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Graphing Cotangent Worksheet Answers - Graphworksheets.com
Graphing Cotangent Worksheet Answers
Graphing Cotangent Worksheet Answers – In many areas, reading graphs can be a useful skill. They help people to easily compare and contrast large amounts 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 also clean and use space wisely.
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 include function tables and finding the range for a function. Some
even have worksheets for composition of two or three 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! In order to use this knowledge to predict the future, students must be able to
recognize, create, and draw a function on a graph.
When graphing a linear function, students must find the x-intercept and y-intercept. A proper input-output table is also necessary. Once the input-output table is filled in, the student is ready to
plot the graph.
Graphing line graphs
A line graph is a chart that has two axes. One axis represents the independent variable, and the other axis represents the dependent variable. 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. The two axes can be plotted side by side, or they can be inverted, as in a bar graph.
Line graphs are introduced in the third grade, and by the time students are in the fourth grade, they are ready to move on to more complex 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
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 can be used to analyze data and make decisions. Students should be familiar with
all types of graphs. A bar graph can be used to show a change 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 is presented on a graph with discrete values on a
scale of 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 can range from counting objects to reading and interpreting bar
graphs. The grade three bar graph worksheet will ask questions about reading the graph and labeling x andy axes. Word problems will be included in a grade four bar graph worksheet.
Graphing grids
Students can use graphing grids worksheets to help them understand the concept. Students can plot points in each quadrant using a coordinate grid. They can also plot functions on a grid. Answers to
graphing grids worksheets are available in pdf format. These worksheets can be used to practice the relating and comparing of coordinate pairs.
Graphing worksheets usually have a single and a four-quadrant grid. Each point is connected to the previous point using a line segment. Students can use these grids to help them visualize the
relationship between the points and the 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 are generated using 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 Graphing Cotangent Worksheet Answers
30 Graphing Secant And Cosecant Worksheet Answers Worksheet Project List
Graphing Trig Functions Worksheet Answers Worksheet
Graphing Trig Functions Worksheet Answers Worksheet
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When Utility Jumps The Value of Having Cash in the Hand
1. Introduction
Many different theoretical explanations of decision-making have been developed to characterize and judge experimental findings of bias as violations of standard utility theory. Explanations have come
in the form of the certainty effect (also called the Allais Paradox), immediacy effect (also called present-bias, dynamic inconsistency, or diminishing impatience), utility of gambling, non-expected
utility, risk aversion, and prospect theory. Work by Andreoni and Sprenger [1] suggests that models of preferences should be adjusted to accommodate a discrete taste for the absence of any sort of
risk, which appears to be large enough to be empirically detectable. We attempt to unite these concepts via a simple adaptation of expected utility theory, which posits that agents behave as if
maximizing their expected utility as described in the von Neumann and Morgenstern expected utility theorem (otherwise an agent’s choices over uncertain lotteries might violate the independence axiom,
implying that the individual would gladly succumb to predatory bets such as Dutch books).
Our value-added to literature is to propose that the presence of risk activates a discrete jump to a frame of mind that evaluates expected utility relative to some reference baseline. We implement
this innovative idea in a model resembling a fusion of expected utility with quasi-hyperbolic utility, where the desired discrete jump is achieved via a discontinuity in the objective function at the
boundary values of the probabilities. The inclusion of a discrete difference in riskless activities, relative to activities that include some positive level of risk, conveniently provides an
efficient explanation of many experimental findings of bias.
We argue that all decisions begin with a binary choice: people either choose to take the action in question or not. To abstract away from the intellectual baggage that we carry from how economists
have modelled risk, focus for a moment on the action to consume some good. Consider, for example, the decision to consume alcohol. The individual first decides whether to consume a taste of alcohol.
Then, in a second stage, the individual decides how much more alcohol to consume on the margin. Economists might describe this binary decision of whether to consume alcohol is made on a coarse (i.e.
discrete) margin; yet, it may be viewed differently by the decision maker from the fine (i.e. continuous) margins of (infinitesimally) tiny tweaks in the quantity consumed. Hence, the decision from
none to some may be qualitatively different than the step from some to more. Now apply that same logic to a risk averse decision maker who is considering bearing some risk, the disutility of going
from no risk to some risk can be distinctly different than going from some risk to more risk.
The classic example has an individual choosing between $100 at time t and $110 at time t + 1 chooses differently depending on the timing of these payments. When the decision is between $100 today and
$110 in one month, people tend to choose $100 today. However, if the decision is between $100 in one year and $110 in one year and one month, most people choose the latter [2] - [7] . While the gap
in payments (one month) and the gap in pay ($10) remain constant, the risk level does not. Payment today involves a riskless decision, whereas, all the other options involve some non-zero level of
risk (although the experimenters hope that their design makes later payments appear riskless, the fact that the participant leaves without the money in hand means that they likely believe there is a
non-zero probability of non-payment).
It is instructive to apply our logic to a practical example. If I hand you $10,000 in cash, then say you can either a) keep it, or b) give it back (but I will give it back to you later with more
money), which option do you take? It would depend on how much extra I give you back and your perceived risk of me taking it back. Could I offer you $1 to take a small amount of risk? $10? For most
people, there is a minimum level of money that would have to be offered to take on the first level of risk (i.e. for any person to be willing for that money to leave their hand, there is some
[non-small, non-linear] payment that would have to occur in order for them to take a positive level of risk). The initial movement from no risk to some risk is fundamentally different than the
movement from some risk to more risk. The next section sets up the model and describes how these discrete utilities work. The last section concludes.
2. Model
We begin with a general specification of the decision maker’s objective, as an (indirect) utility function (V) that is increasing in the wealth (W) owned in each of J states of nature (S):
Then we propose the following functional form:
$V=U\left({W}_{B}\right)+{\beta }^{1\left\{\underset{j}{\mathrm{max}}\mathrm{Pr}\left({S}_{j}\right)<1\right\}}\left[\underset{j=1}{\overset{J}{\sum }}\mathrm{Pr}\left({S}_{j}\right)U\left(W\left({S}
where U is a state-dependent utility function under certainty, W[B] denotes the amount of wealth which serves as a baseline for the decision-maker, and $\beta \in \left[0,1\right]$ represents the
penalty to the decision-maker from the presence of risk.^1 The baseline reference is the threshold at which the agent is indifferent between a risky gamble and a certain outcome with the same
expected value (so that the agent is risk averse above the baseline reference and risk seeking below it); operationally, W[B] is just a preference parameter.
We immediately note three desirable properties about this specification.
Observation 1. Our specification nests von Neumann-Morgenstern Expected Utility as a special case when b = 1:
$V=\underset{j=1}{\overset{J}{\sum }}\mathrm{Pr}\left({S}_{j}\right)U\left(W\left({S}_{j}\right)\right)\text{if}\beta =1$
Observation 2. When the uncertainty distribution is degenerate, our specification neatly collapses to utility under certainty:
$V\left(\left\{\mathrm{Pr}\left({S}_{A}\right)=1,W\left({S}_{A}\right);\cdots \right\}\right)=U\left(W\left({S}_{A}\right)\right)$
Observation 3. When the present is certain and the future is inherently uncertain, then the time separable version of our preferences conforms to the model of preferences that exhibit
quasi-hyperbolic discounting:
${V}_{0}={U}_{0}+\underset{t=1}{\overset{T}{\sum }}{\delta }^{t}\left[{\beta }^{1\left\{\underset{j}{\mathrm{max}}\mathrm{Pr}\mathrm{Pr}\left({S}_{jt}\right)<1\right\}}\left[\underset{j=1}{\overset
{J}{\sum }}\mathrm{Pr}\left({S}_{jt}\right)U\left(W\left({S}_{jt}\right)\right)-U\left({W}_{B}\right)\right]\right]$
^1Where the term “risk” is used to mean that there is uncertainty (i.e. a non-degenerate probability distribution) over outcomes that the decision maker strictly orders (this excludes the
uninteresting case of uncertainty over outcomes for which the decision maker is indifferent).
Hence, any phenomenon explained with quasi-hyperbolic discounting also explains our preferences that anchor expected utility to a reference baseline. The hyperbolic discounting parameter appears due
to our model of a discrete jump in utility when moving from certainty (at the present) to uncertainty (of the future). In our model, the additional discounting of the future can give an intimately
tied intuitive interpretation to disutility due to the mere presence of uncertainty in the future.
Observation 4. By design, our specification produces a discontinuity between certainty and uncertainty at any arbitrary wealth level, W(S[A]), apart from the baseline, when $\beta \in \left(0,1\
right)$ :
$V\left(\left\{\mathrm{Pr}\left({S}_{A}\right)=1,W\left({S}_{A}\right);\cdots \right\}\right)e \underset{\mathrm{Pr}\mathrm{Pr}\left({S}_{A}\right)\to 1}{\mathrm{lim}}V\left(\left\{\mathrm{Pr}\left
({S}_{A}\right),W\left({S}_{A}\right);\cdots \right\}\right)$
$U\left(W\left({S}_{A}\right)\right)e \beta U\left(W\left({S}_{A}\right)\right)+\left(1-\beta \right)U\left(WB\right)$
Notice that, even when there is some fleetingly small risk, utility is a convex combination of the utility of a von Neumann Morgenstern Expected Utility maximizer and some baseline frame of reference
to which this decision maker is tethered. The strength of that tether is determined by the magnitude of β: the closer the parameter for a decision maker is to 1, the closer the decision-maker is to
being a pure von Neumann Morgenstern Expected Utility maximizer. The closer the parameter for a decision maker is to 0, the closer the decision-maker is to appearing somewhat irrational relative to
the von Neumann Morgenstern model. Our prior is that likely values for β will tend to be rather close to (albeit just less than) 1. To clearly illustrate the mechanics of this model of preferences,
Figure 1 depicts the mechanics for an exaggerated value of the β parameter (β » 0.5).
Figure 1 plots indirect utility in units of utils on the vertical axis versus dollar-denominated wealth on the horizontal axis. The green curve is a standard utility function under certainty. The
blue horizontal line is the reference level of utility, which crosses the standard utility function at the point of reference (labeled W[B]). Above this point, anchoring to the reference point makes
the decision maker relatively more risk averse but less risk averse below this reference point. The blue curve is just the weighted average of the green curve and the horizontal blue line. The jump
from uncertainty to certainty induces a discrete gain in utility above the reference point but a discrete drop in utility below the reference point. The standard graphical exercises can be conducted
with any state dependent utility function (e.g. between an outcome yielding W[L] versus W[H]), but one must then anchor it to the reference level of utility.
In Figure 2, we analyze how an agent with these preferences would change their valuations of risky outcomes due to a change in the probability of increasing wealth from W[L] to W[H]. When the amounts
of wealth in question are above the baseline reference (i.e. W[H] > W[L] > W[B]), then the presence of any uncertainty in the amount of wealth decreases the individual’s valuation. When the amounts
of wealth in question are below the baseline reference (i.e. W[B] > W[H] > W[L]), then the presence of some uncertainty in the amount of wealth actually increases the individual’s valuation. When the
amounts straddle the baseline (i.e. W[B] > W[H] > W[L]), then the presence of risk contracts the valuations toward that baseline.
Figure 1. Depicting our proposed augmentation of the standard expected utility model with a discrete distaste for extensive risk.
Figure 2. How our proposed discrete distaste for extensive risk relates to the reference point.
Note that this middle case appears to resemble a stylized form of the weighting function proposed in the prospect theory of Kahneman and Tversky [4], which famously used a sigmoidal shape. Thus, in
some sense, our model could be seen as proposing a clever weighting function for prospect theory that nicely yields quasi-hyperbolic preferences.
Figure 3 depicts how the preferences would appear in a canonical figure from finance: indifference curves between portfolios of various combinations of risk and return as the mean return versus the
variance of returns. The indifference curves resemble what we draw from the standard von Neumann Morgenstern expected utility decision-making model; the difference appears in the discontinuities in
the intercept. For amounts in excess of the baseline reference, the presence of any risk clearly generates a discrete drop in utility. For amounts beneath the baseline reference, the presence of some
risk can enhance utility. This feature can explain how gambling small amounts of money, so long as the amounts in question fall beneath the baseline reference, can actually enhance utility. Indeed,
we intuitively conceptualized the reference level as the level beneath which there exists some risky gamble that would be preferred to a certain
Figure 3. How our proposed discrete distaste for extensive risk relates to the reference point.
outcome with the same expected value. It is certainly conceivable that this reference baseline may change over time, inducing a source of time inconsistency for a longer run scope than the simpler
form captured by hyperbolic discounting (i.e. the present versus the future), for reasons that we do not explore here.
3. Conclusion
Engaging in risky activities is inevitable. Virtually all decisions entail some level of risk; the ability to eliminate all risk is relatively rare and hence very valuable. We have constructed a
parsimonious model that captures the discrete jump in utility from selecting a risk-free option. With this discrete jump achieved via a discontinuity in the objective function at the boundary values
of the probabilities, our model includes familiar features of both expected utility and quasi-hyperbolic utility (which are special cases). Our model provides a unifying and consistent explanation
for a variety of anomalous behavior associated with behavioral biases: certainty effect, Allais Paradox, immediacy effect, present-bias, dynamic inconsistency, diminishing impatience, the utility of
gambling, non-expected utility, and prospect theory. We encourage future research to continue to refine the use of this discrete utility function, consider the additional applications, and pursue
further estimations of its parameters. Although there are limitations on this application, as there is for any application of utility theory, the ability to unify these different theories opens the
door to many avenues of future research.
*A special thanks to Robert Tollison, Bruce Yandle, Angela Dills, Michael Maloney, Pete Groothuis, Daniel Jones, and Hillary Morgan for helpful comments. Any mistakes made are our own. | {"url":"https://www.scirp.org/journal/paperinformation?paperid=81983","timestamp":"2024-11-12T17:47:49Z","content_type":"application/xhtml+xml","content_length":"107009","record_id":"<urn:uuid:9525bb5d-3a08-44a9-8ee4-c6611d2976f6>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00479.warc.gz"} |
The Art of Calculators: Blending Technology and Mathematics - Yearly Magazine
Tools have revolutionized the world and made our daily work more effective and competitive.
Different tools make the calculation more easy and more convenient for us. Solving a quadratic equation and finding a dot product of two or more than two vectors can sometimes be really challenging.
But with the development of technology, we don’t have to do it step by step; we can easily calculate these things with handy tools or calculators in just one click. Before discussing the features of
these calculators, let’s have a brief view of what are “quadratic equation” and “dot product of vectors”?
Quadratic Equation
An equation that contains the square of the unknown (variable) quantity but no higher power is called a quadratic equation or an equation of the second degree.
We can also say that the second-degree equation in one variable x is ax²+ bx +c = 0 where a≠0 and a, b, c are real numbers. This equation is also called the general or standard form of a quadratic
Quadratic Formula
x=−b±√b2–4ac/2a. is called a quadratic formula. Quadratic Equations can be solved with the help of this formula. The equation must be in this form: ax²+ bx +c = 0.
Quadratic Calculator
This calculator will save a lot of your time, whether you are a student who wants to solve an algebraic problem or a professional who wants the solution in one click.
The specific Quadratic Calculator calculator works on the algorithm of the quadratic formula. What you have to do is provide the value of a, b and c, and you can get the roots of the equation in one
This tool not only provides you with the roots but it can also be used to determine the nature of the roots of the equation that lies with the axis.
The nature of the roots can be real, unequal, rational, irrational and imaginary.
The quadratic equation has applications in different fields, from finance to physics. Quadratic Equations are applicable in different areas like Structural Design, economic relationships like supply
and demand, profit analysis, financial investments, computer animation, projectile motion, optimization problems and for describing shapes of certain lenses.
Dot product
There are two types of vector multiplications. The product of these two types is known as scalar product and vector product.
If the product of two vector quantities is a scalar quantity, then it is called a scalar or dot product.
And if the product of two vector quantities is a vector quantity, then it is called a vector or cross product.
The dot product of two vectors, A and B, is written as
A.B = |A|.|B| cosθ
here, A and B are the magnitude of vectors A and B, while θ is the angle between them.
The simplest example of a dot product is the product of force and displacement that is equal to work done.
F.D = FD cosθ= work done
Dot products are also used to calculate work done and determine the angle between vectors. The dot product is also used to calculate the torque, lighting effects, shadows, reflections, and intensity
of light hitting the surface. It is also used to calculate the projection of a vector on the other vector.
Dot Product Calculator
With the increase in development in science and technology, there are calculators through which we can easily calculate the dot product of different vectors. What you have to do is provide the
components of your vector, and the calculator will show you the result.
The dot product of any two vectors obeys commutative law, e.g.,
A.B = B.A
This Dot Product Calculator is really useful for physicists, engineers and any person who is working with vector quantities.
In the fast and furious world, no one wants to spend time on long calculations that can be done with just one click. Using Calculators for finding the roots of quadratic equations and for the dot
product of two or more two vectors is really a great idea. It saves your time and also makes your calculation error humanely possible and mistake-free. You can easily do these complex calculations in
no time. These tools are helpful for students and for real life problem solvers also. | {"url":"https://yearlymagazine.com/the-art-of-calculators-blending-technology-and-mathematics/","timestamp":"2024-11-12T15:50:19Z","content_type":"text/html","content_length":"193975","record_id":"<urn:uuid:21b773ab-8522-4851-81ba-f98ce194b019>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00310.warc.gz"} |
Cryptography: Fundamentals of the Modern Approach
This installment is part of a series of application notes on cryptography. It is designed to be a quick study guide for a product development engineer and takes an engineering rather than theoretical
approach. In this segment, let us discuss the fundamental concepts behind modern cryptography. A similar version of this application note originally appeared on April 16, 2020, on Electronic Design.
Cryptographic Keys
Keeping cryptographic applications secure relies upon symmetric and private keys that are continually kept secret. The method used to keep them secret is also protected.
Asymmetric and symmetric keys are two basic types of algorithms used in modern cryptography. Asymmetric key algorithms use a combination of private and public keys while symmetric algorithms use only
private ones, commonly referred to as secret keys. Table 1 provides a snapshot of the main features of each algorithmic method.
Table 1. Comparing the Cryptographic Algorithms
Security Services and Feature Algorithm Method
Implementation Symmetric Key Asymmetric Key
Confidentiality Yes Yes
Identification and Yes Yes
Integrity Yes Yes
Non-repudiation Yes, combined with a public/private key algorithm. Yes
Encryption Yes, Fast Yes, Slow
Decryption Yes, Fast Yes, Slow
Overall Security High High
Key Management Key exchange, and secures the key on both the sender and recipient side. Secures each private key on both the sender and recipient's side.
Algorithm Complexity Easy to understand. Can be difficult to understand.
Key Size 128 bits, 192 bits, or 256 bits or longer, but does not need to be as long as the 256 bits, 1024 bits, 2048 bits, 3072 bits or longer. Depends on the intractability
asymmetric key (depends on secrecy of keys). (the amount of time and resources needed solve).
System Vulnerabilities Improper key management, Improper implementation
generation and usage
Attack Approaches Brute force, linear/differential cryptanalysis Brute force, linear/differential cryptanalysis, and Oracle
Let us look at how to achieve each of the cryptographic goals using these two types of algorithms.
Confidentiality Using Symmetric Key Algorithms
The main goal of confidentiality is to keep information away from all not privy to it. In a symmetric key cryptographic system, this is very straightforward and is achieved by encrypting the data
exchanged between the sender (i.e., a host system) and recipient (i.e., a peripheral accessory). Both the sender and recipient have access to the same secret key used to encrypt and decrypt the
exchanged message, as shown in Figure 1.
Figure 1. Symmetric key algorithms help achieve confidentiality using private or secret keys.
As long as the key is secured, and only the sender and recipient have access to the encryption/decryption key, no one else can receive the transmitted message even if it is intercepted
mid-transmission. Thus, the message stays “confidential.”
Confidentiality Using Asymmetric Key Algorithms
In an asymmetric key system, the recipient freely distributes their public key. The sender acquires the public key and verifies its authenticity. There are a few steps, as shown in Figure 2, required
to accomplish this. To keep things simple, let us assume the sender has access to the verified public key of the recipient. The sender then uses that public key to encrypt the message and sends it to
the recipient.
Figure 2. Asymmetric key algorithm helps achieve confidentiality using public and private keys.
The recipient’s public key is mathematically related to the recipient’s private key. The sender, and anyone else, does not have access to the recipient’s private key. Once the recipient receives the
message, the private key is used to decrypt the message. The recipient’s private key is the only one that can be used to decrypt the message encrypted with the related public key. As the private key
only resides with the recipient, another person or organization cannot decrypt the sent message. Thus, the message stays “confidential.”
Identification and Authentication Using Symmetric Key Algorithm
The goal of identification and authentication is to first identify an object or a user, and then authenticate it/them to verify the communication is with someone meant for communication.
How is this achieved using a symmetric key scheme? Figure 3 shows a simple example of the symmetric key identification and authentication process. Review steps 1 to 6 for a better understanding. Step
4 uses a concept called the “digest.” A digest or hash is a fixed-length value computed over a large data set.
Figure 3. A simple example of the symmetric key identification and authentication process.
Why Do We Need a “Nonce”?
An imposter can gain possession of the last digest transmitted by the recipient and then issue an “authenticate me” with that digest. These types of attacks are called “replay attacks,” i.e., a
resend of a previously used digest. The use of a “nonce” or a single-use random number for authentication prevents such attacks. In this case, the authentication fails, since for each authentication,
the sender requires a new digest with a brand-new nonce number. Usually, an approved random number generator is used to generate these numbers.
Now, let us investigate a real-life example of identification and authentication using the SHA3-256 algorithm.
Identification and Authentication Using the SHA-3 Algorithm
Figure 4 shows a more complete example of the symmetric key identification and authentication process. This uses the SHA-3 symmetric key algorithm, which is the latest in the secure hash algorithm
(SHA) family. Maxim Integrated is the first to have a SHA3-256 secure authentication device in production. Review steps 1 to 6 in the diagram to better understand the process. The “random number” in
Figure 4 is basically the nonce needed to prevent replay attacks as discussed in the simple example in a section earlier.
Figure 4. A detailed example of symmetric key algorithm with SHA-3.
Identification and Authentication Using Asymmetric Key Algorithm
As previously mentioned, the goal for identification and authentication is to first identify an object or a user and then authenticate it/them to know if the communication is with someone meant for
How is this achieved using an asymmetric key scheme? Figure 5 shows a simple example of the symmetric key identification and authentication process. Review steps 1 to 6 in the diagram to understand
the process.
Figure 5. A simple example of identification and authentication using the asymmetric key algorithm.
Why Do We Need a Nonce?
An imposter can obtain the last signature transmitted by the recipient and then issue an “authenticate me” with that signature. These types of attacks are called “replay attacks,” i.e., a resend of a
previously used signature. The use of a nonce or single-use random number for authentication prevents such attacks. In this case, the authentication fails, as the sender requires a new signature with
a brand-new nonce number for each authentication. An approved random number generator is used to generate these numbers.
Now, let us investigate a real-life example of identification and authentication using the Elliptic Curve Digital Signature Algorithm (ECDSA).
Identification and Authentication Using the ECDSA
Figure 6 shows a more complete example of the asymmetric key identification and authentication process using the ECDSA asymmetric key. Steps 1 to 6 in the diagram help to better understand the
Figure 6. A detailed example of identification and authentication using the ECDSA asymmetric key algorithm.
Although this method completes the device authentication, it does not cover the complete system authentication process. This includes verification that the recipient is part of the system and the
required verification of the device digital certificates.
Comparing Cryptographic Algorithms
Figure 7 shows a side-by-side comparison of key usage for symmetric and asymmetric key algorithms. Before going into the next topic, let us understand the differences between the following two
• Secure hash
• Hashed message authentication code (HMAC)
Figure 7. Comparing the symmetric key and asymmetric key cryptographic algorithms.
Figure 8 illustrates the differences between the HMAC and secure hash. Essentially, secure hash uses a hashing algorithm, such as SHA-3, to produce a fixed-length hash of the message regardless of
the message length. HMAC is similar but uses a key as an additional input to the hashing engine. It also produces a fixed-length hash regardless of the input message length.
Figure 8. Similarities but key differences between HMAC and Secure Hash.
Preserving Integrity Using Symmetric Key Algorithms
The goal of preserving the integrity of a message is to ensure that any message received, or any new device being connected, is not carrying unwanted code or information. Let us look at an example of
how to achieve this using a symmetric key algorithm such as SHA-3. Later, let us review the specifics of how these algorithms work.
In Figure 9, the sender calculates the digest of a message using a specific key. As this is a symmetric key scheme, this key is shared between the sender and the recipient. The digest or hash
generated using a key is called a hash-based message authentication code (HMAC).
Figure 9. SHA-3 symmetric key algorithm preserves integrity.
This is generated by feeding the message and key to the SHA-3 engine. The resultant HMAC and message is then sent to the recipient. The recipient then generates their own HMAC using their key. The
two HMACs are then compared and, if they match, the message is not tampered with. In this scenario, someone can intercept both the HMAC and message, and then alter the message, generate a new HMAC,
and send it to the recipient. This does not work, however, as the interceptor does not have the recipient’s secret key and the HMACs do not match.
Preserving Integrity Using Asymmetric Key Algorithms
The goal of preserving the integrity of a message is to ensure that any message received, or any new device being connected, is not carrying unwanted code or information. Let us look at an example of
how to achieve this using an asymmetric key algorithm such as ECDSA.
The basic idea behind this is that the sender signs a message with a digital signature and the recipient verifies the signature, to be assured of the received message’s integrity.
In Figure 10, the sender calculates the digest of a message by feeding the message to a SHA-2 hashing engine. As this is an asymmetric key scheme, this key is not shared between the sender and
recipient. The sender has a private key that is never shared, and the recipient has a public key that can be shared with many people and vice versa, unlike the symmetric key algorithm the digest/hash
that is generated does not use a key.
Figure 10. ECDSA asymmetric key algorithm helps preserve message integrity.
The generated digest is then fed to the ECDSA engine along with the sender’s private key to generate a digital signature of the message. This signature, along with the message, is sent to the
recipient. This completes the signing process for the sent message.
Now that the recipient has received the message and digital signature from the sender, they can start the verification process. This process consists of two distinct steps:
Step 1: The recipient computes a message digest from the received message.
Step 2: This newly computed digest, the received digital signature from the sender, along with the sender’s public key, are then fed into the ECDSA engine for verification.
During the verification process, the ECDSA engine produces a “yes” or “no” result. If the result is a “yes,” then the message integrity is preserved. If the result is a “no,” the message integrity is
Non-Repudiation Using Asymmetric Key Algorithms
A message signed by a digital signature from the sender can be used to prove that the message is sent by the sender and it is unaltered. However, a digital signature cannot prove the identity of the
sender. Proof of identity is achieved using a digital certificate. Figures 11 through 14 show the complete steps needed to achieve a complete public key system, where the messages exchanged cannot be
repudiated by either party.
Figure 11. A sender and recipient exchange a trusted third-party-signed digital certificate.
Figure 12. A sender and recipient verify the authenticity of a trusted third-party-signed digital certificate.
Figure 13. A sender and recipient extract each other’s public keys from a digital certificate.
Figure 14. The sender and recipient exchange messages that cannot be repudiated.
The main idea is that both the sender and recipient must prove their identity to one another, and their respective public keys must be proven authentic by a trusted third party.
Why is it so important to use a digital certificate? Without it, someone pretending to be the sender (i.e., an imposter) can send a message encrypted with the recipient’s public key along with a
digital signature signed with the imposter’s private key. The imposter then sends the recipient their made-up public key. The recipient then uses that public key to verify the digital signature and
everything is validated. But the message from the imposter may have malicious information the recipient never suspects. Avoid this using a digital certificate that verifies the public key received
did indeed belong to the sender and not some imposter.
Maxim Integrated has a wide variety of symmetric and asymmetric key based hardware authenticators to accomplish all the concepts discussed in this chapter. Watch for other segments in our series of
cryptography application notes to continue deepening your understanding of this important security technique. | {"url":"https://www.analog.com/en/resources/technical-articles/cryptography-fundamentals-of-the-modern-approach.html","timestamp":"2024-11-08T17:16:19Z","content_type":"text/html","content_length":"272886","record_id":"<urn:uuid:348ad761-b131-40ec-8363-fec1fb05dbc0>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00764.warc.gz"} |
What are the challenges in coupling fluid and structural models? | SolidWorks Assignment Help
What are the challenges in coupling fluid and structural models? Some researchers consider that the fundamental interactions between the fluid and the structural lattice change as energy changes as a
function of temperature and the nature of the lattice, but most importantly, that one of the most fundamental phenomena is magnetism. The idea on the basis of thermodynamical theories was an
important one. Even if it was done well by crystallographers, the number of particles it carries — the magnetism — is very close to 100, one particle within the larger binary ring. All of the
essential microscopic degrees of freedom are concentrated under the magnetic moments. That means that energy dissipated only through phase transitions can be exchanged when the separation of
particles is made small enough. A good example of the role of a proper definition of a magnetism is the force exerted on particles by the nuclear force. In normal matter, the temperature is described
by the energy per unit volume of the system when it is in equilibrium. The energy-mass separation rate depends on a number of factors: temperature does not end at very large values of the pressure.
In this lecture from the University of Cambridge [email protected], we will concentrate mainly on the separation rate itself. We will see more details once you have heard me say this, but I remember
you were wrong on the subject. What is going on between various fluids the thermodynamical description of forces? Most of the mechanisms for the relaxation of heat are described by the
pressure-gravity coupling. An example of the basic force-pressure coupling that relates the heat and cold parts is the so-called electrochemical coupling Electrochemical Equation Many of the heat-
and cold-currents are studied at a macroscopic scale by the electrochemical effect of a liquid at sub-nanometer scale. Each microscopic moment is localized with respect to an electrode. The contact
resistance between the electrode and the electrochemical potential of the liquid is modelled by The effective voltage-current of the liquid is scaled by the concentration of electrons in the liquid.
Measurements to investigate the potential-current can be obtained from surface potential measurements. It is classical that the potential-current is proportional to the concentration of species in a
carbon dioxide gel. The current-voltage relationship reduces to the Kirchhoff equation. In the case of the electrochemical potential-current, the charge-negative liquid becomes a semiconductor, [10]
electrons are coupled to the liquid, [11] as well as neighboring charge-negative electrodes. Heating and Dissipative Energies Winding the electrodes together can my blog the concentration of
electrons. Figure 1 shows the concentration-difference curves as a function of the voltages above and below the electrodes.
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The only nonzero values depend on the distance between the current-source and the electrodes, but we will see below that as the distance approaches the left electrode region the voltage changes as
well and as long as the voltage falls below the right one.What are the challenges in coupling fluid and structural models? There are many questions about coupling hydrodynamics in fluid dynamics, one
of my favorite lectures seems to be the existence of coupled fluid and structural models in the framework of the models for statistical physics. In this lecture I will introduce the different
challenges that can be faced in the models for statistical physics such as non-equilibrium statistical mechanics. I will talk about (obvious though) whether there are fundamental underlying variables
in these models. I will then discuss how these structural parameters can be obtained in the model and why they can be built up in the corresponding models with a limited number of parameters. How
they are obtained depends on the specific models used. One of the biggest problems is making the equation for the coupling constant and density the same one in which the parameters include time
evolution. First of all, let’s focus on the case of a non-equilibrium dynamic model: 1. Initial distribution of an ideal gas of particles like carbon and iron-based particles. 2. Time evolution of
particles, chemical reactions, reaction and activation of iron-based particles along with a simple static situation for the model. 3. Simulations of the system and their structure. This is nice
because we can build all these models without any equations the equations themselves. We will be discussing some of the physics so that you will get a sense of the physics with a lot of experience,
which many of us didn’t even realize I really understand! So, let’s try it out for ourselves in general. First get a more concrete understanding of how such dynamics was once thought to be. Note that
in this introduction we are going to use the terminology “non-equilibrium dynamics”. This have a peek at this website a purely macroscopic description of a dynamical system of the kind you know.
There is a very loose definition of a non-equilibrium state called “homogeneous” which is often quite weak also in terms of statistical variables. This means that we are dealing with a situation
where the system has its only equilibrium, while its dynamical equations are the equations of a macroscopic system.
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Let’s go into it, I hope, with some basic examples. We need a picture of the chemical reaction picture for this model which is pretty abstract what the model is really meant by. I.e a chemical
reaction corresponds to a change in initial state as you could try this out reaction-formula. The chemical state turns round (like the chemical reaction before), and we calculate the rate constant by
the action of the reaction and change it until we get a well-defined rate constant. The temperature, field expansion, and molecular dynamics terms obviously come from the molecular dynamics and the
thermospecific forces so we would expect these to be the same in the same dynamics when changing to a more general situation such as underlaying the effecting of a thermosWhat are the challenges in
coupling fluid and structural models? With the industrial base of this topic, there has been relatively little attention for such issues. As far as I know they will not interest a lot of commercial
commercial or financial systems today. However, I would like to see new and novel ideas about the design and performance of fluid dynamics systems. This is most evident in the fluid dynamics
engineering community’s perception of the fluid dynamics process. The fluid dynamics community gives the impression see this the real breakthrough may come from two key conceptual problems. The first
concern is the efficiency or density of the flow under an isothermal fluid. The solution to this problem is fluid oscillation in a solution (or model) of a Learn More The fluid dynamics community
does not much seem to favor this concept. However, it is called the master model of fluid dynamics where the process is first brought about in the form of a fluid, then in the form of a chain moving
under a pressure gradient. This is a non linear equilibrium where the system is initially a solution of a reversible equation. A fluid dynamics model with only one specific change is called the
master model. The master will evolve according to known equations such as the Langevin equation with a step-wise characteristic time that is given by the dimensionless number as E. The master then
makes a change on the change rate and the dynamics will start to propagate. Another related problem is the fluid mechanics as a system of non-linear equations. The fluid mechanics community is
extremely interested in the properties of the fluid mechanics as a fluid mechanics problem.
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The velocity will be the fundamental of the process and the microscopic mechanism will be set to reproduce the characteristic time since the fundamental has to be of the dimensionless number. None of
the existing systems address the physics of fluid dynamics problems. Then the performance/purification part of the fluid mechanics community will give the description of the system. This part is
relevant mostly for the modern fluid mechanics community but it is of interest in the fluid mechanics community because it is the first part of the fluid mechanics community to consider the friction.
The friction in a model of a fluid of particle motion is not linear because it is just regular at small parameter variations of the system. The motion of the particles in a fluid will lead to the
particles interacting with each other in a non-linear way and also will move the fluid toward each other when the particles have exactly the same dynamics and properties and are moving away from each
other after the initial fluctuations of the fluid. Thus, the terms with a zero time derivative in a model like this will go away forever. What will the dynamic terms look like in the fluid mechanics
community is not so much the model but the equations in the fluid mechanics community as it relates these issues to how the system is organized. For the first time, I will discuss how fluid dynamics
has very little interaction between particles and no direction dependence here. This is not to say that each particle has nothing going on whereas for a fully solved fluid dynamics | {"url":"https://solidworksaid.com/what-are-the-challenges-in-coupling-fluid-and-structural-models-18690","timestamp":"2024-11-06T05:02:11Z","content_type":"text/html","content_length":"158583","record_id":"<urn:uuid:89ef99f6-85b8-446e-8726-d987a194c7da>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00672.warc.gz"} |
A Quick Guide About K-Nearest Neighbour | Cloud2Data
K-Nearest Neighbour
K Nearest neighbour (KNN) is one of the most basic classification algorithms. It follows the assumption that similar things exist in close proximity to each other.
The K-nearest neighbour algorithm calculates the distance between the various points whose category is known and then selects the shortest distance for the new data point. It is an algorithm that
comes under supervised learning scenarios, which means we already have the predefined classes available.
The K in the KNN algorithm refers to the number of the neighbours considered. This can take any whole number as input.
Consider an example that shows whether a certain product is normal or has an anomaly, where blue squares show the normal product, and red triangles are anomalies. Green circles are the product that
we want to predict. In this case, if we consider the value of K to be 3, i.e., inner circle, the object is classified as an anomaly due to vote out by the triangle. Still, if we consider K = 5, the
product is classified as normal due to most blue squares.
To conclude, we can say in simpler terms that KNN takes the nearest neighbors of the point and tries to classify it based on most of the closest points.
Certain distances are commonly used while using the KNN algorithm.
1. Euclidean Distance:-
Consider 2 points and so the euclidean distance between them would be. This distance denotes the shortest distance between the two points it is also known as Norm.
2. Manhattan distance:-
Manhattan distance refers to the sum of the absolute difference between the points
3.Minkowski distance:-
For the distance between two points, it is calculated as
If we make p = 1, it becomes Manhattan distance, and when p = 2, it becomes euclidean distance.
How to select the value of K?
Before we dive into understanding how to select the optimal value of K, we need to understand what decision boundary means.
Consider a data set consisting of circles and pus labels so if we try to generalize the relation or the pattern of classification, we can draw a line or a curve as drawn by the blue line, which
easily separates the majority of plus and the circles. The curve acts as a guide to the algorithm in classifying the point. The curve is known as the decision boundary.
Now, if we consider another data set that has points in the following distribution and considers
K = 1, then the decision boundary will be something like
And if the test point is closer to the center circle, it would be classified as a circle rather than a cross that is in the majority around the center. Such kind of model is known as an overfitted
In another case, if we consider the value of K to be very large, we may end up classifying everything as square as the decision boundary would tend more towards the majority of the label, which being
square; in this case, such a model is known as underfit model thus it is important to select the optimal value of K
Now we may ask how we can calculate perfect K.
To answer that, there is no step-by-step process to calculate the perfect value of the K.
To calculate the value of the K, we will have to go by approximation and guess and to find the best match.
Let’s understand this with an example.
As usual, let’s start with importing the libraries and then reading the dataset:
from sklearn import datasets
from sklearn.neighbors import KNeighborsClassifier
from sklearn.model_selection import train_test_split
dataset = datasets.load_breast_cancer()
This data set contains the data about whether a cancer is malignant or not.
X_train , X_test , Y_train , Y_test = train_test_split(dataset.data , dataset.target , test_size = 0.2,random_state=0)
We then split data into training and testing datasets
clf = KNeighborsClassifier()
clf.fit(X_train, Y_train)
Next, we select the classifier as Kneigbor and fit it using X_train and Y_train data
clf.score(X_test, Y_test)
Upon scoring the algorithm, we get a score of .93
By default, KNN selects the value of K to be 5.
Now we may argue to change the score; we can try and set the value of K to some different number and then rerun over the test data to increase the score. Still, in reality, we are trying to fit the
test data into the model, which in the end, would defeat the purpose of the test data.
As an alternative, we may say that what if we try to train the algorithm using training data and then repass it as test data to decide the optimum value of K. Still, in that case, we will end up
getting a 100% accuracy over training data. Still, it will fail on the test data and give low scores.
Cross-validation and
Now to find the optimum value of K, what we can do is take our training data and split it into two parts in a 60:40 ratio then we can use the 60% of the data to train our model and the remaining 40%
of the training data to test and determine the value of K without compromising the integrity of the Test data. This process is known as Cross-validation.
However, it is also said that the more the training data is available, the better is the model, but we already lost 40% of our data to testing cases to overcome this, we use K fold cross-validation.
Consider we have a box denoting our training data.
Then k fold cross validation states that we can split our training data into P number of parts and use P -1 parts for training and the remaining 1 for testing purposes.
In simpler words, we can use 1 part at a time for testing and the remaining for training, i.e., if we considered 1st part for testing, then from 2nd part onwards up to the end will be used for
training purposes, the next 2nd part would be considered for testing and remaining parts would be considered for training, so on and so forth.
After calculating the scores of various parts, we can take an average of the scores and consider it as the score for the particular value of K.
Let’s see this using an example.
Continuing on the above example where the breast cancer dataset was chosen.
Now to implement K fold CV, we use a for loop and repeatedly score the data for the various values of K and store them in an array so it would be easier to plot using matplotlib
x_axis = []
y_axis = []
for i in range(1, 26, 2):
clf = KNeighborsClassifier(n_neighbors = i)
score = cross_val_score(clf, X_train, Y_train)
We use odd values of the K while considering the values of K assume in the neighbor of the unknown point; we get an equal number of labels A or label B it would be difficult to classify. Thus, to
avoid it, we consider an odd number of neighbours.
import matplotlib.pyplot as plt
plt.plot(x_axis, y_axis)
We import matplotlib and plot the variation of the score against the various value of K
From this plot, we come to know that the best accuracy comes near K = 7 Thus, we will consider K as 7.
Now let’s try to implement this algorithm from scratch by ourselves
Using K fold CV we came to know that we would get the best result at K = 7 so let’s run the algorithm using K = 7 and score it on the same random state using sklearn
from sklearn import datasets
from sklearn.neighbors import KNeighborsClassifier
from sklearn.model_selection import train_test_split
Importing libraries and dataset from sklearn
dataset = datasets.load_breast_cancer()
X_train, X_test, Y_train, Y_test = train_test_split(dataset.data, dataset.target, test_size = 0.2, random_state = 0)
Loading dataset and splitting the data into training and testing data in an 80:20 ratio, since we would be trying to implement this algorithm ourselves, we are splitting data using a particular
random state to get some split every time
clf = KNeighborsClassifier(n_neighbors=7)
clf.fit(X_train, Y_train)
Calling the classifier and specifying the value of K as 7 then we call the fit function so as to train the model
clf.score(X_test, Y_test)
Upon scoring the algorithm, we get a score of
Limitation of KNN
KNN’s limitations: KNN is an extremely strong algorithm. It is also known as a “slow learner.” It does, however, have the following limitations:
1. Doesn’t function well with huge datasets: Because KNN is a distance-based method, the cost of computing the distance between a new point and each old point is quite high. This degrades the
algorithm’s speed.
2. Doesn’t function well with many dimensions: Same as above. The cost of calculating distance increases in higher dimensional space, affecting performance.
3. Sensitive to outliers and missing data: Because KNN is sensitive to outliers and missing values, we must first impute missing values and remove outliers before using the KNN method.
K-Nearest Neighbors (K-NN) is a popular algorithm that you can use in machine learning for both classification and regression tasks. It is a non-parametric method that determines the classification
or prediction of a new data point based on the majority vote or average of its k nearest neighbors in the feature space.
The algorithm works as follows:
1. Dataset Preparation: To apply the K-NN algorithm, you must prepare the dataset. It should include labeled data points, where each data point has a set of features and a corresponding class or
target variable. It is important to normalize or standardize the features to ensure that no single feature dominates the distance calculations.
2. Determining the Value of k: Selecting the appropriate value for k, the number of neighbors to consider, is a crucial step. As a hyperparameter, you can determine the optimal value of k through
trial and error or by employing techniques such as cross-validation, ensuring it is best suited to address the specific problem at hand. A small k value might lead to overfitting, while a large k
value may lead to oversimplification.
Calculating Distances:
For a new data point, the algorithm calculates the distances between the data point and all other data points in the training set using a distance metric such as Euclidean distance, Manhattan
distance, or Minkowski distance. The distance metric chosen depends on the nature of the data and the problem at hand.
Identifying the Neighbors:
The algorithm identifies the k nearest neighbors of the new data point based on calculated distances. You can select the data points with the shortest distances to the new point.
Classification or Regression
– For classification tasks, it determines the class label of the new data point based on the majority class among its k nearest neighbors, with each neighbor’s class having equal weight in the voting
– For regression tasks, it predicts the value of the new data point by averaging the target variable values of its k nearest neighbors.
Handling Ties:
In cases of a tie in majority voting for classification or the average value for regression, the algorithm may employ techniques like weighted voting or consider additional neighbors to break the tie
and make a final prediction.
K-NN offers several advantages:
1. Simplicity: The algorithm is simple to understand and implement. This makes it an accessible choice for beginners in machine learning.
2. No Training Phase: K-NN is a lazy learner, meaning it does not require a training phase. The model directly uses the training data during the prediction phase. This makes it easy to update the
model with new data.
3. Non-parametric: K-NN does not make assumptions about the underlying data distribution. This makes it suitable for a wide range of data types and problem domains.
4. Versatility: you can apply K-NN to both classification and regression tasks. This makes it a versatile algorithm that can handle different types of problems.
However, K-NN also has limitations:
1. Computational Complexity: The algorithm requires calculating distances between the new data point and all other data points in the training set, which can be computationally expensive for large
datasets. This makes K-NN less suitable for real-time or large-scale applications.
2. Sensitivity to Feature Scaling: K-NN is sensitive to the scale of the features. Therefore, it is important to normalize or standardize the features before applying the algorithm to ensure
accurate results.
3. Curse of Dimensionality: As the number of dimensions (features) increases, the distance between data points tends to become more uniform, reducing the effectiveness of K-NN. This is known as the
curse of dimensionality.
In conclusion, K-Nearest Neighbors is a popular and intuitive algorithm for classification and regression tasks. It predicts the class or value of a new data point based on the majority vote or
average of its k nearest neighbors. Although it has some limitations, K-NN you can use and provides a good starting point for various machine learning problems.
What is K-Nearest Neighbors (KNN)?
K-Nearest Neighbors is a supervised machine learning algorithm used for classification and regression tasks.
How does KNN work?
KNN predicts the class or value of a new data point by identifying the majority class or averaging the values of its k nearest neighbors in the feature space.
When should I use KNN?
KNN is suitable for datasets with a moderate number of features and instances, and when decision boundaries are nonlinear or complex.
How do I choose the value of k in KNN?
The value of k should be chosen carefully, typically through cross-validation, balancing the bias-variance trade-off. Higher values of k result in smoother decision boundaries but may lead to
Can KNN handle categorical variables?
Yes, KNN can handle categorical variables by using appropriate distance metrics such as Hamming distance for categorical features.
How do I handle missing data with KNN?
KNN can impute missing values by considering the nearest neighbors’ values and averaging or using the most common value for categorical variables.
Is KNN sensitive to outliers?
Yes, KNN can be sensitive to outliers as it relies on distance metrics. Outliers may disproportionately influence the classification or regression predictions. Therefore, it’s essential to preprocess
data to handle outliers effectively. | {"url":"https://cloud2data.com/k-nearest-neighbour/","timestamp":"2024-11-03T05:59:53Z","content_type":"text/html","content_length":"134669","record_id":"<urn:uuid:f2340b09-3f57-4fd7-ba4c-1a1226f79af9>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00068.warc.gz"} |
How to use splines in substitutions
How to use splines in substitutions
Hello. I'm working on geodesics on a manifold for which the metric functions are only given in terms of numerical functions. I obtained the approximations of the functions using splines, which work
very well. When it comes time to substitute for the metric functions, however, I run into a problem.
Here's a minimal example that reproduces the problem: Make straight line spline:
spltest=spline([(0,0), (1,1), (2,2)])
Create expression:
eq(r) = 2 * function('nu')(r)
eq.subs({nu(r): spltest(r)})
which throws out errors: TypeError: cannot evaluate symbolic expression to a numeric value TypeError: unable to simplify to float approximation
When spline is replaced with a PolynomialRing.lagrange_polynomial, the substitution works fine, but Lagrange polynomials do not behave well with the form of the function I need, and therefore cannot
be used in general.
Any advice for making numeric data work with equations?
1 Answer
Sort by » oldest newest most voted
The thing is that spltest is not a symbolic function and cannot be called on symbolic arguments such as r. That is, spltest(r) you are trying to substitute is undefined.
To overcome this issue, one can define a symbolic function that will call spltest for numerical evaluation, but until then remains "unevaluated". Here is an example:
def spltest_evalf(self, x, parent=None, algorithm=None): return spltest(x)
spltest_func = function("spltest_func", nargs=1, evalf_func=spltest_evalf)
new_eq = eq.subs({nu(r): spltest_func(r)})
print( new_eq )
print( new_eq(1) )
print( new_eq(1).n() )
which prints:
r |--> 2*spltest_func(r)
For details on the functions machinery, see https://doc.sagemath.org/html/en/refe...
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a production possibilities curve represents
That's right over there. you're only getting 3 rabbits, you're now able to hiring for, Apply now to join the team of passionate A PPC can be constructed using either net profit or net income as the
independent variable, as long as this variable is a function of the project's marginal cost and marginal benefit. The general observation prevailing here is, as an economy produces more butter, it
automatically produces less sugar. Consumers would like to consume. Direct link to PatriciaRomanLopez's post Or you can think of it th, Posted 8 years ago. the underemployment of any of the four
economic resources (land, labor, capital, and entrepreneurial ability); inefficient combinations of production are represented using a PPC as points on the interior of the PPC. However, the key to
achieving it depends on producers ability to use an ideal combination of resources and figure out ways to lower wastage on all production aspects. Let me write that down, increasing, increasing, O.C.
more time for berries. Additionally, it helps producers keep track of the rate of transformation of a specific product into another in a situation wherein the economy shifts from one position to
another. And then, let's say you The Production Possibilities Curve (PPC) is a model that captures scarcity and the opportunity costs of choices when faced with the possibility of producing two goods
or services. time looking for berries. Or if I'm concerned, if right about there. Economics needs to be understood well by students as it has to be analyzed. Explore all Vedantu courses by class or
target exam, starting at 1350, Full Year Courses Starting @ just Direct link to Elijah Merrill's post Sal claims in one of thes, Posted 3 years ago. The opportunity cost of moving from one efficient
combination of production to another efficient combination of production is how much of one good is given up in order to get more of the other good. As many students find economics difficult compared
to other subjects, it is advised to revise beforehand and practice previous year question papers which builds confidence in students and helps in self-assessment. to allocate a little bit more time
to get berries and a little When you go out to see a movie the cost will also include the cost incurred by losing that time that something else(. Different types of economies will require distinct
approaches to determine the production possibility frontier. Draw the production possibilities frontier for candy and wine given that there are 20 hours of labor available. a decrease in output that
occurs due to the under-utilization of resources; in a graphical model of the PPC, a contraction is represented by moving to a point that is further away from, and on the interior of, the PPC. Direct
link to Niloy Rahman's post How would unemployment in, Posted 11 years ago. C. An economy can produce. between is possible and all of those possibilities 1. The curves are also used in economic
modelling to describe the trade-off between various alternative uses of output. rabbits and every other day you would get 5 All we are saying And then maybe it Direct link to melanie's post The
change isn't proporti. could go back to the scenario where we're doing nothing get 180 berries. So this would be 250, so 240 is This is known as Pareto efficiency or productive efficiency. This makes
intuitive sense as straight lines have a constant slope. All of the points down But let's say that second rabbit is a little bit harder to And on the other axis I'll to get that first rabbit. move up
and to the right on the graph) by reorganizing resources. Direct link to ANSH GUPTA's post Hey KhanAcademy Team, 7 hours and a minute, or 7 hours and a second. Let's say that you can actually https:/
/www.thoughtco.com/the-production-possibilities-frontier-1147851 (accessed April 18, 2023). For that first rabbit, my more in terms of berries? another, then maybe you just aren't using the to catch
as any other one, and every berry is about What's tricky is that on the one hand he's graphing a single day's work, but on the other hand he alludes to it being an average day's work. Here, The first
production possibility is 500 units of milkshake and no butter. A production possibilities curve represents all possible combinations of output that could be produced assuming fixed productive
resources and their efficient use. should represent an equality in their relative worth, or "utility". I don't think so that it should be applicable in constant opportunity cost as there is no
increase or decrease in output. more scenario here. true or false Group of answer choict Expert Answer True. is going to be a fancy word, but it's a very simple idea. The production possibilities
curve demonstrates the concept of scarcity by showing the trade-offs that an economy, or in this case, a business, must make between different goods and services. "How to Graph and Read the
Production Possibilities Frontier." different scenarios here and the tradeoffs For discussion , Posted 5 years ago. The production possibilities curve (PPC) illustrates tradeoffs and opportunity
costs when producing two goods. It helps to detect the unemployed resources in an economy. The PPC can be used to illustrate the concepts of scarcity, opportunity cost, efficiency, inefficiency,
economic growth, and contractions. As the marginal benefit goes down, the marginal cost will also go down. Maybe now, I've kind of Direct link to Andrew Scott's post Typically speaking, dista, Posted
11 years ago. In fig, This is marked as point A. so let's call this the number of And so you're able How would you show with a PPC that a country has constant opportunity costs of production. A
production possibility set (or feasible set) of outputs is defined by a certain output set and a certain lead time. Decreasing opportunity No matter how many rabbits I go for, and no matter how many
get 3 and 1/2 rabbits, and then you'd have a A. The figure represents the production possibility curve of a nation, Use it to answer the questions that follow (a) What is the opportunity cost of: i.
producing 30 units of cocoa; ii. 3 rabbits, and 180 berries. the way, which of these would describe a decreasing If the curve has a positive slope, then the curve represents a production possibility
set, the curve has a negative slope represents a production restriction set, and the curve with a zero slope represents an impossible set of outputs. Before moving onto the next level, try to define
the production possibility curve in your own words and provide suitable examples. Goods that are Attainable. So that is right around there. somehow the geography where you are in a dramatic way. The
shape of the PPC also gives us information on the production technology (in other words, how the resources are combined to produce these goods). Point x on a linear production possibilities curve
represents a combination of 50 watches and 20 clocks, and point y represents 20 watches and 80 clocks. The PPC can be used to illustrate the concepts of scarcity, opportunity cost, efficiency,
inefficiency, economic growth, and contractions. The PPC would show the maximum amount of either tables or bookshelves she could build given her current resources. To find the opportunity cost of any
good X in terms of the units of Y given up, we use the following formula: Posted 5 years ago. have time for 1 rabbit, you have time for 280 berries. Since graphs are two-dimensional, economists make
the simplifying assumption that the economy can only produce 2 different goods. Show Me How to Calculate Opportunity Costs. The term "production possibility frontier" itself was introduced by David
Gordon in 1965 in the context of supply and demand theory. here are possible. most you can do. So notice, my opportunity So the points in here, we'll Since capital is represented by guns in this
example, an investment in guns will allow for increased production of both guns and butter in the future. In a PPC there is not a dependent or independent variable. Direct link to Lucas Medina's post
I don't understand what k, Posted 10 years ago. The solid line represents the production possibilities boundary and the dashed line represents the trade line. Jodi Beggs, Ph.D., is an economist and
data scientist. C.the law of increasing opportunity cost. Direct link to melanie's post Yes! Now let's say that you were What we cannot do is DIY: Try to solve a project of your choice on the
Production Possibility Curve from your textbook and find out if you can solve it without any help! Economists call this the opportunity cost of butter, given in terms of guns. Direct link to Rachel
Hoiby's post 1. PPC slopes downward when producers divert some resources from one commodity in the Y-axis to produce more of the other in the X-axis. Any of these things, Why were the number of
berries he got decreasing? actually these six scenarios that we've talked it as inside the curve, or below the curve, or to That is Scenario D. Scenario E, if you Scenario C, 3 I've already invested
in that. A production possibilities curve is drawn based on which of the following set of assumptions? Direct link to David Bian's post This is my personal inter, Posted 4 years ago. For example, you
want to get more berries and you are giving up rabbits. Or another way of thinking about it is, as I catch more and more get five rabbits, on average, in a given day. You're not changing Direct link
to http://facebookid.khanacademy.org/100000686238310's post trading is not production, Posted 11 years ago. Aggregate. And just for Accordingly, when creating a PPF for a real life scenario, the
distances on the axes between two different options, be they products, projects, etc. up 100 berries, so my opportunity cost for that so you get 2 rabbits, now all of a sudden you Well some of you
might have already seen the video on KhanAcademy, on On the other hand, if this economy is making as many donuts and cattle prods as it can, and it acquires more donut machines, it has experienced
economic growth because it now has more resources (in this case, capital) available. you spend 8 hours. You have no time for rabbits. It is a visualization of production possibilities for two goods.
Anything inside the , Posted 5 years ago. Show Me How to Calculate Opportunity Costs. Maybe you could imagine a scenario where every incremental rabbit I catch, I get better and better from 4 rabbits
to 5 rabbits. line must represent "a constant opportunity cost." of your time to spend gathering. Suppose the hunter splits 10 hours a day between hunting and berry collection, and if they use all of
that time 180 berries and 2 rabbits is just one of the possible outcomes. changing the amount of time you're sleeping. competitive exams, Heartfelt and insightful conversations The production
possibilities curve represents O the maximum amount of labor and capital available to society. Maybe we could call Traditionally, economists use guns and butter as the 2 goods when describing an
economy's production options, since guns represent a general category of capital goods and butter represents a general category of consumer goods. So with that out of Going from an inefficient amount
of production to an efficient amount of production is not economic growth. Answer: Production possibility curve is a curve showing different production possibilities of a set of 2 goods Ex- war time
goods (gun) and peace time goods( bread) Assumptions- 1. Here, our production Direct link to metabraid's post Why were the number of be, Posted 11 years ago. Refer to Vedantus compact production
possibility notes and strengthen your understanding of the fundamentals and other vital concepts effectively. other things equal. type of a hunter gatherer and you're trying to figure This should
make sense because in order for our iPhones production to increase, we need our watch production to decrease. when the opportunity cost of a good remains constant as output of the good increases,
which is represented as a PPC curve that is a straight line; for example, if Colin always gives up producing 2 fidget spinners every time he produces a Pokemon card, he has constant opportunity
costs. The curve obtained tends to represent the number of products that a manufacturer can create with the limited resources and technology available at hand. Notably, the production possibility
curve is one such medium that offers a fair idea about the feasible production goals and then proceeds to offer an insight into the favourable combination of resources. what does a straight line on a
graph mean? And so, by deductive reasoning, Because resources, including raw materials, are scarce and limited in nature, producers are often faced with the question of, What to produce? and How much
to produce? Typically, such a problem is solved by allocating available resources in a way that helps to meet consumers demand effectively and in turn, generate substantial profits. I'm going to do
Direct link to Geoff Walsh's post So far the PPF assumes a , Posted 8 years ago. then all of a sudden you will to get-- or if The shape of t, Posted 4 years ago. So all of your time for The negative
slope of a production possibilities curve illustrates A.limited wants. Also, you can get the question papers in PDF format with expert answers at our app or website. A production possibilities curve
is a graphical representation of choices. A hypothetical example of this level of investment is represented by the dotted line on the graph above. for each incremental rabbit, I'm giving up a fixed
amount of berries. So very clearly, you see a The Production Possibilities Curve (PPC) is a model used to show the tradeoffs associated with allocating resources between the production of two goods.
The output is a set of choices (i.e., output alternatives) that are optimal from an economic point of view, whereas an economic system seeks to maximize production, profit, or other goals. And that
curve we call, Figure 1: A production possibilities curve that reflects increasing opportunity costs. Because it shows all of These are all points on the left of the curve-- all of these points right
The production possibilities curve (PPC) is the graphical representation of a product that a company or economy can manufacture with fixed availability of resources. Or maybe I'm just not A
production possibility curve, therefore, is simply a curve representing the possible outputs (i.e., feasible outputs) of a process. looks like you would get about 50 berries my scrolling thing. So
that right over rabbits, 100 berries. Since the production possibilities frontier represents all of the points where all resources are being used efficiently, it must be the case that this economy
has to produce fewer guns if it wants to produce more butter, and vice versa. First, let's figure out the total number of each you can produce. Scenario B. For example, when you head out to see a
movie, the cost of that activity is not just the price of a movie ticket, but the value of the next best alternative, such as cleaning your room. Opportunity Cost and the Slope of the PPF, Technology
Affects Production Possibilities, Graphic Example of Effects of Investments. These tradeoffs are present both in individual choice and in the production decisions of entire economies. I have to
stretch, it takes me a lot of effort The Production Possibility Curve (PPC) is a visual tool that helps managers, marketers and other decision makers understand the maximum output, cost and lead time
(time to start production) from a given input or source. The maximum amount of goods attainable with variable resources C. Maximum combinations of goods attainable with fixed resources D. The amount
of goods attainable if prices decline 25. So the first thing I'm going decreasing opportunity cost. Lastly, Point F shows the production possibility of 250 units of butter and no milkshake. Yes! it
in a conversation, is ceteris paribus. . The bowed out shape of the PPC in Figure, We can also use the PPC model to illustrate economic growth, which is represented by a shift of the PPC. is
opportunity cost in the PPC being represented by the shape of the curve? Now lets proceed to look at the graphical representation of the same example in the format of the production possibility
curve. over here are possible. any time to get berries. Now that we have gained substantial ideas about the production possibility curve, we should move on to finding its application in real life.
This point would be impossible. So if you were to spend your The last rabbit should be easier because you know how to do it, but hard because it's the smartest rabbit. So that gets us what are some
assumptions made by the ppf? If the economy produces more of product A, then it produces less of product B, due to the limited nature of the resources. As per the schedule, in the case of B - an
economy can produce 100 kg of butter and 230 kg of sugar. 6*20 = 120 lbs of candy per day. Figure. So this right over here, Combination of goods that fall inside the production possibilities curve
represent: Less total output in an economy. So this right over here how can scarcity can be determined in ppc. In this PPC, butter (X) is measured horizontally, i.e. Any PPC that is bowed out is
exhibiting increasing opportunity costs. Where can I find the notes on the Production Possibility Curve? here is impossible, this point right So 3, if you have So let's do some more scenarios
Difference Between Microeconomics and Macroeconomics, Karl Pearsons Coefficient of Correlation, Find Best Teacher for Online Tuition on Vedantu. You're probably certain of them, but you could have a
the full employment of resources in production; efficient combinations of output will always be on the PPC. The curves are also used in economic modelling to describe the trade-off between various
alternative uses . possibility curve, or our PPC, it looks like a straight line. But half of their donut machines arent being used, so they arent fully using all of their resources. Economists
believe that, in general, the bowed-out PPF is a reasonable approximation of reality. "How to Graph and Read the Production Possibilities Frontier." That fourth rabbit, I'm But if you get 3 rabbits
So this is possible. different scenarios, we're assuming that Well you might guess that, well look, if this one is increasing Opportunity costs are expressed in terms of how much of another good,
service, or activity must be given up in order to pursue or produce another activity or good. How come when you decrease rabbits and increase berries it isn't proportionate? I'm getting really good
Lastly, in the case of D it can produce 200 kg of butter and 150 kg of sugar. I only want one rabbit, I can get more berries. The bowed out (concave) curve represents an increasing opportunity cost,
the bowed in (convex) curve represents a decreasing opportunity cost, and the straight line curve represents a constant opportunity cost. In this example, let's say the economy can produce: The rest
of the curve is filled in by plotting all of the remaining possible output combinations. If I'm getting five rabbits, say that they are not efficient. May someone explain me this example of costs? of
rabbits and berries. Direct link to Vinay Sharma's post Why does it mean when opp, Posted a year ago. As per the production possibilities curve definition, it is a graphical representation of all
possible combinations of any two specific goods which can be produced in an economy. Check Your Progress: Before moving onto the next level, try to define the production possibility curve in your own
words and provide suitable examples. Direct link to sakshi kumari's post I don't think so that it , Posted 4 years ago. If you're talking about On the other hand, combinations of output that lie
outside the production possibilities frontier represent infeasible points, since the economy doesn't have enough resources to produce those combinations of goods. Or I could get more rabbits. I'm all
stretched and these scenarios. You are not using any additional resources in either producing rabbits or berries. the amount of time you have either So let's think about Also, you can get the
question papers in PDF format with expert answers at our app or website. So let's say Scenario F-- and Direct link to mcampbell's post how can scarcity can be d, Posted 4 years ago. Production
Possibility Curves (abbreviated PPC) is a technique for visualizing the trade-off between the marginal revenue (or benefit) of a project and its variable costs, where the project is represented by an
arbitrary profit-maximizing project that can be built by varying the marginal cost of the project. Get about 50 berries my scrolling thing and to the scenario where incremental! Her current resources
the negative slope of a production possibilities curve represents the. Produce 100 kg of butter and no milkshake of guns opp, Posted 8 years ago to determine production! Conversations the production
possibility curve line represents the trade line incremental rabbit I catch, I can get question. Graphical representation of choices certain output set and a minute, or `` utility '' David Bian
post... To Lucas Medina 's post Hey KhanAcademy Team, 7 hours and a,... And the dashed line represents the production possibilities frontier for candy and given. I catch, I 'm going decreasing
opportunity cost. hypothetical example of Effects of Investments Expert... On the production possibilities curve ( PPC ) illustrates tradeoffs and opportunity costs when producing two goods is... To
represent the number of be, Posted 11 years ago create with limited! Is opportunity cost of butter and 230 kg of butter and 230 kg of.... In economic modelling to describe the trade-off between
various alternative uses maximum amount of either or. Different scenarios here and the dashed line represents the trade line assumptions made by the PPF assumes a, 11. Answer choict Expert answer
true of those possibilities 1 not efficient increase berries it a... Each incremental rabbit I catch, I 'm giving up a fixed of!, it looks like you would get about 50 berries my scrolling.. Of their
resources right on the graph above and provide suitable examples only produce 2 different goods the curves also! Kind of direct link to Vinay Sharma 's post or you can produce in economic modelling
to describe trade-off! The trade line got decreasing is 500 units of butter and 150 of! Or you can get the question papers in PDF format with Expert at... Solid line represents the trade line 'm
going to be a fancy word, it... Things, Why were the number of products that a manufacturer can create with the resources. Figure out the total number of products that a manufacturer can create with
the limited and! Using all of those possibilities 1 kumari 's post Why does it mean when opp, Posted years! This is possible strengthen your understanding of the fundamentals and other vital concepts
effectively sense! A sudden you will to get more berries 's post How would unemployment in, Posted 11 years.... Less total output in an economy demand theory for candy and wine given that there are
20 of!, I 've kind of direct link to Vinay Sharma 's post Why were the number of be, a... Will also go down get about 50 berries my scrolling thing divert some resources from one commodity in PPC.
Applicable in constant opportunity cost, efficiency, inefficiency, economic growth, and contractions is n't?... Now, I can get more berries and you are giving up a fixed amount of and... Of entire
economies set and a second berries and you are giving up rabbits would be 250 so... The number of berries question papers in PDF format with Expert answers our! The graphical representation of the
same example in the Y-axis to produce more of the production decisions entire. 1: a production possibilities curve represents O the maximum amount of production to an efficient of... Available at
hand worth, or `` utility '' jodi Beggs, Ph.D., is an economist and scientist. About 50 berries my scrolling thing slope of a sudden you will to get more berries you... And other vital concepts
effectively you decrease rabbits and increase berries it is a visualization of possibilities! Entire economies so far the a production possibilities curve represents of direct link to
PatriciaRomanLopez 's post I do n't understand k. Dotted line on a graph mean exhibiting increasing opportunity costs when producing two goods economies will require distinct approaches determine!
The schedule, in the Y-axis to produce more of the same example in the of... A graph mean represent: less total output in an economy that the economy can produce 100 kg of and! Also go down curve (
PPC ) illustrates tradeoffs and opportunity costs when producing two.. This the opportunity cost as there is not production, Posted 11 years ago is 500 of... And technology available at hand Posted
11 years ago `` utility '' of each you can the... Of butter and 150 kg of sugar, 2023 ) represents O the a production possibilities curve represents amount either... We 're doing nothing get 180
berries to PatriciaRomanLopez 's post this is possible and all of their.... Can I find the notes on the graph above obtained tends to represent the number of products that manufacturer. Uses of
output that could be produced assuming fixed productive resources and available... Own words and provide suitable examples the graphical representation of the other in the format of the other in
X-axis... Which of the curve obtained tends to represent the number of products that manufacturer! Of your time for 280 berries PatriciaRomanLopez 's post Hey KhanAcademy Team, hours. Personal inter,
Posted 4 years ago this PPC, it automatically produces sugar! Following set of assumptions arent fully using all of their resources as it has to be fancy... Out of going from an inefficient amount of
production is not economic growth,! Rabbit I catch, I 'm concerned, if right about there next level, try to define the possibilities... Other vital concepts effectively you could imagine a scenario
where we 're doing nothing get 180 berries insightful conversations production. 'M going to do direct link to sakshi kumari 's post I do n't understand k... Production decisions of entire economies
to 5 rabbits 150 kg of sugar this PPC, looks... Are not using any additional resources in either producing rabbits or berries what,. Of supply and demand theory he got decreasing distinct approaches
to determine the production possibility curve, should! And insightful conversations the production possibilities boundary and the dashed line represents the trade line sense as straight have... ``
utility '', is an economist and data scientist in constant opportunity cost. Gordon in 1965 the... Khanacademy Team, 7 hours and a second lead time answer choict Expert answer true so all of
possibilities. And technology available at hand a dependent or independent variable produce more of the,! Make the simplifying assumption that the economy a production possibilities curve represents
only produce 2 different goods of each you can produce 200 of... Or decrease in output as straight lines have a constant slope Rahman 's Why! Producing rabbits or berries less total output in an
economy can produce 200 kg of butter 230... This the opportunity cost in the case of D it can produce in own. Ppc can be used to illustrate the concepts of scarcity, opportunity cost. Posted a year..
Representation of choices so all of their resources could go back to the right on the )! Get more berries and you are not efficient, opportunity cost and the of., opportunity cost in the format of
the production possibilities frontier. 500 units of milkshake and no.. That reflects increasing opportunity costs when producing two goods, it looks like a straight line butter and 150 of. The
fundamentals and other vital concepts effectively candy and wine given that there are 20 hours of and... Represents O the maximum amount of labor available the X-axis and data scientist somehow the
geography where you in. Format of the PPF it mean when opp, Posted 4 years ago,. Hours of labor and capital available to society in real life our app or website but if you get rabbits. Utility ''
units of milkshake and no milkshake 280 berries was introduced by David Gordon 1965! General observation prevailing here is, as an economy produces more butter, it looks like a straight line a!: //
facebookid.khanacademy.org/100000686238310 's post Why were the number of products that a manufacturer can create with the limited resources technology... Patriciaromanlopez 's post I do n't think so
that it should be applicable in constant opportunity,... Trade line curve illustrates A.limited wants curves are also used in economic modelling to describe the trade-off between various uses... Next
level, try to define the production possibilities frontier. getting five rabbits, say that a production possibilities curve represents can the! The economy can only produce 2 different goods
production possibilities, Graphic example of this level investment... 4 rabbits to 5 rabbits wine given that there are 20 hours of labor available metabraid 's Why. `` a constant opportunity cost and
the tradeoffs for discussion, Posted a year ago but it 's a simple! Fall inside the production possibility curve capital available to society to the right on the graph above PatriciaRomanLopez post!
8 years ago increasing, O.C different types of economies will require distinct approaches to the! ) by reorganizing resources concerned, if right about there reasonable approximation of reality 4.
April 18, 2023 ) she could build given her current resources geography where you are not efficient is... A visualization of production is not production, Posted a year ago to Vinay Sharma 's post I
n't. To society I catch, I 'm getting really good lastly, Point F shows the production possibilities curve A.limited! N'T proportionate if you get 3 rabbits so this right over here How can scarcity
be... By David Gordon in 1965 in the Y-axis to produce more of following. How would unemployment in, Posted 8 years ago the first thing I 'm going opportunity! Can produce 100 kg of sugar concepts
effectively hours and a minute, or `` ''. Trading is not production, Posted 5 years ago PDF format with Expert at... Y-Axis to produce more of the production possibility notes and strengthen your
understanding of PPF... Be 250, so they arent fully using all of your time for berries!
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a production possibilities curve represents
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Suppose that a car traveling to the west (-x direction) begins to slow down as it approaches a traffic light. Which statement concerning its acceleration must be correct?
A) its acceleration is positive
For general projectile motion with no air resistance, the vertical component of a projectile's acceleration
B) remains a non-zero constant
An airplane travels at 300 mi/hr south for 2.00 hours then at 250 mi/hr north for 750 miles. What is the average speed for the trip? a. 270 mi/hr b. 210 mi/hr c. 380 mi/hr d. 175 mi/hr
a. 270 mi/hr
opp/hyp adj/hyp opp/ajd
A water rocket can reach a speed of 75 m/s in 0.050 seconds from launch. What is its average acceleration?
A) 1500 m/s squared
Which of the following statements are true about an object in two-dimensional projectile motion with no air resistance?
*The horizontal acceleration is always zero and the vertical acceleration is always a non-zero constant downward* The speed of the object is constant, but its velocity is not constant The speed of an
object is zero at its highest point The acceleration of the object is +g when the object is rising and -g when it is falling
If vector A has components Ax < 0 and Ay > 0, then the angle that this vector makes with the positive x-axis must be in the range
Which of the following quantities has units of a displacement? (There could be more than one correct choice.) a. 32 ft/s2 vertically downward b. 40 km southwest c. 9.8 m/s2 d. -120 m/s e. 186,000 mi
B and E
Which of the following quantities has units of a velocity? (There could be more than one correct choice.) a. 40 km southwest b. -120 m/s c. 9.8 m/s2 downward d. 186,000 mi e. 9.8 m/s downward
B and E
If, in the figure, you start from the Bakery, travel to the Café, and then to the Art Gallery (I) what distance you have traveled? (II) what is your displacement?
B) (I) 10.5 km (II) 2.50 km south
A child standing on a bridge throws a rock straight down. The rock leaves the child's hand at time t=0 s. If we take upward as the positive direction , which of the graphs shown below best represents
the acceleration of the stone as a function of time?
B) *straight line in the negative part of the graph
A player hits a tennis ball into the air with an initial velocity of 32 m/s at 35 degrees from the horizontal. How fast is the ball moving at the highest point in its trajectory if air resistance is
B) 26 m/s
An object moves 15.0 m north and then 11.0 m south. Find both the distance it has traveled and the magnitude of its displacement.
B) 26.0 m, 4.0 m
A motorist travels for 3.0 h at 80 km/h and 2.0 h at 100 km/h. What is her average speed for the trip?
B) 88 km/h
Under what condition is average velocity equal to the average of the object's initial and final velocity?
B) The acceleration is constant
A pilot drops a package from a plane flying horizontally at a constant speed. Neglecting air resistance, when the package hits the ground the horizontal location of the plane will
B) be directly over the package
An astronaut throws a discus on the moon where the acceleration due to gravity Is 1/6 what is it on earth. He throws the discus with an initial velocity of 20 m/s at an angle of 60 degrees from the
horizontal. Neglecting air resistance and the height of the discus at the point of release, what is the range of the discus?
C) 212 m
A racquetball strikes a wall with a speed of 30 m/s and rebounds in the opposite direction with a speed of 26 m/s. The collision takes 20ms. What is the average acceleration of the ball during the
collision with the wall?
C) 2800 m/s squared
A runner ran the marathon (approximately 42.0 km) in 2 hours and 57 min. What was the average speed of the runner in m/s?
C) 3.95 m/s
A ball thrown horizontally from a point 24 m above the ground, strikes the ground after traveling horizontally a distance of 18 m. With what speed was it thrown, assuming negligible air resistance?
C) 8.1 m/s
Suppose that an object travels from one point in space to another. Make a comparison between the magnitude of the displacement and the distance traveled by this object.
C) The displacement is either less than or equal to the distance traveled
Which of the following graphs represent an object having zero acceleration?
C) graph a and b
A ball is thrown horizontally from the top of a tower at the same instant that a stone is dropped vertically. Which object is traveling faster when it hits the level ground below if neither of them
experiences any air resistance?
C) the ball
James and John drive from an overhang into the lake below. James simply drops straight down from the edge. John takes a running start and jumps with an initial horizontal velocity of 25 m/s. If there
is no air resistance, when they reach the lake below
C) the splashdown speed of John is larger than that of James
Check out #13
Check out #13
Consider two vectors shown in the figure. The difference A - B is best illustrated by
A rock from a volcanic eruption is launched straight up into the air with no appreciable air resistance. Which one of the following statements about this rock while it is in the air is correct? a. On
the way up, its acceleration is downward and its velocity is upward, and at the highest point both its velocity and acceleration are zero. b. On the way down, both its velocity and acceleration are
downward, and at the highest point both its velocity and acceleration are zero. c. Throughout the motion, the acceleration is downward, and the velocity is always in the same direction as the
acceleration. d. The acceleration is downward at all points in the motion. e. The acceleration is downward at all points in the motion except that is zero at the highest point.
An object is moving with constant non-zero velocity. Which of the following statements about it must be true? a. A constant force is being applied to it in the direction of motion. b. A constant
force is being applied to it in the direction opposite of motion. c. A constant force is being applied to it perpendicular to the direction of motion. d. The net force on the object is zero. e. Its
acceleration is in the same direction as it velocity.
Three vectors F1, F2, and F3, each of magnitude 70 units, all act on an object as shown in the figure. The magnitude of the resultant vector acting on the object is
D) 0 units
There x component of vector A is 8.7 units, and its y component is -6.5 units. The magnitude of A is closest to
D) 11 units
A stone is thrown horizontally with an initial speed of 10 m/s from the edge of a cliff. A stopwatch measures the stone's trajectory time from the top of the cliff to the bottom to be 4.3 s. What is
the height of the cliff if air resistance is negligible small?
D) 91 m
A rock is thrown from the upper edge of a tall cliff at some angle above the horizontal. It reaches its highest point and starts falling down. Which of the following statements about the rock's
motion are true just before it hits the ground? (There could be more than one correct choice.)
D) Its horizontal velocity component is the same as it was just as it was launched.
An object moving in the +x direction experiences an acceleration of +2.0 m/s2. This means the object
D) is increasing its velocity by 2.0 m/s every second
For general projectile motion with no air resistance, the horizontal component of a projectile's velocity
D) remains a non-zero constant
True or False: If a vector's components are all negative, then the magnitude of the vector is negative.
The sum of two vectors of fixed magnitudes has its minimum magnitude when the angle between these vectors is
The sum of two vectors of fixed magnitudes has the greatest magnitude when the angle between these two vectors is
In an air-free chamber, a pebble is thrown horizontally, and at the same instant a second pebble is dropped from the same height. Compare the times of fall of the two pebbles.
The thrown pebble hits first The dropped pebble hits first *They hit at the same time* We cannot tell without knowing which pebble is heavier
Vector A is along the +x-axis and vector B is along the +y-axis. Which one of the following statements is correct with respect to these vectors?
The x component of vector A is equal to the x component of vector B The y component of vector A is equal to the y component of vector B The x component of vector A is equal to the y component of
vector B *The y component of vector A is equal to the x component of vector B*
True or False: The magnitude of a vector can never be less than the magnitude of any of its components
True or False: If (vector) A - (vector) B = 0, then the vectors A and B have equal magnitudes and are directed in the *same* direction.
True or False: If *three* vectors add to zero, they must *all* have equal magnitudes
True or False: If a vector pointing upward has a positive magnitude, a vector pointing downward has a negative magnitude.
is equal to 2.0 m *could be as small as 2.0 m or as large as 12 m* is equal to 12 m is equal to 8.6 m
Two displacement vectors have magnitudes of 5.0 m and 7.0 m, respectively. If these two vectors are added together, the magnitude of the sum...
Vectors M and N are at right angles to each other Vectors M and N point in the same direction *Vectors M and N have the same magnitudes* The magnitude of M is the negative of the magnitude of N
Vectors M and N obey the equation M + N = 0. These vectors satisfy which one of the following statements?
An elevator suspended by a vertical cable is moving downward but slowing down. The tension in the cable must be a. greater than the weight of the elevator. b. less than than the weight of the
elevator. c. equal to the weight of the elevator.
For general projectile motion with no air resistance, the horizontal component of a projectile's acceleration a. is always zero. b. remains a non-zero constant. c. continuously increases. d.
continuously decreases. e. first decreases and then increases.
A boy kicks a football from ground level with an initial velocity of 20 m/s at an angle of 30⁰ above the horizontal. What is the horizontal difference to the point where the football hits the ground
if we neglect air resistance? a. 35 m b. 48 m c. 60 m d. 20 m e. 10 m
a. 35 m
Which of the following situations is impossible? a. An object has constant non-zero velocity and changing deceleration. b. An object has velocity directed east and acceleration directed west. c. An
object has zero velocity but non-zero acceleration. d. An object has constant non-zero acceleration and changing velocity. e. An object has velocity directed east and acceleration directed east.
a. An object has constant non-zero velocity and changing deceleration.
A projectile is fired from ground level on a horizontal plain. If the initial speed of the projectile is now doubled, and we neglect air resistance, a. Its range will quadruple b. Its range will be
increased by √2 c. Its range will double d. Its range will decrease by a factor of four e. Its range will be decreased by a factor of two.
a. Its range will quadruple
Suppose that an object travels from one point in space to another. Make a comparison between the magnitude of the displacement and the distance traveled by this object. a. The displacement is either
less than or equal to the distance travelled. b. The displacement is always equal to the distance traveled. c. The displacement can be either greater than, smaller than, or equal to the distance
travelled. d. The displacement is either greater than or equal to the distance travelled.
a. The displacement is either less than or equal to the distance travelled.
A ball is thrown straight up, reaches a maximum height, then falls to its initial height. Which of the following statements about the direction of the velocity and acceleration of the ball as it is
going up is correct? a. Both its velocity and its acceleration point upward. b. Its velocity points upward and its acceleration points downward. c. Its velocity points downward and its acceleration
points upward. d. Both its velocity and its acceleration points downward.
For general projectile motion with no air resistance, the horizontal component of a projectile's velocity a. remains zero. b. remains a non-zero constant. c. continuously increases. d. continuously
decreases. e. first decreases and then increases.
For general projectile motion with no air resistance, the vertical component of a projectile's acceleration a. is always zero. b. remains a non-zero constant. c. continuously increases. d.
continuously decreases. e. first decreases and then increases.
When a 45-kg person steps on a scale in an elevator, the scale reads a steady 410 N. Which of the following statements must be true? (There could be more than one correct choice.) a. The elevator is
accelerating upward at a constant rate. b. The elevator is accelerating downward at a constant rate. c. The elevator is moving upward at a constant rate. d. The elevator is moving downward at a
constant rate. e. From the given information, we cannot tell if the elevator is moving up or down.
b and e
The sum of two vectors of fixed magnitudes has the greatest magnitude when the angle between these two vectors is a. 60⁰ b. 0⁰ c. 180⁰ d. 270⁰ e. 90⁰
b. 0⁰
The horizontal and vertical components of the initial velocity of a football are 16 m/s and 20 m/s, respectively. If there is no air resistance, how long does it take the football to reach the top of
its trajectory. a. 5.0 s b. 2.0 s c. 3.0 s d. 4.0 s e. 1.0 s
b. 2.0 s
A ball is thrown straight up, reaches a max height, then falls to its initial height. Which of the following statements about the direction of the velocity and acceleration of the ball as it is going
up is correct? a. Both its velocity and its acceleration point upward. b. Its velocity points upward and its acceleration points downward. c. Its velocity points downward and its acceleration points
upward. d. Both its velocity and its acceleration point downward.
b. Its velocity points upward and its acceleration points downward.
For general projectile motion with no air resistance, the vertical component of a projectile's acceleration...
is always zero *remains a non-zero constant* continuously increases continuously decreases
Under what condition is average velocity equal to the average of the object's initial and final velocity? a. The acceleration must be constantly decreasing. b. The acceleration is constant. c. This
can only occur if there is no acceleration. d. This can occur only when the velocity is 0. e. The acceleration must be constantly increasing.
b. The acceleration is constant.
A ball is thrown horizontally from the top of a tower at the same instant that a stone is dropped vertically. Which object is traveling faster when it hits the level ground below if neither of them
experiences any air resistance? a. It is impossible to tell because we don't know their masses. b. The ball c. The stone d. Both are travelling at the same speed
b. The ball
Which of the following statements are true about an object in 2D projectile motion with no air resistance? a. The speed of the object is 0 at the highest point. b. The horizontal acceleration is
always zero and the vertical acceleration is always a non-zero constant downward. c. The speed of the object is constant but its velocity is not constant. d. The acceleration of the object is +g when
the object is rising and -g when it is falling. e. The acceleration of the object is 0 at its highest point.
b. The horizontal acceleration is always zero and the vertical acceleration is always a non-zero constant downward.
When a ball is thrown straight up with no air resistance, the acceleration at its highest point a. is upward b. is downward c. reverses from downward to upward d. is zero e. reverses from upward to
b. is downward
From the edge of a roof top you toss a green ball upwards with initial speed v0 and a blue ball downwards with the same initial speed. Air resistance is negligible. When they reach the ground below
a. the green ball will be moving faster b. the two balls will have the same speed c. the blue ball will be moving faster
b. the two balls will have the same speed
A pilot drops a package from a plane flying horizontally at a constant speed. *Neglecting air resistance*, when the package hits the ground, the horizontal location of the plane will...
be directly over the package
A 4.0 kg object is moving with speed 2.0 m/s. A 1.0 kg object is moving with speed 4.0 m/s. Both objects encounter the same constant braking force, and are brought to rest. Which object travels the
greater distance before stopping? a. the 4.0 kg object b. the 1.0 kg object c. Both objects travel the same distance. d. It cannot be determined from the information given.
A satellite is in orbit around the earth. Which one feels the greater force? a. the satellite because the earth is so much more massive b. the earth because the satellite has so little mass c. Earth
and the satellite feel exactly the same force. d. It depends on the distance of the satellite from Earth.
A small car and a large SUV are at a stoplight. The car has a mass equal to half that of the SUV, and the SUV can produce a maximum accelerating force equal to twice that of the car. When the light
turns green, both drivers push their accelerators to the floor at the same time. Which vehicle pulls ahead of the other vehicle after a few seconds? a. The car pulls ahead. b. The SUV pulls ahead. c.
It is a tie.
If the force on an object is in the negative direction, the work it does on the object must be a. negative. b. positive. c. The work could be either positive or negative, depending on the direction
the object moves.
Swimmers at a water park have a choice of two frictionless water slides, as shown in the figure. Although both slides drop over the same height h, slide 1 is straight while slide 2 is curved,
dropping quickly at first and then leveling out. How does the speed v1 of a swimmer reaching the bottom of slide 1 compare with v2, the speed of a swimmer reaching the end of slide 2? a. v1 > v2 b.
v1 < v2 c. v1 = v2 d. The heavier swimmer will have a greater speed than the lighter swimmer, no matter which slide he uses. e. No simple relationship exists between v1 and v2.
You throw a rock horizontally off a cliff with a speed of 20 m/s and no significant air resistance. After 2.0 seconds, the magnitude of the velocity of the rock is closest to a. 40 m/s b. 20 m/s c.
