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Enhanced Packages - Maple Help
Home : Support : Online Help : System : Information : Updates : Maple 9.5 : Enhanced Packages
Enhanced Packages in Maple 9.5
For information on new Maple 9.5 packages, see New Packages in Maple 9.5.
Enhanced Packages in Maple 9.5 contains information for the following packages.
The Differential Equation packages: DEtools, PDEtools, and diffalg
Groebner Bases for Toric Ideals
ToInert Options
DEtools and PDEtools
New SolveTools Inequality Submodule
New Options to ToInert
The new CodeGeneration[Save] command allows user-contributed language definitions to be saved in a Maple archive without requiring that you save a copy of the CodeGeneration module.
In connection with important enhancements in the differential equation solvers dsolve and pdsolve (see Updates to Differential Equations (DE) Solvers in Maple 9.5), seven new commands with varied purposes - some of them based on original algorithms - are available in DEtools and PDEtools for this release. For a description of these new commands, see Updates to Differential Equations (DE) Solvers in Maple 9.5 (PDEtools).
The new command Groebner[ToricIdealBasis] implements two algorithms to compute the reduced Groebner basis of a toric ideal.
Three new functions have been added to the LREtools package: AnalyticityConditions, dAlembertiansols, and IsDesingularizable.
This command determines necessary conditions for the solution of a linear recurrence equation to be analytic, in terms of the initial values.
LREtools[AnalyticityConditions](n*E-1,E,f(n));
{\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{0}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}}
LREtools[dAlembertiansols]
This command finds all d'Alembertian solutions of a linear recurrence equation, that is, solutions annihilated by a product of first order operators.
rec := (-n-1)*a(n)+(3+2*n)*a(n+1)+(-n-2)*a(n+2)=1/(n+2):
LREtools[hypergeomsols](rec, a(n), {}, output=basis);
LREtools[dAlembertiansols](rec, a(n), {}, output=basis);
[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\sum }_{\textcolor[rgb]{0,0,1}{\mathrm{n1}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}}^{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{\mathrm{n1}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\left(\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\sum }_{\textcolor[rgb]{0,0,1}{\mathrm{n1}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}}^{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{}\frac{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\sum }_{\textcolor[rgb]{0,0,1}{\mathrm{n2}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}}^{\textcolor[rgb]{0,0,1}{\mathrm{n1}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{\mathrm{n2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}}}{\textcolor[rgb]{0,0,1}{\mathrm{n1}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}}\right)]
This command finds a multiple
M
of a given linear recurrence operator
L
, if possible, such that the leading or trailing coefficient of
M
has no integer roots.
LREtools[IsDesingularizable]((n-1)*E+n,E,n,trailing,output=operator);
\textcolor[rgb]{0,0,1}{\mathrm{true}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{E}}^{\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}{E}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}
Four new functions have been added to the PolynomialTools package: GcdFreeBasis, GreatestFactorialFactorization, ShiftEquivalent, and ShiftlessDecomposition.
PolynomialTools[GcdFreeBasis]
This command factors a given set of polynomials as far as possible by using only gcds. This can be used, for example, to refine several partial factorizations of the same polynomial.
The following is a way to compute the factors in the squarefree decomposition of a polynomial.
f := x^9-x^7-x^5+x^3;
\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{9}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{7}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}
PolynomialTools[GcdFreeBasis]([f, gcd(f,diff(f,x))]);
[\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\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}{1}]
sqrfree(f);
[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]\textcolor[rgb]{0,0,1}{,}[{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}]\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}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}]]]
This command factors a univariate polynomial into a product of falling factorials in a unique way similar to the squarefree factorization.
\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{9}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{7}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}
PolynomialTools[GreatestFactorialFactorization](f,x);
[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}]\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}{2}\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}{3}]]]
This command determines whether a univariate polynomial is a Taylor shift of another polynomial, and if so, returns the shift distance.
PolynomialTools[ShiftEquivalent](2*x+1,2*x+2,x);
\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}
PolynomialTools[ShiftEquivalent](2*x+1,2*x+2,x,'integer');
\textcolor[rgb]{0,0,1}{\mathrm{FAIL}}
\textcolor[rgb]{0,0,1}{2}
PolynomialTools[ShiftEquivalent](x^2-1,x^2+1,x);
\textcolor[rgb]{0,0,1}{\mathrm{FAIL}}
This command computes the coarsest factorization of a univariate polynomial separating the irreducible factors both by their multiplicities and their shift equivalence classes.
\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{9}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{7}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}
PolynomialTools[ShiftlessDecomposition](f,x);
[\textcolor[rgb]{0,0,1}{1}\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}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]\textcolor[rgb]{0,0,1}{,}[\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}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}]]]]]
This package has been extended by four new functions.
QDifferenceEquations[AccurateQSummation] implements the method of accurate q-summation
QDifferenceEquations[ExtendSeries] computes series solutions of a linear q-difference equation
QDifferenceEquations[QHypergeometricSolution] finds all q-hypergeometric solutions of a given linear q-difference equation
QDifferenceEquations[SeriesSolution] computes series solutions of a linear q-difference equation
The SolveTools[Inequality] submodule is intended for solving systems of inequalities. It exports three new procedures.
SolveTools[Inequality][LinearUnivariate] solves a linear inequality with respect to one variable
SolveTools[Inequality][LinearUnivariateSystem] solves a system of linear inequalities with respect to one variable
SolveTools[Inequality][LinearMultivariateSystem] for solving systems of linear inequalities
The StringTools package includes the new Length command that returns the length of a string.
StringTools[MaximalPalindromicSubstring]
The new MaximalPalindromicSubstring command that finds a maximal substring that is equal to itself reversed.
You can use the new StringTools[Randomize] command to seed the random number generator used by the external code underlying the StringTools package.
Most interactive tutors in the Student package now include a field that shows the non-tutor command, which can be used to produce the same result in the worksheet. This can be used as a starting point for further exploration.
The Student[Precalculus] package has the addition of non-tutor commands in allowing exploration of concepts in greater depth. The basic Line command computes lines from various different forms of input and returns information about slope and intercepts, as well as the line itself.
Two new commands have been implemented in this package: SumTools[Hypergeometric][EfficientRepresentation] and SumTools[Hypergeometric][RegularGammaForm].
The two commands SumTools[Hypergeometric][RationalCanonicalForm] and SumTools[Hypergeometric][MultiplicativeDecomposition] have been extended by the ``third'' and the ``fourth'' normal forms.
The ToInert command for converting to an inert representation of a Maple object has been extended to allow for finer control over what objects need to be converted. Addition of exclude and include options let you specify exactly which objects to exclude or convert.
The parse command accepts optional offset and lastread parameters, which make it possible to parse a string containing multiple Maple commands.
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Extreme Points and Strongly Extreme Points in Orlicz Spaces Equipped with the Orlicz Norm | EMS Press
Criteria for extreme points and strongly extreme points of the unit ball in Orlicz spaces with the Orlicz norm are given. These results are applied to a characterization of extreme points of
B(L^1 + L^\infty)
which corresponds to the result obtained by R. Grz\c{a}\'slewicz and H. Schaefer [Indag. Math. 3 (1992) 173--178] and H. Schaefer [Arch. Math. 58 (1992) 160--163]. Moreover, we show that every extreme point of
B(L^1 + L^\infty)
is strongly extreme. We also get criteria for extreme points of
B(L^p \cap L^\infty)
1 \le p < \infty
, using Theorem 1 and for strongly extreme points of
B(L^p \cap L^\infty)
1 \le p <\infty
, applying Theorem 2. Although, criteria for extreme points of
B(L^1 \cap L^\infty)
were known [see H. Hudzik, A. Kami\'nska and M. Masty{\l}o, Arch. Math. 68 (1997) 159--168], we can easily deduce them from our main results and we can extend those results to establish which among extreme points are strongly extreme. The descriptions of the extreme and strongly extreme points of
B(L^p \cap L^\infty)
1 < p < \infty
, are original. Moreover, criteria for extreme points and strongly extreme points of the unit ball in the subspace of finite elements of an Orlicz space are deduced on the basis of our main results.
Henryk Hudzik, Yunan Cui, Ryszard Pluciennik, Extreme Points and Strongly Extreme Points in Orlicz Spaces Equipped with the Orlicz Norm. Z. Anal. Anwend. 22 (2003), no. 4, pp. 789–817
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I shd. be very much obliged for your opinion on enclosed.— You may remember in 3 first vols. tabulated, all orders went right except Labiatæ (By way if by any extraordinary chance you have not thrown away scrap of paper with former results I wish you would return it, for I have lost my copy, & I shall have all the division to do again; but do not hunt for it, for in any case I shd. have gone over calculation again.)2
Now I have done the 3 other vols.—3 You will see that all species in 6 vols together go right, & likewise all orders in the 4 last vols. except Verbenaceæ.4 Is not Verbenaceæ very closely allied to Labiatæ, if so one would think that it was not mere chance this coincidence? The species in Labiatæ & Verbenaceæ together are between
\frac{1}{5}
\frac{1}{6}
of all the species (15,645) which I have now tabulated.
Now bearing in mind the many local Floras which I have tabulated, (belting the whole northern hemisphere) & considering that they (& authors of D. C.Prodromus) would probably take different degrees of care in recording vari-eties; & the genera would be divided on different principles by different men &c, I am much surprised at uniformity of result, & I am satisfied that there must be truth in rule that the small genera vary less than the large.5 What do you think?
Hypothetically I can conjecture how the Labiatæ might fail, namely if some small divisions of the Order were now coming into importance in the world & varying much & making species: this makes me want to know whether you could divide Labiatæ into a few great natural divisions, & then I would tabulate them sep-arately as sub-orders;6 I see Lindley makes so many divisions, that there would not be enough in each for average.7 I send the Table of Labiatæ for the chance of your being able to do this for me: you might draw oblique lines including & separating both large & small genera.
I have, also, divided all the species into two equal masses; & my rule holds good for all the species in a mass in the 6 volumes; but it fails in several (4) large orders viz Labiatæ Scrophulareaceæ, Acanthaceaceæ & Proteaceæ.— But then when the species are divided into two almost exactly equal divisions; the division with large genera are so very few, for instance in Solanaceæ Solanum balances all others. In Labiatæ 7 gigantic genera balance all others (viz 113) & in Proteaceæ 5 genera balance all others. Now according to my hypothetical notions, I am far from supposing that all genera go on increasing for ever, & therefore I am not surprised at this result, when the division is so made that only a very few genera are on one side. But, according to my notions, the sections or sub-genera of the gigantic genera ought to obey my rule (ie supposing a gigantic genus had come to its maximum; whatever increase was still going on ought to be going on in the larger sub-genera.)
Do you think that the sections of the gigantic genera in D. C. Prodromus are generally natural; i.e. not founded on mere artificial characters. If you think that they are generally made as natural as they can be, then I shd. like very much to tabulate the sub-genera, considering them for the time as good genera. In this case, & if you do not think me unreasonable to ask it, I shd. be very glad of loan of Vols. X., XI, XII, & XIV. which include Acanthaceæ, Scrophulariaceæ, Labiatæ & Proteaceæ,—that is the orders, which when divided quite equally do not accord with my rule, & in which a very few genera balance all the others.—
I have written you a tremendous long prose. | Ever yours | C. Darwin.
Dated by the references to CD’s calculations using data from Candolle and Candolle 1824–73 (see nn. 2 and 3, below).
CD had calculated the number of species presenting varieties in large and small genera using volumes 2, 10, and 11 of Candolle and Candolle 1824–73 in December 1857 (see Correspondence vol. 6, letter from J. D. Hooker, [6 December 1857], and letter to J. D. Hooker, 9 December [1857]). His tables and calculations on this work are in DAR 15.2: 35–76.
Hooker had sent CD volumes 12, 13, and 14 of Candolle and Candolle 1824–73 (Correspondence vol. 6, letter from J. D. Hooker, [17–23 December 1857]).
CD’s results are given in a table in Natural selection, pp. 153–4.
For the floras that CD used to tabulate the number of varieties in large and small genera, see Correspondence vol. 6, letter to J. D. Hooker, 30 September [1857].
The order Labiatae was discussed by CD in Natural selection, pp. 155 & 156 n., because the result of his calculations on that group of plants opposed all the others. However, CD did not split the order into sub-orders as suggested here. He focused instead on the gigantic genera.
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AVGWAIT - Anaplan Technical Documentation
Calculation functions
All functions
AVGWAIT(Number of servers, Arrival rate, Average duration)
Number of servers (required) Number The number of servers (for example, call center agents) available to process requests.
Arrival rate (required) Number The interval between the arrival of each request.
Average duration (required) Number The average duration it takes to process each request.
The AVGWAIT function returns a number. This is the average waiting time for a request to be processed, using the same time unit as the Arrival rate and Average duration arguments.
How the AVGWAIT function is calculated
The ANSWERTIME function calculates the solution to this equation:
\text{AVGWAIT}(x,y,z) = \dfrac{(\text{ERLANGC}(x,a)*z)}{x * (1 - p)}
x is the Number of servers.
y is the Arrival rate.
z is the Average duration.
a is the offered load, which is y multiplied by z.
p is the offered load per server, which is a divided by x.
Time unit for arguments
The Arrival rate and Average duration arguments do not have to use a specific time unit. For example, they can use seconds or minutes. However, both arguments must use the same time unit.
The maximum number you can use for the Number of servers argument is five million.
In this example, the Call Centers list is on columns, and line items on rows. The first three line items contain the data for the AVGWAIT function for each call center:
The scheduled number of servers to process requests
The arrival rate, or interval between each request arriving
The average duration it takes to complete requests
The fourth line item uses the AVGWAIT function to calculate what the average waiting time for a request to be processed given the number of servers, arrival rate, and average duration to process requests.
The fifth line item enables you to adjust the arrival rate of requests. The formula in the sixth line item uses the adjusted arrival. This enables you to see how the average waiting time changes given a different arrival rate. A value of Infinity is given for Call Center 2 because the rate of incoming requests is higher than the ability to process them, which means that calls have to wait indefinitely.
Call Center 1 Call Center 2 Call Center 3 Call Center 4
Scheduled Number of Servers 25 45 50 39
Request Arrival Rate 0.84 0.93 0.69 0.68
Average Duration 25 46 45 45
AVGWAIT(Scheduled Number of Servers, Request Arrival Rate, Average Duration)
1.9213063 13.4641512 0.00270453 0.53752576
Adjusted Arrival Rate 0.9 1.1 0.89 0.77
Adjusted Average Waiting Time
AVGWAIT(Scheduled Number of Servers, Adjusted Arrival Rate, Average Duration)
5.07923029 Infinity 0.40000875 3.82731801
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On -Fibrations in Bitopological Semigroups
Suliman Dawood, Adem Kılıçman, "On -Fibrations in Bitopological Semigroups", The Scientific World Journal, vol. 2014, Article ID 675761, 7 pages, 2014. https://doi.org/10.1155/2014/675761
Suliman Dawood1 and Adem Kılıçman2
1Department of Mathematical Sciences, Hodeidah University, Hodeidah, Yemen
We extend the path lifting property in homotopy theory for topological spaces to bitopological semigroups and we show and prove its role in the -fibration property. We give and prove the relationship between the -fibration property and an approximate fibration property. Furthermore, we study the pullback maps for -fibrations.
In homotopy theory for topological space (i.e., spaces), Hurewicz [1] introduced the concepts of fibrations and path lifting property of maps and showed its equivalence with the covering homotopy property. Coram and Duvall [2] introduced approximate fibrations as a generalization of cell-like maps [3] and showed that the uniform limit of a sequence of Hurewicz fibrations is an approximate fibration. In 1963, Kelly [4] introduced the notion of bitopological spaces. Such spaces were equipped with its two (arbitrary) topologies. The reader is suggested to refer to [4] for the detail definitions and notations. The concept of homotopy theory for topological semigroups has been introduced by Cerin in 2002 [5]. In this theory, he introduced -fibrations as extension of Hurewicz fibrations. In [6], we introduced the concepts of bitopological semigroups, -bitopological semigroups, and -fibrations as extension of -fibrations.
This paper is organized as follows. It consists of five sections. After this Introduction, Section 2 is devoted to some preliminaries. In Section 3 we show the pullbacks of -maps which have the -fibration property that will also have this property and the pullbacks of -fibrations are -fibrations under given conditions. In Section 4 we develop and extend path lifting property in homotopy theory for topological semigroups to theory for bitopological semigroups. Some results about Hurewicz fibrations carry over. In Section 5 we give and prove the relationship between the -fibration property and an approximate fibration property.
Throughout this paper, by all we mean all topological spaces which will be assumed Hausdorff spaces. By all we mean all bitopological spaces . For two bitopological spaces and , a p-map is a function from into that is continuous function (i.e., a map) from a space into a space and from into [4].
Recall [5] that a topological semigroup or an S-space is a pair consisting of a topological space and a map from the product space into such that for all . An S-space is called an S-subspace of if is a subspace of and the map takes the product into and for all . We denote the class of all S-spaces by . For every space , by , we mean the space of all paths from the unit closed interval into with the compact-open topology. Recall [5] that, for every S-space , is an S-space where is a map defined by for all , . The shorter notion for this S-space will be . For every space , the natural S-space is an S-space , where is a continuous associative multiplication on given by and for all . We denote the class of all natural S-spaces by , where .
Recall [5] that the function is called an S-map if is a map of a space into and for all . The function of a natural S-space into is an S-map if and only if it is continuous. The S-maps are called S-homotopic and write provided there is an S-map called an S-homotopy such that and for all .
A bitopological semigroup is a pair consisting of a bitopological space and the associative multiplication on such that is an -map from the product bitopological space into . For , by we mean the bitopological subspace of . If the -map takes the product into then the pair will be a bitopological semigroup and will be called an b-subspace of .
The function is called an -map from into provided is an S-map from a function S-space into an S-space , where . We say that is an Sp-map if it is an -map and -map.
An c-bitopological semigroup is a triple consisting of bitopological semigroups and an S-map from an S-space into an S-space . In our work, for any S-space, can be regarded as an -bitopological semigroup where is the identity S-map on . That is, .
An c-map from into is a pair of an -map and -map such that .
Definition 1 (see [6]). Let and be two S-maps. An S-map is said to have the -fibration property by an S-map provided for every and, given two S-maps and with , there exists an S-homotopy such that and for all .
Definition 2 (see [6]). An -map is called an -fibration if an -map has the -fibration property by an S-map . That is, for every and given two S-maps and with , there exists an S-homotopy such that and for all .
Let be an c-bitopological semigroup and let be an -subspace of . The -bitopological semigroup is called an c-subspace of provided for all .
Theorem 3 (see [6]). Let be an c-map and be an S-subspace of such that . Then the triple is an c-subspace of and a pair is an c-map from an c-bitopological semigroup into , where .
Corollary 4 (see [6]). Let be an -fibration and let be an S-subspace of such that . Then the restriction c-map is an -fibration, where .
3. The Pullback c-Maps
In this section, we show that the pullbacks of S-maps which have the -fibration property will also have this property and the pullbacks of -fibrations are -fibrations under given conditions.
Let be an -map and let be an S-map. Let where .
Lemma 5. Let be an c-map and let be an S-map. Then the pair is an b-subspace of the bitopological semigroup , where .
Proof. It is clear that and are subspaces of a bitopological space . Since is an S-map and is an -map, then, for all , This implies for all . That is, is an -subspace of the bitopological semigroup . Similarly, is an -subspace of the bitopological semigroup .
Henceforth, in this paper, by and , we mean the usual first and the second projection S-maps (or maps), respectively.
Theorem 6. Let be an -fibration and let be an S-map. Then the S-map has the -fibration property by an S-map such that for all .
Proof. Since is an -map then, for all , That is, for all . Hence, by the last lemma, is a well-defined S-map taking into .
Now let and let and be two S-maps with .
Take an S-map and an S-homotopy We observe that for all . That is, . Since is an -fibration, then there is an S-homotopy such that and for all .
Define an S-homotopy by for all . We observe that for all and for all . That is, . Hence has the -fibration property by an S-map .
In the last theorem, if (i.e., is an -map), let ; then is a well-defined S-map taking into , where . That is, the triple is an -bitopological semigroup, called a pullback c-bitopological semigroup of induced from by . The pair which is given by for all is an -map, called a pullback c-map of induced by . We observe that for all .
Theorem 7. Let be an -fibration and let be an S-map such that . Then the pullback -map of induced by is an -fibration.
Proof. It is obvious by the last theorem and the second part in Definition 2.
4. The c-Lifting Functions
In this section, we define the path lifting property for -maps by giving the concept of an -lifting property and we show its role in satisfying the -fibration property.
Recall [5] that for an S-map , the map: for all is an S-map from into , denoted by . Then for every -bitopological semigroup , is an S-map from into . That is, the triple is an -bitopological semigroup where and are compact-open topologies on which are induced by and , respectively. The shorter notion for this c-bitopological semigroup will be .
For a map , by , we mean the set
Proposition 8. Let be an S-map. Then is an S-subspace of an S-space , where is a compact-open topology on which is induced by .
Proof. It is clear that is a subspace of a space . We observe that, for all , That is, Hence is an S-subspace of an S-space .
In the last theorem, the shorter notion for the S-space will be .
Definition 9. Let be an -map. An S-map from an S-space into is called an c-lifting function for an -map provided satisfies the following: (1) for all ;(2) for all .
And will be denoted to -lifting function for an -map , if it exists.
Example 10. Let be an -bitopological semigroup. For every S-space , the -map is an -map, where for all . Note that for all , . This -map has an -lifting function which is given by Note that for all , .
The following theorem clarifies the existence property for -lifting function in -fibration theory. That is, it clarifies that the existence of -lifting function for any -fibration is necessary and sufficient condition.
Theorem 11. An -map is an -fibration if and only if there exists an -lifting function for .
Proof. Suppose that is an -fibration. Take . Define two S-maps by and for all , respectively. We observe that Since is an -fibration, then there exists an S-homotopy such that and for all . Define an S-map by We observe that, for all , That is, is an -lifting function for .
Conversely, suppose that there exists an -lifting function for . Let and let and be two given S-maps with . Define an S-homotopy by We observe that for all , . That is, and for all . Hence is an -fibration.
Theorem 12. Let be an -fibration. Then the c-map is an -fibration.
Proof. Since is an -fibration, then there exists -lifting function for such that for all . Let and let and be two given S-maps with for all , , where is a compact-open topology on which is induced by . Define an S-homotopy by We observe that for all , . That is, and for all . Hence is an -fibration.
An -lifting function is called regular if for every , , where is the constant path in (i.e., ), similar for . An -fibration is called regular if it has regular c-lifting function.
Example 13. In Example 10, the -lifting function which is given by is regular. Note that, for every , for all .
The following theorem is an analogue of results of Fadell in Hurewicz fibration theory [7].
Theorem 14. Let be a regular -fibration and let be an S-map defined by for all where . Then (1);(2) preserving projection. That is, there is an S-homotopy between two S-maps and such that for all .
Proof. For the first part, we observe that, for every , That is, .
For the second part, for and , define a path by By the regularity of , we can define an S-homotopy by for all , . Then for all , . That is, . Also we get that for all , . Hence preserving projection.
5. Approximate Fibrations
Coram and Duvall [2] introduced approximate fibrations as a generalization of cell-like maps [3] and showed that the uniform limit of a sequence of Hurewicz fibrations is an approximate fibration. A map of compact metrizable spaces and is called an approximate fibration if, for every space and for given , there exists such that whenever and are maps with , then there is homotopy such that and One notable exception is that the pullback of approximate fibration need not be an approximate fibration.
The following theorem shows the role of the -fibration property in inducing an approximate fibration property.
For an S-map with metrizable spaces and , by and , we mean the metric functions on and , respectively; by we mean the product metrizable space of and with a metric function by we mean the graph of (i.e., ) which is an S-subspace of ; for a positive integer , by , we mean the -neighborhood of in a metrizable space which is also S-subspace of .
Theorem 15. Let be a map with compact metrizable spaces and . Then is an approximate fibration if and only if, for every positive integer , there exists a positive integer such that the S-map has the -fibration property by the inclusion S-map , where for all .
Proof. Let be any positive integer. For , let be given in the definition of approximate fibration. Since and is a continuous function, then let be chosen such that if and , then . Choose a positive integer , such that .
Now let and let and be two given S-maps with . Define a map by and a homotopy by for all and . We get that for all . Since , then there exists such that Then for all . This implies for all . Hence, since is an approximate fibration, there exists a homotopy such that and for all , . Define an S-homotopy by Then we get that for all and for all . Hence has the -fibration property by .
Conversely, let be given. Since is a continuous function, then let be chosen such that if and , then . Choose a positive integer such that . By hypothesis, there exists a positive integer such that has the -fibration property by .
Take . Let be any space and let and be two given maps with for all . Define an S-map by and an S-homotopy by for all and . Since , then there exists an S-homotopy such that and for all . By the last part, we can define a homotopy by We get that . Since , then there exists such that Then This implies for all . Hence is an approximate fibration.
The authors also gratefully acknowledge that this research was partially supported by the University Putra Malaysia under the ERGS Grant Scheme having Project no. 5527068.
W. Hurewicz, “On the concept of fiber space,” Proceedings of the National Academy of Sciences of the United States of America, vol. 41, pp. 956–961, 1955. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
D. S. Coram and J. Duvall, “Approximate fibrations,” The Rocky Mountain Journal of Mathematics, vol. 7, no. 2, pp. 275–288, 1977. View at: Publisher Site | Google Scholar | MathSciNet
R. C. Lacher, “Cell-like mappings. I,” Pacific Journal of Mathematics, vol. 30, pp. 717–731, 1969. View at: Publisher Site | Google Scholar | MathSciNet
J. C. Kelly, “Bitopological spaces,” Proceedings of the London Mathematical Society: Third Series, vol. 13, no. 3, pp. 71–89, 1963. View at: Google Scholar | MathSciNet
Z. Cerin, “Homotopy theory of topological semigroup,” Topology and Its Applications, vol. 123, no. 1, pp. 57–68, 2002. View at: Publisher Site | Google Scholar
S. Dawood and A. Klçman, “On Extending
{S}_{\aleph }
fibrations to
{C}_{\aleph }
-fibrations in bitopological semigroups,” Submitted for publication. View at: Google Scholar
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Copyright © 2014 Suliman Dawood and Adem Kılıçman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Interpolate by a factor of two using polyphase IIR - MATLAB - MathWorks India
H\left(z\right)=0.5*\left[{A}_{1}\left({z}^{2}\right)+{z}^{-1}{A}_{2}\left({z}^{2}\right)\right]
{A}_{1}\left(z\right)=\prod _{k=1}^{{K}_{1}}\frac{{a}_{k}^{\left(1\right)}+{z}^{-1}}{1+{a}_{k}^{\left(1\right)}{z}^{-1}}
{A}_{2}\left(z\right)=\prod _{k=1}^{{K}_{2}}\frac{{a}_{k}^{\left(2\right)}+{z}^{-1}}{1+{a}_{k}^{\left(2\right)}{z}^{-1}}
{A}_{1}\left(z\right)={z}^{-k}
{A}_{2}\left(z\right)=\prod _{K=1}^{{K}_{2}^{\left(1\right)}}\frac{{a}_{k}+{z}^{-1}}{1+{a}_{k}{z}^{-1}}\prod _{K=1}^{{K}_{2}^{\left(2\right)}}\frac{{c}_{k}+{b}_{k}{z}^{-1}+{z}^{-2}}{1+{b}_{k}{z}^{-1}+{c}_{k}{z}^{-2}}
G\left(z\right)=0.5*\left[{A}_{1}\left({z}^{2}\right)-{z}^{-1}{A}_{2}\left({z}^{2}\right)\right]
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Nm Shah 2018 for Class 11 Commerce Economics Chapter 4 - Graphic Presentation
Nm Shah 2018 Solutions for Class 11 Commerce Economics Chapter 4 Graphic Presentation are provided here with simple step-by-step explanations. These solutions for Graphic Presentation are extremely popular among Class 11 Commerce students for Economics Graphic Presentation Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Nm Shah 2018 Book of Class 11 Commerce Economics Chapter 4 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Nm Shah 2018 Solutions. All Nm Shah 2018 Solutions for class Class 11 Commerce Economics are prepared by experts and are 100% accurate.
Marks : 0-10 10-20 20-30 30-40 40-50
No. of Students : 3 10 14 10 3
Draw a histogram to represent the frequency distribution of marks . Comment on the shape of the histogram.
Histogram is a joining rectangular diagram with equal class interval of size 10.
Present the data given in the table below in the form of a Histogram :
Mid-points : 115 125 135 145 155 165 175 185 195
Frequency : 6 25 48 72 116 60 38 22 3
The two-dimensional diagrams that depict the frequency distribution of a continuous series by the means of rectangles are called histograms.
In order to construct a histogram, we first require the class intervals corresponding to the various mid-points, which is calculated using the following formula.
The value obtained is then added to the mid point to obtain the upper limit and subtracted from the mid-point to obtain the lower limit.
For the given data, the class interval is calculated by the following value of adjustment.
\mathrm{Value} \mathrm{of} \mathrm{Adjustment} = \frac{125 -115 }{2} = 5
Thus, we add and subtract 5 to each mid-point to obtain the class interval.
The lower limit of first class = 115 – 5 = 110
Upper limit of first class = 115 + 5 = 120.
Thus, the first class interval is (110-120). Similarly, we can calculate the remaining class intervals.
Mid-points Class Interval Frequency
Make a frequency Polygon and Histogram using the given data:
No. of Students : 5 12 15 22 14 4
Draw Histogram from the following data:
Marks obtained : 10-20 20-30 30-40 40-50 50-70 70-100
No. of Students : 6 10 15 10 6 3
The data is given in the form of unequal class interval. So, we will first make appropriate adjustment in the frequencies to make the class intervals equal. The general formula for the adjustment of the frequency is as follows.
\mathrm{Adjusted} \mathrm{Frequency}=\frac{\mathrm{Required} \mathrm{class} \mathrm{interval} × \mathrm{Frequency}}{\mathrm{Actual} \mathrm{class} \mathrm{interval}}
Marks No. of Students Adjusted Frequency
10−20 6 −
20−30 10 −
\frac{10×6}{20}=3
70−100 3
\frac{10×3}{30}=1
In a certain colony a sample of 40 households was selected . The data on daily income for this sample are given as follows:
(a) Construct a Histogram and a frequency Polygon.
(b) Show that the area under the polygon is equal to the area under the histogram.
Income Tally Marks Frequency Area for each class
0 − 100 2 100 × 2 = 200
100 − 200 14 100 × 14 = 1400
200 − 300 8 100 × 8 = 800
40 Total area =4000
(b) The area under a histogram and under a frequency polygon is the same (i.e equal to 4000) because of the fact that we extend the first class interval to the left by half the size of class interval as the starting point of the frequency polygon. Similarly, the last class interval is extended to the right by the same amount as the end point of the frequency polygon. This ensures that the area that was excluded while joining the mid-points is included in the frequency polygon such that the area under the frequency polygon and the area of histogram is the same.
Present the data given in the table below in Histogram:
Marks : 25-29 30-34 35-39 40-44 45-49 50-54 55-59
Frequency : 4 5 23 31 10 8 5
Before proceeding to construct histogram, we first need to convert the given inclusive series into an exclusive series using the following formula.
The value of adjustment as calculated is then added to the upper limit of each class and subtracted from the lower limit of each class.
\mathrm{Here}, \mathrm{the} \mathrm{value} \mathrm{of} \mathrm{adjustment} = \frac{30 - 29}{2} = 0.5
Therefore, we add 0.5 to the upper limit and subtract 0.5 from the lower limit of each class.
A survey showed that the average daily expenditures (in rupees) of 30 households in a city were:
(a) Prepare a frequency distribution using class intervals:
10-14, 15-19, 20-24, 25-29, 30-34 and 35-39.
(b) What percent of the households spend more than ₹ 29 each day?
(c) Draw a frequency histogram for the above data.
(b) Households that spend more than 29 each day
=\frac{6}{30}×100=20%
(c) To construct histogram, we first need to convert the given inclusive series into an exclusive series using the following formula.
\mathrm{Here}, \mathrm{the} \mathrm{value} \mathrm{of} \mathrm{adjustment} = \frac{15 - 14}{2} = 0.5
Draw Histogram from a given data relating to monthly pocket money allowance of the students of class XII in a school:
Size of classes( in ₹ ) 0-5 5-10 10-15 15-20 20-25 25-30 30-35 35-40
No of students 5 10 15 20 25 15 10 5
Draw a Histogram and Frequency Polygon of the following information.
Draw ogive
—
(a) less than, and (b) more than of the folowing data:
Weekly wages of No of workers : 100-105
Workers (₹) : 200 210 230 320 350 520 410
In order to construct the ogives, we first need to calculate the less than and the more than cumulative frequencies as as follows.
Weekly Wages Cumulative
More than 130 2240
-
-
-
-
-
-
Prepare a less than ogive from the following data:
Class : 0-6 6-12 12-18 18-24 24-30 30-36
Frequency : 4 8 15 20 12 6
In order to prepare a less than ogive, we first need to calculate the less than cumulative frequency distribution as follows:
From the following frequency distribution prepare a 'less than ogive'.
Capital (₹ in lakhs) : 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80
No. of companies : 2 3 7 11 15 7 2 3
For constructing a less than ogive, we convert the frequency distribution into a less than cumulative frequency distribution as follows:
Capital Cumulative Frequency
Arrange the following information on a time-series graph:
Year : 2009-10 2010-11 2011-12 2012-13 2013-14 2014-15 2015-16
NDP(₹ in '000 crores) : 35 36 37 40 41 44 44
Plot the following data of annual profits of a firm on a time series graph.
Profits(₹ in thousand) : 60 72 75 65 80 95
Exports (₹ in crores) 1505 2265 2070 1805 1632 1527 1845
Imports( ₹ in crores) 1005 1145 1980 1335 1547 1435 1740
The given data can be presented in the form of a time series graph as follows:
Here, the smallest value is 1,005 which is far higher than zero. Therefore, in this case we use a false base line starting from 1,000.
Present graphically the following sales of Delhi branch of USHA FANS.
(₹ in '000) : 13 15 12 19 25 31 29 27 35
Prepare a graph showing total cost and total production of a scooter manufacturing company.
Production (in units) : 8500 9990 11700 13300 15600
Total cost (₹ in lakh) : 24 29 34 45 49
The given data can be presented in the form of a time series graph (i.e multiple y-axis graph) as follows:
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q^()_(p)(cos ax-sin bx)^(2) (where a, b are integers) = -Turito
Answer:The correct answer is: 2 The centre of the sphere is (1, 2, –3) so if other extremity of diameter is (x1, y1, z1), then
{\int }_{-\pi /4}^{\pi /4} \frac{{e}^{x}\left(x\mathrm{sin}x\right)}{{e}^{2x}-1}dx
\frac{l}{3}=\frac{m}{-3}=\frac{n}{-9}
\frac{l}{-1}=\frac{m}{+1}=\frac{n}{3}
\frac{x-2}{-1}=\frac{y+1}{1}=\frac{z+1}{3}
{\int }_{-\pi /4}^{\pi /4} \frac{{e}^{x}\left(x\mathrm{sin}x\right)}{{e}^{2x}-1}dx
\frac{l}{3}=\frac{m}{-3}=\frac{n}{-9}
\frac{l}{-1}=\frac{m}{+1}=\frac{n}{3}
\frac{x-2}{-1}=\frac{y+1}{1}=\frac{z+1}{3}
f:R\to R,f\left(x\right)=\left\{\begin{array}{c}|x-\left[x\right]|,\left[x\right]\\ |x-\left[x+1\right]|,\left[x\right]\end{array}\right\
\begin{array}{r}\text{ is odd }\\ 1\text{ is even where [.] }\end{array}
{\int }_{-2}^{4} f\left(x\right)dx
f:R\to R,f\left(x\right)=\left\{\begin{array}{c}|x-\left[x\right]|,\left[x\right]\\ |x-\left[x+1\right]|,\left[x\right]\end{array}\right\
\begin{array}{r}\text{ is odd }\\ 1\text{ is even where [.] }\end{array}
{\int }_{-2}^{4} f\left(x\right)dx
{\int }_{0}^{1} |\mathrm{sin} 2\pi x|\mid dx
{\int }_{0}^{1} |\mathrm{sin} 2\pi x|\mid dx
{\int }_{0}^{100} \left\{\sqrt{x}\right\}dx
{\int }_{0}^{100} \left\{\sqrt{x}\right\}dx
\frac{x-4/3}{2}=\frac{y+6/5}{3}=\frac{z-3/2}{4}
\frac{5y+6}{8}=\frac{2z-3}{9}=\frac{3x-4}{5}
\frac{x-4/3}{2}=\frac{y+6/5}{3}=\frac{z-3/2}{4}
\frac{5y+6}{8}=\frac{2z-3}{9}=\frac{3x-4}{5}
v
\theta
v=\sqrt{5gL}
{\left(\frac{v}{2}\right)}^{2}={v}^{2}-2gh\left(ii\right)
h=L\left(1-\mathrm{cos}\theta \right)\left(iii\right)
Solving Eqs.\left(i\right), \left(ii\right)and \left(iii\right), we get
\mathrm{cos}\theta =-\frac{7}{8}
or \theta ={cos}^{-1}\left(-\frac{7}{8}\right)=151°
v
\theta
v=\sqrt{5gL}
{\left(\frac{v}{2}\right)}^{2}={v}^{2}-2gh\left(ii\right)
h=L\left(1-\mathrm{cos}\theta \right)\left(iii\right)
Solving Eqs.\left(i\right), \left(ii\right)and \left(iii\right), we get
\mathrm{cos}\theta =-\frac{7}{8}
or \theta ={cos}^{-1}\left(-\frac{7}{8}\right)=151°
y=k{x}^{2} \left(y
m
x
a
y
ma\mathrm{cos}\theta =mg\mathrm{cos}\left(90-\theta \right)
⇒\frac{a}{g}=\mathrm{tan}\theta ⇒\frac{a}{g}=\frac{dy}{dx}
⇒\frac{d}{dx}{\left(kx\right)}^{2}=\frac{a}{g}⇒x=\frac{a}{2gk}
y=k{x}^{2} \left(y
m
x
a
y
ma\mathrm{cos}\theta =mg\mathrm{cos}\left(90-\theta \right)
⇒\frac{a}{g}=\mathrm{tan}\theta ⇒\frac{a}{g}=\frac{dy}{dx}
⇒\frac{d}{dx}{\left(kx\right)}^{2}=\frac{a}{g}⇒x=\frac{a}{2gk}
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Sandeep Garg 2018 for Class 11 Commerce Economics Chapter 1 - Organisation Of Data
Sandeep Garg 2018 Solutions for Class 11 Commerce Economics Chapter 1 Organisation Of Data are provided here with simple step-by-step explanations. These solutions for Organisation Of Data are extremely popular among Class 11 Commerce students for Economics Organisation Of Data Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Sandeep Garg 2018 Book of Class 11 Commerce Economics Chapter 1 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Sandeep Garg 2018 Solutions. All Sandeep Garg 2018 Solutions for class Class 11 Commerce Economics are prepared by experts and are 100% accurate.
Following are the figures of marks obtained by 40 students. You ae required to arrange them in ascending and in descending order.
Marks in Ascending Order:
3, 3, 4, 5, 6, 6, 7, 8, 8, 8, 9, 10, 10, 10, 10, 11, 11, 12, 13, 14, 14, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 18, 18 ,19, 19, 21, 22, 22, 22, 25
Marks in Descending Order:
25, 22, 22, 22, 21, 19, 19, 18, 18, 18, 18, 17, 17, 16, 16, 15, 15, 14, 14, 14, 14, 13, 12, 11, 11, 10, 10, 10, 10, 9, 8, 8, 8, 7, 6, 6, 5, 4, 3, 3
Heights (in inches) of 35 students of a class in given below. Classify the following data in a discrete frequency series
In the form of frequency distribution, the given data can be arranged as follows.
(in inches) Tally Marks Frequency
\overline{)\mathrm{ΙΙΙΙ} \mathrm{Ι}}
\overline{)\mathrm{ΙΙΙΙ} \mathrm{Ι}}\mathrm{Ι}
\overline{)\mathrm{ΙΙΙΙ} }
\mathrm{Ι}
\mathrm{Ι}
\mathrm{ΙΙΙ}
\overline{)\mathrm{ΙΙΙΙ} \mathrm{Ι}}
\overline{)\mathrm{ΙΙΙΙ} \mathrm{Ι}}
\sum _{}f=35
The following are the marks of the 30 students in statistics. prepare a frequency distribution taking the class−intervals as 10−20, 20−30 and so on.
\mathrm{ΙΙΙΙ}
\overline{)\mathrm{ΙΙΙΙ} }
\overline{)\mathrm{ΙΙΙΙ} }\mathrm{ΙΙ}
\overline{)\mathrm{ΙΙΙΙ} }\mathrm{ΙΙ}
\mathrm{ΙΙΙΙ}
\mathrm{ΙΙΙ}
\sum _{}f=30
Following are the marks (out of 100) obtained by 50 students in statistics:
make a frequency distribution taking the first class interval as 0-10.
\mathrm{ΙΙ}
\mathrm{ΙΙΙ}
\overline{)\mathrm{ΙΙΙΙ} }\mathrm{ΙΙ}
\overline{)\mathrm{ΙΙΙΙ}} \overline{)\mathrm{ΙΙΙΙ}} \mathrm{ΙΙ}
\overline{)\mathrm{ΙΙΙΙ}} \overline{)\mathrm{ΙΙΙΙ}} \mathrm{ΙΙΙΙ}
\overline{)\mathrm{ΙΙΙΙ}} \mathrm{ΙΙΙΙ}
\mathrm{ΙΙ}
\mathrm{Ι}
\sum _{}f=50
Prepare a frequency table taking class intervals 20−24, 25−29, 30−34 and so on, from the following data:
\mathrm{ΙΙΙ}
\overline{)\mathrm{ΙΙΙΙ}}
\overline{)\mathrm{ΙΙΙΙ}} \mathrm{ΙΙΙ}
\overline{)\mathrm{ΙΙΙΙ}} \overline{)\mathrm{ΙΙΙΙ}} \mathrm{Ι}
\overline{)\mathrm{ΙΙΙΙ}} \mathrm{ΙΙΙ}
\overline{)\mathrm{ΙΙΙΙ}} \mathrm{ΙΙ}
\overline{)\mathrm{ΙΙΙΙ}} \mathrm{ΙΙ}
\mathrm{Ι}
\sum _{}f=50
From the following data, calculate the lower limit of the first class and upper limit of the last class.
Daily Wages Less than 120 120−140 140−160 160−180 Above 180
In the given question, the size of each class is 20. Thus, maintaining the uniformity, we take the first class as 100-120 and the last class as 180-200. Therefore, the lower limit of the first class-interval is 120 − 20 = 100 and the upper limit of the last class is 180 + 20 = 200.
Calculate the missing class-intervals from the following distribution:
Class-Interval Below 20 20−50 50−90 90−140 More than 140
In the given question, the class-intervals are of different width. Thus, the missing class-intervals cannot be determined.
Convert the following 'more than' cumulative frequency distribution into a 'less than' cumulative frequency distribution
Class-Interval (More than) 10 20 30 40 50 60 70 80
Frequency 124 119 107 84 55 31 12 2
The 'more than' cumulative frequency distribution can be presented in the form of a 'simple frequency distribution' as follows.
Class-Interval Frequency
80 - 90 124 − 119 = 5
From the above distribution, we can make the 'less than' cumulative frequency distribution as follows.
Convert the following cumulative frequency series into simple frequency series
The 'less than' cumulative frequency distribution can be presented in the form of a 'simple frequency distribution' as follows.
Prepare a frequency distribution from the following data:
Mid-Points 25 35 45 55 65 75
Frequency 6 10 9 12 8 5
The class interval can be calculated from the mid-points using the following adjustment formula.
\mathrm{Value} \mathrm{of} \mathrm{adjustment} = \frac{35 - 25}{2}= 5
The lower limit of first class = 25 – 5 = 20
Upper limit of first class = 25 + 5 = 30
Thus, the first class interval is (20 − 30). Similarly, we can calculate the remaining class intervals.
Mid Value Class Interval Frequency
{\sum }_{}^{}f=50
The ages of 20 husbands and wives are given below. Form a two-way frequency table showing the relationship between the ages of husbands and wives with the class-intervals 20−25, 25−30, etc.
S.No Age of Husband Age of Wife S. No. Age of Husband Age of Wife
The given data can be presented in the form of a bivariate frequency distribution as follows.
Age of wife (Y) Age of husband (X) Total
20 − 25 25 − 30 30 − 35 35 − 40 40 − 45 45 − 50
The following data give the points scored in a tennis match by two player X and Y at the end of twenty games:
(10,12) (2,11) (7,9) (15,19) (17,21) (12,8) (16,10) (14,14) (22,18) (16,4)
(15,16) (22,20) (19,15) (7,18) (11,11) (12,18) (10,10) (5,13) (11,7) (10,10)
Taking class intervals as: 5−9, 10−14, 15−19..., for both X and Y, construct a Bivariate Frequency Distribution.
Player Y Player X Total
0 − 4 5 − 9 10 − 14 15 − 19 20 − 24
In a survey, it was found that 64 families bought milk in the following quantities in a particular month. Prepare a frequency distribution with classes as 5−9, 10−14 etc.
Quantity Tally Marks Frequency
{\sum }_{}^{}f=64\phantom{\rule{0ex}{0ex}}
You are given below a mid value series, convert it into a continuous series.
Mid-Value 15 25 35 45 55
\mathrm{Value} \mathrm{of} \mathrm{adjustment} = \frac{25 - 15}{2}= 5
{\sum }_{}^{}f=50
For the data given below, prepare a frequency distribution table with classes 100−110, 110-120, etc.
Class Interval Telly Marks Frequency
\sum _{}^{}f=30
Prepare a bivariate frequency distribution for the following data for 20 students:
The data can be presented in the form of a bivariate distribution as follows.
Marks in Statistics Marks in Maths
In a school, no student has scored less than 25 marks or more than 60 marks in an examination. Their cumulative frequencies are as follows:
Less than 60 55 50 45 40 35 30
Total frequency 120 115 90 75 65 45 32
Find the frequencies in the class intervals 25−30, 30−35,....55−60.
This can be presented in the form of a simple frequency distribution as follows.
Marks scored by 50 students are given below:
(a) Arrange the marks in ascending order.
(b) Represent the marks in the form of discrete frequency distribution.
(c) Construct an inclusive frequency distribution with first class as 10−19. Also construct class boundaries.
(d) Construct a frequency distribution with exclusive class-intervals, taking the lowest class as 10−20.
(e) Convert the exclusive series constructed in (d) into 'less than' and 'more than' cumulative frequency distribution.
Marks arranged in ascending order
\sum _{}^{}f=50
Inclusive Frequency Distribution
(Marks) No. of Students
\sum _{}^{}f=50
Exclusive Frequency Distribution
Class Interval No. of Students
\sum _{}^{}f=50
More than frequency distribution
3 − 2 =1
From the following data of the ages of different persons, prepare less than and more than cumulative frequency distribution.
Age (in years) 10−20 20−30 30−40 40−50 50−60 60−70 70−80
No. of Persons 5 12 10 6 4 11 2
\sum _{}f=50
Age Cumulative frequency
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Carbon dioxide - Simple English Wikipedia, the free encyclopedia
Carbon dioxide (CO2) is a chemical compound and is acidic. It is a gas at room temperature. It is made of one carbon and two oxygen atoms. People and animals release carbon dioxide when they breathe out. Also, every time something organic is burnt (or a fire is made), it makes carbon dioxide. Plants use carbon dioxide to make food. This process is called photosynthesis.[1] The properties of carbon dioxide were studied by the Scottish scientist Joseph Black in the 1750s.
Stuctural formula of carbon dioxide. C is carbon and O is oxygen. The double lines represent the double chemical bond between the atoms.
A picture to show simply how the atoms may fill space. The black is carbon and the red is oxygen.
Carbon dioxide is a greenhouse gas.[2] Greenhouse gases trap heat energy. Greenhouse gases change the climate and weather on our planet, Earth. This is called climate change. Greenhouse gases are a cause of global warming, the rise of Earth surface temperature.
Its concentration in Earth's atmosphere since late in the Precambrian was regulated by photosynthetic organisms and geological phenomena (mainly volcanos).
Carbon dioxide is an end product in organisms that obtain energy from breaking down sugars, fats and amino acids with oxygen as part of their metabolism. This is a process known as cellular respiration. This includes all animals, many fungi and some bacteria. In higher animals, the carbon dioxide travels in the blood from the body's tissues to the lungs where it is breathed out. Plants take in carbon dioxide from the atmosphere to use in photosynthesis.
Dry ice, or solid carbon dioxide, is the solid state of CO2 gas below -109.3 °F (-78.5°C). Dry ice does not occur naturally on earth but is man made. It is colorless. People use dry ice to make things cold, and to make drinks fizzy, kill gophers, and freeze warts. The vapor of dry ice causes suffocation and eventually, death. Caution and professional assistance is recommended whenever dry ice is in use.
At usual pressure it will not melt from a solid to a liquid but instead changes directly from a solid to a gas. This is called sublimation. It will change directly from a solid to a gas at any temperature higher than extremely cold temperatures. Dry ice sublimates at normal air temperature. Dry ice exposed to normal air gives off carbon dioxide gas that has no color. Carbon dioxide can be liquified at pressure above 5.1 atmospheres.
Carbon dioxide gas that comes off of dry ice is so cold that when it mixes with air it cools the water vapour in the air to fog, which looks like a thick white smoke. It is often used in the theater to create the appearance of fog or smoke.
Isolation and productionEdit
Chemists can get carbon dioxide from cooling air. They call this air distillation. This method is inefficient because a large amount of air must be refrigerated to extract a small amount of CO2. Chemists can also use several different chemical reactions to separate carbon dioxide. Carbon dioxide is made in the reactions between most acids and most metal carbonates. For example, the reaction between hydrochloric acid and calcium carbonate (limestone or chalk) makes carbon dioxide:
{\displaystyle \mathrm {2\ HCl+CaCO_{3}\longrightarrow CaCl_{2}+H_{2}CO_{3}} }
The carbonic acid (H2CO3) then decomposes to water and CO2. Such reactions cause foaming or bubbling, or both. In industry, such reactions are used many times to neutralize waste acid streams.
Quicklime (CaO), a chemical that has widespread use, can be made heating limestone to about 850 °C. This reaction also makes CO2:
{\displaystyle \mathrm {CaCO_{3}\longrightarrow CaO+CO_{2}} }
Carbon dioxide is also made in the combustion of all carbon-containing fuels, such as methane (natural gas), petroleum distillates (gasoline, diesel, kerosene, propane), coal or wood. In most cases, water is also released. As an example the chemical reaction between methane and oxygen is:
{\displaystyle \mathrm {CH_{4}+2\ O_{2}\longrightarrow CO_{2}+2\ H_{2}O} }
Carbon dioxide is made in steel mills. Iron is reduced from its oxides with coke in a blast furnace, producing pig iron and carbon dioxide:[3]
{\displaystyle \mathrm {Fe_{2}O_{3}+3\ CO\longrightarrow 2\ Fe+3\ CO_{2}} }
{\displaystyle \mathrm {C_{6}H_{12}O_{6}\longrightarrow 2\ CO_{2}+2\ C_{2}H_{5}OH} }
All aerobic organisms produce CO
2 when they oxidize carbohydrates, fatty acids, and proteins in the mitochondria of cells. The large number of reactions involved are exceedingly complex and not described easily. (They include cellular respiration, anaerobic respiration and photosynthesis). Photoautotrophs (i.e. plants, cyanobacteria) use another reaction: Plants absorb CO
2 from the air, and, together with water, react it to form carbohydrates:
{\displaystyle \mathrm {nCO_{2}+nH_{2}O\longrightarrow (CH_{2}O)n+nO_{2}} }
Carbon dioxide is soluble in water, in which it spontaneously interconverts between CO2 and H
3 (carbonic acid). The relative concentrations of CO
3(carbonate) depend on the acidity (pH). In neutral or slightly alkaline water (pH > 6.5), the bicarbonate form predominates (>50%) becoming the most prevalent (>95%) at the pH of seawater, while in very alkaline water (pH > 10.4) the predominant (>50%) form is carbonate. The bicarbonate and carbonate forms are very soluble. So, air-equilibrated ocean water (mildly alkaline with typical pH = 8.2–8.5) contains about 120 mg of bicarbonate per liter.
By capturing natural carbon dioxide springs where it is produced by the action of acidified water on limestone or dolomite.
From combustion of fossil fuels or wood;
From thermal decomposition of limestone, CaCO
3, in the making of lime (Calcium oxide, CaO);
Carbon dioxide can be created with a simple chemical reaction:
{\displaystyle \mathrm {C+O_{2}\longrightarrow CO_{2}} }
↑ Strassburger, Julius H. (1969). Blast Furnace: Theory and Practice. New York: American Institute of Mining, Metallurgical, and Petroleum Engineers. ISBN 9780677104201.
↑ Pierantozzi, Ronald (2001). "Carbon Dioxide". Kirk-Othmer Encyclopedia of Chemical Technology. John Wiley & Sons, Inc. doi:10.1002/0471238961.0301180216090518.a01.pub2.
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Carbon_dioxide&oldid=7865462"
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EUDML | Nonexistence of smooth Levi-flat hypersurfaces in complex projective spaces of dimension . EuDML | Nonexistence of smooth Levi-flat hypersurfaces in complex projective spaces of dimension .
Nonexistence of smooth Levi-flat hypersurfaces in complex projective spaces of dimension
\ge 3
Siu, Yum-Tong. "Nonexistence of smooth Levi-flat hypersurfaces in complex projective spaces of dimension .." Annals of Mathematics. Second Series 151.3 (2000): 1217-1243. <http://eudml.org/doc/121614>.
@article{Siu2000,
keywords = {projective space; Levi-flat; real hypersurface},
title = {Nonexistence of smooth Levi-flat hypersurfaces in complex projective spaces of dimension .},
AU - Siu, Yum-Tong
TI - Nonexistence of smooth Levi-flat hypersurfaces in complex projective spaces of dimension .
KW - projective space; Levi-flat; real hypersurface
Masanori Adachi, Judith Brinkschulte, [unknown]
Giuseppe Della Sala, Liouville-type theorems for foliations with complex leaves
projective space, Levi-flat, real hypersurface
Articles by Siu
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WikiJournal of Medicine/Citation metrics - Wikiversity
WikiJournal of Medicine/Citation metrics
2 Wikipedia readership
4 Estimated impact factor
Altmetrics[edit | edit source]
154 mentions as of 28 March 2018
Wikipedia readership[edit | edit source]
Wikipedia pageviews analysis for last 12 months (note some articles included are less than 12 months old).
Wikipedia articles published in WikiJournal of Medicine
Wikipedia articles with sections published in WikiJournal of Medicine
Estimated impact factor[edit | edit source]
{\displaystyle {\text{IF}}_{y}={{\text{Citations}}_{y-1}+{\text{Citations}}_{y-2} \over {\text{Publications}}_{y-1}+{\text{Publications}}_{y-2}}}
Note: Citation tracking websites will return different results due to different range of publication contents they have access to and interpretation on what is considered as a "citation". Therefore, impact factors may be drastically different between the websites as seen below (Google Scholar vs. Dimensions).
Twelve citable articles were published in 2014 (5 + 7 = 12 publications)
Sixteen citable articles were published in 2014 and 2015 (5 + 7 + 4 = 16 publications)
Citations in 2014 and 2015 (Google Scholar)
Citations in 2014 and 2015 (Dimensions)
Reference ranges for estradiol, progesterone, luteinizing hormone and follicle-stimulating hormone during the menstrual cycle 5 2
An epidemiology-based and a likelihood ratio-based method of differential diagnosis 0 0
Establishment and clinical use of reference ranges 0 0
Allogeneic component to overcome rejection in interspecific pregnancy 0 0
Diagram of the pathways of human steroidogenesis 8 3
Caesarean section photography 0 0
Tubal pregnancy with embryo 0 0
Medical gallery of Mikael Häggström 2014 9 2
Medical gallery of Blausen Medical 2014 1 2
Medical gallery of David Richfield 2014 2 1
Ultrasonography of a cervical pregnancy 0 0
Insights into abdominal pregnancy 0 0
Images of Aerococcus urinae 0 0
Table of pediatric medical conditions and findings named after foods 0 0
The Year of the Elephant 0 0
Estimating the lost benefits of not implementing a visual inspection with acetic acid screen and treat strategy for cervical cancer prevention in South Africa 0 0
2 year estimated impact factor = 25/16 = 1.56 = 10/16 = 0.625
Five citable articles were published in 2015 and 2016 (4 + 1 = 5 publications)
Citations in 2015 and 2016
Images of Aerococcus urinae 0 Google Scholar
Table of pediatric medical conditions and findings named after foods 1 Google Scholar
The Year of the Elephant 0 Google Scholar
Estimating the lost benefits of not implementing a visual inspection with acetic acid screen and treat strategy for cervical cancer prevention in South Africa 0 Google Scholar
The Cerebellum 0 Google Scholar
Total 1 2 year estimated impact factor = 1/5 = 0.2
Eight citable articles were published in 2016 and 2017 (1 + 7 = 8 publications)
Eukaryotic and prokaryotic gene structure 1 Google Scholar
The Hippocampus 1 Google Scholar
Plasmodium falciparum erythrocyte membrane protein 1 1 Google Scholar
Vitamin D as an adjunct for acute community-acquired pneumonia among infants and children: systematic review and meta-analysis 0 Google Scholar
Acute gastrointestinal bleeding from a chronic cause: a teaching case report 0 Google Scholar
Rotavirus 0 Google Scholar
Cell disassembly during apoptosis 0 Google Scholar
Anthracyclines 0 Google Scholar
Total 11 2 year estimated impact factor = 11/8 = 1.375
Nine citable articles were published in 2018 and 2019 (1 + 8 = 9 publications)
Western African Ebola virus epidemic 0 Google Scholar
An overview of Lassa fever 0 Google Scholar
Hepatitis E 0 Google Scholar
Orientia tsutsugamushi, the agent of scrub typhus 0 Google Scholar
Dyslexia 0 Google Scholar
Mealtime difficulty in older people with dementia 1 Google Scholar
Readability of English Wikipedia's health information over time 2 Google Scholar
Dioxins and dioxin-like compounds: toxicity in humans and animals, sources, and behaviour in the environment 10 Google Scholar
Note: The impact factor is a metric whose use is debated.[1]*,**
* Citations during the most recent two year period (2015 and 2016) might be higher because Google Scholar might not have indexed yet articles citing the WikiJournal of Medicine during that period. Because the denominator (number of citable articles is known), and because the numerator (number of citations received) might increase, the estimated impact factor shown might be an underestimate.
** No self-citations were detected.
↑ Rochmyaningsih, Dyna (2017). "The developing world needs more than numbers". Nature 542 (7639): 7–7. doi:10.1038/542007a. ISSN 0028-0836.
Retrieved from "https://en.wikiversity.org/w/index.php?title=WikiJournal_of_Medicine/Citation_metrics&oldid=2374337"
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A bond floor refers to the minimum value that a specific bond, usually a convertible bond, should trade for. The level of the floor is derived from the discounted present value of a bond's coupons, plus its conversion value.
A bond floor may also be used in constant proportion portfolio insurance (CPPI) calculations. When using CPPI calculations, an investor sets a floor on the dollar value of their portfolio and then structures asset allocation around that decision.
Bond floor refers to the minimum value a bond (usually a convertible bond) should trade for and is calculated using the discounted value of its coupons plus redemption value.
Bond floor can also refer to the aspect of constant proportion portfolio insurance (CPPI) that ensures that the value of a given portfolio does not fall below a predefined level.
The difference between the convertible bond price and its bond floor is the risk premium, which is the value that the market places on the option to convert a bond to shares of the underlying stock.
Understanding the Bond Floor
The bond floor is the lowest value that convertible bonds can fall to, given the present value (PV) of the remaining future cash flows and principal repayment. The term can also refer to the aspect of constant proportion portfolio insurance (CPPI) that ensures that the value of a given portfolio does not fall below a predefined level.
Convertible bonds give investors the potential to profit from any appreciation in the price of the issuing company's stock (if they are converted). This added benefit to investors makes a convertible bond more valuable than a straight bond. In effect, a convertible bond is a straight bond plus an embedded call option. The market price of a convertible bond is made up of the straight bond value and the conversion value. (The conversion value is the market value of the underlying equity into which a convertible security may be exchanged.)
When stock prices are high, the price of the convertible is determined by the conversion value. However, when stock prices are low, the convertible bond will trade like a straight bond—given that the straight bond value is the minimum level a convertible bond can trade at and the conversion option is nearly irrelevant when stock prices are low. The straight bond value is, thus, the floor of a convertible bond.
Investors are protected from a downward move in the stock price because the value of the convertible bond will not fall below the value of the traditional or straight bond component. In other words, the bond floor is the value at which the convertible option becomes worthless because the underlying stock price has fallen substantially below the conversion value.
The difference between the convertible bond price and its bond floor is the risk premium. The risk premium can be viewed as the value that the market places on the option to convert a bond to shares of the underlying stock.
Calculating the Bond Floor for a Convertible Bond
\begin{aligned} &\text{Bond Floor} = \sum_{t = 1} ^ {n} \frac{ \text{C} }{ ( 1 + r ) ^ t} + \frac{ \text{P} }{ (1 + r) ^ n }\\ &\textbf{where:} \\ &\text{C} = \text{coupon rate of convertible bond} \\ &\text{P} = \text{par value of convertible bond} \\ &r = \text{rate on straight bond} \\ &n = \text{number of years until maturity} \\ \end{aligned}
Bond Floor=t=1∑n(1+r)tC+(1+r)nPwhere:C=coupon rate of convertible bondP=par value of convertible bondr=rate on straight bondn=number of years until maturity
\begin{aligned} &\text{Bond Floor} = \text{PV}_{\text{coupon} } + \text{PV}_\text{par value} \\ &\textbf{where:} \\ &\text{PV} = \text{present value} \\ \end{aligned}
Bond Floor=PVcoupon+PVpar valuewhere:PV=present value
Example of a Bond Floor
For example, assume a convertible bond with a $1,000 par value has a coupon rate of 3.5% (to be paid annually). The bond matures in 10 years. Consider there is also a comparable straight bond, with the same face value, credit rating, interest payment schedule, and maturity date of the convertible bond, but with a coupon rate of 5%.
To find the bond floor, one must calculate the present value (PV) of the coupon and principal payments discounted at the straight bond interest rate.
\begin{aligned} \text{PV}_\text{factor} &= 1 - \frac{ 1 }{ (1 + r) ^ n } \\ &= 1 - \frac{ 1 }{ 1.05^ {10} } \\ &= 0.3861 \\ \end{aligned}
PVfactor=1−(1+r)n1=1−1.05101=0.3861
\begin{aligned} \text{PV}_\text{coupon} &= \frac {.035 \times \$1,000 }{ 0.05 } \times \text{PV}_\text{factor} \\ &= \$700 \times 0.3861 \\ &= \$270.27 \\ \end{aligned}
PVcoupon=0.05.035×$1,000×PVfactor=$700×0.3861=$270.27
\begin{aligned} \text{PV}_\text{par value} &= \frac {\$1,000 }{ 1.05 ^ {10} } \\ &= \$613.91 \\ \end{aligned}
PVpar value=1.0510$1,000=$613.91
\begin{aligned} \text{Bond Floor} &= \text{PV}_{\text{coupon} } + \text{PV}_\text{par value} \\ &= \$613.91 + \$270.27 \\ &= \$884.18 \\ \end{aligned}
Bond Floor=PVcoupon+PVpar value=$613.91+$270.27=$884.18
So, even if the company's stock price falls, the convertible bond should trade for a minimum of $884.18. Like the value of a regular, non-convertible bond, a convertible bond's floor value fluctuates with market interest rates and various other factors.
