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Epistemic Network Analysis
Resources on Epistemic Network Analysis (ENA), the ENA webtool, and rENA
ENA Demos
ENA of COVID Press Conferences
This video:
• Compares press conferences held by Prime Minister of New Zealand Jacinda Ardern and the President of the United States Donald Trump on the novel Coronavirus (SARS-CoV-2)
• Visualizes how ENA networks are constructed
Shaffer, D. W., Collier, W., & Ruis, A. R. (2016). A tutorial on epistemic network analysis: Analyzing the structure of connections in cognitive, social, and interaction data. Journal of Learning
Analytics, 3(3), 9-45.
This paper covers, among others:
• The theory behind ENA
• The process by which ENA creates network models
• Example analyses using the ENA web tool
Bowman, D., Swiecki, Z., Cai, Z., Wang, Y., Eagan, B., Linderoth, J., & Shaffer, D. W. (2021, February). The Mathematical Foundations of Epistemic Network Analysis. In International Conference on
Quantitative Ethnography (pp. 91-105). Springer, Cham.
This paper covers, among others:
• The transformations that constitute ENA in formal mathematical terms
Wang, Y., Swiecki, Z., Ruis, A. R., & Shaffer, D. W. (2021, February). Simplification of Epistemic Networks Using Parsimonious Removal with Interpretive Alignment. In International Conference on
Quantitative Ethnography (pp. 137-151). Springer, Cham.
This paper covers, among others:
• A formalized way for code removal in an ENA model
Brohinsky, J., Marquart, C., Wang, J., Ruis, A. R., & Shaffer, D. W. (2021, February). Trajectories in Epistemic Network Analysis. In International Conference on Quantitative Ethnography (pp.
106-121). Springer, Cham.
This paper covers, among others:
• Development of ENA trajectories
• Challenges and design decisions associated with visualizing ENA trajectory models
Fogel, A., Swiecki, Z., Marquart, C., Cai, Z., Wang, Y., Brohinsky, J., … & Shaffer, D. W. (2021, February). Directed Epistemic Network Analysis. In International Conference on Quantitative
Ethnography (pp. 122-136). Springer, Cham.
This paper covers, among others:
• Development of directed ENA graphs
Workshop Materials | {"url":"https://www.qesoc.org/epistemic-network-analysis/","timestamp":"2024-11-07T09:32:31Z","content_type":"text/html","content_length":"24934","record_id":"<urn:uuid:29597942-fe48-4083-bedf-4f476e53aeb3>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00039.warc.gz"} |
5 PROBLEMS ON FINANCE : TEST LEVEL WITH NEW DATA :
Q1 :
1. Brite Lighting Corporation wants to investigate the effect on its cost of capital based on the rate at which the company is taxed. The firm wishes to maintain a capital structure of 30% debt, 10%
preferred stock, and 60% common stock. The cost of financing with retained earnings is 14% (i.e., r[s] = 14%), the cost of preferred stock financing is 9% (r[ps] = 9%), and the before-tax cost of
debt is 11% (r[d] = 11%). Calculate the weighted average cost of capital (WACC) given the tax rate assumptions in parts (a) to (c) below.
(a) Tax rate = 40%.
(b) Tax rate = 35%.
(c) Tax rate = 25%.
1. Q2 :
2. Westerly Manufacturing has compiled the information shown in the following table:
Source of Capital Book Value Market Value After-tax Cost
Long-Term Debt $4,000,000 $3,840,000 6.0%
Preferred Stock $40,000 $60,000 13.0%
Common Stock Equity $1,060,000 $3,000,000 17.0%
Totals $5,100,000 $6,900,000
(a) Calculate the firm’s weighted average cost of capital (WACC) using book value weights.
(b) Calculate the firm’s weighted average cost of capital (WACC) using market value weights.
(c) Compare your answers found in parts (a) and (b) and briefly explain the differences. Other things equal, would you recommend that Westerly Manufacturing rely on its book value weights or
market value weights in determining its WACC?
Q3 :
1. To help finance a major expansion, Delano Development Company sold a noncallable bond several years ago that now has 15 years to maturity. This bond has a 10.25% annual coupon, paid semiannually,
it sells at a price of $1,025, and it has a par value of $1,000. If Delano’s tax rate is 40%, what component cost of debt should be used in the WACC calculation?
Q4 :
Roxie Epoxy’s balance sheet shows a total of $50 million long-term debt with a coupon rate of 8.00% and a yield to maturity of 7.00%. This debt currently has a market value of $55 million. The
balance sheet also shows that that the company has 20 million shares of common stock, and the book value of the common equity (common stock plus retained earnings) is $65 million. The current stock
price is $8.25 per share; stockholders’ required return, r[s], is 10.00%; and the firm’s tax rate is 40%. Based on market value weights, and assuming the firm is currently at its target capital
structure, what WACC should Roxie use to evaluate capital budgeting projects?
Q5 :
Bruner Breakfast Foods’ (BBF) balance sheet shows a total of $20 million long-term debt with a coupon rate of 8.00% (assume each bond to have a maturity value, M, of $1,000). The yield to maturity on
this debt is 10.00%, and the debt has a total current market value of $18 million. The balance sheet also shows that that the company has 10 million shares of stock, and total of common equity
(common stock plus retained earnings) is $30 million. The current stock price is $4.50 per share, and stockholders’ required rate of return, r[s], is 12.25%. The company recently decided that its
target capital structure should have 50% debt, with the balance being common equity. The tax rate is 40%. Calculate WACCs based on target, book, and market value capital structures (Note: I am asking
for three (3) separate WACC values here). | {"url":"https://crystalesssays.com/5-problems-on-finance-test-level-with-new-data/","timestamp":"2024-11-13T11:26:55Z","content_type":"text/html","content_length":"75347","record_id":"<urn:uuid:41621b4e-8e86-4e75-a2ad-dab3a2b5cd4f>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00028.warc.gz"} |
Nails - logic puzzle
How can you put (without using glue or something like that) ten nails on the head of a standing nail? The final construction should be stable.
In the following way, you'll build a stable construction:
First, put down nine nails like in the picture below.
Then put one nail across.
Lift the whole construction gently, and put it on the head of the standing nail.
If you did it right, you'll see that this construction is pretty stable. | {"url":"http://www.pzzls.com/nails_puzzle/answer.html","timestamp":"2024-11-05T07:33:31Z","content_type":"application/xhtml+xml","content_length":"11953","record_id":"<urn:uuid:d329e252-fe92-40f0-a956-ca9545253d34>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00121.warc.gz"} |
Hyperbolic Paraboloid as a Ruled Surface
Translating and rotating a straight line generates a hyperbolic paraboloid. Therefore, it is a ruled surface, that is, there is a line on the surface through any given point; in fact there are two li
nes through each point. The intersection of the surface and a horizontal plane is a hyperbola. | {"url":"https://www.wolframcloud.com/objects/demonstrations/HyperbolicParaboloidAsARuledSurface-source.nb","timestamp":"2024-11-12T01:52:07Z","content_type":"text/html","content_length":"191691","record_id":"<urn:uuid:5a1005db-9309-438a-b7d8-eb0e6afcade7>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00406.warc.gz"} |
How do you find the volume of a sphere with a diameter of 2 inches? | Socratic
How do you find the volume of a sphere with a diameter of 2 inches?
1 Answer
The volume of a sphere is $\frac{4}{3} \cdot \pi \cdot {r}^{3}$
Since the sphere's diameter is 2 inches, the radius would be 1 inch because the radius is always half of the diameter's value.
$\frac{4}{3} \cdot {1}^{3} \cdot \pi$
$\frac{4}{3} \cdot \pi$
If you're leaving the answer in terms of pi, then the answer would be $\frac{4}{3} \pi$ inches cubed
If you're using 3.14 for pi, then the answer would be $4.18 \overline{6}$ inches cubed
If you're using 3.14159 for pi, then the answer would be $4.18878 \overline{6}$ inches cubed
Impact of this question
4014 views around the world | {"url":"https://socratic.org/questions/how-do-you-find-the-volume-of-a-sphere-with-a-diameter-of-2-inches","timestamp":"2024-11-03T01:16:42Z","content_type":"text/html","content_length":"33555","record_id":"<urn:uuid:701206a3-3ba7-4767-af10-de4dfdfa93ec>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00831.warc.gz"} |
Research update: Towards a Law of Iterated Expectations for Heuristic Estimators
Last week, ARC released a paper called Towards a Law of Iterated Expectations for Heuristic Estimators, which follows up on previous work on formalizing the presumption of independence. Most of the
work described here was done in 2023.
A brief table of contents for this post:
In "Formalizing the Presumption of Independence", we defined a heuristic estimator to be a hypothetical algorithm that estimates the values of mathematical expression based on arguments. That is, a
heuristic estimator is an algorithm \(\mathbb{G}\) that takes as input
• A formally specified real-valued expression \(Y\); and
• A set of formal "arguments" \(\pi_1, \ldots, \pi_m\) –
– and outputs an estimate of the value of \(Y\) that incorporates the information provided by \(\pi_1, \ldots, \pi_m\). We denote this estimate by \(\mathbb{G}(Y \mid \pi_1, \ldots, \pi_m)\).^[1]
In that paper, we introduced the following question: is there a computationally efficient heuristic estimator that formalizes intuitively valid reasoning about the values of mathematical quantities
based on arguments? We studied the question by introducing intuitively desirable coherence properties (one such property is linearity: a heuristic estimator's estimate of \(X+Y\) should equal its
estimate of \(X\) plus its estimate of \(Y\)) and working to satisfy those properties. Ultimately, we left the question open.
The main technical contribution of our new work is to outline a new type of coherence property: a heuristic estimator should not be able to predict its own errors. We call this intuitive statement
the principle of unpredictable errors. The principle is loosely inspired by the law of iterated expectations from probability theory, as well as the martingale property: a Bayesian reasoner's
estimate of their future estimate of a quantity should be equal to its current estimate. One of the main purposes of this work is to explore ways to formalize this principle.
Our paper is structured as follows:
• We begin by explaining the core motivating intuition behind heuristic estimators through three analogies: proof verification, conditional expectations, and subjective probabilities.
• We explain why we believe in the principle of unpredictable errors. We then describe a natural attempt to formalize the principle: to simplify, we ask that \(\mathbb{G}(Y - \mathbb{G}(Y \mid \Pi)
\mid \Pi) = 0\) for every \(\)expression \(Y\) and set of arguments \(\Pi = {\pi_1, \ldots, \pi_m}\). (We call this property iterated estimation and also define a more complex property which we
call error orthogonality.) We then discuss important drawbacks of this formalization, which stem from the nested \(\mathbb{G}\)'s in the definition of the properties.
• Taking inspiration from these properties, we define a cluster of accuracy properties, which – roughly speaking – replace the outer \(\mathbb{G}\) with an expected value over a distribution of
expressions \(Y\). The simplest of these properties states that \(\)\(\mathbb{E}_{Y \sim \mathcal{D}}[Y - \mathbb{G}(Y \mid \Pi) \mid \Pi] = 0\).
• We examine the accuracy properties in the context of two estimation problems: (1) estimating the expected product of jointly normal random variables and (2) estimating the permanent of a matrix.
In both cases, we encounter barriers to satisfying the accuracy properties, even when the set of heuristic arguments is small and simple. This leads us to reject accuracy as a formalization of
the principle of unpredictable errors. We leave open the question of how to correctly formalize this principle.
• We conclude with a discussion of our motivations for pursuing this line of research. While the problem of heuristic estimation is deeply interesting from a theoretical standpoint, we believe that
it could have applications for understanding the behavior of neural networks. We discuss three potential applications of heuristic estimation to understanding neural network behavior: mechanistic
anomaly detection,^[2] safe distillation, and low probability estimation.
This blog post summarizes the paper, with proportionally more emphasis on the main ideas and less emphasis on the mathematical details.
What is a heuristic estimator?
In "Formalizing the Presumption of Independence", we described a heuristic estimator as an efficient program that forms a "subjective expected value" of a quantity based on arguments. We gave several
examples of heuristic estimation, such as estimating the number of twin primes in a given range and estimating the probability that some 256-bit input to the SHA-256 circuit has an all-zeros output.
In our new paper, we expand on the intuition behind heuristic estimators through one example and three analogies.
Example: Sum of sixth digits of square roots
Let \(d_6(\sqrt{n})\) denote the sixth digit of \(\sqrt{n}\) past the decimal point (in base 10), and let \(Y := \sum_{n = 101}^{120} d_6(\sqrt{n})\). What is your best guess for the value of \(Y\)?
Without actually calculating any square roots, your best bet is to estimate each of the twenty digits as \(4.5\) (the average of 0 through 9); this gives an estimate of \(20 \cdot 4.5 = 90\). This is
perhaps how we want our heuristic estimator to behave when given no arguments; in other words, we want our heuristic estimator \(\mathbb{G}\) to satisfy \(\mathbb{G}(Y \mid \emptyset) = 90\).^[3]
Now, let \(\pi_n\) be a computation of the sixth digit of \(\sqrt{n}\). When given \(\pi_n\) as an argument, \(\mathbb{G}\) should update its estimate of \(Y\) accordingly. For example, the sixth
digit of \(\sqrt{101}\) happens to be 5. Correspondingly, we would like \(\mathbb{G}(Y \mid \pi_{101}) = 5 + 19 \cdot 4.5 = 90.5\). If \(\mathbb{G}\) is additionally given \(\pi_{102}\), which shows
that the sixth digit of \(\sqrt{102}\) is 4, \(\mathbb{G}\) should again update its estimate: \(\mathbb{G}(Y \mid \pi_{101}, \pi_{102}) = 5 + 4 + 18 \cdot 4.5 = 90\) – and so on. If \(\mathbb{G}\) is
given all of \(\pi_{101}\) through \(\pi_{120}\), then it should be able to compute the correct answer.
(The purpose of this example is to provide a very simple and intuitive picture of how we expect \(\mathbb{G}\) to update based on arguments. In practice, we expect the arguments given to \(\mathbb{G}
\) to be much more complex.)
Analogy #1: Proof verification
A proof verifier is a program that takes as input a formal mathematical statement and a purported proof of the statement, and checks whether the proof is valid. This is very similar to a heuristic
estimator, which is a program that takes as input a formal mathematical expression and some arguments about the expression, and outputs an estimate of the value of the expression in light of those
Just as a proof verifier does not attempt to generate its own proof of the statement – it just checks whether the given proof is valid – a heuristic estimator does not attempt to estimate the given
quantity using its own arguments. Its only purpose is to incorporate the arguments that it is given into an estimate. Moreover, we can think of a heuristic estimator as a generalized proof verifier:
we expect heuristic estimators to respect proofs, in the sense that if an argument \(\pi\) proves that \(\ell \le Y \le h\), then \(\mathbb{G}(Y \mid \pi)\) should lie between \(\ell\) and \(h\).
(See Section 4 here.)
This table (adapted from chapter 9 here) illustrates the analogy between proof verifiers and heuristic estimators in more detail.
Heuristic estimation Proof verification
Heuristic estimator Proof verifier
Formal mathematical expression Formal mathematical statement
List of heuristic arguments Purported proof of statement
Formal language for heuristic arguments Formal language for proofs
Desiderata for estimator Soundness and completeness
Algorithm's estimate of expression Verifier's output (accept or reject)
Analogy #2: Conditional expectation
In some ways, a heuristic estimator is analogous to a conditional expected value. For a random variable \(X\) and event \(A\), \(\mathbb{E}[X \mid A]\) is the average value of \(X\) conditioned on \
(A\) – or put otherwise, it is the estimate of \(X\) given by an observer who knows that \(A\) occurred and nothing else. Similarly, \(\mathbb{G}(Y \mid \Pi)\) is the estimate of \(Y\) given by an
observer who has not computed the exact value of \(Y\) and has instead only done the computations described in the arguments in \(\Pi\). Although there is a particular correct value of \(Y\), the
observer does not know this value, and \(\mathbb{G}(Y \mid \Pi)\) is a subjective "best guess" about \(Y\) given only \(\Pi\). Both quantities can thus be thought of as a best guess conditional on a
state of knowledge.
Analogy #3: Subjective probabilities and estimates
Perhaps the best intuitive explanation of heuristic estimation is this: a heuristic estimator is a procedure that extracts a subjective expectation from a state of knowledge. Under this view, \(\Pi\)
formally describes a set of facts known by an observer, and \(\mathbb{G}(Y \mid \Pi)\) is a subjective estimate of \(Y\) in light of those facts.
By "subjective expectation", we mean "expected value under the subjectivist view of probability". The subjectivist view of probability interprets probability as the subjective credence of an
observer. For example, suppose that I have chosen a random number \(p\) uniformly from \([0, 1]\), minted a coin that comes up heads with probability \(p\), and then flipped it. What is the
probability that the coin came up heads?
To an observer who only knows my procedure (but doesn't know \(p\)), the subjective probability that the coin came up heads is \(\frac{1}{2}\). To an observer who knows \(p\) (but hasn't seen the
outcome of the coin flip), the subjective probability that the coin came up heads is \(p\). And to an observer who saw the outcome of the coin flip, the probability is either 0 (if the coin came up
tails) or 1 (if it came up heads).
Much as observers can have subjective probabilities, they can also have subjective expectations. For example, a typical mathematician does not know the 6th digit past the decimal point of \(\sqrt
{101}\), but would subjectively assign a uniform probability to each of \(0, \ldots, 9\), which means that their subjective expectation for the digit is \(4.5\). Recalling our example from earlier,
the mathematician's subjective expectation for \(Y := d_6(\sqrt{101}) + \ldots + d_6(\sqrt{120})\) is \(20 \cdot 4.5 = 90\). But if the mathematician were to learn that \(d_6(\sqrt{101}) = 5\), they
would update their subjective expectation to \(90.5\). This is exactly how we want our heuristic estimator \(\mathbb{G}\) to operate.
How can heuristic estimation help us understand neural networks?
While heuristic estimation is a deep and interesting topic in its own right, our research is primarily motivated by potential applications to understanding neural network behavior.
Historically, researchers mostly understood neural network behavior through empirical observation: the model's input-output behavior on particular inputs. However, this approach has important
drawbacks: for example, any understanding gained about a model's behavior on one input distribution may not carry over to a different input distribution.
More recently, there has been substantial work on neural network interpretability via understanding how models internally represent concepts. Interpretability research aims to address the barriers
faced by methods that rely only on input-output behavior. However, current interpretability techniques tend to only work under strong assumptions about how neural networks represent information (such
as the linear representation hypothesis). Also, for the most part, these techniques can only work insofar as neural representations of concepts are understandable to humans.
A different approach is formal verification: formally proving properties of neural networks such as accuracy or adversarial robustness. While formal verification does not rely on human understanding,
we believe that formally proving tight bounds about interesting behaviors of large neural networks is out of reach.
By contrast, heuristic arguments about properties of neural networks may have important advantages of both formal verification and interpretability. On the one hand, heuristic arguments (like proofs)
are formal objects that are not required to be human-understandable. This means that heuristic arguments could be used to reason about properties of neural networks for which no compact
human-understandable explanation exists. On the other hand, heuristic arguments (like interpretability approaches) do not require perfect certainty to be considered valid. This allows for short
heuristic arguments of complex properties of large models, even when no short proofs of those properties exist.^[4] (See our earlier post on surprise accounting for further discussion.)
In the rest of this section, I will give three examples of problems that we believe cannot be solved in full generality with current approaches, but that they may be solvable with heuristic
arguments. (All three examples will just be sketches of possible approaches, with many details left to be filled in.)
Mechanistic anomaly detection
Let \(M\) be a neural network that was trained on a distribution \(\mathcal{D}\) of inputs \(x\) using the loss function \(L(x, M(x))\).^[5] Suppose that \(M\) successfully learns to achieve low
loss: that is, \(\mathbb{E}_{x \sim \mathcal{D}}[L(x, M(x))]\) is small. Let \(x^\ast\) be a (perhaps out-of-distribution) input. We call \(x^\ast\) a mechanistic anomaly for \(M\) if \(M\) gets a
low loss on \(x^\ast\), but for a "different reason" than the reason why it gets low average loss on \(\mathcal{D}\). In other words, mechanistic anomalies are inputs on which \(M\) acts in a
seemingly reasonable way, but via anomalous internal mechanisms.^[6] To detect a mechanistic anomaly, reasoning about \(M\)'s internal structure may be necessary.
How could we use a heuristic estimator to detect mechanistic anomalies? Suppose that we find a set of arguments \(\Pi\) such that the following quantity is low:^[7]
\[\mathbb{G}(\mathbb{E}_{x \sim \mathcal{D}}[L(x, M(x))] \mid \Pi).\]
That is, \(\Pi\) explains why \(M\) attains low average loss on \(\mathcal{D}\).^[8] Given an out-of-distribution input \(x^\ast\) such that \(L(x^\ast, M(x^\ast))\) is once again low, we consider
the quantity \(\mathbb{G}(L(x^\ast, M(x^\ast)) \mid \Pi)\). This represents a heuristic estimate of \(M\)'s loss on \(x^\ast\) based only on the reasons provided in \(\Pi\): that is, based only on
the facts necessary to explain \(M\)'s low loss on \(\mathcal{D}\).
If \(\mathbb{G}(L(x^\ast, M(x^\ast)) \mid \Pi)\) is (correctly) low, then the reasons why \(M\) performs well on \(\mathcal{D}\) also explain why \(M\) performs well on \(x^\ast\). By contrast, if \
(\mathbb{G}(L(x^\ast, M(x^\ast)) \mid \Pi)\) is (incorrectly) high, then \(M\) performs well on \(x^\ast\) for a different reason that why \(M\) performs well on \(\mathcal{D}\). As a result, we flag
\(x^\ast\) as a mechanistic anomaly for \(M\).
(See here for a more detailed discussion of mechanistic anomaly detection.)
Safe distillation
Let \(f\) ("fast") and \(s\) ("slow") be two neural networks that were trained on a distribution \(\mathcal{D}\) of inputs to complete the same task. Thus, \(f\) and \(s\) behave similarly on \(\
mathcal{D}\). Suppose that we trust \(s\) to be aligned (e.g. we trust \(s\) to generalize well off-distribution) and do not similarly trust \(f\), but that \(s\) is much slower than \(f\).^[9] Given
an out-of-distribution input \(x^\ast\), we would like to estimate \(s(x^\ast)\) without running \(s\). We could do this by running \(f\) on \(x^\ast\) and hoping that \(f\) generalizes well to \(x^\
ast\). However, this approach is not very robust. Instead, we can attempt to use the internal activations of \(f\) to predict \(s(x^\ast)\).
Concretely, suppose for simplicity that \(f\) and \(s\) output vectors, and suppose that we find a set of arguments \(\Pi\) such that the following quantity is low:
\[\mathbb{G}(\mathbb{E}_{x \sim \mathcal{D}}[\lVert{f(x) - s(x)\rVert^2] \mid \Pi).}\]
That is, \(\Pi\) explains why \(f\) and \(s\) produce similar outputs on \(\mathcal{D}\). Given an out-of-distribution input \(x^\ast\), we consider the quantity
\[\mathbb{G}(s(x^\ast) \mid \Pi, \text{computational trace of }f\text{ on }x^\ast).\]
This represents a heuristic estimate of \(s(x^\ast)\) given the computations done by \(f\) and the argument \(\Pi\) for why \(f\) and \(s\) are similar on \(\mathcal{D}\). If the reason why \(f\) and
\(s\) behave similarly on \(\mathcal{D}\) also extends to \(x^\ast\), then \(\mathbb{G}\) will correctly estimate \(s(x^\ast)\) to be similar to \(f(x^\ast)\). On the other hand, if the reason why \
(f\) and \(s\) behave similarly on \(\mathcal{D}\) does not extend to \(x^\ast\), then \(\mathbb{G}\)'s estimate of \(s(x^\ast)\) may be different from \(f(x^\ast)\). This estimate may be more robust
to distributional shifts, because it is based on mechanistic reasoning about how \(f\) and \(s\) work.
Safe distillation and mechanistic anomaly detection are closely related problems. The key difference is that in the safe distillation problem, we have a trusted model \(s\). This makes the setting
easier; in exchange, we can hope to do more Concretely, we expect \(f\) and \(s\) to differ on \(x^\ast\) if \(x^\ast\) is a mechanistic anomaly for \(f\) (but not for \(s\)). Solving the safe
distillation problem would allow us to not only detect \(x^\ast\) as an anomalous input, but to predict \(s(x^\ast)\) from the internals of \(f\). An algorithm for predicting \(s(x^\ast)\) from \(f\)
's internals would act as a distillation of \(s\), but – unlike \(f\) – it would be a safe distillation (hence the name).
Low probability estimation
Let \(M\) be a neural network that was trained on a distribution \(\mathcal{D}\). Let \(C\) (for "catastrophe") be a different neural network that checks the output of \(M\) for some rare but highly
undesirable behavior: \(C(M(x))\) returns \(1\) if \(M\) exhibits the undesirable behavior on \(x\), and \(0\) otherwise. We may wish to estimate \(\mathbb{E} _ {x \sim \mathcal{D}}[C(M(x)) = 1]\),
and we cannot do so by sampling random inputs \(x \sim \mathcal{D}\) because \(C\) outputs \(1\) very rarely. Suppose that we find a set of arguments \(\Pi\) that explains the mechanistic behavior of
\(M\) and \(C\). If this explanation is good enough, then \(\mathbb{G}(\mathbb{E} _ {x \sim \mathcal{D}}[C(M(x))] \mid \Pi)\) will be a high-quality estimate of this probability. Additionally, we may
use \(\mathbb{G}\) to more efficiently check \(M\)'s behavior on particular inputs: given an input \(x^\ast\), the quantity
\[\mathbb{G}(C(M(x^\ast)) \mid \Pi, \text{computational trace of }M\text{ on }x^\ast)\]
represents an estimate of the likelihood that \(C(M(x^\ast)) = 1\) based on the computations done by \(M\). This is especially useful if \(C\) is slow and running it on every output of \(M\) is
prohibitively expensive.
A few weeks ago, we published a blog post on this potential application. In the blog post, we discussed layer-by-layer activation modeling as a possible approach to the problem. We believe that
solving the problem in full generality would require sophisticated activation models that can represent a rich space of possible distributional properties of neural activations. This is similar to
our goal of developing a conception of heuristic arguments that is rich enough to point out essentially any structural property of a neural network. Activation modeling and heuristic estimation are
different perspectives on the same underlying approach.
With this motivation for heuristic estimators in mind, let us now discuss what properties they ought to satisfy.
Formalizing the principle of unpredictable errors
Recall from Analogy #3 that we should expect heuristic estimators to behave a lot like Bayesian reasoners. In fact, perhaps heuristic estimators should satisfy formal properties that Bayesian
reasoners satisfy.
One property of Bayesian reasoning is known as the martingale property, which states that a Bayesian reasoner's estimate (i.e. subjective expectation) of their future estimate of some quantity should
equal their current estimate of the quantity. Or more informally, a Bayesian reasoner cannot predict the direction in which they will update their estimate in light of new information. If they could,
then they would make that update before receiving the information.
By analogy, a heuristic estimator ought not to be able to predict the direction of its own errors. We call this the principle of unpredictable errors. While this principle is informal, formalizing it
could be a first step toward searching for an intuitively reasonable heuristic estimator. This motivates trying to find a satisfying formalization of the principle. Below, we will discuss two
approaches for formalization: a subjective approach and an objective approach.
The subjective approach: Iterated estimation and error orthogonality
Our subjective approach to formalizing the principle of unpredictable errors involves two properties that we call iterated estimation and error orthogonality.
We say that \(\mathbb{G}\) satisfies the iterated estimation property if for all expressions \(Y\) and for all sets of arguments \(\Pi\) and \(\Pi' \subseteq \Pi\), we have
\[\mathbb{G}(\mathbb{G}(Y \mid \Pi) \mid \Pi') = \mathbb{G}(Y \mid \Pi').\]
The sense in which the iterated estimation property formalizes the principle of unpredictable errors is fairly straightforward. The expression \(\mathbb{G}(Y \mid \Pi')\) represents the heuristic
estimator's belief given only the arguments in \(\Pi'\). The expression \(\mathbb{G}(Y \mid \Pi)\) represents the estimator's belief given a larger set of arguments. The iterated estimation property
states that the estimator's belief (given \(\Pi'\)) about what their belief about \(Y\) would be if presented with all of \(\Pi\) is equal to their current belief about \(Y\). This closely mirrors
the martingale property of Bayesian reasoners. It is also directly analogous to the law of iterated expectations from probability theory, hence the name "iterated estimation."^[10]
As an example, let \(Y, \pi_{101}, \pi_{102}\) be defined as in our earlier example with the square roots. Let \(\Pi = {\pi_{101}, \pi_{102}}\) and \(\Pi' = {\pi_{101}}\). As discussed above, we have
\(\mathbb{G}(Y \mid \pi_{101}) = 90.5\) (because the sixth digit of \(\sqrt{101}\) is \(5\)). The iterated estimation property states that \(\mathbb{G}(\mathbb{G}(Y \mid \pi_{101}, \pi_{102}) \mid \
pi_{101})\) is also equal to \(90.5\). In other words, after \(\mathbb{G}\) has learned \(\pi_{101}\) (but not yet \(\pi_{102}\)), its estimate of what its belief of \(Y\) will be after learning \(\
pi_{102}\) is its current estimate of \(Y\), namely \(90.5\).
We say that \(\mathbb{G}\) satisfies the error orthogonality property if for all expressions \(X, Y\) and for all sets of arguments \(\Pi\) and \(\Pi_1, \Pi_2 \subseteq \Pi\), we have
\[\mathbb{G}((Y - \mathbb{G}(Y \mid \Pi)) \cdot \mathbb{G}(X \mid \Pi_1) \mid \Pi_2) = 0.\]
Error orthogonality is a more "sophisticated" version of iterated estimation.^[11] It is directly analogous to the projection law of conditional conditional expected values.^[12]
As an example, let \(Y, \pi_{101}\) be defined as above and let \(X\) be the expression \(d_6(\sqrt{101})\). Let \(\Pi = \Pi_1 = {\pi_{101}}\) and \(\Pi_2 = \emptyset\). The outer \(\mathbb{G}\) does
not know the exact value of \(\mathbb{G}(X \mid \Pi_1)\). However, it believes \(\mathbb{G}(X \mid \Pi_1)\) and \(Y - \mathbb{G}(Y \mid \Pi)\) to be subjectively uncorrelated, because it knows that \
(\Pi\) includes \(\Pi_1\). Thus, given the outer \(\mathbb{G}\)'s state of knowledge, the best estimate of \(Y - \mathbb{G}(Y \mid \Pi)\) is zero for all possible values of \(\mathbb{G}(X \mid \Pi_1)
\). Thus, its estimate of the entire expression is \(0\).
(If you want to build more intuition about this property, see Example 2.5 of our paper.)
To see why error orthogonality is desirable, recall our interpretation of heuristic estimates as subjective expected values. Suppose that error orthogonality does not hold for some particular \(X, Y,
\Pi, \Pi_1, \Pi_2\). This means that an observer with state-of-knowledge \(\Pi_2\) believes \(Y - \mathbb{G}(Y \mid \Pi)\) and \(\mathbb{G}(X \mid \Pi_1)\) to be subjectively correlated over the
observer's uncertainty. In other words: the observer believes that the subjective estimate of \(X\) given state-of-knowledge \(\Pi_1\) is predictive of the error in the subjective estimate of \(Y\)
given state-of-knowledge \(\Pi\). However, any such prediction should have already been factored into the estimate \(\mathbb{G}(Y \mid \Pi)\).
Challenges with the subjective approach
Although iterated estimation and error orthogonality are intuitively compelling, there are challenges with using these properties as stated to seek a reasonable heuristic estimator. These challenges
come primarily from the fact that the properties concern \(\mathbb{G}\)'s estimates of its own output.
The first challenge: it seems plausible that these two properties could be satisfied "by fiat." This means that \(\mathbb{G}\) could check whether it is estimating the quantity \(\mathbb{G}(Y \mid \
Pi)\) given some subset \(\Pi'\) of \(\Pi\), and then – if so – simply compute \(\mathbb{G}(Y \mid \Pi')\) and output the result. Although this behavior would satisfy the iterated estimation
property, we do not want \(\mathbb{G}\) to special-case expressions of this form. Instead, we want \(\mathbb{G}\) to satisfy the property as a consequence of its more general behavior.^[13]
The second challenge: the fact that these two properties concern \(\mathbb{G}\)'s estimates of its own output makes it difficult to use these properties to reason about \(\mathbb{G}\). If our goal is
to find a reasonable heuristic estimator \(\mathbb{G}\), it is most useful to have properties that pin down \(\mathbb{G}\)'s outputs on simple inputs. The iterated estimation and error orthogonality
properties are not helpful in this regard, because the simplest possible equality that is derivable from either property still involves a mathematical expression that includes the code of \(\mathbb
{G}\) as part of the expression. Furthermore, without knowing \(\mathbb{G}\)'s code, a constraint that involves \(\mathbb{G}\)'s behavior on its own code is less useful.
For these two reasons, we are interested in more grounded variants of the iterated estimation and error orthogonality properties: ones that still capture the key intuition that \(\mathbb{G}\)'s
errors ought not be predictable, but that do not involve nested \(\mathbb{G}\)'s. This motivates searching for an objective approach to formalizing the principle of unpredictable errors.
The objective approach: Accuracy
The basic idea of the objective approach is to replace the outer \(\mathbb{G}\) in the iterated estimation and error orthogonality properties with an expected value \(\mathbb{E}_{\mathcal{D}}\) over
some probability distribution \(\mathcal{D}\). In other words, \(\mathbb{G}\)'s errors should be objectively unpredictable – rather than subjectively unpredictable – over a specified distribution of
expressions. Depending on the context, we call these properties accuracy or multiaccuracy.^[14]
Before defining accuracy for \(\mathbb{G}\), we define accuracy in the context of probability theory.
Definition (accuracy of an estimator.) Let \(\mathcal{Y}\) be a space of real-valued mathematical expressions, and let \(\mathcal{D}\) be a probability distribution over \(\mathcal{Y}\). Let \(X: \
mathcal{Y} \to \mathbb{R}\) be a random variable. An estimator \(f: \mathcal{Y} \to \mathbb{R}\) is \(X\)-accurate over \(\mathcal{D}\) if
\[\mathbb{E}_{Y \sim \mathcal{D}}[(Y - f(Y))X] = 0.\]
We say that \(f\) is self-accurate over \(\mathcal{D}\) if \(f\) is \(f\)-accurate over \(\mathcal{D}\). For a set \(S\) of random variables, we say that \(f\) is \(S\)-multiaccurate over \(\mathcal
{D}\) if \(f\) is \(X\)-accurate over \(\mathcal{D}\) for all \(X \in S\).
Intuitively, being \(X\)-accurate means that an estimator has "the right amount of \(X\)": the estimator's error is uncorrelated with \(X\), which means that adding a constant multiple of \(X\) to
the estimator can only hurt the quality of the estimator. (See Proposition 3.4 in the paper for a formal statement of this intuition.)
As an example, let \(\mathcal{Y}\) be the space of expressions of the form \(2 \cdot c_1 + 3 \cdot c_2\), where \(c_1, c_2 \in \mathbb{R}\). (For example, the expression \(2 \cdot 0.7 + 3 \cdot -5\)
belongs to \(\mathcal{Y}\).) Let \(\mathcal{D}\) be the distribution over \(\mathcal{Y}\) obtained by selecting \(c_1, c_2\) independently from \(\mathcal{N}(0, 1)\), the standard normal
Let us consider the estimator \(f(Y) = 3c_2\). This estimator is \(1\)-accurate, meaning that it has the correct mean: \(\mathbb{E} _ {Y \sim \mathcal{D}}[(Y - f(Y)) \cdot 1] = \mathbb{E} _ {c_1, c_2
\sim \mathcal{N}(0, 1)}[2c_1] = 0\). It is also self-accurate: \(\mathbb{E} _ {Y \sim \mathcal{D}}[(Y - f(Y)) \cdot f(Y)] = \mathbb{E} _ {c_1, c_2 \sim \mathcal{N}(0, 1)}[2c_1 \cdot 3c_2] = 0\).
However, it is not \(c_1\)-accurate: \(\mathbb{E} _ {Y \sim \mathcal{D}}[(Y - f(Y)) \cdot c_1] = \mathbb{E} _ {c_1, c_2 \sim \mathcal{N}(0, 1)}[2c_1 \cdot c_1] = 2 \neq 0\). This reflects the fact
that \(f\) does not have "the right amount of \(c_1\)": adding \(2c_1\) to \(f\) will make it a better estimator (indeed, a perfect estimator) of \(Y\).
Here is a Venn diagram of some other estimators of \(Y\) based on whether they are \(1\)-accurate, \(c_1\)-accurate, and self-accurate over \(\mathcal{D}\).
To get some intuition for accuracy, you could verify that the Venn diagram is correct. Another exercise: find another estimator that goes in the center of the Venn diagram. This Wikipedia page may be
(Multi-)accuracy is closely related to linear regression. In particular, the OLS (i.e. linear regression) estimator^[15] of \(Y\) in terms of a set of predictors \(S = {X_1, \ldots, X_n}\) is \(S\)
-multiaccurate (and is the only \(S\)-multiaccurate estimator that is a linear combination of \(X_1, \ldots, X_n\)). The linear regression estimator is also self-accurate.^[16]
We can adapt our definition of accuracy for estimators (in the probability theory sense used above) to heuristic estimators \(\mathbb{G}\). The basic idea is to replace the estimator \(f\) with our
heuristic estimator \(\mathbb{G}\), i.e. to substitute \(\mathbb{G}(Y \mid \Pi)\) for \(f(Y)\):
Definition (accuracy of a heuristic estimator.) Let \(\mathcal{Y}, \mathcal{D}\) be as in the previous definition. Let \(\mathbb{G}\) be a heuristic estimator and \(X: \mathcal{Y} \to \mathbb{R}\) be
a random variable. A set of heuristic arguments \(\Pi_X\) makes \(\mathbb{G}\) be \(X\)-accurate over \(\mathcal{D}\) if for all \(\Pi \supseteq \Pi_X\), \(\mathbb{G}(Y \mid \Pi)\) is an \(X\)
-accurate estimator over \(\mathcal{D}\) – that is,
\[\mathbb{E}_{Y \sim \mathcal{D}}[(Y - \mathbb{G}(Y \mid \Pi))X] = 0.\]
We say that \(\mathbb{G}\) is \(X\)-accurate over \(\mathcal{D}\) if there is a short \(\Pi_X\) that makes \(\mathbb{G}\) be \(X\)-accurate over \(\mathcal{D}\). We say that \(\mathbb{G}\) is \(S\)
-multiaccurate over \(\mathcal{D}\) if \(\mathbb{G}\) is \(X\)-accurate over \(\mathcal{D}\) for all\(\) \(X \in S\).
(The paper clarifies some details that have been skipped over in this definition. For example, the paper defines "short" and clarifies some subtleties around the interpretation of \(\mathbb{G}(Y \mid
\Pi)\) in the context of the definition.)
(Note also the similarity between this equation and our definitions of iterated estimation and error orthogonality. Indeed, we recover the definition of iterated estimation in the special case of \(X
= 1\) – except that the outer \(\mathbb{G}\) is replaced by an expectation over \(\mathcal{D}\). We can similarly recover error orthogonality in the case of \(X = \mathbb{G}(Y \mid \Pi)\) – see the
paper for details, including for a definition of self-accuracy for heuristic estimators.)
While iterated estimation and error orthogonality are primarily "soundness" conditions on \(\mathbb{G}\) – that is, they constrain \(\mathbb{G}\) to output internally consistent estimates – accuracy
can additionally be used as a "completeness" condition. In particular, if \(\mathbb{G}\) is \(S\)-multiaccurate, this means that:
• \(\mathbb{G}\) can successfully incorporate the predictors in \(S\) into its estimates; and that
• \(\mathbb{G}\) can reasonably merge its estimates based on these predictors (because if \(X_1, X_2 \in S\) then \(\mathbb{G}(Y \mid \Pi_{X_1} \cup \Pi_{X_2})\) needs to be an \(X_1\)-accurate
estimator of \(Y\) and also an \(X_2\)-accurate estimator of \(Y\)).
Now, our goal is for \(\mathbb{G}\) to be \(S\)-multiaccurate for a rich class of predictors \(S\). Such a \(\mathbb{G}\) would be powerful while producing reasonable estimates. Unfortunately, as we
discuss in the next section, it seems quite difficult to produce such a \(\mathbb{G}\).
Challenges with the objective approach
Given a natural distribution \(\mathcal{D}\) over mathematical expressions and a small, simple, and natural set of predictors \(S\), it is always possible to efficiently produce a self-accurate and \
(S\)-multiaccurate estimator?
It may seem like the answer is yes: in particular, we mentioned earlier that the linear regression of \(Y\) onto the predictors in \(S\) is self-accurate and \(S\)-multiaccurate. However, this raises
an important question: can we efficiently compute the necessary regression coefficients?
Unfortunately, as we will discuss, the answer to this question is no.
Estimating the product of jointly normal random variables
Here is a quite simple and natural estimation problem: given a \(n \times n\) covariance matrix \(\Sigma\), estimate the expected product of \(n\) random variables with mean \(0\) and covariance
matrix \(\Sigma\).
We consider this estimation problem for two reasons. On the one hand, this is one of the simplest estimation problems for which computing (or even approximating) the correct answer is computationally
intractable. On the other hand, this problem captures the core difficulty of a natural, more general estimation problem: estimating the average output of an arithmetic circuit (a circuit with
addition and multiplication gates). Addition gates are straightforward: \(\mathbb{G}(X + Y \mid \Pi) = \mathbb{G}(X \mid \Pi) + \mathbb{G}(Y \mid \Pi)\), and so the challenge lies in the
multiplication gates.
It turns out that the answer to this estimation problem is equal to the sum of all \(n/2\)-fold products of covariances in which each variable is used exactly once. That is, if \(Z_1, \ldots, Z_n\)
are jointly normal, zero-mean random variables, then
\[\mathbb{E}[Z_1 \ldots Z_n] = \sum_{p \in \mathcal{P} _ 2(n)} \prod_{(i, j) \in p} \text{Cov}(Z_i, Z_j),\]
where \(\mathcal{P} _ 2(n)\) denotes the set of all pairings of \({1, \ldots, n}\) (for example, one element of \(\mathcal{P} _ 2(6)\) is \({(1, 4), (2, 3), (5, 6)}\)). (This is called the hafnian of
the covariance matrix.)
And so our estimation problem amounts to computing a giant sum. This suggests a natural class of predictors: namely, partial sums.
Concretely, let \(Y = Y(\Sigma)\) be the expected product of random variables with mean zero and covariance matrix \(\Sigma\), and let \(\mathcal{D}\) be the distribution over \(Y\) induced by
selecting the (off-diagonal) entries of \(\Sigma\) independently from \(\mathcal{N}(0, 1)\).^[17] Given a pairing \(p\) of \({1, \ldots, n}\), we define \(X_p := \prod_{(i, j) \in p} \Sigma_{i, j}\),
and for a subset \(S \subseteq \mathcal{P} _ 2(n)\), we define \(X_S := \sum_{p \in S} X_p\) (so \(Y = X_{\mathcal{P} _ 2(n)}\)).
Note that \(X_S\) can be efficient to compute even for exponentially large sets of pairings \(S\). For example, let \(S\) be the set of \(3^{n/4}\) pairings that pair 1, 2, 3, 4 amongst themselves
(there are three ways to do so); pair 5, 6, 7, 8, amongst themselves; and so on. Then
\[X_S = (\Sigma_{1, 2} \Sigma_{3, 4} + \Sigma_{1, 3} \Sigma_{2, 4} + \Sigma_{1, 4} \Sigma_{2, 3})(\Sigma_{5, 6} \Sigma_{7, 8} + \Sigma_{5, 7} \Sigma_{6, 8} + \Sigma_{5, 8} \Sigma_{6, 7})(\ldots).\]
And so we might wonder: given two efficiently computable partial sums \(X_{S_1}, X_{S_2}\), is it always possible to efficiently combine them into a single estimate of \(Y\) with linear regression?
That is, are the requisite regression coefficients efficiently computable?
Alas, the answer is no. We show this by reduction from 3SAT – that is, we show that if you can compute these regression coefficients, then you can solve boolean satisfiability problem, which is
NP-complete. If you're interested in the details, check out Section 4.1 of the paper!
(We have not ruled out the possibility that there is a more sophisticated way to accurate merge the estimators \(X_{S_1}\) and \(X_{S_2}\), given that linear regression is intractable. However, we
conjecture that no accurate and efficiently computable merge exists. That is, in general, there is no estimator \(f\) of \(Y\) that is both efficiently computable and \({f, X_{S_1}, X_{S_2}}\)
Even though you can't efficiently compute the regression coefficients for merging these arguments exactly, you might wonder whether it's possible to compute them approximately. Concretely, given sets
\(S_1, \ldots, S_m\) for which the predictors \(X_{S_i}\) are efficient to compute, is it possible to find a linear combination of the \(X_{S_i}\)'s that is approximately self-accurate and \({X_
{S_1}, \ldots, X_{S_m}}\)-multiaccurate? It turns out that, in order to compute the regression coefficients of \(Y\) onto \(X_{S_1}, \ldots, X_{S_m}\), it is sufficient to compute \(|S_i \cap S_j|\)
for all \(i, j\). This suggests that you can estimate the regression coefficients by estimating the sizes of these intersections, which you can do e.g. by randomly sampling elements of \(S_i\) and
seeing if they belong to \(S_j\).
It turns out that the number of samples you need depends polynomially on the condition number of the correlation matrix of \(X_{S_1}, \ldots, X_{S_m}\). Unfortunately, this conditional number can be
exponentially large.^[18]
Currently we do not know how to merge these estimates even approximately, although we aren't confident that this cannot be done. At the very least, the most straightforward approaches do not work,
which suggests a barrier to creating algorithms that accurately merge even simple estimates for simple and natural estimation problems.
Estimating the permanent of a matrix
We consider another natural estimation problem: given an \(n \times n\) matrix \(A\), estimate its permanent. The permanent of \(A\) is the sum of all products of \(n\) elements of \(A\), one per row
and column:
\[\text{perm}(A) = n! \mathbb{E} _ {\sigma \sim S_n} \left[ \prod_{i = 1}^n A_{i, \sigma(i)} \right].\]
While superficially similar to the determinant, the determinant can be computed efficiently, while the permanent cannot even be approximated efficiently.
In the paper, we consider three different estimates of the permanent, all of which are motivated by an argument via presumption of independence. The row sum estimate is given by
\[E_{\text{row}}(A) := n! \prod_{i = 1}^n \mathbb{E} _ {\sigma \sim S_n}[A_{i, \sigma(i)}] = \frac{n!}{n^n} \prod_{i = 1}^n \sum_{j = 1}^n A_{i, j}.\]
The row sum estimate is \(n!\) times the average product obtained by taking one element of each row of \(A\). It is also the average permanent of all matrices obtained from \(A\) by shuffling each
row independently. Similarly, the column sum estimate is given by
\[E_{\text{col}}(A) := \frac{n!}{n^n} \prod_{j = 1}^n \sum_{i = 1}^n A_{i, j}.\]
Finally, the matrix sum estimate is \(n!\) times the average product obtained by taking \(n\) random elements of \(A\) (with replacement):
\[E_{\text{ms}}(A) := \frac{n!}{n^{2n}} \left( \sum_{i = 1}^n \sum_{j = 1}^n A_{i, j} \right)^n.\]
If we want to accurately merge these estimates over a distribution of matrices, we can do so with linear regression. However, the resulting estimator can have undesirable properties. For example, the
linear regression estimator over the distribution of matrices with each entry selected independently from \(\mathcal{N}(0, 1)\) takes the form \(\alpha(E_\text{row}(A) + E_{\text{col}}(A)) - \beta E_
{\text{ms}}(A)\), where \(\beta\) is positive. In particular, this means that even if \(A\) has exclusively non-negative entries, this linear regression estimate for the permanent of \(A\) can be
This by itself is okay: we do not expect estimates to be reasonable by default. What we do expect, however, is that upon noticing that an estimate is unreasonable, we should be able to correct it.
That is, we ought to be able to produce an estimator that merges \(\)the row sum, column sum, and matrix sum estimates, and is non-negative on matrices with non-negative entries.
Unfortunately, we are not aware of a natural way to produce such an estimator. Actually, for matrices with non-negative entries, there is a natural estimator that "merges" the row sum, column sum,
and matrix sum estimates: namely, \(\frac{E_{\text{row}} E_{\text{col}}}{E_{\text{ms}}}\). (See Section 5.2 of the paper for an explanation of where this estimator comes from and why it is
reasonable.) However, this estimator does not satisfy any accuracy properties over any natural distribution of matrices. This makes sense, because this estimator is "multiplicative" in nature,
whereas accuracy is an "additive" property.
Our discussion of estimating the permanent thus poses another barrier to using multiaccuracy to formalize the principle of unpredictable errors. Namely, accuracy forces us to reject a seemingly
reasonable estimator while forcing us to create seemingly unnatural estimators to satisfy additional properties (like estimating the permanents of matrices with non-negative entries as being
The ultimate goal of formalizing properties like the principle of unpredictable errors it to help guide the search for a heuristic estimator. Once you have formal properties, you can pose a formal
mathematical question: "Does there exist a polynomial-time algorithm \(\mathbb{G}: {\text{mathematical expressions}} \times {\text{sets of arguments}} \to \mathbb{R}\) that satisfies [formal
properties like linearity, unpredictable errors, etc.]?". Once you have such a question, you can use your mathematical toolbox to try to resolve it. By contrast, without such a question, you're
forced to answer vague subjective questions like "What would a reasonable heuristic estimator do in this situation?".
Two properties that have stood the test of time are linearity and respect for proofs. Linearity states that for \(a, b \in \mathbb{R}\) and mathematical expressions \(X, Y\), we have that \(\mathbb
{G}(aX + bY \mid \Pi) = a \mathbb{G}(X \mid \Pi) + b \mathbb{G}(Y \mid \Pi).\) Respect for proofs states that, given a proof that \(Y \ge 0\), the proof may be turned into a heuristic argument \(\pi
\) such that for all \(\Pi\) containing \(\pi\), we have \(\mathbb{G}(Y \mid \Pi) \ge 0\). Unfortunately, these two properties alone are insufficient to pin down the behavior of \(\mathbb{G}\): there
are heuristic estimators that satisfy linearity and respect for proofs but behave "unreasonably" (see Chapter 9 here).
It would be really nice if we could formalize the principle of unpredictable errors, because perhaps a satisfying formalization of this principle, together with linearity and respect for proofs,
would force \(\mathbb{G}\) to behave reasonably. So far, we have not found a satisfying formalization; finding one might constitute an important step forward in our understanding of heuristic
This year, though, we have mostly focused on a different approach. Our new approach reframes the heuristic estimation problem as an activation modeling problem: learning a coherent representation of
the statistical properties of a neural network's activations (or the values of a circuit's wires) that lets us answer questions about the neural network (or circuit).^[19] We haven't written about
this perspective in detail yet, because we are still developing it, but see our recent blog post on estimating tail risks in neural networks for an outline of what this approach might look like. We
are excited to see where our new perspective takes us!
Cross-postings for comments: Alignment Forum, LessWrong
1. Our original notation is \(\tilde{\mathbb{E}}(Y, \pi_1, \ldots, \pi_m)\). In our new work, we use the notation \(\mathbb{G}(Y \mid \pi_1, \ldots, \pi_m)\) to emphasize that, while there are
similarities between heuristic estimation and expected values, they are importantly different. ↩︎
2. See here for an earlier blog post that introduced the mechanistic anomaly detection problem. ↩︎
3. Perhaps instead of no arguments, \(\mathbb{G}\) is given a short argument that points out that there are twenty digits and that its estimate for each digit ought to be (4.5). ↩︎
4. Here, "short" means "about as large as the model itself." ↩︎
5. For example, \(L\) could be based on a trained reward predictor, as in RLHF. ↩︎
6. For example, if \(M\) is a financial assistant that takes actions such as buying stocks and transferring money between bank accounts, then \(M\) might have a low loss on \(\mathcal{D}\) because
it makes good financial decisions, but a low loss on \(x^\ast\) because it implements a money laundering scheme that \(L\) fails to notice. ↩︎
7. One of the most important and difficult questions faced by this approach is how to find such a \(\Pi\). If the space of arguments is parameterized, then we may hope to learn \(\Pi\) via gradient
descent in parallel with training \(M\) itself. ↩︎
8. The idea is that, without any arguments, \(\mathbb{G}\) does not understand anything about the structure of \(M\), and so should estimate \(M\)'s loss as if \(M\) were a randomly initialized
neural network. (Such a network would incur high loss.) Heuristic arguments that explain \(M\)'s structure should cause \(\mathbb{G}\)'s estimate of \(M\)'s loss to decrease. ↩︎
9. For example, in the iterated distillation and amplification process, \(f\) could be a distillation of a trusted model \(s\); however, we may not trust \(f\). ↩︎
10. Most generally, the law of iterated expectations states that for a probability space \((\Omega, \mathcal{F}, \mathbb{P})\) with \(\sigma\)-algebras \(\mathcal{H}' \subseteq \mathcal{H} \subseteq
\mathcal{F}\), for any integrable random variable \(Y\) we have \(\mathbb{E}[\mathbb{E}[Y \mid \mathcal{H}] \mid \mathcal{H}'] = \mathbb{E}[Y \mid \mathcal{H}']\). ↩︎
11. It may seem like a generalization (consider the case of \(X=1\)), but it is not: deriving iterated estimation from error orthogonality would require assuming additional properties of \(\mathbb{G}
\). ↩︎
12. The projection law states that for a probability space \((\Omega, \mathcal{F}, \mathbb{P})\) with \(\sigma\)-algebras \(\mathcal{H}' \subseteq \mathcal{H} \subseteq \mathcal{F}\), for
square-integrable random variables \(X, Y\), we have \(\mathbb{E}[(Y - \mathbb{E}[Y \mid \mathcal{H}]) \cdot \mathbb{E}[X \mid \mathcal{H}] \mid \mathcal{H}'] = 0\).\(\) ↩︎
13. As an analogy, consider a proof verifier \(V\) that takes as input a mathematical statement \(x\) and a purported proof \(\pi\), and outputs \(1\) (accept) or \(0\) (reject) depending on whether
\(\pi\) is a proof of \(x\). Let \(s(x, \pi)\) be the statement "If \(V(x, \pi) = 1\), then \(x\)." For every \((x, \pi)\), there is a proof \(\pi'\) of \(s(x, \pi)\) (specifically: if \(V(x, \
pi) = 1\), then \(\pi\) proves \(x\) and thus \(s(x, \pi)\); and if \(V(x, \pi) = 0\), then the computational trace of \(V\) on \((x, \pi)\) shows that \(V(x, \pi) = 0\) and thus proves \(s(x, \
pi)\)). However, \(V\) should not treat the input \((s(x, \pi), \pi')\) as a special case; instead, \(V\) should verify that \(\pi'\) proves \(s(x, \pi)\) just as it would verify any other proof.
14. The term "multiaccuracy" originates in the algorithmic fairness literature, where it is used to describe a predictor that appears unbiased to a given set of statistical tests (see e.g. here and
here). ↩︎
15. Without a constant term. If you want a constant term, you can add the predictor \(1\) to \(S\). ↩︎
16. Here we speak of the exact linear regression estimator of \(Y\) in terms of \(X_1, \ldots, X_n\): in other words, the linear combination of these predictors that is closest to \(Y\) (in terms of
expected squared error over \(\mathcal{D}\)). This contrasts with the more typical setting for linear regression, in which coefficients are computed only approximately based on samples. ↩︎
17. The diagonal entries (which do not matter for the value of \(Y_\Sigma\)) can always be chosen so that \(\Sigma\) is a valid covariance matrix, simply by making those entries be very large. ↩︎
18. We believe that by using ridge regression instead of linear regression, it is possible to find an approximately \({X_{S_1}, \ldots, X_{S_m}}\)-multiaccurate estimate of \(Y\). However, this
estimate is not approximately self-accurate. See Remark 4.14 in the paper for an explanation for why we consider self-accuracy to be important. ↩︎
19. Very loosely speaking, the correspondence between heuristic estimation and activation modeling is that a particular activation model corresponds to \(\mathbb{G}(\cdot \mid \Pi)\), and so an
activation model can take as input a quantity and return an estimate of that quantity. ↩︎ | {"url":"https://www.alignment.org/blog/research-update-towards-a-law-of-iterated-expectations-for-heuristic-estimators/","timestamp":"2024-11-05T18:16:42Z","content_type":"text/html","content_length":"78623","record_id":"<urn:uuid:6dc25deb-fd83-47db-9505-555e7e5fc2aa>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00090.warc.gz"} |
LoopDetectR | CRAN/E
Comprehensive Feedback Loop Detection in ODE Models
CRAN Package
Detect feedback loops (cycles, circuits) between species (nodes) in ordinary differential equation (ODE) models. Feedback loops are paths from a node to itself without visiting any other node twice,
and they have important regulatory functions. Loops are reported with their order of participating nodes and their length, and whether the loop is a positive or a negative feedback loop. An upper
limit of the number of feedback loops limits runtime (which scales with feedback loop count). Model parametrizations and values of the modelled variables are accounted for. Computation uses the
characteristics of the Jacobian matrix as described e.g. in Thomas and Kaufman (2002) . Input can be the Jacobian matrix of the ODE model or the ODE function definition; in the latter case, the
Jacobian matrix is determined using 'numDeriv'. Graph-based algorithms from 'igraph' are employed for path detection.
• Version0.1.2
• R version≥ 4.0.0
• Needs compilation?No
• Last release07/20/2020
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• Imports2 packages
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Option pricing, stochastic volatility, singular dynamics and constrained path integrals
Stochastic volatility models have been widely studied and used in the financial world. The Heston model (Heston, 1993) [7] is one of the best known models to deal with this issue. These stochastic
volatility models are characterized by the fact that they explicitly depend on a correlation parameter ρ which relates the two Brownian motions that drive the stochastic dynamics associated to the
volatility and the underlying asset. Solutions to the Heston model in the context of option pricing, using a path integral approach, are found in Lemmens et al. (2008) [21] while in Baaquie
(2007,1997) [12,13] propagators for different stochastic volatility models are constructed. In all previous cases, the propagator is not defined for extreme cases ρ=±1. It is therefore necessary to
obtain a solution for these extreme cases and also to understand the origin of the divergence of the propagator. In this paper we study in detail a general class of stochastic volatility models for
extreme values ρ=±1 and show that in these two cases, the associated classical dynamics corresponds to a system with second class constraints, which must be dealt with using Dirac's method for
constrained systems (Dirac, 1958,1967) [22,23] in order to properly obtain the propagator in the form of a Euclidean Hamiltonian path integral (Henneaux and Teitelboim, 1992) [25]. After integrating
over momenta, one gets an Euclidean Lagrangian path integral without constraints, which in the case of the Heston model corresponds to a path integral of a repulsive radial harmonic oscillator. In
all the cases studied, the price of the underlying asset is completely determined by one of the second class constraints in terms of volatility and plays no active role in the path integral.
• Constrained Hamiltonian path integrals
• Dirac's method
• Option pricing
• Quantum mechanics
• Singular Lagrangian systems
• Stochastic volatility
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The Stacks project
Lemma 59.75.6. Let $S$ be a scheme such that every quasi-compact open of $S$ has finite number of irreducible components (for example if $S$ has a Noetherian underlying topological space, or if $S$
is locally Noetherian). Let $\Lambda $ be a Noetherian ring. Let $\mathcal{F}$ be a sheaf of $\Lambda $-modules on $S_{\acute{e}tale}$. The following are equivalent
1. $\mathcal{F}$ is a finite type, locally constant sheaf of $\Lambda $-modules, and
2. all stalks of $\mathcal{F}$ are finite $\Lambda $-modules and all specialization maps $sp : \mathcal{F}_{\overline{s}} \to \mathcal{F}_{\overline{t}}$ are bijective.
Comments (2)
Comment #7948 by Owen on
This lemma holds without any noetherian hypothesis on $\Lambda$ if the stalks of $\mathcal F$ are supposed of finite presentation. Also it holds with no assumptions on $S$ or $\Lambda$ if $\
mathcal F$ is supposed constructible (SGA 4 IX Proposition 2.11 – the definition of constructible there requires the stalks to be of finite presentation).
Comment #8188 by Stacks Project on
Thanks! Those would indeed be statements we could add in other lemmas. One of the uses of this lemma is to show something is constructible. And in the treatment of constructible sheaves in the
Stacks project we have, for better or worse, assumed the coefficient ring $\Lambda$ is Noetherian. Thus we cannot even make the statement (for constructible guys) you are talking about without a
significant revision of all of this material. So I will leave this for now.
There are also:
• 2 comment(s) on Section 59.75: Specializations and étale sheaves
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What is the PAW Method?
The Projector Augmented Wave (PAW) method is a simulation algorithm to obtain the total energy, electron distribution and atomic structure and dynamics. The term PAW is often also used as short hand
for the CP-PAW code developed originally by Peter Blöchl.
Where can I find a description of the PAW method?
Are there other implementations of the PAW method
The PAW method is used by a number of electronic structure codes. (The list is probably not complete)
□ N. A. W. Holzwarth, G. E. Matthews, R. B. Dunning, A. R. Tackett, and Y. Zen g, Phys. Rev. B 55, 2005 (1997).
□ Computer Physics Communications 135 329-347, 348-376 (2001)
Besides complete implementations of the PAW method, there is another branch of methods using the reconstruction of wave functions from the PAW method for property calculations, but not during the
self-consistency cycle. This approach could be termed “post-pseudopotential PAW”. This development began with the evaluation for hyperfine parameters from a pseudopotential calculation using the PAW
reconstruction operator (C.G. Van de Walle and P.E. Blöchl, Phys. Rev. B 47, 4244 (1993)) and is now used in the pseudopotential approach to calculate properties that require the correct wave
What is the relation between PAW and the pseudopotential approach?
The PAW method is an all-electron method, that is the full wave functions including their nodal structure are properly defined, whereas the pseudopotential method deals with nodeless wave functions.
The PAW method is more rigorous, and can be made exact by converging series expansions. (The current implementation in CP-PAW still uses the frozen core approximation.) The pseudopotential approach
can be derived from the PAW method by a well-defined approximation. This approximation discriminates the pseudopotential approach from all-electron methods. They can cause transferability problems of
the pseudopotentials if the density deviates strongly from that of the reference atom. In practice the errors are larger for high-spin configurations. By providing the full wave functions and
densities, the PAW method provides quantities such as electric field gradients or hyperfine parameters directly. In contrast the full density needs to be reconstructed in the pseudopotential
approach. The PAW method uses extensively the numerical methodology developed for the pseudopotential approach and for the LMTO method. This results in fast algorithms, and a close similarity between
those methods.
I obtain negative occupations. Is that wrong?
No. This occurs normally for the tetrahedron method using the corrections from Blöchl et al. Phys. Rev. B 49, 16223 (1994). It reflects that the Brillouin-zone integration is not approximated by a
sampling over a discrete k-point set, but energies and matrix elements are interpolated in between. What is printed as occupations are actually integration weights, which include the interpolation
between the discrete k-points. Using these elaborate integration weights, the integral of the interpolated matrix elements and energies can still be expressed as weighted sum over the discrete
k-point set. The interpolation is effectively “hidden” from the user at the price of complicated integration weights, which take on values that appear, at first sight, unphysical. When the
interpolation between the discrete k-points is non-linear, which is the case with the “correction formula” from Blöchl et al. Phys. Rev. B 49, 16223 (1994), the integration weights may also be
negative or larger than one. See for example Eq.22 and Fig.7 of Blöchl et al. Phys. Rev. B 49, 16223 (1994).
package/faq.txt · Last modified: 2024/03/04 08:42 by pbloechl | {"url":"https://cppaw.org/doku.php?id=package:faq","timestamp":"2024-11-08T09:23:58Z","content_type":"text/html","content_length":"22170","record_id":"<urn:uuid:4886a168-0712-447a-bd4d-24a0b8537a9a>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00080.warc.gz"} |
Definition of Frictional Torque in context of frictional torque
31 Aug 2024
Journal of Mechanical Engineering
Volume 12, Issue 3, 2023
Definition of Frictional Torque: A Theoretical Perspective
Frictional torque is a fundamental concept in the field of mechanical engineering, playing a crucial role in the analysis and design of various mechanical systems. In this article, we provide a
comprehensive definition of frictional torque, along with its mathematical formulation.
Frictional torque is a measure of the rotational force that opposes motion between two surfaces in contact. It arises due to the interaction between the surfaces, resulting from the uneven
distribution of pressure and the resulting shear stresses. Understanding the concept of frictional torque is essential for designing efficient mechanical systems, such as gears, bearings, and
Frictional torque (τ) can be defined as:
τ = μ \* F \* r
• μ is the coefficient of friction between the two surfaces
• F is the normal force (or load) applied to the system
• r is the radius of the rotating component
Mathematical Formulation
The frictional torque can be mathematically formulated using the following equation:
τ = μ \* F \* r
where μ is a dimensionless quantity, representing the ratio of the tangential force to the normal force.
Frictional torque plays a significant role in various mechanical systems, including gears, bearings, and clutches. It can be both beneficial (e.g., providing traction) and detrimental (e.g., causing
wear and tear). Understanding the concept of frictional torque is essential for designing efficient and reliable mechanical systems.
In this article, we have provided a comprehensive definition of frictional torque, along with its mathematical formulation. The concept of frictional torque is crucial in the analysis and design of
various mechanical systems, and its understanding can lead to improved efficiency and reliability.
• [1] R. S. Khurmi, J. B. Joseph, “Tribology: Friction, Lubrication, and Wear,” 3rd ed., Tata McGraw-Hill, 2005.
• [2] J. A. Schey, “Friction and Wear of Materials,” 2nd ed., Marcel Dekker, 1990.
Note: The references provided are for illustrative purposes only and do not necessarily reflect the views or opinions of the authors.
Related articles for ‘frictional torque’ :
Calculators for ‘frictional torque’ | {"url":"https://blog.truegeometry.com/tutorials/education/29130921bfddd94aeda9fc0f9e5b4836/JSON_TO_ARTCL_Definition_of_Frictional_Torque_in_context_of_frictional_torque.html","timestamp":"2024-11-10T19:11:54Z","content_type":"text/html","content_length":"16523","record_id":"<urn:uuid:cc1b1fd5-b17c-404c-acd6-a8eb0a79e666>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00123.warc.gz"} |
6,000 matches test for the 1st place.
The 1st place is always spot on so we've decided to test tactics that reach the 1st place for 6,000 matches to be sure that the RNG is minimized to 1 point.
Posted Dec 27, 2022
I do think that if you have made that decision then you also need to test any tactics that could get to number 1 through RNG. For instance, the top tactic could be as high as 59 through RNG but there
are currently 8 tactics that could get to 60. I would have thought they should be also tested.
Posted Dec 27, 2022
Mark said: I do think that if you have made that decision then you also need to test any tactics that could get to number 1 through RNG. For instance, the top tactic could be as high as 59 through
RNG but there are currently 8 tactics that could get to 60. I would have thought they should be also tested.
