text
stringlengths 256
16.4k
|
|---|
{\displaystyle \tau =-\kappa \theta \,}
{\displaystyle \tau \,}
{\displaystyle \theta \,}
{\displaystyle \kappa \,}
{\displaystyle U={\frac {1}{2}}\kappa \theta ^{2}}
{\displaystyle \theta \,}
{\displaystyle I\,}
{\displaystyle C\,}
{\displaystyle \kappa \,}
{\displaystyle \tau \,}
{\displaystyle \mathrm {N\,m} \,}
{\displaystyle f_{n}\,}
{\displaystyle T_{n}\,}
{\displaystyle \omega _{n}\,}
{\displaystyle \mathrm {rad\,s^{-1}} \,}
{\displaystyle f\,}
{\displaystyle \omega \,}
{\displaystyle \mathrm {rad\,s^{-1}} \,}
{\displaystyle \alpha \,}
{\displaystyle \mathrm {s^{-1}} \,}
{\displaystyle \phi \,}
{\displaystyle L\,}
{\displaystyle I{\frac {d^{2}\theta }{dt^{2}}}+C{\frac {d\theta }{dt}}+\kappa \theta =\tau (t)}
{\displaystyle C\ll {\sqrt {\frac {\kappa }{I}}}\,}
{\displaystyle f_{n}={\frac {\omega _{n}}{2\pi }}={\frac {1}{2\pi }}{\sqrt {\frac {\kappa }{I}}}\,}
{\displaystyle T_{n}={\frac {1}{f_{n}}}={\frac {2\pi }{\omega _{n}}}=2\pi {\sqrt {\frac {I}{\kappa }}}\,}
{\displaystyle \tau =0\,}
{\displaystyle \theta =Ae^{-\alpha t}\cos {(\omega t+\phi )}\,}
{\displaystyle \alpha =C/2I\,}
{\displaystyle \omega ={\sqrt {\omega _{n}^{2}-\alpha ^{2}}}={\sqrt {\kappa /I-(C/2I)^{2}}}\,}
{\displaystyle f_{n}\,}
{\displaystyle I\,}
{\displaystyle \kappa \,}
{\displaystyle F\,}
{\displaystyle L\,}
{\displaystyle \tau (t)=FL\,}
{\displaystyle \theta =FL/\kappa \,}
{\displaystyle F\,}
{\displaystyle \kappa \,}
{\displaystyle \kappa =(2\pi f_{n})^{2}I\,}
{\displaystyle C_{c}\,}
{\displaystyle C_{c}=2{\sqrt {\kappa I}}\,}
Coil springGarter springGas springSpring (device)Spring pinMechanical engineeringLeaf springChassisLatch#Spring_latches
This article uses material from the Wikipedia article "Torsion spring", which is released under the Creative Commons Attribution-Share-Alike License 3.0. There is a list of all authors in Wikipedia
|
Monoid - Simple English Wikipedia, the free encyclopedia
In abstract algebra, a monoid is a set of elements along with an operation that has two key properties:
The operation can combine the elements associatively; e.g.
{\displaystyle (A+B)+C=A+(B+C)}
There exists an identity element; e.g.
{\displaystyle 1\times X=X}
{\displaystyle 0+X=X}
The operation need not have the commutative property.
In computing science common monoids include addition, multiplication, or, and. These properties are useful for various problems e.g. they allow a large set of data to be divided, processed in parallel and combined. As each part produces a monoid, the final combined result will be the same. This also works with more complex monoids, such as a map from words to the number of times they appear in a document.
Character strings also form a monoid over concatenation with the empty string "" as the identity element. Examples of the two properties are "abc" + ("def" + "ghi") = ("abc" + "def") + "ghi" and "abc" + "" = ""+ "abc" = "abc". This is true even though "abc" + "def" ≠ "def" + "abc".
Haskell/Monoids at Wikibooks
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Monoid&oldid=7138286"
|
Managing Interest-Rate Risk with Bond Futures - MATLAB & Simulink Example - MathWorks Switzerland
Modifying the Duration of a Portfolio with Bond Futures
Modifying the Key Rate Durations of a Portfolio with Bond Futures
Improving the Performance of a Hedge with Regression
In managing a bond portfolio, you can use a benchmark portfolio to evaluate performance. Sometimes a manager is constrained to keep the portfolio's duration within a particular band of the duration of the benchmark. One way to modify the duration of the portfolio is to buy and sell bonds, however, there may be reasons why a portfolio manager wishes to maintain the existing composition of the portfolio (for example, the current holdings reflect fundamental research/views about future returns). Therefore, another option for modifying the duration is to buy and sell bond futures.
Bond futures are futures contracts where the commodity to be delivered is a government bond that meets the standard outlined in the futures contract (for example, the bond has a specified remaining time to maturity). Since often many bonds are available, and each bond may have a different coupon, you can use a conversion factor to normalize the payment by the long to the short.
There exist well developed markets for government bond futures. Specifically, the Chicago Board of Trade offers futures on the following:
https://www.cmegroup.com/trading/interest-rates/
Eurex offers futures on the following:
Euro-Schatz Futures 1.75 to 2.25
Euro-Bobl Futures 4.5 to 5.5
Euro-Bund Futures 8.5 to 10.5
Euro-Buxl Futures 24.0 to 35
https://www.eurex.com/ex-en/
Bond futures can be used to modify the duration of a portfolio. Since bond futures derive their value from the underlying instrument, the duration of a bond futures contract is related to the duration of the underlying bond.
There are two challenges in computing this duration:
Since there are many available bonds for delivery, the short in the contract has a choice in which bond to deliver.
Some contracts allow the short flexibility in choosing the delivery date.
Typically, the bond used for analysis is the bond that is cheapest for the short to deliver (CTD). One approach is to compute duration measures using the CTD's duration and the conversion factor.
For example, the Present Value of a Basis Point (PVBP) can be computed from the following:
PVB{P}_{Futures}=\frac{PVB{P}_{CTD}}{ConversionFacto{r}_{CTD}}
PVB{P}_{CTD}=\frac{Duratio{n}_{CTD}*Pric{e}_{CTD}}{100}
Note that these definitions of duration for the futures contract are approximate, and do not account for the value of the delivery options for the short.
If the goal is to modify the duration of a portfolio, use the following:
NumContracts=\frac{\left(Du{r}_{Target}-Du{r}_{Initial}\right)*Valu{e}_{Portfolio}}{Du{r}_{CTD}*Pric{e}_{CTD}*ContractSize}*ConvFacto{r}_{CTD}
Note that the contract size is typically for 100,000 face value of a bond -- so the contract size is typically 1000, as the bond face value is 100.
The following example assumes an initial duration, portfolio value, and target duration for a portfolio with exposure to the Euro interest rate. The June Euro-Bund Futures contract is used to modify the duration of the portfolio. Note that typically futures contracts are offered for March, June, September and December.
% Assume the following for the portfolio and target
PortfolioDuration = 6.4;
PortfolioValue = 100000000;
BenchmarkDuration = 4.8;
% Deliverable Bunds -- note that these conversion factors may also be
% computed with the MATLAB(R) function convfactor
BondPrice = [106.46;108.67;104.30];
BondMaturity = datenum({'04-Jan-2018','04-Jul-2018','04-Jan-2019'});
BondCoupon = [.04;.0425;.0375];
ConversionFactor = [.868688;.880218;.839275];
% Futures data -- found from http://www.eurex.com
FuturesPrice = 122.17;
FuturesSettle = '23-Apr-2009';
FuturesDelivery = '10-Jun-2009';
% To find the CTD bond we can compute the implied repo rate
ImpliedRepo = bndfutimprepo(BondPrice,FuturesPrice,FuturesSettle,...
FuturesDelivery,ConversionFactor,BondCoupon,BondMaturity);
% Note that the bond with the highest implied repo rate is the CTD
[CTDImpRepo,CTDIndex] = max(ImpliedRepo);
% Compute the CTD's Duration -- note the period and basis for German Bunds
Duration = bnddurp(BondPrice,BondCoupon,FuturesSettle,BondMaturity,1,8);
ContractSize = 1000;
% Use the formula above to compute the number of contracts to sell
NumContracts = (BenchmarkDuration - PortfolioDuration)*PortfolioValue./...
(BondPrice(CTDIndex)*ContractSize*Duration(CTDIndex))*ConversionFactor(CTDIndex);
disp(['To achieve the target duration, ' num2str(abs(round(NumContracts))) ...
' Euro-Bund Futures must be sold.'])
To achieve the target duration, 180 Euro-Bund Futures must be sold.
One of the shortcomings of using duration as a risk measure is that it assumes parallel shifts in the yield curve. While many studies have shown that this explains roughly 85% of the movement in the yield curve, changes in the slope or shape of the yield curve are not captured by duration, and therefore, hedging strategies are not successful at addressing these dynamics. One approach is to use key rate duration -- this is particularly relevant when using bond futures with multiple maturities, like Treasury futures.
The following example uses 2, 5, 10 and 30 year Treasury Bond futures to hedge the key rate duration of a portfolio. Computing key rate durations requires a zero curve. This example uses the zero curve published by the Treasury and found at the following location:
Note that this zero curve could also be derived using the Interest-Rate Curve functionality found in IRDataCurve and IRFunctionCurve.
% Assume the following for the portfolio and target, where the duration
% vectors are key rate durations at 2, 5, 10, and 30 years.
PortfolioDuration = [.5 1 2 6];
BenchmarkDuration = [.4 .8 1.6 5];
% The following are the CTD Bonds for the 30, 10, 5 and 2 year futures
% contracts -- these were determined using the procedure outlined in the
% previous section.
CTDCoupon = [4.75 3.125 5.125 7.5]'/100;
CTDMaturity = datenum({'3/31/2011','08/31/2013','05/15/2016','11/15/2024'});
CTDConversion = [0.9794 0.8953 0.9519 1.1484]';
CTDPrice = [107.34 105.91 117.00 144.18]';
ZeroRates = [0.07 0.10 0.31 0.50 0.99 1.38 1.96 2.56 3.03 3.99 3.89]'/100;
ZeroDates = daysadd(FuturesSettle,[30 360 360*2 360*3 360*5 ...
% Compute the key rate durations for each of the CTD bonds.
CTDKRD = bndkrdur([ZeroDates ZeroRates], CTDCoupon,FuturesSettle,...
CTDMaturity,'KeyRates',[2 5 10 30]);
% Note that the contract size for the 2 Year Note Future is $200,000
ContractSize = [2000;1000;1000;1000];
NumContracts = (bsxfun(@times,CTDPrice.*ContractSize./CTDConversion,CTDKRD))\...
(BenchmarkDuration - PortfolioDuration)'*PortfolioValue;
sprintf(['To achieve the target duration, \n' ...
num2str(-round(NumContracts(1))) ' 2 Year Treasury Note Futures must be sold, \n' ...
num2str(-round(NumContracts(3))) ' 10 Year Treasury Note Futures must be sold, \n' ...
num2str(-round(NumContracts(4))) ' Treasury Bond Futures must be sold, \n'])
'To achieve the target duration,
24 2 Year Treasury Note Futures must be sold,
68 10 Year Treasury Note Futures must be sold,
120 Treasury Bond Futures must be sold,
An additional component to consider in hedging interest-rate risk with bond futures, again related to movements in the yield curve, is that typically the yield curve moves more at the short end than at the long end.
Therefore, if a position is hedged with a future where the CTD bond has a maturity that is different than the portfolio this could lead to a situation where the hedge under- or over-compensates for the actual interest-rate risk of the portfolio One approach is to perform a regression on historical yields at different maturities to determine a Yield Beta, which is a value that represents how much more the yield changes for different maturities.
This example shows how to use this approach with UK Long Gilt futures and historical data on Gilt Yields.
Market data on Gilt futures is found at the following:
Historical data on gilts is found at the following;
Note that while this approach does offer the possibility of improving the performance of a hedge, any analysis using historical data depends on historical relationships remaining consistent.
Also note that an additional enhancement takes into consideration the correlation between different maturities. While this approach is outside the scope of this example, you can use this to implement a minimum variance hedge.
% This is the CTD Bond for the Long Gilt Futures contract
CTDBondPrice = 113.40;
CTDBondMaturity = datenum('7-Mar-2018');
CTDBondCoupon = .05;
CTDConversionFactor = 0.9325024;
% Market data for the Long Gilt Futures contract
CTDDuration = bnddurp(CTDBondPrice,CTDBondCoupon,FuturesSettle,CTDBondMaturity);
(CTDBondPrice*ContractSize*CTDDuration)*CTDConversionFactor;
disp(['To achieve the target duration with a conventional hedge ' ...
num2str(-round(NumContracts)) ...
' Long Gilt Futures must be sold.'])
To achieve the target duration with a conventional hedge 182 Long Gilt Futures must be sold.
To improve the accuracy of this hedge, historical data is used to determine a relationship between the standard deviation of the yields. Specifically, standard deviation of yields is plotted and regressed vs bond duration. This relationship is then used to compute a Yield Beta for the hedge.
% Load data from XLS spreadsheet
load ukbonddata_20072008
Duration = bnddury(Yield(1,:)',Coupon,Dates(1,:),Maturity);
scatter(Duration,100*std(Yield))
title('Standard Deviation of Yields for UK Gilts 2007-2008')
ylabel('Standard Deviation of Yields (%)')
xlabel('Duration')
annotation(gcf,'textbox',[0.4067 0.685 0.4801 0.0989],...
'String',{'Note that the Standard Deviation',...
'of Yields is greater at shorter maturities.'},...
stats = regstats(std(Yield),Duration);
YieldBeta = (stats.beta'*[1 PortfolioDuration]')./(stats.beta'*[1 CTDDuration]');
Now the Yield Beta is used to compute a new value for the number of contracts to be sold. Note that since the duration of the portfolio was less than the duration of the CTD Gilt, the number of futures to sell is actually greater than in the first case.
(CTDBondPrice*ContractSize*CTDDuration)*CTDConversionFactor*YieldBeta;
disp(['To achieve the target duration using a Yield Beta-modified hedge, ' ...
num2str(abs(round(NumContracts))) ...
To achieve the target duration using a Yield Beta-modified hedge, 193 Long Gilt Futures must be sold.
This example is based on the following books and papers:
[1] Burghardt, G., T. Belton, M. Lane and J. Papa. The Treasury Bond Basis. New York, NY: McGraw-Hill, 2005.
[2] Krgin, D. Handbook of Global Fixed Income Calculations. New York, NY: John Wiley & Sons, 2002.
[3] CFA Program Curriculum, Level III, Volume 4, Reading 31. CFA Institute, 2009.
|
Erratum: “Theoretical Stress Concentration Factors for Short Rectangular Plates With Centered Circular Holes” [ASME J. Mech. Des., 124, No. 1, pp. 126–128] | J. Mech. Des. | ASME Digital Collection
N. Troyani,,
N. Troyani,
C. Gomes, and,
C. Gomes, and
G. Sterlacci
J. Mech. Des. Dec 2002, 124(4): 828 (1 pages)
This is a correction to: Theoretical Stress Concentration Factors for Short Rectangular Plates With Centered Circular Holes
Troyani, , N., Gomes, and , C., and Sterlacci, G. (November 26, 2002). "Erratum: “Theoretical Stress Concentration Factors for Short Rectangular Plates With Centered Circular Holes” [ASME J. Mech. Des., 124, No. 1, pp. 126–128] ." ASME. J. Mech. Des. December 2002; 124(4): 828. https://doi.org/10.1115/1.1507761
structural engineering, stress analysis, tensile strength, finite element analysis
Finite element analysis, Plates (structures), Stress analysis (Engineering), Stress concentration, Structural engineering, Tensile strength
The insert formula in Fig. 2 should read
σnom=Sw/w−a
as indicated in Eq. (2) in the text.
A Method for Ultimate Strength Assessment of Plates in Combined Stresses
|
Round-robin tournament - 3D Model 4 Sport
Round-robin tournament (14037 views - Sports List)
PARTcloud - cricket, football
A round-robin tournament (or all-play-all tournament) is a competition in which each contestant meets all other contestants in turn.[1][2] A round-robin contrasts with an elimination tournament, in which participants are eliminated after a certain number of losses.
3.1 Advantages of the format
3.2 Disadvantages of the format
3.2.1 Tournament length
4 Scheduling algorithm
4.1 Original construction of pairing tables by Richard Schurig (1886)
The term round-robin is derived from the French term ruban, meaning "ribbon". Over a long period of time, the term was corrupted and idiomized to robin.[3][4]
In a single round-robin schedule, each participant plays every other participant once. If each participant plays all others twice, this is frequently called a double round-robin. The term is rarely used when all participants play one another more than twice,[1] and is never used when one participant plays others an unequal number of times (as is the case in almost all of the major United States professional sports leagues – see AFL (1940–41) and All-America Football Conference for exceptions).
In the United Kingdom, a round-robin tournament is often called an American tournament in sports such as tennis or billiards which usually have knockout tournaments.[5][6][7] In Italian it is called girone all'italiana (literally "Italian-style circuit"). In Serbian it is called the Berger system (Бергеров систем, Bergerov sistem), after chess player Johann Berger. A round-robin tournament with four players is sometimes called "quad" or "foursome".[8]
In sports with a large number of competitive matches per season, double round-robins are common. Most association football leagues in the world are organized on a double round-robin basis, in which every team plays all others in its league once at home and once away. This system is also used in qualification for major tournaments such as the FIFA World Cup and the continental tournaments (e.g. UEFA European Championship, CONCACAF Gold Cup). There are also round-robin bridge, chess, draughts, go, curling and Scrabble tournaments. The World Chess Championship decided in 2005 and in 2007 on an eight-player double round-robin tournament where each player faces every other player once as white and once as black.
Group tournaments rankings usually go by number of matches won and drawn, with any of a variety of tiebreaker criteria.
Frequently, pool stages within a wider tournament are conducted on a round-robin basis. Examples with single round-robin scheduling include the FIFA World Cup, UEFA European Football Championship, and UEFA Cup (2004–2009) in football, Super Rugby (rugby union) in the Southern Hemisphere during its past iterations as Super 12 and Super 14 (but not in its later 15- and 18-team formats), the Cricket World Cup along Pakistan Super League & Indian Premier League, the two major Twenty-20 Cricket tournaments, [The International (Dota 2)]] and many American Football college conferences, such as the Big 12 (which currently has 10 members). The group phases of the UEFA Champions League and Copa Libertadores de América are contested as a double round-robin, as are most basketball leagues outside the United States, including the regular-season and Top 16 phases of the Euroleague; the United Football League has used a double round-robin for both its 2009 and 2010 seasons.
Season ending tennis tournaments also use a round robin format prior to the semi on stages
The champion, in a round-robin tournament, is the contestant that wins the most games.
In theory, a round-robin tournament is the fairest way to determine the champion from among a known and fixed number of contestants. Each contestant, whether player or team, has equal chances against all other opponents because there is no prior seeding of contestants that will preclude a match between any given pair. The element of luck is seen to be reduced as compared to a knockout system since one or two bad performances need not ruin a competitor's chance of ultimate victory. Final records of participants are more accurate, in the sense that they represent the results over a longer period against the same opposition.
In team sport the (round-robin) major league champions are generally regarded as the "best" team in the land, rather than the (elimination) cup winners.
The main disadvantage of a round robin tournament is the time needed to complete it. Unlike a knockout tournament where half of the participants are eliminated after each round, a round robin requires one round less than the number of participants multiplied by half the number of participants if the number of participants is even, and as many rounds as participants if the number of participants is odd. For instance, a tournament of 16 teams can be completed in just 4 rounds (i.e. 15 matches) in a knockout (single elimination) format; a double elimination tournament format requires 30 (or 31) matches, but a round-robin would require 15 rounds (i.e. 120 matches) to finish if each competitor faces each other once. Other issues stem from the difference between the theoretical fairness of the round robin format and practice in a real event. Since the victor is gradually arrived at through multiple rounds of play, teams who perform poorly, who might have been quickly eliminated from title contention, are forced to play out their remaining games. Thus games are played late in the competition between competitors with no remaining chance of success. Moreover, some later matches will pair one competitor who has something left to play for against another who does not. It may also be possible for a competitor to play the strongest opponents in a round robin in quick succession while others play them intermittently with weaker opposition. This asymmetry means that playing the same opponents is not necessarily completely equitable: the same opponents in a different order may play harder or easier matches, while other teams are presented with more adversity during periods of the competition.[clarification needed] There is also no scheduled showcase final match. Only by coincidence would two competitors meet in the last match of the tournament where the result of that match determined the championship. A notable instance of such an event was the May 26, 1989 match between Arsenal and Liverpool.
Further issues arise where a round-robin is used as a qualifying round within a larger tournament. A competitor already qualified for the next stage before its last game may either not try hard (in order to conserve resources for the next phase) or even deliberately lose (if the scheduled next-phase opponent for a lower-placed qualifier is perceived to be easier than for a higher-placed one). Four pairs in the 2012 Olympics Women's doubles badminton, having qualified for the next round, were disqualified for attempting to lose in the round robin stage to avoid compatriots and better ranked opponents.[9] The round robin stage at the Olympics were a new introduction and potential problems were readily known prior to the tournament.
Another disadvantage, especially in smaller round-robins, is the "circle of death," where teams cannot be separated on a head-to-head record. In a three-team round-robin, where A defeats B, B defeats C, and C defeats A, all three competitors will have a record of one win and one loss, and a tiebreaker will need to be used to separate the teams.[10] This famously happened during the 1994 FIFA World Cup Group E, where all four teams finished with a record of one win, one draw, and one loss.
{\displaystyle n}
{\displaystyle {\begin{matrix}{\frac {n}{2}}\end{matrix}}(n-1)}
{\displaystyle n}
{\displaystyle (n-1)}
{\displaystyle {\begin{matrix}{\frac {n}{2}}\end{matrix}}}
{\displaystyle n}
{\displaystyle n}
{\displaystyle {\begin{matrix}{\frac {n-1}{2}}\end{matrix}}}
The circle method is the standard algorithm to create a schedule for a round-robin tournament. All competitors are assigned to numbers, and then paired in the first round:
Next, one of the contributors in the first or last column of the table is fixed (number one in this example) and the others rotated clockwise one position
If there are an odd number of competitors, a dummy competitor can be added, whose scheduled opponent in a given round does not play and has a bye. The schedule can therefore be computed as though the dummy were an ordinary player, either fixed or rotating. Instead of rotating one position, any number relatively prime to
{\displaystyle (n-1)}
Alternatively Berger tables,[14] named after the Austrian chess master Johann Berger, are widely used in the planning of tournaments. Berger published the pairing tables in his two Schachjahrbucher,[15][16] with due reference to its inventor Richard Schurig.[17][18]
This constitutes a schedule where player 14 has a fixed position, and all other players are rotated clockwise
{\displaystyle {\begin{matrix}{\frac {n}{2}}\end{matrix}}}
positions. This schedule alternates colours and is easily generated manually. To construct the next round, the last player, number 8 in the first round, moves to the head of the table, followed by player 9 against player 7, player 10 against 6, until player 1 against player 2. Arithmetically, this equates to adding
{\displaystyle {\begin{matrix}{\frac {n}{2}}\end{matrix}}}
{\displaystyle n}
{\displaystyle (n-1)}
{\displaystyle ({\begin{matrix}{\frac {n}{2}}\end{matrix}}-1)}
Both the graph and the schedule were reported by Édouard Lucas in [19] as a recreational mathematics puzzle. Lucas, who describes the method as simple and ingenious, attributes the solution to Felix Walecki, a teacher at Lycée Condorcet. Lucas also included an alternative solution by means of a sliding puzzle.
For 7 or 8 players, Schurig[18] builds a table with
{\displaystyle n/2}
{\displaystyle n-1}
horizontal rows, as follows:
1. Round 1 2 3 4
Then a second table is constructed (with counting from the end) as shown below:
1. Round . 1 . 7 . 6 . 5
1. Round 1, 1 2, 7 3, 6 4, 5
Then the first column is updated: if
{\displaystyle n}
is even, player number
{\displaystyle n}
is alternatingly substituted for the first and second positions, whereas if
{\displaystyle n}
is odd a bye is used instead.
Group tournament ranking system, including details of tie-breaking systems
Combinatorial design, a balanced tournament design of order n (a BTD(n))
Tournament (graph theory), mathematical model of a round-robin tournament
Shaughnessy playoff system, a type of single-elimination tournament featuring four teams
McIntyre System, a series of tournament formats that combine features of single- and double-elimination tournaments
Scheveningen system, where each member of one team plays each member of the other
2018 FIFA World Cup Final2018–19 UEFA Champions League2019 Cricket World CupAmerican footballAssociation footballBall (association football)CricketFlorida Gators footballFootballFootball helmetFootball playerSingle-elimination tournamentProfessional sports
This article uses material from the Wikipedia article "Round-robin tournament", which is released under the Creative Commons Attribution-Share-Alike License 3.0. There is a list of all authors in Wikipedia
|
Electromagnetic Induction - Live Session - NEET 2020
Electromagnetic Induction - Live Session - NEET 2020Contact Number: 9667591930 / 8527521718
A coil having an area of 2 m2 is placed in a magnetic field which changes from 1 Weber/m2 to 4 Weber/m2 in 2 seconds. The e.m.f. induced in the coil will be :–
A flexible wire bent in the form of a circle is placed in a uniform magnetic field perpendicular to the plane of the coil. The radius of the coil changes as shown figure. The induced emf in the coil is :-
A square loop of side 22 cm is changed to a circle in time 0.4 s. The magnetic field present is 0.2 T. The emf induced is :-
1. –6.6 mV
3. +6.6 mV
4. +13.2 mV
Which one of the following can produce maximum induced emf :-
1. 50 ampere dc
2. 50 ampere 50 Hz ac
3. 50 ampere 500 Hz ac
4. 100 ampere dc
A solenoid of 10 henry inductance and 2 ohm resistance, is connected to a 10 volt battery. In how much time the magnetic energy will be increased to 1/4 th of the maximum value?
1. 3.5 sec
An inductance coil have the time constant 4 sec, if it is cut into two equal parts and connected parallel then new time constant of the circuit :-
Which statement is correct from following –
(i) Inductor store energy in the form of magnetic field
(ii) Capacitor store energy in the form of electric field
(iii) Inductor store energy in the form of electric and magnetic field both
(iv) Capacitor store energy in the form of electric and magnetic field both
For a solenoid keeping the turn density constant its lenght makes halved and its cross section radius is doubled then the inductance of the solenoid increased by:–
In the circuit shown in figure bulb will become suddenly bright if:-
1. Key is closed
2. Key is opened
3. Key is opened or closed
4. Would not become bright
A conducting rod of 1m length rotating with a frequency of 50 rev/sec. about its one of end inside the uniform magnetic field of 6.28 mT. The value of induced emf between end of rod is :-
A semicircle loop PQ of radius 'R' is moved with velocity 'v' in transverse magnetic field as shown in figure. The value of induced emf. between the ends of loop is :-
1. Bv (
\mathrm{\pi }
r), end 'P' at high potential
2. 2 BRv, end P at high potential
3. 2 BRv, end Q at high potential
\frac{{\mathrm{\pi R}}^{2}}{2}\mathrm{v}
, end P at high potential
Two long parallel metallic wires with a resistance 'R' form a horizontal plane. A conducting rod AB is on the wires shown in figure. The space has magnetic field pointing vertically downwards. The rod is given an initial velocity 'v0'. There is no friction in the wires and the rod. After a time 't' the velocity v of the rod will be such that:–
1. v > v0
2. v < v0
3. v = v0
If a bar magnet is dropped vertically into a, long copper tube then its final acceleration will be:-
1. a = g
2. a > g
3. a < g
A coil of mean area 500 cm2 and having 1000 turns is held perpendicular to a uniform field of 0.4 gauss. The coil is turned through 180
°
\frac{1}{10}
second. The average induced e.m.f. :–
An emf induced in a coil, the linking magnetic flux
1. Must decrease
2. Must increase
3. Must remain constant
4. Can either increase of decrease
The magnetic flux through a circuit of resistance R changes by an amount
△\mathrm{\varphi }
in time interval
△
t, then the total amount of charge that passes through the circuit is
\frac{△\mathrm{\varphi }}{△\mathrm{t}}
\frac{△\mathrm{\varphi }}{\mathrm{R}}
\frac{△\mathrm{\varphi }}{\mathrm{R}△\mathrm{t}}
\mathrm{R}△\mathrm{t}
The magnetic flux linked with a coil at any instant t (in seconds) is given by
\mathrm{\varphi }
(in Wb) = 8t2 – 16t + 500. The induced emf in the coil at t=2s is
A rectangular loop is in uniform magnetic field such that its plane is perpendicular to the magnetic field as shown in the figure. If the loop is pulled out of the field, then the direction of induced current is
3. Maybe clockwise or anti-clockwise depends on its velocity
4. No current will induce
Two coils each of inductance L are mutually coupled perfectly such that magnetic flux of one coil opposes the flux of other. If the two coils are connected in series then the effective inductance of the combination is
In the circuit shown in the figure the switch is closed at t = 0. The current supplied by the battery at this instant is
\frac{\mathrm{\epsilon }}{{\mathrm{R}}_{1}+{\mathrm{R}}_{2}}
\frac{\mathrm{\epsilon }}{{\mathrm{R}}_{1}+{\mathrm{R}}_{3}}
\frac{\mathrm{\epsilon }\left({\mathrm{R}}_{2}+{\mathrm{R}}_{3}\right)}{{\mathrm{R}}_{1}\left({\mathrm{R}}_{2}+{\mathrm{R}}_{3}\right)+{\mathrm{R}}_{2}{\mathrm{R}}_{3}}
|
Lightweight Detection of a Small Number of Large Errors in a Quantum Circuit | QCS Hub
QCS Hub
- Ion trap processors
- Diamond node chips
- Silicon quantum processors
- Verification, validation and benchmarking
- Architectures, control and emulation
- Algorithms and fundamentals
A paper discussing "Lightweight Detection of a Small Number of Large Errors in a Quantum Circuit" has been published in Quantum, an open-access peer-reviewed journal for quantum
Suppose we want to implement a unitary
U
, for instance a circuit for some quantum algorithm. Suppose our actual implementation is a unitary
\stackrel{~}{U}
, which we can only apply as a black-box. In general it is an exponentially-hard task to decide whether
\stackrel{~}{U}
equals the intended
U
, or is significantly different in a worst-case norm. In this paper we consider two special cases where relatively efficient and lightweight procedures exist for this task.
First, we give an efficient procedure under the assumption that
U
\stackrel{~}{U}
(both of which we can now apply as a black-box) are either equal, or differ significantly in only one
k
-qubit gate, where
k=O\left(1\right)
k
qubits need not be contiguous). Second, we give an even more lightweight procedure under the assumption that
U
\stackrel{~}{U}
\mathit{\text{Clifford}}
circuits which are either equal, or different in arbitrary ways (the specification of
U
is now classically given while
\stackrel{~}{U}
can still only be applied as a black-box). Both procedures only need to run
\stackrel{~}{U}
a constant number of times to detect a constant error in a worst-case norm. We note that the Clifford result also follows from earlier work of Flammia and Liu, and da Silva, Landon-Cardinal, and Poulin.
In the Clifford case, our error-detection procedure also allows us efficiently to learn (and hence correct)
\stackrel{~}{U}
if we have a small list of possible errors that could have happened to
U
; for example if we know that only
O\left(1\right)
of the gates of
\stackrel{~}{U}
are wrong, this list will be polynomially small and we can test each possible erroneous version of
U
for equality with
\stackrel{~}{U}
You can read more here: https://quantum-journal.org/papers/q-2021-04-20-436/
Department of Physics, University of Oxford, Clarendon Laboratory, Parks Road, Oxford, OX1 3PU
We are supported by the EPSRC UK Quantum Technologies Programme under grant EP/T001062/1
|
Isolated power — Wikipedia Republished // WIKI 2
Sabermetric baseball statistic
In baseball, isolated power or ISO is a sabermetric computation used to measure a batter's raw power. One formula is slugging percentage minus batting average.
{\displaystyle ISO=SLG-AVG}
{\displaystyle ={\frac {{\mathit {TB}}-H}{AB}}}
{\displaystyle ={\frac {({\mathit {1B}})+(2\times {\mathit {2B}})+(3\times {\mathit {3B}})+(4\times {\mathit {HR}})}{AB}}-{\frac {H}{AB}}}
{\displaystyle ={\frac {({\mathit {1B}})+(2\times {\mathit {2B}})+(3\times {\mathit {3B}})+(4\times {\mathit {HR}})-({\mathit {1B}}+{\mathit {2B}}+{\mathit {3B}}+{\mathit {HR}})}{AB}}}
{\displaystyle ={\frac {({\mathit {2B}})+(2\times {\mathit {3B}})+(3\times {\mathit {HR}})}{AB}}}
The final result measures how many extra bases a player averages per at bat. A player who hits only singles would thus have an ISO of 0. The maximum ISO is 3.000, and can only be attained by hitting a home run in every at-bat.
The term "isolated power" was coined by Bill James, but the concept dates back to Branch Rickey and his statistician Allan Roth.[1]
^ McCue, Andy. "Allan Roth". Society for American Baseball Research. Society for American Baseball Research. Retrieved 4 June 2016.
Baseball Prospectus Glossary
Baseball Simple : Isolated Power (ISO) Calculator
FanGraphs Sabermetrics Library
|
FIR Gaussian Pulse-Shaping Filter Design - MATLAB & Simulink Example - MathWorks Italia
Continuous-Time Gaussian Filter
Frequency Response for Continuous-Time Gaussian Filter
FIR Approximation of the Gaussian Filter
Frequency Response for FIR Gaussian Filter (oversampling factor=16)
Significance of the Oversampling Factor
Frequency Response for FIR Gaussian Filter (oversampling factor=4)
This example shows how to design a Gaussian pulse-shaping FIR filter and the parameters influencing this design. The FIR Gaussian pulse-shaping filter design is done by truncating a sampled version of the continuous-time impulse response of the Gaussian filter which is given by:
h\left(t\right)=\frac{\sqrt{\pi }}{a}{e}^{-\frac{{\pi }^{2}{t}^{2}}{{a}^{2}}}
The parameter 'a' is related to 3-dB bandwidth-symbol time product (B*Ts) of the Gaussian filter as given by:
a=\frac{1}{B{T}_{s}}\sqrt{\frac{\mathrm{log}2}{2}}
There are two approximation errors in this design: a truncation error and a sampling error. The truncation error is due to a finite-time (FIR) approximation of the theoretically infinite impulse response of the ideal Gaussian filter. The sampling error (aliasing) is due to the fact that a Gaussian frequency response is not really band-limited in a strict sense (i.e. the energy of the Gaussian signal beyond a certain frequency is not exactly zero). This can be noted from the transfer function of the continuous-time Gaussian filter, which is given as below:
H\left(f\right)={e}^{-{a}^{2}{f}^{2}}
As f increases, the frequency response tends to zero, but never is exactly zero, which means that it cannot be sampled without some aliasing occurring.
To design a continuous-time Gaussian filter, let us define the symbol time (Ts) to be 1 micro-second and the number of symbols between the start of the impulse response and its end (filter span) to be 6. From the equations above, we can see that the impulse response and the frequency response of the Gaussian filter depend on the parameter 'a' which is related to the 3 dB bandwidth-symbol time product. To study the effect of this parameter on the Gaussian FIR filter design, we will define various values of 'a' in terms of Ts and compute the corresponding bandwidths. Then, we will plot the impulse response for each 'a' and the magnitude response for each bandwidth.
Ts = 1e-6; % Symbol time (sec)
span = 6; % Filter span in symbols
a = Ts*[.5, .75, 1, 2];
B = sqrt(log(2)/2)./(a);
t = linspace(-span*Ts/2,span*Ts/2,1000)';
hg = zeros(length(t),length(a));
hg(:,k) = sqrt(pi)/a(k)*exp(-(pi*t/a(k)).^2);
plot(t/Ts,hg)
title({'Impulse response of a continuous-time Gaussian filter';...
'for various bandwidths'});
xlabel('Normalized time (t/Ts)')
legend(sprintf('a = %g*Ts',a(1)/Ts),sprintf('a = %g*Ts',a(2)/Ts),...
sprintf('a = %g*Ts',a(3)/Ts),sprintf('a = %g*Ts',a(4)/Ts))
Note that the impulse responses are normalized to the symbol time.
We will compute and plot the frequency response for continuous-time Gaussian filters with different bandwidths. In the graph below, the 3-dB cutoff is indicated by the red circles ('o') on the magnitude response curve. Note that 3-dB bandwidth is between DC and B.
f = linspace(0,32e6,10000)';
Hideal = zeros(length(f),length(a));
Hideal(:,k) = exp(-a(k)^2*f.^2);
plot(f,20*log10(Hideal))
titleStr = {'Ideal magnitude response for a continuous-time ';...
'Gaussian filter for various bandwidths'};
legend(sprintf('B = %g',B(1)),sprintf('B = %g',B(2)),...
sprintf('B = %g',B(3)),sprintf('B = %g',B(4)))
plot(B,20*log10(exp(-a.^2.*B.^2)),'ro','HandleVisibility','off')
axis([0 5*max(B) -50 5])
We will design the FIR Gaussian filter using the gaussdesign function. The inputs to this function are the 3-dB bandwidth-symbol time product, the number of symbol periods between the start and end of the filter impulse response, i.e. filter span in symbols, and the oversampling factor (i.e. the number of samples per symbol).
The oversampling factor (OVSF) determines the sampling frequency and the filter length and hence, plays a significant role in the Gaussian FIR filter design. The approximation errors in the design can be reduced with an appropriate choice of oversampling factor. We illustrate this by comparing the Gaussian FIR filters designed with two different oversampling factors.
First, we will consider an oversampling factor of 16 to design the discrete Gaussian filter.
ovsf = 16; % Oversampling factor (samples/symbol)
h = zeros(97,4);
iz = zeros(97,4);
BT = B(k)*Ts;
h(:,k) = gaussdesign(BT,span,ovsf);
[iz(:,k),t] = impz(h(:,k));
t = (t-t(end)/2)/Ts;
stem(t,iz)
title({'Impulse response of the Gaussian FIR filter for ';...
'various bandwidths, OVSF = 16'});
We will calculate the frequency response for the Gaussian FIR filter with an oversampling factor of 16 and we will compare it with the ideal frequency response (i.e. frequency response of a continuous-time Gaussian filter).
Fs = ovsf/Ts;
fvtool(h(:,1),1,h(:,2),1,h(:,3),1,h(:,4),1,...
'FrequencyRange', 'Specify freq. vector', ...
'FrequencyVector',f,'Fs',Fs,'Color','white');
title('Ideal magnitude responses and FIR approximations, OVSF = 16')
plot(f*Ts,20*log10(Hideal),'--')
axis([0 32 -350 5])
legend(append(["B = " "Ideal, B = "],string(num2str(B','%g'))), ...
'NumColumns',2,'Location','best')
Notice that the first two FIR filters exhibit aliasing errors and the last two FIR filters exhibit truncation errors. Aliasing occurs when the sampling frequency is not greater than the Nyquist frequency. In case of the first two filters, the bandwidth is large enough that the oversampling factor does not separate the spectral replicas enough to avoid aliasing. The amount of aliasing is not very significant however.
On the other hand, the last two FIR filters show the FIR approximation limitation before any aliasing can occur. The magnitude responses of these two filters reach a floor before they can overlap with the spectral replicas.
The aliasing and truncation errors vary according to the oversampling factor. If the oversampling factor is reduced, these errors will be more severe, since this reduces the sampling frequency (thereby moving the replicas closer) and also reduces the filter lengths (increasing the error in the FIR approximation).
For example, if we select an oversampling factor of 4, we will see that all the FIR filters exhibit aliasing errors as the sampling frequency is not large enough to avoid the overlapping of the spectral replicas.
ovsf = 4; % Oversampling factor (samples/symbol)
title({'Impulse response of the Gaussian FIR filter'; 'for various bandwidths, OVSF = 4'});
We will plot and study the frequency response for the Gaussian FIR filter designed with oversampling factor of 4. A smaller oversampling factor means smaller sampling frequency. As a result, this sampling frequency is not enough to avoid the spectral overlap and all the FIR approximation filters exhibit aliasing.
title('Ideal magnitude responses and FIR approximations, OVSF = 4')
'NumColumns',2,'Location','southeast')
FVTool | gaussdesign
|
charfcn - Maple Help
Home : Support : Online Help : Mathematics : Algebra : Expression Manipulation : charfcn
characteristic function for expressions and sets
charfcn[A](x)
specification for a set
The charfcn function is the characteristic function of the "set" A. It is defined to be
\mathrm{charfcn}[A]\left(x\right)={\begin{array}{cc}1& x∈A\\ 0& x\notin A\\ \mathrm{\text{'}charfcn}[A]\left(x\right)\text{'}& \mathrm{otherwise}\end{array}
The set specification A can be a set, a real numeric, a complex numeric, a real numeric range, a complex numeric range, an arbitrary range, or an expression sequence of any of the previous. The meaning of each one of these is as follows
A
"in" (
∈
) means:
a real or complex numeric
a real numeric range,
a..b
a\le x\le b
a complex numeric range,
a..b
\mathrm{ℜ}\left(a\right)\le \mathrm{ℜ}\left(x\right)\le \mathrm{ℜ}\left(b\right)
\mathrm{ℑ}\left(a\right)\le \mathrm{ℑ}\left(x\right)\le \mathrm{ℑ}\left(b\right)
a
is the bottom left corner,
b
is the top right corner)
an arbitrary range,
a..b
a\le x\le b
, as determined by
\mathrm{signum}\left(0\right)
-- note that
\mathrm{ℑ}\left(a\right),\mathrm{ℑ}\left(b\right),\mathrm{ℑ}\left(x\right)
must all evaluate to
0
an expression sequence of any of the above (except set)
"or" of the above conditions
an expression sequence of sets
set membership in the union of the sets
When the specification is a set, the maple function member is used to test set membership, and thus charfcn will always return one of 0 or 1 in this case. In the other cases, charfcn is symbolic, in that it will return unevaluated if the "in" conditions cannot be verified or the specification is not exactly as described above.
a..b
b<a
are treated as empty, and so charfcn will return 0 for all input.
\mathrm{charfcn}[3,5,9]\left(0\right)
\textcolor[rgb]{0,0,1}{0}
\mathrm{charfcn}[3,5,9]\left(3\right)
\textcolor[rgb]{0,0,1}{1}
\mathrm{charfcn}[3,5,9]\left(x\right)
{\textcolor[rgb]{0,0,1}{\mathrm{charfcn}}}_{\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{9}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)
\mathrm{charfcn}[3.13..3.16]\left(\mathrm{\pi }\right)
\textcolor[rgb]{0,0,1}{1}
\mathrm{charfcn}[0..3,5..7]\left(6\right)
\textcolor[rgb]{0,0,1}{1}
\mathrm{charfcn}[{\mathrm{one},\mathrm{three},\mathrm{two}}]\left(\mathrm{four}\right)
\textcolor[rgb]{0,0,1}{0}
\mathrm{charfcn}[{\mathrm{one},\mathrm{three},\mathrm{two}}]\left(\mathrm{two}\right)
\textcolor[rgb]{0,0,1}{1}
\mathrm{charfcn}[1+I..3+2I]\left(2+I\right)
\textcolor[rgb]{0,0,1}{1}
\mathrm{charfcn}[3+I..3+2I,7,\mathrm{\pi }..{\mathrm{\pi }}^{2}]\left(\mathrm{exp}\left({y}^{3}\right)\right)
{\textcolor[rgb]{0,0,1}{\mathrm{charfcn}}}_{\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{I}\textcolor[rgb]{0,0,1}{..}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{I}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{\pi }}\textcolor[rgb]{0,0,1}{..}{\textcolor[rgb]{0,0,1}{\mathrm{\pi }}}^{\textcolor[rgb]{0,0,1}{2}}}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{ⅇ}}^{{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{3}}}\right)
\mathrm{assume}\left(1.1<y,y<1.3\right)
\mathrm{charfcn}[3+I..3+2I,7,\mathrm{\pi }..{\mathrm{\pi }}^{2}]\left(\mathrm{exp}\left({y}^{3}\right)\right)
\textcolor[rgb]{0,0,1}{1}
\mathrm{charfcn}[3+I..-1+2I]\left(x\right)
\textcolor[rgb]{0,0,1}{0}
|
Single Staking Options Vault (SSOV) - Dopex
Similar to single staking vaults, SSOVs allow users to lock up tokens for a specified period of time and earn yield on their staked assets. Users will be able to deposit assets into a contract which then sells your deposits as call options to buyers at fixed strikes that they select for end-of-month expiries.
SSOV call options are either at the money, out of the money, or far out of the money.
https://app.dopex.io/ssov
Prior to the beginning of a new epoch, strikes are set for the month-end.
Users lock assets into this vault and select fixed strikes that you’d like to sell calls at.
The contract deposits the users' tokens into a single staking pool for farming rewards and also earns a yield from selling covered calls.
In essence, users will be selling covered calls at low risk with no need for intensive knowledge on option Greeks.
Buyers will be able to purchase calls from the vaults using the base asset. On the platform frontend, users will be able to use stablecoins or $ETH for purchases but they will be routed through Sushiswap on Arbitrum and swapped to the base asset of the SSOV.
SSOV depositors will receive yield proportional to how close to ATM strikes are being locked into. Users don’t lose any USD notional value, however, they do have a chance of losing a % of staked assets.
SSOV will appeal to USD denominated traders and will also give incentives for users to raise prices higher.
Base Asset (DPX, rDPX etc)
Option Token Standard
All options are auto-exercised on expiry by default and can be settled any time after expiry at your convenience.
Settlements on option exercises happen without requiring the underlying asset and hence are net settlements. The PnL (Profit & Loss) of the option is calculated and if it is +ve then the exercise can go through which burn the option tokens and transfer the PnL in the settlement asset to the user. PnL is calculated in the following manner:
PnL = ((Price - Strike) * Amount) / Price
The settlement price is a TWAP (Time-Weighted Average Price) of the Sushiswap DPX/ETH pool combined with a chainlink pricefeed of ETH/USD to compute the final price of DPX in USD. The time period of the TWAP is 30 minutes.
The settlement price is a TWAP (Time-Weighted Average Price) of the Sushiswap rDPX/ETH pool combined with a chainlink pricefeed of ETH/USD to compute the final price of rDPX in USD. The time period of the TWAP is 30 minutes.
We have introduced an updated fee structure for SSOVs calling it the Fee Multiplier for all our current and upcoming Single Staking Options Vaults (SSOVs). The Fee multiplier is a simple % based multiplier that multiplies fees for Out of the Money (OTM) strikes to account for higher volatilities.
Strike - 2000 Current Price of Asset - 1000
Fee Multiplier = 1 + ((2000/1000) - 1)
Final Fee = Base Fee * Fee Multiplier
The following Fee Multiplier structures are implemented:
ETH SSOV
Purchase → 0.125% of Underlying * Amount
70% to buy back LP for Protocol 30% to Governance Staking
DPX SSOV
rDPX SSOV
Purchase → 0.25% of Underlying * Amount
The above mentioned structure and distribution will be continuously monitored and evaluated for the next couple of months.
For more information check out our blog posts:
Introduction to SSOV
https://blog.dopex.io/single-staking-option-vaults-by-dopex-e7084418736
Tutorial for SSOV
Strategies for SSOV
|
Your friend is taking an algebra class at a different school where she is not allowed to use a graphing calculator.
Explain to her how she can get a good sketch of the graph of the function
y=2(x+3)^2−8
without using a calculator and without having to make an
x→y
table. Be sure to explain how to locate the vertex, whether the parabola should open up or down, and how its shape is related to the shape of the graph of
y=x^2
Where will the vertex be? Does the parabola open up or down? Is it vertically stretched?
Your friend also needs to know the
x
y
-intercepts. Show her how to find them without having to draw an accurate graph or use a graphing calculator.
0
x
y
y
0
y
x
x
|
Work, Energy and Power - Live Session - NEET 2020Contact Number: 9667591930 / 8527521718
When a rubber-band is stretched by a distance x, it exerts a restoring force of magnitude
\mathrm{F}=\mathrm{ax}+{\mathrm{bx}}^{2}
where a and b are constants. The work done in stretching the unstretched rubber-band by L is
{\mathrm{aL}}^{2}+{\mathrm{bL}}^{3}
\frac{1}{2}\left({\mathrm{aL}}^{2}+{\mathrm{bL}}^{3}\right)
\frac{{\mathrm{aL}}^{2}}{2}+\frac{{\mathrm{bL}}^{3}}{3}
\frac{1}{2}\left(\frac{{\mathrm{aL}}^{2}}{2}+\frac{{\mathrm{bL}}^{3}}{3}\right)
The work done on a particle of mass m by a force
\mathrm{K}\left[\frac{\mathrm{x}}{{\left({\mathrm{x}}^{2}+{\mathrm{y}}^{2}\right)}^{3/2}}\stackrel{^}{\mathrm{i}}+\frac{\mathrm{y}}{{\left({\mathrm{x}}^{2}+{\mathrm{y}}^{2}\right)}^{3/2}}\stackrel{^}{\mathrm{j}}\right]
( K being a constant of appropriate dimensions), when the particle is taken from the point (a,0) to the point (0,a) along a circular path of radius a about the origin in the x-y plane is
\frac{2\mathrm{K\pi }}{\mathrm{a}}
\frac{\mathrm{K\pi }}{\mathrm{a}}
\frac{\mathrm{K\pi }}{2\mathrm{a}}
The potential energy function for the force between two atoms in a diatomic molecule is approximately given by
\mathrm{U}\left(\mathrm{x}\right)=\frac{\mathrm{a}}{{\mathrm{x}}^{12}}-\frac{\mathrm{b}}{{\mathrm{x}}^{6}}
, where a and b are constants and x is the distance atoms. If the dissociation energy of the molecule is
\mathrm{D}=\left[\mathrm{U}\left(\mathrm{x}=\infty \right)-{\mathrm{U}}_{\mathrm{at} \mathrm{equilibrium}}\right]
, D is
\frac{{\mathrm{b}}^{2}}{6\mathrm{a}}
\frac{{\mathrm{b}}^{2}}{2\mathrm{a}}
\frac{{\mathrm{b}}^{2}}{12\mathrm{a}}
\frac{{\mathrm{b}}^{2}}{4\mathrm{a}}
A particle in a certain conservative force field has a potential energy given by U =
\frac{20\mathrm{xy}}{\mathrm{z}}
. The force exerted on it is
1. \left(\frac{20\mathrm{y}}{\mathrm{z}}\right)\stackrel{^}{\mathrm{i}}+ \left(\frac{20\mathrm{x}}{\mathrm{z}}\right)\stackrel{^}{\mathrm{j}}+ \left(\frac{20\mathrm{xy}}{{\mathrm{z}}^{2}}\right)\stackrel{^}{\mathrm{k}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}2. -\left(\frac{20\mathrm{y}}{\mathrm{z}}\right)\stackrel{^}{\mathrm{i}}- \left(\frac{20\mathrm{x}}{\mathrm{z}}\right)\stackrel{^}{\mathrm{j}}+ \left(\frac{20\mathrm{xy}}{{\mathrm{z}}^{2}}\right)\stackrel{^}{\mathrm{k}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}3. -\left(\frac{20\mathrm{y}}{\mathrm{z}}\right)\stackrel{^}{\mathrm{i}}- \left(\frac{20\mathrm{x}}{\mathrm{z}}\right)\stackrel{^}{\mathrm{j}}- \left(\frac{20\mathrm{xy}}{{\mathrm{z}}^{2}}\right)\stackrel{^}{\mathrm{k}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}4. \left(\frac{20\mathrm{y}}{\mathrm{z}}\right)\stackrel{^}{\mathrm{i}}+ \left(\frac{20\mathrm{x}}{\mathrm{z}}\right)\stackrel{^}{\mathrm{j}}- \left(\frac{20\mathrm{xy}}{{\mathrm{z}}^{2}}\right)\stackrel{^}{\mathrm{k}}
1. {\mathrm{t}}^{1/2}\phantom{\rule{0ex}{0ex}}2. \mathrm{t}\phantom{\rule{0ex}{0ex}}3. {\mathrm{t}}^{3/2}\phantom{\rule{0ex}{0ex}}4. {\mathrm{t}}^{2}
The particles A and B, move with constant velocities
\overrightarrow{{\mathrm{v}}_{1}}
. At the initial moment their position vector are
\overrightarrow{{\mathrm{r}}_{1}}
\overrightarrow{{\mathrm{r}}_{2}}
respectively. The condition for particles A and b for their collision is
\overrightarrow{{\mathrm{r}}_{1\quad }}.\overrightarrow{{\mathrm{v}}_{1}}\quad =\overrightarrow{{\mathrm{r}}_{2\quad }}.\overrightarrow{{\mathrm{v}}_{2}}
\overrightarrow{{\mathrm{r}}_{1}}\quad \mathrm{x}\quad \overrightarrow{{\mathrm{v}}_{1}}\quad =\overrightarrow{{\mathrm{r}}_{2}}\quad \mathrm{x}\quad \overrightarrow{{\mathrm{v}}_{2}}
\quad \overrightarrow{{\mathrm{r}}_{1}}-\quad \stackrel{\rightharpoonup }{{\mathrm{r}}_{2}}\quad =\overrightarrow{{\mathrm{v}}_{1}}-\quad \overrightarrow{{\mathrm{v}}_{2}}
\quad \frac{\overrightarrow{{\mathrm{r}}_{1}}-\quad \overrightarrow{{\mathrm{r}}_{2}}}{\left|\overrightarrow{{\mathrm{r}}_{1}}-\quad \overrightarrow{{\mathrm{r}}_{2}}\right|}=\quad \frac{\overrightarrow{{\mathrm{v}}_{2}}-\quad \overrightarrow{{\mathrm{v}}_{1}}}{\left|\overrightarrow{{\mathrm{v}}_{2}}-\quad \overrightarrow{{\mathrm{v}}_{1}}\right|}
A particle of mass 'm' is moving in circular path of constant radius 'r' such that centripetal acceleration is varying with time 't' as
{\mathrm{K}}^{2}{\mathrm{rt}}^{2}
where K is a constant. The power delivered to the particle by the force acting on it is
1. {\mathrm{m}}^{2}{\mathrm{K}}^{2}{\mathrm{r}}^{2}{\mathrm{t}}^{2}\phantom{\rule{0ex}{0ex}}2. {\mathrm{mK}}^{2}{\mathrm{r}}^{2}\mathrm{t}\phantom{\rule{0ex}{0ex}}3. {\mathrm{mK}}^{2}{\mathrm{rt}}^{2}\phantom{\rule{0ex}{0ex}}4. {\mathrm{mKr}}^{2}\mathrm{t}
A particle falls from a height h upon a fixed horizontal plane and rebounds. If e is the coefficient of restitution, the total distance travelled before rebounding has stopped is
1. \mathrm{h}\left(\frac{1+ {\mathrm{e}}^{2}}{1- {\mathrm{e}}^{2}}\right)\phantom{\rule{0ex}{0ex}}2. \mathrm{h}\left(\frac{1- {\mathrm{e}}^{2}}{1+ {\mathrm{e}}^{2}}\right)\phantom{\rule{0ex}{0ex}}3. \frac{\mathrm{h}}{2}\left(\frac{1- {\mathrm{e}}^{2}}{1+ {\mathrm{e}}^{2}}\right)\phantom{\rule{0ex}{0ex}}4. \frac{\mathrm{h}}{2}\left(\frac{1+ {\mathrm{e}}^{2}}{1- {\mathrm{e}}^{2}}\right)
{\mathrm{W}}_{1}, {\mathrm{W}}_{2}
{\mathrm{W}}_{3}
represent the work done in moving a particle from A to B along three different paths 1, 2 and 3 respectively (as shown) in the gravitational field of a point mass m, find the correct relation between
{\mathrm{W}}_{1}, {\mathrm{W}}_{2}
{\mathrm{W}}_{3}
1. {\mathrm{W}}_{1}>{\mathrm{W}}_{2}>{\mathrm{W}}_{3}\phantom{\rule{0ex}{0ex}}2. {\mathrm{W}}_{1}= {\mathrm{W}}_{2}= {\mathrm{W}}_{3}\phantom{\rule{0ex}{0ex}}3. {\mathrm{W}}_{1}<{\mathrm{W}}_{2}<{\mathrm{W}}_{3}\phantom{\rule{0ex}{0ex}}4. {\mathrm{W}}_{2}>{\mathrm{W}}_{1}>{\mathrm{W}}_{3}
Consider elastic collision of a particle of a mass m moving with a velocity u with another particle of the same mass at rest. After the collision the projectile and the struck particle move in directions making angle
{\mathrm{\theta }}_{1}
{\mathrm{\theta }}_{2}
respectively with the initial direction of motion. The sum of the angles.
