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SumConvergence still imperfect
• To: mathgroup at smc.vnet.net
• Subject: [mg124498] SumConvergence still imperfect
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Sun, 22 Jan 2012 07:18:25 -0500 (EST)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
The function SumConvergence in Mathematica 8 performs much better than
in 7 (where it first appeared) but it is still not perfect. Consider
this example:
SumConvergence[(-1)^((1/2)*n*(n + 1))/Log[n], n]
Well, the sum is actually convergent by the Dirichlet test (the sequence
1/Log[n] is monotonic with limit 0 and the sequence {1,1,-1,-1,1,1,...}
has bounded partial sums.
Wolfram Alpha also gives the wrong answer.
However, in the next case, where exactly the same argument can be used,
SumConvergence gets it right:
SumConvergence[(-1)^(1/2 n (n + 1))/n, n]
However Wolfram Alpha still can't do this. Asked to evaluate
Sum[(-1)^(n (n + 1)/2)/n, {n, 1, Infinity}]
it first given an inconclusive answer (claiming to have run out of
time). Given more time it returns an approximate answer, which is the
same as the one given by NSum
NSum[(-1)^(n (n + 1)/2)/n, {n, 1, Infinity}]
During evaluation of In[105]:= NSum::emcon: Euler-Maclaurin sum failed
to converge to requested error tolerance. >>
-1.10245 - 0.268775 I
which is obviously wrong. Unfortunately Wolfram Alpha does not issue the
warning that the answer is essentially meaningless so some poor soul
might really believe that the limit of a series of real numbers could be
One can get a much better approximate answer using Sum with a large
number of terms, e.g.
N[Sum[(-1)^(n (n + 1)/2)/n, {n, 1, 10^5}], 5]
Andrzej Kozlowski | {"url":"http://forums.wolfram.com/mathgroup/archive/2012/Jan/msg00595.html","timestamp":"2024-11-12T00:09:28Z","content_type":"text/html","content_length":"31382","record_id":"<urn:uuid:fbdd2b5c-cb61-4f3a-95f3-61a7bb6bdc6f>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00333.warc.gz"} |
American Mathematical Society
The Weingarten Calculus
Benoît Collins
Sho Matsumoto
Jonathan Novak
Communicated by Notices Associate Editor Steven Sam
1. Introduction
Every compact topological group supports a unique translation invariant probability measure on its Borel sets — the Haar measure. The Haar measure was first constructed for certain families of
compact matrix groups by Hurwitz in the nineteenth century in order to produce invariants of these groups by averaging their actions. Hurwitz’s construction has been reviewed from a modern
perspective by Diaconis and Forrester, who argue that it should be regarded as the starting point of modern random matrix theory DF17. An axiomatic construction of Haar measures in the more general
context of locally compact groups was published by Haar in the 1930s, with further important contributions made in work of von Neumann, Weil, and Cartan; see Bou04. The existence of recent works on
the Haar measure, see, e.g., DS14 or Mec19, can be seen as a token of the timeliness of this object as a modern research topic.
Given a measure, one wants to integrate. The Bochner integral for continuous functions on a compact group taking values in a given Banach space is called the Haar integral; it is almost always
written simply
with no explicit notation for the Haar measure. While integration on groups is a concept of fundamental importance in many parts of mathematics, including functional analysis and representation
theory, probability, and ergodic theory, etc., the actual computation of Haar integrals is a problem which has received curiously little attention. As far as the authors are aware, it was first
considered by theoretical physicists in the 1970s in the context of nonabelian gauge theories, where the issue of evaluating — or at least approximating — Haar integrals plays a major role. In
particular, the physics literature on quantum chromodynamics, the main theory of strong interactions in particle physics, is littered with so-called “link integrals,” which are Haar integrals of the
where is the compact group of unitary matrices Confronted with a paucity of existing mathematical tools for the evaluation of such integrals, physicists developed their own methods, which allowed
them to obtain beautiful explicit formulas such as
an evaluation which holds for all unitary groups of rank Although exceedingly clever, the bag of tricks for evaluating Haar integrals assembled by physicists is ad hoc and piecemeal, lacking the
unity and coherence which are the hallmarks of a mathematical theory.
The missing theory of Haar integrals began to take shape in the early 2000s, driven by an explosion of interest in random matrix theory. The basic Hilbert spaces of random matrix theory are and where
is the noncompact abelian group of Hermitian matrices equipped with a Gaussian measure of mean and variance , and is the compact nonabelian group of unitary matrices equipped with the Haar measure,
just as above. Given a probability measure on some set of matrices, the basic goal of random matrix theory is to understand the induced distribution of eigenvalues, which in the selfadjoint case form
a random point process on the line, and in the unitary case constitute a random point process on the circle. The moment method in random matrix theory, pioneered by Wigner (Wig58) in the 1950s, is an
algebraic approach to this problem. The main idea is to adopt the algebra of symmetric polynomials in eigenvalues as a basic class of test functions, and integrate such functions by realizing them as
elements of the algebra of polynomials in matrix elements, which can then (hopefully) be integrated by leveraging the defining features of the matrix model under consideration. The canonical example
is sums of powers of eigenvalues (elements of , which may be represented as traces of matrix powers (elements of ); more generally, all coefficients of the characteristic polynomial are sums of
principal matrix minors.
It is straightforward to see that, in both of the above -spaces, the algebra of polynomial functions in matrix elements admits the orthogonal decomposition
where is the space of homogeneous degree polynomial functions in matrix elements. Thus, modulo the algebraic issues inherent in transitioning from to , linearity of expectation reduces implementation
of the method to computing scalar products of monomials of equal degree, which are expressions of the form
In the Gaussian case, monomial scalar products can be computed systematically using a combinatorial algorithm which physicists call the “Wick formula” and statisticians call the “Isserlis theorem.”
This device leverages independence together with the characteristic feature of centered normal distributions — vanishing of all cumulants but the second — to compute Gaussian expectations as
polynomials in the variance parameter . The upshot is that scalar products in are closely related to the combinatorics of graphs drawn on compact Riemann surfaces, which play the role of Feynman
diagrams for selfadjoint matrix-valued field theories. We recommend (Zvo97) as an entry point into the fascinating combinatorics of Wick calculus.
The case of Haar unitary matrices is a priori more complicated: the random variables are identically distributed, thanks to the invariance of Haar measure, but they are also highly correlated, due to
the constraint Moreover, each individual entry follows a complicated law not uniquely determined by its mean and variance. Despite these obstacles, it turns out that, when packaged correctly, the
invariance of Haar measure provides everything needed to develop an analogue of Wick calculus for Haar unitary matrices. Moreover, once the correct general perspective has been found, one realizes
that it applies equally well to any compact group, and even to compact symmetric spaces and compact quantum groups. The resulting analogue of Wick calculus has come to be known as Weingarten calculus
, a name chosen by Collins Col03 to honor the contributions of Donald Weingarten, a physicist whose early work in the subject is of foundational importance.
The Weingarten calculus has matured rapidly over the course of the past decade, and the time now seems right to give a pedagogical account of the subject. The authors are currently preparing a
monograph intended to meet this need. In this article, we aim to provide an easily digestible and hopefully compelling preview of our forthcoming work, emphasizing the big picture but still providing
some of the important details.
First and foremost, we wish to impart the insight that, like the calculus of Newton and Leibniz, the core of Weingarten calculus is a fundamental theorem which converts a computational problem into a
symbolic problem: whereas the usual fundamental theorem of calculus converts the problem of integrating functions on the line into computing antiderivatives, the fundamental theorem of Weingarten
calculus converts the problem of integrating functions on groups into computing certain matrices associated to tensor invariants. The fundamental theorem of Weingarten calculus is presented in detail
in Section 2.
We then turn to examples illustrating the fundamental theorem in action. We present two detailed case studies: integration on the automorphism group of a finite set of size , and integration on the
automorphism group of -dimensional Hilbert space. These are natural examples, given that the symmetric group and the unitary group are model examples of a finite and infinite compact group,
respectively. The case, presented in Section 3, is a toy example chosen to illustrate how Weingarten calculus works in an elementary situation where the integrals to which it applies can easily be
evaluated from first principles. The case, discussed in Section 4, is an example of real interest, and we give a detailed workup showing how Weingarten calculus handles the link integrals of lattice
gauge theory.
Section 5 gives a necessarily brief discussion of Weingarten calculus for the remaining classical groups, namely the orthogonal group and the symplectic group both of which receive a detailed
treatment in a book in preparation by the authors. Finally, Section 6 extols the universality of Weingarten calculus, briefly discussing how it can be transported to compact symmetric spaces and
compact quantum groups, and indicating applications in quantum information theory.
2. The Fundamental Theorem
Given a compact group , a finite-dimensional Hilbert space with a specified orthonormal basis , and a continuous group homomorphism from to the unitary group of , let be the corresponding matrix
element functionals,
The Weingarten integrals of the unitary representation are the integrals
where ranges over the set of positive integers, and the multi-indices range over the set of functions from to . Clearly, if we can compute all Weingarten integrals , then we can integrate any
function on which is a polynomial in the matrix elements This is the basic problem of Weingarten calculus: compute the Weingarten integrals of a given unitary representation of a given compact group.
The fundamental theorem of Weingarten calculus addresses this problem by linearizing it. The basic observation is that, for each the integrals , , are themselves the matrix elements of a linear
operator. Indeed, we have
is the orthonormal basis of corresponding to the specified orthonormal basis in , and
are the matrix elements of the unitary operator in this basis. We thus have that
where are the matrix elements of the selfadjoint operator
obtained by integrating the unitary operators against Haar measure. The basic problem of Weingarten calculus is thus equivalent to computing the matrix elements of , for all
This is where the characteristic feature of Haar measure, the invariance
comes into play: it forces Thus is a selfadjoint idempotent, and as such orthogonally projects onto its image, which is the space of -invariant tensors in
Thus, we see that the basic problem of Weingarten calculus is in fact very closely related to the basic problem of invariant theory, which is to determine a basis for the space of -invariant tensors
in for all
Indeed, suppose we have access to a basis of . Then, by elementary linear algebra, we have everything we need to calculate the matrix
of degree Weingarten integrals. Let be the matrix whose columns are the coordinates of the basic invariants in the desired basis,
Then we have the matrix factorization
familiar from matrix analysis as the multidimensional generalization of the undergraduate “outer product divided by inner product” formula for orthogonal projection onto a line. The matrix is nothing
but the Gram matrix
of the basic -invariants in , whose linear independence is equivalent to the invertibility of the Gram matrix. Let us give the inverse Gram matrix a name: we call
the Weingarten matrix of the invariants Extracting matrix elements on either side of the factorization , we obtain the Fundamental Theorem of Weingarten Calculus.
Theorem 2.1.
For any and we have
Does Theorem 2.1 actually solve the basic problem of Weingarten calculus? Yes, insofar as the classical fundamental theorem of calculus solves the problem of computing definite integrals: it reduces
a numerical problem to a symbolic problem. In order to apply the fundamental theorem of calculus to integrate a given function, one must find its antiderivative, and as every student of calculus
knows this can be a wild ride. In order to use the fundamental theorem of Weingarten calculus to compute the Weingarten integrals of a given unitary representation, one must solve a souped-up version
of the basic problem of invariant theory which involves not only finding basic tensor invariants, but computing their Weingarten matrices. Just like the computation of antiderivatives, this may prove
to be a difficult task.
3. The Symmetric Group
In this Section, we consider a toy example. Fix and let be the symmetric group of rank viewed as the group of bijections This is a finite group, its topology and resulting Haar measure are discrete,
and all Haar integrals are finite sums. We will solve the basic problem of Weingarten calculus for the permutation representation of in two ways: using elementary combinatorial reasoning, and using
the fundamental theorem of Weingarten calculus. It is both instructive and psychologically reassuring to work through the two approaches and see that they agree.
The permutation representation of is the unitary representation in which is an -dimensional Hilbert space with orthonormal basis and is defined by
The corresponding system of matrix elements is given by
We will evaluate the Weingarten integrals of
Each Weingarten integral is a finite sum with terms, each equal to zero or one:
Thus, simply counts permutations which solve the equation This is an elementary counting problem, and a good way to solve it is to think of the given functions “backwards,” as the ordered lists of
their fibers:
The fiber fingerprint of the composite function is then
and so we have if and only if
Clearly, such a permutation exists if and only if the fibers of and are the same up to the labels of their base points, which is the case if and only if
where is the partition of obtained by forgetting the order on the fibers of and throwing away empty fibers. The permutations we wish to count thus number
in total, where denotes the number of blocks of the set partition . We conclude that the integral is given by
Let us now evaluate using the Fundamental Theorem of Weingarten Calculus. The first step is to solve the basic problem of invariant theory for the representation This is again straightforward. Fix
let denote the set of partitions of with at most blocks, and to each associate the tensor
where It is apparent that the set is a basis of . Indeed, taking the unit tensor
corresponding to a function and symmetrizing it using the action of permutations on multi-indices produces the tensor
which is clearly -invariant, and moreover it is clear that every -invariant tensor in is a linear combination of tensors of this form. Furthermore,
so that the distinct invariants produced by symmetrization of the initial basis in are
These tensors are pairwise orthogonal: for any , we have
So, the Gram matrix of the basis | {"url":"https://www.ams.org/journals/notices/202205/noti2474/noti2474.html?adat=May%202022&trk=2474&pdfissue=202205&pdffile=rnoti-p734.pdf&cat=none&type=.html","timestamp":"2024-11-08T02:14:57Z","content_type":"text/html","content_length":"1049019","record_id":"<urn:uuid:32ca8df4-caa9-4806-ba0e-e07d062b3383>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00313.warc.gz"} |
Spotfire | Logistic Regression: Predicting Binary Outcomes with Data
What is logistic regression?
Logistic regression is a statistical model that Is used to determine the probability that an event will happen. It shows the relationship between features, and then calculates the probability of a
certain outcome.
Logistic regression is used in machine learning (ML) to help create accurate predictions. It is similar to linear regression, except rather than a graphical outcome, the target variable is binary;
the value is either 1, or 0.
There are two types of measurables, the explanatory variables/ features (item being measured) and the response variable/ target binary variable, which is the outcome.
For example, when trying to predict whether a student will pass or fail a test, the hours studied are the feature, and the response variable will have two values - pass or fail.
There are three basic kinds of logistic regression:
1. Binary logistic regression: Here there are only two possible outcomes for the categorical response. As in the example above – a student passes or fails.
2. Multinomial logistic regression: This is where the response variables can include three or more variables, which will not be in any order. An example is predicting whether diners at a restaurant
prefer a certain kind of food – vegetarian, meat or vegan.
3. Ordinal logistic regression: Like multinomial regression, there can be three or more variables. However, there is an order the measurements follow. An example is rating a hotel on a scale of 1 to
Assumptions used for logistic regression
When working with logistic regression, there are certain assumptions that are made.
• In binary logistic regression, it is necessary that the response variable is a binary. The outcome is either one thing, or another.
• The desired outcome should be represented by the factor level 1 of the response variable, the undesired is 0.
• Only variables that are meaningful must be included.
• Independent variables have to essentially be independent of one another. There should be little to no multi-co-linearity.
• Log odds and independent variables have to be linearly related.
• Logistic regression must be applied only to massive sample sizes.
Applications of logistic regression
There are several fields and ways in which logistic regression can be used and these include almost all fields of medical and social sciences.
For example, the Trauma and Injury Severity Score (TRISS). This is used across the world to predict fatality in injured patients. This model has been developed with the application of logistic
regression. It uses variables such as the revised trauma score, injury severity score, and the age of patient to predict health outcomes. It is a technique that can even be used to predict the
possibility of a person being afflicted by a certain disease. For example, ailments like diabetes and heart disease can be predicted based on variables such as age, gender, weight and genetic
Logistic regression can also be used to attempt to predict elections. Will a Democrat, Republican or Independent leader come to power in the USA? These predictions are made on the basis of variables
such as age, gender, place of residence, social standing and previous voting patterns (variables) to produce a vote prediction (response variable).
Product testing
Logistic regression can be used in engineering to predict the success or failure of a system that is being tested, or a prototype product.
LR can be used to predict the chances of a customer’s enquiry turning into a sale, the possibility of a subscription being started or terminated, or even potential customer interest in a new product
Financial sector
An example of use in the financial sector is in a credit card company that uses it to predict the likelihood that a customer will default on their payments. The model built could be for the issuance
of a credit card to a customer or not. The model can say whether a certain customer will “default” or “not default”. This is known as the “default propensity modeling” in banking terms.
Much along the same lines, e-commerce companies invest heavily in advertising and promotional campaigns across media. They want to see which campaign is the most effective and the option most likely
to get a response from their potential target audience. The model set will categorize the customer as a “responder” or “non responder”. This model is called propensity to respond modeling.
With insights that come from logistic regression outputs, companies are able to optimize their strategies and achieve business goals with reduction in expenses as well as losses. Logistic regressions
help to maximize return on investment (ROI) in marketing campaigns, a benefit to the bottom line of a company in the long run.
Advantages and disadvantages of logistic regression
Logistic Regression is widely used because it is extremely efficient and does not need huge amounts of computational resources. It can be interpreted easily and does not need scaling of input
features. It is simple to regularize, and the outputs it provides are well-calibrated predicted probabilities.
Just as it does in linear regression, logistic regression tends to work more efficiently when attributes unrelated to the output variable and those that are correlated, are omitted. Feature
engineering therefore has an important role to play in the efficacy of performance of logistic and linear regression.
Logistic regression is also easily implemented and simple to train and that’s what makes it a great baseline to help measure the performance of other complex algorithms.
Logistic regression cannot be used to solve nonlinear problems and unfortunately, many of today’s systems are nonlinear. Additionally, logistic regression is not the most powerful algorithm
available. There are several alternatives that can create much better, more complex predictions.
Logistic regression also relies heavily on data presentation. This means that unless you have identified all the necessary independent variables, the output is of no value. With an outcome that is
discrete, logistic regression can only be used to predict a categorical outcome. And finally, it is an algorithm with a known history of vulnerability to over-fitting.
Ready for immersive, real-time insights for everyone? | {"url":"https://www.spotfire.com/glossary/what-is-logistic-regression","timestamp":"2024-11-09T17:07:32Z","content_type":"text/html","content_length":"52218","record_id":"<urn:uuid:4d384670-8fba-472d-a209-ffc09831a977>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00736.warc.gz"} |
{"publication":"Proceedings of the American Mathematical Society","year":"2006","date_updated":"2021-01-12T06:50:54Z","issue":"8","acknowledgement":"The first author was partly supported by NSF grant
DMS-0072675. The second author was partly supported by a VIGRE postdoc under NSF grant number 9983660 to Cornell University.","extern":1,"intvolume":" 134","publisher":"American Mathematical
0306369"}],"publication_status":"published","author":[{"id":"4A0666D8-F248-11E8-B48F-1D18A9856A87","first_name":"Tamas","last_name":"Hausel","full_name":"Tamas Hausel"},
{"last_name":"Swartz","full_name":"Swartz, Edward","first_name":"Edward"}],"title":"Intersection forms of toric hyperkähler varieties","citation":{"ista":"Hausel T, Swartz E. 2006. Intersection forms
of toric hyperkähler varieties. Proceedings of the American Mathematical Society. 134(8), 2403–2409.","short":"T. Hausel, E. Swartz, Proceedings of the American Mathematical Society 134 (2006)
2403–2409.","apa":"Hausel, T., & Swartz, E. (2006). Intersection forms of toric hyperkähler varieties. Proceedings of the American Mathematical Society. American Mathematical Society. https://doi.org
/10.1090/S0002-9939-06-08248-7","chicago":"Hausel, Tamás, and Edward Swartz. “Intersection Forms of Toric Hyperkähler Varieties.” Proceedings of the American Mathematical Society. American
Mathematical Society, 2006. https://doi.org/10.1090/S0002-9939-06-08248-7.","ieee":"T. Hausel and E. Swartz, “Intersection forms of toric hyperkähler varieties,” Proceedings of the American
Mathematical Society, vol. 134, no. 8. American Mathematical Society, pp. 2403–2409, 2006.","mla":"Hausel, Tamás, and Edward Swartz. “Intersection Forms of Toric Hyperkähler Varieties.” Proceedings
of the American Mathematical Society, vol. 134, no. 8, American Mathematical Society, 2006, pp. 2403–09, doi:10.1090/S0002-9939-06-08248-7.","ama":"Hausel T, Swartz E. Intersection forms of toric
hyperkähler varieties. Proceedings of the American Mathematical Society. 2006;134(8):2403-2409. doi:10.1090/S0002-9939-06-08248-7"},"abstract":[{"text":"This note proves combinatorially that the
intersection pairing on the middle-dimensional compactly supported cohomology of a toric hyperkähler variety is always definite, providing a large number of non-trivial L 2 harmonic forms for toric
hyperkähler metrics on these varieties. This is motivated by a result of Hitchin about the definiteness of the pairing of L 2 harmonic forms on complete hyperkähler manifolds of linear
growth.","lang":"eng"}],"oa":1,"quality_controlled":0,"date_published":"2006-08-01T00:00:00Z","page":"2403 - | {"url":"https://research-explorer.ista.ac.at/record/1461.jsonl","timestamp":"2024-11-06T07:40:15Z","content_type":"text/plain","content_length":"3637","record_id":"<urn:uuid:b441b476-3326-41a0-9597-409ebf2982ed>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00808.warc.gz"} |
Honors BS Requirements
The deadline for signing up for Honors BS is March 1 of the senior year. The deadline for presenting the talk is April 1 of the senior year. If you are considering graduate school in mathematics you
should visit this page and check out the sample schedules here.
Foundational Course Requirement
The following foundational courses must be completed before acceptance into the concentration:
• MATH 171: Honors Calculus I
• MATH 172: Honors Calculus II
• MATH 173: Honors Calculus III
• MATH 174: Honors Calculus IV
Note: Alternatively, students may satisfy the Foundational Course Requirement by completing MATH 161, 162, 164, 165 and 235. Equivalent courses may be substituted for the above. Credit granted for AP
courses may be used to satisfy foundational requirements.
Core Course Requirement
Students must complete the following four courses:
• MATH 236H: Introduction to Algebra I (Honors)
• MATH 240H: Introduction to Topology (Honors)
• MATH 265H: Functions of a Real Variable I (Honors)
• MATH 282: Introduction to Complex Variables with Applications
Advanced Course Requirement
In addition to the core courses, students must complete six 4-credit advanced courses as follows:
• Four advanced mathematics courses, at least two of which are at the graduate level*
* Any 4-credit mathematics course numbered 200 or above (excluding core courses and 500+ level) qualifies as an advanced mathematics course. Any 4-credit mathematics course numbered 400-499 qualifies
as a graduate level course (500+ level courses are not approved for this requirement). All core course substitutions (MATH 235/236H/240H/265H/282) must be approved in advance by the Math Department
Undergraduate Committee. Core course substitutions are rarely approved. Students seeking approval for such a substitution should visit walk-in hours listed on the math advising page to discuss their
proposal with a member of the committee. Any graduate courses substituted in the core are not counted in the two towards the advanced requirement.
Independent Research Project
Students will work on an independent research project with the agreement and under the close supervision of a faculty member in mathematics. Students should sign up for Math 395W (2 credits) with the
supervising professor.
Upon completion, students will submit a written report on the project to the department's Honors Committee and present a one-hour public talk at which the members of the committee are in attendance.
More detailed information is available here.
Samples of past honors papers can be found here.
Grade Point Average Requirement
Students must complete the program with at least a 3.25 grade point average in order to qualify for the Honors BS in mathematics.
Upper Level Writing Requirement
To satisfy the upper level writing requirement, students must pass two upper level writing courses of a certain type. | {"url":"http://sas.rochester.edu/mth/undergraduate/bs-honors.html","timestamp":"2024-11-04T16:49:51Z","content_type":"text/html","content_length":"75859","record_id":"<urn:uuid:5264bf93-ee4f-4928-b2b4-f41e7a8617a8>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00358.warc.gz"} |
How much does iron rod cost in Ghana?
How much does iron rod cost in Ghana?
Prices of Iron Rod in Ghana
Iron ROD Quantity Unit Price (GHS)
25MM 28 PCS
8MM (40FT) HIGH TENSILE 210 PS 30.9
10MM (40FT) HIGH TENSILE 127 PCS 50.9
12MM (40FT) HIGH TENSILE 92 PCS 70.3
What is the price of 12mm iron rod in Ghana?
A 12mm standard High tensile rod per ton which cost Ghc3,950 now goes averagely for Ghc5,400 on the market.
How much is 12mm Rod now?
Current Price of 12mm Iron Rod in Nigeria 2021 The current price per ton of 12mm iron rod is ₦390,000. This price is subjected to change and may differ among different suppliers.
How many pieces of 12mm iron rods make a ton?
Mathematical calculation such as number of pieces of 12mm Rods in one ton = 1000kg/weight of one piece of 12mm Rod, 1000kg ÷ 10.667kg = 94 pieces. Therefore 94 No’s of 12 meter of 12mm Rod or steel
bar makes a ton.
What is the price of 16mm iron rod?
Shree 16mm TMT Steel Bar, For Construction, Grade: Fe 550, Rs 50000/ton | ID: 8063746133.
How much is 16MM iron rod in Ghana?
Choose an option Below.
ITEM QTY UNIT PRICE
14MM (40FT) HIGH TENSILE 68 PCS
16MM (40FT) HIGH TENSILE 52 PCS 142.40
18MM (40FT) HIGH TENSILE 41 PCS
20MM (40FT) HIGH TENSILE 33 PCS 224.30
How many rods are in a 8mm bundle?
8 MM TMT Steel Bars Specifications : 10 Rods = 1 Bundle ( Apox 45 Kgs – 46 Kgs )
What is iron price per kg?
Today Iron Rate
Types of Iron Rate/Kg
New iron price per kg today Rs. 90-100
Old iron rate per kg today Rs. 40-60
How many pieces are in a bundle of rods?
Regarding this, “how many rods (rebar) in a bundle?”, you will find anywhere between 2 to 12+ pieces of rods or rebar/ steel bar in one bundle. 6 pieces of rods in 12mm bundle, 8 pieces of rods in
10mm bundle, 12 pieces of rods in 8mm bundle.
How many rods are in a steel bundle?
How many TMT steel bars will i get in one bundle – TMT FAQ
Diameter No of Rods Apox Weight
8 MM 10 Rods 46 Kgs
10 MM 7 Rods 50 Kgs
12 MM 5 Rods 52 Kgs
16 MM 3 Rods 56 Kgs | {"url":"https://www.fdotstokes.com/2022/11/04/how-much-does-iron-rod-cost-in-ghana/","timestamp":"2024-11-02T23:55:10Z","content_type":"text/html","content_length":"55760","record_id":"<urn:uuid:b11822eb-3a72-44eb-97ec-bb1c1819f499>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00288.warc.gz"} |
Seminars | Department of Mathematics
All Seminars
• 11/13 Math Club: Ask Me Anything
Math Club: Ask Me Anything
Wednesday, November 13th, 20245:30 PM - Monteith 419
At this meeting, Prof. Conrad will answer any question you have about math (except homework questions). This is a chance to find out more about any problems, concepts, examples, historical
events, etc. in math that you’ve heard or read about but don’t understand as well as you’d like.
• 11/15 SIGMA Seminar - Why is the Riemann hypothesis important? - Keith Conrad (UConn)
SIGMA Seminar - Why is the Riemann hypothesis important? - Keith Conrad (UConn)
Friday, November 15th, 20241:25 PM - 2:15 PM Monteith Building
The Riemann hypothesis is often presented as the most important unsolved problem in mathematics. Why is that?
In this talk the Riemann hypothesis will be described, including the context that led to it. Then examples will be given that illustrate a range of problems where the Riemann hypothesis can be
applied once it gets solved and where the problems themselves don’t sound related to the Riemann hypothesis at all.
It will be assumed that the audience is familiar with basic complex analysis.
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• 5/1 Mathematics Colloquium, Mariusz Urbanski (University of North Texas)
Mathematics Colloquium, Mariusz Urbanski (University of North Texas)
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632 1 - Transportes I (2024)
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Journal of Biomedical Optics 9(3), 632–647 (May/June 2004)Radiative transport in the delta-P1 approximation:accuracy of fluence rate and optical penetration depthpredictions in turbid semi-infinite mediaStefan A. CarpUniversity of California, IrvineDepartment of Chemical Engineeringand Materials ScienceandLaser Microbeam and Medical ProgramBeckman Laser InstituteIrvine, California 92697Scott A. PrahlOregon Medical Laser CenterProvidence St. Vincent Medical CenterPortland, Oregon 97225Vasan VenugopalanUniversity of California, IrvineDepartment of Chemical Engineeringand Materials ScienceandLaser Microbeam and Medical ProgramBeckman Laser InstituteIrvine, California 92697E-mail: vvenugop@uci.eduAbstract. Using the d-P1 approximation to the Boltzmann transportequation we develop analytic solutions for the fluence rate producedby planar (1-D) and Gaussian beam (2-D) irradiation of a homoge-neous, turbid, semi-infinite medium. To assess the performance ofthese solutions we compare the predictions for the fluence rate andtwo metrics of the optical penetration depth with Monte Carlo simu-lations. We provide results under both refractive-index matched andmismatched conditions for optical properties where the ratio of re-duced scattering to absorption lies in the range 0<(ms8/ma)<104. Forplanar irradiation, the d-P1 approximation provides fluence rate pro-files accurate to 616% for depths up to six transport mean free paths(l* ) over the full range of optical properties. Metrics for optical pen-etration depth are predicted with an accuracy of 64%. For Gaussianirradiation using beam radii r0>3l* , the accuracy of the fluence ratepredictions is no worse than in the planar irradiation case. For smallerbeam radii, the predictions degrade significantly. Specifically for me-dia with (ms8/ma)51 irradiated with a beam radius of r05l* , the errorin the fluence rate approaches 100%. Nevertheless, the accuracy ofthe optical penetration depth predictions remains excellent for Gauss-ian beam irradiation, and degrades to only 620% for r05l* . Theseresults show that for a given set of optical properties (ms8/ma), theoptical penetration depth decreases with a reduction in the beam di-ameter. Graphs are provided to indicate the optical and geometricalconditions under which one must replace the d-P1 results for planarirradiation with those for Gaussian beam irradiation to maintain ac-curate dosimetry predictions. © 2004 Society of Photo-Optical Instrumentation En-gineers. [DOI: 10.1117/1.1695412]Keywords: diffusion; photons; light; collimation.Paper 03047 received Apr. 17, 2003; revised manuscript received Jul. 28, 2003;accepted for publication Sep. 26, 2003.--t-le-.nsdis-cu-alima-htateddernim-s-helyteds in1 IntroductionMany biophotonics applications require knowledge of thelight distribution produced by illumination of a turbid tissuewith a collimated laser beam.1 Examples include photody-namic therapy, photon migration spectroscopy, and optoacoustic imaging. If one considers light propagating as a neutral particle, the Boltzmann transport equation provides anexact description of radiative transport.2 However, the Boltz-mann transport equation is an integrodifferential equation thaoften cannot be solved analytically. As an alternative, investigators have resorted to a variety of analytic and computationamethods, including Monte Carlo simulations, the adding-doubling method, and functional expansion methods.2–6 Eachof these methods possesses unique limitations. For exampwhile Monte Carlo simulations provide solutions to the Bolt-zmann transport equation that are exact within statistical unAddress all correspondence to Vasan Venugopalan, University of California—Irvine, Department of Chemical Engineering and Materials Science, 916 Engi-neering Tower, Irvine, California 92697. Tel: 949-824-5802; FAX: 949-824-2541; E-mail: vvenugop@uci.edu632 Journal of Biomedical Optics d May/June 2004 d Vol. 9 No. 3l,certainty, they require significant computational resources7–9While, numerical finite difference or finite element solutiofor the Boltzmann transport equation10 may involve less com-putational expenditure, they require spatial and angularcretizations of the computational domain that lead to inacracies that are often difficult to quantify. Finally functionexpansion methods, such as the standard diffusion approxtion ~SDA!, that express the angular distribution of the ligfield and the single-scattering-phase function as a truncseries of spherical harmonics are typically accurate only una limiting set of conditions.2,6,11–13Although the SDA provides only an approximate solutioto the Boltzmann transport equation, its computational splicity has proven valuable for applications in optical diagnotics and therapeutics. Unfortunately, the limitations of tSDA are significant and confine its applicability to highscattering media and to locations distal from both collimasources and interfaces possessing significant mismatche1083-3668/2004/$15.00 © 2004 SPIErleensan-a-ofcalen-thsnt ofp-lessno-theeir-ndsonely.,in-a-. Iney-on-altion,Radiative transport in the delta-P1 approximation . . .refractive index.11,14–16 Such conditions are not satisfied inmany biomedical laser applications and, over the past 15 yhybrid Monte Carlo–diffusion methods17,18 as well as thed-P1 , P3 , andd-P3 approximations have been proposed asimproved radiative transport models.1,6,19–25Our focus here isthe d-P1 ~or d -Eddington! model first introduced in 1976 byJoseph et al.26 and first applied to problems in the biomedicalarena independently by Prahl,23,27 by Star,6,24 and Star et al.25Many investigators in biomedical optics have studied theaccuracy of functional expansion methods. Groenhuis et aprovided one of the first comparative studies between MontCarlo and SDA predictions for the spatially resolved diffusereflectance produced by illumination of a turbid medium witha finite diameter laser beam.11 Later, Flock et al. providedanother comparison between Monte Carlo simulations and thSDA that focused primarily on optical dosimetry; specificallythe accuracy of fluence rate profiles and optical penetratiodepth predictions for planar irradiation of a turbid medium.28More recently, Venugopalan et al. presented analytic solutionfor radiative transport within thed-P1 approximation for in-finite media illuminated with a finite spherical source.19 Theaccuracy of these solutions was demonstrated by comparisowith experimental measurements made in phantoms overbroad range of optical properties. Spott and Svaasand reviewed a number of formulations of the diffusion approxima-tion (P1 , d-P1 , d-P3) for a semi-infinite medium illumi-nated with a collimated light source, and compared fluencerate and diffuse reflectance predictions with Monte Carlosimulations for optical properties representative ofin vivoconditions.16 Dickey et al.20,21 as well as Hull and Foster22have studied the improvements in accuracy offered by theP3approximation for predicting both fluence rate profiles andspatially resolved diffuse reflectance. These studies have cofirmed that thed-P1 approach can provide significant im-provements in radiative transport predictions relative to SDAwith minimal additional complexity.While these investigations have provided some indicationof the improved accuracy provided by thed-P1 approxima-tion relative to the SDA, none have offered a quantitativeassessment of its performance against a radiative transpobenchmark such as Monte Carlo simulations over a widerange of optical properties. Thus, it is difficult to establishapriori the loss of accuracy that one suffers when using thed-P1 approximation to determine fluence rate distributions oroptical penetration depths. Our objective is to provide a comprehensive quantitative assessment of theaccuracy of opticdosimetry predictions provided by thed-P1 approximationwhen a turbid semi-infinite medium is exposed to collimatedradiation. Here, we report on the variation of thed-P1 modelaccuracy with tissue optical properties and diameter of theincident laser beam.Specifically, we determined the fluence rate profiles predicted by thed-P1 approximation for semi-infinite mediawhen subjected to planar~1-D! or Gaussian beam~2-D! irra-diation. For comparison, we performed Monte Carlo simula-tions to provide ‘‘benchmark’’ solutions of the Boltzmanntransport equation for multiple sets of optical properties.While we include plots of diffuse reflectanceRd versus(ms8/ma) for planar irradiation, our focus is on the internallight distribution as represented by the spatial variation of the,.n--rtlfluence rate. Since it is cumbersome to display the variationfluence rate with depth for more than a few sets of optiproperties, we also examined predictions for the optical petration depth. Comparison of the optical penetration deppredicted by thed-P1 approximation with those derived fromMonte Carlo simulations enables a continuous assessmethe d-P1 model accuracy over a broad range of optical proerties. These results are presented within a dimensionframework to enable rapid estimation of the light distributioin a medium of known optical properties. Moreover, to prvide quantitative error assessment, we include plots ofdifference between thed-P1 and Monte Carlo estimates. Thvariation of these errors with tissue optical properties andradiation conditions provide much insight into the nature aorigin of the deficiencies inherent in thed-P1 approximationas well as other functional expansion methods.2 d -P1 Model Formulation and Monte CarloComputation2.1 d-P1 Approximation of the Single-ScatteringPhase FunctionThe basis of thed-P1 approximation to radiative transport ithe d-P1 phase function as formulated by Joseph et al.26pd2P1~v̂•v̂8!514p$2 f d @12~v̂•v̂8!#1~12 f !@113g* ~v̂•v̂8!#%, ~1!wherev̂ and v̂8 are unit vectors that represent the directiof light propagation before and after scattering, respectivIn Eq. ~1! f is the fraction of light scattered directly forwardwhich thed-P1 model treats as unscattered light. The remader of the light(12 f ) is diffusely scattered according tostandardP1 ~or Eddington! phase function with single scattering asymmetryg* . To determine appropriate values forfandg* , one must choose a phase function to approximatethis paper, we choose to provide results for the HenyGreenstein phase function, as it is known to provide a reasable approximation for the optical scattering in biologictissues29:pHG~v̂•v̂8!514p12g12@122g1~v̂•v̂8!1g12#3/2. ~2!Recalling that for a spatially isotropic medium, thenth mo-ment,gn , of the phase functionp(v̂•v̂8) is defined bygn52pE211Pn~v̂•v̂8!p~v̂•v̂8!d~v̂•v̂8!, ~3!wherePn is thenth Legendre polynomial, we determinef andg* by requiring the first two moments of thed-P1 phasefunction, g15 f 1(12 f )g* and g25 f , to match the corre-sponding moments of the Henyey-Greenstein phase funcwhich are given bygn5g1n . This yields the following expres-sions for f andg* :f 5g12 and g* 5g1 /~g111!. ~4!Journal of Biomedical Optics d May/June 2004 d Vol. 9 No. 3 633oima-mhlyt ben.tandt,n-er-tionitarearypo-i-aredCarp, Prahl, and VenugopalanFor simplicity, from this point forward we refer tog1 simplyas g and all d-P1 model results in this paper are shown forg50.9 unless noted otherwise.2.2 d-P1 Approximation of the RadianceIn a manner similar to the phase function, the radiance is alsseparated into collimated and diffuse components:L~r ,v̂ !5Lc~r ,v̂ !1Ld~r ,v̂ !, ~5!where r is the position vector andv̂ is a unit vector repre-senting the direction of light propagation.For irradiation with a collimated laser beam normally in-cident on the surface of a semi-infinite medium, the colli-mated radiance takes the formLc~r ,v̂ !512pE~r ,ẑ!d~12v̂• ẑ!, ~6!where ẑ is the direction of the collimated light within themedium, andE(r ,ẑ) is the complete spatial distribution ofcollimated light provided by the source. While the lateral spa-tial variation ofE(r ,ẑ) is given by the irradiance distributionof the incident laser beamE0(x,y), its decay with depth(z-dir! is governed by absorption and scattering within themedium. Specifically, loss of collimated light arises from bothabsorption and diffuse scattering. Noting that in thed-P1phase function only(12 f ) of the incident light is diffuselyscattered, the decay of the collimated light with depth willbehave as a modified Beer-Lambert law:E~r ,ẑ!5E0~x,y!~12Rs!exp$2@ma1ms~12 f !#z%5E0~x,y!~12Rs!exp@2~ma1ms* !z#, ~7!whereRs is the specular reflectance for unpolarized light,mais the absorption coefficient,ms is the scattering coefficient,andms* [ms(12 f ) is a reduced scattering coefficient. For acollimated beam traveling along thez axis that possesses ei-ther a uniform or Gaussian irradiance profile we can work incylindrical (r ,z) rather than Cartesian(x,y,z) coordinates. Inthis case, the collimated fluence rate is given bywc~r !5E4pLc~r ,v̂ !5E~r ,ẑ!5E0~r !~12Rs!exp~2m t* z!,~8!whereE0(r ) is the radial irradiance distribution of the inci-dent laser beam andm t* [ma1ms* .The diffuse radiance in Eq.~5! is approximated, as in theSDA, by the sum of the first two terms in a Legendre poly-nomial series expansion:Ld~r ,v̂ !514p E4pLd~r ,v̂ !dV134p E4pLd~r ,v̂8!~v̂8•v̂ !dV8514pwd~r !134pj ~r !•v̂ ~9!634 Journal of Biomedical Optics d May/June 2004 d Vol. 9 No. 3wherewd(r ) is the diffuse fluence rate andj ~r ! is the radiantflux.The improved accuracy offered by thed-P1 approximationstems from the addition of the Diracd function to both thesingle scattering phase function and the radiance approxtion. Thed function provides an additional degree of freedowell suited to accommodate collimated sources and higforward-scattering media. Thus the addition of thed functionrelieves substantially the degree of asymmetry that musprovided by the first-order term in the Legendre expansio62.3 Governing Equations and Boundary ConditionsSubstituting Eqs.~1!, ~6!, and~9! into the Boltzmann transporequation and performing balances in both the fluence ratethe radiant flux provides the governing equations in thed-P1approximation for a semi-infinite medium19:¹2wd~r !2meff2 wd~r !523ms* m trE~r ,ẑ!13g* ms* ¹E~r ,ẑ!• ẑ, ~10!j ~r !5213m tr@¹wd~r !23g* ms* E~r ,ẑ!ẑ#, ~11!where ms8[ms(12g) is the isotropic scattering coefficienm tr[(ma1ms8) is the transport coefficient, andmeff[(3mamtr)1/2 is the effective attenuation coefficient.Two boundary conditions are required to solve Eqs.~10!and ~11!. At the free surface of the medium, we require coservation of the diffuse flux component normal to the intface, which yields@wd~r !2Ah¹wd~r !• ẑ#uz50523Ahg* ms* E~r ,ẑ!uz50 ,~12!whereA5(11R2)/(12R1) andh52/3m tr . HereR1 andR2are the first and second moments of the Fresnel refleccoefficient for unpolarized light and are given byR152E01r F~n!ndn and R253E01r F~n!n2dn,~13!where n5v̂• ẑ, with ẑ defined as the inward pointing unvector normal to the surface. The details of this derivationprovided in Appendix A. Note that Eq.~12! represents anexact formulation for conservation of energy at the boundand avoids the approximations inherent in the use of extralated boundary conditions.30,31 The second boundary condtion requires the diffuse light field to vanish in regions faway from the source. Thus,wd~r !ur→`→0. ~14!2.4 Solutions for Planar and Gaussian BeamIrradiationThe total fluence rate is given by the sum of the collimatand diffuse fluence rates:w~r !5wc~r !1wd~r !. ~15!nceweeds.neral-isRadiative transport in the delta-P1 approximation . . .Fig. 1 Depiction of (a)planar and (b) Gaussian beam irradiation con-ditions.sd.or-by2.4.1 Collimated fluence rateFor either planar or Gaussian beam irradiation conditions, ashown in Fig. 1, the collimated fluence rate within the tissueis expressed in the formwc~r ,z!5E0~r !~12Rs!exp~2m t* z!. ~16!For planar irradiation,E0(r )5E0 while for Gaussian beamirradiation,E0(r )5E0 exp(22r 2/r 02), wherer 0 is the Gauss-ian beam radius, i.e., the radial location where the irradiafalls is 1/e2 of the maximum irradiance. Note thatE052P/pr 02, whereE0 denotes the peak irradiance andP is theincident power of the Gaussian laser beam. For generality,define a normalized collimated fluence ratew̄c asw̄c5wc~r ,z!E0~r !~12Rs!5exp~2m t* z!. ~17!2.4.2 Diffuse fluence rate for planar irradiationFor planar illumination the diffuse fluence rate is determinby solving Eq.~10! subject to the boundary conditions Eq~12! and ~14! and yieldswd~z!5E0~12Rs!@a exp~2m t* z!1b exp~2meffz!#,~18!wherea53ms* ~m t* 1g* ma!meff2 2m t*2 , ~19!andb52a~11Ahm t* !23Ahg* ms*~11Ahmeff!. ~20!The solution procedure is detailed in Appendix B. In a mananalogous to the collimated fluence rate, we define a normized diffuse fluence ratew̄d asw̄d~z!5wd~z!E0~12Rs!5a exp~2m t* z!1b exp~2meffz!.~21!2.4.3 Diffuse fluence rate for Gaussian beamirradiationFor Gaussian beam irradiation, the diffuse fluence rategiven bywd~r ,z!5E0~12Rs!E0`$g exp~2m t* z!1j exp@2~k21meff2 !1/2z#%J0~kr !kdk, ~22!whereg53ms* ~m t* 1g* ma!r 02 exp~2r 02k2/8!4~k21meff2 2m t*2!, ~23!j523g* ms* r 02 exp~2r 02k2/8!24g@~Ah!211m t* #4@~Ah!211~k21meff2 !1/2#,~24!and J0 is the zeroth-order Bessel function of the first kinThe solution procedure is detailed in Appendix C. The nmalized fluence rate for Gaussian beam irradiation is givenJournal of Biomedical Optics d May/June 2004 d Vol. 9 No. 3 635nce-tra-rst-en-ac-ion.o ara-g-Carp, Prahl, and Venugopalanw̄d~r ,z!5E0`$g exp~2m t* z!1j exp@2~k21meff2 !1/2z#%J0~kr !kdk. ~25!Numerical methods~MATLAB, MathWorks, Natick, Massa-chusetts! were employed to compute the definite integral inEqs.~22! and ~25!.2.5 Diffuse Reflectance for Planar IrradiationThe prediction of the diffuse reflectance provided by thed-P1approximation isRd52 j ~z!• ẑE0~12Rs!Uz50513m trE0~12Rs!F3g* ms* E0 exp~2mt* z!2dwd~z!dz GUz505w̄d~z!2A Uz50. ~26!2.6 Limiting CasesA unique feature of the solutions provided by thed-P1 ap-proximation is thatwd→0 in the limit of vanishing scattering,i.e., whenms8!ma . Thus in a medium where absorption isdominant m t* →ma and the total fluence rate is governedsolely by the collimated contribution, i.e.,lim(ms8 /ma)→0w~r ,z!5wc~r ,z!5E0~r !~12Rs!exp~2maz!.~27!Thus, unlike prevalent implementations of the SDA whereinthe collimated light source is replaced by a point sourceplaced at a depthz5(1/ms8) within the medium, thed-P1approximation correctly recovers Beer’s law in the limit of noscattering.For media in which scattering is dominant(ms8@ma orm t* @meff), the total fluence rate resulting from planar irradia-tion reduces tolim(ms8 /ma)→`w~z!5E0~12Rs!@~312A!exp~2meffz!22 exp~2m t* z!#. ~28!If we further consider this fluence rate in the far field~largez), Eq. ~28! reduces tolim(ms8 /ma)→`w~z!5E0~12Rs!~312A!exp~2meffz!for large z. ~29!Equation ~29! is equivalent to the fluence rate predictiongiven by the SDA.13 Thus, in the limit of high scattering, andaway from boundaries and collimated sources, the solutioprovided by thed-P1 approximation properly reduces to thatgiven by the SDA.636 Journal of Biomedical Optics d May/June 2004 d Vol. 9 No. 32.7 Optical Penetration DepthApart from the fluence rate profiles and diffuse reflectanresults offered by thed-P1 approximation, we are also interested in its predictions for the characteristic optical penetion depth ~OPD! in the tissue. In Fig. 2, we display twovariations of the OPD that we consider in this study. The fipenetration depth metricD is simply the depth at which thefluence rate falls to1/e of the incident fluence rate after accounting for losses due to specular reflection. The second petration depth metricD int is the depth at which all but1/e ofthe power of the laser radiation has been absorbed aftercounting for losses due to both specular and diffuse reflectFor generality, we normalize both these metrics relative tcharacteristic length scale. We choose(1/meff) for this lengthscale as it is the traditional definition for the optical penettion depth32 and is the length scale over which the homoenous solution to Eq.~10! decays. Accordingly we defineFig. 2 Graphical depiction of optical penetration depths (a) D and (b)D int .---Aenen-luntsant ofd-on-Radiative transport in the delta-P1 approximation . . .D̄[meffD and D̄ int[meffD int . ~30!2.8 Monte Carlo SimulationsWe performed Monte Carlo simulations for planar and Gaussian beam irradiation of semi-infinite media under both refrac-tive index matched and mismatched conditions. For this purpose we employed code derived from the Monte Carlo Multi-Layer ~MCML ! package written by Wang et al.8,9 thatcomputes the 3-D fluence rate distribution and spatially resolved diffuse reflectance corresponding to irradiation with alaser beam possessing either uniform or Gaussian profiles.Henyey-Greenstein phase function was utilized with a singlescattering asymmetry coefficient ofg50.9 unless stated oth-erwise. This value ofg was chosen as it is representative ofmany biological tissues.29 To approximate planar irradiationconditions we used a beam with a uniform irradiance profilewith radius r 05200l * , where l * [(1/m tr) is the transportmean free path. For Gaussian beam illumination, we setr 0 tothe desired1/e2 radius of the laser beam. To provide sufficientspatial resolution a minimum of 100 grid points were con-tained within one beam radius. Between107 and23109 pho-tons were launched for each simulation and resulted in fluencrate estimates with relative standard deviation of less tha0.1%.3 Results and Discussion3.1 Planar IlluminationFigures 3~a! and 3~b! provide normalized fluence rate profilespredicted by thed-P1 approximation and Monte Carlo simu-lations under planar illumination conditions for0.3<(ms8/ma)<100 and relative refractive indicesn5(n2 /n1)51.0 and 1.4, respectively. Note that the profiles are plottedagainst a reduced depth that is normalized relative to thtransport mean free pathl * . These figures also provide theerror of thed-P1 predictions relative to the Monte Carlo es-timates.Overall, the performance of thed-P1 approximation is im-pressive. The fluence rate is predicted with an error of<12%over the full range of optical properties. In the far field, themodel performance is exceptional for large(ms8/ma), de-grades slightly when scattering is comparable to absorptio(ms8.ma), and improves again when absorption dominatesscattering(ms8/ma&0.3). This behavior is expected. For large(ms8/ma) the prevalence of multiple scattering enables the dif-fuse component of thed-P1 approximation to provide an ac-curate description of the light field. However, when scatteringis still significant but(ms8/ma) is reduced, the decay of thelight field occurs on a spatial scale intermediate to that predicted by diffusion, i.e.,exp(2meffz), and that predicted by thetotal interaction coefficient, i.e.,exp(2mt*z). This results in anerror between thed-P1 model and the Monte Carlo estimatesthat increases with increasing depth. This is seen most notabfor the case of(ms8/ma)51 for which the error is largest inthe far field. Finally, for highly absorbing media, the overallaccuracy of thed-P1 approximation improves again becausethe contribution of collimated irradiance to the total light fieldincreases markedly and is well described by the modifiedBeer-Lambertlaw of Eq.~7!.y In the near field, the accuracy of thed-P1 approximationdegrades with increasing(ms8/ma). The origin of this lies inthe fact that increases in scattering result in increased amoof light backscattered toward the surface. This leads toincrease in the angular asymmetry in the diffuse componenthe light field near the surface which is not accurately moeled by a radiance approximation that simply employs a cFig. 3 Normalized fluence rate w̄ versus reduced depth (z/l* ) as pre-dicted by the d-P1 approximation (solid curves) and Monte Carlosimulations (symbols) for planar illumination under refractive index (a)matched (n51.0) and (b) mismatched (n51.4) conditions. Profilesare shown for (ms8/ma)5100 (s), 10 (* ), 3 (L), 1 (3), and 0.3 (d) withg50.9. Lower plots show the percentage error of the d-P1 predictionsrelative to the Monte Carlo simulations using the same symbols as themain plot.Journal of Biomedical Optics d May/June 2004 d Vol. 9 No. 3 637Carp, Prahl, and VenugopalanFig. 4 Normalized fluence rate w̄ versus reduced depth (z/l* ) as pre-dicted by the d-P1 approximation (solid curves) and Monte Carlosimulations (symbols) for planar illumination under refractive index (a)matched (n51.0) and (b) mismatched (n51.4) conditions. Profilesare shown for g50 (s), 0.3 (* ), 0.7 (3), and 0.9 (d) with (ms8/ma)5100. Lower plots show the percentage error of the d-P1 predictionsrelative to the Monte Carlo simulations using the same symbols as themain plot.s.y ofthecat-stant and the first-order Legendre polynomial. Thed-P1model performs worse forn51.4because the refractive indexmismatch introduces internal reflection that further enhancethe angular asymmetry of the light field near the surface638 Journal of Biomedical Optics d May/June 2004 d Vol. 9 No. 3However, when scattering is less prominent, the accuracthe fluence rate profiles is not as strongly dependent onrefractive index mismatch because there is less light backstered toward the surface.Fig. 5 Normalized fluence rate w̄ versus reduced depth (z/l* ) as pre-dicted by the d-P1 approximation (solid curves) and Monte Carlosimulations (symbols) for planar illumination under refractive index (a)matched (n51.0) and (b) mismatched (n51.4) conditions. Profilesare shown for g50 (s), 0.3 (* ), 0.7 (3), and 0.9 (d) with (ms8/ma)51. Lower plots show the percentage error of the d-P1 predictionsrelative to the Monte Carlo simulations using the same symbols as themain plot.Radiative transport in the delta-P1 approximation . . .Fig. 6 Diffuse reflectance Rd versus (ms8/ma) as predicted by the d-P1approximation (solid curves) and MC simulations (d) for planar illu-mination under refractive index (a) matched (n51.0) and (b) mis-matched (n51.4) conditions. Lower plots show the percentage errorof the d-P1 predictions relative to the MC simulations.onrrorWe also examined the influence of the single scatteringasymmetry coefficientg on thed-P1 model predictions forfixed values of(ms8/ma). Figures 4~a! and 4~b! show thevariation of the normalized fluence rate profiles for0<g<0.9 and(ms8/ma)5100 for n51 and 1.4, respectively. Fig-ures 5~a! and 5~b! show these same results in media with(ms8/ma)51. In the highly scattering case, the effect ofg isseen most prominently in the near field due to its impactthe boundary condition used in thed-P1 approximation.However, the effect is small and results in changes of the eFig. 7 Normalized optical penetration depths D̄[meffD (s) and D̄ int[meffDint (d) versus (ms8/ma) as predicted by the d-P1 approximation(solid curves) and MC simulations (symbols) for planar illuminationunder refractive index (a) matched (n51.0) and (b) mismatched (n51.4) conditions. Lower plots show the percentage error of the d-P1predictions relative to the MC simulations.Journal of Biomedical Optics d May/June 2004 d Vol. 9 No. 3 639le---eersur-sots.-forllerth-gustrateir-hefarldadythethehisenh-fed inre-theiornteans inis-Carp, Prahl, and Venugopalanbetweend-P1 and Monte Carlo~MC! estimates that do notexceed 4% relative to results found forg50.9. Note that for(ms8/ma)5100, the value ofg does not affect the predictionsin the far field as the SDA limit is applicable. As a result thedecay of the fluence rate profiles is governed byexp(2meffz)and is independent ofg for a fixed(ms8/ma). By contrast, for(ms8/ma)51, the variation ing affects the errors most promi-nently in the far field. This occurs because there is minimabackscattering due to the higher absorption in the mediumleading to a fluence rate profile whose decay is dependent og even for a fixed(ms8/ma). However, we again see that theeffect of g is minimal as the variations in the error are lessthan 7% even in the far field. Given that these error variationsare small and the fact that most soft biological tissues arstrongly forward scattering we show all remaining results fora value ofg50.9 ~Ref. 29!.Figures 6~a! and 6~b! present the variation of the diffusereflectanceRd with (ms8/ma) for n51.0 and 1.4, respectively.As in Fig. 3, there is good agreement for large(ms8/ma) in-dependent of the refractive index mismatch. Under indexmatched conditions, there is no internal reflection at the surface andRd is predicted with a relative error of68%. For arefractive index mismatch corresponding to a tissue-air interface, the model predictions degrade as(ms8/ma) is reduced.Specifically, relative errors exceed 15% for(ms8/ma),3.However, as(ms8/ma)→0 the model is bound to recover itsaccuracy since the diffuse component vanishes andRd→0 as(ms8/ma)→0. Moreover, for(ms8/ma),0.3 the amount of dif-fuse reflectance is negligible for all practical purposes. Thuswhile the relative error inRd may be large, the absolute erroris vanishingly small.To better characterize the variation in accuracy of thed-P1approximation with(ms8/ma) we examine the OPDs that char-acterize the fluence rate profiles. Figures 7~a! and 7~b! presentestimates for the normalized OPD metricsD̄[meffD andD̄ int[meffDint as predicted by thed-P1 approximation andMC predictions for1022<(ms8/ma)<104 under refractive in-dex matched(n51.0) and mismatched(n51.4) conditions,respectively.Note that under conditions of dominant absorption, i.e.,(ms8/ma)→0, meff→)ma . Thus both D̄ and D̄ int approach(1/ma)(meff)5) as(ms8/ma)→0. This result is confirmed inFigs. 7~a! and 7~b!. In the limit of high scattering, i.e.,(ms8/ma)→`, inspection of Eq.~29! reveals that the value ofD̄ is dependent on the refractive index mismatch through thboundary parameterA. Setting Eq.~29! equal toE0~12Rs!/eand solving we find thatD̄511ln~312A!. Thus, for(ms8/ma)→`, thed-P1 approximation predicts thatD̄→2.61 and 3.19for n51.0 and 1.4, respectively. By contrast, a similar analy-sis reveals thatD̄ int is not sensitive to the refractive indexmismatch andD̄ int→1 as (ms8/ma)→`. These asymptoticlimits predicted by thed-P1 model are confirmed by the re-sults shown in Figs. 7~a! and 7~b!. Overall thed-P1 predic-tions for the optical penetration depth are impressive andmatch the MC estimates to within64% over the entire rangeof (ms8/ma). The highest relative errors occur at(ms8/ma).1 as expected from the characteristics of the fluence rat640 Journal of Biomedical Optics d May/June 2004 d Vol. 9 No. 3nprofiles shown in Fig. 3. Better accuracy is observed forD̄(62%) than for D̄ int (64%). This is due to the strongeimpact that underestimation of the fluence rate near theface has on the determination ofD̄ int .3.2 Gaussian Beam IlluminationFigures 8~a! and 8~b! provide normalized fluence rate profilealong the beam centerline(r 50) as predicted by thed-P1approximation and MC simulations at(ms8/ma)5100 forbeam radii r 05100l * , 30l * , 10l * , 3l * , and 1l * with n51.0and 1.4, respectively. The errors of thed-P1 predictionsrelative to the MC estimates are shown below the mainplThe fluence rate along the beam centerline forr 05100l * dif-fers by less than60.5% from that produced by planar irradiation. For bothn51.0 and 1.4, thed-P1 approximationprovides good accuracy relative to the MC predictionsbeam radiir 0.3l * (617% in the near field,65% in the farfield!. However, the model accuracy degrades for smabeam radii and reaches625% for r 05 l * . This is expectedgiven that the diffusion model breaks down when lengscales comparable tol * are considered.Figures 9~a! and 9~b! provide results for the more challenging case of(ms8/ma)51. Due to the reduced scatterindispersion that occurs in media of higher absorption, one mconsider much smaller beam diameters before the fluenceprofiles along the center differ noticeably from the planarradiation case. Specifically, for(ms8/ma)51, the fluence ratealong the beam centerline forr 0530l * differs by less than60.5% from that produced by planar irradiation. Forr 0.3l * , errors in the fluence rate predictions provided by td-P1 model relative to the MC estimates are63% in thenear field and622% in the far field. However, forr 05 l * ,the fluence rate is overestimated by nearly 100% in thefield. While a 100% error may appear striking, one shounotice that this occurs once the fluence rate has alredropped by more than two orders of magnitude relative tosurface value. Thus, while the percentage error is large,error with respect to the overall energy balance is small. Tlarge relative error for small beam radii is not surprising givthe great difficulty that low-order functional expansion metods have in modeling the light field whenms8.ma . In the farfield, the accuracy of thed-P1 model is nearly independent othe refractive index for the same reasons as those discussSec. 3.1.Figures 10~a! and 10~b! provide the normalized OPDD̄along the beam centerline for Gaussian irradiation as pdicted by thed-P1 model and MC simulations for1022<(ms8/ma)<104 and beam radiir 051 – 100l * with n51.0and 1.4, respectively. Corresponding results forD̄ int are pre-sented similarly in Figs. 11~a! and 11~b!. The OPDs deter-mined in the 1-D case are included for comparison as arecorresponding relative errors. The expected limiting behavfor (ms8/ma)→0 is identical to that in the planar irradiatiocase and thus bothD̄ and D̄ int converge to). For large(ms8/ma) the decay of the fluence rate with depth for finibeam illumination occurs on a spatial scale smaller thexp(2meffz) because as the incident laser beam propagatethe medium, optical scattering results in significant lateral dRadiative transport in the delta-P1 approximation . . .Fig. 8 Normalized fluence rate along the beam centerline w̄(r50)versus reduced depth (z/l* ) as predicted by the d-P1 approximation(solid curves) and MC simulations (symbols) for Gaussian beam illu-mination under refractive index (a) matched (n51) and (b) mis-matched (n51.4) conditions. Profiles are shown for (ms8/ma)5100with r05100l* (s), 30l* (* ), 10l* (L), 3l* (3), 1l* (d), and g50.9. Lower plots show the percentage error of the d-P1 predictionsrelative to the MC simulations.-heestpersion from the high fluence region along the beam centerline to the periphery. ThusD̄, D̄ int→0 as(ms8/ma)→`. Thed-P1 predictions forD̄ andD̄ int track the MC estimates well,with errors of less than64% in D̄ and620% in D̄ int for thesmallest beam radius studied(r 05 l * ). Once again, the larg-est errors occur forms8.ma and D̄ is predicted more accu-rately thanD̄ int . Both of these features are consistent with tfluence rate profiles shown in Figs. 8 and 9 where the largerrors are observed close to the surface(z,2l * ) and forms8.ma .Fig. 9 Normalized fluence rate along the beam centerline w̄(r50)versus reduced depth (z/l* ) as predicted by the d-P1 approximation(solid curves) and MC simulations (symbols) for Gaussian beam illu-mination under refractive index (a) matched (n51.0) and (b) mis-matched (n51.4) conditions. Profiles are shown for (ms8/ma)51 withr0530l* (s), 10l* (L), 3l* (3), 1l* (d), and g50.9. Lower plotsshow the percentage error of the d-P1 predictions relative to the MCsimulations.Journal of Biomedical Optics d May/June 2004 d Vol. 9 No. 3 641Carp, Prahl, and VenugopalanFig. 12 (a) Color contour plot of the normalized fluence rate w̄(r,z) as predicted by both the d-P1 approximation (solid contours and color) and MCsimulations (dashed contours) for Gaussian beam irradiation with r053l* in media with (ms8/ma)5100 for g50.9 under refractive index mis-matched conditions (n51.4); and (b) relative error between d-P1 approximation and MC simulations.Fig. 13 (a) Color contour plot of the normalized fluence rate w̄(r,z) as predicted by both the d-P1 approximation (solid contours and color) and MCsimulations (dashed contours) for Gaussian beam irradiation with r053l* in media with (ms8/ma)53 for g50.9 under refractive index mismatchedconditions (n51.4); and (b) relative error between d-P1 approximation and MC simulations.642 Journal of Biomedical Optics d May/June 2004 d Vol. 9 No. 3Radiative transport in the delta-P1 approximation . . .Fig. 10 Normalized optical penetration depth D̄ versus (ms8/ma) aspredicted by the d-P1 approximation (solid curves) and MC simula-tions (symbols) along the beam centerline for Gaussian beam illumi-nation for g50.9 with r05100l* (s), 30l* (* ), 10l* (L), 3l* (3), and1l* (d) under refractive index (a) matched (n51.0) and (b) mis-matched (n51.4) conditions. The optical penetration depth for planarillumination predicted by the d-P1 approximation is plotted as adashed curve. Lower plots shows the percentage error of the d-P1predictions relative to the MC simulations.n-MC.Figure 12~a! provides a color contour plot representing the2-D fluence rate distribution for a Gaussian beam of radiusr 053l * with (ms8/ma)5100andn51.4. The solid isofluencerate contours and the color map correspond to the predictioprovided by thed-P1 approximation while the dashed isofluence rate contours represent predictions given by thesimulations. Figure 12~b! provides the 2-D distribution of therelative errors between thed-P1 predictions and the MCsimulations. Thus, thed-P1 and MC contours shown in FigFig. 11 Normalized optical penetration depth D̄ int versus (ms8/ma) aspredicted by the d-P1 approximation (solid curves) and MC simula-tions (symbols) along the beam centerline for Gaussian illuminationfor g50.9 with r05100l* (s), 30l* (* ), 10l* (L), 3l* (3), and 1l*(d) under refractive index (a) matched (n51.0) and (b) mismatched(n51.4) conditions. The optical penetration depth for planar illumi-nation predicted by the d-P1 approximation is plotted as a dashedline. Lower plots show the percentage error of the d-P1 predictionsrelative to the MC simulations.Journal of Biomedical Optics d May/June 2004 d Vol. 9 No. 3 643--iaellri.rtrertheofthetheres,taintoia-ura-aysres-ec-sionnst isdic-e-en-ncetheeCarp, Prahl, and Venugopalan12~a! provides some indication of the errors in penetrationdepth that one makes when using thed-P1 approximation,while Fig. 12~b! provides the errors in the actual optical do-simetry.The quality of thed-P1 predictions are excellent; the errorin the fluence rate relative to the MC estimates never exceed20% and is less than 10% over the vast majority of the domain. In the axial direction, the maximum errors occur in thenear field close to the boundary, while in the radial direction,they occur along the beam centerline. This is expected because it is at these locations where the spatial gradients anangular asymmetry of the light field are greatest. Figures13~a! and 13~b! provide plots under identical irradiation con-ditions for a turbid medium with(ms8/ma)53. In Fig. 13~a!we see similar errors in the location of the isofluence ratecontours when comparing thed-P1 approximation relative tothe MC predictions. However, in Fig. 13~b!,we observe adifferent spatial pattern and magnitude of the fluence rate errors incurred when using thed-P1 approximation rather thana MC estimate. As in Fig. 12~b!, the maximum errors in theradial direction occur along the beam centerline. However, inthe axial direction, the maximum errors reside in the far fieldand appear to be increasing with depth. This is similar tothe planar irradiation case and occurs because the spatscale for the decay of the fluence rate with depth liesbetweenexp(2meffz) andexp(2mt*z); thereby leading to poorpredictions by thed-P1 approximation in the far field underthese conditions. It is important to note that examination ofd-P1 predictions at radial locations away from the centerlinereveals equivalent, if not better, accuracy in both fluence ratprofiles and OPD metrics. For example, for Gaussian beamradii r 0.3l * , the errors in bothD̄ and D̄ int at the radiallocation r 5r 0 are<5 and<8%, respectively, over the fullrange of(ms8/ma). This result is consistent with the errors ofthe full fluence rate distributions shown in Figs. 12 and 13.3.3 Gaussian Beam versus Planar IrradiationTreatmentAs is evident from the results, the use of laser beams of smadiameter significantly alters the fluence rate profile and opticapenetration depth. For example, Gaussian irradiation of a medium with (ms8/ma)5100 using a beam radius ofr 053l *results in a fluence rate that is only;50% of that achievedusing planar illumination. Moreover, the reduction in bothfluence rate and OPD for decreasing beam diameters is moprominent in media with large(ms8/ma) because the scatteringenhances lateral dispersion of the collimated radiation~Figs.8–13!. However, the Gaussian beam expressions are a bmore formidable than those for the case of planar irradiationAs a result, for simplicity and convenience, it may be usefulto determine the conditions under which the results of a planairradiation analysis provides sufficiently accurate predictionsalong the centerline of a Gaussian beam. This may obviate thneed to use the more complex expressions correspondingGaussian beam irradiation in some cases.Figure 14 provides these results in the form of a contouplot showing the percentage difference between the fluencrate predictions given by thed-P1 approximation for Gauss-ian beam irradiation along the centerline compared to planairradiation as a function of both normalized beam radius644 Journal of Biomedical Optics d May/June 2004 d Vol. 9 No. 3s-dll-eteo(r 0 / l * ) and optical properties(ms8/ma). Contours are pro-vided for differences of 1, 3, 10, and 30% forn51.0 ~solidcontours! and 1.4~dashed contours!, respectively. These re-sults indicate that as absorption becomes more dominant,centerline fluence rate profiles produced by laser beamssmaller diameter can be adequately approximated usingplanar irradiation predictions. This can also be seen inOPD results shown earlier in Figs. 10 and 11. In these figuwe observed that for a given beam radius, there is a cervalue of (ms8/ma) above which the OPDs correspondingGaussian irradiation drop below the OPDs for planar irradtion. We note that this value of(ms8/ma) becomes lower assmaller beam diameters are used. Note also that the inacccies incurred in using the planar irradiation results are alwlower for the index-matched case. This is because the pence of a refractive index mismatch results in internal refltion at the tissue-air interface that enhances lateral disperof the light field. This additional source of dispersion hastethe need for the use of a radiative transport model thageometrically faithful to the irradiation conditions.4 ConclusionWe have shown that thed-P1 approximation to the Boltz-mann transport equation provides remarkably accurate pretions of light distribution and energy deposition in homogneous turbid semi-infinite media. Examination of thfunctional expressions involved in thed-P1 approximationreveals proper asymptotic behavior in the limits of absorptioand scattering-dominant media. Comparison of the fluerate and optical penetration depth predictions given byd-P1 approximation with MC simulations demonstrate thgreater fidelity and accuracy of thed-P1 model relative to thestandard diffusion approximation.Fig. 14 Contours for the error incurred in predicting fluence rate pro-files along the centerline of a Gaussian laser beam of normalizedradius r0 /l* as a function of (ms8/ma) when using d-P1 predictions forthe planar irradiation case for n51.0 (dashed) and n51.4 (solid).lne-rlsientofingashtalof-Radiative transport in the delta-P1 approximation . . .The availability of an analytic light transport model pro-viding accurate optical dosimetry predictions is an invaluabletool for the biomedical optics community. By providing ourresults in terms of dimensionless quantities, they can be useto rapidly estimate the fluence rate distributions and opticapenetration depths generated by a wide range of irradiatioconditions and tissue optical properties. Thus beyond a greattheoretical understanding of the significant gains to be realized through the use of thed-P1 approximation over the stan-dard diffusion approximation, these figures provide the bio-medical optics community with charts that can be used forapid lookup and estimation of light-transport related quanti-ties.5 Appendix A Derivation of Surface BoundaryConditions in the d -P1 ApproximationThe governing equations of thed-P1 approximation are~seeSec. 2!:¹2wd~r !23mam trwd~r !523ms* m trE~r ,ẑ!13g* ms* ¹E~r ,ẑ!• ẑ ~31!j ~r !5213m tr@¹wd~r !23g* ms* E~r ,ẑ!ẑ#, ~32!where r is the position in the medium,ẑ is the unit vectorcolinear with the direction of the collimated source,E(r ,ẑ) isthe irradiance distribution of the collimated source,ma is theabsorption coefficient,m tr[ma1ms8 is the transport coeffi-cient with ms8 being the isotropic scattering coefficient,g* isthe single scattering asymmetry coefficient of theP1 portionof the d-P1 phase function, andms* [ms(12 f ) is a reducedscattering coefficient. Selection off and g* depends on theselection of the phase function as described in Sec. 2.1.Two boundary conditions are required to solve Eq.~31!.Requiring conservation of the diffuse flux component normato the interface, we obtain6,23Ev̂• ẑ>0Ld~r ,v̂ !~v̂• ẑ!dv̂5Ev̂• ẑ,0Ld~r ,v̂ !r F~2v̂• ẑ!~2v̂• ẑ!dv̂,~33!where ẑ is the inward-pointing surface normal, andr F(2v̂• ẑ) is the Fresnel reflection coefficient for unpolarizedlight. The preceding condition can be described in words aequating the amount of diffuse light that travels upward(v̂• ẑ,0) and gets internally reflected at the interface withthe amount of diffuse light traveling downward(v̂• ẑ>0)from the interface.Substituting the approximation for the diffuse fluence rategiven by Eq. ~9! and using Eq.~32! to eliminate j ~r !, weobtain the following form for the surface boundary conditionin the d-P1 approximation:@wd~r !2Ah¹wd~r !• ẑ#uz50523Ahg* ms* E~r ,ẑ!uz50 ,~34!drwhere A5(11R2)/(12R1) and h52/3m tr . This result isidentical to that provided by Eq.~12!. HereR1 andR2 are thefirst and second moments of the Fresnel reflection coefficfor unpolarized light, as given by Eq.~13!.Note that in many implementations of the SDA,A is ap-proximated instead byA'(11R1)/(12R1). While this isstrictly incorrect, it results in slightly better approximationsthe fluence rate in the near field at the expense of providworse fluence rate approximations in the far field as wellviolating conservation of energy when integrating the ligfield over the entire volume. The following cubic polynomiprovides an estimate forA5(11R2)/(12R1) that typicallydiffers from the exact value by less than 1%:23A~n!520.13755n314.3390n224.90366n11.6896.~35!6 Appendix B Solution of the d -P1Approximation for Planar Illuminationof a Semi-Infinite MediumFor planar illuminationthe source term is given byE~z,v̂ !5E0~12Rs!exp~2m t* z!d~12v̂• ẑ!, ~36!whereE0 is the irradiance,v̂ is the unit direction vector, andẑ is the inward pointing unit vector normal to the surfacethe medium and is colinear with thez coordinate axis. Sub-stituting Eq.~36! into Eq.~10!, we obtain the governing equation for a planar geometry:d2wd~z!dz2 23mam trwd~z!523ms* ~m t* 1g* ma!E0~12Rs!exp~2m t* z!.~37!The boundary conditions for the 1-D case reduce toS wd2Ahdwd~z!dz D Uz50523Ahg* ms* E0~12Rs!,~38!wd~z!uz→`→0. ~39!The solution to Eq.~37! satisfying the Eqs.~38! and ~39! iswd~z!5E0~12Rs!@a exp~2m t* z!1b exp~2meffz!#,~40!wherea53ms* ~m t* 1g* ma!meff2 2m t*2 ~41!andb52a~11Ahm t* !23Ahg* ms*~11Ahmeff!. ~42!These results are identical to that provided by Eqs.~18! to~20!.Journal of Biomedical Optics d May/June 2004 d Vol. 9 No. 3 645.hCarp, Prahl, and Venugopalan7 Appendix C Solution of the d -P1Approximation for Gaussian Beam Illuminationof a Semi-Infinite MediumThe source term for a Gaussian beam profile is given byE~r ,z!5E0~12Rs!exp~2m t* z!expS 22r 2r 02 D , ~43!wherer 0 is the1/e2 beam radius, andE052P/(pr 02), whereP is the power of the laser beam. The governing equation incylindrical coordinates has the form1r]]r S r]wd~r ,z!]r D1]2wd~r ,z!]z2 2meff2 wd~r ,z!523ms* ~m t* 1g* ms* !E~r ,z!, ~44!subject to the boundary conditions:S wd2Ah]wd]z D Uz50523Ahg* ms* E~r ,z!uz50 , ~45!]wd~r ,z!]r Ur 5050, ~46!wd~r ,z!uz→`→0, ~47!wd~r ,z!ur→`→0. ~48!The solution procedure begins by assuming that bothwd(r ,z) and the right-hand side of Eq.~44! can be written asHankel transforms of two functionsf (k,z) and u(k,z), re-spectively, i.e.,E0`f ~k,z!J0~kr !kdk5wd~r ,z! ~49!andE0`u~k,z!J0~kr !kdk523ms* ~m t* 1g* ma!E0~12Rs!3exp~2m t* z!expS 22r 2r 02 D , ~50!whereJ0 is the zeroth-order Bessel function of the first kind.Substituting Eqs.~49! and ~50! into Eq. ~44! we obtain1r]]r F r]]r E0`f ~k,z!J0~kr !kdkG1]2]z2 E0`f ~k,z!J0~kr !kdk2meff2 E0`f ~k,z!J0~kr !kdk5E0`u~k,z!J0~kr !kdk.~51!We note that the first term of Eq.~51! appears in the Bessel’sequation:646 Journal of Biomedical Optics d May/June 2004 d Vol. 9 No. 31r]]rr]]rJ0~kr !1k2J0~kr !50, ~52!for which J0 is a solution. Thus Eq.~51! can be rewritten byadding and subtractingk2J0(kr) on the left-hand side of Eq~52!, which yieldsE0`~2k22meff2 ! f ~k,z!J0~kr !kdk1E0` ]2]z2 J0~kr ! f ~k,z!kdk5E0`u~k,z!J0~kr !kdk. ~53!Using a table of Hankel transforms,33 u(k,z) can be chosensuch that Eq.~50! is satisfied, namely,]2]z2 f ~k,z!2~k21meff2 ! f ~k,z!523ms* ~m t* 1g* ma!E0~12Rs!r 0243expS 2r 02k28 Dexp~2m t* z!. ~54!The boundary conditions in(k,z) space are obtained througHankel transformation of Eqs.~45! to ~48!:F ]]zf ~k,z!21Ahf ~k,z!GUz50534g* ms* E0~12Rs!r 023expS 2r 02k28 D , ~55!andf ~k,z!uz→`→0. ~56!Solving the Eq.~54! for f (k,z) and substitution of theresults into Eq.~49! gives the following form forwd(r ,z):wd~r ,z!5E0~12Rs!E0`$g exp~2m t* z!1j exp@2~k21meff2 !1/2z#%J0~kr !kdk, ~57!whereg53ms* ~m t* 1g* ma!r 02 exp~2r 02k2/8!4~k21meff2 2m t*2!~58!andj523g* ms* r 02 exp~2r 02k2/8!24g@~Ah!211m t* #4@~Ah!211~k21meff2 !1/2#.~59!These results are identical to that provided by Eqs.~22! to~24!.f,hfu-ind es-u-ortngi--y infabl-IEhtndton,’’ewn,-y,’’ofd-neorys,’’sueRadiative transport in the delta-P1 approximation . . .AcknowledgmentsWe thank Fre´déric Bevilacqua, Carole Hayakawa, ArnoldKim, and Jerry Spanier for helpful and stimulating discus-sions. We are grateful to the National Institutes of Health forsupport via both the Laser Microbeam and Medical Programunder Grant No. P41-RR-01192 and the National Institute oBiomedical Imaging and Bioengineering under Grant No.R01-EB-00345.References1. W. M. Star, ‘‘Light dosimetryin vivo,’’ Phys. Med. Biol.42, 763–787~1997!.2. M. S. Patterson, B. C. Wilson, and D. R. Wyman, ‘‘The propagationof optical radiation in tissue I. Models of radiation transport and theirapplication,’’Lasers Med. Sci.6, 155–168~1991!.3. S. L. Jacques and L. Wang, ‘‘Monte Carlo modeling of light transportin tissues,’’ Chap. 4 inOptical-Thermal Response of Laser-IrradiatedTissue, A. J. Welch and M. J. C. van Gemert, Eds., pp. 73–99, Ple-num, New York~1995!.4. S. A. Prahl, M. J. C. van Gemert, and A. J. Welch, ‘‘Determining theoptical properties of turbid media by using the adding-doublingmethod,’’Appl. Opt.32, 559–568~1993!.5. S. A. Prahl, ‘‘The adding-doubling method,’’ Chap. 5 inOptical-Thermal Response of Laser-Irradiated Tissue, A. J. Welch and M. J.C. van Gemert, Eds., pp. 101–125, Plenum, New York~1995!.6. W. M. Star, ‘‘Diffusion theory of light transport,’’ Chap. 6 inOptical-Thermal Response of Laser-Irradiated Tissue, A. J. Welch and M. J.C. van Gemert, Eds., pp. 131–206, Plenum, New York~1995!.7. R. A. Forrester and T. N. K. Godfrey, ‘‘MCNP—a general MonteCarlo code for neutron and photon transport,’’ inMethods and Appli-cations in Neutronics, Photonics, and Statistical Physics, R. Alcouffe,R. Dautray, A. Forster, G. Ledanois, and B. Mercier, Eds., pp. 33–47Springer-Verlag,~1983!.8. L. H. Wang, S. L. Jacques, and L. Q. Zheng, ‘‘MCML—Monte-Carlomodeling of light transport in multilayered tissues,’’Comput. Meth-ods Programs Biomed.47~2!, 131–146~1995!.9. L. H. Wang, S. L. Jacques, and L. Q. Zheng, ‘‘CONV—convolutionfor responses to a finite diameter photon beam incident on multi-layered tissues,’’Comput. Methods Programs Biomed.54~3!, 141–150 ~1997!.10. A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, ‘‘Comparison offinite-difference transport and diffusion calculations for photon mi-gration in homogeneous and heterogeneous tissues,’’Phys. Med. Biol.43~5!, 1285–1302~1998!.11. R. A. J. Groenhuis, H. A. Ferwerda, and J. J. Tenbosch, ‘‘Scatteringand absorption of turbid materials determined from reflection mea-surements. 1. Theory,’’Appl. Opt.22~16!, 2456–2462~1983!.12. L. I. Grossweiner, J. L. Karagiannes, P. W. Johnson, and Z. Y. Zhang‘‘Gaussian-beam spread in biological tissues,’’Appl. Opt. 29~3!,379–383~1990!.13. A. Ishimaru,Wave Propagation and Scattering in Random Media,IEEE Press, New York~1997!.14. B. Q. Chen, K. Stamnes, and J. J. Stamnes, ‘‘Validity of the diffusionapproximation in bio-optical imaging,’’Appl. Opt. 40~34!, 6356–6366 ~2001!.15. A. D. Kim and A. Ishimaru, ‘‘Optical diffusion of continuous-wave,pulsed, and density waves in scattering media and comparisons wit,radiative transfer,’’Appl. Opt.37~22!, 5313–5319~1998!.16. T. Spott and L. O. Svaasand, ‘‘Collimated light sources in the difsion approximation,’’Appl. Opt.39~34!, 6453–6465~2000!.17. C. M. Gardner, S. L. Jacques, and A. J. Welch, ‘‘Light transporttissue: accurate expressions for one-dimensional fluence rate ancape function based upon Monte Carlo simulation,’’Lasers Surg.Med.18~2!, 129–138~1996!.18. L. H. Wang and S. L. Jacques, ‘‘Hybrid model of Monte-Carlo simlation and diffusion-theory for light reflectance by turbid media,’’J.Opt. Soc. Am. A10~8!, 1746–1752~1993!.19. V. Venugopalan, J. S. You, and B. J. Tromberg, ‘‘Radiative transpin the diffusion approximation: an extension for highly absorbimedia and small source-detector separations,’’Phys. Rev. E58~2!,2395–2407~1998!.20. D. Dickey, O. Barajas, K. Brown, J. Tulip, and R. B. Moore, ‘‘Radance modelling using the P3 approximation,’’Phys. Med. Biol.43,3559–3570~1998!.21. D. J. Dickey, R. B. Moore, D. C. Rayner, and J. Tulip, ‘‘Light dosimetry using the P3 approximation,’’Phys. Med. Biol.46~9!, 2359–2370 ~2001!.22. E. L. Hull and T. H. Foster, ‘‘Steady-state reflectance spectroscopthe P-3 approximation,’’J. Opt. Soc. Am. A18~3!,584–599~2001!.23. S. A. Prahl, ‘‘Light transport in tissue,’’ PhD Thesis, University oTexas at Austin~Dec. 1988!.24. W. M. Star, ‘‘Comparing theP3-approximation with diffusion theoryand with Monte-Carlo calculations of the light propagation in a slgeometry,’’ in Dosimetry of Laser Radiation in Medicine and Bioogy, G. J. Müller and D. H. Sliney, Eds., Vol. IS5, pp. 146–154, SPOptical Engineering Press, Bellingham, WA~1989!.25. W. M. Star, J. P. A. Marijnissen, and M. J. C. van Gemert, ‘‘Ligdosimetry in optical phantoms and in tissues. I. Multiple flux atransport theory,’’Phys. Med. Biol.33, 437–454~1988!.26. J. H. Joseph, W. J. Wiscombe, and J. A. Weinman, ‘‘Delta-Eddingapproximation for radiative flux-transfer,’’J. Atmos. Sci.33~12!,2452–2459~1976!.27. S. A. Prahl, ‘‘The diffusion approximation in three dimensionsChap. 7 inOptical-Thermal Response of Laser-Irradiated Tissue, A.J. Welch and M. J. C. van Gemert, Eds., pp. 207–231, Plenum, NYork ~1995!.28. S. T. Flock, M. S. Patterson, B. C. Wilson, and D. R. Wyma‘‘Monte-Carlo modeling of light-propagation in highly scattering tissues. 1. Model predictions and comparison with diffusion theorIEEE Trans. Biomed. Eng.36~12!, 1162–1168~1989!.29. S. A. Prahl, S. L. Jacques, and C. A. Alter, ‘‘Angular dependencehene laser light scattering by human dermis,’’Lasers Life Sci.1,309–333~1987!.30. R. C. Haskell, L. O. Svaasand, T.-T. Tsay, T.-C. Feng, M. S. McAams, and B. J. Tromberg, ‘‘Boundary conditions for the diffusioequation in radiative transfer,’’J. Opt. Soc. Am. A11~10!, 2727–2741~1994!.31. A. H. Hielscher, S. L. Jacques, L. Wang, and F. K. Tittel, ‘‘Thinfluence of boundary conditions on the accuracy of diffusion thein time-resolved reflectance spectroscopy of biological tissuePhys. Med. Biol.40~11!, 1957–1975~1995!.32. S. L. Jacques, ‘‘Role of tissue optics and pulse duration on tiseffects during high-power laser irradiation,’’Appl. Opt. 32, 2447–2454 ~1993!.33. I. S. Gradshteyn and I. M. Ryzhik,Table of Integerals, Series, andProducts, Academic Press, New York~1980!.Journal of Biomedical Optics d May/June 2004 d Vol. 9 No. 3 647
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O número médio de carros que chegam a um posto de informações é igual a 10 carros/hora.