Bond Floors and Constant Proportion Portfolio Insurance (CPPI)
Constant Proportion Portfolio Insurance (CPPI) is a mixed portfolio allocation of risky and non-risky assets, which varies depending on market conditions. An embedded bond feature ensures that the portfolio does not fall below a certain level, thus acting as a bond floor. The bond floor is the value below which the value of the CPPI portfolio should never fall (in order to ensure the payment of all future due interest and principal payments).
By carrying insurance on the portfolio (through this embedded bond feature), the risk of experiencing more than a certain amount of loss at any given time is kept to a minimum. At the same time, the floor does not inhibit the growth potential of the portfolio, effectively providing the investor with a lot to gain—and only a little to lose.
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Option price by Merton76 model using finite differences - MATLAB optByMertonFD - MathWorks France
\mathrm{max}\left(St-K,0\right)
\mathrm{max}\left(K-St,0\right)
\begin{array}{l}d{S}_{t}=\left(r-q-{\lambda }_{p}{\mu }_{j}\right){S}_{t}dt+\sigma {S}_{t}d{W}_{t}+J{S}_{t}d{P}_{t}\\ \text{prob(}d{P}_{t}=1\right)={\lambda }_{p}dt\end{array}
\mathrm{ln}\left(1+{\mu }_{J}\right)-\frac{{\delta }^{2}}{2}
\frac{1}{\left(1+J\right)\delta \sqrt{2\pi }}\mathrm{exp}\left\{{\frac{-\left[\mathrm{ln}\left(1+J\right)-\left(\mathrm{ln}\left(1+{\mu }_{J}\right)-\frac{{\delta }^{2}}{2}\right]}{2{\delta }^{2}}}^{2}\right\}
|
m
n
Matrix (or 2-dimensional Array), then it is assumed to contain
m
\mathrm{with}\left(\mathrm{SignalProcessing}\right):
\mathrm{f1}≔12.0:
\mathrm{f2}≔24.0:
\mathrm{signal}≔\mathrm{Vector}\left({2}^{10},i↦\mathrm{sin}\left(\frac{\mathrm{f1}\cdot \mathrm{\pi }\cdot i}{50}\right)+1.5\cdot \mathrm{sin}\left(\frac{\mathrm{f2}\cdot \mathrm{\pi }\cdot i}{50}\right),\mathrm{datatype}=\mathrm{float}[8]\right)
{\textcolor[rgb]{0,0,1}{\mathrm{_rtable}}}_{\textcolor[rgb]{0,0,1}{36893628696909549564}}
\mathrm{Periodogram}\left(\mathrm{signal},\mathrm{samplerate}=100\right)
\mathrm{audiofile}≔\mathrm{cat}\left(\mathrm{kernelopts}\left(\mathrm{datadir}\right),"/audio/maplesim.wav"\right):
\mathrm{Periodogram}\left(\mathrm{audiofile},\mathrm{frequencyscale}="kHz"\right)
\mathrm{audiofile2}≔\mathrm{cat}\left(\mathrm{kernelopts}\left(\mathrm{datadir}\right),"/audio/stereo.wav"\right):
\mathrm{Periodogram}\left(\mathrm{audiofile2},\mathrm{compactplot}\right)
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Acoustic location - Wikipedia
(Redirected from Japanese War Tuba)
Use of reflected sound waves to locate objects
This article is about sound localization via mechanical or electrical means. For the biological process, see sound localization.
Acoustic location is the use of sound to determine the distance and direction of its source or reflector. Location can be done actively or passively, and can take place in gases (such as the atmosphere), liquids (such as water), and in solids (such as in the earth).
Active acoustic location involves the creation of sound in order to produce an echo, which is then analyzed to determine the location of the object in question.
Passive acoustic location involves the detection of sound or vibration created by the object being detected, which is then analyzed to determine the location of the object in question.
Both of these techniques, when used in water, are known as sonar; passive sonar and active sonar are both widely used.
Acoustic mirrors and dishes, when using microphones, are a means of passive acoustic localization, but when using speakers are a means of active localization. Typically, more than one device is used, and the location is then triangulated between the several devices.
As a military air defense tool, passive acoustic location was used from mid-World War I[1] to the early years of World War II to detect enemy aircraft by picking up the noise of their engines. It was rendered obsolete before and during World War II by the introduction of radar, which was far more effective (but interceptable). Acoustic techniques had the advantage that they could 'see' around corners and over hills, due to sound diffraction.
The civilian uses include locating wildlife[2] and locating the shooting position of a firearm.[3]
2.1 Particle velocity or intensity vector
2.2 Time difference of arrival
4 Active / passive locators
4.2 Biological echo location
4.3 Time-of-arrival localization
Acoustic source localization[4] is the task of locating a sound source given measurements of the sound field. The sound field can be described using physical quantities like sound pressure and particle velocity. By measuring these properties it is (indirectly) possible to obtain a source direction.
Traditionally sound pressure is measured using microphones. Microphones have a polar pattern describing their sensitivity as a function of the direction of the incident sound. Many microphones have an omnidirectional polar pattern which means their sensitivity is independent of the direction of the incident sound. Microphones with other polar patterns exist that are more sensitive in a certain direction. This however is still no solution for the sound localization problem as one tries to determine either an exact direction, or a point of origin. Besides considering microphones that measure sound pressure, it is also possible to use a particle velocity probe to measure the acoustic particle velocity directly. The particle velocity is another quantity related to acoustic waves however, unlike sound pressure, particle velocity is a vector. By measuring particle velocity one obtains a source direction directly. Other more complicated methods using multiple sensors are also possible. Many of these methods use the time difference of arrival (TDOA) technique.
Some have termed acoustic source localization an "inverse problem" in that the measured sound field is translated to the position of the sound source.
Different methods for obtaining either source direction or source location are possible.
Particle velocity or intensity vector[edit]
The simplest but still a relatively new method is to measure the acoustic particle velocity using a particle velocity probe. The particle velocity is a vector and thus also contains directional information.
Time difference of arrival[edit]
The traditional method to obtain the source direction is using the time difference of arrival (TDOA) method. This method can be used with pressure microphones as well as with particle velocity probes.
With a sensor array (for instance a microphone array) consisting of at least two probes it is possible to obtain the source direction using the cross-correlation function between each probes' signal. The cross-correlation function between two microphones is defined as
{\displaystyle R_{x_{1},x_{2}}(\tau )=\sum _{n=-\infty }^{\infty }x_{1}(n)x_{2}(n+\tau )}
which defines the level of correlation between the outputs of two sensors
{\displaystyle x_{1}}
{\displaystyle x_{2}}
. In general, a higher level of correlation means that the argument
{\displaystyle \tau }
is relatively close to the actual time-difference-of-arrival. For two sensors next to each other the TDOA is given by
{\displaystyle \tau _{\text{true}}={\frac {d_{\text{spacing}}}{c}}}
{\displaystyle c}
is the speed of sound in the medium surrounding the sensors and the source.
A well-known example of TDOA is the interaural time difference. The interaural time difference is the difference in arrival time of a sound between two ears. The interaural time difference is given by
{\displaystyle \Delta t={\frac {x\cos \theta }{c}}}
{\displaystyle \Delta t}
is the time difference in seconds,
{\displaystyle x}
is the distance between the two sensors (ears) in meters,
{\displaystyle \theta }
is the angle between the baseline of the sensors (ears) and the incident sound, in degrees.
In trigonometry and geometry, triangulation is the process of determining the location of a point by measuring angles to it from known points at either end of a fixed baseline, rather than measuring distances to the point directly (trilateration). The point can then be fixed as the third point of a triangle with one known side and two known angles.
For acoustic localization this means that if the source direction is measured at two or more locations in space, it is possible to triangulate its location.
Steered Response Power (SRP) methods are a class of indirect acoustic source localization methods. Instead of estimating a set of time-differences of arrival (TDOAs) between pairs of microphones and combining the acquired estimates to find the source location, indirect methods search for a candidate source location over a grid of spatial points. In this context, methods such as the Steered-Response Power Phase Transform (SRP-PHAT)[5] are usually interpreted as finding the candidate location that maximizes the output of a delay-and-sum beamformer. The method has been shown to be very robust to noise and reverberation, motivating the development of modified approaches aimed at increasing its performance in real-time acoustic processing applications.[6]
See also: Artillery sound ranging
T3 sound locator 1927
Pre-World War II photograph of Japanese Emperor Shōwa (Hirohito) inspecting military acoustic locators mounted on 4-wheel carriages
Military uses have included locating submarines[7] and aircraft.[8] The first use of this type of equipment was claimed by Commander Alfred Rawlinson of the Royal Naval Volunteer Reserve, who in the autumn of 1916 was commanding a mobile anti-aircraft battery on the east coast of England. He needed a means of locating Zeppelins during cloudy conditions and improvised an apparatus from a pair of gramophone horns mounted on a rotating pole. Several of these equipments were able to give a fairly accurate fix on the approaching airships, allowing the guns to be directed at them despite being out of sight.[9] Although no hits were obtained by this method, Rawlinson claimed to have forced a Zeppelin to jettison its bombs on one occasion.[10]
The air-defense instruments usually consisted of large horns or microphones connected to the operators' ears using tubing, much like a very large stethoscope.[11][12]
Sound location equipment in Germany, 1939. It consists of four acoustic horns, a horizontal pair and a vertical pair, connected by rubber tubes to stethoscope type earphones worn by the two technicians left and right. The stereo earphones enabled one technician to determine the direction and the other the elevation of the aircraft.
Most of the work on anti-aircraft sound ranging was done by the British. They developed an extensive network of sound mirrors that were used from World War I through World War II.[13][14] Sound mirrors normally work by using moveable microphones to find the angle that maximizes the amplitude of sound received, which is also the bearing angle to the target. Two sound mirrors at different positions will generate two different bearings, which allows the use of triangulation to determine a sound source's position.
As World War II neared, radar began to become a credible alternative to the sound location of aircraft. For typical aircraft speeds of that time, sound location only gave a few minutes of warning.[8] The acoustic location stations were left in operation as a backup to radar, as exemplified during the Battle of Britain.[15] Today, the abandoned sites are still in existence and are readily accessible.[13][dead link]
After World War II, sound ranging played no further role in anti-aircraft operations.[citation needed]
Active / passive locators[edit]
Active locators have some sort of signal generation device, in addition to a listening device. The two devices do not have to be located together.
SONAR or sonar (sound navigation and ranging) is a technique that uses sound propagation under water (or occasionally in air) to navigate, communicate or to detect other vessels. There are two kinds of sonar – active and passive. A single active sonar can localize in range and bearing as well as measuring radial speed. However, a single passive sonar can only localize in bearing directly, though Target Motion Analysis can be used to localize in range, given time. Multiple passive sonars can be used for range localization by triangulation or correlation, directly.
Biological echo location[edit]
Dolphins, whales and bats use echolocation to detect prey and avoid obstacles.
Time-of-arrival localization[edit]
Having speakers/ultrasonic transmitters emitting sound at known positions and time, the position of a target equipped with a microphone/ultrasonic receiver can be estimated based on the time of arrival of the sound. The accuracy is usually poor under non-line-of-sight conditions, where there are blockages in between the transmitters and the receivers. [16]
Seismic surveys[edit]
A three-dimensional echo-sounding representation of a canyon under the Red Sea by survey vessel HMS Enterprise
Seismic surveys involve the generation of sound waves to measure underground structures. Source waves are generally created by percussion mechanisms located near the ground or water surface, typically dropped weights, vibroseis trucks, or explosives. Data are collected with geophones, then stored and processed by computer. Current technology allows the generation of 3D images of underground rock structures using such equipment.
See also: Reflection seismology
Because the cost of the associated sensors and electronics is dropping, the use of sound ranging technology is becoming accessible for other uses, such as for locating wildlife.[17]
3D sound localization
Acoustic wayfinding, the practice of using auditory cues and sound markers to navigate indoor and outdoor spaces
Animal echolocation, animals emitting sound and listening to the echo in order to locate objects or navigate
Echo sounding, listening to the echo of sound pulses to measure the distance to the bottom of the sea, a special case of sonar
Human echolocation, the use of echolocation by blind people
Medical ultrasonography, the use of ultrasound echoes to look inside the body
^ How Far Off Is That German Gun? How 63 German guns were located by sound waves alone in a single day, Popular Science monthly, December 1918, page 39, Scanned by Google Books: https://books.google.com/books?id=EikDAAAAMBAJ&pg=PA39[permanent dead link]
^ "Selected Projects". Greenridge Sciences Inc. Retrieved 2006-05-16.
^ Lorraine Green Mazerolle; et al. (December 1999). "Random Gunfire Problems and Gunshot Detection Systems" (PDF). National Institute of Justice Research Brief.
^ "Acoustic Source Localization based on independent component analysis". LMS.
^ DiBiase, J. H. (2000). A High Accuracy, Low-Latency Technique for Talker Localization in Reverberant Environments using Microphone Arrays (PDF) (Ph.D.). Brown Univ.
^ Cobos, M.; Marti, A.; Lopez, J. J. (2011). "A Modified SRP-PHAT Functional for Robust Real-Time Sound Source Localization With Scalable Spatial Sampling". IEEE Signal Processing Letters. 18 (1): 71–74. Bibcode:2011ISPL...18...71C. doi:10.1109/LSP.2010.2091502. hdl:10251/55953. S2CID 18207534.
^ Kristian Johanssan; et al. "Submarine tracking using multi-sensor fusion and reactive planning for the positioning of passive sonobuoys" (PDF). Archived from the original (PDF) on 2009-03-27. Retrieved 2006-05-16.
^ a b W.Richmond (2003). "Before RADAR – Acoustic Detection of Aircraft". Archived from the original on 2007-09-28. Retrieved 2013-01-06.
^ Rawlinson, Alfred (1923), Rawlinson, The Defence of London, Andrew Melrose, London & New York, pp. 110–114 Archived May 5, 2016, at the Wayback Machine
^ Rawlinson, pp. 118–119
^ Douglas Self. "Acoustic Location and Sound Mirrors". Archived from the original on 2011-01-12. Retrieved 2006-06-01.
^ Jim Mulligan. "Photo of Sound Locator". Retrieved 2006-05-15.
^ a b Phil Hide (January 2002). "Sound Mirrors on the South Coast". Archived from the original on 2009-05-02. Retrieved 2006-05-13.
^ Andrew Grantham (November 8, 2005). "Early warning sound mirrors".
^ Lee Brimmicombe Woods (7 December 2005). "The Burning Blue: The Battle of Britain 1940" (PDF). GMT Games LLC.
^ Chan, Y.T; Tsui, W. Y.; So, H. C.; Ching, P. C. (2006). "Time-of-arrival based localization under NLOS Conditions". IEEE Trans. Vehicular Technology. 55 (1): 17–24. doi:10.1109/TVT.2005.861207. ISSN 0018-9545. S2CID 6697621.
^ John L. Spiesberger (June 2001). "Hyperbolic location errors due to insufficient numbers of receivers". The Journal of the Acoustical Society of America. 109 (6): 3076–3079. Bibcode:2001ASAJ..109.3076S. doi:10.1121/1.1373442. PMID 11425152.
"Huge Ear Locates Planes and Tells Their Speed" Popular Mechanics, December 1930 article on French aircraft sound detector with photo.
Many references can be found in Beamforming References
Retrieved from "https://en.wikipedia.org/w/index.php?title=Acoustic_location&oldid=1004021782#Military_use"
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(i) This question paper comprises four sections – A, B, C and D. This question paper carries 40 questions. All questions are compulsory:
(ii) Section A : Q. No. 1 to 20 comprises of 20 questions of one mark each.
(iii) Section B : Q. No. 21 to 26 comprises of 6 questions of two marks each.
(v) Section D : Q. No. 35 to 40 comprises of 6 questions of four marks each.
(vi) There is no overall choice in the question paper. However, an internal choice has been provided in 2 questions of one mark each, 2 questions of two marks each, 3 questions of three marks each and 3 questions of four marks each. You have to attempt only one of the choices in such questions.
The median and mode respectively of a frequency distribution are 26 and 29. Then its mean is
(d) 25.8 VIEW SOLUTION
In the given figure on a circle of radius 7 cm, tangent PT is drawn from a point P such that PT = 24 cm. If O is the centre of the circle, then the length of PR is
225 can be expressed as
(b) 52 × 3
(d) 53 × 3 VIEW SOLUTION
The probability that a number selected at random from the numbers 1, 2, 3, ...., 15 is a multiple of 4 is
\frac{4}{15}
\frac{2}{15}
\frac{1}{15}
\frac{1}{5}
If one zero of a quadratic polynomial (kx2 + 3x + k) is 2, then the value of k is
\frac{5}{6}
-\frac{5}{6}
\frac{6}{5}
-\frac{6}{5}
2.\overline{)35}
(d) a natural number VIEW SOLUTION
The graph of a polynomial is shown in figure, then the number of its zeroes is
Distance of point P(3, 4) from x-axis is
(d) 1 unit VIEW SOLUTION
If the distance between the points A(4, p) and B(1, 0) is 5 units, then the value(s) of p is (are)
If the point C(k, 4) divides the line segment joining two points A(2, 6) and B(5, 1) in ratio 2 : 3, the value of k is ____________.
If points A(–3, 12), B(7, 6) and C(x, 9) are collinear, then the value of x is ___________. VIEW SOLUTION
If the equations kx – 2y = 3 and 3x + y = 5 represent two intersecting lines at unique point, then the value of k is ___________.
If quadratic equation 3x2 – 4x + k = 0 has equal roots, then the value of k is _____________. VIEW SOLUTION
The value of (sin 20° cos 70° + sin 70° cos 20°) is _____________. VIEW SOLUTION
If tan (A + B) =
\sqrt{3}
and tan (A – B) =
\frac{1}{\sqrt{3}}
, A > B, then the value of A is ___________. VIEW SOLUTION
The perimeters of two similar triangles are 25 cm and 15 cm respectively. If one side of the first triangle is 9 cm, then the corresponding side of second triangle is _____________. VIEW SOLUTION
If 5tanθ = 3, then what is the value of
\left(\frac{5\mathrm{sin} \mathrm{\theta }-3\mathrm{cos} \mathrm{\theta }}{4\mathrm{sin} \mathrm{\theta }+3\mathrm{cos} \mathrm{\theta }}\right)
? VIEW SOLUTION
The areas of two circles are in the ratio 9 : 4, then what is the ratio of their circumferences? VIEW SOLUTION
If a pair of dice is thrown once, then what is the probability of getting a sum of 8? VIEW SOLUTION
In the given figure in ΔABC, DE || BC such that AD = 2.4 cm, AB = 3.2 cm and AC = 8 cm, then what is the length of AE?
The nth term of an AP is (7 – 4n), then what is its common difference? VIEW SOLUTION
A bag contains 5 red balls and some blue balls. If the probability of drawing a blue ball at random from the bag is three times that of a red ball, find the number of blue balls in the bag. VIEW SOLUTION
\sqrt{\frac{1-\mathrm{sin\theta }}{1+\mathrm{sin\theta }}}=\mathrm{sec\theta }-\mathrm{tan\theta }
\frac{{\mathrm{tan}}^{2} \mathrm{\theta }}{1+{\mathrm{tan}}^{2} \mathrm{\theta }}+\frac{{\mathrm{cot}}^{2} \mathrm{\theta }}{1+{\mathrm{cot}}^{2} \mathrm{\theta }}=1
Two different dice are thrown together, find the probability that the sum a of the numbers appeared is less than 5.
Find the probability that 5 Sundays occur in the month of November of a randomly selected year. VIEW SOLUTION
In the given figure, a circle touches all the four sides of a quadrilateral ABCD. If AB = 6 cm, BC = 9 cm and CD = 8 cm, then find the length of AD.
The perimeter of a sector of a circle with radius 6.5 cm is 31 cm, then find the area of the sector. VIEW SOLUTION
Divide the polynomial (4x2 + 4x + 5) by (2x + 1) and write the quotient and the remainder. VIEW SOLUTION
If α and β are the zeros of the polynomial f(x) = x2 – 4x – 5 then find the value of α2 + β2. VIEW SOLUTION
Draw a circle of radius 4 cm. From a point 7 cm away from the centre of circle. Construct a pair of tangents to the circle.
Draw a line segment of 6 cm and divide it in the ratio 3 : 2. VIEW SOLUTION
A solid metallic cuboid of dimension 24 cm × 11 cm × 7 cm is melted and recast into solid cones of base radius 3.5 cm and height 6 cm. Find the number of cones so formed. VIEW SOLUTION
Prove that (1 + tan A – sec A) × (1 + tan A + sec A) = 2 tan A
\frac{\mathrm{cosec} \mathrm{\theta }}{\mathrm{cosec} \mathrm{\theta }-1}+\frac{\mathrm{cosec} \mathrm{\theta }}{\mathrm{cosec} \mathrm{\theta }+1}=2{\mathrm{sec}}^{2}\mathrm{\theta }
\sqrt{3}
is an irrational number, show that
\left(5+2\sqrt{3}\right)
An army contingent of 612 members is to march behind an army band of 48 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march? VIEW SOLUTION
Prove that, in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. VIEW SOLUTION
Read the following passage carefully and then answer the questions given at the end.
To conduct Sports Day activities, in your rectangular shaped school ground ABCD, lines have been drawn with chalk powder at a distance of 1 m each. 100 flower pots have been placed at a distance of 1 m from each other along AD, as shown in Figure. Niharika runs
\frac{1}{4}\mathrm{th}
the distance AD on the 2nd line and posts a green flag. Preet runs
\frac{1}{5}\mathrm{th}
the distance AD on the eighth line and posts a red flag.
(i) What is the distance between the two flags?
(ii) If Rashmi has to post a blue flag exactly half way between the line segment joining the two flags, where should she post the blue flag? VIEW SOLUTION
Solve graphically: 2x + 3y = 2, x – 2y = 8 VIEW SOLUTION
A two digit number is such that the product of its digits is 14. If 45 is added to the number; the digits interchange their places. Find the number. VIEW SOLUTION
If 4 times the 4th term of an AP is equal to 18 times the 18th term, then find the 22nd term.
How many terms of the AP : 24, 21, 18, ... must be taken so that their sum is 78? VIEW SOLUTION
The angle of elevation of the top of a building from the foot of a tower is 30°. The angle of elevation of the top of the tower from the foot of the building is 60°. If the tower is 60 m high, find the height of the building. VIEW SOLUTION
In the given figure, DEFG is a square in a triangle ABC right angled at A.
(i) ΔAGF ~ ΔDBG
(ii) ΔAGF ~ ΔEFC
In an obtuse ΔABC (∠B is obtuse), AD is perpendicular to CB produced. Then prove that AC2 = AB2 + BC2 + 2BC × BD. VIEW SOLUTION
An open metal bucket is in the shape of a frustum of cone of height 21 cm with radii of its lower and upper ends are 10 cm and 20 cm respectively. Find the cost of milk which can completely fill the bucket at the rate of ₹ 40 per litre.
A solid is in the shape of a cone surmounted on a hemisphere. The radius of each of them being 3.5 cm and the total height of the solid is 9.5 cm. Find the volume of the solid. VIEW SOLUTION
Classes 0 – 20 20 – 40 40 – 60 60 – 80 80 – 100 100 – 120
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Algorithms Dynamic Programming (DP) Greedy Algorithms
Approach 3: Dynamic programming (Top Down)
Optimal Greedy Approach
Prerequisite: Dynamic Programming, Greedy Algorithm, nth Fibonacci
This problem is similar to Leetcode problem 45. Jump Game II.
We are given an array of non-negative integers, initially we are positioned at the first index of the array.
Each element represents the maximum jump length from that position to another.
The goal is to reach the last index in the minimum number of jumps.
Assume that you can always reach the last index.
Ways to reach last index
1. 2 -> 1 -> 1 -> 4, 3 steps
2. 2 -> 3 -> 4, 2 steps
Jump 1 step to 3
Jump 3 steps to 4, final index
1. 1 -> 1 -> 2 -> 2 -> 9, 4 steps
2. 1 -> 1 -> 2 -> 1 -> 2 -> 9, 5 steps
4. 1 -> 1 -> 2 -> 1 -> 2 -> 1 -> 9, 6 steps
Jump 1 step(0 - 1)
Jump 2 steps(2 - 4)
Jump 2 steps(4 - 6) final index
In this approach we will use recursion to solve the problem.
The base case will be triggered when the algorithm reaches the last index, then the algorithm terminates.
The algorithm will recursively call for all elements reachable from the first element. That is, it will explore all branches in the recursion tree and return the minimum number of jumps to reach the last index.
As you can see there are repeated re-computations of same values. We shall see how to avoid them in the next approach.
Initialize jumps to a max value.
Traverse through the list, from start index recursively call for elements reachable from start index until minimum is found;
int minJumps(vector<int> &nums, int l){
if(l >= nums.size() - 1)
int jumps = INT_MAX;
for(int i = l+1; i <= l + nums[l]; i++)
jumps = min(jumps, 1 + minJumps(nums, i));
return minJumps(nums, 0);
O\left({2}^{n}\right)
, that is there are n possible ways to move from an element for each element in the list.
Space complexity is O(1) without including the stack space used for the recursive calls.
In this approach the algorithm will optimize the naive recursive solution to a quadratic time complexity which is a little bit better.
Before we get to it, a little about dynamic programming, we use dynamic programming to optimize recursive problems whereby we call a recursive function for repeated inputs which results in unnecessary re-computations. Generally a dynamic programming problem has the following characteristics
Our problem has overlapping subproblems in that finding the global solution involves solving the same subproblem multiple times. Can you spot the overlapping subproblems from the previous recurrence tree?
The optimal substructure property comes in whereby the overall optimal solution is constructed using optimal solutions of the subproblems. The minimum number of jumps is a combination of optimal steps made to reach the last index.
Dynamic programming approach reduces time complexity for problems with an exponential time complexity to polynomial time. We will see how to optimize the naive approach to a quadratic time complexity, it is not better than the greedy approach which we shall see, but it is a good example to show how dynamic programming works.
Note: There are cases where the greedy approach does not apply and the initial naive solution gives an exponential time complexity. An example is finding the nth fibonacci number, the recursive approach yields an exponential time complexity, when dynamic programming approach is implemented the time complexity reduces to linear time.
Back to the jump game problem, to solve it dynamically, we will store results of subproblems in an array to eliminate repeated work, this is called tabulation, results are stored in a table(array).
We solve a subproblem and using the result from the subproblem we solve another subproblem ans so on building up the final solution step by step.
You can read more on dp approaches at the link provided at the end of this post.
Given the array [2, 3, 1, 1, 4]
We will create an auxiliary array store[]
We fill the first index with 0, store[0, ...], this denotes that we will need 0 steps to reach the first index.
After the first iteration store will have [0, 1, ....] denoting 1 step to reach the second index.
After the second iteration store will have [0, 1, 2, ...] denoting 2 steps to reach the end. We use a conditional statement that will push the index i to the maximum position then break out of the loop.
We return the last element in the store array which will be the minimum number of steps to reach the last index.
Create an auxiliary array to store minimum jumps needed to reach nums[i] from nums[0].
Implement a nested loop, the inner loop traverses the segments and stores the minimum jumps to an index in the auxiliary array. The outer loop traverses the whole list.
At each step in the loop the algorithm finds the minimum number of jumps to reach nums[i] from the start and adds this value to the auxiliary array.
Return the last jump which will be the minimum number of jumps.
vector<int> store(n);
store[i] = INT_MAX;
if(i <= nums[j] + j && store[j] != INT_MAX){
store[i] = min(store[i], store[j] + 1);
return store[n-1];
O\left({n}^{2 }\right)
, that is we perform n jumps for each element in the list of size n.
Space complexity is O(n), additional space is needed to store the auxiliary array.
In this approach we recursively find solutions to smaller subproblems which we shall use to solve the bigger problem.
Whenever we solve a subproblem recursively, we cache its result in an auxiliary array and when it is called again, the recursive call will takes constant time to retrieve the result from the auxiliary array. This is called memorization.
We will use the same recursive algorithm in the naive implementation but additionally implement caching, we use store auxiliary array for that.
Store[] will store all results from previous recursive calls which shall be used in subsequent steps in the algorithm.
Initialize an auxiliary array that will store results from recursive calls.
The algorithm solves the problem recursively just like in the naive approach but instead of repeated computations, results are stored and used for subsequent computations.
int minJumps(vector<int> &nums, int start){
if(start >= nums.size() - 1)
if(store[start])
return store[start];
int minJump = 10000;
for(int i = start + 1; i <= start + nums[start]; i++)
minJump = min(minJump, 1 + minJumps(nums, i));
store[start] = minJump;
store = vector<int>(nums.size());
Note We used 10000 as a constraint for the maximum length for the input list.
The computational time and space complexity is same as the bottom up approach.
In this approach we use a greedy algorithm which makes an optimal choice at each step in the algorithm building the solution piece by piece, that is from a position we determine the next steps that will push the index close to the last index.
Given the array [1, 1, 2, 1, 2, 1, 9]
We initialize left and right pointers, l, r to point to the start and end of a segment and count which will store the number of jumps made.
While r is less than array size.
l = 1, r = 1, [1]
maxReach = max(0, 1)
jumps = 0 + 1
maxReach = max(1, 2) [1, 2]
l = r+1 = 2
r = maxReach = 2
jumps = 1+1 = 2
maxReach = max(2, 4) [2, 1, 2]
The while condition terminates, jumps variable is returned.
Initialize left and right pointers which will be used to point to the start and end of a segment, a segment refers to the steps a particular index can move. Initialize jumps variable that will be used to store the number of jumps made so far which will be the minimum.
While right pointer is not at the end of the list.
loop through the segment and maximize the reach.
Finally update the pointers, left pointer to be at right + 1 and right pointer to be at the maximum reach.
The outer loop terminates when right is at last index, return the return jumps.
int l = 0, r = 0, jumps = 0;
while(r < n-1){
for(int i = l; i < r+1; i++)
maxReach = max(maxReach, nums[i]+i);
l = r+1;
r = maxReach;
jumps += 1;
Time Complexity is linear time O(n). The algorithm traverses the list once, the outer loop goes through the whole list while the inner loop traverses the segments at each step. All loops traverse the list once.
Space complexity is O(1) no extra space is used.
Another simple problem that uses dynamic programming approach is the "nth fibonacci", Can you solve it recursively, come up with a recursion tree, analyze it and optimize using a dynamic programming approach?
With this article at OpenGenus, you must have the complete idea of solving the problem Jump Game II: Minimum number of jumps to last index.
Time & Space Complexity of Dijkstra's Algorithm
In this article, we have explored the Time & Space Complexity of Dijkstra's Algorithm including 3 different variants like naive implementation, Binary Heap + Priority Queue and Fibonacci Heap + Priority Queue.
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The first form of the calling sequence for the readdata function reads
n
columns of numerical data in floating-point format from the file specified by
\mathrm{fileID}
\mathrm{readdata}\left(\mathrm{data},3\right)
[[\textcolor[rgb]{0,0,1}{1.}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{50.}]\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}{55.}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2.}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{55.}]\textcolor[rgb]{0,0,1}{,}[\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}{70.}]]
\mathrm{readdata}\left(\mathrm{data},\mathrm{integer},3\right)
[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{50}]\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}{55}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{55}]\textcolor[rgb]{0,0,1}{,}[\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}{70}]]
\mathrm{readdata}\left(\mathrm{data},\mathrm{integer}\right)
[\textcolor[rgb]{0,0,1}{1}\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}{2}]
\mathrm{readdata}\left(\mathrm{data},\mathrm{float}\right)
[\textcolor[rgb]{0,0,1}{1.}\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}{2.}]
\mathrm{readdata}\left(\mathrm{data},[\mathrm{integer},\mathrm{integer},\mathrm{float}]\right)
[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{50.}]\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}{55.}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{55.}]\textcolor[rgb]{0,0,1}{,}[\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}{70.}]]
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EUDML | Exceptional sets for solutions to quasilinear parabolic equations in weighted Sobolev spaces. EuDML | Exceptional sets for solutions to quasilinear parabolic equations in weighted Sobolev spaces.
Exceptional sets for solutions to quasilinear parabolic equations in weighted Sobolev spaces.
Alborova, M.S.
Alborova, M.S.. "Exceptional sets for solutions to quasilinear parabolic equations in weighted Sobolev spaces.." Vladikavkazskiĭ Matematicheskiĭ Zhurnal 2.3 (2000): 3-12. <http://eudml.org/doc/230046>.
@article{Alborova2000,
author = {Alborova, M.S.},
keywords = {bounded solution; singularity; elimination of singular points},
title = {Exceptional sets for solutions to quasilinear parabolic equations in weighted Sobolev spaces.},
AU - Alborova, M.S.
TI - Exceptional sets for solutions to quasilinear parabolic equations in weighted Sobolev spaces.
KW - bounded solution; singularity; elimination of singular points
bounded solution, singularity, elimination of singular points
Continuation and prolongation of solutions
{L}^{p}
Articles by Alborova
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Unemployment Rate FAQs
The unemployment rate is the percentage of the labor force without a job. It is a lagging indicator, meaning that it generally rises or falls in the wake of changing economic conditions, rather than anticipating them. When the economy is in poor shape and jobs are scarce, the unemployment rate can be expected to rise. When the economy is growing at a healthy rate and jobs are relatively plentiful, it can be expected to fall.
The unemployment rate is the proportion of the labor force that is not currently employed but could be.
There are six different ways the unemployment rate is calculated by the Bureau of Labor Statistics using different criteria.
The most comprehensive statistic reported is called the U-6 rate, but the most widely used and cited is the U-3 rate.
The U-3 unemployment rate for January 2022 was 4%.
The U.S. unemployment rate is released on the first Friday of every month (with a few exceptions) for the preceding month. The current and past editions of the report are available on the website of the Bureau of Labor Statistics (BLS). Users can generate and download tables showing any of the labor market measures named above for a specified date range.
In the U.S., the official and the most commonly cited national unemployment rate is the U-3, which the BLS releases as part of its monthly employment situation report. It defines unemployed people as those who are willing and available to work and who have actively sought work within the past four weeks.
According to the BLS, those with temporary, part-time, or full-time jobs are considered employed, as are those who perform at least 15 hours of unpaid work for a family business or farm. The unemployment rate is seasonally adjusted to account for predictable variations, such as extra hiring during the holidays. The BLS also provides the unadjusted rate.
The unemployment rate for January 2022 increased slightly to 4% from December's 3.9% and remained above its pre-pandemic level of 3.5% in February 2020. The economy added 467,000 nonfarm payrolls during this period, much more than was estimated by economists.
The U-3 is not the only metric available, and it measures unemployment fairly narrowly. The more comprehensive U-6 rate, often called the "real" unemployment rate, is an alternative measure of unemployment that includes groups such as discouraged workers who have stopped looking for a new job and the underemployed who are working part-time because they can't find full-time employment. The U-6 "real unemployment" rate for January 2022 was 7.1%, down from 7.3% in December 2021.
To calculate the U-3 unemployment rate, the number of unemployed people is divided by the number of people in the labor force, which consists of all employed and unemployed people. The ratio is expressed as a percentage. The January 2022 U-3 unemployment rate as reported by the BLS was 4%.
\begin{aligned} &\text{U-3} = \frac { \text{Unemployed} }{ \text{Labor Force} } \times 100 \\ \end{aligned}
U-3=Labor ForceUnemployed×100
Many people who would like to work but cannot (due to a disability, for example) or have become discouraged after looking for work without success, are not considered unemployed under this definition; since they are not employed either, they are categorized as outside the labor force. Critics see this approach as painting an unjustifiably rosy picture of the labor force. U-3 is also criticized for making no distinction between those in temporary, part-time, and full-time jobs, even in cases where part-time or temporary workers would rather work full-time but cannot due to labor market conditions.
In response to concerns that the official rate does not fully convey the health of the labor market, the BLS publishes five alternative measures: U-1, U-2, U-4, U-5, and U-6. Though these are often referred to as unemployment rates (U-6, in particular, is often called the "real" unemployment rate), U-3 is technically the only official unemployment rate. The others are measures of "labor underutilization."
People who have been unemployed for 15 weeks or longer, expressed as a percentage of the labor force.
\begin{aligned} &\text{U-1} = \frac { \text{Unemployed 15+ Weeks} }{ \text{Labor Force} } \times 100 \\ \end{aligned}
U-1=Labor ForceUnemployed 15+ Weeks×100
People who lost their jobs, or whose temporary jobs ended, as a percentage of the labor force.
\begin{aligned} &\text{U-2} = \frac { \text{Job Losers} }{ \text{Labor Force} } \times 100 \\ \end{aligned}
U-2=Labor ForceJob Losers×100
Unemployed people, plus discouraged workers, as a percentage of the labor force (plus discouraged workers).
\begin{aligned} &\text{U-4} = \frac { \text{Unemployed} + \text{Discouraged Workers} }{ \text{Labor Force} + \text{Discouraged Workers} } \times 100 \\ \end{aligned}
U-4=Labor Force+Discouraged WorkersUnemployed+Discouraged Workers×100
Discouraged workers are those who are available to work and would like a job, but have given up actively looking for one. This category includes people who feel they lack the necessary qualifications or education, who believe there is no work available in their field, or who feel they are too young or old to find work.
Those who feel unable to find work due to discrimination also fall under this category. Note that the denominator—normally the labor force—is adjusted to include discouraged workers, who are not technically part of the labor force.
Unemployed people, plus those who are marginally attached to the labor force, as a percentage of the labor force (plus the marginally attached).
\begin{aligned} &\text{U-5} = \frac { \text{Unemployed} + \text{Marginally Attached} }{ \text{Labor Force} + \text{Marginally Attached} } \times 100 \\ \end{aligned}
U-5=Labor Force+Marginally AttachedUnemployed+Marginally Attached×100
People who are marginally attached to the labor force include discouraged workers and anyone else who would like a job and has looked for one in the past 12 months but have given up actively searching. As with U-4, the denominator is expanded to include the marginally attached, who are not technically part of the labor force.
Unemployed people, plus people who are marginally attached to the labor force, plus those who are employed part-time for economic reasons, as a percentage of the labor force (plus marginally attached).
\begin{aligned} &\text{U-6} = \frac { \text{Unemployed} + \text{MA} + \text{PTER} }{ \text{Labor Force} + \text{MA} } \times 100 \\ &\textbf{where:} \\ &\text{MA} = \text{marginally attached} \\ &\text{PTER} = \text{part-time for economic reasons} \\ \end{aligned}
U-6=Labor Force+MAUnemployed+MA+PTER×100where:MA=marginally attachedPTER=part-time for economic reasons
This metric is the BLS's most comprehensive. In addition to the categories included in U-5, it accounts for people who have been forced to settle for part-time work even though they want to work full-time. This category is often referred to as "underemployed," although that label arguably includes full-time workers who are overqualified for their jobs. The denominator for this ratio is the same as in U-5.
Unemployment Rates (seasonally adjusted)
Official U.S. employment statistics are produced by the BLS, an agency within the Department of Labor (DOL). Every month the Census Bureau, part of the Department of Commerce (DOC), conducts the Current Population Survey (CPS) using a sample of approximately 60,000 households, or about 110,000 individuals.
The survey collects data on individuals in these households by race, ethnicity, age, veteran status, and gender (but only allowing for categories of men or women), all of which—along with geography—add nuance to the employment data. The sample is rotated so that 75% of the households remain constant from month to month and 50% from year to year. Interviews are conducted in person or by phone.
The survey excludes individuals under the age of 16 and those who are in the Armed Forces (hence references to the "civilian labor force"). People in correctional facilities, mental healthcare facilities, and similar institutions are also excluded. Interviewers ask a series of questions that determine employment status, but do not ask whether respondents are employed or unemployed. Nor do the interviewers themselves assign employment status; they record the answers for the BLS to analyze.
Interviewers also collect information on industry, occupation, average earnings, union membership, and—for the jobless—whether they quit or were discharged (fired or laid off).
What Are the Other Measures of U.S. Unemployment?
American unemployment rates utilize five measures in addition to the headline H3 figures: U-1, U-2, U-4, U-5, and U-6. Each of these incrementally considers additional groups of individuals and labels them as unemployed (e.g., those "underemployed" or working part-time but seeking full employment, etc.) The U-6 number is sometimes referred to as the "real" unemployment rate since it is the most comprehensive.
What's the Difference Between U-3 and U-6 Unemployment Rates?
U-3 is the headline unemployment number that we see in the news. It looks at those out-of-work Americans who have been looking for a job within the past four weeks. The more comprehensive U-6 includes everyone in U-3 plus those with only temporary work and people who are considered marginally attached to the labor force. These include those who have stopped looking for a job, as well as part-time workers unable to work full-time for economic reasons.
How Is U.S. Unemployment Data Collected?
The U.S. Bureau of Labor Statistics, or BLS, surveys approximately 60,000 households in person or over the phone. The responses are later aggregated by race, ethnicity, age, veteran status, and gender, all of which—along with geography—add greater detail to the employment picture.
U.S. Bureau of Labor Statistics. "Table A-15. Alternative Measures of Labor Underutilization." Accessed Feb. 4, 2022.
U.S. Bureau of Labor Statistics. "Employment Situation." Accessed Feb. 4, 2022.
U.S. Bureau of Labor Statistics. "Schedule of Releases for the Employment Situation." Accessed Feb. 4, 2022.
U.S. Bureau of Labor Statistics. "Home."
U.S. Bureau of Labor Statistics. "Data Retrieval: Labor Force Statistics (CPS)." Accessed Feb. 4, 2022.
U.S. Bureau of Labor Statistics. "Unemployed." Accessed Feb. 4, 2022.
U.S. Bureau of Labor Statistics. "Employed." Accessed Feb. 4, 2022.
U.S. Bureau of Labor Statistics. "How Are Seasonal Fluctuations Taken Into Account?" Accessed Feb. 4, 2022.
U.S. Bureau of Labor Statistics. "The Employment Situation—February 2020." Accessed Feb. 4, 2022.
U.S. Bureau of Labor Statistics. "Not in the Labor Force." Accessed Feb. 4, 2022.
U.S. Bureau of Labor Statistics. "How the Government Measures Unemployment." Accessed Feb. 4, 2022.
The civilian labor force is the U.S. civilian population that the Bureau of Labor Statistics (BLS) classifies as either employed or unemployed.
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Dinitrogen_pentoxide Knowpia
Dinitrogen pentoxide is the chemical compound with the formula N2O5, also known as nitrogen pentoxide or nitric anhydride. It is one of the binary nitrogen oxides, a family of compounds that only contain nitrogen and oxygen. It exists as colourless crystals that melt at 41 °C. Its boiling point is 47 °C, and sublimes slightly above room temperature,[1] yielding a colorless gas.[2]
6XB659ZX2W N
InChI=1S/N2O5/c3-1(4)7-2(5)6 Y
Key: ZWWCURLKEXEFQT-UHFFFAOYSA-N Y
InChI=1/N2O5/c3-1(4)7-2(5)6
Key: ZWWCURLKEXEFQT-UHFFFAOYAN
[O-][N+](=O)O[N+]([O-])=O
Melting point 41 °C (106 °F; 314 K)[1]
Boiling point 47 °C (117 °F; 320 K) sublimes
reacts to give HNO3
Solubility soluble in chloroform
negligible in CCl4
−35.6×10−6 cm3 mol−1 (aq)
planar, C2v (approx. D2h)
N–O–N ≈ 180°
178.2 J K−1 mol−1 (s)
355.6 J K−1 mol−1 (g)
−43.1 kJ/mol (s)
+11.3 kJ/mol (g)
strong oxidizer, forms strong acid in contact with water
Dinitrogen pentoxide is an unstable and potentially dangerous oxidizer that once was used as a reagent when dissolved in chloroform for nitrations but has largely been superseded by nitronium tetrafluoroborate (NO2BF4).
N2O5 is a rare example of a compound that adopts two structures depending on the conditions. The solid is a salt, nitronium nitrate, consisting of separate nitronium cations [NO2]+ and nitrate anions [NO3]−; but in the gas phase and under some other conditions it is a covalently-bound molecule.[3]
N2O5 was first reported by Deville in 1840, who prepared it by treating silver nitrate (AgNO3) with chlorine.[4][5]
Pure solid N2O5 is a salt, consisting of separated linear nitronium ions NO+2 and planar trigonal nitrate anions NO−3. Both nitrogen centers have oxidation state +5. It crystallizes in the space group D4
6h (C6/mmc) with Z = 2, with the NO−3 anions in the D3h sites and the NO+2 cations in D3d sites.[6]
The vapor pressure P (in atm) as a function of temperature T (in kelvin), in the range 211 to 305 K (−62 to 32 °C), is well approximated by the formula
{\displaystyle \ln P=23.2348-{\frac {7098.2}{T}}}
being about 48 torr at 0 °C, 424 torr at 25 °C, and 760 torr at 32 °C (9 °C below the melting point).[7]
In the gas phase, or when dissolved in nonpolar solvents such as carbon tetrachloride, the compound exists as covalently-bonded molecules O2N−O−NO2. In the gas phase, theoretical calculations for the minimum-energy configuration indicate that the O−N−O angle in each −NO2 wing is about 134° and the N−O−N angle is about 112°. In that configuration, the two −NO2 groups are rotated about 35° around the bonds to the central oxygen, away from the N−O−N plane. The molecule thus has a propeller shape, with one axis of 180° rotational symmetry (C2) [8]
When gaseous N2O5 is cooled rapidly ("quenched"), one can obtain the metastable molecular form, which exothermically converts to the ionic form above −70 °C.[9]
Gaseous N2O5 absorbs ultraviolet light with dissociation into the free radicals nitrogen dioxide NO2• and nitrogen trioxide NO3• (uncharged nitrate). The absorption spectrum has a broad band with maximum at wavelength 160 nm.[10]
A recommended laboratory synthesis entails dehydrating nitric acid (HNO3) with phosphorus(V) oxide:[9]
P4O10 + 12 HNO3 → 4 H3PO4 + 6 N2O5
Another laboratory process is the reaction of lithium nitrate LiNO3 and bromine pentafluoride BrF5, in the ratio exceeding 3:1. The reaction first forms nitryl fluoride FNO2 that reacts further with the lithium nitrate:[6]
BrF5 + 3 LiNO3 → 3 LiF + BrONO2 + O2 + 2 FNO2
FNO2 + LiNO3 → LiF + N2O5
The compound can also be created in the gas phase by reacting nitrogen dioxide NO2 or N2O4 with ozone:[11]
2 NO2 + O3 → N2O5 + O2
However, the product catalyzes the rapid decomposition of ozone:[11]
2 O3 + N2O5 → 3 O2 + N2O5
Dinitrogen pentoxide is also formed when a mixture of oxygen and nitrogen is passed through an electric discharge.[6] Another route is the reactions of Phosphoryl chloride POCl3 or nitryl chloride NO2Cl with silver nitrate AgNO3[6][12]
Dinitrogen pentoxide reacts with water (hydrolyses) to produce nitric acid HNO3. Thus, dinitrogen pentoxide is the anhydride of nitric acid:[9]
Solutions of dinitrogen pentoxide in nitric acid can be seen as nitric acid with more than 100% concentration. The phase diagram of the system H2O−N2O5 shows the well-known negative azeotrope at 60% N2O5 (that is, 70% HNO3), a positive azeotrope at 85.7% N2O5 (100% HNO3), and another negative one at 87.5% N2O5 ("102% HNO3").[13]
The reaction with hydrogen chloride HCl also gives nitric acid and nitryl chloride NO2Cl:[14]
N2O5 + HCl → HNO3 + NO2Cl
Dinitrogen pentoxide eventually decomposes at room temperature into NO2 and O2.[15][11] Decomposition is negligible if the solid is kept at 0 °C, in suitably inert containers.[6]
Dinitrogen pentoxide reacts with ammonia NH3 to give several products, including nitrous oxide N2O, ammonium nitrate NH4NO3, nitramide NH2NO2 and ammonium dinitramide NH4N(NO2)2, depending on reaction conditions.[16]
Decomposition of dinitrogen pentoxide at high temperaturesEdit
Dinitrogen pentoxide between high temperatures of 600 and 1,100 K (327–827 °C), is decomposed in two successive stoichiometric steps:
2 NO3 → 2 NO2 + O2
In the shock wave, N2O5 has decomposed stoichiometrically into nitrogen dioxide and oxygen. At temperatures of 600 K and higher, nitrogen dioxide is unstable with respect to nitrogen oxide NO and oxygen. The thermal decomposition of 0.1 mM nitrogen dioxide at 1000 K is known to require about two seconds.[17]
Decomposition of dinitrogen pentoxide in carbon tetrachloride at 30 °CEdit
Apart from the decomposition of N2O5 at high temperatures, it can also be decomposed in carbon tetrachloride CCl4 at 30 °C (303 K).[18] Both N2O5 and NO2 are soluble in CCl4 and remain in solution while oxygen is insoluble and escapes. The volume of the oxygen formed in the reaction can be measured in a gas burette. After this step we can proceed with the decomposition, measuring the quantity of O2 that is produced over time because the only form to obtain O2 is with the N2O5 decomposition. The equation below refers to the decomposition of N2O5 in CCl4:
2 N2O5 → 4 NO2 + O2(g)
And this reaction follows the first order rate law that says:
{\displaystyle -{\frac {d[\mathrm {A} ]}{dt}}=k[\mathrm {A} ]}
Decomposition of nitrogen pentoxide in the presence of nitric oxideEdit
N2O5 can also be decomposed in the presence of nitric oxide NO:
N2O5 + NO → 3 NO2
The rate of the initial reaction between dinitrogen pentoxide and nitric oxide of the elementary unimolecular decomposition.[19]
Nitration of organic compoundsEdit
Dinitrogen pentoxide, for example as a solution in chloroform, has been used as a reagent to introduce the −NO2 functionality in organic compounds. This nitration reaction is represented as follows:
N2O5 + Ar−H → HNO3 + Ar−NO2
where Ar represents an arene moiety.[20] The reactivity of the NO+2 can be further enhanced with strong acids that generate the "super-electrophile" HNO2+2.
In this use, N2O5 has been largely replaced by nitronium tetrafluoroborate [NO2]+[BF4]−. This salt retains the high reactivity of NO+2, but it is thermally stable, decomposing at about 180 °C (into NO2F and BF3).
Dinitrogen pentoxide is relevant to the preparation of explosives.[5][21]
Atmospheric occurrenceEdit
In the atmosphere, dinitrogen pentoxide is an important reservoir of the NOx species that are responsible for ozone depletion: its formation provides a null cycle with which NO and NO2 are temporarily held in an unreactive state.[22] Mixing ratios of several parts per billion by volume have been observed in polluted regions of the nighttime troposphere.[23] Dinitrogen pentoxide has also been observed in the stratosphere[24] at similar levels, the reservoir formation having been postulated in considering the puzzling observations of a sudden drop in stratospheric NO2 levels above 50 °N, the so-called 'Noxon cliff'.
Variations in N2O5 reactivity in aerosols can result in significant losses in tropospheric ozone, hydroxyl radicals, and NOx concentrations.[25] Two important reactions of N2O5 in atmospheric aerosols are hydrolysis to form nitric acid[26] and reaction with halide ions, particularly Cl−, to form ClNO2 molecules which may serve as precursors to reactive chlorine atoms in the atmosphere.[27][28]
N2O5 is a strong oxidizer that forms explosive mixtures with organic compounds and ammonium salts. The decomposition of dinitrogen pentoxide produces the highly toxic nitrogen dioxide gas.
^ a b Emeleus (1 January 1964). Advances in Inorganic Chemistry. Academic Press. pp. 77–. ISBN 978-0-12-023606-0. Retrieved 20 September 2011.
^ Peter Steele Connell The Photochemistry of Dinitrogen Pentoxide. Ph. D. thesis, Lawrence Berkeley National Laboratory.
^ W. Rogie Angus, Richard W. Jones, and Glyn O. Phillips (1949): "Existence of Nitrosyl Ions (NO+) in Dinitrogen Tetroxide and of Nitronium Ions (NO+2) in Liquid Dinitrogen Pentoxide". Nature, volume 164, pages 433–434. doi:10.1038/164433a0
^ M.H. Deville (1849). "Note sur la production de l'acide nitrique anhydre". Compt. Rend. 28: 257–260.
^ a b Jai Prakash Agrawal (19 April 2010). High Energy Materials: Propellants, Explosives and Pyrotechnics. Wiley-VCH. p. 117. ISBN 978-3-527-32610-5. Retrieved 20 September 2011.
^ a b c d e William W. Wilson and Karl O. Christe (1987): "Dinitrogen Pentoxide. New Synthesis and Laser Raman Spectrum". Inorganic Chemistry, volume 26, pages 1631–1633. doi:10.1021/ic00257a033
^ A. H. McDaniel, J. A. Davidson, C. A. Cantrell, R. E. Shetter, and J. G. Calvert (1988): "Enthalpies of formation of dinitrogen pentoxide and the nitrate free radical". Journal of Physical Chemistry, volume 92, issue 14, pages 4172–4175. doi:10.1021/j100325a035
^ S. Parthiban, B. N. Raghunandan, and R.Sumathi (1996): "Structures, energies and vibrational frequencies of dinitrogen pentoxide". Journal of Molecular Structure: THEOCHEM, volume 367, pages 111–118. doi:10.1016/S0166-1280(96)04516-2
^ a b c Holleman, Arnold Frederik; Wiberg, Egon (2001), Wiberg, Nils (ed.), Inorganic Chemistry, translated by Eagleson, Mary; Brewer, William, San Diego/Berlin: Academic Press/De Gruyter, ISBN 0-12-352651-5
^ Bruce A. Osborne, George Marston, L. Kaminski, N. C. Jones, J. M. Gingell, Nigel Mason, Isobel C. Walker, J. Delwiche, and M.-J. Hubin-Franskin (2000): "Vacuum ultraviolet spectrum of dinitrogen pentoxide". Journal of Quantitative Spectroscopy and Radiative Transfer, volume 64, issue 1, pages 67–74. doi:10.1016/S0022-4073(99)00104-1
^ a b c Francis Yao, Ivan Wilson, and Harold Johnston (1982): "Temperature-dependent ultraviolet absorption spectrum for dinitrogen pentoxide". Journal of Physical Chemistry, volume 86, issue 18, pages 3611–3615. doi:10.1021/j100215a023
^ Garry, Schott; Norman, Davidson (1958), "Shock Waves in Chemical Kinetics: The Decomposition of N2O5 at High Temperatures", Journal of the American Chemical Society (published 21 October 1957), 80 (8): 8, doi:10.1021/ja01541a019
^ L. Lloyd and P. A. H. Wyatt (1955): "The vapour pressures of nitric acid solutions. Part I. New azeotropes in the water–dinitrogen pentoxide system". Journal of the Chemical Society (Resumed), volume 1955, pages 2248–2252.doi:10.1039/JR9550002248
^ Robert A. Wilkins Jr. and I. C. Hisatsune (1976): "The Reaction of Dinitrogen Pentoxide with Hydrogen Chloride". Industrial & Engineering Chemistry Fundamentals, volume 15, issue 4, pages 246–248. doi:10.1021/i160060a003
^ Nitrogen(V) Oxide. Inorganic Syntheses. Vol. 3. 1950. pp. 78–81.
^ C. Frenck and W. Weisweiler (2002): "Modeling the Reactions Between Ammonia and Dinitrogen Pentoxide to Synthesize Ammonium Dinitramide (ADN)". Chemical Engineering & Technology, volume 25, issue 2, pages 123–128. doi:10.1002/1521-4125(200202)25:2<123::AID-CEAT123>3.0.CO;2-W
^ Schott, G., & Davidson, N. (1958). Shock Waves in Chemical Kinetics: The Decomposition of N2O5 at High Temperatures. Journal of the American Chemical Society, 80(8), 1841–1853. doi:10.1021/ja01541a019
^ J.,Jaime, R. (2008). Determinación de orden de reacción haciendo uso de integrales definidas. Universidad Nacional Autónoma de Nicaragua, Managua.
^ J. Wilson, David; S. Johnston, Harold (1953). "Decomposition of Nitrogen Pentoxide in the Presence of Nitric Oxide. IV. Effect of Noble Gases". Journal of the American Chemical Society. 75 (22): 5763. doi:10.1021/ja01118a529.
^ Jan M. Bakke and Ingrd Hegbom (1994): "Dinitrogen pentoxide-sulfur dioxide, a new nitration system". Acta chemica scandinavica, volume 48, issue 2, pages 181–182. doi:10.3891/acta.chem.scand.48-0181
^ Talawar, M. B.; et al. (2005). "Establishment of Process Technology for the Manufacture of Dinitrogen Pentoxide and its Utility for the Synthesis of Most Powerful Explosive of Today—CL-20". Journal of Hazardous Materials. 124 (1–3): 153–64. doi:10.1016/j.jhazmat.2005.04.021. PMID 15979786.
^ Finlayson-Pitts, Barbara J.; Pitts, James N. (2000). Chemistry of the upper and lower atmosphere : theory, experiments, and applications. San Diego: Academic Press. ISBN 9780080529073. OCLC 162128929.
^ HaiChao Wang; et al. (2017). "High N2O5 Concentrations Observed in Urban Beijing: Implications of a Large Nitrate Formation Pathway". Environmental Science and Technology Letters. 4 (10): 416–420. doi:10.1021/acs.estlett.7b00341.
^ C.P. Rinsland; et al. (1989). "Stratospheric N2O5 profiles at sunrise and sunset from further analysis of the ATMOS/Spacelab 3 solar spectra". Journal of Geophysical Research. 94: 18341–18349. Bibcode:1989JGR....9418341R. doi:10.1029/JD094iD15p18341.
^ Macintyre, H. L.; Evans, M. J. (2010-08-09). "Sensitivity of a global model to the uptake of N2O5 by tropospheric aerosol". Atmospheric Chemistry and Physics. 10 (15): 7409–7414. Bibcode:2010ACP....10.7409M. doi:10.5194/acp-10-7409-2010. ISSN 1680-7324.
^ Brown, S. S.; Dibb, J. E.; Stark, H.; Aldener, M.; Vozella, M.; Whitlow, S.; Williams, E. J.; Lerner, B. M.; Jakoubek, R. (2004-04-16). "Nighttime removal of NOx in the summer marine boundary layer". Geophysical Research Letters. 31 (7): n/a. Bibcode:2004GeoRL..31.7108B. doi:10.1029/2004GL019412. ISSN 1944-8007.
^ Gerber, R. Benny; Finlayson-Pitts, Barbara J.; Hammerich, Audrey Dell (2015-07-15). "Mechanism for formation of atmospheric Cl atom precursors in the reaction of dinitrogen oxides with HCl/Cl− on aqueous films" (PDF). Physical Chemistry Chemical Physics. 17 (29): 19360–19370. Bibcode:2015PCCP...1719360H. doi:10.1039/C5CP02664D. ISSN 1463-9084. PMID 26140681.
^ Kelleher, Patrick J.; Menges, Fabian S.; DePalma, Joseph W.; Denton, Joanna K.; Johnson, Mark A.; Weddle, Gary H.; Hirshberg, Barak; Gerber, R. Benny (2017-09-18). "Trapping and Structural Characterization of the XNO2·NO3− (X = Cl, Br, I) Exit Channel Complexes in the Water-Mediated X− + N2O5 Reactions with Cryogenic Vibrational Spectroscopy". The Journal of Physical Chemistry Letters. 8 (19): 4710–4715. doi:10.1021/acs.jpclett.7b02120. ISSN 1948-7185. PMID 28898581.
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Carbon Cycle Feedbacks | METEO 469: From Meteorology to Mitigation: Understanding Global Warming
As we saw earlier in the course, the airborne fraction of
{\text{CO}}_{\text{2}}
in the atmosphere has increased by only half as much as it should have given the emissions we have added through fossil fuel burning and deforestation. We know that
{\text{CO}}_{\text{2}}
must be going somewhere.
Figure 8.8: Annual change in atmospheric
{\text{CO}}_{\text{2}}
Click Here for a text alternative for Figure 8.8
Where Did All The CO2 Go?
Emissions and uptake rates in petagrams (1015g) of carbon (C) per year. The graph at left shows carbon emissions from fossil-fuel combustion and cement manufacturing. The graph at right shows the sum of these along with emissions from deforestation and other land-use changes and uptake in terrestrial ecosystems, the atmosphere, and the oceans.
Indeed, it is being absorbed by various reservoirs that exist within the global carbon cycle. As we saw earlier, in Lesson 1, only 55% of the emitted carbon has shown up in the atmosphere, while roughly 30-35% appears to be going into the oceans, and 15-20% into the terrestrial biosphere.
Figure 8.9: Global carbon cycle.[Enlarge]
Click Here for text Alternative of Figure 8.9
The main reservoirs of carbon are the atmsphere, the ocean, and vegetation, soils, and detritus on land. Marine life represents a very small carbon reservoir. On multi-millennial time scales, geologic reservoirs also become important. Various processes transfer carbon between these reservoirs, including photosynthesis and respiration, ocean-atmosphere gas exchange, and ocean mixing. This figure shows carbon movement, weathering and erosion, and human carbon transformation.
The problem is that this pattern of behavior may not continue. There is no guarantee that the ocean and terrestrial biosphere will continue to be able to absorb this same fraction of carbon emissions as time goes on, and that leads us into a discussion of so-called carbon cycle feedbacks.
If we consider the oceans, for example, there are a number of factors that could lead to decreased uptake of carbon as time goes on. Like a warm can of Coke, which loses its carbonation when you warm it up and remove the top, the ocean's
{\text{CO}}_{\text{2}}
solubility decreases as the ocean warms. When we look at the pattern of carbon uptake in the upper ocean, we see that one of the primary regions of uptake is the North Atlantic. This is, in part, due to the formation of carbon-burying deep water in the region. In a scenario we have explored in Lesson 7, the North Atlantic overturning circulation could weaken in the future (though as we have seen, there is quite a bit of uncertainty regarding the magnitude and time frame of this weakening). If that were to happen, it would eliminate one of the ocean's key carbon-burying mechanisms, and allow
{\text{CO}}_{\text{2}}
to accumulate faster in the atmosphere. On the other hand, the biological productivity of the upwelling zone of the cold tongue region of the eastern and central equatorial Pacific is a net source of carbon to the atmosphere, from the ocean. More El Niño-like conditions in the future could suppress this source of carbon, but more La Niña-like conditions could increase this source, further accelerating the buildup of
{\text{CO}}_{\text{2}}
in the atmosphere. So uncertainties in the future course of oceanic uptake abound, but, on balance, it is likely that this uptake will decrease over time, yielding a positive carbon cycle feedback.
Figure 8.10: Ocean
{\text{CO}}_{\text{2}}
fluxes: positive numbers indicate flux out of the ocean.
Other ocean carbon cycle feedbacks relate to the phenomenon of ocean acidification, which results from the fact that increasing atmospheric
{\text{CO}}_{\text{2}}
leads to increased dissolved bicarbonate ion in the ocean (a phenomenon will discuss further in our next lesson on climate change impacts). On the one hand, this process interferes with the productivity of calcite-skeleton forming ocean organisms, such as zooplankton, which bury their calcium carbonate skeletons on the sea floor when they die. This so-called oceanic carbon pump, is a key mechanism by which the ocean buries carbon absorbed from the atmosphere on long timescales. So any decrease in the effectiveness of the ocean's carbon pump would represent a positive carbon cycle feedback. On the other hand, since calcifying organisms release
{\text{CO}}_{\text{2}}
into the water as they build their carbonate skeletons, a decrease in calcite production by these organisms will reduce
{\text{CO}}_{\text{2}}
, amounting to a negative carbon cycle feedback.
There are a number of other carbon cycle feedbacks that apply to the terrestrial biosphere. They vary anywhere from a strong negative to a strong positive feedback. Among them are (a) warmer land increasing microbial activity in soils, which releases
{\text{CO}}_{\text{2}}
(a small positive feedback), (b) increased plant productivity due to higher
{\text{CO}}_{\text{2}}
levels (a strong negative feedback). Finally, there is the negative silicate rock weathering feedback which we know to be a very important regulator of atmospheric
{\text{CO}}_{\text{2}}
levels on very long, geological timescales: a warmer climate, with its more vigorous hydrological cycle, leads to increased physical and chemical weathering (the process of taking
{\text{CO}}_{\text{2}}
out of the atmosphere by reacting it with rocks), through the formation of carbonic acid, which dissolves silicate rocks, producing dissolved salts that run off through river systems, eventually reaching the oceans.