{\mathrm{\theta }}_{1}+ {\mathrm{\theta }}_{2}
1. {45}^{\mathrm{o}}\phantom{\rule{0ex}{0ex}}2. {90}^{\mathrm{o}}\phantom{\rule{0ex}{0ex}}3. {135}^{\mathrm{o}}\phantom{\rule{0ex}{0ex}}4. {180}^{\mathrm{o}}
Two small particles of equal masses moving in opposite directions from a point A in a horizontal circular orbit. Their tangential velocities are v and 2v, respectively, as shown in the figure. Between collisions, the particles move with constant speed. After making how many elastic collisions, other than that at A, these two particles will again reach the point A
A cord is used to lower vertically a block of mass M by a distance d with constant downward acceleration
\frac{\mathrm{g}}{4}
. Work done by the cord on the block is
1. \mathrm{Mg}\frac{\mathrm{d}}{4}\phantom{\rule{0ex}{0ex}}2. 3\mathrm{Mg}\frac{\mathrm{d}}{4}\phantom{\rule{0ex}{0ex}}3. -3\mathrm{Mg}\frac{\mathrm{d}}{4}\phantom{\rule{0ex}{0ex}}4. \mathrm{Mgd}
A frictionless track ABCDE ends in a circular loop of radius R. A body slides down from point A which is at a height h = 5 cm. Maximum value of R for the body to successfully complete the loop is
1. 5 \mathrm{cm}\phantom{\rule{0ex}{0ex}}2. \frac{15}{4} \mathrm{cm}\phantom{\rule{0ex}{0ex}}3. \frac{10}{3} \mathrm{cm}\phantom{\rule{0ex}{0ex}}4. 2 \mathrm{cm}
A body is moved along a straight line by a machine delievering constant power. The distance moved by the body in time t is proportional to
A particle is moving in a circle of radius r under the action of a force F =
{\mathrm{ar}}^{2}
which is directed towards centre of the circle. Total mechanical energy (kinetic energy + potential energy) of the particle is (take potential energy = 0 for r = 0)
1. \frac{1}{2}{\mathrm{\alpha r}}^{3}\phantom{\rule{0ex}{0ex}}2. \frac{5}{6}{\mathrm{\alpha r}}^{3}\phantom{\rule{0ex}{0ex}}3. \frac{4}{3}{\mathrm{\alpha r}}^{3}\phantom{\rule{0ex}{0ex}}4. {\mathrm{\alpha r}}^{3}
A particle is moving in a circular path of radius a under the action of an attractive potential U =
-\frac{\mathrm{k}}{2{\mathrm{r}}^{2}}
. Its total energy is:
1. -\frac{\mathrm{k}}{4{\mathrm{a}}^{2}}\phantom{\rule{0ex}{0ex}}2. \frac{\mathrm{k}}{2{\mathrm{a}}^{2}}\phantom{\rule{0ex}{0ex}}3. \mathrm{zero}\phantom{\rule{0ex}{0ex}}4. -\frac{3}{2}\frac{\mathrm{k}}{{\mathrm{a}}^{2}}
A stone is tied to a string of length
\mathcal{l}
and is whirled in a vertical circle with the other end of the string as the centre. At a certain instant of time, the stone is at its lowest position and has speed u. The magnitude of the change in velocity as it reaches a position where the string is horizontal (g being acceleration due to gravity) is
1. \sqrt{2\mathrm{g}\mathcal{l}}\phantom{\rule{0ex}{0ex}}2. \sqrt{2\left({\mathrm{u}}^{2}- \mathrm{g}\mathcal{l}\right)}\phantom{\rule{0ex}{0ex}}3. \sqrt{{\mathrm{u}}^{2}- \mathrm{g}\mathcal{l}}\phantom{\rule{0ex}{0ex}}4. \mathrm{u}- \sqrt{{\mathrm{u}}^{2}- 2\mathrm{g}\mathcal{l}}
Consider a drop of rain water having mass 1 g falling from a height of 1 km. It hits the ground with a speed of 50 m/s. Take 'g' constant with a value 10
\mathrm{m}/{\mathrm{s}}^{2}
. The work done by the (i) gravitational force and the (ii) resistive force of air
1. (i) 1.25 J (ii) -8.25 J
2. (i) 100 J (ii) 8.75 J
3. (i) 10 J (ii) -8.75 J
4. (i) -10 J (ii) -8.25 J
\mathrm{m}/{\mathrm{s}}^{2}
A body of mass m =
{10}^{-2}
kg is moving in a medium and experiences a frictional force F =
-{\mathrm{kv}}^{2}
. Its initial speed is
{\mathrm{v}}_{0}
{\mathrm{ms}}^{-1}
. If, after 10 s, its energy is
\frac{1}{8}{\mathrm{mv}}_{0}^{2}
, the value of k will be:
1. {10}^{-4} \mathrm{kg} {\mathrm{m}}^{-1}\phantom{\rule{0ex}{0ex}}2. {10}^{-1} \mathrm{kg} {\mathrm{m}}^{-1} {\mathrm{s}}^{-1}\phantom{\rule{0ex}{0ex}}3. {10}^{-3} \mathrm{kg} {\mathrm{m}}^{-1}\phantom{\rule{0ex}{0ex}}4. {10}^{-3} \mathrm{kg} {\mathrm{s}}^{-1}
Figure shows a bob of mass m suspended from a string of length L. The velocity is
{\mathrm{V}}_{0}
at A, then the potential energy of the system is ________ at the lowest point A.
1. \frac{1}{2}{\mathrm{m}}_{0}^{2}\phantom{\rule{0ex}{0ex}}2. \mathrm{mgh}\phantom{\rule{0ex}{0ex}}3. \frac{-1}{2}{\mathrm{mv}}_{0}^{2}\phantom{\rule{0ex}{0ex}}4. \mathrm{zero}
The given plot shows the variation of U, the potential energy of interaction between two particles with the distance separating them, r
1. B and D are equilibrium points
2. C is a point of stable equlibrium
3. The force of interaction between the two particles is attractive between points C and D and repulsive between points D and E on the curve.
4. The force of interaction between the particles is attractive between points E and F on the curve.
|
OpenStax University Physics/E&M/Electromagnetic Induction - Wikiversity
OpenStax University Physics/E&M/Electromagnetic Induction
{\displaystyle {\mathcal {E}}\&{\mathcal {M}}}
1.2 For quiz at QB/d_cp2.13
Electromagnetic Induction[edit | edit source]
{\displaystyle \Phi _{m}=\int _{S}{\vec {B}}\cdot {\hat {n}}dA}
▭ Electromotive force
{\displaystyle \varepsilon =-N{\tfrac {d\Phi _{m}}{dt,}}}
▭ Motional emf
{\displaystyle \varepsilon =B\ell v,}
▭ rotating coil
{\displaystyle NBA\omega \sin \omega t}
▭ Motional emf around circuit
{\displaystyle \varepsilon =\oint {\vec {E}}\cdot d{\vec {\ell }}=-{\tfrac {d\Phi _{m}}{dt}}}
For quiz at QB/d_cp2.13[edit | edit source]
{\displaystyle \Phi _{m}=\int _{S}{\vec {B}}\cdot {\hat {n}}dA}
{\displaystyle \varepsilon =B\ell v}
{\displaystyle {\vec {v}}\perp {\vec {B}}}
{\displaystyle \varepsilon =-N{\tfrac {d\Phi _{m}}{dt}}=\oint {\vec {E}}\cdot d{\vec {\ell }}}
{\displaystyle \varepsilon =NBA\omega \sin \omega t}
Retrieved from "https://en.wikiversity.org/w/index.php?title=OpenStax_University_Physics/E%26M/Electromagnetic_Induction&oldid=1910576"
|
Daniela, Kieu, and Duyen decide to go to the movies one hot summer afternoon. The theater is having a summer special called Three Go Free. They will get free movie tickets if they each buy a large popcorn and a large soft drink. They take the deal and spend
\$22.50
\$8.00
for their ticket, they each get a large soft drink, but they share one large bucket of popcorn. This return trip costs them a total of
\$37.50
Find the price of a large soft drink and the price of a large bucket of popcorn.
3p+3d=22.50\\p+3d+3(8)=37.50\\p=\$4.50\ \text{and } d=\$3.00
|
Classical Mechanics Problem: Rolling and friction - Rohit Gupta | Brilliant
When the velocity of the center
v
and the angular velocity
\omega
of a round object are related as
\vec v = \vec \omega \times \vec r,
\vec r
is the position vector of the particle with respect to the center of the round object, which friction will act on the rolling object at an instant?
The surface is horizontal and rough.
No other force acts in the horizontal direction.
No friction Kinetic friction Rolling friction Static friction
|
Cavalieri's_principle Knowpia
In geometry, Cavalieri's principle, a modern implementation of the method of indivisibles, named after Bonaventura Cavalieri, is as follows:[1]
2-dimensional case: Suppose two regions in a plane are included between two parallel lines in that plane. If every line parallel to these two lines intersects both regions in line segments of equal length, then the two regions have equal areas.
3-dimensional case: Suppose two regions in three-space (solids) are included between two parallel planes. If every plane parallel to these two planes intersects both regions in cross-sections of equal area, then the two regions have equal volumes.
Two stacks of coins with the same volume, illustrating Cavalieri's principle in three dimensions
Today Cavalieri's principle is seen as an early step towards integral calculus, and while it is used in some forms, such as its generalization in Fubini's theorem, results using Cavalieri's principle can often be shown more directly via integration. In the other direction, Cavalieri's principle grew out of the ancient Greek method of exhaustion, which used limits but did not use infinitesimals.
Bonaventura Cavalieri, the mathematician the principle is named after.
Cavalieri's principle was originally called the method of indivisibles, the name it was known by in Renaissance Europe. Cavalieri developed a complete theory of indivisibles, elaborated in his Geometria indivisibilibus continuorum nova quadam ratione promota (Geometry, advanced in a new way by the indivisibles of the continua, 1635) and his Exercitationes geometricae sex (Six geometrical exercises, 1647).[2] While Cavalieri's work established the principle, in his publications he denied that the continuum was composed of indivisibles in an effort to avoid the associated paradoxes and religious controversies, and he did not use it to find previously unknown results.[3]
In the 3rd century BC, Archimedes, using a method resembling Cavalieri's principle,[4] was able to find the volume of a sphere given the volumes of a cone and cylinder in his work The Method of Mechanical Theorems. In the 5th century AD, Zu Chongzhi and his son Zu Gengzhi established a similar method to find a sphere's volume.[5] The transition from Cavalieri's indivisibles to Evangelista Torricelli's and John Wallis's infinitesimals was a major advance in the history of calculus. The indivisibles were entities of codimension 1, so that a plane figure was thought as made out of an infinite number of 1-dimensional lines. Meanwhile, infinitesimals were entities of the same dimension as the figure they make up; thus, a plane figure would be made out of "parallelograms" of infinitesimal width. Applying the formula for the sum of an arithmetic progression, Wallis computed the area of a triangle by partitioning it into infinitesimal parallelograms of width 1/∞.
The disk-shaped cross-section of the sphere has the same area as the ring-shaped cross-section of that part of the cylinder that lies outside the cone.
If one knows that the volume of a cone is
{\textstyle {\frac {1}{3}}\left({\text{base}}\times {\text{height}}\right)}
, then one can use Cavalieri's principle to derive the fact that the volume of a sphere is
{\textstyle {\frac {4}{3}}\pi r^{3}}
{\displaystyle r}
That is done as follows: Consider a sphere of radius
{\displaystyle r}
and a cylinder of radius
{\displaystyle r}
{\displaystyle r}
. Within the cylinder is the cone whose apex is at the center of one base of the cylinder and whose base is the other base of the cylinder. By the Pythagorean theorem, the plane located
{\displaystyle y}
units above the "equator" intersects the sphere in a circle of radius
{\textstyle {\sqrt {r^{2}-y^{2}}}}
{\displaystyle \pi \left(r^{2}-y^{2}\right)}
. The area of the plane's intersection with the part of the cylinder that is outside of the cone is also
{\displaystyle \pi \left(r^{2}-y^{2}\right)}
. As can be seen, the area of the circle defined by the intersection with the sphere of a horizontal plane located at any height
{\displaystyle y}
equals the area of the intersection of that plane with the part of the cylinder that is "outside" of the cone; thus, applying Cavalieri's principle, it could be said that the volume of the half sphere equals the volume of the part of the cylinder that is "outside" the cone. The aforementioned volume of the cone is
{\textstyle {\frac {1}{3}}}
of the volume of the cylinder, thus the volume outside of the cone is
{\textstyle {\frac {2}{3}}}
the volume of the cylinder. Therefore the volume of the upper half of the sphere is
{\textstyle {\frac {2}{3}}}
of the volume of the cylinder. The volume of the cylinder is
{\displaystyle {\text{base}}\times {\text{height}}=\pi r^{2}\cdot r=\pi r^{3}}
("Base" is in units of area; "height" is in units of distance. Area × distance = volume.)
Therefore the volume of the upper half-sphere is
{\textstyle {\frac {2}{3}}\pi r^{3}}
and that of the whole sphere is
{\textstyle {\frac {4}{3}}\pi r^{3}}
Cones and pyramidsEdit
The fact that the volume of any pyramid, regardless of the shape of the base, whether circular as in the case of a cone, or square as in the case of the Egyptian pyramids, or any other shape, is (1/3) × base × height, can be established by Cavalieri's principle if one knows only that it is true in one case. One may initially establish it in a single case by partitioning the interior of a triangular prism into three pyramidal components of equal volumes. One may show the equality of those three volumes by means of Cavalieri's principle.
In fact, Cavalieri's principle or similar infinitesimal argument is necessary to compute the volume of cones and even pyramids, which is essentially the content of Hilbert's third problem – polyhedral pyramids and cones cannot be cut and rearranged into a standard shape, and instead must be compared by infinite (infinitesimal) means. The ancient Greeks used various precursor techniques such as Archimedes's mechanical arguments or method of exhaustion to compute these volumes.
The napkin ring problemEdit
If a hole of height h is drilled straight through the center of a sphere, the volume of the remaining band does not depend on the size of the sphere. For a larger sphere, the band will be thinner but longer.
In what is called the napkin ring problem, one shows by Cavalieri's principle that when a hole is drilled straight through the centre of a sphere where the remaining band has height h, the volume of the remaining material surprisingly does not depend on the size of the sphere. The cross-section of the remaining ring is a plane annulus, whose area is the difference between the areas of two circles. By the Pythagorean theorem, the area of one of the two circles is π times r 2 − y 2, where r is the sphere's radius and y is the distance from the plane of the equator to the cutting plane, and that of the other is π times r 2 − (h/2)2. When these are subtracted, the r 2 cancels; hence the lack of dependence of the bottom-line answer upon r.
CycloidsEdit
The horizontal cross-section of the region bounded by two cycloidal arcs traced by a point on the same circle rolling in one case clockwise on the line below it, and in the other counterclockwise on the line above it, has the same length as the corresponding horizontal cross-section of the circle.
N. Reed has shown[6] how to find the area bounded by a cycloid by using Cavalieri's principle. A circle of radius r can roll in a clockwise direction upon a line below it, or in a counterclockwise direction upon a line above it. A point on the circle thereby traces out two cycloids. When the circle has rolled any particular distance, the angle through which it would have turned clockwise and that through which it would have turned counterclockwise are the same. The two points tracing the cycloids are therefore at equal heights. The line through them is therefore horizontal (i.e. parallel to the two lines on which the circle rolls). Consequently each horizontal cross-section of the circle has the same length as the corresponding horizontal cross-section of the region bounded by the two arcs of cyloids. By Cavalieri's principle, the circle therefore has the same area as that region.
Consider the rectangle bounding a single cycloid arch. From the definition of a cycloid, it has width 2πr and height 2r, so its area is four times the area of the circle. Calculate the area within this rectangle that lies above the cycloid arch by bisecting the rectangle at the midpoint where the arch meets the rectangle, rotate one piece by 180° and overlay the other half of the rectangle with it. The new rectangle, of area twice that of the circle, consists of the "lens" region between two cycloids, whose area was calculated above to be the same as that of the circle, and the two regions that formed the region above the cycloid arch in the original rectangle. Thus, the area bounded by a rectangle above a single complete arch of the cycloid has area equal to the area of the circle, and so, the area bounded by the arch is three times the area of the circle.
Fubini's theorem (Cavalieri's principle is a particular case of Fubini's theorem)
^ Howard Eves, "Two Surprising Theorems on Cavalieri Congruence", The College Mathematics Journal, volume 22, number 2, March, 1991), pages 118–124
^ Katz, Victor J. (1998), A History of Mathematics: An Introduction (2nd ed.), Addison-Wesley, p. 477.
^ Alexander, Amir (2015). Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World. Great Britain: Oneworld. pp. 101–103. ISBN 978-1-78074-642-5.
^ "Archimedes' Lost Method". Encyclopedia Britannica.
^ N. Reed, "Elementary proof of the area under a cycloid", Mathematical Gazette, volume 70, number 454, December, 1986, pages 290–291
Weisstein, Eric W. "Cavalieri's Principle". MathWorld.
(in German) Prinzip von Cavalieri
Cavalieri Integration
|
EUDML | Quasisymmetric embedding of self similar surfaces and origami with rational maps. EuDML | Quasisymmetric embedding of self similar surfaces and origami with rational maps.
Quasisymmetric embedding of self similar surfaces and origami with rational maps.
Meyer, Daniel. "Quasisymmetric embedding of self similar surfaces and origami with rational maps.." Annales Academiae Scientiarum Fennicae. Mathematica 27.2 (2002): 461-484. <http://eudml.org/doc/122788>.
title = {Quasisymmetric embedding of self similar surfaces and origami with rational maps.},
TI - Quasisymmetric embedding of self similar surfaces and origami with rational maps.
{𝐑}^{n}
2
|
6.4 Discussion Activity 1 - Communicating Information to Your Client | EME 810: Solar Resource Assessment and Economics
Given the following table of locations (source data from Duffie and Beckman, 2003) and data presented for the Sept/Oct/Nov climate regime, I'd like you to think about the locale in a broader context and suggest SECS that would enable delivery of solar utility at each site. Note that there are two hemispheres represented, so stay alert!
In each of the four sites, we have presented the average day for each month, in terms of:
daily clearness indices
{\overline{K}}_{t}
Degree Days (DD), and then
average outdoor air temperature (
\overline{T}
, dry bulb in °C).
Of the four locations presented below, we assume the land/roof space is not area constrained. What would you begin suggesting to a client regarding solar technologies, design, orientation, tracking, etc.? Pick at least two locations for your post.
And yet, you do not know your client here! Whoops. What other factors would you add to this discussion if you knew who your client was in each location?
Table of Monthly Metrics for Fall/Spring
Place Sept
{\overline{K}}_{t}
{\overline{K}}_{t}
{\overline{K}}_{t}
Sept DD Oct DD Nov DD Sept
\overline{T}
\overline{T}
\overline{T}
\varphi ={43}^{°}
) 0.49 0.48 0.42 103 232 479 15 11 2
\varphi ={35}^{°}
) 0.70 0.70 0.66 27 142 346 21 15 7
\varphi =-{37.5}^{°}
) 0.48 0.50 0.53 181 133 80 12 14 16
\varphi ={38}^{°}
) 0.57 0.52 0.46 5 37 140 23 19 14
This discussion will take place in the Discussion Activity 6.1 Discussion Forum in Canvas.
‹ 6.3 Engineering Tools to Maximize Solar Utility up 6.5 The Power Grid System ›
|
Wall of numbers - zxc.wiki
A number of walls (even number triangle , number pyramid or number tower ) is a didactic means for learning the basics of addition . An ordinary number wall is pyramid-shaped, and each cell is equal to the sum of the two below. Frequent modifications are the multiplication wall and the subtraction wall.
An ordinary number wall with 4 basic cells
2 walls of numbers in mathematics didactics
a + b b + c
The usual number wall is built so that each cell is the sum of the two lower ones. A wall of numbers can theoretically be any size.
Accordingly, in a wall of four rows with a, b, c, d in the base row (in this order) at the top:
{\ displaystyle a + 3b + 3c + d}
and for six rows with a, b, c, d, e, f in the back row at the top:
{\ displaystyle a + 5b + 10c + 10d + 5e + f}
In general, one can derive a formula for the value in the top depending on the values in the basic series with binomial coefficients as factors.
Number walls in mathematics didactics
The wall of numbers is an operative form of exercise with which different degrees of difficulty can be realized. It is often used in the 1st to 4th grade. Depending on how the values are entered in the cells, the person who solves the wall of numbers has to use different operations. When the basic series is completely filled, only addition is used; if the values are distributed, knowledge of subtraction is also required. These basic math skills and competencies are practiced in this way.
Günter Krauthausen: Wall of numbers in the second school year - a substantial exercise format, in: Grundschulunterricht, Volume 42, Issue 10, 1995, pp. 5–9.
Petra Scherer: Substantielle task formats - cross-year examples for mathematics lessons, 3 parts, in: Grundschulunterricht, Volume 44, 1997, Issue 1 (pp. 34-38), Issue 4 (pp. 36-38), Issue 6 (p. 54 -56).
Rita Schurr, Elisabeth Rathgeb-Schnierer: Wall of numbers with a difference - rediscovering familiar tasks, in: Praxis Förderschule, Volume 2, Issue 2, 2007, pp. 14-18, ISSN 1863-4036 .
Erich Ch. Wittmann , Gerhard N. Müller: Handbook productive arithmetic exercises . tape 1 . Klett, Stuttgart 2017, ISBN 978-3-12-200926-7 , pp. 119-122 .
↑ A. Delius: Walls of numbers. (PDF) In: lehrerfortbildung-bw.de. State Academy for Further Education and Personnel Development in Schools, May 19, 2010, accessed on February 19, 2019 .
↑ Dennis Rudolph: Number pyramid, number tower or number wall. In: gut-explained.de. Dennis Rudolph, December 18, 2017, accessed February 19, 2019 .
↑ Friedhelm Padberg and Christine Benz: Didaktik der Arithmatik . 4th edition. Springer, Heidelberg 2011, ISBN 978-3-8274-1996-5 , pp. 102-103 .
↑ Introduction of the wall of numbers in class 1. In: Hausarbeiten.de. 2012, accessed February 19, 2019 .
↑ Dr. Maria Koth: walls of numbers. (PDF) In: mathe-online.at. Retrieved February 19, 2019 .
This page is based on the copyrighted Wikipedia article "Zahlenmauer" (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.
|
What's a "Perpetual Future"? - UXD Protocol
What's a "Delta"?
What's a "Perpetual Future"?
In our last section "What's a Delta", we explained what a financial derivative is and how to think about the price of financial derivatives through the lens of "Delta", the change in the price of the derivative with respect to the price of the underlying asset.
In this section we will discuss a specific type of derivative in depth, a "Perpetual Future" contract.
Before adding the word "perpetual", let's discuss Future's contracts. A Future's contract between two parties is an agreement to buy or sell some asset at a pre-determined date at a pre-determined price. For example, I agree to buy a Bitcoin from you in one month at a price of $50,000 (today's price is $48,000). Our pre-determined date is one month from now and our pre-determined price is $50,000.
Price Speculation- It would make sense to enter this contract if I think the price of Bitcoin is going to go up to $55,000 in a month, because then I would be able to profit $5,000 by buying it for $50,000 from you. On the other hand, it would make sense for you to enter this contract if you think the price of Bitcoin is going to go down to $45,000 in a month, because then you would be able to sell me a Bitcoin for $50,000, keep $5,000 and buy another Bitcoin for $45,000. The future's contract allows two people to make equal and opposite bets.
Locked in Prices- Suppose my nephew's birthday is in one month, and I know I have to get him one Bitcoin for his birthday, because I promised him way back in 2013 that I would do so when he reached a certain age. Now because I know for a fact I will need it in 1 month, but don't have the money right this second to actually buy a Bitcoin, I can enter into a future's contract with you in order to guarantee that I will be able to buy a Bitcoin for at most $50,000. On the other hand, you want to sell a Bitcoin to buy a new car, but don't need the cash for another month. You're worried about the price going down, so you decide to lock your sale price of $50,000 today with me. The future's contract allows two people to have certainty today about the purchase or sale price of an asset in the future.
In crypto trading, futures contracts tend to be used for Price Speculation as they allow users to take bets on the future value of assets without necessarily having to own the asset today. However, in the above example at the future's expiry date in one month, you have to send me a Bitcoin and I have to send you $50,000 for settlement.
Now that we understand Future's contracts we can add the word "Perpetual". In our above example, we had a contract with an Expiry Date and a Pre-determined Price that required you and I to exchange a Bitcoin and cash on the Expiry Date (settlement). Perpetual Future's are the purest form of price speculation because they (i) remove the Expiry Date (never expires) (ii) replace Pre-determined Price with Market Price and (iii) removes the need for settlement. How does this work?
I decide to go onto a derivative dex like mango markets and enter into a "BTC-USD" perpetual futures position. In order to enter into this position I need to provide some collateral in the form of margin. Margin ensures the exchange that if I lose a lot of money on my trade, I will still have enough money to pay the loss I've incurred.
A Perpetual Future can be thought of as an "Agreement for Deltas": if I want to have exposure to "20 Deltas" of Bitcoin using only 1 BTC worth of collateral (also known as "going long BTC with 20x leverage"), and you want to have exposure to "-20 Deltas" of Bitcoin using only 1 BTC worth of collateral (also known as "going short BTC with 20x leverage"), we can agree today to enter a derivatives contract that we will pay the other 20 times the price change of Bitcoin. We both put up 1 BTC as collateral to guarantee we can pay the other in case our losses grow too large. In this case, because we have 20x leverage, a mere 5% change in the price of Bitcoin will cause one of us to take a 1 BTC loss (20 BTC * 5% = 1 BTC), so if BTC moves 5% one of us will have to take the other's 1 BTC. Any move greater than 5% and we won't be able to guarantee that the other will pay the money owed.
So to prevent losses greater than 1 BTC, we agree at the start of the contract that the Perpetual Futures contract between you and I will continue "forever", until one of two things happens:
One of us sells the derivative contract to someone else- This means you or I want to exit the trade and sell our position to someone else.
Note that in order to do 2. there has to be a settlement. For example, if Bitcoin has gone down 2%, then because I had 20x leverage (a 20 Delta position), I will owe you "2% * 20 BTC = 0.4 BTC. So, I pay you 0.4 BTC and then find someone else to step into the 20 Delta position in my place.
In practice, for a derivatives market place like mango markets the easiest way to create a whole market of perpetual futures is to create a "virtual" BTC trading market. In a normal Bitcoin market, a user enters the market and places an order such as "Buy 3 BTC" and then pays for these 3 BTC and receives the 3 BTC in their wallet.
Imagine instead we set up a "virtual" BTC trading market in which people could submit orders for BTC and rather than paying for these BTC directly (or receiving funds if selling) we could have the "virtual" trading market track everyone's "bought" and "sold" BTC. In order to exit the virtual trading market, users have to settle their gains/losses from the BTC they bought or sold, without ever actually receiving or sending BTC. The market therefore only settles profits and losses from a user's virtual position, rather than having user's exchange BTC directly.
I enter the "virtual" BTC market and deposit 1 BTC as margin collateral.
Even though I don't have 20 BTC worth of cash to buy 20 BTC with, I ask the "virtual" BTC market to buy me 20 BTC at whatever the price of BTC is on this virtual market at that time.
I am required to "leave" the virtual trading market if one of two things happens: (i) BTC goes down by 5%, I will lose (20 BTC * 5%) = 1 BTC and lose my margin collateral, or (ii) I decide to "sell" the 20 BTC I own to someone else on the virtual market.
To summarize, this is exactly what we described above in the "Agreement for Deltas"! I was able to gain 20x exposure to BTC through this virtual market, with the exact same conditions for ending the contract as above: either I get wiped out or decide to sell to someone else. This is how Perpetual Futures contracts are done in practice.
There is, however, a slight issue with the above "virtual" market construction. Each time someone buys or sells on this virtual market, the price of BTC on this virtual market will change. If too many people want to buy or sell, this could cause the virtual price of BTC on this virtual market to deviate from the actual market price. That's no good, because we want the Perpetual Future to track the exact price of the underlying BTC. So, we need a mechanism in order to make sure the price doesn't deviate too far from the actual market price of BTC.
This mechanism for keeping the virtual price close to the market price is called the funding rate. To understand the funding rate, it's helpful to understand how the virtual price would deviate from the market price in the first place.
The funding rate for a virtual market like this one is a periodic payment from the more popular position (buying, for example) to the less popular position (selling). The frequency of this payment varies by derivatives exchange, but let's suppose that it's every 8 hours. Then every 8 hours, if the virtual price is above the market price, people who have "bought" BTC perpetual futures pay people who have "sold" perpetual futures an amount:
\large \text{Funding Rate Payment} = \text{Virtual Price} - \text{Market Price}
For example, if the virtual price of BTC is $48,200 and the market price is $48,000, then anyone who "bought" bitcoin on this virtual market will pay anyone who sold a sum of $200 for this 8 hour payment. If instead the virtual price was $50,000 and the market price was $48,000, then the funding rate payment would be $2,000. As the prices move, so does the funding rate. Note, actual funding rate formulas are a bit more complex, but for illustrative purposes, the above idea is correct.
The idea of the funding rate is to incentivize people to take the opposite position on the exchange and bring the virtual price in order with the market price. For example, if I think BTC likely will not move much over the next few hours, it may make sense for me to "sell" 1 BTC on the exchange in order to collect the $200 payment. My sale of the BTC on the virtual market will bring the price down, and people will likely continue to do so until the "arbitrage" from the funding rate disappears.
So, the funding rate serves the essential purpose of making sure the price of our perpetual future's contract stays in line with the price of the underlying asset.
He goes to the derivatives dex, clicks on the "BTC-PERP". First, he deposits some collateral as margin, say 0.5 BTC.
Then, he "buys" 2.5 BTC (5x his margin) on the virtual perpetual futures market. This represents ~$120,000 of exposure to BTC at a price of ~$48k.
\large (.0003\%)*(\$120,000) = \$0.36
Suppose funding rates stay constant for 30 days, and BTC has gone up 5%. At the end of the 30 day period, what is Bob's total profit?
\text{Profit} = (5\% * \$120,000) - (.0003\%)*(\$120,000)*(24 \text{hours})*(30 \text{days}) = \$5740.80
|
The Complex Partial-Systolic QR Decomposition block uses QR decomposition to compute R and C = Q'B, where QR = A, and A and B are complex-valued matrices. The least-squares solution to Ax = B is x = R\C. R is an upper triangular matrix and Q is an orthogonal matrix. To compute C = Q', set B to be the identity matrix.
When Regularization parameter is nonzero, the Complex Partial-Systolic QR Decomposition block transforms
\left[\begin{array}{c}\lambda {I}_{n}\\ A\end{array}\right]
R=Q\text{'}\left[\begin{array}{c}\lambda {I}_{n}\\ A\end{array}\right]
\left[\begin{array}{c}{0}_{n,p}\\ B\end{array}\right]
C=Q\text{'}\left[\begin{array}{c}{0}_{n,p}\\ B\end{array}\right]
\left[\begin{array}{c}\lambda {I}_{n}\\ A\end{array}\right]
Economy-size QR decomposition matrix R, returned as a matrix. R is an upper triangular matrix. R has the same data type as A.
C — Matrix C=Q'B
Economy-size QR decomposition matrix C=Q'B, returned as a matrix or vector. C has the same number of rows as R. C has the same data type as B.
Whether the output data is valid, returned as a Boolean scalar. This control signal indicates when the data at output ports R and C is valid. When this value is 1 (true), the block has successfully computed the R and C matrices. When this value is 0 (false), the output data is not valid.
Number of rows in matrices A and B — Number of rows in input matrices A and B
The number of rows in input matrices A and B, specified as a positive integer-valued scalar.
Real Partial-Systolic QR Decomposition | Complex Partial-Systolic Q-less QR Decomposition | Complex Burst QR Decomposition
|
Home > Course Outline > Lesson 7 - Solar Finance > 7.2 The Basics: Time Value of Money and Rates
J.R. Brownson, Solar Energy Conversion Systems (SECS), Chapter 10 (Focus on the Introduction and the Time Value of Money.)
W. Short et al. (1995) Manual for the Economic Evaluation of Energy Efficiency and Renewable Energy Technologies [1]. NREL Technical Report TP-462-5173. (Read pp. 1-22: from Introduction through Taxes, skim pp. 10-11.)
Just as in other lessons, we will devote some time to getting everyone up to speed on the basics of financial analysis. Now remember, as a team, you will be working with peers who may have greater expertise than you with financal topics. I would like you to understand the general concepts of cash flows, inflation rates, fixed costs versus variable costs of a system, unit cost for a system, and taxes. This is the pragmatic side of seeking out a high solar utility for our clients, from a financial perspective.
We will be considering cash flows (i.e., revenues - expenses; or savings - costs) in a process called Life Cycle Cost Analysis. As we have seen in the reading, cash flows can be developed for systems operations, for investment decisions, and for financing. We will be representing cash flows in a simple, discrete pattern called end-of-period cash flow, where the periodicity is 1 year and the compounding or discounting that occurs uses an annual rate.
Given: our SECSs will tend to have life-spans that are quite long, often well beyond the 25 years of the PV module warrantees.
Also given: a lot can happen financially in 30 years. The USA has had three recessions since 1988 (list of U.S. recessions) [2], and fluctuations in the rate of inflation between 2-6%.
And finally: SECSs are still "fresh" to many consumers; they're going to be foreign systems to most clients in the beginning of a project development.
The result is energy systems that have a long horizon of life, a long financial period of evaluation to assess, and yet the installed systems exist within a dynamic financial setting: increased uncertainty and risk (without better information available).
Did you see that last bit? Clients will perceive increased uncertainty and risk without better information available. That's your job! To provide better information and transparent project evaluation, which demonstrates an understanding of both the solar resource and the financials associated with a proposed SECS. In Chapter 10 of the textbook, we demonstrate how conveying the financial metrics of the project within a proposal is one way to provide useful information in a transparent manner.
We call the process of evaluating a project the Life Cycle Cost Analysis (LCCA), and one of the important criteria is the period of analysis, or period of evaluation. The "period" conveys a time horizon for your LCCA. If we recall our microeconomic drivers affecting the elasticity of demand, we know that the time horizon is an important factor. In our case, SECS will tend to have long life-spans.
As such, we distinguish between the concept of "value" at various points in time.
Present Value (PV, not photovoltaics this time!): specifies worth for assets like SECSs, for money, or for periodic cash flows, where the worth is in today’s dollars, provided the rate of return is specified (as "d"). The value is processed from year "n" back to "year zero" (meaning the present).
PV=\frac{FV}{{\left(1+d\right)}^{n}}
Future Value (FV): specifies the worth for things as a dollar value in the future. We use FV for Fuel Costs (FC) and Fuel Savings (FS) in our LCCA. Costs are represented as "C" and Savings as "S." The rate of inflation is specified here as "i."
FV=C\cdot {\left(1+i\right)}^{n-1}
Present Worth in year n (PWn): This is the ratio of the future costs with respect to the discount rate over time.
P{W}_{n}=\frac{C\cdot {\left(1+i\right)}^{n-1}}{{\left(1+d\right)}^{n}}
You will notice that the same topics are discussed in detail in the assigned reading of the Manual for Economic Evaluation by Short et al. (1995).
There are two ways to represent discount rates, and you will observe both in the SAM simulation software or similar financial analysis tools. Using these rates, we can produce a discounted cash flow model (DFM) to compare projects.
Nominal Discount Rate ($d_n$): discount rates for time value of money that are not adjusted for the effects of inflation. (Nominal = not inflation-adjusted).
Real Discount Rate ($d_r$): the discount rate where the rate of inflation has been adjusted, by excluding the effect of inflation. As such a real discount rate will be a lower value than the nominal discount rate for inflation. (Real = inflation-adjusted).
Caveat: If the inflation rate is negative (deflation) then the real discount rate would actually be higher than the nominal rate.
\left(1+{d}_{n}\right)=\left(1+{d}_{r}\right)\cdot \left(1+i\right)
{d}_{n}=\left[\left(1+{d}_{r}\right)\cdot \left(1+i\right)\right]-1
{d}_{r}=\left[\frac{\left(1+{d}_{n}\right)}{\left(1+i\right)}\right]-1
You will note in the Short et al. document that the nominal discount rate has a loose approximation of
{d}_{r}\approx {d}_{n}-i
. But I want you to think, will fuel inflation rates be the same as labor inflation rates, and insurance inflation rates? We will have an example in the discussion where we pull apart different inflation rates and use real discount rates in our analysis of a solar hot water system.
We have already seen that the DSIRE website [3] for the states and federal government of the USA is a useful resource for incentives. Part of those incentives is tied in to tax credits, and there is a significant portion of your reading devoted to the concept of depreciation.
Depreciation: the use of income tax deductions to recover the costs of property used in trade/business or for the production of income. Depreciation does not include land.
MACRS: Modified Accelerated Cost Recovery System [4]. You should observe that the Wikipedia site and your reading from Short et al. will be quite similar. MACRS is used in the SAM simulation software.
One of the things that occurs in an LCCA at the end of the Period of Analysis is the question of how to finish the summation. This is like the Monty Python movie, The Holy Grail [5], where the old fellow says: "I'm not dead!" At the end of your 15-25 year evaluation for LCCA, you will no doubt have a fully functional SECS still! They don't just break down and fall apart, and in fact they will likely last for decades beyond your evaluation period. So how do we assess the value of the system at the end of the period?
We assume that the system has a net salvage value (a resale value) that is a fraction of its initial value, translated into present dollars. In our discussion, we will assume a 20-year-old solar hot water system still has 30% of its initial value, framed in present dollars for year 20.
[1] http://www.nrel.gov/docs/legosti/old/5173.pdf
[2] http://en.wikipedia.org/wiki/List_of_recessions_in_the_United_States
[4] http://en.wikipedia.org/wiki/MACRS
[5] http://en.wikiquote.org/wiki/Monty_Python_and_the_Holy_Grail
|
Upskilling: Do Employers Demand Greater Skill When Workers Are Plentiful? | The Review of Economics and Statistics | MIT Press
Daniel Shoag,
Harvard University and Case Western Reserve University
The views expressed in this paper are our own and do not indicate concurrence by the Federal Reserve Bank of Boston or by the principals of the Board of Governors or the Federal Reserve System. We thank seminar participants from the Federal Reserve Bank of Boston, the Harvard Kennedy School, the Institute for Work and Employment Research at MIT, the NBER Summer Institute, Northeastern University, and the Society of Labor Economists for their valuable comments and insights. Two anonymous referees were instrumental in shaping the final paper. Special thanks to Dan Restuccia, Matthew Sigelman, and Bledi Taska of Burning Glass Technologies for supplying the data and providing insights regarding the collection methodology. We also thank the Russell Sage Foundation (award 85-14-05) for their generous support of this work. All remaining errors are our own.
Alicia Sasser Modestino, Daniel Shoag, Joshua Ballance; Upskilling: Do Employers Demand Greater Skill When Workers Are Plentiful?. The Review of Economics and Statistics 2020; 102 (4): 793–805. doi: https://doi.org/10.1162/rest_a_00835
Using a proprietary database of online job postings, we find that education and experience requirements rose during the Great Recession. These increases were larger in states and occupations that experienced greater increases in the supply of available workers. This finding is robust to controlling for local demand conditions and firm
×
job-title fixed effects and using a natural experiment arising from troop withdrawals as an exogenous shock to labor supply. Our results imply that the increase in unemployed workers during the Great Recession can account for 18% to 25% of the increase in skill requirements between 2007 and 2010.
|
Cantilever — Wikipedia Republished // WIKI 2
For the figure skating element, see Cantilever (figure skating).
A schematic image of three types of cantilever. The top example has a full moment connection (like a horizontal flagpole bolted to the side of a building). The middle example is created by an extension of a simple supported beam (such as the way a diving board is anchored and extends over the edge of a swimming pool). The bottom example is created by adding a Robin boundary condition to the beam element, which essentially adds an elastic spring to the end board. The middle and bottom example may be considered structurally equivalent, depending on the effective stiffness of the spring and beam element.
A cantilever is a rigid structural element that extends horizontally and is supported at only one end. Typically it extends from a flat vertical surface such as a wall, to which it must be firmly attached. Like other structural elements, a cantilever can be formed as a beam, plate, truss, or slab.
When subjected to a structural load at its far, unsupported end, the cantilever carries the load to the support where it applies a shear stress and a bending moment.[1]
Physics of Life - Cantilevers
How to Calculate Effective Length of Cantilever Beam | By Learning Technology
Cantilever Slab and Cantilever Beam | Concept and Reinforcement details | Engineering Tactics
Curtailment of steel bars in Cantilever beam || Why cantilever beam are made tapered ?
Depression of Cantilever | in HINDI | EduPoint
1 In bridges, towers, and buildings
2 Aircraft
3 In microelectromechanical systems
4 Chemical sensor applications
5 In storage applications
5.1 Warehouse storage
5.2 Portable storage
Cantilevers are widely found in construction, notably in cantilever bridges and balconies (see corbel). In cantilever bridges, the cantilevers are usually built as pairs, with each cantilever used to support one end of a central section. The Forth Bridge in Scotland is an example of a cantilever truss bridge. A cantilever in a traditionally timber framed building is called a jetty or forebay. In the southern United States, a historic barn type is the cantilever barn of log construction.
Temporary cantilevers are often used in construction. The partially constructed structure creates a cantilever, but the completed structure does not act as a cantilever. This is very helpful when temporary supports, or falsework, cannot be used to support the structure while it is being built (e.g., over a busy roadway or river, or in a deep valley). Therefore, some truss arch bridges (see Navajo Bridge) are built from each side as cantilevers until the spans reach each other and are then jacked apart to stress them in compression before finally joining. Nearly all cable-stayed bridges are built using cantilevers as this is one of their chief advantages. Many box girder bridges are built segmentally, or in short pieces. This type of construction lends itself well to balanced cantilever construction where the bridge is built in both directions from a single support.
In an architectural application, Frank Lloyd Wright's Fallingwater used cantilevers to project large balconies. The East Stand at Elland Road Stadium in Leeds was, when completed, the largest cantilever stand in the world[2] holding 17,000 spectators. The roof built over the stands at Old Trafford uses a cantilever so that no supports will block views of the field. The old (now demolished) Miami Stadium had a similar roof over the spectator area. The largest cantilevered roof in Europe is located at St James' Park in Newcastle-Upon-Tyne, the home stadium of Newcastle United F.C.[3][4]
Less obvious examples of cantilevers are free-standing (vertical) radio towers without guy-wires, and chimneys, which resist being blown over by the wind through cantilever action at their base.
The Forth Bridge, a cantilever truss bridge
This concrete bridge temporarily functions as a set of two balanced cantilevers during construction – with further cantilevers jutting out to support formwork.
Howrah Bridge in India, a cantilever bridge
A cantilevered balcony of the Fallingwater house, by Frank Lloyd Wright
A cantilevered railroad deck and fence on the Canton Viaduct
Cantilever barn at Cades Cove
Cantilever facade of Riverplace Tower in Jacksonville, Florida, by Welton Becket and KBJ Architects
Ronan Point: Structural failure of part of floors cantilevered from a central shaft.
The pioneering Junkers J 1 all-metal monoplane of 1915, the first aircraft to fly with cantilever wings
The cantilever is commonly used in the wings of fixed-wing aircraft. Early aircraft had light structures which were braced with wires and struts. However, these introduced aerodynamic drag which limited performance. While it is heavier, the cantilever avoids this issue and allows the plane to fly faster.
Hugo Junkers pioneered the cantilever wing in 1915. Only a dozen years after the Wright Brothers' initial flights, Junkers endeavored to eliminate virtually all major external bracing members in order to decrease airframe drag in flight. The result of this endeavor was the Junkers J 1 pioneering all-metal monoplane of late 1915, designed from the start with all-metal cantilever wing panels. About a year after the initial success of the Junkers J 1, Reinhold Platz of Fokker also achieved success with a cantilever-winged sesquiplane built instead with wooden materials, the Fokker V.1.
de Havilland DH.88 Comet G-ACSS, winner of the Great Air Race of 1934, showing off its cantilever wing
To resist horizontal shear stress from either drag or engine thrust, the wing must also form a stiff cantilever in the horizontal plane. A single-spar design will usually be fitted with a second smaller drag-spar nearer the trailing edge, braced to the main spar via additional internal members or a stressed skin. The wing must also resist twisting forces, achieved by cross-bracing or otherwise stiffening the main structure.
Cantilever wings require much stronger and heavier spars than would otherwise be needed in a wire-braced design. However, as the speed of the aircraft increases, the drag of the bracing increases sharply, while the wing structure must be strengthened, typically by increasing the strength of the spars and the thickness of the skinning. At speeds of around 200 miles per hour (320 km/h) the drag of the bracing becomes excessive and the wing strong enough to be made a cantilever without excess weight penalty. Increases in engine power through the late 1920s and early 1930s raised speeds through this zone and by the late 1930s cantilever wings had almost wholly superseded braced ones.[5] Other changes such as enclosed cockpits, retractable undercarriage, landing flaps and stressed-skin construction furthered the design revolution, with the pivotal moment widely acknowledged to be the MacRobertson England-Australia air race of 1934, which was won by a de Havilland DH.88 Comet.[6]
Cantilevered beams are the most ubiquitous structures in the field of microelectromechanical systems (MEMS). An early example of a MEMS cantilever is the Resonistor,[7][8] an electromechanical monolithic resonator. MEMS cantilevers are commonly fabricated from silicon (Si), silicon nitride (Si3N4), or polymers. The fabrication process typically involves undercutting the cantilever structure to release it, often with an anisotropic wet or dry etching technique. Without cantilever transducers, atomic force microscopy would not be possible. A large number of research groups are attempting to develop cantilever arrays as biosensors for medical diagnostic applications. MEMS cantilevers are also finding application as radio frequency filters and resonators. The MEMS cantilevers are commonly made as unimorphs or bimorphs.
{\displaystyle \delta ={\frac {3\sigma \left(1-\nu \right)}{E}}{\frac {L^{2}}{t^{2}}}}
{\displaystyle \nu }
is Poisson's ratio,
{\displaystyle E}
is Young's modulus,
{\displaystyle L}
is the beam length and
{\displaystyle t}
is the cantilever thickness. Very sensitive optical and capacitive methods have been developed to measure changes in the static deflection of cantilever beams used in dc-coupled sensors.
The second is the formula relating the cantilever spring constant
{\displaystyle k}
to the cantilever dimensions and material constants:
{\displaystyle k={\frac {F}{\delta }}={\frac {Ewt^{3}}{4L^{3}}}}
{\displaystyle F}
{\displaystyle w}
is the cantilever width. The spring constant is related to the cantilever resonance frequency
{\displaystyle \omega _{0}}
by the usual harmonic oscillator formula
{\displaystyle \omega _{0}={\sqrt {k/m_{\text{equivalent}}}}}
. A change in the force applied to a cantilever can shift the resonance frequency. The frequency shift can be measured with exquisite accuracy using heterodyne techniques and is the basis of ac-coupled cantilever sensors.
The principal advantage of MEMS cantilevers is their cheapness and ease of fabrication in large arrays. The challenge for their practical application lies in the square and cubic dependences of cantilever performance specifications on dimensions. These superlinear dependences mean that cantilevers are quite sensitive to variation in process parameters, particularly the thickness as this is generally difficult to accurately measure.[9] However, it has been shown that microcantilever thicknesses can be precisely measured and that this variation can be quantified.[10] Controlling residual stress can also be difficult.
A chemical sensor can be obtained by coating a recognition receptor layer over the upper side of a microcantilever beam.[12] A typical application is the immunosensor based on an antibody layer that interacts selectively with a particular immunogen and reports about its content in a specimen. In the static mode of operation, the sensor response is represented by the beam bending with respect to a reference microcantilever. Alternatively, microcantilever sensors can be operated in the dynamic mode. In this case, the beam vibrates at its resonance frequency and a variation in this parameter indicates the concentration of the analyte. Recently, microcantilevers have been fabricated that are porous, allowing for a much larger surface area for analyte to bind to, increasing sensitivity by raising the ratio of the analyte mass to the device mass.[13] Surface stress on microcantilever, due to receptor-target binding, which produces cantilever deflection can be analyzed using optical methods like laser interferometry. Zhao et al., also showed that by changing the attachment protocol of the receptor on the microcantilever surface, the sensitivity can be further improved when the surface stress generated on the microcantilever is taken as the sensor signal.[14]
A cantilever rack is a type of warehouse storage system consisting of the vertical column, the base, the arms, and the horizontal and/or cross-bracing. These components are fabricated from both roll formed and structural steel. The horizontal and/or cross bracing are used to connect two or more columns together. They are commonly found in lumber yards, woodworking shops, and plumbing supply warehouses.
Applied mechanics
Cantilever bicycle brakes
Cantilever bicycle frame
Cantilever chair
Cantilever method
Cantilevered stairs
Corbel arch
Grand Canyon Skywalk
Knudsen force in the context of microcantilevers
^ Hool, George A.; Johnson, Nathan Clarke (1920). "Elements of Structural Theory - Definitions". Handbook of Building Construction (Google Books). Vol. 1 (1st ed.). New York: McGraw-Hill. p. 2. Retrieved 2008-10-01. A cantilever beam is a beam having one end rigidly fixed and the other end free.
^ "GMI Construction wins £5.5M Design and Build Contract for Leeds United Football Club's Elland Road East Stand". Construction News. 6 February 1992. Retrieved 24 September 2012.
^ The Architects' Journal Existing stadiums: St James' Park, Newcastle. 1 July 2005
^ Noyce, Steven G.; Vanfleet, Richard R.; Craighead, Harold G.; Davis, Robert C. (1999-02-22). "High surface-area carbon microcantilevers". Nanoscale Advances. 1 (3): 1148–1154. doi:10.1039/C8NA00101D.
Roth, Leland M (1993). Understanding Architecture: Its Elements History and Meaning. Oxford, UK: Westview Press. pp. 23–4. ISBN 0-06-430158-3.
|
Determine fixed-point types for transforming A to R in-place, where R is upper-triangular factor of QR decomposition of A, without computing Q - MATLAB fixed.qlessqrFixedpointTypes - MathWorks Italia
\left[\begin{array}{c}\lambda {I}_{n}\\ A\end{array}\right]
R=Q\text{'}\left[\begin{array}{c}\lambda {I}_{n}\\ A\end{array}\right]
\left[\begin{array}{c}\lambda {I}_{n}\\ A\end{array}\right]
A
A
A
-1
+1
A
A
R={Q}^{\prime }A
\mathit{A}
\mathit{R}={\mathit{Q}}^{\prime }\mathit{A}
R
\mathit{R}={\mathit{Q}}^{\prime }\mathit{A}
Q
{R}^{\prime }R={A}^{\prime }Q{Q}^{\prime }A={A}^{\prime }A
\mathrm{max}\left(|R\left(:\right)|\right)\le \sqrt{m}\mathrm{max}\left(|A\left(:\right)|\right).
|
Option specs - Dopex
Dopex Option Pool options are built to closely mimic Deribit - the market leader in cryptocurrency options as much as possible considering it's dominant market share in the options space. Specifications for Dopex options are as follows:
Base/quote asset
On-chain pricing
Black-scholes based on IV from oracles
combined with a function to estimate volatilty
smile based on past data
Margin requirements (writing)
Fully backed by collateral
Calls - Base asset
Puts - Quote asset (USD stablecoins)
ERC20 Transferrable to other exchanges for arbitrage
Dopex options come in the form ERC20 based tokens and can be found for instant purchase on the Dopex Option Pools (Options AMM). They can be exchanged on any existing AMM (like Uniswap, Sushiswap etc), OTC portals, as well as centralized exchanges in the future considering they are ERC20s - giving exchanges instant adaptability for Dopex option tokens.
Exercise & Settlements
All option expiries are at 8:00 AM UTC and can be exercised during a 1 hour window before the expiry time i.e between 7:00 AM UTC to 8:00 AM UTC.
Settlements on option exercises happen without requiring the underlying asset and hence are net settlements. The PnL (Profit & Loss) of the option is calculated and if it is +ve then the exercise can go through which burn the option tokens (doTokens) and transfer the PnL in the settlement asset to the user. PnL is calculated in the following manner:
For CALLs (settlement is done in the base asset):
PnL = ((Price - Strike) * Amount) / Price
For PUTs (settlement is done in the quote asset):
PnL = (Strike - Price) * Amount
Here Price is the current price of the asset and Amount is the amount of options being exercised.
The settlement fee (exercise fee) is deducted from the final PnL. To learn more about fees, visit the fees section.
ETH Calls
ETH Puts
BTC Puts
|
Otsu's method - Wikipedia
In computer vision and image processing
An example image thresholded using Otsu's algorithm
In computer vision and image processing, Otsu's method, named after Nobuyuki Otsu (大津展之, Ōtsu Nobuyuki), is used to perform automatic image thresholding.[1] In the simplest form, the algorithm returns a single intensity threshold that separate pixels into two classes, foreground and background. This threshold is determined by minimizing intra-class intensity variance, or equivalently, by maximizing inter-class variance.[2] Otsu's method is a one-dimensional discrete analog of Fisher's Discriminant Analysis, is related to Jenks optimization method, and is equivalent to a globally optimal k-means[3] performed on the intensity histogram. The extension to multi-level thresholding was described in the original paper,[2] and computationally efficient implementations have since been proposed.[4][5]
1 Otsu's method
1.2 MATLAB implementation
2 Limitations and variations
2.1 A variation for noisy images
2.1.2 MATLAB implementation
2.2 A variation for unbalanced images
Otsu's method[edit]
Otsu's method visualization
The algorithm exhaustively searches for the threshold that minimizes the intra-class variance, defined as a weighted sum of variances of the two classes:
{\displaystyle \sigma _{w}^{2}(t)=\omega _{0}(t)\sigma _{0}^{2}(t)+\omega _{1}(t)\sigma _{1}^{2}(t)}
{\displaystyle \omega _{0}}
{\displaystyle \omega _{1}}
are the probabilities of the two classes separated by a threshold
{\displaystyle t}
{\displaystyle \sigma _{0}^{2}}
{\displaystyle \sigma _{1}^{2}}
are variances of these two classes.
