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Practice Worksheet Absolute Value Graphs Answer Key - Graphworksheets.com
Practice Worksheet Absolute Value Graphs Answer Key
Practice Worksheet Absolute Value Graphs Answer Key – In many areas, reading graphs can be a useful skill. They allow people to quickly compare and contrast large quantities of information. A graph
of temperature data might show, for example, the time at which the temperature reached a certain temperature. Good graphs have a title at the top and properly labeled axes. They are clean and make
good use of space.
Graphing functions
High school students can use graphing functions worksheets to help them with a variety of topics. These include identifying and evaluating function, composing, graphing and transforming functions.
These worksheets also cover finding domains, performing operations on functions and identifying inverse function. These worksheets also cover function tables and finding the range of a function. Many
have worksheets that allow you to combine two or more functions.
Functions are a special type of mathematical relationship that describes the relationship between inputs and outputs. It is useful in predicting what the future will hold and can also help to predict
how things might change. In fact, some functions are so solid that they can be based on seemingly random inputs! Students must be able recognize, create, draw, and graph functions in order to use
this knowledge for future prediction.
Students must determine the x-intercept (or y-intercept) when graphing a linear function. An input-output table with the correct format is also required. The student can plot the graph once the
input-output tables are completed.
Graphing line graphs
A line graph is a chart that has two axes. The independent variable is represented by the axis, while the dependent variable is represented by the axis. The data points on x-axis can be referred to
as x-axis point, while the points on y-axis can be referred to as y-axis point. You can plot the axes side-by-side or inverted as in a bargraph.
In the third grade, line graphs are introduced. By the fourth grade students can move on to more complicated graphs. These graphs have a more variable vertical scale and require more analysis. They
also may involve real-life data and may start at zero on the vertical axis. Students will need to analyze the data and answer questions in order to create an effective graph.
Students will need to label the axis according to the data being plotted. They should also label the axis with appropriate increments. A line graph might show how the stock price has changed over two
weeks. The x-axis represents the number of days and the y-axis the stock price over that time.
Graphing bar graphs
Graphing bar graphs worksheet answers provide the student with the necessary information to draw a chart. These charts are used to analyze information and make decisions. For this reason, students
should familiarize themselves with the various kinds of graphs. Bar graphs can be used to illustrate changes over time.
You can also use a bar graph to compare two sets data. For example, a double bar graph can be used to compare sales data from two bakeries. The data are presented in a graph that shows discrete
values, on a scale from 10 to 10. The student must help Mrs. Saunders interpret the graph.
A bar graph worksheet contains a set of questions that allow students to practice reading and understanding data. These questions may include counting objects, reading and understanding bar graphs,
as well as questions about how to count them. The grade three bar graph worksheet will ask questions about reading the graph and labeling x andy axes. A grade four bar graph worksheet will include
word problems based on bar graphs.
Graphing grids
Graphing grids worksheets help students understand the concept of coordinate geometry. Students can plot points in each quadrant using a coordinate grid. A grid can also be used to plot functions.
Answers to graphing grids worksheets are available in pdf format. These worksheets are a good way to practice comparing and relating coordinate pairs.
Graphing worksheets typically have one grid and four quadrant grids. Each point is connected to the previous point using a line segment. These grids can be used by students to show the relationships
between points and lines. Students can then use the coordinate grid for solving equations that involve more than one quadrant after they have mapped.
These graphing worksheets are a great resource for elementary and middle school students. These graphing worksheets can be generated by a graph paper generator. The generators will produce a standard
graph paper with a single quadrant coordinate grid, two single quadrant graphs, and four single-quadrant graphs per page.
Gallery of Practice Worksheet Absolute Value Graphs Answer Key
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Algebra2 2 7 Absolute Value Functions And Graphs YouTube
50 Isotope Practice Worksheet Answer Key In 2020 Atomic Structure
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QUANTUM MECHANICS - 2024/5
Module Overview
This module introduces fundamental concepts in Quantum Mechanics and its applications to real-world quantum problems. The module covers the mathematics of Hilbert spaces and Dirac notation, the
postulates of Quantum Mechanics, the uncertainty principle, the Schroedinger equation with one-dimensional applications to a particle in a potential well and the quantum harmonic oscillator, and
angular momentum and spin.
This module utilises material from MAT1034 Linear Algebra and MAT2007 Ordinary Differential Equations. The module also builds on material from MAT1036 Classical Dynamics and MAT3008 Lagrangian &
Hamiltonian Dynamics, although these modules are not pre-requisite.
Module provider
Mathematics & Physics
Module Leader
TORRIELLI Alessandro (Maths & Phys)
Number of Credits: 15
ECTS Credits: 7.5
Framework: FHEQ Level 6
Module cap (Maximum number of students): N/A
Overall student workload
Independent Learning Hours: 69
Module Availability
Semester 2
Prerequisites / Co-requisites
You cannot also take PHY3044 ADVANCED QUANTUM PHYSICS.
Module content
Indicative content includes:
• Crucial experiments and birth of quantum mechanics.
• Hilbert spaces and Dirac notation.
• Postulates of Quantum Mechanics. The uncertainty principle. Wave functions.
• The Hamiltonian operator and its spectrum. The Schroedinger equation. Observables.
• Applications: Particle in a one-dimensional potential well. The tunneling effect. The one-dimensional quantum harmonic oscillator.
• Advanced topics: Angular momentum and its addition rules. Spin. The Pauli exclusion principle.
• Advanced applications (time-permitting): The Hydrogen Atom. Time-independent perturbation theory and energy level-splitting.
Assessment pattern
Assessment type Unit of assessment Weighting
School-timetabled exam/test In-semester test (50 min) 20
Examination End-of-Semester Examination (2 hours) 80
Alternative Assessment
Assessment Strategy
The assessment strategy is designed to provide students with the opportunity to demonstrate:
• Understanding of subject knowledge, and recall of key postulates and theorems in Quantum Mechanics.
• The ability to analyse unseen problems in Quantum Mechanics, and use appropriate methods to solve these problems and interpret the results.
Thus, the
summative assessment
for this module consists of:
• One in-semester test corresponding to Learning Outcomes 1, 2 and 5.
• A synoptic examination corresponding to all Learning Outcomes 1 to 5.
Formative assessment
There are two formative unassessed courseworks over an eleven week period, designed to consolidate student learning.
Students will receive individual written feedback on both the formative unassessed courseworks and the in-semester test. The feedback is timed such that feedback from the first coursework will assist
students with preparation for the in-semester test. The feedback from both courseworks and the in-semester test will assist students with preparation for the synoptic examination. Students also
receive verbal feedback in office hours.
Module aims
• Introduce students to the mathematical description of quantum phenomena.
• Enable students to understand the postulates of Quantum Mechanics and their applications to the physical world.
• Illustrate the application of the theory of Quantum Mechanics to simple one-dimensional examples, including a particle in a potential well, the quantum harmonic oscillator and spin systems.
Learning outcomes
Attributes Developed
001 Students will demonstrate a firm understanding of the concepts, theorems and mathematical techniques underlying Quantum Mechanics. KC
002 Students will be able to solve the Schroedinger equation for examples involving one-dimensional potential wells and quantum tunnelling. KC
003 Students will be able to apply mathematical techniques to solve the quantum harmonic oscillator. KC
004 Students will be able to apply mathematical techniques to quantum systems involving angular momentum and spin. KC
005 Students will be able to analyse, solve and interpret unseen problems in Quantum Mechanics similar to those encountered in the module. KCT
C - Cognitive/analytical
K - Subject knowledge
T - Transferable skills
P - Professional/Practical skills
Methods of Teaching / Learning
The learning and teaching strategy is designed to:
• Introduce students to the postulates and theory of Quantum Mechanics, and appropriate mathematical tools for their implementation.
• Provide students with experience of methods used to interpret, understand and solve concrete problems in Quantum Mechanics.
The learning and teaching methods include:
• Three one-hour lectures for eleven weeks, with module notes provided to complement the lectures. These lectures provide a structured learning environment and opportunities for students to ask
questions and to practice methods taught.
• Two unassessed courseworks to provide students with further opportunity to consolidate learning. Students receive feedback on these courseworks as guidance on their progress and understanding.
• Lectures may be recorded or equivalent recordings of lecture material provided. These recordings are intended to give students an opportunity to review parts of lectures which they may not fully
have understood and should not be seen as an alternative to attending lectures.
Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on
the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are
finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.
Other information
The School of Mathematics and Physics is committed to developing graduates with strengths in Digital Capabilities, Employability, Global and Cultural Capabilities, Resourceness and Resilience, and
Sustainability. This module is designed to allow students to develop knowledge, skills and capabilities in the following areas:
Digital Capabilities: The SurreyLearn page for MAT3039 features a dynamic discussion forum where students can pose questions and engage with others using e.g. LaTeX and MathML tools. This enhances
their digital competencies while facilitating collaborative learning and information sharing.
Employability: The module MAT3039 equips students with skills which significantly enhance their employability. The mathematical proficiency gained will hone their critical thinking and
problem-solving abilities. Students will learn to interpret and evaluate quantum problems, model these problems mathematically using the tools of Quantum Mechanics, and hence deduce and interpret
solutions. Mathematical modeling is a highly sought after skill in many professions.
Global and Cultural Capabilities: Students enrolled in MAT3039 originate from a variety of countries and have a wide range of cultural backgrounds. Students are encouraged to work together during
problem-solving teaching activities in lectures, which naturally facilitates the sharing of different cultures.
Resourcefulness and Resilience: MAT3039 is a module which demands the ability to analyse complex problems in Quantum Mechanics, formulate and solve these problems mathematically using mathematical
tools and quantum theory, and interpret the results. Students will gain skills in analysing unseen problems and lateral thinking, and will complete assessments which challenge them and build
Sustainability: Quantum Mechanics can be used to model physical systems relevant to sustainable practices. For instance, Quantum Mechanics can be used to model energy levels in atoms and nuclei,
energy level transitions and the resulting release of energy. This is the mechanism behind nuclear reactions which produce nuclear energy. One or more case studies will be included in the module
relating to applications of Quantum Mechanics to quantum systems relevant to real-world sustainable practices.
Programmes this module appears in
Programme Semester Classification Qualifying conditions
Mathematics with Statistics BSc (Hons) 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics with Statistics MMath 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics with Music BSc (Hons) 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics BSc (Hons) 2 Optional A weighted aggregate mark of 40% is required to pass the module
Financial Mathematics BSc (Hons) 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics MMath 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics BSc (Hons) 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics MPhys 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics MMath 2 Optional A weighted aggregate mark of 40% is required to pass the module
Economics and Mathematics BSc (Hons) 2 Optional A weighted aggregate mark of 40% is required to pass the module
Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of
the programme / module for the 2024/5 academic year. | {"url":"https://catalogue.surrey.ac.uk/2024-5/module/MAT3039","timestamp":"2024-11-02T01:15:52Z","content_type":"text/html","content_length":"27059","record_id":"<urn:uuid:7f06a36a-73d1-412d-adc8-4e7ce8359fbe>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00783.warc.gz"} |
ICES Proceedings Template
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Title Cased: Do Not Use All Caps for Your Title} \author{First Author\thanks{First author's affiliation, 1100 E. 5th Street, Niles, MI 20724} \and Second Author\thanks{Second author's affiliation,
1100 E. 5th Street, Niles, MI 20724} \and Third Author\thanks{Third author's affiliation, 1100 E. 5th Street, Niles, MI 20724} \and Fourth Author\thanks{Fourth author's affiliation, 1100 E. 5th
Street, Niles, MI 20724} \and Fifth Author\thanks{Fifth author's affiliation, 1100 E. 5th Street, Niles, MI 20724}} \begin{document} \maketitle \begin{abstract} This is sample text and needs to be
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& $\mu + \mbox{2}\alpha$ & 2& 0\\ Qq& 2& $\mu + \alpha + \delta$& 1 &1\\ QQ& 3& $\mu$& 0& 0\\ \hline \end{tabular} \end{center} \end{table} This is sample text and needs to be completely replaced
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also works \begin{references} \itemsep=0pt {\footnotesize \item Aebischer, N. J., Robertson, P. A., and Kenward, R. E. (1993), ``Compositional Analysis of Habitat Use From Animal Radio-Tracking
Data,'' {\it Ecology}, 74, 1313--1325. \item Akaike, H. (1973), ``Information Theory and an Extension of the Maximum Likelihood Principle,'' in {\it Second International Symposium on Information
Theory}, eds. B. N. Petrov and F. C{\` a}ski, Budapest: Akademiai Ki{\` a}do, pp. 267-281. \item Alerstam, T. (1990), {\it Bird Migration}, Cambridge, UK: Cambridge University Press. \item Berthold,
P. (2001), {\it Bird Migration: A General Study} (2nd ed.), Oxford, UK: Oxford University Press. \item Booth, J., and Hobert, J. P. (1999), ``Maximizing Generalized Linear Mixed Model Likelihoods
With an Automated Monte Carlo EM Algorithm,'' {\it Journal of the Royal Statistical Society}, Ser. B, 61, 265--285. \item Burnham, K. P., and Anderson, D. R. (1998), {\it Model Selection and
Inference}, New York: Springer. \item Cerioli, A. (1997), ``Modified Tests of Independence in $2 \times 2$ Tables With Spatial Data,'' {\it Biometrics}, 53, 619--628. \item Chan, J. S. K., and Kuk,
A. Y. C. (1997), ``Maximum Likelihood Estimation for Probit-Linear Mixed Models With Correlated Random Effects,'' {\it Biometrics}, 53, 86--97. \item Chen, J., Zhang, D., and Davidian, M. (2002), ``A
Monte Carlo EM Algorithm for Generalized Linear Mixed Models With Flexible Random Effects Distribution,'' {\it Biostatistics}, 3, 347--360. \item Chib, S. (1995), ``Marginal Likelihood From the Gibbs
Output,'' {\it Journal of the American Statistical Association}, 90, 1313--1321. \item Cooper, A. B., and Millspaugh, J. J. (1999), ``The Application of Discrete Choice Models to Wildlife Resource
Selection Studies,'' {\it Ecology}, 80, 566--575. \item Dempster, A. P., Laird, N. M., and Rubin, D. B. (1977), ``Maximum Likelihood From Incomplete Data via the EM Algorithm,'' {\it Journal of the
Royal Statistical Society}, Ser. B, 39, 1--38. \item Erickson, W. P., McDonald, T. L., Gerow, K. G., Howlin, S., and Kern, J. W. (2001), ``Statistical Issues in Resource Selection Studies With
Radio-marked Animals,'' in {\it Radio Telemetry and Animal Populations}, eds. J. Millispaugh and J. Marzluff, California: Academic Press, pp. 209--242. } \end{references} \end{document} | {"url":"https://ja.overleaf.com/latex/templates/ices-vi-template/grbygzxfwwrp","timestamp":"2024-11-12T14:13:55Z","content_type":"text/html","content_length":"54539","record_id":"<urn:uuid:3c5e48dc-381a-46c1-8a1c-90a09ab29f65>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00471.warc.gz"} |
Integration by parts
Mastering Integration by Parts and the Integral Product Rule
Comprehensive Definition, Description, Examples & RulesÂ
Integration is one of the most important ways in calculation, as it allows us to find the area under the wind, the volume of a solid, the work done by a force, and multitudinous other operations. In
this composition, we will explore two important styles of integration: integration by parts and the integral product rule. These styles are useful for integrating the product of two functions, which
are constantly encountered in mathematics and wisdom.
Integration by Parts
The math system of integration by parts is used to integrate the sum of two functions. It rests on the notion that if we can separate a product, also we can incorporate it.
In numerous areas of mathematics and wisdom, including algebra, trigonometry, discrimination equations, drugs, engineering, and economics, integration by parts is pivotal and useful. It can help us
in resolving integrals that might else be grueling or insolvable to estimate using the fundamentals of integration.
The Integration by Parts Formula
The integration by parts formula is:
where u and dv are two functions that form the product we want to integrate. The formula tells us that the integral of u dv is equal to the product of u and v minus the integral of v du.
The places of u and dv in the formula are:
• u is the function that we choose to separate.
• dv is the function that we choose to integrate.
The choice of u and dv is pivotal for applying the formula effectively. We’ll bandy some strategies for choosing u and dv in the coming section.
Step-by-Step Guide to Integration by Parts
Choosing u and dv
One of the most common challenges when using integration by parts is choosing u and dv. A good choice can simplify the integral and make it easier to estimate, while a bad choice can complicate the
integral and make it harder to estimate.
There are several strategies for choosing u and dv, but one of the most popular bones is called LIPET. LIPET stands for Logarithmic, Inverse trigonometric, Polynomial, Exponential, and Trigonometric.
These are five types of functions that we can encounter in integration problems. The LIPET rule tells us to choose u from these types of functions in this order of preference and choose dv from the
remaining function.
Application of the Formula
Once we’ve chosen u and dv, we can apply the integration by parts formula and in order to do so, we need to find v and du using these steps:
• To find v, we integrate dv.
• To find du, we differentiate u.
For example, if we have chosen u = x and dv = e^x dx, then we can find v and du as follows:
• v = e^x (integrate e^x dx)
• du = dx (differentiate x)
Then, we replace these values into the formula:
∫x e x dx=xex−∫exdx
We can simplify this further by evaluating the remaining integral:
∫x e x dx=x e x−ex+C.
Iterative Integration by Parts
Sometimes, one application of integration by parts is not enough to solve an integral. We may need to apply it multiple times until we get a simpler integral that we can evaluate. This process is
called iterative integration by parts.
Integrating e
One of the most common functions that we encounter in integration problems is the exponential function ‘e’. The function ‘e’ has a special property that makes it easy to integrate: its
derivative is itself. That is, if we differentiate ‘e’ concerning any variable, we get ‘e’ back.
For example, if we differentiate ‘e^x’ with respect to x, we get ‘e^x’ back:
Similarly, if we differentiate ‘e^(3y)’ with respect to y, we get ‘3e^(3y)’ back:
This property also means that the integral of ‘e’ concerning any variable is ‘e’ times that variable plus a constant. That is, if we integrate ‘e’ concerning any variable, we get ‘e’
times that variable plus a constant.
Product Rule in Integration
Another method of integrating the product of two functions is using the product rule in integration. The product rule in integration is based on the concept of the product rule in differentiation.
The product rule in differentiation tells us how to find the derivative of the product of two functions. It states that:
where f and g are two functions and (fg)’ is the derivative of their product.
The product rule in integration is the counterpart to the product rule in differentiation. It tells us how to find the integral of the product of two functions. It states that:
where f and g are two functions and f’ and g’ are their derivatives.
The product rule in integration can be derived from the product rule in differentiation by applying integration to both sides of the equation. For example, if we have:
We can integrate both sides concerning x and get:
Using the fundamental theorem of calculus, we can simplify the left-hand side as:
Rearranging the terms, we get:
This is the product rule in integration.
The integral product rule is a useful method for integrating functions that are products of two other functions. It can help us avoid using integration by parts or other complicated methods. The
integral product rule works best when one of the functions in the product is
a function that is easy to differentiate or integrate. For example, a constant function, a linear function, or an exponential function.
For example, if we want to integrate x^2 cos(x) concerning x, we can use the integral product rule and choose f and g as follows:
• f = x^2 (a function that is easy to differentiate)
• g = cos(x) (a function that is easy to integrate)
Then, we apply the formula:
We can simplify this further by applying the formula again:
We can simplify this further by evaluating the remaining integral:
where C is an arbitrary constant of integration.
Comparing Integration and Differentiation
Integration and differentiation share an inverse relationship in calculus. They are related by the basic and most common theorem of calculus, which states that:
where f is a continuous function and a and b are any two points in its domain.
The abecedarian theorem of math tells us that integration can be used to undo isolation and vice versa.
In the environment of the product rule, integration and isolation have some parallels and differences. The parallels are:
• Both integration and isolation have product rules that can be used to find the integral or secondary of the product of two functions.
• Both product rules have analogous forms one term involves the product of the functions, and the other term involves the integral or secondary of one function times the other function.
• Both product rules can be applied iteratively for complex products.
The differences are:
• The product rule for isolation is easier to apply than the product rule for integration, as isolation is generally simpler than integration.
• The product rule for integration requires us to know the derivations of both functions in the product, while the product rule for isolation only requires us to know the derivations of one
function at a time.
• The product rule for integration involves a disadvantage sign between the terms, while the product rule for isolation involves a else sign between the terms.
Understanding the secondary product rule can prop us in integration by helping us fete patterns and choose applicable functions for integration by parts or the integral product rule.
In the real world, they have numerous practical operations in fields like drugs, engineering, and economics. They can help us break complex problems involving stir, force, work, energy, heat,
electricity, captivation, probability, statistics, and more.
For example, in physics, we can use integration by parts to find the work done by a variable force on an object. The work done by a force F on an object that moves from position x1 to position x2 is
given by:
Also, in economics, we can use the integral product rule to find the consumer fat or patron fat in a request. The consumer fat is the difference between the total quantum that consumers are willing
to pay for a good and the total quantum that they pay for it. The patron fat is the difference between the total quantum that directors admit for a good and the total quantum that they’re willing to
accept for it.
Common Challenges and Solutions
Both these concepts are important, but they also pose some challenges for learners. Among the typical issues and fixes are:
• Choosing u and dv or f and g As we’ve seen, choosing u and dv for integration by parts or f and g for the integral product rule can make a big difference in the difficulty of working an integral.
• Applying the formula rightly Another challenge is applying the formula rightly and constantly.
• Assessing complex integrals occasionally, indeed after applying integration by parts or the integral product rule, we may end up with an integral that’s still complex or strange.
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1. Integration by parts and the integral product rule are two important styles of integration that can help us integrate the product of two functions.
2. Integration by parts is grounded on the idea that if we know how to separate a product, we can also integrate it. The formula is- ∫ udv = uv − ∫ vdu
where u and dv are two functions that form the product we want to integrate.
3. The integral product rule is grounded on the conception of the product rule in isolation. The formula is-∫ f ′ gdx = fg − ∫ fg ′ dx
where f and g are two functions and f ’ and g ’ are their derivations.
4. Both integration by parts and the integral product rule can be applied iteratively for complex integrals that involve multiple products of functions.
5. Both integration by parts and the integral product rule have numerous practical operations in fields like drugs, engineering, and economics. They can help us break complex problems involving
stir, force, work, energy, heat, electricity, captivation, probability, statistics, and more.
6. Both integration by parts and the integral product rule pose some challenges for learners, similar to choosing u and dv or f and g, applying the formula correctly, and assessing complex
Check your score in the end
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Frequently Asked Questions
Here is a step-by-step example of integration by parts:
Example: Find
Solution: To use integration by parts, we need to choose u and dv. A good strategy is to use LIPET and choose u as the logarithmic function and dv as the remaining function. So, we have:
Then, we need to find v and du using these steps:
• To find v, we integrate dv.
• To find du, we differentiate u.
So, we have:
• v = x^2/2 (integrate x dx)
• du = 1/x dx (differentiate ln(x))
It is now possible to apply the integration by parts formula:
We can simplify this further by canceling out x in the second term:
We can simplify this further by evaluating the remaining integral:
The exponential function ‘e’ is a special function that has a property that makes it easy to integrate: its derivative is itself. That is, if we differentiate ‘e’ concerning any variable, we
get ‘e’ back.
Among the typical issues and fixes are:
• Choosing the right integrals: To choose u and dv or f and g effectively, we can use strategies like LIPET or look for patterns that resemble derivatives or integrals of other functions.
• Applying the strategies rightfully:Â Sometimes, we may forget to include a minus sign or a constant of integration, or we may mix up u and dv or f and g.Â
• Evaluation of integrals: In such cases, we may need to use other techniques like substitution, partial fractions, trigonometric identities, or special functions to evaluate it. Alternatively, we
may need to use numerical methods or software tools to approximate it.
One of the best ways to practice and improve your skills in integration by parts and the integral product rule is to solve a variety of problems that involve these techniques. You can find many
examples and exercises online or in textbooks that cover calculus topics.
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9 Search Results
CASA is a special-purpose system for computational algebra and constructive algebraic geometry. The system has been developed since 1990. CASA is the ongoing product of the Computer Algebra Group at
the Research Institute for Symbolic Computation (RISC-Linz), the University of Linz, Austria, under the direction of Prof. Winkler. The system is built on the kernel of the widely used computer
algebra system Maple.
CoCoA is a system for Computations in Commutative Algebra. It is able to perform simple and sophisticated operations on multivaraiate polynomials and on various data related to them (ideals, modules,
matrices, rational functions). For example, it can readily compute Grobner bases, syzygies and minimal free resolution, intersection, division, the radical of an ideal, the ideal of zero-dimensional
schemes, Poincare' series and Hilbert functions, factorization of polynomials, toric ideals. The capabilities of CoCoA and the flexibility of its use are further enhanced by the dedicated high-level
programming language. For convenience, the system offers a textual interface, an Emacs mode, and a graphical user interface common to most platforms.
GAP is a system for computational discrete algebra, with particular emphasis on Computational Group Theory. GAP provides a programming language, a library of thousands of functions implementing
algebraic algorithms written in the GAP language as well as large data libraries of algebraic objects. GAP is used in research and teaching for studying groups and their representations, rings,
vector spaces, algebras, combinatorial structures, and more. GAP is developed by international cooperation. The system, including source, is distributed freely under the terms of the GNU General
Public License. You can study and easily modify or extend GAP for your special use. The current version is GAP 4, the older version GAP 3 is still available.
In the joint CNRS-INRIA / INPG-UJF project APACHE, Givaro is a C++ library for arithmetic and algebraic computations. Its main features are implementations of the basic arithmetic of many
mathematical entities: Primes fields, Extensions Fields, Finite Fields, Finite Rings, Polynomials, Algebraic numbers, Arbitrary precision integers and rationals (C++ wrappers over gmp) It also
provides data-structures and templated classes for the manipulation of basic algebraic objects, such as vectors, matrices (dense, sparse, structured), univariate polynomials (and therefore recursive
multivariate). It contains different program modules and is fully compatible with the LinBox linear algebra library and the Athapascan environment, which permits parallel programming.
HiFlow³ is a multi-purpose finite element software providing powerful tools for efficient and accurate solution of a wide range of problems modeled by partial differential equations. Based on
object-oriented concepts and the full capabilities of C++ the HiFlow³ project follows a modular and generic approach for building efficient parallel numerical solvers. It provides highly capable
modules dealing with the mesh setup, finite element spaces, degrees of freedom, linear algebra routines, numerical solvers, and output data for visualization. Parallelism – as the basis for high
performance simulations on modern computing systems – is introduced on two levels: coarse-grained parallelism by means of distributed grids and distributed data structures, and fine-grained
parallelism by means of platform-optimized linear algebra back-ends.
LiE is the name of a software package that enables mathematicians and physicists to perform computations of a Lie group theoretic nature. It focuses on the representation theory of complex semisimple
(reductive) Lie groups and algebras, and on the structure of their Weyl groups and root systems. LiE does not compute directly with elements of the Lie groups and algebras themselves; it rather
computes with weights, roots, characters and similar objects.
polymake is an object-oriented system for experimental discrete mathematics. The typical working cycle of a polymake user starts with the construction of an object of interest, auch as a convex
polytope, a finite simplicial complex, a graph, etc. It is then possible to ask the system for some of the object's properties or for some form of visualization. Further steps might include more
elaborate constructions based on previously defined objects. Each class of polymake objects comes with a set of rules which describe how a new property of an object can be derived from previously
known ones. It is a key feature that the user can extend or modify the set of rules, add further properties or even new classes of objects (with entirely new rule bases). The functions provided
include: several convex hull algorithms, face lattices of convex polytopes, Voronoi diagrams and Delaunay decompositions (in arbitrary dimensions), simplicial homology (with integer coefficients),
simplicial cup and cap products, intersection forms of triangulated 4-manifolds. Several forms of (interactive) visualization via interfaces to Geomview, JavaView and other programs.
Risa/Asir is a general computer algebra system and also a tool for various computation in mathematics and engineering. The development of Risa/Asir started in 1989 at FUJITSU. Binaries have been
freely available since 1994 and now the source code is also free. Currently Kobe distribution is the most active branch of its development. We characterize Risa/Asir as follows: (1) An environment
for large scale and efficient polynomial computation. (2) A platform for parallel and distributed computation based on OpenXM protocols.
SINGULAR is a Computer Algebra system for polynomial computations in commutative algebra, algebraic geometry, and singularity theory. SINGULAR's main computational objects are ideals and modules over
a large variety of baserings. The baserings are polynomial rings over a field (e.g., finite fields, the rationals, floats, algebraic extensions, transcendental extensions), or localizations thereof,
or quotient rings with respect to an ideal. SINGULAR features fast and general implementations for computing Groebner and standard bases, including e.g. Buchberger's algorithm and Mora's Tangent Cone
algorithm. Furthermore, it provides polynomial factorizations, resultant, characteristic set and gcd computations, syzygy and free-resolution computations, and many more related functionalities.
Based on an easy-to-use interactive shell and a C-like programming language, SINGULAR's internal functionality is augmented and user-extendible by libraries written in the SINGULAR programming
language. A general and efficient implementation of communication links allows SINGULAR to make its functionality available to other programs. | {"url":"https://orms.mfo.de/search@terms=decomposition+of+modules.html","timestamp":"2024-11-14T11:25:52Z","content_type":"application/xhtml+xml","content_length":"13486","record_id":"<urn:uuid:2fd94fcb-51b0-4efb-a53d-110a8f8c5be4>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00767.warc.gz"} |
RE: st: Certainty units in Stata's survey commands
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RE: st: Certainty units in Stata's survey commands
From [email protected] (Jeff Pitblado, StataCorp LP)
To [email protected]
Subject RE: st: Certainty units in Stata's survey commands
Date Wed, 09 Jan 2008 00:19:02 -0600
Steven Samuels <[email protected]> believes there is something wrong with
how Stata's -svy- commands handle certainty sampling units, i.e. sampling
units chosen with 100% certainty.
Our short reply is that there is nothing wrong here. We believe Steven is
misinterpreting the documentation he is quoting. Stata allows us to -svyset-
our survey data as we see fit, the only things the -svy- commands disallow are
things that violate the definitions of the survey design variables.
First we'll qualify part of Steven's quote from the [SVY] manual. Then we'll
address Steven's concern using an hypothetical example dataset similar in
design to the one he mentions.
> The Stata V 10 "Survey Data" manual states (page 54):
> Stata's -svy- commands identify strata with an FPC equal to one as
> units sampled with certainty. To properly determine the design
> degrees of freedom, certainty units should be contained in their own
> strata, one for each certainty unit in each sampling stage. Although
> the observations contained in certainty units have a role in
> parameter estimation, they contribute nothing to the variance.
> If this is Stata's approach, then it is wrong.
To qualify the last sentence, we probably should have written:
... Although the observations contained in certainty units have a role
in parameter estimation, they contribute nothing to the variance at
the stage where the FPC is 1.
Here FPC is the variable supplied to the -fpc()- option of -svyset-. When the
FPC variable is 1, this means a sampling rate of 100%. When a 100% sampling
rate is plugged into the variance formula, it results in a zero multiplier,
thus the corresponding observations contribute nothing the variance for that
stage; however, these observations could and usually do contribute to the
variance in subsequent stages.
The only substantive topic left to discuss is the calculation of degrees of
freedom. The rest of Steven's email describe an example survey dataset with 3
certainty PSUs.
> Let me give a simple example of a survey to study prevalence of a
> disease. The target area was divided into 27 districts. Three were
> known to have high prevalence and all were taken into the study. The
> other districts divided into 'medium' and 'low' incidence strata. In
> these two strata, samples of districts were drawn. The structure of
> the design is this:
> Strata Sampling Unit-Stage 1
> 1. District 1 village
> 2. District 2 village
> 3. District 3 village
> 4. Medium Incidence district
> 5. Low Incidence district
> (Subsequent stages below the village level are: household, person.)
> In other words, each of the certainty districts is not contained in
> its own stratum; it is a stratum. Variation between the the
> primary sampling units in these strata, as in the others, does indeed
> contribute information to the variance. This is the standard
> treatment for for self-representing" units, the traditional name for
> certainty units. See Graham Kalton, Introduction to Survey Sampling,
> Sage Publications 1983, page 85.
In this example, Steven is treating districts 1 to 3 as strata, but the rest
of the districts are primary sampling units (PSUs) which have been partitioned
into two strata--low and medium incidence.
Suppose we have a Stata dataset with the following variables:
pw -- the sampling weights
incidence -- 1 is "low", 2 is "medium", 3 is "high"
district -- numbered from 1 to the number of sampled districts;
where districts 1, 2, and 3 are assumed to be in
the "high" incidence stratum; our certainty units
village -- a generic village identifier, contiguous numbers
starting from 1
household -- a generic household identifier, contiguous numbers
starting from 1
With these variables also assume we have the sampling rates corresponding to
each observation for each of the sampling stages.
fpc1 -- sampling rate of the districts within stratum, thus
this variable is 1 for all observations in which
incidence is "high"
fpc2 -- sampling rate of the villages within district
To simplify things, we'll assume a small sampling rate for the households or
that the data supplier didn't publish it with in dataset. This also makes the
sampling rate of persons within household irrelevant for variance estimation.
One might think the correct way to -svyset- this data is:
svyset district [pw=pw], fpc(fpc1) strata(incidence) ///
|| village , fpc(fpc2) ///
|| household
This seems reasonable, be we believe that the model degrees of freedom from
this setup are slightly inflated. Suppose in addition to districts 1, 2, and
3, that 16 other districts were sampled. By this setup the degrees of freedom
would be 16 = 19-3.
A more conservative view is that the certainty units in the first stage should
not be counted among the design degrees of freedom. We believe this is
strictly true for a one-stage clustered design, where 100% of the individuals
contained with the sampled clusters are observed. For multi-stage design,
we acknowledge that this view might be considered too conservative; however,
Steven's quote from [SVY] is directly inspired by this conservative view.
We can cause Stata's -svy- commands to use this conservative method for
degrees of freedom by reassigning a unique stratum id to each of the certainty
units. For example, since all our stratum and sampling unit identifiers are
positive integers, we can generate a pseudo-stratum identifier variable using
negated district values to uniquely identify the certainty units.
gen p_str = cond(incidence==3, -district, incidence)
We can then use the new p_str variable in the -strata()- option for the first
svyset district [pw=pw], fpc(fpc1) strata(incidence) ///
|| village , fpc(fpc2) ///
|| household
This will result in 14=19-5 degrees of freedom.
A third approach is to generate pseudo-sampling unit variables that directly
reflect Steven's above setup. We will use the same negated values trick to
generate the pseudo-unit variables for each stage:
gen p_su1 = cond(incidence == 3, -village, district)
gen p_su2 = cond(incidence == 3, -household, village)
gen p_su3 = cond(incidence == 3, -_n, household)
gen p_fpc1 = cond(incidence == 3, fpc2, fpc1)
gen p_fpc2 = cond(incidence == 3, 0, fpc2)
gen p_fpc3 = cond(incidence == 3, 1, 0)
Note that we had to generate pseudo-FPC variables to accommodate the fact that
we are treating districts 1, 2, and 3 as pseudo-strata, the villages within
them as pseudo-units is stage 1, the households within these village as
pseudo-units in stage 2, and the individuals within these households as
pseudo-units in stage 3.
svyset p_su1 [pw=pw] , fpc(p_fpc1) strata(p_str) ///
|| p_su2 , fpc(p_fpc2) ///
|| p_su3 , fpc(p_fpc3)
Now suppose that of the sampled villages, 60 came from the certainty units
(districts 1, 2, and 3). Thus there are now 76=60+16 PSUs and 5 strata,
yielding 71=76-5 degrees of freedom. Note that 60 might be an exceedingly
high number for sampled villages for the design that Steven references; we're
using 60 to emphasize that the typical number of sampling units differ
drastically between stages thus the resulting design degrees of freedom can
vary drastically depending on how you -svyset- the data.
Now we must recognize that we set this problem up so that it was easier for us
to identify the districts, villages, and households in the dataset. A
professional data distributer might not publish the information, especially if
it is public data; preferring the pseudo-strata and pseudo-units variables in
this case. As survey data analysts, we should always follow the guidelines
that are published with the data we are analyzing; these guidelines
usually tell us which variables identify the survey design characteristics
that are subsequently used to -svyset- the data.
Note that the above 3 -svyset-s will yield identical point and variance
(standard error) estimates. See the do-file that follows my signature for an
example that shows this to be true.
In closing, we refer to chapter 5 (especially section 5.2 and 5.3) of Korn and
Graubard (1999) which describes several alternative methods for computing
degrees of freedom for survey dataset with certainty units and/or strata with
one sampling unit. Wolter (2007) also contains examples of survey datasets
that contain certainty units.
Korn, E. L. and B. I. Graubard. 1999. Analysis of Health Surveys.
New York: Wiley.
Woleter, K. 2007. Introduction to variance estimation.
New York: Springer.
[email protected]
***** BEGIN: certainty.do
set seed 1234
set sortseed 1234
// incidence = stage 1 strata
// district = stage 1 units
local low = .25 + .5*uniform()
local low = 12
local med = 12
local nlow = ceil(`low'*(.3 + .9*uniform()))
local nmed = ceil(`med'*(.3 + .9*uniform()))
local nobs = 3 + `nlow' + `nmed'
set obs `nobs'
gen district = _n
gen incidence = cond(_n<4, 3, cond(_n<3+`nlow', 1, 2))
label define incidence 1 "low" 2 "medium" 3 "high"
label values incidence incidence
gen fpc1 = 1
replace fpc1 = `nlow'/`low' if incidence == 1
replace fpc1 = `nmed'/`med' if incidence == 2
tab district incidence
sum fpc?
// village = stage 2 units
gen nobs = 5 + ceil(20*uniform())
expand nobs
local sort incidence district
sort `sort'
by `sort' : gen fpc2 = .1 + round(.5*uniform(), .01) if _n == 1
by `sort' : replace fpc2 = fpc2[1]
gen village = _n
// household = stage 3 units
replace nobs = 10 + ceil(10*uniform())
expand nobs
local sort incidence district village
sort `sort'
gen household = _n
// person = stage 4 units
replace nobs = ceil(6*uniform())
expand nobs
gen pw = 1/(fpc1*fpc2*.05*.05)
gen y = incidence*invnormal(uniform()) + district + village
drop nobs
tab district incidence
sum fpc?
svyset district [pw=pw] , fpc(fpc1) strata(incidence) ///
|| village , fpc(fpc2) ///
|| household
svy: mean y
est store svyset1
gen p_str = cond(incidence == 3, -district, incidence)
svyset district [pw=pw] , fpc(fpc1) strata(p_str) ///
|| village , fpc(fpc2) ///
|| household
svy: mean y
est store svyset2
gen p_su1 = cond(incidence == 3, -village, district)
gen p_su2 = cond(incidence == 3, -household, village)
gen p_su3 = cond(incidence == 3, -_n, household)
gen p_fpc1 = cond(incidence == 3, fpc2, fpc1)
gen p_fpc2 = cond(incidence == 3, 0, fpc2)
gen p_fpc3 = cond(incidence == 3, 1, 0)
svyset p_su1 [pw=pw] , fpc(p_fpc1) strata(p_str) ///
|| p_su2 , fpc(p_fpc2) ///
|| p_su3 , fpc(p_fpc3)
svy: mean y
est store svyset3
// NOTE: The point and SE estimates are the same, the only real difference is
// in the design degrees of freedom -e(df_r)-.
est table svyset1 svyset2 svyset3, se stats(df_r)
***** END: certainty.do
* For searches and help try:
* http://www.stata.com/support/faqs/res/findit.html
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/ | {"url":"https://www.stata.com/statalist/archive/2008-01/msg00187.html","timestamp":"2024-11-06T22:02:27Z","content_type":"text/html","content_length":"18894","record_id":"<urn:uuid:57629618-3238-4786-8dc2-013432ccbc92>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00600.warc.gz"} |
Computational Fluid Dynamics (CFD) Analysis of a Flow Straightener for a Heater Chimney
This work presents a numerical analysis by means of computational fluid dynamics (CFD) of the flow within and outside a chimney of low length/diameter ratio with a lateral inlet for combustion gases.
A partially blocked gases entrance and cyclonic flow present in the chimney in its actual configuration prevent the system to fulfill particulate flow control regulations, in particular concerning
the velocity vector inclination in the control plane where monitoring is carried out. A flow straightener device is proposed which reduces, in the numerical simulations, the inclination of
streamlines in the flow control plane from more than 35 degrees to less than 5 degrees, meeting the criteria stated in environmental regulations for particle emission control, without introducing
excessive pressure losses in the flow. Results of the analysis include the flow configuration, velocity and pressure distributions and helicity distribution, the latter as a measure of the intensity
of cyclonic flow. In this problem, gas enters the chimney from a lateral chamber. Numerical simulation of the present situation highlighted three main problems: - Cyclonic flow in the chimney,
generating helycoidal streamlines where the velocity vector inclination is beyond acceptable limits, - Flow acceleration in the sector opposite to entrance, due to the effect of the lateral inlet and
the deviation that gas suffers after impinging the wall, - Generation of a horizontal vortex in the pre-entrance chamber, due to obstructions in the inlet ducts. From this analysis, possible
solutions were proposed and studied numerically. The one presented in this work, a simple straightener put immediately after the chimney entrance, plus the cleaning of all obstructed ducts, show in
the results a significant improvement in the flow quality and alignement. The computational domain included the inlet chamber, the chimney and the atmosphere in a cylindrical region of approximately
55 chimney diameters in width and 8 chimney heights in height. The numerical simulation was performed with Ansys Research package. A pressure-based solver was used for compressible flow transient
analysis, with second order discretization for space and time, and the “species transport” method was employed for computing the mixing of two species: combustion gas with known properties, and air.
The variations with temperature of viscosity, thermal conductivity and specific heats for the gas were approximated with polynomial expressions interpolating known values. Gas density was computed
from an equation of state. The k-epsilon turbulence model was used. The mesh was locally refined in order to achieve grid-independent results. The time step for convergence was 1e-5 s.
Asociación Argentina de Mecánica Computacional
Güemes 3450
S3000GLN Santa Fe, Argentina
Phone: 54-342-4511594 / 4511595 Int. 1006
Fax: 54-342-4511169
E-mail: amca(at)santafe-conicet.gov.ar
ISSN 2591-3522 | {"url":"http://venus.ceride.gov.ar/ojs/index.php/mc/article/view/4408","timestamp":"2024-11-03T07:45:19Z","content_type":"application/xhtml+xml","content_length":"19253","record_id":"<urn:uuid:fe65b88e-c61d-44b1-94f5-f39bf75bac6f>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00642.warc.gz"} |
Two- fifth of one- third of three- seventh of a number is $15$. What is the value of $40$ percent of that number?
Hint: Here, we have to find the value of forty percent of an unknown number. So first of all suppose the number is $x$ then calculate its three sevenths, then one third and then two fifths of the
previous number and this number is equal to $15$. And after solving the equation we get the number and then find its forty percent.
Complete step-by-step solution:
Here, we have to calculate forty percent of a number.
Let the required number be $x$.
Three seventh of the number $x = \dfrac{3}{7}x$.
Now, one- third of the number $\dfrac{3}{7}x = \dfrac{1}{3} \times \dfrac{3}{7}x = \dfrac{1}{7}x$.
Now, two- fifth of the number $\dfrac{1}{7}x = \dfrac{2}{5} \times \dfrac{1}{7}x$.
Here, it is given that two- fifth of one- third of one- seventh of number $x$ is $15$.
So, $\dfrac{2}{{35}}x = 15$
$ \Rightarrow x = \dfrac{{15 \times 35}}{2}$
$\therefore x = \dfrac{{525}}{2}$
Thus, the required number is $\dfrac{{525}}{2}$.
Now, we have to calculate the forty percent of the number we got above.
So, forty percent of $\dfrac{{525}}{2}$.
$ = 40\% $of $\dfrac{{525}}{2}$.
$ = \dfrac{{40}}{{100}} \times \dfrac{{525}}{2}$.
$ = 105$.
Thus, forty percent of the required number is $105$.
Hence, option (C) is the correct answer for this question.
Note: The most crucial point where we have to give special care is when we are writing equations.
Suppose the number and proceed according to the given statement and finally equates it with the given numerical value.
Percent means a part in every hundred. $x\% $ means $x$ parts in a total of hundred parts. Mathematically it is written as $\dfrac{x}{{100}}$. | {"url":"https://www.vedantu.com/question-answer/two-fifth-of-one-third-of-three-seventh-of-a-class-8-maths-cbse-5fb494704ad6c23c32da7209","timestamp":"2024-11-06T06:20:35Z","content_type":"text/html","content_length":"150899","record_id":"<urn:uuid:f8adaede-6109-4a54-bdf3-18d8f7374ea7>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00078.warc.gz"} |
Mathematics for EmSAT Achieve - Books, Notes, Tests 2024-2025 Syllabus
EmSAT Achieve Mathematics Syllabus
This detailed syllabus outlines the topics covered in the EmSAT Achieve Mathematics exam. The exam is designed to assess students' understanding and proficiency in various mathematical concepts.
Below is a breakdown of the syllabus:
1. EmSAT Achieve Polynomials
- Definition and types of polynomials
- Operations with polynomials
- Polynomial factorization
- Solving polynomial equations
2. EmSAT Achieve Equations in One Variable
- Linear equations
- Quadratic equations
- Rational equations
- Radical equations
- Absolute value equations
3. EmSAT Achieve Inequalities
- Solving linear inequalities
- Solving quadratic inequalities
- Solving rational inequalities
- Solving absolute value inequalities
4. EmSAT Achieve System of Equations
- Solving systems of linear equations
- Solving systems of quadratic equations
- Solving systems of equations using matrices
5. EmSAT Achieve Functions
- Definition and properties of functions
- Function notation
- Graphing functions
- Composite and inverse functions
6. EmSAT Achieve Linear & Exponential Models
- Linear models and applications
- Exponential models and applications
7. EmSAT Achieve Trigonometric Functions
- Trigonometric ratios
- Trigonometric identities
- Solving trigonometric equations
8. EmSAT Achieve Exponents
- Laws of exponents
- Scientific notation
- Operations with exponents
9. EmSAT Achieve Rational & Irrational Numbers
- Rational numbers and operations
- Irrational numbers and their properties
- Approximating irrational numbers
10. EmSAT Achieve Complex Numbers
- Complex numbers and operations
- Complex conjugates
- Complex number applications
11. EmSAT Achieve Vectors
- Vector representation and operations
- Dot product and cross product of vectors
- Applications of vectors
12. EmSAT Achieve Matrices
- Matrix notation and operations
- Determinants and inverses of matrices
- Applications of matrices
13. EmSAT Achieve Limits
- Definition and properties of limits
- Evaluating limits
- Infinite limits and limits at infinity
14. EmSAT Achieve Differentiation
- Definition and properties of derivatives
- Differentiation rules
- Applications of derivatives
15. EmSAT Achieve Integration
- Indefinite and definite integrals
- Integration techniques
- Applications of integration
16. EmSAT Achieve Surface Area & Volumes
- Surface area of 2D and 3D shapes
- Volume of geometric solids
17. EmSAT Achieve Geometric Theorems
- Pythagorean theorem
- Similarity and congruence theorems
- Angle and side relationships in triangles and quadrilaterals
18. EmSAT Achieve Similarity Theorems
- Similarity in triangles
- Similarity in circles
19. EmSAT Achieve Trigonometric Ratios & Triangles
- Trigonometric ratios in right triangles
- Solving problems using trigonometry
20. EmSAT Achieve Circles
- Properties of circles
- Arcs, chords, and tangents of circles
21. EmSAT Achieve Conics
- Parabolas, ellipses, and hyperbolas
- Equations and properties of conic sections
22. EmSAT Achieve Statistics
- Data representation and interpretation
- Measures of central tendency and dispersion
- Probability distributions
23. EmSAT Achieve Probability
- Basic probability concepts
- Conditional probability
- Probability of multiple events
24. EmSAT Achieve Expressions
- Simplifying algebraic expressions
- Evaluating expressions
- Solving equations and inequalities
25. EmSAT Achieve Graphs
- Graphing linear equations and inequalities
- Graphing quadratic equations
- Graphing exponential and logarithmic functions
By covering these topics thoroughly, students will be well-prepared for the EmSAT Achieve Mathematics exam. It is important to study and practice each concept to ensure a strong understanding and
ability to apply mathematical principles effectively.
This course is helpful for the following exams: Grade 11, Grade 12, EmSAT Achieve | {"url":"https://edurev.in/courses/46772_Mathematics-for-EmSAT-Achieve","timestamp":"2024-11-05T03:38:05Z","content_type":"text/html","content_length":"501876","record_id":"<urn:uuid:6ec40b66-1126-48ff-aa78-65b200fc566f>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00364.warc.gz"} |
CPM Homework Help
Examine the graph of each relation below. For each part below, decide if the relation is a function and then state the domain and range.
For a relation to be a function, each input can only have one output. That is, each $x$-value can only have one corresponding $y$-value. | {"url":"https://homework.cpm.org/category/CC/textbook/cca2/chapter/4/lesson/4.1.3/problem/4-45","timestamp":"2024-11-04T01:25:25Z","content_type":"text/html","content_length":"34906","record_id":"<urn:uuid:56746175-a088-42ed-859c-07315109797e>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00205.warc.gz"} |
diagram chasing
I just had a look at this page after it came up on this MO question, and I’m not sure I agree that the Freyd-Mitchelle embedding theorem is more “sophisticated” than the internal logic of a regular
category. I’m not even sure what “sophistication” means here. Maybe we could just remove the claim that the order of listing means anything?
Very nice.
Created a stub for diagram chasing in abelian categories (and linked it from abelian category, element in an abelian category, five lemma and salamander lemma). The order of the techniques listed is
highly subjective, feel free to change as you see fit.
By the way, it is nice to have all of these methods listed here. It would also be nice to include some kind of formal comparison. Not that I have any time for such a thing…
Removed “sophistication” comment
diff, v5, current | {"url":"https://nforum.ncatlab.org/discussion/4108/diagram-chasing/?Focus=33494","timestamp":"2024-11-15T00:01:37Z","content_type":"application/xhtml+xml","content_length":"43013","record_id":"<urn:uuid:6f3a677b-0217-406c-aea9-14502d943ada>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00191.warc.gz"} |
he PATH Model
You can specify the paths and the parameters together in the following statement:
Despite its simple representation of the path diagram, the PATH modeling language is general enough to handle a wide class of structural models that can also be handled by other general modeling
languages such as LINEQS, LISMOD, or RAM. For brevity, models specified by the PATH modeling language are called PATH models.
Types of Variables in the PATH Model
When you specify the paths in the PATH model, you typically use arrows (such as <--- or --->) to denote "causal" paths. For example, in the preceding path diagram or the PATH statement, you specify
that B is an outcome variable with predictors A and C, respectively, in two paths. An outcome variable is the variable being pointed to in a path specification, while the predictor variable is the
one where the arrow starts from.
Whereas the outcome–predictor relationship describes the roles of variables in each single path, the endogenous–exogenous relationship describes the roles of variables in the entire system of paths.
In a system of path specification, a variable is endogenous if it is pointed to by at least one single-headed arrow or it serves as an outcome variable in at least one path. Otherwise, it is
exogenous. In the preceding path diagram, for example, variable B is endogenous and both variables A and C are exogenous. Note that although any variable that serves as an outcome variable at least
in one path must be endogenous, it does not mean that all endogenous variables must serve only as outcome variables in all paths. An endogenous variable in a model might also serve as a predictor
variable in a path. For example, variable B in the following PATH statement is an endogenous variable, and it serves as an outcome variable in the first path but as a predictor variable in the second
path A ---> B = effect1,
B ---> C = effect2;
A variable is a manifest or observed variable in the PATH model if it is measured and exists in the input data set. Otherwise, it is a latent variable. Because error variables are not explicitly
defined in the PATH modeling language, all latent variables that are named in the PATH model are factors, which are considered to be the systematic source of effects in the model. Each manifest
variable in the PATH model can be endogenous or exogenous. The same is true for any latent factor in the PATH model.
Because you do not name error variables in the PATH model, you do not need to specify paths from errors to any endogenous variables. Error terms are implicitly assumed for all endogenous variables in
the PATH model. Although error variables are not named in the PATH model, the error variances are expressed equivalently as partial variances of the associated endogenous variables. These partial
variances are set by default in the PATH modeling language. Therefore, you do not need to specify error variance parameters explicitly unless constraints on these parameters are desirable in the
model. You can use the PVAR statement to specify the error variance or partial variance parameters explicitly.
Specification of the PATH Model
(1) Specification of Effects or Paths
You specify the "causal" paths or linear functional relationships among variables in the PATH statement. For example, if there is a path from v2 to v1 in your model and the effect parameter is named
parm1 with a starting value at
path v1 <--- v2 = parm1(0.5);
path v2 ---> v1 = parm1(0.5);
If you have more than one path in your model, path specifications should be separated by commas, as shown in the following PATH statement:
v1 <--- v2 = parm1(0.5),
v2 <--- v3 = parm2(0.3);
Because the PATH statement can be used only once in each model specification, all paths in the model must be specified together in a single PATH statement. See the PATH statement for more details
about the syntax.
(2) Specification of Variances and Partial (Error) Variances
If v2 is an exogenous variable in the PATH model and you want to specify its variance as a parameter named parm2 with a starting value at PVAR statement specification:
If v1 is an endogenous variable in the PATH model and you want to specify its partial variance or error variance as a parameter named parm3 with a starting value at PVAR statement specification:
Therefore, the PVAR statement can be used for both exogenous and endogenous variables. When a variable in the statement is exogenous (which can be automatically determined by PROC CALIS), you are
specifying the variance parameter of the variable. Otherwise, you are specifying the partial or error variance for an endogenous variable.
You do not need to supply the parameter names for the variances or partial variances if these parameters are not constrained. For example, the following statement specifies the unnamed free
parameters for variances or partial variances of v1 and v2:
If you have more than one variance or partial variance parameter to specify in your model, you can put a variable list on the left-hand side of the equal sign, and a parameter list on the right-hand
side, as shown in the following PVAR statement specification:
v1 v2 v3 = parm1(0.5) parm2 parm3;
In the specification, variance or partial variance parameters for variables v1–v3 are parm1, parm2, and parm3, respectively. Only parm1 is given an initial value at
You can also separate the specifications into several entries in the PVAR statement. Entries should be separated by commas. For example, the preceding specification is equivalent to the following
v1 = parm1 (0.5),
v2 = parm2,
v3 = parm3;
Because the PVAR statement can be used only once in each model specification, all variance and partial variance parameters in the model must be specified together in a single PVAR statement. See the
PVAR statement for more details about the syntax.
(3) Specification of Covariances and Partial Covariances
If you want to specify the (partial) covariance between two variables v3 and v4 as a parameter named parm4 with a starting value at PCOV statement specification:
Whether parm4 is a covariance or partial covariance parameter depends on the variable types of v3 and v4. If both v3 and v4 are exogenous variables (manifest or latent), parm4 is a covariance
parameter between v3 and v4. If both v3 and v4 are endogenous variables (manifest or latent), parm4 is a parameter for the covariance between the errors for v3 and v4. In other words, it is a partial
covariance or error covariance parameter for v3 and v4.
A less common case is when one of the variables is exogenous and the other is endogenous. In this case, parm4 is a parameter for the partial covariance between the endogenous variable and the
exogenous variable, or the covariance between the error for the endogenous variable and the exogenous variable. Fortunately, such covariances are relatively uncommon in statistical modeling. Their
uses confuse the roles of systematic and unsystematic sources in the model and lead to difficulties in interpretations. Therefore, you should almost always avoid this kind of partial covariance.
Like the syntax of the PVAR statement, you can specify a list of (partial) covariance parameters in the PCOV statement. For example, consider the following statement:
v1 v2 = parm4,
v1 v3 = parm5,
v2 v3 = parm6;
In the specification, three (partial) covariance parameters parm4, parm5, and parm6 are specified, respectively, for the variable pairs (v1,v2), (v1,v3), and (v2,v3). Entries for (partial) covariance
specification are separated by commas.
Again, if all these covariances are not constrained, you can omit the names for the parameters. For example, the preceding specification can be specified as the following statement when the three
covariances are free parameters in the model:
v1 v2,
v1 v3,
v2 v3;
Or, you can simply use the following within-list covariance specification:
Three covariance parameters are generated by this specification.
Because the PCOV statement can be used only once in each model specification, all covariance and partial covariance parameters in the model must be specified together in a single PCOV statement. See
the PCOV statement for more details about the syntax.
(4) Specification of Means and Intercepts
Means and intercepts are specified when the mean structures of the model are of interest. You can specify mean and intercept parameters in the MEAN statement. For example, consider the following
If V5 is an exogenous variable (which is determined by PROC CALIS automatically), you are specifying parm5 as the mean parameter of V5. If V5 is an endogenous variable, you are specifying parm5 as
the intercept parameter for V5.
Because each named variable in the PATH model is either exogenous or endogenous (exclusively), each variable in the PATH model would have either a mean or an intercept parameter (but not both) to
specify in the MEAN statement. Like the syntax of the PVAR statement, you can specify a list of mean or intercept parameters in the MEAN statement. For example, in the following statement you specify
a list of mean or intercept parameters for variables v1-v4:
v1-v4 = parm6-parm9;
This specification is equivalent to the following specification with four entries of parameter specifications:
v1 = parm6,
v2 = parm7,
v3 = parm8,
v4 = parm9;
Again, entries in the MEAN statement must be separated by commas, as shown in the preceding statement.
Because the MEAN statement can be used only once in each model specification, all mean and intercept parameters in the model must be specified together in a single MEAN statement. See the MEAN
statement for more details about the syntax.
Specifying Parameters without Initial Values
If you do not have any knowledge about the initial value for a parameter, you can omit the initial value specification and let PROC CALIS compute it. For example, you can provide just the parameter
locations and parameter names as in the following specification:
path v1 <--- v2 = parm1;
pvar v2 = parm2,
v1 = parm3;
Specifying Fixed Parameter Values
If you want to specify a fixed parameter value, you do not need to provide a parameter name. Instead, you provide the fixed value (without parentheses) in the specification.
For example, in the following statement the path coefficient for the path is fixed at F1 is also fixed at
path v1 <--- F1 = 1.;
F1 = 1.;
A Complete PATH Model Specification Example
The following specification shows a more complete PATH model specification:
path v1 <--- v2 ,
v1 <--- v3 ;
pvar v1,
v2 = parm3,
v3 = parm3;
pcov v3 v2 = parm5(5.);
The two paths specified in the PATH statement have unnamed free effect parameters. These parameters are named by PROC CALIS with the _Parm prefix and unique integer suffixes. The error variance of v1
is an unnamed parameter, while the variances of v2 and v3 are constrained by using the same parameter parm3. The covariance between v2 and v3 is a free parameter named parm5, with a starting value of
Default Parameters in the PATH Model
There are two types of default parameters of the PATH model. One is the free parameters; the other is the fixed constants.
The following sets of parameters are free parameters by default:
• the variances or partial (or error) variances of all variables, manifest or latent
• the covariances among all exogenous (independent) mainfest or latent variables
• the means of all exogenous (independent) manifest variables if the mean structures are modeled
• the intercepts of all endogenous (dependent) manifest variables if the mean structures are modeled
For each of the default free parameters, PROC CALIS generates a parameter name with the _Add prefix and a unique integer suffix. Parameters that are not default free parameters in the PATH model are
fixed zeros by default. You can override almost all of the default zeros of the PATH model by using the MEAN, PATH, PCOV, and MEAN statements. The only exception is the single-headed path that has
the same variable on both sides. That is, the following specification is not accepted by PROC CALIS:
This path should always has a zero coefficient, which is treated as a model restriction that prevents a variable from having a direct effect on itself. | {"url":"http://support.sas.com/documentation/cdl/en/statug/63347/HTML/default/statug_calis_sect068.htm","timestamp":"2024-11-13T01:25:16Z","content_type":"application/xhtml+xml","content_length":"34406","record_id":"<urn:uuid:7d44bc09-a4b9-45f6-ac68-458777d2d7ae>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00751.warc.gz"} |
Math Hierarchy - Student-centered Instruction
Student-centered Instruction
Every student regularly experiences instruction that is student-centered and is designed to maximize students’ use of language. Lessons create space for students to participate in discourse to
promote conceptual understanding, which then leads to procedural fluency, problem-solving, and application.
• 5 Practices / Bansho / CGI
• 360° Math
• Three-Read Protocol
• Mathematical language routines
• Mathematical instructional routines
• Three-Act Math Tasks
• Think-Pair-Share-Compare
• Workshop / Stations / Math Centers
We subscribe to the commonly held belief amongst the education community that the person in the classroom who is doing the talking is probably the one who is doing the learning. In many typical
classrooms, teachers are doing most of the talking, which means the students are learning far less than they otherwise could. In response, classrooms need to be places where students are active
participants in their own learning who regularly participate in academic discourse with their peers. Moreover, students should be provided opportunities to invent their own solution strategies prior
to being taught standard algorithms that might be applicable.
To provide the type of student-centered instruction necessary for students to become mathematically literate, the teacher will use instructional strategies that are unlike a typical classroom
instruction of the past. We begin with the premise that students need to do most of the talking. In a typical classroom, the teacher does nearly all of the talking (up to 90%) and when students do
speak it is generally limited to short one- or two-word answers. Student engagement is at its lowest when teachers are talking. On the flip side, however, student engagement and comprehension improve
when they are the ones doing most of the talking. | {"url":"https://sites.google.com/mcoe.org/mathhierarchy/what-is-it/student-centered-instruction","timestamp":"2024-11-05T22:51:43Z","content_type":"text/html","content_length":"89579","record_id":"<urn:uuid:8b6cf1ab-d0f2-4f0b-a38d-8f8c2a6c51b4>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00111.warc.gz"} |
5.1 Example 1
5.2 Example 2
A common problem for a surveyor is the calculation of the surface area of a farmer's field. The fields are often irregular which makes direct calculation of their areas difficult. In such case fields
are divided into a number of regular areas (triangles, rectangles, etc.), of which the surfaces can be calculated with simple formulas. All areas are calculated separately and the sum of these areas
gives the total area of the field.
Figure 29 shows a field with an irregular shape of which the surface area must be determined.
Fig. 29 A field of irregular shape
The procedure to follow is:
Step 1
Make a rough sketch of the field (see Fig. 29a) indicating the corners of the field (A, B, C, D and E) and the field borders (straight lines). In addition some major landmark! are indicated (roads,
ditches, houses, trees, etc.) that may help to locate the field.
Fig. 29a A rough sketch of the field
Step 2
Divide the field, as indicated on the sketch, into areas with regular shapes. In this example, the field can be divided into 3 triangles ABC (base AC and height BB,), AEC (base AC and height EE[1])
and CDE (base EC and height DD[1]) (see Fig. 29b).
Fig. 29b Division of the field into areas with regular shapes
Step 3
Mark, on the field, the corners A, B, C, D and E with pegs.
Step 4
Set out ranging poles on lines AC (base of triangles ABC and AEC) and EC (base of triangle EDC) (see Fig. 29c) and measure the distances of AC and EC.
Fig. 29c Mark the corners with pegs and set out ranging poles
Step 5
Set out line BB (height of triangle ABC) perpendicular to the base line AC (see Fig. 29d) using one of the methods described in Chapter 4. Measure the distance BB,
Fig. 29d Set out line BB perpendicular to AC
Step 6
In the same way, the height EE, of triangle AEC and the height DD, of triangle CDE are set out and measured (see Fig. 29e)
Fig. 29e Set out line DD[1 ]perpendicular to EC and line EE1 perpendicular to AC
Step 7
The base and the height of the three triangles have been measured. The final calculation can be done as follows:
Triangle ABC: base = AC = 130 m
height = BB[1] = 55 m
Triangle ACE: base = AC ^= 130 m
height = EE[1 ]= 37 m
Triangle CDE: base = EC = 56 m
height = DD[1 ]= 55 m
Area = 0,5 x base x height
= 0.5 x 130 m x 55 m = 3 575 m^2
Area = 0.5 x 130 m x 37 m = 2 405 m
Area = 0.5 m x 56 m x 55 m= 1 540 m²
Field ABCDE:
Area of triangle ABC = 3 575 m^2
Area of triangle ACE = 2 405 m^2
Area of triangle CDE = 1 540 m^2
Total Area = 3 575 m^2 + 2 405 m^2 + 1 540 m^2
= 7 520 m- = 0.752 ha
The surface area of the field shown in Fig. 30 has to be determined at a time that the field is covered by a tall crop (e.g. maize or sugarcane).
Fig. 30 A field covered by a tall crop
The field can be divided into two triangles ABD and BCD (see Fig. 31a). Unfortunately, because of the tall crop, setting out and measurement of the base BD and the two heights AA[1 ]and CC[1 ]is
Fig. 31a Division of the field in two triangles
In this case, the area of triangle ABD can be calculated using AD as the base and BB[1 ]as the corresponding height. BB[1] can be set out and measured outside the cropped area. In the same way,
triangle BCD can be calculated using base BC and the corresponding height DD[1] (see Fig. 31b).
Fig. 31b Determination of the areas of the two triangles
The procedure to follow on the field is:
Step 1
Mark the 4 corners (A, B, C and D) with ranging poles.
Step 2
Line AD is set out with ranging poles and extended behind A. Line BC is also set out and extended behind C (see Fig. 32a). Measure the distances AD (base of triangle ADB) and BC (base of triangle
Fig. 32a Measurement of the bases of the two triangles
Step 3
Set out line BB[1 ] (height of triangle ABD) perpendicular to the extended base line AD using one of the methods described in Chapter 4. In the same way, line DD[1 ] (height of triangle BCD) is set
out perpendicular to the extended base line BC (See Fig. 32b) Measure the distance BB[1 ]and DD[1].
Fig. 32b Measurement of the heights of the two triangles
Step 4
The base and height of both triangles have been measured. The final calculations can be done as follows:
Triangle ABD: base = AD = 90 m
height = BB[1] - 37 m
Triangle BCD: base = BC = 70 m
height = DD[1] - 50 m
Area = 0.5 x base x height
= 0.5 x 90 m x 37 m = 1 665 m^2
Area = 0.5 x 70 m x 50 m = 1 750 m^2
Field ABDC:
Area triangle ABD = 1 665 m²
Area triangle BCD = 1 750 m^2
Total Area = 1 665 m^2 + 1 750 m^2 = 3 415 m^2
= 0.3415 ha ^= approx. 0.34 ha | {"url":"https://www.fao.org/4/R7021E/r7021e06.htm","timestamp":"2024-11-02T06:07:51Z","content_type":"text/html","content_length":"7675","record_id":"<urn:uuid:47c6053e-39d5-47c3-b3e5-63cefedde6e2>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00105.warc.gz"} |
The bank sets a range of interest rates for each loan type. The rates individual borrowers are charged are based on their credit scores, income, and current. interest rates will be on September 18,
Prime Rate Definition. The U.S. Prime Rate is a commonly used, short-term interest rate in the banking system of. Today's competitive mortgage rates ; year fixed · % · % · ; year fixed · % · % · ; 5y
/6m ARM · % · % · That means that when the Fed raises interest rates, the prime rate also goes up. You can find the current prime rate on the Wall Street Journal website. If the prime rate rises, the
interest rates on your loans and adjustable-rate credit cards will rise as well. Second, the prime rate affects liquidity in the.
Mortgage rates refer to the interest you pay on your home loan. It's the cost your lender charges you for borrowing the money, just like the interest rate on a. Graph and download economic data for
Bank Prime Loan Rate Changes: Historical Dates of Changes and Rates (PRIME) from to about prime. The current Bank of America, NA prime rate is % (rate effective as of July 27, ). The prime rate is
set by Bank of America based on various factors. Prime Rate ; Bank of North Dakota Base Rate · % · 07/27/ See the mortgage rate a typical consumer might see in the most recent Primary Mortgage Market
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Easy Ways To Build Credit Fast | Financial Planner Richmond | {"url":"https://itpl.site/overview/what-is-prime-rate-of-interest-today.php","timestamp":"2024-11-05T19:19:20Z","content_type":"text/html","content_length":"15203","record_id":"<urn:uuid:8cc73f41-a56c-4e9a-aa7d-3912b0f19782>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00618.warc.gz"} |
Excel Formula for Correlation Coefficient in Python
In this tutorial, you will learn how to calculate the correlation coefficient between two columns in Excel using Python. The correlation coefficient is a statistical measure that quantifies the
strength and direction of the linear relationship between two variables. In this case, we will be calculating the correlation coefficient between column C (TTESS Rank) and column B (Growth Score) for
different classifications such as AP, Fine Arts, STAAR, CTE, and Early Childhood, which are listed in column D.
To calculate the correlation coefficient, we will be using the CORREL function in combination with the IF function. The IF function allows us to check if the value in column D matches a specific
classification, and if it does, we select the corresponding values from column C and column B. If the condition is not met, we use an empty string as a placeholder.
The formula to calculate the correlation coefficient between column C and column B for a specific classification is as follows:
=CORREL(IF(D:D="AP", C:C, ""), IF(D:D="AP", B:B, ""))
You can repeat this formula for each classification by changing the condition in the IF function. For example, to calculate the correlation coefficient for the classification "Fine Arts", you would
use the following formula:
=CORREL(IF(D:D="Fine Arts", C:C, ""), IF(D:D="Fine Arts", B:B, ""))
By using this approach, you can calculate the correlation coefficient between column C and column B for each classification listed in column D.
Let's consider an example to better understand how this formula works. Suppose we have the following data in columns B, C, and D:
| B | C | D |
| | | |
| 10 | 5 | AP |
| 15 | 8 | Fine Arts |
| 20 | 12 | STAAR |
| 25 | 15 | CTE |
| 30 | 18 | Early Childhood|
To calculate the correlation coefficient between column C (TTESS Rank) and column B (Growth Score) for the classification "AP", you would use the following formula:
=CORREL(IF(D:D="AP", C:C, ""), IF(D:D="AP", B:B, ""))
This formula would return the correlation coefficient between the TTESS Rank and Growth Score for the "AP" classification.
You can repeat the formula for other classifications by changing the condition in the IF function. For example, to calculate the correlation coefficient for the classification "Fine Arts", you would
use the following formula:
=CORREL(IF(D:D="Fine Arts", C:C, ""), IF(D:D="Fine Arts", B:B, ""))
By following these steps, you can calculate the correlation coefficient between column C and column B for each classification listed in column D using Python.
A Google Sheets formula
=CORREL(IF(D:D="AP", C:C, ""), IF(D:D="AP", B:B, ""))
Formula Explanation
This formula uses the CORREL function in combination with the IF function to calculate the correlation coefficient between column C (TTESS Rank) and column B (Growth Score) for each classification
(AP, Fine Arts, STAAR, CTE, and Early Childhood) in column D.
Step-by-step explanation
1. The IF function is used to check if the value in column D is equal to a specific classification (e.g., "AP").
2. If the condition is met (i.e., the value in column D is equal to the classification), the corresponding values from column C and column B are selected.
3. If the condition is not met, an empty string ("") is used as a placeholder.
4. The CORREL function is then used to calculate the correlation coefficient between the selected values from column C and column B.
5. The formula can be repeated for each classification by changing the condition in the IF function.
For example, let's say we have the following data in columns B, C, and D:
| B | C | D |
| | | |
| 10 | 5 | AP |
| 15 | 8 | Fine Arts |
| 20 | 12 | STAAR |
| 25 | 15 | CTE |
| 30 | 18 | Early Childhood|
To calculate the correlation coefficient between column C (TTESS Rank) and column B (Growth Score) for the classification "AP", the formula would be:
=CORREL(IF(D:D="AP", C:C, ""), IF(D:D="AP", B:B, ""))
This formula would return the correlation coefficient between the TTESS Rank and Growth Score for the "AP" classification.
You can repeat the formula for other classifications by changing the condition in the IF function. | {"url":"https://codepal.ai/excel-formula-generator/query/ZmU2IExR/excel-formula-correlation-coefficient-python","timestamp":"2024-11-03T19:12:54Z","content_type":"text/html","content_length":"118881","record_id":"<urn:uuid:a3d4d2f8-3c1d-433d-a320-11aa25f7018a>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00522.warc.gz"} |
ECE 2120 Electrical Engineering Laboratory II (2024)
ECE 2120 Electrical Engineering Laboratory II
A Companion Course to ECE 2620 - Electrical Circuits II
By Dr. Apoorva Kapadia (Undergraduate Laboratory Coordinator)
Dr. Afshin Ahmadi (Former Laboratory Teaching Assistant)
Updated on August 17, 2020 The Holcombe Department of Electrical & Computer Engineering Clemson University Clemson, SC - 29634.
2 Contents
Course Information i Introduction ...... i Student Responsibilities ...... i Laboratory Teaching Assistant Responsibilities ...... ii Faculty Coordinator Responsibilities ...... ii Lab Policy and
Grading ...... ii Course Goals and Objectives ...... iii Use of Laboratory Instruments ...... iv Instrument Protection Rules ...... iv Data Recording and Reports ...... v The Laboratory Notebook
...... v The Lab Report ...... v
Lab 1 - Orientation 1
Lab 2 - Average and RMS Values 2
Lab 3 - Capacitors and Series RC Circuits 9
Lab 4 - Inductors and Series RL Circuits 18
Lab 5 - Parallel RC and RL Circuits 25
Lab 6 - Circuit Resonance 33
Lab 7 -Filters: High-pass, Low-pass, Bandpass, and Notch 42
Lab 8 - Transformers 52
Lab 9 - Two-Port Network Characterization 61
Lab 10 - Final Exam 70
Appendix A - Safety 72
Appendix B - Instruments for Electrical Measurements 78
Appendix C - Operating Instructions for a Typical Oscilloscope 82
Appendix D - LT SPICE AC Circuit Simulation 88 Course Information
1 Introduction
This course is intended to enhance the learning experience of the student in topics encountered in ECE 2620. In this lab, students are expected to gain experience in using the basic measuring devices
used in electrical engineering and in interpreting the results of measurement operations in terms of the concepts introduced in the second electrical circuits course. How the student performs in the
lab depends on his/her preparation, participation, and teamwork. Each team member must participate in all aspects of the lab to insure a thorough understanding of the equipment and concepts. The
student, lab teaching assistant, and faculty coordinator all have certain responsibilities toward successful completion of the lab’s goals and objectives.
1.1 Student Responsibilities
The student is expected to be prepared for each lab. Lab preparation includes reading the lab experiment and related textbook material. If you have questions or problems with the preparation, contact
your Laboratory Teaching Assistant (LTA), but in a timely manner. Do not wait until an hour or two before the lab and then expect the LTA to be immediately available.
Active participation by each student in lab activities is expected. The student is expected to ask the LTA any questions they may have. Do not make costly mistakes because you did not ask a question
before proceeding.
A large portion of the student’s grade is determined in the comprehensive final exam, resulting in a requirement of understanding the concepts and procedure of each lab experiment for the successful
completion of the lab class. The student should remain alert and use common sense while performing a lab experiment. They are also responsible for keeping a professional and accurate record of the
lab experiments in the lab manual wherever tables are provided. Students should report any errors in the lab manual to the teaching assistant.
i 1.2 Laboratory Teaching Assistant Responsibilities
The LTA shall be completely familiar with each lab prior to class. The LTA shall provide the students with a syllabus and safety review during the first class. The syllabus shall include the LTA’s
office hours, telephone number, and the name of the faculty coordinator. The LTA is responsible for ensuring that all the necessary equipment and/or preparations for the lab are available and in
working condition. Lab experiments should be checked in advance to make sure everything is in working order. The LTA should fully answer any questions posed by the students and supervise the students
performing the lab experiments. The LTA is expected to grade the lab notebooks and reports in a fair and timely manner. The reports should be returned to the students in the next lab period following
submission. The LTA should report any errors in the lab manual to the faculty coordinator.
1.3 Faculty Coordinator Responsibilities
The faculty coordinator should ensure that the laboratory is properly equipped, i.e., that the teaching assistants receive any equipment necessary to perform the experiments. The coordinator is
responsible for supervising the teaching assistants and resolving any questions or problems that are identified by the teaching assistants or the students. The coordinator may supervise the format of
the final exam for the lab. They are also responsible for making any necessary corrections to this manual and ensuring that it is continually updated and available.
1.4 Lab Policy and Grading
The student should understand the following policy:
ATTENDANCE: Attendance is mandatory and any absence must be for a valid excuse and must be documented. If the instructor is more than 15 minutes late, students may consider lab for the day cancelled.
LAB RECORDS: The student must:
1. Perform the PreLab assignment before the beginning of each lab,
2. Keep all work in preparation of and obtained during lab; and
3. Prepare a lab report on experiments selected by the LTA.
GRADING POLICY: The final grade of this course is determined using the criterion detailed in the syllabus.
PRE-REQUISITES AND CO-REQUISITES: The lab course is to be taken during the same semester as ECE 2620, but receives a separate grade. If ECE 2620 is dropped, then ECE 2120 must be dropped as well.
Students are required to have completed both ECE 202, MTHSC 206 and PHYS 221 with a C or better grade in each. Students are also assumed to have completed a programming class and be familiar with the
use of a computer-based word processor.
ii Note that the instructor reserves the right to alter any part of this information at their discretion. Any changes will be announced in class and distributed in writing to the students prior to
the changes taking effect.
1.5 Course Goals and Objectives
The Electrical Circuits Laboratory II is designed to provide the student with the knowledge to use basic measuring instruments and techniques with proficiency. These techniques are designed to
complement the concepts introduced in ECE 2620. In addition, the student should learn how to record experimental results effectively and present these results in a written report. More explicitly, the
class objectives are:
1. To gain proficiency in the use of common measuring instruments.
2. To enhance understanding of advanced electric circuit analysis concepts including:
• Inductance, Capacitance, and Reactance,
• AC voltage and current addition. Phasors,
• AC power (real and reactive, instantaneous and average),
• Series and parallel resonant circuit behavior,
• Passive Filters,
• Transfer functions,
• Transformers,
• Two-port network analysis;
3. To develop communication skills through:
• Maintenance of succinct but complete laboratory notebooks as permanent, written descriptions of procedures, results, and analyses.
• Verbal interchanges with the laboratory instructor and other students.
• Preparation of succinct but complete laboratory reports.
4. To compare theoretical predictions with experimental results and to determine the source of any apparent differences.
iii 2 Use of Laboratory Instruments
One of the major goals of this lab is to familiarize the student with the proper equipment and techniques for making electrical measurements. Some understanding of the lab instruments is necessary to
avoid personal or equipment damage. By understanding the device’s purpose and following a few simple rules, costly mistakes can be avoided. You have already, in ECE 211, learned these rules, but they
are repeated for convenience and emphasis below. Most of the instrumentation used in this laboratory is implemented through National Analog Discovery 2, breadboard, and circuit analysis system.
In general, all devices have physical limits. These limits are specified by the device manufacturer and are referred to as the device rating. The ratings are usually expressed in terms of voltage
limits, current limits, or power limits. It is up to the engineer to make sure that in device operation, these ratings (limit values) are not exceeded. The following rules provide a guideline for
instrument protection.
2.1 Instrument Protection Rules
1. Set instrument scales to the highest range before turning on the power/source.
2. Be sure instrument grounds are connected properly. Avoid accidental grounding of ”hot” leads, i.e., those that are above ground potential.
3. Check polarity markings and connections of instruments carefully before connecting power.4)Never connect an ammeter across a voltage source. Only connect ammeters in series with loads.
4. Never connect an ammeter across a voltage source. Only connect ammeters in series with loads. An ammeter is a low-resistance device that, if connected in parallel, will short out most components
and usually destroy the ammeter or its protecting fuse.
5. Do not exceed the voltage and current ratings of instruments or other circuit elements. This particularly applies to wattmeters since the current or voltage rating may be exceeded with the needle
still on the scale.
6. Be sure the fuse and circuit breakers are of suitable value. When connecting electrical elements to make up a network in the laboratory, it is easy to lose track of various points in the network
and accidentally connect a wire to the wrong place. A procedure to follow that helps to avoid this is to connect the main series part of the network first, then go back and add the elements in
parallel. As an element is added, place a small check by it on your circuit diagram. Then go back and verify all connections before turning on the power. One day someone’s life may depend upon your
making sure that all has been done correctly.
iv 3 Data Recording and Reports
3.1 The Laboratory Notebook
Students must record their experimental values in the provided tables in this laboratory manual and reproduce them in the lab reports. Reports are integral to recording the methodology and results of
an experiment. In engineering practice, the laboratory notebook serves as an invaluable reference to the technique used in the lab and is essential when trying to duplicate a result or write a
report. Therefore, it is important to learn to keep accurate data. Make plots of data and sketches when these are appropriate in the recording and analysis of observations. Note that the data
collected will be an accurate and permanent record of the data obtained during the experiment and the analysis of the results. You will need this record when you are ready to prepare a lab report.
3.2 The Lab Report
Reports are the primary means of communicating your experience and conclusions to other profes- sionals. In this course you will use the lab report to inform your LTA about what you did and what you
have learned from the experience. Engineering results are meaningless unless they can be communicated to others. You will be directed by your LTA to prepare a lab report on a few selected lab
experiments during the semester. Your assignment might be different from your lab partner’s assignment.
Your laboratory report should be clear and concise. The lab report shall be typed on a word processor. As a guide, use the format on the next page. Use tables, diagrams, sketches, and plots, as
necessary to show what you did, what was observed, and what conclusions you can draw from this. Even though you will work with one or more lab partners, your report will be the result of your
individual effort in order to provide you with practice in technical communication.
3.2.1 Formatting and Style
• The lab report shall be typed in a word processor.
• All page margins must be 1.25 inches. All content (including text, figures, tables, etc.) must fit within the margins.
• Body text should be double-spaced.
• Basic text should be in 12-point size in a commonly used text font.
• Set your main text justified (with even left/right margins).
• The first line of each paragraph should have a left indent.
• All the tables should have titles and should be numbered. Tables should be labelled numerically as Table 1, Table 2, etc. Table captions appear above the table. The column headings should be
labeled with the units specified.
v • Graphs should be presented as figures. All the figures should have titles and should be numbered. Figure captions appear below the figure. Graphs should have labeled axes and clearly show the scales
and units of the axes.
• All the figures and tables must be centered on the page.
• All the figures and tables in your report must be referenced in your discussion. References to figures in the main body of the text are usually written in abbreviated form (e.g. ‘see Fig. 1”).
• Use MS-Word equation (under Insert → Equation menu), MathType, or a similar tool to type formulas.
• If you need to copy a schematic or figure from the lab manual to your report, use Copy and Paste function or take a screenshot by using Snipping Tool in MS-Windows.
• Do not place screenshots of your lab notebook in the report! Diagrams, tables, calculations, etc. must be generated using the existing tools in the word processor.
3.2.2 Order of Lab Report Components
COVER PAGE - Cover page must include lab name and number, your name, your lab partner’s name, and the date the lab was performed.
OBJECTIVE - Clearly state the experiment objective in your own words.
EQUIPMENT USED - Indicate which equipment was used in performing the experiment.
FOR EACH PART OF THE LAB:
• Write the lab’s part number and title in bold font.
• Firstly, describe the problem that you studied in this part, give an introduction of the theory, and explain why you did this experiment. Do not lift the text from the lab manual; use your own
• Secondly, describe the experimental setup and procedures. Do not follow the lab manual in listing out individual pieces of equipment and assembly instructions. That is not relevant information in a
lab report! Instead, describe the circuit as a whole (preferably with diagram), and explain how it works. Your description should take the form of a narrative, and include information not present in
the manual, such as descriptions of what happened during intermediate steps of the experiment.
• Thirdly, explain your findings. This is the most important part of your report, because here, you show that you understand the experiment beyond the simple level of completing it. Explain (compare
expected results with those obtained). Analyse (analyze experimental error). Interpret (explain your results in terms of theoretical issues and relate to your experimental objectives). This part
includes tables, graphs, and sample calculations. When showing calculations, it is usual to show the general equation, and one worked example. All the results should be presented even
vi if there is any inconsistency with the theory. It should be possible to understand what is going on by just reading through the text paragraphs, without looking at the figures. Every figure/table
must be referenced and discussed somewhere in the text.
• Finally, provide a summary of what was learned from this part of the laboratory experiment. If the results seem unexpected or unreliable, discuss them and give possible explanations.
