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Circuit containing inductance and resistance in series|Alternating Current
Circuit containing inductance and resistance in series
• Figure below shows pure inductor of inductance L connected in series with a resistor of resistance R through sinusoidal voltage
• An alternating current I flowing in the circuit gives rise to voltage drop V[R] across the resistor and voltage drop V[L] across the coil
• Voltage drop V[R] across R would be in phase with current but voltage drop across the inductor will lead the current by a phase factor π/2
• Now voltage drop across the resistor R is
and across inductor
where I is the value of current in the circuit at a given instant of time
• So voltage phasors diagram is
In figure (10) we have taken current as a reference quantity because same amount of current flows through both the components. Thus fro phasors diagram
is known as impedance of the circuit
• Current in steady state is
and it lags behind applied voltage by an angle φ such that
tanφ=ωL/R ---(16) | {"url":"https://physicscatalyst.com/elecmagnetism/inductance-and-resistance-series-ac-circuit.php","timestamp":"2024-11-03T23:21:04Z","content_type":"text/html","content_length":"67323","record_id":"<urn:uuid:bbabdf0e-9709-48e7-851b-a69f173a5980>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00448.warc.gz"} |
FTION C. SKILL-BASEd Questions
7. Find the angle measure x in e... | Filo
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FTION C. SKILL-BASEd Questions 7. Find the angle measure in each of the following figures: (ii) is not sin. In the given figure, and are the bisectors of and respectively. Find .
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Question Text FTION C. SKILL-BASEd Questions 7. Find the angle measure in each of the following figures: (ii) is not sin. In the given figure, and are the bisectors of and respectively. Find .
Updated On Feb 6, 2023
Topic All topics
Subject Mathematics
Class Class 9
Answer Type Video solution: 1
Upvotes 103
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What is Compensation Theorem? - Circuit Globe
Compensation Theorem
Compensation Theorem states that in a linear time-invariant network when the resistance (R) of an uncoupled branch, carrying a current (I), is changed by (ΔR), then the currents in all the branches
would change and can be obtained by assuming that an ideal voltage source of (V[C]) has been connected such that V[C] = I (ΔR) in series with (R + ΔR) when all other sources in the network are
replaced by their internal resistances.
In Compensation Theorem, the source voltage (V[C]) opposes the original current. In simple words, compensation theorem can be stated as – the resistance of any network can be replaced by a voltage
source, having the same voltage as the voltage drop across the resistance which is replaced.
Let us assume a load R[L] be connected to a DC source network whose Thevenin’s equivalent gives V[0] as the Thevenin’s voltage and R[TH] as the Thevenin’s resistance as shown in the figure below:
Let the load resistance RL be changed to (RL + ΔRL). Since the rest of the circuit remains unchanged, the Thevenin’s equivalent network remains the same as shown in the circuit diagram below:
The change of current being termed as ΔI
Putting the value of I’ and I from the equation (1) and (2) in the equation (3) we will get the following equation:
Now, putting the value of I from the equation (1) in equation (4), we will get the following equation:
As we know, V[C] = I ΔRL and is known as compensating voltage.
Therefore, equation (5) becomes,
Hence, Compensation theorem tells that with the change of branch resistance, branch currents changes and the change is equivalent to an ideal compensating voltage source in series with the branch
opposing the original current, where all other sources in the network being replaced by their internal resistances.
1 thought on “Compensation Theorem”
Much useful. Thank you so much
Leave a Comment | {"url":"https://circuitglobe.com/what-is-compensation-theorem.html","timestamp":"2024-11-04T15:12:31Z","content_type":"text/html","content_length":"163093","record_id":"<urn:uuid:caab4927-ef57-4880-98d9-9a61b032a6e2>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00138.warc.gz"} |
C-E Amp circuit sim using spice
I write the following prog to simulate the circuit shown in Fig.C-E below:
common E Amp freq response
VCC 7 0 DC 12
VS 1 0 AC 1
Q1 4 3 5 Q_2N2222A_N
RS 1 2 1K
C1 2 3 2uF
R1 3 0 10K
R2 7 3 30K
RC 7 4 4.3K
RE 5 0 1.3K
C3 5 0 10uF
C2 4 6 0.1uF
R3 6 0 100K
.MODEL Q_2N2222A_N NPN( IS=11.9F NF=1 NR=1 RE=649M RC=1
+ RB=10 VAF=83.5 VAR=41.7 ISE=350F ISC=350F
+ NE=1.58 NC=1.58 BF=204 BR=5 IKF=149M
+ IKR=149M CJC=13.1P CJE=30P VJC=2.87 VJE=678M
+ MJC=330M MJE=330M TF=531P TR=69N EG=1.11
+ KF=0 AF=1 )
The .op command results are:
NAME Q1
MODEL Q_2N2222A_N
IB 1.25E-05
IC 1.71E-03
VBE 6.66E-01
VBC -1.74E+00
VCE 2.40E+00
BETADC 1.37E+02
GM 6.53E-02
RPI 2.35E+03
RX 1.00E+01
RO 4.90E+04
CBE 8.44E-11
CBC 1.12E-11
CJS 0.00E+00
BETAAC 1.54E+02
CBX 0.00E+00
FT 1.09E+08
And I need to discuss some of these results with you:
(1) How far or close these results to the real life, can i depend on it initially?
(2) RPI is the i/p impedance looking into the base of Q1, so, the total i/p impedance is: R1//R2//RPI = 1.8K ohm. so we can look at the i/p portion of the circuit as: Vs connected in series with Rs
then to C1 then connected inseries to that 1.8k.
if we look at Vs as a MIC and Rs as the impedance of the MIC and C1&Rtot as a high pass filter, is the value of C1 suitable and why?
(3) What exactly these parameters mean: GM, RX, RO, CJS, CBX.
(4) FT is (as I remember) is the freq at which the gain =1, that is at this freq there is no amplification, Is this OK?
(5) In the MODEL Q_2N2222A_N, BF = 204 is this the upper value that this Q can reach?
(6) In the MODEL Q_2N2222A_N, what these mean: NF=1 NR=1 NE=1.58 NC=1.58 IKF=149M IKR=149M VJC=2.87 VJE=678M MJC=330M MJE=330M TF=531P TR=69N EG=1.11 KF=0 AF=1
thats all
thank you in advance
• 2 months later...
(1) How far or close these results to the real life, can i depend on it initially?
Hi Walid,
Doesn't your sim program have built-in spec's for common transistors? A typo can mess it up.
The sim program works with "typical" spec's. The actual results with "real" transistors having a wide range of current gain and Vbe will be much different. Change the current gain in the program to
see if the circuit still works.
(2) RPI is the i/p impedance looking into the base of Q1, so, the total i/p impedance is: R1//R2//RPI = 1.8K ohm. so we can look at the i/p portion of the circuit as: Vs connected in series with
Rs then to C1 then connected inseries to that 1.8k. | {"url":"https://www.electronics-lab.com/community/index.php?/topic/24696-c-e-amp-circuit-sim-using-spice/#comment-106222","timestamp":"2024-11-03T04:14:52Z","content_type":"text/html","content_length":"132164","record_id":"<urn:uuid:275e9efb-d305-4a91-bd98-815b06230079>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00047.warc.gz"} |
How do I plot Isocurves on a HP-50g?
07-10-2015, 10:28 PM
Post: #1
MattiMark Posts: 75
Member Joined: Dec 2014
How do I plot Isocurves on a HP-50g?
I would like to plot isocurves on my 50g, but cannot find any documentation in the manual.
For instance x1^2 + x2^2 + 2 x1*x2 - 4*x1 + 2*x2 - 4.
Have someone done it, how?
07-11-2015, 02:51 PM
(This post was last modified: 07-11-2015 02:52 PM by peacecalc.)
Post: #2
peacecalc Posts: 206
Member Joined: Dec 2013
RE: How do I plot Isocurves on a HP-50g?
Hello Matti,
Quote:I would like to plot isocurves on my 50g, but cannot find any documentation in the manual.
I'm a little bit astonished, because if you have a look in the manual (not AUR) chapter 12 you find a bunch of examples for your problem. For further discussion of these graphic capabilities you'll
be best supported of the ancestor manuals (HP48 (chapter 22 to 24) and HP49 (chapter/s ??)).
Quote:For instance x1^2 + x2^2 + 2 x1*x2 - 4*x1 + 2*x2 - 4.
The second might be you don't know which plotting type you should use for your function. For me it looks like a function which depends on 2 variables (x1 and x2). So the the 3D plotting types
"wireframes" or "fast3D" may be a good choice. But don't be surprised if the display doesn't show what you are expecting, it depends on scale, range and number of steps and the point of view.
When your given expression is a implicit definition of a function, you can use your HP 50g for solving to x1 or x2 (because it is only a quadratic expression in x1 and x2). The solved terms can be
displayed with the plotting type "function".
Sincerely peacecalc
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Calculate Z-scores
In statistics, the z-score (or standard score) of an observation is the number of standard deviations that it is above or below the population mean.
To calculate a z-score you must know the population mean and the population standard deviation. In cases where it is impossible to measure every observation of a population, you can estimate the
standard deviation using a random sample.
Create a z-score visualization to answer questions like the following:
• What percentage of values fall below a specific value?
• What values can be considered exceptional? For example, in an IQ test, what scores represent the top five percent?
• What is the relative score of one distribution versus another? For example, Michael is taller than the average male and Emily is taller than the average female, but who is relatively taller
within their gender?
As a general rule, z-scores lower than -1.96 or higher than 1.96 are considered unusual and interesting. That is, they are statistically significant outliers.
This article demonstrates how to calculate a z-score in Tableau.
1. Connect to the Sample - Superstore data source provided with Tableau Desktop.
2. Create a calculated field to calculate average sales.
Choose Analysis > Create Calculated Field to open the calculation editor. Name the calculation Average Sales and type or paste the following in the formula area:
3. Create another calculated field to calculate the standard deviation. Name the calculation STDEVP Sales and type or paste the following in the formula area:
4. Create one more calculated field, this one to calculate the z-score. Name the calculation Z-score and type or paste the following in the formula area:
(SUM([Sales]) - [Average Sales]) / [STDEVP Sales]
5. Drag Z-Score from the Data pane to Columns and State to Rows.
Notice that the Z-score field on Columns has a table calculation icon on the right side (that is, a small triangle):
The STDEVP Sales function is based on the WINDOW_STDEVP function, which is a table calculation function. The Z-Score function, in turn, is a table calculation function because it includes
STDEVP Sales in its definition. When you use a calculated field that includes a table calculation function in a view, it's the same as adding a table calculation to a field manually. You can edit
the field as a table calculation. In fact, that's what you do next.
6. Click the Z-score field on Columns and choose Compute Using > State.
This causes the z-scores to be computed on a per-state basis.
7. Click the Sort Descending icon on the toolbar:
8. Hold down the Ctrl key and drag the Z-score field from Columns to Color.
Ctrl + Drag copies a field as currently configured to an additional location.
9. Ctrl + Drag Z-score from Columns once again. This time drop it on Label.
You now have a distribution of z-scores broken out by state. California and New York both have z-scores greater than 1.96. You could conclude from this that California and New York have significantly
higher average sales than other states.
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In the xy-plane, the length of the shortest path from (0,0) to ... | Filo
In the -plane, the length of the shortest path from to that does not go inside the circle is:
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Question Text In the -plane, the length of the shortest path from to that does not go inside the circle is:
Updated On Mar 30, 2023
Topic Conic Sections
Subject Mathematics
Class Class 11
Answer Type Text solution:1 Video solution: 3
Upvotes 414
Avg. Video Duration 7 min | {"url":"https://askfilo.com/math-question-answers/in-the-x-y-plane-the-length-of-the-shortest-path-from-00-to-1216-that-does-not","timestamp":"2024-11-13T21:33:39Z","content_type":"text/html","content_length":"532928","record_id":"<urn:uuid:e779b818-6b34-46a1-b76a-a366a9828ceb>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00613.warc.gz"} |
Where will the hour hand of a clock stop if it starts (a) from 6 and turns through 1 right angle? (b) from 8 and turns through 2 right angles? (c) from 10 and turns through 3 right angles? (d) from 7
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
Where will the hour hand of a clock stop if it starts
(a) from 6 and turns through 1 right angle?
(b) from 8 and turns through 2 right angles?
(c) from 10 and turns through 3 right angles?
(d) from 7 and turns through 2 straight angles?
We will be using the concepts of clockwise and Anticlockwise Rotation, right angle to solve this.
(a) Starting from 6 and turning through 1 right angle, the hour hand stops at 9. Refer to the below image.
(b) Starting from 8 and turning through 2 right angles, the hour hand stops at 2.
(c) Starting from 10 and turning through 3 right angles, the hour hand stops at 7.
(d) Starting from 7 and turning through 2 straight angles, the hour hand stops at 7.
NCERT Solutions for Class 6 Maths Chapter 5 Exercise 5.2 Question 7
Where will the hour hand of a clock stop if it starts (a) from 6 and turns through 1 right angle? (b) from 8 and turns through 2 right angles? (c) from 10 and turns through 3 right angles? (d) from 7
and turns through 2 straight angles?
(a) The hour hand stops at 9 if it starts from 6 and turns through 1 right angle. (b) The hour hand stops at 2 if it starts from 8 and turns through 2 right angles. (c) The hour hand stops at 7 if it
starts from 10 and turns through 3 right angles. (d) The hour hand stops at 7 if it starts from 7 and turns through 2 right angles
☛ Related Questions:
Math worksheets and
visual curriculum | {"url":"https://www.cuemath.com/ncert-solutions/where-will-the-hour-hand-of-a-clock-stop-if-it-starts-a-from-6-and-turns-through-1-right-angle-b-from-8-and-turns-through-2-right-angles-c-from-10-and-turns-through-3-right-angles/","timestamp":"2024-11-04T12:15:27Z","content_type":"text/html","content_length":"209685","record_id":"<urn:uuid:ad3725b3-8c7e-46c8-973b-7621096fcaf4>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00099.warc.gz"} |
Tales from the Binomial Tail: Confidence intervals for balanced accuracy | Groundlight
At Groundlight, we put careful thought into measuring the correctness of our machine learning detectors. In the simplest case, this means measuring detector accuracy. But our customers have vastly
different performance needs since our platform allows them to train an ML model for nearly any Yes/No visual question-answering task. A single metric like accuracy is unlikely to provide adequate
resolution for all such problems. Some customers might care more about false positive mistakes (precision) whereas others might care more about false negatives (recall).