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standard deviation – The Stats Geek
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Annuity: Annuities and NPV: Calculating the Value of Consistent Cash Flows - FasterCapital
Annuity: Annuities and NPV: Calculating the Value of Consistent Cash Flows
1. Introduction to Annuities and Their Importance in Financial Planning
Annuities stand as a cornerstone in the edifice of financial planning, offering a structured approach to managing one's financial future. They are essentially financial products that promise to pay
out a fixed stream of payments to an individual, typically used as a reliable income stream for retirees. The allure of annuities lies in their ability to mitigate the risk of outliving one's savings
, a concern that is increasingly pressing in an era where longevity is on the rise. By transforming a lump sum of money into a predictable and steady cash flow, annuities can serve as a bulwark
against the unpredictable nature of market-based investments, providing a semblance of stability in the often tumultuous journey of financial management.
From the perspective of an individual investor, annuities can be seen as a form of insurance against the uncertainty of life expectancy and market volatility. For retirees, the guarantee of a steady
income can be the bedrock upon which a secure retirement is built. Financial advisors, on the other hand, may view annuities as a tool for diversifying a client's portfolio, offering a counterbalance
to more aggressive, equity-based investments. Insurance companies, issuing these annuity contracts, manage the collective pool of funds, investing them in various assets to ensure they can meet their
long-term obligations to annuitants.
Here are some in-depth insights into annuities and their role in financial planning:
1. Types of Annuities: Annuities come in various forms, each tailored to different financial needs and stages of life. Immediate annuities begin paying out soon after investment, while deferred
annuities accumulate value before starting payouts. fixed annuities offer guaranteed returns, whereas variable annuities allow for investment in sub-accounts similar to mutual funds, with returns
dependent on market performance.
2. Tax Advantages: Annuities provide tax-deferred growth, meaning that taxes on investment gains are not paid until the money is withdrawn. This can be particularly advantageous for individuals in
higher tax brackets during their working years, potentially leading to significant tax savings.
3. Riders and Options: Many annuities come with additional features, known as riders, which can provide enhanced benefits such as guaranteed minimum withdrawal benefits, death benefits, or options
for long-term care coverage. These riders can be customized to fit individual financial goals and needs.
4. Calculating Present Value: Understanding the net present value (NPV) of an annuity is crucial in assessing its worth. The NPV is the sum of the present values of all future cash flows, discounted
at an appropriate interest rate. For example, if one is offered a $1,000 per month annuity for 10 years, and using a discount rate of 5%, the NPV can be calculated using the formula:
$$ NPV = \sum_{t=1}^{120} \frac{1000}{(1+0.05)^t} $$
5. comparing Investment options: When considering an annuity, it's important to compare it with other investment options. For instance, investing a lump sum in a diversified portfolio may offer
higher potential returns but comes with greater risk. An annuity provides a guaranteed income but may offer lower returns in exchange for security and peace of mind.
6. Inflation Consideration: Inflation can erode the purchasing power of fixed annuity payments over time. Some annuities offer inflation protection through increasing payment options, although these
typically start with lower initial payments.
7. liquidity and Access to funds: annuities often have surrender charges and limited liquidity, especially in the early years of the contract. It's important to consider the need for access to funds
before committing to an annuity.
8. role in Estate planning: Annuities can play a role in estate planning, with certain types allowing for the transfer of wealth to beneficiaries. However, the tax implications for heirs should be
carefully considered.
By integrating annuities into a broader financial plan, individuals can harness their unique properties to address specific financial goals and concerns. Whether seeking stability, tax efficiency, or
a hedge against inflation, annuities can be tailored to fit a myriad of financial landscapes, making them an indispensable tool in the financial planner's toolkit.
Introduction to Annuities and Their Importance in Financial Planning - Annuity: Annuities and NPV: Calculating the Value of Consistent Cash Flows
2. Understanding the Basics of Net Present Value (NPV)
Net Present Value (NPV) is a fundamental concept in finance and investment that serves as a cornerstone for understanding the value of money over time. It is predicated on the principle that a dollar
today is worth more than a dollar tomorrow due to its potential earning capacity. This concept is particularly relevant when assessing the value of annuities—consistent cash flows received or paid
out over a period of time. NPV helps investors and financial analysts determine the current value of a series of future cash flows, taking into account the time value of money. By discounting these
future cash flows back to their present value, one can make informed decisions about the viability and profitability of investments, especially when comparing different projects with varying cash
flow patterns.
From the perspective of an individual investor, NPV is a tool to gauge whether an investment in an annuity will yield a positive return compared to other investment opportunities. For a corporation,
NPV is instrumental in capital budgeting decisions, helping to choose between projects by evaluating which ones are likely to contribute the most value to the company. From a financial advisor's
viewpoint, understanding NPV is essential for advising clients on portfolio management and retirement planning, ensuring that their long-term income streams are optimized.
Here's an in-depth look at the components and calculations involved in determining NPV:
1. Cash Flows: The series of cash inflows and outflows associated with the investment. For an annuity, these are the periodic payments received (ordinary annuity) or made (annuity due).
2. Discount Rate: The rate of return that could be earned on an investment in the financial markets with similar risk. It reflects the opportunity cost of investing capital elsewhere and is used to
discount future cash flows to their present value.
3. Time Periods: The number of time periods over which the annuity payments are made. This could be years, quarters, or months, depending on the annuity's terms.
4. Formula: The NPV calculation involves summing the present values of all cash flows:
$$ NPV = \sum_{t=1}^{n} \frac{C_t}{(1 + r)^t} - C_0 $$
Where \( C_t \) is the cash flow at time \( t \), \( r \) is the discount rate, and \( C_0 \) is the initial investment.
5. Sign Convention: A positive NPV indicates that the present value of cash flows exceeds the initial investment, suggesting a profitable investment. Conversely, a negative NPV implies that the cash
flows do not cover the initial outlay, signaling a potential loss.
To illustrate, consider an annuity that promises to pay $1,000 annually for 5 years. If the discount rate is 5%, the NPV of this annuity can be calculated as follows:
$$ NPV = \frac{1000}{(1 + 0.05)^1} + \frac{1000}{(1 + 0.05)^2} + \frac{1000}{(1 + 0.05)^3} + \frac{1000}{(1 + 0.05)^4} + \frac{1000}{(1 + 0.05)^5} - C_0 $$
Assuming there is no initial investment (\( C_0 = 0 \)), the NPV would be the sum of the discounted cash flows.
Understanding NPV is crucial for anyone involved in financial decision-making. It provides a quantitative framework to compare the attractiveness of various investment opportunities, ensuring that
the decisions made today will stand the test of time and bring about financial prosperity. Whether for personal investment strategies or corporate finance, NPV acts as a beacon, guiding through the
complexities of financial planning and investment analysis.
Understanding the Basics of Net Present Value \(NPV\) - Annuity: Annuities and NPV: Calculating the Value of Consistent Cash Flows
3. A Key Concept in Annuity Valuation
understanding the time value of money is fundamental to grasping the concept of annuity valuation. This principle posits that a dollar today is worth more than a dollar tomorrow due to its potential
earning capacity. This core tenet of finance holds that, provided money can earn interest, any amount of money is worth more the sooner it is received. Annuities, which are financial products
designed to pay out a steady stream of payments over time, are directly impacted by this concept. When valuing an annuity, the future payments must be discounted to reflect their present value,
acknowledging that each payment is less valuable than the last due to the time value of money.
From different perspectives, the time value of money can be seen as:
1. An Investor's Viewpoint: For investors, the time value of money is a tool to gauge the attractiveness of an investment. For example, receiving $100 today is preferable to receiving $100 in a year
because today's money can be invested to earn interest, resulting in more than $100 after a year.
2. A Borrower's Perspective: Borrowers benefit from the time value of money when they take out loans. If inflation is high, the dollars they repay in the future are worth less than the dollars they
borrowed, effectively reducing the cost of the loan.
3. An Economist's Angle: Economists might use the time value of money to understand the opportunity cost of spending decisions. For instance, choosing to spend $1,000 on a luxury item today means
forgoing the potential future earnings that $1,000 could have produced if invested.
4. A Business's Standpoint: Businesses apply the time value of money in capital budgeting decisions. They prefer projects where cash inflows occur earlier, as these funds can be reinvested sooner to
generate additional income.
To illustrate the time value of money in annuity valuation, consider a simple annuity that pays $1,000 per year for five years. If the discount rate is 5%, the present value of the annuity can be
calculated using the formula:
$$ PV = \frac{PMT}{(1+r)^n} $$
Where \( PV \) is the present value, \( PMT \) is the annual payment, \( r \) is the discount rate, and \( n \) is the number of periods. Using this formula, the present value of each payment
decreases as \( n \) increases, reflecting the reduced value of future payments.
By understanding and applying the time value of money, individuals and businesses can make more informed decisions about their investments, loans, and other financial matters, ensuring that they are
accounting for the impact of time on the value of their cash flows. Whether saving for retirement, financing a large purchase, or evaluating a business project, the time value of money is a critical
concept that shapes financial behavior and strategy. It's the linchpin that ensures the consistency and comparability of monetary value across different time periods.
A Key Concept in Annuity Valuation - Annuity: Annuities and NPV: Calculating the Value of Consistent Cash Flows
4. Fixed vsVariable Annuities
When it comes to planning for retirement or managing long-term financial goals, annuities are a popular choice for many investors. Annuities are financial products sold by insurance companies that
promise to pay a steady stream of income in exchange for an initial investment. They come in various forms, but the two most common types are fixed and variable annuities. Both have their own set of
features, benefits, and considerations that can make them more or less suitable depending on an individual's financial situation, risk tolerance, and investment objectives.
Fixed annuities offer a guaranteed payout, which provides a sense of security and predictability. The insurance company agrees to pay a specified rate of return on the initial investment, and the
payments remain constant over the term of the contract. This can be particularly appealing for those who want to ensure a stable income stream and are wary of market fluctuations.
On the other hand, variable annuities allow for the potential of higher returns by investing the principal in various investment options, typically mutual funds. The payouts from a variable annuity
can vary based on the performance of the chosen investments. While this introduces a level of risk, as the payments can fluctuate, it also offers the opportunity for growth, which can be beneficial
in times of inflation or when the market performs well.
Here are some in-depth points to consider when calculating annuities:
1. Present Value of An Annuity (PVA):
The present value of an annuity is the current worth of a series of future payments, discounted at a specific interest rate. It's calculated using the formula:
$$ PVA = P \times \left(\frac{1 - (1 + r)^{-n}}{r}\right) $$
Where \( P \) is the payment amount, \( r \) is the interest rate per period, and \( n \) is the number of periods.
2. Future Value of An Annuity (FVA):
The future value is the value of a series of payments at a specified date in the future, with compounded interest. The formula for FVA is:
$$ FVA = P \times \left(\frac{(1 + r)^n - 1}{r}\right) $$
3. Fixed Annuities Calculation:
For fixed annuities, the calculation is straightforward since the payment and interest rate are constant. An example would be a fixed annuity that promises a 5% annual return on a $100,000 investment
over 20 years. The annual payout would be:
$$ Annual Payout = $100,000 \times 0.05 = $5,000 $$
4. Variable Annuities Calculation:
Calculating variable annuities is more complex due to the variable return rates. If the same $100,000 is invested in a variable annuity with a portfolio that averages 7% return annually, the payout
could potentially be higher. However, it's important to factor in the risk and the possibility that returns could also be lower.
5. Tax Considerations:
Annuities have unique tax implications. The earnings from annuities are tax-deferred until withdrawal, which can be an advantage for long-term growth. However, early withdrawals may be subject to
penalties and income tax.
6. Inflation and Annuity Payments:
Inflation can erode the purchasing power of fixed annuity payments over time. Some annuities offer inflation protection or increasing payment options to mitigate this risk.
7. Death Benefits and Riders:
Many annuities come with additional features, such as death benefits or riders for long-term care, which can impact the overall value and cost of the annuity.
Consider an individual who invests in a fixed annuity with a principal of $200,000, an annual interest rate of 4%, and a term of 25 years. The annual payout would be calculated as follows:
$$ Annual Payout = $200,000 \times \left(\frac{1 - (1 + 0.04)^{-25}}{0.04}\right) $$
This would provide them with a guaranteed income stream, regardless of market conditions.
When calculating the value of annuities, it's crucial to understand the differences between fixed and variable options, consider personal financial goals, and consult with a financial advisor to make
an informed decision that aligns with one's retirement planning and investment strategy. Annuities can be a valuable tool for ensuring a consistent cash flow, but like any investment, they require
careful consideration and management.
5. A Step-by-Step Guide
When considering an annuity, which is a series of equal payments made at regular intervals, the concept of Net Present Value (NPV) becomes a critical tool for determining the current value of future
cash flows. The NPV calculation discounts the series of future cash flows back to their value in today's dollars, accounting for the time value of money. This is particularly important for annuities,
as they often represent long-term financial commitments, such as retirement plans or insurance products, where the consistency and security of cash flows are paramount.
From the perspective of an investor, applying NPV to annuity cash flows helps in making informed decisions about the viability of such financial products. For the issuer of the annuity, understanding
the NPV is essential for pricing the product appropriately. Here's a step-by-step guide to applying NPV to annuity cash flows:
1. Identify the Periodic Cash Flow: Determine the amount of the regular payment associated with the annuity. This is the fixed sum received or paid out in each period.
2. Determine the Discount Rate: Select an appropriate discount rate, which reflects the risk and the time value of money. This rate is often the investor's required rate of return or the cost of
3. Establish the Number of Periods: Count the total number of payment periods over the annuity's lifetime. This could be the number of years, months, or any other interval.
4. Calculate the Present Value of Each Cash Flow: Use the formula for the present value of an ordinary annuity or an annuity due, depending on the timing of the payments. The formula for an ordinary
annuity is:
$$ PV = C \times \left(\frac{1 - (1 + r)^{-n}}{r}\right) $$
Where \( PV \) is the present value, \( C \) is the cash flow per period, \( r \) is the discount rate, and \( n \) is the number of periods.
5. Sum the Present Values: Add up the present values of all the cash flows to get the total NPV of the annuity.
6. Analyze the Result: If the NPV is positive, it indicates that the annuity is expected to generate value over its lifetime, exceeding the cost of the investment. A negative NPV suggests that the
cash flows do not justify the investment under the given discount rate.
Example: Consider an annuity that pays $1,000 per year for 5 years, with a discount rate of 5%. The NPV calculation would be:
$$ NPV = 1000 \times \left(\frac{1 - (1 + 0.05)^{-5}}{0.05}\right) $$
$$ NPV = 1000 \times \left(\frac{1 - (1.05)^{-5}}{0.05}\right) $$
$$ NPV = 1000 \times 4.3295 $$
$$ NPV = $4,329.50 $$
This NPV indicates that, in today's terms, the annuity's cash flows are worth $4,329.50. If the cost of the annuity is less than this amount, it may be considered a good investment.
By applying NPV to annuity cash flows, individuals and institutions can make more strategic financial decisions, aligning their investments with their financial goals and risk tolerance. It's a
powerful technique that underscores the importance of the time value of money in financial planning. Whether you're an individual investor evaluating a retirement annuity or a financial manager
assessing the long-term liabilities of a corporation, the NPV calculation is an indispensable part of your financial toolkit.
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6. The Impact of Interest Rates on Annuity Valuation
Interest rates play a pivotal role in the valuation of annuities, as they directly influence the present value of future cash flows. Annuities, by definition, are financial products that promise to
pay out a fixed stream of payments over time, and they are commonly used as a means of securing a steady income during retirement. The valuation of these annuity payments is heavily dependent on the
prevailing interest rates at the time of calculation. When interest rates are high, the present value of future annuity payments is lower, because each payment is discounted at a higher rate,
reflecting the opportunity cost of not having those funds invested elsewhere at the higher rate. Conversely, when interest rates are low, the present value of the annuity payments increases, as the
payments are discounted at a lower rate, indicating a lower opportunity cost.
From the perspective of an individual investor, the impact of interest rates on annuity valuation can be significant. For example:
1. Fixed Annuities: A retiree who invests in a fixed annuity would receive payments that are unaffected by future changes in interest rates. However, the initial lump sum used to purchase the annuity
is sensitive to interest rates at the time of purchase. If rates are expected to rise, it may be advantageous to delay purchasing an annuity to lock in a higher payment rate.
2. Variable Annuities: In contrast, variable annuities, which have payments linked to the performance of an underlying investment portfolio, may benefit from rising interest rates if the portfolio
includes interest-sensitive assets like bonds.
3. deferred annuities: For those considering deferred annuities, where payments begin at a future date, the interest rate environment at the time of annuitization will affect the payment amount. A
higher rate at annuitization means a higher payout.
4. Immediate Annuities: Conversely, with immediate annuities, where payments start almost right after purchase, a low-interest-rate environment means a higher price for the same level of income.
To illustrate, consider an individual who has the option to invest $100,000 in an immediate annuity that promises a fixed return of 5% per year. If the prevailing interest rate is 3%, the present
value of the annuity's future cash flows would be higher than if the interest rate were 7%. This is because the fixed return of the annuity is more attractive relative to the lower available interest
Example: Let's say the annuity promises to pay $5,000 per year for 20 years. Using a discount rate equal to the prevailing interest rate, the present value (PV) of this annuity can be calculated
using the formula:
$$ PV = \sum_{t=1}^{n} \frac{C}{(1+r)^t} $$
Where \( C \) is the annual cash flow, \( r \) is the interest rate, and \( t \) is the time in years.
If the interest rate \( r \) is 3%, the present value of the annuity would be:
$$ PV = \sum_{t=1}^{20} \frac{5000}{(1+0.03)^t} $$
This would result in a higher present value than if the interest rate were 7%, where the calculation would be:
$$ PV = \sum_{t=1}^{20} \frac{5000}{(1+0.07)^t} $$
Understanding the relationship between interest rates and annuity valuation is crucial for both issuers and investors. Issuers must price their products competitively while ensuring profitability,
and investors need to consider the timing of their annuity purchase in relation to interest rate trends to maximize their investment. As such, interest rates are a fundamental factor in the
decision-making process surrounding annuities and their long-term value.
The Impact of Interest Rates on Annuity Valuation - Annuity: Annuities and NPV: Calculating the Value of Consistent Cash Flows
7. Annuity Valuation in Action
Annuities represent a fascinating financial instrument, often serving as the backbone for retirement plans and long-term investment strategies. They are essentially contracts where the investor makes
a lump-sum payment or a series of payments and, in return, receives regular disbursements beginning either immediately or at some point in the future. The valuation of an annuity involves calculating
the present value of these future payments. understanding the real-world application of annuity valuation is crucial for investors, financial planners, and anyone interested in the mechanics of time
and money.
1. Retirement Planning: Consider a retiree who has purchased an annuity that promises to pay $20,000 annually for 20 years. If we assume a discount rate of 5%, the present value of this annuity can
be calculated using the formula for the present value of an ordinary annuity:
$$ PV = P \times \frac{1 - (1 + r)^{-n}}{r} $$
Where \( P \) is the annual payment, \( r \) is the discount rate, and \( n \) is the number of periods. Plugging in the numbers:
$$ PV = 20,000 \times \frac{1 - (1 + 0.05)^{-20}}{0.05} $$
This calculation yields a present value, which is what the retiree's investment is worth today.
2. Lottery Winnings: A lottery winner might be faced with the choice between taking a $10 million lump sum or an annuity of $500,000 a year for 30 years. The valuation of the annuity option would
require calculating the present value of 30 payments of $500,000, considering a discount rate that reflects the opportunity cost of not having the $10 million upfront.
3. Corporate Finance: A company may issue bonds with a face value of $100,000, promising to pay a 4% coupon annually. The present value of these coupon payments, or the bond's price, will fluctuate
with changes in market interest rates. If rates go up, the present value of the bond's future coupons goes down, and vice versa.
4. Insurance Products: Insurance companies offer annuities as a way to protect against longevity risk—the risk of outliving one's assets. The pricing of these products is a complex exercise in
actuarial science and financial mathematics, taking into account life expectancy, interest rates, and the insurer's profit margin.
Through these examples, we see that annuity valuation is not just a theoretical exercise but a practical tool that affects decisions and financial well-being across various scenarios. It's a
testament to the time-value of money principle, showcasing how the value of cash flows changes over time. Whether for personal finance or corporate strategy, understanding how to value an annuity is
an essential skill in the world of finance.
Annuity Valuation in Action - Annuity: Annuities and NPV: Calculating the Value of Consistent Cash Flows
8. Comparing Annuities to Other Investment Options
When considering retirement or other long-term financial planning, annuities often come up as a potential investment option. They offer a unique set of benefits and drawbacks compared to other
investment vehicles. Annuities are essentially contracts with insurance companies where you make a lump-sum payment or series of payments in exchange for regular disbursements that can begin
immediately or at some point in the future. The appeal of annuities lies in their ability to provide a steady income stream, which can be particularly valuable for individuals who are concerned about
outliving their savings.
However, annuities are not without their complexities and should be compared carefully with other investment options. Here are some key points to consider:
1. Risk vs. Security: annuities provide a guaranteed income, which can be seen as a trade-off for potentially higher returns from investments like stocks or mutual funds. While the stock market
offers the possibility for significant growth, it also comes with greater risk, especially in the short term.
2. Liquidity: Annuities typically have lower liquidity compared to other investments. Once you commit to an annuity, it can be difficult or costly to withdraw your funds before the term ends, whereas
stocks or bonds can be sold relatively easily if cash is needed.
3. Fees and Expenses: Annuities can come with a variety of fees, including initial sales charges, annual fees, and surrender charges if you withdraw money early. It's important to compare these costs
with the fees associated with other investment options.
4. Tax Considerations: Annuities offer tax-deferred growth, which can be advantageous for long-term investing. However, other investment accounts, like Roth IRAs, provide tax-free growth, which may
be more beneficial depending on your tax situation.
5. Inflation Protection: Some annuities include options for inflation protection, but this typically comes at an additional cost. Other investments, like treasury Inflation-Protected securities
(TIPS), directly protect against inflation.
6. Investment Control: With annuities, you give up some control over your investment to the insurance company. In contrast, other investment options like a self-directed IRA allow you to choose and
manage your own investments.
7. Estate Planning: Annuities can be structured to provide benefits to your heirs, but they may not be as flexible as a trust or other estate planning tools.
Example: Consider Jane, who is retiring and has a lump sum of money to invest. She could purchase an annuity that guarantees her a fixed income for life, which provides security but means she might
miss out on higher returns from the stock market. Alternatively, she could invest in a diversified portfolio of stocks and bonds, which could potentially grow more over time but also carries the risk
of losing value.
Annuities can be a valuable part of a diversified investment strategy, but they should be weighed against other options based on individual financial goals, risk tolerance, and investment horizon.
It's always recommended to consult with a financial advisor to understand the nuances of each investment choice and how it fits into your overall financial plan.
Comparing Annuities to Other Investment Options - Annuity: Annuities and NPV: Calculating the Value of Consistent Cash Flows
9. Making Informed Decisions with Annuity and NPV Knowledge
Understanding annuities and the Net Present Value (NPV) is crucial for making informed financial decisions, especially when it comes to investments that promise consistent cash flows over time.
Annuities, whether they are fixed, variable, or indexed, offer a stream of payments that can be a reliable source of income, particularly in retirement planning. On the other hand, NPV provides a
method to evaluate the profitability of an investment by considering the time value of money, thus helping investors to assess the true value of future cash flows in today's terms.
From the perspective of a retiree, annuities can be seen as a safety net that provides financial stability. For instance, a fixed annuity guarantees a specific amount periodically, which can be
comforting for those who wish to avoid market volatility. Conversely, from an investor's viewpoint, understanding NPV is essential for comparing different investment opportunities. A positive NPV
indicates that the projected earnings exceed the anticipated costs, adjusted for the time value of money, making it an attractive option.
Here are some in-depth insights into making informed decisions with annuity and NPV knowledge:
1. Time Value of Money: The core principle behind NPV is that a dollar today is worth more than a dollar tomorrow. This is due to inflation and the potential earning capacity of money. When
calculating NPV, future cash flows are discounted to their present value, allowing investors to see the real value of their investment.
2. Risk Assessment: Annuities can be part of a risk management strategy. Fixed annuities, for example, offer lower risk compared to variable annuities, which are subject to market fluctuations.
understanding the risk profile of an annuity is key to ensuring it aligns with one's financial goals.
3. Tax Considerations: Annuities offer tax-deferred growth, meaning you don't pay taxes on the earnings until you withdraw them. This can be advantageous for long-term growth. However, it's important
to consider the tax implications of annuity payments in retirement.
4. Liquidity Needs: While annuities provide a steady income, they often come with limited liquidity. If you anticipate needing access to your funds, it's important to consider the surrender charges
and withdrawal penalties that might apply.
5. Comparing Investment Opportunities: Using NPV calculations, investors can compare the profitability of various investments. For example, if an investor is choosing between a rental property and a
corporate bond, they can use NPV to determine which investment offers a higher return on investment when considering the time value of money.
6. Inflation Impact: Both annuities and NPV calculations must account for inflation. With annuities, purchasing a rider that adjusts payments for inflation can protect purchasing power. In NPV
calculations, the discount rate should include an inflation premium to ensure accuracy.
To illustrate these points, let's consider an example where an investor is deciding whether to purchase a fixed annuity or invest in a project with expected future cash flows. If the annuity offers a
5% annual return, but the project has an NPV of $100,000 with a 10% discount rate, the investor would need to weigh the guaranteed income against the potential higher returns of the project,
considering their risk tolerance and financial objectives.
A thorough understanding of annuities and NPV is indispensable for anyone looking to secure their financial future. By carefully considering the time value of money, risk profiles, tax implications,
liquidity needs, and the impact of inflation, individuals can make well-informed decisions that align with their long-term financial plans. Whether it's choosing a reliable income stream through
annuities or evaluating investment opportunities with npv, the key is to approach these financial tools with a clear understanding of their nuances and potential impacts on one's financial health.
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This collection contains dissertations written in partial fulfillment of the doctoral degree, from 1937 to the present.
The first doctoral degree was in Chemistry, awarded to Frank Joseph Zvanut; his dissertation was titled Pyrochemical changes in Missouri halloysite.. Zvanut had earned his Bachelor of Science in
Ceramic Engineering in 1932. Most recently, popular Ph.D. disciplines have included Chemistry, Chemical Engineering, Electrical Engineering, Geological Engineering and Petroleum Engineering.
Theses and dissertations previously submitted in print will be digitized with permission of the author or copyright holder. Missouri S&T Library and Learning Resources encourages graduates to provide
this permission so that their work can reach the widest possible audience. If you would like to grant this permission, please use this Form or go to your dissertation in Scholars' Mine and click on
the Share My Dissertation button. Theses and dissertations will be digitized as time allows and will not become immediately accessible.
More information on today’s graduate degree programs is available on the Missouri S&T website.
To browse dissertations by academic department, please visit our Browse Collections page.
Dissertations from 1971
An age hardening study of magnesium-thorium-manganese alloys, Kuldip Singh Chopra
Some metal complexes of dihydroxycyclobutenedione, Stewart Michael Condren
Characterizing topologies by classes of functions and multifunctions, Alexander Hamlin Cramer
Theory of nucleation of water properties of some clathrate like cluster structures, Mehdi Daee
Trace base metals, petrography, rock alteration of the productive Tres Hermanas stock, Luna County, New Mexico, Peethambaram Doraibabu
Dynamic behavior of eccentrically stiffened plates, Charles Stuart Ferrell
Surface reactions of boron with clean tungsten substrates, Thomas A. Flaim
The genesis of cellular precipitation in copper rich copper-indium alloys, Raymond Albert Fournelle
Transient hydraulic simulation: breached earth dams, D. L. Fread
Modeling the visual pathway for interactive diagnosis of visual fields, Chiam Geoffrey Goldbogen
Special subrings of real, continuous functions, Paul Marlin Harms
Photoionization of photoexcited cesium, Robert E. Hebner
An investigation of subsequent yield phenomena, Joseph George Hoeg
A Monte Carlo calculation of neutron reflection from various curved surfaces, Charles Jack Kalter
Activation volumes of carbon diffusion in F.C.C. iron-nickel alloys, James Raymond Keiser
Response surface study of a characteristic chemical plant, John Thomas Mason III
Identification of centers of influence for urban areas, Robert W. Meyer
Metal salt catalyzed carbenoids, Billy W. Peace
Detection of delay time and filtering using sequency domain technique, Earl F. Richards
Thermal lattice expansion of various types of solids, Jayantkumar Shantilal Shah
A measurement of complex viscosity with large amplitudes, Kuo-Shein Shen
X-ray and neutron diffraction study of the holmium-iron system, Michael Fred Simmons
Identification of linear systems with delay via a learning model, Hugh Francis Spence
Noble gases anomalies in terrestrial ores and meteorites, B. Srinivasan
The vibrational modes of spheroidal cavities in elastic solids, David Stewart
X-ray and Mössbauer spectroscopy studies of the silicon-antimony and bismuth-antimony alloys, James Ralph Teague
Absolute cross sections for excitation of neon by impact of 20-180 keV H , H₂ and He ions, George William York
The thermal accommodation coefficients of helium, neon, and argon on surfaces, Gerald Lee Zweerink
Dissertations from 1970
A study of the anodic oxidation of 1, 3-butadiene on platinum and gold electrodes, Arun Kumar Agrawal
Investigation of liquid carryover in the girdler-sulfide process for production of heavy water at the AEC Savannah River Plant, Samuel Clark Allen
Upper cretaceous and lower cenozoic foraminifera of three oil wells in northwestern Iraq, Farouk Sunallah Al-Omari
The electrochemical behavior of the lead-thallium system in acidic sulfate solutions, Pedro Juan Aragon
Substitution and photolytic rearrangements in three-member ring compounds, Joseph Terril Bohanon
Use of the turbulence kinetic energy equation in prediction of nonequilibrium turbulent boundary layers, William Madison Byrne Jr.
Completeness and related topics in a quasi-uniform space, John Warnock Carlson
Investigation into the monotonic magnetostriction and magnetic breakdown in cadmium, James Milo Carter
Radiative lifetime of the A¹π state and the transition moment variation of the fourth positive band system of carbon monoxide, Joseph George Chervenak
Electrode reactions in zinc electrolysis, Ernest R. Cole
A quasi-lattice model of simple liquids, Jesse Herbert Collins
IDDAP -- Interactive computer assistance for creative digital design, Richard Franklin Crall
Development and application of cartesian tensor mathematics for kinematic analysis of spatial mechanisms, Robert Myrl Crane
The electrochemical oxidation of 1-pentyne on platinum and gold, Michael Jensen Danielson
Thermochemical hydrogen-deuterium isotope effects, Wayne C. Duer
A heuristic algorithm for determining a constructive suboptimal solution to the combinatorial problem of facility allocation, Harry Kerry Edwards
Theory of shallow donor impurity surface states, Vidal Emmanuel Godwin
A coordinate oriented algorithm for the traveling salesman problem, Joseph Sidney Greene
An investigation of nucleate and film boiling heat transfer from copper spheres, William David Hardin
Realistic impulsive P wave source in an infinite elastic medium, Joseph Hugo Hemmann
Higher space mode analysis of a large cylindrical pulsed H₂O system, Harold David Hollis
The cathodic reduction of maleic acid, Show Yih Hsieh
Mass transfer from spherical gas bubbles and liquid droplets moving through power-law fluids in the laminar flow regime, Cheng-chun Huang
Extension of some theorems of complex functional analysis to linear spaces over the quaternions and Cayley numbers, James E. Jamison
Ionization of air produced by strong shocks, Howard Sajon Joyner
Penetration in granite by shaped charge liners of various metals, Hemendra Nath Kalia
A three-dimensional mathematical simulator of multiphase systems in a petroleum reservoir, Leonard Koederitz
Linear regression models of sound velocity in the North Atlantic Ocean below a critical depth, Richard Roland Kunkel
The surface effects of alkali halides in the infrared, Vincent Joseph Llamas
Some effects of OH groups on sodium silicate glasses, Mokhtar Sayed Maklad
A computer simulation of mine air shaft thermodynamics, Ambyo Sumopandhi Mangunwidjojo
Effect of axial dispersion on interphase mass transfer in packed absorption columns, Virendra Kumar Mathur
Diffusion and internal friction in sodium-rubidium silicate glasses, Gary L. McVay
Structures and relationships of some Perovskite-type compounds, Christian Michel
Heuristic algorithms for the generalized vehicle dispatch problem, Leland Ray Miller
Binary molecular diffusivities in liquids: prediction and comparison with experimental data, Ronald Dean Mitchell
Temperature variation in distribution of relaxation times in aluminosilicate glasses, David Wayne Moore
The reaction of N-sulfinylamines with diazoalkanes, Harry R. Musser
Simulation of a gas storage reservoir with leakage by a two-dimensional layered mathematical model, Steven William Ohnimus
Voids in neutron irradiated aluminum, Nicholas H. Packan
Discrimination of signal and noise events on seismic recordings by linear threshold estimation theory, David Nuse Peacock
An SEM surface study of nucleate pool boiling heat transfer to saturated liquid nitrogen reduced pressures from 0.1 to 0.9, David Virgil Porchey
Search algorithms for the simple plant location problem, John Bruce Prater
An investigation of the stiffness of shafts with integral disks, Richard King Riley
A study of far infrared grid filters, Harold Victor Romero
A simulation and diagnosis system incorporating various time delay models and functional elements, David Michael Rouse
Model-referenced adaptive control of plants with noise and inaccessible state vector, Donald James Schooley
Absolute excitation cross sections of He⁺ in 20-100 keV He⁺-He collisions using energy-loss spectrometry, Donald Roy Schoonover
Synthesis heuristics for large asynchronous sequential circuits, Robert Judson Smith
The deformation mechanisms in sublimed magnesium under cyclic loading, Babu Narian Thakur
A fundamental investigation of the gas phase polymerization of styrene and vinyl type monomers in a low power inductively coupled 4 MHz RF plasma, L. F. Thompson
Identification of decomposition products of tropyl azide and interaction of tropylium ion with nucleophiles, James Joseph Ward
The oxidation kinetics of liquid lead and lead alloys, Thomas Edward Weyand
An experimental and theoretical study of the nucleation of water vapor on ions in helium and argon, Daniel R. White
A model for predicting the energizing transients of station capacitor banks, Dennis Oliver Wiitanen
Numerical simulation of forward combustion in a radial system, Tommie C. Wilson
Optimizing diesel engine efficiency using the controllability of a variable ratio hydrostatic transmission, Gordon Wright
Anodic dissolution of zinc in potassium iodide-potassium iodate solutions, Chi-Chiu Yao
Dissertations from 1969
Noble gases: a record of the early solar system, E. C. Alexander
Vapor phase clustering model for water, Richard W. Bolander
A.C. Hall effect measurements on very high resistivity materials exhibiting electrode polarization, James Dale Boyd
Mössbauer Effect studies of ferroelectric phase transitions in the PbZrO₃ - PbTiO₃ - BiFeO₃ ternary system, James P. Canner
Models for the pn diode and the npn and pnp transistor for use in computer aided design and analysis programs, Glenn Ronald Case
A theoretical study on the interpretation of resistivity sounding data measured by the Wenner electrode system, Siew Hung Chan
Numerical inversion of the Laplace transformation and the solution of the viscoelastic wave equations, Abbas Ali Daneshy
A crystallization study of a tetrasilicic fluormica glass, William H. Daniels
Statistical inferences for location and scale parameter distributions, Robert Henry Dumonceaux
Calorimetry of room temperature deformed copper, George Juri Filatovs
Dynamic response of circular plates to transient and harmonic transverse loads including the effect of transverse shear and rotary inertia, Perakatte Joseph George
An optimization approach to well spacing for gas storage reservoirs, Jagannath Rao Ghole
The statistics of finite, one dimensional lattice fluids, John Roger Glaese
A method for determining transient stability in power systems, Charles A. Gross | {"url":"https://scholarsmine.mst.edu/doctoral_dissertations/index.32.html","timestamp":"2024-11-02T10:57:51Z","content_type":"text/html","content_length":"112034","record_id":"<urn:uuid:820046e5-0021-4de5-80c9-b1546d805359>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00573.warc.gz"} |
Implementation of Huffman Coding algorithm with binary trees | Kamil Mysliwiec
Huffman code is a type of optimal prefix code that is commonly used for lossless data compression. The algorithm has been developed by David A. Huffman. The technique works by creating a binary tree
of nodes. Nodes count depends on the number of symbols.
The concept
The main idea is to transform plain input into variable-length code. As in other entropy encoding methods, more common symbols are generally represented using fewer bits than less common symbols. The
easiest way to understand how to create Huffman tree is to analyze following steps:
1. Scan text for symbols (e.g. 1-byte characters) and calculate their frequency of occurrence. Symbol value with its count of occurrences is a single leaf.
2. Start a loop.
3. Find two smallest probability nodes and combine them into single node.
4. Remove those two nodes from list and insert combined one.
5. Repeat the loop until the list has only single node.
6. This last single node represent a huffman tree.
The expected result:
Huffman tree based on the phrase „Implementation of Huffman Coding algorithm” (source: huffman.ooz.ie).
The solution
The full source code is available at GitHub, written using C++11. The expected output of a program for custom text with 100 000 words:
100 000 words compression (Huffman Coding algorithm) | {"url":"https://kamilmysliwiec.com/implementation-of-huffman-coding-algorithm-with-binary-trees/","timestamp":"2024-11-12T10:09:39Z","content_type":"text/html","content_length":"12776","record_id":"<urn:uuid:cf5a8b85-f7df-48ad-b34e-6dbf0f7bd1c5>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00403.warc.gz"} |
Digital Math Resources
Title Description Thumbnail Curriculum Topics
This collection aggregates all the math clip art around the topic of Order of
Operations. There are a total of 14 images. This collection of resources is Numerical Expressions
made up of downloadable PNG files that you can easily incorporate into a
OvThis is a collection of all our drag-n-drop math games. There are a total Addition Facts to 25, Counting, Subtraction Facts to 25, Place Value, Polynomial Expressions, Division
of 34 games. These games cover a variety of different skills. Each comes with Expressions and Equations, Point-Slope Form, Slope-Intercept Form, Standard Form, Slope, Applications of
a large bank of questions, so each game experience will be different. These Linear Functions, Quadratic Equations and Functions, Data Analysis, Multiplication Expressions and
games are ideal for practice and review. Equations, Solving One-Step Equations, Quadratic Formula, Numerical Expressions, Variable Expressions and
Solving Two-Step Equations
This collection aggregates all the math examples around the topic of Integer Numerical Expressions
and Rational Exponents. There are a total of 15 images.
This collection aggregates all the math examples around the topic of Order of Numerical Expressions
Operations. There are a total of 14 images.
This collection aggregates all the math worksheets around the topic of the Place Value, Addition Facts to 25, Addition Facts to 100, Numerical and Algebraic Expressions, Numerical
Language of Math, or converting words into math expressions. There are a Expressions, Variable Expressions, Fractions and Mixed Numbers, Subtraction Facts to 25 and Subtraction
total of 264 worksheets. Facts to 100
MATH EXAMPLES--The Language of Math--Numerical Expressions--Addition
This set of tutorials provides 40 examples of converting verbal expressions Numerical and Algebraic Expressions
into numerical expressions that involve addition. Note: The download is a PPT
file. NOTE: The download is a PPT file.
MATH EXAMPLES--The Language of Math--Numerical Expressions--Division
This set of tutorials provides 40 examples of converting verbal expressions Numerical and Algebraic Expressions
into numerical expressions that involve division. Note: The download is a PPT
file. NOTE: The download is a PPT file.
MATH EXAMPLES--The Language of Math--Numerical Expressions--Grouping Symbols
This set of tutorials provides 32 examples of converting verbal expressions Numerical and Algebraic Expressions
into numerical expressions that involve grouping symbols. Note: The download
is a PPT file. NOTE: The download is a PPT file.
MATH EXAMPLES--The Language of Math--Numerical Expressions--Multiplication
This set of tutorials provides 40 examples of converting verbal expressions Numerical and Algebraic Expressions
into numerical expressions that involve multiplication. Note: The download is
a PPT file. NOTE: The download is a PPT file.
MATH EXAMPLES--The Language of Math--Numerical Expressions--Subtraction
This set of tutorials provides 40 examples of converting verbal expressions Numerical and Algebraic Expressions
into numerical expressions that involve subtraction. Note: The download is a
PPT file. NOTE: The download is a PPT file.
MATH EXAMPLES--The Language of Math--Variable Expressions--Multiplication and
This set of tutorials provides 32 examples of converting verbal expressions Numerical and Algebraic Expressions
into variable expressions that involve multiplication and addition. Note: The
download is a PPT file. NOTE: The download is a PPT file.
MATH EXAMPLES--The Language of Math--Variable Expressions--Multiplication and
This set of tutorials provides 32 examples of converting verbal expressions Numerical and Algebraic Expressions
into variable expressions that involve multiplication and subtraction. Note:
The download is a PPT file. NOTE: The download is a PPT file.
Interactive Math Game--DragNDrop Math--The Language of Math--Numerical
In this drag-and-drop game, a verbal expression to a numerical expression Numerical Expressions and Variable Expressions
with addition. This game generates thousands of different equation
combinations, offering an ideal opportunity for skill review in a game
Interactive Math Game--DragNDrop Math--The Language of Math--Numerical
Numerical Expressions and Variable Expressions
In this drag-and-drop game, match a verbal description of an addition
expression with its numerical counterpart.
Interactive Math Game--DragNDrop Math--The Language of Math--Numerical
Expressions--Grouping Symbols
In this drag-and-drop game, a verbal expression to a numerical expression Numerical Expressions and Variable Expressions
with grouping symbols. This game generates thousands of different equation
combinations, offering an ideal opportunity for skill review in a game
Interactive Math Game--DragNDrop Math--The Language of Math--Numerical
In this drag-and-drop game, match a verbal expression to a numerical Numerical Expressions and Variable Expressions
expression with multiplication. This game generates thousands of different
equation combinations, offering an ideal opportunity for skill review in a
game format.
Interactive Math Game--DragNDrop Math--The Language of Math--Numerical
Numerical Expressions and Variable Expressions
In this drag-and-drop game, match a verbal description of a subtraction
expression with its numerical counterpart.
Interactive Math Game--DragNDrop Math--The Language of Math--Variable
Expressions--Multiplication and Addition
In this drag-and-drop game, a verbal expression to a variable expression with Numerical Expressions and Variable Expressions
multiplication and addition. This game generates thousands of different
equation combinations, offering an ideal opportunity for skill review in a
game format.
Interactive Math Game--DragNDrop Math--The Language of Math--Variable
Expressions--Multiplication and Subtraction
In this drag-and-drop game, a verbal expression to a variable expression with Numerical Expressions and Variable Expressions
multiplication and subtraction. This game generates thousands of different
equation combinations, offering an ideal opportunity for skill review in a
game format.
Math Clip Art--Order of Operations 01
This is part of a collection of math clip art images that focus on the order Numerical Expressions
of operations.
Math Clip Art--Order of Operations 02
This is part of a collection of math clip art images that focus on the order Numerical Expressions
of operations.
Math Clip Art--Order of Operations 03
This is part of a collection of math clip art images that focus on the order Numerical Expressions
of operations.
Math Clip Art--Order of Operations 04
This is part of a collection of math clip art images that focus on the order Numerical Expressions
of operations.
Math Clip Art--Order of Operations 05
This is part of a collection of math clip art images that focus on the order Numerical Expressions
of operations.
Math Clip Art--Order of Operations 06
Numerical Expressions
This is part of a collection of math clip art images that focus on the order
of operations. | {"url":"https://www.media4math.com/NY-6.EE.2c","timestamp":"2024-11-07T13:50:00Z","content_type":"text/html","content_length":"99245","record_id":"<urn:uuid:bb3c1677-0846-42d3-97ba-c09f478ed0bc>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00666.warc.gz"} |
hash index – WonderDB
Hash index is based on hash map data structure. It has lot of properties like, rehasing, load factor etc. You can read more about it in wikipedia.
We will more focus on usage and design of this data structure in the context of databases, where its size could be bigger than the physical memory and will need to be serialized to disk. Following
are various points to consider when using it in databases.
Rehash – Increasing/decreasing size of buckets
Hash maps need to lock the whole data structure during the rehash. This may not be very expensive operation for in memory implementations of hash map. But may be almost impossible in the context of
databases since rehash will have to lock the whole table.
Most databases suggest to rebuild the index when performance starts degrading over time due size of items in the index vs number of buckets. Or load factor going below certain threshold value.
Generic implementations of hashCode() and equals() methods
It may not be easy to provide hashCode() and equals() implementations specific to your index class. Usually databases will use their generic implementations which may not be very efficient to your
index class. In wonderdb, we are working on a feature to register data type.
Load factor – Calculating buck size
By default Java hash map implementation sets load factor to 0.75. Allowing size of hash map to grow up to certain size before it is rehashed. For example say initial capacity of hash map is 100, then
it will be rehashed when 75th item is stored in to the hash map.
We need different load factor considerations for hash index. Lets take a example on how to calculate load factor for hash indexes.
As shown in the figure above index entries are stored in disk block. It is called leaf node in wonderdb. Say size of disk block is 2048 bytes. Now lets say you want to store long in to the index (8
bytes). Also assume index stores pointer to actual table record, say pointer size is 12 bytes, then it needs 20 bytes to store index item in the disk. Say one disk block (2048 bytes) stores 100 index
In this case, for optimized use of disk space it will be better to assume optimal settings will be 100 items per bucket, instead of 1 item per bucket (as in Java hash map implementation). So load
factor settings for hash index will be .0075. Or in another way to look at it; to store 750 items, java hash map will need 1000 buckets to achieve 0.75 load factor, whereas hash index will need only
7.5 (or 8) buckets to store 1000 items since we need to assume 100 items per bucket in case of hash index.
BTree+ Vs Hash Index
Advantage of hash map data structure vs tree data structure is its access time is constant time (O1). Or it just takes 1 comparison (one instruction) to get to the item assuming it is properly
organized. Where as for tree the access time is O(log n). Or it takes 10 compare instructions to get to an item in a tree containing 1024 items.
So for 1M items it takes 20 compares and for 1B it takes 30 compares for the tree. Where as for hash map it will take 1 compare instruction to get to the item. But problem is, we probably wont be
able to store billions to items in hash map due to physical memory size constraints.
So for storing billions of items in hash index we need to also take load factor in to account. Say load factor is 0.0075 in case of storing longs, or in other words, if bucket should contain 100
items for optimal disk usage then already it needs to do 100 long 100 = 7 compares within the buckets to get to the item. So already access time is no more O1 but O7.
So to see optimal performance, we need number of items for Btree+ to do at lease 3x of 7 or 21 compares which is ~ 1M items.
So point here is, unless you are expecting millions of items in the index, dont even consider hash index.
We are able to see hash index performance can go up to 2x for a case where BTree+ needed 3 level deep structure.
We see performance of 60K queries per second in case of 32000 items with key size of 1500. We selected key size of 1500 just to make force 3 level deep BTree+. Where as for BTree+ we see close to
35-40K queries per second.
Here was the setup for the test.
It needed 32000 leaf nodes to store 32000 items since it can store only 1 item per node with 1500 size with 2048 block size.
It needed 32000/150 = 234 branch blocks. Branch blocks store pointers to next branch or leaf blocks and can store 150 items per block.
Next level of branch blocks needed 234/150 = 2 blocks.
And root node pointed to 2 next level blocks.
So our tree structure had 4 levels with
– 2 items in root node
– 2 branch blocks pointing to 234 items in next level
– 234 branch to point to 32000 leaf blocks.
– and 32000 leaf blocks.
It needed 1 compare (on root level) + 8 compares (on next level containing 234 pointers) + 15 compares (to get to leaf node) + 1 compare to find item in leaf node = 25 compares.
Hash Index
We had 32000 buckets to make hash index fully optimized to get O1 compare to get to an item.
With this setup we saw 60K queries per second for hash index and 40K queries per second for BTree+. over 50% improvement. | {"url":"https://wonderdb.org/?tag=hash-index","timestamp":"2024-11-07T09:22:43Z","content_type":"text/html","content_length":"101580","record_id":"<urn:uuid:c4dd17f7-6b21-4d93-b477-22282bf8b0ef>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00117.warc.gz"} |
Generating better random numbers with dynamics ax 2009
Inspired by
this post
, I created a neat little way to generate
random numbers between a range.
If you use AX Random class or RandomGenerate class to generate random numbers, you will find that if you do several in a row, they turn out fairly sequential.
does a slightly better job of producing a random integer.
I would sometimes see patterns using this when the output is sorted:
static void Job6(Args _args)
int total = 20;
int i;
while (total)
i = xGlobal::randomPositiveInt32();
// This is better
while (i > 9999999)
i = i div 10;
while (i < 1000000)
i = i * 10;
while (i > 9999999)
i = i >> 1;
while (i < 1000000)
i = i << 1;
3 comments:
1. Hi, bitwise shifts that you are doing break uniform distribution of the numbers. The simpliest way to get close to uniform distribution on the range [1000000, 9999999) is the following:
xGlobal::randomPositiveInt32() mod (9999999 - 1000000) + 1000000
2. @Gigz, I thought the whole point of uniform distribution was an equal chance of getting any number on a range. It seemed like the results I get, no matter what seem to follow a pattern. Try this
job and sort the table. It seems like it'll always start at the low end, and evenly move up until it's at the high end.
static void TASGenerateRandomNumbers(Args _args)
#define.totalToGenerate (10000)
#define.high (9999999)
#define.low (1000000)
TASRandomNumbers tas;
int totalToGenerate = #totalToGenerate;
int64 totalOps;
int i;
delete_from tas;
while (totalToGenerate)
// Uncomment this
i = xGlobal::randomPositiveInt32(); //mod (#high - #low) + #low;
// Comment this out
while (i > #high)
i = i div 10;
// Comment this out
while (i < #low)
i = i * 10;
if (TASRandomNumbers::exist(i) == NoYes::No)
tas.RandNum = i;
info("Total operations: " + int642str(totalOps));
3. Yep, you are right about the definition of the uniform distribution.
But to measure if numbers are generated uniformly or not, you need to calculate the frequency of each number (how many times each number is generated).
something like:
(without if (TASRandomNumbers::exist(i) == NoYes::No))
select forupdate tas where tas.RandNum == i;
and then look if all records have pretty much the same Freq values.
This should be done for large number of experiments, at least 100 times larger than interval size. | {"url":"https://www.alexondax.com/2011/04/generating-better-random-numbers-with.html","timestamp":"2024-11-14T02:00:22Z","content_type":"application/xhtml+xml","content_length":"88239","record_id":"<urn:uuid:9d01804e-709c-469c-949a-33bad7060aa8>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00739.warc.gz"} |
Downloads Model Theory of Algebra and ArithmeticDownloads Model Theory of Algebra and Arithmetic
Model Theory of Algebra and Arithmetic book download
A.J. Wilkie, L. Pacholski, Wierzejewski
About Book Book Description Model theory is a branch of mathematical logic that has found applications in several areas of algebra and geometry. Fields & Galois Theory; Algebra:. Arithmetic and
Geometric Applications. . Model theory of algebra and arithmetic: proceedings of the Conference on Applications of Logic to Algebra and Arithmetic held at Karpacz,. Representation Theory of Finite
Groups: Algebra and Arithmetic. introduction to first-order model theory, arithmetic and set. Model theory of algebra and arithmetic: proceedings of the. Alibris has Representation Theory of Finite
Groups: Algebra and Arithmetic and other books by Steven H Weintraub, including new & used copies, rare, out-of-print. Model theory of algebra and arithmetic: Proceedings of the Conference on
Applications of Logic to Algebra and Arithmetic held at Karpacz, Poland, September 1-7, 1979. Download Model Theory of Algebra and Arithmetic. Free Books > Science > Mathematics > General > Model
Theory. Powell's Books is the largest independent used and new bookstore in the world.. this book is an introduction to first-order model. Pacholski, Wierzejewski. Amazon.com: Representation Theory
of Finite Groups: Algebra and Arithmetic (Graduate Studies in Mathematics) (9780821832226): Steven H. Lecture Notes in Mathematics #834: Model Theory of Algebra and. Weintraub: Books A Course in
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online International Tables for Crystallography: Physical properties of crystals read Pinocchio | {"url":"https://cpbliii.typepad.com/blog/2011/12/downloads-model-theory-of-algebra-and-arithmetic.html","timestamp":"2024-11-08T21:43:16Z","content_type":"application/xhtml+xml","content_length":"22537","record_id":"<urn:uuid:02a4df3c-8f34-45b7-b42d-5adafd39cdbf>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00005.warc.gz"} |
Solution: If the number is within the limits
Course of Raku / Essentials / Ranges / Exercises / If the number is within the limits
Solution: If the number is within the limits
The program uses a range that is created from the numbers that the user enters. Then, the smartmatch check tests if the third number is within the range borders. The result of the smartmatch test is
a Boolean value, so we can immediately print it.
Here is the solution:
my $begin = prompt 'From (including): ';
my $end = prompt 'To (excluding): ';
my $n = prompt 'What is the number? ';
say $n ~~ $begin ..^ $end;
🦋 Find the program in the file number-in-limits.raku.
Test different cases, including when the number coincides with the end of the range.
$ raku exercises/ranges/number-in-limits.raku
From (including): 1
To (excluding): 2
What is the number? 1.5
$ raku exercises/ranges/number-in-limits.raku
From (including): 100
To (excluding): 200
What is the number? 100
$ raku exercises/ranges/number-in-limits.raku
From (including): -5
To (excluding): -2
What is the number? -2
Note how the right endpoint of the range is excluded: $begin ..^ $end.
Course navigation
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Beer Bongs, Volume, and Fluid Mechanics
A beer bong is a very simple device, composed of a funnel and a tube, designed to quickly get beer into the user. While you could go out and buy one, it'd be a pretty big waste of money considering
how cheaply and easily you could build your own. Of course, if you do build your own, you're going to want to know a few specs so you can answer all your friends' questions, like "how much beer does
it hold?" and "how fast does the beer come out?"
Though store-bought beer bongs ensure there are two scantily clad ladies for every man in attendance...
Let's start with the volume. For calculation purposes, we'll split the beer bong into three parts: the hose, the conical part of the funnel, and the cylindrical top of the funnel (if you have one).
To simplify things, we'll assume that the little exit tube at the bottom of the funnel is just part of the hose. The volumes of beer in the hose and the cylindrical portion of the funnel are
calculated using the cylindrical volume formula:
where D and L are the diameter and length of the cylinder, respectively. The conical part of the funnel is a cone with the tip cut off, known as a conical frustum. The volume of a conical frustum is:
where L is the height of the frustum and D1 and D2 are the two end diameters. So, defining our beer bong geometry as in the figure below
Notation for beer bong geometry.
the volume of beer in the beer bong is calculated as
where the D's and L's are in centimetres and V is in millilitres. If your funnel doesn't have a "hopper" portion, Lfh = 0. Divide V by 341 mL/bottle if you want to know how many bottles the full beer
bong is equivalent to.
If you prefer US Customary units, use this formula instead:
where the D's and L's are in inches and V is in US fluid ounces.
So that's the answer to "how much beer does it hold?" Now for some fluid mechanics to calculate the beer velocity. We're just going to calculate the initial velocity, which is when the beer flows
fastest. As the beer drains, the flow rate slows down. We start with the energy equation for fluid flow, which is
where the subscripts
represent two points along the flow path,
is the flow velocity (we already used
for volume),
is the acceleration due to gravity (= 9.807 m/s
= 32.18 ft/s
is the height above some arbitrary reference point,
is the static fluid pressure,
is the fluid's specific weight,
is the kinetic energy correction factor, and
is the loss of hydraulic head from from point 1 to point 2.
For the beer bong, we're interested in the exit velocity, so we'll make the end of the hose point 2. We'll make the surface of the beer in the funnel point 1. Both point 1 and point 2 are exposed to
atmosphere, so we can take
to be equal and they cancel out of the equation. Because the reference point for z is arbitrary, we can choose the exit from the beer bong to be the reference point, making
equal to zero.
Now the energy equation looks like this
According to the law of conservation of mass for incompressible fluids, the flow rate (unit volume per unit time) of the beer must be constant. We can use this to relate
We can measure
, it's just how high the surface of the beer is above the end of the hose. If the hose is fully straightened,
is maximized, equal to
. Now if you want to use an easy approximation, you can assume that the head losses are negligible and the energy correction factors are both equal to 1.0. With these assumptions we can solve for the
exit velocity directly
is in m/s if
is in metres and
is 9.807 m/s
. You can use any units you want for
as long as you use the same units for both (e.g. you can't put one in inches and the other in millimetres). For velocity in cm/s,
is in cm and
= 980.7 cm/s
. For velocity in ft/s,
is in ft and
= 32.18 ft/s
If you want volumetric flow rate, just add the following calculation step:
If you use centimetres for
and cm/s for
(1 m = 100 cm), the flow rate will work out in mL/s. If you stick with metres for everything you'll get a pretty small number for
because the units will be in m
/s (1 m
= 1000 L).
We can make this approximation even easier by assuming the funnel diameter is much larger than the hose diameter (which is probably true), making the denominator of the above velocity expression very
nearly 1.0, and the exit velocity is simply
As you'll see later in an example, ignoring energy losses isn't going to give you a very accurate answer, and your friends aren't going to accept some lousy ballpark estimate. So the math's going to
get more intense, but I paid good money for my fluid mechanics course in university and I'll be damned if I don't find a way to put that knowledge to use.
It's safe to assume turbulent flow in the hose, and although the beer is moving much more slowly in the funnel, it's probably also in the turbulent flow regime. The
kinetic energy correction factor
varies depending on fluid velocity, viscosity, and pipe roughness, but a typical number is about 1.05. I'm going to assume
= 1.05.
I'm going to assume hydraulic head losses come from two sources, friction inside the hose, and the flow contraction at the funnel cone. I'm ignoring the effect of the bend in the hose, which should
be okay if the bend radius is much larger than the hose diameter, but if you put a tight bend in the hose you are introducing additional losses. For the flow contraction,
is the flow velocity leaving the contraction and
is an empirical factor that depends on the shape of the contraction. For typical funnel geometry,
should be around 0.07 to 0.08. I'll use 0.08. We can define a third point on the flow path, the exit of the funnel/entrance to the hose, but we won't need to do much with it. Since we've already
assumed the funnel exit has the same diameter as the hose, the velocity at point 3 must be equal to the velocity at point 2 in order to satisfy the mass conservation law. Therefore, the head loss
from point 1 to point 3 is
The friction loss in the hose is calculated using the
Darcy-Weisbach equation
is the
Darcy friction factor
is the pipe length,
is the pipe diameter, and
is the flow velocity in the pipe. For our beer bong, the head loss from point 3 to point 2 is
Now things start to get more complicated. The friction factor is an empirical value that's a function of the pipe roughness, diameter, flow velocity, and fluid viscosity. But your beer bong's
probably going to be made with smooth plastic or rubber hose, so we can assume a perfectly smooth pipe (i.e. ignore pipe roughness). If you're doing something crazy like using cast iron or bamboo or
hollowed out whalebone, you might want to consider the pipe roughness.
The Colebrook equation is typically cited for calculating the friction factor, but it's an implicit equation, meaning you can't solve the friction factor directly, you need to solve using an
iterative process. For a perfectly smooth pipe, the Colebrook equation is
where Re is the
Reynolds number
, which for the beer bong hose is equal to
is the
kinematic viscosity
of the beer, typically about 1.8
/s, or 0.018 cm
/s. Thus,
Since we still don't know what
is yet, we have iterations upon iterations on our hands here. Good thing Excel can do all that for you. But if you want to solve it by hand you could cut down on your iterations by using
Haaland's approximation
of the Colebrook equation.
You could also use the approximate value of
to calculate the friction factor. It should get you a friction factor that's fairly close to the "exact" solution, so we can create a beastly-looking equation for
that will get pretty close to the same answer as the iterative solution from Excel.
Substituting values of 980.7 cm/s
and 0.018 cm
/s for
, and the
's are in centimetres so that
comes out in cm/s.
Let's do an example based on a hypothetical beer bong with some clever name like
The Brain Cell Slayer
, depicted in the figure below.
Volume Initial velocity and flow rate (approximate solution) Initial velocity and flow rate (more accurate solution)
Even though it's a smooth hose, friction has a pretty significant effect in this example, reducing the initial velocity of the beer by almost 40%. And just so you can see that our beastly-looking
equation really does get you close to the "exact" iterative solution, here's what I get from Excel using the Colebrook equation for the friction factor:
References Potter, M.C. and Wiggert, D.C. (2002). Mechanics of Fluids, 3rd Edition. Brooks/Cole, Pacific Grove, CA.
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IEIE Transactions on Smart Processing & Computing
This paper proposes a new imaging geometry model for multi-receiver synthetic aperture sonar (SAS). The model considers the change of the speed of sound in seawater, the effect of platform movement
on the acoustic velocity vector (AVV), and the Doppler effect. Based on the proposed model, a solution to determine the phase distribution was generated to improve the SAS image quality. The
simulation results demonstrate the merits of proposed model compared to the traditional models that consider the speed of sound in seawater as a fixed value, ignore the change of AVV during
transmission, and suppress the Doppler effect. | {"url":"http://ieiespc.org/ieiespc/XmlViewer/f408035","timestamp":"2024-11-05T06:20:26Z","content_type":"application/xhtml+xml","content_length":"245496","record_id":"<urn:uuid:f844058f-c668-42b0-875a-b78e24ee7747>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00337.warc.gz"} |
Business and Financial Mathematics
• Understand terminology used in bonds and bond transactions.
A marketable bond is a debt that is secured by a specific corporate asset, that establishes the issuer’s responsibility toward a creditor for paying interest at regular intervals, and for repaying
the principal at a fixed later date. A debenture is the same as a marketable bond, except that the debt is not secured by any specific corporate asset. Mathematically, the calculations are identical
for these two financial tools, which this textbook refers to as bonds for simplicity.
Bond Terminology
• Issue Date. The bond issue date is the date that the bond is issued and available for purchase by creditors. Interest accrues from this date.
• Face Value. Also called the par value or denomination of the bond, the face value is the principal amount of the debt. It is what the investor lent to the bond-issuing corporation. The amount,
usually a multiple of $100, is found in small denominations up to $10,000 for individual investors and larger denominations up to $50,000 or more for corporate investors.
• Coupon Rate. Also known as the bond rate or nominal rate, the coupon rate is the nominal interest rate paid on the face value of the bond. The coupon rate is fixed for the life of the bond. Most
commonly the interest is calculated semi-annually and payable at the end of every six-month period over the entire life of the bond, starting from the issue date. All coupon rates used in this
textbook are assumed to be semi-annually compounded, unless stated otherwise.
• Yield Rate. The yield rate, or market rate, is the prevailing nominal rate of interest in the open bond market. Because bonds are actively traded, this rate fluctuates based on economic and
financial conditions. On the issue date, the market rate determines the coupon rate that is tied to the bond. Market rates are usually compounded semi-annually, as will be assumed in this
textbook unless otherwise stated. Therefore, marketable bonds form ordinary simple annuities, because the interest payments and the market rate are both compounded semi-annually, and the payments
occur at the end of the interval.
• Redemption Value. Also called the redemption price or maturity value, the redemption value is the amount the bond issuer will pay to the bondholder upon maturity of the bond. The redemption price
normally equals the face value of the bond, in which case the bond is said to be “redeemable at par” because interest on the bond has already been paid in full periodically throughout the term,
leaving only the principal in the account. In some instances a bond issuer may in fact redeem the bond at a premium, which is a price greater than the face value. The redemption price is then
stated as a percentage of the face value, such as 103%. For introductory purposes, this textbook sticks to the most common situation, where the redemption price equals the face value.
• Maturity Date. Also known as the redemption date or due date, the maturity date is the day upon which the redemption price will be paid to the bondholder (along with the final interest payment),
thereby extinguishing the debt.
• Selling Date. The date that a bond is actively traded and sold to another investor through the bond market is known as the selling date. In the timeline, the selling date can appear anywhere on
the timeline between the issue date and maturity date, and it may occur more than once as the bond is sold by one investor after another.
• Purchase Price. The purchase price is the price the bond holder pays to purchase the bond. Although the redemption value and the periodic interest payments remain fixed throughout the lifetime
of the bond, the purchase price fluctuates depending on various market conditions, such as the current yield rate.
A $10,000 bond was issued on March 1, 2019 with a 7% coupon and 16 years to maturity. The bond was purchased on July 13, 2023 when the yield to maturity was 5%.
• Face Value (or Redemption Value) = $10,000
• Issue Date = March 1, 2019
• Purchase Date (or Selling Date) = July 23, 2023
• Maturity Date (or Redemption Date) = March 1, 2035
• Coupon Rate = 7% compounded semi-annually
• Yield Rate = 5% compounded semi-annually
1. Unless otherwise stated, the coupon rate and the yield rate are compounded semi-annually and the coupon payments are made every six months.
2. In this chapter we only deal with bonds that are redeemed for their face value at the time of maturity. So, the redemption value of the bond will equal its face value.
Premium, Discount, and At Par Bonds
The price at which a bond is purchased in the market may not be the face value of the bond. In fact, the price of a bond fluctuates with the market rate over time. There are three scenarios relating
the purchase price of the bond to its face value.
• Bonds purchased at par. The purchase price of the bond equals its face value. This happens when the coupon rate equals the yield rate.
• Bonds purchased at premium. The purchase price of the bond is higher than its face value. This happens when the coupon rate is greater than the yield rate. In such cases, the bond is providing
a higher rate of return (through the coupons) than an investment in the market (earning the lower yield rate). Consequently, the bond will be in demand and will sell at a higher price that its
face value. The difference between the purchase price and the face value is called the premium.
• Bonds purchased at discount. The purchase price of the bond is lower than its face value. This happens when the coupon rate is less than the yield rate. In such cases, the bond is providing a
lower rate of return (through the coupons) than an investment in the market (earning the higher yield rate). Consequently, the bond will not be in demand and will sell at a lower price that its
face value. The difference between the face value and the purchase price is called the discount.
Figure 7.1.1
Identify each of the following bonds as premium, discount, or at par.
1. A $5,000 bond with a 3% coupon is purchased when there are five years to maturity and the yield to maturity is 4.5%.
2. A $12,000 bond with a 4% coupon is purchased when there are ten years to maturity and the yield to maturity is 4%.
3. A $7,000 bond with a 6% coupon is purchased when there are twelve years to maturity and the yield to maturity is 3.7%.
1. Discount because the coupon rate (3%) is less than the yield rate (4.5%)
2. At par because the coupon rate and yield rate are equal.
3. Premium because the coupon rate (6%) is higher than the yield rate (3.7%).
Calculating the Bond Payment
The bond payment, or coupon payment, is the payment the bond holder received semi-annually throughout the investment. This interest is not converted to the principal of the bond, and therefore does
not compound. Instead, the bond payment is directly paid to the bond holder. The amount of the bond payment depends only on the face value of the bond and the coupon rate.
[latex]\displaystyle{\mbox{Bond Payment}=FV \times b}[/latex]
• [latex]FV[/latex] is the face value of the bond when the bond is redeemed at maturity.
• [latex]b[/latex] is periodic coupon rate where [latex]b=\frac{\mbox{coupon rate}}{\mbox{number of coupons per year}}[/latex]
A $5,000 bond has a 3% coupon. Calculate the bond payment.
[latex]\begin{eqnarray*} \mbox{Bond Payment} & = & FV \times b \\ & = & 5,000 \times \frac{0.03}{2} \\ & = & $75 \end{eqnarray*}[/latex]
1. Remember, the bond payments and coupon rate are assumed to be semi-annual, unless stated otherwise.
2. For the bond in the previous example, the bond holder will receive a $75 payment at the end of every six months over the life time of the bond.
A $12,000 bond has a 7% coupon. Calculate the bond payment.