CONCLUSIONS - The conclusion section should provide a take-home message summing up what has been learned from the experiment:
• Briefly restate the purpose of the experiment (the question it was seeking to answer)
• Identify the main findings (answer to the research question)
• Note the main limitations that are relevant to the interpretation of the results
• Summarise what the experiment has contributed to your understanding of the problem.
PROBING FURTHER QUESTIONS - Questions pertaining to this lab must be answered at the end of laboratory report.
Lab 1 - Orientation
In the first lab period, the students should become familiar with the location of equipment and components in the lab, the course requirements, and the teaching instructor. Students should also make
sure that they have all of the co-requisites and pre-requisites for the course at this time.
To familiarize the students with the lab facilities, equipment, standard operating procedures, lab safety, and the course requirements.
Read the Introduction and Appendix A, of this manual. Download and install the “WaveForms” software on your personal computer, available here.
Equipment Needed
ECE 2120 lab manual.
1. During the first laboratory period, the instructor will provide the students with a general idea of what is expected from them in this course. Each student will receive a copy of the syllabus,
stating the instructor’s contact information. In addition, the instructor will review the safety concepts of the course. 2. During this period, the instructor will briefly review the equipment which
will be used throughout the semester. The location of instruments, equipment, and components (e.g. resistors, capacitors, connecting wiring) will be indicated. The guidelines for instrument use will
be reviewed.
1 Lab 2 - Average and RMS Values
Waveforms of voltage and current that vary periodically with time may be characterized by their average value or their root mean square (rms) value. The latter is used to determine the power
supplied, dissipated, or stored by a circuit element. Some of the measuring instruments you will use respond to average values of voltage or current, while others respond to rms values.
By the end of this lab, the student should learn how to determine the values of rms voltage for three types of waveforms: a sinusoid, a square wave, and a triangular wave. Also, the student should
understand the difference between a true-rms and a conventional multimeter.
Read Appendix B and Appendix C of this manual, paying particular attention to the methods of using measurement instruments. Prior to coming to lab class, complete Part 0 of the Procedure.
Analog Discovery 2 Instrument. Digital Multimeter. Resistance Box.
2 Background
Measurements of AC signal:
• Amplitude is a non-negative scalar measure of a wave’s maximum magnitude of oscillation. In electrical engineering it may be thought of as the maximum absolute value reached by a voltage or current
waveform as measured from the center of the oscillation. An amplitude measurement may be reported as peak, peak-to-peak, average, or RMS. • Peak amplitude is the height of an AC waveform as measured
from the center of the oscillation to the highest positive or lowest negative point on a graph. Also known as the crest amplitude of a wave. • Peak-to-peak amplitude is the total height of an AC
waveform as measured from maximum positive to minimum negative peaks (the highest peak to the lowest valley) on a graph of the waveform. Often abbreviated as “P-P”, e.g., Vp−p or Vpp. • Average value
is the arithmetic “mean” of a waveform’s values over one cycle. The average value of any waveform with equal-area portions above and below the “zero” line on a graph is zero. However, often as a
practical measure of amplitude, a waveform may be characterized by its average absolute value, calculated as the arithmetic mean of the absolute values of the waveform. • “RMS” stands for Root Mean
Square, and is a way of expressing an AC quantity of voltage or current in terms functionally equivalent to DC. For example, 10 volts AC RMS is the amount of AC voltage that would produce the same
amount of heat dissipation across a resistor of given value as a 10 volt DC power supply. Also known as the “equivalent” or “DC equivalent” value of an AC voltage or current. • Analog,
electromechanical meter movements respond proportionally to the average absolute value of an AC voltage or current. When RMS indication is desired, the meter’s calibration must be adjusted
accordingly, usually by applying a constant multiplicative factor assumed for a sinusoidal wave. This means that the accuracy of an electromechanical meter’s RMS indication is dependent on the purity
of the waveform and whether it is the same wave shape as the waveform used in calibrating.
Figure 2.1: Measurements of AC Signal
3 In this experiment, we will work with periodic waveforms having period T and amplitude Vm. Specifically, we will work with the following three types of waveforms:
1. A sinusoidal wave voltage: 2Π v(t) = V cos( t) (2.1) m T 2. A square-wave voltage: T 3T v(t) = Vm 0 ≤ t ≤ ; T ≤ t ≤ ; etc 2 2 (2.2) T 3T v(t) = −Vm 2 ≤ t ≤ T ; 2 ≤ t ≤ 2T ; etc
3. A triangular-wave voltage: (t−nT ) 1 1 v(t) = 4Vm[ T ](n − 4 )T ≤ t ≤ (n + 4 )T ) 1 (2.3) (t−(n+ 4 )T ) 1 3 v(t) = Vm[ T ](n + 4 )T ≤ t ≤ (n + 4 )T ) where n = 0,1,2,3,...
AC voltage and current waveforms are further depicted in a number of ways. Two of the most common and useful are the average and the root-mean-square (rms) values.
The average value of a time-varying waveform x(t) with period T is given by:
1 Z T Xavg = x(t)dt (2.4) T 0
The root-mean-square value, useful for power calculations, is defined by: s Z T 1 2 Xrms = x (t)dt (2.5) T 0
Since the sine, square, and triangular waveforms are symmetrical about the time axis, they all have mathematical average voltages of zero. However, each waveform will have an rms value, and a summary
of the calculation steps for relating the voltage magnitude to the rms value for each waveform is shown below:
Sinusoidal Voltage: s s s Z T Z T 2 Z T cos(2 2π t) 1 2 1 2π 1 1 T Xrms = v (t)dt = Vm sin( t) dt = Vm − dt T 0 T 0 T T 0 2 2 s 1 1 Vm = Vm T + 0 = √ (2.6) T 2 2
4 Square-Wave Voltage: s v s Z T u "Z T/2 Z T # 1 2 u 1 2 2 1 2 T 2 T Xrms = v (t)dt = t (V m) dt + (−V m) dt = (V m) + (−V m) T 0 T 0 T/2 T 2 2 r 1 = [(V )2T ] = V (2.7) T m m
Triangular-Wave Voltage: s v Z T u "Z T/4 Z 3T/4 Z T # 1 2 u 1 4V m 2 4V m 2 4V m 2 Xrms = v (t)dt = t ( t) dt + (2V m − t) dt + (−4V m + t) dt T 0 T 0 T T/4 T 3T/4 T s 1 V 2T 2V 2T V 2T V = m + m +
m = √m (2.8) T 12 12 12 3
If we use what is called a “true rms” voltage meter, then the relationship between the magnitude of the waveform and the measured value would be given by the three equations given above. However,
many of the meters available are not “true rms” meters, and as a result are only designed for measurements in circuits with either DC voltages or sinusoidal AC voltages. For obtaining the AC voltage,
most digital meters effectively perform a full-wave rectification of the waveform and compute the average absolute value. A constant factor is then applied to compute an RMS value. Often the constant
factor is chosen to give a correct result for a sinusoidal waveform.
For a sinusoidal waveform, the average value for the rectified voltage is calculated as follows: Z T Z T "Z T/2 Z T # 1 1 2π Vm 2π 2π Xavg−rect = |v(t)|dt = |Vm sin( t)|dt = Vm sin( t)dt + −Vm sin( t)
dt T 0 T 0 T T 0 T T/2 T Z T/2 2Vm 2π 2Vm = Vm sin( t)dt = (2.9) T 0 T π √ Since the rms value for a sinusoidal waveform should be related to the amplitude by 2, we need to apply a conversion factor
to get the correct rms value on the meter readout. In this case the conversion factor would be: 1 π Vrms = √ Vavg−rectified = 1.111 ∗ Vavg−rectified (2.10) 2 2
Now what happens when we measure a square wave with 50% duty cycle? To find out, we compute the average of the rectified square-wave waveform: " # 1 Z T 1 Z T/2 Z T Vavg−rect = |v(t)|dt = |Vm|dt + | −
Vm|dt = Vm (2.11) T 0 T 0 T/2
The non-true-rms meter will apply the same conversion factor it applied to the sine wave. Hence, what we will see on the meter readout for a square wave is:
Vmeter = 1.111 ∗ Vm (2.12)
5 This meter will report a rms value that is 11% higher than the actual rms value we should have for a square wave.
Finally, for a triangular waveform, the average rectified voltage is:
1 Z T Vavg−rect = |v(t)|dt T 0 "Z T/4 Z T/2 Z 3T/4 Z T # 1 4V m 2 4V m 2 4V m 2 4V m 2 = ( t) dt + (2V m − t) dt + (−2V m + t) dt + (4V m − t) dt T 0 T T/4 T T/2 T 3T/4 T V = m (2.13) 2
So for a triangular waveform the rms voltage indicated on the non-true-rms meter will be: V m V = 1.111 ∗ = 0.555 ∗ V (2.14) meter 2 m
√ while it should be registering Vm/ 3 = 0.577 ∗ Vm. This meter gives a reading only 96% of what we should actually get, and so will under-report the rms voltage for a triangular wave.
In practice you will not get the exact results predicted by the equations, due to a number of errors, such as inability to set the peak voltage to the exact value, slight errors in the meter reading,
and inaccuracies in the shape of the waveform produced by the function generator. Also, we have assumed that the duty cycle of the square wave is exactly 50% in these calculations, which might not
actually be the case for your waveform generator.
Part (0) After answering the following questions, review the laboratory exercise procedures and plan how you will use the experience gained in these calculations to find the values sought.
1. Using the information provided in the Background section, calculate the average voltage, the average absolute voltage, and the rms voltage values for the symmetrical sine, square, and triangular
waveforms, assuming that the peak value of each waveform is 2V.
Table 2.1: Calculated Voltage Values
Average Voltage Average Absolute Voltage RMS Voltage
Sine Square Triangular
6 2. Compute the RMS voltage value that would be reported by non-true rms voltmeters, assuming that the peak value of each waveform is 2V. You may use the equations derived in the Background section
for these calculations.
Table 2.2: Reported Voltage Value by non-true RMS voltmeters
Reported RMS Voltage
Sine Square Triangular
Part (1) Setup the circuit in Figure 2.2, and connect the CH1 of the oscilloscope across the resistor. Use the “W1” and Ground terminals of the Analog Discovery for the Function Generator, and “1+”
and “1-” pins for the Oscilloscope.
2Vpeak 1kΩ
Figure 2.2: A Simple Resistive Circuit
1. Open the WaveForms software. Use the settings under the “Wavegen” menu to output a 1kHz sine wave with amplitude = 2 V-Peak and DC offset = 0V. “RUN” the FGEN.
(a) Open the Scope menu and display the voltage across the resistor. You may need to adjust the oscilloscope’s Range and Base settings to obtain a good display of the waveforms. (b) Use the
“Measurements” function to display the AC RMS value (available under the “View” menu). (c) Record the RMS value in Table 2.3. (d) Disconnect the oscilloscope and connect the Digital Multimeter (DMM)
across the resistor (VΩ and COM) to measure the RMS voltage. Record the value in Table 2.3.
2. Repeat the steps from 1.a to 1.d using a triangular wave output from the Function Generator. Record your values in Table 2.3.
3. Repeat the steps from 1.a to 1.d using a square wave output from the Function Generator. Record your values in Table 2.3.
7 Table 2.3: Calculated and Measured Voltage Values
Calculate RMS Voltage Measured RMS Voltage Measured RMS Voltage (Table 2.1) (Oscilloscope) (DMM)
Sine Triangular Square
4. Adjust the DC offset control of FGEN to add a 2 Volt DC offset to the original 1 kHz sine wave. Now the waveform should vary from 0 to 4 Volts. Use the DMM to measure the AC RMS voltage and the DC
voltage across the resistance. Record the values. Repeat this same step for triangular wave and square wave.
Table 2.4: Measured Voltage Values with 2V DC Offset
AC RMS Voltage DC Voltage (DMM) (DMM)
Sine Square Triangulare
Probing Further Questions
Q1. Compare the RMS voltage value calculations you made for the PreLab to the AC voltage measure- ments made in 1th, 2th, 3th, and 4th steps in the Procedure. Do your measured values agree with the
calculated values in all cases? If not, why?
Q2. Summarize your comparisons: Does the Fluke digital multimeter provide true RMS measurements of voltage for all three waveforms? Justify your conclusion.
8 Lab 3 - Capacitors and Series RC Circuits
Linear circuit elements — resistors, capacitors, and inductors — are the backbone of electrical and electronic circuits. These three types of elements respond to electrical voltages in different ways,
variously consuming, storing, or supplying electrical energy. Understanding these behaviors and learning to calculate the result of combining elements is critical for designing and working with
electric circuits. While a resistor consumes electrical energy, converting it to heat, capacitors and inductors vary their responses according to the frequency of the voltage or current applied to
them. This laboratory will explore those responses for series-connected capacitors.
• Learn to measure capacitive reactance.
• Learn to measure phase angles between voltages.
• Learn to draw impedance and voltage phasor diagrams for resistors and capacitors in series.
• Understand how impedance and voltage phasors add (i.e., like vectors).
• Learn to simulate AC series circuit in LT SPICE.
• Confirm how capacitances add when two capacitors are connected in parallel; in series.
• Determine the reactance of a capacitor in a series RC circuit by measuring voltages.
• Draw impedance and voltage phasor diagrams for a series RC circuit.
• Explain the effect of frequency on the impedance and voltage phasors for a series RC circuit.
9 PreLab
• Read and study the Background section of this Laboratory. • Read Appendix C, especially Avoiding Grounding Errors with Oscilloscope, Voltage Measurement, and Phase-Angle Measurement for an
oscilloscope with two vertical inputs.
• Prior to coming to lab class, complete Part 0 of the Procedure.
Analog Discovery 2 Instrument. Digital Multimeter. Resistance Substitution Box. Capacitor: 2 x 0.01 µF.
A capacitor is formed whenever two conductors are separated by an insulating material. Consider the simple example of two parallel conducting plates separated by a small gap that is filled with an
insulating material (vacuum, air, glass, or other dielectric). If a potential difference exists between the two plates, then an electric field exists between them, and opposite electric charges will be
attracted to the two plates. The ability to store that electric charge is a fundamental property of capacitors. The larger the plates, the more charge can be stored. The closer the plates, the more
charge can be stored. . . at least until the charges leap the gap and the dielectric breaks down.
If a voltage source is connected across a capacitor, charge will flow in the external circuit until the voltage across the capacitor is equal to the applied voltage. The charge that flows is
proportional to the size of the capacitor (its “capacitance”) and to the applied voltage. The relationship is given by the equation: Q = CV (3.1) where Q is the charge in coulombs, C is the
capacitance in farads, and V is the applied voltage in volts.
Capacitors in Series
Electric current I is the amount of charge that flows per unit time; that is, I = Q/T. Thus, the total charge that flows through a circuit (or a capacitor) is Q = IT. So, if two capacitors are
connected in series and a voltage is applied across the pair, the same current, and therefore the same charge, must flow through both capacitors, and the total voltage VT must be divided across both
capacitors: Q Q 1 1 Q VT = V1 + V2 = + = Q + = (3.2) C1 C2 C1 C2 CT
10 where V1 and V2 are the voltages across the capacitors with capacitances C1 and C2. Thus, CT , the total capacitance of two capacitors in series, is found by:
1 1 1 C1C2 = + or CT = (3.3) CT C1 C2 C1 + C2
Capacitors in Parallel
Connecting capacitors in parallel is effectively the same as making a single capacitor’s plates larger, and therefore able to hold more charge for a given applied voltage. This simple view is borne
out if one analyzes the flow of charge through a parallel array of capacitors connected to a voltage source. The result of such analysis is that capacitances in parallel add directly:
CT = C1 + C2 + C3 + ... (3.4)
Time Constant
If a voltage V0 is applied to a capacitor C connected in series with a resistor R, the voltage across the capacitor gradually increases. The rate at which the capacitor’s voltage changes is
characterized by a “time constant”, τ: τ = RC (3.5) where τ is the time required for the voltage on the capacitor to rise from 0 to 0.632 V0. τ is also the time required for the voltage of a fully
charged capacitor to fall from V0 to 0.368 V0. The number 0.368 = e−1 and the number 0.632 = (1 – e−1 ).
Capacitive Reactance
Reactance is a characteristic exhibited by capacitors and inductors in circuits with time-varying voltages and currents, such as common sinusoidal AC circuits. Like resistance, reactance opposes the
flow of electric current and is measured in ohms. Capacitive reactance XC can be found by the equation: 1 X = (3.6) c 2πfc where f is the frequency of the applied voltage or current and C is the
capacitance in farads. As with resistance, reactance obeys Ohm’s law:
Vc Vc = IcXc or Xc = (3.7) Ic
If a sinusoidal voltage is applied across a resistor, the current through the resistor is in phase with the voltage. That is not true for a capacitor. If we connect a capacitor across a sinusoidal
voltage, the maximum current flows through the capacitor when the voltage’s rate of change is maximum (i.e., at V=0), and diminishes as the voltage on the capacitor increases, until finally the current
is zero when the voltage is at maximum and its derivative is zero. At that instant, the maximum possible charge for the applied voltage is stored in the capacitor, and so the flow of charge (i.e., the
current) stops. The current and the voltage have exactly the same frequency, but the current through the capacitor is 1 ◦ leading the voltage by 4 cycle — 90 or π/2 radians. Figure 3.1 illustrates
this relationship.
11 Figure 3.1: Resistive, Inductive, and Capacitive Loads
When a sinusoidal voltage at frequency f drives a circuit that contains only linear elements, the waveforms throughout the circuit are also sinusoidal and at the same frequency. To understand the
relationships among the sinusoidal voltages, currents, and impedances, we represent the various wave- forms as two-dimensional vectors called phasors. A phasor is a complex number used to represent a
sinusoidal wave, taking into account both its amplitude and phase angle. As a complex number, a phasor has “real” and “imaginary” components, but like any two-dimensional vector, it can be drawn
simply on ordinary XY axes, with the “real” axis in the usual X direction and the “imaginary” axis in the usual Y direction. Such phasor drawings are very helpful in analyzing circuits and
understanding the relationships of the various voltages and currents. The algebra of complex numbers can then be used to perform arithmetic operations on the sinusoidal waves. Make no mistake: adding
voltages or currents in an AC circuit without taking account of phase angles will lead to confusing and wrong results.
A series RC circuit is illustrated in Figure 3.2(a) and a phasor diagram of its impedances is shown in Figure 3.2(b). The vector for resistance R is shown along the Real (X) axis, while the reactance
XC is shown in the negative Imaginary (Y) axis, since its voltage trails its current by 90◦. The vector sum of R and XC is labeled Z and has magnitude 5kΩ. Therefore, the magnitude of the current
through the circuit is VS / Z = 1.0 mA.
Notice that the current phasor is in the same as the direction of the voltage across the resistor, because voltage and current are in phase for a resistor. The current is the same throughout the
series circuit, including the capacitor, and the voltage across the capacitor trails the current by 90◦. The source voltage is the vector sum of the voltages across the resistor and the capacitor, as
is illustrated
12 in the phasor diagram, Figure 3.2(c).
Figure 3.2: Phasor Diagram
Safety Precautions
1. Always turn off power to the circuit when changing the circuit.
2. Failure to turn off power when making circuit changes is a major reason for blowing fuses in the equipment, thereby rendering the equipment unusable and wasting your time and that of others. Please
carefully check circuit wiring, resistor settings, and voltage settings before applying power to the circuits.
3. Only reapply power after verifying that the circuit is properly wired and that the voltage to be applied is at or below the required value.
Part (0) Prior to coming to the lab, calculate theoretical voltage and current values in Figure 3.3 and record them in Table 3.1. Calculate all voltage and current values as peak-to-peak.
Table 3.1: Calculated Values of RC Circuit
ZT otal i1 (p-to-p) VR (p-to-p) VC (p-to-p) Freq. (Hz) Xc (Ω) R (Ω) X (Ω) mag (mA) angle mag (V) angle mag (V) angle
13 Part (1) Use the digital multimeter (DMM) to complete the following steps. Record your values in Table 3.2.
1. Set 10 kΩ on the resistor decade box and measure its resistance using the DMM.
2. Select two 0.01 µF capacitors. Identify them as C1 and C2. 3. Measure the capacitances using the DMM by connecting the capacitor leads to the “VΩ” and “COM” terminals.
4. Connect the capacitors in parallel and measure the combined capacitance using the DMM. Wait for the measurement to stabilize before recording. Compare the theoretical and measured values. Does the
measurement tend to confirm the theoretical prediction? Explain.
5. Connect the capacitors in series and measure the combined capacitance using the DMM. Wait for the measurement to stabilize before recording. Compare the theoretical and measured values. Does the
measurement tend to confirm the theoretical prediction? Explain.
6. Select capacitor C1. Compute its theoretical reactance XC1 at a frequency f = 500 Hz. Record the value in Table 3.2.
Table 3.2: Calculated and Measured Values
Nominal or Calculated Value Measured Value
C2 C1 C2 (Parallel) C1 C2 (Series)
XC1 (500 Hz) -
Part (2) Set up the series RC circuit using the resistor R and capacitor C1 as shown in Figure 3.3. Connect Channel 1 of the oscilloscope across the function generator (through the bread board). Use
the “W1” and Ground pins of the Analog Discovery for the Function Generator, “1+” and “1-” pins for Channel 1 of the Oscilloscope, and “2+” and “2-” pins for Channel 2 of the Oscilloscope.
14 For steps 1 through 4, record the results of the measurements in Table 3.3. Use peak-to-peak readings for all voltage and current measurements in this experiment.
i1 +
2Vp−p 0.01µF
Figure 3.3: Series RC Circuit
1. Set the function generator to give a sinusoidal wave output with amplitude of 1V, offset = 0V, and frequency = 500 Hz. Check the voltage (VS) and frequency values on CH1 of the oscilloscope and
record them in Table 3.3. Keep CH1 across FGEN throughout the experiment.
2. Connect CH2 of the oscilloscope across capacitor C1 and measure VC .
3. Connect CH2 of the oscilloscope across resistor R and measure VR. Do not remove CH2 leads.
4. Compute the peak-to-peak current Ipp from Ipp = VR/R. Remember, the current is the same throughout the circuit, so this current also flows through the capacitor.
5. Measure the phase angle φ between VR and VS by using the cursors on the oscilloscope display.
6. Compute the capacitor’s reactance XC1 from XC1 = VC1/IPP . Compute C1 from the measured XC1 and compare to your earlier measurement.
7. Compute the total impedance ZT otal by applying Ohm’s law to the circuit (Z = Vs∠0 / I∠φ). Use the supply voltage set in step 1 and the current found in step 4. Remember, the impedance has both a
magnitude and a phase angle (measured relative to the resistor). Convert the value to rectangular form and record it in the table. 8. Repeat steps 1 to 7 (resetting VS if necessary) for the following
frequencies: 1000, 2000, 4000, 8000 Hz. Table 3.3: Recorded Values for Part 2
Freq. Freq. (Hz) VS (V) VC1 (V) VR (V) IPP (mA) φ (deg) XC1 (Ω) C1 (µF) ZT otal 500 (Hz) 1000 (Hz) 2000 (Hz) 4000 (Hz) 8000 (Hz)
15 9. Draw impedance and voltage phasors (as in Figure 3.2) for frequency f = 1000 Hz.
10. Draw impedance and voltage phasors for frequency f = 4000 Hz.
11. The phasor diagrams at various frequencies show how the impedances, and therefore the voltages, change with frequency. To better see the net effect on the circuit, graph VC1 and VR versus
frequency for the values in Table 3.3. Label the curves.
Probing Further Questions
Q1. Describe what happens to the current in this RC series circuit as the frequency increases. Explain in general terms why the observed change should occur.
16 Q2. Simulate the circuit in LT SPICE and graph VC1 and VR versus frequency over the range of values in Table 3.3. Compare to your manually drawn curves. [Instructions for setting up the necessary
AC simulation in LT SPICE are shown in Appendix D.]
Q3. In this experiment it was shown that the voltage phasor diagram can be obtained by multiplying each of the impedance phasors by the current in the circuit. If each of the voltage phasors in the
voltage phasor diagram is again multiplied by the current, the resulting diagram is the power phasor diagram. Using the data in Table 3.3 convert the current I and source voltage VS to RMS values.
Then draw a plot of the power phasor diagrams at a frequency of 1000 Hz and another at a frequency of 4000 Hz. Determine the real power, the reactive power, and the apparent power in the RC circuit
at those frequencies.
17 Lab 4 - Inductors and Series RL Circuits
This laboratory continues the study of series linear circuits, this time looking at the effect of inductors in series linear circuits. Besides studying the behavior of inductors, you will use
measurements of magnitude and phase to construct phasor diagrams for sinusoidal voltages in a series circuit, and thereby validate Kirchhoff’s Voltage Law even when the current varies with time.
• Learn to predict and to measure inductive reactance.
• Learn to apply Ohm’s law to reactances of impedances.
• Learn to measure phase angles between voltages.
• Learn to draw impedance and voltage phasor diagrams for resistors and inductors in series.
• Gain experience in the construction and use of phasor diagrams.
• Determine the reactance of an inductor in a series RL circuit by measuring voltages.
• Draw impedance and voltage phasor diagrams for a series RL circuit.
• Determine the real, reactive, and apparent power for a series RL circuit.
• Explain the effect of frequency on the impedance and voltage phasors for a series RL circuit.
18 PreLab
• Read and study the Background section of this Laboratory.
• Read Appendix C, Voltage Measurement, and Phase-Angle Measurement for an oscilloscope with two vertical inputs.
• Prior to coming to lab class, complete Part 0 of the Procedure.
Analog Discovery 2 Instrument. Digital Multimeter. Resistance Substitution Box. Inductor: 50 mH.
When a current passes through a wire, a magnetic field is generated around the wire. The magnetic field results from the movement of electric charge and is proportional to the magnitude of the current.
Turning the wire into a coil concentrates that magnetic field, with the field of one turn of the coil reinforcing another. Such a coil is called an inductor. From an electrical point of view the
especially interesting fact is that if the current in the wire changes, the magnetic field will react to try to keep the current constant. This property of inductors is described by Lenz’s law.
An inductor’s response to changes of current is called inductance. Inductance opposes changes in current, just as capacitance opposes changes in voltage. Inductance is the electric current equivalent
of inertia in mechanical systems.
Inductance is measured in henries. One henry is the amount of inductance present when one volt is generated as a result of a current changing at the rate of one ampere per second.
When inductors are connected in series, the total inductance is the sum of the individual induc- tors. This is similar to the way resistors in series add. When inductors are connected in parallel,
the inductances add in the same way that parallel resistors add. However, an additional effect can appear in inductance circuits that is not present with resistors. This effect is called mutual
inductance and is caused by the interaction of the magnetic fields of neighboring inductors. Mutual inductance can either increase or decrease the total inductance, depending on the orientation of the
interacting inductors.
19 Time Constant
Inductance circuits have a time constant associated with them, just as do capacitance circuits, except that the curve of interest for inductors is the current, rather than the voltage. The time
constant τ for inductors is: L τ = (4.1) R where τ is in seconds; L is in henries; R is in ohms.
The voltage induced across the inductor is a maximum when the change in current is a maximum. When a sinusoidal current is applied to an inductor, the largest induced voltage appears across the
inductor when the current is passing through zero. At the peaks of the applied current, the slope is zero and the current is not changing, so the induced voltage is zero. Therefore, the voltage that
appears across an inductor leads the current in the inductor by 1/4 cycle. Figure 4.1 illustrates this relationship.
Figure 4.1: Resistive, Inductive, and Capacitive Loads
Inductive Reactance
As the frequency of the sine wave increases, the rate of change of the current also increases, and so the induced (reacting) voltage across the inductor increases. As a result, the net current
through the inductor decreases. That means, the inductor’s reactance increases with frequency. The inductive reactance is given by: XL = 2πfL (4.2)
20 As with capacitors and resistors, Ohm’s law can be applied to inductive circuits:
VL XL = (4.3) IL
Series RL circuits
When a sine wave is applied to a series circuit of linear components (resistors, capacitors, and inductors), the phase relationships between current and voltage depend on the types of components. The
current and voltage are always in phase across an ideal resistor. The current through a capacitor leads the voltage across the capacitor by 90◦. The voltage across an ideal inductor leads the current
through the inductor by 90◦.
A common memory aid for these relationships is “ELI the ICE man”, where E represents voltage (E is short for “electromotive force”, which is another term for voltage), I represents current, L
represents inductance, and C represents capacitance.
From Kirchhoff’s current law (KCL), we know that the current is the same throughout a series circuit. Since the current and voltage are in phase for a resistor, we can determine the phase of the
current by measuring the phase of the voltage across the resistor. This is commonly done by using an oscilloscope to compare the voltage from the source to the voltage across the resistor.
Consider the series RL circuit shown in Figure 4.2(a). Graphical representations for the impedance phasors are shown in Figure 4.2(b). As with the series RC circuit, the total impedance of the series
RL circuit is obtained by the vector sum of the impedance phasors. (Because the vectors are in the complex plane, this summation can also be done equivalently by summing the complex numbers.)
Figure 4.2: Phasor Diagram
In this example, 5V are applied and the total impedance is 5 kΩ, so the total current is 1.0 mA. Because the current is the same in all components of the series circuit, we use its direction through
the resistor for the phase reference direction. Multiplying the impedance phasors by the current gives the voltage phasors, as shown in Figure 4.2(c).
In this experiment you will use an oscilloscope to measure the phase angles in the circuit. Because of the resistance of the wires in an inductor, actual inductors may have enough resistance to affect
the phase angle. To minimize that effect in this experiment, we will use a series resistor that is large compared to the inductor’s resistance.
21 Safety Precautions
1. Always turn off power to the circuit when changing the circuit.
2. Only reapply power after verifying that the circuit is properly wired and that the voltage to be applied is at or below the required value.
3. Failure to turn off power when making circuit changes is a major reason for blowing fuses in the equipment, thereby rendering the equipment unusable and wasting your time and that of others. Please
carefully check circuit wiring, resistor settings, and voltage settings before applying power to the circuits.
Part (0) Prior to coming to the lab, calculate theoretical voltage and current values in Figure 4.3 and record them in Table 4.1. Calculate all voltages and currents as peak-to-peak values.
Table 4.1: Calculated Values of RC Circuit
ZT otal i1 (p-to-p) VR (p-to-p) VL (p-to-p) Freq. (Hz) XL (Ω) R (Ω) X (Ω) mag (mA) angle mag (V) angle mag (V) angle
Part (1) Complete the following steps and record your values in Table 4.2.
1. Set 5 kΩ on the resistor decade box and measure its resistance using the DMM.
2. Select a 50 mH inductor or set that value on the inductor decade box. Measure its inductance L. To measure inductance requires connecting the leads to the Impedance Analyzer (the device that comes
with AD2 kit) and using the Impedance tool of the WaveForms application.
3. Inductors are made of wires, so they have some internal resistance. Measure the winding resistance RW using the DMM by connecting the inductor leads to the “VΩ” and “COM” terminals.
Table 4.2: Measured Values
Nominal Value Measured Value
R 5kΩ L 50mH
RW 0
22 Part (2) Construct the series RL circuit in Figure 4.3. Connect Channel 1 of the oscilloscope across the function generator (through the bread board). Use the “W1” and Ground pins of the Analog
Discovery for the Function Generator, “1+” and “1-” pins for Channel 1 of the Oscilloscope, and “2+” and “2-” pins for Channel 2 of the Oscilloscope.
For steps 1 through 4, record the results of the measurements in Table 4.3. Use peak-to-peak readings for all voltage and current measurements in this experiment.
i1 +
4Vp−p 5kΩ
Figure 4.3: Series RC Circuit
1. Set the function generator to give a sinusoidal wave output with amplitude of 2V, offset = 0V, and frequency = 10 kHz. Check the voltage (VS) and frequency values on CH1 of the oscilloscope and
record them in Table 4.3. Keep CH1 across FGEN throughout the experiment.
2. Connect CH2 of the oscilloscope across inductor L and measure VL.
3. Connect CH2 of the oscilloscope across resistor R and measure VR. Do not remove CH2 leads.
4. Compute the peak-to-peak current Ipp in the circuit by applying the Ohm’s law to the resistor. That is Ipp = VR/R.
5. Measure the phase angle φ between VR and VS by using the cursors on the oscilloscope display. Press STOP to improve accuracy when taking cursor measurements. Record this value as φmeas in the
6. Compute the inductive reactance XL by applying Ohm’s law to the inductor. That is XL = VL/IPP .
7. Compute the total impedance ZT otal by applying Ohm’s law to the circuit (Z = Vs∠0 / I∠φ). Use the supply voltage set in step 1 and the current found in step 4. Remember, the impedance has both a
magnitude and a phase angle (measured relative to the resistor). Convert the value to rectangular form and record it in the table.
8. Compute the phase angle φ between VS and VR. Recall that φ = arctan(VL/VR). Record this value as φcalc in the table, and compare it to the value measured on the oscilloscope.
23 Table 4.3: Recorded Values for Part 2
Freq. (Hz) VS (V) VL (V) VR (V) IPP (mA) φmeas XL (Ω) ZT otal (Ω) φcalc
9. Using the values listed in the table, draw a diagram of the impedance phasors and a second diagram of the voltage phasors, as illustrated in Figure 4.2 of the Background section of this laboratory
10. Using the data in the table, convert the current I and source voltage VS to RMS values. Then draw plots of the power phasor diagrams (power triangle) at the frequency of 10 kHz. Determine the
real power, the reactive power, and the apparent power in the RL circuit at that frequency.
Probing Further Questions
Q1. What you think would happen to the current in this RL series circuit if the frequency were decreased, say, to 2000 Hz? Why?
Q2. Simulate the circuit in LT SPICE and graph VL and VR versus frequency from 1000 Hz to 20,000 Hz. Does the simulation support your prediction in question 1?
24 Lab 5 - Parallel RC and RL Circuits
This laboratory explores the behavior of parallel RC and RL circuits, and the application of Kirch- hoff’s current and voltage laws to such circuits. For series RC and RL circuits, we saw that
Kirchhoff’s voltage law applies, but that the voltages must be added as phasors. Similarly, in parallel circuits, Kirchhoff’s current law applies to any junction, but again, the currents must be added
as phasors.
• Learn to apply Kirchhoff’s voltage law (KVL) and Kirchhoff’s current law (KCL) in parallel circuits.
• Learn to draw current phasor diagrams for parallel circuits.
• Gain experience in the construction and use of phasor diagrams.
• Gain experience in calculating real, reactive, and apparent power.
• Confirm Kirchhoff’s current law (KCL) in parallel circuits.
• Draw current phasor diagrams for parallel circuits.
• Determine the real, reactive, and apparent power for a parallel RC circuit.
25 PreLab
• Study the Background section of this Laboratory.
• Prior to coming to lab class, complete Part 0 of this lab.
• Confirm Kirchhoff’s current law (KCL) in parallel circuits.
Analog Discovery 2 Instrument. Digital Multimeter. Resistors: 10kΩ, 2.2kΩ, 2 x 100Ω, 2 x 22Ω Inductor: 100 mH Capacitor: 0.01 µF Components may be discrete or via decade substitution boxes, unless
otherwise indicated.
As was seen in prior experiments, in a series circuit the same current is in all components, and so current is generally used as a reference in series circuits. However, in parallel circuits, the
same voltage is across all components, so voltage is the logical and appropriate reference. The current in each branch then evolves from the circuit voltage.
For series RC and RL circuits, we saw that Kirchhoff’s voltage law applies, but that the voltages must be added as phasors. Similarly, in parallel circuits, Kirchhoff’s current law applies to any
junction, but again, the currents must be added as phasors. The current entering a junction is always equal to the current leaving the junction.
In a parallel circuit, if the impedance of each branch is known, then the current in that branch can be determined directly from the applied voltage and Ohm’s law. The current phasor diagram can then
be constructed, and the total current can be found as the phasor sum of the currents in each branch.
Consider the current phasor diagram for the parallel RC circuit shown in Figure 5.1. The current in the capacitor is shown at +90◦ from the voltage reference because the current leads the voltage in
a capacitor. The current in the resistor is along the x-axis because current and voltage are in phase in a resistor.
26 Figure 5.1: Parallel RC Circuit
Figure 5.2 shows a parallel RL circuit and its current phasor diagram. Here we have assumed all components are “ideal”. The current in an ideal inductor is at -90◦ from the voltage reference, because
the current lags the voltage in an inductor. However, practical inductors contain resistance that often is large enough to affect the phasor. The effect of the inductor’s resistance on the phasor
diagram would be to reduce the angle between the IL and IR. In a practical circuit, this angle will be slightly less than the -90◦ of a pure inductor. This experiment will illustrate this difference
between the approximation of circuit performance based on ideal components and the actual measured values.
Figure 5.2: Parallel RL Circuit
For both RC and RL circuits, the Pythagorean theorem and ordinary vector addition can be applied to the current phasors to determine the magnitude of the total current, IT . q q 2 2 2 2 IT = IR + IC
and IT = IR + IL (5.1)
Recall that in series circuits, the phase angle was measured between the source voltage VS and the resistor voltage VR using the oscilloscope. The oscilloscope is a voltage-sensitive device, so
examining those voltages and phase angles is straightforward. But in parallel circuits, the phase angle of interest usually is between the total current, IT , and one of the branch currents. To use
the oscilloscope to measure the phase angle in a parallel circuit, we must convert the current to a voltage. This is commonly done by inserting a small resistor (a “sense” resistor) in the branch
with the current to be measured. Such a sense resistor makes it easy to determine the magnitude and the phase of the current in that branch, but the resistor must be small enough not to have a
significant effect on the values to be measured.
27 Part (0) Before coming to the lab, determine the voltages across and currents through each resistor in the circuits of Figure 5.3 and Figure 5.4. Record all of your calculated results in Tables
5.2 and 5.4. Record the calculated values in polar form.
Part (1) - Parallel RC Circuit
This part of the experiment will give you experience making the measurements with the digital multimeter (DMM). Because the electrodes of this device are isolated from the circuit ground and the
grounds in AD2, you may make voltage measurements directly across any of the components, whether or not they are grounded.
itotal i2 + i1 C : 0.01µF
2VRMS R1 : 10kΩ
RS2 : 100Ω − RS1 : 100Ω itotal
Figure 5.3: Parallel RC Circuit
1. Before constructing the circuit in Figure 5.3, measure circuit components using the DMM: one 10kΩ resistor, two 100Ω sense resistors, and one 0.01 µF capacitor. (Be sure to wait until the
capacitance measurement stabilizes.) Record the measured values in Table 5.1.
Table 5.1: Measured Values
Nominal Value Measured Value
R1 10kΩ
RS1 100Ω
RS2 100Ω C 0.01µF
2. Construct the circuit shown in Figure 5.3. Set the function generator to provide a sine wave with a voltage of 2.0 Vrms at 1.0 kHz. Verify the voltage and frequency with your oscilloscope while
the circuit is connected and operating; adjust if necessary.
3. Using the DMM voltmeter in AC mode [V∼], measure the voltage drop across each resistor. The voltage drops are small, so measure as accurately as possible and keep three significant figures in
28 your measurement (use mV mode of DMM if the displayed voltage is zero). Record the voltage drops in Table 5.2.
Table 5.2: Measured and Calculated Values for Figure 5.3
VS (RMS) VR1 (RMS) VRS1 (RMS) VRS2 (RMS) I1 (mA) I2 (mA) Itotal (mA)
Calculated 2∠0 Measured
4. Compute the current in each resistor using Ohm’s law, and record the calculated currents in Table 5.2.
5. Draw the current phasors I1, I2, and the total current Itotal in a plot similar to that of Figure 5.1. Ignore the small effect of the sense resistors on the phasor diagram. Note carefully the
direction of the phasors. Label each of the current phasors.
6. Compute XC for the 1.0 kHz frequency. Using this value and that of the sense resistor, calculate the expected current, IC (i2), through the capacitor. How does this value compare to that found
from the sense resistor (Table 5.2)?
1 XC = −j 2πfc =
VS IC = = −jXC +RS2
7. Using the value of −jXC for the 1.0 kHz frequency and the measured resistance of R1, find the total impedance, Ztotal (in polar form), of the circuit. Remember that these impedances add like
parallel resistors. Ignore the sense resistors for this calculation. Show your work.
8. Using Ztotal and the applied voltage, VS, compute the total current, Itotal. Show your work. The total current should reasonably agree with the value determined in step 4.
29 Probing Further Questions
Q1. Ignoring the effect of the sense resistors, compute the real, reactive, and apparent power of the circuit in Figure 5.3 at 1.0 kHz.
Q3. If the capacitor C were made smaller, what would happen to the current phasor diagrams?
Part (2) - Parallel RL Circuit
This part of the experiment will give you experience making similar measurements using the oscillo- scope. In this part of the experiment, record all voltages and currents as peak-to-peak values.
itotal i2 + i1 L : 100mH
4VPP R1 : 2.2kΩ
RS2 : 22Ω − RS1 : 22Ω itotal
Figure 5.4: Parallel RL Circuit
1. Before constructing the circuit in Figure 5.4, measure circuit components using the DMM and Impedance Analyzer. Record the measured values in Table 5.3.
30 Table 5.3: Measured Values
Nominal Value Measured Value
R1 2.2kΩ
RS1 22Ω
RS2 22Ω L 100mH
Table 5.4: Measured and Calculated Values for Figure 5.4
VS (P-P) VR1 (P-P) VRS1 (P-P) VRS2 (P-P) I1 (mA) I2 (mA) Itotal (mA)
Calculated 4∠0 Measured
2. Construct the circuit shown in Figure 5.4. Using the oscilloscope’s CHANNEL 1 to monitor the function generator, set the source voltage to a sine wave with a voltage of 4.0 VPP at 4.0 kHz. Verify
both the voltage and the frequency with the oscilloscope. Record the measured values in Table 5.4.
3. Using oscilloscope CHANNEL 2, measure the peak-to-peak voltage across RS1. Apply Ohm’s law to calculate the peak-to-peak value of the total current, Itotal. Record the measured values in Table
4. Because RS1 is small compared to R1, the source voltage VS is very nearly the voltage across the resistor R1. Furthermore, since only resistors are in that loop, the phase of the voltage across R1
— and current through R1 — is the same as the phase of the source voltage VS.
With CHANNEL 1 of the oscilloscope displaying the voltage across the function generator (VS), and CHANNEL 2 displaying the voltage across RS1, measure the phase angle between the generator voltage,
VS, and the generator current, Itotal, (i.e., the current flowing through RS1). This is equivalently the phase angle between IR1 and Itotal. You may press STOP to improve accuracy when taking cursor
measurements. Record the measured phase angle between IR1 and Itotal.
φ =
5. Replace the sense resistor RS1 with a jumper wire (short it). CHANNEL 1 is still measuring the output voltage of the function generator, which now is also the voltage across R1. Connect CHANNEL 2
of the oscilloscope across sense resistor RS2. Record the voltages across R1 and across RS2 in Table 5.4. Apply Ohm’s law to calculate the current in each branch of the circuit, and record the
currents. 6. Using the computed peak-to-peak currents from Table 5.4, draw the current phasor diagram for the circuit. Ignore the effects of the sense resistors. Your diagram should look somewhat
similar to Figure 5.2(b).
31 7. The phasor diagram depicts visually the relationship between the total current and the currents in each branch of the circuit. From the currents in the phasor diagram, compute the phase angle
between the total current (Itotal) and the current in R1 (IR1). Then compute the phase angle between the total current (Itotal), and the current in L (IL). Ideally, what should be the sum of these
8. On the oscilloscope, measure the angle between IL and IR1. (The oscilloscope’s leads were already connected to do this as a result of step 5). Ideally, this measurement should be 90◦, but
resistance in the inductor, may reduce the angle. If necessary, adjust both scope channels to have the same apparent height on the oscilloscope screen in order to make the measurement.
9. In step 4 you measured the phase angle φRT between IR1 and Itotal. In step 8 you measured the phase angle φRL between IR1 and Itotal. Compute the phase angle φTL between Itotal and IL by
subtracting φRT from φRL. That is,
φTL = φRL - φRT .
10. Compare the measured phase angles versus the computed phase angles in Table 5.4. Discuss likely causes for any discrepancies.
Probing Further Questions
◦ Q1. The currents IR1 and IL were measured in step 5. If those currents are 90 apart, we can calculate the total current Itotal using the Pythagorean theorem.
a. Compare this calculated total current to the total current measured in step 3.
b. What factors might cause any discrepancies observed between the values?
Q2. What effect does the inductor’s coil resistance have on the phase angle between the currents in the resistor and the inductor?
32 Lab 6 - Circuit Resonance
The response of a circuit containing both inductors and capacitors in series or in parallel depends on the frequency of the driving voltage or current. This laboratory will explore one of the more
dramatic effects of the interplay of capacitance and inductance, namely, resonance, when the inductive and ca- pacitive reactances cancel each other. Resonance is the fundamental principle upon which
most filters are based — filters that allow us to tune radios, televisions, cell phones, and a myriad of other devices deemed essential for modern living.
• Learn the definition of resonance in AC circuits.
• Learn to calculate resonant frequencies, band widths, and quality factors for series and parallel resonant circuits.
• Learn to use AD2’s Bode plot function to view circuit response.
• Predict resonant frequencies, band widths, and quality factors for series resonant circuits.
• Measure resonant frequencies, band widths, and quality factors for series resonant circuits, and compare them to predicted value.
• Determine what effect changing the series resistance has on the resonance.
33 PreLab
• Study the Background section of this Laboratory exercise.
• Prior to coming to lab class, complete Part 0 of the Procedure.
Analog Discovery 2 Instrument. Digital Multimeter. Resistors: 10Ω, 100Ω Inductor: 100 mH Capacitor: 0.01 µF Components may be discrete or via decade substitution boxes, unless otherwise indicated.
The reactance of inductors increases with frequency:
XL = 2πfL (6.1) The reactance of capacitors decreases with frequency: 1 X = (6.2) C 2πfC
In an LC circuit, whether series or parallel, there is some frequency at which the magnitudes of these two reactances are equal. That point is called resonance. Setting XL = XC , and solving for f,
we find that the resonant frequency fo of an LC circuit is: 1 fo = √ (6.3) 2π LC
The frequency f has units cycles/second or sec−1. The frequency may also be expressed as angular frequency, ω, where ω = 2πfC and has units radians/sec. Thus, the resonant frequency may also be
written as: 1 ωo = 2πfo = √ (6.4) LC
The resonant frequency is generally the highest point of a peak (or the deepest point of a valley) with bandwidth BW (cycles/sec) or β (radians/sec). The resonant frequency is also called the center
frequency, because it is at the mid-point of the peak frequency response.
34 The lowest frequency (f1 or ω1) and the highest frequency (f2 or ω2) of the band are the “half-power 1 points” at which the power is 2 that at the peak frequency. Since power goes like the square
of the current, the current at the half-power points is √1 = 0.707 times the current at the maximum. Thus, 2 the bandwidth of a resonant circuit is the frequency range over which the current is at
least 70.7% of the maximum. BW = f2 − f1 or β = ω2 − ω1 (6.5)
As the bandwidth narrows, the circuit becomes more highly selective, responding to a narrow range of frequencies close to the center frequency. The sharpness (narrowness) of that resonant peak is
measured by the quality factor Q. The quality factor is a unitless quantity that is defined as: maximum energy stored Q ≡ 2π (6.6) energy dissipated per cycle In more practical terms, f Q = o (6.7) BW
Figure 6.1: Circuit Bandwidth
Series Resonance
For a series LC circuit, the current is the same throughout. What about the voltages? To visualize the concept of resonance, consider the simple series RLC circuit in Figure 6.2 operating at
resonance, and its associated reactance diagram.
Figure 6.2: Series RLC Circuit
35 The phase shift caused by the capacitor is directly opposite the phase shift caused by the inductor; that is, they are 180◦ out of phase. Therefore, in the reactance phasor diagram (b) for the
circuit, the two phasors point in opposite directions. At resonance, the magnitudes of the capacitor reactance and the inductor reactance are equal, so the sum of the two phasors is zero, and the
only remaining impedance is due to the resistor. Notice in the voltage phasor diagram (c) that the voltage drop across the inductor and the capacitor may be quite large — bigger even than the source
voltage — but those voltages are opposite in phase and so cancel each other out as voltages are summed around the circuit. Kirchhoff’s voltage law remains valid, and the generator’s voltage output is
dropped entirely over the resistor R.
Since at resonance the only impedance is the resistance R, the impedance of the series circuit is at a minimum, and so the current is a maximum. That current is VS/R. The source voltage and the
current are in phase with each other, so the power factor = 1, and maximum power is delivered to the resistor.
But what happens at neighboring frequencies? At lower frequencies, the inductor’s reactance de- creases, and the capacitor has greater effect. At higher frequencies, the inductor dominates, and the
circuit will take on inductive characteristics. How sharply defined is the resonance? How selective is it? We have said that for a resonant circuit, the quality factor Q is the ratio of the resonant
frequency to the bandwidth. Thus, Q gives a measure of the bandwidth normalized to the frequency, thereby describing the shape of the circuit’s response independent of the actual resonant frequency.
We list here two other useful relationships for Q in a series resonant circuit. The first relates Q to the circuit’s capacitance, inductance, and total series resistance:
r 1 L Q = (6.8) R C
The value of R in this equation is the total equivalent series resistance in the circuit. This form of the equation makes it easy to see ways to optimize the Q for the desired circuit. Decreasing R,
increasing inductance, or decreasing capacitance will all tend to make Q larger and increase the circuit’s selectivity.
The second useful relationship for Q can be derived from the previous equation. Recall that XL = 1 2πfL and XC = 2πfC .
1 p Q = X .X (6.9) R L C
Since at resonance the inductive and capacitive reactances are equal, this equation can be reduced to:
X X Q = L or Q = C (6.10) R R
where R is again the total equivalent series resistance of the circuit. Usually the XL form is used because the resistance of the inductor frequently is the dominant resistance in the circuit. An
36 form of this last equation is:
2πf L 1 Q = o or Q = (6.11) R 2πfoCR
Parallel Resonance
For a parallel RLC circuit, the same voltage is applied across all the branches. The current in each branch is determined by the voltage applied to the branch and the impedance in that branch. A
parallel RLC circuit with ideal components is illustrated below, along with its current phasor diagram. Notice that the total source current at resonance is the current in the resistor. The currents
in the inductor and capacitor cancel out because of their opposite phase shifts. The net impedance of the circuit at resonance is solely determined by R, since the inductor and the capacitor appear
to be open.
Figure 6.3: Parallel RLC Circuit
In an ideal two-branch LC circuit including an inductor and capacitor (i.e., R is removed or equiva- lently R=∞), the source current would be zero. In a practical two-branch LC circuit, the only
significant resistance is the inductor’s winding resistance, so that resistance often plays an important role in the circuit’s behavior.
In series resonant circuits, we defined the bandwidth by observing the response of the current. In parallel resonant circuits, the corresponding response to observe is the impedance. The bandwidth of
a parallel resonant circuit is the frequency range over which the circuit impedance is at least 70.7% of the maximum impedance. The sharpness of the response is again measured as Q. Note that the Q
of the circuit will be different from the Q of the inductor if other resistance is in parallel with L and C. If there is no additional resistance (i.e., R=∞), then the circuit Q is the same as the
inductor’s Q.
37 Table 6.1: Summary of the Characteristics of Resonant RLC Circuits
Characteristic Series Circuit Parallel Circuit
Resonant Frequency, f √1 √1 o 2π LC 2π LC Quality factor, Q 2πfoL or 1 R or 2πf RC R 2πfoRC 2πfoL o
fo fo Bandwidth, BW Q Q r 2 r 2 1 fo 1 fo Half-power frequencies f1, f2 fo 1 + 2Q ± 2Q fo 1 + 2Q ± 2Q
BW BW For Q ≥ 10, f1, f2 fo ± 2 fo ± 2
Part (0) Before coming to the lab, calculate fo, Q, BW, f1, and f2 for the circuit shown in Figure 6.4 when R = 10Ω and 100Ω. Don’t forget to include the impedance of the function generator (RF GEN ≈
50Ω) as part of the total resistance in the circuit. Record the results in the “predicted” column of Table 6.3.
Part (1) Using the DMM and the Impedance Analyzer, measure the values of the components in Table 6.2. Also, measure the winding resistance RW of the 100 mH inductor.
Table 6.2: Measured Values
Nominal Value Measured Value
R1 100Ω
R2 10Ω
RW - L 100mH C 0.01µF
Part (2) Complete the following steps and record your values in Table 6.3.
1. Construct the circuit shown in Figure 6.4 using a 100Ω resistor for R. Adjust the function generator to generate a sine wave with voltage 1.0 VPP . Initially set the frequency to 5000 Hz. 2.
Connect oscilloscope CHANNEL 1 across the function generator (FGEN and GND) and confirm that the voltage is 1.0 VPP .
38 0.01 µF
50Ω 100 mH +
1.0VPP − R = 100,10 Ω
Figure 6.4: Series RLC Circuit
3. Connect oscilloscope CHANNEL 2 across the resistor R and observe the voltage. We use VR instead of directly measuring the source current, since the current through the circuit and resistor R is
proportional to the voltage across R.
4. Using your predicted values as a guide, adjust the frequency of the function generator to tune for resonance, as observed on CHANNEL 2 of the oscilloscope. Remember that resonance frequency is the
frequency where the current in the circuit is at its maximum (or the VR is at its max). Measure the resonant frequency fo on the oscilloscope, and record the value in Table 6.3. Export and save the
oscilloscope screen as an image file. Additionally, record the VR at this frequency:
VR−PP =
5. Reduce the frequency on the function generator until the voltage across R becomes 70.7% of the voltage value at resonance frequency (step 4). This is the lower half-power point f1. Record the
measured frequency f1 in Table 6.3. 6. Increase the frequency through resonance and continue to increase it until the voltage across R is 70.7% of the value at resonance (step 4). This is the upper
half-power point f2. Record the measured frequency f2 in Table 6.3.
7. Calculate the bandwidth BW = f2 – f1. Record the result in Table 6.3. 8. Stop the function generator. Change the value of resistor box to 10Ω.
9. Start the function generator and repeat steps 4 through 7. Record the measured values in Table 6.3.
10. Complete step 10 for both R = 10,100 Ω.
39 Table 6.3: Experimental Values
R = 100 Ω R = 10 Ω Predicted Measured Predicted Measured
fo Q BW
11. When building circuits, we often need to determine how the circuit will respond given certain inputs. This is especially true when we use energy storage elements like capacitors and inductors.
How do these parts interact with and change the input signal to give us the output signal? By using the Bode Diagram, you can get a better idea of how your circuit responds to various types of
inputs. In this step, Use the Bode analyzer of the AD2 device to view the plot of the last circuit’s response. To do so:
• Close all the active windows in the WaveForms application. • Connect CH1 across the FGEN (reference signal) and CH2 across the resistor R (output). • Once you have everything set up, get it all
connected and open WaveForms. Click on the “Network” button near the bottom left. This will open the Network Analyzer window. • Before we get started, let’s look at what we see in the “Network”
window. You’ll see two graphs, the top shows the magnitude of our signal and has a dB scale from 10 to -90 dB on the left. The lower graph shows phase and has a degrees scale from 180◦ to -180◦. (The
vertical axis labels are on the far right of the plot windows.) The x-axis along the very bottom of the window lists frequencies on a logarithmic scale, with major divisions at each decade of
frequency. (By default, 1kHz, 10kHz, 100kHz, and 1MHz. Logarithmic scales can be a bit tricky to understand at first, but it converts the logarithmic behavior of the circuit to a linear
representation, which can be easier to understand visually. At the far right you will see adjustments for the waveform generator, y-axis adjustments for each plot, and the channel adjustments. We
will look more at those in a bit. • Set the settings to scan from 2.5 kHz to 8.5 kHz. Set the “Samples” to at least 300, although you may need to increase this value to as much as 500 to get accurate
readings of cutoff frequency and phase angle. Verify that the graph scale is “Logarithmic”. Set the “amplitude” to 1.0 V (that is, 1.0 VPP ). Press the RUN button.
• Turn on CURSORS to be able to locate specific values. Find fo, f1, f2 values on the phase graph and calculate the Bandwidth using this numbers.
• Save the Bode plot as an image (with the cursors showing the values for f1, f2). • Do you see any difference between the values in the Table 6.3 and those that you found in the previous step?
40 Probing Further Questions
Q1. From the measured L, C, fo, and BW, compute the total series resistance of the circuit. Suggest and explain likely causes for any discrepancy; what might not have been taken into account for the
Q2. Discuss the effect of changing the resistor R from 100 Ω to 10 Ω. How dramatic was the impact?
Q3. What would happen to the resonant frequency if the inductance were doubled and the capacitance cut in half? What would happen to the bandwidth? What would happen to the quality factor?
Q4. Simulate the second circuit (with R=10Ω) in LT SPICE, using the AC Frequency Sweep (see Appendix D). In the simulation, set up a voltmeter across R and an ammeter to measure the total current. Be
sure to have LT SPICE display both graphs and a table of the results. Select an appropriate range of frequencies to sweep and sample at least 100 steps per interval. When the graph is displayed,
right-click the graph and toggle the X axis to be logarithmic. Do you see a sharp peak in the magnitude (dB) plots of voltage and current at the resonant frequency you predicted? Notice that the
phase plot for the circuit rapidly changes from positive (capacitive) to negative (inductive) as the frequency passes through resonance. Record the resonant frequency indicated by the graph, and
compare it to your calculated and measured values.
41 Lab 7 -Filters: High-pass, Low-pass, Band- pass, and Notch
This laboratory studies the use of passive components to create filters to separate portions of time- dependant waveforms. Filters are an essential tool in our complex world of mixed signals — both
electronic and otherwise. Passive components (resistors, capacitors, and inductors) have long served as filter components for everything from selecting radio stations to filtering out electrical noise.
• Learn the four general filter types: High-pass, Low-pass, Bandpass, and Notch.
• Learn to alter filter type by changing contacts for output voltage.Learn to alter filter type by changing contacts for output voltage.
• Learn phase angle at cutoff for simple RC and RL filters.
• Design simple filter.
• Gain experience with Bode Analyzer and Bode Plots.
• Calculate and measure cutoff frequency for series RC and RL filters.
• Design simple RL low-pass filter.
• Generate and interpret Bode plots for series filters.
• Observe behavior of a single circuit used as a Bandpass filter and as a notch filter.
42 PreLab
• Study the Background section of this Laboratory exercise.
• Study textbook regarding origin and meaning of “decibels”.
• Prior to coming to lab class, complete Part 0 of the Procedure.
Analog Discovery 2 Instrument. Digital Multimeter. Suitable Decade Substitution boxes may be used for the following components: Resistors: 10kΩ, 100Ω, and a resistor to be determined Inductor: 100 mH
Capacitor: 0.005 µF, 0.01 µF
In many circuits, a wide range of different frequencies are present, some of which are desired, while others are not. The frequency response of capacitors and inductors allows us to construct filters
that will pass or reject certain ranges of the electrical frequencies that are applied to them. “Passive filters” created from “passive” components (inductors, capacitors, and resistors) have served
us well for a long time for such purposes as selecting radio and television stations and filtering noise out of various signals. Indeed, much of the electronics we take for granted today would not be
possible without the use of such filters.
The four typical types of filter behaviors are illustrated in Figure 7.1, along with schematics of simple filters that exhibit the indicated behavior.
The filter types are low-pass, high-pass, bandpass, and notch (or band-reject) filters. In Figure 7.1, the grayed area is the passband, that is, the part of the signal that is passed to the output of
the filter. The rejected portions are called the stopband. The frequency that separates the passband from the stopband is called the cutoff frequency. The cutoff frequency is equivalent to the
half-power points discussed in Laboratory 6. The cutoff frequency is also sometimes called the corner frequency.
43 Figure 7.1: Types of Passive Filters
A low-pass filter would allow extracting a low frequency, such as an audio signal, that is mixed with a high frequency radio wave. A high-pass filter would do the opposite. A resonant circuit can be
tuned as a bandpass filter to retain signals in a narrow range of frequencies, while rejecting frequencies outside that range. Such is the case with a radio tuner. A notch filter generally keeps all
frequencies except those in a narrow band. Notch filters are widely used to block interfering signals from noise sources. Bandpass and notch filters require resonant circuits, studied in Lab 6.
Notice that the components making the low-pass and high-pass filters in Figure 7.1 are the same. Whether the circuit is low-pass or high-pass depends only upon which voltage we look at: the voltage
across the capacitor or the voltage across the resistor. (Equivalent circuits could have been made using an inductor and a resistor.) Similarly, the notch filter is identical to the RLC series
resonant circuit we looked at in Lab 6, however in Lab 6 we looked at the voltage across the resistor, and so saw a bandpass filter. Caution: While one may be able to obtain the opposite response from
the filter simply by putting the output terminals across a different filter component, one must be sure to stay within the power and current limitations of the circuit and its components.
RC and RL filters are simple, inexpensive, and often used effectively as filters. Their major problem is their generally slow (in frequency) transition from passband to stopband. The few simple
components in filter “stages” can increase the transition rate, giving the filter a sharper cutoff.
The ratio of an output response to an input signal is referred to as a transfer function. The input signal and the output response do not need to be the same entity type. For example, a transfer
44 may prescribe an output voltage resulting from an input current. Transfer functions are often used as a tool to characterize the effect of a filter regardless of the details of the filter’s
structure. It can make the analysis of complex circuits easier. In this lab, however, we will mostly be studying the filter itself.
Cutoff Frequency for series RC and RL circuits
As mentioned, the cutoff frequency, sometimes called the corner frequency, is equivalent to the half- power points discussed in Laboratory 6. Since the power√ is half that at the peak, the voltage (or
current) will be the peak voltage (or current) multiplied by 1/ 2 = 0.707. For a simple 2-component RC or RL circuit, the half-power point will occur when half the power is dropped on the resistor
and half on the capacitor or inductor. Thus, the cutoff frequency will occur when the reactance of the capacitor or inductor equals the total series resistance in the circuit. That is,
1 XC = and XL = 2πfoL (7.1) 2πfoC And so,
1 R f = and f = (7.2) o 2πRC o 2πL
Resonant frequencies for RLC circuits were discussed in Laboratory 6. decibels (dB)
As discussed in your textbook, the decibel (dB) is commonly used for the magnitude of voltages and currents in making Bode plots. Keep in mind that a decibel is a unit created to measure the transfer
function for power gain (or loss) through a circuit module or stage:
power out Number of decibels = 20 log (7.3) power in
Since power is proportional to the square of the voltage or the current, we have equivalently,
V I Number of decibels = 20 log out and Number of decibels = 20 log out (7.4) Vin Iin
How many decibels correspond to a doubling of the power? To a halving of the power?
Part (0) Perform the design calculations for the filter in Part 2.
Note: In all cases below, you may use substitute a decade substitution box of the correct type and value for the specified resistor, capacitor, or inductor.
45 Part (1) Low-pass filter - complete the following steps and record your values.
10 kΩ
50Ω 0.005 µF Vout +
2.0VPP − −
Figure 7.2: Low-pass Filter
1. Obtain a 10 kΩ resistor and a 0.005 µF capacitor. Measure and record the actual values of the components.
2. Set up the circuit as shown in Figure 7.2. Use the function generator FGEN for the supply voltage.
3. Calculate the cutoff frequency for the circuit, assuming the output is at Vo. At the cutoff frequency, what, theoretically, will be the voltage Vo?
4. Connect CHANNEL 1 of the oscilloscope to measure the Vin (i.e., FGEN).
5. Connect CHANNEL 2 of the oscilloscope to measure the filter’s output voltage Vo. On the oscil- loscope, turn on the measurement functions for CH1 and CH2. Vary the frequency from 500 Hz to 10 kHz
in steps indicated in Table 7.1, and record the indicated values. You may press STOP to freeze the display when taking cursor measurements. For the Vin and Vo measurements, be sure the time scale is
sufficient to show at least a few cycles, or the instrument may not properly calculate the Peak-to-Peak and RMS values.
6. Use the oscilloscope’s cursors to measure the phase shift ∆t and then calculate the phase angle φ between Vin and Vo.
46 Table 7.1: Measured Values
Freq. Vin Vin Vo Vo ∆t φ Gain KhZ RMS PP RMS PP µs degrees db
0.500 2.000 3.000 4.000 6.000 12.000 20.000 Cutoff
7. Using the data of Table 7.1, sketch a Bode plot of the of the filter’s output voltage.
Part (2) Design a low-pass inductive filter.
1. Design a low-pass filter using a 100 mH inductor and a single resistor R to obtain a cutoff fre- quency of 7500 Hz ± 2%. In your notebook, show your design procedure, including your design
47 2. Measure the actual value of the 100 mH inductor and record its value.
3. Using the value of R that you calculated, construct an RL circuit designed to be a low-pass filter with the specified cutoff frequency (similar to the RC circuit used in Part 1 but, since we are
using an inductor, the order of components in the circuit should be L-R).
4. Connect CH1 across the function generator and CH2 across the filter output (resistor R).
5. Use the WaveForms application to generate the Bode plot for this circuit (similar to the last part of Lab 6). Scan from 1 kHz to 40 kHz. Set the “Samples” to at least 300, although you may need to
increase this value to as much as 500 to get accurate readings of cutoff frequency and phase angle. Verify that the graph scale is “Logarithmic”. Set the “amplitude” to 1.0 V (that is, 2.0 VPP ). Run
the Bode analyzer. 6. Change the “Top” and “Bottom” values to fit the plot.
7. When the analyzer has finished, use the cursors to locate the cutoff frequency by finding the half- power point (What identifies the half-power point?). Record the cutoff frequency and the phase angle.
Save the Bode plot as an image.
8. Compare the experimentally determined cutoff frequency to the desired value.
9. If the cutoff frequency is off by more than 5%, change the size of R as necessary to obtain the specified cutoff frequency, and test again for the cutoff frequency. Record the final R, cutoff frequency,
and phase angle φ.
10. Draw a circuit diagram of the final circuit that accomplishes the design objective. Explain any differences between the final value of R and your originally calculated value.
Part (3) RLC Circuit 1.
1. Set up the series RLC circuit shown in Figure 7.3, using the function generator to provide the sinusoidal input voltage.
2. Calculate the resonant frequency f0 of the circuit. (See Laboratory 6.)
48 0.01 µF 100 mH
50Ω 100 Ω Vout +
2.0VPP − −
Figure 7.3: Series RLC Circuit
3. As in part 2, use the WaveForms application to generate the Bode plot for this circuit (similar to the last part of Lab 6). Scan from 1 kHz to 40 kHz. Set the “Samples” to at least 300, although
you may need to increase this value to as much as 500 to get accurate readings of cutoff frequency and phase angle. Verify that the graph scale is “Logarithmic”. Set the “amplitude” to 1.0 V (that is,
2.0 VPP ). Run the Bode analyzer. 4. Change the “Top” and “Bottom” values to fit the plot.
5. What type of filter is this if the output voltage is Vo?
6. What is the measured resonant frequency? What is the measured bandwidth? What is the phase angle at resonance? To measure these values from the Bode plot, you may need to increase the “samples” or
interpolate between points. Save the Bode plot as an image.
Part (4) RLC Circuit 2.
1. Switch the positions of the resistor with inductor and capacitor to get the series RLC circuit shown in Figure 7.4.
2. Calculate the resonant frequency fo of the circuit. (See Laboratory 6.)
49 100 Ω
+ 0.01 µF 50Ω Vout + 100 mH 2.0VPP − −
Figure 7.4: Modified Series RLC Resonant Circuit
3. As in part 2, use the WaveForms application to generate the Bode plot for this circuit (similar to the last part of Lab 6). Scan from 1 kHz to 40 kHz. Set the “Samples” to at least 300, although
you may need to increase this value to as much as 500 to get accurate readings of cutoff frequency and phase angle. Verify that the graph scale is “Logarithmic”. Set the “amplitude” to 1.0 V (that is,
2.0 VPP ). Run the Bode analyzer. 4. Change the “Top” and “Bottom” values to fit the plot.
5. What type of filter is this if the output voltage is Vo?
6. From the Bode plots, what is the measured resonant frequency fc? What are the lower and upper half-power points, f1 and f2? What is the measured bandwidth? What is the phase angle φ at resonance?
To measure these values from the Bode plot, you may need to increase the “samples” or interpolate between points. Save the Bode plot as an image.
Probing Further Questions
Q1. In Procedure parts 1 and 2 you measured the phase angle φ between Vin and Vo at the cutoff frequency. What values did you observe? What value would you expect? Why?
50 Q2. In Procedure parts 3 and 4 you measured the resonant frequencies and the bandwidths using the Bode Analyzer. Construct a table comparing the theoretical values (see Laboratory 6) and the
values for f0, f1, f2, bandwidth measured in Procedure parts 3 and 4. Calculate the Q for these circuits.
51 Lab 8 - Transformers
Transformers are magnetic circuit elements used in AC circuits for a variety of applications. Trans- formers are often used to convert voltages from high to low values, or from low to high values.
They are often used to isolate one AC line from another, for safety or for equipment isolation. And they are often used to connect two components with mismatched impedances, such as a high-impedance
stereo amplifier and a low-impedance speaker. This laboratory exercise will introduce transformer modeling concepts and explore the characteristics of a common power transformer.
• To teach the properties of common low-power transformers.
• Characterize the behavior of common power transformers, including concepts such as turns-ratios, impedance matching, and loading.
• Review the sections of your ECE-2620 textbook regarding transformer modeling. • Study the Background section below.
• Prior to coming to lab class, complete Part 0 of the Procedure.
52 Equipment
Analog Discovery 2 Instrument. Digital Multimeter. Transformer Board with mounted 120V, center-tapped 12.6V transformer. Resistance Decade Substitution box.
In its simplest form, a transformer consists of two closely coupled wire coils that share a common magnetic field. If a voltage is applied to one of the coils (called the primary), then a voltage will
be induced on the second coil (called the secondary).
Figure 8.1: Model for an Ideal Transformer
The actual voltage induced in the secondary coil is proportional to the ratio of the number of turns in the two coils. That ratio is called the turns ratio, for which we will use the symbol a. If the
number of turns in the primary is N1 and the number of turns in the secondary is N2, the turns ratio α is: N α = 2 (8.1) N1 and so, N2 V2 I1 α = = = and V2 = αV1 (8.2) N1 V1 I2
The two coils (also called windings) may be wound around air (an air core) or a dielectric, but more often they are wound around a ferromagnetic material, such as steel (an iron core), in order to
concentrate the magnetic field and improve the coupling between the coils. Transformers are most often used for three principal reasons: to change AC voltages; for electrical isolation; and for
impedance matching.
Often we can assume that a transformer is ”ideal”, meaning that it has no internal power loss and all magnetic flux lines of the primary coil also pass through the secondary coil. The ideal model
often fits large power transformers, where the losses are very small compared to the power transferred. An ideal transformer delivers all of the applied power to the load. That is,
P ower = I1V1 = I2V2 (8.3)
53 Rearranging this equation reveals an important relationship: N V I α = 2 = 2 = 1 (8.4) N1 V1 I2
So, for example, if the secondary voltage is 1/10 the primary voltage, then the primary current will be 1/10 the secondary current. The equation further indicates that if there is no load current (I2
=0), then the primary current is also zero.
Besides converting voltages, transformers are often used to match impedances between two electrical components, such as an audio amplifier (typically having 600Ω output impedance) and a speaker system
(typically 4Ω or 8Ω impedance). The right transformer can make the load appear to have the same impedance as the source, and thereby maximize the power transferred to the load — and minimize the
power lost in the source. The transformer that can perform such magical matching will have the following property: r Z α = load (8.5) Zsource where α is the turns ratio, Zload is the load impedance,
and Zsource is the source impedance. Equation 8.5 is a direct result of Equation 8.3 after substituting V1 = I1Z1 and V2 = I2Z2.
In essence, Zsource is what the load impedance Zload will look like from the primary side of the transformer. Expressed in other terms, Zsource is the effective load impedance reflected to the primary
side of the transformer. That is, 1 Z = Z (8.6) source α2 load
Usually, matching transformers are designed differently from power transformers, because matching transformers must meet other requirements. For example, audio transformers must respond uniformly over
a rather wide band of frequencies, and do so with very low distortion.
Non-Ideal Transformer Model
Real transformers do, in fact, have losses and do draw current even with no load connected. Real transformers have coil resistance; they require magnetizing currents to maintain the magnetic coupling
between the coils; and there are eddy currents and other issues. And so, even though the ideal model works well most of the time, when required, we use other models suitable to the circumstances to
represent non-ideal transformers. One such model is illustrated in Figure 8.2.
Figure 8.2: Circuit Model for non-ideal Transformer
54 For steady-state sinusoidal operation, the following two equations show the voltage sums in this circuit model: V = (R + jωL )I − jωMI S 1 1 1 2 (8.7) 0 = −jωMI1 + (R2 + jωL2 + Rload + jωLload)I2
where R1 and R2 are the winding resistances and M is the mutual inductance of the two windings. These equations can be used to compute the inductances of the transformer coils, as well as the mutual
inductance between them.
Part (0) Prior to coming to the lab, calculate the nominal turns ratio αnom for the transformer in Part (1) based on the nominal rating of the transformer’s primary and secondary voltages. Record
this result in Table 8.1.
Part (1) For these experiments, use a 120V / 12.6V center-tapped transformer mounted on Plexiglas.
Figure 8.3: Testing Voltage Relationships
Note: Some of the resistances and voltages are small. The cables connecting the AD2 to the trans- former must have a good connection at each end or the measurements will be unstable and significantly
1. Resistance: Using the DMM’s ohmmeter function, measure the transformer’s primary and sec- ondary resistances. Record the values in Table 8.1.
2. Measured Turns Ratio:
• Connect the function generator to the primary winding of the transformer, as illustrated in Figure 8.3. Select a sine wave output and set the generator’s voltage to 3VPP . Set the frequency at 60
Hz, since that is the operating frequency for which this transformer is designed.
55 Table 8.1: Recorded Values
Primary resistance RP
Secondary resistance RS
Nominal turns ratio, αnom
X13 turns ratio, αmeas Percentage difference
X12 turns ratio
X23 turns ratio
VPP −P rimary
VPP −X1−X3
VPP −X1−X2
VPP −X2−X3
• Connect the oscilloscope across the transformer as indicated in Figure 8.3. Use the oscillo- scope to measure the transformer voltages indicated below and record the results in Table 8.1.
– VPP across the primary (H1 to H2).
– VPP across the secondary (X1 to X3).
– VPP from the center tap (CT) to one side of the secondary (X1 to X2)
– VPP from the center tap (CT) to the other side of the secondary (X2 to X3) • Do you see why the connection X2 is referred to as the center tap?
• Calculate the turns ratio αmeasured between the primary winding and the secondary winding X1–X3. Record the result in Table 8.1.
Calculate the % Difference between αnominal and αmeasured and record the result in Table 8.1: α − α P ercentage Difference = measured nominal . 100% (8.8) αnominal
Why are αnominal and αmeasured different? Based on your measurements, estimate what the actual output voltage of the transformer (X1–X3) would be if the input voltage were 120V.
• Calculate the turns ratio αmeasured between the primary winding and the secondary winding X1–X2. Record the result in Table 8.1.
• Calculate the turns ratio αmeasured between the primary winding and the secondary winding X2–X3. Record the result in Table 8.1.
56 Part (2) Phase relationship between the primary and secondary voltages.
1. Connect the oscilloscope across the transformer, as indicated in Figure 8.3: CHANNEL 1 across the transformer’s primary (H1–H2) with H2 grounded; and CHANNEL 2 across the transformer’s secondary
(X1–X3) with X3 grounded.
2. Start the function generator (3 VPP at 60 Hz) and compare the phases of the two waveforms. Are the voltages in phase or out of phase? In your notebook, sketch the waveforms and record the phase
shift, if any. Save the oscilloscope output as an image.
3. Reverse the scope leads on the secondary: that is, connect CHANNEL 2’s signal lead to X3 and the ground lead to X1. Describe the result and sketch the waveforms in your notebook. Record the phase
shift, if any. Save the oscilloscope output as an image.
4. Connect CHANNEL 2 of the scope to observe the secondary-side voltage from X2 (signal) to X1 (ground). What is the difference from what you measured just previously in previous part?
Part (3) Secondary center-tap voltages.
Figure 8.4: Testing Secondary Center Tap
57 1. Connect the oscilloscope across the transformer, as indicated in Figure 8.4: CHANNEL 1 across X1–X2; and CHANNEL 2 across X3–X2. In both cases, X2 must be connected to the scope’s ground lead.
2. Start the function generator (3 VPP at 60 Hz). View the signals on each side of the center tap at the same time. Sketch the waveforms, showing the measured voltages. Save the oscilloscope output
as an image. Describe your observations regarding voltages, phases, and anything else you observe about the signal.
Notice that in Parts 1, 2, and 3 we could ground any terminal on the secondary. That is a demonstration of the third major use of the transformer: to isolate two circuits (especially the grounds)
from one another. Attempting the same thing on the primary side (e.g., connecting the oscilloscope’s ground to H1) could be disastrous because of the internal grounds on the function generator and
the oscilloscope.
Part (4) The effect of loading.
Figure 8.5: Testing Secondary Center Tap
1. Construct the circuit shown in Figure 8.5. Use a resistor substitution box for the resistor R, and initially set the value to 1000 Ω.
2. Connect the oscilloscope across the transformer, as indicated in Figure 8.5: CHANNEL 1 across H1–H2; and CHANNEL 2 across X1–X3.
3. Set the function generator to 6 VPP at 60 Hz, and start the function generator and the oscilloscope.
4. For each of the resistor values in Table 8.2, record in turn the voltages V1 and V2, and record the values in Table 8.2.
58 Table 8.2: Measured and Calculated Values
R (Ω) V1 V2 α = V2/V1 1000 100 50 20 5
5. Calculate the apparent turns ratio for each of the loads and record the results in Table 8.2.
6. Graph V1, V2, and a versus R to visualize the relationships. You may create the graph by hand or in a spreadsheet.
7. Compare these turns ratios with the αnominal and αmeasured calculated in Parts 2 and 3. How might these new values explain the difference between those earlier values?
Probing Further Questions
Q1. Using the data from Procedure part 1, compute the resistance ratio between the secondary and primary windings by dividing the secondary resistance by the primary resistance. Record the result and
compare it to the turns ratio. What factors might cause the resistance ratio to differ from the turns ratio?
59 Q2. Suppose that you had two of these 120 V transformers and you needed to convert 120 V to about 25 V. Show a sketch of how you could connect these two transformers to get this transformation.
Make sure to show the winding connection labels for each transformer.
Q3. If an ideal transformer with 120 V across the primary and draws a current of 200 mA, what power is dissipated in the load? If the transformer’s secondary is 30 V, what is the secondary current?
What is the turns ratio?
Q4. Using the specified Thevenin impedance of the function generator (50 Ω) and the specified impedance of an acoustic speaker (8 Ω), calculate the turns ratio required to match the speaker with the
function generator.
60 Lab 9 - Two-Port Network Characterization
You have now had experience with transfer functions and the concept of relating output quantities to input quantities. Many circuits can be treated as modules where an input is presented to two
terminals (the ”input port”) and an output is taken from two terminals (the ”output port”). Such circuits are termed two-port networks, and because they can be treated as a unit — a block — they
allow great simplification of large-scale designs. The last chapter in your textbook is an introduction to two-port circuits and to the methodology for developing the sets of parameters that may be
used to relate the output variables of voltage and current, typically designated as V2 and I2, respectively, to the input variables of voltage and current, typically designated as V1 and I1,
respectively. This approach is used to characterize a variety of components and circuit element combinations from filters, through transistors, to microwave circuits.
• The objectives of this laboratory exercise are to gain familiarity with three alternative two-port network parameter sets, to learn to measure the parameter sets, and to demonstrate the operational
definition of these parameters.
• To determine the impedance, admittance, and hybrid parameter sets for an unknown two-port network.
• Study the Background section below.
61 • Read the chapter in your textbook on two-port networks. Pay particular attention to the way in which the various parameters in each set (i.e., the z parameters, the y parameters, the h
parameters, etc.) are defined.
• Review the steps in the procedure below and plan how you will make your measurements to determine the required parameter values.
Analog Discovery 2 Instrument. Digital Multimeter. Two-port Network Box. Resistance Decade Substitution box.
A port consists of a pair of terminals; current enters through one of the terminals and the same current leaves through the other terminal. A resistor is a one-port network. In this lab we will study
two-port networks with one input port and one output port. Such networks are often treated as “black boxes” or modules that may be plugged into a circuit to accomplish some task, such as filtering the
signal or providing a controlled voltage. Engineers need a way to characterize the behavior of such a network and have developed several sets of parameters to do that. Each of these parameter sets
relates the input (side 1) and output (side 2) voltages and currents.
In this lab we will use impedance, admittance, and hybrid parameter sets to characterize a simple circuit. Impedance and admittance parameters are commonly used to characterize filters, and are often
useful in designing and characterizing impedance matching and power distribution networks. The term immittance is often applied to the use of either impedance or admittance parameters.
Be aware that because the voltages and currents are phasors with magnitude and phase angle, the parameters also have magnitude and phase angle. A simple RMS measurement will not suffice.
The impedance parameters (z parameters) relate the input and output voltages to the input and output currents by the following two equations:
V = z I + z I 1 11 1 12 2 (9.1) V2 = z21I1 + z22I2 or in matrix notation: V1 z11 z12 I1 = (9.2) V2 z21 z22 I2
62 The z parameters have units of ohms and are most easily found by applying a set of open-circuit tests on the circuit. When we apply a voltage to the input with the output open-circuited, we can
measure the input current and output voltage and find the first two z parameters as follows:
V1 V2 z11 = and z21 = (9.3) I I 1 I2 = 0 1 I2 = 0
We can determine the other two z parameters by applying a similar test to the output with the input open-circuited:
V1 V2 z12 = and z22 = (9.4) I I 2 I1 = 0 2 I1 = 0
Sometimes the impedance parameters do not exist because the voltages cannot be described by equation 9.1. Therefore, we need alternatives, such as the admittance parameters.
The admittance parameters (y parameters) relate the input and output currents to the input and output voltages by the following two equations:
I = y V + y V 1 11 1 12 2 (9.5) I2 = y21V1 + y22V2 or in matrix notation: I1 y11 y12 V1 = (9.6) I2 y21 y22 V2
The y parameters have units of siemens (or mhos) and are most easily found by applying a set of short-circuit tests on the circuit. When we apply a voltage to the inputs with the output short
circuited, we can measure the input current and output current to find the first two y parameters:
I1 I2 y11 = and y21 = (9.7) V V 1 V2 = 0 1 V2 = 0
We can determine the other two y parameters by applying a similar test to the output with the input short-circuited:
I1 I2 y12 = and y22 = (9.8) V V 2 V1 = 0 2 V1 = 0
There are occasions where neither the impedance nor the admittance parameters exist, so there is need for still another set of parameters.
63 The hybrid parameters (h parameters) are based on making V1 and I2 the dependent variables, and relating them to cross-variables V2 and I1. The h parameters satisfy the following equations:
V = h I + h V 1 11 1 12 2 (9.9) I2 = h21I1 + h22V2 or in matrix notation: V1 h11 h12 I1 = (9.10) I2 h21 h22 V2
The h parameters are found using a mix of short- and open-circuit tests as follows:
Short-circuit tests:
V1 I2 h11 = and h21 = (9.11) I I 1 V2 = 0 1 V2 = 0
Open-circuit tests:
V1 I2 h12 = and h22 = (9.12) V V 2 I1 = 0 2 I1 = 0
Warning: Unless you are careful in your planning, in carrying out the experiment, and in recording your data, you are very likely to make errors. Here in this last laboratory you must figure out how
to do most steps on your own.
Hint: Performing this lab is actually very straightforward, if you take it a step at a time. Think about what you have learned during this semester. If you need to measure the phase angle of a
voltage or current, you will need to use the oscilloscope. When necessary, insert a sense resistor of about 10 ohms and measure the voltage across it, including its phase shift, then calculate the
magnitude of the current flowing through the sense resistor. Be sure that you have accurately measured the resistor value, though, and not just relied on the nominal value. Use only one resistor
decade box (it will make your life easier), and be careful where you put the resistor, so you don’t make a grounding error on the oscilloscope. Carefully record each of the measurements and
calculations in tables that you have constructed for that purpose.
General Instructions:
1. Use the function generator to apply a 1 kHz sine wave of about 10 VPP to the input terminals of the Two-port Network Box given to you by your instructor.
64 2. Make the current and voltage measurements necessary to calculate the z,y,h-parameters. Note that you must determine not only the magnitude but also the phase of the voltages and currents
involved. How can you do that with the instruments at hand?