The class probability
{\displaystyle \omega _{0,1}(t)}
is computed from the
{\displaystyle L}
bins of the histogram:
{\displaystyle {\begin{aligned}\omega _{0}(t)&=\sum _{i=0}^{t-1}p(i)\\[4pt]\omega _{1}(t)&=\sum _{i=t}^{L-1}p(i)\end{aligned}}}
For 2 classes, minimizing the intra-class variance is equivalent to maximizing inter-class variance:[2]
{\displaystyle {\begin{aligned}\sigma _{b}^{2}(t)&=\sigma ^{2}-\sigma _{w}^{2}(t)=\omega _{0}(t)(\mu _{0}-\mu _{T})^{2}+\omega _{1}(t)(\mu _{1}-\mu _{T})^{2}\\&=\omega _{0}(t)\omega _{1}(t)\left[\mu _{0}(t)-\mu _{1}(t)\right]^{2}\end{aligned}}}
which is expressed in terms of class probabilities
{\displaystyle \omega }
and class means
{\displaystyle \mu }
, where the class means
{\displaystyle \mu _{0}(t)}
{\displaystyle \mu _{1}(t)}
{\displaystyle \mu _{T}}
{\displaystyle {\begin{aligned}\mu _{0}(t)&={\frac {\sum _{i=0}^{t-1}ip(i)}{\omega _{0}(t)}}\\[4pt]\mu _{1}(t)&={\frac {\sum _{i=t}^{L-1}ip(i)}{\omega _{1}(t)}}\\\mu _{T}&=\sum _{i=0}^{L-1}ip(i)\end{aligned}}}
The following relations can be easily verified:
{\displaystyle {\begin{aligned}\omega _{0}\mu _{0}+\omega _{1}\mu _{1}&=\mu _{T}\\\omega _{0}+\omega _{1}&=1\end{aligned}}}
Compute histogram and probabilities of each intensity level
Set up initial
{\displaystyle \omega _{i}(0)}
{\displaystyle \mu _{i}(0)}
Step through all possible thresholds
{\displaystyle t=1,\ldots }
maximum intensity
{\displaystyle \omega _{i}}
{\displaystyle \mu _{i}}
{\displaystyle \sigma _{b}^{2}(t)}
Desired threshold corresponds to the maximum
{\displaystyle \sigma _{b}^{2}(t)}
MATLAB implementation[edit]
histogramCounts is a 256-element histogram of a grayscale image different gray-levels (typical for 8-bit images). level is the threshold for the image (double).
function level = otsu(histogramCounts)
total = sum(histogramCounts); % total number of pixels in the image
%% OTSU automatic thresholding
maximum = 0.0;
sum1 = dot(0:top-1, histogramCounts);
for ii = 1:top
if wB > 0 && wF > 0
mF = (sum1 - sumB) / wF;
val = wB * wF * ((sumB / wB) - mF) * ((sumB / wB) - mF);
if ( val >= maximum )
level = ii;
maximum = val;
wB = wB + histogramCounts(ii);
sumB = sumB + (ii-1) * histogramCounts(ii);
Matlab has built-in functions graythresh() and multithresh() in the Image Processing Toolbox which are implemented with Otsu's method and Multi Otsu's method, respectively.
This implementation requires the NumPy library.
def compute_otsu_criteria(im, th):
# create the thresholded image
thresholded_im = np.zeros(im.shape)
thresholded_im[im >= th] = 1
nb_pixels = im.size
nb_pixels1 = np.count_nonzero(thresholded_im)
weight1 = nb_pixels1 / nb_pixels
weight0 = 1 - weight1
# if one the classes is empty, eg all pixels are below or above the threshold, that threshold will not be considered
# in the search for the best threshold
if weight1 == 0 or weight0 == 0:
# find all pixels belonging to each class
val_pixels1 = im[thresholded_im == 1]
# compute variance of these classes
var0 = np.var(val_pixels0) if len(val_pixels0) > 0 else 0
return weight0 * var0 + weight1 * var1
im = # load your image as a numpy array.
# For testing purposes, one can use for example im = np.random.randint(0,255, size = (50,50))
# testing all thresholds from 0 to the maximum of the image
threshold_range = range(np.max(im)+1)
criterias = [compute_otsu_criteria(im, th) for th in threshold_range]
# best threshold is the one minimizing the Otsu criteria
best_threshold = threshold_range[np.argmin(criterias)]
Python libraries dedicated to image processing such as OpenCV and Scikit-image propose built-in implementations of the algorithm.
Limitations and variations[edit]
Otsu's method performs well when the histogram has a bimodal distribution with a deep and sharp valley between the two peaks.[6]
Like all other global thresholding methods, Otsu's method performs badly in case of heavy noise, small objects size, inhomogeneous lighting and larger intra-class than inter-class variance.[7] In those cases, local adaptations of the Otsu method have been developped.[8]
Moreover, the mathematical grounding of Otsu's method models the histogram of the image as a mixture of two Normal distributions with equal variance and equal size.[9] Otsu's thresholding may however yield satisfying results even when these assumptions are not met, in the same way statistical tests (to which Otsu's method is heavily connected[10]) can perform correctly even when the working assumptions are not fully satsified. However, several variations of Otsu's methods have been proposed to account for more severe deviations from these assumptions,[9] such as the Kittler-Illingworth method.[11]
A variation for noisy images[edit]
A popular local adaptation is the two-dimensional Otsu's method, which performs better for the object segmentation task in noisy images. Here, the intensity value of a given pixel is compared with the average intensity of its immediate neighborhood to improve segmentation results.[8]
At each pixel, the average gray-level value of the neighborhood is calculated. Let the gray level of the given pixel be divided into
{\displaystyle L}
discrete values and the average gray level is also divided into the same
{\displaystyle L}
values. Then a pair is formed: the pixel gray level and the average of the neighborhood
{\displaystyle (i,j)}
. Each pair belongs to one of the
{\displaystyle L\times L}
possible 2-dimensional bins. The total number of occurrences (frequency),
{\displaystyle f_{ij}}
, of a pair
{\displaystyle (i,j)}
, divided by the total number of pixels in the image
{\displaystyle N}
, defines the joint probability mass function in a 2-dimensional histogram:
{\displaystyle P_{ij}={\frac {f_{ij}}{N}},\qquad \sum _{i=0}^{L-1}\sum _{j=0}^{L-1}P_{ij}=1}
And the 2-dimensional Otsu's method is developed based on the 2-dimensional histogram as follows.
The probabilities of two classes can be denoted as:
{\displaystyle {\begin{aligned}\omega _{0}&=\sum _{i=0}^{s-1}\sum _{j=0}^{t-1}P_{ij}\\\omega _{1}&=\sum _{i=s}^{L-1}\sum _{j=t}^{L-1}P_{ij}\end{aligned}}}
The intensity mean value vectors of two classes and total mean vector can be expressed as follows:
{\displaystyle {\begin{aligned}\mu _{0}&=[\mu _{0i},\mu _{0j}]^{T}=\left[\sum _{i=0}^{s-1}\sum _{j=0}^{t-1}i{\frac {P_{ij}}{\omega _{0}}},\sum _{i=0}^{s-1}\sum _{j=0}^{t-1}j{\frac {P_{ij}}{\omega _{0}}}\right]^{T}\\\mu _{1}&=[\mu _{1i},\mu _{1j}]^{T}=\left[\sum _{i=s}^{L-1}\sum _{j=t}^{L-1}i{\frac {P_{ij}}{\omega _{1}}},\sum _{i=s}^{L-1}\sum _{j=t}^{L-1}j{\frac {P_{ij}}{\omega _{1}}}\right]^{T}\\\mu _{T}&=[\mu _{Ti},\mu _{Tj}]^{T}=\left[\sum _{i=0}^{L-1}\sum _{j=0}^{L-1}iP_{ij},\sum _{i=0}^{L-1}\sum _{j=0}^{L-1}jP_{ij}\right]^{T}\end{aligned}}}
In most cases the probability off-diagonal will be negligible, so it is easy to verify:
{\displaystyle \omega _{0}+\omega _{1}\cong 1}
{\displaystyle \omega _{0}\mu _{0}+\omega _{1}\mu _{1}\cong \mu _{T}}
The inter-class discrete matrix is defined as
{\displaystyle S_{b}=\sum _{k=0}^{1}\omega _{k}[(\mu _{k}-\mu _{T})(\mu _{k}-\mu _{T})^{T}]}
The trace of the discrete matrix can be expressed as
{\displaystyle {\begin{aligned}&\operatorname {tr} (S_{b})\\[4pt]={}&\omega _{0}[(\mu _{0i}-\mu _{Ti})^{2}+(\mu _{0j}-\mu _{Tj})^{2}]+\omega _{1}[(\mu _{1i}-\mu _{Ti})^{2}+(\mu _{1j}-\mu _{Tj})^{2}]\\[4pt]={}&{\frac {(\mu _{Ti}\omega _{0}-\mu _{i})^{2}+(\mu _{Tj}\omega _{0}-\mu _{j})^{2}}{\omega _{0}(1-\omega _{0})}}\end{aligned}}}
{\displaystyle \mu _{i}=\sum _{i=0}^{s-1}\sum _{j=0}^{t-1}iP_{ij}}
{\displaystyle \mu _{j}=\sum _{i=0}^{s-1}\sum _{j=0}^{t-1}jP_{ij}}
Similar to one-dimensional Otsu's method, the optimal threshold
{\displaystyle (s,t)}
is obtained by maximizing
{\displaystyle \operatorname {tr} (S_{b})}
{\displaystyle s}
{\displaystyle t}
is obtained iteratively which is similar with one-dimensional Otsu's method. The values of
{\displaystyle s}
{\displaystyle t}
are changed till we obtain the maximum of
{\displaystyle \operatorname {tr} (S_{b})}
max,s,t = 0;
for ss: 0 to L-1 do
for tt: 0 to L-1 do
evaluate tr(S_b);
if tr(S_b) > max
max = tr(S,b);
return s,t;
Notice that for evaluating
{\displaystyle \operatorname {tr} (S_{b})}
, we can use a fast recursive dynamic programming algorithm to improve time performance.[12] However, even with the dynamic programming approach, 2d Otsu's method still has large time complexity. Therefore, much research has been done to reduce the computation cost.[13]
If summed area tables are used to build the 3 tables, sum over
{\displaystyle P_{ij}}
, sum over
{\displaystyle i*P_{ij}}
, and sum over
{\displaystyle j*P_{ij}}
, then the runtime complexity is the maximum of (O(N_pixels), O(N_bins*N_bins)). Note that if only coarse resolution is needed in terms of threshold, N_bins can be reduced.
See also: Summed-area table
function inputs and output:
hists is a
{\displaystyle 256\times 256}
2D-histogram of grayscale value and neighborhood average grayscale value pair.
total is the number of pairs in the given image.it is determined by the number of the bins of 2D-histogram at each direction.
threshold is the threshold obtained.
function threshold = otsu_2D(hists, total)
helperVec = 0:255;
mu_t0 = sum(sum(repmat(helperVec',1,256).*hists));
mu_t1 = sum(sum(repmat(helperVec,256,1).*hists));
p_0 = zeros(256);
mu_i = p_0;
mu_j = p_0;
p_0(1,1) = hists(1,1);
p_0(ii,1) = p_0(ii-1,1) + hists(ii,1);
mu_i(ii,1) = mu_i(ii-1,1)+(ii-1)*hists(ii,1);
mu_j(ii,1) = mu_j(ii-1,1);
p_0(ii,jj) = p_0(ii,jj-1)+p_0(ii-1,jj)-p_0(ii-1,jj-1)+hists(ii,jj); % THERE IS A BUG HERE. INDICES IN MATLAB MUST BE HIGHER THAN 0. ii-1 is not valid
mu_i(ii,jj) = mu_i(ii,jj-1)+mu_i(ii-1,jj)-mu_i(ii-1,jj-1)+(ii-1)*hists(ii,jj);
mu_j(ii,jj) = mu_j(ii,jj-1)+mu_j(ii-1,jj)-mu_j(ii-1,jj-1)+(jj-1)*hists(ii,jj);
if (p_0(ii,jj) == 0)
if (p_0(ii,jj) == total)
tr = ((mu_i(ii,jj)-p_0(ii,jj)*mu_t0)^2 + (mu_j(ii,jj)-p_0(ii,jj)*mu_t1)^2)/(p_0(ii,jj)*(1-p_0(ii,jj)));
if ( tr >= maximum )
threshold = ii;
maximum = tr;
A variation for unbalanced images[edit]
When the levels of gray of the classes of the image can be considered as Normal distributions but with unequal size and/or unequal variances, assumptions for the Otsu algorithm are not met. The Kittler-Illingworth algorithm (also known as Minimum Error thresholding)[11] is a variation of Otsu's method to handle such cases. There are several ways to mathematically describe this algorithm. One of them is to consider that for each threshold being tested, the parameters of the Normal distributions in the resulting binary image are estimated by Maximum likelihood estimation given the data.[9]
While this algorithm could seem superior to Otsu's method, it introduces new parameters to be estimated, and this can result in the algorithm being over-parametrized and thus unstable. In many cases where the assumptions from Otsu's method seem at least partially valid, it may be preferable to favor Otsu's method over the Kittler-Illingworth algorithm, following Occam's razor.[9]
^ M. Sezgin & B. Sankur (2004). "Survey over image thresholding techniques and quantitative performance evaluation". Journal of Electronic Imaging. 13 (1): 146–165. doi:10.1117/1.1631315.
^ a b c Nobuyuki Otsu (1979). "A threshold selection method from gray-level histograms". IEEE Trans. Sys. Man. Cyber. 9 (1): 62–66. doi:10.1109/TSMC.1979.4310076.
^ Liu, Dongju (2009). "Otsu method and K-means". Ninth International Conference on Hybrid Intelligent Systems IEEE. 1: 344–349.
^ Liao, Ping-Sung (2001). "A fast algorithm for multilevel thresholding" (PDF). J. Inf. Sci. Eng. 17 (5): 713–727. Archived from the original (PDF) on 2019-06-24.
^ Huang, Deng-Yuan (2009). "Optimal multi-level thresholding using a two-stage Otsu optimization approach". Pattern Recognition Letters. 30 (3): 275–284. doi:10.1016/j.patrec.2008.10.003.
^ Kittler, J.; Illingworth, J. (September 1985). "On threshold selection using clustering criteria". IEEE Transactions on Systems, Man, and Cybernetics. SMC-15 (5): 652–655. doi:10.1109/tsmc.1985.6313443. ISSN 0018-9472.
^ Lee, Sang Uk and Chung, Seok Yoon and Park, Rae Hong (1990). "A comparative performance study of several global thresholding techniques for segmentation". Computer Vision, Graphics, and Image Processing. 52 (2): 171–190. doi:10.1016/0734-189x(90)90053-x. {{cite journal}}: CS1 maint: multiple names: authors list (link)
^ a b Jianzhuang, Liu and Wenqing, Li and Yupeng, Tian (1991). "Automatic thresholding of gray-level pictures using two-dimension Otsu method". Circuits and Systems, 1991. Conference Proceedings, China., 1991 International Conference on: 325–327. {{cite journal}}: CS1 maint: multiple names: authors list (link)
^ a b c d Kurita, T.; Otsu, N.; Abdelmalek, N. (October 1992). "Maximum likelihood thresholding based on population mixture models". Pattern Recognition. 25 (10): 1231–1240. doi:10.1016/0031-3203(92)90024-d. ISSN 0031-3203.
^ Jing-Hao Xue; Titterington, D. M. (August 2011). "$t$-Tests, $F$-Tests and Otsu's Methods for Image Thresholding". IEEE Transactions on Image Processing. 20 (8): 2392–2396. doi:10.1109/tip.2011.2114358. ISSN 1057-7149.
^ a b Kittler, J.; Illingworth, J. (1986-01-01). "Minimum error thresholding". Pattern Recognition. 19 (1): 41–47. doi:10.1016/0031-3203(86)90030-0. ISSN 0031-3203.
^ Zhang, Jun & Hu, Jinglu (2008). "Image segmentation based on 2D Otsu method with histogram analysis". Computer Science and Software Engineering, 2008 International Conference on. 6: 105–108. doi:10.1109/CSSE.2008.206. ISBN 978-0-7695-3336-0.
^ Zhu, Ningbo and Wang, Gang and Yang, Gaobo and Dai, Weiming (2009). "A fast 2d otsu thresholding algorithm based on improved histogram". Pattern Recognition, 2009. CCPR 2009. Chinese Conference on: 1–5. {{cite journal}}: CS1 maint: multiple names: authors list (link)
Implementation of Otsu's thresholding method as GIMP-plugin using Script-Fu (a Scheme-based language)
Lecture notes on thresholding – covers the Otsu method
A plugin for ImageJ using Otsu's method to do the threshold
A full explanation of Otsu's method with a working example and Java implementation
Implementation of Otsu's method in ITK
Otsu Thresholding in C# – a straightforward C# implementation with explanation
Otsu's method using MATLAB
Otsu Thresholding with scikit-image in Python
Retrieved from "https://en.wikipedia.org/w/index.php?title=Otsu%27s_method&oldid=1086933539"
|
Rapid Prototyping Model Functions - MATLAB & Simulink - MathWorks Switzerland
Rapid prototyping code defines the following functions that interface with the main program (main.c or main.cpp):
Model(): The model registration function. This function initializes the work areas (for example, allocating and setting pointers to various data structures) used by the model. The model registration function calls the MdlInitializeSizes and MdlInitializeSampleTimes functions. These two functions are very similar to the S-function mdlInitializeSizes and mdlInitializeSampleTimes methods.
MdlStart(void): After the model registration functions MdlInitializeSizes and MdlInitializeSampleTimes execute, the main program starts execution by calling MdlStart. This routine is called once at startup.
The function MdlStart has four basic sections:
Code to initialize the states for each block in the root model that has states. A subroutine call is made to the “initialize states” routines of conditionally executed subsystems.
Code generated by the one-time initialization (start) function for each block in the model.
Code to enable the blocks in the root model that have enable methods, and the blocks inside triggered or function-call subsystems residing in the root model. Simulink® blocks can have enable and disable methods. An enable method is called just before a block starts executing, and the disable method is called just after the block stops executing.
Code for each block in the model whose output value is constant. The block code appears in the MdlStart function only if the block parameters are not tunable in the generated code and if the code generator cannot eliminate the block code through constant folding.
MdlOutputs(int_T tid): MdlOutputs updates the output of blocks. The tid (task identifier) parameter identifies the task that in turn maps when to execute blocks based upon their sample time. This routine is invoked by the main program during major and minor time steps. The major time steps are when the main program is taking an actual time step (that is, it is time to execute a specific task). If your model contains continuous states, the minor time steps will be taken. The minor time steps are when the solver is generating integration stages, which are points between major outputs. These integration stages are used to compute the derivatives used in advancing the continuous states.
MdlUpdate(int_T tid): MdlUpdate updates the states and work vector state information (that is, states that are neither continuous nor discrete) saved in work vectors. The tid (task identifier) parameter identifies the task that in turn indicates which sample times are active, allowing you to conditionally update only states of active blocks. This routine is invoked by the interface after the major MdlOutputs has been executed. The solver is also called, and model_Derivatives is called in minor steps by the solver during its integration stages. All blocks that have continuous states have an identical number of derivatives. These blocks are required to compute the derivatives so that the solvers can integrate the states.
MdlTerminate(void): MdlTerminate contains any block shutdown code. MdlTerminate is called by the interface, as part of the termination of the real-time program.
The contents of the above functions are directly related to the blocks in your model. A Simulink block can be generalized to the following set of equations.
y={f}_{0}\left(t,{x}_{c},{x}_{d},u\right)
Output y is a function of continuous state xc, discrete state xd, and input u. Each block writes its specific equation in a section of MdlOutputs.
{x}_{d+1}={f}_{u}\left(t,{x}_{d},u\right)
The discrete states xd are a function of the current state and input. Each block that has a discrete state updates its state in MdlUpdate.
\stackrel{˙}{x}={f}_{d}\left(t,{x}_{c},u\right)
The derivatives x are a function of the current input. Each block that has continuous states provides its derivatives to the solver (for example, ode5) in model_Derivatives. The derivatives are used by the solver to integrate the continuous state to produce the next value.
The output, y, is generally written to the block I/O structure. Root-level Outport blocks write to the external outputs structure. The continuous and discrete states are stored in the states structure. The input, u, can originate from another block's output, which is located in the block I/O structure, an external input (located in the external inputs structure), or a state. These structures are defined in the model.h file that the Simulink Coder™ software generates.
The next example shows the general contents of the rapid prototyping style of C code written to the model.c file.
This figure shows a flow chart describing the execution of the rapid prototyping generated code.
Rapid Prototyping Execution Flow Chart
Each block places code in specific Mdl routines according to the algorithm that it is implementing. Blocks have input, output, parameters, and states, as well as other general items. For example, in general, block inputs and outputs are written to a block I/O structure (model_B). Block inputs can also come from the external input structure (model_U) or the state structure when connected to a state port of an integrator (model_X), or ground (rtGround) if unconnected or grounded. Block outputs can also go to the external output structure (model_Y). This figure shows the general mapping between these items.
Data View of the Generated Code
The following list defines the structures shown in the preceding figure:
Block I/O structure (model_B): This structure consists of persistent block output signals. The number of block output signals is the sum of the widths of the data output ports of the nonvirtual blocks in your model. If you activate block I/O optimizations, the Simulink and Simulink Coder products reduce the size of the model_B structure by
Reusing the entries in the model_B structure
Making other entries local variables
See How Generated Code Stores Internal Signal, State, and Parameter Data for more information on these optimizations.
Structure field names are determined either by the block's output signal name (when present) or by the block name and port number when the output signal is left unlabeled.
Block states structures: The continuous states structure (model_X) contains the continuous state information for blocks in your model that have continuous states. Discrete states are stored in a data structure called the DWork vector (model_DWork).
Block parameters structure (model_P): The parameters structure contains block parameters that can be changed during execution (for example, the parameter of a Gain block).
External inputs structure (model_U): The external inputs structure consists of the root-level Inport block signals. Field names are determined by either the block's output signal name, when present, or by the Inport block's name when the output signal is left unlabeled.
External outputs structure (model_Y): The external outputs structure consists of the root-level Outport blocks. Field names are determined by the root-level Outport block names in your model.
Real work, integer work, and pointer work structures (model_RWork, model_IWork, model_PWork): Blocks might have a need for real, integer, or pointer work areas. For example, the Memory block uses a real work element for each signal. These areas are used to save internal states or similar information.
|
Redox Reaction (Shashi Sir) - Live Session - NEET 2020
Redox Reaction (Shashi Sir) - Live Session - NEET 2020Contact Number: 9667591930 / 8527521718
The equivalent weight of a metal is 4.5 and the molecular weight of its chloride is 80. The atomic weight of the metal is:-
N2H4 + IO3- + 2H+ + Cl-
\to
ICl + N2 + 3H2O
The equivalent masses of N2H4 and KIO3 respectively are :-
1. 8 and 35.6
4. 16 and 53.5
What is the equivalent weight of H3PO3 in the following disproportionation reaction:-
\to
H3PO4 + PH3
\frac{M}{6}
\frac{M}{2}
\frac{2M}{3}
\frac{M}{3}
Al + Fe3O4
\to
Al2O3 + Fe
During the reaction, the total number of electrons transferred is-
25.0 g of FeSO4.7H2O was dissolved in water containing dilute H2SO4, and the volume was made up to 1.0 L. 25.0 mL of this solution required 20 mL of an N/10 KMnO4 solution for complete oxidation. The percentage of FeSO4. 7H2O in the acid solution is
25 mL of a solution containing HCl and H2SO4 required 10 mL of a 1 N NaOH solution for neutralization. 20 mL of the same acid mixture on being treated with an excess of AgNO3 gives 0.1425 g of AgCf. The normality of the HC and the normality of the H2SO4 are respectively
(1) 0.40 N and 0.05 N
(4) 0.40 N and 0.5 N
A brown ring complex compound is formulated as
\left[Fe{\left({H}_{2}O\right)}_{5}NO\right]\text{\hspace{0.17em}}S{O}_{4}.
The oxidation state of iron is
Strongest oxidising agent among the following is -
Br{O}_{3}^{-}/B{r}^{2+},\text{\hspace{0.17em}}{E}^{o}=+1.50
F{e}^{3+}/F{e}^{2+},\text{\hspace{0.17em}}{E}^{o}=+0.76
Mn{O}_{4}^{-}/M{n}^{2+},\text{\hspace{0.17em}}{E}^{o}=+1.52
C{r}_{2}{O}_{7}^{2-}/C{r}^{3+},\text{\hspace{0.17em}}{E}^{o}=+1.33
In alkaline medium ClO2 oxidize H2O2 in O2 and reduced itself in Cl– then how many mole of H2O2 will oxidize by one mole of ClO2 -
Equivalent mass of FeS2 in the half reaction,
{\mathrm{FeS}}_{2}\to {\mathrm{Fe}}_{2}{\mathrm{O}}_{3}+{\mathrm{SO}}_{2}
The equivalent mass of HCl in the given reaction is :
{\mathrm{K}}_{2}{\mathrm{Cr}}_{2}{\mathrm{O}}_{7}+14 \mathrm{HCl}\to 2\mathrm{KCl}+2{\mathrm{CrCl}}_{3}+3{\mathrm{Cl}}_{2}+{\mathrm{H}}_{2}\mathrm{O}
Equivalent mass of
{\mathrm{H}}_{3}{\mathrm{PO}}_{2}
{\mathrm{PH}}_{3}
{\mathrm{H}}_{3}{\mathrm{PO}}_{3}
{\mathrm{As}}_{2}{\mathrm{S}}_{3}+{\mathrm{H}}^{+}+{\mathrm{NO}}_{3}^{-}\to \mathrm{NO}+{\mathrm{H}}_{2}\mathrm{O}+{\mathrm{AsO}}_{4}^{3-}+{\mathrm{SO}}_{4}^{2-}
, the equivalent mass of
{\mathrm{As}}_{2}{\mathrm{S}}_{3}
is related to its molecular mass by:
Sulphur forms the chlorides
{\mathrm{S}}_{2}{\mathrm{Cl}}_{2}
{\mathrm{SCl}}_{2}
. The equivalent mass of sulphur in
{\mathrm{SCl}}_{2}
1. 8 g/mol
2. 16 g/mol
The equivalent mass of an element is 4. Its chloride has a vapour density 59.25. Then, the valency of the elements is :
6×{10}^{-3}
{\mathrm{K}}_{2}{\mathrm{Cr}}_{2}{\mathrm{O}}_{7}
reacts completely with
9×{10}^{-3}
{\mathrm{X}}^{\mathrm{n}+}
{\mathrm{XO}}_{3}^{-}
{\mathrm{Cr}}^{3+}
. The value of n is :
What mass of
{\mathrm{H}}_{2}{\mathrm{C}}_{2}{\mathrm{O}}_{4}.2{\mathrm{H}}_{2}\mathrm{O}
(mol. mass = 126) should be dissolved in water to prepare 250 mL of centinormal solution which act as a reducing agent ?
The equivalent mass of salt,
{\mathrm{KHC}}_{2}{\mathrm{O}}_{4}.{\mathrm{H}}_{2}{\mathrm{C}}_{2}{\mathrm{O}}_{4}.4{\mathrm{H}}_{2}\mathrm{O}
when it act as reducing agent is :
\frac{\mathrm{Mol}. \mathrm{mass}}{1}
\frac{\mathrm{Mol}. \mathrm{mass}}{2}
\frac{\mathrm{Mol}. \mathrm{mass}}{3}
\frac{\mathrm{Mol}. \mathrm{mass}}{4}
The equivalent mass of divalent metal is W. The molecular mass of its chloride is ::
1. W + 35.5
2. W + 71
3. 2W + 71
4. 2W + 35.5
{\mathrm{BrO}}_{3}^{-}
ion reacts with
{\mathrm{Br}}^{-}
{\mathrm{Br}}_{2}
is liberated. The equivalent mass of
{\mathrm{Br}}_{2}
this reaction is :
\frac{5\mathrm{M}}{8}
\frac{5\mathrm{M}}{3}
\frac{3\mathrm{M}}{5}
\frac{4\mathrm{M}}{6}
{\mathrm{m}}_{\mathrm{A}}
gram of a metal displaces
{\mathrm{m}}_{\mathrm{B}}
gram of another metal B from its salt solution and if equivalent mass are
{\mathrm{E}}_{\mathrm{A}}
{\mathrm{E}}_{\mathrm{B}}
respectively then equivalent mass of A can be expressed as :
{\mathrm{E}}_{\mathrm{A}}=\frac{{\mathrm{m}}_{\mathrm{A}}}{{\mathrm{m}}_{\mathrm{B}}}×{\mathrm{E}}_{\mathrm{B}}
{\mathrm{E}}_{\mathrm{A}}=\frac{{\mathrm{m}}_{\mathrm{A}}×{\mathrm{m}}_{\mathrm{B}}}{{\mathrm{E}}_{\mathrm{B}}}
{\mathrm{E}}_{\mathrm{A}}=\frac{{\mathrm{m}}_{\mathrm{B}}}{{\mathrm{m}}_{A}}×{\mathrm{E}}_{\mathrm{B}}
{\mathrm{E}}_{\mathrm{A}}=\sqrt{\frac{{\mathrm{m}}_{\mathrm{A}}}{{\mathrm{m}}_{\mathrm{B}}}×{\mathrm{E}}_{\mathrm{B}}}
|
Uniformity of rational points an up-date and corrections
2022 Uniformity of rational points an up-date and corrections
Lucia Caporaso, Joe Harris, Barry Mazur
Tunisian J. Math. 4(1): 183-201 (2022). DOI: 10.2140/tunis.2022.4.183
The purpose of this note is to correct, and enlarge on, an argument in a paper we published a quarter century ago (J. Amer. Math. Soc. 10:1 (1997), 1–35). The question raised is a simple one to state: given that a curve
C
g\ge 2
K
has only finitely many rational points, we ask if the number of points is bounded as
C
Lucia Caporaso. Joe Harris. Barry Mazur. "Uniformity of rational points an up-date and corrections." Tunisian J. Math. 4 (1) 183 - 201, 2022. https://doi.org/10.2140/tunis.2022.4.183
Received: 14 June 2021; Revised: 13 July 2021; Accepted: 28 July 2021; Published: 2022
Digital Object Identifier: 10.2140/tunis.2022.4.183
Keywords: rational points , uniformity
Lucia Caporaso, Joe Harris, Barry Mazur "Uniformity of rational points an up-date and corrections," Tunisian Journal of Mathematics, Tunisian J. Math. 4(1), 183-201, (2022)
|
Fit uncertain model to set of LTI responses - MATLAB ucover - MathWorks América Latina
Fit Uncertain Model to Array of LTI Responses
ord1ord2
Fit uncertain model to set of LTI responses
usys = ucover(Parray,Pnom,ord)
usys = ucover(Parray,Pnom,ord1,ord2,utype)
[usys,info] = ucover(Parray,___)
[usys,info] = ucover(Pnom,info_in,ord1,ord2)
usys = ucover(Parray,Pnom,ord) returns an uncertain model usys with nominal value Pnom and whose range of behaviors includes all responses in the LTI array Parray. The uncertain model structure is of the form
usys=Pnom\left(I+W\left(s\right)\Delta \left(s\right)\right)
Δ is a ultidyn object that represents uncertain dynamics with unit peak gain.
W is a stable, minimum-phase shaping filter of order ord that adjusts the amount of uncertainty at each frequency. For a MIMO Pnom, W is diagonal, with the orders of the diagonal elements given by ord.
usys = ucover(Parray,Pnom,ord1,ord2,utype) returns an uncertain model with the structure specified by utype.
utype = 'InputMult' — Input multiplicative form, in which usys = Pnom*(I + W1*Delta*W2)
utype = 'OutputMult' — Output multiplicative form, in which usys = (I + W1*Delta*W2)*Pnom
utype = 'Additive' — Additive form, in which usys = Pnom + W1*Delta*W2
Delta represents uncertain dynamics with unit peak gain, and W1 and W2 are diagonal, stable, minimum-phase shaping filters with orders specified by ord1 and ord2, respectively.
[usys,info] = ucover(Parray,___) returns a structure info that contains information about the fit. You can use this syntax with any of the previous input-argument combinations.
[usys,info] = ucover(Pnom,info_in,ord1,ord2) improves the fit using initial filter values in the info result. Supply new orders ord1 and ord1 for W1 and W2. When you are trying different filter orders to improve the result, this syntax speeds up iteration by letting you reuse previously computed information.
Fit an uncertain model to an array of LTI responses. The responses can be, for example, the results of multiple runs to acquire frequency response data from a physical system.
For this example, generate the frequency response data by creating an array of LTI models and sampling the frequency response of those models.
p1 = Pnom*tf(1,[.06 1]);
p2 = Pnom*tf([-.02 1],[.02 1]);
p3 = Pnom*tf(50^2,[1 2*.1*50 50^2]);
array = stack(1,p1,p2,p3);
Parray = frd(array,logspace(-1,3,60));
The frequency response data in Parray represents three separate data acquisition experiments on the system.
Plot the relative errors between the nominal plant response and the three models in the LTI array.
relerr = (Pnom-Parray)/Pnom;
bodemag(relerr)
If you use a multiplicative uncertainty model structure, you want the magnitude of the shaping filter to fit the maximum relative error at each frequency. Use this requirement to help choose the order of the shaping filter. First, try a first-order shaping filter.
[P,Info] = ucover(Parray,Pnom,1);
P is an uncertain state-space (uss) model that captures the uncertainty as a ultidyn uncertain dynamics block.
P.Uncertainty
Parray_InputMultDelta: [1x1 ultidyn]
The Info structure contains other information about the fit, including the resulting shaping filter, Info.W1. Plot the response to see how well the shaping filter fits the relative errors.
W = Info.W1;
bodemag(relerr,'b--',W,'r',{0.1,1000});
The plot shows that the filter W is too conservative and exceeds the maximum relative error at most frequencies. To obtain a tighter fit, rerun the function using a fourth-order filter.
Evaluate the fit by plotting the Bode magnitude plot.
This plot shows that for the fourth-order filter, the magnitude of W closely matches the largest error, yielding the minimum uncertainty that captures all the variation.
Parray — Array of models to cover
array of LTI models
Array of models to cover with a dynamically uncertain model, specified as an array of LTI models such as tf, ss, zpk, or frd models.
Pnom — Nominal model
Nominal model of the uncertain model, specified as an LTI model such as a tf, ss, zpk, or frd model.
ord — Filter order
Filter order, specified as an integer, vector, or []. The values in ord specify the number of states of each diagonal entry of the shaping filter W. Specify ord as:
A single integer, for a SISO Pnom, or to use a scalar filter W for a MIMO Pnom.
A vector of length equal to the number of outputs in Pnom, to specify different orders for each diagonal entry of W.
[], to set W = 1.
ord1, ord2 — Filter orders
Filter orders, specified as integers, vectors, or []. The values in ord1 and ord2 specify the number of states of each diagonal entry of the shaping filters W1, and W2, respectively. Specify ord1 and ord2 as:
A single integer, to use scalar filters for W1 and W2.
A vector, to specify different orders for each diagonal entry of W1 and W2. The lengths of these vectors depend on the uncertainty model you specify in utype. The following table gives the lengths, where Pnom has Nu inputs and Ny outputs.
length(ord1)
'InputMult' Nu Nu
'OutputMult' Ny Ny
'Additive' Ny Nu
[], to set W1 = 1 or W2 = 1.
utype — Uncertainty model
'InputMult' (default) | 'OutputMult' | 'Additive'
Uncertainty model, specified as one of the following.
'InputMult' — Input multiplicative form, in which usys = Pnom*(I + W1*Delta*W2)
'OutputMult' — Output multiplicative form, in which usys = (I + W1*Delta*W2)*Pnom
'Additive' — Additive form, in which usys = Pnom + W1*Delta*W2
Use additive uncertainty to model the absolute gaps between Pnom and Parray, and multiplicative uncertainty to model relative gaps.
For SISO models, input and output multiplicative uncertainty are equivalent. For MIMO systems with more outputs than inputs, the input multiplicative structure might be too restrictive and might not adequately cover the range of models.
info_in — Details from previous ucover run
Details from a previous ucover run, specified as a structure generated as the info output of the previous run. Use this input when calling ucover iteratively to improve fit results by trying different filter orders.
usys — Uncertain model
uss model | ufrd model
Uncertain model, returned as a uss or ufrd model. The returned model is a uss model, unless Parray or Pnom are frequency-response data (frd) models, in which case usys is a ufrd model.
usys has one uncertain element, a ultidyn block with the name given in the DeltaName field of the info output argument.
info — Information about the fit
Information about the fit, returned as a structure containing the following fields.
Fitted shaping filter W or W1, returned as a state-space (ss) model.
Fitted shaping filter W2, returned as a state-space (ss) model.
W1opt
W or W1 evaluated on a frequency grid, returned as an frd model.
W2 evaluated on a frequency grid, returned as an frd model.
Orders of the diagonal elements of W or W1, returned as a scalar or vector. These values are the values you supply with the ord or ord1 input argument.
Orders of the diagonal elements of W2, returned as a scalar or vector. These values are the values you supply with the ord2 input argument.
Uncertainty model used for the fit, returned as 'InputMult', 'OutputMult', or 'Additive'.
DeltaName
Name of the ultidyn block of usys that represents the uncertainty model Delta, returned as a character vector.
Residuals of the fit, returned as an array of frd models with the same array dimensions as Parray.
ucover fits the responses of LTI models in Parray by modeling the gaps between Parray and the nominal response Pnom as uncertainty on the system dynamics. To model the frequency distribution of these unmodeled dynamics, ucover measures the gap between Pnom and Parray at each frequency on a grid, and selects shaping filters whose magnitude approximates the maximum gap.
To design the minimum-phase shaping filters W1 and W2, the ucover command performs two steps:
Compute the optimal values of W1 and W2 on a frequency grid.
Fit W1 and W2 values with the dynamic filters of the specified orders using fitmagfrd.
The model structure
usys=Pnom\left(I+W\left(s\right)\Delta \left(s\right)\right)
that you obtain using usys = ucover(Parray,Pnom,ord) corresponds to W1 = W and W2 = 1.
For instance, the following figure shows the relative gap between the nominal response and six LTI responses, enveloped using a second-order shaping filter and a fourth-order filter.
If you use the single-filter syntax usys = ucover(Parray,Pnom,ord), the software sets the uncertainty to W*Delta, where Delta is a ultidyn object that represents unit-gain uncertain dynamics. Therefore, the amount of uncertainty at each frequency is specified by the magnitude of W and closely tracks the gap between Pnom and Parray. In the above figure, the fourth-order filter tracks the maximum gap more closely and therefore yields a less conservative estimate of uncertainty.
fitmagfrd | ultidyn | uss | ufrd
|
Audio_time_stretching_and_pitch_scaling Knowpia
Time stretching is the process of changing the speed or duration of an audio signal without affecting its pitch. Pitch scaling is the opposite: the process of changing the pitch without affecting the speed. Pitch shift is pitch scaling implemented in an effects unit and intended for live performance. Pitch control is a simpler process which affects pitch and speed simultaneously by slowing down or speeding up a recording.
These processes are often used to match the pitches and tempos of two pre-recorded clips for mixing when the clips cannot be reperformed or resampled. Time stretching is often used to adjust radio commercials[1] and the audio of television advertisements[2] to fit exactly into the 30 or 60 seconds available. It can be used to conform longer material to a designated time slot, such as a 1-hour broadcast.
ResamplingEdit
The simplest way to change the duration or pitch of an audio recording is to change the playback speed. For a digital audio recording, this can be accomplished through sample rate conversion. Unfortunately, the frequencies in the recording are always scaled at the same ratio as the speed, transposing its perceived pitch up or down in the process. Slowing down the recording to increase duration also lowers the pitch, speeding it up for a shorter duration also raises the pitch creating the Chipmunk effect. Thus the two effects cannot be separated when using this method. A drum track containing no pitched instruments can be moderately sample-rate converted to adjust tempo without adverse effects, but a pitched track cannot.
Frequency domainEdit
Phase vocoderEdit
One way of stretching the length of a signal without affecting the pitch is to build a phase vocoder after Flanagan, Golden, and Portnoff.
compute the instantaneous frequency/amplitude relationship of the signal using the STFT, which is the discrete Fourier transform of a short, overlapping and smoothly windowed block of samples;
apply some processing to the Fourier transform magnitudes and phases (like resampling the FFT blocks); and
perform an inverse STFT by taking the inverse Fourier transform on each chunk and adding the resulting waveform chunks, also called overlap and add (OLA).[3]
The phase vocoder handles sinusoid components well, but early implementations introduced considerable smearing on transient ("beat") waveforms at all non-integer compression/expansion rates, which renders the results phasey and diffuse. Recent improvements allow better quality results at all compression/expansion ratios but a residual smearing effect still remains.
The phase vocoder technique can also be used to perform pitch shifting, chorusing, timbre manipulation, harmonizing, and other unusual modifications, all of which can be changed as a function of time.
Sinusoidal analysis/synthesis system (based on McAulay & Quatieri 1988, p. 161)[4]
Sinusoidal spectral modelingEdit
Another method for time stretching relies on a spectral model of the signal. In this method, peaks are identified in frames using the STFT of the signal, and sinusoidal "tracks" are created by connecting peaks in adjacent frames. The tracks are then re-synthesized at a new time scale. This method can yield good results on both polyphonic and percussive material, especially when the signal is separated into sub-bands. However, this method is more computationally demanding than other methods.[citation needed]
Modelling a monophonic sound as observation along a helix of a function with a cylinder domain
Time domainEdit
Rabiner and Schafer in 1978 put forth an alternate solution that works in the time domain: attempt to find the period (or equivalently the fundamental frequency) of a given section of the wave using some pitch detection algorithm (commonly the peak of the signal's autocorrelation, or sometimes cepstral processing), and crossfade one period into another.
This is called time-domain harmonic scaling[5] or the synchronized overlap-add method (SOLA) and performs somewhat faster than the phase vocoder on slower machines but fails when the autocorrelation mis-estimates the period of a signal with complicated harmonics (such as orchestral pieces).
Adobe Audition (formerly Cool Edit Pro) seems to solve this by looking for the period closest to a center period that the user specifies, which should be an integer multiple of the tempo, and between 30 Hz and the lowest bass frequency.
This is much more limited in scope than the phase vocoder based processing, but can be made much less processor intensive, for real-time applications. It provides the most coherent results[citation needed] for single-pitched sounds like voice or musically monophonic instrument recordings.
High-end commercial audio processing packages either combine the two techniques (for example by separating the signal into sinusoid and transient waveforms), or use other techniques based on the wavelet transform, or artificial neural network processing[citation needed], producing the highest-quality time stretching.
Frame-based approachEdit
Frame-based approach of many TSM procedures
In order to preserve an audio signal's pitch when stretching or compressing its duration, many time-scale modification (TSM) procedures follow a frame-based approach.[6] Given an original discrete-time audio signal, this strategy's first step is to split the signal into short analysis frames of fixed length. The analysis frames are spaced by a fixed number of samples, called the analysis hopsize
{\displaystyle H_{a}\in \mathbb {N} }
. To achieve the actual time-scale modification, the analysis frames are then temporally relocated to have a synthesis hopsize
{\displaystyle H_{s}\in \mathbb {N} }
. This frame relocation results in a modification of the signal's duration by a stretching factor of
{\displaystyle \alpha =H_{s}/H_{a}}
. However, simply superimposing the unmodified analysis frames typically results in undesired artifacts such as phase discontinuities or amplitude fluctuations. To prevent these kinds of artifacts, the analysis frames are adapted to form synthesis frames, prior to the reconstruction of the time-scale modified output signal.
The strategy of how to derive the synthesis frames from the analysis frames is a key difference among different TSM procedures.
Speed hearing and speed talkingEdit
For the specific case of speech, time stretching can be performed using PSOLA.
Time-compressed speech is the representation of verbal text in compressed time. While one might expect speeding up to reduce comprehension, Herb Friedman says that "Experiments have shown that the brain works most efficiently if the information rate through the ears—via speech—is the 'average' reading rate, which is about 200–300 wpm (words per minute), yet the average rate of speech is in the neighborhood of 100–150 wpm."[7]
Listening to time-compressed speech is seen as the equivalent of speed reading.[8][9]
Pitch scalingEdit
Pitch shifting (Frequency scaling) is provided on Eventide Harmonizer
Frequency shifting provided by Bode Frequency Shifter does not keep frequency ratio and harmony.
These techniques can also be used to transpose an audio sample while holding speed or duration constant. This may be accomplished by time stretching and then resampling back to the original length. Alternatively, the frequency of the sinusoids in a sinusoidal model may be altered directly, and the signal reconstructed at the appropriate time scale.
Transposing can be called frequency scaling or pitch shifting, depending on perspective.
For example, one could move the pitch of every note up by a perfect fifth, keeping the tempo the same. One can view this transposition as "pitch shifting", "shifting" each note up 7 keys on a piano keyboard, or adding a fixed amount on the Mel scale, or adding a fixed amount in linear pitch space. One can view the same transposition as "frequency scaling", "scaling" (multiplying) the frequency of every note by 3/2.
Musical transposition preserves the ratios of the harmonic frequencies that determine the sound's timbre, unlike the frequency shift performed by amplitude modulation, which adds a fixed frequency offset to the frequency of every note. (In theory one could perform a literal pitch scaling in which the musical pitch space location is scaled [a higher note would be shifted at a greater interval in linear pitch space than a lower note], but that is highly unusual, and not musical.[citation needed])
Time domain processing works much better here, as smearing is less noticeable, but scaling vocal samples distorts the formants into a sort of Alvin and the Chipmunks-like effect, which may be desirable or undesirable. A process that preserves the formants and character of a voice involves analyzing the signal with a channel vocoder or LPC vocoder plus any of several pitch detection algorithms and then resynthesizing it at a different fundamental frequency.
A detailed description of older analog recording techniques for pitch shifting can be found within the Alvin and the Chipmunks entry.
In consumer softwareEdit
Pitch-corrected audio timestretch is found in every modern web browser as part of the HTML standard for media playback.[10] Similar controls are ubiquitous in media applications and frameworks such as GStreamer and Unity.
Dynamic tonality — real-time changes of tuning and timbre
Scrubbing (audio)
^ "Dolby, The Chipmunks And NAB2004". Archived from the original on 2008-05-27. {{cite magazine}}: Cite magazine requires |magazine= (help)
^ "Variable speech". www.atarimagazines.com.
^ Jont B. Allen (June 1977). "Short Time Spectral Analysis, Synthesis, and Modification by Discrete Fourier Transform". IEEE Transactions on Acoustics, Speech, and Signal Processing. ASSP-25 (3): 235–238.
^ McAulay, R. J.; Quatieri, T. F. (1988), "Speech Processing Based on a Sinusoidal Model" (PDF), The Lincoln Laboratory Journal, 1 (2): 153–167, archived from the original (PDF) on 2012-05-21, retrieved 2014-09-07
^ David Malah (April 1979). "Time-domain algorithms for harmonic bandwidth reduction and time scaling of speech signals". IEEE Transactions on Acoustics, Speech, and Signal Processing. ASSP-27 (2): 121–133.
^ Jonathan Driedger and Meinard Müller (2016). "A Review of Time-Scale Modification of Music Signals". Applied Sciences. 6 (2): 57. doi:10.3390/app6020057.
^ Variable Speech, Creative Computing Vol. 9, No. 7 / July 1983 / p. 122
^ "Listen to podcasts in half the time".
^ "Speeding iPods". Archived from the original on 2006-09-02.
^ "HTMLMediaElement.playbackRate - Web APIs". MDN. Retrieved 1 September 2021.
Time Stretching and Pitch Shifting Overview A comprehensive overview of current time and pitch modification techniques by Stephan Bernsee
Stephan Bernsee's smbPitchShift C source code C source code for doing frequency domain pitch manipulation
pitchshift.js from KievII A Javascript pitchshifter based on smbPitchShift code, from the open source KievII library
PICOLA and TDHS
How to build a pitch shifter Theory, equations, figures and performances of a real-time guitar pitch shifter running on a DSP chip
ZTX Time Stretching Library Free and commercial versions of a popular 3rd party time stretching library for iOS, Linux, Windows and Mac OS X
Elastique by zplane commercial cross-platform library, mainly used by DJ and DAW manufacturers
Voice Synth from Qneo - specialized synthesizer for creative voice sculpting
TSM toolbox Free MATLAB implementations of various Time-Scale Modification procedures
PaulStretch, a well-known algorithm for extreme (> 10×) time stretching
|
Electromagnetic Waves, Popular Questions: CBSE Class 12-science SCIENCE, Science - Meritnation
\left[\mathrm{Ans}. {10}^{-2}\mathrm{m}, \mathrm{B}=\left({10}^{-7}\mathrm{T}\right)\mathrm{sin}\left(6\mathrm{\pi }×{10}^{10}\mathrm{t}-200\mathrm{\pi x}\right)\stackrel{^}{\mathrm{k}}\right]
Madhumeet Kaur & 1 other asked a question
why are alkali metals most suitable for photoelectric emission?
which physical quantity has the same value for waves belonging to different parts of electromagnetic spectrum?
|
In this article, we discuss intermediate representations and look at different approaches to IR while considering their properties.
Abstract Syntax Tree(AST).
Directed Acyclic Graph(DAGs).
Static single assignment form.
Linear IR.
Static Machine IR.
Intermediate representations lie between the abstract structure of the source language and the concrete structure of the target assembly language.
These representations are designed to have simple regular structures that facilitate analysis, optimization and efficient code generation.
The Front-end produces the intermediate representation.
The Middle transforms the IR into a more efficient IR.
The Back-end transforms the IR into native code.
Categories of IR.
They are graphically structured.
Used in source-to-source translators.
e.g; AST, DAGs
Simple and compact data structures.
Easier to rearrange.
They have a varied level of abstraction.
They can act as pseudo-code for an abstract machine.
e.g; 3-address code, stack machine code
They are a combination of graphs and linear code.
e.g control flow graph.
An AST is usable as an IR if the goal is to emit assembly language without optimizations or transformations.
The AST of the expression x - 2 * y
In post-fix form -> x 2 y * -
In prefix form -> - * 2 y x
After type checking, optimizations e.g constant folding, strength reduction can be applied to the AST, then post-order traversal is applied to generate assembly language whereby each node of the tree will have corresponding assembly instructions.
In a production compiler, this is not great choice for IR because the structure is too rich, in that, each node has a large number of options and substructure e.g an addition node can represent either floating point or integer addition.
This makes transformations and external representations difficult to perform.
A DAG is an AST with a unique node for each value.
A DAG can have an arbitrary graph structure whereby individual nodes are simplified so that there is little auxilliary information beyond the type and value of each node.
We have the expression x = (a + 10) * (a + 10).
The AST of the expression is shown below.
After type checking, we learn a is a floating point value so 10 must be converted to a floating point value for floating point arithmetic operation to happen.
Additionally, this computation is performed once and this result can be used twice.
The DAG for the expression.
ITOF will be responsible for performing integer-to-float conversion.
FAAD and FMUL perform floating point arithmetic.
DAGs represent address computations related to pointers and arrays so that they are shareable and optimized whenever possible.
An example of an array lookup for x = a[i] represented in an AST
We add the starting address of array a with the index of the item i multiplied by the size of objects in the array(determined by symbol table).
The same lookup in a DAG.
Before code generation, the DAG might be expanded so as to include address computations of local variables. e.g if a and i are both stored on the stack at 16 and 20 bytes past the frame pointer respectively, the DAG is expanded as shown below.
The value-number method can be used to construct a DAG from an AST by building an array where each entry consists of a DAG node type and array index of child nodes, such that each time we need to add a new node, we first search the array for a matching node and reuse it to avoid duplication.
By performing a post-order traversal and adding each element of the AST to an array we are able construct the DAG.
Value-number array representation of the DAG
0 NAME x
1 NAME a
3 ITOF 2
4 FADD 1 3
5 FMUL 4 4
6 ASSIGN 0 5
As long as each expression has its own DAG, we can avoid a polynomial complexity when searching for nodes every time there is a new addition since absolute sizes will remain relatively small.
With DAG representation it is easier to write portable external representations since all information will be encoded into a node.
To optimize a DAG we can use constant folding where by we reduce an expression consisting of constants into a single value.
Constant Folding Algorithm
The algorithm works by examining a DAG recursively and collapsing all operators with two constants into a single constant.
ConstantFold(DAGNode):
If n is leaf:
n.left = ConstantFold(n.left);
n.right = ConstantFold(n.right);
If n.left and n.right are constants:
n.value = n.operator(n.left, n.right);
n.kind = constant;
delete n.left and n.right;
Constant folding in practice
We have the expression secs = days * 24 * 60 * 60 as represented by the programmer which computes the number of seconds in the given number of days, The ConstantFold algorithm descends through the tree and combines IMUL(60, 60) to 3600 and IMUL(3600, 24) to 86400.
A DAG can encode expressions effectively but faces a drawback when it comes to ordered program structures such as controlling flow.
In a DAG common subexpressions are combined assuming they can be evaluated in any ordering and values don't change however this doesn't hold when it comes to multiple statements that modify values and control flow structures that repeat or skip statements hence a control flow graph.
A control flow graph is a directed graph whereby each node will consist of a basic block of sequential statements.
Edges in the graph represent the possible flow of control between basic blocks.
Conditional constructs e.g if or switch statements, define the branches in the graph.
Looping constructs e.g for or while loops, define reverse edges.
For the for loop the control flow graph places each expression in the order it would be executed in practice as compared to the AST which would have each control expression as an intermediate child of the loop node.
The if statements have edges from each branch of the conditional to the following node and thus it it easy to trace the flow of execution from a component to the next.
The idea is to define each name exactly once.
It is a common representation for complex optimizations.
It uses information from the control flow graph and updates each basic block with a new restriction e.g variables cannot change values and when a variable indeed changes a value, it is assigned a version number.
x = 20 * b;
int a_1 = x_1;
int b_1 = a_1 + 10;
x_2 = 20 * b_1;
What if a variable is given a different value in two branches of a conditional? We introduce the phi ϕ-functions to make it work.
ϕ(x, y), indicates that either value x or y could be selected at run-time. This function might not translate into assembly code but serves as a link to the new value to its possible old values.
if(y_1 < 10)
x_2 = a;
x_3 = b;
x_4 = phi(x_2, x_3);
This is an ordered sequence of instructions that are closer to the final goal of an assembly language.
It looses DAG flexibility, but can capture expressions, statements and control flows within one data structure.
It looks like an idealized assembly language with a large/infinite number of registers and common arithmetic and control flow operations.
Linear IR can be easily stored considering each instruction is a fixed size 4-tuple representing the operation and arguments(max of 3).
Assume an IR with LOAD and STOR instructions to move values between memory and registers, and 3-address arithmetic operations that read two registers and write to a third from left to right.
An example expression would look like the following.
1. LOAD a -> %r1
2. LOAD $10 -> %r2
3. ITOF %r2 -> %r3
4. FADD %r1, r3% -> %r4
5. FMUL %r4, r4 -> %r5
6. STOR %r5 -> x
We assume there exists an infinite number of virtual register so as to write each new value to a new register by which we can identify the lifetime of a value by observing the first point where a register is written and the last point where a register is used, e.g the lifetime of %r1 is from instruction 1 to 4.
At any given instruction we can observe virtual registers that are live.
1. LOAD a -> %r1 live: %r1
2. LOAD $10 -> %r2 live: %r1 %r2
3. ITOF %r2 -> %r3 live: %r1 %r2 %r3
4. FADD %r1, r3% -> %r4 live: %r1 %r3 %r4
5. FMUL %r4, r4 -> %r5 live: %r4 %r5
6. STOR %r5 -> x live: %r5
The above makes instruction ordering easier since any instruction can be moved to an earlier position(within on basic block) as long as the values that are read are not moved above their definitions.