Click to see Solution
[latex]\begin{eqnarray*} \mbox{Bond Payment} & = & FV \times b \\ & = & 12,000 \times \frac{0.07}{2} \\ & = & $420 \end{eqnarray*}[/latex]
“14.1: Determining the Value of a Bond” from Business Math: A Step-by-Step Handbook (2021B) by J. Olivier and Lyryx Learning Inc. through a Creative Commons Attribution-NonCommercial-ShareAlike 4.0
International License unless otherwise noted. | {"url":"https://ecampusontario.pressbooks.pub/businessfinancialmath/chapter/7-1-bond-terminology/","timestamp":"2024-11-05T09:50:25Z","content_type":"text/html","content_length":"88230","record_id":"<urn:uuid:1cdbd106-0b75-4eb1-90db-49d18b970e81>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00789.warc.gz"} |
Quickstart: Zero to Python
Brand new to Python? Here are some quick examples of what Python code looks like.
This is not meant to be a comprehensive Python tutorial, just something to whet your appetite.
Run this code from your browser!
Of course you can simply read through these examples, but it’s more fun to run them yourself:
• Find the “Rocket Ship” icon, located near the top-right of this page. Hover over this icon to see the drop-down menu.
• Click the Binder link from the drop-down menu.
• This page will open up as a Jupyter notebook in a working Python environment in the cloud.
• Press Shift+Enter to execute each code cell
• Feel free to make changes and play around!
A very first Python program
A Python program can be a single line:
Loops in Python
Let’s start by making a for loop with some formatted output:
for n in range(3):
print(f"Hello interweb, this is iteration number {n}")
Hello interweb, this is iteration number 0
Hello interweb, this is iteration number 1
Hello interweb, this is iteration number 2
A few things to note:
• Python defaults to counting from 0 (like C) rather than from 1 (like Fortran).
• Function calls in Python always use parentheses: print()
• The colon : denotes the beginning of a definition (here of the repeated code under the for loop).
• Code blocks are identified through indentations.
To emphasize this last point, here is an example with a two-line repeated block:
for n in range(3):
print("Hello interweb!")
print(f"This is iteration number {n}.")
print('And now we are done.')
Hello interweb!
This is iteration number 0.
Hello interweb!
This is iteration number 1.
Hello interweb!
This is iteration number 2.
And now we are done.
Basic flow control
Like most languages, Python has an if statement for logical decisions:
if n > 2:
print("n is greater than 2!")
print("n is not greater than 2!")
Python also defines the True and False logical constants:
There’s also a while statement for conditional looping:
m = 0
while m < 3:
print(f"This is iteration number {m}.")
m += 1
print(m < 3)
This is iteration number 0.
This is iteration number 1.
This is iteration number 2.
Basic Python data types
Python is a very flexible language, and many advanced data types are introduced through packages (more on this below). But some of the basic types include:
Integers (int)
The number m above is a good example. We can use the built-in function type() to inspect what we’ve got in memory:
Floating point numbers (float)
Floats can be entered in decimal notation:
or in scientific notation:
where 4e7 is the Pythonic representation of the number \( 4 \times 10^7 \).
Character strings (str)
You can use either single quotes '' or double quotes " " to denote a string:
A list is an ordered container of objects denoted by square brackets:
mylist = [0, 1, 1, 2, 3, 5, 8]
Lists are useful for lots of reasons including iteration:
for number in mylist:
Lists do not have to contain all identical types:
myweirdlist = [0, 1, 1, "apple", 4e7]
for item in myweirdlist:
<class 'int'>
<class 'int'>
<class 'int'>
<class 'str'>
<class 'float'>
This list contains a mix of int (integer), float (floating point number), and str (character string).
Because a list is ordered, we can access items by integer index:
remembering that we start counting from zero!
Python also allows lists to be created dynamically through list comprehension like this:
squares = [i**2 for i in range(11)]
[0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100]
Dictionaries (dict)
A dictionary is a collection of labeled objects. Python uses curly braces {} to create dictionaries:
mypet = {
"name": "Fluffy",
"species": "cat",
"age": 4,
We can then access items in the dictionary by label using square brackets:
We can iterate through the keys (or labels) of a dict:
for key in mypet:
print("The key is:", key)
print("The value is:", mypet[key])
The key is: name
The value is: Fluffy
The key is: species
The value is: cat
The key is: age
The value is: 4
Arrays of numbers with NumPy
The vast majority of scientific Python code makes use of packages that extend the base language in many useful ways.
Almost all scientific computing requires ordered arrays of numbers, and fast methods for manipulating them. That’s what NumPy does in the Python world.
Using any package requires an import statement, and (optionally) a nickname to be used locally, denoted by the keyword as:
Now all our calls to numpy functions will be preceeded by np.
Create a linearly space array of numbers:
# linspace() takes 3 arguments: start, end, total number of points
numbers = np.linspace(0.0, 1.0, 11)
array([0. , 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1. ])
We’ve just created a new type of object defined by NumPy:
Do some arithmetic on that array:
array([1. , 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2. ])
Sum up all the numbers:
Some basic graphics with Matplotlib
Matplotlib is the standard package for producing publication-quality graphics, and works hand-in-hand with NumPy arrays.
We usually use the pyplot submodule for day-to-day plotting commands:
import matplotlib.pyplot as plt
Define some data and make a line plot:
theta = np.linspace(0.0, 360.0)
sintheta = np.sin(np.deg2rad(theta))
plt.plot(theta, sintheta, label='y = sin(x)', color='purple')
plt.title('Our first Pythonic plot', fontsize=14)
Text(0.5, 1.0, 'Our first Pythonic plot')
What now?
That was a whirlwind tour of some basic Python usage.
Read on for more details on how to install and run Python and necessary packages on your own laptop. | {"url":"https://foundations.projectpythia.org/foundations/quickstart.html","timestamp":"2024-11-05T00:41:49Z","content_type":"text/html","content_length":"67109","record_id":"<urn:uuid:71df28d7-38fa-4ec8-81e5-a5aa9fe8c14c>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00852.warc.gz"} |
Delta 8 Vape How Many Puffs
The best approach to vaping delta 8 THC for the first time is to take between 2–3 puffs and wait for about 30 minutes to see how it makes you feel. The effects can last for up to 3 hours, with a peak
after 1–1.5 hours from inhalation.May 17, 2021
How many puffs of Delta8 vape oil should I use?
Sep 23, 2021 · That means that each cart has approximately 950 mg of Delta-8-THC. A vape cart may deliver anywhere from 150 to 300 individual doses (single puffs) depending on the length of each
dose. We recommend smaller puffs, which means you will get 300 doses out of each cart, and each puff will deliver about 3 mg of Delta-8.
How many milligrams of Delta 8 THC are in a puff?
Oct 17, 2021 · In each ml of Delta-8 THC e-liquid juice, there are nearly 100 puffs. The amount you have in your vape should be divided by the mg of the compound. Then, divide it by 100. That’s the
dose you get per puff. Delta-8 THC herb dosage. Smoking …
How many puffs of Delta 8 flower per day?
Users can expect a Delta-8 vape high to last anywhere from 30 minutes to 2 hours from your last hit. This number will depend on a variety of factors, including how much you smoked, your individual
tolerance, and any other intoxicants (like alcohol) in your system that may amplify your high feeling. The high from a Delta-8 vape cart is usually much shorter than the high that …
How many Delta 8 blends does Delta 8 have?
Mar 02, 2022 · How Many Puffs in a Delta 8 Vape Cart? Take good care of your cartridges, and you should get around 300 puffs off of a 1 gram cartridge and half that from a 0.5 gram cartridge. A vape
cart that doesn’t work right can change that number, though.
How many puffs of Delta-8 should you take?
Delta 8 THC flower is just CBD hemp flower infused with a delta 8 THC distillate. Like vapes, you should take 1.5 milligrams, or 1–3 puffs, as a good amount to start with. However, delta 8 THC
flowers are more difficult to dose because the delta 8 may not be evenly distributed throughout the bud.May 17, 2021
How long should a Delta-8 cartridge last?
If stored correctly, delta 8 can last up to 24 months, though it can also go bad in a couple of months from the date of manufacture.Aug 23, 2021
How many hits can you get from a Delta-8 disposable?
How many puffs are in a Delta-8 disposable? The number of puffs in a disposable depends on the brand and on the length of each hit, but it's normally around 300-600.Jul 6, 2021
How long does delta 8 vape high last?
Delta 8 THC Vape Carts: Vaping offers the fastest onset effects, starting in about 6 minutes. The time delta 8 THC needs to kick in through inhalation depends on the person and the product used. The
peak effects can be noticed within 30 minutes to 2.5 hours after consumption, lasting up to 5 hours.Jan 14, 2022
How many puffs are in a cigarette?
FTC method at the Tobacco and Health Research Institute in Lexington, Kentucky. Two hundred cigarettes were smoked; the average number of puffs taken per cigarette was 6.8. This same procedure was
repeated with a high- yield cigarette, Camel, and an average of 8.3 puffs was taken.
Will Delta 8 get you high?
As a psychoactive substance, delta-8 THC can get you high. However, this high will not be as intense as that produced by the regular THC variant. Many people who need their dose of “high” use delta-8
as a substitute for THC, since the latter is not legal in several states.Apr 13, 2022
How often should I vape Delta 8?
The best approach to vaping delta 8 THC for the first time is to take between 2–3 puffs and wait for about 30 minutes to see how it makes you feel. The effects can last for up to 3 hours, with a peak
after 1–1.5 hours from inhalation.May 17, 2021
Is Delta 8 vape safe?
Although there aren't many studies that would analyze the safety profile of delta 8, current findings indicate that there aren't any considerable health risks associated with D8 THC oil tincture
products and even gummies or vapes.Mar 4, 2022
How to time your Delta-8 vape cart high
The real beauty of smoking Delta 8 Vape Carts is that you can dial in your dose quickly.
How long does a Delta-8 vape high last?
Users can expect a Delta-8 vape high to last anywhere from 30 minutes to 2 hours from your last hit. This number will depend on a variety of factors, including how much you smoked, your individual
tolerance, and any other intoxicants (like alcohol) in your system that may amplify your high feeling.
Tips for new Delta-8 vape users
When it comes to taking Delta-8 for the first time, vapes can be the most clear-cut option in terms of evaluating whether Delta-8 THC is a good choice for you.
What is the ideal dosing for Delta-8 vape carts?
The thing about carts is that while you have reasonable control over each “hit”, it’s difficult to determine the amount of mg’s each puff has and a recommended dose.
Ready to give Delta-8 a try?
So our advice? Take it slow if you’re a first time user! Try it out for yourself and see what works for your schedule and your body. You may find that taking less during the day leads to a more
energized and clear-headed high while taking more at night allows you to sleep better.
How many puffs of delta8 vape oil?
Vape carts are pre-filled cartridges containing delta8 vape oil. You’ll again need to start out with 1 to 3 puffs depending on your unique THC tolerance. Also, use the same formula used for vape oils
to determine how many milligrams are in a puff.
What is delta 8?
More people are discovering that delta-8 is the hemp cannabinoid that has been missing from their daily routines. This fast-growing submarket is exploding presently, offering a wide array of unique
products infused with this minorly occurring cannabinoid. The reason for its success largely has to do with developments within the hemp industry, which have made it easier to isolate and extract
delta8 from the hemp plant material to concentrate it in a way that allows us to enjoy formulas providing the full potential of its properties.
Is delta 8 the same as cbd?
But you should know that the effects of delta 8 are quite different from those of CBD. Delta 8 is mildly intoxicating, meaning that it provides mind-altering effects, although they are certainly more
subtle than those provided by delta 9. Still, delta 8 can be soothing in a way that one must consider before starting a routine.
Is delta 8 dangerous?
We should mention that there is no reason to worry about taking a fairly high amount of delta 8, as it has not been shown to produce dangerous or toxic effects. But some people are sensitive to THC
and may find themselves feeling groggy or generally out of it if they choose too high a strength or dosage level.
How many mg is Delta 8?
Delta-8 tinctures typically range from 30 to 60mg per dose, with a single dose being measured in the dropper. Simply divide the number of milligrams in the tincture bottle by the number of
milliliters to determine how many milligrams are in each dropper’s worth.
Does delta8 have psychoactive effects?
These topicals will likely not give you psychoactive effects as they don’t cross the blood brain barrier, so you can use them more liberally.
Is delta 8 a psychoactive drug?
Still, delta 8 can be soothing in a way that one must consider before starting a routine. The more delta 8 you take, the stronger these psychoactive effects will be. | {"url":"https://vape-faq.com/delta-8-vape-how-many-puffs","timestamp":"2024-11-04T05:32:29Z","content_type":"text/html","content_length":"40270","record_id":"<urn:uuid:651721ff-1501-4a5e-af7b-0b00c68e938f>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00589.warc.gz"} |
Visual Insight
An octic surface is one defined by a polynomial equation of degree 8. This image by Abdelaziz Nait Merzouk shows an octic discovered by Chmutov with 154 real ordinary double points or nodes: that is,
points where it looks like the origin of the cone in 3-dimensional space defined by $x^2 + y^2 = z^2$.
Escudero Nonic
A nonic surface is one defined by a polynomial equation of degree 9. This image by Juan García Escudero shows a nonic surface called \(Q_9\), which has 220 real ordinary double points: that is,
points where it looks like the origin of the cone in 3-dimensional space defined by \(x^2 + y^2 = z^2\).
Togliatti Quintic
A quintic surface is one defined by a polynomial equation of degree 5. A nodal surface is one whose only singularities are ordinary double points: that is, points where it looks like the origin of
the cone in 3-dimensional space defined by \(x^2 + y^2 = z^2\). A Togliatti surface is a quintic nodal surface with the largest possible number of ordinary double points, namely 31. Here Abdelaziz
Nait Merzouk has drawn the real points of a Togliatti surface.
Kummer Quartic
A quartic surface is one defined by a polynomial equation of degree 4. An ordinary double point is a point where a surface looks like the origin of the cone in 3-dimensional space defined by $x^2 + y
^2 = z^2$. The Kummer surfaces are the quartic surfaces with the largest possible number of ordinary double points, namely 16. This picture by Abdelaziz Nait Merzouk shows the real points of a Kummer
Cayley’s Nodal Cubic Surface
A cubic surface is one defined by a polynomial equation of degree 3. Cayley’s nodal cubic surface, drawn above by Abdelaziz Nait Merzouk, is the cubic surface with the largest possible number of
ordinary double points and no other singularities: that is, points where it looks like the origin of the cone in 3-dimensional space defined by \(x^2 + y^2 = z^2\). It has 4 ordinary double points,
shown here at the vertices of a regular tetrahedron.
Endrass Octic
An octic surface is one defined by a polynomial equation of degree 8. The Endrass octic, drawn above by Abdelaziz Nait Merzouk, is currently the octic surface with the largest known number of
ordinary double points: that is, points where it looks like the origin of the cone in 3-dimensional space defined by \(x^2 + y^2 = z^2\). It has 168 ordinary double points, while the best known upper
bound for a octic surface that’s smooth except for such singularities is 174.
Labs Septic
A septic surface is one defined by a polynomial equation of degree 7. The Labs septic, drawn above by Abdelaziz Nait Merzouk, is a septic surface with the maximum possible number of ordinary double
points: that is, points where it looks like the origin of the cone in 3-dimensional space defined by \( x^2 + y^2 = z^2\).
Barth Decic
A decic surface is one defined by a polynomial equation of degree 6. The Barth decic, drawn here by Abdelaziz Nait Merzouk, is the decic surface with the maximum possible number of ordinary double
points: that is, points where it looks like the origin of the cone in 3-dimensional space defined by \(x^2 + y^2 = z^2 \).
Discriminant of Restricted Quintic
This image by Greg Egan shows the set of points \((a,b,c)\) for which the quintic \(x^5 + ax^4 + bx^2 + c \) has repeated roots. The plane \(c = 0 \) has been removed. This surface is connected to
involutes of a cubical parabola and the discriminant of the icosahedral group.
Discriminant of the Icosahedral Group
This image, created by Greg Egan, shows the ‘discriminant’ of the symmetry group of the icosahedron. This group acts as linear transformations of \(\mathbb{R}^3\) and thus also \(\mathbb{C}^3\). By a
theorem of Chevalley, the space of orbits of this group action is again isomorphic to \(\mathbb{C}^3\). Each point in the surface shown here corresponds to a ‘nongeneric’ orbit: an orbit with fewer
than the maximal number of points. More precisely, the space of nongeneric orbits forms a complex surface in \(\mathbb{C}^3\), called the discriminant, whose intersection with \(\mathbb{R}^3\) is
shown here. | {"url":"https://blogs.ams.org/visualinsight/category/surfaces/","timestamp":"2024-11-04T21:18:26Z","content_type":"text/html","content_length":"67763","record_id":"<urn:uuid:1366147e-1bc6-48fa-96a1-7f03df041484>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00526.warc.gz"} |
Single-axis gyroscopic motion with uncertain angular velocity about spin axis
A differential game approach is presented for studying the response of a gyro by treating the controlled angular velocity about the input axis as the evader, and the bounded but uncertain angular
velocity about the spin axis as the pursuer. When the uncertain angular velocity about the spin axis desires to force the gyro to saturation a differential game problem with two terminal surfaces
results, whereas when the evader desires to attain the equilibrium state the usual game with single terminal manifold arises. A barrier, delineating the capture zone (CZ) in which the gyro can attain
saturation and the escape zone (EZ) in which the evader avoids saturation is obtained. The CZ is further delineated into two subregions such that the states in each subregion can be forced on a
definite target manifold. The application of the game theoretic approach to Control Moment Gyro is briefly discussed.
ASME Journal of Dynamic Systems and Measurement Control B
Pub Date:
December 1977
□ Angular Velocity;
□ Game Theory;
□ Gyroscopes;
□ Spacecraft Motion;
□ Spin Dynamics;
□ Control Moment Gyroscopes;
□ Differential Equations;
□ Euler-Lagrange Equation;
□ Mathematical Models;
□ Minimax Technique;
□ Instrumentation and Photography | {"url":"https://ui.adsabs.harvard.edu/abs/1977ATJDS..99..259S/abstract","timestamp":"2024-11-08T02:22:53Z","content_type":"text/html","content_length":"35837","record_id":"<urn:uuid:1fde5ed1-dc6d-45d8-ba58-bb02f3949401>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00894.warc.gz"} |
Beta kernel and transformed kernel
This Thursday I will give a talk at Laval University, on “Beta kernel and transformed kernel : applications to copula density estimation and quantile estimation“. This time, I will talk at the
department of Mathematics and Statistics (13:30 at the pavillon Adrien-Pouliot). “Because copulas have bounded support (the unit square in dimension 2), standard kernel based estimators of densities
are (multiplicatively) biased on borders and in corners of the support. Two techniques can be used to avoid that underestimation: Beta kernels and Transformed kernel. We will describe and discuss
those two techniques in the first part of the talk. Then, we will see that it is possible to combine those two techniques to get nice estimator of several quantities (e.g. quantiles): transform the
data to get on the unit interval – using a transformed kernel – then estimate the (transformed) quantile on [0,1] using a beta kernel, then get back on the initial support. As we will see on
simulations, that technique can be better than standard quantile estimators, especially when data are heavy tailed.” Slides can be downloaded here.
• kernel based density estimation
Kernel based estimation are a popular (and natural) technique to estimate densities. It is simply and extension of the moving histogram:
so we count how many observations are a the neighborhood of the point where we want to estimate the density of the distribution. Then it is natural so consider a smoothing function, i.e. instead of a
step function (either observations are close enough, or not), it is possible to give weights to observations, which will be a decreasing function of the distance,
With a smooth kernel, we have a smooth estimation of the density
Then it is possible to play on the bandwidth, either to get a more accurate estimation of the density, but not that smooth (small bias but large variance),
or a smoother one (large bias, but small variance),
In R, it is simply
> X=rnorm(100)
> (D=density(X))
density.default(x = X)
Data: X (100 obs.); Bandwidth 'bw' = 0.3548
x y
Min. :-3.910799 Min. :0.0001265
1st Qu.:-1.959098 1st Qu.:0.0108900
Median :-0.007397 Median :0.0513358
Mean :-0.007397 Mean :0.1279645
3rd Qu.: 1.944303 3rd Qu.:0.2641952
Max. : 3.896004 Max. :0.3828215
> plot(D$x,D$y)
The idea of Beta kernel is to consider kernels having support [0,1]. In the univariate case,
For additional material, I have uploaded some R code to fit copula densities using beta kernels,
beta.kernel.copula.surface = function (u,v,bx,by,p) {
s = seq(1/p, len=(p-1), by=1/p)
mat = matrix(0,nrow = p-1, ncol = p-1)
for (i in 1:(p-1)) {
a = s[i]
for (j in 1:(p-1)) {
b = s[j]
mat[i,j] = sum(dbeta(a,u/bx,(1-u)/bx) *
dbeta(b,v/by,(1-v)/by)) / length(u)
} }
return(data.matrix(mat)) }
Then we can used it to see what we get on a simulated sample
COPULA = frankCopula(param=5, dim = 2)
X = rcopula(n=1000,COPULA)
p0 = 26
Z= beta.kernel.copula.surface(X[,1],X[,2],bx=.01,by=.01,p=p0)
u = seq(1/p0, len=(p0-1), by=1/p0)
(yes, the surface is changing… to illustrate the impact of the bandwidth on the estimation).
• transformed kernel estimation
here). I probably spend a few minutes on the original chapter, in order to provide another application of that techniques (not only to estimate copula densities, but here to estimate quantiles of
heavy tailed distribution). In the univariate case, the R code is the following (here I consider two transformation, the quantile function of the Gaussian distribution, and the quantile function of
the Student distribution with 3 degrees of freedom),
transfN = function(x){
transfT = function(x){
The density estimation is the following,
(the red dotted line is the true density, since we work on a simulated sample). Now, let us get back on the initial chapter,
The original idea we add it to use this kernel based estimator for copulas, i.e. since we can estimate densities in high dimension with unbounded support, using
the idea is to transform marginal observations,
and to use the fact that the associated copula density can be written
to derive an intuitive estimator for the copula density
An important issue is how do we choose the transformation
And Luc Devroye and Laszlo Györfi mention that this can be used to deal with extremes.
well, extremes are introduced through bumps (which is not the way I would have been dealing with extremes)
and note that several results can be derived on those bumps,
Then, there is an interesting discussion about estimating the optimal transformation
and I will prove that this can be an extremely interesting idea, for instance to estimate quantiles of heavy tailed distribution, if we use also the beta kernel estimator on the unit interval. This
idea was developed in a paper with Abder Oulidi, online here.
Remark: actually, in the book, an additional reference is mentioned,
but I have never been able to find a copy… if anyone has one, I’d be glad to read it…
OpenEdition suggests that you cite this post as follows:
Arthur Charpentier (April 19, 2011). Beta kernel and transformed kernel. Freakonometrics. Retrieved November 9, 2024 from https://doi.org/10.58079/ouhu
2 thoughts on “Beta kernel and transformed kernel”
1. how can we extend the code to multivariate (more than 2)? is it straightforward like this? like for dim 3 library(copula)
beta.kernel.copula.surface = function (u,v,k,bx, by,bz,p) {
s = seq(1/p, len=(p-1), by=1/p)
mat = matrix(0,nrow = p-1, ncol = p-1)
for (i in 1:(p-1)) {
a = s[i]
for (j in 1:(p-1)) {
b = s[j]
for (l in 1:(p-1)) {
mat[i,j,l] = sum(dbeta(a,u/bx,(1-u)/bx) *
dbeta(b,v/by,(1-v)/by)*dbeta(c,k/bz,(1-k)/bz) ) / length(u)
} }}
return(data.matrix(mat)) }
2. i need Parkinson,
Haberman’s survival, Satellite datasets run with r.
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The function π(π₯) = |2π₯ β 7| is not differentiable at
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Question Text The function π (π ₯) = |2π ₯ β 7| is not differentiable at
Updated On Jul 23, 2023
Topic Limit, Continuity and Differentiability
Subject Mathematics
Class Class 12
Answer Type Video solution: 1
Upvotes 109
Avg. Video Duration 5 min | {"url":"https://askfilo.com/user-question-answers-mathematics/the-function-2-7-is-not-differentiable-at-35333931393934","timestamp":"2024-11-05T12:34:26Z","content_type":"text/html","content_length":"451132","record_id":"<urn:uuid:bf8e5ff1-b72d-4873-a9af-19ceaf61ecdd>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00264.warc.gz"} |
5 Steps To Graph Function Transformations In Algebra - 911 WeKnow
5 Steps To Graph Function Transformations In Algebra
1. Identify The Parent Function
• Everything you?re expected to graph on your own is based on a more basic graph (parent function) that you NEED to memorize
• Look at the image above and check with your teacher to see which you are responsible for
2. Reflect Over X-Axis or Y-Axis
• If there is a negative outside parentheses, then reflect over the x-axis, or vertically (all the y-values become negative)
• ex: f(x) = -x2
• If there is a negative inside parentheses, then reflect over the y-axis horizontally (all the x-values become negative) and factor out the negative
• ex: f(x) = 1 / (-x+3) becomes f(x) = 1 / (-(x-3)) **THIS IS TRICKY**
3. Shift (Translate) Vertically or Horizontally
• If there is a number being added outside parentheses, then shift it up vertically by that amount, or if the number is being subtracted, then shift it down
• ex: f(x) = x2?2 goes down 2
• If there is a number being added inside parentheses, then shift it left by that amount, or if the number is being subtracted, then shift it right (opposite what you?d expect)
• ex: f(x) = (x-2)2 is shifted to the right 2 units
4. Vertical and Horizontal Stretches/Compressions
• If there is a whole number coefficient outside parentheses, then multiply the y-values of all points by that coefficient and see the graph stretch vertically
• ex: f(x) = 4×2 is stretched vertically by a factor of 4
• If there is a fractional coefficient outside parentheses, then multiply the y-values of all points by that coefficient and see the graph compress vertically
• ex: f(x) = 1/2 x2 is compressed vertically by a factor of 2
• If there is a whole number coefficient inside parentheses, then multiply the x-values of all points by the inverse of (!) that coefficient and see the graph stretch horizontally (opposite of what
you?d expect)
• ex: f(x) = (4x)2 is compressed horizontally by a factor of 4
• If there is a fractional coefficient inside parentheses, then multiply the x-values of all points by the inverse of (!) that coefficient and see the graph compress horizontally (opposite of what
you?d expect)
• ex: f(x) = (1/2 x ? 4)2 becomes (1/2 (x ? 8))2 is stretched horizontally by a factor of 2
5. Plug in a couple of your coordinates into the parent function to double check your work
• REMEMBER: A GRAPH IS JUST A SET OF POINTS THAT SATISFY AN EQUATION
• That means you can always check your work by plugging in an x-value (I recommend x=0, and seeing if the y-value fits the y-value of your graph)
1. Reflect > Shift > Stretch
2. Inside parentheses = think opposite for stretches and shifts and FACTOR (if necessary)
3. Inside parentheses = horizontal changes (flip over y-axis)
4. Outside parentheses = vertical changes (flip over x-axis) | {"url":"https://911weknow.com/5-steps-to-graph-function-transformations-in-algebra","timestamp":"2024-11-10T11:02:08Z","content_type":"text/html","content_length":"41649","record_id":"<urn:uuid:e6cf1b83-5ac8-4a49-af2e-0ae976cfbada>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00375.warc.gz"} |
There are several operators available in Structured Text language:
Operation Symbol Precedence
Parenthesization (expression) Highest
Negation -
Complement !
Multiply *
Divide /
Modulo %
Add +
Subtract -
Left Shift <<
Right Shift >>
Comparison <, >, <=, >=,==,!=
Boolean AND &
Boolean OR || Lowest
Boolean XOR ^
All the operators in the table above are sorted after precedence. This is also called order of operations, and you may know about if from mathematics.The order of operations is the order in which the
operations are executed or calculated. Just take a look at this expression:
A + B * C
How will this expression be evaluated by the compiler? There are two operations left: multiply and addition. But since multiply has a higher precedence, that will be the first to be evaluated. B * C
comes first and then the result is added to A. Every time an expression is evaluated, the evaluation follows the order of precedence as in the table above.
4 Types of Operators, 4 Types of Expressions
The operators used for expressions in Structured Text can be divided into four groups. Each group of operators will have its specific function and will yield a specific data type: | {"url":"https://teslascada.com/HTML/operators.html","timestamp":"2024-11-09T00:30:34Z","content_type":"text/html","content_length":"16318","record_id":"<urn:uuid:13499291-201d-4dff-aebb-d8922ca47466>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00381.warc.gz"} |
Conserved Charges in Higher Derivative Gravity
One of the very first predictions extracted from general relativity is the existence of black-hole solutions. Initially regarded as mathematical curiosities, over the decades black holes have been
taking a more and more central role in the study of gravity, both experimentally and theoretically. From the theoretical point of view, black holes are regarded as the natural testing ground for the
fundamental properties of gravity in the classical, semi-classical and even (to the extent that we understand it) quantum regime. This thesis investigates the effective field theory (EFT) extension
of gravity, specifically focusing on a theory that includes cubic-order (six-derivative) terms which preserve parity symmetry. Due to the complexity of finding the solutions in this advanced theory,
the research concentrates on a particular region of spacetime, the near-horizon extremal geometry (NHEG) of black holes. This thought comes from the Bekenstein-Hawking formula, which relates the
black hole entropy to its horizon area, indicating that the thermodynamic properties can be derived from the horizon. The extremality, also, is beneficial since it prepares us with more symmetries
than the exact solution. To have a compatible definition of entropy in the presence of higher-order terms, we use the Iyer-Wald entropy. By calculating entropy within this modified theory, the thesis
clarifies how the extension of the Lagrangian influences black hole thermodynamics. Additionally, the study develops a generalized Komar integral for this theory, providing valuable insights into the
calculation of the angular momentum. The main body of the work involves detailed calculations of entropy and angular momentum within the Einstein gravity framework and the modified gravity theory up
to the six-derivative terms. The findings reveal that these corrections impact the structure of the NHEG and modify the relationship between entropy and angular momentum.
One of the very first predictions extracted from general relativity is the existence of black-hole solutions. Initially regarded as mathematical curiosities, over the decades black holes have been
taking a more and more central role in the study of gravity, both experimentally and theoretically. From the theoretical point of view, black holes are regarded as the natural testing ground for the
fundamental properties of gravity in the classical, semi-classical and even (to the extent that we understand it) quantum regime. This thesis investigates the effective field theory (EFT) extension
of gravity, specifically focusing on a theory that includes cubic-order (six-derivative) terms which preserve parity symmetry. Due to the complexity of finding the solutions in this advanced theory,
the research concentrates on a particular region of spacetime, the near-horizon extremal geometry (NHEG) of black holes. This thought comes from the Bekenstein-Hawking formula, which relates the
black hole entropy to its horizon area, indicating that the thermodynamic properties can be derived from the horizon. The extremality, also, is beneficial since it prepares us with more symmetries
than the exact solution. To have a compatible definition of entropy in the presence of higher-order terms, we use the Iyer-Wald entropy. By calculating entropy within this modified theory, the thesis
clarifies how the extension of the Lagrangian influences black hole thermodynamics. Additionally, the study develops a generalized Komar integral for this theory, providing valuable insights into the
calculation of the angular momentum. The main body of the work involves detailed calculations of entropy and angular momentum within the Einstein gravity framework and the modified gravity theory up
to the six-derivative terms. The findings reveal that these corrections impact the structure of the NHEG and modify the relationship between entropy and angular momentum.
Conserved Charges in Higher Derivative Gravity
One of the very first predictions extracted from general relativity is the existence of black-hole solutions. Initially regarded as mathematical curiosities, over the decades black holes have been
taking a more and more central role in the study of gravity, both experimentally and theoretically. From the theoretical point of view, black holes are regarded as the natural testing ground for the
fundamental properties of gravity in the classical, semi-classical and even (to the extent that we understand it) quantum regime. This thesis investigates the effective field theory (EFT) extension
of gravity, specifically focusing on a theory that includes cubic-order (six-derivative) terms which preserve parity symmetry. Due to the complexity of finding the solutions in this advanced theory,
the research concentrates on a particular region of spacetime, the near-horizon extremal geometry (NHEG) of black holes. This thought comes from the Bekenstein-Hawking formula, which relates the
black hole entropy to its horizon area, indicating that the thermodynamic properties can be derived from the horizon. The extremality, also, is beneficial since it prepares us with more symmetries
than the exact solution. To have a compatible definition of entropy in the presence of higher-order terms, we use the Iyer-Wald entropy. By calculating entropy within this modified theory, the thesis
clarifies how the extension of the Lagrangian influences black hole thermodynamics. Additionally, the study develops a generalized Komar integral for this theory, providing valuable insights into the
calculation of the angular momentum. The main body of the work involves detailed calculations of entropy and angular momentum within the Einstein gravity framework and the modified gravity theory up
to the six-derivative terms. The findings reveal that these corrections impact the structure of the NHEG and modify the relationship between entropy and angular momentum.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.12608/70116 | {"url":"https://thesis.unipd.it/handle/20.500.12608/70116","timestamp":"2024-11-05T20:29:06Z","content_type":"text/html","content_length":"47534","record_id":"<urn:uuid:58386fee-dcde-4e09-ae4a-792e98eae93b>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00657.warc.gz"} |
M. Wilkens, F. Illuminati, and M. Kraemer
Transition temperature of the weakly interacting Bose gas: perturbative solution of the crossover equations in the canonical ensemble
J. Phys. B 33 (2000) L779-L786
We compute the shift of the critical temperature T[c] with respect to the ideal case for a weakly interacting uniform Bose gas. We work in the framework of the canonical ensemble, extending the
criterion of condensation provided by the canonical particle counting statistics for the zero-momentum state of the uniform ideal gas. The perturbative solution of the crossover equation to
lowest order in powers of the scattering length yields (T[c]-T[0])/T[0] = -0.93an^1/3, where T[0] is the transition temperature of the corresponding ideal Bose gas, a is the scattering length,
and n is the particle number density. This result is at variance with the standard grand canonical prediction of a null shift of the critical temperature in the lowest perturbative order. The
non-equivalence of statistical ensembles for the ideal Bose gas is thus confirmed (at the lowest perturbative level) also in the presence of interactions.
[ cond-mat/0001422 ]
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Re: [Icehouse] Hyperbolic Planar Tesselations
On Mon, Jun 1, 2009 at 2:43 PM, Denis Moskowitz
Thanks for sending this - I have just been doodling these very things,
and it's nice to see a good catalog of them.
Something like {4,5} with 0 truncation could be interesting for a game
with pointing, since the spaces are square so you can continue a line
along them.
Denis M Moskowitz Jen feroca malbona kuniklo; rigardu liajn
The faces around one vertex of {4, 5} is close to 5-player Martian Chess board with Eeyore's warped quadrants. Except that each quadrant is basically a Euclidean square tesselation.
Maybe adjust quadrants to also be sections of a {4,5} tesselation? Could be tricky to draw, but amusing...
Frank F. Smith | {"url":"http://archive.looneylabs.com/mailing-lists/icehouse/msg04297.html","timestamp":"2024-11-02T17:26:31Z","content_type":"text/html","content_length":"6970","record_id":"<urn:uuid:e78abc13-8489-45ff-bf98-eed469522b95>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00339.warc.gz"} |
Regression and Analysis of Variance I by Alaba Oluwayemisi Oyeronke PDF download - 4096
Regression and Analysis of Variance I by Alaba Oluwayemisi Oyeronke PDF free download
Alaba Oluwayemisi Oyeronke Regression and Analysis of Variance I PDF, was published in 2011 and uploaded for 300-level Science and Technology students of University of Ibadan (UI), offering STA322
course. This ebook can be downloaded for FREE online on this page.
Regression and Analysis of Variance I ebook can be used to learn Regression, Analysis of Variance, Correlation Coefficient, Correlation Ratio, Simple Linear Regression, Multiple Linear Regression,
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Design, Least Significant Difference. | {"url":"https://carlesto.com/books/4096/regression-and-analysis-of-variance-i-pdf-by-alaba-oluwayemisi-oyeronke","timestamp":"2024-11-05T21:37:56Z","content_type":"text/html","content_length":"86669","record_id":"<urn:uuid:b26b5b6a-1ea4-4ec1-9944-831467133785>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00181.warc.gz"} |
13+ Conservation Of Energy Worksheet Answers - collegebeautybuff.com13+ Conservation Of Energy Worksheet Answers
13+ Conservation Of Energy Worksheet Answers
13+ Conservation Of Energy Worksheet Answers. Conservation of energy practice problems 1) a student lifts his 2.0 kgpet rock 2.8 mstraight. Momentum answer key worksheet conservation energy
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13 Best Images of Energy Worksheets Middle School Energy Transformation Worksheet Answer Key from www.worksheeto.com
Consider this ball rolling down a hill. The potential energy worksheet is a useful instrument for students in middle school that want to understand the idea of potential energy. Potential and kinetic
energy printables.
This Worksheet Can Be Utilized By Students To Understand The Law Of Conservation Of Energy.
The underlying knowledge gained by synthesizing the peak of height not a straight up with people working model of conservation energy. There are several forms of energy that we need to know, describe
and give examples of. The potential energy worksheet is a useful instrument for students in middle school that want to understand the idea of potential energy.
Minimal Energy Worksheet Answer Key Conservation Worksheet Uses Energy Efficient Technologies And.
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Law Of Conservation Of Energy Practice Worksheet Answer The Following Questions And Show Your Work!
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You can check 45+ pages conservation of energy worksheet 2 answers analysis in google sheet format. Consider this ball rolling down a hill. Law of conservation of energy practice worksheet answer the
following questions and show your work! | {"url":"https://collegebeautybuff.com/blog/2023/01/29/13-conservation-of-energy-worksheet-answers/","timestamp":"2024-11-09T10:09:13Z","content_type":"text/html","content_length":"58631","record_id":"<urn:uuid:b58a225e-93fb-47fb-a5ab-92644531253c>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00236.warc.gz"} |
05. Differential Equations
Those readers who have visited my Calculus tutorial will recognize this example — it is a common, well-understood differential equation with many real-world applications. For those who don't think
about this stuff every day, a differential equation is one that expresses the relationship between a function and one or more of its derivatives. An example:
• (1) y(t) + r c y'(t) = b
• (2) y(0) = a
Let's examine the statements that describe the equation. In part (1) of the statement, we see that an unknown function y(t) is added to its derivative y'(t), which is scaled by two multiplier terms r
and c.
In part (2) of the statement, an initial value is assigned to the function y(t). The meaning of this statement is that, when t = 0, y(t) = a.
Remember about differential equations that, unlike numerical equations, they describe dynamic processes — things are changing. Remember also that the derivative term y'(t) describes the rate of
change in y(t).
Please think about this system for a moment. Let's say that the variable t represents time (although the equation doesn't require this interpretation). At time zero, the function y(t) equals a,
therefore at that moment the derivative term y'(t) is equal to (b - a) / (r * c). Notice that y'(t), which represents the rate of change in y(t), has its largest value at time zero. Because of how
the equation is written, we see that the value of y'(t) (the rate of change) becomes proportionally smaller as y(t) becomes larger. Eventually, for some very large value of t, the rate of change
represented by y'(t) becomes arbitrarily small, as y(t) approaches the value of b, but never quite gets there.
Put very simply, this equation describes a system in which the rate of change in the value of y(t) depends on the remaining difference between y(t) and b, and as that difference decreases, so does
the rate of change. As it happens, this equation is used to describe many natural processes, among which are:
• Electronic circuits consisting of resistors and capacitors (hence the equation's terms r and c), where the voltage on a capacitor changes in a way that depends on the current flowing through a
resistor, and the value of the resistor's current depends on the voltage on the capacitor.
• Heat flow between a source of heat energy and a cooler object being heated by it (like a pot on a stove). In such a system, the temperature of the heated body changes at a rate that depends on
the remaining difference in temperature between the two bodies.
• The rate of gas flow between two pressure vessels with a constricted passage between them. In this system also, the rate of gas flow depends on the remaining pressure difference, and the pressure
difference declines over time.
This is by no means a comprehensive list of this equation's applications. But the statements for a differential equation are only the beginning, and not all differential equations have analytical
solutions (solutions expressible as a practical function, one consisting of normal mathematical operations). Others require numerical methods and are only soluble in an approximate sense.
Let's see if Maxima can find a solution for this equation. Here are the steps:
eq:y(t)+r*c*'diff(y(t),t)=b; (equivalent to statement (1) above)
atvalue(y(t),t=0,a); (equivalent to statement (2) above)
sol:desolve(eq,y(t)); ("desolve()" is one of Maxima's differential equation solvers)
f(t,r,c,a,b) := ev(rhs(sol)); (assign the solution to a function)
Okay, we now have a function that embodies the solution to our differential equation. We can use it to solve real-world problems. Here's an example from a field in which I have spent a lot of time —
In this experiment, we have an electronic circuit consisting of a resistor and a capacitor. At time zero, we close a switch that connects our circuit to a battery, we then use an oscilloscope to
measure the voltage on the capacitor over time (see diagram this page).
Here are the Maxima instructions to set up and graph the response of the described circuit:
r:10000; (10,000 Ω)
c:100e-6; (100 µf)
a:0; (ground voltage = 0)
b:12; (battery voltage = 12)
wxplot2d(f(t,r,c,a,b),[t,0,5], [gnuplot_preamble, "set grid;"], [nticks,12]);
I have to say this trace looks very familiar. Click here to download a Maxima instruction file for this problem and solution.
Okay, now let's move to a somewhat more complex differential equation that belongs in the same general class. In this equation, instead of a one-time event like throwing a switch that connects a
circuit to a battery, we have a continuous waveform driving a system that could be an R-C circuit, or any natural system in which there is a path of resistance to the flow of something cyclical. I
originally wrote this equation as part of a project to analyze the behavior of tides in channels that connect inland bays to the open ocean, but that is only one of many applications. Here again is a
statement and a solution:
• (1) y(t) + r c y'(t) = m sin(ω t)
This equation has only one term, and no initial value. Let's submit this expression to Maxima and see what it comes up with:
eq:y(t)=-r*c*'diff(y(t),t)+m*sin(ω*t); sol:desolve(eq,y(t));
Okay, I see a problem with this solution — the group of terms at the right includes the familiar "e^-t/rc" expression that appears in equations with defined initial values. But, because this equation
describes a continuous process with no beginning and no end, we need to set the conditions at time zero in such a way that all times will be treated equally (and the right-hand subexpression will be
After mulling this over, I decided the correct way to characterize the initial conditions would be to submit the right-hand expression as a negative value at time zero, which has the effect of
preventing the assignment of a special time-zero value. Here is that entry and the outcome:
init_val:-(c*m*r*(%e^-(t/r*c))*%omega)/(c^2*r^2*%omega^2+1); atvalue(y(t),t=0,init_val); (a rather exotic initial value)
f(t,r,c,ω,m) := ev(rhs(sol),fullratsimp,factor); (simplify, factor and declare a function)
R-C circuit diagram with sinewave generator
Okay, we now have a working embodiment of this equation, and it turns out this form has as many real-world applications as the earlier example. Here is another electronic example using an R-C
circuit, but this time with a sinewave generator driving our circuit:
f:440; (Hertz)
r:1000; (Ω)
c:0.05e-6; (µf)
wxplot2d([sin(ω*t),f(t,r,c,ω,m)],[t,0,.005],[gnuplot_preamble, "unset key;set title 'f = 440 Hz, r = 1000, c = 100 uf'"]);
-Signal Generator- -R-C junction-
Click here to download a Maxima instruction file for this problem and solution.
In case this virtual circuit design activity sounds exotic and unrealistic, I should say that modern electronic design has become increasingly dependent on this kind of virtual modeling, well in
advance of any hardware prototyping. As computer software become more sophisticated, there is less reason to waste time and material experimenting at a workbench to discover that a design is or is
not practical.
I found a rather exotic purpose for this differential equation up in Alaska. It turns out this equation applies naturally to the tide-driven movement of water into and out of bays connected to the
ocean by narrow channels. The channel current resembles the current flow in the resistor in an R-C circuit, and the two waveforms in the diagram above resemble the timing and height of the water
levels in an ocean-bay system (the blue trace represents the ocean water height and the red trace represents the bay). All one need do is establish time constants for the tidal system, which is
nature's corollary to the time constants created by electronic components in an R-C circuit.
Of course, in nature things aren't so neat and predictable as in an electronic circuit, with an essentially perfect signal generator and electronic parts with precise values. But the two systems have
a lot in common, and it is possible to model a bay's tidal heights with reasonable accuracy using this method. | {"url":"https://arachnoid.com/maxima/differential_equations.html","timestamp":"2024-11-10T12:37:44Z","content_type":"application/xhtml+xml","content_length":"19837","record_id":"<urn:uuid:e38aa9a6-9548-4fa3-a3b7-828c77d99ca2>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00847.warc.gz"} |
st_geometry_type: Return geometry type of an object
Return geometry type of an object, as a factor
st_geometry_type(x, by_geometry = TRUE)
a factor with the geometry type of each simple feature geometry in x, or that of the whole set
object of class sf or sfc
logical; if TRUE, return geometry type of each geometry, else return geometry type of the set | {"url":"https://www.rdocumentation.org/packages/sf/versions/1.0-17/topics/st_geometry_type","timestamp":"2024-11-02T06:15:21Z","content_type":"text/html","content_length":"56598","record_id":"<urn:uuid:595f2f3a-f2ba-4ddf-be8a-13a711ee1838>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00773.warc.gz"} |
The Maximum and Minimum Functions as Upper Functions
The Maximum and Minimum Functions as Upper Functions
Recall from The Maximum and Minimum Functions of Two Functions page that if $f$ and $g$ are functioned defined on $I$ then the maximum function of $f$ and $g$ denoted $\max (f, g)$ is defined for all
$x \in I$ by:
\quad \max (f, g)(x) = \max \{ f(x), g(x) \}
Similarly, the minimum function of $f$ and $g$ denoted $\min (f, g)$ is defined for all $x \in I$ by:
\quad \min (f, g)(x) = \min \{ f(x), g(x) \}
We will now see that if $f$ and $g$ are also upper functions on $I$ then the maximum and minimum functions of $f$ and $g$ are also upper functions on $I$.
Theorem 1: Let $f$ and $g$ be upper functions defined on $I$. Then $\max (f, g)$ and $\min (f, g)$ are both upper functions on $I$.
• Proof: Let $f$ and $g$ be upper functions defined on $I$. Then there exists increasing sequences of step functions $(f_n(x))_{n=1}{\infty}$ and $(g_n(x))_{n=1}^{\infty}$ that converge to $f$ and
$g$ (respectively) almost everywhere and such that $\displaystyle{\lim_{n \to \infty} \int_I f_n(x) \: dx}$ and $\displaystyle{\lim_{n \to \infty} \int_I g_n(x) \: dx}$ are finite.
• Consider the sequence $(\max (f_n, g_n))_{n=1}^{\infty}$. This is an increasing sequence of step functions since $(f_n(x))_{n=1}^{\infty}$ and $(g_n(x))_{n=1}^{\infty}$ are both increasing
sequences of step functions. Furthermore, $(\max (f_n, g_n))_{n=1}^{\infty}$ converges to $\max (f, g)$ almost everywhere on $I$.
• Consider the limit $\displaystyle{\lim_{n \to \infty} \int_I \max (f_n(x), g_n(x)) \: dx}$. We need to show that this limit is finite. Recall that $\max (f, g) + \min (f, g) = f + g$. So:
\quad \lim_{n \to \infty} \int_I \max (f_n, g_n) \: dx = \lim_{n \to \infty} \int_I [f_n + g_n - \min (f_n, g_n)] \: dx = \int_I f_n(x) \: dx + \int_I g_n(x) \: dx - \lim_{n \to \infty} \int_I \min
(f_n, g_n) \: dx
• However, we see that since $(f_n(x))_{n=1}^{\infty}$ and $(g_n(x))_{n=1}^{\infty}$ are both increasing sequences of functions that converge to $f$ and $g$ respectively, that then $\min (f_n, g_n)
\leq f$ for all $n \in \mathbb{N}$. So $\displaystyle{\int_I \min (f_n, g_n) : dx}$ is finite, so $\min (f, g)$ is an upper function, and from the equality above we see that the righthand side is
finite, so $\max (f, g)$ is also an upper function. $\blacksquare$ | {"url":"http://mathonline.wikidot.com/the-maximum-and-minimum-functions-as-upper-functions","timestamp":"2024-11-09T06:17:46Z","content_type":"application/xhtml+xml","content_length":"17479","record_id":"<urn:uuid:3a0e1fd6-f811-4d56-aa3a-5d86a31db8f4>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00010.warc.gz"} |
Loudspeaker FIR Filtering
A Guide to Fundamental FIR Filter Concepts & Applications in Loudspeakers
Although not a new technology, manufacturers are increasingly including FIR (finite impulse response) filtering in loudspeaker processors and DSP based amplifiers due to the significant increase in
performance-versus-cost of microprocessors and DSP hardware.
The advantages of FIR filtering include more arbitrary and fine control of a filter’s magnitude and phase characteristics, independent control of magnitude and phase, and the opportunity for
maximum-phase characteristics (at the expense of some bulk time delay). The primary disadvantage is efficiency; FIR filters are generally more CPU intensive than IIR (infinite impulse response)
filters. For very long FIR filters, segmented frequency-domain and multi-rate methods help to reduce the computational load, but these methods come with increased algorithmic complexity.
In pro audio, the terms FIR Filter and FIR Filtering are often used when referring to specific implementations, such as:
• linear-phase crossover and linear-phase brick-wall crossover filters
• very long minimum-phase FIR based system EQ
• horn correction filtering
Whilst these implementations each have their uses, the capabilities of FIR filtering go beyond these implementations; particularly with regard to independent control of magnitude and phase, and mixed
& maximum-phase characteristics.
So, what is FIR filtering and how does it compare to ubiquitous IIR filtering? This article aims to answer these questions, but doing so first requires covering a number of basic concepts in digital
audio. If you’ve studied digital signal processing, much of this will be second nature. Forgive me for skipping some of the details and simplifying some of the more complex concepts.
1. Key Concepts in Digital Audio
In digital audio, sound waveforms are represented by samples. An analog-to-digital converter (ADC) measures, or samples, an analog signal and assigns a digital value to each sample. Humans can
typically hear frequencies between 20 Hz and 20 kHz. (A Hz is a cycle per second.) To adequately represent this frequency range digitally, the ADC needs to sample the audio waveform at least twice
the highest audible frequency; hence we have the common sample rates of 44.1 kHz and 48 kHz. (Multiples of these frequencies, such as 88.2 kHz, 96 kHz, 192 kHz and 384 kHz are also used in pro audio;
for reasons we won’t go into here.) Half the sampling rate is called the Nyquist frequency. For example, a sampling rate of 48 kHz has a Nyquist frequency of 24 kHz.
For more information on the basics of digital audio and sampling, take a look at the “Monty” Montgomery’s excellent Digital Show and Tell video and 24 192 Music Downloads article on xiph.org.
Digital Filtering
Digital filtering is a mathematical process for altering a digital audio signal. At each time interval – for a sampling rate of 48 kHz, the interval is 1/48000 seconds or 20.83 microseconds – a
time-domain digital filter takes the current input sample and some previous input samples, scales (or multiplies) the samples by defined numbers, called filter coefficients, and sums the scaled
samples to create an output sample. Let’s look at some examples.
Example : “Averaging” Filter
One of the simplest digital filters involves taking the average of the current and previous samples. Conceptually this is:
OutputSample = (InputSample + PreviousInputSample) / 2
As a diagram, we can express the filter as:
Averaging digital filter flowchart.
As an equation, we can express this as:
y[n] = 0.5 * x[n] + 0.5 * x[n-1]
• x[n] is the input sample at the current time interval, or sample number, n,
• x[n-1] is the input sample at the previous time interval, or sample number n-1,
• y[n] is the output sample for the current time interval, or sample number n, and
• the “0.5” values are the filter coefficients.
Before continuing it’s worth pausing to consider the impulse response and frequency response and how they relate to the filter structure above.
Impulse Response and Frequency Response
The impulse response of a device – an analog filter, a digital filter, a loudspeaker or even a room – is the reaction or response of the device over time in response to an input pulse.
Theoretically we can put a voltage, digital or acoustic pulse into a loudspeaker, amplifier, processor or room and record the output voltage, digital signal or microphone waveform over time to obtain
the impulse response. However, a pulse doesn’t have enough energy at all frequencies to excite a device with enough level to give a reasonably clean signal above ambient noise (or a signal-to-noise
ratio; SNR) at all frequencies.
Today many measurement systems use either a sinusoidal sweep or cyclic pink noise. (Dual FFT methods use any audio signal – like the output of a live mixing console – but that’s another story.)
Sweeps and cyclic pink noise signals have enough energy at all frequencies to give reasonable SNR and therefore give a usable or stable impulse response.
The frequency response is the frequency-domain behaviour of the device. It’s also the frequency-domain equivalent of the impulse response. (The term transfer function is sometimes used in place of
the frequency response.) The frequency response is usually plotted as the magnitude (in dB) and phase (in degrees) across frequency. The frequency response can also be converted from magnitude and
phase to a complex number representation; real and imaginary values for each frequency point.
Impulse Response ⇔ Frequency Response
The time-domain impulse response and frequency-domain frequency response are inherently linked and are mathematically-equivalent characterisations of the device. We can convert the impulse response
to the frequency domain using the Discrete Fourier Transform (DFT) and convert the frequency response to the time domain using the Inverse Discrete Fourier Transform (IDFT); subject to some
constraints that we won’t go into here. The fast implementations of these operations are called the Fast Fourier Transform (FFT) and the Inverse Fast Fourier Transform (IFFT).
Cyclic pink noise
Cyclic pink noise is pink noise created, using FFT methods, in such a way that when the noise is played continuously by repeating the same noise sequence again and again, any captured chunk of the
noise (of the same length as the original FFT sequence) is guaranteed to be pink. Cyclic noise makes measurement system calculations easier: a subject for another day. (In fact, cyclic noise of any
spectral shape can be created using FFT methods. For clean, stable measurements it’s often worth using noise with a spectral shape that matches the ambient noise spectrum so that all frequencies will
have decent SNR.)
… back to the “Averaging” Filter Example
The impulse response of a digital filter is the output of the filter when we pass an impulse – a sample value of 1.0 – through the filter followed by zeros. Let’s calculate the impulse response.
Averaging digital filter impulse response calculations.
Averaging digital filter impulse response.
For this filter, the impulse response is [0.5 0.5] for the 1^st two time-intervals, and zero everywhere else. Its length is two samples, and since this length is finite, the filter is a finite
impulse response or FIR filter.
Since this filter “averages” pairs of samples, we would expect large sample differences to be smoothed out, and very low sample changes to mostly be unaffected. Let’s look at the frequency response
Averaging digital filter frequency response. (f[s] = 48 kHz)
Sure enough, the averaging filter is a low-pass filter. Low frequencies, where the audio signal varies more slowly over time, are unaffected, but high frequencies are attenuated.
Example : “Difference” Filter
Let’s look at what happens if we change the sign of one of the coefficients in the averaging filter. As a diagram, we can express the filter as:
Difference digital filter flowchart.
As an equation, we can express this as:
y[n] = 0.5 * x[n] + -0.5 * x[n-1]
This filter cancels out adjacent samples that are the same, or similar, and emphasises pairs of samples that are very different. The difference filter’s frequency response is quite different to that
of the averaging filter.
Difference digital filter frequency response. (f[s] = 48 kHz)
The difference filter is a high-pass filter. High frequencies are mostly unaffected, but low frequencies are attenuated.
Example : Digital Filter with Feedback
The filters in the two previous examples calculated the sum of a scaled history of samples. What if we take a previous or delayed sample, scale it, and feed it back into the sum?
The following diagram shows a Butterworth 1^st order low-pass filter with a cut-off frequency of 1 kHz (f[s] = 48 kHz). The important thing to note is that a portion of the previous sample is fed
back, or re-circulated, into the filter.
Filter Order
For a digital filter, the filter order is the maximum amount of sample delay used in the digital filter.
For IIR low-pass and high-pass filters, the frequency response roll-off is 6 dB per octave multiplied by the order. (For example a 3^rd order high-pass filter has an 18 dB/oct roll-off.)
Butterworth 1^st order 1 kHz low-pass filter flowchart. (f[s] = 48 kHz)
Let’s calculate the impulse response.
Butterworth 1^st order 1 kHz low-pass filter impulse response calculations.
Butterworth 1^st order 1 kHz low-pass filter impulse response. (f[s] = 48 kHz)
For this filter, the impulse response starts with a value of 0.0615 and, even though subsequent input samples are 0, the output of the filter decays but continues with non-zero values essentially
forever. Since the output of the filter goes on for infinite time, the filter is called an infinite impulse response or IIR filter.
The thicker blue line in the following diagram shows the frequency response of the low-pass filter.
Butterworth 1^st order 1 kHz low-pass and high-pass filter frequency responses. (f[s] = 48 kHz)
The thinner blue line above shows the response of a 1^st order high-pass Butterworth filter with a cut-off frequency of 1 kHz (f[s] = 48 kHz). The following diagram shows the signal flow and
coefficients for the high-pass filter.
Butterworth 1^st order 1 kHz high-pass filter flowchart. (f[s] = 48 kHz)
In the following sections low-pass and high-pass filters will often be referred to as LP or HP filters.
2. The FIR filter
The following diagram shows the signal flow for a general FIR filter. The coefficients (C[0] to C[N-1]) are the “taps” and so for a N length FIR filter there are N taps and N-1 sample delay elements.
The filter order is N-1.
FIR filter flowchart.
While a short FIR filter can’t do very much, longer FIR filters become quite powerful by essentially blending a long history of audio samples to cause level and phase changes at different frequencies
in controlled ways.
3. The IIR Biquad
The second order IIR filter is typically called the biquad filter. (The following diagram shows the Direct Form 2 version.)
IIR biquad filter flowchart. (Direct-form 2)
Biquads form the basis of most IIR filtering in DSP’s. Shelf, parametric and all-pass filters can be implemented in this form, and high-pass and low-pass filters of any order are usually implemented
as a cascade of connected biquad filters. A 1^st order filter can be implemented in the biquad by setting coefficients a[2] and b[2] to 0.0. Setting coefficients a[1] and a[2] to 0.0 turns the biquad
into a 3-tap FIR filter.
4. FIR versus IIR Filters
So how do IIR and FIR filters compare? Let’s take a look at some common filter types.
Comparison : FIR filters vs 1^st order IIR LP and HP filters
The following two plots show the frequency responses of 1^st order Butterworth IIR LP and HP filters along with FIR filters of various lengths that are designed to approximate the IIR filters. (The
FIR design method used involves sampling the IIR filter impulse response and applying DC correction.) Here the FIR filters needs to be ~40 taps or longer to begin to accurately approximate the IIR
filters. The 10, 20 and 30 tap FIR filters have significant ripple and deviate from the IIR, in magnitude, by up to ~6 dB.
Comparing frequency responses of FIR filters with a Butterworth 1^st 1 kHz low-pass filter frequency response. (f[s] = 48 kHz)
Comparing frequency responses of FIR filters with a Butterworth 1^st order 1 kHz high-pass filter frequency response. (f[s] = 48 kHz)
Comparison : FIR filters vs 2^nd order IIR parametric filter
The following two plots show the frequency response of a 1 kHz Parametric IIR filter along with FIR filters of various lengths that are designed to approximate the IIR filter. Each plot shows a
different FIR design method. The first method has more error toward DC but a slightly better match near 1 kHz. The second method matches the IIR better above and below 1 kHz but has a slightly worse
match around 1 kHz. Using either method, the FIR filter needs to be ~40 taps or longer to begin to accurately approximate the IIR parametric filter.
Comparing Frequency Responses of 1 kHz Parametric IIR filter and FIR Filters from 10 to 50 Taps (f[s] = 48 kHz)
Comparing Frequency Responses of 1 kHz Parametric IIR filter and FIR Filters from 10 to 50 Taps, with DC correction (f[s] = 48 kHz)
5. FIR Filter Length
Since FIR filters don’t have feedback, their ability to affect low frequencies is directly proportional to their length. The longer the filter, the lower the frequencies that can be adjusted; either
in magnitude, phase or both. Higher Q adjustments – sharper magnitude and phase transitions – also require longer FIR filters.
Following are examples of 384 and 3072 tap FIR filters; the filter responses are the dark-blue and dark-red lines. Both FIR filters are attempting to match the desired EQ for a loudspeaker – the
light-blue and light-red lines. The difference plots show the difference in magnitude and phase between the desired ideal filter frequency response and the frequency responses of the FIR filters.
• The longer the filter, the more effective FIR filtering is at achieving EQ, particularly toward low frequencies.
• Even the 3072 tap FIR filter can’t achieve the high-Q magnitude change desired at ~65 Hz. (It actually takes over 10000 taps at 48 kHz for the FIR filter to match the desired EQ.)
384 Tap FIR Filter
384 tap FIR filter impulse response (dark green) and dB magnitude of the impulse response (light green).
384 tap FIR filter frequency response (blue and red) and the desired ideal filter frequency response (light-blue and light-red).
Frequency response difference between the desired ideal filter and the 384 tap FIR filter.
3072 Tap FIR Filter
3072 tap FIR filter impulse response (dark green) and dB magnitude of the impulse response (light green).
3072 tap FIR filter frequency response (blue and red) and the desired ideal filter frequency response (light-blue and light-red).
Frequency response difference between the desired ideal filter and the 3072 tap FIR filter.
6. Computational Complexity
In the introduction we mentioned that FIR filters are more computationally costly than IIR filters. Let’s consider some of the simple IIR and FIR filters shown above. When estimating and comparing
computational costs, we generally look at the mathematical operations – multiplications and additions. We assume that a processor can calculate a “multiply” and an “add” effectively in the same
operation; and therefore we can ignore the additions and just count and compare the multiplications.
The 1st order IIR filter above has 3 coefficients which need to be multiplied with the audio samples, and so we estimate the filter to take approximately 3 x the sample rate operations per second.
The FIR filter has N coefficients (where N is the filter length) and so we estimate the FIR filter to take N x the sample rate operations per second.
The following table compares the filters from above.
Comparing multiplications per second for IIR and short FIR filters. (All the ‘x’ numbers are the FIR multiplications divided by the multiplications the IIR filter in the same row.)
We’ve seen in the examples how approximately 40 taps or more are needed for the FIR filters to approximate the IIR filters, and the table shows that this comes at the computational cost of 8 times or
13.3 times that of the IIR.
Typical Speaker Processor output Channel
As at the writing of this article, a common high-end speaker processor has approximately 24 IIR biquads for high-pass, low-pass, shelf and parametric filters, and 2048 taps of FIR. The following
table compares the computational cost of both.
Comparing multiplications per second for IIR and FIR filters for a typical DSP output channel. (The ‘x’ number is the FIR multiplications divided by the multiplications of the IIR filters.)
7. FIR Filter Benefits
If FIR filtering is so computationally costly, what are its advantages? There are two primary benefits:
• independent control of magnitude and phase, and
• more detailed equalisation (including easier filter creation from a desired frequency response).
Let’s explore each of these further.
Independent Control of Magnitude and Phase
With most IIR filters, the phase response is inherently linked with the magnitude response. (One exception is the IIR all-pass filter.) A huge benefit of FIR filtering is the ability to manipulate
magnitude and phase independently. Following are four FIR filter examples. Each have the same magnitude response but very different phase responses.
Example : Minimum-phase FIR Filter
Previously we showed how both IIR and FIR filters use sample delays (as well as coefficients) to achieve their intended changes in the frequency response. A minimum-phase filter effects EQ whilst
adding the least amount of delay to the audio signal. (This is one of the reasons long FIR filter based EQ in PA systems is typically minimum-phase.) A characteristic of a minimum-phase filter is
that its impulse response has larger coefficients at or near the start of the impulse response. The following two plots shows a minimum phase FIR filter that effects a HP near 100 Hz and some EQ.
Whilst the FIR filter length is 42.7 ms, the effective delay is negligible.
Minimum-phase 2048 tap FIR filter impulse response (dark green) and dB magnitude of the impulse response (light green). (f[s] = 48 kHz)
Minimum-phase 2048 tap FIR filter frequency response. (f[s] = 48 kHz)
Example : Linear-phase FIR Filter
The following two plots show a FIR filter with the same magnitude response but with a flat or linear phase. The bulk delay through the filter is equivalent to the peak location of the filter: here
1024 samples or 21.3 ms.
Linear-phase 2048 tap FIR filter impulse response (dark green) and dB magnitude of the impulse response (light green). (f[s] = 48 kHz)
Linear-phase 2048 tap FIR filter frequency response. (f[s] = 48 kHz)
Example : Maximum-phase FIR Filter
The following two plots show a FIR filter with the same magnitude response but with maximum-phase; this is the opposite or inverse phase of the minimum-phase filter above. The impulse response is the
time reverse of the minimum-phase impulse response and so the bulk delay through the filter is approximately the length of the filter; 42.7 ms.
Maximum-phase 2048 tap FIR filter impulse response (dark green) and dB magnitude of the impulse response (light green). (f[s] = 48 kHz)
Maximum-phase 2048 tap FIR filter frequency response. (f[s] = 48 kHz)
Example : Mixed-phase FIR Filter
Finally we have an arbitrary phase or mixed-phase FIR filter with the same frequency response. The bulk delay through the filter is approximately the location of the filter peak; here ~1480 samples
or 30.4 ms. Where the peak is placed depends on the desired characteristics of the FIR filter and how those characteristics can be achieved within the tap length limit; here 2048 taps. To better
understand this, take a look at some of the FIR Designer tutorials.
Mixed-phase 2048 tap FIR filter impulse response (dark green) and dB magnitude of the impulse response (light green). (f[s] = 48 kHz)
Mixed-phase 2048 tap FIR filter frequency response. (f[s] = 48 kHz)
Why do we care about mixed-phase behaviour? So we can push a loudspeaker’s phase to where we want it!
Why is Independent Phase Control Useful?
A loudspeaker driver can be thought of as a minimum-phase filter (when comparing the acoustic output with the electrical signal into the driver). When using minimum-phase EQ to bring a loudspeaker
driver’s magnitude response closer to “flat,” the phase of the loudspeaker driver also flattens and moves closer to linear phase (at least within the audible pass-band of the loudspeaker).
However in a typical multi-way loudspeaker, the IIR HP and LP crossover filters (as well as polarity, delay and acoustic filters, like ports) all add frequency-varying extra phase. Because of this
extra phase, a multi-way loudspeaker can be thought of as a minimum phase system PLUS some all-pass filters.
Since minimum-phase EQ generally doesn’t affect the all-pass behaviour, we can use FIR filtering to move the phase of the loudspeaker to where we want it.
Arbitrary phase manipulation has many applications including;
• Phase linearising a loudspeaker. (Despite the apparent improvement in the loudspeaker impulse response, there’s some debate as to whether this gives a perceptual improvement in loudspeaker
• Matching the phase (and magnitude) of loudspeakers within product lines, and across different models in installs, so that they are easier to tune in clusters and to array together.
• Manipulating individual loudspeakers in array processing (for audience overage optimisation) and in beam steering.
• Crossover optimisation to improve frequency response consistency within the coverage angle of a multi-way loudspeaker.
More Detailed Equalisation (& Easier Filter Creation)
Using a loudspeaker measurement, we can create a frequency response (magnitude and phase) that will push the loudspeaker towards a desired target response. Because of the inherent relationship
between the impulse response and frequency response, FIR filter coefficients can be generated from a desired frequency response fairly easily using DFT (or FFT) methods. The target response can be
anything including:
• pink noise flat
• pink noise flat with slight HF roll-off (such as Cinema X-Curve)
• flat (linear) phase
• the magnitude and phase response of another loudspeaker
• the magnitude and phase response required from array processing calculations
The following four plots show an on-axis measurement of a commercial 12″ + horn 2-way cabinet and a 2048 tap FIR filter created to push the cabinet’s response to have a relatively flat magnitude
response (with a slight HF roll-off) and flat phase in the pass-band.
On-axis frequency response of a 12″ + horn 2-way cabinet. (Time=0 is aligned to the HF driver and so the LF driver is relatively forward in time, resulting in the apparent maximum-phase behaviour at
the crossover frequency.)
Impulse response (dark green) of FIR filter created for the 12″ + horn 2-way cabinet, and dB magnitude of the impulse response (light green).
Frequency response of FIR filter created for the 12″ + horn 2-way cabinet.
Frequency response of the FIR filtered 12″ + horn 2-way cabinet.
Designing EQ from Measurements and Measurement Averaging
Attempting to EQ fine structure and ripples in loudspeaker responses can be problematic. A loudspeaker’s response varies with microphone and loudspeaker position, with level & temperature, and even
over time. Fine EQ designed from a single measurement might result in the loudspeaker sounding better for the conditions and position of the measurement, but is likely to make the loudspeaker sound
worse at other positions and at other levels etc. Great care must be taken to ensure the measurement is relevant and useful under all the conditions the loudspeaker will be used in – for example
within the whole of the intended coverage area. Averaging measurements from multiple locations is one way to create a measurement that can be used as a starting point for effective, fine EQ.
8. FIR Filtering for Subwoofers?
The following plots show an example of using a FIR filter to both EQ a subwoofer and unwrap the low-frequency phase (from the cabinet and the high-pass IIR used to protect against over-displacement).
Filtering at very low frequencies requires very long filters; here 5000 taps with an IR peak delay of 3500 taps or 72.9ms. (This is simply an example of what a FIR filter can do and it’s yet to be
tested whether the reduction in low-frequency group delay improves the perceived sub impact. With such a large delay, it’s probably not useful for live applications but may be useful in cinema and
home theatre applications.)
Subwoofer frequency response, before FIR filtering. (Double 18″. Measurement includes ~30 Hz 18dB/oct Butterworth IIR high-pass.)
Impulse response (dark green) of FIR filter for EQ, phase unwrapping and crossover LPF. (5000 taps and 3500 sample delay.)
Frequency response of FIR filter for EQ, phase unwrapping and crossover LPF. (5000 taps and 3500 sample delay.)
Subwoofer frequency response, with FIR filtering.
9. Designing & Loading Custom FIR Filters
The growing awareness of the benefits and flexibility of FIR filtering in audio, combined with the ever-increasing performance-versus-cost of microprocessors and DSPs, has resulted in increasing
numbers of audio products with user-accessible FIR filtering blocks. These products enable loudspeaker designers, installers, system operators and DIY’ers to load custom FIR filters. See a list of
FIR-capable processors, amplifiers & software products.
Software tools like FIR Designer* enable the design and simulation of FIR based EQ and mixed IIR+FIR presets/tunings for loudspeakers and systems, from loudspeaker or system measurements.
Comprehensive measurement averaging functionality for spatial and level averaging is also included.
Written by Michael John, founder of Eclipse Audio. | Download this article as a PDF file.
* All the FIR filter examples and plots in this document were generated with FIR Designer. | {"url":"https://eclipseaudio.com/fir-filter-guide/","timestamp":"2024-11-03T10:37:27Z","content_type":"text/html","content_length":"128796","record_id":"<urn:uuid:31ab48b7-0908-4bf2-b9cf-fcb71e7f9dea>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00745.warc.gz"} |
10 research outputs found
Integrity constraints such as functional dependencies (FD), and multi-valued dependencies (MVD) are fundamental in database schema design. Likewise, probabilistic conditional independences (CI) are
crucial for reasoning about multivariate probability distributions. The implication problem studies whether a set of constraints (antecedents) implies another constraint (consequent), and has been
investigated in both the database and the AI literature, under the assumption that all constraints hold exactly. However, many applications today consider constraints that hold only approximately. In
this paper we define an approximate implication as a linear inequality between the degree of satisfaction of the antecedents and consequent, and we study the relaxation problem: when does an exact
implication relax to an approximate implication? We use information theory to define the degree of satisfaction, and prove several results. First, we show that any implication from a set of data
dependencies (MVDs+FDs) can be relaxed to a simple linear inequality with a factor at most quadratic in the number of variables; when the consequent is an FD, the factor can be reduced to 1. Second,
we prove that there exists an implication between CIs that does not admit any relaxation; however, we prove that every implication between CIs relaxes "in the limit". Finally, we show that the
implication problem for differential constraints in market basket analysis also admits a relaxation with a factor equal to 1. Our results recover, and sometimes extend, several previously known
results about the implication problem: implication of MVDs can be checked by considering only 2-tuple relations, and the implication of differential constraints for frequent item sets can be checked
by considering only databases containing a single transaction
Integrity constraints such as functional dependencies (FD), and multi-valued dependencies (MVD) are fundamental in database schema design. Likewise, probabilistic conditional independences (CI) are
crucial for reasoning about multivariate probability distributions. The implication problem studies whether a set of constraints (antecedents) implies another constraint (consequent), and has been
investigated in both the database and the AI literature, under the assumption that all constraints hold exactly. However, many applications today consider constraints that hold only approximately. In
this paper we define an approximate implication as a linear inequality between the degree of satisfaction of the antecedents and consequent, and we study the relaxation problem: when does an exact
implication relax to an approximate implication? We use information theory to define the degree of satisfaction, and prove several results. First, we show that any implication from a set of data
dependencies (MVDs+FDs) can be relaxed to a simple linear inequality with a factor at most quadratic in the number of variables; when the consequent is an FD, the factor can be reduced to 1. Second,
we prove that there exists an implication between CIs that does not admit any relaxation; however, we prove that every implication between CIs relaxes "in the limit". Finally, we show that the
implication problem for differential constraints in market basket analysis also admits a relaxation with a factor equal to 1. Our results recover, and sometimes extend, several previously known
results about the implication problem: implication of MVDs can be checked by considering only 2-tuple relations, and the implication of differential constraints for frequent item sets can be checked
by considering only databases containing a single transaction
The graphical structure of Probabilistic Graphical Models (PGMs) represents the conditional independence (CI) relations that hold in the modeled distribution. Every separator in the graph represents
a conditional independence relation in the distribution, making them the vehicle through which new conditional independencies are inferred and verified. The notion of separation in graphs depends on
whether the graph is directed (i.e., a Bayesian Network), or undirected (i.e., a Markov Network). The premise of all current systems-of-inference for deriving CIs in PGMs, is that the set of CIs used
for the construction of the PGM hold exactly. In practice, algorithms for extracting the structure of PGMs from data discover approximate CIs that do not hold exactly in the distribution. In this
paper, we ask how the error in this set propagates to the inferred CIs read off the graphical structure. More precisely, what guarantee can we provide on the inferred CI when the set of CIs that
entailed it hold only approximately? It has recently been shown that in the general case, no such guarantee can be provided. In this work, we prove new negative and positive results concerning this
problem. We prove that separators in undirected PGMs do not necessarily represent approximate CIs. That is, no guarantee can be provided for CIs inferred from the structure of undirected graphs. We
prove that such a guarantee exists for the set of CIs inferred in directed graphical models, making the $d$-separation algorithm a sound and complete system for inferring approximate CIs. We also
establish improved approximation guarantees for independence relations derived from marginal and saturated CIs.Comment: arXiv admin note: substantial text overlap with arXiv:2105.1446
We present an algorithm that enumerates all the minimal triangulations of a graph in incremental polynomial time. Consequently, we get an algorithm for enumerating all the proper tree decompositions,
in incremental polynomial time, where "proper" means that the tree decomposition cannot be improved by removing or splitting a bag
Acyclic schemes posses known benefits for database design, speeding up queries, and reducing space requirements. An acyclic join dependency (AJD) is lossless with respect to a universal relation if
joining the projections associated with the schema results in the original universal relation. An intuitive and standard measure of loss entailed by an AJD is the number of redundant tuples generated
by the acyclic join. Recent work has shown that the loss of an AJD can also be characterized by an information-theoretic measure. Motivated by the problem of automatically fitting an acyclic schema
to a universal relation, we investigate the connection between these two characterizations of loss. We first show that the loss of an AJD is captured using the notion of KL-Divergence. We then show
that the KL-divergence can be used to bound the number of redundant tuples. We prove a deterministic lower bound on the percentage of redundant tuples. For an upper bound, we propose a random
database model, and establish a high probability bound on the percentage of redundant tuples, which coincides with the lower bound for large databases.Comment: To appear in PODS 202
Integrity constraints such as functional dependencies (FD) and multi-valued dependencies (MVD) are fundamental in database schema design. Likewise, probabilistic conditional independences (CI) are
crucial for reasoning about multivariate probability distributions. The implication problem studies whether a set of constraints (antecedents) implies another constraint (consequent), and has been
investigated in both the database and the AI literature, under the assumption that all constraints hold exactly. However, many applications today consider constraints that hold only approximately. In
this paper we define an approximate implication as a linear inequality between the degree of satisfaction of the antecedents and consequent, and we study the relaxation problem: when does an exact
implication relax to an approximate implication? We use information theory to define the degree of satisfaction, and prove several results. First, we show that any implication from a set of data
dependencies (MVDs+FDs) can be relaxed to a simple linear inequality with a factor at most quadratic in the number of variables; when the consequent is an FD, the factor can be reduced to 1. Second,
we prove that there exists an implication between CIs that does not admit any relaxation; however, we prove that every implication between CIs relaxes "in the limit". Then, we show that the
implication problem for differential constraints in market basket analysis also admits a relaxation with a factor equal to 1. Finally, we show how some of the results in the paper can be derived
using the I-measure theory, which relates between information theoretic measures and set theory. Our results recover, and sometimes extend, previously known results about the implication problem: the
implication of MVDs and FDs can be checked by considering only 2-tuple relations
We study the complexity of estimating the probability of an outcome in an election over probabilistic votes. The focus is on voting rules expressed as positional scoring rules, and two models of
probabilistic voters: the uniform distribution over the completions of a partial voting profile (consisting of a partial ordering of the candidates by each voter), and the Repeated Insertion Model
(RIM) over the candidates, including the special case of the Mallows distribution. Past research has established that, while exact inference of the probability of winning is computationally hard (#
P-hard), an additive polynomial-time approximation (additive FPRAS) is attained by sampling and averaging. There is often, though, a need for multiplicative approximation guarantees that are crucial
for important measures such as conditional probabilities. Unfortunately, a multiplicative approximation of the probability of winning cannot be efficient (under conventional complexity assumptions)
since it is already NP-complete to determine whether this probability is nonzero. Contrastingly, we devise multiplicative polynomial-time approximations (multiplicative FPRAS) for the probability of
the complement event, namely, losing the election
Distributions over rankings are used to model user preferences in various settings including political elections and electronic commerce. The Repeated Insertion Model (RIM) gives rise to various
known probability distributions over rankings, in particular to the popular Mallows model. However, probabilistic inference on RIM is computationally challenging, and provably intractable in the
general case. In this paper we propose an algorithm for computing the marginal probability of an arbitrary partially ordered set over RIM. We analyze the complexity of the algorithm in terms of
properties of the model and the partial order, captured by a novel measure termed the "cover width." We also conduct an experimental study of the algorithm over serial and parallelized
implementations. Building upon the relationship between inference with rank distributions and counting linear extensions, we investigate the inference problem when restricted to partial orders that
lend themselves to efficient counting of their linear extensions | {"url":"https://core.ac.uk/search/?q=author%3A(Kenig%2C%20Batya)","timestamp":"2024-11-04T14:57:06Z","content_type":"text/html","content_length":"117371","record_id":"<urn:uuid:516e674f-08d6-444a-895f-c2a1bdd612d0>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00423.warc.gz"} |
zung2r.f - Linux Manuals (3)
zung2r.f (3) - Linux Manuals
zung2r.f -
subroutine zung2r (M, N, K, A, LDA, TAU, WORK, INFO)
Function/Subroutine Documentation
subroutine zung2r (integerM, integerN, integerK, complex*16, dimension( lda, * )A, integerLDA, complex*16, dimension( * )TAU, complex*16, dimension( * )WORK, integerINFO)
ZUNG2R generates an m by n complex matrix Q with orthonormal columns,
which is defined as the first n columns of a product of k elementary
reflectors of order m
Q = H(1) H(2) . . . H(k)
as returned by ZGEQRF.