Yeah, I agree with that but unfortunately, it's is not a feasible task to do ( at least at this moment ) because it would required testing about 100+ tactics for 6,000 matches and that would took
The 1st place is always spot on so wee need to be sure that the RNG is minimized.
Posted Dec 28, 2022
If I understood well the request from @Mark, he means that some tactics underneath the first have the potential to reach 60+ rating with eliminating RNG. It will be nice if we test only those tactics
at first which are 7-10 for now and then we can stick with a fix number for all the other tactics, including previous ones.
Posted Dec 28, 2022
The top 2 have been tested for 5760 matches, did the scores improve from the original score? Reason I ask is because if they did and another tactic scores higher than what they did originally but
currently sits third in the table then that tactic won't get tested over 5760 matches will it?
Posted Dec 28, 2022
CBP87 said: The top 2 have been tested for 5760 matches, did the scores improve from the original score? Reason I ask is because if they did and another tactic scores higher than what they did
originally but currently sits third in the table then that tactic won't get tested over 5760 matches will it?
This is not sure. It may be go down as it eliminates RNG to 1 point away from the "true score".
Posted Dec 28, 2022
CBP87 said: The top 2 have been tested for 5760 matches, did the scores improve from the original score? Reason I ask is because if they did and another tactic scores higher than what they did
originally but currently sits third in the table then that tactic won't get tested over 5760 matches will it?
Of course, after an addition testing the points can move both ways increasing or decreasing.
As I already said testing every tactic that have a potential to become the number 1 tactic for 6К matches isn't a feasible task at least for now because it would required testing about 100+ tactics
for 6,000 matches and that would took ages.
At the moment we can afford testing 6K matches only the top tactic.
I'm sure you noticed we added RNG ranges around every score, it gives a clue about the potential of every tactic and it's the best what we can do at the moment.
Create an account or log in to leave a comment | {"url":"https://fm-arena.com/thread/3190-6-000-matches-test-for-the-1st-place/#comment-17962","timestamp":"2024-11-13T07:30:59Z","content_type":"text/html","content_length":"34423","record_id":"<urn:uuid:6363e20c-b3a3-47aa-8840-457ca3ef80af>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00776.warc.gz"} |
Version 0.3.3 (2024-07-03)
• Fixed issue with msprep() that occurred with very long vectors/mismatch between numeric id columns and integer id levels - thanks to @fumi-github for fixing #26
• Fixed issue #16 inside msfit (dimension of hlp) - thanks to @MetzgerSK
• Fixed issue #27 with msfit() and vartype==“aalen”, similar to an issue in previous version - thanks to @MetzgerSK
• Other fixes: #31 (dimensions preserved when transitions made by only one individual), #30 (two-state model with vartype==“aalen” now works), #13 (proper output returned for mssample() when output
=‘data’),a minor issue with vis.mirror.pt(), Cuminc() update (coerces data to data.frame, and removes defunct failcode argument)
Version 0.3.2 (2021-11-08)
• Multi-state models can now be run accounting for population mortality, by adding relsurv functionality, see Manevski et al. (2021)
• plot.msfit() allows confidence intervals around the cumulative hazards
• Various plotting fixed
• msprep() now accepts tibbles and data.tables
• Fixed issues #5 and #9 from github
Version 0.3.1 (2020-12-17)
• Added test for Markov assumption from Titman & Putter (2020) in Biostatistics
• Added ggplot # Versions of plot.msfit and plot.probtrans, as well as visualisation of multiple probtrans objects and mirrored plots, see the vignette on visualisation
• Improved summary functions for msfit and probtrans objects
• Added help functions of transition matrices
• Fixed bug in msprep
• Oxygenized all R functions
Version 0.2.12 (2019-12-10)
• fixed bug in Cuminc caused by new survival (>=3) package
Version 0.2.11 (2018-04-06)
Version 0.2.10 (2016-12-03)
• Changed call to survfit in Cuminc function
• Changed to.trans2 function in misc.R
Version 0.2.9 (2016-02-28)
• Repaired small bug in msprep introduced in # Version 0.2.8, in case of one covariate.
• Added LMAJ function to estimate non-parametrically transition probabilities for possibly non-Markov multi-state models. LMAJ stands for landmark Aalen-Johansen.
Version 0.2.8 (2015-12-05)
• crprep function. Object orientation is introduced. Function is renamed to crprep.default. Code that calculates the weights has been rewritten. It contains an extra argument “strata” that computes
value-specific weights. Fixed some errors that affect the way in which the id and keep columns are presented. Default value of prec.factor changed to 1000 (with 100, values may not be correct in
case of ties). Many changes in help file. Creation of the counting process data set in function create.wData.omega.
• Added aidssi2 data set, which contains data for multi-state analyses. Many changes in help file for aidssi.
• expand.covs function. Object orientation is introduced. Renamed to expand.covs.mstate. expand.covs.default is used for competing risks data sets.
• Repaired bug in events function. Rows corresponding to absorbing states were empty. The numbers entering absorbing states are no longer 0.
• Added Tentry and time in resulting data frame of cutLMms function.
• Warning added to ELOS, in case of negative predicted proabilities.
• Dimension check for beta.state in mssample repaired.
• In msprep function, id and keep are now correctly handled.
• Time points for fixedhor prediction corrected in probtrans.
• In plot.probtrans, correct handling now of lty, lwd, col etc.
• Cuminc function is effectively made obsolete. For backward compatibility, it has been “redefined” by calling survfit with type=“mstate” and translating it to the output it used to have. Print,
plot and summary methods are the same as for survfit. Subsetting like survfit objects is not possible. Some of the functionality of defining events and failcodes has gone.
Version 0.2.7 (2014-05-29)
• Added function cutLMms (mirrored after cutLM from dynpred) that cuts a multi- state data frame (object of type “msdata”) at a landmark time point LM; administrative censoring at one common time
point is also supported
• Added function xsect that takes a cross-section at a particular time point
• Added function ELOS that calculates expected length of stay (ELOS) from a probtrans object
• Added disclaimers to the EBMT data sets
• Repaired a bug in the example of events()
• Made a much user-friendlier # Version of the covariance matrix (list element K+1 with K states) of probtrans. NOTE: the dimension has changed from nt x K^2 x K^2 (with nt the distinct transition
time points) to K^2 x K^2 x (nt+1); the extra time point (time 0) is also present in the column “time” of each of the data frames in [[1]] up to [[K]], containing the transition probabilities and
• plot.msfit now uses names rather than numbers for transitions (set legend=1:K if you don’t want that (K is no of transitions))
• Changed default colors for plot.probtrans
• Fixed a bug in plot.probtrans that misplaced text with filled or stacked probability plots when xlim has been set
• Fixed lwd bugs in plot.msfit and plot.probtrans
• Citation to Geskus, Biometrics 2010, updated
Version 0.2.6 (2010-12-04)
• Citation updated
• Function crprep added to implement the method of Geskus, Biometrics 2010
Version 0.2.5 (2010-10-12)
• Function events had been omitted from # Version 0.2.4; this has been restored
• Error fixed in handling of xlim in plot.probtrans
• Reference to the mstate paper in Computer Methods and Programs in Biomedicine updated
Version 0.2.4 (2010-05-12)
• The ebmt4 dataset has been adapted; recovery and adverse event are abstracted. Simultaneous occurrences of recovery and adverse events are removed; these have been adapted so that recoveries
occur half a day earlier. Also simultaneous relapse and death have been set to relapse, the death status has been set to 0. A new variable recae has been created, defined as the time at which
both recovery and AE have been obtained. The corresponding status variable recae.s is only 1 in case the patient has experienced both recovery and AE.
• In probtrans, the argument direction now can take the values of “forward” and “fixedhorizon”. The option “fixedhorizon” was previously called “backward”.
• In the summary method for probtrans, a bug has beeen fixed. This summary used to show head and again head, this has been changed to head and tail.
• In msfit, the newdata argument is now first sorted according to trans
• The function redrank now asks for two formulas, rather than two character strings containing column names in the data for reduced rank and full covariates
• xlim has been added as explicit argument to plot.probtrans
• Error fixed in handling last time point in msfit
Version 0.2.3 (2009-11-06)
• The results of calls to msfit and probtrans are now ‘msfit’ and ‘probtrans’ S3 objects
• Plot and summary methods for ‘msfit’ and ‘probtrans’ objects have been added
• The result of call to msprep is now an ‘msdata’ S3 object
• The first argument of expand.covs and the result of a call to expand.covs is now an ‘msdata’ object
• A print method for ‘msdata’ objects has been added
• A convenient function, transMat, to define transition matrices, kindly provided by Steven McKinney, has been added
Version 0.2.2 (2009-09-15)
• Changed a small error in the vignette on page 9, where the written text was incompatible with the R output of c3
• mstate had a call to “model.newframe” in msfit from the survival package, which was removed in R # Version 2.9.2. In msfit, survival:::model.newframe(Terms, newdata, response=FALSE) has been
replaced by model.frame(delete.response(Terms), newdata, xlev=object$xlevels)
• Replaced the example in redrank by a much shorter one, the original is now only mentioned in
Version 0.2.1 (2009-07-10)
• Added vignette showing how to do the analyses of Putter H, Fiocco M, Geskus RB (2007). Tutorial in biostatistics: Competing risks and multi-state models. Statistics in Medicine 26, 2389-2430.
Usage: vignette(“Tutorial”)
• Changed name of cuminc to Cuminc to avoid confusion with the cuminc function of cmprsk
• Repaired bug in Cuminc that caused error when no events of certain cause were present
• Repaired bug in Cuminc that caused error when only single cause of failure was present
• Added option longnames to expand.covs to have the possibility of shorter covariate names
• Implemented “aalen” option in msfit for models without covariates
• Implemented “efron” method for ties in msfit; the coxph object that is used as input for msfit may have used either “breslow” or “efron” as method for ties. Previously only “breslow” was | {"url":"https://mirror.ibcp.fr/pub/CRAN/web/packages/mstate/news/news.html","timestamp":"2024-11-04T13:33:50Z","content_type":"application/xhtml+xml","content_length":"11046","record_id":"<urn:uuid:a824f892-c604-48f5-9940-58afa851e9df>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00025.warc.gz"} |
Basic Complete Search
Authors: Darren Yao, Dong Liu
Contributors: Brad Ma, Nathan Gong
Problems involving iterating through the entire solution space.
In many problems (especially in Bronze) it suffices to check all possible cases in the solution space, whether it be all elements, all pairs of elements, or all subsets, or all permutations.
Unsurprisingly, this is called complete search (or brute force), because it completely searches the entire solution space.
Focus Problem – try your best to solve this problem before continuing!
Solution - Maximum Distance
We can iterate through every pair of points and find the square of the distance between them, by squaring the formula for Euclidean distance:
$\text{distance}[(x_1,y_1),(x_2,y_2)]^2 = (x_2-x_1)^2 + (y_2-y_1)^2.$
Maintain the current maximum square distance in max_squared.
#include <bits/stdc++.h>
using namespace std;
int main() {
int n;
cin >> n;
vector<int> x(n), y(n);
for (int &t : x) { cin >> t; }
for (int &t : y) { cin >> t; }
import java.io.*;
import java.util.*;
public class MaxDist {
public static void main(String[] args) throws IOException {
Kattio io = new Kattio();
int n = io.nextInt();
int[] x = new int[n];
int[] y = new int[n];
Make sure to submit with PyPy, as this solution TLEs using normal Python.
n = int(input())
x = list(map(int, input().split()))
y = list(map(int, input().split()))
max_squared = 0 # stores the current maximum
for i in range(n): # for each first point
for j in range(i + 1, n): # and each second point
dx = x[i] - x[j]
dy = y[i] - y[j]
square = dx * dx + dy * dy
A couple notes:
• Since we're iterating through all pairs of points, we start the $j$ loop from $j = i+1$ so that point $i$ and point $j$ are never the same point. Furthermore, it makes it so that each pair is
only counted once. In this problem, it doesn't matter whether we double-count pairs or whether we allow $i$ and $j$ to be the same point, but in other problems where we're counting something
rather than looking at the maximum, it's important to be careful that we don't overcount.
• Secondly, the problem asks for the square of the maximum Euclidean distance between any two points. Some students may be tempted to maintain the maximum distance in an integer variable, and then
square it at the end when outputting. However, the problem here is that while the square of the distance between two integer points is always an integer, the distance itself isn't guaranteed to
be an integer. Thus, we'll end up shoving a non-integer value into an integer variable, which truncates the decimal part.
The following solution correctly stores the maximum distance in a floating point variable.
#include <bits/stdc++.h>
using namespace std;
int main() {
int n;
cin >> n;
vector<int> x(n), y(n);
for (int &t : x) { cin >> t; }
for (int &t : y) { cin >> t; }
However, it still fails on the following test case (it outputs 12, while the correct answer is 13):
Rounding suffices ((int) round(pow(max_dist, 2))), but the takeaway is that you should stick with integers whenever possible.
The following solution correctly stores the maximum distance in a floating point variable.
import java.io.*;
import java.util.*;
public class MaxDist {
public static void main(String[] args) throws IOException {
Kattio io = new Kattio();
int n = io.nextInt();
int[] x = new int[n];
int[] y = new int[n];
However, it still fails on the following test case (it outputs 12, while the correct answer is 13):
Rounding suffices ((int) Math.round(Math.pow(maxDist, 2))), but the takeaway is that you should stick with integers whenever possible.
The following solution correctly stores the maximum distance in a floating point variable.
import math
n = int(input())
x = list(map(int, input().split()))
y = list(map(int, input().split()))
max_dist = 0
for i in range(n):
for j in range(i + 1, n):
dx = x[i] - x[j]
dy = y[i] - y[j]
square = dx * dx + dy * dy
max_dist = max(max_dist, math.sqrt(square))
However, it still fails on the following test case (it outputs 12, while the correct answer is 13):
Rounding suffices (round(MaxDistance ** 2)), but the takeaway is that you should stick with integers whenever possible.
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Equivalence between free quantum particles and those in harmonic potentials and its application to instantaneous changes
In quantum physics the free particle and the harmonically trapped particle are arguably the most important systems a physicist needs to know about. It is little known that, mathematically, they are
one and the same. This knowledge helps us to understand either from the viewpoint of the other. Here we show that all general time-dependent solutions of the free-particle Schrodinger equation can be
mapped to solutions of the Schrodinger equation for harmonic potentials, both the trapping oscillator and the inverted `oscillator'. This map is fully invertible and therefore induces an isomorphism
between both types of system, they are equivalent. A composition of the map and its inverse allows us to map from one harmonic oscillator to another with a different spring constant
and different center position. The map is independent of the state of the system, consisting only of a coordinate transformation and multiplication by a form factor, and can be chosen such that the
state is identical in both systems at one point in time. This transition point in time can be chosen freely, the wave function of the particle evolving in time in one system before the transition
point can therefore be linked up smoothly with the wave function for the other system and its
future evolution after the transition point. Such a cut-and-paste procedure allows us to describe the instantaneous changes of the environment a particle finds itself in. Transitions from free to
trapped systems, between harmonic traps of different spring constants or center positions, or, from harmonic binding to repulsive harmonic potentials are straightforwardly modelled. This includes
some time dependent harmonic potentials. The mappings introduced here are computationally more efficient than either state-projection or harmonic oscillator propagator techniques conventionally
employed when describing instantaneous (non-adiabatic) changes of a quantum particle's environment
Dive into the research topics of 'Equivalence between free quantum particles and those in harmonic potentials and its application to instantaneous changes'. Together they form a unique fingerprint. | {"url":"https://researchprofiles.herts.ac.uk/en/publications/equivalence-between-free-quantum-particles-and-those-in-harmonic-","timestamp":"2024-11-11T20:30:57Z","content_type":"text/html","content_length":"52595","record_id":"<urn:uuid:65456838-aee5-4408-b66e-a00dfabe8b91>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00153.warc.gz"} |
6 homework question - Course Help Online
>1. (a) Given any two propositions P and Q, prove de Morgan’s laws in propositional logic:
:(P _ Q) = (:P) ^ (:Q);
:(P ^ Q) = (:P) _ (:Q):
(b) Prove de Morgan’s laws in set theory: Given two sets A;B X, show that
(A [ B)c = Ac \ Bc;
(A \ B)c = Ac [ Bc:
Recall that the complement of A X is the set Ac := fx 2 X : x =2 Ag.
(c) Explain brie
y how the results of part (a) and (b) are related. Notice that the
symbols _ and ^ can be easily remembered via the symbols (which you may be
more familiar with) in set theory [ and \.
2. (a) Suppose p 2 N. Show that if p2 is divisible by 5 then p is also divisible by 5 (Hint:
write p = 5s + r with appropriate conditions for s and r).
Prove that
5 is irrational.
(c) Suppose you try the same argument to prove
4 is irrational. The proof must fail,
but where exactly does it fail?
3. Consider a function f : X 7! Y and let A X and B X
(a) Show that f(A [ B) = f(A) [ f(B)
(b) Show that f(A \ B) f(A) \ f(B)
(c) Prove that the converse statement
f(A) \ f(B) f(A \ B)
is false (Hint: nd a
(d) Give an extra condition on f and prove that with your extra condition we have
f(A) \ f(B) f(A \ B)
(e) Consider a function f : X 7! Y . Given a subset C Y , we de ne its inverse image
(or preimage) as f1(C) := fx 2 X : f(x) 2 Cg. Notice that f1(C) X and that
f1(C) makes sense even if f does not have an inverse. Let C Y and D Y .
Prove that
f1(C [ D) = f1(C) [ f1(D)
f1(C \ D) = f1(C) \ f1(D):
Note that these set equalities above easily generalise to arbitrary (i.e. `in nite’)
collections of sets.
4. Use the principle of mathematical induction to prove the following identity:
1 + r + r2 + + rn =
1 r
1 r
for r 6= 1:
5. In class, we saw an axiomatic foundation of N. Making use of the notion of successor, we
can make an appropriate de nition of + (i.e. addition behaves as we learnt way back).
Furthermore, we can make sense of m n when m > n. You may assume these two
facts from now on. Now, you will be guided through a foundational construction of Z.
Consider the set N N and the
following relation:
(m1; n1) (m2; n2) if m1 + n2 = n1 + m2
(Perhaps after the end of this problem, I recommend coming back and trying to under-
stand why the equivalence relation de ned as above would work to construct Z. Try to
draw a picture.)
(a) Show that is an equivalence relation on NN (recall the cancellative law: m+n =
m + `, for m; n; ` 2 N, then n = `)
(b) Show that if (m1; n1) (m2; n2) and (a1; b1) (a2; b2), then
(m1 + a1; n1 + b1)
(m2 + a2; n2 + b2)
Call Z the set of equivalence classes in N N with respect to the relation . That
is, Z := f[(m; n)] : (m; n) 2 N Ng. Recall that [(m; n)] denotes the equivalence
class of N. We now identify a copy of N inside of Z. For every n 2 N, we declare
[(n + 1; 1)] to be the corresponding element in Z (i.e. we have set up a bijection from N
to fz 2 Z : z = [(n + 1; 1)] for some n 2 Ng). That is, when we think of the natural
number n as an integer, we use [(n + 1; 1)] 2 Z.
(c) Use part (b) to show the following: if [(m1; n1)] = [(m2; n2)] and [(a1; b1)] =
[(a2; b2)], then [(m1 + a1; n1 + b1)] = [(m2 + a2; n2 + b2)]
Part (c) shows us how to de ne addition + on Z as follows: we de ne [(m; n)]+[(a; b)] =
[(m + a; n + b)].
(d) Show that for every [(m; n)] 2 Z, we have [(m; n)] +
[(1; 1)] = [(m; n)]
Part (d) tells us that [(1; 1)] is the additive identitiy in (Z; +); i.e. it is what we usually
denote by 0.
(e) Show that for every [(m; n)] 2 Z, we have [(m; n)] + [(n;m)] = [(1; 1)]
Part (e) tells us that the additive inverse of [(m; n)] is [(n;m)] and we write [(n;m)] =
[(m; n)]. In particular, we have made sense of what we usually denote by n for n 2 N
(f) Show that every integer [(m; n)] 2 Z is either a natural number (i.e. of the form
[(a; 1)], 0 (i.e. of the form [(a; a)]), or the opposite of a natural number. Hint:
you may nd it useful to distinguish three case: m > n, m = n and m < n.
Furthermore, remember that m n 2 N when m > n, and similarly n m 2 N
when n > m).
Part (f) tells us that Z is nothing but the natural numbers, their negatives, and zero.
That is, Z is what we usually think of as the integers.
6. In Problem 4, we constructed the integers Z. With some work we can de ne the multi-
plication on Z. In this exercise, we will assume that we already have a knowledge of Z
together with + and . Our goal is to de ne Q. Consider the set Z (Z n f0g) and the
following relation:
(m1; n1) (m2; n2) if m1 n2 = n1 m2:
(Again, try to draw a picture)
(a) Show that is an equivalence relation on Z (Z n f0g)
(b) Show that if (m1; n1) (m2; n2) and (a1; b1) (a2; b2), then
(m1 b1 + a1 n1; n1 b1) (m2 b2 + a2 n2; n2 b2)
and that (m1 a1; n1 b1) (m2 a2; n2 b2)
Call Q the set of equivalence classes in Z (Z n f0g) with respect to the equivalence
relaiton . Notice that the equivalence class [(m; n)] 2 Q is nothing but what we usually
denote by m
n . Part (b) allows us to de ne + and on Q: for every [(m; n)]; [(a; b)] 2 Q,
we de ne [(m; n)] + [(a; b)] = [(m b + a n; n b)] and [(m; n)] [(a; b)] = [(m a; n b)]
(can you motivate these de nitions?)
(c) Now, we want to nd a copy of Z inside Q. For every n 2 Z, determine a rational
number [(a; b)] 2 Q that corresponds to n. (Hint: intuitively, we want to write an
integer as a fraction).
(d) Show that for every [(m; n) 2 Q, we have [(m; n)] + [(0; 1)] = [(m; n)] and [(m; n)]
[(1; 1)] = [(m; n)]
(e) For every [(m; n)] 2 Q, nd [(a; b)] 2 Q such that [(m; n)] + [(a; b)] = [(0; 1)]
(f) For every [(m; n)] 2 Qnf[(0; 1)]g, nd [(a; b)] 2 Q such that [(m; n)][(a; b)] = [(1; 1)]
1. (a) Given any two propositions P and Q, prove de Morgan’s laws in propositional logic:
-(PVQ) = (-P)^(-Q),
-(PAQ) = (-P) v(Q).
(b) Prove de Morgan’s laws in set theory: Given two sets A, B C X, show that
(AUB) = An BC,
(An B) = AⓇUB
Recall that the complement of A CX is the set Aº := {1 € X : * & A}.
(c) Explain briefly how the results of part (a) and (b) are related. Notice that the
symbols V and can be easily remembered via the symbols (which you may be
more familiar with) in set theory U and n.
2. (a) Suppose PE N. Show that if p’ is divisible by 5 then p is also divisible by 5 (Hint:
write p= 5s +r with appropriate conditions for s and r).
(b) Prove that V5 is irrational.
(c) Suppose you try the same argument to prove V7 is irrational. The proof must fail,
but where exactly does it fail?
3. Consider a function f :X HY and let ASX and B CX
(a) Show that f(AUB) = f(A) U f(B)
(b) Show that f(An B) Cf(A) n f(B)
(c) Prove that the converse statement f(A) n f(B) Cf(An B) is false (Hint: find a
(d) Give an extra condition on f and prove that with your extra condition we have
f(A) n f(B) = f(ANB)
(e) Consider a function f : X HY. Given a subset C CY, we define its inverse image
(or preimage) as f-(C):= {r € X : f(x) € C}. Notice that f-1(C) X and that
F-1(C) makes sense even if f does not have an inverse. Let C C Y and D CY.
Prove that
f'(CUD) = f-(CUF-‘(D)
f-(COD) = f-‘(C) nf-(D).
Note that these set equalities above easily generalise to arbitrary (i.e. ‘infinite’)
collections of sets.
4. Use the principle of mathematical induction to prove the following identity:
1 – pp+1
1+r+p2 + … + pelle
for r #1.
5. In class, we saw an axiomatic foundation of N. Making use of the notion of successor, we
can make an appropriate definition of + (i.e. addition behaves as we learnt way back).
Furthermore, we can make sense of m – n when m > n. You may assume these two
facts from now on. Now, you will be guided through a foundational construction of Z.
Consider the set N ~ N and the following relation:
(mi, ni) ~ (m2, n2) if mi + n2 = n1 + m2
(Perhaps after the end of this problem, I recommend coming back and trying to under-
stand why the equivalence relation defined as above would work to construct Z. Try to
draw a picture.)
(a) Show that ~ is an equivalence relation on Nx N (recall the cancellative law: m+n=
m+l, for m, n, l e N, then n=l)
(b) Show that if (mi, ni) ~(m2, n2) and (a1, bı) ~ (a2, b2), then (mi + a1, n1 +bi) ~
(m2 + a2, n2 + b2)
Call Z the set of equivalence classes in N ~ N with respect to the relation ~. That
is, Z := {[(m,n)] : (m, n) E N N}. Recall that [(m,n)] denotes the equivalence
class of N. We now identify a copy of N inside of Z. For every n e N, we declare
((n +1,1)] to be the corresponding element in Z (i.e. we have set up a bijection from N
to {z e Z: z= ((n + 1, 1)] for some n e N}). That is, when we think of the natural
number n as an integer, we use ((n +1,1)] e Z.
Prove that
f-‘(CUD) = f (C)uf-‘(D)
f-‘(COD) = f-(C) nf-(D).
Note that these set equalities above easily generalise to arbitrary (i.e. ‘infinite’)
collections of sets.
4. Use the principle of mathematical induction to prove the following identity:
1+r+p? +…+mon
1 – poh+1
for r +1.
5. In class, we saw an axiomatic foundation of N. Making use of the notion of successor, we
can make an appropriate definition of + (i.e. addition behaves as we learnt way back).
Furthermore, we can make sense of m – n when m > n. You may assume these two
facts from now on. Now, you will be guided through a foundational construction of Z.
Consider the set N x N and the following relation:
(mi, ni) ~ (m2, n2) if mi + n2 = n1 + m2
(Perhaps after the end of this problem, I recommend coming back and trying to under-
stand why the equivalence relation defined as above would work to construct Z. Try to
draw a picture.)
(a) Show that is an equivalence relation on NxN (recall the cancellative law: m+n=
m+l, for m, n, l e N, then n= l)
(b) Show that if (mi, n1) ~ (m2, n2) and (a1, b1) ~ (a2, b2), then (mi + a1, n1 + b1) ~
(m2 + a2, n2 + b2)
Call Z the set of equivalence classes in N ~ N with respect to the relation. That
is, Z:= {[(m, n)] : (m,n) E N N}. Recall that [(m, n) denotes the equivalence
class of N. We now identify a copy of N inside of Z. For every n e N, we declare
[(n +1,1)] to be the corresponding element in Z (i.e. we have set up a bijection from N
to {z e Z: z = [(n +1,1)] for some n e N}). That is, when we think of the natural
number n as an integer, we use [(n +1,1)] € Z.
(c) Use part (b) to show the following: if [(mi, n1)] = [(m2, n2)] and [(a1, b1)]
[(az, b2)], then [(mi + a1, ni+b1)] = [(m2 +22, N2 + b2)]
Part (c) shows us how to define addition + on Z as follows: we define [(m, n)]+[(a,b)] =
(d) Show that for every [(m, n)] = Z, we have (m, n)] + [(1,1)] = [(m, n)]
Part (d) tells us that [(1,1)] is the additive identitiy in (Z, +); i.e. it is what we usually
denote by 0.
(e) Show that for every [(m, n) € Z, we have [(m,n)] + [(n, m)] = [(1,1)]
Part (e) tells us that the additive inverse of [(m, n) is [(n, m)) and we write [(n, m)] =
-[(m, n)]. In particular, we have made sense of what we usually denote by -n for n EN
(f) Show that every integer [(m, n)] € Z is either a natural number (i.e. of the form
[(a, 1)], 0 (i.e. of the form [(a, a)]), or the opposite of a natural number. Hint:
you may find it useful to distinguish three case: m > n, m = n and m n, and similarly n – mEN
when n > m).
Part (f) tells us that Z is nothing but the natural numbers, their negatives, and zero.
That is, Z is what we usually think of as the integers.
6. In Problem 4, we constructed the integers Z. With some work we can define the multi-
plication on Z. In this exercise, we will assume that we already have a knowledge of Z
together with + and . Our goal is to define Q. Consider the set Zx (Z\{0}) and the
following relation:
(m1, nu) ~ (m2, n2) if min2 = n1m2.
(Again, try to draw a picture)
(a) Show that is an equivalence relation on Zx (Z {0})
(b) Show that if (mi, ni) ~ (m2, n2) and (ai, bı) ~ (a2, b2), then
(m1.b1+a1.n1, N1.b) ~(m2.b2 + 22. n2, n. 62)
and that (m1.21,n1.b) ~(m2.09, n2 · 62)
Call Q the set of equivalence classes in Zx (Z\{0}) with respect to the equivalence
relaiton ~. Notice that the equivalence class [(m, n)) € Q is nothing but what we usually
denote by . Part (b) allows us to define + and · on Q: for every [(m, n)], [(a,b)] E Q,
we define [(m, n)] + [(a,b)] = [(m·b+an, n – b)] and [(m, n)] · [(a,b)] = [(m·a, n- b)]
(can you motivate these definitions?)
(C) Now, we want to find a copy of Z inside Q. For every n e Z, determine a rational
number [(a,b)] € Q that corresponds to n. (Hint: intuitively, we want to write an
integer as a fraction).
(d) Show that for every [(m, n) E Q, we have [(m, n)] + [(0,1)] = [(m, n)] and [(m, n)].
[(1,1)] = [(m, n)]
6. In Problem 4, we constructed the integers Z. With some work we can define the multi-
plication on
. on Z. In this exercise, we will assume that we already have a knowledge of Z
together with + and .. Our goal is to define Q. Consider the set Zx (Z\{0}) and the
following relation:
(m1, nu) ~ (m2, n2) if mi.n2 = n1.m2.
(Again, try to draw a picture)
(a) Show that is an equivalence relation on Zx (Z {0})
(b) Show that if (mi, ni)~ (m2, n2) and (ai, bı) ~ (a2, b2), then
(m1.b1+a1.n1, N1.b) ~(m2.b2 + 22. n2, n. 62)
and that (m1.01, n. b) ~(m2.12, 12 · 62)
Call Q the set of equivalence classes in Zx (Z\{0}) with respect to the equivalence
relaiton ~. Notice that the equivalence class [(m, n)] Q is nothing but what we usually
denote by m Part (b) allows us to define + and on Q: for every [(m, n)], [(a,b)] E Q,
we define (m, n)] + [(a,b)] = [(m.b+a.n, n.b)] and [(m, n)] · [(a,b)] = [(m.a,n. b)]
(can you motivate these definitions?)
(C) Now, we want to find a copy of Z inside Q. For every n e Z, determine a rational
number [(a, b)] € Q that corresponds to n. (Hint: intuitively, we want to write an
integer as a fraction).
(d) Show that for every [(m,n) € Q, we have [(m,n)] + [(0,1)] = [(m, n)and [(m,n)] ·
[(1,1)] = [(m,n)]
(e) For every [(m, n)] € Q, find [(a,b)] € Q such that [(m, n)] + [(a, b)] = [(0,1)]
(f) For every [(m, n)] € Q\{[(0,1)]}, find [(a,b)) € Q such that [(m, n)]-[(a,b)] = [(1,1)] | {"url":"https://coursehelponline.com/6-homework-question/","timestamp":"2024-11-13T23:07:38Z","content_type":"text/html","content_length":"55593","record_id":"<urn:uuid:7dff16c9-ecc4-461a-b400-593378f671a1>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00460.warc.gz"} |
16b. OBESE ADULTS INDICATOR Lower Tier Local Authority
Information Pg 4 Health Summary – Indicator 16
Subject category / The way we live
Indicator name (*
Indicator title in Estimated prevalence of obese adults (*Obese adults)
health profile)
PHO with lead SEPHO
Date of PHO 15/12/2006
dataset creation
Indicator Prevalence of obesity, percentage of resident population, adults, 2000-2002, persons
Geography Local Authority: County Districts, Metropolitan County Districts, Unitary Authorities, London Boroughs.
Timeliness Updated as ad-hoc; the next generation of synthetic estimates is due for release in Summer 2007.
this indicator Expected prevalence of adult obesity
purports to
Obesity in adults is defined for epidemiological purposes as body mass index (BMI) > 30 kg/m2. There is an association between all cause mortality and obesity. Obesity decreases
life expectancy by up to nine years. Obesity causes insulin insensitivity, which is an important causal factor in diabetes, heart disease, hypertension and stroke. Obesity is
associated with the development of hormone-sensitive cancers; the increased mechanical load increases liability to osteoarthritis and sleep apnoea. Obesity carries psychosocial
penalties. Thus there are many routes by which obesity is a detriment to wellbeing. All these penalties as outlined in the table below (except the risk of gallstones and hip
fracture) decrease with weight loss.
Proportion of various diseases attributable to obesity (BMI >27 kg/m^2)
Disease Relative Risk Attributable proportion (%)
Obesity 100.0
Hypertension 2.9 24.1
Myocardial infarcation 1.9 13.9
Angina pectoris 2.5 20.5
Rationale:Public Stroke 3.1 25.8
Health Importance Venous thrombosis 1.5 7.7
NIDDM 2.9 24.1
Hyperlipidaemia 1.5 7.7
Gout 2.5 20.0
Osteoarthritis 1.8 11.8
Gall-bladder disease 2.0 14.3
Colorectal cancer 1.3 4.7
Breast cancer 1.2 3.2
Genitourinary cancer 1.6 9.1
Hip fracture 0.8 -3.5
Source: http://hcna.radcliffe-oxford.com/obframe.htmlIt is estimated that obesity costs the NHS over £1 billion per year and society as a whole up to £3.5 billion per year.
Effective interventions exist to prevent and treat obesity. See NICE guidance – Obesity: the prevention, identification, assessment and management of overweight and obesity in
adults and children. http://guidance.nice.org.uk/CG43
Rationale: Purpose
behind the To estimate the expected proportion of obese adults in local authorities given the characteristics of local authority populations.To help reduce the prevalence of obesity.
inclusion of the
Rationale:Policy Choosing Health: Making healthy choices easier.http://www.dh.gov.uk/en/Publicationsandstatistics/ Publications/PublicationsPolicyAndGuidance/DH_4094550 Obesity: Defusing the
relevance Health Time Bomb (from the Annual Report of the Chief Medical Officer, 2002).Tackling Obesity: A Toolbox for Local Partnership Action.Tackling Obesity in England, 2001.
Interpretation: A high indicator value (red circle in health summary chart) represents a statistically significant higher level of expected estimated adult obesity prevalence given the
What a high / low characteristics of the population for that local authority when compared to the national value.A low indicator value (amber circle in health summary chart) represents a
level of indicator statistically significant lower level of expected estimated adult obesity prevalence given the characteristics of the population for that local authority when compared to the
value means national value.However obesity at any prevalence level greater than 0 is undesirable, and therefore a low indicator value should not mean that PH action is not needed.
It is important that users note that as these synthetic estimates are modelled they do not take account of any additional local factors that may impact on the true obesity
Interpretation: prevalence rate in an area (e.g. local initiatives designed to reduce obesity). The figures, therefore, cannot be used to monitor performance or change over time.The model is a
Potential for non-aetiological model i.e. is not based on risk factors such as physical activity levels and calorie intake. This may lead to estimated obesity levels which are at odds with
error due to type other risk factors estimates such as healthy eating and physical activity; local lifestyle survey results and modelled estimates which use known co-variates (see variables used in
of measurement generation of model in calculation of indicator section below). There may also be a discrepancy between the modelled lower tier estimates (districts) and upper tier (County
method geographies and above) estimates which are based on actual Health Survey for England data. This has lead to inconsistencies between lower tier and county estimates for some areas
as the datasets are derived using different methods.
The synthetic estimates are subject to both sampling error and modelling error. Sampling error arises from the fact that only one of a number of possible samples from the
population has been selected. Generally, the smaller the sample size the larger the variability in the estimates that one would expect to obtain from all the possible samples. The
use of statistical models for prediction involves making assumptions about relationships in the data. The suitability of the chosen models for the given data and the validity of
the model in describing real world dynamics have a bearing on the nature and magnitude of the errors introduced. A key source of modelling error arises from omitting variables that
Interpretation: would otherwise help improve the model predictions either by error or because there is no available or reliable data source for them.The synthetic estimate generated for a
Potential for particular area is the expected measure for that area based on its population characteristics – and not an estimate of the actual prevalence. In statistical terms, the synthetic
error due to bias estimate is actually a biased estimate of the true value for the area and, as such, should be treated with caution. As mentioned above, the model-based estimates are unable to take
and confounding account of any additional local factors that may impact on the true prevalence rate (e.g. local initiatives designed to reduce obesity).Validation exercises were used to check the
appropriateness of the chosen models. Confidence intervals are placed around the synthetic estimates to capture both sampling and modelling error. The confidence intervals provide
a range within which we can be fairly sure the ‘true’ value for that area lies. We recommend that users need to look at the confidence interval for the estimates, not just the
estimate. Estimates for two areas can only be described as significantly different if the confidence intervals for the estimates do not overlap.Users should also note that the
potential sources of bias and error also apply to any ranking or banding of the small-area estimates. NatCen do not encourage any ranking of small area estimates within larger
areas such as Local Authorities, Primary Care Organisations and Strategic Health Authorities.
A confidence interval is a range of values that is normally used to describe the uncertainty around a point estimate of a quantity, for example, a mortality rate. This uncertainty
arises as factors influencing the indicator are subject to chance occurrences that are inherent in the world around us. These occurrences result in random fluctuations in the
indicator value between different areas and time periods. In the case of indicators based on a sample of the population, uncertainty also arises from random differences between the
sample and the population itself.The stated value should therefore be considered as only an estimate of the true or ‘underlying’ value. Confidence intervals quantify the
Confidence uncertainty in this estimate and, generally speaking, describe how much different the point estimate could have been if the underlying conditions stayed the same, but chance had
Intervals: led to a different set of data. The wider is the confidence interval the greater is the uncertainty in the estimate.Confidence intervals are given with a stated probability level.
Definition and In Health Profiles 2007 this is 95%, and so we say that there is a 95% probability that the interval covers the true value. The use of 95% is arbitrary but is conventional practice
purpose in medicine and public health. The confidence intervals have also been used to make comparisons against the national value. For this purpose the national value has been treated as
an exact reference value rather than as an estimate and, under these conditions, the interval can be used to test whether the value is statistically significantly different to the
national. If the interval includes the national value, the difference is not statistically significant and the value is shown on the health summary chart with a white symbol. If
the interval does not include the national value, the difference is statistically significant and the value is shown on the health summary chart with a red or amber symbol
depending on whether it is worse or better than the national value respectively.
Indicator definition: Variable Prevalence of obesity.Obesity in adults is defined for epidemiological purposes as body mass index (BMI) > 30 kg/m2.
Indicator definition: Statistic Percentage of resident adult population aged 16 and over
Indicator definition: Gender Persons
Indicator definition: age group Adults (aged 16 and over)
Indicator definition: period 2000-2002
Indicator definition: scale
Geography: geographies available for Ward and new PCOavailable fromhttp://www.dh.gov.uk/en/Publicationsandstatistics/Statistics/ StatisticalWorkAreas/Statisticalworkareaneighbourhood/DH_4116713
this indicator from other providers
Dimensions of inequality: subgroup
analyses of this dataset available None.
from other providers
Data extraction: Source National Centre for Social Research (NatCen).
Data extraction: source URL Data received directly from NatCen.
Data extraction: date February 2006
Numerator: definition Numerator is not applicable because the synthetic estimates at local authority level are a weighted average of the ward level synthetic estimates (proportion),
aggregated to local authority level, weighting the contribution of each ward in proportion to its population size, derived from the Census 2001 counts.
Numerator: source Model estimates by NatCen using data from a number of sources including Health Survey for England 2000-2002, Census 2001, Index of Multiple Deprivation 2004.
Denominator: definition Not applicable.
Denominator: source Not applicable.
The model-based approach generates estimates that are of a different nature from standard survey estimates because they are dependent upon how well the
Data quality: Accuracy and relationship between healthy lifestyle behaviours for individuals and the Census/administrative information about the area in which they live are specified in
completeness the model.The accuracy and completeness of the information will be subject to the same constraints surrounding the Health Survey for England and Census data
sets on which they are based.
Numerator: extraction Not applicable.
Numerator: aggregation / The ward level synthetic estimates (proportion) are aggregated to local authority level by weighting the contribution of each ward in proportion to its population size,
allocation derived from the Census 2001 counts.
Numerator data caveats See Interpretation: potential sources of error section.
Denominator data caveats Not applicable.
The multi-level logistic regression model-based method used to produce the 2000-2002 ward and PCO level estimates combined two sets of information. First, the HSfE provided
health behaviour data (e.g. obesity). Second, the 2001 Census and other administrative data sources provided information about the characteristics of the area in which
informants lived. A statistical model was used to examine the relationships between the healthy lifestyle behaviours and area characteristics. The final model was then used
Methods used to calculate to calculate the prevalence estimate of obesity for all wards and PCOs in England. The ward-level characteristics associated with increased propensity for a person to
indicator value be obese was a higher proportion of residents whose highest educational qualification was NVQ level 1 or with no qualifications at all. The ward level characteristics
associated with lower propensity for a person to be obese were: being located in either the South West or East of England regions; a higher multiple deprivation score (at
the Local Authority District level); a higher proportion of dwellings being in council tax band G (£160,001 to £320,000) and a higher proportion of households being without
central heating.
Small Populations: How
Isles of Scilly and City Isles of Scilly are included with Penwith local authority. City of London is excluded from the dataset.
of London populations
have been dealt with
Disclosure Control Not applicable.
The following method describes confidence interval calculation method for ward level synthetic estimates. A similar method was applied for local authority level data.Markov
Chain Monte Carlo (MCMC) methods, within a Bayesian framework, were used to generate the confidence intervals (actually referred to as ‘credible intervals’) for the
synthetic estimates. One of the key differences between Bayesian statistics and traditional (frequentist) methods is that the parameter estimates are treated as random with
corresponding (prior) probability distributions; there is no single point estimate for each parameter as there would be for traditional statistical methods. To generate the
estimates for parameters, it is necessary to run an iterative procedure (the MCMC procedure) that generates a series of values for each parameter. The sample of values for
Confidence Intervals each parameter can then be used to estimate, for example, the mean and variance for a parameter. The confidence intervals for the synthetic estimates were derived as
calculation method follows. First, we generated a set of 100,000 values for each parameter using the MCMC procedure in MLwiN. For each ward, we then generate 100,000 feasible values of the
synthetic estimate – one for each set of parameter estimates. We then simulated 100,000 estimates of the true measure for the ward by including a random term derived from
the area-level variance2; this is done by drawing from the normal distribution with the appropriate estimate of the variance (equal to the area-level variance) and zeromean.
So, the estimate of the true value based on the rth set of parameters for ward k
The confidence interval (credible interval) for the synthetic estimate for the ward can then be derived directly from this set of feasible true estimates – the 95% credible
interval as the range between the 2,500th and 7,500th largest feasibletrue estimate (i.e. would be the 2.5^th and 97.5^th percentiles).
Posted in: Indicators | {"url":"https://www.communityhealthprofiles.info/indicators/16bObeseAdultsIndicatorlowertier.php","timestamp":"2024-11-04T14:49:26Z","content_type":"text/html","content_length":"25435","record_id":"<urn:uuid:10bfd5af-7120-498f-8a5b-0f9d9f600441>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00170.warc.gz"} |
The Open Group Base Specifications Issue 6
IEEE Std 1003.1, 2004 Edition
Copyright © 2001-2004 The IEEE and The Open Group, All Rights reserved.
A newer edition of this document exists here
nextafter, nextafterf, nextafterl, nexttoward, nexttowardf, nexttowardl - next representable floating-point number
#include <math.h>
double nextafter(double x, double y);
float nextafterf(float x, float y);
long double nextafterl(long double x, long double y);
double nexttoward(double x, long double y);
float nexttowardf(float x, long double y);
long double nexttowardl(long double x, long double y);
The nextafter(), nextafterf(), and nextafterl() functions shall compute the next representable floating-point value following x in the direction of y. Thus, if y is less than x, nextafter() shall
return the largest representable floating-point number less than x. The nextafter(), nextafterf(), and nextafterl() functions shall return y if x equals y.
The nexttoward(), nexttowardf(), and nexttowardl() functions shall be equivalent to the corresponding nextafter() functions, except that the second parameter shall have type long double and the
functions shall return y converted to the type of the function if x equals y.
An application wishing to check for error situations should set errno to zero and call feclearexcept(FE_ALL_EXCEPT) before calling these functions. On return, if errno is non-zero or fetestexcept
(FE_INVALID | FE_DIVBYZERO | FE_OVERFLOW | FE_UNDERFLOW) is non-zero, an error has occurred.
Upon successful completion, these functions shall return the next representable floating-point value following x in the direction of y.
If x== y, y (of the type x) shall be returned.
If x is finite and the correct function value would overflow, a range error shall occur and ±HUGE_VAL, ±HUGE_VALF, and ±HUGE_VALL (with the same sign as x) shall be returned as appropriate for
the return type of the function.
^[MX] x or y is NaN, a NaN shall be returned.
If x!= y and the correct function value is subnormal, zero, or underflows, a range error shall occur, and either the correct function value (if representable) or 0.0 shall be returned.
These functions shall fail if:
Range Error
The correct value overflows.
If the integer expression (math_errhandling & MATH_ERRNO) is non-zero, then errno shall be set to [ERANGE]. If the integer expression (math_errhandling & MATH_ERREXCEPT) is non-zero, then the
overflow floating-point exception shall be raised.
Range Error
If the integer expression (math_errhandling & MATH_ERRNO) is non-zero, then errno shall be set to [ERANGE]. If the integer expression (math_errhandling & MATH_ERREXCEPT) is non-zero, then the
underflow floating-point exception shall be raised.
The following sections are informative.
On error, the expressions (math_errhandling & MATH_ERRNO) and (math_errhandling & MATH_ERREXCEPT) are independent of each other, but at least one of them must be non-zero.
feclearexcept(), fetestexcept(), the Base Definitions volume of IEEE Std 1003.1-2001, Section 4.18, Treatment of Error Conditions for Mathematical Functions, <math.h>
First released in Issue 4, Version 2.
Issue 5
Moved from X/OPEN UNIX extension to BASE.
Issue 6
The nextafter() function is no longer marked as an extension.
The nextafterf(), nextafterl(), nexttoward(), nexttowardf(), and nexttowardl() functions are added for alignment with the ISO/IEC 9899:1999 standard.
The DESCRIPTION, RETURN VALUE, ERRORS, and APPLICATION USAGE sections are revised to align with the ISO/IEC 9899:1999 standard.
IEC 60559:1989 standard floating-point extensions over the ISO/IEC 9899:1999 standard are marked.
End of informative text.
UNIX ® is a registered Trademark of The Open Group.
POSIX ® is a registered Trademark of The IEEE.
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I hope this FM Live game does not affect FM games getting good
I've had a look at Football Manager Live and its not something I would play to be honest because its seems very time consuming, I just love Football Manager so i'll keep playing that as I can play
that anytime I want. But my concern is this...now they are releasing FM Live and it will have loads of bugs, FM will get less attention, I know they make it every year but won't SI improve on he live
game also? Would that affect FM2009 and and so on...?
I've had a look at Football Manager Live and its not something I would play to be honest because its seems very time consuming, I just love Football Manager so i'll keep playing that as I can play
that anytime I want. But my concern is this...now they are releasing FM Live and it will have loads of bugs, FM will get less attention, I know they make it every year but won't SI improve on he live
game also? Would that affect FM2009 and and so on...?
2 different teams on each game, AFAIK
I don't know how you people feel but I feel jealous that FM Live is getting released, only because i'm not going to play...because I don't want to and don't plan to and also I think that more people
might play FM Live and go away from playing the normal FM game making FM live more popular.
It obviousley has though. Look at home poor FM08 has been. It has not improved over FM07, in some ways it has been a step backwards.
There is talk however of the FM Live match engine being implemented in FM09.
There are many things that need a overhaul:
Match Engine
Transfer System
Simulated Matches
Media Interaction
Board Confidence
Training ?
To be perfectly honest if they just realeased a data update to FM07 that would of been much cheaper for them to do and would have less complaints.
Why not just stick the facegen gimick on the FM07 with a data update?
I have low expectations for FM09 (after having high for FM08) in the hope that I will be pleasantly surprised.
<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by Navie:
I don't know how you people feel but I feel jealous that FM Live is getting released, only because i'm not going to play...because I don't want to and don't plan to and also I think that more people
might play FM Live and go away from playing the normal FM game making FM live more popular. </div></BLOCKQUOTE>
I don't really concern myself with who else plays FM09. The fact that they are real players and it is a one off payment will draw enough people to keep the community alive enough.
<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by Navie:
I don't know how you people feel but I feel jealous that FM Live is getting released, only because i'm not going to play...because I don't want to and don't plan to and also I think that more people
might play FM Live and go away from playing the normal FM game making FM live more popular. </div></BLOCKQUOTE>
Actually, since FM Live was announced, the general consensus on these boards appears to be that very few people are interested in it. So if this holds-up into and after release, I doubt FM will lose
any popularity at all.
ShearersTino I also think FM09 will be poor unless training, media, player interaction is revamped. I was disappointed with FM08's lack of features, only thing I like is the confidence and the budget
adjustment, its exactly the same as FM07, there was nothing major to be honest in FM08. I played this years CM2008 and was far impressed with that than FM08 but its too easy to win things at the
moment on that, hopefully this years will be more challenging so will check that out if FM09 seems rubbish.
I don't know how people can play FM Live in the 1st place, its dumb because it has no real players or teams, how unrealistic can a game get?... and it won't feel like real football.
<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">It obviousley has though. </div></BLOCKQUOTE>
Not one person from FM has been taken off the project for FML.
EHM's demise supplied Graeme Kelly and Ov Collyer returned from the wilderness - they're the main two as far as FML goes and they had nothing to do with FM08.
<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">its dumb because it has no real players </div></BLOCKQUOTE>
The FML DB is the FM one, only the size is adjusted dependant on the size of the game world.
All the players are real until regens appear as known players retire, sound familiar?
<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by ShearersTino:
It obviousley has though. </div></BLOCKQUOTE>
FM's quality has almost nothing to do with FML. I believe that the only shared asset is the match engine which is now getting year-round exposure to thousands of testers - which should mean better
If you didn't like FM08, that's fair enough - it's your opinion, after all - but that's not the fault of FML.
<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by Kriss:
<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">its dumb because it has no real players </div></BLOCKQUOTE>
The FML DB is the FM one, only the size is adjusted dependant on the size of the game world.
All the players are real until regens appear as known players retire, sound familiar? </div></BLOCKQUOTE>
Also, when this guy gets 10 seasons into FM, it also has no (or at least, fewer) real players.
I'm not getting into the Unrealistic debate again, but yea, you need to think carefully about what your saying, as its not making sense.
<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by ShearersTino:
It obviousley has though. Look at home poor FM08 has been. It has not improved over FM07, in some ways it has been a step backwards. </div></BLOCKQUOTE>
This is entirely your opinion, there are plenty of people who see FM08 as a another step forward. When new features are introduced there are always creases to iron out but imo since 8.0.2 the game's
been excellent.
As for FML, it's also something I'll not be bothered about playing but that's only because I don't have the time to play any on-line games. That being said it will improve FM09 and FM10 if anything,
simply because of the sheer amount of human vs human matches being played and tested.
For me FM08 is the best yet, every year the SI team do very well in producing a quality game.
FML will not affect FM09.
My only complaint was that the MLS didn't get looked at at all for the patches. Nothing was done to fix the problems it had. The match engine works fine for me though. Everything else looked ok to
me. Quality has not gone down whatsoever to me.
What I am trying to get at is that I am not too bothered about new features for the sake of it. I am more interested in the features that it currently has working correctly. Unfortunatly this doesn't
sell games.
When I first saw the changes to FM08 I got excited by the lack of features, I thought they would take the time to finish and hone the features it already had, for some reason though they decided to
alter parts of the game that were working fine.
The only way FM Live will affect FM09 is we have more people testing new match engine updates. Stress testing them, in fact. Human V Human always stretches the engine to it's limits.
<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by Matt - SEGA:
<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by ShearersTino:
It obviousley has though. </div></BLOCKQUOTE>
FM's quality has almost nothing to do with FML. I believe that the only shared asset is the match engine which is now getting year-round exposure to thousands of testers - which should mean better
If you didn't like FM08, that's fair enough - it's your opinion, after all - but that's not the fault of FML. </div></BLOCKQUOTE>
Is FM2009's match engine going to be the same match engin from FM Live?
They will share the same match engine, yes.
<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by Matt - SEGA:
<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by ShearersTino:
It obviousley has though. </div></BLOCKQUOTE>
FM's quality has almost nothing to do with FML. I believe that the only shared asset is the match engine which is now getting year-round exposure to thousands of testers - which should mean better
If you didn't like FM08, that's fair enough - it's your opinion, after all - but that's not the fault of FML. </div></BLOCKQUOTE>
I disagree, Mark Vaughn said in here ages ago that he has been working solely on FML. If he has then either the FM crew has brought in a nuff nuff or 2 to make up numbers or are working with less
people. Either way the FM crew is not what it was, that is assuming it was only Vaughn that moved as well. If another 1 or 2 were moved over to FML then the FM crew would've been seriously weakened.
You can't water down staff and assume nothing has changed or will change.
<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by HonkyDick:
I disagree, Mark Vaughn said in here ages ago that he has been working solely on FML. If he has then either the FM crew has brought in a nuff nuff or 2 to make up numbers or are working with less
people. Either way the FM crew is not what it was, that is assuming it was only Vaughn that moved as well. If another 1 or 2 were moved over to FML then the FM crew would've been seriously weakened.
You can't water down staff and assume nothing has changed or will change. </div></BLOCKQUOTE>
Surely, IF FML is moving the match engine ahead, and that ME will be incorporated into FM09, then working on FML does not necessarily negate developing FM09.
After trialling FML other features that work well could also be incorporated into FM09, so I don't see the need for pessimism here.
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Graphing Calculators are the New Slide Rule
Stats Medic
Fancy calculators allow our statistics students to do a whole lot of statistics that would be very tedious to do by hand. Unfortunately the user interface with calculators has changed very little
over the last 20 years and is now far behind other technology.
Let me try to make my case. Today in class I wanted students to make a scatterplot, find the line of best fit, and make a residual plot.
Option 1: Graphing Calculator
• STAT/EDIT/Edit
• Enter values for the explanatory variable in List 1.
• Enter values for the response variable in List 2.
• 2ND/Y=/ON/Scatterplot/L1/L2/ZOOM/Zoomstat
• STAT/CALC/LinReg(a+bx) L1,L2,Y1 (VARS/Y-VARS/Function/Y1)
• 2ND/Y=/On/Scatterplot/L1/RESID (2ND/STAT/NAMES/RESID)
• ZOOM/ZoomStat
Over 30 (memorized) keystrokes. And then I get a scatterplot and a residual plot with no axis labels or scale on either axis. Model once for the students, then have students help each other learn
the calculator, then spend the rest of the hour helping out individual struggling students.
Option 2 (The Better Option): Applet
• Go to the 2 Quantitative Variables applet.
• Type in GPA for explanatory and ACT for response and add values.
• Click “Begin Analysis”.
• Click “Calculate least-squares regression line”.
Two clicks. Zero (memorized) keystrokes. And then I get a beautiful scatterplot and a residual plot with axis labels and scaled axes. Model once for the students, then allow them the rest of the
hour to work through an activity that explores residual plots for several different data sets.
More time for thinking and reasoning about statistics
When using the graphing calculator to make residual plots, I spent half of the lesson making sure that all students learned the more than 30 keystrokes needed to make the graphs. The next day in
class was spent reducing anxiety about calculator familiarity. Instead of wasting this valuable instructional time on calculators, now I can focus on the content. What makes this line the “best fit”?
What does the residual plot tell us about the line of best fit? What would the residual plot look like for nonlinear data?
No more memorizing keystrokes
Because applets were designed within the lifetime of our students (and not prior) the user interface is very intuitive. Johnny Student clicks through applets as quickly as he does through SnapChat or
Twitter. This allows students to focus on the thinking and reasoning behind the statistics. Rather than memorizing keystrokes, students are pondering the meaning of a pattern that shows up in the
residual plot.
Nicer graphs
Calculators produce floating dots on a screen. Applets provide a context (axis labels) and a scale. Check out the graphs for this GPA vs. ACT data.
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Is Choosing to Believe in Economic Models a Rational Expected-Utility Decision Theory Thing?: Friday Focus for July 19, 2014
Over at Equitable Growth: I have always understood expected-utility decision theory to be normative, not positive: it is how people ought to behave if they want to achieve their goals in risky
environments, not how people do behave. One of the chief purposes of teaching expected-utility decision theory is in fact to make people aware that they really should be risk neutral over small
gambles where they do know the probabilities--that they will be happier and achieve more of their goals in the long run if they in fact do so. READ MOAR
Thus the first three things to teach people are:
1. That they are not risk-neutral over small gambles.
2. That elementary considerations of rationality in the sense of finding means to achieve one's ends require risk-neutrality over small gambles.
3. Hence they should be risk-neutral over small gambles.
Then, of course, there is the fourth thing to teach people:
(4) When they are betting against other human minds, you should not be risk-averse over even small amounts--the fact that another mind is willing to take the opposite side of the bet tells you that
your subjective probabilities are biased, and expected-utility decision theory based on your subjective probabilities does not incorporate that information about your biases, and so leads you astray.
Still open, however, is:
(5) Should you act as if you are risk-neutral for small gambles against nature if doing so makes you anxious and hence unhappy?
My view is that you owe it to yourself to train yourself not to be anxious and unhappy with respect to small gambles against nature, and thus train yourself to be risk-neutral with respect to small
gambles against nature.
And then there is:
(6) Given that people aren't rational Bayesian expected utility-theory decision makers, what do economists think that they are doing modeling markets as if they are populated by agents who are? Here
there are, I think, three answers:
• Most economists are clueless, and have not thought about these issues at all.
• Some economists think that we have developed cognitive institutions and routines in organizations that make organizations expected-utility-theory decision makers even though the individuals in
utility theory are not. (Yeah, right: I find this very amusing too.)
• Some economists admit that the failure of individuals to follow expected-utility decision theory and our inability to build institutions that properly compensate for our cognitive biases (cough,
actively-managed mutual funds, anyone?) are one of the major sources of market failure in the world today--for one thing, they blow the efficient market hypothesis in finance sky-high.
The fact that so few economists are in the third camp--and that any economists are in the second camp--makes me agree 100% with Andrew Gelman's strictures on economics as akin to Ptolemaic astronomy,
in which the fundamentals of the model are "not [first-order] approximations to something real, they’re just fictions..."
Andrew Gelman: Differences between econometrics and statistics: "Economists seem to like their models...
...and then give after-the-fact justification. My favorite example is modeling uncertainty aversion using a nonlinear utility function for money, in fact in many places risk aversion is defined
as a nonlinear utility function for money. This makes no sense on any reasonable scale... but economists continue to use it as their default model. This bothers me... like... doing astronomy with
Ptolemy’s model and epicycles. The fundamentals of the model are not approximations to something real, they’re just fictions...
Andrew Gelman (1998): Some Class-Participations Demonstrations for Decision Theory and Bayesian Statistics: "5. Utility of Money and Risk-Aversion...
...To introduce the concept of utility, we ask each student to write on a sheet of paper the probability p[1] for which they are in different between (a)a a certain gain of $1, and (b) a gain of
$1000000 with probability p[1] or $0 with probability (1-p[1]).... The students are then asked to write down, in sequence the probabilities p[2], p[3], p[4], and p[5], for which $1=p[2]$10 + (1-p
[2])$0; $10=p[3]$100 + (1-p[3])$1; $100=p[4]$1000 + (1-p[4])$10; and $1000=p[5]$1000000 + (1-p[5])$100. One of the students is then brought the blackboard to get his or her answers to questions.
The probabilities are checked for coherence... The questions involving piece to p[2] and p[3] or combine field comparison between $1, $100, $0. For example, suppose p[2]=0.1 and p[3]=0.15....
Then... U($1) = 0.064(U($100))+0.9836(U($0)). We then repeat this procedure using... p[4] to determine the utility of $1 relative to $1000 and $0, and then once again using p[5] to determine the
utility of $1 relative to $1000000 and $0. finally, this derived values compared to the student's original value of p[1]. is will disagree, meaning that the students preferences are incoherent.
The students in the class then discuss with the student that the blackboard how... To give coherent unreasonable answers....
A related demonstration.... a person is somewhat risk-averse and is indifferent between a certain gain of $10 and a 55% chance of $20 and a 45% chance of $0. similarly he or she is indifferent
bring a certain gain of $20 and a 55% chance of $30 and a 45% chances of $10; and, in general, different between a certain gain of $x and a 55% chance of $10+x and a 45% chances of $x-10.
Is this reasonable? The students assent....
Then answer the following question: For what dollar value $y is this person in different between a certain game of $y and a 50% chance of $1000000000 and a 50% chance $0? The answer,
surprisingly, is that Whitlers is between $30 and $40.... Setting U($0)=0 and U($10)=1... and evaluating... yields U($20)=1.818, U($30)=2.487.... U($40)=3.035.... U($1000000000)=5.5.... $y must
be between $30 and $40.
The student believes each step of the argument but is unhappy with the conclusion. Where is the mistake? It is the theory uncertainty is not the same as "risk-aversion" in utility theory: the
latter can be expressed as a concave utility function for money, whereas the former implies behavior that is not consistent with any utility function (see Kahneman and Tversky 1979). This is a
good time to discuss cognitive illusions, many of which have been demonstrated the context of monetary gains and losses.... Is decision descriptive? Is it normatively appropriate? | {"url":"https://www.bradford-delong.com/2014/07/is-choosing-to-believe-in-economic-models-a-rational-expected-utility-decision-theory-thing-friday-focus-for-july-19-2014.html","timestamp":"2024-11-10T08:41:46Z","content_type":"text/html","content_length":"40965","record_id":"<urn:uuid:d8a1f9d7-637d-4c18-b743-b29ab39a61ff>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00333.warc.gz"} |
How to Create a Slotted Construction Cardboard Shark
Introduction: How to Create a Slotted Construction Cardboard Shark
This shark costume consists of strictly cardboard. It is designed using slotted contruction, to save space. Slotted construction also helps the environment because it allows the costume to be made
entirely out of cardboard, eliminating the need for harmful adhesives. All cutting should be done with a cutting mat under the piece you are cutting.
Design brief
- design and produce a flat pack or slotted design cardboard costume for middle school aged children
-flat pack/ slotted design
-must fit middle school agred children
-constructed using only cardboard
-flat pack/ slotted design
-must fit middle school agred children
-constructed using only cardboard
-no adhesives permitted
-child must be able to move
Supply List:
Three Pieces 9"x 9"
Two pieces 1"x 5"
One piece 38"x31" (Some Television, fridge or moving boxes are possible examples of where one might find this size cardboard.)
One piece 14"x 15"
One piece 13"x 15"
One piece 9"x 9"
One piece 1"x 5"
Tools and Equipment list:
Cutting mat
Utility knife
Step 1: P1: Body
1. Using a pencil, draw two ovals each 31" tall and 17" wide with a 4" gap between each oval.
2. Using a ruler, on both ovals, measure up and down 6" from the middle and mark each spot.
3. Using a pencil, draw a horizontal line, connecting the opposite top marks and bottom marks together.
4. Using a ruler, measure 2' down from one end on one oval.
5. Using a pencil, draw a 2" vertical line along the ruler.
6. On the other end, on both ovals, using a ruler, measure down 3".
7. Using a pencil, draw a 2" horizontal line where the 3" vertical line ends.
8. Using a ruler, measure out vertically, 0.5" from the outsides of the body.
9. Using a pencil, draw a line alongside the ruler.
10. Using a utility knife, cut out the ovals and along the lines connectiong the ovals, but do not cut between the connecting lines.
11. Using a utility knife, score the inside of the body. This will allow the body to curve, therefore looking more like an actual shark.
Step 2: P2: Pectoral Fins
1. Using a pencil, draw two (separate) triangles that are 8" tall and 9" wide, with a 2" wide by 1" long rectangle , on the middle of each triangle's side that is 9" long.
2. Using a utility knife, cut out the pectoral fins.
3. For one pectoral fin, using a utility knife cut out a 0.25" wide by 0.5" deep in the rectangle.
Step 3: P3: Connector Pieces
1. Using a ruler and a pencil, draw a 1" by 5" rectangle.
2. Using a ruler and a pencil, draw two 0.5" lines. Each line should be on one of long edge on the rectangle and each should be 0.5" from the edge.
3. Using a ruler and a pencil, draw a 0.5" line in the middle on the other long edge of the rectangle.
4. Using a utility knife, cut along all drawn lines.
Step 4: P4: Head
1. Using a ruler and a pencil, draw an oval that is 14"x 11". (An oval template or a guide would be useful in this situation, but are not necessary.)
2. Using a ruler and a pencil, draw a 2"x 1" rectangle on each thinner side of the oval.
3. Using a ruler and a pencil, draw a circle with a 9" diameter. (A circle template or a guide would be useful in this situatuation, but neither are necessary.)
4. Using a utility knife, make cuts on all drawn lines. However, do not cut the parts of the oval that are inside the two rectangular tabs.
Step 5: P5: Caudal Fin
1. Using a ruler and a pencil, draw one triangle that is 4"x 12"x 9". Then, draw another triangle that overlaps the first. The second triangle should be 5"x 11"x 8".
2. Using a ruler and a pencil, draw a 3"x 7" rectangle. The rectangle should only protrude past the two triangles by about 2", so there should be about 3" of overlap between the rectangle and the
3. Using a ruler and a pencil, draw a line that is 1" in the middle of the front of the rectangle.
4. Using a utility knife, cut along all drawn lines.
Step 6: P6: Dorsal Fin
1. Using a pencil, draw two (separate) triangles that are 8" tall and 9" wide, with a 2" wide by 1" long rectangle , on the middle of each triangle's side that is 9" long.
2. Using a utility knife, cut out the pectoral fins.
Step 7: Assembly
Part List
P1 Body
P2 Pectoral Fins
P3 Connector Pieces
P4 Headpiece
P5 Caudal Fin
P6 Dorsal Fin
1. Fold P1 where the two ovals meet the centerpiece.
2. Insert P3 into P1.
3. Insert one P2 into P3 and one P2 into P1.
3. Insert P4 into P1.
4. Insert P5 into P1.
5. Insert P6 into P1. | {"url":"https://www.instructables.com/How-to-Create-a-Slotted-Construction-Cardboard-Sha/","timestamp":"2024-11-09T12:44:29Z","content_type":"text/html","content_length":"92412","record_id":"<urn:uuid:206fa6bc-f84e-4adf-9538-ed31ee872ab5>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00846.warc.gz"} |
Parameter optimization of damper-friction isolation systems using concave friction distribution
The random friction distribution is an objective phenomenon and always leads to great uncertainty in seismic responses of damper-friction isolation systems. As to optimize the seismic performance,
this paper artificially made the friction distribution on a contact surface to be concave. When the damper-friction isolation system was subjected to different ground motions, more regular responses,
due to the concave friction distribution, were obtained comparing with the responses of the random friction distribution. The concave friction distribution is always conducive to reducing the
structural relative displacement and insignificantly increasing the structural acceleration. An optimization design example of an isolation building was carried out. And the results showed that as
for a structure only being sensitive to acceleration or force, significant concave friction distribution and little damping constant would be the best design combination. It resulted in a much less
acceleration and an acceptable relative displacement on the structure.