To provide insight for an endless variety of use cases yet still summarize performance with a single number, Groundlight's accuracy details view displays each detector's balanced accuracy. Balanced
accuracy is the average of recall for all classes and is Groundlight's preferred summary metric. For binary problems, this is just the mean of accuracy on the should-be-YES images and accuracy on the
should-be-NOs. We prefer balanced accuracy because it is easier to understand than metrics like the F1 score or AUROC. And since many commercially interesting problems are highly imbalanced - that is
the answer is almost always YES or always NO - standard accuracy is not a useful performance measure because always predicting the most common class will yield high accuracy but be useless in
Figure 1: the detector accuracy details view shows balanced accuracy and per-class accuracy with exact 95% confidence intervals
However, we've found that just displaying the balanced accuracy is not informative enough, as we do not always have an ample supply of ground truth labeled images to estimate it from. Ground truth
labels are answers to image queries that have been provided by a customer, or customer representative, and are therefore trusted to be correct. With only a few ground truth labels, the estimate of a
detector's balanced accuracy may itself be inaccurate. As such, we find it helpful to quantify and display the degree of possible inaccuracy by constructing confidence intervals for balanced
accuracy, which brings us to the subject of this blog post!
At Groundlight, we compute and display exact confidence intervals in order to upper and lower bound each detector's balanced accuracy, and thereby convey the amount of precision in the reported
metric. The detector's accuracy details view displays these intervals as colored bars surrounding the reported accuracy numbers (see figure 1, above). This blog post describes the mathematics behind
how we compute the intervals using the tails of the binomial distribution, and it also strives to provide a healthy amount of intuition for the math.
Unlike the approximate confidence intervals based on the Gaussian distribution, which you may be familiar with, confidence intervals based on the binomial tails are exact, regardless of the number of
ground truth labels we have available. Our exposition largely follows Langford, 2005 and we use his "program bound" as a primitive to construct confidence intervals for the balanced accuracy metric.
To estimate and construct confidence intervals for balanced accuracy, we first need to understand how to construct confidence intervals for standard "plain old" accuracy. So we'll start here.
Recall that standard accuracy is just the fraction of predictions a classifier makes which happen to be correct. This sounds simple enough, but to define this fraction rigorously, we actually need to
make assumptions. To see why, consider the case that our classifier performs well on daytime images but poorly on nighttime ones. If the stream of images consists mainly of daytime photos, then our
classifier's accuracy will be high, but if it's mainly nighttime images, our classifier's accuracy will be low. Or if the stream of images drifts slowly over time from day to nighttime images, our
classifier won't even have a single accuracy. Its accuracy will be time-period dependent.
Therefore, a classifier's "true accuracy" is inherently a function of the distribution of examples it's applied to. In practice, we almost never know what this distribution is. In fact, it's
something of a mathematical fiction. But it happens to be a useful fiction in so far as it reflects reality, in that it lets us do things like bound the Platonic true accuracy of a classifier and
otherwise reason about out-of-sample performance. Consequently, we make the assumption that there exists a distribution over the set of examples that our classifier sees, and that this distribution
remains fixed over time.
Let's call the distribution over images that our classifier sees, $D$. Each example in $D$ consists of an image, $x \in \mathcal{X}$, and an associated binary label, $y \in$ { YES, NO }, which is the
answer to the query. Let $(x,y) \sim D$ denote the action of sampling an example from $D$. We conceptualize our machine learning classifier as a function, $h$, which maps from the set of images, $\
mathcal{X}$, to the set of labels, $\mathcal{Y}$. We say that $h$ correctly classifies an example $(x,y)$ if $h(x) = y$, and that $h$ misclassifies it otherwise.
For now, our goal is to construct a confidence inverval for the true, but unknown, accuracy of $h$. We define this true accuracy as the probability that $h$ correctly classifies an example drawn from
$\text{acc}_{D}(h) = \Pr_{(x,y) \sim D}[ \,h(x) = y\, ].$
The true accuracy is impossible to compute exactly because $D$ is unknown and the universe of images is impossibly large. However, we can estimate it by evaluating $h$ on a finite set of test
examples, $S$, which have been drawn i.i.d. from $D$. That is,
$S = \{ (x_1, y_1), (x_2, y_2), ..., (x_{n}, y_{n}) \}$
where each $(x_i, y_i) \sim D$ for $i=1,\ldots,n$.
The fraction of images in $S$ that $h$ correctly classifies is called $h$'s empirical accuracy on $S$, and this fraction is computed as
$\widehat{\text{acc}}_{S}(h) = \frac{1}{n} \sum_{i=1}^n \mathbf{1}[\, h(x_i) = y_i \,].$
The notation $\mathbf{1}[\, \texttt{condition} \,]$ is shorthand for the indicator function which equals 1 when the $\texttt{condition}$ is true and 0 otherwise. So the formula above just sums the
number of examples in $S$ that are correctly classified and then multiplies by 1/n.
The egg-shaped infographic below depicts the scenario of estimating $h$'s true accuracy from its performance on a finite test set. The gray ellipse represents the full distribution of examples, $D$.
Each dot corresponds to a single example image, $x$, whose true label, $y$, is represented by the dot's color - red for YES and blue for NO. The classifier, $h$, is represented by the dotted black
line. Here, $h$ is the decision rule that classifies all points to the left of the line as should-be YES and all points to the right as should-be-NO. The points with light gray circles around them
are the ones that have been sampled to form the test set, $S$.
Figure 2: true accuracy can only be estimated from performance on a finite test set. The gray shaded region represents the full distribution. The lightly circled points are examples sampled for the
test set.
In this case, our choice of test set, $S$, was unlucky because $h$'s empirical accuracy on $S$ looks great, appearing to be 9/9 = 100%. But evaluating $h$ on the full distribution of examples, $D$,
reveals that its true accuracy is much lower, only 24/27 = 89%. If our goal is to rarely be fooled into thinking that $h$'s performance is much better than it really is, then this particular test set
was unfortunate in the sense that $h$ performs misleadingly well.
Test Set Accuracy and Coin Flips
It turns out that the problem of determining a classifier's true accuracy from its performance on a finite test set exactly mirrors the problem of determining the bias of a possibly unfair coin after
observing some number of flips. In this analogy, the act of classifying an example corresponds to flipping the coin, and the coin landing heads corresponds to the classifier's prediction being
Usefully, the binomial distribution completely characterizes the probability of observing $k$ heads in $N$ independent tosses of a biased coin whose bias, or propensity to land heads, is known to be
the probability, $p$, through its probability mass function (PMF), defined as
$f_{N,p}(k) = {N \choose k} p^k (1 - p)^{N-k}.$
The cumulative density function (CDF) is the associated function that sums up the PMF probabilities over all outcomes (i.e., number of heads) from 0 through $k$. It tells us the probability of
observing $k$ or fewer heads in $N$ independent tosses when the coin's bias is the probability $p$. The CDF is defined as
$F_{N,p}(k) = \sum_{j = 0}^k f_{N,p}(k).$
Below we've plotted the PMF (left) and CDF (right) functions for a binomial distribution whose parameters are N=30 and p=0.3.
The PMF looks like a symmetric "bell curve". Its x-axis is the number of tosses that are heads, $k$. And its y-axis is the probability of observing $k$ heads in $N$ tosses. The CDF plot shows the
cumulative sum of the PMF probabilities up through $k$ on its y-axis. The CDF is a monotonically increasing function of $k$. Its value is 1.0 on the right side of the plot since the sum of all PMF
probabilities must equal one.
The binomial PMF doesn't always resemble a bell-shaped curve. This is true of the binomial distributions in the two plots below, whose respective bias parameters are p=0.15 and p=0.96.
Upper Bounding the True Accuracy from Test Set Performance
Now that we've examined the probability of coin tossing and seen how the number of heads from tosses of a biased coin mirrors the number of correctly classified examples in a randomly sampled test
set, let's consider the problem of determining an upper bound for the true accuracy of a classifier given its performance on a test set.
Imagine that we've sampled a test set, $S$, from $D$ with 100 examples, and that our classifier, $h$, correctly classified 80 of them. We would like to upper bound $h$'s true accuracy, $\text{acc}_D
(h)$, having observed its empirical accuracy, $\widehat{\text{acc}}_S(h)$ = 80/100 = 80%.
Let's start by considering a very naive choice for the upper bound, taking it to equal the empirical accuracy of 80%.
The figure below plots the PMF of a binomial distribution with parameters N=100 and p=0.80. Here, N is the test set size and p corresponds to the true, but unknown, classifier accuracy. The plot
shows that if our classifier's true accuracy were in fact 80%, there would be a very good chance of observing an even lower empirical accuracy than what we actually observed. This is reflected in the
substantial amount of probability mass lying to the left of the purple vertical line, which is placed at the empirical accuracy point of 80/100 = 80%.
Figure 3: Binomial PMF (top) and CDF (bottom) for N=100 and true accuracy 80.0%. The CDF shows there is a 54% chance of seeing an empirical accuracy of 80% or less.
In fact, the CDF of the binomial tells us that there is a 54% chance of seeing an empirical accuracy of 80% or less when the true accuracy is 80%. And since 54% is fairly good odds, our naive choice
of 80% as an upper bound doesn't appear very safe. It would therefore be wise to increase our upper bound if we want it to be an upper bound!
In contrast, the plot below shows that if the true accuracy were a bit higher, say 83%, we would only have a 1 in 4 chance of observing an empirical accuracy less than or equal to our observed
accuracy of 80%. Or put differently, roughly a quarter of the test sets we could sample from $D$ would yield an empirical accuracy of 80% or lower if $h$'s true accuracy was 83%. This is shown by the
24.8% probability mass located to the left of the purple line at the 80% empirical accuracy point. The red line is positioned at the hypothesized true accuracy of 83%.
Figure 4: Binomial PMF (top) and CDF (bottom) for N=100 and true accuracy 83.0%. The CDF shows there is a 24.8% chance of seeing an empirical accuracy of 80% or less.
Still, events with one in four odds are quite common, so hypothesizing an even larger true accuracy would be wise if we want to ensure it's not less than the actual true accuracy.
The next plot shows that if the true accuracy were higher still, at 86.3%, the empirical accuracy of 80% or less would be observed on only 5% of sampled test sets. This is evidenced by the even
smaller amount of probability mass to the left of the purple line located at the empirical accuracy of 80%. Again, the red line is positioned at the hypothesized true accuracy of 86.3%.
Figure 5: Binomial PMF (top) and CDF (bottom) for N=100 and true accuracy 86.3%. The CDF shows there is a 5% chance of seeing an empirical accuracy of 80% or less.
In other words, if $h$'s true accuracy were 86.3% or greater, we'd observe an empirical accuracy of 80% or lower on just 1 in 20 test sets. Consequently, the hypothesized true accuracy of 86.3% seems
like a pretty safe choice for an upper bound.
Constructing a 95% Upper Confidence Bound
The procedure we just outlined, of increasing the hypothesized true accuracy starting from the observed empirical accuracy until exactly 5% of the binomial's probability mass lies to the left of the
empirical accuracy, is how we construct an exact 95% upper confidence bound for the true accuracy.
Remarkably, if we apply this procedure many times to find 95% accuracy upper confidence bounds for different ML classifiers at Groundlight, the computed upper bounds will in fact be larger than the
respective classifiers' true accuracies in 95% of these encountered cases. This last statement is worth mulling over because it is exactly the right way to think about the guarantees associated with
upper confidence bounds.
Restated, a 95% upper confidence bound procedure for the true accuracy is one that produces a quantity greater than the true accuracy 95% of the time.
Exact Upper Confidence Bounds based on the Binomial CDF
So now that we've intuitively described the procedure used to derive exact upper confidence bounds, we give a more formal treatment that will be useful in discussing confidence intervals for balanced
First, recall that the binomial's CDF function, $F_{N,p}(k)$, gives the probability of observing $k$ or fewer heads in $N$ tosses of a biased coin whose bias is $p$.
Also, recall in the previous section that we decided to put exactly 5% of the probability mass in the lower tail of the PMF, and this yielded a 95% upper confidence bound. But we could have placed 1%
in the lower tail, and doing so would have yielded a 99% upper confidence bound. A 99% upper confidence bound is looser than a 95% upper bound, but it upper bounds the true accuracy on 99% of test
sets sampled as opposed to just 95%.
The tightness of the bound versus the fraction of test sets it holds for is a trade off that we get to make referred to as the coverage. We control the coverage through a parameter named $\delta$.
Above we had set $\delta$ to 5% which gave us a 1 - $\delta$ = 95% upper confidence bound. But we could have picked some other value for $\delta$.
With $\delta$ understood, we are now ready to give our formal definition of upper confidence bounds. Let $\delta$ be given, $N$ be the number of examples in the test set, $k$ be the number of
correctly classified test examples, and $p$ be the true accuracy.
Definition: the 100(1 - $\delta$)% binomial upper confidence bound for $p$ is defined as
$\bar{p}(N, k, \delta) = \max \{ \, p \,:\, F_{N,p}(k) \ge \delta \,\, \}.$
In words, $\bar{p}$ is the maximum accuracy for which there exists at least $\delta$ probability mass in the lower tail lying to the left of the observed number of correct classifications for the
test set. And this definition exactly mirrors the procedure we used above to find the 95% upper confidence bound. We picked $\bar{p}$ to be the max $p$ such that the CDF $F_{N=100,p}(k=80)$ was at
least $\delta$ = 5%.
