.
A manned space station acting as a "staging post" for missions to asteroids or refuge before making the final plunge into Mars' gravitational well.
5261 Eureka on Wikipedia
↑ Lagrangian Point on Wikipedia
Retrieved from "https://marspedia.org/index.php?title=Lagrangian_point&oldid=127193" |
Calculate implied Black volatility using SABR model - MATLAB blackvolbysabr - MathWorks India
blackvolbysabr
Compute the Implied Black Volatility Using the SABR Model
Compute the Shifted Black Volatility Using the Shifted SABR Model
Calculate implied Black volatility using SABR model
outVol = blackvolbysabr(Alpha,Beta,Rho,Nu,Settle,ExerciseDate,ForwardValue,Strike)
outVol = blackvolbysabr(___,Name,Value)
outVol = blackvolbysabr(Alpha,Beta,Rho,Nu,Settle,ExerciseDate,ForwardValue,Strike) calculates the implied Black volatility using the SABR stochastic volatility model.
outVol = blackvolbysabr(___,Name,Value) adds optional name-value pair arguments.
ForwardRate = 0.0357;
ExerciseDate = datenum('15-Sep-2015');
Compute the Black volatility using the SABR model.
ComputedVols = blackvolbysabr(Alpha, Beta, Rho, Nu, Settle, ...
ExerciseDate, ForwardRate, Strike)
Define the model parameters and option data with a negative strike.
Strike = -0.001; % -0.1% strike.
Settle = datenum('1-Mar-2016');
ExerciseDate = datenum('1-Mar-2017');
Compute the Shifted Black volatility using the Shifted SABR model.
ExerciseDate, ForwardRate, Strike, 'Shift', Shift)
Current SABR volatility, specified as a scalar.
SABR CEV exponent, specified as a scalar.
Correlation between forward value and volatility, specified as a scalar.
Volatility of volatility, specified as a scalar.
Current forward value of the underlying asset, specified as a scalar or vector of size NumVols-by-1.
Option strike price values, specified as a scalar value or a vector of size NumVols-by-1.
Example: outVol = blackvolbysabr(Alpha,Beta,Rho,Nu,Settle,ExerciseDate,ForwardValue,Strike,'Basis',2,'Model','Obloj2008')
Day-count basis of the instrument, specified as the comma-separated pair consisting of 'Basis' and a positive integer of the set [1...13].
Model — Version of SABR model
'Hagan2002' (default) | value 'Obloj2008'
Version of SABR model, specified as the comma-separated pair consisting of 'Model' and one of the following values:
'Hagan2002' — Original version by Hagan et al. (2002)
'Obloj2008' — Version by Obloj (2008)
Shift in decimals for the shifted SABR model (to be used with the Shifted Black model), specified as the comma-separated pair consisting of 'Shift' and a scalar positive decimal value. Set this parameter to a positive shift in decimals to add a positive shift to ForwardValue and Strike, which effectively sets a negative lower bound for ForwardValue and Strike. For example, a Shift value of 0.01 is equal to a 1% shift.
outVol — Implied Black volatility computed by SABR model
Implied Black volatility computed by SABR model, returned as a scalar or vector of size NumVols-by-1.
The SABR stochastic volatility model treats the underlying forward
\stackrel{^}{F}
\stackrel{^}{\alpha }
as separate random processes, which are related with correlation
\rho
\begin{array}{l}d\stackrel{^}{F}=\stackrel{^}{\alpha }{\stackrel{^}{F}}^{\beta }d{W}_{1}\\ d\stackrel{^}{\alpha }=v\stackrel{^}{\alpha }d{W}_{2}\\ d{W}_{1}d{W}_{2}=\rho dt\\ \stackrel{^}{F}\left(0\right)=F\\ \stackrel{^}{\alpha }\left(0\right)=\alpha \end{array}
\stackrel{^}{F}
is the underlying forward (a variable).
F
is the current underlying forward (a constant).
\stackrel{^}{\alpha }
is the SABR volatility (a variable).
\alpha
is the current SABR volatility (a constant).
\beta
is the SABR constant elasticity of variance (CEV) exponent.
\upsilon
is the volatility of volatility.
d{W}_{1}
is Brownian motion.
d{W}_{2}
\rho
is the correlation between forward value and volatility.
In contrast, Black's lognormal model assumes a constant volatility,
{\sigma }_{B}
d\stackrel{^}{F}={\sigma }_{B}\stackrel{^}{F}dW
Hagan et al. (2002) derived the following closed-form approximation of implied Black lognormal volatility (
{\sigma }_{B}
) for the SABR model
\begin{array}{l}{\sigma }_{B}\left(F,K\right)=\frac{\alpha \left\{1+\left[\frac{{\left(1-\beta \right)}^{2}}{24}\frac{{\alpha }^{2}}{{\left(FK\right)}^{1-\beta }}+\frac{1}{4}\frac{\rho \beta \upsilon \alpha }{{\left(FK\right)}^{\left(1-\beta \right)/2}}+\frac{2-3{\rho }^{2}}{24}{\upsilon }^{2}\right]T+...\right\}}{{\left(FK\right)}^{\left(1-\beta \right)/2}\left\{1+\frac{{\left(1-\beta \right)}^{2}}{24}{\mathrm{log}}^{2}\left(F/K\right)+\frac{{\left(1-\beta \right)}^{4}}{1920}{\mathrm{log}}^{4}\left(F/K\right)+...\right\}}\left(\frac{z}{x\left(z\right)}\right)\\ z=\frac{\upsilon }{\alpha }{\left(}^{F}\mathrm{log}\left(F/K\right)\\ x\left(z\right)=\mathrm{log}\left\{\frac{\sqrt{1-2\rho z+{z}^{2}}+z-\rho }{1-\rho }\right\}\end{array}
F
is the current forward value of the underlying.
\alpha
is the current SABR volatility.
K
is the strike value.
T
is the time to option maturity.
Obloj (2008) advocated the following closed-form approximation of implied Black lognormal volatility for the SABR model (for
\beta <1
\begin{array}{l}{\sigma }_{B}\left(F,K\right)=\frac{\upsilon \mathrm{log}\left(F/K\right)}{x\left(z\right)}\left\{1+\left[\frac{{\left(1-\beta \right)}^{2}}{24}\frac{{\alpha }^{2}}{{\left(FK\right)}^{1-\beta }}+\frac{1}{4}\frac{\rho \beta \upsilon \alpha }{{\left(FK\right)}^{\left(1-\beta \right)/2}}+\frac{2-3{\rho }^{2}}{24}{\upsilon }^{2}\right]T+...\right\}\\ z=\frac{\upsilon }{\alpha }\frac{{F}^{\left(1-\beta \right)}-{K}^{\left(1-\beta \right)}}{1-\beta }\\ x\left(z\right)=\mathrm{log}\left\{\frac{\sqrt{1-2\rho z+{z}^{2}}+z-\rho }{1-\rho }\right\}\end{array}
These expressions can be simplified in special situations, such as the at-the-money (
F=K
) and stochastic lognormal (
\beta
= 1) cases [1,2].
[1] Hagan, P. S., D. Kumar, A.S. Lesniewski, and D.E. Woodward. “Managing Smile Risk.” Wilmott Magazine, September, pp. 84–108, 2002.
[2] Obloj, J. “Fine-tune your smile: Correction to Hagan et. al.” Wilmott Magazine, 2008. |
Hessian pendulum - zxc.wiki
Hessian pendulum
Simulated movement of a Hessian pendulum with center of gravity axis and angular momentum plane (black), main axes (blue), angular momentum (red) and angular velocity (green)
The Hess pendulum according to Wilhelm Hess is an asymmetrical top in the gyro theory , in which the center of gravity moves like a spherical pendulum , only the gravitational acceleration has to be replaced by a gyro-specific gravitational acceleration, see animation. The speed of a point on the main axis with the medium main moment of inertia always includes the same angle with the center of gravity axis from the base to the center of gravity, which is why the Hess pendulum is also called the loxodromic pendulum . The Hess pendulum is a direct generalization of the spherical pendulum.
The angular momentum is always perpendicular to the center of gravity. In addition, as with the Staude rotations, the center of gravity, the angular momentum and the angular velocity lie in one plane. The main moments of inertia A, B, C around the first, second and third main axis and the center of gravity coordinates s 1,2,3 must meet the conditions
{\ displaystyle s_ {2} = 0, \ quad {\ frac {s_ {1} ^ {2}} {s_ {3} ^ {2}}} = {\ frac {{\ frac {1} {B} } - {\ frac {1} {A}}} {{\ frac {1} {C}} - {\ frac {1} {B}}}} = {\ frac {C (AB)} {A ( BC)}}}
adhere to. Here is wlog A> B> C is assumed.
Hess, a professor at the Lyceum in Bamberg, discovered this analytically describable movement in 1890. Russian mathematicians later deepened his study. The Hessian pendulum could also be transferred to the game top and Mlodzjejowsky found another generalization of the spherical pendulum.
1 Conditions on the top and the initial conditions
1.1 The solid plane containing the angular momentum
1.2 The angular velocity
1.3 The intersection of the plane and the MacCullagh ellipsoid
1.4 The mass distribution in the top
2 Joukowsky's geometrical theorems
2.1 The speed of a point on the 2-axis
2.2 The kinetic energy of the top
2.3 The angular momentum of the body
Conditions at the top and the initial conditions
So that the angular momentum always remains in a fixed plane e during the movement, it must be perpendicular to the center of gravity axis. The angular momentum only remains in plane e if the support point, center of gravity, angular momentum and angular velocity are coplanar. Then the plane e intersects the MacCullagh ellipsoid in a circle, which limits the possible mass distribution in the top.
The solid plane containing the angular momentum
The external torque , formed from the cross product × the center of gravity axis with the weight force, is equal to the speed of the end point of the angular momentum according to the principle of twist . This speed is therefore always perpendicular to the center of gravity axis and must be contained in e. So the plane e is perpendicular to the centroid axis and defined by .
{\ displaystyle {\ vec {s}} \ times {\ vec {G}}}
{\ displaystyle {\ vec {s}}}
{\ displaystyle {\ vec {G}}}
{\ displaystyle {\ vec {L}}}
{\ displaystyle {\ vec {L}} \ cdot {\ vec {s}} = 0}
From the above plane equation follows with the product rule
{\ displaystyle {\ vec {L}} \ cdot {\ vec {s}} = 0 \ quad \ rightarrow \ quad {\ frac {\ mathrm {d}} {\ mathrm {d} t}} ({\ vec {L}} \ cdot {\ vec {s}}) = {\ dot {\ vec {L}}} \ cdot {\ vec {s}} + {\ vec {L}} \ cdot {\ dot {\ vec {s}}} = 0}
in which, like the dot, symbolize the derivative of time . The first summand always disappears. In the second summand the velocity of the center of gravity forms with the angular velocity of the gyro: . So that the second summand is zero at all times, the center of gravity, support point, angular momentum and angular velocity must be coplanar:
{\ displaystyle {\ tfrac {\ mathrm {d}} {\ mathrm {d} t}}}
{\ displaystyle {\ dot {\ vec {L}}} \ cdot {\ vec {s}} = ({\ vec {s}} \ times {\ vec {G}}) \ cdot {\ vec {s} } = 0}
{\ displaystyle {\ vec {\ omega}}}
{\ displaystyle {\ dot {\ vec {s}}} = {\ vec {\ omega}} \ times {\ vec {s}}}
{\ displaystyle {\ vec {L}} \ cdot ({\ vec {\ omega}} \ times {\ vec {s}}) = 0}
The intersection of the plane and the MacCullagh ellipsoid
MacCullagh ellipsoid (yellow) with swirl ball (green), main axes (blue), center of gravity axis (light blue), base point O and center of gravity S.
The body-fixed plane e intersects the MacCullagh ellipsoid in a circle. Because the plane e perpendicular to the center of gravity through the support point intersects the MacCullagh ellipsoid in any case in a conic section (red in the picture). The angular momentum lies in the plane e and touches the ellipsoid at point P and let t be the tangent to the intersection curve in P. The tangent t is contained in e. With a second tangent t 'perpendicular to t, the tangential plane e' is spanned on the ellipsoid in P. This plane is perpendicular to the angular velocity. Because t is the line of intersection of the planes e and e 'which are perpendicular to the axis of the center of gravity or the angular velocity, t is also perpendicular to the plane that is generated by the axis of the center of gravity and the angular velocity. This plane also contains the angular momentum, which is why t is also perpendicular to it. The conic section therefore turns out to be a circle, because its tangents are always perpendicular to the radius vector.
However, the rotational energy is not necessarily constant, which is why the MacCullagh ellipsoid then pulsates in its expansion. The angular momentum does not necessarily follow a circular path in the body-fixed system.
The mass distribution in the top
The angular momentum lies at the momentary rotational energy E rot on the MacCullagh ellipsoid, that in angular momentum space with angular momentum components L 1,2,3 along the main axes by the equation
{\ displaystyle {\ frac {L_ {1} ^ {2}} {A}} + {\ frac {L_ {2} ^ {2}} {B}} + {\ frac {L_ {3} ^ {2 }} {C}} = 2E _ {\ text {red}}}
is defined. This ellipsoid is intersected with a sphere in such a way that the cutting figure is flat . The sphere has the equation
{\ displaystyle {\ frac {1} {B}} (L_ {1} ^ {2} + L_ {2} ^ {2} + L_ {3} ^ {2}) = 2E _ {\ text {red}} }
Subtraction gives:
{\ displaystyle {\ frac {1} {B}} (L_ {1} ^ {2} + L_ {2} ^ {2} + L_ {3} ^ {2}) - {\ frac {L_ {1} ^ {2}} {A}} - {\ frac {L_ {2} ^ {2}} {B}} - {\ frac {L_ {3} ^ {2}} {C}} = \ left ({ \ frac {1} {B}} - {\ frac {1} {A}} \ right) L_ {1} ^ {2} + \ left ({\ frac {1} {B}} - {\ frac { 1} {C}} \ right) L_ {3} ^ {2} = 0}
In the 1-3 plane, this identity defines two straight lines through the origin which create two planes with the 2-axis. So that the centroid axis is perpendicular to one of these planes, must
{\ displaystyle s_ {2} = 0, \ quad \ left ({\ frac {1} {B}} - {\ frac {1} {A}} \ right) s_ {3} ^ {2} + \ left ({\ frac {1} {B}} - {\ frac {1} {C}} \ right) s_ {1} ^ {2} = 0}
The compiled conditions are two for the mass distribution (at s 2 and s 1 / s 3 ) and one for the initial conditions ( ). This degree of specialization is identical to that of the Euler gyro , the Lagrange gyro and the Kowalewskaja gyro , which each also place three conditions on the top, but only on its mass distribution.
{\ displaystyle {\ vec {L}} \ cdot {\ vec {s}} = 0}
It is possible to complete the integration of the Euler-Poisson equations in the Hess case. Joukowsky's geometrical theorems show that the center of gravity moves like a pendulum , only the acceleration due to gravity has to be replaced by a gyro- specific acceleration due to gravity.
Joukowsky's geometrical theorems
N. Joukowsky formulated several sentences that illustrate the movement of Hess' pendulum. The sentences show that
the angular momentum lies in a plane fixed to the body, which was anticipated in the previous section,
the speed of a point on the 2-axis encloses a constant angle with the circular sections,
the kinetic energy of the top is equal to that of a mass point located in the center of gravity of the top, and that too
the angular momentum of the top is equal to the angular momentum of this mass point.
The speed of a point on the 2-axis
Joukowsky's second theorem states that the speed of a point on the 2-axis encloses a constant angle with the circles, which represent successive positions of the circular section.
Because from the condition of the mass distribution in the form and
{\ displaystyle {\ vec {L}} \ cdot {\ vec {s}} = 0}
{\ displaystyle \ left ({\ frac {1} {B}} - {\ frac {1} {A}} \ right) L_ {1} ^ {2} + \ left ({\ frac {1} {B }} - {\ frac {1} {C}} \ right) L_ {3} ^ {2} = 0}
it follows that the ratio of the angular momentum L 3 to L 1 is constant. The ratio ω 3 to ω 1 of the angular velocities proportional to them is therefore also constant. The tangent vector to the circular section on the 2-axis has a constant amount because it is fixed to the body. The vector is because of
{\ displaystyle {\ vec {t}} = {\ vec {s}} \ times {\ hat {e}} _ {2}}
{\ displaystyle {\ dot {\ hat {e}}} _ {2} = {\ vec {\ omega}} \ times {\ hat {e}} _ {2}}
{\ displaystyle | {\ dot {\ hat {e}}} _ {2} | = | \ omega _ {1} {\ hat {e}} _ {3} - \ omega _ {3} {\ has { e}} _ {1} | = | \ omega _ {1} | {\ sqrt {1 + {\ frac {\ omega _ {3} ^ {2}} {\ omega _ {1} ^ {2}} }}}}
proportional to the angular velocity ω 1 . The scalar product
{\ displaystyle {\ dot {\ hat {e}}} _ {2} \ cdot {\ vec {t}} = ({\ vec {\ omega}} \ times {\ hat {e}} _ {2} ) \ cdot ({\ vec {s}} \ times {\ hat {e}} _ {2}) = \ left (s_ {1} + s_ {3} {\ frac {\ omega _ {3}} { \ omega _ {1}}} \ right) \ omega _ {1}}
is also proportional to ω 1 . So the cosine of the angle between and is constant because it is the ratio of the scalar product to the amounts of the vectors involved. Consequently, the angle between the path speed and the tangent or the complementary angle to the center of gravity is always the same.
{\ displaystyle {\ dot {\ hat {e}}} _ {2}}
{\ displaystyle {\ vec {t}}}
The kinetic energy of the top
Joukowsky's third theorem says that the kinetic energy of the top is equal to that of a mass point located at the center of gravity of the top.
The kinetic energy of the top is equal to its rotational energy
{\ displaystyle E _ {\ text {kin}} = E _ {\ text {red}} = {\ frac {1} {2}} {\ vec {L}} \ cdot {\ vec {\ omega}}}
In the case of a mass point with mass M and orbital velocity rotating at the same angular velocity , the kinetic energy is
{\ displaystyle {\ vec {v}} = {\ vec {\ omega}} \ times {\ vec {s}}}
{\ displaystyle E _ {\ text {kin}} = {\ frac {M} {2}} | {\ vec {v}} | ^ {2} = {\ frac {M} {2}} | {\ vec {\ omega}} \ times {\ vec {s}} | ^ {2}}
Combination of the two equations leads to the position of the center of gravity and the orthogonality to the mass under the restrictions specified at the beginning
{\ displaystyle {\ vec {L}} \ cdot {\ vec {s}} = 0}
{\ displaystyle M = {\ frac {2E _ {\ text {kin}}} {| {\ vec {v}} | ^ {2}}} = {\ frac {{\ vec {L}} \ cdot {\ vec {\ omega}}} {| {\ vec {\ omega}} \ times {\ vec {s}} | ^ {2}}} = {\ frac {(AB) C} {(AC) s_ {1 } ^ {2}}} = {\ frac {B} {| {\ vec {s}} | ^ {2}}}}
The angular momentum of the body
Joukowsky's fourth sentence says that the angular momentum of the body is equal to the angular momentum of the mass point from the third movement.
At the roundabout, the angular momentum results from the product of inertia tensor Θ with the angular velocity: . On the other hand, the angular momentum is at the mass point
{\ displaystyle {\ vec {L}}: = \ mathbf {\ Theta} \ cdot {\ vec {\ omega}}}
{\ displaystyle {\ vec {L}} _ {M}: = {\ vec {s}} \ times M {\ vec {v}} = {\ vec {s}} \ times M ({\ vec {\ omega}} \ times {\ vec {s}})}
which with the mass of set 3 and the above-mentioned limitations for the location of the center of gravity and the orthogonality is identical to the angular momentum of the gyroscope: .
{\ displaystyle {\ vec {L}} \ cdot {\ vec {s}} = 0}
{\ displaystyle {\ vec {L}} = {\ vec {L}} _ {M}}
In the Hess pendulum, the rotational energy and the angular momentum are equal to that of a mass point with mass M in the center of gravity. This mass is not necessarily equal to the mass m of the top. A weight acts on this
{\ displaystyle mg = M \ left ({\ frac {m} {M}} g \ right) = M \ left ({\ frac {ms ^ {2}} {B}} g \ right) =: Mg ' }
according to the gravitational acceleration g . The center of gravity of the gyro moves like a spherical pendulum with the mass M under the modified gravitational acceleration g '. In particular, the Hess pendulum adapts to the force-free Euler gyro when the center of gravity moves to the base and the modified gravitational acceleration thus approaches zero.
↑ Magnus (1971), p. 141 ff, Klein and Sommerfeld (2010), p. 197 ff.
↑ a b c Magnus (1971), p. 142 f.
↑ Klein and Sommerfeld (2010), p. 381.
↑ Ulf Hashagen: Walther von Dyck: (1856-1934). Mathematics, technology and science organization at the TH Munich . Franz Steiner Verlag, Stuttgart 2003, ISBN 3-515-08359-6 , pp. 76 f . ( limited preview in Google Book search).
^ Wilhelm Hess (1890), see literature.
↑ see Klein and Sommerfeld (2010), p. 378. For the geometric interpretation, N. Joukowsky (1894) is quoted there (see literature) and for the analytical deepening
P. A. Nekrassoff : Recherches analytiques sur un cas de rotation d'un solid pesant author d'un point fixe . In: Mathematical Annals . tape 47 , 1896 (contains further references). ( eudml.org digitized version )
↑ Grammel (1920), p. 129.
↑ a b Klein and Sommerfeld (2010), p. 379.
↑ Klein and Sommerfeld (2010), p. 381 and Magnus (1971), p. 141.
↑ a b N. Joukowski : Geometric interpretation of the Hessian case of the motion of a heavy rigid body around a fixed point . In: German Mathematicians Association (ed.): Annual report of the German Mathematicians Association . tape 3 . Reimer, 1894, ISSN 0012-0456 , p. 62–70 ( uni-goettingen.de ).
Wilhelm Hess: About Euler's equations of motion and about a new particular solution to the problem of the motion of a rigid body around a fixed point. In: Mathematical Annals . Vol. 37, 1890, pp. 153-181 ( digizeitschriften.de [accessed on May 2, 2018]).
N. Joukowski : Geometric interpretation of the Hessian case of the motion of a heavy rigid body around a fixed point . In: German Mathematicians Association (ed.): Annual report of the German Mathematicians Association . tape 3 . Reimer, 1894, ISSN 0012-0456 , p. 62–70 ( uni-goettingen.de ).
K. Magnus : Kreisel: Theory and Applications . Springer, 1971, ISBN 3-642-52163-0 , pp. 143 ff . ( limited preview in Google Book Search [accessed November 30, 2019]).
F. Klein , A. Sommerfeld : The Theory of the Top . Development of the Theory in the Case of the Heavy Symmetric Top. tape II . Birkhäuser, Boston 2010, ISBN 978-0-8176-4824-4 , pp. 378 ff ., doi : 10.1007 / 978-0-8176-4827-5 (English, symbols are explained on p. 197 ff., in particular p. 200).
This page is based on the copyrighted Wikipedia article "Hesssches_Pendel" (Authors); it is used under the Creative Commons Attribution-ShareAlike 3.0 Unported License. You may redistribute it, verbatim or modified, providing that you comply with the terms of the CC-BY-SA. |
Factor and simplify each expression below.
\frac { x ^ { 2 } - 4 } { x ^ { 2 } + 4 x + 4 }
\frac { 2 x ^ { 2 } - 5 x - 3 } { 4 x ^ { 2 } + 4 x + 1 }
Justify each step in simplifying the expression in part (a).
Factor and then reduce.
For the numerator, what numbers multiply to
-6
-5?
Diamond Pattern, top negative 6, x squared, bottom, negative 5, x.
Put these numbers in a generic rectangle (these create your "middle term") along with the first and last terms. Added to diamond problem, left, negative 6, x right, 1, x. 2 by 2 rectangle added, interior labeled as follows: top left, 2, x squared, top right, 1, x, bottom left, negative 6, x, bottom right, negative 3.
Now find the greatest common factors of each row and column.
See the generic rectangle. Added to rectangle, left edge, x minus 3, bottom edge, 2, x, + 1.
2x^2 - 5x - 3 = (x - 3)(2x + 1)
Now factor the denominator and simplify. |
Experiment_(probability_theory) Knowpia
In probability theory, an experiment or trial (see below) is any procedure that can be infinitely repeated and has a well-defined set of possible outcomes, known as the sample space.[1] An experiment is said to be random if it has more than one possible outcome, and deterministic if it has only one. A random experiment that has exactly two (mutually exclusive) possible outcomes is known as a Bernoulli trial.[2]
Experiments and trialsEdit
Random experiments are often conducted repeatedly, so that the collective results may be subjected to statistical analysis. A fixed number of repetitions of the same experiment can be thought of as a composed experiment, in which case the individual repetitions are called trials. For example, if one were to toss the same coin one hundred times and record each result, each toss would be considered a trial within the experiment composed of all hundred tosses.[3]
A random experiment is described or modeled by a mathematical construct known as a probability space. A probability space is constructed and defined with a specific kind of experiment or trial in mind.
A mathematical description of an experiment consists of three parts:
A sample space, Ω (or S), which is the set of all possible outcomes.
{\displaystyle \scriptstyle {\mathcal {F}}}
, where each event is a set containing zero or more outcomes.
The assignment of probabilities to the events—that is, a function P mapping from events to probabilities.
An outcome is the result of a single execution of the model. Since individual outcomes might be of little practical use, more complicated events are used to characterize groups of outcomes. The collection of all such events is a sigma-algebra
{\displaystyle \scriptstyle {\mathcal {F}}}
. Finally, there is a need to specify each event's likelihood of happening; this is done using the probability measure function, P.
Once an experiment is designed and established, ω, from the sample space Ω. All the events in
{\displaystyle \scriptstyle {\mathcal {F}}}
that contain the selected outcome ω (recall that each event is a subset of Ω) are said to “have occurred”. The probability function P is defined in such a way that, if the experiment were to be repeated an infinite number of times, the relative frequencies of occurrence of each of the events would approach agreement with the values P assigns them.
As a simple experiment, we may flip a coin twice. The sample space (where the order of the two flips is relevant) is {(H, T), (T, H), (T, T), (H, H)} where "H" means "heads" and "T" means "tails". Note that each of (H, T), (T, H), ... are possible outcomes of the experiment. We may define an event which occurs when a "heads" occurs in either of the two flips. This event contains all of the outcomes except (T, T).
^ Albert, Jim (21 January 1998). "Listing All Possible Outcomes (The Sample Space)". Bowling Green State University. Retrieved June 25, 2013.
^ Papoulis, Athanasios (1984). "Bernoulli Trials". Probability, Random Variables, and Stochastic Processes (2nd ed.). New York: McGraw-Hill. pp. 57–63.
^ "Trial, Experiment, Event, Result/Outcome - Probability". Future Accountant. Retrieved 22 July 2013.
Media related to Experiment (probability theory) at Wikimedia Commons |
Linear function (calculus) - Wikipedia
Not to be confused with linear functional or linear map.
This article is missing information about the case of multivariate functions and vector valued functions, which must be considered, as this article is linked to from Jacobian matrix. Please expand the article to include this information. Further details may exist on the talk page. (February 2020)
In calculus and related areas of mathematics, a linear function from the real numbers to the real numbers is a function whose graph (in Cartesian coordinates) is a non-vertical line in the plane.[1] The characteristic property of linear functions is that when the input variable is changed, the change in the output is proportional to the change in the input.
Graph of the linear function:
{\displaystyle y(x)=-x+2}
Linear functions are related to linear equations.
3 Slope-intercept, point-slope, and two-point forms
4 Relationship with linear equations
5 Relationship with other classes of functions
A linear function is a polynomial function in which the variable x has degree at most one:[2]
{\displaystyle f(x)=ax+b}
Such a function is called linear because its graph, the set of all points
{\displaystyle (x,f(x))}
in the Cartesian plane, is a line. The coefficient a is called the slope of the function and of the line (see below).
If the slope is
{\displaystyle a=0}
, this is a constant function
{\displaystyle f(x)=b}
defining a horizontal line, which some authors exclude from the class of linear functions.[3] With this definition, the degree of a linear polynomial would be exactly one, and its graph would be a line that is neither vertical nor horizontal. However, in this article,
{\displaystyle a\neq 0}
is required, so constant functions will be considered linear.
{\displaystyle b=0}
then the linear function is said to be homogeneous. Such function defines a line that passes through the origin of the coordinate system, that is, the point
{\displaystyle (x,y)=(0,0)}
. In advanced mathematics texts, the term linear function often denotes specifically homogeneous linear functions, while the term affine function is used for the general case, which includes
{\displaystyle b\neq 0}
The natural domain of a linear function
{\displaystyle f(x)}
, the set of allowed input values for x, is the entire set of real numbers,
{\displaystyle x\in \mathbb {R} .}
One can also consider such functions with x in an arbitrary field, taking the coefficients a, b in that field.
{\displaystyle y=f(x)=ax+b}
is a non-vertical line having exactly one intersection with the y-axis, its y-intercept point
{\displaystyle (x,y)=(0,b).}
The y-intercept value
{\displaystyle y=f(0)=b}
is also called the initial value of
{\displaystyle f(x).}
{\displaystyle a\neq 0,}
the graph is a non-horizontal line having exactly one intersection with the x-axis, the x-intercept point
{\displaystyle (x,y)=(-{\tfrac {b}{a}},0).}
The x-intercept value
{\displaystyle x=-{\tfrac {b}{a}},}
the solution of the equation
{\displaystyle f(x)=0,}
is also called the root or zero of
{\displaystyle f(x).}
The slope of a line is the ratio
{\displaystyle {\tfrac {\Delta y}{\Delta x}}}
between a change in x, denoted
{\displaystyle \Delta x}
, and the corresponding change in y, denoted
{\displaystyle \Delta y}
The slope of a nonvertical line is a number that measures how steeply the line is slanted (rise-over-run). If the line is the graph of the linear function
{\displaystyle f(x)=ax+b}
, this slope is given by the constant a.
The slope measures the constant rate of change of
{\displaystyle f(x)}
per unit change in x: whenever the input x is increased by one unit, the output changes by a units:
{\displaystyle f(x{+}1)=f(x)+a}
, and more generally
{\displaystyle f(x{+}\Delta x)=f(x)+a\Delta x}
for any number
{\displaystyle \Delta x}
. If the slope is positive,
{\displaystyle a>0}
{\displaystyle f(x)}
is increasing; if
{\displaystyle a<0}
{\displaystyle f(x)}
In calculus, the derivative of a general function measures its rate of change. A linear function
{\displaystyle f(x)=ax+b}
has a constant rate of change equal to its slope a, so its derivative is the constant function
{\displaystyle f\,'(x)=a}
The fundamental idea of differential calculus is that any smooth function
{\displaystyle f(x)}
(not necessarily linear) can be closely approximated near a given point
{\displaystyle x=c}
by a unique linear function. The derivative
{\displaystyle f\,'(c)}
is the slope of this linear function, and the approximation is:
{\displaystyle f(x)\approx f\,'(c)(x{-}c)+f(c)}
{\displaystyle x\approx c}
. The graph of the linear approximation is the tangent line of the graph
{\displaystyle y=f(x)}
{\displaystyle (c,f(c))}
. The derivative slope
{\displaystyle f\,'(c)}
generally varies with the point c. Linear functions can be characterized as the only real functions whose derivative is constant: if
{\displaystyle f\,'(x)=a}
for all x, then
{\displaystyle f(x)=ax+b}
{\displaystyle b=f(0)}
Slope-intercept, point-slope, and two-point formsEdit
A given linear function
{\displaystyle f(x)}
can be written in several standard formulas displaying its various properties. The simplest is the slope-intercept form:
{\displaystyle f(x)=ax+b}
from which one can immediately see the slope a and the initial value
{\displaystyle f(0)=b}
, which is the y-intercept of the graph
{\displaystyle y=f(x)}
Given a slope a and one known value
{\displaystyle f(x_{0})=y_{0}}
, we write the point-slope form:
{\displaystyle f(x)=a(x{-}x_{0})+y_{0}}
In graphical terms, this gives the line
{\displaystyle y=f(x)}
with slope a passing through the point
{\displaystyle (x_{0},y_{0})}
The two-point form starts with two known values
{\displaystyle f(x_{0})=y_{0}}
{\displaystyle f(x_{1})=y_{1}}
. One computes the slope
{\displaystyle a={\tfrac {y_{1}-y_{0}}{x_{1}-x_{0}}}}
and inserts this into the point-slope form:
{\displaystyle f(x)={\tfrac {y_{1}-y_{0}}{x_{1}-x_{0}}}(x{-}x_{0}\!)+y_{0}}
{\displaystyle y=f(x)}
is the unique line passing through the points
{\displaystyle (x_{0},y_{0}\!),(x_{1},y_{1}\!)}
{\displaystyle y=f(x)}
may also be written to emphasize the constant slope:
{\displaystyle {\frac {y-y_{0}}{x-x_{0}}}={\frac {y_{1}-y_{0}}{x_{1}-x_{0}}}}
Relationship with linear equationsEdit
Linear functions commonly arise from practical problems involving variables
{\displaystyle x,y}
with a linear relationship, that is, obeying a linear equation
{\displaystyle Ax+By=C}
{\displaystyle B\neq 0}
, one can solve this equation for y, obtaining
{\displaystyle y=-{\tfrac {A}{B}}x+{\tfrac {C}{B}}=ax+b,}
where we denote
{\displaystyle a=-{\tfrac {A}{B}}}
{\displaystyle b={\tfrac {C}{B}}}
. That is, one may consider y as a dependent variable (output) obtained from the independent variable (input) x via a linear function:
{\displaystyle y=f(x)=ax+b}
. In the xy-coordinate plane, the possible values of
{\displaystyle (x,y)}
form a line, the graph of the function
{\displaystyle f(x)}
{\displaystyle B=0}
in the original equation, the resulting line
{\displaystyle x={\tfrac {C}{A}}}
is vertical, and cannot be written as
{\displaystyle y=f(x)}
The features of the graph
{\displaystyle y=f(x)=ax+b}
can be interpreted in terms of the variables x and y. The y-intercept is the initial value
{\displaystyle y=f(0)=b}
{\displaystyle x=0}
. The slope a measures the rate of change of the output y per unit change in the input x. In the graph, moving one unit to the right (increasing x by 1) moves the y-value up by a: that is,
{\displaystyle f(x{+}1)=f(x)+a}
. Negative slope a indicates a decrease in y for each increase in x.
For example, the linear function
{\displaystyle y=-2x+4}
{\displaystyle a=-2}
, y-intercept point
{\displaystyle (0,b)=(0,4)}
, and x-intercept point
{\displaystyle (2,0)}
Suppose salami and sausage cost €6 and €3 per kilogram, and we wish to buy €12 worth. How much of each can we purchase? If x kilograms of salami and y kilograms of sausage costs a total of €12 then, €6×x + €3×y = €12. Solving for y gives the point-slope form
{\displaystyle y=-2x+4}
, as above. That is, if we first choose the amount of salami x, the amount of sausage can be computed as a function
{\displaystyle y=f(x)=-2x+4}
. Since salami costs twice as much as sausage, adding one kilo of salami decreases the sausage by 2 kilos:
{\displaystyle f(x{+}1)=f(x)-2}
, and the slope is −2. The y-intercept point
{\displaystyle (x,y)=(0,4)}
corresponds to buying only 4 kg of sausage; while the x-intercept point
{\displaystyle (x,y)=(2,0)}
corresponds to buying only 2 kg of salami.
Note that the graph includes points with negative values of x or y, which have no meaning in terms of the original variables (unless we imagine selling meat to the butcher). Thus we should restrict our function
{\displaystyle f(x)}
to the domain
{\displaystyle 0\leq x\leq 2}
Also, we could choose y as the independent variable, and compute x by the inverse linear function:
{\displaystyle x=g(y)=-{\tfrac {1}{2}}y+2}
{\displaystyle 0\leq y\leq 4}
Relationship with other classes of functionsEdit
If the coefficient of the variable is not zero (a ≠ 0), then a linear function is represented by a degree 1 polynomial (also called a linear polynomial), otherwise it is a constant function – also a polynomial function, but of zero degree.
A straight line, when drawn in a different kind of coordinate system may represent other functions.
For example, it may represent an exponential function when its values are expressed in the logarithmic scale. It means that when log(g(x)) is a linear function of x, the function g is exponential. With linear functions, increasing the input by one unit causes the output to increase by a fixed amount, which is the slope of the graph of the function. With exponential functions, increasing the input by one unit causes the output to increase by a fixed multiple, which is known as the base of the exponential function.
If both arguments and values of a function are in the logarithmic scale (i.e., when log(y) is a linear function of log(x)), then the straight line represents a power law:
{\displaystyle \log _{r}y=a\log _{r}x+b\quad \Rightarrow \quad y=r^{b}\cdot x^{a}}
Archimedean spiral defined by the polar equation r = 1⁄2θ + 2
On the other hand, the graph of a linear function in terms of polar coordinates:
{\displaystyle r=f(\theta )=a\theta +b}
is an Archimedean spiral if
{\displaystyle a\neq 0}
and a circle otherwise.
Affine map, a generalization
Arithmetic progression, a linear function of integer argument
Swokowski, Earl W. (1983), Calculus with analytic geometry (Alternate ed.), Boston: Prindle, Weber & Schmidt, ISBN 0871503417
https://web.archive.org/web/20130524101825/http://www.math.okstate.edu/~noell/ebsm/linear.html
Retrieved from "https://en.wikipedia.org/w/index.php?title=Linear_function_(calculus)&oldid=1060912317" |
Mark - Bulbapedia, the community-driven Pokémon encyclopedia
If you were looking for the male player character from Pokémon Trading Card Game and its sequel, see Mark (TCG GB).
If you were looking for the selectable shape markings on a Pokémon's summary screen, see Marking.
If you were looking for the mark on a Pokémon's status screen that indicates which game it came from, see Origin mark.
A mark (Japanese: あかし mark) is a special marking that a Pokémon can have when caught in Pokémon Sword and Shield. Marks are shown on the Pokémon's summary screen and grant a particular title to the Pokémon that is shown when it is sent out in battle, like Ribbons.
A Pokémon can only have one mark. The Pokémon's mark (or lack of mark) is determined when it is encountered—a Pokémon cannot gain or change its mark after being caught.
3 List of marks
Wild Pokémon may have marks when encountered in a random encounter (including fishing) or symbol encounter. Additionally, Pokémon that join while cooking curry at Pokémon Camp always have the Curry Mark.