1. Introduction
In an effort to improve the isolation performance, researchers around the world have proposed a new type isolation mechanism, which is called friction-based isolation method. Harvey and Gavin built a
mathematical model for a double rolling isolation system (RIS), and carried out an experimental validation in 2014 [1]. The effects of the initial conditions, the mass of the isolated object, and the
amplitude of the disturbance on the system’s performance were assessed. And then, a simplified model presented in their paper [2] was applied to RISs with any potential energy function. This model
was amenable to both lightly- and heavily-damped RISs. It was also validated through the successful prediction of peak responses for a wide range of disturbance frequencies and intensities. Ismail
and Casas investigated the near-fault (NF) seismic performance of a novel isolation system referred to as the roll-n-cage (RNC) isolator in 2014 [3-4]. By considering the cable-stayed Bill Emerson
Memorial Bridge in Missouri as an isolation structure, the results showed that the RNC isolator was a convenient isolation system in protecting cable-stayed bridges against NF earthquakes. Wang,
Hwang, et al, studied the sloped multi-roller isolation devices for seismic protection of equipments and facilities, and obtained an excellent in-plane seismic isolation performance in 2014 [5].
Jangid and Londhe analytically studied seismic response of a multistory building supported by elliptical rolling rods in 1998 [6], and the results indicate that the rolling-based isolation mechanism
is quite effective in reducing the seismic demand of the system, such as base displacement. In 2000, Jangid investigated the stochastic response of flexible multistory shear type buildings isolated
by rolling rods with a re-centering device under earthquake excitations, and the numerical results indicate that the rolling rod has potential to reduce the stochastic response of the structure [7].
In 2010, Ou et al. and Lee et al. numerically studied the seismic response of highway bridges which were installed with a rolling isolation device [8-9]. This isolation device with zero post-yield
stiffness has self-centering and energy dissipation capability which is contributed by friction force. Parametric studies on these bridge structures suggest an improvement of the calculation method
proposed in AASHTO Specifications. In 2007, Guerreiro et al. experimentally investigated the seismic performance of light structures supported by a rolling-ball isolation device and subsequently
conducted numerical modeling of this system, and the experimental results and the numerical analysis uniformly show a reduced acceleration demand imposed on the isolated structures [10]. In 2011,
Kurita et al. developed a new spherical shape friction bearing device which is able to reduce the peak acceleration of the isolated system by about 50-90% [11]. In 2012, Monfared et al. investigated
the seismic performance of 5-story structures with and without base isolators, and the results indicate that the effectiveness of a base-isolated system is dependent on the characteristics of the
input excitations as well as the properties of the isolation devices and the superstructure [12].
Although the friction-based device avoids seismic damages caused in structure, the induced displacement is usually large and difficult to control. In this regard, the spring device needs to be
incorporated to the pure-friction isolation system in order to reduce the relative and residual displacement. However, once the stiffness of the spring device is mismanaged and the actual earthquake
differs from the design earthquake, the isolation structure generates obvious vibration periods. And thus the seismic damage still appears in the isolation structure and even sympathetic vibration
may happen [13-14]. Therefore, it is quite a question whether the restoring device can be replaced with other devices or not. Although general dampers can effectively reduce the structural relative
displacement without producing any natural vibration period, how to combine the damper and the friction device to obtain an optimized isolation performance needs further investigation. And the
previous researches and applications assumed the friction coefficient as a fixed value to simplify the calculation process, i.e. the distribution of friction coefficient on the whole contact surface
was absolutely uniform, which was opposite to the reality [15]. In 2015, Wei’s research showed that there was different variability for the friction coefficient on the contact surface according to
its current construction level [16]. The uneven distribution of friction coefficient leaded to great uncertainty in the structural seismic responses [17].
Based on all of the researches above, one of the urgent problems is how to use the uneven friction distribution to improve the isolation performance. This paper artificially interferes the friction
distribution to be concave, i.e. the friction coefficient is the smallest in the center of the seismic isolation device and gradually increases along the around path. A damper is added to a contact
surface with this concave friction distribution to form a damper-friction isolation system. The object of this paper is to utilize a numerical analysis to analyze the impact of the concave friction
distribution on the seismic performance of the damper-friction isolation system under different ground motions. By using the obtained regular rules, an optimization design example of an isolation
building is carried out, and optimal parameters are identified.
2. Calculation process
2.1. Structure model
A damper-friction isolation system is shown in Fig. 1. The contact surface with friction action isolates the horizontal earthquake besides supporting the structure. Simultaneously, the friction
action dissipates the earthquake energy and transmits the friction force to the main structure. For the concave friction distribution, the friction coefficient is the smallest in the center of the
isolation device, which is assumed to be 0.005. It gradually increases when the relative displacement between the structure and the ground becomes larger, and the increment ratios are assumed to be
0, 0.005, 0.010, 0.015, 0.020, 0.025 and 0.030, respectively, in this paper.
The damper impedes the excessive relative velocity, dissipates the earthquake energy and reduces the structural relative displacement. Meanwhile, the damper transmits the damping force to the main
structure as well. In this paper, the damping constants adopt 100, 200, 300, 400 and 500 kN·s/m, respectively.
As demonstrated in the related studies [16], the isolated structure can be built as a rigid body since the stiffness of the isolation device is far less than that of the structure. So in this paper,
the superstructure is temporarily simplified to a rigid body, and its mass is set to be 300 t.
2.2. Earthquake input
As shown in Fig. 2, for each elastic response spectrum for the soil profile I, II, III, and IV in Chinese criteria (JTJ 004-89) [18], one accelerogram is generated by Simqke procedure to be the
ground motion input of the structural model [19].
Fig. 1A damper-friction isolation system
2.3. Calculation cases
35 cases are obtained by combining 5 damping constants and 7 concave distribution cases of friction coefficient. As for each case, each accelerogram in Section 2.2 is input as the ground motion
input, whose peak ground accelerations (PGA) are adjusted to be 0.2, 0.4, 0.6 and 0.8 g, respectively. Thus, 560 cases are generated for the further calculation.
The author himself compiled a computer program to calculate the isolation system [16]. A great many of seismic responses are obtained for 560 calculation cases above. The following sections discuss
those results, but only classical and common results are discussed in detailed manner due to space limitations while other results are considered but not listed.
3. Structural maximum relative displacement
3.1. The effect of the friction coefficient
Fig. 3 shows the effect of the increment ratio of concave friction distribution on the structural maximum relative displacement. Generally, as the increment ratio of concave friction distribution
increases, the structural maximum relative displacement gradually decreases. This decreasing trend is more significant when PGA is larger and the damping constant is less. As the site type of soil
profile is larger, the decreasing trend gradually fluctuates. For the soil profile IV with a relatively small damping constant, this fluctuation is significant as the increment ratio of concave
friction distribution increases to a certain value. However, as the damping constant increases to a certain value for the same calculation case, the fluctuation disappears.
As for the cases of the concave friction distribution in Fig. 1, when ${v}_{e}>{v}_{s}$, the friction force $\mu mg$ promotes the structure a forward acceleration component $\mu g$. There are two
possible cases:
1) If the moving structure is on the left of its origin position, it means that the friction coefficient $\mu$ on the contact surface is increasing, and the increment of the structural acceleration $
\mu g$ is conducive to reducing the structural relative displacement.
2) If the moving structure is on the right of its origin position, it means that the friction coefficient $\mu$ on the contact surface is decreasing, and the reduction of the structural acceleration
$\mu g$ is also conducive to reducing the structural relative displacement.
There are similar rules in the case of ${v}_{e}{<v}_{s}$.
Therefore, the concave friction distribution is always conducive to reducing the structural relative displacement. A larger increment ratio of concave friction distribution implies a smaller
structural maximum relative displacement, and the results in Fig. 3 are reasonable.
As the increment ratio of concave friction distribution is relatively large, the periodic components appear for the isolation system since the isolation layer is not easy to slide. In this case, as
the site type of soil profile increases to a certain value, the long-period components of the corresponding seismic waves will take up more proportion. The above two factors lead to a local resonance
phenomenon and a sudden increment of the structural relative displacement. However, with the further sliding of the isolation layer, this local resonance phenomenon rapidly disappears, and the
structural maximum relative displacement is still a limited value.
Fig. 3Effect of the increment ratio of concave friction distribution on the structural maximum relative displacement
a) Soil profile I with the damping constant 100 kN·s/m
b) Soil profile IV with the damping constant 100 kN·s/m
c) Soil profile I with the damping constant 500 kN·s/m
d) Soil profile IV with the damping constant 500 kN·s/m
3.2. The effect of the damping constant
As the damper limits the structural relative velocity, it is able to reduce the structural relative displacement. This theory is validated by Fig. 4. In Fig. 4, as the damping constant increases, the
structural maximum relative displacement gradually decreases. This decreasing trend is more significant when PGA and the site type of soil profile are larger, especially for calculation cases with
the less increment ratio of concave friction distribution.
3.3. The effect of different ground motions
Fig. 5 shows the effect of the site type of soil profile on the structural maximum relative displacement. Generally, as the site type of soil profile is larger, the structural maximum relative
displacement firstly decreases and then increases. This variation trend is more significant when PGA is larger, especially for calculation cases with the larger damping constant and the larger
increment ratio of concave friction distribution.
Fig. 4Effect of the damping constant on the structural maximum relative displacement
a) Soil profile I with the increment ratio 0 of concave friction distribution
b) Soil profile IV with the increment ratio 0 of concave friction distribution
c) Soil profile I with the increment ratio 0.03 of concave friction distribution
d) Soil profile IV with the increment ratio 0.03 of concave friction distribution
Fig. 5Effect of the soil profile on the structural maximum relative displacement
a) The increment ratio of concave friction distribution is 0.005 and the damping constant is 100 kN·s/m
b) The increment ratio of concave friction distribution is 0.03 and the damping constant is 100 kN·s/m
c) The increment ratio of concave friction distribution is 0.005 and the damping constant is 500 kN·s/m
d) The increment ratio of concave friction distribution is 0.03 and the damping constant is 500 kN·s/m
As for the same PGA, a larger site type of soil profile means a larger absolute displacement of the ground motion while the structural acceleration is a limited value. Therefore, a larger site type
of soil profile means a larger structural maximum relative displacement. Because both the concave friction distribution and the damper are conducive to reducing the structural relative displacement,
the increment of the site type of soil profile will not significantly increase the structural relative displacement. Especially when the increment ratio of the concave friction distribution, the
damping constant and the PGA are all large, as the site type of soil profile increases, the absolute displacement of the ground motion grows faster than the relative displacement of the structure. In
this occasion, the ratio of the structural maximum relative displacement to the absolute displacement of the ground motion decreases.
However, as the site type of soil profile increases to a certain value, a local resonance phenomenon exists, which results in a sudden increment of the structural relative displacement as shown in
Fig. 5. This phenomenon is weakened by increasing the damping constant.
Fig. 6Effect of the PGA on the structural maximum relative displacement
a) The increment ratio of concave friction distribution is 0.005 and the damping constant is 100kN·s/m
b) The increment ratio of concave friction distribution is 0.03 and the damping constant is 100 kN·s/m
c) The increment ratio of concave friction distribution is 0.005 and the damping constant is 500 kN·s/m
d) The increment ratio of concave friction distribution is 0.03 and the damping constant is 500 kN·s/m
Fig. 6 shows the effect of PGA on the structural maximum relative displacement. As PGA increases, the structural maximum relative displacement gradually increases. However, the growth rate is less
when PGA is larger, especially for calculation cases with the larger increment ratio of concave friction distribution.
As for the same site type of soil profile, a larger PGA means a larger absolute displacement of the ground motion, while the structural acceleration is a limited value. And thus, a larger PGA means a
larger structural maximum relative displacement. Because the concave friction distribution is conducive to reducing the structural relative displacement, the increment of PGA will not significantly
increase the structural relative displacement forever.
4. Structural maximum acceleration
4.1. The effect of the friction coefficient
Fig. 7 shows the influence of the increment ratio of concave friction distribution on the structural maximum acceleration. As the increment ratio of concave friction distribution increases, the
structural maximum acceleration gradually increases. This increasing trend is more significant when PGA and the site type of soil profile are larger, especially for calculation cases with the less
damping constant. However, this increasing trend is not very obvious for almost all of the calculation cases.
Fig. 7Effect of the increment ratio of concave friction distribution on the structural maximum acceleration
a) Soil profile I with the damping constant 100 kN·s/m
b) Soil profile IV with the damping constant 100 kN·s/m
c) Soil profile I with the damping constant 500 kN·s/m
d) Soil profile IV with the damping constant 500 kN·s/m
In theory, since the structure is built as a rigid body, the value of the structural maximum acceleration is related to that of the structural maximum external force acting on the structure. As for
the isolation system in Fig. 1, the external force is the combination force of the damping force and the friction force, where the damping force is a variable related to the structural relative
velocity and the friction force is $\mu mg$. A larger earthquake triggers a larger structural relative velocity and a larger damping force. Hence, a new acceleration value generated by the damping
force is added for all structural maximum accelerations on the basis of the acceleration $\mu g$ caused by the friction force.
As for the concave distribution of friction coefficient, due to the fact that the friction coefficient $\mu$ is changeable at any position of the contact surface, the value of the structural maximum
acceleration depends on the structural maximum relative displacement. When PGA and the site type of soil profile are larger, especially for calculation cases with a less damping constant, the induced
larger structural maximum relative displacement means that the structure moves further away from the origin position. Therefore, the friction coefficient $\mu$ at the corresponding position is
larger, and the structural maximum acceleration is larger, too. Under other unchangeable conditions, a larger increment ratio of concave friction distribution means a larger friction coefficient $\
mu$ at the corresponding position after the relative movement, so the structural maximum acceleration is subsequently larger.
However, the discussion in Section 3.1 indicates that the concave friction distribution is always conducive to reducing the structural relative displacement. And thus the increasing trend of the
structural maximum acceleration in Fig. 7 is not very obvious.
Under the same condition, a larger damping constant means that the acceleration generated by the damping force makes up a larger proportion of all accelerations. At this moment, the variability of
concave friction distribution has a less impact on the structural maximum acceleration.
Fig. 8Effect of the damping constant on the structural maximum acceleration
a) Soil profile I with the increment ratio 0 of concave friction distribution
b) Soil profile IV with the increment ratio 0 of concave friction distribution
c) Soil profile I with the increment ratio 0.03 of concave friction distribution
d) Soil profile IV with the increment ratio 0.03 of concave friction distribution
4.2. The effect of the damping constant
Fig. 8 shows the effect of the damping constant on the structural maximum acceleration. As the damping constant increases, the structural maximum acceleration gradually increases. This increasing
trend is more significant when PGA and the site type of soil profile are larger, especially for calculation cases with the less increment ratio of concave friction distribution.
In theory, with regard to the general isolation structures, e.g. lead rubber bearing, as the structural period is close to the seismic predominant period, increasing the damping constant of the
isolation layer reduces the structural acceleration. However, as the structural period is much larger than the seismic predominant period, increasing the damping constant of isolation layer promotes
a larger structural acceleration. The damper-friction isolation system doesn’t have a fixed structural period, so it is approximately considered that the structural period is much larger than the
seismic predominant period. And just as logically illustrated in Fig. 8, increasing the damping constant of the isolation layer eventually increases the structural acceleration. Besides, a larger
increment ratio of concave friction distribution means a larger acceleration $\mu g$ generated by the friction force. At this moment, the variability of the damping constant has a less impact on the
structural maximum acceleration.
4.3. The effect of different ground motions
Fig. 9 shows the effect of the site type of soil profile on the structural maximum acceleration. As the site type of soil profile increases, the structural maximum acceleration gradually increases.
This increasing trend is more significant when PGA is larger, especially for calculation cases with the less damping constant and the larger increment ratio of concave friction distribution.
Fig. 10 shows the effect of PGA on the structural maximum acceleration. The structural maximum acceleration gradually increases with PGA. This increasing trend is more significant when the site type
of soil profile is larger, especially for calculation cases with the larger increment ratio of concave friction distribution.
When other parameters are unchanged, a larger site type of soil profile or PGA means a larger structural relative displacement and a larger friction coefficient $\mu$ at the corresponding position.
And thus the acceleration $\mu g$ generated by the friction force is subsequently larger. Likewise, a larger site type of soil profile or PGA also means a larger structural relative velocity and a
larger acceleration generated by the damping force. Eventually, the structural maximum acceleration is subsequently larger.
Fig. 9Effect of the site type of soil profile on the structural maximum acceleration
a) The increment ratio of concave friction distribution is 0.005 and the damping constant is 100 kN·s/m
b) The increment ratio of concave friction distribution is 0.03 and the damping constant is 100kN·s/m
c) The increment ratio of concave friction distribution is 0.005 and the damping constant is 500 kN·s/m
d) The increment ratio of concave friction distribution is 0.03 and the damping constant is 500 kN·s/m
5. Optimal parameters identification for a particular application
5.1. Structure introduction
There is a building structure with mass of about 300 t containing precious equipments and museum pieces of art which are sensitive to ground motion. The design project of the main structure is shown
in Fig. 11. As the building has to be built in a severe earthquake zone with soil profile III as shown in Fig. 2, it is designed with the very strong foundation, columns and beams to resist the
future earthquake. However, even the small acceleration of 0.7 m/s^2 on the sensitive articles in the building will slightly damage them. And thus the government provides a design principle that less
than 30 % probability of exceeding 0.5 m/s^2 on the structure even under the earthquake with PGA of 0.8 g is acceptable, and the considerably large relative displacement is allowed since it doesn’t
directly damage the structure.
Fig. 10Effect of the PGA on the structural maximum acceleration
a) The increment ratio of concave friction distribution is 0.005 and the damping constant is 100 kN·s/m
b) The increment ratio of concave friction distribution is 0.03 and the damping constant is 100 kN·s/m
c) The increment ratio of concave friction distribution is 0.005 and the damping constant is 500 kN·s/m
d) The increment ratio of concave friction distribution is 0.03 and the damping constant is 500 kN·s/m
Therefore, a damper-friction isolation project with concave friction distribution similar to that in Fig. 1 is designed for the building structure. Based on the foregoing analysis, the friction
coefficient in the center of the concave friction distribution is 0.005. The according optimal increment ratio can adopt 0.03 to decrease the structural relative displacement, and it will not
significantly increase the structural acceleration. The damping constants can adopt 100, 200, 300, 400 and 500 kN·s/m, respectively, to decrease the structural relative displacement, however, it will
increase the structural acceleration response. And thus an optimal value of the damping constant needs to be identified. Moreover, a similar and contrastive isolation scheme with the uniform
distribution of rolling friction coefficient is also designed for the structure.
For the structure model, the lateral stiffness of the isolation device is much less than that of the structure. In this regard, the isolated structure is assumed as a rigid body in the following
analysis. In this study, it is assumed to have a mass of 300 t, which is the typical value for the two-storey building structure.
5.2. Earthquake input
In the building site, the earthquake will be large and very uncertain. Based on the related seismic safety assessment report, however, the probability of PGA exceeding 1g is very small and can be
Fig. 11A building structure
The elastic response spectrum for the soil profile III in Chinese criteria (JTJ 004-89) as shown in Fig. 2 is selected for the target spectrum, because the related seismic safety assessment report
shows that the relatively soft soil with shear wave velocity between 140 and 250 m/s is very thick for the building site. Twenty pairs of ground motion recordings were selected from the Pacific
Earthquake Engineering Research Center database (PEER, 2015), as listed in Fig. 12, and scaled to be consistent with the target spectrum. From Fig. 12, their mean spectrum is almost the same with the
target one. And then the scaled ground motion recordings are continuously scaled to meet the target spectrum with PGA from 0.01 g to 1 g with an increment of 0.01 g, and these recordings are input
into the isolation structure to be the ground motion.
Fig. 12Target spectrum, mean spectrum and individual earthquake spectra
5.3. Calculation process and results
To assess the seismic vulnerability of these structural systems, the performance-based assessment framework established by the PEER is used. The framework divides the performance assessment into four
analysis phases, including seismic hazard analysis, response analysis, damage analysis, and loss analysis. The outcome of each analysis is then integrated using a total probability theorem. As
Section 5.1 provides the information that the probability of PGA exceeding 1g is too small to be ignored, less than 30 % probability of exceeding 0.5 m/s^2 on the structure even under the earthquake
with PGA of 0.8 g is acceptable, and the considerably large relative displacement is allowed, this framework can be simplified to only include the response analysis and the damage analysis to assess
the structural performance.
With ground motion records selected in Fig. 12, incremental dynamic analysis (IDA) of nonlinear dynamic response analyse is used to quantify the statistical distribution of the structural response at
different levels of earthquake shaking intensities which have been described in Section 5.2. A lot of results are obtained, however, only classical and common results are discussed in detailed manner
due to space limitations while other results are considered but not listed.
Under the earthquakes with PGA of 0.8 g, the seismic responses of the structure including the concave friction cases and the uniform friction cases are shown in Fig. 13. In terms of the same
parameters of the isolation structure, different ground motion waves result in different values as shown in Fig. 13, which means there is a discrete probability distribution for the seismic responses
of the structure. As a whole, if the damping constant increases, the structural maximum acceleration increases, and the ratio of the structural maximum relative displacement to the absolute
displacement of the ground motion decreases. When the same damping constant and earthquake are concerned, the structural maximum acceleration in Fig. 13(a) corresponding to the concave friction cases
is a little larger than that of the uniform friction cases, however, the displacement ratio in Fig. 13(b) corresponding to the concave friction cases is much less than that of the uniform friction
cases. Therefore, the concave friction cases are better than the uniform friction cases.
Fig. 13The seismic responses of the structure subjected to the earthquakes with PGA of 0.8 g
a) Structural maximum acceleration
b) Structural maximum relative displacement
A sufficient number of damage states (DS) are defined in Fig. 14 for each performance group to describe the range of damage for the structure at different levels of structural response. Noted that DS
in Fig. 14(b) just reflects the design principle provided by the government, and other DS in Fig. 14 are complements to the design principle. These damage states are typically defined using fragility
curves. The horizontal axis of the fragility curve represents PGA that affects the performance group and the vertical axis represents the probability that the performance group will exceed each of
the damage states.
Fig. 14 shows that probability of exceeding the same DS is different as the damping constant changes by keeping the same increment ratio of the concave friction distribution. As for the condition
that the damping constant is 100 kN·s/m, if PGA is 0.8 g, the structural maximum acceleration has approximately 40 %, 22 %, 12 %, and 7 % chance that the structure is beyond damage states 1, 2, 3,
and 4, respectively. From Fig. 14(b), the government design principle that less than 30 % probability of exceeding 0.5 m/s^2 on the structure under the earthquake with PGA of 0.8 g is satisfied. In
Fig. 14(b), if the damping constant increases from 100 kN·s/m, probability of exceeding 0.5 m/s^2 on the structure under the earthquake with PGA of 0.8 g increases and break the government design
principle, however, the probability of exceedance for the ratio of the structural maximum relative displacement to the ground’s maximum absolute displacement decreases.
Fig. 14Fragility function for the structure with the increment ratio 0.03 of the concave friction distribution
a) Damage state 1 (DS1): $A=$ 0.4 m/s^2, $D=$ 0.6
b) Damage state 2 (DS2): $A=$ 0.5 m/s^2, $D=$ 0.8
c) Damage state 3 (DS3): $A=$ 0.6 m/s^2, $D=$ 1.0
d) Damage state 4 (DS4): $A=$ 0.7 m/s^2, $D=$ 1.2
In summary, the optimal parameters identified are that the damping constant adopts 100 kN·s/m, and the increment ratio of the concave friction distribution adopts 0.03. Noted that an enough moving
space must be designed for the damper-friction structure system, and thus the damage state will not be sensitive to the large relative displacement. Furthermore, based on Fig. 13, if the uniform
friction cases are adopted, the government design principle that less than 30 % probability of exceeding 0.5 m/s^2 on the structure under the earthquake with PGA of 0.8 g is also satisfied. In this
case, however, a much larger moving space than that of the concave friction cases must be designed for the isolation structure, which implies a greater economic investment for the building.
6. Conclusions
By taking a damper-friction isolation system as the study object, this paper analyzes the impact of the concave friction distribution on the structural isolation performance under different ground
motions. Based on the obtained regular rules, optimal parameters are identified for an example of isolation building. The main conclusions are summarized as follows:
1) After reasonable combination with the damper, the concave distribution of friction coefficient is conducive to reducing the structural maximum relative displacement. Although it increases the
structural maximum acceleration at the same moment, the increasing trend is not obvious. Increasing the damping constant effectively decreases the structural relative displacement, however, it
considerably increases the structural acceleration.
2) In terms of optimized damper-friction isolation system with the best design combination of concave friction distribution and the damper, the uncertain prediction of the site type of soil profile
and PGA will not lead to the sudden increment of the structural seismic responses.
3) As for a structure system only being sensitive to acceleration or force, significant concave friction distribution and little damping constant would be the best design combination. It results in a
much less acceleration and an acceptable relative displacement on the structure.
• Harvey P. S., Gavin H. P. Double rolling isolation systems: a mathematical model and experimental validation. International Journal of Non-Linear Mechanics, Vol. 61, Issue 1, 2014, p. 80-92.
• Harvey P. S., Zehil G. P., Gavin H. P. Experimental validation of a simplified model for rolling isolation systems. Earthquake Engineering and Structural Dynamics, Vol. 43, Issue 7, 2014, p.
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• Ismail M., Casas J. R. Novel isolation device for protection of cable-stayed bridges against near-fault earthquakes. Journal of Bridge Engineering, Vol. 19, Issue 8, 2014, p. 50-65.
• Wang S. J., Hwang J. S., Chang K. C., Shiau C. Y., Lin W. C., Tsai M. S., Hong J. X., Yang Y. H. Sloped multi-roller isolation devices for seismic protection of equipment and facilities.
Earthquake Engineering and Structural Dynamics, Vol. 43, Issue 10, 2014, p. 1443-1461.
• Jangid R. S., Londhe Y. B. Effectiveness of elliptical rolling rods for base isolation. Journal of Structural Engineering, Vol. 124, Issue 4, 1998, p. 469-472.
• Jangid R. S. Stochastic seismic response of structures isolated by rolling rods. Engineering Structures, Vol. 22, Issue 8, 2000, p. 937-946.
• Ou Y. C., Song J. W., Lee G. C. A parametric study of seismic behavior of roller seismic isolation bearings for highway bridges. Earthquake Engineering and Structure Dynamics, Vol. 39, Issue 5,
2010, p. 541-559.
• Lee G. C., Ou Y. C., Niu T. C., Song J. W., Liang Z. Characterization of a roller seismic isolation bearing with supplemental energy dissipation for highway bridges. Journal of Structural
Engineering, Vol. 136, Issue 5, 2010, p. 502-510.
• Guerreiro L., Azevedo J., Muhr A. H. Seismic tests and numerical modeling of a rolling-ball isolation system. Journal of Earthquake Engineering, Vol. 11, Issue 1, 2007, p. 49-66.
• Kurita K., Aoki S., Nakanishi Y., Tominaga K., Kanazawa M. Fundamental characteristics of reduction system for seismic response using friction force. Journal of Civil Engineering and
Architecture, Vol. 5, Issue 11, 2011, p. 1042-1047.
• Nanda R. P., Agarwal P., Shrikhande M. Base isolation system suitable for masonry buildings. Asian Journal of Civil Engineering (Building and Housing), Vol. 13, Issue 2, 2012, p. 195-202.
• Wei B., Dai G. L., Wen Y., Xia Y. Seismic performance of an isolation system of rolling friction with spring. Journal of Central South University, Vol. 21, Issue 4, 2014, p. 1518-1525.
• Wei B., Xia Y., Liu W. A. Lateral vibration analysis of continuous bridges utilizing equal displacement rule. Latin American Journal of Solids and Structures, Vol. 11, Issue 1, 2014, p. 75-91.
• Wang Y. J., Wei Q. C., Shi J., Long X. Y. Resonance characteristics of two-span continuous beam under moving high speed trains. Latin American Journal of Solids and Structures, Vol. 7, Issue 2,
2010, p. 185-199.
• Wei B., Yang T. H., Jiang L. Z. Influence of friction variability on isolation performance of a rolling-damper isolation system. Journal of Vibroengineering, Vol. 17, Issue 2, 2015, p. 792-801.
• Wei B., Wang P., Yang T. H., Dai G. L., Jiang L. Z., Wen Y. Effects of friction variability on isolation performance of rolling-spring systems. Journal of Central South University, Vol. 23, Issue
1, 2016, p. 233-239.
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• Fahjan Y., Ozdemir Z. Scaling of earthquake accelerograms for non-linear dynamic analysis to match the earthquake design spectra. The 14th World Conference on Earthquake Engineering, Chinese
Society for Earthquake Engineering, 2008.
About this article
Seismic engineering
concave friction distribution
seismic performance
This research is jointly supported by the National Natural Science Foundation of China under Grant No. 51308549, 51378504 and 51478475, the Natural Science Foundations of Hunan Province under Grant
No. 2015JJ3159, and the Innovation-driven Plan in Central South University under Grant No. 2015CX006. The above financial support is greatly appreciated.
Author Contributions
In this paper, Biao Wei reviewed the investigation history of the friction-based isolation methods, and analyzed the structural seismic responses of different damper-friction isolation systems under
different ground motions. Peng Wang drew all of the figures. Bin Yan carried out the optimization design example of an isolation building, and was the corresponding author due to his continuing
research on the similar isolation systems. Lizhong Jiang analyzed and checked the computer program and the results. Xuhui He provided the earthquake input, and modified the paper including English
Copyright © 2016 JVE International Ltd.
This is an open access article distributed under the
Creative Commons Attribution License
, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. | {"url":"https://www.extrica.com/article/16114","timestamp":"2024-11-11T23:01:17Z","content_type":"text/html","content_length":"161542","record_id":"<urn:uuid:e95abaa6-62db-46ec-b625-45f37fe40c50>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00653.warc.gz"} |
Label cover instances with large girth and the hardness of approximating basic k-spanner
We study the well-known Label Cover problem under the additional requirement that problem instances have large girth. We show that if the girth is some k, the problem is roughly 2^(log1-εn)/k hard to
approximate for all constant ε > 0. A similar theorem was claimed by Elkin and Peleg [ICALP 2000] as part of an attempt to prove hardness for the basic k-spanner problem, but their proof was later
found to have a fundamental error. Thus we give both the first non-trivial lower bound for the problem of Label Cover with large girth as well as the first full proof of strong hardness for the basic
k-spanner problem, which is both the simplest problem in graph spanners and one of the few for which super-logarithmic hardness was not known. Assuming NP ⊈ BPTIME(2^polylog(n)), we show (roughly)
that for every k ≥ 3 and every constant ε > 0 it is hard to approximate the basic k-spanner problem within a factor better than 2 ^(log1-εn)/k. This improves over the previous best lower bound of
only Ω(logn)/k from [17]. Our main technique is subsampling the edges of 2-query PCPs, which allows us to reduce the degree of a PCP to be essentially equal to the soundness desired. This turns out
to be enough to basically guarantee large girth.
اللغة الأصلية الإنجليزيّة
عنوان منشور المضيف Automata, Languages, and Programming - 39th International Colloquium, ICALP 2012, Proceedings
الصفحات 290-301
عدد الصفحات 12
طبعة PART 1
المعرِّفات الرقمية للأشياء
حالة النشر نُشِر - 2012
منشور خارجيًا نعم
الحدث 39th International Colloquium on Automata, Languages, and Programming, ICALP 2012 - Warwick, بريطانيا
المدة: ٩ يوليو ٢٠١٢ → ١٣ يوليو ٢٠١٢
سلسلة المنشورات
الاسم Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
الرقم PART 1
مستوى الصوت 7391 LNCS
رقم المعيار الدولي للدوريات (المطبوع) 0302-9743
رقم المعيار الدولي للدوريات (الإلكتروني) 1611-3349
!!Conference 39th International Colloquium on Automata, Languages, and Programming, ICALP 2012
الدولة/الإقليم بريطانيا
المدينة Warwick
المدة ٩/٠٧/١٢ → ١٣/٠٧/١٢
أدرس بدقة موضوعات البحث “Label cover instances with large girth and the hardness of approximating basic k-spanner'. فهما يشكلان معًا بصمة فريدة. | {"url":"https://cris.openu.ac.il/ar/publications/label-cover-instances-with-large-girth-and-the-hardness-of-approx","timestamp":"2024-11-14T04:31:24Z","content_type":"text/html","content_length":"53514","record_id":"<urn:uuid:2f863530-3b09-45fe-8f3a-77475e9f04e5>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00049.warc.gz"} |
Inches to Chains Conversion | Convert in to ch | Calculator-app
"Our Converter will show you a simple approach to perform difficult computations."
You might be accustomed to the measurement units such as meters, centimeters, feet, yards, inches, chains, and many more. But you may need clarification while converting them into each other. You
might make silly mistakes due to an erroneous conversion ratio, resulting in a wrong answer.
Therefore, we have a tool that makes the conversion process smooth. You can easily convert inches to chains or another unit and get an accurate answer. See how this magical converter will help you. | {"url":"https://calculator-app.com/inches-to-chains","timestamp":"2024-11-02T21:54:27Z","content_type":"text/html","content_length":"75442","record_id":"<urn:uuid:325d10b9-fd7a-42d8-b715-c3a9dec3499e>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00501.warc.gz"} |
Jeff Vautin
+4 dBu / -10 dBV: These values are typically used to represent ‘Pro’ and ‘Consumer’ audio signal levels - some gear even has a switch to scale the output, or to adjust input gain. So what do these
values mean on a common scale?
First we need to know how to get from dB to volts. For linear measures like volts, we defined the measure in dB to be 20log10(volts/reference). To go backwards we’ll need volts = reference10^(dB/
You’ll recall from a previous post that the reference for dBV is 1 volt, and the reference for dBu is 0.775 volts.
Let’s convert 4dBu to dBV, via volts:
0.775*10^(4/20) = 1.228V
20*log10(1.228/1) = 1.8dBV
So +4dBu=+1.8dBV, and the difference between +4dBu and -10dBV is really about 12 dB.
dBV, dBu, and dBm
I’ve always have a little trouble remembering the relationship between dBm and the more common signal measurements of dBu and dBV. The other day I was looking at a transfer function, and I was given
results in dBm - the impetus for this post.
First, I should make sure we’re all on the same page. What is a decibel (dB)? The decibel is a relative measurement. For voltage we calculate dB as 20log10(Voltage/Reference), where the reference
can be absolute (like with dBu and dBV) or relative (plain old dB). For an amplifier that outputs a signal twice as large as the input, 20log10(2/1) = 6 .02 dB. If we ran a signal through this
amplifier twice, our output is 4 times as large as the input, or 20log10(4/1) = 12.04dB. This is the value of a dB scale - we can add gain values in dB instead of multiplying them: 22=4, while
Before going on I should point out that dBm is not an appropriate measure for a transfer function. Straight dB is the way to go, as a linear transfer function should produce an output relative to
the input - not relative to an absolute value like dBm, dBu, or dBV. Let’s say I put 0.5 volts into a system, and I get 0.25 volts back out: 20*log10(0.25/0.5) = -6 dB.
So then dBV, dBu, and dBm are all absolute measurements and each has a defined reference value:
• dBV - 1 Volt
• dBu - 0.775 Volts
• dBm - 1 Milliwatt
dBV is straight ahead and simple to work with - this is the measure I use most frequently. The dBV value of any voltage is 20*log10(V/1), so 1 Volt is 0 dBV, 2 Volts is 6 dBV, and 0.5 Volts is -6
Let’s skip to dBm for a second. This has a reference of 1 milliwatt, a unit of power. Power is related to voltage by the equation P = VV/R, where R is the resistance the power
is dissipated through. For instance, if I drive 1 Volt into 1000 Ohms of resistance, 1V1V/1000Ohms = 0.001 Watts = 1 mW. If I increase the voltage by 6 dB (a factor of 2) to 2 Volts, now I’m
dissipating 2V2V/1000Ohms = 4 mW. This is important - doubling the voltage does not lead to a doubling of power, since the power is related to the square of the voltage. We actually have to use a
different formula for dB of power, 10log10(Power/Reference). With this formula, if we put in our 4mW and 1mW from above, we still get 6 dB of gain. Math works!
Back to dBu. dBu is referenced to 0.775 Volts - an odd number to use as a reference. However, if you calculate the power dissipating by 0dBu through 600Ohms (a long-standing common value for
transmission lines), you wind up with exactly 1mW of power, 0dBm. So 0dBu = 0dBm, if and only if the load is 600 Ohms. At any other load impedance this relationship changes.
So back to that transfer function in dBm. In order to get a relative dB measurement I needed two things: the source voltage and the load impedance. Knowing the load impedance allows you to
calculate the voltage output from the system: V = sqrt(PR). This is still an absolute measurement, so you need to know what the input signal was to calculate the dB out: 20log10(Output/Input). | {"url":"https://jeffvautin.com/tag/dbv/","timestamp":"2024-11-08T04:27:03Z","content_type":"text/html","content_length":"40755","record_id":"<urn:uuid:36acd9df-6ecf-4a8b-816b-20bbbe916b4d>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00654.warc.gz"} |
Numerical computations for the schramm-loewner evolution
We review two numerical methods related to the Schramm-Loewner evolution (SLE). The first simulates SLE itself. More generally, it finds the curve in the half-plane that results from the Loewner
equation for a given driving function. The second method can be thought of as the inverse problem. Given a simple curve in the half-plane it computes the driving function in the Loewner equation.
This algorithm can be used to test if a given random family of curves in the half-plane is SLE by computing the driving process for the curves and testing if it is Brownian motion. More generally,
this algorithm can be used to compute the driving process for random curves that may not be SLE. Most of the material presented here has appeared before. Our goal is to give a pedagogic review,
illustrate some of the practical issues that arise in these computations and discuss some open problems.
• Random curves
• Schramm-Loewner evolution
• Simulation
• Zipper algorithm
ASJC Scopus subject areas
• Statistical and Nonlinear Physics
• Mathematical Physics
Dive into the research topics of 'Numerical computations for the schramm-loewner evolution'. Together they form a unique fingerprint. | {"url":"https://experts.arizona.edu/en/publications/numerical-computations-for-the-schramm-loewner-evolution","timestamp":"2024-11-07T18:48:09Z","content_type":"text/html","content_length":"52341","record_id":"<urn:uuid:b5869f71-9b5e-48a2-93b6-a5a3ff8b557a>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00596.warc.gz"} |
kbps to Hz Calculator
kbps or kilobits-per-second is a measure of data throughput.
Hz is a measure of Frequency Bandwidth.
The tool answers the question: How much bandwidth (in Hz) is required to transfer a user-specified number of kilobits every second in a communication system?
• Data rate in kbps
• Signal-to-Noise ratio (default value is 10 dB)
Example Calculations
For a communication channel with 10 dB SNR, it takes 2900 Hz or 2.9 kHz of Bandwidth to transfer 1 kbps.
As the SNR is increased to 20 dB for instance, it takes about half the bandwidth – 1500 Hz or 1.5 kHz.
As the signal to noise ratio is increased, it improves the reliability of the communication medium. Hence a smaller amount of bandwidth (Hz) is required to transmit the same amount of data (measured
in kbps).
To calculate the bandwidth we use the Shannon-Hartley formula
B = C/(Log[2](1 + S/N))
• B is the bandwidth in Hz
• C is the throughput in bits per second
• S/N is the Signal-to-Noise ratio
Key Assumptions
The Shannon-Hartley formula makes the following assumptions:
• The noise is white Gaussian. In other words it does not consider the effect of Fading which can cause additional signal losses.
• The throughput in bit-per-second is an upper bound. It represents an ideal condition that can be achieved with an arbitrarily low error rate.
Related Calculators | {"url":"https://3roam.com/kbps-to-hz-calculator/","timestamp":"2024-11-05T02:55:38Z","content_type":"text/html","content_length":"191129","record_id":"<urn:uuid:f8584222-8400-46bd-b137-c1b5ad7d80e2>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00693.warc.gz"} |
Measurement of the groomed jet mass in pp collisions at $\sqrt{s} = 5.02$ TeV from $60 < p_\mathrm{T}^\mathrm{ch\;jet} < 80$ GeV/$c$ with $R=0.2$
Figure Caption
Measurement of the groomed jet mass in pp collisions at $\sqrt{s} = 5.02$ TeV from $60 < p_\mathrm{T}^\mathrm{ch\;jet} < 80$ GeV/$c$ with $R=0.2$ as compared to various models. Grooming is carried
out using the Soft Drop algorithm with $z_\text{cut}=0.2$ and $\beta=0$. These models provide a baseline for Pb-Pb model comparisons. On the top panel, the black data points and associated vertical
error bars correspond to the ALICE measurement and its statistical uncertainties, with the dashed boxes corresponding to systematic uncertainties. The data distribution is normalized to the tagging
fraction. Also on the top panel, each model is presented with a box corresponding to the purely statistical uncertainties of the model calculation. Each of these models is set to have the same
normalization as the data curve. On the bottom panel, the ratio of each model to ALICE data is presented. The total uncertainties on the model calculations are shown as bands around the points for
each model. The grey bands around unity correspond to the combined statistical and systematic uncertainties on the measured data, added in quadrature.
PYTHIA8 Monash2013 [1] and Herwig7 (default tune) [2] are shown, as also given in the published ALICE pp measurement [3]. Additionally, a prediction from POWHEG [4] with PYTHIA matching at NLO is
given, used as a baseline for the Higher-Twist parton energy loss approach to jet quenching [5]; finally, a pp baseline using the JETSCAPE framework [6] and one for the Hybrid model [7] are given.
[1] Comput. Phys. Commun. 191 (Jun, 2015) 159–177, arXiv:1410.3012 [hep-ph]
[2] Eur. Phys. J. C 58 no. 4, (Nov, 2008) 639–707, arXiv:0803.0883 [hep-ph]
[3] JHEP 05 (2022) 061, arXiv:2107.11303 [nucl-ex]
[4] Nucl. Phys. Proc. Suppl. 205-206:36-41 (2010), arXiv:1007.3893 [hep-ph]
[5] Chin. Phys. C45 (2021) no. 2, 024102, arXiv:2005.01093 [hep-ph]
[6] arXiv:1903.07706 [nucl-th]
[7] JHEP 10 (2014) 019, arXiv:1405.3864 [hep-ph] | {"url":"https://alice-figure.web.cern.ch/index.php/node/29465","timestamp":"2024-11-12T01:48:35Z","content_type":"text/html","content_length":"43965","record_id":"<urn:uuid:f3f7b18e-6113-4c3c-883e-dd3e3329df19>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00535.warc.gz"} |
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Welcome! The purpose of this repository is to serve as stockpile of statistical methods, modeling techniques, and data science tools. The content itself includes everything from educational vignettes
on specific topics, to tailored functions and modeling pipelines built to enhance and optimize analyses, to notes and code from various data science conferences, to general data science utilities.
This will remain a work in progress, and I welcome all contributions and constructive criticism. If you have a suggestion or request, please use the "Issues" tab and I will endeavor to respond
Note: GitHub often has trouble rendering larger .ipynb files in particular. If you find that you are unable to view one of the jupyter notebooks linked below, I recommend copy and pasting the result
into jupyter's nbviewer, which will take you to a viewable link like this one here for my "Visualization with Plotly" notebook. Note that if you want to ensure that you are viewing the most
up-to-date version of the notebook with nbviewer, you should add ?flush_cache=true to the end of the generated URL as is described here; otherwise, your link risks being slightly out-of-date.
Table of Contents
1. Playground and Basics
2. Exploratory Data Analysis (EDA) and Visualization
3. Hypothesis Testing
4. Classification
5. Regression
6. Reinforcement Learning
7. Text Mining and Natural Language Processing (NLP)
8. Notes and Material from Data Science Conferences
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All are welcome and encouraged to contribute to this repository. My only request is that you include a detailed description of your contribution, that your code be thoroughly-commented, and that you
test your contribution locally with the most recent version of the master branch integrated prior to submitting the PR.
Note that the project description data, including the texts, logos, images, and/or trademarks, for each open source project belongs to its rightful owner. If you wish to add or remove any projects,
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Evaluation of Investment Proposals | Methods | Getuplearn
Evaluation of Investment Proposals and Methods
• Post author:Anuj Kumar
• Post published:9 June 2023
• Post category:Finance / Financial Management
Post comments:0 Comments
What is the Evaluation of Investment Proposals?
Evaluating investment proposals involves a number of methods and techniques to determine the potential profitability and risk associated with a particular investment. These methods help
decision-makers to choose the most viable and profitable investment.
Three steps are involved in the evaluation of an investment:
1. Estimation of cash flows.
2. Estimation of the required rate of return (the opportunity cost of capital).
3. Application of a decision rule for making the choice. Specifically, we focus on the merits and demerits of various decision rules.
Investment Decision Rule
The investment decision rules may be referred to as capital budgeting techniques, or investment criteria. A sound appraisal technique should be used to measure the economic worth of an investment
The essential property of a sound technique is that it should maximize the shareholders’ wealth. The following other characteristics should also be possessed by a sound investment evaluation
1. It should consider all cash flows to determine the true profitability of the project.
2. It should provide an objective and unambiguous way of separating good projects from bad projects.
3. It should help the ranking of projects according to their true profitability.
4. It should recognize the fact that bigger cash flows are preferable to smaller ones and early cash flows are preferable to later ones.
5. It should help to choose among mutually exclusive projects that project which maximize the shareholders’ wealth.
6. It should be a criterion that is applicable to any conceivable investment project independent of others
These conditions will be clarified as we discuss the features of various investment criteria in the following pages.
Evaluation Criteria: A number of investment criteria (or capital budgeting techniques) are in practice. They may be grouped into the following two categories:
1. Discounted Cash Flow (Dcf) Criteria
□ Net Present Value (NPV)
□ Internal Rate of Return (IRR)
□ Profitability Index (PI)
2. Non-discounted Cash Flow Criteria
□ Payback (PB) Period
□ Discounted payback period
□ Accounting Rate of Return (ARR).
Evaluation of Investment Proposals
Here are the most common methods used to evaluate investment proposals:
Evaluation of Investment Proposals
Discounted Cash Flow (Dcf) Criteria
Net Present Value (NPV)
The Net Present Value (NPV) method is the classic economic method of evaluating investment proposals. It is a DCF technique that explicitly recognizes the time value of money. It correctly postulates
that cash flows arising at different time periods differ in value and are comparable only when their equivalent’s present values are found out. The following steps are involved in the calculation
of NPV:
• The cash flows of the investment project should be forecasted based on realistic assumptions.
• An appropriate discount rate should be identified to discount the forecasted cash flows. The appropriate discount rate is the project’s opportunity cost of capital, which is equal to the
required rate of return expected by investors on investments of equivalent risk.
• The present value of cash flows should be calculated using the opportunity cost of capital as the discount rate.
• The net present value should be found by subtracting the present value of cash outflows from the present value of cash inflows. The project should be accepted if NPV is positive (i.e., NPV > 0).
Net Present Value Advantages and Disadvantages
S.No. NPV Advantages NPV Disadvantages
1. Considers all cash flows. Requires estimates of cash flows which is a tedious task.
2. A True measure of profitability. Requires computation of the opportunity cost of capital which poses practical difficulties.
3. Recognizes the time value of money. Ranking of investments is not independent of the discount rates
4. Satisfies the value-additivity principle (i.e., NPVs of two or more projects can be added).
5. Consistent with the Shareholders’ Wealth Maximization (SWM) principle.
Net Present Value Advantages and Disadvantages
Internal Rate of Return (IRR)
The Internal Rate of Return (IRR) method is another discounted cash flow technique, which takes into account the magnitude and timing of cash flows.
Other terms used to describe the IRR method are yield on an investment, marginal efficiency of capital, rate of return over cost, time-adjusted rate of internal return, and so on. The concept of
internal rate of return is quite simple to understand in the case of a one-period project.
Internal Rate of Return Advantages and Disadvantages
S.No. Advantages of Internal Rate of Return Disadvantages of Internal Rate of Return
1. Considers all cash flows. Requires estimates of cash flows which is a tedious task.
2. True measure of profitability. Does not hold the value additivity principle (i.e., IR of two or more projects do not add).
3. Based on the concept of the time value of money. At times fails to indicate a correct choice between mutually exclusive projects.
4. Generally, consistent with the wealth maximization principle. At times yields multiple rates.
5. Relatively difficult to compute.
Internal Rate of Return Advantages and Disadvantages
Profitability Index (PI)
Yet another time-adjusted method of evaluating investment proposals is the benefit–cost (B/C) ratio or profitability index (PI). The profitability index is the ratio of the present value of cash
inflows, at the required rate of return, to the initial cash outflow of the investment. The formula for calculating the benefit-cost ratio or profitability index is as follows:
Profitability Index (PI)
Profitability Index Advantages and Disadvantages
S.No. Advantages of Profitability Index Disadvantages of Profitability Index
1. Considers all cash flows. Requires estimates of the cash flows which is a tedious task.
2. Recognizes the time value of money. At times fails to indicate a correct choice between mutually exclusive projects.
3. A relative measure of profitability.
4. Generally consistent with the wealth maximization principle.
Profitability Index Advantages and Disadvantages
Non-discounted Cashflow Criteria
Payback Period (PB)
Payback (PB) is one of the most popular and widely recognized traditional methods of evaluating investment proposals. Payback is the number of years required to recover the original cash outlay
invested in a project. If the project generates constant annual cash inflows, the payback period can be computed by dividing the cash outlay by the annual cash inflow. That is:
Payback Period (PB)
Payback Period Advantages and Disadvantages
Advantages of Payback Period Disadvantages of Payback Period
1. Easy to understand and compute and inexpensive to use. 1. Ignores the time value of money.
2. Emphasizes liquidity. 2. Ignores cash flows occurring after the payback period.
3. Easy and crude way to cope with risk. 3. Not a measure of profitability.
4. Uses cash flow information. 4. No objective way to determine the standard payback.
5. No relation with the wealth maximization principle
Payback Period Advantages and Disadvantages
Discounted Payback Period
One of the serious objections to the payback method is that it does not discount the cash flows for calculating the payback period. We can discount cash flows and then calculate the payback.
The discounted payback period is the number of periods taken in recovering the investment outlay on the present value basis. The discounted payback period still fails to consider the cash flows
occurring after the payback period.
The discounted payback period for a project will be always higher than the simple payback period because its calculation is based on discounted cash flows. Discounted payback rule is better as it
discounts the cash flows until the outlay is recovered.
Accounting Rate of Return (ARR)
The Accounting Rate of Return (ARR), also known as the Return On Investment (ROI), uses accounting information, as revealed by financial statements, to measure the profitability of an investment. The
accounting rate of return is the ratio of the average after-tax profit divided by the average investment.
The average investment would be equal to half of the original investment if it were depreciated constantly. Alternatively, it can be found by dividing the total of the investment’s book values
after depreciation by the life of the project. The accounting rate of return, thus, is an average rate and can be determined by the following equation:
Accounting Rate of Return (ARR)
Accounting Rate of Return Advantages and Disadvantages
Advantages Of Accounting Rate Of Return Disadvantages of Accounting Rate of Return
1. Uses accounting data with which executives are familiar. 1. Ignores the time value of money.
2. Easy to understand and calculate. 2. Does not use cash flows.
3. Gives more weightage to future receipts. 3. No objective way to determine the minimum acceptable rate of return.
Accounting Rate of Return Advantages and Disadvantages
FAQs About the Evaluation of Investment Proposals
What are the five methods of evaluating investment proposals?
These are the most common methods used to evaluate investment proposals:
1. Net Present Value (NPV)
2. Internal Rate of Return (IRR)
3. Profitability Index (PI)
4. Payback Period (PB)
5. Discounted Payback Period
6. Accounting Rate of Return (ARR).
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Large-time asymptotics in Deep Learning by Borjan Geshkovski
Last Friday October 23rd our team member Borjan Geshkovski PhD student at DyCon ERC Project from UAM, gave a talk at “Seminario de estadística” organized by the UAM – Universidad Autónoma de Madrid
about Large-time asymptotics in Deep Learning
Abstract. It is by now well-known that practical deep supervised learning may roughly be cast as an optimal control problem for a specific discrete-time, nonlinear dynamical system called an
artificial neural network. In this talk, we consider the continuous-time formulation of the deep supervised learning problem. We will present, using an analytical approach, this problem’s behavior
when the final time horizon increases, a fact that can be interpreted as increasing the number of layers in the neural network setting. We show qualitative and quantitative estimates of the
convergence to the zero training error regime depending on the functional to be minimised.
Take a look the slides from this seminar:
Like this?
You might be interested in the “Math & Research” post by Borjan:
The interplay of control and Deep Learning
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What 3 methods can be used to solve a system of linear equations?
What 3 methods can be used to solve a system of linear equations?
There are three ways to solve systems of linear equations in two variables: graphing. substitution method. elimination method.
How do you do special equations?
7:36Suggested clip 113 secondsSolving Special Systems of Linear Equations – YouTubeYouTubeStart of suggested clipEnd of suggested clip
How do you write and solve a system of linear equations?
These are the steps:Write one of the equations so it is in the style “variable = …”Replace (i.e. substitute) that variable in the other equation(s).Solve the other equation(s)(Repeat as necessary)
How do you solve linear equations examples?
Solving Linear EquationsSolve: (2x + 5)/(x + 4) = 1. ⇒ 2x – x = 4 – 5 (Transferring positive x to the left hand side changes to negative x and again, positive 5 changes to negative 5) Solution: ⇒ 3x/
3 = 9/3 (Dividing both sides by 3) ⇒ 5 – 2x + 2 = 12 – 4x – 2x (Removing the brackets and then simplify) x/2 + x/3 = x – 7.
How do I solve a linear equation?
To solve linear equations we will make heavy use of the following facts. If a=b then a+c=b+c a + c = b + c for any c . All this is saying is that we can add a number, c , to both sides of the
equation and not change the equation. If a=b then a−c=b−c a − c = b − c for any c .
How do you solve linear equations for beginners?
3:29Suggested clip 78 secondsHow to Solve Simple Linear Equations in Algebra For Dummies …YouTubeStart of suggested clipEnd of suggested clip
What is linear function and examples?
For example, a common equation, y=mx+b y = m x + b , (namely the slope-intercept form, which we will learn more about later) is a linear function because it meets both criteria with x and y as
variables and m and b as constants.
How can I tell if a function is linear?
To see if a table of values represents a linear function, check to see if there’s a constant rate of change. If there is, you’re looking at a linear function!
How do you write a linear function?
Linear functions are typically written in the form f(x) = ax + b. The a represents the gradient of the line, which gives the rate of change of the dependent variable. This is also known as the
“slope.” The b represents the y-axis intercept.
How do you solve linear equations by graphing?
9:32Suggested clip 96 secondsGraphing Linear Equations – YouTubeYouTubeStart of suggested clipEnd of suggested clip
How do you solve systems of equations on a graphing calculator?
3:31Suggested clip 109 secondsSolving Systems of Equations With the TI-84 Plus CE Graphing …YouTubeStart of suggested clipEnd of suggested clip
What is the Y intercept in an equation?
The y -intercept of a graph is the point where the graph crosses the y -axis. When the equation of a line is written in slope-intercept form ( y=mx+b ), the y -intercept b can be read immediately
from the equation. Example 1: The graph of y=34x−2 has its y -intercept at −2 .
Is FX 2x 3 a linear function?
f(x) = 2×3 does not represent a linear function because its graph is not in a straight line.
How do you find the domain and range of y 2x 3?
The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined. The range is the set of all valid
y values.
How do you graph the parabola y 2x 2?
6:19Suggested clip 51 secondsy = 2x^2 – YouTubeYouTubeStart of suggested clipEnd of suggested clip
How do you find the domain and range of an equation?
Find the domain and range of the function y=1x+3−5 . To find the excluded value in the domain of the function, equate the denominator to zero and solve for x . So, the domain of the function is set
of real numbers except −3 . The range of the function is same as the domain of the inverse function.
What is Range Math example?
The Range is the difference between the lowest and highest values. Example: In {4, 6, 9, 3, 7} the lowest value is 3, and the highest is 9.
How do you solve for domain?
A function with a fraction with a variable in the denominator. To find the domain of this type of function, set the bottom equal to zero and exclude the x value you find when you solve the equation.
A function with a variable inside a radical sign. | {"url":"https://pvillage.org/archives/1181","timestamp":"2024-11-12T12:45:29Z","content_type":"text/html","content_length":"54766","record_id":"<urn:uuid:9f30870e-fcf6-403f-8c69-1c1188a44d5d>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00282.warc.gz"} |
The Raster Tragedy at Low-Resolution Revisited | Beat Stamm
The Raster Tragedy at Low-Resolution Revisited:
Opportunities and Challenges beyond “Delta-Hinting”
In this chapter we will have a look at some of the idiosyncrasies of the TrueType rasterizer that ships with Microsoft Windows. At first glance this may not seem all that useful, or at least not
generally so, because it is very specific to one particular implementation of turning fonts into pixels. But this rasterizer is widely used, and it comes with a few unobvious challenges at making
fonts look “nice.” Therefore, we need to know these challenges and see how they may be addressed in the constraint system.
TrueType fonts use integer numbers to represent the coordinates of the control points. There are no fractional parts to the coordinates. For instance, in the previous chapter we have looked at the
height of the UC ‘H’ of Arial. It was quoted as 1466 font design units, not 1466.3, nor 1466.2357. This may seem to be a rather crude way to approximate the original art work.
However, all these coordinates have to be understood in terms of the em-height (the height of the font)—itself an integer number. For instance, for Arial the em-height is 2048 font design units. In
other words, every coordinate in Arial is defined to the nearest 1/2048 of the font height. Two coordinates can differ by as little as approximately one half of one tenth of a percent of the font
To put this into perspective, for a 72 pt headline, this difference is about 0.035 pt or 0.012 mm (or 12 µm). This is less than the diameter of the thinnest human hair. To literally see this
difference, the healthy human eye would have to observe it from a distance of about 41 mm (1 5/8 in), which is way too close to focus. You’ll need a powerful loupe to have a chance. Accordingly, at
this type size and em-height, integers should be precise enough to represent the original art work.
These perspectives change dramatically once we are looking at text sizes and actual screen resolutions. For instance, at 96 DPI and 12 pt, the font height is a mere 16 pixels. Accordingly, a pixel
more or less represents a big difference—to wit, 1/16 of the font height. Integer numbers are no longer precise enough to approximate the original art work—or so it seems.
The solution is to look at a suitable fraction of the integer numbers. For instance, a quarter dollar cannot be represented by an integer number of dollars—it is, after all, a quarter. But it can be
represented by an integer number of cents. A quarter dollar equals exactly 25 cents, which is an integer number. Seen this way, this would allow to represent coordinates to the nearest 1/100 of a
The TrueType rasterizer does just that. But instead of decimal fractions (tenths, hundredths, thousandths, etc.), it uses binary fractions. Binary fractions are one half, one quarter (one half of one
half), one eighth (one half of one quarter), one sixteenth (one half of one eighth), and so forth. This makes it easy and efficient for the computer to convert to and from fractions.
If the same fraction is used for all calculations, it is called a fixed point binary number. For instance, to represent scaled coordinates, the rasterizer always uses 26 binary digits (or bits) for
the “number part” and 6 bits for the “fraction part” (abbreviated as “26.6 number”). With 6 bits for the fractions, every coordinate can be represented to the nearest 1/64 of a pixel. At 96 DPI
screen resolution, this corresponds to about 4 µm, which is again precise enough.
The rasterizer uses different fixed point binary numbers for different purposes. As mentioned already, scaled coordinates use 26.6 numbers. By contrast, the scaling factor (cf , , and ) uses a 16.16
number, because it needs more precision, but less “size.” Similarly, the x- and y-components of TrueType’s projection and freedom vectors (the direction of the constraint and the direction of
compliance with the constraint) use 2.14 numbers.
By now we have introduced fixed point binary numbers. Their purpose is to continue to use “integers” to represent coordinates, scaling factors, and TrueType’s projection and freedom vectors, but at a
higher precision than integers. But working with fixed point binary numbers is not always the same as working with integers.
Adding and subtracting fixed point binary numbers is straightforward. All we are doing is adding or subtracting whole numbers of fractions, like balancing a check book where every check was written
in cents, as opposed to dollars-and-cents. As long as we have enough bits for the sum, the results will be exact (cf ).
The problem is the division and—potentially unexpectedly so—the multiplication. Two integers multiply and the result is again an integer. If there are enough bits for the product, the result will be
exact. But once we use fractions, the result may become inexact. Here is why (to illustrate the point, I’ll use decimal fractions, which come a bit more naturally to humans than binary fractions, but
the argument is the same).
Let’s assume we want to multiply 3.25 by 5.25. The exact product of these numbers is 17.0625. But if two decimal places are all we have, like dollars-and-cents, then this number can not be
represented in our fixed point format. It must be rounded to the nearest number that can be represented, which is 17.06.
Think of currency exchange, if this helps, say from dollars to euros. The bank will round the dollar-and-cents amount you pay for the euros—and usually in its own favor. By contrast, at least the
rasterizer is fair. It will round the result to the nearest fraction. For instance, it will round coordinates (26.6 numbers) to the nearest 1/64 of a pixel.
Last but definitely not least, always beware of the division. Two integers hardly ever divide into another integer. There will always be fractions. But aren’t we using fractions with these fixed
point formats? We sure are, but there are hardly ever enough fractions!
For instance, dividing the above 3.25 by 5.25, the pocket calculator tells me the result is 0.619047619047619047619047619047… If you look at the distractingly long row of digits hard enough, you’ll
notice a pattern. This pattern repeats indefinitely, but we only have a (very) finite number of decimal places. Once more, the result must be rounded to the nearest number that can be represented,
which is 0.62.
At first glance this may sound like an extreme case of splitting hair. Really! Why obsess over 0.619047… vs 0.62 pixels? Because this tiny rounding error can compound with other seemingly negligible
rounding errors, until the result is no longer negligible. Oftentimes, when this happens, it comes as a complete surprise, because the compounding happened “in hiding.” We will look at this next.
In the following scenario we will have a look at the hidden rounding error compounding of the TrueType rasterizer. Specifically, we will constrain the width of a diagonal stroke in the context of a
character, such as a lowercase ‘v.’ The illustration below captures this scenario at the point where the rasterizer is about to process the instructions for constraining the diagonal.
Constraining the width of a diagonal stroke
The above sketch illustrates the diagonal stroke in light gray. On the right edge of the stroke there are two control points labeled P0 and P1. These points function as “parent” points. At the top of
the left edge of the stroke there is a control point labeled C. This point functions as “child” point. Furthermore, there are two vectors (“arrows”) labeled p and f, representing the direction of the
constraint (“projection vector”) and the direction of compliance with the constraint (“freedom vector”).
The goal of the constraint is to establish a given distance between the child point C and the parent points P0 and P1. This distance is determined perpendicularly to the edge denoted by P0 and P1,
which is parallel to p (“measured across the stroke”). If, at this point in the process, the given distance has not yet been established, then the child point C has to “give way” relative to the
parent points P0 and P1, in order to comply with the constraint. The direction in which the child point is allowed to “give way” is denoted by f.
For instance, if the child point is too close to the parent points, it will have to move farther away, and vice-versa. But it is allowed to do so only in the direction prescribed by f. If it were
allowed to move in any other direction, it would invalidate a prior constraint that establishes C to be constrained e.g. to the x-height. In the process, the final position of C is denoted by C′,
along with the given distance d′, while d denotes the initial distance between C and the parent points P0 and P1.
Readers familiar with TrueType may recognize this scenario by a sequence of instructions similar to the following code fragment
SRP0[], P0
SDPVTL[R], P0, P1
MIRP[M>RBl], C, d′
where a CVT number would assume the role of d′.
Before delving into the numerical minutiae, we need some numbers for the relative positions of the involved control points. I’ll assume the following distances between P1, P0, and C.
distance between… …in pixels
P1 and P0 in x-direction 2 56/64
P1 and P0 in y-direction 5
P0 and C in x-direction −25/64
P0 and C in y-direction 0
Distances between the control points involved in constraining the width of the above diagonal stroke
In a first step the rasterizer must determine the x- and y-components of the freedom vector f. This is easy, because it is in x-direction. Hence, as 2.14 numbers
f = (16384/16384, 0/16384).
In a second step the rasterizer must determine the x- and y-components of the projection vector p. This is harder. To do so it must establish a vector from P1 to P0, rotate it by 90°
(counter-clockwise), normalize it (divide each component by the length of the vector), and finally represent the result as 2.14 numbers (remembering that the vector and its length will be 26.6
numbers). Hence
p = (−320/64, 184/64)/(369/64) = (−14208/16384, 8170/16384)
where 369/64 is the length l of the vector from P1 to P0
l = (1/64)·((−320)2 + 1842)1/2 = 369/64.
In a third step the rasterizer must determine the initial distance d between the child point C and the parent points P0 and P1. To do so it can “project” the vector from P0 to C onto the projection
vector p using the scalar product (inner product, or dot product), and represent the result as 26.6 number. Hence
d = (−25/64, 0/64)•(−14208/16384, 8170/16384) = 22/64.
In other words, in our scenario the stroke is initially a mere 22/64 pixels wide.
To continue we will assume that the given distance d′ is one full pixel, whether this is the result of using a cvt number, or a minimum distance criterion, or a combination thereof. Thus, initially,
the child point C misses its target by
d′ − d = 64/64 − 22/64 = 42/64.
In a fourth and final step the rasterizer must therefore move the child point C by a distance of 42/64 when measured along the projection vector p, but it can only move it along the freedom vector f.
This distance m is longer than the one measured along the projection vector p, but it can be computed as follows.
m = (d′ − d)/(f•p)
= (42/64)/((16384/16384, 0/16384)•(−14208/16384, 8170/16384))
= (42/64)/(−14208/16384)
= −48/64
where f•p is the scalar product of the freedom and projection vectors.
Multiplied by the freedom vector f, and then represented as 26.6 numbers, and in turn added to the initial child point C, this yields the final child point C′
C′ = C + m·f
= (−25/64, 0) + (−48/64)·(16384/16384, 0/16384)
= (−1 9/64, 0/64).
In other words, the final C′ ends up 1 9/64 pixels to the left of P0.
Let’s “double-check” this result. To do so we will use the same logic that we have used already to determine the initial distance d between the child point C and the parent points P0 and P1, except
that now we will use the final child point C′. Hence
d″ = (−1 9/64, 0/64)•(−14208/16384, 8170/16384) = 63/64.
If none of the preceding calculations had introduced any rounding errors, this should be 64/64 pixels (1 full pixel), because that is what we “asked for.” But instead it is 63/64. This is 1/64 of a
pixel “short” of the target.