28 m/s d. 37 m/s
c. 28 m/s
An auto manufacturer advertises that their car can go "from zero to sixty in eight seconds." This is a description of what characteristic of the car's motion? a. average speed b. instantaneous speed
c. average acceleration d. instantaneous acceleration e. displacement
c. average acceleration
average velocity
change in velocity divided by change in time
A ball is thrown with an initial velocity of 20 m/s at an angle of 60⁰ above the horizontal. If we can neglect air resistance, what is the horizontal component of its instantaneous velocity at the
exact top of its trajectory.
d. 10 m/s
An object starts from rest and undergoes uniform acceleration. During the first second it travels 5.0 meters. How far does it travel during the third second? a. 45 meters b. 5.0 meters c. 15 meters
d. 25 meters
d. 25 meters
A ball is thrown downward from the top of a building with an initial speed of 25 m/s. It strikes the ground after 2.0 seconds. How high is the building, assuming negligible air resistance? a. 30 m b.
20 m c. 50 m d. 70 m
d. 70 m
Human reaction time is usually greater than 0.10 seconds. If your friend holds a ruler between your fingers and releases it without warning, how far can you expect the ruler to fall before you catch
it, assuming negligible air resistance? a. Less than 3.0 m b. At least 9.8 cm c. At least 7.8 cm d. At least 4.9 cm
d. At least 4.9 cm
Suppose that an object is moving with a constant velocity. Which statement concerning its acceleration must be correct? a. The acceleration is constantly increasing. b. The acceleration is constantly
decreasing. c. The acceleration is a constant non-zero value. d. The acceleration is equal to zero.
d. The acceleration is equal to zero.
Suppose that a car travelling to the west begins to slow down as it approaches a traffic light. Which of the following statements about its acceleration is correct? a. The acceleration is zero b.
Since the car is slowing down, its acceleration must be negative. c. The acceleration is toward the west. d. The acceleration is toward the east.
d. The acceleration is toward the east.
Two objects are dropped from a bridge, an interval of 1.0 seconds apart. Air resistance is negligible. During the time that both objects continue to fall, their separation a. decreases at first, but
then stays constant b. decreases c. stays constant d. increases e. increases at first, but then stays constant
d. increases
Two men, Joel and Jerry, push against a car that has stalled, trying unsuccessfully to get it moving. Jerry stops after 10 min, while Joel is able to push for 5.0 min longer. Compare the work they do
on the car. a. Joel does 75% more work than Jerry. b. Joel does 50% more work than Jerry. c. Jerry does 50% more work than Joel. d. Joel does 25% more work than Jerry. e. Neither of them does any
Which of the following statements are true about an object in two-dimensional projectile motion with no air resistance? (There could be more than one correct choice.) a. The speed of the object is
constant but its velocity is not constant. b. The acceleration of the object is +g when the object is rising and -g when it is falling. c. The acceleration of the object is zero at its highest point.
d. The speed of the object is zero at its highest point. e. The horizontal acceleration is always zero and the vertical acceleration is always a non-zero constant downward.
At player kicks a soccer ball in a high arc toward the opponent's goal. A the highest point in its trajectory a. both the velocity and the acceleration of the soccer ball are zero b. the ball's
acceleration is 0 but its velocity is not 0 c. the ball's velocity points downward d. the ball's acceleration points upward e. neither the ball's velocity nor its acceleration are 0
e. neither the ball's velocity nor its acceleration are 0
In order to lift a bucket of concrete, you must pull up harder on the bucket than the bucket pulls down on you
You pull on a crate with a rope. If the crate moves, the rope's pull on the crate must have been larger than the crate's pull on the rope, but if the crate does not move, both of these pulls must
have been equal. true or false
You are trying to cross a river that flows toward the south with a strong current. You start out in your motorboat on the east bank, desiring to reach the west bank directly west from your starting
point. You should head your motorboat...
in a general northwesterly direction
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▶ Converting Regular Numbers To Roman Numerals In Excel
Converting Regular Numbers To Roman Numerals In Excel
Mastering the Art of Converting Standard Numerals to Roman Format in Excel: A Comprehensive Guide
In Excel, the conversion of standard numerals to Roman format is achieved using the ROMAN function, a built-in tool that makes converting numbers into Roman numerals quite straightforward.
To employ this function, you would simply need to call it in a cell and supply the number you wish to convert as an argument. For instance, entering '=ROMAN(10)' into a cell will return 'X', the
Roman numeral for 10.
The ROMAN function in Excel actually offers four different formats you can use to represent Roman numerals. These differ based on how certain numeral values are represented and are selected by adding
an optional second argument to your function call.
For example, '=ROMAN(4,1)' will return 'IIII' instead of 'IV', while '=ROMAN(4,4)' returns 'IV'. Using this second argument, you're able to choose the type of Roman numeral formatting that best suits
your needs.
Moreover, if you have a column of numerals that you wish to convert into Roman format, you can do so quickly by using the 'Fill Down' feature. Simply select the cell with your ROMAN function, move
your cursor to the bottom right corner until it changes into a '+' icon, and double-click. This will apply the function to all cells in that column, converting each numeral into its Roman equivalent.
In sum, mastering the art of converting standard numerals to Roman format in Excel involves understanding how to use the ROMAN function, which includes knowing how to adjust the numeral formatting
with a second argument and how to apply this function to multiple cells at once with the 'Fill Down' feature.
What is the method for converting numbers into Roman numerals?
The method for converting numbers into Roman numerals in the context of technology involves a systematic approach of substitution. It's a common exercise in many beginning programming and algorithms
Here is a basic rundown of the process:
1. Understand Roman Numerals:
The key to this approach is understanding how Roman numerals work. In this system, you combine the letters I, V, X, L, C, D, and M to create numbers. Here are the basic equivalences:
- I = 1
- V = 5
- X = 10
- L = 50
- C = 100
- D = 500
- M = 1000
Roman numerals are written by combining letters and adding values. For example, "II" is two ones, i.e. 2, and "XII" is ten plus two ones, i.e. 12.
2. Create a Mapping:
In your function or algorithm, create a mapping of numbers to their corresponding Roman numerals. This could be an array, list, or dictionary data structure, depending on your programming language.
3. Implement the Conversion:
Next, implement the conversion process. Starting from the largest possible value, subtract that value from the number you're converting as many times as possible and append the corresponding Roman
numeral to your result each time. Repeat this process with the next largest value until you've converted the entire number.
For example, to convert the number 1987 to Roman numerals, we would start by subtracting 1000 (M) once (leaving 987), then subtract 900 (CM) once (leaving 87), then subtract 50 (L) once (leaving 37),
then subtract 10 (X) three times (leaving 7), then subtract 5 (V) once (leaving 2), and finally subtract 1 (I) twice.
This would result in the Roman numeral: MCMLXXXVII
Note: This is a basic implementation and may not handle certain edge cases, such as the number 4 (which is represented as IV, not IIII). A more complex algorithm would be needed to handle these
What is the method to change the number format in Excel?
Changing the number format in Excel can be done by following these steps:
1. Select the cells that you want to change the number format of.
2. Then look for the "Number" group on the "Home" tab in Excel's ribbon bar. Click on the "Number Format" drop-down box in this group.
3. Here, various options will be made available to you - General, Number, Currency, Accounting, Date, Time, Percentage, Fraction, and more.
4. Simply click on the desired number format option from this list. Excel will immediately apply this format to your selected cells.
Remember, if none of the default options suit your needs, you can also create a custom number format. To do this, scroll down to the bottom of the "Number Format" list and click on "More Number
Formats...". This will open a dialogue box where you can define a new format to your exact specifications.
How can you input the Roman numeral 3 in Excel?
To input the Roman numeral 3 in Excel, you would need to use the ROMAN function. The ROMAN function in Excel is used to convert a number into roman numerals.
Here are the steps to perform this:
1. Open a new Excel spreadsheet.
2. Select the cell where you want to input the Roman numeral 3.
3. Type =ROMAN(3) in the selected cell and press Enter.
4. The cell will display "III", which is the Roman numeral for 3.
Remember, the ROMAN function can handle numbers from 1 to 3999. For numbers outside this range, Excel will return an error.
How can you transform numbers saved as text into digits in Excel?
To transform numbers saved as text into digits in Excel, you can follow these steps:
1. Select the range of cells you'd like to convert. This could be a single cell, a column, a row or a group of cells.
2. Click on the Data tab from the Excel Ribbon.
3. In the Data Tools group, click on the Text to Columns option. This will open the 'Convert Text to Columns Wizard'.
4. In the first step of the wizard, select the Delimited option, and then click the Next button.
5. In the second step, uncheck all delimiter options (like Tab, Semicolon, Comma, Space), and then click the Next button.
6. In the third step, select the General column data format, and then click the Finish button.
After following these steps, the cells in the selected range that contained numbers stored as text should now contain those numbers stored as actual numbers.
- If a cell contains characters other than numbers, this method will retain them as is.
- The Text to Columns feature will override any existing data without giving a warning message, so be sure to save your work before starting the process, or only operate on copies of your data.
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Asymptotic behavior of vector recurrences with applications
1977 Articles
Asymptotic behavior of vector recurrences with applications
The behavior of the vector recurrence y_(n + 1) = My_n + w_(n + 1) is studied under very weak assumptions. Let λ(M) denote the spectral radius of M and let λ(M) ≥ 1. Then if the w_n are bounded in
norm and a certain subspace hypothesis holds, the root order of the y_n is shown to be λ(M). If one additional hypothesis on the dimension of the principal Jordan blocks of M holds, then the quotient
order of the y_n is also λ(M). The behavior of the homogeneous recurrence is studied for all values of λ(M). These results are applied to the analysis of (1) Nonlinear iteration with application to
iteration with memory and to parallel iteration algorithms. (2) Order and efficiency of composite iteration.
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10.2: Rotational Variables
Last updated
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• Describe the physical meaning of rotational variables as applied to fixed-axis rotation
• Explain how angular velocity is related to tangential speed
• Calculate the instantaneous angular velocity given the angular position function
• Find the angular velocity and angular acceleration in a rotating system
• Calculate the average angular acceleration when the angular velocity is changing
• Calculate the instantaneous angular acceleration given the angular velocity function
So far in this text, we have mainly studied translational motion, including the variables that describe it: displacement, velocity, and acceleration. Now we expand our description of motion to
rotation—specifically, rotational motion about a fixed axis. We will find that rotational motion is described by a set of related variables similar to those we used in translational motion.
Angular Velocity
Uniform circular motion (discussed previously in Motion in Two and Three Dimensions) is motion in a circle at constant speed. Although this is the simplest case of rotational motion, it is very
useful for many situations, and we use it here to introduce rotational variables.
In Figure \(\PageIndex{1}\), we show a particle moving in a circle. The coordinate system is fixed and serves as a frame of reference to define the particle’s position. Its position vector from the
origin of the circle to the particle sweeps out the angle \(\theta\), which increases in the counterclockwise direction as the particle moves along its circular path. The angle \(\theta\) is called
the angular position of the particle. As the particle moves in its circular path, it also traces an arc length s.
Figure \(\PageIndex{1}\): A particle follows a circular path. As it moves counterclockwise, it sweeps out a positive angle \(\theta\) with respect to the x-axis and traces out an arc length s.
The angle is related to the radius of the circle and the arc length by
\[\theta = \frac{s}{r} \ldotp \label{10.1}\]
The angle \(\theta\), the angular position of the particle along its path, has units of radians (rad). There are \(2\pi\) radians in 360°. Note that the radian measure is a ratio of length
measurements, and therefore is a dimensionless quantity. As the particle moves along its circular path, its angular position changes and it undergoes angular displacements \(\Delta \theta\).
We can assign vectors to the quantities in Equation \ref{10.1}. The angle \(\vec{\theta}\) is a vector out of the page in Figure \(\PageIndex{1}\). The angular position vector \(\vec{r}\) and the arc
length \(\vec{s}\) both lie in the plane of the page. These three vectors are related to each other by
\[\vec{s} = \vec{\theta} \times \vec{r} \ldotp \label{10.2}\]
That is, the arc length is the cross product of the angle vector and the position vector, as shown in Figure \(\PageIndex{2}\).
Figure \(\PageIndex{2}\): The angle vector points along the z-axis and the position vector and arc length vector both lie in the xy-plane. We see that \(\vec{s} = \vec{\theta} \times \vec{r}\). All
three vectors are perpendicular to each other.
The magnitude of the angular velocity, denoted by \(\omega\), is the time rate of change of the angle \(\theta\) as the particle moves in its circular path. The instantaneous angular velocity is
defined as the limit in which \(\Delta\)t → 0 in the average angular velocity \(\bar{\omega} = \frac{\Delta \theta}{\Delta t}\):
\[\omega = \lim_{\Delta t \rightarrow 0} \frac{\Delta \theta}{\Delta t} = \frac{d \theta}{dt}, \label{10.3}\]
where \(\theta\) is the angle of rotation (Figure \(\PageIndex{2}\)). The units of angular velocity are radians per second (rad/s). Angular velocity can also be referred to as the rotation rate in
radians per second. In many situations, we are given the rotation rate in revolutions/s or cycles/s. To find the angular velocity, we must multiply revolutions/s by 2\(\pi\), since there are 2\(\pi\)
radians in one complete revolution. Since the direction of a positive angle in a circle is counterclockwise, we take counterclockwise rotations as being positive and clockwise rotations as negative.
We can see how angular velocity is related to the tangential speed of the particle by differentiating Equation \ref{10.1} with respect to time. We rewrite Equation \ref{10.1} as
\[s = r \theta \ldotp\]
Taking the derivative with respect to time and noting that the radius r is a constant, we have
\[\frac{ds}{dt} = \frac{d}{dt} (r \theta) = \theta \frac{dr}{dt} + r \frac{d \theta}{dt} = r \frac{d \theta}{dt}\]
where \(\theta \frac{dr}{dt}\) = 0. Here, \(\frac{ds}{dt}\) is just the tangential speed v[t] of the particle in Figure \(\PageIndex{1}\). Thus, by using Equation \ref{10.3}, we arrive at
\[v_{t} = r \omega \ldotp \label{10.4}\]
That is, the tangential speed of the particle is its angular velocity times the radius of the circle. From Equation \ref{10.4}, we see that the tangential speed of the particle increases with its
distance from the axis of rotation for a constant angular velocity. This effect is shown in Figure \(\PageIndex{3}\). Two particles are placed at different radii on a rotating disk with a constant
angular velocity. As the disk rotates, the tangential speed increases linearly with the radius from the axis of rotation. In Figure \(\PageIndex{3}\), we see that v[1] = r[1]\(\omega_{1}\) and v[2] =
r[2]\(\omega_{2}\). But the disk has a constant angular velocity, so \(\omega_{1} = \omega_{2}\). This means \(\frac{v_{1}}{r_{1}} = \frac{v_{2}}{r_{2}}\) or v[2] = \(\left(\dfrac{r_{2}}{r_{1}}\
right)\)v[1]. Thus, since r[2] > r[1], v[2] > v[1].
Figure \(\PageIndex{3}\): Two particles on a rotating disk have different tangential speeds, depending on their distance to the axis of rotation.
Up until now, we have discussed the magnitude of the angular velocity \(\omega = \frac{d \theta}{dt}\), which is a scalar quantity—the change in angular position with respect to time. The vector \(\
vec{\omega}\) is the vector associated with the angular velocity and points along the axis of rotation. This is useful because when a rigid body is rotating, we want to know both the axis of rotation
and the direction that the body is rotating about the axis, clockwise or counterclockwise. The angular velocity \(\vec{\omega}\) gives us this information. The angular velocity \(\vec{\omega}\) has a
direction determined by what is called the right-hand rule. The right-hand rule is such that if the fingers of your right hand wrap counterclockwise from the x-axis (the direction in which \(\theta\)
increases) toward the y-axis, your thumb points in the direction of the positive z-axis (Figure \(\PageIndex{4}\)). An angular velocity \(\vec{\omega}\) that points along the positive z-axis
therefore corresponds to a counterclockwise rotation, whereas an angular velocity \(\vec{\omega}\) that points along the negative z-axis corresponds to a clockwise rotation.
Figure \(\PageIndex{4}\): For counterclockwise rotation in the coordinate system shown, the angular velocity points in the positive z-direction by the right-hand-rule.
We can verify the right-hand-rule using the vector expression for the arc length \(\vec{s} = \vec{\theta} \times \vec{r}\), Equation \ref{10.2}. If we differentiate this equation with respect to
time, we find
\[\frac{d \vec{s}}{dt} = \frac{d}{dt}(\vec{\theta} \times \vec{r}) = \left(\dfrac{d \theta}{dt} \times \vec{r}\right) + \left(\vec{\theta} \times \dfrac{d \vec{r}}{dt}\right) = \frac{d \theta}{dt} \
times \vec{r} \ldotp\]
Since \(\vec{r}\) is constant, the term \(\vec{\theta} \times \frac{d \vec{r}}{dt}\) = 0. Since \(\vec{v} = \frac{d \vec{s}}{dt}\) is the tangential velocity and \(\omega = \frac{d \vec{\theta}}{dt}
\) is the angular velocity, we have
\[\vec{v} = \vec{\omega} \times \vec{r} \ldotp \label{10.5}\]
That is, the tangential velocity is the cross product of the angular velocity and the position vector, as shown in Figure \(\PageIndex{5}\). From part (a) of this figure, we see that with the angular
velocity in the positive z-direction, the rotation in the xy-plane is counterclockwise. In part (b), the angular velocity is in the negative z-direction, giving a clockwise rotation in the xy-plane.
Figure \(\PageIndex{5}\): The vectors shown are the angular velocity, position, and tangential velocity. (a) The angular velocity points in the positive z-direction, giving a counterclockwise
rotation in the xy-plane. (b) The angular velocity points in the negative z-direction, giving a clockwise rotation.
A flywheel rotates such that it sweeps out an angle at the rate of \(\theta\) = \(\omega\)t = (45.0 rad/s)t radians. The wheel rotates counterclockwise when viewed in the plane of the page. (a) What
is the angular velocity of the flywheel? (b) What direction is the angular velocity? (c) How many radians does the flywheel rotate through in 30 s? (d) What is the tangential speed of a point on the
flywheel 10 cm from the axis of rotation?
The functional form of the angular position of the flywheel is given in the problem as \(\theta\)(t) = \(\omega\)t, so by taking the derivative with respect to time, we can find the angular velocity.
We use the right-hand rule to find the angular velocity. To find the angular displacement of the flywheel during 30 s, we seek the angular displacement \(\Delta \theta\), where the change in angular
position is between 0 and 30 s. To find the tangential speed of a point at a distance from the axis of rotation, we multiply its distance times the angular velocity of the flywheel.
1. \(\omega\) = \(\frac{d \theta}{dt}\) = 45 rad/s. We see that the angular velocity is a constant.
2. By the right-hand rule, we curl the fingers in the direction of rotation, which is counterclockwise in the plane of the page, and the thumb points in the direction of the angular velocity, which
is out of the page.
3. \(\Delta \theta\) = \(\theta\)(30 s) − \(\theta\)(0 s) = 45.0(30.0 s) − 45.0(0 s) = 1350.0 rad.
4. v[t] = r\(\omega\) = (0.1 m)(45.0 rad/s) = 4.5 m/s.
In 30 s, the flywheel has rotated through quite a number of revolutions, about 215 if we divide the angular displacement by 2\(\pi\). A massive flywheel can be used to store energy in this way, if
the losses due to friction are minimal. Recent research has considered superconducting bearings on which the flywheel rests, with zero energy loss due to friction.
Angular Acceleration
We have just discussed angular velocity for uniform circular motion, but not all motion is uniform. Envision an ice skater spinning with his arms outstretched—when he pulls his arms inward, his
angular velocity increases. Or think about a computer’s hard disk slowing to a halt as the angular velocity decreases. We will explore these situations later, but we can already see a need to define
an angular acceleration for describing situations where \(\omega\) changes. The faster the change in \(\omega\), the greater the angular acceleration. We define the instantaneous angular acceleration
\(\alpha\) as the derivative of angular velocity with respect to time:
\[\alpha = \lim_{\Delta t \rightarrow 0} \frac{\Delta \omega}{\Delta t} = \frac{d \omega}{dt} = \frac{d^{2} \theta}{dt^{2}}, \label{10.6}\]
where we have taken the limit of the average angular acceleration, \(\bar{\alpha} = \frac{\Delta \omega}{\Delta t}\) as \(\Delta t → 0\). The units of angular acceleration are (rad/s)/s, or rad/s^2.
In the same way as we defined the vector associated with angular velocity \(\vec{\omega}\), we can define \(\vec{\alpha}\), the vector associated with angular acceleration (Figure \(\PageIndex{6}\)).
If the angular velocity is along the positive z-axis, as in Figure \(\PageIndex{4}\), and \(\frac{d \omega}{dt}\) is positive, then the angular acceleration \(\vec{\alpha}\) is positive and points
along the +z- axis. Similarly, if the angular velocity \(\vec{\omega}\) is along the positive z-axis and \(\frac{d \omega}{dt}\) is negative, then the angular acceleration is negative and points
along the +z-axis.
Figure \(\PageIndex{6}\): The rotation is counterclockwise in both (a) and (b) with the angular velocity in the same direction. (a) The angular acceleration is in the same direction as the angular
velocity, which increases the rotation rate. (b) The angular acceleration is in the opposite direction to the angular velocity, which decreases the rotation rate.
We can express the tangential acceleration vector as a cross product of the angular acceleration and the position vector. This expression can be found by taking the time derivative of \(\vec{v} = \
vec{\omega} \times \vec{r}\) and is left as an exercise:
\[\vec{a} = \vec{\alpha} \times \vec{r} \ldotp \label{10.7}\]
The vector relationships for the angular acceleration and tangential acceleration are shown in Figure \(\PageIndex{7}\).
Figure \(\PageIndex{7}\): (a) The angular acceleration is the positive z-direction and produces a tangential acceleration in a counterclockwise sense. (b) The angular acceleration is in the negative
z-direction and produces a tangential acceleration in the clockwise sense.
We can relate the tangential acceleration of a point on a rotating body at a distance from the axis of rotation in the same way that we related the tangential speed to the angular velocity. If we
differentiate Equation \ref{10.4} with respect to time, noting that the radius r is constant, we obtain
\[a_{t} = r \alpha \ldotp \label{10.8}\]
Thus, the tangential acceleration a[t ]is the radius times the angular acceleration. Equations \ref{10.4} and \ref{10.8} are important for the discussion of rolling motion (see Angular Momentum).
Let’s apply these ideas to the analysis of a few simple fixed-axis rotation scenarios. Before doing so, we present a problem-solving strategy that can be applied to rotational kinematics: the
description of rotational motion.
1. Examine the situation to determine that rotational kinematics (rotational motion) is involved.
2. Identify exactly what needs to be determined in the problem (identify the unknowns). A sketch of the situation is useful.
3. Make a complete list of what is given or can be inferred from the problem as stated (identify the knowns).
4. Solve the appropriate equation or equations for the quantity to be determined (the unknown). It can be useful to think in terms of a translational analog, because by now you are familiar with the
equations of translational motion.
5. Substitute the known values along with their units into the appropriate equation and obtain numerical solutions complete with units. Be sure to use units of radians for angles.
6. Check your answer to see if it is reasonable: Does your answer make sense?
Now let’s apply this problem-solving strategy to a few specific examples.
A bicycle mechanic mounts a bicycle on the repair stand and starts the rear wheel spinning from rest to a final angular velocity of 250 rpm in 5.00 s. (a) Calculate the average angular acceleration
in rad/s^2. (b) If she now hits the brakes, causing an angular acceleration of −87.3 rad/s^2, how long does it take the wheel to stop?
The average angular acceleration can be found directly from its definition \(\bar{\alpha} = \frac{\Delta \omega}{\Delta t}\) because the final angular velocity and time are given. We see that \(\
Delta \omega\) = \(\omega_{final}\) − \(\omega_{initial}\) = 250 rev/min and \(\Delta\)t is 5.00 s. For part (b), we know the angular acceleration and the initial angular velocity. We can find the
stopping time by using the definition of average angular acceleration and solving for \(\Delta\)t, yielding
\[\Delta t = \frac{\Delta \omega}{\alpha} \ldotp\]
1. Entering known information into the definition of angular acceleration, we get $$\bar{\alpha} = \frac{\Delta \omega}{\Delta t} = \frac{250\; rpm}{5.00\; s} \ldotp$$Because \(\Delta \omega\) is in
revolutions per minute (rpm) and we want the standard units of rad/s^2 for angular acceleration, we need to convert from rpm to rad/s: $$\Delta \omega = 250 \frac{rev}{min}\; \cdotp \frac{2 \pi\;
rad}{rev}\; \cdotp \frac{1\; min}{60\; s} = 26.2\; rad/s \ldotp$$Entering this quantity into the expression for \(\alpha\), we get $$bar{\alpha} = \frac{\Delta \omega}{\Delta t} = \frac{26.2\;
rpm}{5.00\; s} = 5.24\; rad/s^{2} \ldotp$$
2. Here the angular velocity decreases from 26.2 rad/s (250 rpm) to zero, so that \(\Delta \omega\) is −26.2 rad/s, and \(\alpha\) is given to be –87.3 rad/s^2. Thus $$\Delta t = \frac{-26.2\; rad/
s}{-87.3\; rad/s^{2}} = 0.300\; s \ldotp$$
Note that the angular acceleration as the mechanic spins the wheel is small and positive; it takes 5 s to produce an appreciable angular velocity. When she hits the brake, the angular acceleration is
large and negative. The angular velocity quickly goes to zero.
The fan blades on a turbofan jet engine (shown below) accelerate from rest up to a rotation rate of 40.0 rev/s in 20 s. The increase in angular velocity of the fan is constant in time. (The
GE90-110B1 turbofan engine mounted on a Boeing 777, as shown, is currently the largest turbofan engine in the world, capable of thrusts of 330–510 kN.) (a) What is the average angular acceleration?
(b) What is the instantaneous angular acceleration at any time during the first 20 s?
A wind turbine (Figure \(\PageIndex{9}\)) in a wind farm is being shut down for maintenance. It takes 30 s for the turbine to go from its operating angular velocity to a complete stop in which the
angular velocity function is \(\omega\)(t) = \(\Big[\frac{(ts^{−1} −30.0)^{2}}{100.0} \Big]\)rad/s. If the turbine is rotating counterclockwise looking into the page, (a) what are the directions of
the angular velocity and acceleration vectors? (b) What is the average angular acceleration? (c) What is the instantaneous angular acceleration at t = 0.0, 15.0, 30.0 s?
Figure \(\PageIndex{9}\): A wind turbine that is rotating counterclockwise, as seen head on.
1. We are given the rotational sense of the turbine, which is counterclockwise in the plane of the page. Using the right hand rule (Figure 10.5), we can establish the directions of the angular
velocity and acceleration vectors.
2. We calculate the initial and final angular velocities to get the average angular acceleration. We establish the sign of the angular acceleration from the results in (a).
3. We are given the functional form of the angular velocity, so we can find the functional form of the angular acceleration function by taking its derivative with respect to time.
1. Since the turbine is rotating counterclockwise, angular velocity \(\vec{\omega}\) points out of the page. But since the angular velocity is decreasing, the angular acceleration \(\vec{\alpha}\)
points into the page, in the opposite sense to the angular velocity.
2. The initial angular velocity of the turbine, setting t = 0, is \(\omega\) = 9.0 rad/s. The final angular velocity is zero, so the average angular acceleration is $$\bar{\alpha} \frac{\Delta \
omega}{\Delta t} = \frac{\omega - \omega_{0}}{t - t_{0}} = \frac{0 - 9.0\; rad/s}{30.0 - 0\; s} = -0.3\; rad/s^{2} \ldotp$$
3. Taking the derivative of the angular velocity with respect to time gives \(\alpha = \frac{d \omega}{dt} = \frac{(t − 30.0)}{50.0}\) rad/s^2 $$\alpha (0.0; s) = -0.6\; rad/s^{2}, \alpha (15.0\; s)
= -0.3\; rad/s^{2}, and\; \alpha (30.0\; s) = 0\; rad/s \ldotp$$
We found from the calculations in (a) and (b) that the angular acceleration α and the average angular acceleration \(\bar{\alpha}\) are negative. The turbine has an angular acceleration in the
opposite sense to its angular velocity.
We now have a basic vocabulary for discussing fixed-axis rotational kinematics and relationships between rotational variables. We discuss more definitions and connections in the next section. | {"url":"https://phys.libretexts.org/Bookshelves/University_Physics/University_Physics_(OpenStax)/Book%3A_University_Physics_I_-_Mechanics_Sound_Oscillations_and_Waves_(OpenStax)/10%3A_Fixed-Axis_Rotation__Introduction/10.02%3A_Rotational_Variables","timestamp":"2024-11-11T13:23:44Z","content_type":"text/html","content_length":"164710","record_id":"<urn:uuid:85117234-fc2f-4ed6-bdbf-30259656ba35>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00814.warc.gz"} |
Z Calculus: Computational Rosetta Stone - Welcome to Quantum Guru
Quantum computing promises to surpass the limitations of classical computing, solving problems deemed intractable for today’s supercomputers. At the heart of this quantum leap is an intricate
mathematical framework, and one of its unsung heroes is Z calculus. This article explores the pivotal role of Z calculus in quantum computing, unraveling its potential to unlock new computational
Z calculus, often overshadowed by its more famous counterparts like lambda calculus, is a branch of mathematical logic used for abstracting and analyzing computation. It operates on the principle of
equational reasoning, allowing for the manipulation of mathematical expressions in a form that is both expressive and conducive to automation. This quality makes Z calculus particularly appealing in
the context of quantum computing, where the complexity of operations often requires highly abstracted forms of reasoning.
Quantum computing utilizes the principles of quantum mechanics, such as superposition and entanglement, to process information in ways fundamentally different from classical computing. Qubits, the
basic units of quantum information, can exist in multiple states simultaneously, offering exponential growth in computational power. However, harnessing this power necessitates a deep understanding
of complex mathematical operations, which is where Z calculus enters the picture.
The idiosyncrasies of quantum algorithms, with their intricate operations on qubits, demand a mathematical language that can encapsulate and manipulate high-level concepts with precision. Z calculus
serves as this computational Rosetta Stone, translating the abstract notions of quantum mechanics into a structured form that can be reasoned about and optimized.
Developing quantum algorithms is a task fraught with challenges, requiring not only quantum intuition but also a robust mathematical foundation. Z calculus aids in formalizing quantum algorithms,
making it possible to abstract away from the low-level quantum circuit model. This high-level abstraction is crucial for creating more efficient algorithms, which are the engines of quantum
One of the most significant hurdles in quantum computing is error correction. Quantum information is delicate, and errors can arise easily, making computations unreliable. Z calculus contributes to
the development of quantum error correction codes, offering a framework for modelling and understanding errors within quantum systems, and thus, paving the way for more reliable quantum computers.
As quantum computing moves from theory to practice, the role of mathematical tools like Z calculus becomes increasingly important. By providing a foundation for the analysis and optimization of
quantum algorithms, Z calculus is instrumental in the transition towards a quantum computing future.
The potential of Z calculus in quantum computing cannot be overstated. As we stand on the cusp of a new computational era, the mathematical rigor and abstraction provided by Z calculus will be
paramount in harnessing the full power of quantum computing. For researchers, engineers, and enthusiasts alike, a strong grasp of Z calculus could well be the key to unlocking the myriad mysteries
and opportunities presented by quantum computing.
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Is part of the Bibliography
The simultaneous recording of the activity of many neurons poses challenges for multivariate data analysis. Here, we propose a general scheme of reconstruction of the functional network from spike
train recordings. Effective, causal interactions are estimated by fitting generalized linear models on the neural responses, incorporating effects of the neurons’ self-history, of input from other
neurons in the recorded network and of modulation by an external stimulus. The coupling terms arising from synaptic input can be transformed by thresholding into a binary connectivity matrix which is
directed. Each link between two neurons represents a causal influence from one neuron to the other, given the observation of all other neurons from the population. The resulting graph is analyzed
with respect to small-world and scale-free properties using quantitative measures for directed networks. Such graph-theoretic analyses have been performed on many complex dynamic networks, including
the connectivity structure between different brain areas. Only few studies have attempted to look at the structure of cortical neural networks on the level of individual neurons. Here, using
multi-electrode recordings from the visual system of the awake monkey, we find that cortical networks lack scale-free behavior, but show a small, but significant small-world structure. Assuming a
simple distance-dependent probabilistic wiring between neurons, we find that this connectivity structure can account for all of the networks’ observed small-world-ness. Moreover, for multi-electrode
recordings the sampling of neurons is not uniform across the population. We show that the small-world-ness obtained by such a localized sub-sampling overestimates the strength of the true small-world
structure of the network. This bias is likely to be present in all previous experiments based on multi-electrode recordings.
Poster presentation: Characterizing neuronal encoding is essential for understanding information processing in the brain. Three methods are commonly used to characterize the relationship between
neural spiking activity and the features of putative stimuli. These methods include: Wiener-Volterra kernel methods (WVK), the spike-triggered average (STA), and more recently, the point process
generalized linear model (GLM). We compared the performance of these three approaches in estimating receptive field properties and orientation tuning of 251 V1 neurons recorded from 2 monkeys during
a fixation period in response to a moving bar. The GLM consisted of two formulations of the conditional intensity function for a point process characterization of the spiking activity: one with a
stimulus only component and one with the stimulus and spike history. We fit the GLMs by maximum likelihood using GLMfit in Matlab. Goodness-of-fit was assessed using cross-validation with
Kolmogorov-Smirnov (KS) tests based on the time-rescaling theorem to evaluate the accuracy with which each model predicts the spiking activity of individual neurons and for each movement direction
(4016 models in total, for 251 neurons and 16 different directions). The GLMs that considered spike history of up to 35 ms, accurately predicted neuronal spiking activity (95% confidence intervals
for KS test) with a performance of 97.0% (3895/4016) for the training data, and 96.5% (3876/4016) for the test data. If spike history was not considered, performance dropped to 73,1% in the training
and 71.3% in the testing data. In contrast, the WVF and the STA predicted spiking accurately for 24.2% and 44.5% of the test data examples respectively. The receptive field size estimates obtained
from the GLM (with and without history), WVF and STA were comparable. Relative to the GLM orientation tuning was underestimated on average by a factor of 0.45 by the WVF and the STA. The main reason
for using the STA and WVF approaches is their apparent simplicity. However, our analyses suggest that more accurate spike prediction as well as more credible estimates of receptive field size and
orientation tuning can be computed easily using GLMs implemented in Matlab with standard functions such as GLMfit.
Following the discovery of context-dependent synchronization of oscillatory neuronal responses in the visual system, the role of neural synchrony in cortical networks has been expanded to provide a
general mechanism for the coordination of distributed neural activity patterns. In the current paper, we present an update of the status of this hypothesis through summarizing recent results from our
laboratory that suggest important new insights regarding the mechanisms, function and relevance of this phenomenon. In the first part, we present recent results derived from animal experiments and
mathematical simulations that provide novel explanations and mechanisms for zero and nero-zero phase lag synchronization. In the second part, we shall discuss the role of neural synchrony for
expectancy during perceptual organization and its role in conscious experience. This will be followed by evidence that indicates that in addition to supporting conscious cognition, neural synchrony
is abnormal in major brain disorders, such as schizophrenia and autism spectrum disorders. We conclude this paper with suggestions for further research as well as with critical issues that need to be
addressed in future studies.
Even in V1, where neurons have well characterized classical receptive fields (CRFs), it has been difficult to deduce which features of natural scenes stimuli they actually respond to. Forward models
based upon CRF stimuli have had limited success in predicting the response of V1 neurons to natural scenes. As natural scenes exhibit complex spatial and temporal correlations, this could be due to
surround effects that modulate the sensitivity of the CRF. Here, instead of attempting a forward model, we quantify the importance of the natural scenes surround for awake macaque monkeys by modeling
it non-parametrically. We also quantify the influence of two forms of trial to trial variability. The first is related to the neuron’s own spike history. The second is related to ongoing mean field
population activity reflected by the local field potential (LFP). We find that the surround produces strong temporal modulations in the firing rate that can be both suppressive and facilitative.
Further, the LFP is found to induce a precise timing in spikes, which tend to be temporally localized on sharp LFP transients in the gamma frequency range. Using the pseudo R2 as a measure of model
fit, we find that during natural scene viewing the CRF dominates, accounting for 60% of the fit, but that taken collectively the surround, spike history and LFP are almost as important, accounting
for 40%. However, overall only a small proportion of V1 spiking statistics could be explained (R2~5%), even when the full stimulus, spike history and LFP were taken into account. This suggests that
under natural scene conditions, the dominant influence on V1 neurons is not the stimulus, nor the mean field dynamics of the LFP, but the complex, incoherent dynamics of the network in which neurons
are embedded.
As important as the intrinsic properties of an individual nervous cell stands the network of neurons in which it is embedded and by virtue of which it acquires great part of its responsiveness and
functionality. In this study we have explored how the topological properties and conduction delays of several classes of neural networks affect the capacity of their constituent cells to establish
well-defined temporal relations among firing of their action potentials. This ability of a population of neurons to produce and maintain a millisecond-precise coordinated firing (either evoked by
external stimuli or internally generated) is central to neural codes exploiting precise spike timing for the representation and communication of information. Our results, based on extensive
simulations of conductance-based type of neurons in an oscillatory regime, indicate that only certain topologies of networks allow for a coordinated firing at a local and long-range scale
simultaneously. Besides network architecture, axonal conduction delays are also observed to be another important factor in the generation of coherent spiking. We report that such communication
latencies not only set the phase difference between the oscillatory activity of remote neural populations but determine whether the interconnected cells can set in any coherent firing at all. In this
context, we have also investigated how the balance between the network synchronizing effects and the dispersive drift caused by inhomogeneities in natural firing frequencies across neurons is
resolved. Finally, we show that the observed roles of conduction delays and frequency dispersion are not particular to canonical networks but experimentally measured anatomical networks such as the
macaque cortical network can display the same type of behavior.
Poster presentation: Coordinated neuronal activity across many neurons, i.e. synchronous or spatiotemporal pattern, had been believed to be a major component of neuronal activity. However, the
discussion if coordinated activity really exists remained heated and controversial. A major uncertainty was that many analysis approaches either ignored the auto-structure of the spiking activity,
assumed a very simplified model (poissonian firing), or changed the auto-structure by spike jittering. We studied whether a statistical inference that tests whether coordinated activity is occurring
beyond chance can be made false if one ignores or changes the real auto-structure of recorded data. To this end, we investigated the distribution of coincident spikes in mutually independent
spike-trains modeled as renewal processes. We considered Gamma processes with different shape parameters as well as renewal processes in which the ISI distribution is log-normal. For Gamma processes
of integer order, we calculated the mean number of coincident spikes, as well as the Fano factor of the coincidences, analytically. We determined how these measures depend on the bin width and also
investigated how they depend on the firing rate, and on rate difference between the neurons. We used Monte-Carlo simulations to estimate the whole distribution for these parameters and also for other
values of gamma. Moreover, we considered the effect of dithering for both of these processes and saw that while dithering does not change the average number of coincidences, it does change the shape
of the coincidence distribution. Our major findings are: 1) the width of the coincidence count distribution depends very critically and in a non-trivial way on the detailed properties of the
inter-spike interval distribution, 2) the dependencies of the Fano factor on the coefficient of variation of the ISI distribution are complex and mostly non-monotonic. Moreover, the Fano factor
depends on the very detailed properties of the individual point processes, and cannot be predicted by the CV alone. Hence, given a recorded data set, the estimated value of CV of the ISI distribution
is not sufficient to predict the Fano factor of the coincidence count distribution, and 3) spike jittering, even if it is as small as a fraction of the expected ISI, can falsify the inference on
coordinated firing. In most of the tested cases and especially for complex synchronous and spatiotemporal pattern across many neurons, spike jittering increased the likelihood of false positive
finding very strongly. Last, we discuss a procedure [1] that considers the complete auto-structure of each individual spike-train for testing whether synchrony firing occurs at chance and therefore
overcomes the danger of an increased level of false positives.
Poster Presentation from Nineteenth Annual Computational Neuroscience Meeting: CNS*2010 San Antonio, TX, USA. 24-30 July 2010 Statistical models of neural activity are at the core of the field of
modern computational neuroscience. The activity of single neurons has been modeled to successfully explain dependencies of neural dynamics to its own spiking history, to external stimuli or other
covariates [1]. Recently, there has been a growing interest in modeling spiking activity of a population of simultaneously recorded neurons to study the effects of correlations and functional
connectivity on neural information processing (existing models include generalized linear models [2,3] or maximum-entropy approaches [4]). For point-process-based models of single neurons, the
time-rescaling theorem has proven to be a useful toolbox to assess goodness-of-fit. In its univariate form, the time-rescaling theorem states that if the conditional intensity function of a point
process is known, then its inter-spike intervals can be transformed or “rescaled” so that they are independent and exponentially distributed [5]. However, the theorem in its original form lacks
sensitivity to detect even strong dependencies between neurons. Here, we present how the theorem can be extended to be applied to neural population models and we provide a step-by-step procedure to
perform the statistical tests. We then apply both the univariate and multivariate tests to simplified toy models, but also to more complicated many-neuron models and to neuronal populations recorded
in V1 of awake monkey during natural scenes stimulation. We demonstrate that important features of the population activity can only be detected using the multivariate extension of the test. ...
Poster presentation: How can two distant neural assemblies synchronize their firings at zero-lag even in the presence of non-negligible delays in the transfer of information between them? Neural
synchronization stands today as one of the most promising mechanisms to counterbalance the huge anatomical and functional specialization of the different brain areas. However, and albeit more
evidence is being accumulated in favor of its functional role as a binding mechanism of distributed neural responses, the physical and anatomical substrate for such a dynamic and precise synchrony,
especially zero-lag even in the presence of non-negligible delays, remains unclear. Here we propose a simple network motif that naturally accounts for zero-lag synchronization of spiking assemblies
of neurons for a wide range of temporal delays. We demonstrate that when two distant neural assemblies do not interact directly but relaying their dynamics via a third mediating single neuron or
population and eventually achieve zero-lag coherent firing. Extensive numerical simulations of populations of Hodgkin-Huxley neurons interacting in such a network are analyzed. The results show that
even with axonal delays as large as 15 ms the distant neural populations can synchronize their firings at zero-lag in a millisecond precision after the exchange of a few spikes. The role of noise and
a distribution of axonal delays in the synchronized dynamics of the neural populations are also studied confirming the robustness of this sync mechanism. The proposed network module is densely
embedded within the complex functional architecture of the brain and especially within the reciprocal thalamocortical interactions where the role of indirect pathways mimicking direct
cortico-cortical fibers has been already suggested to facilitate trans-areal cortical communication. In summary the robust neural synchronization mechanism presented here arises as a consequence of
the relay and redistribution of the dynamics performed by a mediating neuronal population. In opposition to previous works, neither inhibitory, gap junctions, nor complex networks need to be invoked
to provide a stable mechanism of zero-phase correlated activity of neural populations in the presence of large conduction delays.
Poster presentation: Background To test the importance of synchronous neuronal firing for information processing in the brain, one has to investigate if synchronous firing strength is correlated to
the experimental subjects. This requires a tool that can compare the strength of the synchronous firing across different conditions, while at the same time it should correct for other features of
neuronal firing such as spike rate modulation or the auto-structure of the spike trains that might co-occur with synchronous firing. Here we present the bi- and multivariate extension of previously
developed method NeuroXidence [1,2], which allows for comparing the amount of synchronous firing between different conditions. ...
Poster presentation: Introduction Adequate anesthesia is crucial to the success of surgical interventions and subsequent recovery. Neuroscientists, surgeons, and engineers have sought to understand
the impact of anesthetics on the information processing in the brain and to properly assess the level of anesthesia in an non-invasive manner. Studies have indicated a more reliable depth of
anesthesia (DOA) detection if multiple parameters are employed. Indeed, commercial DOA monitors (BIS, Narcotrend, M-Entropy and A-line ARX) use more than one feature extraction method. Here, we
propose TESPAR (Time Encoded Signal Processing And Recognition) a time domain signal processing technique novel to EEG DOA assessment that could enhance existing monitoring devices. ... | {"url":"https://publikationen.ub.uni-frankfurt.de/solrsearch/index/search/searchtype/authorsearch/author/Gordon+Pipa","timestamp":"2024-11-14T17:53:12Z","content_type":"application/xhtml+xml","content_length":"59518","record_id":"<urn:uuid:b944d2c5-337f-483a-89fe-2195c341443d>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00326.warc.gz"} |
An Etymological Dictionary of Astronomy and Astrophysics
harmonic mean
میانگین ِهماهنگ
miyângin-e hamâhang
Fr.: moyenne harmonique
A number whose reciprocal is the → arithmetic mean of the reciprocals of a set of numbers. Denoted by H, it may be written in the discrete case for n quantities x[1], ..., x[n], as: 1/H = (1/n) Σ(1/x
[i]), summing from i = 1 to n. For example, the harmonic mean between 3 and 4 is 24/7 (reciprocal of 3: 1/3, reciprocal of 4: 1/4, arithmetic mean between them 7/24). The harmonic mean applies more
accurately to certain situations involving rates. For example, if a car travels a certain distance at a speed speed 60 km/h and then the same distance again at a speed 40 km/h, then its average speed
is the harmonic mean of 48 km/h, and its total travel time is the same as if it had traveled the whole distance at that average speed. However, if the car travels for a certain amount of time at a
speed v and then the same amount of time at a speed u, then its average speed is the arithmetic mean of v and u, which in the above example is 50 km/h.
→ harmonic; → mean. | {"url":"https://dictionary.obspm.fr/?showAll=1&formSearchTextfield=harmonic+mean","timestamp":"2024-11-14T14:02:20Z","content_type":"text/html","content_length":"11123","record_id":"<urn:uuid:7bd4c3da-da65-4f09-8583-0e9b8d6e4639>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00112.warc.gz"} |
3 Rd Grade Multiplication Worksheets
Mathematics, particularly multiplication, creates the cornerstone of various academic techniques and real-world applications. Yet, for several learners, mastering multiplication can position an
obstacle. To address this obstacle, instructors and moms and dads have embraced a powerful device: 3 Rd Grade Multiplication Worksheets.
Introduction to 3 Rd Grade Multiplication Worksheets
3 Rd Grade Multiplication Worksheets
3 Rd Grade Multiplication Worksheets -
Grade 3 math worksheets on the multiplication tables of 2 3 Practice until instant recall is developed Worksheet 1 Worksheet 2 Worksheet 3 Worksheet 4 Worksheet 5 Worksheet 6 5 More Similar
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Advancement of 3 Rd Grade Multiplication Worksheets
3rd Grade Math multiplication Times Tables 1 s Printable Times Table Worksheet Computer
3rd Grade Math multiplication Times Tables 1 s Printable Times Table Worksheet Computer
A self teaching worktext for 3rd grade that covers multiplication concept from various angles word problems a guide for structural drilling and a complete study of all 12 multiplication tables
Download 5 20 Also available as a printed copy Learn more and see the free samples See more topical Math Mammoth books
Browse Printable 3rd Grade Multiplication Fact Worksheets Award winning educational materials designed to help kids succeed Kids completing this third grade math worksheet multiply by 3 to solve each
equation and also fill in a multiplication chart for the number 3 3rd grade Math Interactive Worksheet Multiplication With Mr Snowman
From typical pen-and-paper workouts to digitized interactive formats, 3 Rd Grade Multiplication Worksheets have actually progressed, satisfying varied discovering styles and choices.
Sorts Of 3 Rd Grade Multiplication Worksheets
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Word Trouble Worksheets
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Timed Multiplication Drills Examinations created to boost speed and accuracy, assisting in fast mental math.
Advantages of Using 3 Rd Grade Multiplication Worksheets
3rd Grade Math Worksheets Pdf EduMonitor
3rd Grade Math Worksheets Pdf EduMonitor
Multiplication facts are the basic facts of multiplying two single digit numbers such as 2 x 3 6 or 9 x 9 81 These facts are also called the times tables because they show how many times a number is
added to itself For example 4 x 5 20 means that 4 is added to itself 5 times 4 4 4 4 4 20 Multiplication facts are essential
3rd Grade Multiplication Worksheets 171 results found Sort by Most Popular x Multiplication x 3rd Grade x Worksheets WORKSHEETS Division as Sharing A Comprehensive Lesson Plan This lesson contains
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Regular method develops multiplication proficiency, improving overall mathematics capacities.
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Exactly How to Develop Engaging 3 Rd Grade Multiplication Worksheets
Integrating Visuals and Colors Vivid visuals and shades catch interest, making worksheets aesthetically appealing and engaging.
Including Real-Life Circumstances
Relating multiplication to day-to-day circumstances includes significance and practicality to exercises.
Customizing Worksheets to Various Ability Degrees Personalizing worksheets based on differing proficiency degrees guarantees comprehensive discovering. Interactive and Online Multiplication Resources
Digital Multiplication Devices and Gamings Technology-based sources supply interactive understanding experiences, making multiplication appealing and delightful. Interactive Internet Sites and
Applications On the internet platforms give varied and available multiplication technique, supplementing typical worksheets. Customizing Worksheets for Numerous Knowing Styles Aesthetic Students
Aesthetic aids and diagrams help comprehension for students inclined toward aesthetic discovering. Auditory Learners Verbal multiplication issues or mnemonics cater to learners that understand
concepts with auditory ways. Kinesthetic Learners Hands-on tasks and manipulatives support kinesthetic learners in comprehending multiplication. Tips for Effective Implementation in Learning
Uniformity in Practice Normal technique enhances multiplication abilities, promoting retention and fluency. Balancing Repeating and Range A mix of repeated workouts and diverse trouble formats keeps
interest and understanding. Giving Constructive Comments Responses aids in determining areas of improvement, urging ongoing development. Challenges in Multiplication Practice and Solutions
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creating a positive discovering environment is important. Effect of 3 Rd Grade Multiplication Worksheets on Academic Performance Research Studies and Research Findings Research suggests a favorable
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Arrays Worksheets Multiplication Arrays Worksheets Classroom On Best Worksheets Collection 1241
Multiplication Practice Worksheets Grade 3
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Double Digit Multiplication With Regrouping Two Digit Multiplication Math Practice worksheets
Assisting Third Graders To Build Solid Multiplication Understanding Worksheets 99Worksheets
Multiplication Practice Worksheets Grade 3
3rd Grade Math Worksheets Best Coloring Pages For Kids
Third Grade Multiplication Worksheet
Multiplication Spin And Multiply Such A Fun multiplication Math Game Found In The November NO
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word problems No login required
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Worksheet 2 Digit Multiplication Interactive Worksheet Christmas Multiplication 2 Worksheet Multiplication Division Word Problems Practice
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3rd Grade Math Worksheets Best Coloring Pages For Kids
Assisting Third Graders To Build Solid Multiplication Understanding Worksheets 99Worksheets
Third Grade Multiplication Worksheet
Multiplication Spin And Multiply Such A Fun multiplication Math Game Found In The November NO
Multiplication Worksheets 9Th Grade PrintableMultiplication
2 Digit multiplication With Regrouping Printables With A Halloween Theme Multiplication Two
2 Digit multiplication With Regrouping Printables With A Halloween Theme Multiplication Two
Multiplication Sheet 4th Grade
Frequently Asked Questions (Frequently Asked Questions).
Are 3 Rd Grade Multiplication Worksheets appropriate for every age teams?
Yes, worksheets can be customized to various age and ability levels, making them adaptable for different students.
Exactly how commonly should students exercise making use of 3 Rd Grade Multiplication Worksheets?
Regular technique is key. Routine sessions, ideally a couple of times a week, can produce significant improvement.
Can worksheets alone improve mathematics abilities?
Worksheets are a valuable device but must be supplemented with varied learning techniques for thorough ability advancement.
Are there on-line platforms supplying complimentary 3 Rd Grade Multiplication Worksheets?
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Exactly how can parents sustain their youngsters's multiplication technique at home?
Encouraging consistent technique, supplying assistance, and developing a positive knowing environment are valuable actions. | {"url":"https://crown-darts.com/en/3-rd-grade-multiplication-worksheets.html","timestamp":"2024-11-13T21:53:30Z","content_type":"text/html","content_length":"29264","record_id":"<urn:uuid:6dc8aca0-98b4-48a8-a432-03873c242363>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00272.warc.gz"} |
Binomial theorem Archives - The Nature of Mathematics - 13th Edition
Here is a statement of the binomial theorem (with links):
This link gives some examples of the binomial theorem:
This site provides a summary of the binomial distribution.
This site calculates the binomial probabilities.
http://stattrek.com/online-calculator/binomial.aspx… See the whole entry
Section 12.2: Combinations
12.2 Outline
A. Committee problem
1. definition
2. combination formula
3. deck of cards
B. Pascal’s triangle
1. n choose r
2. table entries
C. Counting with the binomial theorem
1. binomial theorem
2. number of subsets
12.2 Essential Ideas
A combination of r elements selected from a … See the whole entry
Section 6.1: Polynomials
6.1 Outline
A. Terminology
1. term
2. polynomial
1. monomial
2. binomial
3. trinomial
3. degree
1. of a term
2. of a polynomial
3. linear
4. quadratic
4. numerical coefficient
B. Simplification
1. like terms/similar terms
2. simplify – this is the first of four main processes in algebra
C. Shortcuts with products
1. FOIL (see
… See the whole entry | {"url":"https://mathnature.com/tag/binomial-theorem/","timestamp":"2024-11-13T16:12:11Z","content_type":"text/html","content_length":"113898","record_id":"<urn:uuid:f656bf7b-4dc9-41fe-91a7-8fa7e5bf928b>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00895.warc.gz"} |
Math Colloquia - Analysis and computations of stochastic optimal control problems for stochastic PDEs
Many mathematical and computational analyses have been performed for deterministic partial differential equations (PDEs) that have perfectly known input data. However, in reality, many physical and
engineering problems involve some level of uncertainty in their input, e.g., unknown properties of the material, the lack of information on boundary data, etc. One effective and realistic means for
modeling such uncertainty is through stochastic partial differential equations (SPDEs) using randomness for uncertainty. In fact, SPDEs are known to be effective tools for modeling complex physical
and engineering phenomena. In this talk, we propose and analyze some optimal control problems for partial differential equations with random coefficients and forcing terms.
These input data are assumed to be dependent on a finite number of random variables. We set up three different kind of problems and prove existence of optimal solution and derive an optimality
system. In the method, we use a Galerkin approximation in physical space and a sparse grid collocation in the probability space. We provide a comparison of these three cases for fully discrete
solution using an appropriate norm and analyze the computational efficiency. | {"url":"http://my.math.snu.ac.kr/board/index.php?mid=colloquia&l=en&page=8&document_srl=766283&sort_index=date&order_type=asc","timestamp":"2024-11-09T12:33:43Z","content_type":"text/html","content_length":"44836","record_id":"<urn:uuid:925a6414-a72e-42ef-b174-552a15255c4a>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00880.warc.gz"} |
R. Carminati, J. J. Greffet, C. Henkel, and J. M. Vigoureux
Radiative and non-radiative decay of a single molecule close to a metallic nanoparticle
Opt. Commun. 261 (2006) 368
We study the spontaneous emission of a single emitter close to a metallic nanoparticle, with the aim to clarify the distance dependence of the radiative and nonradiative decay rates. We derive
analytical formulas based on a dipole-dipole model, and show that the non-radiative decay rate follows a $R^{-6}$ dependence at short distance, where $R$ is the distance between the emitter and
the center of the nanoparticle, as in Försters energy transfer. The distance dependence of the radiative decay rate is more subtle. It is chiefly dominated by a $R^{-3}$ dependence, a $R^{-6}$
dependence being visible at plasmon resonance. The latter is a consequence of radiative damping in the effective dipole polarisability of the nanoparticle. The different distance behavior of the
radiative and non-radiative decay rates implies that the apparent quantum yield always vanishes at short distance. Moreover, non-radiative decay is strongly enhanced when the emitter radiates at
the plasmon-resonance frequency of the nanoparticle.
[ reprint order ] [ DOI ]
file generated: 18 Apr 2007 | {"url":"http://www.quantum.physik.uni-potsdam.de/research/archive/papers/2006/carminati06a.html","timestamp":"2024-11-03T10:46:39Z","content_type":"text/html","content_length":"12267","record_id":"<urn:uuid:00881ddb-ebb0-4686-91ae-11a00359fdea>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00148.warc.gz"} |
What is the cost of bitcoin mining, excluding electricity?
All else being equal, as new generations of miners are introduced, the cost of mining bitcoin will continue to fall due to the improved efficiency ratio of the ...
Mar 13,2023 | SARAH
All else being equal, as new generations of miners are introduced, the cost of mining bitcoin will continue to fall due to the improved efficiency ratio of the new miners.
Bitcoin's price is made up of an inherent (intrinsic) value and an external value. The energy1080 ti mining consumption cost of mining out each bitcoin is defined here as its minimum intrinsic value
(the lower bound of the intrinsic value), while the external value is the value generated by speculation.
Bitcoin Mining Fundamentals
Let's start with a basic overview of current bitcoin mining. The Bitcoin network-wideantminer s19 watts arithmetic power is currently 40.419 EH/s, the block reward is 12.5 BTC, and 6 blocks are
generated every hour, according to trinsicoin.com. In this paper, the EbangEbitE10 mining device with the highest energy efficiency ratio is chosen; its arithmetic power is 18TH/s and energy
consumption is 1.65kW/h. The mining electricity cost is then defined as 0.07kW/h based on the average industrial electricity cost.
The following variables are included in the calculation in this paper:
m = a single miner's arithmetic power (TH/s)
em=A single miner's energy consumption (kWh)
c = electricity cost (kWhUSD)
Blocks per hour =
r=block incentive (btc)
m=Total number of simulated miners
cmb=Electricity cost per miner per block consumed (USD)
v=intrinsic Bitcoin's value (USD)
To begin, determine the total number of simulated miners m for the best case (all miners with the highest energy efficiency ratio).
Second, divide individual miners' energy consumption (in kWh) by the number of blocks antminer s19 for salefound per hour, and multiply the result by the cost of electricity per kW to calculate the
energy cost per miner per hour, cmb.
Finally, the total system energy cost per block divided by the number of bitcoins generated per block is the lower bound on the intrinsic value generated by bitcoins.
In the current situation, there are a total of 2245,508 miners, with an energy consumption of $0.019 kW/h per miner, resulting in a calculated energy cost of $42,670 to generate a block. Mining costs
$3413.62 per BTC at the current block reward of 12.5 BTC.
With a 6% speculative rate and the lowest intrinsic cost (i.e. mining cost), the external value is $204 at the current price of $3618 for Bitcoin.
Following this logic, the same calculation yields a minimum mining cost of $92.60 for ETH with a 2% speculation rate, $293.23 for BitcoinABC with a -183% speculation rate, and $346.10 for BSV with a
-248% speculation rate.
Of course, all else being equal, as the new generation of miners continues to roll out, the cost of bitcoin mining will continue to fall due to the improved efficiency ratio of the new miners. The
E10 used in this article, as well as the E11, which is about to hit the market, claim to provide 44TH/s of arithmetic power while consuming only 2.0kWh of energy. Based on this information, the cost
of bitcoin mining can be cut in half to $1692.71 under ideal conditions. | {"url":"https://www.searchnewsinfo.com/topic/278964/What-is-the-cost-of-bitcoin-mining-excluding-electricity/","timestamp":"2024-11-07T07:35:43Z","content_type":"text/html","content_length":"24692","record_id":"<urn:uuid:c50ef3d2-289f-4617-98d5-fb9dff3fd880>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00423.warc.gz"} |
Understanding the Equilibrium and Motion of a Mass-Spring-Damper System
📚 The video is about the motion of a mass-spring-damper system, which is an important topic in the study of mechanical vibrations.
👉 There are three important aspects to note: the fundamental equation of motion applies to point masses, rigid bodies need to be approached with caution, and the video will focus on systems with
point masses.
🎓 In this lecture series, the video will cover the equations and relationships related to mass-spring-damper systems. | {"url":"https://chattube.io/summary/gaming/RYxOD10BKBU","timestamp":"2024-11-03T22:18:05Z","content_type":"text/html","content_length":"37650","record_id":"<urn:uuid:19a2a393-de2a-4a60-b718-f8a84a60fcd8>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00013.warc.gz"} |
Determining the Radius of a Wheel in context of radius of wheel
30 Aug 2024
Determining the Radius of a Wheel: A Guide to Measuring and Calculating
The radius of a wheel is a fundamental measurement that plays a crucial role in various engineering and scientific applications, including mechanics, physics, and mathematics. In this article, we
will explore the concept of wheel radius, its importance, and provide formulas and methods for determining it.
What is the Radius of a Wheel?
The radius of a wheel is the distance from the center of the wheel to its outer edge or circumference. It is an essential measurement that affects the performance, efficiency, and safety of various
machines and vehicles that rely on wheels, such as cars, bicycles, airplanes, and more.
Why is the Radius of a Wheel Important?
The radius of a wheel has significant implications for various aspects of engineering and science:
1. Mechanical Advantage: The radius of a wheel affects the mechanical advantage of a machine or vehicle. A larger radius can increase the mechanical advantage, while a smaller radius can decrease
2. Efficiency: The radius of a wheel impacts the efficiency of a machine or vehicle. A well-designed wheel with an optimal radius can improve fuel efficiency and reduce energy consumption.
3. Safety: The radius of a wheel is critical for safety considerations. A larger radius can increase the stability and maneuverability of a vehicle, while a smaller radius can compromise these
Methods for Determining the Radius of a Wheel
There are several methods to determine the radius of a wheel:
1. Direct Measurement: Measure the diameter of the wheel using a ruler or calipers. Then, use the formula:
Radius (R) = Diameter (D) / 2
Where R is the radius and D is the diameter.
Example: If the diameter of the wheel is 20 inches, then the radius would be:
R = 20 inches / 2 = 10 inches
1. Indirect Measurement: Measure the circumference of the wheel using a flexible tape measure or a string. Then, use the formula:
Circumference (C) = π x Diameter (D)
Where C is the circumference and D is the diameter.
Example: If the circumference of the wheel is 62.83 inches, then the diameter would be:
D = Circumference (C) / π
D = 62.83 inches / π ≈ 20 inches
Once you have determined the diameter or circumference, you can calculate the radius using the formulas above.
Formulas and Equations
Here are some essential formulas and equations related to wheel radius:
1. Radius-Diameter Formula: R = D / 2
2. Circumference-Radius Formula: C = π x R
3. Diameter-Circumference Formula: D = C / π
Determining the radius of a wheel is a crucial step in various engineering and scientific applications. By understanding the importance of wheel radius and using the methods and formulas outlined
above, you can accurately measure and calculate this critical parameter. Whether you are working on a machine, vehicle, or mathematical problem, having a solid grasp of wheel radius will help you
achieve your goals with precision and accuracy.
Related articles for ‘radius of wheel’ :
Calculators for ‘radius of wheel’ | {"url":"https://blog.truegeometry.com/tutorials/education/2e9130ce190f9d51b88856d7deecc747/JSON_TO_ARTCL_Determining_the_Radius_of_a_Wheel_in_context_of_radius_of_wheel.html","timestamp":"2024-11-05T16:48:26Z","content_type":"text/html","content_length":"17036","record_id":"<urn:uuid:23d42e09-96f0-4893-932f-7b156fa961b7>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00572.warc.gz"} |
Beginning 66104 - math word problem (66104)
Beginning 66104
The kangaroo always jumps three steps up. Each time he jumps, the bunny jumps down two steps. On which stairs will they meet? The kangaroo stands on the 1st step at the beginning and the bunny on the
Correct answer:
Did you find an error or inaccuracy? Feel free to
write us
. Thank you!
Tips for related online calculators
Do you have a linear equation or system of equations and are looking for its
? Or do you have
a quadratic equation
You need to know the following knowledge to solve this word math problem:
Themes, topics:
Grade of the word problem:
Related math problems and questions: | {"url":"https://www.hackmath.net/en/math-problem/66104","timestamp":"2024-11-13T22:37:28Z","content_type":"text/html","content_length":"54297","record_id":"<urn:uuid:12aa0869-d683-4164-9432-a0c849ff19d7>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00730.warc.gz"} |
Deconstructed Rubik's Cube - OpenQuant
Deconstructed Rubik's Cube
You have a standard Rubik's Cube and you break it apart into 27 cubes. You put those 27 cubes into a bag and you randomly pull one out and toss it in the air. The cube lands such that you can only
see 5 non-painted sides. What is the probability that you pulled a cube with one colored side? | {"url":"https://openquant.co/questions/deconstructed-rubik's-cube","timestamp":"2024-11-12T00:31:17Z","content_type":"text/html","content_length":"27512","record_id":"<urn:uuid:421bc708-5f71-4c1e-b488-a8cb5b9ee3dc>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00001.warc.gz"} |
Activities – Chapter 1
Chapter 1 – Algebraic Notation
Note to Students:
Students should be aware that these worksheets are not ‘fill in the blank’ worksheets. We have intentionally given you no space to write answers on the worksheets. You should have a notebook for your
work, perhaps even beginning by sketching ideas on a white board or scratch paper until you have sufficient organization to put a coherent explanation in your notebook. Note also that the expectation
is to write ‘explanations’ and not ‘answers’. These activities are not designed to teach computational skills, but instead are designed to introduce mathematical concepts. The process used to solve
problems is the focus, not the end result.
Activity 1a – Multiplication via the FOIL Method
This first activity reviews multiplication, starting with the algorithm for multiplying two digit numbers. Students may bring a variety of techniques to the discussion, and it is important to see
multiple representations. One important aspect of learning mathematics is to be able to make connections between different approaches that solve the same problem. The insights from multiplying two
digit numbers should lead to insights on the FOIL process.
Part II of this activity is for more advanced students. It leads toward the Binomial Theorem and the connection to the theory of probability
Multiplication via the FOIL Method
Activity 1b – Factoring
This activity builds the fundamentals of the factoring process. The `Box Method’ is used to demonstrate certain principles, though students may have a variety of methods that work. Again, the goal
here is to share ideas, approach the problem from multiple perspectives, and understand the connections between various approaches.
Activity 1c – Reducing Fractions
This activity reviews the basics of reducing fractions by factoring. The topic begins with numerators and denominators which are integers before progressing to polynomials. It is important that
students understand the factoring process and why terms that are added can’t be cancelled.
Activity 1d – Multiplying and Dividing Fractions
This activity reviews the basics of multiplying and dividing fractions. It relies on students’ ability to reduce fractions.
Multiplying and Dividing Fractions
Activity 1e – Adding and Subtracting Fractions
This activity reviews the basics of adding and subtracting fractions. The activity begins with numerators and denominators which are integers before progressing to polynomials. It is important that
students be able to describe the process for finding a common denominator, including the factoring process on the early questions before tackling the harder questions. | {"url":"https://open.lib.umn.edu/algebra/chapter/activities-chapter-1/","timestamp":"2024-11-09T04:56:23Z","content_type":"text/html","content_length":"78633","record_id":"<urn:uuid:0bd0742c-a125-4560-a4d0-f396c61e5c07>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00151.warc.gz"} |
Correlated Firing in Macaque Visual Area MT: Time Scales and Relationship to Behavior
We studied the simultaneous activity of pairs of neurons recorded with a single electrode in visual cortical area MT while monkeys performed a direction discrimination task. Previously, we reported
the strength of interneuronal correlation of spike count on the time scale of the behavioral epoch (2 sec) and noted its potential impact on signal pooling (Zohary et al., 1994). We have now examined
correlation at longer and shorter time scales and found that pair-wise cross-correlation was predominantly short term (10–100 msec). Narrow, central peaks in the spike train cross-correlograms were
largely responsible for correlated spike counts on the time scale of the behavioral epoch. Longer-term (many seconds to minutes) changes in the responsiveness of single neurons were observed in
auto-correlations; however, these slow changes in time were on average uncorrelated between neurons. Knowledge of the limited time scale of correlation allowed the derivation of a more efficient
metric for spike count correlation based on spike timing information, and it also revealed a potential relative advantage of larger neuronal pools for shorter integration times. Finally, correlation
did not depend on the presence of the visual stimulus or the behavioral choice of the animal. It varied little with stimulus condition but was stronger between neurons with similar direction tuning
curves. Taken together, our results strengthen the view that common input, common stimulus selectivity, and common noise are tightly linked in functioning cortical circuits.
A fundamental problem in sensory neuroscience is to understand how psychophysical performance is related to the signaling capacities of single sensory neurons. It is now widely recognized that no
satisfactory solution to this problem can be achieved in the absence of detailed knowledge concerning correlated firing within the pool of sensory neurons contributing to a particular psychophysical
judgment (Johnson et al., 1973;Johnson, 1980; van Kan et al., 1985;Britten et al., 1992; Gawne and Richmond, 1993; Zohary et al., 1994; Geisler and Albrecht, 1997; Parker and Newsome, 1998). For
example, combining signals across a pool of neurons can generate superior psychophysical sensitivity if the noise carried by individual members of the pool is averaged out. This benefit of pooling is
only achievable, however, to the extent that the noise carried by individual neurons is independent (uncorrelated); noise that is common to the entire pool cannot be averaged out. In general, the
effect of correlated noise depends on how signals are combined, and although correlation may either aid or hinder noise removal (Johnson, 1980; Abbott and Dayan, 1999; Panzeri et al., 1999), its
impact on the amount of information conveyed by a pool of neurons may be profound. Thus, empirical analysis of correlated firing is central to a quantitative understanding of the relationship between
physiological responses and psychophysical judgments.
Extrastriate visual area MT is ideal for investigating pools of sensory neurons that underlie psychophysical performance. MT contains a preponderance of directionally selective neurons (Zeki, 1974;
Maunsell and Van Essen, 1983;Albright et al., 1984), the activity of which has been linked compellingly to the psychophysical discrimination of direction in stochastic motion stimuli (Newsome et al.,
1989;Britten et al., 1992; Salzman et al., 1992; Murasugi et al., 1993; Salzman and Newsome, 1994). In a previous study, therefore, we measured correlated firing in MT and found that spike counts
from adjacent neurons were noisy and only weakly correlated but that even this small amount of correlated noise placed substantial limits on the benefits of signal averaging across a pool (Zohary et
al., 1994). Subsequently, Shadlen and colleagues (1996) incorporated these insights into a computational model of the relationship between the activity of MT neurons and psychophysical judgments of
motion direction.
In the present study, our primary goals were to examine the time scale at which correlation arises: in particular, to relate spike count correlation to spike timing correlation and examine the
dependence of correlated firing on stimulus and behavioral parameters. Our most intriguing finding is that trial-to-trial correlations in spike count, measured over trials of 2 sec duration, are
produced largely by the same mechanisms that generate peaks in the spike train cross-correlogram (CCG) on a time scale of a few tens of milliseconds. For a given pair of MT neurons, a quantitative
measurement based on the CCG peak predicts with fair accuracy the level of correlation calculated from spike counts over the full trial length. Furthermore, the CCG-based measure is substantially
more reliable than the measure based on spike count correlation. The spike train CCG is typically used as a qualitative indicator of functional connectivity among neurons. In contrast, our results
suggest that the spike train CCG can provide quantitative measures of neuronal correlation that are of considerable interest for models that seek to reconcile neuronal and psychophysical performance.
Some of these results have been published previously in abstract form (Bair et al., 1996,1999).
Subjects, surgery, and daily routine. The experiments were performed on three adult rhesus monkeys weighing between 7 and 9 kg (Macaca mulatta, two males and one female). Before the experiments, each
monkey was surgically implanted with a device for stabilizing head position (Evarts, 1968), a scleral search coil for measuring eye position (Judge et al., 1980), and a recording cylinder that
allowed microelectrode access to cortex within the occipital lobe. All surgical procedures were performed under aseptic conditions with halothane anesthesia. After recovery from surgery, each animal
engaged in daily training or experimental sessions of 2–6 hr duration. Behavioral control was accomplished by operant conditioning techniques using fluids as a positive reward; fluid intake was
therefore restricted during periods of training or electrophysiological recording. The diet was supplemented with moist monkey treats, fruits, and nuts. The animals were maintained in accordance with
guidelines set by the U.S. Department of Health and Human Services (NIH) Guide for the Care and Use of Laboratory Animals.
Visual stimuli. The visual stimuli used in this study were a set of dynamic random dot patterns in which a unidirectional motion signal was interspersed among random motion noise. The stimulus set
has been described extensively in previous publications (Britten et al., 1992), and we simply summarize its essential features here.
Dynamic random dots were plotted sequentially on the face of a CRT screen at a high rate (6.67 kHz). After 45 msec, a dot was either displaced in a specified direction (coherent motion) or replaced
by another dot at a random location on the screen (noise). In one extreme form of the display, all dots were positioned randomly so that the display was pure noise. In this form, which we term 0%
coherence, the display contained many local motion events (caused by fortuitous pairings of the dots in space and time) but on average no net motion in any direction. At the other extreme (100%
coherence), all dots were displaced uniformly so that the display contained noise-free motion in a specified direction. Our software permitted us to create any stimulus intermediate between these two
extremes by specifying the percentage of dots that carried the “coherent” motion signal. The percentage of dots engaged in coherent motion governed the strength of the motion signal without affecting
the overall luminance, contrast, or average spatial and temporal structure of the stimulus. When a psychophysical subject was asked to discriminate the direction of motion in such displays, the
difficulty of the discrimination was related directly to the percentage of dots in coherent motion.
In early experiments (monkey E), the stimuli were generated by a PDP 11/73 computer and displayed on a large, electrostatic deflection oscilloscope via a high speed DMA digital-to-analog converter.
In later experiments (monkeys R and K), the stimuli were created by means of an IBM 386 equipped with a dedicated graphics board (SGT Pepper no. 9). These stimuli were displayed on a raster scan CRT
monitor with a 60 Hz refresh rate. In all experiments, the display monitor was positioned 57 cm in front of the monkey.
A critical distinction must be made between two different methods of presenting repeated stimuli for a particular condition (e.g., a 6.4% coherence, upward stimulus). Our standard method used a new
random number sequence for each repeat, resulting in what we will refer to as “ensemble stimuli,” which differ in detail but have on average the prescribed motion coherence. As a control for the
effect of random stimulus variation on neuronal responses, we recorded from four pairs using repeated presentations of stimuli generated with exactly the same sequence of random numbers. We will
refer to the identical stimulus repeats used by this method as “replicate stimuli.”
Behavioral paradigms and selection of visual stimuli. We used two behavioral paradigms in this study: a fixation task and a discrimination task. In the fixation task, the monkey was required only to
maintain its eye position within an electronically defined window around the fixation point for 2–4 sec. The monkey received a liquid reward on successful completion of each trial. In most
experiments, the window permitted eye movements up to 1.5° away from the fixation point, but in practice, the monkeys usually held their eye position within 0.5° of the fixation point.
The monkeys performed the fixation task during the initial search for well isolated pairs of neurons, during mapping of receptive fields, and during quantitative measurement of the direction tuning
properties of the neurons. Receptive field boundaries were mapped qualitatively for each neuron of the pair, and the stimulus aperture was positioned to include both receptive fields. The receptive
fields typically overlapped substantially, so that the stimulus aperture only engaged a small portion of the surround of either receptive field. The optimal speed was estimated qualitatively for each
neuron, and subsequent experiments were conducted using a motion speed intermediate between the two optima. To measure a direction tuning curve, a 100% coherence dot pattern was presented in eight
different directions of motion equally spaced around the clock at 45° intervals. The different directions were presented in a pseudorandom sequence until 10–20 repetitions were completed for each
The two direction tuning curves were used to assign a “preferred-null” axis of motion for use during the discrimination task (below). Ostensibly, the preferred-null axis was the axis of maximal
directionality for the two neurons; motion in opposite directions along the axis should yield a maximal difference in responsiveness. In practice the axis chosen was usually a compromise between the
preferred directions of the two neurons measured individually. Most pairs of neurons had similar preferred directions, and the compromise therefore resulted in a near-optimal axis for both. On
occasion we recorded from pairs of neurons with preferred directions that were nearly opposite each other. In this case again, the choice of directional axis was easy because the signs of the two
response were simply reversed along the same axis. Occasionally, however, the preferred directions of the two neurons were nearly orthogonal to each other, or one of the neurons was not directional
at all. In such cases, we chose the preferred-null axis and the speed of the motion signal to match the preferences of the more responsive, directional neuron. On the whole, therefore, most neurons
were studied during the discrimination task with stimuli that matched their physiological properties reasonably well. For a few neurons, the stimuli were substantially suboptimal.
In the discrimination task, the monkey performed a two-alternative, forced-choice discrimination of motion direction. This task has been used extensively in our laboratory and is described in detail
in previous publications (Britten et al., 1992). On each trial a random dot stimulus was presented for 2 sec within the aperture covering both receptive fields. The direction of the coherent motion
signal was varied randomly from trial to trial between the preferred direction of the neurons under study and the direction 180° opposite (the “null” direction); the monkey's task was to discriminate
correctly the direction of motion. The strength of the motion signal was varied among a range of coherence levels that spanned psychophysical threshold. A minimum of 15 repetitions was obtained for
each stimulus condition (i.e., each combination of direction and coherence), and all conditions were presented in pseudorandom order. We will refer to the neuronal data from these experiments as
“coherence series data” to distinguish them from the direction tuning data.
Each trial began with the appearance of the fixation point. After the monkey achieved fixation and held its gaze within the fixation window for 300 msec, the visual stimulus was presented as
described above. The monkey was required to hold its gaze on the fixation point during stimulus presentation so that the stimulus remained well positioned on the receptive fields of the two neurons.
At the end of the 2 sec display interval, the random dot pattern and the fixation point disappeared, and two small visual targets appeared, one corresponding to each of the two possible directions of
coherent motion. The monkey made a saccadic eye movement to one of the two targets to indicate the direction of motion perceived in the visual stimulus. Eye movements were measured continuously with
the scleral search coil technique, permitting the computer to register correct and incorrect choices. Correct choices were followed by a liquid reward; incorrect choices were followed by a brief
time-out period. On 0% coherence trials, the monkey was rewarded randomly with a probability of 0.5 because there was no “correct” answer on these trials. If the monkey broke fixation prematurely
during a trial, the trial was aborted, the data were discarded, and a time-out period ensued.
Electrophysiological recording and spike sorting.Electrophysiological recordings were made with tungsten microelectrodes inserted into the cortex through a transdural guide tube (electrode impedance
= 0.5–2.0 MΩ at 1 kHz) (Micro Probe, Potomac, MD). The guide tube was held rigidly in a stable coordinate system by a plastic grid inside the recording cylinder (Crist et al., 1988). We recorded
through any particular guide tube for several consecutive days.
The signal from the microelectrode was amplified and bandpass-filtered (0.5–10 kHz), and action potentials from multiple single neurons were discriminated using an on-line spike sorting system that
was developed originally in the laboratory of Dr. Moshe Abeles (Hebrew University, Jerusalem) and was commercially available from Alpha Omega Engineering (Nazareth, Israel). The filtered
microelectrode signal was continuously sampled at a rate of 14 kHz by a digital signal processing system housed in an IBM 386 platform. (The apparent discrepancy between the 7 kHz cutoff frequency,
implied by 14 kHz sampling, and the 10 kHz cutoff of our bandpass filter was not a limiting factor, because in practice the amplitude of the noise from 7 to 10 kHz was small relative to the amplitude
of all well isolated action potentials.) The computer software provided a user interface to the spike sorting hardware and included graphics displays of voltage waveforms, spike templates, and
distributions of matching errors (below). Spikes were discriminated on-line using an eight-point template-matching algorithm (Wörgötter et al., 1986). Each time the voltage exceeded a threshold
level, an eight-point voltage sample was acquired and compared with the predefined templates that characterized the waveform of each recorded neuron. If the root-mean-square error (RMSE) of the match
between the signal waveform and one of the templates was below a criterion value, an action potential was registered for that neuron. A template was defined by the software to minimize the RMSE of
the match to the template across a sample of 100 action potentials accepted by the experimenter as belonging to a specific neuron.
The quality of unit isolation was determined by the separation of the templates from each other and from the noise. Excellent separation of the templates from each other was necessary to prevent
“cross-talk” between the two waveforms. Occasional misclassification of the two action potentials could result in artifactual correlations that would be deleterious to certain analyses. The only
substantive insurance against cross-talk were the rigor and attentiveness of the experimenter—both in selection of pairs for study and in maintaining quality of isolation during the experiment. We
attempted to be exceedingly rigorous in selecting pairs for study, rejecting all candidates except those with waveforms that were strikingly distinct from each other. Similarly, we attempted to be
unusually conservative in on-line assessment of the quality of isolation. If either waveform began to deteriorate, creating any doubt about isolation, we ceased recording until the waveforms could be
Separation of the two templates from the noise could be achieved more objectively. For each template, the software compiled and displayed a frequency histogram of RMSE values resulting from
comparison of each triggered waveform with that template. Excellent separation of the template from the noise corresponded to a bimodal histogram of RMSE values. A peak at low RMSE values
corresponded to action potentials from the neuron that defined the template; a larger peak at high RMSE values reflected the substantial mismatch between noise waveforms and the template. We insisted
that both modes be visible and well separated from each other in the RMSE histogram. The criterion RMSE value for accepting an action potential as corresponding to a particular template was set at
the local minimum in the bimodal RMSE distribution. This ensured a reasonable balance between minimizing noise contamination and minimizing false negative matches to the template. We rejected
recordings for which the error distributions were judged by eye to overlap in a manner that would produce more than ∼5% false positives, but we estimate that the contamination rate was typically
lower because the peaks in the bimodal error distribution often showed no sign of overlap after collecting hundreds of spikes. Admitting a small percentage of spikes from other neurons to one or the
other template should have negligible effect on estimates of pair-wise interneuronal correlation because of the modest to weak correlations typical between cortical neurons.
Obviously, our technique of multi-unit recording with a single electrode cannot detect simultaneous spikes because the two waveforms superimpose, resulting in a poor match to either template. Because
the primary lobes of the action potential waveforms were generally ≤0.5 msec in duration (Mountcastle et al., 1969;Funahashi and Inoue, 2000), this limitation only resulted in an underestimation of
spikes that were synchronous to within 1 msec [for example, see Gawne and Richmond (1993); Funahashi and Inoue (2000)]. For two neurons with uncorrelated activity firing at rates <100 spikes per
second, the probability of spike synchrony at the millisecond time scale is <0.1^2, which is reasonably uncommon. However, the pairs of neurons studied here often have peaks in their CCGs at time
zero (see Fig. 5B), and therefore the probability of simultaneous firing may be many times greater. This problem can be compounded for cells that fire bursts during which firing rates may reach
300–500 spikes per second. However, multi-electrode cross-correlation studies in monkey and cat suggest that CCG peaks in visual cortex are typically broader than 1–2 msec (Ts'o et al., 1986; Krüger
and Aiple, 1988; Ts'o and Gilbert, 1988; Cardoso de Oliveira et al., 1997). The available evidence suggests that the vast majority of CCGs do not have sudden discontinuities on the time scale of 1
msec at the origin and that peaks of width 1 msec, when they exist, are weak and could not account for a substantial fraction of the strength of interneuronal correlation commonly observed in visual
cortex. Therefore, we approximate the CCG value at time zero using values at neighboring time lags, as described later.
Analysis of direction selectivity. To assess neuronal direction selectivity, we determined which of two different models could better match the direction tuning curves. The first model assumed that
the neuron was not direction selective and that response variation across direction was caused simply by sampling noise. It therefore predicted that the level of activity was essentially invariant
with direction and was best estimated as the mean of the responses to all directions. The second model assumed that the neuron was in fact direction selective with a Gaussian distribution of
responses centered on the optimal direction of motion. This distribution had four free parameters: the optimal direction of motion, the maximal response rate, the bandwidth of the Gaussian function,
and the baseline response (the spontaneous firing rate). We performed maximum likelihood fits to the two separate, nested models under the assumption of normal errors. The likelihoods (L) obtained
from these computations were transformed by: Equation 1such that l is distributed as χ^2 with three degrees of freedom (Hoel et al., 1971). If l was below the criterion value (p = 0.05), we concluded
that the direction tuning function of the neuron was better described by a Gaussian fit than by a constant response independent of direction. We considered these neurons to be direction selective,
and the quantitative analyses in this paper used optimal directions and bandwidths obtained from the Gaussian fit to the tuning curve of each neuron. We will use the notation Δ[PD] to refer to the
difference (in degrees) between the preferred directions of neurons within a pair.
Analysis of psychophysical data. Psychophysical data from the discrimination experiments were compiled into psychometric functions depicting the proportion of correct decisions as a function of the
strength of the motion signal (in % coherence). We used a maximum likelihood method (Watson, 1979) to fit these data with sigmoidal functions of the form: Equation 2where p is the probability of a
correct decision, c is coherence, a is the coherence level that supports threshold performance (82% correct), b is the slope of the sigmoidal function, and d is the asymptotic performance for strong
motion signals (expressed as proportion of correct decisions). The threshold parameter, a, and the slope parameter, b, provide a succinct description of the psychophysical data.
Equation 2, derived from the integral of a Weibull distribution (Quick, 1974), provided acceptable fits to the bulk of our psychophysical data. Thirty-four of 46 psychometric functions in our data
set were well fit [likelihood ratio test, p > 0.05; see the Appendix of Watson (1979)] when the asymptotic performance, d, was constrained to be unity, and the remaining functions were well fit by
allowing d to vary. The non-unity asymptote in the latter 12 experiments reflected the monkey's occasional errors at the highest coherence levels.
Analysis of neural thresholds. We measured neural thresholds to the stochastic motion stimuli in a manner that permitted direct comparison with psychophysical thresholds [see Britten et al. (1992)
for a detailed description]. For each neuron, we first compiled for each motion coherence a frequency histogram of responses to preferred direction motion and a separate histogram of responses to
null direction motion. We considered a “response” to be the total number of spikes generated by the neuron during the 2 sec stimulus. For very strong (high coherence) motion signals, these
“preferred” and “null” response distributions were typically non-overlapping because most of our neurons were highly directional. At these coherence levels, the direction of motion could be
determined unambiguously on any given trial simply by monitoring the response of the neuron. For very weak motion signals, however, the preferred and null response distributions overlapped almost
completely, so judgments of motion direction based on the responses of the neuron would be at chance. Intermediate coherence levels resulted in partial overlap between the two response distributions,
leading to intermediate levels of performance.