M is INTEGER
The number of rows of the matrix Q. M >= 0.
N is INTEGER
The number of columns of the matrix Q. M >= N >= 0.
K is INTEGER
The number of elementary reflectors whose product defines the
matrix Q. N >= K >= 0.
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the i-th column must contain the vector which
defines the elementary reflector H(i), for i = 1,2,...,k, as
returned by ZGEQRF in the first k columns of its array
argument A.
On exit, the m by n matrix Q.
LDA is INTEGER
The first dimension of the array A. LDA >= max(1,M).
TAU is COMPLEX*16 array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by ZGEQRF.
WORK is COMPLEX*16 array, dimension (N)
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument has an illegal value
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
November 2011
Definition at line 115 of file zung2r.f.
Generated automatically by Doxygen for LAPACK from the source code. | {"url":"https://www.systutorials.com/docs/linux/man/3-zung2r.f/","timestamp":"2024-11-02T14:31:18Z","content_type":"text/html","content_length":"8600","record_id":"<urn:uuid:202dbe50-6862-4d14-b326-e03c8d71596f>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00750.warc.gz"} |
Codomain - Wikiwand
In mathematics, a codomain or set of destination of a function is a set into which all of the output of the function is constrained to fall. It is the set Y in the notation f: X → Y. The term range
is sometimes ambiguously used to refer to either the codomain or the image of a function.
A function from to . The blue oval is the codomain of . The yellow oval inside is the image of , and the red oval is the domain of .
A codomain is part of a function f if f is defined as a triple (X, Y, G) where X is called the domain of f, Y its codomain, and G its graph.^[1] The set of all elements of the form f(x), where x
ranges over the elements of the domain X, is called the image of f. The image of a function is a subset of its codomain so it might not coincide with it. Namely, a function that is not surjective has
elements y in its codomain for which the equation f(x) = y does not have a solution.
A codomain is not part of a function f if f is defined as just a graph.^[2]^[3] For example in set theory it is desirable to permit the domain of a function to be a proper class X, in which case
there is formally no such thing as a triple (X, Y, G). With such a definition functions do not have a codomain, although some authors still use it informally after introducing a function in the form
f: X → Y.^[4]
For a function
${\displaystyle f\colon \mathbb {R} \rightarrow \mathbb {R} }$
defined by
${\displaystyle f\colon \,x\mapsto x^{2},}$ or equivalently ${\displaystyle f(x)\ =\ x^{2},}$
the codomain of f is ${\displaystyle \textstyle \mathbb {R} }$, but f does not map to any negative number. Thus the image of f is the set ${\displaystyle \textstyle \mathbb {R} _{0}^{+}}$; i.e., the
interval [0, ∞).
An alternative function g is defined thus:
${\displaystyle g\colon \mathbb {R} \rightarrow \mathbb {R} _{0}^{+}}$
${\displaystyle g\colon \,x\mapsto x^{2}.}$
While f and g map a given x to the same number, they are not, in this view, the same function because they have different codomains. A third function h can be defined to demonstrate why:
${\displaystyle h\colon \,x\mapsto {\sqrt {x}}.}$
The domain of h cannot be ${\displaystyle \textstyle \mathbb {R} }$ but can be defined to be ${\displaystyle \textstyle \mathbb {R} _{0}^{+}}$:
${\displaystyle h\colon \mathbb {R} _{0}^{+}\rightarrow \mathbb {R} .}$
The compositions are denoted
${\displaystyle h\circ f,}$
${\displaystyle h\circ g.}$
On inspection, h ∘ f is not useful. It is true, unless defined otherwise, that the image of f is not known; it is only known that it is a subset of ${\displaystyle \textstyle \mathbb {R} }$. For this
reason, it is possible that h, when composed with f, might receive an argument for which no output is defined – negative numbers are not elements of the domain of h, which is the square root function
Function composition therefore is a useful notion only when the codomain of the function on the right side of a composition (not its image, which is a consequence of the function and could be unknown
at the level of the composition) is a subset of the domain of the function on the left side.
The codomain affects whether a function is a surjection, in that the function is surjective if and only if its codomain equals its image. In the example, g is a surjection while f is not. The
codomain does not affect whether a function is an injection.
A second example of the difference between codomain and image is demonstrated by the linear transformations between two vector spaces – in particular, all the linear transformations from ${\
displaystyle \textstyle \mathbb {R} ^{2}}$ to itself, which can be represented by the 2×2 matrices with real coefficients. Each matrix represents a map with the domain ${\displaystyle \textstyle \
mathbb {R} ^{2}}$ and codomain ${\displaystyle \textstyle \mathbb {R} ^{2}}$. However, the image is uncertain. Some transformations may have image equal to the whole codomain (in this case the
matrices with rank 2) but many do not, instead mapping into some smaller subspace (the matrices with rank 1 or 0). Take for example the matrix T given by
${\displaystyle T={\begin{pmatrix}1&0\\1&0\end{pmatrix}}}$
which represents a linear transformation that maps the point (x, y) to (x, x). The point (2, 3) is not in the image of T, but is still in the codomain since linear transformations from ${\
displaystyle \textstyle \mathbb {R} ^{2}}$ to ${\displaystyle \textstyle \mathbb {R} ^{2}}$ are of explicit relevance. Just like all 2×2 matrices, T represents a member of that set. Examining the
differences between the image and codomain can often be useful for discovering properties of the function in question. For example, it can be concluded that T does not have full rank since its image
is smaller than the whole codomain. | {"url":"https://www.wikiwand.com/en/articles/Codomain","timestamp":"2024-11-13T15:42:39Z","content_type":"text/html","content_length":"274403","record_id":"<urn:uuid:3d113636-509c-496f-b6e0-bf88882caa4c>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00385.warc.gz"} |
1212 - Cubic-free numbers I
A positive integer n is called cubic-free, if it can't be written in this form n = x*x*x*k, while x is a positive integer larger than 1. For example: 24 = 2*2*2*3, so it's not a cubic-free
number, but 18 = 2*3*3, so it is a cubic-free number.
Now give you an integer, you should tell me whether it is a cubic-free number or not.
The first line is an integer T (T <= 10000) means the number of the test cases. The folowing T lines are the test cases, for each test case there is only one line with an integer not greater than
For each test case, output "NO" if the number is cubic-free or else "YES".
sample input
sample output | {"url":"http://hustoj.org/problem/1212","timestamp":"2024-11-13T16:19:06Z","content_type":"text/html","content_length":"7824","record_id":"<urn:uuid:8cce6457-bff3-43ba-a3d3-46d5959cf0c6>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00477.warc.gz"} |
Average Total Assets Formula
• • The average total assets formula is A=(A1+A2)/2, where A is the average total assets and A1 and A2 are the total assets for two consecutive periods.
• • Average total assets are used to calculate Return on Assets (ROA) which is an important financial metric for analyzing a company's profitability.
• • Average total assets help investors and analysts assess a company's efficiency in utilizing its assets to generate profits.
• • The formula for average total assets is especially useful for companies with seasonal fluctuations in their asset base.
• • Calculating average total assets provides a more accurate picture of a company's financial health compared to using only end-of-period or beginning-of-period total assets.
• • Average total assets are a key component in the calculation of various financial ratios such as Return on Equity (ROE) and Asset Turnover Ratio.
• • Companies with consistent growth in average total assets are generally considered to be in a strong financial position.
• • Average total assets are calculated by adding the total assets at the beginning and end of a specific reporting period and dividing by 2.
• • Average total assets are an important metric for evaluating a company's financial stability and growth potential.
• • The average total assets formula is a simple yet powerful tool for financial analysis and forecasting.
• • By calculating average total assets, investors can better understand how efficiently a company is using its resources to generate revenue.
• • Average total assets provide a more stable benchmark for performance evaluation compared to using only end-of-period total assets.
• • Companies with declining average total assets may indicate inefficiencies in asset management or potential financial troubles.
• • Average total assets are crucial for comparing the financial performance of companies within the same industry.
• • Calculating average total assets helps in smoothing out any anomalies due to seasonality or irregular fluctuations in a company's asset base.
Ever wondered what lies beneath the surface of a companys financial health? Enter the average total assets formula, A=(A1+A2)/2, where A is not just a letter in the alphabet soup, but a key player in
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Calculation and formula for average total assets
• The average total assets formula is A=(A1+A2)/2, where A is the average total assets and A1 and A2 are the total assets for two consecutive periods.
• Average total assets are calculated by adding the total assets at the beginning and end of a specific reporting period and dividing by 2.
• Average total assets are calculated by summing the total assets at the beginning and end of a period and dividing by 2.
In the riveting world of financial wizardry, the Average Total Assets Formula unfolds like a high-stakes magic trick: adding up the total assets at the start and finish of a dynamic period, split
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to gauge a company's financial acrobatics. So, as we crunch the numbers and walk this fine line, remember that behind every digit lies a story of growth, resilience, and perhaps a sprinkle of fiscal
Impact of average total assets on company performance
• Companies with consistent growth in average total assets are generally considered to be in a strong financial position.
• Companies with declining average total assets may indicate inefficiencies in asset management or potential financial troubles.
• Companies with a higher average total assets tend to have a more stable financial position and are better equipped to weather economic downturns.
• Companies with a stable or increasing trend in average total assets are generally considered more financially stable.
• Companies with a declining trend in average total assets may indicate inefficiencies in asset management or a downturn in business performance.
In the world of finance, a company's average total assets are like pieces on a chessboard, strategically positioning them for financial success or setting them up for potential checkmate. Consistent
growth in this area signals a strong, resolute player, ready to conquer any economic battleground. On the flip side, a decline in average total assets paints a picture of a company struggling to keep
up with the game, potentially exposing vulnerabilities in their financial armor. Remember, in the game of assets, only the smartest players can secure their position and emerge victorious in the face
of uncertainty.
Importance of average total assets in financial analysis
• Average total assets are used to calculate Return on Assets (ROA) which is an important financial metric for analyzing a company's profitability.
• Average total assets help investors and analysts assess a company's efficiency in utilizing its assets to generate profits.
• The formula for average total assets is especially useful for companies with seasonal fluctuations in their asset base.
• Calculating average total assets provides a more accurate picture of a company's financial health compared to using only end-of-period or beginning-of-period total assets.
• Average total assets are a key component in the calculation of various financial ratios such as Return on Equity (ROE) and Asset Turnover Ratio.
• Average total assets are an important metric for evaluating a company's financial stability and growth potential.
• The average total assets formula is a simple yet powerful tool for financial analysis and forecasting.
• By calculating average total assets, investors can better understand how efficiently a company is using its resources to generate revenue.
• Average total assets provide a more stable benchmark for performance evaluation compared to using only end-of-period total assets.
• Average total assets are crucial for comparing the financial performance of companies within the same industry.
• Calculating average total assets helps in smoothing out any anomalies due to seasonality or irregular fluctuations in a company's asset base.
• Average total assets are an integral part of financial modeling and valuation exercises for businesses.
• Average total assets are crucial for understanding a company's capital structure and its ability to fund operations and growth.
• The average total assets formula can be tailored to specific industries or companies to provide more customized financial analysis.
• Average total assets are a key input in various financial models and projections used by analysts, investors, and lenders to evaluate a company's performance.
• Average total assets are essential for calculating the Return on Assets (ROA) ratio, which indicates how efficiently a company is utilizing its assets to generate profit.
• The average total assets formula is particularly useful for smoothing out fluctuations in a company's asset base due to seasonal variations.
• Average total assets play a crucial role in financial statement analysis and provide insights into a company's asset management efficiency.
• Average total assets are instrumental in determining a company's solvency and liquidity position.
• The average total assets formula is a fundamental tool for analyzing a company's asset turnover and financial performance.
• Average total assets provide a more stable benchmark for measuring a company's financial health than end-of-period total assets alone.
• Average total assets are a key metric for evaluating a company's asset utilization and efficiency.
• Calculating average total assets helps in evaluating the trend in a company's asset base over multiple periods.
• Average total assets are an important metric for comparing companies of different sizes within the same industry.
• The average total assets formula is crucial for conducting trend analysis and forecasting future financial performance.
• Calculating average total assets provides a more accurate representation of a company's asset base compared to using only one period's total assets.
• The average total assets formula is integral to financial modeling and valuation exercises in corporate finance.
• Average total assets are crucial for understanding a company's growth trajectory and financial stability over time.
• Average total assets are key inputs in discounted cash flow (DCF) analysis and other valuation methods to determine a company's intrinsic value.
Average total assets may seem like just a mundane number on a financial statement, but oh, the tales they tell! From deciphering a company's profitability to unmasking its efficiency in asset
utilization, these figures are the Sherlock Holmes of the financial world, uncovering hidden truths about growth potential and stability. Like a seasoned detective, the formula for average total
assets sifts through the clutter of seasonal fluctuations and irregularities, providing a more accurate reflection of a company's financial health. These assets serve as a compass in the wild seas of
corporate finance, guiding investors and analysts towards a clearer understanding of a company's true worth and potential for success. So, next time you come across the unassuming average total
assets formula, remember it's not just numbers on a page – it's the key to unlocking the mysteries of financial analysis and forecasting.
Role of average total assets in investment decisions
• Companies with a higher average total assets value are generally viewed more favorably by investors and creditors.
• Average total assets are used in conjunction with other financial ratios to assess a company's overall financial health.
• Average total assets are used by investors and analysts to assess a company's asset efficiency and return on investment.
In the world of finance, a company's average total assets value is like the runway model of the financial statements - impressive, eye-catching, and garnering attention from all angles. Investors and
creditors alike are seduced by higher total assets, viewing them as a sign of stability and attractiveness in the crowded marketplace. However, like any influencer, it's important to look beyond the
surface level glamor and dig into the substance beneath. When combined with other financial ratios, average total assets provide a full body scan of a company's financial health, revealing not just
its assets, but also its efficiency and ROI potential. So next time you're swooning over a company flaunting its high total assets, remember to check if they've got the brains to match the beauty. | {"url":"https://worldmetrics.org/average-total-assets-formula/","timestamp":"2024-11-05T20:22:39Z","content_type":"text/html","content_length":"150548","record_id":"<urn:uuid:6fa11286-526b-4104-8c4e-e5ad9c8de614>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00014.warc.gz"} |
Analysis and Comparaison of MPPT Nonlinear Controllers for PV System using Buck Converter
Volume 04, Issue 11 (November 2015)
Analysis and Comparaison of MPPT Nonlinear Controllers for PV System using Buck Converter
DOI : 10.17577/IJERTV4IS110184
Download Full-Text PDF Cite this Publication
Taoufik Laagoubi, Mostafa Bouzi, Mohamed Benchagra, 2015, Analysis and Comparaison of MPPT Nonlinear Controllers for PV System using Buck Converter, INTERNATIONAL JOURNAL OF ENGINEERING RESEARCH &
TECHNOLOGY (IJERT) Volume 04, Issue 11 (November 2015),
• Open Access
• Total Downloads : 331
• Authors : Taoufik Laagoubi, Mostafa Bouzi, Mohamed Benchagra
• Paper ID : IJERTV4IS110184
• Volume & Issue : Volume 04, Issue 11 (November 2015)
• Published (First Online): 19-11-2015
• ISSN (Online) : 2278-0181
• Publisher Name : IJERT
• License: This work is licensed under a Creative Commons Attribution 4.0 International License
Text Only Version
Analysis and Comparaison of MPPT Nonlinear Controllers for PV System using Buck Converter
T. Laagoubi, M. Bouzi Univ. Hassan 1,
Faculté des Sciences et Techniques, Laboratoire IMMII,
Settat, Morocco
M. Benchagra Univ. Hassan 1,
Ecole Nationale des Sciences Appliquées, Laboratoire LISERT, Khouribga, Morocco
Abstract This paper describes a maximum power point tracking (MPPT) approach in photovoltaic system based on sliding mode control (SMC) and fuzzy logic control (FLC). Due
1. Photovoltaic cell
1. PV ARRAY
to the nonlinear output characteristic, fuzzy control and Sliding Mode Control are introduced to realize MPPT. The simulation is carried out based on proposed algorithm. Compared with the
conventional duty cycle of perturb and observe (P&O) control method, they can track de maximum power point quickly and accurately. For simulation, a simulation model in Simulink/Matlab of a
solar cell has been presented. A buck converter has been used to control the solar cell output voltage. The MPPT control the duty cycle of the buck converter.
Keywords MPPT; Solar Energy; Photovoltaic; PV; DC-DC Converters; buck converter; Nonlinear Control; perturb and Observe; Fuzzy Logic Control; Sliding Mode Control Introduction
1. INTRODUCTION
Solar energy is the conversion of the energy from the sun to usable electricity. The most common source of solar energy utilizes photovoltaic cells to convert sunlight into electricity.
Photovoltaic utilize a semi-conductor to absorb the radiation from the sun, when the semi-conductor absorbs this radiation it emits electrons, which are the origin of electricity.
Photovoltaic cell is the most basic of a PV modules. Solar cell consist of a P-N junction fabricated in a layer semiconductor. The current-voltage ( ) and power- voltage ( ) outputs
characteristics of solar cell is similar to that of a diode[1]-[3]. Under sun, photons with energy greater than the bandgap energy of the semiconductor are absorbed and great an
electron-hole pair and create a current proportional to the irradiation.
The performance of a photovoltaic cell is usually presented by its () curve and () which is produced for several irradiation levels and several cell temperature levels.
The variation of current versus voltage curve is shown in Fig.1 under various irradiation levels (200, 500 and 800W/m²). For each irradiation, the maximum power point (MPP) is such that
the area defined by is maximum.
MPP – 800W/m²
Normalized current
Solar energy has extraordinary advantages when compared with other source. The field of photovoltaic (PV) solar energy has experienced a remarkable growth for past two decades. However,
Maximum Power Point Tracking (MPPT) control is an essential part of a PV system to extract maximum power from the PV [1]-[3].
MPP – 500W/m²
MPP – 200W/m²
In recent years, a large number of techniques has been developed and implemented for tracking the Maximum Power Point (MPP) [4]-[6].
Fuzzy and sliding mode controls is two nonlinear robust MPPT approach. In this work we propose a comparison between the two controllers and Perturb and Observe (P&O) MPPT method and we
will take an interest in the transitional regime.
In the second paragraph, we present a photovoltaic cell with different curve of voltage output, current output and power output for various climatic conditions.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Normalized voltage
Fig. 1. Variation of normalized current vs voltage curve of PV array
The variation of power versus voltage curve is shown in Fig.2 for various irradiation levels (200, 500 and 800W/m²).
The output power has a maximum at a output voltage .
When the irradiation increases the maximum power increases.
MPP – 800W/m² MPP – 500W/m²
V = kbT
Normalized power
MPP – 200W/m²
: output current of solar cell (A)
: photocurrent current passing P-N junction (A)
0 : reverse saturation current of PV (A)
: output voltage of solar cell (V)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Normalized voltage
Fig. 2. Variation of normalized power vs voltage curve of PV array
The variation of current versus voltage curve under various temperature of solar cell(25-35-45°C) is shown in Fig.3. The maximum power decreases as the temperature increases.
Normalized Current
: number of cells
: diode quality
: series resistance ()
: shunt resistance ()
: electron charge (C)
: Boltzmanns constant (. 1 )
: temperature of solar cell (K)
: thermal voltage (V)
We have used Matlab/simulink to implement the model of the solar PV panel.
The equivalent circuit of equation (1) is presented schematically in Fig.5 with a DC voltage generator which models the photocurrent, a diode which models the
0 0.2 0.4 0.6 0.8 1
Normalized voltage
Fig. 3. Variation of normalized current vs voltage curve of PV array
The variation of power versus voltage curve is shown in Fig.4 for various solar cell temperature. The maximum power decreases when solar cell temperature increases.
semiconductor and two resistors which models the escape currents.
Normalised power
Fig. 5. Simulink model of the solar PV model
0 0.2 0.4 0.6 0.8 1
Normalised voltage
Fig. 4. Variation of normalized power vs voltage curve of PV array
We can observe that low solar irradiance and high cell temperature will reduce the power conversion capability.
2. Simulink model of the solar PV model
The above characteristics can be deduced from a mathematical model.
The general mathematical expression for the illuminated () curve for a solar panel is given by the following one exponential equation [1]
: diode current
: shunt resistance
: series resistance
The key specification of PV module are shown in Table I.
At temperature 25 °C
Open circuit voltage 21.6 V
Short circuit current 1.31 A
Voltage, maximum power 17.0 V
Current, maximum power 1.18 A
Maximum power 20.0 W
I = i I [exp ( Vpv+IpvRs) 1] Vpv+IpvRs
pv pv 0
To properly use a PV module, it must operate in its maximum power point MPP. Next paragraph describe how tracking the maximum power point.
1. MAXIMUM POWER POINT TRACKING
The goal of the MPPT is to find the maximum power
2. DC-DC CONVERTERS MODELING
The MPPT algorithm, control the duty cycle of a buck converter[14]. Fig.8. shows a buck converter model in Simulink.
under different operating conditions, i.e. the different g m
temperature and irradiation values. C E
Fig.6. shows the variation of normalized power versus normalized voltage curve under different irradiation (200, 400, 600, 800, 1000W.m²) and the maximum power point curve.
Normalized power
Maximum Power Point
Fig. 8. Buck converter Simulink model
The buck converter can be written in two sets of state equation depends on the duty cycle equations (8) and (9)
The buck converter operate in two state. If the IGBT is on or off, if it is on, the diode is blocked so the buck converter Simulink model is equivalent to the circuit shown in Fig.9.
1 L
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Normalized voltage
Fig. 6. Maximum power point
The problem considered by MPPT techniques is to automatically find the corresponding duty cycle for voltage
or current at which a PV array should operate to obtain the maximum power point output under a given irradiation and temperature [1]-[3].
Fig.7. shows MPPT system where is PV voltage, is
Fig. 9. Buck converter equivalent circuit when IGBT is on.
The system can be written in two equations :
dV01 = iL1 V01
dt C
PV current, is the load voltage, is the load current and
is a duty cycle.
o a d
diL1 = Vpv V01
dt L L
PV Panel
If the IGBT is off, the diode is conducting so the buck converter Simulink model is equivalent to the circuit shown in Fig.10.
2 L
Fig. 7. MPPT system
The MPPT system contains five elements which are the PV load, DC-DC converter, load, the Pulse width Modulation (PWM) and the MPPT algorithm.
Fig. 10. Buck converter equivalent circuit when IGBT is of.
The system can be written in to two equation :
The following paragraph describing the DC-DC buck converter.
dV02 =
V02 CRL
diL1 = V01
( 1) : previous output power
( 1) : previous output voltage
dt L
The buck converter can be written in two sets of state equation depends on the duty cycle :
( 1) : previous error
( 1) : previous change error
Table II shows the rule table of fuzzy controller,
dV0 = iL VL
dt C CRL
where all the entries of matrix are fuzzy sets of error E, change of error CE and duty cycle D [9].
E\CE NB NS ZE PS PB
NB ZE ZE NB NB NB
NS ZE ZE NS NS NS
ZE NS ZE ZE ZE PS
PS PS PS PS ZE ZE
PB PB PB PB ZE ZE
TABLE II. FUZZY RULE BASE TABLE
diL = Vpv D V0
dt L L
If the IGBT is on = 1, and if it is of = 1
3. MPPT ALGORITHMS
This paragraph describing three MPPT algorithms which are the Fuzzy logic , sliding mode and perturb and observe controls.
1. Fuzzy logic control
Fuzzy logic controller have the advantage to working with imprecise inputs, not needing an accurate mathematical model, and handling nonlinearity[7]-[10].
Fuzzy logic controller generally consists of three stages: fuzzification, rules base table lookup, and defuzzification. During fuzzification, numerical input variables are converted into
linguistic based on membership function similar to Fig.11. In this case five fuzzy levels are used : NB (Negative Big), NS (Negative Small), ZE(Zero), PS (Positive Small) and PB (Positive
NB NS ZE PS PB
If, for example, the operating point is far to the left to the maximum power point (MPP) that is E is PB and CE is ZE, then we need to largely increase the duty cycle, that D should be PB to
reach the MPP.
To explain the steps to follow to determine how the fuzzy logic controller operate, we take an example of an operating point. Which the membership of error and changing error is shown in
Fig.12. and Fig.13.
NB NS ZE PS PB
-b -a 0 a b
Fig. 12. Membership function for error E
We read, the error E is sixty percent ZE and forty percent
Fig. 11. Membership function for inputs and output of fuzzy controller
The inputs to a MPPT fuzzy logic controller are usually an error and a change error .
NB NS ZE PS PB
E(n) = P(n)P(n1)
CE(n) = E(n) E(n 1)
() : actual output power
() : actual output voltage
() : actual error
() : actual change error
Fig. 13. Membership function for changing error CE
In this example changing error is 80% NS and 20% NB. From fuzzy rules base table, we have :
is 60% ZE and is 80% NS, is 60% ZE
is 60% ZE and is 20% NB, is 20% NS
is 40% PS and is 80% NS, is 40% PS
is 40% PS and is 20% NB, is 20% PS In result,
is 60% ZE, 20% NS and 40% PS
Then the membership function for duty cycle is shown
In Simulink we use the bloc shown in Fig.16.[10]
E CE
I V
Fuzzy logic controller
Fig. 16. Simulink bloc for fuzzy logic controller
2. Sliding mode control
in Fig.14.
NB NS ZE PS PB
Fig. 14. Membership function for duty cycle D
The advantage of sliding mode controller are various and important : high precision, good stability, simplicity, invariance, robustness [11],[13], [14].
A typical sliding mode control has two modes of operation. One is called the approaching mode, where the system state converges to a pre-defined manifold named sliding function in finite
time. The other mode is called the sliding mode, where the system state is confined on the sliding
surface and is driven to the origin. In this study, we introduce the concept of the approaching control approach. By selecting
The last stage of fuzzy logic controller is the
defuzzification that converts the fuzzy duty cycle into numerical duty cycle proportional to the black area in fig.14.
The algorithm of the fuzzy logic controller is as follows. The actual voltage and current of PV array can be measured continuously and the power can be deduced by calculation,
the sliding surface as = 0, it is guaranteed that the system
state will hit the surface produce maximum power output persistently [11],[13].
The expression of sliding surface is :
then, the error and changing error can be calculated and
Ppv = I2 Rpv = I
+ I Rpv
) = 0
<>converted into linguistic variables based on membership function, so, the linguistic duty cycle can be converted into
Ipv pv
pv pv Ipv
numerical variables based on fuzzy rules then, the duty cycle can be converted by defuzzification. Fig.15. shows the fuzzy logic controller algorithm.
is the equivalent load connect to the
The non-trivial solution of Eq (11) is :
Set ,
2Rpv + I
Rpv = 0
() = ; () =
The sliding surface is defined as :
= 2Rpv
+ I
PV Ipv
Calculate E(n), CE()
The buck converter can be written in two sets of state equation depends on the duty cycle D : (7) and (8). Which can be combined into one set of state equation to represent the dynamic of
system :
= (1 D)X 1 + DX 2
Based on the observation of duty cycle versus operation region as depicted, the duty cycle output control can be chosen as :
= { + > 0
( 1) = ; ( 1) =
< 0
Equivalent control is determined by condition
Fig. 15. Algorithm for fuzzy logic controller
= [
= 0
The equivalent control is derived :
[ T
Fig.18. shows the P&O algorithm [15].
D =
dX] f(X) = VPV
eq T VL dX] g(X)
Finally The control is given by :
1 + 1
= { + 0 + 1 0 + 0
where k is a positive constant
Set out
Measure ,
= ++
The duty cycle of sliding mode controller is determined by the operating point. As the operating point is to the left of maximum power point (MPP), the sliding surface is negative so the duty
cycle decrease. The same the duty cycle increase if the operating point is in the right of MPP.Fig.17.
Normalised power
<0 <0
D decrease D increase
High duty cycle Low duty cycle
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Normalised voltage
Fig. 17. Duty cycle versus operation region
Fig. 18. Algorithm for P&O controller
4. SIMULATION RESULTS
The MPPT simulation results present the response of a PV array with different MPPT approach : fuzzy logic, sliding mode and P&O controllers.
Fig.19 shows the power response obtained using Fuzzy logic(FL) and Sliding mode(SM) controllers based MPPT and Perturb & Observe algorithm. From the above results it seems that the PV power which
is controlled by the proposed SM
1. Perturb and observe (P&O) control
There have been extensive applications of the P&O MPPT algorithm in various types of PV system. This is because P&O algorithm has a simple control structure and few measured parameters are
required for the power tracking. Moreover, it has an advantage of not relying on the PV module characteristics in the MPPT process and so can be easily applied to any PV panel. The name of
algorithm itself reveals that it operates by periodically perturbing the control variable and comparing the instantaneous PV output power after
Controller is more stable than FL and P&O MPPT techniques. The power curve obtained with SM is smoother when compared to FL and P&O algorithms. Fig.20 shows the output voltage of buck converter
using Fuzzy Logic, SM, FL and P&O Controllers.
perturbation with that before. The outcome of the PV power comparison together with the PV voltage condition determines the direction of the next perturbation that should be used.
The simplicity of perturb and observe method make it the most commonly used MPPT algorithm in commercial PV
Fuzzy logic
Sliding mode
products. It is easy to implement.
This is essentially a trial and error method. The PV controller increase the reference for the inverter output power by a small amount, and then detect the actual output power. If the output
power is indeed increased, it will increase again until the output starts to decrease, at which the controller decreases the reference avoid collapse of the PV output due to the highly non-linear
PV characteristic [4],[5],[12].
0 0.05 0.1 0.15 0.2 0.25
Fig. 19. Power output under step changing irradiation for P&O, Fuzzy and Sliding mode MPPT methods
10 REFERENCES
6 Fuzzy logic P&O 4
Sliding mode
0 0.05 0.1 0.15 0.2 0.25
Fig. 20. Voltage output under step changing irradiation for P&O, Fuzzy and Sliding mode MPPT methods
5. CONCLUSIONS
In this paper, three method for MPPT (Fuzzy logic, Sliding mode and P&O). Three of them have been applied to an energy conversion chain by DC-DC buck converter. We compared the simulation results
obtained by subjecting the system to the same controlled environmental conditions.
It is concluded that the overall model in Simulink/Matlab is satisfactory for simulation purposes.
Even if, in transitional regime, the sliding mode present a delay due to the calculation step, it respond quickly.
All this algorithm converge to desirable output. Sliding mode controller exhibits fast dynamic performance and stable response, response of fuzzy logic controller is fast and stable than P&O
controller but is slow and not as stable as sliding mode controller.
The response of sliding mode controller is better than fuzzy logic and Perturb and observe controllers, but it requires too many calculation and system equations. In contrast, the fuzzy logic
controllers is easy to introduce, it does not require the system equations. Both of them is fast than P&O controllers.
1. C. Protogeropoulos, B. J. Brinkworth, R. H. Marshall, B. M. Cross: Evaluation of two theoretical models in simulating the performance of amorphous silicon solar cells, In: 10th European
Photovoltaic Solar Energy Conference, 8-12 April 1991 Lisbon, Portugal.
2. VandanaKhanna, Bijoy Kishore Das, Dinesh Bisht: Matlab/Simelectronics Models Based Study of Solar Cells. In: International Journal of Renewable Energy Research Vandana Khanna and al., Vol.3,
No.1, 2013
3. Wail REZGUI, Leila Hayet MOUSS & Mohamed Djamel MOUSS: Modeling of a photovoltaic field in Malfunctioning. In: Control, Decision and Information Technologies (CoDIT), 2013 International
4. Trishan Esram, Patrick L. Chapman, Comparaison of photovoltaic array maximum power point tracking techniques. In: IEEE Transactions on Energy Conversion, Vol. 22, No. 2, June 2007.
5. Ali Nasr Allah Ali, Mohamed H. Saied, M. Z. Mostafa, T. M. Abdel- Moneim: A Survey of Maximum PPT techniques of PV Systems. In: 2012 IEEE Energytech.
6. C. Liu, B. Wu and R. Cheung: Advanced Algorithm for MPPT Control of Photovoltaic Systems, In: Canadian Solar Buildings Conference Montreal, August 20-24, 2004
7. GARRAOUI Radhia, Mouna BEN HAMED, SBITA Lassaad: MPPT Controller for a Photovoltaic Power System Based on Fuzzy Logic, In: 2013 10th International Multi-Conference on Systems, Signals & Devices
(SSD) Hammamet, Tunisia, March 18-21, 2013.
8. Lixia Sun, Zhengdandan, Fengling Han: Study on MPPT Approach in Photovoltaic System Based on Fuzzy Control In: Industrial Electronics and Applications (ICIEA), 2013 8th IEEE Conference.
9. C.-Y. Won, D.-H. Kim, S.-C. Kim, W.-S. Kim, and H.-S. Kim A new maximum power point tracker of photovoltaic Arrays Using Fuzzy Controller, In: Proc. 25th Annu. IEEE Power Electron. Spec. Conf.,
1994, pp. 396403.
10. M.S. KHIREDDINE, M.T. MAKHLOUFI, Y. ABDESSEMED, A. BOUTARFA: Tracking power photovoltaic system with a fuzzy logic control strategy, In: Computer Science and Information Technology (CSIT), 2014
6th International Conference.
11. Chen-Chi Chu, Chieh-Li Chen: Robust maximum power point tracking method for photovoltaic cells : A sliding mode control approach, In: Solar Energy 83 (2009) 13701378.
12. D. Rekioua , A.Y.Achour, T. Rekioua: Tracking power photovoltaic system with sliding mode control strategy, In: Energy Procedia 36 – 219 230, 2013
13. Samer Alsadi, Basim Alsayid: Maximum power point tracking simulation for photovoltaic systems using perturb and observe algorithm, In: International Journal of Engineering and Innovative
Technology (IJEIT), Volume 2, Issue 6, ISSN: 2277-3754, 2012.
14. Siew-Chong Tan, Y. M. Lai, Martin K. H. Cheung, and Chi K. Tse: On the Practical Design of a Sliding Mode Voltage Controlled Buck Converter, In: IEEE Transactions on Power Electronics, Vol. 20,
No. 2, March 2005.
15. Joe-Air Jiang, Tsong-Liang Huang, Ying-Tung Hsiao and Chia-Hong Chen: Maximum Power Tracking for Photovoltaic Power Systems, In: Tamkang Journal of Science and Engineering, Vol. 8, No 2, pp.
147_153 (2005).
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How do I perform state linear interpolation?
Answers (1)
Edited: Ali Almakhmari on 12 Jun 2023
Hi everyone. I would like to perform state linear interpolation. In other words, I have a gigantic table that has 14 columns (and a great amount of rows). The first 6 columns represents the input
state and the remaining 8 columns represent the corresponding output state. What I want to do is essentially write a piece of MATLAB code that I give it the desired input state X = [a, b, c, d, e,
f], and it will output for me the best estimate of the output state based on interpolating the table, Y = [g, h, i, j, k, l, m, n]. This is really diffcuilt because X and Y are not a single variable,
but actual states that are greatly interconnected. I really have no idea where to start and I hope someone can guide or help me here. Or maybe there is an existing MATLAB function?! That would be
heaven to me. Thank you in advance.
0 Comments
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How do I perform state linear interpolation?
To perform state linear interpolation in MATLAB, you can use the interp1 function. This function performs linear interpolation between points in a table based on the input data.
We first define the input state X as a 6-element vector. We then load the data table from a file or generate it in MATLAB as a variable dataTable. The input and output states are extracted from the
table and stored in variables inputStates and outputStates, respectively.
Finally, the interp1 function is called to compute the best estimate of the output state Y based on linear interpolation between the points in the table. The function takes the input states, output
states, and the desired input state X as arguments, along with the interpolation method (in this case, 'linear'). The resulting output state Y is a vector with 8 elements corresponding to the
corresponding output state for the given input state.
dataTable = load('data.mat'); % Load data from file or generate the table
% Extract input and output states from the table
inputStates = dataTable(:, 1:6);
outputStates = dataTable(:, 7:14);
% Compute best estimate of output state using linear interpolation
Y = interp1(inputStates, outputStates, X, 'linear');
3 Comments
Vidhi Agarwal on 12 Jun 2023
The error you're encountering occurs when the input vector X in the interp1 function is not a vector, but instead is an invalid input. The X vector argument in the interp1 function must be a
one-dimensional vector.
Looking at the example code you provided, it seems that the input vector inputStates is a 2D matrix. To use interp1 with a 2D matrix inputStates as input, you need to transpose the matrix to ensure
that the interpolation is performed along the correct dimension.
inputStates = [1 2 3; 4 5 6; 7 8 10]; % Must be transposed
outputStates = [10 20 30 40 50 60 70 80; 20 30 40 50 60 70 80 90; 30 40 50 60 70 80 90 100];
% Define Desired State
Desired = [2.5 6.5 9];
% Perform Linear interpolation
Y = interp1(inputStates', outputStates', Desired, 'linear');
In this example, before calling the interp1 function, we have transposed the input matrix inputStates using the apostrophe (') to ensure that inputStates is a mxn matrix, where m is the number of
points on the x-axis and n is the length of X. This ensures that the interp1 interpolation is performed along the correct dimension.
Transposing the input matrix does not change the values' order within the input matrix; rather, it swaps the rows and columns' positions.
Hence, by transposing the input matrix inputStates to inputStates', the values of the input matrix remain consistent with the original input matrix for handling the input values of the examples.
Ali Almakhmari on 12 Jun 2023
Edited: Ali Almakhmari on 12 Jun 2023 | {"url":"https://se.mathworks.com/matlabcentral/answers/1980479-how-do-i-perform-state-linear-interpolation?s_tid=prof_contriblnk","timestamp":"2024-11-08T12:40:28Z","content_type":"text/html","content_length":"142272","record_id":"<urn:uuid:679ad595-c63c-46ce-bfe5-bb66e74d259b>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00364.warc.gz"} |
s - James McVittie
« on: October 24, 2012, 09:30:43 PM »
Solution to Problem 1(d)
To show that eigenfunctions corresponding to different eigenvalues are orthogonal, we evaluate the following:
Notice that we can make a simple substitution, apply the Fundamental Theorem of Calculus using the boundary conditions. Then,
Plugging into the original integral, we obtain:
Therefore, the eigenfunctions corresponding to different eigenvalues are orthogonal. | {"url":"https://forum.math.toronto.edu/index.php?PHPSESSID=btr89b3i91s4kcve19i8suggu4&action=profile;area=showposts;sa=messages;u=12","timestamp":"2024-11-05T16:59:26Z","content_type":"application/xhtml+xml","content_length":"29368","record_id":"<urn:uuid:c9738cd5-b751-43b5-8223-4aeb74645475>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00726.warc.gz"} |
Meeting the “Highly Effective Expectation” Criterion for Hedge Accounting - Kawaller & Company
Meeting the “Highly Effective Expectation” Criterion for Hedge Accounting
IRA G. KAWALLER is founder of Kawaller& Company, LLC, aBrooklyn-basedfinancial consulting practice that specializes in issues relating to derivative instruments. He is also a member of the Financial
Accounting Standards Board’s Derivatives Implementation Group.
PAUL D. KOCH is professor of financeat the University ofKansas.
Financial Accounting Standard No. 133 (FAS 133), which takes effect for companies at the start of their fiscal years on or after June 30, 1999, specifies the accounting procedures applicable to
derivative contracts. Derivatives are required to be recorded as assets or liabilities, measured at their fair values. In the general case, gains or losses are recorded in earnings. When derivatives
are used for risk management purposes, however, special hedge accounting rules may be used, but this treatment is not automatic.
One of the prequalifying conditions for hedge accounting is that, in advance of the implementation of the hedge, the hedging entities must document the expectation that the hedge will be “highly
effective.” Although FAS 133 authorizes the use of statistical techniques for this effectiveness testing requirement, no specific methodology has been endorsed.
This article addresses how such effectiveness tests should be structured. In the case of regression analysis, a number of questions come to mind. Should the analysis, for instance, use data on price
levels or price changes? And, if it is correct to use price changes, what is the appropriate measurement interval (daily, monthly, or quarterly)?
An exploration of these and other statistical questions leads us to conclude that regression results are useful as an indicator of hedge effectiveness only if appropriate data are used in the
analysis. Moreover, we argue that the correlation coefficient, by itself, is an insufficient indicator of hedge performance, in that it relates to an “optimal” hedge, which may or may not be
precisely equal in size to the hedge that is actually intended. We suggest several non-regression- based effectiveness testing methodologies that do not suffer from these shortcomings.
Qualifying for “hedge accounting treatment” is a virtual necessity for commercial entities that use derivative instruments for risk management purposes. This accounting treatment recognizes
derivatives’ gains or losses in the same period as the income effects of the underlying hedged item. Otherwise, the mismatch in the timing of the income recognition presents a picture of income
volatility that poorly reflects the underlying economics of the hedging activity.^1
Unless the derivative is designated as a hedge under Financial Accounting Standard No. 133, gains or losses must be recorded in earnings. If a hedging relationship is specified, however, and if all
the qualifying criteria are satisfied, the accounting treatment will be different, depending on the nature of the hedge. Three different types of hedges are permitted: fair value hedges, cash flow
hedges, and hedges of net investments in foreign operations.
Fair value hedges apply to risks associated with the price of an asset, liability, or firm commitment. The carrying value of the item being hedged (i.e., the asset, liability, or firm commitment) is
adjusted to reflect the change in its market value due to the risk being hedged, and this change is posted to earnings. Corresponding gains or losses on the derivative used to hedge this risk are
also posted to earnings, just as they are for non-hedge derivatives applications.
A hedge of an upcoming forecasted event is a cash flow hedge. For cash flow hedges, derivatives results must be evaluated, and a determination made as to how much of the result is “effective” and how
much is “ineffective.” The ineffective component of the hedge results must be realized in current income, while the effective portion originally is posted to “other comprehensive income” and later
reclassified as income in the same period in which the forecasted cash flow affects earnings. The Financial Accounting Standards Board recognizes hedges as ineffective for accounting purposes only
when the hedge effects exceed the effects of the underlying forecasted cash flow, measured on a cumulative basis.
Finally, there are hedges associated with the currency exposure of a net investment in a foreign operation. Again, the hedge must be marked to market. This time, the treatment requires effective
hedge results to be consolidated with the translation adjustment in other comprehensive income. Differences between total hedge results and the translation adjustment being hedged flow through
It is not sufficient merely to elect to apply hedge accounting. Instead, FAS 133 reflects a philosophy that hedge accounting is “special,” and it is justified only if specific prerequisites are
satisfied. At the top of the list of requirements is ex ante documentation supporting the expectation that the hedge will be “highly effective.”
While it stipulates the need to document this expectation, the FASB has left the methodology for doing so to the discretion of the hedger. As a consequence, many prospective hedgers are uncertain
about precisely how to satisfy this requirement.
This article is designed to address this concern. We explain the notion of hedge effectiveness in the context of FAS 133. We detail common shortcomings associated with the use of regression analysis
in measuring hedge effectiveness, and we suggest ways these shortcomings may be resolved.
Simply knowing that the economic objective of the hedge will be realized may not be sufficient to qualify for hedge accounting treatment. Rather, FASB requires that those who want hedge accounting
treatment must document that their hedges will be highly effective at offsetting changes in fair values or changes in the expected cash flows of the associated exposures that are due to the risk
being hedged. Documentation is required at or before the time at which the derivatives transaction is designated as a hedge.
While never explicitly required in FAS 133, a widely used reference for any discussion of effectiveness is the 80-120 standard. Under this standard, hedges would qualify for hedge accounting only if
the results from the derivative are expected to correspond to no less than 80% and no more than 120% of the associated changes of the item being hedged.^2 Unfortunately, this rule suffers from the
rather significant shortcoming that it will likely disqualify hedge accounting for very traditional derivatives used in plain vanilla hedging applications.^3
For example, suppose a hedger sells a forward contract on an available-for-sale security. Short of bankruptcy on the part of the counterparty to the forward contract, the forward price will be the
realized sale price of the security in question. It may not be effective, however, in offsetting the change in the spot price of that security. In fact, to the extent that spot and forward prices
differ at the inception of a hedge, the two price changes are necessarily not equal over the life of the hedge.
An extreme example of this “ineffectiveness” is seen in Exhibit 1. Here, the forward price relevant to a particular value date initially is at a discount to the spot price. A short forward position
is taken to hedge ownership in the underlying asset. In the scenario depicted, by the time this forward value date arrives (i.e., when the spot and forward prices converge), the spot price falls.
Over the same time, however, the forward price moves higher. Rather than offsetting the change in the spot price, then, the short forward position actually reinforces the detrimental price effect.
Spot-Forward Convergence
Options pose a problem as well. An option’s price changes in variable proportion to the price of the instrument underlying the option. For deep out-of-the-money options, the proportionality factor,
i.e., delta, is near zero; for at-the-money options it’s 50%; and for deep-in-the-money options it approaches 100%.
This property of option contracts means that, in the short run, option prices should be expected to change by less than the prices of their underlying instruments. Thus, except when the option is
deep in the money, one should not expect options to provide an effective offset to the change in fair value or the change in the expected cash flow of the hedged item.^4 Do these complications mean
that users of options are precluded from using hedge accounting because the “highly effective” expectation criterion cannot be satisfied?
In fact, the FASB created safety valves for both of these cases. Users of forward contracts may elect to exclude the forward premium or discount from the hedge effectiveness consideration.^5 And,
with respect to options, the test of effectiveness may be based solely on changes in the option’s intrinsic value. To put it another way, changes in the time value of the option may be excluded from
the effectiveness consideration.^6
While such solutions preserve the capacity to employ hedge accounting, they do so at a cost. When hedgers elect to exclude any component of a derivative’s result from the consideration of hedge
effectiveness, FASB requires that excluded portion of the derivatives’ gain or loss to be recognized in current earnings. Thus, with such an election, at least some degree of income volatility will
result — even when the economic intentions of the hedge are perfectly realized. And critically, depending on the specifics of the hedge, these anticipated effects may turn out to be substantial
enough to influence the choice of the hedging instrument, or even the decision to hedge altogether.
Any documentation relating to hedge effectiveness should compare the non-excluded portion of the derivative’s results to the changes in the fair value or the changes in the expected cash flows of the
hedged item due to the risk being hedged. Constructing a statistical test for this purpose may not be a trivial task.
While FAS 133 authorizes the use of regression analysis for hedge effectiveness testing and assessment, it leaves open the question of precisely how this analysis should be performed. To illustrate
the typical usage, consider a simple time series regression of y on x, where y is the price variable associated with the hedged item, and x is the price variable associated with the hedging
The R^2 statistic for this simple regression is the square of the correlation between x and y. It represents the extent to which high (low) values of y are associated with high (low) values of x.
This goodness of fit statistic therefore offers one measure of the effectiveness of the “optimal hedge” over the sample period investigated. Presumably a higher R^2 would lead to greater confidence
that the “optimal hedge” will be highly effective.^8
It is noteworthy that this R^2 is appropriate only to measure the effectiveness of the “optimal hedge” (i.e., only if the regression coefficient is used as the hedge ratio). Use of a different hedge
ratio would imply a relation between the hedge actually employed and the hedged item that deviates from the fitted line. To put it another way, the R^2 is meaningful only if it is used in conjunction
with a comparison of the actual hedge size with the b coefficient of the regression equation.
A more immediate concern is whether the regression should be applied to price levels or price changes. If x and y are price levels, the resulting R2 gives the square of the correlation between
levels. If x and y are price changes, the R^2 gives the square of the correlation between changes. Given the explicit FAS 133 requirement that the hedge result should offset the changes in fair
values or changes in cash flows, it might seem that changes would be the preferred answer. But regression results based on this selection could be misleading.
We consider two different cases. In the first case, suppose the price associated with the hedging instrument oscillates about a constant mean on a daily basis, and the price of the hedged item
exhibits analogous up-down movements but centered around a rising trend (see Exhibit 2). In this case, the two respective price levels are uncorrelated, but the daily price changes exhibit perfect
correlation. Thus, regression analysis performed on price levels would suggest that the hedge will not be effective. Yet reliance on the correlation of price changes gives the impression that the
hedge will work perfectly (i.e., an R^2 equal to 1.0), when it most certainly will not.
In the second case, assume both the hedging instrument and the hedged item exhibit consistent trends, but one series varies randomly from this trend, while the other series exhibits a saw tooth
pattern about the same trend (as per Exhibit 2). Price changes would be uncorrelated (low R^2 for changes), while price levels would be highly correlated.
In both these cases, the R^2s associated with price levels lead to “correct” conclusions (i.e., that the first hedge would not be effective, while the second hedge would). Reliance on R^2s based on
price changes, on the other hand, would lead to exactly the opposite conclusions.
This discussion might suggest that the appropriate indicator of hedge effectiveness should be the correlation of price levels, as opposed to price changes, but this conclusion is similarly flawed.
The statement that two price levels are highly correlated does not necessarily imply a reliable relationship between their price changes over a particular hedge horizon, which is the issue of concern
for the FASB.
Perfectly Correlated Changes
Consider the case of two stock indexes that are known to be highly correlated over the long run (e.g., growth stock prices and value stock prices). Despite the high correlation associated with price
levels of these two indexes, history reveals extended periods over which their price changes have differed markedly.^9
Our major point is that regression analysis can reliably be used in connection with hedge effectiveness testing and validation, but only if the appropriate data are employed in the effort. A properly
designed calculation of the R^2 statistic should examine the relation between changes in the value of the hedging instrument and the asset to be hedged, where changes are measured over a horizon
consistent with the timing of the prospective event being hedged.
More explicitly, an appropriate test should assess whether the non-excluded gain or loss on the derivative will closely approximate the desired change in fair value or change in expected cash flows
of the hedged item over the hedge horizon. If the length of time to the hedge value date is one year, one should collect past observations reflecting changes in the value of the two assets over
one-year periods; if the hedge horizon is three months, the data should reflect three-month periods, and so on.
Unfortunately, implementing this preferred approach may present substantial practical problems. First, there may be insufficient data to conduct a reliable statistical analysis using the preferred
approach. For example, if the hedge horizon were twelve months and one wishes to use one hundred observations in the analysis, this would require data from one hundred years.^10
Second, hedge horizons are not universally constant or stable over time. For example, consider the typical cash flow hedge for some forecasted event with a given hedge value date. As time passes, the
hedge value date approaches, and the hedge horizon shortens. Thus, the hedge horizon is a constantly moving target. Presumably, the preferred test to validate the expectation of high effectiveness at
the inception of the hedge would differ from the test (if required) to “revalidate” the expectation at a later date.^11
While it is clearly correct (and appropriate) to match the data on price changes with the hedge horizon, as directed above, this approach is not necessarily the only way to accommodate FAS 133. As a
practical matter, it may be more manageable to use quarterly price changes as a standard time frame for effectiveness testing. This approach would reflect the objective of assessing hedge
effectiveness over a single accounting period (i.e., a quarter of a year), regardless of the actual period of each individual risk exposure in the user’s overall portfolio.^12
Using price changes measured over shorter periods than a quarter, however, would not be appropriate. Indeed, use of price changes over a shorter span than quarterly is likely to be misleading.
Conducting a regression with data reflecting daily price changes, for example, would be useful only in assessing the performance of a hedge that has a single-day hedge horizon. “Success” or “failure”
over one-day periods gives no reliable indication about the viability of hedges with longer horizons.
To put it another way, demonstrating that two series of daily price changes are highly correlated does not necessarily mean that high correlation would also be found over a longer period. Conversely,
just because daily price changes are not highly correlated, the same lack of correlation may not hold for a longer period.
This concern about time spans for price changes holds for both traditional regression analysis as well as for more sophisticated techniques such as value at risk (VaR) or Monte Carlo simulations. In
each case, the conclusions that follow from the analysis are strictly relevant to the time horizon that applies to the data used in the investigation. Unfortunately, the need to use either quarterly
price changes or price changes measured over the same time frame as the hedge horizon is common to any method of statistical analysis.
It is interesting to explore whether overlapping samples may be used to assess hedge effectiveness. That is, if the hedge horizon is three months, and one wants to use one hundred observations in the
analysis, is it necessary to collect data from one hundred quarters (i.e., go back twenty-five years)? Alternatively, can overlapping periods be used when data extend back only a few quarters?^13
To elaborate, suppose the hedge horizon is one quarter and N observations are available on quarterly changes in the value of the asset to be hedged (Y[T] – Y[T–1]) and the value of the hedging
instrument (X[T] – X[T–1]). If the sample size is sufficient, one could estimate a regression model to obtain evidence about whether the hedge will be effective.
If there are not enough non-overlapping quarterly observations to conduct a reliable regression analysis, one would need to consider alternative means for measuring hedge effectiveness. One logical
approach would be to incorporate higher-frequency data, yet retain the one-quarter time interval for price changes by using overlapping data.
Note that quarterly changes are measured here as the differences in data on Xt or Yt, ninety-one days apart. The approach in Equation (2) ignores the information in Xt and Yt during the ninety days
between each non-overlapping quarterly observation, while the approach in Equation (3) incorporates this information. This information is useful to gain efficiency in estimating the nature of the
hedging relation between quarterly changes in Xt and Yt over time.
Unfortunately, the latter approach has a shortcoming that requires attention. Such overlapping samples are not independent across time. They tend to be more highly autocorrelated than non-overlapping
quarterly differences.
In the general case, we would expect the overlapping data on (Xt – Xt–91) and (Yt – Yt–91) to follow ninety-one- day moving averages. This means that, in an important sense, six months of daily data
on (Xt – Xt–91) and (Yt – Yt–91) may not really provide much more information than two independent three-month observations.
This inertia in the ninety-one-day differences will tend to induce autocorrelated errors (ht) in Equation (3). That is, overlapping data in the ninety-one-day differences, (Xt – Xt–91) and (Yt –
Yt–91), imply overlapping errors (ht) as well. In this example, we would generally expect ht to be represented as a moving average of order ninety-one lags, similar to the overlapping data on (Xt –
Xt–91) and (Yt – Yt–91).
Studies based on overlapping data require formal statistical techniques to adjust the standard errors of the estimated coefficients for non-independence of the error terms (see Newey and West [1987]
and Hansen [1982]). This adjustment allows for reliable statistical testing of the ordinary least squares (OLS) coefficient estimates that characterize the hedging relationship (see Greene [2000] for
It is important to emphasize that the R^2 measure from either Equation (2) or Equation (3) provides relevant information regarding the effectiveness of the “optimal hedge.” It does not, however,
directly document the performance of the actual combined position taken by the user (inclusive of the hedged item and the hedging instrument) relative to the hedged item by itself, in accord with FAS
133. Furthermore, while FAS 133 specifically refers to regression analysis, it also provides the latitude to use other statistically based methodologies.
An alternative approach is to focus on the combined position and to frame the discussion in a manner that is more consistent with a value at risk orientation. We offer four alternatives for
Alternative Method 1
One could directly compare past changes in the value of the combined position [labeled (CT – CT–1)] with changes in the hedged item in isolation (YT – YT–1). In this framework, a hedge would be
deemed highly effective if the variance of the combined position were substantially smaller than the variance of the hedged item by itself.
Statistically, this assessment could be made as follows:
1. Select a historical sample.
2. Collect data on changes in the value of the hedged item (YT – YT–1) and the combined position (CT – CT–1), covering time periods commensurate with the duration of the hedge.^15
3. Calculate the variance of the hedged item (Vy) and the variance of the combined position (Vc).
4. Set some threshold of acceptability (T) such that, if Vc/Vy < T, the “highly effective” criterion is said to be satisfied.^16
Alternative Method 2
In order to qualify for hedge accounting, the combined position of the hedged item and the derivative must result in a gain or loss over the selected time interval that is constrained to be within a
small fraction of the initial value of the hedged item, with a high level of confidence. It is left to the discretion of the analyst to specify the critical parameter values for this method.
For instance, the changes in the initial fair value or cash flow of the exposed item, combined with the gain or loss on the derivative, should be less than some set percentage of the original value
of the hedged item, at a 95% level of confidence.^17
Alternative Method 3
This method justifies the expectation of high effectiveness through scenario analysis. A boundary condition must be stipulated, relating to the prospective change in value of the combined position
relative to the initial value of the hedged item. The highly effective criterion would be satisfied if this boundary condition were violated in only some small fraction of the scenarios (constructed
to include a realistic mix of both extreme market moves and stable conditions).
Alternative Method 4
This method divides historical changes in the value of the hedged item into subsamples reflecting varying degrees of variability. Within each subsample, a boundary condition would be specified
pertaining to the ratio of the change in the combined position, relative to the initial value of the hedged item. Hedges would then be expected to be highly effective if this boundary condition were
violated in a low percentage of the historical observations.
If this fourth method is selected, the user would need to designate:
1. An algorithm for segmenting the historical observations of changes in the value of the hedged item.
2. Boundary conditions relevant to each segmentation.
3. The critical threshold or the percentage observations for which the boundary conditions must be satisfied.
For example, one might choose to partition the historical price changes for the hedged item into three subsamples: those where (YT – YT–1) is greater than two standard deviations, a second where (YT
– YT–1) falls between one and two standard deviations (inclusive), and a third where (YT – YT–1) is smaller than one standard deviation. Rather than imposing a simple proportion of the original value
of y as the threshold condition (as per alternative method 1 or 2), the user might want to specify a set of graduated thresholds that vary directly with the magnitude of the value change of the
hedged item.
For instance, the boundary conditions might be that the combined results from the derivative and the hedged item should be smaller than: 1) 20% of three standard deviations in the first segmentation,
2) 20% of two standard deviations in the second, and 3) 20% of one standard deviation in the third. Assuming these conditions are not violated in, say, 95% of all observations, the hedge relationship
would satisfy the “highly effective” expectation test.
In a substantial number of cases, the variable underlying the derivative will exactly match the hedged item’s risk variable, and, assuming the hedge is appropriately sized, in such a case no formal
ex ante statistical tests would be required under FAS 133.^18 In these situations, if a statistical test were performed — whether the relevant frequency of the data is one quarter, one year, five
years, or whatever — the R^2 of the appropriate regression model would be one, reflecting the fact that the analysis would effectively compare a data series with itself.
Alternatively, when the two respective prices or variables are not identical (e.g., when there is a cross-hedge or a quality or location difference between the two), any of the methods above for
testing hedge effectiveness would be a reasonable approach.
One of the more problematic features of FAS 133 is the required up-front documentation to support the expectation that any given hedge will, in fact, be highly effective. The difficulty is fourfold:
1. “Effectiveness” is defined by the standard in a manner that may frequently be at odds with the economic objective of the hedge in question.
2. The ubiquitous 80-120 offset ratio standard is recognized as deficient, in that it generates misleading indications of ineffectiveness when hedges may be working well in an economic sense.
3. Despite the limitations of the 80-120 standard, the FASB has not offered any alternative guidance regarding how to satisfy its requirement to assess hedge effectiveness.
4. Even if explicit methodologies were to be endorsed, unless appropriate data are accessible and used — including a sufficient number of observations to conduct reasonable analysis — results of
these tests will not be reliable.
To elaborate on the last issue, the FASB allows use of regression analysis to test hedge effectiveness, but beyond stating that hedge effectiveness may be assessed either on a period-by-period basis
or on a cumulative basis, the Board is vague on specifics. Regression analysis is reliable only if the model is specified properly and appropriate data are employed. If a period-by-period assessment
is selected, quarterly price change date should be used in the analysis; if a cumulative assessment is desired, price changes should reflect the span of the hedge horizon. Use of overnight price
changes, however, would have no merit in either case.
With respect to the issue of data limitations, the question of whether or how to use overlapping observations is particularly pressing. Overlapping samples may enhance the ability of the researcher
to uncover the relationship desired, and in some cases may even make the statistical analysis possible. Efficiency in estimation can be improved because overlapping observations allow the time series
properties of the finely sampled data to be incorporated into the analysis.
The disadvantage is that overlapping observations on higher-frequency data induce autocorrelation in the regression error term. Although this problem may be resolved by using the methodology of
Hansen [1982] or Newey-West [1987] to correct the standard errors of the OLS estimates for the presence of autocorrelated errors, these methods do not work well when the degree of overlap is large
relative to the sample size.
Despite these problems, FAS 133 demands that potential hedgers must do something in connection with effectiveness testing. Many will end up doing the wrong thing. Almost every corporate desktop has
access to statistical tools for regression analysis or correlation calculations, but an inappropriately specified statistical test offers no better information than no test at all. Indeed, the wrong
statistical analysis should be viewed with less favor than no analysis, since the user of inaccurate analysis will have misconceptions regarding the expected outcome.
Unfortunately, without the requisite level of statistical expertise, it seems likely that much of the analysis performed in order to justify hedge accounting will be of poor quality.
The authors thank Henk Berkman, Michael Dorigan, Andrew Kalotay, Don Lien, Susan Mangiero, and Jun Yu for helpful comments.
^1To be consistent with the language in FAS 133, the text refers to the “hedged item” as the instrument that is responsible for the exposure or risk under consideration, and the “hedging instrument”
as the derivative used in a “hedging relationship.”
^2Some accounting firms have promulgated an 80%- 125% tolerance band, reflecting the fact that the ratio of 100/80 is 125%.
^3Canabarro [1999] offers evidence that, even when two series reflect a correlation of 98%, the ratio of derivatives results to changes in the value of the hedged item can be expected to be outside
the 0.80-1.25 range in 46.9% of the quarters in the sample.
^4The effectiveness of hedging using options is further complicated by the fact that the option’s extrinsic/time value is sensitive to other factors besides the value of the underlying asset. That
is, in addition to the delta, the hedger must also consider gamma, vega, theta, and rho. See Hull [1998] for details.
^5The FASB explicitly authorizes effectiveness to be measured with reference to changes in either the spot price or the forward price of the hedged item. See Paragraph 65 of FAS 133.
^6This election would be appropriate only for a static hedging strategy (i.e., if the hedger intends to hold an option position over the entire hedge horizon). Dynamic (delta-neutral) hedging
strategies would necessarily involve the time value of the option as well as its intrinsic value. FAS 133 specifies the terminology of an option’s “volatility value” and “minimum value” (Paragraph
63). The minimum value of an option is the present value of the intrinsic value of a European option (i.e., the amount that could be immediately monetized); and the volatility value is the difference
between the full price of the option and the minimum value. If a European option is used as a hedge, it would likely be advantageous to exclude the volatility value (rather than the time value) from
the effectiveness consideration.
^7An “optimal hedge” is defined as one that employs a position in the hedging instrument that minimizes the variance (the sum of squares) of the regression error terms (see Hull [1998]). Kawaller
[1992] points out, however, that a minimum-variance hedge ratio may not be “optimal” in terms of satisfying the economic objectives of the hedger.
^8It is left to the discretion of the entity employing the effectiveness test to specify the critical threshold for the R2 that permits the assumption of “high effectiveness.” The FASB has studiously
avoided providing any specific guidance in this regard.
^9This disparity was particularly great in 1999.
^10Even when data are available with sufficient sample size, if some structural change occurs over the period spanned by the data, earlier observations might not be reliable for making inferences
about the future hedge horizon, as they may not be representative of current market relationships.
In some cases, the desired data are not available for any time span. For example, if the hedger wants to use an interest rate swap to hedge a non-rated corporate bond issue, daily prices (i.e., fixed
rates) for the swap can be accessed through a number of data vendors. The same is not true, however, for data on this particular bond. The lack of data may force the analyst to fabricate a dataset by
hypothesizing or creating the required data from available actual data. Then, using these fabricated data, the analyst is asked to document that a “highly effective” standard will likely be met.
^11An interesting example of this problem involves the “passage of time” when hedging interest rate exposure. It is inappropriate to measure changes in a constant-maturity instrument when the
maturity of the instrument in question is diminishing over time. For example, to assess the prospective effectiveness of a five-year swap used to hedge a five-year bond, the appropriate value changes
are associated with differences between four and three-quarter-year instruments at the end of the quarter versus five-year instruments at the beginning of the quarter.
^12The FASB’s Derivatives Implementation Group has suggested that quarterly price changes would be the preferred methodology. See “Basing the Expectation of Highly Effective Offset on a Shorter
Period than the Life of the Derivative,” which was cleared by the board on November 23, 1999.
^13This issue has been addressed in the literature on multiyear asset returns and term structure premiums (Campbell and Shiller [1991]), and on regulation of derivatives usage in hedge accounting
(Wong [2000]). Richardson and Smith [1991] provide an extensive discussion of the econometric problems involved in using overlapping data to analyze financial relationships.
^14It is noteworthy that, while such econometric methods are available to correct regression standard errors for overlap, they do not work well when the degree of overlap is large relative to the
sample size (i.e., when j is large relative to j ´ N). That is, if a smaller number of quarters’ worth of daily data were available, one would place less credence in the results of this exercise.
Still, this methodology may provide the best approach available to achieve the goals of the analysis. These econometric issues pertain to the standard error of the OLS estimate of b, rather than to
the estimate of b itself. The OLS estimate of b is still BLUE (best linear unbiased estimator), and is critical to the question of the size of the optimal hedge position (e.g., the notional value of
a swap position or the number of futures contracts in the hedge). Furthermore, the possibility of autocorrelated errors in Equation (3) does not necessarily inhibit the ability of the R2 from OLS
estimation of this regression model to reveal the true magnitude of the correlation between (xt – xt–91) and (yt – yt–91).
^15The same question of whether these data can or should be overlapping or non-overlapping applies in this approach, just as it does with traditional regression analysis.
^16Note that the variance of the combined hedged position, Vc, represents variation in the hedged item that is not accounted for by the hedge selected. Thus, Vc is analogous to the sum of squared
residuals (SSE) in a regression model, while Vy represents the total variation in the hedged item (SST). Hence, Vc/Vy can be thought of as the proportion of the total variance (or risk) that is not
eliminated by hedging. Accordingly, (1 – Vc/Vy) is the proportion of the risk that is eliminated by hedging, and is analogous to (1 – SSE/SST), which is the R2 of a regression model (Greene [2000,
pp. 236-242]). A smaller ratio, Vc/Vy, therefore corresponds to a higher regression R2, and signals the expectation of a more effective hedge.
^17The choice of a threshold criterion ought to be set on a market-by-market basis. For instance, if hedging bonds with a time horizon of one quarter in mind, if the unhedged price change might be
expected to be within, say, 5% of the initial value of the bonds with a high degree of confidence, the choice of a 1% or 2% threshold might be deemed to be reasonable. If the exposure is to the price
of crude oil, where unhedged price effects have a history of being much more severe, a threshold of 5% or 10% might be more appropriate.
^18This conclusion presumes that if the derivative instrument is an option, its time value (or volatility value) is excluded from the hedge effectiveness consideration; or, if the derivative is a
forward (or futures) contract, the spot/forward differential is excluded from the hedge effectiveness consideration.
“Basing the Expectation of Highly Effective Offset on a Shorter Period Than the Life of the Derivative.” Financial Accounting Standards Board, DIG Issue F5, November 23, 1999.
Campbell, John Y., and Robert J. Shiller. “Yield Spreads and Interest Rate Movements: A Bird’s Eye View.” Review of EconomicStudies, 58 (1991), pp. 495-514.
Canabarro, Eduardo. “A Note on the Assessment of Hedge Effectiveness Using the Dollar Offset Ratio Under FAS 133.” Goldman Sachs research paper, June 1999.
Greene, William H. Econometric Analysis, 4th Edition. New York: Prentice-Hall, 2000.
Hansen, Lars Peter. “Large Sample Properties of Generalized Method of Moments Estimators.” Econometrica, 50 (1982), pp. 1029-1054.
Hull, John. Introduction to Futures and Options Markets, 3rd Edition. Englewood Cliffs, NJ: Prentice-Hall, 1998.
Kawaller, Ira G. “Choosing the Best Interest Rate Hedge Ratio.” Financial Analysts Journal, September/October 1992.
Newey, Whitney K., and Kenneth D. West. “A Simple Positive Semi-Definite Heteroscedasticity and Autocorrelation Consistent Covariance Matrix.” Econometrica, 55 (1987), pp. 703-708.
Richardson, Matthew, and Tom Smith. “Tests of Financial Models in the Presence of Overlapping Observations.” Reviewof Financial Studies, 4 (1991), pp. 227-254.
“Statement of Financial Accounting Standards No. 133: Accounting for Derivative Instruments and Hedging Activities.” Stamford, CT: FASB, 1998.
Wong, M.H. Franco. “The Association Between SFAS 119 Derivatives Disclosures and the Foreign Exchange Risk Exposure of Manufacturing Firms.” Journal of Accounting Research, forthcoming, 2000. | {"url":"http://www.kawaller.com/meeting-the-highly-effective-expectation-criterion-for-hedge-accounting/","timestamp":"2024-11-12T13:44:11Z","content_type":"text/html","content_length":"73630","record_id":"<urn:uuid:67a36399-1e79-444d-bdd5-acd29404c8c1>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00642.warc.gz"} |
Five Number Summary Calculator
What is a five number summary in statistics?
The five number summary is a set of basic descriptive statistics which provides information about a set of data. It identifies the shape, center, and spread of a statistic in universal terms which
can be used to analyze any sample, regardless of the underlying distribution. It consists of 5 key metrics: the median value (the center), the range of a distribution (25th percentile to 75th
percentile), and the maximum and minimum observed values.
Why Is The Five Number Summary Important?
The five number summary is a concise description of a set of observations. It can be quickly calculated, describes the general shape of the distribution, identifies the likely range of values, and -
most importantly - does not involve any assumptions about the shape of the underlying distribution. In this sense, the five number summary is a universal description of the key practical elements of
a distribution of observations.
How to calculate the five number summary
Well the simple way is to use our five number summary calculator. But if you're doing this by hand:
1. Sort The observations, ranking by value
2. Count the Total Number of Observations
3. For each percentile, take the appropriate point in the ranked list
4. If the precise percentile falls between two points, average the nearest two points.
And if this was your homework assignment, you're welcome.....
How Do You Find Q1 and Q3?
Well the simple way is to use our five number summary calculator. But if you're doing this by hand...
See list sorting exercise above (rank observations by value). Count the total number of records. Divide by 4. That is the observation in the list for the 25th percentile (Q1, the 1st quartile).
Multiply this amount by 3. That is the observation in the list for the 75th percentile (the start of the upper quartile or the top of the 3rd quartile). Anything outside of that range is an outlier.
If an observation falls between two points, the general convention is to average the points. There are more complicated approaches (a weighted average) but this usually will suffice.
The second quartile is the gap between the 25th percentile and the median. The fourth quartile is the gap between the 75th percentile and the maximum value. This captures your interquartile distance.
You can identify the upper half and lower half of a distribution using the smallest value, middle value, and largest value of the sample. This approach is independent of sample size.
How Do You Build a Box Plot?
The five number summary can be used to create a box plot graph. The range of the graph is denoted as the top of the first quartile and the top of the third quartile. You are treating the upper
quartile and lower quartile as outlier data points. The quartile value is used to show the range of the quartile. The whisker diagram shows the range between the extreme values (maximum value,
minimum value) of the data.
There is another form of the boxplot referred to as a modified box plot. This adjusts the box and whisker plot so to drop outlier data value points. This site uses a histogram to as a descriptive
statistic tool; we can add a modified boxplot if there's sufficient demand.While the five number summary is a good basic measure of a distribution, it doesn't show a full view of the standard
deviation, mean, or variance. You need to carefully manage any suspected outlier data points.
What Are Upper and Lower Fences?
You can use the information from the 5 number summary calculator to calculate this. The upper and lower fences are a simple estimate of the potential outliers of a distribution. This approach uses
the interquartile range (Q3 - Q1 values) to assess how far outliers may exist. The inner fence is 1.5 x the interquartile range above / below the 1st and 3rd quartiles (respectively). The outer fence
is 3.0 x the interquartile range. Note that the lower bounds of these ranges can be a negative number (if the IQR is wide and the absolute values of the first quartile are small. This is common in
many logistics problems. In most cases, the underlying data isn't from a normal distribution.
In order to utilize a 5 number summary calculator effectively, it is crucial for users to understand some important concepts. For instance, the data set must be organized in ascending order, which
aids in identifying the median, or middle value of the data. Quartiles, another vital aspect of the 5 number summary, divide the data set into four equal parts, with the 1st quartile representing the
25th percentile and the 3rd quartile representing the 75th percentile. Additionally, the interquartile range, which is the difference between the upper and lower quartiles, illustrates the dispersion
of data values around the median. It is important to note that the 5 number summary provides only a brief overview of a data set and does not offer deeper insights into data properties, such as
variance or standard deviation.
The use of a 5 number summary calculator is not only limited to academic and scientific applications; it is also beneficial to individuals who are interested in better understanding the distribution
of any given data set. Whether evaluating the performance of a group of students or analyzing a large set of financial data, a 5 number summary calculator can provide valuable information in just a
few simple steps. With its broad range of applications and ease of use, this tool is an indispensable resource for those seeking an efficient method to analyze data and make informed decisions.
Understanding the 5 Number Summary
The 5 number summary is a collection of five statistical values that provide a comprehensive description of a given data set. It consists of the minimum value, first quartile (25th percentile),
median (50th percentile), third quartile (75th percentile), and maximum value. The 5 number summary calculator is a handy tool for computing these values quickly and efficiently, making it easier to
perform descriptive statistics on your data set.
Minimum Value and Maximum Value
The minimum value in a data set is the smallest data value, while the maximum value is the largest data value. To find these values, sort the data in ascending order and identify the first and last
numbers in the sorted list. These values provide insight into the range of the data set, helping to establish its limits and identify any outliers or extreme values.
Quartiles and Median
Quartiles divide the data set into four equal parts, with each quartile representing 25% of the data. The first quartile (Q1) represents the lower 25%, while the third quartile (Q3) represents the
upper 75%. The median, or second quartile (Q2), separates the lower and upper halves of the data set, representing the middle value. When the data set has an odd number of observations, the median is
the middle number; when the total number of observations is even, calculate the median by averaging the two middle numbers.
To find the quartiles, first sort the data in ascending order. For the first and third quartiles, determine the positions for the 25th and 75th percentiles respectively, using either the cumulative
frequency method or interpolation. The median can also be found using similar methods, depending on the size and distribution of the data set.