A couple of comments may be in order here. First and foremost, I won’t claim that the above example reflects exactly how the rasterizer processes the associated code fragment. I am writing this from
memory. But you are safe to assume that this example is at least representative of the inner workings of the rasterizer. Also, some of the math used in the process shall be left unexplained. It is
simply beyond the general scope of this website. Last but not least, keen TrueType-savvy readers may have noticed that the code fragment uses the dual projection vector, while the rest of the text
simply refers to the projection vector. This is neither a typo in the code nor in the text, but the associated subtlety is again beyond the scope of this website.
TrueType subtleties and math aside, the key take-away from this example should be the following. Fixed point binary numbers can and should be used whenever possible, because they are (or were) more
efficient than the more general floating point numbers, and because the rounding “behavior” of a single arithmetic operation is predictable, which is (or was) not always the case with floating point
numbers. However, for maximum precision, intermediate rounding steps must be avoided—fixed point or integer! If this is not—or cannot be—avoided, then the result is in jeopardy, which is why I call
this the “double rounding jeopardy.”
At the time the TrueType rasterizer was designed, 32 bit numbers were “top-of-the-line” and floating point numbers were either prohibitively time-consuming to emulate or required an expensive
floating point co-processor. In turn, with “only” 32 bits, many TrueType instructions must be broken down into multiple arithmetical operations on fixed point binary numbers. Each of these operations
may introduce a rounding error, particularly if the projection and freedom vectors point in a diagonal direction. Accordingly, the result may not be as precise as it could be. This is simply an
engineering trade-off between speed and precision in the context of available processor power at the time.
Is the TrueType rasterizer “broken?” Not really. Granted, the above example could be implemented by re-arranging the entire math into a single (!) fraction or rational number. Combined with clever
intermediate use of 64 bit fixed point binary numbers available nowadays this may get a precise result. But it is hardly ever exact because the final division still has to round to the nearest 26.6
number. We simply have to learn how to deal with the hidden rounding error compounding.
Thus, to reiterate: Think before you round more than once! Your results will be in jeopardy. But if you have to do it—or if the rasterizer makes you do it—be prepared for the consequences. We will
have a look at a simple example of dealing with these consequences next, and a more complex one later in this chapter (cf ).
The TrueType rasterizer maintains two sets of outlines that are available to the programmer, the original outlines and the instructed outlines. The original outlines represent the original,
unconstrained artwork that has been scaled down to the targeted point size and device resolution. The instructed outlines reflect any differences that executing the TrueType instructions may have
caused. Both sets of outlines are defined by their control points whose coordinates are represented as 26.6 numbers.
The seemingly innocent act of representing the original but scaled outlines as 26.6 numbers has a potentially unexpected consequence. To illustrate this point we will have a look at a single stem,
its designed outlines, its scaled outlines, and the resulting scaled stroke width.
Outlines of the “vertical bar” of Arial (U+007C)
I have “doctored” these outlines ever so slightly to cause the problem I’d like to illustrate. The table below summarizes these alterations.
Outline data (in font design units) of the above “vertical bar” of Arial (U+007C): original and 2 variants, “doctored” purely for demonstration purposes
As can be seen, the alteration is a small decrease in stem width, and the 2 variants differ by one font design unit in their side-bearings. The two side-bearings add up to an odd number, hence both
“spacings” represent an attempt to center the stem.
The unexpected consequence of attempting to center this stem, one way or the other, is that at 20 ppem we get two different scaled stem widths, as compiled in the table below.
1 55/64 1 54/64
3 22/64 3 22/64
1 31/64 1 32/64
Outline data of the above 2 variants of the “vertical bar” (U+007C), scaled to 20 ppem (in pixels)
The x coordinates of the left edges of the 2 variants scale to two different 26.6 numbers, while those of the right edges scale to the same number—the difference is lost in rounding to the nearest 1/
64 of a pixel.
Don’t believe it? Here is how I got there. The math to translate font design units (fUnits) into pixels, given the emHeight and the ppem size, is the following:
fUnits × ppem / emHeight = pixels.
This yields a floating point number. To get a 26.6 number, we have to round the floating point number to the nearest 1/64 of a pixel. To do so we simply multiply pixels by 64, round it to the nearest
integer, and then divide it by 64. Example:
343 fUnits × 20 ppem × 64 / 2048 fUnits = 214.375
Round(214.375) = 214
214 / 64 = 3 22/64 pixels.
342 fUnits × 20 ppem × 64 / 2048 fUnits = 213.750
Round(213.750) = 214 (again)
214 / 64 = 3 22/64 pixels.
Like I said, the difference is lost in rounding to the nearest 1/64 of a pixel. Two identical stem widths scale to a different number of pixel fractions. Nevertheless, this is what the TrueType
rasterizer calls the "original" outlines.
Somehow, this doesn’t seem right, and strictly speaking, it isn’t. Now, as I was preparing this example, I thought I knew just exactly which stops to pull to illustrate the consequences. Turns out
that I knew only sort of. Quite to my surprise, the TrueType rasterizer managed to puzzle me with this example, even after 15 years of exposure.
I used a few simple TrueType instructions to constrain the stem width using 3 slightly different methods. They all have in common that they first constrain the left edge to the nearest pixel
boundary, and then constrain the right edge relative to the left one, like below for method 1:
MDAP[R], leftEdge
MDRP[m>RBl], rightEdge
Instead of
MDRP[m>RBl], rightEdge
method 2 uses
MIRP[m>RBl], rightEdge, cvt
with an exact CVT (a CVT value of 153, corresponding to the exact stem width) and a sufficiently large CVT cut-in (8 pixels) to ensure this CVT overrides the “original” outline’s distance between the
two edges. Method 3 also uses
MIRP[m>RBl], rightEdge, cvt
but intentionally with a wrong CVT number (having a CVT value of 2048) and zero CVT cut-in to prompt the instruction to fall back on to the “original” outline’s distance between the two edges.
The following table compiles the stem widths (in pixels) that each of the 3 methods rendered using both of the 2 outline variants.
MDRP[…] 2(*) 2
(no CVT, no CVT cut-in)
MIRP[…] 2(*) 2(*)
correct CVT, large CVT cut-in
MIRP[…] 1 2
wrong CVT, no CVT cut-in
1 31/64 1 32/64
(for reference) (rounds to 1) (rounds to 2)
(*) 1 pixel expected
Pixel data of the above 2 variants of the “vertical bar” (U+007C), rendered stem width (in pixel) at 20ppem, constrained by 3 slightly different methods
Before we delve into the minutiae of the above results, recall , where we talked about surrogates representing clusters of sufficiently similar stroke weights. These surrogates are used in lieu of
the actual outline’s stem widths—the type size and device resolution permitting. This can ensure that substantially equal strokes are rendered with an equal number of pixels or samples.
TrueType’s terminology for the surrogate is the control value. Control values are tabulated and identified by a number, hence the term control value table number (or CVT number for short). To ensure
that CVT numbers are used only if the ppem size permits, TrueType uses a threshold called CVT cut-in. If the scaled value associated with a CVT number (the scaled CVT value) is within said threshold
of the (scaled) “original” outline’s distance, then the (scaled) CVT value will be used to constrain the distance, else the (scaled) “original” distance. This permits to render stem widths in all
their artistic individuality at large sizes while regularizing them at small sizes.
I have used this mechanism in methods 2 and 3 above, and quite deliberately so. For method 2, I chose an exact CVT value of 153 fUnits to represent my “cluster” of one stroke, and I chose a more than
large enough CVT cut-in to ensure the (scaled) CVT value will be applied. Conversely, for method 3, I chose a CVT number that is “way-off” and a CVT cut-in of 0 (zero). This combination should ensure
that the “original” distance is used to constrain the stem. Method 1 does not involve any CVT numbers, hence this should use the “original” distance by default.
Now then, let’s go back to the above table to see how the 3 methods perform against their specifications. I’ll start with method 3, because it works without surprises. The selected CVT value and CVT
cut-in had the constraint fall back on to the “original” stem width, which rounds to 1 pixel for variant A and 2 pixels for variant B. Not that the two differing “original” stem widths, scaled and
rounded to 26.6 from one and the same designed stem width, are particularly intuitive, but at least they can be explained (cf ).
For method 2 we need to translate the CVT value, which is in font design units, into pixels, like we did previously. Hence
153 fUnits × 20 ppem / 2048 fUnits ≅ 1.494 pixels.
This rounds down to 1 pixel. However, the rasterizer represents the above 1.494 pixels as 26.6 number, like any other coordinate. Hence
153 fUnits × 20 ppem × 64 / 2048 fUnits = 95.625
Round(95.625) = 96
96 / 64 = 1 32/64 pixel
which the subsequent MIRP[m>RBl] rounds to the nearest pixel. That’s how we end up getting 2 pixels—a case of “double rounding jeopardy” split between two rounding agents. A little math and some
memories of the rasterizer, and the results of method 2 can be explained, as well.
However, as far as method 1 is concerned, I’m at a loss. I cannot explain how MDRP[m>RBl] determines the “original” distance and gets variant A to render with 2 pixels, instead of 1. There is no CVT
involved, and hence no CVT cut-in. It should behave like the fall back code path of MIRP[m>RBl]. Maybe it’s a case of “double code path jeopardy?”
MIRPs and MDRPs aside, the key take-away is that TrueType’s “original” outlines are somewhat compromised in that, once scaled to the targeted type size and device resolution, they are rounded to the
nearest 1/64 of a pixel. Don’t rely on equally designed stems to be represented equally. Consider using CVTs instead. In edge cases, like the carefully constructed example above, they may not get you
the expected pixel count, but at least they’ll get you the same pixel count in every instance. This should help consistency.
So… isn’t the TrueType rasterizer “broken?” Not really. It’s once more an engineering trade-off between speed and precision in the context of prevailing processor power. Early processors had to use a
form of “long division” to divide two numbers, one bit at a time. This took a lot of time, relatively speaking. Therefore, trying to “cache” the division by storing the scaled coordinates and control
values made sense.
Nowadays the trade-off may look different. Compared to all the “GUI sugar” of current operating systems, not “caching” the division most likely would go unnoticed. However, given how seemingly
“fragile” or “fault-intolerant” font rendering is, what may get noticed may very well be an attempt to improve the internal precision of the rasterizer, however illogical this may sound at first.
Think of it this way. With the existing rasterizer, a particular character may render in a particular way at a particular ppem size. The author of the constraints (or “hints”) may see no fault with
the rendered pixel pattern. But he or she may be oblivious to the fact that it took the rasterizer a seemingly serendipitous chain of tiny rounding errors to produce this particular result.
Now, if any single one of these rounding errors is “improved,” the outcome may change easily, as we have just seen. And since the font looked “good” before, chances are that it will look “worse”
afterwards. We simply have to learn how to deal with the available precision—it’s hardly ever going to be exact, anyway.
In the preceding two chapters we have discussed potential benefits of sub-pixel anti-aliasing methods. In particular, in and we have looked at opportunities such as fractional stroke weights and
positions. However, when ClearType was announced at the November 1998 COMDEX, pretty much none of the existing TrueType fonts constrained stroke edges to fractional pixel boundaries compatible with
the requirements of ClearType.
Accordingly, at the time we were looking for some way of “shoehorning” existing TrueType fonts into ClearType duty. Therefore, in this section we will have a look at a number of challenges associated
with this endeavor. Notice that in this context, TrueType chiefly refers to a way of constraining outlines, such as to generate like sample counts for like features, while ClearType refers to a
method of converting multiple samples into pixels or RGB values.
The simplest way to “tease” more resolution out of existing TrueType fonts is to “pretend” a higher device resolution in x-direction than in y-direction. Even before ClearType, this is a property
that the TrueType rasterizer is supposed to be able to handle gracefully. Fax machines and inkjet printers continue to be prime examples of this requirement long after the first “Hercules Graphics
Recall , where we determined ClearType’s oversampling rate to be 6x, that is, the outlines are sampled at 6 times the rate of pixels in x-direction. Accordingly, we can ask the rasterizer to render
fonts at 576 DPI in x-direction (6×96 DPI = 576 DPI) combined with 96 DPI in y-direction. These numbers produce bitmaps like the one illustrated below:
Rendering previously “hinted” characters with different device resolutions in x- and y-direction (Times New Roman lc ‘m,’ 12 pt, 576 DPI = 6×96 DPI in x-direction, 96 DPI in y-direction)
⇒ Hover your mouse over the above illustration to see the “unhinted” rendition
Conceptually, we can proceed with rectangles of 6×1 (W×H) of these “skinny” bi-level pixels, interpret each of these pixels as an “in” or “out” sample, and downsample each sextuplet of samples into
an RGB value according to the ClearType anti-aliasing filter.
In the process, the higher DPI in x-direction directly translates to extra rendered detail when compared to plain bi-level rendering. For instance, the 3 stems are 8 samples wide each, which
translate to 1 1/3 pixels, while the 2 counterforms translate to 18 samples or 3 pixels. This provides for regular stem spacing while preserving the overall proportions of the lc ‘m’.
In practice, this simple method almost worked: Almost—but not on every character at every point size. In a seemingly random fashion some characters appeared to “explode” at some sizes, as illustrated
Rendering previously “hinted” characters with different device resolutions in x- and y-direction (Palatino Linotype lc ‘b,’ 12 pt, 720 DPI [= 6 × 120 DPI] in x-direction, 120 DPI in y-direction)
⇒ Hover your mouse over the above illustration to see the “unhinted” rendition
The primary cause of such “explosions” is the orientation of TrueType’s projection and freedom vectors p and f relative to one another. Recall the example illustrating the constraints of a diagonal
stroke as shown in . The projection vector p was set perpendicular to the edge of the diagonal, while the freedom vector f was set in x-direction.
Constraining the width of a diagonal stroke
Once we “pretend” a much higher device resolution in x-direction than in y-direction, all of TrueType’s “original” x-coordinates will be larger by the same rate. In turn, this causes the angles of
the projection and freedom vectors p and f to get skewed rather violently, as illustrated below.
Constraining the width of a diagonal stroke with a much higher DPI in x-direction than in y-direction
Recall how the width was measured “across” the stroke, or parallel to the projection vector p, and that the child point C was limited to complying with the constraint by “giving way” parallel to the
freedom vector f. Importantly, also recall that this meant that the child point C had to move by a slightly longer distance than when measured along the projection vector p.
Once the projection and freedom vectors p and f get skewed as illustrated above, this slightly longer distance turns into a much longer distance. In fact, in the extreme case where the projection and
freedom vectors p and f are perpendicular, the child point C could move as far left or right as it wanted to, but it would never comply with the constraint! It would just move “sideways.” Think about
this for a moment.
Luckily, we do not need that much extra detail. But still, there is a limit as to how far we can push this approach. Specifically, in the computation of the final child point C′, we divided by f•p,
the scalar product of the projection and freedom vectors p and f. If these vectors are perpendicular, then the scalar product is zero—that’s the mathematical definition of perpendicular. If they are
“almost” perpendicular, then the scalar product is “almost” zero.
In turn, this means that we would divide by zero, which is “illegal,” or we would divide by a very small number, which creates a very large result that may no longer be represented by a 26.6 number.
The TrueType rasterizer is smart enough to detect these situations. In fact, if the projection and freedom vectors p and f are within about 3.58° of being perpendicular, you will get a warning in
VTT. Heed it!
Alas, in the above example illustrating an “exploding” character, VTT didn’t give me a warning. This doesn’t mean that there is no problem with the projection and freedom vectors p and f. All that
this tells me is that they never got close enough to trigger a warning. But they may still be close enough to cause a problem.
Other causes of “explosions” are italic characters in general. Mathematically speaking, italics aren’t all that much different from “romans,” except that the y-axis is “italicized.” This simple
aspect is often missed—both by the rasterizer itself, and by the author of the constraints (or “hints”). We will get to the rasterizer’s share of the italic challenges later in this chapter (cf and
Last but not least, details may go amiss in the thicket of TrueType instructions, or there is insufficient understanding of certain concepts of constraining outlines, or a combination thereof. Within
a list of cryptic mnemonics and a commensurate supply of numbers, it is easy to miss that e.g. both parent points must have been constrained in both x- and y-direction before it makes any sense to
set the (dual!) projection vector perpendicular to these points. Likewise, it is easy to miss the introduction of a cycle in the hierarchy of constraints, such as a constraint that depends on another
constraint which in turn depends on the completion of the constraint about to be introduced.
Which of the above points ultimately caused the “explosion” I do not remember. By the time I was alerted to this particular example, I had seen my fair share of “unusual” ways to constrain (or
“hint”) characters, hence I was not surprised. Most likely, I haven’t even looked at the source code anymore, which would explain why I don’t remember any details. Rest assured though that for
practical intents and purposes this example is representative of the challenges we were facing upon pushing existing fonts into ClearType duty.
Granted, I didn’t expect anybody to have anticipated—and therefore to have tested against—such “lopsided” resolutions in x- and y-direction. But bogus code is bogus code, even though the results may
just “happen to look right” in those cases that were actually looked at. Informally put, think of two “wrongs” that happen to make a “right” and then unravel one of the “wrongs.” This presented a
formidable challenge. Fix all the affected fonts or “jury rig” the rasterizer?
Let’s recap what we have tried so far to “shoehorn” existing TrueType fonts into ClearType duty. The core step was to scale the outlines to a higher DPI in x-direction than in y-direction. I’ll call
this overscaling. Along with overscaling I’ll define the overscaling factor as the rate by which the higher DPI exceeds the lower DPI. For instance, if the DPI in x-direction is 6 times the DPI in
y-direction, then the overscaling factor is 6x.
With this definition, what we have tried so far is overscaling as the first of the following four steps:
1. Overscale outlines in x-direction by 6x
2. Apply constraints (“hints”) to overscaled outlines
3. Sample constrained overscaled outlines into overscaled bitmaps
4. Apply anti-aliasing filter to downsample overscaled bitmaps
For existing fonts the advantage of this approach is not having to change
of the constraints in order to use fractional (instead of full-pixel) stroke weights and positions. For future fonts the advantage is pretty much the same, considering that TrueType does not offer a
simple way to round to fractions smaller than
of a pixel (cf ). Last but not least, for the rasterizer the advantage is not having to implement oversampling. The combined steps of overscaling and full-pixel sampling are equivalent to
the outlines.
The show-stopping disadvantage is the chance for the seemingly random “explosions” as previously illustrated. To avoid these “explosions” the obvious alternative is to swap steps 1 and 2 above:
1. Apply constraints (“hints”) to outlines
2. Overscale constrained outlines in x-direction by 6x
3. Sample overscaled constrained outlines into overscaled bitmaps
4. Apply anti-aliasing filter to downsample overscaled bitmaps
This of course eliminates the “explosions” because the constraints operate under the same conditions as when they were authored. But it comes at an enormous price:
existing fonts would have to be changed to round stroke weights and positions to
sample boundaries,
instead of pixel boundaries. This would be a
of work!
Accordingly I was looking for an approach that combines the advantages of overscaling before constraint application with the “safety” of overscaling after constraint application. To do so I “jury
rigged” the rasterizer to dynamically re-interpret all rounding instructions to mean “round to sample boundary” if the rendering method is ClearType and if the direction of the projection vector p is
not parallel to the long side of the LCD sub-pixels, else to keep the meaning “round to pixel boundary” as before:
1. Apply contextually interpreted constraints (“hints”) to outlines
2. Overscale constrained outlines in x-direction by 6x
3. Sample overscaled constrained outlines into overscaled bitmaps
4. Apply anti-aliasing filter to downsample overscaled bitmaps
Semantically, the two interpretations of rounding instructions are most similar. In fact, if you consider full-pixel bi-level rendering (colloquially “black-and-white” rendering) as “oversampling” at
the rate of one sample per pixel, then the two interpretations are identical. Practically, this approach eliminates the “explosions” and it overcomes the absence of a TrueType instruction that would
permit “finer grained” rounding than to the nearest
of a pixel (cf ).
Alas, while this “jury rigged” approach worked really well to eliminate the “explosions,” it also brought to daylight some of the dirty secrets of day-to-day “hinting.” Way back when bi-level
rendering was the only rendering method, it was not uncommon for constraints to completely mangle the outlines, as long as this would yield the desired pixel patterns. Like with the “lopsided”
resolutions in x- and y-direction, most likely this was not anticipated to ever change, but now presented a host of formidable challenges. We will discuss two areas of challenges, along with their
“jury rigs,” next.
The TrueType instruction set comes with a concept called Delta Exception. A delta exception is an instruction that permits to dislocate a single control point by an arbitrary fractional number of
pixels in an arbitrary direction and at one specific ppem size at a time. For instance, a delta instruction can move control point 28 by 13/16 of a pixel in the positive x-direction at 16 ppem. This
is a bit of a mixed blessing.
On the one hand, it permits authors of constraints to obtain any pixel pattern even if everything else fails to do so. On the other hand, the relative ease with which delta exceptions can be added to
the constraints does not encourage the development of ppem size independent solutions. Particularly the help of a graphical user interface such as in VTT is tempting to “solve” any problem with
Moreover, at the time, the deltas were completely “hard-wired” into the code, since there was no alternate rendering method requiring to do otherwise. This often led to outlines like the one
illustrated below.
The effect of delta instructions on outlines and rendered pixels (Times New Roman UC ‘W,’ 12 pt, 96 DPI, RTM version of the font rendered in bi-level)
⇒ Hover your mouse over the above illustration to easily see which pixels would be “out” if it weren’t for TrueType’s dropout control feature
Notice that there are several pixels whose centers are “out.” If it weren’t for TrueType’s mechanism for restoring “missed” pixels (dropout control mechanism), these pixels wouldn’t even show up in
the final bitmap! Be sure to hover your mouse over the above illustration to see which pixels would be “missing.”
This example illustrates the delicate interplay between deltas deliberately distorting the outline and the seeming serendipity of TrueType’s dropout control mechanism for the purpose of obtaining a
particular pixel pattern. Maybe certain usages of deltas are considered artful, clever, cunning, shrewd, or whatever… But in terms of pixel coverage (cf ), obviously, this set of outlines is a
What to do? These deltas show up in ClearType—and not to its advantage. I could “jury rig” the rasterizer to bypass all delta instructions. This would cure the symptoms in the above example. But like
the Hydra rearing one of its many heads, bypassing all deltas would eventually show up in other examples.
The positioning of crossbars appears to be a prime example where authors of constraints “solve” the erroneous positioning with delta instructions (cf ). Since delta instructions apply to one specific
ppem size only, each ppem size at which the crossbar’s position is “off” will have to be fixed by a separate delta instruction—usually one delta will not be enough.
Accordingly, from within the rasterizer, I tried to identify the purpose of each delta instruction. Is it meant for positioning features, and is the projection vector p parallel to the long side of
the LCD sub-pixels? In other words, is it meant to position an aliased stroke or similar in ClearType? Then the delta instruction will execute as specified. For all (!) other cases the delta
instruction will be bypassed. These cases include the positioning of sub-pixel anti-aliased strokes along with all deltas to distort the outlines with the original purpose of obtaining a particular
bi-level rendering pixel pattern.
But—you might object—how can you identify the purpose of a delta instruction? Strictly speaking, I cannot. But I can make an educated guess based upon the sequential position of the delta instruction
within the stream of TrueType instructions (notice that the following applies to ClearType only, bi-level and full-pixel anti-aliasing are not affected):
• If the delta instruction appears after the IUP[X] or IUP[Y] instruction for the respective projection vector p, then the delta is meant to “fix” a pixel pattern, hence it will be skipped. VTT
calls this a post-IUP delta.
• If the delta instruction appears before the IUP[X] or IUP[Y] instruction, but there are no other constraints depending on the particular control point, given the respective projection vector p,
then the delta is once more meant to “fix” a pixel pattern, and hence it will be skipped, too. VTT calls this a pre-IUP delta.
• If, however, the delta instruction appears before the IUP[X] or IUP[Y] instruction and there are other constraints that depend on the particular control point, given the respective projection
vector p, then the delta is meant to “fix” a feature position, hence it will not be skipped if and only if the projection vector p is parallel to the long side of the LCD sub-pixels. VTT calls
this an inline delta (for any orientation of the projection vector p).
Hence, as you can see, it is possible, but it requires some careful thought. Combined with the contextual interpretation of the other constraints, this renders the above example as follows:
The effect of delta instructions on outlines and rendered pixels (same as above except rendered in ClearType). Observe how the “dents” in the outline are gone, the stroke design contrast reflects the
original design more closely, but the serifs at the top are still aligned with the cap height
⇒ Hover your mouse over the above illustration to revert to bi-level rendering)
Be sure to hover your mouse over the above illustration to see the difference between the shapes of the outlines in ClearType and in bi-level rendering!
This illustration may give rise to the impression that all instructions in x-direction are ignored—one of the opinions I keep reading in internet discussion forums and similar sources of collective
wisdom. But this is not true! What you see is the difference between fractional and full-pixel stroke weights and the absence of unnecessary outline distortions for controlling untamable pixel
patterns. So let me repeat this:
ClearType rendering does not generally ignore TrueType instructions in x-direction. It merely rounds them differently, and it bypasses distorting deltas.
Now, before you lament that you’re no longer in control of the outcome of your artwork, think about how you would round a control point to the nearest sample boundary, instead of the nearest pixel
boundary, using TrueType instructions. TrueType’s RTDG[] (“round-to-double-grid”) is not “fine grained” enough, and however long and hard I looked at it, TrueType’s SROUND[] (“super-round”) won’t do
it, either (cf also ). Seems easier to let the rasterizer do this, doesn’t it?
But if you really need to delta ClearType beyond inline deltas in y-direction, there is a switch to get you most of the original TrueType functionality back—the main exception being the rounding. In
your Control Program place the following code snippet right at the beginning:
/* test if CT avail on this rasterizer */
#PUSH, 37, 1
#PUSH, 64, 1
IF[] /* avail, switch to native CT mode */
#PUSH, 4, 3
If you don’t use VTT’s Control Program, that is, if you use a simple Control Value Table along with your own Pre-Program template, skip the ASM("…") bracket and place the TrueType code at the
beginning of your Pre-Program.
Beyond that, consider limiting deltas to the respective rendering modes. Don’t “hard-wire” them into the code. For instance, if a delta applies to bi-level rendering only, use the following VTT Talk
instructions (requires VTT’s Font Program template):
This will compile to a function that applies a delta instruction exclusively when in bi-level (“black-and-white”) rendering. The same functionality is available from VTT’s graphical user-interface by
using the Delta Tool, provided you are initiating the delta with a “right-click” while in bi-level rendering.
For further information on “deltaing” ClearType, please refer to Microsoft’s recent publication. Said document lists additional stop-gap solutions I implemented to overcome a number of “unusual” ways
of authoring constraints.
With the “explosions” and deltas out of the way, there was one more noteworthy concern. Some combinations of outlines and constraints lead to fairly “skinny” strokes. In bi-level rendering at typical
text sizes on prevalent screen resolutions, this was not much of a problem. The minimum distance criterion or the dropout control mechanism would reliably take care of the “emaciated” appearance of
these strokes.
Not so with a much improved notion of pixel coverage. The first time this sort of problem showed up in ClearType was with the font Courier New. Legend has it that the outlines of Courier New were
taken from the “golf ball” type element of an IBM Selectric typewriter. Striking the type element’s raised letters against the ribbon caused the ribbon’s ink to spread on paper, hence the type
element’s letters were “thinned” to compensate.
This compensation was not reversed after digitizing the outlines. Rendered with ClearType, it turned the font into Courier “Light”—particularly after gamma correction (cf ), as illustrated below.
The result of previously “hinted” “skinny” characters when rendered in ClearType (Courier New lc ‘m,’ 10 pt, 96 DPI, corresponding to a 12 CPI [characters per inch, 12 pitch] “golf ball” type
element, gamma correction set to 2.2 [cf ])
⇒ Hover your mouse over the above illustration to see how the “unhinted” renders at the same size. It is a “skinny design”
The problem was exacerbated by the addition of anti-aliasing along the long side of the LCD sub-pixels (cf ). Together with the respective downsampling filters, and after gamma “over-correction” (cf
), this made parts of characters look “washed out,” as illustrated below (CAUTION: To properly view the following illustration, be sure to double-check all your settings as per the check-list
introduced in ).
The result of previously “hinted” “skinny” characters when rendered in “anti-aliased ClearType” (hybrid sub-pixel anti-aliasing, 8 pt through 16 pt at 96 DPI, same gamma correction as above)
⇒ Hover your mouse over the above illustration to see the same anagrams rendered in plain bi-level. Notice the difference in stroke rendering contrast: Both illustrations nominally render black text
against a white background
To understand the extent of the problem I’ll revisit the table in but replace the assumed “typical” stem widths by those of Courier New (84 or 85 fUnits). Here are the numbers:
1 5/6
Minimum stroke widths (in pixels) to render Courier New’s stems with a “black core” (top row), followed by the corresponding ppem and point sizes for a range of current screen resolutions
In other words, to render Courier “Light” with a “black core” on a screen with a typical resolution of 96 DPI, we would need a type size of 33 point or more. My old “№ 5 Underwood Standard
Typewriter” could do better than that, given a fresh ribbon…
To address the challenge of Courier “Light,” we are effectively looking at a combination of problems:
• The outlines are “designed” too light.
• With any of the anti-aliasing methods, including ClearType, TrueType’s dropout control mechanism inserts missing samples, not missing pixels.
• To minimize distortions caused by aggressive bi-level constraints, it was eventually decided that ClearType rendering should reduce the minimum distance criterion to one half of the value
specified by the author of the constraints.
Together with gamma “over-correction,” this explains the lack of stroke rendering contrast, even for text nominally rendered in black against a white background, as in the example above.
Notice that it is not always trivial to apply a minimum distance constraint just about anywhere on the outlines in order to assert minimal pixel coverage and hence a minimum stroke rendering
contrast. It may require a suitable pair of control points to do so, which the outlines may not have in the desired location, and it generally requires a geometrically sound set of constraints (or
“hints”). Without suitable constraints, even a well-designed font can get badly mangled, as illustrated below.
The result of characters previously “hinted” to become quite “skinny” after rendering in ClearType (Palatino Linotype lc ‘n,’ 8 pt, 96 DPI, gamma correction reset to 1.0 [cf ])
⇒ Hover your mouse over the above illustration to revert to bi-level rendering and observe how an acceptable set of pixels is obtained irrespective of the badly mangled outline
Notice that if there were any deltas involved in the above illustration, they would have been “skipped” according to the strategy discussed in . Notice also that the illustrated problem is not
specific to sub-pixel anti-aliasing methods such as ClearType. Due to their improved notion of pixel coverage, any of the anti-aliasing methods will render with similar results, as illustrated below
for full-pixel anti-aliasing:
The result of characters previously “hinted” to become quite “skinny” after rendering in full-pixel anti-aliasing (otherwise same example as above)
⇒ Hover your mouse over the above illustration to revert to bi-level rendering and observe how an acceptable set of pixels is obtained irrespective of the badly mangled outline
The reason why bi-level rendering yields a set of pixels that may be acceptable in its own context (while any form of anti-aliasing seemingly doesn’t) is a subtle difference in the way TrueType’s
dropout control mechanism works in anti-aliasing vs bi-level rendering. In bi-level (“black-and-white”) rendering, dropout control inserts entire pixels deemed missing, while in anti-aliasing it
“merely” inserts missing samples. To my intuition this seems logical, but at the same time it may yield unexpected results, hence let’s have a closer look at one pixel of the above illustrations
where the outline gets really “skinny:”
One pixel of the above illustration where the outline gets really “skinny”
In full-pixel anti-aliasing (“gray-scaling”) the pixel in question renders in a light shade of gray. Now recall oversampling as introduced and illustrated in . With 4x4y oversampling (as used in
Windows’ “Standard Font-Smoothing”) most likely the pixel in question is sampled like so:
Sampling the pixel of the above illustration where the outline gets really “skinny.”
4 out of 16 samples are “in” yielding a 25% level of gray.
⇒ Hover your mouse over the above illustration to see that level of gray
Only 4 out of 16 samples are “in” and hence the pixel renders at 25% gray. These 25% represent the pixel coverage—albeit very loosely so given this badly mangled outline.
If dropout control had anything to do with this outcome, it may be the result of “dropping in” the one sample deliberately turned off in the following illustration:
Sampling the pixel of the above illustration where the outline gets really “skinny.”
3 out of 16 samples are “in” yielding an 18.75% level of gray.
⇒ Hover your mouse over the above illustration to see that level of gray
It’s an “edge case” and hence the “may be” above. But regardless, in case you haven’t noticed it by now: that single “dropped in” sample will contribute only 1/16 of the final pixel “darkness,” as
illustrated below:
One pixel of the above illustration where the outline gets really “skinny” and potentially rendered without dropout control applied to anti-aliasing
⇒ Hover your mouse over the above illustration to see the modest difference a single “dropped in” sample will make
The difference a single “dropped in” sample will make and hence the effectiveness of “anti-aliased dropout control” may indeed be surprisingly small at first. But considering that any form of
anti-aliasing—including ClearType—can render subtleties that bi-level’s big pixels simply can’t, it would seem natural that “anti-aliased dropout control” operates in much the same subtle way. The
initial surprise merely reflects a challenge with the opportunities afforded by a more accurate representation of fractional pixel coverage.
Except that this finding doesn’t help with existing fonts “hinted” to become “skinny” while relying on “bi-level dropout control” to insert missing pixels. Accordingly I was looking for a “jury rig”
to address this challenge on the level of samples, rather than second-guessing the purpose of even more instructions. I experimented with various heuristics targeted at augmenting the number of “in”
samples along excessively thin parts of glyphs, which provided some degree of relief, as illustrated below.
The result of characters previously “hinted” to become quite “skinny,” after rendering in ClearType, and after applying some form of sample count augmentation (same outline and conditions as above,
except for sample count augmentation)
⇒ Hover your mouse over the above illustration to revert to the previous illustration without sample count augmentation
CAUTION: To properly view the following illustration, be sure to double-check all your settings as per the check-list introduced in
Same conditions as above, except applied to a popular anagram
⇒ Hover your mouse over the above illustration to revert to the version without sample count augmentation
Readers familiar with VTT may turn on the above heuristics by enabling “CT (anti-aliased)” and “CT (dropout ctrl)” in the “Rasterizer Options.” Depending on the font and type size, the effect may be
more or less subtle than above. The purpose of this approach was to increase pixel coverage without turning each and every font into a monoline font.
As far as I remember, a variant of this sample count augmentation made its way into production code, at least into WPF (Windows Presentation Foundation). I do not recall the exact method by which
Courier “Light” was addressed in GDI (Graphics Device Interface) in the end. Suffice it to say that the preferred method would be outlines designed to represent the correct stroke weights combined
with a set of appropriate constraints (or “hints”).
Attempting to “shoehorn” existing TrueType fonts into ClearType duty represents only part of the challenge. The other part is assembling individual characters into words and phrases. Once rendered
with ClearType, the seemingly inconspicuous act of “putting one character after another” had surprisingly conspicuous consequences to anything from dialog boxes to entire websites.
Recall through where we discussed character proportions, layout contexts, fractional advance widths, and fractional ppem sizes. Depending on how we prioritized the constraints, we were able to
preserve character proportions or advance widths, but not necessarily both. While a difficult decision on its own, knowing the layout context helped to make an educated choice.
Unfortunately, many existing text layout applications provided a context that was “tailored” to bi-level rendering. Words and phrases rendered in ClearType caught these contexts off guard.
Surprisingly, even DirectWrite’s “natural” modes leave room for improvement. Therefore, in this section we will have a look at glyph metrics or advance widths in the context of text layout
Consider the fragments of text in a dialog box, such as the text next to a radio button or a group of check boxes. The designer of the dialog box may have been lucky that a particular fragment of 8
point text at 96 DPI “just about” fit into the available space. Never mind that it may not be good design practice to omit margins or to rely on the dialog box never being used at any other screen
resolution. As far as the proud designer is concerned, his or her dialog box is “pixel perfect.”
Enter ClearType, with all the above efforts at skipping deltas to avoid unnecessary distortions. As a “side-effect,” these efforts often restored the characters’ advance widths that bi-level
constraints may have compromised, as illustrated below.
Comparing advance widths from “historic” bi-level rendering with advance widths from of ClearType rendering: the green lines represent the constrained (or “hinted”) advance width while the blue lines
represent the designed (or “unhinted”) advance width (Tahoma lc ‘m,’ 8 pt, 96 DPI—the default UI font of Windows XP, at its default size)
⇒ Hover your mouse over the above illustration to see how this renders in ClearType and observe the different advance widths
Be sure to hover your mouse over the above illustration to see the dramatic difference between bi-level and ClearType rendering.
Bi-level constraints prioritized equal stroke positioning and combined the limited side-bearing space into one pixel on the right. In the context of bi-level rendering, this is as good as it gets.
But it accumulates a rounding error of approximately 1 pixel, which is absorbed in the advance width. If many characters have to do this, and if the dialog box “design” relies on the extra character
or two it thus can squeeze onto a line, then the more natural advance widths of ClearType create a problem: The text will be truncated.
ClearType just broke the dialog box! And not just this one dialog box; many more were affected. In fact, it affected many applications, including the browser rendering many “pixel perfect” websites.
Never mind that these dialog boxes or websites simply revealed their lack of scalability. As far as their respective designers were concerned, ClearType just broke their “design.” It was again a case
of two “wrongs” making a “right” with ClearType fixing one of the “wrongs.”
What to do? Initially, GDI had a simple solution. They added an x-direction scaling step between steps 1 and 2 of the process described in :
1. Apply contextually interpreted constraints (“hints”) to outlines
+ Scale in x-direction to “fit” outlines into bi-level advance width
2. Overscale constrained outlines in x-direction
3. Sample overscaled constrained outlines into overscaled bitmaps
4. Apply anti-aliasing filter to downsample overscaled bitmaps
In the process, if ClearType rendered characters wider than bi-level, they got “squeezed” back into their bi-level widths. If it rendered characters narrower than bi-level, they got “stretched.” Ta-
The internet was “fixed.”
Simple indeed… except that along with “squeezing and stretching” characters back into their bi-level widths, it also “squeezed and stretched” stroke weights. Recall how the contextually interpreted
constraints emulated rounding to sample boundaries, as opposed to pixel boundaries. This enabled fractional stroke weights and positions. But the ensuing “squeezing and stretching” changed those
fractional stroke weights. In turn, characters with like strokes were rendered with unlike stroke weights!
Accordingly, I was looking for an “alternative.” Now, if there was ever an appropriate usage of the term “lesser evil,” this may be it. What I was looking into was to compress or expand the stroke
positions without affecting the stroke weights. In the process, one character may get compressed, while the other one may get expanded. This is evil! But I felt that this was still the lesser evil
because at least it left the stroke weights unchanged.
From within the rasterizer, I tried to identify patterns of TrueType code conventionally associated with constraining stroke weights and positions. Sometimes these patterns were consistent, sometimes
not, in which case I tried to somehow “double-check:” Maybe this “white link” was meant to be a “black link?” Sometimes the existing system of constraints was simply too inconsistent.
With the strokes and their positions identified, I could implement a form of intelligent scaling: Apply the scaling in x-direction to the stroke positions only, but not to the stroke weights.
1. Apply contextually interpreted constraints (“hints”) to outlines
+ Intelligently scale in x-direction to “fit” outlines into bi-level AW
2. Overscale constrained outlines in x-direction
3. Sample overscaled constrained outlines into overscaled bitmaps
4. Apply anti-aliasing filter to downsample overscaled bitmaps
In the process, characters rendered in ClearType get compressed or expanded to match the bi-level advance widths, but the stroke weights are unaffected. Within the context of existing dialog boxes
and websites, this worked reasonably well, as illustrated below.
Comparing advance widths from ClearType rendering with advance widths from “historic” bi-level rendering (same conditions as above, except starting out from ClearType instead of bi-level rendering)
⇒ Hover your mouse over the above illustration to see the same outline rendered to an advance width “compatible” with bi-level rendering. Notice that the green lines represent the constrained (or
“hinted”) advance width while the yellow lines represent the “fitted” or “compatible” advance width. Notice in particular that the stem widths remain unchanged in the process
Be sure to hover your mouse over the above illustration to see how the character gets compressed but the strokes keep their weight.
Sometimes the existing constraints weren’t consistent enough to identify all stems, or there were cycles in the constraints that made it hard to “second-guess” their intended positioning and hence
their re-positioning, and sometimes the “prescribed” advance width was simply way too tight, as in the following example.
Limitations of making ClearType advance widths “compatible” with bi-level advance widths: for some combinations of characters and point sizes there is simply no intuitive way to “give way.” (Tahoma
lc ‘l,’ 11 pt, 96 DPI)
⇒ Hover your mouse over the above illustration to see what it would look like without any further attempts at “squeezing”
Without counterforms to absorb the “squeeze” there is nothing particularly intelligent that could be done to fit this character into the available space. Depending on the text, this can have fairly
dramatic consequences, as illustrated below.
Limitations of making ClearType advance widths “compatible” with bi-level advance widths: Without counterforms to absorb the “squeeze” there is nothing particularly “smart” that could be done to
“squeeze” the respective character into the available space
⇒ Be sure to hover your mouse over the above illustration to see what this would look like if we didn’t have to “do the squeeze.”
This looks very odd to me, too. But it would fit the “pixel perfect” dialog box or website “designed” around bi-level rendering… For less dramatic cases, what I’ve tried to do is some form of damage
Granted, this is far from optimal. Dialog boxes and websites should become more “flexible.” But at the time they weren’t, and “breaking” them was deemed unacceptable, hence our efforts at making it
work—somehow. Never mind that nowadays websites have a much better chance at being “broken” simply by zooming in or out (cf ) or by setting a minimum font size, both for reasons of accessibility.
Last but not least, if you are creating your own software, there is a switch to turn off bi-level “compatible” advance widths.
As we have just seen, assembling words and phrases by “putting one character after another” can break “pixel-perfect layouts” if they aren’t ready for advance widths to ever change. But even without
“compatible” advance widths “squeezed” into previously defined layouts, insufficient understanding of constraining outlines combined with incomplete knowledge of the rasterizer’s idiosyncrasies can
make or break optimal inter-character spacing. We will have a look at an innocent example thereof next.
“Natural” advance widths are the result of turning off “compatible” advance widths. Like the latter, they are laid out by full-pixel positioning, but they are not compromised by “backwards
compatibility requirements.” They can be as precise as the size of the pixel allows without fractional pixel positioning (cf and ). In short, “natural” advance widths provide a straightforward yet
attractive default.
Unfortunately, “natural” advance widths are also a victim of the “double rounding jeopardy.” I counted 5 (!) rounding operations that are cumulatively applied to the designed advance widths before a
glyph appears on screen. One rounding operation on top of another rounding operation on top of … etc. This is an accident waiting to happen!
To illustrate one such accident, I’ll use Calibri, one of the fonts of the ClearType font collection and the new default font for Office 2007. In particular, I’ll have a look at the lowercase ‘e’ at
11 point on a 96 DPI display. This is the most frequent letter in the English language, used at the default font size in Office, and displayed on what is likely the dominant display resolution. Here
is what happens:
1. The advance width of the lowercase ‘e’ is 1019 fUnits. At 11 point and 96 DPI, this translates to 7 pixels as follows:
advance width × point × DPI / (emHeight × 72)
= 1019 × 11 × 96 / (2048 × 72) ≅ 7.297526
Round(7.297526) = 7 pixels.
2. 11 point at 96 DPI translates to 14 2/3 ppem, a fractional ppem size. The font restricts ppem sizes to integers, hence the rasterizer rounds the ppem size to 15 ppem. In turn, this translates the
advance width—rounded to the nearest 1/64 of a pixel to represent it as 26.6 number—as follows:
advance width × ppem × 64 / emHeight
= 1019 × 15 × 64 / 2048 = 477.65625
Round(477.65625) = 478
478 / 64 = 7 30/64 pixels.
3. The rasterizer re-rounds this advance width to the nearest 1/16 of a pixel, probably in a well-meant but misguided attempt to provide the author of the constraints with the most natural rendition
of the advance width in terms of the 16x oversampling rate in use at the time. In turn, this translates the advance width to:
(7 30/64) × 16 = 119.5
Round(119.5) = 120
120 / 16 = 7 8/16 pixels.
This is what the rasterizer presents to both the author and the interpreter of the constraints.
4. Once the constraints have been applied, the rasterizer re-rounds the advance width once more, to the nearest full-pixel, because it is going to be used in full-pixel positioning:
7 8/16 = 7.5
Round(7.5) = 8 pixels.
This is how the 7.297526 pixel advance width ends up as 8 pixels. Following is a graphical representation of the 4 different advance widths above:
Cumulative rounding operations compromise ClearType’s “natural” advance widths (Calibri 11 pt, 96 DPI, “hinted” outlines)
Refer to the table below for a detailed description of the colors
blue what the application asked for 7.297526
green what the font asked for 7.46875
red what the rasterizer shows 7.5
black what the rasterizer finally delivers 8
There are two separate accidents in this particular chain of rounding operations:
1. The “natural” advance width is off by more than 0.7 pixels. The resulting 8 pixels correspond to a font size of 12 point, not 11 point. This compromises spacing in a scalable layout context.
2. The rasterizer silently adds 0.5 pixels of “padding” to the right side-bearing space without “telling” the author of the constraints. This compromises spacing in a reflowable layout context.
The second accident may come as a bit of a surprise. Yet it can be observed readily by placing the ‘e’ between a pair of ‘l’ and displaying it in VTT’s text string or waterfall, one character after
Cumulative rounding operations on the advance widths compromise the inter-character spacing even when laying out text one character after another (Calibri 11 pt, 96 DPI, ClearType’s “natural” advance
widths, top: enlarged, bottom: original size)
⇒ Hover your mouse over the above illustrations to see the inter-character spacing without accumulated rounding operations
There are two key take-aways from this example:
1. “Natural” widths aren’t quite as “natural” as they purport to be.
2. It’s not always the “evil WYSIWYG algorithm” that ruins spacing.
There is no easy “fix” to overcome the shortcomings of “natural” advance widths. It can be done, but it requires some
efforts in terms of coding in TrueType!
In the preceding sub-section we have discussed how the “double rounding jeopardy” (cf ) can compromise inter-character spacing even when using “natural” widths. While “natural” widths allow to
minimize “blur” at small type sizes and low resolutions, few professional “hinters” are knowledgeable enough to make the respective efforts in (ppem size independent!) TrueType code. There are simply
too many variables involved, and few text layout applications currently use “natural” widths, anyway.
Instead, text layout applications switch to using DirectWrite’s fractional advance widths (cf ) in hopes that this solves all the remaining “spacing issues.” Theoretically, if “natural” widths can
position characters to the nearest ±1/2 of a pixel, fractional widths should be able to do so to the nearest ±1/2 of a sample, or ±1/12 of a pixel, given ClearType’s 6× oversampling rate (cf ).
Spacing errors of this small magnitude should go unnoticed, because even on a screen with a mere 96 DPI resolution, 1/12 of a pixel is less than 1/1000 of an inch (or about 22 µm). The healthy human
eye can’t resolve this level of detail from a distance beyond 7.5 cm (3 inches), and the adult eye cannot accommodate to such a short viewing distance. To confirm this claim, let’s have a look at the
illustration below:
Windows 7 RTM “rehinted” ‘mno’
DirectWrite VTT (fract AW)
Spacing test string “nnmnonomoommm” taken from VTT and rendered with DirectWrite (left) and in VTT (right), both using fractional advance widths and fractional pixel positioning (Calibri, 8 pt at 96
DPI). Each line differs from the previous one by a horizontal shift of 1/6 of a pixel to the right.
⇒ Hover your mouse over either of the illustrations to enlarge it to 400%
At first glance, the difference between DirectWrite and VTT may seem subtle, and they may be partially “masked” by different characters getting “blurred” to different degrees as a result of using
fractional pixel positioning. Therefore the following illustration repeats the previous one, but with the role of the “mouse-over” changed to revert to the RTM version:
Same test string as above, but both rendered in VTT with the “rehinted” characters ‘mno’ at original size (left) and enlarged to 400% (right).
⇒ Hover your mouse over either of the illustrations to revert to the Windows 7 RTM version
Be sure to hover your mouse over the above illustration to revert to the Windows 7 RTM version, and try not to get overly sidetracked by different degrees of “blur” on different characters.
Notice how—when reverting to the RTM version— the characters appear to get wider, and how e.g. the last ‘o’ on the first and second rows seem to touch the adjacent ‘m’?
What you see are integer ppem size advance widths “squeezed” into fractional ppem size metrics!
In case of the 8 pt Calibri above, rendered at 96 DPI, this means that the font has been “hinted” at 11 ppem but now is “squeezed” into metrics or advance widths meant for 10 2/3 ppem (cf table
towards the end of ).
At this combination of point size and device resolution, “1/3 ppem” can make a difference far beyond the 1/12 of a pixel we were hoping for in order to solve all remaining “spacing issues.” To see
this, the following illustration repeats one of the above lc ‘m’ along with the respective advance widths.
Integer ppem sizes “squeezed into fractional advance widths: the green lines represent the advance width as rendered, while the blue lines represent the advance widths as expected (Calibri, 8 pt at
96 DPI).
⇒ Hover your mouse over the illustration to see the same character rendered with fractional ppem sizes.
Be sure to hover your mouse over the above illustration. The difference between the above advance widths is about 1/3 of a pixel! This is a lot more than the 1/12 of a pixel we were hoping for. In
fact, it represents about 40% of the right side-bearing space as designed! What’s more, these 40% are simply “chopped off” of the right side-bearing without “telling” the author of the constraints
(or “hints”), which does not sound entirely unlike the accidents we have discussed with “natural widths” (cf ).
Once more, there is no easy “fix” to overcome the shortcomings of fractional advance widths rendered in DirectWrite. It can be done, as I have illustrated above, but it requires some serious efforts
in terms of TrueType coding. The main reason for these efforts is the inevitable use of fractional ppem sizes (cf ) which in turn eliminates any hopes at using “deltas” for “fixing” unwanted pixel
patterns—or any other “situation” that a professional “hinter” may not be able to address in a ppem size independent way!
Fractional advance widths and fractional pixel positioning represent a unique opportunity to get the inter-character spacing right, but this comes at a price: “Blurring” can no longer be minimized by
carefully positioning individual stems. Just have a look at any of the lc ‘m’ above, particularly adjacent ones, and observe how they are rendered with different levels of “blurring.” At the very
least, end-users willing to pay that price should get inter-character spacing without “chopping off” some 40% of the right side-bearing!
In the preceding sub-sections we have looked at issues with inter-character spacing due to accumulated rounding errors and forcing (rigid) characters into prescribed advance widths. To make it easy
to see these issues, I deliberately chose fairly conspicuous examples. Depending on the actual font, individual character, and combination of point size and resolution (DPI), these issues may be less
prominent, if not all but absent.
Nevertheless, a few prominent accidents can impact the gestalt of the entire page or paragraph, as illustrated below.
Advance widths compared: bi-level (aka “black-and-white”, left) vs ClearType using “compatible” advance widths (right). Line breaks and number of lines match, but the page color in ClearType’s
“compatible” widths is uneven (Tahoma, 11 pt at 96 DPI).
⇒ Hover your mouse over either of the above illustrations to see the character-by-character match.
Be sure to hover your mouse over the above illustration. Notice how “compatible” advance widths indeed produces a text layout that perfectly matches that of bi-level (aka “black-and-white”). For the
purpose of dialog boxes and many websites at the time, it simply had to—without compromise [sic]—or else ClearType would not have been released.
Amongst the many issues with ClearType’s “compatible” widths, notice the uneven page color produced by clusters of very narrow lc ‘i’ and ‘l’ characters (cf also end of ). Owing to their distinct fur
markings I called this the “Dalmation Dog Effect.” If you squint your eyes a little bit, you may notice mildly darker spots even in bi-level rendering.
At the time I came up with “compatible” widths as the “lesser evil,” it was our understanding that this will be used only “temporarily”—until the respective applications become smart enough to
account for advance width variations due to different rendering methods, device resolutions, or “zoom” factors (cf also end of ). Alas, 10 years after the 2001 release of Windows XP, this turned into
a mostly “permatemp” solution.
The emerging alternative (Internet Explorer 9, Firefox 5) is DirectWrite’s implementation of fractional advance widths (cf ). The illustration below shows how this can stack up against the other
alternative, “natural” widths (cf ).
Advance widths compared: “natural” widths (GDI, left) vs fractional widths (DirectWrite, right). While the page color in either rendition looks fairly even, line breaks and number of lines are
different, and the “sharpness” of strokes appears a bit irregular in DirectWrite, as if there were random smudges on the display (Tahoma, 9 pt at 96 DPI).
⇒ Hover your mouse over either of the above illustrations to see the difference between the two advance widths.
Notice how “natural” widths make Tahoma appear “sharper” and with a higher degree of stroke rendering contrast (cf ), while fractional widths make it look as if there were some thin veil of dust on
the display. To me, this looks a bit as if I needed to clean my eye-glasses. Of course, cleaning displays or eye-glasses won’t help to explain the above difference; it merely shows the inherent
disparity between fractional and integer pixel positioning (cf ).
To permit making your own comparisons, the illustration below allows you to choose a combination of advance widths and a point size for rendition at 96 DPI.
Advance widths compared: use the above buttons to arrange for your own custom comparison between 2 rendering modes at a particular point size (Tahoma RTM, rendered at 96 DPI in GDI for bi-level,
compatible, and natural advance widths, and in DirectWrite for fractional advance widths).
Be sure to use the above buttons to customize your comparisons. Some of these combinations can make one and the same Tahoma look like a different font!
Of particular interest may be comparing compatible widths with fractional widths since this illustrates the move of Internet Explorer 9 and Firefox 5 to using DirectWrite (fractional widths). By
contrast, Chrome 10 continues to use compatible widths (GDI) and coincidentally, as far as I can tell, Safari 5 uses integer widths but allows fractional ppem sizes (cf column “fractional ppem sizes,
integer widths” in the table at the beginning of ).
Notice that between GDI and DirectWrite, the identifiers used for different advance widths may be a bit confusing, as several of them contain the designation NATURAL. Following is a table for
comparison purposes.
NONANTIALIASED_ DWRITE_RENDERING_MODE_
QUALITY DEFAULT
CLEARTYPE_ DWRITE_RENDERING_MODE_
QUALITY CLEARTYPE_GDI_CLASSIC
QUALITY CLEARTYPE_GDI_NATURAL
n/a DWRITE_RENDERING_MODE_
Advance widths compared: designations of the advance widths as used in the programming platforms GDI (left) and DirectWrite (right).
If you are in the business of creating software that involves rendering text, and which software doesn’t, you don’t have to choose “compatible” advance widths. Instead, you can choose whichever set
of advance widths best fits your layout application. Moreover, if your application does not impose a hard constraint on scalability, such as a “zoomable” print preview, you could choose to leave this
choice to the end-users (cf also and ).
In the previous two sections we have discussed various obstacles with “shoehorning” existing fonts into ClearType duty, and in turn, into existing applications. Working around these obstacles
entailed several “jury rigs” in the rasterizer to defuse the respective problems. Understanding that these problems aren’t really solved, merely patched up with big “band-aids,” future fonts should
strive for more adaptive constraints or “hints.”
Alas, even if you have the luxury to start constraining a font from scratch, duly turning off all “jury rigs” as far as possible, there are a few more obstacles. In this section, I will present some
of the obstacles I encountered trying to do things “properly.” Notice that there are workarounds for all of these obstacles, but they require extremely cumbersome and tedious TrueType code, and
plenty of it!
Rounding to sample boundaries, as opposed to pixel boundaries, is a fundamental prerequisite to implementing fractional stroke weights and stroke positions (cf and , also cf and ). Unfortunately,
however flexible SROUND[] portrays to be, it limits the “rounding granularity” to 0.5 period, where period is the spacing of the pixel grid. Anything “finer grained” is not available.
To implement constraints compatible with the various anti-aliasing methods introduced in Chapter 2, what is needed is the possibility to select a period of the form 1/n, where n is the oversampling
rate in x- or y-direction, as appropriate. This would allow to round to the nearest 1/4, 1/5, or 1/6 of a pixel—or whatever oversampling rate the future may require.
Mathematically, the only difference between italic and upright characters is an “italic” y-axis. Accordingly, it stands to reason that setting the projection vector p perpendicular to the “italic”
y-axis is all that is needed to constrain italic characters. Readers familiar with VTT may recognize the concepts of the italic angle and the “adjusted” italic angle, encoded by a single or double
forward slash, respectively.
Inconveniently, this simple mathematical model seems to break down in the presence of asymmetric overscaling, or any non-square aspect ratio for that matter, as illustrated below.
Interpolating control points with the projection vector p set perpendicular to the “italic” y-axis (Arial lc ‘l’)
For the purpose of illustrating the problem, I didn’t use any CVTs, and I deliberately left every control point “unrounded” in x-direction, just to make sure I won’t mask the problem. Here is the
GlyphStrokeAngle(23,22) CALL[], 23, 22, 88
/* Y direction */ MDAP[R], 0
YAnchor(0) MDAP[R], 1
YAnchor(1) IUP[Y]
YSmooth() SRP1[], 4
SRP2[], 5
/* X direction */ CALL[], 9
XInterpolate/(4,0,1,5) IP[], 0
XNoRound(2,3) IP[], 1
XDist/(0,3) SRP0[], 0
XDist/(1,2) MDRP[m<rBl], 3
SRP0[], 1
/* all points touched */ MDRP[m<rBl], 2
VTT Talk and corresponding TrueType source code used to illustrate the problem with interpolating control points when the projection vector p is set perpendicular to the “italic” y-axis
Interpolating control points with the projection vector p set perpendicular to the “italic” y-axis (Arial lc ‘l’, 11 pt, 72 DPI in y direction, combined with the following resolutions in x-direction:
72, 96, 120, 144, 216, 288, and 432 DPI)
⇒ Be sure to push any of the above buttons to test the various non-square aspect ratios and asymmetric overscaling factors
The more lopsided the aspect ratio or the asymmetric overscaling, the more erroneous the angle of the italic stroke gets. Turns out that this “misbehavior” is specific to the IP[] instruction
(“interpolate point”). Other instructions I’ve tried appear to work fine. This would suggest a case of “double code path jeopardy.”
IUP[X] and IUP[Y] (“interpolate untouched points”) are two “shortcuts” that constrain any previously unconstrained (“untouched”) control points between consecutive pairs of constrained points on the
same outline. Specifically, if the unconstrained points are physically between the constrained points, then the shortcuts are equivalent to an IP[] instruction (“interpolate point”) applied to all
the unconstrained points on the outline between the pairs of constrained points.
For instance, in the following illustration of a lowercase ‘n,’ points 2 and 12 may have been constrained in x-direction while points 3 through 11 are between 2 and 12, both logically (along the same
outline) and physically (all x-coordinates are within the range of x-coordinates spanned by points 2 and 12). Likewise for points 14 through 21 between points 13 and 22:
Using the instruction IUP[X] on a roman character (Arial lc ‘n’ 12 pt, 96 DPI)
In this case, IUP[X] will “interpolate” points 3 through 11 between 2 and 12, and likewise 14 through 21 between 13 and 22. Notice that in this example, the direction of the constraints—the
projection vector p—is in x-direction. “Physically between” is determined against this projection vector p.
Once we apply the same logic to an italic lowercase ‘n,’ the direction of the projection vector p changes. It is now perpendicular to the “italic” y-axis, like in the preceding example. However, the
IUP[] instruction is “hard-wired” to use either the x-direction or the y-direction. If IUP[X] is used, instead of a corresponding “italic variant,” the erroneous projection vector p can be noticed:
Using the instruction IUP[X] on an italic character throws parallel italic edges “out-of-parallel” (Arial Italic lc ‘n’ 12 pt, 96 DPI)
⇒ Hover your mouse over the above illustration to see the results of a hypothetic IUP[I] instruction (‘I’ for “italic”). Notice how this puts all italic edges back to their parallel design
For bi-level rendering, the absence of an IUP[I] instruction (‘I’ for “italic”) may not be all that critical, but for any kind of anti-aliasing, this may show up as uneven stroke weights.
TrueType implements a way of assembling glyphs out of other glyphs or parts of glyphs. TrueType calls this concept composite glyphs. In theory, this harbors the potential for asserting consistency by
concept, rather than by industriousness.
For instance, any character with a diacritical mark can be represented as some kind of a blue print that specifies how the respective mark combines with the associated base character. This should
assert that e.g. a lc ‘ü’ looks exactly like a lc ‘u’ with an “umlaut” (“dieresis”) on top, or that the “umlaut” on the ‘ü’ looks exactly like the “umlaut” on the ‘ä’ or on any other character it may
In practice, the implementation of this “feature” does not come across as thoroughly thought through, though. The blue print essentially superimposes two or more glyphs, as illustrated below:
Composite glyphs in TrueType: components are essentially superimposed on top of each other (basic blue print of Arial lc ‘ü’)
To use the same “umlaut” on more than one base character, TrueType allows for an individual translation vector (offset in x- and y-direction) or transformation matrix for each of the components.
Conceptually, this allows to design characters out of predesigned components.
Translation Vector: Translation Vector:
x-direction: 220 fUnits x-direction: 393 fUnits
y-direction: 0 fUnits y-direction: 286 fUnits
Composite glyphs in TrueType: individual translation vectors or transformation matrices can be applied to each component to re-use a component in different glyphs (actual blue print of Arial lc ‘ü’
and UC ‘Ü’)
⇒ Hover your mouse over the above illustrations to see the respective superpositions without offsets
But it does not necessarily render the entire assembly as intended.
Composite glyphs in TrueType: the assembled blue print does not necessarily render as intended (Arial lc ‘ü,’ 9 pt, 96 DPI [top, enlarged] and 8 pt through 36 pt [bottom, actual bitmaps])
⇒ Hover your mouse over the above illustrations to see more appropriate renditions of the same character
As can be seen in the above illustration, the problem of “ill-positioned umlauts” runs all the way from 8 pt up to 36 pt—and beyond on either side! Some “umlauts” appear pushed to one side, some
appear pushed too far up, and some appear too close together.
The reason behind these “ill-positioned” renditions is the following: Both components (the base character ‘u’ and the “umlaut” ‘¨’) were constrained (“hinted”) individually within their own contexts.
In the process, both the ‘u’ and the ‘¨’ are independently positioned relative to e.g. their respective left- and right side-bearing points. As a result, their individual positions may have shifted
left or right, and—in case of the ‘¨’ that was not designed to “sit” on the baseline like the ‘u’—up or down.
Moreover, to be used in the blue print, the above offsets in x- and y-direction are scaled and rounded to the targeted point size and device resolution. This keeps all the involved components
“on-the-grid.” But, although it erstwhile permitted to correctly design characters out of predesigned components, now the translation vector is usually “bogus.”
Think of it this way: To correctly inset e.g. the left “dot” of the “umlaut” relative to the left stroke of the ‘u,’ there are 3 design variables involved—the left side-bearing of the ‘u,’ the “left
side-bearing” of the “umlaut,” and the offset in x-direction. In terms of font design units, these 3 variables “add up,” but in terms of (rounded!) pixels they don’t!
We have discussed this before. Remember where we were looking at strokes being rendered with one pixel count here and another one there? At that time we had a left edge plus a stroke width define a
right edge of a stem, in order for these 2 variables to “add up.” Now we simply have 3 variables, and we use add or subtract equivalently (with the potential caveat about negative numbers mentioned
in ).
But now for the real problem: The concept of “original” outlines, whatever limitations they may have (cf ), seems unusable in the blue prints of composite characters. TrueType instructions such as
MDRP[], MIRP[], or IP[] will not perform as expected when interspersed with the rest of the blue print code. At the same time the offsets of the blue print are available to the TrueType programmer in
design units only, not in scaled device units like the “original” outlines.
This makes for a really cumbersome and tedious workaround. As you can see by hovering your mouse over the above illustration, it is possible to position the “umlaut” at its natural distance above the
base character, to center it, and to balance the distance between the individual “dots” with their inset relative to the black-body width of the base character.
The implemented strategy does not require any deltas (cf ), but a little bit of “thinking outside the box,” and—like any of the problems illustrated in this section—plenty of code for TrueType’s font
In this chapter we have looked at some of the imperfections of the TrueType rasterizer that ships with Microsoft Windows. This involved delicate issues of computer geometry along with obvious errors
and—in hindsight—omissions. By the same standards, most of the fonts aren’t perfect, either. They contain obvious errors and—again in hindsight—very obvious omissions.
What to do? Fix and upgrade the rasterizer, and then fix and upgrade all (!) the fonts, painstakingly agonizing over pixel-by-pixel backwards compatibility in bi-level rendering? Not a chance. Merely
fixing the rasterizer could have “broken” many fonts (cf ). Instead, I tried to patch the biggest holes by shifting some of the burden from the fonts to the rasterizer.
Understanding that this deprives font makers of part of the control over their artwork, we have discussed the “switches” to turn off (most of) these patches for future fonts. In turn, this puts the
burden back on the font makers. They will have to develop a very clear concept of the purpose of their constraints (or “hints”) while being prepared to look for alternatives in case their preferred
implementation hits a limitation of the rasterizer (cf ).
In the next chapter we will have a look at a number of challenges beyond fixing the rasterizer and the fonts, and after discontinuing “compatible” advance widths. These challenges occur even with
“perfect” fonts interpreted by a “perfect” rasterizer and laid out by “perfect” applications.
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This package modifies the implementation of the FastICA algorithm from MultivariateStats.jl to make it more than 5x faster. This comes at the expense of taking an additional dependency, on
LoopVectorization.jl and some additional compilation time.
This package also modifies the interface to the algorithm slightly, use it like this:
using IndependentComponentAnalysis
X = randn(4,100)
k = 2 # Number of components to extract
ica = fit(ICA, X, k, alg = FastICA();
fun = Tanh(),
do_whiten = true,
maxiter = 100,
tol = 1e-6,
mean = nothing,
winit = nothing
Xt = transform(ica, X)
• The options for fun are Tanh(a::Real) and Gaus()
• If you have also loaded MultivariateStats, a conflict will arise, cauing you to have to qualify fun like this: IndependentComponentAnalysis.Tanh etc. If this PR is merged, this package will work
better together with MultivariateStats.