While each of these potential carbon cycle feedbacks are uncertain in magnitude—and even in sign in some cases (see the various coloured bars in the figure below), the net result of all of these feedbacks appears to be a net positive carbon cycle feedback (the black bar shown).
Figure 8.11: Estimated magnitudes (including uncertainty ranges) of various potential oceanic and terrestrial carbon cycle feedbacks, expressed in terms of positive or negative estimated change in the airborne fraction of
{\text{CO}}_{\text{2}}
(based on average net increase by 2100 among the various climate models).
Other potential positive carbon cycle feedbacks that are even more uncertain, but could be quite sizeable in magnitude, are methane feedbacks, related to the possible release of frozen methane currently trapped in thawing Arctic permafrost, and so-called "clathrate"—a crystalline form of methane that is found in abundance along the continental shelves of the oceans, which could be destablized by modest ocean warming. Since methane is a very potent greenhouse gas, such releases of potentially large amounts of methane into the atmosphere could further amplify greenhouse warming and associated climate changes.
The key potential implication of a net positive carbon cycle feedback is that current projections of future warming such as those we have explored previously in Lesson 7, may actually underestimate the degree of warming expected from a particular carbon emissions pathway. This is because the assumed relationship between carbon emissions and
{\text{CO}}_{\text{2}}
concentrations would underestimate the actual resulting
{\text{CO}}_{\text{2}}
concentrations because they assume a fixed airborne fraction of emitted
{\text{CO}}_{\text{2}}
, when, it fact, that fraction would instead be increasing over time. While the magnitude of this effect is uncertain, the best estimates suggest an additional 20-30 ppm of
{\text{CO}}_{\text{2}}
per degree C warming, leading to an additional warming of anywhere from 0.1°C to 1.5°C relative to the nominal temperature projections shown in earlier lessons.
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Venn diagram - Simple English Wikipedia, the free encyclopedia
A Venn diagram is a diagram that shows the logical relation between sets. They were popularised by John Venn in the 1880s, and are now widely used. They are used to teach elementary set theory, and to illustrate simple set relationships in probability, logic, statistics, linguistics and computer science. A Venn diagram uses closed curves drawn on a plane to represent sets. Very often, these curves are circles or ellipses.
Stained glass window in Cambridge, where John Venn studied. It shows a Venn diagram.
2 Venn diagrams of common operations on sets
The following example uses two sets, A and B, represented here as coloured circles. The orange circle, set A, represents all living creatures that are two-legged. The blue circle, set B, represents the living creatures that can fly. Each separate type of creature can be imagined as a point somewhere in the diagram. Living creatures that can both fly and have two legs—for example, parrots—are then in both sets, and hence correspond to points in the area where the blue and orange circles overlap. That area contains all such (and only) such living creatures.
Humans and penguins are bipedal, and hence are in the orange circle, but since they cannot fly, they appear in the left part of the orange circle, where it does not overlap with the blue circle. Mosquitoes have six legs, and fly, so the point for mosquitoes is in the part of the blue circle that does not overlap with the orange one. Creatures that are not two-legged and cannot fly (for example, whales and spiders) would all be represented by points outside both circles.
The combined area of sets A and B is called the union of A and B, denoted by A ∪ B.[1] The union in this case contains all living creatures that are either two-legged, or that can fly (or both). The area in both A and B, where the two sets overlap, is called the intersection of A and B, denoted by A ∩ B.[1][2] For example, the intersection of the two sets is not empty, because there are points that represent creatures that are in both the orange and blue circles.
Venn diagrams of common operations on setsEdit
{\displaystyle A\cap B}
{\displaystyle A\cup B}
Complement of A
{\displaystyle A^{\rm {C}}=U\setminus A}
{\displaystyle B\setminus A}
{\displaystyle A\setminus B}
In the illustrations below the left circle shows set
{\displaystyle A}
{\displaystyle B}
. Results of operations are shown as colored areas.
↑ "Set Operations | Union | Intersection | Complement | Difference | Mutually Exclusive | Partitions | De Morgan's Law | Distributive Law | Cartesian Product". www.probabilitycourse.com. Retrieved 2020-09-04.
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Venn_diagram&oldid=7169930"
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EUDML | On weakly symmetric and weakly Ricci-symmetric -contant manifolds. EuDML | On weakly symmetric and weakly Ricci-symmetric -contant manifolds.
On weakly symmetric and weakly Ricci-symmetric
K
-contant manifolds.
De, U.C.; Binh, T.Q.; Shaikh, A.A.
De, U.C., Binh, T.Q., and Shaikh, A.A.. "On weakly symmetric and weakly Ricci-symmetric -contant manifolds.." Acta Mathematica Academiae Paedagogicae Nyí regyháziensis. New Series [electronic only] 16 (2000): 65-71. <http://eudml.org/doc/51289>.
author = {De, U.C., Binh, T.Q., Shaikh, A.A.},
keywords = {weakly symmetric Riemannian manifold; -contact manifold; -contact manifold},
title = {On weakly symmetric and weakly Ricci-symmetric -contant manifolds.},
AU - De, U.C.
AU - Shaikh, A.A.
TI - On weakly symmetric and weakly Ricci-symmetric -contant manifolds.
KW - weakly symmetric Riemannian manifold; -contact manifold; -contact manifold
weakly symmetric Riemannian manifold,
K
-contact manifold,
K
-contact manifold
Articles by De
Articles by Binh
Articles by Shaikh
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Mechanical equilibrium - Wikipedia
In classical mechanics, a particle is in mechanical equilibrium if the net force on that particle is zero.[1]: 39 By extension, a physical system made up of many parts is in mechanical equilibrium if the net force on each of its individual parts is zero.[1]: 45–46 [2]
In addition to defining mechanical equilibrium in terms of force, there are many alternative definitions for mechanical equilibrium which are all mathematically equivalent. In terms of momentum, a system is in equilibrium if the momentum of its parts is all constant. In terms of velocity, the system is in equilibrium if velocity is constant. In a rotational mechanical equilibrium the angular momentum of the object is conserved and the net torque is zero.[2] More generally in conservative systems, equilibrium is established at a point in configuration space where the gradient of the potential energy with respect to the generalized coordinates is zero.
If a particle in equilibrium has zero velocity, that particle is in static equilibrium.[3][4] Since all particles in equilibrium have constant velocity, it is always possible to find an inertial reference frame in which the particle is stationary with respect to the frame.
1.1 Potential energy stability test
1.2 Statically indeterminate system
Potential energy stability testEdit
{\displaystyle V}
The state is neutral to the lowest order and nearly remains in equilibrium if displaced a small amount. To investigate the precise stability of the system, higher order derivatives can be examined. The state is unstable if the lowest nonzero derivative is of odd order or has a negative value, stable if the lowest nonzero derivative is both of even order and has a positive value. If all derivatives are zero then it is impossible to derive any conclusions from the derivatives alone. For example, the function
{\displaystyle e^{-1/x^{2}}}
(defined as 0 in x=0) has all derivatives equal to zero. At the same time, this function has a local minimum in x=0, so it is a stable equilibrium. If you multiply this function by the Sign function, all derivatives will still be zero but it will become an unstable equilibrium.
Function is locally constant
In a truly neutral state the energy does not vary and the state of equilibrium has a finite width. This is sometimes referred to as a state that is marginally stable, or in a state of indifference, or astable equilibrium.
Statically indeterminate systemEdit
^ a b John L Synge & Byron A Griffith (1949). Principles of Mechanics (2nd ed.). McGraw-Hill.
^ a b Beer FP, Johnston ER, Mazurek DF, Cornell PJ, and Eisenberg, ER (2009). Vector Mechanics for Engineers: Statics and Dynamics (9th ed.). McGraw-Hill. p. 158. {{cite book}}: CS1 maint: multiple names: authors list (link)
^ Herbert Charles Corben & Philip Stehle (1994). Classical Mechanics (Reprint of 1960 second ed.). Courier Dover Publications. p. 113. ISBN 0-486-68063-0.
^ Lakshmana C. Rao; J. Lakshminarasimhan; Raju Sethuraman; Srinivasan M. Sivakumar (2004). Engineering Mechanics. PHI Learning Pvt. Ltd. p. 6. ISBN 81-203-2189-8.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Mechanical_equilibrium&oldid=1084915702"
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Ma'am I'm sorry i couldn't attend this class (april 21st trigonometric functions) as i was out of station So i - Maths - Trigonometric Functions - 10315541 | Meritnation.com
Ma'am I'm sorry i couldn't attend this class (april 21st trigonometric functions) as i was out of station.So i watched the recorded video of the class and i did no understand the eg of sec(570) which you did freehand.I haven't understood how it can be written as (5*90+30) .Also since it falls in the 2nd quadrant and since sec is changed to cosec, shouldn't the answer be cosec(30) instead of being -cosec(30)?
P.S the class is very interesting.Thank you.:)
\mathrm{sec} 570°\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=\mathrm{sec} \left(540°+30°\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=\mathrm{sec}\left(6×90°+30°\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=\mathrm{sec}\left(6×\frac{\mathrm{\pi }}{2} + \frac{\mathrm{\pi }}{6}\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=\mathrm{sec}\left(3\mathrm{\pi }+\frac{\mathrm{\pi }}{6}\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=- \mathrm{sec} \left(\frac{\mathrm{\pi }}{6}\right) \left[\mathbf{as}\mathbf{,}\mathbf{ }\mathbf{sec}\mathbf{ }\left(\mathbf{3}\mathbf{\pi }\mathbf{+}\mathbf{\theta }\right)\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{-}\mathbf{ }\mathbf{sec}\mathbf{ }\mathbf{\theta }\right]\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=-\frac{1}{\mathrm{cos}\left(\mathrm{\pi }/6\right)}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=-\frac{1}{\sqrt{3}/2}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=-\frac{2}{\sqrt{3}}
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Mean corpuscular volume - Wikipedia
Average volume of a red blood cell, which sometimes helps in diagnosis
MCV measurement that allows classification as either a microcytic anemia, normocytic anemia or macrocytic anemia
The mean corpuscular volume, or mean cell volume (MCV), is a measure of the average volume of a red blood corpuscle (or red blood cell). The measure is obtained by multiplying a volume of blood by the proportion of blood that is cellular (the hematocrit), and dividing that product by the number of erythrocytes (red blood cells) in that volume. The mean corpuscular volume is a part of a standard complete blood count.
For further specification, it can be used to calculate red blood cell distribution width (RDW). The RDW is a statistical calculation made by automated analyzers that reflects the variability in size and shape of the RBCs.
To calculate MCV, the hematocrit (Hct) is divided by the concentration of RBCs ([RBC]) [1]
{\displaystyle {\textit {MCV}}={\frac {\textit {Hct}}{[{\text{RBC}}]}}}
Normally, MCV is expressed in femtoliters (fL, or 10−15 L), and [RBC] in millions per microliter (106 / μL). The normal range for MCV is 80–100 fL.
If the hematocrit is expressed as a percentage, the red blood cell concentration as millions per microliter, and the MCV in femtoliters, the formula becomes
{\displaystyle {\textit {MCV}}/\mathrm {L} ={\frac {{\mathit {Hct\%}}/100}{[{\text{RBCmmL}}]\times (10^{6}/10^{-6})/\mathrm {L} ^{-1}}}}
{\displaystyle {\textit {MCV}}/\mathrm {fL} ={\textit {MCV}}/(10^{-15}\,\mathrm {L} )=10^{15}{\frac {{\mathit {Hct\%}}/100}{[{\text{RBCmmL}}]\times 10^{12}}}={\frac {{\mathit {Hct\%}}\times 10}{[{\text{RBCmmL}}]}}}
For example, if the Hct = 42.5% and [RBC] = 4.58 million per microliter (4,580,000/μL), then
{\displaystyle {\textit {MCV}}={\frac {0.425}{4.58\cdot 10^{6}/(10^{-6}\,\mathrm {L} )}}=92.8\cdot 10^{-15}\,\mathrm {L} =92.8\,\mathrm {fL} }
Using implied units,
{\displaystyle {\textit {MCV}}/{\textrm {fL}}={\frac {42.5\times 10}{4.58}}=92.8}
The MCV can be determined in a number of ways by automatic analyzers. In volume-sensitive automated blood cell counters, such as the Coulter counter, the red cells pass one-by-one through a small aperture and generate a signal directly proportional to their volume. Other automated counters measure red blood cell volume by means of techniques that measure refracted, diffracted, or scattered light.[2]
The normal reference range is typically 80-100 fL.[3]
In pernicious anemia (macrocytic), MCV can range up to 150 femtolitres. [4] (as are an elevated GGT and an AST/ALT ratio of 2:1). Vitamin B12 and/or folic acid deficiency has also been associated with macrocytic anemia (high MCV numbers).
The most common causes of microcytic anemia are iron deficiency (due to inadequate dietary intake, gastrointestinal blood loss, or menstrual blood loss), thalassemia, sideroblastic anemia or chronic disease. In iron deficiency anemia (microcytic anemia), it can be as low as 60 to 70 femtolitres. In some cases of thalassemia, the MCV may be low even though the patient is not iron deficient.[citation needed]
Hb 100 grams/liter 10 grams/deciliter (deci- is 10−1)
RBC 5E+12 cells/liter 5E+6 cells/μL (micro is 10−6)
MCV = (Hct/100) / RBC 8E-14 liters/cell 80 femtoliters/cell (femto- is 10−15)
MCH = Hb / RBC 2E-11 grams/cell 20 picograms/cell (pico- is 10−12)
MCHC = Hb / (Hct/100) 250 grams/liter 25 grams/deciliter (deci is 10−1)
The MCV can be conceptualized as the total volume of a group of cells divided by the number of cells. For a real world sized example, imagine you had 10 small jellybeans with a combined volume of 10 µL. The mean volume of a jellybean in this group would be 10 µL / 10 jellybeans = 1 µL / jellybean. A similar calculation works for MCV.[citation needed]
1. Measure the RBC index in cells/µL. Take the reciprocal (1/RBC index) to convert it to µL/cell.
{\displaystyle {\frac {1}{5\times 10^{6}}}\ \mathrm {\mu L/cell} =2\times 10^{-7}\ \mathrm {\mu L/cell} }
2. The 1 µL is only made of a proportion of red cells (e.g. 40%) with the rest of the volume composed of plasma. Multiply by the hematocrit (a unitless quantity) to take this into account.
{\displaystyle 2\times 10^{-7}\ \mathrm {\mu L/cell} \times 0.4=8\times 10^{-8}\ \mathrm {\mu L/cell} }
3. Finally, convert the units of µL to fL by multiplying by
{\displaystyle 10^{9}}
. The result would look like this:
{\displaystyle 8\times 10^{-8}\ \mathrm {\mu L/cell} \times {\frac {10^{9}\ \mathrm {fL} }{1\ \mathrm {\mu L} }}=80\ {\frac {\mathrm {fL} }{\mathrm {cell} }}}
Note: the shortcut proposed above just makes the units work out:
{\displaystyle 10\times 40\div 5=80}
^ Mondal, Himel; Budh, Deepa P. (2020). "Hematocrit (HCT)". StatPearls. PMID 31194416.
^ Stanley L Schrier, MD; Stephen A Landaw, MD (30 September 2011). "Mean corpuscular volume". uptodate.com.
^ "RBC indices: MedlinePlus Medical Encyclopedia". medlineplus.gov. Retrieved May 13, 2020.
^ Tønnesen H, Hejberg L, Frobenius S, Andersen J (1986). "Erythrocyte Mean Cell Volume-Correlation to Drinking Pattern in Heavy Alcoholics". Acta Med Scand. 219 (5): 515–8. doi:10.1111/j.0954-6820.1986.tb03348.x. PMID 3739755.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Mean_corpuscular_volume&oldid=1084338569"
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Merkle–Damgård construction - Wikipedia
(Redirected from Merkle-Damgård construction)
Method of building collision-resistant cryptographic hash functions
In cryptography, the Merkle–Damgård construction or Merkle–Damgård hash function is a method of building collision-resistant cryptographic hash functions from collision-resistant one-way compression functions.[1]: 145 This construction was used in the design of many popular hash algorithms such as MD5, SHA-1 and SHA-2.
The Merkle–Damgård construction was described in Ralph Merkle's Ph.D. thesis in 1979.[2] Ralph Merkle and Ivan Damgård independently proved that the structure is sound: that is, if an appropriate padding scheme is used and the compression function is collision-resistant, then the hash function will also be collision-resistant.[3][4]
The Merkle–Damgård hash function first applies an MD-compliant padding function to create an input whose size is a multiple of a fixed number (e.g. 512 or 1024) — this is because compression functions cannot handle inputs of arbitrary size. The hash function then breaks the result into blocks of fixed size, and processes them one at a time with the compression function, each time combining a block of the input with the output of the previous round.[1]: 146 In order to make the construction secure, Merkle and Damgård proposed that messages be padded with a padding that encodes the length of the original message. This is called length padding or Merkle–Damgård strengthening.
Merkle–Damgård hash construction
To harden the hash further, the last result is then sometimes fed through a finalisation function. The finalisation function can have several purposes such as compressing a bigger internal state (the last result) into a smaller output hash size or to guarantee a better mixing and avalanche effect on the bits in the hash sum. The finalisation function is often built by using the compression function.[citation needed] (Note that in some documents a different terminology is used: the act of length padding is called "finalisation".[citation needed])
1 Security characteristics
Security characteristics[edit]
The popularity of this construction is due to the fact, proven by Merkle and Damgård, that if the one-way compression function f is collision resistant, then so is the hash function constructed using it. Unfortunately, this construction also has several undesirable properties:
Second preimage attacks against long messages are always much more efficient than brute force.[5]
Multicollisions (many messages with the same hash) can be found with only a little more work than collisions.[6]
"Herding attacks", which combines the cascaded construction for multicollision finding (similar to the above) with collisions found for a given prefix (chosen-prefix collisions). This allows for constructing highly specific colliding documents, and it can be done for more work than finding a collision, but much less than would be expected to do this for a random oracle.[7][8]
Length extension: Given the hash
{\displaystyle H(X)}
of an unknown input X, it is easy to find the value of
{\displaystyle H({\mathsf {Pad}}(X)\|Y)}
, where pad is the padding function of the hash. That is, it is possible to find hashes of inputs related to X even though X remains unknown.[9] Length extension attacks were actually used to attack a number of commercial web message authentication schemes such as one used by Flickr.[10]
The Wide pipe hash construction. The intermediate chaining values have been doubled.
Due to several structural weaknesses of Merkle–Damgård construction, especially the length extension problem and multicollision attacks, Stefan Lucks proposed the use of the wide-pipe hash[11] instead of Merkle–Damgård construction. The wide-pipe hash is very similar to the Merkle–Damgård construction but has a larger internal state size, meaning that the bit-length that is internally used is larger than the output bit-length. If a hash of n bits is desired, the compression function f takes 2n bits of chaining value and m bits of the message and compresses this to an output of 2n bits.
The Fast wide pipe hash construction. Half of chaining value is used in the compression function.
It has been demonstrated by Mridul Nandi and Souradyuti Paul that the Widepipe hash function can be made approximately twice as fast if the widepipe state can be divided in half in the following manner: one half is input to the succeeding compression function while the other half is combined with the output of that compression function.[12]
As mentioned in the introduction, the padding scheme used in the Merkle–Damgård construction must be chosen carefully to ensure the security of the scheme. Mihir Bellare gives sufficient conditions for a padding scheme to possess to ensure that the MD construction is secure: it suffices that the scheme be "MD-compliant" (the original length-padding scheme used by Merkle is an example of MD-compliant padding).[1]: 145 Conditions:
{\displaystyle M}
{\displaystyle {\mathsf {Pad}}(M).}
{\displaystyle |M_{1}|=|M_{2}|}
{\displaystyle |{\mathsf {Pad}}(M_{1})|=|{\mathsf {Pad}}(M_{2})|.}
{\displaystyle |M_{1}|\neq |M_{2}|}
then the last block of
{\displaystyle {\mathsf {Pad}}(M_{1})}
is different from the last block of
{\displaystyle {\mathsf {Pad}}(M_{2}).}
{\displaystyle |X|}
{\displaystyle X}
. With these conditions in place, we find a collision in the MD hash function exactly when we find a collision in the underlying compression function. Therefore, the Merkle–Damgård construction is provably secure when the underlying compression function is secure.[1]: 147
To be able to feed the message to the compression function, the last block needs to be padded with constant data (generally with zeroes) to a full block. For example, suppose the message to be hashed is "HashInput" (9 octet string, 0x48617368496e707574 in ASCII) and the block size of the compression function is 8 bytes (64 bits). We get two blocks (the padding octets shown with lightblue background color):
48 61 73 68 49 6e 70 75, 74 00 00 00 00 00 00 00
This implies that other messages having the same content but ending with additional zeros at the end will result in the same hash value. In the above example, another almost identical message (0x48617368496e7075 7400) will generate the same hash value as the original message "HashInput" above. In other words, any message having extra zeros at the end makes it indistinguishable with the one without them. To prevent this situation, the first bit of the first padding octet is changed to "1" (0x80), yielding:
To make it resistant against the length extension attack, the message length is added in an extra block at the end (shown with yellow background color):
48 61 73 68 49 6e 70 75, 74 80 00 00 00 00 00 00, 00 00 00 00 00 00 00 09
However, most common implementations use a fixed bit-size (generally 64 or 128 bits in modern algorithms) at a fixed position at the end of the last block for inserting the message length value (see SHA-1 pseudocode). Further improvement can be made by inserting the length value in the last block if there is enough space. Doing so avoids having an extra block for the message length. If we assume the length value is encoded on 5 bytes (40 bits), the message becomes:
Note that storing the message length out-of-band in metadata, or otherwise embedded at the start of the message is an effective mitigation of the length extension attack[citation needed], as long as invalidation of either the message length and checksum are both considered failure of integrity checking.
Handbook of Applied Cryptography by Menezes, van Oorschot and Vanstone (2001), chapter 9.
Introduction to Modern Cryptography, by Jonathan Katz and Yehuda Lindell. Chapman and Hall/CRC Press, August 2007, page 134 (construction 4.13).
Cryptography Made Simple by Nigel Smart (2015), chapter 14.
^ a b c d Goldwasser, S. and Bellare, M. "Lecture Notes on Cryptography". Summer course on cryptography, MIT, 1996-2001
^ R.C. Merkle. Secrecy, authentication, and public key systems. Stanford Ph.D. thesis 1979, pages 13-15.
^ R.C. Merkle. A Certified Digital Signature. In Advances in Cryptology - CRYPTO '89 Proceedings, Lecture Notes in Computer Science Vol. 435, G. Brassard, ed, Springer-Verlag, 1989, pp. 218-238.
^ I. Damgård. A Design Principle for Hash Functions. In Advances in Cryptology – CRYPTO '89 Proceedings, Lecture Notes in Computer Science Vol. 435, G. Brassard, ed, Springer-Verlag, 1989, pp. 416-427.
^ Kelsey, John; Schneier, Bruce (2004). "Second Preimages on n-bit Hash Functions for Much Less than 2^n Work" (PDF) – via Cryptology ePrint Archive: Report 2004/304. {{cite journal}}: Cite journal requires |journal= (help)
^ Antoine Joux. Multicollisions in iterated hash functions. Application to cascaded construction. In Advances in Cryptology - CRYPTO '04 Proceedings, Lecture Notes in Computer Science, Vol. 3152, M. Franklin, ed, Springer-Verlag, 2004, pp. 306–316.
^ John Kelsey and Tadayoshi Kohno. Herding Hash Functions and the Nostradamus Attack In Eurocrypt 2006, Lecture Notes in Computer Science, Vol. 4004, pp. 183–200.
^ Stevens, Marc; Lenstra, Arjen; de Weger, Benne (2007-11-30). "Nostradamus". The HashClash Project. TU/e. Retrieved 2013-03-30.
^ Yevgeniy Dodis, Thomas Ristenpart, Thomas Shrimpton. Salvaging Merkle–Damgård for Practical Applications. Preliminary version in Advances in Cryptology - EUROCRYPT '09 Proceedings, Lecture Notes in Computer Science Vol. 5479, A. Joux, ed, Springer-Verlag, 2009, pp. 371–388.
^ Thai Duong, Juliano Rizzo, Flickr's API Signature Forgery Vulnerability, 2009
^ Lucks, Stefan (2004). "Design Principles for Iterated Hash Functions" – via Cryptology ePrint Archive, Report 2004/253. {{cite journal}}: Cite journal requires |journal= (help)
^ Mridul Nandi and Souradyuti Paul. Speeding Up the Widepipe: Secure and Fast Hashing. In Guang Gong and Kishan Gupta, editor, Indocrypt 2010, Springer, 2010.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Merkle–Damgård_construction&oldid=1081786995"
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Mason–Weaver equation - Wikipedia
Mason–Weaver equation
The Mason–Weaver equation (named after Max Mason and Warren Weaver) describes the sedimentation and diffusion of solutes under a uniform force, usually a gravitational field.[1] Assuming that the gravitational field is aligned in the z direction (Fig. 1), the Mason–Weaver equation may be written
{\displaystyle {\frac {\partial c}{\partial t}}=D{\frac {\partial ^{2}c}{\partial z^{2}}}+sg{\frac {\partial c}{\partial z}}}
where t is the time, c is the solute concentration (moles per unit length in the z-direction), and the parameters D, s, and g represent the solute diffusion constant, sedimentation coefficient and the (presumed constant) acceleration of gravity, respectively.
The Mason–Weaver equation is complemented by the boundary conditions
{\displaystyle D{\frac {\partial c}{\partial z}}+sgc=0}
at the top and bottom of the cell, denoted as
{\displaystyle z_{a}}
{\displaystyle z_{b}}
, respectively (Fig. 1). These boundary conditions correspond to the physical requirement that no solute pass through the top and bottom of the cell, i.e., that the flux there be zero. The cell is assumed to be rectangular and aligned with the Cartesian axes (Fig. 1), so that the net flux through the side walls is likewise zero. Hence, the total amount of solute in the cell
{\displaystyle N_{\text{tot}}=\int _{z_{b}}^{z_{a}}\,dz\ c(z,t)}
is conserved, i.e.,
{\displaystyle dN_{\text{tot}}/dt=0}
1 Derivation of the Mason–Weaver equation
2 The dimensionless Mason–Weaver equation
3 Solution of the Mason–Weaver equation
Derivation of the Mason–Weaver equationEdit
Figure 1: Diagram of Mason–Weaver cell and Forces on Solute
A typical particle of mass m moving with vertical velocity v is acted upon by three forces (Fig. 1): the drag force
{\displaystyle fv}
, the force of gravity
{\displaystyle mg}
and the buoyant force
{\displaystyle \rho Vg}
, where g is the acceleration of gravity, V is the solute particle volume and
{\displaystyle \rho }
is the solvent density. At equilibrium (typically reached in roughly 10 ns for molecular solutes), the particle attains a terminal velocity
{\displaystyle v_{\text{term}}}
where the three forces are balanced. Since V equals the particle mass m times its partial specific volume
{\displaystyle {\bar {\nu }}}
, the equilibrium condition may be written as
{\displaystyle fv_{\text{term}}=m(1-{\bar {\nu }}\rho )g\ {\stackrel {\mathrm {def} }{=}}\ m_{b}g}
{\displaystyle m_{b}}
is the buoyant mass.
We define the Mason–Weaver sedimentation coefficient
{\displaystyle s\ {\stackrel {\mathrm {def} }{=}}\ m_{b}/f=v_{\text{term}}/g}
. Since the drag coefficient f is related to the diffusion constant D by the Einstein relation
{\displaystyle D={\frac {k_{B}T}{f}}}
the ratio of s and D equals
{\displaystyle {\frac {s}{D}}={\frac {m_{b}}{k_{B}T}}}
{\displaystyle k_{B}}
is the Boltzmann constant and T is the temperature in kelvins.
The flux J at any point is given by
{\displaystyle J=-D{\frac {\partial c}{\partial z}}-v_{\text{term}}c=-D{\frac {\partial c}{\partial z}}-sgc.}
The first term describes the flux due to diffusion down a concentration gradient, whereas the second term describes the convective flux due to the average velocity
{\displaystyle v_{\text{term}}}
of the particles. A positive net flux out of a small volume produces a negative change in the local concentration within that volume
{\displaystyle {\frac {\partial c}{\partial t}}=-{\frac {\partial J}{\partial z}}.}
Substituting the equation for the flux J produces the Mason–Weaver equation
{\displaystyle {\frac {\partial c}{\partial t}}=D{\frac {\partial ^{2}c}{\partial z^{2}}}+sg{\frac {\partial c}{\partial z}}.}
The dimensionless Mason–Weaver equationEdit
The parameters D, s and g determine a length scale
{\displaystyle z_{0}}
{\displaystyle z_{0}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {D}{sg}}}
and a time scale
{\displaystyle t_{0}}
{\displaystyle t_{0}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {D}{s^{2}g^{2}}}}
Defining the dimensionless variables
{\displaystyle \zeta \ {\stackrel {\mathrm {def} }{=}}\ z/z_{0}}
{\displaystyle \tau \ {\stackrel {\mathrm {def} }{=}}\ t/t_{0}}
, the Mason–Weaver equation becomes
{\displaystyle {\frac {\partial c}{\partial \tau }}={\frac {\partial ^{2}c}{\partial \zeta ^{2}}}+{\frac {\partial c}{\partial \zeta }}}
{\displaystyle {\frac {\partial c}{\partial \zeta }}+c=0}
at the top and bottom of the cell,
{\displaystyle \zeta _{a}}
{\displaystyle \zeta _{b}}
Solution of the Mason–Weaver equationEdit
This partial differential equation may be solved by separation of variables. Defining
{\displaystyle c(\zeta ,\tau )\ {\stackrel {\mathrm {def} }{=}}\ e^{-\zeta /2}T(\tau )P(\zeta )}
, we obtain two ordinary differential equations coupled by a constant
{\displaystyle \beta }
{\displaystyle {\frac {dT}{d\tau }}+\beta T=0}
{\displaystyle {\frac {d^{2}P}{d\zeta ^{2}}}+\left[\beta -{\frac {1}{4}}\right]P=0}
where acceptable values of
{\displaystyle \beta }
are defined by the boundary conditions
{\displaystyle {\frac {dP}{d\zeta }}+{\frac {1}{2}}P=0}
at the upper and lower boundaries,
{\displaystyle \zeta _{a}}
{\displaystyle \zeta _{b}}
, respectively. Since the T equation has the solution
{\displaystyle T(\tau )=T_{0}e^{-\beta \tau }}
{\displaystyle T_{0}}
is a constant, the Mason–Weaver equation is reduced to solving for the function
{\displaystyle P(\zeta )}
The ordinary differential equation for P and its boundary conditions satisfy the criteria for a Sturm–Liouville problem, from which several conclusions follow. First, there is a discrete set of orthonormal eigenfunctions
{\displaystyle P_{k}(\zeta )}
that satisfy the ordinary differential equation and boundary conditions. Second, the corresponding eigenvalues
{\displaystyle \beta _{k}}
are real, bounded below by a lowest eigenvalue
{\displaystyle \beta _{0}}
and grow asymptotically like
{\displaystyle k^{2}}
where the nonnegative integer k is the rank of the eigenvalue. (In our case, the lowest eigenvalue is zero, corresponding to the equilibrium solution.) Third, the eigenfunctions form a complete set; any solution for
{\displaystyle c(\zeta ,\tau )}
can be expressed as a weighted sum of the eigenfunctions
{\displaystyle c(\zeta ,\tau )=\sum _{k=0}^{\infty }c_{k}P_{k}(\zeta )e^{-\beta _{k}\tau }}
{\displaystyle c_{k}}
are constant coefficients determined from the initial distribution
{\displaystyle c(\zeta ,\tau =0)}
{\displaystyle c_{k}=\int _{\zeta _{a}}^{\zeta _{b}}d\zeta \ c(\zeta ,\tau =0)e^{\zeta /2}P_{k}(\zeta )}
{\displaystyle \beta =0}
(by definition) and the equilibrium concentration distribution is
{\displaystyle e^{-\zeta /2}P_{0}(\zeta )=Be^{-\zeta }=Be^{-m_{b}gz/k_{B}T}}
which agrees with the Boltzmann distribution. The
{\displaystyle P_{0}(\zeta )}
function satisfies the ordinary differential equation and boundary conditions at all values of
{\displaystyle \zeta }
(as may be verified by substitution), and the constant B may be determined from the total amount of solute
{\displaystyle B=N_{\text{tot}}\left({\frac {sg}{D}}\right)\left({\frac {1}{e^{-\zeta _{b}}-e^{-\zeta _{a}}}}\right)}
To find the non-equilibrium values of the eigenvalues
{\displaystyle \beta _{k}}
, we proceed as follows. The P equation has the form of a simple harmonic oscillator with solutions
{\displaystyle P(\zeta )=e^{i\omega _{k}\zeta }}
{\displaystyle \omega _{k}=\pm {\sqrt {\beta _{k}-{\frac {1}{4}}}}}
Depending on the value of
{\displaystyle \beta _{k}}
{\displaystyle \omega _{k}}
is either purely real (
{\displaystyle \beta _{k}\geq {\frac {1}{4}}}
) or purely imaginary (
{\displaystyle \beta _{k}<{\frac {1}{4}}}
). Only one purely imaginary solution can satisfy the boundary conditions, namely, the equilibrium solution. Hence, the non-equilibrium eigenfunctions can be written as
{\displaystyle P(\zeta )=A\cos {\omega _{k}\zeta }+B\sin {\omega _{k}\zeta }}
{\displaystyle \omega }
is real and strictly positive.
By introducing the oscillator amplitude
{\displaystyle \rho }
{\displaystyle \varphi }
as new variables,
{\displaystyle u\ {\stackrel {\mathrm {def} }{=}}\ \rho \sin(\varphi )\ {\stackrel {\mathrm {def} }{=}}\ P}
{\displaystyle v\ {\stackrel {\mathrm {def} }{=}}\ \rho \cos(\varphi )\ {\stackrel {\mathrm {def} }{=}}\ -{\frac {1}{\omega }}\left({\frac {dP}{d\zeta }}\right)}
{\displaystyle \rho \ {\stackrel {\mathrm {def} }{=}}\ u^{2}+v^{2}}
{\displaystyle \tan(\varphi )\ {\stackrel {\mathrm {def} }{=}}\ v/u}
the second-order equation for P is factored into two simple first-order equations
{\displaystyle {\frac {d\rho }{d\zeta }}=0}
{\displaystyle {\frac {d\varphi }{d\zeta }}=\omega }
Remarkably, the transformed boundary conditions are independent of
{\displaystyle \rho }
and the endpoints
{\displaystyle \zeta _{a}}
{\displaystyle \zeta _{b}}
{\displaystyle \tan(\varphi _{a})=\tan(\varphi _{b})={\frac {1}{2\omega _{k}}}}
Therefore, we obtain an equation
{\displaystyle \varphi _{a}-\varphi _{b}+k\pi =k\pi =\int _{\zeta _{b}}^{\zeta _{a}}d\zeta \ {\frac {d\varphi }{d\zeta }}=\omega _{k}(\zeta _{a}-\zeta _{b})}
giving an exact solution for the frequencies
{\displaystyle \omega _{k}}
{\displaystyle \omega _{k}={\frac {k\pi }{\zeta _{a}-\zeta _{b}}}}
The eigenfrequencies
{\displaystyle \omega _{k}}
are positive as required, since
{\displaystyle \zeta _{a}>\zeta _{b}}
, and comprise the set of harmonics of the fundamental frequency
{\displaystyle \omega _{1}\ {\stackrel {\mathrm {def} }{=}}\ \pi /(\zeta _{a}-\zeta _{b})}
. Finally, the eigenvalues
{\displaystyle \beta _{k}}
can be derived from
{\displaystyle \omega _{k}}
{\displaystyle \beta _{k}=\omega _{k}^{2}+{\frac {1}{4}}}
Taken together, the non-equilibrium components of the solution correspond to a Fourier series decomposition of the initial concentration distribution
{\displaystyle c(\zeta ,\tau =0)}
multiplied by the weighting function
{\displaystyle e^{\zeta /2}}
. Each Fourier component decays independently as
{\displaystyle e^{-\beta _{k}\tau }}
{\displaystyle \beta _{k}}
is given above in terms of the Fourier series frequencies
{\displaystyle \omega _{k}}
The Archibald approach, and a simpler presentation of the basic physics of the Mason–Weaver equation than the original.[2]
^ Mason, M; Weaver W (1924). "The Settling of Small Particles in a Fluid". Physical Review. 23: 412–426. Bibcode:1924PhRv...23..412M. doi:10.1103/PhysRev.23.412.
^ Archibald, William J. (1938-05-01). "The Process of Diffusion in a Centrifugal Field of Force". Physical Review. American Physical Society (APS). 53 (9): 746–752. doi:10.1103/physrev.53.746. ISSN 0031-899X.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Mason–Weaver_equation&oldid=1032620870"
|
Implement three-phase transmission line section with lumped parameters - Simulink - MathWorks Nordic
Three-Phase PI Section Line
Implement three-phase transmission line section with lumped parameters
The Three-Phase PI Section Line block implements a balanced three-phase transmission line model with parameters lumped in a PI section.
Contrary to the Distributed Parameters Line model where the resistance, inductance, and capacitance are uniformly distributed along the line, the Three-Phase PI Section Line block lumps the line parameters in a single PI section as shown in the figure below.
The line parameters R, L, and C are specified as positive- and zero-sequence parameters that take into account the inductive and capacitive couplings between the three phase conductors, as well as the ground parameters. This method of specifying line parameters assumes that the three phases are balanced.
The self and mutual resistances (Rs, Rm), self and mutual inductances (Ls, Lm) of the three coupled inductors, as well as phase capacitances Cp and ground capacitances Cg, are deduced from the positive- and zero-sequence RLC parameters as follows.
Let us assume the following line parameters:
r1, r0 Positive- and zero-sequence resistances per unit length (Ω/km)
l1, l0 Positive- and zero-sequence inductances per unit length (H/km)
c1, c0 Positive- and zero-sequence capacitances per unit length (F/km)
lsec Line section length (km)
The total positive- and zero-sequence RLC parameters including hyperbolic corrections are first evaluated:
\begin{array}{l}{R}_{1}={r}_{1}\cdot l\mathrm{sec}\cdot {k}_{r1}\\ {L}_{1}={l}_{1}\cdot l\mathrm{sec}\cdot {k}_{l1}\\ {C}_{1}={c}_{1}\cdot l\mathrm{sec}\cdot {k}_{c1}\\ {R}_{0}={r}_{0}\cdot l\mathrm{sec}\cdot {k}_{r0}\\ {L}_{0}={l}_{0}\cdot l\mathrm{sec}\cdot {k}_{l0}\\ {C}_{0}={c}_{0}\cdot l\mathrm{sec}\cdot {k}_{c0}\end{array}
kr1, kl1, kc1, kr0, kl0, kc0 — positive-sequence and zero-sequence hyperbolic correction factors
See the PI Section Line block reference page for an explanation on how to compute RLC parameters taking into account hyperbolic corrections.
The Powergui block provides a graphical tool for the calculation of resistance, inductance, and capacitance per unit length based on the line geometry and the conductor characteristics. See the power_lineparam reference page to learn how to use this tool.
For a short line section (approximately lsec < 50 km), these correction factors are negligible (close to unity). However, for long lines, these hyperbolic corrections must be taken into account in order to get an exact line model at the specified frequency.
The RLC line section parameters are then computed as follows:
\begin{array}{l}Rs=\left(2{R}_{1}+{R}_{0}\right)/3\\ Ls=\left(2{L}_{1}+{L}_{0}\right)/3\\ {R}_{m}=\left({R}_{0}-{R}_{1}\right)/3\\ {L}_{m}=\left({L}_{0}-{L}_{1}\right)/3\\ {C}_{p}={C}_{1}\\ {C}_{g}=3{C}_{1}{C}_{0}/\left({C}_{1}-{C}_{0}\right)\end{array}
Frequency used for rlc specification
The frequency used for specification of per unit length rlc line parameters, in hertz (Hz). This is usually the nominal system frequency (50 Hz or 60 Hz). Default is 60.
Positive- and zero-sequence resistances
The positive- and zero-sequence resistances in ohms/kilometer (Ω/km). Default is [ 0.01273 0.3864].
Positive- and zero-sequence inductances
The positive- and zero-sequence inductances in henries/kilometer (H/km). The zero-sequence inductance cannot be zero, because it would result in an invalid propagation speed computation. Default is [ 0.9337e-3 4.1264e-3].
Positive- and zero-sequence capacitances
The positive- and zero-sequence capacitances in farads/kilometer (F/km). The zero-sequence capacitance cannot be zero, because it would result in an invalid propagation speed computation. Default is [12.74e-9 7.751e-9].
Line section length
The line section length in kilometers (km). Default is 100.
The power_triphaseline example illustrates voltage transients at the receiving end of a 200-km line when only phase A is energized. Voltages obtained with two line models are compared: 1) the Distributed Parameters Line block and 2) a PI line model using two Three-Phase PI Section Line blocks.
Distributed Parameters Line, PI Section Line
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azdavis.net • Posts • May 22, 2021
In the previous post, we introduced Hatsugen, a small programming language with integer and boolean types.
In this post, we'll add functions to Hatsugen. With this addition, Hatsugen becomes approximately as powerful as the simply-typed lambda calculus.
We update the expression syntax to add variables, function literals, and function application. We also update the type syntax to add function types.
For variables, we just write things like
x
y
x_1
x'
. There are infinitely many variable names to choose from.
\lambda (x: \tau) \ e
is a function literal, taking one argument
x
\tau
e
when applied to an argument.
x
may appear in the expression
e
e_1(e_2)
is an application expression, representing the application of the function
e_1
to the argument
e_2
\tau_1 \rightarrow \tau_2
is the type of functions taking
\tau_1
as input and returning
\tau_2
\begin{aligned} \tau ::= \ & \dots \\ | \ & \tau_1 \rightarrow \tau_2 \\ \\ e ::= \ & \dots \\ | \ & x \\ | \ & \lambda (x: \tau) \ e \\ | \ & e_1(e_2) \end{aligned}
Statics:
\Gamma \vdash e: \tau
\lambda (x: \mathsf{Int}) \ x
. It should have type
\mathsf{Int} \rightarrow \mathsf{Int}
The very similar expression
\lambda (x: \mathsf{Bool}) \ x
should have type
\mathsf{Bool} \rightarrow \mathsf{Bool}
Notice that both of these example expressions contain the sub-expression
x
, as the body of the function. But in each, the type of
x
is different:
\mathsf{Int}
in the first and
\mathsf{Bool}
As this example illustrates, the type of variables is determined by how the variable is declared. This is the first time that the same expression (in this case
x
) can have a different type depending on the context.
We'll need to fundamentally change the structure of the static semantics to account for variables.
First, we'll need to formalize the notion of a context. We'll use
\Gamma
to represent a context, which will just be a list of variable-type pairs.
A context can either be empty, or it can be an existing context augmented with a new variable-type pair.
\begin{aligned} \Gamma ::= \ & \cdot \\ | \ & \Gamma, x: \tau \end{aligned}
The old typing judgement was
e: \tau
, read "
e
\tau
". The new judgement is written
\Gamma \vdash e: \tau
\Gamma
e
\tau
We'll need to update all the statics rules from the first post. All of these rules just use the context without changing it.
\frac {} {\Gamma \vdash \overline{n}: \mathsf{Int}}
\frac {} {\Gamma \vdash \mathsf{true}: \mathsf{Bool}}
\frac {} {\Gamma \vdash \mathsf{false}: \mathsf{Bool}}
\frac { \Gamma \vdash e_1: \mathsf{Bool} \hspace{1em} \Gamma \vdash e_2: \tau \hspace{1em} \Gamma \vdash e_3: \tau } { \Gamma \vdash \mathsf{if} \ e_1 \ \mathsf{then} \ e_2 \ \mathsf{else} \ e_3: \tau }
We can now add the typing rules for the new constructs.
For variables, we need to be able to look up the variable in question in the context to get its type. For that, we'll define a small helper judgement for context lookup, written
\Gamma(x) = \tau
\frac {} {(\Gamma, x: \tau)(x) = \tau}
\frac { x \ne y \hspace{1em} \Gamma(x) = \tau } {(\Gamma, y: \tau')(x) = \tau}
Note that these rules engender shadowing, which is where multiple variable-type pairs with the same variable exist in the context at once, but the one furthest to the right in the context determines the type of the variable under the context. For instance, using the rules we can derive
(\cdot, x: \mathsf{Int}, x: \mathsf{Bool}, y: \mathsf{Int})(x) = \mathsf{Bool}
We can now use this helper lookup judgement in the typing rule for variables.
\frac {\Gamma(x) = \tau} {\Gamma \vdash x: \tau}
For function literals, the type of the parameter is the input type, and the type of the function body is the output type.
However, the bound variable may appear in the function body. So we add the variable and its type to the context when determining the function body's type.
\frac {\Gamma, x: \tau_1 \vdash e: \tau_2} {\Gamma \vdash \lambda (x: \tau_1) \ e: \tau_1 \rightarrow \tau_2}
For application, the parameter and argument types must match.
\frac { \Gamma \vdash e_1: \tau_1 \rightarrow \tau_2 \hspace{1em} \Gamma \vdash e_2: \tau_1 } {\Gamma \vdash e_1(e_2): \tau_2}
Dynamics:
e \ \mathsf{val}
e \mapsto e'
Functions are values, regardless of whether the function body is a value:
\frac {} {\lambda (x: \tau) \ e \ \mathsf{val}}
For application expressions, we first step the function expression. Then, once it's a value, we step the argument to a value as well:
\frac {e_1 \mapsto e_1'} {e_1(e_2) \mapsto e_1'(e_2)}
\frac { e_1 \ \mathsf{val} \hspace{1em} e_2 \mapsto e_2' } {e_1(e_2) \mapsto e_1(e_2')}
Once both expressions are values, we can prove the first will be a function literal, since it had function type.
We will step into the body of the function, substituting all free occurrences of the variable bound by the function with the value of the argument. We write
[x \mapsto e_x] e = e'
to mean "substituting all free occurrences of
x
e_x
e
e'
\frac { e_2 \ \mathsf{val} \hspace{1em} [x \mapsto e_2] e = e' } { (\lambda (x: \tau) \ e) \ e_2 \mapsto e' }
[x \mapsto e_x] e = e'
To define the dynamics for function application, we must now define substitution for expressions.
Substitution does nothing to integer and boolean literals:
\frac {} {[x \mapsto e_x] \overline{n} = \overline{n}}
\frac {} {[x \mapsto e_x] \mathsf{true} = \mathsf{true}}
\frac {} {[x \mapsto e_x] \mathsf{false} = \mathsf{false}}
For conditional and application expressions, we simply recurse on the sub-expressions:
\frac { \begin{aligned} &[x \mapsto e_x] e_1 = e_1' \hspace{1em} \\&[x \mapsto e_x] e_2 = e_2' \hspace{1em} \\&[x \mapsto e_x] e_3 = e_3' \end{aligned} } { \begin{aligned} [x \mapsto e_x] &\mathsf{if} \ e_1 \ \mathsf{then} \ e_2 \ \mathsf{else} \ e_3 = \\&\mathsf{if} \ e_1' \ \mathsf{then} \ e_2' \ \mathsf{else} \ e_3' \end{aligned} }
\frac { [x \mapsto e_x] e_1 = e_1' \hspace{1em} [x \mapsto e_x] e_2 = e_2' } { [x \mapsto e_x] e_1(e_2) = e_1'(e_2') }
For variables, we case on whether the variable in the expression is the variable being substituted. If it is, we replace the variable with
e_x
. If not, we leave it alone.
\frac {} {[x \mapsto e_x] x = e_x}
\frac {x \ne y} {[x \mapsto e_x] y = y}
For function literals, we again case on whether the variables are the same. If they are, we leave the function literal untouched. This is consistent with how we treat variables with the same name in the context
\Gamma
\frac {} {[x \mapsto e_x] \lambda (x: \tau) \ e = \lambda (x: \tau) \ e}
For the case when the variables are different, we must take care to avoid variable capture. For example, if we define
\frac { x \ne y \hspace{1em} [x \mapsto e_x] e = e' } {[x \mapsto e_x] \lambda (y: \tau) \ e = \lambda (y: \tau) \ e'}
we can use the rules to prove that
[x \mapsto y] \lambda (y: \mathsf{Bool}) \ x = \lambda (y: \mathsf{Bool}) \ y
In this case, the variable
y
has been captured by the binding for
y
in the function literal.
To avoid this, we require that the variable bound by the function literal not appear free in
e_x
. We thus revise the rule, writing
\mathsf{fv}(e_x)
to denote the free variables in
e_x
\frac { x \ne y \hspace{1em} y \notin \mathsf{fv}(e_x) \hspace{1em} [x \mapsto e_x] e = e' } {[x \mapsto e_x] \lambda (y: \tau) \ e = \lambda (y: \tau) \ e'}
Free variables:
\mathsf{fv}(e)
We must now define the free variables of an expression.
Integer and boolean literals have no free variables:
\frac {} {\mathsf{fv}(\overline{n}) = \emptyset}
\frac {} {\mathsf{fv}(\mathsf{true}) = \emptyset}
\frac {} {\mathsf{fv}(\mathsf{false}) = \emptyset}
For conditional and application expressions, we simply recurse and union:
\frac { \mathsf{fv}(e_1) = s_1 \hspace{1em} \mathsf{fv}(e_2) = s_2 \hspace{1em} \mathsf{fv}(e_3) = s_3 } { \mathsf{fv}(\mathsf{if} \ e_1 \ \mathsf{then} \ e_2 \ \mathsf{else} \ e_3) = s_1 \cup s_2 \cup s_3 }
\frac { \mathsf{fv}(e_1) = s_1 \hspace{1em} \mathsf{fv}(e_2) = s_2 } { \mathsf{fv}(e_1(e_2)) = s_1 \cup s_2 }
Variables alone are free:
\frac {} {\mathsf{fv}(x) = \{ x \}}
Function literals bind a single variable:
\frac {\mathsf{fv}(e) = s} {\mathsf{fv}(\lambda (x: \tau) \ e) = \{ x \} \setminus s}
Since the judgments have changed a bit, we'll need to restate the theorems slightly.
\Gamma
e
\tau
\Gamma \vdash e: \tau
\mathsf{fv}(e) = \emptyset
e \ \mathsf{val}
e'
e \mapsto e'
We require that
e
have no free variables at all. This will allow us to perform substitution as we evaluate.
To see why this is necessary, consider choosing
e
(\lambda (x: \mathsf{Bool} \rightarrow \mathsf{Bool}) \ x) \ (\lambda (y: \mathsf{Bool}) \ x)
This expression has one free variable,
x
. According to the rules, this expression is neither a value nor can it step. This is because
x
appears free in the argument but is bound by the function being applied.
x
is an even smaller counterexample, since bare variables are neither values nor can they step. Variables are given meaning by substitution.
\Gamma
e
e'
\tau
\Gamma \vdash e: \tau
\mathsf{fv}(e) = \emptyset
e \mapsto e'
\Gamma \vdash e': \tau
\mathsf{fv}(e') = \emptyset
Preservation not only preserves typing, but also the presence of free variables. This is important, since we need there to be no free variables in
e'
in order to be able to feed
e'
back into the progress theorem.
Once again, taken together we have the following safety theorem:
\Gamma
e
\tau
\Gamma \vdash e: \tau
\mathsf{fv}(e) = \emptyset
e \ \mathsf{val}
e'
e \mapsto e'
\Gamma \vdash e': \tau
\mathsf{fv}(e') = \emptyset
In the next post, we'll add product types, also known as structs or tuples.
|
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\frac{1}{x-4}+\frac{2}{x^2-16}=\frac{3}{x+4}
\frac{1}{x-4}+\frac{2}{\left(x-4\right)\left(x+4\right)}=\frac{3}{x+4}
\frac{1}{x-4}\cdot\left(\frac{x+4}{x+4}\right)+\frac{2}{\left(x-4\right)\left(x+4\right)}=\frac{3}{x+4}\cdot\left(\frac{x-4}{x-4}\right)
\frac{1\left(x+4\right)}{\left(x-4\right)\left(x+4\right)}+\frac{2}{\left(x-4\right)\left(x+4\right)}=\frac{3\left(x-4\right)}{\left(x+4\right)\left(x-4\right)}
1\left(x+4\right)+2=3\left(x-4\right)
x+6=3x-12
x+18=3x
18=2x
9=x
\int x\ln xdx
u=\ln x
dv=xdx
du=\frac{1}{x}dx
v=\frac{x^2}{2}
\int x\ln sdx=\frac{x^2}{2}\ln x-\int\frac{x^2}{2}\cdot\frac{1}{x}dx
\frac{x^2}{2}\ln x-\frac{1}{2}\int xdx
\frac{x^2}{2}\ln x-\frac{1}{2}\left(\frac{x^2}{2}\right)+c
\frac{x^2}{2}\ln x-\frac{1}{4}x^2+c
\text{A ball is thrown from 1 m above the ground.}
\text{It is given an initial velocity of 20 m/s}
\text{At an angle of 40 degrees above the horizontal}
\text{Find the maximum height reached}
\text{And velocity at that point}
x\left(t\right)=v\cos\left(\theta\right)t=20\cos\left(40\right)t=15.3t
y\left(t\right)=y_0+v\sin\left(\theta\right)t-\frac{9.8t^2}{2}
y\left(t\right)=1+20\sin\left(40\right)t-4.9t^2
y\left(t\right)=1+12.9t-4.9t^2
v_y\left(t\right)=v\sin\left(\theta\right)-9.8t
v_y\left(t\right)=12.9-9.8t
\max\ height\ at\ v_y\left(t\right)=0
12.9-9.8t=0
-9.8t=-12.9
t=\frac{-12.9}{-9.8}=1.3
y\left(1.3\right)=1+12.9\left(1.3\right)-4.9\left(1.3\right)^2
y\left(1.3\right)=9.5\ m
y\ component\ of\ velocity\ is\ 0\ at\ highest\ pt
total\ velocity\ =v_x=15.3\ \frac{m}{s}
|
Compute RLC parameters of overhead transmission line from its conductor characteristics and tower geometry - MATLAB - MathWorks Nordic
Power Line Parameters
Open the Power Line Parameters App
Load > Typical parameters
Load > User parameters
Phase conductors (bundles)
Ground conductors (bundles)
Phase, Phase Number
Internal conductor inductance evaluated from
Include conductor skin effect
DC res
mu_r
Nb_cond
Frequency-Dependent Line Parameters
Frequency range logspace
Frequency Dependent Model Parameters
Send Sequences to block
Send Frequency-Dependent Parameters to block
Compute RLC parameters of overhead transmission line from its conductor characteristics and tower geometry
The Power Line Parameters app provides a tool to compute the RLC line parameters of the Distributed Parameters Line and PI Section Line blocks and the frequency-dependent parameters of a Distributed Parameters Line (Frequency-Dependent) block. The tool uses the power_lineparam function to compute the line parameters based on the geometry of the line and the type of conductors.
powergui Block Parameters dialog box: On the Tools tab, click Power Line Parameters.
MATLAB® command prompt: Enter powerLineParameters
Comments — Custom comments
Use this text box to type comments that you want to save with the line parameters, for example, voltage level, conductor types and characteristics, etc.
Load > Typical parameters — Load typical parameters
Opens a browser window where you can select examples of line configurations provided with Simscape™ Electrical™ Specialized Power Systems software. Select the desired .mat file.
Selecting Load typical parameters allows you to load one of the following line configurations:
Line_25kV_4wires.mat 25-kV, three-phase distribution feeder with accessible neutral conductor.
Line_315kV_2circ.mat 315-kV, three-phase, double-circuit line using bundles of two conductors. Phase numbering is set to obtain the RLC parameters of the two individual circuits (six-phase line).
Line_450kV.mat Bipolar +/−450-kV DC line using bundles of four conductors.
Line_500kV_2circ.mat 500-kV, three-phase, double-circuit line using bundles of three conductors. Phase numbering is set to obtain the RLC parameters of the three-phase line circuit equivalent to the two circuits connected in parallel.
Line_735kV.mat 735-kV, three-phase line using bundles of four conductors.
Load > User parameters — Load user parameters
Opens a browser window where you can select your own line data. Select the desired .mat file.
Save — Save line data
Saves your line data by generating a .mat file that contains the GUI information and the line data.
Report — Create report
Creates a file containing the line input parameters and the computed RLC parameters. The MATLAB Editor opens to display the contents of the file.
Units — Conductor diameter, GMR, and bundle diameter units
english (default) | metric
Select metric to specify conductor diameter, GMR, and bundle diameter in centimeters and conductor positions in meters. Select english to specify conductor diameter, GMR, and bundle diameter in inches and conductor positions in feet.
Specify the ground resistivity, in ohm-meters. A zero value (perfectly conducting ground) is allowed.
Nominal Frequency — Frequency to evaluate RLC parameters
Specify the frequency, in hertz, to evaluate RLC parameters.
Phase conductors (bundles) — Number of phase conductors (bundles)
Specify the number of phase conductors (single conductors or bundles of subconductors).
Ground conductors (bundles) — Number of ground wires (bundles)
Specify the number of ground wires (single conductors or bundles of subconductors). Ground wires are not usually bundled.
Label — Conductor or bundle identifiers
Lists the conductor or bundle identifiers. Phase conductors are identified as p1, p2,..., pn. Ground wires are identified as g1,g2,..., gn.
Phase, Phase Number — Phase number
Specify the phase number to which the conductor belongs. Several conductors may have the same phase number. All conductors that have the same phase number are lumped together and are considered as a single equivalent conductor in the R, L, and C matrices. For example, if you want to compute the line parameters of a three-phase line equivalent to a double-circuit line such as the one represented in the figure Configuration of a Three-Phase Double-Circuit Line, you specify phase numbers 1, 2, 3 for conductors p1, p2, p3 (circuit 1) and phase numbers 3, 2, 1 for conductors p4, p5, p6 (circuit 2), respectively. If you prefer to simulate this line as two individual circuits and have access to the six phase conductors, you specify phase numbers 1, 2, 3, 6, 5, 4 respectively for conductors p1, p2, p3, p4, p5 and p6.
In three-phase systems, the three phases are usually labeled A, B, and C. The correspondence with the phase number is:
1, 2, 3, 4, 5, 6, 7, 8, 9,.... = A, B, C, A, B, C, A, B, C,...
You can also use the phase number to lump conductors of an asymmetrical bundle.
For ground wires, the phase number is forced to zero. All ground wires are lumped with the ground and they do not contribute to the R, L, and C matrix dimensions. If you need to access the ground wire connections in your model, you must specify these ground wires as normal phase conductors and manually connect them to the ground.
X — Horizontal position of conductor
Specify the horizontal position of the conductor, in meters or feet. The location of the zero reference position is arbitrary. For a symmetrical line, you typically choose X = 0 at the center of the line.
Y tower — Vertical position of conductor at tower
Specify the vertical position of the conductor (at the tower) with respect to ground, in meters or feet.
Y min — Vertical position of the conductor at mid-span
Specify the vertical position of the conductor with respect to ground at mid-span, in meters or feet.
The average height of the conductor (see the figure Configuration of a Three-Phase Double-Circuit Line) is produced by this equation:
{Y}_{average}={Y}_{\mathrm{min}}+\frac{sag}{3}=\frac{2{Y}_{\mathrm{min}}+{Y}_{tower}}{3}
Ytower = height of conductor at tower
Ymin = height of conductor at mid span
sag = Ytower−Ymin
Instead of specifying two different values for Ytower and Ymin, you may specify the same Yaverage value.
Conductor type — Conductor or bundle type numbers
Specify one of the conductor or bundle type numbers listed in the first column of the table of conductor characteristics.
Conductor types — Number of conductor types
Specify the number of conductor types (single conductor or bundle of subconductors). This parameter determines the number of rows in the table of conductors. The phase conductors and ground conductors can be either single conductors or bundles of subconductors. For voltage levels of 230 kV and higher, phase conductors are usually bundled to reduce losses and electromagnetic interferences due to corona effect. Ground wires are usually not bundled.
For a simple AC three-phase line, single- or double-circuit, there are usually two types of conductors: one type for the phase conductors and one type for the ground wires. You need more than two types for several lines in the same corridor, DC bipolar lines or distribution feeders, where neutral and sheaths of TV and telephone cables are represented.
Internal conductor inductance evaluated from — Computation method for conductor internal inductance
T/D ratio (default) | Geometric Mean Radius (GMR) | Reactance Xa at 1-foot spacing | Reactance Xa at 1-meter spacing
Select one of the following three parameters to specify how the conductor internal inductance is computed: T/D ratio, Geometric Mean Radius (GMR), or Reactance Xa at 1-foot spacing (or Reactance Xa at 1-meter spacing if the Units parameter is set to metric).
If you select T/D ratio, the internal inductance is computed from the T/D value specified in the table of conductors, assuming a hollow or solid conductor. D is the conductor diameter and T is the thickness of the conducting material (see the figure Configuration of a Three-Phase Double-Circuit Line). The conductor self-inductance and resistance are computed from the conductor diameter, T/D ratio, DC resistance, and relative permeability of conducting material and specified frequency.
If you select Geometric Mean Radius (GMR), the conductor GMR evaluates the internal inductance. When the conductor inductance is evaluated from the GMR, the specified frequency does not affect the conductor inductance. You must provide the manufacturer's GMR for the desired frequency (usually 50 Hz or 60 Hz). When you are using the T/D ratio option, the corresponding conductor GMR at the specified frequency is displayed in the Conductors table.
Selecting Reactance Xa at 1-foot spacing (or Reactance Xa at 1-meter spacing) uses the positive-sequence reactance at the specified frequency of a three-phase line having 1-foot (or 1-meter) spacing between the three phases to compute the conductor internal inductance.
Include conductor skin effect — Include impact of frequency on conductor AC resistance and inductance
Select this check box to include the impact of frequency on conductor AC resistance and inductance (skin effect). If this parameter is cleared, the resistance is kept constant at the value specified by the Conductor DC resistance parameter and the inductance is kept constant at the value computed in DC, using the D out (conductor outside diameter) and the T/D ratio parameters of the Conductors table. When skin effect is included, the conductor AC resistance and inductance are evaluated considering a hollow conductor with T/D ratio (or solid conductor if T/D = 0.5). The T/D ratio evaluates the AC resistance even if the conductor inductance is evaluated from the GMR or from the reactance at 1-foot spacing or 1-meter spacing. The ground skin effect is always considered and it depends on the ground resistivity.
D out — Conductor outside diameter
Specify the conductor outside diameter, in centimeters or inches.
T/D ratio — Conductor T/D ratio
scalar between 0 and 0.5
Specify the T/D ratio of the hollow conductor. T is the thickness of the conducting material, and D is the outside diameter. This parameter can vary between 0 and 0.5. A T/D value of 0.5 indicates a solid conductor. For Aluminum Cable Steel Reinforced (ACSR) conductors, you can ignore the steel core and consider a hollow aluminum conductor (typical T/D ratios are between 0.3 and 0.4). The T/D ratio is used to compute the conductor AC resistance when the Include conductor skin effect parameter is selected. It is also used to compute the conductor self-inductance when the parameter Internal conductor inductance evaluated from is set to T/D ratio.
GMR — Geometric mean radius
This parameter is accessible only when the parameter Internal conductor inductance evaluated from is set to Geometric Mean Radius (GMR). Specify the GMR in centimeters or inches. The GMR at 60 Hz or 50 Hz is usually provided by conductor manufacturers. When the parameter Internal conductor inductance evaluated from is set to T/D ratio, the value of the corresponding GMR giving the same conductor inductance is displayed. When the parameter Internal conductor inductance evaluated from is set to Reactance Xa at 1-foot spacing or Reactance Xa at 1-meter spacing, the title of the column changes to Xa.