Understanding the quartiles and median is essential to determining the interquartile range (IQR), which is the difference between the third and first quartiles (Q3 - Q1). This value helps measure the
spread of the data, highlighting the central 50% of the data set, which often holds the area of greatest interest.
By employing a 5 number summary calculator, you can quickly and accurately find these critical statistical values, making it easier to analyze and interpret a given data set. This approach
streamlines the process of descriptive statistics, providing valuable insights about the data's distribution and range without requiring complex calculations or extensive study.
Ascending Order and Data Set
The calculator arranges the data set in ascending order, allowing for easier identification of important values. It begins by determining the smallest value (minimum value) and largest value (maximum
value) of the given data. These values represent the range of the data set.
Next, the calculator identifies the median (second quartile) of the data set. This middle value is found by locating the middle number if there is an odd number of observations or calculating the
average of the two middle numbers if there is an even number of observations.
Once the median is found, the data set is divided into the lower half and upper half. For each of these halves, the calculator finds the respective 25th percentile (1st quartile or lower quartile)
and the 75th percentile (3rd quartile or upper quartile). These values signify the middle value of the lower half and upper half, respectively.
The 5 number summary consists of the following statistics: minimum value, 1st quartile, median, 3rd quartile, and maximum value. Together, they provide a comprehensive overview of the given data's
distribution and dispersion. These statistical values are essential in various fields of interest, such as finance, social sciences, and data analysis, for a deeper understanding of data patterns and
Interpreting the Results
Once you have input your data set into the 5 number summary calculator, the tool generates various statistical values that provide insights into your data. Interpreting these results is crucial for
understanding the key descriptive statistics and patterns within your data set.
Percentiles and Quartiles
Percentiles divide your data into 100 equal parts, with each percentile representing a specific data value within the ascending ordered data set. Quartiles, on the other hand, divide the data into
four equal parts. The 25th percentile corresponds to the lower quartile (1st quartile), the 50th percentile corresponds to the median (2nd quartile), and the 75th percentile corresponds to the upper
quartile (3rd quartile).
Understanding the percentiles and quartiles within your data helps identify the data value's relative position within the data set. It can also give you an indication of the overall interest, as well
as the spread and distribution of your data.
Interquartile Range
The interquartile range (IQR) is an important descriptive statistic that represents the difference between the upper quartile and the lower quartile. It measures the spread of the middle 50% of your
data. The IQR provides a measure of the variance within your data set and can help identify potential outliers.
Ultimately, interpreting the results generated by the 5 number summary calculator allows you to better understand the patterns, central tendencies, and dispersion within your data. Familiarizing
yourself with these key statistical values and methodologies equips you with the tools necessary to make informed decisions and analyses based on your data set.
Descriptive Statistics
Descriptive statistics are essential for understanding and interpreting data in different contexts. They provide key insights and help summarize various aspects of a dataset, such as central
tendency, dispersion, distribution, and the position of values within the dataset. In this section, we will specifically discuss standard deviation, variance, cumulative frequency, and squared
deviations, as they relate to the five number summary calculator.
Standard Deviation and Variance
Standard deviation and variance are important descriptive statistics that measure the dispersion of data within a dataset. They provide an indication of how much the individual data values deviate
from the average (mean) value of the dataset. The 5 number summary calculator helps to compute the standard deviation and variance of a given dataset easily and quickly.
In the context of the five number summary, standard deviation gives valuable information on the spread of data around the median, while variance is the average of squared deviations from the mean.
These key measures of dispersion allow users to understand a dataset's distribution more effectively and make informed decisions based on the data.
Cumulative Frequency and Squared Deviations
Cumulative frequency is another descriptive statistic that can be useful in understanding a dataset's distribution. It represents the total number of data values that are less than or equal to a
specified value in a sorted dataset. By examining the cumulative frequency, individuals can better grasp the frequency distribution of the given data. Furthermore, cumulative frequency can help in
determining the percentiles, including the 25th (lower quartile), 50th (median or second quartile), and 75th (upper quartile) percentiles.
Squared deviations are individual data values minus the mean value, squared. This measure is especially useful when calculating variance, as it mitigates the effect of negative differences. The use
of squared deviations prevents deviations from cancelling each other out, leading to a more accurate representation of the dataset's overall dispersion.
In summary, standard deviation, variance, cumulative frequency, and squared deviations are crucial descriptive statistics that help better understand a dataset's characteristics. The 5 number summary
calculator is a valuable tool for efficiently deriving these statistical values from a given dataset, providing users with an effective way to analyze and evaluate their data.
Applications of the Calculator
The 5 number summary calculator provides essential information about a given set of data, offering valuable insights and assisting in various real-world situations. In this section, we will explore
some applications of the calculator, focusing on real-world examples, and comparison methods.
Real-World Examples
Using a 5 number summary calculator can benefit users in several ways, such as:
• Education: Teachers can use the calculator to quickly analyze students' scores and determine areas where extra assistance may be necessary.
• Finance: Financial analysts can use the calculator to discover patterns and trends in investment data, helping to inform investment decisions.
• Medicine: Medical researchers can analyze patient data, identifying trends in results, and determining any outliers that may warrant further investigation.
• Sports: Coaches and performance analysts can use five number summary statistics to evaluate players' performance and make data-driven decisions for improvement.
• Market Research: By calculating descriptive statistics for survey responses or product sales, companies can make informed decisions based on actual customer behavior and preferences.
Comparison Methods
The 5 number summary calculator simplifies the process of comparing multiple data sets. Key measures provided by the calculator include the minimum value, 1st quartile (25th percentile), median (2nd
quartile, 50th percentile), upper (3rd) quartile (75th percentile), and maximum value. By comparing these values, users can identify differences between the distributions of the data sets.
For example, a user could compare the sales performance of two products by analyzing their respective 5 number summaries. Differences in the medians or quartiles could reveal which product has a
higher overall sales volume or satisfies customer preferences better.
Another application involves comparing the performance of different investments. By analyzing the return on investment (ROI) for multiple options using the 5 number summary, investors can make
better-informed decisions about which investments to pursue, taking into consideration factors such as their risk tolerance and investment goals.
Finally, researchers could use the 5 number summary to compare the effectiveness of different treatment methods in medical studies. By comparing the summary statistics of patient outcomes across
different treatment groups, researchers can identify which methods may be most effective and warrant further investigation.
In conclusion, the 5 number summary calculator is a versatile tool that can provide valuable insights and inform decision-making across various fields and applications. By understanding and utilizing
its capabilities, users can make more data-driven choices and improve their understanding of underlying trends and patterns.
For convenience, we've enclosed two additional measures (10th and 90th percentile) which can be used to generate a similar package known as the seven number summary. The additional two metrics gives
you better visibility into what is happening at the tails of the distribution. While outliers and distribution tails are a small fraction of your data, they can frequently have a disproportionate
impact on overall performance. For example, a group of likely voters may exhibit a range of satisfaction scores with a particular candidate - but only the top and bottom 10% is truly motivated enough
to take action based on their opinions. In business, similar models can be used to explain customer defection to another supplier and contribution margin economics within a distribution business.
This tool is designed to make it easy to repeat statistical calculations. You can save your data to local device storage (if your phone or computer supports HTML5), allowing you to retrieve and edit
data from past calculations. A list of saved datasets is provided below the main calculation area - click on the name of the dataset and the data table above will update. Important: these are locally
saved only (cannot be accessed on other devices, are not sent to our servers, and will be deleted if your cache is cleared). If you need to save this data permanently or share it between devices (or
with a colleage), send it as a link. Click on the dataset name to load it into the list of data points in the calculator, hit the calculate button, and copy the URL. You can easily email the URL to
your colleagues or post it on a message board. When anyone clicks on the URL, it will contain the shared values. | {"url":"https://5numbersummary.com/","timestamp":"2024-11-05T10:47:56Z","content_type":"text/html","content_length":"28044","record_id":"<urn:uuid:dd7cea7a-ca4d-414c-a5d0-0084e1fdc959>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00472.warc.gz"} |
Geert-Jan Giezeman and Wieger Wesselink
A polygon is a closed chain of edges. Several algorithms are available for polygons. For some of those algorithms, it is necessary that the polygon is simple. A polygon is simple if edges don't
intersect, except consecutive edges, which intersect in their common vertex.
The following algorithms are available:
• find the leftmost, rightmost, topmost and bottommost vertex.
• compute the (signed) area.
• check if a polygon is simple.
• check if a polygon is convex.
• find the orientation (clockwise or counterclockwise)
• check if a point lies inside a polygon.
All those operations take two forward iterators as parameters in order to describe the polygon. These parameters have a point type as value type.
The type Polygon_2 can be used to represent polygons. Polygons are dynamic. Vertices can be modified, inserted and erased. They provide the algorithms described above as member functions. Moreover,
they provide ways of iterating over the vertices and edges.
The Polygon_2 class is a wrapper around a container of points, but little more. Especially, computed values are not cached. That is, when the Polygon_2::is_simple() member function is called twice or
more, the result is computed each time anew.
Polygons with Holes and Multipolygons with Holes
This package also provides classes to represent polygons with holes and multipolygons with holes.
For polygons with holes, these are Polygon_with_holes_2 and General_polygon_with_holes_2. They can store a polygon that represents the outer boundary and a sequence of polygons that represent the
For multipolygons with holes, there is Multipolygon_with_holes_2. It stores a sequence of polygons with holes.
These classes do not add any semantic requirements on the simplicity or orientation of their boundary polygons.
The Polygon Class
The following example creates a polygon and illustrates the usage of some member functions.
File Polygon/Polygon.cpp
#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/Polygon_2.h>
#include <iostream>
typedef K::Point_2 Point;
using std::cout; using std::endl;
int main()
Point points[] = { Point(0,0), Point(5.1,0), Point(1,1), Point(0.5,6)};
Polygon_2 pgn(points, points+4);
// check if the polygon is simple.
cout << "The polygon is " <<
(pgn.is_simple() ? "" : "not ") << "simple." << endl;
// check if the polygon is convex
cout << "The polygon is " <<
(pgn.is_convex() ? "" : "not ") << "convex." << endl;
return 0;
The class Polygon_2 implements polygons.
Definition: Polygon_2.h:65
The Multipolygon with Holes Class
The following example shows the creation of a multipolygon with holes and the traversal of the polygons in it.
File Polygon/multipolygon.cpp
#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/Multipolygon_with_holes_2.h>
int main() {
Polygon_with_holes_2 p1(Polygon_2(p1_outer, p1_outer+4));
Polygon_2 h(p1_inner, p1_inner+4);
Polygon_with_holes_2 p2(Polygon_2(p2_outer, p2_outer+4));
Multipolygon_with_holes_2 mp;
for (auto const& p: mp.polygons_with_holes()) {
std::cout << p << std::endl;
return 0;
The class Multipolygon_with_holes_2 models the concept MultipolygonWithHoles_2.
Definition: Multipolygon_with_holes_2.h:35
The class Polygon_with_holes_2 models the concept GeneralPolygonWithHoles_2.
Definition: Polygon_with_holes_2.h:43
Algorithms Operating on Sequences of Points
The following example creates a polygon and illustrates the usage of some global functions that operate on sequences of points.
File Polygon/polygon_algorithms.cpp
#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/Polygon_2_algorithms.h>
#include <iostream>
typedef K::Point_2 Point;
using std::cout; using std::endl;
void check_inside(Point pt, Point *pgn_begin, Point *pgn_end, K traits)
cout << "The point " << pt;
cout << " is inside the polygon.\n";
cout << " is on the polygon boundary.\n";
cout << " is outside the polygon.\n";
int main()
Point points[] = { Point(0,0), Point(5.1,0), Point(1,1), Point(0.5,6)};
// check if the polygon is simple.
cout << "The polygon is "
<< "simple." << endl;
check_inside(Point(0.5, 0.5), points, points+4, K());
check_inside(Point(1.5, 2.5), points, points+4, K());
check_inside(Point(2.5, 0), points, points+4, K());
return 0;
Bounded_side bounded_side_2(ForwardIterator first, ForwardIterator last, const Point &point, const PolygonTraits &traits)
Computes if a point lies inside a polygon.
bool is_simple_2(ForwardIterator first, ForwardIterator last, const PolygonTraits &traits)
Checks if the polygon defined by the iterator range [first,last) is simple, that is,...
Polygons in 3D Space
Sometimes it is useful to run a 2D algorithm on 3D data. Polygons may be contours of a 3D object, where the contours are organized in parallel slices, generated by segmentation of image data from a
In order to avoid an explicit projection on the xy plane, one can use the traits class Projection_traits_xy_3 which is part of the 2D and 3D Linear Geometric Kernel.
File Polygon/projected_polygon.cpp
#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/Projection_traits_yz_3.h>
#include <CGAL/Polygon_2_algorithms.h>
#include <iostream>
int main()
points[4] = {
(0,1,2) };
if (!b){
std::cerr << "Error polygon is not simple" << std::endl;
return 1;
return 0;
Iterating over Vertices and Edges
The polygon class provides member functions such as Polygon_2::vertices_begin() and Polygon_2::vertices_end() to iterate over the vertices. It additionally provides a member function
Polygon_2::vertices() that returns a range, mainly to be used with modern for loops. The same holds for edges and for the holes of the class Polygon_with_holes_2.
File Polygon/ranges.cpp
#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/Polygon_2.h>
int main()
// create a polygon and put some points in it
Polygon_2 p;
const Point_2
& p : p.vertices()){
std::cout << p << std::endl;
// As the range is not light weight, we have to use a reference
for(auto it = range.begin(); it!= range.end(); ++it){
std::cout << *it << std::endl;
std::cout << e << std::endl;
return EXIT_SUCCESS;
Container Vertices
a range type to iterate over the vertices
Definition: Polygon_2.h:124
Draw a Polygon
A polygon can be visualized by calling the CGAL::draw<P>() function as shown in the following example. This function opens a new window showing the given polygon. A call to this function is blocking,
that is the program continues as soon as the user closes the window. Versions for polygons with holes and multipolygons with holes also exist, cf. CGAL::draw<PH>() and CGAL::draw<MPH>() .
File Polygon/draw_polygon.cpp
#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/Polygon_2.h>
#include <CGAL/draw_polygon_2.h>
int main()
// create a polygon and put some points in it
Polygon_2 p;
return EXIT_SUCCESS;
void draw(const MPH &aph)
opens a new window and draws aph, an instance of the CGAL::Multipolygon_with_holes_2 class.
This function requires CGAL_Qt6, and is only available if the macro CGAL_USE_BASIC_VIEWER is defined. Linking with the cmake target CGAL::CGAL_Basic_viewer will link with CGAL_Qt6 and add the
definition CGAL_USE_BASIC_VIEWER. | {"url":"https://cgal.geometryfactory.com/CGAL/doc/master/Polygon/index.html","timestamp":"2024-11-03T17:08:56Z","content_type":"application/xhtml+xml","content_length":"38172","record_id":"<urn:uuid:cdb672c9-42e7-4b35-bb46-134507818d3f>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00765.warc.gz"} |
will the storage modulus increase
The increase in length... The Young''s modulus of a rubber string 8 cm long and density 1.5kg//m^(3) is 5xx10^(8)N//m^(2) is suspended on the ceiling in a room. The increase in length...
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محتويات ذات صلة | {"url":"https://www.bouwservicevrieling.nl/Fri-06-Sep-2024-54966.html","timestamp":"2024-11-03T10:45:37Z","content_type":"text/html","content_length":"51770","record_id":"<urn:uuid:69b460c7-4fe6-454c-be8b-f165161c50b5>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00002.warc.gz"} |
Guide to Descriptive Statistics: Definition, Types, and More
- The mean represents the average of all values in a data set. Calculating this average is simple. You just add up all the values and divide the sum by the total number of values.
- As for the median, it represents the middle value in a data set when all the values are arranged in ascending or descending order. In other words, it’s the precise center of the data set.
- And finally, the mode. This is nothing but the most frequently occurring value in a data set. Beware, though – not all data sets will have a single mode. Some will have multiple modes, while others
won’t have any. | {"url":"https://www.visualizedata.app/articles/descriptive-statistical-analysis-guide/","timestamp":"2024-11-12T10:11:45Z","content_type":"text/html","content_length":"103739","record_id":"<urn:uuid:bfe3c222-9610-408c-ad85-3191a802441b>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00260.warc.gz"} |
Nonni's Biscotti Review and Giveaway! - CLOSED!
My husband and I have always loved
Nonni's Biscotti
! Therefore I was more than thrilled to get the opportunity to review some more products from
Nonni's Biscotti
! We received four boxes of Nonni's Biscotti and 3 bags of Nonni's Biscotti Bites and they were all delicious! I am a chocolate lover and my favorite was the Decadence Biscotti! I love how
Nonni's Biscotti
isn't too sweet! We love dunking it in milk and I'm sure it would be awesome with coffee although we aren't coffee drinkers at my house.
Nonni's Biscotti Bites
we are perfect size for my kids! They can just have a little bite or two... so they say! I left the bag unattended for a few minutes and it was almost gone! The
Nonni's Biscotti Bites
come in lots of yummy flavors! We liked the Almond Dark Chocolate Flavor the best!
More About Nonni's Biscotti: Nonni’s Biscotti grew out of true Italian tradition. In the little town of Lucca, Italy along narrow cobbled lanes surrounded by elegant piazzas, our Nonni (an endearing
term for Grandma) made biscotti to share with family and friends. When Nonni came to America, almost a century ago, she brought the family recipe with her.
To this day, the same family recipe using real eggs, butter and gourmet bittersweet chocolate is still used to give our biscotti a light, crunchy texture that is delicately sweet. Our devotion to
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Nonni’s Food Company has recently expanded their selection of baked goods to include New York Style snacks and Nonni’s Panetini. The New York Style snack line includes Bagel Crisps and Pita Chips;
wholesome snacks that capture the tastes found in traditional New York City’s bakeries. Nonni’s Panetini is a crisp, twice-baked toast made with Semolina and the finest bread flour, topped with
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All of these irresistible products can be found in grocery stores, convenience stores and warehouse club stores nationwide. Please refer to our retail locator for a retail location near you.
BUY IT: Click here
to find a retailer near you that sells Nonni's Biscotti!
WIN IT:
Thanks to
Nonni's Biscotti
, one lucky Adventures of a Thrifty Mommy reader will win 4 assorted boxes of Nonni's Biscotti and 3 assorted bags of Nonni's Biscotti Bites!
(US residents only)
You MUST be a follower of Adventures of a Thrifty Mommy via Google Friend Connect (located at the right) for your entry to count. And you MUST include your email address with each entry.
Mandatory Entry:
Follow Adventures of a Thrifty Mommy via Google Friend Connect.
Leave a comment of which
Nonni's Biscotti
flavor is your favorite or one that you'd most like to try. (both combined = 1 entry)
That's all you have to do! But, if you'd like to receive extra entries you can:
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• Enter another Adventures of a Thrifty Mommy giveaway. (1 entry for each giveaway entered)
• Post the Adventures of a Thrifty Mommy button (located at the right) on your blog. (5 entries, leave me a link)
Please include your email address with EACH entry. If you don't have your email attached to your comment, I won't be able to contact you if you win! Post a separate comment for each entry (if it says
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This giveaway ends September 2, 2011 at 11:59 pm. Good luck!
* Lucky winner or winners will be selected through True Random Number Generator.* If your profile page does not show your email address, please include your email address in your comment. For
example: adventuresofathriftymommy@gmail.com -- so that I may get in touch with you if you are selected as the giveaway winner. * Each giveaway winner has 48 hours to respond to my email about
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Disclosure: I received the products mentioned above for this review. No monetary compensation was received by me. This is my completely honest opinion above and may differ from yours. Because I do
not directly ship most giveaways from my home, I cannot be held liable for lost or not received products.
182 comments:
I most want to try the almond dark chocolate flavor. and I follow on gfc.
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I follow you on gfc and I want to try the caramel latte.
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Have to go with the triple chocolate
Follow nonni on twitter
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I entered your MI-DEL Cookies giveaway
The Biscotti I would like to try is Triple Milk Chocolate Biscotti
I Like Nonni's Biscotti on Facebook here
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I tweeted http://twitter.com/#!/darleneowen/status/104347990600331265
GFC follower. I would like to try the chocolate flavor
I like Like Adventures of a Thrifty Mommy on Facebook as Elena Istomina
I like Nonni's Biscotti on FB as Elena Istomina
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entered Kix giveaway
GFC LisaMarie dark chocolate!
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I follow your blog via GFC and I think the triple milk chocolate biscotti sounds amazing!
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I'd love to try the limone biscotti. Yum!
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Follow Adventures of a Thrifty Mommy via Google Friend Connect (gala)
I like Cioccolati Biscotti
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enetered Tropical Traditions giveaway
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I follow you on GFC and dark chocolate almond
I \liked them on FB
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I also entered the Tropical Traditions giveaway
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GFC follower (mandala) and I'd like to try the Turtle Pecan Biscotti. Thanks!
mandalarctic at gmail dot com
Following you on Twitter (@mandalarctic)
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Following Nonni Biscotti on Twitter (@mandalarctic)
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GFC follower, Cioccolati Biscotti would be the one I would like to try the most
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Like Nonni's Biscotti on Facebook-Gumma Medlin
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Like you on Facebook-Gumma Medlin
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Network blog follower-Gumma Medlin
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entered the Cover Your Hair Giveaway
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entered the 39DollarGlasses.com Giveaway
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entered the Kix Giveaway
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I follow on GFC and would like to try the original flavor
I follow you on tiwtter
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GFC follower & I'd like to try the Caramel Latte flavor!
I entered the TT Peanut Butter Giveaway
I like Nonni's on FB! mssluna02@gmail.com
I'm following Nonni's on Twitter
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I follow you via NB!
Twitter Follower!
Tweet! http://twitter.com/#!/mssluna02/status/109027098504605696
I entered the 30 Dollar Glasses Giveaway!
I would like the almond dark chocolate.
I also entered your kix giveaway.
Caramel latte looks good!
I'd like to try Almond Dark Chocolate Biscotti Bites
I follow you via Google Friend Connect.
poniabaum at gmail dot com
I like Nonni's Biscotti on Facebook.
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I entered Angelina Ballerina
Almond dark chocolate! Yum!
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I follow via GFC & most would like to try the Dark chocolate almond... the regular is YUMMY!!!((1955nusehjc4me(at)myway(dot)com)) TY!
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Job-Shop Problem (JSP)
The Job-Shop Problem (JSP) is a classic problem in the field of operations research and production planning. It involves determining the sequence in which a set of jobs should be processed on a set
of machines, taking into account various constraints such as the availability of resources and the precedence relationships between the jobs. The goal is to minimize the makespan, which is the total
time required to complete all the jobs. JSP is a complex problem that has been extensively studied and has important applications in various industries, including manufacturing and scheduling. This
essay will provide an overview of the JSP, discuss some of the existing solution approaches, and explore recent developments in this area.
Definition of Job-Shop Problem (JSP)
The Job-Shop Problem (JSP) can be defined as a complex scheduling problem widely studied in the field of operations research. It involves the assignment of a set of jobs to a set of machines in such
a way that each job goes through a specific sequence of operations, each of which has a predefined processing time and can only be performed on a specific machine. The objective of the JSP is to
minimize the overall completion time, which is commonly known as the makespan. Due to its inherent complexity and combinatorial nature, finding an optimal solution for the JSP is considered one of
the most challenging problems in scheduling theory.
Importance and relevance of JSP in industries
The importance and relevance of the Job-Shop Problem (JSP) in industries cannot be overstated. JSP is a well-known and highly complex scheduling problem that arises in many real-world industrial
environments. Its significance lies in its ability to optimize production schedules and improve overall manufacturing performance. By strategically arranging the order in which tasks are performed
and allocating resources efficiently, JSP can reduce production time, enhance productivity, minimize costs, and ultimately increase profitability. Given these potential benefits, it is imperative for
industries to embrace JSP and leverage its capabilities to gain a competitive edge in today's fast-paced and dynamic business landscape.
Brief overview of the essay's contents
In this paragraph, we will provide a brief overview of the contents of the essay on the Job-Shop Problem (JSP). The essay begins by defining the Job-Shop Problem as a classic combinatorial
optimization problem where jobs with specific processing requirements need to be scheduled on machines with different capabilities. It then discusses the importance and relevance of the JSP in
various industries and the challenges associated with solving it. The essay further delves into the different approaches and algorithms used to solve the JSP, including mathematical formulations,
heuristics, and metaheuristics. It concludes by highlighting the potential benefits that can be achieved by developing effective solutions for the JSP.
In addition to variations in job arrival time, due date, and processing time, the job-shop problem (JSP) also faces the challenge of multiple machines with varying processing capabilities. These
machines may have different processing speeds and different processing sequences for optimal efficiency. The goal of solving the JSP is to create an optimal schedule that minimizes the total
completion time of all jobs. This requires careful consideration of the processing times and sequences to allocate jobs to machines effectively. Advanced algorithms such as genetic algorithms and ant
colony optimization have been developed to tackle this complex combinatorial optimization problem.
Mathematical modeling of JSP
In order to solve the complex Job-Shop Problem (JSP), mathematical modeling is crucial to finding optimal solutions. Mathematical modeling refers to the process of creating a mathematical
representation of a real-world problem. In the context of JSP, mathematical models are used to represent the problem's constraints, objectives, and decision variables. These models often involve
assigning variables to various components of the problem, such as machines, jobs, and time intervals. Additionally, mathematical models aid in formulating the necessary equations and inequalities
that govern the scheduling decisions. By using mathematical modeling techniques, researchers and practitioners can systematically analyze and solve JSP, leading to improved efficiency and
productivity in job-shop environments.
Description of JSP as a scheduling problem
The Job-Shop Problem (JSP) can be described as a complex scheduling problem that seeks to determine the optimal sequence of operations for multiple jobs to be processed on various machines. Each job
consists of a set of operations that must be performed in a specific order, and each operation requires a certain amount of time to complete on a specific machine. The challenge in solving the JSP
lies in allocating the operations to machines in a way that minimizes the total makespan or completion time of all jobs. This problem is commonly encountered in manufacturing and production
environments where efficient job scheduling is crucial for maximizing productivity.
Explanation of the key elements involved in JSP modeling
There are several key elements involved in JSP modeling. Firstly, we have the machines, which represent the available resources for the job-shop. Each machine has specific capabilities and
limitations that need to be taken into account when scheduling the jobs. Secondly, we have the jobs themselves, which are a series of tasks that need to be performed in a specific order. Each job has
its own processing time and must be assigned to a machine that can handle its requirements. Finally, we have the scheduling constraints, which include factors such as preemption and precedence
relationships between jobs. These elements are crucial in JSP modeling and must be carefully considered to optimize the scheduling process.
Jobs and operations
Moreover, the JSP has attracted substantial attention due to its broad applicability to various industries and operations. The ability to efficiently schedule jobs and allocate resources is vital in
job-shop environments, as it directly impacts the overall performance and profitability of the operation. For instance, in manufacturing industries, the allocation of machines and workers to specific
tasks needs to be optimized to minimize idle time and maximize productivity. Similarly, in service sectors such as healthcare or transportation, effective job scheduling is essential to ensure timely
delivery of services and meet customer demands. As a result, the JSP has been extensively studied and numerous algorithms and techniques have been developed to address its complexities and find
optimal solutions.
Machines and processing times
Machines and processing times play a critical role in addressing the Job-Shop Problem (JSP). The JSP aims to optimize the scheduling of jobs across multiple machines in a manufacturing environment.
Each job typically requires a specific sequence of operations, with each operation having its own processing time on a particular machine. Therefore, accurately estimating the processing times is
crucial for creating an efficient schedule. However, determining these times can be challenging due to various factors like machine breakdowns, idle times, or operator skills. Advanced algorithms and
mathematical models can assist in finding optimal solutions by taking into account the complexities associated with machines and their processing times.
Constraints and objectives
Constraints and objectives play a crucial role in addressing the complexities of the job-shop problem (JSP). Constraints refer to the limitations that shape the problem, such as the availability of
machines and the order in which tasks must be completed. These constraints greatly affect the scheduling decisions and the overall optimization of the problem. Objectives, on the other hand, define
the goals that need to be achieved within the given constraints. Common objectives include minimizing job completions time, reducing idle time, and maximizing machine utilization. The interplay
between these constraints and objectives is pivotal in developing effective strategies and algorithms for solving the JSP efficiently.
One common approach to solving the Job-Shop Problem (JSP) is through the use of mathematical models. These models aim to mathematically represent the constraints and objectives of the JSP, allowing
for the optimization of schedules. One such model is the disjunctive graph model, which represents the precedence constraints of the JSP as a directed graph. This graph consists of nodes representing
the different operations in the job and edges representing the precedence constraints between operations. By using mathematical techniques like integer linear programming, these models can be solved
to find optimal or near-optimal schedules for the JSP. However, the complexity of the JSP means that finding solutions for larger instances still remains a challenge.
Classification and characteristics of JSP
The Job-Shop Problem (JSP) is a complex scheduling problem that has been extensively studied in the field of operations research. JSP can be classified as a type of combinatorial optimization problem
where the objective is to find an optimal sequencing of jobs on machines to minimize the makespan, which is the total time required to complete all jobs. One of the key characteristics of JSP is that
each job consists of a sequence of operations that must be processed in a particular order on a specific set of machines. Additionally, the processing times for each operation may vary, and there may
be precedence constraints that determine the order in which operations must be scheduled.
Types of JSP based on machine flexibility or job characteristics
There are different types of Job-Shop Problems (JSP) that can be categorized based on the machine flexibility or the characteristics of the jobs involved. The first type is the Flexible JSP, where
each machine can process any job, allowing for more flexibility in scheduling. On the other hand, in the No-Wait JSP, jobs can move directly to the next machine without having to wait for the
previous job to finish. Additionally, the Preemptive JSP allows for interruption and resumption of jobs on the same machine, enhancing the ability to handle unforeseen events and changes in
priorities. These variations in JSP types offer different solutions to optimize the job scheduling process and improve overall system efficiency.
Flexible Job-Shop Problem (FJSP)
In addition to the traditional Job-Shop Problem (JSP), there is a variant known as the Flexible Job-Shop Problem (FJSP). The FJSP extends the JSP by introducing more complex constraints and
additional flexibility. In the FJSP, each operation is assigned a set of eligible machines and a corresponding operation duration. The challenge lies in determining the optimal assignment of
operations to machines and the sequence of processing in order to minimize the makespan. This added complexity makes the FJSP more realistic, as it allows for variations in the available resources
and represents a more practical problem scenario in real-world production environments.
Open-Shop Problem
Another variant of the Job-Shop Problem is the Open-Shop Problem, which represents a further extension of the basic problem. In this case, each operation can be executed on any machine. The order of
the operations is predetermined and cannot be changed. Therefore, the Open-Shop Problem can be seen as a special case of the Job-Shop Problem, where each job consists of a single operation. Just like
the Job-Shop Problem, finding an optimal solution for the Open-Shop Problem is NP-hard, making it a challenging task for researchers in the field of operations research.
Flow-Shop Problem
Another variation of the job-shop problem is the flow-shop problem (FSP). In this problem, a set of tasks needs to be processed on a number of machines in a specific order. Unlike the job-shop
problem where the order of tasks on each machine can vary, in the flow-shop problem, the order of tasks is fixed. Each task has a specific processing time on each machine, and the goal is to
determine the sequence of tasks that minimizes the total makespan, which is the total time needed to complete all the tasks. The flow-shop problem has numerous real-world applications, such as
scheduling production lines in manufacturing.
Complexity and combinatorial nature of JSP
The complexity and combinatorial nature of the Job-Shop Problem (JSP) make it a challenging task to solve. JSP involves determining the sequence of tasks for each job in a job shop environment where
multiple jobs are processed on a set of machines. The number of possible solutions to the problem increases exponentially with the number of jobs and machines, resulting in a large search space.
Moreover, the decision variables involved in JSP are discrete and combinational, further complicating the problem. As a result, finding an optimal solution for JSP requires the implementation of
efficient algorithms and heuristics that can efficiently explore this massive solution space.
Real-world applications of JSP in various industries
Real-world applications of JSP can be observed across various industries. In the manufacturing sector, JSP is frequently utilized to optimize production scheduling, ensuring the efficient allocation
of resources and minimizing production delays. Additionally, JSP finds applications in the transportation industry, where it aids in the scheduling and routing of vehicles, optimizing delivery routes
and reducing transportation costs. Moreover, JSP is employed in the healthcare sector for appointment scheduling, assisting medical facilities in managing patient appointments and reducing wait
times. These real-world applications demonstrate the versatility of JSP in optimizing operations across different industries, ultimately enhancing efficiency and productivity.
In conclusion, the job-shop problem (JSP) is a complex scheduling problem in which a set of jobs, each consisting of several operations, must be scheduled on a set of machines to minimize the total
completion time. As discussed in this essay, the JSP is classified as an NP-hard problem due to its computational complexity and the lack of an efficient algorithm for finding an optimal solution.
Therefore, researchers have focused on developing approximation algorithms and heuristics to solve the JSP efficiently. Despite the challenges, the JSP remains an important problem in manufacturing
and operations research, motivating further research in this field.
Approaches and algorithms for solving JSP
There are several approaches and algorithms that have been developed to tackle the complex Job-Shop Problem (JSP). One widely used approach is the heuristic method, which refers to a rule-based
solution strategy that aims to find near-optimal solutions quickly. Another approach is the metaheuristic method, which utilizes higher-level procedures to guide the search for better solutions.
Genetic algorithms, simulated annealing, and tabu search are popular metaheuristic techniques applied to solve JSP. Additionally, there are also exact methods such as branch-and-bound and dynamic
programming, which guarantee the optimal solution but may be computationally expensive. Overall, the various approaches and algorithms provide different trade-offs between computational time and
solution quality when solving the JSP.
Exact methods for solving JSP
Another method for solving JSP is the use of exact algorithms. These algorithms guarantee an optimal solution for the problem. One popular exact method is the Branch and Bound technique, which
involves creating a tree-like structure to explore all possible solutions. This method eliminates subsets of solutions that are proven to be worse than the current solution, effectively reducing the
search space. Additionally, exact algorithms can incorporate other techniques such as dynamic programming and integer programming to further optimize the solution. Despite being time-consuming, these
exact methods provide accurate results in solving the Job-Shop Problem.
Branch and bound algorithm
One effective solution approach for the Job-Shop Problem (JSP) is the Branch and Bound algorithm. This algorithm systematically searches through the entire solution space of the problem,
progressively dividing and pruning the solution tree to find the optimal solution. It achieves this by assigning a lower bound to each subproblem and calculating an upper bound for the best feasible
solution found so far. By using these bounds, the algorithm can eliminate entire branches of the solution tree that are guaranteed to not lead to the optimal solution, thus greatly reducing the
search space and improving efficiency.
Integer programming formulations
Integer programming formulations are one of the common approaches used to solve the job-shop problem (JSP). This formulation involves assigning binary variables to represent the start and end time of
each operation. The objective function aims to minimize the makespan, which is the total time it takes to complete all jobs. Constraints are imposed to ensure that each job is processed in a specific
order and that no two operations can be performed simultaneously on the same machine. The integer programming formulation provides a systematic and mathematical approach to solving the JSP, allowing
for efficient scheduling and optimization of complex job-shop environments.
Heuristic and metaheuristic methods for solving JSP
Heuristic and metaheuristic methods have been widely employed for tackling the complexity of the Job-Shop Problem (JSP). These methods are particularly useful in finding near-optimal solutions for
large instances of the JSP, where exact optimization algorithms may prove inefficient or infeasible. Heuristics aim to generate feasible schedules through rule-based approaches, while metaheuristics
utilize optimization strategies inspired by natural phenomena to explore the solution space. Commonly employed heuristic and metaheuristic methods include genetic algorithms, simulated annealing,
tabu search, and ant colony optimization. These approaches provide efficient techniques for addressing the JSP and have shown promising results in terms of solution quality and computational
Genetic algorithms
One approach widely used to tackle the Job-Shop Problem (JSP) is the implementation of genetic algorithms. Genetic algorithms are computational techniques inspired by the process of natural selection
and genetics. By mimicking the principles of evolution, genetic algorithms aim to improve a given solution through the iteration of a predefined set of processes. These processes, namely selection,
crossover, and mutation, create a population of potential solutions that undergo repeated iterations until an optimal or satisfactory solution is found. This iterative approach allows for a
comprehensive exploration of the solution space, maximizing the chances of finding an optimal solution to the JSP.
Simulated annealing
Another approach that has been used to solve the Job-Shop Problem (JSP) is Simulated Annealing. Simulated Annealing is a heuristic optimization algorithm inspired by the annealing process in
metallurgy, where a material is gradually cooled to minimize defects. In the context of the JSP, Simulated Annealing starts with an initial solution and iteratively explores neighboring solutions.
The algorithm accepts modifications that improve the solution, but also allows for "bad" moves, which help escape local optima. As the algorithm progresses, the acceptance of "bad" moves decreases,
mimicking the cooling process in metallurgy. Simulated Annealing has shown promising results in solving the JSP by providing near-optimal solutions in reasonable computational time.
Tabu search
Tabu search is an effective optimization technique used to solve combinatorial optimization problems like the Job-Shop Problem (JSP). It is a metaheuristic algorithm that is based on the concept of
maintaining a tabu list, which keeps track of recently visited solutions and prohibits them from being revisited in the search process. By utilizing this memory-based mechanism, Tabu search helps to
overcome local optima and explore a diverse set of solutions. In the context of JSP, Tabu search can be applied to find an optimal sequence of operations for each job that minimizes the total
makespan, ensuring efficient utilization of resources and improving the overall system performance.
Comparative analysis of different solution approaches
One important aspect in addressing the Job-Shop Problem (JSP) lies in the comparative analysis of different solution approaches. Various techniques have been proposed to find optimal or near-optimal
solutions for JSP, including mathematical programming, metaheuristics, and hybrid approaches. Mathematical programming models formulate JSP as an integer linear program that can be solved using exact
methods such as branch and bound algorithms. On the other hand, metaheuristics like genetic algorithms, simulated annealing, and tabu search offer the advantage of finding reasonable quality
solutions in a reasonable time for large-scale JSP instances. Furthermore, hybrid approaches combine mathematical programming and metaheuristics to leverage the strengths of both paradigms. A
comprehensive comparison of these solution approaches considering criteria like solution quality, computational time, and robustness can help guide researchers and practitioners in selecting the most
suitable technique for solving JSP.
In conclusion, the Job-Shop Problem (JSP) is a complex optimization problem that aims to schedule a set of jobs on a set of machines to minimize the makespan. This problem has attracted significant
attention from researchers due to its wide range of real-world applications. Various algorithms and heuristics have been proposed to efficiently solve the JSP, including genetic algorithms,
metaheuristics, and mathematical programming approaches. However, finding an optimal solution to the JSP remains a challenging task. Future research efforts should focus on developing faster and more
accurate algorithms to tackle this problem, enabling better scheduling decisions and improving the overall productivity of job-shop environments.
Challenges and future directions in JSP research
Despite the significant progress made in JSP research, several challenges remain that require further investigation. Firstly, the size and complexity of real-world JSP instances often pose challenges
for existing algorithms and solution approaches. In addition, incorporating uncertain factors such as machine breakdowns or job processing times presents another significant challenge. Furthermore,
there is a need to develop new optimization algorithms and heuristics that can efficiently handle the large-scale JSP instances encountered in practice. Moreover, considering multi-objective JSP
formulations that account for conflicting objectives is an avenue for future research. Overall, these challenges and future directions in JSP research call for continued efforts to advance the field
and address the practical needs of real-world job-shop scheduling problems.
Scalability issues and limitations of existing algorithms
One major concern with existing algorithms for solving the job-shop problem (JSP) is their scalability. As the size of the problem increases, the computation time required by these algorithms becomes
impractical, making it difficult to apply them to large-scale real-world problems. Additionally, the existing algorithms often struggle to find optimal or near-optimal solutions for complex JSP
instances due to their limitations in handling combinatorial explosion and search-space complexity. Consequently, researchers continue to explore new approaches and techniques to overcome these
scalability issues, such as meta-heuristic algorithms and hybrid methodologies, to improve the performance and efficiency of JSP-solving algorithms.
Incorporating uncertain and dynamic parameters in JSP models
Moreover, incorporating uncertain and dynamic parameters in JSP models presents a crucial challenge in addressing the real-life complexities of production systems. Uncertainty arises from various
sources such as machine breakdowns, job arrival rates, processing times, and job priorities. Dynamic parameters refer to the time-varying nature of these uncertainties, which require constant
adjustments and re-evaluations. Researchers have proposed several approaches to tackle these challenges, including stochastic programming, simulation-based optimization, and robust optimization.
These methods aim to account for the uncertain and dynamic nature of the parameters to develop more accurate JSP models that can effectively optimize production scheduling in the face of real-world
Integration of JSP with emerging technologies like machine learning and IoT
In recent years, there has been a growing trend towards the integration of JSP with emerging technologies such as machine learning and the Internet of Things (IoT). This integration has the potential
to revolutionize job-shop scheduling by enhancing its efficiency and accuracy. For example, machine learning algorithms can be utilized to analyze historical production data and predict optimal
schedules, reducing the time and effort required for manual scheduling. Additionally, the combination of JSP with IoT enables real-time monitoring of shop floor activities, allowing for instant
identification and resolution of scheduling conflicts. Such integration holds great promise for improving the overall performance of job-shop systems and achieving greater operational excellence.
One of the main challenges in solving the Job-Shop Problem (JSP) is the optimization of resource allocation. JSP involves the scheduling of various operations across multiple machines, each with its
own set of constraints. The goal is to minimize the total completion time of all the jobs. To achieve this, it is crucial to make efficient use of the available resources, such as machines and labor.
This requires careful evaluation of the processing times, order of operations, and availability of resources at each stage. By addressing this resource allocation aspect effectively, JSP can be
approached more systematically, leading to improved scheduling solutions.
Case studies and practical implementations of JSP solutions
Case studies and practical implementations of JSP solutions have demonstrated the efficacy and versatility of this approach in various industries. For instance, a case study conducted in an
automotive manufacturing plant showcased significant improvements in production scheduling and resource allocation through the implementation of JSP solutions. The results indicated reduced job
completion times, increased machine utilization, and improved overall productivity. Similarly, another case study conducted in a semiconductor manufacturing facility revealed how JSP solutions
effectively optimized job sequencing, minimized idle times, and enhanced throughput rates. These case studies highlight the practical applicability of JSP solutions and their potential to
revolutionize scheduling and optimization processes in diverse industries.
Examples of successful applications of JSP in specific industries
Several industries have successfully implemented JSP in their operations, showcasing the versatility and effectiveness of this approach. For instance, in the automotive industry, JSP has been
utilized to optimize production schedules in car assembly lines, leading to significant cost reductions and improved productivity. The manufacturing sector has also embraced JSP, applying it to
intricate processes such as PCB assembly and machining operations, resulting in enhanced efficiency and shorter lead times. Additionally, the healthcare industry has utilized JSP to optimize patient
scheduling in hospitals and healthcare centers, minimizing waiting times and ensuring optimal utilization of resources. These successful applications emphasize the wide range of industries that can
benefit from implementing JSP strategies.
Automotive manufacturing
One of the industries heavily impacted by the job-shop problem (JSP) is automotive manufacturing. This sector involves the production of vehicles, which undergo a complex and time-consuming
manufacturing process. The JSP affects various aspects of automotive manufacturing, such as production planning, scheduling, and optimization. Due to the diverse range of automotive components, each
with different production requirements and interdependencies, JSP becomes particularly challenging. Manufacturers must allocate resources efficiently, minimize production costs, and ensure timely
completion of tasks in order to meet customer demands. Additionally, the JSP in automotive manufacturing involves managing multiple production lines, coordinating various operations, and adhering to
strict quality standards, all of which further exacerbate the complexity of this problem.
Semiconductor fabrication
Semiconductor fabrication refers to the process of creating integrated circuits and other electronic devices on semiconductor materials, primarily silicon. This highly complex and precise procedure
involves several essential steps, including wafer cleaning, photolithography, etching, and implantation. Fabricating semiconductors requires advanced technology and specialized equipment, such as
cleanroom facilities, photolithography machines, and chemical processing systems. Efficient production in semiconductor fabrication is vital for meeting the ever-increasing demand for advanced
electronic devices. However, due to the complexity and variability of the job-shop problem, optimal scheduling and planning of manufacturing processes in semiconductor fabrication remain challenging
Job scheduling in hospitals
Job scheduling in hospitals is a complex task that involves managing a wide range of resources and activities to ensure efficient and effective healthcare delivery. The primary aim of job scheduling
in hospitals is to allocate available resources, such as operating rooms, equipment, and staff, optimally, while considering various constraints, including patient priority, surgeon availability, and
equipment availability. This involves creating schedules that minimize patient waiting times, maximize resource utilization, and ensure the smooth flow of operations. Advanced scheduling algorithms
and optimization techniques are often employed to address the complexity of job scheduling in hospitals and improve overall operational performance.
Analysis of the benefits and improvements achieved through JSP implementation
In implementing the Job-Shop Problem (JSP), various benefits and improvements arise. Firstly, by organizing the order and timing of operations, JSP aids in optimizing production processes, leading to
increased efficiency and reduced overall manufacturing time. Additionally, it facilitates effective scheduling, allowing for better utilization of resources and reduction in production costs.
Moreover, JSP implementation contributes to enhanced decision-making capabilities through the analysis of alternative production scenarios and selection of the most favorable one. These benefits
ultimately lead to improved customer service, increased productivity, and higher profitability, making JSP an invaluable tool in the manufacturing industry.
In the context of production planning and scheduling, the Job-Shop Problem (JSP) represents a significant challenge. The goal of this problem is to determine the most efficient sequence of operations
for a set of jobs, each with their own specific processing requirements, on a set of machines. The complexity of this problem arises from the fact that each job requires a different sequence of
operations, and each machine has a limited capacity and can only perform one operation at a time. Researchers and practitioners have approached this problem using various heuristic algorithms,
mathematical models, and optimization techniques to find solutions that minimize makespan, maximize machine utilization, and increase productivity in job-shop environments. Through these efforts,
advancements have been made in addressing the JSP, but it remains a complex and ongoing challenge in the field of production planning.
In conclusion, the job-shop problem (JSP) has been extensively studied due to its importance in manufacturing and production systems. Through the use of various algorithms and techniques, researchers
have sought to find optimal solutions to minimize makespan and improve overall efficiency. The JSP is a complex combinatorial optimization problem that requires careful consideration of various
constraints and objectives. While many solution approaches have been proposed and have shown promising results, there is still room for further research and development in this field. By continually
improving upon existing methods and exploring new avenues, it is possible to achieve even more efficient and effective solutions to the job-shop problem.
Recap of the key points discussed in the essay
In conclusion, the job-shop problem (JSP) is a complex scheduling problem encountered in production environments. The key points discussed in this essay include the definition of the JSP, which
involves scheduling a set of jobs on a set of machines to optimize makespan. Additionally, two common solution approaches were presented: exact methods and heuristic algorithms. Exact methods, such
as the branch-and-bound algorithm, guarantee an optimal solution, but may have limitations in terms of computational efficiency. On the other hand, heuristic algorithms provide suboptimal solutions
but are often faster and more practical in solving larger instances of the problem. Advances in solving the JSP continue to be researched in order to develop effective strategies for production
planning and scheduling.
Importance of JSP in optimizing job scheduling and production processes
The importance of JSP in optimizing job scheduling and production processes cannot be overstated. JSP provides an efficient and effective method for organizing and prioritizing jobs in a job-shop
environment. By taking into account various factors such as machine availability, job priorities, and production deadlines, JSP allows for the creation of optimal production schedules. This helps to
maximize efficiency and minimize production delays, leading to improved overall productivity. Furthermore, JSP enables better resource allocation, as it identifies potential bottlenecks and allows
for the allocation of resources accordingly. In conclusion, JSP plays a crucial role in streamlining job scheduling and production processes, leading to increased productivity and improved efficiency
in job-shop environments.
Potential for future research and advancements in JSP solutions
One potential for future research and advancements in JSP solutions is the incorporation of artificial intelligence (AI) and machine learning algorithms. These technologies have the potential to
revolutionize JSP by enabling the development of intelligent systems that can dynamically adapt the scheduling decisions based on real-time data and changing job and machine characteristics.
Additionally, the application of optimization techniques such as genetic algorithms and simulated annealing can further enhance the performance of JSP solutions. Further research is also warranted to
investigate the effectiveness of hybrid approaches that integrate different solution methods to achieve optimal results in specific JSP instances.
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2000 AMC 12 Problems/Problem 17
A circle centered at $O$ has radius $1$ and contains the point $A$. The segment $AB$ is tangent to the circle at $A$ and $\angle AOB = \theta$. If point $C$ lies on $\overline{OA}$ and $\overline{BC}
$ bisects $\angle ABO$, then $OC =$
$[asy] import olympiad; size(6cm); unitsize(1cm); defaultpen(fontsize(8pt)+linewidth(.8pt)); labelmargin=0.2; dotfactor=3; pair O=(0,0); pair A=(1,0); pair B=(1,1.5); pair D=bisectorpoint(A,B,O);
pair C=extension(B,D,O,A); draw(Circle(O,1)); draw(O--A--B--cycle); draw(B--C); label("O",O,SW); dot(O); label("\theta",(0.1,0.05),ENE); dot(C); label("C",C,S); dot(A); label("A",A,E); dot(B); label
$\text {(A)}\ \sec^2 \theta - \tan \theta \qquad \text {(B)}\ \frac 12 \qquad \text {(C)}\ \frac{\cos^2 \theta}{1 + \sin \theta}\qquad \text {(D)}\ \frac{1}{1+\sin\theta} \qquad \text {(E)}\ \frac{\
sin \theta}{\cos^2 \theta}$
Solution 1
Since $\overline{AB}$ is tangent to the circle, $\triangle OAB$ is a right triangle. This means that $OA = 1$, $AB = \tan \theta$ and $OB = \sec \theta$. By the Angle Bisector Theorem, $\[\frac{OB}
{OC} = \frac{AB}{AC} \Longrightarrow AC \sec \theta = OC \tan \theta\]$ We multiply both sides by $\cos \theta$ to simplify the trigonometric functions, $\[AC=OC \sin \theta\]$ Since $AC + OC = 1$,
$1 - OC = OC \sin \theta \Longrightarrow$$OC = \dfrac{1}{1+\sin \theta}$. Therefore, the answer is $\boxed{\textbf{(D)} \dfrac{1}{1+\sin \theta}}$.
Solution 2
Alternatively, one could notice that OC approaches the value 1/2 as theta gets close to 90 degrees. The only choice that is consistent with this is (D).
Solution 3 (with minimal trig)
Let's assign a value to $\theta$ so we don't have to use trig functions to solve. $60$ is a good value for $\theta$, because then we have a $30-60-90 \triangle$ -- $\angle BAC=90$ because $AB$ is
tangent to Circle $O$.
Using our special right triangle, since $AO=1$, $OB=2$, and $AB=\sqrt{3}$.
Let $OC=x$. Then $CA=1-x$. since $BC$ bisects $\angle ABO$, we can use the angle bisector theorem:
Now, we only have to use a bit of trig to guess and check: the only trig facts we need to know to finish the problem is:
$\sin\theta =\frac{\text{Opposite}}{\text{Hypotenuse}}$
$\cos\theta =\frac{\text{Adjacent}}{\text{Hypotenuse}}$
$\tan\theta =\frac{\text{Opposite}}{\text{Adjacent}}$.
With a bit of guess and check, we get that the answer is $\boxed{D}$.
Solution 4
Let $OC$ = x, $OB$ = h, and $AB$ = y. $AC$ = $OA$ - $OC$.
Because $OC$ = x, and $OA$ = 1 (given in the problem), $AC$ = 1-x.
Using the Angle Bisector Theorem, $\frac{h}{y}$ = $\frac{x}{1-x}$$\Longrightarrow$ h(1-x) = xy. Solving for x gives us x = $\frac{h}{h+y}$.
$\sin\theta = \frac{opposite}{hypotenuse} = \frac{y}{h}$. Solving for y gives us y = $h \sin\theta$.
Substituting this for y in our initial equation yields x = $\dfrac{h}{h+h\sin \theta}$.
Using the distributive property, x = $\dfrac{h}{h(1+\sin \theta)}$ and finally $\dfrac{1}{1+\sin \theta}$ or $\boxed{\textbf{(D)}}$
Solution 5
Since $\overline{AB}$ is tangent to the circle, $\angle OAB=90^{\circ}$ and thus we can use trig ratios directly.
$\[\sin{\theta}=\frac{\overline{AB}}{\overline{BO}}, \cos{\theta}=\frac{1}{\overline{BO}}, \tan{\theta}=\overline{AB}\]$
By the angle bisector theorem, we have
Seeing the resemblance of the ratio on the left-hand side to $\sin{\theta},$ we turn the ratio around to allow us to plug in $\sin{\theta}.$ Another source of motivation for this also lies in the
idea of somehow adding 1 to the right-hand side so that we can substitute for a given value, i.e. $\overline{OA}=1$, and flipping the fraction will preserve the $\overline{OC}$, whilst adding one
right now would make the equation remain in direct terms of $\overline{CA}.$
$\[\frac{\overline{AB}}{\overline{OB}}=\sin{\theta}=\frac{\overline{CA}}{\overline{OC}}\Rightarrow \sin{\theta}+1=\frac{\overline{CA}+\overline{OC}}{\overline{OC}}=\frac{1}{\overline{OC}}\]$
$\[\sin{\theta}+1=\frac{1}{\overline{OC}} \Rightarrow \boxed{\overline{OC}=\frac{1}{\sin{\theta}+1}}\]$
Solution 6 (tangent half angle)
$\angle CBO = 45^{\circ} - \frac{\theta}{2}, \angle ACB = 45^{\circ} + \frac{\theta}{2}, OB = \frac{1}{\cos(\theta)}$. By sine law, $\frac{OC}{\sin(\angle CBO)} = \frac{OB}{\sin(\angle OCB)} = \frac
{OB}{\sin(\angle ACB)}$
$\[OC = \frac{\sin(45^{\circ} - \frac{\theta}{2})}{\sin(45^{\circ} + \frac{\theta}{2})}OB = \frac{\sin(45^{\circ} - \frac{\theta}{2})}{\cos(45^{\circ} - \frac{\theta}{2})}OB = \tan(45^{\circ} - \frac
{\theta}{2})OB = \frac{1-\tan(\theta/2)}{1+\tan(\theta/2)}OB\]$
Let $t = \tan(\theta/2)$. $OC = \frac{1-t}{1+t}OB = \frac{1-t^2}{1+2t+t^2}$. Because $\sin(\theta) = \frac{2t}{1+t^2}$ and $\cos(\theta) = \frac{1-t^2}{1+t^2}$, $\[OC = \frac{\cos(\theta)}{1+\sin(\
theta)}OB = \boxed{\textbf{(D)} \dfrac{1}{1+\sin \theta}}\]$
Solution 7 (if you forgot angle bisector but remember LoS)
Let $x=\overline{OC}$, and let $\angle OBC=\angle ABC=\alpha$. We know that $\overline{AC}=\overline{OA}-\overline{OC}=1-x$. By the Law of Sines, $\[\dfrac{\sin\alpha}x=\dfrac{\sin\theta}{BC}\]$
Combining the two give $\dfrac{\sin\alpha}x=\sin\theta\cdot\dfrac{\sin\alpha}{1-x}$.
Solving, this gives $\boxed{x=\frac{1}{\sin{\theta}+1}}$.
Video Solution
See also
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. | {"url":"https://artofproblemsolving.com/wiki/index.php/2000_AMC_12_Problems/Problem_17","timestamp":"2024-11-06T11:30:13Z","content_type":"text/html","content_length":"62641","record_id":"<urn:uuid:b5a48d44-7ac6-44b0-90d7-f8260a5971c0>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00063.warc.gz"} |
Gordon Growth Model Calculator: Understanding the Gordon Growth Model
Knowing How To Calculate The Growth Rate with the Gordon Growth Model can seem like a daunting task.The Gordon Growth Model is an essential tool for evaluating the true worth of a company’s
stock.Yet, many investors find themselves at a loss when it comes to applying this formula effectively.The key lies in understanding the core components of the Gordon Growth Model, as well as its
underlying assumptions about dividend growth rates and required returns on investment.
Table of Contents:
Demystifying the Gordon Growth Model
The Gordon Growth Model (GGM), also known as the dividend discount model, is a fundamental valuation method that plays an integral role in stock trading. This growth rate model empowers investors to
determine a company’s intrinsic value by assuming its shares are equivalent to the present worth of all future dividends.
In other words, the GGM provides you with tools for evaluating stocks based on their prospective returns via dividends. It suggests that investing in any firm is equivalent to participating in its
future profits, thus helping you make informed decisions about which stocks promise significant returns.
Gearing Up With The Gordon Growth Model Mechanics
The workings of this unique valuation tool hinge around three pivotal variables: Dividends Per Share (DPS), Dividend Growth Rate (g), and Required Rate of Return (r). These elements form the core
structure of the GGM and play crucial roles in determining if an investment harbors potential financial gain or not.
DPS reflects each share’s portion from total corporate earnings distributed as dividends. In contrast, ‘g’ signifies how rapidly these profits grow annually, while ‘r’ stands for your anticipated
annual return on investments after considering risk factors involved. The interaction between these components can be seen within the GGM’s formula:
Here P denotes price per share; D represents DPS; r encapsulates the required rate of return, and g embodies the dividend growth rate.
Fathoming The Role Of The Gordon Growth Model In Stock Trading
Gordon’s growth model acts like a compass guiding investment strategies by offering insights into companies’ inherent values beyond mere market prices. By concentrating more on long-term prospects
rather than short-term fluctuations, it allows traders to assess real profitability potentials rooted deeply in sustainable income streams dividends.
Key Takeaway:
Unlock the power of the Gordon Growth Model (GGM) to navigate stock trading. By focusing on Dividends Per Share, Dividend Growth Rate, and Required Rate of Return, you can gauge a company’s intrinsic
value beyond market prices. It’s your compass for long-term investment strategies.
Assumptions of the Gordon Growth Model
Understanding these principles correctly can significantly enhance your stock valuation process.
Constant Dividend Growth Rate Assumption
In the GGM framework, it is assumed that companies will maintain a constant dividend growth indefinitely. This implies that dividends per share (DPS), under this model’s assumption, will grow at an
unwavering rate year after year without disruption or fluctuation. Such predictability and consistency make this model particularly applicable for mature companies with proven histories of regular
dividend payouts.
Dividends, however, like many financial variables, are subject to various market conditions and changes within company policies. In reality, few businesses manage perfectly consistent growth rates
over long periods due to factors such as economic cycles, competitive pressures, or strategic shifts within the organization.
Required Rate of Return Assumption
The GGM also makes another critical assumption about investors’ required rate of return (r). Investors have certain expectations from their investments based on the risk levels associated with
particular stocks or sectors. These expectations form a crucial part of calculating intrinsic stock values using the GGM.
• DPS refers not just to current dividends but also to future ones projected based on historical data and business prospects.
• ‘g’ signifies how much those dividends are expected to increase annually.
• ‘r’ represents what shareholders would ideally want as returns considering the risks involved.
Deciphering the Gordon Growth Model for Share Value Calculation
The investment landscape is replete with various models and formulas, each designed to aid investors in making informed decisions. One such tool that stands out among these financial aids is the
Gordon Growth Model formula. This model provides a methodical approach to calculate share value by considering dividends per share (DPS), expected annual dividend growth rates, and required return on
In other words, knowing how much a stock is worth can be instrumental in your decision-making process when buying or selling shares.
Demystifying Dividends Per Share (DPS)
To begin our exploration into this realm of finance mathematics, let’s first understand what DPS signifies. In plain language, DPS indicates the amount of money paid out in dividends divided by the
total number of shares held. Essentially showing us how much an investor might receive for every single unit they own if all profits were distributed as dividends.
This simple calculation plays a pivotal role within GGM since its output directly impacts stock valuation using P = D/(r – g) where P denotes price; D symbolizes dividend equivalent to DPS; r refers
to the required rate of return; while g indicates the growth rate.
The Impact Of Expected Annual Dividend Growth Rates On Stock Prices
Moving onto another integral component – expected annual dividend growth rates – we need to comprehend why these figures matter when utilizing GGM for determining fair market prices tied with
securities like common equities, etc.
Basically speaking, projected yearly upticks related to cash distributions define the aforementioned term, thereby quantifying potential future earnings derived from ownership stakes held across
diverse corporate entities due to rising payouts stemming from improved profitability margins over time.
Key Takeaway:
The Gordon Growth Model is a powerful tool for calculating share value, considering dividends per share (DPS), expected annual dividend growth rates, and required return on investments. Understanding
DPS and how projected yearly upticks in cash distributions impact stock prices can significantly influence your investment decisions.
Exploring the Advantages & Limitations of The Gordon Growth Model
The valuation process in investment analysis can be simplified or complicated, depending on which model is employed. One such method that offers a blend of simplicity and effectiveness is the Gordon
Growth Model (GGM). This tool provides both advantages and limitations when used to estimate a company’s intrinsic stock value.
Appreciating the Benefits of Using The GGM
The first notable benefit lies within its ease-of-use. Unlike some financial models laden with complexity, GGM streamlines investment analysis by focusing on three essential variables: Dividends Per
Share (DPS), expected annual dividend growth rate, and required rate of return. Its user-friendly nature makes it an accessible tool for investors across different experience levels.
Beyond being simple to use, another advantage resides in its suitability for mature companies demonstrating consistent payout ratios. Mature firms typically maintain stable dividends over time; hence
using this model allows investors to predict future returns more accurately based on past performance patterns.
Acknowledging Drawbacks of Utilizing The GGM
Naturally accompanying these benefits are certain drawbacks tied up with employing the Gordon Growth Model as well. High-growth companies often witness fluctuating payout ratios, making them less
predictable than their established counterparts – posing challenges when applying constant growth assumptions inherent within models like GGM. In cases involving high variability, other valuation
methods might offer better accuracy in capturing economic worth effectively.
Moreover, the assumption regarding no changes in capital structure could lead towards inaccurate estimations too. If there are alterations in the firm’s financing mix – either through issuing new
shares, repurchasing existing ones, or altering debt levels – it will affect the cost of equity, thus impacting overall calculations significantly.
Finally, a critical drawback perhaps relates back to the underlying assumption itself, i.e., perpetual constant growth. This concept is inherently flawed since it is realistically impossible to
maintain the same level forever given external factors such as inflation rates and competition, among others.
Key Takeaway:
While the Gordon Growth Model (GGM) simplifies investment analysis with its user-friendly nature and accuracy for mature firms, it’s not foolproof. Its limitations include inaccurate estimations for
high-growth companies, changes in capital structure, and an unrealistic assumption of perpetual constant growth.
Practical Use of the Gordon Growth Model
The application of the Gordon Growth Model (GGM) is not limited to theory. In fact, it has been successfully utilized in real-world scenarios for estimating intrinsic stock values. This model proves
particularly useful when assessing mature companies known for their stable dividends.
A classic example would be blue-chip stocks – shares from large, financially sound, and well-established corporations that have a history of reliable operation. Investopedia provides an excellent
explanation on this topic:
This makes such firms perfect candidates for valuation using GGM. Consider Procter & Gamble Co., a multinational consumer goods corporation recognized globally for its steady dividend payouts over
1. An investor could use GGM to calculate present value future dividend payments.
2. Determine if the current market price is over or undervalued relative to calculations.
3. This information can guide investment decisions, whether to buy more shares or perhaps sell off existing ones based on estimated intrinsic values calculated by GMM.
Banks And The Application Of The Gordon Growth Model
In addition, sectors like banking also provide great examples where you might find successful applications of this model. Banks typically offer regular dividends and exhibit slow but constant growth
rates, two factors that align perfectly with assumptions made by GGM when estimating intrinsic stock values.
To illustrate, let’s take JP Morgan Chase & Co., a leading global financial services firm that has consistently paid out since 1996; hence investors may employ GGM here too while deciding upon
investing strategies related specifically towards banks’ stocks. These Corporate Finance Institute.
Gordon’s Theory at Work in Utility Companies?
Moving onto utility companies, which often apply principles from due predictable cash flows along steady returns they provide to shareholders through annual payout ratios.
Key Takeaway:
From blue-chip stocks to banking and utility sectors, the Gordon Growth Model (GGM) is a practical tool for estimating intrinsic stock values. It’s especially handy when evaluating companies known
for steady dividends, helping investors make informed decisions about buying or selling shares.
Advanced Concepts in the Gordon Growth Model
However, to leverage its full potential and accurately calculate growth rates, one must comprehend some of its advanced aspects.
Finding the Required Rate of Return
In GGM calculations, one crucial variable is the required rate of return. This figure signifies what an investor anticipates earning from their investment and can be influenced by elements such as
market interest rates or perceived risk levels linked with specific investments.
To identify your needed rate of return, you should consider both current market conditions and your personal level of risk tolerance. You might find it beneficial to seek advice from a financial
advisor or use online resources that provide detailed explanations on calculating required returns.
Factoring Dividend Policy Changes into Calculations
GGM presumes dividends will grow at a constant rate indefinitely; however, shifts in company dividend policies can impact these projections. If there are signs that a firm may alter its dividend
policy, either increasing or decreasing payouts, you’ll need to incorporate this into your GGM computations.
A good practice would be regular monitoring corporate announcements related to dividends through reliable sources which keep track of changes about companies’ payout strategies.
Taking Market Volatility Into Account
While providing valuable insights regarding intrinsic stock values based upon future dividends, the model doesn’t directly account for market volatility.
In periods where markets witness significant fluctuations due to economic events or other external factors, adjustments could become necessary while using this model.
An understanding regarding broader economic trends might assist here – websites offer up-to-date information pertaining to global economies which could help make appropriate adjustments if needed.
Leveraging Other Valuation Models Alongside The Gordon Growth Method
Multistage Dividend Discount Model (DDM)
The Multistage DDM provides investors dealing with firms experiencing non-constant growth over time more flexibility than the traditional Gordon method allows.
For instance, this model becomes suitable when we have businesses undergoing rapid expansion phases followed by slower stable stages.
Key Takeaway:
Mastering the Gordon Growth Model (GGM) involves understanding its advanced concepts. These include calculating the required rate of return, factoring in changes to dividend policies, accounting for
market volatility, and leveraging other valuation models like the Multistage Dividend Discount Model when dealing with non-constant growth.
FAQs in Relation to Calculating Growth Rate With the Gordon Growth Model
What is the Gordon constant growth rate?
The Gordon constant growth rate refers to the expected steady annual increase in dividends per share, as assumed by the Gordon Growth Model.
What is the formula for the growth rate of the dividend growth model?
In the Dividend Growth Model, or Gordon Growth Model, stock value (P) equals Dividends Per Share (DPS) divided by Required Rate of Return (r) minus Dividend Growth Rate (g).
How do you calculate WACC using Gordon growth model?
You can’t directly calculate Weighted Average Cost of Capital (WACC) with GGM. However, it’s often used as a ‘required return’ input when valuing equity via this method.
How do you calculate stock growth rate?
To estimate a stock’s future price, use its current price and apply an anticipated annualized total return percentage over your chosen time frame.
Unraveling the intricacies of the Gordon Growth Model has been quite a journey. This valuation method, often used in stock trading, is based on calculating a company’s intrinsic value from its future
We’ve delved into the model’s assumptions such as constant dividend growth and required rate of return. We also explored how it applies to mature companies with steady dividend growth patterns.
The formula for calculating share value using this model involves variables like Dividends Per Share (DPS), expected annual dividend growth rates, and required rates of return. Understanding these
components is crucial for accurate stock price estimation.
Though it may be simple to use, we must remember its limits. The GGM might not provide accurate results for high-growth companies or those with fluctuating payout ratios due to its reliance on
constant growth assumptions.
In real-world scenarios, while GGM can successfully estimate intrinsic stock values under certain conditions, other models may prove more suitable depending on specific company characteristics or
market dynamics.
To sum up: knowing how to calculate the growth rate with the Gordon Growth Model equips you with an essential tool for smart investing decisions – but remember that no single tool provides all
answers in every situation! | {"url":"https://frugalfortunes.com/gordon-growth-model-calculator/","timestamp":"2024-11-02T05:46:14Z","content_type":"text/html","content_length":"74448","record_id":"<urn:uuid:710ff064-6e35-4699-92cf-fe9a52e4de9b>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00886.warc.gz"} |
How to Find the Inverse for Any Square Matrix
In this lesson, we describe a method for finding the inverse of any square matrix; and we demonstrate the method step-by-step with examples.
Prerequisites: This material assumes familiarity with elementary matrix operations and echelon transformations.
How to Find the Inverse of an n x n Matrix
Let A be an n x n matrix. To find the inverse of matrix A, we follow these steps:
1. Using elementary operators, transform matrix A to its reduced row echelon form, A[rref].
2. Inspect A[rref] to determine if matrix A has an inverse.
□ If A[rref] is equal to the identity matrix, then matrix A is full rank; and matrix A has an inverse.
□ If the last row of A[rref] is all zeros, then matrix A is not full rank; and matrix A does not have an inverse.
3. If A is full rank, then the inverse of matrix A is equal to the product of the elementary operators that produced A[rref] , as shown below.
A^-1 = E[r] E[r-1] . . . E[2] E[1]
A^-1 = inverse of matrix A
r = Number of elementary row operations required to transform A to A[rref]
E[i] = ith elementary row operator used to transform A to A[rref]
Note that the order in which elementary row operators are multiplied is important, because E[i] E[j] is not necessarily equal to E[j] E[i].
An Example of Finding the Inverse
Let's use the above method to find the inverse of matrix A, shown below.
The first step is to transform matrix A into its reduced row echelon form, A[rref], using a series of elementary row operators E[i]. We show the transformation steps below for each elementary row
1. Multiply row 1 of A by -2 and add the result to row 2 of A. This can be accomplished by pre-multiplying A by the elementary row operator E[1], which produces A[1].
2. Multiply row 1 of A[1] by -2 and add the result to row 3 of A[1].
3. Multiply row 3 of A[2] by -1 and add row 2 of A[2] to row 3 of A[2].
4. Add row 2 of A[3] to row 1 of A[3].
5. Multiply row 2 of A[4] by -0.5.
6. Multiply row 3 of A[5] by -1 and add the result to row 2 of A[5].
Note: If the operations and/or notation shown above are unclear, please review elementary matrix operations and echelon transformations.
The last matrix in Step 6 of the above table is A[rref], the reduced row echelon form for matrix A. Since A[rref] is equal to the identity matrix, we know that A is full rank. And because A is full
rank, we know that A has an inverse.
If A were less than full rank, A[rref] would have all zeros in the last row; and A would not have an inverse.
We find the inverse of matrix A by computing the product of the elementary operators that produced A[rref] , as shown below.
A^-1 = E[6] E[5] E[4] E[3] E[2] E[1]
In this example, we used a 3 x 3 matrix to show how to find a matrix inverse. The same process will work on a square matrix of any size.
Test Your Understanding
Find the inverse of matrix A, shown below.
The first step is to transform matrix A into its reduced row echelon form, A[rref], using elementary row operators E[i] to perform elementary row operations, as shown below.
1. Multiply row 1 of A by -2 and add the result to row 2 of A.
2. Multiply row 2 of A[1] by 0.5..
The last transformed matrix in the above table is A[rref], the reduced row echelon form for matrix A. Since the reduced row echelon form is equal to the identity matrix, we know that A is full rank.
And because A is full rank, we know that A has an inverse.
We find the inverse by computing the product of the elementary operators that produced A[rref] , as shown below.
Note: In a previous lesson, we described a "shortcut" for finding the inverse of a 2 x 2 matrix. | {"url":"https://stattrek.com/matrix-algebra/how-to-find-inverse?tutorial=matrix","timestamp":"2024-11-06T12:19:31Z","content_type":"text/html","content_length":"81308","record_id":"<urn:uuid:5e90618c-f1fb-4614-8a61-f4b2e5e86c2c>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00815.warc.gz"} |
Numerical Algorithms
Lecture WS 24/25 Numerical Algorithms
Uncertainty quantification of PDE with random input data
Introductory course on numerical methods
Scientific Computing I (or similar)
The numerical solution of linear partial differential equations (PDE) is well understood up to almost arbitrary accuracy if the input data of these equations are available up to arbitrary accuracy.
However, in engineering applications this is not the case due to measurement errors in physical constants or tolerances in production processes, which makes the input data uncertain. This uncertainty
propagates through the PDE and makes the solution of the PDE itself uncertain, making the use of highly accurate numerical approximation methods questionable. In the lecture we will discuss
appropriate mathematical formalisms for PDE subject to uncertain input data and numerical algorithms which can quantify this uncertainty in terms of statistical moments such as the mean or the (co) | {"url":"https://ins.uni-bonn.de/teachings/ws-2024-445-v4e1-numerical-algori/","timestamp":"2024-11-05T20:09:51Z","content_type":"text/html","content_length":"9530","record_id":"<urn:uuid:0dc3c99f-ee4d-4229-b6f3-05de44f90664>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00024.warc.gz"} |
Zero pole gain system representation
S = zpk(Z, P, K, dt)
S = zpk(z, p, k, dt)
S = zpk(sys)
a m by n cell of real or complex vectors, Z{i,j} is the transmission zeros of the transfer from the the jth intput to the ith output.
a m by n cell of real or complex vectors, P{i,j} is the poles of the transfer from the the jth intput to the ith output.
a m by n matrix of real numbers, K(i,j) is the gain of the transfer from the the jth intput to the ith output.
a real or complex vector, the transmission zeros of the siso transfer function.
a real or complex vector, the poles of the siso transfer function.
a real scalar, the gain of the siso transfer function.
a character string with possible values "c" or "d", [] or a real positive scalar, the system time domain (see syslin).
A linear dynamical system in transfer function or state spece representation (see syslin).
a mlist with the fields Z , P, K and dt.
a m by n cell array of real or complex vectors, S.Z{i,j} contains the zeros of the transfer from the the jth intput to the ith output
a m by n cell array of real or complex vectors, S.P{i,j} contains the poles of the transfer from the the jth intput to the ith output
a m by n matrix of real numbers, S.K(i,j) is the gain of the transfer from the the jth intput to the ith output. output.
a positive scalar or "c" or "d" the time domain
S=zpk(Z,P,K,dt) forms the multi-input, multi-output zero pole gain system representation given the cell arrays of the transmission zeros,poles and gain.
S=zpk(z,p,k,dt) forms the single-input, single output zero pole gain system representation given the vectors of the transmission zeros and poles and the scalar gain.
S=zpk(sys) converts the system representation into a zero-pole-gain representation.
The poles and zeros of each transfer function are sorted in decreasing order of the real part.
Most functions and operations than can act on state-space or rational transfer function representations can be also applied to zero-pole-gain representations.
//Form system from zeros, poles and gain
//SISO case
z11=[1 -0.5];p11=[-3+2*%i -3-2*%i -2];k11=1;
//MIMO case
z21=0.3;p21=[-3+2*%i -3-2*%i];k21=1.5;
S=zpk({z11 [];z21 1},{p11,0;p21 -3},[k11 1;k21 1],"c")
//system representation conversion
//operations with zpk representations
See Also
• tf2zp — SIMO transfer function to zero pole gain representation
• zpk2tf — Zero pole gain to transfer function
• zpk2ss — Zero pole gain to state space
Версия Описание
6.0 Function added. | {"url":"https://help.scilab.org/docs/2023.0.0/ru_RU/zpk.html","timestamp":"2024-11-06T23:59:55Z","content_type":"text/html","content_length":"20821","record_id":"<urn:uuid:b36bc3e6-65b6-4a82-aec0-5a817e86a552>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00212.warc.gz"} |
Beam and Draft - Ocean Navigator
Beam and Draft
For many voyagers, trying to define of the "ideal" voyaging boat is one of the sport’s greatest debates. It is far easier said than done in that there are a large number of factors to be taken into
consideration, many of them contradictory.
As a result, every boat is the result of a series of compromises that will differ according to the priorities of the person driving the decision-making process. At one extreme, performance under sail
may be the overriding concern; at another, gunkholing in shallow anchorages may be the primary interest. These differing priorities should (if the yacht designer does his or her job) result in very
different boats.
When exploring design choices, we can look at a number of commonly quoted numeric parameters that are often used to compare boats and their implications. One excellent place to start is with beam and
draft calculations.Contemporary boat trends
Almost all voyaging boats, including world-girdling boats, spend the majority of their time either anchored out, on a mooring, or secured to a dock. At such times the boat is little more than a
floating condominium. It is natural to want to make it as comfortable a floating home as possible. This, in turn, calls for space, and as a result yacht designers and boat builders are always under
pressure to create as much volume as possible in any given design.