Pokémon encountered in other ways can never have marks. Pokémon encountered in the following encounter types can never have marks:
Pokémon caught in static encounters, such as Zacian and Zamazenta, or other scripted encounters like the Glimwood Tangle Impidimp
Pokémon caught during Max Raid Battles and Dynamax Adventures
In-game trade Pokémon
Depending on conditions, anywhere from 5% to 10% of Pokémon encountered in the wild will have a mark.
When a Pokémon is encountered in the wild, the game checks whether to apply a mark to it, using the following rates in the following order:[1]
With Mark Charm
Rare Mark 1/1000 3/1000
Rowdy Mark - Slump Mark group (28) 1/100 3/100
Uncommon Mark 1/50 3/50
A weather-based mark (8) 1/50 3/50
A time-based mark (4) 1/50 3/50
Fishing Mark 1/25 3/25
That is, first, the game determines if the Pokémon will have a Rare Mark. If that test fails, it checks the Rowdy - Slump Mark group. If that fails, it checks Uncommon Mark, and so on.
If the Rowdy - Slump Mark group is chosen, the marks within that group all have an even chance of being selected (~3.6% each, relative to each other). If a weather-based mark is chosen and the weather is not clear, the mark corresponding to the weather will be given; if the weather is clear, the game will continue checking the following types of marks. If a time-based mark is chosen, the mark corresponding to the current time will be chosen.
If the player has the Mark Charm in their Bag, wild Pokémon are more likely to have a mark. If a wild Pokémon would not have a mark while the player has the Mark Charm, the game runs through these checks a second and third time. This approximately triples the chance of a wild Pokémon to have a mark.[2]
The weather-based marks and the Fishing Mark are the only random marks that are not part of a group that is always possible (specifically, when the weather is clear or when encountering a Pokémon other than by fishing). This affects the overall chance of encountering a wild Pokémon with a mark, since each chance is separate (as opposed to choosing whether any mark will be received, and then choosing the mark). In clear weather and if not fishing, the overall probability of having any mark (i.e., the complement of not having a mark) is
{\textstyle 1-{999 \over 1000}\cdot {99 \over 100}\cdot {49 \over 50}\cdot {49 \over 50}\approx 5.02\%}
; if fishing during non-clear weather, the overall probability is
{\textstyle 1-{999 \over 1000}\cdot {99 \over 100}\cdot {49 \over 50}\cdot {49 \over 50}\cdot {49 \over 50}\cdot {24 \over 25}\approx 10.64\%}
Lunchtime Mark しょうごのあかし the Peckish はらペコの A mark for a peckish Pokémon. During the afternoon (12:00 p.m to 6:59 p.m.) 1/50
Sleepy-Time Mark しょうしのあかし the Sleepy おねむな A mark for a sleepy Pokémon. At night (8:00 p.m to 5:59 a.m.) 1/50
Dusk Mark たそがれのあかし the Dozy そろそろねむい A mark for a dozy Pokémon. During the evening (7:00 p.m to 7:59 p.m.) 1/50
Dawn Mark あかつきのあかし the Early Riser はやくにめざめた A mark for an early-riser Pokémon. During the morning (6:00 a.m to 11:59 a.m.) 1/50
Cloudy Mark どんてんのあかし the Cloud Watcher くもをみつめる A mark for a cloud-watching Pokémon. Weather is cloudy 1/50
Rainy Mark あめふりのあかし the Sodden あめにむせぶ A mark for a sodden Pokémon. Weather is rain 1/50
Stormy Mark いかづちのあかし the Thunderstruck かみなりにさわぐ A mark for a thunderstruck Pokémon. Weather is thunderstorm 1/50
Snowy Mark こうせつのあかし the Snow Frolicker ゆきにころがる A mark for a snow-frolicking Pokémon. Weather is snow 1/50
Blizzard Mark ごうせつのあかし the Shivering こごえふるえる A mark for a shivering Pokémon. Weather is blizzard 1/50
Dry Mark かんそうのあかし the Parched のどカラカラの A mark for a parched Pokémon. Weather is harsh sunlight 1/50
Sandstorm Mark さじんのあかし the Sandswept すなにまみれる A mark for a sandswept Pokémon. Weather is sandstorm 1/50
Misty Mark のうむのあかし the Mist Drifter きりにとまどう A mark for a mist-drifter Pokémon. Weather is fog 1/50
Destiny Mark うんめいのあかし the Chosen One うんめいかんじる A mark of a chosen Pokémon. Unobtainable 0
Fishing Mark つりあげられたあかし the Catch of the Day つりたてピチピチの A mark for a catch-of-the-day Pokémon. While fishing 1/25
Curry Mark カレーのあかし the Curry Connoisseur カレーずきな A mark for a curry-connoisseur Pokémon. All Pokémon that come to camp after cooking curry 1
Uncommon Mark ときどきみるあかし the Sociable ひとになれてる A mark for a sociable Pokémon. none 1/50
Rare Mark みたことのないあかし the Recluse ひとをしらない A mark for a reclusive Pokémon. none 1/1000
Rowdy Mark わんぱくなあかし the Rowdy あばれんぼうの A mark for a rowdy Pokémon. none (1/28)/100
Absent-Minded Mark のうてんきなあかし the Spacey なにもかんがえてない A mark for a spacey Pokémon. none (1/28)/100
Jittery Mark きんちょうのあかし the Anxious ドキドキしてる A mark for an anxious Pokémon. none (1/28)/100
Excited Mark きたいのあかし the Giddy ワクワクしてる A mark for a giddy Pokémon. none (1/28)/100
Charismatic Mark カリスマのあかし the Radiant オーラをかんじる A mark for a radiant Pokémon. none (1/28)/100
Calmness Mark れいせいのあかし the Serene クールな A mark for a serene Pokémon. none (1/28)/100
Intense Mark じょうねつのあかし the Feisty アグレッシブな A mark for a feisty Pokémon. none (1/28)/100
Zoned-Out Mark ゆだんのあかし the Daydreamer ボーっとしてる A mark for a daydreaming Pokémon. none (1/28)/100
Joyful Mark たこうのあかし the Joyful しあわせそうな A mark for a joyful Pokémon. none (1/28)/100
Angry Mark ふんぬのあかし the Furious プンプンおこる A mark for a furious Pokémon. none (1/28)/100
Smiley Mark びしょうのあかし the Beaming ニコニコわらう A mark for a beaming Pokémon. none (1/28)/100
Teary Mark ひそうのあかし the Teary-Eyed メソメソなく A mark for a teary-eyed Pokémon. none (1/28)/100
Upbeat Mark かいちょうのあかし the Chipper ごきげんな A mark for a chipper Pokémon. none (1/28)/100
Peeved Mark げきはつのあかし the Grumpy ふきげんな A mark for a grumpy Pokémon. none (1/28)/100
Intellectual Mark りせいのあかし the Scholar ちてきな A mark for a scholarly Pokémon. none (1/28)/100
Ferocious Mark ほんのうのあかし the Rampaging あれくるう A mark for a rampaging Pokémon. none (1/28)/100
Crafty Mark こうかつのあかし the Opportunist スキをねらう A mark for an opportunistic Pokémon. none (1/28)/100
Scowling Mark こわもてのあかし the Stern いかつい A mark for a stern Pokémon. none (1/28)/100
Kindly Mark やさがたのあかし the Kindhearted やさしげな A mark for a kindhearted Pokémon. none (1/28)/100
Flustered Mark どうようのあかし the Easily Flustered あわてんぼうの A mark for an easily flustered Pokémon. none (1/28)/100
Pumped-Up Mark こうようのあかし the Driven やるきまんまんの A mark for a driven Pokémon. none (1/28)/100
Zero Energy Mark けんたいのあかし the Apathetic やるきゼロの A mark for an apathetic Pokémon. none (1/28)/100
Prideful Mark じしんのあかし the Arrogant ふんぞりかえった A mark for an arrogant Pokémon. none (1/28)/100
Unsure Mark ふしんのあかし the Reluctant じしんのない A mark for an unsure Pokémon. none (1/28)/100
Humble Mark ぼくとつのあかし the Humble そぼくな A mark for a humble Pokémon. none (1/28)/100
Thorny Mark ふじゅんのあかし the Pompous きどっている A mark for a pompous Pokémon. none (1/28)/100
Vigor Mark げんきのあかし the Lively げんきいっぱいの A mark for a lively Pokémon. none (1/28)/100
Slump Mark ふちょうのあかし the Worn-Out どこかくたびれた A mark for a worn-out Pokémon. none (1/28)/100
Since Pokémon hatched from Eggs or caught in a Max Raid Battle cannot have a mark, the only way for a Pokémon to have both a Hidden Ability and a mark is to use an Ability Patch.
Because the weather is determined by where the player is standing, and weather marks are determined based on the weather when the symbol encounter spawns rather than when it is encountered, it is possible for Pokémon to have weather marks based on the weather in adjacent locations, if they spawned while the player was standing in a different location. This can even allow Pokémon to have marks corresponding to weather that can never occur in the location in which they are caught.[4]
Chinese Cantonese 證章 Jingjēung
Mandarin 證章 / 证章 Zhèngzhāng
French Insigne
Italian Emblema
Korean 증표 Jeungpyo
Spanish Emblema
↑ Reverse-engineered mark logic by SciresM, via Twitter
↑ Mark Logic by SciresM, via Twitter
↑ Actual Mark rates and expected encounters by Anubis/Sibuna_Switch on Twitter
Retrieved from "https://bulbapedia.bulbagarden.net/w/index.php?title=Mark&oldid=3419192" |
Home : Support : Online Help : Programming : Logic : Boolean : testeq
random polynomial-time equivalence tester
testeq(a = b)
testeq(a, b)
testeq(a)
The function testeq tests for equivalence probabilistically. It returns false if the expressions are not equal (or not equal to 0) and true otherwise for the class of expressions that testeq recognizes. The result false is always correct; the result true may be incorrect with very low probability.
This function will succeed over expressions formed with rational constants, independent variables, and I, combined by arithmetic operations, exponentials, trigonometrics and a few others. It may also succeed with some expressions involving algebraic constants and functions and involving Pi as an argument of trigonometrics. If the expressions do not fall in this class, testeq returns FAIL. testeq may also return FAIL if it cannot find an appropriate modulus that works after seven trials.
a≔\left({\mathrm{sin}\left(x\right)}^{2}-\mathrm{cos}\left(x\right)\mathrm{tan}\left(x\right)\right){\left({\mathrm{sin}\left(x\right)}^{2}+\mathrm{cos}\left(x\right)\mathrm{tan}\left(x\right)\right)}^{2}:
b≔\frac{1}{4}{\mathrm{sin}\left(2x\right)}^{2}-\frac{1}{2}\mathrm{sin}\left(2x\right)\mathrm{cos}\left(x\right)-2{\mathrm{cos}\left(x\right)}^{2}+\frac{1}{2}\mathrm{sin}\left(2x\right){\mathrm{cos}\left(x\right)}^{3}+3{\mathrm{cos}\left(x\right)}^{4}-{\mathrm{cos}\left(x\right)}^{6}:
\mathrm{evalb}\left(a=b\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{evalb}\left(\mathrm{expand}\left(a\right)=\mathrm{expand}\left(b\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{testeq}\left(a=b\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
Gonnet, Gaston. "Determining Equivalence of Expressions in Random Polynomial Time." Proceedings of the 16th ACM Symposium on the Theory of Computing. Washington DC. April 1984. pp. 334-341. |
Realization_(probability) Knowpia
Realization (probability)
In probability and statistics, a realization, observation, or observed value, of a random variable is the value that is actually observed (what actually happened). The random variable itself is the process dictating how the observation comes about. Statistical quantities computed from realizations without deploying a statistical model are often called "empirical", as in empirical distribution function or empirical probability.
Conventionally, to avoid confusion, upper case letters denote random variables; the corresponding lower case letters denote their realizations.[1]
In more formal probability theory, a random variable is a function X defined from a sample space Ω to a measurable space called the state space.[2][a] If an element in Ω is mapped to an element in state space by X, then that element in state space is a realization. Elements of the sample space can be thought of as all the different possibilities that could happen; while a realization (an element of the state space) can be thought of as the value X attains when one of the possibilities did happen. Probability is a mapping that assigns numbers between zero and one to certain subsets of the sample space, namely the measurable subsets, known here as events. Subsets of the sample space that contain only one element are called elementary events. The value of the random variable (that is, the function) X at a point ω ∈ Ω,
{\displaystyle x=X(\omega )}
is called a realization of X.[3]
^ A random variable cannot be an arbitrary function; it needs to satisfy other conditions, namely it needs to be measurable with total integral 1.
^ Wilks, Samuel S. (1962). Mathematical Statistics. Wiley. ISBN 9780471946441.
^ Varadhan, S. R. S. (2001). Probability Theory. Courant Lecture Notes in Mathematics. Vol. 7. American Mathematical Society. ISBN 9780821828526.
^ Gubner, John A. (2006). Probability and Random Processes for Electrical and Computer Engineers. Cambridge University Press. p. 383. ISBN 0-521-86470-4. |
Theoretical radiation astronomy/Quiz - Wikiversity
The images show a relativistic jet emanating from the active galactic nucleus of M87 using X-ray, radio, and optical astronomy. Credit: NASA/CXC/VLA/HST.
Theoretical radiation astronomy is the key theory lecture for the course on the principles of radiation astronomy.
Once you’ve read and studied the lecture itself, the links contained within the article/lecture, listed under See also, External links and in the {{principles of radiation astronomy}} template, you should have adequate background to take the quiz and score highly.
1 True or False, An action or process of throwing or sending out a traveling ray in a line, beam, or stream of small cross section may be called radiation.
4 Before the current era and perhaps before 6,000 b2k which classical planet may have been observed as a pole star for the Earth?
6 True or False, Neutrinos emanate from a neutron star because an atomic nucleus the size hypothesized for a neutron star is unstable and the neutrons decompose giving off neutrinos.
7 Which of the following is not a characteristic of a sky?
8 Which of the following are theoretical radiation astronomy phenomena associated with the Earth?
9 A possible solution to the discrepancy between the Spite plateau abundance and the predicted value of the primordial lithium abundance is lithium depletion through?
11 Yes or No, The traditional equation
{\displaystyle E=mc^{2}}
equates energy with matter in their interconversion.
14 A laboratory solution to the discrepancy between the Spite plateau abundance and the predicted value of the primordial lithium abundance is lithium depletion through?
In the radiation physics laboratories here on Earth, the
of radiation is studied and
relative to sources are proven.
16 True or False, The traditional equation
{\displaystyle E=mc^{2}}
equates electromagnetism with mass in their interconversion.
17 Which of the following is not a characteristic of a theory?
19 Capital Greek letters are often used for?
Let the English upper-case (capital) letter
when juxtaposed to
indicate radiation at wavelengths longer than those of radiation that emanates from atomic
to those just above the extreme
{\displaystyle E=mc^{2}}
equates electromagnetism with matter in their interconversion.
22 Which of the following is not a characteristic of a control group?
near the barycenter its system
24 Electromagnetic radiation emitted by accelerating charged particles that are moving at near the speed of light?
26 True or False, A theory may not be right but it should be testable.
27 Which of the following is not a characteristic of background radiation?
29 Electromagnetic radiation emitted by accelerating charged particles?
The emissivity of a perfect
equals 0, that of a perfect black body equals
31 True or False, A theory may not be testable but it should be right.
32 Which of the following is not a characteristic of a black body?
photons go in but they don't come out
a hole or entry much smaller than a cavity
34 Electromagnetic radiation emitted by decelerating charged particles?
A surface's spectral
do not depend on wavelength [for a graybody], so that the emissivity is a
36 True or False, The term meteor is usually associated with visual radiation from an object streaking across an Earth sky at night.
37 Which of the following is not a theoretical characteristic of a meteor?
38 Which of the following are theoretical radiation astronomy phenomena associated with a megacryometeor?
non-water ice
cometary origin
39 Electromagnetic radiation emitted by passing charged particles?
41 True or False, Particle radiation has three main origins: (1) galactic cosmic radiation, (2) solar particle radiation, and (3) geomagnetically trapped particle radiation.
43 Which of the following are theoretical radiation astronomy phenomena associated with space radiation?
nuclear fusion at a star's exposed core
nuclear fusion in a star's chromosphere
44 Measurements from Voyager 1 revealed a steady rise since May in collisions with?
At the same time, in late
, there was a dramatic drop in collisions with
, which are thought to originate from the
47 Which of the following is not a major source of protons within the solar system?
49 The Galactic distribution of 511 keV line emission distribution are the bulge + the thick disk or?
51 True or False, According to relativity theory, although the speed of light in a perfect vacuum cannot be exceeded, since there are no perfect vacuums anywhere in the universe, this speed of light can be exceeded in any medium and produce Cerenkov radiation.
54 The most accurate standard for the metre is conveniently defined so that there are exactly 299,792,458 of them to the distance travelled by light in a standard?
relativity, c is the maximum speed at which all
in the universe can travel.
56 True or False, An electron beam furnace is a type of vacuum furnace employing a high-energy electron beam in vacuum as the means for delivery of heat to the material being melted.
57 Which of the following is not a characteristic of the heliosphere?
58 Which of the following are characteristic of radiation all around us?
biologically functional components of the human body
a component of DNA
59 Background radiation may simply be any radiation that is?
The cosmic microwave background radiation is a
glow that fills the
61 True or False, In a laboratory, background radiation refers to the measured value from any sources that affect an instrument when a radiation source sample is being measured.
63 Which of the following are characteristic of radiation damage?
proton activation
64 High-intensity ionizing radiation in air can produce a visible ionized air glow of telltale?
Like gases,
lack fixed
structure; the effects of
is therefore mainly limited to
67 Which of the following is not a characteristic of geomagnetic polar reversals?
the Earth becomes a monopole
basalts record reversals
reversals in ocean sediment cores
reversals appear to occur at random intervals
69 The diffuse extragalactic background radiation (DEBRA) refers to the diffuse photon field from extragalactic origin that fill our?
DEBRA contains
eV to ~
72 Which of the following is not a characteristic of terrestrial gamma-ray flashes?
74 As of 1977, model calculations cannot reproduce the observed breadth of the Ca II λ3933 line in Da,F stars like Ross 627 without appealing to an?
76 True or False, Observations of X-rays have sometimes found the spectrum to have an upper portion and the lower portion suggesting a two stage acceleration process.
77 Which of the following is not a characteristic of terrestrial X-ray flashes?
78 Which of the following are characteristic of high-velocity stars?
81 True or False, Nitrogen/oxygen abundance ratios may decrease from outer-arm nebulae to inner-arm nebulae in spiral galaxies.
82 Which of the following is not a characteristic of meteoritic lithium abundance?
light elements may have been formed by the irradiation of interstellar matter
closely matches the solar abundance
not diminished by nucleosynthesis
not destroyed by nuclear fission reactions
may have been produced by cosmic-ray spallation
83 Which of the following may be characteristic of hydrogen deficiency in stars?
87 Which of the following is not a characteristic of a globular cluster?
a spherical collection of stars
moves together but unbound by gravity
orbits a galactic core
relatively high stellar density towards its center
may have formed together
88 Which of the following may be characteristic of orbital theory?
a hyperbolic pass
89 A double star with an orbital plane that lies near enough to the line of sight to undergo eclipses is an?
91 True or False, A localized, transient volume that is observed may be called a region.
92 Which of the following is not a characteristic of astrognosy?
93 Which of the following may be characteristic of magnetohydrodynamics?
Theoretical astrohistory concerns
, and experimental facts with respect to observations in the past of the
96 True or False, Although no tubular telescope has been found at ancient archaeological sites, ancient observers may have used air telescopes.
97 Which of the following is not a characteristic of the electrical theory of the corona?
repulsive ejection
auroral-like phenomenon around the Sun
rays and streamers
centrifugal ejection of particles from the solar limb
99 Astronomical radiation mathematics is the laboratory mathematics such as simulations that are generated to try to understand the observations of?
Learn more about Theoretical radiation astronomy
Retrieved from "https://en.wikiversity.org/w/index.php?title=Theoretical_radiation_astronomy/Quiz&oldid=2029624" |
Asset vs. Fixed Asset Turnover
Asset Turnover FAQs
The higher the asset turnover ratio, the more efficient a company is at generating revenue from its assets. Conversely, if a company has a low asset turnover ratio, it indicates it is not efficiently using its assets to generate sales.
Asset turnover is the ratio of total sales or revenue to average assets.
This metric helps investors understand how effectively companies are using their assets to generate sales.
Investors use the asset turnover ratio to compare similar companies in the same sector or group.
A company's asset turnover ratio can be impacted by large asset sales as well as significant asset purchases in a given year.
Formula and Calculation of the Asset Turnover Ratio
Below are the steps as well as the formula for calculating the asset turnover ratio.
\begin{aligned} &\text{Asset Turnover} = \frac{ \text{Total Sales} }{ \frac { \text{Beginning Assets}\ +\ \text{Ending Assets} }{ 2 } } \\ &\textbf{where:}\\ &\text{Total Sales} = \text{Annual sales total} \\ &\text{Beginning Assets} = \text{Assets at start of year} \\ &\text{Ending Assets} = \text{Assets at end of year} \\ \end{aligned}
Asset Turnover=2Beginning Assets + Ending AssetsTotal Saleswhere:Total Sales=Annual sales totalBeginning Assets=Assets at start of yearEnding Assets=Assets at end of year
The asset turnover ratio uses the value of a company's assets in the denominator of the formula. To determine the value of a company's assets, the average value of the assets for the year needs to first be calculated.
Locate the value of the company's assets on the balance sheet as of the start of the year.
Locate the ending balance or value of the company's assets at the end of the year.
Add the beginning asset value to the ending value and divide the sum by two, which will provide an average value of the assets for the year.
Locate total sales—it could be listed as revenue—on the income statement.
Divide total sales or revenue by the average value of the assets for the year.
What the Asset Turnover Ratio Can Tell You
Typically, the asset turnover ratio is calculated on an annual basis. The higher the asset turnover ratio, the better the company is performing, since higher ratios imply that the company is generating more revenue per dollar of assets.
The asset turnover ratio tends to be higher for companies in certain sectors than in others. Retail and consumer staples, for example, have relatively small asset bases but have high sales volume—thus, they have the highest average asset turnover ratio. Conversely, firms in sectors such as utilities and real estate have large asset bases and low asset turnover.
Since this ratio can vary widely from one industry to the next, comparing the asset turnover ratios of a retail company and a telecommunications company would not be very productive. Comparisons are only meaningful when they are made for different companies within the same sector.
Example of How to Use the Asset Turnover Ratio
Let's calculate the asset turnover ratio for four companies in the retail and telecommunication-utilities sectors for FY 2020—Walmart Inc. (WMT), Target Corporation (TGT), AT&T Inc. (T), and Verizon Communications Inc. (VZ).
Asset Turnover Examples
Beginning Assets 219,295 42,779 551,669 291,727
Ending Assets 236,495 51,248 525,761 316,481
Avg. Total Assets 227,895 47,014 538,715 304,104
Asset Turnover 2.3x 2.0x 0.32x 0.42x
AT&T and Verizon have asset turnover ratios of less than one, which is typical for firms in the telecommunications-utilities sector. Since these companies have large asset bases, it is expected that they would slowly turn over their assets through sales. Clearly, it would not make sense to compare the asset turnover ratios for Walmart and AT&T, since they operate in very different industries. But comparing the relative asset turnover ratios for AT&T compared with Verizon may provide a better estimate of which company is using assets more efficiently in that industry. From the table, Verizon turns over its assets at a faster rate than AT&T.
For every dollar in assets, Walmart generated $2.30 in sales, while Target generated $2.00. Target's turnover could indicate that the retail company was experiencing sluggish sales or holding obsolete inventory. Furthermore, its low turnover may also mean that the company has lax collection methods. The firm's collection period may be too long, leading to higher accounts receivable. Target, Inc. could also not be using its assets efficiently: fixed assets such as property or equipment could be sitting idle or not being utilized to their full capacity.
The asset turnover ratio is a key component of DuPont analysis, a system that the DuPont Corporation began using during the 1920s to evaluate performance across corporate divisions. The first step of DuPont analysis breaks down return on equity (ROE) into three components, one of which is asset turnover, the other two being profit margin, and financial leverage. The first step of DuPont analysis can be illustrated as follows:
\begin{aligned} &\text{ROE} = \underbrace{ \left ( \frac{ \text{Net Income} }{ \text{Revenue} } \right ) }_\text{Profit Margin} \times \underbrace{ \left ( \frac{ \text{Revenue} }{ \text{AA} } \right ) }_\text{Asset Turnover} \times \underbrace{ \left ( \frac{ \text{AA} }{ \text{AE} } \right ) }_\text{Financial Leverage} \\ &\textbf{where:}\\ &\text{AA} = \text{Average assets} \\ &\text{AE} = \text{Average equity} \\ \end{aligned}
ROE=Profit Margin(RevenueNet Income)×Asset Turnover(AARevenue)×Financial Leverage(AEAA)where:AA=Average assetsAE=Average equity
The Difference Between Asset Turnover and Fixed Asset Turnover
While the asset turnover ratio considers average total assets in the denominator, the fixed asset turnover ratio looks at only fixed assets. The fixed asset turnover ratio (FAT) is, in general, used by analysts to measure operating performance. This efficiency ratio compares net sales (income statement) to fixed assets (balance sheet) and measures a company's ability to generate net sales from its fixed-asset investments, namely property, plant, and equipment (PP&E).
The fixed asset balance is a used net of accumulated depreciation. Depreciation is the allocation of the cost of a fixed asset, which is spread out—or expensed—each year throughout the asset's useful life. Typically, a higher fixed asset turnover ratio indicates that a company has more effectively utilized its investment in fixed assets to generate revenue.
Limitations of Using the Asset Turnover Ratio
While the asset turnover ratio should be used to compare stocks that are similar, the metric does not provide all of the detail that would be helpful for stock analysis. It is possible that a company's asset turnover ratio in any single year differs substantially from previous or subsequent years. Investors should review the trend in the asset turnover ratio over time to determine whether asset usage is improving or deteriorating.
The asset turnover ratio may be artificially deflated when a company makes large asset purchases in anticipation of higher growth. Likewise, selling off assets to prepare for declining growth will artificially inflate the ratio. Also, many other factors (such as seasonality) can affect a company's asset turnover ratio during periods shorter than a year.
What Is Asset Turnover Measuring?
The asset turnover ratio measures the efficiency of a company's assets in generating revenue or sales. It compares the dollar amount of sales (revenues) to its total assets as an annualized percentage. Thus, to calculate the asset turnover ratio, divide net sales or revenue by the average total assets. One variation on this metric considers only a company's fixed assets (the FAT ratio) instead of total assets.
Is It Better to Have a High or Low Asset Turnover?
Generally, a higher ratio is favored because it implies that the company is efficient in generating sales or revenues from its asset base. A lower ratio indicates that a company is not using its assets efficiently and may have internal problems.
What Is a Good Asset Turnover Value?
Asset turnover ratios vary across different industry sectors, so only the ratios of companies that are in the same sector should be compared. For example, retail or service sector companies have relatively small asset bases combined with high sales volume. This leads to a high average asset turnover ratio. Meanwhile, firms in sectors like utilities or manufacturing tend to have large asset bases, which translates to lower asset turnover.
How Can a Company Improve its Asset Turnover Ratio?
A company may attempt to raise a low asset turnover ratio by stocking its shelves with highly salable items, replenishing inventory only when necessary, and augmenting its hours of operation to increase customer foot traffic and spike sales. Just-in-time (JIT) inventory management, for instance, is a system whereby a firm receives inputs as close as possible to when they are actually needed. So, if a car assembly plant needs to install airbags, it does not keep a stock of airbags on its shelves, but receives them as those cars come onto the assembly line.
Can Asset Turnover be Gamed by a Company?
Like many other accounting figures, a company's management can attempt to make its efficiency seem better on paper than it actually is. Selling off assets to prepare for declining growth, for instance, has the effect of artificially inflating the ratio. Changing depreciation methods for fixed assets can have a similar effect as it will change the accounting value of the firm's assets.
Walmart. "2020 Annual Report."
Target. "2020 Annual Report."
AT&T Inc. "2020 Annual Report."
Verizon. "Annual Report 2020."
Return on Average Assets (ROAA) Definition
Return on average assets (ROAA) is an indicator used to assess the profitability of a firm's assets, and it is most often used by banks. |
Weight: 4,265 lbs (+200 lbs)
Cd: 0.23 | A: 2.27 m2 | Crr:
50 mph NaN mi NaN Wh/mi 3.5 kW NaN kW NaN kW
55 mph NaN mi NaN Wh/mi 4.66 kW NaN kW NaN kW
Battery Charge: 80% 80Speed: 65 mph 65Extra Weight: 200 lbs 200Auxiliary Power: 0.5 kW (Est: 0.5 - 1 kW)
Headwind: 0 mph 0Ambient Temperature: 70 °F 70Elevation: 0 feet 0
Air Pressure: 101,325 Pa
Air Density: 1.2 kg / m^3
This simulation uses Newton's Laws of Physics to calculate all forces acting on the vehicle.
Negative Forces: Air Resistance and Rolling Resistance
Positive Forces: Wheel Force provided by motor(s)
F_d = {1\over2}ρ{v}^2C_dA
The force acting opposite to the relative motion of any object moving with respect to a surrounding fluid (air).
Unlike other resistive forces, air resistance depends on velocity.
Air Density (
ρ
Mass per unit volume of Earth's atmosphere with units
kg/m^3
v
How fast the vehicle is moving (speed) with units
m/s
Coefficient of Drag (
C_d
Dimensionless quantity that is used to quantify the drag or resistance of an object in a fluid environment
Frontal Area (
A
Reference Area or cross-sectional area of vehicle perpendicular to velocity vector with units
m^2
F_{rr} = C_{rr}N
The force resisting the motion when a tire rolls on a surface.
It depends on many factors including tire compound, width, inflation pressure, and more.
Coefficient of Rolling Resistance (
C_{rr}
Force needed to push a wheeled vehicle forward (at constant speed on a level surface with zero grade and air resistance) per unit force of weight
Normal Force (
N
Weight of the object:
F_n=mg
m
g
is the gravitational field strength (about
9.806 m/s^2
on Earth) |
(Redirected from Stars Lecture 9)
{\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^{27}){\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 .\,\!} |
Polynomial/Related Articles - Citizendium
Polynomial/Related Articles
< Polynomial
A list of Citizendium articles, and planned articles, about Polynomial.
See also changes related to Polynomial, or pages that link to Polynomial or to this page or whose text contains "Polynomial".
Auto-populated based on Special:WhatLinksHere/Polynomial. Needs checking by a human.
Algebraic independence [r]: The property of elements of an extension field which satisfy only the trivial polynomial relation. [e]
Binomial theorem [r]:
{\displaystyle \textstyle (x+y)^{n}=\sum _{k=0}^{n}{n \choose k}x^{k}y^{n-k}}
for any natural number n. [e]
Calculus [r]: The elementary study of real (or complex) functions involving derivatives and integration. [e]
Catalog of special functions [r]: Add brief definition or description
Completing the square [r]: Rewriting a quadratic polynomial as a constant multiple of a linear polynomial plus a constant. [e]
Complex number [r]: Numbers of the form a+bi, where a and b are real numbers and i denotes a number satisfying
{\displaystyle i^{2}=-1}
Content (algebra) [r]: The highest common factor of the coefficients of a polynomial. [e]
Cyclotomic polynomial [r]: A polynomial whose roots are primitive roots of unity. [e]
Discriminant of a polynomial [r]: An invariant of a polynomial which vanishes if it has a repeated root: the product of the differences between the roots. [e]
Entire function [r]: is a function that is holomorphic in the whole complex plane. [e]
Exponential growth [r]: Increase of a quantity x with time t according to the equation x = Kat, where K and a are constants, a is greater than 1, and K is greater than 0. [e]
Fibonacci polynomials [r]: Polynomial sequence which can be considered as a generalisation of the Fibonacci numbers. [e]
Field extension [r]: A field containing a given field as a subfield. [e]
Fundamental Theorem of Algebra [r]: Any nonconstant polynomial whose coefficients are complex numbers has at least one complex number as a root. [e]
Galois theory [r]: Add brief definition or description
Gamma function [r]: Add brief definition or description
Graph coloring [r]: Add brief definition or description
Group (mathematics) [r]: Add brief definition or description
Hall polynomial [r]: Add brief definition or description
Holomorphic function [r]: Add brief definition or description
Integer [r]: Add brief definition or description
Legendre-Gauss Quadrature formula [r]: Add brief definition or description
Littlewood polynomial [r]: Add brief definition or description
Multiple (mathematics) [r]: Add brief definition or description
Null set [r]: Add brief definition or description
Reaction rate [r]: Add brief definition or description
Resultant (algebra) [r]: Add brief definition or description
Root of unity [r]: Add brief definition or description
Splitting field [r]: Add brief definition or description
Trigonometric function [r]: Add brief definition or description
Vector space [r]: Add brief definition or description
Retrieved from "https://citizendium.org/wiki/index.php?title=Polynomial/Related_Articles&oldid=655138" |
Rigged Hilbert space - Wikipedia
Construction linking the study of "bound" and continuous eigenvalues in functional analysis
In mathematics, a rigged Hilbert space (Gelfand triple, nested Hilbert space, equipped Hilbert space) is a construction designed to link the distribution and square-integrable aspects of functional analysis. Such spaces were introduced to study spectral theory in the broad sense.[vague] They bring together the 'bound state' (eigenvector) and 'continuous spectrum', in one place.
2 Functional analysis approach
3 Formal definition (Gelfand triple)
A function such as the canonical homomorphism of the real line into the complex plane
{\displaystyle x\mapsto e^{ix},}
is an eigenfunction of the differential operator
{\displaystyle -i{\frac {d}{dx}}}
on the real line R, but isn't square-integrable for the usual Borel measure on R. To properly consider this function as an eigenfunction requires some way of stepping outside the strict confines of the Hilbert space theory. This was supplied by the apparatus of Schwartz distributions, and a generalized eigenfunction theory was developed in the years after 1950.
Functional analysis approach[edit]
The concept of rigged Hilbert space places this idea in an abstract functional-analytic framework. Formally, a rigged Hilbert space consists of a Hilbert space H, together with a subspace Φ which carries a finer topology, that is one for which the natural inclusion
{\displaystyle \Phi \subseteq H}
is continuous. It is no loss to assume that Φ is dense in H for the Hilbert norm. We consider the inclusion of dual spaces H* in Φ*. The latter, dual to Φ in its 'test function' topology, is realised as a space of distributions or generalised functions of some sort, and the linear functionals on the subspace Φ of type
{\displaystyle \phi \mapsto \langle v,\phi \rangle }
for v in H are faithfully represented as distributions (because we assume Φ dense).
Now by applying the Riesz representation theorem we can identify H* with H. Therefore, the definition of rigged Hilbert space is in terms of a sandwich:
{\displaystyle \Phi \subseteq H\subseteq \Phi ^{*}.}
The most significant examples are those for which Φ is a nuclear space; this comment is an abstract expression of the idea that Φ consists of test functions and Φ* of the corresponding distributions. Also, a simple example is given by Sobolev spaces: Here (in the simplest case of Sobolev spaces on
{\displaystyle \mathbb {R} ^{n}}
{\displaystyle H=L^{2}(\mathbb {R} ^{n}),\ \Phi =H^{s}(\mathbb {R} ^{n}),\ \Phi ^{*}=H^{-s}(\mathbb {R} ^{n}),}
{\displaystyle s>0}
Formal definition (Gelfand triple)[edit]
A rigged Hilbert space is a pair (H,Φ) with H a Hilbert space, Φ a dense subspace, such that Φ is given a topological vector space structure for which the inclusion map i is continuous.
Identifying H with its dual space H*, the adjoint to i is the map
{\displaystyle i^{*}:H=H^{*}\to \Phi ^{*}.}
The duality pairing between Φ and Φ* is then compatible with the inner product on H, in the sense that:
{\displaystyle \langle u,v\rangle _{\Phi \times \Phi ^{*}}=(u,v)_{H}}
{\displaystyle u\in \Phi \subset H}
{\displaystyle v\in H=H^{*}\subset \Phi ^{*}}
. In the case of complex Hilbert spaces, we use a Hermitian inner product; it will be complex linear in u (math convention) or v (physics convention), and conjugate-linear (complex anti-linear) in the other variable.
The triple
{\displaystyle (\Phi ,\,\,H,\,\,\Phi ^{*})}
is often named the "Gelfand triple" (after the mathematician Israel Gelfand).
Note that even though Φ is isomorphic to Φ* if it happens that Φ is a Hilbert space in its own right, this isomorphism is not the same as the composition of the inclusion i with its adjoint i*
{\displaystyle i^{*}i:\Phi \subset H=H^{*}\to \Phi ^{*}.}
J.-P. Antoine, Quantum Mechanics Beyond Hilbert Space (1996), appearing in Irreversibility and Causality, Semigroups and Rigged Hilbert Spaces, Arno Bohm, Heinz-Dietrich Doebner, Piotr Kielanowski, eds., Springer-Verlag, ISBN 3-540-64305-2. (Provides a survey overview.)
J. Dieudonné, Éléments d'analyse VII (1978). (See paragraphs 23.8 and 23.32)
I. M. Gelfand and N. Ya. Vilenkin. Generalized Functions, vol. 4: Some Applications of Harmonic Analysis. Rigged Hilbert Spaces. Academic Press, New York, 1964.
K. Maurin, Generalized Eigenfunction Expansions and Unitary Representations of Topological Groups, Polish Scientific Publishers, Warsaw, 1968.
R. de la Madrid, "Quantum Mechanics in Rigged Hilbert Space Language," PhD Thesis (2001).
R. de la Madrid, "The role of the rigged Hilbert space in Quantum Mechanics," Eur. J. Phys. 26, 287 (2005); quant-ph/0502053.