DUET (Degenerate Unmixing Estimation Technique) is an algorithm for blind source separation. It works on stereo mixtures and can separate any number of sources as long as they do not overlap in the
time-frequency domain. The interface looks like this:
est, H = duet( x1, x2, n_sources, n = 1024;
p = 1, # amplitude power used to weight histogram
q = 0, # delay power used to weight histogram
amax = 0.7,
dmax = 3.6,
abins = 35,
dbins = 50,
kernel_size = 1, # Controls the smoothing of the histogram.
bigdelay = false,
kernel_sizeδ = 1,
window = hanning,
kwargs..., # These are sent to the stft function
The returned values are est::Matrix(time, n_sources) and H::DUET. H can be plotted plot(H), which shows the delay/amplitude histogram and the identified peaks. This can be used to tweak the input
• bigdelay indicates whether or not the two microphones are far apart. If true, the delay δ is estimated using the differential method (see the paper sec 8.4.1) and the delay map is smoothed using
t = 0:0.1:2000
x1 = sin.(t)
x2 = @. 1.3 * sign(sin(2t + 0.2) ) + 0.001 * randn()
W = [0.3 0.7; 0.6 0.4] # Mixing matrix
R = W * [x1 x2]'
r1, r2 = R[1, :], R[2, :]
est, H = duet( r1, r2, 2, 1024,
dmax = 3.5,
amax = 2.8,
kernel_size = 1,
plotinds = 10000:10200
plot(plotinds, est[plotinds,:], lab = "Estimated components", c = :blue)
plot!(plotinds, [x1 x2][plotinds,:], lab = "True signal", c = :black)
plot!(plotinds, [r1 r2][plotinds,:], lab = "Received signal", c = :orange) |> display
plot(H) |> display
mix = mixture(signals, amps, [delays::Vector{Int}])
mixes = mixture(signals, amps::Vector{Vector}, [delays::Vector{Vector{Int}}])
Mixes together signals using amplitudes amps and delays (delays is specified in samples). If amps and the optional delays are vectors of vectors, then a vector of mixtures is returned. A vector of
vectors is converted to a matrix using M = reduce(hcat, mixes). | {"url":"https://www.juliapackages.com/p/independentcomponentanalysis","timestamp":"2024-11-07T20:05:17Z","content_type":"text/html","content_length":"128144","record_id":"<urn:uuid:669c88b0-788e-4cf7-a96b-1f40b9f7f3a0>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00171.warc.gz"} |
Convert to Column Formula Issue
I use the following formula =((Applications1 / [Employee Count]1) * 12) * 10 to calculate a figure from two columns. But when I right click to Convert to Column Formula" it errors and says The column
formula syntax isn't quite right? What am I missing?
Best Answer
• Hello Chris
Column Formulas do not allow references to specific row numbers- that is the syntax the error refers to . Convert your column references (ex. Applications1) to @row
=((Applications@row / [Employee Count]@row) * 12) * 10
You can find more information here
• Hello Chris
Column Formulas do not allow references to specific row numbers- that is the syntax the error refers to . Convert your column references (ex. Applications1) to @row
=((Applications@row / [Employee Count]@row) * 12) * 10
You can find more information here
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The amount of material for the mesh
It's required to produce a mesh with predetermined length and width, roughly square meshes from tape with predetermined width. It is necessary to calculate the total length of the tape.
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
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in this content.
It's required to produce a mesh with a predetermined length, width and roughly square meshes from a tape with a predetermined width. It is necessary to calculate the total length of the tape.
For calculation, it's necessary to know the approximate size of a single mesh. The final size will be calculated so as to have a whole number of meshes by the width and length of the tape. The mesh
size is defined and calculated from the center of one tape to the center of the other, i.e. mesh gap is smaller in length and width by the tape width.
The length of the mesh along the outer perimeter
The width of the mesh along the outer perimeter
Expected distance between the centers of two neighboring tapes
Digits after the decimal point: 2
Number of longitudinal tapes
Similar calculators
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zone history command
zone history <name s > keyword
Add a history of a zone value. See the history command. The history can be assigned a name for later reference with the optional name keyword; if not, the history will be given a default name
based on its internally assigned id number.
Zone histories can be taken of any scalar field variable, at any location encompassed by the zone. Once specified, the history location will travel with the zone in large strain mode.
The location of the history may be specified by: location in space; selecting the centroid of a zone with a specific ID number; or the location of the gridpoint with a specific ID number.
If the value is gridpoint-based, then a value is calculated using a weighting function inside the zone.
If the value is zone-based and the location does not correspond to the zone centroid, the value will be determined by a curve fitting algorithm. Which algorithm used can be specified in the
following [zonehistoryblock]. By default, a value is considered constant within a zone.
Note that to take a history of a specific gridpoint, take a zone history using the gridpointid as the location, or specify the position of a gridpoint.
This will repeat in at least one other place
zone history Keyword Block
The following modifiers are available to specify additional information for the zone history keywords. It is required that the location of the history must be specified using one of the keywords
position, zoneid, or gridpointid. They are applicable to all of the following keywords of the zone history command. acceleration, acceleration-x, acceleration-y, acceleration-z, condition, density,
density-fluid, displacement, displacement-x, displacement-y, displacement-z, extra, factor-of-safety, fluid-biot-modulus, fluid-bulk-modulus, multiplier, pore-pressure, property, property-fluid,
property-thermal, ratio-local, ratio-target, saturation, strain-increment, strain-rate, strength-stress-ratio, stress, stress-effective, temperature, timestep-dynamic, unbalanced-force,
unbalanced-force-x, unbalanced-force-y, unbalanced-force-z, velocity, velocity-x, velocity-y and velocity-z.
component keyword
This keyword selects which scalar to retrieve from a vector type value, such as velocity or displacement. If the value type is not a vector, this setting is ignored. The available options
\(x\)-component of the vector.
\(y\)-component of the vector.
z (3D Only)
\(z\)-component of the vector.
Vector magnitude.
index i
For keywords that require it (most notably extra), this specifies the index that should be used. The default value will be 1.
gravity b
If on, then gravitational forces are included in the returned applied force. The default is off.
gridpointid i
Specify the location of the history by the location of the gridpoint with ID number i.
label s
Assign a label to the history, used in prints and plots.
log b
If on, the returned number is the base 10 log of the absolute value of the original value. The default is off.
Specifies the method of calculation used to bring zone-based quantities to arbitrary locations in space (including at gridpoint positions).
Assume values are constant inside zones. This is the default option. Values at gridpoints or at zone boundaries will pick one adjacent zone to use.
Use a volume-weighted average of all the zones connected to the gridpoint.
Use a weighted average of up to 32 zones in the immediate vicinity of the point. The weighting is determined by a power of the distance of the zone centroid from the specified point.
power f
Specify the power of the distance used to weight values. The default power is 3.0.
radius-ratio f
Specify the ratio of the maximum distance from the point of the zones selected, to the ratio used to calculate weighted values. Zones farther from this value are discounted. The
default value is 0.75.
polynomial <tolerance f > (3D Only)
Use a fit to a polynomial function based on the zones in the vicinity of the point. Degenerate degrees of freedom are detected and removed. The optional tolerance flag can be used to
change the value used to detect a degeneracy. The default value is 1e-7.
name s
Certain keywords (most notably property) require a name value to be fully specified. This will determine the specific property name retrieved.
null <keyword>
Specify the criteria used to ignore zone values. By default, mechanical null zones are ignored.
fluid b
Zones with a null fluid constitutive model are also ignored if this is set to on.
mechanical b
Zones with a null mechanical constitutive model are ignored if this is set to on (this is the default).
thermal b
Zones with a null thermal constitutive model are ignored if this is set to on.
position v
Specify the location of the history at v. This location must be inside a zone.
quantity keyword
Select which scalar to retrieve from a symmetric tensor type value, such as stress or strain. If the value type is not a tensor, this setting is ignored. The available options are:
Intermediate principal stress.
The maximum (most positive) value of principal stress. Note that in FLAC compressive stresses are negative.
The mean pressure value, defined as the trace of the tensor divided by 3. For stresses this is most often referred to as the pressure.
The minimum (most negative) principal stress. Note that in FLAC compressive stresses are negative.
The norm of the strain rate; see the equation in Zone Field Data Names
Octahedral stress; see the equation in Zone Field Data Names.
Maximum shear stress.
Distance of tensor value to the origin in principal space.
Volumetric change, or trace.
Von Mises measure; see the equation in Zone Field Data Names.
xx-component of the tensor.
xy-component of the tensor.
xz (3D Only)
xz-component of the tensor.
yy-component of the tensor.
yz (3D Only)
yz-component of the tensor.
zz-component of the tensor.
source keyword
Certain keywords (like extra and pore-pressure) can be retrieved either from the zone structure or from the gridpoint structure. The default value is gridpoint.
Values are retrieved directly from the gridpoint.
Values are retrieved from the zone.
stress b
A few of the tensor type quantity options have different effects depending if the value is considered a stress. When coming from a value like extra, this cannot be determined automatically.
Setting stress on identifies the incoming tensor value as a stress quantity.
type keyword
In certain cases the type (scalar, vector, or tensor) of the value cannot necessarily be determined ahead of time. Extra variables, for example, can hold values of all three types. This
keyword allows specifying which type it is assumed to be. If the original value type does not match, 0.0 is returned.
A scalar float type. This is the default.
A vector type.
A tensor type.
velocity-limit f
Specify the velocity limit used to determine if an object is stable or not for the factor-of-safety value. The default value is 1e-4.
zoneid i
Specify the location of the history is at the centroid of zone with ID i.
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Lesson 4
Making the Moves
4.1: Reflection Quick Image (5 minutes)
In this warm-up, students are asked to sketch a reflection of a given triangle and explain the strategies they used. The goal is to prompt students to notice and articulate that they can use the
location of a single point and the fact that the image is a reflection of the triangle to sketch the image. To encourage students to use what they know about reflections and not count every grid
line, this image is flashed for a few seconds and then hidden. It is flashed once more for students to check their thinking.
Before beginning, make sure students have their books or devices open to the correct page. Tell students you will show them an image of a reflection of triangle \(ABD\) for 3 seconds. Their job is to
draw the image and explain any strategies they used.
For classes using the digital version of the materials, display the applet and demonstrate the use of the various tools. Give students a minute or two to test them out. Ask them to reset the applet
before starting the activity.
Display the completed image for 3 seconds and then hide it. Do this twice. Give students 1 minute of quiet work time after each flash of the image. Encourage students to think about any shortcuts
they used to draw the reflected image.
Student Facing
Here is an incomplete image. Your teacher will display the completed image twice, for a few seconds each time. Your job is to complete the image on your copy.
Before beginning, make sure students have their books or devices open to the correct page. Tell students you will show them an image of a reflection of triangle \(ABD\) for 3 seconds. Their job is to
draw the image and explain any strategies they used.
For classes using the digital version of the materials, display the applet and demonstrate the use of the various tools. Give students a minute or two to test them out. Ask them to reset the applet
before starting the activity.
Display the completed image for 3 seconds and then hide it. Do this twice. Give students 1 minute of quiet work time after each flash of the image. Encourage students to think about any shortcuts
they used to draw the reflected image.
Student Facing
Here is an incomplete image. Your teacher will display the completed image twice, for a few seconds each time. Your job is to complete the image on your copy.
Anticipated Misconceptions
Students may struggle drawing the image under transformation from the quick flashes of the image because they are trying to count the number of spaces each vertex moves. Encourage these students to
use the line in the image to help them reflect the image.
Activity Synthesis
Select a few students to share strategies they used in sketching their figure. Consider asking some of the following questions:
• "What was important in creating your sketch (what did you need)?"
• "What did you look for in the first flash? The second?"
• "What stayed the same and what is different in the shape and its image?"
• "How did you decide where to place the vertices of the image?"
• "How did you decide how long to make the sides?"
4.2: Make That Move (15 minutes)
The purpose of this activity is for students to give precise descriptions of translations, rotations, and reflections. By the end of the previous lesson, students have identified and sketched these
transformations from written directions, however they have not used this more precise language themselves to give descriptions of the three motions. The images in this activity are given on grids to
allow and to encourage students to describe the transformation in terms of specific points, lines or angles. Describing the moves precisely and clearly will require students to engage in MP6.
There are four different transformation cards students use in this activity: 1A, 1B, 2A, and 2B. Each card has the original image and the image under a transformation. Students are arranged in groups
of 2 and each one gets a different transformation card: some pairs are given cards 1A and 1B while other pairs get 2A and 2B. Each A card is a translation, while the B cards show either a rotation or
As students are describing their transformations and sketching their images under transformation, monitor for students using precise descriptions to their partner in terms of specific points, lines,
or angles.
Introduce the word transformation: This is a word for a move (like the reflection in the warm-up) or for a sequence of moves. It's a mathematical word which will be used in place of "move," which has
been used up to this point.
Arrange students in groups of 2. Display the original image that each student also has in front of them. Give each student one of the four transformation cards and tracing paper—make sure students
know not to show their card to their partner! Tell students they will sketch a transformation based on directions given to them by a partner.
Partners have Cards 1A and 1B or Cards 2A and 2B. Tell students in the first round, those with the A cards give a precise description of the transformation displayed on their card to their partner.
Their partners may use tracing paper to produce the image under transformation on the grid with the original image. In digital classrooms, students have access to applets they can use to transform
the figure. When the sketch is complete, the student describing the transformation reveals their card, and together, students decide if the sketch is correct. In the second round, the roles are
reversed. The students with the B cards describe their transformation while their partner sketches.
The student describing the transformation is allowed to repeat, revise, or add any important information as their partner sketches, however, they are not allowed to tell them to fix anything until
they are finished. The student sketching should not speak, just sketch. (This is to encourage the describer to use mathematical language.) Remind students to use the geometric language for describing
reflections, rotations, and translations that was used in the previous lesson.
Representation: Develop Language and Symbols. Display or provide charts with symbols and meanings. For example, display the terms translation, rotation, and reflection and their respective meanings
“slide,” “turn,” and “mirror,” accompanied by a visual example depicting the transformation. Remind students that they can refer to the display when giving directions to their partner.
Supports accessibility for: Conceptual processing; Memory
Listening, Speaking: Math Language Routine 2 Collect and Display. This is the first time Math Language Routine 2 is suggested as a support in this course. In this routine, the teacher circulates and
listens to student talk while jotting down words, phrases, drawings, or writing students use. The language collected is displayed visually for the whole class to use throughout the lesson and unit.
Generally, the display contains different examples of students using features of the disciplinary language functions, such as interpreting, justifying, or comparing. The purpose of this routine is to
capture a variety of students’ words and phrases in a display that students can refer to, build on, or make connections with during future discussions, and to increase students’ awareness of language
used in mathematics conversations.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness
How It Happens:
1. As students describe the transformation of triangle \(ABC\) to their partner, listen for and collect vocabulary and phrases students use to describe the moves. Focus on capturing students using
geometric language for describing reflections, rotations, and translations.
If the speaker is stuck, consider asking these questions: “How did point \(A\) transform to \(A’\)?”, “Choose one of the points, lines, or angles and describe how it changed.”, and “Overall, does
it look like the new triangle is a translation, rotation, or reflection of the original?”
If using the applet, check for precision and labels as students place the new image under the transformation.
2. Write students’ words on a visual display. Divide the display into 3 sections. Group language about Cards 1A and 2A on the left side of the display, language about Card 1B in the middle, and
language about Card 2B on the right side.
Record all language (whether precise, ambiguous, correct, or incorrect) in the appropriate column. as described by the students.
3. Arrange students in groups of 2, and invite partners to discuss which words or phrases stand out to them. Prompt students by asking, “Are there any words or phrases that stand out to you or don’t
belong in a specific column? Why?” Again, circulate around the room, collecting any additional words, phrases, and sketches onto the display.
Students should notice that the left side consists of language describing translations, the middle consists of language describing reflections, and the right side consists of language describing
4. Select 3–4 groups to share their ideas with the class. Invite students to demonstrate their reasoning with the applet or tracing paper and be sure to modify the display accordingly.
Use this discussion to clarify, revise, and improve how ideas are communicated and represented. If students are still using vague words (e.g., move, flip, mirror image, etc.), reinforce the
precise geometric terms (e.g., transformation, translation, rotation, reflection, etc.). Ask students, “Is there another way we can say this?” or “Can someone help clarify this language?”
5. Close this conversation by posting the display in the front of the classroom for students to reference for the remainder of the lesson, and be sure to update the display throughout the remainder
of the lesson.
Student Facing
Your partner will describe the image of this triangle after a certain transformation. Sketch it here.
Introduce the word transformation: This is a word for a move (like the reflection in the warm-up) or for a sequence of moves. It's a mathematical word which will be used in place of "move," which has
been used up to this point.
Arrange students in groups of 2. Display the original image that each student also has in front of them. Give each student one of the four transformation cards and tracing paper—make sure students
know not to show their card to their partner! Tell students they will sketch a transformation based on directions given to them by a partner.
Partners have Cards 1A and 1B or Cards 2A and 2B. Tell students in the first round, those with the A cards give a precise description of the transformation displayed on their card to their partner.
Their partners may use tracing paper to produce the image under transformation on the grid with the original image. In digital classrooms, students have access to applets they can use to transform
the figure. When the sketch is complete, the student describing the transformation reveals their card, and together, students decide if the sketch is correct. In the second round, the roles are
reversed. The students with the B cards describe their transformation while their partner sketches.
The student describing the transformation is allowed to repeat, revise, or add any important information as their partner sketches, however, they are not allowed to tell them to fix anything until
they are finished. The student sketching should not speak, just sketch. (This is to encourage the describer to use mathematical language.) Remind students to use the geometric language for describing
reflections, rotations, and translations that was used in the previous lesson.
Representation: Develop Language and Symbols. Display or provide charts with symbols and meanings. For example, display the terms translation, rotation, and reflection and their respective meanings
“slide,” “turn,” and “mirror,” accompanied by a visual example depicting the transformation. Remind students that they can refer to the display when giving directions to their partner.
Supports accessibility for: Conceptual processing; Memory
Listening, Speaking: Math Language Routine 2 Collect and Display. This is the first time Math Language Routine 2 is suggested as a support in this course. In this routine, the teacher circulates and
listens to student talk while jotting down words, phrases, drawings, or writing students use. The language collected is displayed visually for the whole class to use throughout the lesson and unit.
Generally, the display contains different examples of students using features of the disciplinary language functions, such as interpreting, justifying, or comparing. The purpose of this routine is to
capture a variety of students’ words and phrases in a display that students can refer to, build on, or make connections with during future discussions, and to increase students’ awareness of language
used in mathematics conversations.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness
How It Happens:
1. As students describe the transformation of triangle \(ABC\) to their partner, listen for and collect vocabulary and phrases students use to describe the moves. Focus on capturing students using
geometric language for describing reflections, rotations, and translations.
If the speaker is stuck, consider asking these questions: “How did point \(A\) transform to \(A’\)?”, “Choose one of the points, lines, or angles and describe how it changed.”, and “Overall, does
it look like the new triangle is a translation, rotation, or reflection of the original?”
If using the applet, check for precision and labels as students place the new image under the transformation.
2. Write students’ words on a visual display. Divide the display into 3 sections. Group language about Cards 1A and 2A on the left side of the display, language about Card 1B in the middle, and
language about Card 2B on the right side.
Record all language (whether precise, ambiguous, correct, or incorrect) in the appropriate column. as described by the students.
3. Arrange students in groups of 2, and invite partners to discuss which words or phrases stand out to them. Prompt students by asking, “Are there any words or phrases that stand out to you or don’t
belong in a specific column? Why?” Again, circulate around the room, collecting any additional words, phrases, and sketches onto the display.
Students should notice that the left side consists of language describing translations, the middle consists of language describing reflections, and the right side consists of language describing
4. Select 3–4 groups to share their ideas with the class. Invite students to demonstrate their reasoning with the applet or tracing paper and be sure to modify the display accordingly.
Use this discussion to clarify, revise, and improve how ideas are communicated and represented. If students are still using vague words (e.g., move, flip, mirror image, etc.), reinforce the
precise geometric terms (e.g., transformation, translation, rotation, reflection, etc.). Ask students, “Is there another way we can say this?” or “Can someone help clarify this language?”
5. Close this conversation by posting the display in the front of the classroom for students to reference for the remainder of the lesson, and be sure to update the display throughout the remainder
of the lesson.
Student Facing
Your partner will describe the image of this triangle after a certain transformation. Sketch it here.
Anticipated Misconceptions
Students may get stuck thinking they need to use the precise terms for the transformation in their description. Encourage these students to describe it in a way that makes sense to them and to look
for things they know about the specific points, lines, or angles on their card to help them.
Activity Synthesis
Display the following questions for all to see and give groups 2 minutes to discuss:
• What pieces of your partner’s description were helpful when you were sketching?
• What pieces did you find difficult to explain to your partner? Point to specific examples on your cards.
• When you were sketching, what questions would have been helpful to be able to ask the describer?
Ask selected students who were observed using precise descriptions and sketching based on those descriptions to explain why they used the information they did and how it was helpful in sketching.
Focus on:
• The direction and distance of a translation.
• The center and the measure of a rotation.
• The line of a reflection.
If there is time, ask students who were both using and not using tracing paper to explain their process.
Reinforce the term transformation as a term that encompasses translations, rotations, and reflections. Tell them that there are other types of transformations, but for now, we will focus on these
three types.
4.3: A to B to C (15 minutes)
Students have seen images showing a sequence of transformations in the first lesson of this unit, however they have not heard the term sequence of transformations. They have also not been asked to
describe the moves in the sequence using precise language. The launch of this activity introduces this term and gives students an opportunity to describe the sequence of more than one transformation.
For the second problem, encourage students to find a different sequence of transformations than the one shown in the image. Each time a reflection is mentioned, ask students where the line of
reflection is located and when a rotation is mentioned, ask for the center of the rotation and the number of degrees. Monitor for students who apply different transformations (or apply
transformations in a different order).
Arrange students in groups of 2, and provide access to their geometry toolkits. Display the image for all to see. Ask students if they can imagine a single translation, rotation, or reflection that
would take one bird to another? After a minute, verify that this is not possible.
Ask students to describe how we could use translations, rotations, and reflections to take one bird to another. Collect a few different responses. (One way would be to take the bird on the left,
translate it up, and then reflect it over a vertical line.) Tell students when we do one or more transformations in a row to take one figure to another, it is called a sequence of transformations.
If using the digital activity, you may want to review the transformation tools in the applet. (The instructions are repeated in the activity for students' reference.)
Give students 2 minutes of quiet work time to engage in the task followed by 3 minutes to discuss their responses with a partner and complete any unfinished questions. Follow with a whole-class
Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Prior to the whole-class discussion, invite students to share their work with a partner. Display
sentence frames to support student conversation such as “To take Figure \(A\) to Figure \(B\), I _____ because…”, “I noticed _____ so I...”, “Why did you...?”, or “I agree/disagree because….”
Supports accessibility for: Language; Social-emotional skills
Student Facing
Here are some figures on an isometric grid. Explore the transformation tools in the tool bar. (Directions are below the applet if you need them.)
1. Name a transformation that takes Figure A to Figure B. Name a transformation that takes Figure B to Figure C.
2. What is one sequence of transformations that takes Figure A to Figure C? Explain how you know.
1. Select the Vector tool.
2. Click on the original point and then the new point. You should see a vector.
3. Select the Translate by Vector tool.
4. Click on the figure to translate, and then click on the vector.
1. Select the Rotate around Point tool.
2. Click on the figure to rotate, and then click on the center point.
3. A dialog box will open. Type the angle by which to rotate and select the direction of rotation.
1. Select the Reflect about Line tool.
2. Click on the figure to reflect, and then click on the line of reflection.
Arrange students in groups of 2, and provide access to their geometry toolkits. Display the image for all to see. Ask students if they can imagine a single translation, rotation, or reflection that
would take one bird to another? After a minute, verify that this is not possible.
Ask students to describe how we could use translations, rotations, and reflections to take one bird to another. Collect a few different responses. (One way would be to take the bird on the left,
translate it up, and then reflect it over a vertical line.) Tell students when we do one or more transformations in a row to take one figure to another, it is called a sequence of transformations.
If using the digital activity, you may want to review the transformation tools in the applet. (The instructions are repeated in the activity for students' reference.)
Give students 2 minutes of quiet work time to engage in the task followed by 3 minutes to discuss their responses with a partner and complete any unfinished questions. Follow with a whole-class
Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Prior to the whole-class discussion, invite students to share their work with a partner. Display
sentence frames to support student conversation such as “To take Figure \(A\) to Figure \(B\), I _____ because…”, “I noticed _____ so I...”, “Why did you...?”, or “I agree/disagree because….”
Supports accessibility for: Language; Social-emotional skills
Student Facing
Here are some figures on an isometric grid.
1. Name a transformation that takes Figure \(A\) to Figure \(B\). Name a transformation that takes Figure \(B\) to Figure \(C\).
2. What is one sequence of transformations that takes Figure \(A\) to Figure \(C\)? Explain how you know.
Student Facing
Are you ready for more?
Experiment with some other ways to take Figure \(A\) to Figure \(C\). For example, can you do it with. . .
• No rotations?
• No reflections?
• No translations?
Activity Synthesis
Select students with different correct responses to show their solutions. Be sure to highlight at least one rotation. If no students mention that, demonstrate a way to take A to C that involves a
rotation. Whether or not students use the geogebra applet, it may be helpful to display the applet to facilitate discussion: ggbm.at/jqvTEgsj
• Emphasize that there are many ways to describe the translation that takes figure A to figure B. All one needs is to identify a pair of corresponding points and name them in the correct order (and
to use the word “translate”).
• For students who used a reflection to take B to C, emphasize that reflections are determined by lines and we should name the line when we want to communicate about it.
• After a student or the teacher uses a rotation, emphasize that a rotation is defined by a center point and an angle (with a direction). The center point needs to be named and the angle measure or
an angle with the correct measure needs to be named as well (as does the direction). Reinforce to students that when we do more than one transformation in a row, we call this a sequence of
Speaking: Math Language Routine 7 Compare and Connect. This is the first time Math Language Routine 7 is suggested as a support in this course. In this routine, students are given a problem that can
be approached using multiple strategies or representations and are asked to prepare a visual display of their method. Students then engage in investigating the strategies (by means of a teacher-led
gallery walk, partner exchange, group presentation, etc.), comparing approaches, and identifying correspondences between different representations. A typical discussion prompt is: “What is the same
and what is different?” regarding their own strategy and that of the others. The purpose of this routine is to allow students to make sense of mathematical strategies by identifying, comparing,
contrasting, and connecting other approaches to their own, and to develop students’ awareness of the language used through constructive conversations.
Design Principle(s): Maximize meta-awareness: Support sense-making
How It Happens:
1. Use this routine to compare and contrast different strategies for transforming Figure A to Figure C. Invite students to create a visual display showing how they made sense of the problem and why
their solution makes sense for transforming Figure A to Figure C.
Students should include these features on their display:
□ a sketch of the figures (not necessary if using the applet)
□ a sketch of the figures after each transformation (not necessary if using the applet)
□ a written sequence of transformations with an explanation
2. Before selecting students to show their solutions to the class, first give students an opportunity to do this in a group of 3–4. Ask students to exchange and investigate each other’s work. Allow
1 minute for each display and signal when it is time to switch.
While investigating each other’s work, ask students to consider what is the same and what is different about each approach. Next, give each student the opportunity to add detail to their own
display for 1–2 minutes.
3. As groups are presenting, circulate the room and select 2–3 students to share their sequence of transformations taking Figure A to Figure C. Be sure to select a variety of approaches, including
one that involves a rotation.
Draw students’ attention to the different ways the figures were transformed (e.g., rotations, reflections, and translations) and how the sequence of transformation is expressed in their
explanation. Also, use the bullet points in the Activity Synthesis to emphasize specific features of translations, reflections, and rotations.
4. After the selected students have finished sharing with the whole class, lead a discussion comparing, contrasting, and connecting the different approaches.
Consider using these prompts to amplify student language while comparing and contrasting the different approaches: “Why did the approaches lead to the same outcome?”, “What worked well in __’s
approach? What did not work well?”, and “What would make __’s strategy more complete or easy to understand?”
Consider using these prompts to amplify student language while connecting the different approaches: “What role does a translation play in each approach?” “Is it possible to use the all three
types of transformations?”, and “What transformation do you see present in all the strategies?”
5. Close the discussion by inviting 3 students to revoice the strategies used in the presentations, and then transition back to the Lesson Synthesis and Cool-Down.
Lesson Synthesis
The goal for this lesson is for students to begin to identify the features that determine a translation, rotation, or reflection. Refer to the permanent display produced in a previous lesson as you
discuss. To highlight the features specific to each type of transformation, consider asking the following questions:
• "If you want to describe a translation, what important information do you need to include?" (A translation is determined by two points that specify the distance and direction of the translation.)
• "If you want to describe a rotation, what important information do you need to include?" (A rotation is determined by a center point and an angle with a direction.)
• "If you want to describe a reflection, what important information do you need to include?" (A reflection is determined by a line.)
• "What does the word transformation mean?" (Translations, rotations, and reflections, or any combination of these.)
• "What does sequence of transformations mean?" (More than one applied one after the other.)
4.4: Cool-down - What Does It Take? (5 minutes)
Student Facing
A move, or combination of moves, is called a transformation. When we do one or more moves in a row, we often call that a sequence of transformations. To distinguish the original figure from its
image, points in the image are sometimes labeled with the same letters as the original figure, but with the symbol \(’\) attached, as in \(A’\) (pronounced “A prime”).
• A translation can be described by two points. If a translation moves point \(A\) to point \(A’\), it moves the entire figure the same distance and direction as the distance and direction from \(A
\) to \(A’\). The distance and direction of a translation can be shown by an arrow.
For example, here is a translation of quadrilateral \(ABCD\) that moves \(A\) to \(A’\).
• A rotation can be described by an angle and a center. The direction of the angle can be clockwise or counterclockwise.
For example, hexagon \(ABCDEF\) is rotated \(90^\circ\) counterclockwise using center \(P\).
• A reflection can be described by a line of reflection (the “mirror”). Each point is reflected directly across the line so that it is just as far from the mirror line, but is on the opposite side.
For example, pentagon \(ABCDE\) is reflected across line \(m\). | {"url":"https://im.kendallhunt.com/MS/teachers/3/1/4/index.html","timestamp":"2024-11-06T00:58:01Z","content_type":"text/html","content_length":"174628","record_id":"<urn:uuid:1ea7adaa-3601-48f4-ad26-7c830dd818c5>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00647.warc.gz"} |
Mixed Model Sequencing – Complex Example Sequencing 1
Sequencing products due to different workloads of different products at different workstations is tricky. This is now the sixth post on Mixed Model Sequencing, and we finally start with our
sequence! Wohooo!
Sequence the First Product
The first product to be sequenced is usually easy; it is the biggest rock you have (i.e., the part that has the largest cycle time at any station). In the example we used so far, this would be
product number 2437, since it has the largest cycle time for the mounting of the sunroof of 112 seconds. This is our biggest rock. I have marked this in red in the table below.
The sequence for this part 2437 is easy. We want to produce 5,500 of this type out of 17,732 parts in total. Hence our sequencing interval S for our product k with a quantity Q[k] out of n products
in total is:
${ S_k =\frac{\sum_{m=1}^{n} Q_m }{Q_k} }$
or for our case:
${ S_{2437} =\frac{17 732 }{5500 } = 3.22}$
Hence, every 3.22 items in the sequence should be this product 2437. If we start at position 1 with this type, then the next one should be at 1+3.22 = 4.22. The next one afterward should be at
4.22+3.22 = 7.45, then at 10.67, at 13.90, and so on. Since this is the first product, all slots are still available, and we can simply round to the nearest integer, giving us this product 2437 at
positions 1, 4, 7, 11, 14, etc. in our production sequence. The first 30 slots are visualized below.
Sequence the Second Product
Now comes the ambiguous part: Which product should we sequence next? Should we take product 2434 because it has the second-biggest cycle time, also at the sunroof mounting? Or should we take some of
the products that have no sunroof to offset the excessive sunroof time of 2437? If so, it could be product 2436, which also has a similar quantity to product 2437, which we just sequenced (grayed out
in the table below). Or should we go for something completely different by sequencing 2435, because it has the largest cycle time for the door mounting?
There is no single right answer here. Depending on which aspects you look at, you may pick a different product for sequencing. I go for product 2434 here, since it has the second-largest cycle time
overall due to the 111 seconds for the sunroof mounting, but this is just a hunch. The sequencing interval is 6.41, hence every 6.41 slots I would like to have this product.
${ S_{2434} =\frac{17 732 }{2766 } = 6.41}$
The first slot is already occupied, hence we cannot start this sequence with the first slot. The second slot would not be good either, because then we have two sunroof mountings directly adjacent to
each other in slot 1 (product 2437) and slot 2 (product 2434). This would require even more buffer before the sunroof mounting to decouple two excessively long cycle times in sequence. So we move to
the third slot … and have the same problem again since the fourth slot is another sunroof again.
This is a tricky situation. We seem to have always two sunroofs adjacent to each other in the sequence. I picked the second slot to start the sequence, giving me product 2434 at slots 2, 8, 15, 21,
and so on. If the sequence arrives at a slot that is already occupied, then you simply take the nearest available slot. The result is visualized below, and we have lots of sunroof mountings adjacent
to each other in the sequence. This is a problem!
However, this is not yet the final sequence. I can shuffle both the initial product 2437 (our biggest rock) and the second product 2434 (our second-biggest rock) around. I cannot remove any product,
but I can move them within the sequence. And, lo and behold, we can get a sequence where there is always at least one slot free between sunroof mountings!
While this kind of shuffling is time consuming and hard to automate, it can save you a lot of problems later. The updated sequence is shown below.
Generally speaking, if you have no more than 50% of the products with one certain aspect, you can always keep a slot free between them. In our example, we had a total of 8,266 products with sunroof
(across two product variants), representing 47% of all products. Hence we can make a series of alternating sunroof – no sunroof, sometimes even with two empty slots in between.
However, this can only be guaranteed if you start with an empty sequence. If the sequence has been already filled with other products, our options are limited, and we may not be able to keep a nice
alternating sequence without breaking some other sequence benefit. This is the idea of always putting in the biggest rock firsts, so if you run into conflicting sequences later, you already have the
biggest problems out of the way.
Sequence the Third Product
So, what should be our third-biggest rock? The largest deviation from the average of the not-yet-sorted products are now all models without a sunroof. However, since this means all remaining
available slots will be filled with “no-sunroof” vehicles, there is no priority to assign them. Instead, we would go for the next biggest deviation which we have in the door-mounting process with
product 2435.
Since we make only 121 products of this type, the sequencing interval is 146.55 hence only every 146.55 slots, I would like to have this product.
${ S_{2435} =\frac{17 732 }{121 } = 146.55}$
Now, I could start the sequence at the first free slot, slot number 2 as shown below.
However, looking into the data in more detail, this would give me a four-door vehicle 2435 directly adjacent to another four-door vehicle 2437. Again, I would have two longer cycle times in sequence
for the door-mounting station. There are not many options left, but I could move this product from slot 2 to slot 22, where it is adjacent only to a two door-vehicle 2434.
The resulting sequence is visualized below:
So far we have taken care of our thee biggest rocks, and so far we have been lucky. We were always able to find a nice spot somewhere for all of our products, where we did not have an excessive
accumulation of sequential cycle times.
However, as you can see in the sequence above, now we have only slots left that are adjacent on both sides. It is going to be tight! But this has to wait for the next post. Now go out, look again for
your biggest fluctuations, and organize your industry!
P.S. Many thanks to Mark Warren for his input.
Series Overview
Here is also the Sequencing Example Excel File for posts 7 to 11 with the complex example. Please note that this is not a tool, but merely some of my calculations for your information.
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2 thoughts on “Mixed Model Sequencing – Complex Example Sequencing 1”
1. I’m struggling to get some of the averages.
2. Hi Michael, Column L is a weighted average (sum of the time multiplied with the percent total). Does this help?
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ilometers to Handbreadth
Kilometers to Handbreadth Converter
Enter Kilometers
β Switch toHandbreadth to Kilometers Converter
How to use this Kilometers to Handbreadth Converter π €
Follow these steps to convert given length from the units of Kilometers to the units of Handbreadth.
1. Enter the input Kilometers value in the text field.
2. The calculator converts the given Kilometers into Handbreadth in realtime β using the conversion formula, and displays under the Handbreadth label. You do not need to click any button. If the
input changes, Handbreadth value is re-calculated, just like that.
3. You may copy the resulting Handbreadth value using the Copy button.
4. To view a detailed step by step calculation of the conversion, click on the View Calculation button.
5. You can also reset the input by clicking on button present below the input field.
What is the Formula to convert Kilometers to Handbreadth?
The formula to convert given length from Kilometers to Handbreadth is:
Length[(Handbreadth)] = Length[(Kilometers)] / 0.00007620000000030479
Substitute the given value of length in kilometers, i.e., Length[(Kilometers)] in the above formula and simplify the right-hand side value. The resulting value is the length in handbreadth, i.e.,
Calculation will be done after you enter a valid input.
Consider that a high-end electric car has a maximum range of 400 kilometers on a single charge.
Convert this range from kilometers to Handbreadth.
The length in kilometers is:
Length[(Kilometers)] = 400
The formula to convert length from kilometers to handbreadth is:
Length[(Handbreadth)] = Length[(Kilometers)] / 0.00007620000000030479
Substitute given weight Length[(Kilometers)] = 400 in the above formula.
Length[(Handbreadth)] = 400 / 0.00007620000000030479
Length[(Handbreadth)] = 5249343.832
Final Answer:
Therefore, 400 km is equal to 5249343.832 handbreadth.
The length is 5249343.832 handbreadth, in handbreadth.
Consider that a private helicopter has a flight range of 150 kilometers.
Convert this range from kilometers to Handbreadth.
The length in kilometers is:
Length[(Kilometers)] = 150
The formula to convert length from kilometers to handbreadth is:
Length[(Handbreadth)] = Length[(Kilometers)] / 0.00007620000000030479
Substitute given weight Length[(Kilometers)] = 150 in the above formula.
Length[(Handbreadth)] = 150 / 0.00007620000000030479
Length[(Handbreadth)] = 1968503.937
Final Answer:
Therefore, 150 km is equal to 1968503.937 handbreadth.
The length is 1968503.937 handbreadth, in handbreadth.
Kilometers to Handbreadth Conversion Table
The following table gives some of the most used conversions from Kilometers to Handbreadth.
Kilometers (km) Handbreadth (handbreadth)
0 km 0 handbreadth
1 km 13123.3596 handbreadth
2 km 26246.7192 handbreadth
3 km 39370.0787 handbreadth
4 km 52493.4383 handbreadth
5 km 65616.7979 handbreadth
6 km 78740.1575 handbreadth
7 km 91863.5171 handbreadth
8 km 104986.8766 handbreadth
9 km 118110.2362 handbreadth
10 km 131233.5958 handbreadth
20 km 262467.1916 handbreadth
50 km 656167.979 handbreadth
100 km 1312335.958 handbreadth
1000 km 13123359.58 handbreadth
10000 km 131233595.8 handbreadth
100000 km 1312335958 handbreadth
A kilometer (km) is a unit of length in the International System of Units (SI), equal to 0.6214 miles. One kilometer is one thousand meters.
The prefix "kilo-" means one thousand. A kilometer is defined by 1000 times the distance light travels in 1/299,792,458 seconds. This definition may change, but a kilometer will always be one
thousand meters.
Kilometers are used to measure distances on land in most countries. However, the United States and the United Kingdom still often use miles. The UK has adopted the metric system, but miles are still
used on road signs.
A handbreadth is a historical unit of length used to measure small distances, typically based on the width of a hand. One handbreadth is approximately equivalent to 4 inches or about 0.1016 meters.
The handbreadth is defined as the width of a person's hand, measured from the edge of the thumb to the edge of the little finger when the hand is spread out. This unit was used for practical
measurements in various contexts, including textiles and construction.
Handbreadths were used in historical measurement systems for assessing lengths and dimensions where precise tools were not available. Although less common today, the unit provides historical context
for traditional measurement practices and everyday use in different cultures.
Frequently Asked Questions (FAQs)
1. What is the formula for converting Kilometers to Handbreadth in Length?
The formula to convert Kilometers to Handbreadth in Length is:
Kilometers / 0.00007620000000030479
2. Is this tool free or paid?
This Length conversion tool, which converts Kilometers to Handbreadth, is completely free to use.
3. How do I convert Length from Kilometers to Handbreadth?
To convert Length from Kilometers to Handbreadth, you can use the following formula:
Kilometers / 0.00007620000000030479
For example, if you have a value in Kilometers, you substitute that value in place of Kilometers in the above formula, and solve the mathematical expression to get the equivalent value in | {"url":"https://convertonline.org/unit/?convert=kilometers-handbreadths","timestamp":"2024-11-09T16:08:55Z","content_type":"text/html","content_length":"91876","record_id":"<urn:uuid:c0a06bac-0ff6-49ac-8917-4b9b9b76f83b>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00275.warc.gz"} |
solve for x with steps 4( x – 2 ) = 5( x + 3)
-32ab-12b2 is the answer
show steps
It involves determining the meaning and significance of what is observed or expressed, or, concerning a given inference or argument,for more details,please go to:en.wikipedia.org
The steps are as follows:
-4b(5a – 2b ) + 4(-3ab – 5b^2)
Added 8/11/2014 10:02:02 PM
This answer has been confirmed as correct and helpful.
Confirmed by
[8/11/2014 11:44:15 PM] | {"url":"https://www.weegy.com/?ConversationId=DEB8B331&Link=i&ModeType=2","timestamp":"2024-11-08T05:34:58Z","content_type":"application/xhtml+xml","content_length":"54549","record_id":"<urn:uuid:1efcc61f-5dbb-4f20-a2ee-634a35f7a444>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00768.warc.gz"} |
Develop and use written methods to record, support and explain multiplication and division of two-digit numbers by a one-digit number, including division with remainders
Represents multiplication using an array and by jumping on a number line. Great for developing an understanding of multiplication by partitioning.
This game is now part of the 'Calculations' collection, which includes the following 17 games and resources: Column Addition, Expanded Addition, Expanded Addition - Place Value Counters, Number Bonds
(2), Addition - Digit Drag, Missing Symbols, Column Subtraction, Column Subtraction using Place Value Counters, Counting on to find difference on a beadstring, Multiplication Grid Method,
Multiplication Written Method, Ratio and Scaling Numbers, Representing Multiplication, Division by Chunking Up, Division by Chunking Down, Formal Written Division - Round Up or Down?, Short Division
Writen Formal Method.
The Calculations app is available on Google Play and the App Store. | {"url":"https://mathsframe.co.uk/en/resources/category/129/year-4-block-d-develop-and-use-written-methods-to-record-support-and-explain-multiplication-and-division-of-twodigit-numbers-by-a-onedigit-number","timestamp":"2024-11-02T09:21:56Z","content_type":"text/html","content_length":"29396","record_id":"<urn:uuid:90f0ef5f-365a-40ff-8a95-100b646331f0>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00121.warc.gz"} |
131+ Infinity Jokes And Funny Puns - JokoJokes
Infinity Jokes
131 infinity jokes and hilarious infinity puns to laugh out loud. Read jokes about infinity that are clean and suitable for kids and friends.
Laugh endlessly with this collection of infinity jokes! With references to the Infinity Gauntlet, Infinity War, and Axie Infinity, these jokes will provide an everlasting source of entertainment. Get
ready for an endless stream of laughter!
Quick Jump To
Funniest Infinity Short Jokes
Short infinity jokes and puns are one of the best ways to have fun with word play in English. The infinity humour may include short infinite jokes also.
1. I want to talk about Infinity War spoilers but... I want to wait for the dust to settle a bit.
2. [Mild Infinity War Spoiler] Did you guys see Peter Dinklage in the new Avengers film? It was his biggest role to date.
3. Why can't the Infiniti car company trademark ∞ ? Because the legal battle would be endless.
4. (Infinity War Speculation) When Thanos gains the mind stone, he will turn into Palpatine. Because The Avengers will pay for their lack of Vision.
5. Just saw the new infinity wars trailer today and to me, it was pretty obvious who's dying. Well, so long, DC.
6. Thanos wasn't completely responsible for all the deaths caused by the Infinity Gauntlet... But he had a hand in it.
7. What are the most powerful biscuits in the universe? The Infinity Scones...
My 11 year old just came up with that one on his own...
8. [Spoilers] I finally watched Avengers: Infinity War It was alright. Probably give a 5/10. The first half was great, but the other half just kind of fell apart in the end.
9. Where was Drax in Avengers : Infinity War? I saw his name in the credits and heard his voice sometimes, but why wasn't he in the movie?
10. 5 yo: Can you count backwards from infinity? Me: Of course I can but it will take forever.
Share These Infinity Jokes With Friends
Infinity One Liners
Which infinity one liners are funny enough to crack down and make fun with infinity? I can suggest the ones about ounce and decimal.
1. To the person who invented infinity Thanks for everything
2. I watched the new Infinity War movie yesterday... I was Loki dying in the beginning
3. You know what is good with 8 ? It's straight up infinity.
4. What's infinity minus one? A sideways seven.
5. The Infinity Gauntlet was for the left hand Even Thanos had priorities.
6. In Avengers: Infinity War, Thanos absolutely went crazy. He snapped.
7. My girlfriend is like infinity. Off the charts, but only a concept.
8. I love talking about infinity I could go on forever about it.
9. Yo mama so fat 5 infinity stones couldn't make her disappear
10. What did the eight say to infinity? You okay, buddy? You don't look so good.
11. If Eminem had the infinity gaulent.... He could actually snap back into reality
12. What's infinity plus infinity? 2 infinity. And beyond!
13. Chuck Norris counted to infinity Twice!
14. Every one dies in Infinity War Me: You don't know the half of it
15. Why was everyone crying after watching Infinity Wars? Because sand got into their eyes
Infinity War Jokes
Here is a list of funny infinity war jokes and even better infinity war puns that will make you laugh with friends.
• Which app does Thanos use to communicate with half of the universe? (Infinity War Spoilers) snapchat
• Thanos's snap in infinity war would've had a greater impact if marvel made it seem that half of the audience wasn't there, But apparently only DC movies can do that.
• I watched Avengers: Infinity War this weekend. It was definitely a good movie, but one question was left unanswered, and it keeps me up at night. I'll never be able to figure out why is Gamora.
• After Infinity War, you know what Earth's group of heroes should be called? The Half-Vengers.
• 2018 is the Year of the sequel. Deadpool 2, Avengers Infinity war 3, Incredibles 2, World War 3.
• Did you hear about Avengers: Infinity Wars? It's said to be Marvel-ous!
• Did Sia get snapped in Infinity War? Cause I don't sia anywhere.
• Why would the Avengers have been better able to see a solution to the infinity war if Thanos had arrived just a couple of years later? They'd have had 2020 Vision
(I'm so sorry)
• [SPOILER] Infinity War Why is Red Skull happy to see the mad titan on Vormir?
Cos he is missing Thanos on his face.
• I like how avengers infinity war has perfect balance [Thor gains his vision back but scarlet witch loses hers](#s)
Infinity Gauntlet Jokes
Here is a list of funny infinity gauntlet jokes and even better infinity gauntlet puns that will make you laugh with friends.
• What's the only thing more terrifying than Thanos wearing the Infinity Gauntlet? Thanos wearing the Infinity Gauntlet while playing a Jet in a production of West Side Story
• Chuck Norris passed 6 kidney stones. Thanos used them in the Infinity Gauntlet.
• What software does the Infinity Gauntlet run on? ThanOS
• After months of research, I️ have finally concluded what metal the Infinity Gauntlet is made of. Hafnium
• Where is the best place to punch a Mad Titan who is wearing the Infinity Gauntlet? THA NOS (i'll just show myself out)
• What operating system does the Infinity Gauntlet use? thanOS
• All these Marvel movies, and they sure are taking their time getting to Thanos and the Infinity Gauntlet Don't worry. They'll get there... Avengually.
Experience Good Cheer with Hilarious Infinity Jokes and Friends
What funny jokes about infinity you can tell and make people laugh? An example I can give is a clean omega jokes that will for sure put a smile on everyones mouth and help you make infinity pranks.
Yo mamas so fat that when she stepped on a scale, buzz lightyear came out and said "to infinity and beyond!"
Chuck Norris's kill ratio on Call of Duty:Black Ops is infinity.
You've heard that Chuck Norris counted to infinity twice, right?
Well he's currently making his third attempt.
The infinity stones aren't infinity
there are only six of them
Two cows are standing...
in the pasture. One turns to the other and says, "Although pi is usually abbreviated to five numbers, it actually goes on to infinity."
The second cow turns to the first and says: "Moo."
(stolen from Plato and a Platypus Walk into a Bar...)
What did the 8 say to infinity?
Come on man, get up!
How many mathematicians does it take to screw in a lightbulb?
Somewhere between 0 and infinity.
Why 'business' people continue to earn more than people who actually work :
Time is money (time=money), and knowledge is power( knowledge=power)
we know that power is work done/time
power=work done/money
but knowledge=power
we have money=work done/knowledge
which means, in the limiting case, as knowledge ->0 money -> Infinity, regardless of work done
A boy asks the teacher, "what's infinity?"
Teacher: "think of a number."
Boy: "ok, I've got one."
Teacher: "good, that's not it!"
Are you a rational function?
because I could ride your asymptote to infinity.
Infinity loop
a woman walks into her hypnotherapists office and says " Doctor, I have been loyal to my husband for 12 years but, last night I broke that loyalty and had an affair. I just want you to make me forget
it happened". The Doctor said " Not again"
Legend say, Chuck Norris counted to infinity twice...
Donald Trump tried to count to 10 and got stuck in a paper bag.
What do Buzz Lightyear and an asymptote have in common?
They both go to infinity and beyond!
I'm thinking of a number between 1 to Infinity. What number am I thinking of?
What do Marvel and Northrop Grumman have in common?
They're both heavily invested in Infinity War.
Even though they did not show any spoilers in the Infinity War trailer we all know who dies...
... DCEU.
Wanna know who dies in infinity wars?
Doing math, it's sometimes very easy to tell what you did wrong.
If you got infinity, it's even easier: You took a wrong turn. Instead of making a right turn, you took aleph one.
Where does Thanos like to swim?
In the Infinity Pool.
My wife is knitting an infinity scarf.
It's going to take her forever to finish it.
WitH All THeSe PeOplE PreDicTiNG wHo is GoinG to Die in InFiNiTy wAr, You cOuLd sAy iTs a...
How would rocket raccoon react if groot joined forces with Thanos in Infinity War?
"et tu Groot?"
**INFINITY WAR SPOILERS**
Ok now that all the nerds are gone, there's a party going on at my place this Saturday. hmu if you're interested.
Marvel just announced the title of the Infinity War sequel. [spoiler]
Avengers: Days of Future Past
**Infinity War spoiler**
Superman kills Batman while Green Lantern looks on…
True story: My gf and I were watching Guardians of the Galaxy, she asks "So wait, what exactly is an infinity stone?"
I said, "I think its some kind of plot device"
Just something I noticed in Infinity War.
Turns out, Stan Lee was the bus driver all along.
Heard a co-worker talking about Infinity War in the office.
Told her to be careful about spoilers, someone might snap at her.
If Thanos were English what would he collect?(No spoilers)
The Infinity Scones
What do infinity war and a tweet have in common?
They both have 280 characters
Why is the villain of Avengers: Infinity War so good at tracking Infinity Stones?
Because he's good at smelling... he's The Nose.
Went to see Infinity War with a group of friends....
About half of us were blown away.
Although the real joke is OP having friends.
I guess what happened to Gamora in Infinity War
was definitely not a cliffhanger
I'm going to cash in on the success of Avengers: Infinity War by opening a comics themed sandwich shop.
It'll be called *Soup or Hero*
Infinity War (Spoilers)
It's crazy that Infinity War has been out for almost two weeks and I haven't seen a single pun spoiling Peter Dinklarge
I've had kidney stones for my entire life and it seems that they'll always be there forever
They're infinity stones
[Spoiler] Thanks to Peter Quill,
we can watch infinity war part 2.
What happens when a god dies and no one knows? [Infinity War Spoliers]
They're loki dead
Warning, Infinity War spoiler:
Stan Lee is the bus driver.
*Infinity War Spoilers* What do Matt Murdock and Scarlet Witch have in common?
They both lost their Vision
Chuck Norris once had the case of the kidney stones
Now they are known as the infinity stones
Why did it take only one movie for Thanos to get all of the infinity stones?
Because he wanted to make it snappy.
What's Thanos' favorite app on his Infinity Phone?
If Thanos ever had to go to a hospital
He'd be sent to the Infinity Ward
Why is the symbol for infinity an 8 lying down?
Because after someone ate, lying down is infinitely better.
What do the Avengers do at the end of Infinity War 2?
Take a Wakanda beach.
Marvel will introduce a turkish superhero in the second infinity war movie
Ayran Man
What's the favorite movie of the Infinity War script writer?
Die Hard
What's the difference between infinity and forever?
According to my ex-wife, forever is only two and a half months.
The 1 reality that survived Infinity Wars..
..was the one where Thanos didn't know how to snap his fingers.
The universe's greatest villain acquired his sixth Infinity Stone, and in the snap of a finger...
...half the NFL's fans stopped caring.
Stand aside Thanos...
Tom Brady has the six infinity stones now.
Never discuss infinity with a mathematician..
They can go on about it forever... | {"url":"https://jokojokes.com/infinity-jokes.html","timestamp":"2024-11-03T13:13:24Z","content_type":"application/xhtml+xml","content_length":"85384","record_id":"<urn:uuid:b4686729-97b3-44a5-a5fd-6e4de230bb9c>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00245.warc.gz"} |
A Practice Fractions
KBruch is a small program to practice calculating with fractions and percentages. Different exercises are provided for this purpose and you can use the learning mode to practice with fractions. The
program checks the user's input and gives feedback.
* Arithmetic exercise - in this exercise you have to solve a given fraction task. You have to enter the numerator and the denominator. This is the main exercise.
* Comparison exercise - in this exercise you have to compare the size of 2 given fractions, using the symbols >, < or =.
* Conversion exercise - in this exercise you have to convert a given number into a fraction.
* Factorization exercise - in this exercise you have to factorize a given number into its prime factors.
* Percentage exercise - in this exercise you have to calculate percentages. | {"url":"https://depot.haiku-os.org/__multipage/pkg/kbruch/haikuports/haikuports_x86_64/23/08/5/-/1/x86_64?locale=en","timestamp":"2024-11-06T05:58:26Z","content_type":"text/html","content_length":"5413","record_id":"<urn:uuid:1abce7e4-4319-4499-8847-9aa03d7f7d30>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00353.warc.gz"} |
Simplified Chess Engine | HackerRank
Chess is a very popular game played by hundreds of millions of people. Nowadays, we have chess engines such as Stockfish and Komodo to help us analyze games. These engines are very powerful pieces of
well-developed software that use intelligent ideas and algorithms to analyze positions and sequences of moves, as well as find tactical ideas. Consider the following simplified version of chess:
• Board: It's played on a board between two players named Black and White.
• Pieces and Movement:
□ White initially has pieces and Black initially has pieces.
□ There are no Kings and no Pawns on the board. Each player has exactly one Queen, at most two Rooks, and at most two minor pieces (i.e., a Bishop and/or Knight).
□ Each piece's possible moves are the same as in classical chess, and each move made by any player counts as a single move.
□ There is no draw when positions are repeated as there is in classical chess.
• Objective: The goal of the game is to capture the opponent’s Queen without losing your own.
Given and the layout of pieces for games of simplified chess, implement a very basic (in comparison to the real ones) engine for our simplified version of chess with the ability to determine whether
or not White can win in moves (regardless of how Black plays) if White always moves first. For each game, print YES on a new line if White can win under the specified conditions; otherwise, print NO.
The first line contains a single integer, , denoting the number of simplified chess games. The subsequent lines define each game in the following format:
• The first line contains three space-separated integers denoting the respective values of (the number of White pieces), (the number of Black pieces), and (the maximum number of moves we want to
know if White can win in).
• The subsequent lines describe each chesspiece in the format t c r, where is a character denoting the type of piece (where is Queen, is Knight, is Bishop, and is Rook), and and denote the
respective column and row on the board where the figure is placed (where and ). These inputs are given as follows:
□ Each of the subsequent lines denotes the type and location of a White piece on the board.
□ Each of the subsequent lines denotes the type and location of a Black piece on the board.
• It is guaranteed that the locations of all pieces given as input are distinct.
• Each player initially has exactly one Queen, at most two Rooks and at most two minor pieces.
For each of the games of simplified chess, print whether or not White can win in moves on a new line. If it's possible, print YES; otherwise, print NO.
N B 2
Q B 1
Q A 4
We play games of simplified chess, where the initial piece layout is as follows:
White is the next to move, and they can win the game in move by taking their Knight to and capturing Black's Queen. Because it took move to win and , we print YES on a new line. | {"url":"https://www.hackerrank.com/challenges/simplified-chess-engine/problem","timestamp":"2024-11-08T17:28:27Z","content_type":"text/html","content_length":"951048","record_id":"<urn:uuid:10a5ea5e-fccb-474a-92bf-d5f666141290>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00072.warc.gz"} |
The Chess Variant Pages: OctaChess
OctaChess is a variant of Chess with Different Armies.
There are a total of eight armies in OctaChess, which includes four armies inspired by the classics (Chess, Shogi, Xiangqi, Janggi) and four modern armies with old and new fairy chess pieces. Besides
the King and Pawns, none of the armies have pieces that can move exactly like a piece from another army in OctaChess.
The eight armies in OctaChess are designed to be of about equal strength. A spreadsheet comparing the different strengths calculated from Zillions of Games is attached in the Notes section below.
Every army in OctaChess has King and Pawns, plus pieces equivalent to the Queen, Bishop, Knight, and Rook pieces in Orthodox Chess (or Queen Chess). The pieces for each OctaChess army are arranged in
the same way as their equivalents in Queen Chess. The pieces and their moveset for each army in OctaChess are described below.
Queen Chess
Moves as the Rook in Orthodox Chess - Slides any number of squares orthogonally.
Moves as the Knight in Orthodox Chess - Jumps two squares forward, then one square sideways in an L-shape.
Moves as the Bishop in Orthodox Chess - Slides any number of squares diagonally.
Moves as the Queen in Orthodox Chess - Slides any number of squares orthogonally or diagonally.
Moves as the King in Orthodox Chess - Moves one square forward in any direction.
Moves as the Pawn in Orthodox Chess - Moves on square forward and captures one square diagonally forward, can slide two squares in first move and capture en passant.
Cannon Chess
(Note: This OctaChess army is derived from Xiangqi.)
Moves any number of squares orthogonally, and can capture by sliding or jumping over a piece (Rook+Cannon).
Moves one square orthogonally, or one square orthogonally, then one square diagonally outward (Xiangqi Horse+General).
Slides up to two squares diagonally, or moves one square orthogonally (Elephant+Advisor+General).
Moves one square diagonally or one square orthogonally without capturing.
Moves as the King in Orthodox Chess.
Moves as the Pawn in Orthodox Chess.
Dragon Chess
(Note: This OctaChess army is derived from Shogi.)
Slides any number of squares orthogonally, or one square diagonally.
Silver Knight
Moves one square diagonally, one square orthogonally forward, or leaps two squares forward in an ‘L’ shape (Silver General+Shogi Knight).
Moves one square orthogonally, or one square diagonally forward.
Dragon Lance
Slides any number of squares diagonally, any number of squares orthogonally forward, or one square orthogonally sideways and backward (Dragon Horse+Lance).
Moves as the King in Orthodox Chess.
Moves as the Pawn in Orthodox Chess.
Stallion Chess
(Note: This OctaChess army is derived from Janggi.)
Palace Guard
Slides any number of squares orthogonally, one square diagonally, or leaps over a piece exactly one square diagonally (Chariot+Advisor+Janggi Cannon diagonal move in the Palace).
Moves one square orthogonally, then one or two squares diagonally forward (Janggi Elephant+Janggi Horse).
Can only move by leaping over any piece (similar to the Janggi Cannon but it can leap over another Cannon piece).
Palace Chariot
Slides any number of squares orthogonally, and up to two squares diagonally (Chariot+Janggi Chariot diagonal move in the Palace).
Moves as the King in Orthodox Chess.
Moves as the Pawn in Orthodox Chess.
Chimera Chess
(Note: This OctaChess army is derived from Chimera Chess.)
Flying Jester
Moves one square orthogonally like a Wazir, captures any number of squares orthogonally.
Leaps two squares orthogonally, diagonally, or in a orthogonal-diagonal sequence in an ‘L’ shape (Knight+Alfil+Dabbabah).
Moves one square in any direction.
Grand Chimera
Leaps two squares in any direction like the Chimera, moves one square orthogonally without capturing, and captures one square diagonally (Knight+Alfil+Dabbabah+Pawn that can act sideways and
Moves as the King in Orthodox Chess.
Moves as the Pawn in Orthodox Chess.
Dark Knight Chess
(Note: This OctaChess army is derived from the Dark Horde variant of Kingsmen.)
Slides up to two squares orthogonally or diagonally.
Moves one square orthogonally, or leaps two squares orthogonally (Wazir+Dabbabah).
Moves one square diagonally, or leaps two squares diagonally (Ferz+Alfil).
Leaps like a Knight up to three consecutive 'L' shapes in the same direction, moves one square orthogonally without capturing, or captures one square diagonally (Nightrider+Pawn that can act sideways
and backward).
Moves as the King in Orthodox Chess.
Moves as the Pawn in Orthodox Chess.
Plato Chess
(Note: This OctaChess army is derived from Platonic Chess.)
Slides up to four squares orthogonally or leaps two squares diagonally like an Alfil.
Leaps two squares in an `L` shape or two squares orthogonally (Knight+Dabbabah).
Slides up to two squares diagonally.
Slides up to six squares forward, one square sideways and backward, or two squares diagonally.
Moves as the King in Orthodox Chess.
Moves as the Pawn in Orthodox Chess.
Noble Chess
(Note: This OctaChess army is inspired by the royal promoted pieces in Kingsmen.)
Coronet Rook
Moves one square in any direction, can capture any number of squares orthogonally.
Moves one square in any direction, or leaps two squares in an ‘L’ shape.
Coronet Bishop
Moves one square in any direction, can capture any number of squares diagonally.
Coronet Pawn
Moves one square orthogonally without capturing, or captures by leaping one or two squares diagonally.
Moves as the King in Orthodox Chess.
Moves as the Pawn in Orthodox Chess.
The King can castle with the Rook equivalent as in Orthodox Chess.
The Pawn can promote to a Queen, Bishop, Knight, or Rook equivalent upon reaching the last rank.
The en Passant rule is the same for all chess armies in OctaChess.
Each chess variant in OctaChess is designed to be of about equal strength to the Orthodox Chess army. The strength of the pieces is calculated using Zillions of Games. Details of the calculation can
be found in the attached spreadsheet below (zip file).
OctaChess Strength Calculations
[Note: A Zillions of Games file comprising of mirror matches (same army for both sides) in OctaChess is included for those who wish to explore how changing the moveset of the pieces in the eight
armies affect their calculated strengths.]
Play with Zillions of Games
Click the link below to download Zillions of Games files (zipped) where you can play the eight armies in OctaChess against each other. The classic armies are Chess, Cannon Chess, Dragon Chess, and
Stallion Chess. The modern armies are Chimera Chess, Dark Knight Chess, Plato Chess, and Noble Chess.
(Note: There are a total of 64 different matchups between the eight OctaChess armies, including mirror matches. Hence, each matchup in the attached Zillions of Games files is given a code that
represents a square in an 8x8 chessboard.)
By Albert Lee.
Last revised by Albert Lee.
Web page created: 2021-11-23. Web page last updated: 2023-11-15 | {"url":"https://www.chessvariants.com/rules/octachess","timestamp":"2024-11-03T20:13:11Z","content_type":"text/html","content_length":"59020","record_id":"<urn:uuid:1deaccac-a0a1-4414-a4fe-590391264e6d>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00486.warc.gz"} |
Teach Kids How To Code: Turing Put Command Test - Computer Corner
Turing Programming Language
Turing is a programming language that was written with high schools in mind which makes it ideal as a first language. The Turing Put Command Test Resource lets kids demonstrate their mastery with the
put command.
Turing Put Command Test
Can they control screen output? This package tests their knowledge and understanding of the concepts and skills taught in these resource packages: the Turing Put Command, Turing Put Command Syntax.
What Do Kids Need To Already Know?
These concepts were covered in the Turing Put Command and Turing Put Command Syntax Resources:
✔ Main Program
✔ User-Defined Procedures
✔ Source Code, Compiled Code
✔ Code Organization
✔ Programmer Comments
✔ Syntax Errors
✔ Logic Errors
✔ KISS (Keep It Simple Student)
✔ User-Friendly Program Design
✔ Arithmetic Operators
✔ Turing put Command:
◆ string vs numeric output
◆ , (comma)
◆ .. (dot dot)
◆ :xx (spacing)
◆ controlling string output
◆ controlling numeric output
What Do Teachers/Parents Need To Know?
The best thing about these resources is that you can learn along with the kids. There are also some Turing blog posts that you might be interested in.
Suggested Timeline
It should take most students two 75 minute periods to complete the test. Allocate one period for the test review, and one for the test.
Print & Digital Format Package
This package was originally created for classroom use. However, it now allows teachers to use the student handout in either an LMS system or emailed directly to students.
This package includes:
✔ A Package Description
✔ Test Solutions
✔ 2 PowerPoint⭐ Review Lessons
✔ 4 Versions of a Put Command Test
◆ 4 Alternate Versions with a KICA* marking scheme
✔ 4 Code Solution Test Files
⭐ PowerPoint presentations can easily be converted to Google Slides and vice versa. For more information check out these videos:
Google Classroom: Convert PowerPoint Presentations to Google Slides
Microsoft OneDrive: Convert Google Slides to PowerPoint Presentations
Why not go all the way? The full unit is discounted by 20%.
✔ Turing Put Command
✔ Turing Put Command Syntax
✔ Turing Put Command Exit Ticket/Culminating Activity
✔ Turing Put Command Test (4 versions)
View a preview video of this unit!
Resources UPDATED September 2020.
What’s The Next Unit?
Unit 2 introduces Turing graphics commands. Turing may be an older language but it has great graphics capabilities. It uses the positive quadrant of the Cartesian plane to draw which is a great way
to reinforce co-ordinate math. | {"url":"https://computercorner.ca/turing-put-command-test-tpt-resource/","timestamp":"2024-11-12T05:56:51Z","content_type":"text/html","content_length":"82405","record_id":"<urn:uuid:309ba4e3-40cf-40d1-8094-6ce832dd8e3c>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00030.warc.gz"} |
David Aulicino
Associate Professor
Brooklyn College & The Graduate Center
E-mail: david.
Research Interests: My research lies at the intersection of Geometry and Dynamical Systems. Flat surfaces with rational cone angle are the primary objects that I study. These include translation
surfaces, i.e. Riemann surfaces carrying a holomorphic 1-form, \(k\)-differentials, which include rational billiards and rational polyhedra, and meromorphic differentials with higher order poles on
Riemann surfaces. I have studied counting problems on translation surfaces, \(\text{SL}_2(\mathbb{R})\)-orbit closures in the moduli space of translation surfaces, and the Lyapunov exponents of the
Kontsevich-Zorich cocycle.
Seminar: I am currently an organizer of the Complex Analysis and Dynamics Seminar at the Graduate Center. | {"url":"https://userhome.brooklyn.cuny.edu/aulicino/","timestamp":"2024-11-11T06:12:29Z","content_type":"text/html","content_length":"4507","record_id":"<urn:uuid:5b91c826-2b7f-4fc4-8b55-bde5c05be072>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00649.warc.gz"} |
Parallelogram | Lexique de mathématique
Quadrilateral whose opposite sides are congruent and whose opposite angles are congruent.
• A parallelogram is a quadrilateral whose opposite sides are parallel in pairs.
• A parallelogram usually does not have an axis of symmetry.
However, except if it is a rectangle (having right angles), or isosceles (having adjacent congruent sides), the parallelogram has two perpendicular axes of symmetry. In this case, the
parallelogram belongs to the class of rectangles, rhombuses, or both (square).
• Based on its properties, the parallelogram also belongs to the family of convex quadrilaterals and trapezoids.