We can easily implement this definition in code. The binomial CDF is available in python through the scipy.stats module as binom.cdf. And we can use it to find the largest value of $p$ for which $F_
{N,p}(k) \ge \delta$. However the CDF isn't directly invertible, so we can't just plug in $\delta$ and get $\bar{p}$ out. Instead we need to search over possible values of $p$ until we find the
largest one that satisfies the inequality. This can be done efficiently using the interval bisection method which we implement below.
from scipy.stats import binom
def binomial_upper_bound(N, k, delta):
Returns a 100*(1 - delta)% upper confidence bound on the accuracy
of a classifier that correctly classifies k out of N examples.
def cdf(p):
return binom.cdf(k, N, p)
def search(low, high):
if high - low < 1e-6:
return low # we have converged close enough
mid = (low + high) / 2
if cdf(mid) >= delta:
return search(mid, high)
return search(low, mid)
return search(low=k/N, high=1.0)
Lower Confidence Bounds
Referring back to our discussion of coin flips makes it clear how to construct lower bounds for true accuracy. We had likened a correct classification to a biased coin landing heads and we upper
bounded the probability of heads based on the observed number of heads.
But we could have used the same math to upper bound the probability of tails. And likening tails to misclassifications lets us upper bound the true error rate. Moreover, the error rate equals one
minus the accuracy. And so we immediately get a lower bound on the accuracy by computing an upper bound on the error rate and subtracting it from one.
Again, let $\delta$ be given, $N$ be the number of test examples, $k$ be the number of correctly classified test examples, and let $p$ be the true, but unknown, accuracy.
Definition: the 100(1 - $\delta$)% binomial lower confidence bound for $p$ is defined as
$\underline{p}(N, k, \delta) = 1 - \max \{ \, p \,:\, F_{N,p}(N - k) \ge \delta \,\, \}.$
Here $N - k$ is the number of misclassified examples observed in the test set.
Central Confidence Intervals
Now that we know how to derive upper and lower bounds which hold individually at a given confidence level, we can use our understanding to derive upper and lower bounds which hold simultaneously at
the given confidence level. To do so, we compute what is called a central confidence interval. A 100$\times$(1 - $\delta$)% central confidence interval is computed by running the upper and lower
bound procedures with the adjusted confidence level of 100$\times$(1 - $\delta$/2)%.
For example, if we want to compute a 95% central confidence interval, we compute 97.5% lower and upper confidence bounds. This places $\delta$/2 = 2.5% probability mass in each tail, thereby
providing 95% coverage in the central region.
Pictorially below, you can see that the 95% central confidence interval (top row) produces wider bounds than just using the 95% lower and upper confidence bounds separately (bottom row). The looser
bounds are unfortunate. But naively computing the lower and upper bounds at the original confidence level of 95% sacrifices coverage due to multiple testing.
Figure 6: central confidence intervals produce wider bounds to correct for multiple testing
In the next section, where we compute central confidence intervals for balanced accuracy, we will have to do even more to correct for multiple testing.
Confidence Bounds for Balanced Accuracy
Recall that the balanced accuracy for a binary classifier is the mean of its accuracy on examples from the positive class and its accuracy on examples from the negative class.
To define what we mean by the "true balanced accuracy", we need to define appropriate distributions over examples from each class. To do so, we decompose $D$ into separate class conditional
distributions, $D^+$ and $D^-$, where
$\Pr\left\{ (x,y) \sim D^+ \right\} = \Pr\left\{ (x,y) \sim D \mid y = +1 \right\},$ $\Pr\left\{ (x,y) \sim D^- \right\} = \Pr\left\{ (x,y) \sim D \mid y = -1 \right\}.$
The positive and negative true accuracies are defined with respect to each of these class specific distributions:
$\text{acc}^+(h) = E_{(x,y) \sim D^+} \, \mathbf{1}[ h(x_i) = y_i ],$ $\text{acc}^-(h) = E_{(x,y) \sim D^-} \, \mathbf{1}[ h(x_i) = y_i ].$
The true balanced accuracy is then defined as the average of these,
$\text{acc}_\text{bal}(h) = \frac{\text{acc}^+(h) + \text{acc}^-(h)}{2}.$
Constructing the Bound for Balanced Accuracy
With the above definitions in hand, we can now bound the balanced accuracy of our classifier based on its performance on a test set. Let $S$ be the test set, and let
• $N^+$ denote the number of positive examples in $S$
• $N^-$ denote the number of negative examples in $S$
• $k^+$ denote the number of positive examples in $S$ that $h$ correctly classified
• $k^-$ denote the number of negative examples in $S$ that $h$ correctly classified
From these quantities, we can find lower and upper bounds for the positive and negative accuracies based on the binomial CDF.
Denote these lower and upper bounds on positive and negative accuracy as
$\underline{\text{acc}^+}(h) ,~~ \overline{\text{acc}^+}(h) ,~~ \underline{\text{acc}^-}(h) ,~~ \overline{\text{acc}^-}(h).$
To find a 100(1 - $\delta$)% confidence interval for the $\text{acc}_\text{bal}(h)$, we first compute the quantities
$\underline{\text{acc}^+}(h) = \underline{p}(N^+, k^+, \delta/4) ~~ \text{ and } ~~ \overline{\text{acc}^+}(h) = \overline{p}(N^+, k^+, \delta/4)$ $\underline{\text{acc}^-}(h) = \underline{p}(N^-, k^
-, \delta/4) ~~ \text{ and } ~~ \overline{\text{acc}^-}(h) = \overline{p}(N^-, k^-, \delta/4)$
Importantly, we've used an adjusted delta value of $\delta/4$ to account for mulitple testing. That is, if we desire our overall coverage to be (1 - $\delta$) = 95%, we run our individual bounding
procedures with the substituted delta value of $\delta/4 = 1.25\%$.
The reason why is as follows. By construction, each of the four bounds will fail to hold with probability $\delta/4$. The union bound in appendix A tells us that the probability of at least one of
these four bounds failing is no greater than the sum of the probabilities that each fails. Summing up the failure probabilities for all four bounds, the probability that at least one bound fails is
therefore no greater than $4\cdot(\delta/4) = \delta$. Thus the probability that none of the bounds fails is at least 1 - $\delta$, giving us the desired level of coverage.
Last, we obtain our exact lower and upper bounds for balanced accuracy by averaging the respective lower and upper bounds for the positive and negative class accuracies:
$\underline{\text{acc}_\text{bal}}(h) = (1/2) \left( \underline{\text{acc}^+}(h) + \underline{\text{acc}^-}(h) \right)$ $\overline{\text{acc}_\text{bal}}(h) = (1/2) \left( \overline{\text{acc}^+}(h)
+ \overline{\text{acc}^-}(h) \right)$
Pictorially below, we can see how the averaged lower and upper bounds contain the true balanced accuracy.
Figure 7: the balanced accuracy is bounded by the respective averages of the lower and upper bounds
Comparison with intervals based on the Normal approximation
The main benefit of using bounds derived from the binomial CDF is that they are exact and always contain the true accuracy the desired fraction of the time.
Let's compare this with the commonly used bound obtained by approximating the binomial PMF with a normal distribution. The motivation for the normal approximation comes from the central limit
theorem, which states that for a binomial distribution with parameters $N$ and $p$, the distribution of the empirical accuracy, $\hat{p} = k/N$, converges to a normal distribution as the sample size,
$N$, goes to infinity,
$\hat{p} \stackrel{d}{\longrightarrow} \mathcal{N}\left(p, \frac{p(1-p)}{N}\right).$
This motivates the use of the traditional two-standard deviation confidence interval in which one reports
$\Pr\left\{ | p - \hat{p} | \le 1.96 \,\hat{\sigma} \right\} \ge 95\% ~ ~ ~ \text{where} ~ ~ ~ \hat{\sigma} = \sqrt{ \frac{ \hat{p}(1-\hat{p}) }{N} }.$
But it's well known that the normal distribution poorly approximates the sampling distribution of $\hat{p}$ when $p$ is close to zero or one. For instance, if we observe zero errors on the test set,
then $\hat{p}$ will equal 1.0 (i.e., 100% empirical accuracy), and the sample standard deviation, $\hat{\sigma}$, will equal zero. The estimated lower bound will therefore be equal to the empirical
accuracy of 100%, which is clearly unbelievable.
And since we train classifiers to have as close to 100% accuracy as possible, the regime in which $p$ is close to one is of major interest. Thus, exact confidence intervals based on the binomial CDF
are both more accurate and practically useful than those based on the normal approximation.
At Groundlight, we've put a lot of thought and effort into assessing the performance of our customers' ML models so they can easily understand how their detectors are performing. This includes the
use of balanced accuracy as the summary performance metric and exact confidence intervals to convey the precision of the reported metric.
Here we've provided a detailed tour of the methods we use to estimate confidence intervals around balanced accuracy. The estimated intervals are exact in that they possess the stated coverage, no
matter how many ground truth labeled examples are available for testing. Our aim in this post has been to provide a better understanding of the metrics we display, how to interpret them, and how
they're derived. We hope we've succeeded! If you are interested in reading more about these topics, see the references and brief appendices below.
[Langford, 2005] Tutorial on Practical Prediction Theory for Classification. Journal of Machine Learning Research 6 (2005) 273–306.
[Brodersen et al., 2010] The balanced accuracy and its posterior distribution. Proceedings of the 20th International Conference on Pattern Recognition, 3121-24.
Appendix A - the union bound
Recall that the union bound states that for a collection of events, $A_1, A_2, \ldots, A_n$, the probability that at least one of them occurs is less than the sum of the probabilities that each of
them occurs: $\Pr\left\{ \cup_{i=1}^n A_i \right\} \le \sum_{i=1}^n \Pr(A_i).$
Pictorially, the union bound is understood from the image below which shows that area of the union of the regions is no greater than the sum of the regions' areas.
Figure 8: Visualizing the union bound. The area of each region $A_i$ corresponds to the probability that event $A_i$ occurs. The sum of the total covered area must be less than the sum of the
individual areas.
Appendix B - interpretation of confidence intervals
The semantics around frequentist confidence intervals is subtle and confusing. The construction of a 95% upper confidence bound does NOT imply there is a 95% probability that the true accuracy is
less than the bound. It only guarantees that the true accuracy is less than the upper bound in at least 95% of the cases that we run the the upper confidence bounding procedure (assuming we run the
procedure many many times). For each individual case, however, the true accuracy is either greater than or less than the bound. And thus, for each case, the probability that the true accuracy is less
than the bound equals either 0 or 1, we just don't know which.
If you instead desire more conditional semantics, you need to use Bayesian credible intervals. See Brodersen et al., 2010 for a nice derivation of credible intervals for balanced accuracy. | {"url":"http://code.groundlight.ai/python-sdk/blog/confidence-intervals-for-balanced-accuracy","timestamp":"2024-11-05T17:13:36Z","content_type":"text/html","content_length":"240196","record_id":"<urn:uuid:d43f9530-7dc1-4c98-9909-a08a3dd628e0>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00419.warc.gz"} |
Spline Interpolation
I. Download
This distribution is dated January 5, 2009. It includes the complete set of source files, along with one data file.
II. Related Work
This C program is based on the following paper:
• P. Thévenaz, T. Blu, M. Unser, "Interpolation Revisited," IEEE Transactions on Medical Imaging, vol. 19, no. 7, pp. 739-758, July 2000.
III. Explanations
The purpose of this program is to be a practical and didactic introduction on how to perform spline interpolation. It is a self-contained application that will apply a rigid-body transformation to an
image (rotation and translation). The source code (ANSI-C) is divided into 4 principal files ('.c') and 3 associated header files ('.h'). A standard data file (the famous Lena image) is provided so
that you can test the program as soon as it has been downloaded and compiled.
1. The file demo.c contains the topmost procedure main(). It calls the routines that perform the I/O-associated tasks, calls the routine that converts the image samples into B-spline coefficients,
and cares about geometric issues.
2. The files coeff.c and coeff.h implement the algorithm that converts the image samples into B-spline coefficients. This efficient procedure essentially relies on the paper cited above; data are
processed in-place. Even though this algorithm is robust with respect to quantization, we advocate the use of a floating-point format for the data.
3. The files interpol.c and interpol.h perform the bidimensional interpolation. Given an array of spline coefficients, they return the value of the underlying continuous spline model, sampled at the
location (x, y). The model degree can range from 2 to 9.
4. The files io.c and io.h provide a rudimentary user interface that is supposed to work on any compiler/system/OS that is in compliance with the ANSI-C norm. The image to be processed is read from
file; the file format is binary raw data, with 1 unsigned byte per pixel, organized in raster fashion. The image resulting from executing the program is also stored on file, with the same
conventions. You will need to use your own software to display it. A platform-independent candidate is perhaps ImageJ, a public-domain image-processing package written in Java (use the menu
File:Import:Raw... and supply the requested information about the image).
5. The data file lena.img stores a 256 x 256 image that, along the years, has become the de facto standard in image processing. The format corresponds to that described above. The visualization
convention is to map low numeric values to dark gray-levels (black) and high numeric values to light gray-levels (white).
IV. Underlying Theory
4.1 Interpolation Coefficients
To simplify the explanations, let us examine the one-dimensional case first and let us defer to later the considerations on additional dimensions. Given a set of samples s[j] with j integer, the
fundamental task of interpolation is to compute the following equation for any desired value of x:
ƒ(x) = ∑[k] c[k] g(x - k),
where x is a real number and where k takes on every possible integer value. For a specific integer value j, it is also desired that the interpolated function ƒ(j) takes the same value as the sample s
[j] of the set of available samples we want to interpolate
s[j] = ƒ(j) = ∑[k] c[k] g(j - k).