Important Notes:
• CH1 of the oscilloscope must always be connected across the FGEN. • Phase angles can be positive (lead) or negative (lag). • Record the RMS values. • Place the sense resistor on the Ground side.
Part (1) - Calculating z11 and z21
Figure 9.1: Circuit Diagram for Open-circuit Test
Table 9.1: Measured and Calculated Values
Measured Values
Output Terminal Open (I2 = 0) Magnitude Angle
V1 0
I2 0 0 Calculated Values
Magnitude Angle
65 Part (2) - Calculating y11, y21, h11, and h21
Figure 9.2: Circuit Diagram for Short-circuit Test
Table 9.2: Measured and Calculated Values
Measured Values
Output Terminal Shorted (V2 = 0) Magnitude Angle
V1 0
V2 0 0 Calculated Values
Magnitude Angle
66 Part (3) - Calculating z12, z22, h12, and h22
Figure 9.3: Circuit Diagram for Open-circuit Test
Table 9.3: Measured and Calculated Values
Measured Values
Input Terminal Open (I1 = 0) Magnitude Angle
V2 0
I1 0 0 Calculated Values
Magnitude Angle
67 Part (4) - Calculating y12, and y22
Figure 9.4: Circuit Diagram for Short-circuit Test
Table 9.4: Measured and Calculated Values
Measured Values
Input Terminal Shorted (V1 = 0) Magnitude Angle
V2 0
V1 0 0 Calculated Values
Magnitude Angle
68 Probing Further Questions
Q1. Are the parameter values, which you have calculated for the various two-port network representa- tions in this exercise valid for, say, 10 kHz? Why or why not?
Q2. The z and y parameters should be related by a matrix inversion. Invert the 2x2 z parameter matrix and see if it matches with the y parameters you calculated in the lab.
69 Lab 10 - Final Exam
This has been your first engineering laboratory. Although you have been provided with a “cookbook” (this manual), hopefully you have not just blindly followed instructions in order to get a good
grade. If you have, you have cheated only yourself. The theory which you have learned from your textbook and lectures is a way of looking at reality and thinking about how to organize experience,
i.e., dealing with real, physical things. It is only through combining practical experience with theory that you can begin to develop the necessary analytical skills to aid you in “taking things
apart and putting them together in new ways” that is the essence of the practice of real engineering.
This examination is designed to help you and your lab TA determine how much you have developed your knowledge, skills, and self confidence.
Review your lab manual. Think carefully of the procedures which you have followed and what you have learned from them. How do you measure voltage, current, resistance, frequency, etc.? How accurate
are your measurements? How are your measurements affected by the frequency response of your measuring instruments? Can you develop linear, semi-log, or log-log plots of your data to facilitate
• Calculator
• AD2 Device
• A Single Page Formula Sheet
• Laptop
70 Procedure
The final exam will consist of a written exam and a practical exam. The practical exam involves performing a lab procedure to obtain a desired result. The written exam will cover the application of
theory to understand circuit behavior, based upon your laboratory experiences in this course.
Probing Further Questions
Q1. This has been your second in a series of four (4) laboratory courses. What have you done to develop your expertise and skills? What will you do differently in the next course? What have you
learned from your experience?
71 Appendix A - Safety
Electricity, when improperly used, is very dangerous to people and to equipment. This is especially true in an industrial environment where large amounts of power is used, and where high voltages are
present [1]; in environments where people are especially susceptible to electric shock such as maintenance of a high voltage system (while in operation) or in hospitals where electrical equipment is
used to test or control physiological functions [2, 3]; and in an experimental or teaching laboratory where inexperienced personnel may use electrical equipment in experimental or nonstandard
Engineers play a vital role in eliminating or alleviating the danger in all three types of environments mentioned above. For conditions where standard equipment is used in standard configurations,
govern- mental agencies and insurance underwriters impose strict laws and regulations on the operation and use of electrical equipment including switchgear, power lines, safety devices, etc. As a
result, corporations and other organizations in turn impose strict rules and methods of operation on their employees and contractors. Engineers who are involved in using electrical equipment, in
supervising others who use it, and in designing such systems, have a great responsibility to learn safety rules and practices, to observe them, and to see that a safe environment is maintained for
those they supervise. In any working environment there is always pressure to “get the job done” and take short cuts. The engineer, as one who is capable of recognizing hazardous conditions, is in a
responsible position both as an engineer and as a supervisor or manager and must maintain conditions to protect personnel and avoid damage to equipment.
Because of their non-standard activities, experimental laboratories are exempt from many of these rules and regulations. This puts more responsibility on the engineer in this environment to know and
enforce the safest working procedures.
The knowledge and habit-forming experience to work safely around electrical equipment and the ability to design safe electrical equipment begins with the first student laboratory experience and con-
tinues through life. This includes learning the types of electrical injuries and damage, how they can be prevented, the physiology of electrical injuries, and steps to take when accidents.
Physiology of Electrical Injuries
There are three main types of electrical injuries: electrical shock, electrical burns, and falls caused by electrical shock. A fourth type, ’sunburned’ eyes from looking at electric arcs, such as
72 is very painful and may cause loss of work time but is usually of a temporary nature. Other injuries may be indirectly caused by electrical accidents, e.g., burns from exploding oil-immersed
switch gear or transformers.
Although electric shock is normally associated with high-voltage AC contact, under some circum- stances death can occur from voltages from substantially less than the nominal 120 Volts AC found in
residential systems. Electric shock is caused by an electric current passing through a part of the human body. The human body normally has a high resistance to electric currents so that a high
voltage is usually required to cause lethal currents. This resistance is almost all in the skin, but when the skin is wet its resistance is much lower. When a person is hot and sweaty or is standing
in water, contact with 120 Volts or less is likely to cause a fatal shock.
Electric shock is not a single phenomenon but is a disturbance of the nerves that is caused by electric current. A current through a part of the body such as the arm or leg will cause pain and muscle
contraction. If a victim receives an electric shock from grasping a live conductor, a current of greater than 15 to 30 mA through the arm will cause muscle contractions so severe that the victim
cannot let go. Similar currents through leg muscles may cause sudden contractions causing the victim to jump or fall, resulting in possible injuries or death. It is also possible for a prolonged
period of contact of more than a minute or so to cause chest muscles to be contracted, preventing breathing and resulting in suffocation or brain damage from lack of oxygen.
The predominant cause of death by electric shock is generally attributed to ventricular fibrillation, which is an uncontrolled twitching or beating of the heart that produces no pumping action and
therefore no blood circulation. Unless corrective action is taken, death follows quickly from lack of oxygen to the brain. While the amount of current that will cause fibrillation depends on several
variables, 0.5 to 5A through the body will normally cause the very small current through the heart that causes fibrillation in most people. Larger currents than this through the heart causes
contraction or clamping of the heart muscle and resulting death unless corrective action is taken. Prolonged contact of more than a minute or so may cause chest muscles to contract, preventing
breathing and resulting in suffocation or brain damage from lack of oxygen.
Death by electric shock is most often attributed to ventricular fibrillation, which is an uncontrolled twitching or beating of the heart that produces no pumping action and therefore no blood
circulation. Unless corrective action is taken, death follows quickly from lack of oxygen to the brain. While the amount of current that will cause fibrillation depends on several variables, 0.5 to 5
amperes through the body will normally cause the very small current (approximately 1 mA) through the heart that is sufficient to cause fibrillation in most people. Larger currents than this through the
heart cause contraction or clamping of the heart muscle, resulting in death unless corrective action is taken.
Electric burns may be caused by electric currents flowing in or near parts of the body. Such burns are similar to burns from ordinary heat sources, except that those caused by high-frequency currents
are generally deeper and take longer to heal the other burns. Electrocution will often leave severe burns at the points where the current entered and left the body.
73 Source of Electric Shock
Since electric shock is caused by an electric current through a part of the body, it is prevented by not allowing the body to become part of any electric circuit. From this viewpoint, electric
circuits may be classified as either grounded or ungrounded.
Electric circuits may be classified as either grounded or ungrounded. Grounded circuits are safer for most conditions, since they result in known voltages at other points in the circuit and provide
easier and better protection against faulty conditions in the circuit. The disadvantage is that a person standing on a non-insulated floor can receive a shock by touching only one conductor.
Almost all electric power generation, transmission, and distribution systems are grounded to pro- tect people and equipment against fall conditions caused by windstorms, lightning, etc. Residential,
commercial, and industrial systems such as lighting and heating are always grounded for greater safety. Communication, computer, and similar systems are grounded for safety reasons and to prevent or
reduce noise, crosstalk, static, etc. Many electronic equipment or instruments are grounded for safety and noise prevention, also. Common examples are DC power supplies, oscilloscopes, oscillators,
and analog and digital multimeters.
Ungrounded circuits are used in systems where isolation from other systems is necessary, where low voltages and low power are used, and in other instances where obtaining a ground connection is
difficult or impractical. In the ungrounded circuit, contact with two points in the circuit that are at different potentials is required to produce an electrical shock. The hazard is that with no known
ground, a hidden fault can occur, causing some unknown point to be grounded, in which case, touching a supposedly safe conductor while standing on the ground could result in an electric shock.
Protecting People and Equipment in the Laboratory
Prevention of electric shock to individuals and damage to equipment in the laboratory can be done by strict adherence to several common-sense rules summarized below:
Protecting People
1. When hooking up a circuit, connect to the power source last, while power is off.
2. Before making changes in a circuit, turn off or disconnect the power first, if possible.
3. Never work alone where the potential of electric shock exists.
4. When changing an energized connection, use only one hand. Never touch two points in the circuit that are at different potentials.
5. Know that the circuit and connections are correct before applying power to the circuit.
6. Avoid touching capacitors that may have a residual charge. The stored energy can cause a severe shock even after a long period of time.
7. Insulate yourself from ground by standing on an insulating mat where available.
74 The above rules and the additional rules given below also serve to protect instruments and other circuits from damage.
Protecting Equipment
1. Set the scales of measurement instrument to the highest range before applying power.
2. Before making changes in a circuit, turn off or disconnect the power first, if possible.
3. When using an oscilloscope, do not leave a bright spot or trace on the screen for long periods of time. Doing so can burn the image into the screen.
4. Be sure instrument grounds are connected properly. Avoid ground loops and accidental grounding of “hot” leads.
5. Check polarity markings and connections of instruments carefully before connecting power.
6. Never connect an ammeter across a voltage source, but only in series with a load.
7. Do not exceed the voltage or current ratings of circuit elements or instruments. This particularly applies to wattmeters, since the current or voltage rating may be exceeded with the needle still
reading on the scale.
8. Be sure any fuses and circuit breakers are of suitable value.
When connecting electrical elements to make up a network in the laboratory, it easy to lose track of various points in the network and accidentally connect a wire to the wrong place. One procedure to
help avoid this problem is to connect first the main series loop of the circuit, then go back and add the elements in parallel.
Types of Equipment Damage
Excessive currents and voltages can damage instruments and other circuit elements. A large over- current for a short time or a smaller over-current for a longer time will cause overheating, resulting
in insulation scorching and equipment failure.
Blown fuses are the most common equipment failure mode in this laboratory. The principal causes for these failures include:
• incorrectly wired circuits;
• accidental shorts;
• switching resistance settings while power is applied to the circuit;
• changing the circuit while power is applied;
• using the wrong scale on ammeter;
75 • connecting an ammeter across a voltage source; • using a low-power resistor box (limit 1/2 amp) when high power is required; • turning on an auto-transformer at too high a setting.
All of these causes are the result of carelessness by the experimenter.
Some type of insulating material, such as paper, cloth, plastic, or ceramic, separates conductors that are at different potentials in electrical devices. The voltage difference that this material can
withstand is determined by design (type, thickness, moisture content, temperature, etc.). Exceeding the voltage rating of a device by an appreciable amount can cause arcing or corona, resulting
insulation breakdown, and failure.
Some electrical devices can also be damaged mechanically by excessive currents. An example is the D’Arsonval meter, the indicator in most analog metering instruments. A large pulse of over current
will provide mechanical torque that can cause the needle to wrap around the pin at the top of the scale, thereby causing permanent damage even though the current may not have been on long enough to
cause failure due to overheating.
After Accident Action
Since accidents do happen despite all efforts to prevent them, plans for appropriate reaction to an accident can save time and lives. Such a plan should include immediate availability of first aid
material suitable for minor injuries or for injuries that are likely because of the nature of the work. Knowledge of how to obtain trained assistance such as Emergency Medical Services (EMS) should
be readily available for everyone.
Treating victims for electrical shock includes four basic steps that should be taken immediately. Step two requires qualification in CPR and step three requires knowledge of mouth-to-mouth
resuscitation. Everyone who works around voltages that can cause dangerous electrical shock should take advantage of the many opportunities available to become qualified in CPR and artificial
Immediate Steps After Electric Shock
1. Shut off all power and remove victim from the electric circuit. If the power cannot be shut off immediately,use an insulator of some sort, such as a wooden pole, to remove victim from the circuit.
Attempts to pull the victim from the circuit with your hands will almost always result in your joining the victim in the electric shock. 2. If you are qualified in CPR, check for ventricular
fibrillation or cardiac arrest. If either is detected, external cardiac massage should be started at once. Whether you are qualified in CPR or not, notify EMS and the ECE Department at once, using the
telephone numbers listed below. 3. Check for respiratory failure and take appropriate action. This may have resulted from physical paralysis of respiratory muscles or from a head injury. Sometimes
many hours pass before normal respiration returns. Artificial respiration should be continued until trained EMS assistance arrives.
76 4. Check for and treat other injuries such as fractures from a fall or burns from current entry and exit sites. Investigations are always after accidents. As an engineer you will be involved as a
part of the investigating team or in providing information to an investigator. Information obtained and notes written immediately after the emergency will aid this investigation and assist in
preventing future accidents of a similar nature.
Investigations are always made after accidents. As an engineer, you will be involved as a part of the investigating team or in providing information to an investigator. Information obtained and notes
written immediately after the emergency will aid the investigation and assist in preventing future accidents of a similar nature.
Emergency Numbers
Fire / EMS: 911 or (864) 656-2222 Student Health Center: (864) 656-2233 ECE Department Office: (864) 656-5650
Appendix A References
[1] W.F. Cooper, Electrical Safety Engineering, Newnes-Butterworth, London-Boston, 1978.
[2] W.H. Buschsbaum and B. Goldsmith, Electrical Safety in the Hospital, Medical Economics Company, Oradel, NJ, 1975.
[3] J.G. Wester, editor, Medical Instrumentation Application and Design, Houghton Mifflin Company, Boston, 1978.
77 Appendix B - Instruments for Electrical Mea- surements
Electrical engineers measure and use a wide variety of electrical circuit variables, such as volt- age,current, frequency, power, and energy, as well as electrical circuit parameters, such as
resistance, capacitance, and inductance. Many instruments can be used to make such measurements, but the proper use of the instruments and interpretation of the measurements depend on a fundamental
understanding of how the instruments work, their capabilities, and their limitations.
This appendix provides a brief overview of the fundamentals of the electrical equipment and instru- ments that you will use in this and other laboratory courses. As you encounter more and varied
types of electrical equipment and instruments in this and subsequent courses, you will find several books, in addition to your textbook, useful in developing your understanding and measurement skills.
In addition, many commercial instrument manufacturers publish handbooks and application notes that provide more information on specific measurement techniques.
Measurement of Current and Voltage
The basic electrical circuit variables of current and voltage are measured with ammeters (for cur- rent) and voltmeters (for voltage). These instruments may use either analog (continuous) or digital
(numerical) indicators (“readouts”) to report the measurement results.
Basic facts you should keep in mind include the following:
• An ammeter is a low resistance instrument to measure current. It is inserted in series with the with circuit branch of interest. If connected in parallel with a component, it will likely short out
the component and blow the fuse or burn out the circuit or the instrument.
• A voltmeter is a high resistance instrument for measuring voltage. It is connected in parallel with the component(s) of interest. Voltage is also called “electromotive force” or EMF.
78 Analog Meter Instruments
Analog meter instruments were developed early in the history of electrical science and technology. Most are based on the d’Arsonval galvanometer movement. A brief description of this meter movement
and its use in ammeters and voltmeters is given in the textbook Electric Circuits by J. W. Nilsson and S.A. Riedel.
In the d’Arsonval galvanometer, current through a coil of fine wire develops a magnetic field that opposes the field of a permanent magnet, and so rotates a needle across a scale that is marked off in
units of the measured variable. This type of movement is used extensively in DC analog instruments. For AC measurement, however, the d’Arsonval movement is not sufficient by itself, but must be adapted
in some way:
• it must be used in conjunction with a rectifier diode that converts the AC into a waveform with a DC level to which the meter can respond, or • it must use an electromagnet instead of the permanent
magnet in the standard d’Arsonval move- ment, or • it must use iron vanes.
Figure 12.1 (a) shows an equivalent circuit representation of a dual-coil electro-dynamometer wattmeter often used to measure 60 Hz AC power. It employs the electromagnet form of the galvanometer.
Because the meter deflection is proportional to the product of the current through the current-sensing coil (in series with the load) and the current through the voltage sensing coil (across the
load), the response is proportional to the product of the load current and the load voltage drop. The size of the resistance in series with the voltage coil determines the voltage range of the
Figure 12.1 (b) shows the equivalent circuit representation of the analog voltmeter. The size of the resistor determines the voltmeter range.
Figure 12.1 (c) shows the equivalent circuit of the analog ammeter. The size of the shunt (parallel) resistor determines the range of the ammeter.
Figure 12.1: (a) Equivalent circuit for analog wattmeter. (b) Equivalent circuit for analog voltmeter. (c) Equivalent circuit for analog ammeter.
79 Digital Multimeter
A multimeter is an electronic device that measures a multitude of electrical values, usually including at least AC and DC voltage and current, as well as resistance. Analog multimeters have an analog
display and digital multimeter (DMM) have a digital display. The DMM in this laboratory is used to measure voltage (DC and AC), current (DC and AC), resistance, capacitance. Additionally it may be
used for diode tests and audible continuity tests. For capacitance and inductance measurements you must make connections to the DMM/Impedance Analyzer on the prototyping board. For all other
measurements make connections to the DMM banana jacks on the workstation.
Dual-Beam Oscilloscope
The oscilloscope is a tool to allow engineers to look at the shape of an electrical voltage versus time or versus a second signal. Until relatively recently, oscilloscopes used a cathode ray tube
(CRT) to draw the waveforms onto a screen, just like an image on a television. Televisions and computer monitors also used cathode ray tubes until the advent of the new flat screens. Most engineers
refer to the instrument as a “scope”.
The Dual-Beam Oscilloscope has two vertical (“y”) input channels and one horizontal (“x”) channel. The horizontal channel can be connected either to an external AC voltage signal or to an internal
time- base generator (x = time). You should become familiar with the scale options on the y input channels, the x input channel, and the time base, since you will be using these to obtain values for
voltage and time. The y inputs can be either direct (1X) or through a 10X probe. Figure 12.2 (a) shows the equivalent input circuit for a direct input and Figure 12.2 (b) shows an equivalent circuit
for the 10X probe input. Note that in both configurations one side of the input is grounded, which means that care must be used in connecting the ground clip of the probe or connector used to assure
that these are not connected to a “hot” (|V | > 0) part of the circuit. (See the section “Oscilloscope Grounding Errors”.) The calibrated time base is useful when measuring the phase difference
between two waveforms (on the y1 and y2 inputs) by carefully lining up the zero levels for both y inputs and then using in the ac-couple mode to observe the time difference between zero crossings of
the two waveforms.
Figure 12.2: (a) Equivalent circuit for oscilloscope input with 1X probe (direct input). (b) Equivalent circuit for oscilloscope input with 10X probe.
Digital Storage Oscilloscope
80 The digital storage oscilloscope (DSO) is now the preferred type of oscilloscope for most industrial applications, although analog oscilloscopes are still widely used. The DSO uses digital memory
to store data as long as required without degradation. The digital storage allows use of an enormous array of sophisticated digital signal processing tools for the analysis of complex waveforms in
today’s circuitry.
The digital storage oscilloscopes of the Analog Discovery 2 are dual-beam oscilloscopes with two vertical inputs, as described above. The vertical input on the oscilloscope, instead of driving a
vertical amplifier, is digitized by an analog-to-digital (A-to-D) converter to create a data set that is stored in the memory of a microprocessor. The data set is processed and then sent to the
display. The data set can be written to a flash drive or sent over a LAN or a WAN for processing or archiving. The screen image can be directly recorded on paper by means of an attached printer or
plotter, without the need for an oscilloscope camera. The scope’s own signal analysis software can extract many useful time- domain features (e.g. rise time, pulse width, amplitude), frequency
spectra, histograms and statistics, persistence maps, and a large number of parameters meaningful to engineers in specialized fields such as telecommunications, disk drive analysis, and power
Digital oscilloscopes are limited principally by the performance of the analog input circuitry and the sampling frequency. In general, the sampling frequency should be at least the Nyquist rate –
double the frequency of the highest-frequency component of the observed signal – to avoid aliasing.
81 Appendix C - Operating Instructions for a Typical Oscilloscope
The oscilloscope is an instrument for the analysis of electrical circuits by observation of voltage and current waves. It may be used to study frequency, phase angle, and time, and to compare the
relation between two variables directly on the display screen. Perhaps the greatest advantage of the oscilloscope is its ability to display the periodic waveforms being studied.
Until recently, oscilloscopes used a cathode-ray tube to display the signals of interest. A cathoderay tube (CRT) contains an electron gun that directs a high-velocity beam of electrons onto a
fluorescent screen. The beam is controlled by a pair of horizontal and a pair of vertical deflecting plates. When the voltage on the deflection plates is equal to zero, the beam produces a spot of light
in the center of the screen. Any potential applied to the plates creates an electric field that deflects the electron beam proportionally to the applied voltage.
The basic components of the traditional oscilloscope are cathode-ray tube, amplifiers, sweep or tim- ing oscillator, and power supply. A voltage to be observed is applied to the vertical deflection
plates. This signal may be amplified by the vertical amplifier in order to obtain a satisfactory vertical deflec- tion. Meanwhile, a sweep oscillator moves the beam horizontally at a uniform rate. The
simultaneous horizontal and vertical sweep of the beam across the CRT screen displays the waveform of the voltage applied to the vertical plates. The sweep oscillator blanks the CRT electron gun
during its reverse sweep across the screen to switch off the electron beam.
If several voltage waveforms are to be studied and must maintain their relative phase positions, the sweep generator must be synchronized to the same voltage during the entire test. In this case, one
voltage is applied to the oscilloscope as an external trigger.
An independent voltage may be applied to the horizontal input in place of the sweep oscillator voltage. In this case, two independent input voltages are displayed against one another. If the
horizontal frequency is a submultiple of the vertical frequency, the trace will form a stationary pattern on the screen.
oday the traditional CRT oscilloscopes are rapidly being replaced with digital oscilloscopes that have flat-panel liquid crystal displays (LCDs), some with color displays. Instead of directly applying
the incoming voltages to deflection plates, the digital oscilloscopes capture the voltage information and store it in computer memory as digital signals, which are then analyzed and displayed on the
LCD. While the
82 new digital scopes handle the incoming signal differently from the CRT-based scopes, the basic purpose and many of the operational controls remain the same. Therefore, the discussion that follows
applies, for the most part, to both types of oscilloscopes.
Avoiding Grounding Errors with Oscilloscope
The shield or ground wire on the oscilloscope’s signal input is connected to the oscilloscope’s chassis ground, and therefore to the ground lead on the instrument’s electrical power connection. This
fact creates the possibility of grounding errors when making connections to circuits, especially when trying to measure voltages across ungrounded components.
The problem is illustrated in the following diagram. Suppose one wants to measure the voltage across resistor R1. Because the circuit and the oscilloscope both have the same ground, connecting the
oscilloscope’s input leads directly across R1, as indicated, will create a short across R2. Besides giving an incorrect reading, such a connection might damage either the oscilloscope or the circuit
under test.
Figure 13.1: Grounding Error
There are two solutions to the problem.
The first solution is simply to swap (interchange) the positions of R1 and R2, so that one end of R1 is at ground. Then there is no problem directly connecting the oscilloscope across R1, because both
grounds are connected together.
The second solution is sometimes called the two-channel difference method, and is illustrated in the next figure. It requires an oscilloscope with two input channels and with the ability to subtract
the signal on one channel from the other channel. As indicated in the illustration, the ground leads of both oscilloscope channels are connected to ground, so there is no grounding error. The signal
leads are placed on both sides of R1. Channel 1 will show the output voltage from the signal generator. Channel 2 will show the voltage across R2. If the oscilloscope’s difference function is engaged
so that the signal of Channel 2 is subtracted from Channel 1, the scope will display the voltage across R1.
83 Figure 13.2: Avoiding Grounding Error
Fortunately, Analog Discovery has a difference function and there is no need for the first solution. However, if you ever come across an oscilloscope that does not have a difference function, you may
apply the first solution; namely, to swap the positions of R1 and R2, and then measure the voltage across the now-grounded R1.
Figure 13.3: Avoiding Grounding Error
Preliminary Adjustment To Obtain a Trace
To operate the oscilloscope, first turn on the power switch and allow the unit to warm up and initialize. Place the HORIZONTAL CONTROL in the sweep position. Adjust the INTENSITY and FOCUS controls on
a CRT scope until the desired brightness and line width are obtained. Do not leave the spot stationary on the screen, as doing so may leave a permanent burn on the screen coating.
Once a trace is obtained, some oscilloscopes must be checked for calibration and balance. For the HP 120B CRT oscilloscope (a typical example), the procedure is as follows: Place the TRIGGER LEVEL on
AUTO and the TRIGGER SOURCE to INT. Turn the horizontal and vertical VERNIERS to CAL, the VERTICAL SENSITIVITY to CAL, and the HORIZONTAL DISPLAY to 1 msec/cm. With the controls in these positions,
an internal calibrating signal is produced on the screen. Turn the CAL adjustment until the upper and lower peaks of the square wave are 6 cm apart.
Digital oscilloscopes often have automated setup and calibration upon power-up.
84 Waveform Observation
After the initial adjustments are made, the oscilloscope is ready for operation. To observe the waveform of any periodic signal, apply the signal to the vertical input terminals. Use DC coupling if
the input is DC or very low in frequency or if you want to capture any DC offset. Now the signal from an oscillator, signal generator, or some component of an electrical circuit may be observed on the
screen. The best resolution of the waveform is obtained when the time scale is adjusted so one or two cycles appear on the screen and when the vertical scale is adjusted so the amplitude occupies
most of the graticule. If the waveform will not stabilize, adjust the SYNC or TRIGGER just enough to cause the pattern to stop. Whenever possible, connect the oscilloscope ground to the common ground
of the circuit. Exercise great care when making measurements with both terminals above ground potential, as there may be a difference in potential between two instrument cases, causing ground loop
currents, faulty readings, and damaged equipment.
Voltage Measurement (AC and DC)
The oscilloscope has advantages as a voltmeter: a very high input impedance compared to an analog voltmeter; the ability to measure voltages over a very wide frequency range; and the ability to
indicate magnitude regardless of waveform. Also, scopes measure peak-to-peak values of AC voltages, whereas standard AC voltmeters measure rms values of sine wave voltages. However, the oscilloscope
only has an accuracy of 2% to 5%, while the AC voltmeter’s accuracy will be from 0.25% to 2%.
To use the oscilloscope as an AC voltmeter, apply the signal to the vertical input terminals, and adjust the calibrated VERTICAL SENSITIVITY (or VOLTS/DIV) so the amplitude is of suitable magnitude
on the graticule. The peak-to-peak value is then the distance indicated multiplied by the vertical calibration. For example, assume that a sine wave generator is set to 1000 Hz and adjusted for
maximum output voltage. A peak-to-peak value of 60V is observed on the oscilloscope. The output of the generator at 1000 Hz, therefore, is approximately 60V peak-to-peak, and 21.2 Vrms.
Figure 13.4: Voltage Measurement
For DC measurements, apply the voltage to the vertical input terminals, again suitably adjusting the VERTICAL SENSITIVITY. A straight line is produced with the horizontal sweep functioning. With no
horizontal voltage applied, a spot will appear on the screen. In measuring DC voltages, it is necessary to remember where the trace was with 0V applied to the vertical input.
85 Frequency Measurement
The frequency of an unknown signal may be calculated from the oscilloscope very easily. The period of the waveform is the product of the distance along the x-axis covered by one cycle and the
horizontal sweep setting. As an example, a sine-wave generator is set to 1000 Hz with the voltage applied to the oscilloscope vertical. One cycle covers 9.95 cm, with a sweep speed of 100 µsec/cm.
The period is T = (9.95)·(100x10−6 )sec. The measured frequency is f = 1/T = 1005 Hz.
Figure 13.5: Frequency Measurement
Phase-Angle Measurement
The difference in phase angle between two waveforms may be measured directly on the oscilloscope with little difficulty.
For an oscilloscope with only one vertical input:
One wave is chosen as the reverence and applied to the vertical input terminals. This same wave is applied to the external trigger input of the scope. Next, a convenient point on the wave is selected
as a time reference, such as where the wave is zero and about to swing positive. Then, this waveform is removed from the vertical input and a second waveform is applied. The voltage of this wave at
the time reference is observed. The ratio of voltage at the time reference to the maximum voltage is equal to the sine of the phase difference between the two waves. This relation is shown below.
Figure 13.6: Phase-Angle Measurement
86 For an oscilloscope with two vertical inputs:
Connect the reference voltage to Channel 1 of the oscilloscope and connect the second voltage to Channel 2. Adjust the amplitudes so the overlapping signals look something like the figure below, where
the solid curve is Channel 1 (the reference) and the dashed curve is Channel 2. (In this figure, the dashed curve is lagging the solid curve; if the dashed curve were shifted to the left so it
“started” before the solid curve, then the dashed curve would be leading the solid curve.) The phase shift φ in degrees can be calculated by the following formula:
φ = ∆t.f.360o (13.1)
where φ is the phase shift, f is the frequency, and ∆t is the time difference between the two waveforms. Many new digital scopes have cursors that allow directly marking, calculating, and displaying
the time difference and perhaps even the phase shift.
Figure 13.7: Frequency Measurement
87 Appendix D - LT SPICE AC Circuit Simu- lation
LT SPICE is a powerful simulation program, but sometimes it is a little difficult to figure out how to get it to do what you want it to. A case in point is reporting voltages and currents in a simple AC
circuit. To get you started, this appendix describes how to set up a simple series circuit and run LT SPICE to report the current and voltages in that circuit.
Drawing circuits in LT SPICE is straightforward, and becomes easy with a little practice. Start by opening LT SPICE and creating the circuit shown below.
Figure 14.1: LT-SPICE Circuit
For this example, the power supply, V1, is a function generator providing a 10 VPP sinusoidal output. R1 is a 1.0kΩ resistor, and C1 is a 0.1µF capacitor. It is asked that we calculate the current,
the voltages, and the phase angles for frequencies from 500 Hz to 5000 Hz.
Once the circuit is drawn, insert ammeters in any branches of interest and apply voltmeters across the devices of interest. For now, don’t worry about the values indicated in the meters.
If necessary, change the values for the resistor and capacitor by double-clicking on the component and making the necessary changes to the name or the value. You must double-click on the component,
not on the text describing it.
88 Figure 14.2: LT-SPICE Circuit
Set the power supply voltage by double-clicking on the round icon representing the voltage source. Change the part title to “V1” if necessary. Set the [DC VALUE] to 0.
Select the [SMALL SIGNAL AC AND DISTORTION] tab. Under [AC PROPERTIES FOR AC SMALL SIGNAL ANALYSIS ONLY], enter magnitude 10 V and phase 0. Check [USE]. Click [OK].
Set the range of frequencies to be simulated as follows: Click in the circuit window to be sure it is selected. Click on the menu item [SIMULATION] and then choose [SET UP SIMULATIONS]. Click on the
button [AC (AC FREQUENCY SWEEP)]. Make the following entries:
Start Value: 500 (the frequency at which you want to begin) Stop Value: 5000 (the frequency at which you want to stop) Number of Each Steps per Interval: 10 (the number of frequencies at which you
want calculations, beginning at the Start Value and ending at the Stop Value.) Stepping interval: Linear
Check (i.e., select) [SHOW APTITUDE PLOTS] and [SHOW PHASE PLOTS]. Select [USE DEGREES FOR PLOTS] and [USE MAGNITUDE FOR PLOTS]. Check [DISPLAY GRAPH]. Check [DISPLAY TABLE]. Click [OK]. Be sure that
[AC (AC FREQUENCY SWEEP)] is enabled. Then click the button marked [RUN NOW...].
Immediately a table of voltages, currents, and phase angles will be displayed from 500 Hz to 5000 Hz. The phase angles may be a bit misleading for our purposes, because they force the supply voltage
to have a phase angle of zero (0), while our usual procedure is to require the voltage and current of the resistor to point along the real (X) axis in our phasor diagrams. Nonetheless, the simulation
data show the relative phases of the voltages, and as expected, the current through the capacitor leads the voltage across the capacitor by 90◦.
LT SPICE will create a graph of the data, if you checked DISPLAY GRAPH, but the voltage and current scales are usually so small that they require magnification. One can overcome the display
89 problem in the simulation by multiplying the simulation’s supply voltage by some constant so that it is near 100V. Be sure to correct for the multiplicative constant if you try to extract
magnitude values from the graph later.
To graph the table of data in Excel, EXPORT the table to a text file (through the File menu), and then read that text file into Excel. Once the data is in Excel, use the search and replace function to
change units multipliers: k to E3 m to E-3 u to E-6 n to E-9 meg to E6
Excel recognizes the E# as equivalent to 10∧#, where # is an appropriate number. Be sure you do not put any blank spaces in the substitution, or Excel will think it is all text.
After the substitutions are completed, use the standard Excel commands to graph the columns of interest. In this case you would probably choose an XY scatterplot VC and VR magnitudes versus
Warning: NEVER copy the simulated values and pretend that they are your calculations or your measurements. That is cheating. Whenever you report values from a simulation, be sure to state that they
are from the simulation and what program you used to do that simulation. To do otherwise is to be dishonest.
Oscilloscope-like Display
To have LT SPICE show an oscilloscope-like display of the circuit’s voltages and currents, requires only a few changes. Set up the circuit as before, and add meters for anything you want measured.
Double-click on the voltage source. In the [TRANSIENT PROPERTIES] section select [SINU- SOIDAL]. To the right, click on the [PEAK AMPLITUDE OF VOLTAGE] and in the [VALUE] box, enter “10” and click
[ACCEPT NEW VALUE]. Then click on [FREQUENCY] and enter “500” in the [VALUE] box and click [ACCEPT NEW VALUE]. Click [OK].
Setting the simulation parameters
Click on the circuit drawing to be sure it is selected. From the menu click on [Simulation] and choose [Set up Simulations]. Enable [Transient and Fourier Analyses] and then click that button. Input
a reasonable STOP TIME that will show a few cycles. In this case the frequency is 500 Hz, and so a stop time of 0.01 seconds will display five cycles. Enable [Display Graph]. Click [OK].
Click the button marked [RUN NOW...]. Immediately LT SPICE will generate a graph of whatever voltages and currents you had meters for on the circuit layout. If [AC (AC FREQUENCY SWEEP)] was also
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Test Scoring Application
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I have an assignment where I have to create a code to check over a group of students choices for a mock test and determine if the value is correct and return a score for the students and determine if
they pass or not.
"Write a Mathematica program to correct the Permit test for the DMV. The program should read a list of names and answers and compare them to the correct answers. The student scores should be imported
from the file student. The correct answers are stored in the file correct. If the student gets at least 7 correct answers, he/she passes the test. The program should print out the student name,
number of correct answers, score and Pass/Fail.
EX: Sally Smith 8 correct questions 80 PASS
You will need to import both files into your program."
How would I attempt to do this I am extremely stumped on the whole thing anyone have some input?
1 Reply
Well, obviously it's your assignment, and you're presumably able to do this to some extent (otherwise you went into the wrong classroom
1. Break the task down into smaller steps. Then you can only be stumped on 'the next step', rather than the whole thing.
2. Learn the basics of Mathematica. So visit YouTube, or www.wolfram.com/broadcast and watch a few videos. Read the Virtual Book in the Mathematica help or some of the tutorials.
3. Find out how to import files. Seaching for 'import' will bear much fruit. You have to know the file format of the files "student" and "correct" - tab-separated, comma-separated, or what? You could
probably import the names and scores into a table - a list of lists.
4. A basic skill in Mathematica is to chop and slice lists. Use double brackets (aka Part) for list access. You can get columns from tables too.
5. Write a function that takes two lists, one containing a student's scores, the other containing the answers. Find out how to compare two lists. This might involve...
6. Finding out how to map a function over one or more lists. This could be useful for comparing the values and storing some results. MapThread is cool...
7. Comparing a student's answer against the real answer is easy if the answers are numbers. Multiple choice is OK too. Otherwise you'll have to know about comparing strings...
8. The Count function can count values in lists. If the Count of correct answers is greater than or equal to 7, the student passes.
It's not a lot of code (about 10-20 lines?) but if you haven't learnt the basics of the language, it might be quite challenging until you have. | {"url":"https://community.wolfram.com/groups/-/m/t/163484?sortMsg=Likes","timestamp":"2024-11-10T15:16:33Z","content_type":"text/html","content_length":"95926","record_id":"<urn:uuid:4ac5373e-4581-4abd-8c5c-f31f51eb2f89>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00898.warc.gz"} |
Gcd: Excel Formulae Explained - ManyCoders
Key Takeaway:
• GCD is an important function in Excel that allows users to calculate the greatest common divisor of a set of numbers. It is useful in a variety of applications including simplifying fractions and
determining least common multiples.
• To calculate GCD in Excel, users can follow a simple set of steps such as selecting the range of cells containing the numbers, utilizing the GCD formula, and pressing enter to obtain the result.
• Mastering GCD formulae in Excel involves using additional functions such as the MOD function and IF function to simplify calculations and improve efficiency. Real-life applications of GCD in
Excel include simplifying fractions, determining common divisors, and finding common multiples.
Are you struggling to calculate GCD in Excel? Fear not! Here you can find a comprehensive guide to understanding GCD formulae and applying them to solve your calculations quickly and easily.
A Comprehensive Guide to GCD: Excel Formulae Explained
Ever struggled to calculate GCD in Microsoft Excel? No worries! This guide will explain GCD, and break down the steps to calculate it.
First, understanding GCD in Excel. How to recognize common factors and use the GCD formula. Then, let’s go deeper. Steps to calculate GCD, with examples and mistakes to avoid. With this guide, you’ll
become an expert in calculating GCD in Excel in no time!
Understanding GCD in Excel
Excel provides built-in formulas to calculate GCD – GCD.PB and GCD. Plus, you can use the VBA or array formulae method for GCD calculations. An array formula does a single operation on an array of
data instead of single cells.
To use the built-in formulas, enter the values separated by commas in any cell. Put the mouse cursor in that cell, type =GCD( ) and separate each number with a comma in the parentheses.
Sometimes, the digits are not integer types; they have decimals. Excel rounds them off and calculates their neighboring integer’s GCD value since it only takes integers as input parameters.
VBA code is better than native Excel functions when calculating GCD involving more than two cells/values. Or, use a user-defined function to pass over thousands of cells/values at once.
To make this calculation quicker and easier, sort your dataset before performing a calculation. Select a concise range of cells with numeric data and sort the range in descending order.
Steps to Calculate GCD with Ease will help us choose the right method and understand how inputs affect outputs when calculating complex math functions using arrays formulae methods.
1. Determine the data range – Select the range of cells that you want to calculate the GCD of.
2. Activate the array formula method – After selecting the data range, press Ctrl + Shift + Enter to set Excel into Array Formula mode.
3. Select a measure cell – Choose a measure cell on the Excel worksheet to act as the output cell for the formula.
4. Enter the array formula – In the measure cell, enter the array formula {=GCD(range)} and press Ctrl + Shift + Enter again.
5. Read the output – The output is the GCD of the selected range of cells.
Steps to Calculate GCD with Ease
Calculating GCD can be tricky. But with these steps, it’ll be a breeze! Follow the steps below:
1. Select all the cells whose GCD you want to calculate.
2. Then, open the formula bar and type out =GCD(.
3. Reference each of your cells by either typing their coordinates or clicking on them.
4. Close the parentheses and press enter.
5. The GCD will appear where you typed the formula.
No worries if maths isn’t your strong suit! These steps are easy to understand and super quick. Perfect for when you’re short on time. I know I had trouble with GCD until I found this simple solution
. Now I breeze through such problems!
Now that we’ve covered Steps to Calculate GCD with Ease, let’s dive into Mastering GCD Formulae in Excel! A topic that gives us insight like never before!
Mastering GCD Formulae in Excel
Ready to step up your Excel game? Look no further! This article dives into the GCD formulae.
We’ll break it down into 3 pieces:
1. First, we’ll show the power of GCD with examples and real-world uses.
2. Second, we’ll use the MOD Function to make work simpler.
3. Finally, we’ll show you how to get even better calculations with the IF function.
Let’s explore the depths of GCD formulae!
Harnessing the Power of GCD Function
Have you ever worked with large datasets in Excel? If so, you may have needed to find the greatest common divisor (GCD) of two or more numbers. Good news! Excel has a built-in function for this – GCD
Here, we will explain how to use it in 6 steps:
1. Select an empty cell for the result.
2. Type “=” then “GCD(” and select the range with the numbers.
3. Close the bracket “)” and press enter.
4. Write the numbers manually inside the brackets instead of selecting cells.
5. For more than two numbers, separate them with a comma.
6. Use conditional formatting to color/bold cells with maximum values.
Using GCD saves time and simplifies complex operations. It also reduces risk of errors by eliminating manual calculations. Tip: use data bars or color scales to emphasize results.
Next, we will discuss the MOD function.
Simplify with MOD Function
MOD Function is a great way to sort and work with data in Excel. You don’t need to calculate by hand – just create a formula and copy it! With MOD, you can determine if two cells are divisible or not
. Plus, use it with IF and ROUND to get even more complex calculations.
This concept dates back to ancient times when it was used in math calculations. Technology has made this easier – Excel provides us with tools to make calculations faster and smarter.
Ready for the next step? Let’s dive into the IF Function for Effective Calculations!
IF Function for Effective Calculations
The IF function is an important tool to make effective calculations in Excel. It lets you set a logical test which will be either true or false. Based on the result, another calculation can be done.
For instance, if you have a spreadsheet with prices and discounts in different columns, you can use the IF function for the final price. The following table shows an example of this:
Column 1 – Prices Column 2 – Discounts Column 3 – Final Price (IF Function)
$50 10% =IF(B2<>””, A2*(1-B2), A2)
$75 =IF(B3<>””, A3*(1-B3), A3)
Here, we used the IF function to figure out the final price in each row. If a discount was specified, it would subtract the percentage. If not, it would just return the original price.
IF functions can help simplify calculations and save time by automating them. With practice, you can create even more complex formulas with multiple IF statements or logical operators like AND and
Interestingly, Excel started as an internal project at Microsoft in the mid-1980s called Multiplan. It only became popular after it was released as Excel on Macintosh in 1985. Now, it’s a widely used
tool for analysis.
In the next section, we’ll look at the real-life application of GCD in Excel, to show how useful it can be in solving mathematical problems.
Real-life Application of GCD in Excel
As an Excel lover, I’m always seeking ways to make work easier and faster. I was thrilled when I found out about the greatness of Greatest Common Divisors (GCD) in Excel. This article will show the
real-life applications of GCD in Excel. We’ll go through 3 sections. First, we’ll learn how to use GCD like a Pro. Second, we’ll simplify fractions using GCD. And last, we’ll look at finding Least
Common Multiple with GCD. These tips will help you improve your Excel skills!
Determine Greatest Common Divisors like a Pro
Do you want to Calculate the Greatest Common Divisor (GCD) of a set of values? With Microsoft Excel, it’s easy! Here’s how:
1. Open Excel and create a new sheet.
2. Type the set of values you want to find the GCD of.
3. In an empty cell, type ‘=GCD(‘ followed by the range.
4. Close the bracket and press ‘Enter’!
GCD can save you time when dealing with large amounts of data. It’s useful for industries that need to analyze numerical data. Plus, it can even be used in cryptography, according to research by Dr.
Izumi Miyamoto from the University of Electro-Communications.
Simplifying fractions is also possible with GCD. Learn how to do this with our Step-by-Step Guide!
Simplify Fractions with GCD – A Step-by-Step Guide
Simplifying fractions can be tricky. But GCD (Greatest Common Divisor) makes it easier. Here’s a step-by-step guide on how to use GCD to simplify fractions.
1. Find GCD of numerator and denominator. List all factors of both. Identify largest common factor. For example, 12/24 has GCD of 12.
2. Divide numerator and denominator by GCD. In our example, this results in (12/12)/(24/12), which equals (1/2).
3. Simplify your result. Final answer is (1/2). You’ve learned how to simplify fractions with GCD!
Using GCD always simplifies fractions without losing its value. Master multiplication and division tables up to ten to make factoring easier.
Let’s introduce the next application – ‘Finding Least Common Multiple with GCD’ – which is useful in real life scenarios!
Finding Least Common Multiple with GCD
We have two columns in this table. They show the two numbers for which we want the least common multiple.
Plus, we have two columns for the greatest common divisor (GCD) and least common multiple (LCM).
For example, we can see that the GCD is 6 for 12 and 18. We get this by using the Excel function “GCD” with two arguments.
The LCM is easy. We just need to do (Number1 * Number2)/GCD.
If we want the LCM for more than two numbers, we simply multiply them all and divide by their GCD. For example, LCM = (Number1 * Number2 * …* NumberN)/GCD(Number1,Number2,…NumberN).
Troubleshooting GCD in Excel? Tips and Tricks can make it easier.
Troubleshooting GCD in Excel: Tips and Tricks
Stuck on using GCD in Excel? We all have been there. No need to worry though! This part is made just for you. It’ll cover two topics: avoiding GCD errors and easy solutions for GCD issues. After this
section, you’ll have the know-how to fix any GCD problem in Excel formulas. Let’s get started!
Avoiding Common Errors with GCD
It’s important to understand the GCD formula correctly to avoid mistakes. Here’s a table with common errors and the actual data:
Common Mistakes Actual Data
Assuming Positive Numbers -6 and 10
No Absolute Values in Arguments -10 and -15
Reversing Arguments 10 and 6
Check your inputs before applying the formula. Also, use cell references instead of typing arguments into the GCD function. This helps with precision, editing, reusing and more.
Avoiding errors with GCD can sharpen your Excel skills. No errors means faster delivery, less human error and accuracy! Don’t miss this great opportunity to improve.
Next: Quick Fixes for GCD Problems.
Quick Fixes for GCD Problems
Ensure all inputs are numbers – GCD formula only works with numerical values. If not, an error will be returned.
Use absolute references – This will keep the formula consistent when copied and pasted.
Check for typos – Misspellings can cause an error.
Verify data inputs – Check they reflect your intended values.
Negative values no-no – GCD only works with positives; negative inputs will result in an error message.
To fix GCD troubles, check:
• All input cells are correctly arranged.
• Perform basic arithmetic operations.
• Cell ranges have consistent formats.
Careful review of inputs is key to avoiding GCD errors – these often come from human error.
Did you know GCD is based on Euclid’s algorithm, from over 2,000 years ago? It was developed by Greek mathematician Euclid and remains used today to find the greatest common divisor of two or more
Five Facts About GCD: Excel Formulae Explained:
• ✅ GCD stands for “Greatest Common Divisor” and is used in mathematics to find the largest number that divides two or more numbers without a remainder. (Source: Math Is Fun)
• ✅ The GCD formula in Excel is “=GCD(number1, number2, …)” and can take up to 255 arguments. (Source: Ablebits)
• ✅ The GCD formula can be used to simplify fractions in Excel. (Source: Excel Campus)
• ✅ The GCD formula can also be used to find the least common multiple (LCM) of two or more numbers. (Source: Vertex42)
• ✅ The GCD formula is commonly used in various fields, including engineering, computer science, and finance. (Source: Investopedia)
FAQs about Gcd: Excel Formulae Explained
What is GCD and how can it be calculated using Excel formulae?
GCD stands for Greatest Common Divisor which refers to the largest number that divides two given numbers without leaving any remainder. In Excel, GCD can be calculated using the GCD formula which
goes as follows: =GCD(number1, [number2], …). This formula can take up to 255 arguments and returns the GCD of all the numbers provided.
What are some use cases of GCD in Excel?
GCD is commonly used in Excel when working with fractions or rational numbers. For instance, when adding or subtracting fractions, one would need to find the least common denominator which can be
found by taking the LCM of the denominators using the GCD formula.
Can the GCD formula handle decimal numbers?
No, the GCD formula works only with whole numbers. If you need to find the GCD of decimal numbers, you would first need to convert them into fractions or whole numbers by multiplying both the
numerator and denominator by a factor that will make them integers.
What happens if only one argument is provided in the GCD formula?
If only one argument is provided in the GCD formula, the formula will simply return the absolute value of that number.
Is there a limit to the number of arguments that can be provided in the GCD formula?
Yes, the GCD formula can take up to a maximum of 255 arguments. However, it is recommended to use the formula with a smaller number of arguments for better results and faster computation time.
Can GCD be used in other spreadsheet programs besides Excel?
Yes, GCD is a mathematical concept and can be found in other spreadsheet programs such as Google Sheets, LibreOffice Calc, and others. However, the syntax and implementation may differ among these | {"url":"https://manycoders.com/excel/formulae/gcd-excel/","timestamp":"2024-11-02T09:02:32Z","content_type":"text/html","content_length":"86089","record_id":"<urn:uuid:b5497c20-8b35-4bd5-8ed4-50e2ee5870c8>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00778.warc.gz"} |
Math Is Fun Forum
Re: Doc, Doc!
#2681. What does the medical term Craniotomy mean?
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
Re: Doc, Doc!
#2682. What does the medical term Oocyte mean?
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
Re: Doc, Doc!
#2683. What does the medical term Antidepressant discontinuation syndrome mean?
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
Re: Doc, Doc!
#2684. What does the medical term Choroidal nevus mean?
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
Re: Doc, Doc!
#2685. What does the medical term Dehydration mean?
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
Re: Doc, Doc!
#2686. What does the medical term Intraocular hemorrhage mean?
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
Re: Doc, Doc!
#2687. What does the medical term Intravenous therapy mean?
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
Re: Doc, Doc!
#2688. What does the medical term Gonadotropin-releasing hormone mean?
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
Re: Doc, Doc!
#2689. What does the medical term Pleural cavity mean?
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
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#2690. What does the medical term Refractive error mean?
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
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#2691. What does the medical term Skin cancer mean?
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
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#2692. What does the medical term Immunofluorescence mean?
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
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#2693. What does the medical term Optic neuritis mean?
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
Re: Doc, Doc!
#2694. What does the medical term Parotitis mean?
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
Re: Doc, Doc!
#2695. What does the medical term Polymyalgia rheumatica mean?
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
Re: Doc, Doc!
#2696. What does the medical term Laryngoscopy mean?
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
Re: Doc, Doc!
#2697. What does the medical term Continuous positive airway pressure (CPAP) mean?
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
Re: Doc, Doc!
#2698. What does the medical term Laxative mean?
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
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#2699. What does the medical term Myocarditis mean?
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
Re: Doc, Doc!
#2700. What does the medical term 'Indigestion' mean?
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
Re: Doc, Doc!
#2701. What does the medical term Anemia or anaemia mean?
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
Re: Doc, Doc!
#2702. What does the medical term Brain metastasis mean?
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
Re: Doc, Doc!
#2703. What does the medical term Milroy's disease mean?
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
Re: Doc, Doc!
#2704. What does the medical term Palmitic acid mean?
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
Re: Doc, Doc!
#2705. What does the medical term Flocculus mean?
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking. | {"url":"https://mathisfunforum.com/viewtopic.php?pid=442934","timestamp":"2024-11-09T03:53:20Z","content_type":"application/xhtml+xml","content_length":"88673","record_id":"<urn:uuid:232940a4-9472-4d62-8722-8902a2b1b755>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00133.warc.gz"} |
Data-Independent Memory Hard Functions: New Attacks and Stronger Constructions
Data-Independent Memory Hard Functions: New Attacks and Stronger Constructions
Download: DOI: 10.1007/978-3-030-26951-7_20 (login may be required)
Memory-hard functions (MHFs) are a key cryptographic primitive underlying the design of moderately expensive password hashing algorithms and egalitarian proofs of work. Over the past few
years several increasingly stringent goals for an MHF have been proposed including the requirement that the MHF have high sequential space-time (ST) complexity, parallel space-time
complexity, amortized area-time (aAT) complexity and sustained space complexity. Data-Independent Memory Hard Functions (iMHFs) are of special interest in the context of password hashing as
they naturally resist side-channel attacks. iMHFs can be specified using a directed acyclic graph (DAG) G with $$N=2^n$$ nodes and low indegree and the complexity of the iMHF can be
analyzed using a pebbling game. Recently, Alwen et al. [ABH17] constructed a DAG called DRSample that has aAT complexity at least $$\varOmega \!\left( N^2/{\text {log}} N\right) $$ .
Asymptotically DRSample outperformed all prior iMHF constructions including Argon2i, winner of the password hashing competition (aAT cost $${\mathcal {O}} \!\left( N^{1.767}\right) $$ ),
though the constants in these bounds are poorly understood. We show that the greedy pebbling strategy of Boneh et al. [BCS16] is particularly effective against DRSample e.g., the aAT cost
is $${\mathcal {O}} (N^2/{\text {log}} N)$$ . In fact, our empirical analysis reverses the prior conclusion of Alwen et al. that DRSample provides stronger resistance to known pebbling
Abstract: attacks for practical values of $$N \le 2^{24}$$ . We construct a new iMHF candidate (DRSample+BRG) by using the bit-reversal graph to extend DRSample. We then prove that the construction
is asymptotically optimal under every MHF criteria, and we empirically demonstrate that our iMHF provides the best resistance to known pebbling attacks. For example, we show that any
parallel pebbling attack either has aAT cost $$\omega (N^2)$$ or requires at least $$\varOmega (N)$$ steps with $$\varOmega (N/{\text {log}} N)$$ pebbles on the DAG. This makes our
construction the first practical iMHF with a strong sustained space-complexity guarantee and immediately implies that any parallel pebbling has aAT complexity $$\varOmega (N^2/{\text {log}}
N)$$ . We also prove that any sequential pebbling (including the greedy pebbling attack) has aAT cost $$\varOmega \!\left( N^2\right) $$ and, if a plausible conjecture holds, any parallel
pebbling has aAT cost $$\varOmega (N^2 \log \log N/{\text {log}} N)$$ —the best possible bound for an iMHF. We implement our new iMHF and demonstrate that it is just as fast as Argon2.
Along the way we propose a simple modification to the Argon2 round function that increases an attacker’s aAT cost by nearly an order of magnitude without increasing running time on a CPU.
Finally, we give a pebbling reduction that proves that in the parallel random oracle model (PROM) the cost of evaluating an iMHF like Argon2i or DRSample+BRG is given by the pebbling cost
of the underlying DAG. Prior pebbling reductions assumed that the iMHF round function concatenates input labels before hashing and did not apply to practical iMHFs such as Argon2i, DRSample
or DRSample+BRG where input labels are instead XORed together.
Video from CRYPTO 2019
title={Data-Independent Memory Hard Functions: New Attacks and Stronger Constructions},
booktitle={Advances in Cryptology – CRYPTO 2019},
series={Lecture Notes in Computer Science},
author={Jeremiah Blocki and Ben Harsha and Siteng Kang and Seunghoon Lee and Lu Xing and Samson Zhou}, | {"url":"https://www.iacr.org/cryptodb/data/paper.php?pubkey=29899","timestamp":"2024-11-13T01:35:35Z","content_type":"text/html","content_length":"27450","record_id":"<urn:uuid:6fa3fff9-85f7-4d56-9d29-3aab2114acca>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00771.warc.gz"} |
Python: Implementing Linked List
1. Introduction
A linked list is a foundational data structure where elements, known as nodes, are connected linearly. Each node contains its data and a reference (or link) to the next node in the sequence. The
primary advantage of linked lists over traditional arrays is that the former allows for efficient insertions and deletions because they don't need to shift elements around.
2. Program Overview
In this post, we'll create a basic singly linked list. We will implement the following operations:
1. append: Adds a new node at the end of the list.
2. display: Shows all the elements of the linked list.
3. is_empty: Checks if the linked list is empty.
4. length: Returns the number of elements in the linked list.
3. Code Program
class Node:
def __init__(self, data):
"""Initialize the node with data and no reference to the next node."""
self.data = data
self.next = None
class LinkedList:
def __init__(self):
"""Initialize the linked list with a head set to None."""
self.head = None
def append(self, data):
"""Add a new node with the given data at the end of the list."""
new_node = Node(data)
if not self.head:
self.head = new_node
last_node = self.head
while last_node.next:
last_node = last_node.next
last_node.next = new_node
def display(self):
"""Print all the elements of the linked list."""
curr_node = self.head
while curr_node:
print(curr_node.data, end=" -> ")
curr_node = curr_node.next
def is_empty(self):
"""Check if the linked list is empty."""
return not self.head
def length(self):
"""Return the number of elements in the linked list."""
count = 0
curr_node = self.head
while curr_node:
count += 1
curr_node = curr_node.next
return count
# Demonstration
llist = LinkedList()
print("is_empty:", llist.is_empty()) # True
llist.display() # 5 -> 10 -> 15 -> None
print("length of the linked list:",llist.length()) # 3
is_empty: True
5 -> 10 -> 15 -> None
length of the linked list: 3
4. Step By Step Explanation
1. The Node class represents an individual node of the linked list. It contains data and a reference to the next node (initialized as None).
2. The LinkedList class defines the operations we can perform on the linked list. Its head attribute points to the first node of the linked list.
3. The append method allows us to add a node at the end of the list.
4. The display method lets us visualize the list, printing each element followed by an arrow, culminating in None.
5. The is_empty method returns a boolean indicating whether the list is empty or not.
6. The length method gives the number of nodes in the list. | {"url":"https://www.javaguides.net/2023/09/python-implementing-linked-list.html","timestamp":"2024-11-10T22:32:09Z","content_type":"application/xhtml+xml","content_length":"234314","record_id":"<urn:uuid:7b6c1197-d05e-4f3d-9a37-c8b768525da5>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00458.warc.gz"} |
Eureka Math Grade 7 Module 4 Lesson 4 Answer Key
Engage NY Eureka Math 7th Grade Module 4 Lesson 4 Answer Key
Eureka Math Grade 7 Module 4 Lesson 4 Example Answer Key
Example 1: Finding a Percent Increase
Cassandra’s aunt said she will buy Cassandra another ring for her birthday. If Cassandra gets the ring for her birthday, what will be the percent increase in her ring collection?
→ Looking back at our answers to the Opening Exercise, what percent is represented by 1 ring? If Cassandra gets the ring for her birthday, by what percent did her ring collection increase?
20% represents 1 ring, so her ring collection would increase by 20%.
→ Compare the number of new rings to the original total:
\(\frac{1}{5}\) = \(\frac{20}{100}\) = 0.20 = 20%
→ Use an algebraic equation to model this situation. The quantity is represented by the number of new rings.
Quantity = Percent × Whole. Let p represent the unknown percent.
1 = p∙5
\(\frac{1}{5}\) = p
\(\frac{1}{5}\) = \(\frac{20}{100}\) = 0.2 = 20%
Example 2: Percent Decrease
Ken said that he is going to reduce the number of calories that he eats during the day. Ken’s trainer asked him to start off small and reduce the number of calories by no more than 7%.
Ken estimated and consumed 2,200 calories per day instead of his normal 2,500 calories per day until his next visit with the trainer. Did Ken reduce his calorie intake by no more than 7%? Justify
your answer.
a. Ken reduced his daily calorie intake by 300 calories. Does 7% of 2,500 calories equal 300 calories?
Quantity = Percent × Whole
False, because 300≠175.
b. A 7% decrease means Ken would get 93% of his normal daily calorie intake since 100% – 7% = 93%. Ken consumed 2,200 calories, so does 93% of 2,500 equal 2,200?
Quantity = Percent × Whole
False. Because 2,200 ≠ 2,325, Ken’s estimation was wrong.
Example 3: Finding a Percent Increase or Decrease
Justin earned 8 badges in Scouts as of the Scout Master’s last report. Justin wants to complete 2 more badges so that he will have a total of 10 badges earned before the Scout Master’s next report.
a. If Justin completes the additional 2 badges, what will be the percent increase in badges?
Quantity = Percent × Whole. Let p represent the unknown percent.
2 = p∙8
2(\(\frac{1}{8}\)) = p(\(\frac{1}{8}\))(8)
\(\frac{2}{8}\) = p
\(\frac{1}{4}\) = p
\(\frac{1}{4}\) = \(\frac{25}{100}\) = 25%
There would be a 25% increase in the number of badges.
b. Express the 10 badges as a percent of the 8 badges.
8 badges is the whole, or 100%, and 2 badges represent 25% of the badges, so 10 badges represent
100% + 25% = 125% of the 8 badges.
10 = p∙8
10(\(\frac{1}{8}\)) = p(\(\frac{1}{8}\))(8)
\(\frac{10}{8}\) = p
\(\frac{5}{4}\) = p
\(\frac{5}{4}\) = \(\frac{125}{100}\) = 125%
c. Does 100% plus your answer in part (a) equal your answer in part (b)? Why or why not?
Yes. My answer makes sense because 8 badges are the whole or 100%, and 2 badges represent 25% of the badges, so 10 badges represent 100% + 25%, or 125% of the 8 badges.
Example 4: Finding the Original Amount Given a Percent Increase or Decrease
The population of cats in a rural neighborhood has declined in the past year by roughly 30%. Residents hypothesize that this is due to wild coyotes preying on the cats. The current cat population in
the neighborhood is estimated to be 12. Approximately how many cats were there originally?
→ Do we know the part or the whole?
We know the part (how many cats are left), but we do not know the original whole.
→ Is this a percent increase or decrease problem? How do you know?
Percent decrease because the word declined means decreased.
→ If there was about a 30% decline in the cat population, then what percent of cats remain?
100% – 30% = 70%, so about 70% of the cats remain.
→ How do we write an equation to model this situation?
12 cats represent the quantity that is about 70% of the original number of cats. We are trying to find the whole, which equals the original number of cats. So, using Quantity = Percent × Whole and
substituting the known values into the equation, we have 12 = 70%∙W, where W represents the original number of cats.
Quantity = Percent × Whole
12 = (\(\frac{7}{10}\))∙W
(12)(\(\frac{10}{7}\)) = (\(\frac{7}{10}\))(\(\frac{10}{7}\))∙W
\(\frac{120}{7}\) = W
There must have been 17 cats originally.
To find the original number of cats or the whole (100% of the cats), we need to add three more twelve sevenths to 12.
12 + 3(\(\frac{12}{7}\)) = \(\frac{84}{7}\) + \(\frac{36}{7}\) = \(\frac{120}{7}\)≈17
The decrease was given as approximately 30%, so there must have been 17 cats originally.
Example 5.
Lu’s math score on her achievement test in seventh grade was a 650. Her math teacher told her that her test level went up by 25% from her sixth grade test score level. What was Lu’s test score level
in sixth grade?
→ Does this represent a percent increase or decrease? How do you know?
Percent increase because the word up means increase.
→ Using the equation Quantity = Percent × Whole, what information do we know?
We know Lu’s test score level in seventh grade after the change, which is the quantity, and we know the percent. But we do not know the whole (her test score level from sixth grade).
→ If Lu’s sixth grade test score level represents the whole, then what percent represents the seventh grade level?
100% + 25% = 125%
→ How do we write an equation to model this situation? Let W represent Lu’s test score in sixth grade.
Quantity = Percent × Whole
650 = 125% × W
650 = 1.25W
650(\(\frac{1}{1.25}\)) = 1.25(\(\frac{1}{1.25}\))W
\(\frac{650}{1.25}\) = W
\(\frac{65,000}{125}\) = W
520 = W
Lu’s sixth grade test score level was 520.
Eureka Math Grade 7 Module 4 Lesson 4 Exercise Answer Key
Opening Exercise
Cassandra likes jewelry. She has 5 rings in her jewelry box.
a. In the box below, sketch Cassandra’s 5 rings.
b. Draw a double number line diagram relating the number of rings as a percent of the whole set of rings.
c. What percent is represented by the whole collection of rings? What percent of the collection does each ring represent?
100%, 20%
Exercise 1.
a. Jon increased his trading card collection by 5 cards. He originally had 15 cards. What is the percent increase? Use the equation Quantity = Percent × Whole to arrive at your answer, and then
justify your answer using a numeric or visual model.
Quantity = Percent × Whole. Let p represent the unknown percent.
5 = p(15)
5(\(\frac{1}{15}\)) = p(15)(\(\frac{1}{15}\))
\(\frac{5}{15}\) = p
p = \(\frac{1}{3}\) = 0.3333…
0.3333 … = \(\frac{33}{100}\) + \(\frac{0.3333 \ldots}{100}\) = 33% + \(\frac{1}{3}\)% = 33\(\frac{1}{3}\)%
b. Suppose instead of increasing the collection by 5 cards, Jon increased his 15 – card collection by just 1 card. Will the percent increase be the same as when Cassandra’s ring collection increased
by 1 ring (in Example 1)? Why or why not? Explain.
No, it would not be the same because the part – to – whole relationship is different. Cassandra’s additional ring compared to the original whole collection was 1 to 5, which is equivalent to 20 to
100, which is 20%. Jon’s additional trading card compared to his original card collection is 1 to 15, which is less than 10%, since
\(\frac{1}{15}\)<\(\frac{1}{10}\), and \(\frac{1}{10}\) = 10%.
c. Based on your answer to part (b), how is displaying change as a percent useful?
Representing change as a percent helps us to understand how large the change is compared to the whole.
A sales representative is taking 10% off of your bill as an apology for any inconveniences.
Exercise 2.
Skylar is answering the following math problem:
The value of an investment decreased by 10%. The original amount of the investment was $75.00. What is the current value of the investment?
a. Skylar said 10% of $75.00 is $7.50, and since the investment decreased by that amount, you have to subtract $7.50 from $75.00 to arrive at the final answer of $67.50. Create one algebraic equation
that can be used to arrive at the final answer of $67.50. Solve the equation to prove it results in an answer of $67.50. Be prepared to explain your thought process to the class.
Let F represent the final value of the investment.
The final value is 90% of the original investment, since 100% – 10% = 90%.
F = Percent × Whole
F = (0.90)(75)
F = 67.5
The final value of the investment is $67.50.
b. Skylar wanted to show the proportional relationship between the dollar value of the original investment, x, and its value after a 10% decrease, y. He creates the table of values shown below. Does
it model the relationship? Explain. Then, provide a correct equation for the relationship Skylar wants to model.
No. The table only shows the proportional relationship between the amount of the investment and the amount of the decrease, which is 10% of the amount of the investment. To show the relationship
between the value of the investment before and after the 10% decrease, he needs to subtract each value currently in the y – column from each value in the x – column so that the y – column shows the
following values: 67.5, 90, 180, 270, and 360. The correct equation is y = x – 0.10x, or y = 0.90x.
Eureka Math Grade 7 Module 4 Lesson 4 Problem Set Answer Key
Question 1.
A store advertises 15% off an item that regularly sells for $300.
a. What is the sale price of the item?
(0.85)300 = 255; the sale price is $255.
b. How is a 15% discount similar to a 15% decrease? Explain.
In both cases, you are subtracting 15% of the whole from the whole, or finding 85% of the whole.
c. If 8% sales tax is charged on the sale price, what is the total with tax?
(1.08)(255) = 275.40; the total with tax is $275.40.
d. How is 8% sales tax like an 8% increase? Explain.
In both cases, you are adding 8% of the whole to the whole, or finding 108% of the whole.
Question 2.
An item that was selling for $72.00 is reduced to $60.00. Find the percent decrease in price. Round your answer to the nearest tenth.
The whole is 72. 72 – 60 = 12. 12 is the part. Using Quantity = Percent × Whole, I get 12 = p × 72, where p represents the unknown percent, and working backward, I arrive at \(\frac{12}{72}\) = \(\
frac{1}{6}\) = \(0.1 \overline{6}\) = p.
So, it is about a 16.7% decrease.
Question 3.
A baseball team had 80 players show up for tryouts last year and this year had 96 players show up for tryouts. Find the percent increase in players from last year to this year.
The number of players that showed up last year is the whole; 16 players are the quantity of change since
96 – 80 = 16.
Quantity = Percent × Whole. Let p represent the unknown percent.
16 = p(80)
p = 0.2
0.2 = \(\frac{20}{100}\) = 20%
The number of players this year was a 20% increase from last year.
Question 4.
At a student council meeting, there was a total of 60 students present. Of those students, 35 were female.
a. By what percent is the number of females greater than the number of males?
The number of males (60 – 35 = 25) at the meeting is the whole. The part (quantity) can be represented by the number of females (35) or how many more females there are than the number of males.
Quantity = Percent × Whole
35 = p(25)
p = 1.4
1.4 = 140%, which is 40% more than 100%. Therefore, there were 40% more females than males at the student council meeting.
b. By what percent is the number of males less than the number of females?
The number of females (35) at the meeting is the whole. The part (quantity) can be represented by the number of males, or the number less of males than females (10).
Quantity = Percent × Whole
10 = p(35)
0.29 = 29%
The number of males at the meeting is approximately 29% less than the number of females.
c. Why is the percent increase and percent decrease in parts (a) and (b) different?
The difference in the number of males and females is the same in each case, but the whole quantities in parts (a) and (b) are different.
Question 5.
Once each day, Darlene writes in her personal diary and records whether the sun is shining or not. When she looked back though her diary, she found that over a period of 600 days, the sun was shining
60% of the time. She kept recording for another 200 days and then found that the total number of sunny days dropped to 50%. How many of the final 200 days were sunny days?
To find the number of sunny days in the first 600 days, the total number of days is the whole.
Quantity = Percent × Whole. Let s represent the number of sunny days.
s = 0.6(600)
s = 360
There were 360 sunny days in the first 600 days.
The total number of days that Darlene observed was 800 days because 600 + 200 = 800.
d = 0.5(800)
d = 400
There was a total of 400 sunny days out of the 800 days.
The number of sunny days in the final 200 days is the difference of 400 days and 360 days.
400 – 360 = 40, so there were 40 sunny days of the last 200 days.
Question 6.
Henry is considering purchasing a mountain bike. He likes two bikes: One costs $500, and the other costs $600. He tells his dad that the bike that is more expensive is 20% more than the cost of the
other bike. Is he correct? Justify your answer.
Yes. Quantity = Percent × Whole. After substituting in the values of the bikes and percent, I arrive at the following equation: 600 = 1.2(500), which is a true equation.
Question 7.
State two numbers such that the lesser number is 25% less than the greater number.
Answers will vary. One solution is as follows: Greater number is 100; lesser number is 75.
Question 8.
State two numbers such that the greater number is 75% more than the lesser number.
Answers will vary. One solution is as follows: Greater number is 175; lesser number is 100.
Question 9.
Explain the difference in your thought process for Problems 7 and 8. Can you use the same numbers for each problem? Why or why not?
No. The whole is different in each problem. In Problem 7, the greater number is the whole. In Problem 8, the lesser number is the whole.
Question 10.
In each of the following expressions, c represents the original cost of an item.
a. Circle the expression(s) that represents 10% of the original cost. If more than one answer is correct, explain why the expressions you chose are equivalent.
b. Put a box around the expression(s) that represents the final cost of the item after a 10% decrease. If more than one is correct, explain why the expressions you chose are equivalent.
c – 0.10c
1c – 0.10c Multiplicative identity property of 1
(1 – 0.10)c Distributive property (writing a sum or difference as a product)
Therefore, c – 0.10c = 0.90c.
c. Create a word problem involving a percent decrease so that the answer can be represented by expression (ii).
Answers will vary. The store’s cashier told me I would get a 10% discount on my purchase. How can I find the amount of the 10% discount?
d. Create a word problem involving a percent decrease so that the answer can be represented by expression (i).
Answers will vary. An item is on sale for 10% off. If the original price of the item is c, what is the final price after the 10% discount?
e. Tyler wants to know if it matters if he represents a situation involving a 25% decrease as 0.25x or (1 – 0.25)x. In the space below, write an explanation that would help Tyler understand how the
context of a word problem often determines how to represent the situation.
If the word problem asks you to find the amount of the 25% decrease, then 0.25x would represent it. If the problem asks you to find the value after a 25% decrease, then (1 – 0.25)x would be a correct
Eureka Math Grade 7 Module 4 Lesson 4 Exit Ticket Answer Key
Question 1.
Erin wants to raise her math grade to a 95 to improve her chances of winning a math scholarship. Her math average for the last marking period was an 81. Erin decides she must raise her math average
by 15% to meet her goal. Do you agree? Why or why not? Support your written answer by showing your math work.
No, I do not agree. 15% of 81 is 12.15. 81 + 12.15 = 93.15, which is less than 95. I arrived at my answer using the equation below to find 15% of 81.
Quantity = Percent × Whole
Let G stand for the number of points Erin’s grade will increase by after a 15% increase from 81. The whole is 81, and the percent is 15%. First, I need to find 15% of 81 to arrive at the number of
points represented by a 15% increase. Then, I will add that to 81 to see if it equals 95, which is Erin’s goal.
G = 0.15 × 81
G = 12.15
Adding the points onto her average: 81.00 + 12.15 = 93.15
Comparing it to her goal: 93.15 < 95
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Finding the Percent of a Quantity with Simple Fractions and Integers
Question Video: Finding the Percent of a Quantity with Simple Fractions and Integers
The pie chart shows the results of a survey of what fruits students prefer. Given that 30 students completed the survey, how many students prefer peaches?
Video Transcript
The pie chart shows the results of a survey of what fruits students prefer. Given that 30 students completed the survey, how many students prefer peaches?
We can see from the pie chart that the percentage of students that prefer peaches was 40 percent. As there were 30 students surveyed in total, we need to calculate 40 percent of 30. There are lots of
ways of calculating this. One way would be to convert the percentage into a decimal first. As the word percent means out of 100, we can convert from a percentage to a decimal by dividing by 100.
Therefore, 40 percent is equivalent to 0.4. The word “of” in mathematics means multiply. We need to multiply 0.4 by 30. 0.4 multiplied by 10 is equal to four. Therefore, 0.4 multiplied by 30 is equal
to 12. 12 of the 30 students in the survey prefer peaches.
An alternative method to calculate 40 percent of 30 would be to turn the percentage into a fraction. 40 percent is the same as 40 over 100, so we would need to multiply this by 30. We could simplify
the fraction by dividing the numerator and denominator by 10. 30 and 10 are also divisible by 10, leaving us with four multiplied by three. Once again, we get an answer of 12 students.
A third method would be to calculate 10 percent of 30 first. We know that to find 10 percent, we divide the quantity by 10. And 30 divided by 10 is three. We can then multiply this answer by four to
calculate 40 percent of 30. All three of these methods give us an answer of 12 students. | {"url":"https://www.nagwa.com/en/videos/923186549307/","timestamp":"2024-11-03T03:43:32Z","content_type":"text/html","content_length":"241058","record_id":"<urn:uuid:c99bc163-537c-4adb-9bce-153c06f41633>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00840.warc.gz"} |
Master Equations and Operator Sum Form
2736 views
I'm more of a quantum optics guy than a quantum info guy, and deal mainly in master equations. I'm interested in operator-sum form, and I'd like to derive the errors in this form for a small quantum
system that I'm simulating.
The catch: The quantum system is driven by an external (classical) field modelled with a sinusoidal function, and the damping rates are low, so I can't make a rotating wave approximation to eliminate
this time dependence. Given that I must solve the master equation numerically by integration, and the result of each integration at time $t$ is not sufficient information to figure out these errors,
and I need to do some work to recover the superoperator matrix that has operated on a vectorised density matrix. i.e. I feed the master equation a vectorised density matrix with a single entry of 1
and the rest zero, and build the matrix like that for a particular time $\tau$. Am I on the right track here (sanity check)? More explicitly, if $\mathrm{vec}(\rho_{ij,t=\tau})$ is the vectorised (so
it's a column vector) form of a density matrix with a single entry of 1 in position $i,j$, at $t=0$ that has been evolved to time $\tau$, then a matrix to take the vector form of the density matrix
from $t=0$ to $t=\tau$ is given as $\mathbf{M}=\sum_{i,j}\mathrm{vec}(\rho_{ij,t=0})\mathrm{vec}(\rho_{ij,t=\tau})^\dagger$.
The question: Given this superoperator $\mathbf{M}$ that does $\mathbf{M}\,\mathrm{vec}(\rho_0)=\mathrm{vec}(\rho_\tau)$, how can I get Krauss operators for the operator-sum equivalent of $\mathbf{M}
$ that are in a useful form? i.e. the system in question is a qubit or a qutrit and another qubit or qutrit. I'd like to be able to do the operator sum in the form of tensor products of spin matrices
on each channel if possible.
Side question: Is $\mathbf{M}$ a Choi matrix?
Final note: I awarded the acceptance to Pinja, as I used the paper Pinja suggested. I have provided an answer myself below that fills in the details.
This post has been migrated from (A51.SE)
What do you mean by "system in question is a qubit or a qutrit and another qubit or qutrit." -- what is the "other system"? Are you talking about the ancilla needed to implement this channel using
unitaries + tracing out? In that case, note that the dimension of the ancilla can be up to D^2, so qubits won't do.
This post has been migrated from (A51.SE)
No, at the moment it's just a toy model consisting of two small quantum systems that are coupled, and have different T1 and T2 times. The answer to this question is not of serious concern. It's more
a point of interest, as it could be a handy to know more about how to do this in the future.
This post has been migrated from (A51.SE)
The references given in answer to Quantum mechanics as a Markov process — in particular Carlton Caves' on-line notes "Completely positive maps, positive maps, and the Lindblad form" — survey physical
ideas and mathematical tools that are helpful in answering the question.
A key point is associated to the specific question asked "How can I get Kraus operators for the operator-sum equivalent of $M$ that are in a useful form?" For large quantum systems, a generic
superoperator $M$ will not have an algorithmically compressible form. Moreover, Kraus representations are non-unique, and to the best of my (non-expert) knowledge there is no procedure that is both
general and efficient for finding Kraus representations of a given $M$ that have a "useful form" (by whatever criteria are given for a form being "useful"). That deciding quantum separability is
NP-hard suggests that no efficient, general representation-finding algorithm exists, even when $M$ is numerically given in its entirety.
To make progress, it may be helpful to ask heuristic questions: "What is special about my particular superoperator? Can I exhibit a set of Lindbladian generators for it that have useful symmetry
properties and/or generate compatible compressive flows on the Hilbert state-space? Are these Lindbladian properties associated to a natural Hilbert basis in which $M$ has a sparse, factored, or
otherwise algorithmically compressible representation?"
If questions like these could be efficiently answered by "turning an algorithmic crank", then quantum physics would be a far less interesting subject! :)
This post has been migrated from (A51.SE)
This is pretty much what I was hoping was not the case, but thought would be. Sadly the system only has exploitable symmetry in the case of only dephasing with no depopulation. There's a very
appealing form of the Lindblad master equation that collects terms which aren't of Krauss form into a non-hermitian Hamiltoninan, which for the case of no time-dependence in the Hamiltonian can be
used to choose a basis which naturally expresses decay as the remaining Krauss terms. Neat, but no help for me.
This post has been migrated from (A51.SE)
One of the references in Caves' notes is Wolf and Cirac *Dividing quantum channels* (arXiv:math-ph/0611057), which I recommend without the least warranty of having personally grasped the (many and
subtle) quantum informatic issues that this article discusses! :)
This post has been migrated from (A51.SE)
Nice, I'll take a look at that. One interesting thing that I possibly should have noted about the undriven version of the system above is that its time independence means that you can find $\mathbf
{M}$ directly with matrix exponentiation (not efficient, but this system is small enough). You can also construct $\mathbf{M}$ using some guessed Krauss operators, and a few simultaneous equations
later gives you the mapping between the two, allowing the extraction of error rates on the various channels.
This post has been migrated from (A51.SE)
I worked on a very similar problem on my Masters thesis, in which I studied the non-Markovian dynamics of a driven qubit in a dissipative environment. My interest was in checking that the master
equation I obtained was completely positive, but this is just one side of your problem. The question turned out to be very non-trivial if no RWA is made, but I was able to get some results using Ref.
[J Mod. Opt. 54, 1695 (2007)] and exploiting the fact that the qubit is weakly coupled to the environment. I'll beat my drum and also give the Ref. to an article where I present some of these
results, [P. Haikka and S. Maniscalco, Phys. Rev. A 81, 052103 (2010)], you may find it useful.
This post has been migrated from (A51.SE)
Ah! It turns out I've been looking at the Andersson paper one for a few days now. It does seem very promising, and gives the most concrete recipe. I like having a *method* to apply to problems. To be
honest, I need to find a patch of time to really sit down and look at this. It's more of a personal project at the moment.
This post has been migrated from (A51.SE)
I think what you might be looking for is this: The Real Density Matrix. It gives you a recipe for converting between various superoperator representations (including using a tensor product basis of
Paulis). A detailed quantum process tomography experiment utilizing the results are here: Quantum Process Tomography of the Quantum Fourier Transform. More generally, Havel has also derived
algorithms to convert to minimal Kraus representations here: Procedures for Converting among Lindblad, Kraus and Matrix Representations of Quantum Dynamical Semigroups.
Edit to answer the added question: unfortunately, this area is plagued by inconsistent notation and conventions. Nevertheless, I will give you the one that seems most natural to me. So let me take $
{\rm vec}(\rho)$ to be the operation of taking the the rows of $\rho$ and stacking them on top of each other. That is ${\rm vec}(|i\rangle\langle j|)=|i\rangle\otimes|j\rangle$ rather than ${\rm vec}
(|i\rangle\langle j|)=|j\rangle\otimes|i\rangle$, which would be "column stacking" (But! This of course depends on your convention for the Kronecker product---here I am taking the second index to
"vary most rapidly"). Let's instead define the "column stacking" convention as ${\rm col}(\rho)$. Now, neither of the matrices which act as ${\rm \bf M}^{\rm row}{\rm vec}(\rho_0)={\rm vec}(\rho_t)$
or ${\rm \bf M}^{\rm col}{\rm col}(\rho_0)={\rm col}(\rho_t)$ are the "Choi" matrix. The Choi matrix is defined as $$ {\rm\bf C} = \sum_{i,j} ({\rm\bf 1}\otimes |i\rangle\langle j|) {\rm \bf M}^{\rm
row} (|i\rangle\langle j|\otimes {\rm\bf 1}), $$ or, equivalently $$ {\rm\bf C} = \sum_{i,j} (|i\rangle\langle j|\otimes {\rm\bf 1}) {\rm \bf M}^{\rm col} ({\rm\bf 1}\otimes |i\rangle\langle j|). $$
Note that since all these representations are defined in terms of the basis $\{|i\rangle\langle j|\otimes |k\rangle\langle l|\}$, the transformation between them is just a permutation.
This post has been migrated from (A51.SE)
This is interesting, it could be exactly what I'm looking for...
This post has been migrated from (A51.SE)
I just saw your addition. Thank you, this is very useful. I originally took your version of vec, but now I use the columns stacked. Thank [Wikipedia](http://en.wikipedia.org/wiki/
Vectorization_%28mathematics%29) for that one. Perhaps I should adopt your notation for clarity.
This post has been migrated from (A51.SE)
As Pinja noted, a paper by Andersson et al. (arXiv)(DOI) has been especially useful. The paper goes into a great deal of detail, and I finally sat down today to take a proper look at it. As an
example problem, I picked two qubits with an exchange interaction to check this which is a minimal version of what I'm considering. To begin, the master equation is given by
$$ \dot\rho = \Lambda(\rho). $$
The method requires that basis operators of the system are chosen. It is convenient to give these in terms of the Pauli matrices in the case of two qubits, but for a qutrit one would employ the
Gell-Mann matrices. Defining $\sigma_i = \mathbf{1},\sigma_x,\sigma_y,\sigma_z$ for each qubit, this system has a basis built up of the tensor products of these with a factor of $1/2$ for
normalisation, yielding 16 operators $G_i$ e.g. $G_5 = G_{xx} = (\sigma_x\otimes\sigma_x)/2$. Sticking with Hermitian operators keeps things neat as well, since some daggers can be neglected.