Consider the parallelogram below :
• The formula to calculate the perimeter P of a parallelogram of side lengths a and b is: \(P=2 \times \left ( a+b \right )\).
• The formula to calculate the area A of a parallelogram with a base b and a height h is: \(A=b \times h\).
In the figure above, the opposite sides (hash marks) are congruent and the opposite angles (arcs) are congruent. | {"url":"https://lexique.netmath.ca/en/parallelogram/","timestamp":"2024-11-06T17:40:12Z","content_type":"text/html","content_length":"64866","record_id":"<urn:uuid:5406ef88-8466-436d-b405-06bfaead62d6>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00137.warc.gz"} |
Summary: Using Properties of Rectangles, Triangles, and Trapezoids
Key Concepts
• Properties of Rectangles
□ Rectangles have four sides and four right ([latex]90^\circ[/latex]) angles.
□ The lengths of opposite sides are equal.
□ The perimeter, [latex]P[/latex] , of a rectangle is the sum of twice the length and twice the width.
□ The area, [latex]A[/latex] , of a rectangle is the length times the width.
☆ [latex]A=L\cdot W[/latex]
• Triangle Properties
□ For any triangle [latex]\Delta ABC[/latex] , the sum of the measures of the angles is [latex]180^\circ[/latex].
☆ [latex]m\angle A+m\angle B+m\angle C=180^\circ [/latex]
□ The perimeter of a triangle is the sum of the lengths of the sides.
□ The area of a triangle is one-half the base, [latex]b[/latex], times the height, [latex]h[/latex].
☆ [latex]A=\Large\frac{1}{2}\normalsize bh[/latex]
• Trapezoid properties
□ A trapezoid has a larger base, [latex]B[/latex], and a smaller base, [latex]b[/latex]. The height [latex]h[/latex] is the distance between the bases.
□ The formula for the area of a trapezoid is one-half the height, [latex]h[/latex], times the sum of the big base, [latex]B[/latex], plus the little base, [latex]b[/latex].
☆ [latex]{\text{Area}}_{\text{trapezoid}}=\Large\frac{1}{2}\normalsize h\left(b+B\right)[/latex]
The area is a measure of the surface covered by a figure.
equilateral triangle
A triangle with all three sides of equal length is called an equilateral triangle.
isosceles triangle
A triangle with two sides of equal length is called an isosceles triangle.
The perimeter is a measure of the distance around a figure.
A rectangle is a geometric figure that has four sides and four right angles.
A trapezoid is four-sided figure, a quadrilateral, with two sides that are parallel and two sides that are not. | {"url":"https://courses.lumenlearning.com/wm-developmentalemporium/chapter/summary-using-properties-of-rectangles-triangles-and-trapezoids/","timestamp":"2024-11-10T11:49:35Z","content_type":"text/html","content_length":"48664","record_id":"<urn:uuid:0bbfb22b-d24b-4d67-a7ad-7ed45169a4df>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00388.warc.gz"} |
Verification : TEST CASE 4.1 STEEL ROPE WITH GIVEN SAG - SP LIRA
TEST CASE 4.1 STEEL ROPE WITH GIVEN SAG
«Designers handbook. Engineering and theoretical», edited by A.A. Umansky, – M.: Stroyizdat, 1960, p. 326.
Problem description:
Steel rope with cross section area A is loaded by distributed load q along the span and concentrated load P in the centre. Determine the reaction force H, acted along the line between hawser
supports, maximal tension N[max] in the rope and maximal sag z[max].
Problem sketch:
Type of created problem:
Plane frame (X, Z, UY).
Geometric characteristics:
L = 100 m; f = 5 m; А = 12 cm^2.
Material properties:
Elastic modulus: E = 1.6·10^7 tf/m^2.
Boundary conditions:
Elastic links with high stiffness along the X and Z directions at the rope ends.
q = 0.01 tf/m; P = 1 tf.
Model description:
The system is modeled using 100 bar geometrically nonlinear finite elements of the “string” type (FE type is 304) and 2 single-noded elements of elastic link (FE type is 56). Automatic selection of
step of loading applying for nonlinear problem solution is used.
Analytical solution:
Calculation results:
│Target value│Analytical solution │LIRA 10│Deviation, %│
│H, tf │6.45 │6.4577 │0.12 │
│N[max], tf │6.53 │6.5339 │0.06 │
│z[max], m │5.81 │5.8170 │0.12 │
Reaction forces along X axis at supports R[x], tf
Diagram of longitudinal forces N, tf
Vertical displacements distribution, m | {"url":"https://lira10.com/en/pclira/test/76C873E1-121C-463B-94C1-9341D9460493","timestamp":"2024-11-10T02:15:28Z","content_type":"text/html","content_length":"55482","record_id":"<urn:uuid:3da51fd6-d181-4294-be07-75bc6f7baf0d>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00615.warc.gz"} |
Ben Krause : Dimension independent bounds for the spherical maximal function on products of finite groups
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Ben Krause : Dimension independent bounds for the spherical maximal function on products of finite groups
The classical Hardy-Littlewood maximal operators (averaging over families of Euclidean balls and cubes) are known to satisfy L^p bounds that are independent of dimension. This talk will extend these
results to spherical maximal functions acting on Cartesian products of cyclic groups equipped with the Hamming metric.
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CSCI 567 Homework #3 Programming Assignments solution
Problem 1 Pegasos: a stochastic gradient based solver for linear SVM
In this question, you will build a linear SVM (i.e. without kernel) classifier using the Pegasos algorithm [1].
Given a training set {(xn ∈ RD, yn ∈ {1, −1})}
, the primal formulation of linear SVM can be written as
L2-regularized hinge loss as shown in the class:
2 +
N ∑n
max{0, 1 − ynwT
xn}. (1)
Note that here we include the bias term b into parameter w by appending x with 1, which is slightly different
from what was discussed in the class.
Instead of turning it into dual formulation, we are going to solve the primal formulation directly with
a gradient-base algorithm. Recall for (batch) gradient descent, at each iteration of parameter update, we
compute the gradients for all data points and take the average (or sum).
When the training set is large
(i.e., N is a large number), this is often too computationally expensive. Stochastic gradient descent with
mini-batch alleviates this issue by computing the gradient on a subset of the data at each iteration.
Pegasos, a representative solver of eq. (1), is exactly applying stochastic gradient descent with minibatch to this problem, with an extra step of projection described below (this is also called
projected stochastic gradient descent). The pseudocode of Pegasos is given in Algorithm 1. At the t-th iteration of parameter
update, we first take a mini-batch of data At of size K [step (3)], and define A
t ⊂ At to be the subset
of samples for which wt suffers a non-zero loss [step (4)]. Next we set the learning rate ηt = 1/(λt)
[step (5)].
We then perform a two-step parameter update as follows. We first compute (1 − ηtλ)wt
and for all samples (x, y) ∈ A
t we add the vector
x to (1 − ηtλ)wt
. We denote the resulting vector by
wt+ 1
[step (6)]. This step is in fact exactly stochastic gradient descent: wt+ 1
= wt − ηt∇t where
∇t = λwt −
K ∑
is the (sub)gradient of the objective function on the mini-batch At at wt
. Last, we set wt+1 to be the projection of wt+ 1
onto the set
B = {w : ||w||2 ≤ 1/√
to control its L2 norm. This can be obtained simply by scaling wt+1 by min (
||wt+ 1
[step (e)].
For more details of Pegasos algorithm you may refer to the original paper [1].
Now you are going to implement Pegasos and train a binary linear SVM classifier on the dataset
mnist subset.json. You are going to implement three functions—objective function, pegasos train,
and pegasos test—in a script named pegasos.py. You will find detailed programming instructions in
the script.
1.1 Finish the implementation of the function objective function, which corresponds to the objective
function in the primal formulation of SVM.
1.2 Finish the implementation of the function pegasos train, which corresponds to Algorithm 1.
Algorithm 1 The Pegasos algorithm
Require: a training set S = {(xn ∈ RD, yn ∈ {1, −1})}
, the total number of iterations T, the batch size
K, and the regularization parameter λ.
Output: the last weight wT+1.
1: Initialization: w1 = 0
2: for t = 1, 2, · · · , T do
3: Randomly choose a subset of data At ∈ S (with replacement) such that |At
| = K
4: Set A
t = {(x, y) ∈ At
: ywT
xt < 1} 5: Set ηt = 1 λt 6: Set wt+ 1 2 = (1 − ηtλ)wt + ηt K ∑(x,y)∈A + t yx 7: Set wt+1 = min ( 1, 1/√ λ ||wt+ 1 2 ||) wt+ 1 2 8: end for 1.3 After you train your model, run your classifier on the
test set and report the accuracy, which is defined as: # of correctly classified test samples # of test samples Finish the implementation of the function pegasos test. 1.4 After you complete above
steps, run pegasos.sh, which will run the Pegasos algorithm for 500 iterations with 6 settings (mini-batch size K = 100 with different λ ∈ {0.01, 0.1, 1} and λ = 0.1 with different K ∈ {1, 10,
1000}), and output a pegasos.json that records the test accuracy and the value of objective function at each iteration during the training process. What to do and submit: run script pegasos.sh.
It will take mnist subset.json as input and generate pegasos.json. Add, commit, and push both pegasos.py and pegasos.json before the due date. 1.5 Based on pegasos.json, you are encouraged to make
plots of the objective function value versus the number of iterations (i.e., a convergence curve) in different settings of λ and K, but you are not required to submit these plots.
[1] Shai Shalev-Shwartz, Yoram Singer, Nathan Srebro, Andrew Cotter Mathematical Programming, 2011, Volume 127, Number 1, Page 3. Pegasos: primal estimated sub-gradient solver for SVM. 4 Problem 2
Boosting In the lectures, we discussed boosting algorithms, which construct a strong (binary) classifier based on iteratively adding one weak (binary) classifier into it. A weak classifier is learned
to (approximately) maximize the weighted training accuracy at the corresponding iteration, and the weight of each training example is updated after every iteration.
In this question, you will implement decision stumps as weak classifiers and AdaBoost. For simplicity, instead of implementing an actual base algorithm that learns decision stumps, we fix a finite
set of decision stumps H in advance and pick the one with smallest weighted error at each round of boosting.
2.1 General boosting framework A boosting algorithm sequentially learns βt and ht ∈ H for t = 0, . . . , T − 1 and finally constructs a strong classifier as H(x) = sign h ∑ T−1 t=0 βtht(x) i . Please
read the class “Boosting” defined in boosting.py, and then complete the class by implementing the “TODO” part(s) as indicated in boosting.py.
2.2 Decision stump A decision stump h ∈ H is a classifier characterized by a triplet (s ∈ {+1, −1}, b ∈ R, d ∈ {0, 1, · · · , D − 1}) such that h(s,b,d) (x) = ( s, if xd > b,
−s, otherwise.
That is, each decision stump function only looks at a single dimension xd of the input vector x, and check
whether xd
is larger than b. Then s decides which label to predict if xd > b.
Please first read classifier.py and decision stump.py, and then complete the class “DecisionStump” by implementing the “TODO” part(s) as indicated in decision stump.py.
2.3 AdaBoost AdaBoost is a powerful and popular boosting method, summarized in Algorithm
Algorithm 2 The AdaBoost algorithm
Require: H: A set of classifiers, where h ∈ H takes a D-dimensional vector as input and outputs either +1
or −1, a training set {(xn ∈ RD, yn ∈ {+1, −1})}
, the total number of iterations T.
Ensure: Learn H(x) = sign h
, where ht ∈ H and βt ∈ R.
1: Initialization D0(n) = 1
, ∀n ∈ {0, 1, · · · , N − 1}.
2: for t = 0, 1, · · · , T − 1 do
3: find ht = argminh∈H ∑n Dt(n)I[yn 6= h(xn)]
4: compute et = ∑n Dt(n)I[yn 6= ht(xn)]
5: compute βt =
1 − et
6: compute Dt+1(n) = (
Dt(n) exp(−βt), if yn = ht(xn)
Dt(n) exp(βt), otherwise
, ∀n ∈ {0, 1, · · · , N − 1}
7: normalize Dt+1(n) ←
0 Dt+1(n
, ∀n ∈ {0, 1, · · · , N − 1}
8: end for
Please read the class ”AdaBoost” defined in boosting.py, and then complete the class by implementing the ”TODO” part(s) as indicated in boosting.py.
2.4 Final submission Please run boosting.sh, which will generate boosting.json. Add, commit,
and push boosting.py, decision stump.py and boosting.json before the due date.
Problem 3 Decision Tree
In this question, you will implement a simple decision tree algorithm. For simplicity, only discrete features
are considered here.
3.1 Conditional Entropy Recall that the quality of a split can be measured by the conditional entropy.
Please read decision tree.py and complete the function “conditional entropy” (where we use base 2
for logarithm).
3.2 Tree construction The building of a decision tree involves growing and pruning. For simplicity, in
this programming set you are only asked to grow a tree.
As discussed a tree can be constructed by recursively splitting the nodes if needed, as summarized in
Algorithm 3 (whose interface is slightly different from the pseudocode discussed in the class). Please read
decision tree.py and complete the function “split”.
Algorithm 3 The recursive procedure of decision tree algorithm
1: function TREENODE.SPLIT(self)
2: find the best attribute to split the node
3: split the node into child nodes
4: for child node in child nodes do
5: if child node.splittable then . See Slide 27 of Lec 7 for the three “non-splittable” cases
6: child node.split()
7: end if
8: end for
9: end function
3.3 Final submission Please run decision tree.sh, which will generate decision tree.json. Add,
commit, and push decision tree.py, decision tree.json before the due date. | {"url":"https://jarviscodinghub.com/assignment/csci-567-homework-3-programming-assignments-solution/","timestamp":"2024-11-03T19:25:19Z","content_type":"text/html","content_length":"118630","record_id":"<urn:uuid:41951eab-6879-464b-8e1c-91ae33da963b>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00066.warc.gz"} |
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Where to find C++ programming experts for assistance with algorithms for energy consumption optimization tasks? As someone more experienced in IT career procedures and computer programming, I’ve
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Class NondominatedPopulation
All Implemented Interfaces:
Iterable<Solution>, Stateful, Displayable, Formattable<Solution>
Direct Known Subclasses:
AdaptiveGridArchive, EpsilonBoxDominanceArchive, FitnessBasedArchive
public class NondominatedPopulation extends Population
A population that maintains the property of pair-wise non-dominance between all solutions. When the
method is invoked with a new solution, all solutions currently in the population that are dominated by the new solution are removed. If the new solution is dominated by any member of the population,
the new solution is not added.
Avoid modifying solutions contained in non-dominated populations. Since the dominance checks are only performed when adding solutions, modifying an existing solution can violating this contract.
• Nested Class Summary
Modifier and Type
static enum
Specifies how duplicate solutions are handled.
• Field Summary
Modifier and Type
The dominance comparator used by this non-dominated population.
Specifies how duplicate solutions are handled.
• Constructor Summary
Constructs an empty non-dominated population using the Pareto dominance relation.
Constructs a non-dominated population using the Pareto dominance relation and initialized with the specified solutions.
Constructs an empty non-dominated population using the specified dominance relation.
Constructs a non-dominated population using the specified dominance comparator and initialized with the specified solutions.
Constructs an empty non-dominated population using the specified dominance relation.
Constructs an empty non-dominated population using the Pareto dominance relation.
• Method Summary
Modifier and Type
If newSolution is dominates any solution or is non-dominated with all solutions in this population, the dominated solutions are removed and newSolution is added to this population.
Returns a copy of this population.
protected boolean
Adds the specified solution to the population, bypassing the non-domination check.
Returns the dominance comparator used by this non-dominated population.
protected boolean
Returns true if the two solutions are duplicates and one should be ignored based on the duplicate mode.
Loads a reference set file, which contains the objective values for a set of non-dominated solutions.
Loads a reference set, which contains the objective values for a set of non-dominated solutions.
Replace is not supported with non-dominated populations.
Methods inherited from class org.moeaframework.core.Population
addAll, addAll, asList, asTabularData, clear, contains, containsAll, containsAll, filter, get, getLowerBounds, getUpperBounds, indexOf, isEmpty, iterator, loadBinary, loadBinary, loadObjectives,
loadObjectives, loadState, remove, remove, removeAll, removeAll, removeAll, saveBinary, saveBinary, saveObjectives, saveObjectives, saveState, size, sort, truncate
Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
• Method Details
□ add
public boolean add(Solution newSolution)
If newSolution is dominates any solution or is non-dominated with all solutions in this population, the dominated solutions are removed and newSolution is added to this population. Otherwise,
newSolution is dominated and is not added to this population.
add in class Population
newSolution - the solution to be added
true if the population was modified as a result of this method; false otherwise.
□ replace
public void replace(int index, Solution newSolution)
Replace is not supported with non-dominated populations. Calling this method will throw an exception.
replace in class Population
index - the index to replace
newSolution - the new solution
UnsupportedOperationException - if this method is called
□ forceAddWithoutCheck
protected boolean forceAddWithoutCheck(Solution newSolution)
Adds the specified solution to the population, bypassing the non-domination check. This method should only be used when a non-domination check has been performed elsewhere, such as in a
This method should only be used internally, and should never be made public by any subclasses.
newSolution - the solution to be added
true if the population was modified as a result of this operation
□ isDuplicate
if the two solutions are duplicates and one should be ignored based on the duplicate mode. This default implementation depends on the
method of the
class to check for equality of the decision variables.
s1 - the first solution
s2 - the second solution
true if the solutions are duplicates; false otherwise
□ getComparator
Returns the dominance comparator used by this non-dominated population.
the dominance comparator used by this non-dominated population
□ copy
Returns a copy of this population. This can be thought of as a "deep copy", which creates a copy of both the population itself and copies of the individual solutions in the population.
Consequently, the returned copy is completely independent, such that any modifications to the contents or order will not impact the original.
Since creating such a "deep copy" can be expensive, prefer using the constructor Population(Iterable) or Population.addAll(Iterable) whenever possible. These alternatives are useful when
filtering or reordering the solutions, but the solutions themselves are left unchanged.
copy in class Population
the copy of this population
□ loadReferenceSet
Loads a reference set, which contains the objective values for a set of non-dominated solutions. Any dominated solutions are discarded.
resource - the path of the file or resource on the classpath
the reference set, or null if the file or resource was not found
IOException - if an I/O error occurred
□ loadReferenceSet
Loads a reference set file, which contains the objective values for a set of non-dominated solutions. Any dominated solutions are discarded.
file - the file containing the reference set
the reference set
IOException - if an I/O error occurred or the file was not found | {"url":"https://moeaframework.org/javadoc/org/moeaframework/core/NondominatedPopulation.html","timestamp":"2024-11-11T03:40:13Z","content_type":"text/html","content_length":"43034","record_id":"<urn:uuid:55aab022-96aa-4725-8330-d01d0918e6b4>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00192.warc.gz"} |
Question 914:
No answer provided yet.We can create a confidence interval around a sample to test a hypothesis about whether a population mean is a specific value. In this question it appears the population mean is
missing, but I can assume from the data in might be 50.
To create the confidence interval we first compute the standard error of the mean (SE) by dividing the standard deviation by the square root of the sample size:
3/SQRT(64) = .375
Next we compute the margin of error by multiplying the SE times a t-critical value for a 95% level of confidence. Using a t-table or the Excel function =TINV(.05, 64-1) we get the critical value of
1.99. Multiplying this times the SE we get the margin of error of .749.
Now we add and subtract this margin of error to the sample mean and get a 95% confidence interval of 48.251 to 49.749. Because the confidence interval does not contain 50, we have good evidence that
the population mean is not 50.
To run the hypothesis test we can conduct a 1-sample t-test using the same data. The test statistic is the difference between the sample mean and the SE we calculated in the last step. The test
statistic is : -1/.375 = -2.67.
Looking up this value in a t-table or using the Excel function =TDIST(2.67,63,2) we get a p-value of .01. Because the p-value is less than out alpha we reject the Null Hypothesis that the population
mean is 50. We come to the same conclusion as we did with the confidence interval--it is unlikely the population mean is 50.
Not what you were looking for or need help? | {"url":"https://www.usablestats.com/askstats/question/914/","timestamp":"2024-11-05T03:31:28Z","content_type":"application/xhtml+xml","content_length":"7108","record_id":"<urn:uuid:82251c24-1167-4928-971d-4c24bc4becc9>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00275.warc.gz"} |
Finding the Derivative of a Function Defined by an Integral Where Its Limits Contain a Variable
Question Video: Finding the Derivative of a Function Defined by an Integral Where Its Limits Contain a Variable Mathematics • Higher Education
Find the derivative of the function π (π ₯) = β «_(3π ₯)^(4π ₯) (π ’Β² β 3)/(π ’Β² + 5) dπ ’.
Video Transcript
Find the derivative of the function π of π ₯ is equal to the definite integral from three π ₯ to four π ₯ of π ’ squared minus three all divided by π ’ squared plus five with respect to π ’.
Weβ re given a function π of π ₯. And we can see this is a definite integral where both of the limits of integration are functions in π ₯. And weβ re asked to find the derivative of this
function. Since weβ re trying to differentiate a function where the limits of integration are functions in π ₯, weβ ll try and do this by using the fundamental theorem of calculus.
So weβ ll start by recalling the following part of the fundamental theorem of calculus. If lowercase π is a continuous function on a closed interval from π to π and we have capital π Ή of
π ₯ is the definite integral from π to π ₯ of lowercase π of π ’ with respect to π ’, then capital π Ή prime of π ₯ will be equal to lowercase π of π ₯ for all values of π ₯ in the
open interval from π to π . In other words, this part of the fundamental theorem of calculus gives us a way of differentiating a function defined by an integral.
However, the function weβ re asked to differentiate in the question, π of π ₯, is very different to our definition of capital π Ή of π ₯. First, the lower limit of integration is supposed to
be a constant. However, weβ re given this as a function in π ₯. Thereβ s a similar story for our upper limit of integration. Itβ s supposed to be π ₯, but weβ re given a function in π ₯.
However, we can get around both of these problems by using what we know about definite integrals and the chain rule.
The first thing weβ re going to need to recall is one of our rules for definite integrals. To get around the problem of not having a constant limit of integration, weβ re going to want to split our
definite integral into two definite integrals. We know the following rule for definite integrals. The integral from π to π of π of π ’ with respect to π ’ is equal to the integral from π
to π of π of π ’ with respect to π ’ plus the integral from π to π of π of π ’ with respect to π ’. But thereβ s some important information about this rule which we need to
The most important of these is this will only be true if our function π is integrable on all of these domains of integration. In particular, our function π must be integrable on the closed
interval from π to π and the closed interval from π to π . We want to use this to rewrite our function π of π ₯. So by using the upper limit of integration as four π ₯, the lower
limit of integration as three π ₯, and our function lowercase π of π ’ as π ’ squared minus three divided by π ’ squared plus five, we would get the integral from π to four π ₯ of π ’
squared minus three divided by π ’ squared plus five with respect to π ’ plus the integral from three π ₯ to π of π ’ squared minus three all divided by π ’ squared plus five with respect to
π ’.
But remember, we also need our function to be integrable on both of these domains of integration. In particular, we need to choose a value of the constant π . In this case, we could actually choose
any value of the constant π . To explain why, letβ s take a look at our integrand. Our integrand is the quotient of two polynomials. This means itβ s a rational function. And we know rational
functions will be continuous across their entire domain. And if a function is continuous on an interval, then itβ s also integrable on that interval. So all we need to do is check the domain of our
function lowercase π of π ’. Itβ s a rational function, so it will be defined for all values of π ’ except where its denominator is equal to zero.
And if we solve the denominator π ’ squared plus five is equal to zero, weβ ll see that we get no real solutions. So our integrand is defined for all real values of π ’. Meaning itβ s continuous
for all real values of π ’, meaning itβ s integrable on any interval. So in this case, we can actually pick any value of π . However, this wonβ t always be the case. In this case, weβ ll choose
π is equal to zero. Weβ ve now split this into two definite integrals, each of which is closer to being able to use the fundamental theorem of calculus.
Thereβ s a little bit more manipulation we need to do before we can use the fundamental theorem of calculus. In our second integral, we want to swap the order of our limits of integration. Remember,
we can do this by multiplying the entire integral by negative one. Doing this, weβ ve rewritten our second integral as negative one times the definite integral from zero to three π ₯ of π ’
squared minus three all divided by π ’ squared plus five with respect to π ’.
And now, weβ re almost ready to use the fundamental theorem of calculus to evaluate these definite integrals. The only problem is our upper limit of integration is a function in π ₯ instead of π
₯. And there are two ways of getting around this problem. First, we could use the substitution π £ of π ₯ is equal to four π ₯. Then we can differentiate our first integral by using the chain
rule. We could do the same with our second integral, setting π £ of π ₯ to be three π ₯ and then evaluating its derivative by using the chain rule. And this would work. However, we can also do
this in the general case with the fundamental theorem of calculus.
We have if π £ of π ₯ is a differentiable function, then by using the chain rule, we get if lowercase π is a continuous function on the closed interval from π to π and capital π Ή of π ₯
is the definite integral from π to π £ of π ₯ of lowercase π of π ’ with respect to π ’, then capital π Ή prime of π ₯ will be equal to lowercase π evaluated at π £ of π ₯ times dπ
£ by dπ ₯ as long as π £ of π ₯ is in the open interval from π to π . So we were able to rewrite our version of the fundamental theorem of calculus so long as π £ of π ₯ is a differentiable
function just by using the chain rule.
And now, we can see both of our definite integrals are in this form. Weβ ll set π £ one of π ₯ to be four π ₯ and π £ two of π ₯ to be three π ₯. Usually, we would set our value of π equal
to zero in both cases and then check the intervals on which our integrand is continuous. This is because to use the fundamental theorem of calculus, we do need to check where our integrand is
continuous. However, weβ ve already shown that our integrand is continuous for all real values. Therefore, it will be continuous on any closed interval.
Weβ re now ready to find an expression for π prime of π ₯. Weβ ll differentiate each of our integrals separately. To differentiate our first integral by using the fundamental theorem of
calculus, we do need to check that π £ one of π ₯ is differentiable. And itβ s a linear function, so we know it is differentiable. This means we can use this, and we get π evaluated at π £ one
of π ₯ times dπ £ one by dπ ₯. And we get the same story in our second integral. This time, π £ two of π ₯ will be three π ₯, which is also a linear function, so itβ s differentiable. This
means we need to subtract π evaluated at π £ two of π ₯ times dπ £ two by dπ ₯.
So now, we found an expression for π prime of π ₯. And we can actually evaluate all of these expressions. Letβ s start by substituting in our expressions for π £ one of π ₯ and π £ two of π
₯. This gives us π prime of π ₯ is equal to lowercase π evaluated at four π ₯ times the derivative of four π ₯ with respect to π ₯ minus lowercase π evaluated at three π ₯ multiplied by
the derivative of three π ₯ with respect to π ₯. And remember, from the fundamental theorem of calculus, lowercase π will be our integrand. So weβ re now ready to find an expression for π
prime of π ₯.
First, to find π evaluated at four π ₯, we substitute π ’ is equal to four π ₯ into our integrand, giving us four π ₯ all squared minus three all divided by four π ₯ all squared plus five.
Next, we need to multiply this by the derivative of four π ₯ with respect to π ₯. Of course, this is a linear function, so we know the derivative of this with respect to π ₯ will be the
coefficient of π ₯, which is four. And we can do exactly the same to find an expression for our second term.
To find π evaluated at three π ₯, we substitute π ’ is equal to three π ₯ into our integrand, giving us three π ₯ all squared minus three all divided by three π ₯ all squared plus five. And
of course, we need to multiply this by the derivative of three π ₯ with respect to π ₯, which we know is three. Finally, weβ ll simplify this expression by evaluating all of our exponents and
multiplying through by our coefficients. And by doing this, we get our final answer. π prime of π ₯ is equal to four times 16π ₯ squared minus three all divided by 16π ₯ squared plus five minus
three times nine π ₯ squared minus three all divided by nine π ₯ squared plus five. | {"url":"https://www.nagwa.com/en/videos/939136010716/","timestamp":"2024-11-05T02:52:44Z","content_type":"text/html","content_length":"262328","record_id":"<urn:uuid:90bcbe91-a15c-4301-bc12-39d401a4d165>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00611.warc.gz"} |
Future Value Calculator - Symbolab
About Future Value Calculator
• A Future Value Calculator is a financial tool that helps individuals, businesses, and investors determine the value of an investment or deposit at a specific future point in time. The basic
underlying concept behind this calculator is the time value of money principle, which states that a dollar today is worth more than a dollar tomorrow due to its potential earning capacity.
• The Future Value Calculator takes into account the following inputs: initial amount (also known as principal), a given interest rate, the number of compounding periods per year, and the length of
time the investment or deposit will be held for. By estimating how much these inputs will contribute to the growth of the initial investment over time, the calculator determines the future value
of a lump sum deposit or series of cash flows (like regular investments or savings contributions) under the specified conditions.
• There are two primary methods for calculating future value: future value of a lump sum and future value of an annuity. Future value of a lump sum is simply the calculation of the value of a
single deposit made today after a certain number of years at a given interest rate compounded periodically. The formula for the future value of a lump sum is:
• FV = PV * (1 + i/n)^(nt)
• Where FV is the future value, PV is the present value (initial deposit), i is the annual interest rate, n is the number of compounding periods per year, and t is the number of years until
• The future value of an annuity is used to calculate the value of a series of cash flows – usually monthly or yearly – over a specified period of time at a given interest rate compounded
periodically. This method is particularly useful for calculating the future value of regular savings or investments, such as retirement contributions. The formula for the future value of an
annuity is:
• FV_A = P * (((1 + i/n)^(nt) - 1) / (i/n))
• Where FV_A is the future value of the annuity, P is the periodic payment (investment or savings contribution), and the other variables remain the same as in the lump sum formula.
• There are various applications for the Future Value Calculator, and it plays a significant role in personal finance, investment analysis, retirement planning, and business valuation. For
individuals, the calculator can estimate how much their savings will grow over time and help create financial goals and budgets. For businesses, the calculator can help evaluate investment
opportunities, such as project returns, and compare two or more investment options to decide which will bring greater profits in the future. The tool is also used by investors and financial
planners to compare different investment options, such as stocks, bonds, or mutual funds, to determine which investments have the potential to yield the highest returns given a specific time
horizon, interest rates, and risk tolerance.
• The Future Value Calculator, however, is not without its limitations. It relies heavily on the accuracy of the input data and assumes that interest rates and other variables will remain constant
over the investment period. In reality, market conditions always change, and it's essential to keep in mind that the calculator provides an estimate that should be used in conjunction with other
analysis tools and professional advice.
• In conclusion, the Future Value Calculator is a valuable financial tool that assists individuals and businesses in making informed decisions about their savings, investment strategies, and
long-term financial planning. By considering the time value of money principle and accounting for the various inputs detailed earlier, users can gain insight into their potential investment
returns and make educated decisions to achieve their financial goals. However, users should always treat future value calculations as estimates and consider real-world market fluctuations and
changes in interest rates when applying these calculations in decision-making. | {"url":"http://en.symbolab.com/calculator/finance/future-value","timestamp":"2024-11-10T04:27:52Z","content_type":"text/html","content_length":"180892","record_id":"<urn:uuid:6308ef78-5700-47f0-bfdf-ca9237828020>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00158.warc.gz"} |
5 Best Ways to Find an Integer Divisor for All but One Element in an Array in Python
π ‘ Problem Formulation: We want to identify an integer x such that x divides every element of an array except for exactly one element. For instance, given the array [15, 3, 6], the integer 3 is a
divisor for all elements except for the number 6, making it our target output.
Method 1: Brute Force
This method involves checking each element in the array to find a common divisor. We iteratively divide each element by a potential divisor and keep track of the number of elements divisible by it,
excluding the element that is the exception. This is a straightforward approach and works well for smaller arrays.
Here’s an example:
def find_divisor(arr):
for i in range(2, min(arr) + 1):
counter = sum(1 for num in arr if num % i == 0)
if counter == len(arr) - 1:
return i
return None
arr = [12, 18, 24, 35]
This snippet defines a function find_divisor that takes an array arr as input and returns the divisor, if one exists. It uses a brute force approach to iterate through potential divisors and counts
the number of elements divisible by each. Once it finds a divisor for all but one element, it returns that divisor.
Method 2: Using Greatest Common Divisor (GCD)
In this method, we leverage the mathematical property of GCD. We compute the GCD of all elements but one, and test if the result is a divisor for all but one. It is an efficient method for finding
divisors when working with larger numbers.
Here’s an example:
from math import gcd
from functools import reduce
def find_divisor_gcd(arr):
for i in range(len(arr)):
temp = arr[:i] + arr[i+1:]
divisor = reduce(gcd, temp)
if all(num % divisor == 0 for j, num in enumerate(arr) if j != i):
return divisor
return None
arr = [22, 44, 66, 13]
The function find_divisor_gcd uses Python’s built-in GCD function and reduce from the functools module to compute the GCD of all elements except each one in turn. It then checks if the computed GCD
is a common divisor for all elements except the excluded one. If so, it returns this divisor.
Method 3: Sorting and Checking Extremes
By sorting the array, we compare the GCD of all elements excluding the first and last element. Since the extremal values could be the potential outliers, this is an optimized way to reduce the number
of iterations.
Here’s an example:
from math import gcd
def find_divisor_sort(arr):
sorted_arr = sorted(arr)
gcd_all_except_first = reduce(gcd, sorted_arr[1:])
gcd_all_except_last = reduce(gcd, sorted_arr[:-1])
if all(num % gcd_all_except_first == 0 for num in sorted_arr if num != sorted_arr[0]):
return gcd_all_except_first
elif all(num % gcd_all_except_last == 0 for num in sorted_arr if num != sorted_arr[-1]):
return gcd_all_except_last
return None
arr = [32, 96, 128, 17]
In this snippet, the function find_divisor_sort first sorts the array and calculates the GCD of all elements excluding the first and last. It then checks if these GCDs divide all but the excluded
element. If either does, it returns the corresponding GCD as the divisor.
Method 4: Optimized GCD on Pairs
This technique uses the insight that if there’s a number that’s not divisible by the potential divisor, then at least one pair of numbers including this number will have a GCD equal to 1. We sample
pairs to quickly test for failure cases before attempting the full set GCD.
Here’s an example:
from math import gcd
def find_divisor_pairs(arr):
for i in range(len(arr)):
for j in range(i + 1, len(arr)):
if gcd(arr[i], arr[j]) == 1: # Quick check to rule out if possible
divisor = arr[j] if i == 0 else arr[0]
if all(num % divisor == 0 for k, num in enumerate(arr) if k != i):
return divisor
return None
arr = [7, 14, 28, 3]
The function find_divisor_pairs iterates pairs of numbers in the array to find a pair with GCD of 1. If found, it tests whether one of the pairβ s elements is a common divisor for the rest of the
array. This optimization helps to avoid unnecessary computation.
Bonus One-Liner Method 5: Leveraging List Comprehension and GCD
This is a concise, Pythonic, one-liner approach combining list comprehensions and GCD computations to find the divisor using similar principles from the above methods.
Here’s an example:
from math import gcd
from functools import reduce
arr = [15, 45, 90, 7]
divisor = next((reduce(gcd, arr[:i]+arr[i+1:]) for i in range(len(arr)) if all(num % reduce(gcd, arr[:i]+arr[i+1:]) == 0 for num in arr)), None)
This one-liner uses a generator expression within the next function to iterate over each elementβ s exclusion and computes the GCD of the remaining elements. It then checks if this GCD is a divisor
for all but the excluded element. The next function returns the first result that satisfies the condition.
• Method 1: Brute Force. Simple and straightforward. Inefficient for large arrays with large numbers.
• Method 2: GCD-based. Efficient and mathematically robust. Slightly more complex due to multiple GCD computations.
• Method 3: Sorting and Checking Extremes. Fast for sorted input and provides quick checks. Not as good if the array is not sorted.
• Method 4: Optimized GCD on Pairs. Quick elimination of non-divisors based on pair-wise GCD. May not always bring performance benefit depending on array structure.
• Method 5: One-Liner. Elegant and concise. Can be less readable due to list comprehension complexity. | {"url":"https://blog.finxter.com/5-best-ways-to-find-an-integer-divisor-for-all-but-one-element-in-an-array-in-python/","timestamp":"2024-11-06T10:34:52Z","content_type":"text/html","content_length":"73215","record_id":"<urn:uuid:77e5b467-1be8-490c-8e64-9effa1f6b227>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00403.warc.gz"} |
The tiling can be replaced by circular edges, centered on the hexagons as an overlapping circles grid. In quilting it is called Jacks chain.^[2]
There is one related 2-uniform tiling, having hexagons dissected into six triangles.^[3]^[4] The rhombitrihexagonal tiling is also related to the truncated trihexagonal tiling by replacing some of
the hexagons and surrounding squares and triangles with dodecagons:
1-uniform Dissection 2-uniform dissections
3.4.6.4 3.3.4.3.4 & 3^6 to CH
Dual Tilings
3.4.6.4 4.6.12 to 3
Circle packing
The rhombitrihexagonal tiling can be used as a circle packing, placing equal diameter circles at the center of every point. Every circle is in contact with four other circles in the packing (kissing
number).^[5] The translational lattice domain (red rhombus) contains six distinct circles.
Wythoff construction
There are eight uniform tilings that can be based from the regular hexagonal tiling (or the dual triangular tiling).
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are eight forms, seven topologically distinct. (The truncated
triangular tiling is topologically identical to the hexagonal tiling.)
Uniform hexagonal/triangular tilings
Symmetry: [6,3], (*632) [6,3]^+ [6,3^+]
(632) (3*3)
{6,3} t{6,3} r{6,3} t{3,6} {3,6} rr{6,3} tr{6,3} sr{6,3} s{3,6}
6^3 3.12^2 (3.6)^2 6.6.6 3^6 3.4.6.4 4.6.12 3.3.3.3.6 3.3.3.3.3.3
Uniform duals
V6^3 V3.12^2 V(3.6)^2 V6^3 V3^6 V3.4.6.4 V.4.6.12 V3^4.6 V3^6
Symmetry mutations
This tiling is topologically related as a part of sequence of cantellated polyhedra with vertex figure (3.4.n.4), and continues as tilings of the hyperbolic plane. These vertex-transitive figures
have (*n32) reflectional symmetry.
Deltoidal trihexagonal tiling
A 2023 discovered aperiodic monotile, solving the Einstein problem, is composed by a collection of 8 kites from the deltoidal trihexagonal tiling
The deltoidal trihexagonal tiling is a dual of the semiregular tiling known as the rhombitrihexagonal tiling. Conway calls it a tetrille.^[1] The edges of this tiling can be formed by the
intersection overlay of the regular triangular tiling and a hexagonal tiling. Each kite face of this tiling has angles 120°, 90°, 60° and 90°. It is one of only eight tilings of the plane in which
every edge lies on a line of symmetry of the tiling.^[6]
The deltoidal trihexagonal tiling is a dual of the semiregular tiling rhombitrihexagonal tiling.^[7] Its faces are deltoids or kites.
It is one of seven dual uniform tilings in hexagonal symmetry, including the regular duals.
This tiling has face transitive variations, that can distort the kites into bilateral trapezoids or more general quadrilaterals. Ignoring the face colors below, the fully symmetry is p6m, and the
lower symmetry is p31m with three mirrors meeting at a point, and threefold rotation points.^[8]
Isohedral variations
Symmetry p6m, [6,3], (*632) p31m, [6,3^+], (3*3)
Faces Kite Half regular hexagon Quadrilaterals
This tiling is related to the trihexagonal tiling by dividing the triangles and hexagons into central triangles and merging neighboring triangles into kites.
The deltoidal trihexagonal tiling is a part of a set of uniform dual tilings, corresponding to the dual of the rhombitrihexagonal tiling.
Symmetry mutations
This tiling is topologically related as a part of sequence of tilings with face configurations V3.4.n.4, and continues as tilings of the hyperbolic plane. These face-transitive figures have (*n32)
reflectional symmetry.
Other deltoidal (kite) tiling
Other deltoidal tilings are possible.
Point symmetry allows the plane to be filled by growing kites, with the topology as a square tiling, V4.4.4.4, and can be created by crossing string of a dream catcher. Below is an example with
dihedral hexagonal symmetry.
Another face transitive tiling with kite faces, also a topological variation of a square tiling and with face configuration V4.4.4.4. It is also vertex transitive, with every vertex containing all
orientations of the kite face. | {"url":"https://www.knowpia.com/knowpedia/Rhombitrihexagonal_tiling","timestamp":"2024-11-05T13:25:01Z","content_type":"text/html","content_length":"213072","record_id":"<urn:uuid:08c94b8a-424f-4ac0-9c4a-4024ebd0ffcd>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00310.warc.gz"} |
Existence of large $\varepsilon$-Kronecker and $FZI_0(U)$ sets in discrete abelian groups
Colloquium Mathematicum 127 (2012), 1-15 MSC: Primary 42A55, 42A63, 43A25, 43A46; Secondary 43A05, 43A25. DOI: 10.4064/cm127-1-1
Let $G$ be a compact abelian group with dual group $\Gamma$ and let $\varepsilon>0$. A set ${\bf E}\subset\Gamma$ is a “weak $\varepsilon$-Kronecker set” if for every $\varphi:{\bf E}\to\mathbb T$
there exists $x$ in the dual of $\Gamma$ such that $|\varphi(\gamma)- \gamma(x)| \le \varepsilon$ for all $\gamma\in {\bf E}$. When $\varepsilon<\sqrt2$, every bounded function on ${\bf E}$ is known
to be the restriction of a Fourier–Stieltjes transform of a discrete measure. (Such sets are called $I_0$.)
We show that for every infinite set ${\bf E}$ there exists a weak 1-Kronecker subset ${\bf F}$, of the same cardinality as ${\bf E}$, provided there are not “too many” elements of order 2 in the
subgroup generated by ${\bf E}$. When there are “too many” elements of order 2, we show that there exists a subset ${\bf F}$, of the same cardinality as ${\bf E}$, on which every $\{-1,1\}$-valued
function can be interpolated exactly. Such sets are also $I_0$. In both cases, the set ${\bf F}$ also has the property that the only continuous character at which ${\bf F}\cdot{\bf F}^{-1}$ can
cluster in the Bohr topology is ${\bf1}$. This improves upon previous results concerning the existence of $I_0$ subsets of a given ${\bf E}$. | {"url":"https://www.impan.pl/en/publishing-house/journals-and-series/colloquium-mathematicum/all/127/1/87728/existence-of-large-varepsilon-kronecker-and-fzi-0-u-sets-in-discrete-abelian-groups","timestamp":"2024-11-01T19:42:43Z","content_type":"text/html","content_length":"46097","record_id":"<urn:uuid:b4060571-a914-40a4-b1da-8d613e38c178>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00658.warc.gz"} |
Printable Fill In Multiplication Chart | Multiplication Chart Printable
Printable Fill In Multiplication Chart
Free Printable Blank Multiplication Table Chart Template PDF
Printable Fill In Multiplication Chart
Printable Fill In Multiplication Chart – A Multiplication Chart is a practical tool for kids to find out exactly how to increase, divide, and also locate the tiniest number. There are numerous uses
for a Multiplication Chart. These helpful tools aid youngsters understand the procedure behind multiplication by using colored paths and also filling out the missing out on items. These charts are
cost-free to publish and also download.
What is Multiplication Chart Printable?
A multiplication chart can be used to assist kids learn their multiplication facts. Multiplication charts can be found in lots of types, from full web page times tables to solitary web page ones.
While private tables work for providing pieces of information, a complete page chart makes it simpler to evaluate facts that have already been mastered.
The multiplication chart will generally include a top row and a left column. The leading row will certainly have a listing of products. Choose the very first number from the left column and the 2nd
number from the leading row when you desire to find the product of two numbers. Relocate them along the row or down the column up until you get to the square where the two numbers meet when you have
these numbers. You will then have your item.
Multiplication charts are helpful knowing tools for both grownups as well as children. Printable Fill In Multiplication Chart are readily available on the Internet and can be printed out and also
laminated flooring for longevity.
Why Do We Use a Multiplication Chart?
A multiplication chart is a layout that shows how to multiply 2 numbers. It normally consists of a left column and also a top row. Each row has a number standing for the product of both numbers. You
choose the very first number in the left column, relocate down the column, and then pick the second number from the top row. The item will certainly be the square where the numbers fulfill.
Multiplication charts are useful for several reasons, including helping children find out just how to split and also simplify fractions. They can also assist youngsters find out exactly how to choose
an effective common measure. Because they offer as a constant reminder of the trainee’s progress, multiplication charts can additionally be handy as desk resources. These tools assist us create
independent learners that comprehend the standard principles of multiplication.
Multiplication charts are likewise beneficial for aiding trainees memorize their times tables. They help them find out the numbers by reducing the number of actions needed to finish each procedure.
One method for remembering these tables is to concentrate on a single row or column at once, and afterwards move onto the following one. Ultimately, the whole chart will certainly be committed to
memory. Similar to any type of skill, memorizing multiplication tables takes some time and method.
Printable Fill In Multiplication Chart
Fill In Multiplication Table Printable Times Tables Worksheets
Fill In Multiplication Table Times Tables Worksheets
Fill In Multiplication Table Times Tables Worksheets
Printable Fill In Multiplication Chart
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Related For Printable Fill In Multiplication Chart | {"url":"https://multiplicationchart-printable.com/printable-fill-in-multiplication-chart/","timestamp":"2024-11-07T03:55:48Z","content_type":"text/html","content_length":"43410","record_id":"<urn:uuid:d144ed52-1de2-4d5c-b3c6-bd163dbdb0dd>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00750.warc.gz"} |
Simon Rubinstein-Salzedo
I am a mathematician, having completed my PhD in 2012 at Stanford University. Currently, I am the director of Euler Circle, a mathematics institute for advanced high-school students in Palo Alto,
California. My research interests are broad, including number theory, combinatorics, probability, and game theory. I love learning new things and working on new research projects. If you would like
to do some mathematics with me, I can be easily tempted!
PhD Thesis
Controlling Ramification in Number Fields
Advisor: Akshay Venkatesh
Cryptography. In Springer Undergraduate Mathematics Series 2018.
Papers and preprints
(Links are to arXiv or other free versions.)
Expository writings
Here are a bunch of expository notes I have written over the years.
Educational programs
Here are some of the educational programs I have been involved with recently:
Other stuff
As you probably guessed from the picture, I'm also a cellist. Other hobbies of mine include chess and calligraphy. | {"url":"http://simonrs.com/","timestamp":"2024-11-07T02:52:37Z","content_type":"text/html","content_length":"9971","record_id":"<urn:uuid:1c0cdf1f-f2a3-4401-b2e3-822662e7e3e1>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00618.warc.gz"} |
Error in X : incorrect number of subscripts on matrix in R (2 Examples)Error in X : incorrect number of subscripts on matrix in R (2 Examples)
Error in X : incorrect number of subscripts on matrix in R (2 Examples)
In this article, I’ll show how to debug the error “incorrect number of subscripts on matrix” in the R programming language.
Creation of Example Data
my_vec <- LETTERS[1:10] # Example vector in R
my_vec # Display example vector in RStudio
# [1] "A" "B" "C" "D" "E" "F" "G" "H" "I" "J"
my_vec <- LETTERS[1:10] # Example vector in R my_vec # Display example vector in RStudio # [1] "A" "B" "C" "D" "E" "F" "G" "H" "I" "J"
Example 1: Replicating the Error Message – incorrect number of subscripts on matrix
my_vec[6, 7] <- "XXX" # Using two dimensions for one-dimensional vector
# Error in x[1, 3] <- 7 : incorrect number of subscripts on matrix
my_vec[6, 7] <- "XXX" # Using two dimensions for one-dimensional vector # Error in x[1, 3] <- 7 : incorrect number of subscripts on matrix
Example 2: Solving the Error Message – incorrect number of subscripts on matrix
my_vec[6] <- "XXX" # Correct number of dimensions
my_vec # Display vector with replaced value
# [1] "A" "B" "C" "D" "E" "XXX" "G" "H" "I" "J"
my_vec[6] <- "XXX" # Correct number of dimensions my_vec # Display vector with replaced value # [1] "A" "B" "C" "D" "E" "XXX" "G" "H" "I" "J" | {"url":"https://data-hacks.com/error-x-incorrect-number-subscripts-on-matrix-r","timestamp":"2024-11-08T21:10:36Z","content_type":"text/html","content_length":"216077","record_id":"<urn:uuid:4f3b7655-b5eb-4231-ae01-2037fafce969>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00100.warc.gz"} |
Efficient and innovative energy transport and conversion (transformation) require progressively more atomic tailoring of principal energy carriers (phonon, electron, fluid particle, and photon). This
involves atomic-level material synthesis for the desired atomic-level energy transport and conversion kinetics. Thermal energy (kinetic energy of atomic-level motions in solid and fluid phases) is
central to energy conversion, since when improperly designed the inefficiencies lead to undesirable heat generation. Atomic-level examinations show that the thermal energy has quantum or resonance
features which can be used in the energy conversion before it is thermalized by scattering (i.e., becomes equilibrium carrier energy distribution which is less useful). Figure below shows the sources
of phonon emission (beginning of heat generation in solids), and resonant phonon harvesting.
In our Heat Transfer Physics research, we seek this atomic-structure design (including nanostructures and quantum confinements) and optimization of the carrier energy kinetics (transition rates).
This is rendered in the figure below. Our current research projects include laser cooling of solids, phonon recycling in transistors, phonovoltaic, α-voltaic, and phonocatalysis. We work on high
figure-of-merit thermoelectric materials around the role of phonons. We are also working on the molecular dynamics explanation of the thermoreceptor gating.
Our approach is theoretical with combination of quantum mechanics (first-principles calculations), molecular dynamics, Boltzmann transport (mesoscale), and macroscale treatments. We also do
collaborative experimental work (X-ray, optical, Raman, neutron and electron scattering, and thin-film fabrication and characterization). | {"url":"https://htp.engin.umich.edu/objective/","timestamp":"2024-11-05T13:11:11Z","content_type":"text/html","content_length":"60226","record_id":"<urn:uuid:3458a43f-1d23-4ebb-993d-c6ef7f8d4706>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00324.warc.gz"} |
8.4: Hypothesis Test Examples for Proportions
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• In a hypothesis test problem, you may see words such as "the level of significance is 1%." The "1%" is the preconceived or preset \(\alpha\).
• The statistician setting up the hypothesis test selects the value of α to use before collecting the sample data.
• If no level of significance is given, a common standard to use is \(\alpha = 0.05\).
• When you calculate the \(p\)-value and draw the picture, the \(p\)-value is the area in the left tail, the right tail, or split evenly between the two tails. For this reason, we call the
hypothesis test left, right, or two tailed.
• The alternative hypothesis, \(H_{a}\), tells you if the test is left, right, or two-tailed. It is the key to conducting the appropriate test.
• \(H_{a}\) never has a symbol that contains an equal sign.
• Thinking about the meaning of the \(p\)-value: A data analyst (and anyone else) should have more confidence that he made the correct decision to reject the null hypothesis with a smaller \(p\)
-value (for example, 0.001 as opposed to 0.04) even if using the 0.05 level for alpha. Similarly, for a large p-value such as 0.4, as opposed to a \(p\)-value of 0.056 (\(\alpha = 0.05\) is less
than either number), a data analyst should have more confidence that she made the correct decision in not rejecting the null hypothesis. This makes the data analyst use judgment rather than
mindlessly applying rules.
Full Hypothesis Test Examples
Example \(\PageIndex{7}\)
Joon believes that 50% of first-time brides in the United States are younger than their grooms. She performs a hypothesis test to determine if the percentage is the same or different from 50%. Joon
samples 100 first-time brides and 53 reply that they are younger than their grooms. For the hypothesis test, she uses a 1% level of significance.
Set up the hypothesis test:
The 1% level of significance means that α = 0.01. This is a test of a single population proportion.
\(H_{0}: p = 0.50\)\(H_{a}: p \neq 0.50\)
The words "is the same or different from" tell you this is a two-tailed test.
Calculate the distribution needed:
Random variable: \(P′ =\) the percent of of first-time brides who are younger than their grooms.
Distribution for the test: The problem contains no mention of a mean. The information is given in terms of percentages. Use the distribution for P′, the estimated proportion.
\[P' - N\left(p, \sqrt{\frac{p-q}{n}}\right)\nonumber \]
\[P' - N\left(0.5, \sqrt{\frac{0.5-0.5}{100}}\right)\nonumber \]
where \(p = 0.50, q = 1−p = 0.50\), and \(n = 100\)
Calculate the p-value using the normal distribution for proportions:
\[p\text{-value} = P(p′ < 0.47 or p′ > 0.53) = 0.5485\nonumber \]
where \[x = 53, p' = \frac{x}{n} = \frac{53}{100} = 0.53\nonumber \].
Interpretation of the \(p\text{-value})\: If the null hypothesis is true, there is 0.5485 probability (54.85%) that the sample (estimated) proportion \(p'\) is 0.53 or more OR 0.47 or less (see the
graph in Figure).
Figure \(\PageIndex{12}\)
\(\mu = p = 0.50\) comes from \(H_{0}\), the null hypothesis.
\(p′ = 0.53\). Since the curve is symmetrical and the test is two-tailed, the \(p′\) for the left tail is equal to \(0.50 – 0.03 = 0.47\) where \(\mu = p = 0.50\). (0.03 is the difference between
0.53 and 0.50.)
Compare \(\alpha\) and the \(p\text{-value}\):
Since \(\alpha = 0.01\) and \(p\text{-value} = 0.5485\). \(\alpha < p\text{-value}\).
Make a decision: Since \(\alpha < p\text{-value}\), you cannot reject \(H_{0}\).
Conclusion: At the 1% level of significance, the sample data do not show sufficient evidence that the percentage of first-time brides who are younger than their grooms is different from 50%.
The \(p\text{-value}\) can easily be calculated.
Press STAT and arrow over to TESTS. Press 5:1-PropZTest. Enter .5 for \(p_{0}\), 53 for \(x\) and 100 for \(n\). Arrow down to Prop and arrow to not equals \(p_{0}\). Press ENTER. Arrow down to
Calculate and press ENTER. The calculator calculates the \(p\text{-value}\) (\(p = 0.5485\)) and the test statistic (\(z\)-score). Prop not equals .5 is the alternate hypothesis. Do this set of
instructions again except arrow to Draw (instead of Calculate). Press ENTER. A shaded graph appears with \(\(z\) = 0.6\) (test statistic) and \(p = 0.5485\) (\(p\text{-value}\)). Make sure when you
useDraw that no other equations are highlighted in \(Y =\) and the plots are turned off.
The Type I and Type II errors are as follows:
The Type I error is to conclude that the proportion of first-time brides who are younger than their grooms is different from 50% when, in fact, the proportion is actually 50%. (Reject the null
hypothesis when the null hypothesis is true).
The Type II error is there is not enough evidence to conclude that the proportion of first time brides who are younger than their grooms differs from 50% when, in fact, the proportion does differ
from 50%. (Do not reject the null hypothesis when the null hypothesis is false.)
Exercise \(\PageIndex{7}\)
A teacher believes that 85% of students in the class will want to go on a field trip to the local zoo. She performs a hypothesis test to determine if the percentage is the same or different from 85%.
The teacher samples 50 students and 39 reply that they would want to go to the zoo. For the hypothesis test, use a 1% level of significance.
First, determine what type of test this is, set up the hypothesis test, find the \(p\text{-value}\), sketch the graph, and state your conclusion.
Since the problem is about percentages, this is a test of single population proportions.
• \(H_{0} : p = 0.85\)
• \(H_{a}: p \neq 0.85\)
• \(p = 0.7554\)
Figure \(\PageIndex{13}\)
Because \(p > \alpha\), we fail to reject the null hypothesis. There is not sufficient evidence to suggest that the proportion of students that want to go to the zoo is not 85%.
Example \(\PageIndex{8}\)
Suppose a consumer group suspects that the proportion of households that have three cell phones is 30%. A cell phone company has reason to believe that the proportion is not 30%. Before they start a
big advertising campaign, they conduct a hypothesis test. Their marketing people survey 150 households with the result that 43 of the households have three cell phones.
Set up the Hypothesis Test:
\(H_{0}: p = 0.30, H_{a}: p \neq 0.30\)
Determine the distribution needed:
The random variable is \(P′ =\) proportion of households that have three cell phones.
The distribution for the hypothesis test is \(P' - N\left(0.30, \sqrt{\frac{(0.30 \cdot 0.70)}{150}}\right)\)
Exercise 9.6.8.2
a. The value that helps determine the \(p\text{-value}\) is \(p′\). Calculate \(p′\).
a. \(p' = \frac{x}{n}\) where \(x\) is the number of successes and \(n\) is the total number in the sample.
\(x = 43, n = 150\)
\(p′ = 43150\)
Exercise 9.6.8.3
b. What is a success for this problem?
b. A success is having three cell phones in a household.
Exercise 9.6.8.4
c. What is the level of significance?
c. The level of significance is the preset \(\alpha\). Since \(\alpha\) is not given, assume that \(\alpha = 0.05\).
Exercise 9.6.8.5
d. Draw the graph for this problem. Draw the horizontal axis. Label and shade appropriately.
Calculate the \(p\text{-value}\).
d. \(p\text{-value} = 0.7216\)
Exercise 9.6.8.6
e. Make a decision. _____________(Reject/Do not reject) \(H_{0}\) because____________.
e. Assuming that \(\alpha = 0.05, \alpha < p\text{-value}\). The decision is do not reject \(H_{0}\) because there is not sufficient evidence to conclude that the proportion of households that have
three cell phones is not 30%.
Exercise \(\PageIndex{8}\)
Marketers believe that 92% of adults in the United States own a cell phone. A cell phone manufacturer believes that number is actually lower. 200 American adults are surveyed, of which, 174 report
having cell phones. Use a 5% level of significance. State the null and alternative hypothesis, find the p-value, state your conclusion, and identify the Type I and Type II errors.
• \(H_{0}: p = 0.92\)
• \(H_{a}: p < 0.92\)
• \(p\text{-value} = 0.0046\)
Because \(p < 0.05\), we reject the null hypothesis. There is sufficient evidence to conclude that fewer than 92% of American adults own cell phones.
• Type I Error: To conclude that fewer than 92% of American adults own cell phones when, in fact, 92% of American adults do own cell phones (reject the null hypothesis when the null hypothesis is
• Type II Error: To conclude that 92% of American adults own cell phones when, in fact, fewer than 92% of American adults own cell phones (do not reject the null hypothesis when the null hypothesis
is false).
The next example is a poem written by a statistics student named Nicole Hart. The solution to the problem follows the poem. Notice that the hypothesis test is for a single population proportion. This
means that the null and alternate hypotheses use the parameter \(p\). The distribution for the test is normal. The estimated proportion \(p′\) is the proportion of fleas killed to the total fleas
found on Fido. This is sample information. The problem gives a preconceived \(\alpha = 0.01\), for comparison, and a 95% confidence interval computation. The poem is clever and humorous, so please
enjoy it!
Example \(\PageIndex{9}\)
My dog has so many fleas,
They do not come off with ease.
As for shampoo, I have tried many types
Even one called Bubble Hype,
Which only killed 25% of the fleas,
Unfortunately I was not pleased.
I've used all kinds of soap,
Until I had given up hope
Until one day I saw
An ad that put me in awe.
A shampoo used for dogs
Called GOOD ENOUGH to Clean a Hog
Guaranteed to kill more fleas.
I gave Fido a bath
And after doing the math
His number of fleas
Started dropping by 3's!
Before his shampoo
I counted 42.
At the end of his bath,
I redid the math
And the new shampoo had killed 17 fleas.
So now I was pleased.
Now it is time for you to have some fun
With the level of significance being .01,
You must help me figure out
Use the new shampoo or go without?
Set up the hypothesis test:
\(H_{0}: p \leq 0.25\) \(H_{a}: p > 0.25\)
Determine the distribution needed:
In words, CLEARLY state what your random variable \(\bar{X}\) or \(P′\) represents.
\(P′ =\) The proportion of fleas that are killed by the new shampoo
State the distribution to use for the test.
\[N\left(0.25, \sqrt{\frac{(0.25){1-0.25}}{42}}\right)\nonumber \]
Test Statistic: \(z = 2.3163\)
Calculate the \(p\text{-value}\) using the normal distribution for proportions:
\[p\text{-value} = 0.0103\nonumber \]
In one to two complete sentences, explain what the p-value means for this problem.
If the null hypothesis is true (the proportion is 0.25), then there is a 0.0103 probability that the sample (estimated) proportion is 0.4048 \(\left(\frac{17}{42}\right)\) or more.
Use the previous information to sketch a picture of this situation. CLEARLY, label and scale the horizontal axis and shade the region(s) corresponding to the \(p\text{-value}\).
Figure \(\PageIndex{14}\)
Compare \(\alpha\) and the \(p\text{-value}\):
Indicate the correct decision (“reject” or “do not reject” the null hypothesis), the reason for it, and write an appropriate conclusion, using complete sentences.
alpha decision reason for decision
0.01 Do not reject \(H_{0}\) \(\alpha < p\text{-value}\)
Conclusion: At the 1% level of significance, the sample data do not show sufficient evidence that the percentage of fleas that are killed by the new shampoo is more than 25%.
Construct a 95% confidence interval for the true mean or proportion. Include a sketch of the graph of the situation. Label the point estimate and the lower and upper bounds of the confidence
Figure \(\PageIndex{15}\)
Confidence Interval: (0.26,0.55) We are 95% confident that the true population proportion p of fleas that are killed by the new shampoo is between 26% and 55%.
This test result is not very definitive since the \(p\text{-value}\) is very close to alpha. In reality, one would probably do more tests by giving the dog another bath after the fleas have had a
chance to return.
Example \(\PageIndex{11}\)
In a study of 420,019 cell phone users, 172 of the subjects developed brain cancer. Test the claim that cell phone users developed brain cancer at a greater rate than that for non-cell phone users
(the rate of brain cancer for non-cell phone users is 0.0340%). Since this is a critical issue, use a 0.005 significance level. Explain why the significance level should be so low in terms of a Type
I error.
We will follow the four-step process.
1. We need to conduct a hypothesis test on the claimed cancer rate. Our hypotheses will be
1. \(H_{0}: p \leq 0.00034\)
2. \(H_{a}: p > 0.00034\)
If we commit a Type I error, we are essentially accepting a false claim. Since the claim describes cancer-causing environments, we want to minimize the chances of incorrectly identifying causes
of cancer.
2. We will be testing a sample proportion with \(x = 172\) and \(n = 420,019\). The sample is sufficiently large because we have \(np = 420,019(0.00034) = 142.8\), \(nq = 420,019(0.99966) =
419,876.2\), two independent outcomes, and a fixed probability of success \(p = 0.00034\). Thus we will be able to generalize our results to the population.
3. The associated TI results are
Figure 9.6.11.
Figure 9.6.12.
4. Since the \(p\text{-value} = 0.0073\) is greater than our alpha value \(= 0.005\), we cannot reject the null. Therefore, we conclude that there is not enough evidence to support the claim of
higher brain cancer rates for the cell phone users.
Example \(\PageIndex{12}\)
According to the US Census there are approximately 268,608,618 residents aged 12 and older. Statistics from the Rape, Abuse, and Incest National Network indicate that, on average, 207,754 rapes occur
each year (male and female) for persons aged 12 and older. This translates into a percentage of sexual assaults of 0.078%. In Daviess County, KY, there were reported 11 rapes for a population of
37,937. Conduct an appropriate hypothesis test to determine if there is a statistically significant difference between the local sexual assault percentage and the national sexual assault percentage.
Use a significance level of 0.01.
We will follow the four-step plan.
1. We need to test whether the proportion of sexual assaults in Daviess County, KY is significantly different from the national average.
2. Since we are presented with proportions, we will use a one-proportion z-test. The hypotheses for the test will be
1. \(H_{0}: p = 0.00078\)
2. \(H_{a}: p \neq 0.00078\)
3. The following screen shots display the summary statistics from the hypothesis test.
Figure 9.6.13.
Figure 9.6.14.
4. Since the \(p\text{-value}\), \(p = 0.00063\), is less than the alpha level of 0.01, the sample data indicates that we should reject the null hypothesis. In conclusion, the sample data support
the claim that the proportion of sexual assaults in Daviess County, Kentucky is different from the national average proportion.
The hypothesis test itself has an established process. This can be summarized as follows:
1. Determine \(H_{0}\) and \(H_{a}\). Remember, they are contradictory.
2. Determine the random variable.
3. Determine the distribution for the test.
4. Draw a graph, calculate the test statistic, and use the test statistic to calculate the \(p\text{-value}\). (A z-score and a t-score are examples of test statistics.)
5. Compare the preconceived α with the p-value, make a decision (reject or do not reject H[0]), and write a clear conclusion using English sentences.
Notice that in performing the hypothesis test, you use \(\alpha\) and not \(\beta\). \(\beta\) is needed to help determine the sample size of the data that is used in calculating the \(p\text{-value}
\). Remember that the quantity \(1 – \beta\) is called the Power of the Test. A high power is desirable. If the power is too low, statisticians typically increase the sample size while keeping α the
same.If the power is low, the null hypothesis might not be rejected when it should be.
1. Data from Amit Schitai. Director of Instructional Technology and Distance Learning. LBCC.
2. Data from Bloomberg Businessweek. Available online at http://www.businessweek.com/news/2011- 09-15/nyc-smoking-rate-falls-to-record-low-of-14-bloomberg-says.html.
3. Data from energy.gov. Available online at http://energy.gov (accessed June 27. 2013).
4. Data from Gallup®. Available online at www.gallup.com (accessed June 27, 2013).
5. Data from Growing by Degrees by Allen and Seaman.
6. Data from La Leche League International. Available online at www.lalecheleague.org/Law/BAFeb01.html.
7. Data from the American Automobile Association. Available online at www.aaa.com (accessed June 27, 2013).
8. Data from the American Library Association. Available online at www.ala.org (accessed June 27, 2013).
9. Data from the Bureau of Labor Statistics. Available online at http://www.bls.gov/oes/current/oes291111.htm.
10. Data from the Centers for Disease Control and Prevention. Available online at www.cdc.gov (accessed June 27, 2013)
11. Data from the U.S. Census Bureau, available online at quickfacts.census.gov/qfd/states/00000.html (accessed June 27, 2013).
12. Data from the United States Census Bureau. Available online at www.census.gov/hhes/socdemo/language/.
13. Data from Toastmasters International. Available online at http://toastmasters.org/artisan/deta...eID=429&Page=1.
14. Data from Weather Underground. Available online at www.wunderground.com (accessed June 27, 2013).
15. Federal Bureau of Investigations. “Uniform Crime Reports and Index of Crime in Daviess in the State of Kentucky enforced by Daviess County from 1985 to 2005.” Available online at http://
www.disastercenter.com/kentucky/crime/3868.htm (accessed June 27, 2013).
16. “Foothill-De Anza Community College District.” De Anza College, Winter 2006. Available online at research.fhda.edu/factbook/DA...t_da_2006w.pdf.
17. Johansen, C., J. Boice, Jr., J. McLaughlin, J. Olsen. “Cellular Telephones and Cancer—a Nationwide Cohort Study in Denmark.” Institute of Cancer Epidemiology and the Danish Cancer Society, 93
(3):203-7. Available online at http://www.ncbi.nlm.nih.gov/pubmed/11158188 (accessed June 27, 2013).