Also, instructions may be moved to later positions so long as the values it writes are not moved below their uses.
In a pipelined architecture, moving of instructions reduces the number of physical registers needed for code generation and thus reduced execution time.
Stack machine IR.
This is a more compact version of IR designed to execute on a virtual stack machine without traditional registers except only a stack to hold intermediate registers.
It uses PUSH instruction to push a variable or literal onto the stack and POP instruction to remove an item from the stack and stores it in memory.
FADD and FMUL binary arithmetic operators implicitly pop two values off the stack and push the result on the stack.
ITOF unary operator pops one value off the stack and push another value.
COPY instruction manipulates the stack by pushing duplicate values onto the stack..
A post-order traversal of the AST is used so as to produce a stack machine IR given a DAG and emit a PUSH for each leaf value, an arithmetic instruction for each node and POP instruction which assigns a value to a variable.
An example expression
x=(a + 10)
*
(a + 10)
Instructions in a stack machine
Assuming x is 5, executing thr IR would result in the following.
Stack 5.0 10 10.0 15.0 15.0 225.0 -
State: - 5.0 5.0 - 15.0 - -
Pros of stack machine
It is much more compact compared to a 3-tuple or 4-tuple representation since there is no need for recording register details.
Implementation is straight forward in a simple interpreter.
It is difficult to translate to a conventional register-based assembly language since explicit register names are lost, further transformation and optimization requires transforming implicit information dependencies in the stack-basic IR back to a more explicit DAG or linear IR with explicit register names.
Every compiler or language will have its own intermediate representation with some local features.
Intermediate representations are independent of the source language.
Intermediate representations express operations of a target machine.
IR code generation is not necessary since the semantic analysis phase in compiler design can generate assembly code directly.
Introduction to Compilers and Language Design, 2nd Edition, Prof. Douglas Thain.
|
Comments on tag 01JG—Kerodon
Subsection 4.6.2: Fully Faithful and Essentially Surjective Functors (cite)
Comment #1119 by Daniel Gratzer on November 12, 2021 at 13:16
In Remark 4.6.2.16, I believe after the three assumptions labeled (a), (b), and (c) there should be a sentence stating that under these assumptions
F
is essentially surjective.
Comment #1126 by Kerodon on November 15, 2021 at 20:19
In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 01JG. The letter 'O' is never used.
|
6.5 The Power Grid System | EME 810: Solar Resource Assessment and Economics
J.R. Brownson, Solar Energy Conversion Systems (SECS), Chapter 9: Solar Economics (focus on Managing the Grid)
The main form of energy that we think of in society is power from electricity. As a society, we typically deliver electric power though a complex distribution system called the power grid.
One of the highly visible SECS technologies is photovoltaics, which delivers (generation) electricity to the client, and now pushes excess electricity onto the electricity power grid. The electricity power grid is the physical system that delivers (transmission) electricity from the place where it is generated to the site where it is used (end-use, demand).The electricity leaving the generating station enters a sub-station with a step-up transformer that raises the voltage extremely high for long-distance transmission.
Reminder: Power is Voltage times Current (
W=V×I
When electricity travels through wires (a conductor), some energy is lost, but less energy is lost when the electricity is transmitted at a higher voltage. At a high voltage, the same amount of power can be transmitted, but using a lower current. The amount of energy lost from the conductor is called line loss, and line losses are directly proportional to the current. By reducing current, we reduce losses for the same power transmitted. Typically in the U.S., line losses between generation and end-use are in the 6% to 8% range.
The high-voltage electricity is carried over transmission lines to local substations where a step-down transformer reduces the voltage to levels suitable for customer loads. Distribution lines carry the lower-voltage electricity from the local substations to customer sites.
Figure 6.2 The Electricity Power Grid
Click Here for a text version of The Electricity Power Grid Schematic
On the left, there is a Generating Station with a line going into the “Generator Step Up Transformer”. This is the Generation portion of the diagram.
From there, it goes to the Transmission Customer (138kV or 230kV) and to Transmission Lines (765, 500, 354, 230, and 138kV). This is the transmission portion of the diagram.
From the Transmission Customer and Transmission lines, it goes into a Substation Step Down Transformer for distribution. This is the Distribution portion of the diagram.
From there, it goes to the Subtransmission Customer (26kV and 69kV), the Primary Customer (13kV and 4kV), and the Secondary Customer (120V and 240 V).
Credit: U.S. Department of Energy (Public Domain)
Figure 6.3 Typical substation at a power plant (steps up voltage for transmission)
Figure 6.4 Typical substation that steps down voltage from transmission
The Power Grid is a simulation created by the Cyber Resilient Energy Delivery Consortium for education.
Access the Power Grid animation.
Examine the "Quick-Start Guide" as a helpful resource.
Play with the simulation to understand the relationships.
Figure 6.5 - The Raccoon Mountain project is TVA’s (Tennessee Valley Authority) largest hydroelectric facility. Water is pumped to the reservoir on top of the mountain and then used to generate electricity when additional power is needed by the TVA system. The Raccoon Mountain Pumped-Storage Plant is located in southeast Tennessee on a site that overlooks the Tennessee River near Chattanooga. The plant works like a large storage battery. During periods of low demand, water is pumped from Nickajack Reservoir at the base of the mountain to the reservoir built at the top. It takes 28 hours to fill the upper reservoir. When demand is high, water is released via a tunnel drilled through the center of the mountain to drive generators in the mountain’s underground power plant.
Primitive as it may seem, the energy storage technology that is "grid-tied" and having the largest capacity is accessed by simply pumping water up to a higher elevation, and storing it as potential energy. Called pumped storage, or pumped storage hydroelectricity, the energy is recovered when the water from the higher elevation is used to drive turbines for hydroelectric power conversion.
The Energy Storage Association reports, "Pumped storage hydropower can provide energy-balancing, stability, storage capacity, and ancillary grid services such as network frequency control and reserves." While the US has 20 GW of installed capacity, worldwide over 100 GW of capacity exist. The US figure accounts for roughly 2% of the country's generating capacity, while other areas' figures are as high as 10%.
All in all, however, this process uses more electricity than it produces. So, why do it? When a power plant has extra capacity, it generates electricity used to pump water uphill. Then, when the plant is stretched to capacity and electricity is at its highest price, this pumped storage can be used to generate low-cost hydroelectricity.
Modified from Vera Cole, Power Grid, EGEE 401. Accessed October 2013.
‹ 6.4 Discussion Activity 1 - Communicating Information to Your Client up 6.6 Power Grid Pricing and Capacity ›
|
Electromagnetic Induction, Popular Questions: CBSE Class 12-science SCIENCE, Science - Meritnation
\frac{3g}{4}
\frac{4g}{3}
\frac{2g}{3}
\frac{3g}{2}
\left(1\right) \frac{\sqrt{5}H}{4}\phantom{\rule{0ex}{0ex}}\left(2\right) \frac{\sqrt{3}H}{4}\phantom{\rule{0ex}{0ex}}\left(3\right) \frac{2H}{3}\phantom{\rule{0ex}{0ex}}\left(4\right) \frac{\sqrt{3}H}{2}
\left(1\right) \frac{B{v}_{0}L}{3r},\frac{2}{3},\frac{B{v}_{0}L}{3r},\frac{1}{3}\frac{B{v}_{0}L}{3r}\phantom{\rule{0ex}{0ex}}\left(2\right) \frac{B{v}_{0}L}{3r},\frac{1}{3},\frac{B{v}_{0}L}{3r},\frac{2}{3}\frac{B{v}_{0}L}{3r}\phantom{\rule{0ex}{0ex}}\left(3\right) \frac{B{v}_{0}L}{3r},\frac{1}{3},\frac{B{v}_{0}L}{3r},\frac{1}{3}\frac{B{v}_{0}L}{3r}\phantom{\rule{0ex}{0ex}}\left(4\right) \frac{B{v}_{0}L}{3r},\frac{B{v}_{0}L}{3r},\frac{B{v}_{0}L}{3r}
\omega
\left(1\right) \frac{1}{2}B\omega {l}^{2}\phantom{\rule{0ex}{0ex}}\left(2\right) B{\omega }^{2}\phantom{\rule{0ex}{0ex}}\left(3\right) \frac{3}{2}B\omega {l}^{2}\phantom{\rule{0ex}{0ex}}\left(4\right) Zero
What is the physical significance of parallel plate capacitor?
Box A in the set up below represents an electric device often used/needed to supply electric power from the AC mains to a load. It is known that V0 less than Vi.
a)Identify the device A and draw its symbol.
b) Find the relation between the input and output currents of this device assuming it to be ideal.
Q). The resultant amplitude of a vibrating particle by the superposition of the two waves
{y}_{1}=\mathrm{sin}\left(\omega t+\frac{\mathrm{\pi }}{3}\right) and {y}_{2}=a \mathrm{sin} \omega t
is:-
\sqrt{2}a
\sqrt{3}a
Please give detailed solution for the given question. If the circular conducting loop in the figure is contracting and field is uniform, induced current will be in which direction?
Tisha Grover asked a question
State and proof Reciprocity theorem
HOW IS EDDY CURRENT USEFUL IN ELECTRIC POWER METERS?
Write einstein photoelectric equation. Explain laws of photoelectric emission on the basis of this equation.
a square loop of side 'a' lying in a perpendicular magnetic field to its plane is changed to a circle. if change occurs in 't' seconds in magnetic field B tesla, the induced emf is?
Tanushree Bhanot asked a question
a closed coil of copper whose area is 1mX1m is free to rotate about an axis.The coil is packed perpendicular to a magnetic field of 0.10T.What is the charge flowing through the coil when the coil is rotated through 180 in 0.01 s
\frac{1}{2}B\omega {R}^{2}
\frac{3}{2}B\omega {R}^{2}
\frac{1}{4}B\omega {R}^{2}
Ankit Adil & 3 others asked a question
Show mathematically that potential at a point on the equatorial line of an electric dipole is zero
A storage battery of emf 12 V and internal resistance 0.5 Ω is to be charged by a 120 V d.c. supply of negligible internal resistance. What resistance is required in the circuit for the charging current to be 3 A ? What is the terminal voltage of the battery during charging?
Using Ampere circuital law derive an expression for the magnetic field along the axis of a torodial solenoid
Derive the expression for the self-inductance of a solenoid.
In a given circuit inductor L and resister R have identical resistance. Two similar electric lamps B1and B2are connected as shown. When switch S is closed,
which one of the lamps lights up earlier
Define one coulomb of charge.
The small ozone layer on the top of the stratosphere is crucial for human survival, why?
Why can’t a transformer be used to step up dc voltage?
What is the nature of the magnetic field in a moving coil galvanometer?
The current is set up in a long copper pipe. Is there a magnetic field
outside the pipe?
Sketch three equipotential surfaces for a pt. charge.
How is the coulomb force between two charges affected by the presence of the third charge?
What is the cause of induced emf?
dentify the two specimens X and Y
State the reason for the behaviour of the field lines in X and Y.
Give any one difference between FAX and e-mail systems of communication.
Give the ratio of the number of holes and number of conduction electrons in an intrinsic semiconductor.
How does focal length of a lens change when red light is replaced by blue light?
How does the intensity of central maximum change, if the width of the slit is halved in a single slit diffraction experiment?
How will the photoelectric current change on decreasing the wavelength of incident radiation for a given photosensitive material?
In which series of hydrogen spectra, the transitions involve the longest changes energy?
What is the mass of pion plus (p+)?
Vehicles moving in foggy weather use yellow colour head-lights. Why?
Name the physical quantity whose S.I. unit is :
Volt/ meter.
in faraday's experimental conclusion plzzz tell the direction of flow of current and deflection in galvanometer in all it's 5 conclusions????
34 a ferromagnetic
Matrix and Match
|
Table 1 Effects of NaHS on lung mitochondrial and cytosol Cyt-c protein expression in acute lung injury rats (
\overline{x}
±s, n = 8)
Cyt-c (mitochondria) (Grayscale scan value)
Cyt-c (cytosol) (Grayscale scan value)
β-actin (Grayscale scan value)
control group 178.51 96.992 171.47
LPS injury group 41.177** 293.45** 183.66
L 62.846# 226.91# 182.52
M 88.344## 203.38## 196.02
H 115.27## 111.84## 169.28
**P < 0.01,compared to the control group; # P < 0.05, ## P < 0.01, compared to the LPS injury group; L:LPS + low-dose NaHS group; M:LPS + middle-dose NaHS group;H: LPS + high-dose NaHS group.
The lung mitochondrial Cyt-c protein expression was significantly decreased in the LPS injury group compared to the control group (P < 0.01). In three LPS + NaHS groups, lung mitochondrial Cyt-c protein expression was markedly increased compared to the LPS injury group (P < 0.05 or P < 0.01). The cytosol Cyt-c protein expression was significantly increased in the LPS injury group compared to the control group (P < 0.01). In three LPS + NaHS groups, cytosol Cyt-c protein expression was markedly decreased compared to the LPS injury group (P < 0.05 or P < 0.01).
|
Key Modules - Collar Finance
The Collar core protocol consists of two key modules: Collar Core and Collar Swap. To help explain these modules, Collar's token terminology is discussed first.
Collar Tokens
In Collar, every token is grouped in a lending pool. We use this term to mean they share the same expiry and are associated to the same collateral asset (BOND)/lending asset (WANT) pair.
Expiry: Maturity date of a loan, which is specified in every CALL and COLL token.
CALL Token: The right for borrowers to repay their loan. Before expiry, CALL tokens can be used to unlock BOND tokens.
COLL Token: The right to the debt for lenders. After expiry, COLL tokens can be used to collect repaid WANT tokens and defaulted BOND tokens.
BOND Token: The collateral asset deposited by borrowers.
WANT Token: The asset provided by lenders.
COLLAR: The governance and utility token. In Collar, COLLAR not only acts as the voting power, but will also eventually capture earnings from the platform and the ultimate right to claim failed-to-withdraw BOND tokens (see the CIPs).
CALL and COLL Token Prices
CALL token price info:
holding a CALL token is like holding a call option for BOND tokens
CALL tokens function as insurance for the prices of both BOND tokens and WANT tokens
price of CALL token small compared to BOND/WANT prices (so close to $0 in USD stablecoin cases)
price of CALL token increases when WANT token de-pegs
COLL token price info:
holding a COLL token is like holding a put option for BOND tokens
at expiry, price of COLL token = price of WANT token if WANT token maintains its peg
price of COLL token before expiry acts as WANT token loan interest and should increase over time
View the specification for the Collar core protocol here. A summary of the operations that can be performed using the protocol is provided below for reference.
For each lending pool (group of tokens sharing the same expiry and associated to the same BOND/WANT pair), the Collar Core contains a Collar vault, which is a wallet that stores and ultimately distributes the deposited BOND tokens a repaid WANT tokens. The function of the Collar Core is to mint/burn CALL and COLL tokens whenever BOND and WANT tokens are received/sent by the vault.
MintDual
An account uses MintDual to deposit amount n of BOND tokens into a Collar vault, and then the account will mint both amount n of CALL tokens and amount n of COLL tokens. (before expiry of the lending pool)
MintColl
An account uses MintColl to deposit amount n of WANT tokens into a Collar vault, and then the account receives amount n of COLL tokens. (before expiry of the lending pool)
BurnDual
An account uses BurnDual to burn both amount n of CALL tokens and amount n of COLL tokens, and then the account receives amount n of BOND tokens from a Collar vault. (before expiry of the lending pool)
An account uses BurnCall to burn amount n of CALL tokens and deposits amount n of WANT tokens into a Collar vault, and then the account receives amount n of BOND tokens from the vault. (before expiry of the lending pool)
BurnColl
An account uses BurnColl to burn an amount n of COLL tokens, and then the account receives proportions of BOND tokens and WANT tokens from a Collar vault that are equal to the ratio of burned BOLL tokens to all COLL tokens in existence for that pool. (after expiry of the lending pool)
A loan in Collar is a trade between the lender's WANT tokens and the borrower's COLL tokens. This trading is facilitated by an AMM-based decentralized exchange called Collar Swap. The liquidity providers‘ incentive is a COLLAR token reward.
The AMM used in Collar Swap has an upgraded invariant curve. Since the trading involves tokens which derive their value from the Collar mechanism (which was designed specifically for pegged cryptoasset lending), the invariant was designed to satisfy the following requirements:
Reveals true sentiment of money markets, so that (COLL -WANT)/WANT price ratio represents the market interest rate.
Optimizes liquidity.
Defends against arbitrage trading that manipulates prices away from fair interest rate.
The invariant curve used in the Collar Swap AMM is inspired by Curve's StableSwap and is upgraded to better function as a stablecoin-derivatives liquidity provider. The invariant equation is
(x+K(x, y))(y+aK(x, y)) = (K(x,y))^2
where x is the amount of COLL, y is the amount of WANT,
0 \le a <1
K(x, y)
is a function of the total liquidity in the pool.
Comparison of example invariant curves for different AMMs
For reference, here is the calculation of the function
K(x, y)
function calc_k(uint256 x, uint256 y) public pure returns (uint256) {
uint256 a = swap_sqp();
uint256 ye9 = y * 1e9;
uint256 ax = a * x;
uint256 a_ = 1e9 - a;
uint256 D = (ax + ye9)**2 + 4 * x * ye9 * a_;
return (sqrt(D) + ye9 + ax) / (2 * a_);
function swap_sqp() public pure virtual returns (uint256) {
Next - Collar Core Protocol
|
A sufficient condition for existence of real analytic solutions of P.D.E. with constant coefficients, in open sets of $\mathbb {R}^2$
A sufficient condition for existence of real analytic solutions of P.D.E. with constant coefficients, in open sets of
{ℝ}^{2}
title = {A sufficient condition for existence of real analytic solutions of {P.D.E.} with constant coefficients, in open sets of $\mathbb {R}^2$},
TI - A sufficient condition for existence of real analytic solutions of P.D.E. with constant coefficients, in open sets of $\mathbb {R}^2$
Zampieri, Giuseppe. A sufficient condition for existence of real analytic solutions of P.D.E. with constant coefficients, in open sets of $\mathbb {R}^2$. Rendiconti del Seminario Matematico della Università di Padova, Tome 63 (1980), pp. 83-87. http://www.numdam.org/item/RSMUP_1980__63__83_0/
[1] K.G. Andersson, Global solvability of differential equations in the space of real analytic functions, Coll. on Analysis, Rio de Janeiro, 1972, Analyse fonctionelle, Hermann, 1974. | MR 390466 | Zbl 0307.35017
[2] K.G. Andersson, Propagation of analyticity of solutions of partial differential equations with constant coefficients, Ark. Mat., 8 (1971), pp. 277-302. | MR 299938 | Zbl 0211.40502
[3] L. Cattabriga, Sull'esistenza di soluzioni analitiche reali di equazioni a derivate parziali a coefficienti costanti, Boll. U.M.I. (4), 12 (1975), pp. 221-234. | MR 509089 | Zbl 0328.35009
[4] L. Hörmander, Linear partial differential operators, Springer, 1963. | MR 404822
[5] T. Kawai, On the global existence of real analytic solutions of linear differential equations (I), J. Math. Soc. Japan, 24 (1972), pp. 481-517. | MR 310412 | Zbl 0234.35012
[6] G. Zampieri, On some conjectures by E. De Giorgi relative to the global resolvability of overdetermined systems of differential equations, Rend. Sem. Mat. Univ. Padova 62 (1980). | Numdam | MR 582945 | Zbl 0452.35089
|
Genetic Algorithm Optimization for Determining Fuzzy Measures from Fuzzy Data
2013 Genetic Algorithm Optimization for Determining Fuzzy Measures from Fuzzy Data
Chen Li, Gong Zeng-tai, Duan Gang
Fuzzy measures and fuzzy integrals have been successfully used in many real applications. How to determine fuzzy measures is a very difficult problem in these applications. Though there have existed some methodologies for solving this problem, such as genetic algorithms, gradient descent algorithms, neural networks, and particle swarm algorithm, it is hard to say which one is more appropriate and more feasible. Each method has its advantages. Most of the existed works can only deal with the data consisting of classic numbers which may arise limitations in practical applications. It is not reasonable to assume that all data are real data before we elicit them from practical data. Sometimes, fuzzy data may exist, such as in pharmacological, financial and sociological applications. Thus, we make an attempt to determine a more generalized type of general fuzzy measures from fuzzy data by means of genetic algorithms and Choquet integrals. In this paper, we make the first effort to define the
\sigma \text{-}\lambda
rules. Furthermore we define and characterize the Choquet integrals of interval-valued functions and fuzzy-number-valued functions based on
\sigma \text{-}\lambda
rules. In addition, we design a special genetic algorithm to determine a type of general fuzzy measures from fuzzy data.
Chen Li. Gong Zeng-tai. Duan Gang. "Genetic Algorithm Optimization for Determining Fuzzy Measures from Fuzzy Data." J. Appl. Math. 2013 (SI07) 1 - 11, 2013. https://doi.org/10.1155/2013/542153
Chen Li, Gong Zeng-tai, Duan Gang "Genetic Algorithm Optimization for Determining Fuzzy Measures from Fuzzy Data," Journal of Applied Mathematics, J. Appl. Math. 2013(SI07), 1-11, (2013)
|
Ternary operation - Wikipedia
In mathematics, a ternary operation is an n-ary operation with n = 3. A ternary operation on a set A takes any given three elements of A and combines them to form a single element of A.
In computer science, a ternary operator is an operator that takes three arguments.[1]
Given A, B and point P, geometric construction yields V, the projective harmonic conjugate of P with respect to A and B.
{\displaystyle T(a,b,c)=ab+c}
is an example of a ternary operation on the integers (or on any structure where
{\displaystyle +}
{\displaystyle \times }
are both defined). Properties of this ternary operation have been used to define planar ternary rings in the foundations of projective geometry.
In the Euclidean plane with points a, b, c referred to an origin, the ternary operation
{\displaystyle [a,b,c]=a-b+c}
has been used to define free vectors.[2] Since (abc) = d implies a – b = c – d, these directed segments are equipollent and are associated with the same free vector. Any three points in the plane a, b, c thus determine a parallelogram with d at the fourth vertex.
In projective geometry, the process of finding a projective harmonic conjugate is a ternary operation on three points. In the diagram, points A, B and P determine point V, the harmonic conjugate of P with respect to A and B. Point R and the line through P can be selected arbitrarily, determining C and D. Drawing AC and BD produces the intersection Q, and RQ then yields V.
Suppose A and B are given sets and
{\displaystyle {\mathcal {B}}(A,B)}
is the collection of binary relations between A and B. Composition of relations is always defined when A = B, but otherwise a ternary composition can be defined by
{\displaystyle [p,q,r]=pq^{T}r}
{\displaystyle q^{T}}
is the converse relation of q. Properties of this ternary relation have been used to set the axioms for a heap.[3]
In Boolean algebra,
{\displaystyle T(A,B,C)=AC+(1-A)B}
defines the formula
{\displaystyle (A\lor B)\land (\lnot A\lor C)}
In computer science, a ternary operator is an operator that takes three arguments (or operands).[1] The arguments and result can be of different types. Many programming languages that use C-like syntax[4] feature a ternary operator, ?:, which defines a conditional expression. In some languages, this operator is referred to as the conditional operator.
In Python, the ternary conditional operator reads x if C else y. Python also supports ternary operations called array slicing, e.g. a[b:c] return an array where the first element is a[b] and last element is a[c-1].[5] OCaml expressions provide ternary operations against records, arrays, and strings: a.[b]<-c would mean the string a where index b has value c.[6]
The multiply–accumulate operation is another ternary operator.
Another example of a ternary operator is between, as used in SQL.
The Icon programming language has a "to-by" ternary operator: the expression 1 to 10 by 2 generates the odd integers from 1 through 9.
In Excel formulae, the form is =if(C, x, y).
?: for a list of ternary operators in computer programming languages
^ a b MDN, nmve. "Conditional (ternary) Operator". Mozilla Developer Network. MDN. Retrieved 20 February 2017.
^ Jeremiah Certaine (1943) The ternary operation (abc) = a b−1c of a group, Bulletin of the American Mathematical Society 49: 868–77 MR0009953
^ Hoffer, Alex. "Ternary Operator". Cprogramming.com. Cprogramming.com. Retrieved 20 February 2017.
^ "6. Expressions — Python 3.9.1 documentation". docs.python.org. Retrieved 2021-01-19.
^ "7.7 Expressions". caml.inria.fr. Retrieved 2021-01-19. {{cite web}}: CS1 maint: url-status (link)
Media related to Ternary operations at Wikimedia Commons
Retrieved from "https://en.wikipedia.org/w/index.php?title=Ternary_operation&oldid=1080793976"
|
Magnetic_diffusivity Knowpia
The magnetic diffusivity is a parameter in plasma physics which appears in the magnetic Reynolds number. It has SI units of m²/s and is defined as:[1]
{\displaystyle \eta ={\frac {1}{\mu _{0}\sigma _{0}}}}
while in Gaussian units it can be defined as
{\displaystyle \eta ={\frac {c^{2}}{4\pi \sigma _{0}}}}
{\displaystyle \mu _{0}}
{\displaystyle c}
{\displaystyle \sigma _{0}}
is the electrical conductivity of the material in question. In case of a plasma, this is the conductivity due to Coulomb or neutral collisions:
{\displaystyle \sigma _{0}={\frac {n_{e}e^{2}}{m_{e}\nu _{c}}}}
{\displaystyle n_{e}}
is the electron density.
{\displaystyle e}
is the electron charge.
{\displaystyle m_{e}}
{\displaystyle \nu _{c}}
is the collision frequency.
^ W. Baumjohann and R. A. Treumann, Basic Space Plasma Physics, Imperial College Press, 1997.
|
Generate Simulink filter block - MATLAB filt2block - MathWorks India
Generate Block from FIR Filter
Generate Block from IIR Filter
Generate FIR Filter with Direct Form I Transposed Structure
Generate IIR Filter with Direct Form I Structure
Generate Subsystem Block from Second-Order Section Matrix
Lowpass FIR Filter Block with Sample-Based Processing
New Model with Highpass Elliptic Filter Block
MapCoefficientsToPorts
OptimizeNegativeOnes
Generate Simulink filter block
filt2block(b)
filt2block(b,'subsystem')
filt2block(___,'FilterStructure',structure)
filt2block(b,a)
filt2block(b,a,'subsystem')
filt2block(sos)
filt2block(sos,'subsystem')
filt2block(d)
filt2block(d,'subsystem')
filt2block(___,Name,Value)
filt2block(b) generates a Discrete FIR Filter block with filter coefficients, b.
filt2block(b,'subsystem') generates a Simulink® subsystem block that implements an FIR filter using sum, gain, and delay blocks.
filt2block(___,'FilterStructure',structure) specifies the filter structure for the FIR filter.
filt2block(b,a) generates a Discrete Filter block with numerator coefficients, b, and denominator coefficients, a.
filt2block(b,a,'subsystem') generates a Simulink subsystem block that implements an IIR filter using sum, gain, and delay blocks.
filt2block(___,'FilterStructure',structure) specifies the filter structure for the IIR filter.
filt2block(sos) generates a Biquad Filter block with second order sections matrix, sos. sos is a K-by-6 matrix, where the number of sections, K, must be greater than or equal to 2. You must have the DSP System Toolbox™ software installed to use this syntax.
filt2block(sos,'subsystem') generates a Simulink subsystem block that implements a biquad filter using sum, gain, and delay blocks.
filt2block(___,'FilterStructure',structure) specifies the filter structure for the biquad filter.
filt2block(d) generates a Simulink block that implements a digital filter, d. Use the function designfilt to create d. The block is a Discrete FIR Filter block if d is FIR and a Biquad Filter block if d is IIR.
filt2block(d,'subsystem') generates a Simulink subsystem block that implements d using sum, gain, and delay blocks.
filt2block(___,'FilterStructure',structure) specifies the filter structure to implement d.
filt2block(___,Name,Value) uses additional options specified by one or more Name,Value pair arguments.
Design a 30th-order FIR filter using the window method. Specify a cutoff frequency of π/4 rad/sample. Create a Simulink® block.
Design a 30th-order IIR Butterworth filter. Specify a cutoff frequency of π/4 rad/sample. Create a Simulink® block.
[b,a] = butter(30,0.25);
Design a 30th-order FIR filter using the window method. Specify a cutoff frequency of π/4 rad/sample. Create a Simulink® block with a direct form I transposed structure.
filt2block(b,'FilterStructure','directFormTransposed')
Design a 30th-order IIR Butterworth filter. Specify a cutoff frequency of π/4 rad/sample. Create a Simulink® block with a direct form I structure.
filt2block(b,a,'FilterStructure','directForm1')
Design a 5-th order Butterworth filter with a cutoff frequency of π/5 rad/sample. Obtain the filter in biquad form and generate a Simulink® subsystem block from the second order sections.
Generate a Simulink® subsystem block that implements an FIR lowpass filter using sum, gain, and delay blocks. Specify the input processing to be elements as channels by specifying 'FrameBasedProcessing' as false.
B = fir1(30,.25);
filt2block(B,'subsystem','BlockName','Lowpass FIR',...
'FrameBasedProcessing',false)
Design a highpass elliptic filter with normalized stopband frequency 0.45 and normalized passband frequency 0.55. Specify a stopband attenuation of 40Design a highpass elliptic filter with normalized stopband frequency 0.45 and normalized passband frequency 0.55. Specify a stopband attenuation of 40 dB and a passband ripple of 0.5 dB. Implement the filter as a Direct Form II structure, call it "HP", and place it in a new Simulink® model.
d = designfilt('highpassiir','DesignMethod','ellip', ...
'StopbandFrequency',0.45,'PassbandFrequency',0.55, ...
'StopbandAttenuation',40,'PassbandRipple',0.5);
filt2block(d,'subsystem','FilterStructure','directForm2', ...
'Destination','new','BlockName','HP')
Numerator filter coefficients, specified as a row or column vector. The filter coefficients are ordered in descending powers of z–1 with the first element corresponding to the coefficient for z0.
Example: b = fir1(30,0.25);
Denominator filter coefficients, specified as a row or column vector. The filter coefficients are ordered in descending powers of z–1 with the first element corresponding to the coefficient for z0. The first filter coefficient must be 1.
sos — Second-order section matrix
Second order section matrix, specified as a K-by-2 matrix. Each row of the matrix contains the coefficients for a biquadratic rational function in z–1. The Z-transform of the Kth rational biquadratic system impulse response is
{H}_{k}\left(z\right)=\frac{{B}_{k}\left(1\right)+{B}_{k}\left(2\right){z}^{-1}+{B}_{k}\left(3\right){z}^{-2}}{{A}_{k}\left(1\right)+{A}_{k}\left(2\right){z}^{-1}+{A}_{k}\left(3\right){z}^{-2}}
The coefficients in the Kth row of the matrix, sos, are ordered as follows:
\left[\begin{array}{cccccc}{B}_{k}\left(1\right)& {B}_{k}\left(2\right)& {B}_{k}\left(3\right)& {A}_{k}\left(1\right)& {A}_{k}\left(2\right)& {A}_{k}\left(3\right)\end{array}\right]
The frequency response of the filter is its transfer function evaluated on the unit circle with z = ej2πf.
Filter structure, specified as a character vector or string scalar. Valid options for structure depend on the input arguments. The following table lists the valid filter structures by input.
b 'directForm' (default), 'directFormTransposed', 'directFormSymmetric', 'directFormAntiSymmetric', 'overlapAdd'. The 'overlapAdd' structure is only available when you omit 'subsystem' and requires a DSP System Toolbox software license.
a 'directForm2' (default), 'directForm1', 'directForm1Transposed', 'directForm2', 'directForm2Transposed'
sos 'directForm2Transposed' (default), 'directForm1', 'directForm1Transposed', 'directForm2'
For FIR filters: 'directForm' (default), 'directFormTransposed', 'directFormSymmetric', 'directFormAntiSymmetric', 'overlapAdd'. The 'overlapAdd' structure is only available when you omit 'subsystem' and requires a DSP System Toolbox software license.
For IIR filters: 'directForm2Transposed' (default), 'directForm1', 'directForm1Transposed', 'directForm2'
Example: filt2block(...,'subsystem','BlockName','Lowpass FIR','FrameBasedProcessing',false)
Destination — Destination for Simulink filter block
Destination for the Simulink filter block, specified as a character vector or string scalar. You can add the filter block to your current model with 'current', add the filter block to a new model with 'new', or specify the name of an existing model.
Example: filt2block([1 2 1],'Destination','MyModel','BlockName','New block')
BlockName — Block name
Block name, specified as a character vector or string scalar.
OverwriteBlock — Overwrite block
Overwrite block, specified as a logical false or true. If you use a value for 'BlockName' that is the same as an existing block, the value of 'OverwriteBlock' determines whether the block is overwritten. The default value is false.
MapCoefficientsToPorts — Map coefficients to ports
Map coefficients to ports, specified as a logical false or true.
CoefficientNames — Coefficient variable names
Coefficient variable names, specified as a cell array of character vectors or a string array. This name-value pair is only applicable when 'MapCoefficientsToPorts' is true. The default values are {'Num'}, {'Num','Den'}, and {'Num','Den','g'} for FIR, IIR, and biquad filters.
FrameBasedProcessing — Frame-based or sample-based processing
Frame-based or sample-based processing, specified as a logical true or false. The default is true and frame-based processing is used.
Remove zero-gain blocks, specified as a logical true or false. By default zero-gain blocks are removed.
OptimizeOnes — Replace unity-gain blocks with direct connection
Replace unity-gain blocks with direct connection, specified as a logical true or false. The default is true.
OptimizeNegativeOnes — Replace negative unity-gain blocks with sign change
Replace negative unity-gain blocks with a sign change at the nearest block, specified as a logical true or false. The default is true.
OptimizeDelayChains — Replace cascaded delays with a single delay
Replace cascaded delays with a single delay, specified as a logical true or false. The default is true.
|
Home > Course Outline > Lesson 2 - Tools for Time and Space Relationships > 2.2 Basic Solar Jargon for Energy and Power
We start by reviewing these small sections on language. There are things to measure and symbols for those metrics that we need to agree upon throughout the class.
SECS, Chapter 1, Introduction. Please pay particular attention to the final section: "Communication of Units and a Standard Solar Language." You may also download the original paper from Canvas ("Beckman_etal_1978.pdf").
While reading, consider the following points:
What is the difference between power density and energy density?
What is irradiance? What is the symbol for solar irradiance?
What is irradiation? What are the symbols for irradiation on hourly and daily steps?
Think about the angles that we use to describe spatial and time relationships in solar energy.
Units and symbols in solar energy [1] (Beckman et al., 1978). You can also access this article through Penn State's Electronic Course Reserves.
Solar Energy Journal [2] was established for the International Solar Energy Society (ISES) [3] and has been around for some time now. Solar Energy Journal stands as an important forum for peer to peer sharing of solar research for energy conversion and human applications of solar energy. What I want to establish here is that there is precedent for the complex system of notation used in the solar energy world that has been in use for decades. The original authors have established the following observations:
"Many disciplines are contributing to the literature on solar energy with the result that variations in definitions, symbols and units are appearing for the same terms. These conflicts cause difficulties in understanding which may be reduced by a systematic approach such as is attempted in this paper.
It is recognized that any list of preferred symbols and units will not be permanent nor can it be made mandatory, as new terms will emerge and old ones become less used with the development of the subject. But in the meantime, a list would be appreciated by the many workers who are entering this multi-disciplined field...
...Energy: The S.I. (Systèm International d'Unités) unit is the joule (
J=kg{m}^{2}{s}^{-2}
). The calorie and derivatives, such as the langley (cal cm-2), are not acceptable.
No distinction between the different forms of energy is made in the S.I. system so that mechanical, electrical and heat energy are all measured in joules. However, the watt-hour (Wh) will be used in many countries for commercial metering of electrical energy...
Power: The S.I. unit is the watt (
W=kg{m}^{2}{s}^{-3}=J/s
). The watt will be used to measure power or energy-rate for all forms of energy and should be used wherever instantaneous values of energy flow are involved. Thus, energy flux density will be expressed as W/m2 or specific thermal conductance as
W{m}^{-2}{K}^{-1}
. Energy-rate should not be expressed as $J/h$.
When energy-rate is integrated for a time period, the result is energy which should be expressed in joules, e.g. an energy-rate of 1.2kW would if maintained for one hour produce 4.3 MJ."
W. A. Beckman, et al.
Table 2.1: Preference for Expressing Energy Terms in the Solar Field
It is preferable to say
\text{Hourly energy}=4.3MJ
\text{Energy}=4.3MJ/h
\text{Daily energy}=104MJ
\text{Energy}=104MJ/\text{day}
In summary: received energy flux density (or power density, called irradiance) can be expressed in units of W/m2. We also note that the received radiative energy density (called irradiation) can be expressed in units of J/m2, or in units of Wh/m2. Notice that we did not use radiation, which is an expression of light glowing outward (emitted light, different direction than what we want).
In today's maps of the solar resource, you will often see the units expressed in kWh/m2. You should be aware that these are still only representations of solar light energy density, and not the hourly/daily/annual quantity of potential electricity that could be produced. To find that value, we need a simulation tool like SAM (System Advisor Model, which you should have downloaded at the end of Lesson 1), which takes irradiation data and converts it into power data.
I would like you to now take a short self-quiz to see if you recall the common uses of the notation and descriptions for solar energy (used in particular in this class.).
1. What is the word ending for energy density in light transfer?
-tion like in irradiation
2. What are the two acceptable units for energy density in received light transfer?
J/m2 or Wh/m2
3. What is the word ending for the power density metric in light transfer?
-ance like in irradiance
4. What are the acceptable units for the power density metric in received light transfer?
5. What is the correct way to express hourly irradiation?
You only express the energy density, not the energy density per hour. For example: "The irradiation for the noon hour was 1.4 MJ/m2, while the irradiation from 13h00 to 14h00 was 0.8 MJ/m2."
6. What is the prefix that one uses to express 'received light' transfer?
ir-; as in irradiance and irradiation, rather than radiant exitance, or radiance, or radiation
[1] http://www.sciencedirect.com/science/article/pii/0038092X78901184
[2] http://www.journals.elsevier.com/solar-energy/
[3] http://www.ises.org/home/
|
Motion in a planeContact Number: 9667591930 / 8527521718
Rain is falling vertically downwards with a speed of
4 \mathrm{km} {\mathrm{h}}^{-1}
. A girl moves a straight road with a velocity of
3 \mathrm{km} {\mathrm{h}}^{-1}
. The apparent velocity of rain with respect to the girl is
3 \mathrm{km} {\mathrm{h}}^{-1}
4 \mathrm{km} {\mathrm{h}}^{-1}
5 \mathrm{km} {\mathrm{h}}^{-1}
7 \mathrm{km} {\mathrm{h}}^{-1}
Ship A is travelling with a velocity of
5 \mathrm{km} {\mathrm{h}}^{-1}
due east. A second ship is heading
30°
east of north. What should be the speed of second ship if it is to remain always due north with respect to the first ship?
10 \mathrm{km} {\mathrm{h}}^{-1}
9 \mathrm{km} {\mathrm{h}}^{-1}
8 \mathrm{km} {\mathrm{h}}^{-1}
7 \mathrm{km} {\mathrm{h}}^{-1}
A man swims from a point A on one bank of a river of width 100m. When he swims perpendicular to the water current, he reaches the other bank 50 m downstream. The angle to the bank at which he should swim, to reach the directly opposite point B on the other bank is
10°
20°
30°
60°
A boat is moving with a velocity
3\stackrel{^}{\mathrm{i}}+4\stackrel{^}{\mathrm{j}}
with respect to ground. The water in the river is moving with a velocity
-3\stackrel{^}{\mathrm{i}}-4\stackrel{^}{\mathrm{j}}
with respect to ground. The relative velocity of the boat with respect to water is
8\stackrel{^}{\mathrm{j}}
-6\stackrel{^}{\mathrm{i}}-8\stackrel{^}{\mathrm{j}}
6\stackrel{^}{\mathrm{i}}+8\stackrel{^}{\mathrm{j}}
5\sqrt{2}
The ratio of the distance carried away by the water current, downstream, in crossing a river, by a person, making same angle with downstream and upstream is 2 : 1. The ratio of the speed of person to the water current cannot be less than
A particle is moving in a circle of radius r centred at O with constant speed v. What is the change in velocity in moving from A to B (
\angle \mathrm{AOB}=40°
2\mathrm{v} \mathrm{sin} 20°
4\mathrm{v} \mathrm{sin} 40°
2\mathrm{v} \mathrm{sin} 40°
\mathrm{v} \mathrm{sin} 20°
During a projectile motion, if the maximum height equals the horizontal range, then the angle of projection with the horizontal is
{\mathrm{tan}}^{-1} \left(1\right)
{\mathrm{tan}}^{-1} \left(2\right)
{\mathrm{tan}}^{-1} \left(3\right)
{\mathrm{tan}}^{-1} \left(4\right)
Two bullets are fired horizontally with different velocities from the same height. Which will reach the ground first?
1. Slower one
3. Both will reach simultaneously
A grasshopper can jump a maximum distance 1.6 m. It spends negligible time on the ground. How far can it go in 10s?
5\sqrt{2} \mathrm{m}
10\sqrt{2} \mathrm{m}
20\sqrt{2} \mathrm{m}
40\sqrt{2} \mathrm{m}
A projectile is thrown in the upward direction making an angle of
60°
with the horizontal direction with a velocity of
150 {\mathrm{ms}}^{-1}
. Then the time after which its inclination with the horizontal is
45°
15\left(\sqrt{3}-1\right) \mathrm{s}
15\left(\sqrt{3}+1\right) \mathrm{s}
7.5\left(\sqrt{3}-1\right) \mathrm{s}
7.5\left(\sqrt{3}+1\right) \mathrm{s}
A number of bullets are fired in all possible directions with the same initial velocity
\mathrm{u}
. The maximum area of ground covered by bullets is
\mathrm{\pi }{\left(\frac{{\mathrm{u}}^{2}}{\mathrm{g}}\right)}^{2}
\mathrm{\pi }{\left(\frac{{\mathrm{u}}^{2}}{2\mathrm{g}}\right)}^{2}
\mathrm{\pi }{\left(\frac{\mathrm{u}}{\mathrm{g}}\right)}^{2}
\mathrm{\pi }{\left(\frac{\mathrm{u}}{2\mathrm{g}}\right)}^{2}
A hose lying on the ground shoots a stream of water upward at an angle of
60°
to the horizontal with the velocity of
16 {\mathrm{ms}}^{-1}
. The height at which the water strikes the wall 8m away is
A rifle shoots a bullet with a muzzle velocity of 400 ms-1 at a small target 400 m away. The height above the target at which the bullet must be aimed to hit the target is (g=10 ms-2).
A projectile is given an initial velocity of
\stackrel{^}{\mathrm{i}}+2\stackrel{^}{\mathrm{j}}
. The cartesian equation of its path is (g=10 ms-2).
1. y = 2x - 5x2
2. y = x - 5x2
3. 4y = 2x - 5x2
4. y = 2x - 25x2
In Fig. 5.201, the time taken by the projectile to reach from A to B is t. Then the distance AB is equal to
\frac{\mathrm{ut}}{\sqrt{3}}
\frac{\sqrt{3}\mathrm{ut}}{2}
\sqrt{3}\mathrm{ut}
2\mathrm{ut}
A body is moving in a circle with a speed of 1 ms-1. This speed increases at a constant rate of 2 ms-1 every second. Assume that the radius of the circle described is 25 m. The total acceleration of the body after 2 s is
2 {\mathrm{ms}}^{-2}
25 {\mathrm{ms}}^{-2}
\sqrt{5} {\mathrm{ms}}^{-2}
\sqrt{7} {\mathrm{ms}}^{-2}
In the following question, it contains Statement I (assertion) and Statement II (reason). The question has four choices 1, 2, 3 and 4 out of which only one is correct.
Statement I: A body with constant acceleration always move along a straight line.
Statement II: A body with constant magnitude of acceleration may not speed up.
1. Statement I is true, Statement II is true; Statement II is the correct explanation for Statement I.
2. Statement I is true, Statement II is true; Statement II is not the correct explanation for Statement I.
3. Statement I is true, Statement II is false.
4. Statement I is false, Statement II is true.
A projectile is thrown with velocity v at an angle
\mathrm{\theta }
with the horizontal. When the projectile is at a height equal to half of the maximum height,
The vertical component of the velocity of projectile is
3\mathrm{v} \mathrm{sin} \mathrm{\theta }
\mathrm{v} \mathrm{sin} \mathrm{\theta }
\frac{\mathrm{v} \mathrm{sin} \mathrm{\theta }}{\sqrt{2}}
\frac{\mathrm{v} \mathrm{sin} \mathrm{\theta }}{\sqrt{3}}
\mathrm{\theta }
The velocity of the projectile when it is at a height equal to half of the maximum height is
\mathrm{v}\sqrt{{\mathrm{cos}}^{2} \mathrm{\theta }+\frac{{\mathrm{sin}}^{2} \mathrm{\theta }}{2}}
\sqrt{2} \mathrm{v} \mathrm{cos} \mathrm{\theta }
\sqrt{2} \mathrm{v} \mathrm{sin} \mathrm{\theta }
\mathrm{v} \mathrm{tan} \mathrm{\theta } \mathrm{sec} \mathrm{\theta }
\mathrm{\theta }
What is the angle of projectile with the vertical if the velocity at the highest point is
\sqrt{2/5}
times the velocity when it is at a height equal to half of the maximum height?
15°
30°
45°
60°
|
Cartesian Coordinate Systems >>
— Doctor Who from Meglos (1980)
What you are reading was originally a book. Nowadays, most learning about technical topics such as 3D math and video game programming is done online, so some readers may have a tough time visualizing what a technical book might have looked like. If this describes you, see Figure 1.
Figure 1Physical media was once used to communicate technical knowledge
It contained such anachronistic features as an ISBN number, an index, a bibliography, numbered chapters, sections, and pages, and strings of prose not broken up into 280-character chunks, such as the run-on-sentence you are now reading. This physical manifestation of knowledge weighed over 4 lbs, giving it many advantages over online form factors. It could be used to prop open a door or support a monitor at a desired height. You could collect physical books and display them on a shelf as an expression of your identity—real or aspired—and this worked whether or not you had actually read them! But the greatest advantage of a physical book was that you could tell at a glance how long it was. You are reading an 850 page book.
Also, this book was published in 2011. In 2011, League of Legends was two years old, Skyrim had just been released, and the PS4 was still two years away. But we're proud to say that most of the material in this book, like Bilbo when he still had The Ring, hasn't aged a day. Vectors and matrices work the same way,
F
still equals
ma
, and people still use Blinn-Phong.
However, some parts of the book are starting to showing their age a bit. The two least theoretical and most practical chapters (Chapters 10 and 12) in particular could benefit from an update and a shift of emphasis. We don't cite any publication or website younger than a decade. The jokes and cultural references—borderline to begin with—are now all dated. We'll be working on this; you can help by providing feedback.
This book is about 3D math, the geometry and algebra of 3D space. It is designed to teach you how to describe objects and their positions, orientations, and trajectories in 3D using mathematics. This is not a book about computer graphics, simulation, or even computational geometry, although if you plan on studying those subjects, you will definitely need the information here.
This is not just a book for video game programmers. We do assume that a majority of our readers are learning for the purpose of programming video games, but we expect a wider audience and we have designed the book with a diverse audience in mind. If you're a programmer or interested in learning how to make video games, welcome aboard! If you meet neither of these criteria, there's still plenty for you here. We have made every effort to make the book useful to designers and technical artists. Although there are several code snippets in the book, they are (hopefully) easy to read even for nonprogrammers. Most important, even though it is always necessary to understand the surrounding concepts to make sense of the code, the reverse is never true. We use code samples to illustrate how ideas can be implemented on a computer, not to explain the ideas themselves.
The title of this book says it is for “game development,” but a great deal of the material that we cover is applicable outside of video games. Practically anyone who wants to simulate, render, or understand a three-dimensional world will find this book useful. While we do try to provide motivating examples from the world of video game development, since that is our area of expertise and also our primary target audience, you won't be left out if the last game you completed was Space Quest.1 If your interests lie in more “grown up” things than video games, rest assured that this book is not filled with specific examples from video games about head-shots or severed limbs or how to get the blood spurt to look just right.
This book has many unique features, including its topic, approach, authors, and writing style.
Unique topic. This book fills a gap that has been left by other books on graphics, linear algebra, simulation, and programming. It's an introductory book, meaning we have focused our efforts on providing thorough coverage on fundamental 3D concepts—topics that are normally glossed over in a few quick pages or relegated to an appendix in other books (because, after all, you already know all this stuff). We have found that these very topics are often the sticking points for beginners! In a way, this book is the mirror image of gluing together books on graphics, physics, and curves. Whereas that mythical conglomeration would begin with a brief overview of the mathematical fundamentals, followed by in-depth coverage of the application area, we start with a thorough coverage of the math fundamentals, and then give compact, high-level overviews of the application areas.
This book does try to provide a graceful on-ramp for beginners, but that doesn't mean we'll be stuck in the slow lane forever. There is plenty of material here that is traditionally considered “advanced” and taught in upper-level or graduate classes. In reality, these topics are specialized more than they are difficult, and they have recently become important prerequisites that need to be taught earlier, which is part of what has driven the demand for a book like this.
Unique approach. All authors think that they strike the perfect balance between being pedantic and being chatty in order to best reach their audience, and we are no exception. We recognize, however, that the people who disagree with this glowing self-assessment will mostly find this book too informal (see the index entry for “stickler alert”). We have focused on perspicuous explanations and intuition, and sometimes we have done this at the expense of rigor. Our aim is to simplify, but not to oversimplify. We lead readers to the goal through a path that avoids the trolls and dragons, so why begin the journey by pointing them all out before we've even said what our destination is or why we're going there? However, since we know readers will be crossing the field on their own eventually, after we reach our goal we will turn around to point out where the dangers lie. But we may sometimes need to leave certain troll-slaying to another source, especially if we expect that your usual path won't take you near the danger. Those who intend to be on that land frequently should consult with a local for more intimate knowledge. This is not to say that we think rigor is unimportant; we just think it's easier to get rigor after intuition about the big picture has been established, rather than front-loading every discussion with definitions and axioms needed to handle the edge cases. Frankly, nowadays a reader can pursue concise and formal presentations free on Wikipedia or Wolfram MathWorld, so we don't think any book offers much worth paying for by dwelling excessively on definitions, axioms, proofs, and edge cases, especially for introductory material targeted primarily to engineers.
Unique authors. Our combined experience brings together academic authority with in-the-trenches practical advice. Fletcher Dunn has been making video games professionally since 1996. He worked at Terminal Reality in Dallas, where as principal programmer he was one of the architects of the Infernal engine and lead programmer on BloodRayne. He was a technical director for The Walt Disney Company at Wideload Games in Chicago and the lead programmer for Disney Guilty Party, IGN's E3 2010 Family Game of the Year. He now works for Valve Software in Bellevue, Washington and has contributed to Steam and all of Valve's recent games. He is the primary author of the GameNetworkingSockets networking library and the Steam Datagram Relay service. But his biggest claim to fame by far is as the namesake of Corporal Dunn from Call of Duty: Modern Warfare 2.
Dr. Ian Parberry has more than 35 years of experience in research and teaching in academia. This is his sixth book, his third on game programming. He is currently a tenured full professor in the Department of Computer Science & Engineering at the University of North Texas. He is nationally known as one of the pioneers of game programming in higher education, and has been teaching game programming classes at the University of North Texas continuously since 1993.
Unique writing style. We hope you will enjoy reading this math book (say what?) for two reasons. Most important, we want you to learn from this book, and learning something you are interested in is fun. Secondarily, we want you to enjoy reading this book in the same way that you enjoy reading a work of literature. We have no delusions that we're in the same class as Mark Twain, or that this book is destined to become a classic like, say, The Hitchhikers Guide to the Galaxy. But one can always have aspirations. Honestly, we are just silly people. At the same time, no writing style should stand in the way of the first priority: clear communication of mathematical knowledge about video games.2
We have tried to make the book accessible to as wide an audience as possible; no book, however, can go back all the way to first principles. We expect from the reader the following basic mathematical skills:
Manipulating algebraic expressions, fractions, and basic algebraic laws such as the associative and distributive laws and the quadratic equation.
Understanding what variables are, what a function is, how to graph a function, and so on.
Some very basic 2D Euclidian geometry, such as what a point is, what a line is, what it means for lines to be parallel and perpendicular, and so forth. Some basic formulas for area and circumference are used in a few places. It's OK if you have temporarily forgotten those—you will hopefully recognize them when you see them.
Some prior exposure to trigonometry is best. We give a brief review of trigonometry in the front of this book, but it is not presented with the same level of paced explanation found most elsewhere in this book.
Readers with some prior exposure to calculus will have an advantage, but we have restricted our use of calculus in this book to very basic principles, which we will (attempt to) teach in Chapter 11 for those without this training. Only the most high-level concepts and fundamental laws are needed.
Some programming knowledge is helpful, but not required. In several places, we give brief code snippets to show how the ideas being discussed get translated into code. (Also certain procedures are just easier to explain in code.) These snippets are extremely basic, well commented, and require only the most rudimentary understanding of C language syntax (which has been copied to several other languages). Most technical artists or level designers should be able to interpret these snippets with ease.
Chapter 1 gets warmed up with some groundwork that it is needed in the rest of the book and which you probably already know. It reviews the Cartesian coordinate system in 2D and 3D and discusses how to use the Cartesian coordinate system to locate points in space. Also included is a very quick refresher on trigonometry and summation notation.
Chapter 2 introduces vectors from a mathematical and geometric perspective and investigates the important relationship between points and vectors. It also discusses a number of vector operations, how to do them, what it means geometrically to do them, and situations for which you might find them useful.
Chapter 3 discusses examples of coordinate spaces and how they are nested in a hierarchy. It also introduces the central concepts of basis vectors and coordinate-space transformations.
Chapter 4 introduces matrices from a mathematical and geometric perspective and shows how matrices are a compact notation for the math behind linear transformations.
Chapter 5 surveys different types of linear transformations and their corresponding matrices in detail. It also discusses various ways to classify transformations.
Chapter 6 covers a few more interesting and useful properties of matrices, such as affine transforms and perspective projection, and explains the purpose and workings of four-dimensional vectors and matrices within a three-dimensional world.
Chapter 7 discusses how to use polar coordinates in 2D and 3D, why it is useful to do so, and how to convert between polar and Cartesian representations.
Chapter 8 discusses different techniques for representing orientation and angular displacement in 3D: Euler angles, rotation matrices, exponential maps, and quaternions. For each method, it explains how the method works and presents the advantages and disadvantages of the method and when its use is recommended. It also shows how to convert between different representations.
Chapter 9 surveys a number of commonly used geometric primitives and discusses how to represent and manipulate them mathematically.
Chapter 10 is a whirlwind lesson on graphics, touching on a few selected theoretical as well as modern practical issues. First, it presents a high-level overview of “how graphics works,” leading up to the rendering equation. The chapter then walks through a few theoretical topics of a mathematical nature. Next it discusses two contemporary topics that are often sources of mathematical difficulty and should be of particular interest to the reader: skeletal animation and bump mapping. Finally, the chapter presents an overview of the real-time graphics pipeline, demonstrating how the theories from the first half of the chapter are implemented in the context of current rendering hardware.
Chapter 11 crams two rather large topics into one chapter. It interleaves the highest-level topics from first-semester calculus with a discussion of rigid body kinematics—how to describe and analyze the motion of a rigid body without necessarily understanding its cause or being concerned with orientation or rotation.