Minlos, R.A. (2001) [1994], "Rigged_Hilbert_space", Encyclopedia of Mathematics, EMS Press
Retrieved from "https://en.wikipedia.org/w/index.php?title=Rigged_Hilbert_space&oldid=1060591135"
Template SpringerEOM with broken ref |
the angle of rotation of a figure in 30 degree then order of rotational symmetry is _____ - Maths - Symmetry - 11037081 | Meritnation.com
the angle of rotation of a figure in 30 degree then order of rotational symmetry is _____
We know : Order of rotational symmetry =
\frac{360°}{\mathrm{Angle} \mathrm{of} \mathrm{rotation}}
Order of rotational symmetry for given information =
\frac{360°}{30°}
= 12 ( Ans )
Sreeja Pradeep answered this
Srijita Majumder answered this
order of symmetry of the figure is 12 |
A long current carrying wire, carrying current such that it is -Turito
A long current carrying wire, carrying current such that it is flowing out from the plane of paper, is placed at O A steady state current is flowing in the loop ABCD
as seen from O,the loop will rotate in anticlock wise direction along axis OO’
as seen from O, the loop will rotate in clock wise direction along axis OO’
Answer:The correct answer is: both 1 & 2 are correct
A loop carrying current I lies in the x-y plane as shown in fig The unit vector
\stackrel{^}{k}
is coming out of the plane of the paper The magnetic moment of the current loop is
\stackrel{^}{k}
Oxygen atom of ether is:
A particle originally at rest at the highest point of a smooth vertical circle is slightly displaced. It will leave the circle at a vertical distance
h
below the highest point such that
h
{\mathrm{Lt}}_{x\to 3}\frac{{x}^{3}-8{x}^{2}+45}{2{x}^{2}-3x-9}
{\mathrm{Lt}}_{x\to 3}\frac{{x}^{3}-8{x}^{2}+45}{2{x}^{2}-3x-9}
In figure, a particle is placed at the highest point
A
of a smooth sphere of radius
r
. It is given slight push, and it leaves the sphere at
B
, at a depth
vertically below
A
h
velocity acquired at
B
{v}^{2}=2gh
The particle will leave the sphere at
B
\frac{m{v}^{2}}{r}\ge mg\mathrm{cos}\theta
\frac{2gh}{r}=\frac{g\left(r-h\right)}{r},
h=\frac{r}{3}
A
r
B
, at a depth
A
h
velocity acquired at
B
{v}^{2}=2gh
B
\frac{m{v}^{2}}{r}\ge mg\mathrm{cos}\theta
\frac{2gh}{r}=\frac{g\left(r-h\right)}{r},
h=\frac{r}{3}
The time taken by the projectile to reach from
A
B
t,
then the distance
AB
Horizontal component of velocity at
A
{v}_{H}=u\mathrm{cos}60°=\frac{u}{2} \therefore AC={u}_{H}×t=\frac{ut}{2}
AB=AC\mathrm{sec}30°=\frac{ut}{2}×\frac{2}{\sqrt{3}}=\frac{ut}{2}
A
B
t,
AB
A
{v}_{H}=u\mathrm{cos}60°=\frac{u}{2} \therefore AC={u}_{H}×t=\frac{ut}{2}
AB=AC\mathrm{sec}30°=\frac{ut}{2}×\frac{2}{\sqrt{3}}=\frac{ut}{2}
{\mathrm{Lt}}_{x\to 0}\frac{{e}^{\mathrm{tan} x}-{e}^{x}}{\mathrm{tan} x-x}
{\mathrm{Lt}}_{x\to 0}\frac{{e}^{\mathrm{tan} x}-{e}^{x}}{\mathrm{tan} x-x}
L{t}_{x\to 0}\frac{\sqrt{\left(1+x+{x}^{2}\right)}-1}{x}
L{t}_{x\to 0}\frac{\sqrt{\left(1+x+{x}^{2}\right)}-1}{x}
{\mathrm{Lt}}_{x\to 0} \frac{x\cdot {2}^{x}-x}{1-\mathrm{cos} x}
{\mathrm{Lt}}_{x\to 0} \frac{x\cdot {2}^{x}-x}{1-\mathrm{cos} x}
C-O-C
angle would be maximum in
C-O-C
A circular loop of radius 20cm is placed in a uniform magnetic field
\stackrel{\to }{B}=2T\text{ in }X-Y
plane the loop carries a current 1A in the direction shown in figure The magnitude of torque acting on the loop is nearly
\stackrel{\to }{B}=2T\text{ in }X-Y
Two 1ong wires are placed parallel to each other 10cm apart as shown in fig The magnetic field at piont P is
A proton accelerated by a pd V=500 KV moves through a transverse magnetic field B=0.51 T as shown in figure Then the angle q through which the proton deviates from the initial direction of its motion is (approximately)
A particle of mass m and charge q, moving with velocity V enters Region II normal to the boundary as shown in fig Region II has a uniform magnetic field B perpendicular to the plane of the paper The length of the Region II is
l
l
A small sphere is hung by a string fixed to a wall. The sphere is pushed away from the wall by a stick. The force acting on the sphere are shown in figure. Which of the following statements is wrong?
W=T
cos\theta +sin\theta
<T
P+Q=T
cos\theta +sin\theta
<T
Whereas [a], [b] and [c] are correct and [d] is wrong.
W=T
cos\theta +sin\theta
<T
P+Q=T
cos\theta +sin\theta
<T |
Electric Circuit Analysis/Resistors in Parallel - Wikiversity
Electric Circuit Analysis/Resistors in Parallel
What you need to remember from Resistors in Series. If you ever feel lost, do not be shy to go back to the previous lesson & go through it again. You can learn by repetition.
{\displaystyle R_{E}=R_{1}+R_{2}\!}
The total Resistance of Resistors in series is the sum of all resistors in series.
Voltage Divider Equation 2.3:
{\displaystyle V_{1}={\frac {R_{1}\times V_{s}}{R_{1}+R_{2}}}}
Current through Resistors connected in Series is the same for all resistors.
This Lesson is about Resistors in Parallel. The student/User is expected to understand the following at the end of the lesson.
two resistors connected in Parallel:
{\displaystyle R_{eq}={\frac {R_{1}\times R_{2}}{R_{1}+R_{2}}}}
Current Divider Principle:
{\displaystyle I_{1}=I_{s}\times {\frac {R_{eq}}{R1}}}
Resistors in Parallel← You are here
The best way to understand Parallel circuits is to start with the definition. A circuit is parallel to another circuit or several circuits if and only if they share common terminals; i.e. if both the branches touch each other's endpoints. Here is an example:
Figure 4.1: A Parallel circuit
R1, R2, and the voltage source are all in parallel. To prove this fact consider the top and bottom parts of the circuit.
Figure 4.2: Components in parallel share a common nodes
The areas in yellow are all connected together, as well as the areas in blue. So all the branches have the same terminals, which means that R1, R2, and the source are all in parallel.
If we take this discussion of the water flow analogy. Electric current can be seen as water and the conductors as water pipes.
Something interesting happens as the current reaches the common node of resistors that are connected in parallel: the total current is divided into the parallel branches.
Voltage Rule
If two or more branches are parallel then the voltage across them is equal. So based on this we can conclude that VR1=VR2=5volts. However unlike series resistors, the current across the branches is not necessarily equal.
For series resistors to find the total resistance we simply add them together. For parallel resistors its a little more complicated. Instead we use the following equation:
{\displaystyle R_{eq}={\frac {1}{{\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}+...+{\frac {1}{R_{n}}}}}}
However for the case of only two resistors we can use the following simplified form:
Equation 4.2: Total Parallel Resistance
{\displaystyle R_{eq}={\frac {R_{1}R_{2}}{R_{1}+R_{2}}}}
It is well to note at this point that the total resistance of parallel-connected resistors will always be less than the resistance of smallest of the individual resistors.
In series connection, we deduced that voltage is divided amongst resistors. For parallel-connected resistors, however, current is divided. So, as we did with the voltage division principle, here is the mathematical formula:
Equation 4.3: Current Divider Formula
{\displaystyle I_{1}=I_{s}\times {\frac {R_{eq}}{R1}}}
Using this formula you can work out the currents flowing through individual resistors.
We have spent three lectures hacking on about what and why resistors and& resistive circuits in two connection schemes are used, (i.e. series and parallel connections). The question now is, where & how in real life do these connections happen?
One simple application of these connection schemes is the Shunt Application. In the electric measurement industry, most often enough, we wish to measure currents and voltages of very high magnitudes (in the range of 500kV upwards). The problem is that metering devices have delicate electronic components and usually have small voltage and current ratings.
The solution to the above problem is to have a metering device connected in parallel to a resistor,with the resistor thus called a "shunt" resistor since it is there to protect (shunt) the metering device as shown in Part 4.
Figure 4.3: Application of Parallel Resistive circuits. Shunt connection
If we know the current rating of a device and the total current in the system, we can then work out the shunt current and, thus, the shunt resistance.
Figure 4.4: Example 3.1
Figure 3.4 shows a parallel resistive circuit with the following parameters.
{\displaystyle V_{s}=10Volts}
{\displaystyle R_{1}=3\Omega }
{\displaystyle R_{2}=7\Omega }
; Find
{\displaystyle R_{eq}}
{\displaystyle I_{1}andI_{2}.}
{\displaystyle R_{eq}={\frac {R_{1}R_{2}}{R_{1}+R_{2}}}}
Here are the solutions to the above problem:
{\displaystyle {\begin{matrix}\ R_{eq}&=&{\frac {R_{1}\times R_{2}}{R_{1}+R_{2}}}\\\ \\\ &=&{\frac {(3\Omega \times 7\Omega )}{(3\Omega +7\Omega )}}\\\ \\\ &=&{\frac {21}{10}}\Omega \\\ \\\ &=&2.1\Omega \end{matrix}}}
{\displaystyle {\begin{matrix}\ I_{1}&=&{\frac {V_{s}}{R_{1}}}\\\ \\\ &=&{\frac {10V}{3\Omega }}\\\ \\\ &=&3.33A\end{matrix}}}
{\displaystyle {\begin{matrix}\ I_{2}&=&{\frac {V_{s}}{R_{2}}}\\\ \\\ &=&{\frac {10V}{7\Omega }}\\\ \\\ &=&1.43A\end{matrix}}}
Thus it can be said that the supply current has been divided between R1 and R2.
We know that when solving these problems, we look at the data given and thus we can see how we need to manipulate our equations in order to achieve our objective.The following example highlights this point. See to it that you follow the method used and the reasoning behind it.
{\displaystyle I_{s}=5Amps}
{\displaystyle R_{1}=2\Omega }
{\displaystyle R_{2}=3\Omega }
; Find:
{\displaystyle I_{1};I_{2}}
{\displaystyle V_{s}}
{\displaystyle R_{eq}={\frac {R_{1}R_{2}}{R_{1}+R_{2}}}}
First Find:
{\displaystyle R_{eq}}
{\displaystyle {\begin{matrix}\ R_{eq}&=&{\frac {(R_{1}\times R_{2})}{(R_{1}+R_{2})}}\\\ \\\ &=&{\frac {(2\times 3)}{(2+3)}}\\\ \\\ R_{eq}&=&1.2\Omega \end{matrix}}}
{\displaystyle {\begin{matrix}\ I_{1}&=&I_{s}{\frac {R_{eq}}{R_{1}}}\\\ \\\ &=&(5A){\frac {1.2\Omega }{2\Omega }}\\\ \\\ &=&3A\end{matrix}}}
{\displaystyle {\begin{matrix}\ I_{2}&=&I_{s}{\frac {R_{eq}}{R_{2}}}\\\ \\\ &=&(5A){\frac {1.2\Omega }{3\Omega }}\\\ \\\ &=&2A\end{matrix}}}
{\displaystyle {\begin{matrix}\ V_{s}=V_{1}=V_{2}\\\ \\\ V_{2}=I_{2}\times R_{2}\\\ \\\ =2\times 3\\\ \\\ =6Volts\end{matrix}}}
Let's take some time to Reflect on Material covered thus far. We have learned a great deal about simple resistive circuits and the possible connections they afford us. Here I think you'll want to remember:
{\displaystyle R_{E}=R_{1}+R_{2}\!}
Voltage Divider :
{\displaystyle V_{1}={\frac {R_{1}\times V_{s}}{R_{1}+R_{2}}}}
{\displaystyle R_{eq}={\frac {R_{1}\times R_{2}}{R_{1}+R_{2}}}}
The Current Divider Principle:
{\displaystyle I_{1}=I_{s}\times {\frac {R_{eq}}{R1}}}
Do Exercise 4 in Part 8. After being completely satisfied with your work, you can go on to the next page - for the quiz! Good luck :-)
Resistor Reduction
Given 2 Resistors
{\displaystyle R_{1}=R_{2}=5\Omega }
in parallel find the
{\displaystyle R_{eq}}
{\displaystyle R1=2\Omega }
{\displaystyle \Omega }
{\displaystyle \Omega }
in parallel and supply current as 15A, find
{\displaystyle R_{eq}}
{\displaystyle I_{1}}
{\displaystyle I_{2}}
{\displaystyle I_{3}}
and the supply voltage across these resistors.
{\displaystyle \Omega }
connected in series to a parallel branch of 3 resistors: R2 = 3
{\displaystyle \Omega }
{\displaystyle \Omega }
{\displaystyle \Omega }
Find: total resistance as seen by the voltage source.
Is it possible to effectively connect voltage sources in parallel? If so, what conditions must be met?
Doldham -- 75% & Corrected
Retrieved from "https://en.wikiversity.org/w/index.php?title=Electric_Circuit_Analysis/Resistors_in_Parallel&oldid=2222868" |
Connie and Nora went into Ready Scoop to get ice cream cones, but Nora can’t make up her mind. They have
23
flavors and she wants
3
scoops.
Nora is very particular about the order of the scoops. How many choices does she have if all of the scoops are different?
No repeats, but order DOES matter.
Nora changes her mind. She wants a dish (in which the order does not matter), not a cone, but she still wants three different flavors. How many ways can she order?
No repeats, but order DOES NOT matter.
Connie says, “I want a cone with dark chocolate on the bottom and then any other flavor(s) for the two scoops on top.” How many cones are possible with dark chocolate on the bottom?
Repeats are acceptable:
\frac{1}{bottom}\cdot\frac{22}{middle}\cdot\frac{22}{top}
Vlad came in as they were leaving and saw Connie’s cone. He said, “Oh, that’s what I want, a cone with chocolate on the bottom and then two other different flavors.” The clerk, said, “Okay, but we have four kinds of chocolate.” Vlad replied, “Any kind of chocolate will do.” How many different cones could fill Vlad’s order?
Remember that Vlad has four choices of chocolate for the bottom and not just one.
His next two choices will be different from the bottom and from each other. |
My pupil, Rothrock, now away in vacation, has sent me a brief abstract of his Observations on Houstonia cærula.1 The following are culled from them.
Long-styled: stigmatic hairs are in length —.04 mm.
Short-styled — " " .023"
wide Long-styled pollen .020. x .017
Short-styled " .036 x .02 :
in the fresh plants, but dry. Distended with water became round, by increase of the shorter axis alone, the long diameter unaltered.
Long-styled becoming .026
short-styled " .036.
(In Mitchella, however, the long diameter was shortened as the other enlarged.)
Pollen of Houstonia not wet, seen endwise is quite strongly 3-lobed. In water the reentering angles come out, and so increase the width.
Short-styled had the smaller stigmas, and the largest and best filled anthers.
The above results I overlooked enough to verify substantially. The following I did not. They may be taken as approximately good
A patch of long-styled—a good deal choked by other vegetation, yielded 34 capsules. The 6 largest of them gave 86 seeds,—average of 14
\frac{1}{3}
. Another patch of same better placed 6 best capsules gave 123 seeds average 21
\frac{1}{2}
A patch of short styled: out of 44 capsules, the six best gave gave an average of 15
\frac{1}{2}
each. Out of the 44, 4 were wholly abortive; 8 averaged 2 seeds each. Most of the rest gave about 6 seeds each.
The above was all from wild plants.
Transplanted specimens in the garden.
Long-styled, average no of seeds of best capsules —13
Short-styled —??
Of 13–short-styled pods
11 were wholly sterile
1 had 4 seeds
This was from specimens placed far off by themselves, so that they were not likely to be visited by insects which had left long-styled flowers.
Long-styled flowers fertilised artificially by short-styled pollen.: a patch in what proved to be a most unfavorable situation, much dwarfed. But 6 best capsules gave 77 seeds. In flowers of plants covered with fine netting (coarse gauze) a species of Thrips abounded, also a larva of some small beetle—completely dusted with pollen.2 In many flowers a species of Podura.
I have been looking at the flowers of Rhexia Virginica (I suppose you have examined no true Rhexia)3
Style declined to lower side of the flower. Stamens with their anthers also declined. The small subapical pore of, latter facing inwards. A little pressure on the base of the anther causes puffs of pollen to be blown out through the pore.
In one flower only have I observed the style change its position, in that it bent over towards the upper side of the flower;—accidental?4 A clump in water now 4 days: no stigma has been detected with pollen on it.5
As far as I can see, the likeliest way is that an insect approaching the flower from below, and searching the tube of calyx prolonged above the ovary into a cup (where if he finds any nectar he has sharper eyes than I have),—as he knocks his head against the enlarged bases of the anthers, will get puffs of pollen against his body, I should think on the sides of a humble bee &c. or abdomen underneath.— If he approaches the next flower from the front, below, he will brush against the subcapitate stigma.6
No Orchid examined since my last, except Gymnadenia tridentata,—on which I have a few obs.— It is a congener of Platanthera dilatata,—with the discs formed in two large shallow saucers occupying the whole breadth of the stigma; and the pollen pockets most readily, detached from the caudicle, some pulling off at a touch. It must be fertilised by a very fine proboscis.7
Several more Orchids will soon be sent me by sharp-sighted youngster up in Maine,—to whom I have just sent a copy of your book, to stimulate him.
My latest from you is July 14.8 It leaves me in a state of much anxiety for your boy.9 I will hope for a better account in your next.
I looked to-day for seeds of the little Houstonia for you—in vain. I told Rothrock to gather or save some & hope he has, but know not.— I am worth little now for any commission,—and—now that I am to set down to systematic work, shall be worth still less to you.
Ever Yours most cordially | Asa Gray
1.1 My pupil, … from them. 1.2] crossed blue crayon
1.8 Distended … .036. 1.11] closing square bracket, red crayon
1.12 (In … enlarged.)] closing square bracket, red crayon
5.1 A patch … each. 6.3] crossed red crayon
5.1 patch] underl red crayon
\frac{1}{3}
] underl red crayon
\frac{1}{2}
\frac{1}{2}
7.6 Of 13–short-styled … flowers. 9.11] enclosed in square brackets, red crayon
8.1 Long-styled … 77 seeds. 8.3] ‘6/77 6 — 17 12.8 sq 13 [illeg calculation] ’ ink
8.1 fertilised artificially] underl red crayon
8.1 short-styled] underl red crayon
8.1 pollen.:] ‘Long’ interl red crayon
8.2 6 best capsules gave 77 seeds. 8.3] underl red crayon
9.1 I have … on it. 11.3] crossed red crayon
12.1 the likeliest … stigma. 12.7] crossed red crayon
13.1 No Orchid … to you. 16.4] crossed ink
Top of first page: ‘Rhexia’ ink; ‘Houstonia’ blue crayon; ‘Rubiaceae’ pencil; ‘Rothrock’ pencil
The reference is to Joseph Trimble Rothrock. Gray had informed CD that Houstonia was dimorphic in October 1861, and had promised to look for any differences in the pollen of the two forms the following spring (see Correspondence vol. 9, letter from Asa Gray, 11 October 1861); he sent CD his own observations on differences between the pollen and stigmas of the two forms in his letter of [2 June 1862], and subsequently promised to send further observations by Rothrock (see letter from Asa Gray, 2–3 July 1862). CD included Rothrock’s observations and experimental results in Forms of flowers, pp. 132, 254.
In his letter to CD of 15 July [1862], Gray mentioned that Rothrock found only Thrips in Houstonia.
In the letter to Asa Gray, 16 February [1862], CD asked Gray to help him in his investigation of the possible occurrence of dimorphism in the Melastomataceae, by making observations of the floral anatomy of Rhexia. Gray promised to do so in the summer (see letter to Asa Gray, 16 February [1862], and letter from Asa Gray, 6 March [1862]), but experienced difficulties in obtaining specimens (see letter from Asa Gray, 2–3 July 1862). Meanwhile, CD informed Gray that he was making a number of crosses with a plant of Rhexia glandulosa from the Royal Botanic Gardens, Kew (see letter to Asa Gray, 10–20 June [1862] and n. 10, and letter to Asa Gray, 1 July [1862] and n. 13).
In the letter to Asa Gray, 15 March [1862], CD asked Gray to compare the position of the pistil in young and old Rhexia flowers, noting that in a related genus he had observed that the pistil and stamens changed position over time, and reporting his suspicion that one set of anthers was ‘adapted to pistil in early state, & the other set for it in its later state’.
CD had asked Gray to observe whether Rhexia could be fertilised if insects were excluded (see letter to Asa Gray, 21 April [1862] and n. 13). See also letters to Asa Gray, 10–20 June [1862] and 1 July [1862].
In the letter to Asa Gray, 15 March [1862], CD asked Gray to watch how the anthers and stigma touched bees that visited Rhexia flowers.
Gray had gathered flowers of Gymnadenia tridentata while on holiday in July (see letter from Asa Gray, 29 July 1862); for Gray’s further observations on the species, see the letter from Asa Gray, 18–19 August 1862.
Letter to Asa Gray, 14 July [1862].
Leonard Darwin was suffering from scarlet fever (see letter to Asa Gray, 14 July [1862]). |
Planetary gear train with two meshed planet gear sets - MATLAB - MathWorks í•œêµ
Double-Pinion Planetary Gear
Outer planet (Po) to inner planet (Pi) teeth ratio (NPo/NPi)
Sun-planet, ring-planet and planet-planet ordinary efficiencies
Planet-planet efficiency
Sun-carrier, ring-carrier, and planet-carrier power thresholds
Sun-carrier, ring-carrier, and planet-carrier viscous friction coefficients
Inner planet gear inertia
Outer planet gear inertia
Planetary gear train with two meshed planet gear sets
The Double-Pinion Planetary Gear block represents a planetary gear train with two meshed planet gear sets between the sun gear and the ring gear. A single carrier holds the two planet gear sets at different radii from the sun gear centerline, while allowing the individual gears to rotate with respect to each other. The gear model includes power losses due to friction between meshing gear teeth and viscous damping of the spinning gear shafts.
Structurally, the double-pinion planetary gear resembles a Ravigneaux gear without its second, large, sun gear. The inner planet gears mesh with the sun gear and the outer planet gears mesh with the ring gear. Because it contains two planet gear sets, the double-pinion planetary gear reverses the relative rotation directions of the ring and sun gears.
The teeth ratio of a meshed gear pair fixes the relative angular velocities of the two gears in that pair. In the Double-Pinion Planetary Gear block, you can specify the gear teeth ratios between the ring and sun gears and the outer planet and inner planet gears. A geometric constraint fixes the remaining teeth ratios—the ring gear to the outer planet gear and the inner planet gear to the sun gear. This constrains the ring gear radius to the sum of the sun gear radius and the inner and outer planet gear diameters.
{r}_{r}={r}_{s}+2â
{r}_{pi}+2â
{r}_{po},
rr is the ring gear radius.
rs is the sun gear radius.
rpi is the inner planet gear radius.
rpo is the outer planet gear radius.
The ring to outer planet gear teeth ratio is
\frac{{r}_{r}}{{r}_{po}}=2â
\frac{\frac{{r}_{r}}{{r}_{s}}}{\left(\frac{{r}_{r}}{{r}_{s}}â1\right)}â
\frac{\left(\frac{{r}_{po}}{{r}_{pi}}+1\right)}{\frac{{r}_{po}}{{r}_{pi}}}.
The inner planet-sun teeth ratio is
\frac{{r}_{pi}}{{r}_{s}}=\frac{\left(\frac{{r}_{r}}{{r}_{s}}â1\right)}{2\left(\frac{{r}_{po}}{{r}_{pi}}+1\right)}.
The Double-Pinion Planetary Gear block is a composite component. It contains three underlying blocks—Ring-Planet, Planet-Planet, and Sun-Planet—connected as shown in the figure. Each block connects to a separate drive shaft through a rotational conserving port.
Ring (R) to sun (S) teeth ratio (NR/NS) — Ring to sun gear rotation ratio
Fixed ratio of the ring gear to the sun gear rotations as defined by the number of planet gear teeth divided by the number of sun gear teeth.
Outer planet (Po) to inner planet (Pi) teeth ratio (NPo/NPi) — Outer planet to inner planet gear rotation ratio
Fixed ratio of the outer planet gear to the inner planet gear rotations as defined by the number of planet gear teeth divided by the number of sun gear teeth.
Sun-planet, ring-planet and planet-planet ordinary efficiencies — Torque transfer efficiencies for the
[.98, .98, .99] (default) | vector
Vector of torque transfer efficiencies, [ηSP, ηRP, ηRPP], for sun-planet, and ring-planet, and planet-planet gear wheel pair meshings, respectively. The vector elements must be in the interval (0,1].
Sun-planet efficiency — Torque transfer efficiency from the sun gear to the inner planet gears
[.75, .65, .6] (default) | vector
Vector of output-to-input power ratios that describe the power flow from the sun gear to the inner planet gears, ηSP. The block uses the values to construct a 1-D temperature-efficiency lookup table.
Ring-planet efficiency — Torque transfer efficiency from the ring gear to the outer planet gears
[.5, .45, .4] (default) | vector
Vector of output-to-input power ratios that describe the power flow from the ring gear to the outer planet gears, ηRP. The block uses the values to construct a 1-D temperature-efficiency lookup table.
Planet-planet efficiency — Torque transfer efficiency from the inner planet gears to the outer planet gears
Vector of output-to-input power ratios that describe the power flow from the inner planet gear to the outer planet gear, ηPP. The block uses the values to construct a 1-D temperature-efficiency lookup table.
Sun-carrier, ring-carrier, and planet-carrier power thresholds — Minimum efficiency power threshold for the sun-carrier, ring-carrier, and planet-carrier gear couplings
[.001, .001, .001] W (default) | vector
Vector of power thresholds above which the full efficiency factors apply. Enter the thresholds in the order sun-carrier, ring-carrier, planet-carrier.
The power threshold should be lower than the expected power transmitted during simulation. Higher values may cause the block to underestimate efficiency losses. Very low values may raise the computational cost.
To enable the parameter, set Friction model to Constant efficiency or Temperature-dependent efficiency.
Sun-carrier, ring-carrier, and planet-carrier viscous friction coefficients — Gear viscous friction
[0, 0, 0] N*m/(rad/s) (default) | vector
Vector of viscous friction coefficients, [μS μR μP], for the sun-carrier, ring-carrier, and planet-carrier gear motions, respectively.
Inner planet gear inertia — Inner planet gear inertia
Moment of inertia of the inner planet gears. This value must be positive.
To enable this paramter, set Inertia to On.
Outer planet gear inertia — Outer planet gear inertia
Moment of inertia of the outer planet gears. This value must be positive.
Tho enable this parameter, set Inertia to On.
Planetary Gear | Compound Planetary Gear | Ravigneaux Gear |
Locally collapsed $3$-manifolds
Locally collapsed
3
title = {Locally collapsed $3$-manifolds},
TI - Locally collapsed $3$-manifolds
Kleiner, Bruce; Lott, John. Locally collapsed $3$-manifolds, dans Local collapsing, orbifolds, and geometrization, Astérisque, no. 365 (2014), 93 p. http://archive.numdam.org/item/AST_2014__365__7_0/
[1] M. T. Anderson - "The
{L}^{2}
structure of moduli spaces of Einstein metrics on
4
-manifolds", Geom. Funct. Anal. 2 (1992), no. 1, p. 29-89. | Article | EuDML 58134 | MR 1143663 | Zbl 0768.53021
[2] L. Bessières, G. Besson, M. Boileau, S. Maillot& J. Porti - "Collapsing irreducible
3
-manifolds with nontrivial fundamental group", Invent. Math. 179 (2010), no. 2, p. 435-460. | Article | MR 2570121 | Zbl 1188.57013
[3] D. Burago, Y. Burago & S. Ivanov - A course in metric geometry, Grad. Stud. Math., vol. 33, Amer. Math. Soc., Providence, RI, 2001. | Article | MR 1835418 | Zbl 0981.51016
[4] Y. Burago, M. Gromov & G. Perel'Man - "A. D. Aleksandrov spaces with curvatures bounded below", Uspekhi Mat. Nauk 47 (1992), no. 2(284), p. 3-51, 222. | MR 1185284 | Zbl 0802.53018
[5] J. Cao & J. Ge - "A simple proof of Perelman's collapsing theorem for
3
-manifolds", J. Geom. Anal. 21 (2011), no. 4, p. 807-869. | Article | MR 2836584 | Zbl 1246.53052
[6] J. Cheeger - "Critical points of distance functions and applications to geometry", in Geometric topology: recent developments (Montecatini Terme, 1990), Lecture Notes in Math., vol. 1504, Springer, Berlin, 1991, p. 1-38. | MR 1168042 | Zbl 0771.53015
[7] J. Cheeger & D. Gromoll - "On the structure of complete manifolds of nonnegative curvature", Ann. of Math. (2) 96 (1972), p. 413-443. | Article | MR 309010 | Zbl 0246.53049
[8] J. Cheeger & M. Gromov - "Collapsing Riemannian manifolds while keeping their curvature bounded. I", J. Differential Geom. 23 (1986), no. 3, p. 309-346. | Article | MR 852159 | Zbl 0606.53028
[9] J. Cheeger & M. Gromov, "Collapsing Riemannian manifolds while keeping their curvature bounded. II", J. Differential Geom. 32 (1990), no. 1, p. 269-298. | Article | MR 1064875 | Zbl 0727.53043
[10] J. Cheeger & G. Tian - "Curvature and injectivity radius estimates for Einstein
4
-manifolds", J. Amer. Math. Soc. 19 (2006), no. 2, p. 487-525 (electronic). | Article | MR 2188134 | Zbl 1092.53034
[11] F. H. Clarke - Optimization and nonsmooth analysis, second ed., Classics Appl. Math., vol. 5, Soc. Ind. Appl. Math. (SIAM), Philadelphia, PA, 1990. | MR 1058436 | Zbl 0696.49002
[12] K. Fukaya - "Collapsing Riemannian manifolds to ones of lower dimensions", J. Differential Geom. 25 (1987), no. 1, p. 139-156. | Article | MR 873459 | Zbl 0606.53027
[13] M. Gromov - "Almost flat manifolds", J. Differential Geom. 13 (1978), no. 2, p. 231-241. | Article | MR 540942 | Zbl 0432.53020
[14] M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, Progr. Math., vol. 152, Birkhäuser Boston, Inc., Boston, MA, 1999. | MR 1699320 | Zbl 0953.53002
[15] K. Grove & P. Petersen, V - "Bounding homotopy types by geometry", Ann. of Math. (2) 128 (1988), no. 1, p. 195-206. | Article | MR 951512 | Zbl 0655.53032
[16] K. Grove & K. Shiohama - "A generalized sphere theorem", Ann. of Math. (2) 106 (1977), no. 2, p. 201-211. | Article | MR 500705 | Zbl 0341.53029
[17] R. S. Hamilton - "Three-manifolds with positive Ricci curvature", J. Differential Geom. 17 (1982), no. 2, p. 255-306. | Article | MR 664497 | Zbl 0504.53034
[18] R. S. Hamilton, "Four-manifolds with positive curvature operator", J. Differential Geom. 24 (1986), no. 2, p. 153-179. | Article | MR 862046 | Zbl 0628.53042
[19] V. Kapovitch - "Perelman's stability theorem", in Surveys in differential geometry. Vol. XI, Surv. Differ. Geom., vol. 11, Int. Press, Somerville, MA, 2007, p. 103-136. | Article | MR 2408265 | Zbl 1151.53038
[20] A. Katsuda "Gromov's convergence theorem and its application", Nagoya Math. J. 100 (1985), p. 11-48. | Article | MR 818156 | Zbl 0587.53043
[21] B. Kleiner & J. Lott "Notes on Perelman's papers", Geom. Topol. 12 (2008), no. 5, p. 2587-2855. | Article | MR 2460872 | Zbl 1204.53033
[22] S. Matveev - Algorithmic topology and classification of
3
-manifolds, Algorithms Comput. Math., vol. 9, Springer-Verlag, Berlin, 2003. | Article | MR 1997069 | Zbl 1048.57001
[23] J. Morgan & G. Tian - The geometrization conjecture, Clay Math. Monogr., vol. 5, Amer. Math. Soc./Clay Math. Inst., Providence, RI/Cambridge, MA, 2014. | MR 3186136 | Zbl 1302.53001
[24] G. Perelman - "Ricci Flow with Surgery on Three-Manifolds", http://arxiv.org/abs/math.DG/0303109. | Zbl 1130.53002
[25] G. Perelman, "Alexandrov's Spaces with Curvature Bounded from Below II", preprint, 1991. | Zbl 0802.53018
[26] P. Petersen - Riemannian geometry, second ed., Grad. Texts in Math., vol. 171, Springer, New York, 2006. | Article | MR 2243772 | Zbl 1220.53002
[27] C. C. Pugh - "Smoothing a topological manifold", Topology Appl. 124 (2002), no. 3, p. 487-503. | Article | MR 1930659 | Zbl 1026.57019
[28] E. R. Reifenberg - "Solution of the Plateau Problem for
m
-dimensional surfaces of varying topological type", Acta Math. 104 (1960) , p. 1-92. | Article | MR 114145 | Zbl 0099.08503
[29] T. Shioya & T. Yamaguchi - "Collapsing three-manifolds under a lower curvature bound", J. Differential Geom. 56 (2000), no. 1, p. 1-66. | Article | MR 1863020 | Zbl 1036.53028
[30] T. Shioya & T. Yamaguchi,"Volume collapsed three-manifolds with a lower curvature bound", Math. Ann. 333 (2005), no. 1, p. 131-155. | Article | MR 2169831 | Zbl 1087.53033
[31] M. Simon - "Ricci flow of almost non-negatively curved three manifolds", J. Reine Angew. Math. 630 (2009), p. 177-217. | MR 2526789 | Zbl 1165.53046
[32] J. Stallings "On fibering certain
3
-manifolds", in Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), Prentice-Hall, Englewood Cliffs, N.J. 1962, p. 95-100. | MR 158375 | Zbl 1246.57049
[33] T. Yamaguchi - "Collapsing and pinching under a lower curvature bound", Ann. of Math. (2) 133 (1991) , no. 2, p. 317-357. | Article | MR 1097241 | Zbl 0737.53041 |
To Asa Gray 21 April [1862]1
I am in your debt for two pleasant notes.2 First for business: I should greatly prefer your not returning any of Trübners remittance;3 but you really must not return more than half, as otherwise I shall have gained an immense advantage in having given away many copies of your Pamphlet, gratuitously.4 So add to all your kindness by letting matters remain as they are. I have settled with Trübner.— Trübner has not sold quite all; if copies are quite superfluity rich I shd. certainly like
\frac{1}{2}
a dozen or dozen to give away. I was asked for one but yesterday. I have never met one person who was not delighted with your writing.—
Secondly, in a week or 10 days I shall send
\frac{1}{2}
of my vol. on Orchids, as you desired;5 & other half will soon follow for they set up the whole, before they printed off a sheet.— I fear it can never be popular; but do not judge too severely by first half; for, if I do not deceive myself the two last chapters are better.—6 I believe I have been very foolish in publishing in popular form.—7 When I told Murray that I wanted clean sheets to send you; he thought of some arrangement for American republication, as he said Lyell’s new Book is to appear in America; but with my incomparably less important book it seems to me, as things now are, quite out of the question; but I have thought it as well just to mention what Murray said.—8
The North seems going on grandly victorious;9 & thank God there is distinct ground broken on the Slavery question;10 but we stupid English cannot yet believe that you will ever be a single Union again.—
I hope that you will ask your pupil to look carefully to gradation in sexes in your Hollies.—11 As far as I can yet judge, I am not only wrong, but diametrically wrong about Melastomas, or at least about some of them;12 if a Rhexia grew in a Garden, it would be good to cover up a plant under net & see if it seeded as well as uncovered plants.—13
Thanks for Mill’s pamphet, which is very good & I had not seen it;14 indeed I see hardly any Reviews or Periodicals.—
Hooker has been here for 3 days & we had lots of pleasant talk:15 I am always full of admiration & love for him: I wish he had not so tremendous & dry a job in hand, as the Genera Plantarum.—16
The year is established by the reference to the publication of Orchids (see n. 5, below).
See letters from Asa Gray, 6 March [1862] and 31 March [1862].
The reference is to the London publisher Nicholas Trübner (see letter from Asa Gray, 31 March [1862]). See also n. 4, below.
CD and Gray shared the cost of reprinting and publishing as a pamphlet Gray’s reviews of Origin from the Atlantic Monthly (A. Gray 1861). CD privately distributed over 100 copies of the pamphlet. Trübner acted as London agent for the sale of the remainder of the pamphlets sent to Britain. See Correspondence vol. 9, especially Appendix III.
Gray had asked CD to send him the sheets of Orchids as soon as they were printed because he wished to write an early review (see Correspondence vol. 9, letter from Asa Gray, 31 December 1861). Orchids was published on 15 May 1862 (Freeman 1977, p. 112).
The penultimate chapter of Orchids, chapter 6, was devoted largely to the Catasetidae, described by CD as ‘the most remarkable of all Orchids’ (Orchids, p. 211). The final chapter discussed homologies of orchid flowers and the diversity of flower structure, and presented CD’s general conclusions about the functions of the various contrivances for pollination.
CD originally intended to publish his account of orchid adaptations as a paper in the Proceedings or Transactions of the Linnean Society of London (see Correspondence vol. 9, letter to John Murray, 21 September [1861]).
CD refers to his publisher, John Murray, who also published Charles Lyell’s Antiquity of man (C. Lyell 1863a). Lyell’s book was first published in America in 1863 (NUC); an American edition of Orchids did not appear until 1877 (Freeman 1977, p. 113).
The Union army secured a number of victories in the American Civil War during the spring of 1862—at Fort Henry and Fort Donelson, Tennessee, in February, Pea Ridge, Arkansas, in March, and Shiloh, Tennessee, in April (McPherson 1988, pp. 392–414).
On 6 March 1862, President Abraham Lincoln asked the United States Congress to support a resolution offering Federal compensation for voluntary emancipation of slaves. Congress adopted Lincoln’s resolution on 10 April 1862 (Curry 1968).
In the letter to Asa Gray, 17 September [1861] (Correspondence vol. 9), CD asked whether, unlike the ‘English Holly (& all the cultivated vars.)’, any of the American species of holly showed signs of gradation indicating ‘the steps by which it became dioicous’. In reply, Gray sent some observations, but reported that further observations would be necessary (see ibid., letter from Asa Gray, 11 October 1861, and letter to Asa Gray, [after 11 October 1861]). CD refers to Gray’s ‘zealous pupil’, probably Joseph Trimble Rothrock (see letter from Asa Gray, 6 March [1862] and n. 6).