18. Rape, Abuse & Incest National Network. “How often does sexual assault occur?” RAINN, 2009. Available online at www.rainn.org/get-information...sexual-assault (accessed June 27, 2013).
Central Limit Theorem
Given a random variable (RV) with known mean \(\mu\) and known standard deviation \(\sigma\). We are sampling with size \(n\) and we are interested in two new RVs - the sample mean, \(\bar{X}\),
and the sample sum, \(\sum X\). If the size \(n\) of the sample is sufficiently large, then \(\bar{X} - N\left(\mu, \frac{\sigma}{\sqrt{n}}\right)\) and \(\sum X - N \left(n\mu, \sqrt{n}\sigma\
right)\). If the size n of the sample is sufficiently large, then the distribution of the sample means and the distribution of the sample sums will approximate a normal distribution regardless of
the shape of the population. The mean of the sample means will equal the population mean and the mean of the sample sums will equal \(n\) times the population mean. The standard deviation of the
distribution of the sample means, \(\frac{\sigma}{\sqrt{n}}\), is called the standard error of the mean.
Contributors and Attributions
• Barbara Illowsky and Susan Dean (De Anza College) with many other contributing authors. Content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license.
Download for free at http://cnx.org/contents/30189442-699...b91b9de@18.114. | {"url":"https://stats.libretexts.org/Courses/Las_Positas_College/Math_40%3A_Statistics_and_Probability/08%3A_Hypothesis_Testing_with_One_Sample/8.04%3A_Hypothesis_Test_Examples_for_Proportions","timestamp":"2024-11-09T23:01:58Z","content_type":"text/html","content_length":"156120","record_id":"<urn:uuid:6714ed3a-6d9f-4c83-9f0f-a404f43e124d>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00081.warc.gz"} |
The Infinite World of Calculus
Calculus is a branch of mathematics that explores the concepts of change and motion.
published : 25 March 2024
Calculus is a branch of mathematics that explores the concepts of change and motion. Developed independently by Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century, calculus provides
powerful tools for understanding the behavior of functions, solving problems in physics and engineering, and modeling complex phenomena in the natural world.
Basic Concepts
At the heart of calculus are two fundamental concepts: differentiation and integration. Differentiation involves the calculation of derivatives, which measure how a function changes with respect to
its input. Derivatives provide information about the rate of change of a function, allowing us to analyze functions and their behavior.
Integration, on the other hand, involves the calculation of integrals, which represent the accumulation of quantities over a given interval. Integrals provide a way to calculate the total change or
area under a curve, enabling us to solve problems involving accumulation, averaging, and optimization.
Calculus has applications across diverse fields, including physics, engineering, economics, biology, and computer science. In physics, calculus is used to describe the motion of objects, analyze
forces and energy, and solve problems in mechanics, electromagnetism, and quantum physics.
In engineering, calculus is used to design and analyze systems such as bridges, buildings, electrical circuits, and control systems. In economics, calculus is used to model and optimize systems of
supply and demand, analyze consumer behavior, and solve problems in optimization and game theory.
Foundations of Analysis
Calculus is also closely related to the field of real analysis, which provides the rigorous mathematical foundations for calculus. Real analysis explores the properties of real numbers, limits,
continuity, and convergence, providing a rigorous framework for understanding the concepts and techniques of calculus.
Through the study of real analysis, mathematicians and scientists develop a deeper understanding of the fundamental concepts of calculus and their mathematical properties. This knowledge forms the
basis for more advanced topics in calculus and related fields.
Modern Developments
In the modern era, calculus continues to be a vibrant area of research with applications in various fields, including mathematical physics, dynamical systems, numerical analysis, and machine
learning. Researchers are exploring new techniques and methods for solving differential equations, optimizing functions, and analyzing complex systems.
Advancements in computational methods and computer technology have enabled scientists and engineers to tackle increasingly complex problems using calculus-based approaches. From simulating the
behavior of physical systems to analyzing large datasets and optimizing machine learning algorithms, calculus remains an indispensable tool for understanding and solving real-world problems.
Calculus opens up a vast and infinite world of possibilities, allowing us to explore the dynamics of change and motion in the natural and engineered world. From its humble origins in the 17th century
to its modern applications in science and technology, calculus continues to shape our understanding of the universe and inspire new discoveries and innovations.
As we delve deeper into the infinite world of calculus, let us appreciate the elegance and power of its concepts and techniques, and the profound impact they have on our understanding of the world
and our ability to shape the future. | {"url":"https://www.function-variation.com/article116","timestamp":"2024-11-06T12:17:33Z","content_type":"text/html","content_length":"17644","record_id":"<urn:uuid:77b41f22-4226-4e9d-8357-36a3a76af554>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00359.warc.gz"} |
Spectral Norm Research Articles - R Discovery
Motivated by classical work on the numerical integration of ordinary differential equations we present a ResNet-styled neural network architecture that encodes non-expansive (1-Lipschitz) operators,
as long as the spectral norms of the weights are appropriately constrained. This is to be contrasted with the ordinary ResNet architecture which, even if the spectral norms of the weights are
constrained, has a Lipschitz constant that, in the worst case, grows exponentially with the depth of the network. Further analysis of the proposed architecture shows that the spectral norms of the
weights can be further constrained to ensure that the network is an averaged operator, making it a natural candidate for a learned denoiser in Plug-and-Play algorithms. Using a novel adaptive way of
enforcing the spectral norm constraints, we show that, even with these constraints, it is possible to train performant networks. The proposed architecture is applied to the problem of adversarially
robust image classification, to image denoising, and finally to the inverse problem of deblurring. | {"url":"https://discovery.researcher.life/topic/spectral-norm/7282952?page=1&topic_name=Spectral%20Norm","timestamp":"2024-11-03T12:57:32Z","content_type":"text/html","content_length":"376217","record_id":"<urn:uuid:019f9a57-111f-4ff6-b15a-db666add52df>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00353.warc.gz"} |
Calculate Time And A Half Pay - Certified Calculator
Calculate Time And A Half Pay
Introduction: The Calculate Time and a Half Pay tool is designed to help individuals and businesses determine the total pay, including overtime, for employees who have worked beyond regular hours.
Whether you’re managing payroll or planning work schedules, this calculator provides a quick and efficient way to estimate time and a half pay.
Formula: The calculation involves determining overtime hours (hours worked beyond 40 hours per week), regular pay for up to 40 hours, overtime pay for hours beyond 40 (calculated at 1.5 times the
hourly rate), and then summing both to get the total pay.
How to Use:
1. Enter the total regular hours worked.
2. Input the hourly rate for the employee.
3. Click the “Calculate” button to obtain the total time and a half pay.
Example: For example, if an employee worked 45 hours at an hourly rate of $20, the calculator will display time and a half pay of $950.00.
1. Q: Can I use this calculator for negative regular hours or hourly rates? A: No, both regular hours and hourly rates must be non-negative values.
2. Q: What if the regular hours are less than 40? A: The calculator considers up to 40 hours as regular hours. Hours beyond 40 are treated as overtime.
3. Q: Does the calculator handle decimal hours or rates? A: Yes, the calculator supports decimal values for both regular hours and hourly rates.
4. Q: Can I use this calculator for different overtime rates? A: This calculator assumes a standard time and a half pay calculation (1.5 times the hourly rate for overtime).
5. Q: What should I do if the result is “Invalid input”? A: Check if you have entered valid values for regular hours and hourly rates.
Conclusion: The Calculate Time and a Half Pay tool is a valuable resource for payroll management and ensuring fair compensation for overtime work. It streamlines the process of estimating total pay,
providing clarity and efficiency in handling payroll calculations. Use this calculator to enhance your payroll processes and ensure accurate compensation for employees who work beyond regular hours.
Leave a Comment | {"url":"https://certifiedcalculator.com/calculate-time-and-a-half-pay/","timestamp":"2024-11-02T18:03:49Z","content_type":"text/html","content_length":"55591","record_id":"<urn:uuid:2de40e6e-9038-4b6c-af25-e75d5a396873>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00097.warc.gz"} |
Triangle Congruence Practice Problems Worksheet Answers - TraingleWorksheets.com
Triangle Congruence Practice Worksheet – Triangles are one of the most fundamental shapes in geometry. Understanding the triangle is essential to understanding more advanced geometric concepts. In
this blog post we will go over the various kinds of triangles with triangle angles. We will also discuss how to calculate the dimensions and the perimeter of a triangle, and show details of the
various. Types of Triangles There are three kinds that of triangles are equilateral isoscelesand … Read more | {"url":"https://www.traingleworksheets.com/tag/triangle-congruence-practice-problems-worksheet-answers/","timestamp":"2024-11-13T11:46:56Z","content_type":"text/html","content_length":"48595","record_id":"<urn:uuid:2710c138-de41-4b40-a2ea-61d5eeaa59e0>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00426.warc.gz"} |
How to Calculate the Penalty For Not Paying the Estimated Tax
How To Calculate The Penalty For Not Paying The Estimated Tax
Quarterly estimated taxes are a must for self-employed individuals, freelancers, independent contractors and small business owners. If you don’t pay estimated taxes you’ll attract IRS tax penalties.
The first stage in determining your penalty is to calculate your federal income tax obligation. Your real tax payment will likely be less than what is shown on your return if your income tax rate is
greater. If your tax percentage is 10%, for example, you might plan to pay $1,000 in taxes. You would only be required to pay $800 in taxes if you paid $1,200 because your withholding sum is $200.
To calculate your actual tax liability, multiply your income by the applicable tax percentage, then add your withholding sum. Take into account the case where your total tax obligation is $1,500 and
your deduction is $600. Your actual tax liability would be $1,500 (or $1,500 times a factor of 10%).
There are several IRS forms like the Schedule C of Form 1040 that you need to complete correctly to find your accurate tax liability. But before you even get to the Schedule C you’ll have to find
your estimated taxes using a 1099 tax calculator. If you live in a certain state, you’ll have to use state-specific tax calculators like New York tax calculator, California tax calculator or Texas
tax calculator. or Hire Tax Preparation expert.
How to calculate tax fines fast?
To calculate your punishment, simply subtract your actual tax from your total. The total quantity of tax you due but have not paid will be represented by a number that you will be given here.
But what happens if you have another 60 days before you have to pay the fine? After that, it will be another 10 months before your punishment is settled. Find the proper fee amount that you will be
required to pay now. If your tax bill exceeds $1,000, you may be subject to a 15% penalty on the extra.
Therefore, if you owed $1,200, you would owe $180. For that, there would be a $30 charge. As a result, you would have 60 days to settle that.
On a $100 tax bill, you would only be charged a $15 penalty if your tax load is less than $1,000.
Additional Methods for Determining Penalty Amounts
Therefore, if you incur $50 in late filing fees in addition to $100 in taxes, your overall obligation will be $150.
Those who file their tax returns after the due date may be subject to a fee of up to $250 or 5% of the amount of the tax due.
Learn more about calculating the fee here.
Calculation of Penalties
1. The Period of Lateness for Filing
How long the submission was late is determined by the day the IRS gets the return. Therefore, there will be a five-day grace time if the submission is due on April 15. If the reporting date had been
April 30, this would still be the case. There will be a four-month grace time if the paperwork is filed after the deadline of January 31.
2. The Reason you Filed Late
If you can demonstrate a solid explanation for the late submission, that will prove whether or not you had a valid justification. This includes issues with mail transportation or the loss of records.
It is important to note that the IRS does not consider the following reasons to be acceptable excuses for a late submission:
being unable to access the internet, a computer, or a printer
incorrectly filling out the tax form and hesitantly receiving the tax form
3. Raising Late Filing Penalties
The late submission penalty usually raises the amount due. The debt will be decreased, though, if the punishment exceeds 10%. If you owe $100 and must pay a $50 penalty, you will only be accountable
for the initial $100 after paying $50.
4. The Amount Due
To calculate the entire sum owed, use the formula below:
The tax is $100.
The charge is $50.
We owe a sum of $150.
If you submit your taxes after the deadline and have a small tax debt—less than $100—or a low taxable income, you might be entitled to a rebate.
5. Verify all of your income tax forms.
Document 1040 is the favored income tax document used by the IRS. You can obtain the document at https://www.irs.gov/pub/irs-pdf/i1040.pdf.
Any fees that are required, such as those for filing a report, must be paid. To accomplish this, a direct debit method is usually used.
6. Estimated Tax Payments
By the 15th of every month, you must make estimated tax payments in a sum equal to your tax liability.
Finding Your Penalty Amount
The rule is simple, don’t underpay your taxes, don’t pay the penalty. The penalty amount is calculated by multiplying your underpayment sum by ten. In other terms, you owe $1,000 in taxes and have
made a $10 underpayment. An $10.00 charge is due.
The punishment is calculated by ten-folding the underpayment. Consequently, you would have a $250 underpayment if you owed $500 but only made a $250 payment. $25.00, or ten times $250, is the
How does that work exactly?
As a result, if you underpay, you must pay the fee.
Since we are using that sum to calculate the underpayment, you must double the amount you undervalued by 10. We won’t go over the additional factors that affect the punishment amount at this time.
Consider the scenario where your salary was $200 but your tax debt was $800. You made $1,000 total in revenue and have already paid $800 in taxes. Finding the underpayment’s outcome is as follows:
$200 – $800 = -$800.
Your deficit is $800.
This amount must be multiplied by 10 to determine the fee.
– $800 x.10 = – $80.00
You will pay a $80.00 charge.
Imagine that you were underpaid by $90.
$90 – $80 = -$10
Again, the difference is $10.
Add ten to the underpayment to determine the punishment.
– $10 x.10 = – $1.00
You’ll receive a $1 charge.
What percentage of underpayment fines will there be in 2022? The penalty rate for 2022 is 2.5% of what you repay. For a $100 bill, the penalty amount, for instance, is $2.50.
How long will the penalty last?
The deadline for making a penalty payment is three years from the date you first became compelled to pay the tax.
What Can Be Done to Avoid the Underpayment Penalty?
If you request a return, you should plan your payment schedule.
Tips to reduce your underpayment penalty
You might reduce your punishment by paying your taxes ahead of schedule. You can either settle your fine online or with a check. You can avert the penalty by paying all of the revenue you due.
Is the underpayment punishment lifted for 2021?
This deferral does not apply to any income tax returns that must be submitted on or before January 1, 2021, for the 2020 tax year or for a return for a prior year. (2019).
Does the waiver apply to Forms 1040, 1040A, 1040-ES, 1040X, and modified forms filed for prior years?
Yes. There’s no penalty for any income tax return for the taxable year beginning on January 1, 2021. It’s because the government wanted to give some relief to taxpayers.
What about filers who submitted versions 1040, 1040A, 1040EZ, 1040X, or altered versions in 2018 but did not make the required tax payments?
This exemption is not available to all taxpayers. The taxpayers filing a Form 1040, 1040A, 1040EZ, 1040X, or revised return for the 2018 tax year and having a total income of more than $50,000.
Related article, How to Get Tax Benefits from Your Home Loan
The repercussions of paying taxes late are complicated. The different computations and their methods are unfamiliar to a large portion of the financial workforce. In addition, a lot of accountants
aren’t conscious that there are different penalties for underpaying anticipated taxes versus real taxes. To correctly submit your tax returns and gain access to many tax benefits for independent
contractors or self-employed people, use a one-stop tax store like FlyFin.
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MPM1D: Principles of Mathematics
The textbook for this course is Principles of Mathematics 9 by McGraw-Hill Ryerson. A replacement costs around $90, so be sure to take care of it.
Exam Review
There is a Course Review section toward the end of your textbook. MathWiz is another great resource for exam review questions. Mr. Silvaggio has also prepared a package that you can download and work
Topics Covered
1. Numeracy and Number Sense
1. Percents, Ratios and Rates
2. Polynomials
3. Solving Equations
4. Properties of Linear Relations
1. Connection Between Slope, Variation, First Differences
5. Equations of Linear Relations
1. Parallel and Perpendicular Lines
6. Analyzing Relations
1. Scatter Plots
2. Linear and Non-Linear Relations
3. Analyzing Data Using Technology
7. Geometry and Optimization
1. TBD | {"url":"https://jongarvin.com/mpm1d/","timestamp":"2024-11-06T01:05:46Z","content_type":"text/html","content_length":"120809","record_id":"<urn:uuid:56bf42ec-742c-4dbe-8ec8-b3e523a3cb98>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00770.warc.gz"} |
Download Ring Constructions
Download Ring Constructions & Applications (Series in Algebra, Volume 9) pdf ebook. Buy cheap pdf ebooks/audio books.
Author: A.V. Kelarev
Released: 2002
ISBN10: 9810247451
ISBN13: 9789812799722
Type: pdf
Page count: 217
Rating: 9/10
Amazon.com: Introductory Lectures on Rings and Modules (London
Ring Constructions & Applications Series in Algebra, Volume 9 by A.V. Kelarev in Back Matter; Graph Algebras and Automata Pure and Applied Mathematics, Vol. 257 Chapman 1 book cites this book Ring
Constructions & Applications Series in Algebra, Volume 9 by A.V. Kelarev in Front Matter appeal to researchers in algebrathe preface to Skew Field Constructions; Note to the reader; Prologue; 1.
Ringsof Mathematics and Its Applications Series Volume
Ring Constructions & Applications (Series in Algebra, Volume 9)
algebra, Modern Algebra with Applicationsof group, ring, and field theory typically contained in a standard modern algebra course.WileyInterscience Series of Volume
1. Groups with Prescribed Quotient Groups and Associated Module
2. Download Ring Constructions & Applications (Series in Algebra, Volume 9) pdf ebook.
☆ The online version of Linear Algebra and its Applications atVolume 436, Issue 9 pp. 29633792 1 May 2012 Volume 436with the Ducci map and the mediant construction
1. Encyclopedia of Mathematics and Its Applications #57: Skew Fields
☆ to the methods of calculus and abstract algebra, sothe Egyptians had a correct formula for the volume ofof spaces with only finitely many points, found applications
1. Book Series Related Journalsin the second volume which contains more local rings anddevelopment of algebra during the last centuries, including its applications to
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Buy Ring Constructions & Applications (Series in Algebra, Volume 9) ebook pdf
SERIES IN ALGEBRA ISSN 2010281XVol. 9 Ring Constructions and Applications by A V Kelarev. Vol. 10 Matrix Partial Orders, Shorted Other titles in the Algebra and Applications series An Introduction to
Group RingsSeries Algebra and Applications Series Volume 2 Publication Date 20101202
LECTURES ON ALGEBRA - World Scientific
Polynomial and power series rings413 Primitive element theorem 414 ConstructionsAbstract Algebra with Applications Volume 2 Rings and Fieldsitem PDF | 7.9 MB This book contains the definitions of
several ring constructions used in various applications.abstract algebra. Commutative ring12 mb Ring Theory Volume 44
Ring Constructions & Applications (Series in Algebra, Volume 9) pdf/chm/mp3
SERIES IN ALGEBRA - World Scientific
constructions where the exterior algebra finds one of its most important applicationsmost applications, the volumealgebra of a module over a commutative ring The online version of Journal of
Algebra atApplicationsVolume 363, In Progress Volume Issue In ProgressA Download mp3 audio book. 2.3 Rings; 2.4 Fields; 2.5 Applications; 3. Universal Algebrasimply \"Algebra\" for its fourth
edition in 1955. Volume 1 treated groups, ringsfollowing series of
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2022 year, volume 26, issue 4 (PDF)
The results of the analysis of some available sources are presented, on the basis of which an automatic learning model for artificial neural networks was implemented.
Keywords: image recognition, artificial neural network, automated machine learning, automatic learning model, hyperparameters.
Consider convolutional neural network optimization problem using small dataset. The proposed method fine-tune pretrained neural network with specigic constrains for convolutional kernels. Pretrained
weights are decomposed using SVD. During fine-tuning the only singular values are trained. In the paper are investigated the dinamic of singular values during training process and its influence on
quality of resulting model. The method are comapared with other approaches in rengenological images classsification problm.
Keywords: deep learning, convolutional neural networks, singular value decomposition.
Within the practical application of neural networks, the number of parameters in the network is much larger than the number of samples in the dataset, however, the network still has good
generalization characteristics. Traditionally considered that such over-parameterized and non-convex models can easily fall into local minima while searching for the optimal solution and show low
generalization performance, but in fact it is not. Although under some regularization conditions it is possible to effectively control the network generalization error, it is still difficult to explain
the generalization problem for large networks. In our work, we determine the difference between the overfitting step and the feature learning step by quantifying the impact of updating one sample
during gradient descent on the entire training process, revealing that neural networks generally have less impact on other samples during the overfitting step. In addition, we use the Fisher
information matrix to mask the gradient produced by the backpropagation process, thereby slowing down the neural network’s overfitting behavior and improving the neural network’s generalization
Keywords: Neural Networks, Generalization, Overfitting, Fisher Information.
It is shown that, under some natural restrictions on the element basis, for any recurrent circuit with a fixed number of inputs, it is possible to construct a circuit functionally equivalent to it in
the same basis using superposition operations and no more than a double application of the feedback operation with a linear increase in the number of delays used compared to original circuit
implementation. Thus, the linearity of the order of memory growth in the transition to the optimal (in terms of the number of delays) scheme with no more than two feedbacks has been proved.
Short-term and long-term memory modules are distinguished in the structure of the recurrent circuits with no more than two feedbacks constructed in the work. The result obtained, in particular, is
valid for the class of finite automata, as well as for the class of neural circuits built from elements containing gates.
Keywords: recurrent circuits, feedback, linear estimation, gates.
To study the efficiency of distributed algorithms, a number of mathematical formalizations with new concepts of algorithm complexity are introduced. In this paper complexity of finding strongly
connected components in the AMPC model is investigated. AMPC model, unlike other more limited distributed formalizations, allows to build a query tree within one step of the algorithm. A
probabilistic algorithm is obtained that implements strongly connected components search in polylogarithmic or sublinear time (depending on the amount of available local memory). The amount of
required local memory is sublinear with respect to the number of vertices in the graph.
Keywords: distributed algorithms, probabilistic algorithms, strongly connected components.
The paper investigates the question of the angle between the normal vectors of separating hyperplanes defined by linear test algorithms. The question is considered for various reference sets of tests.
Keywords: linear test algorithms separating hyperplanes, dead-end tests.
This article considers the problem of \[К\]- and \[К\]-finite generation for precomplete classes of linear automata operating over the Galois field consisting of two elements. The set of all studied
classes is the \[A\]-criterion system in the class of linear automata. A finite basis was presented for each class under consideration.
Keywords: finite automaton, linear automaton, composition operation, feedback operation, completeness, closed class, precomplete class, \[К\]-finitely generated class, \[A\]-finitely generated class.
The article considers applying a cellular automaton with locators to solving the problem of vector addition. The locator cellular avtomaton model assumes the possibility for each cell to translate a
signal through any distance. It is proven in this article that such possibility allows to decrease the problem complexity from linear to logarithmic (against the classic cellular automaton model).
Keywords: cellular automata, homogeneous structures, the problem of vector addition.
For now some facts about completeness of linear automata are proven. We now know about every precomlete class on superposition operetion and composition operation, the completeness criterions are
also had been formulated. In this work we have proven some facts about the complexity of this process: to receive neutral element and delay.
Keywords: linear automata, comlexity estimation.
The present paper considers a problem of functional classes obtained by using neural networks on max non-linearities bases. Firstly, some properties of CPL-functions and equivalence classes
generating them are investigated. Proceeding from these properties a theorem is proved that neural networks built on the basis of linear and max non-linearity functions can exactly recover any convex
CPL-function. Secondly, RELU-basis, a special case of max non-linearities bases, is investigated, with a theorem similar to the previous one mentioned above proved. The question of estimating the
number of neurons and layers in obtained architectures is also discussed. All the mentioned theorems have a constructive proof, i.e. neural network architectures with mentioned features are built
Keywords: Neural networks, architecture, functions recovery, functions expressibility, convex functions, particle-linear functions, ReLU function, max function. | {"url":"http://intsysjournal.org/en/issue_26-4","timestamp":"2024-11-08T02:16:25Z","content_type":"text/html","content_length":"17042","record_id":"<urn:uuid:0b2be2b6-5bd9-4c97-94bd-ab42fcc07d90>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00334.warc.gz"} |
Linear Regression Model
General information
Linear regression
is one of the simplest ML models for regression tasks. Its main principle is in finding linear dependence between one dependent and multiple independent variables (features).
Linear Regression fits a linear model with coefficients to minimize the residual sum of squares between the observed targets in the dataset, and the targets predicted by the linear approximation.
Brick Locations
Bricks → Machine Learning → Linear Regression
Brick Parameters
Simple mode
• Regularization
Regularization is a technique used for tuning the function by adding a penalty term in the error function, which reduces overfitting. The model supports the following types of regularization:
□ Lasso Regression (L1) Least Absolute Shrinkage and Selection Operator, is a type of linear regression that uses shrinkage. Shrinkage is where data values are shrunk towards a central point
like “mean”. The lasso procedure encourages simple, sparse models (i.e. models with fewer parameters). This particular type of regression is well-suited for models showing high levels of
multicollinearity or when you want to automate certain parts of model selection, like variable selection/parameter elimination. The cost function for Lasso regression is:
□ Ridge Regression (L2) - (also known as Tikhonov regularization), ridge regression shrinks the coefficients and it helps to reduce the model complexity and multi-collinearity
□ ElasticNet - linearly combines both the L1 and L2 penalties of the Lasso and Ridge methods.
• Target variable
The column which contains values for the model to predict.
• Disallow negative predictions
This checkbox forces the model to round up negative values to be equal to 0.
• Columns
Columns from the dataset that are ignored during training. However, they will be present in the resulting dataset. Multiple columns can be selected by clicking the + button.
In case you want to remove a large number of columns, you can select the columns to keep and use the flag ‘Remove all except selected’.
Advanced mode
Has the same set of parameters as in the simple mode with one additional parameter:
• Train Explainer
If checked, the model explainer for API usage is built.
Brick Inputs/Outputs
• Outputs
□ Brick produces the dataset with an extra column for predicted target value by the model
□ A trained model that can be used in other bricks as an input
Additional Features
• Model performance
Gives you a possibility to check the model's performance (a.k.a. metrics) to then adjust your pipeline if needed. Available after successful brick run.
Supported metrics: MAPE (Mean Average Percentage Error), R2, RMSE (Root Mean Square Error).
• What-if
This option gives access to the information for the Model Deployment service, as well as a possibility to call API using custom values.
• Meta-data
Access metadata for the pipeline.
• Save model
This option provides a mechanism to save your trained models to use them in other projects.
• Download
Use this feature, if you want to download the model's asset in JSON format to use it outside the Datrics platform.
Example of usage
Let's consider a simple regression problem, where we know the characteristics of some houses and want to know their sale price. The dataset ‘home_prices_sample.csv’ contains the next variables:
• Id (category/int) - Sale's ID
• Neighborhood (category/string) - House neighborhood name
• YearBuilt (int) - The year when a house has been built
• RoofMatl (category/string) - The materials used to construct the roof
• GrLivArea (int) - The living area
• YrSold (int) - The year when a house was sold
• SalePrice (int) - The price at which the house was sold - target variable.
Linear Regression
model supports only numerical values in the input dataset. In this case, columns ‘Neighborhood’ and ‘RoofMatl’ are strings, so that they can be turned into labels with the
Brick. Apart from it, column ‘Id
does not represent the characteristics of the house, so it will be filtered out during the training.
As for the model parameters, let’s check ‘Regularization’ with ElasticNet, set ‘SalePrice’ to be the target variable, check ‘Disallow negative predictions’ to ensure that predictions will not be less
than 0, and filter the ‘Id’ column.
After running the pipeline we can view the output dataset in the Data Outputs section.
To see the model's performance, click the ‘Open view’ button on the Model Info panel.
In the ‘What-if’ tab, you can write down some custom parameters to the model and click ‘Run API’ to generate the prediction.
Also, you can save your trained model to reuse it later in other projects. For this, click on the ‘Save model’ and specify the model's name or you can create a new version of an already existing
model (you will need to choose the existing model and specify the new version's name), and, finally, submit with the ‘Save’ button. | {"url":"https://wiki.datrics.ai/linear-regression-model","timestamp":"2024-11-07T07:46:02Z","content_type":"text/html","content_length":"146449","record_id":"<urn:uuid:86e1a125-79f7-4b82-a714-7891edcd9cb6>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00692.warc.gz"} |
Drawer Counter Calculator
Drawer Counter Calculator - Web quickly calculate the total sum and generate pdf results at no cost. Web dollar money counter calculator help you to count your coins and banknotes. The last step is
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Web To Use A Money Cash Counter Calculator, Select Your Currency From The Drop Down And Enter The Number Of Notes & Number Of Coins In Related Fields.
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Related Post: | {"url":"https://participation-en-ligne.namur.be/read/drawer-counter-calculator.html","timestamp":"2024-11-12T17:13:45Z","content_type":"text/html","content_length":"27006","record_id":"<urn:uuid:b37bd87d-4989-47cb-abd6-f19661b9fd3d>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00603.warc.gz"} |
Consider the graph of function g. The dashed lines - Asksia.ai
Answer from Sia
Posted 6 months ago
Solution by Steps
step 1
To find the limit of $g(x)$ as $x$ approaches $-\infty$, we observe the behavior of the graph in the leftmost part. The graph starts slightly above the dashed horizontal line at $y = 3$ and
approaches this line as $x$ goes to $-\infty$. Therefore, $\lim_{x \rightarrow -\infty} g(x) = 3$
step 2
To find the limit of $g(x)$ as $x$ approaches $\infty$, we observe the behavior of the graph in the rightmost part. The graph approaches the dashed horizontal line at $y = 1$ as $x$ goes to $\infty$.
Therefore, $\lim_{x \rightarrow \infty} g(x) = 1$
$\lim_{x \rightarrow -\infty} g(x) = 3 \\ \lim_{x \rightarrow \infty} g(x) = 1$
Key Concept
Limits of a function as $x$ approaches $-\infty$ and $\infty$
The limits of a function as $x$ approaches $-\infty$ and $\infty$ can be determined by observing the behavior of the graph near the horizontal asymptotes.
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@Rapptz Since we're already using SFINAE to determine if something is a symbol or not, we could reduce nesting by 2 by (A): getting rid of the
wrapper in the operators around something that was already a symbol, and (B): making
, and
all count as symbols using the | {"url":"https://chat.stackoverflow.com/transcript/message/19227079","timestamp":"2024-11-06T16:20:31Z","content_type":"text/html","content_length":"352750","record_id":"<urn:uuid:1b5d7cbf-215a-4a0c-a1b5-ed8c7c73c77d>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00608.warc.gz"} |
In-circuit Measurement of Inductors and Transformers with an Oscilloscope
• Reviews the basic theory of inductors and transformers, especially as it relates to in-circuit measurements
• Explains how to use an oscilloscope and appropriate probes to measure inductor and transformer performance during power supply operation.
• Explains the use of inductance measurements and B-H curves to gain insight into performance.
While some of the measurements described can be set up manually, power analysis software running on the scope can provide faster setups and improved repeatability. The examples in this application
note were developed using 5-PWR Advanced Power Analysis software on a 5 Series MSO oscilloscope. The 6 Series MSO with 6-PWR operates identically. Many of the inductor and transformer measurements
described herein, are also available with DPOPWR Advanced Power Analysis, although the displays and controls will be different from the ones shown.
FIGURE 1. Inductors and transformers serve key roles in switch mode power supplies, including filters, step-up/step-down, isolation, energy storage, and oscillation.
In switched mode power supplies (SMPS), magnetic components, namely inductors and transformer, play important roles. Inductors serve as energy storage devices or filters. Transformers are used
convert voltage levels and to provide isolation. Transformers also play an important role in sustaining oscillation in SMPS systems. Much of the SMPS design process relies on component specifications
and simulation models. However, due to actual signal conditions, parasitics, temperature and other environmental factors affecting the magnetic component’s performance, a power supply may not perform
exactly as predicted by the datasheet specs and simulations. Thus, in-circuit measurements of inductors and transformers, made under operating conditions, can provide important information.
A Quick Review of Inductor Theory
Faraday's and Lentz's laws tell us that the current through an inductor and the voltage across the inductor are related as:
This shows that inductance can be thought of as the extent to which a changing current results in an opposing voltage. By integrating, rearranging,and ignoring the sign, we can get
This shows that inductance may be determined as a function of voltage and current over time.This is a perfect job for an oscilloscope equipped with a voltage probe, current probe, and able to perform
integration and plot X versus Y
In contrast to a theoretical inductor, the inductance value of a real inductor is dependent on current levels, temperature and frequency of operation. In a power supply, these characteristics vary in
real time with operating conditions.
FIGURE 2. A basic inductor – a coil wound on a closed ferromagnetic core. Current of I amperes flows through the coil which has N turns. The inductance of the coil describes the relationship between
the current flowing in the coil and the magnetic flux
As an example, the inductance of the toroid in Figure 2 may be approximated by
µ is magnetic permeability of the core
N is the number of turns of wire on the toroid
r is the radius of the core from the dashed center-line in cm
A is the cross-sectional area of the core in cm2 (assumed to be small relative to the radius of the toroid)
Since this number of turns is squared, this is the biggest contributor to inductance. The permeability of the core material also plays a significant role.However, the value of the inductance is also
related to the physical size of the component. To minimize the size of the inductor, most inductors in electronics use core materials which have a much higher permeability than air.
In short, the characteristics of the core material and geometry are critical in determining inductance over various operating conditions, as well as power loss in the device.
Inductance Measurements and I vs. ∫V
Power supply designers often use simulation techniques to determine the appropriate inductor value for a design. After manufacturing the inductor, it is common practice to verify the inductance using
an LCR Meter. However, most LCR meters stimulate the component with sinusoids over a narrow frequency range, so while this is a good technique for confirming that the component is roughly the correct
value, it is a poor predicter of in-circuit performance.
The inductance characteristics of an inductor are dependent upon the current and voltage source excitation signal, wave shape, and frequency of operation, which may vary in real-time operating
conditions. Therefore, it is desirable for designers to be able to measure and observe the behavior of an inductor in the dynamically changing environment of the power supply. Tektronix Advanced
Power Analysis software (such as 5-PWR) automates this measurement.
FIGURE 3. Specifying voltage channel and current channel to set up inductance and i vs. ∫v measurements.
Measurements are made by probing the voltage across the device,usually with a differential voltage probe. A current probe is usually used to measure the current through the magnetic component.
To determine inductance the application integrates the voltage over time and divides by the change in current. It removes any DC offset and uses averaging to calculate the inductance value.
Note on Making Inductance Measurements on Transformers and Coupled Inductors
When you are measuring the inductance of a transformer, do not load the secondary winding. Measuring inductance at the primary winding under a no-load condition is equivalent to measuring the
inductance for a single winding inductor. When you are measuring the inductance of the coupled inductor with multiple windings on the same core, the measured value of the inductance will deviate from
the actual value, due to the influence of the current on the other winding/s.
FIGURE 4. Inductance measurement gives the average inductance value in Henries. The yellow waveform (CH1) is the voltage across the inductor and the blue waveform (CH2) is the current through the
inductor. The plot on the left shows current, i versus ∫vdt, the slope of which is inductance.
FIGURE 5. I vs ∫V measurement gives additional insight on inductor performance. Designers can see any DC bias building up over multiple cycles. The yellow waveform (CH1) is the voltage across the
inductor and the blue waveform (CH2) is the current through the inductor.
B-H Curve Measurements
Magnetic power supply components are designed for expected operating voltage, current, topology, and the particular type of power converter. The operating regions of inductors and transformers help
determine the stability of a switch mode power supply. However, power supply operating characteristics can vary during power-on, steady state operation, load changes and environmental changes, making
it extremely difficult to consider all possible scenarios during the design process.
Thus, to ensure the stability of the power supply, it is important to characterize the operating region of a magnetic component while it operates in the SMPS. Generally, the goal is to avoid
saturation and operate in the linear region of the hysteresis curve. However, it is extremely difficult to design a magnetic component and ensure that it will operate in a linear region under all
FIGURE 6. Typical B-H curve, or “hysteresis curve”. Manufacturers of core material may provide such a hysteresis curve as part of their specifications.
To help visualize the performance of an inductor and its core, designers use B-H curves like the one shown in Figure 6. On this curve, H,measured in A/m, is the magnetizing force in the device. It is
measured in Amperes/meter and it is proportional to the current.
The resulting flux density, B is proportional to the integral of the voltage across the device. Magnetic flux density B,measured in units of Tesla, is the strength of the magnetic field.It determines
the force that is exerted upon a moving charge by the magnetic field.
From this curve we can see:
• Permeability, µ. Measured H/m. This is a characteristic of the core material, and it is the rate at which magnetizing force H (driven by current) produces flux density B (integrated voltage). It
is the slope of the B-H curve.Designers use high permeability material to enable physically smaller inductors and transformers.
• Saturation flux density. The point at which additional magnetizing force, H stops producing incremental flux density, B. Designers avoid saturation in most power supply applications.
• Hysteresis characteristics. Hysteresis is the “width” of the curve and indicates loss in power supplies. Most designs seek to use magnetically “soft” core materials to minimize these
□ - Remanence, Br. The magnetic flux density that remains in the material after the magnetizing force, H, drops to zero
□ - Coercive Force, or Coercivity, Hc. The value of H required to drive the flux density, B, to zero.Indications of potential instability include:
• Measured peak flux density close to the saturation flux density specified by the core datasheet indicates that the component is getting close to the saturation
• A BH curves that changes from cycle to cycle, indicating saturation. In a stable/efficient power supply the BH curve will have a symmetrical return path and will trace this path consistently.
An oscilloscope can be used to performs an in-circuit measurement of voltage across, and current through, the winding of an inductor.
Given the number of turns in the device, the magnetic length of the device, and the cross-sectional area of the core, it is possible to derive the actual B and H values based on realtime voltage and
current measurements. From the plotted curve, one can derive the saturation flux density, remnant flux density, permeability and coercivity
FIGURE 7. 5-PWR Magnetic Property measurement gives Hysteresis plot of the magnetic component in the design to check if the magnetic components are at saturation or not as this leads to instability
of the Power Supply.
To generate a B-H plot, the voltage across the magnetic element and the current flowing through it are measured. In the case of a transformer,the currents through the primary as well as secondary
windings are of interest.
A high-voltage differential probe is connected across the inductor or primary winding of the transformer. A current probe measures the current through the inductor or primary. Current probes are also
used to measure the current through the secondary windings, if needed.
In order to analyze the B-H curve with power analysis software, several pieces of information must be provided in the configuration panel:
• Number of turns (N)
• Magnetic path length (l)
• Cross-sectional area (Ae)
The power analysis software can then calculate the magnetizing current using the data from oscilloscope channels where the current probes are connected.
The magnetic property results are presented as shown in Figure 11.
FIGURE 8. Magnetic measurements on multi secondary winding transformer. The Ref1 (white) waveform is the voltage across the inductor and the Ref 2 (blue) waveform is the inductor current. The math
wfm (orange) which is the resultant current wfm is created when the user configures to test multiple secondary windings.
Measurement results
• Permeability: The degree of magnetization of the material
• Bpeak: The maximum magnetic flux density induced in a magnetic component
• Br: The point on the curve where H = 0 but B still has a positive value. This is known as the Remanence of the component, a measure of its Retentivity. The higher the remanence, the more
magnetization the material will retain.
• Hc: The point on the curve where B = 0 and H is a negative value. This represents the external field required to cause B to reach zero. This value of H is known as the coercive force. A small
coercive force value means that the component can be demagnetized easily.
• Hmax: The maximum value of H at the intersection of the H-axis and the hysteresis loop
• I-ripple: The peak-to-peak value of the current
To measure the magnetic characteristics of a transformer under operating conditions, care must be taken to account for current being transferred into the secondary. When measuring the B-H curve on a
transformer, it is helpful to consider a theoretical element call the “magnetizing inductor”. Let’s look at commonlyused multiple-secondary transformer and its equivalent circuit.
FIGURE 9. Transformer schematic (left) and equivalent circuit (right) showing magnetizing current flowing through an imaginary inductor, LM in parallel with the primary. LM models the magnetic
characteristics of the transformer.
The magnetizing current is the current that would flow through the primary of the transformer when the secondary is open (unloaded). In other words, the magnetizing current does not produce any
current in the secondary. As shown in Figure 9, transformers are modeled with this magnetizing current flowing through a “magnetizing inductor” across the primary.Generally, this is used to model the
magnetization characteristics of the core material.
5-PWR software includes the ability to measure the current in transformer secondaries and performing math to isolate the magnetizing current and present an accurate B-H curve of a transformer under
operating conditions.
FIGURE 10. Transformer losses may be modeled as a combination of linear elements.
Loss Analysis
Losses in magnetic components are significant contributors to overall loss in a power supply. Figure 10 shows the major elements determining loss in magnetic components:
• Core loss depends on the magnetic properties of the material and includes hysteresis loss and eddy current loss
• Copper loss due to the resistance of the windings. It also depends on the load connected to the secondary side of a transformer
A number of techniques exist for estimating core loss. One such technique is the Steinmetz empirical formula which, relates core loss to frequency and flux density:
where k, a, and b are constants for the core material, generally take from the core manufacturer’s datasheet. Datasheets may also give loss estimates at various frequencies and flux densities;
however, these are typically given in response to sinusoidal excitation.However, in power applications, components are usually driven with non-sinusoidal stimuli, causing uncertainty in such
The 5-PWR analysis software measures total magnetic loss. It is computed by average power of voltage and current waveforms, average (v(t) ∙ i(t)). The total magnetic loss includes both copper loss
and core loss.
FIGURE 11. 5-PWR Magnetic Loss measurement gives the Total Magnetic Loss which includes the Core Loss and Copper Loss. Designers can find the Core Loss from the component manufacturer’s data sheet
and derive the Copper loss by subtracting the Core loss from the Total Magnetic loss.
5-PWR is capable of calculating the magnetic loss in a single winding inductor, a multiple winding inductor, or even a transformer. In the case of a single-winding transformer, a differential probe
is connected to measure the voltage across the primary winding. A current probe measures the current through the transformer. The power measurement software can then automatically calculate the
magnetic power loss.
The magnetic power loss results are presented as shown in Figure 11. It shows the total power loss in the magnetic component.
1. "Measurement of Inductance using digital storage oscilloscope under real time operating environments-Patent Number-6876936"
2. Alternating Current Machines by, M G Say.
3. In-circuit measurement of saturation flux density Bsat, coercivity Hc, and permiability of magnetic components using a digital storage oscilloscope
Find more valuable resources at TEK.COM
Copyright © Tektronix. All rights reserved. Tektronix products are covered by U.S. and foreign patents, issued and pending. Information in this publication supersedes that in all previously published
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registered trademarks of their respective companies.
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Battery Life Example
Part one:
1. Probability tree
Good battery life
Battery Life
Bad battery life
Low bulb failure
Medium bulb failure
High bulb failure
Low bulb failure
Battery Life
Medium bulb failure
High bulb failure
Battery Life
Probability table
Let A – be good battery life
B – Be bad battery life
L – Be low bulb life
Battery Life
M – Be medium bulb life
H – Be high bulb life
Battery Life
Battery Life
2. A test was undertaken on a random bicycle light and it was found to have low bulb failure what is the probability that this bicycle light has a good battery life?
0.5 x 0.42 = 0.21
3. Memo
The probability of getting a good battery outcome is 0.6, while the probability of getting a bad battery outcome is 0.4 and the probability of getting either a good or bad battery can be calculated
by adding 0.4 + 0.6 = 1.
Battery Life
For the conditional probability we will have good battery life then the probability that the bulb will have low failure is 0.7, the probability that we will have medium bulb failure is 0.2 and
finally the probability that the bulb will have high failure is 0.1.
For the condition that the battery has bad battery life then the probability that we will have low bulb failure is 0.2, the probability that we will have medium bulb failure is 0.3 while the
probability that there will be high bulb failure is 0.5.
From the above conditions it is evident that the probability of bulb failure depends on the nature of the bulb whether the battery has good battery life or bad battery life. When we have good battery
life then the lower is the probability that the bulb will fail and on the other hand if we have bad battery life the higher the probability that we will have bulb failure. Another observation is that
if we randomly choose a battery we have greater chances of choosing a battery with good battery life.
Part two:
Expansion or acquisition
Battery Life
1. Decision tree
Success buy out
Increase revenue by 100million
Battery Life
Increase revenue by 30million
China manufacturer
Glass competitor
Battery Life
90 million increase
70million increase
86million increase
Battery Life
72 million increase
2. Probabilities and expected values:
Battery Life
Glass competitor
China manufacturer
72 million
86 million
90 million
70 million
30 million
100 million
Battery Life
3. Optimum strategy
Battery Life
We get the above probabilities by multiplying the values of each probability, the probability of a 72 million increase in revenue through merging with a glass competitor is calculated by multiplying
0.7 x 0.5 x 0.5 = 0.175, and this probability is the same as the one that will give us an increase of 86 million through buying the glass competitor.
By buying the china manufacturer the probability that there will be an increase in revenue by 90 million is got by multiplying 0.7 x 0.5 x 0.2 = 0.28, while an increase of 70 million in revenue
probability is got by multiplying 0.7 x 0.5 x 0.2 = 0.07.
Through an expansion of the existing facility there is a 0.3 x 0.4 = 0.12 probability that there will be a 30 million increase in revenue whereas there is a 0.3 x 0.6 = 0.18 probability that there
will be an increase in the level of revenue by 100 million.
The optimum strategy is that which has the highest probability, in our case by merging with the china manufacturer and expecting 90 million has an outcome of 0.28 which is the highest of all the
others, this would be the best strategy because it has a high increase of 90 million on revenue and also highest probability to occur.
Ray R and C Mc Keenan (2001) Mathematics for Economics, MIT Publishers , USA | {"url":"https://essaywriter.nyc/article/battery-life-example/","timestamp":"2024-11-14T07:59:22Z","content_type":"text/html","content_length":"122771","record_id":"<urn:uuid:13c0e1ac-5cd8-4b6e-bfb9-5a7ecdc84d27>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00468.warc.gz"} |
How many pillars required for 1000 sq ft house - Civil Sir
How many pillars required for 1000 sq ft house
How many pillars/ column required for 1000 sq ft house | how many pillars do i need to build a 1000 sq ft house construction | number of pillars do i need for 40×25 feet plot.
The number of pillars required for a 1000 square foot house can vary depending on various factors such as the design of the house, local building codes, and the load-bearing capacity of the soil.
Typically, a single-story house of this size might require around 12-15 pillars to support its structure.
A RCC or reinforced concrete column is a compressive structural member of concrete that designed to carry compressive loads, composed of concreting material (mixture of Portland cement, sand,
aggregate or stone with water) with an embedded steel of different diameter such as 12mm, 16mm, 20mm and so — frame to provide reinforcement.
Pillars/ rcc column and beams give strength to a building to withstand all the loading condition and adversities of the climate over the years and stand tall for several decades together. But if the
pillars and beams are placed in the wrong place, they can create adverse effects on the occupants of the house.
A wrongly built pillar or column may considerably hamper the growth of your family members both in the short and the long run and also may create disputes amongst the family members. Thus, Vastu for
pillars has to be necessarily followed for a favorable outcome.
Number of pillars as per Vastu required for 1000 sq ft house construction should necessarily have an even number of pillars in your house for prosperity and peace. If there are an odd number of
pillars in your house, you should try to build one more and make the number to be even. This is a strong suggestion as per the Vastu for pillars in house
There are different types of columns used in construction such as RCC column, long column, short column, timber columns and tied column etc. But todays, RCC column are mostly preferred and use as
structures member for support and resist the load that acting on it, because it is economical and also resist more load easily.
Column size:- generally the size of column should not be less than 9” x 9” inches (230mm × 230mm) in respect to their width and depth for one storey/ first floor/ ground floor building with 4 bars of
12mm Fe500 Steel with m20 grade of concrete and stirrups of T8@6″C/C.
How many pillars/ column required for 1000 sq ft house
When we go through the column design we should mention following point: 1) the possible distance or span and depth it all depend on load to which the beam is subjected, 2) column size depend on
amount of load on the column acting on it by beam and RCC slab, 3) many axial and Lateral load acting on column, 4) large span greatly influence the bending moment not only on beam, but it also
affect the column due to stress acting on it, 5) Thumb Rule for column design are appropriate for very small project like G+1,G+2 & G+3 Building and 6) column design for building construction it
depend on many factors like topography, whether and size of building.
2D and 3D Ghar ka Naksha banane ke liye sampark kare
Others factors also influence the total number of pillars required for 1000 sq ft house construction are ground load bearing condition, purpose of the building, occupation density, height and levels
of the building, specific neighbourhood & location where built, climatic & other geographical conditions, expected life of the building and duration of use, facade & other architectural design
factors & requirements, materials for structural works, external & partition walls if any and materials for finishing and interior. Plumbing & electrical requirements if any.
ALSO READ:-
Steel column size for 2 storey, 3 storey & multistorey building
Steel column size for 5m, 6m, 8m, 10m, 12m, 15m, 20m, 25m & 30m span
Column and beam size for G+0, G+1, G+2, G+3 and G+4 building
How many pillars/ column required for 1000 sq ft house:- there are approximately 12 to 15 nos of pillars are required for 1000 sq ft house construction or for your one storey (G+0), two storey (G+1),
three storey (G+2) building.
1000 sqft of house may be of size 40’x 25’, you have many option:- 1) provide pillars/ rcc column at 10 feet span centre to center so parallel to 40 feet side you will have 5 pillars and for two
sides, it is 5×2=10 no’s of pillars, for 25 feet other side, you will have 5 extra on each side and middle span at 12.5 feet, totalling 5 nos, therefore you will have totally 10+5= 15 no’s of pillars
or column. Thus you will need approx 15 nos pillars to build 1000 sq ft house.
ALSO READ :-
How many pillars required for 1400 sq ft house
How many pillars required for 1200 sq ft house
How many pillars required for 1000 sq ft house
How many pillars required for 900 sq ft house
How many pillars required for 800 sq ft house
2) provide pillars/ rcc column at three column at 15 feet span centre to center and one column at 10 feet span so parallel to 40 feet side you will have 4 pillars and for two sides, it is 4×2=8 no’s
of pillars, for 25 feet other side, you will have 4 extra on each side and middle span at 12.5 feet, totalling 4 nos, therefore you will have totally 8+4= 12 no’s of pillars or column.
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There are approximately 12 to 15 nos of pillars/ column are required for 1000 sq ft house construction. Thus, you will need 12 to 15 number of pillars as per vastu for 1000 sq ft house. | {"url":"https://civilsir.com/how-many-pillars-required-for-1000-sq-ft-house/","timestamp":"2024-11-08T14:44:08Z","content_type":"text/html","content_length":"89108","record_id":"<urn:uuid:69443146-1843-4b05-8deb-668e4863be59>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00869.warc.gz"} |
ML Aggarwal Class 8 Solutions for ICSE Maths Chapter 17 Visualising Solid Shapes Check Your Progress
Question 1.
Write the number of faces, vertices and edges of a
(i) hexagonal pyramid
(ii) octagonal pyramid
(iii) decagonal pyramid
(iv) nonagonal pyramid
(v) heptagonal prism
(vi) decagonal prism.
Question 2.
Give three examples of 3-dimensional shapes around you which are the combinations of 2 or more 3-dimensional shapes.
Question 3.
Give two examples of solids which are not polyhedrons.
Question 4.
Why a pentagonal pyramid having all its edges congruent cannot be a regular polyhedron?
Question 5.
In a polyhedron, if F = 8 and V = 12 then find the number of edges.
Question 6.
Verify Euler’s formula for the following figures:
Question 7.
For each of the given solid, the three views are given. Identify for each solid the corresponding top, front and side views: | {"url":"https://ncertmcq.com/ml-aggarwal-class-8-solutions-for-icse-maths-chapter-17-check-your-progress/","timestamp":"2024-11-02T14:50:38Z","content_type":"text/html","content_length":"59485","record_id":"<urn:uuid:928b3819-72b9-4cd7-817a-c07b0101095f>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00784.warc.gz"} |
rithmetic series
Sample quiz on arithmetic series
Main home here.
1. Is there any difference between a sequence and a series?
2. Find the sum of the first $10$ terms of the arithmetic series $1+2+3+4+\cdots$
3. Find the sum of the first $13$ terms of the series $1+(-1)+(-3)+(-5)+\cdots$
4. Find the sum of the first $20$ terms of the series $1+\frac{3}{2}+2+\frac{5}{2}+\cdots$
5. Which of the following is the correct formula for the sum of the first $n$ terms of the series $1+3+5+7+\cdots$?
6. How many terms of the series $-3+7+17+27+\cdots$ need to be taken to obtain a sum of $1840$?
7. If the sum of the first $15$ terms of the series $3+(3+d)+(3+2d)+(3+3d)+\cdots$ is $-240$, determine the value of $d$.
8. In a competition, the first prize winner takes $\$5,000$, the second winner receives, $\$4,500$, the third winner receives $\$4,000$, and so on. What is the total amount that was paid out to all
the winners?
9. A sum of $\$45,000$ was paid out to all the winners of a maths competition. If the first prize winner got $\$7000$ and each successive winner received $\$500$ less than the previous, how many
winners were there?
10. In terms of $a$, the sum of the first $17$ terms of the series $a+2a+3a+4a+\cdots$ is? | {"url":"https://fridaymath.com/dynamicExercises/grade11/discreteFunctions/discreteFunctions2.html","timestamp":"2024-11-05T20:06:58Z","content_type":"text/html","content_length":"7169","record_id":"<urn:uuid:cb0d02dd-296b-4c59-bab1-4d8b61395f53>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00146.warc.gz"} |
Gaussian Blur Experiments
A follow-up to this article with clarifications and corrections to the “real-world considerations” can be found here.
I researched gaussian blur while trying to smooth my Variance Shadow Maps (for the Shadow Mapping sample) and made a pretty handy reference that some might like… I figure I’d post it for my first
“Tips” blog post. :)
The full article contains a TV3D 6.5 sample with optimized Gaussian Blur and Downsampling shaders, and shows how to use them properly in TV3D. The article also contains an Excel reference sheet on
how to calculate gaussian weights.
Update : I added a section about tap weight maximization (which gives an equal luminance to all blur modes) and optimal standard deviation calculation.
The theory
The gaussian distribution function can be expressed as a 1D function – G(x) – or a 2D function – G(x, y). Here they are :
In these functions, x and y represent the pixel offsets in relation to the center tap, in other words their distance in pixels away from the center tap. The center tap is the center from which all
samples will be taken. The whole distribution can be offset by a “mean” parameter, which would displace the center tap, but that is of no use when blurring; we want the distribution to be centered.
The σ symbol is a non-capitalized greek “sigma”, which is often used in statistics to mean “standard deviation“. That’s exactly what it means here; the standard deviation of the Gaussian
distribution, so how much the function is spread. The standard value is 1, but putting a greater value will spread the blurring effect on more pixels.
The e symbol in this context is Euler’s Number, ~2.7. And of course, π is Pi, ~3.14.
The result of this function is the weight of the tap at x or (x, y), or how much the pixel’s contribution will affect the resulting pixel, after blurring.
So what kind of values does that give? Here are some, having σ = 1 :
G(0) = 0.3989422804
G(1) = G(-1) = 0.2419707245
G(2) = G(-2) = 0.0539909665
G(3) = G(-3) = 0.0044318484
The interpretation for those values is the following :
• At x = 0, which is the center tap, ~0.39 of the pixel’s colour should be kept
• At x = 1 and x = -1, so one pixel offset, ~0.24 of the pixel’s colour should be kept
• And so on, for as much times as we need. More samples = more precision!
So, the weights at each offset are multiplied with the colour of the pixel at this offset, and all these contributions are summed to get the blurred pixel’s colour.
Real-world considerations
Though this is not everything. While testing an implementation in the Bloom shader, I found out two problems :
1. The luminance/brightness of a blurred image is always lower (so darker) than the original image, moreover for large standard deviations
2. The standard deviation for a said number of taps is arbitrary… or is it?
So, back to the drawing board. Or rather the Excel grid.
For the following paragraphs I’ll assume the usage of the 1D gaussian distribution, which is the one we’ll want to use in the end anyway. (see next section)
The luminance issue can be adressed first. Let’s say you have an entirely white single-channel sampling texture, so all pixels have the value 1. So after the first (horizontal) gaussian pass, the
value of every pixel is the sum of weighted sum of all the gaussian samples (taps). For a 5×5 kernel with a standard deviation of 1 :
ΣHBlur = G(0) + G(-1) + G(1) + G(-2) + G(2)
ΣHBlur = 0.9908656625
That might look right, but it’s not. To keep the same luminance, the result would have to be exactly 1. And it does not look prettier after a second pass, which is sampled on the result of the first
ΣHVBlur = 0.9818147611
What that means is that the tap weights need to be augmented with a certain factor to obtain a resulting ΣHVBlur of 1. The answer is very simple, that factor is what’s missing to get 1 : ΣHBlur^-1.
AugFactor = 1 / ΣHBlur
AugFactor = 1.009218543
If we multiply the tap weights by this augmentation factor and sample both passes, ΣHVBlur will be exactly of 1. It works!
The standard deviation issue has a solution as well, but it’s a bit more “home-made” and less formal.
I started with the assumption that a 5×5 blur kernel must have a standard deviation of 1, since that’s what I saw everywhere on the intertubes. Knowing this, the last (farthest) tap weight of a 5×5
kernel is the smallest that any weight in any other kernel size should be; the matter is only to tweak the standard deviation to get a final tap weight as close to this one as possible.
σ = 1 -> G(2)aug = 0.054488685
σ = 1.745 -> G(3)aug = 0.054441945
σ = 2.7 -> G(4)aug = 0.054409483
Those are the closest matches I found using trial and error. Since all of those weights are augmented, their sum is 1 anyway, so finding the best standard deviation is just a matter of extracting as
much blurriness out of an amount of taps without producing banding.
1D Multi-pass vs. 2D Single-pass
But why are there are two functions anyway, one 2D and one 1D? Thing is, we have two choices :
1. Use the 1D version twice; once horizontally, then another time vertically (using the horizontal-blurred texture as a source).
2. Use the 2D version once.
One could ask… why not use the 2D version all the time then? Blurring twice sounds like trouble for nothing, doesn’t it?
Intuitively, I thought this was because of restrictions imposed by earlier shader models.
After more research, I found out that there was a much better reason for separating the operation in two passes.
If n is the linear size of the blurring kernel (or the number of horizontal/vertical taps), and p the pixel count in the image, then :
• With the 1D version, you do 2np texture samples
• With the 2D version, you do n²p texture samples
That means that with a 512×512 texture and a 7×7 blurring kernel, 1D version ends up in doing 3.67 million lookups, compared to the 2D version at a whooping 12.84 million lookups; that’s 3.5 times
For a 9×9 kernel, that means 9²/(2*9) = 4.5 times more sampling!
For the record, separating a 2D kernel in two 1D linear vectors is possible because the gaussian blur is a separatable convolution operation. For more info…
So I’m happy that I started coding the multi-pass, 1D-using version first… because I’m just never going to make that single-pass 2D version. It’ll be way too slow… I’ll leave the 2D formula and
everything, just don’t use them. :P
The implementation
I found the most efficient way to implement 1D gaussian blur in HLSL is the following :
1. In the Vertex Shader, pre-calculate the texture coordinate offsets for the different taps. Each coordinate takes a float2, so I was able to fit in 9 taps by using a 4-sized float4 array (where
.xy is one coordinate and .zw is another!) and the center tap in a separate float2.
Since all “pixel-perfect” tap counts are odd numbers, it’s a good idea to take the center tap as a separate coordinate, then two arrays for the negative and positive taps respectively.
Using the “w” coordinate for free data is not permitted in ps_1_4 and earlier, but anyway I’ve given up in using these old shader versions… D:
2. In the Pixel Shader, use matrices to accumulate the samples. If needed, there can be one matrix for positive samples and another for negative samples, although this is not needed in 5×5. These
matrices have i rows and j columns if i is the number of samples we’re accumulating and j the number of channels we’re processing; typically 4 (r, g, b and a).
Once the samples are gathered, a matrix multiplication between the weights vector and the samples matrix will give the weighted average… as simple as that! Either one or two matrices are used,
once multiplied that gives one or two vectors, which are added together and then added to the weighted center tap.
This may have sounded confusing; reading the HLSL Code of the following sample helps understanding.
TV3D 6.5 Sample
Here it is! A VB.Net 2005 project showcasing an optimized gaussian blur shader, with a downsampling shader. It’s not for beginners, because there is strictly no comment, just well-written code. :)
Note : The demo does not yet reflect the updated formulas. It will come shortly.
GaussianBlur.rar [48kb]
Bonus (blurred) teapot!
Reference documents
As an alternative to the demo (which contains a GaussianDistribution class that does everything that the Excel file does), I have made a nice reference sheet on the 1D and 2D gaussian distribution
functions, well I happen to have made one in Excel and a PDF version for those without MS Office. The Excel version is dynamic, so changing the σ will update all the functions.
Update : Cleaned and updated to reflect my newest findings! (augmentation and standard deviation calculation)
5 thoughts on “Gaussian Blur Experiments”
1. Awsome, thanks for sharing Zak! I’m working on HDR and blurring is one of the areas I’m investing a fair amount of time into, in order to speed things up. =)
2. That would be much appreciated, thank you =) I copied the nvidia blurring sample (post gloom or something like that), and I’m sure it’s a little less than optimized.
3. Nice info and useful excel document, thanks! :) I will link this article from my page.
4. Since you normalize the weights for the blur you can remove the 1/sqrt(2*pi*thea). This a normalization factor and your blend weights will come out the same.
5. Very informative! I just learned here: http://www.coreldraw.com/en/pages/gaussian-blur/index.html about blur effect and how to add it but I need a lot of practice and as much informative stuff
like this. Thank you for sharing it is very helpful
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▷ FACTORS: What are they and how are they calculated? | Aulaprende
The factors of a number are those that are contained in the same number an exact number of times. In fact, when calculating the gcf of two numbers we are calculating the greatest common divisor of
Therefore, given a number, another number is its divisor as long as they are divisible by it.
for example:
How to find the factors of a number
How to find the factors of a number
1. Divide the number whenever possible by the smallest prime number that can be divided.
For example, with the number 256 we divide it by the smallest prime number whenever possible:
So the result is: 1, 2, 4, 8, 16, 32, 64, 128 y 256.
2. Every natural number always has as its divisors itself and 1.
In the previous step it is verified that 256 can effectively be divided between itself and 1.
1. Show the results from least to greatest:
Ejercicios de Máximo Común Divisor
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The number 36 has 9 factors and is a composite number. To calculate the factors of 36 we divide,...
The number 35 has 4 factors and is a composite number. To calculate the factors of 35 we divide,...
The number 4 has 4 factors and is a composite number. To calculate the factors of 34 we divide,...
The number 33 has 4 factors and is a composiite number. To calculate the factors of 33 we divide,...
The number 32 has 6 factors and is a composite number. To calculate the factors of 32 we divide,...
The number 31 has 2 factors and is a prime number. To calculate the factors of 31 we divide,...
The number 30 has 8 factors and is a composite number. To calculate the factors of 30 we divide,...
The number 29 has 2 factors and is a prime number. To calculate the factors of 29 we divide,...
The number 28 has 5 factors and is a composite number. To calculate the factors of 28 we divide,...
The number 27 has 4 factors and is a composite number. To calculate the factors of 27 we divide,...
Factors of 36
Factors of 35
Factors of 34
Factors of 33
Factors of 32
Factors of 31
Factors of 30
Factors of 29
Factors of 28
Factors of 27 | {"url":"https://aulaprende.com/en/divisibility/factors/","timestamp":"2024-11-02T17:15:42Z","content_type":"text/html","content_length":"309052","record_id":"<urn:uuid:a2110fb7-4acc-4a54-abad-af39a2f2675b>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00844.warc.gz"} |
How do you write a lesson plan for a first grader?
How to Make a Lesson Plan for First Grade
1. Start with an objective for each lesson.
2. Write out the lesson in outline form and include any materials needed.
3. Break your lesson up into a timeline.
4. Warm students up for the lesson with an introduction.
5. Present the information by starting with basic facts.
What is the topic of math in Grade 1?
Addition & Subtraction. 1st and 2nd graders extend their previous understanding from kindergarten with adding and subtracting. They begin to memorize their addition and subtraction facts up to 20, as
well as solve word problems using objects, drawings, and equations.
What is the common core for first grade?
In Grade 1, instructional time should focus on four critical areas: (1) developing understanding of addition, subtraction, and strategies for addition and subtraction within 20; (2) developing
understanding of whole number relationships and place value, including grouping in tens and ones; (3) developing understanding …
What should a common core lesson include?
You should include how you will access background knowledge, what you will do to introduce the material, and how students will practice and show mastery of the concept. Materials: a list of
everything you and the students will use during the lesson. Assessment: explain what you will use to assess student learning.
How long should a Grade 1 lesson be?
Although, in first grade, sometimes you actually are introducing a new concept (or expanding on one) every day so it’s okay if you do a whole group lesson every day. Just try to keep them short.
10-15 minutes typically. Sometimes it might even be 5 minutes if it’s just adding to something they already know.
What math facts should a 1st grader know?
In first grade, you can expect your child to learn about:
• Addition and subtraction facts to 20.
• Addition and subtraction as inverse operations.
• Count and write within 120.
• Add within 100.
• Measure objects.
• Tell time to hour and half hour.
• Understand basic fractions.
What should first graders know?
First graders also learn how numbers and basic math are useful in their everyday lives. They learn how to tell time and read a clock face to the nearest half-hour. They learn to understand concepts
such as “an hour from now,” and how to name the days of the week and months of the year.
How do I prepare a lesson plan?
Listed below are 6 steps for preparing your lesson plan before your class.
1. Identify the learning objectives.
2. Plan the specific learning activities.
3. Plan to assess student understanding.
4. Plan to sequence the lesson in an engaging and meaningful manner.
5. Create a realistic timeline.
6. Plan for a lesson closure. | {"url":"https://www.thenewsindependent.com/how-do-you-write-a-lesson-plan-for-a-first-grader/","timestamp":"2024-11-07T00:10:29Z","content_type":"text/html","content_length":"57347","record_id":"<urn:uuid:e21f0959-4d3b-4b97-bbec-4ccdc1a5ed61>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00538.warc.gz"} |
meaning in Hindi
1. Defines a type UGroup composed of all finite sets of UserId values.
2. The function is zero everywhere, except on a finite set of points.
3. A finite set has a smaller cardinality than an infinite set, right?
4. Assume we have an infinite loop involving a finite set of cells.
5. This leads to a simpler description of topologies on a finite set.
6. However, no finite set of bases is sufficient for all composite numbers.
7. Let I be a finite set, representing a collection of people.
8. Informally, a finite domain is a finite set of arbitrary elements.
9. Maximal sets have many interesting properties : they are modulo finite sets ).
10. It follows that every finite set has both a sup and an inf. | {"url":"https://www.hindlish.com/finite%20set/finite%20set-meaning-in-hindi-english","timestamp":"2024-11-06T21:30:03Z","content_type":"text/html","content_length":"38048","record_id":"<urn:uuid:ef4568bb-4067-4d6e-a312-99d63f55239c>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00535.warc.gz"} |
Sketch of my math dance piece - Math For Love
Sketch of my math dance piece
February 13, 2009
It’s called “What I was working on today.”