Xa — Reactance Xa at 1-meter spacing or 1-foot spacing
This parameter is accessible only when Internal conductor inductance evaluated from is set to Reactance Xa at 1-meter spacing or Reactance Xa at 1-foot spacing. Specify the Xa value in ohms/km or ohms/mile at the specified frequency. The Xa value at 60 Hz or 50 Hz is usually provided by conductor manufacturers.
DC res — Conductor DC resistance
Specify the DC resistance of conductor in ohms/km or ohms/mile.
mu_r — Conductor relative permeability
Specify the relative permeability µr of the conducting material. µr = 1.0 for nonmagnetic conductors (such as aluminum or copper). This parameter is not accessible when the Include conductor skin effect parameter is cleared.
Nb_cond — Number of conductors per bundle
Specify the number of subconductors in the bundle or 1 for single conductors.
Db — Bundle diameter
Specify the bundle diameter, in centimeters or inches. This parameter is not accessible when the Nb_cond is set to 1. When you specify bundled conductors, the subconductors are assumed to be evenly spaced on a circle. If this is not the case, you must enter individual subconductor positions in the Line Geometry table and lump these subconductors by giving them the same Phase Number parameter.
Angle — Angle of conductor 1
Specify an angle, in degrees, that determines the position of the first conductor in the bundle with respect to a horizontal line parallel to ground. This angle determines the bundle orientation. This parameter is not accessible when the Nb_cond is set to 1.
Frequency range logspace — Frequency range for parameter computation
[-2,5,141] (default) | three-element vector
Specify a frequency range for the parameter computation. Enter a vector of three elements, [X1,X2,N]. This parameter defines a frequency vector of N logarithmically equally spaced points between decades 10^X1 and 10^X2.
Line Length — Length of line
Specify the length of the line, in km.
RLC Line Parameters — Compute RLC line parameters
Computes the RLC parameters. After completion of the parameters computation, results are displayed in the Computed Parameters section.
The R, L, and C parameters are always displayed respectively in ohms/km, henries/km, and farads/km, even if the English units specify the input parameters.
If the number of phase conductors is 3 or 6, the symmetrical component parameters are also displayed:
For a three-phase line (one circuit), R10, L10, and C10 vectors of two values are displayed for positive-sequence and zero-sequence RLC values.
For a six-phase line (two coupled three-phase circuits), R10, L10, and C10 are vectors of five values containing the following RLC sequence parameters: the positive-sequence and zero-sequence of circuit 1, the mutual zero-sequence between circuit 1 and circuit 2, and the positive-sequence and zero-sequence of circuit 2.
Frequency Dependent Model Parameters — Compute frequency dependent parameters
Computes the frequency dependent parameters. After completion of the parameters computation, results are displayed in the Computed Parameters section.
Block — Selected block
Select a Distributed Parameters Line block (either to set the matrices or sequence RLC parameters), a Pi Section Line block, or a Three-Phase PI Section Line block in your model, then click the button to confirm the block selection. The name of the selected block appears in the left window.
Send RLC matrices to block — Download RLC matrices to block
Downloads RLC matrices into the selected block. This button is not visible when the selected block is a Distributed Parameters Line (Frequency-Dependent) block.
Send Sequences to block — Download RLC sequence parameters to block
Downloads RLC sequence parameters into the selected block. This button is not visible when the selected block is a Distributed Parameters Line (Frequency-Dependent) block.
Send to workspace — Send matrices and component parameters to MATLAB workspace
Sends the R, L, and C matrices, as well as the symmetrical component parameters, to the MATLAB workspace. The following variables are created in your workspace: R_matrix, L_matrix, C_matrix, and R10, L10, C10 for symmetrical components.
Send Frequency-Dependent Parameters to block — Download frequency-dependent parameters to block
Downloads the frequency-dependent parameters into the selected block. This button is not visible when the block is not a Distributed Parameters Line (Frequency-Dependent) block.
Distributed Parameters Line | Distributed Parameters Line (Frequency-Dependent) | PI Section Line | Three-Phase PI Section Line
power_lineparam
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In this series of posts, we will define a small programming language using formal methods. We will also produce proofs about our definitions using the Lean theorem prover.
There are many programming languages, but most do not have a formal definition. The majority of programming languages are defined by their implementation.
Definition by implementation
Given some programming language
L
, an implementation of
L
is a program, which takes the text of a program
p
in the language
L
, and either produces some result based on
p
, or rejects
p
as invalid.
We could then say
p
is a program in the language
L
if the implementation of
L
produces output for
p
, that is, it does not reject
p
However, a programming language may have multiple implementations. And though these implementations may strive for compatibility between one another, it sometimes happens that different implementations behave differently given the same program as input.
In such a case, which one of these implementations is the one that defines the programming language?
Definition by specification
A fix for this problem is to write a specification for the programming language. Developers then use the specification as a reference when writing implementations of the language.
Languages like C, C++, Go, and JavaScript (technically ECMAScript) are specified in this way.
Specification with mathematics
However, it is possible write a specification not just with words, but with mathematics. Some examples of languages specified in this way are Standard ML, Pony, and WebAssembly.
Formally specifying a language in this way is not a trivial task, but the benefits are real.
For one, a mathematical specification is completely unambiguous. This contrasts with specifications written in human languages, which can be ambiguous. For instance: in that last sentence, what is the thing that I am saying can be ambiguous? Is it specifications or human languages?
Furthermore, using formal methods allows us to state and prove theorems about the specification. This gives us a high degree of confidence that our specification is free of bugs and mistakes.
Let us explore this method of specification by defining our own programming language.
Hatsugen, a small language
First, we will give a name to our language.
There's an excellent series of blog posts about implementing a programming language. As in that series, we'll name our language by translating the English word "utterance" into some other human language. I'll choose the target language to be Japanese, given my interests. Thus, we will call the language "Hatsugen".
For now, Hatsugen will just have integers and booleans.
We'll represent integers with integer literal expressions like
\mathsf{123}
\mathsf{-456}
\mathsf{0}
. There are infinitely many integers, and thus infinitely many such possible literals. Practical considerations like a maximum integer size will be ignored for now.
We'll represent booleans with the literal expressions
\mathsf{true}
\mathsf{false}
As one extra bit, we'll also support a conditional expression
\mathsf{if} \ e_1 \ \mathsf{then} \ e_2 \ \mathsf{else} \ e_3
which will evaluate the condition
e_1
\mathsf{true}
e_2
\mathsf{false}
e_3
It's intentional that Hatsugen is a very small language, at least for now. This will allow us to get comfortable with all the formal stuff without getting bogged down with the complexity of Hatsugen itself.
We'll use a BNF-style grammar to describe the syntax of expressions in Hatsugen.
Since there are infinitely many integers, and writing them all out would take quite a while, we'll represent an arbitrary integer literal with
\overline{n}
\begin{aligned} e ::= \ & \overline{n} \\ | \ & \mathsf{true} \\ | \ & \mathsf{false} \\ | \ & \mathsf{if} \ e_1 \ \mathsf{then} \ e_2 \ \mathsf{else} \ e_3 \end{aligned}
This says that an expression
e
can either be an integer literal, a boolean literal, or an conditional expression.
Note that the conditional expression contains other expressions:
e_1
e_2
e_3
We now define how to evaluate an expression. This is called the dynamic semantics. There are various ways to do this, but the one we'll use for Hatsugen is often called structural operational semantics.
To do this, we will define two judgments. To define these judgments, we will write inference rules.
e \ \mathsf{val}
The first judgement is written
e \ \mathsf{val}
and is read as "
e
is a value". A value is an expression that is "done" evaluating.
We first define that any integer literal is a value. We can express this definition with an inference rule:
\frac {} {\overline{n} \ \mathsf{val}}
An inference rule looks like a big fraction, with the premises on top and the single conclusion on the bottom. There are no premises for this rule, so the top is empty.
We can also define that the boolean literals
\mathsf{true}
\mathsf{false}
are values with two more rules.
\frac {} {\mathsf{true} \ \mathsf{val}}
\frac {} {\mathsf{false} \ \mathsf{val}}
In Hatsugen, these are the only expressions that are values. But what about conditional expressions?
Stepping:
e \mapsto e'
For expressions which are not values, we must now define how to evaluate them. The second dynamics judgement,
e \mapsto e'
, read as "
e
steps to
e'
", holds when the expression
e
takes a single step to another expression,
e'
"Evaluating an expression" is thus defined as repeatedly stepping an expression until it is a value.
In Hatsugen, the only expressions that can take a step are conditional expressions. First, we define that a conditional expression can take a step if its condition
e_1
can take a step to
e_1'
. When it steps, we leave
e_2
e_3
\frac {e_1 \mapsto e_1'} { \begin{aligned} &\mathsf{if} \ e_1 \ \mathsf{then} \ e_2 \ \mathsf{else} \ e_3 \mapsto \\&\mathsf{if} \ e_1' \ \mathsf{then} \ e_2 \ \mathsf{else} \ e_3 \end{aligned} }
Now, we define what happens when
e_1
is a value. If it is
\mathsf{true}
, we step to
e_2
, ignoring
e_3
\frac {} { \mathsf{if} \ \mathsf{true} \ \mathsf{then} \ e_2 \ \mathsf{else} \ e_3 \ \mapsto e_2 }
And if it was
\mathsf{false}
, we do the opposite and step to
e_3
e_2
\frac {} { \mathsf{if} \ \mathsf{false} \ \mathsf{then} \ e_2 \ \mathsf{else} \ e_3 \mapsto e_3 }
With that, we've defined the dynamic semantics.
But we immediately run into a problem.
We earlier informally defined what it means to evaluate an expression:
"Evaluating an expression" is... defined as repeatedly stepping an expression until it is a value.
However, under this definition, there exist expressions that cannot be evaluated. For instance:
\mathsf{if} \ \mathsf{1} \ \mathsf{then} \ \mathsf{2} \ \mathsf{else} \ \mathsf{3}
This expression cannot take a step, because none of the rules for
e \mapsto e'
apply. This is because we only defined how to take a step when
e_1
was either
\mathsf{true}
\mathsf{false}
, or able to take a step itself. We didn't define what to do if
e_1
is an integer literal.
Yet, this expression is also not a value, because none of the rules for
e \ \mathsf{val}
apply. Thus, the expression is "stuck".
There are generally two things we can do here. One is to go back and add more rules to the dynamics to define how to evaluate these kinds of expressions. For instance, we could emulate C-like languages with the following rules:
\frac {} { \mathsf{if} \ \mathsf{0} \ \mathsf{then} \ e_2 \ \mathsf{else} \ e_3 \mapsto e_3 }
\frac {\overline{n} \ne \mathsf{0}} { \mathsf{if} \ \overline{n} \ \mathsf{then} \ e_2 \ \mathsf{else} \ e_3 \mapsto e_2 }
This treats
\mathsf{0}
\mathsf{false}
and any other integer like
\mathsf{true}
The other thing we could do define a notion of a "valid" expression, and only permit evaluation of these valid expressions. This is the approach we will use.
To define which expressions are valid, we will introduce a static semantics.
e: \tau
The static semantics tell us which expressions are valid, and thus permitted to evaluate.
First, we introduce the notion of a type. For now, we have only two: integer and boolean.
\begin{aligned} \tau ::= \ & \mathsf{Int} \\ | \ & \mathsf{Bool} \end{aligned}
Next, we introduce another judgement,
e: \tau
e
\tau
" or "the type of
e
\tau
". This judgement defines what it means for an expression to be valid: an expression
e
is valid if there exists a type
\tau
e: \tau
Now, we define
e: \tau
by writing its rules, starting with rules for the literals. Integers literals have integer type, and boolean literals have boolean type.
\frac {} {\overline{n}: \mathsf{Int}}
\frac {} {\mathsf{true}: \mathsf{Bool}}
\frac {} {\mathsf{false}: \mathsf{Bool}}
For conditional expressions, we require that
e_1
has boolean type. We also require
e_2
e_3
have the same type (but that type could be any type). Then we say the whole conditional expression has that type.
\frac { e_1: \mathsf{Bool} \hspace{1em} e_2: \tau \hspace{1em} e_3: \tau } {\mathsf{if} \ e_1 \ \mathsf{then} \ e_2 \ \mathsf{else} \ e_3: \tau}
Note that because
e_1
must have boolean type, expressions like the previously considered
\mathsf{if} \ \mathsf{1} \ \mathsf{then} \ \mathsf{2} \ \mathsf{else} \ \mathsf{3}
are now disallowed by the static semantics.
This completes the definition of the static semantics.
We can now state and prove theorems about Hatsugen.
The first crucial theorem is that of progress. Progress says that well-typed expressions are either done evaluating or can keep evaluating.
More formally, progress states:
e
\tau
e: \tau
e \ \mathsf{val}
e'
e \mapsto e'
Note that, before we introduced the static semantics, we had the problem of certain expressions neither being values nor being able to step. Now that we have the statics at our disposal, we can use them to strengthen the precondition of the theorem so that it is provable.
Next, we have preservation. Preservation states that well-typed expressions that can keep evaluating preserve their types when evaluating.
Again, more formally:
e
e'
\tau
e: \tau
e \mapsto e'
e': \tau
Note that taken together, we get the following safety theorem:
e
\tau
e: \tau
e \ \mathsf{val}
e'
e \mapsto e'
e': \tau
Crucially, note that in the case where
e \mapsto e'
, we also have that
e': \tau
. This means that we can apply the safety theorem again on
e'
The proofs are available on GitHub.
In the next post, we'll add functions to Hatsugen.
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Table 5 Assay results for the determination of LEV and AMB in their laboratory prepared mixture in 25:6 (w/w) as the case in tablets
LEV/AMB ratio
25:6 20.8 5.0 20.3840 5.0100 98.00 100.20 99.35 100.26
25.0 6.0 24.4750 6.1680 99.00 102.80 100.96 99.53
41.60 10.0 40.9050 10.1450 98.33 101.45 99.49 100.21
\overline{\mathrm{X}}
98.44 101.48 99.93 100.00
± SD 0.61 1.30 0.89 0.41
% Error 0.0.30 0.74 0.52 0.24
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Classification error by resubstitution - MATLAB - MathWorks India
L = resubLoss(obj)
L = resubLoss(obj,Name,Value)
L = resubLoss(obj) returns the resubstitution loss, meaning the loss computed for the data that fitcdiscr used to create obj.
L = resubLoss(obj,Name,Value) returns loss statistics with additional options specified by one or more Name,Value pair arguments.
The following table lists the available loss functions. Specify one using the corresponding character vector or string scalar.
Suppose that n be the number of observations in X and K be the number of distinct classes (numel(obj.ClassNames)). Your function must have this signature
C is an n-by-K logical matrix with rows indicating which class the corresponding observation belongs. The column order corresponds to the class order in obj.ClassNames.
S is an n-by-K numeric matrix of classification scores. The column order corresponds to the class order in obj.ClassNames. S is a matrix of classification scores, similar to the output of predict.
Classification error, a scalar. The meaning of the error depends on the values in weights and lossfun. See Classification Loss.
Compute the resubstituted classification error for the Fisher iris data:
\sum _{j=1}^{n}{w}_{j}=1.
L=\sum _{j=1}^{n}{w}_{j}\mathrm{log}\left\{1+\mathrm{exp}\left[-2{m}_{j}\right]\right\}.
L=\sum _{j=1}^{n}{w}_{j}{c}_{{y}_{j}{\stackrel{^}{y}}_{j}},
{\stackrel{^}{y}}_{j}
{c}_{{y}_{j}{\stackrel{^}{y}}_{j}}
{\stackrel{^}{y}}_{j}
L=\sum _{j=1}^{n}{w}_{j}I\left\{{\stackrel{^}{y}}_{j}\ne {y}_{j}\right\},
L=-\sum _{j=1}^{n}\frac{{\stackrel{˜}{w}}_{j}\mathrm{log}\left({m}_{j}\right)}{Kn},
{\stackrel{˜}{w}}_{j}
L=\sum _{j=1}^{n}{w}_{j}\mathrm{exp}\left(-{m}_{j}\right).
L=\sum _{j=1}^{n}{w}_{j}\mathrm{max}\left\{0,1-{m}_{j}\right\}.
L=\sum _{j=1}^{n}{w}_{j}\mathrm{log}\left(1+\mathrm{exp}\left(-{m}_{j}\right)\right).
{\gamma }_{jk}={\left(f{\left({X}_{j}\right)}^{\prime }C\right)}_{k}.
{\stackrel{^}{y}}_{j}=\underset{k=1,...,K}{\text{argmin}}{\gamma }_{jk}.
L=\sum _{j=1}^{n}{w}_{j}{c}_{j}.
L=\sum _{j=1}^{n}{w}_{j}{\left(1-{m}_{j}\right)}^{2}.
P\left(x|k\right)=\frac{1}{{\left({\left(2\pi \right)}^{d}|{\Sigma }_{k}|\right)}^{1/2}}\mathrm{exp}\left(-\frac{1}{2}\left(x-{\mu }_{k}\right){\Sigma }_{k}^{-1}{\left(x-{\mu }_{k}\right)}^{T}\right),
|{\Sigma }_{k}|
{\Sigma }_{k}^{-1}
\stackrel{^}{P}\left(k|x\right)=\frac{P\left(x|k\right)P\left(k\right)}{P\left(x\right)},
ClassificationDiscriminant | fitcdiscr | loss
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P-Channel metal oxide semiconductor field-effect transistor using either Shichman-Hodges equation or surface-potential-based model - MATLAB - MathWorks í•œêµ
{I}_{DS}=0
{I}_{DS}=âK\left(\left({V}_{GS}â{V}_{th}\right){V}_{DS}â{V}_{DS}{}^{2}/2\right)\left(1+\mathrm{λ}|{V}_{DS}|\right)
{I}_{DS}=â\left(K/2\right){\left({V}_{GS}â{V}_{th}\right)}^{2}\left(1+\mathrm{λ}|{V}_{DS}|\right)
{V}_{BS}â¤0
{V}_{th}={V}_{T0}+\mathrm{γ}\left(\sqrt{2{\mathrm{Ï}}_{B}}\right)â\mathrm{γ}\left(\sqrt{2{\mathrm{Ï}}_{B}+{V}_{BS}}\right)
0<{V}_{BS}â¤4{\mathrm{Ï}}_{B}
{V}_{th}={V}_{T0}â\frac{\mathrm{γ}{V}_{BS}}{\sqrt{2{\mathrm{Ï}}_{B}}}
{V}_{BS}>4{\mathrm{Ï}}_{B}
{V}_{th}={V}_{T0}+\mathrm{γ}\left(\sqrt{2{\mathrm{Ï}}_{B}}\right)
{I}_{dio}={I}_{s}\left[\mathrm{exp}\left(â\frac{{V}_{BD}}{n{\mathrm{Ï}}_{T}}\right)â1\right]
{C}_{j}=\frac{{C}_{j0}}{\sqrt{1+\frac{{V}_{BD}}{{V}_{bi}}}}
{C}_{diff}=\frac{\mathrm{Ï}{I}_{s}}{n{\mathrm{Ï}}_{T}}\mathrm{exp}\left(â\frac{{V}_{BD}}{n{\mathrm{Ï}}_{T}}\right)
{K}_{Ts}={K}_{Tm1}{\left(\frac{{T}_{s}}{{T}_{m1}}\right)}^{BEX}
{V}_{BS}â¤0
\frac{d{V}_{th}}{dT}=\frac{d{V}_{T0}}{dT}â\frac{\mathrm{γ}}{2\sqrt{2{\mathrm{Ï}}_{B}}}\frac{d2{\mathrm{Ï}}_{B}}{dT}+\frac{\mathrm{γ}}{2\sqrt{2{\mathrm{Ï}}_{B}â{V}_{BS}}}\frac{d2{\mathrm{Ï}}_{B}}{dT}
0<{V}_{BS}â¤4{\mathrm{Ï}}_{B}
\frac{d{V}_{th}}{dT}=\frac{d{V}_{T0}}{dT}â\frac{\mathrm{γ}{V}_{BS}}{4}{\left(2{\mathrm{Ï}}_{B}\right)}^{â\frac{3}{2}}\frac{d2{\mathrm{Ï}}_{B}}{dT}
{V}_{BS}>4{\mathrm{Ï}}_{B}
\frac{d{V}_{th}}{dT}=\frac{d{V}_{T0}}{dT}â\frac{\mathrm{γ}}{2\sqrt{2{\mathrm{Ï}}_{B}}}\frac{d2{\mathrm{Ï}}_{B}}{dT}
{\mathrm{Ï}}_{B}=\frac{kT}{q}\mathrm{ln}\left(\frac{{N}_{B}}{{n}_{i}}\right)
\frac{d2{\mathrm{Ï}}_{B}}{dT}=â\frac{1}{T}\left[2{\mathrm{Ï}}_{B}â\left(\frac{{E}_{g}\left(0\right)}{q}+\frac{3kT}{q}\right)\right]
Parameter\left(t\right)=Paramete{r}_{faulted}â\left(Paramete{r}_{faulted}âParamete{r}_{unfaulted}\right)\text{sech}\left(\frac{tâ{t}_{th}}{\mathrm{Ï}}\right),
{G}_{mob}=\sqrt{1+{\left({\mathrm{θ}}_{sr}{V}_{eff}\right)}^{2}}
{I}_{s}={I}_{s,m1}{\left(\frac{{T}_{s}}{{T}_{m1}}\right)}^{{\mathrm{η}}_{Is}}â
\mathrm{exp}\left(\frac{{E}_{G}}{{k}_{B}}â
\left(\frac{1}{{T}_{m1}}â\frac{1}{{T}_{s}}\right)\right).
{I}_{s}={I}_{s,m1}{\left(\frac{{T}_{s}}{{T}_{m1}}\right)}^{{\mathrm{η}}_{Is}}â
\mathrm{exp}\left(\frac{{E}_{G}}{{k}_{B}}â
\left(\frac{1}{{T}_{m1}}â\frac{1}{{T}_{s}}\right)\right).
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Implement inductances with mutual coupling - Simulink - MathWorks Nordic
Two or Three Windings Inductances with Equal Mutual Coupling:
Generalized Mutual Inductance:
Implement inductances with mutual coupling
The Mutual Inductance block can be used to model two- or three-windings inductances with equal mutual coupling, or to model a generalized multi-windings mutual inductance with balanced or unbalanced mutual coupling.
If you choose to model two- or three-windings inductances with equal mutual coupling, you specify the self-resistance and inductance of each winding plus the mutual resistance and inductance. The electrical model for this block in this case is given below:
If you choose to model a general mutual inductance, specify the number of self-windings (not just limited to 2 or 3 windings) plus the resistance and inductance matrices that define the mutual coupling relationship between the windings (balanced or not).
The resistance and inductance matrices are defined as
R=\left[\begin{array}{cc}R1& Rm\\ Rm& R2\end{array}\right]
L=\left[\begin{array}{cc}L1& Lm\\ Lm& L2\end{array}\right],
R1 is the self-resistance of resistor R1.
Rm is the mutual resistance, such that
Rm<R1
Rm<R2
L is the inductance.
L1 is the self-inductance of inductor L1.
Lm is the mutual inductance, such that
Lm\le \sqrt{L1\cdot L2}
Type of mutual inductance
Select Two or Three windings with equal mutual terms to implement a three-phase mutual inductance with equal mutual coupling between the windings. This is the default.
Winding 1 self impedance
The self-resistance and inductance for winding 1, in ohms (Ω) and henries (H). Default is [1.1 1.1e-03].
The self-resistance and inductance for winding 2, in ohms (Ω) and henries (H). Default is [ 1.1 1.1e-03].
Three windings Mutual Inductance
If selected, implements three coupled windings; otherwise, it implements two coupled windings. Default is cleared.
The Winding 3 self impedance parameter is not available if the Three windings Mutual Inductance parameter is not selected. The self-resistance and inductance in ohms (Ω) and henries (H) for winding 3. Default is [ 1.1 1.1e-03].
The mutual resistance and inductance between windings, in ohms (Ω) and henries (H). The mutual resistance and inductance corresponds to the magnetizing resistance and inductance on the standard transformer circuit diagram. If the mutual resistance and reactance are set to [0 0], the block implements three separate inductances with no mutual coupling. Default is [1.0 1.0e-03].
The mutual inductance can be expressed as a relationship between two self inductances as
Lm = k*sqrt(L1*L2),
where k is the coupling coefficient (−1 ≤ k ≤ 1).
Select Winding currents to measure the current flowing through the windings.
Select Winding voltages and currents to measure the winding voltages and currents.
Select Generalized mutual inductance to implement a multi windings mutual inductance with mutual coupling defined by an inductance and a resistance matrix.
The number of self inductances. Default is 3.
Inductance matrix L
The inductance matrix, in Henrys, that define the mutual coupling relationship between the self windings. It must be a N-by-N symmetrical matrix. Default is [1.0 0.9 0.9 ; 0.9 1.0 0.9; 0.9 0.9 1.0 ] * 1e-3.
Resistance matrix R
The resistance matrix, in ohms, that define the mutual coupling relationship between the self windings. It must be a N-by-N symmetrical matrix. Default is [1.0 0.9 0.9 ; 0.9 1.0 0.9; 0.9 0.9 1.0 ] .
If you choose to model two or three windings inductances with equal mutual coupling, the following restrictions apply:
R1, R2, ..., RN ≠ RmL1, L2, ..., LN≠ Lm.
Negative values are allowed for the self- and mutual inductances as long as the self-inductances are different from the mutual inductance.
Windings can be left floating (not connected by an impedance to the rest of the circuit). However an internal resistor between the floating winding and the main circuit is automatically added. This internal connection does not affect voltage and current measurements.
The power_mutual example uses three coupled windings to inject a third harmonic voltage into a circuit fed at 60 Hz.
Linear Transformer, Saturable Transformer, Three-Phase Mutual Inductance Z1-Z0
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match - Maple Help
Home : Support : Online Help : Programming : Logic : Boolean : match
match(expr = pattern, v, 's')
match(expr = pattern, {v1, v2, ..., vN}, 's')
expression of type algebraic to be matched
pattern (also of type algebraic) to match
name of the main variable
name of the return argument
{v1, v2, ..., vN}
names of the main variables
The match(expr = pattern, v, 's') calling sequence returns true if it can match expr to pattern for some values of the variables (excluding the main variable, v). Otherwise, it returns false.
The match(expr = pattern, {v1, v2, ..., vN}, 's') calling sequence returns true if it can match expr to pattern for some values of the variables (excluding the main variables, {v1, v2, ..., vN}). Otherwise, it returns false.
Note: In the multiple main variable case, the pattern specified must be a polynomial in all, or all but one of, the main variables. That is, type(pattern, polynom(anything, vI)) must be true for (N - 1) values of I in the range 1, 2, ..., N. Otherwise, the match command returns false.
If the match is successful, s is assigned a substitution set such that
\mathrm{subs}\left(s,\mathrm{pattern}\right)=\mathrm{expr}
The main variables must be matched exactly in the pattern. In other words, the main variables cannot be substituted for any value.
The match command attempts to compute expressions to satisfy the pattern, as opposed to the typematch command that matches the form of the objects.
\mathrm{match}\left(\frac{\mathrm{ln}\left(k\right)}{{k}^{\frac{1}{2}}}=A{\mathrm{ln}\left(k\right)}^{P}{k}^{Q},k,'s'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
s
{\textcolor[rgb]{0,0,1}{A}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{P}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{Q}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}}
\mathrm{match}\left(5{x}^{2}-3x+zx+y=a{\left(x+b\right)}^{2}+c,x,'s'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
s
{\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{=}\frac{\textcolor[rgb]{0,0,1}{z}}{\textcolor[rgb]{0,0,1}{10}}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{3}}{\textcolor[rgb]{0,0,1}{10}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{20}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{3}}{\textcolor[rgb]{0,0,1}{10}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{9}}{\textcolor[rgb]{0,0,1}{20}}}
In the multiple main variable case, if the pattern is not a polynomial in (all but one of) the main variables, the match command always returns false.
\mathrm{match}\left(\mathrm{sin}\left(x+y\right)=\mathrm{sin}\left(ax+y\right),{x,y},'s'\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{match}\left(5{x}^{2}-3x+zx+y=a{\left(x+b\right)}^{2}+cxz+d,{x,z},'s'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
s
{\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{3}}{\textcolor[rgb]{0,0,1}{10}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{9}}{\textcolor[rgb]{0,0,1}{20}}}
\mathrm{match}\left(5{x}^{2}-3x+zx+y=a{\left(x+b\right)}^{2}+fx-cxz+dy,{x,y,z},'s'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
s
{\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-3}}
The pattern is polynomial in one of the two main variables, and suitable values can be found for all non-main variables in the following example.
\mathrm{match}\left(\mathrm{sin}\left(x\right)+4y=\mathrm{sin}\left(ax\right)+by,{x,y},'s'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
s
{\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{4}}
Take care that the pattern (containing the variables to be matched) appears on the right hand side of the first argument to match.
\mathrm{match}\left(2x+3=ax+b,x,'s'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{match}\left(ax+b=2x+3,x,'s'\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
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Sistem Hijian Internasional - Wikipédia Sunda, énsiklopédi bébas
(dialihkeun ti SI)
Sistim Internasional (lambangna SI, sarua jeung tina asal hartina dina basa Prancis Système International d'Unités), nyaéta sistim unit nu pangumumna dipaké. Iwal di AS jeung Inggris, sistim ieu dipaké dina kahirupan sapopoé (utamana dagang) di sakuliah dunya, ogé dina widang ilmiah. SI was selected as a specific subset of the existing Metre-Kilogram-Second systems of units (MKS), rather than the older Centimetre-Gram-Second system of units (CGS). Various new units were added with the introduction of the SI and at later times. SI is sometimes referred to as the metric system (especially in the United States, which has not widely adopted it, although it has been used more commonly in recent yéars, and the UK, where conversion is incomplete). Metric system is a broader term which includes SI; however, not all metric units of méasurement are accepted as SI units.
There are seven base units and several derived units, together with a set of prefixes. Non-SI units can be converted to SI units (or vice versa) according to the conversion of units.
3 Gaya nulis SI
4.1 Unit dasar
4.6 Obsolete SI prefixes
5 Variasi éjahan
The units of the SI are decided by a series of international conferences organised by the standards organization Bureau International des Poids et Mesures (International Buréau of Weights and Méasures). The SI was first given its name in 1960, and last added to in 1971.
The true origins of the SI or metric system date back to approximately 1640. It was invented by French scientists, and was given a huge boost in popularity by the French Revolution of 1789. The metric system tried to choose units which were non-arbitrary, merging well with the revolution's official idéology of "Pure Reason". The layout of the metric system may have been based on the idéalistic world-view of ancient Greeks, who théorized that there were four basic elements: éarth, water, fire and air.
The most important unit is that of length: one metre was intended to be equal to 1/10,000,000thof the distance from the pole to the equator along the meridian through Paris. This is approximately 10% longer than one yard. Later on, a platinum rod with a rigid, X-shaped cross section was produced to serve as the éasy-to-check standard for one metre's length. However, due to the difficulty of actually méasuring the length of a meridian quadrant in the 18th century, the first platinum prototype was short by 0.2 millimetres. Then a multiple of a specific radiation wavelength was introduced to abstractly define the (unchanged) length of the metre unit, and finally the metre was defined as the distance travelled by light in a vacuum in a specific period of time.
The unit of mass is the kilogram, which was defined by a cube filled with distilled pure water at its densest (+4° Celsius) and having sides equal to 1/10th of a metre. This volume contains one kilogram of water. One kilogram is about 2.2 pounds. This cubic space was also known as one litre (since slightly revised) so volumes of different liquids could be compared. Later on, a platinum-iridium metal cylinder was manufactured to serve as the one kilogram weight standard and remained so ever since.
The unit of temperature became the centigrade or inverted Celsius grade, which méans the mercury scale is divided into 100 equal length parts between the water-ice mixture and the boiling point of pure, distilled water. Boiling water thus becomes one hundred degrees Celsius and freezing is zero degrees Celsius. This is the metric unit of temperature in everyday use. A hundred yéars later, scientists discovered absolute zero. This prompted the establishment of a new temperature scale, called the absolute scale or Kelvin scale, which relocates the zero place but still uses 100 kelvins between the freezing point and boiling point of water.
The metric unit of time remained the second. One definition of day is 86,400 seconds. The formal definition of the second has been changed several times for enhanced scientific requirements (astronomic observations, tuning fork clock, quartz clock and then caesium atomic clock) but wristwatch users remain relatively unaffected.
The swift worldwide adoption of the metric system as a tool of economy and everday commerce was based mainly on the lack of customary systems in many countries to adequately describe some concepts, or as a result of an attempt to standardize the many regional variations in the customary system. International politics also factored into the choice as many countries made the industrial shift when Britain still had empire status, and had various feelings related to its position in the world. Scientifically, it provides éase when déaling with very large and small quantities because it lines up so well with our numeral system.
Cultural differences can be represented in the local everyday uses of metric units. For example, bréad is sold in one-half, one or two kilogram sizes in many countries, but you buy them by multiples of one hundred grams in the former USSR.
Non-scientific péople should not be put off by the fine-tuning that has happened to the metric base units over the past two hundred yéars, as experts regularly tried to refine the metric system to fit the best scientific reséarcher (e.g. MKG to CGS to SI system changes or the invention of Kelvin scale). These changes seldom affect the everyday use of metric units. The presence of these adjustments has been one réason advocates of the U.S. customary units have used against metrication.
Basis[édit | édit sumber]
SI is built on seven SI base units, the kilogram, metre, second, ampere, kelvin, mole, and candela. These are used to define various SI derived units.
SI also defines a number of SI prefixes to be used with the units: these combine with any unit name to give subdivisions and multiples. For example, the prefix kilo denotes a multiple of a thousand, so the kilometre is 1 000 metres, the kilogram 1 000 grams, and so on. Note that a millionth of a kilogram is a milligram, not a microkilogram.
Gaya nulis SI[édit | édit sumber]
Symbols are written in lower case, except the symbols that are derived from the name of a person. This méans that the symbol for the SI unit for pressure, named after Blaise Pascal, is Pa, wheréas the unit itself is written pascal. The official SI brochure lists the symbol for the litre as an allowed exception to the capitalization rules: either capital or lowercase L is acceptable.
Symbols are written in singular, e.g. 25 kg (not "25 kgs").
Symbols, unlike abbreviations, do not have a period (.) at the end.
It is preferable to keep the symbol in upright Roman type (for example, m for metres, L for litres), so as to differentiate from mathematical and physical variables (for example, m for mass, l for length).
A space is left between the numbers and the symbols: 2.21 kg, 7.3·102 m2
SI uses spaces to separate decimal digits in sets of three. e.g. 1 000 000 or 342 142 (in contrast to the commas or dots used in other systems, e.g. 1,000,000 or 1.000.000).
SI used only a comma as the separator for decimal fractions until 1997. The number "twenty four and fifty one hundredths" would be written as "24,51". In 1997 the CIPM decided that the British full stop (the "dot on the line", or period) would be the decimal separator in text whose main language is English ("24.51"); the comma remains the decimal separator in all other languages.
Symbols formed by division of two units are joined with a solidus (/), or given as a negative exponent, e.g. m/s, m s−1, m·s−1 or
{\displaystyle {\frac {\mbox{m}}{\mbox{s}}}}
. A solidus should not be used if the result is ambiguous, e.g. kg·m−1·s−2, not "kg/m/s2".
The system can legally be used in every country in the world, and many countries do not maintain definitions of other units. Those countries that still give official recognition to non-SI units (e.g. the US and UK) have defined the modérn in terms of SI units; for example, the common inch is defined to be exactly 0.0254 metres. Survey distances have, however, not been redefined due to the accumulation of error it would entail. It was adopted by the 11th General Conference on Weights and Measures (CGPM) in 1960. (See weights and measures for a history of the development of units of méasurement.)
Unit[édit | édit sumber]
Unit dasar[édit | édit sumber]
Di handap ieu unit-unit dasar ti mana nu séjén diturunkeun, they are dimensionally independent. The definitions stated below are widely accepted.
Citakan:SI base units
Dimensionless derived units[édit | édit sumber]
The following SI units are derived from the base units and are dimensionless.
Citakan:SI dimensionless units
Derived units with special names[édit | édit sumber]
Base units can be put together to derive units of méasurement for other quantities. Some have been given names.
Citakan:SI special units
Non-SI units accepted for use with SI[édit | édit sumber]
Citakan:SI acceptable units
SI prefixes[édit | édit sumber]
Citakan:SI prefixes
Obsolete SI prefixes[édit | édit sumber]
The following SI prefixes are no longer in use.
Citakan:Obsolete SI prefixes
Variasi éjahan[édit | édit sumber]
Several nations, notably the United States, typically use the spellings 'méter' and 'liter' instéad of 'metre' and 'litre'. This is in keeping with standard American English spelling (for example, Americans also use 'center' rather than 'centre,' using the latter on mostly office buildings; see also American and British English differences). In addition, the official US spelling for the SI prefix 'deca' is 'deka'.
The unit 'gram' is also sometimes spelled 'gramme' in English-spéaking countries other than the United States, though that is an older spelling and use is declining.
Other méasurement systems:
BIPM reference (SI reference)
SI - Its history and use in science and industry
Dicomot ti "https://su.wikipedia.org/w/index.php?title=Sistem_Hijian_Internasional&oldid=488688"
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torch.gradient — PyTorch 1.11.0 documentation
torch.gradient
torch.gradient¶
torch.gradient(input, *, spacing=1, dim=None, edge_order=1) → List of Tensors¶
g : \mathbb{R}^n \rightarrow \mathbb{R}
g
is estimated using samples. By default, when spacing is not specified, the samples are entirely described by input, and the mapping of input coordinates to an output is the same as the tensor’s mapping of indices to values. For example, for a three-dimensional input the function described is
g : \mathbb{R}^3 \rightarrow \mathbb{R}
g(1, 2, 3)\ == input[1, 2, 3]
When spacing is specified, it modifies the relationship between input and input coordinates. This is detailed in the “Keyword Arguments” section below.
The gradient is estimated by estimating each partial derivative of
g
independently. This estimation is accurate if
g
C^3
(it has at least 3 continuous derivatives), and the estimation can be improved by providing closer samples. Mathematically, the value at each interior point of a partial derivative is estimated using Taylor’s theorem with remainder. Letting
x
be an interior point and
x+h_r
be point neighboring it, the partial gradient at
f(x+h_r)
is estimated using:
\begin{aligned} f(x+h_r) = f(x) + h_r f'(x) + {h_r}^2 \frac{f''(x)}{2} + {h_r}^3 \frac{f'''(x_r)}{6} \\ \end{aligned}
x_r
is a number in the interval
[x, x+ h_r]
f \in C^3
we derive :
f'(x) \approx \frac{ {h_l}^2 f(x+h_r) - {h_r}^2 f(x-h_l) + ({h_r}^2-{h_l}^2 ) f(x) }{ {h_r} {h_l}^2 + {h_r}^2 {h_l} }
We estimate the gradient of functions in complex domain
g : \mathbb{C}^n \rightarrow \mathbb{C}
in the same way.
The value of each partial derivative at the boundary points is computed differently. See edge_order below.
input (Tensor) – the tensor that represents the values of the function
spacing (scalar, list of scalar, list of Tensor, optional) – spacing can be used to modify how the input tensor’s indices relate to sample coordinates. If spacing is a scalar then the indices are multiplied by the scalar to produce the coordinates. For example, if spacing=2 the indices (1, 2, 3) become coordinates (2, 4, 6). If spacing is a list of scalars then the corresponding indices are multiplied. For example, if spacing=(2, -1, 3) the indices (1, 2, 3) become coordinates (2, -2, 9). Finally, if spacing is a list of one-dimensional tensors then each tensor specifies the coordinates for the corresponding dimension. For example, if the indices are (1, 2, 3) and the tensors are (t0, t1, t2), then the coordinates are (t0[1], t1[2], t2[3])
dim (int, list of int, optional) – the dimension or dimensions to approximate the gradient over. By default the partial gradient in every dimension is computed. Note that when dim is specified the elements of the spacing argument must correspond with the specified dims.”
edge_order (int, optional) – 1 or 2, for first-order or second-order estimation of the boundary (“edge”) values, respectively.
>>> # Estimates the gradient of f(x)=x^2 at points [-2, -1, 2, 4]
>>> coordinates = (torch.tensor([-2., -1., 1., 4.]),)
>>> values = torch.tensor([4., 1., 1., 16.], )
>>> torch.gradient(values, spacing = coordinates)
(tensor([-3., -2., 2., 5.]),)
>>> # Estimates the gradient of the R^2 -> R function whose samples are
>>> # described by the tensor t. Implicit coordinates are [0, 1] for the outermost
>>> # dimension and [0, 1, 2, 3] for the innermost dimension, and function estimates
>>> # partial derivative for both dimensions.
>>> t = torch.tensor([[1, 2, 4, 8], [10, 20, 40, 80]])
>>> torch.gradient(t)
(tensor([[ 9., 18., 36., 72.],
[ 9., 18., 36., 72.]]),
>>> # A scalar value for spacing modifies the relationship between tensor indices
>>> # and input coordinates by multiplying the indices to find the
>>> # coordinates. For example, below the indices of the innermost
>>> # 0, 1, 2, 3 translate to coordinates of [0, 2, 4, 6], and the indices of
>>> # the outermost dimension 0, 1 translate to coordinates of [0, 2].
>>> torch.gradient(t, spacing = 2.0) # dim = None (implicitly [0, 1])
(tensor([[ 4.5000, 9.0000, 18.0000, 36.0000],
[ 4.5000, 9.0000, 18.0000, 36.0000]]),
[ 5.0000, 7.5000, 15.0000, 20.0000]]))
>>> # doubling the spacing between samples halves the estimated partial gradients.
>>> # Estimates only the partial derivative for dimension 1
>>> torch.gradient(t, dim = 1) # spacing = None (implicitly 1.)
(tensor([[ 1.0000, 1.5000, 3.0000, 4.0000],
[10.0000, 15.0000, 30.0000, 40.0000]]),)
>>> # When spacing is a list of scalars, the relationship between the tensor
>>> # indices and input coordinates changes based on dimension.
>>> # For example, below, the indices of the innermost dimension 0, 1, 2, 3 translate
>>> # to coordinates of [0, 3, 6, 9], and the indices of the outermost dimension
>>> # 0, 1 translate to coordinates of [0, 2].
>>> torch.gradient(t, spacing = [3., 2.])
>>> # The following example is a replication of the previous one with explicit
>>> # coordinates.
>>> coords = (torch.tensor([0, 2]), torch.tensor([0, 3, 6, 9]))
>>> torch.gradient(t, spacing = coords)
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azdavis.net • Posts • Jun 5, 2021
In the previous post, we added functions to Hatsugen.
In this post, we'll add product types. These are often called "structs", "records", or "tuples" in real programming languages.
A product type is a combination of multiple types. For instance, if we want to return multiple values from a function, we can have the function's return type be a product type.
A product type that combines two types is called "pair" and denoted with
\tau_1 \times \tau_2
\langle e_1, e_2 \rangle
is a pair literal expression.
Note that by combining pair types with other pair types, we can effectively construct an product type combining
n
types for any
n > 2
To use a pair, we must be able to extract the values inside. For that, we add the projection expressions
e \cdot \mathsf{L}
e \cdot \mathsf{R}
e
is a pair, these projections extract the left and right value out of the pair respectively.
We will also introduce a product type that combines no other types, called "unit" and denoted with
\mathsf{1}
. There is just 1 value of this type, also often called "unit", and it is written
\langle \rangle
Because the unit type has only one value, it may not seem very useful. However, it can be useful when we want to return "nothing" from a function.
For instance, in most programming languages, functions can perform side effects. Side effects are anything the function does other than return a value, like modify files or access the Internet.
In fact, sometimes functions are only useful because of the side effects they perform, and they don't actually need to return anything useful. In these cases, it is convenient to have these functions return unit.
Different languages call unit different things:
C, C++, Java has void
Python has None
JavaScript has undefined
Ruby has nil
\begin{aligned} \tau ::= \ & \dots \\ | \ & \mathsf{1} \\ | \ & \tau_1 \times \tau_2 \\ \\ e ::= \ & \dots \\ | \ & \langle \rangle \\ | \ & \langle e_1, e_2 \rangle \\ | \ & e \cdot \mathsf{L} \\ | \ & e \cdot \mathsf{R} \end{aligned}
The unit value has unit type.
\frac {} {\Gamma \vdash \langle \rangle: \mathsf{1}}
Given two expressions each with their own type, we can make a pair of those types by assembling the expressions together.
\frac { \Gamma \vdash e_1: \tau_1 \hspace{1em} \Gamma \vdash e_2: \tau_2 } {\Gamma \vdash \langle e_1, e_2 \rangle: \tau_1 \times \tau_2}
We can then project the left or right part out of the pair.
\frac {\Gamma \vdash e: \tau_1 \times \tau_2} {\Gamma \vdash e \cdot \mathsf{L}: \tau_1}
\frac {\Gamma \vdash e: \tau_1 \times \tau_2} {\Gamma \vdash e \cdot \mathsf{R}: \tau_2}
The unit is a value.
\frac {} {\langle \rangle \ \mathsf{val}}
A pair is a value when both constituent expressions are values.
\frac { e_1 \ \mathsf{val} \hspace{1em} e_2 \ \mathsf{val} } {\langle e_1, e_2 \rangle \ \mathsf{val}}
If the left expression in a pair can step, so can the entire pair.
\frac {e_1 \mapsto e_1'} {\langle e_1, e_2 \rangle \mapsto \langle e_1', e_2 \rangle}
After the left expression in a pair is a value, we may step the right expression if possible.
\frac { e_1 \ \mathsf{val} \hspace{1em} e_2 \mapsto e_2' } {\langle e_1, e_2 \rangle \mapsto \langle e_1, e_2' \rangle}
For the projections, we must first step the pair to a value. Then, once it is a value, it will be a pair literal, and we may step to either the left or right value in the pair.
\frac {e \mapsto e'} {e \cdot \mathsf{L} \mapsto e' \cdot \mathsf{L}}
\frac {\langle e_1, e_2 \rangle \ \mathsf{val}} {\langle e_1, e_2 \rangle \cdot \mathsf{L} \mapsto e_1}
\frac {e \mapsto e'} {e \cdot \mathsf{R} \mapsto e' \cdot \mathsf{R}}
\frac {\langle e_1, e_2 \rangle \ \mathsf{val}} {\langle e_1, e_2 \rangle \cdot \mathsf{R} \mapsto e_2}
The helper judgments must also be updated. We can update them rather mechanically.
\frac {} {[x \mapsto e] \langle \rangle = \langle \rangle}
\frac { [x \mapsto e] e_1 = e_1' \hspace{1em} [x \mapsto e] e_2 = e_2' } {[x \mapsto e] \langle e_1, e_2 \rangle = \langle e_1', e_2' \rangle}
\frac {[x \mapsto e] e_1 = e_1'} {[x \mapsto e] e_1 \cdot \mathsf{L} = e_1' \cdot \mathsf{L}}
\frac {[x \mapsto e] e_1 = e_1'} {[x \mapsto e] e_1 \cdot \mathsf{R} = e_1' \cdot \mathsf{R}}
\frac {} {\mathsf{fv}(\langle \rangle) = \emptyset}
\frac { \mathsf{fv}(e_1) = s_1 \hspace{1em} \mathsf{fv}(e_2) = s_2 } {\mathsf{fv}(\langle e_1, e_2 \rangle) = s_1 \cup s_2}
\frac {\mathsf{fv}(e) = s} {\mathsf{fv}(e \cdot \mathsf{L}) = s}
\frac {\mathsf{fv}(e) = s} {\mathsf{fv}(e \cdot \mathsf{R}) = s}
Before we conclude, let us consider the etymology of "product type".
Product types are so named because the number of values in a product type is the product of the number of values in the constituent types.
For instance, consider the type
\mathsf{Bool}
. It has 2 values,
\mathsf{true}
\mathsf{false}
Now consider the type
\mathsf{Bool} \times \mathsf{Bool}
2 \times 2 = 4
\langle \mathsf{true}, \mathsf{true} \rangle
\langle \mathsf{true}, \mathsf{false} \rangle
\langle \mathsf{false}, \mathsf{true} \rangle
\langle \mathsf{false}, \mathsf{false} \rangle
Consider also the type
\mathsf{Bool} \times \mathsf{1}
, the product of
\mathsf{Bool}
and unit. It has
2 \times 1 = 2
\langle \mathsf{true}, \langle \rangle \rangle
\langle \mathsf{false}, \langle \rangle \rangle
This is why it makes sense that the unit type is written
\mathsf{1}
. It is the identity of the operation written
\times
For the integers, we have that for all integers
a
a \times 1 = a
\times
denotes multiplication.
Then similarly, if we write
|\tau|
to mean "the number of values in the type
\tau
", we have that for all types
\tau
|\tau \times \mathsf{1}| = |\tau|
\times
denotes a product type.
And more generally, for all types
\tau_1, \tau_2
|\tau_1 \times \tau_2| = |\tau_1| \times |\tau_2|
where on the left,
\times
denotes a product type, and on the right, it denotes multiplication.
In the next post, we'll add sum types, also known as tagged unions.
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Demodulate PSK-modulated data - Simulink - MathWorks Italia
Log-Likelihood Ratio and Approximate Log-Likelihood Ratio
Demodulate PSK-modulated data
The M-PSK Demodulator Baseband block demodulates a baseband representation of a PSK-modulated signal. The modulation order, M, is equivalent to the number of points in the signal constellation and is determined by the M-ary number parameter. The block accepts scalar or column vector input signals.
Input port accepting a baseband representation of a PSK-modulated signal.
Output signal, returned as a scalar or vector. The output is a demodulated version of the PSK-modulated signal.
Output type — Output signal data type
Specify the elements of the input signal as integers or bits. If Output type is Bit, the number of samples per frame is an integer multiple of the number of bits per symbol, log2(M).
Decision type — Demodulator output
Specify the demodulator output to be hard decision, log-likelihood ratio (LLR), or approximate LLR. The LLR and approximate LLR outputs are used with error decoders that support soft-decision inputs such as a Viterbi Decoder, to achieve superior performance. This parameter is available when Output type is Bit.
See Phase Modulation for algorithm details. The output values for Log-likelihood ratio and Approximate log-likelihood ratio decision types are of the same data type as the input values
Noise variance source — Source of noise variance
Specify the source of the noise variance estimate. This parameter is available when Decision type is Log-likelihood ratio or Approximate log-likelihood ratio.
To specify the noise variance from the dialog box, select Dialog.
To input the noise variance from an input port, select Port.
Noise variance — Estimate of noise variance
Specify the estimate of the noise variance as a positive scalar. This parameter is available when Noise variance source is Dialog.
This parameter is tunable in all simulation modes. If you use the Simulink® Coder™ rapid simulation (RSIM) target to build an RSIM executable, then you can tune the parameter without recompiling the model. Avoiding recompilation is useful for Monte Carlo simulations in which you run the simulation multiple times (perhaps on multiple computers) with different amounts of noise.
The exact LLR algorithm computes exponentials using finite precision arithmetic. Computation of exponentials with very large positive or negative magnitudes might yield:
NaN if both the noise variance and signal power are very small values
When the output returns any of these values, try using the approximate LLR algorithm because it does not compute exponentials.
When Constellation ordering is set to Gray, the output symbol is mapped to the input signal using a Gray-encoded signal constellation.
When Constellation ordering is set to Binary, the modulated symbol is exp(jϕ+j2πm/M), where ϕ is the phase offset in radians, m is the integer output such that 0 ≤ m ≤ M – 1, and M is the modulation order.
When Constellation ordering is User-defined, specify a vector of size M, which has unique integer values in the range [0, M–1]. The first element of this vector corresponds to the constellation point having a value of ejϕ with subsequent elements running counterclockwise.
Inherit via internal rule (default) | Smallest unsigned integer | double | single | int8 | uint8 | int16 | uint16 | int32 | uint32
Specify the data type of the demodulated output signal.
Boolean | double | fixed pointa, b, c | integer | single
a M = 2, 4, 8 only.
b Fixed-point inputs must be signed.
c When ASIC/FPGA is selected in the Hardware Implementation Pane, output is ufix(1) for bit outputs, and ufix(ceil(log2(M))) for integer outputs.
Diagrams for hard-decision demodulation of BPSK signals follow.
Hard-Decision BPSK Demodulator Signal Diagram for Trivial Phase Offset (multiple of π/2)
Hard-Decision BPSK Demodulator Floating-Point Signal Diagram for Nontrivial Phase Offset
Hard-Decision BPSK Demodulator Fixed-Point Signal Diagram for Nontrivial Phase Offset
Diagrams for hard-decision demodulation of QPSK signals follow.
Diagrams for hard-decision demodulation of higher-order (M ≥ 8) signals follow.
Hard-Decision 8-PSK Demodulator Floating-Point Signal Diagram
Hard-Decision 8-PSK Demodulator Fixed-Point Signal Diagram
Hard-Decision M-PSK Demodulator (M > 8) Floating-Point Signal Diagram for Nontrivial Phase Offset
For M > 8, to improve speed and implementation costs, no derotation arithmetic is performed when Phase offset is 0,
\pi /2
\pi
3\pi /2
(that is, when it is trivial).
Also, for M > 8, this block only supports double and single input types.
The exact LLR and approximate LLR algorithms (soft-decision) are described in Phase Modulation.
M-PSK Modulator Baseband | M-DPSK Demodulator Baseband
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CamelCase - Maple Help
Home : Support : Online Help : Programming : Names and Strings : StringTools Package : Case Conversion : CamelCase
capitalize each word in a string
Capitalize( s )
CamelCase( s )
The Capitalize(s) command changes each word in the string s to its capitalized form.
A word is a maximal sequence of alphanumeric characters delimited by another character, or the beginning or end of the string.
The capitalized form of a string is the string obtained by replacing each alphabetic character that begins a word with the corresponding uppercase character, and replacing each non-leading alphabetic character in each word with the corresponding lowercase character. Characters that are not part of a word are not affected.
The CamelCase(s) command converts a string of concatenated words to camel case. The term camel case refers to the capitalization convention used when forming programming language identifiers from multiple words, in which the first letter of each constituent word is capitalized to aid readability. For example, LinearAlgebraicGroups is a camel case string, whereas linearalgebraicgroups and linear_algebraic_groups follow different conventions.
Sometimes, a string like linearAlgebraicGroups is also referred to as camel case, wherein only interior word starts are capitalized. This form of camel case can be obtained from the output of CamelCase by converting the first letter of the result to lowercase.
8
\mathrm{with}\left(\mathrm{StringTools}\right):
\mathrm{Capitalize}\left("This is a test."\right)
\textcolor[rgb]{0,0,1}{"This Is A Test."}
\mathrm{Capitalize}\left("hello"\right)
\textcolor[rgb]{0,0,1}{"Hello"}
\mathrm{Capitalize}\left("foo/bar/baz"\right)
\textcolor[rgb]{0,0,1}{"Foo/Bar/Baz"}
\mathrm{CamelCase}\left("camelcase"\right)
\textcolor[rgb]{0,0,1}{"CamelCase"}
\mathrm{CamelCase}\left("linearalgebra"\right)
\textcolor[rgb]{0,0,1}{"LinearAlgebra"}
\mathrm{CamelCase}\left("linearalgebraicgroups"\right)
\textcolor[rgb]{0,0,1}{"LinearAlgebraicGroups"}
\mathrm{LowerCase}\left(\mathrm{CamelCase}\left("linearalgebraicgroups"\right),1..1\right)
\textcolor[rgb]{0,0,1}{"linearAlgebraicGroups"}
There is not always a unique, correct solution to the camel case problem for a given string. In the following example, the result is probably not what was intended.
\mathrm{CamelCase}\left("thecatsatonthemat"\right)
\textcolor[rgb]{0,0,1}{"TheCatSatOnThemAt"}
Since the apostrophe is a word separator, the result in the next example is not the string "Don't"
\mathrm{Capitalize}\left("DON\text{'}T!"\right)
\textcolor[rgb]{0,0,1}{"Don\text{'}T!"}
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The CAP Theorem Is Not a Theorem
After reading the attempted proof of the CAP conjecture [Lynch], I have come to the same conclusion as Mark Burgess [Burgess]. He writes:
Brewer's original conjecture has not been proven with mathematical rigour -- indeed, the formulation in terms of C, A and P is too imprecise for that to happen. So the status of CAP as a theorem is something of an urban myth [...].
Also, as a piece of scientific work, the paper is somewhat difficult to understand today, because it builds on a very particular set of unstated assumptions, jargon and mode of thinking that was rooted in the time of its writing. Moreover, it provides no precise definitions of any of the quantities referred to, particularly the much-discussed "P" ("tolerance" of packet loss). Part of the problem is that the conjecture is about network fault tolerance, but it never models the network at all.
Despite this, CAP is not useless, quite the contrary. It has spawned numerous discussions on the limitations of the availability and consistency that distributed services can provide.
I would like to see more mathematical rigor in computers science in general and for databases in particular. Data storage is a field where mathematics can be truly helpful. More on that some other time.
[Lynch] Brewer's conjecture and the feasibility of consistent, available, partition-tolerant web services, Nancy Lynch and Seth Gilbert, ACM SIGACT News, Volume 33 Issue 2 (2002), pg. 51-59.
[Burgess] Deconstructing the `CAP theorem' for CM and DevOps, Mark Burgess, http://markburgess.org/blog_cap.html
Labels: CAP, consistency, database
"ACID" Does Not Make Sense
OK, that title is a bit provocative, but it seems to have caught your attention. A more sensible title would have been "A Critique of the ACID Properties".
Even though I have studied databases for some time, I have always had a problem with the commonly referenced ACID concept. It never really made sense to me. At least as four independent properties. So I decided to study the roots of it.
The Transaction Concept in [Haerder] and [Gray1]
"ACID" was first mentioned in [Haerder], 1983 (Principles of Transaction-Oriented Database Recovery). Let me quote:
Atomicity. It must be of the all-or-nothing-type described above, and the user must, whatever happens, know which state he or she is in.
Consistency. A transaction reaching its normal end (EOT, end of transaction), thereby committing its results, preserves the consistency of the database. In other words, each successful transaction by definition commits only legal results. This condition is necessary for the fourth property, durability.
Isolation. Events within a transaction must be hidden from other transactions running concurrently. If this were not the case, a transaction could not be reset to its beginning for the reasons sketched above. The techniques that achieve isolation are known as synchronization, and since Gray et al. [1976] there have been numerous contributions to this topic of database research [Kohler 1981].
Durability. Once a transaction has been completed and has committed its results to the database, the system must guarantee that these results survive any subsequent malfunctions. [...].
These four properties, atomicity, consistency, isolation, and durability (ACID), describe the major highlights of the transaction paradigm, which has influenced many aspects of development in database systems. We therefore consider the question of whether the transaction is supported by a particular system to be the ACID test of the system's quality.
The authors of [Haerder] says that the section quoted above relies on the concept of a transaction in [Gray1]. In that paper, Jim Gray writes: "The transaction concept derives from contract law". And further:
The transaction concept emerges with the following properties:
Consistency: the transaction must obey legal protocols.
Atomicity: either it happens or it does not; either all are bound by the contract or none are.
Durability: once a transaction is committed, it cannot be abrogated.
Haerder added "Isolation" to the previous three properties introduced by Gray.
Neither [Gray1] nor [Haerder], provide an exact definition of the transaction properties and I believe this was never the intention of the authors.
I have two main issues with the standard use of the term "ACID" as defined by [Haerder] (or Wikipedia).
The ACID properties are not well-defined. There are multiple interpretations. A more stringent formal definition would be useful. These properties should and can be defined with mathematical preciseness.
My preferred interpretation of "atomicity" would imply the "isolation" property.
I interpret the all-or-nothing notion of atomicity to mean that a reader of the database will see either all or none of the updates that a transaction makes to the database. Let's call this interpretation Atomic1.
The current text from Wikipedia on Atomicity says:
In an atomic transaction, a series of database operations either all occur, or nothing occurs. A guarantee of atomicity prevents updates to the database occurring only partially, which can cause greater problems than rejecting the whole series outright. In other words, atomicity means indivisibility and irreducibility.
The text says that either all operations occur or nothing occurs and that atomicity means indivisibility. To me, this wording imply isolation of transactions since a transaction cannot see the partial updates made by another transaction.
Another interpretation of atomicity (lets call it Atomic2) is that a transaction either completely survives a database crash or is completely ignored when the database restarts. While the database is running atomicity (Atomic2) means that eventually all updates of the transaction are visible to all readers or none of them is visible to any reader. However, while the transaction is running, readers may see uncommitted updates. With this interpretation of atomicity, the isolation property makes sense. We could then have transactions that have the atomicity property, but not necessarily the isolation property.
With the Atomic2 interpretation, the isolation property is meaningful and the ANSI SQL isolation levels are useful. However, there are multiple problems with those. See [Gray2]. The definitions of the isolation levels are ambiguous and and the set of isolation levels are biased towards pessimistic concurrency control (locking). These isolation levels are sometimes not useful for optimistic conconcurrency control.
The simplest solution would be if databases that claim to support transactions would always use the isolation level SERIALIZABLE. Then, the isolation property can be dropped and ACD (Atomic1 interpretation) would be the only properties of interest. There would be no inconsistent reads: Dirty Reads, Non-Repeatable Reads, and Phantom reads. This would obviously make the world a little bit simpler for us programmers. I will probably write about the feasibility of not relaxing the isolation property of transactions in a later blog post.
The database I am working on, BergDB, only supports the strongest isolation level: SERIALIZABLE. Actually, the only way to change the state of the database is through strong transactions. With optimistic concurrency control (STM), there is little to gain by introducing the read issues that can occur due to lack of isolation between transactions (Dirty Reads, Non-Repeatable Reads, and Phantom reads).
ACD transaction
Let us consider the original proposal by Gray with three essential properties of a transaction: atomicity, consistency, durability. Let us further consider a database with a global state that evolves over time from one state to the next by executing a transaction. Such transactions are easy to define in a precise way. Below, a transaction, f, is defined as a mathematical function that takes the database from one state to the next.
The first "property", atomicity, is part of the definition of the database and transactions. It implies that a reader can only read the distinct database states S0, S1, ... There is no visible state between those states. The other two properties are independent. We may have a transaction that is not consistent or one that is not durable. However, according the definition, it would not be a "transaction" then.
The above definition of durability is intentionally vague. It is an important concept worthy its own discussion. There is no such thing as a 100 per cent durability guarantee. Data could always be lost. Instead, the probability that a transaction result can be read in the future is a more accurate (but more complicated) model.
In my opinion, the original definitions of the ACID properties of database transactions are not well defined and confusing. There are multiple interpretations, and there seems to be no precise mathematical definition of them (please tell me, if you know one). I would prefer to work with the original three properties proposed by Gray: atomicity, consistency and durability. This set of properties can easily be defined in a precise way. An attempt to do so was made in this article.
[Gray1] The Transaction Concept, Jim Gray, 1981.
[Gray2] A Critique of ANSI SQL Isolation Levels, Berenson, Bernstein, Gray et al, 1995.
[Haerder] Principles of Transaction-Oriented Database Recovery, Haerder and Reuter, 1983.
Labels: ACID, database
MathML: A Test
This blog post is to test how well MathLM works with browsers and with Blogger
This expression:
{\int }_{-1}^{1}\sqrt{1-{x}^{2}}dx=\frac{\pi }{2}
is pretty cool to have inline.
{S}_{i+1}={t}_{i}\left({S}_{i}\right)
The MathML code was generated with the editor at
http://www1.chapman.edu/~jipsen/mathml/asciimatheditor/
Java: Check your memory access time
Waiting for data is likely the most common thing your CPU is up to. Even when you CPU is running at 100%, it is likely to waiting for data from RAM. Random access to RAM memory is often a bottleneck in a "CPU-bound" Java program. Such programs should really be called RAM-bound, since their performance is bound by the performance of the RAM, not the CPU.
Out of curiosity, let's measure the performance of memory access from Java. We limit the test to summing values of a single long[] array. The memory size of the array is 0.8 MB when testing the cache and 400 MB for testing the RAM. The array is accesses either sequentially or using random access. So, we have a total of four test cases: cache with sequential access, cache with random access, RAM with sequential access, and RAM with random access.