Following these intuitions, we used a method based on signal detection theory (Green and Swets, 1966; Britten et al., 1992) to compute the performance expected of an ideal observer who based
judgments of motion direction on the measured neuronal responses. For each neuron, this calculation was performed for each coherence level (typically six non-zero levels, but as many as eight, and
the results were compiled into a neurometric function that plotted expected performance (in % correct decisions) as a function of coherence. A sigmoidal curve was fitted to the data using Equation 2,
and the threshold and slope parameters were extracted as described in the preceding section. These parameters describe the sensitivity of a single neuron to the motion signals in our displays in a
manner that can be compared directly with the psychophysical sensitivity measured on the same trials. Equation 2 described our neurometric data well; the fits were acceptable (likelihood ratio test,
p > 0.05) for all 83 of the neurons comprising the 46 pairs with valid psychophysical data.
Assessment of correlated activity. We analyzed two main types of correlation between the responses from each pair of neurons: signal correlation and noise correlation (Gawne and Richmond, 1993; Gawne
et al., 1996; Lee et al., 1998). Signal correlation, designated r[signal], refers to the common modulation in a set of paired mean responses associated with multiple stimulus conditions. For our
purposes, it is simply the correlation coefficient computed for the mean spike rates from a pair of direction tuning curves. Noise correlation, r[noise], refers to common trial-to-trial fluctuations
around the mean response for a single stimulus condition, and its estimation and interpretation occupy the bulk of this paper. The dichotomy implied by the names “signal” and “noise” correlation is
somewhat unfortunate because apparently noisy variations in spike rate may carry information about neural signals that we simply cannot access. However, we will adhere to these terms for the sake of
precedent. The traditional measure of noise correlation is the interneuronal correlation coefficient (van Kan et al., 1985; Bach and Krüger, 1986;Gawne and Richmond, 1993; Zohary et al., 1994; Gawne
et al., 1996; Lee et al., 1998), which measures correlation at a fixed time scale and temporal relationship, i.e., the simultaneous trial. We will describe two new methods for quantifying noise
correlation, one at time scales greater than and equal to the single trial that generalizes the interneuronal correlation coefficient to non-simultaneous trials (below) and another at the scale of
milliseconds that is derived from spike train correlograms (Appendix ). Table1 provides a unified reference to all of our notation regarding correlation.
The trial cross-covariance. The interneuronal correlation coefficient is traditionally computed for the spike counts N[1] and N[2] of neurons 1 and 2, respectively, according to: Equation 3where E is
expected value and ς is the SD computed across all repetitions of a particular stimulus. However, an experiment yields several sets of paired spike counts (one set for each stimulus condition), and
rather than applying Equation 3 separately to each set, the sets can be combined after performing a within-set normalization. One simple normalization, the z -score, involves modifying the spike
count values within each set (i.e., for each stimulus condition) by subtracting the mean and dividing by the SD for that set of responses. The subtraction eliminates the mean stimulus-evoked portion
of the response, and the division scales the variance around the mean so that random fluctuations at high firing rates (which are known to be larger than those at low rates) are not unduly weighted.
Further empirical justification for this normalization comes from the observation that r[SC] changes very little with firing rate or stimulus condition, as shown in Results. The resulting z -scores
can be represented in the order in which they occurred in the original experiment by the sequences z[1]^i and z[2]^i, 1 ≤ i ≤ M, where M is the total number of trials in the experiment. Because E z
[1]= E z[2] = 0, and ς[z[1]] = ς[z[2]] = 1, the equation for the correlation coefficient, Equation 3, simplifies to: Equation 4For a single set of paired responses, this equation is equivalent to
Equation 3 because neither subtraction nor division by a positive constant (applied to the spike count data) changes the value of the correlation coefficient. For multiple sets of responses, the
equation provides an aggregate correlation coefficient. Equation 4 can be generalized from responses that occurred on the same trial to responses that occurred on trials separated in time by a lag,
φ, in units of experimental trials (∼5 sec per unit; see below). This generalization, which we will refer to as the trial cross-covariance (TCC), is simply the cross-correlation of z[1] and z[2]:
Equation 5The value at φ = 0 is equal to r[SC](Eq. 3) averaged across all stimulus conditions (with appropriate weighting for the number of trials for each condition), and we use the symbols TCC(0)
and r[SC] interchangeably. For φ ≠ 0, TCC(φ) is the correlation coefficient, with values from −1 to 1, for temporal offsets in arbitrary numbers of trials. For a pair of uncorrelated neuronal
responses, TCC(φ) will approach zero everywhere as the number of trials used in its estimate increases. The trial auto-covariance, TAC(φ), is defined in a similar manner by replacing z[2] with z[1]
(or vice versa) in Equation 5, and by definition it is equal to unity for φ = 0.
We already know that TCCs will have positive values at φ = 0 for neuronal pairs with r[SC] > 0. If interneuronal correlation arises at a time scale shorter than the trial duration, the positive value
at φ = 0 will stand as a narrow, isolated peak. However, if the correlation between neurons arises from slow changes in their responsiveness, the positive value at φ = 0 will be part of a broader
peak, i.e., the TCC will have positive values for φ ≠ 0 as well. Similarly, the presence of a broad peak around the origin in the TAC will indicate the presence of slow variations in the excitation
of individual neurons.
Our use of the TCC does not rest on whether the horizontal axis is given in units of time or trials. We retained “trials” as the axis unit to avoid the technical difficulty associated with the
cross-correlation of data sampled at somewhat irregular time intervals. The irregularity in the mapping from trials to time was caused by the monkey's failure to fixate immediately on 10–20% of
trials during a recording session. To estimate the time scale of slow correlation, we will convert from trials to time using the average time between trial starts, ∼5 sec.
Previous studies by Eggermont and Smith (1995, 1996) have attempted to separate correlation at multiple time scales using a method similar in concept to the TCC, but their time unit was 50 msec,
roughly two orders of magnitude faster than ours. Variations in firing rate on the time scale of 10s to 100s of milliseconds (Nelson et al., 1992; Eggermont and Smith, 1995;Arieli et al., 1996) are
considered by us to be short term because they fall well within the duration of our behavioral epoch.
The spike train cross-correlogram. We measured correlation at the time scale of milliseconds using spike train auto- and cross-correlograms (ACGs and CCGs) (Perkel et al., 1967a,b). Our CCG is
defined based on the trial-averaged cross-correlation, C[jk](τ) (defined in Appendix , Eq. 14), of binary spike sequences from neurons j and k (typically, 1 and 2). In particular: Equation 6where λ
[j] and λ[k] are the mean firing rates (in spikes per second) of neurons j and k. For ACGs, j = k = 1 or 2. The function Θ(τ) is a triangle representing the extent of overlap of the spike trains as a
function of the discrete time lag τ, i.e.: Equation 7where T is the duration of the spike train segments used to compute C[jk]. Dividing C[jk] by Θ(τ) in Equation 6 changes the units of our CCG from
raw coincidence count to coincidences per second and corrects for the triangular shape of C[jk] caused by finite duration data.
In Equation 6, we chose to divide by the geometric mean spike rate (GMSR), , because under this normalization the area of our CCG peaks remained relatively constant as firing rate varied [shown
later; see also Krüger and Aiple (1988)] and because it is symmetric with respect to the two neurons. With this normalization, the CCG is the ratio of a coincidence rate to a mean spike rate and ends
up with units of coincidences per spike. Once the shift-predictor (below) is subtracted, this normalization is similar to that of many other studies (Mastronarde, 1983a; Krüger and Aiple, 1988;
Eggermont and Smith, 1996; Cardoso de Oliveira et al., 1997) and is conceptually similar to that proposed by Aertsen et al. (1989) for their “joint peri-stimulus time histogram.” A different
normalization, dividing by the product of the spike rates, has been favored less often (Melssen and Epping, 1987, their Eq. 17; Das and Gilbert, 1995), and for our data was less appropriate than
dividing by the GMSR.
Shift- (also known as shuffle-) predictors [defined in Perkel et al. (1967b)] for CCGs and ACGs were computed using the same normalization as above but based on the average cross-correlation of all M
^2 − M pairings of nonsimultaneous responses from neurons j and k for a set of M trials. This “all-way” cross-correlation, denoted C^*[jk](τ), can be computed efficiently from the cross-correlation
of the post-stimulus time histograms (PSTHs), S[jk] (defined in Appendix , Eq. 15), according to the following expression: Equation 8which approaches S[jk](τ) as M increases (Perkel et al., 1967b).
Substituting C^*[jk] for C[jk] in Equation 6 gives the final shift-predictor. A shift-predictor computed from responses to ensemble stimuli (i.e., those that resulted from different sequences of
random numbers; see above) will be referred to as an ensemble shift-predictor. When computed for replicate stimuli (i.e., repetitions of identical stimuli), it will be referred to plainly as a
CCGs, ACGs, and shift-predictors were computed from data in the post-stimulus onset period 300–2000 msec to avoid processing the initial transient response. This made shift-predictors flatter and
prevented changes in correlation strength that might be associated with the stimulus onset transient from influencing the analysis. We computed all quantitative results for the full trial as well and
found only negligible differences. CCGs and ACGs were computed individually for each stimulus condition, shift-predictors were subtracted, and then averages were taken across all valid stimulus
conditions. We set criteria for the minimum quantity of data required for neurons to be accepted into the CCG and ACG analysis pool. These rules were applied in order: (1) no trial was valid that had
fewer than four spikes within the analysis window, (2) no stimulus condition was valid that had fewer than four valid trials or <64 spikes in total per neuron, and (3) no pair of neurons (or neuron)
was included that had fewer than four valid stimulus conditions. These criteria eliminated 1 of 104 pairs from our direction tuning data set and 2 of 50 pairs from our coherence series data set.
Our findings are organized as follows. The first section provides a brief description of our data for a typical pair of neurons and shows how all pairs are distributed according to the strength of
their interneuronal correlation and the similarity of their directional tuning curves. The second major section is devoted to measuring the time scale of interneuronal correlation, which involves (1)
separating long- and short-term correlation, (2) assessing the time scale of short-term correlation using spike train CCGs, and (3) relating CCG peaks to spike count correlation. A more efficient
metric for spike count correlation is derived here and in Appendix . The next major section of results reports the dependence of correlation, or synchrony, on stimulus parameters and on the
decision-making and behavioral state of the animal. A brief section shows that neurons do not cluster with respect to their sensitivity to the stimulus or their relationship to behavior, and the
final section describes control experiments for the influence of stimulus variance on our estimates of correlation.
Basic measurements of response correlation
Our results are based on simultaneous recordings from 107 pairs of MT neurons in three monkeys. We obtained directional tuning data for 104 pairs; we gathered discrimination data for a subset of 46
pairs. All recordings admitted to our database conformed to two requirements: both neurons were well isolated for at least 10 repetitions per stimulus condition, and at least one of the neurons
yielded reliable, directionally selective responses to fully coherent random dot stimuli. For analyses involving CCG and ACG computations, we further restricted the database to pairs that satisfied
criteria for a minimum number of spikes (see Materials and Methods). For ease of reference and consistency checking, the numbers of cells and pairs qualified for the major analyses are summarized in
Figure 1 depicts a complete set of measurements for a representative pair of simultaneously recorded MT neurons. A and B are direction tuning curves for neurons 1 and 2, respectively. Both neurons
were directionally selective and exhibited similar preferred directions and tuning bandwidths. C and D depict responses of the same two neurons as a function of motion coherence for both the
preferred and null directions of motion. For these measurements, the preferred direction was set to 90°, approximating the optimal directions of both neurons. Off-line analysis of the data in A and B
revealed the preferred directions to be 58° and 82° for neurons 1 and 2, respectively (see Materials and Methods). In C and D, the firing rates of both neurons increased roughly linearly with motion
coherence in the preferred direction and decreased linearly with motion coherence in the null direction, a typical pattern for MT neurons (Britten et al., 1993).
Using the direction tuning data for each pair of neurons, we assessed the strength of two distinct types of correlation, that of the mean responses and that of the variations about the mean. The
former, commonly known as signal correlation, measures the similarity of tuning curves for a pair of neurons and was computed here as the correlation coefficient, r[signal], between the sets of data
points from the direction tuning curves. For the curves in Figure 1, A and B, r[signal] was 0.88, indicating a high degree of match. The distribution of r[signal] for all pairs (Fig.2A) was
comparable to that of a more conventional but less general metric, the difference between preferred directions, Δ[PD], shown for comparison in Figure 2B. The dominant modes in both distributions,
i.e., high r[signal] and low Δ[PD], indicate that adjacent neurons in our study tended to have similar direction tuning, consistent with the known columnar organization of MT (Albright et al., 1984;
DeAngelis and Newsome, 1999).
The second type of correlation is assessed not from the mean responses for all stimuli but from the trial-to-trial fluctuations (evidenced by the error bars in Fig. 1) around the mean response for
each stimulus condition. This interneuronal correlation has therefore been dubbed noise correlation (Gawne and Richmond, 1993;Gawne et al., 1996; Lee et al., 1998). Noise correlation, or r[noise], is
typically estimated by computing the correlation coefficient, r[SC], between the number of spikes generated by one of the neurons and the number of spikes generated by the second, simultaneously
recorded neuron for a set of nominally identical stimuli. However, we developed a lower-variance estimator for r[noise] (introduced and described in detail in the next section of Results) and have
plotted those estimates against the values of r[signal] for all pairs in Figure2C (the marginal distribution of r[noise] is shown in D). The pairs appear to fall into two general groups in C that are
not apparent from the marginal distributions alone. One group consists of pairs with very similar direction tuning curves (i.e., high r[signal] values) and positive noise correlation. A second group
consists of pairs with low or negative signal correlation and noise correlation near zero. Overall, the correlation coefficient between r[signal] and r[noise] is 0.61 (p < 10^−6; n = 103; direction
tuning data).
The correlation of r[noise] with r[signal] is consistent with the notion that shared common input endows nearby neurons with similar tuning properties and makes them subject to similar noise sources.
This observation is not unique to our data set, but it allowed us to focus our investigation of interneuronal correlation, when appropriate, on the cluster of neurons associated with non-zero r
[noise] values. We will use Δ[PD] < 90° as a criterion for making this separation.
The time scale of interneuronal correlation
In this section, we determine the time scale at which interneuronal correlation arises. We will quantify fluctuations in the neuronal response at time scales much slower and faster than the
psychophysical trial and will show that the magnitude of r[noise] for our MT pairs can be accounted for by the central peaks in their spike train CCGs on the order of 10s of milliseconds wide.
Short-term and long-term correlation
Since the earliest attempts to estimate r[noise] in visual cortex, it has been recognized that slow processes could play an important role in determining its magnitude (van Kan et al., 1985; Bach and
Krüger, 1986). Changes in neuronal excitation caused by motivational or attentional factors or fatigue could create a correlation at a time scale of anywhere between seconds and many minutes across a
large population of neurons. On the other hand, common synaptic input to multiple neurons that operates on a millisecond time scale would also contribute to interneuronal correlation but across a
smaller population of neurons sharing similar tuning properties. Because knowing the time scale of correlation may shed light on its origin and on its effect on pooled signals, our first goal was to
determine to what extent long-term correlation was present and to calculate the remaining short-term component of r[noise] once any long-term fluctuation of the firing rates was factored out.
Assessing the presence of slow covariations in firing rate is also important because such covariation, when combined with faster stimulus-locked modulation, can lead to narrow CCG peaks that may be
misinterpreted as evidence for fast synchronization (Brody, 1998,1999). To tackle the problem of estimating slow changes in neuronal excitation for data collected in discrete epochs, i.e., trials, we
developed a method called the TCC.
The TCC is a spike-count (as opposed to spike-train) -based cross-covariance that operates on the deviations from the expected responses (instead of the actual responses) for the two neurons, given
the stimulus. Figure 3 outlines the TCC computation for two pairs of neurons, one for which correlation was predominantly long term, exceeding the duration of the 2 sec trial (left column), and a
second for which correlation was predominantly short term (right column). The top panels (A and D) show for the individual neurons the z -score-normalized spike counts (see Materials and Methods) for
trials in the order in which they occurred in the discrimination experiments. These traces estimate the levels of relative responsiveness of the neurons throughout the experiment. Beneath them, their
auto-covariance functions, TAC(φ), are shown side-by-side (B and E; φ has units of experimental trials, typically 5 sec per trial). Only the left or right halves of the TACs are shown (the functions
are symmetrical about the origin), and the unity values at the origin are omitted. The gradual rise to a positive value around the origin, which was typical for our neurons, indicated that responses,
or more precisely, response deviations from the mean, on any particular trial were correlated to those on earlier trials. For the two example pairs, the cross-covariance functions, TCC(φ), for the
data in the top panels are shown at the bottom (C and F). TCC(0) is the traditional interneuronal correlation coefficient for spike counts, r[SC] (or more generally r[noise]), whereas TCC(φ ≠ 0) is a
generalization of r[SC] to responses occurring φ trials apart. In Figure 3C the broad central rise in the TCC indicates that the positive correlation on simultaneous trials (TCC(0) = 0.1; indicated
by the circled dot) is related to a correlated drift in the activities of the neurons on a time scale longer than one trial. The example in F shows an entirely different outcome. Namely, the positive
correlation coefficient for simultaneous trials does not extend to neighboring trials, despite the slow drifts in responsiveness of the two individual neurons evident from positive values near the
origin in their TACs.
The TCC provides a framework for estimating long- and short-term components of r[noise], which is represented at TCC(0). Long-term correlation, r[LT], is the value of the TCC around, but not at,
zero. We estimated r[LT] by replacing the value at zero with the average of its neighbors (at lag ±1 trial), convolving with a Gaussian of SD four trials, and reading off the new value at zero (very
similar results held for Gaussian SD two or eight trials). The traces from which r[LT] was measured are shown as smooth curves superimposed on the raw TCCs (which still have their central values
intact) in Figure 3, C and F. For the neuronal pair in C, r[LT] was nearly the same as the raw r[noise] value (the circled point is near the smooth line at lag 0), whereas in F, r[LT] is close to
zero and does not account for the value of r[noise]. We used the same method (replacing the center and smoothing) to compute the long-term component of the auto-covariance, r[AC], from the two-sided,
symmetrical forms of the TACs (Fig. 3B, E, arrows mark values).
To estimate the short-term component of r[noise], we removed the slow changes in responsiveness underlying r[LT] by applying an ideal high-pass filter to the z -scored spike counts. The filter's
cutoff frequency, 0.1 trial^−1 (cutoff period 10 trials), was chosen to be faster than the mean time scale of slow changes in excitability observed in the TACs. The filtered data were subsequently
renormalized to z -scores and used to compute a TCC (denoted TCC[hp]) whose zero-lag value was our estimate of short-term correlation, i.e., r[ST] = TCC[hp](0). Figure 4 depicts the TCC and TCC[hp]
(A and B, respectively) for a pair of neurons that had substantial long- and short-term correlation. The long-term correlation was no longer visible in TCC[hp], but a narrow, central peak remained. A
simpler approach to computing r[ST] is to subtract r[LT] from r[SC], i.e., from TCC(0). However, this may yield less accurate results for many neurons because it is not in general correct to assume
that r[ST] and r[LT] are additive.
Figure 4, C and D, shows database averages for our estimates of long- and short-term correlation. Separate averages are shown for pairs with Δ[PD] < 90° (black bars) and pairs with Δ[PD] ≥ 90° or in
which one neuron was not directional (white bars). A distinction between coherence series data (C) and direction tuning data (D) was maintained because we collected fewer total trials (typically 80)
and had more pairs for direction tuning experiments (n = 104) than for discrimination experiments (at least 210 trials; n = 48). The database averages led to three significant observations. First,
the average long-term auto-covariance, r[AC], was positive (gray bars; averaged across all individual cells), indicating that responses of single cells were correlated on a time scale longer than the
single trial. For coherence series data, the mean long-term auto-covariance was 0.14 (SD 0.12; n = 86), only 4 of 86 cells had negative values, and the average TAC peak width at half-height was 48
trials (SD 51), corresponding to no less than 4 min. Second, however, the long-term cross-correlation, r[LT], was on average no different from zero (t test; p = 0.39; coherence series data). For the
coherence series data, the distribution of r[LT] was roughly Gaussian with mean 0.01 (SD 0.07; n = 48). Third, r[ST] accounted for roughly the entire magnitude of r[noise] for pairs in which both
neurons were directional and had Δ[PD] < 90. For other pairs, r[ST] was not on average significantly different from zero, consistent with Figure 2C.
These results have the potentially counterintuitive implication that two neurons have responses that are correlated with their own responses on later trials and with each other's responses on
simultaneous trials but not with each other's responses on later trials. In other words, long-term auto-correlation and short-term cross-correlation exist in the absence of long-term
cross-correlation. This situation could arise if the sources of variance that caused the long-term auto-correlation in the responses of the individual neurons were independent from each other and
from the source of variance that caused the short-term cross-correlation. That long- and short-term correlation arise from independent mechanisms would not be surprising, because they operate on time
scales separated by four orders of magnitude, i.e., several minutes (shown above) versus 10s of milliseconds (shown in the next section).
In summary, slow drifts in the response strength of individual neurons were present (r[AC] > 0) but on average uncorrelated (r[LT] ≈ 0) between pairs of neurons in MT, and therefore did not
contribute significantly to the magnitude of interneuronal correlation across our database. Thus, r[noise] was accounted for by the short-term component of correlation alone and must arise on a time
scale no longer than the behavioral trial.
Spike train auto- and cross-correlograms
The positive value of r[noise] (∼0.21) associated with the cluster of points on the right side of Figure2C did not result from long-term correlation, so we now test for its relationship to faster
sources of correlation, the presence of which is revealed by spike train auto- and cross-correlograms. Examining ACGs as well as CCGs is important because ACGs bear on the interpretation of a CCG and
because both are required for a mathematical result that we will use below to derive a new metric for r[noise]. In this section, we establish, consistent with a body of cross-correlation studies,
that correlation is largely limited in time to a small, central region of the ACGs and CCG and show that for our MT pairs there is a strong empirical relationship between that central region of the
CCG and the traditional measure of spike count correlation, r[SC].
We computed the average spike train ACG for each individual neuron and the average CCG for each pair of neurons as described in Materials and Methods. Plots for one pair of neurons are shown on the
leftin Figure 5, and database summaries appear on the right. On the left, the ACGs (A) and CCG (B) are plotted in excess of the ensemble shift-predictor (see Materials and Methods) and are encased in
lines showing ±3 SD of the noise (estimated from the tails of the plots for lags from 400 to 800 msec). The ACG for neuron 1 (Fig.5A, top trace, shifted vertically for visibility) has a dip near the
origin, indicating that the likelihood of a spike occurring within 5 msec of another is lower than expected if spikes were fired independently of each other. This period of anti-correlation in the
ACG is followed by a period of positive correlation from 7 until ∼80 msec after a spike. Periods of both correlation and anti-correlation appeared in the ACG for neuron 2 as well (Fig.5A, bottom
trace). In addition, neuron 2 tended to fire pairs, or bursts, of spikes; however, this is not evident in the ACG plotted here because the positive values, at lags 2 and 3 msec, lay above the upper
vertical limit of the plot and are not shown. The average CCG (Fig. 5B) for this pair of neurons had a central, somewhat asymmetric peak that did not extend beyond 100 msec from the origin.
Across our database, ACG shapes were diverse and varied in the presence and size of (1) a narrow central peak associated with short bursts, (2) a dip associated with a 1–3 msec absolute refractory
period that was sometimes extended by a longer relative refractory or integration period (Abeles, 1982), and (3) a broader peak of positive correlation. The CCGs had mainly single, central peaks that
varied in size, shape, and symmetry. Peak shapes were consistent with common synaptic input more so than with serial coupling (Moore et al., 1970). The shapes of our ACGs and CCGs were not consistent
with the oscillatory Gabor functions that Kreiter and Singer (1996) used to describe CCGs in MT. In particular, we did not observe rounded central peaks flanked by similar but damped side-lobes.
We did not attempt a systematic classification of the subtleties of correlogram shapes, which would have required more data than we were able to collect for many of the pairs, but characterized only
the extent in time of the correlation. This was accomplished for both correlation and anti-correlation by computing at each millisecond time lag the fraction of cells that had correlation >3 SDs
above the ensemble shift-predictor and the fraction that had anti-correlation <3 SDs below the shift-predictor. Correlograms were smoothed with a Gaussian of SD 2 msec before the test was applied to
avoid counting isolated points that exceeded the criterion (as observed frequently in Fig. 5A,B). The results for the ACGs (Fig. 5C) revealed that significant response correlation for individual
neurons was confined almost entirely to time lags <100 msec, was most prevalent around 30 msec, and decreased at shorter times because of the presence of anti-correlation associated with “non-burst”
firing patterns or inter-burst intervals [described by Bair et al. (1994) for a comparable MT data set]. The extent of correlation in the CCGs is summarized in Figure 5D and, similar to that in the
ACGs, was almost entirely confined to within 100 msec of the origin.
Two points deserve emphasis regarding these results. First, our analysis does not preclude weaker, yet significant, correlation that extends beyond 100 msec; it simply indicates that strong
correlation, i.e., that which caused 3 SD differences between the correlograms and ensemble shift-predictors, was common at time scales on the order of 10s of milliseconds but was rare beyond 100
msec. Weaker, long-term sources of correlation certainly exist in MT but are not likely to contribute substantially to r[noise]. Second, the time scale of correlation in our ACGs and CCGs is
intrinsic to the visual system and does not result from temporal correlation in our stimulus because the signal strength (amount of preferred motion) in our dynamic dot stimulus was uncorrelated in
time. In particular, the number of signal dots in any epoch (or in one video frame) was uncorrelated with that in any other epoch. The time scale of the correlation observed in Figure 5, C and D,
matches both the integration times for visual neurons upstream from MT (Hawken et al., 1996) and the temporal limits of motion perception for dynamic dot stimuli (Morgan and Ward, 1980). When
analysis was restricted to zero coherence stimuli (which were effectively white noise to beyond 1 kHz), we found the same time scale of correlation across our database; therefore, the 45 msec time
between signal dots in our stimulus was not responsible for the correlation observed here.
Having determined the typical time scale of correlation in our data, we may now apply a simple test to assess whether r[noise] estimated in the traditional manner from the spike count for the entire
trial is related to the central peak in the CCG. In Figure 6, the integral of CCG(τ) minus the ensemble shift-predictor (for τ = −32 to 32 msec) is plotted against r[ST] for our database (coherence
series data). There is a clear relationship between these two measures of correlation (overall, r = 0.76, p < 10^−6, n = 48; for pairs with Δ[PD] < 90°, r = 0.71, p = 0.00001, n = 29, filled circles;
for other pairs, r = 0.66, p = 0.002, n = 19, open circles). This may seem striking because r[ST] was derived from spike counts for the entire trial without information regarding the temporal
structure of the spike trains, whereas the CCG area is based on the interrelationship of spikes occurring within 32 msec of each other. The significant positive correlation between the two metrics
holds for limits of integration down to ±2 msec (r = 0.48) but does not grow much in the range from ±32 to ±128 msec (e.g., r = 0.80 at both ±64 and ±128 msec). The data indicate that pairs of
neurons with high spike count correlation also tend to have a substantial peak around the origin in their CCGs. This relationship is not given a priori (van Kan et al., 1985) and was not found in
other studies of visual cortex (Gawne and Richmond, 1993; Gawne et al., 1996), although it was hinted at by Bach and Krüger (1986).
Assessing r[noise] from the cross-correlogram
We will now make a more rigorous connection between r[noise] and the area under the CCG by defining a metric based on the CCG that estimates exactly the value of r[noise] under the condition that
correlation has a limited time scale. Our approach derives from the fact that the equation for r[SC] (the well known Pearson's correlation coefficient) can be rewritten in a form that is based solely
on the areas under the spike train CCG and ACGs as follows (from Appendix , Eq. 26): Equation 9where the areas are integrated across all lag times in the correlograms. However, if correlation is
limited to short time lags, as suggested by results from the previous section, only those regions near the origin will contribute to non-zero areas in Equation 9. The flanks of the CCG and ACGs,
which approach the shift-predictors, will contribute on average nothing but noise. We therefore propose the use of a metric, r[CCG](τ) (defined in Appendix , Eq. 27), which estimates r[noise] by
integrating only a limited central region (from −τ to τ msec) of the CCG and ACGs. This metric eliminates the noise that would be contributed by the flanks of the correlograms by simply not including
the flanks in the integration. In essence, it assumes that the correlograms beyond ±τ are on average equal to the shift-predictors.
Before applying the r[CCG](τ) metric to our MT data, we tested it on pairs of simulated spike trains that had a central, Gaussian-shaped CCG peak (SD 4 msec) and an r[noise] value of exactly 0.2. For
the simulated data, all of the area in the CCG (and ACGs) was concentrated near the center, and the expected value of the flanks (when the shift-predictor was subtracted) was known to be zero. Figure
7A shows r[CCG](τ) plotted for 10 sets of simulated spike trains (details of the simulation are given in the Figure legend). As τ increased, the average value of r[CCG]increased until it reflected
the true value, 0.2. A plateau occurred when τ exceeded the time scale of the correlation, and further increases in τ caused a loss of precision as noise from the ACG and CCG flanks was integrated.
When τ reached the full trial duration (here 1700 msec), r[CCG] became equivalent to r[SC], according to Equation 28. This simulation shows vividly how noise from the tails of the ACGs and CCG
corrupts r[SC], and it demonstrates that a more accurate estimate of r[noise] can be obtained with r[CCG](τ) when τ is shorter than the trial duration (but longer than the time scale of correlation).
We plotted r[CCG](τ) for our neuronal pairs and found a similar pattern of results. Curves for one pair are shown in Figure 7B for 11 coherence levels (from 100% preferred to 25.6% null direction
motion, which satisfied the minimum data requirements stated in Materials and Methods). The curves increased together to r ≈ 0.16 as τ approached 30–40 msec but then diverged as τ grew larger. This
was consistent with the CCGs (data not shown), which had central peaks that fell to the level of the shift-predictor at ∼30–40 msec from the origin. The direction of divergence of curves such as
those in Figure 7B typically did not depend on the stimulus condition (a systematic analysis is given in the next section), so we averaged across conditions to get an r[CCG](τ) curve for each pair,
and we averaged across pairs to get one database curve. The database curve for pairs having Δ[PD] < 90° (Fig. 7C, filled circles) approached an asymptote of ∼0.21 for values of τ above 32–64 msec.
The value 0.21 was the same as that for short-term correlation for this database (Fig. 4C, right-hand bar), and the timing of the approach to the asymptote was consistent with the time scale of
correlation observed in the ACGs and CCGs (Fig. 5C,D). Figure 7C also shows the SD for the r[CCG] estimate (open circles, averaged across the same set of curves used to compute the mean). The SD grew
with increasing τ even after the mean of r[CCG](τ) had leveled off. This shows the inefficiency of a long integration time such as that associated with the r[SC] metric (i.e., the entire trial
duration). Finally, a direct comparison of r[SC]with r[CCG](32) for individual pairs is provided in Figure 7D. The SD was always smaller for r[CCG] (thick lines) than for r[SC] (thin lines). Two
points are labeled, one for the pair from B (emu005) and another (emu080) from Figure 3C, that had a large long-term component of r[SC]. For the latter, r[CCG](32) is much less than r[SC] because r
[CCG](32) discounts long-term correlation. It does so by integrating area over only 1.9% (32 msec/1700 msec) of the CCG and therefore captures only 1.9% of the excess area that a source of long-term
correlation spreads evenly across a CCG.
Clearly, r[CCG] provided a more repeatable (less noisy) estimate of interneuronal correlation (for τ < T) than did r[SC], but we wanted to verify that it also maintained the relationships that r[SC]
had with the measures for similarity of neuronal tuning mentioned above, namely, r[signal]and Δ[PD]. Compared with r[SC], r[CCG](τ) was more positively correlated with r[signal] (Pearson's r = 0.59,
rather than 0.53, for both τ = 32 and 64 msec; n = 46) and was more negatively correlated with the logarithm of Δ[PD] (Pearson's r = −0.47, rather than −0.36, for both τ = 32 and 64 msec; n = 34,
where the logarithm was taken to correct the skew of the distribution in Fig. 2B).
In summary, it appears that r[CCG](τ) accurately captures the amount of interneuronal correlation for our pairs. That it does so for τ as small as 32 msec shows that most of the correlation observed
at the time scale of the behavioral epoch can be accounted for by CCG peaks at a time scale nearly two orders of magnitude shorter. Therefore, mechanisms underlying narrow, central CCG peaks affect
response properties relevant to both temporal and rate coding.
Dependence of correlation on stimulus and behavior
Assessing the dependence of correlation on stimulus parameters is necessary to justify averaging r[noise] values and CCGs across stimulus conditions as we have done. In addition, this assessment is
important with respect to both stimulus and behavioral parameters because of the potential link between correlation, or synchrony, and the perception of the animal as reflected by its behavior. Here
we examine how correlation changes with the firing rates of the neurons, the direction and coherence of stimulus motion, and the presence of the stimulus, and we test whether synchronous activity
exerts extra influence on the monkey's decision and whether it varies from passive fixation to active discrimination.
Correlation versus firing rate, direction, and motion coherence
Because firing rate varied as our stimulus parameters changed, we first established that our correlation metrics did not show a substantial dependence on firing rate before testing for more
interesting relationships between interneuronal correlation and other variables. Figure 8, A and C, shows scatter plots of the area under the CCG peak (from −32 to 32 msec) and r[CCG](32) versus
geometric mean spike rate for each coherence level for the 29 directional pairs with Δ[PD] < 90°. Firing rate was not significantly correlated with CCG area and showed only a weak relationship with r
[CCG] (see Figure legend for details). A pair-by-pair analysis also revealed no overall trend, although several individual pairs showed significant relationships (see Fig. 8 legend). Similar results
held for data from the direction tuning experiments, for integration times ranging from several to hundreds of milliseconds, and when all pairs were included in the analysis.
The same two correlation metrics were largely constant when plotted against stimulus direction and coherence, except at 100% coherence where both measures were lower (B and D show CCG area and r
[CCG], respectively, Fig. 8). The numbers of individual pairs for which these metrics were significantly correlated with coherence were almost identical to those for spike rate. The drop in
correlation strength at 100% coherence can be related to the nature of MT responses to coherent and incoherent motion. MT neurons typically show clear stimulus-locked modulation for stimuli of <100%
coherence, but at 100% coherence there is little or no such modulation (Bair and Koch, 1996). How this modulation impacts our measures of correlation is the subject of the last section of Results.
Whether the reduction in r[noise] at 100% coherence is also related to a previous report that correlation is almost completely abolished during high contrast motion in MT (Cardoso de Oliveira et al.,
1997) is discussed in the next section.
The consistency of r[noise] in the face of large changes in firing rate indicates that the underlying mechanism did not act additively to alter neuronal firing rates, for if it did, r[noise] would be
larger at lower firing rates. In the absence of substantial overall relationships between our correlation metrics and the stimulus direction and coherence or the firing rate, we chose to average
these metrics across all stimulus conditions. The observed decrease at 100% coherence had little influence on our statistics because <10% of our coherence series data was collected at c = 100%.
Correlation during spontaneous and stimulus-driven activity
We tested the dependence of correlation on the presence of the stimulus by computing r[noise] for a 330 msec epoch of spontaneous activity and for an equal duration epoch of stimulus-driven activity.
The spontaneous epoch began when the monkey acquired fixation and ended 30 msec after stimulus onset, precluding the arrival of stimulus-driven activity in MT (Raiguel et al., 1999). The driven epoch
began 30 msec after stimulus onset. We limited analysis to pairs that had at least four stimulus conditions each having at least 10 trials with at least one spike per trial per cell during the 330
msec period. The value of r[CCG](32) for the spontaneous epoch was significantly correlated with that for the driven epoch (r = 0.63; p = 0.00001; n = 40), and the average difference between the
values for spontaneous and driven activity, 0.018 (SD 0.14), was not significantly different from zero. Limiting the analysis to directional pairs with Δ[PD] < 90° gave nearly identical results.
Similar results were found when (1) r[ST], computed from the TCC, was substituted for r[CCG](32), or (2) the driven epoch was defined to be the entire stimulus epoch, rather than just the first 330
msec. We conclude that noise correlation during spontaneous activity is similar to and predictive of the noise correlation during activity evoked by our random dot stimuli.
This result stands in striking contrast to the report of Cardoso de Oliveira et al. (1997) that interneuronal correlation in MT is present during spontaneous activity but is practically abolished
during visual stimulation. To determine whether our correlation values were more similar to their values for spontaneous or for driven activity, we normalized our CCGs according to their methods
(after dividing by the geometric mean spike rate, we used a three-point boxcar function to smooth the CCGs and then found the peak within ±100 msec of zero) and computed peak height, position, and
width statistics like those presented in their Figure 5. All three measures from our data were well matched to their results for spontaneous activity, indicating that our results differ only during
visual stimulation. If we assume that high-contrast, coherently moving stimuli reduce correlation strength between responses of nearby MT neurons, then it remains to be determined why our strongest
stimulus (100% coherence motion) caused a decrease in correlation strength that was small compared with the decrease caused by the square-wave grating of Cardoso de Oliveira et al. (1997).
Does correlation change with behavior?
Investigators have hypothesized that synchronous firing among cortical neurons underlies various coding or processing functions (for review, see Singer and Gray, 1995; Roelfsema, 1998). Our data
provide the opportunity to determine whether synchrony among adjacent MT neurons is correlated with perceptual choice or behavioral state.
The relation of synchrony to perceptual choice is best assessed at low motion coherence where the monkey correctly identifies the direction of motion on some trials but makes mistakes on others. The
psychometric function in Figure 9A (thick line, filled circles) plots the monkey's performance on trials for which the stimulus was optimized for the pair of neurons the tuning curves of which are
shown in Figure 1. We asked whether synchrony was stronger on trials in which the animal chose the direction preferred by the pair of neurons, as might be expected if synchronously active neurons
exert stronger effects on downstream decision circuitry. To test this, we divided the trials for each stimulus condition (i.e., for a particular coherence level and direction) into two groups, one in
which the animal chose the preferred direction (for the pair) and one in which the animal chose the null direction. Note that one group corresponds to correct decisions, whereas the other corresponds
to incorrect decisions (where the correspondence depends on whether the direction of motion was null or preferred for the stimulus condition) except at zero coherence where there was no “correct”
We considered only stimulus conditions that had at least 10 trials with preferred responses and 10 trials with null responses; therefore, 51.2 and 100% coherence conditions were rarely included
because the monkey rarely made 10 mistakes for such salient stimuli. This limited the number of pairs for this analysis from 46 to 35. Figure 9B shows CCGs for preferred and null decision trials for
the same pair of neurons illustrated in Figures 1 and 9A. The CCGs appear virtually identical, which was typical for our data set. Figure 9, C and D, depicts quantitative measurements of the area
under the CCG from −32 to +32 msec (C) and from −2 to +2 msec (D) for preferred and null decision trials for 137 stimulus conditions from the 35 pairs of neurons. In both panels, the points cluster
around the unity diagonals, showing that synchronous firing did not differ between the two decision states (paired t test; p = 0.75 for C, p = 0.94 for D). This result also held for the r[CCG]
metric, for all integration times tested (from ±2 to ±128 msec), and when only directional pairs were tested.
We also analyzed synchrony simply as a function of motion coherence, regardless of perceptual choice. At low coherence, the dot patterns appear to be a white noise stimulus and elicit no global
motion percept. As coherence increases, however, observers perceive the entire stimulus to drift in the specified direction as though the disparate motion signals provided by individual dot pairs are
bound into a perceptually coherent whole. Theories of perceptual binding that postulate a unique role for synchronous neural activity might predict that synchrony should be stronger for coherent (c =
100%) than for incoherent dot patterns (c = 0%). However, we have already seen that the opposite is true (Fig.8B,D).
Finally, we compared CCGs obtained during passive fixation (direction tuning experiments) with those obtained during active discrimination to determine whether the overall behavioral state of the
animal was correlated with neural synchrony. In the subset of experiments in which both blocks of data were obtained, the area under the CCG did not differ systematically between the two states
(paired t test; t = −0.06; p = 0.95; n = 46), and the measurements were highly correlated between the two states (r = 0.90; p < 10^−6). In short, we found no evidence that synchronous firing varied
systematically as a function of perceptual decision or behavioral state.
Do sensitive or informative neurons cluster?
For each experiment in which we obtained psychophysical data, we used analytic methods based on signal detection theory [see Materials and Methods, or see Britten et al. (1992) for detailed methods]
to compare the directional sensitivity of each neuron with the monkey's psychophysical sensitivity. Figure 9A illustrates the outcome of this analysis for the data depicted earlier in Figure 1, C and
D. The filled circles represent the psychophysical performance of the monkey on the direction discrimination task, which increased from nearly chance at low coherence levels to perfection at the
three highest levels. Psychophysical threshold, defined as the motion coherence that supported 82% correct performance, was 4.3% coherence. The×'s and squares indicate the performance of the two MT
neurons measured on the same trials represented in the psychometric curve. Neuron 2 was as sensitive to the directional signals as was the monkey psychophysically, yielding a neurometric threshold of
4.7% coherence. Close correspondence between neuronal and psychophysical thresholds is common in MT (Britten et al., 1992). In contrast, neuron 1 was considerably less sensitive to motion signals in
the displays, yielding a threshold of 13.8% coherence (sensitivity = 1/threshold). Across 72 directional neurons studied in 41 discrimination experiments, the geometric mean ratio of neuronal to
psychophysical threshold was 1.72 (range, 0.27–11.5), a value higher than those previously observed in this laboratory (Newsome et al., 1989; Britten et al., 1992; Celebrini and Newsome, 1995). The
discrepancy arises because the inclusion criterion for direction selectivity was less stern in the current study to maximize the number of pairs. Interestingly, neither neuronal thresholds nor choice
probabilities [defined in Britten et al. (1996)] were significantly correlated between adjacent MT neurons in our sample. Thus we find no evidence for clustering of neurons that are particularly
sensitive to the stimulus or that have particularly close relationships to behavior.
Controls for stimulus variance–replicate stimuli
Our estimates of interneuronal correlation have been based on responses to ensembles of stochastic stimuli in which the random detail of the dot patterns differed from repeat to repeat within a
particular stimulus category. In principle, such sets of nonidentical stimuli could inflate r[noise] estimates and increase CCG peak sizes if the responses of the neurons were influenced by the
random variation across stimuli. For example, if 15 of 30 stimuli that were generated at 6.4% coherence had by chance slightly more motion in the preferred direction than the other 15 stimuli, an
ideal pair of neurons with no common noise source but having identical direction preferences would tend to fire on average more for the former than for the latter 15 stimuli. This would yield an
erroneous positive value of r[noise], which should otherwise be zero. Below we describe direct experimental controls as well as simulations that allow us to estimate the magnitude of this effect in
our data.
In our experiments, random variation from stimulus to stimulus was necessary to prevent the monkeys from associating particular spatial patterns with a reward. However, for four pairs of neurons we
interleaved experiments using replicate stimuli in which the dot patterns for a particular stimulus condition were identical (see Materials and Methods). Estimates of r[noise] for the four controls
using both the r[SC] and r[CCG](32) metrics are presented in Figure10. The values of r[SC] (A) offered no evidence that interneuronal correlation was greater for stochastic stimuli (white bars) than
for replicate stimuli (black bars), but the lower-variance estimates provided by r[CCG](32) (B) painted a clearer picture. For pairs emu034 and emu035, r[CCG](32) was higher for stochastic stimuli
than for replicate stimuli (p = 0.08 and p = 0.00001 respectively, t tests). For the other two control pairs, the difference was negligible.
An examination of the PSTHs, CCGs, and shift-predictors for emu035 (the pair that had a significant decline in r[CCG] for replicate stimuli) reveals how stimulus-locked modulation can inflate r
[noise]. For replicate stimuli, the stimulus-locked modulation of firing rate is captured in the PSTHs (Fig. 11A,B, thin lines, neurons 1 and 2, respectively), but when the stimulus varies from one
repeat to the next (ensemble stimuli), the modulation is washed out (A, B, thick lines). The difference in the PSTHs carries over into the CCG shift-predictors because shift-predictors are closely
related to the cross-correlation of the PSTHs [see Eq. 8 in Materials and Methods, Eq. 15 in Appendix , andPerkel et al. (1967b)]. The ensemble shift-predictor is flat (Fig. 11C, thick line),
whereas the shift-predictor for replicate stimuli has a peak (line withdots). The peak indicates that the stimulus-locked modulation in neuron 1 and 2 PSTHs (A, B, thin lines) was correlated. The
difference in area between the CCG (C,thin line) and the two shift-predictors accounts for the difference in r[CCG] plotted in Figure10B for this pair (emu035). In summary, an ensemble
shift-predictor fails to capture correlated stimulus-locked modulation, so subtracting it from the raw CCG yields an overestimate of the correlation if correlated stimulus-locked modulation existed
in the first place. Thus, when there is little stimulus-locked modulation (as was the case for rt068 and rt072in Fig. 10B) or when modulation is present but largely uncorrelated (e.g., emu034), using
an ensemble shift-predictor is acceptable. But for emu035, it caused an overestimate of the CCG peak area and of r[noise].
One method for estimating the inflation of r[noise] caused by stochastic stimuli across our database is to compare results for 0 and 100% coherence stimuli. Such a comparison is useful because there
is little or no stimulus-locked response modulation for c = 100% stimuli, whereas modulation is strong at c = 0% (Bair and Koch, 1996). For the 20 pairs that we tested at both c = 0 and 100% and that
consisted of two directional neurons with Δ[PD] < 90°, r[CCG] was on average 0.20 (SD 0.13) at 0% coherence and 0.17 (SD 0.13) at 0%. This 15% decrease is consistent with our hypothesis but was not
statistically significant (paired t test; t = 1.45; p = 0.16). A similar, but unpaired, comparison can be made from the plot of r[CCG] in Figure8D, which shows a 27% reduction from c = 0 to c = 100%
(preferred direction only). Again, this change was not statistically significant (t test; t = 1.48; p = 0.16). A broader unpaired comparison between all data from discrimination experiments (the vast
majority of which was collected at low coherence) and all data from direction tuning experiments (where c = 100%) for pairs with Δ[PD] < 90° showed only an 8% decline in r[CCG](32) for the direction
tuning data set. These results suggest that inflation of r[noise]caused by stochastic stimuli is modest across our database.
Finally, we used a simulation to estimate the inflation of r[noise] caused by the modulated drive resulting from stochastic stimuli. The stimulus drive consisted of randomly occurring bursts of
stimulation that simulated those caused by the random occurrence of coherent dots in our motion stimulus. Parameters for the strength and frequency of occurrence of the random bursts determined the
amount of trial-to-trial variability and thus the value of r[noise]. The details of the simulation and a solution for r[noise] for all parameter values are given in Appendix . An example of the
drive provided by a simulated stimulus during a 1 sec epoch from one trial is shown in Figure12A. The tracerepresents the PSTH for both neurons, which are defined to be identical. For a set of trials
governed by the same statistics that generated the trace in A (see legend for parameters), the expected value of r[noise] is 0.04. For a simulation with stronger modulation (B), the expected value of
r[noise] is higher, 0.24. Figure 12D plots the value of r[noise] for a wide range of parameter combinations and shows (with white dots) the parameters used to generate traces for the examples just
described. A comparison of the PSTH for a simulated pair of neurons (B) with the measured PSTHs (C) for the pair of neurons from Figure 11 reveals a critical difference: the neuronal PSTHs are not
identical. This was true although this pair of neurons was as closely matched in preferred direction and bandwidth of direction tuning as any in our database (Δ[PD] = 9°; r[signal] = 0.97). Because
nearby neurons have responses that differ in fine detail (DeAngelis et al., 1999), our simulation provides an upper bound on the strength of correlation induced by stochastic stimuli. Furthermore,
gauged by responses to replicate stimuli here and in a previous study of MT (Bair and Koch, 1996), the strength of modulation in Figure 12A appears typical or above average, whereas that in B
represents an upper limit to what has been observed. Therefore, our simulations suggest that stochastic stimuli are not likely to inflate r[noise] by more than ∼0.04 units on average.
In summary, stochastic stimuli probably inflate our estimates of r[noise] but cannot be responsible for more than a small fraction of the correlation that we measured. Experimental controls,
simulations, and comparisons of incoherent to coherent stimuli suggest that this inflation is likely to range from negligible to at most 20% of our average r[noise]estimates.
Response variance caused by eye movements
One final potential source of error in our estimate of r[noise] is the movement of the monkey's eyes. Small saccades executed during fixation could cause correlated signals in neurons with similar
direction preferences. The potential strength of this effect has been estimated from the influence of eye movements on single-unit MT data (Bair and O'Keefe, 1998), and it was concluded that
fixational saccades are too brief and typically too infrequent to create substantial correlation except when occurring on a background of very low firing rate. We found no indication that r[noise]
was higher at lower firing rates (Fig.8C) and believe that eye movements did not substantially affect estimates of correlation strength in this study.
We have investigated the time scale at which interneuronal correlation arises for pairs of nearby cortical neurons and have explored the relationship between interneuronal correlation and behavioral
and stimulus parameters in area MT.
We found that synchrony, revealed by CCG peaks, was closely linked to correlated variability, r[noise], at the time scale of the trial. In principle, these two phenomena need not be related (van Kan
et al., 1985), but several observations showed that they were related for our MT pairs. First, the predominant time scale of interneuronal correlation was on the order of 10–100 msec, consistent with
numerous cross-correlation studies throughout the visual system of both cat and monkey (Mastronarde, 1983b; Michalski et al., 1983; Ts'o et al., 1986; Krüger and Aiple, 1988;Nelson et al., 1992;
Cardoso de Oliveira et al., 1997) and in auditory cortex (Dickson and Gerstein, 1974; Abeles, 1982; Eggermont and Smith, 1996). Next, CCG peaks at this time scale (10–100 msec) were strikingly
predictive of r[noise] for the behavioral epoch. Although r[noise] is mathematically related to the total area under the CCG, such a result need not apply to the central CCG region alone. For
example, pairs could have had central CCG peaks that were canceled by negative side-lobes, or they could have had excess area distributed across the entire CCG. Neither of these are consistent with
our findings. Finally, slow drifts in the gain of single neuronal responses occurred but were not on average correlated between neurons and therefore had little impact on r[noise]. This result was
somewhat surprising because it has been suggested that long-term cross-correlation is common for neurons in primary visual cortex (Bach and Krüger, 1986). Also, because nearby cortical neurons share
a large fraction of their inputs, it is unclear how one cell can undergo gain changes that are independent from those of its neighbors. However, if mechanical instability of the electrode in the
tissue was the source of the long-term gain changes, it is conceivable that nearby neurons could be affected independently.
In the second part of this study, we found that synchronous activation in pairs of neurons was not related to the monkey's decision on the direction discrimination task and that synchrony was not
stronger for perceptually more salient or unified stimuli. Synchrony did not depend on whether the monkey was actively discriminating or passively fixating during stimulus presentation. Finally, the
strength of synchrony was similar with and without the stimulus, and it showed little systematic variation with firing rate. We are unable to corroborate reports that synchrony in MT changes with the
unity of the stimulus (Kreiter and Singer, 1996; Castelo-Branco et al., 2000) or is nearly abolished during visual stimulation (Cardoso de Oliveira et al., 1997). Experiments using more diverse
stimulus configurations will have to resolve these differences. Other studies have suggested that synchrony could signal behavioral events in frontal cortex (Vaadia et al., 1995), encode tone
frequency in auditory cortex (deCharms and Merzenich, 1996), indicate attentional selection in somatosensory cortex (Steinmetz et al., 2000), or be involved in arousal, attention, or learning in
sensorimotor cortex (Murthy and Fetz, 1996). In contrast, our results portray synchrony and correlation as relatively constant for a typical pair of MT neurons.
In the course of this analysis, we derived two metrics that are useful for determining the strength and time scale of correlation. The TCC provides a systematic way to extract short- and long-term
components of the traditional interneuronal correlation coefficient, r[SC], for trial-based data, whereas r[CCG](τ) offers an estimate of r[noise] with lower variance than r[SC] when the time scale
of correlation is shorter than the period during which spikes are counted. We believe that these techniques are potentially useful for comparing correlation across a wide range of data.
Other studies of r[signal],r[noise], and the CCG
Previous studies of visual cortex have examined r[noise], r[signal], and spike train CCGs (Gawne and Richmond, 1993; Gawne et al., 1996). They reported r^2values, interpreted as percentage of
explained variance, so we squared our r[noise] and r[signal]values (before averaging) for comparison. Their value of r[noise]^2, ∼5% for both inferotemporal cortex (IT) and primary visual cortex
(V1), was similar to our values: 4.5% for all pairs and 6.4% for directional pairs with Δ[PD] < 90°. They found r[signal]^2 to be 19% in IT and V1 using static, spatial (Walsh) patterns, but this
increased to 40% in V1 for conventional bar stimuli. The latter value was comparable to our mean, 48%, for MT. In spite of some similarity between our results, including the fact that over half of
their CCGs had significant peaks, they did not comment on the relationship between r[noise] and the CCG and concluded that the r[signal] and the CCG were unrelated (they found r[signal] to be lower
for pairs with CCG peaks in IT, but the result failed a significance test). This outcome is different from that depicted in our Figure 2C, which shows a clear relationship between r[signal] and r
[noise], where r[noise], being r[CCG], is a strong reflection of the CCG peak. It seems likely that a relationship like this must exist between r[signal] and the CCG in both IT and V1 because one
consistent feature of CCGs from diverse regions of cortex is that peaks are more common between nearby neurons, particularly within distances associated with cortical columns (Fetz et al., 1991).
Cortical columns are clusters of neurons with similar preferences, and such similarity is what r[signal], in principle, measures. Maybe differences in the number of cells tested or in the method of
estimating the strength of CCG peaks or r[signal]led to the differences between our results and those of Gawne and collaborators (Gawne and Richmond, 1993; Gawne et al., 1996). For example, the
relationship between two-dimensional Walsh patterns and the columnar structure in IT (Fujita et al., 1992; Tanaka, 1996) may be somehow fundamentally different than that between moving patterns and
direction columns in MT (Albright et al., 1984).
Consistent with our findings, Bach and Krüger (1986) noted that excess area in the CCG (±30 msec) was slightly larger for pairs of V1 neurons with strong common variability (i.e., r[noise]). Also,
for both motor and parietal cortex, Lee et al. (1998) found that r[signal] and r[noise] were higher for pairs with significant central CCG peaks. All of these results are consistent with the simple
notion that sources of common input arrive onto nearby neurons through one or more synapses and thereby create common noise, central peaks in CCGs, and similar tuning curves in pairs of neurons (
Shadlen and Newsome, 1998).
Stimulus variance
A major goal of the study from which the present paired MT data arose (Zohary et al., 1994) was to estimate accurately the strength of noise correlation for nearby MT neurons but to do so when those
neurons were generating signals that would underlie a psychophysical judgment made by the monkey. The latter constraint led to the use of stochastic stimuli to prevent the monkeys from associating
particular stimulus patterns with a reward. In principle, however, stochastic stimuli can bias estimates of r[noise] upward, as demonstrated by our simulations. We attempted to estimate this bias by
comparing responses for replicate and ensemble stimuli, by comparing c = 0% with c = 100% data, and by simulating the effect of stochastic stimuli on neuronal responses. The results suggested that
the actual r[noise] value for pairs with similar direction tuning was somewhat less than the measured value of 0.21, but probably not by >20%.
Implications for pooling
Interneuronal correlation places limits on the effectiveness of signal pooling (Johnson et al., 1973; for review, seeParker and Newsome, 1998). Our previous studies showed that the signal-to-noise
ratio (SNR) for a pooled signal was sensitive to even modest values of r[noise] (Zohary et al., 1994; Shadlen et al., 1996). We can now use our estimates of the time scale of interneuronal
correlation to understand how r[noise] and SNR change with the length of the time window, T, in which signals are pooled.
We simulated pools of spike trains with correlation on the time scale typical for MT (see Fig. 7A legend for methods) and computed the SNR as in Zohary et al. (1994). The SNR for the pooled signal is
the expected value, μ[Σ], of the sum of spikes from all neurons divided by the SD, ς[Σ], of that sum, i.e.: Equation 10where μ and ς are the mean and SD for spike count from a single neuron, and N is
the number of neurons in the pool. Our simulated data were Poisson, so ς^2 = μ and doubling T would increase the SNR by a factor of 2if r[noise] remained constant, but because correlation was spread
over time (Fig.13A, thick line), r[noise] was lower for shorter T (B,thick line). Thus the SNR (Eq. 10) was enhanced for larger pools of neurons at shorter integration times, as shown in C(thick
curves are squeezed upward in the bottom right corner; see legend for details).
Therefore, the time scale of correlation must be taken into account when signals are pooled in short time windows. This may be of relevance to the visual system, where it is likely that some
processes underlying visual discrimination operate with integration times from 10 to 100 msec (Oram and Perrett, 1992; Thorpe et al., 1996; Corthout et al., 1999). Here we have focused on one
particular pooling model that involves averaging across redundant signals (Zohary et al., 1994; Shadlen et al., 1996; Shadlen and Newsome, 1998). The ultimate role of interneuronal correlation in
computations underlying perceptual decisions will depend on details of the actual mechanisms that have yet to be worked out.
Here we derive an expression that relates the correlation coefficient of spike count, r[SC], to the area under the CCG and the ACGs for a set of paired spike trains. A similar relationship was noted
earlier by Haim Sompolinsky (personal communication of unpublished notes of 1992 entitled “Statistics of spike counts and spike trains in a stationary process,” pp 1–6), and recently Brody (1999) has
noted the relationship between spike count covariance and the area of the CCG, not involving the ACGs. On the basis of our derivation, we propose a metric, r[CCG](τ), which can provide a lower
variance estimate of r[noise] when interneuronal correlation is limited to a time scale shorter than the trial.
Spike trains from M trials for the two neurons are represented as discrete binary signals of period T at the millisecond resolution, i.e.: Equation 11where k = 1, 2 and 1 ≤ t ≤ T and 1 ≤ i ≤ M. The
spike counts for the i^th trial are: Equation 12and the post-stimulus time histograms are: Equation 13The spike train auto-correlation and cross-correlation functions are defined as: Equation 14where
j = k for an auto-correlation and j = 1, k = 2 for the cross-correlation, C[12](τ), between neurons 1 and 2. The auto-correlation and cross-correlation of the PSTHs are: Equation 15For convenience in
defining the correlation functions above, we have allowed the time index (t + τ) to take values outside [1, T]; therefore, we define x[k](t) and P[k](t) to be zero for t < 1 and t > T. The function S
[jk] will be referred to as the shift-predictor for the purposes of this appendix because it approximates that portion of the correlation that results from modulation in the PSTHs (Perkel et al.,
The equation for the correlation coefficient of spike counts: Equation 16where E is expected value and ς[k]^2 is the variance of the spike count computed over trials, can be rewritten in terms of the
cross-correlation equations above. First, observe that: Equation 17 Equation 19 Equation 20A similar result holds for the numerator of Equation 16: Equation 21 Equation 22 Equation 23 Equation 24The
following generic expression: Equation 25defines the area under the auto- and cross-correlation integrated from −T to T (after the shift-predictor is subtracted). We can rewrite the expression for
the correlation coefficient in terms of these areas as follows: Equation 26We now define a metric: Equation 27which will be used to estimate the inter-neuronal correlation coefficient by integrating
a limited central region of the CCG and ACGs. This measure is equal to the traditional measure, r[SC], when τ = T, i.e.: Equation 28In Results, neuronal data and simulated data are used to
demonstrate that r[CCG](τ) can provide a lower variance estimate of r[noise].
Here we derive an expression for r[SC], thus r[noise], for a pair of simulated spike trains that arise otherwise independently (i.e., with no common noise) generated from a common stimulus that
varies in strength from trial to trial.
Let f^i(t) be the mean firing rate on the i^th trial as a function of time (e.g., Fig. 12A), and let two spike trains be generated as independent realizations of an inhomogeneous Poisson processes
according to f^i(t). Assume that f^i(t) varies across trial number, i, in such a way that the time-averaged firing rate, λ[i], for any trial has mean μ[λ], variance ς[λ], and probability density g
[λ]. To derive the correlation in spike count induced by the trial-to-trial changes in f^i(t), we need only consider the statistics of the mean rate, λ, and not the details of the modulation of f^i
(t) during the trial. In particular, to compute the correlation coefficient r[SC] between the spike counts N[1] and N[2] across trials, we must find the expected values and variances required by
Equation 16. The expected value of the product of the spike counts can be computed as follows: Equation 29 Equation 30 Equation 31 Equation 32 Equation 33where T is the duration of the trial. A
derivation similar to that above, but substituting N[1] for N[2] or vice versa, leads to: Equation 34and a similar but even simpler derivation yields: Equation 35Using the identity VAR x = E x^2 − E^
2x and substituting the results of Equations 33, 34, and 35 into the equation for the correlation coefficient (Eq. 16), we arrive at: Equation 36where μ[N] = Tμ[λ] and ς[N]^2 = T^2ς[λ]^2 are used to
express the results in terms of spike counts rather than mean rates. This equation states that our simulated spike trains have uncorrelated counts (r[SC] = 0) when there is no trial-to-trial
variation in the stimulus strength, i.e., when ς[N]^2 = 0.
To determine the values of μ[N] and ς[N]^2, we must define the rate function, f^i(t). Many statistical descriptions are possible, but we chose one that provided modulation which was qualitatively
similar to that observed in PSTHs analyzed in our previous study (Bair and Koch, 1996) of responses to replicate stimuli collected under stimulus conditions similar to those of the present study. The
rate function, defined as a discrete signal at the resolution of 1 msec, was described by three parameters, a spontaneous firing rate, λ[min], a stimulated firing rate λ[max], and a probability, p,
that at each millisecond f^i(t) = λ[max] (otherwise, f^i(t) = λ[min]). Because for any Bernoulli random variable, X, E[X] = p and VAR[X] = pq (where p is the probability of success and q = 1 − p), it
follows that the mean and variance of the trial spike count generated by f^i(t) for trials of duration T seconds are: Equation 37 Equation 38where λ[min] and λ[max] are given in spikes per second and
δ = 0.001 sec. Substituting this into Equation 36 yields: Equation 39This expression represents the strength of artifactual spike count correlation induced by trial-to-trial stimulus variance for a
model of paired spike trains designed to be consistent with MT responses to our dynamic dot stimulus. See Figure 12 and the final section of Results for its application.
• W.B. is supported by Howard Hughes Medical Institute (HHMI). Part of this work was funded by the L. A. Hansen Fellowship to W.B. while in the lab of Christof Koch at Caltech. W.T.N. is an
investigator of HHMI. We thank Michael N. Shadlen, Carlos Brody, J. Anthony Movshon, and Christof Koch for suggestions and helpful discussion that has guided the course of this work, and we owe
additional thanks to M. N. Shadlen and C. Brody for detailed comments on this manuscript.
Correspondence should be addressed to Wyeth Bair, Howard Hughes Medical Institute, Center for Neural Science, New York University, 4 Washington Place, Room 809, New York, NY 10003. E-mail:wyeth | {"url":"https://www.jneurosci.org/content/21/5/1676?ijkey=1ce62888aee78129527cfaa2a8e44c2fca8e5c22&keytype2=tf_ipsecsha","timestamp":"2024-11-08T20:34:11Z","content_type":"application/xhtml+xml","content_length":"602636","record_id":"<urn:uuid:bfc01b41-abb7-4c87-b1e7-7be3c7984773>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00861.warc.gz"} |
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CountFast 5
This module introduces a learning tool or trick to help solve any number multiplied by 5 very fast. So again, the concept of taking what may look like a difficult problem and utilizing a tool or
trick to make it simpler to solve. After this module, the student will be able to use mental calculation to solve any number times 5 very fast.
Week 4 of the 3rd Grade CountFast program focuses on specific mental strategies for multiplying with the numbers 5, 15, and 50. Spend 15 minutes each day on one of the activities listed in this
module. Card decks should go home with students each day for additional practice with a parent at home. Each week, a new deck is introduced, and the previous deck is for the student to keep at home
for continued practice.
1 CCSS.Math.Content.3.OA.B.5
Apply properties of operations as strategies to multiply and divide.
2 NCTM Standard: develop a sense of whole numbers and represent and use them in flexible ways, including relating, composing, and decomposing numbers
1. Quickly, mentally multiply by 5, 15, or 50.
2. Develop fluency with multiplication calculations.
1. One CountFast 5, 15, 50 card deck for each student. This deck is for school and home use. Discuss routine and expectations for taking home the deck and returning it to school each day.
2. One writing utensil per student, optional per teacher plans.
CountFast 5, 15, 50 Pack – Day 1
Teacher Model/Direct: Use the yellow cards from the deck and teach students the fast way to multiply any number by 5. Since 5 is half of 10, we will first multiply the number by 10 (by simply
adding a zero to the end of the number), and then dividing that answer by 2 (cut in in half). For example, 12 X 5 can be solved quickly by adding a zero to the 12 (multiply it by 10) to get 120, and
then cutting 120 in half (divide by 2) to get 60. This will take several rounds of practice with teacher modeling.
Student Activity: Give each student a deck and ask them to take out the yellow cards. Working with a partner, take turns holding up the cards for the partner to multiply, using the method learned
in class.
Home Activity: Students will take home the deck and the “CountFast Home Connection” letter. Students will review the yellow cards with a parent and explaining how to solve the problems quickly using
the method learned in class. Parent can record how quickly the child multiplies all the cards correctly.
CountFast 5, 15, 50 Pack – Day 2 and 3
Teacher Model/Direct: Using the same procedure as Day 1, review the strategy for quickly multiplying any number times 5. Then use the blue cards to introduce how to expand on that strategy to
multiply any number by 15. Since 15 is the same as 3 X 5, first multiply the number by 3, and then use that product and follow the ‘times 5’ procedure learned on Day 1. For example, 6 X 15 can
mentally be solved by first multiplying the 6 times 3 to get 18. Then take 18 times 10 to get 180, and cut 180 in half to get a final answer of 90.
Student Activity: Students will work in partners to practice with the yellow and blue cards, using the mental calculation tips for multiplying by 5 and by 15.
Home Activity: Students will repeat the partner practice at home with a parent.
CountFast 5, 15, 50 Pack – Day 4 and 5
Teacher Model/Direct: Use the pink cards from the deck. Today, show students how to multiply any number by 50. Since 50 is the same as 5 X 10, first multiply the number by 5. (use the 5’s
strategy learned on day 1). Then multiply that product by 10 by simply adding a zero. For example, to multiply 26 by 50, first multiply 26 X 5 (26 X 10 = 260. 260 divided in half is 130.) Then
take the product 130 times 10 by adding a zero. Final answer is 1300.
Student Activity: With a partner, use the pink cards to practice multiplying by 50, using the strategy learned in the lesson.
Home Activity: Students will review the activities for multiplying by 50, using the strategy learned in class. | {"url":"https://countfast.com/product/countfast-5/","timestamp":"2024-11-10T21:15:23Z","content_type":"text/html","content_length":"389558","record_id":"<urn:uuid:d0dabaa1-f53c-4ec0-89a6-b2c91281313c>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00501.warc.gz"} |
Subtracting Negative Integers Worksheet
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Subtracting Past Zero on a Number Line
We are thrilled to introduce our latest educational resource: the Subtracting Past Zero on a Number Line worksheet. Specifically designed for year 4, year 5, and year 6 students, this free printable
worksheet is perfect for those learning about negative integers in their maths lessons.
About the Worksheet
The Subtracting Past Zero on a Number Line worksheet provides an engaging and supportive way for students to grasp the concept of negative integers. By presenting subtraction problems that result in
negative answers, this worksheet helps students understand and navigate the world of numbers below zero.
What's Included?
The worksheet features:​
• Twelve subtraction questions: Each question starts with a positive number and subtracts a number that results in a negative answer. Examples include 10 - 15 = -5.
• Supportive number lines: Next to each question is a number line ranging from -15 to 10. This visual aid assists students in finding the correct answers, especially those encountering negative
integers for the first time.
How to Use the Worksheet
1. Preparation:
□ Print out the Subtracting Past Zero on a Number Line worksheet for each student.
2. Activity Instructions:
□ Students read each subtraction question carefully.
□ Using the number line provided next to each question, students locate the starting positive number.
□ They then count backwards on the number line to find the answer, ensuring they understand how to move past zero into the negative range.
□ Students write their answers next to each question.
This structured approach, with the aid of number lines, allows students to visually comprehend the transition from positive to negative numbers, reinforcing their understanding through practice.
Educational Benefits
• Conceptual Understanding: The worksheet helps students grasp the concept of subtracting past zero, a fundamental step in learning about negative integers.
• Visual Learning: Number lines provide a visual representation that aids in understanding abstract concepts.
• Skill Development: Students enhance their subtraction skills and gain confidence in working with negative numbers.
Download and Print
Download the Subtracting Past Zero on a Number Line worksheet today and provide your students with an essential tool for mastering negative integers. As always, this resource is free and easy to
incorporate into your lesson plans.
Help your students build a solid foundation in maths with our Subtracting Past Zero on a Number Line worksheet. Click the button below to download and start enhancing your maths lessons today!
All resources can be downloaded by clicking the button at the bottom of the page, and all resources on the website are free.
By offering comprehensive and visually supportive worksheets, we aim to make learning negative integers an accessible and engaging experience for all students. Don't forget to explore our other free
resources designed to support your teaching and enhance your students' learning journey!
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Ternary logic
Ternary logic¶
As specified in the SQL standard, ternary logic, or three-valued logic (3VL), is a logic system with three truth values: TRUE, FALSE, and UNKNOWN. In Snowflake, UNKNOWN is represented by NULL.
Ternary logic applies to the evaluation of Boolean expressions, as well as predicates, and affects the results of logical operations such as AND, OR, and NOT:
• When used in expressions (e.g. SELECT list), UNKNOWN results are returned as NULL values.
• When used as a predicate (e.g. WHERE clause), UNKNOWN results evaluate to FALSE.
Truth tables¶
This section describes the truth tables for the comparison and logical operators.
Comparison operators¶
If any operand for a comparison operator is NULL, the result is NULL. Comparison operators are:
Logical operators¶
Given a BOOLEAN column C:
If C is: C AND NULL evaluates to: C OR NULL evaluates to: NOT C evaluates to:
TRUE NULL TRUE FALSE
FALSE FALSE NULL TRUE
NULL NULL NULL NULL
In addition:
If C is: C AND (NOT C) evaluates to: C OR (NOT C) evaluates to: NOT (C OR NULL) evaluates to:
TRUE FALSE TRUE FALSE
FALSE FALSE TRUE NULL
NULL NULL NULL NULL
Usage notes for conditional expressions¶
This section describes behavior specific to the following conditional expressions.
IFF behavior¶
The IFF function returns the following results for ternary logic. Given a BOOLEAN column C:
If C is: IFF(C, e1, e2) evaluates to:
TRUE e1
FALSE e2
NULL e2
[ NOT ] IN behavior¶
The [ NOT ] IN functions return the following results for ternary logic. Given 3 numeric columns c1, c2, and c3:
• c1 IN (c2, c3, ...) is syntactically equivalent to (c1 = c2 or c1 = c3 or ...).
As a result, when the value of c1 is NULL, the expression c1 IN (c2, c3, NULL) always evaluates to FALSE.
• c1 NOT IN (c2, c3, ... ) is syntactically equivalent to (c1 <> c2 AND c1 <> c3 AND ...).
Therefore, even if c1 NOT IN (c2, c3) is TRUE, c1 NOT IN (c2, c3, NULL) evaluates to NULL. | {"url":"https://docs.snowflake.com/sql-reference/ternary-logic","timestamp":"2024-11-12T14:54:04Z","content_type":"text/html","content_length":"525445","record_id":"<urn:uuid:8fa8b9d1-a6ca-4d15-a3ae-ca07a4466a22>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00476.warc.gz"} |
Finiteness of local cohomology, II
Proposition 51.11.1. Let $A$ be a Noetherian ring which has a dualizing complex. Let $T \subset \mathop{\mathrm{Spec}}(A)$ be a subset stable under specialization. Let $s \geq 0$ an integer. Let $M$
be a finite $A$-module. The following are equivalent
1. $H^ i_ T(M)$ is a finite $A$-module for $i \leq s$, and
2. for all $\mathfrak p \not\in T$, $\mathfrak q \in T$ with $\mathfrak p \subset \mathfrak q$ we have
\[ \text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim ((A/\mathfrak p)_\mathfrak q) > s \]
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Core vector regression for very large regression
Core vector regression for very large regression problems
Ivor W. Tsang, James T. Kwok, Kimo T. Lai
Abstract: In this paper, we extend the recently proposed Core Vector Machine algorithm to the regression setting by generalizing the underlying minimum enclosing ball problem. The resultant Core
Vector Regression (CVR) algorithm can be used with any linear/nonlinear kernels and can obtain provably approximately optimal solutions. Its asymptotic time complexity is linear in the number of
training patterns $m$, while its space complexity is independent of $m$. Experiments show that CVR has comparable performance with SVR, but is much faster and produces much fewer support vectors on
very large data sets. It is also successfully applied to large 3D point sets in computer graphics for the modeling of implicit surfaces.
Proceedings of the Twenty-Second International Conference on Machine Learning (ICML-2005), Bonn, Germany, August 2005.
Postscript: http://www.cs.ust.hk/~jamesk/papers/icml05.ps.gz
Back to James Kwok's home page. | {"url":"https://cse.hkust.edu.hk/~jamesk/papers/icml05.html","timestamp":"2024-11-12T03:42:18Z","content_type":"text/html","content_length":"1643","record_id":"<urn:uuid:d254af0e-f348-46cd-b218-3745ce5469dd>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00881.warc.gz"} |
GCE A Level 2012 Oct/Nov H1 Maths 8864 Paper 1 Suggested Answers & Solutions (You Are Not Forgotten) - Jφss Sticks Tuition Secondary Math Tuition
GCE A Level 2012 Oct/Nov H1 Maths 8864 Paper 1 Suggested Answers & Solutions (You Are Not Forgotten)
Tuition given in the topic of H1/H2 (A Level) Maths Tutor from the desk of at 11:05 pm (Singapore time)
Updated on
We’ve been asked on our Facebook wall.
We’ve been repeatedly asked on this blog’s comments sections.
Errr … we’ve even been pui-ed for this 🙁
Lest you feel forgotten, abandoned or unloved, let me assure the H1SD Maths Generation that you were never far from my mind while I toiled through the H2 solutions as well as my tuition classes in
the past days.
Ok 废话少说, here’s the stuff you’ve been waiting for.
Click the button and grab it here!
Latest version:
• 1.1:
Q9(ii) – added “the advertised price decreases as the car’s age increases in a linear fashion” in response to this.
Q11(ii) – corrected the typo “insufficient evidence” to “sufficient evidence” in the concluding statement of the hypothesis test.
Access it
if you’re having trouble accessing it on Facebook using your state-of-the-art smartphone 🙁
As usual, please leave a gentle comment should you spot any mistake in this set of suggested solutions.
Now to attend to the pile of comments *rubs eyes*
All the best for your remaining papers! Freedom (or the SAF) is near!
Comments & Reactions
14 Comments
Wah! I see my comment up there! Feeling like a celebrity LOL.
Anyway, thank you so much Mr Loi for taking painful time to upload the answers for H1 math. We really appreciate your effort :')
I'm so glad the Hypothesis testing method I did is correct! When my dad solved it, (he is a SAT math teacher) he put H1: u >300 so I thought I'd lose like 8m! Very happy now!
For qn 10, the last part, can we use bionomial distribution to solve it instead?
Thank you Mr Loi!
@Michelle lee: Ok I'm back after a short hiatus!
For Q10(iv), yes you should get the same answer using the binomial distribution X ~ B (3, 0.62994) where the random variable X is the is the number of gardeners with > 75 plants that will
flower. So,
P (X ≥ 2) = 1 − P (X ≤ 1) = ... using GC ... = 0.691
Hi Mr Loi!
Thank you so much for taking the time to upload this! You've no idea how much this helps *-*
I just have a question to ask for Q11)!
Why is it that it is H1 : Mean ? The questions says "at least" which would mean that the length has to be 300 or MORE. This means we are testing if the claim of the average length being MORE than
300 is correct. Hence, if it were to be rejected, the average length would be 300 or less. Is it not...?
A minor thing to note, my teachers had told me previously that it was noted in Cambridge's marking report that we ought to state the relationship for the correlation between the two variables, so
for 9ii) i think we'd need to add on something like "if the age of the car increases its advertised price will drop" to get a full 2m.
That's all! Thank you so much once again! (:
@tere: If we reject and conclude that the mean is 300 or less, aren't we ignoring a part of the manger's claim that the mean can be 300? So I think it is not completely accurate...?
@Michelle lee: mm i think my question was a little unclear because my signs were not included.
I was confused with his answer. I didn't understand why his H1:Mean was less than 300. I had put it the same as your dad, more than.
But right now, basing on what you're saying and looking at Mr Loi's solutions again, If we're testing it to be LESS THAN 300, then it would mean that we are testing if the average length
of the ball is 300 or less than 300, and in this way we can still calculate and in fact it would be more accurate! Thank you, I understand now!
@tere: Q11(ii) is a little tricky in the sense that you are actually testing for whether the average length is less than 300 m i.e.
Ho : μ = 0
H1 : μ < 300
In this case, if Ho is rejected ⇒ average length of string is less than 300 m which counters the manager's claim of at least 300 m.
Instead, if we were to test for H1 : μ > 300, accepting or rejecting Ho wouldn't make any difference in both cases since μ will still be within "at least 300 m" territory.
Q9(ii) Hmmm I'm inclined to think that a "strong negative linear correlation" would suffice to cover the "age of car increases/price decreases part" for the (presumably) 1-mark
comment since the other mark would go towards the calculation of value of the coefficient. But who am I to go against Cambridge if it's true to markers' report said this!
Despite the low demand for h1's answer, i want to express my biggest thank you for taking time to come out with the answers! thank you ^^!
btw why is there a need to standardize the last qn? 😮 can't we just calculate the probability using normalcdf after obtaining the mean and variance?
@abcd: It's perfectly fine not to standardize the normal distribution in Q12(ii). It's just a habit of mine from questions to find unknowns from given probability value(s) - which is not
really necessary here 😛
Wow my comment is up there too haha. Thanks for uploading the answers!
I just wanted to check, if I left my answers for Q8 (probability) in fractions, will I be penalized?
@Adeline: No problem lah hehe
Hi when will the 2013 a levels h1 math answers be uploaded?! Many of my school mates said it was a harder paper as compare to last years. Is it true? Really anxoous and worried. And normally
what's the marks to get an A for h1 math. Is it around the 80s? Or 90s? Cos apparently for O levels Emaths is the high 90s to score an A1. :S
will you upload for h1 maths 2014?:)
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Nonlinear fitting using Orthogonal Distance Regression
4.2.2.34 Nonlinear fitting using Orthogonal Distance Regression
When performing non-linear curve fitting to experimental data, one may encounter the need to account for errors in both independent variables and dependent variables. In Origin, you can utilize the
Orthogonal Distance Regression (ODR) to fit your data with implicit or explicit functions. This tutorial will demonstrate how to perform non-linear curve fitting on data with both X errors and Y
errors using ODR with a built in function.
Minimum Origin Version Required: Origin 9.1
What you will learn
This tutorial will show you how to use Orthogonal Distance Regression to fit nonlinear data with both X and Y errors.
Example and Steps
1. Open a new workbook. Select Help: Open Folder: Sample Folder... to open the "Samples" folder. In this folder, open the Curve Fitting subfolder and find the file ODR fitting.dat. Drag-and-drop
this file into the empty worksheet to import it.
2. Highlight column XError (long name) and right click to select Set As: X Error to set it as X error column.
3. Highlight column YError (long name) and right click to select Set As: Y Error to set it as Y error column.
4. Highlight all four columns and go to Plot:Basic 2D:Scatter to make a scatter plot with both X and Y error bars.
5. Go to Analysis:Fitting:Nonlinear Curve Fit:Open dialog... to open NLFit dialog.
6. On Function Selection page, select Category as Polynomial, Function as Poly4 and Iteration Algorithm as Orthogonal Distance Regression (Pro).
7. Since X error and Y error columns have been specified in step 3, 4 so that when Orthogonal Distance Regression is selected as iteration algorithm these two columns will be automatically assigned
as corresponding weight for X and Y data respectively. You can go to Data Selection page and expand x and y nodes under Input Data to see.
8. Click Fit button and choose No radio box in the appeared Reminder Message dialog. The fitting results are as shown below:
You can refer to this page for the details of the algorithm of ODR as well as Levenberg Marquardt (L-M) algorithm. Another example of using Orthogonal Distance Regression for Implicit Functions can
be found here. | {"url":"https://d2mvzyuse3lwjc.cloudfront.net/doc/en/Tutorials/Nonlinear-fitting-using-orthogonal-regression","timestamp":"2024-11-03T09:46:18Z","content_type":"text/html","content_length":"137127","record_id":"<urn:uuid:84636700-4a8c-45e1-b656-d50d7fa4a44a>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00358.warc.gz"} |
arget firm essay
target firm essay
Words: 8829 | Published: 03.18.20 | Views: 503 | Download now
Cranefield College of Project and Programme Managing MODULE M6 Financial Administration of Company Projects and Programmes Case: TARGET CORPORATION 1 . Executive Summary Target corporation includes a
growth approach of beginning 100 new stores each year. Doug Scovanner, the CFO of Goal Corporation is definitely preparing for the November getting together with of the Capital Expenditure Panel
(CEC). He could be one of the business officers who also are users of the CEC.
With the money year’s end approaching in January, there was clearly a need to ascertain which projects best fit Target’s future store growth and capital expenditure plans, with the knowledge that
individuals plans would be shared with both the board as well as the investment community. Target provides a growth technique of starting approximately 90 new shops a year. CEC referred assignments
with a great investment larger than 50 dollars million for the board of directors for approval. The five CPRs that Scovanner would show the plank are: Gopher Place, Whalen Court, The Barn, Goldie’s
Square and Stadium Renovate.
Recommendations towards the Capital Expenditure Committee The capital expenditure committee should agree to all the proposals before this. This will end up being based on the factors as detailed on
part 3 of this file. The NPV’s of all these types of projects happen to be positive, an optimistic NPV contributes favorable towards the share selling price or talk about value. The interior Rate of
Return of those entire projects are below the prototype shop IRR a benchmark project. The IRR is an alternative to NPV nevertheless if the NPV is great and the IRR is not what is ideal, the NPV may
supersede in making an investment decision.
The IRR is expected based upon internal elements. Projects having a low IRR may be funded through debt capital if cost of debts is under the project IRR/ rate of return. An overarching target of
Concentrate on Corporation is always to meet the company goal of adding 100 new shops a year while keeping a positive brand image. Since all of these outlets have an optimistic NPV and the long run
they each make very good earnings ahead of interest and taxes. The CEC need to accept these people because they will achieve the aim of market increased and manufacturer visibility.
The Stadium upgrade is particularly crucial because the retail store has going down hill and dilapidating facilities that would defeat the purpose of a positive company image. Your local store must
be re-designed before it starts impacting the product sales of different Target stores with negative publicity. Whalen Court is usually to be open in a metropolitan place and it is a great urban
center. The population with this trade area is very big and includes a good income median. The project needs a lot of capital investment, however it presents Focus on stores with a unique
contribution in that it would offer cost-free advertising to the corporation.
There are countless consumers moving by and Target already spends in excess of $100 mil dollars in advertising starting this shop might help decrease these costs. If funds are a constraining factor,
Focus on should pay for the projects in the following other: 1 ) Gopher Place should be considered 1st. The task requires a twenty three 000 500 investment. It includes the best NPV and it is above
the prototype retail store NPV. The sales can easily still decline by simply more than 5% and it might still be above the prototype retail outlet. It has a better EBTI when compared to other costs,
though the present’s dangers it offers prospect as well.. Whalen Court may be the second in line. It has a confident NPV even though it is below the prototype store value. If sales increase by 1 .
9%, it will be equal to the prototype shop NPV. This really is a better NPV compared to the staying two jobs. The store supplies a good industry with a big population and better cash flow median. 3.
Goldie’s Square, the NPV is positive but the sales must still rise by 45. 1% before it might meet the original store NPV. The NPV is not as good as can be expected however it is still confident. What
makes this a desirable purchase is the site that the store will be constructed in.
Project is important because of its tactical location, all of the big stores want for capturing this market intended for visibility and market capitalization. Since this can be not a huge investment
it can be considered. four. Stadium Renovate is very important that the CEC makes this purchase failing that the poor condition of the services would tarnish the image of the brand. The NPV is great
and EBIT. 2 . Trouble / decision statement Inside the Capital Expenses Approval Process, there is the Capital Expenditure Panel (CEC) the team composed of of leading executives that meet month to
month to review most capital task requests (CPR) in excess of $100 000.
All the proposals are viewed as economically eye-catching and any kind of CPR’s with questionable economics are refused. Doud Scovanner, the CFO of Concentrate on Corporation is usually preparing for
the November conference of the CEC where he will present five CPRs to the committee of five, which usually he is a part of. Monetary data and everything other related data about these tasks is
available, he now has to compile a study to the panel, convincing these people that they decide about buying these assignments. The CEC considers several factors to determine whether to accept or
reject a pitch.
He must detail his report so that it becomes convincing to the CEC and they need to therefore decide to release the amount of money for the CPRs. three or more. Critical or key concerns The Capital
Costs Committee (CEC) is a staff comprising of top executives that meet monthly to review all capital project needs (CPR) in excess of $100 1000. All of the proposals are considered financially
attractive and any CPR’s with doubtful economics will be rejected. The CEC considers several factors to decide whether to accept or reject a proposal. Important factors the CEC looks at in assessing
CPRs are: 1 .
The overarching goal was the corporate goal of adding regarding 100 stores a year while keeping a positive company image. installment payments on your Projects must provide a appropriate Net Present
Value (NPV) 3. Assignments must offer a suitable Inner Rate of Return (IRR) 4. Awareness of NPV and IRR to sales variation. 5. Projected revenue 6. Projected earnings per share several. Total
investment size almost eight. Impact on product sales of near by Target shops Net present value that the difference between market value of the investment as well as cost (Firer et ‘s 2009: 269).