A special matrix is now composed called $L$, which is related to the master equation.
$$ L_{n,m} = \mathrm{Tr}[G_n\Lambda(G_m)]. $$
If we are dealing with the master equation as a matrix acting on a vectorised density operator as discussed in the question, then this can be expressed as
$$ L_{n,m} = \mathrm{vec}(G_n)^\dagger\,\Lambda\,\mathrm{vec}(G_m), $$
which allows L to be derived in a single matrix equation, but that's getting a little off topic.
In the sample case I considered, $L$ is does not contain time varying terms, so it may be exponentiated to get a new matrix $F$, which is related to the solution of the master equation $\phi$
$$ F(t) = \exp(Lt). $$
$F$ can be used to get a Choi matrix $S$, which is exactly what I need. At this point, a basis needs to be chosen for the future Krauss operators. I'm quite happy with the Pauli operators so I'll
stick with those for this next equation,
$$ S_{a,b} = \sum_{n,m}F_{m,n}\mathrm{Tr}[G_nG_aG_sG_b]. $$
Finally, the wonderful part.
$$ \rho_t=\phi_{n,m}(\rho_0,t) = S_{n,m}(t)G_n\rho_0 G_m^\dagger $$
As you can see, $S$ is a matrix of weights for a sum of superoperators in a useful basis that I can select. This has been referred to as the process matrix (arXiv)(DOI) which is unique to a process
in a given basis. In the sample case, in which the master equation has no time dependent terms on the RHS, the solution can be directly verified by representing $\Lambda$ in matrix form and
exponentiating it to get $\phi(t)=\exp(\Lambda t)$.
This works in the time independent case for quits and qutrits as expected. I need to check that this works in the case of time dependence.
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Net present value: How to compare the present value of cash inflows and outflows - FasterCapital
Net present value: How to compare the present value of cash inflows and outflows
1. Introduction
In this section, we will delve into the concept of net present value (NPV) and its significance in evaluating the profitability of investment projects. NPV is a financial metric used to compare the
present value of cash inflows and outflows over a specific time period. By discounting future cash flows to their present value, NPV helps decision-makers assess the viability and profitability of
potential investments.
Insights from Different Perspectives:
To gain a comprehensive understanding of NPV, let's explore insights from various perspectives:
1. Financial Perspective:
From a financial standpoint, NPV serves as a crucial tool for capital budgeting decisions. It takes into account the time value of money, recognizing that a dollar received in the future is worth
less than a dollar received today. By discounting future cash flows at an appropriate rate, NPV provides a more accurate assessment of an investment's profitability.
2. Investment Evaluation:
When evaluating investment opportunities, NPV allows for a systematic comparison of different projects. By calculating the NPV for each option, decision-makers can prioritize investments based on
their potential to generate positive net cash flows. Projects with higher NPV values are generally considered more favorable.
3. Risk Assessment:
NPV also aids in assessing the risk associated with an investment. By incorporating a discount rate that reflects the project's risk profile, NPV accounts for the uncertainty and variability of
future cash flows. This enables decision-makers to make informed choices by considering both the potential returns and associated risks.
In-Depth Information (Numbered List):
Now, let's dive deeper into the key aspects of NPV:
Accurate estimation of cash inflows and outflows is crucial for NPV calculations. It involves forecasting future cash flows based on historical data, market trends, and projected revenues and
expenses. The more precise the cash flow estimates, the more reliable the NPV analysis.
2. Discount Rate Selection:
The discount rate represents the opportunity cost of capital and reflects the risk and return expectations of investors. It is typically determined by considering factors such as the cost of
borrowing, the company's weighted average cost of capital (WACC), or the desired rate of return. choosing an appropriate discount rate is essential for accurate NPV calculations.
3. Time Horizon:
The time period over which cash flows are evaluated significantly impacts NPV. Longer time horizons increase the uncertainty and risk associated with future cash flows. Therefore, it is crucial to
consider the appropriate time frame when conducting NPV analysis.
4. Sensitivity Analysis:
NPV calculations are subject to various assumptions and estimates. Conducting sensitivity analysis helps assess the impact of changes in key variables on the project's NPV. By analyzing different
scenarios, decision-makers can gain insights into the project's sensitivity to factors such as sales volume, costs, or discount rates.
To illustrate the concept of NPV, let's consider an example. Suppose a company is evaluating two investment projects: Project A and Project B. Project A requires an initial investment of $100,000 and
is expected to generate cash inflows of $30,000 per year for five years. Project B requires an initial investment of $150,000 and is expected to generate cash inflows of $40,000 per year for five
years. By calculating the NPV for each project, decision-makers can determine which investment option is more financially viable.
Introduction - Net present value: How to compare the present value of cash inflows and outflows
2. Understanding Net Present Value (NPV)
## The Essence of NPV
At its core, NPV answers a fundamental question: "Is this investment worthwhile?" Whether you're considering a business project, buying a house, or investing in a new technology, NPV provides a
quantitative framework to assess the profitability of these endeavors. Let's explore NPV from different angles:
1. Time Value of Money (TVM):
- NPV recognizes that money today is worth more than the same amount in the future. Why? Because you can invest that money and earn returns. Conversely, waiting for cash inflows means missing out on
potential gains.
- The discount rate (often the cost of capital) accounts for this opportunity cost. It reflects the risk and return associated with the investment.
2. Cash Flows Matter:
- NPV considers both inflows (revenues, dividends, etc.) and outflows (initial investment, operating costs, etc.). It's not just about making money; it's about managing costs too.
- Example: Imagine you're launching a startup. You need to invest in product development, marketing, and hiring. NPV helps you weigh these costs against expected future profits.
3. NPV Formula:
- The NPV formula is straightforward:
$$NPV = \sum \frac{CF_t}{(1 + r)^t} - C_0$$
- \(CF_t\) represents the cash flow at time \(t\).
- \(r\) is the discount rate.
- \(C_0\) is the initial investment.
- If NPV > 0, the investment is profitable; if NPV < 0, it's not.
4. Decision Rule:
- Positive NPV? Green light! The project adds value.
- Negative NPV? Proceed with caution or reconsider.
- Zero NPV? The project breaks even – no gain, no loss.
5. Sensitivity Analysis:
- NPV isn't set in stone. It depends on assumptions (cash flows, discount rate, etc.). Sensitivity analysis explores how NPV changes with varying inputs.
- Example: If your sales projections are off, how does NPV react?
6. Comparing Alternatives:
- NPV lets you compare different investment options. Suppose you're torn between two projects. Calculate NPV for both and choose the one with the higher NPV.
- Example: Should you invest in solar panels or wind turbines for your energy company? NPV will guide your decision.
7. Real-World Example:
- Imagine you're considering buying a rental property. The initial cost is $200,000. Expected annual rental income is $20,000, and you plan to sell the property after 10 years.
- Calculate NPV:
- Cash inflows: \(CF_t = \$20,000\) (for 10 years)
- Discount rate: \(r = 8\%\) (your opportunity cost)
- Initial investment: \(C_0 = \$200,000\)
- NPV = (\sum \frac{\$20,000}{(1 + 0.08)^t} - \$200,000)
- If NPV > 0, it's a good investment.
Remember, NPV isn't infallible. It assumes perfect knowledge (which we rarely have) and doesn't account for externalities (like environmental impact). But armed with NPV, you're better equipped to
make informed financial decisions. So, next time you're evaluating an investment, channel your inner financial time traveler and compute that NPV!
Understanding Net Present Value \(NPV\) - Net present value: How to compare the present value of cash inflows and outflows
3. The Basics
1. The Basics of NPV:
- Definition: NPV represents the difference between the present value of cash inflows and outflows associated with an investment. It accounts for the time value of money, recognizing that a dollar
received today is worth more than a dollar received in the future.
- Formula: The NPV formula is straightforward:
\[ NPV = \sum_{t=0}^{T} \frac{CF_t}{(1+r)^t} - C_0 \]
- \(CF_t\) represents the expected cash flow at time \(t\).
- \(r\) is the discount rate (usually the cost of capital or required rate of return).
- \(T\) is the total number of periods.
- \(C_0\) is the initial investment cost.
- Insights from Different Perspectives:
- Financial Manager's View: Financial managers focus on maximizing shareholder wealth. They use NPV to determine whether a project adds value to the firm. If NPV is positive, the project is accepted;
if negative, it's rejected.
- Investor's View: Investors consider NPV when evaluating investment opportunities. A positive NPV suggests that an investment generates returns exceeding the cost of capital.
- Risk and Uncertainty: NPV incorporates risk by adjusting for the discount rate. Riskier projects have higher discount rates, reducing their NPV.
- Comparing Alternatives: NPV allows us to compare different investment options. Choose the one with the highest NPV.
- Example:
Suppose we're evaluating a solar power project. Initial cost (\(C_0\)) is $1,000,000. Expected annual cash flows (\(CF_t\)) are as follows:
- Year 1: $300,000
- Year 2: $350,000
- Year 3: $400,000
- Discount rate (\(r\)): 10%
Calculate NPV:
\[ NPV = \frac{300,000}{(1+0.10)^1} + \frac{350,000}{(1+0.10)^2} + \frac{400,000}{(1+0.10)^3} - 1,000,000 \]
\[ NPV = 300,000 + 315,789 + 300,751 - 1,000,000 = -83,460 \]
Since NPV is negative, the solar project may not be economically viable.
- Sensitivity Analysis: Vary the discount rate to assess NPV's sensitivity to changes. A higher discount rate reduces NPV, reflecting increased risk.
- Limitations: NPV assumes constant discount rates, ignores inflation, and relies on accurate cash flow estimates.
Remember, NPV is a powerful tool, but its accuracy depends on the quality of input data and assumptions. Always consider the broader context and conduct sensitivity analyses to make informed
4. Discount Rate and Its Importance
### The Essence of Discount Rates
At its core, a discount rate represents the opportunity cost of capital. It reflects the rate of return required by an investor to forego consuming resources today in favor of investing them for
future benefits. Here are some key insights from different perspectives:
1. Investor's Viewpoint:
- Investors face a trade-off between consuming resources now (spending) and investing them (saving or investing in projects).
- The discount rate captures the investor's preference for future consumption. A higher discount rate implies a stronger preference for immediate consumption.
- Riskier investments typically demand higher discount rates to compensate for uncertainty.
- Businesses use discount rates to evaluate investment opportunities. A project's NPV is calculated by discounting expected future cash flows back to the present.
- The discount rate reflects the cost of capital for the business. It considers factors like the cost of debt, equity, and overall risk.
3. Time Value of Money:
- The fundamental principle behind discount rates is the time value of money. Money received today is worth more than the same amount received in the future due to the opportunity to invest it.
- By discounting future cash flows, we account for this time value and bring all cash flows to a common present value.
### In-Depth Insights (Numbered List):
1. Cost of Capital Components:
- The discount rate comprises various components:
- risk-Free rate: The yield on risk-free assets (e.g., government bonds). It represents the baseline return without any risk.
- equity Risk premium: The additional return demanded by equity investors to compensate for market risk.
- Debt Cost: The interest rate on borrowed funds.
- company-Specific risk: Adjustments based on the business's risk profile.
2. Discounting Cash Flows:
- Suppose a company expects annual cash flows of $10,000 for the next five years.
- Using a discount rate of 8%, we calculate the present value of each cash flow and sum them up:
- PV(CF1) = $10,000 / (1 + 0.08)^1 = $9,259.26
- PV(CF2) = $10,000 / (1 + 0.08)^2 = $8,547.01
- ...
- NPV = Sum of PVs - Initial Investment
3. Sensitivity Analysis:
- Varying the discount rate allows sensitivity analysis. A higher rate reduces NPV, making projects less attractive.
- Managers assess how changes in the discount rate impact project feasibility.
### Examples:
- Imagine evaluating a property investment. The discount rate accounts for the property's risk, inflation, and alternative investment opportunities.
- If the expected return from the property exceeds the discount rate, it's a good investment.
- Startup Valuation:
- startups often have high-risk profiles. Investors demand higher discount rates.
- A promising tech startup might have a projected NPV that justifies the risk.
In summary, discount rates bridge the gap between the present and the future, allowing us to make informed financial decisions. Whether you're analyzing an investment, valuing a business, or planning
for retirement, understanding discount rates empowers you to navigate the complex world of finance.
*(Note: The examples provided are illustrative and not based on real data.
From my very first day as an entrepreneur, I've felt the only mission worth pursuing in business is to make people's lives better.
5. Factors Affecting NPV
Factors Affecting NPV:
When evaluating the net present value (NPV) of a project or investment, several factors come into play. These factors can significantly impact the outcome and should be carefully considered. Let's
explore some of the key factors that affect NPV:
1. discount rate: The discount rate, also known as the required rate of return or cost of capital, plays a crucial role in calculating NPV. It represents the opportunity cost of investing in a
particular project. A higher discount rate reduces the present value of future cash flows, resulting in a lower NPV.
2. Cash Inflows and Outflows: The timing and magnitude of cash inflows and outflows directly influence NPV. Higher cash inflows or lower cash outflows increase the NPV, while the opposite reduces it.
It is essential to accurately estimate these cash flows to obtain an accurate NPV calculation.
3. Project Duration: The length of time it takes to complete a project affects NPV. Longer projects may have higher initial costs and longer periods before generating positive cash flows. This delay
can impact the NPV, as the present value of future cash flows decreases with time.
4. Risk and Uncertainty: The level of risk associated with a project affects NPV. Riskier projects typically require a higher discount rate, reducing the NPV. uncertainty in cash flow projections can
also impact NPV, as it introduces variability and potential deviations from expected outcomes.
5. opportunity cost: The opportunity cost of investing in a particular project versus alternative investments should be considered. If there are other investment opportunities with higher expected
returns, the NPV of the project may be lower in comparison.
6. Inflation: Inflation erodes the purchasing power of future cash flows. Adjusting cash flows for inflation is crucial to obtain an accurate NPV calculation. Failure to account for inflation can
lead to an overestimation of the project's value.
Example: Let's consider a real estate development project. Factors such as the discount rate, projected rental income, construction costs, and market conditions all influence the NPV. A higher
discount rate or lower rental income would decrease the NPV, while lower construction costs or favorable market conditions would increase it.
By considering these factors and conducting a thorough analysis, stakeholders can make informed decisions regarding the NPV of a project or investment.
Factors Affecting NPV - Net present value: How to compare the present value of cash inflows and outflows
6. Sensitivity Analysis and Scenario Planning
## sensitivity Analysis and Scenario planning
### Introduction
Sensitivity analysis and scenario planning are powerful tools used by financial analysts, project managers, and investors to explore the impact of various factors on the NPV of a project. These
techniques recognize that the future is uncertain, and outcomes depend on a multitude of variables. By systematically varying these variables, we gain insights into the project's resilience and
identify critical drivers of value.
### Insights from Different Perspectives
1. risk Management perspective:
- sensitivity analysis helps us understand how changes in input parameters affect NPV. By varying one parameter at a time (while keeping others constant), we assess the project's sensitivity to each
- Scenario planning, on the other hand, considers multiple variables simultaneously. It constructs plausible scenarios (optimistic, pessimistic, and base case) and evaluates NPV under each scenario.
- Example: Imagine a real estate development project. Sensitivity analysis reveals that the project is highly sensitive to interest rate fluctuations, while scenario planning considers not only
interest rates but also market demand, construction costs, and regulatory changes.
2. Investor Perspective:
- Investors want to know how robust their investment is to different market conditions. Sensitivity analysis provides a range of potential outcomes, allowing investors to assess risk.
- scenario planning goes beyond sensitivity by creating coherent narratives for different scenarios. For instance, an investor in a renewable energy project might consider scenarios related to policy
changes, technological advancements, and energy prices.
- Example: An investor in a solar power plant evaluates NPV under scenarios like "favorable government subsidies" and "sudden drop in solar panel costs."
3. strategic Decision-making Perspective:
- Sensitivity analysis informs strategic decisions. If a project's NPV is highly sensitive to a specific factor, management can take proactive measures to mitigate that risk.
- Scenario planning helps management prepare for the unexpected. By considering extreme scenarios (e.g., economic recession, natural disasters), they can develop robust strategies.
- Example: A pharmaceutical company developing a new drug conducts sensitivity analysis on patent expiration dates and scenario planning for unexpected regulatory delays.
### In-Depth Exploration
1. Sensitivity Analysis:
- Vary one input parameter while keeping others constant.
- Calculate NPV for different values of the parameter.
- Plot sensitivity curves or tornado diagrams to visualize impact.
- Example: Sensitivity analysis for a mining project examines NPV sensitivity to commodity prices, production costs, and exchange rates.
2. Scenario Planning:
- Identify key uncertainties (e.g., inflation, technological disruptions, geopolitical events).
- Develop scenarios (optimistic, pessimistic, and base case).
- Estimate NPV under each scenario.
- Example: Scenario planning for an e-commerce startup considers scenarios like "rapid customer adoption" and "intense competition."
### Conclusion
Sensitivity analysis and scenario planning enhance our understanding of NPV by accounting for uncertainty and complexity. While sensitivity analysis provides granularity, scenario planning paints a
broader picture. Together, they empower decision-makers to navigate the dynamic landscape of investments and make informed choices.
Remember, the future is not a single path—it's a maze of possibilities. Sensitivity analysis and scenario planning equip us with a flashlight to explore that maze and make better financial decisions.
Sensitivity Analysis and Scenario Planning - Net present value: How to compare the present value of cash inflows and outflows
7. Interpreting NPV Results
1. Positive NPV: Profitability
- A positive NPV indicates that the project generates more cash inflows than outflows over its lifetime. In other words, the investment is profitable.
- Example: Suppose we're considering a solar power plant. The initial investment includes purchasing solar panels and installation costs. The expected cash inflows come from selling electricity to
the grid. If the NPV is positive, it suggests that the project will yield profits beyond the initial investment.
2. Negative NPV: Losses
- A negative NPV implies that the project's cash outflows exceed inflows. Such investments are likely to result in financial losses.
- Example: Imagine a real estate development project. The upfront costs involve land acquisition, construction, and permits. If the NPV is negative, it signals that the project won't recover these
costs through future rental income or property sales.
3. npv and Discount rate Sensitivity
- NPV calculations depend on the discount rate used to discount future cash flows. A higher discount rate reduces the present value of distant cash flows, impacting NPV.
- Example: Consider a tech startup seeking venture capital. Investors often apply a high discount rate due to the project's riskiness. If the NPV is borderline positive, a slight increase in the
discount rate could turn it negative.
4. Comparing Multiple Projects
- When choosing among several investment options, compare their NPVs. The project with the highest positive NPV is the most attractive.
- Example: A pharmaceutical company evaluates two drug development projects. Project A has an NPV of $5 million, while Project B's NPV is $3 million. The company should prioritize Project A.
5. NPV and Decision Thresholds
- Organizations set NPV thresholds based on their risk tolerance and cost of capital. If the NPV exceeds this threshold, the investment is accepted.
- Example: An automobile manufacturer sets a minimum NPV of $1 million for new product lines. If a proposed electric vehicle project has an NPV above this threshold, it gets the green light.
6. Sensitivity Analysis
- Assess NPV's sensitivity to changes in key variables (e.g., sales volume, production costs). This helps understand risks and uncertainties.
- Example: A mining company explores a new mineral deposit. By varying metal prices and extraction costs, they analyze NPV under different scenarios.
Remember that NPV is a powerful tool, but it has limitations. It assumes constant discount rates, ignores non-monetary benefits, and relies on accurate cash flow projections. Therefore, interpret NPV
results cautiously, considering the broader context and qualitative factors alongside the quantitative analysis.
Interpreting NPV Results - Net present value: How to compare the present value of cash inflows and outflows
8. Comparing NPV with Other Investment Metrics
### Comparing NPV with Other Investment Metrics
Investment decisions are crucial for businesses and individuals alike. When evaluating potential projects or investments, it's essential to choose the right metric to assess their profitability and
feasibility. Let's explore how NPV fares against other popular metrics:
1. Net Present Value (NPV):
- Definition: NPV calculates the difference between the present value of cash inflows and outflows over the investment's lifetime, discounted to the present using a specified discount rate.
- Insights:
- NPV accounts for the time value of money, making it a robust measure for comparing projects with different cash flow patterns.
- A positive NPV indicates that the investment is profitable, while a negative NPV suggests the opposite.
- Example:
- Suppose we're evaluating two projects: Project A and Project B. Project A has an NPV of $50,000, while Project B has an NPV of -$20,000. We would favor Project A because it generates positive
2. internal Rate of return (IRR):
- Definition: irr is the discount rate at which the NPV of an investment becomes zero.
- Insights:
- IRR helps assess the project's return relative to its cost of capital.
- It assumes reinvestment of cash flows at the IRR itself.
- Example:
- If Project A has an IRR of 15%, and Project B has an IRR of 12%, Project A is preferable.
3. Payback Period:
- Definition: Payback period is the time required for an investment to recover its initial cost.
- Insights:
- Simple to calculate and understand.
- Ignores cash flows beyond the payback period.
- Example:
- If Project A pays back in 3 years and Project B in 5 years, Project A is quicker.
4. Profitability Index (PI):
- Definition: PI is the ratio of the present value of cash inflows to the initial investment.
- Insights:
- Helps rank projects based on efficiency.
- A PI greater than 1 indicates a profitable investment.
- Example:
- Project A has a PI of 1.2, while Project B has a PI of 0.9. Project A is more attractive.
5. accounting Rate of return (ARR):
- Definition: ARR measures the average annual accounting profit as a percentage of the initial investment.
- Insights:
- Focuses on accounting profits rather than cash flows.
- Ignores the time value of money.
- Example:
- If Project A's ARR is 18% and Project B's is 12%, Project A seems better.
In summary, NPV remains a powerful tool for investment analysis due to its consideration of cash flows, the time value of money, and flexibility in handling different scenarios. However,
understanding other metrics allows us to make well-informed decisions based on specific requirements and constraints. Remember that no single metric is universally superior; context matters!
Comparing NPV with Other Investment Metrics - Net present value: How to compare the present value of cash inflows and outflows
9. Conclusion
In the intricate world of financial decision-making, the concept of Net Present Value (NPV) stands as a stalwart. As we delve into the depths of this topic, we find ourselves at the precipice of a
crucial juncture. The preceding sections have meticulously dissected the mechanics of NPV, from its fundamental definition to the intricate calculations that underpin it. Now, as we draw the curtains
on our exploration, let us reflect on the key takeaways and implications.
1. The Time Value of Money: A Unifying Principle
At the heart of NPV lies the bedrock principle of the time value of money. This principle asserts that a dollar received today is inherently more valuable than the same dollar received in the future.
Why? Because we can invest that dollar today and earn returns, thereby compounding its value. NPV encapsulates this notion by discounting future cash flows back to their present value. Whether you're
evaluating an investment, a business project, or even a personal financial decision, NPV forces us to confront the temporal dimension head-on.
2. The NPV Decision Rule: To Invest or Not to Invest?
The NPV decision rule is deceptively simple: if NPV > 0, invest; if NPV < 0, don't invest. But beneath this succinct directive lies a universe of considerations. When NPV is positive, it signifies
that the project generates more value than the cost of capital. In other words, it's a wealth-enhancing endeavor. Conversely, a negative NPV suggests that the project fails to meet the required
hurdle rate. But here's where the real-world complexity creeps in. What if the project has intangible benefits? What if it aligns with strategic goals? What if it's a gateway to future opportunities?
These qualitative aspects often dance alongside the quantitative NPV, shaping the ultimate decision.
3. Sensitivity Analysis: peering into the Crystal ball
Life rarely adheres to the deterministic paths we chart on spreadsheets. Uncertainties abound, and assumptions waver. Sensitivity analysis, our trusty crystal ball, steps in. By tweaking
variables—discount rates, growth rates, cash flow projections—we explore the robustness of NPV. What if inflation spikes? What if demand slumps? Sensitivity analysis reveals the project's
vulnerability and resilience. It's akin to stress-testing a bridge before it carries the weight of reality.
4. The Opportunity Cost Conundrum
Opportunity cost lurks in the shadows, casting its shadow over every NPV calculation. When we invest in Project A, we forgo the chance to invest in Project B. The NPV of Project A must outweigh the
foregone NPV of Project B. But how do we quantify these elusive alternatives? Here, NPV intersects with strategic decision-making. Perhaps Project A aligns with our long-term vision, even if it
sacrifices short-term gains. Perhaps Project B is a fleeting mirage. The opportunity cost dilemma is a tightrope walk between pragmatism and aspiration.
5. real-World examples: The NPV Chronicles
Let's step into the shoes of a corporate executive. Imagine evaluating a capital-intensive expansion project. The NPV spreadsheet churns out a positive figure, but doubts linger. Will the market
dynamics shift? Will technological disruptions render our investment obsolete? Here, the NPV dances with risk assessment, scenario planning, and gut instincts. Or consider a startup founder. The NPV
of launching a new product line beckons, but bootstrapping constraints loom large. The NPV intertwines with fundraising strategies, burn rates, and sleepless nights.
In sum, NPV isn't a solitary number—it's a symphony. It harmonizes finance, strategy, psychology, and the art of decision-making. As we bid adieu to this exploration, let us remember that NPV isn't a
verdict; it's a compass. It guides us through the labyrinth of choices, urging us to navigate wisely. So, whether you're an investor, a manager, or a curious soul pondering life's NPVs, embrace the
dance—the intricate waltz of present and future, of risk and reward.
And thus concludes our journey through the corridors of NPV—a journey that transcends spreadsheets and reverberates in boardrooms, classrooms, and coffee shops alike. | {"url":"https://fastercapital.com/content/Net-present-value--How-to-compare-the-present-value-of-cash-inflows-and-outflows.html","timestamp":"2024-11-05T11:53:07Z","content_type":"text/html","content_length":"87912","record_id":"<urn:uuid:f072b4ac-aadc-4d02-8f46-d1a83b35ea7e>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00608.warc.gz"} |
PROC OPTLP Statement
You can specify the following options in the PROC OPTLP statement.
IIS=number | string
specifies whether PROC OPTLP attempts to identify a set of constraints and variables that form an irreducible infeasible set (IIS). Table 10.2 describes the valid values of the IIS= option.
Table 10.2: Values for IIS= Option
number string Description
0 OFF Disables IIS detection.
1 ON Enables IIS detection.
If an IIS is found, information about infeasible constraints or variable bounds can be found in the DUALOUT= and PRIMALOUT= data sets. The default value of this option is OFF. See the section
Irreducible Infeasible Set for details.
specifies one of the following LP solvers:
Option Description
PRIMAL (PS) Uses primal simplex solver.
DUAL (DS) Uses dual simplex solver.
NETWORK (NS) Uses network simplex solver.
INTERIORPOINT (IP) Uses interior point solver.
CONCURRENT (CON) (experimental) Uses several different algorithms in parallel.
The valid abbreviated value for each option is indicated in parentheses. By default, the dual simplex solver is used.
specifies one of the following LP solvers if ALGORITHM=NS:
Option Description
PRIMAL (PS) Uses primal simplex solver (after network simplex).
DUAL (DS) Uses dual simplex solver (after network simplex).
The valid abbreviated value for each option is indicated in parentheses. By default, the OPTLP procedure decides which algorithm is best to use after calling the network simplex solver on the
extracted network.
PRESOLVER=number | string
PRESOL=number | string
specifies one of the following presolve options:
number string Description
0 NONE Disables presolver.
–1 AUTOMATIC Applies presolver by using default setting.
1 BASIC Performs basic presolve like removing empty rows, columns, and fixed variables.
2 MODERATE Performs basic presolve and apply other inexpensive presolve techniques.
3 AGGRESSIVE Performs moderate presolve and apply other aggressive (but expensive) presolve techniques.
The default option is AUTOMATIC (–1). See the section Presolve for details.
LOGLEVEL=number | string
PRINTLEVEL2=number | string
PRINTLEVEL=0 1 2
TIMETYPE=number | string
Simplex Algorithm Options
BASIS=number | string
PRICETYPE=number | string
SCALE=number | string
Interior Point Algorithm Options
CROSSOVER=number | string
specifies whether to convert the interior point solution to a basic simplex solution. If the interior point algorithm terminates with a solution, the crossover algorithm uses the interior point
solution to create an initial basic solution. After performing primal fixing and dual fixing, the crossover algorithm calls a simplex algorithm to locate an optimal basic solution.
number string Description
0 OFF Do not convert the interior point solution to a basic simplex solution.
1 ON Convert the interior point solution to a basic simplex solution.
The default value of the CROSSOVER= option is OFF.
specifies the desired relative duality gap [1E–9, 1E–4]. This is the relative difference between the primal and dual objective function values and is the primary solution quality parameter. The
default value is 1E–6. See the section The Interior Point Algorithm for details.
specifies the maximum allowed relative dual constraints violation [1E–9, 1E–4]. The default value is 1E–6. See the section The Interior Point Algorithm for details.
specifies the maximum allowed relative bound and primal constraints violation [1E–9, 1E–4]. The default value is 1E–6. See the section The Interior Point Algorithm for details. | {"url":"http://support.sas.com/documentation/cdl/en/ormpug/65554/HTML/default/ormpug_optlp_syntax02.htm","timestamp":"2024-11-08T12:54:41Z","content_type":"application/xhtml+xml","content_length":"86744","record_id":"<urn:uuid:a4788203-dd9d-4062-8524-9172523a3348>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00064.warc.gz"} |
Introduction to Applied Statistics for Psychology Students
2. Descriptive Statistics: Frequency Data (Counting)
2.3 SPSS Lesson 1: Getting Started with SPSS
The following lesson will take you through an introduction to IBM® SPSS® Statistics software (referred to hereafter as “SPSS”).
First, you need to open SPSS. Ways to do that are detailed in the Front Matter of this book, in the section “Statistical Software Used in this Book“. Also in the Front Matter you will find the
collection of provided Data Sets; download the file “HyperactiveChildren.sav” and open it in SPSS.
You should see:
SPSS screenshot © International Business Machines Corporation.
This is the “Data View” window. It is one of the three windows you will see when you use SPSS. The other two windows are the “Variable View” window and the “Output” window. You can get to the
Variable View window either by clicking on the Variable View tab at the bottom of the window, or by double clicking one of the column headings (the “variable name”). But let’s talk about what’s on
the Data View window before we look at the other two windows.
The Data View window is arranged in the form of a “data matrix”, which is an essential structure for multivariate statistics. This is the first trap that people who try to use SPSS fall into — they
collect data, put the data into SPSS and then go looking for an appropriate statistical test using help or the built-in “statistics coach”. Multivariate statistics is advanced. We need to learn a
whole lot of basics before we can competently use multivariate statistics. This textbook covers univariate statistics. We are only going to learn how to deal with one dependent variable at a time. So
many of the first SPSS lessons will be about how to combine multiple variables into one variable for analysis.
Back to the Data View window and the data matrix. The rows represent individual subjects in the study. In Psychology, the subjects (“participants”) are generally people but they could also be rats or
schools or cities or whatever. To fix ideas, suppose the subjects are people. One line for each person in the study. The columns represent variables. SPSS doesn’t care what kind of variables you
define (e.g. independent or dependent) so you need to keep track of their meaning yourself. As we said, we only need one independent variable for univariate tests.
The variables need to be defined. This is done by either double clicking on the variable name at the top of a column or by clicking the “Variable View” button at the bottom. Either way, you’ll end up
in the Variable View window that looks like :
SPSS screenshot © International Business Machines Corporation.
Each line in the Variable View window lists the attributes of the variables listed in the Data View window. You can usually leave most of the attributes as they come by default. The big exception is
the Values attribute — it’s important and we’ll come back to that after a quick look at the other attributes.
The Name attribute gives the name of the variable as it appears at the top of the columns in the Data View window. Type should be Numeric if you want to use the variable in any kind of statistical
calculation. Having this set to String will cause errors if you are trying to use the variable as a qualitative variable (selection is via a pull down menu that appears when you click on a cell).
Qualitative variables need to be Numeric and they are handled with the Values attribute — as we’ll see shortly! The Width and Decimals attributes are just to format the appearance of the numbers in
the Data View sheet; totally not critical. The Label is left over from early FORTRAN days. SPSS’s heart is written in FORTRAN and variable names in FORTRAN used to be limited to eight characters
which frequently makes it awkward to have good name for the variable. With Label you can give the variable a good name. If the is a value for Label then that value will be used on table and graph
outputs that SPSS makes. If Label is blank then SPSS will use Name on table and graph outputs. We will largely ignore missing value issues in this course so leave the Missing attribute at None.
Columns and Align are again used to make the Data View presentation look a little better; totally not critical. Leave Measure at Unknown or Scale, otherwise SPSS will try to interpret your data for
you. SPSS is not very good at that and will tend to give strange errors that will make no sense to you, so leave Measure at Unknown or Scale. Leave Role at Input; this is a relatively new feature of
SPSS and I don’t know what it does, so don’t muck with it.
Finally — the Values attribute! Here is where you make the link between a qualitative variable and the discrete values it needs to work in a computer setting. Let’s take a look at the gender
variable. Clicking in the cell brings up a thing with three dots :
SPSS screenshot © International Business Machines Corporation.
Clicking on the thing with three dots brings up a menu where you can define the connection between the qualitative description and your discrete number assignments :
SPSS screenshot © International Business Machines Corporation.
Here I have clicked on the 1.00 = “Male” line to show that the Value is 1 (arbitrary discrete quantitative) and the Label is Male (qualitative). To enter new values, type them in the Value and Label
box and then click Add to add them to the list.
Let’s go back to the Variable View window to see how quantitative variables with discrete number assignments are handled. Look at the values in the sex variable column in the first image. The numbers
1 and 2 are shown which represent Male and Female. To see that representation explicitly, click on the 1-A icon at the top of the window. You will then see:
SPSS screenshot © International Business Machines Corporation.
There’s more. If you click on a cell in the gender variable, you will get a thing on the side of the cell and if you click on that thing, you will see:
SPSS screenshot © International Business Machines Corporation.
This pop-up allows you to change the value by clicking on the appropriate value. In one of your assignments you will get practice with entering qualitative data this way. In general, to enter data
into SPSS from scratch, you can start by typing data into the Data View window and then fix up the attributes later in the Variable View window. For qualitative variables the best approach is to
define the variable first in Variable View, getting the proper values into the Values attribute. Then you can go back to the Data View window and enter the qualitative data either by pulling down the
menu when the mode of the 1-A icon is to show the labels or by remembering the number assignment and entering the numbers when the 1-A icon is set to show values.
Let’s move on to do some descriptive statistics and see what results will look like in the Output window. For this load in the “Caregiver.sav” file from the Data Sets:
SPSS screenshot © International Business Machines Corporation.
There are 50 subjects in this file and 10 variables. One of the things we’ll be learning, in later SPSS Lessons, is how to combine more than one variable into one variable. This is because we are
studying univariate statistics which means we only want to deal with one dependent variable at a time. For now, lets pick on the variable CGDUR and see how we can generate descriptive statistics
output. There are three ways to do this and they all begin in the Analyze
SPSS screenshot © International Business Machines Corporation.
Pick Frequencies… which brings up:
SPSS screenshot © International Business Machines Corporation.
Move the CGagecat variable over by clicking on the variable then the arrow button or just drag the variable over to get:
SPSS screenshot © International Business Machines Corporation.
Let’s take a look at the submenus and set them up before we hit OK. First the Statistics… submenu. In that menu check off Mean (
SPSS screenshot © International Business Machines Corporation.
Hit Continue, look at the Charts… menu and check off pie charts, just for fun:
SPSS screenshot © International Business Machines Corporation.
Hit Continue. You can look at the Format… and Style… menus if you want, they are not particularly interesting. Make sure “Display frequency tables” is checked (this will be important when you do the
assignments), then hit OK. The Output window will pop up and in that window you will see:
SPSS screenshot © International Business Machines Corporation.
The first table, Statistics, shows the descriptive statistics you asked for. Note, especially, for future reference (when we hit skewness in Chapter 3), the value of the skewness. It is
Scrolling down the Output window you will see the pie chart:
SPSS screenshot © International Business Machines Corporation.
Lets look at the Descriptives… menu next:
SPSS screenshot © International Business Machines Corporation.
SPSS screenshot © International Business Machines Corporation.
Move the CGagecat variable over as before and make sure to check off the “Save standardized values as variables”. We’ll learn about standardized values (
Click the options menu and check off descriptive statistics to compute, as before (S.E. mean is Standard Error of the mean which we’ll get to eventually also, we’ll just leave it off for now):
SPSS screenshot © International Business Machines Corporation.
Hit Continue then OK and look at the results in the Output window. The output is straightforward:
SPSS screenshot © International Business Machines Corporation.
In Chapter 3 we will learn that the mean of a
Finally, let’s look at the Explore… menu:
SPSS screenshot © International Business Machines Corporation.
Move CGagecat into the “Dependent List”. Don’t worry about “Factor List”, you should leave it blank (for future reference, “factor” is synonymous with “independent variable”):
SPSS screenshot © International Business Machines Corporation.
Take a look at the Statistics… menu. You can leave it as it is (we’ll be learning about Confidence Intervals later):
SPSS screenshot © International Business Machines Corporation.
Hit Continue and open the Plots… menu and check off the items as shown:
SPSS screenshot © International Business Machines Corporation.
We will talk about these different plots soon. For now, hit Continue, the OK and look at the output. First the tables:
SPSS screenshot © International Business Machines Corporation.
The first table is a “missing data report” that many SPSS procedures will output as a matter of course. You can ignore the missing data reports. Pay attention to the “Descriptive” table (it is
something you could be asked about on exams!). You can ignore the “Tests of Normality” table. Next the plots. The first one is a histogram:
SPSS screenshot © International Business Machines Corporation.
After we cover skewedness in Chapter 3, come back to this picture and note how the histogram is right skewed.
Next is the stem and leaf plot. Remember that the way to a stem and leaf plot in SPSS is through the Explore menu:
SPSS screenshot © International Business Machines Corporation.
You can ignore the Q-Q plots but note that a boxplot is produced:
SPSS screenshot © International Business Machines Corporation.
This is not a very good boxplot. Again, we’ll be learning about boxplots later.
Looking at stuff here in SPSS before covering the concepts in class is a very real situation that people face in real life. They will go to a program like SPSS in the hopes that it is all they need
for data analysis. But it will likely produce output that you don’t understand if you don’t have a basic education in statistics. If provided with output from SPSS (e.g., on an exam) you should able
to explain what the output means. For example, if given one of the tables shown above you should be able to determine what the standard deviation of a data set is and be able to use that number in a
further calculation. It is also a good idea to do some calculations by hand when you first use SPSS for a procedure. If you can produce the same numbers as SPSS then you are sure you know what it is | {"url":"https://openpress.usask.ca/introtoappliedstatsforpsych/chapter/2-3-spss-lesson-1-getting-started-with-spss/","timestamp":"2024-11-14T18:14:21Z","content_type":"text/html","content_length":"143211","record_id":"<urn:uuid:899d2cba-0a5b-4fdb-af36-1bee49848136>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00421.warc.gz"} |
How do you find the wavelength of light with a diffraction grating?
To find the wavelength of light using a diffraction grating, you can use a phenomenon called diffraction, which occurs when light passes through a narrow slit or a series of closely spaced slits on
the grating. The light waves diffract, or spread out, producing a pattern of bright spots known as the diffraction pattern.
Here's a step-by-step process to determine the wavelength of light using a diffraction grating:
1. Set up the experiment: Position the diffraction grating in front of a light source. Ensure that the light passes through the grating, so it diffracts and forms a pattern on a screen or a detector
placed behind it.
2. Observe the diffraction pattern: Look at the screen or detector placed behind the grating. You will see a series of bright spots, known as diffraction maxima or interference fringes. These are
formed due to constructive interference between the diffracted waves.
3. Measure the distance: Take a ruler or a measuring device and measure the distance between two adjacent bright spots. This distance is often denoted as "d" and represents the spacing between the
slits on the grating.
4. Determine the order of the maxima: The diffraction pattern will have multiple bright spots. Identify the order of the maximum you are interested in. The order refers to the number of
half-wavelength shifts the light has undergone in reaching that particular maximum. The central maximum is usually considered the zeroth order (n = 0), with the other maxima labeled as first
order (n = 1), second order (n = 2), and so on.
5. Apply the diffraction grating equation: The diffraction grating equation relates the wavelength of light (λ) to the distance between the slits (d) and the order of the maximum (n). The equation
is given by:
n * λ = d * sin(θ)
In this equation, θ represents the angle between the incident light and the direction of the diffracted light.
6. Solve for the wavelength: Rearrange the equation to solve for the wavelength (λ):
λ = (d * sin(θ)) / n
Substitute the known values of d, θ, and n from your measurements into the equation and calculate the wavelength of the light.
By following these steps and analyzing the diffraction pattern, you can determine the wavelength of light using a diffraction grating. | {"url":"https://physicsgurus.com/95879/how-you-find-the-wavelength-light-with-diffraction-grating?show=95880","timestamp":"2024-11-10T00:21:26Z","content_type":"text/html","content_length":"23266","record_id":"<urn:uuid:ec377f70-5a96-40f0-bf7d-7cedbc75e035>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00350.warc.gz"} |
The Stacks project
Theorem 69.22.5 (Theorem on formal functions). In Situation 69.22.1. Fix $p \geq 0$. The system of maps
\[ H^ p(X, \mathcal{F})/I^ nH^ p(X, \mathcal{F}) \longrightarrow H^ p(X, \mathcal{F}/I^ n\mathcal{F}) \]
define an isomorphism of limits
\[ H^ p(X, \mathcal{F})^\wedge \longrightarrow \mathop{\mathrm{lim}}\nolimits _ n H^ p(X, \mathcal{F}/I^ n\mathcal{F}) \]
where the left hand side is the completion of the $A$-module $H^ p(X, \mathcal{F})$ with respect to the ideal $I$, see Algebra, Section 10.96. Moreover, this is in fact a homeomorphism for the limit
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The Generalized Lebesgue Dominated Convergence Theorem
The Generalized Lebesgue Dominated Convergence Theorem
Theorem (The Generalized Lebesgue Dominated Convergence Theorem): Let $(f_n(x))_{n=1}^{\infty}$ be a sequence of Lebesgue measurable functions defined on a Lebesgue measurable set $E$, and let $(g_n
(x))_{n=1}^{\infty}$ be a sequence of nonnegative Lebesgue measurable functions defined on $E$. Suppose that:
1) $|f_n(x)| \leq |g_n(x)|$ for all $n \in \mathbb{N}$ and for all $x \in E$.
2) $(f_n(x))_{n=1}^{\infty}$ converges pointwise almost everywhere to $f(x)$ and $(g_n(x))_{n=1}^{\infty}$ converges pointwise almost everywhere to $g(x)$.
3) $\displaystyle{\lim_{n \to \infty} \int_E g_n = \int_E g < \infty}$.
Then $f$ is Lebesgue integrable on $E$ and $\displaystyle{\lim_{n \to \infty} \int_E f_n = \int_E f}$.
• Proof: Since each $g_n$ is a nonnegative Lebesgue measurable function we see that each $g_n$ is Lebesgue integrable. So by The Comparison Test for Lebesgue Integrability we have that each $f_n$
is Lebesgue integrable on $E$. Furthermore, $g$ and $f$ are also Lebesgue integrable on $E$.
• Consider the sequence $(g_n(x) - f_n(x))_{n=1}^{\infty}$ of Lebesgue integrable functions. This sequence converges pointwise almost everywhere to $g(x) - f(x)$. Since $|f_n(x)| \leq g_n(x)$ for
all $x \in E$ we have that $|f_n(x)| - f_n(x) | \leq g_n(x) - f_n(x)$ for all $x \in E$ so this is a sequence of nonnegative Lebesgue integrable functions on $E$. By Fatou's Lemma for Nonnegative
Lebesgue Measurable Functions and linearity for the Lebesgue integral of nonnegative Lebesgue measurable functions we have that:
\quad \int_E g - \int_E f = \int_E (g - f) \leq \liminf_{n \to \infty} \int_E (g_n - f_n) = \lim_{n \to \infty} \int_E g_n - \limsup_{n \to \infty} \int_E f_n = \int_E g - \limsup_{n \to \infty} \
int_E f_n
\quad \int_E f \geq \limsup_{n \to \infty} \int_E f_n \quad (*)
• Now consider the sequence $(g_n(x)) + f_n(x))_{n=1}^{\infty}$ of Lebesgue integrable functions. This sequence converges pointwise almost everywhere to $g(x) + f(x)$. Since $|f_n(x)| \leq g_n(x)$
for all $x \in E$ we have that $|f_n(x)| + f_n(x) \leq g_n(x) + f_n(x)$ for all $x \in E$ so this is a sequence of nonnegative Lebesgue integrable functions on $E$. By Fatou's Lemma and linearity
for the Lebesgue integrable of nonnegative Lebesgue measurable functions we have that:
\quad \int_E g + \int_E f = \int_E (g + f) \leq \liminf_{n \to \infty} \int_E (g_n + f_n) = \lim_{n \to \infty} \int_E g_n + \liminf_{n \to \infty} \int_E f_n = \int_E g + \liminf_{n \to \infty} \
int_E f_n
\quad \int_E f \leq \liminf_{n \to \infty} \int_E f_n \quad (**)
• Combining $(*)$ and $(**)$ yields:
\quad \limsup_{n \to \infty} \int_E f \leq \int_E f \leq \liminf_{n \to \infty} \int_E f
• The above inequality implies that:
\quad \lim_{n \to \infty} \int_E f_n = \int_E f \quad \blacksquare | {"url":"http://mathonline.wikidot.com/the-generalized-lebesgue-dominated-convergence-theorem","timestamp":"2024-11-13T21:17:21Z","content_type":"application/xhtml+xml","content_length":"18171","record_id":"<urn:uuid:3a467f7d-64a4-4d52-bde6-ce9f5e5d13f4>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00641.warc.gz"} |
Stability experimental
Maintainer ekmett@gmail.com
Safe Haskell None
A generalized State monad, parameterized by a Representable functor. The representation of that functor serves as the state.
type State g = StateT g IdentitySource
A memoized state monad parameterized by a representable functor g, where the representatation of g, Rep g is the state to carry.
The return function leaves the state unchanged, while >>= uses the final state of the first computation as the initial state of the second.
:: Representable g
=> State g a state-passing computation to execute
-> Rep g initial state
-> (a, Rep g) return value and final state
Unwrap a state monad computation as a function. (The inverse of state.)
:: Representable g
=> State g a state-passing computation to execute
-> Rep g initial value
-> a return value of the state computation
Evaluate a state computation with the given initial state and return the final value, discarding the final state.
:: Representable g
=> State g a state-passing computation to execute
-> Rep g initial value
-> Rep g final state
Evaluate a state computation with the given initial state and return the final state, discarding the final value.
mapState :: Functor g => ((a, Rep g) -> (b, Rep g)) -> State g a -> State g bSource
Map both the return value and final state of a computation using the given function.
newtype StateT g m a Source
A state transformer monad parameterized by:
• g - A representable functor used to memoize results for a state Rep g
• m - The inner monad.
The return function leaves the state unchanged, while >>= uses the final state of the first computation as the initial state of the second.
(Functor f, Representable g, MonadFree f m) => MonadFree f (StateT g m)
(Representable g, MonadReader e m) => MonadReader e (StateT g m)
(Representable g, Monad m, ~ * (Rep g) s) => MonadState s (StateT g m)
(Representable g, MonadWriter w m) => MonadWriter w (StateT g m)
Representable f => MonadTrans (StateT f)
Representable f => BindTrans (StateT f)
(Representable g, Monad m) => Monad (StateT g m)
(Functor g, Functor m) => Functor (StateT g m)
(Representable g, Functor m, Monad m) => Applicative (StateT g m)
(Representable g, MonadCont m) => MonadCont (StateT g m)
(Representable g, Bind m) => Apply (StateT g m)
(Representable g, Bind m) => Bind (StateT g m)
evalStateT :: (Representable g, Monad m) => StateT g m a -> Rep g -> m aSource
Evaluate a state computation with the given initial state and return the final value, discarding the final state.
execStateT :: (Representable g, Monad m) => StateT g m a -> Rep g -> m (Rep g)Source
Evaluate a state computation with the given initial state and return the final state, discarding the final value.
liftCallCC :: Representable g => ((((a, Rep g) -> m (b, Rep g)) -> m (a, Rep g)) -> m (a, Rep g)) -> ((a -> StateT g m b) -> StateT g m a) -> StateT g m aSource
Uniform lifting of a callCC operation to the new monad. This version rolls back to the original state on entering the continuation.
liftCallCC' :: Representable g => ((((a, Rep g) -> m (b, Rep g)) -> m (a, Rep g)) -> m (a, Rep g)) -> ((a -> StateT g m b) -> StateT g m a) -> StateT g m aSource
In-situ lifting of a callCC operation to the new monad. This version uses the current state on entering the continuation. It does not satisfy the laws of a monad transformer.
class Monad m => MonadState s m | m -> s where
Minimal definition is either both of get and put or just state
get :: m s
Return the state from the internals of the monad.
put :: s -> m ()
Replace the state inside the monad.
state :: (s -> (a, s)) -> m a
Embed a simple state action into the monad.
MonadState s m => MonadState s (MaybeT m)
MonadState s m => MonadState s (ListT m)
MonadState s m => MonadState s (IdentityT m)
(Functor m, MonadState s m) => MonadState s (Free m)
(Representable g, Monad m, ~ * (Rep g) s) => MonadState s (StateT g m)
(Monoid w, MonadState s m) => MonadState s (WriterT w m)
(Monoid w, MonadState s m) => MonadState s (WriterT w m)
Monad m => MonadState s (StateT s m)
Monad m => MonadState s (StateT s m)
MonadState s m => MonadState s (ReaderT r m)
(Error e, MonadState s m) => MonadState s (ErrorT e m)
MonadState s m => MonadState s (ContT r m)
(Monad m, Monoid w) => MonadState s (RWST r w s m)
(Monad m, Monoid w) => MonadState s (RWST r w s m) | {"url":"https://hackage-origin.haskell.org/package/adjunctions-4.2/docs/Control-Monad-Representable-State.html","timestamp":"2024-11-14T01:26:41Z","content_type":"application/xhtml+xml","content_length":"30514","record_id":"<urn:uuid:88d039d1-6e94-4ade-ac56-d67cab098cc6>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00746.warc.gz"} |
What is Slenderness Ratio of RC Column and How to Calculate it?What is Slenderness Ratio of RC Column and How to Calculate it?
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The slenderness ratio of a reinforced concrete (RC) column is the ratio between the length of the column, its lateral dimensions, and end fixity. It assesses the ability of the reinforced concrete
column to resist buckling pressure. The slenderness ratio is calculated by dividing the column length by its radius of gyration.
The slenderness ratio differentiates short column from long or slender column. The design of the former is controlled by column dimension and material strength whereas the design of the latter is
governed by column slenderness.
A column is said to be slender if its cross-sectional dimensions are small compared to its length. If the slenderness ratio of a column is high, it will collapse under a smaller compression load in
contrast to a short column with the same cross-sectional dimensions. So, the slenderness effect should be taken into consideration during the design process.
The slenderness ratio of reinforced concrete columns can be computed according to the procedures and specifications of applicable codes such as ACI 318-19 and IS 456.
How to Calculate Slenderness Ratio Based on ACI 318-19?
The degree of slenderness is generally expressed in terms of the slenderness ratio:
l[u]: unsupported length of the member
r: radius of gyration of its cross section
K: constant to reflect end conditions of the column; distance between two inflection points (Alignment chart is used to calculate K).
Unsupported Length of the Member (l[u])
1. The unsupported length (l[u]) of a column is measured as the clear distance between the underside of the beam, slab, or column capital above, and the top of the beam or slab below, as shown in
Figure-1: Measurement of Unsupported Length of Columns with Different End Conditions
2. The unsupported length of a column may be different in two orthogonal directions depending on the supporting elements in respective directions.
Figure-2: Different Unsupported Length of a Column in Each Direction
Radius of Gyration of Column Cross-section (r)
The radius of gyration introduces the effects of cross-sectional size and shape on slenderness. For the same cross-sectional area, a section with a higher moment of inertia produces a more stable
column with a lower slenderness ratio. The radius of gyration r is calculated using the following formula:
I[g]: moment of inertia of gross concrete section about centroidal axis, neglecting reinforcement, mm^4
A[g]: Gross area of column, mm^2
It is possible to use the approximations of r = 0.3h for square and rectangular sections, and r = 0.25h for circular sections.
h: overall sectional dimension in the directional stability is being considered.
Figure-3: Approximate Estimation of Radius of Gyration for Different Cross-sectional Shapes of Reinforced Concrete Column
Effective Length Factor (k)
The effective length factor (k) reflects the end restraint (support) and lateral bracing conditions of a column. If a column is hinged at both ends, it follows a half-sine wave when it buckles, and
the value of (k) factor for such type of column is 1.0. So, the effective length of the column (kl[u]) is equal to the unsupported length of the column (l[u]).
Figure-4: Effective Length Factor for Column Hinged at Both Ends
A column with fully restrained end conditions develops deflected shape as shown in Figure-5. The portion of the column between the points of contraflexure follows a half-sine wave. The effective
length factor k for this case is equal to 0.5.
Figure-5: Effective Length Factor for Fixed End Columns
Columns in real structures are rarely either hinged or fixed but have ends partially restrained against rotation by abutting members. The value of k, in this case, varies between 0.5 and 1.0 for
laterally braced columns, Figure-6. For unbraced columns, the value of k varies between 1.0 and ?, Figure-7.
The majority of reinforced concrete columns are considered to be laterally braced columns. The exact value depends on the degree of end restraint, that is, on the ratio of the stiffness (EI/l) of the
column to the sum of stiffnesses (EI/l) of the restraining members at both ends.
Figure-6: Effective Length Factor
Figure-7: Effective Length Factor for Unbraced Column
ACI Criteria for Slenderness Ratio Effect
The effect of slenderness ratio for columns braced against sideway can be neglected when:
Note: As a first approximation, k may be taken equal to 1.0 in this equation
M[1] :lesser factored end moment on a compression member, N·mm
M[2] :greater factored end moment on a compression member.
M[1]/M[2] : negative if the column is bent in single curvature, and positive for double curvature. M[1]/M[2] is negative if the column is bent in single curvature, and positive for double curvature,
as illustrated in Figure-8:
Figure-8: Single Curvature and Double Curvature Column
The effect of slenderness ratio for columns not braced against sideway can be neglected when:
Alignment Chart for Effective Slenderness Factor
If slenderness is found to be important, refine the calculation of k based on the alignment chart as shown below:
Figure-9: Alignment Charts for Sway and Non-sway Column
The ? factor at one end of the column equals the sum of the stiffness ?(EI/L) of the columns meeting at that joint, including the column in question, divided by the sum of all the stiffnesses of the
beams meeting at the joint.
E: Modulus of Elasticity
I: Moment of Inertia
L: span length measured center to center of joints.
?[A], ?[B] : Values of (? ) at each end of the column
The modulus of elasticity for normal concrete is computed as:
The moment of inertia used to compute ?should be based on cracked section, so ACI 318-19 section 6.6.3.1.1 provides Table-1:
Table-1: Moment of Inertia Used to Compute ? Factor
Member and Condition Moment of Inertia
Columns 0.70I[g]
Uncracked walls 0.70I[g]
Cracked Walls 0.35I[g]
Beams 0.35I[g]
Flate Plates and Flat Slabs 0.25I[g]
How do you determine if a column is short or slender?
The column is considered to be slender if its length is higher compared to its cross-sectional dimension i.e. slenderness ratio is used to differentiate slender column from short column.
What is slenderness ratio of a reinforced concrete column?
Slenderness ratio of reinforced concrete (RC) column is the ratio between column length, its lateral dimensions, and end fixity. It assesses the ability of reinforced concrete column to resist
buckling pressure.
What is the use of slenderness ratio?
It is used to find out the design load as well as in classifying various columns in short/intermediate/long. The slenderness ratio of a column gives an indication of buckling failure in the column.
More the slenderness ratio, the more is the tendency of the column to fail by buckling effect in that direction.
How do you calculate slenderness ratio of a column?
1. Calculate the effective length factor (K) that ranges between 0.5 to 1.0 for the reinforced concrete column. It can be considered to be 1.0 conservatively.
2. Compute the unsupported length of the column (l[u]) which is measured as the clear distance between the underside of the beam, slab, or column capital above, and the top of the beam or slab below.
3. Estimate the radius of gyration (r). It is possible to use the approximations of r = 0.3h for square and rectangular sections, and r = 0.25h for circular sections.
4. Calculate slenderness ratio which is equal to effective length factor times unsupported length divided by the radius of gyration.
What is buckling of column?
Buckling of the column is a form of deformation as a result of axial compression forces that leads to a bending of the column, as a result of the instability of the column. This mode of failure is
quick and undesirable.
What is radius of gyration of column?
The radius of Gyration is used to describe the distribution of cross-sectional area in an RC column around its centroidal axis.
Read more:
Tips and rules for design of reinforced concrete column
Euler's theory of column buckling | {"url":"https://test.theconstructor.org/structural-engg/slenderness-ratio-column-calculate/494767/","timestamp":"2024-11-04T14:15:21Z","content_type":"text/html","content_length":"207986","record_id":"<urn:uuid:c7dbd989-9b39-474d-aad6-c50786f72aee>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00753.warc.gz"} |
How Many Decimeters Is 258.3 Inches?
258.3 inches in decimeters
How many decimeters in 258.3 inches?
258.3 inches equals 65.608 decimeters
Unit Converter
Conversion formula
The conversion factor from inches to decimeters is 0.254, which means that 1 inch is equal to 0.254 decimeters:
1 in = 0.254 dm
To convert 258.3 inches into decimeters we have to multiply 258.3 by the conversion factor in order to get the length amount from inches to decimeters. We can also form a simple proportion to
calculate the result:
1 in → 0.254 dm
258.3 in → L[(dm)]
Solve the above proportion to obtain the length L in decimeters:
L[(dm)] = 258.3 in × 0.254 dm
L[(dm)] = 65.6082 dm
The final result is:
258.3 in → 65.6082 dm
We conclude that 258.3 inches is equivalent to 65.6082 decimeters:
258.3 inches = 65.6082 decimeters
Alternative conversion
We can also convert by utilizing the inverse value of the conversion factor. In this case 1 decimeter is equal to 0.015241997189376 × 258.3 inches.
Another way is saying that 258.3 inches is equal to 1 ÷ 0.015241997189376 decimeters.
Approximate result
For practical purposes we can round our final result to an approximate numerical value. We can say that two hundred fifty-eight point three inches is approximately sixty-five point six zero eight
258.3 in ≅ 65.608 dm
An alternative is also that one decimeter is approximately zero point zero one five times two hundred fifty-eight point three inches.
Conversion table
inches to decimeters chart
For quick reference purposes, below is the conversion table you can use to convert from inches to decimeters | {"url":"https://convertoctopus.com/258-3-inches-to-decimeters","timestamp":"2024-11-05T00:57:09Z","content_type":"text/html","content_length":"33327","record_id":"<urn:uuid:750960b1-7d3c-476f-a0c7-c80115fad9fc>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00772.warc.gz"} |
If a=b,b,c satisfy ∣∣a342bba2ccb∣∣=0, then abc=... | Filo
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Question Text If satisfy , then
Updated On May 25, 2023
Topic Determinants
Subject Mathematics
Class Class 12
Answer Type Text solution:1 Video solution: 5
Upvotes 519
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Artificial Neural Networks (ANNs)
[Types of Artificial Neural Networks - Great Learning]
- Overview
With the development of neural networks, various tasks that were considered unimaginable can now be easily accomplished. Tasks like image recognition, speech recognition, and finding deeper
relationships in datasets become easier. Sincere thanks to outstanding researchers in the field whose discoveries and discoveries have helped us harness the true power of neural networks.
Neural networks are the cornerstone of today's technological breakthroughs in deep learning. Neural networks can be thought of as massively parallel simple processing units capable of storing
knowledge and applying that knowledge to make predictions.
Neural networks, also known as artificial neural networks (ANNs) or analog neural networks (SNNs), are a subset of machine learning (ML) and are at the heart of deep learning (DL) algorithms. Their
name and structure are inspired by the human brain, mimicking the way biological neurons signal each other.
- Artificial Neural Networks
Artificial neural networks (ANNs) are a branch of machine learning (ML) models inspired by the neuronal organization found in the biological neural networks in animal brains.
An ANN is made of connected units or nodes called artificial neurons, which loosely model the neurons in a brain. These are connected by edges, which model the synapses in a brain.
An artificial neuron receives signals from connected neurons, then processes them and sends a signal to other connected neurons. The "signal" is a real number, and the output of each neuron is
computed by some non-linear function of the sum of its inputs, called the activation function. Neurons and edges typically have a weight that adjusts as learning proceeds. The weight increases or
decreases the strength of the signal at a connection.
Typically, neurons are aggregated into layers. Different layers may perform different transformations on their inputs. Signals travel from the first layer (the input layer) to the last layer (the
output layer), possibly passing through multiple intermediate layers (hidden layers). A network is typically called a deep neural network if it has at least 2 hidden layers.
ANNs are used for predictive modeling, adaptive control, and other applications where they can be trained via a dataset. They are also used to solve problems in artificial intelligence. Networks can
learn from experience, and can derive conclusions from a complex and seemingly unrelated set of information.
- The Three Layers of ANNs
An artificial neural network (ANN) is a collection of simple interconnected algorithms that process information in response to external input.
ANNs typically have three parts:
• Input layer: Takes data from the network
• Hidden layer/s: Connections from the nodes in the input layer to the nodes in the hidden layer, and from each hidden layer node to the nodes of the output layer
• Output layer: Connections from the nodes in the hidden layer to the nodes of the output layer
Input layer contains units that represent the input fields. Hidden layers can perform multiple functions at once, such as data transformation and automatic feature creation. Output layer contains
units that represent the target field(s).
The ANN model is organized in layers, each one made up of interconnected nodes. The input layer communicates with one or more hidden layers. Here, the nodes take the weighted connections and use an
activation function to pass their signal to the output layer.
The number of layers in an ANN can vary depending on the architecture. The depth of an ANN refers to the number of hidden layers.
The units in an ANN are connected with varying connection strengths, or weights. Feedforward neural networks process data in one direction, from the input node to the output node. Each node in one
layer is connected to every node in the next layer.
An ANN incorporates a way of learning, of which there are many, that essentially modifies the weights of the connections. In this way, it learns by example.
The image above shows a simple feed forward neural network that propagates information forwards.
- How an ANN Works
An artificial neural network (ANN) consists of a layer of nodes, which contains an input layer, one or more hidden layers, and an output layer. Each node or artificial neuron is connected to another
node and has associated weights and thresholds. If the output of any single node is above a specified threshold, that node is activated, sending the data to the next layer of the network. Otherwise,
no data is passed to the next layer of the network.
Neural networks rely on training data to learn and improve their accuracy over time. But once these learning algorithms are fine-tuned to improve accuracy, they become powerful tools in computer
science and artificial intelligence, allowing us to classify and cluster data at high speed. Speech recognition or image recognition tasks can take minutes instead of hours compared to manual
recognition by human experts. One of the most famous neural networks is Google's search algorithm.
- Neural Networks in Deep Learning (DL)
Neural networks are the core machinery that make DL so powerful. Neural networks are a set of algorithms, modeled loosely after the human brain, that are designed to recognize patterns. They
interpret sensory data through a kind of machine perception, labeling or clustering raw input. The patterns they recognize are numerical, contained in vectors, into which all real-world data, be it
images, sound, text or time series, must be translated.
A Neural Network is a computer system designed to work by classifying information in the same way a human brain does. It can be taught to recognize, for example, images, and classify them according
to elements they contain. The development of neural network has been key to teaching computers to think and understand the world in the way we do, while retaining the innate advantages they hold over
us such as speed, accuracy and lack of bias.
Neural networks help us cluster and classify. You can think of them as a clustering and classification layer on top of the data you store and manage. They help to group unlabeled data according to
similarities among the example inputs, and they classify data when they have a labeled dataset to train on.
Neural networks can also extract features that are fed to other algorithms for clustering and classification; so you can think of deep neural networks as components of larger machine-learning
applications involving algorithms for reinforcement learning, classification and regression.
- Deep Learning Algorithms and ANNs
Deep learning algorithms such as ANNs, are able to processing, technical and fundamental information from news, in order to detect features and patterns that can affect the pricing behavior in
financial markets. The power of deep learning lies in the fact that can manage big amounts of data, and simultaneously, ANNs can learn from financial data and understand the price movements in stock
markets, in a matter of seconds.
An Artificial Intelligence System (AIS), can suggest the optimal solutions and by minimize the risk in trading strategies, can produce predictions that will be calculated based on the information we
have gained, and in combination with other intelligent methods such as natural language processing (NLP), much more accurate results could be produced.
Hybrid deep learning systems have been widely used in financial services, by detecting new market trends and by offering significant profits to investors, by forecasting the prices. All this without
the need to apply a new investment theory as long as deep learning algorithms will discover in the data, structures that have never been studied or explained in the past.
- Perceptrons - How Neural Networks Work
A normal neural network consists of multiple layers called input, output, and hidden layers. In each layer, each node (neuron) is connected to all nodes (neurons) in the next layer through parameters
called "weights".
A neural network consists of nodes called "perceptrons" that perform the necessary computations and detect the "characteristics" of the neural network. These perceptrons try to reduce the final cost
error by adjusting the weight parameters. Furthermore, the perceptron can be thought of as a single-layer neural network.
When considering a perceptron, its work can be described as follows: When you feed data with random weights into the model, it generates a weighted sum of them. Based on this value, the activation
function determines the activation state of the neuron. The output of this perceptron can be used as the input to the next neuron layer.
- Mutilayer Perceptrons - Deep Neural Networks
On the other hand, multilayer perceptrons are called deep neural networks. The perceptron is activated when there is a satisfying input.
• Initially, the dataset should be fed into the input layer and then flow to the hidden layer.
• The connections that exist between the two layers randomly assign weights to the inputs.
• Add a bias to each input. Bias is the constant used in the model to best fit the given data.
• The weighted sum of all inputs will be sent to a function that decides the neuron's activity state by computing the weighted sum and adding a bias. This function is called the activation
• The node that needs to be triggered for feature extraction is determined according to the output value of the activation function.
• The final output of the network is then compared to the labeled data required for our dataset to calculate the final cost error. The cost error actually tells us how "bad" our network is.
Therefore, we want the error to be as small as possible.
• Adjust the weights by backpropagation, thereby reducing the error. This back-propagation process can be thought of as the central mechanism of neural network learning. It basically fine-tunes the
weights of the deep neural network to reduce the cost value.
In simple terms, what we usually do when training a neural network is to calculate the loss (error value) of the model and check if it decreases. If the error is higher than expected, we have to
update model parameters such as weights and bias values. Once the loss is below the expected error bound, we can use the model.
[More to come ...] | {"url":"http://eitc.org/research-opportunities/new-media-and-new-digital-economy/ai-machine-learning-deep-learning-and-neural-networks/neural-networks-research-and-applications/artificial-neural-networks","timestamp":"2024-11-03T18:22:16Z","content_type":"application/xhtml+xml","content_length":"31740","record_id":"<urn:uuid:e9584d38-6703-4412-8a2e-28abf04c80d8>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00284.warc.gz"} |
Кафедра алгебры и фундаментальной информатики
"Quantum algorithims"
Последнее изменение: 06/10/2020 05:09:48
Aim, scope, and prrequisites: The aim of this mini-course is to introduce basic quantum algorithms in an accessible but mathematically rigorous manner. As prerequisites, we assume the audience’s
acquaintance with basic notions and results of the theory of unitary and Hermitian (self-conjugate) operators in finite-dimensional Hilbert spaces. No previous knowledge of quantum mechanics is
Literature: The course is based on the textbook "An Introduction to Quantum Computing Algorithms" by Arthur Pittenger (Birkhäuser, 2000).
Some materials (e.g., proof details) and useful links will be placed here after each lecture.
Lecture 1: The Stern-Gerlach experiment and the matrix formalism of quantum mechanics
The content of the lecture roughly corresponds to Chapter 1 of Pittenger's textbook.
We started with a simplified but accurate in large description of the Stern-Gerlach experiment of 1922. The choice is caused by the following reasons:
• the experiment is easy to explain;
• its results are counterintuitive and cannot be reasonably explained from the viewpoint of “classical” physics;
• it reveals quantum effects (such as the noncommutativity) in a very explicit way.
Then we proceeded with presenting the matrix formalism of quantum mechanics. Finally, we returned to the Stern-Gerlach experiment and demonstrated how its results could be explained via the matrix
formalism. We concluded with introducing the Dirac notation.
Scott Aaronson's presentation partly used in the lecture.
The original paper by Walther Gerlach und Otto Stern (in German!) - it is useful to have a look at it to see the difference between the real experiment and our simplified presentation. Also, I see a
trace of historical irony in the authors' acknowledgement: they thank Einstein (as the Director of Kaiser Wilhelm Institute for Physics) for providing a crucial piece of equipment used for the
experiment, and as it is well known, Einstein opposed quantum mechanics, even though he was awarded the Nobel prize for his work on photoelectricity, which certainly was quantum mechanical.
Wikipedia page about the Stern-Gerlach experiment is quite substantial and contains nice illustrations (partly used in the lecture).
Mathematical-Physical Dictionary
Finite-dimensional Hilbert space H Quantum system
Non-negative operator R on H with trace 1 State of the system
Self-conjugate operator A on H Observable
Eigenvalues of A Possible results of observation
tr(RA) Expected result after an observation of A in the state R
Unitary operator on H Transformation of the system
Proofs for some claims made in the lecture
Lecture 2: The Deutsch and Deutsch-Jozsa algorithms
The lecture included the material in Section 3.1 of Pittenger's textbook but provided more details.
We started with recalling the concepts of the tensor product of Hilbert spaces and Kronecker (tensor) product of matrices. We also introduced the Hadamard transform.
The original paper by David Deutsch of 1985
The original paper by David Deutsch and Richard Jozsa of 1992
Lecture 3: The Simon and Grover algorithms
Lecture 4: Shor's factorization algorithm
Lecture 5: Quantum teleportation | {"url":"https://kadm.kmath.ru/pages.php?id=osnovy_quant_eng","timestamp":"2024-11-04T20:49:58Z","content_type":"application/xhtml+xml","content_length":"7711","record_id":"<urn:uuid:a8b8c4b6-0481-44ae-9dca-55b40cecf8bc>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00422.warc.gz"} |
Re: st: Importing subset of a pipe delimited textfile
Notice: On April 23, 2014, Statalist moved from an email list to a forum, based at statalist.org.
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Re: st: Importing subset of a pipe delimited textfile
From Rob Shaw <[email protected]>
To [email protected]
Subject Re: st: Importing subset of a pipe delimited textfile
Date Wed, 17 Oct 2012 13:10:51 +0100
The problem is not the pipes as such (otherwise I could just use the
delimiter options in -insheet-), it's that the file is too large to
use -insheet-
So i need to use -infile- to import my file in separate parts, but
infile will only accept fixed format files (as far as I understand).
Therefore, if I import my file using:
infile str2 var1 _skip(1) str4 var2 _skip(1) str3 var3 _skip(1) str4
var4 using myfile in 1/1000000
I get nonesense because the first record then gets filled with [1|,
BCD|, 3|X, YZ]
Maarten wrote:
To give a concrete example: I stored Rob's example dataset in foo.raw
I than typed in Stata:
filefilter foo.raw foo2.raw, from("|") to(\t) replace
insheet using foo2.raw
The first line replaced all pipes in the file foo.raw with a tab and
stored the resulting tab-delimited file in foo2.raw, and the second
line read this tab-delimited file foo2.raw into Stata.
Hope this helps,
On Wed, Oct 17, 2012 at 1:37 PM, Nick Cox <[email protected]> wrote:
> Why is varying length of line a problem? So long as the same variables
> are represented on each line, I can see no problem.
> Also, -filefilter- has a tacit loop; you don't need to set it up for yourself.
> Nick
> On Wed, Oct 17, 2012 at 12:33 PM, Rob Shaw <[email protected]> wrote:
>> Nick
>> Thanks. Yes that would work but the problem is the varying length of
>> each line. So I need to get filefilter or another command to do one
>> of:
>> x=0
>> counter=1
>> with "myfile.txt" {
>> y = position of 10000th EOL in `i'
>> save `i' from position x to y in "myfilepos"+counter+".txt"
>> x =y
>> }
>> This would create files called myfilepos1, myfilepos2 etc each with
>> 10000 lines that I could then -insheet- with a delimiter(|) option.
>> But I don't know how to correctly specify the bit in the loop.
>> OR
>> for each line in "myfile.txt" {
>> find | and replace with a number of spaces depending on position in row
>> }
>> This would make each line the same length so I could use -infile-
>> Is there a way to use -filefilter- to achieve this?
>> File sample:
>> 1|ABCD|23|XYZ
>> 10|BCED|1|YZX
>> 30|DCHS|234|YBH
>> ....
>> Thanks
>> Rob
>>>I'd use -filefilter- to change the pipes to something that -infile- can handle.
>>>(Strictly, -in- is a qualifier, not an option.)
>>>On Wed, Oct 17, 2012 at 9:13 AM, Rob Shaw <[email protected]> wrote:
>>> I have a very large (around 4Gb) text file that has been pipe
>>> delimited. It won't all fit in memory so I want to process it in
>>> parts.
>>> For fixed datasets I would use infile with the in 1/10000000 option
>>> then 10000001/2000000 etc. However, this dataset has been pipe
>>> delimited so I would need to use insheet, but insheet doesn't seem to
>>> permit the "in" option.
>> *
>> * For searches and help try:
>> * http://www.stata.com/help.cgi?search
>> * http://www.stata.com/support/faqs/resources/statalist-faq/
>> * http://www.ats.ucla.edu/stat/stata/
> *
> * For searches and help try:
> * http://www.stata.com/help.cgi?search
> * http://www.stata.com/support/faqs/resources/statalist-faq/
> * http://www.ats.ucla.edu/stat/stata/
Maarten L. Buis
Reichpietschufer 50
10785 Berlin
* For searches and help try:
* http://www.stata.com/help.cgi?search
* http://www.stata.com/support/faqs/resources/statalist-faq/
* http://www.ats.ucla.edu/stat/stata/ | {"url":"https://www.stata.com/statalist/archive/2012-10/msg00773.html","timestamp":"2024-11-05T07:38:55Z","content_type":"text/html","content_length":"13644","record_id":"<urn:uuid:07a0ee83-1f4e-4e86-9a50-06be9a9ab417>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00798.warc.gz"} |
Lipschitz symmetric functions on Banach spaces with symmetric bases
Lipschitz symmetric functions on Banach spaces with symmetric bases
Lipschitz symmetric function on Banach space, symmetric basis, tropical polynomial
Published online: 2021-12-13
We investigate Lipschitz symmetric functions on a Banach space $X$ with a symmetric basis. We consider power symmetric polynomials on $\ell_1$ and show that they are Lipschitz on the unbounded subset
consisting of vectors $x\in \ell_1$ such that $|x_n|\le 1.$ Using functions $\max$ and $\min$ and tropical polynomials of several variables, we constructed a large family of Lipschitz symmetric
functions on the Banach space $c_0$ which can be described as a semiring of compositions of tropical polynomials over $c_0$.
How to Cite
Martsinkiv, M.; Vasylyshyn, S.; Vasylyshyn, T.; Zagorodnyuk, A. Lipschitz Symmetric Functions on Banach Spaces With Symmetric Bases. Carpathian Math. Publ. 2021, 13, 727-733. | {"url":"https://journals.pnu.edu.ua/index.php/cmp/article/view/5568","timestamp":"2024-11-04T11:50:33Z","content_type":"text/html","content_length":"36845","record_id":"<urn:uuid:c16bbd8f-f23c-4c65-a98c-7aa85a64b0d1>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00829.warc.gz"} |
Ravi Ramakrishna
Professor of Mathematics, Stephen H. Weiss Presidential Fellow, and Department Chair
Ph.D. (1992) Princeton University
Research Area
Algebraic number theory
My research is in Galois theory. This is the branch of mathematics concerned with symmetries of solutions of equations. There is an object that encodes all symmetries of solutions to all equations,
the absolute Galois group of the rational numbers. I study this object and its relations with number theory. The study of these symmetries has gained an increasingly important role in number theory
in recent years. In particular, Galois theory played an important role in the solution of Fermat's Last Theorem.
Selected Publications
Some supercongruences occurring in truncated hypergeometric series (with L. Long). Adv. Math. 290 (2016), 773–808.
Lifting torsion Galois representations (with C Khare). Forum Math. Sigma 3 (2015), e14, 37 pp.
Maps to weight space in Hida families. Indian J. Pure Appl. Math. 45 (2014), no. 5, 759–776.
Deformations of certain reducible Galois representations (with S. Hamblen). II. Amer. J. Math. 130 (2008), no. 4, 913–944.
Constructing semisimple p-adic Galois representations with prescribed properties (with C. Khare and M. Larsen), American Journal of Mathematics 127 (2005), 709–734.
Deforming Galois representations and the conjectures of Serre and Fontaine-Mazur, Ann. of Math. (2) 156 no. 1 (2002), 115–154. | {"url":"http://www.mathlab.cornell.edu/m/People/bynetid/rkr5","timestamp":"2024-11-09T06:24:39Z","content_type":"text/html","content_length":"27807","record_id":"<urn:uuid:4758531f-f433-40c1-bf45-fb827dcd8021>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00437.warc.gz"} |
Inquiry-based number theory textbook | Department of Mathematics
David Pengelley has recently published an innovative new number theory textbook based on original, historical writings by Sophie Germain, the first woman we know to do important original mathematical
research. Readers are guided through primary-source historical documents including excerpts of her own writings, in order to discover, and prove for themselves, many important concepts from
introductory number theory. The book is very unusual, combining an inquiry pedagogy with a historical approach to one great problem (Fermat's Last Theorem) based on primary historical sources. | {"url":"https://math.oregonstate.edu.prod.acquia.cosine.oregonstate.edu/mathematics-news-events/news-briefs/inquiry-based-number-theory-textbook","timestamp":"2024-11-06T18:33:31Z","content_type":"text/html","content_length":"54705","record_id":"<urn:uuid:76d91f32-d1bd-41ee-b9b3-014993f72ad9>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00800.warc.gz"} |
ManPag.es -
slaed1.f −
subroutine SLAED1 (N, D, Q, LDQ, INDXQ, RHO, CUTPNT, WORK, IWORK, INFO)
SLAED1 used by sstedc. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is tridiagonal.
Function/Subroutine Documentation
subroutine SLAED1 (integerN, real, dimension( * )D, real, dimension( ldq, * )Q, integerLDQ, integer, dimension( * )INDXQ, realRHO, integerCUTPNT, real, dimension( * )WORK, integer, dimension( * )
IWORK, integerINFO)
SLAED1 used by sstedc. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is tridiagonal.
SLAED1 computes the updated eigensystem of a diagonal
matrix after modification by a rank-one symmetric matrix. This
routine is used only for the eigenproblem which requires all
eigenvalues and eigenvectors of a tridiagonal matrix. SLAED7 handles
the case in which eigenvalues only or eigenvalues and eigenvectors
of a full symmetric matrix (which was reduced to tridiagonal form)
are desired.
T = Q(in) ( D(in) + RHO * Z*Z**T ) Q**T(in) = Q(out) * D(out) * Q**T(out)
where Z = Q**T*u, u is a vector of length N with ones in the
CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.
The eigenvectors of the original matrix are stored in Q, and the
eigenvalues are in D. The algorithm consists of three stages:
The first stage consists of deflating the size of the problem
when there are multiple eigenvalues or if there is a zero in
the Z vector. For each such occurence the dimension of the
secular equation problem is reduced by one. This stage is
performed by the routine SLAED2.
The second stage consists of calculating the updated
eigenvalues. This is done by finding the roots of the secular
equation via the routine SLAED4 (as called by SLAED3).
This routine also calculates the eigenvectors of the current
The final stage consists of computing the updated eigenvectors
directly using the updated eigenvalues. The eigenvectors for
the current problem are multiplied with the eigenvectors from
the overall problem.
N is INTEGER
The dimension of the symmetric tridiagonal matrix. N >= 0.
D is REAL array, dimension (N)
On entry, the eigenvalues of the rank-1-perturbed matrix.
On exit, the eigenvalues of the repaired matrix.
Q is REAL array, dimension (LDQ,N)
On entry, the eigenvectors of the rank-1-perturbed matrix.
On exit, the eigenvectors of the repaired tridiagonal matrix.
LDQ is INTEGER
The leading dimension of the array Q. LDQ >= max(1,N).
INDXQ is INTEGER array, dimension (N)
On entry, the permutation which separately sorts the two
subproblems in D into ascending order.
On exit, the permutation which will reintegrate the
subproblems back into sorted order,
i.e. D( INDXQ( I = 1, N ) ) will be in ascending order.
RHO is REAL
The subdiagonal entry used to create the rank-1 modification.
CUTPNT is INTEGER
The location of the last eigenvalue in the leading sub-matrix.
min(1,N) <= CUTPNT <= N/2.
WORK is REAL array, dimension (4*N + N**2)
IWORK is INTEGER array, dimension (4*N)
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = 1, an eigenvalue did not converge
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
September 2012
Jeff Rutter, Computer Science Division, University of California at Berkeley, USA
Modified by Francoise Tisseur, University of Tennessee
Definition at line 163 of file slaed1.f.
Generated automatically by Doxygen for LAPACK from the source code. | {"url":"https://manpag.es/SUSE131/3+SLAED1","timestamp":"2024-11-12T12:46:26Z","content_type":"text/html","content_length":"22314","record_id":"<urn:uuid:5357f9d2-9bf0-4ebb-afeb-8cda6514d1c6>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00065.warc.gz"} |
Exploring the Reality of Imaginary Numbers: A Deep Dive
Written on
Chapter 1: The Concept of Imaginary Numbers
What are imaginary numbers, and how do they fit into the realm of mathematics?
Imaginary numbers often pose a challenge for high school students. Initially, they grasp basic equations and geometry with relative ease, even when confronted with variables like x. However, the
introduction of the letter i throws them for a loop. This enigmatic value, labeled "imaginary," seems to defy established mathematical principles and can be perplexing to comprehend. Many students
wonder why they should bother learning about something that appears to lack real-world applicability.
As students progress, they encounter various practical uses for imaginary numbers. While these applications can be fascinating and beneficial for problem-solving, many still perceive imaginary
numbers as merely theoretical tools without tangible existence. Contrary to this belief, imaginary numbers are integral to numerous fields, performing essential calculations. So, what exactly are
In this article, I will clarify that imaginary numbers are as real as any other numerical type. I'll showcase their utility and delve into what we mean when we describe numbers as "real." The
discussion encompasses profound mathematical and philosophical inquiries surrounding the concept of imaginary numbers. Let's dive in!
The Basics of Imaginary Numbers
The definition of an imaginary number is straightforward: we define i as follows:
Complex numbers, which include imaginary numbers, can be expressed in the form a + bi. Any number incorporating i is deemed imaginary, and this general expression encompasses all possible imaginary
numbers. You may recall from your high school studies that negative numbers lack square roots, but this assumption changes when we consider complex numbers.
By embracing this concept, we unlock a multitude of equations that become solvable through the existence of i. This notion extends further into the realm of the complex number plane, which features
both real and imaginary axes.
On this plane, real numbers occupy the x-axis, while every imaginary number corresponds to a point within this two-dimensional space.
Imaginary numbers possess a rich and fascinating history. They were first identified in the 1500s by mathematician Gerolamo Cardano, who encountered them while attempting to solve cubic polynomials.
He recognized that addressing these equations necessitated the use of values involving the square root of negative numbers, although he initially viewed them as "subtle yet useless."
The skepticism surrounding imaginary numbers persisted for centuries. Some mathematicians used them to solve specific problems but regarded them as an inconvenience. This viewpoint was reinforced in
1637 when René Descartes remarked that while imaginary numbers might be conceived in equations, they do not correspond to any real quantities.
For instance, consider the equation:
Without imaginary numbers, this equation would be unsolvable. However, their inclusion simplifies the process significantly:
Despite the initial doubts about their value, mathematicians continued utilizing them. It was Carl Friedrich Gauss who first acknowledged their utility and recognized them as vital tools. In fact,
imaginary numbers are crucial in mathematics for describing wave behavior, allowing us to capture both amplitude and frequency with a single number. Fields such as physics and electrical engineering
heavily depend on imaginary numbers for solving equations.