CD had asked Gray to carry out some observations on Rhexia in the letters to Asa Gray, 16 February [1862] and 15 March [1862].
Mill 1862. See letter from Asa Gray, 6 March [1862] and n. 8.
Joseph Dalton Hooker was a guest at Down House from 17 to 21 April 1862 (Emma Darwin’s diary (DAR 242)).
Bentham and Hooker 1862–83.
Curry, Leonard P. 1968. Blueprint for modern America: nonmilitary legislation of the first Civil War congress. Nashville: Vanderbilt University Press.
NUC: The national union catalog. Pre-1956 imprints. 685 vols. and supplement (69 vols.). London and Chicago: Mansell. 1968–81.
Is sending first half of orchid book.
Feels he is wrong about Melastoma. |
UniformRandomVariable - Maple Help
Home : Support : Online Help : Education : Student Packages : Statistics : Random Variable Distributions : UniformRandomVariable
uniform (rectangular) random variable
UniformRandomVariable(a, b)
The uniform distribution is a continuous probability random variable with probability density function given by:
f\left(t\right)={\begin{array}{cc}0& t<a\\ \frac{1}{b-a}& t<b\\ 0& \mathrm{otherwise}\end{array}
a<b
\mathrm{with}\left(\mathrm{Student}[\mathrm{Statistics}]\right):
X≔\mathrm{UniformRandomVariable}\left(a,b\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}{a}\\ \frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{a}}& \textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{<}\textcolor[rgb]{0,0,1}{b}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{\mathrm{otherwise}}\end{array}
\mathrm{PDF}\left(X,0.5\right)
{\begin{array}{cc}\textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{0.5}\textcolor[rgb]{0,0,1}{<}\textcolor[rgb]{0,0,1}{a}\\ \frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1.}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{a}}& \textcolor[rgb]{0,0,1}{0.5}\textcolor[rgb]{0,0,1}{<}\textcolor[rgb]{0,0,1}{b}\\ \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{\mathrm{otherwise}}\end{array}
\mathrm{Mean}\left(X\right)
\frac{\textcolor[rgb]{0,0,1}{a}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{b}}{\textcolor[rgb]{0,0,1}{2}}
\mathrm{Variance}\left(X\right)
\frac{{\left(\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{a}\right)}^{\textcolor[rgb]{0,0,1}{2}}}{\textcolor[rgb]{0,0,1}{12}}
Y≔\mathrm{UniformRandomVariable}\left(-1,1\right):
\mathrm{PDF}\left(Y,x,\mathrm{output}=\mathrm{plot}\right)
\mathrm{CDF}\left(Y,x\right)
{\begin{array}{cc}\textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{<}\textcolor[rgb]{0,0,1}{-1}\\ \frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{x}}{\textcolor[rgb]{0,0,1}{2}}& \textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{<}\textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{\mathrm{otherwise}}\end{array}
\mathrm{CDF}\left(Y,0.5,\mathrm{output}=\mathrm{plot}\right)
The Student[Statistics][UniformRandomVariable] command was introduced in Maple 18. |
OMTEX CLASSES (k): Chapter 1 - Relations And Functions NCERT Solutions for Class 12 Science Math Exercise No. 1.4
Chapter 1 - Relations And Functions NCERT Solutions for Class 12 Science Math Exercise No. 1.4
Question 1:Determine whether or not each of the definition of given below gives a binary operation. In the event that * is not a binary operation, give justification for this.(i) On Z+, define * by a * b = a − b(ii) On Z+, define * by a * b = ab(iii) On R, define * by a * b = ab2(iv) On Z+, define * by a * b = |a − b|(v) On Z+, define * by a * b = a
Question 2:For each binary operation * defined below, determine whether * is commutative or associative.(i) On Z, define a * b = a − b(ii) On Q, define a * b = ab + 1(iii) On Q, define a * b (iv) On Z+, define a * b = 2ab(v) On Z+, define a * b = ab(vi) On R − {−1}, define
Question 3:Consider the binary operation ∨ on the set {1, 2, 3, 4, 5} defined by a ∨b = min {a, b}. Write the operation table of the operation∨.
Question 4:Consider a binary operation * on the set {1, 2, 3, 4, 5} given by the following multiplication table. (i) Compute (2 * 3) * 4 and 2 * (3 * 4)(ii) Is * commutative?(iii) Compute (2 * 3) * (4 * 5).(Hint: use the following table)
Question 5:Let*′ be the binary operation on the set {1, 2, 3, 4, 5} defined by a *′ b = H.C.F. of a and b. Is the operation *′ same as the operation * defined in Exercise 4 above? Justify your answer.
Question 6:Let * be the binary operation on N given by a * b = L.C.M. of a and b. Find(i) 5 * 7, 20 * 16 (ii) Is * commutative?(iii) Is * associative? (iv) Find the identity of * in N(v) Which elements of N are invertible for the operation *?
Question 7:Is * defined on the set {1, 2, 3, 4, 5} by a * b = L.C.M. of a and b a binary operation? Justify your answer.
Question 8:Let * be the binary operation on N defined by a * b = H.C.F. of a and b. Is * commutative? Is * associative? Does there exist identity for this binary operation on N?
Question 9:Let * be a binary operation on the set Q of rational numbers as follows:(i) a * b = a − b (ii) a * b = a2 + b2(iii) a * b = a + ab (iv) a * b = (a − b)2(v) (vi) a * b = ab2Find which of the binary operations are commutative and which are associative.
{}^{}{}^{}{}^{}{}^{}
{}^{}{}^{}{}^{}{}^{}
Question 10:Find which of the operations given above has identity.
\frac{}{}
\phantom{\rule{0ex}{0ex}}\frac{}{}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}
Question 11:Let A = N × N and * be the binary operation on A defined by(a, b) * (c, d) = (a + c, b + d)Show that * is commutative and associative. Find the identity element for * on A, if any.
State whether the following statements are true or false. Justify.(i) For an arbitrary binary operation * on a set N, a * a = a a * N.(ii) If * is a commutative binary operation on N, then a * (b * c) = (c * b) * a
Question 13:Consider a binary operation * on N defined as a * b = a3 + b3. Choose the correct answer.(A) Is * both associative and commutative?(B) Is * commutative but not associative?(C) Is * associative but not commutative?(D) Is * neither commutative nor associative?
லேபிள்கள்: Chapter 1 - Relations And Functions NCERT Solutions for Class 12 Science Math Exercise No. 1.4 |
Equivalent airspeed - Wikipedia
Find sources: "Equivalent airspeed" – news · newspapers · books · scholar · JSTOR (January 2009) (Learn how and when to remove this template message)
Equivalent airspeed (EAS) is calibrated airspeed (CAS) corrected for the compressibility of air at a non-trivial Mach number. It is also the airspeed at sea level in the International Standard Atmosphere at which the dynamic pressure is the same as the dynamic pressure at the true airspeed (TAS) and altitude at which the aircraft is flying.[1][2] In low-speed flight, it is the speed which would be shown by an airspeed indicator with zero error.[3] It is useful for predicting aircraft handling, aerodynamic loads, stalling etc.
{\displaystyle EAS=TAS\times {\sqrt {\frac {\rho }{\rho _{0}}}}}
{\displaystyle \rho \,}
is actual air density.
{\displaystyle \rho _{0}\,}
is standard sea level density (1.225 kg/m3 or 0.00237 slug/ft3).
EAS is a function of dynamic pressure.
{\displaystyle EAS={\sqrt {\frac {2q}{\rho _{0}}}}}
{\displaystyle {q}\,}
is dynamic pressure
{\displaystyle q={\tfrac {1}{2}}\,\rho \,v^{2},}
EAS can also be obtained from the aircraft Mach number and static pressure.
{\displaystyle EAS={a_{0}}M{\sqrt {P \over P_{0}}}}
{\displaystyle {a_{0}}\,}
is 1,225 km/h (661.45 kn), the standard speed of sound at 15 °C
{\displaystyle M\,}
is Mach number
{\displaystyle P\,}
is static pressure
{\displaystyle P_{0}\,}
is standard sea level pressure (1013.25 hPa)
Combining the above with the expression for Mach number gives EAS as a function of impact pressure and static pressure (valid for subsonic flow):
{\displaystyle EAS={a_{0}}{\sqrt {{5P \over P_{0}}\left[\left({\frac {q_{c}}{P}}+1\right)^{\frac {2}{7}}-1\right]}}}
{\displaystyle {q_{c}}\,}
is impact pressure.
At standard sea level, EAS is the same as calibrated airspeed (CAS) and true airspeed (TAS). At any other altitude, EAS may be obtained from CAS by correcting for compressibility error.
The following simplified formula allows calculation of CAS from EAS:
{\displaystyle CAS={EAS\times \left[1+{\frac {1}{8}}(1-\delta )M^{2}+{\frac {3}{640}}(1-10\delta +9\delta ^{2})M^{4}\right]}}
pressure ratio:
{\displaystyle \delta ={\frac {P}{P_{0}}}}
{\displaystyle CAS\,}
{\displaystyle EAS\,}
are airspeeds and can be measured in knots, km/h, mph or any other appropriate unit.
The above formula is accurate within 1% up to Mach 1.2 and useful with acceptable error up to Mach 1.5. The 4th order Mach term can be neglected for speeds below Mach 0.85.
^ Clancy, L.J. (1975), Aerodynamics, Section 3.8, Pitman Publishing Limited, London. ISBN 0-273-01120-0
^ Anderson, John D. (2007), Fundamentals of Aerodynamics, p.215 (4th edition), McGraw-Hill, New York USA. ISBN 978-0-07-295046-5
^ Houghton, E.L. and Carpenter, P.W. (1993), Aerodynamics for Engineering Students, Section 2.3.3, Butterworth-Heinemann, Oxford UK. ISBN 0-340-54847-9
Anderson, John D. (2007), Fundamentals of Aerodynamics, Section 3.4 (4th edition), McGraw-Hill, New York USA. ISBN 978-0-07-295046-5
Equivalent airspeed calculator
Retrieved from "https://en.wikipedia.org/w/index.php?title=Equivalent_airspeed&oldid=923202685" |
Options Guide - Delta Exchange - User Guide
Options are a class of derivtive contracts that give a buyer a right to buy or sell an underlying asset at a specified price prior to or on a specified date. The seller of the options contract has the corresponding obligation to fulfill the transaction (i.e. to sell or buy) if the buyer “exercises” the option.
At Delta Exchange, we offer the following three categories of options contracts:
In this category we offer call and put options on BTC, ETH, BNB, LINK, XRP, LTC, BCH, SOL and ADA. All these are European options (i.e. they can only be exercised on expiry) and are available for a multitude of strikes and expiry dates.
MOVE options are a direct way to speculate on the volatility of the underlying assets. A MOVE contract is essentially a straddle, i.e. an at-the-money call & put option pair. Because of this, the price of a MOVE contract is proportional to the price swings in the underlying, instead of the direction of underlying’s price movement. More details on MOVE options are available here.
Turbo Options (deprecated)
Turbo options are exotic options in which a knockout barrier is attached to vanilla call/ put options. Just like a call option, a Turbo call option increases in value when price of the underlying goes up. And, like put options, a Turbo put option increase in value when price of the underlying goes down. It is the knockout barrier which is differentiates Turbo options from vanilla options. Vanilla options have a fixed expiry date, whereas a Turbo option may expire (or get knocked out) before its expiry date if the underlying’s price touches the knockout barrier.More details on Turbo options are available here.
Options Symbology
The symbols of all options contracts on Delta Exchange are based on the following scheme:
ProductSymbol-UnderlyingSymbol-StrikePrice-ExpiryDate
ProductSymbol: specifies which product category a given contract belongs to. Currently, it can take the following values:
ProductSymbol
Trubo Put
UnderlyingSymbol: is the symbol of the underlying asset of the options contract. Currently, we offer options on BTC, ETH, BNB, LINK, XRP, LTC, BCH, SOL and ADA.
Strike Price: is the strike price of the option.
ExpiryDate: is the date in ddmmyy format at which the option expires. On Delta, all options expire at 12pm UTC.
C-BTC-50000-200821: this is the symbol for a BTC call option which has a strike price of 50000 and expires on 20th August 2021
MV-BNB-200-300421: this is the symbol for a BNB contract which has a strike price of 200 and expires on 30th April 2021
Open positions in option contracts are marked at fair mark price. The fair mark price is computed by averaging the bid and offer price from the order book for a pre-specified order size, aka impact size. The mark price thus obtained is constrained within a band defined by the risk engine of Delta Exchange. The risk engine maintains a proprietary model for implied volatility (IV). The fair mark price band is this computed as:
Fair\ Mark\ Price\ Min = Black Scholes Price (IV: model\ IV - 25\%)
Fair\ Mark\ Price\ Max = Black Scholes Price (IV: model\ IV + 25\%)
If the fair mark price computed from the order book lies outside the mark price band, it will be capped at either Fair Price Min or Fair Price Max, whichever is relevant in the situation. The fair mark price band is enforced to prevent manipulation of mark price.
Please note that mark price does not impact realised profit/loss. When you close an open position, a trade happens by matching your close order against orders in the order book. The execution price of this trade determines your realised profit/ loss. However, mark price is used for the calculation of unrealised profit/ loss and consequently is used for decisions on liquidation of short positions.
All options contract settle at 12pm UTC. The settlement price of an options contract is computed using its strike price and the 30 minute TWAP (Time Weighted Average Price) of the index (i.e. spot) price of the underlying asset. The settlement price is determined using the following formulae:
Settlement\ price = max (30minTWAP(\ index\ price) - strike\ price,0)
Settlement\ price = max (strike\ price - 30minTWAP(\ index\ price),0)
At expiry, all open positions are closed at the settlement price. Settlement prices of expired contracts are available on this page.
Options Expiry and Launch Schedule
We offer daily, weekly (current week and the next week), monthy (current month and the next month) and quarterly (current quarter and the next quarter) options. Please note that for some of the underlying, we currently offer only daily options.
Daily options are lauched at 10am UTC everyday. For other maturities, options with new expiries are launched on Fridays at 10am UTC. For existing maturities, options with new strike prices are launched as the underlying index price moves.
Turbo Options |
Logical quantifier - Simple English Wikipedia, the free encyclopedia
In logic, a quantifier is a way to state that a certain number of elements fulfill some criteria. For example, every natural number has another natural number larger than it. In this example, the word "every" is a quantifier. Therefore, the sentence "every natural number has another natural number larger than it" is a quantified expression.
Quantifiers and quantified expressions are a useful part of formal languages. They are useful because they let rigorous statements claim how widespread a criteria is. Two basic kinds of quantifiers used in predicate logic are universal and existential quantifiers. A universal quantifier states that all the elements considered fulfill the criteria. The universal quantifier is symbolized with "∀", an upside down "A", to stand for "all". An existence quantifier (symbolized with "∃") states that at least one element considered fits the criteria. The existential quantifier is symbolized with "∃", a backwards "E", to stand for "exists".[1][2][3]
Quantifiers are also used in natural languages. Examples of quantifiers in English include for all, for some, many, few, a lot, and no.
This statement is infinitely long:
This is a problem for formal languages, since a formal statement needs to be finite in length. These problems can be avoided by using universal quantification. This results in the following compact statement:
In the same way, we can shorten an infinite sequence of statements joined by or such as:
by rewriting it using existential quantification:
For at least one natural number n, n is equal to 5+5.
The two quantifiers most widely used are the universal quantifier and the existence quantifier.
The universal quantifier is used to claim that for elements in a set, the elements all match some criteria. Usually, this statement "for all elements" is shortened to an "A" flipped upside down, which is "∀".[1][2][3]
The existential quantifier is used to claim that for elements in a set, there exists at least one element that matches some criteria. Usually, this statement "there exists an element" is shortened to an "E" flipped upside down, which is "∃".[1][2][3]
English statements can be often rewritten using symbols, predicates representing criteria, and quantifiers. One example is "Each of Peter's friends either likes to dance or likes to go to the beach". Let X be the set of all Peter's friends. Let P(x) be the predicate "x likes to dance". Let Q(x) be the predicate "x likes to go to the beach". We can rewrite the example using formal notation as
{\displaystyle \forall {x}{\in }X,P(x)\lor Q(x)}
. The statement can be read as "For every x that is a member of X, P applies to x or Q applies to x."
There are other ways to use quantifiers in formal language. Each of the following statements below says the same thing as
{\displaystyle \exists {x}{\in }X,P(x)}
{\displaystyle \exists {x}P}
{\displaystyle (\exists {x})P}
{\displaystyle (\exists x\ .\ P)}
{\displaystyle \exists x\ \cdot \ P}
{\displaystyle (\exists x:P)}
{\displaystyle \exists {x}{\in }X\,P}
{\displaystyle \exists \,x{:}X\,P}
There are a few more ways to represent the universal quantifier:
{\displaystyle (x)\,P}
{\displaystyle \bigwedge _{x}P}
Several statements above explicitly include X, the set of elements that the quantifier applies to. This set of elements is also known as the range of quantification, or the universe of discourse. Some of the statements above do not include such a set. In this case, the set will have to be specified before the statement. For example, "x is an apple" must be stated before
{\displaystyle \exists {x}P(x)}
. In this case, we are making a statement that at least one apple fits the predicate P.
Using quantifiers formally does not require using the symbol x. In fact, other symbols, such as y and z, can also be used. However, one must be careful not to use the same symbol to refer to two different things.
Nesting[change | change source]
It is important to put quantifiers in the right order. Below is an example of an English sentence showing how meaning changes with order:
This statement is true. It states that every natural number has a square. However, if we turn the order of the quantifiers around:
There exists a natural number s, such that for every natural number n, s = n2.
then the statement is false. It claims that there is one natural number s that is the square of every natural number.
In certain circumstances, changing the order of quantifiers does not change the meaning of the statement. For instance:
There exists a natural number x, and there exists a natural number y such that x = y2.
Other quantifiers[change | change source]
There are also less common quantifiers used by mathematicians.
An example is the solution quantifier.[4] It is used to state which elements solve a particular equation. The solution quantifier is represented by a § (section sign). For example, the following statement claims the squares of 0, 1 and 2 are smaller than 4. :
{\displaystyle \left[\S n\in \mathbb {N} \quad n^{2}\leq 4\right]=\left\{0,1,2\right\}}
Other quantifiers include:
There are many elements such that...
There are few elements such that...
Term logic was developed by Aristotle. It was an early form of logic, and included quantification. The use of quantification was closer to that of natural language. This meant that statements in term logic with quantifiers were less suited for formal analysis. Term logic included quantifiers for All, Some and No (none) in 4th century BC.
In 1879, Gottlob Frege created a notation for universal quantification. Unlike the modern notation, he would represent a universal quantification by writing a variable over a dimple in an otherwise straight line. Frege did not create a notation for existential quantification. Instead, he combined universal quantification and a number of negations to make an equivalent statement. Frege's use of quantification was not widely known until Bertrand Russell's 1903 Principles of Mathematics.
In 1885, Charles Sanders Peirce and his student Oscar Howard Mitchell also created a notation for universal and existential quantifiers. They wrote Πx and Σx where we now write ∀x and ∃x. Pierce's notation was used by many mathematicians into the 1950s.
In 1897, William Ernest Johnson and Giuseppe Peano created another notation for universal and existential quantification. They were influenced by Pierce's previous quantification notation. Johnson and Peano used the simple (x) for universal quantification, and ∃x for existential quantification. Peano's influence on math spread this notation across Europe.
In 1935, Gerhard Gentzen created the ∀ symbol for universal quantification. It was not widely used until the 1960s.
The Simple English Wiktionary has a definition for: quantification.
↑ 1.0 1.1 1.2 "Comprehensive List of Logic Symbols". Math Vault. 2020-04-06. Retrieved 2020-09-04.
↑ 2.0 2.1 2.2 "Predicates and Quantifiers". www.csm.ornl.gov. Retrieved 2020-09-04. {{cite web}}: CS1 maint: url-status (link)
↑ 3.0 3.1 3.2 "1.2 Quantifiers". www.whitman.edu. Retrieved 2020-09-04.
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Logical_quantifier&oldid=7415771" |
Incentive Pendulum - THORChain Docs
THORChain's Incentive Pendulum keeps the network in a balanced state.
The capital on THORChain can lose its balance over time. Sometimes there will be too much capital in liquidity pools; sometimes there will be too much bonded by nodes. If there is too much capital in liquidity pools, the network is unsafe. If there is too much capital bonded by nodes, the network is inefficient.
If the network becomes unsafe, it increases rewards (block rewards and liquidity fees) for node operators and reduces rewards for liquidity providers. If the network becomes inefficient, it boosts rewards for liquidity providers and reduces rewards for node operators.
Balancing System States
THORChain can be in 1 of 5 main states—
Under-Bonded
Over-Bonded
These different states can be seen in the relationship between bonded Rune and pooled Rune. The amount of Rune which has been bonded by node operators, and the amount which has been added to liquidity pools by liquidity providers.
In the optimal state, bonded capital is roughly equal to pooled capital. Bonded capital is 100% Rune; pooled capital half Rune and half external assets.
67% of Rune in the system is bonded and 33% is pooled. This is the desired state. The system makes no changes to the incentives for node operators or liquidity providers.
The system may become unsafe. In this case, pooled capital is higher than bonded capital. Pooled Rune is now equal to bonded Rune – a 50/50 split.
This is undesirable because it means that it's become profitable for node operators to work together to steal assets.
To fix this, the system increases the amount of rewards going to node operators and lowers the rewards going to liquidity providers. This leads to more node operators getting involved, bonding more Rune and increasing bonded capital. This also disincentivises liquidity providers from taking part. They receive less return on their investment, so they pull assets out and reduce the amount of pooled capital. With time, this restores balance and the system moves back towards the optimal state.
Inefficient State
The system can also become inefficient. In this case, pooled capital would be much lower in value than bonded capital. This is a problem because it means that much more capital is being put into securing pooled assets than those assets are actually worth.
To fix this, the system increases rewards for liquidity providers and decreases rewards for node operators. This attracts more liquidity providers to the system, and fewer node operators. Liquidity providers add more capital to receive more rewards, increasing pooled capital. Some node operators remove their bonded Rune, seeking more profitable places to put their capital. Bonded capital falls. In this way, the system returns to the optimal state.
Under and Over-Bonded States
The under- and over-bonded states are less severe intermediary states. Being under-bonded is not a threat in itself because it is not yet profitable for node operators to steal. Being over-bonded is not a problem in itself because the system is still operating quite well.
The THORChain team does not expect the unsafe or inefficient states to come up often. The system will be in the over-bonded state most of the time, particularly as it gets easier for people to run nodes.
Try this interactive model of the Incentive Pendulum.
The algorithm that controls the Incentive Pendulum is as follows:
shareFactor = \frac{b - s}{b + s / ic}
b = totalBonded, s = totalPooled, ic = Incentive Curve
In a stable state of 67m RUNE bonded and 33m RUNE pooled:
shareFactor = \frac{67 - 33}{67 + 33/1} = 0.33
Thus, 33% of the rewards go to Liquidity Providers and 67% go to Node Operators.
In the under-bonded state of 60m RUNE bonded and 40m RUNE pooled
shareFactor = \frac{60 - 40}{60 + 40/1} = 0.2
Thus, 20% of the rewards go to Liquidity Providers and 80% to the Node Operators, incentivising more node operators to be created.
In a very under-bonded state of 55m RUNE bonded and 45m RUNE pooled
shareFactor = \frac{55 - 45}{55 + 45/1} = 0.10
Thus, Liquidity providers will receive 10% of the rewards. This may drive Liquidity providers away.
In an over-bonded state of 80m RUNE bonded and 20m RUNE pooled:
shareFactor = \frac{80 - 20}{80 + 20/1} = 0.6
Incentive Curve
In the stable state of 67m RUNE bonded and 33m RUNE pooled, Liquidity providers are providing half the assets TVL (RUNE and Assets) and bonders are providing the other half.
The Incentive Curve can be used to adjust the reward flow with respect to the Incentive Pendulum.
In a stable situation with the Incentive Curve set to 1.
shareFactor = \frac{67 - 33}{67 + 33/1} = 0.33
shareFactor = \frac{67 - 33}{67 + 33/2} = 0.4
shareFactor = \frac{67 - 33}{67 + 33/4} = 0.45
In a stable situation with the Incentive Curve set to 100.
shareFactor = \frac{67 - 33}{67 + 33/100} = 0.5
The Incentive Pendulum is a linear path between Liquidity providers and Node Operators. The Incentive Curve can smooth the effect of the Incentive Pendulum as it moves. With the Incentive Pendulum in a set position, as Incentive Curve increases so do the rewards that are paid to Liquidity providers.
In a stable situation, it can be set to 100 to ensure Liquidity providers and Node Operator's are rewarded equally.
In an under-bond state, it can be increased so Liquidity providers are not adversely affected by the reduction of rewards due to the Incentive Pendulum.
In an over-bonded state, it can be reduced to ensure Liquidity Providers are not being paid too much.
Incentive Curve value set in constants however can be overridden in Mimir by Node Operators.
Driving Capital Allocation
As a by-product of the Incentive Pendulum's aggressive re-targeting of 67:33 split of BONDED:POOLED RUNE, it means that in an equilibrium, the value of BONDED RUNE will always be double the value of POOLED RUNE. Since POOLED RUNE is 1:1 bonded with POOLED Capital (due to liquidity pools), then the total market value of RUNE is targeted to be 3 times the value of pooled assets.
If there is any disruption to this, then it means capital will be re-allocated by Nodes and Liquidity providers to pursue maximum yield, and thus correct the imbalance. |
Revision as of 12:07, 1 May 2017 by C207 (talk | contribs)
{\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 .\,\!} |
Write (and Run) Python in Blog - Allen's Whiteboard
\require{sansmath}
- matplotlib - pandas - numpy - scipy - statsmodels
Write (and Run) Python in Blog
Trying out PyScript, a new tool that embeds (runnable) Python code inside HTML. Cool huh? Even better, it’s developed and maintained by Anaconda team which makes it sound more promising and trustworthy, at least in the long run.
import numpy as np from types import SimpleNamespace import scipy.stats as ss import statsmodels.api as sm import matplotlib.pyplot as plt n = 1000 u = ss.pareto.rvs(1.5, size=n) v = ss.norm.rvs(0, 1, size=n) x = np.random.randn(n) y = x * 12.5 + .25 + u + v reg = sm.OLS(y, sm.add_constant(x.reshape(-1, 1))).fit() x_fit = np.linspace(x.min(), x.max(), 10) y_fit = sm.add_constant(x_fit.reshape(-1, 1)) @ reg.params colors = dict(black=(0, 0, 0), red=(187, 34, 34), blue=(51, 102, 153)) for c, v in colors.items(): colors[c] = tuple(i / 255 for i in v) colors = SimpleNamespace(**colors) fig, (ax1, ax2) = plt.subplots(figsize=(8, 4), ncols=2) ax1.scatter(x, y, s=1, c=[colors.black]) ax1.plot(x_fit, y_fit, c=colors.red) ax2.hist(reg.resid, fc=colors.blue, density=True, bins=30, ec=colors.black) plt.tight_layout() fig
Charts above are randomly generated when this page loads, by scripts below. Specificaly, what’s shown on the left is OLS fitting on a Pareto/Normal mixture model, and on the right the residual distribution.
u = ss.pareto.rvs(1.5, size=n)
v = ss.norm.rvs(0, 1, size=n)
y = x * 12.5 + .25 + u + v
reg = sm.OLS(y, sm.add_constant(x.reshape(-1, 1))).fit()
x_fit = np.linspace(x.min(), x.max(), 10)
y_fit = sm.add_constant(x_fit.reshape(-1, 1)) @ reg.params
colors = dict(black=(0, 0, 0), red=(187, 34, 34), blue=(51, 102, 153))
for c, v in colors.items(): colors[c] = tuple(i / 255 for i in v)
colors = SimpleNamespace(**colors)
ax1.scatter(x, y, s=1, c=[colors.black])
ax1.plot(x_fit, y_fit, c=colors.red)
ax2.hist(reg.resid, fc=colors.blue, density=True, bins=30, ec=colors.black)
As you may have noticed, I’m taking an extra effort to define colors as RGB tuples in my custom namespace, and the charts lack anything textual - axis labels, titles, legends. This is because whenever I try to use single or double quotation marks, there comes a SyntaxError that prevents the code from proceeding. Also, it seems to be the case that codes inside <py-script> tags do not support empty lines, which is kinda important (and maybe not so). Unfortunately as PyScript is still an experimental repo, I haven’t seen anyone posting about these on their issues page. Maybe I should be the one.
← Two Pretty Shots |
What is APY? - Bird Docs
Annual Percentage Yield as known as APY gives you an estimate of how much your money would earn in a year.
APY standardizes the rate of return. It does this by stating the real percentage of growth that will be earned in compound interest assuming that the money is deposited for one year.
Formula for APY:
APY = (1+r/n)^n-1
For example, if you deposited $100 for one year at 5% interest and your deposit was compounded quarterly, at the end of the year you would have $105.09. If you had been paid simple interest, you would have had $105.
The APY would be (1 + .05/4)4 - 1 = .05095 = 5.095%.
Now for Lending - Earn APY, it frequently changes because it depends on the number and amount of borrowers using our Lending platform.
Net APY Calculation:
1. Convert all supplied and borrowed asset amounts to a single asset (like USD or ETH).
2. Calculate the sum of (suppliedAmount x supplyApyAsDecimal - borrowedAmount x borrowApyAsDecimal) for all underlying assets.
3. If the calculated sum from the previous step is > 0, then
Net APY = 100 (sum / totalSuppliedValue)
If the calculation from the previous step is < 0, then
Net APY = 100 (sum / totalBorrowedValue)
If the calculation from the previous step is 0, then Net APY = 0. |
Two extensions of Ramsey’s theorem
1 December 2013 Two extensions of Ramsey’s theorem
David Conlon, Jacob Fox, Benny Sudakov
Ramsey’s theorem, in the version of Erdős and Szekeres, states that every
2
-coloring of the edges of the complete graph on
\left\{1,2,\dots ,n\right\}
contains a monochromatic clique of order
\left(1/2\right)logn
. In this article, we consider two well-studied extensions of Ramsey’s theorem. Improving a result of Rödl, we show that there is a constant
c>0
2
\left\{2,3,\dots ,n\right\}
contains a monochromatic clique
S
for which the sum of
1/logi
over all vertices
i\in S
clogloglogn
. This is tight up to the constant factor
c
and answers a question of Erdős from 1981. Motivated by a problem in model theory, Väänänen asked whether for every
k
there is an
such that the following holds: for every permutation
\pi
1,\dots ,k-1
2
\left\{1,2,\dots ,n\right\}
{a}_{1}<\cdots <{a}_{k}
{a}_{\pi \left(1\right)+1}-{a}_{\pi \left(1\right)}>{a}_{\pi \left(2\right)+1}-{a}_{\pi \left(2\right)}>\cdots >{a}_{\pi \left(k-1\right)+1}-{a}_{\pi \left(k-1\right)}.
That is, not only do we want a monochromatic clique, but the differences between consecutive vertices must satisfy a prescribed order. Alon and, independently, Erdős, Hajnal, and Pach answered this question affirmatively. Alon further conjectured that the true growth rate should be exponential in
k
. We make progress towards this conjecture, obtaining an upper bound on
which is exponential in a power of
k
. This improves a result of Shelah, who showed that
n
is at most double-exponential in
k
David Conlon. Jacob Fox. Benny Sudakov. "Two extensions of Ramsey’s theorem." Duke Math. J. 162 (15) 2903 - 2927, 1 December 2013. https://doi.org/10.1215/00127094-2382566
Secondary: 05D10 , 05D40
David Conlon, Jacob Fox, Benny Sudakov "Two extensions of Ramsey’s theorem," Duke Mathematical Journal, Duke Math. J. 162(15), 2903-2927, (1 December 2013) |
WACC Calculator (Weighted Average Cost of Capital) | Dash Calculator
Our WACC calculator outputs the weighted average cost of capital given the cost of equity and the cost of debt. It can be used to determine what discount rate to use in capital budgeting decisions, and to evaluate potential investments.
Total equity
(E)
(D)
(r_e)
(r_d)
The weighted average cost of capital is 7.17%.
Show calculations Hide calculations
Let's calculate the weighted average cost of capital. Continue reading below for a detailed explanation of the WACC formula.
\textrm{WACC} = \textrm{Cost of equity} \times \frac{\textrm{Equity}}{\textrm{Debt + Equity}} + \textrm{Cost of debt (after tax)} \times \frac{\textrm{Debt}}{\textrm{Debt + Equity}}
\textrm{WACC} = r_e \times \frac{\textrm{Equity}}{\textrm{Debt + Equity}} + r_d \times (1 - \textrm{Corporate tax rate}) \times \frac{\textrm{Debt}}{\textrm{Debt + Equity}}
\textrm{WACC} = 14\% \times \frac{5000}{10000 + 5000} + 5\% \times (1 - 25\%) \times \frac{10000}{10000 + 5000}
\textrm{WACC} = 7.17\%
The WACC equation is:
is the cost of equity
is the cost of debt
The cost of equity and the cost of debt are percentages that are weighted by how much debt and equity a company is using to finance itself.
The total financing for a company is its Debt + Equity.
Equity / Debt + Equity is the proportion of equity financing and Debt / Debt + Equity is the proportion of debt financing.
What is the WACC?
The weighted average cost of capital (WACC) is the rate companies pay lenders and shareholders for the use of their funds. It is a weighted average of the cost of debt (paid to lenders) and the cost of equity (paid to shareholders).
Use the WACC to determine what discount rate to use in capital budgeting decisions (e.g. in an NPV calculation). Use the WACC to evaluate potential investments.
Companies are financed through two ways:
Debt: By borrowing from lenders such as banks.
Equity: By selling shares in the company to investors.
Examples include IPOs and selling shares in a start-up to venture capitalists (VCs).
Both ways have a cost associated with them. Lenders expect to be paid an interest on their loan. Shareholders expect their shares in the company to increase in value and may receive dividend payments.
The WACC is weighted by the proportion of debt and equity in the capital structure. If more debt is used, then the WACC is closer to the cost of debt. Similarly if more equity is used, the WACC will be more similar to the cost of equity.
The WACC is often used as the discount rate in a Net Present Value (NPV) calculation or a discounted cash flow (DCF) model.
Because the WACC is a weighted average of the cost of equity and the cost of debt, we need to first calculate the cost of equity and then the cost of debt separately.
Cost of debt: The cost of debt is simply the interest rate a company pays on its debt. This is relatively easy to calculate. Look at a company's interest on its debt or its corporate bond yield.
Cost of equity: The cost of equity is harder to calculate. Use CAPM or the dividend growth model.
The cost of debt is relatively easier to measure. There are a few methods you can use to find company’s cost of debt.
Interest payments: If a company already has outstanding loans, you can use the current interest rates on those loans.
Bonds: Companies can issue bonds to raise money. Bonds are a form of debt. The cost of debt for these bonds is the yield to maturity. If the company’s corporate bonds are publicly traded, you can observe their market prices and calculate their yield to maturity.
Credit worthiness: The yield of maturity of corporate bonds cannot be calculated if the bonds are not traded in the public market. In this case, you can consider what kinds corporate bond yields a company could get based on its credit worthiness. Credit worthiness for companies is measured by three primary credit rating agencies:
S&P, and
Companies with better credit — a lower probability of default — tend to have the lowest yield to maturities, and a lower cost of debt. Those with higher probability of default will have higher yield of maturities, and a higher cost of debt.
Because interest expenses on debt is often tax-deductible, we use the after tax cost of debt.
The cost of equity is the rate of return investors expect to get from investing in a company’s stock. This return comes in the form of cash distributions, which are cash proceeds from sale of the stock and any dividend payouts.
The cost of equity is more difficult to calculate.
There are two common methods of calculating the cost of equity:
Dividend growth model: Discounted cash flow approach to calculating the cost of equity
The dividend growth model estimates the future dividends a shareholder expects to receive. Using (1) the current stock price and (2) the future stream of dividends, we can calculate the internal rate of return (IRR). This is the cost of equity.
Most analysts assume that dividends will grow forever at a constant rate. This tends to be a low single-digit number between 1% and 4%.
Capital Asset Pricing Model (CAPM) approach to calculating the cost of equity
CAPM estimates the cost of equity by adjusting for the risk premium of a company’s stock. It is a little bit easier to calculate than the dividend growth model.
Risk-free rate: The rate of return an investment without any risk expects to return. This is typically the 10-year US Treasury.
Equity beta coefficient: Measurement of how the company’s stock correlates with the market.
Market risk premium: The difference between the return of the market (say the S&P 500) and the risk-free rate.
The cost of capital varies across companies and industries due to differences in risk, capital structures, and type of business.
Principles of Corporate Finance by Richard Brealey, Stewart Myers, and Franklin Allen
Investments by Zvi Bodie, Alex Kane, and Alan J. Marcus
"Investment and the weighted average cost of capital" by Murray Z. Frank and Tao Shen
"The Weighted Average Cost of Capital: Some Questions on its Definition, Interpretation, and Use" by Fred D. Arditti |
The use of e provides a shortcut for calculation of continuously compounded interest. The formula for continuous compounding is A = Pert, where A is the total amount at any time, P is the original principal, r is the rate of interest, and t is the time period.
Use this new formula to calculate the amount Tabitha had after investing her $10,000 for one year at 3% annual interest, compounded continuously. How does this compare with her hourly compounding result of $10,304.54?
A = 10{,}000e^{(0.03)(1)}
Suppose a large investment group invested 10 billion dollars at 8% annual interest. Compare the amount they would have in one year if the interest were compounded continuously with the amount they would earn if the interest were compounded daily.
For the interest compounded daily, use
A=10,000,000,000\left( 1+\frac{0.08}{365} \right)^{365}.
Which investors might benefit from continuous compounding, rather than daily compounding? |
This problem is a checkpoint for average and instantaneous rates of change.
f(x) = 3x^2 − 4
x = 10
x = 12
x = 11
x = 11 + h
Evaluate the expression in part (b) as
h → 0
. What does this tell you about the function at
x = 11
Calculate the instantaneous rate of change at
x = 1
Ideally at this point you are comfortable working with these types of problems and can complete them correctly. If you feel that you need more confidence when completing these types of problems, then review the Rates of Change Checkpoint materials available at cpm.org and try the practice problems provided. From this point on, you will be expected to do problems like these correctly and with confidence.
Click the following link: Rates of Change |
Radiated power - zxc.wiki
The radiant power or radiant flux or is that differential amount of energy ( is the radiant energy ) that is transported by electromagnetic waves per period of time :
{\ displaystyle \ Phi}
{\ displaystyle \ Phi _ {\ mathrm {e}}}
{\ displaystyle \ mathrm {d} Q}
{\ displaystyle Q}
{\ displaystyle \ mathrm {d} t}
{\ displaystyle \ Phi = {\ frac {\ mathrm {d} Q} {\ mathrm {d} t}}}
Your unit is W ( watt ).