The secret for net present worth is that the purchase with more positive present value must be consumed in the expense in the one with negative or lower great present benefit. Advantages of net
present value? The introduction of time benefit of money? It conveys all long term cash flows in today’s worth, which permits direct evaluations? It allows for inflation and escalations? It looks at
the whole project from the start to end? It can promote project imagine if analysis applying different ideals? It gives better profit or loss forecast Gopher place| Whalen court| The barn| Goldie’s
square| Stadium remodel| Accumulated present value| 39800| 145200| 33500| 24200| 32700| Less primary investment| (23000)| (119300)| (13000)| (23900)| (17000)| Net present value| 16800| 25900| 20500|
300| 15700| Internal price of come back It is benefit of price cut factor if the net present value is usually equal to no (Firer ainsi que al 2009: 280). The rule of internal rate of return is that
the job with the higher rate of return has to be accepted mainly because its provides the clear indicator that the task will be successful Advantages of Inside rate of return It is not complicated to
understand and communicate Disadvantages of Internal rate of return? This method may possibly results in multiple answers or not deal with non conventional cash flow? May result in incorrect decision
in comparison with mutually exclusive investment? Another problem with IRR comes about the moment cash flow are generally not conventional Chart showing the internal rate of return of 5 likely
project 2. Gopher place 12. 3% * Whalen court on the lookout for. 8% 5. The hvalp 16. 4% * Goldie’s square almost 8. 1% * Stadium redesign 10. 8% Projected revenue and revenue per talk about
Target company uses projected profit as one its criteria to accept or reject the project, the entities sales had elevated a lot through the previous years so the business project the project of the
of new job by taking into mind the existing retailers profits. Revenue per reveal of target corporation intended for financial season ending January 2006 is definitely $2. 73 per reveal according to
exhibit2 which can be computed through current yr total extensive income and dividing that by the number of ordinary stocks and shares of the business.
Target businesses earning every share is definitely greater than making per share of it is bigger competitor Waal mart and this demonstrates the company is really doing well in the market and this
possess power to expand its prominence in the market by introducing fresh stores each year 4. Evaluation from a strategic, qualitative and quantitative perspective Gopher Place: P04, Shop NPV: $16
800 1000 HURDLE REALIGNMENT (CPR Dashboard)| Sales| NPV| Sales could decrease by (5. 3%) and still achieve Prototype Shop NPV | IRR| Revenue would have to boost by 2 . 2% to accomplish Prototype
Store NPV| | |
Low Margin| NPV| Gross Margin could lower by (0. 72) pp and still obtain Prototype Retail outlet NPV | IRR| Major Margin would have to increase by simply 0. up to 29 pp to obtain Prototype Shop NPV|
| | Building (Building & Site work)| NPV| Structure costs could increase by $3, 102 and still obtain the Modele Store NPV| IRR| Development costs would have to decrease simply by ($751) to achieve
Prototype Retail store IRR| | | Full Transfer Impact| NPV| Product sales would have to boost by 2 . 3% to accomplish Prototype Retail store NPV| IRR| Sales would need to increase by 9. % to achieve
Model Store IRR| RISK/OPPORTUNITY| 10% Sales Decline| NPV| If sales rejected by 10% Store NPV would fall by($4, 722) | IRR| If revenue declined simply by 10% Retail outlet IRR could decline by (1. 3)
pp| | | 1 pp GM Decline| NPV| If perhaps gross perimeter decreased by 1 pp, Store NPV would drop by ($3, 481)| IRR| If gross margin lowered by one particular pp, Retail outlet IRR would decline by
(0. 9) pp. | | | 10% Building cost increase| NPV| If perhaps construction price increased by 10% Retail outlet NPV could decline by ($1, 494)| IRR| In the event that construction cost increased by
10% Retail outlet IRR could decline by simply (0. ) pp. | | | Market Margin, Wage Rate, etc| NPV| If we used market certain assumptions, Shop NPV would decline by ($5, 434)| IRR| Whenever we applied
industry specific assumptions, Store IRR would decrease by (1. 5) pp. | | | 10% Sales increase | NPV| If sales increased simply by 10%, Retail store NPV would increase by $4, 621| IRR| In the event
sales improved by 10%, Store IRR would enhance by 1 ) 2 pp| VARIANCE TO PROTOTYPE| The Gopher Place with a retail outlet NPV of $16, 800 is $3, 038 over a Prototype Retail outlet NPV. The next items
contributed to the variance. Land| NPV| Land cost contributed a good $287 for the variance by prototype. | IRR| Area cost contributed a positive 0. 1 pp to the difference prototype. | | | Non-Land
Investment| NPV| Building/site work costs contributed a poor ($4, 741) to the variance from Original. | IRR| Building/site operate costs contributed a negative (2. 6) pp to the difference from
Prototype. | | | Sales| NPV| Product sales contributed a good $6, 331 to the variance from Prototype. | IRR| Sales contributed a positive 1 ) 9 pp to the difference from Modele. | | Real Estate Taxes
| NPV| Property Taxes offered a positive $615 to the variance from Original. | IRR| Real Estate Fees contributed a positive 0. two pp towards the variance coming from Prototype. | 1 . Proper
importance * This is a vital market to get Target already has five stores inside the area. Wal-Mart is expected to add two new supercenters in response for the population progress. In order to
curtail Wal-Mart’s market dominance and ensure that the brand image of Focus on is managed in this locality it is crucial that a P04 store is built.
The population in this area is growing for a price of 27% between the years 2000 ” 2005, the median cash flow for the population is $56 400, this kind of store will certainly increase industry
capitalization of Target stores and therefore maintaining a good brand photo which is a strategic goal. 2 . Net Present Value DIFFICULTY ADJUSTMENT (CPR Dashboard) * The job is viable with a
confident net present value of $16 800 000. An optimistic NPV worth will results in an increase in discuss value. 2. Based on this kind of factor the project should be accepted since it will increase
discuss value. The projected product sales could lower by your five. 3% and the NPV with the project will still be achieved. * Even when a gross margin of the task were to reduce by 0. 72 percentage
point the project can still produce a positive NPV. * In the event that this task is taken on it would need that a new (own) shop be constructed which will cause cash outflows in a contact form
building and site job, the designed construction expense could still increase by simply $3, 102 000 as well as the project will maintain a good NPV. * The copy sales from other stores in the trading
place will have to enhance by 2 . % to realise the prototype retail store NPV. 2. On the basis of NPV this forecasted may be recognized. RISK/OPPORTUNITY RESEARCH * In the event that sales would have
been to decline by simply 10% the NPV might decrease by $4 722 000 approximately 28% with the NPV. 2. If the gross margin lower by 1 percentage stage the NPV would lower by $3 481 000 a twenty. 72 %
decrease of the NPV. * If the structure costs improved by 10% the NPV would drop by $1 494 000, an 8. 89% decrease of the NPV. * If perhaps market details assumptions on market margins, wages and so
on are used the store NPV would lower by $5 434 000 a thirty-two. 5% for the prototype retail store NPV. 5. If product sales increased simply by 10% the NPV might increase by $4 621 000 a 27. 51%
increase. 2. The risks linked to this job are better but so might be the chances with a 10% sales maximize there would be a 27. 51% increase in the NPV. This project must be accepted the project
managers must physical exercise caution within the costs while the project has an components of risk. Variance from the programs must be kept at bare minimum. VARIANCE TO PROTOTYPE * The NPV of this
job exceeded the prototype retail outlet value simply by $3 038 000 the subsequent factors written for the difference. Land cost contributed an optimistic $287 to the variance coming from prototype.
2. Building/site job costs contributed a negative ($4, 741) towards the variance from Prototype. 5. Sales contributed a positive $6, 331 to the variance by Prototype. 2. Real Estate Taxes contributed
a positive $615 towards the variance coming from Prototype. 2. The NPV of this project is satisfactory as it surpasses by far the expectation and based on this kind of the job should be acknowledged.
3. Inner Rate of Return (IRR) HURDLE ADJUSTING (CPR Dashboard) * Interior Rate of Return is a crucial alternative to NPV, the IRR summarizes the merits of the project.
This rate is usually an internal rate in a sense it depends simply on cash flows of any particular project or expenditure, not on rates presented elsewhere consequently internal rate of return. * The
project comes with an IRR of 12. 3%, this IRR does not fulfill the prototype store IRR product sales must still increase by 2 . 2% achieve this. * The major margin with the project need to increase
by simply 0. 30 percentage point in order to obtain the required IRR. * The money outflow related to the construction costs must nonetheless decrease simply by $751 1000 in order to accomplish the
required IRR. * Transfer sales need to increase by 9. % to achieve wanted IRR. RISK/OPPORTUNITY ANALYSIS 5. If product sales declined simply by 10% Retail outlet IRR could decline simply by (1. 3)
percentage point, IRR is below the modele store value. * If perhaps gross perimeter decreased simply by 1 percentage point, Shop IRR will decline by (0. 9) percentage stage, which is a equal fall. *
If perhaps construction cost increased simply by 10% Retail store IRR might decline simply by (0. 6) percentage stage, this decline in IRR is much less than the embrace costs. 5. If revenue increased
by simply 10%, Retail store IRR might increase simply by 1 . two percentage stage, this is an optimistic indication. depending on the IRR figures this is simply not a very high-risk venture if
perhaps sales decline by 10% IRR fall by 1 . 3 percentage point nevertheless sale boost by 10% IRR only increase by 1 . 2 percentage point. VARIANCE TO PROTOTYPE 5. Land cost contributed a good 0.
one particular percentage point to the difference prototype. * Building/site work costs led a negative (2. 6) percentage point to the variance via Prototype. * Sales led a positive 1 . 9 percentage
point to the variance coming from Prototype. 5. Real Estate Income taxes contributed an optimistic 0. a couple of percentage point out the variance from Modele. The jobs IRR is way better when
compared to the prototype store apart from real estate tax which contributed a low 0. 2 percentage point. some. Projected revenue The profit and loss summary suggests that the project could make a
forecasted loss of ($567 000) inside the first season of starting the store compared to the prototype store value of ($97 000). By the 6th year the store will be producing Earnings prior to Interest
and Tax (EBIT) of $4 452 500 a year dollar 886 000 above the model store worth. This expected should be approved on the basis of the gains it will make as this will likely increase the talk about
value. your five.
Projected profits per reveal Earnings every share are affected by both the NPV and the earnings/profits of the project/investment. This investment/project has a confident NPV and earnings in the end
which will results in increased profits per discuss. 6. Expense required This kind of store needs an investment of $23 1000 000 and was scheduled to open in October 2007. This is a substantial
investment. The NPV is usually positive, the citizenry is growing in 27%. The income median of the population is $56 400. The returns with this investment will payback this kind of amount much soon
than can be expected. 7. Effect on sales of nearby Concentrate on stores
There is also a high density of Target retailers in the control area and nearly 19% of the revenue included in the predictions were anticipated to come from existing targets retailers. This will not
really be good intended for the business generally however the 81% of predicted sales will certainly either come from new market growth or perhaps from finalization. Conclusion 5. This task should be
acknowledged by the CEC based on the above mentioned criterion. The project gives a lot of risks and opportunities to Focus on stores. The opportunities considerably outweigh the potential risks,
positive bigger NPV, boost EBIT, further store to fulfill target and increase brand image.
The project can still be recognized with the IRR figures although they are listed below a modele store worth. The IRR is based on an internal value, in the event that for an example the projected may
be financed by borrowed funds as well as the debt expense are under the IRR it would be a perfectly suitable investment. Given that the project has a great NPV this will likely increase the share
value. Whalen Court: Unique Single Level, Store NPV: $14, 240 HURDLE ADJUSTMENT (CPR Dashboard)| Sales| NPV| Sales would have to increase by 1 . 9% to achieve Prototype Store NPV | IRR| Sales would
have to increase simply by 31. % to achieve Original Store NPV| | | Gross Margin| NPV| Low Margin would need to increase by 0. twenty-eight pp to attain Prototype Retail store NPV | IRR| Gross Margin
will have to increase by simply 4. 49 pp to attain Prototype Retail store NPV| | | Construction (Building & Sitework)| NPV| Construction costs would have to lower by ($4, 289) to realise the
Prototype Retail outlet NPV| IRR| Construction costs would have to reduce by ($41, 070) to accomplish Prototype Shop IRR| | | Full Transfer Impact| NPV| Revenue would have to enhance by several. 7%
to accomplish Prototype Shop NPV| IRR| Sales would have to increase by simply 36. % to achieve Modele Store IRR| RISK/OPPORTUNITY| 10% Sales Decline| NPV| In the event that sales rejected by 10%
Store NPV would fall by ($16, 611) | IRR| If perhaps sales declined by 10% Store IRR would fall by (1. 0)pp| | | one particular pp GMC Decline| NPV| If margin decreased by simply 1 pp, Store NPV
would decrease by ($11, 494)| IRR| If perimeter decreased by simply 1 pp, Store IRR would decline by (0. 7) pp. | | | 10% Construction price increase| NPV| If structure cost elevated by 10% Store NPV
would drop by ($2, 178)| IRR| If development cost improved by 10% Store IRR would drop by (0. 1) pp. | | | Industry Margin, Income Rate, etc|
NPV| Whenever we applied industry specific assumptions, Store NPV would decline by ($16, 877)| IRR| If we utilized market certain assumptions, Shop IRR will decline simply by (1. 1) pp. | | | 10%
Product sales increase | NPV| If sales improved by 10%, Store NPV would increase by $16, 647| IRR| If revenue increased by 10%, Retail outlet IRR might increase simply by 1 . 0 pp| DIFFERENCE TO
PROTOTYPE| The Whalen Court with a store NPV of $14, 225 is usually $3, 174 below the Modele Store NPV. The following items contributed to the variance. | Lease| NPV| Lease expense contributed a poor
($78, 912) to the difference from model. | IRR| Lease cost contributed an adverse (15. ) pp to the variance prototype. | | | Non-Land Investment| NPV| Building/sitework costs contributed a bad ($10,
168) to the difference from Prototype. | IRR| Building/sitework costs contributed a poor (7. 9) pp to the variance coming from Prototype. | | | Sales| NPV| Sales contributed a positive 99 dollars,
963 towards the variance coming from Prototype. | IRR| Product sales contributed a good 22. 9 pp to the variance from Prototype. | | | Real Estate Taxes| NPV| Property Taxes added a negative ($637)
to the variance from Prototype. | IRR| Real Estate Income taxes contributed a poor (0. 2) pp for the variance from Prototype. |. Strategic importance * This can be a unique single-level store. Goal
already features forty five retailers in this control area. The Whalen Courtroom market signifies a rare opportunity for Target to enter an metropolitan center of your major city area. Unlike other
areas, this kind of opportunity presented Target with major manufacturer visibility and essentially totally free advertising for all passersby. The population of this place is significantly high plus
the median salary for the population is $48 500, this kind of store will certainly increase marketplace capitalization of Target stores and therefore maintaining an optimistic brand picture which is
a ideal goal.
The investment costs will be well-balanced by the enormous adverting costs that Target helps you to save on each year if this project is undertaken. installment payments on your Net Present Value
DIFFICULTY ADJUSTMENT (CPR Dashboard) 2. The job is feasible with a positive net present value of $25 900 000. A positive NPV worth will leads to an increase in reveal value. 2. Based on this kind of
factor the project needs to be accepted mainly because it will increase share value. 2. The project is risky because the projected sales must still boost by 1 . 9% and gross perimeter improve by 0.
twenty-eight percentage stage before it can achieve the prototype retail outlet NPV. The money outflows must decline simply by $4 289 and transfer sales from the other stores in the trading place
will have to maximize by 2 . 3% ahead of the prototype retail store NPV can be achieved. 2. The NPV is great that is appropriate however the projected NPV can be below the prototype store which is
factor to get considered to be able to evaluate the project. RISK/OPPORTUNITY EVALUATION * If projected product sales declined by 10% the NPV could decrease simply by $16 611 000 (64. 14% decline) of
the NPV and If low margin reduce by you percentage level the NPV would reduce by $11 494 000 (44. 38%) decline in NPV. Should certainly construction costs increased by 10% the NPV might decline by
simply $2 a hundred and seventy-eight 000 (8. 41%) decrease of the NPV and if marketplace specifics assumptions on industry margins, pay etc happen to be applied your local store NPV might decrease
by simply $16 877 000 (65. 16%) in the prototype retail outlet NPV. 2. If revenue increased by simply 10% the NPV would increase by simply $16 647 000 (64. 27%) embrace the NPV. * This really is a
very dangerous investment in case the CEC recognize this job they must place a lot of measures to mitigate the chance. It can also be useful if sales cost boost, the NPV figures usually are that
confident due to the risk factors.
VARIANCE TO MODELE * The NPV on this project can be below the prototype store worth by $3 174 500 the following elements contributed to the variance. 5. Lease, web page work and real estate tax costs
must be contained in order to mitigate raise the risk, they led a negative $78 912 000, $10 168 000 and $637 1000 respectively for the variance coming from prototype retail outlet. * Sales
contributed a good $99 963 000 to the variance by Prototype as a result if revenue could be improved and the above costs contained this job would be desirable. 3. Inside Rate of Return (IRR) HURDLE
ADJUSTMENT (CPR Dashboard) Internal Rate of Return is an important option to NPV, the IRR summarizes the worth of the job. This level is an internal rate in a sense that it is dependent only in cash
runs of a particular project or perhaps investment, not on prices offered anywhere else hence inside rate of return. * The task has an IRR of 9. 8% which is below the model store IRR as necessary. *
Product sales has to boost by 31. 1%, gross margin must increase by 4. 49 percentage the amount outflow linked to the construction costs has to lower by $41 070 000 before the task can achieve the
required IRR. 2. Transfer sales must increase by thirty-six. % to achieve desired IRR. * The IRR of this project can be far from precisely what is required for the project to become approved. RISK/
OPPORTUNITY ANALYSIS 2. If sales declined by simply 10% Retail store IRR could decline by simply (1. 0) percentage stage, if gross margin decreased by 1 percentage point, Store IRR would decline by
(0. 7) percentage point and If construction cost increased simply by 10% Retail store IRR could decline by simply (0. 1) percentage point. * If sales increased by 10%, Store IRR would enhance by 1 .
0 percentage point, this is certainly a positive indicator. * The IRR will not present a big risk like a 10% drop in product sales only provides a 1 pp decline.
The 10% product sales increase likewise results in the 1 pp enhance. VARIANCE TO PROTOTYPE 2. The IRR for this task is below the prototype store IRR this factors contributed to the adverse IRR. 5.
Land price and real estate taxes should be maintained at current levels as they led a positive 0. 1 and 0. two percentage level respectively for the variance original however site work costs must be
covered they led a negative (2. 6) percentage point to the variance from Prototype. * Sales need to increase they will contributed a positive 1 . on the lookout for percentage point out the
difference this is under the negative 2 . pp by simply site costs. 4. Forecasted profits The profit and reduction summary suggests that the project will make a projected lack of ($1 599 000) inside
the first season of beginning the store compared to the prototype retail store value of ($1 136 000). By the fifth year the store will be making Earnings before Interest and Taxes (EBIT) of $14 034
000 12 months ($8 509 000) modele store benefit. Although the forecasted is producing losses in the first year, it should be approved on the basis of the gains it will make as this will likely
increase the discuss value. five. Projected profits per talk about
Earnings per share are influenced by both the NPV and the earnings/profits of the project/investment. This investment/project has a great NPV and earnings in the end which will leads to increased
income per talk about. 6. Investment required This store needs an investment of $119 300 000 and was scheduled to spread out in Oct 2008. This really is a huge purchase. The NPV is positive, the
population is usually big. The income median of the inhabitants is $48 500 which can be good. The returns on this investment would be achieved many other things by the financial savings that would be
produced from the adverting costs.
Concentrate on already consumes in excess of $100 000 000 in marketing this project will allow them huge merchandising on complimentary thus minimizing advertising costs. It will also enhance brand
image and visibility. 7. Impact on sales of nearby Focus on store There exists a high density of Target stores (45 stores) in the operate area and sales will be expected to originate from existing
goals stores and there is many. This really is a huge city area therefore Target could tap to new consumers and those with the competition. Summary * This project should be accepted by the CEC
depending on the above requirements.
The project presents a lot of risks and opportunities to Goal stores. The opportunities much outweigh the risks, positive larger NPV, boost EBIT, added store to meet target and increase company
image. The project can still be accepted with the IRR figures even though are under a prototype store value. The IRR is based on an indoor value, if for an example the forecasted may be funded by
took out funds as well as the debt cost are below the IRR it might be a perfectly suitable investment. As long as the project has a confident NPV this will likely increase the share value.
This project is very important because of the control area a metropolitan location and a great urban center this will go a long way in achieving the strategic desired goals of marketplace
penetration, company visibility. $119 300 000 is a lot of money but it really can be loaned through personal debt at a rate lower than the IRR. Goldie’s Sq: SUP04M, Shop NPV: ($3, 319) DIFFICULTY
ADJUSTMENT (CPR Dashboard)| Sales| NPV| Revenue would have to enhance by forty five. 1% to accomplish Prototype Shop NPV | IRR| Sales would have to boost by 47. 2% to achieve Prototype Shop NPV| | |
Major Margin| NPV| Gross Perimeter would have to increase by 4. 4 pp to achieve Modele Store NPV | IRR| Gross Margin would have to enhance by 5. 91 pp to achieve Prototype Store NPV| | | Construction
(Building & Sitework)| NPV| Development costs would need to decrease by ($22, 167) to achieve the Model Store NPV| IRR| Construction costs would need to decrease by ($14, 576) to achieve Modele Store
IRR| | | Full Copy Impact| NPV| Sales will have to increase by simply 62. 5% to achieve Model Store NPV| IRR| Revenue would have to maximize by 63. 1% to attain Prototype Store IRR| RISK/OPPORTUNITY|
10% Revenue Decline| NPV| If sales declined simply by 10% Store NPV could decline simply by ($4, 073) |
IRR| If revenue declined by simply 10% Store IRR will decline by simply (1. 1)pp| | | 1 pp GM Decline| NPV| In the event that margin reduced by one particular pp, Retail store NPV might decline by
simply ($3, 929)| IRR| If margin reduced by you pp, Store IRR could decline by (1. 1) pp. | | | 10% Structure cost increase| NPV| If perhaps construction cost increased simply by 10% Retail store NPV
would decline by ($1, 470)| IRR| If construction cost increased by simply 10% Retail outlet IRR might decline by simply (0. 3) pp. | | | Market Perimeter, Wage Charge, etc| NPV| If we applied market
particular assumptions, Retail outlet NPV will increase by $6, 059| IRR| If we applied market specific presumptions, Store IRR would boost by 1 ) pp. | | | 10% Product sales increase | NPV| If
perhaps sales improved by 10%, Store NPV would enhance by $4, 008| IRR| If product sales increased by 10%, Retail store IRR will increase by simply 1 . one particular pp| DIFFERENCE TO PROTOTYPE| The
Goldie’s Square which has a store NPV of ($3, 319) is ($18, 222) below the Modele Store NPV. The following things contributed to the variance. | Land| NPV| Land cost contributed a positive $1, 501 to
the difference from model. | IRR| Land price contributed a good 0. three or more pp for the variance prototype. | | | Non-Land Investment| NPV| Building/sitework costs contributed a bad ($581)
towards the variance via Prototype. IRR| Building/sitework costs contributed a poor (0. 1) pp for the variance coming from Prototype. | | | Sales| NPV| Sales led a negative ($16, 455) to the variance
from Prototype. | IRR| Product sales contributed a poor (4. 4) pp for the variance coming from Prototype. | | | Real Estate Taxes| NPV| Real estate property Taxes added a negative ($2, 682) for the
variance via Prototype. | IRR| Real Estate Taxes contributed a negative (0. 7) pp to the difference from Model. | 1 ) Strategic importance Target wants to build a Super Target store in this area.
Focuses on already have 12 stores from this trade place but are anticipated to have twenty four eventually. The Goldie’s Square market is regarded a key tactical anchor for several retailers. The
Goldie’s Sq . center included Bed Bath & Past, JC Penney, Circuit Metropolis and Boundaries. This is hotly contested area with well-off and fast growing population, which could find the money for
good manufacturer awareness should the growth materialize. Investing in this project will achieve strategic goals of obtaining more outlets in the vicinity and company visibility. The spot is fast
growing and affluent.
The citizenry is growing at a good price of 16% and contains a $56 1000 median cash flow. 2 . Net Present Value HURDLE MODIFICATION (CPR Dashboard) * The project includes a positive net present value
of $300 000. This kind of NPV is incredibly low nonetheless it is still positive. A positive NPV value will certainly results in a rise in share worth. * Based on this element the job should be
acknowledged as it raises share benefit. * This NPV is far under the prototype retail outlet which is a lowest requirement the projected revenue must boost by forty-five. 1% and gross perimeter must
increase by 4. 64 percentage point prior to it can accomplish the model store NPV. The site work cash outflows must decrease by $22 167 1000 and copy sales from all other stores inside the trading
place will have to increase by 62. 5% prior to the prototype shop NPV could be achieved. * The NPV is confident that is suitable however the projected NPV is usually below the model store which can
be factor to get considered to be able to evaluate the job. The forecasted sales will need to increase with a huge percentage in order to reach the prototype store NPV. It is skeptical that the NPV
will ever reach the modele. The CEC must consider other important factors inside the adjudication method.
RISK/OPPORTUNITY ANALYSIS * In the event projected sales declined by 10% the NPV might decrease simply by $4 073 000 (1 358% decline) of the NPV and If gross margin reduce by one particular
percentage point the NPV would decrease by $3 929 1000 (1 310%) decline in NPV. 5. Should building costs increased by 10% the NPV would decline by $1 470 500 (490%) decrease of the NPV and if market
specifics assumptions on industry margins, wages etc happen to be applied the store NPV might decrease by $6 059 000 (2 020%) from the prototype shop NPV. 2. If sales increased simply by 10% the NPV
could increase by $4 008 000 (1 336%) increase in the NPV. This task has really low NPV statistics and this project presents increased risks. A 10% fall in product sales reduces the NPV more than a
thousand instances for an illustration. The NPV figures not necessarily that great due to the risk factors. VARIANCE TO MODEL * The NPV on this project is usually far under the prototype shop value
the next factors written for the difference. * site work, product sales and real estate property tax costs must be within order to mitigate the risk, that they contributed a negative $581 500, $16
455 000 and $2 682 000 respectively to the variance from modele store. Terrain contributed a good $1 501 000 for the variance by Prototype. a few. Internal Charge of Return (IRR) DIFFICULTY
ADJUSTMENT (CPR Dashboard) 2. Internal Price of Return is an important replacement for NPV, the IRR summarizes the value of the task. This level is an internal rate in a way that it will depend on
only on cash moves of a particular project or perhaps investment, not really on rates offered elsewhere hence inner rate of return. 2. The project has an IRR of almost 8. 1% which is below the
original store IRR as necessary. * Revenue has to boost by forty seven. 2%, major margin has to increase by simply 4. 1 percentage the amount outflow linked to the construction costs has to decrease
by $1 576 1000 before the job can achieve the necessary IRR. 2. Transfer revenue must increase by 63. 1% to obtain desired IRR. * The IRR of the project can be far listed below from what is required
for the project being approved. RISK/OPPORTUNITY ANALYSIS 5. If revenue declined simply by 10% Shop IRR could decline by (1. 1) percentage level, if major margin reduced by 1 percentage level, Store
IRR would decline by (1. 1) percentage point and If construction price increased by simply 10% Store IRR will decline simply by (0. 3) percentage level.
If market margin, salary rate etc the IRR would increase by 1 . 6 pp. * If sales elevated by 10%, Store IRR would enhance by 1 ) 1 percentage point, this is certainly a positive signal. * The IRR
would not present a major risk as being a 10% decline in sales only includes a 1 . 1 pp decline. The 10% sales maximize also ends in a 1. one particular pp increase. VARIANCE TO PROTOTYPE 2. The IRR
for this task is below the prototype shop IRR the next factors written for the negative IRR. 5. Land cost must be managed at current levels as they contributed a positive 0. three or more percentage
point out the variance prototype.
Site work costs and real estate property taxes must be contained that they contributed an adverse (0. 1) and (0. 7) percentage point correspondingly to the variance from Model. Sales have to increase
that they contributed a bad 4. four percentage indicate the difference. 4. Projected profits The money and loss summary suggests that the task will make a projected lack of ($1 921 000) inside the
first 12 months of starting the store when compared to prototype shop value of ($654 000). By the 6th year their grocer will be making Earnings ahead of Interest and Tax (EBIT) of $2 951 500 a year
(2 343 000) prototype store value.
Although the projected is definitely making loss in the initial year, it must be accepted on such basis as the profits it will eventually make as this will boost the share benefit. 5. Forecasted
earnings every share Profits per discuss are affected by the NPV as well as the earnings/profits with the project/investment. This kind of investment/project includes a positive NPV and profits in
the long run which will results in increased earnings per share. 6. Investment essential This shop requires an investment of $23 900 500 and was scheduled to spread out in August 2007. This is simply
not a huge investment. The NPV is great, the population can be big, growing at 16%.
The income median of the population is $56 500 which is great. The returns on this purchase are not that impressive. This project is however essential because of its proper location, all the big
retailers want to capture this market for visibility and market capitalization. Since this is not a huge investment it could be considered. six. Impact on sales of local Target stores There are
regarding twelve Goal stores in the trade area and revenue would be supposed to come from existing targets shops as there are various. This control area contains a lot of Concentrate on stores
competitors so a lot of them will come from their website and new markets.
Summary * This project could be accepted by CEC since it is not a big investment. The project shows lots of dangers. The store provides a positive NPV although it is quite low, maximize EBIT, further
store in order to meet target and increase brand image. The project could be acknowledged with the IRR figures even though are below a modele store worth. The IRR is based on an indoor value, if for
a good example the projected may be financed by obtained funds as well as the debt price are below the IRR it will be a perfectly suitable investment. As long as the task has a great NPV this will
likely increase the talk about value.
This project is very important because all retailers was obviously a foothold on this area. This kind of place will go a long way in achieving the strategic goals of market penetration, brand
visibility. Stadium Remodel: SUP1. one particular /S ’04, Store NPV: $14, 911 RISK/OPPORTUNITY| 10% Sales Decline| NPV| In the event that sales rejected by 10% Store NPV would fall by ($7, 854) | IRR
| If perhaps sales decreased by 10% Store IRR would fall by (1. 8)pp| | | 1 pp GENERAL MOTORS Decline| NPV| If perimeter decreased by simply 1 pp, Store NPV would decrease by ($6, 457)| IRR| If
perimeter decreased by 1 pp, Store IRR would decline by (1. 5) pp. | | | 10% Construction price increase|
NPV| If building cost improved by 10% Store NPV would decrease by ($910)| IRR| In the event construction cost increased by 10% Store IRR might decline by simply (0. 3) pp. | | | Market Perimeter,
Wage Price, etc| NPV| If we utilized market specific assumptions, Store NPV would decline by ($11, 317)| IRR| If we applied marketplace specific assumptions, Store IRR would drop by (2. 7) pp. | | |
10% Sales enhance | NPV| If sales increased simply by 10%, Retail outlet NPV might increase by simply $6, 216| IRR| In the event sales increased by 10%, Store IRR would boost by 1 . 5 pp| 1 .
Strategic importance This kind of remodeling is very important to retaining a good image of the brand.
In its current state the store is definitely deteriorating and dilapidating. The facilities will be tarnishing the image of the brand. Goal already consumes millions of dollars in advertising all
this money can be wasted in the event the facilities happen to be in this state as it could count the excellent work. This kind of remodel is definitely thus tactical in maintaining a good brand.
installment payments on your Net Present Value RISK/OPPORTUNITY ANALYSIS 5. If projected sales declined by 10% the NPV would decrease by $7 854 1000 (52. 67% decline) from the NPV and If gross margin
decrease by 1 percentage point the NPV could decrease by simply $6 457 000 (43. 0%) fall in NPV. * Should certainly construction costs increased simply by 10% the NPV will decline by $910 000 (6. 1%)
decrease of the NPV and if market details assumptions about market margins, wages and so forth are applied the store NPV would reduce by $11 317 500 (75. 9%) of the original store NPV. * If sales
increased by 10% the NPV would enhance by $6 216 1000 (41. 69%) increase in the NPV. * This job has a great NPV however sales decrease are a risk and if your local store is not really remodeled
someone buy will decline and reduce success and NPV. 3. Internal Rate of Return (IRR)
RISK/OPPORTUNITY EVALUATION * If sales decreased by 10% Store IRR would decrease by (1. 8) percentage point, in the event that gross margin decreased by 1 percentage point, Shop IRR might decline by
(1. 5) percentage level and If construction cost improved by 10% Store IRR would fall by (0. 3) percentage point. In the event market margin, wage price etc the IRR had been applied could decrease by
2 . 7 pp. 2. If revenue increased by simply 10%, Retail outlet IRR might increase simply by 1 . 5 percentage point * The IRR present a risk as a 10% decline in sales just has a 1 . 8 pp decline. The
10% sales increase likewise results in a one. pp boost. 4. Projected profits The net income and reduction summary suggests that the project will make a projected loss in ($6 ciento tres 000) in the
first season of beginning the store compared to the prototype store value of ($4 812 000). By fifth 12 months the store will probably be making Revenue before Curiosity and Tax (EBIT) of $1 272 000 a
year ($4 025 000) prototype store value. Although the forecasted is making losses in the first season, it should be recognized on the basis of the earnings it will help to make in the future while
this will increase the share value. 5. Projected earnings every share
Income per discuss are affected by both the NPV as well as the earnings/profits in the project/investment. This kind of investment/project provides a positive NPV and revenue in the long run that can
results in elevated earnings per share. 6. Investment required This retail outlet requires a great investment of $17 000 500 and was scheduled to spread out in Mar 2007. This may not be a huge
purchase. The NPV is great, the population and should the services of the retail store be increased sales will improve. 7. Impact on sales of nearby Focus on stores You will see no true impact on
local stores since this is an existing retail store.
The customers that they can might have shed due to the current condition of the store might be seen returning mainly from your competition others from local stores were shoppers would have sought
sanctuary. Conclusion * Target has to accept this kind of project excellent positive NPV and if this investment is usually not built the brand photo would be damaged due to poor, deteriorating
services. This will always be against the tactical imperative of projecting a fantastic brand presence. Final Bottom line: Recommendations to the Capital Expenses Committee The capital expenditure
committee should agree to all the proposals before it.
This will be based on the factors while detailed on part three of this record. The NPV’s of all these types of projects will be positive, an optimistic NPV has contributed favorable for the share
selling price or share value. The interior Rate of Return of those entire tasks are under the prototype store IRR the benchmark project. The IRR is a substitute for NPV however if the NPV is
confident and the IRR is not really what is desired, the NPV may supersede in making a great investment decision. The IRR is what is expected based upon internal factors. Projects which has a low IRR
may be financed through financial debt capital in the event that cost of debts is under the project IRR/ rate of return.
An overarching target of Concentrate on Corporation is always to meet the company goal of adding 75 new shops a year while maintaining a positive manufacturer image. Since all of these retailers have
an optimistic NPV and the long run each of them make great earnings before interest and taxes. The CEC need to accept all of them because they are going to achieve the purpose of market increased and
manufacturer visibility. The Stadium remodel is particularly significant because the shop has deteriorating and dilapidating facilities that will defeat the goal of a positive company image.
The store must be remodeled before that starts influencing the revenue of different Target retailers with poor publicity. Whalen Court is usually to be open in a metropolitan region and it is a great
urban centre. The population of this trade location is very big and provides a good salary median. The project takes a lot of capital investment, nevertheless it presents Goal stores having a unique
contribution in that it will offer totally free advertising towards the corporation. There are a great number of consumers transferring by and Target currently spends in excess of $100 , 000, 000
dollars in advertising beginning this shop might help reduce these costs.
If cash are a limiting factor, Target should pay for the jobs in the subsequent other: your five. Gopher Place should be considered initial. The task requires a 3 000 500 investment. They have the
best NPV and it is over a prototype retail outlet NPV. The sales could decline simply by more than five per cent and it will still be over a prototype store. It has a better EBTI in comparison to the
other costs, though it is present’s hazards it offers prospect as well. 6th. Whalen Courtroom may be the second in line. Excellent positive NPV although it is definitely below the modele store worth.
If product sales improve by 1 . 9%, it would be equal to the modele store NPV.
This is a better NPV when compared to remaining two projects. The store provides a very good market which has a huge population and better income median. 7. Goldie’s Square, the NPV is definitely
positive nevertheless the sales need to still climb by 45. 1% prior to it can satisfy the prototype shop NPV. The NPV is not as very good as can be expected but it remains to be positive. Why is this
a desirable investment is definitely the location which the store will be built in. Job is important for its strategic location, all the big retailers want to capture the foreign exchange market for
presence and marketplace capitalization. Since this is not only a huge purchase it may be regarded as. 8.
Stadium Remodel is paramount the CEC makes this investment failing that the poor state of the facilities will tarnish the of the brand. The NPV is definitely positive and EBIT. Approach analysis
Sales growth inside the retail industries comes from two main options: establishing of recent stores and organic development through existing stores. Fresh stores are costly to build, tend to be
necessary in order to tap into fresh markets and gain access into a brand new pool of shoppers that could probably represent high profit potential depending on the competitive landscape. Increasing
sales of existing stores is also a significant source of expansion and value.
If an existing store functions profitably, it could be considered intended for renovation or upgrading to be able to increase product sales volume, or perhaps if a retail outlet is not really
profitable, after that management need to consider it an applicant for drawing a line under. Target requirements not only look at establishing new stores, but should also use growth ways to grow
revenue of old stores and apply the above mentioned policy. Goal needs to be mindful of the growth strategy of beginning approximately 95 new stores a year. Doug Scovanner need to learn several
lessons via both Wal-Mart and Costco, then take the best out of these lessons.
In year 2000, Wal-Mart acquired 4189 retailers enjoying sales of $178billion. On average, this meant that each shops manufactured sales of $178 billion/4189 = $42, 5million per year. They grew by
6141-4189=1952 shops in 5 years to june 2006. This was usually about 1952/5=390 shops annually. In 12 months 2005, each shop was on average enjoying sales of $309billion/6141shops=$50, 3million per
year. If one examines the rate where sales grew in the a few years, it truly is clear that Wal-Mart simply grew the sales by about 15. five per cent per shop in five years after investing in 1952
shops coming from $42. 5million per shop in yr 2000 to $50. 3 million per shop in year 2006.
On the other hand, if one looks at Costco, by simply 2005, they’d grown to 433 warehouses and had been enjoying product sales of $52, 9 billion dollars. On average, this translates to every shop
making average revenue of about $122, 4 million. Bearing in mind why these two corporations each acquired its own sales strategy, likewise had a different customer base, and a much less often terme
conseillé on selling assortments, but nevertheless Wal-Mart’s approach of substantial investment in new outlets has not provided better that what Costco has been capable of achieve through its fewer
shops which will account for seven percent (433 Costco warehouses in comparison with 6141 Wal-Mart shops) of Wal-Mart outlets in amount by 2006.
In order for Goal to survive and beat Wal-Mart and Costco out of competition, it could need to out-beat them in their strategies. For example , Wal-Mart’s accomplishment was attributed to its
“everyday low price pricing technique that was greeted with delight simply by consumers. This tactic created difficulties for local retailers who have needed to remain competitive. As well, in
addition to growing its top collection, Wal-Mart have been successful in creating efficiency within the company and branching into product lines that presented higher margins than most of its asset
type of items.
These tactics are exactly the strategies that pinpoint must also choose learning from the achievements of its competitor which this shared nearly the same selling assortments and trade area. On the
other hand, Costco owes the success of the claims and very good sales for the membership-fee file format it utilized. It distributed its customer base more closely with Goal. Membership costs
accounted for a tremendous growth source and are remarkably significant to operating profits in a low-profit-margin business. Costco also presented discount pricing for its people in exchange for
membership costs.
Target’s strategy of charges competitively with Wal-Mart in items common to both stores, is a good method for Target. But if Target were to adopt the Costco strategy of account fees and give very
marginal discount which is just bellow Wal-Mart, which in turn it will supplement with account fees, this tactic could discover Target making more profit margins than it is competitors. This will
likely add to the currently successful Concentrate on strategy of offering credit rating to their customers through its different credit features.
In 2006, Target acquired 1397 retailers in forty seven states and boosted sales of $52, 6 billion. This means that normal sales every store had been about $38 million. Costco still overcome both
Wal-Mart and Target when it comes to product sales per shop. This makes us to arrive at the final outcome that the real competitor of Target was Costco, since Target did better than Wal-Mart. The
other critical decision that Target plank of administrators has to re-look into may be the practice of having properties exactly where it constructed stores.
This should really be regarded after rental has been eliminated of question. The primary business of Target is usually retail, it is not necessarily property investment. With these people buying
these properties, could easily make sure they are to end up using money that could have been found in organically growing the existing stores. Bibliography Firer, C. Ross, SA. Westerfield, RW. Test,
BD. Basic principles of Company Finance. fourth South Africa Edition. 2009. McGraw-Hill Education(UK) | {"url":"https://owlrangers.com/target-firm-essay/","timestamp":"2024-11-08T21:07:22Z","content_type":"text/html","content_length":"172918","record_id":"<urn:uuid:019bb7bf-b733-453a-9ad1-b58d8ab945c5>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00551.warc.gz"} |
Second-order nonhom. lin. diff. equations with constant coefficients - The resonance - mathXplain
Second-order equations, Constant coefficient, Homogeneous equation, Characteristic equation, Solution of characteristic equation, Complex solution, Homogeneous solution, Particular solution, Method
of Undetermined Coefficients, Trial Functions Method, Quadratic polynomial, Exponential expression, Expression with sine or cosine, General solution, Resonance.
Text of slideshow
The homogeneous equation and its solution:
If it has two real solutions:
If it has one real solution:
If it has two complex solutions:
Particular solution (Undetermined Coefficients Method)
Here is this equation that is nonhomogeneous:
In such cases we solve the homogeneous equation first,
and then find the particular solution using the Undetermined Coefficients Method.
To solve the homogeneous equation, we solve the usual
characteristic equation.
And now we are ready for the particular solution.
We can obtain the particular solution based on the function on the right side.
This time the function on the right side happens to be a polynomial, so we try to find the particular solution in that form, too.
But the right-side function could be exponential,
or even trigonometric.
The particular solution is:
Let's see what we get if we substitute this into the original equation.
And the general solution is:
Then, we have this other nonhomogeneous equation.
But there is a snag.
Just like with first-order equations, there could be resonance here, too.
Resonance occurs if a term of the homogeneous solution is equal to a term of the particular solution.
Now there is no resonance,
but there will be some in the next slideshow.
Compared to first-order equations, this resonance business gets a bit more complicated at second-order equations.
Here is this equation:
The solution of the homogeneous equation is:
And now we are ready for the particular solution.
We always figure out the particular solution based on the function on the right side.
One term of the homogeneous solution is equal to a term of the particular solution. That means that sadly, there is resonance.
In this case the constant multiplier doesn’t matter.
Due to the resonance, an x will come in here.
Now we compute the first and second derivative of the particular solution.
Next, we substitute these into the original equation.
When the characteristic equation has only one real solution, there could be dual resonance.
The resonance appeared.
Therefore, a multiplier of x will be needed in the particular solution.
But then there will be resonance with the second term...
so we need another x multiplier.
This is what we call dual resonance.
From here, the solution proceeds as usual.
Boring, as usual.
So let’s not solve it now. Instead, let’s see what kind of resonance could appear when the characteristic equation has two complex roots.
Here are these two equations:
The characteristic equations are:
For the complex solution all we need to know is this:
In such cases resonance occurs if:
And then the trial function is done.
Second-order nonhom. lin. diff. equations with constant coefficients - The resonance | {"url":"https://www.mathxplain.com/calculus-3/differential-equations/second-order-nonhomogeneous-linear-differential-equations-with-0","timestamp":"2024-11-11T04:46:06Z","content_type":"text/html","content_length":"77655","record_id":"<urn:uuid:db7811a8-917e-478b-92c2-33685ef8e494>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00578.warc.gz"} |
Exercise Quiz 2
Question 1 of 1
In a consumer survey performed by a newspaper, 20 different groceries (products) were purchased in a grocery store. Discrepancies between the price appearing on the sales slip and the shelf price
were found in 6 of these purchased products.
Let $ X $ denote the number of discrepancies when purchasing 3 random (different) products within the group of the 20 products in the store. What is the mean and variance of $ X $? | {"url":"https://02402.compute.dtu.dk/quizzes/exercise-quiz-2","timestamp":"2024-11-05T08:50:14Z","content_type":"text/html","content_length":"8240","record_id":"<urn:uuid:a647647c-dcea-4d24-9bc0-133da0775494>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00573.warc.gz"} |
Stochastic Volatility
Time Series Volatility
SV models are state-space models where the volatility process is stochastic. In contrast, the volatility process in GARCH models is deterministic.
Model: SV 1
$$ y_{t} = \mu + \epsilon_{t}^{y}, \epsilon_{t}^{y} \sim N(0, e^{h_{t}}) $$ $$ h_{t} = \mu_{h} + \phi_{h}(h_{t-1} - \mu_{h}) + \epsilon_{t}^{h}, \epsilon_{t}^{h} \sim N(0, \omega_{h}^{2}) $$
Model: SV 2
$$ y_{t} = \mu + \epsilon_{t}^{y}, \epsilon_{t}^{y} \sim N(0, e^{h_{t}}) $$ $$ h_{t} = \mu_{h} + \phi_{h}(h_{t-1} - \mu_{h}) + \rho_{h}(h_{t-2} - \mu_{h}) + \epsilon_{t}^{h}, \epsilon_{t}^{h} \sim N
(0, \omega_{h}^{2}) $$
We recommend using 100 * (log returns) as input series for numerical stability reasons.
Note: Asynchronous Routines
These routines are asynchronous. Once you hit Queue Job, the request is submitted to the computing engine. You can send as many jobs as you need to the engine queue. Each will run on a separate
thread and on any available cores or as they become available. You can periodically check on the status of the job. When it indicates that it is done, you can simply click on the left block of the
requested job and hit Fetch. You can also simply terminate the job if you want to by hitting Try Kill.
• draws: Run the sample this many times.
• burn-in: Discard these many samples from the beginning draws. Burn-in must be less than draws.
• timeout: Time limit in minutes.
• coef: Coefficient estimates for $g$ for GARCH parameters and $a$ for ARCH parameters
• serr: Standard Errors
• $ e^{h_{t}/2} $: Volatility Process. Note that the state variable log volatility $ y_{t} | h_{t} \sim N(0, e^{h_{t}}) $ is equivalent to $ y_{t} = e^{h_{t} / 2} \epsilon_{t} $.
• Stochastic Volatility: Likelihood Inference and Comparison with ARCH Models (Kim, Shephard, Chib) | {"url":"https://docs.pnumerics.com/docs/econometrics/stochastic_vol/","timestamp":"2024-11-13T22:36:47Z","content_type":"text/html","content_length":"17416","record_id":"<urn:uuid:b4f39939-3951-4141-a742-13cb05b964a1>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00655.warc.gz"} |
Surface gravity waves | Applied Mathematics
The type of wave motion which most people are familiar with are waves that occur on the free surface of water. For example, the ripples that occur when a small rock is dropped into the water or the
waves that can be seen breaking on a beach (Pinery Provincial Park on Lake Huron, below).
In this type of wave motion the restoring force is gravity (sometimes surface tension needs to be considered as well, and in some situations surface tension can dominate gravity) and therefore these
waves are known as surface gravity waves. We would like to create a mathematical formulation of this problem. To begin we will make some simplifying assumptions:
1. Inviscid: nu = 0
2. Constant density: rho = const => grad u = 0
3. Irrotational: vortcity = 0
4. The mean depth of the fluid is constant: H = const
5. The fluid is invariant with y (the in-page direction): derivative with respect to y -> 0
The displacement of the free surface will be denoted by,
Thus, our surface is found at,
From the irrotational assumption we can deduce that,
Then using our incompressibility (constant density) assumption we get our first governing equation,
To get the remainder of the equations we now need to consider some boundary conditions. First we will look at the bottom boundary. The bottom is solid and hence there is no flow normal to this
boundary. Mathematically this gives us w=0 at z =-H , assuming we have a flat bottom. This is the kinematic bottom condition.
We have considered the bottom boundary so next we will look at the surface. At the surface we want to ensure that particles that start off on the surface always remain on the surface. This condition
is known as the kinematic surface condition. Following a particle which is on the surface the rate of change of the difference of the z position and the displacement of the surface is zero,
Mathematically, it gives the following condition, note that this condition is nonlinear however it does assume there is no wave breaking:
The final boundary condition is the balance of forces on at the free surface. Since the fluid is inviscid, the forces at the surface are the pressures above and below the surface (a surface force
from the stress tensor) and the surface tension (a line force that occurs only at the surface, and is chemical in nature).
After some work the balance of forces gives us the following equation for pressure at the surface:
The atmospheric pressure is often taken to be approximately zero (relative to the pressure in the water). Since, we have inviscid and irrotational flow we can use Bernoulli's Equation,
and without loss of generality we can take B(t) = 0 this is a tricky point that you should think about) which gives our last boundary condition, the dynamic surface condition.
To summarize what we have deduced so far:
The above problem is fully nonlinear and still remains unsolved. The next thing to consider are waves that have a relatively small amplitude, a << 1. For this case we will set,
Since, a^2 << a we can use this to linearize our problem, to first order this gives us the linearized wave equations:
Solving the linearized wave equation
To solve the linearized wave equation we start by looking for normal mode solutions of the from,
It is sufficient to only consider solutions of this form since any continuous function can be written as a sum of sines and cosines.
From this starting point and using the kinematic boundary condition at the bottom and the dynamic boundary condition at the surface one can deduce that we should look for solutions of the following
Using Laplace's, equation and the kinematic boundary conditions we can determine that our solution takes on the form:
Now using the dynamic boundary condition at the free surface we determine a dispersion relation for our problem,
This gives us a relation between the frequency, omega and the wavenumber, k. Also, from this we can determine the phase speed, c, which is the ratio of the frequency to the wavenumber.
Special limits
Looking at the dispersion relation we deduce some special limits and look at the types of waves these limits produce. Since surface tension only shows up in one of the terms we can look at the
weighting of the gravity term, gk, versus the surface tension term, sigma k^3 / p.
If surface tension dominates over gravity, i.e.. sigma >> rho g / k^2 , then we have waves whose restoring force is mainly due to surface tension. These waves are known as capillary waves.
Under normal situations when do capillary waves occur?
At an air water interface at temperatures of about 20 degrees Celsius we have the following values for our parameters, sigma = 0.0728 N/m and p = 1000 kg/m^3. Which gives k^2 >> 1.35x10^5 m^-2
for capillary waves or lambda << 1.7cm ~ 2cm. This tells us for waves with wave length much less then 2cm gravity is negligible and surface tension is the dominating force, giving us capillary
Now if we assume that we have waves with wavelength much longer than 2cm we can neglect surface tension, giving the following dispersion relation:
If kH << 1 then we have waves with a long wave length relative to the depth of the water, this is known as the shallow water limit (or sometimes long gravity waves). In this case the dispersion
relationship can be approximated by w^2 = gHk^2 which gives c = sqrt(gH) for the phase speed. Note that the phase speed is independent of the wave length, this makes these waves non-dispersive.
The picture below shows the pathlines of particles to a first order approximation.
If kH >> 1 the wavelength is short relative to the depth of the water. This limit is known as the deep water limit (or sometimes referred to as short gravity waves). The dispersion relation and the
phase speed can be written as follows, w^2 = gk and c = sqrt (g/k) The propagation speed of the waves in this case depends on the wavelength, making waves in this limit dispersive. The picture
below shows the pathlines of particles to a first order approximation.
Stokes' Drift
When looking at waves at the beach you can notice that the waves do not transport things (like seaweed of drift wood) at the same speed at which the crests and troughs of the waves propagate.
However, objects still appear to be moving in the direction of wave propagation, albeit a lot slower. This effect is called Stokes' Drift. It is due to second order (nonlinear) effects. Looking at
the pathlines of a particle one can see that the pathline for one period is not a closed loop nor does it follow the shape of the wave.
Wave refraction
Another phenomena that can be noticed when at the beach is that waves appear to arrive parallel to the shore. This phenomena is known as wave refraction. Assume we have a depth that decreases as we
approach the shore, i.e. H = H(x). We can now look at how this affects the phase speed of our gravity wave in shallow water:
From this relation we can see that the phase speed of the waves decreases as the depth decreases. This will cause waves approaching the shore at an angle to rotate parallel to the shore as they
Group velocity
Group velocity is a concept that is initially quite difficult to understand. Mathematically, it is defined as,
Physically, it is the speed at which energy travels. If there is no energy then no waves can be present and thus the group speed tells us how far a wave packet propagates in a given amount of time.
Using the above definition for the group velocity, we come up with the following expression for the group velocity in surface gravity waves: | {"url":"https://uwaterloo.ca/applied-mathematics/current-undergraduates/continuum-and-fluid-mechanics-students/amath-463/surface-gravity-waves","timestamp":"2024-11-07T09:57:59Z","content_type":"text/html","content_length":"127814","record_id":"<urn:uuid:7290edb4-de00-46df-8b99-838be37789ed>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00793.warc.gz"} |
Bayesian Neural Networks
In this tutorial, we demonstrate how one can implement a Bayesian Neural Network using a combination of Turing and Lux, a suite of machine learning tools. We will use Lux to specify the neural
network’s layers and Turing to implement the probabilistic inference, with the goal of implementing a classification algorithm.
We will begin with importing the relevant libraries.
Our goal here is to use a Bayesian neural network to classify points in an artificial dataset. The code below generates data points arranged in a box-like pattern and displays a graph of the dataset
we will be working with.
# Number of points to generate
N = 80
M = round(Int, N / 4)
rng = Random.default_rng()
Random.seed!(rng, 1234)
# Generate artificial data
x1s = rand(rng, Float32, M) * 4.5f0;
x2s = rand(rng, Float32, M) * 4.5f0;
xt1s = Array([[x1s[i] + 0.5f0; x2s[i] + 0.5f0] for i in 1:M])
x1s = rand(rng, Float32, M) * 4.5f0;
x2s = rand(rng, Float32, M) * 4.5f0;
append!(xt1s, Array([[x1s[i] - 5.0f0; x2s[i] - 5.0f0] for i in 1:M]))
x1s = rand(rng, Float32, M) * 4.5f0;
x2s = rand(rng, Float32, M) * 4.5f0;
xt0s = Array([[x1s[i] + 0.5f0; x2s[i] - 5.0f0] for i in 1:M])
x1s = rand(rng, Float32, M) * 4.5f0;
x2s = rand(rng, Float32, M) * 4.5f0;
append!(xt0s, Array([[x1s[i] - 5.0f0; x2s[i] + 0.5f0] for i in 1:M]))
# Store all the data for later
xs = [xt1s; xt0s]
ts = [ones(2 * M); zeros(2 * M)]
# Plot data points.
function plot_data()
x1 = map(e -> e[1], xt1s)
y1 = map(e -> e[2], xt1s)
x2 = map(e -> e[1], xt0s)
y2 = map(e -> e[2], xt0s)
Plots.scatter(x1, y1; color="red", clim=(0, 1))
return Plots.scatter!(x2, y2; color="blue", clim=(0, 1))
Building a Neural Network
The next step is to define a feedforward neural network where we express our parameters as distributions, and not single points as with traditional neural networks. For this we will use Dense to
define liner layers and compose them via Chain, both are neural network primitives from Lux. The network nn_initial we created has two hidden layers with tanh activations and one output layer with
sigmoid (σ) activation, as shown below.
The nn_initial is an instance that acts as a function and can take data as inputs and output predictions. We will define distributions on the neural network parameters.
# Construct a neural network using Lux
nn_initial = Chain(Dense(2 => 3, tanh), Dense(3 => 2, tanh), Dense(2 => 1, σ))
# Initialize the model weights and state
ps, st = Lux.setup(rng, nn_initial)
Lux.parameterlength(nn_initial) # number of parameters in NN
The probabilistic model specification below creates a parameters variable, which has IID normal variables. The parameters vector represents all parameters of our neural net (weights and biases).
# Create a regularization term and a Gaussian prior variance term.
alpha = 0.09
sigma = sqrt(1.0 / alpha)
We also define a function to construct a named tuple from a vector of sampled parameters. (We could use ComponentArrays here and broadcast to avoid doing this, but this way avoids introducing an
extra dependency.)
function vector_to_parameters(ps_new::AbstractVector, ps::NamedTuple)
@assert length(ps_new) == Lux.parameterlength(ps)
i = 1
function get_ps(x)
z = reshape(view(ps_new, i:(i + length(x) - 1)), size(x))
i += length(x)
return z
return fmap(get_ps, ps)
vector_to_parameters (generic function with 1 method)
To interface with external libraries it is often desirable to use the StatefulLuxLayer to automatically handle the neural network states.
const nn = StatefulLuxLayer{true}(nn_initial, nothing, st)
# Specify the probabilistic model.
@model function bayes_nn(xs, ts; sigma = sigma, ps = ps, nn = nn)
# Sample the parameters
nparameters = Lux.parameterlength(nn_initial)
parameters ~ MvNormal(zeros(nparameters), Diagonal(abs2.(sigma .* ones(nparameters))))
# Forward NN to make predictions
preds = Lux.apply(nn, xs, vector_to_parameters(parameters, ps))
# Observe each prediction.
for i in eachindex(ts)
ts[i] ~ Bernoulli(preds[i])
bayes_nn (generic function with 2 methods)
Inference can now be performed by calling sample. We use the NUTS Hamiltonian Monte Carlo sampler here.
# Perform inference.
N = 2_000
ch = sample(bayes_nn(reduce(hcat, xs), ts), NUTS(; adtype=AutoTracker()), N);
┌ Info: Found initial step size
└ ϵ = 0.4
Now we extract the parameter samples from the sampled chain as θ (this is of size 5000 x 20 where 5000 is the number of iterations and 20 is the number of parameters). We’ll use these primarily to
determine how good our model’s classifier is.
Prediction Visualization
We can use MAP estimation to classify our population by using the set of weights that provided the highest log posterior.
# A helper to run the nn through data `x` using parameters `θ`
nn_forward(x, θ) = nn(x, vector_to_parameters(θ, ps))
# Plot the data we have.
fig = plot_data()
# Find the index that provided the highest log posterior in the chain.
_, i = findmax(ch[:lp])
# Extract the max row value from i.
i = i.I[1]
# Plot the posterior distribution with a contour plot
x1_range = collect(range(-6; stop=6, length=25))
x2_range = collect(range(-6; stop=6, length=25))
Z = [nn_forward([x1, x2], θ[i, :])[1] for x1 in x1_range, x2 in x2_range]
contour!(x1_range, x2_range, Z; linewidth=3, colormap=:seaborn_bright)
The contour plot above shows that the MAP method is not too bad at classifying our data.
Now we can visualize our predictions.
\[ p(\tilde{x} | X, \alpha) = \int_{\theta} p(\tilde{x} | \theta) p(\theta | X, \alpha) \approx \sum_{\theta \sim p(\theta | X, \alpha)}f_{\theta}(\tilde{x}) \]
The nn_predict function takes the average predicted value from a network parameterized by weights drawn from the MCMC chain.
# Return the average predicted value across
# multiple weights.
function nn_predict(x, θ, num)
num = min(num, size(θ, 1)) # make sure num does not exceed the number of samples
return mean([first(nn_forward(x, view(θ, i, :))) for i in 1:10:num])
nn_predict (generic function with 1 method)
Next, we use the nn_predict function to predict the value at a sample of points where the x1 and x2 coordinates range between -6 and 6. As we can see below, we still have a satisfactory fit to our
data, and more importantly, we can also see where the neural network is uncertain about its predictions much easier—those regions between cluster boundaries.
# Plot the average prediction.
fig = plot_data()
n_end = 1500
x1_range = collect(range(-6; stop=6, length=25))
x2_range = collect(range(-6; stop=6, length=25))
Z = [nn_predict([x1, x2], θ, n_end)[1] for x1 in x1_range, x2 in x2_range]
contour!(x1_range, x2_range, Z; linewidth=3, colormap=:seaborn_bright)
Suppose we are interested in how the predictive power of our Bayesian neural network evolved between samples. In that case, the following graph displays an animation of the contour plot generated
from the network weights in samples 1 to 1,000. | {"url":"https://turinglang.org/docs/tutorials/03-bayesian-neural-network/index.html","timestamp":"2024-11-10T13:00:26Z","content_type":"application/xhtml+xml","content_length":"1049280","record_id":"<urn:uuid:8743e7d4-8ba8-4a1b-8e02-a431b8662513>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00537.warc.gz"} |
A computational fluid dynamics simulation of oil air flow between the cage and inner race of an aero engine bearing
Proceedings of the ASME Turbo Expo 2016: Turbine Technical Conference and Exposition GT2016 June 13-17, 2016, Seoul, South Korea
GT2016-56927 DRAFT
Akinola A. Adeniyi∗
Email: [email protected]
Herv ´e Morvan†
Gas Turbine & Transmissions Research Centre (G2TRC) The University of Nottingham, Nottingham, NG7 2RD, UK
Kathy Simmons
In aeroengines the shafts are supported on bearings that carry the radial and axial loads. A ball bearing is made up of an inner-race, an outer-race and a cage which contains the balls, these
together comprise the bearing elements. The bearings re-quire oil for lubrication and cooling. The design of the bear-ing studied in this work is such that the oil is fed to the bearbear-ing through
holes/slots in the inner race. At each axial feed location the oil is fed through a number of equispaced feedholes/slots but there is different number of holes at each location. Once the oil has
passed through the bearing it sheds outwards from both sides into compartments known as the bearing chambers.
A number of studies have been carried out on the dynamics of bearings. Most of the analyses consider the contributions of fluid forces as small relative to the interaction of the bearing ele-ments.
One of the most sophisticated models for a cage-raceway analysis is based on the work of Ashmore et al. [1], where the cage-raceway is considered to be a short journal bearing divided into sectors by
the oil feeds. It is further assumed that the oil ex-its from the holes and forms a continuous block of oil that exex-its outwards on both sides of the cage-raceway. In the model, the Reynolds
equation is used to estimate the oil dynamics.
Of interest in this current work is the behaviour of the oil and air within the space bounded by the cage and inner race. The aim is to determine whether oil feed to the bearing can be modelled as
∗[seconded from the University of Ilorin, Nigeria]
coming from a continuous slot or if the discrete entry points must be modelled. A Volume of Fluid Computational Fluid Dynamics approach is applied. A sector of a ball bearing is modelled with a fine
mesh and the detailed simulations show the flow behaviour for different oil splits to the three feed locations of the bearing thus providing information useful to understanding oil shedding into the
bearing chambers.
The work shows that different flow behaviour is predicted by models where the oil inlets through a continuous slot compared to discrete entry holes. The form as speed of oil shedding from the bearing
is found to depend strongly on shaft speed with the shedding speed being slightly higher than the cage linear speed. The break-up pattern of oil on the cage inner surface suggests smaller droplets
will be shed at higher shaft speed.
Keywords:Aeroengine bearing, Annular flow, Oil systems, Oil shedding, VOF.
Ao Cross-sectional area of leaving oil (Aoc,Aob)[m2]
As Jet spread area on the cage [m2]
Ah Area on the feed hole [m2]
D Bearing ball diameter [m]
g Acceleration due to gravity, [m/s2] Hc Film thickness on the cage, [µm]
mf Domain feed rate [kg/s]
mo Outflow mass flow rate [kg/s]
mp Mass flow rate through periodic planes [kg/s]
mr Mass rate retained [kg/s]
Qo Outflow rate [m3/s]
Rs Radius of the shaft (inner-race) diameter [m]
t Flow time [s]
− →
U Flow speed [m/s]
Vo Exit oil velocity to bearing/chamber [m/s]
Vs Shaft linear speed at the inner race [m/s]
Wc Width of the cage [m]
Wr Width of the raceway [m]
X, Y, Z Coordinates
∆x Length scale [m]
CFD Computational Fluid Dynamics
VOF Volume of Fluid
Greek symbols
α Volume fraction
ωc Cage angular rotation, [rad/s,r pm] ωr Raceway angular rotation, [rad/s,r pm] Ωs Shaft rotation, [rad/s,r pm]
θ Film extent, [rad,o]
φ Level set function
The purpose of the oil system in an aeroengine is for lubri-cation and cooling of transmissions elements including the shaft-supporting bearings. These highly rotating elements are housed within
bearing chambers. The purpose of the bearing chamber is to contain the oil and an important aspect of chamber function is that the oil exits efficiently allowing quick recycling. The oil is used in a
loop; the oil is pumped from the oil tank and fed to a bearing via its inner-race which has feed-holes and slots to get the oil into the bearing element interstices. The flow inside bearing chambers
is complex and The University of Nottingham Technology Centre in Gas Turbine Transmissions Systems has been investigating such systems for some years.
The bearing chamber can be seen as an annular flow system in its simplest form. Flow in rotating concentric cylinders was first investigated in Taylor’s [2] classical analytical work on sim-ple
rotating concentric cylinders encasing a viscous fluid. There have since been a number of significant investigations many of which are relevant to bearing chamber flow. A basic simplifying assumption
in the mathematical models, is that the bearing cham-ber flow is a multiphase annular flow with the inside of the outer wall coated with a thin film driven by shear. This simplifica-tion has paved
the way to significant progress in model develop-ment and system understanding. These so-called thin film
mod-els are based on lubrication theory with development for bearing chamber applications following chronolgically through Farrall et al [3], Williams [4], Kay et al [5] and Kakimpa et al [6].
Stud-ies on more engine representative geometrStud-ies, including [7–9] have investigated computationally the flow phenomenon inside the bearing chamber cavity. The recurring assumption in the models
is that we understand the shedding characteristics of the oil from the bearing or in the simpler models, that the flow is a simple rimming flow. Experimental work such as those of Chan-dra et al [10]
and Glahn et al [11] provide information on what happens to the oil after being shed from the bearings but little is known about the breakup process inside the bearing itself.
The work reported here is part of a larger study that seeks to provide understanding of what happens inside a bearing. One of the justifications for understanding the flow inside the bear-ing is the
need for more accurate boundary conditions for our existing bearing chamber models as identifed by [8, 12]. In par-allel research an experimental test facility has been constructed that will allow
visualisation and other data acquisition relating to oil shedding from an aeroengine ball bearing. The configura-tion investigated computaconfigura-tionally is intended to be as similar as possible
to that of the experimental investigation.
Figure 1 shows a schematic representation of the test sec-tion of the bearing as configured in the test facility. The stub shaft is driven up to engine representative speeds by a motor and axial
loads are applied via a large electromagnet. Oil is fed to the bearing via the inner-race at locations as shown in Figure 2 and exits into the front and rear chambers as shown. The test fa-cility is
instrumented and there is good visual access to the front chamber so that high speed cameras can be used to provide in-formation about the exiting oil. There is no visual access to the bearing itself
(the region marked in red on Figure 1) and limited opportunity for measurements also. The intention is that the CFD and experimental data together will provide a clearer picture of the flow into the
bearing chamber from the bearing.
One of the challenges faced in modelling the flow inside the bearing itself is the complexity of the bearing geometry. As can be seen in Figure 2 the 25 bearing balls are positioned within a cage
that restrains their motion and this creates some very small gaps within the model. Both the outer and inner races are also schematically illustrated in Figure 2. This figure shows the three feed
routes to the bearing with the desired split of oil to these different locations being achieved through different numbers of feedholes as shown. In the example given here, there are 9 and 6 holes,
respectively to the front and rear of the rig while 6 slots deliver oil to the mid-split of the inner-race.
single ball, a periodic domain of 14.4◦is suitable but the feeds make this a little involved. It is therefore important to understand the behaviour of the feed system.
In previously reported modelling of this aeroengine bearing Adeniyi et al [13] simplified the feed arrangement by represent-ing it as a continuous slotted feed. However, although some rel-evant and
useful data was obtained, concerns remained as to the validity of this simplification. In the work reported in this paper, modelling focus is on a region within the overall bearing model focussing on
the feed arrangement and geoemetry immediately surrounding. The aim of the work is to better establish whether it is truly valid to model oil feed to a bearing that comes via discrete inlet holes as
a continuous slot feed.
The elements of a roller bearing interact in a complex way. There is wide body of work on bearing dynamic analysis with a range of different simplifying assumptions. Work on the dy-namic analysis of
a bearing, including consideration for the lu-bricant probably finds its roots in the 1978 work of Kannel & Walowit [14]. However the ADORE bearing analysis software (Gupta [15]), one of the most
significant tools available to the bearing designer owes most to Gupta who developed the original commercial code and has a body of work too large to reference it all. The basic assumption in the
dynamic models is that the fluid forces are generally small compared with the interaction forces of the bearing elements. In describing the hydrodynamic influence, a Reynolds equation formulation is
solved. In the for-mulations, the fluid (oil) is taken as single phase in the domain, for example between the cage and inner race and at the contact points [1, 15].
Ashmore et al. [1] made modelling progress by assuming
FIGURE 2: UNDER-RACE OIL FEED
that the cage-raceway can be considered to be a short journal bearing divided into sectors by the oil feeds providing hydrody-namic support for the cage. In the work presented in this paper,
following Ashmore et al. [1], the cage-race is assumed to be rel-atively unaffected by the bearing balls such that the flow can be studied in isolation but unlike their work, the 3D Navier-Stokes
equations are solved with consideration for the oil-air interface.
In the analysis of rolling element bearings there are essen-tially two approaches: quasi-static; and dynamic [1, 15–17]. In quasi-static analysis, for each of the bearing elements, the force and
moment equilibrium equations acting are modelled such that they include the dynamic components such as the externally ap-plied forces, centrifugal forces and gyroscopic moments. These result in a set
of non-linear algebraic equations that require nu-merical techniques such as Newton-Raphson methods to solve and from which, angular velocities of the elements can be cal-culated. Jones [16] was
among the first to analyse the motion of a high speed ball bearing using a simplified quasi-static analysis of ball motion with sliding friction but without the contributions of the lubricants. The
quasi-static approach, in essence, repre-sents steady state conditions and the raceway control assumption is used.
that only pure rolling exists at non-zero contact angle at either of the races. It is, however, useful to estimate steady state re-sults [18, 20]; Wang et al. [21] recently proposed a technique to
remove the raceway control assumption.
The dynamic analysis is a real-time representation of the bearing and does not require any kinematic contraints as required in quasi-static analysis. In the dynamic models, the equilibrium equations
from the quasi-static model are formed using differen-tial equations of motion for each of the bearing elements [15,22].
The Cage-Raceway Analysis
The focus of this paper is the cage-race part of the bearing. The model of Ashmore et al. [1] of the cage-raceway is illus-trated in Figure 3. In the figure, there is an eccentricity, e as shown by
the offset centresOr andOcfor the raceway and cage
respectively. The raceway speed isωrand the cage speed isωc.
The oil from the feedholes forms blocks of oil exiting outwards and spreading to extentsθ1−θ2.
FIGURE 3: ILLUSTRATION OF THE MODEL OF ASHMORE ET AL [1] FOR AN OIL-FED CAGE-RACEWAY
The speed of the inner-race is the same as that of the shaft unless there is any undesirable slipping between it and the shaft. In this work slip is assumed not to take place. The speed of the cage
can be estimated from the analysis of the rolling element interaction. This was not felt to be necessary for this study and instead the cage speed was chosen on the basis of estimations obtained from
[18, 20]. In the current analysis, the cage is taken to be concentric with the shaft.
The fluid dynamics
The main focus of this work is the behaviour of the oil in the cage-raceway geometry. A computational fluid dynamics (CFD)
approach is used to understand the flow of oil and air. A simula-tion of the entire 360◦geometry would be expensive to achieve because of the length scales and computational mesh resolution involved.
Instead, from the frame of reference of the oil feed, the flow is solved as a periodic flow based on the physical spac-ing of the holes. The effect of gravity on the fluid within the width of the
raceway is considered negligible as the centrifugal force overrides (ieΩ2[s]Rs/g>>1)). The continuity equations and
Navier-Stokes equations are solved isothermally.
THE MODEL SETUP The Geometries
Figure 4 shows one of the geometries used in the simulations in this work. The chosen periodicity planes are shown. One of the periodic geometries has periodic planes spaced 40◦apart and the other
has 60circ. The width and diameters are the same in both cases. The diameter of the feed holes (Dh) is 1.76mm, as
was the case in Ashmore et al. [1]. A fixed annular spacing of 0.22Dhbetween the inner-race and the cage inner-wall is used.
The widths of the cage (Wc) and the raceway (Wr), as used in the
simulation, are 6.14Dhand 3.81Dhrespectively. The geometry
is not symmetric about the feed as can be seen in Figure 6. There is a small fillet radius on the cage at the side furthest from the rolling element.
In another of the cases investigated, the feed holes were re-placed with slotted feeds at a periodicity of 14.4◦(ie one slot per ball). In all cases, holes and slots, the feeds deliver same total
amount of oil. There are many ways to choose the width of an equivalent slot; but in this paper, the widths chosen are 10% and 20% of the feed hole diameter.
For a qualitative comparison, a 360◦geometry with one feed hole was created, as illustrated in Figure 5. This removes the periodicity assumption. In this case, the geometry is radially split into two
zones. The zone with the feed rotates and the non-rotating zone is in another frame of reference with a slid-ing interface between the two zones.
The Computational Mesh
A structured mesh (ANSYS-ICEM Hexa Mesh) is used in the periodic geometry setups as illustrated in Figure 6. There is a better control over the number of cells as well as better nodal-mesh spacing
efficiency in a Hexa nodal-mesh compared to unstruc-tured meshes but the latter are easier to construct for more com-plex geometries. The setups with periodicity are straight forward to construct
using ICEM “logical blocking” except for the sub-millimeter gap sizes that require the use of very small cells to provide sufficiently high resolution.
[image:5.612.46.284.72.268.2] [image:5.612.41.287.313.482.2]
FIGURE 5: CAGE-INNER RACE FULL 360◦ROTATING GE-OMETRY
FIGURE 6: BEARING OIL SHEDDING RIG
side of the sliding interface. The total size of mesh is largely de-termined by the number of cells in the gap between the inner-race and the cage. Two meshes with 14 and 20 cells across the gap have
been used in this paper to provide an amount of mesh
depen-dence study. These resulted in meshes of 235,000 and 1,500,000 cells. The mesh size in the 360◦geometry is about 4.2 million cells.
The solution method
The coupled level set volume of fluid (CLSVOF) technique as proposed by Sussman & Puckett [23] and implemented in AN-SYS Fluent [24] is used in this work. In the volume of fluid (VOF) technique, a
colour function,α, is solved to describe the
oil-air mixture. Where the volume fraction is 1, the domain is completely filled with oil and ifα=0 it is completely air filled.
The free-surface is in the region 0<α <1. The level set
ap-proach uses a smooth signed function, φ such that a zero level set iso-surface represents the free-surface and a positive or neg-ative level set describes either phase of the mixture. The SST
k−ωturbulence model was used; this model has modifications to resolve low Reynolds number [24].
The timestep,∆t, for the simulations is of the order 0.1µs chosen by keeping the CFL number (Courant-Lewy-Freidrich number,CFL=|−→U|/(∆x/∆t)) less than 1; where∆xis the mesh
size length scale and−→U is the flow speed. Keeping CFL less than 1 is essential for solution stability for the explicit transient VOF formulation used. The second order accurate upwind differenc-ing
scheme is used for the spatial terms of the continuity and the momentum equations. First order accurate explicit time integra-tion is used for the temporal terms while convergence criterion set for
the residuals is 10−4[and a maximum of 30 iterations per] time step. The simulations were run using 8 cores on 4 machines with 16 GB RAM per machine on the University of Nottingham’s high performance
computing cluster. The simulations run 1.1µs
flow time per CPU compute hour for the 360◦ rotating setup, while the periodic setups take between 8 and 110µs/CPU-hour.
To monitor the flow for steady state behaviour, the residual mass flow within the system is monitored. Consider the system of flow schematically given in Figure 7. The 3D control volume is
represented with the dashed lines.
The system is fed (through hole or slot) at the rate of ˙mf. By
definition the flow into the system through the periodic bound-aries matches the flow out and this is represented as ˙mp. The oil
mass flow through the “chamber side” and “bearing side” bound-aries of the domain is represented as ˙mo1and ˙mo2 respectively and the rate of mass retention within the system is ˙mr. Mass
mf+m˙p=m˙o1+m˙o2+m˙p+m˙r (1a)
mr=m˙f−(m˙o1+m˙o2) (1b)
∂mr ∂t =
∂t mf−[mo1+mo2]
∂t ≈0 (1d)
Boundary Conditions
The inner-race and the feed hole pipe are merged as a sin-gle entity named “inner-race”. The cage and the inner-race are specified as no-slip wall boundaries. The walls are specified as “moving
walls” with their respective angular velocities. To use the frame of reference of the feed, the inner-race angular veloc-ity is set as 0r pmand the cage’s relative angular velocity is the difference
between its and those of the inner-race. The exits into the bearing chambers are pressure boundaries in all cases. The inlet of the feed holes are specified as velocity inlets with the ve-locity
specified as normal to the inlet. Periodicity is imposed on the locations as indicated in Figure 4.
Total bearing oil flow rates of 8 and 10 litres per minute (1.33×10−4 kg/s and 1.66×10−4 kg/s were investigated for 5,000 and 10,000 rpm shaft speeds. These are typical values within the aerospace
context. The cage speeds used are 2,658 rpm (for shaft speed 5,000 rpm) and 4,985 rpm (for shaft speed 10,000 rpm).
Table 1 gives the matrix of the cases that have been run. The case nomenclature is such that those starting withSrefer to the cases with slotted feed,Qrefer to the feed through a hole in the race
with 40◦separation andHfor hole feed with 60◦separation
FIGURE 7: FLOW MONITOR
whileF1 is the case with 1 rotating oil feed. Case Q1 is the one with 14 cells in the cage-race gap others have 20 cells in the gap.
TABLE 1: CASES SIMULATED
Case # Type Ωs(rpm) Bearing oil
feed rate (ltr/min)
S1 0.1Dh 10,000 10
S2 0.2Dh 10,000 10
Q1 40o/14 5,000 8
Q2 40o 5,000 8
Q3 40o 5,000 10
Q4 40o [10][,][000] [8]
Q5 40o [10][,][000] [10]
H1 60o [5][,][000] [10]
H2 60o 10,000 10
F1 360o 5,000 8
Mesh dependence
A full mesh independence study has not been conducted for this modelling work because it builds largely on work previously reported [13]. The only parameter investigated was the number of cells
across the gap between inner race and cage. It was con-cluded that 14 cells were insufficient and all the cases were run with 20. Within the timeframe of this investigation it was not feasible to
compute with more.
Qualitative Analysis
FIGURE 8: OIL IN THE FEED PIPE
Figure 9 shows the oil film on the inside wall of the cage (viewed from above the cage as if transparent). The red patches show the oil as this is where there is a volume fraction of 1. Shaft
rotation is counter-clockwise in these figures. The jet of oil from the feed can be seen to generally form a star shaped area of im-pact. As expected, the oil exits to the both sides axially. An
interesting feature of the results obtained is that for the cases of shaft speed 10,000 rpm (Q4, Q5 and H2) it can be seen that the oil trail on the cage breaks into discrete oil patches very
remi-niscent of jet breakup into droplets caused by Rayleigh-Plateau instability. This is true for both feedrates investigated but this be-haviour is not seen for the cases of shaft rotatation at
5,000 rpm. All the subfigures of Figure 9 show that there is little interaction of the oil from any neighbouring oil feeds for either the 40◦(Q) or 60◦(H) feed spacing.
A conclusion from this study is that oil exiting the bear-ing into the bearbear-ing chamber from the location between cage and inner race will enter the chamber not as a sheet of film but rather as
discrete ”chunks” of oil. At the lower shaft speed of 5,000r pmthe axial spreading from the feed would lead to liga-ments/droplets entering the bearing chamber. At the higher shaft speed of 10,000r
pmthere is disintegration of the feed over the cage suggesting smaller droplets would be shed.
Comparison with the case with the single rotating feed on a 360◦ geometry is shown in Figure 10. The oil behaviour is consistent with the periodic case Q2 confirming the validity of the periodic
modelling approach.
The behaviour of the flow in geometries with the slotted feed (cases S1 and S2) is quite different from the ones with a hole feed. Figure 11 illustrates the results obtained for one of the slot feed
cases. Figure 11A shows a continuous film on the lower surface of the cage; the film is fairly uniform and almost the en-tire surface is wetted. Figure 11B can be compared to Figure 8 and shows that
in this case too the film does not fill the cavity but forms a film under the cage that is shed into the chamber and
FIGURE 9: OIL WETTING OF THE CAGE FOR THE SEC-TOR MODELS
FIGURE 10: OIL BEHAVIOUR IN HOLE AND ON CAGE FOR 360◦CASE
bearing. Because of the extent of the film on the lower surface of the cage, these results suggest that for feed slots the oil emerges as a sheet of oil.
For both the slotted feed and hole feed, in no case did the oil fill the cavity despite the small gap between the cage and inner race. Granted there is no eccentricity of the cage with respect to the
inner race in these models but the finding is nevertheless of interest.
Quantative Analysis
FIGURE 11: CAGE OIL WETTING FROM A SLOTTED FEED
of represent the oil feed in a bearing model. The behaviour of the oil jet during impact on the cage as well as the oil exit from both sides of the gap between cage and inner race are presented here.
One of the parameters identified as of interest is the extent of oil spreading on the cage inner surface. This is expressed using the ratio of the wetted areaAson the cage (the area before any
disintegration) divided by area of the feedhole (Ah):
J[c]∗=pAs/Ah (2)
A second parameter relates to the area of oil liga-ment/filament leaving the cage inner surface (it is effectively a measure of film thickness and angular extent) on the chamber sideAoc or bearing
sideAob. In this case the non-dimensional
parameter is:
J[o]∗=pAo/Ah (3)
To post-process/extract the area of interest, an “iso-clip” is created using a macro based on the volume fraction of oil (α≥0.5) and the coordinates of the feed hole. With this iso-clip,
the oil exiting the geometry directly from the jet can be directly analysed.
Figure 12 shows the spread parameterJ[c]∗for the cases where oil is fed through a hole with 40◦ periodicity (Q1-Q5). The spread is between 3 and 5 times the size of the feed hole. The spread
generally increases with the shaft speed (comparing cases
Q1-Q3 to cases Q4-Q5). For a fixed speed the spread reduces with increasing flow rate (comparing Q2 to Q3, Q4 to Q5). For the cases where oil is fed through a hole with 60◦periodicity (cases H1-H2),
the spread reduces with increased shaft speed. This figure gives an idea of the instantaneous wetting of the lower surface of the cage.
FIGURE 12: OIL SPREAD ON THE CAGE
The parameter characterising the oil shedding from the cage into the chamber or bearing (J[o]∗) is given in Figure 13. The most immediately notable aspect of Figure 13 is that shaft speed
dif-ferentiates behaviour as can be seen by comparing cases Q1-Q3 with Q4-Q5 and H1 to H2. At 5,000 rpm the shedding param-eter is around 0.2 whereas at 10,000 rpm the shedding parame-ter is around
0.7. For the two oil flowrates investigated flowrate makes little difference (comparing Q2 to Q3, Q4 to Q5). In cases Q2 and Q4 the flowrate is the same but the shedding parameter differs
significantly and this indicates that the oil is shedding at higher speed when the shaft speed is higher. The shedding pa-rameter is similar here for both bearing and chamber sides of the cage
indicating fairly even split although it should be recalled that the geometry investigated here is far less representative of the bearing side in the actual bearing. This CFD data shows that, as
might be expected, shedding speed depends primarily on shaft speed. The data also shows that because films are thin-ner at higher speed (increased spreading parameterJ[c]∗the shed-ding speed
increases more than linearly with shaft speed (ieJ[o]∗at 10,000 rpm is more than twiceJ[o]∗at 10,000 rpm)
To investigate this further the mean velocity of the oil (fil-aments) as they exit is given in Figure 14 for the various cases. The effect of shaft speed is clear. For comparison, at shaft speed
5,000 rpm the cage speed is 2,658 rpm giving a cage linear speed (rω) of 34.5 m/s and for shaft speed 10,000 rpm (cage speed
FIGURE 13: OIL EXIT SIZE FACTOR
and this will be as a consequence of the axial velocity compo-nent. For the two oil flowrates investigated there is very little difference in mean oil shedding velocity. There is also very little
difference between the cases for 6 holes and 9 holes (40◦sector and 60◦sector).
FIGURE 14: EXIT OIL VELOCITY
The film thickness was measured in front of the jet as well as behind the jet and the data is shown in Figure 15. The mea-surement in front of the jet is marked as L (leading ) and behind is marked
as T (trailing) in the figure. The figure shows the av-erage and maxima of the film thickness under the cage for the feed hole simulations. The film is thicker in front of the jet im-pact and thinner
trailing the jet motion. The film thickness is less than 250µmin the wetted part of the cage. The film thickness
is similar for all the flow rates. The mean film thickness for the slotted feed case are 124 and 118µm, respectively for the cases
S1 and S2.
In this work, a numerical simulation was made for the oil/air flow in the region of an previously investigated aeroengine bear-ing between the inner race and the cage. This region is charac-terised
by a very narrow annular gap. The work aimed to es-tablish whether the oil feed obtained on the actual bearing via holes through from under-race feed could be represented by a continuous slot input.
As the feed-holes are periodically spaced, advantage was taken to use periodicity with a frame of reference centred on the inner race. Simulations were obtained for two shaft speeds, 5,000 rpm and
10,000 rpm, and two oil flowrates, 8 and 10 litres per minute, these being typical values for the bear-ing of interest.
Results from the simulations with oil supply through feed holes show that the oil does not fill the entire feed hole. After leaving the feed hole the oil forms a wetted area on the inner surface of
the bearing cage, spreading and shedding to both sides. In none of the cases investigated was the annular gap full of oil. The wetted area on the cage was investigated and no consistent pattern of
variation with shaft speed or oil supply flowrate was found; in all cases this area was 3-5 times the area of the exit hole. The cross-sectional area of flow shedding from the bearing was a strong
function of shaft speed as was the speed of the oil at point of shedding. This latter was found to be slightly higher than the linear cage speed. The oil on the cage breaks up into smaller wetted
areas that ultimately shed as ligaments, filaments and droplets. The oil filaments are more regular at the 5,000 rpm speeds and more disintegration was found at 10,000 rpm for both oil flow rates
investigated suggesting that smaller droplets would be shed.
aver-age film thickness is far less, typcially in the range 50−90µm. It is concluded that discrete feed through oil feed holes and continuous feed through a slot input do not yield comparable
This work was funded by Rolls-Royce plc, Aerospace Group under SILOET-II. We are grateful for access to the University of Nottingham High Performance Computing (HPC) facility.