We discourage the traditional approach one usually follows in order to satisfy this equation. Traditionally, one makes sure that two conditions are met. Firstly, set c[k] = s[k] for all k. Secondly,
select a function g() that takes value 0 for all integer arguments, except at the origin where it must equal unity (for any non-integer arguments, g() is free to take any value it may like). This
would do the trick, and would be the classical approach to interpolation.
Now, B-splines have very attractive properties, which makes them very good candidates in the role of the function g(). Unfortunately, they don't have all the qualities in the world; in particular,
they may take a non-zero value for integer arguments, and they may differ from unity at the origin. Thus, one must depart from traditional interpolation so as to use B-splines for interpolating data.
Instead, one must determine the suitable set of coefficients c[k] ≠ s[k] = ƒ(k) such that ƒ(j) = ∑ c[k] g(j - k) still holds true for any integer j. This time, setting c[k] = s[k] doesn't do the
trick. We need instead some routine that produces a set of coefficients c[] out of a set of samples s[]. This is the role of the routine available in the file coeff.c. The core of this routine
consists in a forward-backward recursive filter (for more details, see the reference [1]).
4.2 Several Dimensions
To process an image, one must consider more than a single dimension. The traditional extension--that we follow here--goes by the name of tensor product. It consists in weighting the interpolation in
one direction by that of the remaining directions (this is also called a separable approach). More precisely, in two dimensions we compute
ƒ(x, y) = ∑[j] ∑[i] c[i, j] g(x - i) g(y - j),
where the notations should be obvious. The only difficulty is that we now have to deal with a bidimensional array of coefficients c[i, j]. Fortunately, these coefficients can be obtained by
processing the bidimensional set of samples in one direction first, and then by processing the remaining direction. This separable approach is the most efficient way to process data with more than
one dimension. Provided that the set of coefficients c[i, j] has already been obtained from coeff.c, the role of the part of the program available in the file interpol.c is to carry out the
computation of the tensor product and of the function g(). More specifically, this latter is a B-spline.
4.3 Border Conditions
So far, we have taken for granted that the number of samples s[] was infinite, same with the number of coefficients c[]. Obviously, the amount of RAM in your computer is only so much, which asks for
a finite number of data. Reconciliation of these contradictory requirements is the object of the present section.
Let us examine the question in one dimension, and let us suppose you have N samples at your disposal. Given only those samples you know of, if we find a way to predict (extrapolate) the value of any
coefficient outside the range of your known data, then we are back to the first (infinite) formulation, without its cost. This is known as an implicit scheme. Extrapolating the samples s[] is less
relevant than extrapolating the coefficients c[], for the former do not explicitly appear in what we compute in practice, that is, ∑ c[k] g(x - k).
There is no limit to the number of ways one may select to perform extrapolation. After all, any single one is but a wild guess, even though some are more informed than others. We advocate here an
extrapolation scheme where coefficients are mirrored around their extremities, namely, around c[0] and around c[N-1]. This double mirroring ensures that every c[k] takes a definite value, no matter
how far k is from the interval [0, N-1]. Moreover, it imposes the same doubly-mirrored structure on the samples s[k], and also on the whole curve ƒ(x). An important benefit of this scheme is that
there is no need to "invent" special values (e.g., zero); all extrapolated values are replicated from those of the known interval. Moreover, this scheme ensures that no discontinuities (or strong
changes of value) of ƒ() arise at the extremities of the known interval.
V. Examples
The following pair of images is typical of the kind of results you can obtain with the present program. The image has been rotated by some angle around an arbitrary origin, while an additional
translation has been thrown in for good measure. Please observe the influence of mirroring the data (the program allows for the masking out of the extrapolated data, if desired).
Left: 256x256
Lena input
image. Right:
resulting from
an arbitrary
(no masking)
Here is a list of additional examples that illustrate the use of our interpolation routine. You may want to compare your own results to the ones below so as to ensure that your implementation is
correct. In terms of ease of check, we have ordered the examples from most easy to more difficult. The images that we provide here are subject to lossless coding. | {"url":"https://bigwww.epfl.ch/thevenaz/interpolation/","timestamp":"2024-11-08T04:09:26Z","content_type":"text/html","content_length":"13153","record_id":"<urn:uuid:f1c962ee-b967-452b-aa22-289a93a8fe57>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00755.warc.gz"} |
Stata: Set Up
Introduction to Stata windows: restoring closed windows, personalizing window sizes, font sizes, and saving preferences.
There are 4 main windows in STATA
1. Variables - Lists variables and labels. Double click on a variable in this window to paste it into the command window
2. Command - Type in commands here
3. Results - Displays results generated from commands
4. Review - Keeps a history of all past commands executed. Click on a past command to paste it into the command window.
Exploring data layout, variable types, and use of variable and value labels
• Variable labels give a fuller description of variables
• Text variables stored in string format, numerical variables stores as int, float, byte, etc.
• Store a variable with numerical value and attach text labels to each value by creating a value label and applying it to the variable
Dummy Variables for each category take values "0" or "1"
Stata: Data Management & Analysis
One-way and two-way tables, incorporating missing data, and including relative frequencies
• Tabulations best used for categorical variables
Two-group mean comparison test
• Plotting graphs gives an idea of distributions of data
• ttest gives results for one-tail and two-tail tests
Stata: Graphs
Using Graphs to understand your data through histogram, scatterplot, and bar graph
• Use graphs to compare statistics across categories
Using the Graph Editor to customize labels, colour schemes, design, etc.
• Saving graphs and pasting to Word documents
• Generate separate graphs by category
Record and save customizations to generate a standard format for all your graphs
Stata: Regression Analysis
Review of Ordinary Least Squares (OLS) simple linear regression and the constant coefficient
• STATA generates the constant term by regressing on x=1
Regressing with dummy variables and choosing a reference group
• Generate dummies to capture different effects of each category
• Choose one reference group (or constant) to omit to avoid multicollinearity
• Coefficients measure effects relative to reference group | {"url":"https://blogs.ubc.ca/datawithstata/videos/","timestamp":"2024-11-11T06:59:15Z","content_type":"text/html","content_length":"55358","record_id":"<urn:uuid:ed81a381-a949-488d-88d0-8c62bbb2ec83>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00301.warc.gz"} |
Worked Examples: Units, Conversion Factors & Dimensional Analysis
63 Worked Examples: Units, Conversion Factors & Dimensional Analysis
These worked examples have been fielded as homework problems or exam questions.
Determine the base dimensions of each of the following variables:
(a) Plane angle
(b) Specific volume
(c) Force
(d) Stress
(e) Angular velocity
(a) Plane angle: A plane angle radian. Because it is the ratio of an arc length to the radius, then the plane angle is dimensionless, i.e., a radian is one measurement unit that is already
dimensionless, i.e.,
(b) Specific volume: The specific volume
(c) Force: Force
(d) Stress: Stress
(e) Angular velocity: Angular velocity
High school students become curious about why some insects can walk on water. They discover that a fluid property of importance in this problem is called surface tension, which is given the symbol
The units of surface tension,
Write the primary dimensions of each of the following variables from the field of thermodynamics:
(a) Energy,
(b) Specific energy,
(c) Power,
(a) Energy has units of force times distance, i.e.,
(b) Specific energy has units of energy per unit mass, i.e.,
(c) Power is the rate of doing work, so a force times distance per unit time, i.e.,
Determine the primary (base) dimensions of each of the following parameters from thermodynamics:
1. Energy,
2. Work,
3. Power,
4. Heat,
1. Energy,
2. Work,
3. Power,
4. Heat
Given that jet fuel’s average energy density is 43 to 45 Mega-Joules per kilogram), calculate the equivalent energy density in kilowatt-hours per kilogram (kWh/kg).
A Watt is a Joule per second. So, one Watt hour is the equivalent of 3,600 Joules per hour. Therefore, one kilo-watt-hour (kWh) = 3.6 Mega-Joules (MJ). The energy density (in units of kWh/kg) =
energy density (in units of MJ/kg)/3.6. Therefore, the energy density of jet fuel in kilowatt-hours per kilogram is approximately 11.9 to 12.5 kWh/kg.
Write down the Bernoulli equation and explain the meaning of each term. Verify that each of the terms in the Bernoulli equation has the exact fundamental dimensions.
The Bernoulli equation can be written as
The first term is the local static pressure, the second is dynamic pressure, and the third is the hydrostatic pressure. The sum of the three terms is called total pressure. The Bernoulli equation is
a surrogate for the energy equation in a steady, incompressible flow without losses or energy addition.
Each of the terms in the Bernoulli equation has units of pressure. In terms of fundamental dimensions, then
So, all terms have the same fundamental dimensions of M L
In each case, convert the units given into base units (can only include mass, length, and time). Show all the steps.
1. N (Newton)
2. Pa (Pascal) or N m
3. lb (pound)
4. J (Joule) or N m
5. W (Watt) or J s
Notes: A Newton (N) is an SI unit of force but not a base unit. However, a force is equivalent to a mass times an acceleration, so in terms of base units, a Newton is equivalent to units of M L T
In each case, convert the numerical values in the units given into equivalent numerical values in base units (mass, length, and time). Show all the steps and state the conversion factor(s) you used.
1. 3.3 l (liters) = 3.3/1,000 = 0.0033 m
2. 1.2 hrs (hours) = 3,600
3. 15.6 hp (horsepower) = 15.6
4. 12.8 gals (US gallons) = 12.8
5. 12.4 kN cm
Notes: Converting units takes some work, but it is essential to do it correctly. The first step is to realize and accept that many quantities measured daily are not base quantities but need to be
converted to base quantities for engineering calculations. Volume is often measured in liters but is not a base unit. There are 1,000 liters in a cubic meter; a meter is a base SI unit. James Watt
figured out from experiments with Scottish farm horses that one horsepower (hp) was equivalent to a rate of doing work of 550 foot-pounds per second or 550 ft-lb s
In a particular fluids problem, the flow rate,
In terms of the dimensions
and so for each parameter, then
To obtain dimensional homogeneity, then
A small jet airplane has a mean chord of 1.5 m and flies at a Mach number of 0.7 in conditions equivalent to those in the ISA. If the Reynolds number, based on mean chord length, is
This problem gives minimal information besides the Mach number, Reynolds number, and wing chord. The Mach number at altitude (alt) is
and the corresponding Reynolds number by
so the ratio
Also, both
therefore the ratio
The ratio
Therefore, the pressure altitude corresponding to this standard temperature in the ISA is
which, by rechecking the Reynolds number and Mach number, is a reasonably good estimate of the pressure altitude at which the airplane is flying.
A student wants to know the drag on a full-scale race car with a length
In the wind tunnel, we want to match the Reynolds numbers between the actual car and the model to achieve flow similarity. The Reynolds number for the actual vehicle can be written as
and for the model
Therefore, we have that
if the flow conditions (e.g., temperature and pressure, hence its density and viscosity) are to be the same in the wind tunnel as those for the actual car. The ratio
The drag force on the actual car can be written as
where the ratio
So, the student is correct! This is an interesting and useful outcome that follows from the scaling relationships. The force on a body of a particular shape at a given Reynolds number is the same
regardless of the combination of size and speed used to produce that given Reynolds number. Of course, the flow conditions (e.g., temperature and pressure, hence density and viscosity) must also be
the same.
Consider the drag of a sphere problem using the Buckingham
The relationship between
For each dependency, then
Setting up the dimensional matrix gives
For the first
The values of the coefficients
By inspection
which is still dimensionless, but it is a different grouping to what was obtained with
This is an interesting outcome because this parameter is a form of Stoke’s Law, which Sir George Stokes determined in the 1840s. He found that the drag force
Comments: This drag force is proportional to the sphere’s radius. This outcome is not obvious because, based on what was done before, it might be thought that drag would be proportional to the
cross-section area, which would vary as the square of the radius. The drag force is also directly proportional to the speed and
Therefore, in this case, then
For the second
and setting the sum of the powers to zero gives
In this case
which will be recognized as the Reynolds number.
Therefore, as a result of the dimensional analysis of the sphere, then
or in explicit form
Based on experiments performed in a low-speed wind tunnel, it is determined that the power required at the shaft to drive a propeller forward is a function of the thrust the propeller produces,
The power required for the propeller (the dependent variable),
In this case, there are six variables (
The functional dependence can also be written in the form
Choose the variables
where in each case, the powers
Now, the base dimensions of each variable can be written down. For this problem, then
and so the dimensional matrix is
Examining the determinants of the submatrix formed by each of the elected repeating variables quickly confirms that they are indeed linearly independent.
Considering the first
In terms of the base dimensions of the parameters, then
Making the equation dimensionally homogeneous by equating the exponents for each of the dimensions, in turn, gives
These simultaneous equations have the solution that
Which is a form of thrust coefficient, i.e., a dimensionless measure of thrust.
Considering now the second
where new values for
Making this latter equation dimensionally homogeneous gives
These equations have the solution that
which is a form of power coefficient, i.e., a dimensionless measure of power.
Finally, for the third
and in terms of dimensions, then
Making this final equation dimensionally homogeneous gives
These latter equations have the solution that
which is a dimensionless airspeed called a tip speed ratio or advance ratio.
Therefore, for this propeller problem, then
or finally as
This outcome allows us to evaluate the propeller’s performance in terms of power coefficient as a function of the thrust coefficient and the tip speed ratio.
Consider the internal flow through a rough pipe. The objective is to determine the dimensionless groupings that will describe this problem. The dependencies include the average flow velocity
Proceeding using the Buckingham
In this case, there are six variables (
The dimensional matrix is
Considering the first
and in terms of dimensions, then
Making the equation dimensionally homogeneous by equating the exponents for each of the dimensions, in turn, gives
These simultaneous equations have the solution that
which is the reciprocal of the Reynolds number, but as discussed before, this grouping can also be inverted to get the first
For the second
and in terms of dimensions, then
and the exponents
These simultaneous equations have the solution that
which is a measure of the relative surface roughness.