My setup: an i7-3687U CPU (2.10 GHz), Ubuntu 12.04 64-bit, 4 MiB cache, Oracle JVM 1.7.0_25 (64-Bit, build 23.25-b01). The "-server" flag is used when running Java. Computer: Toshiba Portege Z930-14L.
My Java code. Consider it public domain, do whatever you like with it. It's all in one class and it has no dependencies besides a JVM. There are deficiences, of course. Code style has sometime been sacrificed for performance. Loop unrolling did not improve performance, however. There are many other things that happens then just accessing memory. Computation of the next index is a significant part of the cacheRan test.
Test Access time
cacheSeq 0.33 ns, 0.69 cycles
cacheRan 3.16 ns, 6.64 cycles
ramSeq 0.56 ns, 1.37 cycles
ramRan 20.07 ns, 42.16 cycles
Sequential access to the long[] array is surprisingly fast. It is interesting to note that accessing data from the cache in sequence results in less than a cycle per access. I believe this could be because SSE instructions are used. I have not confirmed this. It could also be because of Intel's turbo boost technology. The cycle time assumed is based on a frequency of 2.10 GHz.
Sequential access in RAM is on par with the result for sequential access to cached data. Random access to the cache is significantly slower (3 ns) and random access to the big RAM memory is many times slower (20 ns).
Cache-aware algorithms are important. More on that later.
Check the memory access time on your computer! Just run the Java class provided. Add the result as a comment, please.
UPDATE: Needless to say (?). It is very seldom a programmer would ever need to consider making his code cache-aware. Other things (disk/network IO) usually limits application performance. And the value of the product is seldom limited by its CPU performance anyway. Cache-aware algorithms are however interesting for me while creating BergDB. I aim at very high performance for data that fits into RAM - it should be on par with databases that assumes all data fits into RAM. It is also interesting for those who writes code for basic data structures: maps, set, lists and so on.
Welcome to my blog! I am Frans Lundberg, a curious software engineer and physicist who have spent quite some time on developing algorithms and software for the BergDB database. In this blog, I will share some thoughts and facts on databases and data storage in general. Everything from theory to measurements of disk performance is of interest.
I enjoy scientific papers on the subject and aim to keep a scientific approach. If there are experiments (performance tests, for example), I intend to make sure they are possible to repeat. Facts, knowledge, figures should have references. Tell me if I slip. This blog is hopefully useful for others, but it is also written for myself as a way of storing thoughts, results and useful references for future use.
Don't expect a post per day, or per week. I prefer quality over quantity and hope I can avoid polluting the Internet with even more garbage. Use your RSS reader to get updated when new posts are published.
Thoughts and opinions will be included in the blog, but I am not writing to preach or convince the reader of anything in particular. I am just a curious learner who wants to learn more about data storage. Hopefully, I can learn more together with you, my dear reader.
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Quantile loss using bag of regression trees - MATLAB - MathWorks Nordic
quantileError
Estimate In-Sample Quantile Regression Error
Find Appropriate Ensemble Size Using Quantile Regression Error
Quantile loss using bag of regression trees
err = quantileError(Mdl,X)
err = quantileError(Mdl,X,ResponseVarName)
err = quantileError(Mdl,X,Y)
err = quantileError(___,Name,Value)
err = quantileError(Mdl,X) returns half of the mean absolute deviation (MAD) from comparing the true responses in the table X to the predicted medians resulting from applying the bag of regression trees Mdl to the observations of the predictor data in X.
Mdl must be a TreeBagger model object.
The response variable name in X must have the same name as the response variable in the table containing the training data.
err = quantileError(Mdl,X,ResponseVarName) uses the true response and predictor variables contained in the table X. ResponseVarName is the name of the response variable and Mdl.PredictorNames contain the names of the predictor variables.
err = quantileError(Mdl,X,Y) uses the predictor data in the table or matrix X and the response data in the vector Y.
err = quantileError(___,Name,Value) uses any of the previous syntaxes and additional options specified by one or more Name,Value pair arguments. For example, specify quantile probabilities, the error type, or which trees to include in the quantile-regression-error estimation.
Sample data used to estimate quantiles, specified as a numeric matrix or table.
Each row of X corresponds to one observation, and each column corresponds to one variable. If you specify Y, then the number of rows in X must be equal to the length of Y.
The variables making up the columns of X must have the same order as the predictor variables that trained Mdl (stored in Mdl.PredictorNames).
If you trained Mdl using a table (for example, Tbl), then X can be a numeric matrix if Tbl contains all numeric predictor variables. If Tbl contains heterogeneous predictor variables (for example, numeric and categorical data types), then quantileError throws an error.
Specify Y for the true responses.
quantileError does not support multicolumn variables or cell arrays other than cell arrays of character vectors.
If you trained Mdl using a table (for example, Tbl), then all predictor variables in X must have the same variable names and data types as those variables that trained Mdl (stored in Mdl.PredictorNames). However, the column order of X does not need to correspond to the column order of Tbl. Tbl and X can contain additional variables (response variables, observation weights, etc.).
If you trained Mdl using a numeric matrix, then the predictor names in Mdl.PredictorNames and corresponding predictor variable names in X must be the same. To specify predictor names during training, see the PredictorNames name-value pair argument of TreeBagger. All predictor variables in X must be numeric vectors. X can contain additional variables (response variables, observation weights, etc.).
If X contains the response variable:
If the response variable has the same name as the response variable that trained Mdl, then you do not have to supply the response variable name or vector of true responses. quantileError uses that variable for the true responses by default.
You can specify ResponseVarName or Y for the true responses.
Response variable name, specified as a character vector or string scalar. ResponseVarName must be the name of the response variable in the table of sample data X.
If the table X contains the response variable, and it has the same name as the response variable used to train Mdl, then you do not have to specify ResponseVarName. quantileError uses that variable for the true responses by default.
Y — True responses
True responses, specified as a numeric vector. The number of rows in X must be equal to the length of Y.
For 'cumulative' and 'individual', if you include fewer trees in quantile estimation using Trees or UseInstanceForTree, then the number of rows in err decreases from Mdl.NumTrees.
ones(size(X,1),1) (default) | numeric vector of positive values
Observation weights, specified as the comma-separated pair consisting of 'Weights' and a numeric vector of positive values with length equal to size(X,1). quantileError uses Weights to compute the weighted average of the deviations when estimating the quantile regression error.
By default, quantileError attributes a weight of 1 to each observation, which yields an unweighted average of the deviations.
Quantile probability, specified as the comma-separated pair consisting of 'Quantile' and a numeric vector containing values in the interval [0,1]. For each element in Quantile, quantileError returns corresponding quantile regression errors for all probabilities in Quantile.
For 'all', quantileError uses all trees in the ensemble (that is, the indices 1:Mdl.NumTrees).
If you specify 'Mode','individual', then quantileError ignores TreeWeights.
If UseInstanceForTree(j,k) = true, then quantileError uses the tree in Mdl.Trees(k) when it predicts the response for the observation X(j,:).
You can estimate quantiles using the response data in Mdl.Y directly instead of using the predictions from the random forest by specifying a row composed entirely of false values. For example, to estimate the quantile for observation j using the response data, and to use the predictions from the random forest for all other observations, specify this matrix:
Values other than the default can affect the number of rows in err. Also, the value of Trees affects the value of UseInstanceForTree. Suppose that U is the value of UseInstanceForTree. quantileError ignores the columns of U corresponding to trees not being used in estimation from the specification of Trees. That is, quantileError resets the value of 'UseInstanceForTree' to U(:,trees), where trees is the value of 'Trees'.
err — Half of quantile regression error
Half of the quantile regression error, returned as a numeric scalar or T-by-numel(tau) matrix. tau is the value of Quantile.
T depends on the values of Mode, Trees, UseInstanceForTree, and Quantile. Suppose that you specify 'Trees',trees and you use the default value of 'UseInstanceForTree'.
For 'Mode','cumulative', err is a numel(trees)-by-numel(tau) numeric matrix. err(j,k) is the tau(k) cumulative quantile regression error using the learners in Mdl.Trees(trees(1:j)).
For 'Mode','ensemble', err is a 1-by-numel(tau) numeric vector. err(k) is the tau(k) cumulative quantile regression error using the learners in Mdl.Trees(trees).
For 'Mode','individual', err is a numel(trees)-by-numel(tau) numeric matrix. err(j,k) is the tau(k) quantile regression error using the learner in Mdl.Trees(trees(j)).
Mdl = TreeBagger(100,X,'MPG','Method','regression');
Perform quantile regression, and estimate the MAD of the entire ensemble using the predicted conditional medians.
Because X is a table containing the response and commensurate variable names, you do not have to specify the response variable name or data. However, you can specify the response using this syntax.
err = quantileError(Mdl,X,'MPG')
Randomly split the data into two sets: 75% training and 25% testing. Extract the subset indices.
cvp = cvpartition(size(X,1),'Holdout',0.25);
idxTrn = training(cvp);
Train an ensemble of bagged regression trees using the training set. Specify 250 weak learners.
Mdl = TreeBagger(250,X(idxTrn,:),'MPG','Method','regression');
Estimate the cumulative 0.25, 0.5, and 0.75 quantile regression errors for the test set. Pass the predictor data in as a numeric matrix, and the response data in as a vector.
err = quantileError(Mdl,X{idxTest,1:3},MPG(idxTest),'Quantile',[0.25 0.5 0.75],...
'Mode','cumulative');
err is a 250-by-3 matrix of cumulative quantile regression errors. Columns correspond to quantile probabilities and rows correspond to trees in the ensemble. The errors are cumulative, so they incorporate aggregated predictions from previous trees. Although, Mdl was trained using a table, if all predictor variables in the table are numeric, then you can supply a matrix of predictor data instead.
Plot the cumulative quantile errors on the same plot.
ylabel('Quantile error');
title('Cumulative Quantile Regression Error')
Training using about 60 trees appears to be enough for the first two quartiles, but the third quartile requires about 150 trees.
{L}_{\tau }=\tau \frac{\sum _{\left\{j:{y}_{j}\ge {\stackrel{^}{y}}_{\tau ,j}\right\}}{w}_{j}\left({y}_{j}-{\stackrel{^}{y}}_{\tau ,j}\right)}{\sum _{j=1}^{n}{w}_{j}}+\left(1-\tau \right)\frac{\sum _{\left\{j:{y}_{j}<{\stackrel{^}{y}}_{\tau ,j}\right\}}{w}_{j}\left({\stackrel{^}{y}}_{\tau ,j}-{y}_{j}\right)}{\sum _{j=1}^{n}{w}_{j}}.
{\stackrel{^}{y}}_{\tau ,j}
To tune the number of trees in the ensemble, set 'Mode','cumulative' and plot the quantile regression errors with respect to tree indices. The maximal number of required trees is the tree index where the quantile regression error appears to level off.
To investigate the performance of a model when the training sample is small, use oobQuantileError instead.
[1] Breiman, L. Random Forests. Machine Learning 45, pp. 5–32, 2001.
error | oobQuantileError | quantilePredict | TreeBagger
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Compute RLC parameters of radial copper cables with single screen, based on conductor and insulator characteristics - MATLAB power_cableparam - MathWorks Nordic
power_cableparam
[r l c z]
Building the RLC Matrices
Compute RLC parameters of radial copper cables with single screen, based on conductor and insulator characteristics
[r,l,c,z] = power_cableparam(CableData)
power_cableparam opens the Cable Parameters Tool with the default cable parameter values provided by Simscape™ Electrical™ Specialized Power Systems software.
[r,l,c,z] = power_cableparam(CableData) computes the impedances and capacitances of a structure, CableData, that represents a set of cables that have a screen conductor.
CableData — Conductor and insulator characteristics
Conductor and insulator characteristics for a set of cables that have a screen conductor, specified as a structure with the following fields:
the frequency in hertz to be used to evaluate RLC parameters
rh0_e
the ground resistivity (in ohm.meters)
n_ba
the number of strands contained in one phase conductor
d_ba
diameter of one strand (in m)
rho_ba
DC resistivity of conductor in ohms*m.
mu_r_ba
phase conductor outside diameter (in m)
DC resistivity of the screen conductor in ohms*m.
Total section of screen conductor (in m^2)
screen conductor internal diameter (in m)
screen conductor external diameter (in m)
GMD_phi
d_iax
phase-screen insulator internal diameter (in m)
phase-screen insulator external diameter (in m)
epsilon_iax
relative permittivity of the phase-screen insulator material.
d_ixe
outer screen insulator internal diameter (in m)
outer screen insulator external diameter (in m)
epsilon_ixe
relative permittivity of the outer screen insulator material.
[r l c z] — Parameters of radial copper cables
Parameters of the radial copper cables, returned as a structure with the following fields:
Variable, Field
Self resistance of phase conductor, in Ohm/Km
r.xx
Self resistance of screen conductor, in Ohm/Km
Mutual resistance between the phase conductors, in Ohm/Km
Mutual resistance between phase and screen conductors, in Ohm/Km
Self inductance of phase conductor, in Henries/Km
Self inductance of screen conductor, in Henries/Km
Mutual inductance between the phase conductors, in Henries/Km
Mutual inductance between phase and screen conductor, in Henries/Km
c.ax
Capacitance between the phase conductor and its screen conductor, in Farad/Km
c.xe
Capacitance between the screen conductor and the ground, in Farad/Km
Self impedance of phase conductor, in Ohm/Km
Self impedance of screen conductor, in Ohm/Km
z.ab
Mutual impedance between phase conductors, in Ohm/Km
z.ax
Mutual impedance between phase and corresponding screen conductors, in Ohm/Km
These computed resistances, impedances, and capacitances need to be organized into 2N-by-2N matrices that can be directly used in the block you selected to model your cable. See the 4 Cables with screen (PI model) block in the power_cable example.
The RLC matrices are defined as follows (the example is given for a 3-cable configuration):
\begin{array}{cc}R=\left[\begin{array}{cccccc}{r}_{aa}& {r}_{ax}& {r}_{ab}& {r}_{ab}& {r}_{ab}& {r}_{ab}\\ {r}_{ax}& {r}_{xx}& {r}_{ab}& {r}_{ab}& {r}_{ab}& {r}_{ab}\\ {r}_{ab}& {r}_{ab}& {r}_{aa}& {r}_{ax}& {r}_{ab}& {r}_{ab}\\ {r}_{ab}& {r}_{ab}& {r}_{ax}& {r}_{xx}& {r}_{ab}& {r}_{ab}\\ {r}_{ab}& {r}_{ab}& {r}_{ab}& {r}_{ab}& {r}_{aa}& {r}_{ax}\\ {r}_{ab}& {r}_{ab}& {r}_{ab}& {r}_{ab}& {r}_{ax}& {r}_{xx}\end{array}\right]& L=\left[\begin{array}{cccccc}{l}_{aa}& {l}_{ax}& {l}_{ab}& {l}_{ab}& {l}_{ab}& {l}_{ab}\\ {l}_{ax}& {l}_{xx}& {l}_{ab}& {l}_{ab}& {l}_{ab}& {l}_{ab}\\ {l}_{ab}& {l}_{ab}& {l}_{aa}& {l}_{ax}& {l}_{ab}& {l}_{ab}\\ {l}_{ab}& {l}_{ab}& {l}_{ax}& {l}_{xx}& {l}_{ab}& {l}_{ab}\\ {l}_{ab}& {l}_{ab}& {l}_{ab}& {l}_{ab}& {l}_{aa}& {l}_{ax}\\ {l}_{ab}& {l}_{ab}& {l}_{ab}& {l}_{ab}& {l}_{ax}& {l}_{xx}\end{array}\right]\end{array}
C=\left[\begin{array}{cccccc}{c}_{ax}& -{c}_{ax}& 0& 0& 0& 0\\ -{c}_{ax}& {c}_{ax}+{c}_{xe}& 0& 0& 0& 0\\ 0& 0& {c}_{ax}& -{c}_{ax}& 0& 0\\ 0& 0& -{c}_{ax}& {c}_{ax}+{c}_{xe}& 0& 0\\ 0& 0& 0& 0& {c}_{ax}& -{c}_{ax}\\ 0& 0& 0& 0& -{c}_{ax}& {c}_{ax}+{c}_{xe}\end{array}\right]
For an example using the power_cableparam function, see the power_cable model.
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Write an equation and solve it to find the answer to the question below. Use the 5-D Process to help you write the equation, if needed. Remember to define your variable and to write your answer in a complete sentence.
25
more red tiles than beige tiles, and three times as many navy-blue tiles as beige tiles. If Susan buys
435
Choose a number for the beige tiles.
25
to the number of beige tiles to give you the number of red tiles.
3
Add the beige, red, and navy-blue tiles together to get the total number of tiles Susan bought. Did your total come out to
435
? If not, try another number for the beige tiles.
Use the eTool below to input the points into the table and graph the results.
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seminars - Completely Positive Noncommutative Kernels
Completely Positive Noncommutative Kernels
It is well known that for a function
K:OmegatimesOmegatomathcalL\left(mathcalY\right)
mathcalL\left(mathcalY\right)
denotes the set of all bounded linear operators on a Hilbert space
mathcalY
) the following are equivalent:
K
is a positive kernel in the sense of Aronszajn (i.e.
su{m}_{i,j=1}^{N}langleK\left(omeg{a}_{i},omeg{a}_{j}\right){y}_{j},{y}_{i}ranglegeq0
omeg{a}_{1},dots,omeg{a}_{N}inOmega
{y}_{1},dots,{y}_{N}inmathcalY
N=1,2,dots
K
is the reproducing kernel for a reproducing kernel Hilbert space
mathcalH\left(K\right)
K
has a Kolmogorov decomposition: There exists an operator-valued function
H:OmegatomathcalL\left(mathcalX,mathcalY\right)
mathcalX
is an auxiliary Hilbert space) such that
K\left(omega,zeta\right)=H\left(omega\right)H\left(zeta{\right)}^{\ast }
In work with Joe Ball and Victor Vinnikov, we extend this result to the setting of free noncommutative function theory with the target set
mathcalL\left(mathcalY\right)
K
mathcalL\left(mathcalA,mathcalL\left(mathcalY\right)\right)
mathcalA
{C}^{\ast }
-algebra. In my talk, I will start with a brief introduction to free noncommutative function theory and follow up with a sketch of our proof. Afterwards, I will discuss some well-known results (e.g. Stinespring's dilation theorem for completely positive maps) which follow as corollaries and talk about more recent work.
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Implement sinusoidal voltage source - Simulink - MathWorks Nordic
Implement sinusoidal voltage source
The AC Voltage Source block implements an ideal AC voltage source. The generated voltage U is described by the following relationship:
U=A\mathrm{sin}\left(\omega t+\varphi \right),\text{ }\omega =2\pi f,\text{ }\varphi =\text{Phase in radians}\text{.}
Negative values are allowed for amplitude and phase. A frequency of 0 and phase equal to 90 degrees specify a DC voltage source. Negative frequency is not allowed; otherwise the software signals an error, and the block displays a question mark in the block icon.
The peak amplitude of the generated voltage, in volts (V). Default is 100.
The phase in degrees (deg). Default is 0.
The source frequency in hertz (Hz). Default is 60.
Select Voltage to measure the voltage across the terminals of the AC Voltage Source block. Default is None.
Specify the generator type of the voltage source. The default value is swing.
Select swing to implement a generator controlling the magnitude and phase angle of its terminal voltage. Specify the reference voltage magnitude and angle in the Swing bus or PV bus voltage and Swing bus voltage angle parameters of the Load Flow Bus block connected to the voltage source terminals.
Select PV to implement a generator controlling its output active power P and voltage magnitude V. Specify P in the Active power generation P parameter of the block. Specify V in the Swing bus or PV bus voltage parameter of the Load Flow Bus block connected to the voltage source terminals. You can control the minimum and maximum reactive power generated by the block by using the Minimum reactive power Qmin and Maximum reactive power Qmax parameters.
Select PQ to implement a generator controlling its output active power P and reactive power Q. Specify P and Q in the Active power generation P and Reactive power generation Q parameters of the block, respectively.
Specify the desired active power generated by the source, in watts. Default is 10e3. This parameter is available if you specify Generator type as PV or PQ.
Specify the desired reactive power generated by the source, in vars. Default is 0. This parameter is available only if you specify Generator type as PQ.
This parameter is available only if you specify Generator type as PV. This parameter indicates the minimum reactive power the source can generate while keeping the terminal voltage at its reference value. Specify the reference voltage as the Swing bus or PV bus voltage parameter of the Load Flow Bus block connected to the source terminals. The default value is -inf, which means that there is no lower limit on the reactive power output.
This parameter is available only if you specify Generator type as PV. This parameter indicates the maximum reactive power the source can generate while keeping the terminal voltage at its reference value. Specify the reference voltage with the Swing bus or PV bus voltage parameter of the Load Flow Bus block connected to the source terminals. The default value is inf, which means that there is no upper limit on the reactive power output.
The power_acvoltage example uses two AC Voltage Source blocks at different frequencies connected in series across a resistor. The sum of the two voltages is read by a Voltage Measurement block.
Controlled Voltage Source, DC Voltage Source, Multimeter
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The optional filter parameter, passed as the index to the Map or Map2 command, restricts the application of
\mathrm{with}\left(\mathrm{LinearAlgebra}\right):
A≔\mathrm{Matrix}\left([[1,2,3],[0,1,4]],\mathrm{shape}=\mathrm{triangular}[\mathrm{upper},\mathrm{unit}]\right)
\textcolor[rgb]{0,0,1}{A}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{4}\end{array}]
M≔\mathrm{Map}\left(x↦x+1,A\right)
\textcolor[rgb]{0,0,1}{M}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{5}\end{array}]
\mathrm{evalb}\left(\mathrm{addressof}\left(A\right)=\mathrm{addressof}\left(M\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
B≔〈〈1,2,3〉|〈4,5,6〉〉
\textcolor[rgb]{0,0,1}{B}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{5}\\ \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{6}\end{array}]
\mathrm{Map2}[\left(i,j\right)↦\mathrm{evalb}\left(i=1\right)]\left(\left(x,a\right)↦a\cdot x,3,B\right)
[\begin{array}{cc}\textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{12}\\ \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{5}\\ \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{6}\end{array}]
\mathrm{Map}\left(x↦x+1,g\left(3,A\right)\right)
\textcolor[rgb]{0,0,1}{g}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{5}\end{array}]\right)
C≔\mathrm{Matrix}\left([[1,2],[3]],\mathrm{scan}=\mathrm{triangular}[\mathrm{upper}],\mathrm{shape}=\mathrm{symmetric}\right)
\textcolor[rgb]{0,0,1}{C}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}\end{array}]
\mathrm{Map}\left(x↦x+1,C\right)
[\begin{array}{cc}\textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{4}\end{array}]
[\begin{array}{cc}\textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{9}\\ \textcolor[rgb]{0,0,1}{9}& \textcolor[rgb]{0,0,1}{4}\end{array}]
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Running a Social Media Promotion - OFN User Guide
To retain loyal customers and/or attract new ones you may from time to time like to run a social media promotion. For example, you might post on Facebook and Instagram that for all orders received in the coming week over the value of £20, customers can claim a 5% discount on their shopping the following week.
Below is a step-by-step guide to implementing the above example. The process relies on the highly flexible Tag and Tag rules facility on the OFN platform. At the end of this page are tips on how to adapt this promotional offer.
Identifying eligible customers who purchased above the threshold value.
Enabling these customers to get a discount the following week.
Removing this discount after one week.
Review your Orders when the current order cycle has closed. You can identify all the customers who have purchased above the threshold using the following:
Filter by Order Cycle name
Sort by total high to low (click on ‘Total’ twice)
Visit your Customers page.
Use the ‘Quick Search’ box to find customers who spent above your threshold on last week’s order cycle.
Add the tag ‘week2reward’ to the customer’s entry.
Visit Enterprises -> Settings and then select ‘Payment Methods’ from the left hand menu. Click + New Payment Method.
Name: Thank you Discount Description: a message of your choice (for example ‘As a small thank you for your support we would like to offer you 5% off this week’s shop’) Display: Both Checkout and Back Office Active: yes Tags: Add the tag ‘week2reward’ into this space. Provider: choose the most appropriate method for your business. Fee Calculator: Flat Percent
After selecting Create, add ‘-5’ * to the ‘Amount’ field of the ‘Fee Calculator’ Section. (Negative sign results in a discount) *-5 will result in a 5% discount if your enterprise does not use Enterprise Fees.
All percentage fees are calculated on a percentage of product costs only.
If your business adds a flat percent Enterprise Fee to all products then the amount you need to enter into the 'Flat Percent' field for this discount payment method is:
= (100 + Enterprise Fee)*Desired Discount/100
eg. for a business with an enterprise fee of 20% who would like to offer a 5% discount to volunteers, the amount to enter in the flat percent of this payment method is:
= -(100 + 20) *5/100 = -6
Visit your Enterprise -> Settings page and select ‘Tag Rules’ from the left hand menu. Set up the following Tag rules:
Default: Payment Methods tagged ‘week2reward’ are not visible. For customers tagged ‘week2reward’ payment methods tagged ‘week2reward’ are visible.
Bingo! Only those customers who spent more than your threshold amount last week will be offered a 5% discount when they shop with you this week
Checkout view for eligible customer
When the order cycle closes, as a hub, you may not want that particular set of customers to be eligible for a 5% discount on their future purchases (ie continue to have a discount for more than one week). In which case you will need to either:
Remove the ‘week2reward’ tag from all the customer names Or
Change the tag rule to: For Customers tagged ‘week2reward’ Payment Methods tagged ‘week2reward’ are NOT VISIBLE.
You might also like to consider rewarding repeat customers with a small discount.
Rewarding only those customers who spend over a threshold amount may exclude those who live on their own or have a more limited household budget. You may like to run a social media campaign to encourage customer loyalty- whatever the value of their weekly/monthly spends are.
To do this you will need to keep an external record of how often each customer purchases from your OFN shopfront. The support team in your local instance maybe able to help compile this data for you.
With this information these are some ideas of campaigns you could run:
5% discount off their next order to customers who order x number weeks/order cycles in a row.
In this case you would use the same steps outlined above to tag both the customer and the payment method (with a negative fee calculator) for one order cycle.
Free or discounted delivery of their next order to customers who have ordered x number of weeks in a row (akin to buy 3 lots of delivery and get the next free).
In this case you would tag the customer (say ‘thankyouweek1’) and then create a shipping method with the same tag (which was at a discounted rate).
The default and complementary tag rules would be:
Default: Shipping Methods tagged thankyouweek1 are NOT VISIBLE Rule: For customers tagged thankyouweek1 Shipping Methods tagged thakyouweek1 are VISIBLE
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In a poultry farm, every 100 birds, provide an avearage of 72 eggs how many birds will produce 324 - Maths - Comparing Quantities - 7083004 | Meritnation.com
In a poultry farm, every 100 birds, provide an avearage of 72 eggs. how many birds will produce 324 eggs?
Since 72 eggs are produced by 100 birds.
So 324 eggs are produced by
\left(\frac{100}{72}\times 324\right)\quad
= 450 birds.
Therefore 450 birds will produce 324 eggs.
birds=b
eggs=e
yb=324e
=y=100*324/72
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Change Datatype of Column Entries - Maple Help
Home : Support : Online Help : Statistics and Data Analysis : DataFrames and DataSeries : DataFrame Commands : Change Datatype of Column Entries
change the datatype for a column in a DataFrame
SubsDatatype( DF, index, newdatatype, options )
newdatatype
type; the new datatype for the given column
conversion : procedure; specifies a procedure to be mapped onto the elements in the given column. This option is entered in the form conversion = procedure.
The SubsDatatype command changes the datatype of the entries in a given column of a DataFrame as well as the indicated datatype of the column.
Internally, the DataFrame/SubsDatatype command uses the DataSeries/SubsDatatype command to change the datatype.
If the conversion option is given, then the values in the DataSeries are converted by this conversion procedure. Otherwise, they are typically not modified, but there are exceptions. Internally, a new Array is constructed with the given datatype, and the current values are used to initialize this Array; this is the step where an automatic conversion could take place.
\mathrm{genus}≔〈"Rubus","Vitis","Fragaria"〉:
\mathrm{energy}≔〈220,288,136〉:
\mathrm{carbohydrates}≔〈11.94,18.1,7.68〉:
\mathrm{top_producer}≔〈\mathrm{Russia},\mathrm{China},\mathrm{USA}〉:
\mathrm{berries}≔\mathrm{DataFrame}\left(〈\mathrm{energy}|\mathrm{carbohydrates}|\mathrm{top_producer}|\mathrm{genus}〉,\mathrm{columns}=[\mathrm{Energy},\mathrm{Carbohydrates},\mathrm{`Top Producer`},\mathrm{Genus}],\mathrm{rows}=[\mathrm{Raspberry},\mathrm{Grape},\mathrm{Strawberry}],\mathrm{datatypes}=[\mathrm{integer},\mathrm{float},\mathrm{anything},\mathrm{string}]\right)
\textcolor[rgb]{0,0,1}{\mathrm{berries}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccccc}\textcolor[rgb]{0,0,1}{}& \textcolor[rgb]{0,0,1}{\mathrm{Energy}}& \textcolor[rgb]{0,0,1}{\mathrm{Carbohydrates}}& \textcolor[rgb]{0,0,1}{\mathrm{Top Producer}}& \textcolor[rgb]{0,0,1}{\mathrm{Genus}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Raspberry}}& \textcolor[rgb]{0,0,1}{220}& \textcolor[rgb]{0,0,1}{11.9400000000000}& \textcolor[rgb]{0,0,1}{\mathrm{Russia}}& \textcolor[rgb]{0,0,1}{"Rubus"}\\ \textcolor[rgb]{0,0,1}{\mathrm{Grape}}& \textcolor[rgb]{0,0,1}{288}& \textcolor[rgb]{0,0,1}{18.1000000000000}& \textcolor[rgb]{0,0,1}{\mathrm{China}}& \textcolor[rgb]{0,0,1}{"Vitis"}\\ \textcolor[rgb]{0,0,1}{\mathrm{Strawberry}}& \textcolor[rgb]{0,0,1}{136}& \textcolor[rgb]{0,0,1}{7.68000000000000}& \textcolor[rgb]{0,0,1}{\mathrm{USA}}& \textcolor[rgb]{0,0,1}{"Fragaria"}\end{array}]
You can get the data types for the columns with the Datatypes command.
\mathrm{Datatypes}\left(\mathrm{berries}\right)
[\textcolor[rgb]{0,0,1}{\mathrm{integer}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathrm{float}}}_{\textcolor[rgb]{0,0,1}{8}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{anything}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{string}}]
You can change the datatype of the Energy column to float:
\mathrm{SubsDatatype}\left(\mathrm{berries},\mathrm{Energy},\mathrm{float}\right)
[\begin{array}{ccccc}\textcolor[rgb]{0,0,1}{}& \textcolor[rgb]{0,0,1}{\mathrm{Energy}}& \textcolor[rgb]{0,0,1}{\mathrm{Carbohydrates}}& \textcolor[rgb]{0,0,1}{\mathrm{Top Producer}}& \textcolor[rgb]{0,0,1}{\mathrm{Genus}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Raspberry}}& \textcolor[rgb]{0,0,1}{220.}& \textcolor[rgb]{0,0,1}{11.9400000000000}& \textcolor[rgb]{0,0,1}{\mathrm{Russia}}& \textcolor[rgb]{0,0,1}{"Rubus"}\\ \textcolor[rgb]{0,0,1}{\mathrm{Grape}}& \textcolor[rgb]{0,0,1}{288.}& \textcolor[rgb]{0,0,1}{18.1000000000000}& \textcolor[rgb]{0,0,1}{\mathrm{China}}& \textcolor[rgb]{0,0,1}{"Vitis"}\\ \textcolor[rgb]{0,0,1}{\mathrm{Strawberry}}& \textcolor[rgb]{0,0,1}{136.}& \textcolor[rgb]{0,0,1}{7.68000000000000}& \textcolor[rgb]{0,0,1}{\mathrm{USA}}& \textcolor[rgb]{0,0,1}{"Fragaria"}\end{array}]
This does not change the datatype in place. To permanently change the datatype:
\mathrm{berries}≔\mathrm{SubsDatatype}\left(\mathrm{berries},\mathrm{Energy},\mathrm{float}\right)
\textcolor[rgb]{0,0,1}{\mathrm{berries}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccccc}\textcolor[rgb]{0,0,1}{}& \textcolor[rgb]{0,0,1}{\mathrm{Energy}}& \textcolor[rgb]{0,0,1}{\mathrm{Carbohydrates}}& \textcolor[rgb]{0,0,1}{\mathrm{Top Producer}}& \textcolor[rgb]{0,0,1}{\mathrm{Genus}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Raspberry}}& \textcolor[rgb]{0,0,1}{220.}& \textcolor[rgb]{0,0,1}{11.9400000000000}& \textcolor[rgb]{0,0,1}{\mathrm{Russia}}& \textcolor[rgb]{0,0,1}{"Rubus"}\\ \textcolor[rgb]{0,0,1}{\mathrm{Grape}}& \textcolor[rgb]{0,0,1}{288.}& \textcolor[rgb]{0,0,1}{18.1000000000000}& \textcolor[rgb]{0,0,1}{\mathrm{China}}& \textcolor[rgb]{0,0,1}{"Vitis"}\\ \textcolor[rgb]{0,0,1}{\mathrm{Strawberry}}& \textcolor[rgb]{0,0,1}{136.}& \textcolor[rgb]{0,0,1}{7.68000000000000}& \textcolor[rgb]{0,0,1}{\mathrm{USA}}& \textcolor[rgb]{0,0,1}{"Fragaria"}\end{array}]
\mathrm{Datatype}\left(\mathrm{berries}[\mathrm{Energy}]\right)
{\textcolor[rgb]{0,0,1}{\mathrm{float}}}_{\textcolor[rgb]{0,0,1}{8}}
When working with strings or name conversions, it may be necessary to supply an explicit conversion for values in the column of the data frame:
\mathrm{berries}≔\mathrm{SubsDatatype}\left(\mathrm{berries},\mathrm{Genus},\mathrm{name},\mathrm{`=`}\left(\mathrm{conversion},x↦\mathrm{convert}\left(x,\mathrm{name}\right)\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{berries}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccccc}\textcolor[rgb]{0,0,1}{}& \textcolor[rgb]{0,0,1}{\mathrm{Energy}}& \textcolor[rgb]{0,0,1}{\mathrm{Carbohydrates}}& \textcolor[rgb]{0,0,1}{\mathrm{Top Producer}}& \textcolor[rgb]{0,0,1}{\mathrm{Genus}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Raspberry}}& \textcolor[rgb]{0,0,1}{220.}& \textcolor[rgb]{0,0,1}{11.9400000000000}& \textcolor[rgb]{0,0,1}{\mathrm{Russia}}& \textcolor[rgb]{0,0,1}{\mathrm{Rubus}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Grape}}& \textcolor[rgb]{0,0,1}{288.}& \textcolor[rgb]{0,0,1}{18.1000000000000}& \textcolor[rgb]{0,0,1}{\mathrm{China}}& \textcolor[rgb]{0,0,1}{\mathrm{Vitis}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Strawberry}}& \textcolor[rgb]{0,0,1}{136.}& \textcolor[rgb]{0,0,1}{7.68000000000000}& \textcolor[rgb]{0,0,1}{\mathrm{USA}}& \textcolor[rgb]{0,0,1}{\mathrm{Fragaria}}\end{array}]
\mathrm{berries}≔\mathrm{SubsDatatype}\left(\mathrm{berries},\mathrm{`Top Producer`},\mathrm{string},\mathrm{`=`}\left(\mathrm{conversion},x↦\mathrm{convert}\left(x,\mathrm{string}\right)\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{berries}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccccc}\textcolor[rgb]{0,0,1}{}& \textcolor[rgb]{0,0,1}{\mathrm{Energy}}& \textcolor[rgb]{0,0,1}{\mathrm{Carbohydrates}}& \textcolor[rgb]{0,0,1}{\mathrm{Top Producer}}& \textcolor[rgb]{0,0,1}{\mathrm{Genus}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Raspberry}}& \textcolor[rgb]{0,0,1}{220.}& \textcolor[rgb]{0,0,1}{11.9400000000000}& \textcolor[rgb]{0,0,1}{"Russia"}& \textcolor[rgb]{0,0,1}{\mathrm{Rubus}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Grape}}& \textcolor[rgb]{0,0,1}{288.}& \textcolor[rgb]{0,0,1}{18.1000000000000}& \textcolor[rgb]{0,0,1}{"China"}& \textcolor[rgb]{0,0,1}{\mathrm{Vitis}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Strawberry}}& \textcolor[rgb]{0,0,1}{136.}& \textcolor[rgb]{0,0,1}{7.68000000000000}& \textcolor[rgb]{0,0,1}{"USA"}& \textcolor[rgb]{0,0,1}{\mathrm{Fragaria}}\end{array}]
\mathrm{Datatypes}\left(\mathrm{berries}\right)
[{\textcolor[rgb]{0,0,1}{\mathrm{float}}}_{\textcolor[rgb]{0,0,1}{8}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathrm{float}}}_{\textcolor[rgb]{0,0,1}{8}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{string}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{name}}]
The DataFrame/SubsDatatype command was introduced in Maple 2017.
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Tensile testing - 3D CAD Models & 2D Drawings
Tensile testing (12456 views - Mechanical Engineering)
Tensile testing, also known as tension testing,[1] is a fundamental materials science and engineering test in which a sample is subjected to a controlled tension until failure. Properties that are directly measured via a tensile test are ultimate tensile strength, breaking strength, maximum elongation and reduction in area.[2] From these measurements the following properties can also be determined: Young's modulus, Poisson's ratio, yield strength, and strain-hardening characteristics.[3] Uniaxial tensile testing is the most commonly used for obtaining the mechanical characteristics of isotropic materials. Some materials use biaxial tensile testing.
2 Tensile specimen
5.3 Flexible materials
Tensile testing might have a variety of purposes, such as:
Select a material or item for an application
Predict how a material will perform in use: normal and extreme forces.
Determine if, or verify that, the requirements of a specification, regulation, or contract are met
Decide if a new product development program is on track: Demonstrate proof of concept
Demonstrate the utility of a proposed patent
Provide evidence in legal proceedings
The preparation of test specimens depends on the purposes of testing and on the governing test method or specification. A tensile specimens is usually a standardized sample cross-section. It has two shoulders and a gage (section) in between. The shoulders are large so they can be readily gripped, whereas the gauge section has a smaller cross-section so that the deformation and failure can occur in this area.[2][4]
The shoulders of the test specimen can be manufactured in various ways to mate to various grips in the testing machine (see the image below). Each system has advantages and disadvantages; for example, shoulders designed for serrated grips are easy and cheap to manufacture, but the alignment of the specimen is dependent on the skill of the technician. On the other hand, a pinned grip assures good alignment. Threaded shoulders and grips also assure good alignment, but the technician must know to thread each shoulder into the grip at least one diameter's length, otherwise the threads can strip before the specimen fractures.[5]
In large castings and forgings it is common to add extra material, which is designed to be removed from the casting so that test specimens can be made from it. These specimens may not be exact representation of the whole workpiece because the grain structure may be different throughout. In smaller workpieces or when critical parts of the casting must be tested, a workpiece may be sacrificed to make the test specimens.[6] For workpieces that are machined from bar stock, the test specimen can be made from the same piece as the bar stock.
Various shoulder styles for tensile specimens. Keys A through C are for round specimens, whereas keys D and E are for flat specimens. Key:
A. A Threaded shoulder for use with a thread
B. A round shoulder for use with serrated grips
C. A butt end shoulder for use with a split collar
D. A flat shoulder for used with serrated grips
E. A flat shoulder with a through hole for a pinned grip
Test specimen nomenclature
The repeatability of a testing machine can be found by using special test specimens meticulously made to be as similar as possible.[6]
A standard specimen is prepared in a round or a square section along the gauge length, depending on the standard used. Both ends of the specimens should have sufficient length and a surface condition such that they are firmly gripped during testing. The initial gauge length Lo is standardized (in several countries) and varies with the diameter (Do) or the cross-sectional area (Ao) of the specimen as listed
United States(ASTM)
Sheet ( Lo / √Ao) 4.5 5.65 11.3
Rod ( Lo / Do) 4.0 5.00 10.0
The following tables gives examples of test specimen dimensions and tolerances per standard ASTM E8.
Flat test specimen[7]
Plate type (1.5 in. wide)
Sheet type (0.5 in. wide)
Sub-size specimen (0.25 in. wide)
Gauge length 8.00±0.01 2.00±0.005 1.000±0.003
Width 1.5 +0.125–0.25 0.500±0.010 0.250±0.005
Thickness 0.188 ≤ T 0.005 ≤ T ≤ 0.75 0.005 ≤ T ≤ 0.25
Fillet radius (min.) 1 0.25 0.25
Overall length (min.) 18 8 4
Length of reduced section (min.) 9 2.25 1.25
Length of grip section (min.) 3 2 1.25
Width of grip section (approx.) 2 0.75 3⁄8
Round test specimen[7]
Standard specimen at nominal diameter:
Small specimen at nominal diameter:
Gauge length 2.00±0.005 1.400±0.005 1.000±0.005 0.640±0.005 0.450±0.005
Diameter tolerance ±0.010 ±0.007 ±0.005 ±0.003 ±0.002
Fillet radius (min.) 3⁄8 0.25 5⁄16 5⁄32 3⁄32
Length of reduced section (min.) 2.5 1.75 1.25 0.75 5⁄8
Tensile testing is most often carried out at a material testing laboratory. The ASTM D638 is among the most common tensile testing protocols. The ASTM D638 measures plastics tensile properties including ultimate tensile strength, yield strength, elongation and Poisson’s ratio.
The most common testing machine used in tensile testing is the universal testing machine. This type of machine has two crossheads; one is adjusted for the length of the specimen and the other is driven to apply tension to the test specimen. There are two types: hydraulic powered and electromagnetically powered machines.[4]
The machine must have the proper capabilities for the test specimen being tested. There are four main parameters: force capacity, speed, precision and accuracy. Force capacity refers to the fact that the machine must be able to generate enough force to fracture the specimen. The machine must be able to apply the force quickly or slowly enough to properly mimic the actual application. Finally, the machine must be able to accurately and precisely measure the gauge length and forces applied; for instance, a large machine that is designed to measure long elongations may not work with a brittle material that experiences short elongations prior to fracturing.[5]
Alignment of the test specimen in the testing machine is critical, because if the specimen is misaligned, either at an angle or offset to one side, the machine will exert a bending force on the specimen. This is especially bad for brittle materials, because it will dramatically skew the results. This situation can be minimized by using spherical seats or U-joints between the grips and the test machine.[5] If the initial portion of the stress–strain curve is curved and not linear, it indicates the specimen is misaligned in the testing machine.[8]
The strain measurements are most commonly measured with an extensometer, but strain gauges are also frequently used on small test specimen or when Poisson's ratio is being measured.[5] Newer test machines have digital time, force, and elongation measurement systems consisting of electronic sensors connected to a data collection device (often a computer) and software to manipulate and output the data. However, analog machines continue to meet and exceed ASTM, NIST, and ASM metal tensile testing accuracy requirements, continuing to be used today.[citation needed]
The test process involves placing the test specimen in the testing machine and slowly extending it until it fractures. During this process, the elongation of the gauge section is recorded against the applied force. The data is manipulated so that it is not specific to the geometry of the test sample. The elongation measurement is used to calculate the engineering strain, ε, using the following equation:[4]
{\displaystyle \varepsilon ={\frac {\Delta L}{L_{0}}}={\frac {L-L_{0}}{L_{0}}}}
where ΔL is the change in gauge length, L0 is the initial gauge length, and L is the final length. The force measurement is used to calculate the engineering stress, σ, using the following equation:[4]
{\displaystyle \sigma ={\frac {F_{n}}{A}}}
where F is the tensile force and A is the nominal cross-section of the specimen. The machine does these calculations as the force increases, so that the data points can be graphed into a stress–strain curve.[4]
^ Czichos, Horst (2006). Springer Handbook of Materials Measurement Methods. Berlin: Springer. pp. 303–304. ISBN 978-3-540-20785-6.
^ a b Davis, Joseph R. (2004). Tensile testing (2nd ed.). ASM International. ISBN 978-0-87170-806-9.
^ a b c d e Davis 2004, p. 2.
^ a b c d Davis 2004, p. 9.
Video on the tensile test
Determining the properties of a material by use of Tensile Testing
Learn more about the ASTM D638 Tensile Test
Control panel (engineering)Key (engineering)Mechanical engineeringShaft (mechanical engineering)Universal testing machineFixture (tool)Jig (tool)
This article uses material from the Wikipedia article "Tensile testing", which is released under the Creative Commons Attribution-Share-Alike License 3.0. There is a list of all authors in Wikipedia
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[[Image:Bettencourt.jpg|200px|left]]
'''Luis Bettencourt''', Research Professor, Santa Fe Institute; Staff Researcher, Theoretical Division, LANL
'''Luis Bettencourt''', External Professor, Santa Fe Institute; Staff Researcher, Theoretical Division, LANL
Luís M. A Bettencourt carries research in the structure and dynamics of several complex systems, with an emphasis on dynamical problems in biology and society. Currently
Luis Bettencourt, External Professor, Santa Fe Institute; Staff Researcher, Theoretical Division, LANL
Luís M. A Bettencourt carries research in the structure and dynamics of several complex systems, with an emphasis on dynamical problems in biology and society. Currently he works on information processing in neural systems, information theoretic optimization in collective behavior, urban organization and dynamics, and the development of science and technology. Luis obtained his PhD from Imperial College, University of London for work on critical phenomena in the early Universe, and associated mathematical techniques of Statistical Physics, Field Theory and Non-linear Dynamics. He held postdoctoral positions at the University of Heidelberg, Germany, as a Director’s Fellow in the Theoretical Division at LANL, and at the Center for Theoretical Physics at MIT. In 2000 he was awarded the distinguished Slansky Fellowship at Los Alamos National Laboratory for excellence in interdisciplinary research. He has been a scientist at LANL since the spring of 2003, first at the Computer and Computational Sciences Division (CCS), and since September 2005 in the Theoretical Division (T-5: Mathematical Modeling and Analysis). He is also External Professor at the Santa Fe Institute.
{\displaystyle Omega}
"Dominos, Ergodic Flows": We present a model, developed with Norman Packard, of a simple discrete open flow system. Dimers are created at one edge of a two-dimensional lattice, diffuse across, and are removed at the opposite side. A steady-state flow is established, under various kinetic rules. In the equilibrium case, the system reduces to the classical monomer-dimer tiling problem, whose entropy as a function of density is known. This entropy density is reproduced locally in the flow system, as shown by statistics over local templates. The goal is to clarify informational aspects of a flowing pattern.
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azdavis.net • Posts • Jul 15, 2021
(Translations: English • 日本語)
In the previous post, we added product type to Hatsugen.
In this post, we'll add sum types.
A sum type is a choice between multiple types. For instance, if we have a function from which we would sometimes like to return one type, and sometimes another, we can have the function return a sum type of those two types.
Or, suppose we wish to represent sometimes having a value of some type, and sometimes having "nothing". We can use a sum type of the actual type and the unit type, and use unit to mean "nothing".
Surprisingly, although many programming languages have product types, not as many have sum types as we will present them here. But, many programming languages that support a "functional" style, like Haskell, OCaml, Standard ML, and Rust, do have sum types.
First we have the binary sum type, denoted with
\tau_1 + \tau_2
and called "either". As the name suggests, values of this type can either be from the left type
\tau_1
or the right type
\tau_2
However, the values must be "tagged" with whether they are the left or right type. We add two new expressions,
\mathsf{L} \ e
\mathsf{R} \ e
, to do this tagging. These are sometimes called "injections".
The sum type that is a choice between no types is written
\mathsf{0}
and called "never", because we can never have a value of this type.
Although it may seem odd to have a type with no values, the never type can be useful. For instance, imagine we added the ability to "exit" a Hatsugen program with a special function. This is a common feature in programming languages:
C/C++ has exit
Python has sys.exit
Java has System.exit
Rust has std::process::exit
Note that this function has the property that it never returns to the caller, since calling it exits the program.
To communicate this, we can have the return type of this function be the never type. When we see that a function returns the never type, we know that the function will indeed never return. Since, if it returned, we would have produced a value of type never, which is impossible.
Indeed, in Sorbet, the static type-checker for Ruby developed at Stripe, the equivalent of the never type is called T.noreturn.
Finally, we add the ability to actually use a sum type with case expressions. These case expressions must handle all of the possibilities that the value of sum type could be.
For the either type, there are two cases: left or right. For the never type, there are no cases to handle.
\begin{aligned} \tau ::= \ & \dots \\ | \ & \mathsf{0} \\ | \ & \tau_1 + \tau_2 \\ \\ e ::= \ & \dots \\ | \ & \mathsf{L} \ e \\ | \ & \mathsf{R} \ e \\ | \ & \mathsf{case} \ e \ \{ \} \\ | \ & \mathsf{case} \ e \ \{ \mathsf{L} \ x_1 . e_1, \mathsf{R} \ x_2 . e_2 \} \end{aligned}
The left injection requires its argument be the left type in the either, but the right type can be any type.
\frac {\Gamma \vdash e: \tau_1} {\Gamma \vdash \mathsf{L} \ e: \tau_1 + \tau_2}
Similar with the right injection.
\frac {\Gamma \vdash e: \tau_2} {\Gamma \vdash \mathsf{R} \ e: \tau_1 + \tau_2}
The empty case requires that its "head" be never type, but permits the result of the case to be any type. This is because we know it is impossible to have a value of never type, so if we achieved the impossible by producing a never, we can do anything we want.
\frac {\Gamma \vdash e: \mathsf{0}} {\Gamma \vdash \mathsf{case} \ e \ \{ \}: \tau}
The either case requires its head be an either type. Then, each respective case binds a variable with the contents of the head. These variables are allowed to be used when evaluating the respective subexpressions for each case. Each of these subexpressions must be the same type. Then, the whole case expression evaluates to that type.
\frac { \begin{aligned} &\Gamma \vdash e: \tau_1 + \tau_2 \\&\Gamma, x_1: \tau_1 \vdash e_1: \tau \\&\Gamma, x_2: \tau_2 \vdash e_2: \tau \end{aligned} } { \Gamma \vdash \mathsf{case} \ e \ \{ \mathsf{L} \ x_1 . e_1, \mathsf{R} \ x_2 . e_2 \}: \tau }
The left injection is a value if its argument is a value.
\frac {e \ \mathsf{val}} {\mathsf{L} \ e \ \mathsf{val}}
Same with the right.
\frac {e \ \mathsf{val}} {\mathsf{R} \ e \ \mathsf{val}}
If the left injection's argument steps, then the whole injection steps.
\frac {e \mapsto e'} {\mathsf{L} \ e \mapsto \mathsf{L} \ e'}
And same as the right.
\frac {e \mapsto e'} {\mathsf{R} \ e \mapsto \mathsf{R} \ e'}
If the empty case's head can step, so can the whole case.
\frac {e \mapsto e'} {\mathsf{case} \ e \ \{ \} \mapsto \mathsf{case} \ e' \ \{ \}}
And same with the binary case.
\frac {e \mapsto e'} { \begin{aligned} & \mathsf{case} \ e \ \{ \mathsf{L} \ x_1 . e_1, \mathsf{R} \ x_2 . e_2 \} \mapsto \\ & \mathsf{case} \ e' \ \{ \mathsf{L} \ x_1 . e_1, \mathsf{R} \ x_2 . e_2 \} \end{aligned} }
When the binary case's head is a value, and it is the left injection, we extract the argument, bind it to the left variable, and step into the left expression.
\frac { \mathsf{L} \ e \ \mathsf{val} \hspace{1em} [x_1 \mapsto e] e_1 = e' } { \mathsf{case} \ \mathsf{L} \ e \ \{ \mathsf{L} \ x_1 . e_1, \mathsf{R} \ x_2 . e_2 \} \mapsto e' }
And similarly with the right.
\frac { \mathsf{R} \ e \ \mathsf{val} \hspace{1em} [x_2 \mapsto e] e_2 = e' } { \mathsf{case} \ \mathsf{R} \ e \ \{ \mathsf{L} \ x_1 . e_1, \mathsf{R} \ x_2 . e_2 \} \mapsto e' }
Just as last time, updating the helper judgments is a mostly mechanical process.
\frac {[x \mapsto e_x] e = e'} {[x \mapsto e_x] \mathsf{L} \ e = \mathsf{L} \ e'}
\frac {[x \mapsto e_x] e = e'} {[x \mapsto e_x] \mathsf{R} \ e = \mathsf{R} \ e'}
\frac {[x \mapsto e_x] e = e'} {[x \mapsto e_x] \mathsf{case} \ e \ \{\} = \mathsf{case} \ e' \ \{\}}
For the binary case, we cheat a bit and re-use the definition of substitution for function literals.
This is convenient since each of the two cases behave similarly to a function, since they bind one variable and evaluate to one expression each. It also lets us re-use the logic for function literals that handles checking whether the bound variable shadows (aka is the same as) the variable to be substituted.
\tau_1
\tau_2
for the type of the arguments on the functions we construct, but it doesn't matter for the purposes of substitution.
\frac { \begin{aligned} & [x \mapsto e_x] e = e' \\& [x \mapsto e_x] \lambda (x_1: \tau_1) \ e_1 = \lambda (x_1: \tau_1) \ e_1' \\& [x \mapsto e_x] \lambda (x_2: \tau_2) \ e_2 = \lambda (x_2: \tau_2) \ e_2' \end{aligned} } { \begin{aligned} & [x \mapsto e_x] \\& \mathsf{case} \ e \ \{ \mathsf{L} \ x_1 . e_1, \mathsf{R} \ x_2 . e_2 \} = \\ & \mathsf{case} \ e' \ \{ \mathsf{L} \ x_1 . e_1', \mathsf{R} \ x_2 . e_2' \} \end{aligned} }
\frac {\mathsf{fv}(e) = s} {\mathsf{fv}(\mathsf{L} \ e) = s}
\frac {\mathsf{fv}(e) = s} {\mathsf{fv}(\mathsf{R} \ e) = s}
\frac {\mathsf{fv}(e) = s} {\mathsf{fv}(\mathsf{case} \ e \ \{\}) = s}
\frac { \mathsf{fv}(e) = s \hspace{1em} \mathsf{fv}(e_1) = s_1 \hspace{1em} \mathsf{fv}(e_2) = s_2 } { \begin{aligned} \mathsf{fv}(&\mathsf{case} \ e \ \{ \mathsf{L} \ x_1 . e_1, \mathsf{R} \ x_2 . e_2 \}) = \\ & s \cup (s_1 \setminus \{ x_1 \}) \cup (s_2 \setminus \{ x_2 \}) \end{aligned} }
Sum types, similarly to product types, are so named because of how the number of values in the sum or product type relates to the number of values in the other types.
|\mathsf{Bool} + \mathsf{1}| = 2 + 1 = 3
\mathsf{L} \ \mathsf{true}
\mathsf{L} \ \mathsf{false}
\mathsf{R} \ \langle \rangle
|\mathsf{0}| = 0
|\tau_1 + \tau_2| = |\tau_1| + |\tau_2|
Duality of sums and products
Sums and products are duals.
To construct a pair of two types, one must provide a value of both types. Then, when using a pair, one may get out either one of the two types.
To construct an either of two types, one may provide a value of either one of the two types. Then, when using an either, one must handle both types.
The proofs are once again on GitHub.
I think this is the end of this little series, at least for now. I'll probably continue to write about programming language related topics, but not specifically by adding to Hatsugen.
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EUDML | Double categories, 2-categories, thin structures and connections. EuDML | Double categories, 2-categories, thin structures and connections.
Double categories, 2-categories, thin structures and connections.
Brown, Ronald, and Mosa, Ghafar H.. "Double categories, 2-categories, thin structures and connections.." Theory and Applications of Categories [electronic only] 5 (1999): 163-175. <http://eudml.org/doc/119853>.
author = {Brown, Ronald, Mosa, Ghafar H.},
keywords = {double category; 2-category; thin structure; connection},
title = {Double categories, 2-categories, thin structures and connections.},
AU - Mosa, Ghafar H.
TI - Double categories, 2-categories, thin structures and connections.
KW - double category; 2-category; thin structure; connection
Marco Grandis, Robert Pare, Adjoint for double categories
R. J. M. Dawson, R. Pare, D. A. Pronk, Free extensions of double categories
double category, 2-category, thin structure, connection
2
Articles by Mosa
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What Is Variable Overhead Efficiency Variance
Variable overhead efficiency variance refers to the difference between the true time it takes to manufacture a product and the time budgeted for it, as well as the impact of that difference. It arises from variance in productive efficiency.
For example, the number of labor hours taken to manufacture a certain amount of product may differ significantly from the standard or budgeted number of hours. Variable overhead efficiency variance is one of the two components of total variable overhead variance, the other being variable overhead spending variance.
Understanding Variable Overhead Efficiency Variance
In numerical terms, variable overhead efficiency variance is defined as:
\begin{aligned} &\text{VOEV} = ( \text{ALH} - \text{BLH} ) \times \text{Hourly Rate} \\ &\textbf{where:} \\ &\text{VOEV} = \text{Variable overhead efficiency variance} \\ &\text{ALH} = \text{Actual labor hours} \\ &\text{BLH} = \text{Budgeted labor hours} \\ &\text{Hourly Rate} = \text{Rate for standard variable overhead} \\ \end{aligned}
VOEV=(ALH−BLH)×Hourly Ratewhere:VOEV=Variable overhead efficiency varianceALH=Actual labor hoursBLH=Budgeted labor hoursHourly Rate=Rate for standard variable overhead
The hourly rate in this formula includes such indirect labor costs as shop foreman and security. If actual labor hours are less than the budgeted or standard amount, the variable overhead efficiency variance is favorable; if actual labor hours are more than the budgeted or standard amount, the variance is unfavorable.
Example of Variable Overhead Efficiency Variance
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Non-perturbative - Wikipedia
Functions that can't be described by perturbation theory
The function e−1/x2. The MacLaurin series is identically zero, but the function is not.
In mathematics and physics, a non-perturbative function or process is one that cannot be described by perturbation theory. An example is the function
{\displaystyle f(x)=e^{-1/x^{2}},}
which does not have a Taylor series at x = 0. Every coefficient of the Taylor expansion around x = 0 is exactly zero, but the function is non-zero if x ≠ 0.
In physics, such functions arise for phenomena which are impossible to understand by perturbation theory, at any finite order. In quantum field theory, 't Hooft–Polyakov monopoles, domain walls, flux tubes, and instantons are examples.[1] A concrete, physical example is given by the Schwinger effect,[2] whereby a strong electric field may spontaneously decay into electron-positron pairs. For not too strong fields, the rate per unit volume of this process is given by,
{\displaystyle \Gamma ={\frac {(eE)^{2}}{4\pi ^{3}}}\mathrm {e} ^{-{\frac {\pi m^{2}}{eE}}}}
which cannot be expanded in a Taylor series in the electric charg{\displaystyle e}
, or the electric field strength
{\displaystyle E}
{\displaystyle m}
is the mass of an electron and we have used units where
{\displaystyle c=\hbar =1}
In theoretical physics, a non-perturbative solution is one that cannot be described in terms of perturbations about some simple background, such as empty space. For this reason, non-perturbative solutions and theories yield insights into areas and subjects that perturbative methods cannot reveal.
Nonperturbative vacuum
^ Shifman, M. (2012). Advanced Topics in Quantum Field Theory. Cambridge University Press. ISBN 978-0-521-19084-8.
^ Schwinger, Julian (1951-06-01). "On Gauge Invariance and Vacuum Polarization". Physical Review. American Physical Society (APS). 82 (5): 664–679. doi:10.1103/physrev.82.664. ISSN 0031-899X.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Non-perturbative&oldid=1085406961"
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Axiom_of_dependent_choice Knowpia
In mathematics, the axiom of dependent choice, denoted by
{\displaystyle {\mathsf {DC}}}
, is a weak form of the axiom of choice (
{\displaystyle {\mathsf {AC}}}
) that is still sufficient to develop most of real analysis. It was introduced by Paul Bernays in a 1942 article that explores which set-theoretic axioms are needed to develop analysis.[a]
A homogeneous relation
{\displaystyle R}
{\displaystyle X}
is called a total relation if for every
{\displaystyle a\in X,}
{\displaystyle b\in X}
{\displaystyle a\,R~b}
The axiom of dependent choice can be stated as follows: For every nonempty set
{\displaystyle X}
and every total relation
{\displaystyle R}
{\displaystyle X,}
there exists a sequence
{\displaystyle (x_{n})_{n\in \mathbb {N} }}
{\displaystyle X}
{\displaystyle x_{n}\,R~x_{n+1}}
{\displaystyle n\in \mathbb {N} .}
{\displaystyle X}
above is restricted to be the set of all real numbers, then the resulting axiom is denoted by
{\displaystyle {\mathsf {DC}}_{\mathbb {R} }.}
Even without such an axiom, for any
{\displaystyle n}
, one can use ordinary mathematical induction to form the first
{\displaystyle n}
terms of such a sequence. The axiom of dependent choice says that we can form a whole (countably infinite) sequence this way.
The axiom
{\displaystyle {\mathsf {DC}}}
is the fragment of
{\displaystyle {\mathsf {AC}}}
that is required to show the existence of a sequence constructed by transfinite recursion of countable length, if it is necessary to make a choice at each step and if some of those choices cannot be made independently of previous choices.
Over Zermelo–Fraenkel set theory
{\displaystyle {\mathsf {ZF}}}
{\displaystyle {\mathsf {DC}}}
is equivalent to the Baire category theorem for complete metric spaces.[1]
It is also equivalent over
{\displaystyle {\mathsf {ZF}}}
to the Löwenheim–Skolem theorem.[b][2]
{\displaystyle {\mathsf {DC}}}
is also equivalent over
{\displaystyle {\mathsf {ZF}}}
to the statement that every pruned tree with
{\displaystyle \omega }
levels has a branch (proof below).
{\displaystyle {\mathsf {DC}}}
is equivalent to a weakened form of Zorn's lemma; specifically
{\displaystyle {\mathsf {DC}}}
is equivalent to the statement that any partial order such that every well-ordered chain is finite and bounded, must have a maximal element.[3]
{\displaystyle \,{\mathsf {DC}}\iff }
Every pruned tree with ω levels has a branch
{\displaystyle (\,\Longleftarrow \,)}
{\displaystyle R}
be an entire binary relation on
{\displaystyle X}
. The strategy is to define a tree
{\displaystyle T}
{\displaystyle X}
of finite sequences whose neighboring elements satisfy
{\displaystyle R.}
Then a branch through
{\displaystyle T}
is an infinite sequence whose neighboring elements satisfy
{\displaystyle R.}
Start by defining
{\displaystyle \langle x_{0},\dots ,x_{n}\rangle \in T}
{\displaystyle x_{k}R\,x_{k+1}}
{\displaystyle 0\leq k<n.}
{\displaystyle R}
is entire,
{\displaystyle T}
is a pruned tree with
{\displaystyle \omega }
levels. Thus,
{\displaystyle T}
has a branch
{\displaystyle \langle x_{0},\dots ,x_{n},\dots \rangle .}
So, for all
{\displaystyle n\geq 0\!:}
{\displaystyle \langle x_{0},\dots ,x_{n},x_{n+1}\rangle \in T,}
{\displaystyle x_{n}R\,x_{n+1}.}
{\displaystyle {\mathsf {DC}}}
{\displaystyle (\,\Longrightarrow \,)}
{\displaystyle T}
be a pruned tree on
{\displaystyle X}
{\displaystyle \omega }
levels. The strategy is to define a binary relation
{\displaystyle R}
{\displaystyle T}
{\displaystyle {\mathsf {DC}}}
produces a sequence
{\displaystyle t_{n}=\langle x_{0},\dots ,x_{f(n)}\rangle }
{\displaystyle t_{n}R\,t_{n+1}}
{\displaystyle f(n)}
is a strictly increasing function. Then the infinite sequence
{\displaystyle \langle x_{0},\dots ,x_{k},\dots \rangle }
is a branch. (This proof only needs to prove this for
{\displaystyle f(n)=m+n.}
) Start by defining
{\displaystyle u\,R\,v}
{\displaystyle u}
is an initial subsequence of
{\displaystyle v,\operatorname {length} (u)>0,\,}
{\displaystyle \operatorname {length} (v)=}
{\displaystyle \operatorname {length} (u)+1.}
{\displaystyle T}
{\displaystyle \omega }
levels,
{\displaystyle R}
is entire. Therefore,
{\displaystyle {\mathsf {DC}}}
implies that there is an infinite sequence
{\displaystyle t_{n}}
{\displaystyle t_{n}\,R\,t_{n+1}.}
{\displaystyle t_{0}=\langle x_{0},\dots ,x_{m}\rangle }
{\displaystyle m\geq 0.}
{\displaystyle x_{m+n}}
be the last element of
{\displaystyle t_{n}.}
{\displaystyle t_{n}=\langle x_{0},\dots ,x_{m},\dots ,x_{m+n}\rangle .}
{\displaystyle k\geq 0,}
{\displaystyle \langle x_{0},\dots ,x_{k}\rangle }
{\displaystyle T}
because it is an initial subsequence of
{\displaystyle t_{0}\,(k\leq m)}
or it is a
{\displaystyle t_{n}\,(k\geq m).}
{\displaystyle \langle x_{0},\dots ,x_{k},\dots \rangle }
is a branch.