Chapter 12 continues the discussion of rigid body mechanics. It starts with a condensed explanation of classical mechanics, including Newton's laws of motion and basic concepts such as inertia, mass, force, and momentum. It reviews a few basic force laws, such as gravity, springs, and friction. The chapter also considers the rotational analogs of all of the linear ideas discussed up to this point. Due attention is paid to the important topic of collisions. The chapter ends with a discussion of issues that arise when using a computer to simulate rigid bodies.
Chapter 13 explains parametric curves in 3D. The first half of the chapter explains how a relatively short curve is represented in some common, important forms: monomial, Bézier, and Hermite. The second half is concerned with fitting together these shorter pieces into a longer curve, called a spline. In understanding each system, the chapter considers what controls the system presents to a designer of curves, how to take a description of a curve made by a designer and recreate the curve, and how these controls can be used to construct a curve with specific properties.
Chapter 14 inspires the reader to pursue greatness in video games.
Appendix A is an assortment of useful tests that can be performed on geometric primitives. We intend it to be a helpful reference, but it can also make for interesting browsing.
Appendix B has all the answers.3
We calculated the odds that we could write an 800+ page math book free of mistakes. The result was a negative number, which we know can't be right, but is probably pretty close. If you find a bug in this book, or have feedback of any kind, send an email to feedback@gamemath.com or reach out to Fletcher on twitter (@ZPostFacto).
— Bill Watterson (1958–) from Calvin and Hobbes
Well, you may be left out of a few jokes, like that one. Sorry.
Which is why we've put most of the jokes and useless trivia in footnotes like this. Somehow, we felt like we could get away with more that way.
To the exercises, that is.
|
torch.trapezoid — PyTorch 1.11.0 documentation
torch.trapezoid
torch.trapezoid¶
torch.trapezoid(y, x=None, *, dx=None, dim=- 1) → Tensor¶
Computes the trapezoidal rule along dim. By default the spacing between elements is assumed to be 1, but dx can be used to specify a different constant spacing, and x can be used to specify arbitrary spacing along dim.
Assuming y is a one-dimensional tensor with elements
{y_0, y_1, ..., y_n}
, the default computation is
\begin{aligned} \sum_{i = 1}^{n-1} \frac{1}{2} (y_i + y_{i-1}) \end{aligned}
When dx is specified the computation becomes
\begin{aligned} \sum_{i = 1}^{n-1} \frac{\Delta x}{2} (y_i + y_{i-1}) \end{aligned}
effectively multiplying the result by dx. When x is specified, assuming x is also a one-dimensional tensor with elements
{x_0, x_1, ..., x_n}
, the computation becomes
\begin{aligned} \sum_{i = 1}^{n-1} \frac{(x_i - x_{i-1})}{2} (y_i + y_{i-1}) \end{aligned}
When x and y have the same size, the computation is as described above and no broadcasting is needed. The broadcasting behavior of this function is as follows when their sizes are different. For both x and y, the function computes the difference between consecutive elements along dimension dim. This effectively creates two tensors, x_diff and y_diff , that have the same shape as the original tensors except their lengths along the dimension dim is reduced by 1. After that, those two tensors are broadcast together to compute final output as part of the trapezoidal rule. See the examples below for details.
The trapezoidal rule is a technique for approximating the definite integral of a function by averaging its left and right Riemann sums. The approximation becomes more accurate as the resolution of the partition increases.
y (Tensor) – Values to use when computing the trapezoidal rule.
x (Tensor) – If specified, defines spacing between values as specified above.
dx (float) – constant spacing between values. If neither x or dx are specified then this defaults to 1. Effectively multiplies the result by its value.
dim (int) – The dimension along which to compute the trapezoidal rule. The last (inner-most) dimension by default.
>>> # Computes the trapezoidal rule in 1D, spacing is implicitly 1
>>> y = torch.tensor([1, 5, 10])
>>> torch.trapezoid(y)
tensor(10.5)
>>> # Computes the same trapezoidal rule directly to verify
>>> (1 + 10 + 10) / 2
>>> # Computes the trapezoidal rule in 1D with constant spacing of 2
>>> # NOTE: the result is the same as before, but multiplied by 2
>>> torch.trapezoid(y, dx=2)
>>> # Computes the trapezoidal rule in 1D with arbitrary spacing
>>> torch.trapezoid(y, x)
>>> ((3 - 1) * (1 + 5) + (6 - 3) * (5 + 10)) / 2
>>> # Computes the trapezoidal rule for each row of a 3x3 matrix
>>> y = torch.arange(9).reshape(3, 3)
tensor([ 2., 8., 14.])
>>> # Computes the trapezoidal rule for each column of the matrix
>>> torch.trapezoid(y, dim=0)
>>> # Computes the trapezoidal rule for each row of a 3x3 ones matrix
>>> # with the same arbitrary spacing
>>> # with different arbitrary spacing per row
>>> x = torch.tensor([[1, 2, 3], [1, 3, 5], [1, 4, 7]])
|
Rail fence cipher - Wikipedia
(Redirected from Rail Fence Cipher)
Type of transposition cipher
"Rail fence" redirects here. For the actual fence, see split-rail fence.
Find sources: "Rail fence cipher" – news · newspapers · books · scholar · JSTOR (January 2017) (Learn how and when to remove this template message)
The rail fence cipher (also called a zigzag cipher) is a classical type of transposition cipher. It derives its name from the manner in which encryption is performed.
4 Zigzag cipher
In the rail fence cipher, the plaintext is written downwards diagonally on successive "rails" of an imaginary fence, then moving up when the bottom rail is reached, down again when the top rail is reached, and so on until the whole plaintext is written out. The ciphertext is then read off in rows.
For example, to encrypt the message 'WE ARE DISCOVERED. RUN AT ONCE.' with 3 "rails", write the text as:
(Note that spaces and punctuation are omitted.) Then read off the text horizontally to get the ciphertext:
WECRUO ERDSOEERNTNE AIVDAC
{\displaystyle N}
be the number of rails used during encryption. Observe that as the plaintext is written, the sequence of each letter's vertical position on the rails varies up and down in a repeating cycle. In the above example (where
{\displaystyle N=3}
) the vertical position repeats with a period of 4. In general the sequence repeats with a period of
{\displaystyle 2(N-1)}
{\displaystyle L}
be the length of the string to be decrypted. Suppose for a moment that
{\displaystyle L}
{\displaystyle 2(N-1)}
{\displaystyle K={L \over {2(N-1)}}}
. One begins by splitting the ciphertext into strings such that the length of the first and last string is
{\displaystyle K}
and the length of each intermediate string is
{\displaystyle 2K}
. For the above example with
{\displaystyle L=24}
{\displaystyle K=6}
, so we split the ciphertext as follows:
Write each string on a separate line with spaces after each letter in the first and last line:
W E C R U O
E R D S O E E R N T N E
A I V D A C
Then one can read off the plaintext down the first column, diagonally up, down the next column, and so on.
{\displaystyle L}
{\displaystyle 2(N-1)}
, the determination of how to split up the ciphertext is slightly more complicated than as described above, but the basic approach is the same. Alternatively, for simplicity in decrypting, one can pad the plaintext with extra letters to make its length a multiple of
{\displaystyle 2(N-1)}
If the ciphertext has not been padded, but you either know or are willing to brute-force the number of rails used, you can decrypt it using the following steps.
As above, let
{\displaystyle L}
be the length of the string to be decrypted and let
{\displaystyle N}
be the number of rails used during encryption. We will add two variables,
{\displaystyle x}
{\displaystyle y}
{\displaystyle x+1}
= the number of diagonals in the decrypted Rail Fence, and
{\displaystyle y}
= the number of empty spaces in the last diagonal.
{\displaystyle 1={\frac {L+y}{N+((N-1)*x)}}}
Next solve for
{\displaystyle x}
{\displaystyle y}
algebraically, where both values are the smallest number possible. This is easily done by incrementing
{\displaystyle x}
by 1 until the denominator is larger than
{\displaystyle L}
, and then simply solving for
{\displaystyle y}
. Consider the example cipher, modified to use 6 rails instead of 3.
W.........V.........O
.E.......O.E.......T.N
..A.....C...R.....A...C
...R...S.....E...N.....E
....E.I.......D.U.......
.....D.........R........
The resulting cipher text is:
WVO EOETN ACRAC RSENE EIDU DR
{\displaystyle L=24}
, and if we use
{\displaystyle N=6}
we can solve the equation above.
{\displaystyle 1={\frac {24+y}{6+(5*x)}}}
{\displaystyle 1={\frac {18+y}{5*x}}}
{\displaystyle 1={\frac {18+y}{5*4}}}
{\displaystyle x}
{\displaystyle 1={\frac {18+2}{20}}}
{\displaystyle y}
{\displaystyle N=6}
{\displaystyle x=4}
{\displaystyle y=2}
. Or, 6 rails, 5 diagonals (4+1), and 2 empty spaces at the end. By blocking out the empty spaces at the end of the last diagonal, we can simply fill in the Rail Fence line by line using the ciphertext.
_ _ _ _ _
_ _ _ _ _
_ _ _ _ X
_ _ X
W V O
E O E T N
A C R A C
The cipher's key is
{\displaystyle N}
, the number of rails. If
{\displaystyle N}
is known, the ciphertext can be decrypted by using the above algorithm. Values of
{\displaystyle N}
equal to or greater than
{\displaystyle L}
, the length of the ciphertext, are not usable, since then the ciphertext is the same as the plaintext. Therefore the number of usable keys is low, allowing the brute-force attack of trying all possible keys. As a result, the rail-fence cipher is considered weak.[citation needed]
Zigzag cipher[edit]
The term zigzag cipher may refer to the rail fence cipher as described above. However, it may also refer to a different type of cipher described by Fletcher Pratt in Secret and Urgent. It is "written by ruling a sheet of paper in vertical columns, with a letter at the head of each column. A dot is made for each letter of the message in the proper column, reading from top to bottom of the sheet. The letters at the head of the columns are then cut off, the ruling erased and the message of dots sent along to the recipient, who, knowing the width of the columns and the arrangement of the letters at the top, reconstitutes the diagram and reads what it has to say."[1]
^ Pratt, Fletcher (1939). Secret and Urgent: The story of codes and ciphers. Aegean Park Press. pp. 143–144. ISBN 0-89412-261-4.
Helen Fouché Gaines, Cryptanalysis, a study of ciphers and their solution, Dover, 1956, ISBN 0-486-20097-3
Black Chamber page for encrypting and decrypting the Rail Fence cipher
Rumkin page for encrypting and decrypting the Rail Fence cipher
Cryptii page for encrypting and decypting the Rail Fence cipher
Retrieved from "https://en.wikipedia.org/w/index.php?title=Rail_fence_cipher&oldid=1081234945"
|
Compare accuracies of two classification models by repeated cross-validation - MATLAB testckfold - MathWorks Switzerland
{e}_{crk}
{\stackrel{^}{\delta }}_{rk}={e}_{1rk}-{e}_{2rk}.
{\overline{\delta }}_{r}=\frac{1}{K}\sum _{k=1}^{K}{\stackrel{^}{\delta }}_{kr}.
\overline{\delta }=\frac{1}{KR}\sum _{r=1}^{R}\sum _{k=1}^{K}{\stackrel{^}{\delta }}_{rk}.
{s}_{r}^{2}=\frac{1}{K}\sum _{k=1}^{K}{\left({\stackrel{^}{\delta }}_{rk}-{\overline{\delta }}_{r}\right)}^{2}.
{\overline{s}}^{2}=\frac{1}{R}\sum _{r=1}^{R}{s}_{r}^{2}.
{S}^{2}=\frac{1}{KR-1}\sum _{r=1}^{R}\sum _{k=1}^{K}{\left({\stackrel{^}{\delta }}_{rk}-\overline{\delta }\right)}^{2}.
{t}_{paired}^{\ast }=\frac{{\stackrel{^}{\delta }}_{11}}{\sqrt{{\overline{s}}^{2}}}.
{t}_{paired}^{\ast }
{\stackrel{^}{\delta }}_{11}
\overline{\delta }
{F}_{paired}^{\ast }=\frac{\frac{1}{RK}\sum _{r=1}^{R}\sum _{k=1}^{K}{\left({\stackrel{^}{\delta }}_{rk}\right)}^{2}}{{\overline{s}}^{2}}.
{F}_{paired}^{\ast }
{t}_{CV}^{\ast }=\frac{\overline{\delta }}{S/\sqrt{\nu +1}}.
{t}_{CV}^{\ast }
{\stackrel{^}{p}}_{1j}
{e}_{1}=\frac{\sum _{j=1}^{{n}_{test}}{w}_{j}\mathrm{log}\left(1+\mathrm{exp}\left(-2{y}_{j}^{\prime }f\left({X}_{j}\right)\right)\right)}{\sum _{j=1}^{{n}_{test}}{w}_{j}}
f\left({X}_{j}\right)
{e}_{1}=\frac{\sum _{j=1}^{{n}_{test}}{w}_{j}\mathrm{exp}\left(-{y}_{j}f\left({X}_{j}\right)\right)}{\sum _{j=1}^{{n}_{test}}{w}_{j}}.
f\left({X}_{j}\right)
{e}_{1}=\frac{\sum _{j=1}^{n}{w}_{j}\mathrm{max}\left\{0,1-{y}_{j}\prime f\left({X}_{j}\right)\right\}}{\sum _{j=1}^{n}{w}_{j}},
f\left({X}_{j}\right)
{e}_{1}=\frac{\sum _{j=1}^{{n}_{test}}{w}_{j}I\left({\stackrel{^}{p}}_{1j}\ne {y}_{j}\right)}{\sum _{j=1}^{{n}_{test}}{w}_{j}}.
|
Controller-driven bidirectional DC-DC step-up and step-down voltage regulator - MATLAB - MathWorks Deutschland
Switching device HV
Forward voltage HV
Drain-source on resistance HV
On-state resistance HV
Off-state conductance HV
Threshold voltage HV
Gate trigger voltage HV, Vgt_hv
Gate turn-off voltage HV, Vgt_off_hv
Holding current HV
Switching device LV
Forward voltage LV
Drain-source on resistance LV
On-state resistance LV
Off-state conductance LV
Threshold voltage LV
Gate trigger voltage LV, Vgt_lv
Gate turn-off voltage LV, Vgt_off_lv
Holding current LV
Transformer inductance L1
Transformer coefficient of coupling
Capacitance, C1
C1 effective series resistance
Snubber HV
Snubber resistance HV
Snubber capacitance HV
Snubber LV
Snubber resistance LV
Snubber capacitance LV
Controller-driven bidirectional DC-DC step-up and step-down voltage regulator
The Bidirectional DC-DC Converter block represents a converter that steps up or steps down DC voltage from either side of the converter to the other as driven by an attached controller and gate-signal generator. Bidirectional DC-DC converters are useful for switching between energy storage and use, for example, in electric vehicles.
The Bidirectional DC-DC Converter block allows you to model a nonisolated converter with two switching devices, an isolated converter with six switching devices, or a dual active bridge converter with eight switching devices. Options for the type of switching devices are:
You can model three different types of bidirectional DC-DC converter. To access the different options, double-click the block and set the Modeling option parameter to either:
Nonisolated converter — Bidirectional DC-DC converter without an electrical barrier. This converter contains an inductor, two capacitors, and two switches that are of the same device type.
Isolated converter — Bidirectional DC-DC converter with an electrical barrier. This converter contains four additional switches that form a full bridge. The full bridge is on the input or high-voltage (HV) side of the converter. The other two switches are on the output or low-voltage (LV) side of the converter. You can select different semiconductor types for the HV and LV switching devices. For example, you can use a GTO for the HV switching devices and an IGBT for the LV switching devices. To provide separation between the input and output voltages, the model uses a high-frequency transformer.
Dual Active Bridge converter — This bidirectional DC-DC converter contains two full-bridges. The left bridge is the input or high-voltage (HV) side of the converter. The right bridge is the output or low-voltage (LV) side of the converter. You can select different semiconductor types for the HV and LV switching devices. For example, you can use a GTO for the HV switching devices and an IGBT for the LV switching devices. To provide separation between the input and output voltages, the model uses a high-frequency transformer.
The block contains an integral protection diode for each switching device. The integral diode protects the semiconductor device by providing a conduction path for reverse current. An inductive load can produce a high reverse-voltage spike when the semiconductor device suddenly switches off the voltage supply to the load.
To configure the internal protection diode block, use the Protection Diode parameters. This table shows how to set the Model dynamics parameter based on your goals.
To connect Simulink® gate-control voltage signals to the gate ports of the switching devices:
Multiplex the converted gate signals into a single vector.
For a nonisolated converter model, use a Two-Pulse Gate Multiplexer block.
For an isolated converter model, use a Six-Pulse Gate Multiplexer block.
For a dual active bridge converter model, use a Eight-Pulse Gate Multiplexer block.
A source impedance or a nonzero equivalent-series resistance (ESR) is connected to the left side of the Bidirectional DC-DC Converter block.
Modeling option — Converter model
Nonisolated converter (default) | Isolated converter | Dual Active Bridge converter
Whether to model a nonisolated converter with two switching devices, an isolated converter with six switching devices, or a dual active bridge converter with eight switching devices.
These tables show how the visibility of Switching Devices parameters depends on the converter model and switching devices that you select. To learn how to read the table, see Parameter Dependencies.
Nonisolated Converter Switching Devices Parameter Dependencies
Isolated Converter and Dual Active Bridge Converter Switching Devices Parameter Dependencies
On-state resistance HV Forward voltage HV Forward voltage HV Drain-source on resistance HV Forward voltage HV On-state resistance HV
Off-state conductance HV On-state resistance HV On-state resistance HV Off-state conductance HV On-state resistance HV
Threshold voltage HV Off-state conductance HV Off-state conductance HV Threshold voltage HV Off-state conductance HV
Gate trigger voltage HV, Vgt_hv Threshold voltage HV Gate trigger voltage HV, Vgt_hv
Gate turn-off voltage HV, Vgt_off_hv Gate turn-off voltage HV, Vgt_off_hv
Holding current HV Holding current HV
On-state resistance LV Forward voltage LV Forward voltage LV Drain-source on resistance LV Forward voltage LV On-state resistance LV
Off-state conductance LV On-state resistance LV On-state resistance LV Off-state conductance LV On-state resistance LV
Threshold voltage LV Off-state conductance LV Off-state conductance LV Threshold voltage LV Off-state conductance LV
Gate trigger voltage LV, Vgt Threshold voltage LV Gate trigger voltage LV, Vgt
Gate turn-off voltage LV, Vgt_off_lv Gate turn-off voltage LV, Vgt_off_lv
Holding current LV Holding current LV
Switching device type for the nonisolated converter model.
See the Nonisolated Converter Switching Devices Parameter Dependencies table.
Switching device HV — Switch
Switching device type for the high-voltage side of the isolated converter model.
See the Isolated Converter and Dual Active Bridge Converter Switching Devices Parameter Dependencies table.
Forward voltage HV — Voltage
0.8 Ohm (default) | scalar
For the different switching device types, the Forward voltage HV is taken as:
Drain-source on resistance HV — Resistance
On-state resistance HV — Resistance
For the different switching device types, the On-state resistance HV is taken as:
Off-state conductance HV — Conductance
Conductance when the device is off. The value must be less than 1/R, where R is the value of On-state resistance HV.
Threshold voltage HV — Voltage threshold
Gate trigger voltage HV, Vgt_hv — Voltage threshold
Gate turn-off voltage HV, Vgt_off_hv — Voltage threshold
Holding current HV — Current threshold
Switching device LV — Switch
Switching device type for the low-voltage side of the isolated converter model.
Forward voltage LV — Voltage
For the different switching device types, the Forward voltage LV is taken as:
Drain-source on resistance LV — Resistance
On-state resistance LV — Resistance
For the different switching device types, the On-state resistance LV is taken as:
Off-state conductance LV — Conductance
Conductance when the device is off. The value must be less than 1/R, where R is the value of On-state resistance LV.
Threshold voltage LV — Voltage threshold
Gate trigger voltage LV, Vgt_lv — Voltage threshold
Gate turn-off voltage LV, Vgt_off_lv — Voltage threshold
Holding current LV — Current threshold
The visibility of Protection Diode parameters depends on how you configure the protection diode Model dynamics and Reverse recovery time parameterization parameters. To learn how to read this table, see Parameter Dependencies.
Protection Diode Parameter Dependencies
See the Protection Diode Parameter Dependencies table.
Model for parameterizing the recovery time. When you select Specify stretch factor or Specify reverse recovery charge, you can specify a value that the block uses to derive the reverse recovery time. For more information on these options, see How the Block Calculates TM and Tau.
-\frac{{i}^{2}{}_{RM}}{2a},
To enable these parameters, set Modeling option to Isolated converter.
Transformer inductance L1 — Inductance
10 H (default) | positive scalar
Self-inductance of the first winding of the transformer.
To enable this parameter, set Modeling option to Isolated converter.
0.1 H (default) | positive scalar
Self-inductance of the second winding of the transformer.
Transformer coefficient of coupling — Coupling coefficient
0.9 (default) | positive scalar greater than zero and less than 1
Defines the mutual inductance of the transformer.
Inductance, L — Inductance
Converter inductance. If you set the Modeling option parameter to Isolated converter, the two inductors are identical.
Capacitance, C1 — Capacitance
Capacitance of the first DC terminal.
Capacitance of the second DC terminal.
C1 effective series resistance — Resistance
Series resistance of capacitor C1.
Nonisolated Converter Isolated Converter
Snubber Snubber HV
None RC Snubber None RC Snubber
Snubber resistance Snubber resistance HV
Snubber capacitance Snubber capacitance HV
Snubber for each switching device.
Resistance of the snubbers.
1e-7 F (default) | scalar
Capacitance of the snubbers.
Snubber HV — Snubber model
HV snubber for each switching device.
Snubber resistance HV — Resistance
Resistance of the high-voltage snubbers.
Snubber capacitance HV — Capacitance
Capacitance of the high-voltage snubbers.
Snubber LV — Snubber model
LV snubber for each switching device.
Snubber resistance LV — Resistance
Resistance of the low-voltage snubbers.
Snubber capacitance LV — Capacitance
Capacitance of the low-voltage snubbers.
[1] Saleh, M., Y. Esa, Y. Mhandi, W. Brandauer, and A. Mohamed. Design and implementation of CCNY DC microgrid testbed. Industry Applications Society Annual Meeting. Portland, OR: 2016, pp 1-7.
[2] Kutkut, N. H., and G. Luckjiff. Current mode control of a full bridge DC-to-DC converter with a two inductor rectifier. Power Electronics Specialists Conference. Saint Louis, MO: 1997, pp 203-209.
[3] Nene, H. Digital control of a bi-directional DC-DC converter for automotive applications. Twenty-Eighth Annual IEEE Applied Power Electronics Conference and Exposition (APEC). Long Beach, CA: 2013, pp 1360-1365.
Average-Value DC-DC Converter | Buck Converter | Buck-Boost Converter | Boost Converter | Converter (Three-Phase) | GTO | IGBT (Ideal, Switching) | MOSFET (Ideal, Switching) | Ideal Semiconductor Switch | PWM Generator | PWM Generator (Three-phase, Two-level) | Six-Pulse Gate Multiplexer | Three-Level Converter (Three-Phase) | Thyristor (Piecewise Linear)
|
torch.cov — PyTorch 1.11.0 documentation
torch.cov
torch.cov¶
torch.cov(input, *, correction=1, fweights=None, aweights=None) → Tensor¶
A covariance matrix is a square matrix giving the covariance of each pair of variables. The diagonal contains the variance of each variable (covariance of a variable with itself). By definition, if input represents a single variable (Scalar or 1D) then its variance is returned.
The unbiased sample covariance of the variables
x
y
\text{cov}_w(x,y) = \frac{\sum^{N}_{i = 1}(x_{i} - \bar{x})(y_{i} - \bar{y})}{N~-~1}
\bar{x}
\bar{y}
are the simple means of the
x
y
If fweights and/or aweights are provided, the unbiased weighted covariance is calculated, which is given by:
\text{cov}_w(x,y) = \frac{\sum^{N}_{i = 1}w_i(x_{i} - \mu_x^*)(y_{i} - \mu_y^*)}{\sum^{N}_{i = 1}w_i~-~1}
w
denotes fweights or aweights based on whichever is provided, or
w = fweights \times aweights
if both are provided, and
\mu_x^* = \frac{\sum^{N}_{i = 1}w_ix_{i} }{\sum^{N}_{i = 1}w_i}
is the weighted mean of the variable.
correction (int, optional) – difference between the sample size and sample degrees of freedom. Defaults to Bessel’s correction, correction = 1 which returns the unbiased estimate, even if both fweights and aweights are specified. correction = 0 will return the simple average. Defaults to 1.
fweights (tensor, optional) – A Scalar or 1D tensor of observation vector frequencies representing the number of times each observation should be repeated. Its numel must equal the number of columns of input. Must have integral dtype. Ignored if None. Defaults to ``None` .
aweights (tensor, optional) – A Scalar or 1D array of observation vector weights. These relative weights are typically large for observations considered “important” and smaller for observations considered less “important”. Its numel must equal the number of columns of input. Must have floating point dtype. Ignored if None. Defaults to ``None` .
(Tensor) The covariance matrix of the variables.
torch.corrcoef() normalized covariance matrix.
>>> x = torch.tensor([[0, 2], [1, 1], [2, 0]]).T
>>> torch.cov(x)
>>> torch.cov(x, correction=0)
>>> fw = torch.randint(1, 10, (3,))
>>> fw
>>> aw = torch.rand(3)
>>> torch.cov(x, fweights=fw, aweights=aw)
|
Cast coefficients of digital filter to single precision - MATLAB single - MathWorks Italia
Lowpass FIR Filter in Double and Single Precision
Cast coefficients of digital filter to single precision
f2 = single(f1)
f2 = single(f1) casts coefficients in a digital filter, f1, to single precision and returns a new digital filter, f2, that contains these coefficients. This is the only way that you can create single-precision digitalFilter objects.
0.2\pi
0.55\pi
rad/sample. Cast the filter coefficients to single precision.
classd = class(d.Coefficients)
classd =
classe = class(e.Coefficients)
f1 — Digital filter
Digital filter, specified as a digitalFilter object. Use designfilt to generate f1 based on frequency-response specifications.
Single-precision digital filter, returned as a digitalFilter object.
designfilt | digitalFilter | double | isdouble | issingle
|
EUDML | Synthesis of fixed-architecture, robust and controllers. EuDML | Synthesis of fixed-architecture, robust and controllers.
Synthesis of fixed-architecture, robust
{H}_{2}
{H}_{\infty }
controllers.
Collins, Emmanuel G.; Sadhukhan, Debashis
Collins, Emmanuel G., and Sadhukhan, Debashis. "Synthesis of fixed-architecture, robust and controllers.." Mathematical Problems in Engineering 6.2-3 (2000): 125-144. <http://eudml.org/doc/49483>.
author = {Collins, Emmanuel G., Sadhukhan, Debashis},
keywords = {controller synthesis; Popov multiplier; homotopy algorithms; robust controllers; performance; performance; performance; performance},
title = {Synthesis of fixed-architecture, robust and controllers.},
AU - Collins, Emmanuel G.
AU - Sadhukhan, Debashis
TI - Synthesis of fixed-architecture, robust and controllers.
KW - controller synthesis; Popov multiplier; homotopy algorithms; robust controllers; performance; performance; performance; performance
controller synthesis, Popov multiplier, homotopy algorithms, robust controllers,
{H}_{\infty }
{H}_{2}
{H}_{\infty }
{H}_{2}
{H}^{\infty }
Articles by Collins
Articles by Sadhukhan
|
RMSprop — PyTorch 1.11.0 documentation
class torch.optim.RMSprop(params, lr=0.01, alpha=0.99, eps=1e-08, weight_decay=0, momentum=0, centered=False)[source]¶
\begin{aligned} &\rule{110mm}{0.4pt} \\ &\textbf{input} : \alpha \text{ (alpha)},\: \gamma \text{ (lr)}, \: \theta_0 \text{ (params)}, \: f(\theta) \text{ (objective)} \\ &\hspace{13mm} \lambda \text{ (weight decay)},\: \mu \text{ (momentum)},\: centered\\ &\textbf{initialize} : v_0 \leftarrow 0 \text{ (square average)}, \: \textbf{b}_0 \leftarrow 0 \text{ (buffer)}, \: g^{ave}_0 \leftarrow 0 \\[-1.ex] &\rule{110mm}{0.4pt} \\ &\textbf{for} \: t=1 \: \textbf{to} \: \ldots \: \textbf{do} \\ &\hspace{5mm}g_t \leftarrow \nabla_{\theta} f_t (\theta_{t-1}) \\ &\hspace{5mm}if \: \lambda \neq 0 \\ &\hspace{10mm} g_t \leftarrow g_t + \lambda \theta_{t-1} \\ &\hspace{5mm}v_t \leftarrow \alpha v_{t-1} + (1 - \alpha) g^2_t \hspace{8mm} \\ &\hspace{5mm} \tilde{v_t} \leftarrow v_t \\ &\hspace{5mm}if \: centered \\ &\hspace{10mm} g^{ave}_t \leftarrow g^{ave}_{t-1} \alpha + (1-\alpha) g_t \\ &\hspace{10mm} \tilde{v_t} \leftarrow \tilde{v_t} - \big(g^{ave}_{t} \big)^2 \\ &\hspace{5mm}if \: \mu > 0 \\ &\hspace{10mm} \textbf{b}_t\leftarrow \mu \textbf{b}_{t-1} + g_t/ \big(\sqrt{\tilde{v_t}} + \epsilon \big) \\ &\hspace{10mm} \theta_t \leftarrow \theta_{t-1} - \gamma \textbf{b}_t \\ &\hspace{5mm} else \\ &\hspace{10mm}\theta_t \leftarrow \theta_{t-1} - \gamma g_t/ \big(\sqrt{\tilde{v_t}} + \epsilon \big) \hspace{3mm} \\ &\rule{110mm}{0.4pt} \\[-1.ex] &\bf{return} \: \theta_t \\[-1.ex] &\rule{110mm}{0.4pt} \\[-1.ex] \end{aligned}
For further details regarding the algorithm we refer to lecture notes by G. Hinton. and centered version Generating Sequences With Recurrent Neural Networks. The implementation here takes the square root of the gradient average before adding epsilon (note that TensorFlow interchanges these two operations). The effective learning rate is thus
\gamma/(\sqrt{v} + \epsilon)
\gamma
is the scheduled learning rate and
v
is the weighted moving average of the squared gradient.
|
Heredity and Evolution - Revision Notes
Mendel's Experiments : Mendel conducted a series of experiments inwhich he crossed the pollinated plants to study one character (at a time)
\left(\begin{array}{l}TT\text{ }:\text{ }Tt\text{ }:\text{ }tt\\ \phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}1\text{ }:\text{ }2\text{ }\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}:\text{ }1\end{array}\right)
3. Characters/Traits like 'T' are called dominant trait (because it express itself) and ‘t’ are recessive trait (because it remains suppressed)
\underset{Generation}{Parent}\to \underset{Green\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}seeds}{Round}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}×\underset{Yellow\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}seeds}{Wrinkled}
Situation - I
\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}↓
\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}↓
\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}↓
\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}↓
\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}↓
1.These are the traits which are developed in an individual due to special conditions
1. These are the traits which are passed from one generation to the next.
They cannot be transferred to the progeny
2. They get transferred to the progeny.
They cannot direct evolutioneg. Low weight of starving beetles
3. They are helpful in evolution.eg. Colour of eyes and hair
↓
Sub population
{Z}_{1}
{Z}_{{2}_{}}
↓
↓
Forelimb of Horse
Same basic structural plan, but different functions perform.
Winds of bat
Paw of a cat
(walk/scratch/attack)
Wings of bat
\stackrel{}{\to }
Elongated fingers with skin folds
Different basic structure, but perform similar function i.e., flight.
Wings of bird
\stackrel{}{\to }
Feathery covering along the arm
AMMONITE - Fossil-invertebrate
TRILOBITE - Fossil-invertebrate
KNIGHTIA - Fossil-fish
RAJASAURUS - Fossil dinosaur skull
x\phantom{\rule{thickmathspace}{0ex}}rightarrow
Insects have compound eyes
x\phantom{\rule{thickmathspace}{0ex}}rightarrow
Humans have binocular eyes
Variations arising during the process of reproduction can be inherited.
These variations may lead to increased survival of the individuals.
Sexually reproducing individuals have two copies of genes for the same trait. If the copies are not identical, the trait that gets expressed is called the dominant trait and the other is called the recessive trait.
Traits in one individual may be inherited separately, giving rise to new combinations of traits in the offspring of sexual reproduction.
Sex is determined by different factors in various species. In human beings, the sex of the child depends on whether the paternal chromosome is X (for girls) or Y (for boys).
Variations in the species may confer survival advantages or merely contribute to the genetic drift.
Changes in the non-reproductive tissues caused by environmental factors are not inheritable.
Speciation may take place when variation is combined with geographical isolation.
Evolutionary relationships are traced in the classification of organisms.
Tracing common ancestors back in time leads us to the idea that at some point of time, non-living material must have given rise to life.
Evolution can be worked out by the study of not just living species, but also fossils.
Complex organs may have evolved because of the survival advantage of even the intermediate stages.
Organs or features may be adapted to new functions during the course of evolution.
For example, feathers are thought to have been initially evolved for warmth and later adapted for flight.
Evolution cannot be said to ‘progress’ from ‘lower’ forms to ‘higher’ forms. Rather, evolution seems to have given rise to more complex body designs even while the simpler body designs continue to flourish.
Study of the evolution of human beings indicates that all of us belong to a single species that evolved in Africa and spread across the world in stages.
|
Power of a source of 2mW and wavelength 400nm light is directed on a photoelectric cell If the current in - Physics - Semiconductor Electronics Materials Devices And Simple Circuits - 16912209 | Meritnation.com
Power of a source of 2mW and wavelength 400nm light is directed on a photoelectric cell.If the current in the cell is 0.40 microampere,then the percentage of the incident photons that produce photoelectrons is
Please solve this question and explain it.
P =2mW =2×{10}^{-3}W\phantom{\rule{0ex}{0ex}}Energy of each photon =\frac{hc}{\lambda }=\frac{6.62×{10}^{-34}×3×{10}^{8}}{4×{10}^{-7}}\phantom{\rule{0ex}{0ex}}=4.96×{10}^{-19}J\phantom{\rule{0ex}{0ex}}Number of photons incident per second =\frac{P}{Energy of each photon}\phantom{\rule{0ex}{0ex}}=\frac{2×{10}^{-3}}{4.96×{10}^{-19}} \simeq 4×{10}^{15}\phantom{\rule{0ex}{0ex}} Let n percentage of the incident photons that produce photoelectrons \phantom{\rule{0ex}{0ex}}The number of photo electrons produced\phantom{\rule{0ex}{0ex}}=n%×4×{10}^{15} \phantom{\rule{0ex}{0ex}}Current i =n%×4×{10}^{15}×1.6×{10}^{-19}\phantom{\rule{0ex}{0ex}}0.40×{10}^{-6} =n%×4×{10}^{15}×1.6×{10}^{-19}\phantom{\rule{0ex}{0ex}}n% =\frac{4×{10}^{-7}}{4×{10}^{15}×1.6×{10}^{-19}}=0.625×{10}^{-3}\phantom{\rule{0ex}{0ex}}or n =0.06%\phantom{\rule{0ex}{0ex}}
|
Multiple Plots - Maple Help
Home : Support : Online Help : Graphics : 2-D : Multiple Plots
plot({f1, f2,...}, h, v, options)
plot([f1, f2,...], h, v, options)
plot3d([f1, f2,...], h, v, options)
plot3d({f1, f2,...}, h, v, options)
functions to be plotted
vertical range, optional for plot
any of the optional arguments specified in plot/options
When plot or plot3d is passed a set or list of functions, it plots all of these functions on the same graph.
When a list of three functions is passed to plot3d, Maple displays a 3-D parametric plot. For more information, see plot3d. To avoid this, add the three functions as a set.
When a list of functions is plotted, options may be associated with each function. Options are specified as option=[value1, value2, ..., valuen] with one value for each curve plotted.
Unless otherwise specified, the curves are colored using the current list of plotting colors (see plots[setcolors]).
\mathrm{plot}\left([x-\frac{{x}^{3}}{3},\mathrm{sin}\left(x\right)],x=0..1,\mathrm{linestyle}=[\mathrm{dot},\mathrm{dash}]\right)
\mathrm{plot}\left({\mathrm{exp},\mathrm{ln}},-\mathrm{\infty }..\mathrm{\infty },\mathrm{color}=["DarkGreen","CornflowerBlue"]\right)
\mathrm{plot}\left({x,[{t}^{2},t,t=-1..1]},x=0..1\right)
\mathrm{plot3d}\left([\mathrm{cos}\left(x\right)-2\mathrm{cos}\left(0.4y\right),\mathrm{sin}\left(xy\right)],x=-10..10,y=-1..1,\mathrm{style}=[\mathrm{PATCH},\mathrm{PATCHNOGRID}],\mathrm{shading}=[\mathrm{DEFAULT},\mathrm{ZGRAYSCALE}],\mathrm{lightmodel}=\mathrm{light1}\right)
|
Injective_object Knowpia
In mathematics, especially in the field of category theory, the concept of injective object is a generalization of the concept of injective module. This concept is important in cohomology, in homotopy theory and in the theory of model categories. The dual notion is that of a projective object.
An object Q is injective if, given a monomorphism f : X → Y, any g : X → Q can be extended to Y.
{\displaystyle Q}
in a category
{\displaystyle \mathbf {C} }
is said to be injective if for every monomorphism
{\displaystyle f:X\to Y}
{\displaystyle g:X\to Q}
there exists a morphism
{\displaystyle h:Y\to Q}
extendin{\displaystyle g}
{\displaystyle Y}
{\displaystyle h\circ f=g}
That is, every morphism
{\displaystyle X\to Q}
factors through every monomorphism
{\displaystyle X\hookrightarrow Y}
The morphism
{\displaystyle h}
in the above definition is not required to be uniquely determined by
{\displaystyle f}
{\displaystyle g}
In a locally small category, it is equivalent to require that the hom functor
{\displaystyle \operatorname {Hom} _{\mathbf {C} }(-,Q)}
carries monomorphisms in
{\displaystyle \mathbf {C} }
to surjective set maps.
In Abelian categoriesEdit
The notion of injectivity was first formulated for abelian categories, and this is still one of its primary areas of application. When
{\displaystyle \mathbf {C} }
is an abelian category, an object Q of
{\displaystyle \mathbf {C} }
is injective if and only if its hom functor HomC(–,Q) is exact.
{\displaystyle 0\to Q\to U\to V\to 0}
is an exact sequence in
{\displaystyle \mathbf {C} }
such that Q is injective, then the sequence splits.
Enough injectives and injective hullsEdit
{\displaystyle \mathbf {C} }
is said to have enough injectives if for every object X of
{\displaystyle \mathbf {C} }
, there exists a monomorphism from X to an injective object.
A monomorphism g in
{\displaystyle \mathbf {C} }
is called an essential monomorphism if for any morphism f, the composite fg is a monomorphism only if f is a monomorphism.
If g is an essential monomorphism with domain X and an injective codomain G, then G is called an injective hull of X. The injective hull is then uniquely determined by X up to a non-canonical isomorphism.
In the category of abelian groups and group homomorphisms, Ab, an injective object is necessarily a divisible group. Assuming the axiom of choice, the notions are equivalent.
In the category of (left) modules and module homomorphisms, R-Mod, an injective object is an injective module. R-Mod has injective hulls (as a consequence, R-Mod has enough injectives).
In the category of metric spaces, Met, an injective object is an injective metric space, and the injective hull of a metric space is its tight span.
In the category of T0 spaces and continuous mappings, an injective object is always a Scott topology on a continuous lattice, and therefore it is always sober and locally compact.
If an abelian category has enough injectives, we can form injective resolutions, i.e. for a given object X we can form a long exact sequence
{\displaystyle 0\to X\to Q^{0}\to Q^{1}\to Q^{2}\to \cdots }
and one can then define the derived functors of a given functor F by applying F to this sequence and computing the homology of the resulting (not necessarily exact) sequence. This approach is used to define Ext, and Tor functors and also the various cohomology theories in group theory, algebraic topology and algebraic geometry. The categories being used are typically functor categories or categories of sheaves of OX modules over some ringed space (X, OX) or, more generally, any Grothendieck category.
An object Q is H-injective if, given h : A → B in H, any f : A → Q factors through h.
{\displaystyle \mathbf {C} }
be a category and let
{\displaystyle {\mathcal {H}}}
be a class of morphisms of
{\displaystyle \mathbf {C} }
{\displaystyle Q}
{\displaystyle \mathbf {C} }
{\displaystyle {\mathcal {H}}}
-injective if for every morphism
{\displaystyle f:A\to Q}
{\displaystyle h:A\to B}
{\displaystyle {\mathcal {H}}}
{\displaystyle g:B\to Q}
{\displaystyle g\circ h=f}
{\displaystyle {\mathcal {H}}}
is the class of monomorphisms, we are back to the injective objects that were treated above.
{\displaystyle \mathbf {C} }
is said to have enough
{\displaystyle {\mathcal {H}}}
-injectives if for every object X of
{\displaystyle \mathbf {C} }
{\displaystyle {\mathcal {H}}}
-morphism from X to an
{\displaystyle {\mathcal {H}}}
-injective object.
{\displaystyle {\mathcal {H}}}
-morphism g in
{\displaystyle \mathbf {C} }
{\displaystyle {\mathcal {H}}}
-essential if for any morphism f, the composite fg is in
{\displaystyle {\mathcal {H}}}
only if f is in
{\displaystyle {\mathcal {H}}}
If g is a
{\displaystyle {\mathcal {H}}}
-essential morphism with domain X and an
{\displaystyle {\mathcal {H}}}
-injective codomain G, then G is called an
{\displaystyle {\mathcal {H}}}
-injective hull of X.
Examples of H-injective objectsEdit
In the category of simplicial sets, the injective objects with respect to the class
{\displaystyle {\mathcal {H}}}
of anodyne extensions are Kan complexes.
In the category of partially ordered sets and monotone maps, the complete lattices form the injective objects for the class
{\displaystyle {\mathcal {H}}}
of order-embeddings, and the Dedekind–MacNeille completion of a partially ordered set is its
{\displaystyle {\mathcal {H}}}
-injective hull.
|
Common Fixed Point Theorems in Modified Intuitionistic Fuzzy Metric Spaces
2013 Common Fixed Point Theorems in Modified Intuitionistic Fuzzy Metric Spaces
Saurabh Manro, Sanjay Kumar, S. S. Bhatia, Kenan Tas
This paper consists of main two sections. In the first section, we prove a common fixed point theorem in modified intuitionistic fuzzy metric space by combining the ideas of pointwise
R
-weak commutativity and reciprocal continuity of mappings satisfying contractive conditions. In the second section, we prove common fixed point theorems in modified intuitionistic fuzzy metric space from the class of compatible continuous mappings to noncompatible and discontinuous mappings. Lastly, as an application, we prove fixed point theorems using weakly reciprocally continuous noncompatible self-mappings on modified intuitionistic fuzzy metric space satisfying some implicit relations.
Erratum to “Common Fixed Point Theorems in Modified Intuitionistic Fuzzy Metric Spaces”dx.doi.org/10.1155/2014/486102
Saurabh Manro. Sanjay Kumar. S. S. Bhatia. Kenan Tas. "Common Fixed Point Theorems in Modified Intuitionistic Fuzzy Metric Spaces." J. Appl. Math. 2013 1 - 13, 2013. https://doi.org/10.1155/2013/189321
Saurabh Manro, Sanjay Kumar, S. S. Bhatia, Kenan Tas "Common Fixed Point Theorems in Modified Intuitionistic Fuzzy Metric Spaces," Journal of Applied Mathematics, J. Appl. Math. 2013(none), 1-13, (2013)
|
Galois points on quartic surfaces
July, 2001 Galois points on quartic surfaces
S
be a smooth hypersurface in the projective three space and consider a projection of
S
P\in S
to a plane
H
. This projection induces an extension of fields
k\left(S\right)/k\left(H\right)
P
is called a Galois point if the extension is Galois. We study structures of quartic surfaces focusing on Galois points. We will show that the number of the Galois points is zero, one, two, four or eight and the existence of some rule of distribution of the Galois points.
Hisao YOSHIHARA. "Galois points on quartic surfaces." J. Math. Soc. Japan 53 (3) 731 - 743, July, 2001. https://doi.org/10.2969/jmsj/05330731
Keywords: elliptic surface , Galois point , Projective transformation , Quartic surface
Hisao YOSHIHARA "Galois points on quartic surfaces," Journal of the Mathematical Society of Japan, J. Math. Soc. Japan 53(3), 731-743, (July, 2001)
|
Bertie placed a transparent grid made up of unit squares over each of the shapes she was measuring below. Using her grid, approximate the area of each region.
Approximate the area by counting the squares. Count squares completely filled as one unit squared, and those only partially filled as one-half a unit squared.
Count filled squares (a):
10
Count partially filled squares (a):
10
10(1\text{ unit})+10\left ( \frac{1}{2} \text{ units}\right)=15\text{ units}^2
|
Honeycomb_(geometry) Knowpia
In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions. Its dimension can be clarified as n-honeycomb for a honeycomb of n-dimensional space.
Honeycombs are usually constructed in ordinary Euclidean ("flat") space. They may also be constructed in non-Euclidean spaces, such as hyperbolic honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.
It is possible to fill the plane with polygons which do not meet at their corners, for example using rectangles, as in a brick wall pattern: this is not a proper tiling because corners lie part way along the edge of a neighbouring polygon. Similarly, in a proper honeycomb, there must be no edges or vertices lying part way along the face of a neighbouring cell. Interpreting each brick face as a hexagon having two interior angles of 180 degrees allows the pattern to be considered as a proper tiling. However, not all geometers accept such hexagons.
There are infinitely many honeycombs, which have only been partially classified. The more regular ones have attracted the most interest, while a rich and varied assortment of others continue to be discovered.
The simplest honeycombs to build are formed from stacked layers or slabs of prisms based on some tessellations of the plane. In particular, for every parallelepiped, copies can fill space, with the cubic honeycomb being special because it is the only regular honeycomb in ordinary (Euclidean) space. Another interesting family is the Hill tetrahedra and their generalizations, which can also tile the space.
Uniform 3-honeycombsEdit
A 3-dimensional uniform honeycomb is a honeycomb in 3-space composed of uniform polyhedral cells, and having all vertices the same (i.e., the group of [isometries of 3-space that preserve the tiling] is transitive on vertices). There are 28 convex examples in Euclidean 3-space,[1] also called the Archimedean honeycombs.
A honeycomb is called regular if the group of isometries preserving the tiling acts transitively on flags, where a flag is a vertex lying on an edge lying on a face lying on a cell. Every regular honeycomb is automatically uniform. However, there is just one regular honeycomb in Euclidean 3-space, the cubic honeycomb. Two are quasiregular (made from two types of regular cells):
Regular cubic honeycomb
Quasiregular honeycombs
The tetrahedral-octahedral honeycomb and gyrated tetrahedral-octahedral honeycombs are generated by 3 or 2 positions of slab layer of cells, each alternating tetrahedra and octahedra. An infinite number of unique honeycombs can be created by higher order of patterns of repeating these slab layers.
A honeycomb having all cells identical within its symmetries is said to be cell-transitive or isochoric. In the 3-dimensional euclidean space, a cell of such a honeycomb is said to be a space-filling polyhedron.[2] A necessary condition for a polyhedron to be a space-filling polyhedron is that its Dehn invariant must be zero,[3][4] ruling out any of the Platonic solids other than the cube.
Five space-filling polyhedra can tessellate 3-dimensional euclidean space using translations only. They are called parallelohedra:
Cubic honeycomb (or variations: cuboid, rhombic hexahedron or parallelepiped)
Hexagonal prismatic honeycomb[5]
Elongated dodecahedral honeycomb[6]
Bitruncated cubic honeycomb or truncated octahedra[7]
Elongated dodecahedra
(parallelepiped)
3 edge-lengths
3+1 edge-lengths
Other known examples of space-filling polyhedra include:
The triangular prismatic honeycomb
The gyrated triangular prismatic honeycomb
The triakis truncated tetrahedral honeycomb. The Voronoi cells of the carbon atoms in diamond are this shape.[8]
The trapezo-rhombic dodecahedral honeycomb[9]
Isohedral tilings[10]
Other honeycombs with two or more polyhedraEdit
Sometimes, two [11] or more different polyhedra may be combined to fill space. Besides many of the uniform honeycombs, another well known example is the Weaire–Phelan structure, adopted from the structure of clathrate hydrate crystals [12]
The periodic unit of the Weaire–Phelan structure.
A honeycomb by left and right-handed versions of the same polyhedron.
Non-convex 3-honeycombsEdit
Documented examples are rare. Two classes can be distinguished:
Non-convex cells which pack without overlapping, analogous to tilings of concave polygons. These include a packing of the small stellated rhombic dodecahedron, as in the Yoshimoto Cube.
Overlapping of cells whose positive and negative densities 'cancel out' to form a uniformly dense continuum, analogous to overlapping tilings of the plane.
Hyperbolic honeycombsEdit
In 3-dimensional hyperbolic space, the dihedral angle of a polyhedron depends on its size. The regular hyperbolic honeycombs thus include two with four or five dodecahedra meeting at each edge; their dihedral angles thus are π/2 and 2π/5, both of which are less than that of a Euclidean dodecahedron. Apart from this effect, the hyperbolic honeycombs obey the same topological constraints as Euclidean honeycombs and polychora.
The 4 compact and 11 paracompact regular hyperbolic honeycombs and many compact and paracompact uniform hyperbolic honeycombs have been enumerated.
11 paracompact regular honeycombs
Duality of 3-honeycombsEdit
For every honeycomb there is a dual honeycomb, which may be obtained by exchanging:
cells for vertices.
faces for edges.
These are just the rules for dualising four-dimensional 4-polytopes, except that the usual finite method of reciprocation about a concentric hypersphere can run into problems.
The more regular honeycombs dualise neatly:
The cubic honeycomb is self-dual.
That of octahedra and tetrahedra is dual to that of rhombic dodecahedra.
The slab honeycombs derived from uniform plane tilings are dual to each other in the same way that the tilings are.
The duals of the remaining Archimedean honeycombs are all cell-transitive and have been described by Inchbald.[13]
Self-dual honeycombsEdit
Honeycombs can also be self-dual. All n-dimensional hypercubic honeycombs with Schläfli symbols {4,3n−2,4}, are self-dual.
Wikimedia Commons has media related to Honeycombs (geometry).
Regular honeycombs
^ Grünbaum (1994). "Uniform tilings of 3-space". Geombinatorics 4(2)
^ Weisstein, Eric W. "Space-filling polyhedron". MathWorld.
^ Lagarias, J. C.; Moews, D. (1995), "Polytopes that fill
{\displaystyle \mathbb {R} ^{n}}
and scissors congruence", Discrete and Computational Geometry, 13 (3–4): 573–583, doi:10.1007/BF02574064, MR 1318797 .
^ [1] Uniform space-filling using triangular, square, and hexagonal prisms
^ [2] Uniform space-filling using only rhombo-hexagonal dodecahedra
^ [3] Uniform space-filling using only truncated octahedra
^ John Conway (2003-12-22). "Voronoi Polyhedron. geometry.puzzles". Newsgroup: geometry.puzzles. Usenet: Pine.LNX.4.44.0312221226380.25139-100000@fine318a.math.Princeton.EDU.
^ X. Qian, D. Strahs and T. Schlick, J. Comput. Chem. 22(15) 1843–1850 (2001)
^ [4] O. Delgado-Friedrichs and M. O'Keeffe. Isohedral simple tilings: binodal and by tiles with <16 faces. Acta Crystallogr. (2005) A61, 358-362
^ [5] Archived 2015-06-30 at the Wayback Machine Gabbrielli, Ruggero. A thirteen-sided polyhedron which fills space with its chiral copy.
^ Pauling, Linus. The Nature of the Chemical Bond. Cornell University Press, 1960
^ Inchbald, Guy (July 1997), "The Archimedean honeycomb duals", The Mathematical Gazette, 81 (491): 213–219, doi:10.2307/3619198, JSTOR 3619198 .
Coxeter, H. S. M.: Regular Polytopes.
Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. pp. 164–199. ISBN 0-486-23729-X. Chapter 5: Polyhedra packing and space filling
Critchlow, K.: Order in space.
Pearce, P.: Structure in nature is a strategy for design.
Goldberg, Michael Three Infinite Families of Tetrahedral Space-Fillers Journal of Combinatorial Theory A, 16, pp. 348–354, 1974.
Goldberg, Michael (1972). "The space-filling pentahedra". Journal of Combinatorial Theory, Series A. 13 (3): 437–443. doi:10.1016/0097-3165(72)90077-5.
Goldberg, Michael The Space-filling Pentahedra II, Journal of Combinatorial Theory 17 (1974), 375–378.
Goldberg, Michael (1977). "On the space-filling hexahedra". Geometriae Dedicata. 6. doi:10.1007/BF00181585. S2CID 189889869.
Goldberg, Michael (1978). "On the space-filling heptahedra". Geometriae Dedicata. 7 (2): 175–184. doi:10.1007/BF00181630. S2CID 120562040.
Goldberg, Michael Convex Polyhedral Space-Fillers of More than Twelve Faces. Geom. Dedicata 8, 491-500, 1979.
Goldberg, Michael (1981). "On the space-filling octahedra". Geometriae Dedicata. 10 (1–4): 323–335. doi:10.1007/BF01447431. S2CID 189876836.
Goldberg, Michael (1982). "On the Space-filling Decahedra". {{cite journal}}: Cite journal requires |journal= (help)
Goldberg, Michael (1982). "On the space-filling enneahedra". Geometriae Dedicata. 12 (3). doi:10.1007/BF00147314. S2CID 120914105.
Olshevsky, George. "Honeycomb". Glossary for Hyperspace. Archived from the original on 4 February 2007.
Five space-filling polyhedra, Guy Inchbald, The Mathematical Gazette 80, November 1996, p.p. 466-475.
Raumfueller (Space filling polyhedra) by T.E. Dorozinski
Weisstein, Eric W. "Space-Filling Polyhedron". MathWorld.
{\displaystyle {\tilde {A}}_{n-1}}
{\displaystyle {\tilde {C}}_{n-1}}
{\displaystyle {\tilde {B}}_{n-1}}
{\displaystyle {\tilde {D}}_{n-1}}
{\displaystyle {\tilde {G}}_{2}}
{\displaystyle {\tilde {F}}_{4}}
{\displaystyle {\tilde {E}}_{n-1}}
|
dodecahedron - Maple Help
Home : Support : Online Help : Graphics : Packages : Plot Tools : dodecahedron
generate 3-D plot object for a dodecahedron
dodecahedron([x, y, z], s, options)
location of the dodecahedron
(optional) scale of the dodecahedron; default is 1
The dodecahedron command creates a three-dimensional plot data object, which when displayed is a scaled dodecahedron located at point [x, y, z]. Note that the scale factor is applied in each dimension. This command is an interface to the plots[polyhedraplot] routine.
The plot data object produced by the dodecahedron command can be used in a PLOT3D data structure, or displayed using the plots[display] command.