Finally, for the third
and in terms of dimensions, then
Making the equation dimensionally homogeneous by equating the exponents for each of the dimensions, in turn, gives
These simultaneous equations have the solution that
which is a dimensionless pressure drop or “head” drop. Usually, this latter grouping is expressed in terms of a friction factor, i.e.,
Therefore, the final result is
This result shows that the frictional pressure drop along the pipe will be a function of the Reynolds number and the pipe’s effective dimensionless roughness.
For this problem, the drag
In implicit form, then
Setting up the dimensional matrix gives
Now, the repeating variable must be chosen, for which the standard choice is
and for the second
Continuing with the first
and terms of the dimensions, then
which is the Froude number,
For the second
and terms of the dimensions, then
which is a drag coefficient,
Therefore, the final result is
i.e., the drag coefficient on the hull is some function of the Froude number.
A tiny spherical particle of diameter
The information given is that
so, in implicit form, then
Setting down the dimensional matrix gives
For the
and in terms of dimensions, then
and so
Therefore, the
which is a viscous drag coefficient applicable, in this case, to what is known as a Stokes flow, which is a flow corresponding to Reynolds numbers near unity.
A flow experiment with a circular cylinder shows that at a specific condition, a vortex shedding phenomenon at frequency
The frequency of shedding
or in the implicit form, then
Setting up the dimensional matrix, i.e.,
So, the first grouping is
and terms of the dimensions, then
which is the Strouhal number
A result could also have been obtained by recognizing that the frequency of shedding may also be a function of the flow density
or in the implicit form, then
In this case,
Setting up the dimensional matrix gives
Proceeding as usual with the selection of the repeating variables (again, the standard choice is
For the first
and terms of the dimensions, then
which is the Strouhal number
For the second
and terms of the dimensions, then
which is the Reynolds number
Therefore, the frequency of shedding is expected to be a function of Reynolds number, i.e.,
Define Reynolds number and explain its meaning. Show that the Reynolds number represents a ratio of the relative magnitude of inertial effects to viscous effects in the flow. Hint: Multiply both the
numerator and denominator of the equation for the Reynolds number by a velocity and a length scale.
The Reynolds number is a dimensionless grouping formed in terms of fluid density,
Reynolds number represents “the ratio of the relative effects of inertial effects to viscous effects,” which can be seen by writing
On the numerator,
The distance traveled by a dimpled golf ball depends on its aerodynamic drag,
The drag of the golf ball (the dependent variable)
In this case, there are six variables (
The functional dependence can also be written in the form
Choose the standard aerodynamic repeating variables
where in each case the values of the exponents
For this problem, then
and so the dimensional matrix is
Considering the first
and in terms of dimensions, then
Making the equation dimensionally homogeneous by equating the exponents for each of the dimensions, in turn, gives
These simultaneous equations have the solution that
which is a force coefficient.
Considering the second
and in terms of dimensions, then
Making the equation dimensionally homogeneous by equating the exponents for each of the dimensions, in turn, gives
These simultaneous equations have the solution that
which is a dimensionless length scale, i.e., the ratio of the diameter of the dimples to the diameter of the golf ball.
Considering the third
and in terms of dimensions, then
Making the equation dimensionally homogeneous by equating the exponents for each of the dimensions, in turn,n gives
These simultaneous equations have the solution that
or inverting the grouping (it is still dimensionless)
which is a Reynolds number.
Finally, then
or just
The sound intensity
The relationship between
A hint is given that the sound intensity
Also, pressure is force per unit area, so
In this problem,
Proceeding with selecting the repeating variables, one choice is
For the first
and terms of the dimensions, then
For the second
and terms of the dimensions, then
Interestingly, the factor
Consider a liquid in a cylindrical container where both the container and the liquid rotate as a rigid body (called solid-body rotation), as shown in the figure below.
In this problem, the objective is to find the effects of the elevation difference
Proceeding with selecting the repeating variables, one choice is
For the first
and terms of the dimensions, then
For the second
and terms of the dimensions, then
Therefore, the final result is
A liquid of density
The objective is to find the effects on the exit flow velocity
Proceeding with selecting the repeating variables, the only choice, in this case, is
For the first
and terms of the dimensions, then
which is recognized as a Reynolds number.
For the second
and terms of the dimensions, then
For the third
which quickly follows as per the
Therefore, in this case, the dimensionless groupings involved are such that
The AIAA Design Build & Fly (DBF) team must determine the factors influencing the aerodynamic drag on a rectangular banner being towed behind their airplane.
The relationship between the drag on the banner
where the size of the banner is represented by its length,
Counting the variables gives
The dimensional matrix is
Following the Buckingham
The values of the coefficients
By inspection
i.e., a form of the drag coefficient. Aerodynamic force coefficients are usually defined in terms of the dynamic pressure, i.e.,
It would also be legitimate to write the drag coefficient as
where the banner area
Therefore, in this case
Inverting the grouping gives
which, in the latter case, is a Reynolds number based on the banner length. Notice that the grouping can be inverted by following established conventions for a similarity parameter or because it is
otherwise convenient.
Therefore, in this case
or again, this grouping can be inverted (for convenience), giving
which is a length-to-height ratio or what would be called an aspect ratio
As a result of the dimensional analysis, then
Finally, in explicit form, the drag coefficient can be written as a function of the Reynolds number based on the banner length and the aspect ratio of the banner, i.e.,
Note: Try this problem again using
The DBF team has observed that the banner in the previous problem begins to flutter at some critical airspeed, which results in a much higher drag on the banner. The flutter speed of the banner
appears to depend on the length of the banner,
The relationship between the flutter speed
or in implicit form as
Hence, in this problem
For each variable, the dimensions are
Setting up the dimensional matrix gives
Again, as in most aerodynamic problems,
In terms of the dimensions of the parameters, then
By inspection
which is a speed ratio or a dimensionless flutter speed.
In terms of the dimensions of the parameters, then
By inspection,
and, once again, a Reynolds number comes into the problem.
In terms of the dimensions of the parameters, then
By inspection
which is a form of structural dimensionless frequency or a structural reduced frequency.
As a result of the dimensional analysis, then
or in explicit form
Therefore, the dimensional analysis tells us that the dimensionless flutter speed of the banner will depend on the Reynolds number and its structural reduced frequency.
A spherical projectile of diameter
The relationship between the drag on the sphere and the given variables can be written in a general functional form as
or in implicit form as
For each variable, the units are
The dimensional matrix is
Following the Buckingham
In terms of the dimensions of the parameters, then
By inspection
which is a drag coefficient, which can be expressed in the conventional way as
In terms of the dimensions of the parameters, then
By inspection
More conventionally, this ratio is written as
Notice that
which is the familiar ratio of specific heats. This ratio would have been a product of the dimensional analysis if
In terms of the dimensions of the parameters, then
By inspection
or just
which is the Reynolds number.
As a result of the dimensional analysis, then
or in explicit form
But the critical grouping that comes out of this problem is
A force
The relationship between the force and the tip deflection can be written in the general functional form as
or in implicit form as
For each variable, the units are
The dimensional matrix is
This problem poses a dilemma because the choice of the repeating variables here is not apparent. Suppose
The accepted solution to this dilemma is to reduce the number of repeating variables by one and create a third
In terms of the dimensions of the parameters, then
By inspection
i.e., a dimensionless displacement is an expected, if not obvious, grouping.
In terms of the dimensions of the parameters, then
By inspection
and so
which is a dimensionless form of the second moment of area.
In terms of the dimensions of the parameters, then
By inspection
which is a form of dimensionless force or force coefficient.
Finally, all three
The ERAU wind tunnel uses tiny oil-based aerosol particles of characteristic size,
The information given is
and the units of
which confirms that the units of
A dimensionless form of
This outcome is interesting because it involves a Reynolds number based on particle diameter and the particle diameter ratio to the length scale. Therefore, the higher the Reynolds number and/or the
bigger the particle, the longer it will take to adjust to any changes in the flow conditions.
Based on experiments performed with a wind turbine, it is determined that its power output is a function of the size of the wind turbine as characterized by its radius
For this problem, the power output
In implicit form, then
The dimensional matrix is
Notice the base units of power are M L
Now, the repeating variable must be chosen. In this case, a good choice is
For the first
and for the second
Continuing with the first
and terms of the dimensions, then
which is a form of power coefficient, i.e.,
Considering now the second
and terms of the dimensions, then
which is a form of an advance ratio or tip speed ratio, i.e.,
Therefore, based on the information given, the power output of the wind turbine in terms of a power coefficient
A sphere is located in a pipe through which a liquid flows. The drag force
2. Write down the dimensional matrix for this problem in terms of base units M, L, and T.
3. Determine the relevant
4. If the drag force on the sphere with
1. In explicit form, then
or in the implicit form, then
2. Setting up the dimensional matrix for this problem gives
3. In this case,
4. Use
So the first
and terms of the dimensions, then
which is a force coefficient.
For the second
and terms of the dimensions, then
which is a dimensionless length. Therefore,
and so finally, in explicit form, then
5. To examine dimensional similitude, for both cases, the force coefficients must be the same so
so by rearrangement, then
Confirming geometric similarity gives
The singing sounds produced by power lines in the wind are called Aeolian tones, caused by vortex shedding behind the lines. The frequency of the sound,
1. Write down the functional relationship for the frequency in terms of the other parameters in implicit and explicit form.
2. How many base dimensions and
3. Write down the dimensional matrix for the problem.
4. Use the Buckingham
5. Rewrite the functional relationship in terms of the dimensionless parameter(s).
1. The frequency of the sound can be written explicitly as
or implicitly as
2. The number of variables is five, so
3. The dimensional matrix is
4. Choose the variables as
Raising the repeating variables to unknown powers gives
In terms of dimensions, then
Solving the equations gives
which is the inverse of the Reynolds number, so the grouping can be inverted (still having a dimensionless grouping), giving
Solving for
In terms of dimensions, then
Solving the equations gives
which is a Strouhal number, i.e.,
5. Therefore, based on the preceding analysis, then
So, the Strouhal number is a function of the Reynolds number.
Using Worked Example #30 as a basis, it is desired to replicate the physics of the singing sound and study it in a low-speed wind tunnel. The actual power wires have a diameter of 2.2 cm and are
known to sing at wind speeds between 25 mph and 70 mph. How would you develop a wind tunnel test plan to study this problem? The equivalent wire available for the wind tunnel test is 1.1 cm in
diameter, and the wires are strung across the test section’s width. The tunnel can reach a maximum flow speed of 75 ft/s. Is it possible to obtain the dynamic similarity of this problem in the wind
tunnel test? If not, why not, and what other consideration might be given to the wind tunnel test?
Based on the previous problem, the two relevant similarity parameters in t, in this case, the Reynolds number and the Strouhal number, and the Strouhal number is a function of the Reynolds number.
For the actual power wires, the Reynolds number based on diameter
using the highest wind speed of 70 mph and MSL ISA values for air. Notice that 70 mph is equivalent to 102.67 ft/s.
The wire available in the wind tunnel is only 1.1 cm in diameter, i.e.,
One solution would be to use a wire of a diameter of, say, 3.3 cm in the wind tunnel, which would need a flow speed of
to match the Reynolds numbers, and this is easily achievable. It is a factor of 0.67 of the actual wind speed.
Therefore, if the Reynolds number is matched by increasing the wire diameter, can the Strouhal number also be matched? In this case, the same sound frequency would be obtained if
But in this case, then
So even though the Reynolds number could be matched, the same frequency of the Aeolian sounds would not be obtained. Nevertheless, by matching the Reynolds number in the wind tunnel, the same
Strouhal number would be obtained, so the frequency obtained in this case would be higher by a factor of 2.24. In principle, it would be possible to study the singing sound behavior of the wires in
the wind tunnel. It is just that the frequencies obtained would be higher.
This is another example of the challenges in sub-scale testing to study fundamental problems. However, it can be seen that with a bit of ingenuity, the problem can be studied by matching or matching
the similarity parameters that govern the physics as closely as possible.
A Covid-19 particle has a density
1. Write down the functional dependency of
2. How many base dimensions and
3. Write down the dimensions of each of the variables involved.
4. Form the dimensional matrix for this problem.
5. Choose
1. Let
and in implicit form
3. For each variable, the dimensions are
4. The dimensional matrix is
5. The parameters
In terms of the dimensions of the parameters, then
By inspection
A weir is an obstruction in an open channel water flow. The volume flow rate
1. Write out the functional relationship in explicit and implicit form.
2. How many fundamental (base) dimensions are involved?
3. Write out the dimensional matrix for this problem.
4. Determine the
1. The explicit form of the relationship is
and in implicit form, then
2. In this problem, there are only two base dimensions, length
The units involved are
3. The dimensional matrix is
4. There will be two
For the first
In terms of the dimensions of the parameters, then
By inspection
For the second
In terms of the dimensions of the parameters, then
By inspection
Therefore, the result is
Sir Geoffrey I. Taylor (1886–1975) was a British physicist and engineer. He used dimensional analysis to estimate an explosion’s blast wave propagation characteristics. Taylor assumed that the radius
of the wave
The relationship may be written in a general functional form as
or in implicit form, then
Following the Buckingham
where the specific values of the coefficients
In this case,
In the final form, then
In the second part, the question is how the radius of the blast wave changes by a doubling of
An ocean surface wave is a sinusoidal-like disturbance propagating along the ocean’s surface, as shown in the figure below. The speed of the wave,
2. How many base dimensions and
3. Write down the dimensions of all the parameters in base units. Notice: The units of surface tension are force per unit length.