In astronomy , the radiant power of astronomical objects is called luminosity .
3 Relation to other sizes
From the photon stream (number of photons per unit of time) , the radiation power for monochromatic light results as:
{\ displaystyle \ phi = {\ tfrac {\ mathrm {d} N} {\ mathrm {d} t}}}
{\ displaystyle \ Phi = h \ cdot \ phi \ cdot \ nu}
{\ displaystyle h}
the Planck's constant
{\ displaystyle \ nu}
the light frequency .
For electromagnetic radiation with a frequency of 540 THz (green light with a wavelength of 555 nm), a photon current of 2 corresponds .79e18 s −1 of a radiation power of 1 W.
For polychromatic light one has to form the integral over all frequencies:
{\ displaystyle \ Phi = h \ cdot \ int _ {0} ^ {\ infty} {\ frac {\ mathrm {d} \ phi} {\ mathrm {d} \ nu}} \ cdot \ nu \ cdot \ mathrm {d} \ nu}
Explanatory graphic for the basic photometric law
In order to determine the dependence of the radiant power on a surface element of a radiator surface of the luminance of a Lambert radiator (constant surface brightness) on a surface element located at a distance , the so-called fundamental photometric law can be used, which combines Lambert's law of cosines and the law of photometric distance .
{\ displaystyle \ mathrm {d} ^ {2} \ Phi _ {\ mathrm {1 \ rightarrow 2}}}
{\ displaystyle \ mathrm {d} A_ {1}}
{\ displaystyle A_ {1}}
{\ displaystyle L_ {1}}
{\ displaystyle r_ {12}}
{\ displaystyle \ mathrm {d} A_ {2}}
{\ displaystyle \ mathrm {d} ^ {2} \ Phi _ {\ mathrm {1 \ rightarrow 2}} = L_ {1} \ cdot {\ frac {\ mathrm {d} A_ {1} \ cos \ beta _ {1} \ cdot \ mathrm {d} A_ {2} \ cos \ beta _ {2}} {r_ {12} ^ {2}}}}
This depends, among other things, on the mutual position of the two surfaces in space, which is taken into account by the angle and between the beam direction and the surface normals.
{\ displaystyle \ beta _ {1}}
{\ displaystyle \ beta _ {2}}
Relation to other sizes
If the radiation output is related to the size of the irradiated area, the irradiance is obtained (unit: W / m²):
{\ displaystyle E}
{\ displaystyle E = {\ frac {\ mathrm {d} \ Phi} {\ mathrm {d} A}}}
If, on the other hand, it is related to the solid angle into which a light beam emanating from a light source falls, then the radiation intensity is obtained
{\ displaystyle \ Omega}
{\ displaystyle I = {\ frac {\ mathrm {d} \ Phi} {\ mathrm {d} \ Omega}}}
with the unit W / sr.
In photometry (lighting technology) the corresponding measurand is the luminous flux , measured in the unit lumen . While the radiant power (mostly written in this context ) is an energetic , i.e. objective, measured variable, the spectral sensitivity of the human eye is included in the luminous flux ( V-lambda curve ). The link between the two quantities is the photometric radiation equivalent of the light source
{\ displaystyle \ Phi _ {\ mathrm {v}}}
{\ displaystyle \ Phi _ {\ mathrm {e}}}
{\ displaystyle K}
{\ displaystyle K \, = \, {\ frac {\ Phi _ {\ mathrm {v}}} {\ Phi _ {\ mathrm {e}}}}}
which depends on their wavelength spectrum.
The following table gives an overview of quantities and units in radiometry and photometry:
{\ displaystyle \ Phi _ {\ mathrm {e}}}
{\ displaystyle \ Phi _ {\ mathrm {v}}}
{\ displaystyle I _ {\ mathrm {e}}}
{\ displaystyle I _ {\ mathrm {v}}}
{\ displaystyle E _ {\ mathrm {e}}}
{\ displaystyle E _ {\ mathrm {v}}}
{\ displaystyle M _ {\ mathrm {e}}}
{\ displaystyle M _ {\ mathrm {v}}}
{\ displaystyle L _ {\ mathrm {e}}}
{\ displaystyle L _ {\ mathrm {v}}}
{\ displaystyle Q _ {\ mathrm {e}}}
{\ displaystyle Q _ {\ mathrm {v}}}
{\ displaystyle H _ {\ mathrm {e}}}
{\ displaystyle H _ {\ mathrm {v}}}
{\ displaystyle \ eta _ {\ mathrm {e}}}
{\ displaystyle \ eta _ {\ mathrm {v}}}
F. Pedrotti, L. Pedrotti, W. Bausch, H. Schmidt: Optics for engineers: Fundamentals . 2nd Edition. Springer, Berlin 2001, ISBN 3-540-67379-2 .
↑ electropedia , International Electrotechnical Dictionary (IEV) of the International Electrotechnical Commission : Entry 845-01-24 (area of lighting) is synonymous with: radiant flux = radiant power = "radiation power" = "radiation flux"
This page is based on the copyrighted Wikipedia article "Strahlungsleistung" (Authors); it is used under the Creative Commons Attribution-ShareAlike 3.0 Unported License. You may redistribute it, verbatim or modified, providing that you comply with the terms of the CC-BY-SA. |
Convection - MapleSim Help
Home : Support : Online Help : MapleSim : MapleSim Component Library : Thermal and Thermal Fluids : Heat Transfer Components : Convection
The Convection component models a linear heat convection between solid geometries and fluid flow over the solid using a convective thermal conductance controlled by an input signal.
{Q}_{\mathrm{flow}}={Q}_{{\mathrm{flow}}_{\mathrm{solid}}}=-{Q}_{{\mathrm{flow}}_{\mathrm{fluid}}}={G}_{c}\Delta T
\Delta T={T}_{\mathrm{solid}}-{T}_{\mathrm{fluid}}
{Q}_{\mathrm{flow}}
W
Heat flow rate from solid -> fluid
\Delta T
K
= solid.T - fluid.T
{G}_{c}
Real input; convective thermal conductance in
\frac{W}{K}
\mathrm{solid}
\mathrm{fluid} |
Generate fault frequency bands for spectral feature extraction - MATLAB faultBands - MathWorks Nordic
faultBands
Frequency Bands of Electrical Mains Supply
Frequency Bands of Faulty Induction Motor
Visualize Frequency Bands and Harmonics of the Electrical Mains Supply
Folding Negative Fault Frequencies
Generate fault frequency bands for spectral feature extraction
FB = faultBands(F0,N0)
FB = faultBands(F0,N0,F1,N1)
___ = faultBands(___,Name,Value)
[FB,info] = faultBands(___)
faultBands(___)
FB = faultBands(F0,N0) generates fault frequency bands FB, using the fundamental frequency of interest F0 and the array of harmonics N0. For instance, to construct fault bands for an induction motor, the mains frequency of 60 Hz is the fundamental frequency of interest.
FB = faultBands(F0,N0,F1,N1) constructs fault frequency bands FB, using the distance of the first sideband F1 from the fundamental frequency F0. N1 is the array of the sidebands around F0. If F1 is not specified, then faultBands sets F1 to 10 percent of F0 by default. N1 is equivalent to the 'Sidebands' name-value pair. You can use the 'Type' name-value pair to specify separation between successive sidebands.
___ = faultBands(___,Name,Value) allows you to specify additional parameters using one or more name-value pair arguments.
[FB,info] = faultBands(___) also returns the structure info containing information about the generated fault frequency bands FB.
faultBands(___) with no output arguments plots a bar chart of the generated fault frequency bands FB.
For this example, generate frequency bands for analyzing the signal components around the first 5 harmonics of the mains supply frequency.
With the fundamental frequency of 60 Hz, the frequency of the alternating current in the mains power supply, use faultBands to generate the first 5 harmonics of the mains supply.
N0 = 1:5;
FB is returned as a 5x2 array with default frequency band width of 5% of F0 which is 3 Hz. The first column in FB contains the values of
F-\frac{W}{2}
, while the second column contains all the values of
F+\frac{W}{2}
for each harmonic.
For this example, consider an induction motor with broken rotor bars. Under normal operation with load, the rotor speed always lags the speed of the magnetic field allowing the rotor bars to cut magnetic lines of force and produce useful torque. This difference is called slip. Considering a slip value of 0.03 in the system with broken rotors, construct frequency bands for sideband components around the fundamental frequency of 60 Hz.
slip = 0.03;
F1 = 2*slip*F0;
[FB,info] = faultBands(F0,N0,F1,N1)
FB = 12×2
Centers: [49.2000 52.8000 56.4000 63.6000 67.2000 70.8000 ... ]
Labels: ["1F0-3F1" "1F0-2F1" "1F0-1F1" ... ]
HarmonicGroups: [1 1 1 1 1 1 2 2 2 2 2 2]
Construct frequency bands for analyzing the signal components around the first three harmonics of the electrical mains supply frequency.
With the fundamental frequency of 60 Hz, the alternating current in the mains power supply, use faultBands to visualize the first 3 harmonics of the mains supply.
faultBands(F0,N0)
From the plot, observe the following:
The fundamental frequency, which is also the first harmonic, 1F0 at 60 Hz
The second harmonic, 2F0 at 120 Hz
The third harmonic, 3F0 at 180 Hz
To better capture the expected variations of the actual system signals around the nominal fault frequencies, set the widths of each band to 10 Hz.
faultBands(F0,N0,'Width',10)
For this example, consider an induction motor with static and dynamic rotor eccentricities. Construct and visualize the frequency bands for the 4 sideband components of an induction motor with 4 pole pairs around the fundamental frequency due to the rotor eccentricities.
slip = 0.029;
polePairs = 4;
F1 = 2*F0*(1-slip)/polePairs
F1 = 29.1300
faultBands(F0,N0,F1,N1)
Warning: Truncated or removed negative fault frequency bands.
To avoid truncating negative fault frequency bands, set 'Folding' to true to fold them onto the positive frequency axis.
faultBands(F0,N0,F1,N1,'Folding',true)
Observe that the sideband frequencies 1F0-3F1 and 1F0-4F1 are now visible on the positive axis.
F0 — Fundamental frequency of interest
Fundamental frequency of interest, specified as a positive scalar. faultBands constructs the fault frequency bands around the fundamental frequency F0. For instance, to construct fault bands for a faulty induction motor, the mains frequency of 60 Hz is the fundamental frequency of interest. Similarly, to generate fault bands for a faulty gear train, the input shaft frequency is the fundamental frequency.
You can specify F0 in either hertz or orders.
N0 — Harmonics of the fundamental frequency
Harmonics of the fundamental frequency, specified as a vector of positive integers. Specify fault bands around the fundamental frequency F0 and its harmonics by N0. N0 is equivalent to the 'Harmonics' name-value pair with a default value of 1.
F1 — Distance of the first sideband from the fundamental frequency
0.1*F0 (default) | positive scalar
Distance of the first sideband from the fundamental frequency, specified as a positive scalar. If F1 is not specified, then faultBands assumes a value of 10 percent of the fundamental frequency for F1.
N1 — Sidebands of the fundamental frequency and its harmonics
Sidebands of the fundamental frequency and its harmonics, specified as a vector of nonnegative integers. N1 is equivalent to the 'Sidebands' name-value pair with a default value of 0.
Example: ...,'Harmonics',[1,3,5]
Harmonics — Harmonics of the fundamental frequency to be included
Harmonics of the fundamental frequency to be included, specified as the comma-separated pair consisting of 'Harmonics' and a vector of positive integers. The default value is 1. Specify 'Harmonics' when you want to construct the frequency bands with more harmonics of the fundamental frequency.
Sidebands — Sidebands around the fundamental frequency and its harmonics to be included
0 (default) | vector of nonnegative integers
Sidebands around the fundamental frequency and its harmonics to be included, specified as the comma-separated pair consisting of 'Sidebands' and a vector of nonnegative integers. The default value is 0. Specify 'Sidebands' when you want to construct the frequency bands with sidebands around the fundamental frequency and its harmonics.
Width — Width of the frequency bands centered at the nominal fault frequencies
5 percent of the fundamental frequency (default) | positive scalar
Width of the frequency bands centered at the nominal fault frequencies, specified as the comma-separated pair consisting of 'Width' and a positive scalar. The default value is 5 percent of the fundamental frequency. Avoid specifying 'Width' with a large value so that the fault bands do not overlap.
Type — Separation value between successive sidebands
'additive' (default) | 'multiplicative'
Separation value between successive sidebands, specified as the comma-separated pair consisting of 'Type' and either 'additive' or 'multiplicative'. Specify 'Type' as:
'additive', to set the separation between successive sidebands to F1.
'multiplicative', to set the separation between successive sidebands proportional to both the harmonic order and the sideband value.
Folding — Logical value specifying whether negative nominal fault frequencies have to be folded about the frequency origin
Logical value specifying whether negative nominal fault frequencies have to be folded about the frequency origin, specified as the comma-separated pair consisting of 'Folding' and either true or false. If you set 'Folding' to true, then faultBands folds the negative nominal fault frequencies about the frequency origin by taking their absolute values such that the folded fault bands always fall in the positive frequency intervals. The folded fault bands are computed as
\left[\text{max}\left(0,\text{ }|F|-\frac{W}{2}\right),\text{ }|F|+\frac{W}{2}\right]
, where W is the 'Width' name-value pair and F is one of the nominal fault frequencies.
FB — Fault frequency bands
Nx2 array
Fault frequency bands, returned as an Nx2 array, where N is the number of fault frequencies. FB is returned in the same units as F0, in either Hertz or orders. The generated fault bands,
\left[F-\frac{W}{2},\text{ }F+\frac{W}{2}\right]
, are centered depending on the sideband specification as follows:
If you do not specify the sidebands, then the fault bands are centered at
F={n}_{0}{F}_{0}
, where the integer n0 ranges through the elements of the array of harmonics, N0.
If you specify sidebands using N1 or the 'Sidebands' name-value pair, then fault bands are centered at:
F={n}_{0}{F}_{0}±{n}_{1}{F}_{1}
, when 'Type' is specified as 'additive'. Here, the integer n1 ranges through the elements of the array of sidebands, N1.
F={n}_{0}\left({F}_{0}±{n}_{1}{F}_{1}\right)
, when 'Type' is specified as 'multiplicative'.
info — Information about the fault frequency bands
Information about the fault frequency bands in FB, returned as a structure with the following fields:
Centers — Center fault frequencies
Labels — Labels describing each frequency
HarmonicGroups — Harmonic group numbers equal to the harmonic order of each frequency band to be able to identify fault bands associated with the nominal fault frequency
F={n}_{0}{F}_{0}
, where the integer n0 ranges through the elements of the array of harmonics, N0
faultBandMetrics | bearingFaultBands | gearMeshFaultBands |
Nickel–hydrogen battery - Wikipedia
Schematics of a nickel-hydrogen battery
55-75 W·h/kg[1][2]
~60 W·h/L[2]
~220 W/kg[3]
>20,000 cycles[4]
A nickel–hydrogen battery (NiH2 or Ni–H2) is a rechargeable electrochemical power source based on nickel and hydrogen.[5] It differs from a nickel–metal hydride (NiMH) battery by the use of hydrogen in gaseous form, stored in a pressurized cell at up to 1200 psi (82.7 bar) pressure.[6] The nickel–hydrogen battery was patented on February 25, 1971 by Alexandr Ilich Kloss and Boris Ioselevich Tsenter in the United States.[7]
NiH2 cells using 26% potassium hydroxide (KOH) as an electrolyte have shown a service life of 15 years or more at 80% depth of discharge (DOD)[8] The energy density is 75 Wh/kg, 60 Wh/dm3[2] specific power 220 W/kg.[3] The open-circuit voltage is 1.55 V, the average voltage during discharge is 1.25 V.[9]
While the energy density is only around one third as that of a lithium battery, the distinctive virtue of the nickel–hydrogen battery is its long life: the cells handle more than 20,000 charge cycles[4] with 85% energy efficiency and 100% faradaic efficiency.
NiH2 rechargeable batteries possess properties which make them attractive for the energy storage of electrical energy in satellites[10] and space probes. For example, the ISS,[11] Mercury Messenger,[12] Mars Odyssey[13] and the Mars Global Surveyor[14] are equipped with nickel–hydrogen batteries. The Hubble Space Telescope, when its original batteries were changed in May 2009 more than 19 years after launch, led with the highest number of charge and discharge cycles of any NiH2[15] battery in low Earth orbit.[16]
Main article: History of the battery § Nickel-hydrogen and nickel metal-hydride
The development of the nickel hydrogen battery started in 1970 at Comsat[17] and was used for the first time in 1977 aboard the U.S. Navy's Navigation technology satellite-2 (NTS-2).[18] Currently, the major manufacturers of nickel-hydrogen batteries are Eagle-Picher Technologies and Johnson Controls, Inc.
Nickel-hydrogen batteries for Hubble[19]
The nickel-hydrogen battery combines the positive nickel electrode of a nickel-cadmium battery and the negative electrode, including the catalyst and gas diffusion elements, of a fuel cell. During discharge, hydrogen contained in the pressure vessel is oxidized into water while the nickel oxyhydroxide electrode is reduced to nickel hydroxide. Water is consumed at the nickel electrode and produced at the hydrogen electrode, so the concentration of the potassium hydroxide electrolyte does not change. As the battery discharges, the hydrogen pressure drops, providing a reliable state of charge indicator. In one communication satellite battery, the pressure at full charge was over 500 pounds/square inch (3.4 MPa), dropping to only about 15 PSI (0.1 MPa) at full discharge.
If the cell is over-charged, the oxygen produced at the nickel electrode reacts with the hydrogen present in the cell and forms water; as a consequence the cells can withstand overcharging as long as the heat generated can be dissipated.[dubious – discuss]
The cells have the disadvantage of relatively high self-discharge rate, i.e. chemical reduction of Ni(III) into Ni(II) in the cathode:
{\displaystyle {\ce {NiOOH + 1/2H2 <=> Ni(OH)2.}}}
which is proportional to the pressure of hydrogen in the cell; in some designs, 50% of the capacity can be lost after only a few days' storage. Self-discharge is less at lower temperature.[1]
Compared with other rechargeable batteries, a nickel-hydrogen battery provides good specific energy of 55-60 watt-hours/kg, and very long cycle life (40,000 cycles at 40% DOD) and operating life (> 15 years) in satellite applications. The cells can tolerate overcharging and accidental polarity reversal, and the hydrogen pressure in the cell provides a good indication of the state of charge. However, the gaseous nature of hydrogen means that the volume efficiency is relatively low (60-100 Wh/L for an IPV (individual pressure vessel) cell), and the high pressure required makes for high-cost pressure vessels.[1]
The positive electrode is made up of a dry sintered[20] porous nickel plaque, which contains nickel hydroxide. The negative hydrogen electrode utilises a teflon-bonded platinum black catalyst at a loading of 7 mg/cm2 and the separator is knit zirconia cloth (ZYK-15 Zircar).[21][22]
The Hubble replacement batteries are produced with a wet slurry process where a binder agent and powdered metallic materials are molded and heated to boil off the liquid.[23]
Individual pressure vessel (IPV) design consists of a single unit of NiH2 cells in a pressure vessel.[24]
Common pressure vessel (CPV) design consist of two NiH2 cell stacks in series in a common pressure vessel. The CPV provides a slightly higher specific energy than the IPV.
Single pressure vessel (SPV) design combines up to 22 cells in series in a single pressure vessel.
Bipolar design is based on thick electrodes, positive-to-negative back-to-back stacked in a SPV.[25]
Dependent pressure vessel (DPV) cell design offers higher specific energy and reduced cost.[26]
Common/dependent pressure vessel (C/DPV) is a hybrid of the common pressure vessel (CPV) and the dependent pressure vessel (DPV) with a high volumetric efficiency.[27]
^ a b c David Linden, Thomas Reddy (ed.) Handbook of Batteries Third Edition, McGraw-Hill, 2002 ISBN 0-07-135978-8 Chapter 32, "Nickel Hydrogen Batteries"
^ a b c Spacecraft Power Systems Pag.9
^ a b NASA/CR—2001-210563/PART2 -Pag.10 Archived 2008-12-19 at the Wayback Machine
^ a b Five-year update: nickel hydrogen industry survey
^ A simplified physics-based model for nickel hydrogen battery
^ Hermetically sealed nickel-hydrogen storage cell US Patent 3669744
^ Potassium hydroxide electrolyte for long-term nickel-hydrogen geosynchronous missions
^ Optimization of spacecraft electrical power subsystems -Pag.40
^ Ni-H2 Cell Characterization for INTELSAT Programs
^ Validation of International Space Station electrical performance model via on-orbit telemetry Archived 2009-02-18 at the Wayback Machine
^ Mars Global Surveyor Archived 2009-08-10 at the Wayback Machine
^ Hubble space telescope servicing mission 4 batteries
^ "Nickel-Hydrogen Battery Technology—Development and Status" (PDF). Archived from the original (PDF) on 2009-03-18. Retrieved 2012-08-29.
^ NTS-2 Nickel-Hydrogen Battery Performance 31 Archived 2009-08-10 at the Wayback Machine
^ Performance comparison between NiH2 dry sinter and slurry electrode cells
^ Nickel hydrogen batteries-an overview Archived 2009-04-12 at the Wayback Machine
^ 1995–dependent pressure vessel (DPV)
Albert H. Zimmerman (ed), Nickel-Hydrogen Batteries Principles and Practice, The Aerospace Press, El Segundo, California. ISBN 1-884989-20-9.
Overview of the design, development, and application of nickel-hydrogen batteries
Implantable nickel hydrogen batteries for bio-power applications
NASA handbook for nickel-hydrogen batteries
A nickel/hydrogen battery for terrestrial PV systems
Retrieved from "https://en.wikipedia.org/w/index.php?title=Nickel–hydrogen_battery&oldid=1076191313" |
Random Greedy Forest tutorial – The Kernel Trip
A not so famous algorithm
Regularized Greedy Forest is a quite recent algorithm in machine learning, the article “Learning Nonlinear Functions Using Regularized Greedy Forest” has been published in 2014.
If we want to compare it to gradient boosting, which seems to have been studied back in 1999, it took a while before this algorithm received its many reliable and fast implementations (xgboost, catboost, LightGBM).
It seems to be a very good candidate in terms of performance. However, as we will see in the parameters section, it requires some tuning (regularization and number of leaves can be application critical).
As stated by the authors, Regularized Greedy Forest can achieve better performance than gradient boosting approaches:
In contrast to these traditional boosting algorithms that treat a tree learner as a black box, the method we propose directly learns decision forests via fully-corrective regularized greedy search using the underlying forest structure. Our method achieves higher accuracy and smaller models than gradient boosting on many of the datasets we have tested on.
And if you are skeptical about benchmarks proposed by the authors, check out the following posts, related to data science competitions:
1st Place Solution (TReNDS Neuroimaging)
1st Place Solution (Home Credit Default Risk)
1st Place Solution (Allstate Claims Severity)
2nd Place Solution (Home Credit Default Risk)
3rd Place Solution (Santander Customer Satisfaction)
13th Place Solution (IEEE-CIS Fraud Detection)
18th Place Solution (Porto Seguro’s Safe Driver Prediction)
Kernel (Porto Seguro’s Safe Driver Prediction)
On top of that, I noted that on some dataset I have comparable performance between xgboost and RGF after a careful tuning of the parameters for both the models.
As for gradient boosting and random forests, the idea is to train a collection of decision trees. However, the key difference is that you are allowed to modify previously trained trees and weights attributed to each tree if that improves the performance of the overall model.
To make things more clear:
In the case of random forests, all the trees are trained simultaneously and regardless of each other performance.
In the case of gradient boosting, a new tree is trained on the residuals of the previous trees.
In the case of random greedy forest, things are more complicated :) At each step we may either start a new tree, or split an existing leaf node. Then, the weights of each leaf are adjusted, to optimize the loss function.
To put it in equations, when we try to learn some objective function
F
what we do is to solve this type of optimization program.
The space of functions where
F
lives being quite large, we usually rely on heuristics to make the above problem solvable in an acceptable amount of time.
Per example, in the case of gradient boosting, we solve a series of training of decision trees, with the following induction:
F_m(x) = F_{m-1}(x) + \underset{h_m \in \mathcal{H}}{\operatorname{arg\,min}} \left[{\sum_{i=1}^n {L(y_i, F_{m-1}(x_i) + h_m(x_i))}}\right]
So that each step is as simple as training a decision tree, and the training time is, roughly, the number of trees times the training time of a tree.
In the case of the Regularized Greedy Forest, the procedure is as follows:
Fix the weights, and change the structure of the forest (which changes basis functions) so that the loss $Q(F)$ is reduced the most.
Fix the structure of the forest, and change the weights so that lossQ(F) is minimized
Cross validating a random greedy forest
The cross validation is usual, here are the roles of the different parameters.
The parameters are sorted by order of importance in terms of their impact on the accuracy of the model.
max_leaf : the total number of leaves in the forest. Note that given the training procedure, we never have to specify the total number of trees needed in the forest. Beware, the larger this parameter, the longer the training. By default, the value is 1000 for RGFClassifier and 500 RGFRegressor.
l2 : the penalty. This parameter has to be tuned to obtain a good performance. By default, the value is 0.1 but smaller values usually improve performance.
n_tree_search : (1 by default) Number of trees to be searched for the nodes to split.
algorithm one of (“RGF”, “RGF_Opt”, “RGF_Sib”), where the algorithm are the following:
RGF: RGF with L2 regularization on leaf-only models.
RGF Opt: RGF with min-penalty regularization.
RGF Sib: RGF with min-penalty regularization with the sum-to-zero sibling constraints.
By default, the algorithm is “RGF” for both RGFClassifier() and RGFRegressor().
reg_depth : Must be no smaller than 1.0. Meant for being used with algorithm="RGF_Opt"|"RGF_Sib". A larger value penalizes deeper nodes more severely.
loss one of ("LS", "Expo", "Log", "Abs"), by default this is LS for regressions and Log for classification.
LS: Square loss,
Expo: Exponential loss,
Log: Logistic loss,
Abs: Absolute error loss.
n_iter : Number of iterations of coordinate descent to optimize weights. If None, 10 is used for loss=”LS” and 5 for loss=”Expo” or “Log”. Not critical to improve accuracy.
calc_prob : One of (“sigmoid”, “softmax”), by default “sigmoid”. I guess it will not affect accuracy or roc_auc, but may affect logloss.
Fortunately, the implementation comes with a scikit learn interface, therefore you can use the usual .fit() and .transform() methods, so there is not much to say about how to use it. Per example, the following will cross validate a random greedy forest.
perm = rng.permutation(iris.target.size)
rgf = RGFClassifier(max_leaf=400,
algorithm="RGF_Sib",
test_interval=100,
rgf_scores = cross_val_score(rgf,
cv=StratifiedKFold(n_folds))
rgf_score = sum(rgf_scores)/n_folds
print('RGF Classifier score: {0:.5f}'.format(rgf_score))
Learning Nonlinear Functions Using Regularized Greedy Forest is the main article on the topic.
The RGF implementation for the basic implementation, and for the sparse implementation (FastRGF).
pythonmachine-learning |
Periodic Points and Fixed Points for the Weaker ( ϕ , φ ) -Contractive Mappings in Complete Generalized Metric Spaces
2012 Periodic Points and Fixed Points for the Weaker
\left(\varphi ,\phi \right)
-Contractive Mappings in Complete Generalized Metric Spaces
Chi-Ming Chen, W. Y. Sun
We introduce the notion of weaker
\left(\varphi ,\phi \right)
-contractive mapping in complete metric spaces and prove the periodic points and fixed points for this type of contraction. Our results generalize or improve many recent fixed point theorems in the literature.
Chi-Ming Chen. W. Y. Sun. "Periodic Points and Fixed Points for the Weaker
\left(\varphi ,\phi \right)
-Contractive Mappings in Complete Generalized Metric Spaces." J. Appl. Math. 2012 1 - 7, 2012. https://doi.org/10.1155/2012/856974
Chi-Ming Chen, W. Y. Sun "Periodic Points and Fixed Points for the Weaker
\left(\varphi ,\phi \right)
-Contractive Mappings in Complete Generalized Metric Spaces," Journal of Applied Mathematics, J. Appl. Math. 2012(none), 1-7, (2012) |
From Francis Boott 26 December 1862
My faith is an orthodoxy of its own, embracing a much wider revelation than the church admits of, & the reverence I feel for those “who still sway our spirits from their irons”,1 rests with a fond complacency on you as one of Natures true interpreters. I also hold to the privilege of openly professing worship to all who have enlightened & quickened my spirit. I have not been in a church but once for the last 25 years, & I never yet comprehended the possibility of prayer beyond that emotional habit of the mind, when one contemplates the beauty of nature or the glories of genius. I place you in a nitch in my self-consecrated Temple, associated with a few of my Divinities. Your altar close to that of Linneus & Gilbert White,—& the Dear Hookers2—& not far from those imputed sinners Byron, Burns & Charles Lambe3—& those earlier Saints Milton, Shakespeare4 & that still earlier Bard whose spirit was more simply divine than the church tries to make it.—5 I make this confession to justify my last note to you, & you are too liberal to object to the worship of any man.6
But my especial object in writing is to beg you not to take the trouble of sending me any American Newspapers. If Dear Gray asks you to do so, silently appropriate them to the only use I make of them.7 I never look at them, nor do I read a syllable about this horrid war. I cannot see any desirable issue to it or to the fate of the poor blacks. I have had a nephew shot thro’ the abdomen,8 & my last remittance was at the rate of exchange 146
\frac{1}{2}
—9 These evils I consent to, for I have no power to redress them, but I cannot read details of hopeless bloodshed—
Many happy new years to you & Mrs Darwin & your family— | Yrs sincerely | F. Boott
Charles Darwin Esqre | &c &c
The source of the quotation has not been identified.
Carolus Linnaeus, Gilbert White, William Jackson Hooker, and Joseph Dalton Hooker. See also letter from Francis Boott, 22 December [1862].
George Gordon Byron, Robert Burns, and Charles Lamb.
John Milton and William Shakespeare.
See letter from Francis Boott, 22 December [1862]. CD’s reply has not been found
Asa Gray had apparently enclosed copies of the Boston Advertiser with his letter to CD of 24 November 1862, requesting that he send them on to Boott. CD may have forwarded them with his reply to Boott’s letter of 22 December [1862], which has not been found. See also letter to J. D. Hooker, 29 [December 1862].
Boott’s nephew has not been identified.
See letter from Francis Boott, 22 December [1862] and n. 3.
On his particular spiritual faith; worships great naturalists and authors.
Does not wish to see American newspapers that Asa Gray offers to send, or hear about Civil War.
London, Gower St |
Get a Bird Loan - Bird Docs
Using Supply and Borrow on our Lending Platform
What is a Bird Loan?
Like all DeFi loans, Bird loans allow users to use crypto assets as collateral to borrow additional funds.
Unlike any other DeFi loans, Bird treats every borrower uniquely. Our lending platform allows the best borrowers greater access to capital.
Collateral is just fancy finance speak for a guarantee that you'll pay back any money you've borrowed. When you take out a home mortgage, your collateral is the house. Crypto loans are even easier because the crypto tokens you already have are your collateral.
The amount you can borrow is determined by your collateral ratio. Your collateral ratio is calculated as:
Loan Amount / Collateral Amount
This means that if the value of your collateral asset decreases, your collateral ratio will increase. If it increases too much, your loan can be liquidated. If your loan is liquidated, the asset you supplied as collateral is seized and used to repay the amount you borrowed and did not repay. This is how all DeFi loans work.
How do I get a Bird loan?
Bird loans are made for everyone. Getting one is easy. Here's how:
Before taking flight, make sure you have ETH in your wallet to pay the gas fees associated with receiving a Bird loan.
Step 1: Fly over to the Bird Lending App
Bird.Money | Non-Custodial Digital Asset Lending Platform
Currently, only Metamask and WalletConnect wallets are supported. When you arrive at the app, you'll be prompted to connect. If not, just click the "Connect Wallet" button in the upper left.
To change wallets, select the new wallet in the Metamask UI. The app will then automatically refresh to connect to the new wallet.
Step 3: Connect to Ethereum Mainnet
Check that Metamask (or WalletConnect) is connected to the correct network. If it isn't, you'll likely see a message like this:
To connect to the mainnet, click the network button on Metamask.
Then select "Ethereum Mainnet".
Step 4: Enable an asset as collateral
When supplying an asset as collateral for the first time, you'll first need to approve that asset for spending. This is the same process you've likely experienced when using Uniswap and other web3 apps. To approve an asset, click the slider bar next to that asset under the "Supply Markets" section.
You'll then be prompted to this asset as collateral. This happens by sending a transaction using Metamask. To do this, click the "Use ____ as Collateral" collateral button. In this example, we're enabling ETH.
When approving collateral and supplying or withdrawing collateral, you can check the status of a pending transaction within the Metamask interface. Check the Metamask docs for more information.
Any assets that have been approved will show a red check in the slider and slide to the right. Here we've approved ETH
Step 5: Supply collateral
Supplying collateral is the last step before you'll be able to receive a Bird loan. The value of the collateral you supply will determine the maximum amount you can borrow.
To supply collateral you must first have approved that asset as collateral (see the previous step). Once that's done, simply click the asset under the "Supply Markets" section, enter the amount you wish to use as collateral, then click "Supply".
If the supply doesn't appear and shows No Funds Available, then try deleting few decimals.
You'll then be prompted to approve the transaction in Metamask.
Step 6: Borrow assets against your supplied collateral
You made it! You're ready to receive a Bird loan. Wasn't too hard, right? Now that you've supplied collateral, you'll see a few changes in the Bird Lending App interface. In the dashboard on the left, you'll see the value of the collateral you supplied and the borrow limit your collateral allows.
In this example, $182 of supplied collateral allows up to $136 in borrowing. Next, click the asset you wish to borrow from the "Borrow Markets" on the right.
A screen to configure your loan will appear next. First, make sure you've selected the "Borrow" toggle and not the "Repay" toggle under the borrow amount field. Next, enter the amount you wish to borrow. Finally, click the "Borrow" button at the bottom of the screen.
You'll then be prompted to approve the transaction in Metamask. That's it! Once the transaction is confirmed, you will have received your first Bird Loan!
Use the "SAFE MAX" button to automatically populate the borrow amount field with a borrow amount equal to 80% of your total limit.
Like most DeFi projects, every step you just took can be tracked on the blockchain. Let's have a look.
Enable Asset as Collateral
This transaction typically requires the lowest gas fee of all transactions required to issue a Bird loan. You only need to enable each asset one time for a given account.
The transaction will appear as a 0 ETH transfer.
If you switch to the Logs view on Etherscan, you'll see that it was a call to the "MarketEntered" function for one of Bird's smart contracts.
Supply Collateral to the Platform
When supplying collateral, you're actually transferring custody of that collateral to Bird's smart contracts and receiving 'bTokens' in exchange.
Because you're sending tokens to a smart contract, they'll appear in the transaction. In our example, the 0.1 ETH has been transferred and 5 bETH has been returned.
You can also see the bTokens you're currently holding if you click the "Tokens" drop-down on your account page in Etherscan.
Borrow Action
When your loan is originated, you'll see the amount reflected in your total balance for that asset. To see the transfer, you'll either need to review the "Internal Txns" tab or the "Erc20 Token Txns" tab on Etherscan. Since the loan in this example was issued in ETH, we can see the transfer on the "Internal Txns" tab. The value matches the loan amount. |
First post: what I'm doing in summer 2021
VasBlog
I am happy to have been accepted to Google Summer of Code 2021 with the project "Efficient Spatial Simulations in DiffEqJump." I will be writing code under the mentorship of Samuel Isaacson and Chris Rackauckas. The goal of the project is to expand DiffEqJump with new, optimized spatial solvers and a consistent interface, enabling the study of large spatial systems of jump processes.
Jump processes are a fundamental component in stochastic models throughout engineering, medicine and the sciences. For example, biology is rich with chemical networks consisting of hundreds or thousands of chemical reactions such as the B-cell network (Barua et al. (2012)). Often they are modelled as systems of differential equations, which has multiple drawbacks:
in general the solutions to the DE's do not coincide with the mean of the stochastic solutions;
variance and other moments cannot be gathered from solutions to DE's;
when the number of species (e.g. molecules or individuals) is low, treating it as a continuous random variable introduces inaccuracies.
Stochastic Simulation Algorithms (SSAs) are used to stochastically simulate the system. While there are approximate SSAs, the focus of this project is on exact SSAs – those that keep all statistics exact modulo the sampling error.
DiffEqJump is a part of the SciML ecosystem, which contains infrastructure to perform such simulations. DiffEqJump is a package in Julia, and is a part of the SciML ecosystem. It contains infrastructure to perform such simulations.
While SSAs for spatially non-homogeneous systems exist in academic literature (Elf & Ehrenberg (2004), Sanft & Othmer (2015)), they are not currently supported in DiffEqJump. The goal of this project is to expand DiffEqJump with new, optimized spatial solvers and a consistent interface, enabling the study of large spatial systems of jump processes and the use of jump processes within other SciML tooling.
My main motivation for working on DiffEqJump is to take part in the effort of building better models to describe the world around us. DiffEqJump is used by scientists in the fields of biology, medicine, applied mathematics to stochastically simulate chemical and other jump process-based network models. One application of a non-chemical network, whose evolution can be stochastically simulated, is epidemic dynamics, the simplest of which is the SIR model (1) (SIR stands for Susceptible, Infected and Recovered/Removed). Since the assumption of people being well-mixed is violated in this scenario, the model would be more realistic if people were allowed to "diffuse" between different locales, and "react" when they are within the same locale. As the year of 2020 showed, epidemic dynamics is one of the most important applications to which mathematical models can provide critical insights.
\begin{aligned} S + I &\xrightarrow{\alpha} 2I\\ I &\xrightarrow{\beta} R \end{aligned}
S+IIα2IβR
Another big motivation for working on DiffEqJump comes from knowing that it is freely available for any modeler, for example, being actively used by researchers in Quantitative Systems Pharmacology working on drug development. This open source project helps biologists, chemists, epidemiologists and other scientists simulate stochastic networks efficiently and accurately.
In the coming weeks I will be designing an interface for spatial simulations in DiffEqJump. Follow my Github and DiffEqJump to see the progress this summer.