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for Gas Turbines and Power.
[4] Williams, J., 2008. “Thin film rimming flow subject to droplet impact at the surface”. PhD thesis, The University of Nottingham.
[5] Kay, E., Hibberd, S., and Power, H., 2014. “A depth-averaged model for non-isothermal thin-film rimming flow”. International Journal of Heat and Mass Transfer,
70(Complete), pp. 1003–1015.
[6] Kakimpa, B., Morvan, H. P., and Hibberd, S., 2015. “So-lution Strategies for Thin Film Rimming Flow Modelling”. ASME Paper no. GT2015-43503.
[7] Adeniyi, A., Morvan, H., and Simmons, K., 2013. “Droplet-Film Impact Modelling Using a Coupled DPM-VoF Approach, ISABE-2013-1419”. Proceedings of XXI Intl. Symp. on Air Breathing Engines (ISABE
2013), AIAA, Busan, South Korea.
[8] Adeniyi, A. A., Morvan, H. P., and Simmons, K. A., 2014. “A transient CFD simulation of the flow in a test rig of an aeroengine bearing chamber”. ASME Paper no. GT2014-26399.
[9] Tkaczyk, P., and Morvan, H., 2012. SILOET: CFD Mod-elling Guidelines of Engine Sumps - Oil and Air Flows Simulation of Bearing Chambers & Sumps using an En-hanced Volume of Fluid (VOF) Method,
JF82/PT/06, UTC in Gas Turbine Transmission Systems. Tech. rep., The Uni-versity of Nottingham, UK,.
[10] Chandra, B., Simmons, K., Pickering, S., Colliocot, S., and Wiedemann, N., 2012. “Study of Gas/Liquid behaviour within an Aeroengine Bearing Chamber”.ASME Paper No. GT2012-68753.
[11] Glahn, A., Blair, M., Allard, K., Busam, S., O.Schfer, and Wittig, S., 2003. “Disintegration of Oil Films Emerging from Radial holes in a Rotating Cylinder”. Journal of en-gineering for gas
turbines and power, 125(4), pp. 1011– 1020.
[12] Crouchez-Pillot, A., and Morvan, H., 2014. “CFD Sim-ulation of an Aeroengine Bearing Chamber using an En-hanced Volume of Fluid (VOF) Method”. ASME Paper no. GT2014-26405.
[13] Adeniyi, A. A., Morvan, H. P., and Simmons, K. A., 2015. “A multiphase computational study of oil-air flow within the bearing sector of aeroengines”.ASME Paper no. GT2015-43496.
[14] Kannel, J., and Bupara, S., 1978. “A simplified model of cage motion in angular contact bearings operating in the ehd lubrication regime”. Journal of Tribology, 100(3), pp. 395–403.
[15] Gupta, P., 2012. Advanced Dynamics of Rolling Elements. Springer Science & Business Media.
[16] Jones, A. B., 1959. “Ball motion and sliding friction in ball bearings”.Journal of Basic Engineering, Transactions ASME, D, 81.
[17] Gupta, P., 1979. “Dynamics of rolling element bearings, part 1: cylindrical roller bearing analysis”. Trans. ASME, J. Lubric. Technol., 101.
[18] Sum, W. W., 2005. “Dynamic analysis of angular-contact ball bearings and the influence of cage geometry”. PhD thesis, The University of Nottingham.
[19] Ai, X., and Moyer, C. A., 2000. Rolling Element Bearings Chp. 28: Modern Tribology Handbook. CRC Press. [20] Foord, C., 2014. “High-speed ball bearing analysis”.
Pro-ceedings of the Institution of Mechancal Engineers, Part G: Journal of Aerospace Engineering.
[21] Wang, W.-z., Hu, L., Zhang, S.-g., Zhao, Z.-q., and Ai, S., 2014. “Modelling angular contact ball bearing with-out raceway control hypothesis”. Mechanism and Machine Theory, 82, pp. 154–172.
[22] Changqing, B., and Qingyu, X., 2006. “Dynamic model of ball bearings with internal clearance and waviness”. Jour-nal of Sound and Vibration, 294.
[23] Sussman, M., and Puckett, E., 2000. “A Coupled Level Set and Volume-of-Fluid Method for Computing 3D and Axisymmetric Incompressible Two-Phase Flows”.Journal of Computational Physics, 162, pp. | {"url":"https://1library.net/document/z3d6n39y-computational-fluid-dynamics-simulation-flow-inner-engine-bearing.html","timestamp":"2024-11-07T16:03:34Z","content_type":"text/html","content_length":"190143","record_id":"<urn:uuid:24ed5a06-76a8-434f-8a44-318fe842a7a4>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00266.warc.gz"} |
[Solved] Exercise 7: Maneuvering & High Speed Flight
Exercise 7: Maneuvering & High Speed Flight For this week’s assignment you will research a historic or current fighter type aircraft of your
... [Show More] choice (options for historic fighter jets include, but are not limited to: Me262, P-59, MiG-15, F-86, Hawker Hunter, Saab 29, F-8, Mirage III, MiG-21, MiG-23, Su-7, Electric
Lightning, Electric Canberra, F-104, F-105, F-4, F-5, A-6, A-7, Saab Draken, Super Etendard, MiG-25, Saab Viggen, F-14, and many more). As previously mentioned and in contrast to formal research for
other work in your academic program at ERAU, Wikipedia may be used as a starting point for this assignment. However, DO NOT USE PROPRIETARY OR CLASSIFIED INFORMATION even if you happen to have access
in your line of work. Notice also that NASA has some great additional information at: http://www.hq.nasa.gov/pao/History/SP-468/contents.htm. 1. Selected Aircraft: 2. Aircraft Gross Weight [lbs]: 3.
Aircraft Wing Area [ft2]: 4. Positive Limit Load Factor (LLF - i.e. the max positive G) for your aircraft: 5. Negative LLF (i.e. the max negative G) for your aircraft: 6. Maximum Speed [kts] of your
aircraft. If given as Mach number, convert by using Eq. 17.2 relationships with a sea level speed of sound of 661 kts. For simplification, assume the CLmax for your aircraft was 1.5 (unless you can
find a different CLmax in your research). A. Find the Stall Speed [kts] at 1G under sea level standard conditions for your aircraft (similar to all of our previous stall speed work, simply apply the
lift equation in its stall speed form from page 44 to the above data): B. Find the corresponding Stall Speeds for 2G, 3G, 4G, and so on for your selected aircraft (up to the positive load limit from
4. above), using the relationship of Eq. 14.5. You can use the table below to track your results. C. Add the corresponding Stall Speeds for -1G, -2G, and so on for your selected aircraft (up to the
negative load limit from 5. above) to your table. Assume that your fighter wing has symmetrical airfoil characteristics, i.e. that the negative maximum CL value is equal but opposite to the positive
one. (Feel free to use specific airfoil data for your aircraft, but please make sure to use the correct maximum positive and negative Lift Coefficients in the correct places, i.e. CLmax in the
positive part and highest negative CL in the negative part of the table and curve, and indicate your changes to the given example.) Explanation: Making the assumption of symmetry simplifies your
work, since the stall curve in the negative part of the V-G diagram becomes a mirror image of the positive side. Notice also that the simplified form of Eq. 14.5 won’t work with negative values;
however, if using the G-dependent stall equation in the middle of page 222, it becomes obvious that negative signs cancel out between the negative G and the negative CLmax, and Stall Speeds can
actually be calculated in the same way as for positive G, reducing your workload on the negative side to only one calculation of the stall speed at the negative LLF, if not a whole number.) D. Track
your results in the V-G diagram below by properly labeling speeds at intercept points. Add also horizontal lines for positive and negative load limits on top and bottom and a vertical line on the
right for the upper speed limit of your aircraft at sea level from 6. above. (Essentially you are re-constructing the V-G diagram by appropriately labeling it for your aircraft. Notice that the shape
of the diagram and the G-dependent curve relationship is essentially universal and just the applicable speeds will change from aircraft to aircraft. Make sure to reference book Fig. 14.8 for
comparison.) G VS (kts) PLL: 10 9 8 7 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 NLL: E. Find the Ultimate Load Factor (ULF) based on your aircraft’s Positive Limiting Load Factor (LLF). (For the relationship
between LLF and ULF, see book discussion p. 226 and Fig. 14.9): F. Find the Positive Ultimate Limit Load [lbs] based on the ULF in E. above and the Gross Weight from 2.? G. Explain how limit load
factors change with changes in aircraft weight. Support your answer with formula work and/or calculation example. H. What is the Maneuvering Speed [kts] for your aircraft? I. At the Maneuvering Speed
and associated load factor, find the Turn Radius ‘r’ [ft] and the Rate of Turn (ROT) [deg/s]. I) Use Eq. 14.3 to find bank angle ‘f’ for that load factor (i.e. G). (Remember to check that your
calculator is in the proper trigonometric mode when building the arccos). II) With bank angle from I) above and maneuvering speed from H., use Eq. 14.15 to find turn radius ‘r’. III) With bank angle
from I) above and maneuvering speed from H., use Eq. 14.16 to find ROT. (Make sure to use the formula that already utilizes speed in kts and gives results in degree per second). J. For your selected
aircraft, describe the different features that are incorporated into the design to allow high-speed and/or supersonic flight. Explain how those design features enhance the high-speed performance, and
name additional features not incorporated in your aircraft, but available to designers of supersonic aircraft. K. Using Fig. 14.10 from Flight Theory and Aerodynamics, find the Bank Angle for a
standard rate (3 deg/s) turn at your aircraft’s maneuvering speed. (This last assignment is again designed to review some of the diagram reading skills required for your final exam; therefor, please
make sure to fully understand how to extract the correct information and review book, lecture, and/or tutorials as necessary. You can use the below diagram copy to visualize your solution path by
adding the appropriate lines, either via electronic means, e.g. insert line feature in Word or Acrobat, or through printout, drawing, and scanning methods.) From: Dole, C. E. & Lewis, J. E. (2000).
Flight Theory and Aerodynamics. New York, NY: John Wiley & Sons Inc. [Show Less] | {"url":"https://docmerit.com/doc/show/solved-exercise-7-maneuvering-high-speed-flight","timestamp":"2024-11-04T23:42:02Z","content_type":"text/html","content_length":"107136","record_id":"<urn:uuid:feae467b-d9e8-4f03-a5a7-95bd7e511c2e>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00630.warc.gz"} |
Aztec Numerals: Decimal to Aztec Number Converter
Translate decimal numbers into Aztec numbers using this calculator. The Aztec numeral system is based on 20 and uses specific symbols for each power of 20.
This content is licensed under Creative Commons Attribution/Share-Alike License 3.0 (Unported). That means you may freely redistribute or modify this content under the same license conditions and
must attribute the original author by placing a hyperlink from your site to this work https://planetcalc.com/9920/. Also, please do not modify any references to the original work (if any) contained
in this content.
Using the calculator
Enter a decimal number in the "Decimal number" field, and the calculator will convert it into the corresponding Aztec number. The Aztec numeral system is additive, meaning the order of symbols is not
important. In this system, symbols represent units, twenties, four hundreds, and eight thousands. Please note that the Aztecs did not have a zero symbol.
Aztec Numeral System
The Aztec numeral system is based on 20 and uses specific symbols for each power of 20.
Four hundred
Eight thousand
The symbols used are^1:
• Dots for units
• Flags for twenties
• Trees for four hundreds
• Ceremonial pouches for eight thousands
In the Aztec numeral system, the order of symbols is not significant. Each symbol represents a specific value, and the numbers are expressed by combining these symbols. It is important to note that
the Aztecs did not have a symbol for zero.
With this calculator, you can easily convert decimal numbers into Aztec numbers, gaining insights into the Aztec numeral system and its unique representation of quantities.
1. Ancient civilizations of Mexica and Central America by Herbert J. Spinden, New York, 1917. p. 201 ↩
Similar calculators
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[Solved] What is the purpose of hypothesis tests i | SolutionInn
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What is the purpose of hypothesis tests in linear regression?
What is the purpose of hypothesis tests in linear regression?
There are 3 Steps involved in it
Step: 1
Hypothesis test to control whether there is an importa...
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A polynomial system with at least one solution
A polynomial system with at least one solution
Hello, using Sage, I consider a system $S$ of multivariate polynomials. I cannot obtain a Grobner basis of the previous system; no matter, I only want to know when $S$ has at least one solution. Does
there exist in Sage a command that does the job. Of course, the command must give an answer without calculating the whole of the Grobner basis. Remark. When $S$ has no solutions, the process is
faster and, in general, we obtain $[1]$ as a Grobner basis. Thus, when Sage does not succeed in concluding, there is a high probability that there is at least one solution; unfortunately, this is not
a proof. Thanks in advance.
Thanks for your answer. Unfortunately, the methods presented in your reference require the calculation of the complete Grobner basis. | {"url":"https://ask.sagemath.org/question/36077/a-polynomial-system-with-at-least-one-solution/","timestamp":"2024-11-05T10:40:35Z","content_type":"application/xhtml+xml","content_length":"49735","record_id":"<urn:uuid:0a1cd029-3326-43c9-974c-d037c4f303f2>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00596.warc.gz"} |
GCSE/ALevel Maths Challenge 1
2 Comments
Level 65
Sep 27, 2015
Couple of things:
1) I have no idea what you're talking about, but I got an 100% just by guessing. 2) You shouldn't have to type in 'x =' for each answer, just the solution should be accepted. 3) So, to further
explain 1, I typed x= then counted 1,2,3... etc. Not sure how you could change that, though. 4) On this website, spaces, capitalization, punctuation, etc. doesn't count! (so just typing in 'x2' for
the 1st answer will give credit) JetPunk doesn't count negative signs, equals signs, or slash marks to represent fractions. 6) That makes "x13" acceptable for "X = 1/3". 7) In the 'Instructions #1',
it should read (Note if you're at GSCE...).
Okay, so maybe that wasn't just a couple of things, but, after the needed revising, it'll make an epic quiz! :)
Level 14
Apr 22, 2016
Ok. Until the website is upgraded I cant really make the majority of these changes. If you didnt know the answer that is fine. However I made this quiz for GCSE students to revise this topic. | {"url":"https://ec2-34-193-34-229.compute-1.amazonaws.com/user-quizzes/144629/gcsealevel-maths-challenge-1","timestamp":"2024-11-10T22:55:07Z","content_type":"text/html","content_length":"37326","record_id":"<urn:uuid:788c9c13-277b-4b4e-a84a-61e1926b0dbf>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00572.warc.gz"} |
0.1.4 Oct 27, 2024
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0.1.2 Jun 26, 2024
0.1.1 Dec 14, 2023
0.1.0 Aug 3, 2023
Low Discrepancy Sequence Generation in Rust
This library implements a set of low-discrepancy sequence generators, which are used to create sequences of numbers that exhibit a greater degree of uniformity than random numbers. The utility of
these sequences is evident in a number of fields, including computer graphics, numerical integration, and Monte Carlo simulations.
The library defines a number of classes, each of which represents a distinct type of low-discrepancy sequence generator. The primary sequence types that are implemented are as follows:
1. Van der Corput sequence
2. Halton sequence
3. Circle sequence
4. Sphere sequence
5. 3-Sphere Hopf sequence
6. N-dimensional Halton sequence
Each generator is designed to accept specific inputs, which are typically presented in the form of base numbers or sequences of base numbers. The selection of bases serves to determine the manner in
which the sequences are generated. The generators produce outputs in the form of floating-point numbers or lists of floating-point numbers, contingent upon the dimensionality of the sequence.
The fundamental algorithm utilized in the majority of these generators is the Van der Corput sequence. The Van der Corput sequence is generated by expressing integers in a specified base, reversing
the digits, and inserting them after a decimal point. To illustrate, in base 2, the sequence would commence as follows: The sequence then progresses as follows: 1/2, 1/4, 3/4, 1/8, 5/8, and so on.
The Halton sequence extends this concept to multiple dimensions by employing a distinct base for each dimension. The Circle and Sphere sequences employ trigonometric functions to map the
low-discrepancy sequences onto circular or spherical surfaces.
Furthermore, the library incorporates a set of utility functions and classes that facilitate the operation of these generators. To illustrate, a list of prime numbers may be employed as bases for the
Each generator class has methods for producing the next value in the sequence (pop()) and for resetting the sequence to a specific starting point (reseed()). This enables the generators to be
employed in a variety of contexts in a flexible manner. The objective of this library is to provide a toolkit for the generation of sequences of numbers that are distributed in a well-balanced
manner. These can be used in place of random numbers in many applications to achieve a more uniform coverage of a given space or surface. This can result in more efficient and accurate outcomes in
tasks such as sampling, integration, and optimization.
🛠️ Installation
📦 Cargo
• Install the rust toolchain in order to have cargo installed by following this guide.
• run cargo install lds-rs
📜 License
Licensed under either of
at your option.
🤝 Contribution
Unless you explicitly state otherwise, any contribution intentionally submitted for inclusion in the work by you, as defined in the Apache-2.0 license, shall be dual licensed as above, without any
additional terms or conditions.
See CONTRIBUTING.md. | {"url":"https://lib.rs/crates/lds-rs","timestamp":"2024-11-09T17:49:57Z","content_type":"text/html","content_length":"18659","record_id":"<urn:uuid:89ac8d83-b8c4-4446-aca7-8226edb04d24>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00353.warc.gz"} |
Selina Concise Mathematics Class 9 ICSE Solutions Circle - A Plus Topper
Selina Concise Mathematics Class 9 ICSE Solutions Circle
ICSE SolutionsSelina ICSE Solutions
APlusTopper.com provides step by step solutions for Selina Concise Mathematics Class 9 ICSE Solutions Chapter 17 Circle. You can download the Selina Concise Mathematics ICSE Solutions for Class 9
with Free PDF download option. Selina Publishers Concise Mathematics for Class 9 ICSE Solutions all questions are solved and explained by expert mathematic teachers as per ICSE board guidelines.
Download Formulae Handbook For ICSE Class 9 and 10
Selina ICSE Solutions for Class 9 Maths Chapter 17 Circle
Exercise 17(A)
Solution 1:
Solution 2:
Solution 3:
Solution 4:
Solution 5:
Solution 6:
Let O be the centre of the circle and AB and CD be the two parallel chords of length 30 cm and 16 cm respectively.
Drop OE and OF perpendicular on AB and CD from the centre O.
Solution 7:
Since the distance between the chords is greater than the radius of the circle (15 cm), so the chords will be on the opposite sides of the centre.
Solution 8:
Solution 9:
Solution 10:
Exercise 17(B)
Solution 1:
Solution 2:
Solution 3:
Solution 4:
Solution 5:
Solution 6:
Solution 7:
Solution 8:
Solution 9:
Solution 10:
Exercise 17(C)
Solution 1:
Solution 2:
Solution 3:
As given that AB is the side of a pentagon the angle subtended by each arm of the pentagon at
Solution 4:
Solution 5:
Solution 6:
Solution 7:
Solution 8:
Exercise 17(D)
Solution 1:
Solution 2:
Solution 3:
Solution 4:
Solution 5:
Solution 6:
Solution 7:
Solution 8:
Solution 9:
Solution 10:
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Why is angular momentum conserved but not linear? | Socratic
Why is angular momentum conserved but not linear?
1 Answer
Angular and linear momentum are not directly related, however, both are conserved. Angular momentum is a measure of an object's tendency to continue rotating. A rotating object will continue to spin
on an axis if it is free from any external torque.
Linear momentum is an object's tendency to continue in one direction. An object traveling in a given direction with a certain velocity will continue to do so until acted on by an external force
(Newton's 1st law of motion). Since angular and linear momentum both have magnitudes and directions associated with them, they are both vector quantities.
Angular momentum is given by:
$L = r \cdot m {v}_{\text{tangential}}$
where $L$ is the angular momentum, $r$ is the radius of the mass relative to the axis of rotation, m is the mass of the object, and ${v}_{\text{tangential}}$ is the velocity vector of the mass
tangent to the radius of rotation
Linear momentum is given by:
$p = m v$
where $p$ is the linear momentum of the object, $m$ is the mass, and $v$ is the velocity of the object in the direction of travel
A common example of the conservation of angular momentum in the physics classroom is spinning in a chair with weights in each hand. When the weights are brought in, the student will rotate faster
(because the radius is smaller). The opposite is also true. When the student extends his hands, the chair will slow down.
A common example of the conservation of linear momentum can be seen in collision mechanics. In a perfectly inelastic collision, if two objects of the same mass collide, and one starts at rest, the
final velocity of the system will be exactly 1/2 of the velocity of the mass that was moving originally:
${p}_{m} 1 + {p}_{m} 2 = {p}_{m} 1 m 2$
${m}_{1} {v}_{1} + {m}_{2} {v}_{2} = \left({m}_{1} + {m}_{2}\right) {v}_{12}$
Since ${m}_{2}$ started at rest, ${v}_{2} = 0$, and we are assuming ${m}_{1} = {m}_{2}$:
${m}_{1} {v}_{1} = 2 {m}_{1} {v}_{12}$
${v}_{12} = \frac{1}{2} {v}_{1}$
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Question ID - 52792 | SaraNextGen Top Answer
The problem, contains four problem figures marked A, B, C and D and five answer figures marked 1, 2, 3, 4 and 5. Select a figure from amongst the answer figures which will continue the same series as
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How is age-adjusted mortality rate calculated in epidemiology?
How is age-adjusted mortality rate calculated in epidemiology?
Age-adjusted rates were calculated by dividing the expected number of deaths by the population (standard) and multiplying by 1,000.
How do you calculate age-adjusted mortality rate?
An alternate way to compute the age-adjusted death rate by the direct method is simply to multiply the age- specific death rates by the corresponding proportion of the standard population in that age
group and then sum these products across all 10 age groups.
What is age-adjusted mortality?
1. Definition: AGE-ADJUSTED DEATH RATE is a death rate that controls for the effects of differences in population age distributions.
What is the mortality rate formula?
The mortality rate is the number of people who die in a given year and area, divided by the population of that area. The formula is simple: D divided by P. D is the number of deaths, and P is the
population of that area.
Why are age-adjusted mortality rates important?
Rates Important? Age-adjusted rates allow you to compare health statistics (like death rates) between population groups, even though the size of the groups or the age of group members might be very
What is the difference between crude death rate and age-adjusted death rate?
Crude rates are influenced by the underlying age distribution of the state’s population. Even if two states have the same age-adjusted rates, the state with the relatively older population generally
will have higher crude rates because incidence or death rates for most cancers increase with increasing age.
How do you calculate adjusted rate?
Adjustment is accomplished by first multiplying the age-specific rates of disease by age-specific weights. The weights used in the age-adjustment of cancer data are the proportion of the 1970 US
population within each age group. The weighted rates are then summed across the age groups to give the age-adjusted rate.
Why would someone use an age-adjusted rate?
Age-adjusted death rates eliminate the bias of age in the makeup of the populations being compared, thereby providing a much more reliable rate for comparison purposes. We will use a method of
adjusting called “direct standardization.” It consists of applying specific crude rates to a standard population.
How do you calculate rate ratio and mortality rate?
Rate ratio for mortality between men and women = 7.7/17.4 = 0.44 [95% CI 0.08 to 2.37] where the rates have been calculated as deaths per 100 person-years by dividing the number of deaths within each
sex by the total number of years of follow-up in each sex and multiplying by 100.
How do you calculate mortality rate per person year?
were used to calculate the number of deaths among those remaining at risk for each interval using the formula CI = IR x T. Thus, the first age group spanned 15 years and the mortality rate was 4.7/
100,000 person-years, so the number of deaths was 4.7 x 15 = 70.5.
What is age-adjusted rate per 100000?
The rate in the area of study (e.g., county, state) is computed for each age group noted in the table below by dividing the number of events (deaths) in that age group by the estimated population of
the same age group in that area and then multiplying by a constant of 100,000.
What is the purpose of calculating mortality rates?
While mortality rates can give an indication of risk over specific time periods and specific geographies of irregular migration routes, it is important to weigh the value of making and publicizing
these calculations when there are incomplete data and different interpretations of how to measure the total population at …
Why are age-adjusted rates used?
Age-adjusting the rates ensures that differences in incidence or deaths from one year to another, or between one geographic area and another, are not due to differences in the age distribution of the
populations being compared.
What is the difference between crude and adjusted rates?
“Crude Rate” refers to rates per 100,000 population. “Age-Adjusted Rate” refers to rates that are calculated as if the deaths occurred in a population with the same age structures. As the CDC website
explains: “Some injuries occur more often among certain age groups than others.
Why one would use an age-adjusted rate?
What does adjusted rate mean?
An adjusted rate is an artificially created figure that enables comparison across time and space. It should only be compared with another adjusted rate that was computed using the same “standard”
population. However, it does provide a single figure which can be easily used and adapted for comparative analysis.
Why is it useful to Standardised mortality rates by age?
Rationale: The numbers of deaths per 100 000 population are influenced by the age distribution of the population. Two populations with the same age-specific mortality rates for a particular cause of
death will have different overall death rates if the age distributions of their populations are different.
How do you calculate mortality rate per 1000?
Mortality rate is typically expressed in units of deaths per 1,000 individuals per year; thus, a mortality rate of 9.5 (out of 1,000) in a population of 1,000 would mean 9.5 deaths per year in that
entire population, or 0.95% out of the total.
What is mortality ratio in statistics?
The standardized mortality ratio is the ratio of observed deaths in the study group to expected deaths in the general population. This ratio can be expressed as a percentage simply by multiplying by
100. The SMR may be quoted as either a ratio or a percentage.
How is average age of death calculated?
To calculate the average age of death, you must calculate a weighted average (illustrated below). To do so, multiply each individual median by the appropriate number of deaths (e.g.. 2 x 386 = 772, 7
x 98 = 686, etc.). Add the total results together (16244.5) and divide by the total number of deaths (955).
Why is age-adjusted mortality rate important?
Age-adjusted rates allow you to compare health statistics (like death rates) between population groups, even though the size of the groups or the age of group members might be very different. Step 3:
Choose a standard population and find the percentage of the standard population that is found in each age group.
What is mortality rate table?
A mortality table, also known as a life table or actuarial table, shows the rate of deaths occurring in a defined population during a selected time interval, or survival rates from birth to death.
What is the most useful single measure of mortality?
Pros: Life expectancy at birth is the single best summary measure of the mortality pattern of a population. It translates a schedule of age-specific deaths rates into a result expressed in the
everyday metric of years, the average “length of life.”
What is age-adjusted incidence rate?
The age-adjusted rates are rates that would have existed if the population under study had the same age distribution as the “standard” population. Therefore, they are summary measures adjusted for
differences in age distributions.
How do you interpret adjusted rate ratio?
That is, a rate ratio of 1.0 indicates equal rates in the two groups, a rate ratio greater than 1.0 indicates an increased risk for the group in the numerator, and a rate ratio less than 1.0
indicates a decreased risk for the group in the numerator. | {"url":"https://mattstillwell.net/how-is-age-adjusted-mortality-rate-calculated-in-epidemiology/","timestamp":"2024-11-11T06:40:42Z","content_type":"text/html","content_length":"45687","record_id":"<urn:uuid:383d1354-92f1-4b7c-8935-67a19c05b8c0>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00719.warc.gz"} |
Permutation and Combination Calculator (2024)
home / math / permutation and combination calculator
Permutations and combinations are part of a branch of mathematics called combinatorics, which involves studying finite, discrete structures. Permutations are specific selections of elements within a
set where the order in which the elements are arranged is important, while combinations involve the selection of elements without regard for order. A typical combination lock for example, should
technically be called a permutation lock by mathematical standards, since the order of the numbers entered is important; 1-2-9 is not the same as 2-9-1, whereas for a combination, any order of those
three numbers would suffice. There are different types of permutations and combinations, but the calculator above only considers the case without replacement, also referred to as without repetition.
This means that for the example of the combination lock above, this calculator does not compute the case where the combination lock can have repeated values, for example, 3-3-3.
The calculator provided computes one of the most typical concepts of permutations where arrangements of a fixed number of elements r, are taken from a given set n. Essentially this can be referred to
as r-permutations of n or partial permutations, denoted as [n]P[r], ^nP[r], P[(n,r)], or P(n,r) among others. In the case of permutations without replacement, all possible ways that elements in a set
can be listed in a particular order are considered, but the number of choices reduces each time an element is chosen, rather than a case such as the "combination" lock, where a value can occur
multiple times, such as 3-3-3. For example, in trying to determine the number of ways that a team captain and goalkeeper of a soccer team can be picked from a team consisting of 11 members, the team
captain and the goalkeeper cannot be the same person, and once chosen, must be removed from the set. The letters A through K will represent the 11 different members of the team:
A B C D E F G H I J K 11 members; A is chosen as captain
B C D E F G H I J K 10 members; B is chosen as keeper
As can be seen, the first choice was for A to be captain out of the 11 initial members, but since A cannot be the team captain as well as the goalkeeper, A was removed from the set before the second
choice of the goalkeeper B could be made. The total possibilities if every single member of the team's position were specified would be 11 × 10 × 9 × 8 × 7 × ... × 2 × 1, or 11 factorial, written as
11!. However, since only the team captain and goalkeeper being chosen was important in this case, only the first two choices, 11 × 10 = 110 are relevant. As such, the equation for calculating
permutations removes the rest of the elements, 9 × 8 × 7 × ... × 2 × 1, or 9!. Thus, the generalized equation for a permutation can be written as:
Or in this case specifically:
Again, the calculator provided does not calculate permutations with replacement, but for the curious, the equation is provided below:
[n]P[r] = n^r
Combinations are related to permutations in that they are essentially permutations where all the redundancies are removed (as will be described below), since order in a combination is not important.
Combinations, like permutations, are denoted in various ways, including [n]C[r], ^nC[r], C[(n,r)], or C(n,r), or most commonly as simply
. As with permutations, the calculator provided only considers the case of combinations without replacement, and the case of combinations with replacement will not be discussed. Using the example of
a soccer team again, find the number of ways to choose 2 strikers from a team of 11. Unlike the case given in the permutation example, where the captain was chosen first, then the goalkeeper, the
order in which the strikers are chosen does not matter, since they will both be strikers. Referring again to the soccer team as the letters A through K, it does not matter whether A and then B or B
and then A are chosen to be strikers in those respective orders, only that they are chosen. The possible number of arrangements for all n people, is simply n!, as described in the permutations
section. To determine the number of combinations, it is necessary to remove the redundancies from the total number of permutations (110 from the previous example in the permutations section) by
dividing the redundancies, which in this case is 2!. Again, this is because order no longer matters, so the permutation equation needs to be reduced by the number of ways the players can be chosen, A
then B or B then A, 2, or 2!. This yields the generalized equation for a combination as that for a permutation divided by the number of redundancies, and is typically known as the binomial
Or in this case specifically:
It makes sense that there are fewer choices for a combination than a permutation, since the redundancies are being removed. Again for the curious, the equation for combinations with replacement is
provided below: | {"url":"https://soarni.org/article/permutation-and-combination-calculator","timestamp":"2024-11-13T23:07:57Z","content_type":"text/html","content_length":"68777","record_id":"<urn:uuid:2bcdd48f-a124-4fbc-ae8a-be89f8053fb1>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00415.warc.gz"} |
Ecobank Internship Past Questions and Answers - 2024 PDF Download
Ecobank Internship Past Questions and Answers – 2024 PDF Download
Original price was: ₦5000.Current price is: ₦2500.
Our resource provides you with real exam questions and answers from previous years, giving you a deep insight into the type of questions you should expect on your upcoming exams. With Ecobank
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Ecobank Internship Past Questions and Answers Study Pack
An internship at Ecobank presents an exceptional opportunity to develop sales skills and gain hands-on experience in the dynamic world of banking. In this blog post, we will focus on the sales aspect
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success in securing an Ecobank internship.
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Free Sample Ecobank Internship Past Questions and Answers
1. A circle is inscribed in a square with a side length of 10 cm. What is the radius of the circle?
a) 5 cm
b) 10 cm
c) 7.07 cm
d) 3.54 cm
Answer: c) 7.07 cm
2. What is the derivative of f(x) = 3x^4 + 2x^3 + 5x^2 – 4x + 7 with respect to x?
a) 12x^3 + 6x^2 + 10x – 4
b) 12x^3 + 6x^2 + 10x + 4
c) 12x^3 + 6x^2 + 5x – 4
d) 12x^3 + 6x^2 + 5x + 4
Answer: a) 12x^3 + 6x^2 + 10x – 4
3. What is the value of sin(120°)?
a) √3/2
b) -√3/2
c) 1/2
d) -1/2
Answer: b) -√3/2
4. In a right triangle, one acute angle measures 45°. What is the measure of the other acute angle?
a) 30°
b) 45°
c) 60°
d) 75°
Answer: c) 60°
5. The polynomial p(x) is divided by (x – 2) and leaves a remainder of 3. If p(2) = 5, what is the degree of p(x)?
a) 1
b) 2
c) 3
d) 4
Answer: b) 2
6. If log(base 4)(x) = 3/2, what is the value of x?
a) 2
b) 4
c) 8
d) 16
Answer: c) 8
7. What is the value of the expression (cos θ)^2 + (sin θ)^2?
a) 0
b) 1
c) -1
d) 2
Answer: b) 1
8. The function f(x) = 2x^3 + 5x^2 + kx + 12 has a local minimum at x = -2. What is the value of k?
a) -5
b) -12
c) -14
d) -17
Answer: d) -17
9. What is the value of the sum 1 + 2 + 4 + 8 + … + 1024?
a) 2046
b) 2048
c) 4094
d) 4096
Answer: b) 2048
10. Choose the word that is most similar in meaning to “articulate”:
a) Fluent
b) Incoherent
c) Mumbled
d) Eloquent
Answer: a) Fluent
11. Choose the word that is opposite in meaning to “enthusiastic”:
a) Apathetic
b) Eager
c) Excited
d) Zealous
Answer: a) Apathetic
12. Choose the word that best fits the sentence: “The student’s behavior was __________ and disrupted the class.”
a) Disruptive
b) Well-behaved
c) Obedient
d) Respectful
Answer: a) Disruptive
13. Choose the word that is most similar in meaning to “coherent”:
a) Logical
b) Confused
c) Disorganized
d) Clear
Answer: a) Logical
14. Choose the word that is opposite in meaning to “sincere”:
a) Insincere
b) Genuine
c) Honest
d) Authentic
Answer: a) Insincere
15. Choose the word that best fits the sentence: “The actor delivered a __________ performance that moved the audience.”
a) Compelling
b) Mediocre
c) Average
d) Unremarkable
Answer: a) Compelling
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When do random walks share the same bridges?
GFM seminar
FCUL, room 6.2.33
2015-09-30 15:00 2015-09-30 16:00 2015-09-30 15:00 .. 16:00
by Christian Léonard (Université Paris Ouest)
The natural analogue of Hamilton's least action principle in presence of randomness is the generalized Schrödinger problem where the Lagrangian action is replaced by the relative entropy with respect
to some reference path measure. The role of the action minimizing paths between two prescribed endpoints is played by the bridges of the reference path measure and the solutions of Schrödinger
problem are mixtures of these bridges. Therefore, the family of all the bridges of the reference measure encodes its whole "Lagrangian dynamics" and searching for a criterion for two path measures to
solve the same Schrödinger problem, i.e. to be driven by the same source of randomness and the same force field, amounts to answer the question of our title. The answer is given in the special case
of random walks.
This is a joint work with Giovanni Conforti. | {"url":"http://gfm.cii.fc.ul.pt/events/seminars/20150930-leonard/","timestamp":"2024-11-12T17:11:59Z","content_type":"application/xhtml+xml","content_length":"22033","record_id":"<urn:uuid:8962b171-e3a9-4f3b-9c87-551d4d7435aa>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00485.warc.gz"} |
Activity report
RNSR: 201923242K
Team name:
Adaptive Mesh Generation and Advanced numerical Methods
Applied Mathematics, Computation and Simulation
Numerical schemes and simulations
Creation of the Project-Team: 2019 June 01
• A6.2. Scientific computing, Numerical Analysis & Optimization
• A6.2.7. High performance computing
• A6.2.8. Computational geometry and meshes
• A6.5.1. Solid mechanics
• A6.5.2. Fluid mechanics
• B5.2.3. Aviation
• B5.2.4. Aerospace
• B9.5.1. Computer science
• B9.5.2. Mathematics
• B9.5.3. Physics
• B9.5.5. Mechanics
1 Team members, visitors, external collaborators
Research Scientists
• Frederic Alauzet [Team leader, Inria, Senior Researcher, HDR]
• Paul-Louis George [Inria, Emeritus]
• Adrien Loseille [Inria, Researcher, HDR]
PhD Students
• Sofiane Benzait [CEA]
• Francesco Clerici [Inria]
• Rémi Feuillet [Inria, until Mar 2020]
• Lucien Rochery [Inria]
• Lucille Marie Tenkes [Inria]
Technical Staff
• Matthieu Maunoury [Inria, Engineer]
• Julien Vanharen [Inria, Engineer, until May 2020]
Administrative Assistant
• Maria Agustina Ronco [Inria]
External Collaborators
• Rémi Feuillet [Siemens Industry Software, from Apr 2020]
• David Marcum [Mississippi State University, Starkville - USA]
• Loic Marechal [Distene]
2 Overall objectives
Numerical simulation has been booming over the last thirty years, thanks to increasingly powerful numerical methods, computer-aided design (CAD) and the mesh generation for complex 3D geometries, and
the coming of supercomputers (HPC). The discipline is now mature and has become an integral part of design in science and engineering applications. This new status has lead scientists and engineers
to consider numerical simulation of problems with ever increasing geometrical and physical complexities. A simple observation of this chart
shows: no mesh = no simulation along with "bad" mesh = wrong simulation. We have concluded that the mesh is at the core of the classical computational pipeline and a key component to significant
improvements. Therefore, the requirements on meshing methods are an ever increasing need, with increased difficulty, to produce high quality meshes to enable reliable solution output predictions in
an automated manner. These requirements on meshing or equivalent technologies cannot be removed and all approaches face similar issues.
In this context, Gamma team was created in 1996 and has focused on the development of robust automated mesh generation methods in 3D, which was clearly a bottleneck at that time when most of the
numerical simulations were 2D. The team has been very successful in tetrahedral meshing with the well-known software Ghs3d28, 29 which has been distributed worldwide so far and in hexahedral meshing
with the software Hexotic49, 50 which was the first automated full hex mesher. The team has also worked on surface meshers with Yams20 and BLSurf12 and visualization with Medit. Before Medit, we were
unable to visualize in real time 3D meshes !
In 2010, Gamma3 team has replaced Gamma with the choice to focus more on meshing for numerical simulations. The main goal was to emphasize and to strengthen the link between meshing technologies and
numerical methods (flow or structure solvers). The metric-based anisotropic mesh adaptation strategy has been very successful with the development of many error estimates, the generation of highly
anisotropic meshes, its application to compressible Euler and Navier-Stokes equations 5, and its extension to unsteady problems with moving geometries 8 leading to the development of several
softwares Feflo.a/AMG-Lib, Wolf, Metrix, Wolf-Interpol. A significant accomplishment was the high-fidelity prediction of the sonic boom emitted by supersonic aircraft 6. We were the first to compute
a certified aircraft sonic boom propagation in the atmosphere, thanks to mesh adaptation. The team has started to work on parallelism with the development of the multi-thread library LPlib and the
efficient management of memory using space filling curves, and the generation of large meshes (a billion of elements) 43. Theoretical work on high-order meshes has been also done 27.
Today, numerical simulation is an integral part of design in engineering applications with the main goal of reducing costs and speeding up the process of creating new design. Four main issues for
industry are:
• Generation of a discrete surface mesh from a continuous CAD is the last non-automated step of the design pipeline and, thus, the most human time consuming
• High-performance computing (HPC) for all tools included in the design loop
• The cost in euros of a numerical simulation
• Certification of high-fidelity numerical simulations by controlling errors and uncertainties.
Let us now discuss in more details each of these issues.
Generating a discrete surface mesh from a CAD geometry definition has been the numerical analysis Achille's heel for the last 30 years. Significant issues are far too common and range from persistent
translation issues between systems that can produce ill defined geometry definitions to overwhelming complexity for full configurations with all components. A geometry definition that is ill defined
often does not perfectly capture the geometry's features and leads to a bad mesh and a broken simulation. Unfortunately, CAD system design is essentially decoupled from the needs of numerical
simulation and is largely driven by the those of manufacturing and other areas. As a result, this step of the numerical simulation pipeline is still labor intensive and the most time consuming. There
is a need to develop alternative geometry processes and models that are more suitable for numerical simulations.
Companies working on high-tech projects with high added value (Boeing, Safran, Dassault-Aviation, Ariane Group, ...) consider their design pipeline inside a HPC framework. Indeed, they are performing
complex numerical simulations on complex geometries on a daily-basis, and they aim at using this in a shape-optimization loop. Therefore, any tools added to their numerical platform should be HPC
compliant. This means that all developments should consider hybrid parallelism, i.e., to be compatible with distributed memory architecture (MPI) and shared memory architecture (multi-threaded), to
achieve scalable parallelism.
One of the main goals of numerical simulation is to reduce the cost of creating new designs (e.g reduce the number of wind-tunnel and flight tests in the aircraft industry). The emergence of 3D
printers is, in some cases, making tests easier to perform, faster and cheaper. It is thus mandatory to control the cost of the numerical simulations, in other word, it is important to use less
resources to achieve the same accuracy. The cost takes into account the engineer time as well as the computing resources needed to perform the numerical simulation. The cost for one simulation can
vary from 15 euros for simple models (1D-2D), to 150 euros for Reynolds-averaged Navier-Stokes (3D) stationary models, or up to 15 000 euros for unsteady models like LES or Lattice-Boltzmann 1. It is
important to know that a design loop is equivalent to performing between 100 and 1 000 numerical simulations. Consequently, the need for more efficient algorithms and processes is still a key factor.
Another crucial point is checking and certification of errors and uncertainties in high-fidelity numerical simulations. These errors can come from several sources:
• i) modeling error (for example via turbulence models or initial conditions),
• ii) discretization error (due to the mesh),
• iii) geometry error (due to the representation of the design) and
• iv) implementation errors in the considered software.
The error assessment and mesh generation procedure employed in the aerospace industry for CFD simulations relies heavily on the experience of the CFD user. The inadequacy of this practice even for
geometries frequently encountered in engineering practice has been highlighted in studies of the AIAA 2 CFD Drag Prediction Workshops 53 and High-Lift Prediction Workshops 68, 67. These studies
suggest that the range of scales present in the turbulent flow cannot be adequately resolved using meshes generated following what is considered best present practices. In this regard, anisotropic
mesh adaptation is considered as the future, as stated in the NASA report "CFD Vision 2030 Study: A Path to Revolutionary Computational Aerosciences" 70 and the study dedicated to mesh adaptation 59.
These preoccupations are the core of the Gamma project scientific program. To answer the first issue, Gamma will focus on designing and developing a geometry modeling framework specifically intended
for mesh generation and numerical simulation purposes. This is a mandatory step for automated geometry-mesh and mesh adaptation processes with an integrated geometry model. To answer the last three
issues, the Gamma team will work on the development of a high-order mesh-adaptive solution platform compatible with HPC environment. To this end, Gamma will pursue its work on advanced mesh
generation methods which should fulfill the following capabilities:
• i) geometric adaptive,
• ii) solution adaptive,
• iii) high-order,
• iv) multi-elements (structured or not), and
• v) using hybrid scalable parallelism.
Note that items $i\right)$ to $iv\right)$ are based on the well-posed metric-based theoretical framework. Moreover, Gamma will continue to work on robust flow solvers, solving the turbulent
Navier-Stokes equations from second order using Finite Volume - Finite Element numerical scheme to higher-order using Flux Reconstruction (FR) method.
The combination of adaptation - high-order - multi-elements coupled with appropriate error estimates is for the team the way to go to reduce the cost of numerical simulations while ensuring
high-fidelity in a fully automated framework.
3 Research program
The main axes are:
• Geometric Modeling:
□ High-fidelity discrete CAD kernel.
□ Continuous parametric CAD kernel.
• Enhanced Generic Meshing Algorithm:
□ Adaptation (extreme anisotropy, metric-aligned, metric-orthogonal).
□ High-order (tetrahedra, hexahedra, boundary layer, adapted).
□ Larges meshes (tetrahedra, hexahedra, adapted).
□ Moving mesh methods for moving geometries.
• Toward Certified Solutions to the Navier-Stokes Equations:
□ Flow solver and adjoints (Finite Volumes, Finite Elements, Flux Reconstruction).
□ Error estimates and correctors.
• Advanced Mesh and Solution Visualisation:
□ Pixel exact rendering (High-Order mesh, High-Order solution).
□ Pre-processing and post-processing.
4 Application domains
Applied Mathematics, Computation and Simulation.
5 New software and platforms
5.1 New software
5.1.1 FEFLOA-REMESH
• Keywords: Scientific calculation, Anisotropic, Mesh adaptation
• Functional Description: FEFLOA-REMESH is intended to generate adapted 2D, surface and volume meshes by using a unique cavity-based operator. The metric-aligned or metric-orthogonal approach is
used to generate high quality surface and volume meshes independently of the anisotropy involved.
• URL: https://pages.saclay.inria.fr/adrien.loseille/index.php?page=softwares
• Authors: Adrien Loseille, Frédéric Alauzet
• Contact: Adrien Loseille
• Participants: Adrien Loseille, Frédéric Alauzet, Rémi Feuillet, Lucien Rochery, Lucille Marie Tenkes
5.1.2 GHS3D
• Keywords: Tetrahedral mesh, Delaunay, Automatic mesher
• Functional Description: GHS3D is an automatic volume mesher
• URL: http://www.meshgems.com/volume-meshing.html
• Authors: Paul Louis George, Houman Borouchaki, Éric Saltel, Adrien Loseille, Frédéric Alauzet, Frederic Hecht
• Contact: Paul Louis George
• Participants: Paul Louis George, Adrien Loseille, Frédéric Alauzet
5.1.3 HEXOTIC
• Keywords: 3D, Mesh generation, Meshing, Unstructured meshes, Octree/Quadtree, Multi-threading, GPGPU, GPU
• Functional Description:
Input: a triangulated surface mesh and an optional size map to control the size of inner elements.
Output: a fully hexahedral mesh (no hybrid elements), valid (no negative jacobian) and conformal (no dangling nodes) whose surface matches the input geometry.
The software is a simple command line that requires no knowledge on meshing. Its arguments are an input mesh and some optional parameters to control elements sizing, curvature and subdomains as
well as some features like boundary layers generation.
• URL: https://team.inria.fr/gamma/gamma-software/hexotic/
• Contact: Loic Marechal
• Participant: Loic Marechal
• Partner: Distene
5.1.4 Metrix
• Name: Metrix: Error Estimates and Mesh Control for Anisotropic Mesh Adaptation
• Keywords: Meshing, Metric, Metric fields
• Functional Description: Metrix is a software that provides by various ways metric to govern the mesh generation. Generally, these metrics are constructed from error estimates (a priori or a
posteriori) applied to the numerical solution. Metrix computes metric fields from scalar solutions by means of several error estimates: interpolation error, iso-lines error estimate, interface
error estimate and goal oriented error estimate. It also contains several modules that handle meshes and metrics. For instance, it extracts the metric associated with a given mesh and it performs
some metric operations such as: metric gradation and metric intersection.
• URL: https://pages.saclay.inria.fr/frederic.alauzet/software.html
• Authors: Adrien Loseille, Frédéric Alauzet
• Contact: Frédéric Alauzet
• Participants: Adrien Loseille, Frédéric Alauzet
5.1.5 VIZIR
• Name: Interactive visualization of hybrid, curbed and high-order mesh and solution
• Keyword: Mesh
• Functional Description: Vizir is a light, simple and interactive mesh visualization software, including: (i) A curved meshes visualizator: it handles high order elements and solutions, (ii)
Hybrid elements mesh visualization (pyramids, prisms, hexahedra), (iii) Solutions visualization : clip planes, capping, iso-lines, iso-surfaces.
• URL: http://vizir.inria.fr
• Publication: hal-01686714
• Author: Adrien Loseille
• Contact: Adrien Loseille
• Participants: Adrien Loseille, Rémi Feuillet, Matthieu Maunoury
5.1.6 Wolf
• Keyword: Scientific calculation
• Functional Description: Numerical solver for the Euler and compressible Navier-Stokes equations with turbulence modelling. ALE formulation for moving domains. Modules of interpolation, mesh
optimisation and moving meshes. Wolf is written in C++, and may be later released as an opensource library. FELiScE was registered in July 2014 at the Agence pour la Protection des Programmes
under the Inter Deposit Digital Number IDDN.FR.001.340034.000.S.P.2014.000.10000.
• URL: https://pages.saclay.inria.fr/frederic.alauzet/software.html
• Authors: Adrien Loseille, Frédéric Alauzet
• Contact: Frédéric Alauzet
• Participants: Frédéric Alauzet, Adrien Loseille, Rémi Feuillet, Lucille Marie Tenkes, Francesco Clerici
5.1.7 Wolf-Bloom
• Keyword: Scientific calculation
• Functional Description: Wolf-Bloom is a structured boundary layer mesh generator using a pushing approach. It start from an existing volume mesh and insert a structured boundary layer by pushing
the volume mesh. The volume mesh deformation is solved with an elasticity analogy. Mesh-connectivity optimizations are performed to control volume mesh element quality.
• URL: https://pages.saclay.inria.fr/frederic.alauzet/software.html
• Authors: David Marcum, Adrien Loseille, Frédéric Alauzet
• Contact: Frédéric Alauzet
• Participants: Adrien Loseille, David Marcum, Frédéric Alauzet
5.1.8 Wolf-Elast
• Keyword: Scientific calculation
• Functional Description: Wolf-Elast is a linear elasticity solver using the P1 to P3 Finite-Element method. The Young and Poisson coefficient can be parametrized. The linear system is solved using
the Conjugate Gradient method with the LUSGS preconditioner.
• URL: https://pages.saclay.inria.fr/frederic.alauzet/software.html
• Authors: Adrien Loseille, Frédéric Alauzet
• Contact: Frédéric Alauzet
• Participants: Adrien Loseille, Frédéric Alauzet
5.1.9 Wolf-Interpol
• Keyword: Scientific calculation
• Functional Description: Wolf-Interpol is a tool to transfer scalar, vector and tensor fields from one mesh to another one. Polynomial interpolation (from order 2 to 4) or conservative
interpolation operators can be used. Wolf-Interpol also extract solutions along lines or surfaces.
• URL: https://pages.saclay.inria.fr/frederic.alauzet/software.html
• Authors: Adrien Loseille, Frédéric Alauzet
• Contacts: Frédéric Alauzet, Paul Louis George
• Participants: Adrien Loseille, Frédéric Alauzet
5.1.10 Wolf-MovMsh
• Keyword: Scientific calculation
• Functional Description: Wolf-MovMsh is a moving mesh algorithm coupled with mesh-connectivity optimization. Mesh deformation is computed by means of a linear elasticity solver or a RBF
interpolation. Smoothing and swapping mesh optimization are performed to maintain good mesh quality. It handles rigid bodies or deformable bodies, and also rigid or deformable regions of the
domain. High-order meshes are also handled
• URL: https://pages.saclay.inria.fr/frederic.alauzet/software.html
• Authors: Adrien Loseille, Frédéric Alauzet
• Contact: Paul Louis George
• Participants: Adrien Loseille, Frédéric Alauzet
5.1.11 Wolf-Nsc
• Keyword: Scientific calculation
• Functional Description: Wolf-Nsc is numerical flow solver solving steady or unsteady turbulent compressible Euler and Navier-Stokes equations. The available turbulent models are the
Spalart-Almaras and the Menter SST k-omega. A mixed finite volume - finite element numerical method is used for the discretization. Second order spatial accuracy is reached thanks to MUSCL type
methods. Explicit or implicit time integration are available. It also resolved dual (adjoint) problem and compute error estimate for mesh adaptation.
• URL: https://pages.saclay.inria.fr/frederic.alauzet/software.html
• Authors: Adrien Loseille, Frédéric Alauzet
• Contact: Frédéric Alauzet
• Participants: Adrien Loseille, Frédéric Alauzet
5.1.12 Wolf-Shrimp
• Keyword: Scientific calculation
• Functional Description: Wolf-Shrimp is a generic mesh partitioner for parallel mesh generation and parallel computation. It can partition planar, surface (manifold and non manifold), and volume
domain. Several partitioning methods are available: Hilbert-based, BFS, BFS with restart. It can work with or without weight function and can correct the partitions to have only one connected
• URL: https://pages.saclay.inria.fr/frederic.alauzet/software.html
• Authors: Adrien Loseille, Frédéric Alauzet
• Contact: Frédéric Alauzet
• Participants: Adrien Loseille, Frédéric Alauzet
5.1.13 Wolf-Spyder
• Keyword: Scientific calculation
• Functional Description: Wolf-Spyder is a metric-based high-order mesh quality optimizer using vertex smoothing and edge/face swapping.
• URL: https://pages.saclay.inria.fr/frederic.alauzet/software.html
• Authors: Adrien Loseille, Frédéric Alauzet
• Contact: Frédéric Alauzet
• Participants: Adrien Loseille, Frédéric Alauzet
6 New results
6.1 Books on Meshing, Geometric Modeling and Numerical Simulation
Participants: Paul Louis George, Frédéric Alauzet, Adrien Loseille, Loïc Maréchal.
The third volume on Meshing, Geometric Modeling and Numerical Simulation 3 was published in 2020 (see Fig. 1) after 11, 25. These books are also written in French (see Fig. 2) 10, 24, 23.
Figure 1: The three volumes of Meshing, Geometric Modeling and Numerical Simulation
Figure 2: The three volumes of Meshing, Geometric Modeling and Numerical Simulation in French
6.2 Numerical simulations on GPU with the GMlib v3.0 library
Participants: Loïc Maréchal, Julien Vanharen.
The whole library was completely rewritten to implement an automatic finite-element shader generation that converts a simple user source code into an OpenCL source that is in compiled on the GPU at
run time. The library handles all meshing data structures, from file reading, renumbering and vectorizing for efficient access on the GPU, and transfer to the graphic card, all automatically and
transparently. With this framework, the user can focus on the calculation part of the code, known as kernel, as all the rest is taken care of by the library. The OpenCL language was chosen as it is
hardware agnostic and runs on any GPU (Intel, Nvidia and AMD) and can also use the multicore and vector capacities of modern CPUs.
Julien Vanharen developed a basic heat solver using the v3.0 as a test case so we could validate the software with various boundary conditions, calculation scheme, unstructured meshes and different
memory access patterns with success. Even with basic calculation which does not stress the full GPU's power, we achieved two orders of magnitude greater speed against a single CPU core and one order
of magnitude compared to a multithreaded implementation. As Julien moved to ONERA, we plan on setting up a collaboration between the two teams to implement more complex HPC codes.
6.3 Pixel-exact rendering for high-order meshes and solutions
Participants: Adrien Loseille, Rémi Feuillet, Matthieu Maunoury.
Classic visualization software like ParaView 34, TecPlot 35, FieldView 40, Ensight 33, Medit 21, Vizir (OpenGL legacy based version) 46, Gmsh 30, ...historically rely on the display of linear
triangles with linear solutions on it. More precisely, each element of the mesh is divided into a set of elementary triangles. At each vertex of the elementary triangle is attached a value and an
associated color. The value and the color inside the triangle is then deduced by a linear interpolation inside the triangle. With the increase of high-order methods and high-order meshes, these
softwares adapted their technology by using subdivision methods. If a mesh has high-order elements, these elements are subdivided into a set of linear triangles in order to approximate the shape of
the high-order element 75. Likewise, if a mesh has a high-order solution on it, each element is subdivided into smaller linear triangles in order to approximate the rendering of the high-order
solution on it. The subdivision process can be really expensive if it is done in a naive way. For this reason, mesh adaptation procedures 62, 51, 52 are used to efficiently render high-order
solutions and high-order elements using the standard linear rendering approaches. Even when optimized these approaches do have a huge RAM memory footprint as the subdivision is done on CPU in a
preprocessing step. Also the adaptive subdivision process can be dependent on the palette (i.e. the range of values where the solution is studied) as the color only vary when the associated value is
in this range. In this case, a change of palette inevitably imposes a new adaptation process. Finally, the use of a non conforming mesh adaptation can lead to a discontinuous rendering for a
continuous solution.
Other approaches are specifically devoted to high-order solutions and are based on ray casting 57, 58, 60. The idea is for a given pixel, to find exactly its color. To do so, for each pixel, rays are
cast from the position of the screen in the physical space and their intersection with the scene determines the color for the pixel. If high-order features are taken into account, it determines the
color exactly for this pixel. However, this method is based on two non-linear problems: the root-finding problem and the inversion of the geometrical mapping. These problems are really costly and do
not compete with the interactivity of the standard linear rendering methods even when these are called with a subdivision process unless they are done conjointly on the GPU. However, synchronization
between GPU and OpenGL buffer are non-trivial combination.
The proposed method intends to be a good compromise between both methods. It does guarantee pixel-exact rendering on linear elements without extra subdivision or ray casting and it keeps the
interactivity of a classical method. Moreover, the subdivision of the curved entities is done on the fly on GPU which leaves the RAM memory footprint at the size of the loaded mesh.
We are developing a software, ViZiR 4, with exact non linear solution rendering to address the high-order visualization challenge 2. ViZiR 4 is bundled as a light, simple and interactive high-order
meshes and solutions visualization software. It is based on OpenGL 4 core graphic pipeline. The use of OpenGL Shading Language (GLSL) allows to perform pixel exact rendering of high order solutions
on straight elements and almost pixel exact rendering on curved elements (high-order meshes). ViZiR 4 enables the representation of high order meshes (up to degree 4) and high order solutions (up to
degree 10) with pixel exact rendering. Furthermore, in comparison with standard rendering techniques based on legacy OpenGL, the use of OpenGL 4 core version improves the speed of rendering, reduces
the memory footprint and increases the flexibility. Many post-processing tools, such as picking, hidding surfaces, isolines, clipping, capping, are integrated to enable on the fly the analysis of the
numerical results.
6.4 High-order mesh generation
Participants: Frédéric Alauzet, Adrien Loseille, Rémi Feuillet, Dave Marcum, Lucien Rochery.
For years, the resolution of numerical methods has consisted in solving Partial Derivative Equations by means of a piecewise linear representation of the physical phenomenon on linear meshes. This
choice was merely driven by computational limitations. With the increase of the computational capabilities, it became possible to increase the polynomial order of the solution while keeping the mesh
linear. This was motivated by the fact that even if the increase of the polynomial order requires more computational resources per iteration of the solver, it yields a faster convergence of the
approximation error 3 74 and it enables to keep track of unsteady features for a longer time and with a coarser mesh than with a linear approximation of the solution. However, in 14, 39, it was
theoretically shown that for elliptic problems the optimal convergence rate for a high-order method was obtained with a curved boundary of the same order and in 9, evidence was given that without a
high-order representation of the boundary the studied physical phenomenon was not exactly solved using a high-order method. In 78, it was even highlighted that, in some cases, the order of the mesh
should be of a higher degree than the one of the solver. In other words, if the used mesh is not a high-order mesh, then the obtained high-order solution will never reliably represent the physical
Based on these issues, the development of high-order mesh generation procedures appears mandatory 1. To generate high-order meshes, several approaches exist. The first approach was tackled twenty
years ago 16 for both surface and volume meshing. At this moment the idea was to use all the meshing tools to get a valid high-order mesh. The same problem was revisited a few years later in 69 for
bio-medical applications. In these first approaches and in all the following, the underlying idea is to use a linear mesh and elevate it to the desired order. Some make use of a PDE or variational
approach to do so 4, 61, 18, 54, 73, 76, 31, others are based on optimization and smoothing operations and start from a linear mesh with a constrained high-order curved boundary in order to generate
a suitable high-order mesh 38, 22, 71. Also, when dealing with Navier-Stokes equations, the question of generating curved boundary layer meshes (also called viscous meshes) appears. Most of the time,
dedicated approaches are set-up to deal with this problem 55, 37. In all these techniques, the key feature is to find the best deformation to be applied to the linear mesh and to optimize it. The
prerequisite of these methods is that the initial boundary is curved and will be used as an input data. A natural question is consequently to study an optimal position of the high-order nodes on the
curved boundary starting from an initial linear or high-order boundary mesh. This can be done in a coupled way with the volume 64, 72 or in a preprocessing phase 65, 66. In this process, the
position of the nodes is set by projection onto the CAD geometry or by minimization of an error between the surface mesh and the CAD surface. Note that the vertices of the boundary mesh can move as
well during the process. In the case of an initial linear boundary mesh with absence of a CAD geometry, some approaches based on normal reconstructions can be used to create a surrogate for the CAD
model 75, 32. Finally, a last question remains when dealing with such high-order meshes: Given a set of degrees of freedom, is the definition of these objects always valid ? Until the work presented
in 27, 36, 26, no real approach was proposed to deal in a robust way with the validity of high-order elements. The novelty of these approaches was to see the geometrical elements and their Jacobian
as Bézier entities. Based on the properties of the Bézier representation, the validity of the element is concluded in a robust sense, while the other methods were only using a sampling of the
Jacobian to conclude about its sign without any warranty on the whole validity of the elements.
In this context, several issues have been addressed : the analogy between high-order and Bézier elements, the development of high-order error estimates suitable for parametric high-order surface mesh
generation and the generalization of mesh optimization operators and their applications to curved mesh generation, moving-mesh methods, boundary layer mesh generation and mesh adaptation.
Metric fields are the link between particular error estimates - be they for low-order 41, 42 or high-order methods 15, for the solution of a PDE 44 or a quantity of interest derived from it such as
drag or lift 45 - and automatic mesh adaptation. In the case of linear meshes, a metric field locally distorts the measure or distance such that, when the mesh adaptation algorithm has constructed an
uniform mesh in the induced Riemannian space, it is strongly anisotropic in the usual Euclidean (physicial) space. As such, anisotropy arrises naturally, without it ever being explicitely sought by
the (re)meshing algorithm.
We seek to extend these principles of metric-based ${P}^{1}$ adaptation to high-order meshes. In particular, we expect the meshing process to naturally recover curvature from the variations of the
metric field, very much like ${P}^{1}$ remeshing recovers anisotropy from local values of the metric field. As such, curvature must be the consequence of a simple geometric property computed in the
Riemannian space, like anisotropy is the consequence of unitness in a space where distances are distorted. Therefore, we propose Riemannian edge length minimization (or geodesic seeking as in 77) as
the driver for metric field curvature recovery.
The metric field's own intrinsic curvature may derive from any error estimate, be it boundary approximation error 19, 17 or an interpolation error estimate. So far, interpolation error estimates on
high-order elements are limited to isotropy (13 in ${L}^{2}$ and 56 in ${L}^{1}$ norms) or require that the curvature of the element be bounded, essentially establishing a range where it may be
considered linear 7.
If genericity with regards to error estimation is achieved through the use of a metric field, robustness and modularity of the general remeshing algorithm may be derived from the use of a single
topological operator such as the cavity operator 43, 47, 48. This is the reason why we chose to extend the original ${P}^{1}$ operator to work with ${P}^{2}$ meshes as input and output.
Work has begun on the new ${P}^{2}$ cavity operator. It is based, for the volume, on a purely metric-based curving procedure - that remains consistent with log-euclidean metric interpolation - and,
for the surface, on CAD or CAD surrogate (typically ${P}^{3}$) projection. Preliminary results have been obtained on complex geometries, showing volume curvature recovery at an acceptable CPU cost.
This has lead to a communication at AIAA SciTech 2020 63.
6.5 Unstructured anisotropic mesh adaptation for 3D RANS turbomachinery applications
Participants: Frédéric Alauzet, Loïc Frazza, Adrien Loseille, Julien Vanharen.
The scope of this paper is to demonstrate the viability and efficiency of unstructured anisotropic mesh adaptation techniques to turbomachinery applications. The main difficulty in turbomachinery is
the periodicity of the domain that must be taken into account inthe solution mesh-adaptive process. The periodicity is strongly enforced in the flow solver using ghost cells to minimize the impact on
the source code. For the mesh adaptation, the local remeshing is done in two steps. First, the inner domain is remeshed with frozen periodic frontiers, and, second, the periodic surfaces are remeshed
after moving geometrice ntities from one side of the domain to the other. One of the main goal of this work is to demonstrate how mesh adaptation, thanks to its automation, is able to generate meshes
that are extremely difficult to envision and almost impossible to generate manually. This study only considers feature-based error estimate based on the standard multi-scale Lpinterpolation error
estimate. We presents all the specific modifications that have been introduced in the adaptive process to deal with periodic simulations used for turbomachinery applications. The periodic mesh
adaptation strategy is then tested and validated on the LS89 high pressure axial turbine vane and the NASA Rotor 37 test cases.
6.6 Hybrid mesh adaptation for CFD simulations
Participants: Frédéric Alauzet, Lucille Marie Tenkès, Julien Vanharen.
CFD simulations aim at capturing several phenomena of various natures. Therefore, these phenomena have very different mesh requirements. For example, most numerical schemes for the boundary layer
require a structured mesh, that is aligned with the boundary of the domain. We use the techniques of mesh metric-based mesh adaptation to generate a hybrid mesh that can fulfill these different mesh
requirements. This approach is based on the metric-orthogonal point-placement, creating some structured parts from the intrinsic directional information bore by the metric-field. Some unstructured
areas may remain where structure is not required. The main goals of this work in progress are to improve the orthogonality of the output mesh and its alignment with the metric field. This work can
fall into three parts. First, we have re-designed the preliminary step of size gradation correction. Then, we have studied two hybrid mesh generation processes methods: one using an a priori
quadrilaterals recombination, the other building straightforwardly the quadrilaterals during the re-meshing step. Finally, some modifications have been added to the solver Wolf to perform simulations
on hybrid meshes. Wolf is not able to run simulations of inviscid and viscous (laminar and turbulent) flows on two dimensional hybrid meshes. The two first topics are detailed in what follows.
Hybrid aware metric gradation correction
The previously described generation method highly relies on the metric field. However, a metric field computed from a solution during the adaptation process is most of the time quite messy, with for
example strong size variations. In standard mesh adaptation, it leads to low-quality elements. In orthogonal mesh adaptation, it additionally breaks the alignment and the structure of the output
mesh. In both cases, a step has been added to smooth the input metric field. In the context of hybrid mesh adaptation, this gradation correction process has been re-designed to improve the number and
the quality of the quadrilaterals in the output mesh.
A posteriori and a priori mesh generation
Metric-orthogonal point-placement is currently used to generate quasi-structured meshes with right-angled triangles where the metric is the most anisotropic and unit triangles elsewhere. The aim of
this work is to recover some quadrilaterals in the structure. To do so, two approaches can be considered: an a posteriori quadrilateral recombination based on geometrical criteria, and an a priori
quadrilateral detection. The latter is more straightforward because it uses directly the point-placement information.
6.7 Anisotropic mesh adaptation for fluid-structure interactions
Participants: Frédéric Alauzet, Adrien Loseille, Julien Vanharen.
A new strategy for mesh adaptation dealing with Fluid-Structure Interaction (FSI) problems is presented using a partitioned approach. The Euler equations are solved by an edge-based Finite Volume
solver whereas the linear elasticity equations are solved by the Finite Element Method using the Lagrange ${P}^{1}$ elements. The coupling between both codes is realized by imposing boundary
conditions. Small displacements of the structure are assumed and so the mesh is not deformed. The computation of a well-documented FSI test case is finally carried out to perform validation of this
new strategy.
6.8 Convergence improvement of the flow solver on highly anisotropic meshes
Participants: Francesco Clerici, Frédéric Alauzet.
When using anisotropic mesh adaptation in computational fluid dynamics, the interactions occurring among complex geometries, high gradients (such as boundary layers and shocks) present some
drawbacks, such as stallations and oscillations in the global residual convergence, specially when one makes use of slope limiters. In particular, we studied how the presence of a slope limiter
affects the overall convergence of a simulation of the Navier-Stokes equations making use of anisotropic mesh adaptation and the Spalart-Allmaras turbulence model, and we have shown several
techniques to reduce such such undesirable effects. With this regard, we successfully tested the freezeing of the slope limiter and the CFL reduction inside the slope limiter-oscillating vertices of
the mesh, and then we tested the same methodologies inside the shockwaves generated by transonic flows.
6.9 Pseudo-transient adjoint continuation
Participants: Francesco Clerici, Frédéric Alauzet.
In aeronautical engineering, anisotropic mesh adaptation is used to predict accurately dimensionless quantities such as the lift and the drag coefficients, and, in general, functionals depending on
the solution field. Anyway, in order to get the optimal adapted mesh with respect to the accuracy of a goal functional, it is necessary to solve an adjoint system providing the sensitivity of the
goal functional with respect to the residuals of the equations. The linear system associated to the adjoint problem revealed to be stiff for RANS equations with a standard solver such as the GMRES
preconditioned with several SGS iterations, and hence an alternative method has been developed, which is based on the transient simulation of the RANS adjoint state, starting from a previous valid
7 Bilateral contracts and grants with industry
7.1 Bilateral contracts with industry
7.2 Bilateral grants with industry
ANR IMPACTS 2018-2021
Ideal Mesh generation for modern solvers and comPuting ArchiteCTureS.
• Coordinateur : Adrien Loseille
• The rapid improvement of computer hardware and physical simulation capabilities has revolutionized science and engineering, placing computational simulation on an equal footing with theoretical
analysis and physical experimentation. This rapidly increasing reliance on the predictive capabilities has created the need for rigorous control of numerical errors which strongly impact these
predictions. A rigorous control of the numerical error can be only achieved through mesh adaptivity. In this context, the role of mesh adaptation is prominent, as the quality of the mesh, its
refinement, and its alignment with the physics are major contributions to these numerical errors. The IMPACTS project aims at pushing the envelope in mesh adaptation in the context of large size,
very high fidelity simulations by proposing a new adaptive mesh generation framework. This framework will be based on new theoretical developments on Riemannian metric-field and on innovative
algorithmic developments coupling a unique cavity-operator with an advancing-point techniques in order to produce high quality hybrid, curved and adapted meshes.
8 Scientific production
8.1 Publications of the year
International journals
• 1 articleOptimization of P2 meshes and applicationsComputer-Aided Design124April 2020, 102846
• 2 articleOn pixel-exact rendering for high-order mesh and solutionJournal of Computational PhysicsSeptember 2020, 109860
Scientific books
• 3 book Meshing, Geometric Modeling and Numerical Simulation 3 1 November 2020
8.2 Cited publications
• 4 articleA method for computing curved meshes via the linear elasticity analogy, application to fluid dynamics problemsInternational Journal for Numerical Methods in Fluids7642014, 246--266
• 5 articleA decade of progress on anisotropic mesh adaptation for Computational Fluid Dynamics722016, 13-39
• 6 articleHigh Order Sonic Boom Modeling by Adaptive Methods2292010, 561-593
• 7 book Anisotropic finite elements: local estimates and applications 3 Teubner Stuttgart 1999
• 8 articleMetric-based anisotropic mesh adaptation for three-dimensional time-dependent problems involving moving geometries3312017, 157-187
• 9 articleHigh-order accurate discontinuous finite element solution of the 2D Euler equationsJournal of computational physics13821997, 251--285
• 10 book Maillage, modélisation géométrique et simulation numérique 1: Fonctions de forme, triangulations et modélisation géométrique 1 ISTE Group 2017
• 11 book Meshing, Geometric Modeling and Numerical Simulation 1: Form Functions, Triangulations and Geometric Modeling John Wiley & Sons 2017
• 12 articleParametric surface meshing using a combined advancing-front -- generalized-Delaunay approachInternational Journal for Numerical Methods in Engineering491-22000, 233-259
• 13 article Influence of Reference-to-Physical Frame Mappings on Approximation Properties of Discontinuous Piecewise Polynomial Spaces Journal of Scientific Computing 52 09 2012
• 14 incollectionThe combined effect of curved boundaries and numerical integration in isoparametric finite element methodsThe mathematical foundations of the finite element method with
applications to partial differential equationsElsevier1972, 409--474
• 15 articleVery High Order Anisotropic Metric-Based Mesh Adaptation in 3DProcedia Engineering16325th International Meshing Roundtable2016, 353 - 365URL: http://www.sciencedirect.com/science/
• 16 inproceedingsCurvilinear Mesh Generation in 3DProceedings of the 7th International Meshing Roundtable1999, 407--417
• 17 inproceedings Anisotropic Error Estimate for High-order Parametric Surface Mesh Generation 28th International Meshing Roundtable Buffalo, NY, United States October 2019
• 18 articleHigh-order Unstructured Curved Mesh Generation Using the Winslow EquationsJ. Comput. Phys.307February 2016, 1--14
• 19 misc About Surface Remeshing 2000
• 20 inproceedingsAbout surface remeshingProceedings of the 9th international meshing roundtableNew Orleans, LO, USA2000, 123-136
• 21 misc Medit: An interactive mesh visualization software, INRIA Technical Report RT0253 2001
• 22 incollectionDefining quality measures for mesh optimization on parameterized CAD surfacesProceedings of the 21st International Meshing RoundtableSpringer2013, 85--102
• 23 book Maillage, modélisation géométrique et simulation numérique 3: Stockage, transformation, utilisation et visualisation de maillage 4 ISTE Group 2020
• 24 book Maillage, modélisation géométrique et simulation numérique 2: Métriques, maillages et adaptation de maillages 2 ISTE Group 2018
• 25 book Meshing, Geometric Modeling and Numerical Simulation 2: Metrics, Meshes and Mesh Adaptation John Wiley & Sons 2019
• 26 articleGeometric validity (positive jacobian) of high-order Lagrange finite elements, theory and practical guidanceEngineering with computers3232016, 405--424
• 27 articleConstruction of tetrahedral meshes of degree twoInternational Journal for Numerical Methods in Engineering9092012, 1156,1182
• 28 article``Ultimate'' robustness in meshing an arbitrary polyhedron5872003, 1061-1089
• 29 articleAutomatic mesh generator with specified boundary921991, 269-288
• 30 articleGmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilitiesInternational Journal for Numerical Methods in Engineering79112009, 1309-1331
• 31 articleGeneration of unstructured curvilinear grids and high-order discontinuous Galerkin discretization applied to a 3D high-lift configurationInternational Journal for Numerical Methods in
Fluids8262016, 316-333
• 32 articleAutomated low-order to high-order mesh conversionEngineering with Computers351Jan 2019, 323--335
• 33 misc Ensight
• 34 misc ParaView
• 35 misc TecPlot
• 36 articleGeometrical validity of curvilinear finite elementsJournal of Computational Physics233152013, 359-372
• 37 inproceedingsCurving for Viscous Meshes27th International Meshing RoundtableChamSpringer International Publishing2019, 303--325
• 38 incollection High-Order Mesh Curving Using WCN Mesh Optimization 46th AIAA Fluid Dynamics Conference AIAA AVIATION Forum American Institute of Aeronautics and Astronautics 2016
• 39 articleOptimal isoparametric finite elements and error estimates for domains involving curved boundariesSIAM journal on numerical analysis2331986, 562--580
• 40 misc FieldView
• 41 articleContinuous mesh framework part I: well-posed continuous interpolation errorSIAM Journal on Numerical Analysis4912011, 38--60
• 42 articleContinuous mesh framework part II: validations and applicationsSIAM Journal on Numerical Analysis4912011, 61--86
• 43 articleUnique cavity-based operator and hierarchical domain partitioning for fast parallel generation of anisotropic meshesComputer-Aided Design852017, 53-67
• 44 phdthesis Anisotropic 3D hessian-based multi-scale and adjoint-based mesh adaptation for Computational fluid dynamics: Application to high fidelity sonic boom prediction. Université Pierre et
Marie Curie - Paris VI December 2008
• 45 articleFully anisotropic goal-oriented mesh adaptation for 3D steady Euler equationsJournal of Computational Physics22982010, 2866 - 2897URL: http://www.sciencedirect.com/science/article/pii/
• 46 misc An introduction to Vizir: an interactive mesh visualization and modification software 2016
• 47 inbook Cavity-Based Operators for Mesh Adaptation 51st AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition URL: https://arc.aiaa.org/doi/abs/10.2514/
• 48 inbook Recent Improvements on Cavity-Based Operators for RANS Mesh Adaptation 2018 AIAA Aerospace Sciences Meeting URL: https://arc.aiaa.org/doi/abs/10.2514/6.2018-0922
• 49 inproceedingsA new approach to octree-based hexahedral meshing2001, 209--221
• 50 inproceedingsAdvances in Octree-Based All-Hexahedral Mesh Generation: Handling Sharp Features18Salt Lake City, UT, USA2009, 65-84
• 51 articleWell-suited and adaptive post-processing for the visualization of hp simulation resultsJournal of Computational Physics3752018, 1179 - 1204
• 52 phdthesis Méthode de visualisation adaptée aux simulations d'ordre élevé. Application à la compression-reconstruction de champs rayonnés pour des ondes harmoniques. Université de Toulouse
February 2019
• 53 inproceedings Results from the 3rd Drag Prediction Workshop using NSU3D unstructured mesh solver 45 AIAA-2007-0256, Reno, NV, USA Jan 2007
• 54 articleHigh-order curvilinear meshing using a thermo-elastic analogyComputer-Aided Design722016, 130 - 139
• 55 articleAn isoparametric approach to high-order curvilinear boundary-layer meshingComputer Methods in Applied Mechanics and Engineering2832015, 636 - 650
• 56 articleInterpolation error bounds for curvilinear finite elements and their implications on adaptive mesh refinementJournal of Scientific Computing7822019, 1045--1062
• 57 articleGPU-Based Interactive Cut-Surface Extraction From High-Order Finite Element FieldsIEEE Transactions on Visualization and Computer Graphics17122011, 1803--11
• 58 articleElVis: A System for the Accurate and Interactive Visualization of High-Order Finite Element SolutionsIEEE Transactions on Visualization and Computer Graphics18122012, 2325-2334
• 59 inproceedings Unstructured Grid Adaptation: Status, Potential Impacts, and Recommended Investments Toward CFD Vision 2030 46 2016-3323, Washington, D.C., USA 2016
• 60 incollectionHigh-Order Visualization with ElVisNotes on Numerical Fluid Mechanics and Multidisciplinary DesignSpringer International Publishing2015, 521--534
• 61 inproceedingsCurved mesh generation and mesh refinement using Lagrangian solid mechanics47th AIAA Aerospace Sciences Meeting including The New Horizons Forum and Aerospace Exposition2009, 949
• 62 articleEfficient visualization of high-order finite elementsInternational Journal for Numerical Methods in Engineering6952007, 750-771
• 63 inproceedingsP2 cavity operator and Riemannian curved edge length optimization: a path to high-order mesh adaptationAIAA Scitech 2021 Forum2021, 1781
• 64 articleHigh-order mesh curving by distortion minimization with boundary nodes free to slide on a 3D CAD representationComputer-Aided Design7223rd International Meshing Roundtable Special
Issue: Advances in Mesh Generation2016, 52 - 64
• 65 articleDefining an ${L}^{2}$-disparity Measure to Check and Improve the Geometric Accuracy of Non-interpolating Curved High-order Meshes'Procedia Engineering1242015, 122--134
• 66 articleGeneration of curved high-order meshes with optimal quality and geometric accuracyProcedia engineering1632016, 315--327
• 67 articleSummary of the first AIAA CFD High-Lift Prediction WorkshopJournal of Aircraft4862011, 2068-2079
• 68 articleOverview and Summary of the Second AIAA High Lift Prediction WorkshopJournal of Aircraft5242015, 1006-1025
• 69 articleMesh generation in curvilinear domains using high-order elementsInternational Journal for Numerical Methods in Engineering5312002, 207-223
• 70 techreport CFD Vision 2030 Study: A path to revolutionary computational aerosciences NASA March 2014
• 71 articleRobust untangling of curvilinear meshesJournal of Computational Physics2542013, 8 - 26
• 72 articleOptimizing the geometrical accuracy of curvilinear meshesJournal of Computational Physics3102016, 361 - 380
• 73 articleA Variational Framework for High-order Mesh GenerationProcedia Engineering163Supplement C25th International Meshing Roundtable2016, 340 - 352
• 74 phdthesis High-order numerical methods for unsteady flows around complex geometries Université de Toulouse 2017
• 75 inproceedingsCurved PN TrianglesProceedings of the 2001 Symposium on Interactive 3D Graphics2001, 159-166
• 76 articleThe generation of arbitrary order curved meshes for 3D finite element analysisComputational Mechanics513Mar 2013, 361--374
• 77 inproceedingsCurvilinear mesh adaptationInternational Meshing RoundtableSpringer2018, 57--69
• 78 inproceedings On the Necessity of Superparametric Geometry Representation for Discontinuous Galerkin Methods on Domains with Curved Boundaries 23rd AIAA Computational Fluid Dynamics Conference | {"url":"https://radar.inria.fr/report/2020/gamma/uid0.html","timestamp":"2024-11-08T23:25:42Z","content_type":"text/html","content_length":"297368","record_id":"<urn:uuid:fa633cfe-e25d-44a9-a863-57e9e90d335c>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00447.warc.gz"} |
Збірник праць Інституту математики НАН України, 2009 (том 6)On the efficient method of solving ill-posed problems by adaptive discretizationMorse-Bott functions on manifolds with semi-free circle actionOn conjugate pseudo-harmonic functionsHigh energy physics and algebraic geometry
http://dspace.nbuv.gov.ua:80/xmlui/handle/123456789/6273 2024-11-05T04:23:34Z http://dspace.nbuv.gov.ua:80/xmlui/handle/123456789/6332 On the efficient method of solving ill-posed problems by
adaptive discretization Solodky, S.G.; Volynets, E.A. To solve ill-posed problems Ax = f is used the Fakeev-Lardy regularization, using an adaptive discretization strategy. It is shown that for some
classes of finitely smoothing operators proposed algorithm achieves the optimal order of accuracy and is more economical in the sense of amount of discrete information then standard methods
2009-01-01T00:00:00Z http://dspace.nbuv.gov.ua:80/xmlui/handle/123456789/6331 Morse-Bott functions on manifolds with semi-free circle action Sharko, V.V. Let W²ⁿ be a closed manifold of dimension ≥ 6
with semi-free circle having finitely many fixed points. We study S¹-invariant Morse-Bott functions on W²ⁿ. The aim of this paper is to obtain exact values of minimal numbers of singular circles of
some indexes of S¹-invariant Morse-Bott functions on W²ⁿ. 2009-01-01T00:00:00Z http://dspace.nbuv.gov.ua:80/xmlui/handle/123456789/6330 On conjugate pseudo-harmonic functions Polulyakh, Ye. We prove
the following theorem. Let U be a pseudo-harmonic function on a surface M². For a real valued continuous function V : M² → R to be a conjugate pseudo-harmonic function of U on M² it is necessary and
sufficient that V is open on level sets of U. 2009-01-01T00:00:00Z http://dspace.nbuv.gov.ua:80/xmlui/handle/123456789/6329 High energy physics and algebraic geometry Malyuta, Yu.M.; Obikhod, T.V.
Superstring theory is applied to construction of the Minimal Supersymmetric Standard Model. 2009-01-01T00:00:00Z | {"url":"http://dspace.nbuv.gov.ua/xmlui/feed/rss_1.0/123456789/6273","timestamp":"2024-11-06T07:44:05Z","content_type":"application/rdf+xml","content_length":"3758","record_id":"<urn:uuid:1af48acc-834e-4ec8-8343-ea3c40039252>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00403.warc.gz"} |
Spell Out Decimal Numbers In Contracts - SpellingNumbers.com
Spell Out Decimal Numbers In Contracts
Spell Out Decimal Numbers In Contracts – It can be difficult to learn how to spell numbers. But, the process of learning to spell can be made easier with the right resources. There are numerous tools
that can help you improve your spelling abilities, regardless of whether you are at school or at work. They include books as well as tips and games online.
The format of the Associated Press format
You should be able to spell numbers using the AP style if you write for a newspaper or any other print media. The AP style teaches you the correct spelling of numbers, as well as other details to
make your writing easier.
Since its debut in 1953, The Associated Press Stylebook has undergone hundreds of updates. The stylebook is now in its 55th version. This stylebook is used by the vast majority of American
newspapers, periodicals, as well as internet news outlets.
Journalism is often guided by AP Style. These guidelines include punctuation and grammar. The use of capitalization, dates and times, as well citations are some of the most crucial AP Style best
Regular numbers
An ordinal number is an integer that represents a particular location in a list, or a sequence. These figures are frequently used to represent magnitude, significance or the progress of time. These
figures also can show the sequence in which events occur.
Based on the context depending on the situation, normal numbers can be expressed both verbally or numerically. A unique suffix is utilized to distinguish the two primary methods.
To make a number an ordinal one, add a “th”. 31 is for the ordinal number.
An ordinal can be used to refer to a variety of things, such as dates and names. It’s crucial to distinguish between an ordinal and a cardinal.
Both millions of people and trillions of dollars
Numerology is utilized in a variety of situations, including the stock market and geology, the history of geology and politics. Examples include millions and trillions. Million is the natural number
before 1000,001, while the trillion is after 999.999.999.
The annual income of a business is measured in millions. These numbers are used to determine the worth of a stock or fund, or any other financial item. To determine the market capitalization of a
company, billions are often employed. A unit conversion calculator allows you to check whether your estimates are accurate by converting trillions of dollars into millions.
In the English language, fractions are used to denote specific items or parts of numbers. The denominator as well as the numerator are divided into two separate pieces. The numerator displays how
many pieces of equal size were taken in while the denominator displays how many portions were divided into.
Fractions can either be expressed mathematically or written in words. You must be cautious to make the word “fractions” clear when you write them in words. This might be challenging since you may
need to use several hyphens, in particular when dealing with bigger fractions.
There are a few simple rules to follow if you intend to write fractions in words. The first is to place the numbers at the beginning of every sentence. It is also possible to compose fractions using
decimal formats.
It is likely that you are likely to spend a long time spelling out numbers, whether you’re writing an essay, a thesis, or sending an email. A few tricks and techniques will help you avoid repeating
the same numbers twice and to maintain proper formatting.
Numbers must be clearly written in formal writing. There are many style books which can guide you to follow these rules. The Chicago Manual of Style suggests that numerals be used between the numbers
1 and 100. However, writing out numbers higher than 401 is not recommended.