Relation with other axiomsEdit
Unlike full
{\displaystyle {\mathsf {AC}}}
{\displaystyle {\mathsf {DC}}}
is insufficient to prove (given
{\displaystyle {\mathsf {ZF}}}
) that there is a non-measurable set of real numbers, or that there is a set of real numbers without the property of Baire or without the perfect set property. This follows because the Solovay model satisfies
{\displaystyle {\mathsf {ZF}}+{\mathsf {DC}}}
, and every set of real numbers in this model is Lebesgue measurable, has the Baire property and has the perfect set property.
The axiom of dependent choice implies the axiom of countable choice and is strictly stronger.[4][5]
^ "The foundation of analysis does not require the full generality of set theory but can be accomplished within a more restricted frame." Bernays, Paul (1942). "Part III. Infinity and enumerability. Analysis" (PDF). Journal of Symbolic Logic. A system of axiomatic set theory. 7 (2): 65–89. doi:10.2307/2266303. JSTOR 2266303. MR 0006333. The axiom of dependent choice is stated on p. 86.
^ Moore states that "Principle of Dependent Choices
{\displaystyle \Rightarrow }
Löwenheim–Skolem theorem" — that is,
{\displaystyle {\mathsf {DC}}}
implies the Löwenheim–Skolem theorem. See table Moore, Gregory H. (1982). Zermelo's Axiom of Choice: Its origins, development, and influence. Springer. p. 325. ISBN 0-387-90670-3.
^ "The Baire category theorem implies the principle of dependent choices." Blair, Charles E. (1977). "The Baire category theorem implies the principle of dependent choices". Bull. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys. 25 (10): 933–934.
^ The converse is proved in Boolos, George S.; Jeffrey, Richard C. (1989). Computability and Logic (3rd ed.). Cambridge University Press. pp. 155–156. ISBN 0-521-38026-X.
^ Wolk, Elliot S. (1983), On the principle of dependent choices and some forms of Zorn's lemma, vol. 26, Canadian Mathematical Bulletin, pp. 365–367, doi:10.4153/CMB-1983-062-5
^ Bernays proved that the axiom of dependent choice implies the axiom of countable choice See esp. p. 86 in Bernays, Paul (1942). "Part III. Infinity and enumerability. Analysis" (PDF). Journal of Symbolic Logic. A system of axiomatic set theory. 7 (2): 65–89. doi:10.2307/2266303. JSTOR 2266303. MR 0006333.
^ For a proof that the Axiom of Countable Choice does not imply the Axiom of Dependent Choice see Jech, Thomas (1973), The Axiom of Choice, North Holland, pp. 130–131, ISBN 978-0-486-46624-8
Jech, Thomas (2003). Set Theory (Third Millennium ed.). Springer-Verlag. ISBN 3-540-44085-2. OCLC 174929965. Zbl 1007.03002.
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EUDML | Permutation separations and complete bipartite factorisations of . EuDML | Permutation separations and complete bipartite factorisations of .
Permutation separations and complete bipartite factorisations of
{K}_{n,n}
Martin, Nigel, and Stong, Richard. "Permutation separations and complete bipartite factorisations of .." The Electronic Journal of Combinatorics [electronic only] 10.1 (2003): Research paper R37, 26 p.-Research paper R37, 26 p.. <http://eudml.org/doc/122971>.
author = {Martin, Nigel, Stong, Richard},
title = {Permutation separations and complete bipartite factorisations of .},
TI - Permutation separations and complete bipartite factorisations of .
Articles by Stong
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EUDML | Generalizing Narayana and Schröder numbers to higher dimensions. EuDML | Generalizing Narayana and Schröder numbers to higher dimensions.
Generalizing Narayana and Schröder numbers to higher dimensions.
Sulanke, Robert A.. "Generalizing Narayana and Schröder numbers to higher dimensions.." The Electronic Journal of Combinatorics [electronic only] 11.1 (2004): Research paper R54, 20 p.-Research paper R54, 20 p.. <http://eudml.org/doc/124258>.
@article{Sulanke2004,
author = {Sulanke, Robert A.},
keywords = {lattice paths; Narayana number; Sister Celine's (Wilf-Zeilberger) method; Schröder numbers},
title = {Generalizing Narayana and Schröder numbers to higher dimensions.},
AU - Sulanke, Robert A.
TI - Generalizing Narayana and Schröder numbers to higher dimensions.
KW - lattice paths; Narayana number; Sister Celine's (Wilf-Zeilberger) method; Schröder numbers
Lukas Braun, Hilbert series of the Grassmannian and
k
-Narayana numbers
lattice paths, Narayana number, Sister Celine's (Wilf-Zeilberger) method, Schröder numbers
Articles by Sulanke
|
EUDML | On the k-analogue of a result in the theory of the Riemann zeta function EuDML | On the k-analogue of a result in the theory of the Riemann zeta function
On the k-analogue of a result in the theory of the Riemann zeta function
Chowla, S.. "On the k-analogue of a result in the theory of the Riemann zeta function." Mathematische Zeitschrift 38 (1934): 483-487. <http://eudml.org/doc/168511>.
author = {Chowla, S.},
title = {On the k-analogue of a result in the theory of the Riemann zeta function},
AU - Chowla, S.
TI - On the k-analogue of a result in the theory of the Riemann zeta function
Laurent Habsieger, Emmanuel Royer,
L
-functions of automorphic forms and combinatorics: Dyck paths
L
\zeta \left(s\right)
L\left(s,\chi \right)
Articles by S. Chowla
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Home : Support : Online Help : Mathematics : Geometry : Polyhedral Sets : Visualizing Sets : PrintLevel
control level of detail when printing polyhedral sets
PrintLevel(n)
non-negative integer, maximum number of constraints to print
This command limits how many relations will be printed when evaluating a polyhedral set, returning the previous value. The default value is 10.
A value of zero disables the structured printing of polyhedral sets.
\mathrm{with}\left(\mathrm{PolyhedralSets}\right):
If PrintLevel is less than the number of relations for a set,
\mathrm{PrintLevel}\left(4\right);
Q≔\mathrm{PolyhedralSet}\left([x+y+z\le 1,0\le x,0\le y,0\le z,\frac{1}{2}\le x+y+z,\frac{1}{2}\le x+y+2z,3\le w+x,-10\le 3w-2x],[w,x,y,z]\right)
\textcolor[rgb]{0,0,1}{10}
\textcolor[rgb]{0,0,1}{Q}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{{}\begin{array}{lll}\textcolor[rgb]{0,0,1}{\mathrm{Coordinates}}& \textcolor[rgb]{0,0,1}{:}& [\textcolor[rgb]{0,0,1}{w}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{z}]\\ \textcolor[rgb]{0,0,1}{\mathrm{Relations}}& \textcolor[rgb]{0,0,1}{:}& [\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{\le }\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{\le }\textcolor[rgb]{0,0,1}{0}\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}{y}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{\le }\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{\le }\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{and 2 more constraints}}]\end{array}
\mathrm{PrintLevel}\left(10\right)
\textcolor[rgb]{0,0,1}{4}
Q
\textcolor[rgb]{0,0,1}{{}\begin{array}{lll}\textcolor[rgb]{0,0,1}{\mathrm{Coordinates}}& \textcolor[rgb]{0,0,1}{:}& [\textcolor[rgb]{0,0,1}{w}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{z}]\\ \textcolor[rgb]{0,0,1}{\mathrm{Relations}}& \textcolor[rgb]{0,0,1}{:}& [\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{\le }\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{\le }\textcolor[rgb]{0,0,1}{0}\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}{y}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{\le }\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{\le }\textcolor[rgb]{0,0,1}{0}\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}{z}\textcolor[rgb]{0,0,1}{\le }\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{w}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{\le }\textcolor[rgb]{0,0,1}{-3}]\end{array}
\mathrm{PrintLevel}\left(0\right)
\textcolor[rgb]{0,0,1}{10}
Q
\textcolor[rgb]{0,0,1}{\mathrm{Object<<PolyhedralSet,140122312386752>>}}
The PolyhedralSets[PrintLevel] command was introduced in Maple 2015.
|
3 Ways to Calculate the Harmonic Mean - wikiHow
How to Calculate the Harmonic Mean
1 Setting Up the Formula
2 Calculating the Harmonic Mean by Hand or Calculator
3 Calculating the Harmonic Mean using Excel
The harmonic mean is a way to calculate the mean, or average, of a set of numbers. Using the harmonic mean is most appropriate when the set of numbers contains outliers that might skew the result. Most people are familiar with calculating the arithmetic mean, in which the sum of values is divided by the number of values. Calculating the harmonic mean is a little more complicated. If working with a small set of numbers you may be able to solve by hand using the formula. Otherwise, you can easily use Microsoft Excel to find the harmonic mean.
Setting Up the Formula
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Set up the formula for the harmonic mean. The formula is
{\displaystyle {\frac {n}{{\frac {1}{a_{1}}}+{\frac {1}{a_{2}}}+{\frac {1}{a_{3}}}+...+{\frac {1}{a_{n}}}}}}
{\displaystyle n}
is the number of values in the set of numbers, and
{\displaystyle a_{1}}
{\displaystyle a_{2}}
{\displaystyle a_{3}...}
are the values in the set.[1] X Research source
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Determine the values you need to find the harmonic mean for. This can be any set of numbers.
For example, you may need to find the harmonic mean for the numbers 10, 12, 16, and 8.
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<b class="whb">{\displaystyle n}</b>
into the formula. This will equal the number of values in your set.
For example, if you are finding the harmonic mean of the numbers 10, 12, 16, and 8, you are working with 4 values, the numerator of your formula will be 4:
{\displaystyle {\frac {4}{{\frac {1}{a_{1}}}+{\frac {1}{a_{2}}}+{\frac {1}{a_{3}}}+...+{\frac {1}{a_{n}}}}}}
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Plug the values your are averaging into your formula. You will take the reciprocal of each number and add them in the denominator of the formula.[2] X Research source Remember, when you take the reciprocal of a whole number, you turn the number into a fraction by placing a 1 in the numerator and the whole number in the denominator.
For example, if the values in your set are 10, 12, 16, and 8, you would place the fractions
{\displaystyle {\frac {1}{10}}}
{\displaystyle {\frac {1}{12}}}
{\displaystyle {\frac {1}{16}}}
{\displaystyle {\frac {1}{8}}}
in your denominator:
{\displaystyle {\frac {4}{{\frac {1}{10}}+{\frac {1}{12}}+{\frac {1}{16}}+{\frac {1}{8}}}}}
Calculating the Harmonic Mean by Hand or Calculator
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Add the fractions in the denominator. You can use a calculator, or add them up by hand. If you are not using a calculator, remember to find a common denominator first. To learn more about adding fractions, read Add Fractions.
{\displaystyle {\frac {4}{{\frac {1}{10}}+{\frac {1}{12}}+{\frac {1}{16}}+{\frac {1}{8}}}}={\frac {4}{\frac {89}{240}}}}
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Divide the numerator by the denominator. Remember that dividing by a fraction is the same as multiplying by its reciprocal.[3] X Research source
{\displaystyle {\frac {4}{\frac {89}{240}}}}
{\displaystyle =4\times {\frac {240}{89}}}
{\displaystyle ={\frac {4}{1}}\times {\frac {240}{89}}}
{\displaystyle ={\frac {960}{89}}}
Convert to a decimal to find the harmonic mean of your set of numbers. To convert a fraction to a decimal, divide the numerator by the denominator.[4] X Research source
{\displaystyle {\frac {960}{89}}}
{\displaystyle =960\div 89}
{\displaystyle =10.79}
So, the harmonic mean of the numbers 10, 12, 16, and 8 is 10.79.
Calculating the Harmonic Mean using Excel
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Enter the values into your spreadsheet. Make sure to only place one value in each cell.
For example, if you need to find the harmonic mean of 10, 12, 16, and 8, you might type each of these values into a separate cell in the spreadsheet, cells A1-A4.
Enter the function for the harmonic mean. The function is HARMEAN(number 1, [number 2]...).[5] X Research source To select the function, begin typing "=HARMEAN" into a blank cell of the spreadsheet, then double-click on the function when it pops up.
For example, type "=HARMEAN" into cell A5 of your spreadsheet and double-click on the function.
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Highlight the cells containing the values you are averaging. Hit the enter key. Excel will calculate the harmonic mean for you and display it in the spreadsheet.
For example, highlight cells A1-A4 of your spreadsheet and hit enter. Excel will calculate 10.78652 as the harmonic mean.
What are the harmonic means of 5, 6 and 7?
3/(1/5+1/6+1/7) = 3/(42/210 + 35/210 + 30/210 ) = 3/( 107/210) = 3 x 210/107= 630/107 = 5.89
What is the harmonic mean of x1,x2,x3.xn?
This is just asking for the formula: n/(1/x1 + 1/x2 + 1/x3 +...+1/xn)
Can you tell me the harmonic mean of 0 2 4 6?
There's no harmonic mean when zero is one of the values in the set. That's because the formula would have 1/0 as one of the terms in the denominator, and 1/0 is infinity and not usable.
Obtain the harmonic mean of the following data 3, 2, 6, 4,
Use the formula. 4/(1/3 + 1/2 + 1/6 + 1/4) = 4/(4/12 + 6/12 + 2/12 + 3/12) = 4/(15/12) = 4 x 12/15 = 48/15 = 3.2
If we have the values of 1/x, then how will find the values of x?
The value of x is the reciprocal of the value of 1/x. For example, if 1/x equals 3, x equals 1/3.
What is the harmonic mean of the numbers 3,4,5,6?
Use the formula. 4/(1/3 + 1/4 + 1/5 + 1/6) = 4/(20/60 + 15/60 + 12/60 + 10/60) = 4/(57/50) = 4 x 50/57 = 200/57 = 3.5.
How do I find the harmonic mean of two numbers?
Use the above formula, with n=2.
The sum of 20 reciprocals (1 over the number) is found to equal 5. What is the harmonic mean of the original (non-reciprocated) scores?
The average of the reciprocals is 5/20 = 1/4. Thus the harmonic mean of the original numbers is the reciprocal of that or 4.
What is harmonic mean of 2, 3, 5?
It's 3 / (1/2 + 1/3 + 1/5) = 3 / (15/30 + 10/30 + 6/30) = 3 / (31/30) = (3)(30/31) = 90/31 = 2 30/31. In other words, the harmonic mean of 2, 3 and 5 is slightly less than 3.
↑ http://www.ck12.org/book/CK-12-Probability-and-Statistics-Concepts/section/5.3/
↑ http://www.mathwords.com/h/harmonic_mean.htm
↑ http://www.purplemath.com/modules/percents2.htm
↑ https://support.office.com/en-us/article/HARMEAN-function-5efd9184-fab5-42f9-b1d3-57883a1d3bc6
To calculate the harmonic meaning, start by determining the number of values in your set of numbers. For example, if you're working with 10, 12, 16, and 8, you have 4 numbers, so the value is 4. Then, rewrite the numbers you're working with as denominators over the number 1. For example, if you're working with 10, 12, 16, and 8, write them as 1/10, 1/12, 1/16, and 1/8. Then, divide 4 by the sum of the fractions to find the harmonic mean. To calculate the harmonic meaning using a calculator, keep reading!
Español:calcular la media armónica
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Board Paper Solutions for CBSE Class 12-commerce MATHS Board Paper 2021 Delhi Set 4
\mathrm{log} \left[\mathrm{log} \left(\mathrm{log} {x}^{5}\right)\right]
\frac{5}{x \mathrm{log} \left({x}^{5}\right) \mathrm{log} \left(\mathrm{log} {x}^{5}\right)}
\frac{5}{x \mathrm{log} \left(\mathrm{log} {x}^{5}\right)}
\frac{5{x}^{4}}{\mathrm{log} \left({x}^{5}\right) \mathrm{log} \left(\mathrm{log} {x}^{5}\right)}
\frac{5{x}^{4}}{\mathrm{log} {x}^{5} \mathrm{log} \left(\mathrm{log} {x}^{5}\right)}
\to
{x}^{3}+1
\frac{dx}{dy}
\frac{\mathrm{cos} a}{{\mathrm{cos}}^{2} \left(a+y\right)}
\frac{-\mathrm{cos} a}{{\mathrm{cos}}^{2} \left(a+y\right)}
\frac{\mathrm{cos} a}{{\mathrm{sin}}^{2} y}
\frac{-\mathrm{cos} a}{{\mathrm{sin}}^{2} y}
\frac{{x}^{2}}{9}+\frac{{y}^{2}}{25}=1,
\left(±5, 0\right)
\left(0, ±5\right)
\left(0, ±3\right)
\left(±3, 0\right)
{\mathrm{cos}}^{-1}\left(\frac{1}{2}\right)+{\mathrm{sin}}^{-1}\left(-\frac{1}{\sqrt{2}}\right)
\frac{\mathrm{\pi }}{12}
\frac{\mathrm{\pi }}{3}
\mathrm{\pi }
\frac{\mathrm{\pi }}{6}
\frac{dy}{dx}
\frac{y+4x\left({x}^{2}+{y}^{2}\right)}{4y\left({x}^{2}+{y}^{2}\right)-x}
\frac{y-4x\left({x}^{2}+{y}^{2}\right)}{x+4\left({x}^{2}+{y}^{2}\right)}
\frac{y-4x\left({x}^{2}+{y}^{2}\right)}{4y\left({x}^{2}+{y}^{2}\right)-x}
\frac{4y\left({x}^{2}+{y}^{2}\right)-x}{y-4x\left({x}^{2}+{y}^{2}\right)}
f\left(x\right)=\left\{\begin{array}{l}\frac{{e}^{3x}-{e}^{-5x}}{x}, \mathrm{if} x\ne 0\\ k , \mathrm{if} x=0\end{array}\right\
\mathrm{P}=\left[\begin{array}{ccc}1& -1& 2\\ 0& 2& -3\\ 3& 2& 4\end{array}\right]
\ne
\ne
\left[\begin{array}{cc}3c+6& a-d\\ a+d& 2-3b\end{array}\right]=\left[\begin{array}{cc}12& 2\\ -8& -4\end{array}\right]
{\mathrm{tan}}^{-1}\left(\mathrm{tan}\frac{9\mathrm{\pi }}{8}\right)
\frac{\mathrm{\pi }}{8}
\frac{3\mathrm{\pi }}{8}
-\frac{\mathrm{\pi }}{8}
-\frac{3\mathrm{\pi }}{8}
\left[\begin{array}{cc}3& 4\\ -1& 2\\ 0& 1\end{array}\right]
\left[\begin{array}{ccc}-1& 2& 1\\ 1& 2& 3\end{array}\right]
\left[\begin{array}{cc}2& 3\\ -3& 0\\ 0& -3\end{array}\right]
\left[\begin{array}{cc}4& 3\\ -3& 0\\ -1& -2\end{array}\right]
\left[\begin{array}{cc}4& 3\\ 0& -3\\ -1& -2\end{array}\right]
\left[\begin{array}{cc}2& 3\\ 0& -3\\ 0& -3\end{array}\right]
\frac{\mathrm{dy}}{\mathrm{dx}}
\frac{\mathrm{cos\theta }+\mathrm{cos} 2\mathrm{\theta }}{\mathrm{sin\theta }-\mathrm{sin} 2\mathrm{\theta }}
\frac{\mathrm{cos\theta }-\mathrm{cos} 2\mathrm{\theta }}{\mathrm{sin} 2\mathrm{\theta }-\mathrm{sin\theta }}
\frac{\mathrm{cos\theta }-\mathrm{cos} 2\mathrm{\theta }}{\mathrm{sin\theta }-\mathrm{sin} 2\mathrm{\theta }}
\frac{\mathrm{cos} 2\mathrm{\theta }-\mathrm{cos\theta }}{\mathrm{sin} 2\mathrm{\theta }+\mathrm{sin\theta }}
{a}_{\mathit{i}\mathit{j}}=\left\{\begin{array}{ll}2i+3j,& i<j\\ 5,& i=j\\ 3i-2j,& i>j\end{array}\right\
f\left(x\right)=\left\{\begin{array}{ll}\frac{k \mathrm{cos}x}{\mathrm{\pi }-2x},& \mathrm{if} x\ne \frac{\mathrm{\pi }}{2}\\ 3,& \mathrm{if} x=\frac{\mathrm{\pi }}{2}\end{array}\right\
x=\frac{\mathrm{\pi }}{2},
\mathrm{X}=\left[\begin{array}{ccc}0& 1& 1\\ 1& 0& 1\\ 1& 1& 0\end{array}\right]
\left[\frac{\sqrt{1+x}-\sqrt{1-x}}{\sqrt{1+x}+\sqrt{1-x}}\right]
\frac{\mathrm{\pi }}{4}-\frac{x}{2}
\frac{\mathrm{\pi }}{4}+\frac{x}{2}
\frac{\mathrm{\pi }}{4}-\frac{1}{2}{\mathrm{cos}}^{-1}x
\frac{\mathrm{\pi }}{4}+\frac{1}{2}{\mathrm{cos}}^{-1}x
\left[\begin{array}{cc}\mathrm{\alpha }& -2\\ -2& \mathrm{\alpha }\end{array}\right]
\left|{\mathrm{A}}^{3}\right|=125,
\mathrm{\alpha }
±
±
\left(1-{x}^{2}\right)\frac{{d}^{2}y}{d{x}^{2}}+x\frac{dy}{dx}+{m}^{2}y=0
\left(1-{x}^{2}\right)\frac{{d}^{2}y}{d{x}^{2}}-x\frac{dy}{dx}+{m}^{2}y=0
\left(1+{x}^{2}\right)\frac{{d}^{2}y}{d{x}^{2}}-x\frac{dy}{dx}-{m}^{2}y=0
\left(1+{x}^{2}\right)\frac{{d}^{2}y}{d{x}^{2}}+x\frac{dy}{dx}-{m}^{2}x=0
\left[{\mathrm{tan}}^{-1}\sqrt{3}-{\mathrm{cot}}^{-1}\left(-\sqrt{3}\right)\right]
\mathrm{\pi }
-\frac{\mathrm{\pi }}{2}
2\sqrt{3}
{\left(\frac{1}{x}\right)}^{x}
{e}^{\frac{1}{e}}
{\left(\frac{1}{e}\right)}^{\frac{1}{e}}
\mathrm{X} = \left[\begin{array}{ccc}1& –1& 2\\ 3& 4& –5\\ 2& –1& 3\end{array}\right]
\mathrm{X} = \left[\begin{array}{ccc}7& –5& –3\\ 19& 1& –11\\ –11& 1& 7\end{array}\right]
\mathrm{X} = \left[\begin{array}{ccc}7& –19& –11\\ 5& –1& –1\\ 3& 11& 7\end{array}\right]
\mathrm{X} = \left[\begin{array}{ccc}7& 19& –11\\ –3& 11& 7\\ –5& –1& –1\end{array}\right]
\mathrm{X} = \left[\begin{array}{ccc}7& 19& –11\\ –1& –1& 1\\ –3& –11& 7\end{array}\right]
{\left|\begin{array}{ccc}x& 2& 3\\ 1& x& 1\\ 3& 2& x\end{array}\right|}_{ }=0,
\mathrm{f}\left(x\right)=4x–\frac{1}{2}{x}^{2}
\left[–2,\frac{9}{2}\right]
4\sqrt{2}
2\sqrt{2}
-4\sqrt{2}
\mathrm{X}=\left[\begin{array}{ccc}2& 0& 0\\ 0& 3& 0\\ 0& 0& 4\end{array}\right]
24\left[\begin{array}{ccc}1/2& 0& 0\\ 0& 1/3& 0\\ 0& 0& 1/4\end{array}\right]
\frac{1}{24}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]
\frac{1}{24}\left[\begin{array}{ccc}2& 0& 0\\ 0& 3& 0\\ 0& 0& 4\end{array}\right]
\left[\begin{array}{ccc}1/2& 0& 0\\ 0& 1/3& 0\\ 0& 0& 1/4\end{array}\right]
\frac{50}{h}+400 {h}^{2}
\frac{12500}{h}+400 {h}^{2}
\frac{250}{h}+{h}^{2}
\frac{250}{h}+400 {h}^{2}
\frac{dc}{dh}=0
\frac{{d}^{2}c}{d{h}^{2}}
\frac{25000}{{h}^{3}}+800
\frac{500}{{h}^{3}}+800
\frac{100}{{h}^{3}}+800
\frac{500}{{h}^{3}}+2
10\sqrt{\frac{5}{3}}
5\sqrt{5}
More Board Paper Solutions for Class 12 Commerce Math
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Cairo_pentagonal_tiling Knowpia
In geometry, a Cairo pentagonal tiling is a tessellation of the Euclidean plane by congruent convex pentagons, formed by overlaying two tessellations of the plane by hexagons and named for its use as a paving design in Cairo. It is also called MacMahon's net[1] after Percy Alexander MacMahon, who depicted it in his 1921 publication New Mathematical Pastimes.[2] John Horton Conway called it a 4-fold pentille.[3]
Equilateral form of the Cairo tiling
face-transitive
Infinitely many different pentagons can form this pattern, belonging to two of the 15 families of convex pentagons that can tile the plane. Their tilings have varying symmetries; all are face-symmetric. One particular form of the tiling, dual to the snub square tiling, has tiles with the minimum possible perimeter among all pentagonal tilings. Another, overlaying two flattened tilings by regular hexagons, is the form used in Cairo and has the property that every edge is collinear with infinitely many other edges.
In architecture, beyond Cairo, the Cairo tiling has been used in Mughal architecture in 18th-century India, in the early 20th-century Laeiszhalle in Germany, and in many modern buildings and installations. It has also been studied as a crystal structure and appears in the art of M. C. Escher.
The union of all edges of a Cairo tiling is the same as the union of two tilings of the plane by hexagons. Each hexagon of one tiling surrounds two vertices of the other tiling, and is divided by the hexagons of the other tiling into four of the pentagons in the Cairo tiling.[4] Infinitely many different pentagons can form Cairo tilings, all with the same pattern of adjacencies between tiles and with the same decomposition into hexagons, but with varying edge lengths, angles, and symmetries. The pentagons that form these tilings can be grouped into two different infinite families, drawn from the 15 families of convex pentagons that can tile the plane,[5] and the five families of pentagon found by Karl Reinhardt in 1918 that can tile the plane isohedrally (all tiles symmetric to each other).[6]
One of these two families consists of pentagons that have two non-adjacent right angles, with a pair of sides of equal length meeting at each of these right angles. Any pentagon meeting these requirements tiles the plane by copies that, at the chosen right angled corners, are rotated by a right angle with respect to each other. At the pentagon sides that are not adjacent to one of these two right angles, two tiles meet, rotated by a 180° angle with respect to each other. The result is an isohedral tiling, meaning that any pentagon in the tiling can be transformed into any other pentagon by a symmetry of the tiling. These pentagons and their tiling are often listed as "type 4" in the listing of types of pentagon that can tile.[4] For any type 4 Cairo tiling, twelve of the same tiles can also cover the surface of a cube, with one tile folded across each cube edge and three right angles of tiles meeting at each cube vertex, to form the same combinatorial structure as a regular dodecahedron.[7][8]
The other family of pentagons forming the Cairo tiling are pentagons that have two complementary angles at non-adjacent vertices, each having the same two side lengths incident to it. In their tilings, the vertices with complementary angles alternate around each degree-four vertex. The pentagons meeting these constraints are not generally listed as one of the 15 families of pentagons that tile; rather, they are part of a larger family of pentagons (the "type 2" pentagons) that tile the plane isohedrally in a different way.[4]
Bilaterally symmetric Cairo tilings are formed by pentagons that belong to both the type 2 and type 4 families.[4] The basketweave brick paving pattern can be seen as a degenerate case of the bilaterally symmetric Cairo tilings, with each brick (a
{\displaystyle 1\times 2}
rectangle) interpreted as a pentagon with four right angles and one 180° angle.[9]
Type 2 Cairo tiles have non-adjacent complementary angles, with the same two adjacent side lengths
Type 4 tiles have non-adjacent right angles between pairs of equal-length sides
Bilaterally symmetric tilings (belonging to both types) use tiles with non-adjacent right angles and four equal edges
Type 2 Cairo tiling, with coloring showing reflected and non-reflected tiles
In a type 4 chiral tiling, the pentagons can be bilaterally symmetric even when the tiling isn't
The basketweave, a degenerate bilaterally symmetric tiling, with non-degenerate tiling overlaid
It is possible to assign six-dimensional half-integer coordinates to the pentagons of the tiling, in such a way that the number of edge-to-edge steps between any two pentagons equals the L1 distance between their coordinates. The six coordinates of each pentagon can be grouped into two triples of coordinates, in which each triple gives the coordinates of a hexagon in an analogous three-dimensional coordinate system for each of the two overlaid hexagon tilings.[10] The number of tiles that are
{\displaystyle i}
steps away from any given tile, for
{\displaystyle i=0,1,2,\dots }
, is given by the coordination sequence
{\displaystyle 1,5,11,16,21,27,32,37,\dots }
in which, after the first three terms, each term differs by 16 from the term three steps back in the sequence. One can also define analogous coordination sequences for the vertices of the tiling instead of for its tiles, but because there are two types of vertices (of degree three and degree four) there are two different coordination sequences arising in this way. The degree-four sequence is the same as for the square grid.[11][12]
Catalan tilingEdit
Cairo tiling as the dual of the snub square tiling
Geometry of pentagons for the dual snub square tiling
The snub square tiling, made of two squares and three equilateral triangles around each vertex, has a bilaterally symmetric Cairo tiling as its dual tiling.[13] The Cairo tiling can be formed from the snub square tiling by placing a vertex of the Cairo tiling at the center of each square or triangle of the snub square tiling, and connecting these vertices by edges when they come from adjacent tiles.[14] Its pentagons can be circumscribed around a circle. They have four long edges and one short one with lengths in the ratio
{\displaystyle 1:{\sqrt {3}}-1}
. The angles of these pentagons form the sequence 120°, 120°, 90°, 120°, 90°.[15]
The snub square tiling is an Archimedean tiling, and as the dual to an Archimedean tiling this form of the Cairo pentagonal tiling is a Catalan tiling or Laves tiling.[14] It is one of two monohedral pentagonal tilings that, when the tiles have unit area, minimizes the perimeter of the tiles. The other is also a tiling by circumscribed pentagons with two right angles and three 120° angles, but with the two right angles adjacent; there are also infinitely many tilings formed by combining both kinds of pentagon.[15]
Tilings with collinear edgesEdit
Collinear form of Cairo tiling, with integer-coordinate pentagons, formed by flattening two perpendicular regular hexagonal tilings in perpendicular directions
Pentagons with integer vertex coordinates
{\displaystyle (\pm 2,0)}
{\displaystyle (\pm 3,3)}
{\displaystyle (0,4)}
, with four equal sides shorter than the remaining side, form a Cairo tiling whose two hexagonal tilings can be formed by flattening two perpendicular tilings by regular hexagons in perpendicular directions, by a ratio of
{\displaystyle {\sqrt {3}}}
. This form of the Cairo tiling inherits the property of the tilings by regular hexagons (unchanged by the flattening), that every edge is collinear with infinitely many other edges.[9][16]
Tilings with equal side lengthsEdit
The regular pentagon cannot form Cairo tilings, as it does not tile the plane without gaps. There is a unique equilateral pentagon that can form a type 4 Cairo tiling; it has five equal sides but its angles are unequal, and its tiling is bilaterally symmetric.[4][13] Infinitely many other equilateral pentagons can form type 2 Cairo tilings.[4]
Several streets in Cairo have been paved with the collinear form of the Cairo tiling;[9][17] this application is the origin of the name of the tiling.[18][19] As of 2019 this pattern can still be seen as a surface decoration for square tiles near the Qasr El Nil Bridge and the El Behoos Metro station; other versions of the tiling are visible elsewhere in the city.[20] Some authors including Martin Gardner have written that this pattern is used more widely in Islamic architecture, and although this claim appears to have been based on a misunderstanding, patterns resembling the Cairo tiling are visible on the 17th-century Tomb of I'timād-ud-Daulah in India, and the Cairo tiling itself has been found on a 17th-century Mughal jali.[16]
Tomb of I'timād-ud-Daulah, with rectangular side panels resembling the Cairo tiling
Centar Zamet, with the Cairo tiling visible on its walls
Cairo tiling in Hørsholm, Denmark
Penta-graphane
One of the earliest publications on the Cairo tiling as a decorative pattern occurs in a book on textile design from 1906.[21] Inventor H. C. Moore filed a US patent on tiles forming this pattern in 1908.[22] At roughly the same time, Villeroy & Boch created a line of ceramic floor tiles in the Cairo tiling pattern, used in the foyer of the Laeiszhalle in Hamburg, Germany. The Cairo tiling has been used as a decorative pattern in many recent architectural designs; for instance, the city center of Hørsholm, Denmark, is paved with this pattern, and the Centar Zamet, a sports hall in Croatia, uses it both for its exterior walls and its paving tiles.[16]
In crystallography, this tiling has been studied at least since 1911.[23] It has been proposed as the structure for layered hydrate crystals,[24] certain compounds of bismuth and iron,[25] and penta-graphene, a hypothetical compound of pure carbon. In the penta-graphene structure, the edges of the tiling incident to degree-four vertices form single bonds, while the remaining edges form double bonds. In its hydrogenated form, penta-graphane, all bonds are single bonds and the carbon atoms at the degree-three vertices of the structure have a fourth bond connecting them to hydrogen atoms.[26]
The Cairo tiling has been described as one of M. C. Escher's "favorite geometric patterns".[7] He used it as the basis for his drawing Shells and Starfish (1941), in the bees-on-flowers segment of his Metamorphosis III (1967–1968), and in several other drawings from 1967–1968. An image of this tessellation has also been used as the cover art for the 1974 first edition of H. S. M. Coxeter's book Regular Complex Polytopes.[4][16]
^ O'Keeffe, M.; Hyde, B. G. (1980), "Plane nets in crystal chemistry", Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 295 (1417): 553–618, Bibcode:1980RSPTA.295..553O, doi:10.1098/rsta.1980.0150, JSTOR 36648, S2CID 121456259 .
^ Macmahon, Major P. A. (1921), New Mathematical Pastimes, University Press, p. 101
^ Conway, John H.; Burgiel, Heidi; Goodman-Strass, Chaim (2008), The Symmetries of Things, AK Peters, p. 288, ISBN 978-1-56881-220-5
^ a b c d e f g Schattschneider, Doris (1978), "Tiling the plane with congruent pentagons", Mathematics Magazine, 51 (1): 29–44, doi:10.1080/0025570X.1978.11976672, JSTOR 2689644, MR 0493766
^ Rao, Michaël (2017), Exhaustive search of convex pentagons which tile the plane (PDF), arXiv:1708.00274
^ Reinhardt, Karl (1918), Über die Zerlegung der Ebene in Polygone (Doctoral dissertation) (in German), Borna-Leipzig: Druck von Robert Noske, "Vierter Typus", p. 78, and Figure 24, p. 81
^ a b Schattschneider, Doris; Walker, Wallace (1977), "Dodecahedron", M. C. Escher Kaleidocycles, Ballantine Books, p. 22 ; reprinted by Taschen, 2015
^ Thomas, B.G.; Hann, M.A. (2008), "Patterning by projection: Tiling the dodecahedron and other solids", in Sarhangi, Reza; Séquin, Carlo H. (eds.), Bridges Leeuwarden: Mathematics, Music, Art, Architecture, Culture, London: Tarquin Publications, pp. 101–108, ISBN 9780966520194
^ a b c Macmillan, R. H. (December 1979), "Pyramids and pavements: Some thoughts from Cairo", The Mathematical Gazette, 63 (426): 251–255, doi:10.2307/3618038, JSTOR 3618038
^ Kovács, Gergely; Nagy, Benedek; Turgay, Neşet Deniz (May 2021), "Distance on the Cairo pattern", Pattern Recognition Letters, 145: 141–146, Bibcode:2021PaReL.145..141K, doi:10.1016/j.patrec.2021.02.002, S2CID 233375125
^ Coordination sequences for the Cairo pentagonal tiling in the On-Line Encyclopedia of Integer Sequences: A219529 for pentagons, A296368 for degree-three vertices, and A008574 for degree-four vertices, retrieved 2021-06-17
^ Goodman-Strauss, C.; Sloane, N. J. A. (2019), "A coloring-book approach to finding coordination sequences" (PDF), Acta Crystallographica Section A, 75 (1): 121–134, arXiv:1803.08530, doi:10.1107/s2053273318014481, MR 3896412, PMID 30575590, S2CID 4553572
^ a b Rollett, A. P. (September 1955), "2530. A pentagonal tessellation", Mathematical Notes, The Mathematical Gazette, 39 (329): 209, doi:10.2307/3608750, JSTOR 3608750
^ a b Steurer, Walter; Dshemuchadse, Julia (2016), Intermetallics: Structures, Properties, and Statistics, International Union of Crystallography Monographs on Crystallography, vol. 26, Oxford University Press, p. 42, ISBN 9780191023927
^ a b Chung, Ping Ngai; Fernandez, Miguel A.; Li, Yifei; Mara, Michael; Morgan, Frank; Plata, Isamar Rosa; Shah, Niralee; Vieira, Luis Sordo; Wikner, Elena (2012), "Isoperimetric pentagonal tilings", Notices of the American Mathematical Society, 59 (5): 632–640, doi:10.1090/noti838, MR 2954290
^ a b c d Bailey, David, "Cairo tiling", David Bailey's World of Escher-like Tessellations, retrieved 2020-12-06
^ Dunn, J. A. (December 1971), "Tessellations with pentagons", The Mathematical Gazette, 55 (394): 366–369, doi:10.2307/3612359, JSTOR 3612359 . Although Dunn writes that the equilateral form of the tiling was used in Cairo, this appears to be a mistake.
^ Alsina, Claudi; Nelsen, Roger B. (2010), Charming proofs: a journey into elegant mathematics, Dolciani mathematical expositions, vol. 42, Mathematical Association of America, p. 164, ISBN 978-0-88385-348-1 .
^ Martin, George Edward (1982), Transformation Geometry: An Introduction to Symmetry, Undergraduate Texts in Mathematics, Springer, p. 119, ISBN 978-0-387-90636-2 .
^ Morgan, Frank (2019), "My undercover mission to find Cairo tilings", The Mathematical Intelligencer, 41 (3): 19–22, doi:10.1007/s00283-019-09906-7, MR 3995312
^ Nisbet, Harry (1906), Grammar of Textile Design, London: Scott, Greenwood & Son, p. 101
^ Moore, H. C. (July 20, 1909), Tile (US Patent 928,320)
^ Haag, F. (1911), "Die regelmäßigen Planteilungen", Zeitschrift für Kristallographie, Kristallgeometrie, Kristallphysik, Kristallchemie, 49: 360–369, hdl:2027/uc1.b3327994 See in particular Figures 2b, p. 361, and 4a, p. 363.
^ Banaru, A. M.; Banaru, G. A. (August 2011), "Cairo tiling and the topology of layered hydrates", Moscow University Chemistry Bulletin, 66 (3), Article 159, doi:10.3103/S0027131411030023, S2CID 96002269
^ Ressouche, E.; Simonet, V.; Canals, B.; Gospodinov, M.; Skumryev, V. (December 2009), "Magnetic frustration in an iron-based Cairo pentagonal lattice", Physical Review Letters, 103 (26): 267204, arXiv:1001.0710, Bibcode:2009PhRvL.103z7204R, doi:10.1103/physrevlett.103.267204, PMID 20366341, S2CID 20752605
^ Zhang, Shunhong; Zhou, Jian; Wang, Qian; Chen, Xiaoshuang; Kawazoe, Yoshiyuki; Jena, Puru (February 2015), "Penta-graphene: A new carbon allotrope", Proceedings of the National Academy of Sciences of the United States of America, 112 (8): 2372–2377, Bibcode:2015PNAS..112.2372Z, doi:10.1073/pnas.1416591112, PMC 4345574, PMID 25646451
Wikimedia Commons has media related to Cairo pentagonal tiling.
Weisstein, Eric W., "Cairo Tessellation", MathWorld
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How Commodities (Brent, Crude and Nat.Gas) prices are derived - Help Center
PrimeXBT Commodities trading has many advantages for investors who are only interested in price speculation. We offer hassle-free trading access to these contracts with no need to worry about the expiration and ‘rollover’ to the next month. A Commodity's (BRENT, CRUDE and NAT.GAS) price is derived as a combination of the first and second nearby month future contract. The price for each of these instruments is derived from a weighted average between the 1st and 2nd month Future Contracts (explained in further detail below) and follows the business day convention from New York.
This pricing method diminishes the level of volatility when the first nearby futures contract is near expiration, since there is often lower liquidity. Furthermore, rolling your position from the 1st to 2nd nearby month happens in smaller daily increments, instead of paying the full difference when close to expiration.
Pricing of PrimeXBT Commodities (BRENT, CRUDE, NAT.GAS)
The price of PrimeXBT Commodities is determined in accordance with the following formula:
1 - \\ \dfrac{D}{NumDays} * Relevant\, Price\, of\, First\, Nearby\, Month \\ + \\ \dfrac{D}{NumDays}*Relevant\, Price\, of\, Second\, Nearby\, Month
The number of Commodity Business Days from (and including) Previous Expiration Date to (but excluding) the Roll Date.
The number of Commodity Business Days from (and including) the Previous Expiration Date to (but excluding) the Next Expiration Date
The date of expiration of the First Nearby Month
The date of expiration of the Previous Nearby Month that expired Immediately prior to the Roll Date.
The second Commodity Business Day after the current Business Day
CRUDE: Underlying asset exchange is CME and the product is NYMEX WTI Light Sweet Crude Oil Futures; NAT.GAS: Underlying asset exchange is CME and the product is NYMEX Henry Hub Natural Gas Futures; BRENT: Underlying asset exchange is ICE (ICE Futures Europe) and the product is Brent Crude Futures
D = 11 NumDays = 20 Relevant price of First Nearby Month = 20 USD Relevant price of Second Nearby Month = 25 USD
Commodity\, Price = \left( 1 – \tfrac{11}{20}\right) * 20 + \tfrac{11}{20} * 25 \atop = 0.45 * 20 + 0.55 * 25 = 22.75
Financing Charges of Commodities
Commodities are a margined product; therefore, the traded value is financed through an overnight financing charge. If a position is opened and closed within the same trading day, it will not be subject to overnight financing charges.
More info on fees: https://primexbt.com/fees
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Boltzmann constant - Wikipedia
(Redirected from Boltzmann's constant)
Not to be confused with Stefan–Boltzmann constant.
The Boltzmann constant (kB or k) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas.[2] It occurs in the definitions of the kelvin and the gas constant, and in Planck's law of black-body radiation and Boltzmann's entropy formula, and is used in calculating thermal noise in resistors. The Boltzmann constant has dimensions of energy divided by temperature, the same as entropy. It is named after the Austrian scientist Ludwig Boltzmann.
The proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas
Value in joules per kelvin:
1.380649×10−23 J K-1[1]
As part of the 2019 redefinition of SI base units, the Boltzmann constant is one of the seven "defining constants" that have been given exact definitions. They are used in various combinations to define the seven SI base units. The Boltzmann constant is defined to be exactly 1.380649×10−23 J K-1.[1]
1 Roles of the Boltzmann constant
1.1 Role in the equipartition of energy
1.2 Role in Boltzmann factors
1.3 Role in the statistical definition of entropy
1.4 The thermal voltage
3 Value in different units
Roles of the Boltzmann constantEdit
Macroscopically, the ideal gas law states that, for an ideal gas, the product of pressure p and volume V is proportional to the product of amount of substance n (in moles) and absolute temperature T:
{\displaystyle pV=nRT,}
where R is the molar gas constant (8.31446261815324 J⋅K−1⋅mol−1).[3] Introducing the Boltzmann constant as the gas constant per molecule[4] k = R/NA transforms the ideal gas law into an alternative form:
{\displaystyle pV=NkT,}
where N is the number of molecules of gas. For n = 1 mol, N is equal to the number of particles in one mole (the Avogadro number).
Role in the equipartition of energyEdit
Given a thermodynamic system at an absolute temperature T, the average thermal energy carried by each microscopic degree of freedom in the system is 1/2kT (i.e., about 2.07×10−21 J, or 0.013 eV, at room temperature). It's important to note that this is generally true only for classical systems with a large number of particles, and in which quantum effects are negligible.
In classical statistical mechanics, this average is predicted to hold exactly for homogeneous ideal gases. Monatomic ideal gases (the six noble gases) possess three degrees of freedom per atom, corresponding to the three spatial directions. According to the equipartition of energy this means that there is a thermal energy of 3/2kT per atom. This corresponds very well with experimental data. The thermal energy can be used to calculate the root-mean-square speed of the atoms, which turns out to be inversely proportional to the square root of the atomic mass. The root mean square speeds found at room temperature accurately reflect this, ranging from 1370 m/s for helium, down to 240 m/s for xenon.
{\displaystyle p={\frac {1}{3}}{\frac {N}{V}}m{\overline {v^{2}}}.}
{\displaystyle pV=NkT}
{\displaystyle {\tfrac {1}{2}}m{\overline {v^{2}}}={\tfrac {3}{2}}kT.}
Considering that the translational motion velocity vector v has three degrees of freedom (one for each dimension) gives the average energy per degree of freedom equal to one third of that, i.e. 1/2kT.
Role in Boltzmann factorsEdit
More generally, systems in equilibrium at temperature T have probability Pi of occupying a state i with energy E weighted by the corresponding Boltzmann factor:
{\displaystyle P_{i}\propto {\frac {\exp \left(-{\frac {E}{kT}}\right)}{Z}},}
where Z is the partition function. Again, it is the energy-like quantity kT that takes central importance.
Role in the statistical definition of entropyEdit
{\displaystyle S=k\,\ln W.}
{\displaystyle \Delta S=\int {\frac {{\rm {d}}Q}{T}}.}
{\displaystyle {S'=\ln W},\quad \Delta S'=\int {\frac {\mathrm {d} Q}{kT}}.}
The characteristic energy kT is thus the energy required to increase the rescaled entropy by one nat.
The thermal voltageEdit
In semiconductors, the Shockley diode equation—the relationship between the flow of electric current and the electrostatic potential across a p–n junction—depends on a characteristic voltage called the thermal voltage, denoted by VT. The thermal voltage depends on absolute temperature T as
{\displaystyle V_{\mathrm {T} }={kT \over q},}
where q is the magnitude of the electrical charge on the electron with a value 1.602176634×10−19 C[5] Equivalently,
{\displaystyle {V_{\mathrm {T} } \over T}={k \over q}\approx 8.61733034\times 10^{-5}\ \mathrm {V/K} .}
At room temperature 300 K (27 °C; 80 °F), VT is approximately 25.85 mV[6][7] which can be derived by plugging in the values as follows:
{\displaystyle V_{\mathrm {T} }={kT \over q}={\frac {1.38\times 10^{-23}\mathrm {J} \cdot k^{-1}\times 300\mathrm {K} }{1.6\times 10^{-19}\mathrm {C} }}\simeq 25.85\mathrm {mV} }
At the standard state temperature of 298.15 K (25.00 °C; 77.00 °F), it is approximately 25.69 mV. The thermal voltage is also important in plasmas and electrolyte solutions (e.g. the Nernst equation); in both cases it provides a measure of how much the spatial distribution of electrons or ions is affected by a boundary held at a fixed voltage.[8][9]
The Boltzmann constant is named after its 19th century Austrian discoverer, Ludwig Boltzmann. Although Boltzmann first linked entropy and probability in 1877, the relation was never expressed with a specific constant until Max Planck first introduced k, and gave a more precise value for it (1.346×10−23 J/K, about 2.5% lower than today's figure), in his derivation of the law of black-body radiation in 1900–1901.[10] Before 1900, equations involving Boltzmann factors were not written using the energies per molecule and the Boltzmann constant, but rather using a form of the gas constant R, and macroscopic energies for macroscopic quantities of the substance. The iconic terse form of the equation S = k ln W on Boltzmann's tombstone is in fact due to Planck, not Boltzmann. Planck actually introduced it in the same work as his eponymous h.[11]
In 1920, Planck wrote in his Nobel Prize lecture:[12]
This "peculiar state of affairs" is illustrated by reference to one of the great scientific debates of the time. There was considerable disagreement in the second half of the nineteenth century as to whether atoms and molecules were real or whether they were simply a heuristic tool for solving problems. There was no agreement whether chemical molecules, as measured by atomic weights, were the same as physical molecules, as measured by kinetic theory. Planck's 1920 lecture continued:[12]
In versions of SI prior to the 2019 redefinition of the SI base units, the Boltzmann constant was a measured quantity rather than a fixed value. Its exact definition also varied over the years due to redefinitions of the kelvin (see Kelvin § History) and other SI base units (see Joule § History).
In 2017, the most accurate measures of the Boltzmann constant were obtained by acoustic gas thermometry, which determines the speed of sound of a monatomic gas in a triaxial ellipsoid chamber using microwave and acoustic resonances.[13][14] This decade-long effort was undertaken with different techniques by several laboratories;[a] it is one of the cornerstones of the 2019 redefinition of SI base units. Based on these measurements, the CODATA recommended 1.380649×10−23 J/K to be the final fixed value of the Boltzmann constant to be used for the International System of Units.[15]
Value in different unitsEdit
Values of kB
1.380649×10−23 J/K SI by definition, J/K = m2⋅kg/(s2⋅K) in SI base units
8.617333262×10−5 eV/K [note 1]
2.083661912×1010 Hz/K (k/h) [note 1]
1.380649×10−16 erg/K CGS system, 1 erg = 1×10−7 J
3.297623483×10−24 cal/K [note 1] 1 calorie = 4.1868 J
1.832013046×10−24 cal/°R [note 1]
5.657302466×10−24 ft lb/°R [note 1]
0.695034800 cm−1/K (k/(hc)) [note 1]
3.166811563×10−6 Eh/K (Eh = Hartree)
1.987204259×10−3 kcal/(mol⋅K) (kNA) [note 1]
8.314462618×10−3 kJ/(mol⋅K) (kNA) [note 1]
−228.5991672 dB(W/K/Hz) 10 log10(k/(1 W/K/Hz)),[note 1] used for thermal noise calculations
Since k is a proportionality factor between temperature and energy, its numerical value depends on the choice of units for energy and temperature. The small numerical value of the Boltzmann constant in SI units means a change in temperature by 1 K only changes a particle's energy by a small amount. A change of 1 °C is defined to be the same as a change of 1 K. The characteristic energy kT is a term encountered in many physical relationships.
The Boltzmann constant sets up a relationship between wavelength and temperature (dividing hc/k by a wavelength gives a temperature) with one micrometer being related to 14387.777 K, and also a relationship between voltage and temperature (multiplying the voltage by k in units of eV/K) with one volt being related to 11604.518 K. The ratio of these two temperatures, 14387.777 K / 11604.518 K ≈ 1.239842, is the numerical value of hc in units of eV⋅μm.
Natural unitsEdit
The Boltzmann constant provides a mapping from this characteristic microscopic energy E to the macroscopic temperature scale T = E/k. In physics research another definition is often encountered in setting k to unity, resulting in temperature and energy quantities of the same type. In this context temperature is measured effectively in units of energy and the Boltzmann constant is not explicitly needed.[16]
{\displaystyle E_{\mathrm {dof} }={\tfrac {1}{2}}T}
{\displaystyle S=-\sum _{i}P_{i}\ln P_{i}.}
^ Independent techniques exploited: acoustic gas thermometry, dielectric constant gas thermometry, johnson noise thermometry. Involved laboratories cited by CODATA in 2017: LNE-Cnam (France), NPL (UK), INRIM (Italy), PTB (Germany), NIST (USA), NIM (China).
^ a b c d e f g h i The value is exact but not expressible as a finite decimal; approximated to 9 decimal places only.
^ a b Newell, David B.; Tiesinga, Eite (2019). The International System of Units (SI). NIST Special Publication 330. Gaithersburg, Maryland: National Institute of Standards and Technology. doi:10.6028/nist.sp.330-2019.
^ Richard Feynman (1970). The Feynman Lectures on Physics Vol I. Addison Wesley Longman. ISBN 978-0-201-02115-8.
^ "Proceedings of the 106th meeting" (PDF). 16–20 October 2017.
^ Petrucci, Ralph H.; Harwood, William S.; Herring, F. Geoffrey (2002). GENERAL CHEMISTRY: Principles and Modern Applications (8th ed.). Prentice Hall. p. 785. ISBN 0-13-014329-4.
^ "2018 CODATA Value: elementary charge". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 20 May 2019.
^ Rashid, Muhammad H. (2016). Microelectronic circuits: analysis and design (Third ed.). Cengage Learning. pp. 183–184. ISBN 9781305635166.
^ Cataldo, Enrico; Lieto, Alberto Di; Maccarrone, Francesco; Paffuti, Giampiero (18 August 2016). "Measurements and analysis of current-voltage characteristic of a pn diode for an undergraduate physics laboratory". arXiv:1608.05638v1 [physics.ed-ph].
^ Kirby, Brian J. (2009). Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices (PDF). Cambridge University Press. ISBN 978-0-521-11903-0.
^ Tabeling, Patrick (2006). Introduction to Microfluidics. Oxford University Press. ISBN 978-0-19-856864-3.
^ Planck, Max (1901), "Ueber das Gesetz der Energieverteilung im Normalspectrum", Ann. Phys., 309 (3): 553–63, Bibcode:1901AnP...309..553P, doi:10.1002/andp.19013090310 . English translation: "On the Law of Distribution of Energy in the Normal Spectrum". Archived from the original on 17 December 2008.
^ Gearhart, Clayton A. (2002). "Planck, the Quantum, and the Historians". Physics in Perspective. 4 (2): 177. doi:10.1007/s00016-002-8363-7. ISSN 1422-6944.
^ Pitre, L; Sparasci, F; Risegari, L; Guianvarc’h, C; Martin, C; Himbert, M E; Plimmer, M D; Allard, A; Marty, B; Giuliano Albo, P A; Gao, B; Moldover, M R; Mehl, J B (1 December 2017). "New measurement of the Boltzmann constant by acoustic thermometry of helium-4 gas" (PDF). Metrologia. 54 (6): 856–873. Bibcode:2017Metro..54..856P. doi:10.1088/1681-7575/aa7bf5. S2CID 53680647. Archived from the original (PDF) on 5 March 2019.
^ de Podesta, Michael; Mark, Darren F; Dymock, Ross C; Underwood, Robin; Bacquart, Thomas; Sutton, Gavin; Davidson, Stuart; Machin, Graham (1 October 2017). "Re-estimation of argon isotope ratios leading to a revised estimate of the Boltzmann constant" (PDF). Metrologia. 54 (5): 683–692. Bibcode:2017Metro..54..683D. doi:10.1088/1681-7575/aa7880.
^ Newell, D. B.; Cabiati, F.; Fischer, J.; Fujii, K.; Karshenboim, S. G.; Margolis, H. S.; Mirandés, E. de; Mohr, P. J.; Nez, F. (2018). "The CODATA 2017 values of h, e, k, and N A for the revision of the SI". Metrologia. 55 (1): L13. Bibcode:2018Metro..55L..13N. doi:10.1088/1681-7575/aa950a. ISSN 0026-1394.
^ Kalinin, M; Kononogov, S (2005), "Boltzmann's Constant, the Energy Meaning of Temperature, and Thermodynamic Irreversibility", Measurement Techniques, 48 (7): 632–36, doi:10.1007/s11018-005-0195-9, S2CID 118726162
Retrieved from "https://en.wikipedia.org/w/index.php?title=Boltzmann_constant&oldid=1088819505"
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Periodic Classification of Elements - Revision Notes
Mendeleev's Periodic Law : The properties of elements are the periodic function of their atomic mass.
Mendeleev's periodic table based on the chemical properties of elements. It contains vertical columns called groups and horizontal rows called periods.
Maximum no. of electrons that can be accommodated in a shell depend on the formula
2{n}^{2}
where n is the no. of the given shell. e.g. K shell – 2 × (1)
{}^{2}
= 2 elements in the first period L shell – 2 ×(2)
{}^{2}
= 8 elements in the second period.
- Metallic character decreases across a period because the effective nuclear charge increases that means the tendency to lose electrons decreases.
- Metals are electro-positive as they tend to lose electrons while forming bonds.
- Metallic character increases as we go down a group as the effective nuclear charge is decreasing. Non metals are electro-negative. They tend to form bonds by gaining electrons.
- Metals are found on the left side of the period table while non-metals are towards the right hand side of the periodic table.
- In the middle we have semi-metals or metalloid because they exhibit some properties of both metals and non metals.
- Oxides of metals are basic in nature while oxides of non-metals are acidic in nature.
across period
Variation along group
Due to increase in nuclear charge
due to addition of new shells distance between outermost electron and nucleus increases due to addition of new shells.
Due to increase In effective nuclear charge, tendency to lose valence electrons decreases.
decrease in effective nuclear charge experienced by valence electrons Tendency to lose electron s (metallic character) increases.
Non-Metallic Increases Character (electro- negativity)
due to increase in effective nuclear charge tendency to gain electrons increases
due to decrease in effective nuclear charge experienced by valence electron (due to addition of new shell), tendency to gain electrons decreases.
Dobereiner grouped the elements into triads and Newlands gave the Law of Octaves.
Mendeleev arranged the elements in increasing order of their atomic masses and according to their chemical properties.
Mendeleev even predicted the existence of some yet to be discovered elements on the basis of gaps in his Periodic Table.
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Quasicomplementary foliations and the Mather–Thurston theorem
2021 Quasicomplementary foliations and the Mather–Thurston theorem
We establish a form of the h–principle for the existence of foliations of codimension at least
2
which are quasicomplementary to a given one. Roughly, “quasicomplementary” means that they are complementary except on the boundaries of some kind of Reeb components. The construction involves an adaptation of W Thurston’s “inflation” process. The same methods also provide a proof of the classical Mather–Thurston theorem.
Gaël Meigniez. "Quasicomplementary foliations and the Mather–Thurston theorem." Geom. Topol. 25 (2) 643 - 710, 2021. https://doi.org/10.2140/gt.2021.25.643
Primary: 57R30 , 57R32 , 58H10
Keywords: Foliation , Haefliger structure , h–principle , Mather–Thurston theorem , Thurston's inflation
Gaël Meigniez "Quasicomplementary foliations and the Mather–Thurston theorem," Geometry & Topology, Geom. Topol. 25(2), 643-710, (2021)
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Improved stability for $SK_1$ and $WMS_d$ of a non-singular affine algebra
Improved stability for
S{K}_{1}
WM{S}_{d}
of a non-singular affine algebra
Rao, Ravi A. ; van der Kallen, Wilberd
K
author = {Rao, Ravi A. and van der Kallen, Wilberd},
title = {Improved stability for $SK_1$ and $WMS_d$ of a non-singular affine algebra},
AU - Rao, Ravi A.
AU - van der Kallen, Wilberd
TI - Improved stability for $SK_1$ and $WMS_d$ of a non-singular affine algebra
Rao, Ravi A.; van der Kallen, Wilberd. Improved stability for $SK_1$ and $WMS_d$ of a non-singular affine algebra, dans $K$-theory - Strasbourg, 1992, Astérisque, no. 226 (1994), 10 p. http://archive.numdam.org/item/AST_1994__226__411_0/
[BT] R. Bott and L. W. Tu, Differential Forms in Algebraic Topology, Graduate Text in Mathematics 82, Springer 1982.
[KM] N. Mohan Kumar and M. P. Murthy, Algebraic cycles and vector bundles over affine three folds, Annals of Math. 116 (1982) 579-591.
[Li] H. Lindel, On the Bass-Quillen conjecture concerning projective modules over polynomial rings, Invent. Math. 65 (1981) 319-323.
[MS] M. P. Murthy and R. G. Swan, Vector bundles over affine surfaces, Invent. Math. 36 (1976) 125-165.
[Qu] D. Quillen, Projective modules over polynomial rings, Invent. Math. 36 (1976) 167-171.
[Ra1] R. A. Rao, An elementary transformation of a special unimodular vector to its top coefficient vector, Proc. Amer. Math. Soc. 93 (1985) 21-24.
[Ra2] R. A. Rao, The Bass-Quillen conjecture in dimension three but char-acteristic
\ne 2,3
via a question of A. Suslin, Invent. Math. 93 (1988) 609-618.
[Ro] M. Roitman, On stably extended projective modules over polynomial rings, Proc. Amer. Math. Soc. 97 (1986) 585-589.
[Se] J.-P. Serre, Cohomologie Galoisienne, Lecture Notes in Math. 5, Springer 1973.
[SV] A. A. Suslin and L. N. Vaserstein, Serre's problem on projective modules over polynomial rings, and Algebraic
K
-theory, Math. USSR Izv. 10 (1976) 937-1001.
[Su1] A. A. Suslin, On the structure of the special linear group over polynomial rings, Math. USSR Izv. 11, No. 2 (1977) 221-238.
[Su2] A. A. Suslin, On stably free modules, Math. USSR Sbornik 31 (1977) 479-491.
[Su3] A. A. Suslin, Cancellation over affine varieties, LOMI AN SSR 114 (1982) 187-195, Leningrad
[Su3] A. A. Suslin, Cancellation over affine varieties, (Journal of Soviet Math. 27 (1984) 2974-2980).
[Su4] A. A. Suslin, Mennicke symbols and their applications in the
K
-theory of fields, Lecture Notes in Math. 966, 334-356, Springer 1982.
[Sw] R. G. Swan, A cancellation theorem for projective modules in the metastable range, Invent. Math. 27 (1974) 23-43.
[vdK1] W. Van Der Kallen, A group structure on certain orbit sets of unimodular rows, Journal of Algebra 82 (1983) 363-397.
[vdK2] W. Van Der Kallen, A module structure on certain orbit sets of unimodular rows, Journal of Pure and Appt. Algebra 57 (1989) 281-316.
[Va1] L. N. Vaserstein, On the stabilization of the general linear group over a ring, Math. USSR Sbornik 8 (1969) 383-400.
[Va2] L. N. Vaserstein, Stable rank of rings and dimensionality of topological spaces, Functional. Anal. i Prilozhen. 5 (1971) 17-27
[Va2] L. N. Vaserstein, Stable rank of rings and dimensionality of topological spaces (Functional Anal Appl. 5 (1971) 102-110).
[Va3] L. N. Vaserstein, The structure of classical arithmetic groups of rank greater than 1, Math. USSR Sbornik 20 (1973) 465-492.
[Va4] L. N. Vaserstein, Operations on orbits of unimodular vectors, J. Algebra 100 (1986) 456-461.