4. Create the dimensional matrix for this problem.
5. Determine the dimensionless parameter(s) that describe this problem.
1. In explicit form, the speed of the wave is
2. The base dimensions and
3. Below are the dimensions of all the parameters in this problem in terms of base units:
The speed of the wave,
Surface tension,
4. From the previous results, then the dimensional matrix is
5. The wave speed
In terms of dimensions, then
Alternatively, the preceding can be written as
For this equation to be mathematically balanced on the left and right sides, i.e., to be dimensionally homogeneous, then
Solving the foregoing equations gives
As a final check, it is easy to show that this grouping is dimensionless because
The flow rate in a pipe is to be measured with an orifice plate, as shown in the figure below. The static pressure before and after the plate is measured using two pressure gauges. The volumetric
flow rate
1. Write down the functional relationship for the volumetric flow rate
2. Write down the base units of the parameters involved in this problem.
3. How many base dimensions and dimensionless groupings are involved in this problem?
4. Write out the dimensional matrix for this problem.
5. Choose the repeating variables and explain your choice.
6. Determine the dimensionless grouping(s) for the parameters involved.
7. Write down the final dimensionless functional relationship(s).
1. The volumetric flow rate of water can be written in an explicit form as
2. The dimensions of the parameters involved are:
3. The number of base dimensions and groupings involved in this problem:
• Number of variables:
• Number of base dimensions (mass, length, and time are all involved):
• Number of
4. The dimensional matrix is
5. The only possible choice of repeating variables are
6. The
The dimensionless groupings must now be determined.
and inserting the dimensions for each parameter gives
Solving the equations gives
To check if this is a dimensionless grouping, substitute the units of the parameters, i.e.,
so confirming that the grouping is indeed dimensionless.
and inserting the dimensions for each parameter gives
Solving the equations gives
In this case, the ratio of one length to another is dimensionless.
7. Finally, the dimensionless relationship between the volumetric flow rate
Worked Example #37 – Air jet
An air jet holds a ball in vertical equilibrium, as shown in the figure below. The equilibrium height,
The relationship may be written in a general functional form as
or in the implicit form as
The base units are:
Setting up the dimensional matrix gives
The parameters collectively involve mass, length, and time, so
The four
The information given is that
so only the two groupings involving
Raising the repeating variables to unknown powers gives for the grouping
In terms of dimensions, then
By inspection
which is a dimensionless length.
Raising the repeating variables to unknown powers gives for the grouping
In terms of dimensions, then
By inspection | {"url":"https://eaglepubs.erau.edu/introductiontoaerospaceflightvehicles/chapter/worked-examples-in-dimensional-analysis/","timestamp":"2024-11-05T15:48:48Z","content_type":"text/html","content_length":"1050144","record_id":"<urn:uuid:6c18cd43-e599-4376-b40d-b59a76b57def>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00590.warc.gz"} |
Reflections of a High School Math Teacher
I would encourage you to use visual patterns in your class as soon as possible. There is an excellent website to get yourself started at
I gave this problem to my students in the middle of the quadratics unit for Algebra 1. Any student has a chance at this. That is what makes this so wonderful. The playing field is leveled. This
particular problem is very cool because there are so many ways to view it. So the beauty of visual patterns is that each student can look at the problem differently and still get the same answer. The
key is having them JUSTIFY their work. You might want to try it yourself before looking below.
Here is my original problem
Some things that I try to do when using this kind of problem.
1. Have students work on their own first. Then after some alone time, give time for collaboration.
2. Have students try to work multiple solutions if they finish with one.
3. Push students to visualize the problem in some way. (manipulatives or drawing or sketch or computer based image...)
4. Push students to give some meaning to the problem with algebraic symbols.
5. I try to remember that this problem might take 20 or more minutes to work out.
Here are a few examples of what my students created with this problem.
This one saw the perfect squares involved and then just subtracted the two missing pieces off at the end.
This group saw the smaller perfect square in the pattern. Then dealt with the rest in a linear way.
I love this one because it incorporated a graph to make sure of their answer
This one is detailed. The recognized the perfect square in the middle and then dealt with the other stuff as linear.
I made these visualizations for the problem. However, many of the students had these types of things on their papers before they wrote up the equations. You can also see by the video below how the
students were visualizing this problem. Also, next year I'm planning on having the students do this type of visualization with a spreadsheet that @alicekeeler made.
I found it in her blog post:
More questions were asked for this problem.
Are the algebraic results all the same?
How do we know that the algebraic results are all the same?
Can we use DESMOS to see if they are the same?
Can we simplify to see if the algebraic results are the same?
Can you imagine your students wanting to know these questions? It was so much fun.
Here is the reward of the day. One student who has trouble with the algebraic concepts and who almost never wants to talk about it got up and did this....magical. | {"url":"https://teachhighschoolmath.blogspot.com/2016/07/","timestamp":"2024-11-03T20:31:08Z","content_type":"application/xhtml+xml","content_length":"190855","record_id":"<urn:uuid:4d3e3057-6a1e-4f21-8b2f-8d8a136acc53>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00793.warc.gz"} |
Church–Turing–Deutsch principle
Short description: Principle in computer science
In computer science and quantum physics, the Church–Turing–Deutsch principle (CTD principle) is a stronger, physical form of the Church–Turing thesis formulated by David Deutsch in 1985.^[1] The
principle states that a universal computing device can simulate every physical process.
The principle was stated by Deutsch in 1985 with respect to finitary machines and processes. He observed that classical physics, which makes use of the concept of real numbers, cannot be simulated by
a Turing machine, which can only represent computable reals. Deutsch proposed that quantum computers may actually obey the CTD principle, assuming that the laws of quantum physics can completely
describe every physical process.
An earlier version of this thesis for classical computers was stated by Alan Turing's friend and student Robin Gandy in 1980.^[2]^[3]
See also
Further reading
• Deutsch, D. (1997). "6: Universality and the Limits of Computation". The Fabric of Reality. New York: Allan Lane. ISBN 978-0-14-027541-4.
• Christopher G. Timpson Quantum Computers: the Church-Turing Hypothesis Versus the Turing Principle in Christof Teuscher, Douglas Hofstadter (eds.) Alan Turing: life and legacy of a great thinker,
Springer, 2004, ISBN:3-540-20020-7, pp. 213–240
External links
Original source: https://en.wikipedia.org/wiki/Church–Turing–Deutsch principle. Read more | {"url":"https://handwiki.org/wiki/Church%E2%80%93Turing%E2%80%93Deutsch_principle","timestamp":"2024-11-10T11:18:52Z","content_type":"text/html","content_length":"37770","record_id":"<urn:uuid:ec250ad4-9335-43fe-a349-1adaf398bea8>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00866.warc.gz"} |
Frequency distribution charts and graphs
This is called a frequency distribution. This leads to the second difference from bar graphs. In a bar graph, the categories that you made in the frequency table were organize qualitative data in
tables; construct bar graphs; construct pie charts. For a quick A frequency distribution lists each category of data and the number of
A frequency distribution table showing categorical variables. Frequency distribution tables give you a snapshot of the data to allow you to find patterns. A quick By counting frequencies we can make
a Frequency Distribution table. like to make a Bar Graph or a Pie Chart using the Data Graphs (Bar, Line and Pie) page . The graph below shows a frequency distribution on the left, and a cumulative
distribution of the same data on the right, both plotting the number of values in each Frequency distribution examples with charts and graphs; Central tendency; Percentiles; and Correlations.
History Midterm for fifty students - scores and They should be chosen so that the shape of the graph resembles a distribution curve similar to the histograms shown above. Bar charts and histograms:
cussion of frequency distribution tables and concluding with graphs of distributions for 2.2 FREqUENCY DISTRIBUTIONS FOR GROUPED DATA. Frequency
A cumulative relative frequency graph of a quantitative variable is a curve graphically showing the cumulative relative frequency distribution. In this statistics tutorial, learn to visualize
frequency distributions using different types of graphs, and understand when to use each kind of graph. Knowledge in statistics provides you with the necessary tools and conceptual foundations in
quantitative reasoning to extract information intelligently from this Notes: Bar charts can also be created in the Chart platform. (Graph > Chart). For more details on creating bar charts, see the
book Basic Analysis and Graphing ( Graphs of Frequency Distributions. Frequency Histogram. • A bar graph that represents the frequency distribution. • The horizontal scale is quantitative and A
frequency distribution as a term used in statistics is a graph or table that displays the frequency of different outcomes in one sample. Each of the entries in the Ans: false Response: See section
2.1 Frequency Distributions Difficulty: Medium 12. In a histogram, the tallest bar represents the class with the highest cumulative
Start studying Frequency Distributions and Graphs. Learn vocabulary, terms, and more with flashcards, games, and other study tools. drawing charts and graphs. T/F. To determine the appropriate width
of each class interval in a grouped frequency distribution, we:
Frequency Tables, Pie Charts, and Bar Charts Frequency tables, pie charts, and bar charts can be used to display the distribution of a single categorical variable . These displays show all possible
values of the variable along with either the frequency (count) or relative frequency (percentage). A frequency distribution is an overview of all distinct values in some variable and the number of
times they occur. That is, a frequency distribution tells how frequencies are distributed over values. Frequency distributions are mostly used for summarizing categorical variables. That's because
metric variables tend to have many distinct values. These result in huge tables and charts that don't give insight into your data. Frequency Graphs Histograms and bar charts are both visual displays
of frequencies using columns plotted on a graph. The Y-axis (vertical axis) generally represents the frequency count, while the X-axis (horizontal axis) generally represents the variable being
measured. Some of the graphs that can be used with frequency distributions are histograms, line charts, bar charts and pie charts. Frequency distributions are used for both qualitative and
quantitative data. Construction [ edit ] Decide the number of classes. Optionally, select the Chart Output check box to have Excel include a histogram chart with the frequency distribution. If you
don’t select this check box, you don’t get the histogram — only the frequency distribution. Click OK. Excel creates the frequency distribution and, optionally, the histogram. • Create and interpret
frequency distribution tables, bar graphs, histograms, and line graphs • Explain when to use a bar graph, histogram, and line graph • Enter data into SPSS and generate frequency distribution tables
and graphs. HOW TO BE SUCCESSFUL IN THIS COURSE. Have you ever read a few pages of a textbook and realized Pie charts are a great way to graphically show a frequency distribution. In a pie chart, the
frequency or percentage is represented both visually and numerically, so it is typically quick for readers to understand the data and what the researcher is conveying.
In a frequency distribution graph, frequencies are presented on the, and the scores (categories) are listed on the . Y axis; X axis Histograms, bar graphs, and polygons Polygons and bar graphs
Histograms and bar graphs. Histograms and polygons. A set of scores ranges from a high of X = 72 to a low of X = 28. If these scores were put in a
In this statistics tutorial, learn to visualize frequency distributions using different types of graphs, and understand when to use each kind of graph. Knowledge in statistics provides you with the
necessary tools and conceptual foundations in quantitative reasoning to extract information intelligently from this Notes: Bar charts can also be created in the Chart platform. (Graph > Chart). For
more details on creating bar charts, see the book Basic Analysis and Graphing ( Graphs of Frequency Distributions. Frequency Histogram. • A bar graph that represents the frequency distribution. •
The horizontal scale is quantitative and A frequency distribution as a term used in statistics is a graph or table that displays the frequency of different outcomes in one sample. Each of the
entries in the
Chapter 2: Frequency Distribution and Graphs 1. Chapter 2
Frequency Distributions
and Graphs
2. A frequency distribution is the organization of raw data in table from, using classes and frequency.
3. The number of miles that the employees of a large department store traveled to work each day
Represent data using bar chart, Pareto chart, pie graph and time series graph. □ Draw and interpret a stem and leaf plot. Objectives. A frequency distribution table showing categorical variables.
Frequency distribution tables give you a snapshot of the data to allow you to find patterns. A quick By counting frequencies we can make a Frequency Distribution table. like to make a Bar Graph or a
Pie Chart using the Data Graphs (Bar, Line and Pie) page .
Absolute, relative or cumulative frequencies can be represented in the graphs, A bar chart graphs the frequency distribution of the data on an x-y coordinate | {"url":"https://bestbinlmgsv.netlify.app/mclaren44288jyw/frequency-distribution-charts-and-graphs-116.html","timestamp":"2024-11-05T09:15:26Z","content_type":"text/html","content_length":"34049","record_id":"<urn:uuid:97593b7b-1d80-4c6b-94f7-55e44ebea2f8>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00865.warc.gz"} |
Allocating fair shares of land - ΑΙhub
Consider a large piece of land that is to be split in a fair manner among several farmers, who all have an equal entitlement to a share of this land. They all have different plans for their allotted
pieces – growing a variety of crops, using the land as a pasture, or putting up a solar farm – so each of them has their own preferences over the land, depending on the type of soil, incline, access
to water, etc. There may also be constraints on the shape of each individual piece: e.g., it is probably a bad idea to partition the land into pieces that are 800m long and 2m wide, even if such a
partition is perfectly fair.
The problem of allocating the land in a fair manner under these constraints has been considered in prior work (Segal-Halevi et al., Fair and square: Cake-cutting in two dimensions, Journal of
Mathematical Economics 2017; Segal-Halevi et al., Envy-free division of land, Mathematics of Operations Research 2020), for two classic notions of fairness, namely, proportionality (if there are N
agents, each of them should value their piece at least as highly as V/N, where V is the value they assign to the entire piece of land) and envy-freeness (no agent considers another agent’s piece to
be more valuable than their own).
In our work, we consider a variant of this problem where, in addition to geometric constraints on the shapes of the individual pieces, we require the pieces to be separated: there is a separation
parameter s such that any two pieces belonging to two different agents have to be at distance at least s from each other. Such a constraint is motivated by practical considerations, e.g., providing
access or avoiding cross-pollination; if the “land” to be divided is, say, an exhibition hall or a market square, the separation requirement can be used to capture social distancing constraints. In
our earlier paper, which was published in AAAI’21, we considered this question in the context of dividing a one-dimensional resource, commonly referred to as “cake”; however, it turns out that we
need an entirely new set of techniques to handle the two-dimensional scenario.