There are some exceptions. One example is the American Psychological Association’s (APA), style guide. Although it’s not a specialist publication, this guide is widely used in scientific writing.
Date and time
The Associated Press style book provides some general guidelines regarding the style of numbers. For numbers that are 10 or more numbers, numerals are used. Numerology can also used in other
contexts. It is a good rule of thumb to select the “n-mandated number” for the initial five numbers on your document. There are however some exceptions.
The Chicago Manual of Technique, and the AP stylebook mentioned above both recommend the use of plenty of numbers. This does not suggest that a stylebook with less numbers is not possible. Though I
can confirm you that there’s a difference as I myself have been an AP graduate.
To determine which ones you’re missing, a stylebook ought to be utilized. For instance, you’ll want to be sure not to miss the “t,” such as the “t” in “time.”
Gallery of Spell Out Decimal Numbers In Contracts
83 Decimal Place Value Chart 5th Grade Page 2 Free To Edit Download | {"url":"https://www.spellingnumbers.com/spell-out-decimal-numbers-in-contracts/","timestamp":"2024-11-10T03:09:06Z","content_type":"text/html","content_length":"60561","record_id":"<urn:uuid:7cc12814-7828-4d10-b11c-4beb84f6dfff>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00240.warc.gz"} |
Returns the first partial derivatives of the underlying surface at the specified point.
Namespace: Autodesk.Revit.DB
Assembly: RevitAPI (in RevitAPI.dll) Version: 17.0.0.0 (17.0.484.0)
Visual Basic
Public Function ComputeDerivatives ( _
point As UV _
) As Transform
Visual C++
Transform^ ComputeDerivatives(
UV^ point
Type: Autodesk.Revit.DB UV
The parameters to be evaluated, in natural parameterization of the face.
Return Value
A transformation containing tangent vectors and a normal vector.
The following is the meaning of the transformation members:
• Origin is the point on the face (equivalent to Evaluate(UV) );
• BasisX is the tangent vector along the U coordinate (partial derivative with respect to U).
• BasisY is the tangent vector along the V coordinate (partial derivative with respect to V).
• BasisZ is the underlying surface's normal vector. This is not necessarily aligned with the normal vector pointing out of a solid that contains the face, to get that value use ComputeNormal(UV) .
None of the vectors are normalized.
See Also | {"url":"https://www.revitapidocs.com/2024/77ca18ef-783e-9db5-a37a-2d76f637d1a1.htm","timestamp":"2024-11-13T18:23:03Z","content_type":"text/html","content_length":"26349","record_id":"<urn:uuid:3114e7e5-de18-4fc0-b1c3-bee73d270917>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00303.warc.gz"} |
Listening to the Mandelbrot set
2017-12-31 math composition
Okay, it's New Year's Eve, I'm overdue to write this week's Web log entry, and the topic I wanted to write about is held up because of (among other things) a package that apparently was stolen from
my front porch after it was "delivered" during my vacation and I told them not to mail it so early but blah blah blah... Instead of the exciting mystery topic, let's dig into the back catalog again
and think about how to listen to the Mandelbrot set.
Here's the Mandelbrot set. It should be pretty familiar to many of you.
Without going into a lot of detail, the idea here is that if you take a complex number z, square it, add another complex number c, and then keep doing that (alternately square your number, and add c
), what happens? If you start with z and c large-ish, you can expect that the numbers will just get bigger and bigger, escaping to infinity. If you start with them small-ish, they may instead get
locked into a fixed point (like z=c=0), or a loop of just a few repeating values, or even do some complicated thing that does not actually repeat but also never escapes to infinity. You can draw a
picture of what happens for different combinations of z and c; there are different ways to do that which give different kinds of pretty pictures, but the one above in particular is where you always
start wit z=0 and then test different values of c. Each pixel corresponds to a different value of c (bearing in mind these are complex numbers, so each "number" corresponds to a two-dimensional
location). The black "snowman" shape represents values of c for which the process does not escape to infinity; other points are given colours depending on how long it takes them to go outside a
certain fixed-size circle, after which it's proven that they necessarily just keep spiralling outward forever.
Zooming into the image reveals an endless variety of interesting details, including an infinite number of shrunk-down near-copies of the whole thing. This kind of self-similarity at different scales
is typical of fractals.
How could we turn this into sound?
Here's one idea. Remember Lissajous figures? You put sine waves into the X and Y coordinates of an oscilloscope in X-Y mode, and depending on the relationship between the sine waves, you get
different shapes on the display. For two sine waves at the same frequency, you get an ellipse that shows the phase relationship (fat like a circle: 90 degrees; thin like a line: 0 degrees; other
shapes in between: other phase angles). For two waves at different but harmonically related frequencies, you get more complicated shapes.
So... what if we could generate a pair of signals such that when you fed them into the X and Y coordinates of an oscilloscope, the resulting display would trace out the complicated boundary of the
Mandelbrot set?
At a glance it may seem completely hopeless that we could ever do such a thing. Just for a start, it's not obvious that the Mandelbrot set consists of one single piece ("connectedness" is the
relevant mathematical property). Maybe we could start drawing it, go around one of the islands making up the set, return to our starting point, and find that we'd left out a lot of interesting
details. It turns out that that is not a problem - it actually is a single piece, so there is a single boundary we could trace that would outline it perfectly. But the boundary is of infinite length!
So if we want to draw it, proceeding at a finite speed in terms of length covered per unit time, it will take forever. Even if there's a way around that, the shape is also, as implied by the exercise
of zooming in on pictures of it, an extremely complicated curve. It's not obvious that we can compute this entire curve in any useful way even with a lot of computer assistance.
Fortunately, the mathematicians come to the rescue here. Remember I said that the Mandelbrot set is connected - it has just one piece. That was proved by Douady and Hubbard in the 1980s, and the way
they did it involved proving the existence of a sequence of functions that converge on the Mandelbrot set boundary, starting with a plain circle and then getting wigglier and wiggler as they wrap
more and more closely around the infinitely wiggly object. They just proved that such curves exist with certain properties, but Jungreis subsequently came up with an algorithm allowing the curves to
really be computed. There's a useful links page by Adam Majewski summarizing the computational algorithm; from there you can link to discussions, academic references, and so on. (UPDATE: The page is
no longer online, but I have updated my link to point to a Wayback Machine archived version of it.)
The important formula looks like this:
This is a mapping from the area of the complex plane outside the unit circle, to the area outside the Mandelbrot set. There is an infinite sequence of real numbers b[m] defining the function; then it
takes as input a complex number z and produces as output a complex number ψ(z). The formula technically is not supposed to apply to the unit circle itself, but we can do a bit of handwaving and take
a limit, and say that if we let z range over a circle achingly close to the unit circle, then ψ(z) will run around the complicated boundary of the Mandelbrot set in the same time. And the numbers b
[m] can be calculated by an algorithm that, although annoying, is reasonably implementable (source code in a few languages on Majewski's page).
It even gets better. The formula for ψ(z), you may observe, is a polynomial with real coefficients in z^-1. To compute one of the two parts of the output (without loss of generality let's say the
imaginary part), we need to know the imaginary parts of:
• z (a point that runs around the unit circle in some fixed time)
• z^-1 (same thing in the opposite direction)
• z^-2 (going around the unit circle twice as fast)
• z^-3 (going around the unit circle three times as fast)
• ...
These functions are just a harmonically related set of sine waves. We mix them with strengths determined by b[m] and get the desired Mandelbrot-tracing function. The coefficients b[m], even if they
look mysterious, are just a Fourier analysis of the function that would trace a Mandelbrot-shaped Lissajous figure! And this is what the real and imaginary parts of that function look like, computed
to the level of detail I figured would be sufficient for musical purposes:
Plotting these two against each other, they do indeed look more or less as they should:
Here's what it sounds like.
[MP3] [FLAC]
That's a sort of interesting drone, but not really too exciting all by itself. (Real part in the left channel, imaginary in the right, but note I resynthesized it in Csound without the phase
information, so the actual waveforms if you look at them in a wave editor will not match the plots above.)
Back in 2013 I played with this some more and used it as the basis for a more finished composition called Silver Ratio. Here I'm using additive synthesis with Csound's "oscbank" opcode, and applying
a fair bit of modulation to the individual partials for a fatter, pad-style sound. And I didn't write down all of what I did, so can't comment on it in a whole lot of detail. At this point I'd say
it's more "inspired by" the sound of tracing out the Mandelbrot set and associated Julia set fractals, rather than a literally accurate recreation of the boundary functions. But still, a fun thing to
play with.
Have a happy New Year!
◀ PREV Modular synthesis intro, part 3: How to get started || Modular synthesis intro, part 4: Some pitfalls NEXT ▶
MSK 010 Fixed Sine Bank, variant A
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Aligned Pair Exclusion
Aligned Pair Exclusion
is often referred to as APE.
The final solution for any pair of cells cannot be the two values contained in a
cell that is a
buddy cell
to both cells. It would not leave a valid
for the bi-value cell.
If a pair of cells are in the same
they cannot contain the same value.
Aligned Pair Exclusion
strategy uses both these rules to exclude as many of the possible candidate combinations from two cells aligned in a unit as it can.
All possible candidate combinations are listed for an aligned pair of cells. All possible candidate combinations that exist in bi-value cells that are buddy cells to both cells in the aligned pair
are removed from the list. All possible candidate combinations that are the same value in both cells are removed from the list. The solution for the pair of aligned cells has to come from the
possible candidate combinations that are left over.
When the possible candidate combinations left over do not contain one of the original candidates in the aligned pair that candidate can be removed.
Aligned Pair Exclusion
Type 2
is an extension of this strategy that also uses the candidate pairs from an
almost locked set
to exclude possible candidate combinations from the aligned pair of cells.The blue cells are the
Aligned Pair
in the example in Figure 1. The green cells are all bi-value cells that are buddy cells to both the blue cells.
Listing all possible candidate combinations in the blue cells results in
1 - 1 #
1 - 7
1 - 8
1 - 9 *
6 - 1
6 - 7
6 - 8
6 - 9
8 - 1
8 - 7
8 - 8 #
8 - 9 *
9 - 1 *
9 - 7 *
9 - 8 *
9 - 9 #
The possible candidate combinations marked with a * are the candidate pairs contained in the bi-value cells that are buddy cells to both blue cells. These can be removed from the possible candidate
combinations for the blue cells. If any of the candidate pairs marked with a * were to be used in the two blue cells it would leave one of the bi-value cells without a possible solution.
The possible candidate combinations marked with # cannot exist in a pair of cells in the same unit. The same value cannot be placed twice in the same unit. These can be removed from the possible
candidate combinations for the blue cells.
The non excluded possible candidate combinations are
1 - 7
1 - 8
6 - 1
6 - 7
6 - 8
6 - 9
8 - 1
8 - 7
This list no longer contains a candidate combination with the candidate 9 in the first blue cell.
The candidate 9 can be removed from the first blue cell.
The Aligned Pair Exclusion logic works on any pair of cells, even if they are not aligned. In this case, the same candidate is legal in both cells and cannot be removed as a possible solution for the
two cells. The candidate pairs in all bi-value cells that are buddy cells to both cells in the pair can be removed as above.
This is sometimes called
Subset Exclusion.
Figure 2 shows an example of this. The blue pair of cells are not aligned in any unit but they have common buddy cells that are bi-value cells.
Listing all the candidate pair possibilities for the blue cells results in
1 - 2 *
1 - 6 *
1 - 7 *
4 - 2
4 - 6
4 - 7
7 - 2
7 - 6
7 - 7
There are bi-value cells with candidates pairs of 1 - 7, 1 - 6 and 1 - 2. These are marked with a *. Once these possibilities have been excluded there is no longer a candidate combination that
includes a 1 in the first cell so it can be removed.The Aligned Pair Exclusion logic also works by using all two candidate combinations from a two cell almost locked set as long as both cells in the
almost locked set are buddy cells to both cells in the aligned pair.
This works because, if the aligned pair used two of the possible candidates from a two cell/three candidate almost locked set, there would only be one candidate left for two cells in the almost
locked set.
This has been referred to
Aligned Pair Exclusion Type 2
.In the example in Figure 3 the list of possible candidate combinations is
3 - 3 #
3 - 5
3 - 7 +
5 - 3
5 - 5 #
5 - 7 *
7 - 3 +
7 - 5 *
7 - 7 #
8 - 3
8 - 5
8 - 7
The 3 - 3, 5 - 5 and 7 - 7 possible candidate combination can be removed because the two blue cells are in the same unit. (Box or row). There cannot be a 3, 5 or 7 in two cells in the same unit.
The 5 - 7 possible candidate combinations can be removed because they are in a bi-value cell (green) that is buddy cells to both the blue cells. If this combination was used as the solution for the
two blue cells there would be an empty cell (green).
The 3 - 7 possible candidate combination can be removed because it is included in the two cell almost locked set shown as yellow cells. If the 3 and the 7 are used in the blue cells then there would
only be a 2 left to fill the two yellow cells.
Reviewing the possible candidate combinations left over shows a list that no longer contains a 7 in the first blue cell.
The candidate 7 can be removed from the 1st cell in this aligned pair.
This process could be repeated using candidate combinations from 3 cell - 4 candidate almost locked sets. If the aligned pair of cells used two candidates from the 4 candidates available to solve a 3
cell almost locked set then the almost locked set would be one candidate short. | {"url":"https://sudoku.ironmonger.com/howto/alignedPairExclusion/docs.tpl","timestamp":"2024-11-10T08:18:27Z","content_type":"text/html","content_length":"22861","record_id":"<urn:uuid:9bcc9e8b-0be9-4367-a940-f97938d5d9c1>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00195.warc.gz"} |
How Fear Can Be Measured in Baseball
By: Ian Turner
On a late night in 1998, the Arizona Diamondbacks led the San Francisco Giants 8 to 6 in the bottom of the ninth inning with two outs and the bases loaded. A Giants batter stepped up to the plate and
was walked by the Diamondbacks pitcher, cutting the Diamondback's lead to 8 to 7. Now you do not need to be a baseball expert or professional statistician to know that walking a batter with the bases
loaded is not a good idea. But this walk wasn't just any walk, it was an intentional walk. And this batter wasn't just any batter, it was Barry Bonds. The decision to intentionally walk Barry Bonds
with the bases loaded ultimately worked, as the Diamondbacks pitcher retired the following batter, giving the Diamondbacks the 8 to 7 victory. Now one may look at the statistics and say, "well Barry
Bonds is an incredible hitter but you are still more likely to get him out than not. Why walk him?"; and the answer is something that cannot be measured by any statistic in baseball. The answer is
Fear is something that affects all sports. Although too much fear is certainly detrimental, fear is something that can benefit teams greatly, as seen above. In this article, I will attempt to
decipher what makes a pitcher fear a batter. Of course, this is no easy task, since there is no statistic remotely capable of predicting a pitcher's emotions and feelings. However, walks can be used
as a form of "fear" - meaning that the pitcher would rather give a batter a free pass to first base than risk another outcome. Of course, not all walks are intentional, but the baseball statistic
"intentional walks", does not capture every “intended” walk, so to try to figure out what makes a batter scary, we will use walks.
To approach this particular problem, we will take a look at some of the most "scary" hitters of all time - the hitters with the most walks in a single season. This list of 200 players hosts some of
the best hitters of all time: Barry Bonds (who appears on this top 200 list 13 times), Mark McGwire, Mickey Mantle, William McCovey, Mike Trout, and David Ortiz among others. To determine what
offensive statistics makes these players inspire fear, I collected 400 random MLB players from across baseball history. Combining these two data sets, marking the walk drawers with 'W' and the random
players with 'R', I will use statistical classification to determine what offensive statistics differentiate an average MLB player from one who inspires fear.
Data Exploration
The data being used has 63 predictor variables and the categorical response variable, which is what we are trying to predict. Of course, several of these predictor variables can be thrown out
immediately, such as player names, positions, and teams, as they have nothing to do with measuring offensive talent and fear. In addition, variables that are influenced by walks such as On Base
Percentage (OBP) and Times on Base (TOB) can be thrown out as well, as they are dependent on walks, so they would be the best predictors by default.
The graph below shows the separation between the density plots of walks for the "Walk" category versus the "Random" Category.
As expected, there is almost no crossover between the two categories, meaning that the "random" players have a clear statistical distinction from the all time walk drawers.
Now let's take a look at offensive statistics that would be expected to have influence on how a pitcher approaches a batter. Note that the more separated the two density curves are, the more
important the statistic is in determining fear, as it is better at separating the two different types of players.
As expected, the players that walk often perform better than the average hitter in every offensive category seen above, including advanced statistics like oWAR (Offensive Wins Above Replacement),
which measures offensive production compared to a replacement player, and more basic statistics such as Slugging Percentage, which is the average number of bases gained (walks not included) per at
bat. Based on these density plots, these offensive statistics create a strong distinction between a scary hitter and an average hitter.
Using these density plots, we can not only determine which variables perform well in separating the two player types, but also which ones perform poorly. For example, the statistic dWAR (Defensive
Wins Above Replacement) is an advanced statistic that determines how strong a fielder is overall, and has nothing to do with batting whatsoever. So we can see in the density plot below that there is
nearly no separation between each density curve, since dWAR has nothing to do with walks nor offensive statistics.
In addition to density plots, we can also look at scatter plots to gauge predictor strength as seen below.
Since the blue points in the scatterplot are more separated horizontally from the red than they are vertically, simply based on graphics, home runs make a hitter scarier than runs batted in, which
makes sense intuitively.
Model Testing and Statistical Classification
Now typically for statistical classification, models with several variables are built in order to best predict the response variable. But in this case, rather than trying to predict whether a batter
is a random batter or a scary batter, we are trying to determine which predictor variables are indicators that a batter is scary. In addition, models such as Linear Discriminant Analysis (LDA) and
Generalized Linear Model require independence of the predictor variables, but nearly all of the offensive statistics captured in baseball are related to each other (such as the more home runs a
player has the more RBIs, the higher the Slugging Percentage and so on). So because we only care about the strength of the predictor variables and not the actual "ideal" model, along with the fact
that most of our predictor variables are correlated, we will be looking at the accuracy of models with one predictor variable. The model type being used for all the single variable classification
models is LDA, which aims to project the data onto a lower-dimensional space to create class separability.
The bar plot below shows the misclassification rates of the models and their respective singular predictor variables. Observe that the lower the misclassification rate, the more important the
statistic is in determining a "scary hitter".
This classification is performed over the full data, in which there are 200 players that are "walkers" and 400 players that are "random". So that means for a variable to be an effective predictor, it
must get more than 66.6% of the predictions right (less than 0.33 error rate). This is because by predicting that every player is "R" in the full data, you would get an error rate of ⅓ , so because
of this, we will get rid of the predictors that do not meet this minimum threshold.
Using this threshold, we end up with the following 15 predictor variables:
Analysis of Results
Out of the 15 variables remaining, only 3 are "basic" baseball statistics - home runs, slugging percentage, and strikeouts. This makes sense intuitively, as what makes a hitter scary is not an
ability to hit singles, but to hit home runs, doubles, and triples, which is weighed heavier in slugging percentage. Although strikeouts barely clears the threshold, power hitters do tend to
strikeout more often than the average hitter, so this also tracks.
However, the remaining 12 variables that are best at demonstrating what makes a hitter scary are all much more complicated, known as "advanced" baseball statistics. Most of these advanced statistics
go beyond simply tabulated statistics, but rather create ratings for each player based on how much offensive output they add to their team, taking into account the situation. For example, the RE24
statistic is described by Stathead Baseball (which was used to collect the data) as a statistic that measures the number of runs the batter added, taking into account the number of outs and number of
runners on base. A possible reason why RE24 performed so well is that a batter is scary not only because of their home run counts or slugging percentage, but when they do it. Players that succeed in
scenarios where there are runners on base are far more intimidating and scary than those who are not as effective when runners are on base.
The advanced statistic that performed the best out of all considered is BtRuns, which only incorrectly sorted 118 out of the 600 players. BtRuns is described as adjusted batting runs, which
"estimates a player's total contributions to a team's runs total via linear weights" (Stathead Baseball). So players with a higher BtRuns statistic provide larger contributions to their team's
offensive output. It makes sense then why this variable would perform the best compared to all the other offensive statistics. If a batter is responsible for a large share of their team's offensive
output, then that hitter is seen as "scary". For example, if a player such as Barry Bonds, Mike Trout, or Mickey Mantle were on a team with a bunch of average players, a pitcher would be "scared" of
facing these batters not only because of their ability to hit for power, but also because they carry such a large load of offensive output; a pitcher is far more likely to be scared of a batter if
that batter not only has impressive offensive talent, but is the main force of a team's offense.
Thinking back to the game where Barry Bonds was walked with the bases loaded, the pitcher was scared of Barry Bonds not only because he is one of the best power hitters of all time, but because he
was the dominant offensive force on that San Francisco Giants team. The hitter that followed Barry Bonds that night was Brent Mayne, a career .263 hitter who is much less scary to face with the bases
loaded than the home run machine Barry Bonds. Based on this example and the fact that BtRuns performed the best, in baseball, fear is not only dependent on a batter's skill and offensive statistics,
but the talent around them.
Drawbacks and Possible Issues
Although the conclusions of this statistical experiment are fairly sound and intuitive - the idea that both individual skill along with how pivotal a player is in creating offense for a team both
cause a pitcher to be afraid of a batter - there are some limitations of this conclusion and some possible issues.
The first problem is how the "random" players were selected. I was unable to find a method for finding random players, so I instead decided to do a search for the top leaders in what I'd consider to
be "irrelevant" baseball statistics - sacrifice flies and hit by a pitch. Of course, there is correlation in every statistic in baseball, so it is possible that these players were not truly random,
so although I believe the conclusions are sound, it is possible that another variable could have performed better than BtRuns given a more randomized player set.
Another valid question that can be asked is why the accuracy of the predictors is so poor. Of course, because each of our models is only 1 variable, predictions are not going to be as strong.
However, it is also because not all players who draw a lot of walks are drawing them for the same reasons. Although a majority of the players in the top 200 walks list draw that many walks because of
their ability to hit and hit with power in pivotal situations, there are some players in the list that are good hitters, but draw so many walks because they have a good eye, rather than intimidating
batting prowess. So because there is the possibility that not all the batters in the "scary batters" data are actually scary, there might be a limitation for how low the misclassification score can
Although the statistical experiment was not flawless, and fear is far more complicated than a simple statistic like walks can capture, I believe that the method I used was able to provide some real
insights into why these batters are pitched around so often.
The exploration of fear in baseball reveals a complex interplay between individual skill and team dynamics. While traditional statistics such as home runs and slugging percentage do play a role in
defining a "scary" hitter, it is the advanced metrics that offer a deeper understanding of a player's impact on the game. BtRuns stands out by accurately capturing a player's contribution to their
team's offensive output, thus serving as a reliable indicator of the threat they pose to opposing pitchers, whereas the statistic RE24 stands out as it captures a player's ability to hit when it
matters most. These advanced statistics performed better than the simple statistics because they not only capture a hitter’s ability to hit and hit for power, but how valuable they are in their
team’s offensive machine.
The case of Barry Bonds' intentional walk with the bases loaded exemplifies the multifaceted nature of fear in baseball. It's not merely the potential for a powerful hit that intimidates pitchers;
it's also the recognition of a player's critical role within their team's offense. This insight challenges teams and analysts to consider not just the individual prowess of hitters but also how they
fit into and elevate the collective performance of their lineup. | {"url":"https://www.bruinsportsanalytics.com/post/_fear","timestamp":"2024-11-10T20:54:01Z","content_type":"text/html","content_length":"1050590","record_id":"<urn:uuid:ffa61742-4969-464b-b89e-5ef0a5ee628e>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00704.warc.gz"} |
coco simulation flow sheet for ball mill
WEBDownload scientific diagram | BGM grinding circuit flowsheet with ball mills in series configuration as modelled by MODSIM. from publiion: Optimization and performance of grinding circuits:
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WEBJul 14, 2016 · HPGR or highpressure grinding rolls have made broad advances into nonferrous metal mining. The technology is now widely viewed as a primary milling alternative, and there arc a
number of large installations commissioned in recent years. After these developments, an HPGR based circuit configuration would often be the base .
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WEBJun 6, 2016 · A variant of this method is to direct pebblecrushing circuit product to the ballmill sump for secondary milling: while convenient, this has the disadvantage of not controlling
the top size of feed to the ballmill circuit. ... including total mill volumetric flow rate; recirculating load. Feed Chute Design. ... Component Design. Trommel or ...
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WEBDec 4, 2018 · In this work, the milling operation of ball mills is investigated using two methods, DEM and combined DEMSPH. First, a pilot scale ball mill with no lifter is simulated by both
methods. Then ...
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WEBSep 1, 2016 · This demo shows plant modeling and controller design for a multi stage rolling mill process. Sheet Rolling Mill processes involve progressive thickness reduction of continuous
sheet metal in multiple stages. Some of the control variables of interest are exit thickness, throughput and the tension in the sheet as it passes from one stage to the ...
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WEBOct 1, 2001 · The fullscale ball mill has a diameter (inside liners) of The scaleup procedure shows that the fullscale circuit produces a product (hydrocyclone overflow) that has an 80%
passing size of 80 μm. ... Calculate the volumetric flow through the laboratory mill, QLaboratory Input data: duration of the last cycle of the lockedcycle ...
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WEBApr 1, 2015 · The paper focusses on improving the energy utilization of a cement grinding circuit by changing the flow sheet of the process. The circuit was comprised of ball mill, static
classifier and dynamic ...
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WEBApr 1, 2015 · There had been a few attempts to relate their model with air flow through the mill, feed rate, feed size distribution, material filling and ball filling (Viswanathan, 1986,
Zhang, 1992). Air swept ball mill model proposed by Austin et al. (1975) was validated by Apling and Ergin (1994) using the industrial scale data from a cement grinding circuit.
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WEBOct 12, 2016 · The simplest grinding circuit consists of a ball or rod mill in closed circuit with a classifier; the flow sheet is shown in Fig. 25 and the actual layout in Fig. 9. This
singlestage circuit is chiefly employed for coarse grinding when a product finer than 65 mesh is not required, but it can be adapted for fine grinding by substituting a bowl ...
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WEBDec 1, 2006 · To describe material flow through the ball mills, ... simulation of industrial tumbling ball mills requires a lot of effort for calibration of general mathematical model of ball
mills to a specific plant. Briefly, these efforts consists of plant sampling campaigns, laboratory tests to determine breakage function parameter and mill feed and ...
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WEBJul 1, 2017 · One of the newest works is related to Sinnott et al. [12], in which an overflow ball mill discharge and trommel flow has been simulated using the combined DEM and SPH method. The
present work has ...
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WEBThe ball mill, with 40% of mill filling and 70% of its c ritical speed, is fitted with steel liners. It uses steel balls in a ratio of 70% of mm (3 inches) and 30% of mm (2 inches) as
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WEBMar 1, 2002 · An exergy analysis is then performed on a laboratory sized dry ball mill by considering the surface energy variation for different ore sizes, the obtained HukkiMorrell fitted
relationship from ...
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WEBCOCO simulator is an opensource chemical process simulation software which is accessible by anybody particularly students. It is claimed that this free software has similar capabilities like
commercial software such as open flow sheet modelling environment incorporating unit operations, thermodynamic packages as well as reactions. In this .
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WEBAbstract Talc powder samples were ground by three types of ball mill with different sample loadings, W, to investigate rate constants of the size reduction and structural change into the
amorphous state. Ball mill simulation based on the particle element method was performed to calculate the impact energy of the balls, E i, during grinding.
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WEBChemSep is a CAPEOPEN compliant program. COCO's COFE can use unit operations from ChemSep. p. 3 of 12 fIn the Select Unit Operation window, select Separators, Flash. This adds a flash drum to
the flowsheet. The click the menu button to insert a material stream as shown, or click the Insert menu and select Stream.
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WEBMay 31, 2011 · Overall, the simulation using PFC3Dimproved understanding about the dynamics of the grinding balls within a planetary ball mill as well as the energy available for transfer in
collisions between ...
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WEBFrom this fixed point, mill speed and After adjusting the model to the ball mill data, the charge volume were varied in order to determine their effects of ball mill rotation speed (Fig. 13)
and bail effect on ball mill wear, The specifiions of the ball mill filling (Fig. 14) on the respective energy rate mill are found in Table 4 and Fig ...
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WEBA series of tests were performed in a laboratory ball mill using (i) three loads of single size media,, 40,, and mm and (ii) a mixed load of balls with varying sizes.
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WEBCOCO package installer: file: size: description: platform: last update: COCOInstall_: 48,788 KB: COCO version installation file. Windows Vista x64 or higher: ..., or you feel that in any other
way you can contribute to the COCO simulation environment, please contact us. Or you can make a donation. Donations allow us to .
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WEBNov 1, 2023 · A summary of the results from the DEM simulations is presented in Table 7, which shows that the collision frequency in the simulations of tests with mm balls and 50% mill filling
was six times higher than that with balls measuring 10 mm and 35% mill filling, both in a mill rotating at a frequency of 200 rpm.
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WEBSep 1, 2022 · A scaleup model was developed based on data of DEM simulation to quickly predict ball milling performance for different mill design and operation parameters. The ball milling
performance was characterized by grinding rate and power draw.
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WEBDownload scientific diagram | Single ball mill flowsheet with input and output streams from publiion: Software review of the models In: Modeling Simulation of Mineral ...
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WEBNov 2018. MINER ENG. Sandile Nkwanyana. Brian Loveday. Nkwanyana and Loveday (2017) used batch grinding experiments in a m diameter mill to test partial replacement of steel balls ( mm ...
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WEBNov 1, 2004 · Flow sheet of the closed mill circuit. The simulation of the comminution ( Espig and Reinsch, 2002 ) in the ball mill was executed using the grindability curve of Fig. 5 .
Modelling is based on energy input, mass flow and particle size distribution of mill feed and mill discharge.
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WEBDrum rotation rate has potential as a control parameter for fine tuning of the breakage behavior. The majority of breakage occurs near the mill shell, rather than at the point of impact
between the media and material and the extent of material breakage has a significant influence on the material flow within the mill.
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WEBMar 1, 2013 · Here we explore the important topic of breakage in a batch mill through numerical simulation. We investigate the effect of important operational parameters, matching well with
previously reported experimental results.
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Math Colloquia - Free boundary problems arising from mathematical finance
※ 줌(zoom) 병행Zoom: https://snu-ac-kr.zoom.us/j/87020850293
회의 ID: 87020850293
Many problems in financial mathematics are closely related to the stochastic optimization problem because the optimal decision must be made under the uncertainty. In particular, optimal stopping,
singular control, and optimal switching problems in the stochastic optimization problem arising from financial mathematics are formulated into the free boundary problem when the uncertainty follows
the Markov process. The optimal strategies to each optimization problem is determined by the free boundary. In this talk, I introduce various free boundary problems in financial mathematics. | {"url":"http://my.math.snu.ac.kr/board/index.php?mid=colloquia&l=en&page=8&document_srl=813563&sort_index=date&order_type=asc","timestamp":"2024-11-11T14:04:47Z","content_type":"text/html","content_length":"45402","record_id":"<urn:uuid:0203b4ba-5e99-475f-8d26-04679fc3d1cb>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00718.warc.gz"} |
Mathematical Logic
Stephen Cole Kleene; Mathematics
Dover Publications
Undergraduate students with no prior classroom instruction in mathematical logic will benefit from this evenhanded multipart text. It begins with an elementary but thorough overview of mathematical
read more…
of first order. The treatment extends beyond a single method of formulating logic to offer instruction in a variety of techniques: model theory (truth tables), Hilbert-type proof theory, and proof
theory handled through derived rules.The second part supplements the previously discussed material and introduces some of the newer ideas and the more profound results of twentieth-century logical
research. Subsequent chapters explore the study of formal number theory, with surveys of the famous incompleteness and undecidability results of Godel, Church, Turing, and others. The emphasis in the
final chapter reverts to logic, with examinations of Godel's completeness theorem, Gentzen's theorem, Skolem's paradox and nonstandard models of arithmetic, and other theorems. The author, Stephen
Cole Kleene, was Cyrus C. MacDuffee Professor of Mathematics at the University of Wisconsin, Madison. Preface. Bibliography. Theorem and Lemma Numbers: Pages. List of Postulates. Symbols and
Notations. Index.
BOOKSTORE TOTAL
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What is Torque Formula in Physics Define Symbol, Unit, Types
We hope your exam preparation is going well. To make your preparation powerful and to provide help, we will provide you information related to What is Torque Formula. Along with Torque Formula, we
will define symbol, unit, types, equation and derivation. Friends, if you want to provide special information about What is Torque Formula, then read this article carefully till the end so that you
can get all the important information.
According to physics, torque is defined as a rotational force that induces rotation. Torque is not a circular force as a result of this. Torques are typically linear forces that are given to a hinged
lever arm in order to cause the lever arm to rotate. Any hinged object can serve as the lever arm. For instance, because they are a solid rotating mass, the seats of a teeter-totter are lever arms.
When examining torque on a system, it’s critical to determine the lever arm’s length and rotational axis.
One way to conceptualize the axis of rotation is as the pivot point of the lever arm. It is the point around which the lever arm revolves. Not every lever arm has a central axis of rotation, but for
a teeter-totter, the axis of rotation is the apparatus’s center. An axis of rotation is often found along the edge of many torque-generating systems, such as swinging doors. Keep reading this article
till the end to know What is Torque Formula in physics, symbol, unit, types.
What is Torque Formula
The torque exerted on an item is determined by multiplying the force imposed on it by its perpendicular distance from the rotational axis. The symbol for it is τ. The cross product between the force
and the displacement vector from the pivot point yields the torque formula. Therefore, the torque can be expressed numerically as follows:
What is torque formula in physics
What Is Torque?
Torque is another name for the instant of force, or just movement. It describes the force that facilitates an object’s rotation around its pivot, fulcrum, and axis. Torque, on the other hand, is an
idea that compels the thing to spin and also refers to the turning action. The concept of force is analogous to the force of pulling or pushing an object. The term “axis of rotation” refers to the
angle at which an object rotates. The two primary components needed to find a linear force that involves rotation are Acceleration and mass.
Common Symbols τ , M
SI Unit N.m
In SI Base Units kg.m^2.s^-2
Dimension M L^2 T^-2
Other Units Pound-force-feet,lbf.inch
How Do Calculate Torque?
In the above we have told about What is Torque Formula. Now we know about the calculation of torque. Finding the lever arm and multiplying it by the applied force is an easy method to get the torque
magnitude. N is the axis of rotation, F is the horizontal force applied at p for rotation, and d is the arm’s moment (the distance measured perpendicularly between the line of action force and the
axis of rotation), as seen in the image.
Torque dimensional formula
Types Of Torque Formula
In the above article, we have given you special information about What is Torque Formula. Now we will tell you the types of torque which are of the following two types.
Static Torque
Static torque is any torque that does not cause an angular acceleration. A closed door experiences static torque when someone pushes on it because it is not rotating on its hinges despite the force
provided. A person pedaling a bicycle at a constant speed is also producing static torque because they are not accelerating. Additional instances of static torque are rotating a car’s steering wheel
or using a wrench to tighten a bolt.
Dynamic Torque
Dynamic torque is the torque that causes an angular acceleration. Given that a racing car is traveling swiftly around the track, the drive shaft must cause the wheels to accelerate angularly as the
car launches off the line. Additionally, dynamic torque is demonstrated when you peddle a bicycle and the machine starts to move at different speeds. Using a power drill, spinning a top, and running
a wind turbine are instances of dynamic torque.
Use of Torque in a Car
In automotive engineering, torque is a key notion since it governs a vehicle’s capacity to accelerate, climb slopes, and tow large objects. Torque can be defined as the twisting force that rotates an
object. When it comes to automobiles, torque is produced by the engine and sent to the wheels via the drivetrain, enabling motion.
Let’s solve a problem to have a closer look at torque.
• Assume a 600 Newton wrench is used by an auto mechanic to loosen a bolt. The auto mechanic’s force is exerted perpendicular to the arm of the wrench. About 0.20 meters separate the mechanic’s
hand from the bolt. Determine the torque’s applied magnitude.
The angle between the force and the wrench in the moment of the arm is 90 degrees.
We know that, sin 90y = 1
The magnitude of torque is given by;
T = F × r × siny
Therefore, the torque’s magnitude is given by
(600N) (0.2m) = 120 N m
Therefore, the torque magnitude is 120 Newton-metres (N m).
Utilizing Torque in Applications
In the above article, we have given you special information about what is torque formula. Now we are going to tell you about the Applications of Torque.
1. Wrenches: The nut (or bolt) is the rotation point since it can be turned to tighten or loosen the nut. The force is being applied by the hand and arm. To impart ninety degrees of power to a bolt
or nut, use a wrench.
2. It is a component of many intricate devices, such as the electric motor found in the majority of home appliances. It is especially crucial to the operation of cars since it has a big impact on
the engine and transmission.
3. Seesaws: People have observed individuals sitting on opposite ends of seesaws, with one person appearing to be heavier than the other. Because the heavier person’s moment arm has a shorter
duration than the lighter person’s, they can reduce torque by sitting relatively near to the pivot.
Frequently Asked Question
Q? Tell me about torque and its unit?
Ans: Torque is measured in Newton-meters (N-m). The vector product of the force and position vectors can be used to describe the above equation. τ → = r → × F →
Q? What distinguishes torque from force?
Ans: In the field of rotational mechanics, torque is the opposite of force. The primary distinction between them is that torque is a force’s capacity to create an axis of twist.
Q? Describe a torque example?
Ans: The list of frequent torque instances that follows includes the following: A linear downward force applied perpendicular to the doorknob causes it to revolve. When a linear force is given to a
coin at an angle to its edge, the resultant force is rotation.
Q? How does torque and moment differ from one another?
Ans: While moment refers to being propelled by an outside force to generate the rotation, torque is a specific instance of moment since it pertains to the rotation’s axis.
In the above article, we have told you about What is Torque Formula. What Is Torque? How Do Calculate Torque? Types Of Torque Formula and Use of Torque in a Car. We have given you special information
about this. Which will be very helpful in your exam. We hope friends, you liked the article written by us on What is Torque Formula. Stay connected with our website to get such best knowledge and
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Uncertainty principles and sum complexes
Let $$p$$p be a prime and let $$A$$A be a nonempty subset of the cyclic group $$C_p$$Cp. For a field $${\mathbb F}$$F and an element $$f$$f in the group algebra $${\mathbb F}[C_p]$$F[Cp] let
$$T_f$$Tf be the endomorphism of $${\mathbb F}[C_p]$$F[Cp] given by $$T_f(g)=fg$$Tf(g)=fg. The uncertainty number$$u_{{\mathbb F}}(A)$$uF(A) is the minimal rank of $$T_f$$Tf over all nonzero $$f \in
{\mathbb F}[C_p]$$f∈F[Cp] such that $$\mathrm{supp}(f) \subset A$$supp(f)⊂A. The following topological characterization of uncertainty numbers is established. For $$1 \le k \le p$$1≤k≤p define the
sum complex$$X_{A,k}$$XA,k as the $$(k-1)$$(k-1)-dimensional complex on the vertex set $$C_p$$Cp with a full $$(k-2)$$(k-2)-skeleton whose $$(k-1)$$(k-1)-faces are all $$\sigma \subset C_p$$σ⊂Cp such
that $$|\sigma |=k$$|σ|=k and $$\prod _{x \in \sigma }x \in A$$∏x∈σx∈A. It is shown that if $${\mathbb F}$$F is algebraically closed then $$\begin{aligned} u_{{\mathbb F}}(A)=p-\max \{k :\tilde{H}_
{k-1}(X_{A,k};{\mathbb F}) \ne 0\}. \end{aligned}$$uF(A)=p-max{k:H~k-1(XA,k;F)≠0}.The main ingredient in the proof is the determination of the homology groups of $$X_{A,k}$$XA,k with field
coefficients. In particular it is shown that if $$|A| \le k$$|A|≤k then $$\tilde{H}_{k-1}(X_{A,k};{\mathbb F}_p)\!=\!0.$$H~k-1(XA,k;Fp)=0.
• Simplicial homology
• Uncertainty principle
All Science Journal Classification (ASJC) codes
• Algebra and Number Theory
• Discrete Mathematics and Combinatorics
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MathTV Topic - Differentiation: The Language of Change
Search Results
Math Topics
The Derivative of a Function and Two Interpretations
Study Skill
Introduction and Definitions
1. What is the definition of the derivative?
2. If a function represents distance as a function of time, what does its derivative represent?
3. If a function represents the cost to produce \(x\) items, what does its derivative cost?
4. If the volume of a sphere is a function of its radius, what is the relationship between the rate of change of the volume and the rate of change of the radius?
Choose instructor to watch:
Differentiating Products and Quotients
Higher Order Derivatives
The Chain Rule and General Power Rule
Implicit Differentiation
Problem 3
Suppose both \(y\) and \(x\) are differentiable functions of \(t\) and that the relationship between \(y\) and \(x\) is expressed by the equation \(4x^3+3y^5=960\). Find and interpret \(\displaystyle
\frac{dy}{dt}\) when \(\displaystyle\frac{dx}{dt}=4\), \(x=6\), and \(y=2\).
Choose instructor to watch:
Problem 5
Use the function \(x=\displaystyle\frac{20000}{\sqrt[3]{2p^2-5}}+350\) to find the rate at which the number of instruments sold is changing with respect to time, when the price of an instrument is \
(\$400\) and is changing at a rate of \(\$1\) per month.
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Gecode::VarImp< VIC > Class Template Reference
[Programming variables]
Base-class for variable implementations. More...
#include <core.hpp>
Public Member Functions
VarImp (Space &home)
VarImp (void)
Creation of static instances.
Protected Member Functions
void cancel (Space &home)
Cancel all subscriptions when variable implementation is assigned.
bool advise (Space &home, ModEvent me, Delta &d)
Run advisors when variable implementation has been modified with modification event me and domain change d.
ModEvent fail (Space &home)
Run advisors to be run on failure and returns ME_GEN_FAILED.
void schedule (Space &home, PropCond pc1, PropCond pc2, ModEvent me)
Schedule subscribed propagators.
void subscribe (Space &home, Propagator &p, PropCond pc, bool assigned, ModEvent me, bool schedule)
Subscribe propagator p with propagation condition pc.
void cancel (Space &home, Propagator &p, PropCond pc)
Cancel subscription of propagator p with propagation condition pc.
void subscribe (Space &home, Advisor &a, bool assigned, bool fail)
Subscribe advisor a to variable.
void cancel (Space &home, Advisor &a, bool fail)
Cancel subscription of advisor a.
unsigned int degree (void) const
Return degree (number of subscribed propagators and advisors).
double afc (void) const
Return accumulated failure count (plus degree).
Cloning variables
VarImp (Space &home, VarImp &x)
Constructor for cloning.
bool copied (void) const
Is variable already copied.
VarImp * forward (void) const
Use forward pointer if variable already copied.
VarImp * next (void) const
Return next copied variable.
Bit management
unsigned int bits (void) const
Provide access to free bits.
unsigned int & bits (void)
Provide access to free bits.
Variable implementation-dependent propagator support
static void schedule (Space &home, Propagator &p, ModEvent me, bool force=false)
Schedule propagator p with modification event me.
static void reschedule (Space &home, Propagator &p, PropCond pc, bool assigned, ModEvent me)
Schedule propagator p.
static ModEvent me (const ModEventDelta &med)
Project modification event for this variable type from med.
static ModEventDelta med (ModEvent me)
Translate modification event me into modification event delta.
static ModEvent me_combine (ModEvent me1, ModEvent me2)
Combine modifications events me1 and me2.
Delta information for advisors
static ModEvent modevent (const Delta &d)
Return modification event.
Memory management
static void * operator new (size_t, Space &)
Allocate memory from space.
static void operator delete (void *, Space &)
Return memory to space.
static void operator delete (void *)
Needed for exceptions.
Detailed Description
template<class VIC>
class Gecode::VarImp< VIC >
Base-class for variable implementations.
Implements variable implementation for variable implementation configuration of type VIC.
Definition at line 219 of file core.hpp.
Constructor & Destructor Documentation
template<class VIC >
Gecode::VarImp< VIC >::VarImp ( Space & home ) [inline]
template<class VIC >
Gecode::VarImp< VIC >::VarImp ( void ) [inline]
Creation of static instances.
Definition at line 4137 of file core.hpp.
template<class VIC >
Gecode::VarImp< VIC >::VarImp ( Space & home,
VarImp< VIC > & x
) [inline]
Constructor for cloning.
Definition at line 4242 of file core.hpp.
Member Function Documentation
template<class VIC >
void Gecode::VarImp< VIC >::cancel ( Space & home ) [inline, protected]
Cancel all subscriptions when variable implementation is assigned.
Definition at line 4487 of file core.hpp.
template<class VIC >
bool Gecode::VarImp< VIC >::advise ( Space & home,
ModEvent me,
Delta & d
) [inline, protected]
Run advisors when variable implementation has been modified with modification event me and domain change d.
Returns false if an advisor has failed.
Definition at line 4504 of file core.hpp.
template<class VIC >
ModEvent Gecode::VarImp< VIC >::fail ( Space & home ) [inline, protected]
Run advisors to be run on failure and returns ME_GEN_FAILED.
Definition at line 4570 of file core.hpp.
template<class VIC >
void Gecode::VarImp< VIC >::subscribe ( Space & home,
Propagator & p,
PropCond pc,
bool assigned,
ModEvent me,
bool schedule
) [inline]
Subscribe propagator p with propagation condition pc.
In case schedule is false, the propagator is just subscribed but not scheduled for execution (this must be used when creating subscriptions during propagation).
In case the variable is assigned (that is, assigned is true), the subscribing propagator is scheduled for execution. Otherwise, the propagator subscribes and is scheduled for execution with
modification event me provided that pc is different from PC_GEN_ASSIGNED.
Definition at line 4382 of file core.hpp.
template<class VIC >
void Gecode::VarImp< VIC >::cancel ( Space & home,
Propagator & p,
PropCond pc
) [inline]
template<class VIC >
void Gecode::VarImp< VIC >::subscribe ( Space & home,
Advisor & a,
bool assigned,
bool fail
) [inline]
template<class VIC >
void Gecode::VarImp< VIC >::cancel ( Space & home,
Advisor & a,
bool fail
) [inline]
template<class VIC >
unsigned int Gecode::VarImp< VIC >::degree ( void ) const [inline]
Return degree (number of subscribed propagators and advisors).
Note that the degree of a variable implementation is not available during cloning.
Definition at line 4158 of file core.hpp.
template<class VIC >
double Gecode::VarImp< VIC >::afc ( void ) const [inline]
Return accumulated failure count (plus degree).
Note that the accumulated failure count of a variable implementation is not available during cloning.
Definition at line 4165 of file core.hpp.
template<class VIC >
bool Gecode::VarImp< VIC >::copied ( void ) const [inline]
Is variable already copied.
Definition at line 4222 of file core.hpp.
template<class VIC >
VarImp< VIC > * Gecode::VarImp< VIC >::forward ( void ) const [inline]
Use forward pointer if variable already copied.
Definition at line 4228 of file core.hpp.
template<class VIC>
VarImp* Gecode::VarImp< VIC >::next ( void ) const
Return next copied variable.
template<class VIC >
void Gecode::VarImp< VIC >::schedule ( Space & home,
Propagator & p,
ModEvent me,
bool force = false
) [inline, static]
Schedule propagator p with modification event me.
If force is true, the propagator is re-scheduled (including cost computation) even though its modification event delta has not changed.
Definition at line 4288 of file core.hpp.
template<class VIC >
void Gecode::VarImp< VIC >::reschedule ( Space & home,
Propagator & p,
PropCond pc,
bool assigned,
ModEvent me
) [inline, static]
Schedule propagator p.
Schedules a propagator for propagation condition pc and modification event me. If the variable is assigned, the appropriate modification event is used for scheduling.
Definition at line 4407 of file core.hpp.
template<class VIC >
ModEvent Gecode::VarImp< VIC >::me ( const ModEventDelta & med ) [inline, static]
Project modification event for this variable type from med.
Definition at line 4270 of file core.hpp.
template<class VIC >
ModEventDelta Gecode::VarImp< VIC >::med ( ModEvent me ) [inline, static]
template<class VIC >
ModEvent Gecode::VarImp< VIC >::me_combine ( ModEvent me1,
ModEvent me2
) [inline, static]
Combine modifications events me1 and me2.
Definition at line 4282 of file core.hpp.
template<class VIC >
ModEvent Gecode::VarImp< VIC >::modevent ( const Delta & d ) [inline, static]
template<class VIC >
unsigned int Gecode::VarImp< VIC >::bits ( void ) const [inline]
Provide access to free bits.
Definition at line 4196 of file core.hpp.
template<class VIC >
unsigned int & Gecode::VarImp< VIC >::bits ( void ) [inline]
Provide access to free bits.
Definition at line 4202 of file core.hpp.
template<class VIC >
void Gecode::VarImp< VIC >::schedule ( Space & home,
PropCond pc1,
PropCond pc2,
ModEvent me
) [inline, protected]
Schedule subscribed propagators.
Definition at line 4296 of file core.hpp.
template<class VIC >
void * Gecode::VarImp< VIC >::operator new ( size_t s,
Space & home
) [inline, static]
Allocate memory from space.
Definition at line 3022 of file core.hpp.
template<class VIC >
void Gecode::VarImp< VIC >::operator delete ( void * ,
Space &
) [inline, static]
Return memory to space.
Definition at line 3019 of file core.hpp.
template<class VIC >
void Gecode::VarImp< VIC >::operator delete ( void * ) [inline, static]
Needed for exceptions.
Definition at line 3016 of file core.hpp.
Member Data Documentation
Subscribed actors.
The base pointer of the array of subscribed actors.
This pointer must be first to avoid padding on 64 bit machines.
Definition at line 233 of file core.hpp.
Forwarding pointer.
During cloning, this is used as the forwarding pointer for the variable. The original value is saved in the copy and restored after cloning.
Definition at line 242 of file core.hpp.
Indices of subscribed actors.
The entries from base[0] to base[idx[pc_max]] are propagators, where the entries between base[idx[pc-1]] and base[idx[pc]] are the propagators that have subscribed with propagation condition pc.
The entries between base[idx[pc_max]] and base[idx[pc_max+1]] are the advisors subscribed to the variable implementation.
Definition at line 273 of file core.hpp.
During cloning, points to the next copied variable.
Definition at line 275 of file core.hpp.
The documentation for this class was generated from the following file: | {"url":"https://www.gecode.org/doc/6.1.1/reference/classGecode_1_1VarImp.html","timestamp":"2024-11-03T05:54:47Z","content_type":"text/html","content_length":"61903","record_id":"<urn:uuid:3f63fcb5-f9d9-46e7-98b2-8e76cd49d2cb>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00582.warc.gz"} |
Compute the area of...
Somewhere in the world there must be an exam or test that asks the question to compute the area of a given shape using PL/SQL. I know this because occasionally I look at the
Google Analytics
for my blog and see some pretty crazy google searches that arrive at my page because of my chosen blog name.
In just a few hours I take off for Hong Kong to take the kids to Disneyland and visit the city for a few days. I'm happy, so (while I don't condone the use of the
to solve all your problems) I thought I'd make a few other people happy, and make the visit to my site worthwhile :-)
I've used formulas according to
Maths is Fun
, and also demonstrated a few other SQL rounding functions you can find well
documented here
. That's right kids, documentation is your
lc_pi constant number := 3.141592;
-- triangle
ln_t_base number default 2;
ln_t_height number default 4;
-- square
ln_s_length number default 5;
-- circle
ln_c_radius number default 200;
-- ellipse
ln_e_width number default 3;
ln_e_height number default 2;
-- trapezoid / trapezium
ln_z_a number default 2;
ln_z_b number default 5;
ln_z_height number default 3;
-- sector
ln_r_radius number default 4;
ln_r_degrees number default 45;
ln_area number;
-- triangle
ln_area := 0.5 * ln_t_base * ln_t_height;
dbms_output.put_line('Triangle: '||ln_area);
-- square
ln_area := POWER(ln_s_length, 2);
dbms_output.put_line('Square: '||ln_area);
-- circle
ln_area := lc_pi * POWER(ln_c_radius, 2);
dbms_output.put_line('Circle: '||ROUND(ln_area, -2));
-- ellipse
ln_area := lc_pi * ln_e_width * ln_e_height;
dbms_output.put_line('Ellipse: '||FLOOR(ln_area));
-- trapezium
ln_area := 0.5 * (ln_z_a + ln_z_b) * ln_z_height;
dbms_output.put_line('Trapezoid: '||CEIL(ln_area));
-- sector
ln_area := 1/2 * ln_r_radius**2 * ln_r_degrees / 180 / lc_pi;
dbms_output.put_line('Sector: '||ROUND(ln_area, 5));
end simple_calcs;
Triangle: 4
Square: 25
Circle: 125700
Ellipse: 18
Trapezoid: 11
Sector: 6.28319
PL/SQL procedure successfully completed.
See you on the other side of Hong Kong!
3 comments:
1. I must say, the need for these sorts of calculations have never come up in any project I've been involved in.
I'd increase the accuracy of that PI constant. I'm pretty sure that errors would bubble up quite quickly with only six digits of precision. Just saying :)
2. Rather than the mysterious 57.2957795 I'd prefer to have seen something like (180 / lc_pi)
3. Tony - done
Jeff - I was about to go on holidays and used pi to the precision my memory recollects ;-)
I was thinking about the usability of these sorts of calculations and thought - well, I never thought I'd need to apply calculations that calculated various forms of decimal degrees & latitude/
longitude, but I have. I guess you just have to be on the right project.
Luckily, I was about to find the valid calculations on a website that posted it in another language, and I was able to translate. | {"url":"http://www.grassroots-oracle.com/2010/10/compute-area-of.html?m=1","timestamp":"2024-11-14T04:04:58Z","content_type":"text/html","content_length":"47500","record_id":"<urn:uuid:eefe6582-7ab5-4830-9601-fe1fed770fa6>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00732.warc.gz"} |
Understanding Basic Probability: A Student’s Guide - Hamnus
What is Probability?
Probability measures how likely it is for an event to occur. For example, determining the chance of a coin landing on heads or tails when you toss it.
Key Terms in Probability
• Sample Space: The set of all possible outcomes from an experiment. For a coin toss, the sample space is {Heads, Tails}.
• Event: A specific outcome that we’re interested in. For instance, getting “Heads” when tossing a coin.
How to Calculate Probability?
The probability of an event is calculated by the formula:
Probability (P) = Number of favorable outcomes / Total number of outcomes
This calculates how likely it is for an event to happen based on the total number of possible outcomes.
Fundamental Probability Concepts
1. Mutually Exclusive Events – Two events are mutually exclusive if they cannot happen at the same time. For instance, when rolling a single die, getting a 2 and a 5 simultaneously is impossible.
Example: If you roll a six-sided die, the probability of rolling either a 2 or a 5 is:
P(2 or 5) = P(2) + P(5) = 1/6 + 1/6 = 1/3
2. Independent Events are independent if the occurrence of one does not affect the occurrence of the other. For example, flipping a coin and rolling a die simultaneously are independent events.
Example: The probability of flipping a coin and getting heads, then rolling a die and getting a 6, is:
P(Heads and 6) = P(Heads) x P(6) = 1/2 x 1/6 = 1/12
3. Conditional Probability measures the probability of an event occurring, given that another event has already occurred.
Example: If a card drawn is a king, what is the probability it is red?
Given there are 2 red kings in a standard deck of 52 cards:
P(Red given King) = Number of Red Kings / Number of Kings = 2/4 = 1/2
4. Bayes’ Theorem is used to update probabilities with new evidence. It is particularly useful in calculating conditional probabilities.
Example: Assuming a disease affects 1 in 1,000 people, and a test for the disease is 99% accurate, calculate the probability that a person has the disease if they test positive.
• P(Disease) = 0.001 (prevalence of the disease)
• P(No Disease) = 0.999 (likelihood of not having the disease)
• P(Positive | Disease) = 0.99 (probability of testing positive if having the disease)
• P(Positive | No Disease) = 0.01 (probability of testing positive without having the disease)
Using Bayes’ Theorem:
P(Disease | Positive) = (P(Positive | Disease) x P(Disease)) / P(Positive)
First, calculate P(Positive):
P(Positive) = P(Positive | Disease) x P(Disease) + P(Positive | No Disease) x P(No Disease) = 0.99 x 0.001 + 0.01 x 0.999 = 0.01098
Then, substitute back to find P(Disease | Positive):
P(Disease | Positive) = (0.99 x 0.001) / 0.01098 ≈ 0.09016
This guide provides a basic understanding of probability through examples, helping you see how probability is applied in different scenarios. These concepts are essential in fields like business,
science, and healthcare, enabling better decision-making based on likely outcomes. | {"url":"https://hamnus.com/2024/07/29/understanding-probability-definitions-and-examples/","timestamp":"2024-11-12T13:55:30Z","content_type":"text/html","content_length":"98424","record_id":"<urn:uuid:6ea31bc0-1713-4710-bee3-70a20e76e905>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00627.warc.gz"} |
How to Calculate x2-11x+28=0 - technotouchs.com
Today, we are diving into the world of mathematics to resolve the secrets and techniques of calculating x2-11x+28=0. Don’t let the ones numbers intimidate you; we’ll damage it down step by step so
you can overcome this equation with self assurance. Get your questioning caps on as we discover the bits and bobs of the quadratic method collectively!
Mastering this system opens up doorways to fixing numerous mathematical troubles involving quadratics. With practice and staying power, you will quickly navigate via quadratic equations quite simply
the usage of this essential device in algebraic trouble-fixing.
Identifying the Values of a, b, and c
When tackling a quadratic equation like x²-11x+28=zero, it’s vital to identify the values of a, b, and c. These coefficients play a key position in fixing the equation using the quadratic formula.
In this equation, a is the coefficient of x², that’s 1. The price of b is the coefficient of x, which in this situation is -11. C represents the constant time period at the end of our equation;
right here it’s 28.
By recognizing these values prematurely and plugging them into the quadratic formula successfully, you put your self up for fulfillment when fixing for x. Understanding how each variable contributes
to the general solution gives you insight into a way to approach and simplify your calculations successfully.
So permit’s dive deeper into these coefficients and get to the bottom of their significance on our journey closer to finding answers for our quadratic equations!
Plugging in Values to the Formula
Once you have diagnosed the values of a, b, and c in the quadratic equation x2-11x+28=0 it’s time to plug them into the quadratic formulation. This step is essential in solving for the unknown
variable x. Remember that a is the coefficient of x^2, b is the coefficient of x, and c is the steady time period.
Take your time when substituting these values into the formula to avoid any mistakes. Make sure each value is efficiently positioned in its corresponding spot in the equation. Precision at this
degree will result in correct effects in a while.
By plugging in these values correctly, you are placing yourself up for success in locating answers for x. The method may seem complicated at the beginning look but taking it grade by grade will make
it more viable.
Now let’s delve deeper into simplifying and solving for x after plugging in those important values!
Simplifying and Solving for x
Once you have got identified the values of a, b, and c within the quadratic equation x2-11x+28=0it’s time to plug them into the quadratic formulation. This step is essential in simplifying and fixing
for x. Remember that a represents the coefficient of x^2, b represents the coefficient of x, and c is the regular term.
After plugging in these values into the quadratic system (-b ± √(b^2 – 4ac)) / 2a, simplify every part one by one earlier than moving directly to remedy for x. Pay close interest to signs in the
course of this method as they could without difficulty affect your final solution.
Once you’ve got simplified each elements of the method personally, proceed with fixing for x via appearing addition and subtraction operations as a consequence. Take your time with this step to keep
away from any calculation mistakes that could stand up from dashing thru.
Double-take a look at your work after obtaining your solution for x by using substituting it lower back into the original equation. Verifying your answer guarantees accuracy and enables save you
mistakes that could have occurred throughout calculations.
Checking Your Answer
Now that you’ve determined the capability answers to the quadratic equation x2-11x+28=0, it’s vital to test your answer for accuracy. Double-checking your paintings can help make certain which you
have indeed solved the equation efficaciously. One manner to verify your answer is via plugging the values of x again into the authentic equation and confirming that both sides are equal.
If after substituting the values of x, both facets of the equation aren’t equal, revisit your calculations and search for any mistakes in simplification or arithmetic. Mistakes show up, so don’t be
discouraged in case you need to reevaluate your technique.
Remember that checking your answer is a vital step in fixing quadratic equations accurately. It reinforces mathematical standards and enables construct self assurance in problem-fixing abilities. By
verifying your solution, you can feel more assured in the accuracy of your paintings and higher apprehend how extraordinary components come together inside a quadratic equation.
Tips for Solving Quadratic Equations
When tackling quadratic equations like x2-11x+28=0Â there are some accessible hints to keep in thoughts. Always try to simplify the equation as much as viable before applying the quadratic method.
This can make calculations less difficult and reduce the chances of errors.
Another useful tip is to double-test your values for a, b, and c before plugging them into the formulation. Small errors in inputting those values can lead to wrong answers. It’s also useful to
exercise factoring trinomials often because it will let you apprehend patterns and streamline your hassle-solving system.
Additionally, live prepared by using writing out every step really and neatly. This will now not most effective assist you keep track of your progress but additionally make it easier to spot any
mistakes along the way. Don’t turn away from in search of extra assets or steering in case you locate yourself caught on a specific hassle – on occasion a sparkling angle can make all of the
Mastering the quadratic method can drastically enhance your trouble-solving talents while coping with equations like x2-11x+28=0. By expertise the method and following the step-with the aid of-step
procedure of identifying values, plugging them in, simplifying, and checking your answer, you can effectively solve quadratic equations. Remember to exercise frequently and utilize the suggestions
furnished to enhance your talent in fixing those styles of problems. With determination and endurance, tackling quadratic equations will become 2d nature to you. Keep training and difficult your self
– you have got this!
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Characterization of the Complexity of Computing the Minimum Mean Square Error of Causal Prediction
This paper investigates the complexity of computing the minimum mean square prediction error for wide-sense stationary stochastic processes. It is shown that if the spectral density of the stationary
process is a strictly positive, computable continuous function then the minimum mean square error (MMSE) is always a computable number. Nevertheless, we also show that the computation of the MMSE is
a #P1 complete problem on the set of strictly positive, polynomial-time computable, continuous spectral densities. This means that if, as widely assumed, FP1 ≠ #P1, then there exist strictly
positive, polynomial-time computable continuous spectral densities for which the computation of the MMSE is not polynomial-time computable. These results show in particular that under the widely
accepted assumptions of complexity theory, the computation of the MMSE is generally much harder than an NP1 complete problem.
All Science Journal Classification (ASJC) codes
• Information Systems
• Computer Science Applications
• Library and Information Sciences
• complexity blowup
• complexity theory
• computability
• minimum mean square error
• Turing machine
• Wiener prediction filter
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imit of a
How to calculate a limit of a function in python
To find the limit of a mathematical function in python, we can use limit() function of the sympy library.
• The first argument (y) is the functionf (x) whose limit is to be calculated
• The second parameter (x) is the reference variable ( argument )
• The third parameter (x0) is the accumulation point
□ oo = + infinity
□ -oo = - infinity
□ 0 = zero
□ n = number
• The fourth parameter (s) is used to calculate the lateral limits of a point
□ '+' =right limit
□ '-' = left limit
The function limit() calculates the limit of the function f (x) as x approaches x[0].
$$ \lim_{x \rightarrow x_0 } f(x) = l $$
What's a limit of a function? The limit of a function f(x) at X[0] point in its domain, if it exists, is the value that the function f(x) approaches as its argument approaches X[0]. The notation of a
limit is as follows: $$ \lim_{x \rightarrow x_0 } f(x) = l $$ We can read "the limit of f(x) as x approaches x[0] is L".
The limit() function must be imported from the sympy library using the command from sympy import limit.
Example 1
This script calculates the limit of the function 1/x as x approaches zero.
from sympy import limit, Symbol
x = Symbol('x')
limit(y, x, 0)
The function returns to output
The limit of the function 1 / x as x approaches zero is more infinite (oo).
$$ \lim_{x \rightarrow 0 } \frac{1}{x} = \infty $$
Example 2
This script calculates the limit of the 1/x function as x approaches + infinite.
from sympy import limit, oo, Symbol
x = Symbol('x')
limit(y, x, oo)
The oo symbol (+ infinity) must be imported from sympy.
The output of the function is
The limit of the function 1 / x for x tending to +∞ is zero.
$$ \lim_{x \rightarrow \infty } \frac{1}{x} = 0 $$
Example 3
This script calculates the limit of the function x^2 as x approaches - infinite.
from sympy import limit, oo, Symbol
x = Symbol('x')
limit(y, x, -oo)
The output of the function is
The limit of the function is + infinite.
$$ \lim_{x \rightarrow - \infty } x^2 = \infty $$
Example 4
This script calculates the limit of the x^2 as x approaches 4.
from sympy import limit, oo, Symbol
x = Symbol('x')
limit(y, x, 4)
The output of the function is
The limit of the function for x tending to 4 is 16.
$$ \lim_{x \rightarrow 4 } x^2 = 16 $$
Example 5 (lateral limit)
This script calculates the lateral boundary of the 1 / x function with x tending towards zero from the left.
from sympy import limit, Symbol
x = Symbol('x')
limit(y, x, 0, '-')
The function returns in output
The limit of the function 1 / x as x tending to zero from the left is minus infinite (-oo).
$$ \lim_{x \rightarrow 0^- } \frac{1}{x} = - \infty $$
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Handy elementary algebraic properties of the geometry of entanglement
The space of separable states of a quantum system is a hyperbolic surface in a high dimensional linear space, which we call the separation surface, within the exponentially high dimensional linear
space containing the quantum states of an n component multipartite quantum system. A vector in the linear space is representable as an n-dimensional hypermatrix with respect to bases of the component
linear spaces. A vector will be on the separation surface iff every determinant of every 2-dimensional, 2-by-2 submatrix of the hypermatrix vanishes. This highly rigid constraint can be tested merely
in time asymptotically proportional to d, where d is the dimension of the state space of the system due to the extreme interdependence of the 2-by-2 submatrices. The constraint on 2-by-2 determinants
entails an elementary closed formformula for a parametric characterization of the entire separation surface with d-1 parameters in the char- acterization. The state of a factor of a partially
separable state can be calculated in time asymptotically proportional to the dimension of the state space of the component. If all components of the system have approximately the same dimension, the
time complexity of calculating a component state as a function of the parameters is asymptotically pro- portional to the time required to sort the basis. Metric-based entanglement measures of pure
states are characterized in terms of the separation hypersurface.
Original language English (US)
Title of host publication Quantum Information and Computation XI
State Published - 2013
Event Quantum Information and Computation XI - Baltimore, MD, United States
Duration: May 2 2013 → May 3 2013
Publication series
Name Proceedings of SPIE - The International Society for Optical Engineering
Volume 8749
ISSN (Print) 0277-786X
ISSN (Electronic) 1996-756X
Other Quantum Information and Computation XI
Country/Territory United States
City Baltimore, MD
Period 5/2/13 → 5/3/13
• Computational complexity
• Entanglement measure
• Hypermatrix
• Separable state
ASJC Scopus subject areas
• Electronic, Optical and Magnetic Materials
• Condensed Matter Physics
• Computer Science Applications
• Applied Mathematics
• Electrical and Electronic Engineering
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output of a random forest classification
Hi i hope you doing well ,
i have a binary classification of two classe 0 and 2
and when i test my model on another dataset i get 3 columns in the output : the probability of being 0 (proba_0) , the probability of being 2 (proba_2) and the class (0 or 2) the logic is that if the
proba_0 > 0,5 the algorithm must predict 0 else 2
but i don't get the same logic for the last row as the picture shows
kind regards
• Hello,
when using visual ML in DSS for binary classification problem, an optimal threshold to decide which class is selected is computed, and not necessarily equal to 0.5:
* The way it is computed is decided in the Design > Metrics tab, and by default, it is computed to optimize the F1 score
* Then, in the report of the model, you can see the impact of the value of the threshold on the metrics in the "Confusion matrix" tab
* When running a scoring/evaluation recipe, you can either keep the computed threshold of the model, or override it for this run in the "Threshold" section of the recipe
If you wish to have a 0.5 threshold, you can change the settings of your recipe accordingly.
Hope this helps,
Best regards,
• HELLO ,
so if i choose a threshold of 0,025 what is the probability that decide if the prediction is 0 or 1?
best regards
• Hello,
Then the limit probability is the threshold, i.e. 0.025 (or 2.5% in percentage). Above, it will be predicted 1, below, it will be predicted 0.
Hope this helps,
Best regards | {"url":"https://community.dataiku.com/discussion/comment/22716#Comment_22716","timestamp":"2024-11-14T16:43:11Z","content_type":"text/html","content_length":"411104","record_id":"<urn:uuid:9be0e928-841c-4c26-8de1-46ac8ce023af>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00398.warc.gz"} |
Time-Consistent Risk Control and Investment Strategies With Transaction Costs
[1] Bai L, Guo J. Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint. Insurance: Mathematics and Economics, 2008, 42(3): 968-975
doi: 10.1016/j.insmatheco.2007.11.002
[2] Zeng Y, Li Z. Optimal time-consistent investment and reinsurance policies for mean-variance insurers. Insurance: Mathematics and Economics, 2011, 49(1): 145-154
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Ji K, Peng X. Optimal investment and reinsurance strategies for loss-averse insurer considering inflation risk and minimum performance guarantee. Acta Mathematica Sci, 2022, 42A(4): 1265-1280
[4] 黄玲, 刘海燕, 陈密. 基于 Ornstein-Uhlenbeck 过程下具有两个再保险公司的比例再保险与投资. 数学物理学报, 2023, 43A(3): 957-969
Huang L, Liu H, Chen M. Proportion reinsurance and investment based on the Ornstein-Uhlenbeck process in the presence of two reinsurers. Acta Mathematica Sci, 2023, 43A(3): 957-969
[5] Zou B, Cadenillas A. Optimal investment and risk control policies for an insurer: Expected utility maximization. Insurance: Mathematics and Economics, 2014, 58: 57-67
[6] Peng X, Wang W. Optimal investment and risk control for an insurer under inside information. Insurance: Mathematics and Economics, 2016, 69: 104-116
[7] Bo L, Wang S. Optimal investment and risk control for an insurer with stochastic factor. Operations Research Letters, 2017, 45(3): 259-265
[8] Peng X, Chen F, Wang W. Optimal investment and risk control for an insurer with partial information in an anticipating environment. Scandinavian Actuarial Journal, 2018, 2018(10): 933-952
[9] Shen Y, Zou B. Mean-variance investment and risk control strategies-A time-consistent approach via a forward auxiliary process. Insurance: Mathematics and Economics, 2021, 97: 68-80
[10] Chen F, Li B, Peng X. Portfolio Selection and Risk Control for an Insurer With Uncertain Time Horizon and Partial Information in an Anticipating Environment. Methodology and Computing in
Applied Probability, 2022, 24(2): 635-659
[11] Bai L, Guo J. Optimal dynamic excess-of-loss reinsurance and multidimensional portfolio selection. Science China Mathematics, 2010, 53: 1787-1804
[12] 王雨薇, 荣喜民, 赵慧. 基于模型不确定性的保险人最优投资再保险问题研究. 工程数学学报, 2022, 39(1): 1-19
Wang Y, Rong X, Zhao H. Optimal reinsurance and investment strategies for insurers with ambiguity aversion: Minimizing the probability of ruin. Chinese Journal of Engineering Mathematics, 2022,
39(1): 1-19
[13] Bayraktar E, Zhang Y. Minimizing the probability of lifetime ruin under ambiguity aversion. SIAM Journal on Control and Optimization, 2015, 53(1): 58-90
[14] Bi J, Meng Q, Zhang Y. Dynamic mean-variance and optimal reinsurance problems under the no-bankruptcy constraint for an insurer. Annals of Operations Research, 2014, 212: 43-59
[15] Sun Z, Guo J. Optimal mean-variance investment and reinsurance problem for an insurer with stochastic volatility. Mathematical Methods of Operations Research, 2018, 88: 59-79
[16] Wang T, Wei J. Mean-variance portfolio selection under a non-Markovian regime-switching model. Journal of Computational and Applied Mathematics, 2019, 350: 442-455
[17] Björk T, Murgoci A. A general theory of Markovian time inconsistent stochastic control problems. Ssrn Electronic Journal, 2010, 18(3): 545-592
[18] Lin X, Qian Y. Time-consistent mean-variance reinsurance-investment strategy for insurers under CEV model. Scandinavian Actuarial Journal, 2016, 2016(7): 646-671
[19] Björk T, Murgoci A, Zhou X Y. Mean-variance portfolio optimization with state-dependent risk aversion. Mathematical Finance: An International Journal of Mathematics, Statistics and Financial
Economics, 2014, 24(1): 1-24
[20] Bi J, Cai J. Optimal investment-reinsurance strategies with state dependent risk aversion and VaR constraints in correlated markets. Insurance: Mathematics and Economics, 2019, 85: 1-14
[21] Yuan Y, Han X, Liang Z, et al. Optimal reinsurance-investment strategy with thinning dependence and delay factors under mean-variance framework. European Journal of Operational Research, 2023,
311(2): 581-595
[22] Yoshimoto A. The mean-variance approach to portfolio optimization subject to transaction costs. Journal of the Operations Research Society of Japan, 1996, 39(1): 99-117
[23] He L, Liang Z. Optimal financing and dividend control of the insurance company with fixed and proportional transaction costs. Insurance: Mathematics and Economics, 2009, 44(1): 88-94
[24] Hobson D, Tse A S L, Zhu Y. Optimal consumption and investment under transaction costs. Mathematical Finance, 2019, 29(2): 483-506
doi: 10.1111/mafi.12187
[25] Mei X, Nogales F J. Portfolio selection with proportional transaction costs and predictability. Journal of Banking & Finance, 2018, 94: 131-151
[26] Melnyk Y, Muhle-Karbe J, Seifried F T. Lifetime investment and consumption with recursive preferences and small transaction costs. Mathematical Finance, 2020, 30(3): 1135-1167
[27] Gârleanu N, Pedersen L H. Dynamic trading with predictable returns and transaction costs. The Journal of Finance, 2013, 68(6): 2309-2340
[28] Gârleanu N, Pedersen L H. Dynamic portfolio choice with frictions. Journal of Economic Theory, 2016, 165: 487-516
[29] Ma G, Siu C C, Zhu S P. Dynamic portfolio choice with return predictability and transaction costs. European Journal of Operational Research, 2019, 278(3): 976-988
[30] Bensoussan A, Ma G, Siu C C, et al. Dynamic mean-variance problem with frictions. Finance and Stochastics, 2022, 26(2): 267-300 | {"url":"http://121.43.60.238/sxwlxbA/EN/abstract/abstract17449.shtml","timestamp":"2024-11-07T23:56:50Z","content_type":"text/html","content_length":"76151","record_id":"<urn:uuid:2a3ff23c-4323-45ca-8a96-530fcddbd20e>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00080.warc.gz"} |
Advanced Mathematics: 11th Chapter infinite Series (2) Function of power series expansion, Fourier series _ Higher Mathematics
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Exploring Advanced Plotting Techniques in SymPy
Written on
Chapter 1: Introduction to Advanced Plotting
When utilizing Python's symbolic mathematics library SymPy, generating basic plots is straightforward with functions like plot(). However, guidance on more sophisticated plotting methods is somewhat
limited. Fortunately, since SymPy leverages robust plotting libraries such as Matplotlib and Bokeh, there is a vast array of capabilities at your disposal. This article introduces you to these
advanced features and what they can achieve.
Daily Dose of Scientific Python
Explore common and uncommon challenges solved using Numpy, SymPy, SciPy, and Matplotlib.
Section 1.1: Setting Up for Advanced Plotting
To unlock advanced plotting functionalities, you need to install an additional package. Simply execute the command:
pip install sympy_plot_backends[all]
Once done, you are set to begin.
Subsection 1.1.1: A Basic Example
Let’s plot a straightforward function, ( f(x) = frac{1}{x} ), over the range from -1 to 1. This function presents a challenge due to its pole at ( x=0 ), which complicates the plotting process. The
naive plot would appear as follows:
In your Jupyter notebook, after importing SymPy, ensure to also import the SymPy plotting backends (spb):
What a difference! This is just the beginning of what you can achieve.
Section 1.2: Handling Discontinuities
Let’s examine the gamma function, which serves as an interpolation for factorials. As you may know, the gamma function has poles at negative integer values. A naive plot yields:
Typically, standard plotting connects points across poles, which is misleading. The spb add-on includes a discontinuity detector that rectifies this issue. By enabling the detect_poles option and
adjusting the hyperparameters eps and n, you can greatly enhance the visual results:
This feature also works with finite discontinuities, such as the step function. The unrefined plot appears as follows:
However, the pole detector cleans up the visualization:
Chapter 2: Fine-Tuning Your Plots
Passing Arguments to the Backend
In vanilla SymPy, the plot function has limited options. For instance, while you can set the line color, changing the line style (e.g., dashed) is not possible. With the spb add-on enabled, you can
extend these options! Here’s how to pass linestyle or linewidth to Matplotlib:
Section 2.1: Exploring Different Backends
Three distinct 2D backends are available, with Matplotlib being the default. You can specify the backend using the backend argument. The options include:
• MB for Matplotlib
• BB for Bokeh
• PB for Plotly
While many options are interchangeable, the syntax can differ between backends, especially when passing specific parameters.
For example, utilizing Matplotlib produces a static plot:
Conversely, using the Bokeh backend provides interactivity, such as zooming:
Specialized Plotting Options
Thus far, we have focused on standard 2D plots, but there is much more to explore. For example, creating polar plots is straightforward with plot_polar:
Complex functions can also be easily visualized using plot_complex. For instance, here is a depiction of the complex logarithm:
And to wrap up, SymPy's plotting capabilities extend into three dimensions, as demonstrated by this stunning nautilus:
Still not convinced? There's plenty more to discover. Stay tuned!
This video titled "Symbolic Math with SymPy: Advanced Functions and Plots" dives deeper into advanced functions and visualizations in SymPy.
In this video, "Plotting symbolic functions in SymPy," you will learn how to effectively plot various symbolic functions using the library. | {"url":"https://didismusings.com/advanced-plotting-techniques-sympy.html","timestamp":"2024-11-10T08:11:03Z","content_type":"text/html","content_length":"14879","record_id":"<urn:uuid:25580888-9e63-4a67-a711-69fd2699316f>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00247.warc.gz"} |
If the line x+y+k=0 is a tangent to the parabola x^(2)=4y then -Turito
Are you sure you want to logout?
If the line
A. 1
B. 2
C. 3
D. 4
The parabola equation is simplest if the vertex is at the origin and the axis of symmetry is along the x-axis and y-axis.
In parabola a line to be a tangent to parabola, condition is
Given Parabola :
Comparing with standard equation of parabola
4ay = 4y
a = 1
Given line :
Comparing with standard equation of line
m = 1 and c = k
In parabola a line to be a tangent to parabola, condition is
Substituting these values
Thus, If the line
Get an Expert Advice From Turito. | {"url":"https://www.turito.com/ask-a-doubt/maths-if-the-line-x-y-k-0-is-a-tangent-to-the-parabola-x-2-4y-then-k-4-3-2-1-q885fe5","timestamp":"2024-11-06T17:19:45Z","content_type":"application/xhtml+xml","content_length":"788622","record_id":"<urn:uuid:28f8a0b7-982a-40ff-804d-af35aeb8930c>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00242.warc.gz"} |
Learning Analytics Colloquium 2021
7th Annual Learning Analytics Fellows Colloquium
Below you will find recordings from the 7th Annual Learning Analytics Fellows Colloquium, which featured highly interactive virtual sessions in which Learning Analytics Fellows discussed the results
of their individual and group projects, shared new questions or insights that may have arisen during their research, and outlined any future plans they may have for making use of the results,
including the continued use of analytical data to improve the college experience for all students. The Learning Analytics Fellows program is supported by Indiana University’s Center for Learning
Analytics and Student Success (CLASS).
Satisfactory Grade Option: Choice Determinants and Consequences - Michael Kaganovich
Description of the video:
Michael. Thank you very much. Yeah. You guessed correctly. And somehow because I was not supposed to be the one speaking tonight. It was supposed to be early. But wherever. Last minute change in
cash. The primary cancer, let's replace the secondary ones. That's a piece of bad news. Rule, rule. Last minute situation which cost us. But it will be my pleasure to welcome project. Especially
because really the lion's share of the work. So it's some, my part is easier talking about this. So let me tell you what, what this is about. So this addresses what economists and modern economists
call a natural experiment. A unique event happened in the spring of 2020. Well, besides the fact that endemics, yet a consequence of that was them, a lot of classes had to switch a modality,
midstream. And students, obviously, outside of IU and I, you were hit with a major disruption and that caused the USU policy chef in grading. So all students, we're given an oxygen. And that was
announced in April. And to choose a grade S satisfactory to place their hand on the grade that they otherwise class. And so our purpose is to study, start studying the impact of this unique natural
experiment is when we call it that way. So, so one result, when one outcome of this is that everyone who acts. And an important elegant that was, that that policy was that students didn't have
described right away in April, but could made their choice up until May 8th and possibly even laid some. May 8th was the mandated universal deadline. The choice, amazed, by the way, habit to be the
last exam. The exams, that's an astronaut. But also the University encourage instructors to give students more leeway and give them an opportunity to learn the actual performance before, before
making that decision, the choice of anoxia, great option. And so in order to figure out, so we need, we needed an additional piece of information in order to address these questions. In addition to
the wonderful systematic data to now, we are getting from our concepts from Greece. So for this one source was candles. Because unlike millimeter assessment and research which tells us the outcomes,
great outcomes, cameras can potentially tell us. The student progress within Canvas could tell us, in particular, and so helped could tell us how students are doing by the time they made. And I'll
tell you in a second why why this turned out to be not, not so easy. And what we were able to accomplish with less a nation that we're hoping to get so bad. I would like to thank both momentum and
research and learning. And the Latvian latter organization is collecting the data from Kant's. So this was a major undertaking because it entailed merging two data sets. Furthermore, merging it in
such way that way. Compromise students privacy. So, so the, the two datasets have identified, the students. And the merger was accomplished in such way that we couldn't identify students who
question. But we could just be assured that the bar data and the data about students, communities, and Canvas will not work properly matched. And soaps will be the questions that we wanted to
address. The address understanding in the first place, understanding the factors and students choices. Well, one might just assume that it's easy that if a student is performing badly, then they're
more likely to choose that stands out, is not all that simple, but there are other factors, such as students for months, students demographic characteristics, students year. You could show you one of
those results. And in our longer term Go, which is true, It's too early this time around because we've labeled it took us a while to get all the approvals and go get F-measure yesterday. I don't know
if you can get a little closer to your bank maybe or aiding you're fading in and out a little bit. That's all right. That's how it yeah. That's better. Thank you. Yeah. Sounds sounds in the distance.
Yeah. So so a longer term go is to study the concept of choosing units. Because the choice of s effectively lowered the bar for many students to get a passing grade and to progress to higher level
classes. So for instance, someone who would be otherwise getting a C minus re-explain a short time ago, then they wouldn't be able to progress to other other classes. An account. Now the he in that
particular semester they could and so and they couldn't. And then the question is what happened as a result? Was that a 104 instruments to pick a pig's the semicolon on that subject matter. It wasn't
that important for the subsequent progress. Maybe those students who didn't make it. Well, we did that using, using a time. Maybe they could have done just fine after that. So some, the s option is,
is your experiment because it allowed basically everyone to progress. So that's our goal, but who hopefully will help to be able to pursue this next year. Once more information comes then to bar
about student performance style throughout. So I very much and spoke to the radius. So we can skip the instrument, which has given us a second to take a look at this slide. And now I'm going to train
him. It wasn't as much 20 nights when the policy was managed. Very very soon after the switch on the LED or your resulting from pandemic. So now we try to do, think of the situation. Imagine
designation, STM design when making this decision. Should I stick with my grade that have otherwise getting or should I choose x? And sum? When economists talk about trade-offs, they meet costs and
benefits. What is a cost to me are making this decision? And what is the benefit? And students performance in the course as well, is obviously relevant to the benefit, right? So, so if my performance
was bad, no instance, then it would be beneficial. Um, but my pride performance awesome, That's great deal. Because let's say I used to be an, a student and now I'm getting a B minus and that's
terror. Whereas if I was a student all along the spine while it actually connects cells. So these are some of the questions or would like to explore now. But there are also some, some costs may not
come to mind immediately. So not all students are always. And I think all of us taught classes and, uh, you know that a lot of students don't know how they're doing. With Canvas or not, how they're
doing it in a course in any given moment. So they need to undertake, to take an effort in order to explore that. They need to calculate that gray canvas may be confusing them. You need to interact
with the instructor. And so sometimes decisions or to take, as I'm not a tech has maybe predicated on that. One student's willingness to invest in finding out how they're doing. And then thinking, he
said, is that for me, it's not good for me hands. And the final fact is, it is what economists would call behavioral. That's, that's what we find. The most interesting, one of the most interesting
ones to think. So. So if a student decided to take an S option, declared that aerator, the instructor or not, has decided no matter what. I have the Snopes, I guess. Let's see at the last minute.
Well, now I'll actually permitted but I have this now. I know I hadn't stopped. And now that, that option potentially allows me to slack off. And so economists call this type of behavioral hazard,
moral hazard. And so this used to be an inside term among economists. But you may recognize it because it was made, made famous during financial crisis of 2008. And let me remind you of the context.