[Va5] L. N. Vaserstein, Computation of
{K}_{1}
via Mennicke Symbols, Communications in Algebra 15 (1987) 611-656.
[Vo] A. C. F. Vorst, The general linear group of polynomial rings over regular rings, Commun. Algebra 9 (1981) 499-509.
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8
\mathrm{with}\left(\mathrm{StringTools}\right):
\mathrm{text1}≔"See Spot. See spot run. Spot is a dog. Look Jill! Look! Look! Look at Spot! Tristan, see spot. See spot run. Spot runs and runs. Spot is funny. Spot is nice. Tristan is nice. Jill is nice. We are all nice. Spot is special. Spot is my special dog. My special dog Spot runs and runs. He is so special. Tristan is special. Jill is special. I am special. We are all special. Spot is black. Black dogs are special. So Spot is special. Jill is a big girl. Big girls are special. So Jill is special. Tristan is a big boy. Big boys are special. So Tristan is special. I am a little girl. Little girls are special. So I am special. We are all special. We are all very, very special.":
\mathrm{text2}≔"Long ago two Indian boys lived in the Canadian forest with their parents. One boy was much older and larger and stronger than the other. He forced his little brother to do all the hard work about the place. He stole from him all the good things his parents gave him and often he beat him until he cried with pain. If the little boy told his parents of his brother\text{'}s cruelty, his brother beat him all the harder, and the little boy found that it was more to his comfort not to complain. But at last he could stand the cruelty no longer, and he decided to run away from home. So one morning he took his bow and arrows and an extra pair of moccasins, and set out alone to seek his fortune and to find a kinder world.":
\mathrm{text3}≔"Not all that Mrs. Bennet, however, with the assistance of her five daughters, could ask on the subject, was sufficient to draw from her husband any satisfactory description of Mr. Bingley. They attacked him in various ways--with barefaced questions, ingenious suppositions, and distant surmises; but he eluded the skill of them all, and they were at last obliged to accept the second-hand intelligence of their neighbour, Lady Lucas. Her report was highly favourable. Sir William had been delighted with him. He was quite young, wonderfully handsome, extremely agreeable, and, to crown the whole, he meant to be at the next assembly with a large party. Nothing could be more delightful! To be fond of dancing was a certain step towards falling in love; and very lively hopes of Mr. Bingley\text{'}s heart were entertained.":
\mathrm{map}\left(\mathrm{nops},\mathrm{map}\left(\mathrm{Words},[\mathrm{text1},\mathrm{text2},\mathrm{text3}]\right)\right)
[\textcolor[rgb]{0,0,1}{134}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{141}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{135}]
\mathrm{Readability}\left(\mathrm{text1}\right)
\textcolor[rgb]{0,0,1}{3.0}
\mathrm{Readability}\left(\mathrm{text2}\right)
\textcolor[rgb]{0,0,1}{6.585685828}
\mathrm{Readability}\left(\mathrm{text3}\right)
\textcolor[rgb]{0,0,1}{12.48683298}
\mathrm{Readability}\left(\mathrm{text1},':-\mathrm{method}'=':-\mathrm{GunningFog}'\right)
\textcolor[rgb]{0,0,1}{1.448648649}
\mathrm{Readability}\left(\mathrm{text3},':-\mathrm{method}'=':-\mathrm{GunningFog}'\right)
\textcolor[rgb]{0,0,1}{13.93650794}
\mathrm{Readability}\left(\mathrm{text2},':-\mathrm{method}'=':-\mathrm{FRES}'\right)
\textcolor[rgb]{0,0,1}{80.7900000}
\mathrm{Readability}\left(\mathrm{text3},':-\mathrm{method}'=':-\mathrm{FRES}'\right)
\textcolor[rgb]{0,0,1}{53.7800000}
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Implement series RLC branch - Simulink - MathWorks Nordic
Series RLC Branch
Implement series RLC branch
The Series RLC Branch block implements a single resistor, inductor, or capacitor, or a series combination of these. Use the Branch type parameter to select elements you want to include in the branch.
The branch resistance, in ohms (Ω). Default is 1. The Resistance parameter is not visible if the resistor element is not specified in the Branch type parameter.
The branch inductance, in henries (H). Default is 1e-3. The Inductance parameter is not visible if the inductor element is not specified in the Branch type parameter.
The Set the initial inductor current parameter is not visible and has no effect on the block if the inductor element is not specified in the Branch type parameter.
The initial inductor current used at the start of the simulation. This parameter is not visible and has no effect on the block if the inductor is not modeled and if the Set the initial inductor current parameter is not selected. Default is 0.
The branch capacitance, in farads (F). Default is 1e-6. The Capacitance parameter is not visible if the capacitance element is not specified in the Branch type parameter.
The Set the initial capacitor voltage parameter is not visible and has no effect on the block if the capacitor element is not specified in the Branch type parameter.
The initial capacitor voltage used at the start of the simulation. The Capacitor initial voltage parameter is not visible and has no effect on the block if the capacitor is not modeled and if the Set the initial capacitor voltage parameter is not selected.
Select Branch voltage to measure the voltage across the Series RLC Branch block terminals.
Select Branch current to measure the current flowing through the Series RLC Branch block.
Select Branch voltage and current to measure the voltage and the current of the Series RLC Branch block.
Obtain the frequency response of a fifth-harmonic filter (tuned frequency = 300 Hz) connected on a 60 Hz power system. This example is available in the power_seriesbranch model.
The network impedance in the Laplace domain is
Z\left(s\right)=\frac{V\left(s\right)}{I\left(s\right)}=\frac{LC{s}^{2}+RCs+1}{Cs}.
To obtain the frequency response of the impedance you have to get the state-space model (A B C D matrices) of the system.
This system is a one-input (Vsource) and one-output (Current Measurement block) system.
If you have Control System Toolbox™ software installed, you can use the bode function to get the transfer function Z(s) from the state-space matrices as follows:
[A,B,C,D] = power_analyze('power_seriesbranch');
[Ymag,Yphase] = bode(A,B,C,D,1,w);
% invert Y(s) to get Z(s)
Zmag = 1./Ymag;
Zphase = -Yphase;
loglog(freq,Zmag)
title('5th harmonic filter')
ylabel('Impedance Zmag')
semilogx(freq,Zphase)
ylabel('phase Z')
You can also use the Impedance Measurement block and the Powergui block to plot the impedance as a function of frequency. In order to measure the impedance you must disconnect the voltage source.
Multimeter, Parallel RLC Branch, Parallel RLC Load, Series RLC Load
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Handbook of Management Scales - Wikibooks, open books for an open world
The Handbook of Management Scales is a collection of previously used multi-item scales to measure constructs in empirical management research literature.
1 Critical Introduction
2 Unidimensional constructs
3 Multidimensional constructs
7 Related Handbooks
Critical IntroductionEdit
The Handbook of Management Scales was first edited by A. Wieland in 2010 and has since grown. It contains a collection of measurement scales, which are the basis for empirical research. Unfortunately, management researchers often neglect the importance of good scales. This leads to models with a high goodness-of-fit but with poor reliabiliy and validity. Construct definition and content validity are probably the most important and most neglected criteria to ensure that a scale really measures what it is intended to measure. Expert judges can help to improve content validity by capturing all important aspects that together encircle a construct.
The deletion of scale items (often called “scale purification”) may improve tau-equivalent reliability (
{\displaystyle \rho _{T}}
) (= Cronbach's
{\displaystyle \alpha }
) or congeneric reliability (
{\displaystyle \rho _{C}}
) (= composite reliability) or the statistical performance indicators of a model. However, important aspects to be measured may then disappear and this can bias the scale into a new direction. Content validity may then be destroyed and must therefore carefully be observed all along the scale purification process. Judges could also be asked to label a new construct giving them the items retained after scale purification to compare the judgement to the intended construct. Content validity may also be improved by adding missing aspects to an existing scale or by merging two existing scales.
Reflective scales often prevail in management research. But researchers should know when to use reflective and when to use formative scales. Too often do researchers specify a model with a reflective scale that is actually formative. Likert scales and semantic differential scales are probably the most common scales in management research. However, researchers often fail to weigh the pros and cons of such scales. Researchers should take more time to think about the appropriate measurement. The Handbook of Management Scales helps to find previously used scales, but will not release the researcher from carefully testing the scales in terms of reliability and validity before using them.
You are invited to contribute by adding new multi-item metrics (edit this page) to this Scales Handbook. Scales from high-ranked journals are preferred that are developed in a systematic scale development process and that are tested to measure a construct in terms of specification (reflective vs. formative), dimensionality, reliability, and validity (including content, convergent, discriminant, and nomological validity). For each scale at least its objective items, source, and, if available, reliability (e.g. tau-equivalent reliability, congeneric reliability, item reliability, average variance extracted) are listed.
Unidimensional constructsEdit
Acquiring tacit knowledge through offshore outsourcing
Advanced manufacturing technology: administrative
Advanced manufacturing technology: design
Advanced manufacturing technology: manufacturing
Advocacy participation
Agreement on vision and goals
Alliance portfolio coordination
Alliance portfolio performance
Alliance proactiveness
Alliance structures
Buyer dependence
Buyer financial performance
Buyer operational performance
Buyer's knowledge of supplier's past performance
Buying firm's expectation of relationship continuity
Buying firm's top management support
Change disposition
Change initiation - Change success
Change initiation - The degree of dissatisfaction with the status quo
Change initiation - Clarity and strength of vision of change
Change initiation - Clarity of actions towards preferred future state
Change initiation - Resistance to change
Communication of manufacturing strategy
Community-oriented CSR
Comparative organizational performance
Competence-enhancing, competence-destroying innovation
Competitive priority: cost
Competitive priority: delivery
Competitive priority: flexibility
Competitive priority: quality
Consensus on appropriation
Contractual safeguarding
Cooperation-based firm performance
Cooperation-based learning
Coordination flexibility in contingent worker skills and behaviors
Coordination flexibility in employee skills and behaviors
Coordination flexibility in HR practices
Coordination of decision making
Customer-oriented CSR
Distribution service performance
Employee-oriented CSR
Exposure to external turbulence
Faithfulness of appropriation
Feedback self-efficacy
Firm risk preference
Firm's financial performance over the past 5 years
Flexibility performance
Fluid partnering
Generational consolidation
Group esteem
Information and communication technologies implementation
Information sharing quality
Interfunctional design process
Internal quality information usage
Internal task routines
Interorganizational altruism
Interorganizational compliance
Interorganizational learning
Interorganizational loyalty
Interorganizational tolerance
Leadership involvement in quality
Manufacturing process innovation implementation performance
Market acuity
Multi-functional employees
Natural environment-oriented CSR
New application technology
New competence acquisition
New product quality
New service development culture
New service development process focus
New service development strategy
Offline store atmosphere
Offline store merchandise
Offline store layout
Online store atmosphere
Opportunity for sustainable advantage
Outsourcing intent
Partner's value proposition
Perceived delegation
Perceived product commercial success
Performance measurement system design for turbulence (PMS design for turbulence)
Private benefit, competition
Private benefit, cooperation
Proactive cost improvement
Proactive performance improvement
Processes and equipment development
Product design simplicity
Quality data and reporting
Quality improvement rewards
Quality of information exchanged
Recruiting and selection for flexible employees
Reliable and financially strong company
Resource flexibility in employee skills and behaviors
Resource flexibility in HR practices
Risk and reward sharing
Search for new technologies
Selection for teamwork potential
Shareholder-oriented CSR
Statistical process control usage
Supervisee trust
Supervisor accessibility
Supervisory interaction facilitation
Supplier evaluation systems
Supplier operational performance
Supplier-oriented CSR
Supplier's technical influence
Supply base availability
Supply chain disruption orientation
Task and work responsibilities
Threat of commercial failure
Threat of opportunism
Total productive and preventive maintenance
Transaction-specific supplier development activities
Willingness to transact
Multidimensional constructsEdit
The list in the previous section contains scales to measure constructs that were often conceptualized as dimensions of multidimensional constructs (mostly second-order constructs). The following list contains the name of multidimensional constructs and the names of their dimensions.
Absorptive Capacity: administrative, design, manufacturing
Advanced manufacturing technology: administrative, design, manufacturing
Alliance management capability: Interorganizational coordination, Alliance portfolio coordination, Interorganizational learning, Alliance proactiveness, Alliance transformation
Authentic leadership: self-awareness, relational transparency, internalized moral perspective, balanced processing
Business-to-business seller competence (B2B-SC): technical skills, change disposition, conflict management, market acuity, coordinated logistics, knowledge channels, fluid partnering
Competitive priority: cost, delivery, flexibility, quality
Control: formal control, social control
Corporate stakeholder responsibility (CStR): community-oriented CSR, natural environment-oriented CSR, employee-oriented CSR, supplier-oriented CSR, customer-oriented CSR, shareholder-oriented CSR
Customer-based corporate reputation: customer orientation, good employer, reliable and financially strong company, product and service quality, social and environmental responsibility
Driving forces: Customer focus, Competitive priorities, Strategic purchasing, Top management support, Information technology
Emotional Intelligence: Outlook, Resilience, Social Intuition, Self-Awareness, Sensitivity to Context, Attention
Enterprise resource planning (ERP) competence: strategic IT planning, executive commitment, project management, IT skills, business process skills, ERP training, learning, change readiness
Entrepreneurial orientation: innovativeness, proactiveness
Environmental uncertainty: Supply uncertainty, Demand uncertainty, Technology uncertainty
Family supportive supervisor behaviors (FSSB): emotional support, instrumental support, role modeling behaviors, creative work-family management
Feedback orientation scale (FOS): utility, accountability, social awareness, feedback self-efficacy
Group identification: relational identification, collective identification
HR flexibility: resource flexibility in HR practices, resource flexibility in employee skills and behaviors, coordination flexibility in HR practices, coordination flexibility in contingent worker skills and behaviors, coordination flexibility in employee skills and behaviors
Human resource policy: human capital, reward system
Idiosyncratic deals (i-deals): task and work responsibilities, schedule flexibility, location flexibility, financial incentives
Integrated quality management: top management commitment, customer focus, supplier quality management, design quality management, benchmarking, SPC usage, internal quality information usage, employee empowerment, employee involvement, employee training, product quality, supplier performance
Intellectual capital: Worker expertise, information sharing quality, team psychological safety
Interorganizational citizenship behaviors (ICBs): interorganizational altruism, interorganizational tolerance, interorganizational loyalty, interorganizational compliance
Justice: procedual, distributive, interpersonal, informational justice
Job burnout: Emotional Exhaustion, Depersonalization, Reduced Personal Accomplishment
Leader-member exchange (LMX): affect, loyalty, contribution, professional respect
Lean production: continuous flow, customer involvement, employee involvement, JIT delivery by suppliers, pull, set up time reduction, statistical process control, supplier development, supplier feedback, total productive/preventive maintenance
Machiavellian personality scale (MPS): distrust of others, amorality, desire for control, desire for status
Market focus: market acuity, relationship management, new service development
Market orientation scale (MO): intelligence generation, intelligence dissemination, responsiveness
Metrics and standards: measurement system, service standards
New service development competence: new service development process focus, market acuity, new service development strategy, new service development culture, information technology experience
Organizational citizenship behavior (OCB): loyalty, obedience, social participation, advocacy participation, functional participation
Organizational learning: information acquisition, information distribution, information interpretation, information integration, organizational memory
Organizational performance: liquidity, profitability, growth, stock market performance
Organizational change success prediction: Change initiation - Change success, Change initiation - The degree of dissatisfaction with the status quo, Change initiation - Clarity and strength of vision of change, Change initiation - Clarity of actions towards preferred future state, Change initiation - Resistance to change
Perceived environmental uncertainty (PEU): state uncertainty, effect uncertainty, response uncertainty
Performance: quality performance, delivery performance, flexibility performance, cost performance
Performance measurement system use focus: Attention focusing, Score keeping
Personal cultural orientations (PCO): independence, interdependence, power, social inequality, masculinity, gender equality, risk aversion, ambiguity intolerance, tradition, prudence
Political skill inventory (PSI): social astuteness, interpersonal influence, networking ability, apparent sincerity
Process management: information and communication technologies implementation, service processes
Psychological empowerment in the workplace: meaning, competence, self-determination, impact
Quality management practices: customer focus, employee relations, management leadership, process management, product/service design, quality data and reporting, supplier quality management, training
Relational competencies: communication, cooperation, integration
Resilience: agility, robustness
Service climate: leadership, service culture, value orientation
Service orientation (third-order construct): service climate, market focus, process management, human resource policy, metrics and standard
Supply chain: Supply network structure, Supply base reduction, Long-term relationship, Communication, Cross-functional teams, Supplier involvement, Logistics integration
Supply chain performance: Supplier operational performance, Buyer operational performance, Buyer financial performance
Team learning: intuition, interpretation, integration, codification
Technology acceptance model (TAM): ease of use, usefulness
Trustworthiness: ability, benevolence, integrity
Value creation in interfirm alliances: common benefit, private benefit, cooperation, private benefit, competition
The following scale example can be used, if you want to add a new scale (click edit this page to add a new one).
Rankings are recommended to assess the quality of a journal. The better the quality of a journal the more likely is a good quality of a scale published in it. Widely accepted rankings are the German VHB-JOURQUAL and the British CABS Academic Journal Guide, although the latter ranking has previously been criticized (McKinnon, 2013). In this handbook scales are used from various high-ranked management journals, e.g.
Recommended literatureEdit
MacKenzie et al. (2011): Construct Measurement and Validation Procedures in MIS and Behavioral Research: Integrating New and Existing Techniques. MIS Quarterly, Vol. 35, No. 2, pp. 293-334.
Related HandbooksEdit
Bearden et al. (2010): Handbook of Marketing Scales: Multi-Item Measures for Marketing and Consumer Behavior Research. Sage. ISBN 1412980186
Bruner II, Gordon C. (2013): Marketing Scales Handbook, Volume 7. GCBII Productions, LLC. ISBN 0615846068 [1]
Keller et al. (2002): A Summary and Analysis of Multi-Item Scales Used in Logistics Research. Journal of Business Logistics, Vol. 23, No. 2, pp. 83–119.
Keller et al. (2013): A Compendium of Multi-Item Scales Utilized in Logistics Research (2001–10): Progress Achieved Since Publication of the 1973–2000 Compendium. Journal of Business Logistics, Vol. 34, No. 2, pp. 85–93.
Roth et al. (2007): Handbook of Metrics for Research in Operations Management: Multi-item Measurement Scales and Objective Items. Sage. ISBN 1412954517
Schäffer (2008): Management Accounting & Control Scales Handbook. Research in Management Accounting & Control Series. Deutscher Universitätsverlag. ISBN 3835005251
Supply Chain Management Research. Understanding complex and dynamic networks of organizations.
Retrieved from "https://en.wikibooks.org/w/index.php?title=Handbook_of_Management_Scales&oldid=4027479"
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Ambiguity and crossambiguity function - MATLAB ambgfun - MathWorks Australia
ambgfun
Plot Ambiguity Function of Rectangular Pulse
Plot Autocorrelation Sequences of Rectangular and Linear FM Pulses
Plot Nonzero-Doppler Cuts of Autocorrelation Sequences
Plot Crossambiguity Function
Normalized Ambiguity Function
Ambiguity and crossambiguity function
afmag = ambgfun(x,Fs,PRF)
afmag = ambgfun(x,y,Fs,PRF)
[afmag,delay,doppler] = ambgfun(___)
[afmag,delay,doppler] = ambgfun(___,'Cut','2D')
[afmag,delay] = ambgfun(___,'Cut','Doppler')
[afmag,delay] = ambgfun(___,'Cut','Doppler','CutValue',V)
[afmag,doppler] = ambgfun(___,'Cut','Delay')
[afmag,doppler] = ambgfun(___,'Cut','Delay','CutValue',V)
ambgfun(___)
afmag = ambgfun(x,Fs,PRF) returns the magnitude of the normalized ambiguity function for the vector x. Fs is the sampling rate. PRF is the pulse repetition rate.
afmag = ambgfun(x,y,Fs,PRF) returns the magnitude of the normalized crossambiguity function between the pulse x and the pulse y.
[afmag,delay,doppler] = ambgfun(___) or [afmag,delay,doppler] = ambgfun(___,'Cut','2D') returns the time delay vector, delay, and the Doppler frequency vector, doppler.
[afmag,delay] = ambgfun(___,'Cut','Doppler') returns delays from a zero-Doppler cut through the 2-D normalized ambiguity function magnitude.
[afmag,delay] = ambgfun(___,'Cut','Doppler','CutValue',V) returns delays from a nonzero Doppler cut through the 2-D normalized ambiguity function magnitude at Doppler value, V.
[afmag,doppler] = ambgfun(___,'Cut','Delay') returns the Doppler values from zero-delay cut through the 2-D normalized ambiguity function magnitude.
[afmag,doppler] = ambgfun(___,'Cut','Delay','CutValue',V) returns the Doppler values from a one-dimensional cut through the 2-D normalized ambiguity function magnitude at a delay value of V.
ambgfun(___), with no output arguments, plots the ambiguity or crossambiguity function. When 'Cut' is '2D', the function produces a contour plot of the periodic ambiguity function. When 'Cut' is 'Delay' or 'Doppler', the function produces a line plot of the periodic ambiguity function cut.
x — Input pulse waveform
complex-valued row or column vector
Input pulse waveform.
y — Second input pulse waveform
Second input pulse waveform.
Pulse repetition frequency in hertz.
Example: 'Cut','Doppler','CutValue',10 specifies that a vector of ambiguity function values be produced at a Doppler shift of 10 Hz.
Cut — Direction of one-dimensional cut through ambiguity function
'2D' (default) | 'Delay' | 'Doppler'
Used to generate an ambiguity surface or one-dimensional cut through the ambiguity diagram. The value '2D' generates a surface plot of the two-dimensional ambiguity function. The direction of the one-dimensional cut is determined by setting the value of 'Cut' to 'Delay' or 'Doppler'.
The choice of 'Delay' generates a cut at zero time delay. In this case, the second output argument of ambgfuncontains the ambiguity function values at Doppler shifted values. You can create a cut at nonzero time delay using the name-value pair 'CutValue'.
The choice of 'Doppler' generates a cut at zero Doppler shift. In this case, the second output argument of ambgfun contains the ambiguity function values at time-delayed values. You can create cut at nonzero Doppler using the name-value pair 'CutValue'.
CutValue — Optional time delay or Doppler shift at which ambiguity function cut is taken
When setting the name-value pair 'Cut' to 'Delay' or 'Doppler', you can set 'CutValue' to specify a cross-section that may not coincide with either zero time delay or zero Doppler shift. However, 'CutValue' cannot be used when 'Cut' is set to '2D'.
When 'Cut' is set to 'Delay' ,'CutValue' is the time delay at which the cut is taken. Time delay units are in seconds.
When 'Cut' is set to 'Doppler', 'CutValue' is the Doppler shift at which the cut is taken. Doppler units are in hertz.
Example: 'CutValue',10.0
Normalized ambiguity or crossambiguity function magnitudes. afmag is an M-by-N matrix where M is the number of Doppler frequencies and N is the number of time delays.
Time delay vector.
delay is an N-by-1 vector of time delays.
For the ambiguity function, if Nx is the length of signal x, then the delay vector consist of N = 2Nx – 1 samples in the range, –(Nx/2) – 1,...,(Nx/2) – 1).
For the crossambiguity function, let Ny be the length of the second signal. The time delay vector consists of N = Nx+ Ny– 1 equally spaced samples. For an even number of delays, the delay sample times are –(N/2 – 1)/Fs,...,(N/2 – 1))/Fs. For an odd number of delays, if Nf = floor(N/2), the delay sample times are –Nf /Fs,...,(Nf – 1)/Fs.
Doppler frequency vector.
doppler is an M-by-1 vector of Doppler frequencies. The Doppler frequency vector consists of M = 2ceil(log2 N) equally-spaced samples. Frequencies are (–(M/2)Fs,...,(M/2–1)Fs).
Plot the ambiguity function magnitude of a rectangular pulse.
waveform = phased.RectangularWaveform;
[afmag,delay,doppler] = ambgfun(x,waveform.SampleRate,PRF);
contour(delay,doppler,afmag)
xlabel('Delay (seconds)')
ylabel('Doppler Shift (hertz)')
This example shows how to plot zero-Doppler cuts of the autocorrelation sequences of rectangular and linear FM pulses of equal duration. Note the pulse compression exhibited in the autocorrelation sequence of the linear FM pulse.
hrect = phased.RectangularWaveform('PRF',2e4);
hfm = phased.LinearFMWaveform('PRF',2e4);
xrect = step(hrect);
xfm = step(hfm);
[ambrect,delayrect] = ambgfun(xrect,hrect.SampleRate,...,
hrect.PRF,'Cut','Doppler');
[ambfm,delayfm] = ambgfun(xfm,hfm.SampleRate,...,
hfm.PRF,'Cut','Doppler');
stem(delayrect,ambrect);
title('Autocorrelation of Rectangular Pulse');
stem(delayfm,ambfm)
xlabel('Delay (seconds)');
title('Autocorrelation of Linear FM Pulse');
Plot nonzero-Doppler cuts of the autocorrelation sequences of rectangular and linear FM pulses of equal duration. Both cuts are taken at a 5 kHz Doppler shift. Besides the reduction of the peak value, there is a strong shift in the position of the linear FM peak, evidence of range-doppler coupling.
hrect.PRF,'Cut','Doppler','CutValue',fd);
hfm.PRF,'Cut','Doppler','CutValue',fd);
stem(delayrect*10^6,ambrect);
title('Autocorrelation of Rectangular Pulse at 5 kHz Doppler Shift');
stem(delayfm*10^6,ambfm)
xlabel('Delay (\mu sec)');
title('Autocorrelation of Linear FM Pulse at 5 kHz Doppler Shift');
Plot the crossambiguity function between an LFM pulse and a delayed replica. Compare the crossambiguity function with the original ambiguity function. Set the sampling rate to 100 Hz, the pulse width to 0.5 sec, and the pulse repetition frequency to 1 Hz. The delay or lag is 10 samples equal to 100 ms. The bandwidth of the LFM signal is 10 Hz.
fs = 100.0;
bw1 = 10.0;
pw = 0.5;
nlag = 10;
Create the original waveform and its delayed replica.
waveform1 = phased.LinearFMWaveform('SampleRate',fs,'PulseWidth',1,...
'SweepBandwidth',bw1,'SweepDirection','Up','PulseWidth',pw,'PRF',prf);
wav1 = waveform1();
wav2 = [zeros(nlag,1);wav1(1:(end-nlag))];
Plot the ambiguity and crossambiguity functions.
ambgfun(wav1,fs,prf,'Cut','Doppler','CutVal',5)
ambgfun(wav1,wav2,fs,[prf,prf],'Cut','Doppler','CutVal',5)
legend('Signal ambiguity', 'Crossambiguity')
The normalized ambiguity function is
\begin{array}{l}A\left(t,{f}_{d}\right)=\frac{1}{{E}_{x}}|{\int }_{-\infty }^{\infty }x\left(u\right){e}^{j2\pi {f}_{d}u}{x}^{*}\left(u-t\right)\text{ }\text{ }du|\text{ }\\ {E}_{x}={\int }_{-\infty }^{\infty }x\left(u\right){x}^{*}\left(u\right)\text{ }\text{ }du\end{array}
where Ex is the squared norm of the signal, x(t), t is the time delay, and fd is the Doppler shift. The asterisk (*) denotes the complex conjugate. The ambiguity function describes the effects of time delays and Doppler shifts on the output of a matched filter.
The magnitude of the ambiguity function achieves maximum value at (0,0). At this point, there is perfect correspondence between the received waveform and the matched filter. The maximum value of the normalized ambiguity function is one.
The magnitude of the ambiguity function at zero time delay and Doppler shift,
|A\left(0,0\right)|,
is the matched filter output when the received waveform exhibits the time delay and Doppler shift for which the matched filter is designed. Nonzero values of the time delay and Doppler shift variables indicate that the received waveform exhibits mismatches in time delay and Doppler shift from the matched filter.
The crossambiguity function between two different signals is
\begin{array}{l}A\left(t,{f}_{d}\right)=\frac{1}{\sqrt{{E}_{x}{E}_{y}}}|{\int }_{-\infty }^{\infty }x\left(u\right){e}^{j2\pi {f}_{d}u}{y}^{*}\left(u-t\right)\text{ }\text{ }du|\text{ }\\ {E}_{x}={\int }_{-\infty }^{\infty }x\left(u\right){x}^{*}\left(u\right)\text{ }\text{ }du\\ {E}_{x}={\int }_{-\infty }^{\infty }y\left(u\right){y}^{*}\left(u\right)\text{ }\text{ }du\end{array}
The peak of the crossambiguity function is not necessarily unity.
[2] Mahafza, B. R., and A. Z. Elsherbeni. MATLAB® Simulations for Radar Systems Design. Boca Raton, FL: CRC Press, 2004.
pambgfun
phased.LinearFMWaveform | phased.MatchedFilter | phased.PhaseCodedWaveform | phased.RectangularWaveform | phased.SteppedFMWaveform
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Reconstruct a Signal from Irregularly Sampled Data - MATLAB & Simulink - MathWorks Deutschland
Resample the data to make the INR readings uniformly spaced. The first reading was taken at 11:28 a.m. on a Friday. Use resample to estimate the patient's INR at that time on every subsequent Friday. Specify a sample rate of one reading per week, or equivalently,
1/\left(7×86400\right)
readings per second. Use spline interpolation for the resampling.
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Classification loss for cross-validated ECOC model - MATLAB kfoldLoss - MathWorks Nordic
L=\sum _{j=1}^{n}{w}_{j}{e}_{j},
L=\sum _{j=1}^{n}{w}_{j}{c}_{{y}_{j}{\stackrel{^}{y}}_{j}},
{c}_{{y}_{j}{\stackrel{^}{y}}_{j}}
{\stackrel{^}{y}}_{j}
\stackrel{^}{k}
\stackrel{^}{k}=\underset{k}{\text{argmin}}\frac{1}{B}\sum _{j=1}^{B}|{m}_{kj}|g\left({m}_{kj},{s}_{j}\right).
\stackrel{^}{k}=\underset{k}{\text{argmin}}\frac{\sum _{j=1}^{B}|{m}_{kj}|g\left({m}_{kj},{s}_{j}\right)}{\sum _{j=1}^{B}|{m}_{kj}|}.
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electricityMap – Practices for real-world data science
As I write this introduction, I have been working as a data scientist for electricityMap for about 8 months. Overall, I am responsible, together with the other data scientist in the team, for delivering high data quality at the end of the entire data processing pipeline. That data can be real-time, historical or forecasted.
In a nutshell, the generation of electricityMap’s data is a multi-stage process. Firstly, varied data points from numerous public data sources about electricity are aggregated. Secondly, they are validated and standardised against a reference schema. Finally, they are run through our flow-tracing algorithm for the generation of worldwide real-time hourly electricity consumption figures; and their associated greenhouse gas emissions.
A simplified overview of the electricityMap infrastructure. Thanks Felix for the figure!
This data is at the core of our mission: to organise the world’s electricity data to drive the transition towards a truly decarbonised electricity system. This global ambition has two important consequences. We must first be able to overcome data sources becoming erroneous or unavailable at any moment. Secondly, we must come up with clever ways to generate truthful data for regions of the world where reliable public electrical data sources are unavailable.
These consequences reveal my other current responsibilities; develop and maintain a wide range of models to capture the dynamics of electrical production per factor and exchanges in vastly different areas of the world, and have enough domain expertise to ensure that they behave according to what is physically possible.
The electricityMap team. (PS: yes, thank you Nick! See here the announcement for a 1,000,000€ grant from Google.)
These responsibilities are far-reaching and evolve rapidly as electricityMap scales up. It is impossible to be highly specialised when only 11 (10 brights + me) people are fighting for something that is way too big for them only. The good news is that as electricityMap grows, I am constantly able to redefine my role as a data scientist, and the practices I should adopt to help the team be successful.
Recently, we opened a position to find a brilliant mind that can become the most knowledgeable about our data quality. At the same time, I started delivering on tasks whose scope overflowed into the realm of data engineering. The former event, because it will most likely reduce the scope of my responsibilities, pushed me to redefine what I, as a data scientist, should focus my efforts on. The latter, because it is much more common within software/data engineering, revealed to me the necessity of defining and implementing good practices for successfully delivering on that newly defined scope.
Hence this blog post, which aims at capturing my current thoughts around the data scientist role and practices that should be more widely adopted within the profession. This opinion piece is a reflection of the prism through which I perceive my work. It is therefore highly influenced by my work for electricityMap, my previous interrogations between research and industry, and my relative freshness within this position. For that reason, it might get updated occasionally.
Data scientists are often described as part mathematician, part computer scientist and part trend-spotter
^1
by corporates that don’t understand that finding someone that can be a stellar mathematician, computer scientist and analyst at once might be extremely hard. I would tend to think that a company that looks for all three competencies in a single individual has an immature data infrastructure and lacks the experience to realise that asking a single individual to do it all might be counterproductive.
This thought pattern has probably been harmful to many recent graduates who have been developing and researching state-of-the-art machine learning and optimisation models during their studies to end up having to spend most of their time researching data sources, and frying their brains over complex software architecture problems.
The previous description could correspond to three different jobs; the expertise in computer science could be delegated to a data engineer, the expertise in analysis delegated to a data analyst, while the mathematics and modelling should be the realm of data scientists.
This means that in most cases, recruiting a data scientist should be preceded by hiring a data engineer, who will be much better suited for building up the infrastructure that will support the forthcoming modelling efforts. With libraries like scikit-learn or PyTorch, any talented data engineer would already be able to build up a decent pipeline for predictions or estimations without a deep understanding of the models themselves.
With the infrastructure already built up, a data scientist can then generate tremendous value, by capturing and accumulating marginal gains from guiding the creation of automated Q&A systems, ensuring consistency between data distributions (distributional shifts are common with real-world dynamic data), and researching and putting in production new models
^2
Meanwhile, the visualisations generated, trends identified and domain expertise accumulated by a data analyst through its constant interaction with the end data can increase the engagement with and actionability of the data, and generate customer-side insights. This will help improve the overall quality of the generated data, and make sure that it fits the business requirements.
In the end, it is the compound efforts of a unified data team that should produce the data visualisations, insights generation and domain knowledge required for any given task.
Within this understanding of the data scientist role, and its strong ties to the other data positions, I tend to define the objectives of the data scientist as:
Ensuring consistency of data fed through the pipeline. Their knowledge of statistics and understanding of distributions makes them suited for defining rules to check the consistency of the data flowing through the pipeline. These verifications are of paramount importance before feeding data through any model. Data that does not respect the expected properties can have disastrous consequences on any model’s output. Typically, data distributions can drift and display a significant shift when compared to the data a model has been trained on, or some inputs might be periodically unavailable, affecting the computability of key features for the models.
Setting up a modular modelisation infrastructure with individual components that automate repetitive tasks, training, validation, monitoring etc. Ideally, this infrastructure should also offer the bindings that make the experimentation with new models seamless. For example, as is conceptualised in PyTorch Lightning, creating an independent module for data loading, and a trainer instance accelerates the setup time of any new model.
Researching (and implementing) new models. One needs to deserve the scientist part of his title, and any data scientist should leverage his modelling expertise to unlock varied benefits from different model classes. Furthermore, the field of machine learning has been receiving a lot of attention (no pun intended) these last years, leading to fast developments and paradigm changes (think transformers for example). The consequence is that the expertise a data scientist has over a certain class of models can quickly become outdated, and the time spent keeping up with new models will be rewarded in terms of increased model performance later on.
The data scientist role can further be broken down between pure modelling (leaning towards research scientist) and the practical implementation of models (machine learning scientist).
In much larger organisations, these different responsibilities might be too large for one person alone. In this case, the data scientist role can be further split up into machine learning engineers, who will inherit the practical implementation of models within the data pipeline, and research scientist, who can focus only on the modelling and research aspects.
Better understanding one’s responsibilities is essential to prioritise work efficiently, but it does not help in actually delivering on the work that has been prioritised. Learning a lot from my colleagues, I’ve come to realise that the adoption of simple principles does.
There is an inherent risk within data science related to the hybrid position it occupies between machine learning research and commercial applications. As a data scientist, one can naturally tend to perfect a model’s performance, tweaking hyperparameters, fine-tuning pre-processing steps, and running multiple experiments with incremental improvements, as we’ve been taught throughout school or in a research environment. The problem with that process is how time and resource intensive it can be - Tracking the results of multiple experiments at once is arguably one of the most strenuous tasks in machine learning, and providing time estimates for when quantitative improvements can be secured is seldom more accurate than the output of a random number generator -, while the business requirements for a model might not require such a high performance model. I would therefore strongly advise in favour of making requirements clear before starting to research and implement any model.
The requirements should cover expected accuracy, robustness, scalability and model freshness.
A common pitfall.
Tooling is crucial as it defines how easy it will be to take a model from an experimentation phase to a production-ready application. The way machine learning is commonly taught, with an emphasis on creating one-off notebooks or scripts that do not need to take into consideration dependencies and inter-operability, does not translate well to real-world problems tackled by cross-functional teams. Having proper tooling shared by all in the team is essential, yet mostly invisible, to allow fast iterations and adoption of models performing better.
On the opposite, the absence of proper tooling can easily lead to a mountain of technical debt where the combined efforts of poor code quality, naming conventions and architecture will seriously undermine the maintainability, testability and scalability of any model. (Just look at any graduate student’s thesis code for a good example, especially the dataprocessingfinalv2.py_ file)
Another common pitfall. Credit Joel Grus.
At the very basis of this effort are the adoption of common code style (think automatic linting with black & isort in python for example) and versioning systems (besides e.g GitHub, investing in a functional versioning system for datasets and metadata might be worthwhile).
Next, the adoption of internal tools that facilitate the development of any new project has far-reaching benefits for aligning the different team members’ output, making reviewing each other’s code far easier and freeing up thinking space to focus on the actual deliverable. These can include for example creating a collection of libraries that each offers a specific set of services (e.g DB access, configuration, plotting style etc), creating domain-related toolboxes, or even setting up a well documented templating system.
Finally, using the tools already set up should simplify the separation of concerns for the various essential parts of the data science workflow. Time invested into converting each of these steps (data fetching, data preparation, features generation, training, validation, evaluation, serving etc) into their own independent module should be rewarded pretty quickly in terms of adaptability, testability and scalability of the system.
Test first, then debug
In software engineering, it is very common practice to test every piece of production-ready code. From unit testing, to end-to-end testing, frameworks for reliably testing individual logic and their global cooperation ensure that the produced software (mostly) behaves as expected. In data science, whether due to the sometimes non-deterministic nature of the models used, the additional burden of testing the data itself, or to the hubris of the data scientist (that’s what you get when a job gets so much hype), testing is not as strictly mandated. The consequence is that important parts of the modelling infrastructure can get adopted in production without full test coverage. This directly affects the reliability and hence the trustworthiness of the models’ output.
I don’t want here to expand precisely on how to test data science systems in practice, as it would extend beyond the scope of this post, and also because good resources already exist out there
^3
. Instead, I want to provide two simple, but sticky rules. Firstly, as often as possible when debugging, write first a test that isolates and replicates the bug under investigation. This has multiple benefits; speeding up the investigation by focusing it, providing a direct way to verify that the fix indeed fixes the bug without introducing any regression somewhere else, and expanding test coverage without much effort! Secondly, always test the data itself. Machine learning systems have an additional dependency on the data compared to standard software systems, and the testing setup must account for it.
I hope that you found this post instructive. As hinted earlier, I mostly wrote up this post to help me structure my thoughts around the development of my role working for electricityMap, but I would be thrilled to hear your opinion on the definition of the data scientist role and what principles he/she should adopt to successfully get his new ideas and models implemented in the real-world. I’m only beginning my journey and still have a lot to learn from the community ✌️.
1: source, SAS
2: Some might argue that putting models in production, because it is mostly an engineering task, should be delegated to someone else that has a stronger emphasis on software engineering. In the line of what Eugene Yan defends in his Unpopular opinion - Data scientists should be more end-to-end, I tend to think that letting data scientists own the entire process from getting data to delivering value based on identified patterns result in faster execution and output that is more in line with the context and the business requirements.
3: See for example the extensive tutorial from MadeWithML
Written by Pierre Segonne
Data Scientist @ electricityMap
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Receiver operating characteristic (ROC) curve or other performance curve for classifier output - MATLAB perfcurve - MathWorks India
score\left(:,2\right)-\mathrm{max}\left(score\left(:,1\right),score\left(:,3\right)\right)
score\left(:,2\right)-score\left(:,3\right)
S=\frac{\text{Cost}\left(P|N\right)-\text{Cost}\left(N|N\right)}{\text{Cost}\left(N|P\right)-\text{Cost}\left(P|P\right)}*\frac{N}{P}
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Trapezoidal numerical integration - MATLAB trapz - MathWorks América Latina
Y contains function values for
f\left(x\right)={x}^{2}
in the domain [1, 5].
This approximate integration yields a value of 42. In this case, the exact answer is a little less,
41\frac{1}{3}
. The trapz function overestimates the value of the integral because f(x) is concave up.
Calculate the function
f\left(x,y\right)={x}^{2}+{y}^{2}
on the grid.
I={\int }_{-5}^{5}{\int }_{-3}^{3}\left({x}^{2}+{y}^{2}\right)dx\phantom{\rule{0.2222222222222222em}{0ex}}dy
\begin{array}{c}\underset{a}{\overset{b}{\int }}f\left(x\right)dx\text{\hspace{0.17em}}\text{\hspace{0.17em}}\approx \text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{b-a}{2N}\sum _{n=1}^{N}\left(f\left({x}_{n}\right)+f\left({x}_{n+1}\right)\right)\\ =\frac{b-a}{2N}\left[f\left({x}_{1}\right)+2f\left({x}_{2}\right)+...+2f\left({x}_{N}\right)+f\left({x}_{N+1}\right)\right]\text{\hspace{0.17em}}\text{\hspace{0.17em}},\end{array}
where the spacing between each point is equal to the scalar value
\frac{b-a}{N}
. By default MATLAB® uses a spacing of 1.
\underset{a}{\overset{b}{\int }}f\left(x\right)dx\text{\hspace{0.17em}}\text{\hspace{0.17em}}\approx \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{1}{2}\sum _{n=1}^{N}\left({x}_{n+1}-{x}_{n}\right)\left[f\left({x}_{n}\right)+f\left({x}_{n+1}\right)\right]\text{\hspace{0.17em}},
a={x}_{1}<{x}_{2}<\text{\hspace{0.17em}}\text{\hspace{0.17em}}...\text{\hspace{0.17em}}\text{\hspace{0.17em}}<{x}_{N}<{x}_{N+1}=b
\left({x}_{n+1}-{x}_{n}\right)
is the spacing between each consecutive pair of points.
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Discrete nonlinear resistor - Simulink - MathWorks Nordic
Voltage-Current characteristics [(V);(A)]
Discrete nonlinear resistor
The Nonlinear Resistor block represents a time-varying resistor. It implements a discrete variable resistor as a current source or a voltage source. The resistance is specified by a monotonically increasing or decreasing voltage-current characteristic.
When the nonlinear resistor is implemented as a voltage source, the block uses the following equation for the relationship between the voltage, v, across the device and the current through the resistor, i, for the resistance characteristic specified by R:
v=R\ast i.
When the nonlinear resistor is implemented as a current source, the block uses the following equation for the relationship:
i=\frac{1}{R}\ast v.
When you use a Nonlinear Resistor block in your model, set the powergui block Simulation type to Discrete and select the Automatically handle Discrete solver and Advanced tab solver settings of blocks parameter in the Preferences tab. The robust discrete solver is used to discretize the electrical model. Simulink® signals an error if the robust discrete solver is not used.
Specialized electrical conserving port associated with the resistor positive voltage.
Specialized electrical conserving port associated with the resistor negative voltage.
Type — Source type
Current source (default) | Voltage source
Whether the nonlinear resistor is implemented as a current source or a voltage source.
Voltage-Current characteristics [(V);(A)] — Resistance characteristic, R
[0,0 ; 1,1] (default) | 2-by-n matrix
Resistance specified by a monotonically increasing or decreasing voltage-current characteristic, specified as a 2-by-n matrix where n is greater than or equal to 2.
Nonlinear Inductor | Variable Capacitor | Variable Inductor | Variable Resistor | Variable-Ratio Transformer
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When you want to convince someone that
8x+(2x−3)
(8x+2x)−3
, you can support your claim with the Associative Property of Addition.
Determine if the following statements are true or false. If true, justify your conclusion by stating the appropriate algebraic property. If false, explain how you know.
(81)(38)=(38)(81)
Notice that the only difference between the left side and the right side of the equation is the placement of the factors.
Does this change the product? How do you know?
True; Commutative Property of Multiplication
27+0=27
Does adding zero to a number ever change its value? What is this property called?
3−5=5−3
Unlike addition and multiplication, subtraction is not commutative.
Meaning that the order of the terms matters.
19.4·1=19.4
Hint: (d)
\left(a\right)\left(1\right)=a
True; Identity Property of Multiplication
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Discard support vectors of linear SVM binary learners in ECOC model - MATLAB discardSupportVectors - MathWorks Nordic
Linear SVM Binary Learner
Discard support vectors of linear SVM binary learners in ECOC model
Mdl = discardSupportVectors(MdlSV) returns a trained multiclass error-correcting output codes (ECOC) model (Mdl) from the trained multiclass ECOC model (MdlSV), which contains at least one linear CompactClassificationSVM binary learner. Both Mdl and MdlSV are objects of the same type, either ClassificationECOC objects or CompactClassificationECOC objects.
Mdl has these characteristics:
The Alpha, SupportVectors, and SupportVectorLabels properties of all the linear SVM binary learners are empty ([]).
If you display any linear SVM binary learners stored in the cell array of trained models Mdl.BinaryLearners, the software lists the Beta property instead of Alpha.
\alpha
\alpha
MdlSV — Full or compact, trained multiclass ECOC model
ClassificationECOC model | CompactClassificationECOC model
Full or compact, trained multiclass ECOC model containing at least one linear SVM binary learner, specified as a ClassificationECOC or CompactClassificationECOC model.
In the context of this page, a linear support vector machine (SVM) binary learner is a binary SVM classifier created using a linear kernel function. If the jth binary learner in an ECOC model Mdl is a linear SVM binary learner, then Mdl.BinaryLearners{j} is a CompactClassificationSVM object, where Mdl.BinaryLearners{j}.KernelParameters.Function is 'linear'.
predict and resubPredict estimate SVM scores f(x) for each linear SVM binary learner in an ECOC model using
f\left(x\right)=x\prime \beta +b.
β is the Beta property and b is the Bias property of the binary learners. You can access these properties for each linear SVM binary learner in the cell array Mdl.BinaryLearners. For more details on the SVM score calculation, see Support Vector Machines for Binary Classification.
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Means of projecting three-dimensional objects in two dimensions
For the orthographic projection as a map projection, see Orthographic projection in cartography. For mathematical discussion in terms of linear algebra, see Projection (linear algebra).
Orthographic projection (sometimes referred to as orthogonal projection, used to be called analemma[a]) is a means of representing three-dimensional objects in two dimensions. It is a form of parallel projection, in which all the projection lines are orthogonal to the projection plane,[2] resulting in every plane of the scene appearing in affine transformation on the viewing surface. The obverse of an orthographic projection is an oblique projection, which is a parallel projection in which the projection lines are not orthogonal to the projection plane.
The term orthographic is sometimes reserved specifically for depictions of objects where the principal axes or planes of the object are also parallel with the projection plane.[2] However, these are better known as primary views in multiview projection. Furthermore, when the principal planes or axes of an object in an orthographic projection are not parallel with the projection plane, the depictions are sometimes referred to as axonometric. However, these are better known as auxiliary views. (Axonometric projection might be more accurately described as being synonymous with parallel projection.) Sub-types of primary views include plans, elevations and sections. Sub-types of auxiliary views might include isometric, dimetric and trimetric projections.
A lens providing an orthographic projection is known as an object-space telecentric lens.
3 Multiview projection
Comparison of several types of graphical projection
The three views. The percentages show the amount of foreshortening.
A simple orthographic projection onto the plane z = 0 can be defined by the following matrix:
{\displaystyle P={\begin{bmatrix}1&0&0\\0&1&0\\0&0&0\\\end{bmatrix}}}
For each point v = (vx, vy, vz), the transformed point Pv would be
{\displaystyle Pv={\begin{bmatrix}1&0&0\\0&1&0\\0&0&0\\\end{bmatrix}}{\begin{bmatrix}v_{x}\\v_{y}\\v_{z}\end{bmatrix}}={\begin{bmatrix}v_{x}\\v_{y}\\0\end{bmatrix}}}
Often, it is more useful to use homogeneous coordinates. The transformation above can be represented for homogeneous coordinates as
{\displaystyle P={\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&0&0\\0&0&0&1\end{bmatrix}}}
For each homogeneous vector v = (vx, vy, vz, 1), the transformed vector Pv would be
{\displaystyle Pv={\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&0&0\\0&0&0&1\end{bmatrix}}{\begin{bmatrix}v_{x}\\v_{y}\\v_{z}\\1\end{bmatrix}}={\begin{bmatrix}v_{x}\\v_{y}\\0\\1\end{bmatrix}}}
In computer graphics, one of the most common matrices used for orthographic projection can be defined by a 6-tuple, (left, right, bottom, top, near, far), which defines the clipping planes. These planes form a box with the minimum corner at (left, bottom, -near) and the maximum corner at (right, top, -far).[3]
The box is translated so that its center is at the origin, then it is scaled to the unit cube which is defined by having a minimum corner at (−1,−1,−1) and a maximum corner at (1,1,1).
The orthographic transform can be given by the following matrix:
{\displaystyle P={\begin{bmatrix}{\frac {2}{{\text{right}}-{\text{left}}}}&0&0&-{\frac {{\text{right}}+{\text{left}}}{{\text{right}}-{\text{left}}}}\\0&{\frac {2}{{\text{top}}-{\text{bottom}}}}&0&-{\frac {{\text{top}}+{\text{bottom}}}{{\text{top}}-{\text{bottom}}}}\\0&0&{\frac {-2}{{\text{far}}-{\text{near}}}}&-{\frac {{\text{far}}+{\text{near}}}{{\text{far}}-{\text{near}}}}\\0&0&0&1\end{bmatrix}}}
which can be given as a scaling S followed by a translation T of the form
{\displaystyle P=ST={\begin{bmatrix}{\frac {2}{{\text{right}}-{\text{left}}}}&0&0&0\\0&{\frac {2}{{\text{top}}-{\text{bottom}}}}&0&0\\0&0&{\frac {2}{{\text{far}}-{\text{near}}}}&0\\0&0&0&1\end{bmatrix}}{\begin{bmatrix}1&0&0&-{\frac {{\text{left}}+{\text{right}}}{2}}\\0&1&0&-{\frac {{\text{top}}+{\text{bottom}}}{2}}\\0&0&-1&-{\frac {{\text{far}}+{\text{near}}}{2}}\\0&0&0&1\end{bmatrix}}}
The inversion of the projection matrix P−1, which can be used as the unprojection matrix is defined:
{\displaystyle P^{-1}={\begin{bmatrix}{\frac {{\text{right}}-{\text{left}}}{2}}&0&0&{\frac {{\text{left}}+{\text{right}}}{2}}\\0&{\frac {{\text{top}}-{\text{bottom}}}{2}}&0&{\frac {{\text{top}}+{\text{bottom}}}{2}}\\0&0&{\frac {{\text{far}}-{\text{near}}}{-2}}&-{\frac {{\text{far}}+{\text{near}}}{2}}\\0&0&0&1\end{bmatrix}}}
Classification of Orthographic projection and some 3D projections
Three sub-types of orthographic projection are isometric projection, dimetric projection, and trimetric projection, depending on the exact angle at which the view deviates from the orthogonal.[2][4] Typically in axonometric drawing, as in other types of pictorials, one axis of space is shown to be vertical.
In isometric projection, the most commonly used form of axonometric projection in engineering drawing,[5] the direction of viewing is such that the three axes of space appear equally foreshortened, and there is a common angle of 120° between them. As the distortion caused by foreshortening is uniform, the proportionality between lengths is preserved, and the axes share a common scale; this eases one's ability to take measurements directly from the drawing. Another advantage is that 120° angles are easily constructed using only a compass and straightedge.
In dimetric projection, the direction of viewing is such that two of the three axes of space appear equally foreshortened, of which the attendant scale and angles of presentation are determined according to the angle of viewing; the scale of the third direction is determined separately. Dimensional approximations are common in dimetric drawings.[clarification needed]
In trimetric projection, the direction of viewing is such that all of the three axes of space appear unequally foreshortened. The scale along each of the three axes and the angles among them are determined separately as dictated by the angle of viewing. Dimensional approximations in trimetric drawings are common,[clarification needed] and trimetric perspective is seldom used in technical drawings.[4]
Symbols used to define whether a multiview projection is either third-angle (right) or first-angle (left).
In multiview projection, up to six pictures of an object are produced, called primary views, with each projection plane parallel to one of the coordinate axes of the object. The views are positioned relative to each other according to either of two schemes: first-angle or third-angle projection. In each, the appearances of views may be thought of as being projected onto planes that form a six-sided box around the object. Although six different sides can be drawn, usually three views of a drawing give enough information to make a three-dimensional object. These views are known as front view, top view and end view. Other names for these views include plan, elevation and section. When the plane or axis of the object depicted is not parallel to the projection plane, and where multiple sides of an object are visible in the same image, it is called an auxiliary view. Thus isometric projection, dimetric projection and trimetric projection would be considered auxiliary views in multiview projection. A typical characteristic of multiview projection is that one axis of space is usually displayed as vertical.
Main article: Orthographic projection in cartography
Orthographic projection (equatorial aspect) of eastern hemisphere 30°W–150°E
An orthographic projection map is a map projection of cartography. Like the stereographic projection and gnomonic projection, orthographic projection is a perspective (or azimuthal) projection, in which the sphere is projected onto a tangent plane or secant plane. The point of perspective for the orthographic projection is at infinite distance. It depicts a hemisphere of the globe as it appears from outer space, where the horizon is a great circle. The shapes and areas are distorted, particularly near the edges.[6][7]
The orthographic projection has been known since antiquity, with its cartographic uses being well documented. Hipparchus used the projection in the 2nd century BC to determine the places of star-rise and star-set. In about 14 BC, Roman engineer Marcus Vitruvius Pollio used the projection to construct sundials and to compute sun positions.[7]
Vitruvius also seems to have devised the term orthographic (from the Greek orthos (= “straight”) and graphē (= “drawing”) for the projection. However, the name analemma, which also meant a sundial showing latitude and longitude, was the common name until François d'Aguilon of Antwerp promoted its present name in 1613.[7]
The earliest surviving maps on the projection appear as woodcut drawings of terrestrial globes of 1509 (anonymous), 1533 and 1551 (Johannes Schöner), and 1524 and 1551 (Apian).[7]
^ Today the word analemma is more commonly used in its more specific meaning of a diagram showing the position of the Sun from the earth.[1]
^ a b c Maynard, Patric (2005). Drawing distinctions: the varieties of graphic expression. Cornell University Press. p. 22. ISBN 0-8014-7280-6.
^ Thormählen, Thorsten (November 26, 2021). "Graphics Programming – Cameras: Parallel Projection – Part 6, Chapter 2". Mathematik Uni Marburg. pp. 8 ff. Retrieved 2022-04-22. {{cite web}}: CS1 maint: url-status (link)
^ a b McReynolds, Tom; David Blythe (2005). Advanced graphics programming using openGL. Elsevier. p. 502. ISBN 1-55860-659-9.
^ Godse, A. P. (1984). Computer graphics. Technical Publications. p. 29. ISBN 81-8431-558-9.
^ Snyder, J. P. (1987). Map Projections—A Working Manual (US Geologic Survey Professional Paper 1395). Washington, D.C.: US Government Printing Office. pp. 145–153.
^ a b c d Snyder, John P. (1993). Flattening the Earth: Two Thousand Years of Map Projections pp. 16–18. Chicago and London: The University of Chicago Press. ISBN 0-226-76746-9.
Wikimedia Commons has media related to Orthographic projections.
Normale (orthogonale) Axonometrie (in German)
Orthographic Projection Video and mathematics
Retrieved from "https://en.wikipedia.org/w/index.php?title=Orthographic_projection&oldid=1084105946"
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1911 Encyclopædia Britannica/Chalk - Wikisource, the free online library
1911 Encyclopædia Britannica/Chalk
See also Chalk on Wikipedia; and our 1911 Encyclopædia Britannica disclaimer.
20514911911 Encyclopædia Britannica, Volume 5 — Chalk
CHALK, the name given to any soft, pulverulent, pure white limestone. The word is an old one, having its origin in the Saxon cealc, and the hard form “kalk” is still in use amongst the country folk of Lincolnshire. The German Kalk comprehends all forms of limestone; therefore a special term, Kreide, is employed for chalk—French craie. From being used as a common name, denoting a particular material, the word was subsequently utilized by geologists as an appellation for the Chalk formation; and so prominent was this formation in the eyes of the earlier workers that it imposed its name upon a whole system of rocks, the Cretaceous (Lat. creta, chalk), although this rock itself is by no means generally characteristic of the system as a whole.
The Chalk formation, in addition to the typical chalk material—creta scriptoria—comprises several variations; argillaceous kinds—creta marga of Linnaeus—known locally as malm, marl, clunch, &c.; and harder, more stony kinds, called rag, freestone, rock, hurlock or harrock in different districts. In certain parts of the formation layers of nodular flints (q.v.) abound; in parts, it is inclined to be sandy, or to contain grains of glauconite which was originally confounded with another green mineral, chlorite, hence the name “chloritic marl” applied to one of the subdivisions of the chalk. In its purest form chalk consists of from 95 to 99% of calcium carbonate (carbonate of lime); in this condition it is composed of a mass of fine granular particles held together by a somewhat feeble calcareous cement. The particles are mostly the broken tests of foraminifera, along with the débris of echinoderm and molluscan shells, and many minute bodies, like coccoliths, of somewhat obscure nature.
The earliest attempts at subdivision of the Chalk formation initiated by Wm. Phillips were based upon lithological characters, and such a classification as “Upper Chalk with Flints,” “Lower Chalk without Flints,” “Chalk marl or Grey chalk,” was generally in use in England until W. Whitaker established the following order in 1865:—
Upper Chalk, with flints
Lower Chalk
{\displaystyle \scriptstyle {\left\{{\begin{matrix}\ \\\\\ \ \end{matrix}}\right.}}
chalk rock
chalk with few flints
chalk without flints
Chalk Marl
{\displaystyle \scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}}
Totternhoe stone
Totternhoe marl
In France, a similar system of classification was in vogue, the subdivisions being craie blanche, craie tufan, craie chloritée, until 1843 when d’Orbigny proposed the term Senonien for the Upper Chalk and Turonien for the Lower; later he divided the Turonien, giving the name Cénomanien to the lower portion. The subdivisions of d’Orbigny were based upon the fossil contents and not upon the lithological characters of the rocks. In 1876 Prof. Ch. Barrois showed how d’Orbigny’s classification might be applied to the British chalk rocks; and this scheme has been generally adopted by geologists, although there is some divergence of opinion as to the exact position of the base line of the Cenomanian.
The accompanying table shows the classification now adopted in England, with the zonal fossils and the continental names of the substages:—
Zonal fossils used in Britain. Stages. N. France
Belgium. S.E. and
S. France.
{\displaystyle \scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right.}}
Ostrea lunata (Norfolk)
Actinocamax quadratus
= Inoceramus lingua in Yorkshire
Marsupites testudinarium
{Marsupites, Uintacrinus}
Danian?
(Trimingham)
Craie blanche Flint-
chalk.* Marls,
(not chalky)
Hippurites.
{\displaystyle \scriptstyle {\left\{{\begin{matrix}\ \\\\\ \ \end{matrix}}\right.}}
Micraster cor-anguinum
Micraster cor-testudinarium
Holaster planus, Chalk rock
Terebratulina gracilis
Rhynchonella Cuvieri, Melbourne rock
Craie marneuse
Lower Chalk,
Chalk Marl and
Cambridge Greensand Marly
chalk.*
Actinocamax plenus
Holaster subglobosus, Totternhoe stone.
Schloenbachia varians.
Craie glauconieuse
*(See table in article Cretaceous System.)
Since Prof. Barrois introduced the zonal system of subdivision (C. Evans had used a similar scheme six years earlier), our knowledge of the English chalk has been greatly increased by the work of Jukes-Browne and William Hill, and particularly by the laborious studies of Dr A. W. Rowe. Instead of employing the mixed assemblage of animals indicated as zone fossils in the table, A. de Grossouvre proposed a scheme for the north of France based upon ammonite faunas alone, which he contended would be of more general applicability (Recherches sur la Craie Supérieure, Paris, 1901).
The Upper Chalk has a maximum thickness in England of about 1000 ft., but post-cretaceous erosion has removed much of it in many districts. It is more constant in character, and more typically chalky than the lower stages; flints are abundant, and harder nodular beds are limited to the lower portions, where some of the compact limestones are known as “chalk rock.” The thickness of the Middle Chalk varies from about 100 to 240 ft.; flints become scarcer in descending from the upper to the lower portions. The whole is more compact than the upper stage, and nodular layers are more frequent—the “chalk rock” of Dorset and the Isle of Wight belong to this stage. At the base is the hard “Melbourne rock.” The thickness of the Lower Chalk in England varies from 60 to 240 ft. This stage includes part of the “white chalk without flints,” the “chalk marl,” and the “grey chalk.” The Totternhoe stone is a hard freestone found locally in this stage. The basement bed in Norfolk is a pure limestone, but very frequently it is marly with grains of sand and glauconite, and often contains phosphatic nodules; this facies is equivalent to the “Cambridge Greensand” of some districts and the “chloritic marl” of others. In Devonshire the Lower Chalk has become thin sandy calcareous series.
The chalk can be traced in England from Flamborough Head in Yorkshire, in a south-westerly direction, to the coast of Dorset; and it not only underlies the whole of the S.E. corner, where it is often obscured by Tertiary deposits, but it can be followed across the Channel into northern France. Rocks of the same age as the chalk are widespread (see Cretaceous System); but the variety of limestone properly called by this name is almost confined to the Anglo-Parisian basin. Some chalk occurs in the great Cretaceous deposits of Russia, and in Kansas, Iowa, Nebraska and S. Dakota in the United States. Hard white chalk occurs in Ireland in Antrim, and on the opposite shore of Scotland in Mull and Morven.
Economic Products of the Chalk.—Common chalk has been frequently used for rough building purposes, but the more important building stones are “Beer stone,” from Beer Head in Devonshire, “Sutton stone” from a little north of Beer, and the “Totternhoe stone.” It is burned for lime, and when mixed with some form of clay is used for the manufacture of cement; chalk marl has been used alone for this purpose. As a manure, it has been much used as a dressing for clayey land. Flints from the chalk are used for road metal and concrete, and have been employed in building as a facing for walls. Phosphatic nodules for manure have been worked from the chloritic marl and Cambridge Greensand, and to some extent from the Middle Chalk. The same material is worked at Ciply in Belgium and Picardy in France. Chalk is employed in the manufacture of carbonate of soda, in the preparation of carbon dioxide, and in many other chemical processes; also for making paints, crayons and tooth-powder. Whiting or Spanish white, used to polish glass and metal, is purified chalk prepared by triturating common chalk with a large quantity of water, which is then decanted and allowed to deposit the finely-divided particles it holds in suspension.
Chalk Scenery.—Where exposed at the surface, chalk produces rounded, smooth, grass-covered hills as in the Downs of southern England and the Wolds of Yorkshire and Lincolnshire. The hills are often intersected by clean-cut dry valleys. It forms fine cliffs on the coast of Kent, Yorkshire and Devonshire.
Chalk is employed medicinally as a very mild astringent either alone or more usually with other astringents. It is more often used, however, for a purely mechanical action, as in the preparation hydrargyrum cum creta. As an antacid its use has been replaced by other drugs.