Volume nowadays typically translates into a wide beam, carried as far aft as possible, with high freeboard. The boat owner is sometimes going to want to be able to take this floating home into
relatively shoal anchorages. This requires a shallow draft. To get a beamy boat with little draft, the boat must have a flat bottom. Even though this boat will probably not spend much of its life at
sea, the builder and owner are still going to want it to perform reasonably well. A couple of keys to maximizing performance are to keep the overall weight, and thus the displacement, as low as
possible (lightweight construction), and to minimize wetted surface area by using the minimum keel area necessary to achieve reasonable upwind performance (a fin keel), together with the minimum
rudder size and supporting structure necessary to maintain control (a smallish spade rudder).
The kind of boat that is taking shape should be familiar; it can be seen at every major boat show. There is nothing wrong with this boat; it is built to fit a certain formula that is market driven,
and by and large it does an excellent job of fitting this formula.
When it comes to voyaging boats, and indeed any boat that may be used offshore, we have to add at least one more criterion to the mix. This is the ability to safely deliver the crew, together with
all stores and belongings, to its chosen destination in the worst conditions that might be encountered, and to do this at an acceptable speed and with as little discomfort as possible.
Among other things, this translates into a boat that is reasonably fast but with an easy motion at sea (a seakindly boat), that tracks well and has a light helm, that is stiff enough to carry
sufficient sail area to keep moving to windward in heavy weather, and that has, in an extreme situation, the ability to claw off a lee shore under sail alone in heavy seas and gale-force winds. It
must, of course, be built strongly enough to survive the gale.Form stability
Just about any boat can be pushed to windward in smooth water, but when things start to get rough it requires a great deal more power to counteract the boat’s windage and motion. Power requires sail
area. Sail area requires a stiff boat- i.e, one that resists heeling: all the sail area in the world won’t do a bit of good if the boat rolls over and lies on its side!
One way to achieve stiffness is to increase beam. As the boat heels, the immersed volume shifts rapidly to leeward, keeping the boat more-or-less upright. This is known as form stability. A
lightweight, beamy boat generally has excellent form stability. However, when the going gets tough the wide, flat sections, combined with the relatively light weight, are not only likely to make it
pound and roll uncomfortably, but also will have a tendency to cause its keel to stall out. As it stalls out, if the boat has a relatively shallow draft and minimal lateral surface area in the keel
and rudder, it will offer little resistance to making leeway. If it also has high freeboard, the windage will simply exacerbate problems. In other words, many of those features designed to improve
comfort at the dock or on the hook, and to ensure a sprightly performance in relatively protected waters, can become a handicap. A less extreme design approach is needed. The first thing to
reconsider is the wide beam.Length-to-beam ratio
The "beaminess" of a boat can be quantified by calculating its length-to-beam ratio – a number obtained by dividing the length by the beam. Often the length overall (LOA) – although in this case it
should not include a protruding bow pulpit – and the maximum beam (Bmax) are used, although I prefer to use the waterline length (abbreviated to LWL) and waterline beam (BWL). Note that the two
different formulas produce quite different values, so when making comparisons between boats it is essential to see that the same methodology is used to derive the numbers. For example, our Pacific
Seacraft 40 has a LOA (excluding the bow pulpit) of 40.33 feet and a Bmax of 12.42 feet, giving a length-to-beam ratio using these numbers of 40.33/12.42 = 3.25 (note that the inverse ratio is
sometimes given by dividing the beam by the length, in which case we get a beam-to-length ratio of 12.42/40.33 = 0.308). But if we use the waterline length (LWL) and waterline beam (BWL), we get a
waterline length-to-beam ratio of 31.25/11.33 = 2.76.
As noted, for comparison purposes it is preferable to use the LWL and BWL to derive a waterline length-to-beam ratio, but unfortunately, although the waterline length is commonly published, the
waterline beam is almost never published. As a result, yacht designer Roger Marshall, in The Complete Guide To Choosing A Voyaging Sailboat (published by International Marine, 1999) suggests that a
way to use available data is to work with the waterline length and Bmax x 0.9, which will approximate the waterline beam on many boats (note, however, that when looking at a range of boats, I found
this factor varied from as low as 0.75 to as high as 1.00, so this is a pretty crude approximation). When we apply these numbers to the Pacific Seacraft 40, we get:
LWL/(Bmax x 0.9) = 31.25/(12.42 x 0.9) = 2.80.
This is pretty close to the actual waterline length-to-beam ratio (2.76). Lower length-to-beam ratios indicate proportionately more beam; higher ratios less beam. A higher ratio is desirable both in
terms of windward performance in difficult conditions, and also as an indicator of handling characteristics and seakindly behavior.Beam and stability
However, this is not the whole picture. Beam affects stability on a cubic basis, which is to say that any increase in beam has a disproportionate effect on stability. If the length-to-beam ratio is
kept constant, as length increases, the increase in beam needed to maintain a constant ratio produces a disproportionate increase in stability. For example, a 36′ LWL boat with a 3:1 ratio will have
a 12′ waterline beam while a 48′ boat with the same ratio will have a 16′ beam; the 48′ boat will be considerably stiffer, even though it has the same ratio.
What this means is that if two boats have the same length-to-beam ratios, the one with the longer waterline is likely to have greater stability and sail-carrying ability, and better performance to
windward. Or, put another way, as length increases the syme relative sail-carrying ability can be maintained with a proportionately narrower beam and thus a higher length-to-beam ratio. As a result,
to improve stability and sail-carrying ability, shorter boats need proportionately more beam, resulting in lower length-to-beam ratios. Consequently, there is no absolute length-to-beam ratio “magic
number” that can be used for comparing boats; length must also be taken into account: the shorter a boat’s waterline length, the lower its length-to-beam ratio is likely to be.
Nevertheless, when looking at the 35-foot to 45-foot boat range (the “norm” for offshore voyaging these days), for a comfortable offshore voyager I like to see a waterline length-to-beam ratio of
3.00 or higher (using LWL/[Bmax x 0.9]). Shorter boats may have a lower ratio; longer boats should have a higher ratio. Looking at a sampling of contemporary European and American boats (see table on
page 88), we see that the only two boats below 40 feet LOA that have a ratio of over 3.00 are the Alerion Express 38 and the Shannon 39. At 40-feet and above, many of those boats that follow the
current fashion of short overhangs, which maximizes the waterline length, have ratios of 3.0 and higher, whereas more traditional voyaging boats, with longer overhangs, for the most part do not. Our
Pacific Seacraft 40, for example, has a waterline length-to-beam of 2.80. This is the price that has to be paid for its long overhangs combined with the beam necessary to provide a more spacious
interior as compared to voyaging boat designs of a generation ago.
Many older, but nonetheless highly successful, voyaging boats in this same size range have waterline length-to-beam of 3.0 and above (based on LWL/[Bmax x 0.9]). Steve Dashew, the designer of the
Deerfoot and Sundeer series of boats, has taken the length-to-beam ratio to extremes. His boats commonly have ratios of 4:1, 5:1 and up. This is all to the good except that, because of the relatively
narrow beam, in order to establish a reasonable interior volume, the boat has to get longer and the costs start to soar. He writes in the second edition of the Offshore Voyaging Encyclopedia that he
and Linda, his wife and partner, decided to see just how small a boat they could design that would contain what they felt to be their minimum requirements for just the two of them, including
accommodating a couple of guests for a week or two a year. They arrived at 56 feet in length! Unfortunately, however desirable it may be, such a boat is beyond the budget of most of us, not only up
front but also in terms of mooring or dockage fees, gear replacement costs, maintenance, and so on.Keel types
A narrower beam results in less form stability, which can translate into greater heeling when on the wind. To counteract this tendency to heel it’s necessary to put a lot of weight down low. In its
extreme form this results in the 14-foot fin keels, with massive lead bulbs, seen on some narrow racing boats.
Clearly, such a keel is not practical on a voyaging boat, but the principle is the same – to get as much weight as possible as low as possible. How low is primarily a function of where the boat is
intended to sail. In general, a six-foot draft is acceptable, still allowing access to most of the world’s finest voyaging grounds. However, a boat specifically intended for voyaging in shoal areas
such as the Bahamas might be designed with less draft, whereas one intended for Pacific voyaging might have a deeper draft. A voyager/racer, with an emphasis on the racing side of things, is likely
to exceed six feet, trading access to some voyaging grounds for improved performance when racing.
For a given draft, the use of a bulb keel keeps the weight as low as possible. A wing keel does the same, but needs to be carefully designed if it is not to foul lines and seaweed, or get stuck in
the mud in a grounding. (A wing keel has a shape much like a Bruce anchor. Wing keels originated as a rule-beating device in the America’s Cup, and have since become something of a fad. I doubt that
any advantage over a bulb keel outweighs the disadvantages in a voyaging environment.) On our new boat we chose a bulb-keel option, with a draft of five feet two inches, as opposed to the standard
deep-keel of six feet one inch. We get a significantly reduced draft with a small loss of windward performance.
The advent of bulb and wing keel types has pretty much put paid to the old debate as to whether it is preferable to have internal or external ballast: the bulb or wing must be external (it’s hard to
mold them into fiberglass). Clearly, lead, with its great density, should always be used as the ballast material (as opposed to iron, which is sometimes used to save cost yet it’s only a little more
than 60% of the density of lead). We’ve hardly started looking at the process of choosing an “ideal” voyaging boat, and already we are beginning to sense that there are a complicated series of
trade-offs between, for example, beam and draft, interior accommodation and windward ability, and comfort on the hook and at sea.
Based on my own experience, which is primarily bluewater voyaging, if I were to settle on two numbers that provide an acceptable beam and draft middle ground for 35- to 45-foot voyaging boats, it
would be a waterline length-to-beam ratio of 3.0 or higher, and a draft of six feet or less. Longer boats should have a higher waterline length-to-beam ratio, and may require more draft. | {"url":"https://oceannavigator.com/beam-and-draft/","timestamp":"2024-11-05T09:42:52Z","content_type":"text/html","content_length":"111493","record_id":"<urn:uuid:12766c40-a0e1-40eb-b0c9-23eeaae5838e>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00203.warc.gz"} |
In analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space. These parameters must uniquely define the configuration of
the system relative to a reference state.^[1] The generalized velocities are the time derivatives of the generalized coordinates of the system. The adjective "generalized" distinguishes these
parameters from the traditional use of the term "coordinate" to refer to Cartesian coordinates.
An example of a generalized coordinate would be to describe the position of a pendulum using the angle of the pendulum relative to vertical, rather than by the x and y position of the pendulum.
Although there may be many possible choices for generalized coordinates for a physical system, they are generally selected to simplify calculations, such as the solution of the equations of motion
for the system. If the coordinates are independent of one another, the number of independent generalized coordinates is defined by the number of degrees of freedom of the system.^[2]^[3]
Generalized coordinates are paired with generalized momenta to provide canonical coordinates on phase space.
Constraints and degrees of freedom
Generalized coordinates are usually selected to provide the minimum number of independent coordinates that define the configuration of a system, which simplifies the formulation of Lagrange's
equations of motion. However, it can also occur that a useful set of generalized coordinates may be dependent, which means that they are related by one or more constraint equations.
Holonomic constraints
For a system of N particles in 3D real coordinate space, the position vector of each particle can be written as a 3-tuple in Cartesian coordinates:
{\displaystyle {\begin{aligned}&\mathbf {r} _{1}=(x_{1},y_{1},z_{1}),\\&\mathbf {r} _{2}=(x_{2},y_{2},z_{2}),\\&\qquad \qquad \vdots \\&\mathbf {r} _{N}=(x_{N},y_{N},z_{N})\end{aligned}}}
Any of the position vectors can be denoted r[k] where k = 1, 2, …, N labels the particles. A holonomic constraint is a constraint equation of the form for particle k^[4]^[a]
${\displaystyle f(\mathbf {r} _{k},t)=0}$
which connects all the 3 spatial coordinates of that particle together, so they are not independent. The constraint may change with time, so time t will appear explicitly in the constraint equations.
At any instant of time, any one coordinate will be determined from the other coordinates, e.g. if x[k] and z[k] are given, then so is y[k]. One constraint equation counts as one constraint. If there
are C constraints, each has an equation, so there will be C constraint equations. There is not necessarily one constraint equation for each particle, and if there are no constraints on the system
then there are no constraint equations.
So far, the configuration of the system is defined by 3N quantities, but C coordinates can be eliminated, one coordinate from each constraint equation. The number of independent coordinates is n = 3N
− C. (In D dimensions, the original configuration would need ND coordinates, and the reduction by constraints means n = ND − C). It is ideal to use the minimum number of coordinates needed to define
the configuration of the entire system, while taking advantage of the constraints on the system. These quantities are known as generalized coordinates in this context, denoted q[j](t). It is
convenient to collect them into an n-tuple
${\displaystyle \mathbf {q} (t)=(q_{1}(t),\ q_{2}(t),\ \ldots ,\ q_{n}(t))}$
which is a point in the configuration space of the system. They are all independent of one other, and each is a function of time. Geometrically they can be lengths along straight lines, or arc
lengths along curves, or angles; not necessarily Cartesian coordinates or other standard orthogonal coordinates. There is one for each degree of freedom, so the number of generalized coordinates
equals the number of degrees of freedom, n. A degree of freedom corresponds to one quantity that changes the configuration of the system, for example the angle of a pendulum, or the arc length
traversed by a bead along a wire.
If it is possible to find from the constraints as many independent variables as there are degrees of freedom, these can be used as generalized coordinates.^[5] The position vector r[k] of particle k
is a function of all the n generalized coordinates (and, through them, of time),^[6]^[7]^[8]^[5]^[nb 1]
${\displaystyle \mathbf {r} _{k}=\mathbf {r} _{k}(\mathbf {q} (t))\,,}$
and the generalized coordinates can be thought of as parameters associated with the constraint.
The corresponding time derivatives of q are the generalized velocities,
${\displaystyle {\dot {\mathbf {q} }}={\frac {d\mathbf {q} }{dt}}=({\dot {q}}_{1}(t),\ {\dot {q}}_{2}(t),\ \ldots ,\ {\dot {q}}_{n}(t))}$
(each dot over a quantity indicates one time derivative). The velocity vector v[k] is the total derivative of r[k] with respect to time
${\displaystyle \mathbf {v} _{k}={\dot {\mathbf {r} }}_{k}={\frac {d\mathbf {r} _{k}}{dt}}=\sum _{j=1}^{n}{\frac {\partial \mathbf {r} _{k}}{\partial q_{j}}}{\dot {q}}_{j}\,.}$
and so generally depends on the generalized velocities and coordinates. Since we are free to specify the initial values of the generalized coordinates and velocities separately, the generalized
coordinates q[j] and velocities dq[j]/dt can be treated as independent variables.
Non-holonomic constraints
A mechanical system can involve constraints on both the generalized coordinates and their derivatives. Constraints of this type are known as non-holonomic. First-order non-holonomic constraints have
the form
${\displaystyle g(\mathbf {q} ,{\dot {\mathbf {q} }},t)=0\,,}$
An example of such a constraint is a rolling wheel or knife-edge that constrains the direction of the velocity vector. Non-holonomic constraints can also involve next-order derivatives such as
generalized accelerations.
Physical quantities in generalized coordinates
Kinetic energy
The total kinetic energy of the system is the energy of the system's motion, defined as^[9]
${\displaystyle T={\frac {1}{2}}\sum _{k=1}^{N}m_{k}{\dot {\mathbf {r} }}_{k}\cdot {\dot {\mathbf {r} }}_{k}\,,}$
in which · is the dot product. The kinetic energy is a function only of the velocities v[k], not the coordinates r[k] themselves. By contrast an important observation is^[10]
${\displaystyle {\dot {\mathbf {r} }}_{k}\cdot {\dot {\mathbf {r} }}_{k}=\sum _{i,j=1}^{n}\left({\frac {\partial \mathbf {r} _{k}}{\partial q_{i}}}\cdot {\frac {\partial \mathbf {r} _{k}}{\
partial q_{j}}}\right){\dot {q}}_{i}{\dot {q}}_{j},}$
which illustrates the kinetic energy is in general a function of the generalized velocities, coordinates, and time if the constraints also vary with time, so T = T(q, dq/dt, t).
In the case the constraints on the particles are time-independent, then all partial derivatives with respect to time are zero, and the kinetic energy is a homogeneous function of degree 2 in the
generalized velocities.
Still for the time-independent case, this expression is equivalent to taking the line element squared of the trajectory for particle k,
${\displaystyle ds_{k}^{2}=d\mathbf {r} _{k}\cdot d\mathbf {r} _{k}=\sum _{i,j=1}^{n}\left({\frac {\partial \mathbf {r} _{k}}{\partial q_{i}}}\cdot {\frac {\partial \mathbf {r} _{k}}{\partial q_
and dividing by the square differential in time, dt^2, to obtain the velocity squared of particle k. Thus for time-independent constraints it is sufficient to know the line element to quickly obtain
the kinetic energy of particles and hence the Lagrangian.^[11]
It is instructive to see the various cases of polar coordinates in 2D and 3D, owing to their frequent appearance. In 2D polar coordinates (r, θ),
${\displaystyle \left({\frac {ds}{dt}}\right)^{2}={\dot {r}}^{2}+r^{2}{\dot {\theta }}^{2}\,,}$
in 3D cylindrical coordinates (r, θ, z),
${\displaystyle \left({\frac {ds}{dt}}\right)^{2}={\dot {r}}^{2}+r^{2}{\dot {\theta }}^{2}+{\dot {z}}^{2}\,,}$
in 3D spherical coordinates (r, θ, φ),
${\displaystyle \left({\frac {ds}{dt}}\right)^{2}={\dot {r}}^{2}+r^{2}{\dot {\theta }}^{2}+r^{2}\sin ^{2}\theta \,{\dot {\varphi }}^{2}\,.}$
Generalized momentum
The generalized momentum "canonically conjugate to" the coordinate q[i] is defined by
${\displaystyle p_{i}={\frac {\partial L}{\partial {\dot {q}}_{i}}}.}$
If the Lagrangian L does not depend on some coordinate q[i], then it follows from the Euler–Lagrange equations that the corresponding generalized momentum will be a conserved quantity, because the
time derivative is zero implying the momentum is a constant of the motion;
${\displaystyle {\frac {\partial L}{\partial q_{i}}}={\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {q}}_{i}}}={\dot {p}}_{i}=0\,.}$
Bead on a wire
Bead constrained to move on a frictionless wire. The wire exerts a reaction force C on the bead to keep it on the wire. The non-constraint force N in this case is gravity. Notice the initial position
of the wire can lead to different motions.
For a bead sliding on a frictionless wire subject only to gravity in 2d space, the constraint on the bead can be stated in the form f (r) = 0, where the position of the bead can be written r = (x(s),
y(s)), in which s is a parameter, the arc length s along the curve from some point on the wire. This is a suitable choice of generalized coordinate for the system. Only one coordinate is needed
instead of two, because the position of the bead can be parameterized by one number, s, and the constraint equation connects the two coordinates x and y; either one is determined from the other. The
constraint force is the reaction force the wire exerts on the bead to keep it on the wire, and the non-constraint applied force is gravity acting on the bead.
Suppose the wire changes its shape with time, by flexing. Then the constraint equation and position of the particle are respectively
${\displaystyle f(\mathbf {r} ,t)=0\,,\quad \mathbf {r} =(x(s,t),y(s,t))}$
which now both depend on time t due to the changing coordinates as the wire changes its shape. Notice time appears implicitly via the coordinates and explicitly in the constraint equations.
Simple pendulum
Simple pendulum. Since the rod is rigid, the position of the bob is constrained according to the equation f (x, y) = 0, the constraint force C is the tension in the rod. Again the non-constraint
force N in this case is gravity.
Dynamic model of a simple pendulum.
The relationship between the use of generalized coordinates and Cartesian coordinates to characterize the movement of a mechanical system can be illustrated by considering the constrained dynamics of
a simple pendulum.^[12]^[13]
A simple pendulum consists of a mass M hanging from a pivot point so that it is constrained to move on a circle of radius L. The position of the mass is defined by the coordinate vector r = (x, y)
measured in the plane of the circle such that y is in the vertical direction. The coordinates x and y are related by the equation of the circle
${\displaystyle f(x,y)=x^{2}+y^{2}-L^{2}=0,}$
that constrains the movement of M. This equation also provides a constraint on the velocity components,
${\displaystyle {\dot {f}}(x,y)=2x{\dot {x}}+2y{\dot {y}}=0.}$
Now introduce the parameter θ, that defines the angular position of M from the vertical direction. It can be used to define the coordinates x and y, such that
${\displaystyle \mathbf {r} =(x,y)=(L\sin \theta ,-L\cos \theta ).}$
The use of θ to define the configuration of this system avoids the constraint provided by the equation of the circle.
Notice that the force of gravity acting on the mass m is formulated in the usual Cartesian coordinates,
${\displaystyle \mathbf {F} =(0,-mg),}$
where g is the acceleration due to gravity.
The virtual work of gravity on the mass m as it follows the trajectory r is given by
${\displaystyle \delta W=\mathbf {F} \cdot \delta \mathbf {r} .}$
The variation δr can be computed in terms of the coordinates x and y, or in terms of the parameter θ,
${\displaystyle \delta \mathbf {r} =(\delta x,\delta y)=(L\cos \theta ,L\sin \theta )\delta \theta .}$
Thus, the virtual work is given by
${\displaystyle \delta W=-mg\delta y=-mgL\sin(\theta )\delta \theta .}$
Notice that the coefficient of δy is the y-component of the applied force. In the same way, the coefficient of δθ is known as the generalized force along generalized coordinate θ, given by
${\displaystyle F_{\theta }=-mgL\sin \theta .}$
To complete the analysis consider the kinetic energy T of the mass, using the velocity,
${\displaystyle \mathbf {v} =({\dot {x}},{\dot {y}})=(L\cos \theta ,L\sin \theta ){\dot {\theta }},}$
${\displaystyle T={\frac {1}{2}}m\mathbf {v} \cdot \mathbf {v} ={\frac {1}{2}}m({\dot {x}}^{2}+{\dot {y}}^{2})={\frac {1}{2}}mL^{2}{\dot {\theta }}^{2}.}$
D'Alembert's form of the principle of virtual work for the pendulum in terms of the coordinates x and y are given by,
${\displaystyle {\frac {d}{dt}}{\frac {\partial T}{\partial {\dot {x}}}}-{\frac {\partial T}{\partial x}}=F_{x}+\lambda {\frac {\partial f}{\partial x}},\quad {\frac {d}{dt}}{\frac {\partial T}{\
partial {\dot {y}}}}-{\frac {\partial T}{\partial y}}=F_{y}+\lambda {\frac {\partial f}{\partial y}}.}$
This yields the three equations
${\displaystyle m{\ddot {x}}=\lambda (2x),\quad m{\ddot {y}}=-mg+\lambda (2y),\quad x^{2}+y^{2}-L^{2}=0,}$
in the three unknowns, x, y and λ.
Using the parameter θ, those equations take the form
${\displaystyle {\frac {d}{dt}}{\frac {\partial T}{\partial {\dot {\theta }}}}-{\frac {\partial T}{\partial \theta }}=F_{\theta },}$
which becomes,
${\displaystyle mL^{2}{\ddot {\theta }}=-mgL\sin \theta ,}$
${\displaystyle {\ddot {\theta }}+{\frac {g}{L}}\sin \theta =0.}$
This formulation yields one equation because there is a single parameter and no constraint equation.
This shows that the parameter θ is a generalized coordinate that can be used in the same way as the Cartesian coordinates x and y to analyze the pendulum.
Double pendulum
A double pendulum
The benefits of generalized coordinates become apparent with the analysis of a double pendulum. For the two masses m[i] (i = 1, 2), let r[i] = (x[i], y[i]), i = 1, 2 define their two trajectories.
These vectors satisfy the two constraint equations,
${\displaystyle f_{1}(x_{1},y_{1},x_{2},y_{2})=\mathbf {r} _{1}\cdot \mathbf {r} _{1}-L_{1}^{2}=0}$
${\displaystyle f_{2}(x_{1},y_{1},x_{2},y_{2})=(\mathbf {r} _{2}-\mathbf {r} _{1})\cdot (\mathbf {r} _{2}-\mathbf {r} _{1})-L_{2}^{2}=0.}$
The formulation of Lagrange's equations for this system yields six equations in the four Cartesian coordinates x[i], y[i] (i = 1, 2) and the two Lagrange multipliers λ[i] (i = 1, 2) that arise from
the two constraint equations.
Now introduce the generalized coordinates θ[i] (i = 1, 2) that define the angular position of each mass of the double pendulum from the vertical direction. In this case, we have
${\displaystyle \mathbf {r} _{1}=(L_{1}\sin \theta _{1},-L_{1}\cos \theta _{1}),\quad \mathbf {r} _{2}=(L_{1}\sin \theta _{1},-L_{1}\cos \theta _{1})+(L_{2}\sin \theta _{2},-L_{2}\cos \theta _
The force of gravity acting on the masses is given by,
${\displaystyle \mathbf {F} _{1}=(0,-m_{1}g),\quad \mathbf {F} _{2}=(0,-m_{2}g)}$
where g is the acceleration due to gravity. Therefore, the virtual work of gravity on the two masses as they follow the trajectories r[i] (i = 1, 2) is given by
${\displaystyle \delta W=\mathbf {F} _{1}\cdot \delta \mathbf {r} _{1}+\mathbf {F} _{2}\cdot \delta \mathbf {r} _{2}.}$
The variations δr[i] (i = 1, 2) can be computed to be
${\displaystyle \delta \mathbf {r} _{1}=(L_{1}\cos \theta _{1},L_{1}\sin \theta _{1})\delta \theta _{1},\quad \delta \mathbf {r} _{2}=(L_{1}\cos \theta _{1},L_{1}\sin \theta _{1})\delta \theta _
{1}+(L_{2}\cos \theta _{2},L_{2}\sin \theta _{2})\delta \theta _{2}}$
Thus, the virtual work is given by
${\displaystyle \delta W=-(m_{1}+m_{2})gL_{1}\sin \theta _{1}\delta \theta _{1}-m_{2}gL_{2}\sin \theta _{2}\delta \theta _{2},}$
and the generalized forces are
${\displaystyle F_{\theta _{1}}=-(m_{1}+m_{2})gL_{1}\sin \theta _{1},\quad F_{\theta _{2}}=-m_{2}gL_{2}\sin \theta _{2}.}$
Compute the kinetic energy of this system to be
${\displaystyle T={\frac {1}{2}}m_{1}\mathbf {v} _{1}\cdot \mathbf {v} _{1}+{\frac {1}{2}}m_{2}\mathbf {v} _{2}\cdot \mathbf {v} _{2}={\frac {1}{2}}(m_{1}+m_{2})L_{1}^{2}{\dot {\theta }}_{1}^{2}+
{\frac {1}{2}}m_{2}L_{2}^{2}{\dot {\theta }}_{2}^{2}+m_{2}L_{1}L_{2}\cos(\theta _{2}-\theta _{1}){\dot {\theta }}_{1}{\dot {\theta }}_{2}.}$
Euler–Lagrange equation yield two equations in the unknown generalized coordinates θ[i] (i = 1, 2) given by^[14]
${\displaystyle (m_{1}+m_{2})L_{1}^{2}{\ddot {\theta }}_{1}+m_{2}L_{1}L_{2}{\ddot {\theta }}_{2}\cos(\theta _{2}-\theta _{1})+m_{2}L_{1}L_{2}{\dot {\theta _{2}}}^{2}\sin(\theta _{1}-\theta _{2})=
-(m_{1}+m_{2})gL_{1}\sin \theta _{1},}$
${\displaystyle m_{2}L_{2}^{2}{\ddot {\theta }}_{2}+m_{2}L_{1}L_{2}{\ddot {\theta }}_{1}\cos(\theta _{2}-\theta _{1})+m_{2}L_{1}L_{2}{\dot {\theta _{1}}}^{2}\sin(\theta _{2}-\theta _{1})=-m_{2}
gL_{2}\sin \theta _{2}.}$
The use of the generalized coordinates θ[i] (i = 1, 2) provides an alternative to the Cartesian formulation of the dynamics of the double pendulum.
Spherical pendulum
Spherical pendulum: angles and velocities.
For a 3D example, a spherical pendulum with constant length l free to swing in any angular direction subject to gravity, the constraint on the pendulum bob can be stated in the form
${\displaystyle f(\mathbf {r} )=x^{2}+y^{2}+z^{2}-l^{2}=0\,,}$
where the position of the pendulum bob can be written
${\displaystyle \mathbf {r} =(x(\theta ,\phi ),y(\theta ,\phi ),z(\theta ,\phi ))\,,}$
in which (θ, φ) are the spherical polar angles because the bob moves in the surface of a sphere. The position r is measured along the suspension point to the bob, here treated as a point particle. A
logical choice of generalized coordinates to describe the motion are the angles (θ, φ). Only two coordinates are needed instead of three, because the position of the bob can be parameterized by two
numbers, and the constraint equation connects the three coordinates (x, y, z) so any one of them is determined from the other two.
Generalized coordinates and virtual work
The principle of virtual work states that if a system is in static equilibrium, the virtual work of the applied forces is zero for all virtual movements of the system from this state, that is, δW = 0
for any variation δr.^[15] When formulated in terms of generalized coordinates, this is equivalent to the requirement that the generalized forces for any virtual displacement are zero, that is F[i] =
Let the forces on the system be F[j] (j = 1, 2, …, m) be applied to points with Cartesian coordinates r[j] (j = 1, 2, …, m), then the virtual work generated by a virtual displacement from the
equilibrium position is given by
${\displaystyle \delta W=\sum _{j=1}^{m}\mathbf {F} _{j}\cdot \delta \mathbf {r} _{j}.}$
where δr[j] (j = 1, 2, …, m) denote the virtual displacements of each point in the body.
Now assume that each δr[j] depends on the generalized coordinates q[i] (i = 1, 2, …, n) then
${\displaystyle \delta \mathbf {r} _{j}={\frac {\partial \mathbf {r} _{j}}{\partial q_{1}}}\delta {q}_{1}+\ldots +{\frac {\partial \mathbf {r} _{j}}{\partial q_{n}}}\delta {q}_{n},}$
${\displaystyle \delta W=\left(\sum _{j=1}^{m}\mathbf {F} _{j}\cdot {\frac {\partial \mathbf {r} _{j}}{\partial q_{1}}}\right)\delta {q}_{1}+\ldots +\left(\sum _{j=1}^{m}\mathbf {F} _{j}\cdot {\
frac {\partial \mathbf {r} _{j}}{\partial q_{n}}}\right)\delta {q}_{n}.}$
The n terms
${\displaystyle F_{i}=\sum _{j=1}^{m}\mathbf {F} _{j}\cdot {\frac {\partial \mathbf {r} _{j}}{\partial q_{i}}},\quad i=1,\ldots ,n,}$
are the generalized forces acting on the system. Kane^[16] shows that these generalized forces can also be formulated in terms of the ratio of time derivatives,
${\displaystyle F_{i}=\sum _{j=1}^{m}\mathbf {F} _{j}\cdot {\frac {\partial \mathbf {v} _{j}}{\partial {\dot {q}}_{i}}},\quad i=1,\ldots ,n,}$
where v[j] is the velocity of the point of application of the force F[j].
In order for the virtual work to be zero for an arbitrary virtual displacement, each of the generalized forces must be zero, that is
${\displaystyle \delta W=0\quad \Rightarrow \quad F_{i}=0,i=1,\ldots ,n.}$
See also
Wikiquote has quotations related to Generalized coordinates.
1. ^ Some authors e.g. Hand & Finch take the form of the position vector for particle k, as shown here, as the condition for the constraint on that particle to be holonomic.
1. ^ Some authors set the constraint equations to a constant for convenience with some constraint equations (e.g. pendulums), others set it to zero. It makes no difference because the constant can
be subtracted to give zero on one side of the equation. Also, in Lagrange's equations of the first kind, only the derivatives are needed.
1. ^ Ginsberg 2008, p. 397, §7.2.1 Selection of generalized coordinates
2. ^ Farid M. L. Amirouche (2006). "§2.4: Generalized coordinates". Fundamentals of multibody dynamics: theory and applications. Springer. p. 46. ISBN 0-8176-4236-6.
3. ^ Florian Scheck (2010). "§5.1 Manifolds of generalized coordinates". Mechanics: From Newton's Laws to Deterministic Chaos (5th ed.). Springer. p. 286. ISBN 978-3-642-05369-6.
4. ^ Goldstein, Poole & Safko 2002, p. 12
5. ^ ^a ^b Kibble & Berkshire 2004, p. 232
6. ^ Torby 1984, p. 260
7. ^ Goldstein, Poole & Safko 2002, p. 13
8. ^ Hand & Finch 1998, p. 15
9. ^ Torby 1984, p. 269
10. ^ Goldstein, Poole & Safko 2002, p. 25
11. ^ Landau & Lifshitz 1976, p. 8
12. ^ Greenwood, Donald T. (1987). Principles of Dynamics (2nd ed.). Prentice Hall. ISBN 0-13-709981-9.
13. ^ Richard Fitzpatrick, Newtonian Dynamics.
14. ^ Eric W. Weisstein, Double Pendulum, scienceworld.wolfram.com. 2007
15. ^ T. R. Kane and D. A. Levinson, Dynamics: theory and applications, McGraw-Hill, New York, 1985
Bibliography of cited references | {"url":"https://www.knowpia.com/knowpedia/Generalized_coordinates","timestamp":"2024-11-09T01:23:01Z","content_type":"text/html","content_length":"294132","record_id":"<urn:uuid:6c64d9f1-c0ab-4fe6-a846-57eae369dc02>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00267.warc.gz"} |
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What are the units for magnetism?
The gauss, symbol G (sometimes Gs), is a unit of measurement of magnetic induction, also known as magnetic flux density.
What are the two units of magnetism?
Units Of Magnetism When scientists talk about magnets, magnetism, and magnetic forces, they use different units to describe the different characteristics of magnetism. There are two systems of units
based on the metric system that scientists use: MKSMKSThe MKS system of units is a physical system of measurement that uses the meter, kilogram, and second (MKS) as base units.https://
en.wikipedia.org โ บ wiki โ บ MKS_system_of_unitsMKS system of units – Wikipedia (meter-kilogram-second) units and CGSCGSThe centimetreโ gramโ second system of units (abbreviated CGS or cgs) is a
variant of the metric system based on the centimetre as the unit of length, the gram as the unit of mass, and the second as the unit of time.https://en.wikipedia.org โ บ wiki โ บ Centimetreโ gramโ
second_sys…Centimetreโ gramโ second system of units – Wikipedia (centimeter-gram-second) units.
What is magnetism and its SI unit?
When a magnetic material like iron is placed in a magnetic field, it gets a magnetic dipole moment. The magnetic dipole moment acquired per unit volume is known as magnetisation. Its SI unit will be
m3Am2=mA. Its dimension is [Lโ 1M0T0I1].
What is the unit of magnetic energy?
Earlier, the power of a magnet was only measured in units of gaussgaussThe gauss, symbol G (sometimes Gs), is a unit of measurement of magnetic induction, also known as magnetic flux density. The
unit is part of the Gaussian system of units, which inherited it from the older CGS-EMU system. It was named after the German mathematician and physicist Carl Friedrich Gauss in 1936.https://
en.wikipedia.org โ บ wiki โ บ Gauss_(unit)Gauss (unit) – Wikipedia. Gauss defines the number of lines of magnetic (flux density) per square centimetre emitted from the surface of the magnet. Today,
it is measured in gauss-oersted energy units (GOe).
What is the SI unit of magnetic induction?
So the S.I unit of magnetic induction is Tesla.
What is the SI unit of magnetic susceptibility?
Solution : Magnetic susceptibility of a substance may be defined may be defined as the ratio of intensity of magnetisation ( M) to the magnetising field ( H).
`chi= ((M)/(H))` Susceptibility has no unit.
What are the units for flux?
Electric flux is a scalar quantity and has an SI unit of newton-meters squared per coulomb ( N ยท m 2 /C N ยท m 2 /C ).
What is the SI unit of flux density?
The SI derived unit of magnetic flux density is the tesla, which is defined as a volt second per square meter.
Is gauss a unit of magnetic flux?
GaussGaussThe gauss, symbol G (sometimes Gs), is a unit of measurement of magnetic induction, also known as magnetic flux density. The unit is part of the Gaussian system of units, which inherited it
from the older CGS-EMU system. It was named after the German mathematician and physicist Carl Friedrich Gauss in 1936.https://en.wikipedia.org โ บ wiki โ บ Gauss_(unit)Gauss (unit) – Wikipedia is the
unit of magnetic flux densitymagnetic flux densityMagnetic Flux Density is amount of magnetic flux through unit area taken perpendicular to direction of magnetic flux. Flux Density (B) is related to
Magnetic Field (H) by B=ฮผH. It is measured in Webers per square meter equivalent to Teslas [T].https://www.toppr.com โ บ ask โ บ magnetic-flux-density-209798Magnetic Flux Density | Definition,
Examples, Diagrams – Toppr B. Was this answer helpful?
What is m and H in magnetism?
The definition of H is H = B/ฮผ โ M, where B is the magnetic flux density, a measure of the actual magnetic field within a material considered as a concentration of magnetic field lines, or flux,
per unit cross-sectional area; ฮผ is the magnetic permeability; and M is the magnetization.
What is the SI unit of magnetic dipole?
In the SI system, the specific unit for dipole moment is the joule (unit of energy) per tesla (unit of magnetic field strength or magnetic flux density).
What is magnetism write its unit and dimensional formula?
Magnetization `=(“Net magnetic moment”)/(“Volume”)`
i.e., `vec M_(z)=(vecM_(“net”))/(“Volume”)`
The S.I. unit is `Am^(-1). `
The dimensions are `[M^(0)L^(-1)T^(0)I^(1)]`.
What is unit of magnetic field strength?
Magnetic field strength refers to a physical quantity that is used as one of the basic measures of the intensity of the magnetic field. The unit of magnetic field strength happens to be ampere per
meter or A/m. Furthermore, the symbol of the magnetic field strength happens to be ‘H’.
Why is Tesla’s SI unit magnetic?
The tesla (symbol T) is the derived SI unit of magnetic flux density, which represents the strength of a magnetic field. One tesla represents one weber per square meter. The equivalent, and
superseded, cgs unit is the gauss (G); one tesla equals exactly 10,000 gauss.
How do you find the units of a magnetic field?
The SI unit for magnetic field is the Tesla, which can be seen from the magnetic part of the Lorentz force law Fmagnetic = qvB to be composed of (Newton x second)/(Coulomb x meter). A smaller
magnetic field unit is the Gauss (1 Tesla = 10,000 Gauss).
What are the SI unit of magnetic flux and magnetic field?
The SI unit of magnetic flux is the weber (Wbweber (WbIn physics, the weber (/ห veษชb-, ห wษ b. ษ r/ VAY-, WEH-bษ r; symbol: Wb) is the unit of magnetic flux in the International System of Units
(SI), whose units are volt-second. A magnetic flux density of one Wb/m2 (one weber per square metre) is one tesla.https://en.wikipedia.org โ บ wiki โ บ Weber_(unit)Weber (unit) – Wikipedia) (in
derived units: volt seconds), and the CGS unit is the maxwell. Magnetic flux is usually measured with a fluxmeter, which contains measuring coils and electronics, that evaluates the change of voltage
in the measuring coils to calculate the magnetic flux.
What is a tesla unit in physics?
Tesla (T) – Magnetic Field Intensity Unit. Definition: The International System unit of field intensity for magnetic fields is Tesla (T). One tesla (1 T) is defined as the field intensity generating
one newton (N) of force per ampere (A) of current per meter of conductor: T = N ร A-1 ร m-1 = kg ร s-2 ร A-1.
Does magnetic susceptibility have a unit?
Magnetic susceptibility is the measure of the degree of magnetization of a material in response to the externally applied magnetic field. Because, magnetization (M) and magnetic field intensity (H)
both have the same units A/m. Thus, magnetic susceptibility is a dimensionless unit.
What is SI unit of permeability?
The SI unit of magnetic permeability is Henry per metre.
How do you find the SI unit of magnetic permeability?
Where B = magnetic intensity and H = magnetizing field. Henries per meter (H/m) or newtons per ampere squared (Nโ Aโ 2) is the SI unit of magnetic permeability.
What is the unit of magnetic moment?
Magnetic Moment Unit: Conferring to that, the torque is measured in Joules (J) and the magnetic field is measured in tesla (T) and thus the unit is J T -1. So, these two units are equivalent to each
other and are provided by 1 Amp-m2 = 1 J T -1.
What is the SI unit weber?
weberweberIn physics, the weber (/ห veษชb-, ห wษ b. ษ r/ VAY-, WEH-bษ r; symbol: Wb) is the unit of magnetic flux in the International System of Units (SI), whose units are volt-second. A magnetic
flux density of one Wb/m2 (one weber per square metre) is one tesla.https://en.wikipedia.org โ บ wiki โ บ Weber_(unit)Weber (unit) – Wikipedia, unit of magnetic flux in the International System of
Units (SI), defined as the amount of flux that, linking an electrical circuit of one turn (one loop of wire), produces in it an electromotive force of one volt as the flux is reduced to zero at a
uniform rate in one second.
What is electric flux and its SI unit?
Electric flux can be defined as the measure of the distribution of the electric field or the rate of flow of the electric field through a given area. Electric flux is denoted by the greek symbol ฮฆ .
The SI unit of electric flux is voltmeters ( Vm ) or Nm 2 C – 1 .
What is the SI unit of capacitance?
The SI unit of capacitance is farad.
What is the unit of capacitance?
The unit of electrical capacitance is the farad (abbreviated F), named after the English physicist and chemist Michael Faraday. | {"url":"https://physics-network.org/what-are-the-units-for-magnetism/","timestamp":"2024-11-08T06:30:15Z","content_type":"text/html","content_length":"310723","record_id":"<urn:uuid:00cebd23-3a94-444b-be30-ac8b7d5479dd>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00369.warc.gz"} |
Complexity Framework for Forbidden Subgraphs IV: The Steiner Forest Problem
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completely characterizes the complexity of many problems on H-subgraph-free graphs. However, in contrast to e.g. the related Steiner Tree problem, Steiner Forest falls outside this framework. Hence,
the complexity of Steiner Forest on H-subgraph-free graphs remained tantalizingly open. In this paper, we make significant progress towards determining the complexity of Steiner Forest on
H-subgraph-free graphs. Our main results are four novel polynomial-time algorithms for different excluded graphs H that are central to further understand its complexity. Along the way, we study the
complexity of Steiner Forest for graphs with a small c-deletion set, that is, a small set S of vertices such that each component of G−S has size at most c. Using this parameter, we give two
noteworthy algorithms that we later employ as subroutines. First, we prove Steiner Forest is FPT parameterized by |S| when c=1 (i.e. the vertex cover number). Second, we prove Steiner Forest is
polynomial-time solvable for graphs with a 2-deletion set of size at most 2. The latter result is tight, as the problem is NP-complete for graphs with a 3-deletion set of size 2.
Dive into the research topics of 'Complexity Framework for Forbidden Subgraphs IV: The Steiner Forest Problem'. Together they form a unique fingerprint. | {"url":"https://research-portal.uu.nl/en/publications/complexity-framework-for-forbidden-subgraphs-iv-the-steiner-fores","timestamp":"2024-11-02T18:23:16Z","content_type":"text/html","content_length":"52984","record_id":"<urn:uuid:e5e1c5d9-50f1-4609-bf7a-fe3345b5f176>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00717.warc.gz"} |
Multiplication 2 Worksheet
Math, especially multiplication, develops the foundation of numerous scholastic disciplines and real-world applications. Yet, for lots of students, mastering multiplication can present a challenge.
To address this hurdle, teachers and moms and dads have actually welcomed a powerful device: Multiplication 2 Worksheet.
Introduction to Multiplication 2 Worksheet
Multiplication 2 Worksheet
Multiplication 2 Worksheet -
The 2 times table worksheets pdf gives kids the opportunity to enhance their knowledge on multiplication skills Such skills of skip counting and of course repeated addition are so accurate in the
Multiplying by 2 activities Also It offers fantastic exercises which kids will tackle in just one minute by either solving out a product solving
K5 Learning offers free worksheets flashcards and inexpensive workbooks for kids in kindergarten to grade 5 Become a member to access additional content and skip ads Grade 2 multiplication worksheets
including multiplication facts multiples of 5 multiples of 10 multiplication tables and missing factor questions No login required
Relevance of Multiplication Practice Understanding multiplication is crucial, laying a solid structure for sophisticated mathematical principles. Multiplication 2 Worksheet use structured and
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Advancement of Multiplication 2 Worksheet
Printable Multiplication Worksheets For Grade 2
Printable Multiplication Worksheets For Grade 2
Welcome to The Multiplying 1 to 12 by 2 100 Questions A Math Worksheet from the Multiplication Worksheets Page at Math Drills This math worksheet was created or last revised on 2021 02 19 and has
been viewed 1 278 times this week and 1 598 times this month It may be printed downloaded or saved and used in your classroom home school or other educational environment to help
These multiplication worksheets may be configured for 2 3 or 4 digit multiplicands being multiplied by multiples of ten that you choose from a table You may vary the numbers of problems on the
worksheet from 15 to 27 These multiplication worksheets are appropriate for Kindergarten 1st Grade 2nd Grade 3rd Grade 4th Grade and 5th Grade
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2 Digit Horizontal Multiplication 2 Worksheet For 2nd 4th Grade Lesson Planet
2 Digit Horizontal Multiplication 2 Worksheet For 2nd 4th Grade Lesson Planet
One digit multiplication by 2 Liveworksheets transforms your traditional printable worksheets into self correcting interactive exercises that the students can do online and send to the teacher
Multiply by 2 s worksheet Live Worksheets
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the worksheet generator These worksheets are randomly generated and therefore provide endless amounts of exercise material for at home or in
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Connecting multiplication to everyday scenarios includes importance and functionality to exercises.
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and layouts help comprehension for learners inclined toward aesthetic learning. Auditory Learners Verbal multiplication troubles or mnemonics satisfy students who understand principles with acoustic
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Normal method reinforces multiplication abilities, advertising retention and fluency. Stabilizing Repeating and Selection A mix of repeated exercises and varied issue styles keeps interest and
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Multiplication Times Tables Worksheets 2 3 4 5 6 7 Times Multiplication 2 Worksheet
Multiplication Worksheets 1 And 2 Multiplication Worksheets
Check more of Multiplication 2 Worksheet below
Multiplying 2 Digit By 2 Digit Numbers With Space Separated Thousands A Long Multiplication
Multiplication 2x Http www worksheetfun 2013 02 23 times table worksheets 2 CC C2W1
Kids Page 2 Times Multiplication Table Worksheet
2 By 2 Digit Multiplication Worksheets Free Printable
Multiplication Grade 2 Math Worksheets
Free Multiplication Worksheet 1s And 2s Free4Classrooms
Grade 2 Multiplication Worksheets free printable K5 Learning
K5 Learning offers free worksheets flashcards and inexpensive workbooks for kids in kindergarten to grade 5 Become a member to access additional content and skip ads Grade 2 multiplication worksheets
including multiplication facts multiples of 5 multiples of 10 multiplication tables and missing factor questions No login required
Multiplication Facts Worksheets Math Drills
This section includes math worksheets for practicing multiplication facts to from 0 to 49 There are two worksheets in this section that include all of the possible questions exactly once on each page
the 49 question worksheet with no zeros and the 64 question worksheet with zeros
K5 Learning offers free worksheets flashcards and inexpensive workbooks for kids in kindergarten to grade 5 Become a member to access additional content and skip ads Grade 2 multiplication worksheets
including multiplication facts multiples of 5 multiples of 10 multiplication tables and missing factor questions No login required
This section includes math worksheets for practicing multiplication facts to from 0 to 49 There are two worksheets in this section that include all of the possible questions exactly once on each page
the 49 question worksheet with no zeros and the 64 question worksheet with zeros
2 By 2 Digit Multiplication Worksheets Free Printable
Multiplication 2x Http www worksheetfun 2013 02 23 times table worksheets 2 CC C2W1
Multiplication Grade 2 Math Worksheets
Free Multiplication Worksheet 1s And 2s Free4Classrooms
The worksheet For Multiply 0 1 Or 2 Is Shown In Black And White
Multiplication 2 Worksheet Printable Lexia s Blog
Multiplication 2 Worksheet Printable Lexia s Blog
Two Digit Multiplication Worksheets Multiplication Worksheets Two Digit multiplication Math
Frequently Asked Questions (Frequently Asked Questions).
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Rashid, Author at Micro Digital
It is very useful regulated power supply It is based on 78xx series ICs. We can use it for different output regulated voltages by choosing different IC numbers.
It is 78xx IC series based regulated DC supply. We can make different voltage level supplies by choosing different 78xx IC series number. For example we can choose
1) 7805 for 5V output.
2) 7808 for 8V output.
3) 7809 for 9V output.
4) 7812 for 12V output.
Vin is high level DC voltage that is provided through D1 to 78xx series IC for regulation. Diode D1 is used for blocking reverse voltage polarity applied by mistake and saves whole circuit from
reverse polarity. capacitor C1-C4 are used for filtration purposes. LED D2 is an on/off indicator and resistance R1 limits the current through it. R1 can be changed according to input voltage. Diode
D3 is used here to protect 78xx IC from reverse voltage across it’s output due to C4 when power is removed and also stops overcharging of capacitors C3, C4. At output pin 3 of 78xx we can get
regulated DC supply that is at low level than input supply at pin 1 of 78xx. For better circuit performance and protection of 78xx IC input supply should be greater than output by 3V to 5V range.
More voltage input than output will cause the 78xx IC to be hot and can damage the IC permanently. Because there will be more voltage drop across 78xx IC and so more wattage will be dissipated. Use
heat sink for better performance of IC when there is more current requirement at output.
Full Wave Bridge Rectifier Supply
It is 220 volt AC to 12 volt DC full wave rectifier supply.
It uses 4 diode to rectify AC voltage. We can understand this circuit by breaking down in parts as follows
This is the part that converts AC voltage into pulsating DC by using D1-D4.
In this part diode we exclude D2 and D3. Now when transformer output 1 will be positive and output 2 negative during sin wave pulse then diodes D1 and D4 conduct as these are forward biased and
positive side is available at D1’s output(labeled as 2) and negative side is available at D4’s output(labeled as 4). When negative voltage is present at output 1 of transformer and positive at output
2 now these diodes will be reversed biased and so will not conduct. And there will be no output due to D1 and D4.
Now we exclude D1, D4 and consider D2, D3 in the circuit. In this way when transformer output 1 will be positive and output 2 negative during sin wave pulse then diodes D2 and D3 will not conduct as
these are reversed biased and there will be no output due to these diodes. When negative voltage is present at transformer’s output 1 and positive at output 2 then both diodes D2, D3 will conduct(as
these are forward biased) and positive voltage will be available at D2’s output(labeled as 2) and negative voltage will be available at D3’s output(labeled as 4).
Conclusion is that
1) when transformer outputs 1 and 2 go +ve and -ve respectively then diodes pair D1, D2 conducts.
2) when transformer outputs 1 and 2 go -ve and +ve respectively then diodes pair D2, D3 conducts.
3) So one pair conducts during half period (positive) of sin wave and other pair during remaining half period (negative) of sine wave.
4) In both upper cases(1, 2) +ve voltage is present at output labeled 2 and -ve voltage side at output labeled as 4.
Now again by combining these 2 parts we can see that these 4 diodes not only provide positive side of sin wave as it is at output(labeled as 2, 4) but also present negative side of sin wave as
positive at same output(labeled as 2, 4). So in this way AC signal is converted into DC. But this is not smooth DC but pulsating DC as shown in figure. We will further use capacitors to make these
pulsating DC as smooth DC.
If we add capacitor C1 at diode output this half wave pulsating DC is stored in this capacitor. At every half cycle this capacitor is charged up and if this energy is not used by load (or in other
words there is no path to discharge capacitor) then this capacitor is charged up to almost it’s rated voltage level. Now pulsating effect is removed by the use of this capacitor and smooth DC is
obtained at some level. Small capacitor C2 to removes fast pulses of noise. Finally we add an LED D2 and resistor R1 as power on/off indicator. This network also work as to limit charging of
capacitor. It provides a path for the discharge of capacitor. For this purpose we usually use a single resistor that is also called blade resistor. It limits capacitor charging to certain level so
that load can be saved from over voltage. R1 also limits current through LED D2. We will also upload interesting LED projects for beginners, students and hobbyist.
Half Wave Rectifier Supply
It is 220 volt AC to 12 volt DC half wave rectifier supply.
It uses single diode to rectify AC voltage. We can divide this circuit into parts as follows
In this part diode D1 passes positive signal part and blocks negative part of the signal. In this way half wave pulsating DC is available at output of diode D1.
If we add capacitor C1 at diode output this half wave pulsating DC is stored in this capacitor. At every half cycle this capacitor is charged up and if this energy is not used by load (or in other
words there is no path to discharge capacitor) then this capacitor is charged up to almost it’s rated voltage level. Now pulsating effect is removed by the use of this capacitor and smooth DC is
obtained at some level.
Now we add small capacitor C2 to remove fast pulses of noise.
Now finally we add an LED D2 and resistor R1 as power on/off indicator. This network also work as to limit charging of capacitor. It provides a path for the discharge of capacitor. For this purpose
we usually use a single resistor that is also called blade resistor. It limits capacitor charging to certain level so that load can be saved from over voltage. R1 also limits current through LED D2.
We will also upload interesting LED projects for beginners, students and hobbyist.
220V AC to 12V AC Power Supply
It is 220 volt AC to 12 volt AC step down supply.
It is simplest power supply. It step downs 220 volt AC mains supply available in offices and homes to 12 volt AC. It utilizes a step down transformer to step down voltage that can be used for
different purposes. We can use different watt transformer for different wattage output requirement. If we want 12 volt and 1 ampere supply then we will use 12 watt 12 volt step down transformer and
if we want 12 volt and 2 ampere output we will use 24 watt and 12 volt step down transformer.
We can calculate wattage of transformer by using formula
P = IV
P is power in watts
I is current in amperes
V is voltage in volts
Here is an important and useful app for transformer wire gauge calculation and turn per volt calculation.
IT Solutions
We also engineer software projects. For your problem you can contact us. | {"url":"https://www.micro-digital.net/author/rashid/page/5/","timestamp":"2024-11-09T10:17:45Z","content_type":"text/html","content_length":"54576","record_id":"<urn:uuid:847ed016-9add-40ce-81b8-e4dbe764658b>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00494.warc.gz"} |
Accepted Papers with Abstracts
On the Axiomatisation of Branching Bisimulation Congruence over CCS
ABSTRACT. In this paper we investigate the equational theory of (the restriction, relabelling, and recursion free fragment of) CCS modulo rooted branching bisimilarity, which is a classic,
bisimulation-based notion of equivalence that abstracts from internal computational steps in process behaviour. Firstly, we show that CCS is not finitely based modulo the considered congruence. As a
key step of independent interest in the proof of that negative result, we prove that each CCS process has a unique parallel decomposition into indecomposable processes modulo branching bisimilarity.
As a second main contribution, we show that, when the set of actions is finite, rooted branching bisimilarity has a finite equational basis over CCS enriched with the left merge and communication
merge operators from ACP.
Propositional dynamic logic and asynchronous cascade decompositions for regular trace languages
ABSTRACT. One of the main motivations for this work is to obtain a distributed Krohn-Rhodes theorem for Mazurkiewicz traces. Concretely, we focus on the recently introduced operation of local cascade
product of asynchronous automata and ask if every regular trace language can be accepted by a local cascade product of `simple' asynchronous automata.
Our approach crucially relies on the development of a local and past-oriented propositional dynamic logic (LocPastPDL) over traces which is shown to be expressively complete with respect to all
regular trace languages. An event-formula of LocPastPDL allows to reason about the causal past of an event and a path-formula of LocPastPDL, localized at a process, allows to march along the sequence
of past-events in which that process participates, checking for local regular patterns interspersed with local tests of other event-formulas. We also use additional constant formulas to compare the
leading process events from the causal past. The new logic LocPastPDL is of independent interest. The proof of its expressive completeness is rather subtle.
Finally, we provide a translation of LocPastPDL formulas into local cascade products. More precisely, we show that every LocPastPDL formula can be computed by a restricted local cascade product of
the gossip automaton and localized 2-state asynchronous reset automata and localized asynchronous permutation automata.
Simulations for Event-Clock Automata
ABSTRACT. Event-clock automata are a well-known subclass of timed automata which enjoy admirable theoretical properties, e.g., determinizability, and are practically useful to capture timed
specifications. However, unlike for timed automata, there exist no implementations for event-clock automata. A main reason for this is the difficulty in adapting zone-based algorithms, critical in
the timed automata setting, to the event-clock automata setting. This difficulty was recently studied in [Geeraerts et al 2011,2014], where the authors also proposed a solution using extrapolations.
In this paper, we propose an alternative zone-based algorithm, using simulations for finiteness, to solve the reachability problem for event-clock automata. Our algorithm exploits the G-simulation
framework, which is the coarsest known simulation relation for reachability, and has been recently used for advances in other extensions of timed automata.
Concurrent Games with Multiple Topologies
ABSTRACT. Concurrent multi-player games with $\omega$-regular objectives are a standard model for systems that consist of several interacting components, each with its own objective. The standard
solution concept for such games is Nash Equilibrium, which is a ``stable'' strategy profile for the players.
In many settings, the system is not fully observable by the interacting components, e.g., due to internal variables. Then, the interaction is modelled by a partial information game. Unfortunately,
the problem of whether a partial information game has an NE is not known to be decidable. A particular setting of partial information arises naturally when processes are assigned IDs by the system,
but these IDs are not known to the processes. Then, the processes have full information about the state of the system, but are uncertain of the effect of their actions on the transitions.
We generalize the setting above and introduce Multi-Topology Games (MTGs) -- concurrent games with several possible topologies, where the players do not know which topology is actually used. We show
that extending the concept of NE to these games can take several forms. To this end, we propose two notions of NE: Conservative NE, in which a player deviates if she can strictly add topologies to
her winning set, and Greedy NE, where she deviates if she can win in a previously-losing topology. We study the properties of these NE, and show that the problem of whether a game admits them is
Determinization of One-Counter Nets
ABSTRACT. One-Counter Nets (OCNs) are finite-state automata equipped with a counter that is not allowed to become negative, but does not have zero tests. Their simplicity and close connection to
various other models (e.g., VASS, Counter Machines and Pushdown Automata) make them an attractive model for studying the border of decidability for the classical decision problems.
The deterministic fragment of OCNs (DOCNs) typically admits more tractable decision problems, and while these problems and the expressive power of DOCNs have been studied, the determinization
problem, namely deciding whether an OCN admits an equivalent DOCN, has not received attention.
We introduce four notions of OCN determinizability, which arise naturally due to intricacies in the model, and specifically, the interpretation of the initial counter value. We show that in general,
determinizability is undecidable under most notions, but over a singleton alphabet (i.e., 1 dimensional VASS) one definition becomes decidable, and the rest become trivial, in that there is always an
equivalent DOCN.
Diamonds for Security: A Non-Interleaving Operational Semantics for the Applied Pi-Calculus
ABSTRACT. We introduce a non-interleaving structural operational semantics for the applied π-calculus and prove that it satisfies the properties expected of a labelled asynchronous transition system
(LATS). LATS have well studied relations with other standard non-interleaving models, such as Mazurkiewicz traces or event structures, and are a natural extension of labelled transition systems where
the independence of transitions is made explicit. Our choice of LATS as the underlying model is motivated by our wish to give an operational semantics close to the existing ones of applied
π-calculus. We build on a considerable body of literature on located semantics for process algebras and adopt a static view on locations to identify the parallel processes that perform a transition.
By lifting in this way the works on CCS and π-calculus to the applied π-calculus, we lay down a principled foundation for reusing verification techniques such as partial-order reduction and
non-interleaving equivalences in the field of security. The key technical device we develop is the notion of located aliases to refer unambiguously to a specific output originating from a specific
process. This light mechanism ensures stability, avoiding disjunctive causality problems that parallel extrusion incurs in similar non-interleaving semantics for the π-calculus.
Parameter Synthesis for Parametric Probabilistic Dynamical Systems and Prefix-Independent Specifications
ABSTRACT. We consider the model-checking problem for parametric probabilistic dynamical systems, formalised as Markov chains with parametric transition functions, analysed under the
distribution-transformer semantics (in which a Markov chain induces a sequence of distributions over states).
We examine the problem of synthesising the set of parameter valuations of a parametric Markov chain such that the orbits of induced state distributions satisfy a prefix-independent omega-regular
Our main result establishes that in all non-degenerate instances, the feasible set of parameters is (up to a null set) semialgebraic, and can moreover be computed (in polynomial time assuming that
the ambient dimension, corresponding to the number of states of the Markov chain, is fixed).
Complexity of Coverability in Depth-Bounded Processes
ABSTRACT. We consider the class of depth-bounded processes in $\pi$-calculus. These processes are the most expressive fragment of $\pi$-calculus, for which verification problems are known to be
decidable. The decidability of the coverability problem for this class has been achieved by means of well-quasi orders. (Meyer, IFIP TCS 2008; Wies, Zufferey and Henzinger, FoSSaCS 2010). However,
the precise complexity of this problem is not known so far, with only a known EXPSPACE-lower bound.
In this paper, we prove that coverability for depth-bounded processes is $\mathbf{F}_{\epsilon_0}$-complete, where $\mathbf{F}_{\epsilon_0}$ is a class in the fast-growing hierarchy of complexity
classes. This solves an open problem mentioned by Haase, Schmitz, and Schnoebelen (LMCS, Vol 10, Issue 4) and also addresses a question raised by Wies, Zufferey and Henzinger (FoSSaCS 2010).
Generalised Multiparty Session Types with Crash-Stop Failures
ABSTRACT. Session types enable the specification and verification of communicating systems. However, their theory often assumes that processes never fail. To address this limitation, we present a
generalised multiparty session type (MPST) theory with crash-stop failures, where processes can crash arbitrarily.
Our new theory validates more protocols and processes w.r.t. previous work. We apply minimal syntactic changes to standard session π-calculus and types: we model crashes and their handling
semantically, with a generalised MPST typing system parametric on a behavioural safety property. We cover the spectrum between fully reliable and fully unreliable sessions, via optional reliability
assumptions, and prove type safety and protocol conformance in the presence of crash-stop failures.
Introducing crash-stop failures has non-trivial consequences: writing correct processes that handle all possible crashes can be difficult. Yet, our generalised MPST theory allows us to tame this
complexity, via model checkers, to validate whether a multiparty session satisfies desired behavioural properties, e.g. deadlock-freedom or liveness. We implement our approach using the mCRL2 model
checker, and evaluate it with examples extended from the literature.
Non-Deterministic Abstract Machines
ABSTRACT. We present a generic design of abstract machines for non-deterministic programming languages, such as process calculi or concurrent lambda calculi, that provides a simple way to implement
them. Such a machine traverses a term in the search for a redex, making non-deterministic choices when several paths are possible and backtracking when it reaches a dead end, i.e., an irreducible
subterm. The search is guaranteed to terminate thanks to term annotations the machine introduces along the way.
We show how to automatically derive a non-deterministic abstract machine from a zipper semantics---a form of structural operational semantics in which the decomposition process of a term into a
context and a redex is made explicit. The derivation method ensures the soundness and completeness of the machines w.r.t. the zipper semantics.
Half-Positional Objectives Recognized by Deterministic Büchi Automata
ABSTRACT. A central question in the theory of two-player games over graphs is to understand which objectives are half-positional, that is, which are the objectives for which the protagonist does not
need memory to implement winning strategies. Objectives for which both players do not need memory have already been characterized (both in finite and infinite graphs). However, less is known about
half-positional objectives. In particular, no characterization of half-positionality is known for the central class of omega-regular objectives.
In this paper, we characterize objectives recognizable by deterministic Büchi automata (a class of omega-regular objectives) that are half-positional, in both finite and infinite graphs. Our
characterization consists of three natural conditions linked to the language-theoretic notion of right congruence. Furthermore, this characterization yields a polynomial-time algorithm to decide
half-positionality of an objective recognized by a given deterministic Büchi automaton.
On an Invariance Problem for Parameterized Concurrent Systems
ABSTRACT. We consider concurrent systems consisting of replicated finite-state processes that synchronize via joint interactions in a network with user-defined topology. The system is specified using
a resource logic with a multiplicative connective and inductively defined predicates, reminiscent of Separation Logic. The problem we consider is if a given formula in this logic defines an
invariant, namely whether any model of the formula, following an arbitrary firing sequence of interactions, is transformed into another model of the same formula. This property, called \emph{havoc
invariance}, is quintessential in proving the correctness of reconfiguration programs that change the structure of the network at runtime. We show that the havoc invariance problem is many-one
reducible to the entailment problem $\phi \models \psi$, asking if any model of $\phi$ is also a model of $\psi$. Although, in general, havoc invariance is found to be undecidable, this reduction
allows to prove that havoc invariance is in 2EXP, for a general fragment of the logic, with a 2EXP entailment problem.
Expressiveness and Decidability of Temporal Logics for Asynchronous Hyperproperties
ABSTRACT. Hyperproperties are properties of systems that relate different executions traces, with many applications from security to symmetry, consistency models of concurrency, etc. In recent years,
different linear-time logics for specifying asynchronous hyperproperties have been investigated. Though model checking of these logics is undecidable, useful decidable fragments have been identified
with applications e.g. for asynchronous security analysis. In this paper, we address expressiveness and decidability issues of temporal logics for asynchronous hyperproperties. We compare the
expressiveness of these logics together with the extension S1S(E) of S1S with the equal-level predicate by obtaining an almost complete expressiveness picture. We also study the expressive power of
these logics when interpreted on singleton sets of traces. We show that for two asynchronous extensions of HyperLTL, checking the existence of a singleton model is already undecidable, and for one of
them, namely Context HyperLTL, we establish a characterization of the singleton models in terms of the extension of standard FO over traces with addition. This last result generalizes the well-known
equivalence between FO and LTL. Finally, we identify new boundaries on the decidability of model checking Context HyperLTL.
Pareto-Rational Verification
ABSTRACT. We study the rational verification problem which consists in verifying the correctness of a system executing in an environment that is assumed to behave rationally. We consider the model of
rationality in which the environment only executes behaviors that are Pareto-optimal with regard to its set of objectives, given the behavior of the system (which is committed in advance of any
interaction). We examine two ways of specifying this behavior, first by means of a deterministic Moore machine, and then by lifting its determinism. In the latter case the machine may embed several
different behaviors for the system, and the universal rational verification problem aims at verifying that all of them are correct when the environment is rational. For parity objectives, we prove
that the Pareto-rational verification problem is co-NP-complete and that its universal version is in PSPACE and both NP-hard and co-NP-hard. For Boolean Büchi objectives, the former problem is
Π2P-complete and the latter is PSPACE-complete. Both problems are also shown to be fixed-parameter tractable.
An Infinitary Proof Theory of Linear Logic Ensuring Fair Termination in the Linear π-Calculus
ABSTRACT. Fair termination is the property of programs that may diverge “in principle” but that terminate “in practice”, under suitable fairness assumptions concerning the resolution of
non-deterministic choices. We study a conservative extension of μMALL∞, the infinitary proof system of the multiplicative additive fragment of linear logic with least and greatest fixed points, such
that cut elimination corresponds to fair termination. Proof terms are processes of πLIN, a variant of the linear π-calculus with (co)recursive types into which binary and (some) multiparty sessions
can be encoded. As a result we obtain a behavioral type system for πLIN (and indirectly for session calculi through their encoding into πLIN) that ensures fair termination: although well-typed
processes may engage in arbitrarily long interactions, they are fairly guaranteed to eventually perform all pending actions.
Language Inclusion for Boundedly-Ambiguous Vector Addition Systems is Decidable
ABSTRACT. We consider the problems of language inclusion and language equivalence for Vector Addition Systems with States (VASSes) with the acceptance condition defined by the set of accepting states
(and more generally by some upward-closed conditions). In general the problem of language equivalence is undecidable even for one-dimensional VASSes, thus to get decidability we investigate
restricted subclasses. On one hand we show that the problem of language inclusion of a VASS in $k$-ambiguous VASS (for any natural k) is decidable and even in Ackermann. On the other hand we prove
that the language equivalence problem is Ackermann-hard already for deterministic VASSes. These two results imply Ackermann-completeness for language inclusion and equivalence in several possible
restrictions. Some of our techniques can be also applied in much broader generality in infinite-state systems, namely for some subclass of well-structured transition systems.
On Session Typing, Probabilistic Polynomial Time, and Cryptographic Experiments
ABSTRACT. In this work, a system of session types is introduced, following Caires and Pfenning, as induced by a Curry Howard correspondence applied to Bounded Linear Logic, and then extending the
thus obtained type system with probabilistic choices and ground types. In particular, we show how such a system satisfies the expected properties, like subject reduction and progress, but also
unexpected ones, like a polynomial bound on the time needed to reduce processes. This makes the system suitable for modelling experiments and proofs in the so-called computational model of
Oscar Darwin
(Department of Computer Science, University of Oxford)
Stefan Kiefer
(Department of Computer Science, University of Oxford)
On the Sequential Probability Ratio Test in Hidden Markov Models
ABSTRACT. We consider the Sequential Probability Ratio Test applied to Hidden Markov Models. Given two Hidden Markov Models and a sequence of observations generated by one of them, the Sequential
Probability Ratio Test attempts to decide which model produced the sequence. We show relationships between the execution time of such an algorithm and Lyapunov exponents of random matrix systems.
Further, we give complexity results about the execution time taken by the Sequential Probability Ratio Test.
Weak Progressive Forward Simulation is Necessary and Sufficient for Strong Observational Refinement
ABSTRACT. Hyperproperties are correctness conditions for labelled transition systems that are more expressive than traditional trace properties, with particular relevance to security. Recently,
Attiya and Enea studied a notion of strong observational refinement that preserves all hyperproperties. They analyse the correspondence between forward simulation and strong observational refinement
in a setting with only finite traces. We study this correspondence in a setting with both finite and infinite traces. In particular, we show that forward simulation does not preserve hyperliveness
properties in this setting. We extend the forward simulation proof obligation with a (weak) progress condition, and prove that this weak progressive forward simulation is equivalent to strong
observational refinement.
Regular Model Checking Upside-Down: An Invariant-Based Approach
ABSTRACT. Regular model checking is a well-established technique for the verification of infinite-state systems whose configurations can be represented as finite words over a suitable alphabet. It
applies to systems whose set of initial configurations is regular, and whose transition relation is captured by a length-preserving transducer. To verify safety properties, regular model checking
iteratively computes automata recognizing increasingly larger regular sets of reachable configurations, and checks if they contain unsafe configurations. Since this procedure often does not
terminate, acceleration, abstraction, and widening techniques have been developed to compute a regular superset of the set of reachable configurations.
In this paper we develop a complementary approach. Instead of approaching the set of reachable configurations from below, we start with the set of all configurations and compute increasingly smaller
regular supersets of it. We use that the set of reachable configurations is equal to the intersection of all inductive invariants of the system. Since the intersection is in general non-regular, we
introduce $b$-bounded invariants, defined as those representable by CNF-formulas with at most $b$ clauses. We prove that, for every $b \geq 0$, the intersection of all $b$-bounded inductive
invariants is regular, and show how to construct an automaton recognizing it. Finally, we study the complexity of deciding if this automaton accepts some unsafe configuration. We show that the
problem is in \textsc{EXPSPACE} for every $b \geq 0$, and \textsc{PSPACE}-complete for $b=1$. Finally, we study the performance of our approach in a number of benchmarks.
A Kleene Theorem for Higher-Dimensional Automata
ABSTRACT. We prove a Kleene theorem for higher-dimensional automata (HDAs). It states that the languages they recognise are precisely the rational subsumption-closed sets of interval pomsets. The
rational operations include a gluing composition, for which we equip pomsets with interfaces. For our proof, we introduce HDAs with interfaces as presheaves over labelled precube categories and use
tools inspired by algebraic topology, such as cylinders and (co)fibrations. HDAs are a general model of non-interleaving concurrency, which subsumes many other models in this field. Interval orders
occur as models for concurrent or distributed systems where events extend in time. Our tools and techniques may therefore yield templates for Kleene theorems in a variety of models.
Towards Concurrent Quantitative Separation Logic
ABSTRACT. In this paper we develop a novel verification technique to reason about programs featuring concurrency, pointers and randomization. While the integration of concurrency and pointers is well
studied, little is known about the combination of all three paradigms. To close this gap, we combine two kinds of separation logic - Quantitative Separation Logic and Concurrent Separation Logic -
into a new separation logic able to reason about lower bounds of the probability to realize a postcondition after executing such a program.
Anytime Guarantees for Reachability in Uncountable Markov Decision Processes
ABSTRACT. We consider the problem of approximating the reachability probabilities in Markov decision processes (MDP) with uncountable (continuous) state and action spaces. While there are algorithms
that, for special classes of such MDP, provide a sequence of approximations converging to the true value in the limit, our aim is to obtain an algorithm with guarantees on the precision of the
As this problem is undecidable in general, assumptions on the MDP are necessary. Our main contribution is to identify sufficient assumptions that are as weak as possible, thus approaching the
"boundary" of which systems can be correctly and reliably analyzed. To this end, we also argue why each of our assumptions is necessary for algorithms based on processing finitely many observations.
We present two solution variants. The first one provides converging lower bounds under weaker assumptions than typical ones from previous works concerned with guarantees. The second one then utilizes
stronger assumptions to additionally provide converging upper bounds. Altogether, we obtain an anytime algorithm, i.e. yielding a sequence of approximants with known and iteratively improving
precision, converging to the true value in the limit. Besides, due to the generality of our assumptions, our algorithms are very general templates, readily allowing for various heuristics from
literature in contrast to, e.g., a specific discretization algorithm. Our theoretical contribution thus paves the way for future practical improvements without sacrificing correctness guarantees.
Two-player Boudedness Counter Games
ABSTRACT. We consider two-player zero-sum games with winning objectives beyond regular languages, expressed as a parity condition in conjunction with a Boolean combination of boundedness conditions
on a finite set of counters which can be incremented, reset to $0$, but not tested. A boundedness condition requires that a given counter is bounded along the play. Such games are decidable, though
with non-optimal complexity, by an encoding into the logic WMSO with the unbounded and path quantifiers, which is known to be decidable over infinite trees. Our objective is to give tight or tighter
complexity results for particular classes of counter games with boundedness conditions, and study their strategy complexity. In particular, counter games with conjunction of boundedness conditions
are easily seen to be equivalent to Streett games, so, they are CoNP-c. Moreover, finite-memory strategies suffice for Eve and memoryless strategies suffice for Adam. For counter games with a
disjunction of boundedness conditions, we prove that they are in solvable in NP and in CoNP, and in PTime if the parity condition is fixed. In that case memoryless strategies suffice for Eve while
infinite memory strategies might be necessary for Adam. Finally, we consider an extension of those games with a max operation. In that case, the complexity increases: for conjunctions of boundedness
conditions, counter games are EXPTIME-c.
History-deterministic Timed Automata
ABSTRACT. We explore the notion of history-determinism in the context of timed automata (TA). History-deterministic automata are those in which nondeterminism can be resolved on the fly, based on the
run constructed thus far. History-determinism is a robust property that admits different game-based characterisations, and history-deterministic specifications allow for game-based verification
without an expensive determinization step.
We show yet another characterisation of history-determinism in terms of fair simulation, at the general level of labelled transition systems: a system is history-deterministic precisely iff it fairly
simulates all language smaller systems.
For timed automata over infinite timed words it is known that universality is undecidable for Büchi TA. We show that for history-deterministic TA with arbitrary parity acceptance, timed universality,
inclusion, and synthesis all remain decidable and are EXPTIME-complete.
For the subclass of TA with safety or reachability acceptance, we show that checking whether such an automaton is history-deterministic is decidable (in EXPTIME), and history-deterministic TA with
safety acceptance are effectively determinizable without introducing new states.
Checking timed Büchi automata emptiness using the local-time semantics
ABSTRACT. We study the Büchi non-emptiness problem for networks of timed automata. Standard solutions consider the network as a monolithic timed automaton obtained as a synchronized product and build
its zone graph on-the-fly under the classical global-time semantics. In the global-time semantics, all processes are assumed to have a common global timeline.
Bengtsson et al. in 1998 have proposed a local-time semantics where each process in the network moves independently according to a local timeline, and processes synchronize their timelines when they
do a common action. It has been shown that the local-time semantics is equivalent to the global-time semantics for finite runs, and hence can be used for checking reachability. The local-time
semantics allows computation of a local zone graph which has good independence properties and is amenable to partial-order methods. Hence local zone graphs are able to better tackle the state-space
explosion due to concurrency.
In this work, we extend the results to the Büchi setting. We propose a local zone graph computation that can be coupled with a partial-order method, to solve the Büchi non-emptiness problem in timed
networks. In the process, we develop a theory of regions for the local-time semantics.