Under the separation constraint, proportionality and envy-freeness become very challenging, so we focus on another fairness concept, known as maximin fair share. This notion of fairness is based on
the following idea, which is a generalisation of the classic cut-and-choose protocol. Each of the N agents executes the following mental experiment: she splits the land into N pieces that are s-
separated, and then lets the other N-1 agents pick a piece for themselves, so that she ends up with the last piece. Her goal is to maximise the value of the piece she gets in the worst-case scenario,
i.e., when she ends up with the piece that she finds the least valuable among the N pieces in her partition. The value that she can guarantee to herself in this fashion is called her maximin fair
share. Then, an allocation is considered fair if each agent receives a piece that she values at least as much as her maximin fair share.
Maximin fair share is generally viewed as a less demanding concept than proportionality or envy-freeness, but it turns out that, in the setting with separation, an allocation that guarantees each
agent her maximin fair share may fail to exist. Therefore, we further relax this solution concept by asking agents to divide the land into k > N pieces when running their mental experiment for
computing their share. Naturally, we expect the least valuable of the k pieces to be less valuable than the least valuable of the N pieces, so the larger k is, the easier it is to satisfy all agents.
(Of course, in the actual allocation we still divide the land into N pieces.)
We refer to the resulting solution concept as 1-out-of-k fair share.
In our work, we ask what is the smallest value of k such that we can guarantee to each agent her 1-out-of-k fair share, in the presence of separation constraints. Now, it turns out that the answer to
this question depends on the constraints on the shapes of individual pieces. In particular, if each agent is to receive a square-shaped piece of land, it suffices to set k = 4N – 5. However, if
agents’ pieces can be arbitrary axis-aligned rectangles (and the land itself is an axis-aligned rectangle), we get a much weaker upper bound of k = 2^N+2, and converting it into a finite algorithm
comes at an additional cost. The proof is constructive, in the sense that, given agents’ fair shares, we explicitly construct an allocation that satisfies all agents; however, the fair shares
themselves are difficult to compute, so we need to use an approximation algorithm.
We do not know if our bounds on k (as a function of N) are tight; improving them, or, alternatively, proving matching lower bounds, is a challenge for future work.
Edith Elkind, Erel Segal-Halevi and Warut Suksompong recently won an IJCAI 2021 distinguished paper award for the work covered in this post. The title of their winning paper is Keep your distance:
land division with separation.
Edith Elkind
is a Professor of Computer Science at University of Oxford.
Edith Elkind is a Professor of Computer Science at University of Oxford. | {"url":"https://aihub.org/2021/09/21/allocating-fair-shares-of-land/","timestamp":"2024-11-03T13:10:40Z","content_type":"text/html","content_length":"60278","record_id":"<urn:uuid:7a96b486-eca1-4827-ac99-f620e1d1baa1>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00269.warc.gz"} |
Relativistic Runaway Electron Avalanches Inside the High Field Regions of Thunderclouds
Document Type
Degree Name
Doctor of Philosophy (PhD)
Aerospace, Physics, and Space Sciences
First Advisor
Hamid K. Rassoul
Second Advisor
Joseph R. Dwyer
Fourth Advisor
Ming Zhang
In this dissertation, simplified equations describing the transport and energy spectrum of runaway electrons are derived from the basic kinematics of the continuity equations. These equations are
useful in modeling the energy distribution of energetic electrons in strong electric fields, such as those found inside thunderstorms. Dwyer and Babich [2011] investigated the generation of
low-energy electrons in relativistic runaway electron avalanches. The paper also developed simple analytical expressions to describe the detailed physics of Monte Carlo simulations of relativistic
runaway electrons in air. In this work, the energy spectra of the runaway electron population are studied in detail. Dependence of electron avalanche development on properties such as the avalanche
length, radiation length, and the effective Møller scattering efficiency factor, are discussed in detail. To describe the shapes of the electron energy spectra for a wide range of electric field
strengths, the random deviation of electron energy loss from the mean value is added to the solutions. We find that this effect helps maintain an exponential energy spectrum for electric fields that
approach the runaway electron threshold field. We also investigate the source mechanisms of Terrestrial Gamma-ray Flashes, which are a result of relativistic runaway electron avalanches in air. In
this study, the bremsstrahlung photons are propagated through the atmosphere, where they undergo Compton scattering, pair production, and photoelectric absorption. We model these interactions with a
Monte Carlo simulation from the TGF source location (assumed to vary between 8 and 20 km) and the edge of the atmosphere (≈ 100 km). We then propagate these photons to a satellite plane at 568 km in
order to compare with measurements. In collaboration with the GBM instrument team in Huntsville, AL, we were able to model spectral and temporal properties of observed TGFs. Although the analysis of
individual TGF photon spectra was qualitative, we were able to put some constraints, i.e. source altitude and beaming angle, on a sample of observed GBM TGFs. However, assuming a height of 15 km, we
were able to model the softening in the spectrum observed as the satellite moves off-axis from the TGF source location [Fitzpatrick et al., 2014]. The conclusion of this analysis shows that Compton
scattering alone can not explain the temporal dispersion observed. This suggests that an intrinsic time variation exists at the source of the TGF.
Recommended Citation
Cramer, Eric Scott, "Relativistic Runaway Electron Avalanches Inside the High Field Regions of Thunderclouds" (2015). Theses and Dissertations. 434. | {"url":"https://repository.fit.edu/etd/434/","timestamp":"2024-11-01T19:04:01Z","content_type":"text/html","content_length":"40326","record_id":"<urn:uuid:c37ad46f-8241-41fd-b5ba-0c83d3e50f5c>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00066.warc.gz"} |
Needed Math for the 21st Century STEM Workplace
There are three kinds of mathematics: the math that’s taught, the math that’s learned, and the math that’s needed in the 21st century STEM workplace. With support from the Advanced Technological
Education Program at the National Science Foundation, Michael Hacker, Co-Director of the Center for STEM Research at Hofstra University, and I organized a conference to study why those three “maths”
are not the same.
Held in Baltimore from January 12th through the 15th, the conference attracted 46 attendees drawn from three groups: math educators, STEM content instructors, and STEM employers. Three fields of STEM
employment were represented: Information and Communication Technology, Biotechnology, and Advanced Manufacturing.
There is ample evidence (see, for example, “Still Searching: Job Vacancies and STEM Skills”) that companies in these and other STEM-related fields are finding it difficult to find qualified employees
for entry-level jobs. This is due in part to the poor math skills of prospective candidates, and – perhaps even more telling – their lack of confidence in their ability to “do” math. In this context,
the objective of the meeting was to solicit from employers examples of problems that prospective recruits often could not solve. The meeting would then collectively examine those problems, identify
the underlying relevant mathematical concepts and skills, and explore possible explanations for why high school and even two- and four-year college graduates find the problems so challenging.
A complete reporting of the findings of the conference must await our analysis of the data we collected over the course of two days of intense discussion. However, it is already evident that
real-world challenges, such as those described by the employers, differ from the math problems that most students encounter in formal school settings.
An example – one of many – may illuminate these differences.
An employee in a communications technology firm is tasked with providing a commercial space, consisting of several offices as well as other rooms, with wireless Internet access. The tools available
consist mainly of access points and routers, the former connecting multiple devices using radio frequency communication, the latter directing information between those devices and an Internet service
provider (ISP). Access points have a limited range and their locations must be selected so that those ranges overlap, providing connectivity to every device on the network as well as to one or more
routers. Routers, in turn, require connectivity to the ISP.
At first glance, the problem seems simple enough: just place the access points close enough to one another so that their ranges overlap. But real-world complications soon arise.
To save money, the number of access points should be minimized. Further, they require power so installing them in some locations may result in wiring expenses. The range of each access point may be
affected by the materials used for interior walls or by metallic structures such as elevators or vaults. Privacy and security concerns dictate that access to the network be restricted, as much as
possible, to the premises of the customer. Some locations within those premises – e.g., conference rooms – may require greater bandwidth than others.
These and other real-world considerations are not, strictly speaking, mathematical in nature, but insofar as they constrain the set of acceptable solutions, they require mathematical skills – e.g.,
modeling – that may be foreign to many would-be network technicians. Moreover, although the calculations required consist primarily of arithmetic operations on numbers (signed integers, decimals, and
fractions), the semantics behind these calculations – unit conversions, use of the Pythagorean Theorem to compute point-to-point distances, algorithms for computing overall costs – are not explicitly
called out in the statement of the problem.
Thus, even though the problem appears to require no more than middle school math and Algebra 1, it differs from the problems commonly encountered in traditional classes in those subjects.
• The statement of the problem does not contain all the information required to solve it and may in fact contain irrelevant information.
• The mathematical concepts and skills required are not spelled out (in contrast to the problems found at the end of the chapter in a math textbook, all of which involve the specific concept
covered in that chapter).
• The problem is multi-step and involves multiple variables.
• The problem may have many solutions of varying utility, rather than a single “right” one.
A major finding of the conference was that the kind of mathematics encountered in each of the three domains represented (ICT, biotech, and manufacturing) involved contextualized problems similar to
the one described above. Thus, an important barrier to success in these fields may arise from the features of such problems that we have identified.
Are there ways in which educational technologies such as those pioneered, deployed, and investigated by the Concord Consortium could help students to acquire the relevant, contextualized
problem-solving skills? A major outcome of the conference may turn out to be a number of proposals aimed at answering that question. | {"url":"https://concord.org/blog/needed-math-for-the-21st-century-stem-workplace/","timestamp":"2024-11-04T04:43:13Z","content_type":"text/html","content_length":"63385","record_id":"<urn:uuid:3cfb0649-4bd7-489d-9ca7-1696b6b8d011>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00816.warc.gz"} |
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Similar search terms for Area:
• What is the difference between base area, top area, and lateral surface area?
The base area is the area of the bottom surface of a three-dimensional shape, such as a cylinder or a prism. The top area is the area of the top surface of the shape. The lateral surface area is
the combined area of all the side surfaces of the shape, excluding the top and bottom surfaces. In summary, the base area and top area refer to the individual areas of the bottom and top
surfaces, while the lateral surface area refers to the combined area of all the side surfaces.
• Is there a difference between area and surface area?
Yes, there is a difference between area and surface area. Area refers to the measurement of the space inside a two-dimensional shape, such as a square or a circle. Surface area, on the other
hand, refers to the measurement of the total area of all the surfaces of a three-dimensional object, such as a cube or a sphere. In other words, while area is a two-dimensional measurement,
surface area is a three-dimensional measurement that includes the total area of all the sides of an object.
• What is the difference between area and area balance?
Area refers to the physical space or region occupied by an object or a specific area of land. It is a measure of the size of a surface. On the other hand, area balance refers to the equitable
distribution or allocation of resources, benefits, or opportunities within a specific area. It involves ensuring that all parts of the area are given fair and equal consideration in terms of
development, investment, and access to resources. In summary, area is a measure of physical space, while area balance is about ensuring fairness and equity in the distribution of resources and
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• Can you calculate the area of the colored area?
No, I cannot calculate the area of the colored area without specific measurements or dimensions provided. In order to calculate the area, I would need to know the length and width of the colored
area or have a clear outline of its shape. Once the necessary information is provided, I can help calculate the area using the appropriate formula for that shape.
• What is the difference between surface area and area?
Surface area refers to the total area that covers the surface of a three-dimensional object, including all its faces and curved surfaces. On the other hand, area refers to the measure of the
space enclosed within the boundaries of a two-dimensional shape, such as a square or circle. In simple terms, surface area deals with the total area of all surfaces of a 3D object, while area
deals with the space inside a 2D shape.
• How do I calculate the area of this area?
To calculate the area of a given area, you would need to determine the shape of the area. If it is a square or rectangle, you would multiply the length by the width. If it is a triangle, you
would multiply the base by the height and then divide by 2. If it is a circle, you would use the formula πr^2, where r is the radius. If the area is irregularly shaped, you may need to break it
down into smaller, more manageable shapes and calculate their individual areas before adding them together.
• What area code?
The area code is a three-digit code used to identify a specific geographic region within a country. It is typically dialed before the phone number when making a call to a different area. The
specific area code you need will depend on the location you are trying to reach.
• How do you calculate the area of a colored area?
To calculate the area of a colored area, you first need to identify the shape of the colored area, such as a square, rectangle, triangle, or circle. Once you know the shape, you can use the
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for a circle, you use the formula π x radius^2.
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To calculate the area of the marked area, you would need to first identify the shape of the marked area. If it is a rectangle or square, you would multiply the length by the width. If it is a
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To convert from ceiling area to wall area, you need to consider the dimensions of the room. First, calculate the perimeter of the room by adding the lengths of all four walls. Then, multiply the
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No, the surface area is not the same as the surface area. It seems like there might be a typo in the question. If you meant to ask if the surface area is the same as the volume, then the answer
is no. Surface area refers to the total area of the outer surface of an object, while volume refers to the amount of space that an object occupies. They are two different measurements that
describe different aspects of an object's size and shape.
• What is the surface area and the lateral surface area?
Surface area is the total area of all the surfaces of a three-dimensional object, including the area of its faces, bases, and any curved surfaces. Lateral surface area, on the other hand, refers
to the total area of the side surfaces of a three-dimensional object, excluding the area of its bases. In other words, the lateral surface area is the surface area of the sides of the object,
while the surface area includes the lateral surface area as well as the area of the bases.
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Unveiling Angle Relationships: Essential for Mastering Geometry
Unveiling Angle Relationships: Essential For Mastering Geometry
The relationship between angles a and b depends on their position relative to each other. If angles a and b are vertical angles, they are congruent, meaning A = B. If they are adjacent angles, they
form a straight line, making A + B = 180 degrees. If they are supplementary angles, they are adjacent and their sum is 180 degrees, so A + B = 180 degrees. Finally, if they are complementary angles,
their sum is 90 degrees, so A + B = 90 degrees. Understanding these angle relationships is crucial for solving geometry problems, as it allows one to determine unknown angle measures based on known
• Definition of angles and their classification
• Importance of understanding angle relationships
Unlocking the Secrets of Angles: A Comprehensive Guide for Beginners
In the realm of geometry, angles play a pivotal role in shaping our understanding of the world around us. From the majestic pyramids to the intricate designs of snowflakes, angles are the building
blocks of countless structures and phenomena. Embark on a journey with us as we delve into the captivating world of angles, exploring their definitions, classifications, and the crucial relationships
that connect them.