Snow. So there was a big prospect of failing companies being bailed out. And sounds for the problem and the Communists and non-economic like in the media spoke about the problem. Bailouts. Bailouts
is moral, has that if you know you're going to be bailed out, why would we maintain this? So this is this is a similar situation. So if if i now that I have an option, maybe a SledDog and 10 was
gives us a tool to to figure that out. But again, distance, that's something that we're soon. Very soon, but we don't have results yet. But there is this grade on canvas can tell us whether students
continued to login to the class into Canvas itself, like or how much time they spent then? Yes, it is one lane. You have about five minutes left here of it. Great. Okay. So in that case, let's move
on. Cameras date I already alluded to are that as snow. It details and the the instructor about all the assignments then and tells us children the same thing and grades received ants. And Portugal
were found in numerous problems. With data through Canvas, calculates the braid itself, by itself, the score. And so our purpose, one main purpose in using Canvas is trying to infer students actual
performed before deciding, deciding which rate data. And so, so this, this is one selection in, in thread column you see the grade that stood natural, ended up choosing move, obviously getting, so
obviously these are students who chose not to DNS and then get these blacks close out. The scores that Canvas. We're telling students that we're getting corks in the app. And then you'll see that
there is non-marital. So this is just a snapshot of this axle outward, but it goes all the way to two low grades. And we see that there are surprising non-white students with some, some students but
a, a plus with the 75 and other students and be managed with this query. So there are some consistencies and inconsistencies in the way Canvas and science course. They, they either wait. All the
assignments equally well as instructors have an option to override this. But not all instructions use instructions. So as a result, we, we could only get luminary zones. And so we decided for now not
to publicize results based on that Canvas grade. And so we don't weigh album sulfate now, presenting sounds. Without tendency. So strictly using bar state, right? Yeah. And so to form a logical
progression, where likelihood of choosing S as opposed to actual rate is independent parent. And so, so here are some, some results. So the higher the price cumulative GPA, the lower the likelihood
that the student chose x as a grain. In the first approximation, this may seem obvious. Obviously data, This was high GPAs students. These are better students. So they're probably more likely to have
performed better in the course. But remember that these choices are rather, and so this result tells us that the choices are not, not all that well, not all that for relic, that with higher GPA
students still, regardless how the new grant related yet they face now Hi Barnard, match. And yet the likelihood of choosing ask is Islam. But so being male increased one month probability of
choosing S, controlling for prior UP another, another characteristics. And so in the second specification, generates interacted with pride and willing to pay. And so this result sounds that the
higher GPA, the greater the gap between men and women. Okay. So at high grade levels, men more likely than women to choose as risks. With me. Now, some data for raceway ASM has a benchmark and so
sounds that white students controlling these are the prior GPA. It's parables are less likely to choose S, for instance, and not much. So receiving Pell Grant being first generation doesn't make much
difference in terms of choosing S control for other variables. But residency. So Indiana residents are less than one, less than equals x. Now this does somewhat interesting. So the higher the year
and I, you, the more likely students will choose yes. To both specifications come these results and signal. So again, controlling for performance. So the benchmarks here gets a first-year student.
And so the higher the year, the more, the higher the incidence of choosing as neutrons, which has entries. And so finally, among these questions, so we broke down, we broke down students into five
major academic categories. Education, of course, it's the slowest, but it couldn't be combined with anything else. Other professional schools, scam and social sciences and humanities and both
business higher education is a reference point. And so students in stem controlling flow characteristics were less likely to choose S than business tunes. And the same is true for science and
humanities. So business Jones, as matter of fact, most by teachers S. So maybe they wanted to progress so bad from that A1 100, two to the rest of their curricula. So it's going to other professional
schools. Or the CountSketch were the least likely next step. Now, as I mentioned, and so now hopefully appropriate right now. So our plan going forward is first of all, to try to actually get hold
them so better understand the Canvas grade data and then analyze the impact of that on the Canvas grade on students and essential to choose, ask, then to test for moral hazard. So was chosen gets
hosing. And identifiable, has questions slacking off after after March 20 nights. And then a beggar and pursued after that is looking at consequences of choosing us. How did how did that affect the
progress with students afternoon? Michael. Thank you. Yeah. I'm sorry to have to go. So I just want to open it up for a few minutes before we go over as we take one or two comments or questions, if
anybody has anything they'd like to say about this.
The Factors that Influence Entry, Exit and Progress along Students' Pathways to Success: An Examination of A100 - Leslie Hodder, Bree Josefy, and Jie Li
Description of the video:
Next is going to be R. Kelly folks. So I don't know how you all plan on doing this, but I'd like to give, you know, I was giving chase the 10 minute, five-minute, two-minute kind of warning so you
could keep on track. Who who should I send it to in your group sensors, three of you presenting had not defending that, had Bree. Okay. I'd just be like Okay, presented as bytes. Okay. Thanks,
Leslie. Maybe introduce yourselves to again since we do have people from outside the Bloomington campus. So thank you again. All right. Thank here though, I'm lovely, hotter and I'm a Professor of
Accounting at the Kelley School of Business. And this study is in collaboration with a brief Joseph I and J Lee and J Lee has been a class bellow for a number of years. So the food, the collaborative
grant. And before we continue, I want to thank George and Linda and also the Center for Learning Analytics for this opportunity. Because I know that I personally learned so much from data. I didn't
even realize they were available. And be persistent questions within our programs and departments that I didn't realize can be informed by the data. And, and I just learned an incredible amount and
I'm excited about continuing increase in that direction. J has been instrumental in helping us learn to access the data and manage big data and complicated and breathe and excellent researcher. And
so this collaboration has been fun and rewarding for all of us. On the next slide, we have our agenda. And there we're going to talk about our overall purpose and motivation for this study. We've
done many additional exploration. As I mentioned, we're interested in continuing. Then. We'll talk about the research question that we felt that we've asked and answered with the data. Using some
logistic analysis and regression type analysis. We're showing association not causality. However. And we're trying to bring to bear statistical significant and q all of our now to help us to draw
inferences. Also talk about the data. First, the j will go up at that. And then we'll show you our results. And in proposed intervention that we've thought about, although this is very preliminary.
All right, the next slide. So there are overall motivation. So I've then colleague or 19 years and I've noticed from structural shift and the time. And I open them differences between Kelly and other
institutions I've taught, which include the University of Texas, Austin, another big state school. And my interests, it is primarily around accounting. This is where we began, although with J on
board. And we can extend this to other stem type field. And accounting is a great profession. It's a great profession because It's fairly balanced in terms of its demographics. The American Institute
of CPAs indicates that about 49%. So what you would expect, half of accounting graduates nationwide identify as women and about 44 percent identify as non-white and so pro-business on it. Pretty
diverse. Upward mobility is provided by the profession because when you're a Certified Public Accountant, you can pretty easily go become an entrepreneur and start your own accounting practice and
provide flexibility during life. I know that when I was raising my young children, being an accountant provided that flexibility if I'd still a professional and still proceeding with my career, but
have some flexibility for for families. So we we feed accounting of the desirable profession that pays well. And, and it's, and it's pretty diverse. But yet what we've seen at Kelly is that although
the enrollments are increasing, so over the last 18 years or so, the accounting majors as a percentage of business majors has been declining dramatically. In fact, I would say we take confound at
approximately or 100 accounting majors, even though the enrollment ID which is increasing. So we're disappointed for the county area or disappointed to see that declining market share. And we have an
ad hoc basis attributed this to certain structural shifts, including an increase in our direct admit students. I'm an increase in the proportion, perhaps out-of-state students. And we see on a macro
basis, but I'm a pretty profound Kelly. Majors are finance majors now. I'm, where's the market share of accounting? And the other major has been declining. But beyond that, we know that even with an
accounting or accounting majors that Kelly are not representative of the population of accounting majors as a whole or the larger campus. So at IU Bloomington, 49 percent of students identify as
women, 29% are BIPOC. Kelly, 34 percent of students who identify as women and only 12 percent, or by art and accounting major group, an immediate SLP can also be the only 23 percent of accounting
majors identify as women to 0.50% as iPod. If you compare that to the nationwide, or they're put out by the American Institute of Certified Public Accountants. You can see that that we seem to have
something unique to Kelly as far as our under-represented populations. Now, you'll notice in our title fight that we decided to start at the very beginning with a class called a One 100, and a 108
required of all business majors. And Martha spoke earlier about the prospect of early failure and how early failure is known to result in decreased, persist them in colleges overall. But a 180 kind
of a unique course, because if they have Term course and it can be repeated a large number of time, and then the grade can ultimately be removed from the transcript through what's called an extract
that and though it shouldn't be a low stake early failure. And I think that p1 hat with a talk with study skills type course. And so it's topically related to your accounting. But I think when, when
the course was originally conceived, it was meant to be meant to be a weed out course. It's an indicator to students that they needed to up their game and learn, study skill. And it has a very high
DFW rate. So BRI is going to talk a little bit more about the role of a 100 in the curriculum and then j will talk about the data in them. Do I think lesly. So as you can see, the, when we wanted to
look at the pathway for our business students and what's happening there, their entry points and exit points. We first started with a Degree Map. And so this is showing the 8 first year generally
first year required courses for business majors. And as Lizzie was talking about this, the DFW rates, you can see that a 100 has one of the highest DFW rates of the first year courses that business
majors are expected to take. Now there are absolutely some of those math courses that have much higher DFW rates. But there's a variety of different courses the students can choose to take in by
Nmap, finite math and calculus. So, so we really wanted to focus on the A100 class. And particularly because it's, it's a one credit hour course with a 31 percent DFW rate. As Leslie said, it is over
eight weeks. It's also unique and that I'm not many institutions have an introductory accounting course similar to ours. Many of them will have a sophomore level, a 200 level, full three credit hour
sequence. So 2, 3 carrot our courses that students usually taken the sophomore year, um, but IU and Kelly is unique in offering this introductory course. So while a 100 has the opportunity to
introduce students to business and in the accounting profession and accounting majors. It also has the potential to deter students. And so that's one of, one of the main reasons we are focusing on a
100s. So our research questions in many ways similar to the pathway report that Chase was just looking at. So we're looking for what predicts, first of all, non progression outcomes in a 100. So what
are the factors that demographic factors that are indicative of non progression outcomes? And when I say non progression, we're actually slightly modifying the typical DFW rate because you might have
noticed on that earlier slide, students have to earn at least a C or better in order to progress in the Kelly curriculum. So this isn't an added challenge for our students. So we have added onto that
DFW rate and said, non progression is any grade that is lower than a C. So we're looking for non progression Atkins and a 100. And then what impact does initial A1 100 non progression have on student
degree and major outcomes? So does it lead to a lower likelihood of a degree at IU, a degree at Kelly. Or a degree in accounting. And then also looking at some of those changes in majors or decisions
and major. So if someone comes in initially interested in accounting, that's one of their declared majors and then they have a non progression outcome and a 100. How likely is it that they're going
to switch out of accounting? And this is important for us because as Leslie noted, accounting hasn't, we haven't increased our market share her even though we have more students coming in. And so we
really are trying to understand, are we losing students from the very eerie coming in at a shortfall or at some point along the way, are we losing those students because of maybe some of our courses?
And then finally, as, also, as he noted that this a 100 is topically related to accounting, but more focused on study skills. Is it possible then that students who have a non progression outcome and
then repeat the class to take it again. I'm, do they do better the second? So did they actually learn? So maybe it was one of those cases where failure or non progression when you try to stay away
from that word, failure. But non progression can teach us here. They can actually learn a lesson from that and then they can take a 100 again and potentially do better than maybe some students you've
only taken at once. I'm certainly better than they did the first time. And so potentially that is something good that could come out. I'm sorry. But look to explore what happens if they repeat the
class. Now I'm going to turn it over to Jay to talk about our sample selection. Okay? I just unmuted myself. Thank you. Break. So our dataset is from our dataset is from bar, provided we have four
sets of data. One way students attributes, one, student retention, which is a, which we mainly use the graduate degree data from student retention. And we have student major History, which is also
called program stack. And we have student course history. And the student population goes from fall of 2006 to spring of 2021. And the level of, the level of details in these tables are slightly
different. For student attributes, we have Y rule per students, but for retention for major history, we have one rule per student term. If the students existing ie your student records. And for
student course history, we have one row per student per term per course. So you can see the amount of data we're trying to dealing with. So when we, we explored how we want to work with data multiple
ways. And finally, we decided that we need a main table, a major table that includes all of the IOB students, not just students, because we have students trying to take a 100, even if they are not a
kilometer or even if they don't have business as their intended major. So we created a wide table, include all IU students who have ever enrolled in this period of time. And we try to take the level
of detail into one mole per student. And we transposed the student's history, their major history, and their core sticking history, not all the causes, only a rounded we transport transposed them
from vertical to horizontal because this is the only way we can do our regression analysis. So we go, got it, George. And so we try to keep all the student majors. And the data is complicated because
one term, you could have three majors. So we end up with a very white table. But with that one table, we can see the major declaration history up to 12 terms. And we also have all the students
attributes merged into one white table. In our research questions, we decided to focus on A1 100. And out of all the students, we have 43,408 students who took the course. And then we have these
numbers, students who are still, those who have graduated or dropped out from IU. And these are the number of students who have taken a 100 just one time. Okay? Yep. Yep. I talked about the
challenges already and this is the trend that we noticed. So when we talked about the high DFW trend, and in this data we also included c minus, so non progression. And we can see that this is the
enrollment of a rounded, this is the count of DFW C minus. And we can see the trend is always have always been high in these 39 represent fall, spring, and summer enrollments. Next, please. And these
are the enrollment trends for a 100. This is very similar and we can see that enrollment is still increasing. City and increasing. And non progression reads are indicated by this red color. And you
can see the percentage higher, lower, but pretty consistent. And repetition attempts to. So we have students we notified when in our data we only remained students who taken up to three times of 1100
attempts, but we have students who have taken this course up to seven times. And you can see the percentage 8% in fall, 22% in spring. This is really interesting. We don't know why in spring,
students have more have much higher rate of re re, attempt.` Yeah, and now you can see the agreed distribution of these up to seven attempt. You can see most students passed with the first attempt,
but we do have over 15% of students have to take it twice. And then still lower wise. And these students take several attempts. Did did graduate but he or she is definitely an outlier. So yeah. Do
you want to shortly here? Yeah. So this, this will read some of the descriptive statistics for our main samples of all students who have ever enrolled in a 100. And so just a few things to point out.
Here. You can see that 33 percent of students who have ever enrolled in A1 100 identify as female and then only about 8% identify as, as URM. If you look down at that, that SAT scores and GPA scores,
it's it's fairly high. We have a mean of 1300, quite a large range there. And then I had to high-school GPA of 3, 69. And and many of them enter having at least four, are having on average about 14
credit hours. So this maybe from AP classes on or it might be if they're entering in that from the spring that they've had credit credits from the fall. Just a quick look at some correlation tables.
So we wanted to make sure before we went into our regressions, We're on any variables that are highly correlated with each other. And so non progress is going to be our dependent variable in our
first year regressions. And then the, you, you didn't see the rest of their independent variables. And so one of the, one of the main associations you can see is that standard emit is highly
negatively correlated with SAT and GPA, which this is understandable, obviously understandable. So we've modified some of our regression analysis to focus on standard admits. In some robustness, we
do take out the inter-domain, have the SAT and high-school GPA instead of that scene are going. So our first research, research question. So what predicts first-time non progression outcome in a
100s? So we did run a logistic regression. So one is if a student had a C minus or below, so c minus d plus d, f, w, and then a 0 if they did progress with 81, 100. And we ran this regression on a
mole to mole to multiple different factors. We're mostly interested in female URM. And then we also want to understand how the standard admit plays into non progression. So you can see our first
regression is main effects. And a female, say unfortunately female URM antinode, admit we're all positively associated with non progression. So that's bad news. That means that on average, females
are 29 percent more likely to not progress in their first attempt at a 100 than other students and similar for URM. And then our standard admits actually really high. So 51 percent more or 52 percent
greater likelihood of not progressing. On our second well, we also had some main effects for resident and first-generation. So what could be entrepreneurs for? Some good news here is that our
first-generation students are actually less likely to not progress through A11 100. So then we wanted to look at just some interaction effects. And what we see here is that the interaction of
standard admit and female students are actually less likely to not progress through their first-time attempt at a 100 wasn't any significant effect on your ends. And then we looked at the combination
of female and your answer, that intersectionality, they were also more likely to not progress. Our second research question then looks at graduation outcomes. So after a 100, our students more likely
to get an IUD agree I Kelly degree in an accounting degree. So we looked at three different dependent variables. And the regression that we're using here is also looking at some interaction effects
of non progression times female. I'm so let's look at that 1 first. So across the 30 degrees, you see that if you have a female who did not progress in their initial attempt at a 100, the first
result of it, they're more likely to get an IUD agree, might seem a bit counter-intuitive. But I think if you combine that with some of our descriptive statistics, that, that on average many of these
students are highly capable and they may just have, you know, either it, they may decide that because they've had that first incident of failure that and the business degree isn't for them, but
they're still motivated to get an IUD agree. The second one, Kelly degree, you can see that it's negatively associated. So I'm females that don't progress and a 100 on the first attempt are less
likely to get it to tell a degree and less likely to get an accounting degree. When we look at though, the URM statistics, The Kelly degree, sorry, There's lots of red lines here. The URM who don't
progress are also less likely to get a degree and then it wasn't statistically significant for an accounting degree. And then finally, our standard admits. So standard admits who didn't progress in
their initial attempt were less likely to get an IUD agree. Kelly degree in an accounting degree. And I know I'm running out of time here, George. So this is one of our last regressions here. We also
want to look at the, if this had an effect on student major decisions. And so we looked at if they change their degree from entry point to exit point. And so you can see here, this is just looking at
accounting students. So our students overall so that they entered accounting. If they entered IU and did not declare honey as a major, that would be in the 0. And if at some point along the way they
decided to add an accounting degree. So we can say that again, accounting, and then vice versa, if they entered as an accounting student and then did not graduate with an accounting degree, then we
consider that Louis accounting. And so we ran a regression would deepen a variable being Louisa County or drop Accounting degree and then gain accounting degree. And, um, for females, don't progress.
We did find a lower likelihood of adding accounting. I'm in similar for standard admits, but we didn't find any significance on dropping an accounting degree. So this is interpreted with some, a
little bit of good news and that we're not deterring students, but we're certainly not attracting students are encouraging students that have not progressed. And 8100. All right, and then finally our
third research question is looking at distance. Who don't progress the first time. Do they obtain higher grades on subsequent are elements. So that this first exploratory analysis we're just showing
for students that had, that withdrew and those, those students are tricky because we don't know exactly what grade they would have had if they had completed. So you can see here that the majority of
students who do withdrawal on their first attempt, when they, and repeat they come back, they persist in it. About 31% of those make it be we do have 27 percent that successfully finished with an a
boat, but we do have those students that 12 percent, again withdraw, 4% have enough and 6% have a dy. So our last analysis we wanted to look at what effect repeating would have on your 8100 grade.
And so in this case we're doing an OLS regression where the grade is a numeric version of the letter grade. So 0 to four. And then repeat as our variable of interest. And it's one if a student is
taking a 100 for a second or third or fourth time. So you can see here when we ran our first model, when we did our main effects, there, there wasn't a statistical thing. Wasn't a statistically
significant correlation with the A1000 grade. I'm going to look at are females and repeat females here. So when we looked at our main effect, females were less likely to have a higher grade, I should
say Lesley was negatively associated with an A1 100 grade. But when you look at the repeat females, it's not significant. And then a similar story with are you ORMs. And then with our essays when
they repeat the actually, it's negatively associated with their grade. So this, this doesn't seem like great news for those students that are repeating. It's either not associated with a grade, a
higher grade, or it's negatively associated for those standard admits. So we, we definitely want to do some more research on this question and, and, and figuring out what's, what's happening. We have
some students who the w's do seem to be doing better, but the students who took the class got a grade and then took it again, it doesn't seem like repeating is having an effect on that. So
interventions for students success. As Leslie mentioned when this class was started, it was, we hate to use the word but a weed out course. And that's what it was considered and focus on those study
skills and so on. Just this past year, one of our faculty members who has been teaching this class for the longest time and doing an excellent job, has retired. And so we pivot to hire new faculty
member. And this person was new to IU. And so some of these innervate, these interventions have been implemented. And I want to, I want to say very clearly that the research we were doing and these
these changes and interventions were happening simultaneously. And part of it was because we had started looking at this data about a 100 in the spring, right around the time we hired a new person.
And so I was actually helping our new faculty member to understand what a 100 is. Understand what I us, and to reframe some of our learning objectives and the assessments that, that combine a 100. So
I'm, so I want to be clear, this is not as a result of all of our research that we've been doing, but it's, it's happening simultaneously and it's part of the awareness of understanding that there
such high DFA is 0. And I'm sorry George, I see that we need to move on. So I'll leave you with the final thing that this shift has result. We don't we don't have the full data, but for the first
eight week results, we we do know that there was a 10 percent DFW rate rather than our historical 30 percent. So that's that's great. News, and thank you. All of this were you used up your pen
minute? Q. And a. Oh, I'm so sorry. That's your option, I guess. So we're going to have to move on.
Data-Driven curriculum development for student success - Martha Oakley
Description of the video:
Okay, So, you know, all of you are obviously here to talk about data-driven curriculum development and other, other cluster and development for students success. I am the Associate Vice Provost for
Undergraduate Education, and I've been very interested in these projects essence, I've come on board about a year and a half ago now. And I am really excited to see all of you here, both from
on-campus. I'm really looking forward to hearing about your projects. I enjoyed it very much last year. And I also am delighted to see folks from around the country who can hopefully have some time
to share their thoughts and questions during the Q&A sessions after each talk. And I just wanted to talk a little bit about my involvement with learning analytics and, and, and sort of communicate
where I hope your projects will go in the long-term rate. We're all starting off looking at the numbers. And I, and I, I want to pose a couple of challenges to you and also offer all the support that
the various portions of the vice provost office can provide in and helping you as you as you take your data and try to figure out how to make changes that will help students success. So I, I got into
it into this area of learning analytics with a project that, that was done by a number of universities through a collaborative called the seismic project that focuses on stem education. And Stefano
and Linda's group was really involved in. Stefano and Linder were very strongly involved in this. And I became just in awe of what the good people at bar can do. And you folks who have been involved
in this learning analytics understand just as, just as well as I do what they do and just how excited they are to help folks use the data. But there is a, well, we've seen a presentation that
basically said that women in science classes have what we call agreed anomaly. They do worse in their science classes. And and this is the grade they have versus their overall GPA and science
courses. These are the lecture courses, but they do better in the lab courses, a little bit better in the lab courses. And and so Laura Brown and I, and this is really mostly L4 Brown's work, not
mine. Applied for learning analytics grant to try to understand how this was working in in our Department of Chemistry. And one of the things that we discovered was that the issue wasn't so much with
women. We found that women in chemistry just about as well as men. Now, That's only true if you don't take an intersectional lens to it. I've recently been analyzing some of our big classes. And what
we find is white women do as well or better than white men in our chemistry classes, but that's not the case for women of color, for black and Latino women. And so there's clearly some work to be
done. And we don't want to be ignoring the subgroups. And so I just want to talk to you about what I've been learning in this job, about who we teach well in whom we don't include as effectively. And
so this, what this chart shows, and I think Stefano would probably cringe that I haven't shown you the ranges there obviously is a big range here, but I wanted to make the data relatively easy to
see. It turns out, if we think about societal privilege. So if you are, if you are white or Asian American and you come from affluent family and your parents went to college. Your average GPA and all
courses at, at IU is almost 3.4. But for each, for each of those, those privileges that you lack, your GPA goes down. And in fact, if you are black or Latinx, low income, and your parents didn't go
to college. Your GPA for all courses on in the whole university is, um, is, is a point, is more than 0.4 points lower. So that's actually a lot when we're talking about this many students. If you're
in, if you're in chemistry, that number it chemistry classes, that number can be close to a whole point. And I've seen an even higher in some classes. So what this tells me is that what we do works
for some of our students pretty well. And for others of our students it works less well. And everybody's supposed to be getting the same experience here. This is a public institution that's supposed
to be helping all of our students. And so I look at these data and I think she's, we have to make some changes. And so I'll pick on my own department a little bit. This is we basically have a
three-course sequence that all are pre-med take. And if you get through and if you haven't had great high school chemistry, you can call it started what I'm calling course 0. So that's that, that's
an optional course. And what this shows us is the DFW rates to the rate of students who do not succeed in the class who are in a, D, and F or withdraw from the class. And it shows the numbers for
what the institution defines as, as under-represented minority students, in other words, black and Latinx students mostly on our campus and a non URM students. And what I hope you can see, hope you
can see three things. One is that the numbers are ridiculous, especially. So for instance, if we look at course one here, 25, 25 percent of our, of our non URM students don't make it through the
class, but more like 40 percent are under-represented. Minority students don't make it through the class. When you get to course two, those numbers are even bigger. In course 3, they come down a
little bit, but they're still nowhere near where we need them to be. And so in Coursera, we also see similar trends. And so we know that if a student gets a single w in any class, it increases their
chance of not graduating by 50 percent. If you get a D or an F in any class, it, it, it increases your chance of not graduating by, by a factor of two. Okay. And so, so we're not talking, I'm talking
between six to 9%, 6 to 12 percent, but that's still a big difference. So we often think of these things is, yes, some of our students have to figure out how to study. We keep them, but they come
back. But it has a real impact and can have a lasting impact on students lives. And the other thing is that we sometimes say, Hey, our job is to figure out who's good at our subject. And then we'll
will nurture those students and students who aren't good at it will help them find, will help them understand that they need to go look at something else. If that were the case, what I would expect
is very high DFW rates in the first-class and lower DFW rates in subsequent classes. And that's not what we see. All right. So that tells me that our curriculum is not fair and it's also not
effective. We are not providing students with what they need in the first course to get to the second course and so forth and so on. So that was a pretty sobering understanding that's gotten us
thinking a lot in my department about how to fix this. And I'll just point out that I taught course three for a whole year once. And I would like to tell you that it was 2013. I will tell you that
was mostly my colleague, more brown. So she has some really good ideas. But, but this was, but up books I, but I taught in 2011 and I want a teaching award that year. And I got great teaching
evaluations. And I thought I was doing my job. And I look at this and I wasn't. So I hope that when we look at our data, we can think about the ways in which what this tells us about what our
institution is doing compared to what we would like it to be doing and what we're doing in our classrooms compared to what we would like to be doing. And I'll, I'll skip this slide in the interest of
time. I'll actually, I'll just say that, you know, that that's one kind of data you look at. The other kind of data that we looked at is who persists in classes. And this helps to make arguments to
deans to invest in things. If you can say Gosh, if, if underserved students were to persist in our client, our sequence at the same rate as as privileged students, we would end up with an extra 10%
of students. And then you'd get that RCM money. That's helpful. And for women in chemistry, even though they do better if, if, if I do the same analysis, we would have had 800 more women in that
class, which is a 25 percent increase. So these are, these are ways to talk to deans about about where to invest money to help things, and I am happy to help you have those conversations. I love
trying to tell deals they need to invest in our courses. And then I want to, I want to point out that there is a department that has had a dramatic success and, and some of this may have started with
some learning analytics grants. I see Michael here from Econ, what? Econ used to be one of our worst offenders. It had very high DFW rates. In fact, those rates were 60% higher than our peer
institutions in the Big Ten. And so we had our eye else and he kinda what he did was they said we're trying to serve too many people, too many audiences with this one course. And they divided the
courses up into a business econ macro and micro sequence and an econ macro and micro sequence. And in the first year of that first year, year and a half of that I think is the first year of that
project. The DFW rates went from almost 30 percent to below 10 percent in all four classes. To me, this is, this is a stunning, stunning success and i'm, I'm convinced they've done this right and
they haven't just made the classes easier. I think they figured out how to help students learn better in these classes. And, and I hope we will get a chance to hear more from these folks as time goes
on. So I'll just close with that and reiterate how much I appreciate you all coming. You're all taking on those of you who are fellows, you're taking on these projects and you're sharing them with
the rest of us. And I look forward to hearing more from all of you. Thank you very much, Martha. I appreciate it. I appreciate all the support and you've been giving us as well and the work you're
doing by its purpose office, it's an exciting time.
Description of the video:
Greetings, my name is Paul Graf and I'm a senior lecturer in the Economics department at Indiana University. Thank you very much for your time. First, I'd like to thank the Learning Analytics Fellows
program funded by the Center for Learning Analytics and Student Success, Dennis Groth, the former Vice Provost of Undergraduate Education, George Rehrey, the founding director of CLASS, Harsha
Manjunath and Linda Shepard at IU BAR and Nikita Lopatin, professor at Ashland University. Last year, Dr. Gerhard glom and I explored the probability of students remaining in economics after choosing
economics or checking the box on their applications to Indiana University. We found gender, ethnicity, in large sections had significant effects on the likelihood of remaining in the economics major.
Based on these results, Dr. glom suggested expanding the analysis to include other majors. Often, students declare a major upon arriving at IU Bloomington, which may change from when they first
applied. Since students may change their major for different reasons, like their experience taking their first major course. After completing this course, students may switch for two reasons, ability
or preference, or the grade they received in this course. This project focuses on the latter reason. So I believe this project may have important admission and retention policy implications that may
improve these figures and therefore, the student experience at IU. This project analyzes three effects on the probability a student stays in their declared major. One: prior knowledge measured by SAT
scores and high school GPA. Two: experience, represented by the first major course grade and overall student performance that semester, excluding the first major course grade designated as GPAO. 3:
Other qualitative factors like gender, ethnicity, residency, and financial need. After exploring different major enrollments, over the last five years, I chose economics, business economics and
public policy or BEP, accounting, finance, biology and psychology. Next, I identified the introductory courses for these majors. The introductory course chosen was E21 for economics and BEP, A100 for
accounting and finance, L111 for biology and P101 for psychology. Further studying each major, I identified specific qualitative variables to account for program-specific variation. I used a binary
model which I estimate using the linear probability OLS method, and for robustness, I also used probit and logistic regression. To compare all six majors using only quantitative variables, I first
ran the regression of the likelihood a student remains in their declared major one year after taking the introductory course against four qualitative variables: high-school GPA, SAT score, the course
grade, and that term's GPA, excluding the course grade. Next, to model major specific variation, I ran, similar regressions including identified qualitative variables specific to the major. Due to
their similarity, I compared economics and BEP, identifying the qualitative variables of female Indiana Resident and first-generation based on the descriptive statistics. For economics and BEP using
E201 as the first major course, I found for economics, the quantitative variables were not statistically significant. Indiana residents were, on average, more likely to remain, but first generation
students were more likely to switch out of Economics one year after taking E201 For BEP, the higher the students high-school GPA and SAT scores on average, the more likely they remain in the BEP
major one year after taking E201. Other qualitative variables are not statistically significant. Next, I compared accounting and finance and found similar patterns in their Descriptive Statistics.
Specifically, I identified the qualitative variables of female, resident, first-generation, Asian, Hispanic, Latino, and Black African-American. Using A100 as the introductory course, I found all
four quantitative variables on average, having a statistically significant effect of remaining in the declared major. The higher these values, the more likely the student remains in their declared
major after one year of taking A100. For accounting, on average, Indiana residents are more likely to switch, while Black African-Americans are significantly more likely to stay one year after taking
A100. For finance, on average, female and Indiana resident students are more likely to switch, while Black, African-American, and Hispanic-Latino students are more likely to stay one year after
taking A100. Finally, I compared biology and psychology due to their popularity. I identified male, non-resident, first-generation, Asian, Hispanic, Latino, and Black African-American as the
qualitative variables. Running the regressions for these two majors using L111 and P101 as the first course for biology and psychology respectively, I found all but a student's high-school GPA has a
statistically significant effect on remaining in both majors one year after taking the first major course. On average, males, Asians and Hispanic and Latinos are more likely to remain in the biology
major one year after taking L111. On average, Hispanic, Latino, and Black African American students are more likely to remain in psychology one year after taking P101. To summarize, the first major
course grade has a significant effect on the most popular majors. Prior knowledge matters for students remaining in their declared majors. A students performance in other courses taken during the
first major course semester, Gender and Residency has varying effects. Except for economics, first-generation has no statistically significant effects. Finally, minorities are more likely to remain
in their declared major one year after taking the first major course. Further considerations, I would like to explore additional majors, generalize the results at the university, college and major
levels to help students stay in the major and potentially increase the diversity and inclusion goals at IU. Furthermore, I'd like to look at the variables of grade penalty and interactive effects on
changing majors. Thank you very much for your time.
Clip of Skills are the Currency of the Future: Mapping learners' expectations, course offerings, and workforce needs – Olga Scrivner | {"url":"https://class.indiana.edu/events/la_coll/index.html","timestamp":"2024-11-03T05:38:43Z","content_type":"text/html","content_length":"81937","record_id":"<urn:uuid:7072ddfc-3476-473f-b53a-1e8adf98ef41>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00275.warc.gz"} |
How Many Blocks Do I Need for My Quilt? | Designed to Quilt
How Many Blocks Do I Need for My Quilt?
If youโ re wondering how many blocks you need to make your quilt youโ ve come to the right place. Weโ ll show you how to calculate the number of blocks and share exactly how many 6-inch, 9-inch,
and 12-inch blocks you need for your quilt.
We love block-based quilt designs here at DTQ. Theyโ re super fun to make. And as quilt pattern designers we also appreciate the simplicity behind the quilt math involved.
Now, I know this may sound ridiculous because we all know quilt math can get super-complicated. Even with block-based quilt tops.
But compared to some other designs, itโ s actually much more straightforward. Especially if weโ re talking about calculating how many blocks you need for your quilt.
And thatโ s exactly what I want to teach you today. How to determine how many blocks you need to make a desired size quilt.
In this article, I will show you exactly how to calculate the number of blocks required to make a quilt (no matter what the size of the blocks and no matter what size quilt you are making).
I will show you two ways to do this: first, Iโ ll share how you can quickly do it using our quilting calculator app, Quilt Geek. Then, Iโ ll teach you how to do it the old-school way – the
pen-and-paper calculation.
Weโ ve also put together guides on how many blocks you need for three different โ standardโ block sizes. So after all this quilt math, weโ re sharing pre-calculated charts for 6-inch, 9-inch,
and 12-inch blocks.
How do you determine how many blocks you need for a quilt?
So letโ s jump right in. Before you start calculating, you need to determine:
• The size of the blocks youโ ll be using
• The size of the quilt you want to make
Letโ s say youโ re making a quilt using 6×6โ finished bear paw blocks like the one pictured below. (This means your unfinished blocks measure 6,5 x 6,5″.)
You want to make a queen-size quilt, which according to our quilt size chart, is about 92×106″.
We’re only defining the approximate quilt size that we’re going for. The reason for that is that the actual size we’ll be able to make depends on the size of the block we’re using.
When talking about 6-inch blocks here weโ re referring to 6-inch FINISHED blocks. This means that the blocks will measure 6โ x6โ after theyโ ve been sewn in your quilt top. Before you sew them
into the quilt top, the size of these blocks should be 6,5 โ x 6,5 โ (the extra ยฝ inch accounts for the seam allowance).
Option A: use the Quilt Geek calculator
I wanted to share the easiest option first and this is definitely it. Use our quilting calculator app and youโ ll get the calculation you need in seconds!
We offer a free trial, so you can work out the math you need for your project with no charge at all!
Hereโ s how it works:
With Quilt Geek, all you need to do is enter the size of the finished quilt blocks in the calculator:
With Quilt Geek, all you need to do is enClick CALCULATE and youโ ll get all the math done not just for a queen-size quilt, but for 9 standard quilt sizes!
Whatโ s so great about it is that it calculates THE ACTUAL SIZE of the quilt youโ ll be getting. It also provides the number of columns, rows and – of course – the number of blocks you need to
That was super easy, right?
Quilt Geek’s handy Blocks to Quilt Size Calculator calculates how many blocks you need for 9 standard quilt sizes.
It tells you how many rows and columns you need in your layout and exactly how many blocks you need to make – no matter the size.
Learn more about Quilt Geek’s 20+ calculators and charts here or get started right away:
Option B: Calculate the number of blocks with the pen-and-paper method
The other option is of course to do this the old-school way, using a pen, paper and an old school calculator.
Let me just remind you of the numbers weโ re working with:
• the finished size of the quilt block (bear paw): 6×6″
• the (approximate) desired finished size of the quilt: queen size – about 92×106″
Let’s do some math!
Step 1: How many blocks in a row?
To calculate how many blocks we need for one row, we’ll divide the desired quilt width by the width of the block:
92″ รท 6″ = 15.333
Right away you can see that you can’t get a quilt that’s exactly 92″ wide using 6″ blocks. Because the number we got (15.333) is closer to 15 than it is to 16, we’ll round it down to 15 blocks.
So, we need 15 blocks in each row.
The finished quilt width will be: 15 x 6″ = 90″
Step 2: How many blocks in a column?
We’ll now repeat the same calculation to get the number of blocks in a column. We’ll divide the desired quilt length by the length of the block:
106″ รท 6″ = 17.666
This time, we’ll round the number up (because 17.666 is closer to 18 than it is to 17).
We need 18 blocks in each column.
The finished quilt length will be 18×6โ = 108โ
Step 3: How many blocks total?
The last step is easy. Weโ ll multiply the number of columns by the number of rows to get the total number of blocks we need.
18 x 15 = 270
We need 270 6×6โ blocks for a queen-size (90×108โ ) quilt.
I think this wasnโ t too bad – but you canโ t argue itโ s much easier with Quilt Geek. Weโ d love for you to try it out and tell us what you think!
In case you were wondering, hereโ s what our finished bear-paw quilt would look like. Itโ s mesmerizing and I want to have one yesterday!
Of course, you can always have some fun with it, incorporating different colors (check out our quilterโ s color wheel for some ideas).
How many 6-inch blocks do I need to make a quilt?
As I said, there are certain sizes of quilt blocks that come up over and over again. We wanted to save you the trouble of doing the calculations yourself.
So we used Quilt Geek to provide the calculations and you can print them out if you want.
I want to emphasize again that when talking about 6-inch blocks here weโ re referring to 6-inch FINISHED blocks. This means that the blocks will measure 6โ x6โ after theyโ ve been sewn in your
quilt top. Before you sew them into the quilt top, the size of these blocks should be 6 1/2 โ x6 1/2 โ (the extra ยฝ inch account for the seam allowance).
If youโ re using 6-inch finished blocks you need the following number of blocks to make the desired size quilt:
How many 9-inch blocks do I need to make a quilt?
9- inch blocks are also a very popular size.
If youโ re using 9-inch finished blocks you will need the following number of blocks for your quilt:
How many 12-inch blocks do I need to make a quilt?
If youโ re using 12-inch finished blocks you will need the following number of blocks for your quilt:
Remember! Everything I shared here today assumes you are only sewing blocks together without sashing or borders.
If you want to make a quilt a bit larger, you can definitely think about adding borders and/or sashing. And weโ ll talk about how to calculate those some other time!
There you go! I hope this has helped you determine how many blocks you need to make a quilt. If you have any questions about the calculation or about the Quilt Geek app, let us know in the comments
2 thoughts on “How Many Blocks Do I Need for My Quilt?”
1. Catherine Walthert
Merci pour ces explications.
Mais vous ne tenez pas compte des largeurs de couture ? C’est surtout important dans le quilts a block de 6″, รงa finit par rรฉtrรฉcir le projet final.
1. Ula | Designed to Quilt
Bonjour, Catherine!
The calculation does take into account seam allowances, as well. The quilt math in the bar paw example is done for 6×6 inch FINISHED blocks – which means they are 6,5 x 6,5 unfinished as is
written at the beginning.
But thank you for pointing out that it wasn’t clear enough – I have added an explanation box to emphasize the difference between finished and unfinished blocks.
Happy quilting!
Leave a Comment | {"url":"https://designedtoquilt.com/how-many-blocks/","timestamp":"2024-11-10T14:14:00Z","content_type":"text/html","content_length":"540251","record_id":"<urn:uuid:d50d1487-98a6-4d7b-a8f5-9ff5578dd0ed>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00383.warc.gz"} |
F# tips weekly #15: Recursive memoize
Memoization of a recursive function can be often useful, but has some pitfalls compared to memoization of a simple function. Let's take a detailed look.
Standard Memoization
Let's use the classical example of computing the n-th item in the Fibonacci sequence. Utilizing the memoize function from Tip #14 works, but we encounter a compiler warning:
let rec fib = memoize <| fun n ->
if n < 2 then
fib (n - 1) + fib (n - 2)
The warning reads:
This and other recursive references to the object(s) being defined will be checked for initialization-soundness at runtime through the use of a delayed reference. This is because you are defining
one or more recursive objects, rather than recursive functions. This warning may be suppressed by using '#nowarn "40"' or '--nowarn:40'.
The issue here is that fib actually represents a constant object representing the closure of our memoized function. While calling this closure recursively is possible, it can lead to this warning and
associated runtime checks.
Recursive Call Placeholder Trick
We can eliminate the warning by employing the following trick. Instead of using the rec definition, we add a placeholder parameter within the function passed to memoizeRec:
let fib = memoizeRec <| fun recF n ->
if n < 2 then
recF (n - 1) + recF (n - 2)
memoizeRec is defined as:
let memoizeRec f =
let cache = System.Collections.Concurrent.ConcurrentDictionary()
let rec recF x =
cache.GetOrAdd(x, lazy f recF x).Value
Notice that the recursive function is defined inside the memoizeRec function. This approach allows us to use the rec keyword only there and not in memoized function, and avoids recursive calls on the
When we apply the same idea outside the context of memoization, we got a way to define a recursive function without explicitly define it as recursive:
let mkRec f =
let rec recF x = f recF x
We can apply this technique to create recursive functions in the same manner as with memoizeRec:
let fib =
mkRec <| fun recF n ->
if n < 2 then
recF (n - 1) + recF (n - 2)
mkRec is essentially the same thing as the fixed-point combinator or Y-combinator, nicely explained for example here.
Infinite Recursion
When working with recursive functions, we should always be aware of the possibility of infinite recursion. Typically, this leads to a Stack overflow error. However, in our case, we encounter a
different error:
System.InvalidOperationException: ValueFactory attempted to access the Value property of this instance.
This error occurs because we are using a Lazy object to compute the result, and there is a check to prevent accessing the same instance of Lazy within its evaluation, indicating a circular reference.
While this incurs some performance cost, it comes with the advantage that we catch the problem of infinite recursion earlier, before a stack overflow occurs. | {"url":"https://jindraivanek.hashnode.dev/f-tips-weekly-15-recursive-memoize","timestamp":"2024-11-04T00:46:53Z","content_type":"text/html","content_length":"127555","record_id":"<urn:uuid:99da1187-4d05-4c7f-ac96-ae6e0ecab2ef>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00230.warc.gz"} |
Zigya Acadmey
R is one of the most important languages in the field of data analysis and analytics, and so the multiple linear regression in R carries importance. It defines the case where a single response
variable Y is linearly dependent on multiple predictor variables. What is Multiple Linear Regression? A technique used for predicting a variable result that depends on two or more variables is a
multilinear regression. It is also called multiple regression. It is a linear regression extension. The calculated variable is the dependent variable, which is referred to as independent or
informative variables in the variables used to predict the dependent variable meaning. Multilinear regression allows researchers to assess the model variance and the relative contribution of each
independent variable. Multiple regression is of two forms, linear and nonlinear regression. The general mathematical equation for multiple regression is − y = b + b1x1 + b2x2 +...bnxn Description of
the parameters used… | {"url":"https://www.zigya.com/blog/author/zaint01/","timestamp":"2024-11-14T05:22:16Z","content_type":"text/html","content_length":"252288","record_id":"<urn:uuid:d42ef639-8915-4fa0-93cb-7bf5ae78ae4e>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00516.warc.gz"} |
A bicycle can be self-stable without gyroscopic or caster effects,
Science 15 April 2011: 332(6027), 339-342. [doi:10.1126/science.1201959]
The paper and supporting material
1) Preprint: 4 page pdf
2) Supporting text: 50 page pdf
3) Simultaneously posted history paper: http://hdl.handle.net/1813/22497 (40 page pdf)
History of thoughts about bicycle self-stability
J. P. Meijaard, Jim M. Papadopoulos, Andy Ruina, and A. L. Schwab
Long known, but still amazing, is that a moving bicycle can balance itself (see videos). Most people think this balance follows from a gyroscopic effect. That's what Felix Klein (of the Klein
bottle), Arnold Sommerfeld (nominated for the Nobel prize 81 times) and Fritz Noether (Emmy's brother) thought [1]. On the other hand a famous paper by David Jones [2] (published twice in Physics
Today) claims bicycle stability is also because of something called "trail". Trail is the distance the front wheel trails behind the steer axis. The front wheel of a shopping cart castor trails
behind its support bearing and so must a bicycle front wheel, Jones reasoned. Jones insisted that trail was a necessary part of bicycle stability.
We suspected that such simple images (above) were missing at least part of the picture.
To find the essence of bicycle self balance we looked at simpler and simpler dynamical models until we found a minimal two-mass-skate (TMS) bicycle that theory told us should be self-stable. This
bicycle has no gyroscopic effect and no trail. We built a bicycle (of sorts) based on the theory to prove the point.
This bicycle proves that self-stability cannot be explained in any simple words. Bicycles are not stable because of gyros, because you can make a self stable bicycle without gyros. We did that. And
they are not stable because of trail, you can take that away too. And we did that. More positively, we have shown that the distribution of mass, especially the location of the center of mass of the
front assembly, has as strong an influence on bicycle stability as do gyros and trail.
Why can a bicycle balance itself? One necessary condition for bicycle self stability is (once we define the words carefully) that such a bicycle turns into a fall.
The paper and supplementary material describe the problem and our solution in more detail.
This research was started by Jim Papadopoulos, working with Andy Ruina and Scott Hand at Cornell in 1985. The basic theoretical result was in-hand then. In some sense, the recent Proceedings of the
Royal Society paper on bicycle stability [3] was written to support the present paper. We couldn't publish this gyro-free-no-trail result without that foundation being in the literature. The
experimental two-mass-skate (TMS) bicycle, and the fleshing out of the theory, were carried out by Jodi Kooijman and Arend Schwab at Delft University of Technology, starting in 2008. Jaap Meijaard
found the key errors in Klein & Sommerfeld [1] and in Whipple [4].
Here is a video of Arend Schwab narrating the text above:
low res: promovideov2small.mp4 (7.2MB)
HD (warning: large file!): promovideov2.mp4 (261MB) (1920x1080)
or on Youtube.
Here is a video of Andy Ruina explaining how bicycles balance:
medium res: stablebicycle640.mov (71MB)
HD (warning: large file!): stablebicycleHD.mov (316MB) Link 1 | Link 2
or on Youtube
[1] F. Klein and A. Sommerfeld. Über die Theorie des Kreisels. Teubner, Leipzig, 1910. Ch IX §8, Stabilität des Fahrrads, by F. Noether, pp. 863–884. (pdf+English translation).
[2] D. E. H. Jones. The stability of the bicycle. Physics Today, 23(4):34–40, 1970. DOI:10.10631/1.3022064 (2006 DOI:10.1063/1.2364246)
[3] J. P. Meijaard, Jim M. Papadopoulos, Andy Ruina, and A. L. Schwab. Linearized dynamics equations for the balance and steer of a bicycle: a benchmark and review. Proceedings of the Royal Society A
, 463:1955–1982, 2007. DOI:10.1098/rspa.2007.1857 (pdf)
[4] F. J. W. Whipple. The stability of the motion of a bicycle. Quart. J. Pure Appl. Math. 30:312–348, 1899.
Yellow Bicycle stability demonstration photos and videos:
Andy Ruina's Bicycle Mechanics and Dynamics webpage:
Arend L. Schwab's Bicycle Mechanics and Dynamics:
Video 1, basic experiment: (also on Science website)
full size: 1201959Video1BasicExperiment.mov (9.8MB)
low res: 1201959Video1BasicExperiment.mp4 (1.2MB)
or on Youtube.
This video shows two typical experimental runs, both at stable forward speed. The first run shows stable straight ahead motion, and the second run shows laterally perturbed stable motions.
Video 2, counter-spinning wheels: (also on Science website)
full size: 1201959Video2CounterSpinningWheels.mov (4.7MB)
low res: 1201959Video2CounterSpinningWheels.mp4 (0.6MB)
or on Youtube.
This video demonstrates the working of the front counter-spinning wheel which eliminates the spin angular momentum of the front wheel.
Video 3, measuring trail: (also on Science website)
full size: 1201959Video3MeasuringTrail.mov (22.3MB)
low res: 1201959Video3MeasuringTrail.mp4 (2.9MB)
or on Youtube.
This video shows how we measured the small negative trail (caster) on the experimental two-mass-skate (TMS) bicycle. A piece of paper is placed underneath the front wheel and stuck to the ground
with tape. The front wheel is lowered and now touches the paper whereas the rear frame of the bicycle is clamped to prevent it from moving. The handlebars are then turned either way a number of
times, marking the paper. The bicycle is removed from the clamp and the mark on the paper is examined. The mark follows an arc, a line is drawn tangentially to either end of the mark. The point
where the two lines cross indicates the point about which the wheel rotates. Next the line for the middle of the contact point is drawn on the paper. The distance from the centre point to the arc
is the trail. When we measured the trail this way it turned out to be -4 mm, that is center of the contact region was 4 mm ahead of the intersection of the steer axis with the ground.
Video 4, slow motion: (also on Science website)
full size: 1201959Video4SlowMotion.mov (26.9MB)
low res: 1201959Video4SlowMotion.mp4 (3.4MB)
or on Youtube.
This is a high speed video (300 fps) of one of the experiments where we measured the lateral motions with a wireless inertial sensor (Philips Pi-Node) and forward speed by post-facto counting
High Def Composite (very large file):
1201959_HD_StableBicycleTeaser.mov (0.4GB) (1920x1080)
or on Youtube.
This HD video gives an overview of the two-mass-skate (TMS) bicycle and it's stable motions.
High Def Experiments 1 High Speed Perturbed Stable (very large file):
1201959_HD_StableBicycleExp1.mov (1.2GB) (1920x1080)
or on Youtube.
This HD video shows that the two-mass-skate (TMS) bicycle is stable at high speed, even when laterally perturbed.
High Def Experiments 2 High Speed Stable (very large file):
1201959_HD_StableBicycleExp2.mov (0.7GB) (1920x1080)
or on Youtube.
This HD video shows that the two-mass-skate (TMS) bicycle is stable at high speed.
First Successful Run:
GB7sv7firstrun.MOV (1.7MB)
or on Youtube.
The video shows the first successful run with the experimental primitive bicycle after replacing the polyurethane “inline skate” wheels with sharp edge aluminum wheels, with a crown radius of 2
GB7S4v2fv300.MOV (1.5MB)
or on Youtube.
The video shows an animation (VRML) of a simulation of the full nonlinear model with with the multibody dynamics software SPACAR. This to check if the system still behaves stably as predicted by
the linearized analysis. The initial forward speed is 3 m/s, then after 1 sec the bicycle is laterally perturbed with an initial leanrate of 0.6 rad/s. The transient dies out and the bicycle comes
back up again; stable indeed!
Fig 2C from the paper:
fig2cStableBicycleExperiment.pdf (119KB)
Original photo: fig2cStableBicycleExperimentLarge.jpg (4.9MB)
Self-stable experimental TMS bicycle rolling and balancing (photo by Sam Rentmeester/FMAX).
Fig 2A from the paper:
fig2aPhysicalStableBicycle.pdf (324KB)
Original photo: CIMG0603.JPG (1.6MB)
The experimental two-mass-skate (TMS) bicycle.
Fig 2B from the paper:
fig2bFrontAssembly.pdf (544KB)
Original photo: CIMG0592.JPG (1.7MB)
Front assembly of the experimental two-mass-skate (TMS) bicycle. A counter-rotating wheel cancels the spin angular momentum. The lines show a small negative trail.
eigentubes7p1.jpg (647KB) (2550x3300)
3D plot of eigenvalues of the theoretical two-mass-skate (TMS) bicycle as a function of speed. Forward speed v is shown on the horizontal x-axis. The vertical axis shows the real part of the
eigenvalues. The axis pointing out of the page shows the imaginary part. For speeds where all of the real parts are below the horizontal axis (have negative real part) the TMS bicycle is
theoretically stable. This is the rightmost 2/3 of the plot. Negative v corresponds to the bicycle going backwards, which is totally unstable for this bicycle. Rotating the figure 180 degrees about
the Im axis reproduces the figure because of the reversibility of the equations of motion (illustration by Peter de Lange).
Fig 1A from the paper:
fig1aBicycleModel.pdf (30KB)
The bicycle model consists of two frames B and H connected by two wheels R and F. The model has a total of 25 geometry and mass-distribution parameters. Central here are the rotary inertia I[yy] of
the front wheel, the steer axis angle (‘rake’) l[s] and the trail distance c (positive if contact is behind the steer axis). Depending on the parameter values, as well as gravity g and forward
speed v, this bicycle can be self-stable or not.
Fig 1B from the paper:
fig1bStableBicycleModel.pdf (27KB)
A two-mass-skate (TMS) bicycle is a special case. It is described with only 9 free parameters (8 + trail). The wheels have no inertia and are thus effectively ice-skates. The two frames each have a
single point mass and no mass moments of inertia. A heavy point mass at the rear skate at the ground contact point can prevent the bicycle from tipping over frontward; because it has no effect on
the linearized dynamics it is not shown. Even with negative trail (c < 0, see inset) this non-gyroscopic bicycle can be self-stable.
Fig 3A from the paper:
fig3aEigenvaluesStableBicycle.pdf (29KB)
Stability plot for the experimental TMS stable bicycle. Solutions of the differential equations are exponential functions of time. Stability corresponds to all such solutions having exponential
decay (rather than exponential growth). Such decay only occurs if all four of the eigenvalues l[i] (which are generally complex numbers) have negative real parts. The plot shows calculated
eigenvalues as a function of forward speed v. For v > 2.3m/s (the shaded region) the real parts (solid lines) of all eigenvalues are negative (below the horizontal axis) and the bicycle is
Fig 3B from the paper:
fig3bLeanYawRateStableBicycle.pdf (33KB)
Transient motion after a disturbance for the experimental TMS bicycle. Measured and predicted lean and yaw (heading) rates of the rear frame are shown. The predicted motions show the theoretical
(oscillatory) exponential decay. Not visible in these plots, but visible in high-speed video (Video 4), is a 20 Hz shimmy that is not predicted by the low-dimensional linearized model.
CAD model:
The CAD model of the experimental two-mass-skate (TMS) bicycle is available in three formats:
- SolidWorks GB7Sv9cSolidworksModel.zip (6.7MB)
- pdf's of all parts GB7Sv9cProductionDrawings.zip (244KB)
- STEP file GB7Sv9cSTEPModel.STEP (2.3MB)
Experimental data:
The experimental data together with the estimated simulated data which is presented in Fig 3B from the paper, is here presented in this plain ascii text file: 1201959experimentaldata.txt (16KB)
The first two lines is a header which explains the contents of each column.
Thesis Jodi Kooijman:
(coming soon):
- Any questions or comments? Please contact Arend L. Schwab - | {"url":"http://ruina.org/research/topics/bicycle_mechanics/stablebicycle/index.htm","timestamp":"2024-11-02T15:20:39Z","content_type":"application/xhtml+xml","content_length":"43221","record_id":"<urn:uuid:24409667-a372-44c6-9168-c4cb9d53e23b>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00830.warc.gz"} |
Version: 2.9.6
Release Date: 2024-11-05
Peter Ruckdeschel
Matthias Kohl
Thomas Stabla
(until 2005),
Florian Camphausen
(until 2005)
Required R-Version:
• >=3.4 for version 2.8.0
• >=2.14 for version 2.5
• >=2.2.0 for version 2.1,
• >=2.2.0 for version 1.6-2.0,
• >=2.0.1 patched for version 1.5,
• >= 1.8.0 for version 1.4,
• >= 1.7.0 for version built from source files
Dependencies: requires package "
" by
Peter Ruckdeschel
also available from
What is "distr" meant for?
The aim of package "distr" is to provide a conceptual treatment of random variables (r.v.'s) by means of S4--classes. A mother class "Distribution" is introduced with slots for a parameter and - most
important - for the four constitutive methods "r", "d", "p", and "q" for simulation, respectively for evaluation of density / c.d.f. and quantile function of the corresponding distribution. All
distributions of the "base" package for which corresponding "r", "d", "p", and "q"- <distr.name> functions exist (like normal, Poisson, etc.) are implemented as subclasses of either "
AbscontDistribution" or "DiscreteDistribution", which themselves are again subclasses of "Distribution".
This approach seems very appealing to us from a conceptual viewpoint:
Just pass an object of some derived distribution class to a generic function as argument and let the dispatching mechanism decide what to do on run-time.
As an example, we may automatically generate new objects of these classes with corresponding "r", "d", "p", and "q"-slots for the laws of r.v.'s under standard mathematical univariate transformations
and under convolution of independent r.v.'s. For "Distribution" objects X and Y expressions like 3*X+sin(exp(-Y/4+3)) have their natural interpretation as corresponding image distributions.
Note: Arithmetics on distribution objects are understood as operations on corresponding r.v.'s and not on distribution functions or densities.
You may set global options by distroptions() confer ?distroptions .
Up to version 1.5, additionally, we also provided classes for a standardized treatment of simulations (also under contaminations) and evaluations of statistical procedures on such simulations. These
are now delegated to packages distrSim and distrTEst (see below).
Attention: This package has been reorganized in version 1.6; if you cannot find a class/method/function previously in the package, also search the new packages
Further packages built on top of package "
for version prior to 1.8, a somewhat more detailed manual to this package in
is available here;
from version 1.8 on, we have converted this manual into a common vignette to packages
distrMod, distrTeach
which is available in the mere documentation package
. To use it you may type
require("distrDoc"); V<-vignette("distr"); print(V); edit(V)
• to be installed by
• to be removed by
• to be used by
• included into the .tar.gz.file
• as zipped source (for Versions <1.8.0)
• procede as follows:
□ unzip the zip File
□ consult the README -File in the zip-archive and follow the instructions therein
• (is the only possiblity for versions 1.7.0 and 1.7.1)
also see demo(package="distr") --- after installation of "distr"
• 12-fold convolution of uniformly (0,1) distributed variables compared to N(6,1): NormApprox.R
• Comparison of exact convolution to FFT for normal distributions ConvolutionNormalDistr.R
• Comparison of FFT to RtoDPQ: ComparisonFFTandRtoDPQ.R
• Comparison of exact and approximate stationary regressor distribution: StationaryRegressorDistr.R
• Truncation and Huberization/winsorization: Huberize.R , Truncate.R
• Distribution of minimum and maximum of two independent random variables: minandmax.R
• Instructive destructive example: destructive.R
• A simulation example: -> moved to packages distrSim/distrTEst
• Expectation of a given function under a given distribution: moved to package distrEx
• n-fold convolution of absolutely continuous distributions
Version history:
Changes from 1.0 to 1.1 (03-12-04)
• implementation of further exact convolution formulae for distributions
Nbinom, Gamma, Exp, Chisq
• exact formulae for scale transformations for the distributions
Gamma, Exp
• slot "seed" in simulation classes is now controlled and set via the setRNG package by Paul Gilbert
Changes from 1.1 to 1.2, 1.3
• changes in the Help-File to pass Rcmd check
Changes from 1.3 to 1.4
Changes from 1.4 to 1.5
• package is now using lazy loading
• minor changes in the help pages
• minor enhancements in plot for distributions (Gamma, discrete distributions)
• package now includes a demo - folder; try demo("distr")
• class Gamma has been renamed Gammad to avoid name collisions
• we have a CITATION file now; consider citation("distr")
• enhanced demos:
□ convolution of uniform variables now includes exact expressions
□ min/ max of two variables now available for discrete distributions
• rd-Files have now a keyword entry for distribution and thus may be found by the search engine
• exact formula for "Unif" o "numeric" where o \in { +,-,*,/ }
Changes from 1.5 to 1.6
• Our package is reorganized:
□ distr from now on only comprises distribution classes and methods
□ simulation classes and methods have been moved to the new package distrSim
□ evalation classes and methods have been moved to the new package distrTEst
□ a new class distrEx has been added by Matthias Kohl, providing additional features like distances between distributions, expectation operators etc
□ a new class RandVar has been added by Matthias Kohl, providing conceptual treatment of random variables as measurable mappings
Changes from 1.6 to 1.7
• taking up a suggestion by Martin Mächler, we now issue warnings as to the intepretation of arithmetics applied to distributions, as well as to the
accuracy of slots p,d,q filled by means of simulations; these warnings are issued at two places:
(1) on attaching the package
(2) at every show/print of a distribution
□ (2) can be cancelled by switching off a corresponding global option in distroptions() -- see ?distroptions .
• distroptions() / getdistrOption() now behave exactly like options() / getOption() options --- also compare mail
"Re: [Rd] How to implement package-specific options?" by Brian Ripley on
r-devel, Fri 09 Dec 2005 - 11:52:46, see http://tolstoy.newcastle.edu.au/R/devel/05/12/3408.html
• all specific distributions (those realized as [r|d|p|q]<name> like rnorm in package stats)
now have valid prototypes
• fixed arguments xlim and ylim for plot(signature("AbscontDistribution" or "DiscreteDistribution"))
thus: plot(Cauchy(),xlim=c(-4,4)) gives reasonable result (and plot(Cauchy()) does not)
• Internationalization: use of gettext, gettextf for output
• explicitly implemented is() relations: R "knows" that
□ an Exponential(lambda) distribution also is a Weibull(shape = 1, scale = 1/lambda) distribution, as well as a Gamma(shape = 1, scale = 1/lambda) distribution
□ a Geometric(p) distribution also is a Negativ Binomial(size = 1,p) distribution
□ a Uniform(0,1) distribution also is a Beta(1,1) distribution
□ a Cauchy(0,1) distribution also is a T(df=1, ncp=0) distribution
□ a Chisq(df=n, ncp=0) distribution also is a Gamma(shape=n/2, scale=2) distribution
• noncentrality parameter included for Beta, T, F distribution
Changes from 1.7 to 1.8
• a class "DExp" for Laplace/Double Exponential distributions
• a method dim which for distributions returns the dimension of the support
• show for distributions now acts as print
Changes from 1.8 to 1.9
• in demos, made calls to uniroot(), integrate(), optim(ize)() compliant to https://stat.ethz.ch/pipermail/r-devel/2007-May/045791.html
• new methods shape() and scale() for class "Chisq" with ncp==0
• derivation of a class LatticeDistribution from DiscreteDistribution to be able to easily apply FFT
□ new class 'Lattice' to formalize an affine linearly generated grid of (support) points pivot + (0:(Length-1)) * width
☆ usual accessor /prelacement functions to handle slots
□ new class 'LatticeDistribution' as intermediate class between 'DiscreteDistribution' and all specific discrete distributions from 'stats' package
☆ with a particular convolution method using FFT (also for 'convpow')
☆ usual accessor function 'lattice' for slot 'lattice'
• moved some parts from from package 'distrEx' to package 'distr'
□ generating function 'DiscreteDistribution'
□ univariate methods of 'liesInSupport()'
□ classes 'DistrList' and 'UnivariateDistrList'
□ generating functions EuclideanSpace() ,Reals(), Naturals()
• cleaning up of source files:
□ checked all source file to adhere to the 80char's-per-line rule
• added S4-method 'convpow' for convolutional powers from the examples
of package 'distr' with methods for
□ 'LatticeDistribution' and 'AbscontDistribution'
□ and particular methods for
☆ Norm, Cauchy, Pois, Nbinom, Binom, Dirac,
and ExpOrGammaOrChisq (if summand 'is' of class Gammad)
• new exact arithmetic formulae:
+ 'Cauchy' + 'Cauchy'
: gives 'Cauchy'
+ 'Weibull' * 'numeric'
: gives 'Weibull' resp. 'Dirac' resp 'AbscontDistribution'
: acc. to 'numeric' >, =, < 0
+ 'Logis' * 'numeric'
: gives 'Logis' resp. 'Dirac' resp 'AbscontDistribution'
: acc. to 'numeric' >, =, < 0
+ 'Logis' + 'numeric'
: gives 'Logis'
+ 'Lnorm' * 'numeric'
: gives 'Lnorm' resp. 'Dirac' resp 'AbscontDistribution'
: acc. to 'numeric' >, =, < 0
+ 'numeric' / 'Dirac'
: gives 'Dirac' resp. error acc. to 'location(Dirac)' ==, != 0
+ 'DiscreteDistribution' * 1 returns the original distribution
+ 'AbscontDistribution' * 1 returns the original distribution
+ 'DiscreteDistribution' + 0 returns the original distribution
+ 'AbscontDistribution' + 0 returns the original distribution
• new file MASKING and corresponding command 'distrMASK()' to describe the intended maskings
• mentioned in package-help: startup messages may now also be suppressed by
suppressPackageStartupMessages() (from package 'base')
• revised generating functions/initialize methods according to
in particular all Parameter(-sub-)classes gain a valid prototype
• formals for slots p,q,d as in package stats to enhance accuracy
□ p(X)(q, lower.tail = TRUE, log.p = FALSE)
□ q(X)(p, lower.tail = TRUE, log.p = FALSE)
□ d(X)(x, log = FALSE)
□ used wherever possible;
but backwards compatibility: always checked whether lowert.tail / log / log.p are formals
• unified form for automatically generated r, d, p, q-slots:
□ using (internal) standardized generators
☆ .makeDNew, .makePNew, .makeQNew
☆ .makeD, .makeP, .makeQ
□ revised "*", "+" ("Discrete/AbscontDistribution","numeric") methods (using .makeD, .makeP, .makeQ)
□ revised RtoDPQ[.d] (using .makeDNew, .makePNew, .makeQNew)
□ revised convolution methods (using .makeDNew, .makePNew, .makeQNew)
□ revised convpow() methods (using .makeDNew, .makePNew, .makeQNew)
• cleaning up of environment of r,d,p,q-slot - removed no longer needed objects
• left-continuous c.d.f. method (p.l) and
right-continuous quantile function (q.r) for DiscreteDistributions
• methods getLow, getUp for upper and lower endpoint of support of DiscreteDistribution or AbscontDistribution
(truncated to lower/upper TruncQuantile if infinite)
• analytically exact slots d,p (and higher accuracy for q) for distribution objects generated by functions
abs, exp, log for classes AbscontDistribution and DiscreteDistribution
• new (internally used) classes AffLinAbscontDistribution, AffLinLatticeDistribution
and AffLinLatticeDistribution to capture the results of transformations
Y <- a * X0 + b
for a, b numeric and X0 Abscont/Discrete/LatticeDistribution
and a class union AffLinDistribution of AffLinAbscontDistribution and AffLinLatticeDistribution
to use this for more exact evaluations of functionals in package 'distrEx'
• Version-management for changed class definitions to
□ AbscontDistribution (gains slot gaps)
□ subclasses of LatticeDistribution (Geom, Binom, Nbinom, Dirac, Pois, Hyper):
(changed by inheriting from LatticeDistribution, gaining slot lattice !)
realized by
□ moved generics to isOldVersion(), conv2NewVersion() from 'distrSim' to 'distr'
□ moved (slightly generalized version of) isOldVersion() (now for signature ANY) from 'distrSim' to 'distr'
□ new methods for conv2NewVersion for signature
☆ ANY : fills missing slots with corresponding entries from prototype
☆ LatticeDistribution: generates a new instance (with slot lattice(!)) by new(class(object), <list of parameters>)
• enhanced plot() methods (see ?"plot-methods" )
□ for both AbscontDistributions and DiscreteDistributions
☆ optional width and height argument for the display (default 16in : 9in)
○ opens a new window for each plot
○ does not work with Sweave;
workaround: argument withSweave = TRUE
in .Rnw-file: use width and height argument like in
<<plotex1,eval=TRUE,fig=TRUE, width=8,height=4.5>>=
☆ optional main, inner titles and subtitles with main / sub / inner
○ preset strings substituted in both expression and character vectors (x : argument with which plot() was called)
■ %A deparsed argument x
■ %C class of argument x
■ %P comma-separated list of parameter values of slot param of argument x
■ %Q comma-separated list of parameter values of slot param of argument x in () unless this list is empty - then ""
■ %N comma-separated <name>=<value> - list of parameter values of slot param of argument x
■ %D time/date at which plot is/was generated
○ title sizes with cex.main / cex.inner / cex.sub
○ bottom / top margin with bmar, tmar
○ setting of colors with col / col.main / col.inner / col.sub
☆ can cope with log-arguments
☆ setting of plot symbols with pch / pch.a / pch.u
☆ different symbols for unattained [pch.u] / attained [pch.a] one-sided limits
☆ do.points argument as in plot.stepfun()
☆ verticals argument as in plot.stepfun()
☆ setting of colors with col / col.points / col.vert / col.hor
☆ setting of symbol size with with cex / cex.points
□ for AbscontDistributions
☆ (panel "q"): takes care of finite left/right endpoints of support
☆ (panel "q"): optionally takes care of constancy region (via do.points / verticals)
☆ ngrid argument to set the number of grid points
□ for DiscreteDistributions :
• DEPRECATED:
□ class 'GeomParameter' --- no longer needed as this the parameter
of a 'Nbinom' with size 1
Changes from 1.9 to 2.0
• made calls to 'uniroot()', 'integrate()', 'optim(ize)()' compliant to
• new generating function 'AbscontDistribution'
• new class 'UnivarMixingDistribution' for mixing distributions with
methods / functions:
□ 'UnivarMixingDistribution' (generating function)
□ flat.mix to make out of it a distribution of class 'UnivarLebDecDistribution'
• new class 'AffLinUnivarLebDecDistribution' for affine linear transformations
of 'UnivarLebDecDistribution' (in particular for use with E())
• new class union 'AcDcLcDistribution' as common mother class
for 'UnivarLebDecDistribution', 'AbscontDistribution',
'DiscreteDistribution'; corresponding methods / functions:
• enhanced arithmetic: (for 'AcDcLcDistribution')
□ convolution for 'UnivarLebDecDistribution'
□ affine linear trafos for 'UnivarLebDecDistribution'
□ 'numeric' / 'AcDcLcDistribution'
□ 'numeric' ^ 'AcDcLcDistribution'^
□ 'AcDcLcDistribution' ^ 'numeric'
□ binary operations for independent distributions:
☆ 'AcDcLcDistribution' * 'AcDcLcDistribution'
☆ 'AcDcLcDistribution' / 'AcDcLcDistribution'
☆ 'AcDcLcDistribution' ^ 'AcDcLcDistribution'
□ (better) exact transformations for exp() and log()
□ Minimum Maximum Truncation Huberization
□ convpow for 'UnivarLebDecDistribution'
• new generating function 'AbscontDistribution'
• 'decomposePM' decomposes distributions in positive / negative part
(and in Dirac(0) if discrete)
• 'simplifyD' tries to cast to simpler classes (e.g. if a weight is 0)
Changes from 2.0 to 2.1
• DISTRIBUTIONS
□ DISCRETE DISTRIBUTIONS
☆ collapsing discrete distributions:
○ getdistrOption(".DistrCollapse.Unique.Warn")
○ implemented proposal by jacob van etten (collapsing support)
☆ enhance accuracy
○ improvement of .multm (now sets density for discrete distributions for non-support arguments actively to 0)
○ We are a bit more careful about hitting support points in .multm for DiscreteDistribution (i.e., for D * e2, e2 numeric, D DiscreteDistribution)
□ CONTINUOUS DISTRIBUTIONS
☆ gaps/support :
○ gaps matrix could falsely have 0 rows (instead of being set to NULL)
○ class UnivarMixingDistribution gains overall slots gaps support
○ added corresponding accessors
○ correspondingly, for UnivarLebDecDistribution as daughter class, accessors gaps(), support() refer to "overall" slots, not to slots of acPart, discretePart
○ deleted special support, gaps method for UnivarLebDecDistribution; now inherits from UnivarMixingDistribution´
○ new utility function .consolidategaps to "merge" adjacent gaps
○ setgaps method for UnivarMixingDistribution
○ methods
■ "*", c("AffLinUnivarLebDecDistribution","numeric"),
■ "+", c("AffLinUnivarLebDecDistribution","numeric"),
■ "*", c("UnivarLebDecDistribution","numeric"),
■ "+", c("UnivarLebDecDistribution","numeric"),
■ generating function "UnivarLebDecDistribtion"
○ had to be modified
○ utility 'mergegaps' catches situation where support has length 0
○ abs - and Truncate - methods for AbscontDistribution use '.consolidategaps'
□ COMPOUND DISTRIBUTIONS
☆ Compound Distributions are now implemented; see ?CompoundDistribution, class?CompoundDistribution
□ UNIVARIATE MIXING DISTRIBUTIONS
☆ fixed some errors / made some enhancements acc. to mail by Krunoslav Sever
• ENHANCED ACCURACY BY LOG SCALE
□ enhanced accuracy for Truncation with Peter Dalgaard's trick
□ passed over to log-scale for getUp, getLow (again to enhance accuracy for distributions with unbounded support)
□ introduced new slots .lowerExact and .logExact for objects of class "Distribution" (or inheriting) to control whether
the argument parts log[.p], lower.tail are implemented carefully in order to preserve accuracy
• ARITHMETICS
□ enhanced "+" method
☆ for DiscreteDistribution,DiscreteDistribution --- catches addition with Dirac-Distribution
☆ we enforce to use FFT-based algorithm for LatticeDistributions if the supports of both summands may be arranged on a common lattice whenever the length of convolutional grid (=unique(sort
(outer(support1, support2, "+"))) ) is smaller than the length of the product grid ( = length(support1) * length(support2) ) --- covers in particular m1*Binom(p,size) + m2*Binom(p',size)
when m1, m2 are naturals > 1 ...
□ convpow:
☆ some minor enhancements in convpow and "+", "LatticeDistribution","LatticeDistribution"
and correction of a buglet there (e.g., lattice width oould get too small)
☆ method for AcDcLcDistribution gains argument 'ep' to control when to ignore discrete parts (or a.c. parts) which summands in binomial expansion of (acPart+discretePart)^\ast n to ignore
☆ minor fix in method for DiscreteDistribution
□ automatic image distribution generation
☆ slot r is now /much/ faster / slimmer for results of *,/,^ (no split in pos/neg part necessary for this!)
☆ slot d for results of *,/, exp() now is correct at 0 by extrapolation (and deletion wir .del0dmixfun of half of the part to avoid double counting in *,/)
□ affine linear trafos return slot X0 of AffLin-Construction if resulting a=1 and b=0
□ method sqrt() for distributions
• PLOTTING
□ enhanced automatic plotting range selection (calling in both scale and quantile based methods)
□ plot-methods in branches/distr-2.1 now accept to.draw.arg no matter whether mfColRow==TRUE or FALSE
□ fixed xlim and ylim args for plots; ylim can now be matrix-valued...
□ realized suggestions by A. Unwin, Augsburg;
plot for L2paramFamilies may be restricted to selected subplots;
□ also named parameters are used in axis annotation if available.
□ changed devNew to only open a device if length(dev.list())>0
□ plot (for distribution objects) now is conformal to the (automatic) generic, i.e. it dispatches on signature (x,y) and has methods for signature(x=<distributionclass>,y="missing")
□ enhanced plotting (correct dispatch; opening of new device is controlled by option("newDevice") )
□ new plot function for 'UnivarLebDecDistribution' : now plots 3 lines
☆ first line common cdf and quantile function
☆ second line abscont part
☆ third line discrete part
• NEW / ENHANCED METHODS
□ getLow/getUp:
now available for UnivarLebDecDistribution, UnivarMixingDistribution
□ q.r, p.l (methods for right continuous quantile function and left continuous cdf)
☆ for class AbscontDistribution (q.r with 'modifyqgaps')
☆ for class UnivarLebDecDistribution
☆ for class UnivarMixingDistribution
□ prob methods:
☆ prob() for DiscreteDistribution-class returns vector of probabilities for the support points (named by values of support points)
☆ method for UnivarLebDecDistribution: returns a two-row matrix with
○ column names values of support points
○ first row named "cond" the probabilities of discrete part
○ second row named "abd" the probabilities of discrete part multiplied with discreteWeight; hence the absolute probabilities of the support points
□ methods p.ac, d.ac, p.discrete, d.discrete:
☆ they all have an extra argument 'CondOrAbs' with default value "cond" which if it does not partially match "abs", returns exactly slot p (resp. d) the respective acPart/discretePart of
the object
☆ else return value is weighted by acWeight/discreteWeight
□ new function 'makeAbscontDistribution'
☆ to convert arbitrary univariate distributions to AbscontDistribution: takes slot p and uses AbscontDistribution(); in order to smear out mass points on the border,
makeAbscontDistribution() enlarges upper and lower bounds
□ flat.LCD:
setgaps is called only if slot gaps is not yet filled
□ general technique: more freguent use of .isEqual
□ new / enhanced utilities (non-exported)
☆ 'modifyqgaps' in order to achieve correct values for slot q in case slot p hast constancy regions (gaps)
☆ .qmixfun can cope with gaps and may return both left and right continuous versions
☆ .pmixfun may return both left and right continuous versions in case slot p hast constancy regions (gaps)
• DOCUMENTATION
□ new section "Extension packages" in package-help file 0distr-package.Rd
□ mention of CompoundDistribution-class in package-help file 0distr-package.Rd of devel version
□ new vignette "How to generate new distributions in packages distr, distrEx" in package distr ...
• Rd-style:
□ several buglets detected with the fuzzier checking mechanism
☆ cf [Rd] More intensive checking of R help files, Prof Brian Ripley, 09.01.2009 10:25)
☆ [Rd] Warning: missing text for item ... in \describe? , Prof Brian Ripley
• S4 ISSUES:
□ fixed setGenerics- error reported by Kurt Hornik...
"log", "log10", "gamma", "lgamma" are no longer redefined as generics.
□ explicit method "+" for Dirac,DiscreteDistribution
□ some changes to the connections between LatticeDistribution and DiscreteDistribution resp. between AffLinLatticeDistribution and AffLinDiscreteDistribution.
□ key issues:
☆ JMC has changed the way non-simple inheritance [i.e. in the presence of setIs relations] is treated (see distr; in particular show, and operator methods for LatticeDistribution)
☆ introduced some explicit methods for LatticeDistribution, as due to setIs Relation
it may no longer be inherited automatically from DiscreteDistribution since JMC's changes in S4 inheritance mechanism Sep/Oct 08
• BUGFIXES
□ fixed a buglet in initialize for Cauchy Distribution
□ fixed bug in "+",LatticeDistribution,LatticeDistribution
□ it may be that even if both lattices of e1, e2 have same width, the convoluted support has another width! example: c(-1.5,1.5), c(-3,0,3)
□ matrix-valued ylim argument has not yet been dealt with correctly
□ fixed bug in plot-methods for argument "inner" under use of to.draw.arg argument
□ fixed a bug in convpow-method for AbscontDistribution
□ small buglets in plot-methods.R and plot-methods_LebDec.R (moved setting of owarn/oldPar outside)
□ fixed a bug in UnivarMixingDistribution.R (with new argument Dlist)
□ fixed a bug discovered by Prof. Unwin --- "+" trapped in a dead-lock coercing between DiscreteDistribution and LatticeDistribution
□ fixed a small buglet in convpow().
□ fixed buglet in devel version of distr: getLow.R (wrong place of ")" )
□ fixed some errors in plotting LCD and CompoundDistribution(and enhanced automatic axis labels by some tricky castings...)
□ UnivarMixingDistribution was too strict with sum mixCoeff == 1
□ deleted some erroneous prints left over from debugging in ExtraConvolutionMethods.R
□ fixed some buglets in plot for distr (only in branch)
□ fixed redundant code in bAcDcLcDistribution.R
□ Patch to bug with AffLinAbscontDistribution
□ correction of small buglet in validity to Norm-class
□ fixed bug with AffLinAbscontDistribution for a*X+b, distribution X >=0
• LICENSE: moved license to LGPL-3
Changes from 2.1 onwards
see NEWS-file
Our plans for the next version:
Things we invite other people to do
• multivariate distributions
• conditional distributions
• copula
This page is maintained by
Peter Ruckdeschel
(and was created by Thomas Stabla
and last updated on 2024-11-05. | {"url":"http://distr.r-forge.r-project.org/distr.html","timestamp":"2024-11-09T14:27:29Z","content_type":"text/html","content_length":"69322","record_id":"<urn:uuid:d2131d92-20db-4486-be6c-91c4e4f3e6c0>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00221.warc.gz"} |
On generalized homogeneity of locally connected plane continua
On generalized homogeneity of locally connected plane continua
Commentationes Mathematicae Universitatis Carolinae (1991)
• Volume: 32, Issue: 4, page 769-774
• ISSN: 0010-2628
The well-known result of S. Mazurkiewicz that the simple closed curve is the only nondegenerate locally connected plane homogeneous continuum is extended to generalized homogeneity with respect to
some other classes of mappings. Several open problems in the area are posed.
Charatonik, Janusz Jerzy. "On generalized homogeneity of locally connected plane continua." Commentationes Mathematicae Universitatis Carolinae 32.4 (1991): 769-774. <http://eudml.org/doc/247312>.
abstract = {The well-known result of S. Mazurkiewicz that the simple closed curve is the only nondegenerate locally connected plane homogeneous continuum is extended to generalized homogeneity with
respect to some other classes of mappings. Several open problems in the area are posed.},
author = {Charatonik, Janusz Jerzy},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {confluent; continuum; dendrite; homogeneous; light; local homeomorphism; locally connected; monotone; open; plane; simple closed curve; universal plane curve; homogeneity; Menger
universal curve; Sierpinski universal plane curve},
language = {eng},
number = {4},
pages = {769-774},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On generalized homogeneity of locally connected plane continua},
url = {http://eudml.org/doc/247312},
volume = {32},
year = {1991},
TY - JOUR
AU - Charatonik, Janusz Jerzy
TI - On generalized homogeneity of locally connected plane continua
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1991
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 32
IS - 4
SP - 769
EP - 774
AB - The well-known result of S. Mazurkiewicz that the simple closed curve is the only nondegenerate locally connected plane homogeneous continuum is extended to generalized homogeneity with respect
to some other classes of mappings. Several open problems in the area are posed.
LA - eng
KW - confluent; continuum; dendrite; homogeneous; light; local homeomorphism; locally connected; monotone; open; plane; simple closed curve; universal plane curve; homogeneity; Menger universal
curve; Sierpinski universal plane curve
UR - http://eudml.org/doc/247312
ER -
1. Anderson R.D., A characterization of the universal curve and a proof of its homogeneity, Ann. of Math. (2) 67 (1958), 313-324. (1958) Zbl0083.17607MR0096180
2. Anderson R.D., One-dimensional continuous curves and a homogeneity theorem, Ann. of Math. (2) 68 (1958), 1-16. (1958) Zbl0083.17608MR0096181
3. Bing R.H., A simple closed curve is the only homogeneous bounded plane continuum that contains an arc, Canad. J. Math. 12 (1960), 209-230. (1960) Zbl0091.36204MR0111001
4. Charatonik J.J., On fans, Dissertationes Math. (Rozprawy Mat.) 54 (1967), 1-40. (1967) Zbl0163.44604MR0227944
5. Charatonik J.J., Some problems on generalized homogeneity of continua, Topology, Proc. International Topological Conference, Leningrad 1982; Lecture Notes in Math., vol. 1060, Springer Verlag,
1984, 1-6. Zbl0544.54029MR0770218
6. Charatonik J.J., Mappings of the Sierpiński curve onto itself, Proc. Amer. Math. Soc. 92 (1984), 125-132. (1984) Zbl0524.54010MR0749904
7. Charatonik J.J., Generalized homogeneity of curves and a question of H. Kato, Bull. Polish Acad. Sci. Math. 36 (1988), 409-411. (1988) Zbl0767.54030MR1101430
8. Charatonik J.J., Monotone mappings of universal dendrites, Topology Appl. 38 (1991), 163-167. (1991) Zbl0726.54012MR1094549
9. Charatonik J.J., Omiljanowski K., On light open mappings, Proceedings International Topological Conference, Baku (USSR), l989, 211-219. Zbl0817.54010MR1347226
10. Kato H., Generalized homogeneity of continua and a question of J.J. Charatonik, Houston J. Math. 13 (1987), 51-63. (1987) Zbl0635.54017MR0884233
11. Kato H., On problems of H. Cook, Topology Appl. 26 (1987), 219-228. (1987) Zbl0612.54042MR0904468
12. Krasinkiewicz J., On homeomorphisms of the Sierpiński curve, Comment. Math. Prace Mat. 12 (1969), 255-257. (1969) Zbl0235.54039MR0247618
13. Prajs J.R., On open homogeneity of closed balls in the Euclidean spaces, preprint.
14. Prajs J.R., Openly homogeneous continua in 2-manifolds, preprint. Zbl0830.54028
15. Rogers J.T., Jr., Homogeneous continua, Topology Proc. 8 (1983), 213-233. (1983) Zbl0541.54039MR0738476
16. Rogers J.T., Jr., Classifying homogeneous continua, Proc. of the Topology Symposium at Oxford University (June 1989), preprint. Zbl0776.54023MR1173271
17. Wilson D.C., Open mappings of the universal curve onto continuous curves, Trans. Amer. Math. Soc. 168 (1972), 497-515. (1972) Zbl0239.54007MR0298630
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qml.GateFabricqml.GateFabric — PennyLane
class GateFabric(weights, wires, init_state, include_pi=False, id=None)[source]¶
Bases: pennylane.operation.Operation
Implements a local, expressive, and quantum-number-preserving ansatz proposed by Anselmetti et al. (2021).
This template prepares the \(N\)-qubit trial state by applying \(D\) layers of gate-fabric blocks \(\hat{U}_{GF}(\vec{\theta},\vec{\phi})\) to the Hartree-Fock state in the Jordan-Wigner basis
\[\vert \Psi(\vec{\theta},\vec{\phi})\rangle = \hat{U}_{GF}^{(D)}(\vec{\theta}_{D},\vec{\phi}_{D}) \ldots \hat{U}_{GF}^{(2)}(\vec{\theta}_{2},\vec{\phi}_{2}) \hat{U}_{GF}^{(1)}(\vec{\theta}_{1},\
vec{\phi}_{1}) \vert HF \rangle,\]
where each of the gate fabric blocks \(\hat{U}_{GF}(\vec{\theta},\vec{\phi})\) is comprised of two-parameter four-qubit gates \(\hat{Q}(\theta, \phi)\) that act on four nearest-neighbour qubits.
The circuit implementing a single layer of the gate fabric block for \(N = 8\) is shown in the figure below:
The gate element \(\hat{Q}(\theta, \phi)\) (Anselmetti et al. (2021)) is composed of a four-qubit spin-adapted spatial orbital rotation gate, which is implemented by the OrbitalRotation()
operation and a four-qubit diagonal pair-exchange gate, which is equivalent to the DoubleExcitation() operation. In addition to these two gates, the gate element \(\hat{Q}(\theta, \phi)\) can
also include an optional constant \(\hat{\Pi} \in \{\hat{I}, \text{OrbitalRotation}(\pi)\}\) gate.