\mathrm{with}\left(\mathrm{plottools}\right):
\mathrm{with}\left(\mathrm{plots}\right):
\mathrm{display}\left(\mathrm{dodecahedron}\left([0,0,0],0.8\right),\mathrm{lightmodel}=\mathrm{light2},\mathrm{shading}=\mathrm{xy}\right)
\mathrm{display}\left(\mathrm{dodecahedron}\left([0,0,0],0.8\right),\mathrm{dodecahedron}\left([1,1,1],0.5\right),\mathrm{orientation}=[45,0]\right)
|
Section 59.79 (09XP): Cohomology with support in a closed subscheme—The Stacks project
Section 59.79: Cohomology with support in a closed subscheme (cite)
59.79 Cohomology with support in a closed subscheme
Let $X$ be a scheme and let $Z \subset X$ be a closed subscheme. Let $\mathcal{F}$ be an abelian sheaf on $X_{\acute{e}tale}$. We let
\[ \Gamma _ Z(X, \mathcal{F}) = \{ s \in \mathcal{F}(X) \mid \text{Supp}(s) \subset Z\} \]
be the sections with support in $Z$ (Definition 59.31.3). This is a left exact functor which is not exact in general. Hence we obtain a derived functor
\[ R\Gamma _ Z(X, -) : D(X_{\acute{e}tale}) \longrightarrow D(\textit{Ab}) \]
and cohomology groups with support in $Z$ defined by $H^ q_ Z(X, \mathcal{F}) = R^ q\Gamma _ Z(X, \mathcal{F})$.
Let $\mathcal{I}$ be an injective abelian sheaf on $X_{\acute{e}tale}$. Let $U = X \setminus Z$. Then the restriction map $\mathcal{I}(X) \to \mathcal{I}(U)$ is surjective (Cohomology on Sites, Lemma 21.12.6) with kernel $\Gamma _ Z(X, \mathcal{I})$. It immediately follows that for $K \in D(X_{\acute{e}tale})$ there is a distinguished triangle
\[ R\Gamma _ Z(X, K) \to R\Gamma (X, K) \to R\Gamma (U, K) \to R\Gamma _ Z(X, K)[1] \]
in $D(\textit{Ab})$. As a consequence we obtain a long exact cohomology sequence
\[ \ldots \to H^ i_ Z(X, K) \to H^ i(X, K) \to H^ i(U, K) \to H^{i + 1}_ Z(X, K) \to \ldots \]
for any $K$ in $D(X_{\acute{e}tale})$.
For an abelian sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$ we can consider the subsheaf of sections with support in $Z$, denoted $\mathcal{H}_ Z(\mathcal{F})$, defined by the rule
\[ \mathcal{H}_ Z(\mathcal{F})(U) = \{ s \in \mathcal{F}(U) \mid \text{Supp}(s) \subset U \times _ X Z\} \]
Here we use the support of a section from Definition 59.31.3. Using the equivalence of Proposition 59.46.4 we may view $\mathcal{H}_ Z(\mathcal{F})$ as an abelian sheaf on $Z_{\acute{e}tale}$. Thus we obtain a functor
\[ \textit{Ab}(X_{\acute{e}tale}) \longrightarrow \textit{Ab}(Z_{\acute{e}tale}),\quad \mathcal{F} \longmapsto \mathcal{H}_ Z(\mathcal{F}) \]
which is left exact, but in general not exact.
Lemma 59.79.1. Let $i : Z \to X$ be a closed immersion of schemes. Let $\mathcal{I}$ be an injective abelian sheaf on $X_{\acute{e}tale}$. Then $\mathcal{H}_ Z(\mathcal{I})$ is an injective abelian sheaf on $Z_{\acute{e}tale}$.
Proof. Observe that for any abelian sheaf $\mathcal{G}$ on $Z_{\acute{e}tale}$ we have
\[ \mathop{\mathrm{Hom}}\nolimits _ Z(\mathcal{G}, \mathcal{H}_ Z(\mathcal{F})) = \mathop{\mathrm{Hom}}\nolimits _ X(i_*\mathcal{G}, \mathcal{F}) \]
because after all any section of $i_*\mathcal{G}$ has support in $Z$. Since $i_*$ is exact (Section 59.46) and as $\mathcal{I}$ is injective on $X_{\acute{e}tale}$ we conclude that $\mathcal{H}_ Z(\mathcal{I})$ is injective on $Z_{\acute{e}tale}$. $\square$
\[ R\mathcal{H}_ Z : D(X_{\acute{e}tale}) \longrightarrow D(Z_{\acute{e}tale}) \]
the derived functor. We set $\mathcal{H}^ q_ Z(\mathcal{F}) = R^ q\mathcal{H}_ Z(\mathcal{F})$ so that $\mathcal{H}^0_ Z(\mathcal{F}) = \mathcal{H}_ Z(\mathcal{F})$. By the lemma above we have a Grothendieck spectral sequence
\[ E_2^{p, q} = H^ p(Z, \mathcal{H}^ q_ Z(\mathcal{F})) \Rightarrow H^{p + q}_ Z(X, \mathcal{F}) \]
Lemma 59.79.2. Let $i : Z \to X$ be a closed immersion of schemes. Let $\mathcal{G}$ be an injective abelian sheaf on $Z_{\acute{e}tale}$. Then $\mathcal{H}^ p_ Z(i_*\mathcal{G}) = 0$ for $p > 0$.
Proof. This is true because the functor $i_*$ is exact and transforms injective abelian sheaves into injective abelian sheaves (Cohomology on Sites, Lemma 21.14.2). $\square$
Lemma 59.79.3. Let $i : Z \to X$ be a closed immersion of schemes. Let $j : U \to X$ be the inclusion of the complement of $Z$. Let $\mathcal{F}$ be an abelian sheaf on $X_{\acute{e}tale}$. There is a distinguished triangle
\[ i_*R\mathcal{H}_ Z(\mathcal{F}) \to \mathcal{F} \to Rj_*(\mathcal{F}|_ U) \to i_*R\mathcal{H}_ Z(\mathcal{F})[1] \]
in $D(X_{\acute{e}tale})$. This produces an exact sequence
\[ 0 \to i_*\mathcal{H}_ Z(\mathcal{F}) \to \mathcal{F} \to j_*(\mathcal{F}|_ U) \to i_*\mathcal{H}^1_ Z(\mathcal{F}) \to 0 \]
and isomorphisms $R^ pj_*(\mathcal{F}|_ U) \cong i_*\mathcal{H}^{p + 1}_ Z(\mathcal{F})$ for $p \geq 1$.
Proof. To get the distinguished triangle, choose an injective resolution $\mathcal{F} \to \mathcal{I}^\bullet $. Then we obtain a short exact sequence of complexes
\[ 0 \to i_*\mathcal{H}_ Z(\mathcal{I}^\bullet ) \to \mathcal{I}^\bullet \to j_*(\mathcal{I}^\bullet |_ U) \to 0 \]
by the discussion above. Thus the distinguished triangle by Derived Categories, Section 13.12. $\square$
Let $X$ be a scheme and let $Z \subset X$ be a closed subscheme. We denote $D_ Z(X_{\acute{e}tale})$ the strictly full saturated triangulated subcategory of $D(X_{\acute{e}tale})$ consisting of complexes whose cohomology sheaves are supported on $Z$. Note that $D_ Z(X_{\acute{e}tale})$ only depends on the underlying closed subset of $X$.
Lemma 59.79.4. Let $i : Z \to X$ be a closed immersion of schemes. The map $Ri_{small, *} = i_{small, *} : D(Z_{\acute{e}tale}) \to D(X_{\acute{e}tale})$ induces an equivalence $D(Z_{\acute{e}tale}) \to D_ Z(X_{\acute{e}tale})$ with quasi-inverse
\[ i_{small}^{-1}|_{D_ Z(X_{\acute{e}tale})} = R\mathcal{H}_ Z|_{D_ Z(X_{\acute{e}tale})} \]
Proof. Recall that $i_{small}^{-1}$ and $i_{small, *}$ is an adjoint pair of exact functors such that $i_{small}^{-1}i_{small, *}$ is isomorphic to the identify functor on abelian sheaves. See Proposition 59.46.4 and Lemma 59.36.2. Thus $i_{small, *} : D(Z_{\acute{e}tale}) \to D_ Z(X_{\acute{e}tale})$ is fully faithful and $i_{small}^{-1}$ determines a left inverse. On the other hand, suppose that $K$ is an object of $D_ Z(X_{\acute{e}tale})$ and consider the adjunction map $K \to i_{small, *}i_{small}^{-1}K$. Using exactness of $i_{small, *}$ and $i_{small}^{-1}$ this induces the adjunction maps $H^ n(K) \to i_{small, *}i_{small}^{-1}H^ n(K)$ on cohomology sheaves. Since these cohomology sheaves are supported on $Z$ we see these adjunction maps are isomorphisms and we conclude that $D(Z_{\acute{e}tale}) \to D_ Z(X_{\acute{e}tale})$ is an equivalence.
To finish the proof we have to show that $R\mathcal{H}_ Z(K) = i_{small}^{-1}K$ if $K$ is an object of $D_ Z(X_{\acute{e}tale})$. To do this we can use that $K = i_{small, *}i_{small}^{-1}K$ as we've just proved this is the case. Then we can choose a K-injective representative $\mathcal{I}^\bullet $ for $i_{small}^{-1}K$. Since $i_{small, *}$ is the right adjoint to the exact functor $i_{small}^{-1}$, the complex $i_{small, *}\mathcal{I}^\bullet $ is K-injective (Derived Categories, Lemma 13.31.9). We see that $R\mathcal{H}_ Z(K)$ is computed by $\mathcal{H}_ Z(i_{small, *}\mathcal{I}^\bullet ) = \mathcal{I}^\bullet $ as desired. $\square$
Lemma 59.79.5. Let $X$ be a scheme. Let $Z \subset X$ be a closed subscheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module and denote $\mathcal{F}^ a$ the associated quasi-coherent sheaf on the small étale site of $X$ (Proposition 59.17.1). Then
$H^ q_ Z(X, \mathcal{F})$ agrees with $H^ q_ Z(X_{\acute{e}tale}, \mathcal{F}^ a)$,
if the complement of $Z$ is retrocompact in $X$, then $i_*\mathcal{H}^ q_ Z(\mathcal{F}^ a)$ is a quasi-coherent sheaf of $\mathcal{O}_ X$-modules equal to $(i_*\mathcal{H}^ q_ Z(\mathcal{F}))^ a$.
Proof. Let $j : U \to X$ be the inclusion of the complement of $Z$. The statement (1) on cohomology groups follows from the long exact sequences for cohomology with supports and the agreements $H^ q(X_{\acute{e}tale}, \mathcal{F}^ a) = H^ q(X, \mathcal{F})$ and $H^ q(U_{\acute{e}tale}, \mathcal{F}^ a) = H^ q(U, \mathcal{F})$, see Theorem 59.22.4. If $j : U \to X$ is a quasi-compact morphism, i.e., if $U \subset X$ is retrocompact, then $R^ qj_*$ transforms quasi-coherent sheaves into quasi-coherent sheaves (Cohomology of Schemes, Lemma 30.4.5) and commutes with taking associated sheaf on étale sites (Descent, Lemma 35.9.5). We conclude by applying Lemma 59.79.3. $\square$
Comment #2150 by Katha on July 31, 2016 at 12:13
In Lemma 50.72.5(2) it should be
i_* \mathcal{H}^q_Z(\mathcal{F})
i_* \mathcal{H}^q(\mathcal{F})
Thanks. If you want to be listed as a contributor, then leave a first and last name in your next comment. Fixed here.
|
Frequency - Wikiversity
Frequency is the number of repeating events per unit of time. In physics, the frequency of a wave is the number of wave crests (the peaks of the wave) that pass a point in one second.
An animation of wave functions with increasing frequency f.
{\displaystyle f=v/\lambda }
{\displaystyle \lambda }
{\displaystyle f=c/\lambda }
Three flashing lights, from the lowest frequency (top) to the highest one (bottom).
Different types of waves have different frequencies. One way to visualize this is if there were two trains traveling at the same speed, but the size of the train cars was smaller on one train than the other. If someone picked something that was not moving, like a signpost, and then counted how many train cars passed the sign post in one second for each train, they would know the frequency of cars passing in each train. The number and frequency of train cars passing the sign post would be different, because the train with smaller train cars would have more train cars passing the sign post in a second than the train with larger train cars. Knowing how many cars passed the sign post in one second, and knowing the speed of the train, one could figure out mathematically the size of each train car for each train.
Wikimedia Commons has media related to Frequency.
↑ ""Frequency of a wave"". Science Scope. Retrieved 2009-03-04.
Retrieved from "https://en.wikiversity.org/w/index.php?title=Frequency&oldid=1875232"
|
Section 60.18 (07JI): General remarks on cohomology—The Stacks project
Section 60.18: General remarks on cohomology (cite)
60.18 General remarks on cohomology
In this section we do a bit of work to translate the cohomology of modules on the cristalline site of an affine scheme into an algebraic question.
Lemma 60.18.1. In Situation 60.7.5. Let $\mathcal{F}$ be a locally quasi-coherent $\mathcal{O}_{X/S}$-module on $\text{Cris}(X/S)$. Then we have
\[ H^ p((U, T, \delta ), \mathcal{F}) = 0 \]
for all $p > 0$ and all $(U, T, \delta )$ with $T$ or $U$ affine.
Proof. As $U \to T$ is a thickening we see that $U$ is affine if and only if $T$ is affine, see Limits, Lemma 32.11.1. Having said this, let us apply Cohomology on Sites, Lemma 21.10.9 to the collection $\mathcal{B}$ of affine objects $(U, T, \delta )$ and the collection $\text{Cov}$ of affine open coverings $\mathcal{U} = \{ (U_ i, T_ i, \delta _ i) \to (U, T, \delta )\} $. The Čech complex ${\check C}^*(\mathcal{U}, \mathcal{F})$ for such a covering is simply the Čech complex of the quasi-coherent $\mathcal{O}_ T$-module $\mathcal{F}_ T$ (here we are using the assumption that $\mathcal{F}$ is locally quasi-coherent) with respect to the affine open covering $\{ T_ i \to T\} $ of the affine scheme $T$. Hence the Čech cohomology is zero by Cohomology of Schemes, Lemma 30.2.6 and 30.2.2. Thus the hypothesis of Cohomology on Sites, Lemma 21.10.9 are satisfied and we win. $\square$
Lemma 60.18.2. In Situation 60.7.5. Assume moreover $X$ and $S$ are affine schemes. Consider the full subcategory $\mathcal{C} \subset \text{Cris}(X/S)$ consisting of divided power thickenings $(X, T, \delta )$ endowed with the chaotic topology (see Sites, Example 7.6.6). For any locally quasi-coherent $\mathcal{O}_{X/S}$-module $\mathcal{F}$ we have
\[ R\Gamma (\mathcal{C}, \mathcal{F}|_\mathcal {C}) = R\Gamma (\text{Cris}(X/S), \mathcal{F}) \]
Proof. Denote $\text{AffineCris}(X/S)$ the fully subcategory of $\text{Cris}(X/S)$ consisting of those objects $(U, T, \delta )$ with $U$ and $T$ affine. We turn this into a site by saying a family of morphisms $\{ (U_ i, T_ i, \delta _ i) \to (U, T, \delta )\} _{i \in I}$ of $\text{AffineCris}(X/S)$ is a covering if and only if it is a covering of $\text{Cris}(X/S)$. With this definition the inclusion functor
\[ \text{AffineCris}(X/S) \longrightarrow \text{Cris}(X/S) \]
is a special cocontinuous functor as defined in Sites, Definition 7.29.2. The proof of this is exactly the same as the proof of Topologies, Lemma 34.3.10. Thus we see that the topos of sheaves on $\text{Cris}(X/S)$ is the same as the topos of sheaves on $\text{AffineCris}(X/S)$ via restriction by the displayed inclusion functor. Therefore we have to prove the corresponding statement for the inclusion $\mathcal{C} \subset \text{AffineCris}(X/S)$.
We will use without further mention that $\mathcal{C}$ and $\text{AffineCris}(X/S)$ have products and fibre products (details omitted, see Lemma 60.8.2). The inclusion functor $u : \mathcal{C} \to \text{AffineCris}(X/S)$ is fully faithful, continuous, and commutes with products and fibre products. We claim it defines a morphism of ringed sites
\[ f : (\text{AffineCris}(X/S), \mathcal{O}_{X/S}) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_{X/S}|_\mathcal {C}) \]
To see this we will use Sites, Lemma 7.14.6. Note that $\mathcal{C}$ has fibre products and $u$ commutes with them so the categories $\mathcal{I}^ u_{(U, T, \delta )}$ are disjoint unions of directed categories (by Sites, Lemma 7.5.1 and Categories, Lemma 4.19.8). Hence it suffices to show that $\mathcal{I}^ u_{(U, T, \delta )}$ is connected. Nonempty follows from Lemma 60.5.6: since $U$ and $T$ are affine that lemma says there is at least one object $(X, T', \delta ')$ of $\mathcal{C}$ and a morphism $(U, T, \delta ) \to (X, T', \delta ')$ of divided power thickenings. Connectedness follows from the fact that $\mathcal{C}$ has products and that $u$ commutes with them (compare with the proof of Sites, Lemma 7.5.2).
Note that $f_*\mathcal{F} = \mathcal{F}|_\mathcal {C}$. Hence the lemma follows if $R^ pf_*\mathcal{F} = 0$ for $p > 0$, see Cohomology on Sites, Lemma 21.14.6. By Cohomology on Sites, Lemma 21.7.4 it suffices to show that $H^ p(\text{AffineCris}(X/S)/(X, T, \delta ), \mathcal{F}) = 0$ for all $(X, T, \delta )$. This follows from Lemma 60.18.1 because the topos of the site $\text{AffineCris}(X/S)/(X, T, \delta )$ is equivalent to the topos of the site $\text{Cris}(X/S)/(X, T, \delta )$ used in the lemma. $\square$
Lemma 60.18.3. In Situation 60.5.1. Set $\mathcal{C} = (\text{Cris}(C/A))^{opp}$ and $\mathcal{C}^\wedge = (\text{Cris}^\wedge (C/A))^{opp}$ endowed with the chaotic topology, see Remark 60.5.4 for notation. There is a morphism of topoi
\[ g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \longrightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{C}^\wedge ) \]
such that if $\mathcal{F}$ is a sheaf of abelian groups on $\mathcal{C}$, then
\[ R^ pg_*\mathcal{F}(B \to C, \delta ) = \left\{ \begin{matrix} \mathop{\mathrm{lim}}\nolimits _ e \mathcal{F}(B_ e \to C, \delta ) & \text{if }p = 0 \\ R^1\mathop{\mathrm{lim}}\nolimits _ e \mathcal{F}(B_ e \to C, \delta ) & \text{if }p = 1 \\ 0 & \text{else} \end{matrix} \right. \]
where $B_ e = B/p^ eB$ for $e \gg 0$.
Proof. Any functor between categories defines a morphism between chaotic topoi in the same direction, for example because such a functor can be considered as a cocontinuous functor between sites, see Sites, Section 7.21. Proof of the description of $g_*\mathcal{F}$ is omitted. Note that in the statement we take $(B_ e \to C, \delta )$ is an object of $\text{Cris}(C/A)$ only for $e$ large enough. Let $\mathcal{I}$ be an injective abelian sheaf on $\mathcal{C}$. Then the transition maps
\[ \mathcal{I}(B_ e \to C, \delta ) \leftarrow \mathcal{I}(B_{e + 1} \to C, \delta ) \]
are surjective as the morphisms
\[ (B_ e \to C, \delta ) \longrightarrow (B_{e + 1} \to C, \delta ) \]
are monomorphisms in the category $\mathcal{C}$. Hence for an injective abelian sheaf both sides of the displayed formula of the lemma agree. Taking an injective resolution of $\mathcal{F}$ one easily obtains the result (sheaves are presheaves, so exactness is measured on the level of groups of sections over objects). $\square$
Lemma 60.18.4. Let $\mathcal{C}$ be a category endowed with the chaotic topology. Let $X$ be an object of $\mathcal{C}$ such that every object of $\mathcal{C}$ has a morphism towards $X$. Assume that $\mathcal{C}$ has products of pairs. Then for every abelian sheaf $\mathcal{F}$ on $\mathcal{C}$ the total cohomology $R\Gamma (\mathcal{C}, \mathcal{F})$ is represented by the complex
\[ \mathcal{F}(X) \to \mathcal{F}(X \times X) \to \mathcal{F}(X \times X \times X) \to \ldots \]
associated to the cosimplicial abelian group $[n] \mapsto \mathcal{F}(X^ n)$.
Proof. Note that $H^ q(X^ p, \mathcal{F}) = 0$ for all $q > 0$ as any presheaf is a sheaf on $\mathcal{C}$. The assumption on $X$ is that $h_ X \to *$ is surjective. Using that $H^ q(X, \mathcal{F}) = H^ q(h_ X, \mathcal{F})$ and $H^ q(\mathcal{C}, \mathcal{F}) = H^ q(*, \mathcal{F})$ we see that our statement is a special case of Cohomology on Sites, Lemma 21.13.2. $\square$
Comment #6254 by Hao Peng on May 24, 2021 at 15:56
I think the proof of tag07JK cannot go through where you claim that
I^u_{(U, T, \delta)}
is non-empty: the tag07HP cannot be used because
T
may not be affine so
p
is only locally nilpotent on
T
, thus may not exist an
e
p^e=0
T
However, the same argument in fact shows the
(\varinjlim_eh_{(X, T_e, \delta)})^\sharp\to
is surjective as sheaves. So I think this tag and the introduction of the chaotic site
\mathcal C
is in fact unnecessary: as this tag is used for the tag07JN, we can prove the tag07JN directly using tag079Z for
K^\prime=\varinjlim_eh_{(X, T_e, \delta)}, K=*
instead of using tag07JM. The shifified product
(K^\prime_n)^\sharp
are in fact represeted by
(X, D(n), \delta)
as an ind-object and the hypothesis of tag07JN is strong enough to prove the spectral sequence of tag079Z degenrate at
E_1
, thus we are done.
OK, I think the lemma is correct, but you are correct in saying that the argument does not work as given. Thanks for pointing this out! I do want to keep this lemma as I think it is very useful psychologically! The fact that one can use the chaotic topology is sort of marvelous and plays a role als in other similar settings (prismatic cohomology, etc).
To fix the proof we can look at the full subcategory
\text{AffineCris}(X/S)
\text{Cris}(X/S)
consisting of the objects
(U, T, \delta)
U
T
affine and endowed with the Zariski topology (defined in the obvious manner). Then we first prove that
\text{AffineCris}(X/S)
\text{Cris}(X/S)
define the same topos by the usual method, see for example Lemma 34.4.11. Finally, we prove the inclusion functor
u : \mathcal{C} \to \text{affineCris}(X/S)
does work by the arguments given in the proof of lemma. OK?
(In fact, I think I replaced Cris by AffineCris in my head when I wrote the proof of this lemma.)
Thanks! I think this will do. Btw chaotic site simplifies things so much, and it can be used for the the prismatic cohomology is for the same reason that higher cohomology of
\mathcal O_\Delta
\overline{\mathcal O_\Delta}
all vanishes. But I'm not sure when we consider
\mathcal O_\Delta
-modules with structures(if someday) will still be so lucky. For example Scholze's calculus of condensed cohomology used hypercovering in an essential way... But I will take care of the simplification of cohomologies.
OK, I fixed it as I indicated. See changes here.
|
Centrifugal force - Simple English Wikipedia, the free encyclopedia
This is the force that acts on the body in a direction away from the centre, which contributes to making the body try to fly away. When you hold a rope with a heavy object attached to it, and rotate it around, the rope becomes tight and keeps the body from flying away. This is caused by centripetal force.
In physics, centrifugal force (from Latin centrum "center" and fugere "to flee") is a fictitious force that appears when describing physics in a rotating reference frame; it acts on anything with mass considered in such a frame. Centrifugal force is fictitious because although it may feel to a person like a certain force is being exerted on them, someone outside the scene will see something different.
Example: If John is in a car that takes a sharp right turn, he will feel as though he is being pushed to his left. This is an imaginary force, called a centrifugal force, or a "running away from the center" force. John feels it because he is inside the car and is affected by it. However, if John's friend, Andy, is on the side of the road facing the front of John's car and watches John's car take a sharp right turn, Andy will see the car push John to the right with the car as it changes direction. This is a real force called centripetal force (or an "aiming towards the center" force) and acts towards the center of the circle of rotation.
{\displaystyle F=ma=m{\frac {v^{2}}{r}}}
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Centrifugal_force&oldid=6843237"
|
SimilarityTransformation - Maple Help
Home : Support : Online Help : Mathematics : Differential Equations : DEtools : Lie Symmetry Method : Commands for PDEs (and ODEs) : SimilarityTransformation
computes a transformation reducing by one the number of independent variables of PDE systems possessing a given symmetry
SimilarityTransformation(S, DepVars, NewVars, 'options'='value')
Given a list with the infinitesimals S of a generator of symmetry transformations leaving invariant a PDE system (PDESYS), or the corresponding infinitesimal generator differential operator, the SimilarityTransformation command computes a transformation that reduces by one the number of independent variables of PDESYS. The output consists of a sequence of two sets respectively containing the transformation and inverse transformation equations.
These similarity transformations are special cases of group invariant transformations able to reduce the number of independent variables by many in one go, computed with the InvariantTransformation command.
The process of computing similarity transformations implies computing the invariants associated to the given infinitesimals. The typical formulation of these transformations in textbooks, however, sometimes avoids those wordings and instead presents these transformations as the introduction of variables
\mathrm{\psi }
{\mathrm{\phi }}_{k}
k=n+m-1
n
m
are, respectively, the number of independent and dependent variables of the problem, such that the infinitesimals of the symmetry generator used to construct the transformation assume the form
[\mathrm{...}{\mathrm{\xi }}_{r}=1,\mathrm{...},{\mathrm{\eta }}_{1}=0,\mathrm{...},{\mathrm{\eta }}_{m}=0]
0\le r
\le n
. Hence, by applying this similarity transformation to any PDE invariant under S you obtain a PDE not depending on the rth independent variable to which corresponds
{\mathrm{\xi }}_{r}=1
- see the examples below.
When there is only one dependent variable, DepVars and NewVars can be a function; otherwise they must be a list of functions representing dependent variables. If NewVars are not given, SimilarityTransformation will generate a list of globals to represent them.
You can optionally specify a simplifier, to be used instead of the default which is simplify/size, as well as requesting the output to be in jet notation by respectively using the optional arguments simplifier = ... and jetnotation. Note that the option simplifier = ... can be used not just to "simplify" the output but also to post-process this output in the way you want, for instance using a procedure written by you, to discard, change or do what you find necessary with the transformation.
\mathrm{with}\left(\mathrm{PDEtools},\mathrm{SimilarityTransformation},\mathrm{ChangeSymmetry},\mathrm{InfinitesimalGenerator}\right)
[\textcolor[rgb]{0,0,1}{\mathrm{SimilarityTransformation}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{ChangeSymmetry}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{InfinitesimalGenerator}}]
Consider a PDE problem, for example PDESYS, with two independent variables and one dependent variable,
u\left(x,t\right)
, and consider the list of infinitesimals of a symmetry group assumed to be admitted by PDESYS
S≔[\mathrm{_ξ}[x]=x,\mathrm{_ξ}[t]=1,\mathrm{_η}[u]=u]
\textcolor[rgb]{0,0,1}{S}\textcolor[rgb]{0,0,1}{≔}[{\textcolor[rgb]{0,0,1}{\mathrm{_ξ}}}_{\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathrm{_ξ}}}_{\textcolor[rgb]{0,0,1}{t}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathrm{_η}}}_{\textcolor[rgb]{0,0,1}{u}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{u}]
S
[x,1,u]
G≔\mathrm{InfinitesimalGenerator}\left(S,u\left(x,t\right)\right)
\textcolor[rgb]{0,0,1}{G}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{→}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0,0,1}{∂}}{\textcolor[rgb]{0,0,1}{∂}\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{f}\right)\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{∂}}{\textcolor[rgb]{0,0,1}{∂}\textcolor[rgb]{0,0,1}{t}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0,0,1}{∂}}{\textcolor[rgb]{0,0,1}{∂}\textcolor[rgb]{0,0,1}{u}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{f}\right)
We say that PDESYS is invariant under the transformations generated by G in that G(PDESYS) = 0 were in this formula G represents the prolongation necessary to act on PDESYS (see InfinitesimalGenerator). The similarity transformation relating the original variables
{t,x,u\left(x,t\right)}
to new variables - say
{r,s,v\left(r,s\right)}
, that reduces by one the number of independent variables of a PDE system invariant under G above is obtained via
\mathrm{ITR},\mathrm{TR}≔\mathrm{SimilarityTransformation}\left(S,u\left(x,t\right),v\left(r,s\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{ITR}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{TR}}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{r}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{ln}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{s}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{ln}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{v}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{r}\right)\textcolor[rgb]{0,0,1}{=}\frac{\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{t}\right)}{\textcolor[rgb]{0,0,1}{x}}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{r}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{s}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{=}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{s}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{t}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{v}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{r}\right)\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{s}}}
Note these transformation sets are returned with
v\left(r,s\right)
↦v\left(r\right)
, making explicit that the unknown of the problem you obtain when you change variables does not depend on s.
To express these transformations using jet notation use
\mathrm{SimilarityTransformation}\left(S,u\left(x,t\right),v\left(r,s\right),\mathrm{jetnotation}\right)
{\textcolor[rgb]{0,0,1}{r}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{ln}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{s}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{ln}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{v}\textcolor[rgb]{0,0,1}{=}\frac{\textcolor[rgb]{0,0,1}{u}}{\textcolor[rgb]{0,0,1}{x}}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{r}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{s}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{v}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{s}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{=}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{s}}}
\mathrm{SimilarityTransformation}\left(S,u\left(x,t\right),v\left(r,s\right),\mathrm{jetnotation}=\mathrm{jetnumbers}\right)
\textcolor[rgb]{0,0,1}{\mathrm{jet notation: \left[1 = x, 2 = t, 3 = r\right]}}
\textcolor[rgb]{0,0,1}{\mathrm{________________________________}}
{\textcolor[rgb]{0,0,1}{r}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{ln}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{s}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{ln}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{v}[]\textcolor[rgb]{0,0,1}{=}\frac{\textcolor[rgb]{0,0,1}{u}[]}{\textcolor[rgb]{0,0,1}{x}}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{r}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{s}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{=}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{s}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{u}[]\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{v}[]\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{s}}}
That this transformation TR reduces the number of independent variables of any PDE system invariant under G above is visible in the fact that it transforms the given infinitesimals
[{\mathrm{_ξ}}_{x}=x,{\mathrm{_ξ}}_{t}=1,{\mathrm{_η}}_{u}=u]
{t,x,u\left(x,t\right)}
) into
[{\mathrm{_ξ}}_{r}=0,{\mathrm{_ξ}}_{s}=1,{\mathrm{_η}}_{v}=0]
{r,s,v\left(r,s\right)}
). To verify this you can use ChangeSymmetry
\mathrm{NewVars}≔\mathrm{map}\left(\mathrm{lhs},\mathrm{ITR}\right)
\textcolor[rgb]{0,0,1}{\mathrm{NewVars}}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{r}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{s}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{v}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{r}\right)}
\mathrm{ChangeSymmetry}\left(\mathrm{TR},S,u\left(x,t\right),\mathrm{NewVars}\right)
[{\textcolor[rgb]{0,0,1}{\mathrm{_ξ}}}_{\textcolor[rgb]{0,0,1}{r}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathrm{_ξ}}}_{\textcolor[rgb]{0,0,1}{s}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathrm{_η}}}_{\textcolor[rgb]{0,0,1}{v}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}]
v\left(r,s\right)
\mathrm{InfinitesimalGenerator}\left(,v\left(r,s\right)\right)
\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{→}\frac{\textcolor[rgb]{0,0,1}{∂}}{\textcolor[rgb]{0,0,1}{∂}\textcolor[rgb]{0,0,1}{s}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{f}
Any PDESYS invariant under G will also be invariant under the operator above, that is, PDESYS will be independent of r after you change variables in it using TR computed with SimilarityTransformation lines above.
If the new variables, here
v\left(r,s\right)
, are not indicated, variables
\mathrm{_ψ}
and _phi[k] prefixed by an underscore _ to represent the new variables are introduced
\mathrm{SimilarityTransformation}\left(S,u\left(x,t\right)\right)
{\textcolor[rgb]{0,0,1}{\mathrm{_ψ}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{ln}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathrm{_φ}}}_{\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{ln}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathrm{_φ}}}_{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{\mathrm{_φ}}}_{\textcolor[rgb]{0,0,1}{1}}\right)\textcolor[rgb]{0,0,1}{=}\frac{\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{t}\right)}{\textcolor[rgb]{0,0,1}{x}}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{=}{\textcolor[rgb]{0,0,1}{\mathrm{_φ}}}_{\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{_ψ}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{=}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{\mathrm{_ψ}}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{t}\right)\textcolor[rgb]{0,0,1}{=}{\textcolor[rgb]{0,0,1}{\mathrm{_φ}}}_{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{\mathrm{_φ}}}_{\textcolor[rgb]{0,0,1}{1}}\right)\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{\mathrm{_ψ}}}}
|
Cubic smoothing spline - MATLAB csaps - MathWorks Benelux
Fit Splines with Different Smoothing Parameters
Adjust Smoothing Parameters and Weights
Smooth Bivariate Data
Cubic smoothing spline
For a simpler but less flexible method to generate smoothing splines, try the Curve Fitter app or the fit function.
pp = csaps(x,y) returns the cubic smoothing spline interpolation to the given data (x,y) in ppform. The value of spline f at data site x(j) approximates the data value y(:,j) for j = 1:length(x).
The smoothing spline f minimizes
p\underset{\text{error measure}}{\underbrace{\sum _{j=1}^{n}{w}_{j}|{y}_{j}-f\left({x}_{j}\right){|}^{2}}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+\text{ }\left(1-p\right)\underset{\text{roughness measure}}{\underbrace{\underset{}{\overset{}{\int }}\lambda \left(t\right)|{D}^{2}f\left(t\right){|}^{2}dt}}
Here, n is the number of entries of x and the integral is over the smallest interval containing all the entries of x. yj and xj refer to the jth entries of y and x, respectively. D2f denotes the second derivative of the function f.
The default values for the error measure weights wj are 1. The default value for the piecewise constant weight function λ in the roughness measure is the constant function 1. By default, csaps chooses a value for the smoothing parameter p based on the given data sites x.
To evaluate a smoothing spline outside its basic interval, you must first extrapolate it. Use the command pp = fnxtr(pp) to ensure that the second derivative is zero outside the interval spanned by the data sites.
pp = csaps(x,y,p) specifies the smoothing parameter p. You can also supply the roughness measure weights λ by providing p as a vector whose first entry is p and ith entry is the value of λ on the interval (x(i-1),x(i)).
pp = csaps(x,y,p,[],w) also specifies the weights w in the error measure.
values = csaps(x,y,p,xx) uses the smoothing parameter p and returns the values of the smoothing spline evaluated at the points xx. This syntax is the same as fnval(csaps(x,y,p),xx).
values = csaps(x,y,p,xx,w) uses the smoothing parameter p and the error measure weights w, and returns the values of the smoothing spline evaluated at the points xx. This syntax is the same as fnval(csaps(x,y,p,[],w),xx)
[___] = csaps({x1,...,xm},y,___) provides the ppform of an m-variate tensor-product smoothing spline to data on the rectangular grid described by {x1,...,xm}. You can use this syntax with any of the arguments in the previous syntaxes.
[___,P] = csaps(___) also returns the value of the smoothing parameter used in the final spline result whether or not you specify p. This syntax is useful for experimentation in which you can start with [pp,P] = csaps(x,y) and obtain a reasonable first guess for p.
Fit smoothing splines using the csaps function with different values for the smoothing parameter p. Use values of p between the extremes of 0 and 1 to see how they affect the shape and closeness of the fitted spline.
Load the titanium data set.
When p = 0, s0 is the least-squares straight line fit to the data. When p = 1, s1 is the variational, or natural, cubic spline interpolant.
For 0 < p < 1, sp is a smoothing spline that is a trade-off between the two extremes: smoother than the interpolant s1 and closer to the data than the straight line s0.
Adjust the smoothing parameter, error measure weights, and roughness measure weights.
Create a sine curve with noise.
Fit a smoothing spline to the data. Specify the smoothing parameter p = 0.4 and error measure weights w that vary across the data.
The function returns a smooth fit to the noisy data that is much closer to the data in the right half because of the much larger error measure weight there. Note that the error weighting of zero for the last data point excludes this point from the fit.
Now fit a smoothing spline using the same data, smoothing parameter and error measure weights, but with adjusted roughness measure weights.
The roughness measure weight is only 0.2 in the right half of the interval. Correspondingly, the fit is rougher but closer on the right side of the data (except for the last data point, which is ignored).
Plot both fits for comparison.
Fit a smoothing spline to bivariate data generated by the peaks function with added uniform noise. Use csaps to obtain the new, smoothed data points and the smoothing parameters csaps determines for the fit.
Create the grid. For this example, the grid is a 51-by-61 uniform grid.
Generate the noisy data using the peaks function and random numbers in the interval
\left[-\frac{1}{2},\frac{1}{2}\right]
Fit the data. Use csaps to obtain the smoothed data values evaluated over the grid x and the default smoothing parameter used in the fit.
The plot of the fit shows that some roughness remains. Note that you must transpose the array sval.
For a somewhat smoother approximation, specify a value for p that is slightly smaller than the csaps default value.
Data sites of data values y to be fit, specified as a vector or as a cell array for multivariate data. Spline f is created with knots at each data site x such that f(x(j)) = y(:,j) for all values of j.
For multivariate, gridded data, you can specify x as a cell array that specifies the data site in each variable dimension: f(x1(i),x2(j),...xn(k)) = y(:,i,j,...,k).
y — Data values to fit
Data values to fit during creation of the spline, specified as a vector, matrix, or array. Data values y(:,j) can be scalars, matrices, or n-dimensional arrays. Data values given at the same data site x are averaged.
p — Smoothing parameter
scalar in the range [0,1] | vector | cell array | empty array
Smoothing parameter, specified as a scalar value between 0 and 1 or as a cell array of values for multivariate data. You can also specify values for the roughness measure weights λ by providing p as a vector. To provide roughness measure weights for multivariate data, use a cell array of vectors. If you provide an empty array, the function chooses a default value for p based on the data sites x and the default value of 1 for the roughness measure weight λ.
The smoothing parameter determines the relative weight to place on the contradictory demands of having f be smooth or having f be close to the data. For p = 0, f is the least-squares straight-line fit to the data. For p = 1, f is the variational, or natural, cubic spline interpolant. As p moves from 0 to 1, the smoothing spline changes from one extreme to the other.
The favorable range for p is often near 1/(1 + h3/6), where h is the average spacing of the data sites. The function chooses a default value for p within this range. For uniformly spaced data, you can expect a close fit with p = 1(1 + h3/60) and some satisfactory smoothing with p = 1/(1 + h3/0.6). You can input p > 1, but this choice leads to a smoothing spline even rougher than the variational cubic spline interpolant.
If the input p is negative or empty, then the function uses the default value for p.
You can specify the roughness measure weights λ alongside the smoothing parameter by providing p as a vector. This vector must be the same size as x, with the ith entry the value of λ on the interval (x(i-1)...x(i)), for i = 2:length(x). The first entry of the input vector p is the desired value of the smoothness parameter p. By providing roughness measure weights, you can make the resulting smoothing spline smoother (with larger weight values) or closer to the data (with smaller weight values) in different parts of the interval. Roughness measure weights must be nonnegative.
If you have difficulty choosing p but have some feeling for the size of the noise in y, consider using spaps(x,y,tol) instead. This function chooses p such that the roughness measure is as small as possible, subject to the condition that the error measure does not exceed tol. In this case, the error measure usually equals the specified value for tol.
w — Error measure weights
Error measure weights w in the error measure, specified as a vector of nonnegative entries of the same size as x.
The default value for the weight vector w in the error measure is ones(size(x)).
xx — Evaluation points
Evaluation points over which the spline is evaluated, specified as a vector or as a cell array of vectors for multivariate data. Spline evaluation is performed using fnval.
values — Evaluated spline
Evaluated spline, returned as a vector or as a matrix or array for multivariate data. The spline is evaluated at the given evaluation points xx.
Smoothing parameter used to calculate the spline, returned as a scalar or as a cell array of scalar values for multivariate data. P is between 0 and 1.
csaps is an implementation of the Fortran routine SMOOTH from PGS.
The calculation of the smoothing spline requires solving a linear system whose coefficient matrix has the form p*A + (1-p)*B, with the matrices A and B depending on the data sites x. The default value of p makes p*trace(A) equal (1-p)*trace(B).
|
Lemma 37.44.7 (09Z0)—The Stacks project
In the parenthetical early in the 3rd paragraph of the proof, replace "quasi-compact etale" with "quasi-compact separated etale" (it is implicit in the Lemma invoked there, and also necessary as well), so earlier when
U
is made affine it should have been noted (with cross-reference) that
U \to X
is thereby separated. In the 2nd to last line of this same paragraph, don't call the neighborhood of
z
Z
by the name
V
, since the notation
V
already has an entirely different meaning in the overall proof.
Comment #5556 by Harry Gindi on October 25, 2020 at 18:12
Separatedness of the morphism U'→X doesn't seem to follow from the referenced lemma. We know that U' is affine, but not necessarily that the map U'→X is affine (which is the hypothesis of 01S7). This would be true if X itself were separated (or at least has affine diagonal). Is the reference incorrect?
Hey, I found the correct reference for the claim: It should be 01KN (a morphism from an affine scheme to a scheme is separated).
|
This article is about a generalized derivative of a multivariate function. For another use in mathematics, see Slope. For a similarly spelled unit of angle, see Gradian. For other uses, see Gradient (disambiguation).
Find sources: "Gradient" – news · newspapers · books · scholar · JSTOR (January 2018) (Learn how and when to remove this template message)
In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function)
{\displaystyle \nabla f}
whose value at a point
{\displaystyle p}
is the vector[a] whose components are the partial derivatives o{\displaystyle f}
{\displaystyle p}
.[1] That is, for
{\displaystyle f\colon \mathbb {R} ^{n}\to \mathbb {R} }
, its gradient
{\displaystyle \nabla f\colon \mathbb {R} ^{n}\to \mathbb {R} ^{n}}
is defined at the point
{\displaystyle p=(x_{1},\ldots ,x_{n})}
in n-dimensional space as the vector[b]
{\displaystyle \nabla f(p)={\begin{bmatrix}{\frac {\partial f}{\partial x_{1}}}(p)\\\vdots \\{\frac {\partial f}{\partial x_{n}}}(p)\end{bmatrix}}.}
{\displaystyle \nabla }
, written as an upside-down triangle and pronounced "del", denotes the vector differential operator.
The gradient is dual to the total derivative
{\displaystyle df}
: the value of the gradient at a point is a tangent vector – a vector at each point; while the value of the derivative at a point is a cotangent vector – a linear function on vectors.[c] They are related in that the dot product of the gradient of f at a point p with another tangent vector v equals the directional derivative of f at p of the function along v; that is,
{\textstyle \nabla f(p)\cdot \mathbf {v} ={\frac {\partial f}{\partial \mathbf {v} }}(p)=df_{p}(\mathbf {v} )}
. The gradient admits multiple generalizations to more general functions on manifolds; see § Generalizations.
3.3 General coordinates
4 Relationship with derivative
4.1 Relationship with total derivative
4.1.1 Differential or (exterior) derivative
4.1.2 Linear approximation to a function
4.2 Relationship with Fréchet derivative
6.1 Jacobian
6.2 Gradient of a vector field
Gradient of the 2D function f(x, y) = xe−(x2 + y2) is plotted as blue arrows over the pseudocolor plot of the function.
The gradient of a function
{\displaystyle f}
{\displaystyle a}
{\displaystyle \nabla f(a)}
. It may also be denoted by any of the following:
{\displaystyle {\vec {\nabla }}f(a)}
: to emphasize the vector nature of the result.
{\displaystyle \partial _{i}f}
{\displaystyle f_{i}}
: Einstein notation.
{\displaystyle {\big (}\nabla f(x){\big )}\cdot \mathbf {v} =D_{\mathbf {v} }f(x)}
where the right-side hand is the directional derivative and there are many ways to represent it. Formally, the gradient is dual to the derivative; see relationship with derivative.
{\displaystyle \nabla f={\frac {\partial f}{\partial x}}\mathbf {i} +{\frac {\partial f}{\partial y}}\mathbf {j} +{\frac {\partial f}{\partial z}}\mathbf {k} ,}
{\displaystyle f(x,y,z)=2x+3y^{2}-\sin(z)}
{\displaystyle \nabla f=2\mathbf {i} +6y\mathbf {j} -\cos(z)\mathbf {k} .}
Cylindrical and spherical coordinatesEdit
{\displaystyle \nabla f(\rho ,\varphi ,z)={\frac {\partial f}{\partial \rho }}\mathbf {e} _{\rho }+{\frac {1}{\rho }}{\frac {\partial f}{\partial \varphi }}\mathbf {e} _{\varphi }+{\frac {\partial f}{\partial z}}\mathbf {e} _{z},}
In spherical coordinates, the gradient is given by:[5]
{\displaystyle \nabla f(r,\theta ,\varphi )={\frac {\partial f}{\partial r}}\mathbf {e} _{r}+{\frac {1}{r}}{\frac {\partial f}{\partial \theta }}\mathbf {e} _{\theta }+{\frac {1}{r\sin \theta }}{\frac {\partial f}{\partial \varphi }}\mathbf {e} _{\varphi },}
General coordinatesEdit
{\displaystyle \nabla f={\frac {\partial f}{\partial x^{i}}}g^{ij}\mathbf {e} _{j}}
(Note that its dual is
{\textstyle \mathrm {d} f={\frac {\partial f}{\partial x^{i}}}\mathbf {e} ^{i}}
{\displaystyle \mathbf {e} _{i}=\partial \mathbf {x} /\partial x^{i}}
{\displaystyle \mathbf {e} ^{i}=\mathrm {d} x^{i}}
refer to the unnormalized local covariant and contravariant bases respectively,
{\displaystyle g^{ij}}
is the inverse metric tensor, and the Einstein summation convention implies summation over i and j.
If the coordinates are orthogonal we can easily express the gradient (and the differential) in terms of the normalized bases, which we refer to as
{\displaystyle {\hat {\mathbf {e} }}_{i}}
{\displaystyle {\hat {\mathbf {e} }}^{i}}
, using the scale factors (also known as Lamé coefficients)
{\displaystyle h_{i}=\lVert \mathbf {e} _{i}\rVert =1\,/\lVert \mathbf {e} ^{i}\rVert }
{\displaystyle \nabla f=\sum _{i=1}^{n}\,{\frac {\partial f}{\partial x^{i}}}{\frac {1}{h_{i}}}\mathbf {\hat {e}} _{i}}
{\textstyle \mathrm {d} f=\sum _{i=1}^{n}\,{\frac {\partial f}{\partial x^{i}}}{\frac {1}{h_{i}}}\mathbf {\hat {e}} ^{i}}
where we cannot use Einstein notation, since it is impossible to avoid the repetition of more than two indices. Despite the use of upper and lower indices,
{\displaystyle \mathbf {\hat {e}} _{i}}
{\displaystyle \mathbf {\hat {e}} ^{i}}
{\displaystyle h_{i}}
are neither contravariant nor covariant.
Relationship with derivativeEdit
Relationship with total derivativeEdit
The gradient is closely related to the total derivative (total differential)
{\displaystyle df}
: they are transpose (dual) to each other. Using the convention that vectors in
{\displaystyle \mathbb {R} ^{n}}
are represented by column vectors, and that covectors (linear maps
{\displaystyle \mathbb {R} ^{n}\to \mathbb {R} }
) are represented by row vectors,[a] the gradient
{\displaystyle \nabla f}
and the derivative
{\displaystyle df}
are expressed as a column and row vector, respectively, with the same components, but transpose of each other:
{\displaystyle \nabla f(p)={\begin{bmatrix}{\frac {\partial f}{\partial x_{1}}}(p)\\\vdots \\{\frac {\partial f}{\partial x_{n}}}(p)\end{bmatrix}};}
{\displaystyle df_{p}={\begin{bmatrix}{\frac {\partial f}{\partial x_{1}}}(p)&\cdots &{\frac {\partial f}{\partial x_{n}}}(p)\end{bmatrix}}.}
While these both have the same components, they differ in what kind of mathematical object they represent: at each point, the derivative is a cotangent vector, a linear form (covector) which expresses how much the (scalar) output changes for a given infinitesimal change in (vector) input, while at each point, the gradient is a tangent vector, which represents an infinitesimal change in (vector) input. In symbols, the gradient is an element of the tangent space at a point,
{\displaystyle \nabla f(p)\in T_{p}\mathbb {R} ^{n}}
, while the derivative is a map from the tangent space to the real numbers,
{\displaystyle df_{p}\colon T_{p}\mathbb {R} ^{n}\to \mathbb {R} }
. The tangent spaces at each point of
{\displaystyle \mathbb {R} ^{n}}
can be "naturally" identified[d] with the vector space
{\displaystyle \mathbb {R} ^{n}}
itself, and similarly the cotangent space at each point can be naturally identified with the dual vector space
{\displaystyle (\mathbb {R} ^{n})^{*}}
of covectors; thus the value of the gradient at a point can be thought of a vector in the original
{\displaystyle \mathbb {R} ^{n}}
, not just as a tangent vector.
{\displaystyle (df_{p})(v)={\begin{bmatrix}{\frac {\partial f}{\partial x_{1}}}(p)&\cdots &{\frac {\partial f}{\partial x_{n}}}(p)\end{bmatrix}}{\begin{bmatrix}v_{1}\\\vdots \\v_{n}\end{bmatrix}}=\sum _{i=1}^{n}{\frac {\partial f}{\partial x_{i}}}(p)v_{i}={\begin{bmatrix}{\frac {\partial f}{\partial x_{1}}}(p)\\\vdots \\{\frac {\partial f}{\partial x_{n}}}(p)\end{bmatrix}}\cdot {\begin{bmatrix}v_{1}\\\vdots \\v_{n}\end{bmatrix}}=\nabla f(p)\cdot v}
Differential or (exterior) derivativeEdit
The best linear approximation to a differentiable function
{\displaystyle f\colon \mathbb {R} ^{n}\to \mathbb {R} }
The gradient is related to the differential by the formula
{\displaystyle (\nabla f)_{x}\cdot v=df_{x}(v)}
for any v ∈ Rn, where
{\displaystyle \cdot }
is the dot product: taking the dot product of a vector with the gradient is the same as taking the directional derivative along the vector.
{\displaystyle \left({\frac {\partial f}{\partial x_{1}}},\dots ,{\frac {\partial f}{\partial x_{n}}}\right),}
{\displaystyle (\nabla f)_{i}=df_{i}^{\mathsf {T}}.}
Linear approximation to a functionEdit
{\displaystyle f(x)\approx f(x_{0})+(\nabla f)_{x_{0}}\cdot (x-x_{0})}
Relationship with Fréchet derivativeEdit
{\displaystyle \lim _{h\to 0}{\frac {|f(x+h)-f(x)-\nabla f(x)\cdot h|}{\|h\|}}=0,}
where · is the dot product.
{\displaystyle \nabla \left(\alpha f+\beta g\right)(a)=\alpha \nabla f(a)+\beta \nabla g(a).}
{\displaystyle \nabla (fg)(a)=f(a)\nabla g(a)+g(a)\nabla f(a).}
{\displaystyle (f\circ g)'(c)=\nabla f(a)\cdot g'(c),}
where ∘ is the composition operator: (f ∘ g)(x) = f(g(x)).
{\displaystyle \nabla (f\circ g)(c)={\big (}Dg(c){\big )}^{\mathsf {T}}{\big (}\nabla f(a){\big )},}
{\displaystyle \nabla (h\circ f)(a)=h'{\big (}f(a){\big )}\nabla f(a).}
Further properties and applicationsEdit
Level setsEdit
Conservative vector fields and the gradient theoremEdit
Main article: Jacobian matrix and determinant
Suppose f : Rn → Rm is a function such that each of its first-order partial derivatives exist on ℝn. Then the Jacobian matrix of f is defined to be an m×n matrix, denoted by
{\displaystyle \mathbf {J} _{\mathbb {f} }(\mathbb {x} )}
{\displaystyle \mathbf {J} }
. The (i,j)th entry is
{\displaystyle \mathbf {J} _{ij}={\frac {\partial f_{i}}{\partial x_{j}}}}
{\displaystyle \mathbf {J} ={\begin{bmatrix}{\dfrac {\partial \mathbf {f} }{\partial x_{1}}}&\cdots &{\dfrac {\partial \mathbf {f} }{\partial x_{n}}}\end{bmatrix}}={\begin{bmatrix}\nabla ^{\mathsf {T}}f_{1}\\\vdots \\\nabla ^{\mathsf {T}}f_{m}\end{bmatrix}}={\begin{bmatrix}{\dfrac {\partial f_{1}}{\partial x_{1}}}&\cdots &{\dfrac {\partial f_{1}}{\partial x_{n}}}\\\vdots &\ddots &\vdots \\{\dfrac {\partial f_{m}}{\partial x_{1}}}&\cdots &{\dfrac {\partial f_{m}}{\partial x_{n}}}\end{bmatrix}}.}
Gradient of a vector fieldEdit
{\displaystyle \nabla \mathbf {f} =g^{jk}{\frac {\partial f^{i}}{\partial x^{j}}}\mathbf {e} _{i}\otimes \mathbf {e} _{k},}
{\displaystyle {\frac {\partial f^{i}}{\partial x^{j}}}={\frac {\partial (f^{1},f^{2},f^{3})}{\partial (x^{1},x^{2},x^{3})}}.}
{\displaystyle \nabla \mathbf {f} =g^{jk}\left({\frac {\partial f^{i}}{\partial x^{j}}}+{\Gamma ^{i}}_{jl}f^{l}\right)\mathbf {e} _{i}\otimes \mathbf {e} _{k},}
{\displaystyle \nabla ^{a}f^{b}=g^{ac}\nabla _{c}f^{b},}
where ∇c is the connection.
Riemannian manifoldsEdit
{\displaystyle g(\nabla f,X)=\partial _{X}f,}
{\displaystyle g_{x}{\big (}(\nabla f)_{x},X_{x}{\big )}=(\partial _{X}f)(x),}
{\displaystyle \sum _{j=1}^{n}X^{j}{\big (}\varphi (x){\big )}{\frac {\partial }{\partial x_{j}}}(f\circ \varphi ^{-1}){\Bigg |}_{\varphi (x)},}
{\displaystyle \nabla f=g^{ik}{\frac {\partial f}{\partial x^{k}}}{\textbf {e}}_{i}.}
{\displaystyle (\partial _{X}f)(x)=(df)_{x}(X_{x}).}
{\displaystyle \sharp =\sharp ^{g}\colon T^{*}M\to TM}
Wikimedia Commons has media related to Gradient fields.
^ a b This article uses the convention that column vectors represent vectors, and row vectors represent covectors, but the opposite convention is also common.
^ Strictly speaking, the gradient is a vector field
{\displaystyle f\colon \mathbb {R} ^{n}\to T\mathbb {R} ^{n}}
, and the value of the gradient at a point is a tangent vector in the tangent space at that point,
{\displaystyle T_{p}\mathbb {R} ^{n}}
, not a vector in the original space
{\displaystyle \mathbb {R} ^{n}}
. However, all the tangent spaces can be naturally identified with the original space
{\displaystyle \mathbb {R} ^{n}}
, so these do not need to be distinguished; see § Definition and relationship with the derivative.