Dipak Barua, William S Hlavacek, and Tomasz Lipniacki. “A computational model for early events in B cell antigen receptor signaling: analysis of the roles of Lyn and Fyn”. In: The Journal of Immunology 189.2 (2012),pp. 646–658.
Johan Elf and M ̊ans Ehrenberg. “Spontaneous separation of bi-stable biochemical systems into spatial domains of opposite phases”. In: Systems biology 1.2 (2004), pp. 230–236.
Kevin R Sanft and Hans G Othmer. “Constant-complexity stochastic simulation algorithm with optimal binning”. In: The Journal of chemical physics 143.7 (2015), 08B6091.
© Vasily Ilin. Last modified: November 25, 2021. Website built with Franklin.jl and the Julia programming language. |
Expressive React Component APIs with Discriminated Unions ・ Andrew Branch
Expressive React Component APIs with Discriminated Unions
When I wrote this piece, I realized it would be much easier to explain if I could show you how the TypeScript compiler handles the code I’m talking about, as if we were both sitting in front of your favorite code editor with language support. So, I built this blog to imitate that experience. Hover (or tap) on code underlined in red to see errors, hover identifiers to see type information, and even edit the samples if you want. I also wrote more about why I did this.
Disclaimer: I wrote this before joining the TypeScript team. I’m not speaking officially for TypeScript or Microsoft in any post on this blog, but I’m especially not in this one.
One of TypeScript’s most underrated features is discriminated union types. Borrowed primarily from functional programming (FP) languages, they match an elegant FP concept to a pattern people intuitively write in JavaScript. Discriminated unions also enable a useful pattern for typing complex React component props more safely and expressively. But first, we’ll review what discriminated unions look like independent of React.
A simple union type in TypeScript looks like this:
let x: string | number = 42;
x = 0; // fine
x = 'Hiya!'; // also fine
x = true; // not fine
Riveting, right? Things get more interesting with object types:
numberOfSides: number;
sideLengths: number[];
enum TriangleKind {
Accute = 'Accute',
Obtuse = 'Obtuse'
interface Triangle extends Polygon {
numberOfSides: 3;
triangleKind: TriangleKind;
interface Quadrilateral extends Polygon {
isRectangle: boolean;
We have a base type Polygon, and two specializations that specify a number literal type for numberOfSides, along with some extra properties that are specific to polygons of their kind. This allows us to write a function that accepts either a Triangle or Quadrilateral and discriminate between them based on the shape’s numberOfSides:
function addShape(shape: Triangle | Quadrilateral) {
if (shape.numberOfSides === 3) {
// In here, the compiler knows that `shape` is a `Triangle`,
// so we can access triangle-specific properties.
// See for yourself: hover each occurance of “shape” and
// compare the typing info.
console.log(shape.triangleKind);
// In here, the compiler knows that `shape` is a `Quadrilateral`.
console.log(shape.isRectangle);
When we have a union (like Triangle | Quadrilateral) that can be narrowed by a literal member (like numberOfSides), that union is called a discriminated union and that property is called the discriminant property.
The Problem: Overly Permissive Props
You’re writing a Select component (i.e., a fancy replacement for an HTMLSelectElement) with React and TypeScript. You want it to support both single-selection and multiple-selection, just like a native select element. Perhaps you look at the SelectHTMLAttributes interface from @types/react for inspiration, and notice that a native select element, in React, can have a value of type string | string[] | number. From TypeScript’s perspective, you can pass a single value or an array of values indiscriminately, but you know that an array of values is really only meaningful if the multiple prop is set. Nonetheless, you try this approach for your component:
interface SelectProps {
onChange: (newValue: string | string[]) => void;
class Select extends React.Component<SelectProps> {
The idea is that when multiple is true, the consumer should set value to an array and expect an array back as newValue in onChange. You’ll quickly realize that this looseness of your API allows for some invalid configurations and headaches for your consumers:
// Value is an array, but it’s missing the `multiple`
// prop, but no compiler error
options={['Red', 'Green', 'Blue']}
value={['Red', 'Blue']}
// Value should be an array, but no compiler error
// Everything is right, but the compiler complains
// because technically `newValue` could be an array
console.log(newValue.toLowerCase())
Sure, you could add some validation in your runtime code, like fancy custom propTypes validators, but wouldn’t it be nice if TypeScript could infer the correct types based on the component’s usage? After all, a type system isn’t just for catching bugs early, it should also guide developers unfamiliar with your API as they type, surfacing correct patterns and hiding invalid ones—a developer experience that runtime validation can’t provide.
Props Unions to the Rescue
Since you care deeply about developer experience, you decide to iterate on your initial API by applying what you know about union types to these props. It occurs to you that where you initially wrote multiple unions within a single interface, your intent is actually better expressed by one union of multiple interfaces:
interface CommonSelectProps {
interface SingleSelectProps extends CommonSelectProps {
multiple?: false;
interface MultipleSelectProps extends CommonSelectProps {
onChange: (newValue: string[]) => void;
type SelectProps = SingleSelectProps | MultipleSelectProps;
As triangles and quadrilaterals can be distinguished by their number of sides, the union type SelectProps can be discriminated by its multiple property. And as luck would have it, TypeScript will do exactly that when you pass (or don’t pass) the multiple prop to your new and improved component: 1
// Compiler knows that `value` shouldn’t be an array
// Compiler knows that `value` should be an array
// Compiler knows that `newValue` will be a string
Whoa, this is a bazillion times better! Nice work; consumers of your component will thank you for coaching them down the right path before they run their code in a browser. 🎉
Going Deeper with the Distributive Law of Sets
Time goes by. Your Select component was a big hit with the other developers who were using it. But then, the design team shows you specs for a Select component with groups of options, with customizable titles for each group. You start prototyping the props you’ll have to add in your head:
type OptionGroup = {
interface YourMentalModelOfChangesToSelectProps {
grouped?: boolean;
options: string[] | OptionGroup[];
renderGroupTitle?: (group: OptionGroup) => React.ReactNode;
Does this feel familiar? You have two distinct subsets of functionality, manifested over multiple props, discriminated by a single prop. The value of grouped controls the type of options and the validity of having a renderGroupTitle prop at all. You recognize that you could make these buckets of functionality a discriminated union of two separate interfaces, but how do you reconcile that with the API you already have, which is a discriminated union on multiple?
With two different choices to make (multiple and grouped), each with two options (true and false), there are four distinct options:
single selection, not grouped
multiple selection, not grouped
single selection, grouped
multiple selection, grouped.
Writing each of those options out as a complete interface of possible Select props and creating a union of all four isn’t unthinkably tedious, but the relationship is exponential: three boolean choices makes a union of
2^3 = 8
, four choices is 16, and so on. Rather sooner than later, it becomes unwieldy to express every combination of essentially unrelated choices explicitly.
You can avoid repeating yourself and writing out every combination by taking advantage of some set theory. Instead of writing four complete interfaces that repeat props from each other, you can write interfaces for each discrete piece of functionality and combine them via intersection:
interface SingleSelectPropsFragment {
multiple: false;
interface MultipleSelectPropsFragment {
interface UngroupedSelectPropsFragment {
grouped?: false;
interface GroupedSelectPropsFragment {
grouped: true;
options: OptionGroup[];
renderGroup: (group: OptionGroup) => React.ReactNode;
// All together now...
type SelectProps = CommonSelectProps &
(SingleSelectPropsFragment | MultipleSelectPropsFragment) &
(UngroupedSelectPropsFragment | GroupedSelectPropsFragment);
For each constituent in the union, we removed its extends clause so the interface reflects only a discrete subset of functionality that can be intersected cleanly with anything else. (In this example, that’s not strictly necessary, but I think it’s cleaner, and I have an unverified theory that it’s less work for the compiler.2) To reflect this change in our naming, we also suffixed each interface with Fragment to be clear that it’s not a complete working set of Select props.
We broke down grouped and non-grouped selects into two interfaces discriminated on grouped, just like we did before with multiple.
We combined everything together with an intersection of unions. In plain English, SelectProps is made up of:
CommonSelectProps, along with
either SingleSelectPropsFragment or MultipleSelectPropsFragment, along with
either UngroupedSelectPropsFragment or GroupedSelectPropsFragment.
The expression is evaluated according to set theory’s distributive law, which in a nutshell says that unions are like adding numbers and intersections are like multiplying numbers. In algebra, the distributive properties of multiplication and addition give us
Z(A + B)(C + D) = ZAC + ZAD + ZBC + ZBD
and set theory says the exact same thing about unions and intersections:
Z \cap (A \cup B) \cap (C \cup D) = (Z \cap A \cap B) \cup (Z \cap A \cap D) \cup (Z \cap B \cap C) \cup (Z \cap B \cap D)
If, like me, you haven’t studied computer science in an academic setting, this may look intimidatingly theoretical, but quickly make the following mental substitutions:
Set theory’s union operator,
\cup
, is written as | in TypeScript
Set theory’s intersection operator,
\cap
, is written as & in TypeScript3
Z =
CommonSelectProps
A =
SingleSelectPropsFragment
B =
MultipleSelectPropsFragment
C =
UngroupedSelectPropsFragment
D =
GroupedSelectPropsFragment
So, the resulting type of SelectProps expands to every possible combination that we outlined earlier. And TypeScript will discriminate between each of those four constituents based on the props you pass to Select3:
// `renderGroupTitle` doesn’t exist unless `grouped` is set
renderGroup={group => group.title}
// Everything together, looking good 👍🏽
options={[{
options: ['Red', 'Green', 'Blue']
value={['Red']}
// `multiple` still works, `newValue` is an array
newValue.forEach(value => {
Discriminated unions can be a powerful tool for writing better React component typings, but it’s not always the only way or the best way to write safe and expressive APIs. Swapping between string and string[] in multiple type positions, like we did with multiple, could be done with generics. But more poignantly, building a component with tons of unions could be a sign that the component is getting bloated and should be split into multiple components that can be composed via render props, higher order components, or any other means of component composition.
Discriminated Unions · TypeScript Deep Dive
Tagged union - Wikipedia
Interestingly, in the final case here, the explicit value multiple={false} is required not to pass type checking, but to get accurate inference on the argument to onChange. This seems like a limitation/bug to me.
My hypothesis is that in calculating the intersection of N types that all include common properties, the compiler must calculate for each of n common properties of type T that T intersected with itself N times is still T. This is surely not a computationally expensive code path, but unless there’s a clever short circuit early in the calculation, it still has to happen N ⨉ n times, all of which are unnecessary. This is purely unscientific speculation, and I would be happy for someone to correct or corroborate this theory.
This statement applies only in the type declaration space. | and & are bitwise operators in the variable declaration space. E.g., | is the union operator in var x: string | number but the bitwise or operator in var x = 0xF0 | 0x0F.
TypeScript does successfully discriminate between these constituents, but type inference is currently broken for properties that have different function signatures in different constituents when any of those constituents are an intersection type. |
Twelve people signed up to play darts during lunch. How many ways can a three-person dart team be chosen? For help, refer to the Math Notes box in this lesson.
The order in which they are selected does not matter.
\frac{_{12}P_3}{3!}=\frac{12!}{3!(12-3)!}
220 |
1,000,000 Knowpia
{\displaystyle {\stackrel {\rho }{\mathrm {M} }}}
Visualizing one millionEdit
Length: There are one million millimetres in a kilometre, and roughly a million sixteenths of an inch in a mile (1 sixteenth = 0.0625). A typical car tire might rotate a million times in a 1,900-kilometre (1,200 mi) trip, while the engine would do several times that number of revolutions.
Fingers: If the width of a human finger is 22 mm (7⁄8 in), then a million fingers lined up would cover a distance of 22 km (14 mi). If a person walks at a speed of 4 km/h (2.5 mph), it would take them approximately five and a half hours to reach the end of the fingers.
Volume: The cube root of one million is one hundred, so a million objects or cubic units is contained in a cube a hundred objects or linear units on a side. A million grains of table salt or granulated sugar occupies about 64 mL (2.3 imp fl oz; 2.2 US fl oz), the volume of a cube one hundred grains on a side. One million cubic inches would be the volume of a small room 8+1⁄3 feet long by 8+1⁄3 feet wide by 8+1⁄3 feet high.
Mass: A million cubic millimetres (small droplets) of water would have a volume of one litre and a mass of one kilogram. A million millilitres or cubic centimetres (one cubic metre) of water has a mass of a million grams or one tonne.
Landscape: A pyramidal hill 600 feet (180 m) wide at the base and 100 feet (30 m) high would weigh about a million short tons.
Money: A USD bill of any denomination weighs 1 gram (0.035 oz). There are 454 grams in a pound. One million USD bills would weigh 1 megagram (1,000 kg; 2,200 lb) or 1 tonne (just over 1 short ton).
Time: A million seconds, 1 megasecond, is 11.57 days.
In Indian English and Pakistani English, it is also expressed as 10 lakh. Lakh is derived from lakṣa for 100,000 in Sanskrit.
One million black dots (pixels) – each tile with white or grey background contains 1000 dots (full image)
Selected 7-digit numbers (1,000,001–9,999,999)Edit
1,000,001 to 1,999,999Edit
1,000,003 = Smallest 7-digit prime number
1,000,405 = Smallest triangular number with 7 digits and the 1,414th triangular number
1,002,001 = 10012, palindromic square
1,006,301 = First number of the first pair of prime quadruplets occurring thirty apart ({1006301, 1006303, 1006307, 1006309} and {1006331, 1006333, 1006337, 1006339})[9]
1,024,000 = Sometimes, the number of bytes in a megabyte[10]
1,030,301 = 1013, palindromic cube
1,048,576 = 10242 = 324 = 165 = 410 = 220, the number of bytes in a mebibyte (or often, a megabyte)
1,048,976 = smallest 7 digit Leyland number
1,058,576 = Leyland number
1,084,051 = fifth Keith prime[11]
1,089,270 = harmonic divisor number[12]
1,111,111 = repunit
1,136,689 = Pell number,[13] Markov number
1,185,921 = 10892 = 334
1,203,623 = smallest unprimeable number ending in 3[14][15]
1,278,818 = Markov number
1,299,709 = 100,000th prime number
1,346,269 = Fibonacci number,[16] Markov number
1,413,721 = square triangular number[17]
1,419,857 = 175
1,441,440 = colossally abundant number,[18] superior highly composite number[19]
1,563,372 = Wedderburn-Etherington number[20]
1,607,521/1,136,689 ≈ √2
1,671,800 = Initial number of first century xx00 to xx99 consisting entirely of composite numbers[21]
1,679,616 = 12962 = 364 = 68
1,686,049 = Markov prime
1,741,725 = equal to the sum of the seventh power of its digits
1,771,561 = 13312 = 1213 = 116, also, Commander Spock's estimate for the tribble population in the Star Trek episode "The Trouble with Tribbles"
1,928,934 = 2 x 39 x 72
1,953,125 = 1253 = 59
2,000,376 = 1263
2,097,152 = 1283 = 87 = 221
2,097,593 = Leyland prime[22]
2,124,679 = largest known Wolstenholme prime[23]
2,178,309 = Fibonacci number[16]
2,222,222 = repdigit
2,356,779 = Motzkin number[24]
2,674,440 = Catalan number[25]
2,692,537 = Leonardo prime
2,744,210 = Pell number[13]
2,796,203 = Wagstaff prime,[26] Jacobsthal prime
2,890,625 = 1-automorphic number[27]
2,985,984 = 17282 = 1443 = 126 = 1,000,00012 AKA a great-great-gross
3,263,442 = product of the first five terms of Sylvester's sequence
3,263,443 = sixth term of Sylvester's sequence[28]
3,301,819 = alternating factorial[29]
3,360,633 = palindromic in 3 consecutive bases: 62818269 = 336063310 = 199599111
3,426,576 = number of free 15-ominoes
3,626,149 = Wedderburn–Etherington prime[20]
3,628,800 = 10!
4,037,913 = sum of the first ten factorials
4,194,304 = 20482 = 411 = 222
4,210,818 = equal to the sum of the seventh powers of its digits
4,213,597 = Bell number[31]
4,324,320 = colossally abundant number,[18] superior highly composite number,[19] pronic number
4,782,969 = 21872 = 97 = 314
4,826,809 = 21972 = 1693 = 136
5,134,240 = the largest number that cannot be expressed as the sum of distinct fourth powers
5,496,925 = first cyclic number in base 6
5,882,353 = 5882 + 23532
7,652,413 = Largest n-digit pandigital prime
7,779,311 = A hit song written by Prince and released in 1982 by The Time
7,906,276 = pentagonal triangular number
7,913,837 = Keith number[11]
8,000,000 = Used to represent infinity in Japanese mythology
8,108,731 = repunit prime in base 14
8,388,607 = second composite Mersenne number with a prime exponent
8,675,309 = A hit song for Tommy Tutone (also a twin prime with 8,675,311)
8,675,311 = Twin prime with 8,675,309
8,946,176 = self-descriptive number in base 8
9,369,319 = Newman–Shanks–Williams prime[32]
9,699,690 = eighth primorial
9,938,375 = 2153, the largest 7-digit cube
9,997,156 = largest triangular number with 7 digits and the 4,471st triangular number
9,998,244 = 31622, the largest 7-digit square
9,999,991 = Largest 7-digit prime number
^ "million". Dictionary.com Unabridged. Random House, Inc. Retrieved 4 October 2010.
^ "m". Oxford Dictionaries. Oxford University Press. Retrieved 2015-06-30.
^ "6.7 Abbreviating 'million' and 'billion'". English Style Guide. A handbook for authors and translators in the European Commission (PDF) (2019 ed.). 26 February 2019. p. 37.
^ "m". Merriam-Webster. Merriam-Webster Inc. Retrieved 2015-06-30.
^ "Definition of 'M'". Collins English Dictionary. HarperCollins Publishers. Retrieved 2015-06-30.
^ Averkamp, Harold. "Q&A: What Does M and MM Stand For?". AccountingCoach.com. AccountingCoach, LLC. Retrieved 25 June 2015.
^ David Wells (1987). The Penguin Dictionary of Curious and Interesting Numbers. London: Penguin Group. p. 185. 1,000,000 = 106
^ Sloane, N. J. A. (ed.). "Sequence A059925 (Initial members of two prime quadruples (A007530) with the smallest possible difference of 30.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2019-01-27.
^ Tracing the History of the Computer - History of the Floppy Disk
^ a b "Sloane's A007629 : Repfigit (REPetitive FIbonacci-like diGIT) numbers (or Keith numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-17.
^ a b c "Sloane's A001599 : Harmonic or Ore numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-17.
^ a b c "Sloane's A000129 : Pell numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-17.
^ Collins, Julia (2019). Numbers in Minutes. United Kingdom: Quercus. p. 140. ISBN 978-1635061772.
^ Sloane, N. J. A. (ed.). "Sequence A143641 (Odd prime-proof numbers not ending in 5)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
^ a b c d e "Sloane's A000045 : Fibonacci numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-17.
^ "Sloane's A001110 : Square triangular numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-17.
^ a b "Sloane's A004490 : Colossally abundant numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-17.
^ a b "Sloane's A002201 : Superior highly composite numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-17.
^ a b c "Sloane's A001190 : Wedderburn-Etherington numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-17.
^ Sloane, N. J. A. (ed.). "Sequence A181098 (Primefree centuries)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2019-01-27.
^ "Sloane's A094133 : Leyland primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-17.
^ "Wolstenholme primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-17.
^ a b "Sloane's A001006 : Motzkin numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-17.
^ a b "Sloane's A000108 : Catalan numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-17.
^ "Sloane's A000979 : Wagstaff primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-17.
^ "Sloane's A000058 : Sylvester's sequence". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-17.
^ "Sloane's A005165 : Alternating factorials". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-17.
^ Sloane, N. J. A. (ed.). "Sequence A030984 (2-automorphic numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2021-09-01.
^ "Sloane's A000110 : Bell or exponential numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-17. |
Template for Package Command Help Page - Maple Help
Home : Support : Online Help : Applications and Example Worksheets : Language and System : Template for Package Command Help Page
<Package Command Template>
<For more information on using this template, refer to the ?Templates/PackageCommand help page. Delete this text and the title.>
<Package_Name>[<Command_Name>] <one line description of the command >
The <Command_Name>(<param1>, <param2>) calling sequence ...
This command is part of the <Package_Name> package, so it can be used in the form <Command_Name>(..) only after executing the command with(<Package_Name>). However, it can always be accessed through the long form of the command by using <Package_Name>[<Command_Name>](..).
\mathrm{with}\left(〈\mathrm{Package_Name}〉\right):
\mathrm{example1}
\mathrm{example2}
<related help topic>, <related help topic>, <related help topic>, <Package Overview> |
Worksheet/TableOfContents - Maple Help
Home : Support : Online Help : Worksheet/TableOfContents
create a table of contents at the beginning of a Maple document
TableOfContents( target )
TableOfContents( target , opts )
TableOfContents( target , destination )
TableOfContents( target , destination , opts )
string; name of file to read in .mw format
sting; name of file to write in .mw format
options; any number of options for the formatting of the table of contents
The command TableOfContents adds a table of contents of the section headings to the beginning of a Maple document (.mw). All headings in the table of contents are links to the corresponding section in the Maple document.
If the second argument destination is omitted then the generated file with a table of contents is opened in the Maple GUI. Otherwise, the generated file is written with a table of contents to the destination file in .mw format.
The opts argument can be one or more options to TableOfContents which are as follows:
The option depth indicates the number of subsections (including the first level) that are included in the table of contents. The default value is 2.
The option startingsize indicates the size of font of headings of the first level sections in the table of contents. The default value is 12.
The option sizeincrement indicates the decrease in font size between each level of section title in the table of contents. The default value is 2.
When the option bullets is specified the table of contents is created with indented bullet points instead of nested sections.
When the option nosection is specified the table is not contained in section titled Table of contents and instead has a header of the same title.
The option columns specifies the number of columns (as a positive integer) the table containing section titles has. The default is 0 which creates the table of contents without a table surrounding it. A value of columns 2 or greater splits the information across multiple columns to make the table of contents take less vertical space. A value of 1 creates the same table of contents as a value of 0 except contained within a table.
The option color changes the color of the text of all the section titles of the table of contents. The default is the color of hyperlinks. The value can be specified in any form that ColorTools can recognize.
When the option nounderline is specified all the hyperlink titles in the table of contents are formatted without underlines.
\mathrm{file}≔\mathrm{cat}\left(\mathrm{kernelopts}\left(\mathrm{datadir}\right),"/help/Worksheet/SimpleSectionDocument.mw"\right):
\mathrm{Worksheet}:-\mathrm{TableOfContents}\left(\mathrm{file}\right):
\mathrm{Worksheet}:-\mathrm{TableOfContents}\left(\mathrm{file},\mathrm{depth}=3,\mathrm{columns}=3,\mathrm{color}="Black",\mathrm{bullets},\mathrm{nounderline},\mathrm{startingsize}=14,\mathrm{sizeincrement}=3,\mathrm{nosection}\right):
The Worksheet:-TableOfContents command was introduced in Maple 2020. |
Double Entry Book Keeping Ts Grewal 2018 for Class 11 Commerce Accountancy Chapter 3 - Accounting Procedures Rules Of Debit And Credit
accounting procedures rules of debit and credit
Double Entry Book Keeping Ts Grewal 2018 Solutions for Class 11 Commerce Accountancy Chapter 3 Accounting Procedures Rules Of Debit And Credit are provided here with simple step-by-step explanations. These solutions for Accounting Procedures Rules Of Debit And Credit are extremely popular among Class 11 Commerce students for Accountancy Accounting Procedures Rules Of Debit And Credit Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Double Entry Book Keeping Ts Grewal 2018 Book of Class 11 Commerce Accountancy Chapter 3 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Double Entry Book Keeping Ts Grewal 2018 Solutions. All Double Entry Book Keeping Ts Grewal 2018 Solutions for class Class 11 Commerce Accountancy are prepared by experts and are 100% accurate.
(i) Land and Building (vii) Investments (xiv) Ramesh, a debtor
(ii) Excise Duty (viii) Salary (xv) Interest Received
(iii) Creditors (ix) Debtors (xvi) Bank Overdraft
(iv) Capital (x) Bad Debts (xvii) Purchase Returns
(v) Motor Vehicles (xi) Depreciation (xviii) Drawings
(vi) Goodwill (xii) Wages (xix) Freight
(xiii) Repairs (xx) Return Inwards.
Item Nature of Account
Land & Building Real A/c
Excise Duty Nominal A/c
Creditors Personal A/c
Capital Personal A/c
Motor Vehicles Real A/c
Goodwill Real A/c
Investments Real A/c
Salary Nominal A/c
Debtors Personal A/c
Bad Debts Nominal A/c
Depreciation Nominal A/c
Wages Nominal A/c
Repair Nominal A/c
Ramesh, a debtor Personal A/c
Interest Received Nominal A/c
Bank Overdraft Personal A/c
Purchase Returns Nominal A/c
Drawings Personal A/c
Freight Nominal A/c
Return Inwards Nominal A/c
Classify the following into Assets, Liabilities, Capital, Expenses and Revenue:
(i) Land; (ii) Investments; (iii) Building; (iv) Interest Received;
(v) Salary; (vi) Bank Overdraft; (vii) Debtors; (viii) Creditors;
(ix) Bad Debts; (x) Capital; (xi) Depreciation; (xii) Motor Vehicles;
(xiii) Freight; (xiv) Wages; (xv) Goodwill; (xvi) Repairs.
(i) Plant and Machinery; (ii) Bank Loan; (iii) Sales; (iv) Rent;
(v) Discount Received; (vi) Carriage Inwards; (vii) Carriage outwards; (viii) Purchases;
(ix) Bills Payable; (x) Wages; (xi) Advance Income; (xii) Accrued Income;
(xiii) Goodwill; (xiv) Furniture and Fixtures; (xv) Outstanding Expenses; (xvi) Capital.
On which side will the increase in the following accounts be recorded? Also, state the nature of the account:
(i) Furniture A/c (ii) Mohan (proprietor)
(iii) Salary A/c (iv) Purchases A/c
(v) Sales A/c (vi) Interest Paid A/c
(vii) Sohan (Creditor) (viii) Ram (Debtor)
Consequence of increase
Sohan (Creditor)
Ram (Debtor)
On which side will the decrease in the following accounts be recorded? Also, state the nature of the account:
(i) Cash (ii) Bank Overdraft
(iii) Outstanding Salary paid (iv) Outstanding Rent
(v) Prepaid Insurance (vi) Mohan, Proprietor of the business
Consequence of decrease
Outstanding Salary Paid
Mohan, proprietor of the business
(i) Manu started business with cash 1,00,000
(ii) He purchased furniture for business 20,000
(iii) Purchase goods on credit from Anshul 6,000
(iv) Paid to his creditor, Anshul 2,000
(v) Paid salary to his clerk 1,000
(vi) Paid rent 500
(vii) Received interest 200
Transactions Nature of Account
Manu started business with cash Cash A/c- Debit Capital A/c- Credit
He purchased furniture for business Furniture A/c- Debit Cash A/c- Credit
Purchased goods on credit from Anshul Purchases A/c- Debit Creditor A/c- Credit
Paid to his creditor, Anshul Creditor A/c- Debit Cash A/c- Credit
Paid salary to his clerk Salary A/c- Debit Cash A/c- Credit
Paid rent Rent A/c- Debit Cash A/c- Credit
Received interest Cash A/c- Debit Interest A/c- Credit
Analyse the following transactions , state the nature of accounts and state which account will be debited and which account credited:
(i) Lal started business with cash 3,00,000
(ii) Purchased furniture for cash from Shanti Furniture House 75,000
(iii) Purchase goods for cash 55,000
(iv) Sold goods for cash to Pal 35,000
(v) Sold goods to Om on credit 60,000
(vi) Deposited cash in bank for opening an account 70,000
(vii) Received a cheque from Om 20,000
(viii) Deposited Om's cheque the next day
(ix) Borrowed from Sohan 1,00,000
(x) Purchased furniture from Haryana safe 50,000
(xi) Paid interest on loan 10,000
(xii) Paid rent by cheque 4,000
(xiii) Paid salary to staff 14,000
(xiv) Witrhdrew cash for personal use 5,000
Lal started business with cash Cash A/c- Debit Capital A/c- Credit
Purchased furniture for cash from Shanti Furniture House Furniture A/c- Debit Cash A/c- Credit
Purchased goods for cash Purchases A/c- Debit Cash A/c- Credit
Sold goods for cash to Pal Cash A/c- Debit Sales A/c- Credit
Sold goods to Om on credit Om A/c- Debit Sales A/c- Credit
Deposited cash in bank for opening an account Bank A/c- Debit Cash A/c- Credit
Received a cheque from Om Cheques in Hand A/c- Debit Om A/c- Credit
Deposited Om’s cheque the next day Bank A/c- Debit Cheques in Hand A/c- Credit
Borrowed from Sohan Cash A/c- Debit Loan from Sohan A/c- Credit
Purchased furniture from Haryana Safe Furniture A/c- Debit Haryana Safe A/c- Credit
Paid interest on loan Interest on Loan A/c- Debit Cash A/c- Credit
Paid rent by cheque Rent A/c- Debit Bank A/c- Credit
Paid salary to staff Salary A/c- Debit Cash A/c- Credit
Withdrew cash for personal use Drawings A/c- Debit Cash A/c- Credit
Open a 'T' shape account for machinery and put the following transactions on the proper side:
(i) Machinery purchased 40,000
(ii) Machinery sold 10,000
(iii) Machinery purchased 8,000
(iv) Machinery discarded 14,000
(v) Depreciation on machinery 1,000
Machinery(Asset) Account
Bank (Purchased)
Bank (Sale)
Bank (Discarded)
Open a 'T' shape Cash Account with the following transactions:
(i) Mohan started business with cash 40,000
(ii) Purchased Goods 20,000
(iii) Sold Goods 24,000
(iv) Paid Rent 400
(v) Paid salaries 600
(vi) Drew for personal use 1,000
Open a 'T' shape account of creditor, 'Rakesh', and write the following transactions on the proper side:
(i) Goods purchased from Rakesh on credit 50,000
(ii) Goods returned to Rakesh for 5,000
(iii) Paid to Rakesh 20,000
(iv) Purchase goods from Rakesh on credit 10,000
Open a 'T' shape account of debtor 'Brij' and write the following transactions on the proper side:
(i) Sold goods to Brij on credit ₹ 25,000
(ii) Cash received from Brij
Discount allowed to him
(iii) Goods returned by Brij ₹ 5,000
Brij (Debtors)
Put the following on the proper side of a Cash Account, a Debtor's Account and a Creditor's Account:
(i) Sold goods to Sanjay on credit 50,000
(ii) Sold goods to Mohan for cash 20,000
(iii) Purchased goods from Ram on credit 25,000
(iv) Cash received from Sanjay 19,000
(v) Goods returned by Sanjay 2,000
(vii) Cash paid to Ram 15,000
Sanjay (Debtors)
Ram (Creditors)
From the following particulars, prepare the proprietor's Capital Account:
1st April, 2017
—
Commenced business with cash 2,00,000
—
Net Loss as per Profit and Loss Account 18,000
—
Drawings during the period 15,000
Profit and Loss A/c (Net Loss)
Note: As per the calculation Capital Balance as on 31st March, 2018 should be Rs.1,67,000 but the answer given in the text book is Rs.16,700.
April 1 Started business with 45,000
May 10 Withdrew from business for personal use 10,000
July 15 Further Capital introduced 55,000
Nov. 30 Income tax paid 5,000
Mar. 31 Profit for the year 30,000
Proprietor’s Capital Account
Drawings A/c ( Income Tax) |
How Vortex works - Smartlink
Vortex is an automated market maker (AMM) liquidity protocol. Instead of a classic order book, a factory smart contract creates liquidity pool smart contracts for each pair.
This means the first person that adds liquidity to a pair also creates the unique pool for that particular pair.
The pools keep track of the liquidity added and removed at all times.
For a given liquidity pool, each pool uses the function x * y = k to maintain a curve along which trades can happen, with x and y indicating the quantity of the tokens, and k a scalar (invariant).
Fig.1: The function x * y = k
When someone trades, it is equivalent to moving on the curve and the prices adjust accordingly to maintain the value of ‘k.’
When you swap a token, you’re trading against the pool and it doesn’t require a matching order from another user.
1000
A tokens and
1000
XTZ in the pool,
k=1000*1000=1000000.
If a trader wants to trade
20
A tokens against some XTZ, without fees he would receive
x
XTZ such that :
(1000+20)*(1000-x)=k=1000000
x = \frac{20000}{1020}≈19.608
But since there are fees (see Price and Fees), the trader will actually receive:
{\bar {x}}=\frac{9972*1000*20}{10000*1020}=\frac{9972}{10000} x≈19.553
Since the fees remain in the pool as long as liquidity providers do not withdraw liquidity besides the fees to the Smartlink Treasury and towards $SMAK token buy back and burn which together are worth:
f=\frac{3x}{10000}=0.006,
k
slightly increases to:
(1000 + 20)*(1000 − {\bar {x}} − f ) = 1000050
However, still with
1000
1000
XTZ in the reserve, if a trader wants to trade 20 XTZ against some A tokens, he would receive y A tokens such that:
(1000 + 0.9972* 20)*(1000 − y) = k .
y=\frac{0,9972*20*1000 }{1000 + 0,9972*20}≈19.554
k
(1000 + 0.9997* 20)*(1000 − y) ≈ 1000049
Click on the picture to zoom in:
Each pool is decentralized and includes functionalities to swap tokens and manage liquidity, including adding and removing liquidity. |
GNUnet - Wikipedia
(Redirected from Gnunet)
Framework for decentralized, peer-to-peer networking which is part of the GNU Project
Not to be confused with Gnutella.
Find sources: "GNUnet" – news · newspapers · books · scholar · JSTOR (April 2019) (Learn how and when to remove this template message)
GNUnet with the GTK+ user interface
GNUnet e.V.[1]
official: Free software operating systems (Linux, FreeBSD, NetBSD, OpenBSD);
unofficial: Other operating systems (OS X, Windows)
Spanish, English, Russian, German, French
Anonymous P2P, Friend-to-friend
2018: AGPL-3.0-or-later[a][4]
Christian Grothoff, maintainer of GNUnet, in Berlin on August 1, 2013 at the "#youbroketheinternet. We'll make ourselves a GNU one." event.
GNUnet is a software framework for decentralized, peer-to-peer networking and an official GNU package. The framework offers link encryption, peer discovery, resource allocation, communication over many transports (such as TCP, UDP, HTTP, HTTPS, WLAN and Bluetooth) and various basic peer-to-peer algorithms for routing, multicast and network size estimation.[5][6]
GNUnet's basic network topology is that of a mesh network. GNUnet includes a distributed hash table (DHT) which is a randomized variant of Kademlia that can still efficiently route in small-world networks. GNUnet offers a "F2F topology" option for restricting connections to only the users' trusted friends. The users' friends' own friends (and so on) can then indirectly exchange files with the users' computer, never using its IP address directly.
GNUnet uses Uniform resource identifiers (not approved by IANA, although an application has been made).[when?] GNUnet URIs consist of two major parts: the module and the module specific identifier. A GNUnet URI is of form
gnunet://module/identifier where module is the module name and identifier is a module specific string.
The primary codebase is written in C, but there are bindings in other languages to produce an API for developing extensions in those languages. GNUnet is part of the GNU Project. It has gained interest in the hacker community after the PRISM revelations.[7]
GNUnet consists of several subsystems, of which essential ones are Transport and Core subsystems.[8] Transport subsystem provides insecure link-layer communications, while Core provides peer discovery and encryption.[9] On top of the core subsystem various applications are built.
GNUnet includes various P2P applications in the main distribution of the framework, including filesharing, chat and VPN; additionally, a few external projects (such as secushare) are also extending the GNUnet infrastructure.
GNUnet is unrelated to the older Gnutella P2P protocol. Gnutella is not an official GNU project, while GNUnet is.[10]
3.1 File encoding
3.2 Queries and replies
3.3 File sharing URIs
4 GNU Name System
5 Protocol translation
6 Social API
Originally, GNUnet used UDP for underlying transport.[11] Now GNUnet transport subsystem provides multiple options, such as TCP and SMTP.[12]
The communication port, officially registered at IANA, is 2086 (tcp + udp).[13]
Trust system[edit]
GNUnet provides trust system based on excess-based economic model.[14] The idea of employing economic system is taken from MojoNation network.[15]
GNUnet network has no trusted entities so it is impossible to maintain global reputation. Instead, each peer maintains its own trust for each of its local links.
When resources, such as bandwidth and CPU time, are in excess, peer provides them to all requesting neighbors without reducing trust or otherwise charging them. When a node is under stress it drops requests from its neighbor nodes having lower internal trust value. However, when peer has less resources than enough to fulfill everyone's requests, it denies requests of those neighbors that it trusts less and charges others by reducing their trust.
The primary application at this point is anonymous, censorship-resistant file-sharing, allowing users to anonymously publish or retrieve information of all kinds. The GNUnet protocol which provides anonymity is called GAP (GNUnet anonymity protocol).[16] GNUnet FS can additionally make use of GNU libextractor to automatically annotate shared files with metadata.
File encoding[edit]
Files shared with GNUnet are ECRS (An Encoding for Censorship-Resistant Sharing) coded.[17]
All content is represented as GBlocks. Each GBlock contains 1024 bytes. There are several types of GBlocks, each of them serves a particular purpose. Any GBlock
{\displaystyle B}
is uniquely identified by its RIPEMD-160 hash
{\displaystyle H(B)}
DBlocks store actual file contents and nothing else. File is split at 1024 byte boundaries and resulting chunks are stored in DBlocks. DBlocks are linked together into Merkle tree by means of IBlocks that store DBlock identifiers.
Blocks are encrypted with a symmetric key derived from
{\displaystyle H(B)}
when they are stored in the network.
Queries and replies[edit]
GNUnet Anonymity Protocol consists of queries and replies. Depending on load of the forwarding node, messages are forwarded to zero or more nodes.
Queries are used to search for content and request data blocks.
Query contains resource identifier, reply address, priority and TTL (Time-to-Live).
Resource identifier of datum
{\displaystyle Q}
is a triple-hash
{\displaystyle H(H(H(Q)))}
.[18] Peer that replies to query provides
{\displaystyle H(H(Q))}
to prove that it indeed has the requested resource without providing
{\displaystyle H(Q)}
to intermediate nodes, so intermediate nodes can't decrypt
{\displaystyle Q}
Reply address is the major difference compared to Freenet protocol. While in Freenet reply always propagates back using the same path as the query, in GNUnet the path may be shorter. Peer receiving a query may drop it, forward it without rewriting reply address or indirect it by replacing reply address with its own address. By indirecting queries peer provides cover traffic for its own queries, while by forwarding them peer avoids being a link in reply propagation and preserves its bandwidth. This feature allows the user to trade anonymity for efficiency. User can specify an anonymity level for each publish, search and download operation. An anonymity level of zero can be used to select non-anonymous file-sharing. GNUnet's DHT infrastructure is only used if non-anonymous file-sharing is specified. The anonymity level determines how much cover traffic a peer must have to hide the user's own actions.