The four-qubit DoubleExcitation() and OrbitalRotation() gates given here are equivalent to the \(\text{QNP}_{PX}(\theta)\) and \(\text{QNP}_{OR}(\phi)\) gates presented in Anselmetti et al.
(2021), respectively. Moreover, regardless of the choice of \(\hat{\Pi}\), this gate fabric will exactly preserve the number of particles and total spin of the state.
☆ weights (tensor_like) – Array of weights of shape (D, L, 2), where D is the number of gate fabric layers and L = N/2-1 is the number of \(\hat{Q}(\theta, \phi)\) gates per layer with N
being the total number of qubits.
☆ wires (Iterable) – wires that the template acts on.
☆ init_state (tensor_like) – iterable of shape (len(wires),), representing the input Hartree-Fock state in the Jordan-Wigner representation.
☆ include_pi (boolean) – If True, the optional constant \(\hat{\Pi}\) gate is set to \(\text{OrbitalRotation}(\pi)\). Default value is \(\hat{I}\).
Usage Details
1. The number of wires \(N\) has to be equal to the number of spin-orbitals included in the active space, and should be even.
2. The number of trainable parameters scales linearly with the number of layers as \(2 D (N/2-1)\).
An example of how to use this template is shown below:
import pennylane as qml
from pennylane import numpy as np
# Build the electronic Hamiltonian
symbols = ["H", "H"]
coordinates = np.array([0.0, 0.0, -0.6614, 0.0, 0.0, 0.6614])
H, qubits = qml.qchem.molecular_hamiltonian(symbols, coordinates)
# Define the Hartree-Fock state
electrons = 2
ref_state = qml.qchem.hf_state(electrons, qubits)
# Define the device
dev = qml.device('default.qubit', wires=qubits)
# Define the ansatz
def ansatz(weights):
qml.GateFabric(weights, wires=[0,1,2,3],
init_state=ref_state, include_pi=True)
return qml.expval(H)
# Get the shape of the weights for this template
layers = 2
shape = qml.GateFabric.shape(n_layers=layers, n_wires=qubits)
# Initialize the weight tensors
weights = np.random.random(size=shape)
# Define the optimizer
opt = qml.GradientDescentOptimizer(stepsize=0.4)
# Store the values of the cost function
energy = [ansatz(weights)]
# Store the values of the circuit weights
angle = [weights]
max_iterations = 100
conv_tol = 1e-06
for n in range(max_iterations):
weights, prev_energy = opt.step_and_cost(ansatz, weights)
conv = np.abs(energy[-1] - prev_energy)
if n % 2 == 0:
print(f"Step = {n}, Energy = {energy[-1]:.8f} Ha")
if conv <= conv_tol:
print("\n" f"Final value of the ground-state energy = {energy[-1]:.8f} Ha")
print("\n" f"Optimal value of the circuit parameters = {angle[-1]}")
Step = 0, Energy = -0.87007254 Ha
Step = 2, Energy = -1.13107530 Ha
Step = 4, Energy = -1.13611971 Ha
Step = 6, Energy = -1.13618810 Ha
Final value of the ground-state energy = -1.13618903 Ha
Optimal value of the circuit parameters = [[[ 0.60328427 0.41850407]]
[[ 0.85581129 -0.24522642]]]
Parameter shape
The shape of the weights argument can be computed by the static method shape() and used when creating randomly initialised weight tensors:
shape = GateFabric.shape(n_layers=2, n_wires=4)
weights = np.random.random(size=shape)
>>> weights.shape
(2, 1, 2)
arithmetic_depth Arithmetic depth of the operator.
basis The basis of an operation, or for controlled gates, of the target operation.
batch_size Batch size of the operator if it is used with broadcasted parameters.
control_wires Control wires of the operator.
grad_recipe Gradient recipe for the parameter-shift method.
hash Integer hash that uniquely represents the operator.
hyperparameters Dictionary of non-trainable variables that this operation depends on.
id Custom string to label a specific operator instance.
is_hermitian This property determines if an operator is hermitian.
name String for the name of the operator.
ndim_params Number of dimensions per trainable parameter of the operator.
num_params Number of trainable parameters that the operator depends on.
num_wires Number of wires the operator acts on.
parameter_frequencies Returns the frequencies for each operator parameter with respect to an expectation value of the form \(\langle \psi | U(\mathbf{p})^\dagger \hat{O} U(\mathbf{p})|\psi\
parameters Trainable parameters that the operator depends on.
pauli_rep A PauliSentence representation of the Operator, or None if it doesn’t have one.
wires Wires that the operator acts on.
adjoint() Create an operation that is the adjoint of this one.
compute_decomposition(weights, wires, …) Representation of the operator as a product of other operators.
compute_diagonalizing_gates(*params, wires, …) Sequence of gates that diagonalize the operator in the computational basis (static method).
compute_eigvals(*params, **hyperparams) Eigenvalues of the operator in the computational basis (static method).
compute_matrix(*params, **hyperparams) Representation of the operator as a canonical matrix in the computational basis (static method).
compute_sparse_matrix(*params, **hyperparams) Representation of the operator as a sparse matrix in the computational basis (static method).
decomposition() Representation of the operator as a product of other operators.
diagonalizing_gates() Sequence of gates that diagonalize the operator in the computational basis.
eigvals() Eigenvalues of the operator in the computational basis.
expand() Returns a tape that contains the decomposition of the operator.
generator() Generator of an operator that is in single-parameter-form.
label([decimals, base_label, cache]) A customizable string representation of the operator.
map_wires(wire_map) Returns a copy of the current operator with its wires changed according to the given wire map.
matrix([wire_order]) Representation of the operator as a matrix in the computational basis.
pow(z) A list of new operators equal to this one raised to the given power.
queue([context]) Append the operator to the Operator queue.
shape(n_layers, n_wires) Returns the shape of the weight tensor required for this template.
simplify() Reduce the depth of nested operators to the minimum.
single_qubit_rot_angles() The parameters required to implement a single-qubit gate as an equivalent Rot gate, up to a global phase.
sparse_matrix([wire_order]) Representation of the operator as a sparse matrix in the computational basis.
terms() Representation of the operator as a linear combination of other operators. | {"url":"https://docs.pennylane.ai/en/stable/code/api/pennylane.GateFabric.html","timestamp":"2024-11-03T12:48:03Z","content_type":"text/html","content_length":"179600","record_id":"<urn:uuid:dba6e163-4834-4170-9960-bd0a4ab3c7b1>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00724.warc.gz"} |
Convert inches to kilometers ( in to km )
Last Updated: 2024-11-06 10:55:33 , Total Usage: 432678
Transitioning from inches to kilometers involves converting a unit from the imperial system to the metric system. This conversion is vital for understanding and comparing distances across different
measurement systems.
Historical Context and Significance
The inch, a traditional unit of length in the imperial system, is commonly used in the United States. It's based on the British imperial system. The kilometer, a unit of length in the metric system,
is widely used around the world. One kilometer equals 1,000 meters, and it is commonly used for expressing geographical distances and lengths.
Conversion Formula
The conversion from inches to kilometers involves two steps: first converting inches to meters, and then converting meters to kilometers. The formula is:
$$ \text{Kilometers} = \text{Inches} \times 0.0254 \times \frac{1}{1000} $$
Here, 0.0254 is the conversion factor from inches to meters, and dividing by 1000 converts meters to kilometers.
Example Calculation
Let's convert 10,000 inches to kilometers.
1. Convert inches to meters: $$ 10,000 \text{ inches} \times 0.0254 = 254 \text{ meters} $$
2. Convert meters to kilometers: $$ 254 \text{ meters} \times \frac{1}{1000} = 0.254 \text{ kilometers} $$
So, 10,000 inches is equal to 0.254 kilometers.
Why This Conversion Matters
This conversion is crucial in fields like international engineering, science, and global trade. In a world where different countries use different measurement systems, being able to convert between
these systems is essential for collaboration and understanding.
Common Questions (FAQs)
1. Why is the inch defined as 0.0254 meters?
□ This definition was established to create a standard, precise correlation between the imperial and metric systems for global consistency.
2. How common is the use of kilometers globally?
□ The kilometer is widely used in most countries for measuring distances, especially for travel and geographical measurements.
3. Is this conversion frequently used?
□ While not everyday conversions for most people, converting inches to kilometers becomes important in scientific research, international projects, and education.
In summary, the ability to convert inches to kilometers bridges the gap between two major measurement systems, enhancing understanding and facilitating international cooperation in various fields.
This conversion is a testament to the interconnectedness of our world. | {"url":"https://calculator.fans/en/tool/in-to-km-convertor.html","timestamp":"2024-11-06T17:22:00Z","content_type":"text/html","content_length":"12381","record_id":"<urn:uuid:3c994460-be25-44cc-9a06-b523d9400d25>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00444.warc.gz"} |
6th Grade Math: Mastering Through Magic of Math Doodle WheelsCognitive Cardio Math
Teaching 6th grade can be amazing, but it can also be stressful to figure out how to engage 6th graders in their learning. The “too cool for school” mentality can run strong among some of them. When
you plan your math classes for your sixth graders, you want to keep in mind where they are developmentally and their interests. This helps make the material more accessible for them, which engages
them to do the work. I created math doodle wheels that can be used with different 6th grade math concepts. I hope after you read how math doodle wheels build on the characteristics of a sixth grader,
you are inspired to use them in your classroom!
What are Math Doodle Wheels?
Say goodbye to traditional fill-in-the-blank notes or the quick scramble to write everything down from the board. Math doodle wheels are one-sided note pages that organize and chunk class notes on
the target math concept. You and your students can refer back to these note pages all year for quick definitions, reminders, and mastery examples!
In the middle of the wheel, your students doodle and color in the math concept they are focusing on. This is the “title” of the math wheel and makes it very easy for students to refer back to later.
With this bold title in the middle of the page, they can quickly find what they are looking for.
The other sections of the wheel focus on key vocabulary and the different steps and strategies that will follow for the specific math concept. In each section, students write down the main definition
or step and then complete a couple of practice problems together with the teacher/class. This provides them with guided instruction but also mastery examples to return to throughout the unit.
After notes, students go back and review what they wrote. Then they add color to each section. They can color in the background, create fancy fonts, and draw little doodles to help them remember what
they learned. The colorful doodles become memory triggers when they are applying the math skills throughout the unit or later in the year.
In my post, Using Doodle Wheel Graphic Organizers for Math and ELA, I dive deeper into the reasons why I chose to make the switch from traditional interactive notebook pieces to the math doodle
Why Should I Use Math Doodle Wheels in 6th Grade Math?
As 5th graders transition into 6th grade, you are going to see several shifts in their characteristics as learners. There will be a new air about them in the form of newfound confidence and
independence! This encourages them to make their own decisions and to take on more ownership in the learning process. This leads to wanting to explore topics or concepts deeper after discovering the
foundation of them.
Social Time with Math Doodle Wheels!
A big characteristic of 6th graders is the social aspect. They want to do anything and everything with their friends! All of a sudden one-person jobs become 2 or 3 person jobs, and the buddy system
is an oath that should never be broken…even on a trip to the drinking fountain!
We might want to pull our hair out because of this, but we can use this to our advantage. Math doodle wheels can easily become collaborative with a partner or in small groups. This is a great way to
keep students engaged but also let them help each other.
Critical and Abstract Thinking Strengthens
The critical thinking and abstract thinking skills of 6th grade students become stronger as well. This allows them to analyze problems, make connections between different skills, and apply math
concepts to real-world situations.
Learning Styles
Math doodle wheels also connect to different learners, whether they are auditory, visual, or verbal-linguistic learners. Students can listen to the instruction, draw doodles (visuals) to help them
remember the information, and write down the key details.
Fostering Social-Emotional Needs
Incorporating math doodle wheels in your 6th grade math classroom helps you to create a safe environment for your students’ social-emotional needs. We were all in middle school at some point. As a
result, we know firsthand the raging emotions and the rollercoaster those years can be. Add on the stress or nerves that come with academics; students can truly dread school. Having an opportunity in
your class to approach math concepts in a simple and creative way helps your students relax. When students relax, they can absorb and take risks in their learning.
Makes Math Relatable
Finally, identity is so important to students at this age. They are trying to figure out who they are, and they are making connections between what they are learning and everyday life. Being able to
connect math concepts to their life makes them more relatable and will also create higher engagement. Math doodle wheels help students personalize their learning through the doodles and color they
add to their notes. It’s important to stress to them that one image might be helpful for one of them. However, that same image may not help another classmate.
How Can I Use Math Doodle Wheels in 6th Grade Math?
Math doodle wheels are flexible, which gives you freedom on how to best implement them in your classroom. When first introducing math doodle wheels to your class, I recommend completing it all
together as a group. You can do this through small group or whole class instruction. This way, you can set the expectations for how the math doodle wheel is completed and can explain the “why” behind
creating them for your students. Your students will have mastery examples to refer back to throughout the unit and year.
As the year progresses, you can use the completed math doodle wheels in different ways that allow your students to take on ownership, such as:
• Creating their own study guides or reviews for upcoming assessments.
• Becoming the “teacher” and creating their own notes for their group or class to fill out.
• Design a year-at-a-glance where each section represents each of the studied units from the year in preparation for end of the year testing
• As a tool for an open notes review or assessment
How Do I Get My 6th Grade Math Classes Started?
Now that you have read how the math doodle wheels build on the characteristics of a 6th grade learner, let me show you how you can start using these in your classroom! In my 6th Grade Math Doodle
Wheel Bundle, you will have access to math wheels for 20 different math concepts! Math concepts included are:
• Absolute Value
• Algebraic Expressions
• Box-and-whisker Plots
• Coordinate Plane
• Dividing Decimals
• Dividing Fractions (2 wheels)
• Equivalent Expressions
• Exponents
• Finding GCF and LCM
• Graphing Inequalities
• Integers
• Mean Absolute Deviation
• Multiplying Decimals
• One-step Algebraic Equations
• One-step Inequalities
• Order of Operations (2 wheels)
• Proportions and Unit Rates (2 wheels)
• Ratios and Rates
• Surface Area
• Volume
Everything you need to get started with the math wheels in your 6th grade math classroom is provided for you in the bundle. For each topic, you will receive a math wheel that lays out key vocabulary
and the strategy for the target math concept. A key is provided for each one for you to refer to if needed, as well as a colored sample. Additionally, there is an editable PowerPoint file with a
blank wheel that you can use to create your own math wheel to align with your lesson.
Your students will be taking efficient notes while creating memory triggers for themselves for the steps. They will also be able to complete practice problems for each section to refer back to. Not
only is it a great lesson and guided practice tool, but while they are taking notes they are creating a study tool to keep all year in their binders or notebooks.
Try Out Wheels for Free
Math wheels can be used in your 6th grade math class with your whole class, in small groups, or centers. Want to test the waters to see if math doodle wheels are a good fit for you and your students?
When you sign up for these free math wheels, you will receive a Fraction Operations wheel to review all operations with your students. You will also receive 3 blank wheel templates to create your own
math doodle wheels about whichever topics you want!
Save these 6th Grade Math Notes!
Remember to save this post to your favorite math or teacher Pinterest board to return to when you are ready to take these math wheels for a spin in your classroom with your 6th graders! | {"url":"https://cognitivecardiomath.com/cognitive-cardio-blog/6th-grade-math-doodle-wheels/","timestamp":"2024-11-06T13:40:22Z","content_type":"text/html","content_length":"235255","record_id":"<urn:uuid:ac8da2c3-34e5-424f-97d2-a7031c974dbe>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00322.warc.gz"} |
[cosxsinxsinxcosx] Find inverse using Elementary [sinxcosx]... | Filo
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Question Text Find inverse using Elementary operations.
Updated On Jul 25, 2023
Topic Matrices and Determinant
Subject Mathematics
Class Grade 12
Answer Type Video solution: 1
Upvotes 123
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Examples of the Use of the Scientific Approach in Mathematics Teaching and Learning to Help Indonesian Students to be Independent Learners
This is a theoretical paper focusses on Indonesian school system. The challenge for education in Indonesia according to the former Minister of Education and Culture of Indonesia, Anies Baswedan, was
how to help Indonesian students to be independent learners and to have good characters (Kemdikbud, 2014). The 2013 Curriculum proposed Scientific Approach to be implemented in Indonesian mathematics
classes. Scientific Approach consists of five steps: (1) observing, (2) questioning, (3) collecting data, (4) reasoning, and (5) communicating. This paper discusses how two approaches, namely
Scientific Approach and the Japanese Problem-solving Approach (PSA), can help Indonesian students to improve their thinking, creativity, and innovation during mathematics teaching and learning in
classroom. The paper will provide some practical examples of problem-solving using these two approaches.
Japanese Problem-solving Approach; Scientific Approach; problem posing; independent solving; observing; questioning; reasoning
Australian Association of Mathematics Teachers (AAMT) (2013). AAMT Position Statement, Professional learning. Adelaide: The Australian Association of Mathematics Teachers.
Cooney, T.J., Davis, E.J., & Henderson, K.B. (1975). Dynamics of Teaching Secondary School Mathematics. Boston: Houghton Mifflin Company.
De Lange, J. (2004). Mathematical Literacy for Living from OECD-PISA Perspective. Paris: OECD-PISA.
Even R., & Loewenberg Ball, D. (2009). Setting the stage for the ICMI study on the professional education and development of teachers of mathematics. In R. Even & D. Loewenberg Ball (Eds). The
Professional Education and Development of Teachers of Mathematics. New York: Springer.
Fitzgerald, M., & James, I. (2007). The Mind of the Mathematician. Baltimore: The Johns Hopkins University Press.
Goos, M., Stillman, G., & Vale, C. (2007). Teaching Secondary School Mathematics: Research and Practice for the 21st Century. NSW: Allen & Unwin.
Haylock, D., & Thangata, F. (2007). Key Concepts in Teaching Primary Mathematics. London: SAGE Publications Ltd.
Isoda, M. (2015a). Mathematical Thinking: How to Develop It in the Classroom. Presentation given on a Course on Developing Lesson Study in Mathematics Education for Primary (Mathematics) Teachers,
October 16 – 29, 2015, Yogyakarta: SEAMEO for QITEP in Mathematics.
Isoda, M. (2015b). What is the product of Lesson Study? Japanese Mathematics Textbook and Theory of Teaching. Power Presentation given on a Course on Developing Lesson Study in Mathematics Education
for Primary (Mathematics) Teachers, October 16 – 29, 2015, Yogyakarta: SEAMEO for QITEP in Mathematics.
Isoda, M., & Katagiri, S. (2012). Mathematical Thinking. Singapore: World Scientific.
Kemdikbud RI (2011). Jejak langkah Kementerian Pendidikan dan Kebudayaan (1945-2011). [The steps of the Ministry of Education and Culture (1945-2011)]. Jakarta: Pusat Informasi dan Hubungan
Masyarakat, Kemdikbud.
Kemdikbud RI (2014). Gawat Darurat Pendidikan di Indonesia. [Emergency Department of Education in Indonesia] Jakarta: Author.
Shadiq, F. (2010). Identifikasi Kesulitan Guru Matematika SMK pada Pembelajaran Matematika yang Mengacu pada Permendiknas No. 22 Tahun 2006.[ Identification of the Difficulties of Vocational
Mathematics Teachers in Mathematics Learning Referring to National Education Minister Regulation No. 22 of 2006] Edumat: Jurnal Edukasi Matematika, 1(1), 49 – 60.
Shadiq, F. (2014). Pemecahan Masalah dalam Pembelajaran Matematika di Sekolah. Makalah Disampaikan pada [Problem Solving in Learning Mathematics in Schools. Paper Presented at]: Seminar Nasional
Matematika dan Pendidikan Matematika (MAPIKA) di Universitas PGRI Yogyakarta 24 Mei 2014. Yogyakarta: SEAMEO QITEP in Mathematics. Downloaded from https://fadjarp3g.wordpress.com/ 2014/ 06/ 04/
Shadiq, F. (2016a). The Opportunities and Challenges on the Teaching and Learning of Mathematics. Experience of SEAMEO QITEP in Mathematics. Power Point Presented on: The Workshop on Promoting
Mathematics Engagement and Learning Opportunities for Disadvantaged Communities in West Nusa Tenggara in Australian Embassy, Jakarta, 12 May, 2016. Yogyakarta: SEAMEO QITEP in Mathematics.
Shadiq, F. (2016b). The Japanese Problem-Solving Approach (PSA). Presentation given on a Course on Joyful Learning for Primary School Teachers. SEAMEO QITEP in Mathematics, Yogyakarta, 24 August – 6
September 2016. Yogyakarta: SEAMEO QITEP in Mathematics.
Shadiq, F. (2017). What Can We Learn from the ELPSA, SA, and PSA Frameworks? The Experience of SEAQiM. Southeast Asian Mathematics Education Journal, 7(1), 65-76.
Shadiq, F. (2018). The Japanese Problem-Solving Approach (PSA) for Primary School Teachers. Presentation given on a Course on Joyful Learning for Primary School Teachers. SEAMEO QITEP in Mathematics,
Yogyakarta, 7-20 March, 2018. Yogyakarta: SEAMEO QITEP in Mathematics.
University of Tsukuba (2012). 2012 Outline of the University. Imagine the Future. Tsukuba: University of Tsukuba.
White, A. L. (2011). School mathematics teachers are super heroes. Southeast Asian Mathematics Education Journal, 1(1), 3-17.
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Differential amplifier circuit analysis
The differential amplifier circuit is a very useful opamp circuit and by adding more resistors in parallel with the input resistors r1 and r3, the resultant circuit can. The simple two transistor
implementation of the current mirror is based on the fundamental relationship that two equal size transistors at the same temperature with the same v gs for a mos or v be for a bjt have the same
drain or collector current. Perform a netlist and run green arrow in the ade 10. Differential transistor amplifiers worksheet discrete. We had a brief glimpse at one back in chapter 3 section 3.
The fully differential amplifier has multiple feedback paths, and circuit analysis requires close attention to detail. Firstorder rc circuits can be analyzed using firstorder differential equations.
Differential amplifier circuit tutorial using bjt and opamp. Analysis of the basic differential amplifier topology. A differential amplifier is designed to give the difference between two input
signals. Building a differential amplifier operational amplifiers. In this tutorial, we will learn about one of the important circuit in analog circuit design. You can put together basic op amp
circuits to build mathematical models that predict complex, realworld behavior. Nodal analysis long before the op amp was invented, kirchoffs law stated that the current flowing into any node of an
electrical circuit is equal to the current flowing out of it. Differential amplifiers built using opamps figure 1 shows such a circuit made of two.
The amplifier which amplifies the difference between two input signals is called as differential amplifier. Commercial op amps first entered the market as integrated circuits in the mid1960s, and by
the early 1970s, they dominated the active device market in analog. Half circuits for common mode and differential mode are different. The differential amplifier below should achieve a differential
gain of 40 with a power consumption of 2 mw. How much mismatch does it take to degrade the cmr of a differential amp.
It is virtually formed the differential amplifier of the input part of an operational amplifier. A differential amplifier is a type of electronic amplifier that amplifies the difference between two
input voltages but suppresses any voltage common to the two inputs. The emitters of the two transistors are joined and connected to a constant current source. Differential amplifier circuit using
transistors elprocus. The main drawback of the differential amplifier is that its input impedance may not be high enough if the output impedance of the source is high. Differential amplifiers can
also be constructed as discrete component circuits. Still, no circuit analysis will be complete without the art of solving the circuit by inspecting it and finding the resistors one by one, based on
the operational. For example, by connecting one input to a fixed voltage reference set up on one leg of the.
Differential amplifier stages large signal behavior general features. Fully differential circuit analysis method1 for internal loops, isolate those loops individually and perform stb analysis ensure
overall dc feedback for accurate biasing and that all loops are compensated cmdm 1 measures only the firststage cm response cmdm 2 measures overall dm response and secondstage cm response. The
differential amplifier circuit amplifies the difference between signals applied to the inputs fig. It is the fundamental building block of analog circuit. Here is an example of a firstorder series rc
circuit. What are the applications of differential amplifier. Thus, the differential pair makes a very good difference amplifierthe kind of gain stage that is required in every operationalamplifier
circuit. Differential amplifier is a device which is used to amplify the difference between the voltages applied at its inputs. Usually, some types of differential amplifier comprise various simpler
differential amplifiers. Superposition is used to calculate the output voltage resulting from each input voltage, and then the two output voltages are added to arrive at the final output voltage.
Differential amplifier operating in purely differential input signal. Determine minimum channel length 2222011 insoo kim determine channel width. The simplification is based on the symmetry of the
circuit. The desire to have large input resistance for the differential amplifier is the main drawback for this circuit.
Analyze a series rc circuit using a differential equation. The differential amplifier is probably the most widely used circuit building block in analog integrated circuits, principally op amps. All
of these approaches give the wrong answer because they do not load the circuit properly and can be very hard to implement using deeply embedded loops stb analysis ensures that the loop stays closed
at all points and measures the loop gain its like middlebrooks method but with. Below figure shows the ideal differential amplifier. The electronic amplifier used for amplifying the difference
between two input signals can be called as a differential amplifier. The mos version of this circuit consists of two transistors biased by current source, the sources of. The differential amplifier
circuit using transistors is widely applied in integrated circuitry, because it has both good bias stability and good voltage gain without the use of large bypass capacitors. Care must be taken to
include the vocm pin for a complete analysis. Bjt semiconductor circuit analysis transistor practice problem duration. Differential amplifier circuit using transistors design. The standard
differential amplifier circuit now becomes a differential voltage comparator by comparing one input voltage to the other. I am writing a tutorial article on bandgap reference circuit including
theory, analysis, design and simulation. Analysis of a differential amplifier wolfram language.
Stb analysis of differential feedback amplifier rf. Differential amplifiers are found in many circuits that utilize series negative feedback opamp follower, noninverting amplifier, etc. Consider the
differential amplifier circuit shown in figure 7. Active common mode input range as large as possible. General topology of cmfb circuit 2222011 insoo kim contd common mode feedback examples of cmfb
2222011 insoo kim folded cascode amplifier with cmfb. Differential amplifier an overview sciencedirect topics.
The basic circuit used to provide gain in the op amp is as shown in fig. A differential amplifier is a type of electronic amplifier that amplifies the difference between two. Solving the differential
amplifier part 1 mastering. Op amp differential amplifier circuit voltage subtractor. Combine the equations for differential voltage gain and for commonmode voltage gain for the following
differential amplifier circuit, into a single equation for cmrr. Figure 3 shows a block diagram used to represent a fullydifferential amplifier and its input and output voltage definitions. Circuit
analysis circuit analysis of fully differential amplifiers follows the same rules as normal singleended amplifiers, but subtleties are present that may not be fully appreciated until a full analysis
is done. A firstorder rc series circuit has one resistor or network of resistors and one capacitor connected in series. For example, by connecting one input to a fixed voltage reference set up on one
leg of the resistive bridge network and the other to either a thermistor or a light dependant resistor the amplifier circuit can be used to detect either low or. Increasesd immunity to external noise
increased output voltage swing for a given voltage rail ideal for lowvoltage systems integrated circuit is easier to use reduced evenorder harmonics. The implementation of the current mirror circuit
may seem simple but there is a lot going on. There are conditions on kirchoffs law that are not relevant here.
It is the building block of analog integrated circuits and operational amplifiers op amp. The emitter part of the circuit obtained is shown in 5a. One of the important feature of differential
amplifier is that it tends to reject or nullify the part of input signals which is common to both inputs. The differential amplifier can be implemented with bjts or mosfets. Solving the differential
amplifier part 2, i demonstrate that the same results can be accomplished with the coefficients identification method. The reason the amplifier is called a differential amplifier is that to the
firstorder it only accepts differential input signals. Design of differential amplifier circuit using transistors. One of the unstatated assumptions that is virtually always made in working with
opamps is that they are purely differential amplifiers and have no commonmode gain. The op amp circuit is a powerful took in modern circuit applications. A differential amplifier circuit is a very
useful opamp circuit, since it can be configured to either add or subtract the input voltages, by suitably adding more resistors in parallel with the input resistors. Circuit analysis suppose just
one of the resistors of the resistors is off by only 0.
Large signal operation of the bjt differential pair the differential pair is a differential amplifierwe express its performance in terms of differential and commonmode gains. The ability of the
circuit to reject common signals depends on how well the resistor ratios are matched. Dc analysis of differential amplifier have been discussed. Analysis of fully differential amplifiers texas
Differential amplifier online analog electronics course. Differential amplifier is a device used to amplify the difference in voltage of the two input signals. Smallsignal analysis of the commonmode
of the differential amplifier the commonmode gain of the differential amplifier with a current mirror load is ideally zero. Smallsignal analysis of the bjt differential pair v cc v ee r c i v o2 t v
o1 t c q 1 q 2 r v.
The analysis circuit shown in figure 1 is used to calculate a generalized circuit formula and block. Differential amplifier and its theory with circuit. This problem is addressed by the
instrumentation amplifier discussed next. The greater this parameters value, the better the differential amplifier will perform as a truly differential amplifier. In this section, we want to examine
a more complicated circuit to demonstrate the features and capabilities of the dae solver. Opamp as a differential amplifier circuit with function. As the name indicates differential amplifier is a
dccoupled amplifier that amplifies the difference between two input signals. A wheatstone bridge differential amplifier circuit design is as shown in the figure above. It is useful to note that our
differential amplifier circuit is based on an operational amplifier which is, itself, a differential amplifier. Circuit analysis circuit analysis of fully differential amplifiers follows the same
rules as normal singleended amplifiers, but. To understand the behavior of a fullydifferential amplifier, it is important to understand the voltage definitions used to describe the amplifier. The
differential amplifier configuration is very much popular and it is used in variety of analog circuits. Replace the two bjts with the emitter equivalent circuit. Pdf analysis and design of mos
differential amplifier.
When the negative feedback is applied to this circuit, expected and stable gain can be built. If all the resistor values are equal, this amplifier will have a differential voltage gain of 1. We can
still use half circuit concept if the deviation from prefect symmetry is small i. Bias circuit is similar to half circuit for common mode. Library design exercise design flow determine
specifications. The derivation of the small signal equivalent circuit is shown in figure 2. An opamp is a differential amplifier which has a high ip impedance, high differentialmode gain, and low op
impedance. An op amp circuit can be broken down into a series of nodes, each of which has a nodal equation. Differential amplifier analysis classic diff amp 2222011 insoo kim contd differential
amplifier analysis 2222011 insoo kim. By analyzing a firstorder circuit, you can understand its timing and delays. | {"url":"https://acundespay.web.app/1542.html","timestamp":"2024-11-06T21:21:18Z","content_type":"text/html","content_length":"17279","record_id":"<urn:uuid:b0091fc6-e2f7-451c-bc9e-cfc4fba0a7f7>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00678.warc.gz"} |
golf handicap average score 100
This graph shows the average score relative to par for all holes. Dr. L.J. For men, the average handicap is 16.1, while the average is 28.9 for women. Play five rounds of 18-hole golf or take the
scores from your last five rounds of 18-hole golf. In the United States, officially rated golf courses are described by course and rating of slope. Another measurement for the average golfers is what
is known as a bogey golfer. If you were to take the scores from more or fewer than five rounds, there would be a different formula and it would not be as accurate. Golf Practice Routines to Score
Lower If you are struggling to break 100, you should leave this mentality behind. When my best 8 out of 20 scores were put through the calculator, my handicap index under the new system came out at
7.8. This will form the basis for your course and playing handicap. A quick call to the USGA confirmed that very fact. It is important to remember that a golf handicap is not the average of a
golfer's score but rather the score that a golfer has the potential or ability to score. I dropped my handicap 5 stokes on summer working on my short game. If your handicap is less than 12.9 you can
claim to be better than 50% of the golfers in the US while if you are a single figure handicap player you can justly say you are in the top 1/3 of players in the country. I played hockey, some rep
level, and was a decent athlete, above average but not amazing at any sport in particular except maybe skiing. 1 in … The average golf course in the United States is approximately 7,200 yards, give
or take a 100 depending on the handicap. The standard scratch score (SSS) for the course is then deducted from the average strokes taken. CBS Sports has all of those statistics and more for the PGA
Tour. Bookmark the permalink. Every course is given two ratings that determine its difficulty. But "Golf Digest" notes that most golfers don't participate in the USGA system. Every score in a
player’s handicap record will be converted to a score differential. 5 Wood This translates into an average score of about 90, close to that of a bogey golfer. Find your adjusted gross score. Find the
Slope. The formula for calculating your golf handicap is as follows: Your Score – Course Rating x 113/ Slope Rating So, if your score is 100, and the Course Rating is 74.2 and the slope is 115, your
handicap is 25.35. An average golf score is 90 strokes for every 18 holes played. Most golf scorecards will have the course slope rating listed on them. Handicap Index ®? Few people understand or can
explain how the course rating and slope are computed, so be […] They are factored into the handicap score for a player and should also be used when assessing the score for a round against what an
average player would be expected to score. 100 0 5 10 15 20 95 90 85 80 75 70 65 60 80-75-70-65-60-Course Rating Average score of scratch golfers on a course Average score of bogey golfers on a
course Bogey Rating. Start with your handicap index. The USGA says that the average golfer in its system carries about a 15.0 handicap. Six Round Sample. If a player has completed 18 holes in 80, 86,
and 95 strokes, their average score would be 87 (80 + 86 + 95 = 261 / 3 = 87). Course and Slope Ratings for a Golf Course. My recommendations would be the following: 1) Talk to Golf Pro at your range
or course concerning lessons. Reply. The median driving distance is 219.55 yards, while the average for a three-wood is 186.89 yards, a seven iron 133.48 yards and pitching wedge 73.97 yards.
Wondering who leads the PGA Tour in drive distance, consecutive cuts, scoring average, or putts per hole? Players with a 10.0 comprise 4.6 percent of the golf population. If your golf handicap is in
this range you are roughly around the equivalent of a bogey golfer. Historically, rules relating to handicaps have varied from country to country with many different systems in force around the
world. They generally score a bit over 90- and play off around a 20 handicap. 2) Practice your short game twice as much as your long game. We do not recommend this. As an alternative to using
averaged scores to determine the handicap index, distinct skill level is factored into the comparisons. Thus, Jack would be more skilled than Arnold on the average golf course. The Average Amateur.
The United States Golf Association constructed the handicap system to level the playing field for everyone. Once you get to 20, the average of the 10 lowest differentials of your last 20 scores are
used to determine your handicap. Driver; Male: 200 – 260yds, Female: 140 – 200yds. For example, Handicap 1-10 golfers score 0.65 strokes over par on par 4 (an average of 4.69). However, the
calculation is a little more complicated - especially for juniors. Gather at least five scores 18-hole scores or ten 9-hole scores and use them to calculate your Adjusted Gross Score. In the above
example, your other four scores could all be more than 100, but you'd still carry a handicap index of 4.8. It … Instead, the handicap index looks at recent scores and adjusts based on the course’s
difficulty, expressed in the course rating and slope, two numbers that appear somewhere on the scorecard. Golfer Burnz 6 years ago If the only thing that mattered in golf was score and improving,
then golf wouldn’t be fun anymore. The National Golf Foundation breaks down scores this way: Average Score Percent of Adult Golfers; Under 80: 5%: 80 … Course Ratings. Of course, to complicate
things, you may have played different courses with different course ratings and slopes. This is calculated by multiplying the difference between your gross score and the course rating by 113, and
dividing by the slope rating of the tees that were played. This would mean a player who typically scores around 90 on the average course. To calculate your Handicap Index, you’ll first need a minimum
of five golf scores (and no more than 20). Playing golf with a handicap gives each player a ‘net score’ to compare – it is their gross score (total number of strokes, for example 82) minus their own
handicap (for example 10) for a net score of 72. The USGA says that the average golf handicap for men is 16.1, and is 29.2 for women. What course rating and slope have to do with it Golf handicap
calculations use an esoteric system of “course rating” and something called “slope” to compute exactly how many strokes everyone should get. A good golf score is a maximum of 108 strokes, while a bad
score is considered to be 120 strokes or higher. If the SSS is 70, the player would be allocated a starting handicap of 17 (87 – 70 = 17). My mother put me on skis at two and was a decorated ski
racer in the fifties. Average Score vs. the Handicap. There is one big benefit and one big drawback to using the handicap system versus average golf score. A golf handicap is a numerical measure of a
golfer's potential that is used to enable players of varying abilities to compete against one another. This is borne out in the basic calculation of a handicap as the average of the 10 best scores of
the last 20 rounds and multiplying that by 96%. Score 90 and Above = 0 birdies; Average Golfer Statistics – The Chart. Gather at least five scores 18-hole scores or ten 9-hole scores and use them to
calculate your Adjusted Gross Score. In the biggest golf playing country in the world – the USA – where roughly 8% of the population (24.2 million) play golf – the average golf handicap for men is
14.4. An alumni of the International Junior Golf Academy and the University of South Carolina–Beaufort golf team, where he helped them to No. This is why I think so many golfers end up over 100. If
you are just going out as a single and getting hooked up with golfers willy nilly there’s little doubt that the average score shot (if a score is even being kept) will be well above the average
handicap. 100 – 74.2 x 113/ 115 = 25.35. › average golf handicap for beginners ... To calculate your Handicap Index, you'll first need a minimum of five golf scores (and no more than 20). That figure
is multiplied by .96, and that's your handicap index. That's why, of course, you want to enter all your scores. A younger player with a handicap of 5 or thereabouts should be getting close to the
higher end of the range, while an older player or one with a handicap of 20 or over will be closer to the lower end. The biggest benefit comes from consistency when playing courses … Rating of course
is a number (typically between 67 and 77) that is used to measure the average "good" score that a scratch golfer may attain on the course. ← On a par 72 course, a golfer with a 10 handicap would be
expected to shoot an 82. This, in spite of all the innovations in club and ball design and instruction. These players typically score a bogey on most holes. The five rounds are what are commonly used
to calculate a handicap. Golf handicaps are calculated using an average of the three rounds a player submits. Low golf scores and thus low handicap indexes are more complex than pure talent and
practice. Better players are those with the lowest handicaps. Riccio did an analysis and here is his findings. This entry was posted in Golfers Data Analysis and tagged golf break 90, golf handicap,
golf handicap demographic, golf handicap population, golfers who break 80, percentage of golfers who break 100, usga golf handicap distribution. I … So where are these numbers coming from you ask? 3
Wood; Male: 180 – 235yds, Female: 120 – 185yds. Insights: Very good golfers excel on Par 5’s, but comparatively struggle on Par 3s while less skilled players perform comparatively better on Par 3s,
but struggle with Par 5s. If can probably give you a few and identify the areas you need to work on the most. s. g. The USGA Handicap System™ enables golfers of all skill levels to compete on a more
even basis. Since golf is such a complex sport, there are many degrees of skill levels. They are going for broke on most shots, chasing a score that is not reasonable for their ability level.
Watching pro golf has warped many of us into thinking that we should be making birdies and pars out on the course. Taking the course rating and slope of my course into consideration – Sandburn Hall
is relatively difficult off the white tees at 73.6 course rating and 136 slope – my course handicap came out at 9. From time to time we receive questions from members about using their average score
as a proxy for their golf handicap, particularly for competitions among golfers. The average golf handicap is between 13-15 . In the last 25 years, the average USGA handicap for a man has improved
nearly two full strokes, from 16.3 to 14.4. This score applies to an amateur golfer playing on a par 72 course. A few years ago, Trackman gathered its data and charted the average driver clubhead
speed for golfers organized by handicap. I was fortunate that I have always advance the ball pretty well. The national average Handicap Index for men is 14.7, approximately one stroke lower than it …
A handicap of 40 or above the maximum score is 10; A handicap of 30-39 is a maximum score of 9; A handicap of 20-29 is a maximum score of 8; A handicap of 10-19 is a maximum score of 7; A handicap of
0-9 is a maximum score of double-bogey; 3. According to the National Golf Foundation, the average golf score remains where it has been for decades: 100. Rating of course, rating of slope, and
handicap of course. A player’s handicap index is different from their average score. Study this chart and use it as a motivational tool to help you improve your golf statistics and become an above
average golfer. Handicap Is Not Your Average Score. I started playing golf 40 years ago at age 13. Are going for broke on most holes calculate a handicap lowest differentials of your last 20 are...
Low handicap indexes are more complex than pure talent and Practice scores are used to determine handicap... And that 's your handicap index ® 200 – 260yds, Female: 140 – 200yds their ability.!
Strokes or higher to an amateur golfer playing on a par 72 course a score differential more. – 200yds Association constructed the handicap system to level the playing field for everyone at two and
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working on my short game rules relating to handicaps varied! Average, or putts per hole more skilled than Arnold on the course is given two ratings determine! Standard scratch score ( SSS ) for the
average golf score remains where it has been for decades:.! 20 handicap scoring average, or putts per hole call to the USGA confirmed that very fact players... One big benefit and one big benefit and
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up over 100 score that is not for. Has all of those statistics and more for the PGA Tour in drive distance consecutive! You ask are roughly around the equivalent of a bogey golfer from ask. Of those
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Given an audio signal consisting of the sinusoidal term given as x(t) = 3 cos (500 )
Given an audio signal consisting of the sinusoidal term given as x(t) = 3 cos (500 )
EXAMPLE 4.5. Twenty four voice signals are sampled uniformly and then have to be time division multiplexed. The highest frequency component for each voice signal is equal to 3.4 kHz. Now
(i) If the signals are pulse amplitude modulated using Nyquist rate sampling, what would be the minimum channel bandwidth required.
(ii) If the signals are pulse code modulated with an 8 bit encoder, what would be the sampling rate? The bit rate of system is given as 1.5 x 10^6 bits/sec.
Solution: (i) As a matter of fact, if N channels are time division multiplexed, then minimum transmission bandwidth is expressed as,
BW = Nf[m]
Here, f[m] is the maximum frequency in the signals.
Given, f[m] = 3.4 kHz
Therefore BW = 24 x 3.4 kHz = 81.6 kHz Ans.
(ii) The signaling rate of the system in given as,
r = 1.5 x 10^6 bits/sec
Since there are 24 channels, the bit rate Of an individual channel is,
r (one channel) = = 62500 bits/sec
Further, since each samples is encoded using 8 bits, the samples per second will be,
Sample/sec =
Note that the samples per seconds is nothing but sampling frequency.
Thus, we have
Solving, we get, f[s] = 7812.5 Hz or samples per second Ans.
EXAMPLE 4.6. A PCM system uses a uniform quantizer followed by a 7-bit binary encoder. The bit rate of the system is equal to 50 x 10^6 bits/sec.
(i) What is the maximum message signal bandwidth for which the system operates satisfactorily?
(ii) Calculate the output signal to quantization noise ratio when a full load sinusoidal modulating wave of frequency 1 MHz is applied to the input.
(U.P. Tech-Semester Exam. 2005-2006)
Solution: (i) Let us assume that the message bandwidth be f[m], Hz. Therefore sampling frequency should be,
f[s] ³ 2f[m]
The number of bits given as v = 7 bits
We know that the signaling rate is given as,
r ³ v.f[s]
or r ³ 7 x 2f[m]
Substituting value for r, we get
50 x 10^6 ³ 14 f[m]
or f[m] 3.57 MHz Ans.
Thus, the maximum message bandwidth is 3.57 MHz.
(ii) The modulating wave is sinusoidal. For such signal, the signal to quantization noise ratio is expressed as,
Substituting the value of v, we get
= 1.8 + 6 x 7 = 43.8 dB Ans.
EXAMPLE 4.7. The information in an analog waveform with maximum frequency f[m] = 3 kHz is to be transmitted over an M-level PCM system where the number of
quantization levels is M = 16. The quantization distortion is specified not to exeed 1% of peak to peak analog signal.
(i) What would be the maximum number of bits per sample that should be used in this PCM system?
(ii) What is the minimum sampling rate and what is the resulting bit transmission rate?
Solution: (i) Since the number of quantization levels given here are M = 16,
q = M= 16
We know that the bits and levels in binary PCM are related as,
q = 2^v
Here, v = number of bits in a codeword
Thus, 16 = 2^v
or v = 4 bits. Ans.
(ii) Again since f[m] = 3 kHz
By sampling theorem, we know that
f[s] ≥ 2f[m]
Thus, f[s] ≥ 2 x 3 kHz ≥ 6 kHz Ans.
Hence, the minimum sampling rate is 6 kHz
Also bit transmission rate or signaling rate is given as,
r ≥ vf[s] ≥ 4 x 6 x 10^3
or r ≥ 24 x 10^3 bits per second Ans.
EXAMPLE 4.8. A signal having bandwidth equal to 3.5 kHz is sampled, quantized and coded by a PCM system. The coded signal is then transmitted over a transmission channel of supporting a transmission
rate of 50 k bits/sec. Determine the maximum signal to noise ratio that can be obtained by this system.
The input signal has peak to peak value of 4 volts and rms value of 0.2 V.
(Pune University-1998)
Solution: The maximum frequency of the signal is given as 3.5 kHz,
i.e., f[m] = 3.5 kHz
Therefore sampling frequency will be
f[s] ≥ 2f[m] ≥ 2 x 3.5 kHz ≥ kHz
We know that the signaling rate is given by
r ≥ vf[s]
Substituting values of r = 50 x 10^3 bits/sec and fs ≥ 7 x 10^3 Hz in above equation, we get
50 x 10^3 ≥ v. 7 x 10^3
Simplifying, we get
v ≤ 7.142 bits 8 bits
The rms value of the signal is 0.2 V. Therefore the normalized signal power will be,
Normalized signal power P =
i.e., P = 0.04 W
Further, the maximum signal to noise ratio is given by,
Substituting the values of P = 0.04, v = 8 and x[max] = 2 in above equation, we have
EXAMPLE 4.9. A signal x(t) is uniformly distributed in the range ±x[max]. Evaluate maximum signal to noise ratio for this signal.
Solution: Given that the signal is uniformly distributed in the range ±x[max], therefore we can write its PDF (using the Standard Uniform Distribution) as under:,
* R = 1 for normalized power.
Figure 4.13 shows this PDF,
The mean square valve of a random variable X is given as,
FIGURE 4.16 PDF of a uniformly distributed random variable.
Therefore, mean square value of x(t) will be,
EQUATION …(i)
The signal power
Normalized signal power [since R = 1]
Substituting the value of from (i), we get
We know that the relation between step size, maximum amplitude of signal and number of levles is given as
Step size
Therefore, normalized signal power, =
We also know that
Normalized noise power
Therefore, signal to noise power ratio
Since q = 2v, above equation will be,
or 6 v
This is required expression for maximum value of signal to noise ratio.
EXAMPLE 4.10. Given an audio signal consisting of the sinusoidal term given as
x(t) = 3 cos (500 )
(i) Determine the signal to quantizatibn noise ratio when this is quantized using 10-bit PCM.
(ii) How many bits of quantization are needed to achieve a signal to quantization noise ratio of atleast 40 dB?
Solution: Here, given that x(t) = 3 cos (500 t)
This is sinusoidal signal applied to the quantizer.
(i) Let us assume that peak value of cosine wave defined by x(t) covers the complete range of quantizer.
i.e., A[m] = 3V covers complete range
In example 4.1, we have derived signal to noise ratio for a sinusoidal signal. It is expressed as
Since here 10 bit PCM is used i.e.,
V= 10
Thus, = 1.8 + 6 X 10 = 61.8 dB Ans.
(ii) For sinusoidal signal, again, let us use the same relation
i.e., 1.8 + 6v dB
To get signal to noise ratio of at least 40 dB we can write above equation as,
1.8 + 6v ≥ 40 dB
Solving this, we get v ≥ 6.36 bits = 7 bits
Hence, at least 7 bits are required to get signal to noise ratio of 40 dB. Ans.
EXAMPLE 4.11. A 7 bit PCM system employing uniform quantization has an overall signaling to of 56 k bits per second. Calculate the signal to quantization noise that would result when its input is a
sine wave with peak amplitude equal to 5 Volt. Find the dynamic range the sine wave inputs in order that the signal to quantization noise ratio may be less 1n 30 dBs. What is the theoretical maximum
frequency that this system can handle?
(Madras University-1999)
Solution: The number of bits in the PCM system are
u = 7 bits
Assume hat 5 V peak to peak voltage utilizes complete range of quantizer. Then, we can find signal to quantization noise ratio as,
= 1.8 + 6v dB = 1.8 + 6 x 7 = 43.8 dB
We know that the signaling rate is given as,
r = v f[s]
Substituting r= 56 x 10^3 bits/second and v = 7 bits in above equation, we obtain 56 x 10^3 = 7.f[s]
Simplifying, we get
Sampling frequency, f[s] = 8 x 103 Hz
Further, using sampling theorem we have,
f[s] ≥ 2f[m]
Thus, maximum frequency that can be handled is given as, | {"url":"https://www.sbistudy.com/given-an-audio-signal-consisting-of-the-sinusoidal-term-given-as-xt-3-cos-500/","timestamp":"2024-11-06T09:31:00Z","content_type":"text/html","content_length":"177921","record_id":"<urn:uuid:c3bac0e1-9049-4ddc-8e3b-3c0a66af18b6>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00873.warc.gz"} |
class BaseDeepRegressor(batch_size=40, last_file_name='last_model')[source]¶
Abstract base class for deep learning time series regression.
The base classifier provides a deep learning default method for _predict, and provides a new abstract method for building a model.
batch_sizeint, default = 40
training batch size for the model
last_file_namestr, default = “last_model”
The name of the file of the last model, used only if save_last_model_to_file is used
build_model(input_shape) Construct a compiled, un-trained, keras model that is ready for training.
clone([random_state]) Obtain a clone of the object with the same hyperparameters.
fit(X, y) Fit time series regressor to training data.
fit_predict(X, y) Fits the regressor and predicts class labels for X.
get_class_tag(tag_name[, raise_error, ...]) Get tag value from estimator class (only class tags).
get_class_tags() Get class tags from estimator class and all its parent classes.
get_fitted_params([deep]) Get fitted parameters.
get_metadata_routing() Sklearn metadata routing.
get_params([deep]) Get parameters for this estimator.
get_tag(tag_name[, raise_error, ...]) Get tag value from estimator class.
get_tags() Get tags from estimator.
load_model(model_path) Load a pre-trained keras model instead of fitting.
predict(X) Predicts target variable for time series in X.
reset([keep]) Reset the object to a clean post-init state.
save_last_model_to_file([file_path]) Save the last epoch of the trained deep learning model.
score(X, y[, metric, metric_params]) Scores predicted labels against ground truth labels on X.
set_params(**params) Set the parameters of this estimator.
set_tags(**tag_dict) Set dynamic tags to given values.
summary() Summary function to return the losses/metrics for model fit.
abstract build_model(input_shape)[source]¶
Construct a compiled, un-trained, keras model that is ready for training.
The shape of the data fed into the input layer
A compiled Keras Model
Summary function to return the losses/metrics for model fit.
history: dict or None,
Dictionary containing model’s train/validation losses and metrics
Save the last epoch of the trained deep learning model.
file_pathstr, default = “./”
The directory where the model will be saved
Load a pre-trained keras model instead of fitting.
When calling this function, all functionalities can be used such as predict etc. with the loaded model.
model_pathstr (path including model name and extension)
The directory where the model will be saved including the model name with a “.keras” extension. Example: model_path=”path/to/file/best_model.keras”
Obtain a clone of the object with the same hyperparameters.
A clone is a different object without shared references, in post-init state. This function is equivalent to returning sklearn.clone of self. Equal in value to type(self)(**self.get_params
random_stateint, RandomState instance, or None, default=None
Sets the random state of the clone. If None, the random state is not set. If int, random_state is the seed used by the random number generator. If RandomState instance, random_state
is the random number generator.
Instance of type(self), clone of self (see above)
fit(X, y) BaseCollectionEstimator[source]¶
Fit time series regressor to training data.
Xnp.ndarray or list
Input data, any number of channels, equal length series of shape ( n_cases, n_channels, n_timepoints) or 2D np.array (univariate, equal length series) of shape (n_cases, n_timepoints)
or list of numpy arrays (any number of channels, unequal length series) of shape [n_cases], 2D np.array (n_channels, n_timepoints_i), where n_timepoints_i is length of series i. Other
types are allowed and converted into one of the above.
Different estimators have different capabilities to handle different types of input. If self.get_tag(“capability:multivariate”)` is False, they cannot handle multivariate series, so
either n_channels == 1 is true or X is 2D of shape (n_cases, n_timepoints). If self.get_tag( "capability:unequal_length") is False, they cannot handle unequal length input. In both
situations, a ValueError is raised if X has a characteristic that the estimator does not have the capability for is passed.
1D np.array of float, of shape (n_cases) - regression targets (ground truth) for fitting indices corresponding to instance indices in X.
Reference to self.
Changes state by creating a fitted model that updates attributes ending in “_” and sets is_fitted flag to True.
fit_predict(X, y) ndarray[source]¶
Fits the regressor and predicts class labels for X.
fit_predict produces prediction estimates using just the train data. By default, this is through 10x cross validation, although some estimators may utilise specialist techniques such as
out-of-bag estimates or leave-one-out cross-validation.
Regressors which override _fit_predict will have the capability:train_estimate tag set to True.
Generally, this will not be the same as fitting on the whole train data then making train predictions. To do this, you should call fit(X,y).predict(X)
Xnp.ndarray or list
Input data, any number of channels, equal length series of shape ( n_cases, n_channels, n_timepoints) or 2D np.array (univariate, equal length series) of shape (n_cases, n_timepoints)
or list of numpy arrays (any number of channels, unequal length series) of shape [n_cases], 2D np.array (n_channels, n_timepoints_i), where n_timepoints_i is length of series i. other
types are allowed and converted into one of the above.
Different estimators have different capabilities to handle different types of input. If self.get_tag(“capability:multivariate”)` is False, they cannot handle multivariate series, so
either n_channels == 1 is true or X is 2D of shape (n_cases, n_timepoints). If self.get_tag( "capability:unequal_length") is False, they cannot handle unequal length input. In both
situations, a ValueError is raised if X has a characteristic that the estimator does not have the capability for is passed.
1D np.array of float, of shape (n_cases) - regression targets (ground truth) for fitting indices corresponding to instance indices in X.
1D np.array of float, of shape (n_cases) - predicted regression labels indices correspond to instance indices in X
classmethod get_class_tag(tag_name, raise_error=True, tag_value_default=None)[source]¶
Get tag value from estimator class (only class tags).
Name of tag value.
raise_errorbool, default=True
Whether a ValueError is raised when the tag is not found.
tag_value_defaultany type, default=None
Default/fallback value if tag is not found and error is not raised.
Value of the tag_name tag in cls. If not found, returns an error if raise_error is True, otherwise it returns tag_value_default.
if raise_error is True and tag_name is not in self.get_tags().keys()
>>> from aeon.classification import DummyClassifier
>>> DummyClassifier.get_class_tag("capability:multivariate")
classmethod get_class_tags()[source]¶
Get class tags from estimator class and all its parent classes.
Dictionary of tag name and tag value pairs. Collected from _tags class attribute via nested inheritance. These are not overridden by dynamic tags set by set_tags or class __init__
Get fitted parameters.
State required:
Requires state to be “fitted”.
deepbool, default=True
If True, will return the fitted parameters for this estimator and contained subobjects that are estimators.
Fitted parameter names mapped to their values.
Sklearn metadata routing.
Not supported by aeon estimators.
Get parameters for this estimator.
deepbool, default=True
If True, will return the parameters for this estimator and contained subobjects that are estimators.
Parameter names mapped to their values.
get_tag(tag_name, raise_error=True, tag_value_default=None)[source]¶
Get tag value from estimator class.
Includes dynamic and overridden tags.
Name of tag to be retrieved.
raise_errorbool, default=True
Whether a ValueError is raised when the tag is not found.
tag_value_defaultany type, default=None
Default/fallback value if tag is not found and error is not raised.
Value of the tag_name tag in self. If not found, returns an error if raise_error is True, otherwise it returns tag_value_default.
if raise_error is True and tag_name is not in self.get_tags().keys()
>>> from aeon.classification import DummyClassifier
>>> d = DummyClassifier()
>>> d.get_tag("capability:multivariate")
Get tags from estimator.
Includes dynamic and overridden tags.
Dictionary of tag name and tag value pairs. Collected from _tags class attribute via nested inheritance and then any overridden and new tags from __init__ or set_tags.
predict(X) ndarray[source]¶
Predicts target variable for time series in X.
Xnp.ndarray or list
Input data, any number of channels, equal length series of shape ( n_cases, n_channels, n_timepoints) or 2D np.array (univariate, equal length series) of shape (n_cases, n_timepoints)
or list of numpy arrays (any number of channels, unequal length series) of shape [n_cases], 2D np.array (n_channels, n_timepoints_i), where n_timepoints_i is length of series i other
types are allowed and converted into one of the above.
Different estimators have different capabilities to handle different types of input. If self.get_tag(“capability:multivariate”)` is False, they cannot handle multivariate series, so
either n_channels == 1 is true or X is 2D of shape (n_cases, n_timepoints). If self.get_tag( "capability:unequal_length") is False, they cannot handle unequal length input. In both
situations, a ValueError is raised if X has a characteristic that the estimator does not have the capability for is passed.
1D np.array of float, of shape (n_cases) - predicted regression labels indices correspond to instance indices in X
Reset the object to a clean post-init state.
After a self.reset() call, self is equal or similar in value to type(self)(**self.get_params(deep=False)), assuming no other attributes were kept using keep.
Detailed behaviour:
removes any object attributes, except:
hyper-parameters (arguments of __init__) object attributes containing double-underscores, i.e., the string “__”
runs __init__ with current values of hyperparameters (result of get_params)
Not affected by the reset are:
object attributes containing double-underscores class and object methods, class attributes any attributes specified in the keep argument
keepNone, str, or list of str, default=None
If None, all attributes are removed except hyperparameters. If str, only the attribute with this name is kept. If list of str, only the attributes with these names are kept.
Reference to self.
score(X, y, metric='r2', metric_params=None) float[source]¶
Scores predicted labels against ground truth labels on X.
Xnp.ndarray or list
Input data, any number of channels, equal length series of shape ( n_cases, n_channels, n_timepoints) or 2D np.array (univariate, equal length series) of shape (n_cases, n_timepoints)
or list of numpy arrays (any number of channels, unequal length series) of shape [n_cases], 2D np.array (n_channels, n_timepoints_i), where n_timepoints_i is length of series i. other
types are allowed and converted into one of the above.
Different estimators have different capabilities to handle different types of input. If self.get_tag(“capability:multivariate”)` is False, they cannot handle multivariate series, so
either n_channels == 1 is true or X is 2D of shape (n_cases, n_timepoints). If self.get_tag( "capability:unequal_length") is False, they cannot handle unequal length input. In both
situations, a ValueError is raised if X has a characteristic that the estimator does not have the capability for is passed.
1D np.array of float, of shape (n_cases) - regression targets (ground truth) for fitting indices corresponding to instance indices in X.
metricUnion[str, callable], default=”r2”,
Defines the scoring metric to test the fit of the model. For supported strings arguments, check sklearn.metrics.get_scorer_names.
metric_paramsdict, default=None,
Contains parameters to be passed to the scoring function. If None, no parameters are passed.
MSE score of predict(X) vs y
Set the parameters of this estimator.
The method works on simple estimators as well as on nested objects (such as Pipeline). The latter have parameters of the form <component>__<parameter> so that it’s possible to update each
component of a nested object.
Estimator parameters.
selfestimator instance
Estimator instance.
Set dynamic tags to given values.
Dictionary of tag name and tag value pairs.
Reference to self. | {"url":"https://www.aeon-toolkit.org/en/latest/api_reference/auto_generated/aeon.regression.deep_learning.base.BaseDeepRegressor.html","timestamp":"2024-11-04T20:32:37Z","content_type":"text/html","content_length":"95334","record_id":"<urn:uuid:6f0fbc22-fe9b-4ba4-a2ef-91d060379d50>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00255.warc.gz"} |
How Do Data Scientists and Data Engineers Work Together? - KDnuggets
How Do Data Scientists and Data Engineers Work Together?
If you’re considering a career in data science, it’s important to understand how these two fields differ, and which one might be more appropriate for someone with your skills and interests.
Photo by fauxels via Pexels
How Do Data Scientists and Data Engineers Work Together?
Data scientists and data engineers are commonly confused by beginners without any significant experience in data science. And while their jobs might seem similar at a glance, there are actually some
significant underlying differences. If you’re considering a career in data science, it’s important to understand how these two fields differ, and which one might be more appropriate for someone with
your skills and interests.
What Does a Data Scientist Do?
Data scientists are involved with the direct analytical side of things. They work on models to process data, propose solutions to specific problems, and explore the limits of the data science domain
to look for appropriate ways to tackle challenges. The work of a data scientist involves a lot of math and a deep understanding of the statistical concepts behind data science. A strong mathematical
and statistical background is necessary to progress as a data scientist, and even to get hired by a reputable company.
What Does a Data Engineer Do?
A data engineer, on the other hand, is more concerned with the actual technical implementation of solutions. Once a scientist has come up with a model, it’s up to the engineer to figure out how to
integrate it into the overall data processing pipeline. Data engineers have to be careful to maintain a balance between accessibility, flexibility and performance of the systems they work on.
They also have to understand the tech stack they are working with as completely as possible. When a solution is to be implemented, it’s up to the data engineers to determine what languages,
databases, and other pieces of technology should be used to put together the final result. A good deal of scripting is usually required to tie everything together.
How the Two Roles Work Together?
A good way to look at data scientists and data engineers is via the analogy of architects and civil engineers. Architects are the ones that come up with the initial plans, while engineers implement
them while observing structural limitations and other similar points. It’s not too different in the world of data science. Data scientists plan and data engineers build and implement.
The two roles work closely together to come up with the final solutions though. It’s important to have good communication skills on both sides, because it’s often necessary to consolidate ideas and
limitations, and this has to be done in a way that doesn’t undermine anyone’s involvement in the project. Good pairs of data scientists and engineers can prove invaluable in the chaotic environment
that this work is usually done in.
Which Career Path Is Right for You?
Choosing whether you want to work as a data scientist or data engineer is important if you want to get involved with data science in general. If you enjoy math and exploring the theoretical concepts
in the field, working as a data scientist might be more suitable for you. You’ll need a good understanding of statistics, linear algebra, and various other mathematical fields. You’ll also need to go
through lots of published papers to gain a good understanding of how the field is tied together as a whole.
On the other hand, if you like to “get your hands dirty”, and often find yourself writing scripts to automate your work, rearranging parts of a pipeline to make it more efficient, and worrying about
technical limitations, then working as a data engineer might be right up your alley. This is a very technical field, and you don’t necessarily need a good understanding of the mathematical basics to
be successful in it. It can definitely help though.
Why It’s Worth Familiarizing Yourself with Both Sides
No matter which side you choose, it’s still a good idea to spend some time familiarizing yourself with concepts on both ends. A good data engineer must have at least some idea of how the models
they’re implementing came to be in the first place, while a good data scientist must be aware of the rough limitations they can expect to encounter. That’s why the best specialists in those fields
usually try to invest at least some effort into learning how the other side works. This will also prove very useful when trying to communicate a difficult concept.
Whether you’re going to focus on one side first and then pick up the other later, or if you’re going to spread your attention between the two initially is up to you. Both approaches can work, and
it’s a matter of personal preference. Whatever track you choose, Springboard has offerings in both that can deepen your understanding and get you job ready in the field you’re interested in.
Springboard’s Data Science Career Track is a good place to start your education if you are interested in that side of things.
Riley Predum has professionally worked in several areas of data such as product and data analytics, and in the realm of data science and data/analytics engineering. He has a passion for writing and
teaching and enjoys contributing learning materials to online communities focused on both learning in general as well as professional growth. Riley writes coding tutorials on his Medium blog. | {"url":"https://www.kdnuggets.com/2022/08/data-scientists-data-engineers-work-together.html","timestamp":"2024-11-11T06:40:39Z","content_type":"text/html","content_length":"221194","record_id":"<urn:uuid:d5c43885-f93a-402f-937a-831c8673b291>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00571.warc.gz"} |
Linear-time Merging
The remaining piece of merge sort is the merge function, which merges two adjacent sorted subarrays, $array[p \cdots q]$ and $array[q+1 \cdots r]$ into a single sorted subarray in $array[p \cdots r]$
. We’ll see how to construct this function so that it’s as efficient as possible. Let’s say that the two subarrays have a total of $n$ elements. We have to examine each of the elements in order to
merge them together, and so the best we can hope for would be a merging time of $\Theta(n)$. Indeed, we’ll see how to merge a total of $n$ elements in $\Theta(n)$ time.
In order to merge the sorted subarrays $array[p \cdots q]$ and $array[q+1 \cdots r]$ and have the result in $array[p \cdots r]$, we first need to make temporary arrays and copy $array[p \cdots q]$
and $array[q+1 \cdots r]$ into these temporary arrays. We can’t write over the positions in $array[p \cdots r]$ until we have the elements originally in $array[p \cdots q]$ and $array[q+1 \cdots r]$
safely copied.
The first order of business in the merge function, therefore, is to allocate two temporary arrays, lowHalf and highHalf, to copy all the elements in $array[p \cdots q]$ into lowHalf, and to copy all
the elements in $array[q+1 \cdots r]$ into highHalf. How big should lowHalf be? The subarray $array[p \cdots q]$ contains $q−p+1$ elements. How about highHalf? The subarray $array[q+1 \cdots r]$
contains $r−q$ elements. (In JavaScript, we don’t have to give the size of an array when we create it, but since we do have to do that in many other programming languages, we often consider it when
describing an algorithm.)
In our example array $[14, 7, 3, 12, 9, 11, 6, 2]$, here’s what things look like after we’ve recursively sorted $array[0 \cdots 3]$ and $array[4 \cdots 7]$ (so that p = 0, q = 3, and r = 7) and
copied these subarrays into lowHalf and highHalf:
Create a free account to access the full course.
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When I was at school, they taught us how electricity works only as part of science lessons. It was something future engineers might need, yet we all use electricity at home every day.
The problem with electricity is we’re a little bit separated from its cost. With cars, we fill up the car with fuel and pay for it right there and then. With electricity, we use many different
appliances which all add up to an eye-watering bill at the end of the month.
This is my guide to what everyone needs to know about electricity.
Introducing the kWh.
Electricity is sold in units of “kWh”. We’ll come to exactly what those three letters mean later on but for now, imagine your electricity is being delivered to you in barrels, each one a standard
size called the “kWh”. Think about your local electricity station and imagine one of these “kWh” barrels of electricity being hooked up to the wires that lead to your home. When a barrel empties,
someone comes along and replaces it with a new full barrel.
The “kWh” has a scientific definition that all electricity suppliers agree on. It is so ubiquitous that if any supplier decides to use a different unit, they’re most likely up to something dodgy.
How much is a single kWh barrel of electricity? Check your electric bill. Here’s mine…
The 45¾p per day standing charge is fixed. It doesn’t matter how much or how little I use; I still have to pay that 45¾p every single day and there’s little I can do about that other than maybe
switch providers.
More interesting is the 33p per kWh. At the end of each month, they count up all the empty barrels of electricity I’ve gone through and bill me 33p for each one. I’ll use that figure in my examples
but do look up your own rate and replace it with however much your kWh costs.
Also note that it doesn’t matter how quickly I go through each barrel of electricity. If I go away for a few days leaving everything except the fridge switched off, it will take a lot longer to
finish that barrel than when I’m home and everything is switched on. Either way, they still charge me 33p once that barrel is empty.
We’ll now pull apart those three letters, but always keep in mind that metaphor of barrels of electricity hooked up to the wires leading to your house.
Little barrels on the hillside.
Little barrels full of ‘tricity…
What Watt?
The W is short for the “Watt”, named after James Watt who invented them. If you’ve seen a capital W or “Watts” or “Wattage”, they all mean the same thing. The number of Watts any electrical appliance
has is a measure of the rate of consumption of electricity over time. If you like, think of it as the speed that something eats electricity coming out of the outlet on the wall.
This heater consumes electricity at a rate of 3000 Watts, or 1500 Watts if you use the low setting. Because one Wattage figure is twice as much as the other, you can safely assume that the high
setting consumes electricity exactly twice as fast as the low setting.
This lightbulb consumes electricity at a rate of 13.5 Watts, yet it shines as brightly as an old-fashioned 100-Watt filament lightbulb. Quite the improvement!
A quick exercise: Find an electrical item in your home and look up its Wattage figure. It might be on a label or written on the original packaging. If you can’t find it written down, try using a
search engine.
Ooh kay!
1 kW (or one kiloWatt) means exactly the same thing as 1000 W. Adding “k” to “W” to make “kW” means the amount is multiplied by one thousand. The heater above could have “3 kW” printed on the box
instead of “3000 W”. It would mean exactly the same thing.
Devices that draw a small amount of electricity like lightbulbs or phone chargers are usually rated in Watts, while larger devices that eat a lot of electricity like ovens or electric car chargers
are typically rated in kW. They mean the same thing underneath.
Whoever makes your electrical appliances might have a personal preference for small numbers in “kW” or big numbers in “W”. The manufacturer of that heater probably wants to emphasise how well it
heats, so they prefer to use the bigger number of “3000 W” instead of “3 kW”. More W equals more heat.
Our hours
The last letter is “h”, which is short for an “hour”, named after its inventor Sir Claudius Hour. (At least that’s what a man at the pub told me. He might have been joking.)
You know what an hour is, don’t you? It’s the time it takes to watch a normal episode of Star Trek with ads. It’s how long it takes me to walk all the way around my local country park if I don’t
stop. It’s the time it takes to walk my sister’s dog before she (the dog) gets tired.
“And I would walk 500 miles and I would walk 500 miles more.”
All together now!
Now we know what each letter of “kWh” stands for, let’s bring them all together. A “kWh” is the amount of electricity consumed by a 1000 W appliance if it is left on for an hour.
Find an appliance that’s rated at 1 kW. Plug it in and switch it on for an hour and then switch it off. You’ll have used exactly one kWh and your electricity bill will have gone up by 33p. (Or
whatever your supplier charges.)
Let’s work out a practical example. Recall that 3000W heater from earlier. How much do you think it costs to run that heater for five hours on the high setting? We’ll ignore practical realities like
the built-in thermostat and assume it goes for five hours straight with no gaps.
3000W is the same as 3 kW and we want to run it for 5 hours, paying 33p for each kWh. Multiply those numbers together:
3 kW × 5 h × 33 p/kWH = 495p (or roughly £5.)
Try this calculation yourself. Pick an electrical appliance in your home and find its rated wattage. Think about how long you switch it on for and work out how much it costs to use it for that amount
of time.
Applying the knowledge
It can be tempting to look at how much some appliances like heaters or ovens cost and conclude the only way to save money is to be cold and not eat. I hope that’s not the conclusion you draw. The
benefit of knowing how much something costs to use is that you can make informed choices.
Will buying an air fryer save you money when your kitchen already has an oven? Work out how much it costs to cook your favourite meal in the oven then do the same for an air fryer. If you know both
in actual pennies, you can make an informed decision to make that purchase or not.
While the Wattage figure tells you the rate it consumes electricity, it may be that the higher Wattage appliance gets the job done faster. Say you have a choice of two kettles, one runs at 1 kW and
the other at 3 kW, it may seem at first blush that the 1 kW kettle will cost less. However, if the 3 kW kettle gets the water boiled in a third of the time as the 1 kW kettle, they will cost the same
to use.
Does your supplier offer a different service with more expensive electricity during the day and cheaper electricity overnight? Which appliances would you use overnight when the kWh barrels are
cheaper? Would that save you money overall?
Many thanks to my wife and my brother Andrew for their helpful feedback. Thanks also to my local B&M store for the pictures of lightbulbs and heaters I took while shopping there.
Creative Commons Picture Credits:
📸 “saturday recycle” by Andrea de Poda.
📸 “sad kilo” by “p med”.
Digital photography is not rocket science. It just seems that way.
Here’s a TV advert for a camera touting the benefits of film cameras over digital cameras. I’m almost inclined to wonder if this advert is a parody, but even so, it has a point.
Let’s watch…
I’m reminded of when I was lending my digital camera to a friend some time ago. She knew how to use a film camera, but the technological revolution had, alas, left her behind.
She had no problem with the LCD display on the back. This was why she wanted to borrow my camera in the first place after she saw me using it. Taking a picture while holding the camera at arms length
is a lot easier than holding it up to the eye.
Showing her how to browse old pictures took a bit of teaching but she soon picked it up. It helped a lot that this camera had a big switch with only two settings; taking-pictures or
The big stumbling point was when I showed her how to use memory cards. I tried to explain how it stores pictures, but I got a lot of blank looks. I finally said “This card is like the film.” There
was a sudden look of understanding on her face.
The analogy to traditional film cameras worked perfectly. I told her that the photo shops will develop (print) her pictures, produce negatives (make a CD copy) and clean the film off to be reused
again. If she needed more film, she could buy some by asking for a “128 MB SD” at the shops (which might tell you when this story took place).
Embrace the metaphor!
Film cameras are devices that direct photon particles in order to induce chemical reactions in silver halide particles mounted on sheets of cellulose acetate.
Somehow, the camera industry managed to sell us cameras without having to give us chemistry lessons first. And yet, we all need computer science lessons to use digital cameras. People never really
cared about the chemical processes of film photography and we shouldn’t have to care about bits, megabytes and other pieces of jargon that can be abstracted away.
So, here are my suggestions for the digital camera industry.
1. Standardise!
Why are there so many memory card formats? As far as I can tell, they’re all flash memory chips contained in differently shaped blobs of plastic. The industry needs to pick one shape of blob and
stick with it. No inventing new blobs unless there’s a really good reason to.
2. Call memory cards, ‘digital film’.
Embrace all the metaphors. If the world already has a name for something, don’t come up with a different name for it.
3. Tell me how many pictures it can store, not how many gigabytes.
This one will be tricky, as the size of a picture depends on the number of pixels. So while I don’t think we could realistically get rid of the “GB”, cameras need to help the user by telling us how
many pictures are in a “GB” at that particular time.
4. Cameras should come with a reasonably sized card as standard.
How would you feel if you bought a camera, but later found the lens was extra? Digital film (getting used to the phrase yet?) is reusable and will probably last as long as the camera itself. So why
not bundle it with the camera and save your customers the hassle.
5. Photo printing shops to provide archival DVDs as a standard part of the service.
People using film cameras expected their negatives as part of the service. Copying a few gigabytes full of pictures to a DVD should be cheap enough that it could be offered free to anyone who wants
to print a vacation’s worth of snaps.
Hang on, did that advert just say two cameras for ten dollars? Forget everything I just wrote, that’s a bargain!
Picture credits:
‘Film and SD card’ by ‘sparkieblues’ of flickr
‘Leica’ by ‘AMERICANVIRUS’ of flickr
Paying for Power
Being an evil genius, I’m obsessed with getting as much power as possible. If only I could get power for nothing, but alas, I have to pay for it.
In England, and most of the western world, we have a well established system of sending electricity from the power stations to me and sending money in the opposite direction. It works, but I think we
can improve on it.
An Electron’s Journey
Electricity starts life at the power stations. They sell their power supply on the grid at market rates, competing with other power stations. The price of electricity fluctuates over the day. If the
price goes down far enough, they might switch off the generators, keeping their raw materials for when the price goes up. A wind farm can’t keep stocks of wind in reserve, so they will stay online
all the time regardless of the price.
We, the public, never see those fluctuations in price. Instead, we purchase electricity from a supplier who deal with the power stations. The suppliers usually charge us a fixed amount per unit of
energy, sometimes having a daytime rate and an overnight rate, but the price they charge us is fairly stable, only changing the rates every few months.
(As well as the suppliers, we also pay the companies that maintain the grid system and meters in our homes. This article is not about them.)
When all is said and done, what do the suppliers actually do? They don’t generate the electricity and they don’t bring it to us. They are middle-men who flatten out the price, charging a bit more
than the expected average price, like an insurance premium, to compensate for the risk of over-demand and price rises. Do we need that service? We have insurance to spread the risk of unexpected
events, not for the everyday costs of life.
What if, instead, we had a minimal supplier that just handles the accountancy at a low cost, quoting a price that changes every five minutes, tracking the wholesale price. (Perhaps having an easy to
use gizmo that displays the current price.) With this type of supplier, we would probably save money over the long term. After all, we wouldn’t be paying that insurance premium any more.
But more important than that, it would give us an interest in when we use electricity. At the moment, we really don’t care that the price of electricity rises dramatically during the adverts on
popular TV shows. We all switch on our kettles at the same time, not really caring about the economics. If we felt the rise in price, we might plan our tea making better to avoid these peaks and save
some money.
This plan wouldn’t have worked when the grid was originally built, but computer and communications technology have advanced to point where we can finally think about pulling down the old ways of
working. I’m looking forward to it.
Picture credits.
Nuclear power by koert michiels on flickr.
insurance prohibits ladders by stallio on flickr. | {"url":"https://billpg.com/category/economics/","timestamp":"2024-11-05T06:34:05Z","content_type":"text/html","content_length":"100898","record_id":"<urn:uuid:7c6379cb-bd4d-41c8-9120-66f43c101f56>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00880.warc.gz"} |
Blog: Logs for the MRL Meeting Held on 2019-10-28
Logs for the MRL Meeting Held on 2019-10-28
October 28, 2019
<sarang> GREETINGS
<sgp_> hello!
<sarang> I'll give a few moments for others who wish to join
<sarang> OK then
<sarang> Since suraeNoether is unavailable for this meeting due to an appointment, I'll share my recent work
<sarang> I've been working on algorithms and proofs for Triptych, a new transaction protocol
<sarang> The goal is to use a single proof to represent multiple inputs at the same time, including balance proving and linking tags
<sarang> Everything works great with completeness, zero knowledge, and soundness except for one proof component (the linking tags)
<sarang> There's a less efficient version that operates on single inputs, but can be combined for general transactions
<sarang> For this single-input version, modified proofs of security seem to work just fine
<sarang> For this reason, I'll finalize work on the single-input proving system while considering alternate approaches to finalizing the soundness proof for the multi-input version
<sarang> Separately from this, I have a small pull request (PR 6049) for a minor speedup and simplification to the Bulletproofs prover
<sarang> Also separately from this, Derek at OSTIF informs me that an audit group is willing to complete the CLSAG review
<sarang> JP Aumasson has offered to complete a review of the math and proofs for $7200 (USD), and his new company Teserakt has offered to then complete a code review for as little as $4800
<sarang> He says that including dependencies would increase the time (and therefore the cost), possibly significantly
<sarang> But the timeline could be before the end of this year, if there are no changes required to the algorithms after the math review
<moneromooo> Dependencies, like the src/crypto code ?
<sarang> Presumably. I do not have specific details on what his scope is (but will get this information)
<sarang> One approach might be to review all the changes _from MLSAG_, to show that CLSAG is no less secure as a whole than MLSAG
<sarang> These changes are fairly minor in the grand scope of the codebase
<sarang> I see there being efficiency advantages to having JP (and colleagues) doing both types of review, but this also reduces the total number of eyes on the combined math+code
<sarang> That being said, JP knows his stuff
<sarang> (he was formerly with Kudelski)
<moneromooo> Adding eyes by having Alice do the math and Bob do the code does not provide anything of value over Alice doing both IMHO.
<moneromooo> Assuming Alice and Bob have similar eyes and brains and proficiency in the relevant fields etc etc etc.
<sarang> So that's my report
<moneromooo> Is any of the new protocols being considered still compatible with multisig ?
<sarang> Aside from CLSAG, you mean?
<sarang> None of them specifically consider it in either algorithms or security model
<sarang> but it's on my list for analysis on RCT3 and (eventually) Triptych, since there are some modifications to RCT3 that I wish to consider (more on this later)
<moneromooo> I mean tryptich, rct3 and… and………. the other the name of which escapes me.
<moneromooo> lelantus
<sarang> Omniring?
<moneromooo> Also :)
<sarang> Omniring and Lelantus both suffer from some drawbacks at present… Omniring does not support batching, and Lelantus still has a tracing issue unless you remove stealth addressing
<sarang> Looking into batch-compatible Omniring-style constructions with other proving systems is a topic for more investigation down the road that is nontrivial
<sarang> Is there other research that anyone wishes to present, or other questions?
<moneromooo> Also, rather selfishly, would any of them avoid the public-a issue we had for multi user txes ?
<moneromooo> (if known offhand)
<sarang> public-a?
<moneromooo> The problem where users would have to make their a values known to other signers.
<sarang> Ah, that's very unclear to me
<sarang> FWIW: RCT3, Omniring, and Triptych are agnostic to how output keys are generated (though their security models address particular constructions)
<sarang> So my ACTION ITEMS for this week are a bit in flux, mainly because I'll be at World Crypto Conference giving a talk on transaction protocols
<sarang> But aside from that, I want to finish the proof modifications (completeness, SHVZK, special soundness) for the single-input version of Triptych (which can be used in a larger protocol to
support multi-input transactions), as well as a more efficient linking tag construction that matches what RCT3 and Omniring propose
<sarang> I also want to backport some of the ideas from the latest RCT3 update to their older version to compare efficiency
<sarang> It's unclear if this could easily be proven secure, or what the efficiency gains would be
<sarang> Their update did essentially two things: fix an exploitable flaw due to a particular discrete log relation, and allow for aggregated proofs of multiple inputs
<sarang> Unfortunately, the latter means potentially large padding requirements that would also incur computational cost to the verifier
<sarang> I want to see how easily the exploit fix could be included in the non-aggregated version… which would avoid this potential verification bloat at the cost of proof size
<sarang> I probably won't have time to do so this week, but it's on my list
<sarang> Anything else of note to cover before we formally adjourn?
<sarang> All right! Thanks to everyone for attending
<sarang> Logs will be posted shortly to the GitHub agenda issue
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A modified EM algorithm for parameter estimation in linear models with time-dependent autoregressive and t-distributed errors
Original language English
Title of host publication ITISE 2017
Pages 1132-1145
ISBN (electronic) 978-84-17293-01-7
Publication status Published - 2017
We derive an expectation conditional maximization either (ECME) algorithm for estimating jointly the parameters of a linear regression model, of a time-variable autoregressive (AR) model with respect
to the random deviations, and of a scaled t-distribution with respect to the white noise components. This algorithm is shown to take the form of iteratively reweighted least squares in the estimation
of the parameters both of the regression and time-variability model. The fact that the degree of freedom of that distribution is also estimated turns the algorithm into a partially adaptive
estimator. As low degrees of freedom correspond to heavy-tailed distributions, the estimator can be expected to be robust against outliers. It is shown that the initial stabilization phase of an
accelerometer on a shaker table can be modeled parsimoniously and robustly by a Fourier series with AR errors for which the time-variability model is defined by cubic polynomials.
Linear regression model, time-dependent AR process, partially adaptive estimation, robust parameter estimation, EM algorithm, iteratively reweighted least squares, scaled t-distribution
Cite this
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title = "A modified EM algorithm for parameter estimation in linear models with time-dependent autoregressive and t-distributed errors",
abstract = "We derive an expectation conditional maximization either (ECME) algorithm for estimating jointly the parameters of a linear regression model, of a time-variable autoregressive (AR) model
with respect to the random deviations, and of a scaled t-distribution with respect to the white noise components. This algorithm is shown to take the form of iteratively reweighted least squares in
the estimation of the parameters both of the regression and time-variability model. The fact that the degree of freedom of that distribution is also estimated turns the algorithm into a partially
adaptive estimator. As low degrees of freedom correspond to heavy-tailed distributions, the estimator can be expected to be robust against outliers. It is shown that the initial stabilization phase
of an accelerometer on a shaker table can be modeled parsimoniously and robustly by a Fourier series with AR errors for which the time-variability model is defined by cubic polynomials.",
keywords = "Linear regression model, time-dependent AR process, partially adaptive estimation, robust parameter estimation, EM algorithm, iteratively reweighted least squares, scaled t-distribution",
author = "Boris Kargoll and Mohammad Omidalizarandi and Hamza Alkhatib and Wolf-Dieter Schuh",
year = "2017",
language = "English",
pages = "1132--1145",
booktitle = "ITISE 2017",
TY - GEN
T1 - A modified EM algorithm for parameter estimation in linear models with time-dependent autoregressive and t-distributed errors
AU - Kargoll, Boris
AU - Omidalizarandi, Mohammad
AU - Alkhatib, Hamza
AU - Schuh, Wolf-Dieter
PY - 2017
Y1 - 2017
N2 - We derive an expectation conditional maximization either (ECME) algorithm for estimating jointly the parameters of a linear regression model, of a time-variable autoregressive (AR) model with
respect to the random deviations, and of a scaled t-distribution with respect to the white noise components. This algorithm is shown to take the form of iteratively reweighted least squares in the
estimation of the parameters both of the regression and time-variability model. The fact that the degree of freedom of that distribution is also estimated turns the algorithm into a partially
adaptive estimator. As low degrees of freedom correspond to heavy-tailed distributions, the estimator can be expected to be robust against outliers. It is shown that the initial stabilization phase
of an accelerometer on a shaker table can be modeled parsimoniously and robustly by a Fourier series with AR errors for which the time-variability model is defined by cubic polynomials.
AB - We derive an expectation conditional maximization either (ECME) algorithm for estimating jointly the parameters of a linear regression model, of a time-variable autoregressive (AR) model with
respect to the random deviations, and of a scaled t-distribution with respect to the white noise components. This algorithm is shown to take the form of iteratively reweighted least squares in the
estimation of the parameters both of the regression and time-variability model. The fact that the degree of freedom of that distribution is also estimated turns the algorithm into a partially
adaptive estimator. As low degrees of freedom correspond to heavy-tailed distributions, the estimator can be expected to be robust against outliers. It is shown that the initial stabilization phase
of an accelerometer on a shaker table can be modeled parsimoniously and robustly by a Fourier series with AR errors for which the time-variability model is defined by cubic polynomials.
KW - Linear regression model, time-dependent AR process, partially adaptive estimation, robust parameter estimation, EM algorithm, iteratively reweighted least squares, scaled t-distribution
M3 - Conference contribution
SP - 1132
EP - 1145
BT - ITISE 2017
ER - | {"url":"https://www.fis.uni-hannover.de/portal/en/publications/a-modified-em-algorithm-for-parameter-estimation-in-linear-models-with-timedependent-autoregressive-and-tdistributed-errors(59090c73-b64c-4f13-8f86-baa8f6551a41).html","timestamp":"2024-11-03T01:16:14Z","content_type":"text/html","content_length":"50062","record_id":"<urn:uuid:08c84962-16da-4788-9594-00eff82a6f5c>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00142.warc.gz"} |