^ The value of the gradient at a point can be thought of as a vector in the original space
{\displaystyle \mathbb {R} ^{n}}
, while the value of the derivative at a point can be thought of as a covector on the original space: a linear map
{\displaystyle \mathbb {R} ^{n}\to \mathbb {R} }
^ Informally, "naturally" identified means that this can be done without making any arbitrary choices. This can be formalized with a natural transformation.
Bachman (2007, p. 76)
Beauregard & Fraleigh (1973, p. 84)
Downing (2010, p. 316)
Harper (1976, p. 15)
Kreyszig (1972, p. 307)
McGraw-Hill (2007, p. 196)
Moise (1967, p. 683)
Protter & Morrey, Jr. (1970, p. 714)
Swokowski et al. (1994, p. 1038)
Downing (2010, pp. 316–317)
Swokowski et al. (1994, pp. 1036, 1038–1039)
^ Kreyszig (1972, pp. 308–309)
^ Stoker (1969, p. 292)
^ a b Schey 1992, pp. 139–142.
^ Protter & Morrey, Jr. (1970, pp. 21, 88)
^ Beauregard & Fraleigh (1973, pp. 87, 248)
^ Kreyszig (1972, pp. 333, 353, 496)
^ Dubrovin, Fomenko & Novikov 1991, pp. 348–349.
Bachman, David (2007), Advanced Calculus Demystified, New York: McGraw-Hill, ISBN 978-0-07-148121-2
Downing, Douglas, Ph.D. (2010), Barron's E-Z Calculus, New York: Barron's, ISBN 978-0-7641-4461-5
Dubrovin, B. A.; Fomenko, A. T.; Novikov, S. P. (1991). Modern Geometry—Methods and Applications: Part I: The Geometry of Surfaces, Transformation Groups, and Fields. Graduate Texts in Mathematics (2nd ed.). Springer. ISBN 978-0-387-97663-1.
Kreyszig, Erwin (1972), Advanced Engineering Mathematics (3rd ed.), New York: Wiley, ISBN 0-471-50728-8
"McGraw Hill Encyclopedia of Science & Technology". McGraw-Hill Encyclopedia of Science & Technology (10th ed.). New York: McGraw-Hill. 2007. ISBN 978-0-07-144143-8.
Schey, H. M. (1992). Div, Grad, Curl, and All That (2nd ed.). W. W. Norton. ISBN 0-393-96251-2. OCLC 25048561.
Stoker, J. J. (1969), Differential Geometry, New York: Wiley, ISBN 0-471-82825-4
Swokowski, Earl W.; Olinick, Michael; Pence, Dennis; Cole, Jeffery A. (1994), Calculus (6th ed.), Boston: PWS Publishing Company, ISBN 0-534-93624-5
Korn, Theresa M.; Korn, Granino Arthur (2000). Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review. Dover Publications. pp. 157–160. ISBN 0-486-41147-8. OCLC 43864234.
Kuptsov, L.P. (2001) [1994], "Gradient", Encyclopedia of Mathematics, EMS Press .
Weisstein, Eric W. "Gradient". MathWorld.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Gradient&oldid=1088990827"
|
Implement vehicle in 3D environment - Simulink - MathWorks United Kingdom
Translation=\left[\begin{array}{ccc}{X}_{v}& {Y}_{v}& {Z}_{v}\\ {X}_{FL}& {Y}_{FL}& {Z}_{FL}\\ {X}_{FR}& {Y}_{FR}& {Z}_{FR}\\ {X}_{RL}& {Y}_{RL}& {Z}_{RL}\\ {X}_{RR}& {Y}_{RR}& {Z}_{RR}\end{array}\right]
Rotation=\left[\begin{array}{ccc}Rol{l}_{v}& Pitc{h}_{v}& Ya{w}_{v}\\ Rol{l}_{FL}& Pitc{h}_{FL}& Ya{w}_{FL}\\ Rol{l}_{FR}& Pitc{h}_{FR}& Ya{w}_{FR}\\ Rol{l}_{RL}& Pitc{h}_{RL}& Ya{w}_{RL}\\ Rol{l}_{RR}& Pitc{h}_{RR}& Ya{w}_{RR}\end{array}\right]
Scale=\left[\begin{array}{ccc}{X}_{{V}_{scale}}& {Y}_{{V}_{scale}}& {Z}_{{V}_{scale}}\\ {X}_{F{L}_{scale}}& {Y}_{F{L}_{scale}}& {Z}_{F{L}_{scale}}\\ {X}_{F{R}_{scale}}& {Y}_{F{R}_{scale}}& {Z}_{F{R}_{scale}}\\ {X}_{R{L}_{scale}}& {Y}_{R{L}_{scale}}& {Z}_{R{L}_{scale}}\\ {X}_{R{R}_{scale}}& {Y}_{R{R}_{scale}}& {Z}_{R{R}_{scale}}\end{array}\right]
Translation=\left[\begin{array}{ccc}{X}_{v}& {Y}_{v}& {Z}_{v}\\ {X}_{FL}& {Y}_{FL}& {Z}_{FL}\\ {X}_{FR}& {Y}_{FR}& {Z}_{FR}\\ {X}_{RL}& {Y}_{RL}& {Z}_{RL}\\ {X}_{RR}& {Y}_{RR}& {Z}_{RR}\end{array}\right]
Rotation=\left[\begin{array}{ccc}Rol{l}_{v}& Pitc{h}_{v}& Ya{w}_{v}\\ Rol{l}_{FL}& Pitc{h}_{FL}& Ya{w}_{FL}\\ Rol{l}_{FR}& Pitc{h}_{FR}& Ya{w}_{FR}\\ Rol{l}_{RL}& Pitc{h}_{RL}& Ya{w}_{RL}\\ Rol{l}_{RR}& Pitc{h}_{RR}& Ya{w}_{RR}\end{array}\right]
Scale=\left[\begin{array}{ccc}{X}_{{V}_{scale}}& {Y}_{{V}_{scale}}& {Z}_{{V}_{scale}}\\ {X}_{F{L}_{scale}}& {Y}_{F{L}_{scale}}& {Z}_{F{L}_{scale}}\\ {X}_{F{R}_{scale}}& {Y}_{F{R}_{scale}}& {Z}_{F{R}_{scale}}\\ {X}_{R{L}_{scale}}& {Y}_{R{L}_{scale}}& {Z}_{R{L}_{scale}}\\ {X}_{R{R}_{scale}}& {Y}_{R{R}_{scale}}& {Z}_{R{R}_{scale}}\end{array}\right]
|
LMI Applications - MATLAB & Simulink - MathWorks United Kingdom
RMS Gain
LQG Performance
Finding a solution x to the LMI system
A(x) < 0 (1)
is called the feasibility problem. Minimizing a convex objective under LMI constraints is also a convex problem. In particular, the linear objective minimization problem:
Minimize cTx subject to
plays an important role in LMI-based design. Finally, the generalized eigenvalue minimization problem
Minimize λ subject to
\begin{array}{l}A\left(x\right)<\lambda B\left(x\right)\\ B\left(x\right)>0\\ C\left(x\right)>0\end{array}
is quasi-convex and can be solved by similar techniques. It owes its name to the fact that is related to the largest generalized eigenvalue of the pencil (A(x),B(x)).
Many control problems and design specifications have LMI formulations [9]. This is especially true for Lyapunov-based analysis and design, but also for optimal LQG control, H∞ control, covariance control, etc. Further applications of LMIs arise in estimation, identification, optimal design, structural design [6], [7], matrix scaling problems, and so on. The main strength of LMI formulations is the ability to combine various design constraints or objectives in a numerically tractable manner.
A nonexhaustive list of problems addressed by LMI techniques includes the following:
Robust stability of systems with LTI uncertainty (µ-analysis) ([24], [21], [27])
Robust stability in the face of sector-bounded nonlinearities (Popov criterion) ([22], [28], [13], [16])
Quadratic stability of differential inclusions ([15], [8])
Lyapunov stability of parameter-dependent systems ([12])
Input/state/output properties of LTI systems (invariant ellipsoids, decay rate, etc.) ([9])
Multi-model/multi-objective state feedback design ([4], [17], [3], [9], [10])
Optimal LQG control ([9])
Robust H∞ control ([11], [14])
Multi-objective H∞ synthesis ([18], [23], [10], [18])
Design of robust gain-scheduled controllers ([5], [2])
Control of stochastic systems ([9])
Weighted interpolation problems ([9])
To hint at the principles underlying LMI design, let's review the LMI formulations of a few typical design objectives.
The stability of the dynamic system
\stackrel{˙}{x}=Ax
is equivalent to the feasibility of the following problem:
Find P = PT such that AT P + P A < 0, P > I.
This can be generalized to linear differential inclusions (LDI)
\stackrel{˙}{x}=A\left(t\right)x
where A(t) varies in the convex envelope of a set of LTI models:
A\left(t\right)\in C\text{o}\left\{{A}_{\text{1}},\dots ,{A}_{n}\right\}=\left\{\sum _{i=1}^{n}{a}_{i}{A}_{i}:{a}_{i}\ge 0,\sum _{i=1}^{N}{a}_{i}=1\right\}.
A sufficient condition for the asymptotic stability of this LDI is the feasibility of
Find P = PT such that
{A}_{i}^{T}P+P{A}_{i}<0,\text{ }P>I
The random-mean-squares (RMS) gain of a stable LTI system
\left\{\begin{array}{c}\stackrel{˙}{x}=Ax+Bu\\ y=Cx+Du\end{array}
is the largest input/output gain over all bounded inputs u(t). This gain is the global minimum of the following linear objective minimization problem [1], [25], [26].
Minimize γ over X = XT and γ such that
\left(\begin{array}{ccc}{A}^{T}X+XA& XB& {C}^{T}\\ {B}^{T}X& -\gamma I& {D}^{T}\\ C& D& -\gamma I\end{array}\right)<0
X>0.
For a stable LTI system
G\text{\hspace{0.17em}}\left\{\begin{array}{c}\stackrel{˙}{x}=Ax+Bw\\ y=Cx\end{array}
where w is a white noise disturbance with unit covariance, the LQG or H2 performance ∥G∥2 is defined by
\begin{array}{c}{‖G‖}_{2}^{2}:\text{\hspace{0.17em}}=\underset{T\to \infty }{\mathrm{lim}}E\left\{\frac{1}{T}\underset{0}{\overset{T}{\int }}{y}^{T}\left(t\right)y\left(t\right)dt\right\}\\ =\frac{1}{2\pi }\underset{-\infty }{\overset{\infty }{\int }}{G}^{H}\left(j\omega \right)G\left(j\omega \right)d\omega .\end{array}
{‖G‖}_{2}^{2}=\mathrm{inf}\left\{\text{Trace}\left({\text{CPC}}^{T}\right):AP+P{A}^{T}+B{B}^{T}<0\right\}.
{‖G‖}_{2}^{2}
is the global minimum of the LMI problem. Minimize Trace (Q) over the symmetric matrices P,Q such that
AP+P{A}^{T}+B{B}^{T}<0
\left(\begin{array}{cc}Q& CP\\ P{C}^{T}& P\end{array}\right)>0.
Again this is a linear objective minimization problem since the objective Trace (Q) is linear in the decision variables (free entries of P,Q).
|
The Rational(f, k) command computes a closed form of the indefinite sum of
k
f\left(k\right)
s\left(k\right)
t\left(k\right)
f\left(k\right)=s\left(k+1\right)-s\left(k\right)+t\left(k\right)
t\left(k\right)
k
\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\sum }_{k}t\left(k\right)
g,[p,q]
g
is the closed form of the indefinite sum of
k
p
is a list containing the integer poles of
q
s
that are not poles of
\mathrm{with}\left(\mathrm{SumTools}[\mathrm{IndefiniteSum}]\right):
f≔\frac{1}{{n}^{2}+\mathrm{sqrt}\left(5\right)n-1}
\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{≔}\frac{\textcolor[rgb]{0,0,1}{1}}{{\textcolor[rgb]{0,0,1}{n}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\sqrt{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}}
g≔\mathrm{Rational}\left(f,n\right)
\textcolor[rgb]{0,0,1}{g}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{3}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\frac{\sqrt{\textcolor[rgb]{0,0,1}{5}}}{\textcolor[rgb]{0,0,1}{2}}\right)}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\frac{\sqrt{\textcolor[rgb]{0,0,1}{5}}}{\textcolor[rgb]{0,0,1}{2}}\right)}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\frac{\sqrt{\textcolor[rgb]{0,0,1}{5}}}{\textcolor[rgb]{0,0,1}{2}}\right)}
\mathrm{evala}\left(\mathrm{Normal}\left(\mathrm{eval}\left(g,n=n+1\right)-g\right),\mathrm{expanded}\right)
\frac{\textcolor[rgb]{0,0,1}{1}}{{\textcolor[rgb]{0,0,1}{n}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\sqrt{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}}
f≔\frac{13-57x+2y+20{x}^{2}-18xy+10{y}^{2}}{15+10x-26y-25{x}^{2}+10xy+8{y}^{2}}
\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{≔}\frac{\textcolor[rgb]{0,0,1}{20}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{18}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{10}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{57}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{13}}{\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{25}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{10}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{8}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{10}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{26}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{15}}
g≔\mathrm{Rational}\left(f,x\right)
\textcolor[rgb]{0,0,1}{g}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}}{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{+}\left(\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{7}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}}{\textcolor[rgb]{0,0,1}{25}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{34}}{\textcolor[rgb]{0,0,1}{25}}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{\Psi }}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}}{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{3}}{\textcolor[rgb]{0,0,1}{5}}\right)\textcolor[rgb]{0,0,1}{+}\left(\frac{\textcolor[rgb]{0,0,1}{17}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}}{\textcolor[rgb]{0,0,1}{25}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{3}}{\textcolor[rgb]{0,0,1}{5}}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{\Psi }}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}}{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\right)
\mathrm{simplify}\left(\mathrm{combine}\left(f-\left(\mathrm{eval}\left(g,x=x+1\right)-g\right),\mathrm{\Psi }\right)\right)
\textcolor[rgb]{0,0,1}{0}
f≔\frac{1}{n}-\frac{2}{n-3}+\frac{1}{n-5}
\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{≔}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{n}}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{2}}{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{5}}
g,\mathrm{fp}≔\mathrm{Rational}\left(f,n,'\mathrm{failpoints}'\right)
\textcolor[rgb]{0,0,1}{g}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{fp}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{..}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{..}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{..}\textcolor[rgb]{0,0,1}{5}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}]]
f
n=0,3,5
g
n=1,2,4
|
The Bubble sort is a simple sorting algorithm that continues to swap adjacent values, if they are in the incorrect order, until all the values are sorted.
The steps in the algorithm
To sort an array of size n in ascending order, it performs n-1 iterations and in each iteration:
It scans through the entire array.
It compares element at index i with the element at i+1.
If the element at position i is greater than the element at i+1, then the elements are swapped.
On average, this algorithm takes
O(n^2)
time (where n is the size of the array). The algorithm can be optimized to take
O(n)
, if an array is already sorted in the required order, by breaking the outer for-loop if no element is swapped in the inner for-loop.
The following illustration shows the algorithm in action:
int arrayA[] = {3, 2, 1, 0};
// Finding the size of the array
auto n = end(arrayA) - begin(arrayA);
// Sorting array_A in ascending order
for(int it = 0; it < (n - 1); it++){
for(int i = 0; i < (n - 1); i++){
// Comparing adjacent elements
if(arrayA[i] > arrayA[i + 1]){
// Since the element at i is greater,
//swap elements at index i and i+1
int temp = arrayA[i];
arrayA[i] = arrayA[i + 1];
arrayA[i + 1] = temp;
// Optimization. Break the outer for loop
// if no element is swapped in the inner for loop.
cout<< arrayA[i] << " ";
|
Americium-241 - Wikipedia
Radioactive isotope of Americium
3.1 Ionization-type smoke detector
3.2 RTG (radioisotope thermoelectric generator) power generation
3.4 Production of other elements
Nucleosynthesis[edit]
{\displaystyle \mathrm {^{238}_{\ 92}U\ {\xrightarrow {(n,\gamma )}}\ _{\ 92}^{239}U\ {\xrightarrow[{23.5\ min}]{\beta ^{-}}}\ _{\ 93}^{239}Np\ {\xrightarrow[{2.3565\ d}]{\beta ^{-}}}\ _{\ 94}^{239}Pu} }
{\displaystyle \mathrm {^{239}_{\ 94}Pu\ {\xrightarrow {2~(n,\gamma )}}\ _{\ 94}^{241}Pu\ {\xrightarrow[{14.35\ yr}]{\beta ^{-}}}\ _{\ 95}^{241}Am} }
{\displaystyle \mathrm {^{241}_{\ 95}Am\ {\xrightarrow {(n,\gamma )}}\ _{\ 95}^{242}Am} }
{\displaystyle \mathrm {^{241\!\,}_{\ 95}Am\ {\overset {432.2y}{\longrightarrow }}\ _{\ 93}^{237}Np~+~_{2}^{4}\alpha ^{2+}+\gamma ~59.5409~keV} }
{\displaystyle \mathrm {^{241}_{\ 95}Am\longrightarrow ~_{\ 95}^{241}Am^{*}\longrightarrow 3_{0}^{1}n~+~fission~products~+energy~(\gamma )} }
{\displaystyle \mathrm {^{241\!\,}_{\ 95}Am\longrightarrow _{\ 81}^{207}Tl+_{14}^{34}Si} }
Ionization-type smoke detector[edit]
Main article: Smoke detector
RTG (radioisotope thermoelectric generator) power generation[edit]
{\displaystyle \mathrm {^{241\!\,}_{\ 95}Am\ {\overset {432.2y}{\longrightarrow }}\ _{\ 93}^{237}Np\ +\ _{2}^{4}\alpha ^{2+}+\ \gamma ~59.5~keV} }
{\textstyle \mathrm {^{9}_{4}Be\ +\ _{2}^{4}\alpha ^{2+}\longrightarrow \ _{\ 6}^{12}C\ +\ _{0}^{1}n\ +\ \gamma } }
Production of other elements[edit]
{\displaystyle \mathrm {^{241}_{\ 95}Am\ {\xrightarrow {(n,\gamma )}}\ _{\ 95}^{242}Am} }
{\displaystyle \mathrm {^{241}_{\ 95}Am\ {\xrightarrow {(n,\gamma )}}\ _{\ 95}^{242}Am\ {\xrightarrow[{16.02\ h}]{\beta ^{-}}}\ _{\ 96}^{242}Cm} }
{\displaystyle \mathrm {^{241}_{\ 95}Am\ {\xrightarrow {(n,\gamma )}}\ _{\ 95}^{242}Am\ {\xrightarrow[{16.02\ h}]{\beta ^{+}}}\ _{\ 94}^{242}Pu} }
{\displaystyle \mathrm {^{242}_{\ 95}Am{\xrightarrow {(n,\gamma )}}~_{\ 95}^{243}Am\ {\xrightarrow {(n,\gamma )}}\ _{\ 95}^{244}Am\ {\xrightarrow[{10.1\ h}]{\beta ^{-}}}\ _{\ 96}^{244}Cm} }
Spectrometer[edit]
This section's factual accuracy is disputed. Relevant discussion may be found on Talk:Americium-241. Please help to ensure that disputed statements are reliably sourced. (February 2020) (Learn how and when to remove this template message)
Retrieved from "https://en.wikipedia.org/w/index.php?title=Americium-241&oldid=1088623792"
|
Coulomb's Law - Maple Help
Coulomb's inverse square law is a law of physics that describes the electrostatic interaction between two electrically charged particles.
F = \frac{{k}_{e} {q}_{1}{q}_{2}}{{r}^{2}}
{k}_{e}
is a proportionality constant approximately equal to
8.987551 {\mathrm{x10}}^{9}
\frac{N\cdot {m}^{2}}{ {C}^{2}}
{q}_{1}
{q}_{2}
are the charges of the two particles,
r
is the distance between the two particles.
If the force is positive the particles are being repulsed. If the force is negative, the particles are being attracted.
Adjust the sliders to change the charge and distance of each particle.
{q}_{1}=
×{10}^{-5} C
{q}_{2}=
×{10}^{-5} C
r=
m
|
Put together function in stform - MATLAB stmak - MathWorks Italia
stmak
Put together function in stform
stmak(centers,coefs)
st = stmak(centers,x,type)
st = stmak(centers,coefs,type,interv)
stmak(centers,coefs) returns the stform of the function f given by
f\left(x\right)=\sum _{j=1}^{n}\text{coefs}\left(:,j\right)\cdot \psi \left(x-\text{centers}\left(:,j\right)\right)
\psi \left(x\right)={|x|}^{2}\mathrm{log}{|x|}^{2}
the thin-plate spline basis function, and with |x| denoting the Euclidean norm of the vector x.
centers and coefs must be matrices with the same number of columns.
st = stmak(centers,x,type) stores in st the stform of the function f given by
f\left(x\right)=\sum _{j=1}^{n}\text{coefs}\left(:,j\right)\cdot {\psi }_{j}\left(x\right)
with the ψj as indicated by the character vector or string scalar type, which can be one of the following:
'tp00', for the thin-plate spline;
'tp10', for the first derivative of a thin-plate spline with respect to its first argument;
'tp01', for the first derivative of a thin-plate spline with respect to its second argument;
'tp', the default.
'tp00'
ψj(x) = φ(|x – cj|2), cj =centers(:,j), j=1:n-3
with φ(t) = tlog(t)
ψn–2(x) = x(1)
ψn(x) = 1
with φ(t) = (D1t)(logt + 1), and D1t the partial derivative of t = t(x) = |x – cj|2
with respect to x(1)
'tp' (default)
ψj(x) = φ(|x – cj|2), cj =centers(:,j), j=1:n
st = stmak(centers,coefs,type,interv) also specifies the basic interval for the stform, with interv{j} specifying, in the form [a,b], the range of the jth variable. The default for interv is the smallest such box that contains all the given centers.
Example 1. The following generates the figure below, of the thin-plate spline basis function,
\psi \left(x\right)={|x|}^{2}\mathrm{log}{|x|}^{2},
but suitably restricted to show that this function is negative near the origin. For this, the extra lines are there to indicate the zero level.
inx = [-1.5 1.5]; iny = [0 1.2];
fnplt(stmak([0;0],1),{inx,iny})
hold on, plot(inx,repmat(linspace(iny(1),iny(2),11),2,1),'r')
view([25,20]),axis off, hold off
Example 2. We now also generate and plot, on the very same domain, the first partial derivative D2ψ of the thin-plate spline basis function, with respect to its second argument.
fnplt(stmak([0;0],[1 0],'tp01',{inx,iny}))
view([13,10]),shading flat,axis off
Note that, this time, we have explicitly set the basic interval for the stform.
The resulting figure, below, shows a very strong variation near the origin. This reflects the fact that the second derivatives of ψ have a logarithmic singularity there.
|
Merton jump diffusion model - MATLAB - MathWorks 日本
d{X}_{t}=B\left(t,{X}_{t}\right){X}_{t}dt+D\left(t,{X}_{t}\right)V\left(t,{x}_{t}\right)d{W}_{t}+Y\left(t,{X}_{t},{N}_{t}\right){X}_{t}d{N}_{t}
Instantaneous mean of random percentage jump sizes J, where log(1+J) is normally distributed with mean (log(1+JumpMean) - 0.5 × JumpVol2) and standard deviation JumpVol, specified as an array, a deterministic function of time, or a deterministic function of time and state.
F\left(t,{X}_{t}\right)=A\left(t\right)+B\left(t\right){X}_{t}
G\left(t,{X}_{t}\right)=D\left(t,{X}_{t}^{\mathrm{α}\left(t\right)}\right)V\left(t\right)
\begin{array}{l}d{S}_{t}=\left(\mathrm{γ}âqâ{\mathrm{λ}}_{p}{\mathrm{μ}}_{j}\right){S}_{t}dt+{\mathrm{Ï}}_{M}{S}_{t}d{W}_{t}+J{S}_{t}d{P}_{t}\\ \text{prob}\left(d{P}_{t}=1\right)={\mathrm{λ}}_{p}dt\end{array}
ᵞ is the continuous risk-free rate.
\mathrm{ln}\left(1+J\right)~N\left(\text{ln(1+}{u}_{j}\right)â\frac{{\mathrm{δ}}^{2}}{2},{\mathrm{δ}}^{2}
\frac{1}{\left(1+J\right)\mathrm{δ}\sqrt{2\mathrm{Ï}}}\text{exp}\left\{\frac{â{\left[\mathrm{ln}\left(1+J\right)â\left(\text{ln(1+}{\mathrm{μ}}_{j}\right)â\frac{{\mathrm{δ}}^{2}}{2}\right]}^{2}}{2{\mathrm{δ}}^{2}}\right\}
μj is the mean of J(μj > -1).
ƛp is the annual frequency (intensity) of the Poisson process Pt (ƛp ≥ 0).
σM is the volatility of the asset price (σM> 0).
[1] Aït-Sahalia, Yacine. “Testing Continuous-Time Models of the Spot Interest Rate.†Review of Financial Studies 9, no. 2 ( Apr. 1996): 385–426.
[2] Aït-Sahalia, Yacine. “Transition Densities for Interest Rate and Other Nonlinear Diffusions.†The Journal of Finance 54, no. 4 (Aug. 1999): 1361–95.
|
i\le j
\mathrm{Hemingway}≔"Nothing happened. The fish just moved away slowly and the old man could not raise him an inch. His line was strong and made for heavy fish and he held it against his hack until it was so taut that beads of water were jumping from it. Then it began to make a slow hissing sound in the water and he still held it, bracing himself against the thwart and leaning back against the pull. The boat began to move slowly off toward the north-west.":
\mathrm{Melville}≔"I see in him outrageous strength, with an inscrutable malice sinewing it. That inscrutable thing is chiefly what I hate; and be the white whale agent, or be the white whale principal, I will wreak that hate upon him.":
\mathrm{Anderson}≔"They were six beautiful children; but the youngest was the prettiest of them all; her skin was as clear and delicate as a rose-leaf, and her eyes as blue as the deepest sea; but, like all the others, she had no feet, and her body ended in a fish\text{'}s tail.":
\mathrm{with}\left(\mathrm{EssayTools}\right):
\mathrm{SimilarityScore}\left(\mathrm{Hemingway},\mathrm{Melville}\right)
[\begin{array}{c}\textcolor[rgb]{0,0,1}{0.1025641026}\end{array}]
\mathrm{SimilarityScore}\left([\mathrm{Hemingway},\mathrm{Melville},\mathrm{Anderson}]\right)
[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{1.}& \textcolor[rgb]{0,0,1}{0.1025641026}& \textcolor[rgb]{0,0,1}{0.08139534884}\\ \textcolor[rgb]{0,0,1}{0.1025641026}& \textcolor[rgb]{0,0,1}{1.}& \textcolor[rgb]{0,0,1}{0.08333333333}\\ \textcolor[rgb]{0,0,1}{0.08139534884}& \textcolor[rgb]{0,0,1}{0.08333333333}& \textcolor[rgb]{0,0,1}{1.}\end{array}]
\mathrm{SimilarityScore}\left([\mathrm{Hemingway},\mathrm{Melville},\mathrm{Anderson}],\mathrm{methods}=[\mathrm{EssayTools}:-\mathrm{CosineCoefficient}]\right)
[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{1.000000000}& \textcolor[rgb]{0,0,1}{0.3583566001}& \textcolor[rgb]{0,0,1}{0.3811595642}\\ \textcolor[rgb]{0,0,1}{0.3583566001}& \textcolor[rgb]{0,0,1}{1.000000000}& \textcolor[rgb]{0,0,1}{0.2912325843}\\ \textcolor[rgb]{0,0,1}{0.3811595642}& \textcolor[rgb]{0,0,1}{0.2912325843}& \textcolor[rgb]{0,0,1}{1.000000000}\end{array}]
\mathrm{SimilarityScore}\left(["a b c"],["a b c","a a a","b c d","x y z"]\right)
[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{1.}& \textcolor[rgb]{0,0,1}{0.3333333333}& \textcolor[rgb]{0,0,1}{0.5000000000}& \textcolor[rgb]{0,0,1}{0.}\end{array}]
|
Limit (mathematics) - Wikiversity
The concept of the limit is the cornerstone of calculus, analysis, and topology. For starters, the limit of a function at a point is, intuitively, the value that the function "approaches" as its argument "approaches" that point. That idea needs to be refined carefully to get a satisfactory definition.
1 Limit of a Function
1.1 For Every Epsilon ...
1.2 ... There Exists a Delta
1.3 The Full Definition
2 The Limit of f(x) is Infinity
3 The limit of f(x) as x Goes to Infinity
4 The limit of f(x) as x Goes to Infinity is Infinity
6 Infinite Sums
Limit of a Function[edit | edit source]
Here is an example. Suppose a function is
{\displaystyle f(x)={\frac {x^{2}-x-6}{x^{2}-2x-3}}\,}
What is the limit of f(x) as x approaches 3? This could be written
{\displaystyle \lim _{x\to 3}f(x)\,}
We can't just evaluate f(3), because the numerator and denominator are both zero. But there's a trick here—we can divide (x-3) into both numerator and denominator, getting
{\displaystyle f(x)={\frac {x+2}{x+1}}\,}
so the limit is 5/4. (That does't actually prove that the limit is 5/4. Once we have defined the limit correctly, we will need a few theorems about limits and continuous functions to establish this result. It is nevertheless true.)
Now try an example that isn't trivial. Let
The function x^x, for x > 0.
{\displaystyle f(x)=x^{x}\,}
This function is well-defined for x>0, using the general definition that involves the exponential and natural logarithm functions. We can calculate f(x) for various values:
f(0.5) = 0.7071
It got smaller, but now it's getting bigger. What's happening?
f(0.0001) = 0.999079
f(0.000001) = 0.99998618
It looks as though it's approaching 1. Is it? And what does that mean?
f(1 trillionth) = 0.999999999972369
Remember that f(0) doesn't exist. So we really have to be careful.
We are going to say that the limit of f(x), as x approaches 0, is 1. What that means is this:
We can get f(x) arbitrarily close to 1 if we choose an x sufficiently close to zero. If we want f(x) within one quadrillionth of 1,
{\displaystyle x=10^{-17}\,}
will do. We never have to set x to zero exactly, and we never have to get f(x) = 1 exactly.
In the general case, for arbitrary functions, we might want to say something like
{\displaystyle \lim _{x\to X}f(x)=Y\,}
For Every Epsilon ...[edit | edit source]
Stated precisely, given any tolerance ε (by tradition, the letter ε is always used) x being sufficiently close to X will get f(x) within ε of Y. That condition is formally written:
{\displaystyle |f(x)-Y|<\varepsilon \,}
In this example (
{\displaystyle f(x)=x^{x}\,}
, X=0, Y=1), when ε is
{\displaystyle 10^{-15}\,}
(one quadrillionth),
{\displaystyle x<10^{-17}\,}
satisfies the condition.
... There Exists a Delta[edit | edit source]
The way we formalize the notion of x being very close to X is:
"There is a number δ (by tradition it's always δ) such that, whenever x is within δ of X, f(x) is within ε of Y".
{\displaystyle 0<|x-X|<\delta ,|f(x)-Y|<\varepsilon \,}
In our example, if ε is
{\displaystyle 10^{-15}\,}
, δ =
{\displaystyle 10^{-17}\,}
works. That is, any
{\displaystyle x<10^{-17}\,}
will guarantee
{\displaystyle |f(x)-1|<10^{-15}\,}
The Full Definition[edit | edit source]
So here is the full definition:
{\displaystyle \lim _{x\to X}f(x)=Y\,}
For every ε > 0, there is a δ > 0 such that, whenever
{\displaystyle 0<|x-X|<\delta ,|f(x)-Y|<\varepsilon \,}
So the definition is sort of like a bet or a contract—"For any epsilon you can give me, I can come up with a delta."
We require ε > 0. Specifying a required tolerance of zero is not allowed. We only have to be able to get f(x) within an arbitrarily close but nonzero tolerance of Y. We don't ever have to get it exactly equal to Y.
We have 0 < |x-X| < δ, not just |x-X| < δ. That is, we never have to calculate f(X) exactly. f(X) doesn't need to be defined. In the example we are considering,
{\displaystyle 0^{0}\,}
isn't defined.
This definition, and variations of it, are the central point of calculus, analysis, and topology. The phrase "For every ε there is a δ" is ingrained into the consciousness of every mathematics student. This notion of a "bet" could be considered to set the branches of mathematics that follow (calculus, topology, ...) apart from the earlier branches (arithmetic, algebra, geometry, ...) Students who have mastered the notion of "For every ε there is a δ" are ready for higher mathematics.
In our example of
{\displaystyle f(x)=x^{x}\,}
, we haven't actually satisfied the definition of the limit, because f(x) isn't defined for negative x. There are more restrictive notions of "limit from the left" and "limit from the right". We have the limit from the right of
{\displaystyle x^{x}=1\,}
For every ε > 0, there is a δ > 0 such that, whenever 0 < x-X < δ, |f(x)-Y| < ε
We still haven't proved that the limit is actually 1. We just gave some accurate calculations strongly suggesting that it is. In fact it is, and the proof requires a few theorems about limits, continuous functions, and the exponential and natural logarithm functions.
The Limit of f(x) is Infinity[edit | edit source]
With this precise (or, as mathematicians say, rigorous) definition of a limit, we can examine variations that involve "infinity". Remember, Infinity is not a number. It is only through the magic of the epsilon-delta formulation that we can make sense of it.
We might say something like "The limit of f(x), as x approaches X, is infinity." What that means is that we replace the
{\displaystyle |f(x)-Y|<\varepsilon \,}
{\displaystyle f(x)>M\,}
For every M > 0, there is a δ > 0 such that, whenever
{\displaystyle 0<|x-X|<\delta ,f(x)>M\,}
{\displaystyle f(x)={\frac {1}{(x-3)^{2}}}\,}
If you graph this, there is an "infinitely high" peak at x=3. We have:
{\displaystyle \lim _{x\to 3}f(x)=\infty \,}
{\displaystyle M=10^{20}\,}
{\displaystyle \delta =10^{-10}\,}
will win the bet.
For a limit of minus infinity, we set M to some huge negative number, so
{\displaystyle \lim _{x\to X}f(x)=-\infty \,}
For every M < 0, there is a δ > 0 such that, whenever
{\displaystyle 0<|x-X|<\delta ,f(x)<M\,}
The limit of f(x) as x Goes to Infinity[edit | edit source]
Infinity can also make an appearance in the function's domain, as in "The limit of f(x), as x approaches infinity, is 3." What that means is that we replace the
{\displaystyle 0<|x-X|<\delta \,}
{\displaystyle x>M\,}
For every ε > 0, there is an M such that, whenever
{\displaystyle x>M,|f(x)-Y|<\varepsilon \,}
{\displaystyle f(x)={\frac {3x+5}{x}}\,}
If you graph this, it goes off to a value of 3 toward the right. We have:
{\displaystyle \lim _{x\to \infty }f(x)=3\,}
{\displaystyle \varepsilon =10^{-10}\,}
{\displaystyle M=10^{11}\,}
We can do a similar thing for the limit as x goes to minus infinity.
The limit of f(x) as x Goes to Infinity is Infinity[edit | edit source]
A function like
{\displaystyle f(x)=e^{x}\,}
combines both inifinites. We write
{\displaystyle \lim _{x\to \infty }e^{x}=\infty \,}
Here we replace both δ and ε:
For every N > 0, there is an M such that, whenever
{\displaystyle x>M,f(x)>N\,}
Limit of a Sequence[edit | edit source]
An important type of limit is the limit of a sequence of numbers
{\displaystyle (a_{n})_{n\in N}=(a_{1},a_{2},a_{3},...)}
. When we say that this sequence has a limit A, we are essentially using the definition above for x going to infinity, but using n instead of x:
{\displaystyle \lim _{n\to \infty }a_{n}=A,\,}
{\displaystyle n>M,|a_{n}-A|<\varepsilon \,}
When a sequence has a limit like that, we say that the sequence converges to that limit.
Infinite Sums[edit | edit source]
It is important to distinguish between the limit of the sequence, as defined above, and the much more commonly used limit of the partial sums of the sequence. If we have a sequence
{\displaystyle (a_{n})=(a_{1},a_{2},a_{3},...)}
, we also have a sequence of "partial sums", the nth item of which is the sum of the first n items in the original sequence:
{\displaystyle s_{n}=\sum _{i=1}^{n}a_{n}\,}
{\displaystyle s_{n}\,}
converges to A, we say that the infinite sum
{\displaystyle a_{n}\,}
converges to A, written like this:
{\displaystyle \sum _{i=1}^{\infty }a_{n}=A\,}
{\displaystyle a_{n}={\frac {1}{n!}}\,}
(where the exclamation point is the factorial operation) converges to the number known as "e":
{\displaystyle \sum _{i=0}^{\infty }{\frac {1}{n!}}=e\,}
This is the only sense in which a summation going to infinity is meaningful.
Retrieved from "https://en.wikiversity.org/w/index.php?title=Limit_(mathematics)&oldid=2252598"
|
Triviality (mathematics) - WikiMili, The Best Wikipedia Reader
In mathematics, the adjective trivial is often used to refer to a claim or a case which can be readily obtained from context, or an object which possesses a simple structure (e.g., groups, topological spaces). [1] [2] The noun triviality usually refers to a simple technical aspect of some proof or definition. The origin of the term in mathematical language comes from the medieval trivium curriculum, which distinguishes from the more difficult quadrivium curriculum. [1] [3] The opposite of trivial is nontrivial, which is commonly used to indicate that an example or a solution is not simple, or that a statement or a theorem is not easy to prove. [2]
In mathematical reasoning
In mathematics, the term "trivial" is often used to refer to objects (e.g., groups, topological spaces) with a very simple structure. These include, among others
Empty set: the set containing no or null members
Trivial group: the mathematical group containing only the identity element
Trivial ring: a ring defined on a singleton set
"Trivial" can also be used to describe solutions to an equation that have a very simple structure, but for the sake of completeness cannot be omitted. These solutions are called the trivial solutions. For example, consider the differential equation
{\displaystyle y'=y}
{\displaystyle y=y(x)}
{\displaystyle y'}
. The trivial solution is the zero function
{\displaystyle y(x)=0}
while a nontrivial solution is the exponential function
{\displaystyle y(x)=e^{x}.}
{\displaystyle f''(x)=-\lambda f(x)}
{\displaystyle f(0)=f(L)=0}
is important in math and physics, as it could be used to describe a particle in a box in quantum mechanics, or a standing wave on a string. It always includes the solution
{\displaystyle f(x)=0}
, which is considered obvious and hence is called the "trivial" solution. In some cases, there may be other solutions (sinusoids), which are called "nontrivial" solutions. [4]
Similarly, mathematicians often describe Fermat's Last Theorem as asserting that there are no nontrivial integer solutions to the equation
{\displaystyle a^{n}+b^{n}=c^{n}}
, where n is greater than 2. Clearly, there are some solutions to the equation. For example,
{\displaystyle a=b=c=0}
is a solution for any n, but such solutions are obvious and obtainable with little effort, and hence "trivial".
Trivial may also refer to any easy case of a proof, which for the sake of completeness cannot be ignored. For instance, proofs by mathematical induction have two parts: the "base case" which shows that the theorem is true for a particular initial value (such as n = 0 or n = 1), and the inductive step which shows that if the theorem is true for a certain value of n, then it is also true for the value n + 1. The base case is often trivial and is identified as such, although there are situations where the base case is difficult but the inductive step is trivial. Similarly, one might want to prove that some property is possessed by all the members of a certain set. The main part of the proof will consider the case of a nonempty set, and examine the members in detail; in the case where the set is empty, the property is trivially possessed by all the members, since there are none (see vacuous truth for more).
A common joke in the mathematical community is to say that "trivial" is synonymous with "proved"—that is, any theorem can be considered "trivial" once it is known to be true. [1]
Another joke concerns two mathematicians who are discussing a theorem: the first mathematician says that the theorem is "trivial". In response to the other's request for an explanation, he then proceeds with twenty minutes of exposition. At the end of the explanation, the second mathematician agrees that the theorem is trivial. These jokes point out the subjectivity of judgments about triviality. The joke also applies when the first mathematician says the theorem is trivial, but is unable to prove it himself. Often, as a joke, the theorem is then referred to as "intuitively obvious". Someone experienced in calculus, for example, would consider the following statement trivial:
{\displaystyle \int _{0}^{1}x^{2}\,dx={\frac {1}{3}}}
However, to someone with no knowledge of integral calculus, this is not obvious at all.
Triviality also depends on context. A proof in functional analysis would probably, given a number, trivially assume the existence of a larger number. However, when proving basic results about the natural numbers in elementary number theory, the proof may very well hinge on the remark that any natural number has a successor—a statement which should itself be proved or be taken as an axiom (for more, see Peano's axioms).
In some texts, a trivial proof refers to a statement involving a material implication P→Q, where the consequent, Q, is always true. [5] Here, the proof follows immediately by virtue of the definition of material implication, as the implication is true regardless of the truth value of the antecedent P. [5]
A related concept is a vacuous truth, where the antecedent P in the material implication P→Q is always false. [5] Here, the implication is always true regardless of the truth value of the consequent Q—again by virtue of the definition of material implication. [5]
In number theory, it is often important to find factors of an integer number N. Any number N has four obvious factors: ±1 and ±N. These are called "trivial factors". Any other factor, if it exists, would be called "nontrivial". [6]
The homogeneous matrix equation
{\displaystyle A\mathbf {x} =\mathbf {0} }
{\displaystyle A}
is a fixed matrix,
{\displaystyle \mathbf {x} }
is an unknown vector, and
{\displaystyle \mathbf {0} }
is the zero vector, has an obvious solution
{\displaystyle \mathbf {x} =\mathbf {0} }
. This is called the "trivial solution". If it has other solutions
{\displaystyle \mathbf {x} \neq \mathbf {0} }
, then they would be called "nontrivial" [7]
In group theory, there is a very simple group with just one element in it; this is often called the "trivial group". All other groups, which are more complicated, are called "nontrivial".
In graph theory, the trivial graph is a graph which has only 1 vertex and no edge.
Database theory has a concept called functional dependency, written
{\displaystyle X\to Y}
. The dependence
{\displaystyle X\to Y}
is true if Y is a subset of X, so this type of dependence is called "trivial". All other dependences, which are less obvious, are called "nontrivial".
It can be shown that Riemann's zeta function has zeros at the negative even numbers −2, −4, … Though the proof is comparatively easy, this result would still not normally be called trivial; however, it is in this case, for its other zeros are generally unknown and have important applications and involve open questions (such as the Riemann hypothesis). Accordingly, the negative even numbers are called the trivial zeros of the function, while any other zeros are considered to be non-trivial.
Trivial measure
In computability theory, Rice's theorem states that all non-trivial semantic properties of programs are undecidable. A semantic property is one about the program's behavior, unlike a syntactic property. A property is non-trivial if it is neither true for every partial computable function, nor false for every partial computable function.
In mathematics, physics, and engineering, a vector space is a set of objects called vectors, which may be added together and multiplied ("scaled") by numbers called scalars. Scalars are often real numbers, but some vector spaces have scalar multiplication by complex numbers or, generally, by a scalar from any mathematic field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called vector axioms. To specify whether the scalars in a particular vector space are real numbers or complex numbers, the terms real vector space and complex vector space are often used.
Hilbert's tenth problem is the tenth on the list of mathematical problems that the German mathematician David Hilbert posed in 1900. It is the challenge to provide a general algorithm which, for any given Diophantine equation, can decide whether the equation has a solution with all unknowns taking integer values.
In mathematics, the Abel–Ruffini theorem states that there is no solution in radicals to general polynomial equations of degree five or higher with arbitrary coefficients. Here, general means that the coefficients of the equation are viewed and manipulated as indeterminates.
An instanton is a notion appearing in theoretical and mathematical physics. An instanton is a classical solution to equations of motion with a finite, non-zero action, either in quantum mechanics or in quantum field theory. More precisely, it is a solution to the equations of motion of the classical field theory on a Euclidean spacetime.
In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form
In mathematics, the homotopy principle is a very general way to solve partial differential equations (PDEs), and more generally partial differential relations (PDRs). The h-principle is good for underdetermined PDEs or PDRs, such as occur in the immersion problem, isometric immersion problem, fluid dynamics, and other areas.
In universal algebra, a variety of algebras or equational class is the class of all algebraic structures of a given signature satisfying a given set of identities. For example, the groups form a variety of algebras, as do the abelian groups, the rings, the monoids etc. According to Birkhoff's theorem, a class of algebraic structures of the same signature is a variety if and only if it is closed under the taking of homomorphic images, subalgebras and (direct) products. In the context of category theory, a variety of algebras, together with its homomorphisms, forms a category; these are usually called finitary algebraic categories.
Farkas' lemma is a solvability theorem for a finite system of linear inequalities in mathematics. It was originally proven by the Hungarian mathematician Gyula Farkas. Farkas' lemma is the key result underpinning the linear programming duality and has played a central role in the development of mathematical optimization. It is used amongst other things in the proof of the Karush–Kuhn–Tucker theorem in nonlinear programming. Remarkably, in the area of the foundations of quantum theory, the lemma also underlies the complete set of Bell inequalities in the form of necessary and sufficient conditions for the existence of a local hidden-variable theory, given data from any specific set of measurements.
In mathematics, the Fredholm alternative, named after Ivar Fredholm, is one of Fredholm's theorems and is a result in Fredholm theory. It may be expressed in several ways, as a theorem of linear algebra, a theorem of integral equations, or as a theorem on Fredholm operators. Part of the result states that a non-zero complex number in the spectrum of a compact operator is an eigenvalue.
In linear algebra, eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. When the matrix being factorized is a normal or real symmetric matrix, the decomposition is called "spectral decomposition", derived from the spectral theorem.
In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known since antiquity to have infinitely many solutions.
Wiles's proof of Fermat's Last Theorem is a proof by British mathematician Andrew Wiles of a special case of the modularity theorem for elliptic curves. Together with Ribet's theorem, it provides a proof for Fermat's Last Theorem. Both Fermat's Last Theorem and the modularity theorem were almost universally considered inaccessible to proof by contemporaneous mathematicians, meaning that they were believed to be impossible to prove using current knowledge.
In mathematics, the Pythagorean theorem, or Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares on the other two sides. This theorem can be written as an equation relating the lengths of the sides a, b and c, often called the Pythagorean equation:
The Millennium Prize Problems are seven unsolved problems in mathematics that were stated by the Clay Mathematics Institute on May 24, 2000. The problems are the Birch and Swinnerton-Dyer conjecture, Hodge conjecture, Navier–Stokes existence and smoothness, P versus NP problem, Poincaré conjecture, Riemann hypothesis, and Yang–Mills existence and mass gap. A correct solution to any of the problems results in a US$1 million prize being awarded by the institute to the discoverer(s).
In algebraic geometry, the main theorem of elimination theory states that every projective scheme is proper. A version of this theorem predates the existence of scheme theory. It can be stated, proved, and applied in the following more classical setting. Let k be a field, denote by the n-dimensional projective space over k. The main theorem of elimination theory is the statement that for any n and any algebraic variety V defined over k, the projection map sends Zariski-closed subsets to Zariski-closed subsets.
1 2 3 Weisstein, Eric W. "Trivial". mathworld.wolfram.com. Retrieved 2019-12-14.
1 2 "Mathwords: Trivial". www.mathwords.com. Retrieved 2019-12-14.
↑ Ayto, John (1990). Dictionary of word origins. University of Texas Press. p. 542. ISBN 1-55970-214-1. OCLC 33022699.
↑ Zachmanoglou, E. C.; Thoe, Dale W. (1986). Introduction to Partial Differential Equations with Applications. p. 309. ISBN 9780486652511.
1 2 3 4 Chartrand, Gary; Polimeni, Albert D.; Zhang, Ping (2008). Mathematical proofs : a transition to advanced mathematics (2nd ed.). Boston: Pearson/Addison Wesley. p. 68. ISBN 978-0-3-2139053-0.
↑ Yan, Song Y. (2002). Number Theory for Computing (2nd, illustrated ed.). Berlin: Springer. p. 250. ISBN 3-540-43072-5.
↑ Jeffrey, Alan (2004). Mathematics for Engineers and Scientists (Sixth ed.). CRC Press. p. 502. ISBN 1-58488-488-6.
|
Tools for Specifying and Solving LMIs - MATLAB & Simulink - MathWorks United Kingdom
Overview of the LMI Lab
Specification of a System of LMIs
Solvers for LMI Optimization Problems
Modification of a System of LMIs
The LMI Lab is a high-performance package for solving general LMI problems. It blends simple tools for the specification and manipulation of LMIs with powerful LMI solvers for three generic LMI problems. Thanks to a structure-oriented representation of LMIs, the various LMI constraints can be described in their natural block-matrix form. Similarly, the optimization variables are specified directly as matrix variables with some given structure. Once an LMI problem is specified, it can be solved numerically by calling the appropriate LMI solver. The three solvers feasp, mincx, and gevp constitute the computational engine of the LMI portion of Robust Control Toolbox™ software. Their high performance is achieved through C-MEX implementation and by taking advantage of the particular structure of each LMI.
The LMI Lab offers tools to
Specify LMI systems either symbolically with the LMI Editor or incrementally with the lmivar and lmiterm commands
Retrieve information about existing systems of LMIs
Modify existing systems of LMIs
Solve the three generic LMI problems (feasibility problem, linear objective minimization, and generalized eigenvalue minimization)
This chapter gives a tutorial introduction to the LMI Lab as well as more advanced tips for making the most out of its potential.
Any linear matrix inequality can be expressed in the canonical form
L(x) = L0 + x1L1 + . . . + xNLN < 0
L0, L1, . . . , LN are given symmetric matrices
x = (x1, . . . , xN)T ∊ RN is the vector of scalar variables to be determined. We refer to x1, . . . , xN as the decision variables. The names “design variables” and “optimization variables” are also found in the literature.
Even though this canonical expression is generic, LMIs rarely arise in this form in control applications. Consider for instance the Lyapunov inequality
{A}^{T}X+XA<0
A=\left(\begin{array}{cc}-1& 2\\ 0& -2\end{array}\right)
X=\left(\begin{array}{cc}{x}_{1}& {x}_{2}\\ {x}_{2}& {x}_{3}\end{array}\right)
is a symmetric matrix. Here the decision variables are the free entries x1, x2, x3 of X and the canonical form of this LMI reads
{x}_{1}\left(\begin{array}{cc}-2& 2\\ 2& 0\end{array}\right)+{x}_{2}\left(\begin{array}{cc}0& -3\\ -3& 4\end{array}\right)+{x}_{3}\left(\begin{array}{cc}0& 0\\ 0& -4\end{array}\right)<0.
Clearly this expression is less intuitive and transparent than Equation 1. Moreover, the number of matrices involved in Equation 2 grows roughly as n2 /2 if n is the size of the A matrix. Hence, the canonical form is very inefficient from a storage viewpoint since it requires storing o(n2 /2) matrices of size n when the single n-by-n matrix A would be sufficient. Finally, working with the canonical form is also detrimental to the efficiency of the LMI solvers. For these various reasons, the LMI Lab uses a structured representation of LMIs. For instance, the expression ATX + XA in the Lyapunov inequality Equation 1 is explicitly described as a function of the matrix variable X, and only the A matrix is stored.
In general, LMIs assume a block matrix form where each block is an affine combination of the matrix variables. As a fairly typical illustration, consider the following LMI drawn from H∞ theory
{N}^{T}\left(\begin{array}{ccc}{A}^{T}X+XA& X{C}^{T}& B\\ CX& -\gamma I& D\\ {B}^{T}& {D}^{T}& -\gamma I\end{array}\right)N<0
where A, B, C, D, and N are given matrices and the problem variables are X = XT ∊ Rn×n and γ ∊ R. We use the following terminology to describe such LMIs:
N is called the outer factor, and the block matrix
L\left(X,\gamma \right)=\left(\begin{array}{ccc}{A}^{T}X+XA& X{C}^{T}& B\\ CX& -\gamma I& D\\ {B}^{T}& {D}^{T}& -\gamma I\end{array}\right)
is called the inner factor. The outer factor needs not be square and is often absent.
X and γ are the matrix variables of the problem. Note that scalars are considered as 1-by-1 matrices.
The inner factor L(X, γ) is a symmetric block matrix, its block structure being characterized by the sizes of its diagonal blocks. By symmetry, L(X, γ) is entirely specified by the blocks on or above the diagonal.
Each block of L(X, γ) is an affine expression in the matrix variables X and γ. This expression can be broken down into a sum of elementary terms. For instance, the block (1,1) contains two elementary terms: ATX and XA.
Terms are either constant or variable. Constant terms are fixed matrices like B and D above. Variable terms involve one of the matrix variables, like XA, XCT, and –γI above.
The LMI (Equation 3) is specified by the list of terms in each block, as is any LMI regardless of its complexity.
As for the matrix variables X and γ, they are characterized by their dimensions and structure. Common structures include rectangular unstructured, symmetric, skew-symmetric, and scalar. More sophisticated structures are sometimes encountered in control problems. For instance, the matrix variable X could be constrained to the block-diagonal structure:
X=\left(\begin{array}{ccc}{x}_{1}& 0& 0\\ 0& {x}_{2}& {x}_{3}\\ 0& {x}_{3}& {x}_{4}\end{array}\right).
Another possibility is the symmetric Toeplitz structure:
X=\left(\begin{array}{ccc}{x}_{1}& {x}_{2}& {x}_{3}\\ {x}_{2}& {x}_{1}& {x}_{2}\\ {x}_{3}& {x}_{2}& {x}_{1}\end{array}\right).
Summing up, structured LMI problems are specified by declaring the matrix variables and describing the term content of each LMI. This term-oriented description is systematic and accurately reflects the specific structure of the LMI constraints. There is no built-in limitation on the number of LMIs that you can specify or on the number of blocks and terms in any given LMI. LMI systems of arbitrary complexity can therefore, be defined in the LMI Lab.
The LMI Lab offers tools to specify, manipulate, and numerically solve LMIs. Its main purpose is to
Allow for straightforward description of LMIs in their natural block-matrix form
Provide easy access to the LMI solvers (optimization codes)
Facilitate result validation and problem modification
The structure-oriented description of a given LMI system is stored as a single vector called the internal representation and generically denoted by LMISYS in the sequel. This vector encodes the structure and dimensions of the LMIs and matrix variables, a description of all LMI terms, and the related numerical data. It must be stressed that you need not attempt to read or understand the content of LMISYS since all manipulations involving this internal representation can be performed in a transparent manner with LMI-Lab tools.
The LMI Lab supports the following functionalities:
LMI systems can be either specified as symbolic matrix expressions with the interactive graphical user interface lmiedit, or assembled incrementally with the two commands lmivar and lmiterm. The first option is more intuitive and transparent while the second option is more powerful and flexible.
The interactive function lmiinfo answers qualitative queries about LMI systems created with lmiedit or lmivar and lmiterm. You can also use lmiedit to visualize the LMI system produced by a particular sequence of lmivar/lmiterm commands.
General-purpose LMI solvers are provided for the three generic LMI problems defined in LMI Applications. These solvers can handle very general LMI systems and matrix variable structures. They return a feasible or optimal vector of decision variables x*. The corresponding values
{X}_{1}^{*},\dots ,{X}_{K}^{*}
of the matrix variables are given by the function dec2mat.
The solution x* produced by the LMI solvers is easily validated with the functions evallmi and showlmi. This allows a fast check and/or analysis of the results. With evallmi, all variable terms in the LMI system are evaluated for the value x* of the decision variables. The left and right sides of each LMI then become constant matrices that can be displayed with showlmi.