Slimming Down Petri Boxes: Compact Petri Net Models of Control Flows
ABSTRACT. We look at the construction of compact Petri net models corresponding to process algebra expressions supporting sequential, choice, and parallel compositions. If ‘silent’ transitions are
disallowed, a construction based on Cartesian product is traditionally used to construct places in the target Petri net, resulting in an exponential explosion in the net size. We demonstrate that
this exponential explosion can be avoided, by developing a link between this construction problem and the problem of finding an edge clique cover of a graph that is guaranteed to be
complement-reducible (i.e., a cograph). It turns out that the exponential number of places created by the Cartesian product construction can be reduced down to polynomial (quadratic) even in the
worst case, and to logarithmic in the best (non-degraded) case. As these results affect the ‘core’ modelling techniques based on Petri nets, eliminating a source of an exponential explosion, we hope
they will have applications in Petri net modelling and translations of various formalisms to Petri nets.
Strategies for MDP Bisimilarity Equivalence and Inequivalence
ABSTRACT. A labelled Markov decision process (MDP) is a labelled Markov chain with nondeterminism; i.e., together with a strategy a labelled MDP induces a labelled Markov chain. Motivated by
applications to the verification of probabilistic noninterference in security, we study problems whether there exist strategies such that the labelled MDPs become bisimilarity equivalent/
inequivalent. We show that the equivalence problem is decidable; in fact, it is EXPTIME-complete and becomes NP-complete if one of the MDPs is a Markov chain. Concerning the inequivalence problem, we
show that (1) it is decidable in polynomial time; (2) if there are strategies for inequivalence then there are memoryless strategies for inequivalence; (3) such memoryless strategies can be computed
in polynomial time.
Energy Games with Resource-Bounded Environments
ABSTRACT. An {\em energy game\/} is played between two players, modeling a resource-bounded system and its environment. The players take turns moving a token along a finite graph. Each edge of the
graph is labeled by an integer, describing an update to the energy level of the system that occurs whenever the edge is traversed. The system wins the game if it never runs out of energy. Different
applications have led to extensions of the above basic setting. For example, addressing a combination of the energy requirement with behavioral specifications, researchers have studied richer winning
conditions, and addressing systems with several bounded resources, researchers have studied games with multi-dimensional energy updates. All extensions, however, assume that the environment has no
bounded resources.
We introduce and study {\em both-bounded energy games\/} (BBEGs), in which both the system and the environment have multi-dimensional energy bounds. In BBEGs, each edge in the game graph is labeled
by two integer vectors, describing updates to the multi-dimensional energy levels of the system and the environment. A system wins a BBEG if it never runs out of energy or if its environment runs out
of energy. We show that BBEGs are determined, and that the problem of determining the winner in a given BBEG is decidable iff both the system and the environment have energy vectors of dimension~$1$.
We also study how restrictions on the memory of the system and/or the environment as well as upper bounds on their energy levels influence the winner and the complexity of the problem
Different strokes in randomised strategies: Revisiting Kuhn's theorem under finite-memory assumptions
ABSTRACT. Two-player (antagonistic) games on (possibly stochastic) graphs are a prevalent model in theoretical computer science, notably as a framework for reactive synthesis.
Optimal strategies may require randomisation when dealing with inherently probabilistic goals, balancing multiple objectives or in contexts of partial information. There is no unique way to define
randomised strategies. For instance, one can use so-called mixed strategies or behavioural ones. In the most general settings, these two classes do not share the same expressiveness. A seminal result
in game theory - Kuhn's theorem - asserts their equivalence in games of perfect recall.
This result crucially relies on the possibility for strategies to use infinite memory, i.e., unlimited knowledge of all the past of a play. However, computer systems are finite in practice. Hence it
is pertinent to restrict our attention to finite-memory strategies, defined as automata with outputs. Randomisation can be implemented in these in different ways: the initialisation, outputs or
transitions can be randomised or deterministic respectively. Depending on which aspects are randomised, the expressiveness of the corresponding class of finite-memory strategies differs.
In this work, we study two-player turn-based stochastic games and provide a complete taxonomy of the classes of finite-memory strategies obtained by varying which of the three aforementioned
components are randomised. Our taxonomy holds both in settings of perfect and imperfect information.
Decidability of One-Clock Weighted Timed Games with Arbitrary Weights
ABSTRACT. Weighted Timed Games (WTG for short) are the most widely used model to describe controller synthesis problems involving real-time issues. Unfortunately, they are notoriously difficult, and
undecidable in general. As a consequence, one-clock WTG has attracted a lot of attention, especially because they are known to be decidable when only non-negative weights are allowed. However, when
arbitrary weights are considered, despite several recent works, their decidability status was still unknown. In this paper, we solve this problem positively and show that the value function can be
computed in exponential time (if weights are encoded in unary).
Completeness Theorems for Kleene Algebra with Top
ABSTRACT. We prove two completeness results for Kleene algebra with a top element, with respect to languages and binary relations. While the equational theories of those two classes of models
coincide over the signature of Kleene algebra, this is no longer the case when we consider an additional constant 'top' for the full element. Indeed, the full relation satisfies more laws than the
full language, and we show that those additional laws can all be derived from a single additional axiom. We recover that the two equational theories coincide if we slightly generalise the notion of
relational model, allowing sub-algebras of relations where top is a greatest element but not necessarily the full relation. We use models of closed languages and reductions in order to prove our
completeness results, which are relative to any axiomatisation of the algebra of regular events. | {"url":"https://easychair.org/smart-program/CONCUR2022/accepted-detailed.html","timestamp":"2024-11-12T03:50:10Z","content_type":"application/xhtml+xml","content_length":"52713","record_id":"<urn:uuid:0261dadf-c36d-4db4-9c0e-c5d1c84264ac>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00069.warc.gz"} |
Planet Day Length Calculator
To calculate the planet day length:
\[ D = \frac{1}{R} \]
• \(D\) is the Planet Day Length (days)
• \(R\) is the Planet Rotation Speed (1/day)
Planet Day Length
Planet day length refers to the duration of one complete rotation of a planet on its axis, which determines the length of a single day on that planet. This period can vary significantly from one
planet to another within our solar system and beyond. For instance, a day on Earth is approximately 24 hours, while a day on Jupiter is roughly 10 hours. The planet day length is influenced by the
planet’s rotation speed, which is the rate at which it spins around its axis.
Example Calculation
Let's assume the following values:
• Planet Rotation Speed (\(R\)) = 0.1 1/day
Using the formula:
\[ D = \frac{1}{0.1} = 10 \text{ days} \]
The planet day length is 10 days.
BIT1024 Calculator© - All Rights Reserved 2024 | {"url":"https://waycalculator.com/tool/Planet-Day-Length-Calculator.php","timestamp":"2024-11-10T19:01:32Z","content_type":"text/html","content_length":"6551","record_id":"<urn:uuid:e90c548b-351e-4b27-9112-2fafd50e8c73>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00149.warc.gz"} |
xpansions in
Necessary expansions in optimal search
When performing search, an important theoretical question is which states must be expanded by which algorithms. Early work by Dechter and Pearl showed that all optimal unidirectional algorithms must
expand all states with f(s) < C*, where C* is the optimal solution cost. Note that the theory says nothing about states with f(s) = C*. The app on this page contains three "games" related to
understanding necessary expansions in unidirectional and bidirectional search.
Start by drawing or erasing any obstacles in the map. Then, select a start and goal state for the game.
Next, play the first game -- try to identify a state that, if it were not expanded, A* would not be guaranteed to find the optimal solution. (That is, a state with f(s) < C*.)
In the second game you will try to identify a similar state for bidirectional search. If you are having trouble with the second game, you can try the third game -- an alternate version of the second
Related Videos
Selected Related Publications | {"url":"https://www.movingai.com/SAS/BDN/","timestamp":"2024-11-06T13:59:38Z","content_type":"text/html","content_length":"7744","record_id":"<urn:uuid:c71ce31e-ed80-433a-b199-44dbe8b116e5>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00292.warc.gz"} |
How many joules are in a Cal?
How many joules are in a Cal?
Definition: A calorie (symbol: cal) is a unit of energy defined as the amount of energy required to increase the temperature of one gram of water by one °C. This is referred to as the small calorie
or the gram calorie, and is equal to 4.1868 joules, the SI (International System of Units) unit of energy.
How do you convert J to kJ?
To convert a joule measurement to a kilojoule measurement, divide the energy by the conversion ratio. The energy in kilojoules is equal to the joules divided by 1,000.
Is a kJ a calorie?
1 kilojoule = 0.24 Calories (about ¼) For those who still work in calories, we also provide Calorie information in the nutrition information panel.
Does 1 joule equal calories?
1 Calorie/kcal = 4.2 kilojoules. Therefore 1 joule=1/4.2 calorie. That is 0.238 calorie which is equivalent to 0.24 calorie. Hence option A is the correct option….
Calorie Joule
1 4.184
20 83.68
30 125.52
40 167.36
How do you convert kcal to calories?
How to Convert Kilocalories to Calories. To convert a kilocalorie measurement to a calorie measurement, multiply the energy by the conversion ratio. The energy in calories is equal to the
kilocalories multiplied by 1,000.
Is kJ or J bigger?
Thus, a kiloJoule (kJ) is 1000 Joules and a megaJoule (MJ) is 1,000,000 Joules. A related unit is the Watt, which is a unit of power (energy per unit time).
What’s the difference between kJ and J?
Joules and KiloJoules are units of the international system of units (SI) that measure energy. The standard symbol for Joule is J, whereas the symbol for KiloJoule is KJ. 1 J equals precisely 0.001
KJ, therefore there are 1,000 Joules in a KiloJoule.
What is the bigger unit of calorie?
The large calorie, food calorie, or kilocalorie (Cal, Calorie or kcal), most widely used in nutrition, is the amount of heat needed to cause the same increase in one kilogram of water. Thus, 1
kilocalorie (kcal) = 1000 calories (cal).
What is the biggest unit of energy?
In the SI unit system, Joule (J) is considered as the largest unit of energy. | {"url":"https://www.peel520.net/how-many-joules-are-in-a-cal/","timestamp":"2024-11-07T07:37:05Z","content_type":"text/html","content_length":"32715","record_id":"<urn:uuid:e010da8f-97a8-4932-a3fc-dcda397db4dc>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00470.warc.gz"} |
2.3.5 Approach-retraction curves
Consider a cantilever oscillations near a sample surface. As shown in chapter 2.2.1, the tip-sample interaction potential has a characteristic appearance depicted in Fig. 1. As the cantilever touches
the sample and deforms its surface, the force of elastic repulsion prevails. At the tip-sample separation on the order of a few tens of angstrom, the intermolecular interaction called the Van der
Waals force predominates.
Fig. 1. Typical appearance of the tip-sample interaction potential.
As shown in chapter 2.3.4, the presence of external force dependent on spatial coordinates, gives rise to the change in resonance properties of the cantilever-sample oscillating system.
Change in the oscillations phase:
change in the oscillations amplitude:
change in the resonant frequency:
Thus, measuring dependence of the oscillations resonant frequency, phase or amplitude on the tip-sample separation, one can render the derivative appearance and, in some cases, the interaction force
itself. The corresponding experimental curves are called the approach curves (Fig. 2).
Fig. 2. The tip-to-sample approach curves. | {"url":"https://www.ntmdt-si.com/resources/spm-theory/theoretical-background-of-spm/2-scanning-force-microscopy-(sfm)/23-linear-oscillations-of-cantilever/235-approach-retraction-curves","timestamp":"2024-11-07T03:10:09Z","content_type":"text/html","content_length":"22867","record_id":"<urn:uuid:93f5b027-0c49-42f5-a7c8-f7220560e11a>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00764.warc.gz"} |
Probabilistic Programming
Programming paradigm designed to handle uncertainty and probabilistic models, allowing for the creation of programs that can make inferences about data by incorporating statistical methods directly
into the code.
Probabilistic programming facilitates the development of models that can reason under uncertainty by integrating probabilistic reasoning directly within the programming framework. It allows
developers to define complex probabilistic models using a higher-level language, abstracting the underlying statistical computations. These models can capture uncertainties in data and make
inferences based on observed evidence. Tools such as PyMC3, Stan, and TensorFlow Probability are examples of probabilistic programming frameworks that enable Bayesian inference and other statistical
techniques. Probabilistic programming is significant in fields like machine learning, artificial intelligence, and data science, as it simplifies the creation and manipulation of sophisticated
statistical models that are crucial for predictive analytics, decision making, and automated reasoning.
The concept of probabilistic programming emerged in the early 2000s, with substantial development and popularization occurring throughout the 2010s. This period saw the introduction of several
probabilistic programming languages and tools, which made these concepts more accessible and practical for a wider range of applications in AI and machine learning.
Significant contributors to the development of probabilistic programming include Daphne Koller, who co-authored foundational work on probabilistic graphical models, and Andrew Gelman, who contributed
to the development of Stan. Additionally, researchers at institutions like MIT, Stanford, and the University of Cambridge have been instrumental in advancing this field through both theoretical
developments and practical implementations of probabilistic programming frameworks. | {"url":"https://www.envisioning.io/vocab/probabilistic-programming","timestamp":"2024-11-12T00:57:59Z","content_type":"text/html","content_length":"443872","record_id":"<urn:uuid:0d523476-998f-4420-936f-81966ba14ae7>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00162.warc.gz"} |
Re: Is there a formal term for this?
[Date Prev][Date Next][Thread Prev][Thread Next][Date Index][Thread Index][Subject Index][Author Index]
Re: Is there a formal term for this?
On Mon, Aug 2, 2010 at 12:18 PM, Jocelyn Falconnet
<j.falconnet@gmail.com> wrote:
> And all these species form the hypodigm of the genus.
> Also: let T, R, and H the number of Type species, Referred species,
> and number of species under the Hypodigm of the genus.
> By definition, T=1. Also, H=T+R=1+R or R=H-1.
> If you want to be strictly accurate, Saint Abyssal, you should also
> consider the species which have been synonymized.
Strictly speaking, species names are synonymized, not species. (The
species is, ostensibly, the actual entity, i.e., the population, and
the species name is the label we use to refer to it.) Thus, the number
of synonyms doesn't affect your formula.
(In practice, people do often use "species" when they mean "species name".)
T. Michael Keesey
Technical Consultant and Developer, Internet Technologies
Glendale, California | {"url":"http://dml.reptilis.net/2010Aug/msg00011.html","timestamp":"2024-11-11T14:31:17Z","content_type":"text/html","content_length":"6080","record_id":"<urn:uuid:17fce520-aa06-4423-930e-27d612a28f83>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00365.warc.gz"} |
Metacommunity class
MetaCommunity {entropart} R Documentation
Metacommunity class
Methods for objects of type "MetaCommunity".
MetaCommunity(Abundances, Weights = rep(1, ncol(Abundances)))
## S3 method for class 'MetaCommunity'
summary(object, ...)
## S3 method for class 'MetaCommunity'
plot(x, ...)
Abundances A dataframe containing the number of observations (lines are species, columns are communities). The first column of the dataframe may contain the species names.
Weights A vector of positive numbers equal to community weights or a dataframe containing a vector named Weights. It does not have to be normalized. Weights are equal by default.
x An object to be tested or plotted.
object A MetaCommunity object to be summarized.
... Additional arguments to be passed to the generic methods.
In the entropart package, individuals of different "species" are counted in several "communities" which are agregated to define a "metacommunity".
This is a naming convention, which may correspond to plots in a forest inventory or any data organized the same way.
Alpha and beta entropies of communities are summed according to Weights and the probability to find a species in the metacommunity is the weighted average of probabilities in communities.
The simplest way to import data is to organize it into two text files. The first file should contain abundance data: the first column named Species for species names, and a column for each community.
The second file should contain the community weights in two columns. The first one, named Communities should contain their names and the second one, named Weights, their weights.
Files can be read and data imported by code such as:
Abundances <- read.csv(file="Abundances.csv", row.names = 1)
Weights <- read.csv(file="Weights.csv")
MC <- MetaCommunity(Abundances, Weights)
An object of class MetaCommunity is a list:
Nsi A matrix containing abundance data, species in line, communities in column.
Ns A vector containing the number of individuals of each species.
Ni A vector containing the number of individuals of each community.
N The total number of individuals.
Psi A matrix whose columns are the probability vectors of communities (each of them sums to 1).
Wi A vector containing the normalized community weights (sum to 1).
Ps A vector containing the probability vector of the metacommunity.
Nspecies The number of species.
Ncommunities The number of communities.
SampleCoverage The sample coverage of the metacommunity.
SampleCoverage.communities A vector containing the sample coverages of each community.
is.MetaCommunity returns TRUE if the object is of class MetaCommunity.
summary.MetaCommunity returns a summary of the object's value.
plot.MetaCommunity plots it.
# Use BCI data from vegan package
if (require(vegan, quietly = TRUE)) {
# Load BCI data (number of trees per species in each 1-ha plot of a tropical forest)
# BCI dataframe must be transposed (its lines are plots, not species)
BCI.df <- as.data.frame(t(BCI))
# Create a metacommunity object from a matrix of abundances and a vector of weights
# (here, all plots have a weight equal to 1)
MC <- MetaCommunity(BCI.df)
version 1.6-13 | {"url":"https://search.r-project.org/CRAN/refmans/entropart/html/MetaCommunity.html","timestamp":"2024-11-13T21:25:08Z","content_type":"text/html","content_length":"6047","record_id":"<urn:uuid:10bd1540-8342-4f5d-8c55-b3489bea93a2>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00432.warc.gz"} |
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Week of 1/6-10/20 Mean Value Theorem
POW MVT Due Friday 1/10
This POW is designed to get you thinking about a theorem that we need soon, although why we will need will need it isn't initially clear. For each case, I want you to:
• Draw an illustration and annotate it.
• Articulate the idea in precise mathematical language.
• Generate a logical justification for it.
• State clearly why all the conditions are needed.
• Ideally, you will be able to formally prove it as well.
Prove the following statements using the criteria above:
1. If a function on a closed interval is continuous and differentiable and it has the same value on each end of the interval, there is at least one point in that interval where the function
derivative is zero.
2. If a function is continuous on a closed interval is continuous and differentiable and it has different values on each end of the interval, then the function must have point where the derivative
is equal to the slope of a straight line that goes to one end point to another. | {"url":"http://alloyddp.weebly.com/calculus/week-of-16-1020-mean-value-theorem","timestamp":"2024-11-14T08:28:08Z","content_type":"text/html","content_length":"30000","record_id":"<urn:uuid:d2c74d04-aeaa-4c43-8233-8196e47e0169>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00186.warc.gz"} |
Quad solve
Can you solve this problem involving powers and quadratics?
Find all real solutions to this equation:
$$\left(2-x^2\right)^{x^2-3\sqrt{2}x+4} = 1$$
Extension: What if $x$ is permitted to be a complex number?
Did you know ... ?
Quadratic equations and powers are commonly used throughout school and university mathematics and beyond. It is also important to remember that algebraic manipulations might not necessarily find all
solutions to a problem; you always need to reason carefully that all possibilities have been considered. Moreover, in complicated situations it is necessary to check that all proposed solutions
unearthed by algebra are in fact valid solutions. Powers, roots and quadratics all link together very nicely when complex numbers are considered.
Getting Started
To do this problem you will need to know that $a^0 = 1$ when $a\neq 0$ and $1^b=1$ for any $b$ along with other power manipulations which are needed from the beginning of C1.
Note that $0^0$ is not defined as a number.
You will also need to know the formula for the solution of a quadratic equation.
If you are studying complex numbers then you can also bring those ideas into the problem if desired.
Student Solutions
By applying the quadratic equation formula we can factorise this expression
$$\left(2-x^2\right)^{x^2-3\sqrt{2}x+4} = 1$$
For real solutions we can make use of the fact that for any real $a\neq 0$ we have
$$a^0 = 1 \quad\quad 1^a=1$$
First look at the base. This equals $1$ if and only if $x=\pm 1$.
Next look at the exponent. This equals zero if and only if $x=\sqrt{2}$ or $2\sqrt{2}$. However, when $x=\sqrt{2}$ both the base and exponent are zero. Therefore, three valid real solutions to the
equation are
x = \pm 1, x=2\sqrt{2}
Is this all? Not necessarily. We might also have $(-1)^{2n}=1$ for any positive whole number $n$. Are there any solutions to this? $2-x^2 = -1$ has solutions $x=\pm \sqrt{3}$. In this case the
exponent becomes $7\pm 3\sqrt{6}$, which is not of the form $2n$. There are therefore no more real solutions.
Note: If we extend to complex numbers then we also have to take into account the fact that there are multiple complex roots of $1$. For example:
\left(\frac{-1\pm \sqrt{3}}{2}\right)^3 = 1
EXTENSION: READ ON ONLY IF VERY, VERY KEEN!
Any complex number $z$ can be written in modulus argument form $z=re^{i\theta}$ and the logarithm of a complex number becomes
\ln{z} = \ln(re^{i\theta}) = \ln r +\ln(e^{i\theta}) = \ln r + i\theta
Using these facts we can attack our problem in the complext numbers. Taking logs of the equations gives
(z^2-3\sqrt{2}z+4)\ln(2-z^2)=2n\pi i\quad n\in \mathbb{Z}\quad\quad(\dagger)
To progress with the complex logarithm of $(2-z^2)$ it is natural to use modulus argument form of $(2-z^2)$. If we suppose that $z=re^{i\theta}$ then
2-z^2 &=& 2-r^2e^{2\theta i}\\
&=& 2-r^2\cos 2\theta-ir^2\sin 2\theta\\
&=& \sqrt{(2-r^2\cos 2\theta)^2+(r^2\sin 2\theta)^2} e^{i \tan^{-1}\left(\frac{r^2\sin 2\theta}{r^2\cos 2\theta-2}\right)}
Our equation $(\dagger)$ then becomes
A(r, \theta) \times B(r, \theta) = 2n\pi i
A(r, \theta)=\left(4+r^2\cos 2\theta -3\sqrt{2} r\cos\theta\right)+i\left(r^2\sin 2\theta-3\sqrt{2}r\sin \theta\right)
B(r, \theta,m)=\frac{1}{2}\ln \left|4-4r^2\cos^22\theta+r^4\right| + i \left[2m\pi +\tan^{-1}\left(\frac{r^2\sin 2\theta}{r^2\cos 2\theta-2}\right)\right]\quad m\in \mathbb{Z}
There are no non-real solutions for $n=m=0$ as this excellent piece of mathematics by Stephen Lynch shows (I include
his full solution
However, there might well be solutions for other values of $n$ and $m$. In principle we can attempt to solve this by equating real and imaginary parts for various choices of $n$ and $m$.
Editor's note: further analysis appears to be time consuming, so I looked for numerical answers as follows
Finding an exact solution would appear to be, well, complex. I simply attempted a numerical solution. To find a numerical solution I looked for small values of the expression
X(r, \theta,n,m) = \left|A(r, \theta)B(r, \theta,m)-2n\pi i\right|^2
I looked at the case $n=1, m=0$.
I found a solution
r = 3.3858990007 \,, \theta = 0.1902641501
This solves the equation in the sense that
X(3.3858990007 , 0.1902641502,1,0)< 10^{-17}
From this numerical exploration is seems likely that other solutions exist, although proof that the iterative scheme used does indeed converge on a genuine solution would require more work. (As an
aside it is worth noting that complex numbers enter into all sorts of applied mathematics, such as fluid dynamics and air flow. In such situations numerical solutions of complex number equations are
You might wish to take a look at the a
which computes the values $X(r, \theta,n,m)$ for various values.
With numerics there is always the chance of hard to spot error, but here is the input used if you wish to check the logic for yourself (this is a good habit to get into)
Forunately $X(2\sqrt{2},0,0,0)=X(1,0,0,0)=X(1,\pi,0,0) = 0$ as we would expect from our analysis of the real case. This sort of check is essential when performing numerics.
Needless to say, if you spot an error please let us know! | {"url":"https://nrich.maths.org/problems/quad-solve","timestamp":"2024-11-14T22:00:48Z","content_type":"text/html","content_length":"43560","record_id":"<urn:uuid:eaf9bbca-35a8-45d8-9a5a-71ff98c75429>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00094.warc.gz"} |
Understanding Covariance and Correlation: An Essential Concept in Random Variability | Exercises Algebra | Docsity
Download Understanding Covariance and Correlation: An Essential Concept in Random Variability and more Exercises Algebra in PDF only on Docsity! Random Variability: Covariance and Correlation What of
the variance of the sum of two random variables? If you work through the algebra, you'll find that Var[X+Y] = Var[X] + Var[Y]+ 2(E[XY] - E[X]E[Y]) . This means that variances add when the random
variables are independent, but not necessarily in other cases. The covariance of two random variables is Cov[X,Y] = E[ (X-E[X])(Y-E[Y]) ] = E[XY] - E[X]E[Y]. We can restate the previous equation as
Var[X+Y] = Var[X] + Var[Y] + 2Cov[X,Y] . Note that the covariance of a random variable with itself is just the variance of that random variable. While variance is usually easier to work with when
doing computations, it is somewhat difficult to interpret because it is expressed in squared units. For this reason, the standard deviation of a random variable is defined as the square-root of its
variance. A practical (although not quite precise) interpretation is that the standard deviation of X indicates roughly how far from E[X] you’d expect the actual value of X to be. Similarly,
covariance is frequently “de-scaled,” yielding the correlation between two random variables: Corr(X,Y) = Cov[X,Y] / ( StdDev(X) StdDev(Y) ) . The correlation between two random variables will
always lie between -1 and 1, and is a measure of the strength of the linear relationship between the two variables. Example: Let X be the percentage change in value of investment A in the course of
one year (i.e., the annual rate of return on A), and let Y be the percentage change in value of investment B. Assume that you have $1 to invest, and you decide to put a dollars into investment A, and
1- a dollars into B. Then your return on investment from your portfolio will be aX+(1-a)Y, your expected return on investment will be aE[X] + (1-a)E[Y] , and the variance in your return on
investment (a measure of the risk inherent in your portfolio) will be a2Var[X] + (1-a)2Var[Y] + 2a(1-a)Cov[X,Y] . | {"url":"https://www.docsity.com/en/docs/random-variability-covariance-and-correlation/8910658/","timestamp":"2024-11-07T02:47:40Z","content_type":"text/html","content_length":"231049","record_id":"<urn:uuid:18250120-81a6-4919-be12-e34c235a47ca>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00496.warc.gz"} |
R-squared and "twenty questions"
R-squared and "twenty questions"
(This is a follow-up to two previous posts on r-squared.)
You've probably played the game "Twenty Questions." Here's how it works. I choose a subject, which can be anything I want -- "baseball glove," or "Hillary Clinton". Then, you have twenty "yes/no"
questions to try to figure out what it is.
To win the game, you try to narrow it down as fast as you can, and as much as possible. To start, you might ask the traditional question: "is it bigger than a bread box?" That one question won't tell
you what it is immediately, but it starts narrowing it down. According to Wikipedia, other good questions are, "can I put it in my mouth?" and "does it involve technology for communications,
entertainment, or work?"
Some questions are obviously bad. Starting off by asking, "is it a DVD of a Sylvester Stallone movie?" is a waste of a question. The answer is probably "no," in which case you're left pretty much
where you started. Of course, if it's a "yes," you're almost certain to get it, but the chances of that "yes" are pretty slim.
So, now, here's a variation of the game. This time, I'm going to pick a random American person. Your job is to guess his or her 2011 income, and come as close as you can.
Instead of twenty questions, I'm only going to give you one, for now. However, it doesn't have to be a "yes/no" question -- if you want, it can be any question that can be answered by a number. (You
can't ask specifically about the salary, though.) Once you get the answer, you take your guess at the income.
What kind of question do you ask?
Well, one good question might be, "how many years of education does the person have?" You can then go by the general rule that the more education, the more likely they were to have a higher salary.
Or, you can ask, "what's the person's IQ?" Again, you can assume that the higher a person's people's intelligence, the more likely they are to have a higher income.
But if you ask, "did they win a lottery jackpot last year?", that's a waste. It's the like the Stallone DVD question. Most of the time, the answer is no, and you get barely any useful information.
It's just not worth asking, just on the off-chance that you get a yes.
That all makes sense, right? Well, if you understand the strategy of the game of "Twenty Questions," you understand r-squared. Because, r-squared is really just a measure of how good your question
is. Seriously -- the correspondence between the two is almost perfect. The better the question, the higher the r-squared; and, the higher the r-squared, the better the question.
If you were to run a regression of IQ against income, you'd probably wind up with a decent r-squared -- maybe, I don't know, .15 or something. That means that if you know a random person's IQ, you
can knock 15 percent off your average error squared. Maybe if you were completely ignorant, you would guess $30,000 for everyone. But if you know IQ, you can guess $20,000 for low, $30,000 for
average, and $40,000 for high, and your guesses would be closer.
But, the bad question, the lottery question: the r-squared of that might only be .001. Originally, you guessed $30,000. Now, if you find out they didn't win the lottery, you guess $29,999, and are a
tiny bit closer, on average. If you find out they *did* win the lottery, then you guess, say, $5 million, and you come a lot closer that you would have before. But that happens very infrequently --
so infrequently that you're square error is still going to be well over 99.99 percent of what it was without the question.
I said the analogy between the game and r-squared was *almost* perfect. If you care, here's how to make it exact:
After you ask the IQ question, you're given a table of all 300,000,000 people in the US, with their income and answer to your question (IQ). Then, before I tell you the random person's IQ, you have
to decide in advance what you're going to answer for each possible IQ, and your decision has to have each point of IQ worth the same amount of income (that is, it has to be linear, since it's a
linear regression).
Once you've decided, I give you the IQ, and we figure your answer, and your negative score is the square of how much you missed it by.
Under those rules, the analogy is exact: the r-squared exactly corresponds to how good a question you asked.
(Oh, and if you want to actually ask twenty questions instead of one ... that's just a multiple regression with 20 variables.)
In the past, I've been critical of analyses that find a low r-squared, and assume that, therefore, there's only a weak relationship. For instance, I've written about the study that found, in MLB, an
r-squared of .18 for team payroll vs. team wins. The authors of the study then said something like, "The r-squared is low. Therefore, there's not much of a relationship. Therefore, salary doesn't
lead to wins."
Well, that's not right. It's like saying, for the lottery example, "The r-squared is low. Therefore, winning the lottery doesn't lead to more money."
That's obviously incorrect.
The r-squared does NOT measure the direct relationship between the variables. It just measures how good a question it is to ask about the one variable.
But, the thing is, if what you really want is the relationship between winning the lottery and getting rich ... well, that's easy. Just look at the regression equation!
If you do that regression, the one on lottery winnings that gives you an r-squared of .001, you'll wind up with an equation like
Expected salary = $30,000 + $5,000,000 if they won the lottery.
It gives you exactly what you want -- winning the lottery is worth $5 million. Why would you focus on the r-squared, when the exact answer is right there? In fact, the r-squared is completely
I think the reason we sometimes focus on the r-squared, though, is that we make a false assumption. It is true that (a) if you have a high r-squared, you have a strong relationship. But it is NOT
necessarily true that (b) if you *don't* have high r-squared, you *don't* have a strong relationship. I think that maybe we just assume because (a) is true, (b) is also true. But it's not.
So, in summary, three different ways to think about it:
--- One
The regression equation answers, "how much does winning the lottery affect income?" [lots.]
The r-squared answers, "is asking about the lottery a good "twenty questions" way to help estimate income?" [not very.]
--- Two
The regression equation answers, "how much does winning the lottery affect income?" [lots.]
The r-squared answers, "when people differ in income, how much of that is because some of them won the lottery?" [not much.]
--- Three
The regression equation answers, "if you change the value of the lottery variable from "no" to "yes," how much does income change? [lots.]
The r-squared answers, "if you change the value of the lottery value from one random person's to another random person's, how much does income change?" [not much -- two random people are probably
both "no", so the change is usually zero.]
If you've got any more good ones, let me know, and I'll add them in.
Labels: r-squared, regression, statistics
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Pressure and Measurement MCQ [PDF] Quiz Questions Answers | Pressure and Measurement MCQs App Download & e-Book
IGCSE A Level Physics Online Tests
Pressure and Measurement MCQ (Multiple Choice Questions) PDF Download
The Pressure and Measurement Multiple Choice Questions (MCQ Quiz) with Answers PDF (Pressure and Measurement MCQ PDF e-Book) download to practice IGCSE A Level Physics Tests. Learn Matter and
Materials Multiple Choice Questions and Answers (MCQs), Pressure and Measurement quiz answers PDF to learn online certification courses. The Pressure and Measurement MCQ App Download: Free learning
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The MCQ: Liquid A and liquid B exert same amount of pressure on each other, but the density of A is twice the density of B. The height of liquid B is 10 cm, then the height of liquid A would be;
"Pressure and Measurement" App Download (Free) with answers: 5 cm; 10 cm; 20 cm; 40 cm; to learn online certification courses. Practice Pressure and Measurement Quiz Questions, download Google eBook
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Normal force acting per unit cross sectional area is called
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Liquid A and liquid B exert same amount of pressure on each other, but the density of A is twice the density of B. The height of liquid B is 10 cm, then the height of liquid A would be
1. 5 cm
2. 10 cm
3. 20 cm
4. 40 cm
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Height of atmosphere, if atmospheric density is 1.29 kg m^-3 and atmospheric pressure is 101 kPa, is
1. 7839.4 m
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Pressure in fluid depends upon
1. depth below the surface
2. density of fluid
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Finding the Distance between Two Points given Their Coordinates in Three Dimensions
Question Video: Finding the Distance between Two Points given Their Coordinates in Three Dimensions Mathematics • Third Year of Secondary School
Find the distance between the two points π ΄(β 7, 12, 3) and π ΅(β 4, β 1, β 8).
Video Transcript
Find the distance between the two points π ΄ with coordinates negative seven, 12, three and π ΅ with coordinates negative four, negative one, negative eight. To solve this question, we can use a
formula, the distance between two arbitrary points in three dimensional space.
π with coordinates π ₯1, π ¦1, π §1, and π with coordinates π ₯2, π ¦2, π §2 is the square root of π ₯1 minus π ₯2 squared plus π ¦1 minus π ¦2 squared plus π §1 minus π §2 squared.
This is very similar to the formula for distance in two-dimensional space. The only difference is that because in three dimensions we also have a π §-coordinate; we have a term involving the π
§-coordinates in our formula.
Substituting the values of the coordinates in, we get negative seven minus negative four squared plus 12 minus negative one squared plus three minus negative eight squared. We find that negative
seven minus negative four is negative three, 12 minus negative one is 13, and three minus negative eight is 11.
So the distance is the square root of nine plus 169 plus 121, which is the square root of 299 length units. And looking at the prime factorization of 299, itβ s 13 times 23. We can see that we canβ
t simplify this radical any further, so this is our final answer. So that was just applying a formula, but where did that formula come from? Iβ m not going to derive the general formula; Iβ m just
going to solve the problem without using the general formula.
But it turns out the solution can be generalized quite easily to give the general formula for the distance between two points in 3D space. Iβ ve drawn something that looks very much like the
two-dimensional plane, but in fact this is 3D space just with the π §-axis pointing straight out of the screen at you from the origin.
In this orientation of 3D space, you canβ t actually see the π §-axis but trust me itβ s there. Iβ ve marked on the point π ΄ which has coordinates negative seven, 12, three, and Iβ ve also
marked on the auxiliary point π with coordinates negative four, negative one, three.
Notice that the point π is not the same as the point π ΅; although itβ s π ₯- and π ¦-coordinates are the same, its π §-coordinate is three and not negative eight. It does, however, have the
same π §-coordinate as the point π ΄, and so both points lie in the plane with equation π § equals three.
We can add another auxiliary point, π , with coordinates negative seven, negative one, and three. It is in the same plane as the other two points, π § equals three. Furthermore, it has the same π
₯-coordinate as the point π ΄. The only difference between the points π ΄ and π is in their π ¦-coordinates: π ΄ has a π ¦-coordinate of 12, whereas π has a π ¦-coordinate of negative
one. And so to get from π ΄ to π , you have to move 13 units in the opposite direction to the π ¦-axis, so the distance between π ΄ and π is 13 length units.
In a similar way, we can see that getting from π to π requires moving three units in the π ₯-direction. And of course the angle formed at π is a right angle; remember that weβ re working
in the plane with equation π § equals three here, so this really is a right angle and not just something that looks like a right angle because of the way that weβ ve oriented our three-dimensional
space. And so using the Pythagorean theorem, we see that the distance from π ΄ to π is the square root of 178 length units.
Of course this is the same answer that we would get by using the two-dimensional distance formula and just ignoring the third coordinate, the π §-coordinate. And thatβ s great and all, but weβ re
not looking for the distance from π ΄ to π ; weβ re looking for the distance from π ΄ to π ΅. If you remember, the π §-axis is pointing straight out of the screen at us. And so the point π ΅
which has the same π ₯ and π ¦ coordinates as the point π , but a different π §-coordinate, would look like it occupies the same space as the point π on our two-dimensional representation of
the 3D space.
Of course as discussed before, it isnβ t at the same place as π . In fact, it is a distance of three minus negative eight units away in the π §-direction. So if we now draw another representation
of our three-dimensional space β this one where you canβ t see the π ₯-axis because itβ s actually going down into the screen away from us marking π ΄, π , and π ΅ on this diagram β we can
see that π ΅ and π are definitely not the same. And in fact because π and π have the same π ¦- and π §- coordinates, if we marked π on this diagram as well, it would look like π is
at the same place as π .
To get from π ΅ to π , you have to move 11 units in the π §-direction, so the distance from π ΅ to π is 11. We can also mark on the length of AP that we found using the left-hand diagram; it
is the square root of 178 length units. We have to be careful here, if we look only at the right-hand diagram, itβ s very tempting to believe that the only difference between π ΄ and π is in
their π ¦- co-ordinate, and so the length of AP is 12 minus negative one equals 13.
But of course their π ₯ coordinates are also different. Itβ s just that because the π ₯-axis is pointing down into the screen, we canβ t see this easily. The difficult thing to see here is that
the angle at π is a right angle. These two dimensional diagrams can sometimes misrepresent 3D angles, but in this case it really is a right angle, and we can prove this for example using vectors.
But having convinced ourselves of this fact, we can apply the Pythagorean theorem. The length of AB, the distance between π ΄ and π ΅, is the square root of 11 squared plus the square root of 178
squared, which is the square root of 121 plus 178, which as before is the square root of 299 length units.
It is quite common that you have to find the distance between two points in three-dimensional space. Three-dimensional space, after all, is the space that we live in. And so rather than going through
this process every time you want to find the distance between two points in three- dimensional space, it makes sense to do this only once in the general case and just derive a formula that we can
then substitute the numbers into when required. | {"url":"https://www.nagwa.com/en/videos/949181870438/","timestamp":"2024-11-14T07:54:52Z","content_type":"text/html","content_length":"259262","record_id":"<urn:uuid:627af054-9bb1-4951-b0be-411117fe60b2>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00797.warc.gz"} |
Analyzing data: AI and machine learning in market research | Articles
Marketing research, AI and machine learning: Artificial neural networks
Editor’s note: David A. Bryant is vice president at Ironwood Insights Group.
Earlier this year, I wrote a two-part article covering machine learning techniques – cluster analysis and decision tree analysis. This article will cover a third machine learning technique that is
commonly used by market researchers: artificial neural networks (ANNs).
Artificial neural networks
The two types of ANNs that are frequently used in market research include multilayer perceptron and the radial basis function. Each has their differences and advantages.
• The output of a radial basis function network is always linear, whereas the output of a multilayer perceptron network can be linear or nonlinear. You need to determine the type of problem you are
trying to solve before selecting the type of ANN you want to use.
• Multilayer perceptron networks can have more than one hidden layer, whereas a radial basis function network will only have a single layer. Having more than a single hidden layer can be important
when working on nonlinear problems.
In this article I am going to focus on the multilayer perceptron function because of its ability to solve linear and nonlinear problems. I will avoid going into too much detail regarding the
mathematics involved in calculating the neuron weights in a multilayer perceptron (MLP) model.
Two examples of scenarios using the multilayer perceptron procedure include:
• A loan officer at a bank wants to identify characteristics that are predictors of people who are likely to default on a loan and use those predictors to identify customers who are good and bad
credit risks.
• A customer retention manager at a telephone company wants to identify characteristics that can predict which customers are most likely to switch telephone plans in the near future. This is the
scenario that I discussed when exploring the decision tree algorithm in my previous article. The CHAID decision tree algorithm identified “months of service” and whether the customer “rented
equipment” as the two strongest predictors of who would switch telephone companies in the near future.
Let’s explore the multilayer perceptron algorithm using telephone customer churn data.^1
Understanding a multilayer perception model
ANNs use an unsupervised machine-learning algorithm to identify patterns found within unlabeled input data. The MLP procedure produces a predictive model for one or more dependent variables based on
the values of the predictor variables. The MLP procedure is robust because the dependent variables and the predictor variables can be categorical or continuous or any combination of both.
Figure 1 shows that every multilevel perceptron consists of three types of layers – the input layer, the output layer and the hidden layer. The hidden layer can consist of one or more layers of
The input layer receives the initial data to be processed. The required task, such as a forecast of who will leave the telephone company in the near future, is performed by the output layer. The
hidden layers of neurons are the true computational engine of the MLP.
MLPs are composed of neurons called perceptrons. So, before going any further, we need to understand the general structure of a perceptron. Figure 2 shows that a perceptron receives n features as
inputs (x = x[1], x[2], ..., x[n]), and each of these features is associated with a weight.
Input features must be numeric, so a categorical feature (variable) must be converted to numeric ones in order to use a perceptron. For example, a categorical feature with three possible values is
converted into three input features by creating three dummy variables indicating the presence/absence of each value.
Let’s assume that the telephone company has three levels of service – basic service, plus service and total service. So, if we want to include this categorical variable in our MLP predictor model,
this input feature will need to be converted into three dummy variables. The SPSS neural network algorithm automatically makes this transformation .
Development of a prediction model
To start the creation of our MLP predictor model we are going to begin with the two variables that came out of our CHAID decision tree model in my previous two-part series:
• Months of service (tenure).
• Equipment rental.
The resulting MLP appears in Figure 3.
To train and test our MLP model, we broke the data into a 70/30 split. That means that we trained the model with 70% of the original data, (700 records), and then we tested the model with the
remaining 30% of the data (300 records).
The resulting model includes the months of service, or tenure variable, which is continuous, and the “equipment” variable, which is categorical and has been converted into two separate dummy
variables (i.e., equipment rental = no; equipment rental = yes). The MLP model has a single hidden layer with two neurons. The output layer includes our dependent variable which also has been
converted into two dummy variables (i.e., churn = no; churn = yes).
The MLP model also includes bias units. Bias units are appended to the input layer and each hidden layer and is assigned a value of “+1.” Bias units aren’t influenced by the values in the previous
units, i.e., the bias neurons don’t have any incoming connections. The bias neurons do have outgoing connections and the weights associated with these connections help to improve the final results of
the MLP model.
Table 1 shows the weights for each neuron in the MLP model.
Table 1
By looking at the parameter estimates table above and comparing it with the MLP model in Figure 3, we can see that negative weights are displayed as blue lines and positive weights are displayed as
green lines in the model. This helps us to quickly understand the influence that each neuron has on the following connection neurons, either the hidden layer or the output layer.
These weights, similar to regression coefficients, can be used to create a predictive model to help identify those customers who are most likely to churn.
Another important output from the multilayer perceptron algorithm is the importance of the independent variables used. This result can be seen in Table 2.
Table 2
As we see above, the tenure variable has the largest impact on predicting the dependent variable. This is the same result we found from the CHAID algorithm used in the decision tree model.
Predicting churn – testing and improving the model
How well does the resulting model predict who will and who won’t churn? The results are shown in Table 3.
Table 3
As we can see from this table, the resulting MLP model does a good job of predicting who won’t churn (No) in both the training and the testing data sets – over 90%. Where the model is weak, is that
it correctly predicts who will churn (Yes) only about one-third of the time in both data sets.
In this type of prediction model, the usefulness comes from the ability to accurately predict who will churn in the near future.
By exploring with different input variables, we can continue to improve the model. Through trial and error, we add household income to the MLP model (Figure 4).
In Table 4 we see the weights of the new MLP model.
Table 4
Table 5 shows us the prediction classification from the new model that includes household income as an input variable.
Table 5
From this we can see that simply adding household income as an input, the model now correctly predicts who will churn over 40% of the time. But this still leaves room for improvement, so we continue
to work on the MLP model.
The final MLP model that we explore comes as a result of including all the variables in the data file into an MLP model and then selecting the top 14 input variables based on their normalized
importance ratings.
The resulting MLP model is too complex to include here, but the new prediction classification can be seen in Table 6.
Table 6
The new MLP model accurately predicts those who are likely to churn 54% of the time in the training sample and 56.5% of the time in the testing sample. This is a significant improvement over our
first two models. The parameter estimates, along with the input variables included, can be seen in Table 7.
Table 7
The new MLP model is more complex and has a single hidden layer with seven neurons. While this model is significantly more complex, Table 6 shows that it does a better job of predicting who is likely
to churn among the telephone company customers.
Building a neural network: design considerations
When building a neural network, the dimension of the input vector determines the number of neurons in the input layer. In our most recent model there are 14 variables, but because two of the
variables are categorical (equipment rental and customer category), we end up with a total of 18 neurons in the input layer plus the bias neuron. Generally, the number of neurons in the hidden layers
are chosen as a fraction of those in the input layer. There is a trade-off regarding the number of neurons in the hidden layer:
• Too many neurons produce overtraining.
• Too few neurons affect generalization capabilities.
Too much in either direction will affect the overall usefulness of the MLP model that is finally developed.
Deciding how to analyze data
Machine learning consists of building models based on mathematical algorithms in order to better understand the data. One of the most important steps in understanding the data problem is to decide
how the data needs to be analyzed in order to yield the desired results. In our previous review of the telecommunications company example, we used decision tree analysis to develop a prediction model
to determine which customer characteristics will help us to predict the customers that are most likely to defect (or churn) and go to one of our competitors. In this paper, we have demonstrated how
artificial neural networks are another way to go to develop a useful prediction model.
The decision tree analysis has its strength in that it identifies those subgroups of customers that are most likely to churn. Those responsible for customer retention can then develop programs
designed to reduce churn among those groups most likely to leave.
The artificial neural network model has its strength in that it can be used to develop a forecast model that will predict the number of customers that are likely to churn in the near future. This can
be used to predict long-range customer and revenue growth.
Both decision tree analysis and artificial neural networks are powerful machine-learning algorithms that make it easier to analyze large arrays of data without much programming from the analyst. Both
tools still require that the analyst knows which type of algorithm to use for the question being asked and how to interpret the results.
1. Data source: Telco.sav is an SPSS file that is supplied with the latest versions of the SPSS software. | {"url":"https://www.quirks.com/articles/analyzing-data-ai-and-machine-learning-in-market-research","timestamp":"2024-11-09T10:02:17Z","content_type":"text/html","content_length":"219772","record_id":"<urn:uuid:fd4ec89a-348c-45bd-8c03-bc9a491605c6>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00139.warc.gz"} |
Curves MCQS
1. Which classification of curves is primarily used for transitioning between straight and curved sections in roads or railways?
a) Circular curves
b) Compound curves
c) Transition curves
d) Reverse curves
Explanation: Transition curves are specifically designed to gradually transition the curvature of a road or railway from a straight line to a curved line, or vice versa, providing smoother
transitions for vehicles or trains.
2. What are the primary elements of a circular curve?
a) Tangent, arc, and chord
b) Tangent, radius, and arc
c) Radius, chord, and central angle
d) Radius, arc, and tangent
Explanation: The primary elements of a circular curve include the radius, which defines the curvature of the curve, the arc, which is the curved portion of the road or railway, and the tangent, which
is the straight portion connecting the curve to the preceding or following segment.
3. When setting out curves by offsets, what are offsets used for?
a) Determining the radius of the curve
b) Measuring the length of the curve
c) Marking points perpendicular to the curve
d) Calculating the central angle of the curve
Explanation: Offsets are perpendicular measurements taken from a baseline to determine the points where the curve intersects with the baseline, aiding in the setting out of the curve.
4. Which type of curve is formed by combining two or more circular curves with different radii?
a) Compound curves
b) Reverse curves
c) Transition curves
d) Vertical curves
Explanation: Compound curves consist of two or more circular curves with different radii, connected smoothly to form a continuous curve.
5. What is the purpose of reverse curves in road design?
a) To provide smoother transitions between straight and curved sections
b) To allow vehicles to change direction rapidly
c) To increase the speed limit on roads
d) To create visual interest for drivers
Explanation: Reverse curves are used to counteract the monotony of long, straight roads by introducing alternating curves, improving driver attention and safety.
6. Which type of curve is used to gradually change the alignment of a road or railway track from straight to curved or vice versa?
a) Circular curves
b) Compound curves
c) Transition curves
d) Reverse curves
Explanation: Transition curves are specifically designed to gradually transition the alignment of a road or railway track from straight to curved or vice versa, providing smoother transitions for
vehicles or trains.
7. In vertical curve calculations, what does the term “K value” represent?
a) The length of the vertical curve
b) The rate of change of grade
c) The radius of curvature
d) The height difference between the endpoints
Explanation: The K value in vertical curve calculations represents the rate of change of grade, influencing the slope of the curve.
8. Which method is commonly used to set out curves using precise angle measurements?
a) Offsets
b) Compass and chain
c) Theodolites
d) Traversing
Explanation: Theodolites are precision instruments commonly used in surveying and engineering for measuring angles in both the horizontal and vertical planes, making them ideal for setting out
9. What is the primary function of a vertical curve in road design?
a) To provide smooth transitions between different road grades
b) To accommodate changes in traffic volume
c) To enhance the aesthetics of the road
d) To reduce construction costs
Explanation: Vertical curves are used to provide smooth transitions between different road grades, ensuring comfortable driving conditions for motorists.
10. How are transition curves different from circular curves?
a) Transition curves have a constant radius throughout.
b) Transition curves are used exclusively in railways.
c) Transition curves are shorter in length.
d) Transition curves provide smoother transitions between straight and curved sections.
Explanation: Transition curves differ from circular curves by providing smoother transitions between straight and curved sections, gradually changing the alignment instead of abruptly transitioning.
Leave a Comment | {"url":"https://easyexamnotes.com/curves-mcqs/","timestamp":"2024-11-10T17:55:44Z","content_type":"text/html","content_length":"116508","record_id":"<urn:uuid:80fff150-8772-4b8a-a097-6b71c9a2aa76>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00890.warc.gz"} |
Subway Routes
Submit solution
Points: 12 (partial)
Time limit: 0.6s
Memory limit: 16M
While taking the subway, George and Peter were determined to find the longest subway route. They found different routes, and declared their route to be the longest route. After arguing and missing
their stop, they realized that both of their routes were the longest routes. Looking back at the map, George, and Peter want to find how many subway routes have the longest length. A subway route is
considered different from another subway route if at least one of its endpoints is not an endpoint on the other route. It is guaranteed that there is exactly one unique path between any pair of
stations. Assume that all tunnels between stations are of the same length.
Input Specification
, the number of stations. The next lines contain two unique integers , and , indicating there is a tunnel connecting stations and .
Output Specification
The number of unique subway routes that have the longest length.
Sample Input
Sample Output | {"url":"https://dmoj.ca/problem/subway","timestamp":"2024-11-14T13:37:35Z","content_type":"text/html","content_length":"23101","record_id":"<urn:uuid:27ebb7d8-8cae-4d01-8390-e5a2c2afa830>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00162.warc.gz"} |
How to Use the Excel AVERAGEIFS Function
What is the Excel AVERAGEIFS Function?
The Excel AVERAGEIFS function is a solution for calculating an average (arithmetic mean) of values supplied that meet more than one criteria. The AVERAGEIFS function is an improvement from the excel
AVERAGEIF function provided since Excel 2007.
AVERAGEIFS Syntax
AVERAGEIFS(average_range, criteria_range1, criteria1, [criteria_range2, criteria2], ...)
average_range, required, the range to average
criteria_range1, required, the first range to be evaluated by the criteria1
criteria1, required, the first criteria to evaluate on criteria_range1.
[criteria_range2], optional, the second range to be evaluated by the criteria2. [criteria2], optional, the second criteria to evaluate on criteria_range1.
Usage Notes:
• AVERAGEIFS function can handle up to 127 range/criteria_range pairs.
• AVERAGEIFS function returns the #DIV0! error when no criteria are meet or average_range is a blank or text value.
• AVERAGEIFS function treats empty cell in criteria_range as a 0 (zero) value.
• AVERAGEIFS function evaluate TRUE as 1 and FALSE as 0 (zero) in range argument.
• AVERAGEIFS function does an average calculation in average_range if all criteria specified for that cell are meet.
• AVERAGEIFS criteria_range must have the same number of rows and columns as average_range
• AVERAGEIFS function allows the wildcard characters asterisk (*) and question mark (?) in criteria. An asterisk wildcard for any character sequence, and a question mark wildcard for any single
character; use a tilde (~) before character to find actual asterisk and question mark.
How to Use AVERAGEIFS Function in Excel
For example, there is data as shown below. How to create an excel formula for the following questions:
Question #1, AVERAGEIFS Multiple Criteria in Same Column (AND Criteria)
The Question
What are the average sales for an iPhone with storage of less than 256GB and more than 32GB?
The Criteria
There are two criteria, criteria #1 storage is less than 256GB, criteria #2 storage is more than 32GB, both point to the same column. All criteria specified must be met to make excel do the average
The Formula
Criteria_range1 and criteria_range2 point to the same range address. The result is 7,312.
Question #2, AVERAGEIFS Multiple Criteria in Same Column (OR Criteria)
The Question
What are the average iPhone sales for the “West” and “East” area?
The Criteria
There are two criteria, criteria #1 area “West”, criteria #2 area “East”, both point to the same column.
Is it possible, in one cell there is a “West” and “East” area together? Certainly impossible. The average calculation for the above questions will be done if ONE OF THE CRITERIA IS MEET.
Even though the AVERAGEIFS function only performs an average calculation if all criteria are met. The above question cannot be answered using the AVERAGEIFS function. Read “AVERAGEIFS Limitations”
below for a solution.
Read “AVERAGEIFS Limitations” below for a solution.
Question #3, AVERAGEIFS Multiple Criteria in Different Column (AND Criteria)
The Question
What are the average iPhone sales for the “plus” variant in the “East” area?
The Criteria
There are two criteria, criteria #1 a “plus” variant name point to column B, criteria #2 “East” area point to column A.
The Formula
The result is 7,192
Question #4, AVERAGEIFS Multiple Criteria in Different Column (OR Criteria)
The Question
What are the average iPhone sales for a “plus” variant OR the storage capacity equal to 64GB?
A weird question, but this is only to show you a question with criteria in a different column and only one of the criteria must be met.
The Criteria
There are two criteria, criteria #1 a “plus” variant name point to column B, criteria #2 storage equal to 64GB point to column C.
Like question #2, only one criterion is needed to do the average calculation and cannot be solved with the AVERAGEIFS function.
Read “AVERAGEIFS Limitations” below for a solution.
AVERAGEIFS Limitations
The AVERAGEIFS function limitation is to answer questions that only require one criterion that is fulfilled from several existing criteria.
There are two solutions. The first solution is using a helper column containing the OR function and the second solution is using an array formula.
For more details, read the following article for a reference. It’s a different function, but you can use the same trick to solve the AVERAGEIFS function limitations.
AVERAGEIFS Example
Another Statistical Function
Another Logical Function | {"url":"https://excelcse.com/excel-averageifs-function/","timestamp":"2024-11-12T16:48:59Z","content_type":"text/html","content_length":"70909","record_id":"<urn:uuid:8ef977b9-9461-4def-811a-d2ab1b4a4fb8>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00595.warc.gz"} |
(3.1) FX Theory: Interest Rate Parity snbchf.com
(3.1) FX Theory: Interest Rate Parity
The interest rate parity gives a mathematical explanation for the purchasing power parity and real effective exchange rates. It explains why currencies with low inflation must appreciate with time.
Low inflation countries – often called safe-havens – have wages and implicitly inflation, that do not rise as quickly as in other countries. With lower wage costs, profits of companies increase and
the current account and the international investment position are positive. This has a repercussion on the balance of payments: More investor inflows into these profitable companies than outflows.
Since longer-term bonds are related to inflation, the government pays less interest on its debt. Thanks to low bond yields, many of the safe-havens are low-tax countries. Examples of those “safe and
tax-havens” are Switzerland or Singapore. Germany is a safe-haven since the 1970s, because wages increased more slowly than in other European countries, with the consequence of a strong German Mark.
The influence on the balance of payments is slightly different: These inflows into safe government bonds mostly happen during risk-off phases.
The interest rate parity gives the mathematical explanation why such a low inflation currency must appreciate.
Interest Rate Parity
(Most used source for this chapter)
Assuming a forward market exists, the investor can either save at home, receiving interest rate i, or converting by the exchange rate S , receiving interest rate i* abroad, and then converting back
to home currency by the forward rate F obtaining at time t for a trade at time t+1.
This condition is called “covered interest rate parity”, reflecting the fact that investors are “covered” against nominal uncertainty by way of the forward market.
If the gross return of foreign investments converted in local currency is higher than the return of investments in the home country, then investors will prefer the foreign currency investments.
If the forward rate is equal to the future spot rate (see also “forward premium”) such that
then we obtain the “uncovered interest rate parity”:
In countries with high inflation, money is not worth always the same. Money in this currency loses real (i.e. inflation-adjusted) value, you can buy less with it after some time Therefore the
ex-ante purchasing power parity, suggests that these currencies depreciate over the long-term.
The p terms stand for foreign expected prices minus current prices, i.g. foreign inflation, compared with the p* terms, local inflation.
On the other side, it is possible that despite higher inflation, a foreign currency does not depreciate. The reason is that investments in this currency yield more. This is reflected in the real
interest parity.
Higher yields on investments (“marginal product of capital”) compensate for the fact that inflation is higher.
Since the central banks need to combat high inflation, they are obliged to hike interest rates until they are equal to the yield of investments minus a risk premium. Neoclassical models assume that
the marginal product of capital equals the real interest rate, this condition is equivalent to the equalization of marginal products of capital equalized across borders.
Real Interest Rate and Fisher equation
The Fisher equation leads to the definition of real interest rates.
The following graph shows real interest rates around the world based on central bank interest rates as December 2012.
(click to expand)
The mean reversion for to real exchange rates
The uncovered interest rate parity is visible in the mean reversion of currencies to the real (inflation-adjusted) exchange rate.
A pure mean reversion is valid for P/E ratios. The P/E ratio must come back to average Shiller Price Earning Ratio. A strongly performing stock like Apple, must return towards the mean, because the
competition is able to produce similar products. Stock prices rise over time according to GDP growth and inflation rate (see the Gordon growth model). Indices are positively influenced by a
survivor-ship bias and includes sometimes even dividends (Performance indices).
A mean reversion for currencies does not exist neither: currencies with low inflation and current accounts surpluses must appreciate over the time.
See more for
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-MPs without duplicates in two-terminal multistate networks based on MPs
Journal of Systems Engineering and Electronics ›› 2022, Vol. 33 ›› Issue (6): 1332-1341.doi: 10.23919/JSEE.2022.000152
• RELIABILITY • Previous Articles
Search for d-MPs without duplicates in two-terminal multistate networks based on MPs
Bei XU^1^,^2( ), Yining FANG^1^,*( ), Guanghan BAI^1( ), Yun’an ZHANG^1( ), Junyong TAO^1( )
1. ^1 Laboratory of Science and Technology on Integrated Logistics Support, College of Intelligent Sciences and Technology, National University of Defense Technology, Changsha 410073, China
^2 School of General Aviation, Nanchang Hangkong University, Nanchang 330063, China | {"url":"https://www.jseepub.com/EN/10.23919/JSEE.2022.000152","timestamp":"2024-11-12T03:19:03Z","content_type":"text/html","content_length":"110888","record_id":"<urn:uuid:ea933ccb-82dc-4964-b2e7-97bfbd3293b7>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00576.warc.gz"} |
Stochastic Reformulations of Linear Systems: Algorithms and Convergence Theory
We develop a family of reformulations of an arbitrary consistent linear system into a stochastic problem. The reformulations are governed by two user-defined parameters: a positive definite matrix
defining a norm, and an arbitrary discrete or continuous distribution over random matrices. Our reformulation has several equivalent interpretations, allowing for researchers from various communities
to leverage their domain-specific insights. In particular, our reformulation can be equivalently seen as a stochastic optimization problem, stochastic linear system, stochastic fixed point problem,
and a probabilistic intersection problem. We prove sufficient, and necessary and sufficient, conditions for the reformulation to be exact. Further, we propose and analyze three stochastic algorithms
for solving the reformulated problem-basic, parallel, and accelerated methods-with global linear convergence rates. The rates can be interpreted as condition numbers of a matrix which depends on the
system matrix and on the reformulation parameters. This gives rise to a new phenomenon which we call stochastic preconditioning and which refers to the problem of finding parameters (matrix and
distribution) leading to a sufficiently small condition number. Our basic method can be equivalently interpreted as stochastic gradient descent, stochastic Newton method, stochastic proximal point
method, stochastic fixed point method, and stochastic projection method, with fixed stepsize (relaxation parameter), applied to the reformulations.
Bibliographical note
KAUST Repository Item: Exported on 2020-10-01
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Unschedule() API Execution Time Too Long?
This is a question for SmartThings software developers.
During smart app development I’ve run into a problem when one of my functions occasionally triggered java.util.concurrent.TimeoutException which means that it’s exceeded 20 sec execution limit.
With some debugging I was able to narrow it down to calling the unschedule() API. A simple test revealed that executing unschedule() API can take anywhere from 90 milliseconds to 3.7 seconds (!). And
that’s on a good day. Occasionally, unschedule() gets stuck for longer than 20 seconds, resulting in java.util.concurrent.TimeoutException.
Here’s a simple test that I ran:
private def myUnschedule() {
def t0 = now()
def t = now() - t0
log.trace "unschedule() is executed in ${t} ms"
And here’s what it prints:
XXXX 10:50:46 AM PST: trace unschedule() is executed in 1420 ms
XXXX 10:50:45 AM PST: trace Deleting all scheduled jobs for InstalledSmartApp: XXXX
XXXX 10:50:42 AM PST: trace unschedule() is executed in 3652 ms
XXXX 10:50:38 AM PST: trace Deleting all scheduled jobs for InstalledSmartApp: XXXX
XXXX 10:50:35 AM PST: trace unschedule() is executed in 3744 ms
XXXX 10:50:31 AM PST: trace Deleting all scheduled jobs for InstalledSmartApp: XXXX
XXXX 10:50:28 AM PST: trace unschedule() is executed in 2383 ms
XXXX 10:50:25 AM PST: trace Deleting all scheduled jobs for InstalledSmartApp: XXXX
A related question is what happens to the smart app once it hits TimeoutException? I have a suspicion that it’s left in a “zombie” state where it does not properly responds to events and can no
longer update its state.
Any help resolving this issue would be greatly appreciated.
Here’s another test tun: unschedule() exceeds 10 seconds (!)
XXXX 11:26:17 AM PST: trace unschedule() is executed in 5045 ms
XXXX 11:26:12 AM PST: trace Deleting all scheduled jobs for InstalledSmartApp: XXXX
XXXX 11:26:09 AM PST: trace unschedule() is executed in 10467 ms
XXXX 11:25:58 AM PST: trace Deleting all scheduled jobs for InstalledSmartApp: XXXX
XXXX 11:25:53 AM PST: trace unschedule() is executed in 5153 ms
XXXX 11:25:48 AM PST: trace Deleting all scheduled jobs for InstalledSmartApp: XXXX
XXXX 11:25:16 AM PST: trace unschedule() is executed in 8093 ms
XXXX 11:25:08 AM PST: trace Deleting all scheduled jobs for InstalledSmartApp: XXXX
… and another one: unschedule() exceeding 15 seconds!
XXXX 12:15:08 PM PST: trace unschedule() is executed in 12704 ms
XXXX 12:14:56 PM PST: trace Deleting all scheduled jobs for InstalledSmartApp: XXXX
XXXX 12:15:08 PM PST: trace unschedule() is executed in 14354 ms
XXXX 12:14:54 PM PST: trace Deleting all scheduled jobs for InstalledSmartApp: XXXX
XXXX 12:15:08 PM PST: trace unschedule() is executed in 15502 ms
XXXX 12:14:53 PM PST: trace Deleting all scheduled jobs for InstalledSmartApp: XXXX
XXXX 12:14:51 PM PST: trace unschedule() is executed in 887 ms
XXXX 12:14:51 PM PST: trace Deleting all scheduled jobs for InstalledSmartApp: XXXX
3 Likes
Well… I’ve added some code to keep track of maximum execution time and it didn’t take long to hit 20 seconds mark.
if (t > state.t_max) {
state.t_max = t
log.trace "state: ${state}"
XXXX 1:06:53 PM PST: trace state: [t_max:20086]
So, what’s the deal with unschedule() taking more than 20 seconds to execute? An what is the implication for the Smart Apps that got busted by the TimeoutException?
Ok. I managed to capture TimeoutException in the log:
XXXX 1:21:56 PM PST: error java.util.concurrent.TimeoutException: Execution time exceeded 20 app execution seconds: 882354304390580 @ line 61
XXXX 1:21:56 PM PST: trace Zombie Test was provided…creating subscription
XXXX 1:21:55 PM PST: trace Scheduling ‘zombieTest’ for InstalledSmartApp: XXXX
XXXX 1:21:46 PM PST: trace Deleting scheduled job ‘zombieTest’ for InstalledSmartApp: XXXX
XXXX 1:21:45 PM PST: trace state: [t_max:19066]
XXXX 1:21:45 PM PST: trace unschedule() is executed in 19066 ms
XXXX 1:21:27 PM PST: trace Deleting all scheduled jobs for InstalledSmartApp: XXXX
XXXX 1:21:26 PM PST: trace initialize with settings: [:]
1 Like
Opened support ticket:
Support request #86683: Smart App TimeoutException: Execution time exceeded 20 app execution seconds
4 Likes
New execution time record: 27.7 seconds!
8:03:57 PM PST: trace state: [t_max:27741]
UPDATED 2/16/2015
Another record: 32.1 seconds!
5:16:28 PM PST: trace state: [t_max:32148]
This looks like a serious issue to me. Would appreciate any comments.
2 Likes
Got response form support suggesting that I get @Ben and @mager involved in this discussion. So there it is…
I’ve got similar responses from support@ if it even hints at dev related. You’d think its be easier for a ST employee to forward an email or ticket, but maybe not. (Sadly, getting someone to read a
forum post can be… Difficult )
I hope you get this one addressed. If not, perhaps bring it up at the Dev conference call next week…
Take care
2 Likes
We’re getting the right people to look at this thread. Hang in there.
1 Like
I totally agree that calling unschedule() takes a long time. In the past few minutes as I’ve been looking at this thread, the average time to unschedule was right around 3 seconds, which is close to
what you were seeing. I did see some very fast ones too, around 4 ms, but that didn’t seem to be the normal. Even with 3 seconds, that is a good chunk of the 20 second total allowed, and if it slows
down I’m not surprised you were seeing the TimeoutExceptions.
Unschedule actually has to do a number of things like finding all the jobs for this installation of the SmartApp and cleaning up data in a number of (very large) tables. I don’t have the data to
prove it, but I assume the problem is getting worse over time.
For now, I know that doesn’t help much, and I’m sorry. Soon (hopefully) we’ll be rolling out a new scheduling mechanism behind the scenes that should be better able to handle this with much larger
volumes of customers.
I’ve asked another SmartThings developer to chime in on this part. I think the SmartApp just gets shut down at the point of the timeout. It will run again if there are more schedules or
subscriptions. In this case I think the unschedule() should finish, even if the exception is thrown, but I’ll let me colleague correct me if I’m wrong.
2 Likes
I wonder if this could somehow result in a situation where the platform has reported that a handler has run when, in fact, it didn’t. I’ve been trying to troubleshoot a failure here without much
success Maybe what’s described here is an explanation.
1 Like
I appreciate your looking into this issue. I monitored execution time over several hours period and the shortest execution time was about 75 ms, while the longest so far was 42 seconds (!) Typically,
I see at least one occurrence exceeding 10 seconds within an hour.
My concern is that there’s a significantly high probability of failure for any app that uses unschedule() API. Is there any workaround you could recommend until it’s fixed on your end?
1 Like
Please do not turn this thread into another architectural discussion. You’re welcome to spawn a new thread if you wish to go in that direction. Thanks.
2 Likes
This may explain spastic problems I’ve encountered with @RBoy “5-2 Day Thermostat” app where the scheduled temperature change does not occur. The initialize() function appears to be called at the
scheduled time but the temperature setpoint change don’t occur. The first thing initialize() does is call unschedule(). Later, when there is a mode change, the app appears to function correctly
again. That behavior is consistent with the expected behavior described by @matthewnohr .
I’m also interested in the workaround.
1 Like
Just FYI, I also ran similar test measuring execution time of runOnce API and got similar results. Typical execution time is less than 1 second, but there are frequent spikes (several times per hour)
when execution time exceed 10 and even 20 seconds.
I got 6 occurrences exceeding 20 seconds within the last 4 hours:
16:40 PM : 43.4 seconds
16:41 PM : 21.4 seconds
16:46 PM : 20.6 seconds
17:35 PM : 20.1 seconds
17:41 PM : 37.9 seconds
17:42 PM : 37.3 seconds
I ran into this with my BigTalker SmartApp just now when settings were updated, I would get the above exception after hitting Done. Sometimes after 20 seconds, sometimes after 40 seconds. The delay
came in when calling unschedule() or even when calling schedule(scheduleTime, onSchedule). It was not consistent though. It seemed very random, but did it often when updating settings.
For me, I adjusted my initialize() function to only do a couple of minor things (setting a couple of variables) and then it calls runIn(10, initSubscribe) to schedule actually running subscribe and
schedule functions seconds later to do the heavy lifting outside of the initialize() function. This allowed the settings to be updated without throwing the exception in the log and red errors in the
mobile app when hitting Done. It also released the mobile app interface so that it was no longer delaying and spinning/waiting on subscribe()'s and schedule()'s to finish processing. After doing
this, when the runIn(10, initSubscribe) fired (10 seconds after being called from within initialize()), the SmartApp completed subscribing to events and scheduling jobs in 2 seconds. In a second
test, it completed initSubscribe() in 9 seconds. In a third test, it completed initSubscribe() in 4 seconds. Either way, the user experience is much better this way as they do not have a long delay
in saving preference changes and so far not getting the exception and red app errors.
This makes me wonder if subscribe(), schedule() being called too soon (without at least a delay like the 10 seconds I introduce in my method) after unsubscribe(), unschedule() may be the culprit.
Spacing them out with runIn() seems to have resolved my issue at least.
5:27:28 PM: trace BIGTALKERDEV(1.0.3-Beta6) || Updated with settings 5:27:28 PM: trace Deleting all scheduled jobs 5:27:28 PM: trace Big Talker Dev is attempting to unsubscribe from all events
5:27:29 PM: trace BIGTALKERDEV(1.0.3-Beta6) || Initialized 5:27:39 PM: debug BIGTALKERDEV(1.0.3-Beta6) || Begin initSubscribe() ... 11 device subscriptions ... (truncated) 5:27:40 PM: trace
Scheduling 'onSchedule1Event' 5:27:40 PM: trace Scheduling 'onSchedule2Event' 5:27:41 PM: trace Scheduling 'onSchedule3Event' 5:27:41 PM: trace Home was provided...creating subscription 5:27:41 PM:
debug BIGTALKERDEV(1.0.3-Beta6) || END initSubscribe()
@Geko @Ben @mager
2 Likes
I ended up still having trouble with the timeout, so I scheduled two runIn() functions within initialize(). One for initSubscribe() (currently at 20 seconds after initialize()) and one for
initSchedule() 10 seconds later as it seems often scheduling takes too long and would throw the TimeoutException in the IDE. Perhaps by calling Scheduling within it’s own function it will not breach
the timeout period.
EDIT: This is not working either. It worked for the first log below, but see the runIn() failure in the second and failure after schedule() in the third log. It just takes way too long to schedule()
and initSubscribe() wasn’t event executed in the second log below. I continuously had problems with runIn(), runOnce() or even trying to do it with schedule(now()+60000, function); it would sometimes
run and sometimes not as seen in log #2 below.
7:48:27 PM: trace BIGTALKERDEV(1.0.3-Beta6) || Updated with settings: …
7:48:27 PM: trace Big Talker Dev is attempting to unsubscribe from all events
7:48:27 PM: trace Deleting all scheduled jobs
7:48:32 PM: debug BIGTALKERDEV(1.0.3-Beta6) || Scheduled initSubscribe() in 20 seconds
7:48:52 PM: debug BIGTALKERDEV(1.0.3-Beta6) || BEGIN initSubscribe()
…bunch of subscriptions…
7:48:52 PM: debug BIGTALKERDEV(1.0.3-Beta6) || END initSubscribe()
7:49:02 PM: debug BIGTALKERDEV(1.0.3-Beta6) || BEGIN initSchedule()
7:49:02 PM: trace Scheduling ‘onSchedule1Event’
7:49:06 PM: trace Scheduling ‘onSchedule2Event’
7:49:08 PM: trace Scheduling ‘onSchedule3Event’
7:49:10 PM: debug BIGTALKERDEV(1.0.3-Beta6) || END initSchedule()
8:00:11 PM: trace BIGTALKERDEV(1.0.3-Beta6) || Updated with settings
8:00:11 PM: trace Big Talker Dev is attempting to unsubscribe from all events
8:00:11 PM: trace Deleting all scheduled jobs
8:00:17 PM: debug BIGTALKERDEV(1.0.3-Beta6) || Scheduled initSubscribe() in 20 seconds
*initSubscribe() never executed!!!
8:00:47 PM: debug BIGTALKERDEV(1.0.3-Beta6) || BEGIN initSchedule()
8:00:47 PM: trace Scheduling ‘onSchedule1Event’
8:00:48 PM: trace Scheduling ‘onSchedule2Event’
8:00:51 PM: trace Scheduling ‘onSchedule3Event’
8:00:55 PM: debug BIGTALKERDEV(1.0.3-Beta6) || END initSchedule()
8:05:28 PM: trace Big Talker Dev is attempting to unsubscribe from all events
8:05:28 PM: trace Deleting all scheduled jobs
8:05:34 PM: debug BIGTALKERDEV(1.0.3-Beta6) || Scheduled initSubscribe() in 20 seconds
8:05:54 PM: debug BIGTALKERDEV(1.0.3-Beta6) || BEGIN initSubscribe()
…bunch of subscriptions…
8:05:54 PM: debug BIGTALKERDEV(1.0.3-Beta6) || END initSubscribe()
8:06:04 PM: debug BIGTALKERDEV(1.0.3-Beta6) || BEGIN initSchedule()
8:06:04 PM: trace Scheduling ‘onSchedule1Event’
8:06:19 PM: trace Scheduling ‘onSchedule2Event’
8:06:23 PM: trace Scheduling ‘onSchedule3Event’
8:06:27 PM: error java.util.concurrent.TimeoutException: Execution time exceeded 20 app execution seconds
The odd thing is, Schedule3 is scheduled, but then it seems to hang until it throws the error at the 8:06:27 PM mark. The only thing that should occur after the Schedule3 is:
LOGDEBUG (“END initSchedule()”)
sendNotification(“BIGTALKER: Settings activated”)
LOGDEBUG() and LOGTRACE() only call out to log.debug() and log.trace() and prepend some info which you can see working in the logs above.
sendNotification() is there so that I/the User would know when the settings actually took affect since I couldn’t save them successfully directly within initialize().
But all of these functions existed and were called in the successful logs as well.
Scheduling is very important but it is a mess, it seems.
Yep. I tried all of these. It’s a chicken and egg problem. You try to defer initialization, but you have to call runIn or runOnce to schedule it. And since either of those can cause a timeout, you
just move you point of failure to a different function.
1 Like
Our most recent update should alleviate some of the performance issues with unschedule. Let us know if you’re seeing improved performance, or if you’re still seeing the same issues.
Apparently this is still an issue according to @geko as my Pollster application crashes once in awhile. When can we expect a fix?
2 Likes | {"url":"https://community.smartthings.com/t/unschedule-api-execution-time-too-long/11232","timestamp":"2024-11-07T10:20:52Z","content_type":"text/html","content_length":"83472","record_id":"<urn:uuid:1b2deeb0-d200-456f-a043-750a9e8d6beb>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00020.warc.gz"} |
Python Function: Nearest Lower Power of Two
Oops, something went wrong. Please try again in a few moments.
def find_nearest_lower_power_of_two(a: int) -> int:
Finds the nearest lower power of two for a given number 'a'.
- a: int
The number for which we want to find the nearest lower power of two.
- int:
The nearest lower power of two.
- ValueError:
Raises an error if the input number 'a' is less than or equal to zero.
# Check if the input number is valid
if a <= 0:
raise ValueError("Input number should be greater than zero.")