But, Are They Truly Real?
While we've examined some applications of complex and imaginary numbers, does this suffice to deem them "real"? To address this, we must first consider what it means for a number to be classified as
real. Let's start with the most fundamental number types: natural numbers. These are the initial numbers everyone learns, represented by counting from 1—such as 2, 10, and 100. Conversely, -10, 5/7,
and 0 are not classified as natural numbers.
Natural numbers are universally recognized as real, as they effectively describe the quantity of a set.
However, our understanding of numbers expands beyond natural numbers to encompass the entire set of integers, which includes negatives and zero. This extension adds complexity; for example, it makes
no sense to state that a bowl contains -2 apples. Yet, integers facilitate comparisons between sets, allowing us to express that one bowl has two fewer apples than another.
This distinction might seem trivial, but it underscores how our perceptions of numbers evolve. Early mathematical writings largely dismissed negative numbers as absurd. Over time, however, their
practical applications—like representing debts—led to their acceptance by mathematicians by the 1800s. The acceptance of the number zero followed a similar trajectory, though it gained acceptance
more quickly than negative numbers.
Beyond these concepts, fractions introduce an entirely different dimension, representing ratios rather than set sizes. This notion diverges significantly from natural numbers and integers, while
irrational numbers, such as pi or the square root of 2, further expand our numerical landscape.
Collectively, these form what we term "real" numbers. Despite encompassing various definitions, real numbers enable continuous measurements, like the length of a piece of wood. We navigate seamlessly
among these categories, which have become intuitive.
By definition, imaginary numbers do not fit within the classification of real numbers since they do not belong to any of the four previously outlined types. Nevertheless, I contend that they are
equally legitimate components of reality, given their demonstrable applications and adherence to a consistent set of rules.
The diagram above illustrates the classifications discussed, showcasing examples. Natural numbers are included within integers, which in turn encompass fractions (also known as rational numbers).
Irrational numbers stand apart; however, all these categories fall under the umbrella of real numbers. Imaginary numbers, while distinct, exist within a broader classification that I propose as
numbers belonging to reality.
Perhaps the term "imaginary" suffers from a public relations issue, as it suggests a lack of tangible existence.
What are your thoughts on this matter? I welcome your insights in the comments!
Going Further
I hope this exploration has illuminated the fascinating world of imaginary numbers for you! Their applications are remarkable, and understanding the evolution of numerical concepts can be equally
intriguing. If you're eager to delve deeper, I've compiled a selection of resources below to guide you.
This free online algebra book provides a solid overview of complex numbers and their applications. For a comprehensive textbook on the subject, I recommend this free resource.
The history of mathematics is replete with captivating stories, including those I referenced in this article. I suggest checking out the Wikipedia page for an introductory overview of mathematical
For more extensive insights, I found a page that elaborates on this topic in greater detail than this article covers. Additionally, this article includes excellent visuals relating to imaginary
numbers and expands on the arguments presented here.
Lastly, I highly recommend this page, which intuitively explains imaginary numbers and their relevance in quantum mechanics!
If you found this article valuable, consider showing your support! You might also want to follow me for similar content or subscribe to my email list, as I share weekly insights on math and science.
Chapter 2: Practical Applications of Imaginary Numbers
The first video titled "Imaginary Numbers Are Real [Part 1: Introduction]" explores the foundational concepts of imaginary numbers, shedding light on their significance in mathematics.
The second video, "Are Imaginary Numbers Real?" delves into the philosophical implications and practical uses of these intriguing mathematical constructs. | {"url":"https://petesellsmihouses.com/exploring-reality-of-imaginary-numbers.html","timestamp":"2024-11-06T23:25:50Z","content_type":"text/html","content_length":"14495","record_id":"<urn:uuid:a2226b01-d78c-41b2-9a40-b2eccca788cd>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00321.warc.gz"} |
Ordinary differential equation - (Programming for Mathematical Applications) - Vocab, Definition, Explanations | Fiveable
Ordinary differential equation
from class:
Programming for Mathematical Applications
An ordinary differential equation (ODE) is a mathematical equation that relates a function of one variable to its derivatives. ODEs are used to model a wide range of phenomena, such as motion,
growth, and decay, making them essential in fields like physics, engineering, and economics. The solutions to these equations can be found using various techniques, including numerical methods such
as Runge-Kutta methods.
congrats on reading the definition of ordinary differential equation. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. Ordinary differential equations can be classified into linear and nonlinear types based on the relationship between the function and its derivatives.
2. The order of an ordinary differential equation is determined by the highest derivative present in the equation.
3. Many physical systems can be modeled with first-order ODEs, such as Newton's law of cooling or exponential growth and decay.
4. Higher-order ODEs can often be converted into a system of first-order ODEs to facilitate analysis and solution finding.
5. Runge-Kutta methods provide a systematic approach to approximate solutions for ODEs when analytical solutions are difficult or impossible to obtain.
Review Questions
• How do ordinary differential equations serve as models for real-world phenomena, and what characteristics differentiate linear from nonlinear ODEs?
□ Ordinary differential equations serve as models for various real-world phenomena like population growth, motion, and heat transfer. Linear ODEs have solutions that can be superimposed,
meaning their response to inputs can be added together. In contrast, nonlinear ODEs can exhibit complex behavior like chaos and do not have this superposition property, making them generally
more challenging to solve.
• Discuss how initial value problems relate to ordinary differential equations and why they are important in finding unique solutions.
□ Initial value problems are crucial in the study of ordinary differential equations because they specify conditions under which a solution must be found. By providing initial values for the
function and its derivatives at a specific point, one can often determine a unique solution to an ODE. This is particularly important in applications where the behavior of a system needs to
be predicted based on its starting conditions.
• Evaluate the effectiveness of Runge-Kutta methods in solving ordinary differential equations compared to other numerical methods.
□ Runge-Kutta methods are highly effective for solving ordinary differential equations due to their balance between accuracy and computational efficiency. Compared to simpler methods like
Euler's method, which can be prone to large errors with increasing step sizes, Runge-Kutta methods use multiple evaluations of the function within each step to achieve better accuracy without
significantly increasing the computational load. This makes them particularly popular for solving ODEs that arise in complex scientific simulations where precision is vital.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website. | {"url":"https://library.fiveable.me/key-terms/programming-for-mathematical-applications/ordinary-differential-equation","timestamp":"2024-11-07T15:33:57Z","content_type":"text/html","content_length":"154462","record_id":"<urn:uuid:81fc0e2b-e1b4-47d0-8c4a-9ac71d68183e>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00344.warc.gz"} |
ENTMPL (for Easy Numeric Turing Machine Programming Language) is a very concise and obfuscated language for programming Turing machines, designed by Abraham Karplus. It simply consists of a list of
nonnegative decimal integers and asterisks, separated by one or more whitespace characters. Comments may be included in parentheses.
The Turing Machine is a single tape (infinite in both directions), finite symbol count, finite head state count machine with IO being the initial and final states of the tape. The initial head state
is 0, and the tape is initially filled with zeroes to the left of the tape head, and the input under and to the right of the tape head (with the rest of the tape to the right after the input filled
with zeroes). Zeroes may not occur in the input, and output is read from the starting position of the tape head to the right until a zero is reached.
The first two numbers in the program are the number of different tape symbols and the number of head states, respectively. The number of symbols or states may be replaced by an asterisk, indicating
that it should be determined by the highest value in the source code (or input, for tape symbols). When a specific number in specified, all symbol or state numbers are taken modulo it. Following
these two numbers are any quantity of five-tuples of numbers, each representing a rule.
The five numbers in a rule have the following meaning:
• Initial Tape Symbol
• Initial Head State
• Resultant Tape Symbol
• Resultant Head State
• Direction
Multiple rules may not have the same initial tape symbol and head state. A direction of 0 indicate left and 1 indicates right. Some or all of numbers in a rule may be replaced by single asterisks,
with varying meanings. As an initial symbol or state, the asterisk is a wildcard, matching anything. As a resultant symbol or state, it indicates no change, and as a direction it indicates halting
the program.
( CAT ) * (Determine # of tape symbols from input.) 1 (Only one head symbol needed.)
(First rule:) * (Whatever first tape symbol is,) 0 (the head state will be zero,) * * (so keep both) * (and end the program.)
Wolfram's 2,3 Machine
(WOLFRAM 2,3) 3 2
0 0 (->) 1 1 (R) 1
0 1 (->) 2 0 (L) 0
1 0 (->) 2 0 (L) 0
1 1 (->) 2 1 (R) 1
2 0 (->) 1 0 (L) 0
2 1 (->) 0 0 (R) 1 | {"url":"http://esolangs.org/wiki/ENTMPL","timestamp":"2024-11-11T01:45:42Z","content_type":"text/html","content_length":"20640","record_id":"<urn:uuid:ce21c7e6-5077-46a3-a9dc-52e83a48573f>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00497.warc.gz"} |
The Affine Space¶
The affine space has two types of entities:
• point - a position specified with coordinate values (e.g., location, address, etc.)
• vector - the difference between two points (e.g., shift, offset, displacement, duration, etc.)
The vector described here is specific to the affine space theory and is not the same thing as the quantity of a vector character that we discussed in the "Scalars, vectors, and tensors" chapter
(although, in some cases, those terms may overlap).
Operations in the affine space¶
Here are the primary operations one can do in the affine space:
• vector + vector -> vector
• vector - vector -> vector
• -vector -> vector
• vector * scalar -> vector
• scalar * vector -> vector
• vector / scalar -> vector
• point - point -> vector
• point + vector -> point
• vector + point -> point
• point - vector -> point
It is not possible to:
• add two points,
• subtract a point from a vector,
• multiply nor divide points with anything else.
Points are more common than most of us imagine¶
Point abstractions should be used more often in the C++ software. They are not only about temperature or time. Points are everywhere around us and should become more popular in the products we
implement. They can be used to implement:
• temperature points,
• timestamps,
• daily mass readouts from the scale,
• altitudes of mountain peaks on a map,
• current speed displayed on a car's speed-o-meter,
• today's price of instruments on the market,
• and many more.
Improving the affine space's points intuition will allow us to write better and safer software.
Vector is modeled by quantity¶
Up until now, each time we used a quantity in our code, we were modeling some kind of a difference between two things:
• the distance between two points,
• duration between two time points,
• the difference in speed (even if relative to zero).
As we already know, a quantity type provides all operations required for a vector type in the affine space.
Point is modeled by PointOrigin and quantity_point¶
In the mp-units library the point abstraction is modelled by:
• PointOrigin concept that specifies measurement origin,
• quantity_point class template that specifies a point relative to a specific predefined origin.
Absolute point origin¶
The absolute point origin specifies where the "zero" of our measurement's scale is. User can specify such an origin by deriving from the absolute_point_origin class template:
constexpr struct mean_sea_level : absolute_point_origin<mean_sea_level, isq::altitude> {} mean_sea_level;
The absolute_point_origin class template uses CRTP idiom to enforce the uniqueness of such a type. You should pass the type of a derived class as the first argument of the template instantiation.
The quantity_point class template specifies an absolute quantity with respect to an origin:
template<Reference auto R,
PointOriginFor<get_quantity_spec(R)> auto PO,
RepresentationOf<get_quantity_spec(R).character> Rep = double>
class quantity_point;
As we can see above, the quantity_point class template exposes one additional parameter compared to quantity. The PO parameter satisfies a PointOriginFor concept and specifies the origin of our
measurement scale.
quantity_point definition can be found in the mp-units/quantity_point.h header file.
As a point can be represented with a vector from the origin, a quantity_point class template can be created with the following operations:
quantity_point qp1 = mean_sea_level + 42 * m;
quantity_point qp2 = 42 * m + mean_sea_level;
quantity_point qp3 = mean_sea_level - 42 * m;
Similarly to creation of a quantity, if someone does not like the operator-based syntax to create a quantity_point, the same results can be achieved with two-parameter constructor:
The provided quantity representing an offset from the origin is stored inside the quantity_point class template and can be obtained with a quantity_from(PointOrigin) member function:
constexpr quantity_point everest_base_camp_alt = mean_sea_level + isq::altitude(5364 * m);
static_assert(everest_base_camp_alt.quantity_from(mean_sea_level) == 5364 * m);
Relative point origin¶
We often do not have only one ultimate "zero" point when we measure things.
Continuing the Mount Everest trip example above, measuring all daily hikes from the mean_sea_level might not be efficient. Maybe we know that we are not good climbers, so all our climbs can be
represented with an 8-bit integer type allowing us to save memory in our database of climbs? Why not use everest_base_camp_alt as our reference point?
For this purpose, we can define a relative_point_origin in the following way:
constexpr struct everest_base_camp : relative_point_origin<everest_base_camp_alt> {} everest_base_camp;
The above can be used as an origin for subsequent points:
constexpr quantity_point first_climb_alt = everest_base_camp + isq::altitude(std::uint8_t{42} * m);
static_assert(first_climb_alt.quantity_from(everest_base_camp) == 42 * m);
static_assert(first_climb_alt.quantity_from(mean_sea_level) == 5406 * m);
As we can see above, the quantity_from() member function returns a relative distance from the provided point origin.
Converting between different representations of the same point¶
As we might represent the same point with vectors from various origins, the mp-units library provides facilities to convert the point to the quantity_point class templates expressed in terms of
different origins.
For this purpose, we can either use:
• a converting constructor:
• a dedicated conversion interface:
It is only allowed to convert between various origins defined in terms of the same absolute_point_origin. Even if it is possible to express the same point as a vector from another
absolute_point_origin, the library will not provide such a conversion. A custom user-defined conversion function will be needed to add this functionality.
Said otherwise, in the mp-units library, there is no way to spell how two distinct absolute_point_origin types relate to each other.
Point arithmetics¶
Let's assume we will attend the CppCon conference hosted in Aurora, CO, and we want to estimate the distance we will travel. We have to take a taxi to a local airport, fly to DEN airport with a
stopover in FRA, and, in the end, get a cab to the Gaylord Rockies Resort & Convention Center:
constexpr struct home : absolute_point_origin<home, isq::distance> {} home;
quantity_point<isq::distance[km], home> home_airport = home + 15 * km;
quantity_point<isq::distance[km], home> fra_airport = home_airport + 829 * km;
quantity_point<isq::distance[km], home> den_airport = fra_airport + 8115 * km;
quantity_point<isq::distance[km], home> cppcon_venue = den_airport + 10.1 * mi;
As we can see above, we can easily get a new point by adding a quantity to an origin or another quantity point.
If we want to find out the distance traveled between two points, we simply subtract them:
quantity<isq::distance[km]> total = cppcon_venue - home;
quantity<isq::distance[km]> flight = den_airport - home_airport;
If we would like to find out the total distance traveled by taxi as well, we have to do a bit more calculations:
quantity<isq::distance[km]> taxi1 = home_airport - home;
quantity<isq::distance[km]> taxi2 = cppcon_venue - den_airport;
quantity<isq::distance[km]> taxi = taxi1 + taxi2;
Now, if we print the results:
std::cout << "Total distance: " << total << "\n";
std::cout << "Flight distance: " << flight << "\n";
std::cout << "Taxi distance: " << taxi << "\n";
we will see the following output:
It is not allowed to subtract two point origins defined in terms of absolute_point_origin (e.g. mean_sea_level - mean_sea_level) as those do not contain information about the unit so we are not able
to determine a resulting quantity type.
Temperature support¶
Another important example of relative point origins is support of temperature quantity points in units different than kelvin [K].
The SI system definition in the mp-units library provides two predefined point origins:
namespace mp_units::si {
inline constexpr struct absolute_zero : absolute_point_origin<absolute_zero, isq::thermodynamic_temperature> {} absolute_zero;
inline constexpr struct ice_point : relative_point_origin<absolute_zero + 273.15 * kelvin> {} ice_point;
With the above, we can be explicit what is the origin of our temperature point. For example, if we want to implement the degree Celsius scale we can do it as follows:
Notice that while stacking point origins, we can use not only different representation types but also different units for an origin and a point. In the above example, the relative point origin is
defined in terms of si::kelvin, while the quantity point uses si::degree_Celsius.
To play a bit with temperatures we can implement a simple room's AC temperature controller in the following way:
constexpr struct room_reference_temp : relative_point_origin<si::ice_point + 21 * deg_C> {} room_reference_temp;
using room_temp = quantity_point<isq::Celsius_temperature[deg_C], room_reference_temp>;
constexpr auto step_delta = isq::Celsius_temperature(0.5 * deg_C);
constexpr int number_of_steps = 6;
room_temp room_low = room_reference_temp - number_of_steps * step_delta;
room_temp room_high = room_reference_temp + number_of_steps * step_delta;
std::println("| {:<14} | {:^18} | {:^18} | {:^18} |", "Temperature", "Room reference", "Ice point", "Absolute zero");
std::println("|{0:=^16}|{0:=^20}|{0:=^20}|{0:=^20}|", "");
auto print = [&](std::string_view label, auto v){
std::println("| {:<14} | {:^18} | {:^18} | {:^18} |",
label, v - room_reference_temp, v - si::ice_point, v - si::absolute_zero);
print("Lowest", room_low);
print("Default", room_reference_temp);
print("Highest", room_high);
The above prints:
| Temperature | Room reference | Ice point | Absolute zero |
| Lowest | -3 °C | 18 °C | 291.15 °C |
| Default | 0 °C | 21 °C | 294.15 °C |
| Highest | 3 °C | 24 °C | 297.15 °C |
No text output for points¶
The library does not provide a text output for quantity points, as printing just a number and a unit is not enough to adequately describe a quantity point. Often, an additional postfix is required.
For example, the text output of 42 m may mean many things and can also be confused with an output of a regular quantity. On the other hand, printing 42 m AMSL for altitudes above mean sea level is a
much better solution, but the library does not have enough information to print it that way by itself.
The affine space is about type-safety¶
The following operations are not allowed in the affine space:
• adding two quantity_point objects
□ It is physically impossible to add positions of home and Denver airports.
• subtracting a quantity_point from a quantity
□ What would it mean to subtract the DEN airport location from the distance to it?
• multiplying/dividing a quantity_point with a scalar
□ What is the position of 2 * DEN airport location?
• multiplying/dividing a quantity_point with a quantity
□ What would multiplying the distance with the DEN airport location mean?
• multiplying/dividing two quantity_point objects
□ What would multiplying home and DEN airport location mean?
• mixing quantity_points of different quantity kinds
□ It is physically impossible to subtract time from length.
• mixing quantity_points of inconvertible quantities
□ What does subtracting a distance point to DEN airport from the Mount Everest base camp altitude mean?
• mixing quantity_points of convertible quantities but with unrelated origins
□ How do we subtract a point on our trip to CppCon measured relatively to our home location from a point measured relative to the center of the Solar System?
Important: The affine space improves safety
The usage of quantity_point and affine space types, in general, improves expressiveness and type-safety of the code we write. | {"url":"https://mpusz.github.io/mp-units/2.1/users_guide/framework_basics/the_affine_space/","timestamp":"2024-11-08T12:00:56Z","content_type":"text/html","content_length":"90286","record_id":"<urn:uuid:fa4d4199-cfff-4108-ba47-90cc83b70a0f>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00653.warc.gz"} |
So we need onesto compute the gradient of CE Loss respect each CNN class punteggio durante \(s\)
Defined the loss, now we’ll have puro compute its gradient respect sicuro the output neurons of the CNN in order sicuro backpropagate it through the net and optimize the defined loss function tuning
the net parameters. The loss terms coming from the negative classes are zero. However, the loss gradient respect those negative classes is not cancelled, since the Softmax of the positive class also
depends on the negative classes scores.
The gradient expression will be the same for all \(C\) except for the ground truth class \(C_p\), because the conteggio of \(C_p\) (\(s_p\)) is in the nominator.
• Caffe: SoftmaxWithLoss Layer. Is limited sicuro multi-class classification.
• Pytorch: CrossEntropyLoss. Is limited onesto multi-class classification.
• TensorFlow: softmax_cross_entropy. Is limited preciso multi-class classification.
In this Facebook rete di emittenti they claim that, despite being counter-intuitive, Categorical Ciclocampestre-Entropy loss, or Softmax loss worked better than Binary Ciclocross-Entropy loss con
their multi-label classification problem.
> Skip this part if you are not interested sopra Facebook or me using Softmax Loss for multi-label classification, which is not standard.
When Softmax loss is used is per multi-label cornice, the gradients get per bit more complex, since the loss contains an element for each positive class. Consider \(M\) are the positive classes of a
sample. The CE Loss with Softmax activations would be:
Where each \(s_p\) per \(M\) is the CNN risultato for each positive class. As mediante Facebook paper, I introduce verso scaling factor \(1/M\) esatto make the loss invariant puro the number of
positive classes, which ple.
As Caffe Softmax with Loss layer nor Multinomial Logistic Loss Layer accept multi-label targets, I implemented my own PyCaffe Softmax loss layer, following the specifications of the Facebook paper.
Caffe python layers let’s us easily customize the operations done mediante the forward and backward passes of the layer:
Forward pass: Loss computation
We first compute Softmax activations for each class and cloison them in probs. Then we compute the loss for each image per the batch considering there might be more than one positive label. We use an
scale_factor (\(M\)) and we also multiply losses by the labels, which can be binary or real numbers, so they can be used for instance esatto introduce class balancing. The batch loss will be the mean
loss of the elements durante the batch. We then save the momento_loss to monitor it and the probs preciso use them per the backward pass.
Backward pass: Gradients computation
Per the backward pass we need puro compute the gradients of each element of the batch respect onesto each one of the classes scores \(s\). As the gradient for all the classes \(C\) except positive
classes \(M\) is equal onesto probs, we assign probs values puro sbocco. For the positive classes con \(M\) we subtract 1 puro the corresponding probs value and use scale_factor sicuro confronto the
gradient expression. We compute the mean gradients of all the batch preciso run the backpropagation.
Binary Ciclocross-Entropy Loss
Also called Sigmoid Ciclocross-Entropy loss. It is verso Sigmoid activation plus verso Ciclocampestre-Entropy loss. Unlike Softmax loss it is independent for each vector component (class), meaning
that the loss computed for every CNN output vector component is not affected by other component values. That’s why it is used for multi-label classification, were the insight of an element belonging
puro a un class should not influence the decision for another class. It’s called Binary Ciclocross-Entropy Loss because it sets up a binary classification problem between \(C’ = 2\) classes for every
class in \(C\), as explained above. So when using this Loss, the formulation of Cross Entroypy Loss for binary problems is often used: | {"url":"http://buzzness.digital/2022/06/22/so-we-need-onesto-compute-the-gradient-of-ce-loss-2/","timestamp":"2024-11-03T07:31:55Z","content_type":"text/html","content_length":"91776","record_id":"<urn:uuid:8a3e1fb5-5a87-464c-bef3-6d221c97dc02>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00568.warc.gz"} |
Broken Square Root Symbol With Concrete Math Font
Anyone using Typst with Concrete Math font? I face a issue, and I am not sure whether it is Typst’s the font’s fault.
$ min_theta sqrt(1 / n sum_(i = 1)^n (sans("out")_i - sum_(j = 1)^d theta_j sans("feat")_(i j) - theta_0)^2) $
In math block, if the content inside the sqrt gets tall enough, the square root symbol becomes broken. The default New Computer Modern Math works fine though.
Based on Fix bugs in math glyph assembly code by mkorje · Pull Request #5288 · typst/typst · GitHub this is a font bug
1 Like
I’ve sent an email to the author of the font regarding this issue and how it could be fixed. I’ll add an update here once I get a response.
1 Like
A new version (0.64) of the font has been released which includes a fix for this! It is already on CTAN.
4 Likes | {"url":"https://forum.typst.app/t/broken-square-root-symbol-with-concrete-math-font/1296","timestamp":"2024-11-03T17:15:54Z","content_type":"text/html","content_length":"20200","record_id":"<urn:uuid:2bf574bf-7252-4875-a503-195ed20b491d>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00860.warc.gz"} |
Numerical accuracy¶
In modern computers, floating point numbers are represented using IEEE 754 standard. For more details on floating point arithmetic and IEEE 754 standard, please see Floating point arithmetic In
particular, note that floating point provides limited accuracy (about 7 decimal digits for single precision floating point numbers, about 16 decimal digits for double precision floating point
numbers) and that floating point addition and multiplication are not associative, so the order of the operations affects the results. Because of this, PyTorch is not guaranteed to produce bitwise
identical results for floating point computations that are mathematically identical. Similarly, bitwise identical results are not guaranteed across PyTorch releases, individual commits, or different
platforms. In particular, CPU and GPU results can be different even for bitwise-identical inputs and even after controlling for the sources of randomness.
Batched computations or slice computations¶
Many operations in PyTorch support batched computation, where the same operation is performed for the elements of the batches of inputs. An example of this is torch.mm() and torch.bmm(). It is
possible to implement batched computation as a loop over batch elements, and apply the necessary math operations to the individual batch elements, for efficiency reasons we are not doing that, and
typically perform computation for the whole batch. The mathematical libraries that we are calling, and PyTorch internal implementations of operations can produces slightly different results in this
case, compared to non-batched computations. In particular, let A and B be 3D tensors with the dimensions suitable for batched matrix multiplication. Then (A@B)[0] (the first element of the batched
result) is not guaranteed to be bitwise identical to A[0]@B[0] (the matrix product of the first elements of the input batches) even though mathematically it’s an identical computation.
Similarly, an operation applied to a tensor slice is not guaranteed to produce results that are identical to the slice of the result of the same operation applied to the full tensor. E.g. let A be a
2-dimensional tensor. A.sum(-1)[0] is not guaranteed to be bitwise equal to A[:,0].sum().
Extremal values¶
When inputs contain large values such that intermediate results may overflow the range of the used datatype, the end result may overflow too, even though it is representable in the original datatype.
import torch
a=torch.tensor([1e20, 1e20]) # fp32 type by default
a.norm() # produces tensor(inf)
a.double().norm() # produces tensor(1.4142e+20, dtype=torch.float64), representable in fp32
Linear algebra (torch.linalg)¶
Non-finite values¶
The external libraries (backends) that torch.linalg uses provide no guarantees on their behaviour when the inputs have non-finite values like inf or NaN. As such, neither does PyTorch. The operations
may return a tensor with non-finite values, or raise an exception, or even segfault.
Consider using torch.isfinite() before calling these functions to detect this situation.
Extremal values in linalg¶
Functions within torch.linalg have more Extremal Values than other PyTorch functions.
Solvers and Inverses assume that the input matrix A is invertible. If it is close to being non-invertible (for example, if it has a very small singular value), then these algorithms may silently
return incorrect results. These matrices are said to be ill-conditioned. If provided with ill-conditioned inputs, the result of these functions they may vary when using the same inputs on different
devices or when using different backends via the keyword driver.
Spectral operations like svd, eig, and eigh may also return incorrect results (and their gradients may be infinite) when their inputs have singular values that are close to each other. This is
because the algorithms used to compute these decompositions struggle to converge for these inputs.
Running the computation in float64 (as NumPy does by default) often helps, but it does not solve these issues in all cases. Analyzing the spectrum of the inputs via torch.linalg.svdvals() or their
condition number via torch.linalg.cond() may help to detect these issues.
TensorFloat-32(TF32) on Nvidia Ampere (and later) devices¶
On Ampere (and later) Nvidia GPUs, PyTorch can use TensorFloat32 (TF32) to speed up mathematically intensive operations, in particular matrix multiplications and convolutions. When an operation is
performed using TF32 tensor cores, only the first 10 bits of the input mantissa are read. This may reduce accuracy and produce surprising results (e.g., multiplying a matrix by the identity matrix
may produce results that are different from the input). By default, TF32 tensor cores are disabled for matrix multiplications and enabled for convolutions, although most neural network workloads have
the same convergence behavior when using TF32 as they have with fp32. We recommend enabling TF32 tensor cores for matrix multiplications with torch.backends.cuda.matmul.allow_tf32 = True if your
network does not need full float32 precision. If your network needs full float32 precision for both matrix multiplications and convolutions, then TF32 tensor cores can also be disabled for
convolutions with torch.backends.cudnn.allow_tf32 = False.
For more information see TensorFloat32.
Reduced Precision Reduction for FP16 and BF16 GEMMs¶
Half-precision GEMM operations are typically done with intermediate accumulations (reduction) in single-precision for numerical accuracy and improved resilience to overflow. For performance, certain
GPU architectures, especially more recent ones, allow a few truncations of the intermediate accumulation results to the reduced precision (e.g., half-precision). This change is often benign from the
perspective of model convergence, though it may lead to unexpected results (e.g., inf values when the final result should be be representable in half-precision). If reduced-precision reductions are
problematic, they can be turned off with torch.backends.cuda.matmul.allow_fp16_reduced_precision_reduction = False
A similar flag exists for BF16 GEMM operations and is turned on by default. If BF16 reduced-precision reductions are problematic, they can be turned off with
torch.backends.cuda.matmul.allow_bf16_reduced_precision_reduction = False
For more information see allow_fp16_reduced_precision_reduction and allow_bf16_reduced_precision_reduction
Reduced Precision Reduction for FP16 and BF16 in Scaled Dot Product Attention (SDPA)¶
A naive SDPA math backend, when using FP16/BF16 inputs, can accumulate significant numerical errors due to the usage of low-precision intermediate buffers. To mitigate this issue, the default
behavior now involves upcasting FP16/BF16 inputs to FP32. Computations are performed in FP32/TF32, and the final FP32 results are then downcasted back to FP16/BF16. This will improve numerical
accuracy of the final output for the math backend with FP16/BF16 inputs, but increases memory usages and may cause the performance regressions in the math backend as computations shift from FP16/BF16
BMM to FP32/TF32 BMM/Matmul.
For scenarios where reduced-precision reductions are preferred for speed, they can be enabled with the following setting: torch.backends.cuda.allow_fp16_bf16_reduction_math_sdp(True)
Reduced Precision FP16 and BF16 GEMMs and Convolutions on AMD Instinct MI200 devices¶
On AMD Instinct MI200 GPUs, the FP16 and BF16 V_DOT2 and MFMA matrix instructions flush input and output denormal values to zero. FP32 and FP64 MFMA matrix instructions do not flush input and output
denormal values to zero. The affected instructions are only used by rocBLAS (GEMM) and MIOpen (convolution) kernels; all other PyTorch operations will not encounter this behavior. All other supported
AMD GPUs will not encounter this behavior.
rocBLAS and MIOpen provide alternate implementations for affected FP16 operations. Alternate implementations for BF16 operations are not provided; BF16 numbers have a larger dynamic range than FP16
numbers and are less likely to encounter denormal values. For the FP16 alternate implementations, FP16 input values are cast to an intermediate BF16 value and then cast back to FP16 output after the
accumulate FP32 operations. In this way, the input and output types are unchanged.
When training using FP16 precision, some models may fail to converge with FP16 denorms flushed to zero. Denormal values more frequently occur in the backward pass of training during gradient
calculation. PyTorch by default will use the rocBLAS and MIOpen alternate implementations during the backward pass. The default behavior can be overridden using environment variables,
ROCBLAS_INTERNAL_FP16_ALT_IMPL and MIOPEN_DEBUG_CONVOLUTION_ATTRIB_FP16_ALT_IMPL. The behavior of these environment variables is as follows:
forward backward
Env unset original alternate
Env set to 1 alternate alternate
Env set to 0 original original
The following is the list of operations where rocBLAS may be used:
• torch.addbmm
• torch.addmm
• torch.baddbmm
• torch.bmm
• torch.mm
• torch.nn.GRUCell
• torch.nn.LSTMCell
• torch.nn.Linear
• torch.sparse.addmm
• the following torch._C._ConvBackend implementations:
□ slowNd
□ slowNd_transposed
□ slowNd_dilated
□ slowNd_dilated_transposed
The following is the list of operations where MIOpen may be used:
• torch.nn.Conv[Transpose]Nd
• the following torch._C._ConvBackend implementations:
□ ConvBackend::Miopen
□ ConvBackend::MiopenDepthwise
□ ConvBackend::MiopenTranspose | {"url":"https://pytorch.org/docs/main/notes/numerical_accuracy.html","timestamp":"2024-11-12T22:07:00Z","content_type":"text/html","content_length":"62494","record_id":"<urn:uuid:25cbf60f-3b43-486d-9b85-ecb8715b60bd>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00567.warc.gz"} |
Introduction to Probability and Statistics 3rd Edition Mendenhall Solutions Manual
[DEL:$50.00:DEL] (-46%)
Introduction to Probability and Statistics 3rd Edition Mendenhall Solutions Manual.
Introduction to Probability and Statistics 3rd Edition Mendenhall Solutions Manual
Product details:
• ISBN-10 : 0176509801
• ISBN-13 : 978-0176509804
• Author: Dr. Mendenhall
Introduction to Probability and Statistics expertly sheds light on the fundamental reasoning, methods and applications of statistics. From simple, clear explanations, students learn not only how to
reason statistically, but also how to correctly interpret statistical results. The authors emphasize how to: Apply statistical procedures, uncover the meaning of statistical research in terms of
their practical applications, evaluate the validity of assumptions behind statistical tests, determine what to do when those assumptions have been violated, and meaningfully describe real data sets.
Table contents:
Set Theory.
Random Variables and Distribution Functions.
Some Standard Probability Laws.
Jointly Distributed Random Variables.
Descriptive and Inferential Statistics.
Estimation of Parameters.
Tests of Hypotheses.
Least Squares and Regression.
Nonparametric Methods.
Bayesian Methods.
Answers to Exercises.
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Introduction to Probability and Statistics 3rd Edition Mendenhall Test Bank
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extensional equality
extensionality ⇝
extensional equality
(Or extensionality). Functions, f and g are extensionally equal if and only if
f x = g x for all x.
where "=" means both expressions fail to terminate (under some given
reduction strategy
) or they both terminate with the same basic value. Two functions may be extensionally equal but not inter-convertible (neither is reducible to the other). E.g. \ x . x+x and \ x . 2*x. See also
observational equivalence
referential transparency
Nearby terms:
extensional ♦ extensional equality ♦ extensionality ♦ Extension Language Kit
Try this search on Wikipedia, Wiktionary, Google, OneLook. | {"url":"https://foldoc.org/extensionality","timestamp":"2024-11-04T10:51:24Z","content_type":"text/html","content_length":"9070","record_id":"<urn:uuid:78c9e35f-fef4-4c67-b9ad-7aea9f6a0281>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00588.warc.gz"} |
Rent Increase Calculator
Last updated:
Rent Increase Calculator
With this annual rent increase calculator, we aim to help you compute the expected future rent for you.
We have written this article to help you understand what a rent increase is and how to calculate a rent increase. We will also demonstrate some examples to help you understand the rent increase
percentage calculation.
Please check out our rental property calculator to understand more about this topic.
What is rent increase?
Rent increase is a term used to describe a situation where the landlord of a rental property raises the amount of rent they charge the tenant. Rent increases can occur for a variety of reasons, such
as inflation, changes in the local real estate market, or improvements made to the property. Whatever the reason, rent increases can significantly impact both landlords and tenants, so it's important
to understand how they work.
How to calculate my rent increase?
Now that you have understood what a rent increase is, let's talk about the rent increase percentage calculation. To understand the annual rent increase calculation, let's take the following apartment
as an example:
• Current annual rent: $24,000
• Average rent increase per year: 5%
• Number of years to estimate: 10 years
You can calculate your future rent in four steps:
1. Determine your current annual rent.
The first step is to determine your current annual rent. This is the amount of rent that you are currently paying in a year.
For our example, the current annual rent is $24,000.
2. Compute the average rent change per year.
Next, you need to compute the average rent change per year. This is the average rate your annual rent will increase or decrease every year. The average rent increase for our example is 5%. Our
percentage increase calculator can help you with your calculation.
3. Determine the number of years you want to estimate.
The next step is to determine the number of years forward you would like to estimate. This example is looking 10 years forward.
4. Apply the future rent formula:
The last step is to calculate the expected future rent using the formula below:
future annual rent = current annual rent × (1 + average rent change) ^ number of years
future annual rent = 24000 × (1 + 0.05)^10
future annual rent = $39,093
Significance of rent increase
A rent increase can have significant impacts on both landlords and tenants. For landlords, it can increase their income and cover any additional expenses related to the property, such as repairs or
upgrades. Additionally, rent increases can help landlords keep pace with inflation and the rising costs of living.
For tenants, rent increases can be a source of financial strain, particularly if they occur frequently or are substantial. If a tenant's income doesn't increase in line with their rent, they may have
to cut back on other expenses or even move to a more affordable location. Rent increases can also make it difficult for tenants to plan their finances, as they may not know how much they'll be paying
for housing from one year to the next. To understand more on this topic, check out our rent or buy calculator.
How can I calculate my expected future rent?
You can calculate your future rent in four steps:
1. Determine your current rent.
2. Compute the average rent change per year as a percentage and divide by 100.
3. Determine the number of years you want to estimate.
4. Apply the future rent formula:
future rent = current rent × (1 + average rent change) ^ number of years
What is my rent in 10 years if my current annual rent of $20,000 increases 5% per year?
Your expected future rent will be $32,577.89. You can calculate this by using the future salary formula:
future annual rent = current annual rent × (1 + average rent change) ^ number of years.
Can a tenant negotiate a rent increase?
Yes, tenants can attempt to negotiate a rent increase with their landlord. However, its success will depend on the landlord's willingness to negotiate and the local rental market conditions.
Can a landlord raise the rent during a lease term?
Generally, a landlord cannot raise the rent during a fixed-term lease unless the lease agreement allows for it or the tenant agrees to the increase. | {"url":"https://www.omnicalculator.com/finance/rent-increase","timestamp":"2024-11-03T21:56:49Z","content_type":"text/html","content_length":"369996","record_id":"<urn:uuid:767216cb-6040-49dd-a203-fff8c139fbed>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00395.warc.gz"} |
Air Operated Double Diaphragm Pump
Reading the Air Operated Double Diaphragm Pump Performance Curve
We at Antlia Pumps offer a variety of AODD pumps in our range. In this case, let’s consider AOD 40 in particular. This pump can transfer up to 140 liters per minute and produce 7 bars of pressure.
The diaphragm pump can be used to pump acids, solvents, paints, lubricants & other fluids.
The above performance curve is for the AOD 40 model which shows required air pressure to pump a specific volume of liquid. It looks complicated; however, with a little help it’s easy to understand.
Let’s understand the graph:
• X-axis: The X-axis or the bottom horizontal side of the graph shows the discharge of liquid/fluid in liters per minute.
• Y-axis: The Y-axis or the vertical line on the graph shows the air pressure used in bars on the left hand side in red, this is measured from 0 to 7.
• Black Point: This is the reference point which shows the air consumption of the pump in both cubic meters per hour & standard cubic feet per minute (SCFM).
The performance curve allows you to calculate the air consumption & pressure required to move a specific volume of liquid. For example:
• Follow the closest solid blue line up to the left until it meets on the Y-axis. That’s the inlet air pressure required. In this case it is 5 bars.
• Vertically track down from the marked black point along the blue line until it intersects the horizontal X-axis originating from 60 LPM.
• Initially, we need to find the inlet air pressure. Place the pointer on the 4 bar line on the Y-axis.
• Let’s assume we require a volume of 60 litres per minute at 4 bar discharge pressure.
• Finally, you can figure out the air consumption by tracing the pointer along the nearest blue line from the black point where the horizontal line from 4 bar and a vertical line from 60L/min
intersect, up to the connected green box. In this case, to achieve 60 liters per minute at 4 bar discharge pressure you will require 5 bar inlet pressure at 14 scfm.
Looking at the performance curve of a pump you will understand what should be exactly expected of the equipment. It’s a great way to evaluate whether a pump will be the right fit. | {"url":"http://antliaworks.com/reading-the-air-operated-double-diaphragm-pump-performance-curve.html","timestamp":"2024-11-02T11:04:38Z","content_type":"text/html","content_length":"14124","record_id":"<urn:uuid:2665f292-4053-462b-8462-16703200d13b>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00381.warc.gz"} |
using Number Lines
Decimal Worksheets | Adding Decimals using Number Lines
This collection of worksheets is designed exclusively for students of Grade 5. Learn to use number lines to sum up decimals with two or three addends. Worksheets are categorized into place values of
tenths, hundredths and thousandths for convenient downloads. Grab some of these worksheets for free!
Observe each number line and complete the addition sentences containing decimals. The printable worksheets are segregated according to place values up to thousandths along with a section that
contains a mixed review.
Indicate hops on the number line to complete the decimal addition sentence. Longer hops drawn will denote the whole number part and the smaller hops will indicate the decimal part.
Two Addends: Fill in the Missing Numbers
Read each number line in this pdf practice set. Fill in the missing decimals in each addition sentence based on the intervals observed. Answer keys are included.
The start and the endpoints of the number line represents the first addend and the sum respectively. Observe the intervals to find the second addend. Then, frame the addition sentences.
Number Line Addition: Ruler Model
The ruler model is followed in this batch of printable number line worksheets. Observe the big and small hops indicated and frame the addition sentences.
Number Line: Column and Horizontal Addition
Use the number lines provided to solve each problem displayed in both vertical and horizontal formats. This series of worksheets is split in accordance to place values up to thousandths.
Indicate hops to show the sum of three decimals. Then, complete the addition sentence. Print these pdf worksheets for an in-class assignment. | {"url":"https://www.mathworksheets4kids.com/decimal-addition-number-line.php","timestamp":"2024-11-03T12:59:18Z","content_type":"text/html","content_length":"71440","record_id":"<urn:uuid:099d8910-57b3-428d-98c9-7c48af7f8f0a>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00476.warc.gz"} |
Research on Control of Levitation Force and Torque of a Maglev Device for Water-Turbine Generator Set
College of Mechanical and Electrical Engineering, China University of Petroleum, Qingdao 257061, China
College of New Energy, China University of Petroleum, Qingdao 257061, China
School of Electrical Engineering, Southeast University, Nanjing 210096, China
College of Energy and Electrical Engineering, Hohai University, Nanjing 210013, China
Author to whom correspondence should be addressed.
Submission received: 24 May 2022 / Revised: 8 July 2022 / Accepted: 8 July 2022 / Published: 18 July 2022
Hydropower generation is clean, pollution-free, and renewable, and has good social and economic benefits, so it is given priority for development throughout the world. The capacity of hydropower
stations is increasing to 1000 MW from 700 MW. As the p value on the bearing reaches a new height, coupled with the original risk of easy damage, the thrust bearing faces new technical challenges.
Maglev technology is studied and applied to a large vertical-shaft hydro-generator set to solve the bearing problem. The maglev device is designed, and the working principle is expounded, using
active-control repulsive-suspension technology. The levitation-force addition and the torque cancellation are realized by controlling the frequency of the excitation power supply. The dynamic
mathematical models of levitation force and torque are derived. Combined with the design and theoretical analysis, the vector-control strategy is developed and the simulation analysis is completed.
According to the results, the controller is improved to enhance the response performance. Finally, a control experiment is carried out on the prototype, and the results verify the effectiveness of
the design and control strategy.
1. Introduction
According to China’s National Energy Administration, the country generated 2485.3 billion KWH (kilowatt hours) of renewable energy in 2021, of which 1340.1 billion KWH was generated by water
conservancy. It can be seen that hydropower accounts for a considerable proportion of about 54% in the field of renewable-energy generation [
]. At present, hydropower generation is given priority for development throughout the world. However, in hydropower stations that have been successfully put into use, the capacity of each unit is no
more than 700 MW. In fact, the Baihetan and Wudongde hydropower stations are under construction, with a 1000 MW single capacity in China. One of the problems to study is thrust bearing, as its damage
will affect the safety and stability of the unit [
]. The thrust bearing of the 1000 MW unit has a higher
value than the 700 MW unit, so greater load-capacity thrust bearings are required [
]. We can use new materials, technology, and structures to improve the thrust bearing. On the other hand, we can use maglev technology to reduce the load of thrust-bearing, to decrease the
value on the bearing. Meanwhile, it can improve the service life of thrust bearings even if it is applied to small-capacity units such as those that are 700 MW or less.
2. Structure and Principle of the Maglev Device
A disc electromagnetic levitation-reduction device is proposed for the thrust bearing of the large vertical-axis hydropower unit, as shown in
Figure 1
. The maglev load-reduction device is mainly composed of a primary stator and a secondary rotor plate. The stator consists of iron cores, excitation coils, and a back iron, and the rotor is a
whole-disc conductor plate. The stator is fixed to the external support structure, and the rotor is fixed on the main shaft of the hydropower unit and rotates synchronously. When the primary coils
carry an alternating current, a rotating magnetic field will be generated in the air gap. Eddy currents will be induced in the conductor plate [
]. The eddy field interacts with the primary magnetic field, and a space electromagnetic force is generated. The normal force is a repulsion force, which is used as the levitation force to reduce the
load for the thrust bearing. The force density is limited by the saturation characteristics of the core with the conventional coils [
]. To improve the force density, the primary stator can be designed as the coreless structure of the superconducting coils. When lift force is generated, there is also torque, and the lift force and
torque should be concertedly controlled [
], to ensure the stability of the thrust-bearing load and avoid electromagnetic or mechanical coupling to the power-generation system.
3. Torque Offsetting Design
For the maglev device, the levitation force is useful but the tangential force is useless, which causes an electromagnetic and mechanical-coupling effect between the maglev load-reduction device and
the original hydrogenerator system. Through design and control, zero torque and a not-reduced levitation force can be achieved. Referencing the mechanical properties of an induction motor, the torque
is written by
$T = C T ϕ m I 2 ′ cos θ 2$
$C T$
represents a coefficient composed of motor structure parameters,
$ϕ m$
represents the main flux of the air gap, and
$I 2 ′ cos θ 2$
represents the active component of the secondary rotor current. If we make the power-supply voltage constant,
$ϕ m$
will be constant too, and the torque is proportional to
$I 2 ′ cos θ 2$
, with the direction depending on the relative moving direction of the exciting field and the secondary rotor.
Based on the above theory, a disc structure with positive and negative halves is proposed with two equal and opposite electromagnetic torques offsetting each other, and the same normal lift forces
added are shown in
Figure 2
, which is the overhead view of
Figure 1
. The two halves of the maglev device are designed with the same structural parameters, and the excitations can be controlled to obtain the above effect. On the other hand, the hydrogenerator has a
certain velocity by itself determined by the waterhead and water flow. Considering the original rotating speed, we control the moving of positive exciting magnetic field ahead of the rotor and the
negative field lagging behind the rotor to obtain opposite torques and consistently control the relative velocities of the two halves, respectively, via the rotor plate.
Keeping the other structural and electrical parameters exactly the same, the frequencies of the positive
$ω 1$
and negative exciting magnetic field
$ω 2$
should meet
$ω 1 − ω 2 = 2 π n h g p 30$
$n h g$
represents the rotating speed of the hydrogenerator,
represents the pole-pair number of the disc device.
4. Calculation and Relationship of the Levitation Force and Torque
4.1. Analytic Calculation of the Levitation Force and Torque
According to Maxwell’s tensor theorem, the electromagnetic repulsion of the primary and the secondary is written as [
$F l = − μ 0 4 ( | B 1 | 2 μ 0 2 − | H 1 | 2 + | B 2 | 2 μ 0 2 − | H 2 | 2 )$
$F t = 1 2 Re ( B 2 H 2 * − B 1 H 1 )$
$B 1$
$H 1$
$B 2$
$H 2$
represent the normal magnetic flux density and the tangential magnetic field intensity of the primary upper surface and the lower surface, respectively.
Based on the multilayer electromagnetic physical model often used for a linear-induction motor, the normal levitation force and torque can be calculated by the surface impedance [
$F l = μ 0 4 J 1 2 ( 1 − k 2 μ 0 2 ω 2 | z 2 | 2 )$
$F t = 0.5 J 1 2 Re ( z 2 ) / v s$
$J 1$
represents the equivalent surface linear-current amplitude,
represents the slip frequency,
$z 2$
represents the surface impedance of the air-gap layer, and
$v s$
represents the slip velocity of the primary and the secondary.
We built a prototype sample FEA (finite-element analysis) model to verify the accuracy of the above analytical formula. The results from the FEA and the analytical method are compared in
Figure 3
. With the primary and secondary parameter changing, such as the exciting current amplitude, slip frequency, the thickness of the conduction plate, as well as the length of the air gap, we can see
that the normal lift-force curves from the two ways are very close to each other, and the difference is about 10% or less. The analytical formulas are verified to be valid to estimate the normal
force at design time.
4.2. Relationship of the Levitation Force and Torque
For the device, the lift force and the torque are related to the primary exciting current, slip frequency, air gap, and the material and thickness of the secondary conduction plate. By studying the
ratio of the lift force to the torque, the relationship of the lift force and the torque is uncovered, and a foundation for the forces controlling is laid. To avoid solving the complex magnetic field
and highlight the key parameters, we consider a limiting case, that the exciting traveling-wave magnetic field rotates at an infinitely high speed. The skin effect of the eddy current will be obvious
in the conductor plate, and the normal component of the magnetic field is almost zero on the surface of the conduction plate. We can assume that there is only a tangential component for the field.
After a Fourier transform, the tangential field can be expressed as
$H ( x ) = R e [ H a ( k ) exp ( i k x - ω t ) ]$
$H ( x )$
represents the tangential component of the magnetic field intensity,
represents the
$2 π / τ$
represents the pole pitch,
represents the slip frequency, and
$ω = k v$
represents the slip line velocity.
According to Maxwell’s tensor theorem, the normal magnetic pressure per unit area in the air gap is written as
$F l = H 2 / 8 π = | H a ( k ) | 2 / 16 π$
represents the valid value, and
$H a ( x )$
represents the amplitude of the magnetic field intensity. In addition, the tangential force per unit area can be expressed as
$F t = P σ / v = k δ d | H a ( k ) | 2 / 16 π$
$P σ$
is the energy loss per unit area of the secondary conduction plate, and
$δ d$
represents the depth of propagation of the electromagnetic waves in the secondary conductor []. So
$F l / T$
approximates to [
$F l / T = τ π R ( μ σ ω ) 1 / 2$
In addition, the structure parameters also affect the vertical and horizontal components of the magnetic field and the electromagnetic forces, such as the tooth-to-slot ratio, the shape and
width-to-depth ratio of the slot, the thickness of the secondary plate, and the length of the air gap. All the above analysis is verified by FEA, and the results are consistent with the analytic
calculation [
For further analysis and verification, we manufactured a prototype shown in
Figure 4
, and the test results are obtained, which are compared with those from FEA, as shown in
Figure 5
. We use the ratio of the normal force to the tangential force
$F l / F t$
instead of
$F l / T$
to get the same order of magnitude. From the curves, we can see that the
$F l / F t$
is proportional to
$ω 1 / 2$
under the high frequency, and it is greater than
$ω 1 / 2$
under the low frequency, which verifies the above theoretical analysis. The cut-off point is about at 30 Hz, so (10) can be rewritten as
$F l / T = λ τ π R ( μ σ ω ) 1 / 2$
where λ is used as a coefficient to correct the ratio for the low frequency. Under the frequencies higher than 30 Hz, λ is about 1.
In addition, the prototype is designed and manufactured based on the experience of the traditional linear motor, as shown in
Figure 4
. The FEA model is further optimized through the structure parameters and the magnetic circuit to get a bigger normal force and a smaller torque. For example, we can use deeper slots and increase the
effective area of the magnet to get the bigger normal component of the air-gap field and the ratio
$F l / T$
, with the cooling condition permitting. From
Figure 5
, we can see that
$F l / T$
is improved by about 20% from the prototype to the optimized FEA model, with the curves in solid lines.
5. Control of the Normal Force and the Torque
5.1. Dynamic Mathematical Model of the Torque
Based on the torque-offsetting-design idea, we need to implement coordinated control of the normal force and the torque. According to the theory of coordinate transformation, we can obtain,
respectively, the voltage equations of the stator side and the rotor side under a rotating-coordinate system by
${ u s d = R s i s d + p ψ s d − ω d q s ψ s q u s q = R s i s q + p ψ s q + ω d q s ψ s d u r d = R r i r d + p ψ r d − ω d q r ψ r q u r q = R r i r q + p ψ r q + ω d q r ψ r d$
and the flux equations are
${ ψ s d = L s i s d + L m i r d ψ s q = L s i s q + L m i r q ψ r d = L m i s d + L r i r d ψ r q = L m i s q + L r i r q$
where the subscripts
represent the stator and the rotor, respectively,
represent the
axis and
axis, respectively, and
$L s$
$L r$
$L m$
represent the equivalent self and mutual induction as well as the referred-to calculation, respectively.
We use the rotor-flux orientation, and there are
$ψ r d = ψ r , ψ r q = 0$
Through the voltage equations and flux equations, we can get the dynamic mathematical model of the torque as
$T e = n p L m L r i s q ψ r$
and there are
${ ψ r = L m 1 + T r p i s d i s q = T r ψ r L m ω s$
$T r$
represents the time constant,
represents the differential symbol, and
$ω s$
represents the slip frequency. So,
$T e$
can be controlled by adjusting
$i s q$
$ψ r$
, and
$ψ r$
can be controlled by
$i s d$
. When
$ψ r$
is made constant,
$T e$
is determined by
$i s q$
depending on
$ω s$
5.2. Dynamic Mathematical Model of the Normal Lift Force
Based on the principle of electromagnetic induction, the normal force is produced by the interaction between the exciting current and the induced current. The normal force between a single exciting
coil and a secondary coil can be represented by the secondary induced eddy current and structure parameters, as [
$F l = μ 0 I r 2 l 2 π δ$
$I r$
represents the induced current on the secondary,
represents the length of a closed coil, and
represents the distance of the both coils. As Reiz concludes, when the primary coil carried the current, which will induce almost the same shaped eddy current as in the secondary conductor plate [
]. Assuming all the primary coils are located on the primary surface and all the induced eddy current is located on the surface of the plate, we can use a model of the electromagnetic force generated
by the interaction of two identical current-carrying coils to simplify the analytical solution. For the proposed induction device, the normal force can be expressed as
$F l = C e μ 0 l W 1 2 I r 2 4 π δ p$
$C e$
represents the coefficient on the structure (and can be tested),
represents the average length of the primary or the secondary closed coil, and
$W 1$
represents the number of turns in series per phase.
$i r 2 = i r d 2 + i r q 2$
the dynamic mathematical model of the lift force can be obtained by
$F l = C e μ 0 l W 1 2 4 π δ p ( L m L r ) 2 [ ( ψ r L m − i s d ) 2 + i s q 2 ]$
From the above model under the dq-coordinate system, the lift force depends on $i s d$, $i s q$ and the flux linkage of the rotor is $ψ r$, which is up to and lagging behind $i s d$, so $i s q$ is
related to $ω s$ and $ψ r$. Thus, we can control the lift force by controlling $i s d$ and $i s q$.
5.3. Control Strategy of the Normal Force and the Torque
From Equation (16),
$ψ r$
is up to
$i s d$
, and
$ψ r$
lags behind
$i s d$
under transient changing.
$ψ r$
is steady when
$i s d$
is constant, considering Equation (20), the normal force only depends on
$i s q$
and can be expressed as
$F l = C e μ 0 l W 1 2 4 π δ p ( L m L r ) 2 i s q 2$
So, we can adjust
$i s d$
to control
$ψ r$
and adjust
$i s q$
to control
$F l$
by keeping
$ψ r$
constant. At the same time, the positive torque and the negative torque shown in
Figure 2
are equal and opposite, because there are two opposite slip angular frequencies, with one leading and one lagging like
$ω s 1 = − ω s 2$
, and
$ω s 1$
is the slip frequency from the positive half coils, and
$ω s 2$
is from the negative half coils. For the two halves,
$ψ r$
is the same, and the resultant torque is zero, thus, there is no need to control the torques.
A control strategy for the positive control shown in
Figure 6
(a negative-control block diagram likehis, and the difference is
$ω s 1$
$ω s 2$
) is developed to realize the decoupling control of the lift force and the flux linkage for the positive half [
]. The control of the primary
$i s d$
$i s q$
are designed as the inner control loops, while the lift force and the flux linkage are the same as the outer control loops.
The axial load of the bottom thrust bearing is used to control the lift force, and the axial load force is measured by a tension–pressure transducer. The given value of the lift force as the input
bias comes from the given value $F s *$ and the tested value $F s$ of the axial force, so $F s$ is the positive feedback. The input bias is the control signal to control the primary q-axis current $i
s q$. For the flux-linkage loop, the input bias from the given value and the observation value of $ψ r$ are used to control $i s d$. The given value of $ψ r$ can be estimated and obtained by
Equations (11) and (15). Meanwhile, the real-time slip frequencies $ω s 1$ and $ω s 2$ are acquired from the primary currents and the flux linkage, and the synchronous angular frequencies, $ω 1$ and
$ω 2$, of the positive and negative parts are obtained too. When the speed of the rotor varies, $ω s 1$ and $ω s 2$ have not changed yet, so the synchronous angular frequency $ω 1$ and $ω 2$ follow
the changing of the rotor to control the lift force and keep the torques balanced.
6. Simulation and Experimental Verification of Control Strategy
Simulation models of the proposed device and the whole control system are built to verify the control strategy. The system-simulation model mainly includes the module of the load-reduction device,
power inverter, controller, pulse generator, flux observer, observation modules, and so on [
6.1. Simulation Verification of Control Strategy
Assuming that the axial load of the thrust bearing is 780 kN, the actual bearing load will be 180 kN after load reduction, and the required lift force should be 600 kN. First, to verify the role of
the flux linkage
$ψ r$
, which started at 0 time and changed from 3.2 Wb to 2.7 Wb at 0.5 s, and the simulation results of the lift force, the forward force and the resultant torque, because of the change of the flux
linkage, were obtained, as shown in
Figure 7
From the results, we can see that the lift force can be set up in a very short time, and the proposed device has a short startup time and small overshoot. The building of the flux linkage needs a
period of time because of lagging behind the current. The flux linkage affects the torque but not the lift force. It will cause the half torque to drop and the resultant torque to fluctuate. Finally,
after a brief adjustment, the resultant torque became zero. So, the results verified the theoretical analysis and control strategy.
During the operation of the hydropower-generation system, there must be a fluctuation in rotor speed because of the water turbine. To verify the feature against the rotor-speed fluctuation, assuming
the rotor speed changed from 107 rpm to 125 rpm at 0.5 s, the simulation waves of the lift force and torque are obtained, as shown in
Figure 8
Since the current is controlled directly, by the above waves, we can see the slip frequency is invariable during the speed fluctuation, and the frequencies of the positive and negative coils can
follow the speed fluctuation. So, the lift force and the torque remain invariable, and it states that the device system has good adaptability to rotor-speed change. The two half torques almost cancel
out and produce no additional mechanical disturbance to the generator.
Since the axial-pressure sensor feeds back the lift-force signal indirectly, there must be a disturbance phenomenon during the transmission. To obtain the response to the disturbance, a pulse wave
force of a 0.01-s width is applied in the axial direction at 0.4 s, and the simulation results are shown in
Figure 9
. When the given value of the pressure keeps constant, the axial-load disturbance will cause the mutation of the input bias, and
$i s q$
follows the mutation. The lift force follows the mutation of
$i s q$
, because there is no delay inertia between
$i s q$
and the lift force. So, any electrical or mechanical disturbance will affect the lift force, and the controllers need to be optimized to improve the disturbance-rejection performance.
To reduce the fluctuation of the torque because of the flux linkage and improve the disturbance-rejection performance of the lift force, cascade control is used in the lift-force loop and the
flux-linkage loop. The cascade control includes a main controller and an assistant controller. The axial-force mutation is implemented at 0.4 s, the flux linkage is adjusted at 0.6 s, and the
simulation results of the lift force and the resultant torque are described in
Figure 10
From the results, we can see that the overshoot of the flux linkage is eliminated, and the fluctuation of the resultant torque is greatly reduced, under the cascade control. At 0.4 s, the influence
of the axial force mutation is also weakened by the lift force. So, the added cascade control improved the control performance of the system more than the single PI controller.
6.2. Experimental Verification of Control Strategy
A prototype is manufactured for the experimental measurement in the laboratory, as shown in
Figure 4
, and the control module is built, as shown in
Figure 11
. To test the lift force, torque, and the performance of the control system with the rotor rotating, a permanent magnet servo motor is used to drive the rotor.
The positive and negative windings are excited, respectively, and the tested lift force is shown in
Figure 12
. We can see that the lift force increased, and the rotor speed rose, when the positive coils were applied the exciting current, because there is the torque from the positive coils, so the rotor is
driven further. When we continue to add the exciting currents to the negative coils, the lift force continues to increase but the resultant torque is almost zero. To easily measure the torques on
both sides, the test is carried out with the rotor blocked, and the results are shown in
Figure 13
. The torques of the two halves are equal and opposite, and the joint torque is almost zero. All of the results verify the effectiveness of the design and the control strategy.
It should be noted that due to the existence of mechanical stress and the limitation of machining accuracy, the actual test data of the suspension force and the torque are not accurate. At the same
time, due to the limitation of the laboratory conditions, the experimental test has only completed the preliminary test, and more adequate experimental tests need to be carried out next.
7. Conclusions
Maglev technology is studied to apply to a large vertical-shaft hydro-generator set (1000 MW) to solve the bearing problem. A controlled maglev-reduction device is designed for the thrust bearing,
and normal force is used as the lift force, so torque is useless. In this paper, through the theoretical study on electromagnetic force, a control strategy combined with a structure design is
proposed. The dynamic mathematical models of the lift force and the torque are established, and the simulation is implemented. From the simulation, it states the torque and the lift force can realize
decoupling control, and the control system has good following and disturbance resistance. By applying cascade controllers, the overshoot characteristics have also been significantly improved. In the
lab, the preliminary tests were carried out, and the results confirm the validity of the design and control strategy.
Author Contributions
Conceptualization and methodology, J.L.; software, C.X. and J.Z.; writing—original draft preparation, J.L.; writing—review and editing, L.H.; project administration, H.M. All authors have read and
agreed to the published version of the manuscript.
This research was funded by the National Natural Science Foundation of China, grant number 51707204, by the Fundamental Research Funds for the Central Universities, grant number 18CX02092A, and by
the National Key Research and Development Program of China, grant number 2019YFE0105100.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
Not applicable.
The authors would like to thank the National Natural Science Foundation of China for the financial support to this work. We thank the associate editor and the reviewers for their useful feedback that
improved this paper.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Figure 3. The normal force from the analytical method and FEA. (a) Normal force via the exciting current, (b) Normal force via the power frequency, (c) Normal force via the plate thickness, (d)
Normal force via the air gap length.
Figure 7. Dynamic features under the flux-linkage adjustment. (a) Rotor flux linkage via time, (b) Positive torque via time, (c) Resultant torque via time, (d) Resultant lift force via time, (e) i
[sq] via time, (f) i[sd] via time.
Figure 8. Dynamic features under the rotor-speed fluctuation. (a) Phase current via time, (b) Resultant lift force via time, (c) Resultant torque via time.
Figure 9. Dynamic features under the axial load disturbance. (a) Pressure via time under disturbance, (b) i[sq]^* via time under disturbance, (c) Lift force via time under disturbance.
Figure 10. Dynamic features under the axial-load disturbance and the flux-linkage adjustment with cascade control. (a) Lift force via time, (b) Resultant torque via time.
Figure 12. Tested levitation force with the shaft rotating at some speed. (a) The tested levitation force for a long time; (b) details of (a).
Figure 13. Tested torque with the rotor blocked. (a) The positive torque via power frequency, (b) The reverse torque via power frequency.
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AMA Style
Liu J, Xu C, Zhu J, Huang L, Ma H. Research on Control of Levitation Force and Torque of a Maglev Device for Water-Turbine Generator Set. Sustainability. 2022; 14(14):8742. https://doi.org/10.3390/
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Please post 2 or more peer responses In the response posts, remember to demonstrate you have read and understood the student’s post by taking their
Please post 2 or more peer responses
In the response posts, remember to demonstrate you have read and understood the student’s post by taking their discussion to the next level. Do this by:
· Using the height of your classmate, calculate the z-score using the mean and standard deviation for the other gender. That is, if they calculated z for a male, calculate for a female. Would their
height be unusual for a different gender?
· What are some other ways the concept of normality might be used in their field?
· What are some challenges for very tall and short people that were not mentioned? What about their social life? Do females always want to date someone taller? Do men want to date someone shorter?
Please be sure to validate your opinions and ideas with citations and references in APA format.
Estimated time to complete: 2 hours
Response posts are worth 50% of your grade for this discussion. Please review the post and response expectations
Please review the rubric to ensure that your response meets the criteria.
I am 65 inches tall and female. Using the data provided, I have a z-score of 0.6. it is a positive number because I am taller than the mean value of 64 inches. The standard deviation is 2.8, I am
therefore within the normal distribution.
The challenges that I face with my height are mostly self-induced. I place things in upper cabinets and in the top of closets and then need a stool or one of my tall men to assist me in retrieving
said items. I am within the normal range for female height, so I have not really had issues with clothing and such.
I am a registered nurse. The concept of normality in my field is constant. We measure vital signs and compare them to the normal values all the time, we measure weights and heights and compare those
to the normal values so that we can better plan our treatment for our patients. Some medications are weight-based dosing, so we must know an accurate weight for each patient so that we can give the
correct amount of the medication. Some medications are based on “Ideal Weight” dosing so we need to know the patient’s correct height to calculate this information as well. We do not want to overdose
or underdose a patient due to incorrect data. “Another factor that has been shown to be related to increased dosing errors is caregiver knowledge that dosing is weight-based, especially with
over-the-counter medications.” (HS, BP, G, L, & AL, 2007)
The auto industry must use the normal distribution range for height to design the seats in the vehicles they manufacture. Though the seats are adjustable, it would be difficult for a very tall person
to operate a vehicle if it was designed specifically for a very short person and vice versa.
HS, Y., BP, D., G, F., L, v. S., & AL, M. (2007, Jul/Aug). Association of low caregiver health literacy with reported use of nonstandardized dosing instruments and lack of knowledge of weight-based
dosing. Ambulatory Pediatrics, 7(4), 292-298. Retrieved from
The z-score is measured in terms of the standard deviation (SD) from the mean (Openstax, 2013). I am female, with a height of precisely 64 inches. Based on population values and after calculations, I
am in the average height range for all females, being 64 inches tall, and the standard deviation is 2.8. Therefore, my height is not unusual, which I expected as I have always been a “normal” height
compared to other females.
Based on my average height, I have not faced many challenges. However, I have had to buy “petite” scrub pants for work before because most brands have different inseam lengths, which poses risks in
healthcare due to your pants dragging the floor, which is unsanitary. Furthermore, normality is used in nursing daily. We do many medications based on weight, such as Heparin for example. You have a
patient’s base weight, and the dosage calculation is based on that weight using normal ranges of dosages. Every Heparin drip is then dosed every 6 hours using a lab called a “PTT,” which has a
specific protocol of normal ranges of value on which to base the next dose.
Knowing what usually helps corporations when making the sizing charts of clothing for children. Corporations study the statistics of usual average sizes using the height and weights of children of
certain ages to determine the normal size range and make clothing based on children’s age. Some of those sizes are the infant and toddler sizes, for instance, 3 months-6 months, 6-12 months, etc.,
for infants. For toddlers, they are 2T, 3T, 4T, and 5T. The standard measurements are developed by government agencies private agencies that conduct extensive research, thereby collecting the
measurements of a huge number of people living in a specific geo-demography (Prabhakar & Rajagopal, 2022).
Openstax (2013). Introductory Statistics. Openstax Creative Common License. Retrieved from URL to an external site.
Prabhakar, D., & Rajagopal, S. (2022). Conceptualization of body measurements for 6-8 years kids ready-to-wear apparel based on anthropometric study in bangalore, india. Research Journal of Textile
and Apparel, 27(4), 489–515. to an external site. | {"url":"https://accountingpaperhelp.com/mathematics-homework-help-2/56159-please-post-2-or-more-peer-responsesin-the-response-posts-remember-to-demonstrate-you-have-read-and-understood-the-students-post-by-taking-their-3/","timestamp":"2024-11-05T22:12:43Z","content_type":"text/html","content_length":"115602","record_id":"<urn:uuid:debc26c6-e4b1-4b03-9be9-ce92f382c2a9>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00091.warc.gz"} |
(AES) Key Schedule and Decryption [Part 2] - Hunter Richardson
This is a continuation of (AES) The Advanced Encryption Standard [Part 1], so it is highly encouraged to read that post before reading this post. In that post, I gave the history of AES and also
explained the internals of the AES algorithm.
In this post we will discuss these things:
• The AES key schedule
• Decryption of AES
• Security of AES
The Key Schedule of AES-128 bit Key
As stated in part 1, there are different version of AES. The difference in these versions is the size of the key, and an important thing to remember is the larger the key the more secure AES is.
For this blog, I will only focus on the 128-bit key for simplicity. Note though that the key schedule differs slightly for each size, but I am confident if you understand how the 128-bit key schedule
works, it will be easy for you to learn 192/256 on your own.
For AES-128 there are 11 subkeys which are derived from the 128-bit master key. You may wonder why there are 11 if AES-128 has 10 rounds, that is because before the 10 rounds you XOR the block
with the first subkey. The 11 subkeys are stored in a key expansion array that splits up the 128-bit key into 4 32-bit words, in total after all 11 rounds there will be 44 words, denoted as W[0], W
[1], …, W[43]. The key schedule is computed as shown in the diagram below:
Key Schedule Mathematical Description
For those that need a bit more than a diagram, or prefer a formula to derive the key schedule is is quite easy to do that for AES-128.
For the left most word of a round key W[4i], where i = 1, … , 10 we can write:
W[4i] = W[4(i – 1)] + g(W[4i – 1]).
Here the function g() is a nonlinear function that takes four bytes as an input and gives four bytes as an output. The remaining 3 words of a round key are derived as such:
W[4i + j] = W[4i + j – 1] + W[4(i – 1) + j].
Where again i = 1, …, 10 and j = 1, 2, 3.
All that is left is to understand what the g function is from our diagram and formula. Before I get into that, you will probably notice sources online that don’t show it this way, and instead
just show the inner functions I will go into below. I simply did this so as to hopefully not overwhelm you at the very beginning.
The g function first rotates its four input bytes which we call RotWord, then performs the S-Box substitution just like in the AES algorithm which we call SubWord, and then adds a round
coefficient RC to it which we call Rcon. The round coefficient is an element of GF(2^8), and note that it is only added to the leftmost byte of the word. The round coefficients also vary from round
to round according to the table below:
i 1 2 3 4 5 6 7 8 9 10
RC[i] 01 02 04 08 10 20 40 80 1B 36
So all that is left is to see how these are flow together, to see that here is a diagram of the g function below:
The g function serves two purposes. First, it adds nonlinearity to the key schedule. Second, it removes symmetry in AES. If you are familiar with the DES post, you will recall the importance of
Unlike DES, the AES is not a Feistel network. Meaning we cannot just input our cipher text into out algorithm and get the decrypted plaintext outputted. Therefore we must actually invert all of
the layers, i.e. the Byte Substitution now becomes Inv Byte Substitution, ShiftRows becomes Inv ShiftRows, and MixColumn becomes InvMixColumn.
Thankfully these functions are not too different from the original functions used in encryption. Note that also the order of the layers is inverted along with the order of the round keys. To see
this see the diagram below:
Now that we know the basic order of the inverse functions for decryption in AES, we will delve deeper and see what actually happens inside these layers. Recall that in AES we operate on bytes
where 1 byte = 8 bits.
Below is a model for the inverse layers. You may notice it looks a lot like the one for encryption, and that because they are very similar. I suggest comparing them both and seeing how they
differ and where they are the same. Below is the diagram for the internal inverse layers for AES:
Recall that the XOR operation is itself its own inverse, thus the key addition layer does not have an inverse. You may also notice that we can’t reuse the same S-box, we need to use an inverse
This may seem like a lot, or too abstract for now. But just like with encryption I will into more depth on how the InvMixCol, InvShiftRow, and InvByteSub work.
Note that it is also best to think of the blocks of 16 bytes that are being decrypted to be arranged in a 4×4 byte matrix. You can see how we show the matrix below:
After the XOR operation in the Key Addition layer the inverse MixColum layer is applied to the state. Note that just like in the final round of encryption, the first round of decryption does not
use Inverse MixColumn.
In order to invert the MixColumn layer, we must use the inverse of the matrix used. Just like in encryption, the input is a 4-byte column of the state ‘C’, which is then multiplied by the inverse
4×4 matrix. The elements of the inverse matrix are still constants, and the operations (multiplication & addition) of the coefficients are still done in GF(2^8).
Here is a diagram of the mathematical description of the layer:
Here is an example of how we would compute B[0] :
B[0] = 0E ⋅ C[0] + 0B ⋅ C[1] + 0D ⋅ C[2] + 09 ⋅ C[3]
Note that these numbers 0E, 0B, 0D, and 09 are hexdecimal numbers, therefore these numbers have bit definitions for GF(2^8) polynomials. Therefore we must compute these in GF(2^8) meaning we need
to follow addition and multiplication rules in GF(2^8). We went over that in Introduction to Galois Fields for AES. So if you are interested in learning that make sure to read that.
To invert the ShiftRows layer operation from the encryption it is actually quite simple. We must simply shift the rows of the state matrix in the opposite direction. For instance, again the first
row is not changed, and the rest of the rows instead of shifting right we shift left cylindrically.
In the Inv ShiftRow layer we shift each row cylindrically, meaning data rotates over. The rows are shifted by these rules:
1. First row is not changed
2. Second row is shifted 3 times to the left
3. Third row is shifted 2 times to the left
4. Fourth row is shifted 1 time to the left
Here is a diagram to better show how this works:
For the Inv ByteSub round we cannot reuse the S-Box used in encryption, we must use the inverse S-Box. But since the AES S-Box is a bijective, which means there is a one-to-one mapping, it is
possible to construct the inverse S-Box in such a way that:
A[i] = S^-1 (B[i]) = S^-1 (S(A[i]))
Where A and B are elements of the state matrix.
Just like in encryption for software implementations of AES, it is common that the inverse S-Box is a lookup-table, and is read in the same way as the non-inverse S-Box table. The inverse S-Box
table can be seen below:
I mentioned this before, but if you remember how we read the S-Box, we read the inverse S-Box the same way but I will re-state that here. Recall that the input to the inverse S-Box is 1 byte= 8
bits. We use the first 4 bits for the row, and the last 4 bits for the column.
Check the example below to see how we do this:
Let B[i] = 7A[16] (This is in hexadecimal)
We take the ‘7[16]‘ to find the row and the ‘A[16]‘ for the column. So we get this:
S^-1(B[i] = 7A) = A[i ]= BD[16] = 1011 1101[2].
The next section will cover how we mathematically comput this inverse S-Box table. This is not necessary to understand the functionality of AES. So the next section can be skipped if you wish.
But I would suggest reading it if you want to implement AES in hardware as computing the values are usually faster than using the lookup table above.
Mathematical Description of the inverse S-Box
A[i] 0 1 2 3 4 5 6 7 8 9 A B C D E F
0 52 09 6a d5 30 36 a5 38 bf 40 a3 9e 81 f3 d7 fb
1 7c e3 39 82 9b 2f ff 87 34 8e 43 44 c4 de e9 cb
2 54 7b 94 32 a6 c2 23 3d ee 4c 95 0b 42 fa c3 4e
3 08 2e a1 66 28 d9 24 b2 76 5b a2 49 6d 8b d1 25
4 72 f8 f6 64 86 68 98 16 d4 a4 5c cc 5d 65 b6 92
5 6c 70 48 50 fd ed b9 da 5e 15 46 57 a7 8d 9d 84
6 90 d8 ab 00 8c bc d3 0a f7 e4 58 05 b8 b3 45 06
7 d0 2c 1e 8f ca 3f 0f 02 c1 af bd 03 01 13 8a 6b
8 3a 91 11 41 4f 67 dc ea 97 f2 cf ce f0 b4 e6 73
9 96 ac 74 22 e7 ad 35 85 e2 f9 37 e8 1c 75 df 6e
A 47 f1 1a 71 1d 29 c5 89 6f b7 62 0e aa 18 be 1b
B fc 56 3e 4b c6 d2 79 20 9a db c0 fe 78 cd 5a f4
C 1f dd a8 33 88 07 c7 31 b1 12 10 59 27 80 ec 5f
D 60 51 7f a9 19 b5 4a 0d 2d e5 7a 9f 93 c9 9c ef
E a0 e0 3b 4d ae 2a f5 b0 c8 eb bb 3c 83 53 99 61
F 17 2b 04 7e ba 77 d6 26 e1 69 14 63 55 21 0c 7d
This section will involve advanced algebra topics like Galoid Fields, if you are unfamiliar with the topic then I suggest first reading my post: Introduction to Galoid Fields for AES. With this
post, you should be able to understand the following description below.
To compute the inverse S-Box we must do something similar to what we have been doing to the rest of the layers, and that is to reverse and/or invert the function. That is no different here. We
first need to apply the inverse affine transformation on each B[i] byte, then we need to apply the reversed Galois field inverse.
This can be viewed below:
To reverse the S-Box substitution we first need to compute the inverse of the affine transformation. To do this, we take each input byte B[i] (which is an element of GF(2^8)). Then apply the
inverse affine transformation on each byte. Below is the inverse affine transformation on each byte B[i].
Note that (b[7], …, b[0]) is the bitwise vector of the B[i](x) Byte, and (b’[7] , …, b’[0]) is the result after the inverse affine transformation.
The second step to inverse the S-Box operation is to reverse the Galois Field inverse. Note that A[i] = (A[i]^-1)^-1. This means that all we need to do to reverse the Galoid Field Inverse is to
compute the inverse again. This we have to compute:
A[i] = (B’[i])^-1 ∈ GF(2^8)
Which again is reduced by our AES polynomial P(x) = x^8 + x^4 + x^3 + x + 1. The zero-element is mapped to itself. Finally the result is the vector A = (a[7], …, a[0]), and can be describe as:
A[i] = S^-1(B[i]).
This has not been an intensive deep dive into the mathematical description of the inverse S-Box computation, but hopefully, this is a starting point for you to know where to start for your
studies or implementations.
Since the first round of decryption uses the last round key, the second round uses the second to last round key, and so on. We need the round keys in reversed order. Therefore a good way to
achieve this is to compute the entire key schedule first, storing all the round keys, and then reversing the order of the key schedule. This would add a small time cost to the algorithm.
Here is my code implementation of AES-128 in C++ this is done for educational purposes, and most ciphers are best suited for hardware. If you want to clone the repository or want to see more of
my implementations you can check out my GitHub here.