Priority specifies how much of its trust user wants to spend in case of resource shortage.
TTL is used to prevent queries from staying in the network for too long.
File sharing URIs[edit]
The fs module identifier consists of either chk, sks, ksk or loc followed by a slash and a category specific value. Most URIs contain hashes, which are encoded in base32hex.[19]
chk identifies files, typically: gnunet://fs/chk/[file hash].[query hash].[file size in bytes]
File hash is the hash of the plaintext file, which allows decrypting it once it is downloaded. Query hash is the hash of topmost GBlock which allows downloading the whole tree of GBlocks that contain encrypted file. File size is required to determine the shape of the tree.
sks identifies files within namespaces, typically: gnunet://fs/sks/NAMESPACE/IDENTIFIER
ksk identifies search queries, typically: gnunet://fs/ksk/KEYWORD[+KEYWORD]*
loc identifies a datum on a specific machine, typically: gnunet://fs/loc/PEER/QUERY.TYPE.KEY.SIZE
A type of GNUnet filesharing URI pointing to a specific copy of GNU GPL license text: gnunet://fs/chk/9E4MDN4VULE8KJG6U1C8FKH5HA8C5CHSJTILRTTPGK8MJ6VHORERHE68JU8Q0FDTOH1DGLUJ3NLE99N0ML0N9PIBAGKG7MNPBTT6UKG.1I823C58O3LKS24LLI9KB384LH82LGF9GUQRJHACCUINSCQH36SI4NF88CMAET3T3BHI93D4S0M5CC6MVDL1K8GFKVBN69Q6T307U6O.17992
Another type of GNUnet filesharing URI, pointing to the search results of a search with keyword "gpl": gnunet://fs/ksk/gpl
GNU Name System[edit]
GNUnet includes an implementation of the GNU Name System (GNS), a decentralized and censorship-resistant replacement for DNS. In GNS, each user manages their own zones and can delegate subdomains to zones managed by other users. Lookups of records defined by other users are performed using GNUnet's DHT.[20]
Protocol translation[edit]
GNUnet can tunnel IP traffic over the peer-to-peer network. If necessary, GNUnet can perform IPv4-IPv6 protocol translation in the process. GNUnet provides a DNS Application-level gateway to proxy DNS requests and map addresses to the desired address family as necessary. This way, GNUnet offers a possible technology to facilitate IPv6 transition. Furthermore, in combination with GNS, GNUnet's protocol translation system can be used to access hidden services — IP-based services that run locally at some peer in the network and which can only be accessed by resolving a GNS name.
Social API[edit]
Gabor X Toth published in early September 2013 a thesis[21] to present the design of a social messaging service for the GNUnet peer-to-peer framework that offers scalability, extensibility, and end-to-end encrypted communication. The scalability property is achieved through multicast message delivery, while extensibility is made possible by using PSYC (Protocol for SYnchronous Conferencing), which provides an extensible RPC (Remote Procedure Call) syntax that can evolve over time without having to upgrade the software on all nodes in the network. Another key feature provided by the PSYC layer are stateful multicast channels, which are used to store e.g. user profiles. End-to-end encrypted communication is provided by the mesh service of GNUnet, upon which the multicast channels are built. Pseudonymous users and social places in the system have cryptographical identities — identified by their public key — these are mapped to human memorable names using GNS (GNU Name System), where each pseudonym has a zone pointing to its places.
That is the required building block for turning the GNUnet framework into a fully peer-to-peer social networking platform.
A chat has been implemented in the CADET module,[22] for which a GTK interface for GNOME exists,[23] specifically designed for the emerging Linux phones (such as the Librem 5 or the PinePhone).[24]
^ AGPL-3.0-or-later since 2018-06-05.
^ GPL-2.0-or-later from 2001 until 2007-07-02.
^ GNUnet.org – About GNUnet e.V.
^ "GNUnet 0.16.3 released"; author name string: Martin Schanzenbach; publication date: 29 March 2022; retrieved: 30 March 2022.
^ GNUnet Source Code
^ "license notice placed at the top in one of the source files of the project's repository, probably in each of its source files". Retrieved June 8, 2018. GNUnet is free software: you can redistribute it and/or modify it under the terms of the GNU Affero General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.
^ EvansPolotGrothoff 2012.
^ Grothoff, Christian (2017-10-10). The GNUnet System (habilitation thesis). Université de Rennes 1.
^ Grothoff 2013.
^ FerreiraGrothoffRuth 2003, Figure 1.
^ FerreiraGrothoffRuth 2003, II. GNUNET AND THE TRANSPORT LAYER: "The core is responsible for link-to-link encryption, binding of node identities to routable addresses, and peer discovery."
^ "GNU Software". Software - GNU Project - Free Software Foundation. Free Software Foundation, Inc. Retrieved 25 January 2020.
^ GrothoffPatrascuBennettStef 2002, 3.1.1 UDP.
^ FerreiraGrothoffRuth 2003.
^ Service Name and Transport Protocol Port Number Registry, p. 38
^ GrothoffPatrascuBennettStef 2002, 2.4 Mojo Nation.
^ BennettGrothoff 2003.
^ BennettGrothoffHorozovPatrascu 2002.
^ GrothoffPatrascuBennettStef 2002, 5.5 More on queries.
^ Grothoff, Christian. "File-Sharing URIs". www.gnunet.org. Retrieved 15 July 2016.
^ Wachs 2014. sfn error: no target: CITEREFWachs2014 (help)
^ Toth, Gabor X (2013-09-05), Design of a Social Messaging System Using Stateful Multicast Archived 2014-02-22 at the Wayback Machine - Retrieved 2013-09-28
^ GNUnet.org documentation (2019-11-14) Chatting with a (simple) client - Retrieved 2019-11-14
^ cadet-gtk on GitLab
^ GNUnet CADET for mobile Linux – Reddit
Grothoff, Christian; Patrascu, Ioana; Bennett, Krista; Stef, Tiberiu; Horozov, Tzvetan (2002-06-13). The GNet whitepaper (PDF) (Technical report). Purdue University.
Bennett, Krista; Grothoff, Christian; Horozov, Tzvetan; Patrascu, Ioana (2002-07-03). Batten, Lynn; Seberry, Jennifer (eds.). Efficient Sharing of Encrypted Data. Lecture Notes in Computer Science. Springer Berlin Heidelberg. pp. 107–120. CiteSeerX 10.1.1.19.9837. doi:10.1007/3-540-45450-0_8. ISBN 9783540438618.
Ferreira, Ronaldo; Grothoff, Christian; Ruth, Paul (2003-05-01). "A transport layer abstraction for peer-to-peer networks" (PDF). CCGrid 2003. 3rd IEEE/ACM International Symposium on Cluster Computing and the Grid, 2003. Proceedings. IEEE. pp. 398–405. CiteSeerX 10.1.1.13.5086. doi:10.1109/CCGRID.2003.1199393. ISBN 978-0-7695-1919-7. S2CID 1928711.
Grothoff, Dipl-Math Christian (2003-06-01). "Resource allocation in peer-to-peer networks". Wirtschaftsinformatik. 45 (3): 285–292. doi:10.1007/BF03254946. ISSN 0937-6429. S2CID 4479637.
Wachs, Matthias; Schanzenbach, Martin; Grothoff, Christian (2014). "A Censorship-Resistant, Privacy-Enhancing and Fully Decentralized Name System" (PDF). 13th International Conference on Cryptology and Network Security (CANS 2014). Lecture Notes in Computer Science. 8813 (13): 127–142. doi:10.1007/978-3-319-12280-9_9. ISBN 978-3-319-12279-3.
Bennett, Krista; Grothoff, Christian (2003-03-26). Dingledine, Roger (ed.). gap – Practical Anonymous Networking. Lecture Notes in Computer Science. Springer Berlin Heidelberg. pp. 141–160. CiteSeerX 10.1.1.125.9673. doi:10.1007/978-3-540-40956-4_10. ISBN 9783540206101.
Evans, Nathan; Polot, Bartlomiej; Grothoff, Christian (2012-05-21). Bestak, Robert; Kencl, Lukas; Li, Li Erran; Widmer, Joerg; Yin, Hao (eds.). Efficient and Secure Decentralized Network Size Estimation. Lecture Notes in Computer Science. Springer Berlin Heidelberg. pp. 304–317. doi:10.1007/978-3-642-30045-5_23. ISBN 9783642300448.
Grothoff, Christian (2013-08-01). "Video: You broke the Internet. We're making ourselves a GNU one". gnunet.org. Pirate Party Germany, Berlin. Retrieved 4 October 2013.
Grothoff, Christian (2015). "The Architecture of the GNUnet: 45 Subsystems in 45 Minutes" (PDF). Retrieved 2016-07-14.
Retrieved from "https://en.wikipedia.org/w/index.php?title=GNUnet&oldid=1083007955" |
Ashley Sheridan - How Readable is Your Content?
How Readable is Your Content?
The readability of your content is one of the most important accessibility issues you will encounter, but is often one of the least thought about. Readability affects everyone who browses the Web, and can mean the difference between retaining or scaring your site visitors.
The readability of your content is how easily understood your content is. One of the most useful measures of this is the Flesch-Kinkaid test, a formula that scores content and produces a reading age for English text.
206.835-1.015\left(\frac{\mathrm{total words}}{\mathrm{total sentences}}\right)-84.6\left(\frac{\mathrm{total syllables}}{\mathrm{total words}}\right)
The higher the score, the easier the content is to read, while lower scores indicate content that is more difficult. To give the scores some context, the values produced from running the equation can be looked up against this table (taken from the Wikipedia article on the FRES test which uses American school terms for comparisons):
n ≥ 90 5th grade Very easy to read. Easily understood by an average 11-year-old student.
90 > n ≥ 80 6th grade Easy to read. Conversational English for consumers.
80 > n ≥ 70 7th grade Fairly easy to read.
70 > n ≥ 60 8th & 9th grade Plain English. Easily understood by 13 to 15 year-old students.
60 > n ≥ 50 10th to 12th grade Fairly diffficult to read.
50 > n ≥ 30 College Difficult to read.
n < 30 College graduate Very difficult to read. Best understood by university graduates.
Hopefully, this should be obvious. As well as their actual age, there are a plethora of other reasons why somebody might find content difficult to read, and they don't just relate to problems that are typically considered disabilities under the law. Issues like Dyslexia, or Down's syndrome, will obviously have an impact on a persons ability to learn and thus impact their reading. But what if English isn't their language? Tiredness might impact how easily a person can read too.
If your visitors aren't able to comprehend your content, then they're more likely to leave your website. If you rely on this digital footfall then you want to keep those people. Ensuring they understand you is essential in retaining them.
I've put together a simple checker in Javascript that will calculate the score and from that, select the corresponding reading age from the examples in the above table.
This is not perfect, and the syllable counting code could probably do with a little extra work around the syllable counting code, but it seems to do the job to within a fair degree of accuracy.
The Counting Functions
At the heart of this is the code to produce the counts for sentences, words, and syllables:
function get_sentence_count(sample_text) { let sentences = sample_text.match(/.[.?!](\s|$)/gm); if(sentences) return sentences.length; return 1; }
This is used to get the number of sentences in a piece of text using a regular expression to count, and identifies the end of a sentence with a period, exclamation mark, or question mark. If I was really going to flesh this out fully, I'd look at using other sentence endings, like the infrequently used but beautiful interrobang "‽". I end it by returning a count of 1, which assumes that there's always at least one sentence in-case the regular expression doesn't match anything. A simple check of the sample string length before attempting any text metrics will prevent false positives.
Word count is done a little more simply. I trim whitespace, replace multiple occurrences of whitespace within the sample text with a single space, and then break the string into an array of "words" using the single space as a delimiter. It's crude, but should do the job:
let all_words = sample_text.trim().replace(/\s+/gi, ' ').split(' '); let word_count = all_words.length;
This isn't in a function of its own, as I needed the list of words to loop through and count the syllables:
function get_syllables_in_text(word) { word = word.toLowerCase(); if(word.length <= 3 && !word.match(/^[aeiou].y$/)) return 1; let vowel_count = word.match(/[aiouy]+e*|e(?!d$|ly).|[td]ed|le$/g); return (vowel_count) ? vowel_count.length : 1; }
Firstly we check to see if the passed in word is 3 letters or less, and if it doesn't begin with a vowel and end in a letter y (which is a vowel in some circumstances).
After that, I find all matches for a a regular expression that looks for any of the following:
One or more of the letters "a, i, o, u, y", followed by zero or more letter e's
The letter "e" followed by anything that *isn't* a "d" and end of the word, or *isn't* an "ly" and any other character
Either the letter "t" or "d" followed by "ed"
Or finally "le" and the end of the word
If no matches can be found, I return 1, as all words should have at least one syllable.
After that, it's just about putting it all together in a single formula:
let reading_level = (206.835 - 1.015 * (word_count / sentence_count) - 84.6 * (syllable_count / word_count) );
So, you've run the test, and you have your website contents reading level. Now what? It's important to think about your target audience and the subjects you're covering.
For example, the majority of the articles on this site are technical in nature, and are aimed at other developers. One side-effect of this subject is that there are a lot of acronyms and technical terms which will influence the Flesch-Kinkaid algorithm towards the difficult end of the spectrum. As the subject requires a higher level of technical knowledge and understanding, I fully expect the content to reflect this, so I accept that the reading age will be higher than average.
However, a website that's aimed at home bakers and offers recipes would likely expect a much more varied audience. So it would be prudent to aim the content at a lower reading level, in order to make it accessible to as wide an audience as possible.
The Flesch-Kinkaid Test and Non-English Languages
The Flesch-Kinkaid algorithm was specifically created for English, so it doesn't really work for other languages. The main problem is that of average syllables in a given language, usually indicated as an average per 100 words in a given text.
This test should work with other romance languages with some tweaking to the formula seed values. However, depending on the specific language, there might be better choices specifilly created, such as the Fry Readability Graph for Spanish.
Outside of the romance languages, there are languages in which the syllable count doesn't make much sense as a metric. The AARI looks at word length of Arabic content after removing ال from the beginning of words.
The Affects of Covid on Web Accessibility
Building a Web-Based Character Picker
Picking The Right HTML Tag |
Difference between revisions of "Tables" - Boolean Functions
* [[Known infinite families of quadratic APN polynomials over GF(2^n)]]
* [[Known switching classes of APN functions over GF(2^n) for n = 5,6,7,8]]
* [[CCZ-inequivalent APN functions from the known APN classes over GF(2^n) (for n is equal or larger than 6 and equal or smaller than 11)]]
* [[CCZ-inequivalent APN functions from the known APN classes over GF(2^n) (for n between 6 and 11)]]
* [[Known quadratic APN polynomial functions over GF(2^7)]]
Known instances of APN functions over
{\displaystyle \mathbb {F} _{2^{n}}}
Known APN power functions over GF(2^n) with n less than or equal to 13
Differential uniformity of all power functions over GF(2^n) with n less than or equal to 13
Inverses of APN power permutations over GF(2^n) with n less than or equal to 129
Known quadratic APN polynomial functions over GF(2^7) |
Order (relation) - Citizendium
Order (relation)
In mathematics, an order relation is a relation on a set which generalises the notion of comparison between numbers and magnitudes, or inclusion between sets or algebraic structures.
Throughout the discussion of various forms of order, it is necessary to distinguish between a strict or strong form and a weak form of an order: the difference being that the weak form includes the possibility that the objects being compared are equal. We shall usually denote a general order by the traditional symbols < or > for the strict form and ≤ or ≥ for the weak form, but notations such as
{\displaystyle {\subset },{\supset }}
{\displaystyle {\subseteq },{\supseteq }}
{\displaystyle {\prec },{\succ }}
{\displaystyle {\preceq },{\succeq }}
{\displaystyle {\sqsubset },{\sqsupset }}
{\displaystyle {\sqsubseteq },{\sqsupseteq }}
are also common. We also use the traditional notational convention that
{\displaystyle x<y\Leftrightarrow y>x}
An ordered set is a pair (X,<) consisting of a set and an order relation.
1 Partial order
3.1 Mappings of ordered sets
3.3 Dilworth's theorem
4.1 Semi-modular lattices
4.2 Modular lattices
4.4 Complemented lattices
4.5 Subjunctive lattice
4.6 Boolean lattice
4.7 Lattice homomorphisms
4.8 Ideals and filters
The most general form of order is the (weak) partial order, a relation ≤ on a set satisfying:
{\displaystyle x\leq x;\,}
Antisymmetric:
{\displaystyle x\leq y,y\leq x\Rightarrow x=y;\,}
{\displaystyle x\leq y,y\leq z\Rightarrow x\leq z.\,}
The strict form < of an order satisfies the variant conditions:
Irreflexive:
{\displaystyle x\not <x;\,}
{\displaystyle x<y\Rightarrow y\not <x;\,}
{\displaystyle x<y,y<z\Rightarrow x<z.\,}
Weak and strict partial orders are equivalent via the following translations:
{\displaystyle x\leq y}
{\displaystyle x<y}
{\displaystyle x=y;}
{\displaystyle x<y}
{\displaystyle x\leq y}
{\displaystyle x\neq y.}
A reflexive and transitive relation is called a preorder. In a preorder the relation defined by
{\displaystyle x\leq y\leq x}
is an equivalence relation, and the preorder gives rise to a partial order on the corresponding equivalence classes.
A total or linear order is one which has the trichotomy property: for any x, y exactly one of the three statements
{\displaystyle x<y}
{\displaystyle x=y}
{\displaystyle x>y}
If a ≤ b in an ordered set (X,<) then the interval
{\displaystyle [a,b]=\{x\in X:a\leq x\leq b\}.\,}
We say that b covers a if the interval
{\displaystyle [a,b]=\{a,b\}}
: that is, there is no x strictly between a and b. We write
{\displaystyle a\prec b}
{\displaystyle b\succ a}
Let S be a subset of a ordered set (X,<). An upper bound for S is an element U of X such that
{\displaystyle U\geq s}
{\displaystyle s\in S}
. A lower bound for S is an element L of X such that
{\displaystyle L\leq s}
{\displaystyle s\in S}
. A set is bounded if it has both lower and upper bounds. In general a set need not have either an upper or a lower bound. A directed set is one in which any finite set has an upper bound.
The set of upper bounds for S is denoted UB(S); the set of lower bounds is LB(S).
A supremum for S is an upper bound which is less than or equal to any other upper bound for S; an infimum is a lower bound for S which is greater than or equal to any other lower bound for S. In general a set with upper bounds need not have a supremum; a set with lower bounds need not have an infimum. The supremum or infimum of S, if one exists, is unique
A maximum for S is an upper bound which is in S; a minimum for S is a lower bound which is in S. A maximum is necessarily a supremum, but a supremum for a set need not be a maximum (that is, need not be an element of the set); similarly an infimum need not be a minimum.
A maximum element for the whole set may be termed top, one or true and denoted by
{\displaystyle \top }
or 1; a minimum element for the whole set may be termed bottom, zero or false and denoted
{\displaystyle \bot }
or 0. An ordered set with a 0 and 1 is bounded.
In a bounded order, an atom is an element that covers 0.
An antichain is a subset of an ordered set in which no two elements are comparable. The width of a partially ordered set is the largest cardinality of an antichain.
A subset S of an ordered set X is downward closed or a lower set if it satisfies
{\displaystyle x\leq s\in S\Rightarrow x\in S.\,}
Similarly, a subset S of an ordered set X is upward closed or an upper set if it satisfies
{\displaystyle x\geq s\in S\Rightarrow x\in S.\,}
A (Dedekind) cut in an ordered set X is a pair (A,B) of subsets of X such that B is the set of upper bounds of A and A is the set of lower bounds of B: B = UB(A) and A = LB(B). We may equivalently define a cut by A = LB(UB(A)), whereas in general A is merely a subset of LB(UB(A)).
Mappings of ordered sets
A function from an ordered set (X,<) to (Y,<) is monotonic or monotone increasing if it preserves order: that is, when x and y satisfy
{\displaystyle x\leq y}
{\displaystyle f(x)\leq f(y)}
. A monotone decreasing function similarly reverses the order. A function is strictly monotonic if
{\displaystyle x<y}
{\displaystyle f(x)<f(y)}
: such a function is necessarily injective.
An order isomorphism, or simply isomorphism between ordered sets is a monotonic bijection.
A chain is a subset of an ordered set for which the induced order is total. An ordered set satisfies the ascending chain condition (ACC) if every strictly increasing chain is finite, and the descending chain condition (DCC) if every strictly decreasing chain is finite. An order relation satisfying the DCC is also termed well-founded.
A maximal chain is a chain which cannot be extended by any element and still be linearly ordered (it is maximal within the family of chains ordered by set-theoretic inclusion).
The dimension of an element x in an ordered set with 0 is the length d(x) of a longest maximal chain from 0 to x.
Dilworth's theorem states that the width of an ordered set, the maximal size of an antichain, is equal to the minimal number of chains which together covers the set.
A lattice is an ordered set in which any two element set
{\displaystyle \{a,b\}}
has a supremum and an infimum. We call the supremum the join and the infimum the meet of the elements a and b, denoted
{\displaystyle a\vee b}
{\displaystyle a\wedge b}
The join and meet satisfy the properties:
Idempotence:
{\displaystyle x\vee x=x=x\wedge x;\,}
{\displaystyle x\vee y=y\vee x;~x\wedge y=y\wedge x;\,}
{\displaystyle x\vee (y\vee z)=(x\vee y)\vee z;~x\wedge (y\wedge z)=(x\wedge y)\wedge z;\,}
Absorption (mathematics):
{\displaystyle x\wedge (x\vee y)=x;~x\vee (x\wedge y)=x;\,}
These four properties characterize a lattice. The order relation may be recovered from the join and meet by
{\displaystyle a\vee b=b\Leftrightarrow a\leq b\Leftrightarrow a\wedge b=a.\,}
Semi-modular lattices
An upper semi-modular lattice satisfies the further property:
Upper semi-modularity: If
{\displaystyle a\wedge b\prec a}
{\displaystyle b\prec a\vee b}
Dually, a lower semi-modular lattice satisfies
Lower semi-modularity: If
{\displaystyle b\prec a\vee b}
{\displaystyle a\wedge b\prec a}
The Jordan-Dedekind chain condition holds in a semi-modular (lower or upper) lattice: all finite maximal chains between two given elements have the same length.
A modular lattice satisfies the further property:
Modularity: If
{\displaystyle x\geq y}
{\displaystyle x\wedge (y\vee z)=y\vee (x\wedge z).\,}
A pair of intervals of the form
{\displaystyle [b,a\vee b]}
{\displaystyle [a\wedge b,a]}
are said to be in perspective. In a modular lattice, perspective intervals are isomorphic: the maps
{\displaystyle x\mapsto a\vee x}
{\displaystyle y\mapsto y\wedge b}
are order-isomorphisms.
Modularity implies both forms of semi-modularity and hence the Jordan-Dedekind chain condition. In a modular lattice with 0, if an element x has finite dimension d, then all maximal chains from 0 to x have the same length d.
The dimension is related to the join and meet in a modular lattice by
{\displaystyle d(x)+d(y)=d(x\wedge y)+d(x\vee y).\,}
A distributive lattice satisfies the further property:
{\displaystyle x\vee (y\wedge z)=(x\vee y)\wedge (x\vee z);~x\wedge (y\vee z)=(x\wedge y)\vee (x\wedge z).\,}
Distributivity implies modularity for a lattice.
A complete lattice is one in which every set has a supremum and an infimum. In particular the lattice must be a bounded order, with bottom and top elements, usually denoted 0 and 1.
A complemented lattice is a lattice with 0 and 1 with the property that for every element a there is some element b such that
{\displaystyle a\wedge b=\mathbf {0} }
{\displaystyle a\vee b=\mathbf {1} }
. If the lattice is distributive then the complement of a, denoted
{\displaystyle {\bar {a}}}
{\displaystyle a'}
Subjunctive lattice
A subjunctive or Brouwerian lattice has the property that for any two elements a,b, there exists an element a→b with the properties
{\displaystyle x\wedge a\leq b\Leftrightarrow x\leq (a\rightarrow b).\,}
This element is the pseudo-complement of a relative to b and is unique. We note that a→a = 1.
A Heyting algebra is a bounded subjunctive lattice. The pseudo-complement ~a is the relative pseudo-complement a→0. We have
{\displaystyle \sim a\wedge a=\mathbf {0} }
{\displaystyle \sim a\vee a}
need not be 1. A Heyting algebra is necessarily distributive.
A Boolean lattice is a distributive complemented lattice, and hence with a uniquely defined complement.
A Boolean lattice is subjunctive.
Lattice homomorphisms
A lattice homomorphism is a map between lattices which preserves join and meet. It is necessarily montone, but not every monotone map is a lattice homomorphism. A lattice isomorphism is just an order isomorphism.
Ideals and filters
An ideal in a lattice is a non-empty join-closed downward-closed subset. A filter is a non-empty meet-closed upward-closed subset. Every cut defines an ideal, but not conversely. The downset
{\displaystyle {\downarrow }a=\{x\leq a\}}
is the principal ideal on a; the upset
{\displaystyle {\uparrow }a=\{x\geq a\}}
is the principal filter on a.
Retrieved from "https://citizendium.org/wiki/index.php?title=Order_(relation)&oldid=550107" |
Angle histogram plot - MATLAB rose - MathWorks Australia
Create Angle Histogram
Angle histogram plot
rose is not recommended. Use polarhistogram instead.
rose(theta)
rose(theta,x)
rose(theta,nbins)
rose(ax,...)
h = rose(...)
[tout,rout] = rose(...)
rose(theta) creates an angle histogram, which is a polar plot showing the distribution of values grouped according to their numeric range, showing the distribution of theta in 20 angle bins or less. The vector theta, expressed in radians, determines the angle of each bin from the origin. The length of each bin reflects the number of elements in theta that fall within a group, which ranges from 0 to the greatest number of elements deposited in any one bin.
rose(theta,x) uses the vector x to specify the number and the locations of bins. length(x) is the number of bins and the values of x specify the center angle of each bin. For example, if x is a five-element vector, rose distributes the elements of theta in five bins centered at the specified x values.
rose(theta,nbins) plots nbins equally spaced bins in the range [0,2*pi]. The default is 20.
rose(ax,...) plots into the axes ax instead of the current axes (gca).
h = rose(...) returns the handle of the line object used to create the graph.
[tout,rout] = rose(...) returns the vectors tout and rout so polar(tout,rout) generates the histogram for the data. This syntax does not generate a plot.
Create an angle histogram of values between 0 and
2\pi
. Distribute the data among 10 bins.
theta = [0.4 1.4 3.1 2.3 0.4 2.5 3.9 2.8 2.3 1.6 4.6 4.5 6.1 3.9 5.1];
rose(theta,10)
compass | histogram | polarhistogram | polarplot |
(Redirected from Halflife)
Scientific and mathematical term
This article is about the scientific and mathematical concept. For the video game series, see Half-Life (series). For other uses, see Half-Life (disambiguation).
3 1⁄8 12 .5
4 1⁄16 6 .25
5 1⁄32 3 .125
6 1⁄64 1 .5625
7 1⁄128 0 .78125
n 1⁄2n 100⁄2n
1 Probabilistic nature
2 Formulas for half-life in exponential decay
2.1 Half-life and reaction orders
2.2 Decay by two or more processes
3 In non-exponential decay
4 In biology and pharmacology
Probabilistic nature[edit]
Simulation of many identical atoms undergoing radioactive decay, starting with either 4 atoms per box (left) or 400 (right). The number at the top is how many half-lives have elapsed. Note the consequence of the law of large numbers: with more atoms, the overall decay is more regular and more predictable.
Various simple exercises can demonstrate probabilistic decay, for example involving flipping coins or running a statistical computer program.[3][4][5]
Formulas for half-life in exponential decay[edit]
Main article: Exponential decay
An exponential decay can be described by any of the following four equivalent formulas:[6]: 109–112
{\displaystyle {\begin{aligned}N(t)&=N_{0}\left({\frac {1}{2}}\right)^{\frac {t}{t_{1/2}}}\\N(t)&=N_{0}2^{-{\frac {t}{t_{1/2}}}}\\N(t)&=N_{0}e^{-{\frac {t}{\tau }}}\\N(t)&=N_{0}e^{-\lambda t}\end{aligned}}}
t1⁄2 is the half-life of the decaying quantity,
τ is a positive number called the mean lifetime of the decaying quantity,
λ is a positive number called the decay constant of the decaying quantity.
The three parameters t1⁄2, τ, and λ are directly related in the following way:
{\displaystyle t_{1/2}={\frac {\ln(2)}{\lambda }}=\tau \ln(2)}
where ln(2) is the natural logarithm of 2 (approximately 0.693).[6]: 112
Half-life and reaction orders[edit]
Zero order kinetics: The rate of this kind of reaction does not depend on the substrate concentration, [A]:
{\displaystyle d[{\ce {A}}]/dt=-k}
The integrated rate law of zero order kinetics is:
{\displaystyle [{\ce {A}}]=[{\ce {A}}]_{0}-kt}
In order to find the half-life, we have to replace the concentration value for the initial concentration divided by 2:
{\displaystyle [{\ce {A}}]/2=[{\ce {A}}]_{0}-kt_{1/2}}
and isolate the time:
{\displaystyle t_{1/2}={\frac {[{\ce {A}}]_{0}}{2k}}}
This t1/2 formula indicates that the half-life for a zero order reaction depends on the initial concentration and the rate constant.
First order kinetics: In first order reactions, the concentration of the reactant will decrease exponentially
{\displaystyle [{\ce {A}}]=[{\ce {A}}]_{0}exp(-kt)}
as time progresses until it reaches zero, and the half-life will be constant, independent of concentration.
The time t1/2 for [A] to decrease from [A]0 to 1/2 [A]0 in a first-order reaction is given by the following equation:
{\displaystyle [{\ce {A}}]_{0}/2=[{\ce {A}}]_{0}exp(-kt_{1/2})}
It can be solved for
{\displaystyle kt_{1/2}=-\ln \left({\frac {[{\ce {A}}]_{0}/2}{[{\ce {A}}]_{0}}}\right)=-\ln {\frac {1}{2}}=\ln 2}
For a first-order reaction, the half-life of a reactant is independent of its initial concentration. Therefore, if the concentration of A at some arbitrary stage of the reaction is [A], then it will have fallen to 1/2 [A] after a further interval of (ln 2)/k. Hence, the half-life of a first order reaction is given as the following:
{\displaystyle t_{1/2}={\frac {\ln 2}{k}}}
The half-life of a first order reaction is independent of its initial concentration and depends solely on the reaction rate constant, k.
{\displaystyle {\frac {1}{[{\ce {A}}]}}=kt+{\frac {1}{[{\ce {A}}]_{0}}}}
We replace [A] for 1/2 [A]0 in order to calculate the half-life of the reactant A
{\displaystyle {\frac {1}{[{\ce {A}}]_{0}/2}}=kt_{1/2}+{\frac {1}{[{\ce {A}}]_{0}}}}
and isolate the time of the half-life (t1/2):
{\displaystyle t_{1/2}={\frac {1}{[{\ce {A}}]_{0}k}}}
This shows that the half-life of second order reactions depends on the initial concentration and rate constant.
Decay by two or more processes[edit]
Some quantities decay by two exponential-decay processes simultaneously. In this case, the actual half-life T1⁄2 can be related to the half-lives t1 and t2 that the quantity would have if each of the decay processes acted in isolation:
{\displaystyle {\frac {1}{T_{1/2}}}={\frac {1}{t_{1}}}+{\frac {1}{t_{2}}}}
For three or more processes, the analogous formula is:
{\displaystyle {\frac {1}{T_{1/2}}}={\frac {1}{t_{1}}}+{\frac {1}{t_{2}}}+{\frac {1}{t_{3}}}+\cdots }
For a proof of these formulas, see Exponential decay § Decay by two or more processes.
Half-life demonstrated using dice in a classroom experiment
Further information: Exponential decay § Applications and examples
In a chemical reaction, the half-life of a species is the time it takes for the concentration of that substance to fall to half of its initial value. In a first-order reaction the half-life of the reactant is ln(2)/λ, where λ (also denoted as k) is the reaction rate constant.
In non-exponential decay[edit]
In biology and pharmacology[edit]
A biological half-life or elimination half-life is the time it takes for a substance (drug, radioactive nuclide, or other) to lose one-half of its pharmacologic, physiologic, or radiological activity. In a medical context, the half-life may also describe the time that it takes for the concentration of a substance in blood plasma to reach one-half of its steady-state value (the "plasma half-life").
In epidemiology, the concept of half-life can refer to the length of time for the number of incident cases in a disease outbreak to drop by half, particularly if the dynamics of the outbreak can be modeled exponentially.[12][13]
^ John Ayto, 20th Century Words (1989), Cambridge University Press.
^ Muller, Richard A. (April 12, 2010). Physics and Technology for Future Presidents. Princeton University Press. pp. 128–129. ISBN 9780691135045.
^ Chivers, Sidney (March 16, 2003). "Re: What happens during half-lifes [sic] when there is only one atom left?". MADSCI.org.
^ "Radioactive-Decay Model". Exploratorium.edu. Retrieved 2012-04-25.
^ Wallin, John (September 1996). "Assignment #2: Data, Simulations, and Analytic Science in Decay". Astro.GLU.edu. Archived from the original on 2011-09-29. {{cite web}}: CS1 maint: unfit URL (link)
^ a b Rösch, Frank (September 12, 2014). Nuclear- and Radiochemistry: Introduction. Vol. 1. Walter de Gruyter. ISBN 978-3-11-022191-6.
^ Jonathan Crowe; Tony Bradshaw (2014). Chemistry for the Biosciences: The Essential Concepts. p. 568. ISBN 9780199662883.
^ Lin VW; Cardenas DD (2003). Spinal cord medicine. Demos Medical Publishing, LLC. p. 251. ISBN 978-1-888799-61-3.
^ Pang, Xiao-Feng (2014). Water: Molecular Structure and Properties. New Jersey: World Scientific. p. 451. ISBN 9789814440424.
^ Australian Pesticides and Veterinary Medicines Authority (31 March 2015). "Tebufenozide in the product Mimic 700 WP Insecticide, Mimic 240 SC Insecticide". Australian Government. Retrieved 30 April 2018.
^ Fantke, Peter; Gillespie, Brenda W.; Juraske, Ronnie; Jolliet, Olivier (11 July 2014). "Estimating Half-Lives for Pesticide Dissipation from Plants". Environmental Science & Technology. 48 (15): 8588–8602. Bibcode:2014EnST...48.8588F. doi:10.1021/es500434p. PMID 24968074.
^ Balkew, Teshome Mogessie (December 2010). The SIR Model When S(t) is a Multi-Exponential Function (Thesis). East Tennessee State University.
^ Ireland, MW, ed. (1928). The Medical Department of the United States Army in the World War, vol. IX: Communicable and Other Diseases. Washington: U.S.: U.S. Government Printing Office. pp. 116–7.
Look up half-life in Wiktionary, the free dictionary.
Wikimedia Commons has media related to Half times.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Half-life&oldid=1081230051"
Temporal exponentials |
Write Simple Test Case Using Classes - MATLAB & Simulink - MathWorks Australia
Create SolverTest Class
Run Tests in SolverTest Class
You can test your MATLAB® program by defining unit tests within a test class that inherits from the matlab.unittest.TestCase class. A unit test in a class-based test is a method that determines the correctness of a unit of software. It is defined within a methods block with the Test attribute and can use qualifications for testing values and responding to failures. For more information about class-based tests, see Author Class-Based Unit Tests in MATLAB.
This example shows how to write class-based unit tests to qualify the correctness of a function defined in a file in your current folder. The quadraticSolver function takes as inputs the coefficients of a quadratic polynomial and returns the roots of that polynomial. If the coefficients are specified as nonnumeric values, the function throws an error.
In a file named SolverTest.m in your current folder, create the SolverTest class by subclassing the matlab.unittest.TestCase class. This class provides a place for tests for the quadraticSolver function. Add three unit tests inside a methods block with the Test attribute. These test the quadraticSolver function against real solutions, imaginary solutions, and error conditions. Each Test method must accept a TestCase instance as an input. The order of the tests within the block does not matter.
First, create a Test method realSolution to verify that quadraticSolver returns the correct real solutions for specific coefficients. For example, the equation
{x}^{2}-3x+2=0
has real solutions
x=1
x=2
. The method calls quadraticSolver with the coefficients of this equation. Then, it uses the verifyEqual method of matlab.unittest.TestCase to compare the actual output actSolution to the expected output expSolution.
Create a second Test method imaginarySolution to verify that quadraticSolver returns the correct imaginary solutions for specific coefficients. For example, the equation
{x}^{2}+2x+10=0
has imaginary solutions
x=-1+3i
x=-1-3i
. Just like the previous method, this method calls quadraticSolver with the coefficients of this equation, and then uses the verifyEqual method to compare the actual output actSolution to the expected output expSolution.
Finally, add a Test method nonnumericInput to verify that quadraticSolver produces an error for nonnumeric coefficients. Use the verifyError method of matlab.unittest.TestCase to test that the function throws the error specified by 'quadraticSolver:InputMustBeNumeric' when it is called with inputs 1, '-3', and 2.
testCase.verifyError(@()quadraticSolver(1,'-3',2), ...