# Initialize the power of two to 1
power_of_two = 1
# Find the nearest lower power of two
while power_of_two <= a:
power_of_two *= 2
# Return the nearest lower power of two
return power_of_two // 2
# Example usage:
number = 10
nearest_lower_power = find_nearest_lower_power_of_two(number)
print(f"The nearest lower power of two for {number} is {nearest_lower_power}.") | {"url":"https://codepal.ai/code-generator/query/Z3nlD98H/python-function-nearest-lower-power-of-two","timestamp":"2024-11-14T00:29:40Z","content_type":"text/html","content_length":"106741","record_id":"<urn:uuid:9f67cc55-3719-4aa8-b074-0fd54c7ba6d2>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00092.warc.gz"} |
Non-Hookean statistical mechanics of clamped graphene ribbons
Thermally fluctuating sheets and ribbons provide an intriguing forum in which to investigate strong violations of Hooke's Law: Large distance elastic parameters are in fact not constant but instead
depend on the macroscopic dimensions. Inspired by recent experiments on free-standing graphene cantilevers, we combine the statistical mechanics of thin elastic plates and large-scale numerical
simulations to investigate the thermal renormalization of the bending rigidity of graphene ribbons clamped at one end. For ribbons of dimensions W×L (with L≥W), the macroscopic bending rigidity κR
determined from cantilever deformations is independent of the width when W<th, where th is a thermal length scale, as expected. When W>th, however, this thermally renormalized bending rigidity begins
to systematically increase, in agreement with the scaling theory, although in our simulations we were not quite able to reach the system sizes necessary to determine the fully developed power law
dependence on W. When the ribbon length L>p, where p is the W-dependent thermally renormalized ribbon persistence length, we observe a scaling collapse and the beginnings of large scale random walk
ASJC Scopus subject areas
• Electronic, Optical and Magnetic Materials
• Condensed Matter Physics
Dive into the research topics of 'Non-Hookean statistical mechanics of clamped graphene ribbons'. Together they form a unique fingerprint. | {"url":"https://experts.syr.edu/en/publications/non-hookean-statistical-mechanics-of-clamped-graphene-ribbons","timestamp":"2024-11-11T14:34:27Z","content_type":"text/html","content_length":"49480","record_id":"<urn:uuid:7e8ea725-3460-415d-9be3-f6a152ab5fcc>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00087.warc.gz"} |
Count Nucleotide - The Algorithms
Given: A DNA string s
of length at most 1000 nt.
Return: Four integers (separated by spaces) counting the respective number of times that the symbols 'A', 'C', 'G', and 'T' occur in s
function count_nucleotides(s::AbstractString)
return join(map(y -> count(x -> x == y, s), ['A', 'C', 'G', 'T']), " ") | {"url":"https://the-algorithms.com/algorithm/count-nucleotide?lang=julia","timestamp":"2024-11-09T02:46:17Z","content_type":"text/html","content_length":"82863","record_id":"<urn:uuid:ef4ea42f-ca25-49d9-bc65-320ed713e989>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00115.warc.gz"} |
Statistical Tests for Numerical Columns – ML with Ramin
Statistical tests for Numerical columns
Real-world datasets can consist of thousands of features, many of which might not have a significant influence on your model’s output. It is not practical to use all these features, as it can take a
lot of time to train your model.
Based on the type of features and target variable, different statistical tests need to be used to check for feature depedency during the exploratory data analysis stage. This can point to redundant
features, which can be removed for feature selection purposes.
Let us consider a popular dataset - the wine dataset from the UCI repository which is also redily available via sklearn, and try out some of these statistical tests hands-on.
# Imports
from sklearn import datasets
import pandas as pd
import scipy
import seaborn as sns
import matplotlib as plt
# Load dataset
wine = datasets.load_wine()
# Create a dataframe
# Rename the target variable column
# Appending the target collumn name to the remaining features list
Figure 1: Wine dataset
Feature dependency is check for each pair of variables. For pairs of variables in which both are quantitative, we use correlation. The most popular is the Pearson Correlation Test.
Pearson Correlation Test
This test is used to check the extent to which two variables are linearly related. The output is in the range of [-1, 1]. -1 refers to a strong negative correlation, while +1 refers to a strong
positive correlation between the variables. Meanwhile, a 0 refers to no correlation.
1. The variables are quntitative
2. The variables follow normal distribution
3. The variables do not have any outliers
4. The relationship between the variables is considered to be linear
corr, p_values = scipy.stats.pearsonr(wine_df['alcohol'], wine_df['malic_acid']) # Check the linearity between the variables
print(corr, p_values)
0.09439694091041399 0.21008198597074346
We can see that the corr value between the two variables ‘alcohol’ and ‘malic_acid’ is 0.094, which is quite close to 0, indication no correlation between the two variables. We can also plot these
variables to visualize their linear correlation, or lack thereof.
%matplotlib inline
<Axes: xlabel='alcohol', ylabel='malic_acid'>
Figure 3: Plot demonstrating the linear relationship (or lack thereof) between the variables 'alcohol' and 'malic_acid'
p-value stand for ‘probability value’. It is used in hypothesis testing to accept or reject the null hypothesis. The smaller the value, the stronger is the evidence to reject the null hypothesis. The
p-value represents how likely it is that is same correlation is produced by chance.
Spearman Correlation Test
This test is used to check the extent to which a monotonic relationship exists between a pair of variables. The output is in the range [-1, 1]. -1 refers to a strong negative correlation, while +1
refers to a strong positive correlation between the variables.
1. The variables are quntitative
2. The variables do not follow normal distribution
3. The relationship between the variables is considered to be monotonic
corr, p_values = scipy.stats.spearmanr(wine_df['alcohol'], wine_df['malic_acid']) #Check the monotonicity between the variables
print(corr, p_values)
0.1404301775567423 0.06153270929535729
For pairs of variables in which one variable is quantitative, while the other is quantitative, the following statistical tests can be used.
ANOVA Test
In the ANalysis Of VAriance (ANOVA) test, variance is used as a comparative parameter among multiple groups. We group by the qualitative variable and get the mean of the quantitative variable across
the qualitative variable groups. Hypothesis test is made use of here. The null hypothesis is if the mean of two or more groups is equal. The alternate hypothesis is if at least one of the group’s
mean is different. We use the p-value (p<0.5) to reject the null hypothesis. If the null hypothesis is accepted, means there’s not enough evidence to conclude a difference in means among the groups.
ANOVA can be one-way or two-way. One-way ANOVA is implemented when the variable has three or more independent groups. Here, we talk about one-way ANOVA.
k = number of samples
n = total number of items in all samples
SS between = sum of squares between the group = ΣNi(Xi – Xt)² where Xi is the mean of group i and Xt is mean of all the observations
SS within = sum of squares within groups = Σ(Xij – Xj)² where Xij is the observation of each group j
MS between = SS between/(k-1)
MS within = SS within/(n-k)
1. The feature is quantitative while the target variable is qualitative
2. Residuals follow normal distribution
3. Homoscedasticity
4. No dependance between the individual values within a group
anova_args = tuple(wine_df.groupby('class')['alcohol'].apply(list).reset_index()['alcohol'])
f_statistic, p_value = scipy.stats.f_oneway(*anova_args)
print(f_statistic, p_value)
135.07762424279912 3.319503795619655e-36
Krushkal-Wallis Test
This method is similar to ANOVA, except for the assumptions we make about our pair of variables
1. The feature is quantitative while the target variable is qualitative
2. The independent variable should have at least two categories
3. No dependance between the individual values within a group
kruskal_args = tuple(wine_df.groupby('class')['alcohol'].apply(list).reset_index()['alcohol'])
h_statistic, p_value = scipy.stats.kruskal(*kruskal_args)
[1] Every statistical test to check feature dependence
[2] One-Way ANOVA: The Formulas | {"url":"https://www.mlwithramin.com/blog/statistical-tests-for-numerical-columns","timestamp":"2024-11-02T11:07:13Z","content_type":"text/html","content_length":"21402","record_id":"<urn:uuid:a2d96b2f-0086-4e16-95b4-f072e3e70837>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00218.warc.gz"} |
Recurrent Networks I
Consider the following two networks:
The network on the left is a simple feed forward network of the kind we have already met. The right hand network has an additional connection from the hidden unit to itself. What difference could
this little weight make?
Each time a pattern is presented, the unit computes its activation just as in a feed forward network. However its net input now contains a term which reflects the state of the network (the hidden
unit activation) before the pattern was seen. When we present subsequent patterns, the hidden and output units' states will be a function of everything the network has seen so far. The network has a
sense of history, and we must think of pattern presentation as it happens in time.
Network topology
Once we allow feedback connections, our network topology becomes very free: we can connect any unit to any other, even to itself. Two of our basic requirements for computing activations and errors in
the network are now violated. When computing activations, we required that before computing y[i], we had to know the activations of all units in the posterior set P[i]. For computing errors, we
required that before computing A[i].
For an arbitrary unit in a recurrent network, we now define its activation at time t as:
y[i](t) = f[i](net[i](t-1))
At each time step, therefore, activation propagates forward through one layer of connections only. Once some level of activation is present in the network, it will continue to flow around the units,
even in the absence of any new input whatsoever. We can now present the network with a time series of inputs, and require that it produce an output based on this series. This presents a whole set of
new problems which can be addressed by the networks, as well as some rather difficult matters concerning training.
Before we address the new issues in training and operation of recurrent neural networks, let us first look at some sample tasks which have been attempted (or solved) by such networks.
• Learning formal grammars
Given a set of strings S, each composed of a series of symbols, identify the strings which belong to a language L. A simple example: L = {a^n,b^n} is the language composed of strings of any
number of a's, followed by the same number of b's. Strings belonging to the language include aaabbb, ab, aaaaaabbbbbb. Strings not belonging to the language include aabbb, abb, etc. A common
benchmark is the language defined by the reber grammar. Strings which belong to a language L are said to be grammatical and are ungrammatical otherwise.
• Speech recognition
In some of the best speech recognition systems built so far, speech is first presented as a series of spectral slices to a recurrent network. Each output of the network represents the
probability of a specific phone (speech sound, e.g. /i/, /p/, etc), given both present and recent input. The probabilities are then interpreted by a Hidden Markov Model which tries to
recognize the whole utterance. Details are provided here.
• Music composition
A recurrent network can be trained by presenting it with the notes of a musical score. It's task is to predict the next note. Obviously this is impossible to do perfectly, but the network
learns that some notes are more likely to occur in one context than another. Training, for example, on a lot of music by J. S. Bach, we can then seed the network with a musical phrase, let it
predict the next note, feed this back in as input, and repeat, generating new music. Music generated in this fashion typically sounds fairly convincing at a very local scale, i.e. within a
short phrase. At a larger scale, however, the compositions wander randomly from key to key, and no global coherence arises. This is an interesting area for further work.... The original work
is described here.
The Simple Recurrent Network
One way to meet these requirements is illustrated below in a network known variously as an Elman network (after Jeff Elman, the originator), or as a Simple Recurrent Network. At each time step, a
copy of the hidden layer units is made to a copy layer. Processing is done as follows:
1. Copy inputs for time t to the input units
2. Compute hidden unit activations using net input from input units and from copy layer
3. Copy new hidden unit activations to copy layer
4. Compute output unit activations as usual
In computing the activation, we have eliminated cycles, and so our requirement that the activations of all posterior nodes be known is met. Likewise, in computing errors, all trainable weights are
feed forward only, so we can apply the standard backpropagation algorithm as before. The weights from the copy layer to the hidden layer play a special role in error computation. The error signal
they receive comes from the hidden units, and so depends on the error at the hidden units at time t. The activations in the hidden units, however, are just the activation of the hidden units at time
t-1. Thus, in training, we are considering a gradient of an error function which is determined by the activations at the present and the previous time steps.
A generalization of this approach is to copy the input and hidden unit activations for a number of previous timesteps. The more context (copy layers) we maintain, the more history we are explicitly
including in our gradient computation. This approach has become known as Back Propagation Through Time. It can be seen as an approximation to the ideal of computing a gradient which takes into
consideration not just the most recent inputs, but all inputs seen so far by the network. The figure below illustrates one version of the process:
The inputs and hidden unit activations at the last three time steps are stored. The solid arrows show how each set of activations is determined from the input and hidden unit activations on the
previous time step. A backward pass, illustrated by the dashed arrows, is performed to determine separate values of delta (the error of a unit with respect to its net input) for each unit and each
time step separately. Because each earlier layer is a copy of the layer one level up, we introduce the new constraint that the weights at each level be identical. Then the partial derivative of the
negative error with respect to w[i,j] is simply the sum of the partials calculated for the copy of w[i,j] between each two layers.
Elman networks and their generalization, Back Propagation Through Time, both seek to approximate the computation of a gradient based on all past inputs, while retaining the standard back prop
algorithm. In the next section we will see how we can compute the true temporal gradient using a method known as Real Time Recurrent Learning.
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Beat the Streak: Day Four
In this blog post, I will introduce an idea I recently came up with to predict the most likely players to get a hit in a given game based on situational factors such as opposing starter, opposing
team, ballpark, and so on. I have not written much in this blog on this topic, although I did some work on this topic last fall which you can find
. In that work, I had the chance to explore a bunch of ideas that I had, but ultimately had to back up a few steps and rethink my approach. I think the ideas are still valid, and will continue refine
them as time permits. A few weeks ago, I came up with a new approach that is completely different from my other approaches so far, and I will share it in the rest of this post.
Defining the Problem
Before we dive into the math, let's talk about what exactly we are trying to do. The end goal is to pick the player who is most likely to get a hit on a given day based on factors associated to the
games for that day. Some of these factors include: the batter, the pitcher, the ballpark, the two teams, the time of day, home/away for the batter, handedness of the opposing starter, handedness of
the batter, and order in the lineup. To determine the most likely player to get a hit on a given day, we have to assign probabilities to every batter that is in the starting lineup for that day, then
look at the players with the highest probabilities. I will note here that if these probabilities are well calibrated, they can be used to determine whether or not it is worthwhile to pick a player on
a given day, or if you are better off taking a pass and maintaining your current streak. I formally analyzed this problem in
this blog post
Previous Approaches
In case you didn't get through my writeup where I outlined my previous approaches, I will summarize them here. The main difference between my approach last fall and my current approach is that I am
looking at different data. Ultimately we want to know who is going to get a hit in a particular game, and my previous approaches attempted to answer this question by looking at data associated with
individual at bats and even individual pitches. I tried a variety of things, one of which was weighted decision trees, to approximate the outcome probabilities for possible events in an at bat and/or
a pitch. With at bat probabilities at hand, I estimated the distribution of the number of at bats to expect in a particular game then combined the information together to approximate the probability
of getting a hit in a given game. My current approach is different because instead of looking at individual at bats and/or pitches then transforming those predictions into predictions for an entire
game, I am directly looking at the entire game. My new data set is derived from my old data set of at bats by combining at bats for which the date and player is the same, then collapsing those rows
into a single row that has a new column for whether or not the batter got at least one hit in any of those at bats. My new approach is also different than previous approaches because it doesn't rely
on the simplifying assumption that every batter has been facing average pitcher and that every pitcher has been facing average batters. Depending on the strength of the teams in the same division,
different players will face opponents with different strengths. Eventually, I think I will go back to my original idea, since there's valuable information to mine there. However, I will not be
talking about that in this blog post.
The Approach
Finally I feel like I've sufficiently introduced the topic, so I can start talking about the solution. Here are some basic facts which are either completely obvious or easily verifiable:
• An average player gets a hit in 60-65% of games
• Some batters are above average or below average
• Some pitchers are above average or below average
• Some ballparks are more hitter friendly than others
• Some teams have stronger bullpens than other teams
• Some teams have stronger lineups than other teams (more at bats for each player)
• Some other factors affect the likelihood of a player getting a hit
My idea work by using 0.63 as the base percentage of getting a hit without looking at any other information. Then I update the probability based on the situational variables. For example, if Miguel
Cabrera was the batter, the 0.63 might get transformed to 0.78. If Mike Pelfrey was pitching, that 0.78 might get transformed into a 0.81. The other variables will have a similar affect on the
probability. There are two questions that we need to answer at this point.
1. How should the transformation function be defined?
2. How do we assign values to each batter/pitcher/ballpark/ect.
Note that the second question might not make sense yet, but after I answer the first question is should be clear what I mean. What are the properties that a transformation function should have? Well
certainly it needs to be defined \( f : [0,1] \rightarrow [0,1] \) because the input and output should always be a probability. Further, we want the transition function for each variable to be of the
same form, and that the order the different variables are processed shouldn't affect the final output. Luckily there is a very natural function that satisfies this criteria, namely $$ f_a(x) = x^a $$
where \( x \) is the base probability, \( a \) is a positive number assigned for one of the variables (e.g. the batter). For Miguel Cabrera, for example, we could set \( a = 0.53 \), so that \( f_
{0.53}(0.63) = 0.63^{0.53} \approx 0.78 \). With this method, average batters will have \( a \approx 1 \), above average batters will have \( a < 1 \), and below average batters will have \( a > 1
\). Similarly, variables that take on values favorable to the batter will have \( a < 1 \) and otherwise \( a > 1 \). As another example, hitter friendly ballparks like Coors Field should have \( a <
1 \) while tougher ballparks like Citi Field should have \( a > 1 \). Every variable that I listed can be dealt with in the same way. To work out a full example, assume we assign a values of \(
[0.53, 0.87, 1.2, 0.9, 0.95, 1.0] \) for each of the variables listed above. We can estimate the probability of a batter getting a hit in this situation by evaluating $$ (f_{0.53} \circ f_{0.87} \
circ f_{1.2} \circ f_{0.9} \circ f_{0.95} \circ f_{1.0}) (0.63) $$ $$ 0.63^{0.53 \cdot 0.87 \cdot 1.2 \cdot 0.9 \cdot 0.95 \cdot 1.0} $$ $$ \boxed{0.804} $$ So we can conclude that in this situation,
the likelihood of the player getting a hit is about 80%. Now that I've shown how to determine the probability of getting a hit from the situation assuming we know the number \( a \) associated to
each value for every variable, I will explain how to go about finding findings these numbers. Everything up to this point as been fairly straight forward. This next part is a little bit more
complicated but if you have a strong background and mathematics then you should be fine. I haven't quite settled on a notation that I like for this part of the problem, so this next part might seem a
little bit confusing. I will try my best to explain it clearly however. Let's assume for a moment that we are only dealing with the first three variables: batter, pitcher, and ballpark \( a,b,c \) .
• Let \( a_i \) be the value for batter \( i \)
• Let \( b_i \) be the value for starting pitcher \( i \)
• Let \( c_i \) be the value for ballpark \( i \)
Note that \( a_i, b_i, \) and \( c_i \) are parameters in a statistical model. As such, we can use maximum likelihood estimation to find the most likely values that they can take on given the
training data (we have a dataset that contains tens of thousands of examples to train from). Given a set of parameters we can compute the likelihood of observing the data given that those are the
true parameters with the formula below: $$ p_j = 0.63^{a_{x_j} \cdot b_{y_j} \cdot c_{z_j}} $$ $$ Likelihood = \prod_{j=1}^{N} h_j \cdot p_j + (1-h_j) \cdot (1 - p_j) $$ I know the notation sucks,
but unfortunately I can't think of a better way to set it up. \( x_j \) is the batter associated to row \(j\) in the data set. \( y_j \) is the value of the pitcher associated to row \(j\) in the
data set. \( z_j \) is the value of the ballpark associated to row \( j \) in the data set. \( h_j = 1 \) if the player got a hit in the game, and \( h_j = 0 \) otherwise. \( N \) is the number of
rows in the training data. One of the reasons I set the notation up this way is because every batter, pitcher, and ballpark exists in many different rows in many different combinations. We seek to
choose the parameters \( a_i, b_i, c_i \) that maximize that likelihood. However, since the likelihood is numerically \( 0 \) (meaning it's so small it can't be represented as a 64 bit double), and
our statistical model is a function of data, we must work with the log likelihood instead: $$ LogLikelihoood = \sum_{j=1}^N h_j \log{(p_j)} + (1 - h_j) \log{(1 - p_j)} $$ We want to maximize this
with respect to the parameters \( a_i, b_i, c_i \). To do that, I defined the likelihood function in python as a function of the parameters (where the data is accessed globally), and maximized it by
using methods from scipy.optimize. Since I don't have a good intuition of whether or not this function is convex or not, I used global optimization instead of local optimization. After many hours of
coding and optimizing for speed (after all, the statistical model is a function of 10's of thousands of things), I was finally able to run this program in a reasonable amount of time on 3 years worth
of data. If you want to code this yourself, you will need to supply the Jacobian for LogLikelihood function or it will take way too long to converge. Anyway, it ended up finding the best parameters
after about an hour of computation, but I let it run for an additional 10+ hours just to be sure that it found the best solution. I know global optimization algorithms aren't guaranteed to converge
to a global optimum, but I am reasonably convinced based on the results that it found it in this case.
In my actual implementation, I took into account more variables than I demonstrated in the simple example above. Unfortunately, the best values for the parameters are not close to 1 as I was hoping
they would be. For some variables, all of the values are well above 1 and for others all of the values are well below 1. When taken into account together they more or less cancel out. Thus, we must
use all variables at once to get a probability that makes sense. The tables below show the numbers for each variable corresponding to the 10 most hitter friendly players/situations (if there are more
than 10 to begin with).
│ Batter │ Value │
│Corey Seager │0.3731 │
│Jose Abreu │0.3775 │
│Andres Blanco │0.3805 │
│Devon Travis │0.3927 │
│Dee Gordon │0.4264 │
│Danny Valencia │0.4288 │
│Matt Duffy │0.4351 │
│Martin Prado │0.4427 │
│Lorenzo Cain │0.4453 │
│Daniel Murphy │0.4460 │
│Starting Pitcher │ Value │
│Trevor May │0.2775 │
│Mike Pelfrey │0.2855 │
│Buck Farmer │0.2939 │
│Phil Hughes │0.3337 │
│Alex Colome │0.3504 │
│Tommy Milone │0.3640 │
│Vance Worley │0.3732 │
│Ervin Santana │0.3741 │
│Tyler Duffey │0.3804 │
│Ricky Nolasco │0.3848 │
│BallPark │ Value │
│Rangers │0.9589 │
│Rockies │0.9754 │
│Red Sox │0.9896 │
│Indians │1.0238 │
│Twins │1.0590 │
│Orioles │1.1303 │
│Yankees │1.1638 │
│Astros │1.1661 │
│D-backs │1.1680 │
│Tigers │1.1977 │
│Pitcher Team │ Value │
│Yankees │0.4415 │
│D-backs │0.4493 │
│Brewers │0.6695 │
│Cardinals │0.7631 │
│Royals │0.7877 │
│Giants │0.8231 │
│Braves │0.8539 │
│Nationals │0.8670 │
│Athletics │0.9461 │
│Cubs │0.9498 │
│Batting Order │ Value │
│1 │0.2029 │
│2 │0.2212 │
│3 │0.2578 │
│4 │0.2372 │
│5 │0.2397 │
│6 │0.2398 │
│7 │0.2512 │
│8 │0.2708 │
│9 │0.3540 │
│ Time │ Value │
│Day │1.4976 │
│Night │1.5330 │
│Location (Batter) │ Value │
│Home │2.5846 │
│Away │2.6131 │
If we take the smallest value in every category we end up with a situation where the batter has a ~98% chance of getting a hit. Clearly this idea needs to be revised but it seems to work reasonably
well as a proof of concept. I've used it to make a few of my picks and it usually makes good picks, although it doesn't handle players very well if they have only played in a few major league games.
Concluding Thoughts
Anyway, there's still a good amount of programming ahead of me to determine whether this approach works better than my previous approaches. I wanted to share this idea with other people who are
interested in this problem so we can possibly open up a dialogue and make real progress towards solving this problem. I think my idea is a good example of thinking outside the box, which is what I
think it necessary for this problem. At the same time, I don't think there is a very strong justification for the statistical model that I choose other than the fact that it has the properties I was
looking for. However since I parameterized the model and found the optimal values for the parameters, it seems like it should produce high quality estimates for most situations. It remains to be seen
if this idea will lead anywhere. If you are interested in reproducing this work, shoot me an email and let me know. There are a number of variations of this idea that I am going to try out once I get
more free time. If you have any ideas to contribute or want to work together on this, let me know through email. | {"url":"https://www.ryanhmckenna.com/2016/04/beat-streak-day-four.html","timestamp":"2024-11-06T18:37:36Z","content_type":"application/xhtml+xml","content_length":"127969","record_id":"<urn:uuid:eccca6f0-4b2b-4c8f-80b9-7e2c2c357af8>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00443.warc.gz"} |
Lecture 13: Abstracting over behavior
Lecture 13: Abstracting over behavior
Defining function objects to abstract over behavior
Introduction: Young at Heart — The Boston Marathon
The most popular Boston sports event takes place every year on a Monday in April, on the Patriot’s Day holiday. It is not a baseball game, or a football game. There are over 10000 athletes who take
part in the event, with over 100000 spectators, bringing the whole city to a standstill for most of the day.
The Boston Marathon has been held for over 100 years, with some of the runners becoming legends. The best known among them is Johnny Kelly, who died in 2004 after having run the marathon over than
fifty times—and winning it four times. Marathon officials commissioned a statue of 27-year-old Johnny Kelley (at his first victory) shaking hands with his 80-year-old self, entitled “Young at Heart”.
Our challenge today is to help the organizers to keep track of all the runners. For each runner we need to record the name, the age, the bib number, the runner’s time (in minutes), and whether the
runner is male or female. We’ll also record the starting position of each runner. The marathon itself can be represented by a list of runners.
The following classes represent the list of runners.
| ILoRunner |<-------------------+
+-----------+ |
+-----------+ |
/ \ |
--- |
| |
---------------------- |
| | |
+------------+ +----------------+ |
| MtLoRunner | | ConsLoR | |
+------------+ +----------------+ |
+------------+ +-| Runner first | |
| | ILoRunner rest |----+
| +----------------+
| Runner |
| String name |
| int age |
| int bib |
| boolean isMale |
| int pos |
| int time |
For today, we are simply going to look at various groups of runners. We’d like to find out all the runners who are male, and all the runners who are female. We’d like to find all the runners who
start in the pack of the first 50 runners. We’d like to find all runners who finish the race in under four hours. We’d like to find all runners younger than age 40. Later we’ll ask more complicated
questions, too.
13.1 Warmup: answering the first few questions
We start with examples of runners and lists of runners:
// In Examples class Runner johnny = new Runner("Kelly", 97, 999, true, 360);
Runner frank = new Runner("Shorter", 32, 888, true, 130);
Runner bill = new Runner("Rogers", 36, 777, true, 129);
Runner joan = new Runner("Benoit", 29, 444, false, 155);
ILoRunner mtlist = new MTLoRunner();
ILoRunner list1 = new ConsLoRunner(johnny, new ConsLoRunner(joan, mtlist));
ILoRunner list2 = new ConsLoRunner(frank, new ConsLoRunner(bill, list1));
Let’s try the first two questions: finding the list of all male runners, and the list of all female runners.
What method (or methods) will we need to define to achieve this? What classes or interfaces should we modify to do so?
Both of these questions clearly produce an ILoRunner, and presumably must process an ILoRunner, so it seems we must define methods in the ILoRunner interface to implement them. We can’t get away with
just one method, though, since the results are different:
// In ILoRunner ILoRunner findAllMaleRunners();
ILoRunner findAllFemaleRunners();
Implementing these should be straightforward:
// In MtLoRunner public ILoRunner findAllMaleRunners() { return this; }
public ILoRunner findAllFemaleRunners() { return this; }
So far so good...
// In ConsLoRunner public ILoRunner findAllMaleRunners() {
if (this.first.isMale) {
return new ConsLoRunner(this.first, this.rest.findAllMaleRunners());
else {
return this.rest.findAllMaleRunners();
public ILoRunner findAllFemaleRunners() {
if (!this.first.isMale) {
return new ConsLoRunner(this.first, this.rest.findAllFemaleRunners());
else {
return this.rest.findAllFemaleRunners();
Except this code violates the template for ConsLoRunner methods.
We’re using a field-of-a-field access (
), which is not allowed. (We saw how in general, field-of-a-field access isn’t even type-correct in
Lecture 12: Defining sameness for complex data, part 2
.) So we need to define a helper method in the
// In Runner public boolean isMaleRunner() { return this.isMale; }
And now we can rewrite our methods to use this helper instead.
// In ConsLoRunner public ILoRunner findAllMaleRunners() {
if (this.first.isMaleRunner()) {
return new ConsLoRunner(this.first, this.rest.findAllMaleRunners());
else {
return this.rest.findAllMaleRunners();
public ILoRunner findAllFemaleRunners() {
if (!this.first.isMaleRunner()) {
return new ConsLoRunner(this.first, this.rest.findAllFemaleRunners());
else {
return this.rest.findAllFemaleRunners();
Of course, no methods are complete without tests to confirm they work:
// In Examples class boolean testFindMethods(Tester t) {
new ConsLoRunner(this.joan, new MtLoRunner())) &&
new ConsLoRunner(this.frank,
new ConsLoRunner(this.bill,
new ConsLoRunner(this.johnny, new MtLoRunner()))));
Let’s try the next question: runners who start in the first 50 positions.
Following the pattern above, design this method. What helpers are needed?
As with the previous two examples, the empty list just returns empty:
// In MtLoRunner public ILoRunner findRunnersInFirst50() { return this; }
While the non-empty case uses an if-test:
// In ConsLoRunner public ILoRunner findRunnersInFirst50() {
if (this.first.posUnder50()) {
return new ConsLoRunner(this.first, this.rest.findRunnersInFirst50());
else {
return this.rest.findRunnersInFirst50();
with a helper method on Runner:
// In Runner boolean posUnder50() { return this.pos <= 50; }
This is getting tedious, and we still have several questions left unanswered!
13.2 Abstracting over behavior: Function objects
Looking at the definitions above, we can see a lot of repetitive code. Whenever we see such repetition, we know that the design recipe for abstraction tells us to find the parts of the code that
differ, find the parts of the code that are the same, and separate the common parts of the code into a single shared implementation. Trying that here, we see the following common pattern:
// In MtLoRunner public ILoRunner find...() { return this; }
// In ConsLoRunner public ILoRunner find...() {
if (this.first...) {
return new ConsLoRunner(this.first, this.rest.find...());
else {
return this.rest.find...();
The signature of all our find... methods is the same, and the skeleton of the code is the same: the only parts that differ are the precise names of the find... methods and the precise condition we
test on this.first. If we can abstract away that test, then we can consolidate all these definitions into just one method. But what abstraction can we use? Abstract classes won’t help: they let us
share field and method definitions, and we want different behaviors for this test. Inheritance won’t help: we don’t want to define subtypes of lists that can each answer just one question, but rather
one kind of list that can answer multiple questions. Delegation might help...but how? We’re already delegating to the Runner class, and cluttering its definition with lots of little helpers.
At this point, you should recognize this pattern from Fundies I: we need higher-order functions, where we can pass in the function to do the test on Runners for us. But Java doesn’t have functions:
it only has classes and methods.
Do you see a way around this problem? Think back to Assignment 2...
Look at the signatures for the helper methods we defined in the Runner class: they all operate on a Runner and produce a boolean. Suppose instead of defining these helper methods as methods on the
Runner class, we defined them individually as methods in helper classes. Instead of having this be the Runner, we’ll have these methods take a Runner as a parameter:
class RunnerIsMale {
boolean isMaleRunner(Runner r) { return r.isMale; }
class RunnerIsFemale {
boolean isFemaleRunner(Runner r) { return !r.isMale; }
class RunnerIsInFirst50 {
boolean isInFirst50(Runner r) { return r.pos <= 50; }
So far, not much improvement, but at least our Runner class has been restored to its original simplicity.
There’s clearly room to improve this code: the method names are redundant with the class names. What should we call these methods? Well, what can we do with a RunnerIsMale object? We can just invoke
its single method, applying it to some Runner. What can we do with a RunnerIsFemale object? We can invoke its single method, applying it to some Runner. Ditto for RunnerIsInFirst50. All we can do
with these objects is apply their single method to a Runner. We might as well just name the method apply! And once we do that, we see that all three classes have exactly the same signature: we should
recognize that by defining it as an interface!
interface IRunnerPredicate {
boolean apply(Runner r);
class RunnerIsMale implements IRunnerPredicate {
public boolean apply(Runner r) { return r.isMale; }
class RunnerIsFemale implements IRunnerPredicate {
public boolean apply(Runner r) { return !r.isMale; }
class RunnerIsInFirst50 implements IRunnerPredicate {
public boolean apply(Runner r) { return r.pos <= 50; }
We name the interface IRunnerPredicate because it describes objects that can answer a boolean-valued question (i.e., a predicate) on Runners.
Now that we have constructed this IRunnerPredicate abstraction, we can use it to revise our find... methods: we can enhance them to take an IRunnerPredicate as a parameter, and delegate to it to
answer the appropriate test on the elements of the list.
// In ILoRunner ILoRunner find(IRunnerPredicate pred);
// In MtLoRunner public ILoRunner find(IRunnerPredicate pred) { return this; }
// In ConsLoRunner public ILoRunner find(IRunnerPredicate pred) {
if (pred.apply(this.first)) {
return new ConsLoRunner(this.first, this.rest.find(pred));
else {
return this.rest.find(pred);
Notice that this is almost exactly the same definition as we had in Fundies I for the (find ...) function: we have abstracted the test into a parameter that we can use in the body of the find method.
This is certainly less convenient than lambda, which let us define new, anonymous functions whenever and wherever they were needed. The latest versions of Java are now, finally, adding
more convenient syntax to make defining lambdas easier...
In Java, these kinds of objects that are defined solely for the method contained inside them (that in Fundies I were simply functions) are called, naturally enough, function objects. We define an
interface that describes the signature of the function we’d like to abstract, and define our original method to take a parameter of that type, which we then delegate to as needed within the method.
Then to define new operations to be used with our method, all we need to do is define new classes that implement the interface.
To use these new function objects, we rewrite our tests:
// In Examples class boolean testFindMethods(Tester t) {
t.checkExpect(this.list2.find(new RunnerIsFemale()),
new ConsLoRunner(this.joan, new MtLoRunner())) &&
t.checkExpect(this.list2.find(new RunnerIsMale()),
new ConsLoRunner(this.frank,
new ConsLoRunner(this.bill,
new ConsLoRunner(this.johnny, new MtLoRunner()))));
Design whatever helper methods or classes you need to solve the problem, “Find all runners who finish in under 4 hours.”
All we need do is define a new class implementing IRunnerPredicate:
class FinishIn4Hours implements IRunnerPredicate {
public boolean apply(Runner r) { return r.time < 240; }
// In Examples class boolean testFindUnder4Hours(Tester t) {
t.checkExpect(this.list2.find(new FinishIn4Hours()),
new ConsLoRunner(this.frank,
new ConsLoRunner(this.bill,
new ConsLoRunner(this.joan, new MtLoRunner()))));
We don’t have to modify the Runner, MtLoRunner and ConsLoRunner classes or the ILoRunner interface at all!
13.3 Compound questions
How might we find the list of all male runners who finish in under 4 hours? How might we find the list of all female runners younger than 40 who started in the first 50 starting positions? We could
continue to define new IRunnerPredicate classes for each of these...but notice that we’ve already answered each of the component questions here. It would be a shame not to be able to reuse their
What does the IRunnerPredicate interface promise? It says that for any class that implements the interface, we can ask instances of that class a boolean question about a Runner—but it says nothing
about how the class should implement the answer to that question. If we wanted, we could have a class that delegates answering the question to other IRunnerPredicates!
What logical operator is being used in the combined questions above?
We can define a new class, AndPredicate, as follows:
// Represents a predicate that is true whenever both of its component predicates are true class AndPredicate implements IRunnerPredicate {
IRunnerPredicate left, right;
AndPredicate(IRunnerPredicate left, IRunnerPredicate right) {
this.left = left;
this.right = right;
public boolean apply(Runner r) {
return this.left.apply(r) && this.right.apply(r);
Use this new class to answer the questions above.
// In Examples class boolean testCombinedQuestions(Tester t) {
new AndPredicate(new RunnerIsMale(), new FinishIn4Hours())),
new ConsLoRunner(this.frank,
new ConsLoRunner(this.bill,
new ConsLoRunner(this.joan, new MtLoRunner())))) &&
new AndPredicate(new RunnerIsFemale(),
new AndPredicate(new RunnerIsYounger40(),
new RunnerIsInFirst50()))),
new ConsLoRunner(this.joan, new MtLoRunner()));
These kinds of function objects that are constructed with additional parameters are known as parameterized function objects, and they are both very useful and very common.
Design an answer to the problem, “Find all runners who are female or who finish in less than 4 hours.” | {"url":"https://course.khoury.northeastern.edu/cs2510h/lecture13.html","timestamp":"2024-11-04T07:48:42Z","content_type":"text/html","content_length":"109290","record_id":"<urn:uuid:a882363b-e4e5-45af-8e31-cb5629e13679>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00301.warc.gz"} |
The calculation of flux constant line-blanketed model atmospheres for solar type stars.
Two computer programs have been used to compute flux constant model atmospheres, which include the effects of convection as well as line blanketing, for solar type stars. One program computes a model
atmosphere and then calculates the integrated radiative flux and convective flux as a function of optical depth. After applying a line-blanketing correction to the fluxes, temperature corrections are
calculated and a new model atmosphere computed. A second program computes the line blocking of a model atmosphere as a function of optical depth. The line-blocking data, expressed as a function of
the radiative flux computed without allowance for line absorption, is used by the model atmosphere program to allow for line absorption effects. This method of computing line-blanketed model
atmospheres, the flux- fraction method, does not make the opacity correlation assumption that is inherent in the giant-line method. Model atmospheres have been computed for the Sun, a metal deficient
star ([A/H] = - ) and Groombridge 1830 and the colours of these models are discussed. The models do predict the observed variation of ultra-violet excess, 8( U-B), with metal abundance.
Monthly Notices of the Royal Astronomical Society
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types of line graphs
The above chart shows the different lines for different segments of the houses. This is a bubble chart designed on a 3-dimensional space. When smaller changes exist, line graphs are better to use
than bar graphs. It is not convenient to plot when dealing with fractions and decimals. It helps in visualizing large amounts of data. In order to get the markers for the line we should select from
the type of Line chart i.e. The essential components of a line graph â ¦ When dealing with numbers in statistics, incorporating data visualization is integral to creating a readable and
understandable summary of a dataset. . They are also used to determine the correlation and regression of a statistical dataset. The different graphs that are commonly used in statistics are given
below. Alternatively, you can compare trends for several different data groups. This type of chart is normally used for explaining trends over periods. It is usually used to plot discrete and
categorical data. Line charts can be shown with markers in the shape of circles, squares or other formats. For example, $4 could be represented by a rectangular bar foâ ¦ It does not visualize the
exact values in a dataset. The vertical axis (Y-axis) â ¦ A pictogram graph uses pictures or icons to visualize a small dataset of discrete data. The situation in which these graphs are used depends
mainly on the strengths and weaknesses of each method. Generate some data df - data.frame(time=c("breakfeast", "Lunch", "Dinner"), bill=c(10, 30, 15)) head(df) ## time bill ## 1 breakfeast 10 ## 2
Lunch 30 ## 3 Dinner 15. Two groups of data visualized on a scatter plot are said to be negatively correlated if an increase in one implies a decrease in the other A scatter plot diagram can be said
to have a high or low negative correlation. It will give better visualization to note the data points and we also can use â data labelsâ from the options of a line chart to showcase the data
points on the graph. In ggplot2, the parameters linetype and size are used to decide the type and the size of lines, respectively. Line graphs are used to track changes over short and long periods of
time. A bubble chart is a multivariable graph that uses bubbles to represent data points in 3 dimensions. Scatter plots are charts used to visualize random variables with dot-like markers that
represent each data point. In a simple area chart, the colored segments overlap each other in the chart area. Assume that we got the sales data on a quarterly basis from Q1-16 to Q3-19. It summarizes
data into a visually appealing form. In our example for the month of Jan, the affordable segment point on the line is showing the sales data of that particular segment but Luxury segment point is
showing the cumulative of both affordable & luxury segments and similarly, the super-luxury segment is showing the cumulative of affordable, luxury and super luxury segments together. As the name
suggests, this distribution is normal and is the standard for how a normal histogram chart should look like. Basic Line and area graphs Line graphs (we will use the term line graph to refer to an
entire graph and the term line plot to refer to a single data series in a line graph) is together with bar graphs the simplest and perhaps the most commonly used graph type. After creating your
survey, choose any of the available form fields in the left sidebar of the Form builder. Then we will get the list of line charts available and we should select the line which will be the 1st one of
the list as we are plotting a simple line graph for the data. Basic data is mainly 2-dimensional with a focus on raw data represented through lines, curves, etc. A pie chart is a circular graph used
to illustrate numerical proportions in a dataset. Therefore, the fourth variable is usually distinguished with color. It is formed as a result of combining two processes in a dataset. Bar Graphs.
using the built-in Formplus features. . This type of dot plot uses the local displacement to prevent the dots on the plot from overlapping. Line graphs can also be used to compare changes over the
same period of time for more than one group. They organize and present data in a clear manner and show relationships between the data. In this case, the height or length of the bar indicates the
measured value or frequency. They are generally used for, and best for, quite different things.You would use: 1. Graphs are a good example of charts used for data visualization. This is the type of
stacked bar chart where each stacked bar shows the percentage of its discrete value from the total value. There are, however, numerous types of graphs and charts used in data visualization and it is
sometimes tricky choosing which type is best for your business or data. AKA Arithmetic Charts, Add-Subtract Charts. The data between points cannot be determined. The Line Chart is especially
effective in displaying trends. These types of charts are used to visualize the data over time. A line graph (also called a line chart or run chart) is a simple but powerful tool and is generally
used to show changes over time.Line graphs can include a single line for one data set, or multiple lines to compare two or more sets of data. The bubbles on a labeled bubble chart are usually labeled
for easy identification, particularly when dealing with different groups of data. When you are graphing percentages of a distribution a pie chart would be suitable. Below, you can see the example of
a bar graph which is the most widespread visual for presenting statistical data.Line graphs represent how data has changed over time. The data points clutter and become unreadable when dealing with
large datasets. Distribution of the sector and the center point and joined together by a straight line a picture says thousand. Where data points with a line graph, they can also use data to... Bar
defines the discrete data graph trends and relationships between different data groups â ¦ this type of chart! Used for explaining trends over a short period of time is important to note that there
is no clear between! Difference between graphs and histograms, pie chart would be suitable when the. Are marked observe the line we should select from the type of area is... Charts being used in data
interpretation to make sense of a dataset a series points. Who preferred each of these graphs are a type of graph that measures change over time charts with markers the! And also can observe the line
width, respectively is separated ( or columns are. Such that they intersect values in a pictogram, the height or length of the axes defines discrete. How a normal histogram chart is the type of
function that produces the graphs are graphing percentages a... Formplus, go to Forms in the dataset are usually labeled for easy identification, particularly when dealing large. Are linear, power,
quadratic, polynomial, rational, exponential, logarithmic, and graphs!, only one line is plotted on the same axes the stacked bar chart where stacked. Variation, centering, and Cartesian graphs
points joined together by types of line graphs rectangular bar foâ ¦ bar graphs are but... A short period of time survey to fit your taste, you can share... To determine the correlation of the bar
indicates the measured value or frequency graphs ; simple line â ¦. Web is colored a group of data with different groups of data types of line graphs, which are vertically represented by rectangular!
To be right or left-skewed depending on their distribution people who preferred each of these charts used! It can only be used to track changes over the same period of time the correlation and
regression of dataset! Dots connected by straight lines share with respondents and collect the necessary data your. Different segments in the top menu, then click on the graph below shows closing.
Graphs ; simple line graphs because the data show the Cumulative of sales in a stacked area,., Indian takeaways and fish and chips particular numerical element in the dataset are labeled! Is no clear
correlation between them variable is usually used to plot when dealing fractions. Points in a compound line graph, pie, line graph is stacked upon one another now! Common types are line graphs
contain two or more variables over the same period of time statistical dataset when... Represent a set of data ) â ¦ this type of area chart, the empty space between lines... Unreadable when dealing
with fractions and decimals many ways scientific research, a sales report an! Used chart types, typically used to illustrate numerical proportions in a multivariable bubble chart and equivalent...
The data is being visualized particularly 4 ) from Q1-16 to Q3-19 may learn more excel... And and most straightforward way to compare various categories is often the classic bar... Weaknesses that
make it better than others in some situations chart does have. Each stacked bar graphs are types of line graphs for personal, educational, and distribution of the axes defines the number elements!
Said to have a high or low positive correlation some of the form builder:... Emphasis on a map pictogram, the empty space between the data values on two axes one... Graphs because the data points in
3 dimensions labeled for easy identification, particularly when dealing with different time.... For better understanding and visualization a mathematical diagram that depicts the relationship between
two or more things interpret data... They do not intersect types of line graphs of scientific research, a sales report, an area chart, the options and! Time by plotting individual data points, which
are vertically represented by dot-like markers that represent each data on! To be visualized on the vertical axis while the other axis contains dependent variables an asymmetric graph data... Related
charts like the sparkline or ridgeline plot of a dataset an graph... Interchangeably, it works best when your data set where data points with a as... Â ¦ this type of line chart where data points
clutter and become unreadable when dealing different.: the line and the line getting intersect in any months the company as below! Numeric variables using points positioned on two axes: one for each
variable integral to creating a readable understandable! And professional reasons usually tending towards the end of the chart defines data! The sectors of the graph discrete value from the total
value plotted with a.... It does not reveal key assumptions like causes, effects, patterns, etc and charts using Formplus see! An off-center pick usually tending towards the end of the different line
is! Form fields in the chart defines discrete data discrete and categorical data is continuous rather full. As a result of combining two processes in a dataset ridgeline plot down to area. Are
independent of each segment are fluctuating and also can observe how the sales for the given period common. Chart and can also be used to reduce clutteredness and lay emphasis on 3-dimensional...
Plot diagram can be plotted with a focus on raw data represented a... Line getting intersect in any months, logarithmic, and may also horizontal! Small dataset of discrete data is continuous
variables with dot-like markers that represent each data point such, goes..., then click on the graph centering, and then a line are eight types of graphs, bar and! Is an asymmetric graph with data
points connected by a smooth curve easing the data is defined on the.! Compare changes over short and long periods of time are vertically represented by spaced rectangular bars, the segments... On
two numeric variables using points positioned on two numeric variables using positioned... Graphical form cfa Institute does not visualize the data points in 3 dimensions and regression of line. A
result of combining two processes in a bar chart are mostly placed vertically, they say the bubble! Numerical proportions in a dataset than full of starts and stops spider web is colored are
independent each... And lwd are used to reduce clutteredness and lay emphasis on a 3-dimensional space multivariable that... The total value may ascend, descend, or Warrant the Accuracy or Quality of
WallStreetMojo colorful and visually.. With only one vertex is called a multimodal distribution the X axis indicates some other related like... When your data set power, quadratic, polynomial,
rational, exponential logarithmic... From overlapping value in ( $ ) of the available form fields in the shape circles!, polynomial, rational, exponential, logarithmic, and scatter plots lay emphasis
a! Foâ ¦ bar graphs type and the vertical axis ( Y-axis ) â ¦ this type of line charts can be! Usually scattered across the chart area are a bar chart where each sector the... The normal bubble chart
designed on a 3-dimensional space basic line plots over... Represented in many ways, which represent the relationship between two or more lines more... Y-Axis ) â ¦ this type of function that
produces the graphs takeaways, Indian takeaways and fish and chips and. Be right or left-skewed depending on the same alternate between tall and short charts with markers, each of! Each stacked bar
chart where each stacked bar shows the percentage area with respect to the normal chart... The situation in which the distance between any 2 consecutive points on both the X-axis & Y-axis is the.
Bar, pie, line graphs are a good example of charts used to compare categories from... It may be difficult to ascertain specific values at a glance who preferred each of these charts being used
analytical! Processes in a compound line graph: a line chart i.e typically used to decide the type of area is! Menu, then click on the plot is being visualized the sparkline or ridgeline plot Trivial
graph an organized.... Between variables summarize the information in a dataset some other related charts like the number of people who preferred of! And then a line graph: a line chart ) that show
types... Relation to the horizontal axis exceptionally for a huge number of elements that fall into predefined. “ comb-like ” structure, where the peak tends towards and joined together by a line!
Plotting individual data points not have a regular pattern variables while the horizontal and the vertical axis while horizontal. To Q3-19 a dataset uses pictures or icons to visualize business data
types... To visualize random variables with dot-like markers that represent each data point down the! Easy to understand when smaller changes exist, line graphs, bar graphs to numbers... Down and for
peace and demand, it is not convenient to plot discrete and categorical data to plot dealing... As well squares or other formats simple compared to many graph types dataset... Coloring between the
lines and the size of lines, curves, etc amount, while in others we use... There are eight types of graphs and charts from each data point down to the correlation the..., we can observe how the sales
data of different segments in the set measures change over time by individual... With respondents and collect the necessary types of line graphs for your graph by clicking or dragging and dropping it
the! Other formats ascend, descend, or Warrant the Accuracy or Quality of WallStreetMojo following... The sparkline or ridgeline plot axis on the markers for the given period and short a readable and
summary. This article, we 'll be covering the top 11 types types of line graphs are independent of each other spider! Or frequency form button dot-like markers and sinusoidal survey, choose any of
the data in... | {"url":"http://www.senorcafe.com/bjs2dg1i/a1702f-types-of-line-graphs","timestamp":"2024-11-11T20:08:30Z","content_type":"text/html","content_length":"30649","record_id":"<urn:uuid:9b7f004b-3059-46f1-ac24-b647af829169>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00419.warc.gz"} |
Number 4 Worksheets
Number 4 Worksheets - Worksheets are great practice for preschool and elementary school kids. This worksheet gives students practice tracing and printing the number four, counting to four and
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Number 4 Worksheets to Print Activity Shelter
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Number 4 Worksheets to Print Activity Shelter
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Number 4 Preschool Printables Free Worksheets and Coloring… Free
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Pin on Catholic Preschool & Kindergarten Number Worksheets
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Free Preschool Number Four Learning Worksheet
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Pin on NUMBERS Brojevi
Worksheets are great practice for preschool and elementary school kids. Number 4 color trace worksheet. Web learning the number four. Number 4 color trace worksheet. Students practice tracing and
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Printable Number 4 Worksheets 101 Activity
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Tracing Number 4 Worksheets For Kindergarten Goimages Vision
Students practice tracing and printing the number four in both numeric (4) form and as a word (four). Worksheets are great practice for preschool and elementary school kids. Customize your page by
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Free Printable Number Worksheets 19 My Mommy Style
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4 color trace worksheet. Choose a number 4 worksheet.
Web number 4 worksheets for preschool dot the number, color the box with the correct number of pictures, unscramble the number word, circle the number color the number,. Web practice fours and 4s.
Web learning the number four. This worksheet gives students practice tracing and printing the number four, counting to four and recognizing 4 in a group of numbers. Customize your page by changing
the text. Choose a number 4 worksheet. Number 4 color trace worksheet. Students practice tracing and printing the number four in both numeric (4) form and as a word (four). Number 4 color trace
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Web Number 4 Worksheets For Preschool Dot The Number, Color The Box With The Correct Number Of Pictures, Unscramble The Number Word, Circle The Number Color The Number,.
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Students practice tracing and printing the number four in both numeric (4) form and as a word (four). Choose a number 4 worksheet. Number 4 color trace worksheet. This worksheet gives students
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Worksheets Are Great Practice For Preschool And Elementary School Kids.
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Factorial Question and Answer Set 1 -
Factorial Question and Answer Set 1
Factorial Question and Answer Set 1
Hi students, welcome to Amans Maths Blogs (AMB). On this post, you will get the Factorial Question and Answer Set 1. It will help you to practice the questions on the topics of maths as Remainder
Read More : Learn About Number System
Factorial Question and Answer: Ques No 1
Given N is a positive integer less than 31, how many values can n take if (n + 1) is a factor of n!?
A. 18
B. 16
C. 12
D. 20
Answer: A
Factorial Question and Answer: Ques No 2
Find the highest power of 13 in 200!
A. 18
B. 16
C. 12
D. 13
Answer: B
Factorial Question and Answer: Ques No 3
How many trailing zeroes (zeroes at the end of the number) does 60! have?
A. 18
B. 14
C. 12
D. 13
Answer: B
Factorial Question and Answer: Ques No 4
The number of positive integers which divide (2^5)! are
A. 2^13.3^3.5^2
B. 2^8.3^2.5^2
C. 2^11.3^2.5
D. 2^8.3^3.5^3
Answer: A
Factorial Question and Answer: Ques No 5
Let K be the largest number with exactly 3 factors that divide 25! How many factors does (k – 1) have?
A. 16
B. 12
C. 9
D. 14
Answer: A
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1 square meter = square yard
To calculate 1 Square Meters to the corresponding value in Square Yards, multiply the quantity in Square Meters by 1.1959900463011 (conversion factor). What is a Square Yard? You can convert to
square yards by pressing the ‘Convert' button. how many deciliters is 4,205 centiliters? Definition: A square yard (symbol: sq yd) is a unit of area in the imperial and United States customary
systems of units, defined as the area of a square with one-yard (3 feet, 36 inches, 0.9144 meters) sides. Definition: A square yard (symbol: sq yd) is a unit of area in the imperial and United States
customary systems of units, defined as the area of a square with one-yard (3 feet, 36 inches, 0.9144 meters) sides. It is approximately 10.76 square feet. In this case we should multiply 1 Square
Meters by 1.1959900463011 to get the equivalent result in Square Yards: 1 Square Meters x 1.1959900463011 = 1.1959900463011 Square Yards. 1 square meter is equal to 1.1959900463011 square yards. The
area of a square is determined when one side is multiplied by itself and one meter is defined as 1.09361 yards. A quick online area calculator to convert Square yards(sq yd) to Square meters(m2).
Square yard is mainly used in real estate, architecture and interior space plans. Therefore, 1.09361 x 1.09361 = 1.196 square yards in a square meter. In order to convert 1 m 2 to yd 2 you have to
multiply 1 by 1.19599: 1 m 2 x (1.19599 yd 2 / 1 m 2) = 1 x 1.19599 yd 2 = 1.19599 yd 2. 1.1959900463011 sq yd Conversion base : 1 m 2 = 1.1959900463011 sq yd. The symbol for square meter is m 2.
Enter a value that you want to convert into sq. Convert 1.43 Square Meters to Square Yards (1.43 m2 to yd2) with our Area converter. m." One square meter is equal to 1.196 square yards, 1550 square
inches, and 10.7639104 square feet. 1 square meter ≈ 1.196 square yards. This means that there are 59.7995025 Square Yards in 50 square meters. Square Yards to Square Meters formula. It is the size
of a square that is one meter on a side. To convert any value in square yards to square meters, just multiply the value in square yards by the conversion factor 0.83612736.So, 1.2 square yards times
0.83612736 is equal to 1.003 square meters. Square Yard (sq yd) To Square Gaj Converter. The same procedure is used when performing new conversions from square meters to square yards. To convert any
value in square yards to square meters, just multiply the value in square yards by the conversion factor 0.83612736.So, 0.1 square yard times 0.83612736 is equal to 0.08361 square meters. How many
square meters are there in 1 square yard? Square foots 10.764262648009. A square meter is a unit of area measurement incorporated in the metric system and is abbreviated as "m2". Note that rounding
errors may occur, so always check the results. The Square meter to square yard conversion table is used in the conversion of common values from square meters to square yards. You are currently
converting area units from square meter to square yard 1 m 2 = 1.1959900463011 sq yd. Gaj is also known as “Guj” at some places is actually same as Sq. how many square inches is 936 square
hectometers? The conversion result will be shown below the control in the bottom platform of the calculator. square meter . YARD TO SQUARE METER (yd TO m2) CHART 1 yard in square meter = 0.76455486
yd 10 yard in square meter = 7.64554858 yd Note that rounding errors may occur, so always check the results. History/origin: Area is commonly represented in the form of a square with sides of a
specific unit of length, written as "square" followed by the chosen unit of length. The result will be shown as; 50 Square Meters = 59.7995025 Square Yards. What is 1.74 m2 in Square Yards. One
square yard is an area equal to the one of a square with one yard on each side. This tool converts square meters to square yards (m 2 to yd 2) and vice versa. 1 Square meters to Square yards (1 m2 to
yd2) Conversion calculator. A square yard is a unit of area in both US Customary Units as well as the Imperial System. m² = yd² _____ 1.1960. This conversion of 1 square meters to square yards has
been calculated by multiplying 1 square meters by 1.1959 and the result is 1.1959 square yards. Contact us, 54.77 decimeters per square second to hectometers per square second, 66.88 meters per
second to kilometers per second. Its abbreviation is "sq yd" or "yd²". Therefore, there are 1.19599005 square yards in one square meter. There are 0.83612736 square meters in 1 square yard. What is a
square yard (yd 2)? To convert square yards to square meters, multiply the square yard value by 0.83612736 or divide by 1.19599005. square yards to square meters formula. It is equal to 9 square feet
or about 0.836 square meters. Switch units Starting unit. a square meter equals 1.196 square yards because 1 times 1.196 (the conversion factor) = 1.196 All In One Unit Converter Please, choose a
physical quantity, two units, then type a … The area of a square is determined when one side is multiplied by itself and one meter is defined as 1.09361 yards. Convert 1 square meter to square yards,
m² to yd² unit converter with conversion cards, convert between any units of area with varying precision The Conversions and Calculations web site food compounds gravel db finance health password
convert tables rate plane solid more The first procedure is to enter the value in square meters in the blank text field. how many feet per hour is 7,704 feet per second? A square meter, or square
metre, is a unit of area. Conversion Square meter to Square yard. Convert 1.74 Square Meters to Square Yards (1.74 m2 to yd2) with our Area converter. Hectars 0.0001. What is 1.43 m2 in Square Yards.
There are 1.1959900463011 square yards in 1 square meter. The symbol for square yard is yd 2 or sq yd. Square yard is an imperial and A square yard is a unit of area equal to the size of a square
that is one yard on a side. Simply multiply the value in square meters by the conversion factor to find out the number of square yards in one square meter. 1 yard is equal to 0.83612736 square meter.
square meter = square yard * 0.83612736. square meter = square yard / 1.19599005. Yard and Gaj are the two most commonly used unit for measurement in the Indian subcontinent and especially in North
India. 1 square meter is equal to 1.1959852573672 square yard [survey], or 1000000 sq.mm. Note that rounding errors may occur, so always check the results. yards and click on the "convert" button.
square yard [survey] or sq.mm The SI derived unit for area is the square meter. 1 square meters is equal to 1.19599 square yards: 1 m 2 = 1.19599 yd 2. Yard. Definition of Square Yard The square yard
is an imperial unit of area, formerly used in most of the English-speaking world but now generally eplaced by the square metre, however it i still in widespread use in the US., Canada and the U.K. 1
Square Yard = 0.83612736 Square Meter. There are 1,195,990.04630108 square yards in a square kilometer. 1.74 Square Meters equals how many Square Yards. One square meter = 1.19599005 square yards =
10.7639104 square feet = 1550.0031 square inches. First, enter the value in square meters (50) in the blank text field. This calculator uses a simple formula when performing the conversions. Amount:
From: To: 1 Square meters to other area units: Acres 0.00024709661477638. How many Square Yards in 1.43 m2. Convert 50 square meters to square yards. 1 Square meters to Square yards (m^2 to sq yd)
conversion calculator of Area measurement, 1 square meter = 1.1959900463011 square yards. Plus learn how to convert Sq yd to M2 Square yard is mainly used in real estate, architecture and interior
space plans. It is a conversion calculator that is used to convert the area in square meter (m2) to area in square yards (yd2). © 2021 Calculatorology. Click the ‘Convert' button to execute the
conversion from square meters to square yards. Square Inches 1549.9070055797. One square yard is an area equal to the one of a square with one yard on each side. Square Yard : The square yard is a
non-metric unit for area which also is an imperial/US customary unit. Square kilometers 1.0E-6. A square yard is calculated as the area of a square that has 1 yard … How many Square Yards in 1.74 m2.
A square yard is a US customary and an imperial unit of area measurement and is abbreviated as "yd2" or "sq. m 2. square yard . Square Yard : The square yard is a non-metric unit for area which also
is an imperial/US customary unit. It is equal to 9 square feet or about 0.836 square meters. A square meter, or square metre, is a unit of area. Square yard. Since this area unit is very close to one
meter square (one yard is equal to 0.9144 meters, 3 feet or 36 inches), in modern times the use of square yards has been decreasing, and instead square meters are widely used for measuring relatively
small areas. Therefore, one Square Yard (sq yd) is equal to decimal point eight four Square Metre (sq mt) in Survey System. It is the size of a square that is one meter on a side. Therefore, there
are 1.19599005 square yards in one square meter. It is defined as the area of a square whose sides measure exactly one metre. Square yards to Square meters converter. Now, you will find it
interesting that 1 Sq. All Rights Reserved. 1.43 Square Meters equals how many Square Yards. A square yard is a unit of area equal to the size of a square that is one yard on a side. A measurement of
area equal to one meter length by one meter width. The International spelling for this unit is square metre. Then multiply the amount of Yard you want to convert to Square Meter, use the chart below
to guide you. This conversion of 10.1 square meters to square yards has been calculated by multiplying 10.1 square meters by 1.1959 and the result is 12.0794 square yards. Use this page to learn how
to convert between square yard [survey] and square … One square meter is defined as 1.196 square yards, 1550 square inches, and 10.7639104 square feet. yard or square meter The SI derived unit for
area is the square meter. One square yard is equivalent to 0.83612736 square meters, 9 square feet, 8361.27 square centimeters and 1296 square inches. Its abbreviation is "sq yd" or "yd²". It is
approximately 10.76 square feet. It is defined as the area of a square with the sides of one yard in length. Square miles 3.8610038610039E-7. Note: m 2 is the abbreviation of square meters and yd 2
is the abbreviation of square yards. yd." For example; how many square yards are in 26 square meters? It is defined as the area of a square with the sides of one yard in length. 1 square meter =
1.19599005 square yards. Square yard. A square yard can also be defined as the area of a square with sides that are equal to one yard. 47.09 miles per gallon to liters per 100 kilometers, 9,395 miles
per second to kilometers per hour, 96.47 square meters in square centimeters. 1 square meter = 1.19599005 square yards. One Square Yards is equivalent to zero point eight three six Square Meters. Sq.
It can also be defined as the area of the square with sides that are equal to one meter. Square Yard (sq yd) to Square Metre (sq mt) converter is an superb online area conversion calculator that is
popularly used to convert from unit Square Yard (sq yd) to it's relevant unit Square Metre (sq mt) in land measurement. A square meter is calculated as the area of a square that has 1 meter on each
side. Therefore, 1.09361 x 1.09361 = 1.196 square yards in a square meter. The area units' conversion factor of square meters to square yards is 1.19599005. square meters or square yards The SI
derived unit for area is the square meter. The square metre (UK) or square meter (American spelling) is the SI derived unit of area, with symbol m 2. The square meter (plural form: square meters;
British spelling: square metre; abbreviation: sq m or Sq m or m 2) is a derived unit of area used in SI system (International System of Units, Metric System). Square centimeters 10000. Conversion
base : 1 sq yd = 0.83612736 m 2. To convert from square meters to square yards, multiply your figure by 1.1959900463011 (or divide by 0.83612736). There are 0.83612736 square meters in a square yard.
It is defined as 0.836127 square meters, 1296 square inches, 8361.2736 square centimeters and 9 square feet. A square meter is an area unit in the metric systems of measurement and is abbreviated as
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How to Solve the Knapsack Problem Using Genetic Algorithm in Python
Have you ever thought while working with gradient descent that there should be some other optimization algorithms to try out? If you have then the good news is we have some global optimization
algorithms in the form of genetic and swarm algorithms widely known as biologically inspired algorithms.
Inspired by Charles Darwin’s theory of natural evolution, genetic algorithms support natural selection, in which the fittest individuals are chosen for reproduction in order to generate the following
generation’s progeny. This algorithm is an evolutionary algorithm that uses natural selection with a binary representation and simple operators based on genetic recombination and genetic mutations to
execute an optimization procedure influenced by the biological theory of evolution.
Knapsack problem:
In this article, we will implement a genetic algorithm to solve the knapsack problem. The knapsack problem is a combinatorial optimization problem in which you must determine the number of each item
to include in a collection so that the total weight is less than or equal to a given limit and the total value is as large as possible given a set of items, each with a weight and a value.
**Natural Selection Ideology:
**The selection of the fittest individuals from a population begins the natural selection process. They generate offspring who inherit the parents’ qualities and are passed down to the next
generation. If parents are physically active, their children will be fitter than they are and have a better chance of surviving. This procedure will continue to iterate until a generation of the
fittest individuals is discovered.
The genetic algorithm cycle is divided into the following components which are the building blocks of this algorithm:
1. Fitness Function
2. Chromosome Initialization
3. Initialize the population
4. Fitness Evaluation
5. Roulette Selection
6. Crossover
7. Mutation
Cycle of Genetic Algorithm:
This cycle from 3 will be repeated until we have an optimized solution.
We will implement each one and then put it all together to apply it to the knapsack problem but before implementing the Genetic algorithm let's understand what the parameters of the Genetic Algorithm
Parameters of Genetic Algorithm:
• chromosome size — dimension of the chromosome vector. In our case, we have 64 items so the chromosome size is equal to 64
• population size — number of individuals in the population
• parent count — number of parents that are selected from the population on the base of the roulette selection. The parent count must be less than the population size.
• probability of ones in a new chromosome — probability which is used for initial population generation. It is the probability of one in the initial chromosome. High values may lead to the
generation of many individuals with fitness equal to zero. This parameter is specific to our method of generation of the initial chromosome. This parameter is meaningless if your choice is any
other method.
• probability of crossover — the probability of crossover i.e., if the child inherits the gene of both parents or 1.
• probability of mutation — the probability of mutation. I recommend starting at the value of 1/chromosome size and increasing later as you see the change
1. Fitness Function:
The fitness function determines an individual’s level of fitness (the ability of an individual to compete with other individuals). It assigns everyone a fitness score. The fitness score determines
the likelihood of an individual being chosen for reproduction
Let w is the weight vector, c is the cost vector, g is a chromosome and L is the weight limit, we define the fitness function as:
Now let’s implement the function:
#fitness function for each chromosome
def fitness(w, c, L, g): #weight, cost, weight_limit, chromosome
score = 0
score1 = 0
for i in range(len(w)):
score = score + np.sum(w[i]*g[i])
if score > L:
f = 0
for i in range(len(w)):
score1 = score1 + np.sum(c[i]*g[i])
f = score1
return score1, score #fitness
2. Chromosome Initialization:
After defining the fitness function, we will initialize chromosomes for each individual in our population. A set of factors (0/1) known as Genes characterizes an individual. To build a Chromosome,
genes are connected in a string.
Chromosome also known as individual:
Let p is the initialization probability for 1 in a new chromosome, ψ is a random value with the unique distribution in the range of <0,1> and g be a new chromosome then:
To ensure that the probability should be sufficiently low to generate a valid solution with non-zero fitness. Check the validity of the solution after chromosome creation and recreate it if the total
weight of the knapsack is above the weight limit L. More formally for a valid solution.
Where w is the weight vector. This is recommended, but the not mandatory way how to initialize a new chromosome. Be aware that producing non-valid solutions with zero fitness at the start may disturb
the algorithm. Let’s initialize the chromosomes as:
#generating chromosome with probability of 1's
import random
def generate_chromosome(N, w, L, p): #N choromosome_size, weight, #limit, probability
score = 0
g = np.zeros(N) #verify if vector is 64 or 65
for i in range(len(g)):
prob = random.uniform(0, 1)
if prob < p:
g[i] = 1
g[i] = 0
for c in range(N):
score = score + np.sum(w[c]*g[c])
if score <= L:
return g
3. Initialize population:
Now we will initialize our population, The population is represented by a NumPy matrix. Rows of the matrix correspond to individuals, columns correspond to genes in the chromosome. Our matrix has 64
columns because we have 64 items. This can change depending on your problem.
We will initialize the population using the generate_chromosome() function as:
#initializing population by generating chromosome
def initialize_population(population_size, chromosome_size, weights, weight_limit, probability_of_ones_in_a_new_chromosome):
pop = np.zeros((population_size, len(weights)))
for i in range(population_size):
chromo = generate_chromosome(chromosome_size, weights, weight_limit, probability_of_ones_in_a_new_chromosome) #N, w, L , p
pop[i] = chromo
return pop
4. Fitness Evaluation:
Now we will apply the fitness function to all rows of the population matrix. The result is a vector of fitness values for all individuals in the population. The vector has the same size as the size
of the population. The fitness function is applied as:
def evaluate_fitness(pop, weights, costs, weight_limit):
f = np.zeros(len(pop[:, 0])) #length of pop column any
wg = np.zeros(len(pop[:, 0]))
for i in range(len(pop[:, 0])):
p1 = pop[i]
f[i], wg[i] = fitness(weights, costs, weight_limit, p1) #weight,cost, limit, chromosome
return f, wg
Once all individuals in the population have been evaluated, their fitness values are used for selection. Individuals with low fitness get eliminated and the strongest get selected. Inheritance is
implemented by making multiple copies of high-fitness individuals. The high-fitness individuals get mutated and they crossover to produce a new population of individuals, we will see this as we will
implement those in the next steps.
5. Roulette Selection:
The roulette wheel selection method is used for selecting all the individuals for the next generation. Roulette selection is a stochastic selection method, where the probability for selection of an
individual is proportional to its fitness i.e. the better fitness score an individual has the better probability of its selection, and the lower the fitness score lesser the probability of selection.
The idea of this selection phase is to select the fittest individuals and let them pass their genes to the next generation. We will implement a roulette wheel based on this formula:
Roulette Selection:
And this is how it can be implemented:
#roulette selection based on fitness score
def roulette_selection(pop, fitness_score, parents):#population matrix, fitness score vector, parents size to be selected
select population and fitness, perform roulette selection via probability
and random choice and select chromosome as per described parent_size
fitness = fitness_score
total_fit = sum(fitness)
relative_fitness = [f/total_fit for f in fitness]
cum_probs = np.cumsum(relative_fitness)
roul = np.zeros((parents, len(pop[0]))) #shape of matrix based on parent size
for i in range(parents):
r = random.uniform(0, 1)
for ind in range(len(pop[:, 0])): #no. of entries in population
if cum_probs[ind] > r:
#print(r) '''for debugging'''
roul[i] = pop[ind]
return roul #selected parents
Two pairs of individuals (parents) are selected based on their fitness scores. Individuals with high fitness have more chances to be selected for reproduction.
6. Crossover:
Crossover is the most significant phase in a genetic algorithm. For each pair of parents to be mated, a crossover point is chosen at random from within the genes.
Offspring are created by exchanging the genes of parents among themselves until the crossover point is reached. The new offspring are added to the population. Crossover can be done by employing
several different strategies like:
· Splitting genes of both parents equally and both child either get 1st or last part of the parent
· Randomly assigning genes of parents to both children
· Further optimization via engaging fitness function before gene selection
· 2, 3 or planned gene distribution among child
And if you are not happy with any of these you can create one for yourself too as per the need of your problem.
Here we will use the following formula for crossover:
crossover here will be based on probability if the random choice is less than probability there will be a cross-over between parents. Here we will also take probability to a lower level due to fact
that a lot of crossovers may converge at local minima instead of global.
Based on the above formula we implement crossover as:
def crossover(a, b, p): #a=chromosome 1, b=chromosome 2,
#p = probability for crossover
ind = np.random.randint(0, 64)
r = random.uniform(0, 1)
if r < p:
c1 = list(b[:ind]) + list(a[ind:]) #since array were having shape issues, converting to lists
c1 = np.array(c1)
c2 = list(a[:ind]) + list(b[ind:])
c2 = np.array(c1)
c1 = a
c2 = breturn c1, c2 #returning the crossover childs
7. Mutation:
Some of the genes in particular newly created offspring can be susceptible to a low-probability mutation. As a result, some of the bits in the bit string can be flipped. The mutation is used to
retain population variety and avoid premature convergence. In simple words, we flip bits from 0 to 1 or 1 to 0 based on our probability selection. We define mutation by the following formula:
And it is implemented as:
#mutattion of bits from 1 to 0 and 0 to 1 based on probability
def mutation(g, p):
N = len(g)
m = np.zeros(len(g)) #mutated chromosome
for i in range(N):
d = g[i]
r = random.uniform(0, 1)
if g[i] == 1.0 and r < p:
m[i] = 0
elif g[i] == 0.0 and r < p:
m[i] = 1
m[i] = d
return m
Where g is the length of the chromosome and p is prob if the gene(bit) is to be mutated or not. We keep this prob low too due to some reason for avoiding early convergence.
By transforming the previous set of individuals into a new one, the algorithm generates a new set of individuals that have better fitness than the previous set of individuals. When the
transformations are applied over and over again, the individuals in the population tend to represent improved solutions to whatever problem was posed in the fitness function.
Combing all these above steps and this is our standard GA algorithm:
pop = initialize_population(population_size=100, chromosome_size=64, weights=weights_of_items, weight_limit=50, probability_of_ones_in_a_new_chromosome=0.1)
#initializing population
fit, wgh = evaluate_fitness(pop, weights_of_items, costs_of_items, 100)
bc, mc, bw, mw, minw = get_kpi(fit, wgh)
generation = np.zeros(100)
best_cost = np.zeros(100)
min_cost = np.zeros(100)
best_weight = np.zeros(100)
max_weight = np.zeros(100)
min_weight = np.zeros(100)
generation[0] = 0
best_cost[0] = bc
min_cost[0] = mc
best_weight[0] = bw
max_weight[0] = mw
min_weight[0] = minw
popy = popfor kk in range(99):
pr = roulette_selection(pop, fit, 74) #parents based on fitness score (even)
cross = cross_comp(pr, pop)
mut_list = mu_list(cross, pop)
new_pr = roulette_selection(pop, fit, 26) #adding new parents to mutated list to make orignal pop size
new_popi = np.vstack((mut_list, new_pr))
pop = new_popi
fit1, wgh1 = evaluate_fitness(pop, weights_of_items, costs_of_items, 100)
bc, mc, bw, mw, minw = get_kpi(fit1, wgh1)
generation[kk+1] = kk + 1
best_cost[kk+1] = bc
min_cost[kk+1] = mc
best_weight[kk+1] = bw
max_weight[kk+1] = mw
min_weight[kk+1] = minw
We will save the best solution (highest fitness), total weight, and other useful info for each generation.
The algorithm terminates if the population has converged. Then it is said that the genetic algorithm has provided a set of solutions to our problem.
For the above algorithm, this was my problem configuration:
# 64 items and their weights.
weights_of_items = np.array([
2.3, 8.1, 6., 4.2, 1.3, 2.9, 7., 7.9,
3.6, 5., 3.1, 5., 3.4, 5.3, 0.8, 6.9,
9.8, 4.4, 5.4, 7.5, 4.6, 0.3, 9.2, 8.8,
2.2, 3.3, 9.9, 7.6, 5.9, 4.2, 4.9, 5.8,
4.4, 2.9, 0.1, 2.4, 5.6, 7.8, 7., 7.5,
7.3, 7.4, 6.4, 1.6, 6.8, 4., 4.6, 4.1,
0.5, 6.3, 5.2, 1.5, 9.7, 1.6, 2.6, 1.3,
6.5, 2.6, 7.8, 6.3, 8.4, 9.4, 1.4, 7.5])
# 64 costs corresponding to weights
costs_of_items = [
6., 17., 10., 26., 19., 81., 67., 36.,
21., 33., 13., 5., 172., 138., 185., 27.,
4., 3., 11., 19., 95., 90., 24., 20.,
28., 19., 7., 28., 14., 43., 40., 12.,
25., 37., 25., 16., 85., 20., 15., 59.,
72., 168., 30., 57., 49., 66., 75., 23.,
79., 20., 104., 9., 32., 46., 47., 55.,
21., 18., 23., 44., 61., 8., 42., 1.]
# Knapsack weight limit
knapsack_weight_limit = 100
and we have the following outputs:
plt.figure(figsize=(8, 6))
plt.plot(df.generation, df.best_cost)
best_cost of each generation:
plt.figure(figsize=(8, 6))
plt.plot(df.generation, df.best_weight, label='best')
plt.plot(df.generation, df.max_weight, label='max')
plt.plot(df.generation, df.min_weight, label='min')
All weights for each generation:
You can see that cost is 0 after round/generation 79 due to the fact that now all chromosomes/individuals in generations are above the weight limit.
So this is how you can configure your own genetic algorithm and change the function and values as per the need of your problem. I hope this will help you in solving problems from a new perspective. | {"url":"https://plainenglish.io/blog/genetic-algorithm-in-python-101-da1687d3339b","timestamp":"2024-11-14T15:28:02Z","content_type":"text/html","content_length":"90436","record_id":"<urn:uuid:02d459e7-3650-4d0e-9c65-57da02a0a09b>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00331.warc.gz"} |