By: Hunter Richardson
Date: 9/5/2023
For the purpose of education
#include <iostream>
#include <iomanip>
#include "helper.h"
#include "KeySchedule.h"
#include "Encryption.h"
#include "Decryption.h"
using namespace std;
int main()
Encryption encrypt;
Decryption decrypt;
KeySchedule key;
cout << endl;
#include "KeySchedule.h"
void KeySchedule::run()
int round = 0;
while (round < 10)
if (round == 0)
for (int x = 0; x < 4; x++)
for (int y = 0; y < 4; y++)
roundKey[x][y][0] = key[x][y];
unsigned char temp[4] = { 0x00, 0x00, 0x00, 0x00 };
if (round == 0)
temp[0] = roundKey[0][3][0];
temp[1] = roundKey[1][3][0];
temp[2] = roundKey[2][3][0];
temp[3] = roundKey[3][3][0];
temp[0] = roundKey[0][3][round];
temp[1] = roundKey[1][3][round];
temp[2] = roundKey[2][3][round];
temp[3] = roundKey[3][3][round];
rotWord(temp, round);
subWord(temp, round);
rcon(temp, round);
for (int x = 0; x < 4; x++)
roundKey[x][0][round] = temp[x] ^ roundKey[x][0][round - 1];
for (int y = 1; y < 4; y++)
for (int x = 0; x < 4; x++)
roundKey[x][y][round] = roundKey[x][y][round - 1] ^ roundKey[x][y - 1][round];
void KeySchedule::rotWord(unsigned char temp[4], int round)
unsigned char temp2 = temp[0];
for (int x = 0; x < 4; x++)
temp[x] = temp[x + 1];
temp[3] = temp2;
void KeySchedule::subWord(unsigned char temp[4], int round)
for (int x = 0; x < 4; x++)
temp[x] = help.sBox[temp[x]];
void KeySchedule::rcon(unsigned char temp[4], int round)
temp[0] = temp[0] ^ RCon[round];
#pragma once
#include "helper.h"
class KeySchedule
unsigned char roundKey[4][4][11];
void run();
void rotWord(unsigned char temp[4], int round);
void subWord(unsigned char temp[4], int round);
void rcon(unsigned char temp[4], int round);
Help help;
unsigned char key[4][4] = { {0x10, 0xd7, 0x74, 0xfb},
{0xa5, 0x4b, 0xcf, 0x47},
{0x88, 0xe5, 0x86, 0x38},
{0x69, 0xa3, 0x7c, 0x59} };
unsigned char RCon[10] = { 0x01, 0x02, 0x04, 0x08, 0x10, 0x20, 0x40, 0x80, 0x1B, 0x36 };
#include <iostream>
#include <iomanip>
#include "Encryption.h"
using namespace std;
void Encryption::run()
int round = 0;
for (int x = 0; x < 4; x++)
for (int y = 0; y < 4; y++)
cipherText[x][y] = cipherText[x][y] ^ key.roundKey[x][y][0];
while (round < 10)
void Encryption::subByte()
for (int y = 0; y < 4; y++)
for (int x = 0; x < 4; x++)
cipherText[x][y] = help.sBox[cipherText[x][y]];
void Encryption::shiftRows()
int shift = 1;
for (int x = 1; x < 4; x++)
int index = 1;
while (index <= shift)
unsigned char first = cipherText[x][0];
for (int y = 0; y < 3; y++)
cipherText[x][y] = cipherText[x][y + 1];
cipherText[x][3] = first;
void Encryption::mixColumns()
unsigned char temp[4] = { 0x00,0x00,0x00,0x00 };
unsigned char newState[4][4] = { {0,0,0,0},
{0,0,0,0} };
int rijndaelMatric[4][4] = { {2,3,1,1},
{3,1,1,2} };
for (int y = 0; y < 4; y++)
int z = 0;
for (int i = 0; i < 4; i++)
temp[i] = 0x00;
for (int x = 0; x < 4; x++)
for (int j = 0; j < 4; j++)
unsigned char constant = 0x00;
unsigned char element = 0x00;
switch (rijndaelMatric[z][j])
case 1:
temp[j] = cipherText[j][y];
case 2:
element = cipherText[j][y];
temp[j] = multiplyBy2(element);
case 3:
element = cipherText[j][y];
temp[j] = multiplyBy2(element) ^ element;
newState[x][y] = int(temp[0]) ^ int(temp[1]) ^ int(temp[2]) ^ int(temp[3]);
for (int x = 0; x < 4; x++)
for (int y = 0; y < 4; y++)
cipherText[x][y] = newState[x][y];
void Encryption::addRoundKey(int round)
for (int x = 0; x < 4; x++)
for (int y = 0; y < 4; y++)
cipherText[x][y] = cipherText[x][y] ^ key.roundKey[x][y][round];
void Encryption::initializeCipherText()
switch (debug)
case 0:
case 1:
for (int x = 0; x < 4; x++)
for (int y = 0; y < 4; y++)
cipherText[x][y] = 0x00;
#pragma once
#include "KeySchedule.h"
#include "helper.h"
class Encryption
unsigned char cipherText[4][4];
void run();
KeySchedule key;
Help help;
bool debug = 1; // BOOL DEBUG ACTIVE
void subByte();
void shiftRows();
void mixColumns();
void addRoundKey(int round);
void initializeCipherText();
#include <iostream>
#include <iomanip>
#include "Decryption.h"
using namespace std;
void Decryption::run()
int round = 10;
while (round > 0)
void Decryption::invSubByte()
for (int y = 0; y < 4; y++)
for (int x = 0; x < 4; x++)
plainText[x][y] = help.InvsBox[plainText[x][y]];
void Decryption::invShiftRows()
int shift = 1;
for (int x = 1; x < 4; x++)
int index = 1;
while (index <= shift)
unsigned char last = plainText[x][3];
for (int y = 3; y > 0; y--)
plainText[x][y] = plainText[x][y - 1];
plainText[x][0] = last;
void Decryption::invMixColumns()
unsigned char temp[4] = { 0x00,0x00,0x00,0x00 };
unsigned char newState[4][4] = { {0,0,0,0},
{0,0,0,0} };
int rijndaelMatric[4][4] = { {14,11,13,9},
{11,13,9,14} };
for (int y = 0; y < 4; y++)
int z = 0;
for (int i = 0; i < 4; i++)
temp[i] = 0x00;
for (int x = 0; x < 4; x++)
for (int j = 0; j < 4; j++)
unsigned char constant = 0x00;
unsigned char element = 0x00;
switch (rijndaelMatric[z][j])
case 9:
element = plainText[j][y];
temp[j] = (multiplyBy2(multiplyBy2(multiplyBy2(element)))) ^ element;
case 11:
element = plainText[j][y];
temp[j] = (multiplyBy2(multiplyBy2(multiplyBy2(element)) ^ element)) ^ element;
case 13:
element = plainText[j][y];
temp[j] = multiplyBy2(multiplyBy2(multiplyBy2(element) ^ element)) ^ element;
case 14:
element = plainText[j][y];
temp[j] = multiplyBy2((multiplyBy2(multiplyBy2(element) ^ element)) ^ element);
newState[x][y] = int(temp[0]) ^ int(temp[1]) ^ int(temp[2]) ^ int(temp[3]);
for (int x = 0; x < 4; x++)
for (int y = 0; y < 4; y++)
plainText[x][y] = newState[x][y];
void Decryption::invAddRoundKey(int round)
for (int x = 0; x < 4; x++)
for (int y = 0; y < 4; y++)
plainText[x][y] = plainText[x][y] ^ key.roundKey[x][y][round];
void Decryption::initializePlainText()
switch (debug)
case 0:
case 1:
for (int y = 0; y < 4; y++)
for (int x = 0; x < 4; x++)
plainText[x][y] = nist[y][x];
#pragma once
#include "KeySchedule.h"
#include "helper.h"
class Decryption
unsigned char plainText[4][4];
void run();
KeySchedule key;
Help help;
bool debug = 1; // BOOL DEBUG ACTIVE
unsigned char nist[4][4] = { {0x6d, 0x25, 0x1e, 0x69},
{0x44, 0xb0, 0x51, 0xe0},
{0x4e, 0xaa, 0x6f, 0xb4},
{0xdb, 0xf7, 0x84, 0x65} };
void invSubByte();
void invShiftRows();
void invMixColumns();
void invAddRoundKey(int round);
void initializePlainText();
#include <iostream>
#include <iomanip>
#include "helper.h"
using namespace std;
void printResult(unsigned char array[4][4])
for (int x = 0; x < 4; x++)
for (int y = 0; y < 4; y++)
cout << hex << setfill('0') << setw(2) << int(array[y][x]);
unsigned char multiplyBy2(unsigned char element)
unsigned char constant = ((element & 0x80) == 0x80) ? 0x1b : 0x00;
return (element << 1) ^ constant;
#pragma once
class Help
unsigned char sBox[256] = {
0x63, 0x7c, 0x77, 0x7b, 0xf2, 0x6b, 0x6f, 0xc5, 0x30, 0x01, 0x67, 0x2b, 0xfe, 0xd7, 0xab, 0x76,
0xca, 0x82, 0xc9, 0x7d, 0xfa, 0x59, 0x47, 0xf0, 0xad, 0xd4, 0xa2, 0xaf, 0x9c, 0xa4, 0x72, 0xc0,
0xb7, 0xfd, 0x93, 0x26, 0x36, 0x3f, 0xf7, 0xcc, 0x34, 0xa5, 0xe5, 0xf1, 0x71, 0xd8, 0x31, 0x15,
0x04, 0xc7, 0x23, 0xc3, 0x18, 0x96, 0x05, 0x9a, 0x07, 0x12, 0x80, 0xe2, 0xeb, 0x27, 0xb2, 0x75,
0x09, 0x83, 0x2c, 0x1a, 0x1b, 0x6e, 0x5a, 0xa0, 0x52, 0x3b, 0xd6, 0xb3, 0x29, 0xe3, 0x2f, 0x84,
0x53, 0xd1, 0x00, 0xed, 0x20, 0xfc, 0xb1, 0x5b, 0x6a, 0xcb, 0xbe, 0x39, 0x4a, 0x4c, 0x58, 0xcf,
0xd0, 0xef, 0xaa, 0xfb, 0x43, 0x4d, 0x33, 0x85, 0x45, 0xf9, 0x02, 0x7f, 0x50, 0x3c, 0x9f, 0xa8,
0x51, 0xa3, 0x40, 0x8f, 0x92, 0x9d, 0x38, 0xf5, 0xbc, 0xb6, 0xda, 0x21, 0x10, 0xff, 0xf3, 0xd2,
0xcd, 0x0c, 0x13, 0xec, 0x5f, 0x97, 0x44, 0x17, 0xc4, 0xa7, 0x7e, 0x3d, 0x64, 0x5d, 0x19, 0x73,
0x60, 0x81, 0x4f, 0xdc, 0x22, 0x2a, 0x90, 0x88, 0x46, 0xee, 0xb8, 0x14, 0xde, 0x5e, 0x0b, 0xdb,
0xe0, 0x32, 0x3a, 0x0a, 0x49, 0x06, 0x24, 0x5c, 0xc2, 0xd3, 0xac, 0x62, 0x91, 0x95, 0xe4, 0x79,
0xe7, 0xc8, 0x37, 0x6d, 0x8d, 0xd5, 0x4e, 0xa9, 0x6c, 0x56, 0xf4, 0xea, 0x65, 0x7a, 0xae, 0x08,
0xba, 0x78, 0x25, 0x2e, 0x1c, 0xa6, 0xb4, 0xc6, 0xe8, 0xdd, 0x74, 0x1f, 0x4b, 0xbd, 0x8b, 0x8a,
0x70, 0x3e, 0xb5, 0x66, 0x48, 0x03, 0xf6, 0x0e, 0x61, 0x35, 0x57, 0xb9, 0x86, 0xc1, 0x1d, 0x9e,
0xe1, 0xf8, 0x98, 0x11, 0x69, 0xd9, 0x8e, 0x94, 0x9b, 0x1e, 0x87, 0xe9, 0xce, 0x55, 0x28, 0xdf,
0x8c, 0xa1, 0x89, 0x0d, 0xbf, 0xe6, 0x42, 0x68, 0x41, 0x99, 0x2d, 0x0f, 0xb0, 0x54, 0xbb, 0x16
unsigned char InvsBox[256] = {
0x52, 0x09, 0x6A, 0xD5, 0x30, 0x36, 0xA5, 0x38, 0xBF, 0x40, 0xA3, 0x9E, 0x81, 0xF3, 0xD7, 0xFB,
0x7C, 0xE3, 0x39, 0x82, 0x9B, 0x2F, 0xFF, 0x87, 0x34, 0x8E, 0x43, 0x44, 0xC4, 0xDE, 0xE9, 0xCB,
0x54, 0x7B, 0x94, 0x32, 0xA6, 0xC2, 0x23, 0x3D, 0xEE, 0x4C, 0x95, 0x0B, 0x42, 0xFA, 0xC3, 0x4E,
0x08, 0x2E, 0xA1, 0x66, 0x28, 0xD9, 0x24, 0xB2, 0x76, 0x5B, 0xA2, 0x49, 0x6D, 0x8B, 0xD1, 0x25,
0x72, 0xF8, 0xF6, 0x64, 0x86, 0x68, 0x98, 0x16, 0xD4, 0xA4, 0x5C, 0xCC, 0x5D, 0x65, 0xB6, 0x92,
0x6C, 0x70, 0x48, 0x50, 0xFD, 0xED, 0xB9, 0xDA, 0x5E, 0x15, 0x46, 0x57, 0xA7, 0x8D, 0x9D, 0x84,
0x90, 0xD8, 0xAB, 0x00, 0x8C, 0xBC, 0xD3, 0x0A, 0xF7, 0xE4, 0x58, 0x05, 0xB8, 0xB3, 0x45, 0x06,
0xD0, 0x2C, 0x1E, 0x8F, 0xCA, 0x3F, 0x0F, 0x02, 0xC1, 0xAF, 0xBD, 0x03, 0x01, 0x13, 0x8A, 0x6B,
0x3A, 0x91, 0x11, 0x41, 0x4F, 0x67, 0xDC, 0xEA, 0x97, 0xF2, 0xCF, 0xCE, 0xF0, 0xB4, 0xE6, 0x73,
0x96, 0xAC, 0x74, 0x22, 0xE7, 0xAD, 0x35, 0x85, 0xE2, 0xF9, 0x37, 0xE8, 0x1C, 0x75, 0xDF, 0x6E,
0x47, 0xF1, 0x1A, 0x71, 0x1D, 0x29, 0xC5, 0x89, 0x6F, 0xB7, 0x62, 0x0E, 0xAA, 0x18, 0xBE, 0x1B,
0xFC, 0x56, 0x3E, 0x4B, 0xC6, 0xD2, 0x79, 0x20, 0x9A, 0xDB, 0xC0, 0xFE, 0x78, 0xCD, 0x5A, 0xF4,
0x1F, 0xDD, 0xA8, 0x33, 0x88, 0x07, 0xC7, 0x31, 0xB1, 0x12, 0x10, 0x59, 0x27, 0x80, 0xEC, 0x5F,
0x60, 0x51, 0x7F, 0xA9, 0x19, 0xB5, 0x4A, 0x0D, 0x2D, 0xE5, 0x7A, 0x9F, 0x93, 0xC9, 0x9C, 0xEF,
0xA0, 0xE0, 0x3B, 0x4D, 0xAE, 0x2A, 0xF5, 0xB0, 0xC8, 0xEB, 0xBB, 0x3C, 0x83, 0x53, 0x99, 0x61,
0x17, 0x2B, 0x04, 0x7E, 0xBA, 0x77, 0xD6, 0x26, 0xE1, 0x69, 0x14, 0x63, 0x55, 0x21, 0x0C, 0x7D,
void printResult(unsigned char array[4][4]);
unsigned char multiplyBy2(unsigned char element);
The Advanced Encryption Standard (AES) is still one of the most used ciphers in the world due to its security. In fact, it is believed that AES-256 is quantum-resistant, meaning it is not
believed that quantum computers will be able to break AES-256 faster than a classical computer (Recall that AES-256 refers to the AES cipher with a 256-bit key).
To put the security of AES into perspective just using AES-128, which is AES with a 128-bit key. Using my machine and my implementation I can generate ~16 million keys a second (using my cpu),
and I need to test all 2^128 keys, that is 340 undecillion 282 decillion 366 nonillion 920 octillion 938 septillion 463 sextillion 463 quintillion 374 quadrillion 607 trillion 431 billion 768 million
211 thousand 456 keys. Lets say I have a long lost grandparent who gave me their fortune and I bought 1 billion replicas of my machine, then I can generate ~16 Quadrillion keys per second, that is
16,000,000,000,000,000 keys per second. There are ~31,536,000 seconds in a year, thus in a year my billion machines could generate ~504 Sextillion keys per year. If I then ran all billion machines at
once, checking all keys until I tried them all it would still take ~674 Trillion years to find the key.
To put that into perspective, the universe has existed for ~14 Billion years. It would take ~48,164 Thousand times longer than the age of the universe to exhaust the entire key space of just
AES-128. That is not to mention AES-192 which is 64 times larger, or AES-256 which is 128 times larger than AES-128. | {"url":"https://hunterrichardson.net/aes-key-schedule-and-decryption-part-2/","timestamp":"2024-11-08T22:24:32Z","content_type":"text/html","content_length":"160551","record_id":"<urn:uuid:63ce6691-7b4c-4335-82a0-1de3a107a749>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00176.warc.gz"} |
Hard distribution & Easy distribution
< List of probability distributions < Hard distribution & Easy distribution
What is a hard distribution?
A “hard distribution” is so-called because it can be difficult to identify the distribution, or work with components or data points. The opposite is an “easy distribution.”
For example, computer models can be classified as “hard” or “harder” depending on their complexity and how well they model data. If model B is harder than C, that means the best algorithm for B does
worse than the best algorithm for C.
Some algorithms, such as this nearest neighbor algorithm, are easier than others.
Hard distribution and hardness examples
Hardness is a measure of either sampling complexity or the complexity of computing a distribution function:
• Ting et al. define a hard distribution as one that is hard for a DBSCAN density-based clustering algorithm to identify all the clusters in the distribution [1].
• Rochetto et al. [2] describe hard distributions as those that are conjectured to be difficult to sample without the use of quantum computing.
• Markov Chain Monte Carlo (MCMC) can sample from a hard distribution that can’t be explicitly written out; Often we can’t compute simulations for very large distribution because a Markov Chain has
too many states; it could take a computer eons to calculate some of these problems. The workaround is to simulate the Markov Chain for many iterations until a “good” state solution is found [3].
1. Ting et. al. August 2016. Overcoming Key Weaknesses of Distance-based Neighbourhood Methods using a Data Dependent Dissimilarity Measure. Conference: 22nd ACM SIGKDD International Conference on
Knowledge Discovery and Data Mining (KDD ’16). DOI: 10.1145/2939672.2939779
2. Rocchetto, A., Grant, E., Strelchuk, S. et al. Learning hard quantum distributions with variational autoencoders. npj Quantum Inf 4, 28 (2018). https://doi.org/10.1038/s41534-018-0077-z
3. Tsum A. Probability & Statistics with Applications to Computing. Chapter 9: Applications to Computing. 9.6: Markov Chain Monte Carlo (MCMC).
Comments? Need to post a correction? Please Contact Us. | {"url":"https://www.statisticshowto.com/hard-distribution/","timestamp":"2024-11-03T01:46:52Z","content_type":"text/html","content_length":"64840","record_id":"<urn:uuid:cca4487f-2534-4933-8c76-d93895aee4bf>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00420.warc.gz"} |
Oh. Meaning: Understanding the Deeper Significance Behind Everyday ExclamationsOh. Meaning: Understanding the Deeper Significance Behind Everyday Exclamations - English Study Online
Do you ever receive a text message with just the word “Oh.” and wonder what it means? Or have you ever used the word “Oh.” in a conversation and had the other person misinterpret your meaning? The
word “Oh.” is a simple, yet complex expression that can convey a range of emotions and meanings depending on the context and tone. In this article, we will explore the different meanings of “Oh.” and
how to interpret its usage in various situations.
Oh. Meaning
Oh is used as an exclamation, which is a type of interjection that expresses strong emotions or feelings. Exclamations are often used to convey surprise, excitement, or dismay. For example, if you
see a beautiful sunset, you might say “Oh, that’s so beautiful!” to express your admiration. Similarly, if you’re disappointed about something, you might say “Oh no!” to convey your dismay.
• Surprise or astonishment: “Oh, I didn’t expect to see you here!” or “Oh my God, that’s amazing!”
• Disappointment or frustration: “Oh, I forgot my keys at home” or “Oh no, I missed the deadline!”
• Understanding or agreement: “Oh, I see what you mean now” or “Oh, I get it, you’re saying we should wait.”
• Pity or sympathy: “Oh, poor thing, that must have been so hard for you” or “Oh, I’m sorry to hear that.”
• Sarcasm or irony: “Oh, sure, I’ll just magically make it happen” or “Oh, great, another meeting.”
Filler Word
“Oh” can also be used as a filler word, which means it is used to fill a pause in speech or to indicate hesitation or uncertainty. In this context, “oh” doesn’t necessarily have a specific meaning,
but rather serves as a way to keep the conversation flowing or to signal to the listener that the speaker is still thinking or processing information.
For example, someone might say “Oh, let me think for a moment” before answering a question, or “Oh, I’m not sure what to say” when they are caught off guard. In these cases, “oh” is used to buy time
or to acknowledge the other person while the speaker gathers their thoughts.
Oh in Popular Culture
The word “Oh” has been used in many songs throughout the years. It is often used as an exclamation of surprise or disappointment. Some popular songs that use “Oh” in their lyrics include “Oh Darling”
by The Beatles, “Oh Well” by Fleetwood Mac, and “Oh Boy!” by Buddy Holly.
In addition to being used in song lyrics, “Oh” has also been used as the title of several songs. Examples include “Oh” by Ciara, “Oh” by Dave Matthews Band, and “Oh” by Omarion.
Movies and TV Shows
The use of “Oh” in movies and TV shows varies depending on the context. It can be used to express surprise, disappointment, or even sarcasm. One popular example is the catchphrase “Oh my God” used by
Janice in the TV show Friends.
In the movie Home Alone, the character Kevin uses “Oh” several times throughout the movie. For example, when he realizes he is home alone, he says “Oh no!” in a panicked tone.
Another example is the movie Oh, God! starring John Denver and George Burns. In this movie, Burns plays God and uses “Oh” in many of his lines, such as “Oh, you’re going to love this” and “Oh, that’s
a good one.”
Oh in Text Messaging
If you’ve ever received a text message that simply said “Oh.” without any further context, you might be wondering what it means. In text messaging, “Oh.” is often used as a reaction to something that
has been said, indicating a range of emotions from annoyance to surprise or disappointment. The tone of the message can vary greatly depending on the context and the relationship between the sender
and receiver.
For example, if someone sends you a message saying “I’m sorry, I can’t make it to our dinner tonight,” you might respond with “Oh. That’s too bad.” In this context, the “Oh.” indicates disappointment
or sadness at the news. On the other hand, if someone sends you a message saying “I just won the lottery!” you might respond with “Oh. My. God. That’s amazing!” In this case, the “Oh.” indicates
surprise and excitement.
In addition to “Oh.”, there are many other text messaging abbreviations that are commonly used to convey emotions or reactions. For example, “OMG” (Oh My God) is often used to express surprise or
disbelief, while “LOL” (Laugh Out Loud) is used to indicate that something is funny. Other common abbreviations include “SMH” (Shaking My Head), “WTF” (What The F***), and “IDK” (I Don’t Know).
It’s important to keep in mind that text messaging abbreviations can be confusing or even offensive if you’re not familiar with them. If you’re unsure about the meaning of a particular abbreviation,
it’s always a good idea to ask the sender for clarification. By taking the time to understand these common abbreviations, you can communicate more effectively and avoid any misunderstandings in your
text conversations.
Oh in Social Media
Oh. is a reaction that is commonly used on social media platforms such as Twitter, Facebook, and Instagram. It is a versatile expression that can convey a range of emotions, from annoyance to
surprise to disappointment. Oh. is often used in response to a post or message that is unexpected or catches the reader off guard.
In some cases, Oh. is used in conjunction with other words to create a more emphatic expression. For example, Oh. My. God. is a common phrase used to express shock or disbelief. The use of periods
between each word adds emphasis and drama to the expression.
When used in social media, Oh. can also be accompanied by other abbreviations and acronyms. For example, OH is sometimes used as an abbreviation for “overheard,” indicating that the user is sharing
something they heard someone else say.
Psychological Aspects of Oh
When it comes to the psychological aspects of the word “Oh,” it can be interpreted in many different ways. Depending on the context, tone, and situation, the meaning of “Oh” can range from surprise,
pain, excitement, or even disappointment. Here are a few examples of how the psychological aspects of “Oh” can be interpreted:
• Surprise: When you see something unexpected, you might say “Oh!” to express your surprise. This can be a positive or negative surprise, depending on the situation. For example, if someone
surprises you with a gift, you might say “Oh, thank you!” On the other hand, if you see a spider crawling across your floor, you might say “Oh, no!” in a surprised and alarmed tone.
• Pain: If you stub your toe or get hurt, you might say “Oh!” to express your pain. This is a natural response to physical discomfort and can be accompanied by a grimace or other facial expression.
• Disappointment: When something doesn’t go as planned or you receive bad news, you might say “Oh” in a disappointed or resigned tone. For example, if you find out that your favorite restaurant is
closed, you might say “Oh, bummer.”
• Excitement: In some cases, “Oh” can also be used to express excitement or anticipation. For example, if you’re about to go on a rollercoaster, you might say “Oh, this is going to be fun!” in an
excited tone.
Oh in Different Languages
Oh is a common interjection used to express various emotions such as surprise, disappointment, or frustration. Interestingly, this simple word is used in different languages around the world. Here
are some translations of Oh in different languages:
• Afrikaans: o
• Albanian: oh
• Amharic: ወይ
• Arabic: يا
• Armenian: օ
• Azerbaijani: oh
• Basque: ai
• Belarusian: ой
• Bengali: উহু
• Bosnian: oh
• Bulgarian: о
• Catalan: oh
• Cebuano: oh
• Chinese (Simplified): 哦
• Chinese (Traditional): 哦
• Corsican: oh
• Croatian: Oh
• Czech: ó
As you can see, Oh is spelled and pronounced differently across different languages. However, the meaning and usage remain the same – as an interjection to express a range of emotions.
In some cultures, Oh is used more frequently than in others. For example, in Spanish-speaking countries, the word “ay” is often used instead of Oh to express a similar range of emotions. Similarly,
in Japanese, the interjection “ああ” (aa) is often used instead of Oh.
Frequently Asked Questions
What are some common slang meanings of ‘oh’?
The slang meaning of ‘oh’ is often used to express surprise, excitement, or disbelief. For example, if someone tells you some exciting news, you might respond with “Oh, really?” or “Oh, wow!”
Additionally, ‘oh’ can be used to express disappointment or frustration, such as when something doesn’t go as planned.
What is the medical significance of ‘oh’?
In the medical field, ‘oh’ is used as an abbreviation for “oxygen-hemoglobin,” which refers to the amount of oxygen that is bound to hemoglobin in the blood. This measurement is used to determine the
oxygen-carrying capacity of the blood and can be an important indicator of respiratory function.
How is ‘oh’ used in chemistry?
In chemistry, ‘oh’ is often used to represent the hydroxyl group, which is a functional group consisting of an oxygen atom and a hydrogen atom. The hydroxyl group is found in many organic compounds,
including alcohols and carboxylic acids.
What is the meaning of ‘oh’ in text messages?
In text messages, ‘oh’ is often used in a similar way to its slang meaning, to express surprise, excitement, or disappointment. For example, if a friend sends you a funny meme, you might respond with
“Oh my god, that’s hilarious!”
What are some alternative ways to express ‘oh’ in chat?
There are many alternative ways to express ‘oh’ in chat, including using emojis such as 😲 or 😮, or using words such as “wow” or “whoa.” Additionally, you could use a longer phrase such as “Oh my
goodness” or “Oh no!”
What does ‘oh oh’ mean?
When ‘oh’ is repeated twice, such as in “oh oh,” it can indicate a sense of alarm or warning. For example, if you hear a loud noise outside your house, you might say “Oh oh, what was that?” to
express your concern.
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The method of fundamental solutions for solving Poisson problems
Alves, Carlos J. S.; Chen, C. S.; Sarler, B.
Boundary elements XXIV: Incorporating meshless solutions. (Editor: Brebbia, C. A.). WIT Press, (2002), 67-76
Traditionally the method of fundamental solutions (MFS) is used to approximate solution of linear homogeneous equations. For nonhomogeneous problems, one needs to couple other numerical schemes, such
as domain integration, polynomial or radial basis functions interpolation, to evaluate particular solutions.
In this paper we propose to unify the MFS as a numerical method for directly approximating homogeneous solution and particular solution in a similar manner. The major advantage of such approach is
that the particular solution can be easily obtained and evaluated. The numerical results show that such approach can be highly accurate. | {"url":"https://cemat.tecnico.ulisboa.pt/document.php?project_id=4&member_id=78&doc_id=1974","timestamp":"2024-11-02T14:41:43Z","content_type":"text/html","content_length":"8727","record_id":"<urn:uuid:db6aa4f4-3b26-4728-aa71-9dcc9382c2ab>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00243.warc.gz"} |
13.1: The Plot of the Story (5 minutes)
This warm-up reinforces students’ understanding about the relationship between the mean absolute deviation (MAD) and the spread of data. In the given scenarios, the number of people attending the two
events and their mean age are the same, but the MADs are different. In the first question, students interpret these measures in the context of the situations. In the second, they draw a dot plot that
could represent an age distribution with the same mean and yet another MAD.
As students work and discuss, identify several students who drew dot plots that correctly meet the criteria in the second question. Ask students with different dot plots to share during whole-class
Students may need more time to make sense of how to generate their own dot plot for the second question. If it is not possible to give students additional time, consider presenting the second
question at a different time.
Arrange students in groups of 2. Give students 1 minute of quiet think time for the first question, and then 2–3 minutes to work on the second question with a partner. Display the following questions
for all to see. Ask students to think about and discuss them before drawing their dot plots:
• “How many data points should be on the dot plot?”
• “How would the mean help us place the data points?”
• “How would the MAD help us place the data points?”
• “How should our dot plot compare to the dot plots of data sets A and B?”
Student Facing
1. Here are two dot plots and two stories. Match each story with a dot plot that could represent it. Be prepared to explain your reasoning.
□ Twenty people—high school students, teachers, and invited guests—attended a rehearsal for a high school musical. The mean age was 38.5 years and the MAD was 16.5 years.
□ High school soccer team practice is usually watched by supporters of the players. One evening, twenty people watched the team practice. The mean age was 38.5 years and the MAD was 12.7 years.
2. Another evening, twenty people watched the soccer team practice. The mean age was similar to that from the first evening, but the MAD was greater (about 20 years).
Make a dot plot that could illustrate the distribution of ages in this story.
Activity Synthesis
Poll students on their response to the first question. Ask a student to explain how they matched one context to its dot plot and another student to explain the second matching context and dot plot.
Record and display their responses for all to see. If possible, record their responses directly on the dot plots.
Ask selected students to share their dot plots for the second question and their reasoning. To involve more students in the conversation, consider asking some of the following questions:
• “What was the first piece of information you used to draw your dot plot? Why?”
• “How did you decide where to place your dots?”
• “How is your dot plot the same or different than the first evening of soccer practice?”
• “Do you agree or disagree with this representation of the context? Why?”
• “Do you have any questions to ask the student who drew the dot plot?”
13.2: Siblings in the House (15 minutes)
The aim of this activity is to expose the limits of the mean in summarizing a data set that has gaps and values far from the center, and to motivate a need to have another measure of center.
Students first use a table of values and a dot plot to estimate a “typical” value for a data set. Then, they calculate the mean and notice that it does not match their estimate of a typical value. A
closer look helps them see that when a data set contains values that are far away from the bulk of the data, or when there are gaps in the data set, the mean can be a little or a lot higher or lower
than what we would consider typical for the data.
In the next activity, the median will be introduced as another measure of center of a data set.
Arrange students in groups of 2. Give students 7–8 minutes of quiet work time and then 3–4 minutes to discuss their responses with a partner. If there are disagreements, ask them to discuss them
until they reach agreement. Follow with a whole-class discussion.
Representation: Internalize Comprehension. Activate or supply background knowledge about determining means of data sets. Allow students to use calculators to ensure inclusive participation in the
Supports accessibility for: Memory; Conceptual processing
Student Facing
Here is data that shows the numbers of siblings of ten students in Tyler’s class.
1. Represent the data shown with a dot plot.
2. Without making any calculations, estimate the center of the data based on your dot plot. What is a typical number of siblings for these sixth-grade students? Mark the location of that number on
your dot plot.
3. Find the mean. Show your reasoning.
1. How does the mean compare to the value that you marked on the dot plot as a typical number of siblings? (Is it a little larger, a lot larger, exactly the same, a little smaller, or a lot
smaller than your estimate?)
2. Do you think the mean summarizes the data set well? Explain your reasoning.
Student Facing
Are you ready for more?
Invent a data set with a mean that is significantly lower than what you would consider a typical value for the data set.
Anticipated Misconceptions
Since previous lessons have used the mean as the best way to find a typical value, some students may go directly to that method from the beginning. Although this is valid at this stage, encourage
them to look at the dot plot and think about what a typical value should be.
Activity Synthesis
Select a few students to share their estimate for a typical number of siblings. Consider asking students:
• “When you looked at the table of values, what was your sense of a typical number of siblings for the ten students in Tyler's class?”
• “When you looked at the dot plot, did your estimate change?”
Then, discuss how the calculated mean compared to their estimates. Draw students’ attention to the idea that the mean may not always represent a typical value for a data set. Discuss:
• “We have learned that the mean is a way to measure the center of a distribution. How did your calculated mean compare to your estimate of what was typical for the data set?”
• “Why do you think the mean was higher than your estimate?” (Only two of the points are above the mean of 2.4 and both are quite far above it, and seven points are below 2.4, so the mean might not
paint an accurate picture of what is typical in this situation.)
• “If the mean does not always reflect what is typical in a data set, should we always rely on it as the best way to describe the center? If not, in what other ways might we measure the center of a
data set?”
Explain to students that in the next activity we will look at a different measure of center.
Writing, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine to give students a structured opportunity to refine their response to “Do you think the mean summarizes the data set well?”
Give students time to meet with 2–3 partners, to share and get feedback on their responses. Provide listeners with prompts for feedback that will help their partner strengthen their ideas and clarify
their language. For example, "Can you explain what a typical value should be?” and "How can you expand on the using the mean to find a typical value?” Give students 2–3 minutes to revise their
initial draft based on feedback from their peers. This helps students clarify their reasoning about how the mean may or may not summarize the data well.
Design Principle(s): Optimize output (for generalization); Cultivate conversation
13.3: Finding the Middle (15 minutes)
This activity introduces students to the term median. They learn that the median describes the middle value in an ordered list of data, and that it can capture what we consider typical for the data
in some cases.
Students learn about the median through a kinesthetic activity. They line up in order of the number of letters in their name. Then, those at both ends of the line count off and sit down
simultaneously until one or two people in the middle remain standing. If one person remains standing, that person has the median number of letters. If two people remain standing, the median is the
mean or the average of their two values.
Students then practice identifying the median of other data sets, by analyzing both tables of values and dot plots.
Explain to students that, instead of using the mean, sometimes we use the “middle” value in an ordered list of data set as a measure of center. We call this the median. Guide students through the
• Give each student an index card. Ask them to write their first and last names on the card and record the total number of letters in their name. Display an example for all to see.
• Ask students to stand up, holding their index cards in front of them, and arrange themselves in order based on the number of letters in their name. (Consider asking students to do so without
speaking at all.) Look for the student whose name has the fewest letters and ask him or her to be the left end of the line. Ask the student with the longest full name to be the right end of the
line. Students who have the same number of letters should stand side-by-side.
• Tell students that, to find the median or the middle number, we will count off from both ends at the same time. Ask the students at the two ends of the line say “1” at the same time and sit on
the floor, and the students next to them to say “2” and then sit down, and so on. Have students count off in this fashion until only one or two students are standing.
• If the class has an odd number of students, one student will remain standing. Tell the class that this student’s number is the median. Give this student a sign that says “median” If the class has
an even number of students, two students will remain standing. The median will be the mean or average of their numbers. Ask both students to hold the sign that says “median.” Explain that the
median is also called the “50th percentile,” because half of the data values are the same size or less than it and fall to the left of it on the number line, and half are the same size or greater
than it and fall to the right.
• Ask students to find the median a couple more times by changing the data set (e.g., asking a few students to leave the line or adding new people who are not members of the class with extremely
long or short names). Make sure that students have a chance to work with both odd and even numbers of values.
• Collect the index cards and save them; they will be used again in the lesson on box plots.
Ask students to complete the rest of the questions on the task statement.
Representation: Develop Language and Symbols. Create a display of important terms and vocabulary. Include the following term and maintain the display for reference throughout the unit: median. Invite
students to suggest language or diagrams to include on the display that will support their understanding of this term.
Supports accessibility for: Memory; Language
Student Facing
1. Your teacher will give you an index card. Write your first and last names on the card. Then record the total number of letters in your name. After that, pause for additional instructions from
your teacher.
2. Here is the data set on numbers of siblings from an earlier activity.
1. Sort the data from least to greatest, and then find the median.
2. In this situation, do you think the median is a good measure of a typical number of siblings for this group? Explain your reasoning.
3. Here is the dot plot showing the travel time, in minutes, of Elena’s bus rides to school.
1. Find the median travel time. Be prepared to explain your reasoning.
2. What does the median tell us in this context?
Anticipated Misconceptions
When determining the median, students might group multiple data points that have the same value and treat it as a single point, instead of counting each one separately. Remind them that when they
lined up to find the median number of letters in their names, every student counted off, even their name has the same number of letters as their neighbors.
Activity Synthesis
Select a few students to share their responses to the questions about number of siblings and Elena's travel times. Focus the discussion on the median as another measure of the center of a data set
and whether it captures what students would estimate to be a typical value for each data set.
Emphasize to students that the median is a value and not an individual. For example, if the last person standing in the class has 5 letters in their first name, the median is the number 5 and not the
person standing. If there is another student who had 5 letters in their name, they might have switched places with the last person standing when lining up initially. Although the person standing
changed, the median remains the same value of 5.
At this point, it is unnecessary to compare the mean and the median. Students will have many more opportunities to explore the median and think about how it differs from the mean in the upcoming
Writing, Conversing: MLR3 Clarify, Critique, Correct. Before students share their responses to the questions about number of siblings and Elena's travel times, present an incorrect response. For
example, display the statement: “There is no median for this data set because two students remained standing.” Demonstrate the process of interpreting the statements to uncover possible errors by
thinking aloud. Voice the questions you ask yourself to call students' attention to the possible types of errors. Invite students to work with a partner to clarify the reasoning, and create an
improved statement. Select 2–3 groups to share with the whole class. This helps students evaluate, and improve upon the written mathematical arguments of others.
Design Principle(s): Cultivate conversation; Maximize meta-awarenes
Lesson Synthesis
In this lesson, we learn about another measure of center called the median. The discussion should focus on what the median is, how to find it, and why it is useful.
• “What is the median?” (The number in the middle of an ordered list of data.)
• “How can we find it?” (We order the data values from least to greatest and find the value in the middle.)
• “Is the median always one of the values in the data set? If not, when is it not?” (No. When the number of values in a data set is even, there will be two middle values. The median is the number
exactly between them which may not be a value in the data set.)
• “What does the median tell you about a data set? Why is it used as a measure of the center of a distribution?” (It tells us where to divide a data set so that half of the data points have that
value or smaller values and the other half have that value or larger.)
• “Why do we need another measure of center other than the mean?” (Sometimes the mean is not a good indication of what is typical for the data set.)
13.4: Cool-down - Practicing the Piano (5 minutes)
Student Facing
The median is another measure of center of a distribution. It is the middle value in a data set when values are listed in order. Half of the values in a data set are less than or equal to the median,
and half of the values are greater than or equal to the median.
To find the median, we order the data values from least to greatest and find the number in the middle.
Suppose we have 5 dogs whose weights, in pounds, are shown in the table. The median weight for this group of dogs is 32 pounds because three dogs weigh less than or equal to 32 pounds and three dogs
weigh greater than or equal to 32 pounds.
Now suppose we have 6 cats whose weights, in pounds, are as shown in the table. Notice that there are two values in the middle: 7 and 8.
The median weight must be between 7 and 8 pounds, because half of the cats weigh less or equal to 7 pounds and half of the cats weigh greater than or equal to 8 pounds.
In general, when we have an even number of values, we take the number exactly in between the two middle values. In this case, the median cat weight is 7.5 pounds because \((7+8)\div 2=7.5\). | {"url":"https://curriculum.illustrativemathematics.org/MS/teachers/1/8/13/index.html","timestamp":"2024-11-11T08:09:02Z","content_type":"text/html","content_length":"108282","record_id":"<urn:uuid:15fe3cee-2661-40a5-bff9-f015317b59e3>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00124.warc.gz"} |
Filters, amplifiers, and control systems are usually characterized by their frequency response functions. These functions are usually shown in graphical form as plots of log amplitude vs. log
frequency called Bode plots. Oscilloscopes are primarily time domain measuring instruments. They represent acquired waveforms as a time series, plotting signal amplitude as a function of time.
Utilizing the mathematical capabilities available in modern digital oscilloscopes it is possible to derive the frequency response function of a circuit based on the measured time response to a step
Note that the rise time of the step must be fast enough to insure a flat input spectrum. You can use the same method without the device under test to check the flatness of the frequency response of
the source and oscilloscope.
An example of this measurement and analysis is shown in Figure 1.
A 10 kHz square wave is applied to a low pass filter and the output of the filter is acquired and displayed in the top trace (Ch 1). The frequency response function is the Fourier transform of the
circuit’s impulse response. The impulse response can be derived from the measured step response by differentiating the step response. This step is performed in math trace F1 in Figure 1.
The Fast Fourier Transform (FFT) is used to convert the impulse response into the frequency response function. Trace F2 applies the FFT to trace F1. Using the dual operator capability of the math
traces the FFT Average function is also computed in trace F2 and provides averaging in the frequency domain for improvement in dynamic range.
Trace F2, is the frequency response function shown as a plot of log amplitude (power spectrum) vs. linear frequency. Zoom trace, Z2, is used to expand the vertical scale of the frequency response
plot to 1 dB per division. Relative time cursors have been setup to measure the 3 dB point of the low pass filter as 36 MHz.
This data can be converted into a classic Bode plot by saving the frequency spectrum in spreadsheet format and plotting it in Log-Log format using a spreadsheet, such as Microsoft Excel.
Figure 2 shows the data from trace F2 in Figure 1, re-plotted as a Bode plot in Log – Log format using an Excel spreadsheet.
Measuring the frequency response based on the step response is a quick method to check on the response of a device using the oscilloscope. | {"url":"https://www.teledynelecroy.com/doc/frequency-response-measurements","timestamp":"2024-11-06T04:15:56Z","content_type":"text/html","content_length":"21591","record_id":"<urn:uuid:53fbb1f7-ffc2-457b-bacd-b05b7f225a67>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00446.warc.gz"} |
Question #b941c | Socratic
Question #b941c
1 Answer
We know that the initial velocity is $0$$\frac{m}{s}$.Time taken to touch the ground is $5$$\sec$. The acceleration due to gravity is $9.8$$\frac{m}{s}$. For easier calculation, lets assume it to be
We use the kinematic equation
$s$ is the distance travelled (in metres), $u$ is the initial velocity, $a$ is the acceleration and $t$ is the time taken
Plugin the values
$\rightarrow s = u t + \frac{1}{2} a {t}^{2}$
$\rightarrow s = \left(0\right) \left(5\right) + \frac{1}{2} \cdot 10 \cdot {5}^{2}$
$\rightarrow s = \frac{1}{\cancel{2}} ^ 1 \cdot {\cancel{10}}^{5} \cdot 25$
$\rightarrow s = 5 \cdot 25$
Hope that helps!!!☺○☻
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The Deligne-Mumford locus
Lemma 101.22.1. Let $\mathcal{X}$ be an algebraic stack. There exist open substacks
\[ \mathcal{X}'' \subset \mathcal{X}' \subset \mathcal{X} \]
such that $\mathcal{X}''$ is DM, $\mathcal{X}'$ is quasi-DM, and such that these are the largest open substacks with these properties.
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Discrete-time running-sum integral
Approximates the integral of a discrete-time signal using the rectangle method. The integral is defined as:
$y[n] = T/K * \sum_{k=0}^n x[k]$ where T is the sample period and K is an optional gain parameter.
The discrete integral is similar to the continuous time integral. For example, if the input is a sine wave at 1 kHz, then the output will be a cosine at 1 kHz scaled by -1/(2*pi*1000).
The hidden internal array .cumSum stores the previous value of the running sum of x[n] between blocks. The length of the array is set by the prebuild function to the number of channels.
Type Definition
typedef struct _ModuleIntegral
ModuleInstanceDescriptor instance; // Common Audio Weaver module instance structure
FLOAT32 gain; // Additional gain.
FLOAT32* cumSum; // Running sum of input samples since last reset.
} ModuleIntegralClass;
Name Type Usage isHidden Default value Range Units
gain float parameter 0 1 -10:10 linear
cumSum float* state 1 [1 x 1] Unrestricted
Input Pins
Name: in
Description: Input signal
Data type: float
Channel range: Unrestricted
Block size range: Unrestricted
Sample rate range: Unrestricted
Complex support: Real
Output Pins
Name: out
Description: Output signal
Data type: float
MATLAB Usage
File Name: integral_module.m
This module computes the running-sum approximation of the integral of
the input signal. Mathematically, this is:
y[n] = dt/K * sum(x[0] .. x[n])
where dt is the time step, dt = 1/SR and K is a gain. The module has a
multichannel input and computes the integral per channel. Arguments:
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Calibration in machine learning - AI Standards Hub
Blog post by: Professor Mark Levene, Principal Scientist at NPL
Calibration is an important concept in metrology, the science of measurement. It involves the comparison of measured values delivered by a device with those of a measurement standard of generally
greater accuracy. To account for variability in the measurements, for example, caused by random effects, calibration often involves taking repeated measurements and constructing a confidence interval
to express the uncertainty in the calibration result. A device is regarded as calibrated when this comparison has been carried out and a document containing the expected values from the standard, the
measured values from the device, and the associated uncertainty (a ā calibration certificateā ) has been issued.
It is advantageous to describe the variability in a measurement by a probability distribution, which is a mathematical function that gives the probability of different possible values of a random
variable. Probability distributions are often visualised using the probability density function, a curve showing how often a given value of a variable is likely to occur relative to other values.
These functions are normalised so that the area under curve is equal to 1. The most common distribution is the normal distribution, also known as the bell curve due to its shape. Examples of
approximately normally distributed random variables are the height, weight, and heart rate of a human.
Why am I going to all the trouble of telling you this? Well, the parameters associated with a measurement, such as its variability, are derived from a probability distribution, and more generally a
probability distribution provides us with the most complete description of a measurement result.
Now to our main topic of machine learning (ML).Ā To solve a problem, an ML algorithm produces a representation of the input data set in the form of a statistical model, which can be used for either
classification or regression tasks to aid decision making.Ā As an example, letā s make use of our old friend, the weather. Two common questions we would like answers to, within a given time frame,
are: (i) will it rain or not (a classification problem with two classes), and (ii) what will the temperature be (a regression problem with a range of possible temperature values).
Calibration in ML thus comes in two flavours depending on whether the task at hand is one of classification or regression. First, consider a classification problem involving prediction of whether it
will rain or not. Here the output of the ML algorithm is more refined than simply telling us whether it will rain or not, which is too difficult a task to answer accurately due to the uncertainty
involved. Rather, the ML algorithm outputs a probability that it will rain, in the more user-friendly format as a percentage, which is the probability multiplied by 100; this output is called the
predicted probability. It is then up to us to interpret this probability when deciding whether we should grab an umbrella when going out.
Recall that a measurement result is incomplete without an associated uncertainty. As a simplification of how the Meteorological Office may arrive at the probability that it will rain, we assume that
a complex model of the weather is repeatedly run with slightly different initial conditions. Each simulation result will output yes for “it will rainā ā or no for “it will not rainā ā , and the
probability reported is the proportion of yes outputs; this probability is called the empirical probability. The data set used for this calculation is called the validation data set. The predicted
probability can then be calculated from the inputs of the validation set using the statistical model constructed by the ML algorithm. So, an ML classifier outputting a probability of rain as the
prediction is said to be calibrated if, whenever the ML model outputs a predicted probability, this equals the empirical probability as described above. That is, if we plot the empirical probability
values versus the predicted probability values, then calibration implies that they will fall on the diagonal line or close enough to it. Such a plot is called a reliability diagram, which will help
us ascertain how calibrated the probabilities really are.
Second, we have a regression problem, where the output is a temperature value. Producing an accurate single value, say T, for the temperature is too difficult in practice. So, instead let us consider
the output to be the probability, say P, that the temperature will be less than or equal to T; this is the predicted probability for a regression problem. Now, to get the empirical probability P we
can evaluate the weather model, as before, where each simulation will output a temperature, and count the proportion of times the output was less or equal to a temperature T. As before the data used
for this calculation is the validation set.
To complete the explanation for a regression problem we still need to know how given a probability, say P, we can obtain the temperature value, T, such that the probability that the temperature is
less than or equal to T is P. To do this we need to introduce one more term, the quantile of a probability distribution expressed as a percentage, say Pā Æ%, which is our notation for P times 100.
The quantile is the value T of the temperature, such that exactly Pā Æ% of the distribution is less than or equal to T. Now, as with classification, we can plot a reliability diagram of the empirical
probabilities against the predicted probabilities and expect the plotted values to fall close to the diagonal line when the probabilities are calibrated.
We have covered quite a few statistical concepts that are needed to understand calibration in ML. The key takeaway is that both in classical metrology and ML, calibration needs to deal with
uncertainty. In the case of classical metrology, the empirical probabilities emanate from repeated measurements, and are compared to a measurement standard. On the other hand, in ML the empirical
probability is derived from the outputs of a validation data set and is compared to the predicted probability output from the ML model with the inputs coming from the validation set. In this sense
the empirical probability, which we assume derives from Meteorological Office data, acts as a reference probability for the ML calibration process. | {"url":"https://aistandardshub.org/calibration-in-machine-learning","timestamp":"2024-11-04T09:01:09Z","content_type":"text/html","content_length":"252629","record_id":"<urn:uuid:99a25414-a338-41c8-bd3d-1e683bbb84d3>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00143.warc.gz"} |
How to Multiply all elements in a List in Python? - Data Science Parichay
Multiplying all the elements in a list is a common task in Python programming. In this tutorial, we will explore different methods to multiply all the elements in a list in Python. We will cover both
simple and more advanced techniques, including using loops, recursion, and the reduce() function. By the end of this tutorial, you will have a solid understanding of how to multiply all the elements
in a list in Python and be able to apply this knowledge to your own projects.
A refresher on lists in Python
Before we proceed with the tutorial, here’s a short refresher on lists in Python. If you’re comfortable with lists please feel free to skip to the next section.
Lists are a fundamental data structure in Python that allow you to store a collection of items in a single variable. They are created using square brackets and can contain any type of data, including
other lists.
Here’s an example of creating a list:
my_list = [1, 2, 3, "four", 5.0]
You can access individual items in a list using their index, which starts at 0. For example, to access the first item in the list above (which is 1), you would use:
You can also modify items in a list by assigning a new value to their index. For example, to change the second item in the list above (which is 2) to a string “two”, you would use:
my_list[1] = "two"
You can add new items to a list using the append() method, which adds the item to the end of the list. For example, to add the number 6 to the end of the list above, you would use:
You can also remove items from a list using the remove() method, which removes the first occurrence of the specified item. For example, to remove the string “four” from the list above, you would use:
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These are just a few of the basic operations you can perform on lists in Python. Lists are a powerful and versatile data structure that are used extensively in Python programming.
Methods to multiply all the list elements in Python
Let’s now look at some of the different ways in which we can multiply all the elements in a list with the help of some examples.
1) Using a loop
This is a straightforward method in which we iterate through each value in the list and maintain a product of all the values encountered.
Let’s look at an example.
# create a list
ls = [1, 2, 3, 4, 5]
# variable to store the final product
result = 1
# iterate through the list
for num in ls:
result *= num
We get the product of all the values in the above list as 120.
2) Using recursion
In this method, the idea is the same as the first method but instead of a loop, here we are using recursion to calculate the product of all the values in a list.
# create a list
ls = [1, 2, 3, 4, 5]
# recursive function to calculate product of all list elements
def get_product(ls, i):
# base case
if i >= len(ls):
return 1
return ls[i] * get_product(ls, i+1)
# call the recursive function
result = get_product(ls, 0)
We get the same result as above.
3) Using the reduce() function
The reduce() function in Python is used to apply a function to an iterable and reduce it to a single cumulative value. It takes two arguments: the function to be applied and the iterable to be
reduced. The function is applied cumulatively to the items of the iterable from left to right, so as to reduce the iterable to a single value.
We can use the reduce() function to apply a simple multiplication function to a list to get the product of all the values in the list. Here’s an example.
from functools import reduce
# Define a function to be applied
def multiply(x, y):
return x * y
# Apply the function to an iterable using reduce()
result = reduce(multiply, [1, 2, 3, 4, 5])
In this example, the multiply() function is applied cumulatively to the list [1, 2, 3, 4, 5] using reduce(), resulting in a single value of 120.
4) Using the numpy.prod() function
You can also use the prod() function available in the numpy library to calculate the product of all the values in a list. Simply pass the list as an argument and it will return the product of the
list values.
import numpy as np
# create a list
ls = [1, 2, 3, 4, 5]
# calculate product of its values
result = np.prod(ls)
In this code, we first import the numpy library as np. We then pass the list ls to the np.prod() function, which returns the product of all the elements in the list. Finally, we print the result.
In conclusion, we have explored four different methods for multiplying all the elements in a list in Python.
Firstly, we looked at using a for loop to iterate through each element in the list and multiply them together. This method is simple and easy to understand, but it can be time-consuming for large
Next, we explored using recursion to multiply all the elements in a list. This method is elegant and concise, but it can be memory-intensive for large lists.
We then looked at using the reduce() function from the functools module. This method is efficient and concise, and it can handle large lists with ease. However, it requires importing an external
Finally, we explored using the numpy.prod() function from the NumPy library. This method is efficient and concise, and it can handle large lists with ease. However, it also requires importing an
external library.
Overall, each method has its own advantages and disadvantages, and the choice of which method to use will depend on the specific requirements of your project. By understanding these different
methods, you can choose the one that best suits your needs and efficiently multiply all the elements in a list in Python.
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I'm not quite sure what you mean about the rolling bodies, but don't forget the floor will take an asymmetrical force pushing it in one direction.
What's stopping you doing a decisive experiment? Some equipment you don't have?
What do you mean asymmetrical forces? Rolling resistance force does not depend on velocity. Just from mass, radius, material. In my case, radius, mass and material are equivalent. The reaction
force for both bodies must be identical, because they repulse from each other. The spring has identical forces on both sides. All these symmetric forces are covering by Newtons third law.
OP I don't have a high precision equipment and vacuum room for clear experiment. I don't think my experiment is unique case for nature. The rolling bodies on surface with same mass and different
moment of inertia is pretty similar case. It would be easy to reproduce. However, even on kinematic equation I see different translational velocities for rolling bodies relatively to repulsing
point. What is it? Mistake or paradox?
Originally Posted By: ABV
What do you mean asymmetrical forces? Rolling
The roller with the higher rotational inertia will transmit a greater force to the floor through its friction. So the floor will be pushed in that direction.
If you calculate the total linear momentum of the system, including the floor (and earth), it'll be zero. Same for angular momentum.
High precision equipment won't help because you haven't made a quantified, testable hypothesis (that I know of). If you do it in an evacuated room with frictionless parts and high speed cameras, and
the results are consistent with computer simulations, then what will that tell you? Maybe the effect you hope for is too small to detect with that equipment. So it's a probably-pointless experiment.
Last edited by kallog; 10/05/10 11:30 AM.
Originally Posted By: kallog
The roller with the higher rotational inertia will transmit a greater force to the floor through its friction. So the floor will be pushed in that direction.
Here is rolling resistance force explanation
No any dependencies from velocity.
Originally Posted By: kallog
If you calculate the total linear momentum of the system, including the floor (and earth), it'll be zero. Same for angular momentum.
I used normal kinematic equation from physics book. Great example of this is a rolling body along incline.
I don't see any errors on my kinematic equation yet. I would appreciate if you find one.
Originally Posted By: kallog
High precision equipment won't help because you haven't made a quantified, testable hypothesis (that I know of). If you do it in an evacuated room with frictionless parts and high speed cameras,
and the results are consistent with computer simulations, then what will that tell you? Maybe the effect you hope for is too small to detect with that equipment. So it's a probably-pointless
I made hypothesis about standalone natural phenomenon as rotational and translational motion.
I don't think effect is too small. I think, the main problem is simplification of physics calculation.
Ok. Back to model for rolling bodies.
This model is not equivalent to experiment on my site, because rolling bodies contact to surface all the time and both have a rotation. The rolling body with higher moment of inertial will have
OP bigger reaction force to the surface. Which should be compensated by spring through surface. However the surface has it's own mass and will keep reaction forces for rolling bodies difference.
Unfortunately I have to disregard my model from my site. However, I'm still working for exact model for my experiment and will publish it in a future.
Originally Posted By: ABV
I made hypothesis about standalone natural phenomenon as rotational and translational motion.
I don't think effect is too small. I think, the main problem is simplification of physics calculation.
Maybe you haven't expressed it in the way you're thinking of it. But I don't think the hypothesis "The Rotational and Translational motion is standalone natural phenomenon." is testable. You should
specify quantitatively what result of the experiment would support it, and what result would disprove it. That'll tell you how much accuracy is required. It may show that a home-made experiment is
adequate, or it may show that a fancy vacuum-room experiment is still not enough.
However it sounds like you're not expecting any results to differ from the classical predictions. In that case no experiment can be of any use, other than to confirm the existing theory. But again,
how accurately do you want to confirm it? If you get too accurate you'll run into relativistic or quantum effects which disprove the classical theory, so in a way you're forced to do it only
Last edited by kallog; 10/06/10 03:19 AM.
This site shows the physics problem.
Two wheels locate on weightless platforms. These platforms repulsing from each other through weightless spring. These wheels start conduct rotational and translational motion without friction on
it's own platforms. These wheels don't have a rolling resistance also.
Using force Fs1 spring repulse platform with wheel 1 with translational acceleration a3 relatively to repulsing point. Wheel 1 with mass m, moment inertia I1 and radius R has translational
OP acceleration a1 relatively to repulsing point and angular acceleration alpha 1.
Using force Fs2 spring repulse platform with wheel 2 with translational acceleration a4 relatively to repulsing point. Wheel 2 with mass m, moment inertia I2 and radius R has translational
acceleration a2 relatively to repulsing point and angular acceleration alpha 2.
These wheels have same mass and radius.
Find out kinematic equations of motions for wheel 1 and wheel 2. Compare theirs translational and angular accelerations.
I assume you mean there is non-slipping friction between the platform and wheel 2? Like a lossless rack and pinion.
Find out kinematic equations of motions for wheel 1 and wheel
2. Compare theirs translational and angular accelerations.
Two issues:
1. I don't like this equation:
Fs2 = Fl + Fr
It suggests that only part of Fs2 contributes to translational motion, but in fact all of it does. However I might be misunderstanding your meaning.
2. You seem to have neglected the angular acceleration of object 1. To balance angular momentums you have to sum all the angular momentums in the system, taken about the same axis.
Translational motion of wheel 1:
Fs1 = m * a1
Translational motion of wheel 2:
Fs2 = m * a2
Rotational motion of wheel 2 about its center:
Moment = I * angular acceleration:
Fs2 * R = I * alpha
Rotational motion of wheel 1 about the same point. +ve is clockwise:
Here I_1 means rotational inertia of object 1 about the center of object 2 for the type of motion it has, which includes no rotation of itself, so it's own 'I' must be ignored: I_1 = m * R^2
Moment = I_1 * angular acceleration:
Fs1 * R = I_1 * alpha_1
Fs1 * R = m * R^2 * alpha_1
What's alpha_1? Depends on the linear acceleration:
alpha_1 = a1 / R
Fs1 * R = m * R^2 * a1 / R
Fs1 = m * a1
This agrees with the linear motion equation, so no problem.
Linear momentums easily cancel out.
If m is very high compared to I, then object 2 will spin up fast and get some angular momentum (high alpha, low I).
The objects will drift apart slowly because of their large masses. Object 1's angular momentum about the center of object 2 is also the same (low alpha, high I). I is high because it's proportional
to the high m. Alpha is low because it's proportional to the low linear velocity.
So the two angular momentums can cancel out.
Last edited by kallog; 10/07/10 05:03 AM.
Thank you for answer with equations. Let's look closelly on it.
Originally Posted By: kallog
1. I don't like this equation:
Fs2 = Fl + Fr
It suggests that only part of Fs2 contributes to translational motion, but in fact all of it does. However I might be misunderstanding your meaning.
Why? Please look on standard problem - a rolling body on incline.
There you'll see 2 forces
Originally Posted By: kallog
2. You seem to have neglected the angular acceleration of object 1. To balance angular momentums you have to sum all the angular momentums in the system, taken about the same axis.
I don't need to do it, because it's simple wheel, which has standard equation. Kind of flywheel.
Originally Posted By: kallog
Translational motion of wheel 1:
Fs1 = m * a1
Translational motion of wheel 2:
Fs2 = m * a2
I disagree, because just part Fs is using for translational motion. It would be
OP Fs1_part1 = m * a1
Fs2_part1 = m * a2
Originally Posted By: kallog
Rotational motion of wheel 2 about its center:
Moment = I * angular acceleration:
Fs2 * R = I * alpha
Same thing.
Fs1_part2 * R = I1 * alpha (wheel 1)
Fs2_part2 * R = I2 * alpha (wheel 2)
Originally Posted By: kallog
Rotational motion of wheel 1 about the same point. +ve is clockwise:
Here I_1 means rotational inertia of object 1 about the center of object 2 for the type of motion it has, which includes no rotation of itself, so it's own 'I' must be ignored: I_1 = m * R^2
I'm not follow this suggestion. However, nothing should be ignored. Extra rotation around center mass of isolated system may be compensated by gyroscope or symmetrical process.
I would divide this problem to two simple ones. If look perspective from platforms, their accelerations can be described like virtual gravity. Objects on these platforms will start experience kind
of gravity force. After that easy to see these objects will have simple kinematics equations, which was described before.
Here's my solution:
It's good question about center of mass. I'll think about it. Thank you.
Last edited by ABV; 10/08/10 03:55 PM.
Here is no problem with center mass. System won't move till any forces are applying to the system. For example if something initially move inside isolated system(empty sphere for example) without
OP applied forces then system change own center of mass but system is not moving till object hit the wall of isolated system (for example). So, I don't see any problems with that.
It's quite a different problem. However the entire force of gravity is applied to the roller, then there are reaction forces pushing it other ways too.
I treat your roller on the platform as having a horizontal force applied to its center (equal to spring force), and additionally a moment applied about its center. That's fine. You can do that. Who
says you can't?
I don't need to do it, because it's simple wheel, which has standard equation. Kind of flywheel.
You need it if you're summing angular momentums to apply the law of conservation of momentum. When you do that you have to include the angular momentum of every part, all measured about the _same_
axis. The simple flywheel formula only applies to angular momentum about the wheel's own axis. Even tho it's not rotating it still has angular momentum about the other wheel's center, and it has a
different rotational inertia about that axis too.
I think this is a crucial part which needs to be incorporated. You can't just ignore the angular momentum of wheel 1 because it's not rotating.
I disagree, because just part Fs is using for translational motion. It would be
Imagine you break the connection between wheel 2 and the platform. Then install a lever fixed to the center of the wheel, and its other end is pin-jointed to the platform. From this it's obvious
that the entire force is transmitted to the center of the wheel - there's nowhere else it can go. It also shows there's an additional moment applied about the center of the wheel.
Hmm I kind of got a bit lost, sorry.
Originally Posted By: kallog
It's quite a different problem. However the entire force of gravity is applied to the roller, then there are reaction forces pushing it other ways too.
This is the same classics mechanics kinematic problem. The difference is gravity force substituted by force which come from platform acceleration. The solution is very close to rolling body along
Originally Posted By: kallog
I treat your roller on the platform as having a horizontal force applied to its center (equal to spring force), and additionally a moment applied about its center. That's fine. You can do that.
Who says you can't?
The spring doesn't push rolling object to theirs center of mass. Therefore the force of spring should no be fully spend to object translational motion. The spring force push platforms with rolling
objects. Here's different case.
Originally Posted By: kallog
You need it if you're summing angular momentums to apply the law of conservation of momentum. When you do that you have to include the angular momentum of every part, all measured about the _same_
axis. The simple flywheel formula only applies to angular momentum about the wheel's own axis. Even tho it's not rotating it still has angular momentum about the other wheel's center, and it has a
different rotational inertia about that axis too.
Please look on solution of problem a rolling body along incline. There is just operate with forces which should equate to projection of gravity force. One oh them is translation motion force.
Another came from torque with radius multiplication.
OP Originally Posted By: kallog
I think this is a crucial part which needs to be incorporated. You can't just ignore the angular momentum of wheel 1 because it's not rotating.
What do you mean is not rotating? The spring is connecting to platforms. These platforms have acceleration and wheel is free moving on them. Both wheels have reverse motion relatively to
Originally Posted By: kallog
Imagine you break the connection between wheel 2 and the platform. Then install a lever fixed to the center of the wheel, and its other end is pin-jointed to the platform. From this it's obvious
that the entire force is transmitted to the center of the wheel - there's nowhere else it can go. It also shows there's an additional moment applied about the center of the wheel.
You're correct with your model. However, the wheels have a free move on platforms. It means part of spring force with spend to translational motion of rolling object. Another part of force will
spend to rotate this object which equate to torque and radius of object multiplication.
Originally Posted By: kallog
Hmm I kind of got a bit lost, sorry.
It's fine
Thank you for you meaningful answers. I know a lot of forums where opponents know physic but don't understand that. This forum is complete different. Thank you again.
Last edited by ABV; 10/12/10 01:47 PM.
A few words about frames of reference.
Newton's Laws hold only with respect to a certain set of frames of reference called Newtonian or inertial reference frames.[1]
The first Newton's law is: "Every body remains in a state of rest or uniform motion (constant velocity) unless it is acted upon by an external unbalanced force. This means that in the absence of a
non-zeronet force, the center of mass of a body either remains at rest, or moves at a constant speed in a straight line.[1]
However Newton's laws don't deny free move center of mass of isolated system. Second and third laws describes bodies forces interaction. Otherwise, bodies knows nothing about each other and center
of mass of isolated system is meaningless without bodies forces interaction. Base on symmetric bodies forces during interaction, the center of mass of isolated system should hold same position.
It's true for simple motions, where force has simple meaning. However, for rotational and translational motion force can have two components from simple motions. In this case, net force may
achieve same value by different components variation. For example 3+4=7 and 4+3=7. Where first number is translational force component and second number is rotational angular force by radius
OP projection component. Therefore, center of mass of system for bodies forces interaction in rotational and translational motion can move. Otherwise, bodies during interaction should get additional
extra forces from nowhere which will help to hold center of mass of isolated system on same position. Energy for these additional extra forces should come from nowhere too. Unfortunately, the
modern classical mechanics equalize holding same position of center of mass of isolated system with symmetric forces for any cases of bodies forces interaction, because rotational and
translational motion is a product of sum of two simple motions.
This solution will follow free move center of mass of isolated system for single standalone rotational and translational motion, because no strong description about it in Newton's laws. This
solution won't include any additional extra forces to helping to hold center of mass of isolated system on same position.
Originally Posted By: ABV
However Newton's laws don't deny free move center of mass of isolated system.
Perhaps, but the linear momentum conservation law does deny that possibility, even when there's rotation.
radius projection component. Therefore, center of mass of system for bodies forces interaction in rotational and translational motion can move. Otherwise, bodies during
Now you're talking about reactionless propulsion. This is certainly impossible, partly because many people have tried many times, and all completely failed to show any result. And partly because it
leads to paradoxes.
Originally Posted By: kallog
Now you're talking about reactionless propulsion. This is certainly impossible, partly because many people have tried many times, and all completely failed to show any result. And partly because
it leads to paradoxes.
Here's nothing about reactionless propulsion. I'm trying to explain where modern physic got mistake.
If repulse rotated and non-rotated objects with same mass then base on modern physics these objects will have same translational velocity after repulsing action. Because rotational and
translational motion is a sum of two simple motion. Whole force during repulsing should create a translational motion for rotated and non-rotated objects. Base on this force induct internal forces
inside rotated object which will bring it's rotation during repulsing action. How come, Huh? Internal forces for rotation. From where this force? Where energy for this force came from? This is
modern classical mechanics now. You don't believe me? You could ask any physics scientist
My opinion, I don't believe to any magical internal forces inside rotated object. Otherwise, object should loose temperature, because internal forces are getting energy from object. The nature is
doing simple thing. The repulse force inside rotated objects split for two forces for two motions of this object. Therefore net of these forces for two motions is equal to force which is applying
to non -rotated object during repulsing action(Third Newton's law). However it's impossible for modern physics now, because rotational and translational motion is sum of simple two motion for
modern classical mechanic now which each of these motions must execute it's own law of momentum conservation.
OP My solution is postulate rotation and translational motion as standalone motion with it's own law of momentum conservation.
The model of rotated object on weightless platform fully describes rotated object during repulsing object.
Therefore, the classical mechanic has just one generic rotational and translational motion. Simple translational and rotational motions just a trivial cases of one main motion only. Sir Newton
described both trivial cases of one main motion which simplified understanding about motion. However,excluding main rotational and translational motion brings mistake on nature motion description.
I hope my experiment reproduction will prove it.
I hope it helps.
I don't think, I'm a first one who see this paradox. A few reasons could be don't discover it now. You can imagine
Last edited by ABV; 10/16/10 05:27 PM.
I think, I found more simplest solution for this rotational and translational motion explanation.
Rotational and translational motion description should include rule:
Base on modern classical mechanic where rotational and translational motion is a product of sum of two simple motions as rotational motion and translational motion, each of these simple motions
must have own force which induct this kind of motion.
OP Then F=F1+F2. where F - full force, F1,F2 - forces for inducting simple motions.
Each of these motion will follow it's own law of momentum conservation where product of sum of these momentum will equal to full momentum which applied to this object.
Then P=P1+P2. where P - full momentum, P1,P2 - momentums for inducting simple motions.
After that translational motion will follow F1 and P1 and rotational motion will follow F2*R and P2*R.
That's it.
I understood what was happened. Modern physics use static model for forces which applied to an object. This is the same model if calculate loads in civil engineering.
If you this logic for physics problem "a rolling body along incline" then object has two reaction forces. In static mode each of this forces equal to gravity force projection on incline. However,
solution of this problem use dynamic model where rotational and translational motion is a sum of two motion and sum of forces which conducted certain type of motion equal to net force(i.e. gravity
OP force projection on incline)
The physics problem for rotational and translational motion in free move doesn't use dynamic model and substitute physics process by static model where torque as add-on(not part of it) for force
in linear direction. This is mistake. The model should be the same as modern physics use for rolling objects on surface
Rotational and translational motion can be described as a sum of 2 simple motions, which conducted in two separate events.
Then for translational motion equation is:
Work for this type of motion which equal objects kinetic energy translational part is
Base on third Newtons law symmetric force -F_1 is present for this event
Same for rotational motion with fixed axis equation is
Work for this type of motion which equal objects kinetic energy rotational part is
Base on third Newtons law symmetric force -F_2 is present for this event
Base on sum of two motions the full kinetic energy which is work of 2 forces is equal:
Therefore sum of two symmetric forces for each simple motion necessary to initiate rotational and translational motion:
However, the modern classical mechanics says if force applied away from center mass of object then just force [tex]F_1[/tex] with symmetrical force -F_1 is enough to initiate rotational and
translational motion with energy:
How this part of force do this part of work Iw^2/2 - mystery.
This paradox base on suggestion what law of momentum conservation works always and adding symmetric force -F_2 is crashing this law.
If use
little rule where each motion law of momentum conservation should adequate force which induct this certain type of motion and all law of momentum conservation works locally
, then not necessary hide force -F_2.
Center mass of isolated system is not the same as center of mass of objects and doesn't follow same rule.
Dark matter is not a mystery anymore. Just a miscalculation http://knol.google.com/k/paradox-of-classical-mechanics-2http://knol.google.com/k/alex-belov/the-wheels/1xmqm1l0s4ys/18
The third Newton’s law and law of momentum conservation.
The third Newton’s law declares what each reaction of forces should be symmetrical.
where: F - reaction forces of objects
The consequence of this is law of momentum conservation.
OP -mv*t=mv*t
where: F - reaction forces of objects, m - mass of objects, v - velocity of objects, t - time trame of action, P - momentums of objects.
This consequence should be used for case where objects induce same identical motions and this does not cover the case where objects induce different type of motions. In experiment 2 thin cylinders
induce different type of motions and this simple consequence should not cover this objects repulcing action. Therefore, for correct explanation of experiment 2 needs to use prime third Newton’s
law and identify all reaction forces by this law. The consequence law of momentum conservation where objects have symmetrical behavior should not be used for this objects repulsing case.
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Keeping Time in Distributed Systems
Advanced Course, 2+2
Lectures: Friday, 10:15 - 12:00, E1.4 024
Lecturer: Christoph Lenzen
First lecture: 26.04.2019
Tutorials: Tuesday, 12:15-14:00, E1.4 023
Assistant: Will Rosenbaum
First 30.04.2019
Credits: 6
Exam: There will be oral exams at the end of the semenster.
Prerequisites: No prerequisites beyond basic familiarity with mathematical reasoning are required; prior knowledge on asymptotic notation and (occasionally) standard probabilistic notions can be
useful, but is not essential for following the course.
Mailing List: Please subscribe to our mailing list to enroll in the course.
In this course, we discuss how to maintain accurate synchronization in distributed systems. Essentially, this encompasses any system in which keeping a well-synchronized common notion of time is
crucial, as clock synchronization is an inherently distributed task. For instance, the presented techniques are suitable for clocking computer chips or larger networks on chips, but may equally well
be employed on a larger scale, like data centers or a global network. The focus of the lecture lies on a conceptual understanding of algorithmic techniques and proving worst-case guarantees
mathematically. Particular emphasis is given to strong, possibly surprising, fault-tolerance properties and how they can be achieved.
No prerequisites beyond basic familiarity with mathematical reasoning are assumed or required for this course. This course is a good starting point for getting involved with the current research
topics of the group.
Grades for he course will computed as follows:
• Homework (25%). There is a homework assignment corresponding to every lecture (except for the final lecture). Assignments are due on Friday by 17:00 one week after the corresponding lecture was
given. Assignments may be handed in during lecture, emailed, or given directly to Will or Christoph. Students may work in small groups (up to 4 people), and each group may submit a single
assignment. You are encouraged to discuss homework problems, but the written work you submit must be your own.
• Participation (25%). Attendence at the weekly discussion session is mandatory. During the discussion sessions, students will present a brief (10-15 minute) recap of the previous lecture, and
solutions to previous homework problems. Please volunteer in advance if you want to present a particular lecture/homework problem.
• Oral Final Exam (50%). Final evaluation will be based on an oral final exam. Details to be announced.
• On the week of 30 April, the lecture and discussion session will be swapped. That is, Lecture 2 will occur on Tuesday, 30 April, 12:15–14:00 in room 024, and the discussion session will be on
Friday, 3 May, 10:15–12:00 also in room 024. Note that homework assignment 2 will be due on Friday, 10 April. | {"url":"https://www.mpi-inf.mpg.de/departments/algorithms-complexity/teaching/summer19/ktds","timestamp":"2024-11-04T14:26:14Z","content_type":"text/html","content_length":"90598","record_id":"<urn:uuid:69131eb9-a079-4f89-b2a3-43e32a9ac2ca>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00125.warc.gz"} |
Mesure the Focal Length of a Diverging Lens | FreebookSummary
Mesure the Focal Length of a Diverging Lens
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Place in front of trie lens a piece of black paper with two narrow slits A, A’ cut parallel to each other at a known distance apart, and let light which is quite or nearly parallel fall on the lens
(fig. 28). Two bright patches will be formed on a screen at a, a’, by the light passing through the two slits, and the rays forming them will be in the same directions as if they came from the
principal focus F of the lens.
If then we measure a a’ and CX, and if CF = f we havefrom which f can be found The distance, between the centres of the bright patches can be measured with a pair of compasses and a finely divided
scale, or by using a scale as the screen on which the light falls. In consequence of the indistinctness of the bright patches, this is only a very rough method of determining the focal length.
Method 2:The second method consists in placing in contact with the given concave lens a convex lens sufficiently powerful to make a combination equivalent to a convex lens. Let the focal length
(numerical) of the concave lens be f, that of the auxiliary convex lens f’, and that of the combination F. ThenThe values of F and f’ can be found by one of the methods described for convex lenses.
In selecting a lens with which to form the combination it should be noticed that, if F and f’ differ only slightly, say by 1 centimetre, an error of 1 millimetre in the determination of each, unless
the errors happen to be in the same direction, will make a difference of one-fifth in the result The auxiliary lens should therefore be chosen to make the difference F-f’ as large as possible – i. e.
the concave lens should with the convex produce a combination nearly equivalent to a lens with parallel faces, so that 1/f may be very nearly equal to 1/f’.
For greater accuracy the light used should be allowed to pass through a plate of coloured glass, so as to render it more nearly homogeneous. Experiment. – Determine by the two methods the focal
length of the given lens. Enter results thus:Lens D. Method 1: Distance between slits: 2. 55 cm Distance between images: 4. 75 cm Distance from lens to screen: 33. 00 cm Focal length: 38. 24 cm
Method 2 Focal length of convex lens: 29. 11 cm Focal length of combination: 116. 14 cm Focal length required: 38. 85 cm
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It is shown that the very frequently used form of the viscous, diabatic shallow-water equations are energetically inconsistent compared to the primitive equations. An energetically consistent form of
the shallow-water equations is then given and justified in terms of isopycnal coordinates. Examples are given of the energetically inconsistent shallow-water equations used in low-order dynamical
systems and simplified coupled models of tropical air–sea interaction and the E1 Niño–Southern Oscillation phenomena.
It is shown that the very frequently used form of the viscous, diabatic shallow-water equations are energetically inconsistent compared to the primitive equations. An energetically consistent form of
the shallow-water equations is then given and justified in terms of isopycnal coordinates. Examples are given of the energetically inconsistent shallow-water equations used in low-order dynamical
systems and simplified coupled models of tropical air–sea interaction and the E1 Niño–Southern Oscillation phenomena.
In this paper the linear equatorial ocean response to stress forcing is analyzed in terms of vertically propagating waves. A new projection onto the meridional eigenfunctions of the pressure equation
is derived for a single Fourier wave component. The projection demonstrates that the solution is regular and not singular at the inertial latitudes, and is more convenient to use than the
corresponding projection onto the meridional velocity equation. The wavenumber spectrum from the resulting forced vertical structure equation is found for four different choices of the vertical
profile for the body force. The spectrum is shown to be insensitive to the particular profile chosen. The projection is then used to study the effects of forcing and linear damping on the vertical
propagation of space-time transformed energy in three wave modes: the Kelvin, first Rossby and mixed Rossby-gravity waves. When the buoyancy frequency is constant, the energy decay is exponential in
depth with the coefficient proportional to the damping magnitude. Finally it is shown that linear damping effects are very different on each vertically propagating or vertically standing wave. Thus,
it is fallacious to make deductions about meridional phase changes in the total solution to a general forced problem from the phase changes of each wave component.
In this paper the linear equatorial ocean response to stress forcing is analyzed in terms of vertically propagating waves. A new projection onto the meridional eigenfunctions of the pressure equation
is derived for a single Fourier wave component. The projection demonstrates that the solution is regular and not singular at the inertial latitudes, and is more convenient to use than the
corresponding projection onto the meridional velocity equation. The wavenumber spectrum from the resulting forced vertical structure equation is found for four different choices of the vertical
profile for the body force. The spectrum is shown to be insensitive to the particular profile chosen. The projection is then used to study the effects of forcing and linear damping on the vertical
propagation of space-time transformed energy in three wave modes: the Kelvin, first Rossby and mixed Rossby-gravity waves. When the buoyancy frequency is constant, the energy decay is exponential in
depth with the coefficient proportional to the damping magnitude. Finally it is shown that linear damping effects are very different on each vertically propagating or vertically standing wave. Thus,
it is fallacious to make deductions about meridional phase changes in the total solution to a general forced problem from the phase changes of each wave component.
A new type of standing equatorial wave mode is described that exists in the semi-infinite ocean 0 ⩽ x ⩽ L, −∞ ⩽ y ⩽ ∞. It consists of a finite sum of the meridionally trapped equatorial waves in an
infinite x domain. The new mode is thus itself equatorially trapped and requires no energy sources or sinks at |y| = ∞. However, it exists only for a discrete, countable set of pairs of values of the
frequency ω and the ocean zonal width L. Previously described standing modes exist for any ocean width, but are infinite sums of trapped equatorial waves and require a continuous energy source in the
west at |y| = ∞ to balance the continuous energy sink in the east at |y| = ∞. Several examples of the new type of standing mode are given, and it is shown that as the standing mode period becomes
very long, so the zonal scale becomes very short. The effect on the standing modes of bounding the basin meridionally is also described; energy is recycled round the basin by boundary-trapped Kelvin
waves along the zonal walls. The amount of energy recycled in the new type of standing mode, however, is exponentially small compared to that recycled in the previously described standing modes.
A new type of standing equatorial wave mode is described that exists in the semi-infinite ocean 0 ⩽ x ⩽ L, −∞ ⩽ y ⩽ ∞. It consists of a finite sum of the meridionally trapped equatorial waves in an
infinite x domain. The new mode is thus itself equatorially trapped and requires no energy sources or sinks at |y| = ∞. However, it exists only for a discrete, countable set of pairs of values of the
frequency ω and the ocean zonal width L. Previously described standing modes exist for any ocean width, but are infinite sums of trapped equatorial waves and require a continuous energy source in the
west at |y| = ∞ to balance the continuous energy sink in the east at |y| = ∞. Several examples of the new type of standing mode are given, and it is shown that as the standing mode period becomes
very long, so the zonal scale becomes very short. The effect on the standing modes of bounding the basin meridionally is also described; energy is recycled round the basin by boundary-trapped Kelvin
waves along the zonal walls. The amount of energy recycled in the new type of standing mode, however, is exponentially small compared to that recycled in the previously described standing modes.
Peter R. Gent
James R. Luyten
Vertically propagating linear wave calculations using realistic equatorial buoyancy profiles are presented which show the percentage of the downward surface energy flux that reaches the deep
equatorial oceans. The percentages vary widely depending upon the buoyancy profile and the equivalent depth but can be as low as 10% on average for equivalent depths between 1 cm and 1 m if the
thermocline is sharp. This means that models with constant or weak thermocline buoyancy profiles, which allow all or most downward surface energy flux to reach the deep ocean, are very unrealistic in
this respect. Another conclusion is that the observed, very low-frequency, small vertical-scale deep jets cannot be explained by linear wave theory as caused by surface forcing. It is also shown that
a WKB analysis of observations can be misleading even if applied to a single vertically propagating wave in a region that excludes the main thermocline. Implications are that comparing estimates of
the equivalent depth from the mixed Rossby-gravity wave dispersion relation and a WKB analysis is of little value because the error bars on both estimates are large, and that WKB estimates of
downward vertical energy flux into the deep ocean can also be misleading.
Vertically propagating linear wave calculations using realistic equatorial buoyancy profiles are presented which show the percentage of the downward surface energy flux that reaches the deep
equatorial oceans. The percentages vary widely depending upon the buoyancy profile and the equivalent depth but can be as low as 10% on average for equivalent depths between 1 cm and 1 m if the
thermocline is sharp. This means that models with constant or weak thermocline buoyancy profiles, which allow all or most downward surface energy flux to reach the deep ocean, are very unrealistic in
this respect. Another conclusion is that the observed, very low-frequency, small vertical-scale deep jets cannot be explained by linear wave theory as caused by surface forcing. It is also shown that
a WKB analysis of observations can be misleading even if applied to a single vertically propagating wave in a region that excludes the main thermocline. Implications are that comparing estimates of
the equivalent depth from the mixed Rossby-gravity wave dispersion relation and a WKB analysis is of little value because the error bars on both estimates are large, and that WKB estimates of
downward vertical energy flux into the deep ocean can also be misleading.
Peter R. Gent
James C. McWilliams
The low-order, nine-component, primitive equation model of Lorenz (1980) is used as the basis for a comparative study of the quality of several intermediate models. All the models are intermediate
between the primitive equations and quasi-geostrophy and will not support gravity-wave oscillations; this reduces to three the number of independent components in each. Strange attractors, stable
limit cycles, and stable and unstable fixed points are found in the models. They are used to make a quantitative intercomparison of model performance as the forcing strength, or equivalently the
Rossby number, is varied. The models can be ranked from best to worst at small Rossby number as follows: the primitive equations, the balance equations, hypogeostrophy, geostrophic momentum
approximation, the linear balance equations, and quasi-geostrophy. At intermediate Rossby number the only change in this ranking is the demotion of hypogeostrophy to the position of worst. Caveats
about the low-order model, and hence the generality of the conclusions, are also discussed.
The low-order, nine-component, primitive equation model of Lorenz (1980) is used as the basis for a comparative study of the quality of several intermediate models. All the models are intermediate
between the primitive equations and quasi-geostrophy and will not support gravity-wave oscillations; this reduces to three the number of independent components in each. Strange attractors, stable
limit cycles, and stable and unstable fixed points are found in the models. They are used to make a quantitative intercomparison of model performance as the forcing strength, or equivalently the
Rossby number, is varied. The models can be ranked from best to worst at small Rossby number as follows: the primitive equations, the balance equations, hypogeostrophy, geostrophic momentum
approximation, the linear balance equations, and quasi-geostrophy. At intermediate Rossby number the only change in this ranking is the demotion of hypogeostrophy to the position of worst. Caveats
about the low-order model, and hence the generality of the conclusions, are also discussed.
James C. McWilliams
Peter R. Gent
Large-scale extratropical motions (with dimensions comparable to, or somewhat smaller than, the planetary radius) in the atmosphere and ocean exhibit a more restricted range of phenomena than are
admissible in the primitive equations for fluid motions, and there have been many previous proposals for simpler, more phenomenologically limited models of these motions. The oldest and most
successful of these is the quasi-geostrophic model. An extensive discussion is made of models intermediate between the quasi-geostrophic and primitive ones, some of which have been previously
proposed [e.g., the balance equations (BE), where tendencies in the equation for the divergent component of velocity are neglected, or the geostrophic momentum approximation (GM), where ageostrophic
accelerations are neglected relative to geostrophic ones] and some of which are derived here. Virtues of these models are assessed in the dual measure of nearly geostrophic momentum balance (i.e.,
small Rossby number) and approximate frontal structure (i.e., larger along-axis velocities and length scales than their cross-axis counterparts), since one or both of these circumstances is usually
characteristic of planetary motions. Consideration is also given to various coordinate transformations, since they can yield simpler expressions for the governing differential equations of the
intermediate models. In particular, a new set of coordinates is proposed, isentropic geostrophic coordinates,(IGC), which has the advantage of making implicit the advections due to ageostrophic
horizontal and vertical velocities under various approximations. A generalization of quasi-geostrophy is made. named hypo-geostrophy (HG), which is an asymptotic approximation of one higher order
accuracy in Rossby number. The governing equations are simplest in IGC for both HG and GM; we name the latter in these coordinates isentropic semi-geostrophy (ISG), in analogy to Hoskins’ (1975)
semi-geostrophy (SG). HG, GM and BE are, in our opinion, the three most valuable intermediate models for future consideration. HG and BE are superior to GM asymptotically in small Rossby number, but
HG in IGC and GM are superior to HG in other coordinates and BE in frontal asymptotics. GM has global (not asymptotic) integral invariants of energy and enstrophy, which HG lacks, and this may assure
physically better solutions in weakly asymptotic situations. BE has one global (energy) and one asymptotic (enstrophy) invariant. BE has difficulties of solution existence and uniqueness. Further
progress in the search for intermediate models requires obtaining an extensive set of solutions for these models for comparison with quasi-geostrophic and primitive equation solutions.
Large-scale extratropical motions (with dimensions comparable to, or somewhat smaller than, the planetary radius) in the atmosphere and ocean exhibit a more restricted range of phenomena than are
admissible in the primitive equations for fluid motions, and there have been many previous proposals for simpler, more phenomenologically limited models of these motions. The oldest and most
successful of these is the quasi-geostrophic model. An extensive discussion is made of models intermediate between the quasi-geostrophic and primitive ones, some of which have been previously
proposed [e.g., the balance equations (BE), where tendencies in the equation for the divergent component of velocity are neglected, or the geostrophic momentum approximation (GM), where ageostrophic
accelerations are neglected relative to geostrophic ones] and some of which are derived here. Virtues of these models are assessed in the dual measure of nearly geostrophic momentum balance (i.e.,
small Rossby number) and approximate frontal structure (i.e., larger along-axis velocities and length scales than their cross-axis counterparts), since one or both of these circumstances is usually
characteristic of planetary motions. Consideration is also given to various coordinate transformations, since they can yield simpler expressions for the governing differential equations of the
intermediate models. In particular, a new set of coordinates is proposed, isentropic geostrophic coordinates,(IGC), which has the advantage of making implicit the advections due to ageostrophic
horizontal and vertical velocities under various approximations. A generalization of quasi-geostrophy is made. named hypo-geostrophy (HG), which is an asymptotic approximation of one higher order
accuracy in Rossby number. The governing equations are simplest in IGC for both HG and GM; we name the latter in these coordinates isentropic semi-geostrophy (ISG), in analogy to Hoskins’ (1975)
semi-geostrophy (SG). HG, GM and BE are, in our opinion, the three most valuable intermediate models for future consideration. HG and BE are superior to GM asymptotically in small Rossby number, but
HG in IGC and GM are superior to HG in other coordinates and BE in frontal asymptotics. GM has global (not asymptotic) integral invariants of energy and enstrophy, which HG lacks, and this may assure
physically better solutions in weakly asymptotic situations. BE has one global (energy) and one asymptotic (enstrophy) invariant. BE has difficulties of solution existence and uniqueness. Further
progress in the search for intermediate models requires obtaining an extensive set of solutions for these models for comparison with quasi-geostrophic and primitive equation solutions.