'quadraticSolver:InputMustBeNumeric')
To run all of the tests in the SolverTest class, create a TestCase object from the class and then call the run method on the object. In this example, all three tests pass.
testCase = SolverTest;
results = testCase.run
You also can run a single test specified by one of the Test methods. To run a specific Test method, pass the name of the method to run. For example, run the realSolution method.
result = run(testCase,'realSolution')
Name: 'SolverTest/realSolution' |
Raman Thermometry Measurements and Thermal Simulations for MEMS Bridges at Pressures From 0.05 Torr to 625 Torr | J. Heat Transfer | ASME Digital Collection
Raman Thermometry Measurements and Thermal Simulations for MEMS Bridges at Pressures From 0.05 Torr to 625 Torr
Justin R. Serrano,
Edward S. Piekos,
Edward S. Piekos
Michael A. Gallis,
Michael A. Gallis
Allen D. Gorby
Phinney, L. M., Serrano, J. R., Piekos, E. S., Torczynski, J. R., Gallis, M. A., and Gorby, A. D. (April 28, 2010). "Raman Thermometry Measurements and Thermal Simulations for MEMS Bridges at Pressures From 0.05 Torr to 625 Torr." ASME. J. Heat Transfer. July 2010; 132(7): 072402. https://doi.org/10.1115/1.4000965
This paper reports on experimental and computational investigations into the thermal performance of microelectromechanical systems (MEMS) as a function of the pressure of the surrounding gas. High spatial resolution Raman thermometry was used to measure the temperature profiles on electrically heated, polycrystalline silicon bridges that are nominally
10 μm
wide,
2.25 μm
thick, and either
200 μm
400 μm
long in nitrogen atmospheres with pressures ranging from 0.05 Torr to 625 Torr (6.67 Pa–83.3 kPa). Finite element modeling of the thermal behavior of the MEMS bridges is performed and compared with the experimental results. Noncontinuum gas effects are incorporated into the continuum finite element model by imposing temperature discontinuities at gas-solid interfaces that are determined from noncontinuum simulations. The results indicate that gas-phase heat transfer is significant for devices of this size at ambient pressures but becomes minimal as the pressure is reduced below 5 Torr. The model and experimental results are in qualitative agreement, and better quantitative agreement requires increased accuracy in the geometrical and material property values.
elemental semiconductors, finite element analysis, heat conduction, micromechanical devices, Raman spectra, silicon, thermometers, Raman thermometry, MEMS, finite element heat conduction simulations, noncontinuum gas-phase heat-transfer model, low-pressure effects, suspended microbridge
Bridges (Structures), Finite element analysis, Heat transfer, Microelectromechanical systems, Pressure, Simulation, Temperature, Temperature measurement, Heat conduction, Temperature profiles, Polysilicon, Engineering simulation
K. -D.
Khuri-Yakob
Fabrication and Characterization of Surface Micromachined Capacitive Ultrasonic Immersion Transducers
Thermal Characterization of Surface-Micromachined Silicon Nitride Membranes for Thermal Infrared Detectors
Effect of Ambient on the Thermal Parameters of a Micromachined Bolometer
A Computational Investigation of Noncontinuum Gas-Phase Heat Transfer Between a Heated Microbeam and the Adjacent Ambient Substrate
MEMS Technologies Department
SUMMiT V™ Five Level Surface Micromachining Technology Design Manual, Version 3.1a
,” Sandia Report No. SAND2008-0659P, Sandia National Laboratories, Albuquerque, NM.
Spatially Resolved Temperature Mapping of Electrothermal Actuators by Surface Raman Scattering
Aspnes
Simultaneous Mapping of Temperature and Stress in Microdevices Using Micro-Raman Spectroscopy
Calore: A Computational Heat Transfer Program, Volume 1: Theory Manual
,” Sandia Report No. SAND2006, Sandia National Laboratories, Albuquerque, NM.
Calore Development Team
Calore: A Computational Heat Transfer Program, Volume 2: User Reference Manual, Version 4.6
Modeling Microscale Heat Transfer Using Calore
,” Sandia Report No. SAND2005-5979, Sandia National Laboratories, Albuquerque, NM.
Thermal Conductivity Measurements of SUMMiT V Polycrystalline Silicon
Thermal Accommodation and Adsorption Coefficients of Gases
CINDAS Data Series on Material Properties
Measurement of Gas-Surface Accommodation
Rarefied Gas Dynamics: 26th International Symposium
Design and Test of Carbon Nanotube Biwick Structure for High-Heat-Flux Phase Change Heat Transfer
Raman Thermometry Measurements and Thermal Simulations for MEMS Bridges at Pressures From 0.05 to 625 Torr |
Difference between revisions of "Lower bounds on APN-distance for all known APN functions" - Boolean Functions
Difference between revisions of "Lower bounds on APN-distance for all known APN functions"
<th colspan="4">DIMENSION 10</th>
<th><math>\Pi_F^0</math></th>
<th><math>m_F</math></th>
<th>lower bound</th>
<td>495<sup>682</sup>, 543<sup>341</sup>, 1024</td>
<td>477<sup>40</sup>, 483<sup>30</sup>, 489<sup>60</sup>, 495<sup>57</sup>, 501<sup>220</sup>, 507<sup>70</sup>, 513<sup>160</sup>, 519<sup>105</sup>, 525<sup>150</sup>, 531<sup>40</sup>, 537<sup>50</sup>, 543<sup>1</sup>, 549<sup>40</sup>, 1024</td>
The following tables list a lower bound on the Hamming distance between all known CCZ-inequivalent APN representatives up to dimension 11 using the methods described used in [1]. Note that the lower bound is a CCZ-invariant (unlike the exact minimum distance itself) and it can be calculated via the formula
{\displaystyle l(F)=\lceil {\frac {m_{F}}{3}}\rceil +1}
{\displaystyle l(F)}
is the lower bound on the Hamming distance between an
{\displaystyle (n,n)}
{\displaystyle F}
and the closest APN function, and
{\displaystyle m_{F}}
{\displaystyle m_{F}=\min _{b,\beta \in \mathbb {F} _{2^{n}}}|\{a\in \mathbb {F} _{2^{n}}:(\exists x\in \mathbb {F} _{2^{n}})(F(x)+F(a+x)+F(a+\beta )=b)\}|}
{\displaystyle m_{F}}
for the CCZ-inequivalent representatives are provided in the tables for convenience. The representatives for dimensions 7 and 8 are taken from the list ofKnown quadratic APN polynomial functions over GF(2^7) and Known quadratic APN polynomial functions over GF(2^8), respectively, while the rest are taken from the table of CCZ-inequivalent APN functions from the known APN classes over GF(2^n) (for n between 6 and 11).
The tables for dimensions 7 and 8 can be found under Lower bounds on APN-distance for all known APN functions in dimension 7 and Lower bounds on APN-distance for all known APN functions in dimension 8, respectively, due to their large size.
{\displaystyle \Pi _{F}^{0}}
{\displaystyle m_{F}}
1 255511, 512 255 86
7 2313, 23745, 24027, 24336, 24654, 24936, 25236, 25537, 25827, 26145, 26454, 26745, 27036, 2739, 27618, 2793, 512 231 78
10 255511, 512 255 86
{\displaystyle \Pi _{F}^{0}}
{\displaystyle m_{F}}
1 495682, 543341, 1024 495 166
4 47740, 48330, 48960, 49557, 501220, 50770, 513160, 519105, 525150, 53140, 53750, 5431, 54940, 1024 477 160
↑ Budaghyan L, Carlet C, Helleseth T, Kaleyski N. Changing Points in APN Functions. IACR Cryptology ePrint Archive. 2018;2018:1217.
Retrieved from "https://boolean.h.uib.no/mediawiki/index.php?title=Lower_bounds_on_APN-distance_for_all_known_APN_functions&oldid=308" |
Walsh spectra of all known APN functions over GF(2^8) - Boolean Functions
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:
{\displaystyle \{-32^{2380},-16^{20400},0^{16320},16^{23120},32^{3060}\}}
(same as the Gold functions)
{\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}\}}
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 Gold-like Walsh spectrum, a Walsh spectrum of type 1 and a Walsh spectrum of type 2 is available.
Functions with Walsh spectrum of type 2
{\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}}
Retrieved from "https://boolean.h.uib.no/mediawiki/index.php?title=Walsh_spectra_of_all_known_APN_functions_over_GF(2%5E8)&oldid=504" |
Branching processes, and random-cluster measures on trees | EMS Press
Branching processes, and random-cluster measures on trees
Random-cluster measures on infinite regular trees are studied in conjunction with a general type of `boundary condition', namely an equivalence relation on the set of infinite paths of the tree. The uniqueness and non-uniqueness of \rc\ measures are explored for certain classes of equivalence relations. In proving uniqueness, the following problem concerning branching processes is encountered and answered. Consider bond percolation on the family-tree
T
of a branching process. What is the probability that every infinite path of
T
, beginning at its root, contains some vertex which is itself the root of an infinite open sub-tree?
Geoffrey R. Grimmett, Svante Janson, Branching processes, and random-cluster measures on trees. J. Eur. Math. Soc. 7 (2005), no. 2, pp. 253–281 |
Parking Valet Using Nonlinear Model Predictive Control - MATLAB & Simulink - MathWorks Switzerland
Generate a Trajectory Using Nonlinear Model Predictive Controller
Track Reference Trajectory in Simulink Model
This example shows how to generate a reference trajectory and track the trajectory for a parking valet using nonlinear model predictive control (NLMPC).
In this example, the parking garage contains an ego vehicle and eight static obstacles. The obstacles are given by six parked vehicles, a reserved parking area, and the garage border. The goal of the ego vehicle is to park at a target pose without colliding with any of the obstacles. The reference point of the ego pose is located at the center of the rear axle.
Define the parameters of the ego vehicle.
Specify the initial ego vehicle pose.
% Ego initial pose: x(m), y(m) and yaw angle (rad)
egoInitialPose = [4,12,0];
Define the target pose for the ego vehicle. In this example, there are two possible parking directions. To park facing north, set parkNorth to true. To park facing south, set parkNorth to false.
parkNorth = true;
if parkNorth
egoTargetPose = [36,45,pi/2];
egoTargetPose = [27.2,4.7,-pi/2];
The helperSLCreateCostmap function creates a static map of the parking lot that contains information about stationary obstacles, road markings, and parked cars. For more details, see the Automated Parking Valet in Simulink (Automated Driving Toolbox) example.
costmap = helperSLCreateCostmap();
centerToFront = distToCenter;
centerToRear = distToCenter;
helperSLCreateUtilityBus;
costmapStruct = helperSLCreateUtilityStruct(costmap);
Visualize the parking environment. Use a sample time of 0.1 for the visualizer.
helperSLVisualizeParkingValet(egoInitialPose, 0, costmapStruct);
The six parked vehicles are orange boxes on the top and bottom of the figure. The middle area represents the reserved parking area. The left border of the garage is also modeled as a static obstacle. The ego vehicle in blue has two axles and four wheels. The two green boxes represent the target parking spots for the ego vehicle, with the top spot facing north.
In this example, a kinematic bicycle model with front steering angle is used. The motion of the ego vehicle can be described by the following equations.
\begin{array}{l}\underset{}{\overset{˙}{x}}=v\cdot \mathrm{cos}\left(\psi \right)\\ \underset{}{\overset{˙}{y}}=v\cdot \mathrm{sin}\left(\psi \right)\\ \underset{}{\overset{˙}{\psi }}=\frac{v}{b}\cdot \mathrm{tan}\left(\delta \right)\end{array}
\left(\mathit{x},\mathit{y}\right)
\psi
b
\left(x,y,\psi \right)
are the state variables of the vehicle state functions. The speed
\mathit{v}
\delta
are the control variables of the vehicle state functions.
The parking valet trajectory from the NLMPC controller for is designed based on the analysis similar to Parallel Parking Using Nonlinear Model Predictive Control example. The design of controller is implemented in the createMPCForParkingValet script.
The speed of the ego vehicle is constrained to be within [-6.5,6.5] m/s (approximately with speed limit as 15 mph) and the steering angle of the ego vehicle is constrained to be within [-45,45] degrees.
The cost function for nlmpc controller object is a custom cost function defined in a manner similar to a quadratic tracking cost plus a terminal cost. In the following custom cost function,
s\left(t\right)
t
d
represents the duration of simulation.
{s}_{ref}
is given by the target pose for the ego vehicle. The matrices
{Q}_{p}
{R}_{p}
{Q}_{t}
{R}_{t}
J={\int }_{0}^{d}\left(s\left(t\right)-{s}_{ref}{\right)}^{T}{Q}_{p}\left(s\left(t\right)-{s}_{ref}\right)+u\left(t{\right)}^{T}{R}_{p}u\left(t\right)dt+\left(s\left(d\right)-{s}_{ref}{\right)}^{T}{Q}_{t}\left(s\left(d\right)-{s}_{ref}\right)+u\left(d{\right)}^{T}{R}_{t}u\left(d\right)
To avoid collision with obstacles, the NLMPC controller must satisfy the following inequality constraints, where minimum distance to all obstacles
dis{t}_{min}
dis{t}_{safe}
. In this example, the ego vehicle and obstacles are modeled as collisionBox (Robotics System Toolbox) objects and the distance from the ego vehicle to obstacles is computed by the checkCollision (Robotics System Toolbox) function.
dis{t}_{min}\ge dis{t}_{safe}
The initial guess for the solution path is given by two straight lines. The first line is from the initial ego vehicle pose to a middle point, and the second line is from the middle point to the ego vehicle target pose.
Select a middle point for the initial solution path guess.
midPoint = [4,34,pi/2];
midPoint = [27,12,0];
Configure the parameters of the NLMPC controller. To plan an optimal trajectory over the entire prediction horizon, set the control horizon equal to the prediction horizon.
% Prediction horizon
% Control horizon
% Weight matrices for terminal cost
Qt = 0.5*diag([10 5 20]);
Rt = 0.1*diag([1 2]);
% Weight matrices for tracking cost
Qp = 1e-6*diag([2 2 0]);
Rp = 1e-4*diag([1 15]);
Qp = 0*diag([2 2 0]);
Rp = 1e-2*diag([1 5]);
% Safety distance to obstacles (m)
% Maximum iteration number
% Disable message display
Create the NLMPC controller using the specified parameters.
[nlobj,opt,paras] = createMPCForParkingValet(p,c,Ts,egoInitialPose,egoTargetPose,...
maxIter,Qp,Rp,Qt,Rt,distToCenter,safetyDistance,midPoint);
Set the initial conditions for the ego vehicle.
x0 = egoInitialPose';
Generate the reference trajectory using the nlmpcmove function.
[mv,nloptions,info] = nlmpcmove(nlobj,x0,u0,[],[],opt);
timeVal = toc;
Obtain the reference trajectories for the states (xRef) and the control actions (uRef), which are the optimal trajectories computed of the prediction horizon.
xRef = info.Xopt;
uRef = info.MVopt;
Analyze the planned trajectory.
analyzeParkingValetResults(nlobj,info,egoTargetPose,Qp,Rp,Qt,Rt,...
distToCenter,safetyDistance,timeVal)
5) Final states error in x (m), y (m) and theta (deg): -0.0004, 0.0412, -1.9488
6) Final control inputs speed (m/s) and steering angle (deg): -0.2190, 4.5549
As shown in the following plots, the planned trajectory successfully parks the ego vehicle in the target pose. The final control input values are close to zero.
plotTrajectoryParkingValet(xRef,uRef)
Design an NLMPC controller to track the reference trajectory.
First, set the simulation duration and update the reference trajectory based on the duration.
Create an NLMPC controller with a tracking prediction horizon (pTracking) of 10.
nlobjTracking = createMPCForTrackingParkingValet(pTracking,Xref);
mdl = 'mpcAutoParkingValet';
Close the animation plots before running the simulation.
f = findobj('Name','Automated Parking Valet');
The animation shows that the ego vehicle parks at the target pose successfully without any obstacle collisions. You can also view the ego vehicle and pose trajectories using the Ego Vehicle Pose and Controls scopes.
This example shows how to generate a reference trajectory and track the trajectory for parking valet using nonlinear model predictive control. The controller navigates the ego vehicle to the target parking spot without colliding with any obstacles. |
Origin and evolution of Humans - Sinfronteras
Origin and evolution of Humans
(Redirected from Origin and evolution of Homo sapiens)
https://www.youtube.com/watch?v=H-fwl6pMVLA
The origin and evolution of Humans
{\displaystyle \color {White}{\mathbf {-42,000}}}
The Neanderthals became extinct around 40,000 years ago. This date, which is based on research published in Nature in 2014, is much earlier than previous estimates, and it was established through improved radiocarbon dating methods analyzing 40 sites from Spain to Russia..[1] Evidence for continued Neanderthal presence in the Iberian Peninsula at 37,000 years ago was published in 2017.
Hypotheses on the fate of the Neanderthals include violence from encroaching anatomically modern humans,[3] parasites and pathogens, competitive replacement,[4] competitive exclusion, extinction by interbreeding with early modern human populations,[5] natural catastrophes, and failure or inability to adapt to climate change. It is unlikely that any one of these hypotheses is sufficient on its own; rather, multiple factors probably contributed to the demise of an already low population. https://en.wikipedia.org/wiki/Neanderthal_extinction
{\displaystyle \color {White}{\mathbf {-300,000}}}
The origin of the Homo sapiens
Several early hominins used fire and occupied much of Eurasia. Homo sapiens (sometimes also known as "modern humans") are thought to have diverged in Africa from an earlier hominin around 300,000 years ago, with the earliest fossil evidence of Homo sapiens also appearing around 300,000 years ago in Africa. https://en.wikipedia.org/wiki/Human#Rise_of_Homo_sapiens
Documentary about the earliest fossil evidence of Homo sapiens. La datation par thermoluminescence, plus fiable que le carbone 14, fait remonter leur origine à 300 000 ans: https://www.youtube.com/watch?v=-XUsBUDoB08
{\displaystyle \color {White}{\mathbf {-342,000}}}
The origin of the Neanderthal
Disparu il y a plus de 40 000 ans, l’homme de Neandertal a peuplé pendant quelque 300,000 ans une large partie du continent eurasiatique, de l’Atlantique à la Sibérie. Majoritairement présents dans le nord-ouest de l'Europe actuelle.
Retrieved from "http://wiki.sinfronteras.ws/index.php?title=Origin_and_evolution_of_Humans&oldid=16952" |
Towards the second PR
In the previous post I described the Next Subvolume Method. Once that PR was merged, I wrote an issue outlining what needs to be done next. My next PR will complete the top 6 TODOs on the list. Most importantly, it introduces more comprehensive tests and three optimized CartesianGrid structs.
The previous PR only tested the NSM on 1D grids, which left room for bugs to go undetected. In this PR I added a way to test pure diffusion on arbitrary graphs. First, given a graph and a hopping rate (probability per time of hopping to a neighboring site) we must contruct the discrete Laplacian matrix
L. The following function does that.
function discrete_laplacian_from_spatial_system(spatial_system, hopping_rate)
sites = 1:DiffEqJump.num_sites(spatial_system)
laplacian = zeros(Int, length(sites), length(sites))
laplacian[site,site] = -DiffEqJump.num_neighbors(spatial_system, site)
for nb in DiffEqJump.neighbors(spatial_system, site)
laplacian[site, nb] = 1
laplacian*hopping_rate
We now have the system of differential equations
\frac{dy}{dt} = Ly
dtdy=Ly, where
y[s](t) is the expected number of species
s at time
t. The solution is given by
y(t) = e^{Lt}
y(t)=eLt, which is computed by the following script.
lap = discrete_laplacian_from_spatial_system(LightGraphs.grid(dims), hopping_rate)
evals, B = eigen(lap) # lap == B*diagm(evals)*B'
Bt = B'
analytic_solution(t) = B*diagm(ℯ.^(t*evals))*Bt * reshape(prob.u0, num_nodes, 1)
A neat fact from linear algebra is that a symmetric matrix is always diagonalizable, and
L is symmetric. So the procedure above will always work.
By the weak law of large numbers we know that the average of many simulations should converge to the expected value, which is what we test.
Nsims = 10000
rel_tol = 0.01
grids = [DiffEqJump.CartesianGrid1(dims), DiffEqJump.CartesianGrid2(dims), DiffEqJump.CartesianGrid3(dims), LightGraphs.grid(dims)]
for grid in grids
spatial_jump_prob = JumpProblem(prob, alg, majumps, hopping_constants=hopping_constants, spatial_system=grid, save_positions=(false,false)) #set up the jump problem
mean_sol = get_mean_sol(spatial_jump_prob, Nsims, tf/num_time_points) # average of 10000 runs
for (i,t) in enumerate(times)
local diff = analytic_solution(t) - reshape(mean_sol[i], num_nodes, 1)
@test abs(sum(diff[1:center_node])/sum(analytic_solution(t)[1:center_node])) < rel_tol
Optimizing Cartesian grid
A 3 by 3 Cartesian grid:
A frequent use-case for spatial SSAss is to do simulation on a 1D, 2D or 3D rectangular grid in Euclidean space. For example, we might want be interested in some molecule diffusing on a small patch of a cell's membrane, which is basically a two-dimensional rectangle. While this is a special case of a graph, it is common enough to implement it separately and efficiently. I implemented three versions of a Cartesian grid in utils.jl. The most important function required of a graph or a Cartesian grid is rand_nbr(grid, site), which outputs a random neighbor of site with uniform probability. For an arbitrary graph it is simply
rand_nbr(graph::AbstractGraph, site) = rand(neighbors(graph, site))
where neighbors(graph, site) is a pre-computed list of neighbors.
For a Cartesian grid there is no need to pre-compute the list of neighbors for each site because of strong regularity of the neighbor list. For example, in a 2D grid, each site has 4 neighbors (up, down, left, right) except those neighbors that are outside the grid. For example, site 4 has neighbors 1, 7, and 5. In a
100 \times 100 \times 100
100×100×100 grid, we would need an array of length
10^6
106 to store the list of all neighbors, which is not cache-friendly. So what are the alternatives to pre-computing all neighbors?
We can rejection-sample a neighbor in the following fashion. Given a site, assume that it has all 4 (or 6 in case of 3D) neighbors and pick a random one. If it is inside the grid, we have our neighbor. Otherwise, do the same thing again. Since most sites have all 4 (6)neighbors, the probability of having to try again is small. This turns out to be the fastest sampling method for large grids, according to my benchmarking.
We can sample a neighbor by iterating. Pre-compute the number of neighbors for each site, and given a site, draw a random number from 1 to the number of neighbors of the site. Then iterate to that neighbor, discarding those potential neighbors that are outside the grid.
In order to find out which method (pre-computing neighbors, rejection-sampling or iteration-sampling) is the fastest I imlpemented three version of a Cartesian grid and benchmarked them. The rejection-sampling method came out to be the winner for larger (
64\times 64 \times 64
64×64×64) grids.
Benchmarking pure diffusion on a 64x64x64 grid.
The next step is to add support of various forms of hopping rates. Currently, the assumption is that hopping rates are of form
D_{s,i}
Ds,i, where
s is a species and
i is a site (i.e. hopping rates depend only on the species and site). Other common forms of hopping rates include
L_{s,i,j}
Ls,i,j,
D_s \cdot L_{i,j}
Ds⋅Li,j, and
D_{s,i}\cdot L_{i,j}
Ds,i⋅Li,j. Each of these is a special case of
L_{s,i,j}
Ls,i,j, so implementing this is the priority, but each special case allows for more efficient memory usage and thus better performance.
At the moment there are two data main structures used in spatial simulation: AbstractSpatialSystem and AbstractSpatialRates. The role of the first is to contain all information about the topology (number of sites, number of neighbors for each site, sampling a random neighbor). The role of the second is to contain all information about the current rates of reactions and jumps. In order to improve orthogonality AbstractSpatialRates can be split into AbstractRxRates and AbstractHoppingRates. Depending on the form that hopping or reaction rates take, the user will pass in different objects, making it easy to "assemble" the spatial simulation. |
RandomGraph - Maple Help
Home : Support : Online Help : Mathematics : Discrete Mathematics : Graph Theory : GraphTheory Package : RandomGraphs : RandomGraph
generate random graph
RandomGraph(V,p,options)
RandomGraph(V,m,options)
RandomGraph(n,p,options)
RandomGraph(n,m,options)
sequence of options (see below)
connected : truefalse
If the option connected is specified, the graph created is connected, and hence has at least n-1 edges.
For RandomGraph(n,m,connected), m must be at least n-1. A random tree is first created, then the remaining m-n+1 edges are
For RandomGraph(n,p,connected), a random tree is first created then each remaining edge is present with probability p.
degree : nonnegint
If the option degree=d is specified, and d-regular n vertex graph is possible, then a random d-regular graph having n vertices will be returned. Note that this option cannot be present with the directed option. This is equivalent to using the RandomRegularGraph command.
If the option directed is specified, a random directed graph is chosen. This is equivalent to using the RandomDigraph command. Default value is false.
weights : range
m\le n
are integers, the graph is a weighted graph with integer edge weights chosen from [m,n] uniformly at random. The weight matrix W in the graph has datatype=integer, and if the edge from vertex i to j is not in the graph then W[i,j] = 0.
x\le y
RandomGraph(n,p) creates an undirected unweighted graph on n vertices where each possible edge is present with probability p where
0.0\le p\le 1.0
RandomGraph(n,m) creates an undirected unweighted graph on n vertices and m edges where the m edges are chosen uniformly at random. The value of m must satisfy
0\le m\le \mathrm{binomial}\left(n,2\right)=n\frac{n-1}{2}
If the first input is a positive integer n, then the vertices are labeled 1,2,...,n. Alternatively, you may specify the vertex labels in a list.
This model of random graph generation, in which edges are selected with uniform probability from all possible edges in a graph on the specified vertices, is known as the Erdős–Rényi model.
\mathrm{with}\left(\mathrm{GraphTheory}\right):
\mathrm{with}\left(\mathrm{RandomGraphs}\right):
G≔\mathrm{RandomGraph}\left(8,0.5\right)
\textcolor[rgb]{0,0,1}{G}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{Graph 1: an undirected unweighted graph with 8 vertices and 10 edge\left(s\right)}}
G≔\mathrm{RandomGraph}\left(8,10\right)
\textcolor[rgb]{0,0,1}{G}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{Graph 2: an undirected unweighted graph with 8 vertices and 10 edge\left(s\right)}}
G≔\mathrm{RandomGraph}\left(8,10,\mathrm{connected}\right)
\textcolor[rgb]{0,0,1}{G}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{Graph 3: an undirected unweighted graph with 8 vertices and 10 edge\left(s\right)}}
\mathrm{IsConnected}\left(G\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
G≔\mathrm{RandomGraph}\left(6,\mathrm{degree}=3\right)
\textcolor[rgb]{0,0,1}{G}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{Graph 4: an undirected unweighted graph with 6 vertices and 9 edge\left(s\right)}}
\mathrm{IsRegular}\left(G\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
H≔\mathrm{RandomGraph}\left(4,1.0,\mathrm{weights}=0...1.0\right)
\textcolor[rgb]{0,0,1}{H}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{Graph 5: an undirected weighted graph with 4 vertices and 6 edge\left(s\right)}}
\mathrm{WeightMatrix}\left(H\right)
[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{0.809734551911930}& \textcolor[rgb]{0,0,1}{0.230156065952094}& \textcolor[rgb]{0,0,1}{0.761731208483085}\\ \textcolor[rgb]{0,0,1}{0.809734551911930}& \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{0.158057578940872}& \textcolor[rgb]{0,0,1}{0.580956679189321}\\ \textcolor[rgb]{0,0,1}{0.230156065952094}& \textcolor[rgb]{0,0,1}{0.158057578940872}& \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{0.423165119881119}\\ \textcolor[rgb]{0,0,1}{0.761731208483085}& \textcolor[rgb]{0,0,1}{0.580956679189321}& \textcolor[rgb]{0,0,1}{0.423165119881119}& \textcolor[rgb]{0,0,1}{0.}\end{array}]
H≔\mathrm{RandomGraph}\left(8,10,\mathrm{connected},\mathrm{weights}=1..4\right)
\textcolor[rgb]{0,0,1}{H}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{Graph 6: an undirected weighted graph with 8 vertices and 10 edge\left(s\right)}}
\mathrm{WeightMatrix}\left(H\right)
[\begin{array}{cccccccc}\textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{4}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{4}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{4}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{4}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{4}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{4}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\end{array}]
U≔\mathrm{rand}\left(1..4\right):
f := proc() local x; x := U(); if x=1 then 1 else 2 end if; end proc:
H≔\mathrm{RandomGraph}\left(6,1.0,\mathrm{weights}=f\right)
\textcolor[rgb]{0,0,1}{H}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{Graph 7: an undirected weighted graph with 6 vertices and 15 edge\left(s\right)}}
\mathrm{WeightMatrix}\left(H\right)
[\begin{array}{cccccc}\textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{0}\end{array}]
GraphTheory:-IsConnected |
Ask Answer - Ray optics Optical instrument, Dual Nature Of Radiation And Matter, Nuclei, Atoms, Semiconductor Electronics Materials Devices And Simple Circuits, Electric Charges And Fields, Wave Optics, Physics, Electromagnetic Waves - Popular Questions for School Students
m=\frac{M}{2}
m=\frac{M}{3}
Pradumn Upadhyay
A Uranium-235 reactor is producing nuclear energy at a rate of 32,000kW.How many nuclei of Uranium -235 is fissioned per second? How much Uranium-235 would be consumed in 1000 hours?Assume an average energy of 200Mev released per fission.Take Avagadro's number as 6*10^23 mol^-1.
Q.30. A refrigerator based on Carnot cycle receives heat from 227
°
C and rejects heat at 727
°
C. Let the work input required per unit heat received is x. If the same machine is used as a Carnot engine, then its efficiency is y. The ratio of x and y is
\frac{1}{2}
Draw the circuit diagram of an illuminated photodiode in reverse bias. How is photodiode used to measure light intensity?
The circuit diagram of an illuminated photodiode in reverse bias can be represented as
The greater the intensity of light, the greater is the number of photons falling per second per unit area. Thus, the greater the intensity of light, the greater is the number of electron−hole pairs produced at the junction. The photocurrent is, thus, directly proportional to the intensity of light. This can be used for measuring the intensity of incident light.
Diagram is incorrect please check...
39. For a liquid and glass interface, if adhesive force is greater than cohesive force, then the angle of contact is
(1) Greater than 90o (2) Equal to 90o
(3) Less than 900 (4) Both (1)&(2)
Difference between interference and diffraction: Interference is due to superposition of two distinct waves coming from two coherent sources. Diffraction is produced as a result of superposition of the secondary wavelets coming from different parts of the same wavefront.
Numerical: Here, λ = 600 nm = 600 × 10−19 = 6 × 10−7 m
D = 0.8 m, x = 15 mm = 1.5 × 10−3 m,
n = 2, a = ?
Please check something is wrong in solution...
Define TIR and write the conditions for TIR. Derive a relation between critical angle and the refractive index of the medium. Also explain the working of isosceles triangle and optical fiber.
Name the waves that are often refered to as Heat Waves. Name the physical quantity that has a lower, higher and same valuefor these waves as compared to its value for x rays
Q). A particle is thrown up with speed
\sqrt{\frac{3gR}{2}}
from surface of earth. The maximum height attained by
the particle is
(2) 2 R
If light strikes a surface with a force of 1.2 X 10-6 N, then how is the result modified if the surface is a perfect reflector?
Alisha Sarah
Two metals X and Y ,when illuminated by appropriate radiation emit photoelectrons. The work function of X is greater than that of Y.Which metal will have higher value of threshold frequency and why?
Meenakshy Kishore
Name the constituent radiation of electromagnetic spectrum which is used for
Studying crystal structure |
Decompose signals into subbands with smaller bandwidths and slower sample rates or compute discrete wavelet transform (DWT) - Simulink - MathWorks 日本
Dyadic Analysis Filter Bank
Decompose signals into subbands with smaller bandwidths and slower sample rates or compute discrete wavelet transform (DWT)
This block always interprets input signals as frames. The frame size of the input signal must be a multiple of 2n, where n is the value of the Number of levels parameter. The block decomposes the input signal into either n+1 or 2n subbands. To decompose signals with a frame size that is not a multiple of 2n, use the Two-Channel Analysis Subband Filter block. (You can connect multiple copies of the Two-Channel Analysis Subband Filter block to create a multilevel dyadic analysis filter bank.)
You can configure this block to compute the Discrete Wavelet Transform (DWT) or decompose a broadband signal into a collection of subbands with smaller bandwidths and slower sample rates. The block uses a series of highpass and lowpass FIR filters to repeatedly divide the input frequency range, as illustrated in Wavelet Filter Banks (the Asymmetric one).
For the same input, the DWT configuration of this block does not produce the same results as the Wavelet Toolbox dwt function. Because DSP System Toolbox™ is designed for real-time implementation and Wavelet Toolbox is designed for analysis, the products handle boundary conditions and filter states differently. To make the output of the dwt function match the DWT output of this block, complete the following steps:
Set the boundary condition of the dwt function to zero-padding. To do so, type dwtmode('zpd') at the MATLAB® command line.
To match the latency of the block (implemented using FIR filters), add zeros to the input of the dwt function. The number of zeros you add must be equal to the half-length of the filter.
Input must be a vector or matrix.
The input frame size must be a multiple of 2n, where n is the number of filter bank levels. For example, a frame size of 16 would be appropriate for a three-level tree (16 is a multiple of 23).
The block always treats input signals as frames and operates along the columns.
For an illustration of why the above input requirements exist, see the figure Outputs of a 3-Level Asymmetric Dyadic Analysis Filter Bank.
The output characteristics vary depending on the block's parameter settings, as summarized in the following list and figure:
Number of levels parameter set to n
Tree structure parameter setting:
Asymmetric — Block produces n+1 output subbands
Symmetric — Block produces 2n output subbands
Output parameter setting can be Multiple ports or Single port. When you set the Output parameter to Single port, the block outputs one vector or matrix of concatenated subbands. The following figure illustrates the difference between the two settings for a 3-level asymmetric dyadic analysis filter bank. For an explanation of the illustrated output characteristics, see the table Output Characteristics for an n-Level Dyadic Analysis Filter Bank.
For more information about the filter bank levels and structures, see Dyadic Analysis Filter Banks.
Outputs of a 3-Level Asymmetric Dyadic Analysis Filter Bank
The following table summarizes the different output characteristics of the block when it is set to output from single or multiple ports.
Output Characteristics for an n-Level Dyadic Analysis Filter Bank
Block concatenates all the subbands into one vector or matrix, and outputs the concatenated subbands from a single output port. Each output column contains subbands of the corresponding input channel.
Block outputs each subband from a separate output port. The topmost port outputs the subband with the highest frequencies. Each output column contains a subband for the corresponding input channel.
Same as input frame rate
(However, the output frame sizes can vary, so the output sample rates can vary.)
Output Dimensions (Frame Size)
Same number of rows and columns as the input.
The output has the same number of columns as the input. The number of output rows is the output frame size. For an input with frame size Mi output yk has frame size Mo,k:
Symmetric — All outputs have the frame size, Mi / 2n.
Asymmetric — The frame size of each output (except the last) is half that of the output from the previous level. The outputs from the last two output ports have the same frame size since they originate from the same level in the filter bank.
{M}_{o,k}=\left\{\begin{array}{cc}{M}_{i}/{2}^{k}& \left(1â¤kâ¤n\right)\\ {M}_{i}/{2}^{n}& \left(k=n+1\right)\end{array}
Same as input sample rate.
Though the outputs have the same frame rate as the input, they have different frame sizes than the input. Thus, the output sample rates, Fso, k, are different from the input sample rate, Fsi:
Symmetric — All outputs have the sample rate Fsi / 2n.
Asymmetric —
{F}_{so,k}=\left\{\begin{array}{cc}{F}_{si}/{2}^{k}& \left(1â¤kâ¤n\right)\\ {F}_{si}/{2}^{n}& \left(k=n+1\right)\end{array}
User defined — Allows you to explicitly specify the filters with two vectors of filter coefficients in the Lowpass FIR filter coefficients and Highpass FIR filter coefficients parameters. The block uses the same lowpass and highpass filters throughout the filter bank. The two filters should be halfband filters, where each filter passes the frequency band that the other filter stops.
Wavelet such as Biorthogonal or Daubechies — The block uses the specified wavelet to construct the lowpass and highpass filters using the Wavelet Toolbox wfilters function. Depending on the wavelet, the block might enable either the Wavelet order or Filter order [synthesis / analysis] parameter. (The latter parameter allows you to specify different wavelet orders for the analysis and synthesis filter stages.) You must have a Wavelet Toolbox license to use wavelets.
The primary application for dyadic analysis filter banks and dyadic synthesis filter banks is coding for data compression using wavelets.
At the transmitting end, the output of the dyadic analysis filter bank is fed to a lossy compression scheme, which typically assigns the number of bits for each filter bank output in proportion to the relative energy in that frequency band. This represents the more powerful signal components by a greater number of bits than the less powerful signal components.
At the receiving end, the transmission is decoded and fed to a dyadic synthesis filter bank to reconstruct the original signal. The filter coefficients of the complementary analysis and synthesis stages are designed to cancel aliasing introduced by the filtering and resampling.
See Calculate Channel Latencies Required for Wavelet Reconstruction for an example using the Dyadic Analysis and Dyadic Synthesis Filter Bank blocks.
See the floating-point frame-based version of the DSP System Toolbox Wavelet Reconstruction and Noise Reduction example, which uses the Dyadic Analysis Filter Bank and Dyadic Synthesis Filter Bank blocks.
Select a wavelet such as Biorthogonal or Daubechies to specify a wavelet-based filter. The block uses the Wavelet Toolbox wfilters function to construct the filters. Extra parameters such as Wavelet order or Filter order [synthesis / analysis] might become enabled. For a list of the supported wavelets, see Specifying Filters with the Filter Parameter and Related Parameters.
A vector of filter coefficients (descending powers of z) that specifies coefficients used by all the lowpass filters in the filter bank. This parameter is enabled when you set Filter to User defined. The lowpass filter should be a half-band filter that passes the frequency band stopped by the filter specified in the Highpass FIR filter coefficients parameter. The default values of this parameter specify a filter based on a Daubechies wavelet with wavelet order 3.
A vector of filter coefficients (descending powers of z) that specifies coefficients used by all the highpass filters in the filter bank. This parameter is enabled when you set Filter to User defined. The highpass filter should be a half-band filter that passes the frequency band stopped by the filter specified in the Lowpass FIR filter coefficients parameter. The default values of this parameter specify a filter based on a Daubechies wavelet with wavelet order 3.
The order of the wavelet selected in the Filter parameter. This parameter is enabled only when you set Filter to certain types of wavelets, as shown in the Specifying Filters with the Filter Parameter and Related Parameters table.
The number of filter bank levels. An n-level asymmetric structure has n+1 outputs, and an n-level symmetric structure has 2n outputs, as shown in Wavelet Filter Banks. The block's icon changes depending on the value of this parameter.
The default setting of this parameter is Asymmetric for the Dyadic Analysis Filter Bank block, and Symmetric for the DWT block.
Set to Multiple ports to output each output subband on a separate port (the topmost port outputs the subband with the highest frequency band). Set to Single port to concatenate the subbands into one vector or matrix and output the concatenated subbands on a single port. For more information, see Output Characteristics.
The default setting of this parameter is Multiple ports for the Dyadic Analysis Filter Bank block, and Single port for the DWT block.
DWT | Dyadic Synthesis Filter Bank | Two-Channel Analysis Subband Filter |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.