Black chalk or drawing slate is a soft carbonaceous schist, which gives a black streak, so that it can be used for drawing or writing. Brown chalk is a kind of umber. Red chalk or reddle is an impure earthy variety of haematite. French chalk is a soft variety of steatite, a hydrated magnesium silicate.
The most comprehensive account of the British chalk is contained in the Memoirs of the Geological Survey of the United Kingdom, “The Cretaceous Rocks of Britain,” vol. ii. 1903, vol. iii. 1904 (with bibliography), by Jukes-Browne and Hill. See also “The White Chalk of the English Coast,” several papers in the Proceedings of the Geologists’ Association, London, (1) Kent and Sussex, xvi. 1900, (2) Dorset, xvii., 1901, (3) Devon, xviii., 1903, (4) Yorkshire, xviii., 1904. (J. A. H.)
Retrieved from "https://en.wikisource.org/w/index.php?title=1911_Encyclopædia_Britannica/Chalk&oldid=6247022"
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Mean, Median, Mode, Range Calculator ✔️ ConvertBinary.com
Calculate Mean, Median, Mode, and Range!
Use this statistics calculator to find the Mean, Median, Mode and Range measurements for a set of values.
How to use the Mean, Median, Mode and Range Calculator?
Enter the Data Set
Write or paste a set of at least two numbers separated by commas, spaces, tabs, or newlines into the first field.
Press the Calculate button below the data set field.
Copy or Save the result
🔢 Data set length: Unlimited
What is the Mean? Mean definition in statistics
The mean is the arithmetic average of a given set of numbers. It is a measure of central tendency.
The mean is tipically denoted as x̄, pronounced “x bar.
The mean is simply the sum of all the values, divided by the total number of values.
Follow these steps to calculate the mean of a set of values:
Add up all data values to get the sum.
Count the number of values in your data set.
Let’s find the mean of 3, 7, 11, 17:
Now let’s divide 38 by the number of values, which in our case is 4.
You can use the mean calculator at the top of this page to easily find the mean of a set of values.
\text{mean}=\overline{x}=\frac{\sum _{i=1}^{n}{x}_{i}}{n}
Median, or Middle Value
What is the Median? Median definition in statistics
The median is the middle of a sorted set of number. It divides a data set into two halves.
The median is a measure of central tendency, and it represents the point in relation to which half of the values are lower and half are higher. In other words, it is the central point of an ordered set of values.
How to Calculate the Median?
Sort the data set arranging its values from lowest to highest.
If the data set has an odd number of values, find the value in the middle of the set (the median data point separating the upper half of the data values from the lower half)
If the data set has an even number of values, find the two values in the middle of the set (the median data points separating the upper half of the data set from the lower half) and then calculate their mean, which will be the median.
Let’s find the median of an odd data set: 1, 2, 3, 8, 9:
The median value is 3.
Now let’s find the median of an even data set: 1, 2, 3, 8, 9, 10:
Take the mean of the two values in the middle: ( 3 + 8 ) / 2 = 5.5.
You can use the median calculator at the top of this page to easily find the median of a set of values.
Median Formula (for data sets of Odd size)
If the size of the data set n is odd the median is the value at position p where
p=\frac{n+1}{2}
\stackrel{~}{x}={x}_{p}
Median Formula (for data sets of Even size)
p=\frac{n}{2}
\stackrel{~}{x}=\frac{{x}_{p}+{x}_{p+1}}{2}
Mode, or Most Frequent Value
What is the Mode? Mode definition in statistics
The mode is the value that appears the most often in a given data set.
The mode represents the value that is most likely to be observed, i.e. the typical values.
A data set may have:
No mode (when there are no repeating values, or when all values occur the same number of times), or
One mode (when one single value occurs most of the times), or
More than one mode (when two or more single values occur most of the times).
How to Calculate the Mode
Count how many times each individual value appears in the data set.
The value(s) that occur most frequently is (are) the mode(s).
The mode for the data set 1, 2, 3, 3, 4, 5 is 3.
The modes for the data set 1, 2, 3, 3, 4, 4, 5 are 3 and 4.
Both the data sets 1, 1, 1 and 1, 2, 3, 4, 5 have no mode.
You can use the mode calculator at the top of this page to easily find the mode(s) of a set of values.
\frac{\left({f}_{m}-{f}_{1}\right)}{\left({f}_{m}-{f}_{1}\right)+\left({f}_{m}-{f}_{2}\right)}
L is the lower limit of the modal class.
h is the size of the class interval.
fm is the frequency of the modal class.
f1 is the frequency of the class preceding the modal class.
f2 is the frequency of the class succeeding the modal class
What is the Range? Range definition in statistics
The range is simply the difference between the largest and the smallest number within a given set of data.
The range of a data set is a measure of dispersion of the data set itself, and it represents by how much the values in the data set are likely to differ from their mean.
Find the highest value in the data set.
Find the lowest value in the data set.
Let’s find the range for 1, 2, 3, 4, 5:
The highest value is 5, and the lowest is 1.
In this case the range is 5 – 1 = 4.
You can use the range calculator at the top of this page to easily find the range of a set of values.
Range = maximum(xi) – minimum(xi)
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Heaviside step function - MATLAB heaviside - MathWorks Benelux
Evaluate Heaviside Function for Symbolic and Numeric Arguments
Plot Heaviside Function
Evaluate Heaviside Function for Symbolic Matrix
Differentiate and Integrate Expressions Involving Heaviside Function
Find Fourier and Laplace Transforms of Heaviside Function
H = heaviside(x)
H = heaviside(x) evaluates the Heaviside step function (also known as the unit step function) at x. The Heaviside function is a discontinuous function that returns 0 for x < 0, 1/2 for x = 0, and 1 for x > 0.
The heaviside function returns 0, 1/2, or 1 depending on the argument value. If the argument is a floating-point number (not a symbolic object), then heaviside returns floating-point results.
Evaluate the Heaviside step function for a symbolic input sym(-3). The function heaviside(x) returns 0 for x < 0.
H = heaviside(sym(-3))
0
Evaluate the Heaviside step function for a symbolic input sym(3). The function heaviside(x) returns 1 for x > 0.
1
Evaluate the Heaviside step function for a symbolic input sym(0). The function heaviside(x) returns 1/2 for x = 0.
\frac{1}{2}
For a numeric input x = 0, the function heaviside(x) returns floating-point results.
H = heaviside(0)
heaviside takes into account assumptions on variables.
Create a symbolic variable x and assume that it is less than 0.
Evaluate the Heaviside step function for the symbolic input x.
0
Plot the Heaviside step function for x and x - 1.
fplot(heaviside(x), [-2, 2])
fplot(heaviside(x - 1), [-2, 2])
Evaluate the Heaviside function for a symbolic matrix. When the input argument is a matrix, heaviside computes the Heaviside function for each element.
H = heaviside(sym([-1 0; 1/2 x]))
\left(\begin{array}{cc}0& \frac{1}{2}\\ 1& \mathrm{heaviside}\left(x\right)\end{array}\right)
Compute derivatives and integrals of expressions involving the Heaviside function.
Find the first derivative of the Heaviside function. The first derivative of the Heaviside function is the Dirac delta function.
diff_H = diff(heaviside(x),x)
diff_H =
\delta \text{dirac}\left(x\right)
{\int }_{-\infty }^{\infty }{\mathit{e}}^{-\mathit{x}}\mathit{H}\left(\mathit{x}\right)\text{\hspace{0.17em}}\mathit{dx}
int_H = int(exp(-x)*heaviside(x),x,-Inf,Inf)
int_H =
1
Find the fourier transform of the Heaviside function.
F = fourier(heaviside(x))
\pi \delta \text{dirac}\left(w\right)-\frac{\mathrm{i}}{w}
Find the laplace transform of the Heaviside function.
L = laplace(heaviside(x))
\frac{1}{s}
The default value of the Heaviside function at the origin is 1/2.
\frac{1}{2}
Other common values for the Heaviside function at the origin are 0 and 1. To change the value of heaviside at the origin, use sympref to set the value of the 'HeavisideAtOrigin' preference. Store the previous parameter value returned by sympref, so that you can restore it later.
oldparam = sympref('HeavisideAtOrigin',1);
Check the new value of heaviside at 0.
1
The preferences set by sympref persist throughout your current and future MATLAB® sessions. To restore the previous value of heaviside at the origin, use the value stored in oldparam.
sympref('HeavisideAtOrigin',oldparam);
Alternatively, you can restore the default value of 'HeavisideAtOrigin' by using the 'default' setting.
Input, specified as a number, symbolic number, variable, expression, function, vector, or matrix.
dirac | laplace | sympref
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In this post, we have discussed how to implement Quick Sort algorithm parallely using 5 different approaches including HyperQuickSort, Parallel quicksort by regular sampling and many more.
Sequential Quick Sort Algorithm
Approach 1: Naive Parallel Quick Sort
Approach 2: Optimized Parallel Quick Sort
Approach 3: using both sequential and parallel approaches
Approach 4: Hyperquicksort
Approach 5: Parallel quicksort by regular sampling
Prerequisite: Quick Sort, OpenMP, CPU vs GPU
Let us get started with Parallel Quick Sort.
Sorting is a very important building block in most useful algorithms. We need to sort large amounts of data so we can process it efficiently.
The normal way to implement quick sort on serialized processors is whereby steps are executed sequentially until the program terminates when the task is completed. One process is started in the CPU which executes the code line by line.
Parallel programming is whereby a program is broken down into concurrent programs which are executed concurrently on multiple threads on a processor. Here coordination is required.
Less time taken to complete tasks.
Lower work loads per processor.
You can learn more on the link at the end of this post.
Generally, there are cases whereby performance is prioritized over costs and in these cases, parallel processing is implemented. We can see cases of it being used in cryptocurrency mining and video rendering.
While it is straightforward to implement most algorithms using serial processors, a need may arise whereby we need to implement these algorithms parallelly. Some of the commonly used algorithms especially those recursive in nature(quick sort) do not sit well with GPUs(parallel processors).
In serial CPUs, there is a stack to store recursive calls, while in GPUs there is no "stack" of stored values but an emulation of a large contiguous memory whereby the pointer to the top of the "stack" is tracked.
Find a random pivot p.
Partition the list in accordance with this pivot, elements less than pivot to the left of pivot, elements greater than pivot to the right of pivot and elements equal to pivot in the middle. <p =p >p.
That is initialize i to first element in list and j to last element.
Increment i until list[i] > pivot
Decrement j until list[j] < pivot
Repeat the above steps until i > j.
Replace pivot element with list[j]
Recurse on each partition.
When the list size is 1, it terminates. This acts as the base case. At this point the partitions are in sorted order so it merges them forming a complete sorted list.
The above algorithm runs on one process executing each step after another. One step has to finish before the next starts.
Sequential quick sort analysis:
The time complexity is O(nlogn) in the average case.
The space complexity is O(logn).
In this naive approach the algorithm starts a process at each step to concurrently process the partitions.
Start a process to find pivot and partition the list into two, p < =p > p.
At each step, start p processes proportional to n partitions.
Each process finds the pivot element and divides the list based on the selected pivot.
Finally processes values are merged, a sorted list is returned.
\theta \left(n\right)
to choose pivot and rearrange list. There are n processes in each step.
Total time complexity is
\theta \left({n}^{2}\right)
In this approach we change a small detail in the number of processes used at each step. Instead of doubling the number of processes at each step, this approach uses n number of processes throughout the whole algorithm to find pivot element and rearrange the list. All these processes run concurrently at each step sorting the lists.
Start n processes which will partition the list and sort it using selected pivot element.
n processes will work on all partitions from the start of the algorithm till the list is sorted.
Each processes finds a pivot and partitions the list based on selected pivot.
Finally the list is merged forming a sorted list.
Here is the algorithm pictorially,
class ParallelQuickSort{
//keep count of threads
//partitioning procedure
int partition(int arr[], int l, int r){
int key = arr[l];
while(i < r && key >= arr[i])
arr[l] = arr[j];
void quickSort(int arr[], int l, int r){
cout << "pivot " << p << " found by thread no. " << k << endl;
quickSort(arr, l, p-1);
quickSort(arr, p+1, r);
void printArr(int arr[], int n){
//run the whole procedure
int arr[] = {9, 6, 3, 7, 2, 12, 5, 1};
ParallelQuickSort pqs;
pqs.run();
#include<omp.h> is a header file for openmp so we can be able to use its functions.
We use openmp(open multiprocessing) , an open source library for multi-threading which will enable use to implement the algorithm concurrently.
You can learn more about it on the link at the end of this post.
#pragma omp parallel sections defines a parallel region containing the code that we will execute using multiple threads in parallel. This code will be divided among all threads.
Variable k stores the thread number which we print out just for understanding what is happening under the hood.
Compile and run code using g++ -fopenmp fileName.cpp -o outputFile && ./outputFile.
Parallel quick sort analysis
At each step n processes process log(n) lists in constant time O(1).
The parallel execution time is O(logn) and there are n processes. Total time complxity is
\theta \left(nlogn\right)
This complexity did not change from the sequential one but we have a achieved an algorithm that can run on parallel processors, meaning it will execute much faster at a larger scale.
Space complexity is O(logn).
In this approach, the initial list is divided into n smaller sublists that are explicitly dispatched to p remote processors for parallel execution and once the execution is finished at a distributed processor, the sorted sublist is sent back to the central processing node which merges the results from the processes giving a fully sorted list.
Divide a list of size n to create a number of sublists compatible with the number of available processors p.
Create p threads according to the number of available processors.
Assign the sublist to each of the p threads so that each has n/p consecutive elements from the original list.
Randomly select a pivot element and broadcast it to all partner processes.
Each process will partition its elements and divide them into two groups according to the selected pivot. group1 <= pivot <= group2. This happens parallelly across all processes concurrently.
Each process in upper half of the process list sends its "low list" to a partner process in the lower half of the process list and it receives the "high list" in return.
The lower half will have values less than pivot and upper half elements will have a values greater than pivot.
After logP recursions, each process has an unsorted sublist disjoint from values of other processes.
The largest value of process i will be less the the smallest value of process i+1.
At this point the sublist is small enough, each process sorts its values sequentially and the main process combines the sorted results for each process.
Here is the procedure pictorially,
In above approach, p number of processes work on n number of elements of a list, each p process works on n/p sub-list elements for(logn) steps.
The total time complexity is O(N/P * logN).
Pivot; This algorithm does a poor job of load balancing, that is, we need to choose a suitable median value as a pivot element for the algorithm so as to divide the list into at-least equal partitions and maintain balance.
Finding a median value is an expensive operation on a parallel processor.
The solution for the above problem is to find a median value that is close to the true median.
Combining Processes; We can concatenate each block in process order, that is find where each block starts and ends and join its end to the start of the next process.
Assuming a machine has 64 threads, a list of size n can be distributed among all the threads n/64 and processing can happen in parallel until the last step where the sublist is sorted sequentially and combined, as n increases the size of list each thread handles increases.
Each threads processes a list in constant time O(1) and there are logn steps.
Assuming an unlimited threads can run in parallel the time and space complexity is O(logn).
This approach is an improvement of the previous approach. Previously we had a problem of load balancing.
This process improves the chances of finding a true median by sorting the sublists sequentially using one pivot that is broadcasted to all processes at the beginning of the algorithm.
A list of size n is divided among n processes. Assume list of size 16 and 4 processes, each process will handle 4 elements.
A process among the four responsible for finding the pivot element, finds a pivot and broadcasts it to all processes which sort their sublists sequentially using the broadcasted pivot element. This step will improve chances of finding pivots close to the true median.
We repeat steps 4-6 from the previous approach,
Pivot selection and broadcasting to partner processes.
Sublist partitioning of low and high values.
Swapping of values between partner processes.
The remaining top half from one partner process and the received top half from the other partner process are merged into local sublist for each process.
Recurse the upper half and lower half of each subprocess to achieve a sorted list.
Finally merge the processes in order to get a fully sorted list.
There is a communication overhead in that values are passed between partner processes.
Load imbalance may still occur but the algorithm is better as compared to the previous approach which is much worse at load balancing.
There are logn steps and n processes, the total time complexity is
\theta \left(nlogn\right)
In this approach the algorithm sorts the list sequentially at the beginning and then selects a range of samples which will be used for further partitioning and swapping of elements in subsequent processes.
Original list is divided among n processes.
Each process sorts its sublist using sequential quicksort.
Each process selects regular samples from its sorted sublist.
A single process gathers the samples, sorts them and broadcasts selected pivots to other processes.
All processes use selected pivots to divide their sublists into partitions according to selected pivots concurrently.
Processes communicate to swap sorted values with other partner processes.
The sorted processes values are merged to return a fully sorted list.
Advantages of PSRS
Better load balancing is achieved but not perfect.
Repeated swappings of same values are avoided, no overhead.
The number of processes don't have to be a power of 2, during the algorithm some processes can be freed up depending on the order of elements and pivots selected.
Initial quicksort is
\theta \left({\frac{n}{n}}^{ }\mathrm{log} \frac{n}{p}\right)
Sorting samples is
\theta \left({p}^{2 }\mathrm{log} p\right)
Merging sub-arrays is
\theta \left({\frac{n}{p}}^{ }\mathrm{log} p\right)
Total time complexity is O(nlogn).
Space complexity is O(logn)
We discussed 3 main parallel quicksort algorithms.
Parallel quicksort, poor.
Hyperquicksort, better.
PRSR algorithm, best.
In some cases we may not be able to store the data we are processing to memory in this case we need an efficient external sorting algorithm, which is this algorithm, can you parallelize it?
With this article at OpenGenus, you must have the complete idea of Parallel Quick Sort.
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Board Paper Solutions for CBSE Class 10 SCIENCE Board Paper 2021 Delhi Set 4
(iii) Section – A consists of 24 questions. Attempt any 20 questions from Q.No. 1 to 24.
(iv) Section – B also consists 24 questions. Attempt any 20 questions from Q.No. 25 to 48.
(v) Section – C consists of three Case Study containing 12 questions and 4 questions in each case. Attempt any 10 from Q.No. 49 to 60.
(vi) There is only one correct option for every Multiple Choice Question (MCQ).
A student took Sodium Sulphate solution in a test tube and added Barium Chloride solution to it. He observed that an insoluble substance has formed. The colour and molecular formula of the insoluble substance is:
Which of the following oxide(s) is/are soluble in water to form alkalies?
Study the diagram given below and identify the gas formed in the reaction.
(d) Hydrogen which while burning produces a popping sound. VIEW SOLUTION
Sodium reacts with water to form sodium hydroxide and hydrogen gas.
The balanced equation which represents the above reaction is;
(b) 2Na(s) + 2H2O(l) → 2NaOH(aq) + H2(g)
(d) 2Na(s) + H2O(l) → 2NaOH(aq) + 2H2(g) VIEW SOLUTION
Which of the options in the given table are correct?
C6H12O6(aq) + 6O2(aq) → 6CO2(aq) + 6H2O(l)
(d) neutralisation reaction VIEW SOLUTION
Which of the following statements about the reaction given below are correct?
Select from the following the statement which is true for bases.
(d) Bases turn pink when a drop of phenolphthalein is added to them. VIEW SOLUTION
Study the following table and choose the correct option:
Parent Base
Nature of Salt
Solidum Chloride
It is important to balance the chemical equations to satisfy the law of conservation of mass. Which of the following statements of the law is incorrect?
(d) Mass can neither be created nor can it be destroyed in a chemical reaction. VIEW SOLUTION
Consider the following statements in connection with the functions of the blood vessels marked A and B in the diagram of a human heart as shown.
(i) Blood vessel A - It carries carbon dioxide-rich blood to the lungs.
(ii) Blood vessel B - It carries oxygen-rich blood from the lungs.
(iii) Blood vessel B - Left atrium relaxes as it receives blood from this blood vessel.
(iv) Blood vessel A - Right atrium has a thick muscular wall as it has to pump blood to this blood vessel.
In living organisms during respiration which of the following products are not formed if oxygen is not available?
(d) Carbon dioxide + Lactic Acid VIEW SOLUTION
The correct statements with reference to single-celled organisms are
Which one among the following is not removed as a waste product from the body of a plant?
(d) Excess Water VIEW SOLUTION
Which of the following statements are correct in reference to the role of A (shown in the given diagram) during a breathing cycle in human beings?
(d) (i), (ii) and (iv) VIEW SOLUTION
Which one of the following conditions is true for the state of stomata of a green leaf shown in the given diagram?
(d) Large amount of sugar collects in the guard cells. VIEW SOLUTION
In which of the following is a concave mirror used?
(d) In viewing full size image of distant tall buildings. VIEW SOLUTION
A student wants to obtain magnified image of an object AB as on a screen. Which one of the following arrangements shows the correct position of AB for him/her to be successful?
(c) (d) VIEW SOLUTION
The following diagram shows the use of an optical device to perform an experiment of light. As per the arrangement shown, the optical device is likely to be a;
(d) Convex lens VIEW SOLUTION
A ray of light starting from air passes through medium A of refractive index 1.50, enters medium B of refractive index 1.33 and finally enters medium C of refractive index 2.42. If this ray emerges out in air from C, then for which of the following pairs of media the bending of light is least?
(d) C-air VIEW SOLUTION
Which of the following statements is not true for scattering of light?
(c) Scattering of light takes place as various colours of white light travel with different speed in air.
(d) The fine particles in the atmospheric air scatter the blue light more strongly than red. So the scattered blue light enters our eyes. VIEW SOLUTION
For the diagram shown, according to the new cartesian sign convention the magnification of the image formed will have the following specifications:
(a) Sign - Positive, Value - Less than 1
(b) Sign - Positive, Value - More than 1
(c) Sign - Negative, Value - Less than 1
(d) Sign - Negative, Value - More than 1 VIEW SOLUTION
\angle
\angle
\angle
\angle
\angle
\angle
\angle
\angle
In the diagram given below, X and Y are the end colours of the spectrum of white light. The colour of 'Y' represents the
(d) Colour of sun at the time of noon. VIEW SOLUTION
Which one of the following reactions is categorised as thermal decomposition reaction?
(a) 2H2O(l) → 2H2(g) + O2(g)
(b) 2AgBr(s) → 2Ag(s) + Br2(l)
(d) CaCO3(s) → CaO(s) + CO2(g) VIEW SOLUTION
Consider the pH value of the following acidic samples:
Study the experimental set up shown in given figure and choose the correct option from the following:
P Q Change Observed in calcium hydroxide solution
(a) K2CO3 Cl2 gas No change
(b) KHCO3 CO2 gas No change
(c) KHCO3 H2 gas Turns milky
(d) K2CO3 CO2 gas Turns milky
Which one of the following structures correctly depicts the compound CaCl2?
The pair(s) which will show displacement reaction is/are
{\mathrm{AgNO}}_{3}
solution and copper metal
{\mathrm{Al}}_{2}{\left({\mathrm{SO}}_{4}\right)}_{3}
solution and magnesium metal
{\mathrm{ZnSO}}_{4}
solution and iron metal
(d) (i) and (ii) VIEW SOLUTION
Which of the following salts do not have the water of crystallisation?
Assertion (A) : Sodium hydrogen carbonate is used as an ingredient in antacids.
Reason (R) :
{\mathrm{NaHCO}}_{3}
is a mild non-corrosive basic salt.
Assertion (A) : Burning of Natural gas is an endothermic process.
Reason (R) : Methane gas combines with oxygen to produce carbon dioxide and water.
Assertion (A) : Nitrogen is an essential element for plant growth and is taken up by plants in the form of inorganic nitrates or nitrites.
Assertion (A) : Sun appears reddish at the time of Sunrise and Sunset.
Assertion (A) : Hydrochloric acid helps in the digestion of food in the stomach.
A student was asked to write a stepwise procedure to demonstrate that carbon dioxide is necessary for photosynthesis. He wrote the following steps, the wrongly worded step is -
(b) Bottom of the bell jars is sealed to make them air tight.
(d) A leaf from both the plants is taken to test the presence of starch. VIEW SOLUTION
Respiratory structures of two different animals-a fish and a human being are as shown.
(a) Both are placed internally in the body of animal.
Observe the diagram of an activity given below. What does it help to conclude, when the person exhales into the test-tube?
(d) Fermentation occurs in the presence of carbon dioxide. VIEW SOLUTION
If a lens can converge the sun rays at a point 20 cm away from its optical centre, the power of this lens is -
(a) + 2 D
(b) – 2 D
(c) + 5 D
(d) – 5 D VIEW SOLUTION
The radius of curvature of a converging mirror is 30 cm. At what distance from the mirror should an object be placed so as to obtain a virtual image?
(d) Between 0 cm and 15 cm VIEW SOLUTION
The length of small intestine in a deer is more as compared to the length of small intestine of a tiger. The reason for this is -
(d) Presence or absence of digestive enzymes. VIEW SOLUTION
Identify the two components of Phloem tissue that help in transportation of food in plants.
(d) Phloem fibres and sieve tubes VIEW SOLUTION
A converging lens forms a three times magnified image of an object, which can be take
on a screen. If the focal length of the lens is 30 cm, then the distance of the object from the lens is
(a) –55 cm (b) –50 cm (c) –45 cm (d) –40 cm VIEW SOLUTION
Which of the following statements is not true in reference to the diagram shown above ?
(d) Image formed is inverted. VIEW SOLUTION
In the diagram shown above n1, n2 and n3 are refractive indices of the media 1, 2 and 3 respectively. Which one of the following is true in this case ?
(a) n1 = n2 (b) n1 > n2 (c) n2 > n3 (d) n3 > n1 VIEW SOLUTION
The refractive index of medium A is 1.5 and that of medium B is 1.33. If the speed of light in air is 3 × 108 m/s, what is the speed of light in medium A and B respectively ?
(d) 2 × 108 m/s and 2.25 × 108 m/s VIEW SOLUTION
An object of height 4 cm is kept at a distance of 30 cm from the pole of a diverging mirror. If the focal length of the mirror is 10 cm, the height of the image formed is
(a) + 3.0 cm (b) + 2.5 cm (c) + 1.0 cm (d) + 0.75 cm VIEW SOLUTION
50.0 ml of tap water was taken in a beaker. Hydrochloric acid was added drop by drop to water. The temperature and pH of the solution was noted. The following graph was obtained. Choose the correct statements related to this activity.
(ii) The pH of the solution increases rapidly on addition of acid.
(c) (iii) and (iv) (d) (ii) and (iv) VIEW SOLUTION
Case-I:
Out of the given metals, the one which needs to be stored used Kerosene is
(d) Q VIEW SOLUTION
Out of the given metals, the metal Q is
(d) Magnesium VIEW SOLUTION
Metal which forms amphoteric oxides is
The increasing order of the reactivity of the four metals is;
(d) P < R < Q < S VIEW SOLUTION
Case-II :
The figure shown below represents a common type of dialysis called as Haemodialysis. It removes waste products from the blood. Such as excess salts, and urea which are insufficiently removed by the kidney in patients with kidney failure. During the procedure, the patient's blood is cleaned by filtration through a series of semi-permeable membranes before being returned to the blood of the patient. On the basis of this, answer the following questions :
The haemodialyzer has semi-permeable lining of tubes which help to :
(d) To pump purified blood back into the body of the patient. VIEW SOLUTION
Which one of the following is not a function of Artificial Kidney ?
(d) To filter and purify the blood. VIEW SOLUTION
The 'used dialysing' solution is rich in;
(d) Proteins VIEW SOLUTION
Which part of the nephron in human kidney, serves the function of reabsorption of certain substances?
(d) Collecting duct VIEW SOLUTION
Case-III :
A compound microscope is an instrument which consists of two lenses L1 and L2. The lens L1 called objective, forms a real, inverted and magnified image of the given object. This serves as the object for the second lens L2; the eye piece. The eye piece functions like a simple microscope or magnifier. It produces the final image, which is inverted with respect to the original object, enlarged and virtual.
What types of lenses must be L1 and L2?
(d) L1 – convex and L2 – concave VIEW SOLUTION
What is the value and sign of magnification (according to the new Cartesian sign convention) of the image formed by L1?
(d) Value = More than 1 and Sign = Negative VIEW SOLUTION
What is the value and sign of (according to new Cartesian sign convention) magnification of the image formed by L2?
If power of the eyepiece (L2) is 5 diopters and it forms an image at a distance of 80 cm from its optical centre, at what distance should the object be?
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Home : Support : Online Help : Mathematics : Number Theory : Radical
calculate the radical of an integer
Radical( n )
The Radical( n ) function computes the radical of the nonzero integer n. The radical of a nonzero integer is the product of its prime divisors. The radical of any nonzero integer is, therefore, a square-free integer.
The radical of
0
is not defined; an exception is raised if 0 is passed.
A positive integer is equal to its radical precisely when it is square-free.
\mathrm{with}\left(\mathrm{NumberTheory}\right):
\mathrm{Radical}\left(12\right)
\textcolor[rgb]{0,0,1}{6}
\mathrm{Radical}\left(6\right)
\textcolor[rgb]{0,0,1}{6}
\mathrm{Radical}\left(-12\right)
\textcolor[rgb]{0,0,1}{6}
The NumberTheory[Radical] command was introduced in Maple 2017.
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EUDML | Immersions with a parallel normal field. EuDML | Immersions with a parallel normal field.
Immersions with a parallel normal field.
Carter, Sheila; Kaya, Yusuf
Carter, Sheila, and Kaya, Yusuf. "Immersions with a parallel normal field.." Beiträge zur Algebra und Geometrie 41.2 (2000): 359-370. <http://eudml.org/doc/121615>.
author = {Carter, Sheila, Kaya, Yusuf},
keywords = {immersion; focal point; parallel normal field; normal bundle},
title = {Immersions with a parallel normal field.},
AU - Carter, Sheila
TI - Immersions with a parallel normal field.
KW - immersion; focal point; parallel normal field; normal bundle
Yusuf Kaya, On the Components of the Push-out Space with Certain Indices
immersion, focal point, parallel normal field, normal bundle
Higher-dimensional and -codimensional surfaces in Euclidean
Articles by Kaya
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Cretaceous age for Ir‐rich Deccan intertrappean deposits: palaeontological evidence from Anjar, western India
S. BAJPAI; G. V. R. PRASAD
Journal of the Geological Society March 01, 2000, Vol.157, 257-260. doi:https://doi.org/10.1144/jgs.157.2.257
Geochemical transitions in the ancestral Iceland plume: evidence from the Isle of Mull Tertiary volcano, Scotland
L. M. CHAMBERS; J. G. FITTON
New U–Pb monazite and zircon data from the Sudetes Mountains in SW Poland: evidence for a single‐cycle Variscan orogeny
H. TIMMERMANN; R. R. PARRISH; S. R. NOBLE; R. KRYZA
Kinetics of precipitation of gypsum and implications for pressure‐solution creep
SIESE DE MEER; CHRISTOPHER J. SPIERS; COLIN J. PEACH
Macrofabric fingerprints of Late Devonian–Early Carboniferous subduction in the Polish Variscides, the Kaczawa complex, Sudetes
ALAN S. COLLINS; RYSZARD KRYZA; JAN ZALASIEWICZ
H. M. GRIFFITHS; R. A. CLARK; K. M. THORP; S. SPENCER
Response of Plio‐Pleistocene alluvial systems to tectonically induced base‐level changes, Vera Basin, SE Spain
M. STOKES; A. E. MATHER
P. TURNER; R. SHELTON; A. RUFFELL; J. PUGH
FABIO SPERANZA; MASSIMO MATTEI; LEONARDO SAGNOTTI; FABIO GRASSO
Isotopic evidence for temperature variation during the early Cretaceous (late Ryazanian–mid‐Hauterivian)
GREGORY D. PRICE; ALASTAIR H. RUFFELL; CHARLES E. JONES; ROBERT M. KALIN; JÖRG MUTTERLOSE
Thermochronology of northern Murihiku Terrane, New Zealand, derived from apatite FT analysis
PETER J. J. KAMP; IVAN J. LIDDELL
Evidence of hydrocarbon and metalliferous fluid migration in the Palaeoproterozoic Earaheedy Basin of Western Australia
B. RASMUSSEN; B. KRAPEZ
An oxygen and carbon isotopic study of multiple episodes of fluid flow in the Dalradian and Highland Border Complex, Stonehaven, Scotland
R. L. MASTERS; J. J. AGUE; D. M. RYE
P. T. S. ROSE; A. L. HARRIS
A. R. PRAVE; L. G. KESSLER, II; M. MALO; W. V. BLOECHL; J. RIVA
KAREL SCHULMANN; RODNEY GAYER
Continental rift to back‐arc basin: Jurassic–Cretaceous stratigraphical and structural evolution of the Larsen Basin, Antarctic Peninsula
Sequence stratigraphy of a tidally dominated carbonate–siliciclastic ramp; the Tithonian–Early Berriasian of the Southern Neuquén Basin, Argentina
LUIS A. SPALLETTI; JUAN R. FRANZESE; SERGIO D. MATHEOS; ERNESTO SCHWARZ
Late Cretaceous sea‐level changes in Tunisia: a multi‐disciplinary approach
Liangquan Li; Gerta Keller; Thierry Adatte; Wolfgang Stinnesbeck
Cambro‐Ordovician stratigraphy of Bjørnøya and North Greenland: constraints on tectonic models for the Arctic Caledonides and the Tertiary opening of the Greenland Sea
Subsidence and erosion in the Pennine Carboniferous Basin, England: lithological and thermal constraints on maturity modelling
MICHAEL J. PEARSON; MARIE A. RUSSELL
Structural evolution of the Li Basin, northern Thailand
C. K. MORLEY; N. SANGKUMARN; T. B. HOON; C. CHONGLAKMANI; J. LAMBIASE
Silurian K‐bentonites of the Dnestr Basin, Podolia, Ukraine
WARREN D. HUFF; STIG M. BERGSTRÖM; DENNIS R. KOLATA
Discussion on late Quaternary evolution of the upper reaches of the Solent River, Southern England, based upon marine geophysical evidence Journal, Vol. 156, 1999, pp. 73–87
D. A. G. NOWELL
Discussion on the influence of country rock structural architecture during pluton emplacement: the Loch Loyal syenites, ScotlandJournal, Vol. 156, 1999, 163–175
P. J. CAREY; I. M. PLATTEN
Discussion on Lazarus taxa and fossil abundance at times of biotic crisis Journal, Vol. 156, 1999, pp. 453–456
RICHARD J. TWITCHETT; P. B. WIGNALL; M. J. BENTON
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Implement three-phase two-winding transformer with configurable winding connections and core geometry - Simulink - MathWorks Nordic
Three-Phase Transformer Inductance Matrix Type (Two Windings)
Implement three-phase two-winding transformer with configurable winding connections and core geometry
The Three-Phase Transformer Inductance Matrix Type (Two Windings) block is a three-phase transformer with a three-limb core and two windings per phase. Unlike the Three-Phase Transformer (Two Windings) block, which is modeled by three separate single-phase transformers, this block takes into account the couplings between windings of different phases. The transformer core and windings are shown in the following illustration.
1 and 4 on phase A
2 and 5 on phase B
3 and 6 on phase C
This core geometry implies that phase winding 1 is coupled to all other phase windings (2 to 6), whereas in Three-Phase Transformer (Two Windings) block (a three-phase transformer using three independent cores) winding 1 is coupled only with winding 4.
The phase winding numbers 1 and 2 should not be confused with the numbers used to identify the three-phase windings of the transformer. Three-phase winding 1 consists of phase windings 1,2,3, and three-phase winding 2 consists of phase windings 4,5,6.
The Three-Phase Transformer Inductance Matrix Type (Two Windings) block implements the following matrix relationship:
\left[\begin{array}{c}{V}_{1}\\ {V}_{2}\\ ⋮\\ {V}_{6}\end{array}\right]=\left[\begin{array}{cccc}{R}_{1}& 0& \dots & 0\\ 0& {R}_{2}& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & {R}_{6}\end{array}\right]\cdot \left[\begin{array}{c}{I}_{1}\\ {I}_{2}\\ ⋮\\ {I}_{6}\end{array}\right]+\left[\begin{array}{cccc}{L}_{11}& {L}_{12}& \dots & {L}_{16}\\ {L}_{21}& {L}_{22}& \dots & {L}_{26}\\ ⋮& ⋮& \ddots & ⋮\\ {L}_{61}& {L}_{62}& \dots & {L}_{66}\end{array}\right]\cdot \frac{d}{dt}\left[\begin{array}{c}{I}_{1}\\ {I}_{2}\\ ⋮\\ {I}_{6}\end{array}\right].
When the parameter Core type is set to Three single-phase cores, the model uses two independent circuits with (3x3) R and L matrices. In this condition, the positive-sequence and zero-sequence parameters are identical and you only specify positive-sequence values.
The self and mutual terms of the (6x6) L matrix are obtained from excitation currents (one three-phase winding is excited and the other three-phase winding is left open) and from positive- and zero-sequence short-circuit reactances X112 and X012 measured with three-phase winding 1 excited and three-phase winding 2 short-circuited.
Assuming the following positive-sequence parameters:
Q11= Three-phase reactive power absorbed by winding 1 at no load when winding 1 is excited by a positive-sequence voltage Vnom1 with winding 2 open
X112= Positive-sequence short-circuit reactance seen from winding 1
when winding 2 is short-circuited
Vnom1, Vnom2= Nominal line-line voltages of windings 1 and 2
\begin{array}{c}{X}_{1}\left(1,1\right)=\frac{{V}_{{\text{nom}}_{1}}^{2}}{Q{1}_{1}}\\ {X}_{1}\left(2,2\right)=\frac{{V}_{{\text{nom}}_{2}}^{2}}{Q{1}_{2}}\\ {X}_{1}\left(1,2\right)={X}_{1}\left(2,1\right)=\sqrt{{X}_{1}\left(2,2\right)\cdot \left({X}_{1}\left(1,1\right)-X{1}_{12}}\right).\end{array}
The zero-sequence self-reactances X0(1,1), X0(2,2), and mutual reactance X0(1,2) = X0(2,1) are also computed using similar equations.
\left[\begin{array}{cc}{X}_{1}\left(1,1\right)& {X}_{1}\left(1,2\right)\\ {X}_{1}\left(2,1\right)& {X}_{1}\left(2,2\right)\end{array}\right]\text{ }\left[\begin{array}{cc}{X}_{0}\left(1,1\right)& {X}_{0}\left(1,2\right)\\ {X}_{0}\left(2,1\right)& {X}_{0}\left(2,2\right)\end{array}\right]
to a (6x6) matrix, is performed by replacing each of the four [X1 X0] pairs by a (3x3) submatrix of the form:
\left[\begin{array}{ccc}{X}_{s}& {X}_{m}& {X}_{m}\\ {X}_{m}& {X}_{s}& {X}_{m}\\ {X}_{m}& {X}_{m}& {X}_{s}\end{array}\right]
In order to model the core losses (active power P1 and P0 in positive- and zero-sequences), additional shunt resistances are also connected to terminals of one of the three-phase windings. If winding 1 is selected, the resistances are computed as:
R{1}_{1}=\frac{{V}_{{\text{nom}}_{1}}^{2}}{P{1}_{1}}\text{ }R{0}_{1}=\frac{{V}_{{\text{nom}}_{1}}^{2}}{P{0}_{1}}.
The block takes into account the connection type you select, and the icon of the block is automatically updated. An input port labeled N is added to the block if you select the Y connection with accessible neutral for winding 1. If you ask for an accessible neutral on winding 2, an extra outport port labeled n2 is generated.
The following figure shows a three-limb core with a single three-phase winding. Only phase B is excited and voltage is measured on phase A and phase C. The flux Φ produced by phase B shares equally between phase A and phase C so that Φ/2 is flowing in limb A and in limb C. Therefore, in this particular case, if leakage inductance of winding B would be zero, voltage induced on phases A an C would be -k.VB=-VB/2. In fact, because of the leakage inductance of the three windings, the average value of induced voltage ratio k when windings A, B, and C are successively excited must be slightly lower than 0.5.
\begin{array}{c}{V}_{B}={Z}_{s}{I}_{B}\\ {V}_{A}={Z}_{m}{I}_{B}=-{V}_{B}/2\\ {V}_{C}={Z}_{m}{I}_{B}=-{V}_{B}/2\\ {Z}_{s}=\frac{2{Z}_{1}+{Z}_{0}}{3}\\ {Z}_{m}=\frac{{Z}_{0}-{Z}_{1}}{3}\\ {V}_{A}={V}_{C}=\frac{{Z}_{m}}{{Z}_{s}}{V}_{B}=-\frac{\frac{{Z}_{1}}{{Z}_{0}}-1}{2\frac{{Z}_{1}}{{Z}_{0}}+1}{V}_{B}=-\frac{\frac{{I}_{0}}{{I}_{1}}-1}{2\frac{{I}_{0}}{{I}_{1}}+1}{V}_{B}=-k{V}_{B},\end{array}
\frac{{I}_{0}}{{I}_{1}}=\frac{1+k}{1-2k}.
Obviously k cannot be exactly 0.5 because this would lead to an infinite zero-sequence current. Also, when the three windings are excited with a zero-sequence voltage, the flux path should return through the air and tank surrounding the iron core. The high reluctance of the zero-sequence flux path results in a high zero-sequence current.
Let us assume I1= 0.5%. A reasonable value for I0 could be 100%. Therefore I0/I1=200. According to the equation for I0/I1 given above, one can deduce the value of k. k=(200-1)/(2*200+1)= 199/401= 0.496.
Zero-sequence losses should be also higher than the positive-sequence losses because of the additional eddy current losses in the tank.
The three-phase windings of the transformer can be connected in the following manner:
Select the core geometry: Three single-phase cores or Three-limb or five-limb core (default). If you select the first option, only the positive-sequence parameters are used to compute the inductance matrix. If you select the second option, both the positive- and zero-sequence parameters are used.
The following figure illustrates winding connections for one phase of an autotransformer when the three-phase windings are both connected in Yg.
Phase A: W1=1, W2=4
Phase B: W1=2, W2=5
Phase C: W1=3, W2=6
The same labels apply for three-phase winding 2, except that 1 is replaced by 2 in the labels.
Nominal line-line voltages [V1 V2]
The phase-to-phase nominal voltages of windings 1 and 2 in volts RMS. Default is [2400, 600].
Winding resistances [R1 R2]
The resistances in pu for windings 1 and 2. Default is [0.01, 0.01].
The no-load excitation current in percent of the nominal current when positive-sequence nominal voltage is applied at any three-phase winding terminals (ABC or abc2). Default is 2.
The core losses plus winding losses at no-load, in watts (W), when positive-sequence nominal voltage is applied at any three-phase winding terminals (ABC or abc2). Default is 1000.
Positive-sequence short-circuit reactance
The positive-sequence short-circuit reactances X12 in pu. X12 is the reactance measured from winding 1 when winding 2 is short-circuited. Default is 0.06.
When the Connect windings 1 and 2 in autotransformer parameter is selected, the short-circuit reactances is labeled XHL. H and L indicate respectively the high voltage winding (either winding 1 or winding 2) and the low voltage winding (either winding 1 or winding 2).
The no-load excitation current in percent of the nominal current when zero-sequence nominal voltage is applied at any three-phase winding terminals (ABC or abc2) connected in Yg or Yn. Default is 100.
The core losses plus winding losses at no-load, in watts (W), when zero-sequence nominal voltage is applied at any group of winding terminals (ABC or abc2) connected in Yg or Yn. The Delta winding must be temporarily open to measure these losses. Default is 1500.
Zero-sequence short-circuit reactance
The zero-sequence short-circuit reactance X12 in pu. X12 is the reactance measured from winding 1 when winding 2 is short-circuited. Default is 0.03.
This transformer model does not include saturation. If you need modeling saturation, connect the primary winding of a saturable Three-Phase Transformer (Two Windings) in parallel with the primary winding of your model. Use the same connection (Yg, D1 or D11) and same winding resistance for the two windings connected in parallel. Specify the Y or Yg connection for the secondary winding and leave it open. Specify appropriate voltage, power ratings, and saturation characteristics that you want. The saturation characteristic is obtained when the transformer is excited by a positive-sequence voltage.
For a three-limb core, this saturation model still produces acceptable results, even if zero-sequence flux circulates outside of the core and returns through the air and the transformer tank surrounding the iron core. As the zero-sequence flux circulates in the air, the magnetic circuit is mainly linear and its reluctance is high (high magnetizing currents). These high zero-sequence currents (100% and more of nominal current) required to magnetize the air path are already taken into account in the linear model. Connecting a saturable transformer outside the three-limb linear model with a flux-current characteristic obtained in positive sequence produces currents required for magnetization of the iron core. This model gives acceptable results whether the three-limb transformer has a delta or not.
Linear Transformer, Multimeter, Three-Phase Transformer (Two Windings), Three-Phase Transformer (Three Windings), Three-Phase Transformer Inductance Matrix Type (Three Windings)
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torch.linalg.pinv — PyTorch 1.11.0 documentation
torch.linalg.pinv
torch.linalg.pinv¶
torch.linalg.pinv(A, *, atol=None, rtol=None, hermitian=False, out=None) → Tensor¶
Computes the pseudoinverse (Moore-Penrose inverse) of a matrix.
The pseudoinverse may be defined algebraically but it is more computationally convenient to understand it through the SVD
If hermitian = True , A is assumed to be Hermitian if complex or symmetric if real, but this is not checked internally. Instead, just the lower triangular part of the matrix is used in the computations.
The singular values (or the norm of the eigenvalues when hermitian = True ) that are below
\max(\text{atol}, \sigma_1 \cdot \text{rtol})
threshold are treated as zero and discarded in the computation, where
\sigma_1
is the largest singular value (or eigenvalue).
If rtol is not specified and A is a matrix of dimensions (m, n) , the relative tolerance is set to be
\text{rtol} = \max(m, n) \varepsilon
\varepsilon
is the epsilon value for the dtype of A (see finfo). If rtol is not specified and atol is specified to be larger than zero then rtol is set to zero.
If atol or rtol is a torch.Tensor, its shape must be broadcastable to that of the singular values of A as returned by torch.linalg.svd().
This function uses torch.linalg.svd() if hermitian = False and torch.linalg.eigh() if hermitian = True . For CUDA inputs, this function synchronizes that device with the CPU.
Consider using torch.linalg.lstsq() if possible for multiplying a matrix on the left by the pseudoinverse, as:
torch.linalg.lstsq(A, B).solution == A.pinv() @ B
It is always prefered to use lstsq() when possible, as it is faster and more numerically stable than computing the pseudoinverse explicitly.
This function has NumPy compatible variant linalg.pinv(A, rcond, hermitian=False) . However, use of the positional argument rcond is deprecated in favor of rtol.
This function uses internally torch.linalg.svd() (or torch.linalg.eigh() when hermitian = True ), so its derivative has the same problems as those of these functions. See the warnings in torch.linalg.svd() and torch.linalg.eigh() for more details.
torch.linalg.inv() computes the inverse of a square matrix.
torch.linalg.lstsq() computes A .pinv() @ B with a numerically stable algorithm.
A (Tensor) – tensor of shape (*, m, n) where * is zero or more batch dimensions.
rcond (float, Tensor, optional) – [NumPy Compat]. Alias for rtol. Default: None .
atol (float, Tensor, optional) – the absolute tolerance value. When None it’s considered to be zero. Default: None .
rtol (float, Tensor, optional) – the relative tolerance value. See above for the value it takes when None . Default: None .
hermitian (bool, optional) – indicates whether A is Hermitian if complex or symmetric if real. Default: False .
tensor([[ 0.5495, 0.0979, -1.4092, -0.1128, 0.4132],
[-0.3269, -0.5745, -0.0382, -0.5922, -0.6759]])
>>> torch.linalg.pinv(A)
>>> Apinv = torch.linalg.pinv(A)
>>> torch.dist(Apinv @ A, torch.eye(3))
>>> A = torch.randn(3, 3, dtype=torch.complex64)
>>> A = A + A.T.conj() # creates a Hermitian matrix
>>> Apinv = torch.linalg.pinv(A, hermitian=True)
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Lower and upper matrix bandwidth - MATLAB bandwidth - MathWorks Italia
Find Bandwidth of Triangular Matrix
Find Bandwidth of Sparse Block Matrix
Upper and Lower Bandwidth
Lower and upper matrix bandwidth
B = bandwidth(A,type)
[lower,upper] = bandwidth(A)
B = bandwidth(A,type) returns the bandwidth of matrix A specified by type. Specify type as 'lower' for the lower bandwidth, or 'upper' for the upper bandwidth.
[lower,upper] = bandwidth(A) returns the lower bandwidth, lower, and upper bandwidth, upper, of matrix A.
Create a 6-by-6 lower triangular matrix.
A = tril(magic(6))
30 5 34 12 14 0
Find the lower bandwidth of A by specifying type as 'lower'. The result is 5 since every diagonal below the main diagonal has nonzero elements.
B = bandwidth(A,'lower')
Find the upper bandwidth of A by specifying type as 'upper'. The result is 0 since there are no nonzero elements above the main diagonal.
B = bandwidth(A,'upper')
View a 10-by-10 section of elements from the top left of B.
full(B(1:10,1:10))
B has 4-by-4 blocks of ones centered on the main diagonal.
Find both the lower and upper bandwidths of B by specifying two output arguments.
[lower,upper] = bandwidth(B)
Input matrix, specified as a 2-D numeric matrix. A can be either full or sparse.
type — Bandwidth type
'lower' | 'upper'
Bandwidth type, specified as 'lower' or 'upper'.
Specify 'lower' for the lower bandwidth (below the main diagonal).
Specify 'upper' for the upper bandwidth (above the main diagonal).
B — Lower or upper bandwidth
Lower or upper bandwidth, returned as a nonnegative integer scalar.
If type is 'lower', then 0 ≤ B ≤ size(A,1)-1.
If type is 'upper', then 0 ≤ B ≤ size(A,2)-1.
Lower bandwidth, returned as a nonnegative integer scalar. lower is in the range 0 ≤ lower ≤ size(A,1)-1.
Upper bandwidth, returned as a nonnegative integer scalar. upper is in the range 0 ≤ upper ≤ size(A,2)-1.
The upper and lower bandwidths of a matrix are measured by finding the last diagonal (above or below the main diagonal, respectively) that contains nonzero values.
That is, for a matrix A with elements Aij:
The upper bandwidth B1 is the smallest number such that
{A}_{ij}=0
j-i>{B}_{1}
The lower bandwidth B2 is the smallest number such that
{A}_{ij}=0
i-j>{B}_{2}
Note that this measurement does not disallow intermediate diagonals in a band from being all zero, but instead focuses on the location of the last diagonal containing nonzeros. By convention, the upper and lower bandwidths of an empty matrix are both zero.
Use the isbanded function to test if a matrix is within a specific lower and upper bandwidth.
isbanded | isdiag | istriu | istril | diag
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Create pixel classification layer using generalized Dice loss for semantic segmentation - MATLAB - MathWorks 日本
The Dice loss function is based on the Sørensen-Dice similarity coefficient for measuring overlap between two segmented images.
L=1â\frac{2{â}_{k=1}^{K}{w}_{k}{â}_{m=1}^{M}{Y}_{km}{T}_{km}}{{â}_{k=1}^{K}{w}_{k}{â}_{m=1}^{M}{Y}_{km}^{2}+{T}_{km}^{2}}
{w}_{k}=\frac{1}{{\left({â}_{m=1}^{M}{T}_{km}\right)}^{2}}
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Metals and Non-metals - Revision Notes
About 118 elements are known today. There are more than 90 metals,22 non metals and a few metalloids.
Sodium (Na), potassium (K), magnesium(Mg), aluminium(Al),calcium(Ca), Iron(Fe), Barium(Ba) are some metals.
Oxygen(O), hydrogen(H), nitrogen(N), sulphur(S), hosphorus(P),fluorine(F), chlorine(Cl), bromine(Br), iodine(l) are some non-metals.
1. Generally solid except Hg(present in liquid form). 1. Can be solid, liquid organs e.g., C is solid, Br (liq), H
{}_{2}
(gas)
2. Ductile, Malleable (drawn into wires) (beaten into sheets) 2. Non-ductile, non-Malleable
3. Sonorous (produces sound) 3. Non-sonorous
4. Lustrous (have natural shine) 4. Non-lustrous except Iodine.
5. High Melting Point except Ce and Ga 5. Lower M.P. than metals.
6. Generally good conductors of heat and electricity except Pb and Hg. 6. Bad conductors of heat and electricity except Graphite (form of C)
7. High density except Na and K 7. Low densities except Diamond (form of C)
8. Reactive 8. Not very reactive.
9. Ionic bonding is present, 9. Covalent/Hydrogen bonding is present
Metals form basic oxides e.g., Magnesium oxide(MgO), while non-metals form acidic oxides e.g.,
S{O}_{2},C{O}_{2}
Ag and Cu are best conductors of electricity.
Metals and Non-metals can be distinguished on the basis of their physical and chemical properties.
Some elements show the properties of both metals and non-metals and are called metalloids.
Chemical Properties of Metals Reaction with air : Different metals show different reactivities towards oxygen present in air.
Metal+oxygen\stackrel{}{\to }
Some metals like Na and K are kept immersed in kerosene oil as they react vigorously with air and catch fire.
Some metals like Mg, Al, Zn, Pb react slowly with air and form a protective layer.
Mg can also burning air with a white dazzling light to form its oxide
Fe and Cu don’t burn in air but combine with oxygen to form oxide.Iron filings burn when sprinkled over flame.
Metals like silver, platinum and gold don't burn or react with air.
e.g..
2Na+\frac{1}{2}{O}_{2}\stackrel{}{\to }N{a}_{2}O
2Mg+{O}_{2}\stackrel{}{\to }2MgO
2Cu+{O}_{2}\stackrel{}{\to }2CuO
4Al+3{O}_{2}\stackrel{}{\to }2A{l}_{2}{O}_{3}
Usually metal oxides are basic in nature,but some metal oxides show both acidic and basic nature. Amphoteric Oxides :metal oxides which react with both acids as well as bases to form salt and water e.g.
A{l}_{2}{O}_{3,\phantom{\rule{thickmathspace}{0ex}}}ZnO.
eg..
A{l}_{2}{O}_{3}+HCl\stackrel{}{\to }AlC{l}_{3}+{H}_{2}O
A{l}_{2}{O}_{3}+NaOH\stackrel{}{\to }NaAl{O}_{3}+{H}_{2}O
REACTION WITH WATER : Metal oxides on reaction with water form alkalis.
Na+H{O}_{2}\stackrel{}{\to }NaOH+{H}_{2}
K+H{O}_{2}\stackrel{}{\to }KOH+{H}_{2}
Ca+H{O}_{2}\stackrel{}{\to }Ca\left(OH{\right)}_{2}+{H}_{2}
Mg+{H}_{2}O\stackrel{}{\to }Mg\left(OH{\right)}_{2}+{H}_{2}
In case of Ca and Mg, the metal starts floating due to bubbles of hydrogen gas sticking to its surface.
Al+{H}_{2}O\stackrel{}{\to }A{l}_{2}{O}_{3}+{H}_{2}
Fe+{H}_{2}O\stackrel{}{\to }F{e}_{3}{O}_{4}+{H}_{2}
Inert metals like Au and Ag do not react with water.
Metal\text{ }oxide+water\stackrel{}{\to }MetalHydroxide
Note: Try Balancing the above Chemical equations yourself
Metal +dilute acid
\stackrel{}{\to }
Salt + Hydrogen gas
metals react with dilute hydrochloric acid and dilute sulphuric acid to form chlorides.
Fe+2HCL\to FeC{l}_{2}+{H}_{2}
Mg+2HCl\to MgC{l}_{2}+{H}_{2}
Zn+2HCl\to ZnC{l}_{2}+{H}_{2}
2Al+6HCl\to 2AlC{l}_{3}+3{H}_{2}
Note : Copper, mercury and silver don’t react with dilute acids.
Hydrogen gas produced is oxidised to water. This happens because
HN{O}_{3}
is a strong oxidising agent when metals react with nitric acid
\left(HN{O}_{3}\right).
But Mg and Mn, react with very dilute nitric acid to evolve hydrogen gas. 4. Reaction of metals with other metal salts :
Metal\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}A+Salt\phantom{\rule{thinmathspace}{0ex}}solution\phantom{\rule{thinmathspace}{0ex}}of\phantom{\rule{thinmathspace}{0ex}}B\to \text{ }Salt\phantom{\rule{thinmathspace}{0ex}}Solution\phantom{\rule{thinmathspace}{0ex}}of\phantom{\rule{thinmathspace}{0ex}}A+Metal\phantom{\rule{thinmathspace}{0ex}}B
All metals are not equally reactive. Reactive metals can displace less reactive metals from their compounds in solution. This forms the basis of reactivity series of metals. Reactivity series is a list of metals arranged in order of their decreasing activities.
Fe+CuS{O}_{4}\to FeS{O}_{4}+Cu
Zn+CuS{O}_{4}\to ZnS{O}_{4}+Cu
REACTION OF NON-METALS :
reaction with oxygen non-metals form acidic oxides Eg:C+O2 ->CO2
reaction with water non-metals do not react with water because they cannot release electrons.
reaction with dilute acids no reaction
reaction with salt solutions a more reactive non-metal will displace less reactive non-metal from its salt solution.
reaction with chlorine chloride is formed. Eg;H2(g)+Cl2->2HCl
reactions with hydrogen hydrides are formed. H2 + S(l)->H2S
Reaction between Metals and Non-Metals
Reactivity of elements can be understood as a tendency to attain a completely filled valence shell.
Atom of metals can lose electrons from valence shells to form cations(+ve ions).
Atom of non-metals gain electrons in valence shell to form anions (–ve ions).
Oppositely charged ions attract each other and are held by strong electrostatic forces of attraction forming ionic compounds.
Formation of
MgC{l}_{2}
Mg\stackrel{}{\to }M{g}^{2+}+2{e}^{-}
2,8, 22,8 (Magnesium ion) Properties of Ionic Compounds
Are solid and mostly brittle.
Have high melting and boiling points. More energy is required to break the strong inter-ionic attraction.
Generally soluble in water and insoluble in kerosene, petrol.
Conduct electricity in solution and in molten state. In both cases, free ions are formed and conduct electricity.
Occurrence of Metals Minerals : Elements or compounds occurring naturally are minerals. ORES : Mineral from which metal can be profitably extracted is an ore. For example, sulphide ore, oxide ore, carbonate ore.
Metals at the bottom of activity series like gold, platinum, silver, copper generally occur in free state. But copper and silver also occur as sulphide and oxide ores.
Metals of medium reactivity (Zn, Fe, Pb etc.) occur mainly as oxides,sulphides or carbonates.
Metals of high reactivity (K, Na, Ca, Mg and Al) are very reactive and are thus found in combined state.
GANGUE : the commercially valueless materiallike soil,sand, etc. in which ore is found.called gangue. The gangue is removed from the ore. Various Methods to remove gangue: 1.GRAVITY SEPARATION 2.FROTH FLOATATION 3.MAGNETIC SEPARATION METALLURGY : Step-wise process of obtaining metal from its ore.
I. *Enrichment of ore
II. *Obtaining metal from enriched ore.
III. *Refining of impure metal to obtain pure metal.
Enrichment of Ores : It is the process of the removal of impurities such as soil, sand etc. from the ore prior to extraction of the metal. Different separation techniques are used based on physical or chemical properties of ore. Extracting Metals from the Enriched Ore
Extracting Metals Low in the Activity Series : By heating the ores in air at high temperature. e.g.*Mercury from cinnabar
2HgS+3{O}_{2}\stackrel{Heat}{\to }2HgO+2S{O}_{2}
2HgO\stackrel{Heat}{\to }2Hg+{O}_{2}
e.g. *Copper from copper sulphide
C{u}_{2}S+3{O}_{2}\stackrel{Heat}{\to }2C{u}_{2}{O}_{2}S{O}_{2}
2C{u}_{2}S+C{u}_{2}S\stackrel{Heat}{\to }6Cu+S{O}_{2}
Extracting Metals in the Middle of Activity Series : *Metals are easier to obtain from oxide ores, thus, sulphide and carbonate ores are converted into oxides. *Metal ore heated strongly in excess of air (Roasting) e.g..
2ZnS+3{O}_{2}\stackrel{Heat}{\to }2ZnO+2S{O}_{2}
Metal ore heated strongly in limited or no supply of air (Calcination) e.g..
ZnS{O}_{3}\stackrel{Heat}{\to }ZnO+C{O}_{2}
Reduction of Metal Oxide :
USING COKE : Coke as a reducing agent.
ZnO+C\stackrel{Heat}{\to }Zn+CO
USING DISPLACEMENT REACTION : highly reactive metal like Na, Ca and Al are used to displace metals of lower reactivity from their compounds. These displacement reactions are highly exothermic.
Mn{O}_{2}+4Al\stackrel{Heat}{\to }3Mn+2A{l}_{2}{O}_{3}+heat
F{e}_{2}{O}_{3}+2Al\stackrel{Heat}{\to }2Fe+A{l}_{2}{O}_{3}+heat
Thermite Reaction : Reduction of a metal oxide to form metal by using Al powder as a reducing agent. This process is used to join broken pieces of heavy iron objects or welding. Extracting Metals at the Top of Activity Series
These metals have more affinity for oxygen than carbon so they cannot be obtained from their compounds by reducing with carbon.
So are obtained by electrolytic reduction. e.g.Sodium is obtained by electrolysis of its molten chloride
NaCl\stackrel{}{\to }N{a}^{+}\phantom{\rule{thickmathspace}{0ex}}+\phantom{\rule{thickmathspace}{0ex}}C{l}^{-}
As electricity is passed through the solution metal gets deposited at cathode and non-metal at anode. At cathode : e.g.
N{a}^{+}+{e}^{-}\stackrel{}{\to }Na
At anode :
2C{l}^{-}\stackrel{}{\to }C{l}_{2}+2{e}^{-}
III. Refining of Metals Impurities present in the obtained metal can be removed by electrolytic refining. Copper is obtained using this method. Following are present inside the electrolytic tank. Anode – slab of impure copper Cathode– slab of pure copper Solution – aqueous solution of copper sulphate with some dilute sulphuric acid From anode copper ions are released in the solution and equivalent amount of copper from solution is deposited at cathode. Insoluble impurities containing silver and gold gets deposited at the bottom of anode as anode mud.
Corrosion Metals are attacked by substances in surroundings like moisture and acids. Silver - it reacts with sulphur in air to our form silver sulphide and articles become black. Copper - reacts with moist carbon dioxide in air and gains a green coat of copper carbonate. Iron-acquires a coating of a brown flaky substance called rust. Both air and moisture are necessary for rusting of iron. Rust is hydrated Iron (III) oxide i.e.
F{e}_{2}{O}_{{3}^{.}}x{H}_{2}O
Prevention of Corrosion Rusting of iron is prevented by painting, oiling, greasing, galvanizing, chrome plating, anodising and making alloys.
In galvanization, iron or steel is coated with a layer of zinc because oxide thus formed is impervious to air and moisture thus protects further layers from corrosion.
Alloys: These are homogeneous mixture of metals with metals or non-metals. Adding small amount of carbon makes iron hard and strong.
Some examples of alloys are following ;
1. Steel : Hard Iron and carbon.Used for construction of roads, railways, other infrastructure, appliances
2. Stainless steel :Hard Rust Free Iron, nickel and chromium. Used in utensils.
3. Brass :Low electrical conductivity Copper and zinc.used for decoration for its bright gold-like appearance and in locks,gears ,plumbing and electrical appliances.
4. Bronze: than pure metal Copper and tin. used to make coins, springs, turbines and blades. 5. Solder : Low MP, used to weld wires Lead and tin. used to create a permanent bond between metal work pieces
6. Amalgam :Used by dentists. Mercury and any other metal
Metals combine with oxygen to form basic oxides. Aluminium oxide and zinc oxide show the properties of both basic as well as acidic oxides. These oxides are know as amphoteric oxides.
The extraction of metals from their ores and then refining them for use is know as metallurgy.
An alloy is a homogeneous mixture of two or more metals, or a metal and anon-metal.
Non-metals have properties opposite to that of metals. They are neither malleable nor ductile. They are bad conductors of heat and electricity, except for graphite,which conducts electricity.
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