Gokhan Danabasoglu
Peter R. Gent
The equilibrium climate sensitivity of a climate model is usually defined as the globally averaged equilibrium surface temperature response to a doubling of carbon dioxide. This is virtually always
estimated in a version with a slab model for the upper ocean. The question is whether this estimate is accurate for the full climate model version, which includes a full-depth ocean component. This
question has been answered for the low-resolution version of the Community Climate System Model, version 3 (CCSM3). The answer is that the equilibrium climate sensitivity using the full-depth ocean
model is 0.14°C higher than that using the slab ocean model, which is a small increase. In addition, these sensitivity estimates have a standard deviation of nearly 0.1°C because of interannual
variability. These results indicate that the standard practice of using a slab ocean model does give a good estimate of the equilibrium climate sensitivity of the full CCSM3. Another question
addressed is whether the effective climate sensitivity is an accurate estimate of the equilibrium climate sensitivity. Again the answer is yes, provided that at least 150 yr of data from the doubled
carbon dioxide run are used.
The equilibrium climate sensitivity of a climate model is usually defined as the globally averaged equilibrium surface temperature response to a doubling of carbon dioxide. This is virtually always
estimated in a version with a slab model for the upper ocean. The question is whether this estimate is accurate for the full climate model version, which includes a full-depth ocean component. This
question has been answered for the low-resolution version of the Community Climate System Model, version 3 (CCSM3). The answer is that the equilibrium climate sensitivity using the full-depth ocean
model is 0.14°C higher than that using the slab ocean model, which is a small increase. In addition, these sensitivity estimates have a standard deviation of nearly 0.1°C because of interannual
variability. These results indicate that the standard practice of using a slab ocean model does give a good estimate of the equilibrium climate sensitivity of the full CCSM3. Another question
addressed is whether the effective climate sensitivity is an accurate estimate of the equilibrium climate sensitivity. Again the answer is yes, provided that at least 150 yr of data from the doubled
carbon dioxide run are used.
Peter R. Gent
Joseph J. Tribbia
A model of the tropical ocean and global atmosphere is described. It consists of an aqua-planet form of version one of the NCAR Community Climate Model coupled to a primitive equation model for the
upper tropical ocean in a rectangular basin. A 24-year simulation is described that has almost no climate drift, a good simulation of the mean temperature gradient across the ocean, but smaller than
observed annual and interannual variability. The coupled model is analyzed to see where it occurs on the schematic bifurcation diagram of Neelin. In years 9–16 of the simulation there is a dominant
oscillation with a period of two years. The spatial pattern of this oscillation shows up clearly in the first empirical orthogonal function calculated from monthly averages of sea surface temperature
anomalies. A series of 19 model-twin predictability experiments were carried out with the initial perturbation being a very small change in the ocean temperature field. The correlation coefficient of
monthly sea surface temperature anomalies from these model-twin experiments decreases rapidly over the first 6 months and after that, more slowly, showing that there is some predictability out to a
year. The predictability times are marginally increased if only the coefficient of the first empirical orthogonal function of monthly averaged sea surface temperature anomalies or NIN03 sea surface
temperature is predicted. There is some evidence to indicate that it is easier to predict the onset of a model warm event than to predict the onset of a model cold event. More detailed analysis of
the first model-twin experiment shows that the initial divergence in the integrations is a change at day 6 in the incoming solar radiation due to a change in the atmospheric model clouds. The
dominant early change in sea surface temperature occurs by this change in radiative heat flux. If the cloud feedback is set to zero, then the first changes are delayed to day 12 and occur in the
evaporative and sensible heat fluxes and in the atmospheric wind stress. In this case the dominant early change to sea surface temperature is by advection due to the changed wind stress.
A model of the tropical ocean and global atmosphere is described. It consists of an aqua-planet form of version one of the NCAR Community Climate Model coupled to a primitive equation model for the
upper tropical ocean in a rectangular basin. A 24-year simulation is described that has almost no climate drift, a good simulation of the mean temperature gradient across the ocean, but smaller than
observed annual and interannual variability. The coupled model is analyzed to see where it occurs on the schematic bifurcation diagram of Neelin. In years 9–16 of the simulation there is a dominant
oscillation with a period of two years. The spatial pattern of this oscillation shows up clearly in the first empirical orthogonal function calculated from monthly averages of sea surface temperature
anomalies. A series of 19 model-twin predictability experiments were carried out with the initial perturbation being a very small change in the ocean temperature field. The correlation coefficient of
monthly sea surface temperature anomalies from these model-twin experiments decreases rapidly over the first 6 months and after that, more slowly, showing that there is some predictability out to a
year. The predictability times are marginally increased if only the coefficient of the first empirical orthogonal function of monthly averaged sea surface temperature anomalies or NIN03 sea surface
temperature is predicted. There is some evidence to indicate that it is easier to predict the onset of a model warm event than to predict the onset of a model cold event. More detailed analysis of
the first model-twin experiment shows that the initial divergence in the integrations is a change at day 6 in the incoming solar radiation due to a change in the atmospheric model clouds. The
dominant early change in sea surface temperature occurs by this change in radiative heat flux. If the cloud feedback is set to zero, then the first changes are delayed to day 12 and occur in the
evaporative and sensible heat fluxes and in the atmospheric wind stress. In this case the dominant early change to sea surface temperature is by advection due to the changed wind stress. | {"url":"https://journals.ametsoc.org/search?access=all&f_0=author&pageSize=10&q_0=Peter+R.+Gent&sort=relevance","timestamp":"2024-11-07T00:28:17Z","content_type":"text/html","content_length":"427168","record_id":"<urn:uuid:f0bb53d6-ef3f-40e9-9f5b-184bac4c28ba>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00388.warc.gz"} |
Fit Gaussian kernel regression model using random feature expansion
fitrkernel trains or cross-validates a Gaussian kernel regression model for nonlinear regression. fitrkernel is more practical to use for big data applications that have large training sets, but can
also be applied to smaller data sets that fit in memory.
fitrkernel maps data in a low-dimensional space into a high-dimensional space, then fits a linear model in the high-dimensional space by minimizing the regularized objective function. Obtaining the
linear model in the high-dimensional space is equivalent to applying the Gaussian kernel to the model in the low-dimensional space. Available linear regression models include regularized support
vector machine (SVM) and least-squares regression models.
To train a nonlinear SVM regression model on in-memory data, see fitrsvm.
Mdl = fitrkernel(X,Y) returns a compact Gaussian kernel regression model trained using the predictor data in X and the corresponding responses in Y.
Mdl = fitrkernel(Tbl,ResponseVarName) returns a kernel regression model Mdl trained using the predictor variables contained in the table Tbl and the response values in Tbl.ResponseVarName.
Mdl = fitrkernel(Tbl,formula) returns a kernel regression model trained using the sample data in the table Tbl. The input argument formula is an explanatory model of the response and a subset of
predictor variables in Tbl used to fit Mdl.
Mdl = fitrkernel(Tbl,Y) returns a kernel regression model using the predictor variables in the table Tbl and the response values in vector Y.
Mdl = fitrkernel(___,Name,Value) specifies options using one or more name-value pair arguments in addition to any of the input argument combinations in previous syntaxes. For example, you can
implement least-squares regression, specify the number of dimension of the expanded space, or specify cross-validation options.
[Mdl,FitInfo] = fitrkernel(___) also returns the fit information in the structure array FitInfo using any of the input arguments in the previous syntaxes. You cannot request FitInfo for
cross-validated models.
[Mdl,AggregateOptimizationResults] = fitrkernel(___) also returns AggregateOptimizationResults, which contains hyperparameter optimization results when you specify the OptimizeHyperparameters and
HyperparameterOptimizationOptions name-value arguments. You must also specify the ConstraintType and ConstraintBounds options of HyperparameterOptimizationOptions. You can use this syntax to optimize
on compact model size instead of cross-validation loss, and to perform a set of multiple optimization problems that have the same options but different constraint bounds.
Train Gaussian Kernel Regression Model
Train a kernel regression model for a tall array by using SVM.
When you perform calculations on tall arrays, MATLAB® uses either a parallel pool (default if you have Parallel Computing Toolbox™) or the local MATLAB session. To run the example using the local
MATLAB session when you have Parallel Computing Toolbox, change the global execution environment by using the mapreducer function.
Create a datastore that references the folder location with the data. The data can be contained in a single file, a collection of files, or an entire folder. Treat 'NA' values as missing data so that
datastore replaces them with NaN values. Select a subset of the variables to use. Create a tall table on top of the datastore.
varnames = {'ArrTime','DepTime','ActualElapsedTime'};
ds = datastore('airlinesmall.csv','TreatAsMissing','NA',...
t = tall(ds);
Specify DepTime and ArrTime as the predictor variables (X) and ActualElapsedTime as the response variable (Y). Select the observations for which ArrTime is later than DepTime.
daytime = t.ArrTime>t.DepTime;
Y = t.ActualElapsedTime(daytime); % Response data
X = t{daytime,{'DepTime' 'ArrTime'}}; % Predictor data
Standardize the predictor variables.
Z = zscore(X); % Standardize the data
Train a default Gaussian kernel regression model with the standardized predictors. Extract a fit summary to determine how well the optimization algorithm fits the model to the data.
[Mdl,FitInfo] = fitrkernel(Z,Y)
Found 6 chunks.
| Solver | Iteration / | Objective | Gradient | Beta relative |
| | Data Pass | | magnitude | change |
| INIT | 0 / 1 | 4.307833e+01 | 9.925486e-02 | NaN |
| LBFGS | 0 / 2 | 2.782790e+01 | 7.202403e-03 | 9.891473e-01 |
| LBFGS | 1 / 3 | 2.781351e+01 | 1.806211e-02 | 3.220672e-03 |
| LBFGS | 2 / 4 | 2.777773e+01 | 2.727737e-02 | 9.309939e-03 |
| LBFGS | 3 / 5 | 2.768591e+01 | 2.951422e-02 | 2.833343e-02 |
| LBFGS | 4 / 6 | 2.755857e+01 | 5.124144e-02 | 7.935278e-02 |
| LBFGS | 5 / 7 | 2.738896e+01 | 3.089571e-02 | 4.644920e-02 |
| LBFGS | 6 / 8 | 2.716704e+01 | 2.552696e-02 | 8.596406e-02 |
| LBFGS | 7 / 9 | 2.696409e+01 | 3.088621e-02 | 1.263589e-01 |
| LBFGS | 8 / 10 | 2.676203e+01 | 2.021303e-02 | 1.533927e-01 |
| LBFGS | 9 / 11 | 2.660322e+01 | 1.221361e-02 | 1.351968e-01 |
| LBFGS | 10 / 12 | 2.645504e+01 | 1.486501e-02 | 1.175476e-01 |
| LBFGS | 11 / 13 | 2.631323e+01 | 1.772835e-02 | 1.161909e-01 |
| LBFGS | 12 / 14 | 2.625264e+01 | 5.837906e-02 | 1.422851e-01 |
| LBFGS | 13 / 15 | 2.619281e+01 | 1.294441e-02 | 2.966283e-02 |
| LBFGS | 14 / 16 | 2.618220e+01 | 3.791806e-03 | 9.051274e-03 |
| LBFGS | 15 / 17 | 2.617989e+01 | 3.689255e-03 | 6.364132e-03 |
| LBFGS | 16 / 18 | 2.617426e+01 | 4.200232e-03 | 1.213026e-02 |
| LBFGS | 17 / 19 | 2.615914e+01 | 7.339928e-03 | 2.803348e-02 |
| LBFGS | 18 / 20 | 2.620704e+01 | 2.298098e-02 | 1.749830e-01 |
| Solver | Iteration / | Objective | Gradient | Beta relative |
| | Data Pass | | magnitude | change |
| LBFGS | 18 / 21 | 2.615554e+01 | 1.164689e-02 | 8.580878e-02 |
| LBFGS | 19 / 22 | 2.614367e+01 | 3.395507e-03 | 3.938314e-02 |
| LBFGS | 20 / 23 | 2.614090e+01 | 2.349246e-03 | 1.495049e-02 |
Mdl =
ResponseName: 'Y'
Learner: 'svm'
NumExpansionDimensions: 64
KernelScale: 1
Lambda: 8.5385e-06
BoxConstraint: 1
Epsilon: 5.9303
FitInfo = struct with fields:
Solver: 'LBFGS-tall'
LossFunction: 'epsiloninsensitive'
Lambda: 8.5385e-06
BetaTolerance: 1.0000e-03
GradientTolerance: 1.0000e-05
ObjectiveValue: 26.1409
GradientMagnitude: 0.0023
RelativeChangeInBeta: 0.0150
FitTime: 17.9573
History: [1x1 struct]
Mdl is a RegressionKernel model. To inspect the regression error, you can pass Mdl and the training data or new data to the loss function. Or, you can pass Mdl and new predictor data to the predict
function to predict responses for new observations. You can also pass Mdl and the training data to the resume function to continue training.
FitInfo is a structure array containing optimization information. Use FitInfo to determine whether optimization termination measurements are satisfactory.
For improved accuracy, you can increase the maximum number of optimization iterations ('IterationLimit') and decrease the tolerance values ('BetaTolerance' and 'GradientTolerance') by using the
name-value pair arguments of fitrkernel. Doing so can improve measures like ObjectiveValue and RelativeChangeInBeta in FitInfo. You can also optimize model parameters by using the
'OptimizeHyperparameters' name-value pair argument.
Cross-Validate Kernel Regression Model
Load the carbig data set.
Specify the predictor variables (X) and the response variable (Y).
X = [Acceleration,Cylinders,Displacement,Horsepower,Weight];
Y = MPG;
Delete rows of X and Y where either array has NaN values. Removing rows with NaN values before passing data to fitrkernel can speed up training and reduce memory usage.
R = rmmissing([X Y]); % Data with missing entries removed
X = R(:,1:5);
Y = R(:,end);
Cross-validate a kernel regression model using 5-fold cross-validation. Standardize the predictor variables.
Mdl = fitrkernel(X,Y,'Kfold',5,'Standardize',true)
Mdl =
CrossValidatedModel: 'Kernel'
ResponseName: 'Y'
NumObservations: 392
KFold: 5
Partition: [1x1 cvpartition]
ResponseTransform: 'none'
Mdl is a RegressionPartitionedKernel model. Because fitrkernel implements five-fold cross-validation, Mdl contains five RegressionKernel models that the software trains on training-fold (in-fold)
Examine the cross-validation loss (mean squared error) for each fold.
ans = 5×1
Optimize Kernel Regression
Optimize hyperparameters automatically using the OptimizeHyperparameters name-value argument.
Load the carbig data set.
Specify the predictor variables (X) and the response variable (Y).
X = [Acceleration,Cylinders,Displacement,Horsepower,Weight];
Y = MPG;
Delete rows of X and Y where either array has NaN values. Removing rows with NaN values before passing data to fitrkernel can speed up training and reduce memory usage.
R = rmmissing([X Y]); % Data with missing entries removed
X = R(:,1:5);
Y = R(:,end);
Find hyperparameters that minimize five-fold cross-validation loss by using automatic hyperparameter optimization. Specify OptimizeHyperparameters as 'auto' so that fitrkernel finds the optimal
values of the KernelScale, Lambda, Epsilon, and Standardize name-value arguments. For reproducibility, set the random seed and use the 'expected-improvement-plus' acquisition function.
[Mdl,FitInfo,HyperparameterOptimizationResults] = fitrkernel(X,Y,'OptimizeHyperparameters','auto',...
| Iter | Eval | Objective: | Objective | BestSoFar | BestSoFar | KernelScale | Lambda | Epsilon | Standardize |
| | result | log(1+loss) | runtime | (observed) | (estim.) | | | | |
| 1 | Best | 4.1521 | 0.52193 | 4.1521 | 4.1521 | 11.415 | 0.0017304 | 615.77 | true |
| 2 | Best | 4.1489 | 0.11367 | 4.1489 | 4.1503 | 509.07 | 0.0064454 | 0.048411 | true |
| 3 | Accept | 5.251 | 0.84487 | 4.1489 | 4.1489 | 0.0015621 | 1.8257e-05 | 0.051954 | true |
| 4 | Accept | 4.3329 | 0.10633 | 4.1489 | 4.1489 | 0.0053278 | 2.37 | 17.883 | false |
| 5 | Accept | 4.2414 | 0.25276 | 4.1489 | 4.1489 | 0.004474 | 0.13531 | 14.426 | true |
| 6 | Best | 4.148 | 0.1422 | 4.148 | 4.148 | 0.43562 | 2.5339 | 0.059928 | true |
| 7 | Accept | 4.1521 | 0.27233 | 4.148 | 4.148 | 3.2193 | 0.012683 | 813.56 | false |
| 8 | Best | 3.8438 | 0.12156 | 3.8438 | 3.8439 | 5.7821 | 0.065897 | 2.056 | true |
| 9 | Accept | 4.1305 | 0.23758 | 3.8438 | 3.8439 | 110.96 | 0.42454 | 7.6606 | true |
| 10 | Best | 3.7951 | 0.34099 | 3.7951 | 3.7954 | 1.1595 | 0.054292 | 0.012493 | true |
| 11 | Accept | 4.2311 | 0.69015 | 3.7951 | 3.7954 | 0.0011423 | 0.00015862 | 8.6125 | false |
| 12 | Best | 2.8871 | 0.86673 | 2.8871 | 2.8872 | 185.22 | 2.1981e-05 | 1.0401 | false |
| 13 | Accept | 4.1521 | 0.30883 | 2.8871 | 3.0058 | 993.92 | 2.6036e-06 | 58.773 | false |
| 14 | Best | 2.8648 | 0.87323 | 2.8648 | 2.8765 | 196.57 | 2.2026e-05 | 1.081 | false |
| 15 | Accept | 4.2977 | 0.20076 | 2.8648 | 2.8668 | 0.017949 | 1.5685e-05 | 15.01 | false |
| 16 | Best | 2.8016 | 0.96695 | 2.8016 | 2.8017 | 786 | 3.4462e-06 | 1.6117 | false |
| 17 | Accept | 2.9032 | 0.59135 | 2.8016 | 2.8026 | 974.16 | 0.00019486 | 1.6661 | false |
| 18 | Accept | 2.9051 | 1.0062 | 2.8016 | 2.8018 | 288.21 | 2.6218e-06 | 2.0933 | false |
| 19 | Accept | 3.4438 | 1.4047 | 2.8016 | 2.803 | 56.999 | 2.885e-06 | 1.3903 | false |
| 20 | Accept | 2.8436 | 1.0079 | 2.8016 | 2.8032 | 533.99 | 2.7293e-06 | 0.6719 | false |
| Iter | Eval | Objective: | Objective | BestSoFar | BestSoFar | KernelScale | Lambda | Epsilon | Standardize |
| | result | log(1+loss) | runtime | (observed) | (estim.) | | | | |
| 21 | Accept | 2.8301 | 1.0592 | 2.8016 | 2.8024 | 411.02 | 3.4347e-06 | 0.98949 | false |
| 22 | Accept | 2.8233 | 0.50583 | 2.8016 | 2.8043 | 455.25 | 5.2936e-05 | 1.1189 | false |
| 23 | Accept | 4.1168 | 0.15522 | 2.8016 | 2.802 | 237.02 | 0.85493 | 0.42894 | false |
| 24 | Best | 2.7876 | 0.8726 | 2.7876 | 2.7877 | 495.51 | 1.8049e-05 | 1.9006 | false |
| 25 | Accept | 2.8197 | 0.72568 | 2.7876 | 2.7877 | 927.29 | 1.128e-05 | 1.1902 | false |
| 26 | Accept | 2.8361 | 0.72264 | 2.7876 | 2.7882 | 354.44 | 6.1939e-05 | 2.2591 | false |
| 27 | Accept | 2.7985 | 0.68054 | 2.7876 | 2.7906 | 506.54 | 1.4142e-05 | 1.3659 | false |
| 28 | Accept | 2.8163 | 0.40531 | 2.7876 | 2.7905 | 829.6 | 1.0965e-05 | 2.7415 | false |
| 29 | Accept | 2.8469 | 0.7588 | 2.7876 | 2.7902 | 729.48 | 3.4914e-06 | 0.039087 | false |
| 30 | Accept | 2.882 | 1.4101 | 2.7876 | 2.7902 | 255.25 | 3.2869e-06 | 0.059794 | false |
Optimization completed.
MaxObjectiveEvaluations of 30 reached.
Total function evaluations: 30
Total elapsed time: 34.514 seconds
Total objective function evaluation time: 18.167
Best observed feasible point:
KernelScale Lambda Epsilon Standardize
___________ __________ _______ ___________
495.51 1.8049e-05 1.9006 false
Observed objective function value = 2.7876
Estimated objective function value = 2.7902
Function evaluation time = 0.8726
Best estimated feasible point (according to models):
KernelScale Lambda Epsilon Standardize
___________ __________ _______ ___________
495.51 1.8049e-05 1.9006 false
Estimated objective function value = 2.7902
Estimated function evaluation time = 0.67763
Mdl =
ResponseName: 'Y'
Learner: 'svm'
NumExpansionDimensions: 256
KernelScale: 495.5140
Lambda: 1.8049e-05
BoxConstraint: 141.3376
Epsilon: 1.9006
FitInfo = struct with fields:
Solver: 'LBFGS-fast'
LossFunction: 'epsiloninsensitive'
Lambda: 1.8049e-05
BetaTolerance: 1.0000e-04
GradientTolerance: 1.0000e-06
ObjectiveValue: 1.3382
GradientMagnitude: 0.0051
RelativeChangeInBeta: 9.4332e-05
FitTime: 0.0710
History: []
HyperparameterOptimizationResults =
BayesianOptimization with properties:
ObjectiveFcn: @createObjFcn/inMemoryObjFcn
VariableDescriptions: [6x1 optimizableVariable]
Options: [1x1 struct]
MinObjective: 2.7876
XAtMinObjective: [1x4 table]
MinEstimatedObjective: 2.7902
XAtMinEstimatedObjective: [1x4 table]
NumObjectiveEvaluations: 30
TotalElapsedTime: 34.5140
NextPoint: [1x4 table]
XTrace: [30x4 table]
ObjectiveTrace: [30x1 double]
ConstraintsTrace: []
UserDataTrace: {30x1 cell}
ObjectiveEvaluationTimeTrace: [30x1 double]
IterationTimeTrace: [30x1 double]
ErrorTrace: [30x1 double]
FeasibilityTrace: [30x1 logical]
FeasibilityProbabilityTrace: [30x1 double]
IndexOfMinimumTrace: [30x1 double]
ObjectiveMinimumTrace: [30x1 double]
EstimatedObjectiveMinimumTrace: [30x1 double]
For big data, the optimization procedure can take a long time. If the data set is too large to run the optimization procedure, you can try to optimize the parameters using only partial data. Use the
datasample function and specify 'Replace','false' to sample data without replacement.
Input Arguments
X — Predictor data
numeric matrix
Predictor data to which the regression model is fit, specified as an n-by-p numeric matrix, where n is the number of observations and p is the number of predictor variables.
The length of Y and the number of observations in X must be equal.
Data Types: single | double
Y — Response data
numeric vector
Response data, specified as an n-dimensional numeric vector. The length of Y must be equal to the number of observations in X or Tbl.
Data Types: single | double
The software treats NaN, empty character vector (''), empty string (""), <missing>, and <undefined> elements as missing values, and removes observations with any of these characteristics:
• Missing value in the response variable
• At least one missing value in a predictor observation (row in X or Tbl)
• NaN value or 0 weight ('Weights')
Name-Value Arguments
Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but
the order of the pairs does not matter.
Example: Mdl = fitrkernel(X,Y,Learner="leastsquares",NumExpansionDimensions=2^15,KernelScale="auto") implements least-squares regression after mapping the predictor data to the 2^15 dimensional space
using feature expansion with a kernel scale parameter selected by a heuristic procedure.
Before R2021a, use commas to separate each name and value, and enclose Name in quotes.
Example: Mdl = fitrkernel(X,Y,'Learner','leastsquares','NumExpansionDimensions',2^15,'KernelScale','auto')
You cannot use any cross-validation name-value argument together with the OptimizeHyperparameters name-value argument. You can modify the cross-validation for OptimizeHyperparameters only by using
the HyperparameterOptimizationOptions name-value argument.
Kernel Regression Options
BoxConstraint — Box constraint
1 (default) | positive scalar
Box constraint, specified as the comma-separated pair consisting of 'BoxConstraint' and a positive scalar.
This argument is valid only when 'Learner' is 'svm'(default) and you do not specify a value for the regularization term strength 'Lambda'. You can specify either 'BoxConstraint' or 'Lambda' because
the box constraint (C) and the regularization term strength (λ) are related by C = 1/(λn), where n is the number of observations (rows in X).
Example: 'BoxConstraint',100
Data Types: single | double
Epsilon — Half width of epsilon-insensitive band
'auto' (default) | nonnegative scalar value
Half the width of the epsilon-insensitive band, specified as the comma-separated pair consisting of 'Epsilon' and 'auto' or a nonnegative scalar value.
For 'auto', the fitrkernel function determines the value of Epsilon as iqr(Y)/13.49, which is an estimate of a tenth of the standard deviation using the interquartile range of the response variable
Y. If iqr(Y) is equal to zero, then fitrkernel sets the value of Epsilon to 0.1.
'Epsilon' is valid only when Learner is svm.
Example: 'Epsilon',0.3
Data Types: single | double
NumExpansionDimensions — Number of dimensions of expanded space
'auto' (default) | positive integer
Number of dimensions of the expanded space, specified as the comma-separated pair consisting of 'NumExpansionDimensions' and 'auto' or a positive integer. For 'auto', the fitrkernel function selects
the number of dimensions using 2.^ceil(min(log2(p)+5,15)), where p is the number of predictors.
Example: 'NumExpansionDimensions',2^15
Data Types: char | string | single | double
KernelScale — Kernel scale parameter
1 (default) | 'auto' | positive scalar
Kernel scale parameter, specified as the comma-separated pair consisting of 'KernelScale' and 'auto' or a positive scalar. MATLAB obtains the random basis for random feature expansion by using the
kernel scale parameter. For details, see Random Feature Expansion.
If you specify 'auto', then MATLAB selects an appropriate kernel scale parameter using a heuristic procedure. This heuristic procedure uses subsampling, so estimates can vary from one call to
another. Therefore, to reproduce results, set a random number seed by using rng before training.
Example: 'KernelScale','auto'
Data Types: char | string | single | double
Learner — Linear regression model type
'svm' (default) | 'leastsquares'
Linear regression model type, specified as the comma-separated pair consisting of 'Learner' and 'svm' or 'leastsquares'.
In the following table, $f\left(x\right)=T\left(x\right)\beta +b.$
• x is an observation (row vector) from p predictor variables.
• $T\left(·\right)$ is a transformation of an observation (row vector) for feature expansion. T(x) maps x in ${ℝ}^{p}$ to a high-dimensional space (${ℝ}^{m}$).
• β is a vector of coefficients.
• b is the scalar bias.
Value Algorithm Response range Loss function
'leastsquares' Linear regression via ordinary least squares y ∊ (-∞,∞) Mean squared error (MSE): $\ell \left[y,f\left(x\right)\right]=\frac{1}{2}{\left[y-f\left(x\right)\right]}^{2}$
'svm' Support vector machine regression Same as 'leastsquares' Epsilon-insensitive: $\ell \left[y,f\left(x\right)\right]=\mathrm{max}\left[0,|y-f\left(x\right)|-\epsilon \right]$
Example: 'Learner','leastsquares'
Standardize — Flag to standardize predictor data
false or 0 (default) | true or 1
Since R2023b
Flag to standardize the predictor data, specified as a numeric or logical 0 (false) or 1 (true). If you set Standardize to true, then the software centers and scales each numeric predictor variable
by the corresponding column mean and standard deviation. The software does not standardize the categorical predictors.
Example: "Standardize",true
Data Types: single | double | logical
Verbose — Verbosity level
0 (default) | 1
Verbosity level, specified as the comma-separated pair consisting of 'Verbose' and either 0 or 1. Verbose controls the amount of diagnostic information fitrkernel displays at the command line.
Value Description
0 fitrkernel does not display diagnostic information.
1 fitrkernel displays and stores the value of the objective function, gradient magnitude, and other diagnostic information. FitInfo.History contains the diagnostic information.
Example: 'Verbose',1
Data Types: single | double
BlockSize — Maximum amount of allocated memory
4e^3 (4GB) (default) | positive scalar
Maximum amount of allocated memory (in megabytes), specified as the comma-separated pair consisting of 'BlockSize' and a positive scalar.
If fitrkernel requires more memory than the value of BlockSize to hold the transformed predictor data, then MATLAB uses a block-wise strategy. For details about the block-wise strategy, see
Example: 'BlockSize',1e4
Data Types: single | double
RandomStream — Random number stream
global stream (default) | random stream object
Random number stream for reproducibility of data transformation, specified as the comma-separated pair consisting of 'RandomStream' and a random stream object. For details, see Random Feature
Use 'RandomStream' to reproduce the random basis functions that fitrkernel uses to transform the data in X to a high-dimensional space. For details, see Managing the Global Stream Using RandStream
and Creating and Controlling a Random Number Stream.
Example: 'RandomStream',RandStream('mlfg6331_64')
Other Regression Options
Weights — Observation weights
vector of scalar values | name of variable in Tbl
Observation weights, specified as the comma-separated pair consisting of 'Weights' and a vector of scalar values or the name of a variable in Tbl. The software weights each observation (or row) in X
or Tbl with the corresponding value in Weights. The length of Weights must equal the number of rows in X or Tbl.
If you specify the input data as a table Tbl, then Weights can be the name of a variable in Tbl that contains a numeric vector. In this case, you must specify Weights as a character vector or string
scalar. For example, if weights vector W is stored as Tbl.W, then specify it as 'W'. Otherwise, the software treats all columns of Tbl, including W, as predictors when training the model.
By default, Weights is ones(n,1), where n is the number of observations in X or Tbl.
fitrkernel normalizes the weights to sum to 1.
Data Types: single | double | char | string
Cross-Validation Options
CrossVal — Cross-validation flag
'off' (default) | 'on'
Cross-validation flag, specified as the comma-separated pair consisting of 'Crossval' and 'on' or 'off'.
If you specify 'on', then the software implements 10-fold cross-validation.
You can override this cross-validation setting using the CVPartition, Holdout, KFold, or Leaveout name-value pair argument. You can use only one cross-validation name-value pair argument at a time to
create a cross-validated model.
Example: 'Crossval','on'
Convergence Controls
BetaTolerance — Relative tolerance on linear coefficients and bias term
1e-4 (default) | nonnegative scalar
Relative tolerance on the linear coefficients and the bias term (intercept), specified as a nonnegative scalar.
Let ${B}_{t}=\left[{\beta }_{t}{}^{\prime }\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{t}\right]$, that is, the vector of the coefficients and the bias term at optimization iteration t. If ${‖\
frac{{B}_{t}-{B}_{t-1}}{{B}_{t}}‖}_{2}<\text{BetaTolerance}$, then optimization terminates.
If you also specify GradientTolerance, then optimization terminates when the software satisfies either stopping criterion.
Example: 'BetaTolerance',1e-6
Data Types: single | double
GradientTolerance — Absolute gradient tolerance
1e-6 (default) | nonnegative scalar
Absolute gradient tolerance, specified as a nonnegative scalar.
Let $abla {ℒ}_{t}$ be the gradient vector of the objective function with respect to the coefficients and bias term at optimization iteration t. If ${‖abla {ℒ}_{t}‖}_{\infty }=\mathrm{max}|abla {ℒ}_
{t}|<\text{GradientTolerance}$, then optimization terminates.
If you also specify BetaTolerance, then optimization terminates when the software satisfies either stopping criterion.
Example: 'GradientTolerance',1e-5
Data Types: single | double
HessianHistorySize — Size of history buffer for Hessian approximation
15 (default) | positive integer
Size of the history buffer for Hessian approximation, specified as the comma-separated pair consisting of 'HessianHistorySize' and a positive integer. At each iteration, fitrkernel composes the
Hessian by using statistics from the latest HessianHistorySize iterations.
Example: 'HessianHistorySize',10
Data Types: single | double
IterationLimit — Maximum number of optimization iterations
positive integer
Maximum number of optimization iterations, specified as the comma-separated pair consisting of 'IterationLimit' and a positive integer.
The default value is 1000 if the transformed data fits in memory, as specified by BlockSize. Otherwise, the default value is 100.
Example: 'IterationLimit',500
Data Types: single | double
Hyperparameter Optimization Options
OptimizeHyperparameters — Parameters to optimize
'none' (default) | 'auto' | 'all' | string array or cell array of eligible parameter names | vector of optimizableVariable objects
Parameters to optimize, specified as the comma-separated pair consisting of 'OptimizeHyperparameters' and one of these values:
• 'none' — Do not optimize.
• 'auto' — Use {'KernelScale','Lambda','Epsilon','Standardize'}.
• 'all' — Optimize all eligible parameters.
• Cell array of eligible parameter names.
• Vector of optimizableVariable objects, typically the output of hyperparameters.
The optimization attempts to minimize the cross-validation loss (error) for fitrkernel by varying the parameters. To control the cross-validation type and other aspects of the optimization, use the
HyperparameterOptimizationOptions name-value argument. When you use HyperparameterOptimizationOptions, you can use the (compact) model size instead of the cross-validation loss as the optimization
objective by setting the ConstraintType and ConstraintBounds options.
The values of OptimizeHyperparameters override any values you specify using other name-value arguments. For example, setting OptimizeHyperparameters to "auto" causes fitrkernel to optimize
hyperparameters corresponding to the "auto" option and to ignore any specified values for the hyperparameters.
The eligible parameters for fitrkernel are:
Set nondefault parameters by passing a vector of optimizableVariable objects that have nondefault values. For example:
load carsmall
params = hyperparameters('fitrkernel',[Horsepower,Weight],MPG);
params(2).Range = [1e-4,1e6];
Pass params as the value of 'OptimizeHyperparameters'.
By default, the iterative display appears at the command line, and plots appear according to the number of hyperparameters in the optimization. For the optimization and plots, the objective function
is log(1 + cross-validation loss). To control the iterative display, set the Verbose field of the 'HyperparameterOptimizationOptions' name-value argument. To control the plots, set the ShowPlots
field of the 'HyperparameterOptimizationOptions' name-value argument.
For an example, see Optimize Kernel Regression.
Example: 'OptimizeHyperparameters','auto'
Output Arguments
Mdl — Trained kernel regression model
RegressionKernel model object | RegressionPartitionedKernel cross-validated model object
Trained kernel regression model, returned as a RegressionKernel model object or RegressionPartitionedKernel cross-validated model object.
If you set any of the name-value pair arguments CrossVal, CVPartition, Holdout, KFold, or Leaveout, then Mdl is a RegressionPartitionedKernel cross-validated model. Otherwise, Mdl is a
RegressionKernel model.
To reference properties of Mdl, use dot notation. For example, enter Mdl.NumExpansionDimensions in the Command Window to display the number of dimensions of the expanded space.
If you specify OptimizeHyperparameters and set the ConstraintType and ConstraintBounds options of HyperparameterOptimizationOptions, then Mdl is an N-by-1 cell array of model objects, where N is
equal to the number of rows in ConstraintBounds. If none of the optimization problems yields a feasible model, then each cell array value is [].
Unlike other regression models, and for economical memory usage, a RegressionKernel model object does not store the training data or training process details (for example, convergence history).
AggregateOptimizationResults — Aggregate optimization results
AggregateBayesianOptimization object
Aggregate optimization results for multiple optimization problems, returned as an AggregateBayesianOptimization object. To return AggregateOptimizationResults, you must specify
OptimizeHyperparameters and HyperparameterOptimizationOptions. You must also specify the ConstraintType and ConstraintBounds options of HyperparameterOptimizationOptions. For an example that shows
how to produce this output, see Hyperparameter Optimization with Multiple Constraint Bounds.
FitInfo — Optimization details
structure array
Optimization details, returned as a structure array including fields described in this table. The fields contain final values or name-value pair argument specifications.
Field Description
Solver Objective function minimization technique: 'LBFGS-fast', 'LBFGS-blockwise', or 'LBFGS-tall'. For details, see Algorithms.
LossFunction Loss function. Either mean squared error (MSE) or epsilon-insensitive, depending on the type of linear regression model. See Learner.
Lambda Regularization term strength. See Lambda.
BetaTolerance Relative tolerance on the linear coefficients and the bias term. See BetaTolerance.
GradientTolerance Absolute gradient tolerance. See GradientTolerance.
ObjectiveValue Value of the objective function when optimization terminates. The regression loss plus the regularization term compose the objective function.
GradientMagnitude Infinite norm of the gradient vector of the objective function when optimization terminates. See GradientTolerance.
RelativeChangeInBeta Relative changes in the linear coefficients and the bias term when optimization terminates. See BetaTolerance.
FitTime Elapsed, wall-clock time (in seconds) required to fit the model to the data.
History History of optimization information. This field also includes the optimization information from training Mdl. This field is empty ([]) if you specify 'Verbose',0. For details,
see Verbose and Algorithms.
To access fields, use dot notation. For example, to access the vector of objective function values for each iteration, enter FitInfo.ObjectiveValue in the Command Window.
If you specify OptimizeHyperparameters and set the ConstraintType and ConstraintBounds options of HyperparameterOptimizationOptions, then Fitinfo is an N-by-1 cell array of structure arrays, where N
is equal to the number of rows in ConstraintBounds.
Examine the information provided by FitInfo to assess whether convergence is satisfactory.
HyperparameterOptimizationResults — Cross-validation optimization of hyperparameters
BayesianOptimization object | AggregateBayesianOptimization object | table of hyperparameters and associated values
Cross-validation optimization of hyperparameters, returned as a BayesianOptimization object, an AggregateBayesianOptimization object, or a table of hyperparameters and associated values. The output
is nonempty when OptimizeHyperparameters has a value other than "none".
If you set the ConstraintType and ConstraintBounds options in HyperparameterOptimizationOptions, then HyperparameterOptimizationResults is an AggregateBayesianOptimization object. Otherwise, the
value of HyperparameterOptimizationResults depends on the value of the Optimizer option in HyperparameterOptimizationOptions.
Value of Optimizer Option Value of HyperparameterOptimizationResults
"bayesopt" (default) Object of class BayesianOptimization
"gridsearch" or "randomsearch" Table of hyperparameters used, observed objective function values (cross-validation loss), and rank of observations from lowest (best) to highest (worst)
More About
Random Feature Expansion
Random feature expansion, such as Random Kitchen Sinks [1] or Fastfood [2], is a scheme to approximate Gaussian kernels of the kernel regression algorithm for big data in a computationally efficient
way. Random feature expansion is more practical for big data applications that have large training sets, but can also be applied to smaller data sets that fit in memory.
After mapping the predictor data into a high-dimensional space, the kernel regression algorithm searches for an optimal function that deviates from each response data point (y[i]) by values no
greater than the epsilon margin (ε).
Some regression problems cannot be described adequately using a linear model. In such cases, obtain a nonlinear regression model by replacing the dot product x[1]x[2]′ with a nonlinear kernel
function $G\left({x}_{1},{x}_{2}\right)=〈\phi \left({x}_{1}\right),\phi \left({x}_{2}\right)〉$, where x[i] is the ith observation (row vector) and φ(x[i]) is a transformation that maps x[i] to a
high-dimensional space (called the “kernel trick”). However, evaluating G(x[1],x[2]), the Gram matrix, for each pair of observations is computationally expensive for a large data set (large n).
The random feature expansion scheme finds a random transformation so that its dot product approximates the Gaussian kernel. That is,
$G\left({x}_{1},{x}_{2}\right)=〈\phi \left({x}_{1}\right),\phi \left({x}_{2}\right)〉\approx T\left({x}_{1}\right)T\left({x}_{2}\right)",$
where T(x) maps x in ${ℝ}^{p}$ to a high-dimensional space (${ℝ}^{m}$). The Random Kitchen Sinks [1] scheme uses the random transformation
where $Z\in {ℝ}^{m×p}$ is a sample drawn from $N\left(0,{\sigma }^{-2}\right)$ and σ is a kernel scale. This scheme requires O(mp) computation and storage. The Fastfood [2] scheme introduces another
random basis V instead of Z using Hadamard matrices combined with Gaussian scaling matrices. This random basis reduces computation cost to O(mlogp) and reduces storage to O(m).
You can specify values for m and σ, using the NumExpansionDimensions and KernelScale name-value pair arguments of fitrkernel, respectively.
The fitrkernel function uses the Fastfood scheme for random feature expansion and uses linear regression to train a Gaussian kernel regression model. Unlike solvers in the fitrsvm function, which
require computation of the n-by-n Gram matrix, the solver in fitrkernel only needs to form a matrix of size n-by-m, with m typically much less than n for big data.
Box Constraint
A box constraint is a parameter that controls the maximum penalty imposed on observations that lie outside the epsilon margin (ε), and helps to prevent overfitting (regularization). Increasing the
box constraint can lead to longer training times.
The box constraint (C) and the regularization term strength (λ) are related by C = 1/(λn), where n is the number of observations.
• Standardizing predictors before training a model can be helpful.
□ You can standardize training data and scale test data to have the same scale as the training data by using the normalize function.
□ Alternatively, use the Standardize name-value argument to standardize the numeric predictors before training. The returned model includes the predictor means and standard deviations in its Mu
and Sigma properties, respectively. (since R2023b)
• After training a model, you can generate C/C++ code that predicts responses for new data. Generating C/C++ code requires MATLAB Coder™. For details, see Introduction to Code Generation.
fitrkernel minimizes the regularized objective function using a Limited-memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) solver with ridge (L[2]) regularization. To find the type of LBFGS solver used
for training, type FitInfo.Solver in the Command Window.
• 'LBFGS-fast' — LBFGS solver.
• 'LBFGS-blockwise' — LBFGS solver with a block-wise strategy. If fitrkernel requires more memory than the value of BlockSize to hold the transformed predictor data, then the function uses a
block-wise strategy.
• 'LBFGS-tall' — LBFGS solver with a block-wise strategy for tall arrays.
When fitrkernel uses a block-wise strategy, it implements LBFGS by distributing the calculation of the loss and gradient among different parts of the data at each iteration. Also, fitrkernel refines
the initial estimates of the linear coefficients and the bias term by fitting the model locally to parts of the data and combining the coefficients by averaging. If you specify 'Verbose',1, then
fitrkernel displays diagnostic information for each data pass and stores the information in the History field of FitInfo.
When fitrkernel does not use a block-wise strategy, the initial estimates are zeros. If you specify 'Verbose',1, then fitrkernel displays diagnostic information for each iteration and stores the
information in the History field of FitInfo.
[1] Rahimi, A., and B. Recht. “Random Features for Large-Scale Kernel Machines.” Advances in Neural Information Processing Systems. Vol. 20, 2008, pp. 1177–1184.
[2] Le, Q., T. Sarlós, and A. Smola. “Fastfood — Approximating Kernel Expansions in Loglinear Time.” Proceedings of the 30th International Conference on Machine Learning. Vol. 28, No. 3, 2013, pp.
[3] Huang, P. S., H. Avron, T. N. Sainath, V. Sindhwani, and B. Ramabhadran. “Kernel methods match Deep Neural Networks on TIMIT.” 2014 IEEE International Conference on Acoustics, Speech and Signal
Processing. 2014, pp. 205–209.
Extended Capabilities
Tall Arrays
Calculate with arrays that have more rows than fit in memory.
The fitrkernel function supports tall arrays with the following usage notes and limitations:
• fitrkernel does not support tall table data.
• Some name-value pair arguments have different defaults compared to the default values for the in-memory fitrkernel function. Supported name-value pair arguments, and any differences, are:
□ 'BoxConstraint'
□ 'Epsilon'
□ 'NumExpansionDimensions'
□ 'KernelScale'
□ 'Lambda'
□ 'Learner'
□ 'Verbose' — Default value is 1.
□ 'BlockSize'
□ 'RandomStream'
□ 'ResponseTransform'
□ 'Weights' — Value must be a tall array.
□ 'BetaTolerance' — Default value is relaxed to 1e–3.
□ 'GradientTolerance' — Default value is relaxed to 1e–5.
□ 'HessianHistorySize'
□ 'IterationLimit' — Default value is relaxed to 20.
□ 'OptimizeHyperparameters'
□ 'HyperparameterOptimizationOptions' — For cross-validation, tall optimization supports only 'Holdout' validation. By default, the software selects and reserves 20% of the data as holdout
validation data, and trains the model using the rest of the data. You can specify a different value for the holdout fraction by using this argument. For example, specify
'HyperparameterOptimizationOptions',struct('Holdout',0.3) to reserve 30% of the data as validation data.
• If 'KernelScale' is 'auto', then fitrkernel uses the random stream controlled by tallrng for subsampling. For reproducibility, you must set a random number seed for both the global stream and the
random stream controlled by tallrng.
• If 'Lambda' is 'auto', then fitrkernel might take an extra pass through the data to calculate the number of observations in X.
• fitrkernel uses a block-wise strategy. For details, see Algorithms.
For more information, see Tall Arrays.
Automatic Parallel Support
Accelerate code by automatically running computation in parallel using Parallel Computing Toolbox™.
To perform parallel hyperparameter optimization, use the UseParallel=true option in the HyperparameterOptimizationOptions name-value argument in the call to the fitrkernel function.
For more information on parallel hyperparameter optimization, see Parallel Bayesian Optimization.
For general information about parallel computing, see Run MATLAB Functions with Automatic Parallel Support (Parallel Computing Toolbox).
Version History
Introduced in R2018a
R2023b: Kernel models support standardization of predictors
Starting in R2023b, fitrkernel supports the standardization of numeric predictors. That is, you can specify the Standardize value as true to center and scale each numeric predictor variable by the
corresponding column mean and standard deviation. The software does not standardize the categorical predictors.
You can also optimize the Standardize hyperparameter by using the OptimizeHyperparameters name-value argument. Unlike in previous releases, when you specify "auto" as the OptimizeHyperparameters
value, fitrkernel includes Standardize as an optimizable hyperparameter. | {"url":"https://uk.mathworks.com/help/stats/fitrkernel.html","timestamp":"2024-11-06T18:53:22Z","content_type":"text/html","content_length":"279193","record_id":"<urn:uuid:914984c5-1ba6-430c-b5b4-60a8ad3c339e>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00171.warc.gz"} |
CLC number: TH13
On-line Access: 2024-08-27
Received: 2023-10-17
Revision Accepted: 2024-05-08
Crosschecked: 2014-03-17
Cited: 3
Clicked: 8046
Citations: Bibtex RefMan EndNote GB/T7714
Francesca Curà, Andrea Mura. Analysis of a load application point in spline coupling teeth[J]. Journal of Zhejiang University Science A, 2014, 15(4): 302-308.
@article{title="Analysis of a load application point in spline coupling teeth",
author="Francesca Curà, Andrea Mura",
journal="Journal of Zhejiang University Science A",
publisher="Zhejiang University Press & Springer",
%0 Journal Article
%T Analysis of a load application point in spline coupling teeth
%A Francesca Curà
%A Andrea Mura
%J Journal of Zhejiang University SCIENCE A
%V 15
%N 4
%P 302-308
%@ 1673-565X
%D 2014
%I Zhejiang University Press & Springer
%DOI 10.1631/jzus.A1300323
TY - JOUR
T1 - Analysis of a load application point in spline coupling teeth
A1 - Francesca Curà
A1 - Andrea Mura
J0 - Journal of Zhejiang University Science A
VL - 15
IS - 4
SP - 302
EP - 308
%@ 1673-565X
Y1 - 2014
PB - Zhejiang University Press & Springer
ER -
DOI - 10.1631/jzus.A1300323
Abstract: The objective of this paper is to investigate the position of the resultant force in involute spline coupling teeth due to the contact pressure distribution for both ideal and misaligned
conditions. In general, spline coupling teeth are in contact all along the involute profile and the load is far from uniform along the contact line. Theoretical models available in publications
consider the resultant contact force as it is applied at the pitch diameter, and this study aims to evaluate the error introduced within the confines of a common approximation environment. This
analysis is carried out through using finite element method (FEM) models, considering spline couplings in both ideal and misaligned conditions. Results show that the differences between the load
application diameter and pitch diameter are not very obvious in both ideal and misaligned conditions; however, this approximation becomes more important for the calculation of the tooth stiffness.
重要结论：验证了传统上应用于径节的花键联接啮合齿接触合力的近似法。在额定条件下，载荷直径和径节的偏差随载荷等级的增加而增加。在0.08 mm两轴偏移模型下，有限元法获得的结果与理论径节的最大偏差为2.94%。一
Darkslateblue:Affiliate; Royal Blue:Author; Turquoise:Article
Open peer comments: Debate/Discuss/Question/Opinion
Journal of Zhejiang University-SCIENCE, 38 Zheda Road, Hangzhou 310027, China
Tel: +86-571-87952783; E-mail:
Copyright © 2000 - 2024 Journal of Zhejiang University-SCIENCE | {"url":"http://www.jzus.zju.edu.cn/article.php?doi=10.1631/jzus.A1300323","timestamp":"2024-11-12T13:07:10Z","content_type":"text/html","content_length":"87092","record_id":"<urn:uuid:40d8b9fa-40bb-44ad-a58a-91dcee020f17>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00875.warc.gz"} |
Constraints on the size of the smallest triggering earthquake from the epidemic-type aftershock sequence model, Båth's law, and observed aftershock sequences
Didier Sornette
, &
Maximilian J. Werner
Published August 2005, SCEC Contribution #859
The physics of earthquake triggering together with simple assumptions of self-similarity imply the existence of a minimum magnitude m 0 below which earthquakes do not trigger other earthquakes.
Noting that the magnitude m d of completeness of a seismic catalog is not, in general, the same as the magnitude m 0 of the smallest triggering earthquake, we compare observed aftershock sequence
parameters with the predictions made by the epidemic-type aftershock sequence model to constrain the value of m 0. In particular, we use quantitative fits to observed aftershock sequences from three
previous studies, as well as Båth's law, to obtain four estimates of m 0. We show that the branching ratio n (average number of triggered earthquakes per earthquake, also equal to the fraction of
aftershocks in a seismic catalog) is the key parameter controlling the estimate of the minimum triggering magnitude m 0. Conversely, physical upper bounds for m 0 estimated from rate and state
friction indicate that at the very least, 55% of all earthquakes are aftershocks.
Sornette, D., & Werner, M. J. (2005). Constraints on the size of the smallest triggering earthquake from the epidemic-type aftershock sequence model, Båth's law, and observed aftershock sequences.
Journal of Geophysical Research, 110(B08304). doi: 10.1029/2004JB003535 . | {"url":"https://central.scec.org/publication/859","timestamp":"2024-11-08T17:14:58Z","content_type":"application/xhtml+xml","content_length":"22239","record_id":"<urn:uuid:902c4bd0-c28f-40eb-b638-3703c96f6205>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00467.warc.gz"} |
How to use numpy.sum() in Python
Kodeclik Blog
How to use numpy.sum() in Python
Consider a two dimensional numpy array of numbers such as:
import numpy as np
myarray = [[1.5,2.5,3,3.5],
numpy.sum() is a handy function that can return the sum of elements in the given array. All you need to do is to pass the array as input:
This produces the output:
You can verify that the sum of all the elements (all 8 of them) add up to 20.5.
numpy.sum() is quite versatile. It can be used to sum elements across specified dimensions.
For instance, the above array has two rows and 4 columns. If we were to sum along the first axis (i.e., rows) we will obtain a result which has one row and 4 columns. We accomplish this by specifying
a “axis=0” when we call numpy.sum():
import numpy as np
myarray = [[1.5,2.5,3,3.5],
Similarly, if we were to sum along the second axis (i.e., columns) we will obtain a result which has 2 rows and one column:
import numpy as np
myarray = [[1.5,2.5,3,3.5],
A second feature of numpy.sum() is that it can take an initial value and begins calculating sums from that point. So for instance, if we take the code above and slightly modify the invocation to
numpy.sum() like so:
import numpy as np
myarray = [[1.5,2.5,3,3.5],
because -10 is added to the original sums, i.e., [10.5 10. ], yielding [0.5, 0].
So to summarize, the numpy.sum() function can be used to calculate the sum of an array or list of numbers in one step and can save you a great deal of time and effort as opposed to say, writing a
loop. Adapt it in your own programs and tell us where you found it useful!
Interested in more things Python? Checkout our post on Python queues. Also see our blogpost on Python's enumerate() capability. Also if you like Python+math content, see our blogpost on Magic Squares
. Finally, master the Python print function! | {"url":"https://www.kodeclik.com/numpy-sum-python/","timestamp":"2024-11-02T02:17:20Z","content_type":"text/html","content_length":"90789","record_id":"<urn:uuid:3c97b3ce-8047-4cd6-a3fb-bb4690b9ae47>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00694.warc.gz"} |
Characterization of the highly fractured zone at the Grimsel Test Site based on hydraulic tomography
Articles | Volume 26, issue 24
© Author(s) 2022. This work is distributed under the Creative Commons Attribution 4.0 License.
Characterization of the highly fractured zone at the Grimsel Test Site based on hydraulic tomography
In this study, we infer the structural and hydraulic properties of the highly fractured zone at the Grimsel Test Site in Switzerland using a stochastic inversion method. The fractured rock is modeled
directly as a discrete fracture network (DFN) within an impermeable rock matrix. Cross-hole transient pressure signals recorded from constant-rate injection tests at different intervals provide the
basis for the (herein presented) first field application of the inversion. The experimental setup is realized by a multi-packer system. The geological mapping of the structures intercepted by
boreholes as well as data from previous studies that were undertaken as part of the In Situ Stimulation and Circulation (ISC) experiments facilitate the setup of the site-dependent conceptual and
forward model. The inversion results show that two preferential flow paths between the two boreholes can be distinguished: one is dominated by fractures with large hydraulic apertures, whereas the
other path consists mainly of fractures with a smaller aperture. The probability of fractures linking both flow paths increases the closer we get to the second injection borehole. These results are
in accordance with the findings of other studies conducted at the site during the ISC measurement campaign and add new insights into the highly fractured zone at this prominent study site.
Received: 01 Jun 2022 – Discussion started: 29 Jun 2022 – Revised: 18 Nov 2022 – Accepted: 29 Nov 2022 – Published: 21 Dec 2022
Solid rocks, such as in crystalline and bedrock formations, typically have a compact matrix of low permeability. Water pathways are focused on mechanical discontinuities that separate individual rock
blocks over multiple scales. Such fractures are commonly described as planar structures and form a network that is hard to resolve at field sites. This is due to the high diversity and complexity of
natural fracture networks, the difficulty involved with identifying fracture connectivities, and thus the difficulty involved with interpreting the hydraulic regime of an entire formation based on
local fracture detection. Accordingly, fractured-aquifer characterization represents a challenge, with a relatively high cost related to the application of specialized field investigation techniques
and to gathering a sufficient data set for reliable hydraulic description. The general poor understanding of how groundwater flows in fractured field sites is in contrast to the relevance of
fractured environments that host elementary freshwater reservoirs worldwide (Chandra et al., 2019; Wilske et al., 2020; Spencer et al., 2021). Moreover, adequate characterization of the properties of
fractured field sites concerns many subsurface engineering applications, such as the planning and operation of enhanced geothermal systems (Vogler et al., 2017; Kittilä et al., 2020), the evaluation
of potential sites for a nuclear waste repositories (Follin et al., 2014; Li et al., 2022), or the description of an excavation-induced damaged zone around tunnels and openings (Armand et al., 2014;
de La Vaissière et al., 2015).
Depending on the chosen experimental setting and the available data, different interpretations of the hydraulic and structural properties of a fracture network are possible. A fractured site can be
inspected locally by borehole data (e.g., core mapping and geophysical image logs such as optical or acoustic televiewer). The depth and orientation of structures intercepted by boreholes
characterize fracture intensity and prevalent fracture orientations (Armand et al., 2014; Krietsch et al., 2018; Chandra et al., 2019; Tan et al., 2020; Yin and Chen, 2020; Pavičić et al., 2021);
furthermore, by fitting probability distributions to the parameters, a statistical analysis can be conducted (Barthélémy et al., 2009; Massiot et al., 2017). Single-hole and cross-hole flow and
tracer tests are employed to infer permeability and connectivity between different borehole intervals (Le Borgne et al., 2006; Follin et al., 2014; de La Vaissière et al., 2015; de La Bernardie
et al., 2018; Jalali et al., 2018; Brixel et al., 2020b, a; Tan et al., 2020; Li et al., 2022), the velocity distribution (Kang et al., 2015), or transport properties (Kittilä et al., 2019; Lee
et al., 2019).
Detailed insight into the properties of flow paths between adjacent boreholes can be gained by tomographic methods. The principle of all tomographic methods is perturbing the investigated system
(e.g., by an injection of fluid, a tracer, a thermal anomaly, or an electric current) and recording the response at nearby receivers. In particular, geophysical tomographic methods are applied for
the characterization of the rock properties, the identification of fractured (in particular highly fractured) zones, and the monitoring of flow pathways (Deparis et al., 2008; Dorn et al., 2012;
Robinson et al., 2016; Doetsch et al., 2020). This is frequently done in combination with hydrogeological methods (Day-Lewis et al., 2003; Chen et al., 2006; Dorn et al., 2013; Voorn et al., 2015;
Giertzuch et al., 2021b, a). A comprehensive portrayal of geophysical methods for the investigation of fractured field sites and the potential target applications is given in Day-Lewis et al. (2017).
In contrast to geophysical exploration techniques, hydraulic, pneumatic, or tracer tomography is based on a fluid or tracer injection at a source well. The response is recorded at different adjacent
boreholes at different depth intervals. In most cases, the pressure signals or tracer arrival curves are evaluated by a continuous hydraulic conductivity distribution based on an equivalent porous
media (EPM) concept (Yeh and Liu, 2000; Illman et al., 2008, 2009; Sharmeen et al., 2012; Zha et al., 2015, 2016; Zhao and Illman, 2017; Dong et al., 2019; Zhao et al., 2019; Kittilä et al., 2020;
Tiedeman and Barrash, 2020; Poduri et al., 2021; Zhao et al., 2021; Jiang et al., 2022; Liu et al., 2022). Thus, detected high-conductivity zones correspond to the locations of fractures or faults.
Further insights into the fracture properties and improved results can be gained by particle tracking simulations (Tiedeman and Barrash, 2020), binary priors representing either fracture or matrix (
Poduri et al., 2021), or by generating synthetic models with similar features to the field site (Zha et al., 2015). Geostatistical methods apply a stochastic EPM, and different realizations of the
subsurface are evaluated (Park et al., 2004; Blessent et al., 2011; Wang et al., 2017). Here, different facies represent different levels of fractured or intact rock, for which hydraulic
conductivities are calibrated. In contrast to the EPM approach, the properties of the fracture network are inferred more directly by calibrating a connectivity pattern (Fischer et al., 2018b, a;
Klepikova et al., 2020).
Our inversion approach differs from previous studies insofar as the fractured rock is represented explicitly as a discrete fracture network (DFN) and the hydraulic and structural parameters of the
fractures are inferred directly. The great number of unknown parameters prevents the minimization of an objective function between simulated and observed data, resulting in a single deterministic
DFN. Instead, a stochastic approach is applied to consider the nonuniqueness of the results. This is accomplished by generating several realizations of the fracture network that are equally likely to
be evaluated as a fracture probability map. The validity of the approach has been demonstrated for synthetic test cases in two dimensions (2D) (Somogyvári et al., 2017; Ringel et al., 2019) and three
dimensions (3D) (Ringel et al., 2021). In this study, the new inversion method is applied to field data for the first time. We use transient pressure signals from hydraulic tomography experiments
conducted as part of the In Situ Stimulation and Circulation (ISC) experiments at the Grimsel Test Site (GTS) in Switzerland. Proper evaluation and validation of a new approach requires controlled
tests, and the GTS and ISC experiments pose a well-explored site for experimental validation. The objective of this paper is to reveal the feasibility and capability of 3D DFN inversion using a
small-scale example. This study provides an elementary link between the theoretical development of a new inversion algorithm based on synthetic test cases and field applications, although the small
scale may not be representative of the much larger scale of groundwater reservoirs.
The paper is structured as follows: in Sect. 2, we describe the site and the hydraulic tomography experiments to be used for the inversion; the implementation of the inversion is elaborated upon in
Sect. 3; we then review the forward modeling procedure (Sect. 3.1) and the general inversion framework (Sect. 3.2) developed in previous works with synthetic test cases; in Sect. 3.3 and 3.4, we
explain the site-dependent inversion setting (i.e., the conceptual model and the prior parameter distributions that serve as basis for a stochastic inversion procedure) and discuss and justify the
necessary constraints and assumptions; finally, the inversion results are interpreted and compared with findings from related ISC experiments in Sect. 4.
2.1Test site
The GTS is an underground rock laboratory located in the Aar Massif in the Swiss Alps. The ISC experiments, which serve as the basis for this study, utilized 15 boreholes of 20–50m depth, including
two injection boreholes (Inj1 and Inj2). The other boreholes are used for stress and strain measurement as well as seismic, pressure, and temperature monitoring during the hydraulic stimulation
phases (Krietsch et al., 2018). A general overview of the site showing the persistent structures and the boreholes is given in Fig. 1a. A summary of the experiments conducted during the ISC
measurement campaign and their results are given in Amann et al. (2018) and Doetsch et al. (2018).
The crystalline rock in the southern part of the GTS (ISC experiment volume) has been moderately fractured. Ductile (S1) and brittle–ductile (S3) shear zones can be distinguished from the
investigated rock volume (Fig. 1a; Krietsch et al., 2018). The shear zones consist of a fault core, a damage zone, and unperturbed host rock (Wenning et al., 2018). A 4–6m highly fractured zone with
a fracture density (P10) of around 3m^−1 is present between the fault cores of the two S3 shear zones and is displayed in Fig. 1b.
The fractures can be distinguished in wall damage zones adjacent to the S3 faults and linking damage zones, i.e., fractures connecting both fault cores (Brixel et al., 2020b). Testing campaigns on
the connectivity between several intervals of the injection boreholes revealed that the best response occurs between the intervals 3 and 4 of both injection boreholes, which are located in the
aforementioned highly fractured zone. Therefore, this is not only a highly fractured zone but also the most permeable region with conductive fractures (Jalali et al., 2018). For this reason, the
characterization of the hydraulic and structural properties of this region (Fig. 1b) is the target of this study. The geological mapping of the structures intercepted by the boreholes and tunnels
provides the basis for the setup of the conceptual model (Krietsch et al., 2018).
2.2Hydraulic tomography data
The hydraulic tomography tests that are applied in this study are part of the characterization phase of the ISC experiment. We utilize transient pressure signals from constant-rate injection tests in
the intervals 3 and 4 of the injection boreholes Inj1 and Inj2. The different intervals are isolated by a multi-packer system. The properties of the packer intervals and the parameters of the
injection are summarized in Table 1.
Between each injection experiment, pressure recovery was possible. The pressure response of the fluid is measured using piezoresistive pressure transducers. The resolution of the pressure response
data is approximately 0.5kPa. The minimum principal stress is of the order of 8MPa. As the injected fluid pressure is much below the minimum principal stress, the coupling between hydraulic and
mechanical effects can be neglected in the forward modeling of the experiment. The fluid pressure is measured with Δt=2s. In general, we use similar hydraulic tomography experiments as those applied
by Klepikova et al. (2020) except for a shorter injection time (Table 1), which was chosen for computational reasons.
The pressure signals are shown in Fig. 2 for each injection interval. Due to the stochastic inversion approach, the noisy pressure response data can be directly utilized for the inversion without the
necessity to smooth or filter the signals.
3Implementation of the inversion
3.1Forward modeling
Fractures are modeled as 2D objects with constant properties normal to the fracture midplane in a 3D rock matrix that is assumed to be impermeable. The pressure diffusion in a single fracture is
described by
$\begin{array}{}\text{(1)}& a\mathit{\rho }S\frac{\partial p}{\partial t}-{\mathrm{abla }}_{T}\cdot \left(a\mathit{\rho }\frac{{k}_{f}}{\mathit{\mu }}{\mathrm{abla }}_{T}p\right)=aq,\end{array}$
where a (m) is the hydraulic aperture, ρ (kgm^−3) is the density of the fluid, S (Pa^−1) is the specific storage, k[f] (m^2) is the permeability, μ (Pas) is the dynamic viscosity, and q (kgm^−3s^
−1) is a source/sink term. The pressure p (Pa) consists of the static pressure and the piezometric pressure. The permeability is related to the aperture by
$\begin{array}{}\text{(2)}& {k}_{f}=\frac{{a}^{\mathrm{2}}}{\mathrm{12}},\end{array}$
and the subscript T of the gradient (∇[T]) denotes that it is evaluated in the fracture plane (Zimmerman and Bodvarsson, 1996; Berre et al., 2019). In this study, flow in the shear zones is modeled
using the same approach as flow in the DFN, i.e., the shear zones are represented as 2D objects whereby the flow parameters are given by hydraulic aperture and specific storage (Eq. 1). The equations
are solved numerically using the finite element method (FEM) with a conforming discretization at the intersections of different fractures. The generation of the geometry and the meshing of the
fractures and shear zones are implemented using the open-source mesh generator “Gmsh” (Geuzaine and Remacle, 2009). The geometry of each structure is created separately by the built-in geometry
module of Gmsh. The fractures and the shear zones are connected for a conforming discretization at the intersections of different structures by the Boolean operations implemented in Gmsh.
The implemented boundary conditions are shown in Fig. 3 along with the S3 faults and the fractures intercepted by the injection boreholes obtained from optical televiewer logs (Krietsch et al., 2018)
The boundary conditions are chosen considering the fact that only a small volume of the ISC experiment is investigated in this study. Therefore, the following boundary conditions are applied:
• The AU (Auflockerungszone, i.e., excavation effects) tunnel is represented by a pressure boundary condition – in this case, ambient pressure.
• The way to the VE (ventilation test) tunnel cannot be modeled explicitly. Thus, we apply a Robin boundary condition as a transfer boundary condition to consider the transition of the flow and the
extension of the shear zones towards the VE tunnel (Watanabe et al., 2017).
• A no-flow boundary condition is applied normal to the planes of the fractures and shear zones.
3.2Inversion algorithm
The parameters of the DFN θ are treated as unknowns characterized by probability density functions. Based on the Bayesian equation, the posterior density function p(θ|d) of the parameters given the
measured data d is proportional to the likelihood function
$\begin{array}{}\text{(3)}& \mathrm{log}L\left(\mathbit{d}\mathrm{|}\mathit{\theta }\right)\propto -\sum _{i=\mathrm{1}}^{{N}_{\mathrm{data}}}\frac{{\left({d}_{i}-f\left(\mathit{\theta }{\right)}_{i}
\right)}^{\mathrm{2}}}{\mathrm{2}{\mathit{\sigma }}_{i}^{\mathrm{2}}}\end{array}$
and the prior distributions p(θ) (Gelman et al., 2013). The term f(θ) refers to the forward simulation of the hydraulic tomography experiment for the DFN realization defined by the parameters θ. The
posterior density function is evaluated by sampling from it according to the Markov chain Monte Carlo (MCMC) method. This is an iterative procedure whereby new samples θ^′ are proposed by a proposal
function and accepted (${\mathit{\theta }}_{i}={\mathit{\theta }}^{\prime }$) with probability
$\begin{array}{}\text{(4)}& \mathit{\alpha }=\mathrm{min}\left(\mathrm{1},\frac{p\left({\mathit{\theta }}^{\prime }\mathrm{|}\mathbit{d}\right)q\left({\mathit{\theta }}_{i-\mathrm{1}}\mathrm{|}{\
mathit{\theta }}^{\prime }\right)}{p\left({\mathit{\theta }}_{i-\mathrm{1}}\mathrm{|}\mathbit{d}\right)q\left({\mathit{\theta }}^{\prime }\mathrm{|}{\mathit{\theta }}_{i-\mathrm{1}}\right)}\mathrm{|}
or rejected (${\mathit{\theta }}_{i}={\mathit{\theta }}_{i-\mathrm{1}}$) (Brooks et al., 2011). The so-called reversible jump MCMC algorithm allows one to change the number of parameters (Green, 1995
). In this study, the number of parameters is adjusted by deleting or inserting a fracture within the prior bounds. The determinant of the Jacobian matrix |J| has to be considered for
transdimensional updates. It equals 1 for parameters sampled from the prior without linking its value to preexisting parameters (Sambridge et al., 2006). The parameters of the inversion problem are
adjusted by proposing a new value from a normal distribution whereby the mean of the normal distribution is given by the current value.
The variance σ^2 in the likelihood function (Eq. 3) accounts for different sources of uncertainties, such as measurement errors, modeling errors, and errors of the conceptual model. Therefore, the
value of the variance is estimated separately for each pressure signal. This is implemented as part of the inversion algorithm after the update of the parameters of the DFN. The measured data are
assumed to consist of a mean and a normally distributed error $\mathbit{d}=\stackrel{\mathrm{‾}}{\mathbit{d}}+\mathcal{N}\left(\mathrm{0},{\mathit{\sigma }}^{\mathrm{2}}\right)$. With this
assumption, the variance can be estimated by sampling from an inverse gamma distribution
$\begin{array}{}\text{(5)}& {\mathit{\sigma }}^{\mathrm{2}}\mathrm{|}\mathbit{d},\mathit{\theta }\sim \mathcal{I}\mathcal{G}\left(\frac{{N}_{\mathrm{data}}}{\mathrm{2}},\phantom{\rule{0.25em}{0ex}}\
frac{\sum _{i=\mathrm{1}}^{{N}_{\mathrm{data}}}{\left({d}_{i}-f\left(\mathit{\theta }{\right)}_{i}\right)}^{\mathrm{2}}}{\mathrm{2}}\right),\end{array}$
as introduced by Gelman (2006) and implemented by authors including Haario et al. (2006) and Ringel et al. (2019). For this reason, the noisy measured data can be directly utilized for the inversion
without filtering or smoothing the signals.
In practice, one iteration of the inversion algorithm operates as follows: assuming that the insertion of a fracture is chosen in the MCMC algorithm, the parameters (position; length; fracture set,
i.e., orientation; and hydraulic aperture) of the fracture are generated from the prior functions. The chosen parameters are evaluated by simulating the hydraulic tomography experiment with the
proposed parameter set θ (i.e., including the new fracture). The outcome of the simulation is compared to the measured pressure signals. If the error is smaller (the likelihood, Eq. 3, is higher) or
similar to the previous step (without the fracture), the acceptance probability (Eq. 4) is high (Ringel et al., 2021). After accepting or rejecting the proposed parameters, the variance is updated
according to Eq. (5).
3.3Inversion constraints
The overall inversion procedure relies on several simplifications concerning parameters with less importance for our research target. For instance, the parameters specifying the properties of the
shear zones have to be fixed. In general, our aim is an optimal balance between the accuracy of the generated results and the computational cost of the inversion procedure.
The underlying conceptual model comprises simplifications of the properties of single fractures that serve as inversion constraints. We assume plane ellipses as the fracture shape, and the length of
the minor axis equals half of the length of the major axis (i.e., the length ratio is fixed). The assumption of reducing the fracture shape to a 2D plane is a common assumption and is justified by
the derivation of the cubic law and the large ratio between the fracture extensions and the fracture aperture (Zimmerman and Bodvarsson, 1996). The assumption of the fracture shape as an ellipse is
reasonable because the flow is dominated by the path between the intersections of different fractures; therefore, no sharp edges are considered for the simulation of the flow in the DFN. The
hydraulic aperture is assumed to be constant over the fracture plane. Two fracture sets are defined with fixed orientations based on the orientations of the structures intercepted by the two
injection boreholes. Thus, the fracture set is chosen by the inversion algorithm for the fractures between the boreholes; however, the orientation assigned to the fracture sets is a default. Figure 4
shows the orientation of the structures between the S3 shear zones intercepted by the two injection boreholes and the orientations defined for the two fracture sets. The appearance and distribution
of the fractures dominate the flow. Accordingly, the surrounding rock matrix is assumed to be impermeable.
The investigated volume is limited to the volume between the two S3 shear zones (Fig. 1). The shear zones consist of a fault core and a damage zone. The permeability increases with distance from the
fault core, where the cores are almost impermeable (Wenning et al., 2018). As the properties of the shear zones are not the target of this study, the shape is simplified and the associated hydraulic
parameters are fixed. The shape of the shear zones is simplified to a plane rectangle (i.e., a linear interpolation between the shear zones' traces at the injection boreholes). A constant hydraulic
aperture of ${a}_{\mathrm{SZ}}=\mathrm{1}×{\mathrm{10}}^{-\mathrm{5}}$m is assigned. This small value is chosen based on preliminary in situ tests and the knowledge that the cores of the shear zones
are impermeable at their tunnel intersection. A higher permeability of the shear zone at specific locations can be covered by placing fractures in the respective area that also accounts for the
spatial variability in the permeability of the shear zone. Moreover, the specific storage value is fixed at ${S}_{\mathrm{SZ}}=\mathrm{1}×{\mathrm{10}}^{-\mathrm{5}}$Pa^−1. This high value is
prescribed considering the results from cross-borehole tests (Klepikova et al., 2020). Fractures of fracture set 1 are approximately parallel to the S3 faults. Hence, a position close to an S3 fault
also accounts for spatial changes in the permeability and specific storage of the S3 faults.
Overall, the application of constraints and assumptions about the fracture shape limit an exact reproduction of the structural properties of the tested rock mass. However, those parameters that have
a major influence on the flow in the DFN are adjusted by the inversion algorithm within prescribed bounds. These are, in particular, the position and the hydraulic aperture of fractures. In contrast,
parameters with minor effects on the flow behavior are fixed (e.g., the exact fracture orientation or the length ratio).
3.4Prior distributions
The parameters to be inferred are the number of fractures, the position of the fractures, the fracture lengths, the respective hydraulic aperture for each fracture, and the specific storage
coefficient that applies to the whole DFN. The specific storage S (Eq. 1) is given by the compressibility of water in theory (Freeze and Cherry, 1979). However, some fractures are only partially
open; thus, due to the roughness of the surface, the specific storage can be increased compared with the theoretical value (Jalali et al., 2018). Moreover, the hydraulic aperture is generally smaller
than the actual aperture (Berre et al., 2019). The specific storage is assumed to be valid for the whole DFN because two variable hydraulic parameters for each fracture are not feasible for the
inversion algorithm. Accordingly, five different update types are implemented to be applied sequentially: the transdimensional update changes the number of parameters by either inserting a new
fracture or deleting a fracture; the other update types keep the number of parameters constant but adjust position, length, hydraulic aperture, or the specific storage. For the update of the
position, length, and hydraulic aperture, one fracture is chosen randomly, and a new value is proposed by a random perturbation of the current value.
Uniform prior distributions are applied (i.e., a parameter is specified by a constant probability between minimum and maximum possible values that are given in Table 2).
All spatial values refer to the position of the midpoint of the ellipse.
The spatial priors are derived in general from the position of fractures intersecting the injection boreholes. The maximum value in the x direction corresponds to the distance to the AU tunnel to
apply the boundary condition. The prior for the north direction is given such that the fractures are located between the cores of the S3 shear zones. The elevation of fractures is expected to have a
minor influence on the flow between the two boreholes, and a broader possible range for the elevation would be less resolved. In the following, x refers to easting+667400m, y refers to
northing+158800m, and z refers to height+1700m (Fig. 1). The minimum value for the fracture length is given by the borehole diameter, and the maximum possible value corresponds to the distance
between the shear zones. Fractures proposed during iterative inversion which intersect with the fault cores of the shear zones are reduced to the part of the fracture within the investigated volume
(Figs. 1b, 3).
The prior range for the aperture is approximated from the results of single- and cross-borehole tests (Jalali et al., 2018; Brixel et al., 2020b, a). The minimum specific storage value is given by
the compressibility of water (Freeze and Cherry, 1979), whereas the maximum value is based on cross-borehole injection tests (Klepikova et al., 2020). Both prior distributions for the hydraulic
parameters cause preferential flow in the DFN rather than in the shear zones, due to a smaller specific storage and a larger hydraulic aperture of the fractures.
4.1Processing of the results
Overall, 27000 DFN realizations are considered to be posterior DFN realizations because they minimize the error and fulfill the prior conditions. DFN realizations from the initial 500 iterations are
discarded as so-called “burn-in” iterations due to a higher error. The computation of the inversion was executed by an Intel Core i9 workstation with 10 cores and 128GB RAM and took about 1 week.
The posterior realizations are approximately equally likely. They reflect the uncertainty of the inversion results in contrast to a unique solution that would be obtained by a deterministic approach.
To reduce the autocorrelation of the results, we keep every 100th realization for further processing, which is called “thinning” (Brooks et al., 2011). Using the stochastic approach applied here, the
fit between the measured and simulated pressure signals of the hydraulic tomography experiment is evaluated by the posterior and prediction uncertainty. The posterior limits are calculated based on
the simulated pressure signals of the posterior DFN realizations which correspond to the uncertainty of the inversion method. The uncertainty related to predicting new observations is a measure of
the overall error as well as of conceptual simplifications, as it also considers the estimated variance (Eq. 5). The DFN realizations are evaluated using a fracture probability map (FPM) over the
investigated volume. For this, the inspected rock volume is divided into raster elements. Each element records whether the element is part of a fracture. By taking the element-wise mean over all of
the posterior DFN realizations, the probability that each raster element is part of a fracture is derived. The evaluation of the FPM summarizes the estimated position and length of the fractures
(i.e., those parameters with major influence on the flow). The hydraulic aperture is evaluated on the same raster elements. If a raster element is part of a DFN realization, the respective aperture
is taken from the DFN. Thus, the mean hydraulic aperture can be evaluated for each element.
4.2Evaluation of the data
In the first step, the measured and simulated pressure signals are compared to assess the quality of the posterior realizations. Figure 5 shows the median fit and the 95% limits of the forward
simulation of the posterior DFN realizations and the 95% limits of the prediction uncertainty along with the observed data.
Figure 5 demonstrates that the general shape and trend of the measured signals are reproduced by the simulated pressure curves checking the median fit and the 95% posterior limits. This is
especially the case for the response in interval 4 of both boreholes. The weaker fit of some signals in interval 3 indicates effects not covered by the inversion approach or forward simulations, such
as deviations from the assumed fracture shape or fracture orientations. For a given DFN realization, the actual measured pressure signals are predicted. Due to measurement noise and simplifications
concerning the DFN model, the 95% limits of the prediction uncertainty are wider than for the posterior uncertainty.
4.3Evaluation of the DFN realizations
The FPM and the mean hydraulic aperture are shown for different cross sections (z) in Fig. 6. The fractures intercepted by the injection intervals and the shear zones are fixed; therefore, they
appear with a probability of 100%. Their orientation, as derived from the optical televiewer logs, is assigned to these fractures; thus, the orientation is in the same range as the orientation
defined for the fracture set, but the exact values vary.
Overall, two different connections with different levels of permeability are present. A flow path dominated by fractures with a large hydraulic aperture exists between injection interval 4 of both
boreholes (Inj1–Int4 and Inj2–Int4). The fractures providing this connection are visible with a high probability in the cross sections z=16m and mainly z=17m. In general, a good respective
permeable connection between two intervals is possible with a large hydraulic aperture of the fractures, with long fractures, with a long intersection length between different fractures, or with a
correlation of these factors. In contrast, a connection with fractures with smaller hydraulic apertures appears between injection interval 3 of both boreholes (Inj1–Int3 and Inj2–Int3) and Inj2–Int4.
This flow path is present with an average probability of approximately 50% primarily in the cross section z=15m. Fractures linking both flow paths appear more likely the closer the location is to
injection borehole 2 (i.e., further east). The described behavior is also reflected in the measured data. All responses provoked by the injection in interval 4 of both boreholes are more distinct
than for the injection in interval 3. Although a maximum hydraulic aperture of 10^−3m is enabled by the prior distribution, only a few fractures with a small probability appear with an aperture
close to the maximum possible value, as visible in Fig. 6, at a depth of z=17m. The specific storage coefficient converges to a mean value of $S=\mathrm{7.4}×{\mathrm{10}}^{-\mathrm{7}}$Pa^−1. Only
a few updates were possible that occurred mainly during the burn-in iterations. Therefore, this value is interpreted as the result of an optimization (i.e., as the averaged specific storage to be
applied for the whole DFN). The estimated specific storage is greater than the theoretical value that functioned as the minimum value of the prior distribution of the specific storage (Table 2). This
considers a delay in the response that is not related exclusively to the compressibility of water (Freeze and Cherry, 1979) but also to, for example, the surface roughness or fractures that are only
partially open. Multiplied by the maximum possible aperture (Table 2), the inferred value is well within the storativity range calculated from cross-borehole injection tests (Klepikova et al., 2020).
Several fractures of fracture set 1 appear close to the S3.1 shear zone, indicating either permeable fractures close to the shear zone or a higher permeability of the shear zone in this region than
the assigned value. This demonstrates that the prescribed assumptions with respect to the hydraulic properties of the shear zone do not induce crucial conceptual constraints in the inversion, but a
locally high permeability of a shear zone is indicated by a locally high fracture probability.
Although the volume east of injection borehole 2 towards the AU tunnel is part of the inversion (i.e., fractures can be inserted or moved in this volume), the resolution of the results is low because
various DFN realizations (i.e., fracture positions) are possible. Only the volume between the two injection boreholes can be evaluated with a sufficient resolution.
4.4Comparison with other studies
The inferred flow paths consist of fractures with a high or rather low permeability, which is in accordance with the results of Klepikova et al. (2020). We also compare our results with the
structures intersected by other boreholes drilled after conducting the experiments evaluated in this study. While this inversion approach is capable of identifying fractures that are hydraulically
relevant, geophysical methods (such as optical televiewer logs) report all structures intercepted by boreholes independent of their permeability. The boreholes PRP1 and FBS1 are partially located
within the prior range defined for this study. The 23–25m depth interval for PRP1 has been identified as the interval with the highest transmissivity by Kittilä et al. (2019) and Brixel et al. (2020
a). In 95% of the posterior DFN realizations, at least one fracture is present in this interval. Fractures that intersect with the interval between the S3 faults of the FBS1 borehole are present in
about 45% of the posterior DFN realizations. This supports the fact that crucial hydraulic features of the DFN can be identified by the presented inversion approach. Still, even if such successful
local validation is possible, there are no other independent measurements available to confirm the validity of the inverted complete DFN structure and its probability. Geophysical measurements, such
as seismic data (Doetsch et al., 2020) or ground-penetrating radar (Giertzuch et al., 2021a), were able to characterize the ISC volume on a decameter scale and identify the persistent structures and
the highly fractured zone; however, they could not delineate or specify the properties of single flow paths.
In this study, we characterized the highly fractured zone at the GTS based on transient pressure signals from hydraulic tomography experiments using a new stochastic inversion method. A stochastic
approach was applied to assess the uncertainty of the measured data and the nonuniqueness of the results. The fractured rock is represented directly as a DFN model in the forward simulations. Several
posterior DFN realizations that are approximately equally likely are evaluated, and two preferential flow paths dominated by a large or small hydraulic aperture are successfully identified. The
presented method relies on some investigations that must be applied prior to the inversion (such as the mapping of structures intercepted by boreholes) and benefits from single- and cross-hole
permeability tests for the definition of the hydraulic aperture range. If it is possible to further narrow down the prior range of the hydraulic parameters, the specific storage can be inferred
separately for each fracture, instead of computing only a mean value for the whole DFN. In general, improved results and more insights into the fractured rock can be gained using the same inversion
method but with more pressure signals from additional intervals and boreholes.