An existing system of LMIs can be modified in two ways:
An LMI can be removed from the system with dellmi.
A matrix variable X can be deleted using delmvar. It can also be instantiated, that is, set to some given matrix value. This operation is performed by setmvar and allows, for example, to fix some variables and solve the LMI problem with respect to the remaining ones.
|
Using your knowledge of exponents, rewrite each expression below so that there are no negative exponents or parentheses remaining.
\frac { 4 x ^ { 18 } } { 2 x ^ { 22 } }
$ \frac{ 2 }{x^{4} } $
\left(s^4tu^2\right)\left(s^7t^{-1}\right)
How many factors of s are multiplied?
How many of
t
u
s^{11}u^{2}
\left(3w^{-2}\right)^4
3w^{-2}
multiplied four times then follow the same pattern as seen in part (b).
\frac{ 81 }{w^{8} }
m^{-3}
Move the exponent and its base into the denominator
of a fraction to make the exponent positive.
\frac{ 1 }{m^{3} }
|
Analytic Number Theory/Printable version - Wikibooks, open books for an open world
Analytic Number Theory/Printable version
This is the print version of Analytic Number Theory
https://en.wikibooks.org/wiki/Analytic_Number_Theory
1 The Chebychev ψ and ϑ functions
Proposition (the Chebychev ψ function may be written as the sum of Chebyshev ϑ functions):
We have the identity
{\displaystyle \psi (x)=\sum _{m=1}^{\log _{2}(x)}\vartheta (x^{1/m})}
Proposition (estimate of the distance between the Chebychev ψ and ϑ functions):
{\displaystyle x\geq 3,594,641}
{\displaystyle \psi (x)-\vartheta (x)={\sqrt {x}}+O\left({\frac {\sqrt {x}}{\ln(x)^{2}}}\right)}
Note: The current proof gives an inferior error term. A subsequent version will redeem this issue. (Given the Riemann hypothesis, the error term can be made even smaller.)
Proof: We know that the formula
{\displaystyle \psi (x)=\sum _{m=1}^{\log _{2}(x)}\vartheta (x^{1/m})}
holds. Hence,
{\displaystyle \psi (x)-\vartheta (x)=\sum _{m=2}^{\log _{2}(x)}\vartheta (x^{1/m})}
By a result obtained by Pierre Dusart (based upon the computational verification of the Riemann hypothesis for small moduli), we have
{\displaystyle \left|\theta (x)-x\right|\leq {\frac {0.2}{\ln(x)^{2}}}x}
{\displaystyle x\geq 3,594,641}
{\displaystyle x}
is in that range, we hence conclude
{\displaystyle \psi (x)-\vartheta (x)=\sum _{m=2}^{\log _{2}(x)}\vartheta (x^{1/m})\leq \left(1+{\frac {0.2}{\ln(x)^{2}}}\right)\sum _{m=2}^{\log _{2}(x)}x^{1/m}}
By Euler's summation formula, we have
{\displaystyle \sum _{m=2}^{\log _{2}(x)}x^{1/m}=\int _{2}^{\log _{2}(x)}x^{1/t}dt+\int _{2}^{\log _{2}(x)}\{t\}\left({\frac {d}{dt}}x^{1/t}\right)dt+\{\log _{2}(x)\}x^{1/\log _{2}(x)}-\{2\}x^{1/2}}
Certainly
{\displaystyle \{2\}=0}
{\displaystyle \{\log _{2}(x)\}\leq 1}
{\displaystyle x^{1/\log _{2}(x)}=2}
. Now derivation shows that
{\displaystyle -\exp \left({\frac {\ln(x)}{t}}\right){\frac {t^{2}}{\ln(x)}}}
is an anti-derivative of the function
{\displaystyle x^{1/t}-{\frac {2t}{\ln(x)}}x^{1/t}}
{\displaystyle t}
. By the fundamental theorem of calculus, it follows that
{\displaystyle \int _{a}^{b}\left(1-{\frac {2t}{\ln(x)}}\right)x^{1/t}dt=\left[-\exp \left({\frac {\ln(x)}{t}}\right){\frac {t^{2}}{\ln(x)}}\right]_{t=a}^{t=b}}
{\displaystyle a,b\in \mathbb {R} }
{\displaystyle a<b}
. This integral is not precisely the one we want to estimate. Hence, some analytical trickery will be necessary in order to obtain the estimate we want.
We start by noting that if only the bracketed term in the integral were absent, we would have the estimate we desire. In order to proceed, we replace
{\displaystyle x}
by the more general expression
{\displaystyle xy}
{\displaystyle y\geq 1}
), and obtain
{\displaystyle \int _{a}^{b}\left(1-{\frac {2t}{\ln(xy)}}\right)x^{1/t}y^{1/t}dt=\left[-\exp \left({\frac {\ln(xy)}{t}}\right){\frac {t^{2}}{\ln(xy)}}\right]_{t=a}^{t=b}}
The integrand is non-negative so long as
{\displaystyle t\leq {\frac {\ln(xy)}{2}}}
{\displaystyle t_{0}}
is strictly within that range, we obtain
{\displaystyle \int _{2}^{t_{0}}x^{1/t}y^{1/t}dt\leq \left(1-{\frac {2t_{0}}{\ln(xy)}}\right)^{-1}\int _{2}^{t_{0}}\left(1-{\frac {2t}{\ln(xy)}}\right)x^{1/t}y^{1/t}dt=\left[-\exp \left({\frac {\ln(xy)}{t}}\right){\frac {t^{2}}{\ln(xy)}}\right]_{t=2}^{t=t_{0}}}
We now introduce a constant
{\displaystyle K\in (2,t_{0})}
and obtain the integrals
{\displaystyle \int _{2}^{K}x^{1/t}y^{1/t}dt}
{\displaystyle \int _{K}^{t_{0}}x^{1/t}y^{1/t}dt}
The first integral majorises the integral
{\displaystyle y^{1/K}\int _{2}^{K}x^{1/t}dt}
whereas the second integral majorises the integral
{\displaystyle \int _{K}^{t_{0}}x^{1/t}dt}
{\displaystyle \int _{2}^{t_{0}}x^{1/t}dt\leq {\frac {1}{y^{1/K}}}\int _{2}^{K}x^{1/t}y_{1}^{1/t}dt+\int _{K}^{t_{0}}x^{1/t}y_{2}^{1/t}dt}
Now we would like to set
{\displaystyle t_{0}=\log _{2}(x)}
. To do so, we must ensure that
{\displaystyle y}
is sufficiently large so that
{\displaystyle K}
{\displaystyle t_{0}}
is strictly within the admissible interval.
The two summands on the left are now estimated using our computation above, where
{\displaystyle t_{0}}
{\displaystyle K}
for the first computation: Indeed,
{\displaystyle \int _{2}^{K}x^{1/t}y_{1}^{1/t}dt\leq \left(1-{\frac {2K}{\ln(xy_{1})}}\right)^{-1}\left[-\exp \left({\frac {\ln(xy_{1})}{t}}\right){\frac {t^{2}}{\ln(xy_{1})}}\right]_{t=2}^{t=K}}
{\displaystyle \int _{K}^{t_{0}}x^{1/t}y_{2}^{1/t}dt\leq \left(1-{\frac {2t_{0}}{\ln(xy_{2})}}\right)^{-1}\left[-\exp \left({\frac {\ln(xy_{2})}{t}}\right){\frac {t^{2}}{\ln(xy_{2})}}\right]_{t=K}^{t=t_{0}}}
Putting the estimates together and setting
{\displaystyle t_{0}=\log _{2}(x)}
{\displaystyle \int _{2}^{\log _{2}(x)}x^{1/t}dt\leq {\frac {1}{y_{1}^{1/K}}}\left(1-{\frac {2K}{\ln(xy_{1})}}\right)^{-1}\left[-\exp \left({\frac {\ln(xy_{1})}{t}}\right){\frac {t^{2}}{\ln(xy_{1})}}\right]_{t=2}^{t=K}+\left(1-{\frac {2t_{0}}{\ln(xy_{2})}}\right)^{-1}\left[-\exp \left({\frac {\ln(xy_{2})}{t}}\right){\frac {t^{2}}{\ln(xy_{2})}}\right]_{t=K}^{t=\log _{2}(x)}}
{\displaystyle K\leq {\frac {\ln(xy_{1})}{2}}}
{\displaystyle \log _{2}(x)\leq {\frac {\ln(xy_{2})}{2}}}
We now choose the ansatz
{\displaystyle 1-{\frac {2K}{\ln(xy_{1})}}=C}
{\displaystyle 1-{\frac {2t_{0}}{\ln(xy_{2})}}=D}
{\displaystyle C}
{\displaystyle D}
. These equations are readily seen to imply
{\displaystyle y_{1}={\frac {1}{x}}\exp \left({\frac {2K}{1-C}}\right)}
{\displaystyle y_{2}={\frac {1}{x}}\exp \left({\frac {2\log _{2}(x)}{1-D}}\right)}
Note though that
{\displaystyle y_{1}\geq 1}
{\displaystyle y_{2}\geq 1}
is needed. The first condition yields
{\displaystyle K\geq {\frac {1-C}{2}}\ln(x)}
{\displaystyle y_{1}}
{\displaystyle y_{2}}
may be inserted into the above constraints on
{\displaystyle K}
{\displaystyle \log _{2}(x)}
; this yields
{\displaystyle K\leq {\frac {2K}{1-C}}}
{\displaystyle \log _{2}(x)\leq {\frac {2\log _{2}(x)}{1-D}}}
{\displaystyle C\geq {\frac {1}{2}}}
{\displaystyle D\geq {\frac {1}{2}}}
If all these conditions are true, the ansatz immediately yields
{\displaystyle \int _{2}^{\log _{2}(x)}x^{1/t}dt\leq {\frac {C^{-1}}{y_{1}^{1/K}}}\left[-\exp \left({\frac {\ln(xy_{1})}{t}}\right){\frac {t^{2}}{\ln(xy_{1})}}\right]_{t=2}^{t=K}+D^{-1}\left[-\exp \left({\frac {\ln(xy_{2})}{t}}\right){\frac {t^{2}}{\ln(xy_{2})}}\right]_{t=K}^{t=\log _{2}(x)}}
We now amend our ansatz by further postulating
{\displaystyle K=\left({\frac {1-C}{2}}+\alpha \right)\ln(x)}
{\displaystyle y_{1}={\frac {1}{x}}\exp \left({\frac {(1-C+2\alpha )\ln(x)}{1-C}}\right)}
{\displaystyle {\frac {C^{-1}}{y_{1}^{1/K}}}\left[-\exp \left({\frac {\ln(xy_{1})}{t}}\right){\frac {t^{2}}{\ln(xy_{1})}}\right]_{t=2}^{t=K}={\frac {C^{-1}}{y_{1}^{1/K}}}\left[-\exp \left({\frac {\frac {(1-C+2\alpha )\ln(x)}{1-C}}{t}}\right){\frac {t^{2}}{\ln(xy_{1})}}\right]_{t=2}^{t=K}}
From this we deduce that in order to obtain an asymptotically sharp error term, we need to set
{\displaystyle \alpha =0}
. But doing so yields the desired result.
{\displaystyle \Box }
Retrieved from "https://en.wikibooks.org/w/index.php?title=Analytic_Number_Theory/Printable_version&oldid=4008941"
|
Symmetry, tessellations, golden mean, 17, and patterns » Loren on the Art of MATLAB - MATLAB & Simulink
< What Do MATLAB and Games... < Previous
Benefits of Refactoring Code >Next >
Symmetry, tessellations, golden mean, 17, and patterns
Seventeen? Why 17? Well, as a high school student, I attended HCSSIM, a summer program for students interested in math. There we learned all kinds of math you don't typically learn about until much later in your studies. One of the reference books was Calculus on Manifolds by Michael Spivak. Inside, you learn some of the mysteries of algebra, and, if you read carefully, you will find references to both yellow pigs and the number 17. I leave it as a challenge to you to learn more about either or both if you are interested.
As I went to college, the number 17 was a part of my life. Looking through the course catalogue before my first semester, I saw an offering something like "the seventeen regular tilings of the plane", and I signed up. And isn't cool that all of these patterns are displayed in tiles within the Alhambra! I leave you to search the many sites with pictures and drawings of these.
I enjoy the artwork of Rafael Araujo. If you have watched any webinars I have delivered during 2020-2021, you may notice a piece of Araujo's hanging in the background. The basis for much of his work is the golden mean (or golden ratio). Here's a place where you can explore the influence of math on art.
So what is the golden mean?
It's defined as the solution to
And the value, typically denoted by the Greek letter
\varphi \text{\hspace{0.17em}}=\frac{1+\sqrt{5}}{2}\text{\hspace{0.17em}}
And there are claims that this ratio is universally(?) pleasing. You can see approximations to it show up in everyday life. In the US, we use note cards that are 5x3".
ratio5to3 = 5/3
ratio5to3 = 1.6667
plot(0:(5/3):5,0:3,'.')
title("Not quite the golden ratio: " + ratio5to3)
I have written several blogs that show ways to compute Fibonacci numbers, also related to the golden mean. Why? Because ratios of successive Fibonacci numbers converge to the golden mean
\underset{\mathit{n}\to \infty }{\mathrm{lim}}\frac{{\mathit{F}}_{\mathit{n}+1}}{{\mathit{F}}_{\mathit{n}}}=\varphi \text{\hspace{0.17em}}
In the 1990s, we held several MATLAB User Conferences. In 1997, I gave a talk on Programming Patterns in MATLAB. I had 17 of them available, but time to only discuss 6 of them. The regular tilings of the plane seemed like a cool way to categorize and clump together some of programming patterns I wanted to talk about. I thought it would be interesting to revisit many of these and see how well they held up over time. So that's my plan for some of the upcoming posts though I feel no compulsion to do all of them or in the order they showed up in my original talk.
First pattern - data duplication in service of mathematical operations
Steve Eddins wrote two posts on this topic in 2016: one and two. And I wrote one as well, on performance implications.
When I first started at MathWorks (1987), MATLAB had only double matrices and no other data types or dimensions. If I wanted to remove the mean of each column of data in a matrix, I would do something like this.
A(:) = randperm(16)
7 3 11 1 6 10 5 13 12 8 4 2 14 15 16 9
Here I'll calculate the mean of each column.
meanAc = mean(A)
meanAc = 1×4
and then I needed to create an array from meanAc that was the same size as A in order to subtract the means. Originally, we did this by matrix multiplication.
Ameans1 = ones(4,1)*meanAc
Ameans1 = 4×4
And now I can do the subtraction.
Ameanless1 = A-Ameans1
Ameanless1 = 4×4
-2.7500 -6.0000 2.0000 -5.2500 -3.7500 1.0000 -4.0000 6.7500 2.2500 -1.0000 -5.0000 -4.2500 4.2500 6.0000 7.0000 2.7500
I then met a customer at my first ICASSP conference (in Phoenix, AZ), Tony, and he asked why I was not using indexing instead - because I never thought about it! This is cool because I didn't need to do arithmetic to get my expanded mean matrix.
Ameans2 = meanAc(ones(1,4),:)
isequal(Ameans1, Ameans2)
That was all well and good - but potentially not so easy to remember each time you might need it.
In 1996, we had heard plenty from customers that we were making something simple a little too difficult. And, we were very close to introducing ND arrays, where we wanted to be able to do similar operations in any chosen dimension(s). So we introduced a new function, repmat.
Now I can find the matrix mean with easier to read code, in my opinion.
Ameanlessr = A - repmat(mean(A),[4,1])
Ameanlessr = 4×4
isequal(Ameanless1, Ameanlessr)
By 2006, we had a lot of evidence that handling really large data was important for many of our customers, and likely to be an increasing demand. Up until then, we always created an intermediate matrix the same size as our original one, A, in order to calculate the result. But this wasn't strictly necessary -- we just need some syntax -- a way to express that all the rows (or columns) would be the same. Now, of course we need a matrix the same size as A for the answer. But how many more arrays of that size did we need along the way? Along came the function, gloriously named bsxfun (standing for binary singleton expansion), and we could perform the computation without fully forming the mxn matrix to subtract from the original.
Ameanlessb = bsxfun(@minus, A, mean(A))
Ameanlessb = 4×4
isequal(Ameanless1, Ameanlessb)
Finally, in 2016, we decided that the meaning was clear even if it wasn't strictly linear algebra, and we now allow many operations to take advantage of implicit expansion of singleton dimensions. What this means for us with this problem is now we can simply say
Ameanless2016 = A - mean(A)
Ameanless2016 = 4×4
isequal(Ameanless1, Ameanless2016)
I do not expect a Part 5 to come along in 2026, though of course I could be wrong!
Did any of you attend the conferences in 1993, 1995, and 1997? Share your memories here!
|
The choir is planning a trip to the water park, and the parents are trying to determine how much to charge each child. The cost to use a school bus is $350. Complete the table at right, graph your results, and then answer parts (a) and (b) below.
Think back to your work with proportions. Is this a proportional relationship?
To complete the table, divide
350
by the number of students.
Graphs of proportional relationships can pass through the point
\left(0, 0\right)
. Does your graph look like it will pass through
\left(0, 0\right)
Is there an association between the number of students and the cost per student? If so, describe it. Consider the form, direction, and outliers in your description.
For every student added, the trip costs less for everyone.
There is a non-linear negative association.
Bus Cost per
Student ($)
10
15
20
35
Use the eTool below to help complete the table and graph the problem.
|
Transpose - Simple English Wikipedia, the free encyclopedia
This article is about the transpose of a matrix. For other uses, see Transposition (disambiguation).
The transpose of a matrix A is another matrix where the rows of A are written as columns. Vectors can be transposed in the same way. We can write the transpose of A using different symbols such as AT,[1][2] A′ [3] Atr and At.
Here is the vector
{\displaystyle {\begin{bmatrix}1&2\end{bmatrix}}}
being transposed:
{\displaystyle {\begin{bmatrix}1&2\end{bmatrix}}^{\mathrm {T} }\!\!\;\!=\,{\begin{bmatrix}1\\2\end{bmatrix}}.}
Here are a few matrices being transposed:
{\displaystyle {\begin{bmatrix}1&2\\3&4\end{bmatrix}}^{\mathrm {T} }\!\!\;\!=\,{\begin{bmatrix}1&3\\2&4\end{bmatrix}}.}
{\displaystyle {\begin{bmatrix}1&2\\3&4\\5&6\end{bmatrix}}^{\mathrm {T} }\!\!\;\!=\,{\begin{bmatrix}1&3&5\\2&4&6\end{bmatrix}}.\;}
{\displaystyle {\begin{bmatrix}1&2&8\\3&4&3\\5&6&1\end{bmatrix}}^{\mathrm {T} }\!\!\;\!=\,{\begin{bmatrix}1&3&5\\2&4&6\\8&3&1\end{bmatrix}}.\;}
Given two matrices A and B, the following properties related to the transpose are true:[3]
{\displaystyle (A^{T})^{-1}=(A^{-1})^{T}}
{\displaystyle (AB)^{T}=B^{T}A^{T}}
↑ Nykamp, Duane. "The transpose of a matrix". Math Insight. Retrieved September 8, 2020. {{cite web}}: CS1 maint: url-status (link)
↑ 3.0 3.1 Weisstein, Eric W. "Transpose". mathworld.wolfram.com. Retrieved 2020-09-08.
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Transpose&oldid=8146147"
|
Universal_instantiation Knowpia
In predicate logic, universal instantiation[1][2][3] (UI; also called universal specification or universal elimination, and sometimes confused with dictum de omni) is a valid rule of inference from a truth about each member of a class of individuals to the truth about a particular individual of that class. It is generally given as a quantification rule for the universal quantifier but it can also be encoded in an axiom schema. It is one of the basic principles used in quantification theory.
Example: "All dogs are mammals. Fido is a dog. Therefore Fido is a mammal."
Formally, the rule as an axiom schema is given as
{\displaystyle \forall x\,A\Rightarrow A\{x\mapsto a\},}
for every formula A and every term a, where
{\displaystyle A\{x\mapsto a\}}
is the result of substituting a for each free occurrence of x in A.
{\displaystyle \,A\{x\mapsto a\}}
{\displaystyle \forall x\,A.}
And as a rule of inference it is
from ⊢ ∀x A infer ⊢ A{x↦a}.
Irving Copi noted that universal instantiation "...follows from variants of rules for 'natural deduction', which were devised independently by Gerhard Gentzen and Stanisław Jaśkowski in 1934."[4]
According to Willard Van Orman Quine, universal instantiation and existential generalization are two aspects of a single principle, for instead of saying that "∀x x = x" implies "Socrates = Socrates", we could as well say that the denial "Socrates ≠ Socrates" implies "∃x x ≠ x". The principle embodied in these two operations is the link between quantifications and the singular statements that are related to them as instances. Yet it is a principle only by courtesy. It holds only in the case where a term names and, furthermore, occurs referentially.[5]
^ Irving M. Copi; Carl Cohen; Kenneth McMahon (Nov 2010). Introduction to Logic. Pearson Education. ISBN 978-0205820375. [page needed]
^ Hurley, Patrick. A Concise Introduction to Logic. Wadsworth Pub Co, 2008.
^ Copi, Irving M. (1979). Symbolic Logic, 5th edition, Prentice Hall, Upper Saddle River, NJ
^ Willard Van Orman Quine; Roger F. Gibson (2008). "V.24. Reference and Modality". Quintessence. Cambridge, Mass: Belknap Press of Harvard University Press. OCLC 728954096. Here: p. 366.
|
FactoredMinimalAnnihilator - Maple Help
Home : Support : Online Help : Mathematics : Factorization and Solving Equations : LinearOperators : FactoredMinimalAnnihilator
construct the minimal annihilator in the completely factored form
FactoredMinimalAnnihilator(expr, x, case)
Given a d'Alembertian term expr, the LinearOperators[FactoredMinimalAnnihilator] function returns a factored Ore operator that is the minimal annihilator in the completely factored form for expr. That is, applying this operator to expr yields zero.
A completely factored Ore operator is an operator that can be factored into a product of linear factors.
\left(-1+\mathrm{xD}\right)\left(x\right)\left({x}^{2}\mathrm{D}+4\right)\left(\mathrm{D}\right)
\mathrm{expr}≔x\mathrm{ln}\left(x\right)-x+1
\textcolor[rgb]{0,0,1}{\mathrm{expr}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{ln}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}
L≔\mathrm{LinearOperators}[\mathrm{FactoredMinimalAnnihilator}]\left(\mathrm{expr},x,'\mathrm{differential}'\right)
\textcolor[rgb]{0,0,1}{L}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{FactoredOrePoly}}\textcolor[rgb]{0,0,1}{}\left([\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}]\right)
\mathrm{LinearOperators}[\mathrm{Apply}]\left(L,\mathrm{expr},x,'\mathrm{differential}'\right)
\textcolor[rgb]{0,0,1}{0}
\mathrm{expr}≔\mathrm{\Gamma }\left(n\right){n}^{2}
\textcolor[rgb]{0,0,1}{\mathrm{expr}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{\Gamma }}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{n}\right)\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{n}}^{\textcolor[rgb]{0,0,1}{2}}
L≔\mathrm{LinearOperators}[\mathrm{FactoredMinimalAnnihilator}]\left(\mathrm{expr},n,'\mathrm{shift}'\right)
\textcolor[rgb]{0,0,1}{L}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{FactoredOrePoly}}\textcolor[rgb]{0,0,1}{}\left([\textcolor[rgb]{0,0,1}{-}\frac{{\textcolor[rgb]{0,0,1}{n}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{n}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}]\right)
\mathrm{simplify}\left(\mathrm{LinearOperators}[\mathrm{Apply}]\left(L,\mathrm{expr},n,'\mathrm{shift}'\right)\right)
\textcolor[rgb]{0,0,1}{0}
|
Retracted: Three Homoclinic Solutions for Second-Order -Laplacian Differential System
Abstract and Applied Analysis, "Retracted: Three Homoclinic Solutions for Second-Order -Laplacian Differential System", Abstract and Applied Analysis, vol. 2014, Article ID 798572, 1 page, 2014. https://doi.org/10.1155/2014/798572
This paper titled “Three Homoclinic Solutions for Second-Order -Laplacian Differential System” [1], published in Abstract and Applied Analysis, has been retracted since the limits of the three sequences included in the paper are likely to be the same, which cannot be ruled out in the proof. So, the three nontrivial homoclinic solutions in Theorem 15 are likely to be the same one.
J. Guo and B. Dai, “Three homoclinic solutions for second-order
p
-Laplacian differential system,” Abstract and Applied Analysis, vol. 2013, Article ID 183585, 10 pages, 2013. View at: Publisher Site | Google Scholar | MathSciNet
Copyright © 2014 Abstract and Applied Analysis. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
|
Evaluate translation or summarization with BLEU similarity score - MATLAB bleuEvaluationScore - MathWorks España
bleuEvaluationScore
Specify N-Gram Weights
ngramWeights
Evaluate translation or summarization with BLEU similarity score
score = bleuEvaluationScore(candidate,references)
score = bleuEvaluationScore(candidate,references,'NgramWeights',ngramWeights)
The BiLingual Evaluation Understudy (BLEU) scoring algorithm evaluates the similarity between a candidate document and a collection of reference documents. Use the BLEU score to evaluate the quality of document translation and summarization models.
score = bleuEvaluationScore(candidate,references) returns the BLEU similarity score between the specified candidate document and the reference documents. The function computes n-gram overlaps between candidate and references for n-gram lengths one through four, with equal weighting. For more information, see BLEU Score.
score = bleuEvaluationScore(candidate,references,'NgramWeights',ngramWeights) uses the specified n-gram weighting, where ngramWeights(i) corresponds to the weight for n-grams of length i. The length of the weight vector determines the range of n-gram lengths to use for the BLEU score evaluation.
Create an array of tokenized documents and extract a summary using the extractSummary function.
"The fast brown fox jumped over the lazy dog."
10 tokens: The fast brown fox jumped over the lazy dog .
Specify the reference documents as a tokenizedDocument array.
"The quick brown animal jumped over the lazy dog."
"The quick brown fox jumped over the lazy dog."];
references = tokenizedDocument(str);
Calculate the BLEU score between the summary and the reference documents using the bleuEvaluationScore function.
score = bleuEvaluationScore(summary,references)
This score indicates a fairly good similarity. A BLEU score close to one indicates strong similarity.
Calculate the BLEU score between the candidate document and the reference documents using the default options. The bleuEvaluationScore function, by default, uses n-grams of length one through four with equal weights.
Given that the summary document differs only by one word to one of the reference documents, this score might suggest a lower similarity than might be expected. This behavior is due to the function using n-grams which are too large for the short document length.
To address this, use shorter n-grams by setting the 'NgramWeights' option to a shorter vector. Calculate the BLEU score again using only unigrams and bigrams by setting the 'NgramWeights' option to a two-element vector. Treat unigrams and bigrams equally by specifying equal weights.
score = bleuEvaluationScore(summary,references,'NgramWeights',[0.5 0.5])
This score suggests a better similarity than before.
candidate — Candidate document
Candidate document, specified as a tokenizedDocument scalar, a string array, or a cell array of character vectors. If candidate is not a tokenizedDocument scalar, then it must be a row vector representing a single document, where each element is a word.
references — Reference documents
tokenizedDocument array | string array | cell array of character vectors
Reference documents, specified as a tokenizedDocument array, a string array, or a cell array of character vectors. If references is not a tokenizedDocument array, then it must be a row vector representing a single document, where each element is a word. To evaluate against multiple reference documents, use a tokenizedDocument array.
ngramWeights — N-gram weights
[0.25 0.25 0.25 0.25] (default) | row vector of finite nonnegative values
N-gram weights, specified as a row vector of finite nonnegative values, where ngramWeights(i) corresponds to the weight for n-grams of length i. The length of the weight vector determines the range of n-gram lengths to use for the BLEU score evaluation. The function normalizes the n-gram weights to sum to one.
If the number of words in candidate is smaller than the number of elements in ngramWeights, then the resulting BLEU score is zero. To ensure that bleuEvaluationScore returns nonzero scores for very short documents, set ngramWeights to a vector with fewer elements than the number of words in candidate.
score — BLEU score
BLEU score, returned as a scalar value in the range [0,1] or NaN.
A BLEU score close to zero indicates poor similarity between candidate and references. A BLEU score close to one indicates strong similarity. If candidate is identical to one of the reference documents, then score is 1. If candidate and references are both empty documents, then score is NaN. For more information, see BLEU Score.
The BiLingual Evaluation Understudy (BLEU) scoring algorithm [1] evaluates the similarity between a candidate document and a collection of reference documents. Use the BLEU score to evaluate the quality of document translation and summarization models.
To compute the BLEU score, the algorithm uses n-gram counts, clipped n-gram counts, modified n-gram precision scores, and a brevity penalty.
The clipped n-gram counts function
{\text{Count}}_{\text{clip}}
, if necessary, truncates the n-gram count for each n-gram so that it does not exceed the largest count observed in any single reference for that n-gram. The clipped counts function is given by
{\text{Count}}_{\text{clip}}\left(\text{n-gram}\right)=\text{min}\left(\text{Count}\left(\text{n-gram}\right),\text{MaxRefCount}\left(\text{n-gram}\right)\right),
\text{Count}\left(\text{n-gram}\right)
denotes the n-gram counts and
\text{MaxRefCount}\left(\text{n-gram}\right)
is the largest n-gram count observed in a single reference document for that n-gram.
The modified n-gram precision scores are given by
{p}_{n}=\frac{\sum _{C\in \left\{\text{Candidates}\right\}}\sum _{\text{n-gram}\in C}{\text{Count}}_{\text{clip}}\left(\text{n-gram}\right)}{\sum _{C\text{'}\in \left\{\text{Candidates}\right\}}\sum _{{\text{n-gram}}^{\prime }\in {C}^{\prime }}\text{Count}\left({\text{n-gram}}^{\prime }\right)},
where n corresponds to the n-gram length and
\left\{\text{candidates}\right\}
is the set of sentences in the candidate documents.
Given a vector of n-gram weights w, the BLEU score is given by
\text{bleuScore}=\text{BP}·\mathrm{exp}\left(\sum _{n=1}^{N}{w}_{n}\mathrm{log}{\overline{p}}_{n}\right),
where N is the largest n-gram length, the entries in
\overline{p}
correspond to the geometric averages of the modified n-gram precisions, and
\text{BP}
is the brevity penalty given by
\text{BP}=\left\{\begin{array}{cc}1& \text{if }c>r\\ {e}^{1-\frac{r}{c}}& \text{if }c\le r\end{array}
where c is the length of the candidate document and r is the length of the reference document with length closest to the candidate length.
[1] Papineni, Kishore, Salim Roukos, Todd Ward, and Wei-Jing Zhu. "BLEU: A Method for Automatic Evaluation of Machine Translation." In Proceedings of the 40th annual meeting on association for computational linguistics, pp. 311-318. Association for Computational Linguistics, 2002.
tokenizedDocument | rougeEvaluationScore | bm25Similarity | cosineSimilarity | textrankScores | lexrankScores | mmrScores | extractSummary
|
Home : Support : Online Help : Mathematics : Differential Equations : Differential Algebra : Tools : Tail
returns the tail of a differential polynomial
Tail(ideal, v, opts)
Tail(p, v, R, opts)
Tail(L, v, R, opts)
fullset = boolean. In the case of the function call Tail(ideal,v), applies the function also over the differential polynomials which state that the derivatives of the parameters are zero. Default value is false.
The function call Tail(p,v,R) returns the tail of p regarded as a univariate polynomial in v, that is the differential polynomial p, regarded as a univariate polynomial in v, minus its leading monomial with respect to this variable, If p does not depend on v then the function call returns
0
The function call Tail(L,v,R) returns the list or the set of the tails of the elements of L with respect to v.
If ideal is a regular differential chain, the function call Tail(ideal,v) returns the list of the tails of the chain elements. If ideal is a list of regular differential chains, the function call Tail(ideal,v) returns a list of lists of tails.
When the parameter v is omitted, it is understood to be the leading derivative of the processed differential polynomial with respect to the ranking of R, or the one of its embedding polynomial ring, if R is an ideal. In that case, p must be non-numeric.
This command is part of the DifferentialAlgebra:-Tools package. It can be called using the form Tail(...) after executing the command with(DifferentialAlgebra:-Tools). It can also be directly called using the form DifferentialAlgebra[Tools][Tail](...).
\mathrm{with}\left(\mathrm{DifferentialAlgebra}\right):
\mathrm{with}\left(\mathrm{Tools}\right):
R≔\mathrm{DifferentialRing}\left(\mathrm{derivations}=[x,y],\mathrm{blocks}=[[v,u],p],\mathrm{parameters}=[p]\right)
\textcolor[rgb]{0,0,1}{R}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{differential_ring}}
The tail, with respect to the leading derivative
\mathrm{Tail}\left(u[x,y]v[y]-u+p,R\right)
\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{p}
\mathrm{ideal}≔\mathrm{RosenfeldGroebner}\left([{u[x]}^{2}-4u,u[x,y]v[y]-u+p,v[x,x]-u[x]],R\right)
\textcolor[rgb]{0,0,1}{\mathrm{ideal}}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{\mathrm{regular_differential_chain}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{regular_differential_chain}}]
\mathrm{Equations}\left(\mathrm{ideal}[1]\right)
[{\textcolor[rgb]{0,0,1}{v}}_{\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{p}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{y}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{y}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{v}}_{\textcolor[rgb]{0,0,1}{y}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}]
The tails of the equations, with respect to
{u}_{x}
\mathrm{Tail}\left(\mathrm{ideal}[1],u[x]\right)
[{\textcolor[rgb]{0,0,1}{v}}_{\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{v}}_{\textcolor[rgb]{0,0,1}{y}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}]
|
Section 60.22 (07LI): Two counter examples—The Stacks project
Section 60.22: Two counter examples (cite)
60.22 Two counter examples
Before we turn to some of the successes of crystalline cohomology, let us give two examples which explain why crystalline cohomology does not work very well if the schemes in question are either not proper over the base, or singular. The first example can be found in [BO].
Example 60.22.1. Let $A = \mathbf{Z}_ p$ with divided power ideal $(p)$ endowed with its unique divided powers $\gamma $. Let $C = \mathbf{F}_ p[x, y]/(x^2, xy, y^2)$. We choose the presentation
\[ C = P/J = \mathbf{Z}_ p[x, y]/(x^2, xy, y^2, p) \]
Let $D = D_{P, \gamma }(J)^\wedge $ with divided power ideal $(\bar J, \bar\gamma )$ as in Section 60.17. We will denote $x, y$ also the images of $x$ and $y$ in $D$. Consider the element
\[ \tau = \bar\gamma _ p(x^2)\bar\gamma _ p(y^2) - \bar\gamma _ p(xy)^2 \in D \]
We note that $p\tau = 0$ as
\[ p! \bar\gamma _ p(x^2) \bar\gamma _ p(y^2) = x^{2p} \bar\gamma _ p(y^2) = \bar\gamma _ p(x^2y^2) = x^ py^ p \bar\gamma _ p(xy) = p! \bar\gamma _ p(xy)^2 \]
in $D$. We also note that $\text{d}\tau = 0$ in $\Omega _ D$ as
\begin{align*} \text{d}(\bar\gamma _ p(x^2) \bar\gamma _ p(y^2)) & = \bar\gamma _{p - 1}(x^2)\bar\gamma _ p(y^2)\text{d}x^2 + \bar\gamma _ p(x^2)\bar\gamma _{p - 1}(y^2)\text{d}y^2 \\ & = 2 x \bar\gamma _{p - 1}(x^2)\bar\gamma _ p(y^2)\text{d}x + 2 y \bar\gamma _ p(x^2)\bar\gamma _{p - 1}(y^2)\text{d}y \\ & = 2/(p - 1)!( x^{2p - 1} \bar\gamma _ p(y^2)\text{d}x + y^{2p - 1} \bar\gamma _ p(x^2)\text{d}y ) \\ & = 2/(p - 1)! (x^{p - 1} \bar\gamma _ p(xy^2)\text{d}x + y^{p - 1} \bar\gamma _ p(x^2y)\text{d}y) \\ & = 2/(p - 1)! (x^{p - 1}y^ p \bar\gamma _ p(xy)\text{d}x + x^ py^{p - 1} \bar\gamma _ p(xy)\text{d}y) \\ & = 2 \bar\gamma _{p - 1}(xy) \bar\gamma _ p(xy)(y\text{d}x + x \text{d}y) \\ & = \text{d}(\bar\gamma _ p(xy)^2) \end{align*}
Finally, we claim that $\tau \not= 0$ in $D$. To see this it suffices to produce an object $(B \to \mathbf{F}_ p[x, y]/(x^2, xy, y^2), \delta )$ of $\text{Cris}(C/S)$ such that $\tau $ does not map to zero in $B$. To do this take
\[ B = \mathbf{F}_ p[x, y, u, v]/(x^3, x^2y, xy^2, y^3, xu, yu, xv, yv, u^2, v^2) \]
with the obvious surjection to $C$. Let $K = \mathop{\mathrm{Ker}}(B \to C)$ and consider the map
\[ \delta _ p : K \longrightarrow K,\quad ax^2 + bxy + cy^2 + du + ev + fuv \longmapsto a^ pu + c^ pv \]
One checks this satisfies the assumptions (1), (2), (3) of Divided Power Algebra, Lemma 23.5.3 and hence defines a divided power structure. Moreover, we see that $\tau $ maps to $uv$ which is not zero in $B$. Set $X = \mathop{\mathrm{Spec}}(C)$ and $S = \mathop{\mathrm{Spec}}(A)$. We draw the following conclusions
$H^0(\text{Cris}(X/S), \mathcal{O}_{X/S})$ has $p$-torsion, and
pulling back by Frobenius $F^* : H^0(\text{Cris}(X/S), \mathcal{O}_{X/S}) \to H^0(\text{Cris}(X/S), \mathcal{O}_{X/S})$ is not injective.
Namely, $\tau $ defines a nonzero torsion element of $H^0(\text{Cris}(X/S), \mathcal{O}_{X/S})$ by Proposition 60.21.3. Similarly, $F^*(\tau ) = \sigma (\tau )$ where $\sigma : D \to D$ is the map induced by any lift of Frobenius on $P$. If we choose $\sigma (x) = x^ p$ and $\sigma (y) = y^ p$, then an easy computation shows that $F^*(\tau ) = 0$.
The next example shows that even for affine $n$-space crystalline cohomology does not give the correct thing.
Example 60.22.2. Let $A = \mathbf{Z}_ p$ with divided power ideal $(p)$ endowed with its unique divided powers $\gamma $. Let $C = \mathbf{F}_ p[x_1, \ldots , x_ r]$. We choose the presentation
\[ C = P/J = P/pP\quad \text{with}\quad P = \mathbf{Z}_ p[x_1, \ldots , x_ r] \]
Note that $pP$ has divided powers by Divided Power Algebra, Lemma 23.4.2. Hence setting $D = P^\wedge $ with divided power ideal $(p)$ we obtain a situation as in Section 60.17. We conclude that $R\Gamma (\text{Cris}(X/S), \mathcal{O}_{X/S})$ is represented by the complex
\[ D \to \Omega ^1_ D \to \Omega ^2_ D \to \ldots \to \Omega ^ r_ D \]
see Proposition 60.21.3. Assuming $r > 0$ we conclude the following
The cristalline cohomology of the cristalline structure sheaf of $X = \mathbf{A}^ r_{\mathbf{F}_ p}$ over $S = \mathop{\mathrm{Spec}}(\mathbf{Z}_ p)$ is zero except in degrees $0, \ldots , r$.
We have $H^0(\text{Cris}(X/S), \mathcal{O}_{X/S}) = \mathbf{Z}_ p$.
The cohomology group $H^ r(\text{Cris}(X/S), \mathcal{O}_{X/S})$ is infinite and is not a torsion abelian group.
The cohomology group $H^ r(\text{Cris}(X/S), \mathcal{O}_{X/S})$ is not separated for the $p$-adic topology.
While the first two statements are reasonable, parts (3) and (4) are disconcerting! The truth of these statements follows immediately from working out what the complex displayed above looks like. Let's just do this in case $r = 1$. Then we are just looking at the two term complex of $p$-adically complete modules
\[ \text{d} : D = \left( \bigoplus \nolimits _{n \geq 0} \mathbf{Z}_ p x^ n \right)^\wedge \longrightarrow \Omega ^1_ D = \left( \bigoplus \nolimits _{n \geq 1} \mathbf{Z}_ p x^{n - 1}\text{d}x \right)^\wedge \]
The map is given by $\text{diag}(0, 1, 2, 3, 4, \ldots )$ except that the first summand is missing on the right hand side. Now it is clear that $\bigoplus _{n > 0} \mathbf{Z}_ p/n\mathbf{Z}_ p$ is a subgroup of the cokernel, hence the cokernel is infinite. In fact, the element
\[ \omega = \sum \nolimits _{e > 0} p^ e x^{p^{2e} - 1}\text{d}x \]
is clearly not a torsion element of the cokernel. But it gets worse. Namely, consider the element
\[ \eta = \sum \nolimits _{e > 0} p^ e x^{p^ e - 1}\text{d}x \]
For every $t > 0$ the element $\eta $ is congruent to $\sum _{e > t} p^ e x^{p^ e - 1}\text{d}x$ modulo the image of $\text{d}$ which is divisible by $p^ t$. But $\eta $ is not in the image of $\text{d}$ because it would have to be the image of $a + \sum _{e > 0} x^{p^ e}$ for some $a \in \mathbf{Z}_ p$ which is not an element of the left hand side. In fact, $p^ N\eta $ is similarly not in the image of $\text{d}$ for any integer $N$. This implies that $\eta $ “generates” a copy of $\mathbf{Q}_ p$ inside of $H^1_{\text{cris}}(\mathbf{A}_{\mathbf{F}_ p}^1/\mathop{\mathrm{Spec}}(\mathbf{Z}_ p))$.
In example 49.22.1. in the definition of the
\delta_p: K\to K
the input should be corrected to
ax^2+bxy+cy^2+du+ev+fuv
|
Tropine acyltransferase - WikiMili, The Best Wikipedia Reader
162535-29-7&title= 138440-79-6, 162535-29-7
Tropine acyltransferase (EC 2.3.1.185, tropine:acyl-CoA transferase, acetyl-CoA:tropan-3-ol acyltransferase, tropine acetyltransferase, tropine tigloyltransferase, TAT) is an enzyme with systematic name acyl-CoA:tropine O-acyltransferase. [1] [2] [3] [4] This enzyme catalyses the following chemical reaction
acyl-CoA + tropine
{\displaystyle \rightleftharpoons }
CoA + O-acyltropine
This enzyme exhibits absolute specificity for the endo/3alpha configuration found in tropine as pseudotropine.
In biochemistry and metabolism, beta-oxidation is the catabolic process by which fatty acid molecules are broken down in the cytosol in prokaryotes and in the mitochondria in eukaryotes to generate acetyl-CoA, which enters the citric acid cycle, and NADH and FADH2, which are co-enzymes used in the electron transport chain. It is named as such because the beta carbon of the fatty acid undergoes oxidation to a carbonyl group. Beta-oxidation is primarily facilitated by the mitochondrial trifunctional protein, an enzyme complex associated with the inner mitochondrial membrane, although very long chain fatty acids are oxidized in peroxisomes.
Acyl-CoA is a group of coenzymes that metabolize fatty acids. Acyl-CoA's are susceptible to beta oxidation, forming, ultimately, acetyl-CoA. The acetyl-CoA enters the citric acid cycle, eventually forming several equivalents of ATP. In this way, fats are converted to ATP, the universal biochemical energy carrier.
In enzymology, a 1-acylglycerol-3-phosphate O-acyltransferase is an enzyme that catalyzes the chemical reaction
In enzymology, a 2-ethylmalate synthase (EC 2.3.3.6) is an enzyme that catalyzes the chemical reaction
In enzymology, a [acyl-carrier-protein] S-acetyltransferase is an enzyme that catalyzes the reversible chemical reaction
In enzymology, a [acyl-carrier-protein] S-malonyltransferase is an enzyme that catalyzes the chemical reaction
In enzymology, a beta-ketoacyl-acyl-carrier-protein synthase I is an enzyme that catalyzes the chemical reaction
In enzymology, a citrate (Re)-synthase (EC 2.3.3.3) is an enzyme that catalyzes the chemical reaction
In enzymology, a homocitrate synthase (EC 2.3.3.14) is an enzyme that catalyzes the chemical reaction
In enzymology, a salutaridinol 7-O-acetyltransferase is an enzyme that catalyzes the chemical reaction
Sterol O-acyltransferase is an intracellular protein located in the endoplasmic reticulum that forms cholesteryl esters from cholesterol.
In enzymology, a vinorine synthase is an enzyme that catalyzes the chemical reaction
Pseudotropine acyltransferase is an enzyme with systematic name acyl-CoA:pseudotropine O-acyltransferase. This enzyme catalyses the following chemical reaction
UDP-3-O-(3-hydroxymyristoyl)glucosamine N-acyltransferase is an enzyme with systematic name (3R)-3-hydroxymyristoyl-(acyl-carrier protein):UDP-3-O-( -3-hydroxymyristoyl)-alpha-D-glucosamine N-acetyltransferase. This enzyme catalyses the following chemical reaction
An O-acylpseudotropine is any derivative of pseudotropine in which the alcohol group is substituted with an acyl group.
↑ Robins RJ, Bachmann P, Robinson T, Rhodes MJ, Yamada Y (November 1991). "The formation of 3 alpha- and 3 beta-acetoxytropanes by Datura stramonium transformed root cultures involves two acetyl-CoA-dependent acyltransferases". FEBS Letters. 292 (1–2): 293–7. doi:10.1016/0014-5793(91)80887-9. PMID 1959620.
↑ Robins, R.J.; Bachmann,P.; Peerless, A.C.J.; Rabot, S. (1994). "Esterification reactions in the biosynthesis of tropane alkaloids in transformed root cultures". Plant Cell Tissue Organ Cult. 38: 241–247. doi:10.1007/bf00033883.
↑ Boswell HD, Dräger B, McLauchlan WR, Portsteffen A, Robins DJ, Robins RJ, Walton NJ (November 1999). "Specificities of the enzymes of N-alkyltropane biosynthesis in Brugmansia and Datura". Phytochemistry. 52 (5): 871–8. doi:10.1016/S0031-9422(99)00293-9. PMID 10626376.
↑ Li R, Reed DW, Liu E, Nowak J, Pelcher LE, Page JE, Covello PS (May 2006). "Functional genomic analysis of alkaloid biosynthesis in Hyoscyamus niger reveals a cytochrome P450 involved in littorine rearrangement". Chemistry & Biology. 13 (5): 513–20. doi: 10.1016/j.chembiol.2006.03.005 . PMID 16720272.
Tropine+acyltransferase at the US National Library of Medicine Medical Subject Headings (MeSH)
|
Clearing Price Examples
Clearing Examples
OneChronos periodic auctions run throughout the trading day, ~10x per second. Instead of matching orders one by one using price-time priority, we use mathematical optimization to seek optimal matches across all orders for all securities within an auction.
Each auction looks for the configuration of potential order matches and per-security clearing prices that will result in the most price improvement dollars cleared within the auction. Price improvement dollars refers to the difference between the limit price on an order and the auction clearing price (i.e., the price at which an order fills), times the quantity filled.
The following examples illustrate the OneChronos order matching process. They are non-exhaustive, demonstrating general and simple outcomes in which the OneChronos optimizer finds the optimal outcome. Our Form ATS-N details the process, and you can always reach us at [email protected].
Example 1: Scenario
Order 1: Buy 100 @ $10.01
Order 2: Sell 100 @ $10.00
Example 1: Outcome
Orders 1 and 2 fill 100 shares @ $10.005.
Example 1: Key Takeaways
Any price in the interval
[\$10.000000,\ \$10.010000]
is a valid clearing price. In the absence of other constraints, OneChronos fills at the midpoint of the clearing price range.
Orders 1 and 2 buy 100 shares @ $10.005. Order 3 sells 200 shares @ $10.005.
Within an auction, all orders for a specific trading symbol will fill at a uniform price. If there is no imbalance, all orders will fill in their entirety.
Orders 1 and 2 have a 50/50 chance of receiving a fill for 100 shares @ $10.005. Order 3 fills 100 shares @ $10.005. Orders on OneChronos may only be entered and executed in round lots.
If there is an imbalance where two or more orders contribute the same number of price improvement dollars (Orders 1 and 2), (round-lot) allocation is round-robin from a randomized starting point.
Orders 1 and 3 receive a fill for 100 shares @ $10.01.
Order 1 has a higher match priority than Order 2 given that it contributes more price improvement dollars to the auction:
100\cdot(10.02-10.01) > 100\cdot(10.01-10.01)
Orders 1 and 2 receive a fill for 100 shares @ $10.005. Order 3 receives a fill for 200 shares @ $10.005.
Filling 100 shares of Order 1 and 2 results in the same price improvement dollars as filling 200 shares of Order 2:
100\cdot(10.01-10.005)+100\cdot(10.01-10.005) = 0.50 = 200\cdot(10.01-10.005)
. Because the two outcomes are pari passu with respect to price improvement dollars and volume cleared, round-robin allocation comes into play as it did for Example 3. Unlike Example 3, there are enough shares available to ensure that both Orders 1 and 2 receive a round lot fill regardless of which randomly selected order serves as the starting point for the round-robin procedure.
Orders 1–4 fill 100 shares @ $10.02.
Filling all orders is both possible and optimal, and the only price at which all orders can fill is $10.02. Neither the least aggressive buy (Order 1) nor the least aggressive sell (Order 4) receives price improvement; Orders 2 and 3 do.
Order 1: Buy 200 all-or-nothing @ $10.01 (Expressive Bidding constraint)
Orders 1 and 3 fill 200 @ $10.005.
Order 1 is a simple Expressive Bid. Expressive Bids allow Users to express complements and substitutes and control execution risk. Order 1 is an Expressive Bid that ensures that the order fills in entirety or not at all; examples involving more sophisticated Expressive Bids will follow.
Matching Order 1 against Order 3 yields the same price improvement dollars as matching Order 2 against Order 3:
200(10.01-10.00) = 2.00 = 100(10.02-10.00)
. However, matching Order 1 against Order 3 also maximizes liquidity (trade volume), which is the optimization's secondary objective. The clearing price range given the match is
[\$10.00,\ \$10.01]
, so $10.005 is the midpoint.
Order 1: Buy 100 ABC @ 10.01 EXCLUSIVE OR Buy 100 XYZ @ 11.00 (Expressive Bidding constraint)
Order 2: Buy 100 XYZ @ 11.00
Order 3: Sell 100 ABC @ 10.00
Order 4: Sell 100 XYZ @ 10.98
Orders 1 and 2 have a 50/50 chance of filling 100 XYZ @ 10.99; Order 4 fills @ 10.99. Order 3 does not receive a fill.
Expressive Bids are neither advantaged or disadvantaged with respect to match priority, but their ability to express substitutability benefits limit orders and other Expressive Bids alike. If Order 1 fills, the optimizer will choose the XYZ leg, given that it yields more price improvement dollars to the auction than the ABC leg.
Order 1: Buy 100 ABC AND Sell 100 XYZ at a spread of $0.05 or better (Expressive Bidding constraint)
Order 2: Sell 100 ABC @ $10.00
Order 3: Buy 100 XYZ @ $10.08
Order 1 fills 100 ABC @ $10.015 and 100 XYZ @ $10.065 resulting in a spread of $0.05. Order 2 fills @ $10.015. Order 3 fills @ $10.065.
A system of inequalities dictates the clearing price range for ABC and XYZ:
\begin{aligned} p_\textrm{xyz} - p_\textrm{abc} &\geq 0.05 \\ p_\textrm{abc} &\geq 10.00 \\ p_\textrm{xyz} &\leq 10.08. \end{aligned}
These inequalities form a triangle with vertices
(p_\textrm{xyz}, p_\textrm{abc})
(10.05, 10.00)
(10.08, 10.00)
(10.08, 10.03)
Thus, the clearing price range for
p_\textrm{xyz}
[\$10.05,\, \$10.08]
and the midpoint is $10.065. Similarly, the clearing price range for
p_\textrm{abc}
[\$10.00,\, \$10.03]
and the midpoint is $10.015.
In this scenario, Orders 2 and 3 receive $0.015/share price improvement, and Order 1 trades at its threshold spread. In this scenario, the auction dynamics resulted in Order 1 receiving no price improvement and Orders 2 and 3 "splitting the difference." The following example shows that this is not always the case. Both Expressive Bids and unconstrained (e.g., limit) orders are eligible for and receive price improvement in the same way.
Example 10: Scenario
Order 1: Buy 200 ABC AND Sell 100 XYZ paying at most $998 net (Expressive Bidding constraint)
Order 2: Sell 100 XYZ @ $10.07
Example 10: Outcome
Order 1 fills 200 ABC @ $10.015 and 100 XYZ @ $10.075 resulting in a spread of $0.06. Order 2 fills @ $10.075. Order 3 fills @ $10.015. Order 4 fills @ $10.075.
Example 10: Key Takeaways
Order 1 results in the constraint
200\cdot p_\textrm{abc} - 100\cdot p_\textrm{xyz} \leq 998
. Order 2 introduces a new inequality into the system of equations from Example 9:
p_\textrm{xyz} \geq 10.07
. These constraints results in a clearing price range of
[\$10.07, \$10.08]
with a midpoint of $10.075, and all orders receive price improvement.
|
Armando has an A average for his first three test scores. His percentages on these tests were
96,92,
97
What is the mean of his three scores?
95
Armando was absent for the fourth test, so he has a
0
on it until he takes it. His teacher has included the
0
as part of his weekly progress report. What is the mean for the four test scores
96,92,97,
0
Geanna did it this way:
96+92+97+0=285
Then she divided
285
by three because Armando only took
3
tests. Joey reminded her that she still has to include the
0
when she divides because it is the average of Armando's scores, not the average of the tests he has taken.
Using Joey's logic, what is Armando's new mean?
Since Armando was doing so well before missing the test, he thinks that just one missing test will not hurt his grade very much. Write him a short note explaining the effect that this one zero has on his grade.
Tell him how many percentage points his average will drop.
|
VRML Preparation for Robot Simulation - FreeCAD Documentation
This page is a translated version of the page VRML Preparation for Robot Simulation and the translation is 4% complete.
In order to build the Denavit-Hartenberg table (see Robot 6-Axis) and prepare the vrml file, you need to get characteristics of the robot. For now, the measurement tool of FreeCAD is not ready, you can use the axes included in TX40_HB007 (the co-ordinates are indicated on the bottom left when you point an object with the mouse) or you have to use the Python console to get some information about the geometry. Note that the DH-table is only required if you need to use the inverse kinematics, i.e. get the Cartesian coordinates or drive the robot with Cartesian coordinates. The DH-table for this robot is the following (mm, deg and deg/s):
{\displaystyle \pi }
{\displaystyle -\pi }
after it. Also, a translation of (-xd, -yd, -zd) is needed just before the Group corresponding to the definition of FOREARM to express it in the relative reference frame centered at D. This means that a translation of (xd, yd, zd) must be inserted before the first rotation. At the end, the VRML file from the definition of ELBOW to the definition of FOREARM looks like this:
At the end of the document, the appropriate closing brackets must be inserted: ]}}}}, for each of the 6 axes. Eventually, the document looks like this (I don't know if I can link the file here because of copyrights):
Retrieved from "http://wiki.freecadweb.org/index.php?title=VRML_Preparation_for_Robot_Simulation/cs&oldid=633039"
Robot/cs
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.