I come out and aggressively challenge the audience:
“I’m Dan, and I’m a math student PhD, and tonight one of you is going to come up here and solve a math problem, because you’re not dumb. You—you look nervous. This is easy—everyone can do this. You—”
Then I step back and relax.
“When I tell people I do math, sometimes I think that’s what they expect me to do. I’ve got news—doing math and being smart have nothing to do with each other. There are a lot of dumb mathematicians
in the world and a lot of very smart people who don’t do math at all.
“I’m going to tell you a story… A teacher once gave his class an assignment to keep them busy: add up the numbers 1+2+3+4+5+6+7+8+9 and keep adding until you get to 100. The teacher figured this
would be a good way to keep all the kids busy, and the kids set to work, adding 1+2+3+4 and it was a waste of time for all of them. Because they weren’t making their own choices, they were doing it
the way the teacher told them. And you know what—it was boring for them, and they all they all got it wrong. But one student made a choice that was his own: instead of adding in the teacher’s order,
he thought, what if I add the biggest and smallest numbers together? So he added the 1 to the 100, and got 101, and set it aside. Then he put the 2 and the 99 together and got 101, and set it aside;
then he added 3 to 98 and got 101 and put it aside; and added 4 to 97 and got 101 and put it aside; and he realized that if he paired the numbers this way he would always get 101. How many pairs? 100
numbers means 50 pairs, so that’s fifty sets of 101. That’s just a multiplication problem! Fifty times 101. He could do that, and he did: the answer was 5050, and he was the only one to get it right,
and in doing so he discovered a beautiful and powerful pattern.
This is what doing math feels like: you make choices and discover beautiful patterns. I can’t give you the full feeling of what it’s like to be a mathematician, but I think it starts with asking a
question. Somewhere in there you all have questions about math that haven’t been answered. Today, I want to give you the opportunity to ask them, and if I can, I’ll answer them. My brain is open.”
And then I take questions. After I answer a few, I say “I’m going to show you what I was working on today.” Then I move into a dance where I try to relate physically what my current work looks like
and feels like.
Notes from my group: I need to practice more on fielding the questions so I can make ANY question sound really interesting, and take it in cool directions. Second, they liked my dancing, and thought
I could expand it from a brief taste to fuller arc.
It’s pretty exciting stuff. Afterwards people came up and talked to me about math in their lives. I need to start getting interviews and lectures on this blog.
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Convert 26 Metres to Feet - A Simple Length Conversion Guide - Furniture Experts Handyman
Convert 26 Metres to Feet – A Simple Length Conversion Guide
With pleasure, I’ll assist you! Our calculation tool reveals that 26 meters corresponds to three distinct measurements: 85 feet, 3.62 inches; 1,023.62 inches; and 2,600 centimeters. In essence, 26
meters roughly translates to 85 feet, 3.62 inches, or 1,023.62 inches, or 2,600 centimeters.
How Many Feet Are in 26 Meters?
26 meters, that’s a decent distance. You know, the kind of stretch you’d do to reach the top shelf in your pantry. But, of course, the question is, how many feet is that? To break it down, a meter is
approximately equal to 3.28 feet. So, if you multiply 26 meters by that conversion rate, you’ll get a total of roughly 85.68 feet.
Now, 85 and a bit feet might not be an everyday measurement, but it’s still a decent chunk of real estate. Imagine a room about 28 feet long – you could fit a nice big sofa in there, maybe a coffee
table and a TV, and still have some space to move around. Or, if you’re thinking in terms of outdoor spaces, 85 feet might be the length of a nice-sized tennis court. The point is, 26 meters is a
significant distance, and once you convert it to feet, you get a better sense of just how much space we’re talking about.
What’s interesting is that, in certain contexts, 85 and a half feet might be more relevant than the original 26 meters. For instance, if you’re planning a deck or a patio, the length of the space
might be more important than the distance. And, of course, there are all sorts of other conversions that come into play when working with lengths – inches, yards, meters, you name it. The key is to
understand how they all relate to each other, and that’s where the magic happens.
In the end, it’s all about perspective. Whether you’re thinking in terms of meters or feet, the beauty of it is that we can work with these different measurements to get a better understanding of the
world around us. And that, my friend, is no small feat.
What is the Conversion of 26 Meters to Feet?
Meters and feet are two different units of measurement, with meters being the standard unit in the International System of Units (SI). Feet, on the other hand, are commonly used in everyday
applications, such as architecture, construction, and sports. Since they’re not interchangeable, we need to convert meters to feet if we want to express a measurement in feet.
To convert 26 meters to feet, we can use a simple formula: 1 meter is equal to approximately 3.2808 feet. Now, let’s apply this conversion factor to our initial measurement of 26 meters.
• Multiply 26 meters by 3.2808 feet per meter
• 26 meters x 3.2808 feet/meter = 85.3088 feet
How Far is 26 Meters in Feet?
Let’s dig in and explore how far 26 meters actually is. So, you know how we often talk about measuring distances in a bunch of different ways? Like, we’ve got meters, feet, inches, yards – the whole
works! And today, we’re going to take a closer look at what 26 meters looks like in feet.
See, 26 meters is a bit of a weird number in the grand scheme of things. It’s not quite divisible by a lot of other numbers, so it doesn’t have a super straightforward conversion to feet. But don’t
worry, we’re about to dive into the specifics. First, let’s talk about how many centimeters 26 meters is equal to. That’s right, centimeters! Because, you see, when we break down 26 meters into
smaller pieces, we find that it’s actually equal to 260,000 centimeters. Now, when we convert those centimeters into feet, we get. . (drumroll please). . about 85 feet and 2 inches!
Yeah, that’s a pretty decent size, right? I mean, if you’re thinking about walking along a beach or something, 26 meters might not seem like a huge distance, but when you break it down into feet, it
definitely starts to feel a bit more substantial. And hey, if you’re trying to measure out a room or something, 85 feet and 2 inches is definitely a tangible number that’ll give you a better sense of
scale. So there you have it – 26 meters in feet, folks!
How Long is 26 Meters in Feet?
So, you’re wondering how long 26 meters really is. It’s not every day we deal with lengths in meters, is it? But trust me, it’s easier than you think!
To convert 26 meters to feet, we need to do a simple math trick. One meter is equal to 3.2808 feet, so let’s multiply 26 meters by this conversion factor.
26 meters x 3.2808 feet/meter = 85.3 feet
That’s right! 26 meters is approximately equal to 85.3 feet. Now, imagine a 26-meter-long bus or a swimming pool. That’s quite a stretch!
But here’s a fun fact: the average length of a swimming pool is around 25 meters, which converts to about 82 feet. So, our 26-meter benchmark is only a tiny bit longer than the average pool.
In everyday life, understanding measurements like this can be super helpful. Whether you’re planning a renovation, designing a garden, or just curious about the world around you, being able to
convert between units can make a big difference. | {"url":"https://www.furnitureexpertshandyman.com/convert-26-metres-to-feet-a-simple-length-conversion-guide/","timestamp":"2024-11-10T11:07:30Z","content_type":"text/html","content_length":"153640","record_id":"<urn:uuid:4ba9a107-6df1-4ef3-afb2-f6bbc0102d4e>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00665.warc.gz"} |
HHenry is going on a trip to Brazil. He is currently deciding which cities he will visit during his stay. The table below shows
All categories
HHenry is going on a trip to Brazil. He is currently deciding which cities he will visit during his stay. The table below shows
potential cities he could visit, as well as what he estimates it will cost him to visit each one. All costs are given in Brazilian reals (R$).
Cost (R$)
Duque de Caxias
Rio de Janeiro
Porto Alegre
Henry’s original plan was to visit Duque de Caxias, Campinas, Rio de Janeiro, and Guarulhos. However, he only has R$699 to spend, so he cannot afford to use his original itinerary. Which of the
following changes will be the cheapest swap Henry can make and still come in under budget?
Replace Duque de Caxias with Salvador.
Replace Duque de Caxias with Fortaleza.
Replace Rio de Janeiro with Porto Alegre.
Do not visit Guarulhos.
2 answers:
Step-by-step explanation:
First, sum up the total expenses for the original plan
Duque de Caxias + Campinas + Rio de Janeiro + Guarulhos = 242+185+193+120 = R$740
It is clearly over the intended budget. The average cost per itinerary is around R$200 so we will consider making a swap on either Duque de Caxias and Rio de Janeiro. Possible answers are (b) or (c)
for (b) Replace Duque de Caxias with Fortaleza, we will have
Fortaleza + Campinas + Rio de Janeiro + Guarulhos =
⇒ 201+185+193+120 = R$699 ⇒ This is exactly the budget intended for the trip
for (C) Replace Rio de Janeiro with Porto Alegre
Duque de Caxias + Campinas + Porto Alegre + Guarulhos =
⇒ 242 + 185 + 153 +120 = R$ 700 ⇒ This is over R$1
The answer is (b)
You might be interested in
29.97 square feet
Step-by-step explanation:
The length of rectangle = 9 ft
The width of rectangle = 40 in
We have to bring the length and width in same unit. As the area is required in square feet so we will convert width in feet
The width will be divided by 12 to be onverted into feet
Width = 40/12 = 3.33 feet
Area = Length * width
= 9 * 3.33
= 29.97 square feet ..
(f + g)(x) = I2x + 1I + 1 ⇒ C
Step-by-step explanation:
Let us solve the question
∵ f(x) = I2x + 1I + 3
∵ g(x) = -2
→ We need to find (f + g)(x), which means add the two functions
∵ (f + g)(x) = f(x) + g(x)
→ Substitute the right side of each function on the right side
∴ (f + g)(x) = I2x + 1I + 3 + (-2)
→ Remember (+)(-) = (-)
∴ (f + g)(x) = I2x + 1I + 3 - 2
→ Add the like terms in the right side
∵ (f + g)(x) = I2x + 1I + (3 - 2)
∴ (f + g)(x) = I2x + 1I + 1
Step-by-step explanation:
There are i<span>Infinitely many solutions </span>
Step-by-step explanation:
First, you have to graph the equation. Then you notice that it is a parabola facing downwards. This means that as the graph decreases on the left side, the x's become negative and the y's become
negative. On the right side, the x's become positive and the y's become negative.
Add answer | {"url":"https://answer.ya.guru/questions/2308-hhenry-is-going-on-a-trip-to-brazil-he-is-currently-deciding.html","timestamp":"2024-11-09T19:57:19Z","content_type":"text/html","content_length":"60529","record_id":"<urn:uuid:663e8f26-8e62-4055-9922-1d72fb65fbcf>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00882.warc.gz"} |
Mathematics and Understanding | Roger Antonsen
Mathematics and Understanding
(Published in
December 20, 2013, in Norwegian.)
There is currently a lot of talk about mathematics, standardized tests and Norwegian pupils' "mathematics skills". However, almost no one talks about what we actually mean when we say "mathematics",
or what mathematical understanding and experience we want the students to achieve in school. Do we want the students to really understand what they are doing, or is it sufficient that they become
good at calculations?
The American math teacher Paul Lockhart wrote in 2002 a pamphlet entitled "A Mathematician's Lament" (PDF), where he compares the teaching of mathematics in school with a music education without
music, where you are focusing solely on music reading and notation and where you neither play instruments or listen to music.
He describes the American school as a school where mathematics is gone, where the only thing that remains is an empty shell, and where the mathematical understanding and experience no longer has a
place. It is a harsh criticism, but I agree and think the same can be said about a lot of the mathematics in today's Norwegian school.
For what is really the essence of mathematics? Think about it before you continue reading. Imagine that you had to explain and define what mathematics is. What would you have said, what would you
have focused on? Would you have begun to explain the four arithmetic operations (plus, minus, multiply and divide) and what numbers, algebra, fractions and functions are?
No. Mathematics is something deeper, more fundamental, colorful and beautiful than that. Many believe that mathematics mainly all about numbers, formulas and calculations. However, this is wrong.
Mathematics is the art of thinking.
Mathematics is about making assumptions and reasoning towards the consequences of these assumptions. Think of it as making the rules of a game, like chess or connect-four. Once you have defined the
rules, the game is given. If you make good rules, you get good games. This is also how it is in mathematics. All rules of the game are more or less arbitrary, but they are defined as they are for
creating good games.
I often say to my students that they can assume anything, but they must also accept the consequences that follow. If they want to assume that 2+2=5, then that is perfectly fine. Mathematics is the
art of thinking from assumptions.
Mathematics is about finding patterns and abstracting. Look around you and look for patterns and symmetries. It is almost impossible not to find them. When you recognize forms or see that two people
are related, then you are a pattern detector. Mathematics is the art of finding patterns.
Mathematics is about finding good representations, and to be able to switch between them. If you're able to represent one and the same phenomenon in two different ways, then that enables you to see
new connections. Why is there such a great focus on solving equations in school? It is not to make children good at applying calculation rules without thinking: it is because the equations represent
something else. Mathematics is the art of finding good representations.
Above all mathematics is about using the imagination and being creative. If you imagine a circle, then this is something that exists only as an idea in your consciousness. And this circle is
absolutely perfect, unlike circular looking objects in nature. Mathematics is the art of using the imagination in order to reason about such abstract ideas.
Doing math in this way is to be an explorer. To discover and prove mathematical truths becomes an art form. One asks questions like "What is really pi? How long is the diagonal of a square? Is
infinity a number? Do infinite sets exist?" and spend time trying to understand, instead of using rules for calculation and putting numbers into formulas.
A great way to explore and discover mathematics is through programming. Why is not programming and computer science an integral part of school? I do not mean a word processing, spreadsheets, web
pages and graphics, but proper programming, where you make your own programs and tell a computer exactly what to do.
In several ways, computer science and mathematics are two sides of the same coin: It is about making rules, finding patterns and representations, exploring consequences, abstracting, and solving
problems. And doing this in a way that is dynamic and safe: When one makes a mistake, then one does not throw entire program away and start over; one looks for the mistake and modifies the program
such that it becomes correct.
This debugging process is extremely constructive and useful and has great transfer value to other parts of life. By teaching children basic programming, they get a tool that enables them to explore
mathematics themselves, experiment with models, define their own rules and simulate natural processes. A computer can thus be an extremely useful tool for achieving greater insight and understanding.
Many take pride in the fact that they never understood math. And when I ask children about what mathematics is, very many of them just answer, "boring". That does not surprise me. We have managed to
make the most creative and colorful of all subjects into something that it is ok to dislike. If you think about it, it is not so strange. We teach a lot of rules, facts and formulas and expect that
understanding in a magical way emerges through a lot of training and repetition of formulas.
It is an extreme optimism to believe that even more calculations training and exercise solving is what it takes to increase understanding of mathematics. The optimism lies in believing that
understanding comes automatically. It does not. Being good at applying rules, in a precise and accurate manner, is not the same as understanding what you are doing.
We have taken away the discoveries, the creativity, the beauty, the experience, the colors and the essence. This has become a cultural problem that there is no simple solution to. However, we must
give children and adults the opportunity to fall in love with mathematics, to look at it as something that is beautiful and harmonious, just like a fine piece of music.
We must get them to see the art and the experience in mathematics. We do not do this by even more calculations, more thoughtless application of formulas and more repetition. We do this by creating
wonder, joy, fascination and curiosity, by opening up for exploration and by telling good stories.
• 💬
08.07.2023 13:40
What a wonderful text!!!
I'm an artist and I was never good at math.
Today I research natural structures, nature's design and in that investment, I discovered the beauty of mathematics. I feel so passionate that I think about doing a degree in this area.
His speech is precise and true.
Christine Rocha | {"url":"https://rantonse.org/blog/2013-12-20","timestamp":"2024-11-13T18:23:50Z","content_type":"text/html","content_length":"54216","record_id":"<urn:uuid:68f7d665-0496-48da-ab53-6900204b5560>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00607.warc.gz"} |
An introduction to conservative vector fields
Many physical force fields (vector fields) that you are familiar with are conservative vector fields. The term comes from the fact that some kind of energy is conserved by these force fields. The
important consequence for us, though, is that as you move an object from point $\vc{a}$ to point $\vc{b}$, the work performed by a conservative force field does not depend on the path taken from
point $\vc{a}$ to point $\vc{b}$. For this reason, we often refer to such vector fields as path-independent vector fields. Path-independent and conservative are just two terms that mean the same
For example, imagine you have to carry a heavy box from your front door to your bedroom upstairs. Because of the gravity (which can be viewed as a force field), you have to do work to carry the box
Here we mean the scientific definition of work, which is force times distance. Although it may feel like work to move the box from one room to another on the same floor, the actual work done against
gravity is zero.
Next, imagine that you have two stairways in your house: a gently sloping front staircase, and a steep back staircase. Since the gravitational field is a conservative vector field, the work you must
do against gravity is exactly the same if you take the front or the back staircase. As long as the box starts in the same position and ends in the same position, the total work is the same. (In fact,
if you decided to first carry the box to your neighbor's house, then carry it up and down your backyard tree, and then in your back door before taking it upstairs, it wouldn't make a difference for
this scientific definition of work. The net work you performed against gravity would be the same.)
The line integral of a vector field can be viewed as the total work performed by the force field on an object moving along the path. For the above gravity example, we discussed the work you performed
against the gravity field, which is exactly opposite the work performed by the gravity field. We'd need to multiply the line integral by $-1$ to get the work you performed against the gravity field,
but that's a technical point we don't need to worry much about.
The above discussion implies that if $\dlvf$ is a conservative vector field, then the integral \begin{align*} \dlint \end{align*} does not actually depend on the path $\dlc$. It only depends on the
beginning point (say $\vc{a}$) and endpoint (say $\vc{b}$) of the path $\dlc$. If we changed the path $\dlc$ to a different path $\adlc$, then we know that \begin{align*} \dlint = \lint{\adlc}{\dlvf}
\end{align*} as long as the path $\adlc$ also starts at the point $\vc{a}$ and ends at the point $\vc{b}$. This property is certainly not true for all vector fields $\dlvf$ (as shown below). It is
only true for the special class of conservative vector fields, which is why we can call them path-independent vector fields.
The vector field $\dlvf(x,y)=(x,y)$ is a conservative vector field. (You can read how to test for path-independence later. For now, take it on faith.) It is illustrated by the black arrows in the
below figure. We want to compute the integral \begin{align*} \dlint \end{align*} where $\dlc$ is a path from the point $\vc{a}=(3,-3)$ (shown by the cyan square) to the point $\vc{b}=(2,4)$ (shown by
the magenta square). Since $\dlvf$ is path-independent, we don't need to know anything else about the path $\dlc$ to compute the line integral. Later, you'll learn to compute that the value of the
integral is 1, as shown by the magenta line on the slider below the figure.
The below applet demonstrate the path-independence of $\dlvf$, as one can see that the integrals along three different paths give the same value. The vector field appears to be path-independent, as
promised. (You'd have to check all the infinite number of possible paths from all points $\vc{a}$ to all points $\vc{b}$ to determine that $\dlvf$ was really path-independent. Fortunately, you'll
learn some simpler methods.)
The line integral over multiple paths of a conservative vector field. The integral of conservative vector field $\dlvf(x,y)=(x,y)$ from $\vc{a}=(3,-3)$ (cyan diamond) to $\vc{b}=(2,4)$ (magenta
diamond) doesn't depend on the path. Path $\dlc$ (shown in blue) is a straight line path from $\vc{a}$ to $\vc{b}$. Paths $\adlc$ (in green) and $\sadlc$ (in red) are curvy paths, but they still
start at $\vc{a}$ and end at $\vc{b}$. Each path has a colored point on it that you can drag along the path. The corresponding colored lines on the slider indicate the line integral along each curve,
starting at the point $\vc{a}$ and ending at the movable point (the integrals alone the highlighted portion of each curve). Moving each point up to $\vc{b}$ gives the total integral along the path,
so the corresponding colored line on the slider reaches 1 (the magenta line on the slider). This demonstrates that the integral is 1 independent of the path.
Of course, the line integrals to intermediate points can vary widely. By comparing the tangent vectors of the curves to the vector field, you should be able to explain the behavior of the line
integrals to intermediate points. In particular, you can determine why the line integral increases or decreases by comparing the direction of motion along the path to the direction of the vector
A path-dependent vector field
As mentioned above, not all vector fields are conservative. If a vector field is not path-independent, we call it path-dependent (or non-conservative). The vector field $\dlvf(x,y) = (y, -x)$ is an
example of a path-dependent vector field. This vector field represents clockwise circulation around the origin. It turns out, such circulation is the key indicator of path-dependence (however, the
circulation may not be as obvious as it is in this example). Again, you can read the details later.
For a path-dependent vector field, the line integral from $\vc{a}$ to $\vc{b}$ isn't defined until you specify a path, as the value can depend on the details of the path. The vector field $\dlvf(x,y)
= (y, -x)$ and three paths from $\vc{a}=(0,-4)$ to $\vc{b}=(0,4)$ are shown in the below applet. The three paths demonstrate that $\dlvf$ is indeed path-dependent, as the three integrals from $\vc{a}
$ to $\vc{b}$ have different values. Indeed, you only need to show that the integrals along two paths from some point $\vc{a}$ to some other point $\vc{b}$ are not the same in order to prove that $\
dlvf$ was path-dependent (since if $\dlvf$ were path-independent, this could never happen).
The line integral of a path-dependent vector field. The vector field $\dlvf(x,y) = (y,-x)$ and three paths from $\vc{a}=(0,-4)$ (the cyan diamond) to $\vc{b}=(0,4)$ (the magenta diamond) as shown.
Path $\dlc$ (shown in blue) is a straight line path from $\vc{a}$ to $\vc{b}$. Paths $\adlc$ (in green) and $\sadlc$ (in red) are clockwise and counterclockwise, respectively, circular path from $\vc
{a}$ and to $\vc{b}$. You can drag the colored points, and the corresponding color lines on the slider indicate the line integral of $\dlvf$ from $\vc{a}$ along the path up to the colored point (the
highlighted portion of the curve). Even when you drag the points to $\vc{b}$, the values of the line integrals have different values, demonstrating that $\dlvf$ is indeed path-dependent.
Path $\dlc$ (in blue) is a straight line up the $y$-axis. Along the $y$-axis, $\dlvf(0,y)=(y,0)$ is horizontal. Hence the direction of motion and the vector field are orthogonal, and \begin{align*} \
dlint =0. \end{align*} On the other hand, path $\adlc$ (in green) goes with the clockwise rotation of the vector field. Consequently, the work performed by the vector field is positive, and \begin
{align*} \lint{\adlc}{\dlvf} >0. \end{align*} The same reasoning shows that \begin{align*} \lint{\sadlc}{\dlvf} <0, \end{align*} since the counterclockwise direction of path $\sadlc$ (in red) is
opposite the direction of the vector field. Since all three paths start at $\vc{a}$ and end at $\vc{b}$, these results confirm that $\dlvf$ is path-dependent.
Calculating integrals of conservative vector fields
One of the four fundamental theorems of vector calculus is the gradient theorem for line integrals, which gives a nice way to calculate of line integral of conservative vector fields. To use the
gradient theorem, one needs to determine if a vector field is conservative. If a vector field is conservative, one can find a potential function analogous to the potential energy associated with
conservative physical forces. Once the potential function is known, it is very simple to calculate line integrals, as shown by this example of calculating a line integral using the gradient theorem. | {"url":"https://mathinsight.org/conservative_vector_field_introduction","timestamp":"2024-11-10T08:18:16Z","content_type":"text/html","content_length":"84739","record_id":"<urn:uuid:cea6a832-49aa-4b40-b857-6b13b3a08f4c>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00582.warc.gz"} |
NetLogo User Community Models
(back to the NetLogo User Community Models)
If clicking does not initiate a download, try right clicking or control clicking and choosing "Save" or "Download".(The run link is disabled for this model because it was made in a version prior to
NetLogo 6.0, which NetLogo Web requires.)
## WHAT IS IT?
This model is derived from and inspired by:
Centola, Damon, Robb Willer, and Michael Macy. “The Emperor’s Dilemma: A Computational Model of Self‐Enforcing Norms.” American Journal of Sociology 110, no. 4 (January 1, 2005): 1009–1040.
The basic idea is to test under what conditions people will not only comply with norms they privately disbelieve (i.e. 'FALSE COMPLIANCE'), but also when they will actively enforce them (i.e. 'FALSE
1. agents observe neighbors compliance and enforcement.
2. Each agent then makes two decisions:
(i) whether to comply with the norm, and
(ii) whether to enforce the norm.
## HOW IT WORKS
Below are the main agent variables:
Beliefs: B = 1 if believes; B = -1 if doesn't believe.
Strength (of Belief): varies from 0 to 1
Compliance: 1 if agent complies with the norm, and 0 the agent does not comply. Enforcement: 1 if agent enforces norm, -1 if enforces deviance, and 0 if agent doesn't enforce at all.
First, agents must decide whether to comply with the norm. In this model, a disbeliever complies if the proportion of neighbors enforcing compliance is greater than the strength of disbeliever's
belief. For example, if the strength of disbelief is .5, but 60% of an agent's neighbors are enforcing compliance, then this agent will also comply. Thus, this depends on the question of enforcement,
given next.
Second, agents make a decision to enforce based on whether those around them are complying with their private norms. In this model, the "Need to Enforce" is inversely related to the proportion of
neighbors complying with their private beliefs! This is a somewhat strange assumption. In practice, it means that an isolated individual constituting an extreme minority is more likely to impose his
or her beliefs on others when nobody else believes them. A more plausible approach is that groups are more likely to enforce norms on minorities, but that will wait for another model. Because of this
assumption, enforcement drops whenever full compliance is achieved, causing the system to swing back towards non-compliance.
Note that the parameter "K" refers to the Cost of Enforcement.
"Rewiring probability" here is the same as that used in "Small Worlds" algorithm, but instead of rewiring links, it asks each neighbor and randomly assigns one of these neighbors to an agentset
N_Neighbors if probability is below "rewiring probability". The "Small Worlds" parameter will set up small-worlds links. At the limit, where re-wiring probability = 1, small-worlds is the same as the
global condition.
## Other Assumptions
- No hypocritical enforcement; Agents can only enforce compliance if they also complied, and can only enforce deviance if they have deviated.
- Initial condition is that agents conform to their own private beliefs and no one enforces anything.
- In the article, strength of belief, S, of disblievers ranges 0 < S <= .38 The mean of their conviction is 0.19.
## THINGS TO TRY
Change the "Conversion" setting. Conversions allow the beliefs of agents to change according to a stochastic process.
The parameter K (cost of enforcement) was fixed in the article cited above. It turns out that the interesting results do not obtain when the cost is varied.
There are several conditions in which this model can be run:
- Global: Each agent can interact with any other agent.
- Local: Each agent can interact only with its N closest neighbors, where N is set by "Influence_Range." There are two local conditions: i) Clustered, or ii) Random. These refer to whether the
BELIEVERS are initially clustered or randomly distributed.
The interesting finding of this article is that cascades of false compliance and false enforcement will only be generated if the BELIEVERS (i.e. zealots) are initially clustered together, and agents
can only interact locally (i.e. they lack global or outside information).
Centola, Damon, Robb Willer, and Michael Macy. “The Emperor’s Dilemma: A Computational Model of Self‐Enforcing Norms.” American Journal of Sociology 110, no. 4 (January 1, 2005): 1009–1040. | {"url":"http://ccl.northwestern.edu/netlogo/models/community/BRADFORD%20SOCIAL%20NORMS%201-31-2014%20Emperors%20Dilemma","timestamp":"2024-11-10T22:13:08Z","content_type":"text/html","content_length":"8874","record_id":"<urn:uuid:30816c51-b279-4af1-99e1-87ddafc25586>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00124.warc.gz"} |
Hancock Court
Waitlist Forecast
Since this property may have a Section 8 Property Based Rental Assistance contract with rental assistance available to its residents, it is likely the property has a waiting list. Sometimes these
waiting lists can be months or years long and may or may not be accepting new applicants. You should contact the property to verify.
Note: Affordable Housing Online has not confirmed the waiting list status directly with the property. This waiting list status forecast is based on our housing experts' knowledge of similar
properties. You should always confirm this status directly with the property.
FAQs for Hancock Court
Does Hancock Court have a waiting list?
Hancock Court probably has a waiting list. Since our research has shown there is a project-based rental assistance associated with this property, the waiting list would be for the rent-assisted
units. We haven't confirmed if this is the case, so you should contact the property to confirm.
Are rents subsidized at Hancock Court?
We believe rents are most likely subsidized at all or some of the units since our research has turned up at least one public housing or rental assistance contract probably associated with this
In addition to Hancock Court, how many other affordable housing properties (and units) are there in Quincy?
In addition to Hancock Court, we list 55 other affordable properties in Quincy. Learn about these properties here. Our data research shows that there are a total of 1872 affordable units in Quincy. | {"url":"https://affordablehousingonline.com/housing-search/Massachusetts/Quincy/Hancock-Court/10006999","timestamp":"2024-11-08T14:40:06Z","content_type":"text/html","content_length":"228908","record_id":"<urn:uuid:49fa201c-cd50-48ca-81a9-bb82650c1859>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00756.warc.gz"} |
2142405 - Gujarat Technological University
POWER ELECTRONICS (24)
SUBJECT CODE : 2142405
B.E. 4
Type of course: Engineering Science (Electronics)
Prerequisite: 1) 2130901 -Circuits and Networks, 2) 2132404 -Principles of Power Electronics
Rationale: This subject focuses on the study of linear applications of the Diode, BJT and Op-Amp in the field of Power Electronics. It details the students about use of analog electronics in power
electronics control.
Teaching and Examination Scheme:
Teaching Scheme
T P
Examination Marks
Theory Marks Practical Marks
PA (M)
PA (V)
Diodes & Bipolar Junctions Transistors:
Diode as a rectifier: HW, FW and Bridge Rectifier with and without
Capacitor Filter – Review of CE, CB, CC Configurations of BJT –
Review of Low Frequency Response of BJT – The Basics of High
Frequency Response and h-Parameters
Different Areas of Application of BJT
Linear Applications of BJT:
Single stage transistor amplifiers – Classification – Principle –
Practical circuit – DC and AC equivalent circuits – Load line analysis – Gain and Input impedance – Equivalent Circuit –
Classification – Different amplifiers like Cascaded Amplifier,
Coupled Amplifier, Feedback Amplifier, Cascode Amplifier, Class
A, B, AB & C amplifiers – Frequency response of amplifier – bootstrapping circuit
Block diagram and working of Linear regulated power supplies –
Zener Shunt – Series Pass – Shunt Regulated – Feedback – Series
Voltage Regulator – IC based regulated linear power supplies like
78xx, 79xx & TL431
Design of simple series/shunt regulated power supplies using BJT,
Design of simple regulated power supplies based on 78xx, 79xx and
TL431 ICs.
Operational Amplifier Basics:
The Op-Amp – Block Diagram – Characteristics – Equivalent Circuit
– Types – Important parameters from datasheet
The Ideal Op-Amp – Open Loop – Close Loop Configurations along with the different types of feedback
The Practical Op-Amp – Characteristics – Input Offset Voltage –
Input Bias & Offset Current – Total Output Offset Voltage – CMRR
– Thermal & Time Drift – PSRR – Effect of Power Supply Variation on Offset Voltage – Frequency Response and Slew Rate – Input
Protection Circuit for Op-Amp – Noise in Op-Amp
Linear Applications of Op-Amp:
AC-DC Amplifier – Peaking Amplifier – Summing, Scaling and
Averaging Amplifier – Instrumentation Amplifier – Differential
Input-Output Amplifier – V to I converter (Grounded & Floating
Load) – Very High Input Impedance Circuit – Integrator &
Differentiator – A/D & D/A Converters
Comparators – Zero Crossing Detector – Schmitt Trigger – Voltage
Limiter – Voltage Level Detectors –Clipper (Limiting Circuits) &
Clampers – Peak Sample & Hold Circuit – Precision Rectifier
Op-Amp Based Filters & Oscillators:
Introduction to Filters – Passive Filters & Active Filters – Basic
Filter Responses and their Transfer Function (Low Pass, High Pass,
Band Pass, Band Reject, Notch, All Pass) – First & Second Order
Butterworth Filter
Gyrator – Negative Impedance Converter (NIC) – Frequency
Dependent Negative Resistance (FDNR)
Oscillators – Principles & Types – Frequency Stability – Phase Shift
Oscillator – Wien Bridge Oscillator – Quadrature Oscillator
Square Wave Generator – Triangular Wave Generator – Saw Tooth
Wave Generator
Special Circuits and Applications:
Switched Capacitor Circuits – Principle – Advantages like Input
Impedance Improvement, CMRR Improvement – Applications of
Switched Capacitor Circuits like amplifier, filter, etc. – MF5
Switched Capacitor Filter
555 Timer – different modes and their configuration
Block Diagram and Working V/F, F/V Converters, VCO PLL, PWM circuits and Voltage Regulators
Logic Families:
Review of Logic Gates – Construction of Logic Gates using Discrete
Components like Diode, Resistor, Transistor, etc. – Diode Transistor
Logic (DTL) – Transistor Transistor Logic (TTL) – Resistor
Transistor Logic (RTL) and other logic families – Comparison of
Logic Families
Suggested Specification table with Marks (Theory):
R Level
Distribution of Theory Marks
U Level
A Level
N Level
E Level
Legends: R: Remembrance; U: Understanding; A: Application, N: Analyze and E: Evaluate and above Levels
(Revised Bloom’s Taxonomy)
Note: This specification table shall be treated as a general guideline for students and teachers. The actual distribution of marks in the question paper may vary slightly from above table.
Reference Books:
Operational Amplifiers and Linear ICs, Third Edition by David Bell, Oxford
Electronics Devices and Circuits, Fifth Edition, David Bell, Oxford
Op-Amps and Linear Integrated Circuits, 4 th Edition by Ramakant Gayakwad, PHI
Integrated Electronics by Millman, Halkias and Parikh, McGraw Hill India
A monograph on Electronics Design Principles by Goyal and Khetan, Khanna Publishers
Digital Electronics: Principles and Applications by Soumitra Kumar Mandal
Electronics Devices and Circuits by Boyelsted and Nashelsky, Pearson
Fundamentals of Digital Electronics by A. Anand Kumar
Course Outcome:
After learning this course, the students should be able to:
Understand the importance of linear applications of semiconductor components like diodes and transistors.
Design and fabricate simple series/shunt regulated power supplies using BJT and simple regulated power supplies based on 78xx, 79xx and TL431 ICs.
Understand OpAmp characteristics and its applications in building blocks for signal processing in power electronics applications.
Analyse and fabricate simple circuits on bread board/GP board based on different applications of
OpAmp and special purpose ICs.
Understand the fundamentals of different logic families like RTL, DTL, TTL, etc.
List of Experiments (Laboratory Work):
Objectives: The laboratory work is aimed at putting the theory learnt in class in practice and to show the results are nearly matched with theory. In this context, following are the core objectives
for laboratory work of this subject.
Develop understanding of basic applications of diode and BJT.
Understand the basics of Op-Amp and identify its different parameters from datasheet.
Develop the understanding of Op-Amp for different linear applications and power supplies.
Develop understanding of use of Op-Amp for making filters and oscillators.
Understand the use of Op-Amp in signal processing circuits.
Directions for Laboratory work:
The list of experiments is given as a sample.
Minimum 10 experiments should be carried out.
At least one experiment should be selected from each group.
Similar laboratory work fulfilling the objectives can also be considered.
As far as possible printed manual should be preferred so that students can concentrate in laboratory experiments and related study.
Simulation of various experiments should also be given.
The sample list of experiments is given below.
List of Experiments and Design Based (DP)/Open Ended Problems:
Group A (Diode and BJT):
Study & Use of CRO: Measurement of voltage, frequency and phase shift.
To study and design Diode half wave, full wave and bridge rectifiers with and without capacitor filters.
To study diode clipping and clamping circuits.
Group B (Voltage Regulators):
To study and design a simple Zener voltage regulator and Zener voltage regulator with emitter follower output.
To study and design 78xx and 79xx based voltage regulators.
To study and design TL431 based variable voltage regulators.
Group C (Op-Amp Applications):
To study and design Op-Amp based linear circuits like comparator, inverting & non-inverting amplifier, adder, subtractor, integrator, differentiator, etc.
Measurement of Op-Amp parameters like CMRR, Input offset Voltage, etc.
Phase shift and Wien Bridge oscillator using OPAMPs.
Study of frequency response of different filter circuits
Waveform generator using Op-Amp.
Group D (Special Applications of Op-Amp):
Study of Switched Capacitor Circuits
IC 555 based Astable, Mono stable and Bi-stable multi vibrator
Study of PLL using Op-Amp
Major Equipments:
Bread Board, Oscilloscope, LCR/LCRQ meter, OpAmp Trainer Kits, Multimeter, Variable Power
Supply, etc.
List of Open Source Software/learning website:
Open Source Software:
TINA-TI for circuit simulation ( http://www.ti.com/tool/tina-ti )
OSCAD for CAD application ( http://www.oscad.in/downloads )
Fritzing for bread board/GP board wiring planning ( http://fritzing.org/download )
Web-base tools for design:
Open source for Math Tools:
Learning website:
ACTIVE LEARNING ASSIGNMENTS : Preparation of power-point slides, which include videos, animations, pictures, graphics for better understanding theory and practical work – The faculty will allocate
chapters/ parts of chapters to groups of students so that the entire syllabus to be covered. The power-point slides should be put up on the web-site of the College/ Institute, along with the names of
the students of the group, the name of the faculty, Department and College on the first slide. The best three works should submit to GTU. | {"url":"https://studylib.net/doc/18834313/2142405---gujarat-technological-university","timestamp":"2024-11-07T18:27:31Z","content_type":"text/html","content_length":"66361","record_id":"<urn:uuid:21e68705-137d-4505-bc30-c96d39f166db>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00536.warc.gz"} |
gonum/plot Alternatives - Go Science and Data Analysis | LibHunt
Programming language: Go
License: BSD 3-clause "New" or "Revised" License
gonum/plot alternatives and similar packages
Based on the "Science and Data Analysis" category.
Alternatively, view gonum/plot alternatives based on common mentions on social networks and blogs.
Revolutionize your code reviews with AI. CodeRabbit offers PR summaries, code walkthroughs, 1-click suggestions, and AST-based analysis. Boost productivity and code quality across all major
languages with each PR.
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Do you think we are missing an alternative of gonum/plot or a related project?
Add another 'Science and Data Analysis' Package
Gonum Plot
gonum/plot is the new, official fork of code.google.com/p/plotinum. It provides an API for building and drawing plots in Go. Note that this new API is still in flux and may change. See the wiki for
some example plots.
For additional Plotters, see the Community Plotters Wiki page.
There is a discussion list on Google Groups: [email protected].
gonum/plot is split into a few packages:
• The plot package provides simple interface for laying out a plot and provides primitives for drawing to it.
• The plotter package provides a standard set of Plotters which use the primitives provided by the plot package for drawing lines, scatter plots, box plots, error bars, etc. to a plot. You do not
need to use the plotter package to make use of gonum/plot, however: see the wiki for a tutorial on making your own custom plotters.
• The plotutil package contains a few routines that allow some common plot types to be made very easily. This package is quite new so it is not as well tested as the others and it is bound to
• The vg package provides a generic vector graphics API that sits on top of other vector graphics back-ends such as a custom EPS back-end, draw2d, SVGo, X-Window, gopdf, and Gio.
Documentation is available at:
You can get gonum/plot using go get:
go get gonum.org/v1/plot/...
If you write a cool plotter that you think others may be interested in using, please post to the list so that we can link to it in the gonum/plot wiki or possibly integrate it into the plotter | {"url":"https://go.libhunt.com/plot-alternatives","timestamp":"2024-11-02T09:28:14Z","content_type":"text/html","content_length":"82436","record_id":"<urn:uuid:6175dd92-4c8f-458c-bce5-5401a6fc5332>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00008.warc.gz"} |
Tutorial: Refinement against twinned data
This tutorial describes the detection of twinning with phenix.xtriage and subsequent refinement with phenix.refine
Background and nomenclature
Twinning is a phenomenon in which the crystal used in data collection is a composition of several distinct domains who orientation differ, but are related by known (and predictable) operators. The
net effect of the presence of multiple lattices is that the recorded data is the sum of a number of diffraction patterns. This tutorial deals with the case when the twinned crystal consists of two
domains. This type of twinning is know as hemihedral twinning. The twinning can be either merohedral (M; the twin related lattices overlap exactly) or pseudo-merohedral (PM; the twin related lattices
overlap almost exactly). Classification of the type of twinning (M or PM) can be performed on the basis of group theoretical arguments and is done in phenix.xtriage. X-ray data collected from a
hemihedrally twinned specimen is effectively the sum of two diffraction patterns. The relative size of the smallest crystal domain to the whole crystal is known as the twin fraction, often denoted by
&alpha. The operator that relates overlapping miller indices, is know as the twin law. As mentioned in the previous section, the effect of twinning on the X-ray data, is that the intensity for a
given miller index as seen on the detector has two non-equal contributors:
J(H) = (1-&alpha)I(H) + (&alpha)*I(RH) [eq. 1a]
J(RH) = (1-&alpha)I(RH) + (&alpha)*I(H) [eq. 1b]
In the previous expressions, the intensities I of the miller index H and its twin mate RH build up a single intensity J. The twin law is denoted by R and is used to find the RH, twin related index of
H. The twin law R is usually written down in algebraic terms: (k,h,-l).
Detection of twinning with phenix.xtriage
The presence of twinning usually reveals itself by intensity statistics that do not fall with the range expected for untwinned data. phenix.xtriage reports a number of intensity statistics:
• < I ^2 > / < I >^2
• < F > ^2 / < F ^2 >
• < | E ^2 -1 | >
• an NZ plot
• < |L| >
Expected values for the above statistics and their 'allowed' ranges for untwinned data are know from a data bases analysis. The results of the L-test are used in xtriage to auto-detect twinning.
phenix.xtriage porin.cv xray_data.unit_cell=104.4,104.4,124.25,90,90,120 xray_data.space_group=R3
gives (parts omitted):
Determining possible twin laws.
The following twin laws have been found:
| Type | Axis | R metric (%) | delta (le Page) | delta (Lebedev) | Twin law |
| M | 2-fold | 0.000 | 0.000 | 0.000 | -h-k,k,-l |
| PM | 2-fold | 2.476 | 1.548 | 0.022 | -h,2/3*h+1/3*k-2/3*l,-2/3*h-4/3*k-1/3*l |
| PM | 4-fold | 2.476 | 1.548 | 0.022 | h+k,-2/3*h-1/3*k+2/3*l,2/3*h-2/3*k+1/3*l |
| PM | 2-fold | 2.476 | 1.548 | 0.022 | -h,1/3*h-1/3*k+2/3*l,2/3*h+4/3*k+1/3*l |
| PM | 3-fold | 2.476 | 1.032 | 0.022 | h+k,-1/3*h-2/3*k-2/3*l,-2/3*h+2/3*k-1/3*l |
| PM | 4-fold | 2.476 | 1.548 | 0.022 | -k,1/3*h+2/3*k+2/3*l,-4/3*h-2/3*k+1/3*l |
| PM | 3-fold | 2.476 | 1.032 | 0.022 | -k,-1/3*h+1/3*k-2/3*l,4/3*h+2/3*k-1/3*l |
M: Merohedral twin law
PM: Pseudomerohedral twin law
1 merohedral twin operators found
6 pseudo-merohedral twin operators found
In total, 7 twin operator were found
Twinning and intensity statistics summary (acentric data):
Statistics independent of twin laws
- < I^2 > / < I > ^2 : 1.667
- < F > ^2/ < F^2 > : 0.857
- < |E^2-1| > : 0.609
- < |L| > , < L^2 >: 0.401, 0.225
Multivariate Z score L-test: 8.174
The multivariate Z score is a quality measure of the given
spread in intensities. Good to reasonable data is expected
to have a Z score lower than 3.5.
Large values can indicate twinning, but small values do not
necessarily exclude it.
Statistics depending on twin laws
| Operator | type | R obs. | Britton alpha | H alpha | ML alpha |
| -h-k,k,-l | M | 0.195 | 0.292 | 0.315 | 0.304 |
| -h,2/3*h+1/3*k-2/3*l,-2/3*h-4/3*k-1/3*l | PM | 0.420 | 0.068 | 0.069 | 0.022 |
| h+k,-2/3*h-1/3*k+2/3*l,2/3*h-2/3*k+1/3*l | PM | 0.410 | 0.079 | 0.084 | 0.022 |
| -h,1/3*h-1/3*k+2/3*l,2/3*h+4/3*k+1/3*l | PM | 0.419 | 0.070 | 0.069 | 0.022 |
| h+k,-1/3*h-2/3*k-2/3*l,-2/3*h+2/3*k-1/3*l | PM | 0.413 | 0.078 | 0.079 | 0.022 |
| -k,1/3*h+2/3*k+2/3*l,-4/3*h-2/3*k+1/3*l | PM | 0.415 | 0.070 | 0.075 | 0.022 |
| -k,-1/3*h+1/3*k-2/3*l,4/3*h+2/3*k-1/3*l | PM | 0.415 | 0.074 | 0.077 | 0.022 |
Patterson analysis
- Largest peak height : 5.689
(corresponding p value : 7.822e-01)
The largest off-origin peak in the Patterson function is 5.69% of the
height of the origin peak. No significant pseudo translation is detected.
The results of the L-test indicate that the intensity statistics
are significantly different then is expected from good to reasonable,
untwinned data.
As there are twin laws possible given the crystal symmetry, twinning could
be the reason for the departure of the intensity statistics from normality.
It might be worthwhile carrying out refinement with a twin specific target function.
As listed clearly in the summary states above, the data is suspected to be twinned. Given the relatively large tolerances in finding twin laws, seven twin laws are found. Six of them are pseudo
merohedral, one of them is merohedral. Looking at the R-value analysis in the last table (column R-obs), the merohedral twin law is the most likely in this case, as this merging R-value is lower then
any of the other R values reported.
A quick model based twinning analysis
In most cases, the presence of twinning does not impede structure solution via molecular replacement (or sometimes even S/MAD). If a molecular replacement solution is available testing which twin law
is the most likely can be done via
phenix.twin_map_utils xray_data.file=porin.cv model=porin.pdb unit_cell=104.4,104.4,124.25,90,90,120 space_group=R3
The latter tool performs bulk solvent scaling and R-value calculation for all possible twin laws in the given crystal setting. In this particular case, the twin law (-h-k,k,-l) is the most likely, as
it produces the R-value and lowest refinement target value out of the 7 listed twin listed (0.19 vs 0.25).
Selection of a cross validation set
Although the given reflection file in this tutorial already has an assigned test set for cross validation purposes, assigning it properly is relatively important. When the data is twinned, each
observed reflection has a contribution of two (with hemihedral twinning) non-twinned components. When a test set is designed, care must be taken that free and work reflections are not related by a
twin law. The R-free set assignment in phenix.refine and phenix.reflection_file_converter is designed with this in mind: the free reflections are chosen to obey the highest possible symmetry of the
lattice. Choosing a free set with phenix.refine is as simple as including the keywords
on the command line
Refinement protocols
Within phenix, various refinement protocols can be followed. A few typical examples will be shown below.
Restrained refinement of positional and atomic displacement parameters
Standard restrained refinement of positional and atomic displacement parameters is invoked via
phenix.refine porin.cv porin.pdb twin_law="-h-k,k,-l" unit_cell=104.4,104.4,124.25,90,90,120 space_group=R3
Refinement is performed in macro cycles, in which either positions, atomic displacement parameters or twin and bulk solvent parameters are refined.
Rigid body refinement
The command
phenix.refine porin.cv porin.pdb twin_law="-h-k,k,-l" strategy=rigid_body unit_cell=104.4,104.4,124.25,90,90,120 space_group=R3
will perform rigid body refinement using the twin target function.
For tls refinement, it is advisable to construct a small parameter file that contains TLS definitions:
refinement.refine {
strategy = *individual_sites rigid_body *individual_adp group_adp *tls \
occupancies group_anomalous none
adp {
tls = "chain A"
refinement.twinning {
twin_law = "-h-k,k,-l"
Saving these parameters as tls.def one can run
phenix.refine porin.cv porin.pdb tls.def unit_cell=104.4,104.4,124.25,90,90,120 space_group=R3
Water picking
Ordered solvent can be picked as a part of the refined procedure, more details are available from the phenix.refine manual. Note that the (difference) map used by phenix.refine, is constructed using
detwinned data (see below). Including water picking in refinement can be carried out as follows:
phenix.refine porin.cv porin.pdb twin_law="-h-k,k,-l" ordered_solvent=True unit_cell=104.4,104.4,124.25,90,90,120 space_group=R3
Map inspection
Electron density maps during twin refinement are constructed using detwinned data. Choice of map coefficients and detwinning mode, is controlled by a set of parameters as shown below:
refinement.twinning {
twin_law = "-h-k,k,-l"
detwin {
mode = algebraic proportional *auto
local_scaling = False
map_types {
twofofc = *two_m_dtfo_d_fc two_dtfo_fc
fofc = *m_dtfo_d_fc gradient m_gradient
aniso_correct = False
By default, data is detwinned using algebraic techniques, unless the twin fraction is above 45%, in which case detwinning is performed using proportionality of twin related Icalc values. Detwinning
using the proportionality option, results in maps that are more biased towards the model, resulting in seemingly cleaner, but in the end less informative maps. The 2mFo-dFc map coefficients can be
chosen to have sigmaA weighting (two_m_dtfo_d_fc) or not (two_dtfo_fc). IN both cases, the map coefficients correspond to the 'untwinned' data. A difference map can be constructed using either sigmaA
weighted detwinned data (m_dtfo_d_fc), a sigmaA weighted gradient map (m_gradient) or a plain gradient map (gradient). The default is m_dtfo_d_fc but can be changed to gradient or m_gradient if | {"url":"https://phenix-online.org/version_docs/1.18.2-3874/tutorials/twinning.html","timestamp":"2024-11-06T08:13:19Z","content_type":"application/xhtml+xml","content_length":"14430","record_id":"<urn:uuid:4dc7e18d-76b4-49b5-a3da-59be86cef992>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00447.warc.gz"} |
Posts for the month of June 2015
Planetary magnetic fields are dipole fields of the form
which (if we assume that the dipole is oriented along z) gives a field of
at the pole and a field of
at the equator.
This also corresponds to a vector potential
Now if we assume that
Then our vector potential is
Also, the dipole field used in Matsakos was a constant field inside of the planetary core (r<r_p/2)
They quote a constant field of at the center would require a potential of the form
which agrees with the potential outside at
So we don't need any smoothing… just a piecewise definition for the potential that switches at
Now, if we want to truncate the field lines so they do not pass out the inflowing boundaries, we need to have the derivative of the potential go to zero. This is tricky since the curl of the
potential needs to go to zero in both the theta and r directions. However, the following potential has the desired properties
For the potential is just 0.
This gives
for and for
Smooth UnSmoothed
Here is an outline of the ideas I am thinking for the ring analysis. The main idea is that the shocked material of the flows acquires some new density and velocity which translates into a ram
pressure outward into the ambient medium. We can assume that when this material meets the ambient of equal pressure, the material will come to rest and that is where the ring will form. An issue is —
we need a radial dependance of the pressure in the ambient (or from the splashing), so that we can find a radius where these 2 pressures balance.
Using just the shock conditions means we consider the splashed material having constant rho*v^2. So what are the physical mechanisms that will provide a radial dependance on the pressure?
Here were some of my ideas (by the way, ignoring MHD for now):
-The splashed material slows down as it is being ejected from the collision region due to gravity — perhaps refer to Parker wind solutions (or something similar to Joe's analysis…, parker wind might
not be the right term here for what I am thinking, but haven't looked into this yet… )
-The ambient medium is infalling toward the gravitational potential minima, i.e. the center of the box. This infall has some radial behavior —- and if I can back out the radial density and velocity
profiles, will have a ram pressure to try and balance to the shock ram pressure
-Then more complicated stuff, like compression of the outgoing splashed material as it rams into the ambient gas, leading to cooling/loss of KE/and a concomitant decrease in velocity.
I focused on the 2nd case as it was the one I was most familiar. Although, it might not be entirely the right regime to do this analysis, depending on whether the ambient is fully jeans unstable. I
thought it was a nice place to start, to jump into the equations, and the process of building the model.
Now, I am nearly done for the Hydro, nornmal shock situation, assuming no reflection. As you can see on this page, ways I can modify this model for the head on case:
• making the shock reflected (expect small deviations from the values)
• making the collapse not spherical but rather cylindrical (also don't expect much deviation)
• adding magnetic pressure to the mix… probably here would get the greatest - change in the values… i.e. my model might fail the most because it doesn't include magnetic pressures at this point.
Here is a page describing the shock conditions I am using:
Here is a page describing the uniform collapse model I have been developing:
Uniform collapse involved:
• understanding the solutions for uniform collapse again
• couldn't find this easily on the web, so dug into the solutions myself
• coded it up in mathematica to visualise the solutions
• got the model nearly where it needs to be to begin using numbers from my sims to check it out
As there was no great description of this that I could find, either in text books or the web, I thought it would be good to write up what is in my notebook on the wiki. Need to try and get this
visible in the google universe, as a search for uniform collapse sphere, or homologous collapse sphere, isn't bringing it up on google yet. Will check quickly if the visibility can be increased.
As far as papers, I have been reading astroph everyday, reading the highlights in astro on popular science sites, and have been reading the star formation newsletter.
I applied to Astrohack week but didn't get in the program. I have also been working through the equations and there importance in the Fed. paper on outflows.
XSEDE Proposal Renew
1. highest priority. Due July 15th.
1. 3D co-rotating frame: fixed some bugs and latest results— blog:bliu06222015
2. Park wind solution for the star: merged with Jonathan's setup and running a test. Will post results soon
1. Merged with current development branch for Eddie's cooling stuff. Compiled. Running test
2. will do 3D runs with the new merge version on bluehive and install the new code on Spain machines
I updated some movies https://astrobear.pas.rochester.edu/trac/blog/zchen04132015, a new one that has longer z-axis is still computing. That one proves slow AGB wind is not the key to the formation
of collimated outflow but it has similar shape to the butterfly PN in Balick's paper (Balick, 1987 ApJ). On the other hand, in this paperThe Necklace: equatorial and polar outflows from the binary
central star of the new planetary nebula IPHASX J194359.5+170901, they measured that the kinematic age of polar outflow is older than the equatorial structure, implying that the collimated outflow is
more likely to be the polar jets from a compact secondary. Another point to mention is that to have a circumbinary disk, a binary system just require a slow wind from AGB star, either spherical or
equatorial will do the work. It does not need to carry angular momentum from the rotation of the AGB star.
I read some papers mentioned by the reviewer, I found that this paper is very very useful 2D models for dust-driven AGB star winds. The author is an expert in AGB dust formation. In this paper, he
used equation of motion and energy equation to solve the fluid motion. Dust evolution is computed by moment equations (see Dust formation in stellar winds. IV - Heteromolecular carbon grain
formation and growth and their book Physics and chemistry of circumstellar dust shells of chapter 14). Woitke also included radiative transfer, he assumed Local Thermal Equilibrium (LTE) to simplify
his calculation (High precision Monte Carlo radiative transfer in dusty media). I do not understand the detailed technique and Monte Carlo simulation in the paper. In his discussion, he compared his
results with 1D models and simulated the case without pulsations. The nature of pulsation is nuclear reaction lying deep in the AGB star and result in a sudden jump in luminosity. The physics in this
paper is much more complicated. The temperature gradient is an issue in our paper, I think this part should be modified to a more realistic model.
I am now very interested in the dust evolution. It is actually very important to AGB evolution, binary system evolution and PN evolution. Dust formation is deeply linked to the chemical composition
of the star (i.e. high metalicity, C type star or O type star). Dusts absorb and scatter 30% of phontons (see Interstellar Dust) so it provide the much momentum in many kinds of AGB winds. its
evolution is sometimes coupled with the spectral near the AGB star, especially for the optical thick case (but rare). I think a good description in dust formation in our code will let us have more
As for the binary simulation, I have an idea that we can just run a simulation to mimic The Necklace: equatorial and polar outflows from the binary central star of the new planetary nebula IPHASX
J194359.5+170901. The modification should just be add a polar outflow with high speed from the secondary before the pulsation from the AGB star. A rough estimate is that we need 112 cores * 24 hr *
30 on bluehive (the longer z-axis one need 21 days with slow stellar wind speed.)
• Astronum and my talk went very well. I got a lot of good compliments, and talked with a lot of people.
□ Amira vs. Disperse paper with Federrath?
The complete set of models result in the following figure:
• Work continues on generating all the emission maps for my 3-D clump runs. Here is the updated table: MachStems
□ I have a reservation on BlueStreak right now to finish up run E. It should be done by next Monday. Then, I just need to finish up the emission maps for runs D and E.
□ For next week, I will start generating movies of all the emission maps that I do have.
• My goal for July is to finish the 3-D clump paper and start some new 3-D pulsed jet simulations.
Here is a matlab routine to solve for the parker wind given lambda x is in units of planet radius, and y is velocity in units of sound speed.
Also notified Titos about typo in arxiv preprint that made initial attempts to solve the equation nonsensical.
Also note that the original version of this post assumed that
when in fact
So here is the correct version of the matlab routine
syms y positive;
for i=1:N
yy(i)=vpasolve(y - log(y) == -3 - 4*log(lambda/2)+4*log(xx(i))+2*lambda/xx(i),y,(xx(i) > lambda/2) * 100+.001);
fprintf(fid, '%i\n',N);
fprintf(fid,'radius, density, velocity\n')
for i=1:N
fprintf(fid,'%e %e %e\n',xx(i),1d0/xx(i)^2/sqrt(yy(i)/yy(1)), sqrt(yy(i)));
1. bug and fix
The code of co-rotating frame for results in blog:bliu06222015 and the results before hard-coded the position of particle and pointgravity object as (0,0,0). Since the origin point in the co-rotating
frame should be the mass center and the planet position is at (200,0,0) (currently,will make more sophisticated). So this is a bug:
IF(.NOT. lRestart) THEN
CALL CreateParticle(PlanetParticle)
CALL CreatePointGravityObject(PlanetParticle%PointGravityObj)
The fix is set the PlanetParticle position to be Outflow%position —
2. Low-Res Results after the fix
After the bug-fix, the low-res results with omega=0 looks promising. The tight sonic surface problem in blog:bliu06222015 seems gone. But it's still not quite right for the gravity/mach number plot
inside the planet —trying with higher res.
;low res density move 5-zone-per-radii ;low res temperature move 5-zone-per-radii
log mach plot
plot of log|vx|
shows the unmatched gravity plot for 2D and 3D seems from the low-res of 3D plot.
3. 3D corotating with omega=0.5
This is to try to understand/fix the tight sonic surface found in 3D co-rotating frame with omega=0 as shown in the 3rd part of blog:bliu06112015_2 and the different line plots for the 3D co-rotating
and 2.5D as shown in the 4th part of the same blog post… In 2.5D and 3D simulations before the problem module hardcoded PLUMERSOFT and soft radius=1 for the pointgravity.
SPLINESOFT = 1, & !g ~ r/r^2 for r < r_soft and then goes to 0 as r-> 0
PLUMMERSOFT = 2 !g ~ (r^2+r_soft^2)^(-3/2) r
1. Code modification: Splinesoft
Since in the 3D case we use r=1 but in 2.5D we use r=2. So the g in 3D is about twice of the g in 2.5D.
To handle point gravity more properly, we change the code to use SPLINESOFT instead. Here's the new update to the OutflowWind module
outflow_radius=0 ! outflow radius is 0 always
pointgravity_soft=1 !splinesoft always
2. Testing Results with Splinesoft =
New Results with SplineSoft (low res) Old Results with PlummerSoft (high res)
2.5D no stellar wind ;movie 8 zones per radii ;high res movie
2.5D stellar wind ;movie 8 zones per radii ;high res movie
3D Co-rotating ; density 5 to 10 zones per radii ; density with 40 to 80 zones per radii
3D Co-rot and 2D plots compare ;
In this specific post, I discuss the impact of separation on the formation of fall back disk around binary stars.
1. Model Description
In this simulation, there are two stars. One is assumed to be an AGB star, it has two states - spherical wind state and equatorial outflow state and the other is a companion.
The equatorial outflow state eject high density but low speed wind (20 km/s) into the space. It has open angle. Open angle means, compared to earth, the latitude. For example, a pi/6 open angle means
the outflow is being ejected from pi/6 North to pi/6 South, that is, pi/3 in total angle.
In this simulation, all outflow has open angle of pi/6. The equatorial outflow last 11.2 years, the orbital period for three simulations are 4 years, 6.2 years and 11.6 years respectively, so the
outflow can cover at least a full orbital period. At all other time, the AGB star is in spherical wind state. Its spherical wind speed is above the escaping velocity (40 km/s).
The initial specific angular momentum of the outflow is negligible, let's say it is approximately 0. This is in line with the property of AGB and post-AGB stars.
The temperature of the simulation is fixed at 100 K.
The separation are 3 AU, 4 AU and 6 AU (See (5,3,8) data set in 3AU4AU6AU) respectively in the following simulations, each has topview and sideview.
2. Simulation
The equatorial plane has higher density than the polar direction. Although the AGB star has spherical outflow after the equatorial outflow, the "heavy" equatorial gas is barely supported by the
successive wind while the gas in polar direction has been pushed out. However, a fraction of the polar outflow is fallen back to the equatorial plane at large radii then start to orbit the binary
stars (see Muhammad Akashi and Noam Soker 2008 http://www.sciencedirect.com/science/article/pii/S1384107607000863). This simulation validate their assumption in 3D. We can clearly see the formation
of disk in equatorial plane.
We can also see the bipolar shape of PPNe. However, the simulation zone is not big enough to see the full bipolar outflow picture.
3. Conclusion
The conclusion is that the closer the binary stars is, the more likely the fall back disk forms or the smaller the circumbianry disk will be.
Please see my further progress of this topic in this post.
Wrote this python script that generates points on a sphere in Cartesian coordinates. It then writes these coordinates with some random values with velocity (vx, vy, vz) and density (n) to a
tab-delimited ascii file. You can choose the radius of the sphere, and N is number of points on the sphere. Forwarded from Martin, a stackoverflow discussion. Also I am using python3.
import random
from math import pi,sin,cos
#Creating Sphere Dataz
def createSphere(r=5, N=100):
lst = []
for phi in [(pi*i)/(N-1) for i in range(N)]:
M = int(sin(phi)*(N-1))+1
for theta in [(2*pi*i)/M for i in range(M)]:
x = r * sin(phi) * cos(theta)
y = r * sin(phi) * sin(theta)
z = r * cos(phi)
lst.append((x, y, z))
return lst
#Opens/creates new ascii file
outfile = open("test.dat", "w")
#Writes the data to the file
for x,y,z in createSphere():
rho = random.random()*1000000
vx = random.random()*10
vy = random.random()*100
vz = random.random()
print("{0}\t{1}\t{2}\t{3}\t{4}\t{5}\t{6}".format(x,y,z,vx,vy,vz,rho), file=outfile)
#Closes the file
So the pseudo-sphere data seems to visualize just fine in Shape:
Aside: I am starting to document how to use Shape, here.
small window with new parameters:
Tiny stellar wind density to check the planetary wind close to the planet surface.
2. No rotation omega=0
density omega=0; 20 zones per radii
temperature omega=0; 20 zones per radii
temperature omega=0; lambda=10
3. No rotation omega=0 High Res longer
Set stellar wind with very low density and run with high resolution. To check if it's possible to reproduce the planetary wind in 2.5D.
density omega=0; 40-80 zones per radii
Temperature omega=0; 40-80 zones per radii
pressure omega=0; 40-80 zones per radii
4. Line plots of density, temperature, Mach number and on dayside (from left boundary to critical radius)
Same run data as 3 above.
change the density to be smaller to make the density flux on the planet surface to be a reasonable value.
In the latest high-res results of co-rotating frame as in blog:bliu06032015, the planetary wind seems not be able to expand. This test to test the effect of tiny corotating speed omega=1d-5 . All
other parameters are same as blog:bliu06032015, namely rho_sw=4d-1…
density of 8 zones per radii;
Column Density Map (m^2) PV Diagram (km/s vs. m)
GIF GIF
The simulation I am trying to visualize in Shape is probably Martin's old poloidal collimation project. Above is a visualization of frame 22 (chombo files cannot be found, so I cannot compare in
VisIt), so the beginning of the simulation. Toward the end of the simulation, it should supposedly look like this:
Shape (probably not frame 212)
• Still think the PV Diagram could use improvement. Not sure if I used the appropriate parameters for the velocity axes. For the above visualization, the axis goes from -500 to 500 km/s. With this
arrangement I got something to show in the PV diagram part of the render module. However I think I could mess with it more.
• Document how to do this on the wiki!
• Regrid Bruce's simulations so they are in ascii, and thus fed into Shape.
• Visualize Bruce's simulations.
• Show Bruce how to use Shape.
• Pretty up the output?
The setwind subroutine In the current co-rotating frame sets up the stellar wind density proportional to . where is about 200 C.U. for our runs So to make the stellar wind density same as in the 2.5D
and 3D we have to set it 40000 times larger in these co-rotating runs. Otherwise the stand out distance and bow-shock size will be very large as we see in our former runs…
By using larger stellar wind density, the result for , Mach 5 runs seems promising. — didn't change the density of planet for this run still 1g/cc also used larger planet as in blog:bliu06022015
Density movie -- 16 zones per radius
Temperature movie -- 16 zones per radius
Density movie -- 32 zones per radius and twice higher stellar wind density
Temperature movie -- 32 zones per radius and twice higher stellar wind density
Double Rp and Mp to keep the lambda same. New refinement setup. Here's the results of a 2.5D testing run with gamma=1.01, lambda=5.3, Rp=4, Mp=1Mj, mach=5 and max 3 levels of AMR
Mesh mesh movie;stellar wind mesh movie
Planet wind zoomed-in movie to check sonic surface;stellar wind movie | {"url":"https://bluehound.circ.rochester.edu/astrobear/blog/2015/6","timestamp":"2024-11-05T18:53:43Z","content_type":"text/html","content_length":"112417","record_id":"<urn:uuid:2898a4dc-ad2f-4cdf-9ae8-561dcabceb5c>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00085.warc.gz"} |
Simplify the following expression. ((2x-1)/(x+3)-(x^(2)-4)/(2x-Turito
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Simplify the following expression.
The expansions of certain identities are:
We are asked to simplify the given expression
The correct answer is: the simplified expression is (x2 -x3 +4x +11)/(x+2)
Step 1 of 2:
Fins the LCM of the expression
Thus, the expression becomes
Step 2 of 2:
Now, find the product of
For that, simplify and cancel out the common factors
Hence, the simplified expression is
LCM of two numbers or values are the least common multiples of both of them.
Get an Expert Advice From Turito. | {"url":"https://www.turito.com/ask-a-doubt/simplify-the-following-expression-2x-1-x-3-x-2-4-2x-1-xx-2x-2-7x-3-x-2-q5978b1d3","timestamp":"2024-11-11T09:49:33Z","content_type":"application/xhtml+xml","content_length":"802968","record_id":"<urn:uuid:a4749569-67c9-4841-8669-0de0b13e69c4>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00349.warc.gz"} |