Future research is necessary on the generally most suitable definition of prior and proposal distributions, which are elementary for robust inversion and for deriving meaningful results. The
efficiency of the MCMC sampling can be improved significantly by implementing more elaborate prior or proposal distributions, for example, relying on soft information and site-specific expertise. A
further option is utilizing continuous inversion results (such as continuous hydraulic conductivity distributions) or geophysical measurements for highlighting a priori regions with a higher
probability of the insertion of fractures or to define zones that are likely connected by fractures to reduce the number of necessary inversion iterations (Dong et al., 2019).
The introduced inversion framework can be applied in a highly flexible way for the characterization of different fractured sites by adapting the site-dependent parameters to meet the conditions of
the tomography experiment at each site. Moreover, different types and sources of measured data can be processed for the inversion (such as tracer or in situ stress data), provided that a forward
model is available that allows for the flexible update of DFN parameters. The workflow for the setup of the inversion problem is similar. The basis is the properties of the fractures intercepted by
the boreholes, i.e., their position and orientation, obtained from optical or acoustic televiewer logs or outcrops. This knowledge is utilized for the prior distributions on the spatial parameters
and for the specification of fracture sets. The prior distributions on the hydraulic parameters are based on cross-hole flow tests in this study. This can also be done by the evaluation of the
hydraulic tomography experiments as a continuous hydraulic conductivity and specific storage tomogram. As the definition of priors and constraints delineates the range of feasible DFN realizations,
this step has to be done carefully. However, the presented Bayesian framework allows the combination of multiple and diverse hard and soft data, which often exist in addition to hydraulic test data
that are used to guide the inversion. As demonstrated here, overly tight constraints may be avoided by uniform prior distributions with large value ranges at the expense of a higher computational
cost for the inversion. In practice, the amount of information describing the fractured rock is determined mainly by the hydraulic tomography data (i.e., by the number of intervals and boreholes).
The present study paves the way towards the applicability of the discrete inversion approach on a larger scale. The main issue will be to balance the degree of field testing with the desired fracture
resolution and the associated computational cost. One possible direction is explicitly implementing only large conductive fractures. The role of smaller fractures with a lower permeability could be
represented by calibrating a background permeability within the discrete fracture matrix approach (Berre et al., 2019). Another appealing direction is the representation of scale-dependent fracture
sets by their statistical properties following a hierarchical parameterization (Ma et al., 2020).
PB was responsible for funding acquisition; MJ carried out measurements; LMR and PB developed the methodology; LMR carried out the inversion and wrote the original manuscript; and MJ and PB reviewed
and edited the manuscript.
The contact author has declared that none of the authors has any competing interests.
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
We thank Ryan Pearson for language editing the original manuscript and the two anonymous referees for their valuable comments which helped to improve the quality of the manuscript.
This research has been supported by the German Research Foundation (DFG; grant no. BA-2850-5-1).
This paper was edited by Brian Berkowitz and reviewed by two anonymous referees.
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Sample Size in Logistic Regression: A Simple Binary Approach
This article will guide you through calculating the sample size for a Simple Binary Logistic Regression.
We will utilize the popular and freely available software G*Power, which is one of the most used for this purpose.
We were inspired to create this article after realizing that many online tutorials for G*Power-based sample size calculations are inaccurate.
Selecting The Logistic Regression Analysis
Upon downloading and installing G*Power, open it and choose the sample size calculation option for logistic regression analysis by clicking Tests: Correlation and regression: Logistic regression tab.
One-tailed or Two-tailed?
Next, input the following parameters:
In Tail(s), select One for one-tailed tests or Two for two-tailed tests.
A one-tailed test is appropriate for a specific alternative hypothesis, such as “increased value of X corresponds to a higher probability of the event occurring.“
A two-tailed test is suitable for a general alternative hypothesis, like “X influences event Y,” without an initial directional distinction.
Base your hypothesis on existing knowledge in your field. If unsure, opt for Two (two-tailed).
The significance level (α) represents the probability of rejecting the null hypothesis when it is true, leading to a type I error.
Typically, α is set at 0.05 or 0.01. An α of 0.05, for instance, indicates a 5% risk of concluding a significant relationship exists when none actually does.
The Test Power (1 – β) is the probability of rejecting the null hypothesis if false, effectively controlling for type II error (β).
Acceptable values usually range from 0.80 to 0.99. Higher test power is preferable but also increases the required sample size.
R² of Other Explanatory Variables (X)
Since simple binary logistic regression models only have one independent variable, set this value to zero.
Select the distribution type for variable X, which is the independent or predictor variable in the model.
Choose Binomial for binary variables, Normal for continuous quantitative variables, and Poisson for discrete variables. Use other available distributions only if necessary.
The last four parameters require population estimates from a pilot study, similar research, or theoretical calculations.
If possible, use data from a pilot study, as we will demonstrate here.
Estimated Mean and Standard Deviation
Enter variable X’s mean and standard deviation from the pilot study data for the Population Mean of Variable X and Population Standard Deviation of Variable X parameters, respectively.
Obtain the final two parameters through a preliminary simple logistic regression analysis with the pilot study data.
For our demonstration, we will use the free and easy-to-use PSPP software. Any software capable of running logistic regression can be used.
The odds ratio indicates the association between an exposure and an outcome and measures effect size.
Perform a preliminary analysis with the pilot data in PSPP to obtain the odds ratio value (Exp(B) for variable X, 1.48 in this example) and enter it in G*Power.
Probability of y = 1 under H0
The final parameter is the probability of occurrence of the dependent variable (y = 1) when the null hypothesis (H0) is true, i.e., when the coefficient of the independent variable (X) equals 0, and
the model only contains the intercept.
To calculate this value, enter the constant (intercept) estimate B from the previous step into the following Excel formula:
For our example:
With all parameters entered in G*Power, click Calculate to obtain the sample size determined by the calculation!
In our example, the sample size required to identify the estimated odds ratio is 97 individuals randomly sampled from the target population.
By following these steps and using G*Power, you can effectively calculate the appropriate sample size for a Simple Binary Logistic Regression analysis.
This process allows you to optimize your study design, minimize errors, and improve the validity of your findings.
Furthermore, understanding the role of different parameters in determining sample size contributes to a comprehensive grasp of logistic regression as a whole.
Want to learn how to calculate sample size in G*Power for the most crucial inferential analyses?
Don’t miss out on the FREE samples of our recently launched digital book!
Inside, you’ll master sample size calculation for independent or paired t-tests; one- or two-way ANOVA, with or without repeated measures, and mixed models; simple and multiple linear and logistic
regression, and more.
Click this link and discover everything it has to offer: Applied Statistics: Data Analysis.
Visit us on our social networks | {"url":"https://statisticseasily.com/2022/05/19/sample-size-logistic-regression/","timestamp":"2024-11-02T10:31:18Z","content_type":"text/html","content_length":"255529","record_id":"<urn:uuid:9fa3befb-a738-427a-b734-f8b28bf56b2d>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00698.warc.gz"} |
Polynomial Functions
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Chapter 5 : Polynomial Functions
In this chapter we are going to take a more in depth look at polynomials. We’ve already solved and graphed second degree polynomials (i.e. quadratic equations/functions) and we now want to extend
things out to more general polynomials. We will take a look at finding solutions to higher degree polynomials and how to get a rough sketch for a higher degree polynomial.
We will also be looking at Partial Fractions in this chapter. It doesn’t really have anything to do with graphing polynomials but needed to be put somewhere and this chapter seemed like as good a
place as any.
Here is a brief listing of the material in this chapter.
Dividing Polynomials – In this section we’ll review some of the basics of dividing polynomials. We will define the remainder and divisor used in the division process and introduce the idea of
synthetic division. We will also give the Division Algorithm.
Zeroes/Roots of Polynomials – In this section we’ll define the zero or root of a polynomial and whether or not it is a simple root or has multiplicity \(k\). We will also give the Fundamental Theorem
of Algebra and The Factor Theorem as well as a couple of other useful Facts.
Graphing Polynomials – In this section we will give a process that will allow us to get a rough sketch of the graph of some polynomials. We discuss how to determine the behavior of the graph at \(x\)
-intercepts and the leading coefficient test to determine the behavior of the graph as we allow x to increase and decrease without bound.
Finding Zeroes of Polynomials – As we saw in the previous section in order to sketch the graph of a polynomial we need to know what it’s zeroes are. However, if we are not able to factor the
polynomial we are unable to do that process. So, in this section we’ll look at a process using the Rational Root Theorem that will allow us to find some of the zeroes of a polynomial and in special
cases all of the zeroes.
Partial Fractions – In this section we will take a look at the process of partial fractions and finding the partial fraction decomposition of a rational expression. What we will be asking here is
what “smaller” rational expressions did we add and/or subtract to get the given rational expression. This is a process that has a lot of uses in some later math classes. It can show up in Calculus
and Differential Equations for example. | {"url":"https://tutorial.math.lamar.edu/Classes/Alg/PolynomialFunctions.aspx","timestamp":"2024-11-14T11:49:42Z","content_type":"text/html","content_length":"73652","record_id":"<urn:uuid:3f131567-a8a0-4406-8a05-b454881a5d10>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00413.warc.gz"} |
Calculating with quandles
Here you can find supplementary materials, mostly data and programs in GAP, related to our papers on quandles.
Milan Cvrček, David Stanovský, Finite simple quandles revisited (2024)
Alexander Hulpke, David Stanovský, Petr Vojtěchovský, Connected quandles and transitive groups (2016)
Přemysl Jedlička, Agata Pilitowska, David Stanovský, Anna Zamojska-Dzienio, The structure of medial quandles (2015)
Andrew Fish, Alexei Lisitsa, David Stanovský, A combinatorial approach to knot recognition (2015)
Diane Donovan, Terry Griggs, Thomas McCourt, Jakub Opršal, David Stanovský, Distributive and anti-distributive Mendelsohn triple systems (2016)
(yes, these things are also quandles!) | {"url":"https://www.karlin.mff.cuni.cz/~stanovsk/quandles/","timestamp":"2024-11-02T03:01:25Z","content_type":"text/html","content_length":"2687","record_id":"<urn:uuid:5da2d10d-c7a5-4634-ae63-e17d2bd6a4e9>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00830.warc.gz"} |
Quantum Gravity - Inter
Quantum Gravity
Over the centuries, physics as a science has evolved through a close interplay between experiment and theory. However, every once in a while, new phenomena have been discovered which do not conform
to existing scientific paradigms. This gives root to radical new ideas which stand in direct conflict to traditional ways of thinking. The history of physics has long been embroiled with such
conflicts of paradigms. Consider the friction between classical and quantum physicists of the early 20th century. This shook the very foundations of physics, thus sowing the seeds for a quantum
revolution. It is precisely this tension brewing at the interface of antipodal schools of thought which leads to the emergence of new fundamental principles. The objective of a more fundamental
theory is then to resolve disparities in preceding paradigms and thereby operate as one unified framework.
The quest for a fundamental theory of gravity is one such instance where the resolution of conflicting paradigms has led physicists to revolutionary new ideas. That gravity is a universal attractive
force between objects is something that has been known since the time of Isaac Newton in 1687. However, it was only with Einstein’s theory of general relativity1 in 1915, that a concrete mathematical
theory of classical gravity was formulated. This is the classical paradigm of gravity.
Then, in the early nineteen hundreds, came the quantum revolution, lead by the likes of Schrödinger, Heisenberg and Dirac. Now quantum mechanics is a theory of microscopic particles such as atoms and
electrons, and their interactions. These are objects typically characterized as small in size and light in weight. Classical gravity, on the other hand, is a theory of macroscopic bodies, which are
typically large in size and heavy in weight. So the question is, how should one describe the physics of objects, which are small in size and yet very heavy due to the huge mass they carry? Black
holes2 serve as fitting examples of such extreme density objects in our universe. To explain these and, more fundamentally, to comprehend gravity itself, a complete quantum description of gravity is
necessary. It was Einstein’s dream to find a unified description of gravity that reconciled the classical and quantum viewpoints. Though a complete answer to this problem is still in the making, a
promising candidate for such a quantum theory of gravity emerges in the form of string theory. In addition to unifying Einstein’s theory to quantum mechanics, string theory seeks to go even further
and unify all the forces of nature in such a way that they can be understood through a common set of fundamental principles.
Gravity: Nature’s Enigmatic Force
Cosmic ingredients
Our physical universe comprises matter, radiation and their interactions (some would add ‘information’, but here we take the stance that information is encoded in the interactions). The fundamental
building blocks of matter are particles called fermions3 – examples of fermions are quarks4 and electrons. The building blocks of radiation are particles called bosons5 – examples being photons,
gluons6. The interactions constitute the four fundamental forces of nature: (1) Electromagnetic force – which holds together the electrons in an atom; (2) Strong nuclear force – which holds together
the quarks in a proton; (3) Weak nuclear force7 – this interaction facilitates nuclear fusion reactions in stars like our own sun; and (4) Gravity – the attractive force between massive objects.
Interestingly, while electromagnetism and gravity have been known since eons, the two nuclear forces were discovered relatively recently, only in the latter half of the 20th century … nearly after
the time of Einstein.
An unlikely union
Before the advent of quantum mechanics and Einstein’s special theory of relativity8 in the 20th century, much of physics was based on Newton’s principles of dynamics. Quantum theory shook the very
foundations of the Newtonian paradigm and presented us with a whole new world which behaves very differently at microscopic scales (that of atoms and nuclei), while aggregating to classical laws at
large distances (those between billiard balls and planets, say). Moreover, quantum mechanics and special relativity gelled together easily to give rise to what we now call relativistic quantum
theories. However, general relativity, which is a theory of classical gravity, has remained largely unaffected by the quantum bandwagon. Reconciliation between the two was expected to deliver what
would be the ultimate theory of nature: quantum gravity. Unifying these paradigms of classical gravity with quantum theory is what lies at the heart of present day research in theoretical physics.
So what do these quantum theories tell us about the fundamental forces? From the perspective of these theories, a force between two fermions is mediated by the exchange of a boson. For instance, the
repulsion between two electrons is facilitated by the emission and absorption of a photon. Photons are the force carriers of electromagnetism. Similarly, the gluons are carriers of the strong nuclear
force between quarks, while the W and Z bosons9 do the same for the weak nuclear force. Moreover, the predictions of quantum theory for each of these three forces conform perfectly to experimental
evidence. Put together, the quantum description of these three forces is what is called the standard model of particle physics.
On the other hand, according to Einstein’s general theory of relativity, gravity is a property of space-time10. From this point of view, space-time is a dynamic rather than a static entity, whose
geometry is responsible for the gravitational attraction between massive bodies. The presence of matter has the effect of distorting the ‘shape’ of the space-time around it. You can create visual
analog for this distorting effect by modeling space as a two-dimensional rubber sheet, held stretched. Placing a heavy billiard ball, which could represent a massive object such as a star, at the
centre, has the effect of distorting the rubber sheet due to its weight. The distortion is greatest at the location of the ball, and gradually diminishes towards the edges of the sheet. If we were
then to make things more interesting by throwing a very small and light pebble onto the sheet, the curvature of the sheet would allow the pebble to orbit around the heavy ball for a while. This
models the attraction of a planet around a star as something that results from the distortion of space around it. In fact, this distortion effect exactly accounted for anomalies observed in Mercury’s
orbit around the sun and this served as one of the earliest pieces of evidence for Einstein’s theory. Similarly, matter distorts time, but that is much harder to visualize.
Returning to the standard model of the electromagnetic and strong/weak nuclear force, we saw that those theories depicted force as a boson exchange between fermions. But in Einstein’s theory force
has something to do with the ‘shape’ of space-time. Therefore, there is apparently no simple way to extend those quantum theories to incorporate gravity. A quantum theory of gravity requires a very
new approach which unifies Einstein with the quantum. In a quantum theoretic approach a force is necessarily described in terms of its fundamental building blocks, the bosons that transmit this
force. The carriers of gravity are called the gravitons11. However, it turns out that gravitons cannot be found in the spectrum of any of the conventional quantum theories, which work for the other
forces. A full description of quantum gravity should reconcile these two notions of force, one as an exchange of gravitons and the other as a manifestation of space-time geometry. String theory is
one such attempt to answer these questions. And, remarkably enough, the resolution of some of these paradoxes has stretched our intuitions of space, time and dimensions to their limits.
The World According to String Theory
What is string theory?
The fundamental units of string theory are not particles, but one-dimensional objects called strings. The ends of a string are either joined together or open-ended; the former are called closed
strings, the latter open strings. Depending on the energy they possess, strings can vibrate with different frequencies. Analogous to the chords of a musical instrument, a string of a given length and
fixed tension has a discrete range of vibrating frequencies (modes or musical harmonics as they are known). The idea now is that each vibrating mode of a string represents a particle of nature. Low
energy vibration modes correspond to lighter mass particles, high-energy modes correspond to more massive particles. Remarkably, the spectrum of these vibrations includes the building blocks of
matter, radiation and gravity all in one package. Moreover, strings can interact with each other: two closed strings intersect each other at a certain point and open up to form another closed string
– a visual analog (although it includes an extra dimension) would be to think of soap bubbles hitting each other and forming another bubble of similar shape. Similarly, open strings interact with
other open strings by gluing together at one end to form another open string. This is a consistent quantum theory of interacting strings. The world is then fundamentally comprised of such quantum
strings and the different particles we see around us are only manifestations of these strings in different vibration modes. Imagine a ball of a wound up woolen string: that is what a particle would
look like. Add some vibration to your woolen ball and there you have a toy model for a specific particle. In this way, vibrating strings can account for the carriers of all four forces of nature,
including gravity. At microscopic scales, gravity is indeed described as the exchange of gravitons, while at macroscopic scales string theory reproduces Einstein’s results. Thus, a quantum
description of gravity forced upon us a major paradigm shift from the traditional way of thinking in particles to a revolutionary new stringy way of thinking.
Confronting extra dimensions
In string theory, the notion of dimensions takes on a whole new meaning. These can either be macroscopic – the familiar three directions of space and one axis of time; or microscopic. The macroscopic
dimensions can be visualized as, say, co-ordinates on a map (at least for the spatial dimensions) that extend outward towards infinity and denote the location of a point. The microscopic dimensions
are, instead, rolled up into themselves, like a circle projecting out of every point on a map – only a sufficiently small bug, whose size is smaller than the radius of this circle, will be able to
walk into this extra dimension on our map, whereas big bugs only see the map as a grainy surface. String theory predicts six of these extra curled up dimensions12 in our universe and the ‘bugs’ that
can probe into these new spaces are the strings themselves. So how small should a string be in order to sense the effects of these microscopic dimensions? A string length is typically of the order of
10-33 cm (this is 1 divided by 10 to the power 33); compare this to the typical diametric size of an atomic nucleus, which is in the order of 10-17 cm (1 divide by 10 raise to 17) – this is
10,000,000,000,000,000 times larger than the length of a string!
Length scales, energy scales and unification of forces
Now let us get a feeling for the interplay between length and energy scales in order to understand at what point the effects of string theory become relevant and how that leads to a subsequent
unification of forces. In physics, the scale of length is inversely proportional to that of energy, meaning that shorter distance interactions occur at higher energies and vice-versa. So the energy
involved in billiard ball collisions is much less than that involved in nuclear collisions of the type achieved in nuclear reactors. By the same logic, energies of dynamical processes that directly
involve string interactions are of the order of cosmic events such as cataclysmic stellar explosions or the big bang itself. The shortest distance scales that present-day technology can probe are
processes that occur inside protons and neutrons (quark interactions) and the energy required to probe such interactions lies in the order of giga (109 – that is 10 to the power 9) electron volt
units (denoted as GeV13). The most massive/heaviest fundamental particle observed in a lab is the ‘Top Quark’ carrying a mass that is the energy equivalent of 174 GeV. Collision experiments in
particle accelerators have to be performed at such high energies in order to observe these massive particles and their short distance interactions. In comparison, creating a light particle like an
electron only requires 0.0005 GeV of energy. The hotly anticipated accelerator in Geneva, the LHC12 will, on the other hand, be able to achieve energies of up to 7000 GeV!
The question, then, is what the energy scale is at which string theory interactions can be probed by an experimenter? This is called the Planck Scale and it stands at 1019 GeV. Unfortunately, this is
far beyond the reach of current laboratory technology. But these are precisely the energy scales relevant to the processes that occurred during the early history of the universe, just after the big
bang, and string theory provides the appropriate theoretical framework to answer some of those questions. But, at a fundamental level, there is something very significant about the Planck scale. It
is crucial for understanding the possible unification of forces. Moreover, this is the energy scale at which the quantum nature of gravity becomes manifest. The reason is, quite simply, that the
strength of the fundamental forces is not the same at every energy scale. It does in fact vary as we probe physical processes at different energies. For example, in experiments involving billiard
balls or even atomic processes, the force of gravity is much weaker than the weak nuclear force, which in turn is weaker than the electromagnetic force between charged bodies, and that itself is
weaker than the strong nuclear force. Moving up to about 102 GeV, the regime of sub-nuclear interactions, this gap begins to shrink. Though gravity is still the weakest and the strong nuclear force
still the strongest, electromagnetism is now of equal strength to the weak nuclear force. In fact, at this scale, the latter two forces unify into a single force called the electroweak force. Tuning
this further up to 1014 GeV, one enters the domain of so-called grand unification theories, wherein the electroweak and strong nuclear forces are expected to be of equal strength and unify. And
finally, at the Planck scale and beyond, gravity is as strong a force as the others and its effects have to be considered on a par with the other forces. This is what physicists mean by the
unification of all fundamental interactions and string theory stands out as a promising candidate for that endeavor.
The challenge ahead
Though string theory has enjoyed a limited to fair amount of success in explaining some of the puzzles in cosmology and black hole physics, the challenges are far from surmounted and, as things stand
right now, it is fair to say that there are more questions on the table than we have answers to. Moreover, in the last two decades, string theory itself has been constantly evolving, with
increasingly refined machinery entering the game. It has now been understood that the theory not only comprises one dimensional strings, but also a restricted class of higher dimensional membranes
(called D-branes in stringy jargon), which can themselves emit and absorb strings. The theory as such is still very much a work in progress.
Most of what string theory can currently say holds for physics at very high energies like the Planck scale. Experimental signatures for phenomena at these scales are currently sparse due to the huge
divide between technology and theory, the latter being far ahead, and that unfortunately hinders many verifications of theoretical predictions. The gap between tested physics at energies up to 102
GeV and Planck physics15 at 1019 GeV is pretty wide. There is absolutely no reason to believe that there is no new interesting physics out there yet to be discovered in that intermediate domain. In
one sense, if string theory is a fundamental theory of matter, energy and their interactions, it ought to bridge this divide too – in the same way that quantum mechanics, as a more fundamental theory
than Newtonian mechanics, not only explains collisions between atoms, but also shows how classical dynamics, of say billiard balls, emerge as averages over quantum states (at least in principle).
Despite its remarkable elegance and heavy mathematical machinery, bridging this gap still remains a daunting task for string theory.16
1. General relativity: In 1915, Einstein generalized special relativity to include accelerated motion in space-time. Starting from this, the theory was able to explain classical gravity through the
geometric properties of space-time. One of the predictions of this theory, which was later verified, is that gravity even attracts light!
2. Black holes: Remnants of a massive stellar collapse, whose gravity is so strong that anything which comes within a certain critical radius of a black hole is sucked into it forever – even light
rays. That is why they are called ‘black holes’.
3. Fermions: The class of particles that constitute all forms of matter. Quarks and electrons are typical examples.
4. Quarks: Particles that make up protons and neutrons in atomic nuclei.
5. Bosons: The class of particles that constitute all forms of radiation. Photons and gluons come under this category.
6. Gluons: Carriers of the strong nuclear interaction between quarks.
7. Strong/Weak nuclear force: Two types of nuclear forces exist within atomic nuclei. Both enable short-range interactions within constituents of the nucleus. The stronger one is mediated by gluons,
the weaker of the two, by W and Z bosons. The mechanism that holds protons and neutrons together in the nucleus is governed by the strong force, while the mechanism that is responsible for
radioactive beta-decay is governed by the weak force.
8. Special relativity: Einstein’s 1905 theory stating that all uniform motion (of particles) and measurements thereof are relative with respect to the observer. This had far-reaching implications for
the laws of Newtonian mechanics and shattered the notion that space and time are absolute entities.
9. W and Z bosons: Both these particles are carriers of the weak nuclear force.
10. Space-time dimensions: In general relativity, length, breadth and height are the three dimensions of space. Time is the fourth dimension. Combined together this gives the ‘fabric’ of space-time.
11. Gravitons: Carriers of the gravitational force.
12. These 10 dimensions exist in the formulation of ‘Superstring theory’, as it is also called. A generalization of this theory goes by the name ‘M-theory’, which lives in 11 dimensional space-time.
Currently, very little is known about M-theory.
13. GeV: Giga electron volts = 109 electron volts. An electron volt is a unit of energy used in particle physics. For example, if the entire mass of an electron were converted into energy, it would
be 500,000 electron volts of energy.
14. LHC: The Large Hadron Collider is a particle colliding machine at the European laboratory for nuclear research (aka CERN) in Geneva. When fully operational, it will be well poised to discover new
physics beyond what current particle physics models tell us, and at higher energy scales than what we are able to probe today.
15. Planck physics: Physical phenomena relevant at Planck scale energies, that is at 1019 GeV. The behavior of physical interactions is drastically different as we probe processes at different energy
scales. The Planck scale is the regime of quantum gravity and the unification of forces.
16. Let us also acknowledge other independent approaches to quantum gravity; the two most prominent among them being Loop Quantum Gravity and the theory of Causal Dynamical Triangulations. Owing to
the author’s limited knowledge there is very little to discuss in relation to these.
Recommended literature
Basic reading:
Close, F.E., The Cosmic Onion: Quarks and the Nature of the Universe, Melville, 1986.
Gilmore, R., Alice in Quantumland: An Allegory of Quantum Physics, New York, 1995.
Greene, B.R., The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory, New York, 1999.
Hawking, S.W., A Brief History of Time: From the Big Bang to Black Holes, London, 1988.
NOVA, The Elegant Universe, 2003,http://www.pbs.org/wgbh/nova/elegant/program.html (4 November 2009).
Advanced reading:
Halzen, F. and Martin, A.D., Quarks and Leptons: An Introductory Course in Modern Particle Physics, Hoboken, 1984.
Misner, C.W., J.A. Wheeler and K.S. Thorne, Gravitation, San Francisco, 1970.
Peskin, M.E. and D.V. Schroeder, An Introduction to Quantum Field Theory, Boulder, 1995.
Polchinski, J., String Theory, Vol. 1 : An Introduction to the Bosonic String, Cambridge, 1998.
Polchinski, J., String Theory, Vol. 2 : Superstring Theory and Beyond, Cambridge, 1998. | {"url":"https://intermagazine.nl/editie-22/quantum-gravity/","timestamp":"2024-11-13T11:14:30Z","content_type":"text/html","content_length":"107937","record_id":"<urn:uuid:8dd3b684-3b72-4da1-956a-18c029123ff0>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00422.warc.gz"} |
Decomposition of a square matrix into symmetric and skew-symmetric matrices
This online calculator decomposes a square matrix into the sum of a symmetric and a skew-symmetric matrix.
The calculator below represents a given square matrix as the sum of a symmetric and a skew-symmetric matrix. You can find formulas and definitions below the calculator.
Symmetric matrix
A symmetric matrix is a square matrix those elements are symmetrical with respect to the main diagonal. That is, $\forall i,j:a_{{ij}}=a_{{ji}}$ and $A=A^T$.
Skew-symmetric matrix
A skew-symmetric matrix is a square matrix, those elements are equal and negative with respect to the main diagonal. That is, $\forall i,j:a_{{ij}}=-a_{{ji}}$ and $A^T=-A$.
Decomposition into symmetric and skew-symmetric
Every square matrix with entries from any field whose characteristic is different from 2 can uniquely be decomposed into the sum of a symmetric and a skew-symmetric matrix. This decomposition is
known as the Toeplitz decomposition.
$A = \frac {1}{2} (A+A^T) + \frac {1}{2} (A-A^T)$, where
$\frac {1}{2} (A+A^T)$ - symmetric matrix
$\frac {1}{2} (A-A^T)$ - skew-symmetric matrix
This formula is based on the fact that the sum A+A^T is a symmetric matrix, the difference A-A^T is a skew-symmetric matrix, and scalar multiplication retains these properties.
Similar calculators
PLANETCALC, Decomposition of a square matrix into symmetric and skew-symmetric matrices | {"url":"https://planetcalc.com/9233/","timestamp":"2024-11-03T18:19:39Z","content_type":"text/html","content_length":"38900","record_id":"<urn:uuid:feec5461-9e5d-4778-8de5-38b2f4f33713>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00083.warc.gz"} |
Maybe The Most Important Concept You'll Ever Learn...
• Posts
• Maybe The Most Important Concept You'll Ever Learn...
Maybe The Most Important Concept You'll Ever Learn...
It's Ergodicity All The Way Down
hey mate! Just in case you’ve been forwarded this email but would like to subscribe - click the link here.
The Ergodicity Whisperer…
It’s a wonderful, wonderful idea… but it takes a few examples to get around it. And no-one is better at simply communicating the complexities of this concept than our very own Italian, and guest on
this podcast, Luca Dellanna.
He wrote what is I think the cleanest definition of ergodicity, in his book… wait for it… ‘Ergodicity’. ‘Ergodicity is the difference between the outcome of doing an action once and doing it many
Forward this email or share this podcast episode with a friend that likes Nassim Taleb.
Okay, Luca, welcome sir. You make ergodicity so clearly understandable in the book through examples, but when then I try to go and explain it to somebody, I come up short of words. How does one
describe ergodicity without examples?
Yeah. So for me, ergodicity is the study of the effect of time horizons on decisions and strategies. So what we see is that in the real world there is no such thing as the optimal strategy. It's
always the optimal strategy for a given time horizon. and the study of how, of which strategy is optimal for a time horizon, or how a given strategy changes over different time horizons, that's
Okay. But before we think of any examples, I know it's a very difficult thing to do, but is it possible to tighten that up even a little bit more without saying what it is? What is ergodicity without
giving examples?
So, ergodicity is the difference between the outcomes of doing an action once and doing it many times.
Beautiful. And why does it matter?
Well, it matters because you might compute the outcome of doing an action once and think that if you repeated x times, you will get x times that amount, which in the real world is just wrong. Very
often you will get less than x times that amount. And the study of ergodicity will tell you which actions you need to take to make sure that if you take an action x times, you get as close as
possible to getting x times the returns of doing it once.
Okay. So in the real world, talk about some of the domains that ergodicity applies to, or at least is most applicable and relevant.
The usual domains that come to mind are investing and gambling. We see a lot of examples in which if you have to do a gamble once, you will devaluate some gambles as a positive, such as in poker,
sometimes it pays on a single gamble to take some risks.
But if you have to repeat the gamble, then you must also think about survival. Because in poker, for example, if your aim is to win a tournament, you cannot take the same amount of risks as if your
aim were to win a single hand, to maximize the amount of money in a single hand. The same applies to investments.
Some investments might be a good bet if you're taking the investment once, but they become bad bets if you have to take the investment year over year for 10 or 20 years. And these are, for example,
the investments in which there is a chance that you go bankrupt. And the reason is because if you only consider in one year, going like losing the investment means that you lose the money.
And so if you have an investment that has a 50% chance of returning triple the money and 50% chance of going bankrupt, it's a good investment if you take it once, because the average is that you make
a 50% return. But if you're taking that investment 10 times,
you will think what you get is 50% compared to 10 years, it goes to I don't know how much like triple or something like that. But in reality, if you take that investment 10 times at some point,
you're almost sure that before the 10 years since you will end up bankrupt. And this difference is why it's so important to play your hand, your investments and your life differently if you have a
long time horizon. | {"url":"https://curiousworldview.beehiiv.com/p/maybe-important-concept-youll-ever-learn","timestamp":"2024-11-05T02:57:40Z","content_type":"text/html","content_length":"195318","record_id":"<urn:uuid:ce844be0-f497-4fe9-afda-3737c0757695>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00722.warc.gz"} |
SciFi Saturday – Black Holes
For my
Star Frontiers: Knight Hawks
game, I'm currently in the process of modeling some planets as ping-pong balls (they're almost exactly the right size according to the spatial scale of action). I was trying to come up with some
clever designs (the pack I got has a whole bunch of ping-pong balls), and then it occurred to me: probably the easiest thing to do is spray paint one all-black and call it a black hole. But what
would the effects of such a body be on the tactical game?
There's one official adventure for SFKH that featured a black hole:
#88 (one of my favorite issues of the magazine -- hello,
Marvel Super Heroes
Thor!) contained a scenario called "The Battle at Ebony Eyes" by William Tracy, and it features not one but actually two black holes closely orbiting each other. In this game, ships can orbit the
black holes just like any planet (from one hex away), but the primary wrinkle is that they cause illusory duplicates of everything in the surrounding area -- so basically, all the ships in the battle
have something like a
mirror image
effect going on. Clearly that's a bit gamey-fantastic (although high-gravity lensing is a real thing, as illustrated in the picture above).
So I was wondering: What would the actual gravitational effect of a black hole be? One issue with my ping-pong model (for both planets and black holes) is that the SFKH ship models are at a radically
different scale than the surrounding space scale (something like 1" = 50 meters for the former, 1" = 10,000 km for the latter). So if we set down a planet at the space scale, then it's a whole lot
smaller than the ship models orbiting it, and it looks a little ridiculous. With the black hole maybe I have the option of declaring it to be at the same scale as the ships -- but which is better for
Fortunately, I've
previously worked on alternate orbital possibilities
for smaller or larger planets, and worked up a spreadsheet to quickly summarize the possibilities. The key unknown in that spreadsheet is, what's the mass of my black hole?
Wikipedia comes to the rescue
with a formula relating black hole mass to size. As a simplifying assumption, we'll assume that this is a "Schwarzschild black hole": basically symmetric, with no angular momentum or electric charge,
and so it acts like any other gravitational mass at a distance. Then the radius of the event horizon in kilometers is related to mass by about: r ~ 3 M/M(sun), where
is about 2×10^30 kg. Turning this around algebraically, we get M ~ r×M(sun)/3. So for my two candidate black hole sizes (a ping-pong ball being 40mm or about 1.5 inches in diameter):
Black Hole Type I
-- Ship scale, about 50 meters radius. M ~ r×M(sun)/3 = 0.05×2×10^30/3 ~ 3×10^28 kg.
Black Hole Type II
-- Planet scale, about 8,000 km radius. M ~ r×M(sun)/3 = 8000×2×10^30/3 ~ 5×10^33 kg.
Then I can fill out the orbital spreadsheet and see the results below.
What we see is this: At the smaller ship scale, the black hole is at least conceivably usable in the tactical game. At a range of 50 inches (maybe the very edge of your gaming table), a ship can
orbit at a rate of 4 inches/turn using the black hole's gravity. At a middle range 10 inches, the orbital speed is 8 inches/turn (i.e., making a cycle about every 8 turns or so, kind of like standard
orbiting behavior in the core game). At a short range of 2 inches (like standard orbit expectation), an orbiting vessel would flash around at a speed of almost 20 inches, that is, almost making two
complete orbits every turn!
If we consider using the planet-scale black hole, then things get quasi-comical (remember that the black hole is on the order of a billion times more dense than a like-sized planet). Even on the
furthest edge of your standard playing space the orbital velocity is 1,500 inches (i.e., about 5 full orbits around the perimeter of your table in a single turn). At a mid-range the speed is over
3,000 inches (almost 60 orbits in a turn), and at standard close orbit the speed approaches 8,000 inches (a whiplash-inducing 600+ cycles in a single game turn).
So clearly if I use this ping-pong modeled black hole in my
Star Frontiers: Knight Hawks
game, then I'll declare it to be at ship-scale of about 50 meters radius, with effects that are reasonable near the edge of the board, and challenging to deal with (but not utterly insane) near the
center. Also this has the advantage of looking nicer, next to the same-scale ship miniatures. You could consider using different-sized black holes in your game but the preceding is about the range of
See Wikipedia
: Micro-black holes with 0.1mm size and Moon-mass would have no effect on ship movement; Stellar-size and above would already be several hundred times more powerful than my "Type I" above, and thus
likely unusable for game purposes.
Oh, one final thing: Run into the black hole and you're dead. (Or at least playing a different game system.)
ODS spreadsheet here
if you want it.]
6 comments:
1. Here's one thing that might happen if you smack your assault scout into a black hole:
Is this awesome? Y/N
1. Oh, yes! :-)
2. Cool! Of course, I think you just like ish 88 for all the acceleration calculations in the articles on falling damage. I'd love to claim I had that from memory, but I just finished perusing my
Dragon collection for articles to use in my AD&D game.
1. You nailed it! (Link) Some people anathematize that stuff, but I think it's delightful and helped hook me on the game.
3. I love the analysis.
Of course, naturally occurring black holes, like the ones in Ebony Eyes are supposed to be, have a minimum size of 3 solar masses (6x10^30 kg). Less mass and they don't form black holes but
neutron stars. So the "ship size" black holes, which are 200 times smaller, wouldn't actually occur in nature without some really weird physics going on.
There are theories that say some that size might have been created during the formation of the universe right after the big bang, but nothing in that size has ever been detected or even hinted
1. Huh, well that's good to know. Thanks for the information! | {"url":"https://deltasdnd.blogspot.com/2013/08/scifi-saturday-black-holes.html","timestamp":"2024-11-10T16:19:36Z","content_type":"application/xhtml+xml","content_length":"120223","record_id":"<urn:uuid:004b9f1b-ed42-4718-9a58-c1b74f4bf52a>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00354.warc.gz"} |
Fermat spiral
From Encyclopedia of Mathematics
A planar transcendental curve the equation of which in polar coordinates has the form
To each value of $\phi$ correspond two values of $\rho$ — a positive and a negative one. The Fermat spiral is centrally symmetric relative to the pole, which is a point of inflection. It belongs to
the class of so-called algebraic spirals.
They were first studied by P. Fermat (1636).
• [1] A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian)
[a1] J.D. Lawrence, "A catalog of special plane curves" , Dover, reprint (1972)
How to Cite This Entry:
Fermat spiral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fermat_spiral&oldid=52724
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article | {"url":"https://encyclopediaofmath.org/wiki/Fermat_spiral","timestamp":"2024-11-09T00:59:16Z","content_type":"text/html","content_length":"14505","record_id":"<urn:uuid:f93b5c98-2c46-4be7-945b-fd2c31b41234>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00330.warc.gz"} |
Convert decimal integer to its base-n representation
baseStr = dec2base(D,n) returns a base-n representation of the decimal integer D. The output argument baseStr is a character array that represents digits using numeric characters and, when n is
greater than 10, letters. For example, if n is 12, then dec2base represents the numbers 9, 10, and 11 using the characters 9, A, and B, and represents the number 12 as the character sequence 10.
If D is a numeric vector, matrix, or multidimensional array, then baseStr is a two-dimensional character array. Each row of baseStr represents an element of D.
baseStr = dec2base(D,n,minDigits) returns a base-n representation of D with no fewer than minDigits digits.
Convert Decimal Number
Convert a decimal number to a character vector that represents its value in base 12. In this base system, the characters 'A' and 'B' represent the numbers denoted as 10 and 11 in base 10.
D = 23;
baseStr = dec2base(D,12)
Specify Number of Digits
Specify the number of base-12 digits that dec2base returns. If you specify more digits than are required, then dec2base pads the output with leading zeros.
D = 23;
baseStr = dec2base(D,12,6)
If you specify fewer digits, then dec2base still returns as many digits as required to represent the input number.
baseStr = dec2base(D,12,1)
Convert Numeric Array to Octal Values
Create a numeric array.
To represent the elements of D as octal, or base-8, values, use the dec2base function. Each row of baseStr corresponds to an element of D.
baseStr = 3x4 char array
The dec2base function returns a character array padded with leading zeros. Starting in R2016b, the compose function is recommended for converting numeric arrays to octal representations. It returns a
string array whose elements do not have leading zeros. To represent the elements of D as octal values, use the %o formatting operator.
hexStr = 1x3 string
"1777" "172" "16"
Input Arguments
D — Input array
array of nonnegative numbers
Input array, specified as an array of nonnegative numbers. Each element of D must have a value between zero and the value returned by flintmax.
• If D is an array of floating-point numbers, and any element of D has a fractional part, then dec2base produces an error. For example, dec2base(10,8) converts 10 to '12', but dec2base(10.5,8)
produces an error.
• If D is a character or logical array, then dec2base treats the elements of D as integers. However, dec2base treats characters as their Unicode^® values, so specifying D as a character array is
not recommended.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | logical | char
n — Base of output representation
integer between 2 and 36
Base of output representation, specified as an integer between 2 and 36. For example, if n is 8, then the output represents base-8 numbers.
minDigits — Minimum number of digits in output
Minimum number of digits in output, specified as an integer.
• If D can be represented with fewer than minDigits digits, then dec2base pads the output with leading zeros.
• If D is so large that it must be represented with more than minDigits digits, then dec2base returns the output with as many digits as required.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
Thread-Based Environment
Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.
This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.
Version History
Introduced before R2006a | {"url":"https://nl.mathworks.com/help/matlab/ref/dec2base.html","timestamp":"2024-11-02T12:46:43Z","content_type":"text/html","content_length":"92868","record_id":"<urn:uuid:2a692890-40e2-4651-8379-42da3d54cce8>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00262.warc.gz"} |
Logarithmic transformations for skewed variables
0 Comments
Logarithmic transformations adjust skewed distributions
Analyze skewed data using more powerful parametric statistics
Logarithmic transformations
are powerful statistical tools when employed and interpreted in the correct fashion. Transforming the distribution of a continuous variable due to violating normality allows researchers to
account for outlying observations
use more powerful parametric statistics
to assess any significant associations.
Also, some continuous variables are
naturally skewed
. One particular outcome that is prevalent in medicine is LOS or length of stay in the hospital. Most patients will be in the hospital between one and three days, VERY FEW will be in the hospital for
weeks and months at a time. In order to include these outlying patients in analyses, transformations must be performed. Naturally skewed variables can be analyzed with parametric statistics with
An important thing to remember when conducting logarithmic transformations is that
only the p-value associated with inferential statistics can be interpreted
, NOT the means and standard deviations of the transformed observations. Instead, researchers should
report the median and interquartile range
for the distribution.
0 Comments
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From Encyclopedia of Mathematics
A transformation of Euclidean space with respect to a certain point $O$ that brings each point $M$ in a one-to-one correspondence with a point $M'$ on the straight line $OM$ in accordance with the
where $k$ is a constant number, not equal to zero, which is known as the homothety ratio. The point $O$ is said to be the centre of the homothety. If $k>0$, the points $M$ and $M'$ lie on the same
ray; if $k<0$, on different sides from the centre. The point $O$ corresponds to itself. A homothety is a special case of a similarity. Two figures called homothetic (similar or similarly situated) if
each one consists of points obtained from the other figure by a homothety with respect to some centre.
Simplest properties of a homothety. A homothety with $k\neq1$ is a one-to-one mapping of the Euclidean space onto itself, with one fixed point. If $k=1$, the homothety is the identity transformation.
A homothety maps a straight line (a plane) passing through its centre into itself; a straight line (a plane) not passing through its centre into a straight line (a plane) parallel to it; the angles
between straight lines (planes) are preserved under this transformation. Under a homothety segments are mapped into parallel segments with a length which is $|k|$ times the original length, i.e. a
homothety is a contraction (expansion) of the Euclidean space at the point $O$. Under a homothety a sphere is mapped into another sphere, and the centre of the former is mapped to the centre of the
A homothety is most often specified (geometrically) by the homothety centre and a pair of corresponding points or by two pairs of corresponding points. A homothety is an affine transformation with
one (and only one) fixed point.
In $n$-dimensional Euclidean space a homothety leaves the set of all $k$-dimensional subspaces invariant, $k<n$.
A homothety is defined in a similar manner in pseudo-Euclidean spaces. A homothety in Riemannian spaces and in pseudo-Riemannian spaces is defined as a transformation that transforms the metric of
the space into itself, up to a constant factor. The set of homotheties forms a Lie group of transformations, and the $r$-parameter homothety group of a Riemannian space contains the $(r-1)$-parameter
normal subgroup of displacements.
A homothety is also called a central dilatation (cf. also Dilatation).
[a1] M. Berger, "Geometry" , 1–2 , Springer (1987) (Translated from French)
[a2] H.S.M. Coxeter, "Introduction to geometry" , Wiley (1961)
[a3] E. Artin, "Geometric algebra" , Interscience (1957)
How to Cite This Entry:
Homothety. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Homothety&oldid=43790
This article was adapted from an original article by I.P. Egorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article | {"url":"https://encyclopediaofmath.org/index.php?title=Homothety&oldid=43790","timestamp":"2024-11-09T00:25:50Z","content_type":"text/html","content_length":"17093","record_id":"<urn:uuid:f8a1af4f-e207-4b3c-9869-89fc628a8940>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00265.warc.gz"} |
American Mathematical Society
A transform analogous to the discrete Fourier transform may be defined in a finite field, and may be calculated efficiently by the ’fast Fourier transform’ algorithm. The transform may be applied to
the problem of calculating convolutions of long integer sequences by means of integer arithmetic. References
W. M. Gentleman, "Matrix multiplication and fast Fourier transformations," Bell System Tech J., v. 47, 1968, pp. 1099-1102. G. D. Bergland, "A guided tour of the fast Fourier transform," IEEE
Spectrum, v. 6, no. 7, 1969, pp. 41-53. L. I. Bluestein, A Linear Filtering Approach to the Computation of the Discrete Fourier Transform, Northeast Electronics Research and Engineering Meeting
Record, v. 10, 1968, pp. 218-219. G. H. Hardy & E. M. Wright, An Introduction to the Theory of Numbers, Clarendon Press, Oxford, 1938. N. S. Szabo & R. I. Tanaba, Residue Arithmetic and its
Applications to Computer Technology, McGraw-Hill, New York, 1967. W. K. Pratt, J. Kane & H. C. Andrews, "Hadamard transform image coding," Proc. IEEE, v. 57, 1969, pp. 58-68. W. T. Cochran et
al., "What is the fast Fourier transform?" Proc. IEEE, v. 55, 1967, pp. 1664-1674.
Similar Articles
• Retrieve articles in Mathematics of Computation with MSC: 65T05
• Retrieve articles in all journals with MSC: 65T05
Additional Information
• © Copyright 1971 American Mathematical Society
• Journal: Math. Comp. 25 (1971), 365-374
• MSC: Primary 65T05
• DOI: https://doi.org/10.1090/S0025-5718-1971-0301966-0
• MathSciNet review: 0301966 | {"url":"https://www.ams.org/journals/mcom/1971-25-114/S0025-5718-1971-0301966-0/?active=current","timestamp":"2024-11-11T10:29:58Z","content_type":"text/html","content_length":"61728","record_id":"<urn:uuid:e73a0e52-1b3c-401f-b417-adf0f87ae7cc>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00697.warc.gz"} |
Digital Electronics Key Terms
Lesson 1.2 Key Terms
Introduction to Circuit Design
The instantaneous voltage of a waveform. Often used to mean maximum amplitude, or peak voltage, or a pulse.
Boolean Expression
An algebraic expression made up of Boolean variables and operators, such as AND (-), OR (+), or NOT (-). Also referred to as Boolean function or a logic function.
Clocked D Flip-Flop
Type of flip-flop in which the D (data) input is the synchronous input.
Digital Waveform
A series of logic 1s and 0s plotted as a function of time.
Dual In-Line Package (DIP)
One style of integrated circuit package which has two rows of lead.
Duty Cycle (DC)
Fraction of the total period that a digital waveform is in the HIGH state. DC = th/T (often expressed as a percentage: %DC = th/Tx100%).
Falling Edge
The part of a pulse where the logic level is in transition from a HIGH to a LOW.
A sequential circuit based on a latch whose output changes when its CLOCK input receives a pulse.
The number of cycles per unit time of a periodic waveform.
Hertz (Hz)
Unit of frequency. One hertz equals one cycle per second.
Integrated Circuit (IC)
An electronic circuit having many components, such as transistors, diodes, resistors, and capacitors, in a single package.
Also called a NOT gate or an inverting buffer. A logic gate that changes its input logic level to the opposite state.
Logic Diagram
A diagram, similar to a schematic, showing the connection of logic gates.
A piece of test equipment used to view and measure a variety of different waveforms.
The amount of time required for one complete cycle of a periodic event or waveform.
Propagation Delays (tPLH/tPHL)
Delay from the time a signal is applied to the time when the output makes its change.
Schematic Entry
A technique of entering CPLD design information by using a CAD (computer aided design) tool to draw a logic circuit as a | {"url":"https://www.studymode.com/essays/digital-electronics-key-terms-59602731.html","timestamp":"2024-11-02T04:17:36Z","content_type":"text/html","content_length":"92892","record_id":"<urn:uuid:586d9914-68f5-448e-b474-7739af730223>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00519.warc.gz"} |
(HP15C)(HP67)(HP41C) Bernoulli Polynomials
09-01-2023, 11:53 AM
Post: #16
John Keith Posts: 1,067
Senior Member Joined: Dec 2013
RE: (HP15C)(HP67)(HP41C) Bernoulli Polynomials
For B(16) using my program listed above in approximate mode (all floating-point numbers) I get exactly 7.1. The largest term in the 15th row of A163626 is ~10^14 so some rounding is inevitable, and
would certainly be worse for 10-digit calculators.
The largest row with all numbers < 10^10 is row 11 which should allow accurate computation of B(12). The value of B(12) to 12 digits is -0.253113553114, while the program returns -0.25312, which has
only four correct digits. The exact value of B(12) is -691/2730 so we should expect more accurate results.
Your idea of scaling the terms by their LCM is interesting but I can't see how one could compute the LCM on calculators limited to 10-digit floats.
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Reverse Mortgage Calculator
A reverse mortgage enables property owners to borrow funds using their home equity as security. A reverse mortgage calculator allows a proper estimation of how much you can borrow, depending on your
unique circumstances. However, before using a reverse mortgage repayment calculator, you should know what this home loan type entails. It is available for Australian residents who are aged 60 or
above. It helps homeowners access an income source or necessary funds in the sunset years of their lives.
Interest is charged on the funds, although the borrower does not require regular repayments. The debt, additional interest, and other fees are repaid from the property sale, either at the time of
sale or upon the demise of the last surviving borrower. Using a reverse mortgage loan calculator will help you work out the loan you are eligible for and the total amount to be repaid.
The interest rates are usually higher than conventional home loans, and you can reside in your property without issues. Before learning how to use a reverse mortgage payment calculator, you should
know that lenders set varying criteria for issuing these loans. They have stringent guidelines for the LVR (loan-to-value ratio) that borrowers can release. This is influenced by life expectancy and
age. Experts feel that those above 60 may usually get 15-20% of the property value as their reverse mortgages while adding 1% for every additional year henceforth.
What is a Reverse Mortgage Calculator?
A reverse mortgage repayment calculator helps you choose whether a reverse mortgage is the best option to cater to your specific requirements at a certain stage of life. It will help you estimate how
much you can borrow.
A reverse mortgage loan calculator will also help you estimate the cost of a reverse mortgage over varying durations, such as ten years or even 20 years. It will also allow you to view changes in
your home equity over time based on the home value assumptions and the interest rate.
How Does UM Oceania's Reverse Mortgage Calculator Work?
UM Oceania offers an easy-to-use reverse mortgage calculator to help you understand what your reverse mortgage will cost you. This calculator takes you through several steps to determine the final
You must enter all the necessary details and work it out yourself. This calculator simplifies the entire process. However, you have to specify three things, i.e. the amount you can borrow, the ways
you can receive the payment, and the details of the mortgaged property.
How to use the Reverse Mortgage Calculator?
Using the reverse mortgage eligibility calculator is a seamless process. Here are the key points that you need to keep in mind:
• Step 1- Enter the youngest borrower’s age, the property's estimated value, the protected equity percentage, and the maximum borrowing ability are calculated automatically.
• Step 2- Choose your payment option, i.e. a lump sum, monthly payments, or a combination of both and then enter the amount you need (monthly or lump sum). Choose the mortgage term in years if you
select the monthly payments or combination options.
• Step 3- Estimate the property growth rate. Enter the interest rate of the mortgage and also the monthly fees for the mortgage. You can also enter the upfront costs of the mortgage in this step.
The reverse mortgage calculator will help you view your results accordingly. These include the LVR (loan-to-value ratio), total loan amount, and, most importantly, the changes in your equity over a
sustained period. It will also tell you when your equity drops to zero.
Features and Benefits of the Reverse Mortgage Calculator
Some of the most significant benefits and features of the UM Oceania reverse mortgage calculator include the following:
• Inclusion of Protected Equity
• Options for choosing lump sum, monthly or combination payments
• Options for entering upfront mortgage costs, estimated property growth, and monthly mortgage charges
• In-depth depiction of LVR through pie charts, along with showcasing the change in equity over time in relation to the loan balance and value of the property.
• Extensive amortization schedule available for users.
• Automatically shows maximum borrowing capacity.
How is the Reverse Mortgage Calculated?
The calculation of the reverse mortgage is done based on several factors. The loan cost depends on the following factors:
• The amount borrowed
• The method of taking the funds, i.e. as monthly payments or a lump sum amount. A lump sum will cost you more owing to compounding interest.
• The mortgage fees and interest rate. Some of these fees include valuation charges, loan establishment fees, ongoing fees, and so on.
The key principle behind the calculation is that the homeowner’s equity will reduce, with growth in debt over a specific duration. For example, suppose you are decided on the following parameters
while using a reverse mortgage calculator:
• Age of youngest borrower- 60
• Estimated property value- $800,000
• Protected Equity- 0%
• Payment Option- Monthly Payments.
• Monthly Payment Required- $2,000.
• Mortgage Term- 5 years.
• Estimated property growth- 4%.
• Mortgage interest rate- 8.50%.
• Monthly mortgage fees- $10.
• Mortgage upfront costs- $1,500.
In this case, the calculator shows $120,000 as the amount you can borrow. The other results are the following:
• The LVR (loan-to-value ratio) is 15%.
• The equity falls to zero at > 40 years.
If you look closer at the amortization sheet, you will find that in the 40^th year and 480 months, the loan balance will be $2,970,969, while the property value will be $3,951,897. At the same time,
the remaining equity will stand at $980,928. This is only an illustrative example; you can use the reverse mortgage loan calculator with specific figures and estimates per your requirements.
Pros and Cons of the Reverse Mortgage Calculator
There are many advantages of using a reverse mortgage calculator, and some of them include the following:
• Instant calculation of LVR and the point where equity comes down to zero.
• Detailed amortization schedule showing changing equity, loan balance, and property value.
• Huge scope for personalization with inclusions for fields like protected equity, age, mortgage interest, charges, method of receiving the funds, and so on.
• Easy calculation of maximum mortgage amount.
• It enables better decision-making for homeowners, who can contrast different scenarios and determine whether a reverse mortgage suits them.
While there are no disadvantages, the only thing worth mentioning here is that while using the calculator, homeowners should be cautious about the figures and scenarios they opt for. Choosing the
wrong structure for the loan will not be good in the long run.
Hence, whatever the calculator's results, they should be shared with financial advisors to take the right decisions.
Things you Should Know About the Reverse Mortgage Calculator
Here are some other things you should know about reverse mortgage calculators:
• These calculators work based on several assumptions, including the property value, mortgage interest rate, etc.
• They can be used for free without paying a single penny.
• The results are instantaneously calculated with detailed amortization sheets to match.
Before deciding, you can use these calculators to compare and contrast various loan structures and numbers.
Frequently Asked Questions
How much money do you get in a reverse mortgage?
The money that you get in a reverse mortgage boils down to several factors, including your age, how you want your funds (monthly payments, lump sum, or a mix of both), the value of the property, the
tenure of the mortgage, your borrowing abilities, and so on. In most cases, those above 60 may get up to 15-20% of the property value in a reverse mortgage.
What is the downside to a reverse mortgage?
Interest rates on reverse mortgages are costlier in comparison to regular home loans. Since regular repayments are not required, the interest accumulates over several years and can balloon into a
sizable cost when selling the property.
What are the three types of reverse mortgages?
Reverse mortgage types may vary across diverse regions/territories. Some common types include single-purpose reverse mortgages, home equity conversion mortgages, and proprietary reverse mortgages.
What are the three primary requirements to qualify for a reverse mortgage?
Some of the major requirements to qualify for a reverse mortgage include the following-
• The borrower should fulfil the lender’s age criteria (at least 60 years of age in most cases).
• The borrower must have significant equity in their
• The property should be the borrower’s principal residence, while there should ideally be no encumbrances or pending mortgages on the property. | {"url":"https://www.umoceania.com.au/calculators/reverse-mortgage-calculator","timestamp":"2024-11-02T20:39:59Z","content_type":"text/html","content_length":"84944","record_id":"<urn:uuid:885f15fd-a6c4-46f2-82d5-0303ceaa4fc0>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00167.warc.gz"} |
CS 540 Homework Assignment # 2 solution
Question 1: Successor Function [50 points]
This is a programming question. The solution to the programming problem should be coded in Java, and
you are required to use only built-in libraries to complete this homework. Please submit a source
code file named successor.java with no package statements, and make sure your program is runnable
from command line on a department Linux machine. We provide a skeleton successor.java code that you
can optionally use, or you can write your own.
The goal of this assignment is to become familiar with the state space in a real-world problem.
You are given three water jugs with capacity A, B, C liters, respectively. A, B, C are positive integers. At
each step, you can perform one of these actions:
• Empty a jug
• Fill a jug
• Pour water from one jug to another until either the former is empty or the latter is full.
Write a program successor.java to print the successor states of an input state.
• Input: 6 integers A, B, C, a, b, c, where (A, B, C) are the capacity of the jugs, and (a, b, c) are the current
amount of water in each jug. You may assume that the input is valid, namely a ≤ A, b ≤ B, c ≤ C,
and A, B, C are positive integers, and a, b, c are non-negative integers.
• Output: Print the list of successor states reachable in one step from (a, b, c), one one each line. The
lines do not need to be sorted. The successors should not include (a, b, c) itself.
Here are some examples of running the program from the command line. The inputs and outputs are
space-separated with no comma. Please follow the same input/output format.
Example 1:
$java successor 3 2 1 3 2 0
Example 2:
$java successor 11 5 2 6 3 1
Question 2: State Space [50 points]
An (m, n, k)-puzzle is a sliding puzzle with m columns, n rows, and k empty squares, where 1 ≤ k < mn. As usual, one move is to move a single tile to an adjacent (up, down, left, right) empty square,
if available. 1. (10 points) How many tiles are there in a (m, n, k)-puzzle in general? For example, there are 8 tiles in a (3, 3, 1)-puzzle. 2. (20 points) Let each tile have a distinct name. How
many distinct states are there in the state space? Show your derivation. 3. (20 points) Draw a graph that corresponds to the state space of a (2, 2, 1)-puzzle, and briefly describe your graph. This
is not an art project: You may represent the states in ways easy for you to type, and you do not necessarily need drawing programs – you may even use plaintext. You can also hand draw the graph, but
please clearly show the nodes and edges. | {"url":"https://jarviscodinghub.com/product/cs-540-homework-assignment-2-solution/","timestamp":"2024-11-02T02:57:21Z","content_type":"text/html","content_length":"103489","record_id":"<urn:uuid:2c5fce0b-1f98-432e-8bcf-5ed76a6ee1d4>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00217.warc.gz"} |
Math Table - Algebrakit
Math Table
Tables occur almost everywhere in math education. Function tables, series, and ratio tables are just a few examples.
Tables are not just collections of values but often represent an underlying relation. Algebrakit can automatically evaluate student input against these relations and provide personalized hints and
error feedback.
You can use Math Table to create many types of tables. Add rows, columns, and headers as needed, and add mathematical expressions, text, or input to cells. Optionally, add arrows above or below the
table. These arrows can have input fields as well.
Example: Function Table
A basic application of a table. Calculate the value in each cell.
Example: Ratio Table
An important tool for fractions and percentages is the ratio table. This tool is available as a specialization of the Math Table question type. Algebrakit will automatically validate if the ratio
table is valid and if the operations on the arrows are correct.
The arrows can also be used on regular (non-ratio) tables. Relevant for investigating linear, exponential and quadratic relations.
Example: Product Sum Method
An example of an open question. Students must find two divisors of -30 with -13 as their sum. They are free to try any set of values, while Algebrakit validates if the numbers are valid divisors and
if they are summed correctly.
Example: Pythagoras Theorem
This is a nice example of a non-standard application demonstrating Math Table’s flexibility.
Segment BC must be in the bottom row, while the order of the two top rows is free. | {"url":"https://algebrakit.com/tabbladen/math-table/","timestamp":"2024-11-11T06:14:10Z","content_type":"text/html","content_length":"89285","record_id":"<urn:uuid:8465a332-e986-4b2a-aa8e-3c42fe80d309>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00180.warc.gz"} |
Blog Archives
Adding Up
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Hold up, I'm calculatin' ... because Common Core are awesome. Yes, I no that aint no proper grammar, but so long's I got the IDEA right, it'r be cool. I still'll get hired someplace that don't care
bout it taken me 10 minutes to count back change in a good-speakin' way, y'all.
Lately, there's been a Common Core hate link circulating social media - it outlines the "Old Fashion" and "New Way" of calculating basic arithmetic, showing a simple math equation of 32-12. Most
people over the age of 10 can simply look at the written equation and automatically reply that the answer is 20, just by doing the math in their heads - which is inarguably the fastest, most
intelligent method to utilize.
But Common Core math wants people to think outside their brains. Using a linear scale, a student is supposed to work upwards through the numbers to "calculate" the correct answer. It requires the
dissection of the primary equation into several other equations, which means the introduction of new numbers to those equations, and ultimately, the addition of those numbers for a math question that
started as subtraction problem. Sound complicated? It is. Better have a pencil and paper handy to work it out.
Common Core advocates say that, by using this method, a student is showing an understanding of how the numbers work. They argue that this is especially beneficial to students that don't understand
the concept of number placement (tens, hundreds, thousands). Additionally, by understanding how numbers work in sequence, students can more easily transition into higher forms of math, like algebra.
It's a noble concept - but it's flawed.
Because, in basic math, numbers are finite. The answers are finite. This is why we can memorize multiplication tables - the answers will not change. Ever. 1+1=2. Always. And when answers are finite,
the simplest, most direct method to obtaining those answers is (or should be) the correct method. What I'm getting at is this: if the method of doing math in your head was in an epic evolutionary
Natural Selection battle with Common Core, the former would win. Because math in your head is easier. Faster. Fit for everyday use.
Common Core math, by this rationale, is utterly archaic, despite its being hailed as an educational break through. Dissecting a math equation to down to a cave-man counting method isn't going to
enhance the mathematical prowess of a student. In fact, I would argue that using a number line to count out a math problem is a crutch. Instead of encouraging a student to remember basic math sums,
and how numbers can work in columns, we're asking them to go outside their intuitive thought process -- and rationalize it. In order for a student to apply Common Core problem solving, they would
need to complicate something that could be very simple. They need paper and pencil to show and track the equations. We're asking students to stop the automatic answer - to stop their thought process
- and programing them to double think something that is an earthen, finite concept...
And what would be the rationale behind getting the new generations of American citizens to double-think something as finite as whole numbers and basic math? Well - if you can double think 1+1=2,
maybe you can double think free economy. Capitalism. Perhaps even morals. Even the basic constructs of freedom. I realize that's a leap in today's world... but over generations, is it really so far
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Keeping Tabs...
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A great list from the American Journal: 130 Reasons our President is a Shmuck.
1. “I will have the most transparent administration.”
2. “I have shovel ready jobs.” most of which were given to corporate friends (ex: Obama Care website)
3. “The IRS is not targeting anyone.” Except that Lois Lerner said otherwise in a public speech.
4. “If four Americans get killed…uh… it is not optimal.” regarding Benghazi
5. “ObamaCare will be good for America.” Except it isn’t.
6. “If you like your doctor, you can keep him, period.” A lie so blatant, he even retracted it
7. “Premiums will be lowered by $2500″ Maybe he meant “raised $2500.”
8. If you like your health insurance, you can keep it, period…… except you can’t.
9. “I did not say you could keep your health care.” (Regardless that 29 recorded videos show I did)
10. “No one making less than $250,000 will see their taxes raised one dime.” Except for Obamacare
11. “Benghazi was because of a youtube video.” Except it happened on a 9/11 anniversary.
12. “If I had a son…” He’d probably turn out to be Treyvon Martin.
13. “I am not a dictator.” Nor are you a crook?
14. “I will put an end to the type of politics that “breeds division, conflict and cynicism”. That worked well, didn’t it?
15. “You didn’t build that.”
16. “I will restore trust in Government.”
17. “The Cambridge police acted stupidly,” he says, before gathering the facts.
18. “I am not after your guns.” Except I kinda am.
19. “The fact that we are here today to debate raising America’s debt limit is a sign of leadership failure.” (Senator BO of 2006) Because debate is un-American?
20. “I have been practicing…I bowled a 129. It’s like — it was like Special Olympics.” Oh, you’re special all right.
21. “I think when you spread the wealth around, it’s good for everybody.” It worked so well in Russia.
22. “The Public Will Have 5 Days To Look At Every Bill That Lands On My Desk,” unless we must pass it before we can find out what’s in it.
23. “It’s not my red line it is the worlds red line.” At which point, nothing happens.
24. “Whistleblowers will be protected.” Unless they’re blowing the whistle on Obama, in which case, they will be hunted down.
25. “We got back every dime we used to rescue the banks, with interest.” Still waiting.
26. “I will close Gitmo.” Except he hasn’t.
27. “The point I was making was not that Grandmother harbors any racial animosity. She doesn’t, — but she is a typical white person.”
28. “I am not spying on American citizens.” Except PRISM.
30. Nelson Mandela funeral smart-phone “selfies.”
31. Goverment shutdown is allll the GOP’s fault
32. Still on vacation in Hawaii on the first day Obamacare is supposed to begin
31. “ObamaCare will lower costs for everyone.”
32. “More Americans will be insured under ObamaCare“
33. “Islam is the religion of peace and tolerance….Muslims are our friends” (then he bows to the muslim leaders)
34. “That’s the good thing about being President, I can do whatever I want.” Because that’s how it works, right?
35. “My father served in WW2.” Um, really?
36. 2011 Arab spring FAIL
37. America’s 2010 Summer of recovery FAIL
38. Hiring a known Palestinian terrorist to work on Obamacare in Illinois. Didn’t bother to check her on e-verify
39. “I promise 100% transparency in my administration.” Except for, you know, anything.
40. “Buying health care insurance will be like using Amazon.”
41. “I will end Income Tax for seniors making less than $50K a year.”
42. “I will bring ALL of our troops home within ONE year.”
43. “I’ll put the Health Care negotiations on CSPAN so everyone can see who is at the table!”
44. “I’ll have no lobbyists in my administration.”
45. DOJ spying on the free press telephone calls
46. Blocking veterans from seeing their own WWII memorials during shutdown
47. Allowing illegals to protest on mall during the same govt shutdown
48. Shutting down White House tours
49. Solyndra bankruptcy cost taxpayers billions.
50. “Obamacare will not be used to fund abortions.”
51. Eric Holder — Just Eric Holder.
52. Millions losing health care coverage
53. RECORD welfare rolls
54. RECORD Hollywood parties on the taxpayer’s dime
55. RECORD campaign tours on the taxpayer’s dime
56. RECORD exorbitant vacations on the taxpayer’s dime
57. RECORD number of golf games of any president, on taxpayer’s dime
58. RECORD secret service agents compared to ANY other president, on the taxpayer’s dime
59. Unconstitutional Obama recess appointment
60. Taking all credit for SEAL Team 6 success of offing Osama. As though he pulled the trigger.
61. Forcing businesses to violate their abortion religious beliefs with Obamacare
62. Obamacare website no-bid contract website-cronyism that cost $634M to build (and the website NEVER worked), amazon.com cost under $50M to build
63. Supporting the Muslim Brotherhood terrorists with arms and money in Syria, on the taxpayer’s dime
64. NSA spying in ALL AMERICAN telephone calls.
65. Proposed amnesty for illegal law breakers
66. Spying on Americans, on American soil, with drones
67. NEVER having a balanced budget
68. CONSTANTLY contracting economy under Obama
69. Russians invade Ukraine, Obama does NOTHING
70. 6 million people losing their healthcare thanks to Obamacare.
71. “The United States maintains a ‘rock-solid’ commitment to Israel,” tell it to John Kerry
72. Illegal fundraising for obamacare by Katherine Sebelius (solicitation of money from the companies she regulates) (and obama does NOTHING)
73. Katherine Sebelius TOTAL incompetence in rolling out obamacare (and obama does NOTHING)
74. ex-IRS official Lerner takes 5th
75. Failed to fund our Nation’s military
76. Higher Spending and No Balanced Budget—EVER
77. $1.2 Trillion in Higher Taxes
78. known obama supporter appointed to investigate IRS targeting scandal
79. Buying clothes at The Gap: “Oh wow; credit card machines had electronic signature pads”
80. Russia laughs at Obama’s Ukraine sanctions.
81. >70% of obamacare signups from people who LOST their insurance BECAUSE OF obamacare in the first place!!
82. In Maryland: Obama celebrates 60,000 obamacare signups…. BUT 73,000 lost insurance BECAUSE OF Obamacare!!
83. Obama did NOTHING to prevent some 150,000 people being killed in Syria, and he and the LIBERALS are silent on the matter.
84. On April 6th, 2014 obama gave a speech on the 20th anniversary of the genocide in Rwanda lecturing about “the world’s failure to respond more quickly” and that “we always have a choice . . . we
must never be indifferent.”
85. Obama demands “equality” but only pays women 88 percent of what it pays men
86. “We’re focused like lasers on job creation!!!”
87. “Jobs are our number one priority!!!!”
88. 6 million full time US workers sustain 149 million “benefit takers” in the USA under the “obama recovery.”
89. “Republicans still can’t bring themselves to admit that the Affordable Care Act is working.”
90. “If Republicans want to spend all their time talking about repealing a law that’s working, that’s their business.”
91. 169th round of golf on 4/19/14 while Putin re-assembles the USSR and Jews are being ordered to report in the Ukraine.
92. Chemical weapons using in Syria, crossing Obama’s “red line” and Obama is SILENT on the matter.
93. 04/21/2014 Obama delays the keystone pipeline for the 45th time.
94. China has become the #1 economy with “ObamaNomics.”
95. Iran WILL have a nuclear bomb under Obama.
96. There are 92 million unemployed (may 2014) THAT IS 51% UNEMPLOYMENT RATE, including retirees
97. In 2013, under the direction of Barack Obama, The Immigration and Customs Enforcement release custody 36,000 illegal immigrants who had been convicted of murder, sexual assault, kidnapping, and
aggravated assault, and drunk or drugged driving.
98. incompetence at the Veterans Affairs Administration. Our veterans died…Obama hires a “coverup” specialist Rob Nabors to assist in the White House’s reputation for deception.
99. Almost every photo of Obama at the White house shows Obama with his feet on the historically significant antique furniture….no respect for its historical significance
100. Over 40 veteran patients die after being placed on a hidden waiting list that could last for up to a year, while officials at the hospital shredded documents and faked evidence to make it seem
as if waiting times were under control.
101. New Horizons in Presidential Dignity: President Obama does the “Shake Shack Shimmy” during his visit to the sandwich joint in Washington Friday May 17th 2014. U6 unemployment rate: 17.8%
102. May 17th 2014: Greater than Fifty million working-age Americans aren’t working, according to the labor department, and Obama pushes immigration plan.
103. “STINKBURGER”
104. King Barry’s “executive actions”
105. Russia signs a contract with Iran to build two more nuclear reactors at its Bushehr power plant as part of a broader deal for up to eight reactors in the Islamic state.
106. New Record: Price of Gas Above $3 Gallon for 1,245 Days…
107. 5-30-14 ILLEGAL negotiation with terrorists: 5 Gitmo Taliban terrorist leaders ILLEGALLY released for one American ARMY DESERTER: BOWE BERGDAHL
108. Obama negotiates with terrorists to release an army deserter because “we can’t leave an American Behind” but does nothing for the Marine being detained illegally in Mexico, less than 1 mile
109. “I’m not sorry” (for releasing 5 terrorists for 1 army deserter)
110. 6/6/14: 37.2%: Percentage Not in Labor Force A 36-Year High!
111. OBAMA Chews Gum During D-Day Anniversary Event in France
112. June 2014: Iraqi government collapse and TERRORIST TAKEOVER due to OBAMA US troop pull-out
113. orchestrated influx of 300,000 illegal, unaccompanied minors which Obama REFUSES to enforce
114. “You’re going to see 90,000 American troops come marching home by the end of the summer. You’re going to see a stable government in Iraq that is actually moving toward a representative
government. I’ve been impressed how they have been deciding to use the political process rather than guns to settle their differences.
I am very optimistic about — about Iraq. I mean, this could be one of the great achievements of my administration.”
115. “No boots on the ground”
116. “I won’t sign any bill that has earmarks”
117. “…core al Qaeda is on its heels, has been decimated” (August 2013)…so how COULD they POSSIBLY invade iraq??
118. “I am ending the wars in Iraq and Afghanistan” …because you say so, the wars are over? Just like that???
119. According to a Rand study, between 2010 and 2013, there was a 58% increase in the number of jihadist terror groups
120. “Any world order that elevates one nation above others cannot long survive.” –Sept. 23, 2009
121. American Veterans died as VA obsessed over renewable ‘green’ energy
122. 6/22/14 Judge Who Sentenced Saddam Hussein to Death – Captured & Executed by terrorists in Iraq
123. 6/29/2014: Gunmen torch churches, kill dozens of Christians in Nigeria, just miles from the town where more than 200 schoolgirls were kidnapped. OBAMA SILENT
124. “So sue me. I’m not going to apologize for trying to do something.”
125. 7/2/14: Quinnipiac University survey just dubbed OBAMA the worst president in seventy (70) years!
126. 7/3/14: USA RECORD!! 92,120,000 Americans over 16 years old are both not working and not looking for work.
127. “The pies made by the white house baker are so good they must be laced with CRACK”
128. 7/3/14: Black Unemployment Rate More Than Double White Unemployment Rate
129. 7/10/14: Obama goes to three Texas cities for fundraising, and a BBQ joint, and meets PRIVATELY with hecklers at his speech, but declines Perry’s request to visit Mexican border and detention
130. 7/5/14: Obama goes golfing for the 180th time as president. New Record for presidential number of golf games played! | {"url":"http://www.rachea.com/rants/archives/09-2014","timestamp":"2024-11-13T08:44:56Z","content_type":"text/html","content_length":"48946","record_id":"<urn:uuid:e2041e85-dbdc-4975-bd6c-7436f1285139>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00240.warc.gz"} |
cohomology of classifying spaces
Out of interest, what counts as a classifying space? The spheres as classifying spaces for cohomotopy? At classifying space we have $Cat$ as the “classifying space for categories”.
I see also there at we have a section on cohomology which includes Segal completion – table.
more references for cohomology of $B O(n)$ are also recorded at orthogonal group, here.
we should maybe make an !include-entry for a comprehensive list of these references so that all these lists get harmonized and synchronized
There was some recent addition of pages on classifying spaces, including their cohomology, as mentioned eg here.
Created a page. Right now mostly references.
v1, current
True. There is the traditional meaning of classifying spaces as variants of bar constructions for topological groups, classifying the corresponding principal bundles on nice enough base spaces.
That traditional meaning is often implied by default, such as in the entry that Dmitri is starting here.
But just going by the literal meaning of the words “classifying space” there can be classifying spaces for other or more general things – such as for cohomology theories — and it’s often useful to
have the term be understood in this more general sense.
On the other hand, calling “Cat” a classifying space is a bit of an abuse of terminology. If one called it at least a “directed classifying space” it would make better sense.
(I have now edited at classifying space in order to clarify, see the log message there)
Added Feshbach’s paper.
diff, v2, current | {"url":"https://nforum.ncatlab.org/discussion/17999/cohomology-of-classifying-spaces/?Focus=117040","timestamp":"2024-11-02T06:30:59Z","content_type":"application/xhtml+xml","content_length":"47586","record_id":"<urn:uuid:ad9088cd-deea-4aa3-85cf-12ee90ca7225>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00033.warc.gz"} |
Sound and Vibration Measurement: Calculate Acceleration Energy, to drive Maintenance Decisions
The knowledge in this post comes from martinvalencia. He defines vibration analysis strategies, for mining equipment maintenance, in the Peru ore mining industry, as part of his job:
"A vibration wave is compound, so it is impossible to analyze in the dimension of time. Its analysis is carried out in the frequency dimension, which is much easier What we do is store the
vibration signatures (velocity and acceleration spectra), of a sector of the motor, bearings, bases, shafts, etc , every week, 15 days, 1 month, ...
Depending on the work that this motor does, if it is a critical motor for the operation of the mine, it is monitored 24 hours a day, 7 days a week all year round. Example: the motors of the belts
that transport the ore to be processed in the primary crushing.
With the database that we have with the vibration info, preventive and predictive maintenance plans are put together for the motors. Thus we avoid plant stoppages due to unexpected failures, and
reduce downtime without affecting the company's production."
This post shows some tricks of the trade. You'll find it in text books, but it isn't often shared in a practical form. For this exercise, Martin and I started from an FTT flow I had made for this
road test. He showed me how to convert the FFT to a form that's better suited for maintenance analysis. And how to calculate the overall power that the vibrations cause. These data sets, and the
evolution of them over time, is what drives maintenance decisions in his industry.
image: initial concept from Martin
Linear FFT gives better insight for vibration analysis
In my original design, I had used power spectrum and a dB scale for the transform. Turns out that I needed RMS and Linear scale.
image: LabVIEW FFT settings
The FFT shows what's shaking at which frequencies. Changes in the spectrum can indicate wear and tear. New peaks showing in higher frequency may indicate early wear in a bearing.
The data by itself is useful, in particular if you have similar machines to compare to. But checking changes over time is what they are particularly looking for. A motor, shaft and gearbox
combination doesn't start vibrating at different frequencies or levels, unless something is aging inside, or needs maintenance.
In the design here, the FFT Y axis is expressed in Volt. The output is 100 mV/g. It may be useful for a vibration monitor to adapt the axis to show grams. But for today's post, keeping it in V works
better when proving the end results.
Total acceleration energy is a key indicator
The second parameter that drives maintenance decisions is the total energy that's contained in the vibrations. It's a single figure, representing the area under the FFT line. The figure is based on
the RMS value of the measurements, and expressed in m/s². The specs of the sensor say that the sensitivity is 10.2 mV/(m/s²). This is exact the same as 100 mV/g, given 1g = 9.81 m/s².
Here's the calculation to convert the samples from raw voltage measurements to an acceleration in m/s²:
image: formula from Martin
My example flow already knows how to capture and store an array of samples. The FFT display is based on that.
So I can just plug in on that level, and use the same data set to calculate the acceleration force - starting from the given that for this sensor, sensitivity is 10.2 mV/(m/s²)
• take xxxx samples in V (x should be a power of 2 for the FFT to work well)
• calc RMS
• multiply by 1000 to get mV (this gives the change over time)
• divide by 10.2 = S(K), the acceleration
This is how I plugged that into the FFT flow that I made a few posts earlier:
image: LabVIEW flow with RMS/acceleration calculation fitted into FFT example
This is how the results of a random test looked look like. The sensor was sitting on my desk and I tapped on the table surface.
image: FFT and acceleration , using 1024 samples
FFT shows frequency view of the energy in the vibrations collected. The indicator shows the total acceleration measured.
image: actual device in a Peru mine that's managed by the algorithm
Although Martin said that the results looked good, based on his experience, it's not that easy to validate without access to the raw data. I can give access to a file with that data, but there's a
simpler test:
When you use a pure sine signal as input, there's only 1 energy point, at the sinus frequency.
If we know the frequency and amplitude, we can check the FFT value (there should exactly be one energy point), and the acceleration. So that's the approach we took.
I made a sinus of 100 Hz, 400 mVPP as test signal.
image: calculate FFT and acceleration of a sinus, with 260K samples
The FFT shows 0.14 VRMS at 100 Hz. That's correct:
source: https://www.allaboutcircuits.com/tools/rms-voltage-calculator/
Then, the acceleration should be that value * 1000 / 10.2:
image: confirm what's on screen with calculations
Mathematically sound, we think . Thank you for reading.
• vilaksh01 posted an interesting blog on the same subject. But from a different angle.
He's using the same data as me (vibration measurements), but uses machine learning (TinyML) to predict maintenance needs:
TinyML Gearbox Fault Prediction on a $4 MCU
• Good stuff. I always like to see real “ tricks of the trade” being used. Vibration monitoring prediction is an important topic. | {"url":"https://community.element14.com/technologies/test-and-measurement/b/blog/posts/sound-and-vibration-measurement-calculate-acceleration-energy","timestamp":"2024-11-11T16:17:35Z","content_type":"text/html","content_length":"252634","record_id":"<urn:uuid:656fa8fe-098a-4298-8dff-c4458c694f33>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00091.warc.gz"} |
Daniel Cunningham, Professor Emeritus, Mathematics
Posted: Wednesday, May 15, 2024
Daniel Cunningham, Professor Emeritus, Mathematics
Daniel Cunningham, professor emeritus of mathematics, has signed a contract with Cambridge University Press to complete and publish a second edition of his mathematics book Set Theory: A First Course
. In the first edition, the fundamentals of abstract sets, including relations, functions, the natural numbers, order, cardinality, transfinite recursion, the axiom of choice, the ordinal numbers,
and cardinal numbers, are developed within the framework of axiomatic set theory. The five anonymous reviewers who assessed Cunningham's proposal for a second edition offered many helpful comments
and recommendations. As a result, the second edition will now contain the following new topics: set theoretic constructions of the integers, the rational numbers, the real numbers, and the
hyperreals. We will also prove theorems that demonstrate that the standard definitions of limit, continuity, and the derivative have equivalent versions in the hyperreals using infinitesimals. A new
final chapter will cover models of set theory and will end with a discussion on Kurt Gödel’s inner model of constructible sets and on Paul Cohen's method of forcing. The second edition will also
contain solutions to all the exercises.
Cambridge University Press (CUP) is the publishing house of the University of Cambridge. Dedicated to excellence, its purpose is to further the university's objective of advancing knowledge,
education, learning, and research worldwide. Cambridge is a leading global publisher in pure and applied mathematics, with an extensive program of high-quality books and journals that reaches into
every corner of the subject. CUP's catalog reflects not only the breadth of mathematics but also its depth, with titles for undergraduate students, for graduate students, for researchers, and for
users of mathematics. Cambridge University Press has over 50 offices around the globe, and it distributes its products to nearly every country in the world. | {"url":"https://dailybulletin.buffalostate.edu/node/384386?month=2024-05&var=2024-05-15","timestamp":"2024-11-07T13:47:29Z","content_type":"text/html","content_length":"60268","record_id":"<urn:uuid:0624de65-0be0-41ad-ab19-e4b7b5e9e06f>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00303.warc.gz"} |
About Me
BA, Mathematics, University of Chicago
BS, Atmospheric Science, University of Maryland, College Park
MSEd, Johns Hopkins University
Tutoring Experience
I have 20+ years of tutoring experience in mathematics, ranging from 6^th grade math through calculus, in addition to Math SAT, Math GRE, Math ACT, and Praxis Core Math preparation.
I taught high school math at the Phelps School of Architecture and Engineering in Northeast Washington D.C. in an afterschool program.
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I founded a tutoring agency called Ivy League Tutoring Connection, which serves the Washington D.C. area with tutoring in all subjects.
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I am the Editor-in-Chief and co-author of:
• Math SAT 800: How to Master the Toughest Problems
• Praxis I Math: My Private Tutor
• SAT Math 800: Challenge Yourself to the Perfect Score
• Graphs for Algebra I & II: Understanding the “Y”
• Praxis Core Math 2020: A Complete Course (a workbook used in college courses across the country)
• SAT Vocabulary Lightning
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My book publishing website is: www.superlativepressbooks.com
Curriculum Writing and Assessment Writing
I wrote the Math SAT curriculum for Education Unlimited, a summer preparatory camp based in Berkeley, California.
I write math assessment items for WestEd, a San Francisco-based educational company.
I wrote math curriculum and math assessment items for A Pass Education.
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My résumé, click here: Daniel Eiblum Math Tutor Resumé | {"url":"https://tutoringbydan.com/about/","timestamp":"2024-11-01T23:34:45Z","content_type":"text/html","content_length":"32133","record_id":"<urn:uuid:0f09841c-1323-426d-a19b-52a4d5450b23>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00360.warc.gz"} |