What Are Angles?
Imagine a meeting point between two intersecting lines. That meeting point, where the lines diverge, defines an angle. Angles are measured in degrees, and they can be classified into various types
based on their measure. The most common types of angles include:
• Acute angles: Angles less than 90 degrees
• Right angles: Angles exactly equal to 90 degrees
• Obtuse angles: Angles greater than 90 degrees but less than 180 degrees
• Straight angles: Angles exactly equal to 180 degrees
Why Angle Relationships Matter
Understanding the relationships between angles is essential for problem-solving in geometry. These relationships allow us to determine the measures of unknown angles, solve complex puzzles, and
navigate the intricate world of shapes and patterns.
Vertical Angles:
• Definition and properties of vertical angles
• Relationship between vertical angles (A = B)
Unveiling the Secrets of Vertical Angles: A Geometrical Journey
In the realm of geometry, angles hold a significant place. Among them, vertical angles stand out as a fascinating concept with intriguing properties. So, let’s dive into the world of vertical angles
and discover their intriguing nature.
Definition of Vertical Angles
Vertical angles are formed when two intersecting lines create four angles. Specifically, when two lines intersect, the angles that are opposite each other are called vertical angles. They are denoted
by the same letter with a small sub-number (e.g., ∠A1 and ∠A2).
Properties of Vertical Angles
The defining characteristic of vertical angles is their unique and equal measure. **No matter the orientation or size of the intersecting lines, the vertical angles always have the same measure_.
This fundamental property makes vertical angles invaluable in various geometrical applications.
Relationship Between Vertical Angles (A = B)
The most significant property of vertical angles is their equal measure. This relationship is expressed by the equation: ∠A1 = ∠A2. This means that the measure of one vertical angle is identical to
the measure of the other vertical angle.
This property holds true regardless of the shape or orientation of the intersecting lines. It’s a fundamental theorem in geometry that simplifies complex calculations and aids in solving geometrical
Vertical angles emerge as a fundamental concept in geometry, characterized by their unique and equal measure. Their relationship (A = B) plays a pivotal role in problem-solving and serves as a
cornerstone for further geometrical explorations. By understanding the properties of vertical angles, we unlock the door to a deeper comprehension of the fascinating world of geometry.
Adjacent Angles: An Intriguing Puzzle in the World of Geometry
In the captivating realm of geometry, angles play a pivotal role, forming the building blocks of various shapes and structures. Among these angles, adjacent angles stand out as a fascinating concept
that has captivated mathematicians and geometry enthusiasts alike.
Understanding Adjacent Angles
Adjacent angles are a pair of angles that share a common vertex and a common side. Imagine two intersecting lines forming four angles. The pair of angles that are next to each other, sharing the same
vertex and one side, are known as adjacent angles.
Properties of Adjacent Angles
One of the most intriguing properties of adjacent angles is that they are supplementary, meaning they add up to 180 degrees. This extraordinary property can be visualized by imagining adjacent angles
as a straight line. When you combine two adjacent angles, you end up with a complete straight line, which measures exactly 180 degrees.
Another important property of adjacent angles is that they are coplanar, meaning they lie in the same plane. This property is essential for understanding the spatial relationships between adjacent
Importance of Adjacent Angles
Understanding the properties of adjacent angles is crucial in geometry. It allows mathematicians to solve various angle-related problems with ease. For instance, if you know the measure of one
adjacent angle, you can quickly find the measure of the other adjacent angle by subtracting it from 180 degrees.
Adjacent angles also play a vital role in proving theorems and solving constructions. They are instrumental in understanding the relationship between parallel lines and perpendicular lines, as well
as in deducing angle measures in polygons and other geometric shapes.
Adjacent angles are an intriguing concept that offers a glimpse into the captivating world of geometry. Their properties and relationships are fundamental to solving angle-related problems and
comprehending the spatial relationships between lines and angles. By unraveling the mysteries of adjacent angles, we unlock a door to a world of geometric wonders.
Understanding Supplementary Angles: A Geometric Perspective
In the realm of geometry, angles are the building blocks of polygons and other geometric shapes. Comprehending their properties and relationships is crucial for solving complex geometry problems. One
type of angle that plays a significant role is the supplementary angle.
Defining Supplementary Angles
Supplementary angles are a pair of angles that, when added together, form a straight angle, which measures 180 degrees. They are often represented using the symbol “∠”. For example, if ∠A and ∠B are
supplementary angles, we can write:
∠A + ∠B = 180°
Identifying Supplementary Angles
Supplementary angles can be found in various geometric configurations. Common examples include:
• Two angles that share a common vertex and a side, known as adjacent angles.
• Two non-adjacent angles that lie on the same side of a transversal line and are on opposite sides of an intersecting line.
• Two angles that are formed by intersecting chords in a circle.
Properties of Supplementary Angles
The key property of supplementary angles is their sum:
• The sum of two supplementary angles is always 180 degrees.
This property makes it easy to calculate the measure of one supplementary angle if the measure of the other is known. For instance, if ∠A = 70°, then its supplementary angle, ∠B, must measure 110°:
∠A = 70°
∠B = 180° - ∠A
∠B = 180° - 70°
∠B = 110°
Importance of Supplementary Angles
Understanding supplementary angles is essential for solving geometry problems involving:
• Parallel lines: When two lines are parallel, the angles formed by a transversal are supplementary. This property is used to prove that lines are parallel.
• Polygon angles: The sum of the interior angles of a polygon with n sides is (n – 2) * 180°. This formula relies on the property that adjacent angles around a vertex are supplementary.
• Trigonometry: In trigonometry, supplementary angles are used to find missing angles and solve problems involving triangles.
Mastering the concept of supplementary angles is a cornerstone of geometry. By understanding their definition, properties, and importance, you can confidently tackle geometric challenges and unlock
the mysteries of this fascinating subject.
Complementary Angles: Understanding the Perfect Pair
In the realm of geometry, angles hold a significant place. Among the various types, complementary angles stand out as a special duo with a unique relationship.
What’s a Complementary Angle?
Imagine holding two straight lines side by side. When you form an angle on one line and then an adjacent angle on the other, opening outward, you create a pair of complementary angles. These angles
are like two pieces of a puzzle, fitting together perfectly to form a 90-degree right angle.
The Magic of 90 Degrees
The defining characteristic of complementary angles is their sum. When you add together the measures of two complementary angles, you always get 90 degrees. This means that the two angles share an
equal amount of the space around the intersection point of their lines.
Practical Applications
Complementary angles play a crucial role in geometry. They help us determine the dimensions of shapes, create architectural designs, and even measure angles in the real world. For example, if you
know that one angle of a triangle is 60 degrees, you can conclude that the other two angles must be complementary, adding up to 90 degrees.
Complementary angles are like two peas in a pod, inseparable and working together to form a perfect whole. Understanding their properties and their special 90-degree sum is essential for anyone
wanting to delve into the fascinating world of geometry. So, next time you encounter complementary angles, embrace their special bond and appreciate the harmonious balance they bring to your
geometric endeavors.
The Interplay between Angles A and B: A Gateway to Geometric Problem-Solving
In the realm of geometry, angles play a pivotal role, influencing the shape and properties of figures. Understanding the relationships between angles, particularly between angles A and B, is
fundamental to navigating the intricacies of geometric problems.
Vertical Angles: Mirrors in the Geometric World
Imagine two intersecting lines, creating four angles. The angles that are opposite each other are known as vertical angles. They share a common vertex and are congruent, meaning they have the same
measure. This unwavering relationship is captured by the equation: . This simple yet profound fact provides a crucial tool for solving angle problems.
Adjacent Angles: Partners in Crime
Adjacent angles are those that share a common vertex and a common side. They are like two neighboring apartments, side by side. The sum of adjacent angles is always a constant, . This rule, known as
the , is a cornerstone of geometry. By knowing one angle, you can effortlessly determine its adjacent partner.
Supplementary Angles: A Harmonious Duet
When two angles add up to , they are considered . Like a musical harmony, they complement each other to create a perfect balance. Just as two notes can form a pleasing chord, two supplementary angles
create a geometric harmony.
Complementary Angles: A Perfect Pair
Complementary angles, on the other hand, are a more intimate pair, adding up to . They form a , a cornerstone of geometric constructions and测量. Complementary angles create a sense of completeness
and precision, like two puzzle pieces that fit together perfectly.
The Significance of Angle Relationships
These angle relationships are not mere abstract concepts; they are the building blocks of geometric problem-solving. By understanding how angles interact, we can tackle complex geometric problems
with confidence. From determining the missing angles in a polygon to solving intricate geometric puzzles, angle relationships are the key to unlocking the secrets of geometry.
In the tapestry of geometric knowledge, the interplay between angles A and B stands as a vital thread. By unraveling their intricate connections, we gain a deeper appreciation for the beauty and
logic that governs the geometric world. | {"url":"https://rectangles.cc/angle-relationships-mastering-geometry/","timestamp":"2024-11-11T21:13:11Z","content_type":"text/html","content_length":"151502","record_id":"<urn:uuid:482bb9f5-0984-4142-b687-69ed4574e628>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00803.warc.gz"} |
average functions
The AVERAGE function in Microsoft Excel is a commonly used statistical function that calculates the average value of a given set of numbers. It is a useful tool for quickly finding the mean of a set
of data, which is often useful in statistical analysis or for comparing data sets.
To use the AVERAGE function in Excel, you first need to select the cells that contain the numbers you want to average. You can then click on the “Formulas” tab in the ribbon and choose the “AVERAGE”
function from the “Statistical” group. This will open a dialog box where you can enter the range of cells you want to average.
Alternatively, you can type the AVERAGE function directly into the formula bar, using the syntax “=AVERAGE(range)”. For example, if you want to find the average of the numbers in cells A1 through
A10, you would type “=AVERAGE(A1:A10)”.
In addition to these options, the AVERAGE function also has several arguments you can use to customize the way it calculates the average. These include the “ignore_nums” argument, which allows you to
exclude specific numbers from the average, and the “average_range” argument, which allows you to specify a range of cells to be averaged.
One useful application of the AVERAGE function is in financial analysis, where it can be used to calculate the average return on investment (ROI) of a portfolio. To do this, you would need to enter
the data for each investment in a separate cell, and then use the AVERAGE function to calculate the average ROI.
Another application of the AVERAGE function is in quality control, where it can be used to calculate the average defect rate of a product. To do this, you would need to enter the data for each
product in a separate cell, and then use the AVERAGE function to calculate the average defect rate.
There are many other uses for the AVERAGE function in Excel, and it is a powerful tool for quickly calculating statistical averages. Whether you are analyzing data for financial or quality control
purposes, or simply want to find the mean of a set of numbers, the AVERAGE function is a valuable tool to have in your toolkit. | {"url":"https://excelguru.pk/average-function/","timestamp":"2024-11-06T08:13:34Z","content_type":"text/html","content_length":"63084","record_id":"<urn:uuid:f62bc29b-fe0f-4915-907c-4779f8e6f760>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00841.warc.gz"} |
How are the averages calculated in the reviews? | Knowledge Base
Where do the averages appear?
The Performance Review tool consists of computable and non-computable questions. Computable questions allow, for example, decision-making support by ranking employees based on scores in the team
results screen and distributing them in the quadrants of the 9Box.
Computable questions are grouped by topics, which correspond to each page in a performance review form.
Throughout the review tool, we can find average calculations in two contexts: individual averages and collective averages.
Context 1: Individual Averages (1 reviewer to 1 reviewee)
Since each question (from now on, let's refer to computable questions as simply "questions") is within a topic, when Dora (reviewer) completes a review for Santiago (reviewee), we calculate the
average of a topic by averaging the scores of its questions, weighted by their respective weights.
Similarly, the system calculates the average of the topics, weighted by their weights, to obtain the overall average of the review conducted by Dora for Santiago.
Consider a review with the following weight distribution:
This calculation is performed as follows:
These averages, both for the topics and overall, can appear in the following situations:
• Summary when answering the review
1. Summary when answering the review
When completing a review form, on the final screen, a summary of the responses is displayed. In this summary, the averages are presented according to the previously chosen configuration, which can be
either numerical values or a concept scale.
2. Participant details
When the manager or administrator clicks on "View Details" on the Tracking screen or on the name of the evaluated individual in Team Results, the participant details are displayed. In these details,
the averages are shown based on the chosen configuration, which can be either numerical values or a concept scale.
Context 2: Collective averages (n reviewers for 1 reviewed individual)
Here we deal with average scores given by multiple reviewers to a specific reviewed individual.
To understand this topic, it is important that you first understand how relationships work in the Qulture.Rocks system. Click here to read the article.
These averages appear on the following screens:
Let's take the following hierarchy as an example, where each colored rectangle represents a person:
Let's consider the following review structure, with the table on the right indicating the weights per relationship, configured by an admin during the review setup.
Now, let's understand how the two types of collective average calculations work.
When creating a review, the admin chooses the type of average calculation per relationship: not grouped by relationship or grouped by relationship.
1. Not Grouped by Relationship Averages
This option causes the system to calculate averages simply by computing each reviewer with their respective weight.
Thus, in our example, we have the following average calculation:
2. Grouped by relationship average
In the case of averaging grouped by relationship, the system first calculates the simple average within a group (for example, the simple average of all peers) and then calculates the average of each
group of relationships with its respective weight.
Please note that the system treats all relationships as a group, even if there is only one participant (e.g., self-evaluation).
For this first step, we have the averages by groups:
With these averages and the weight table, we can calculate the grouped averages.
🛑⚠️ It is important to note that when the N/A option is active and marked, the questions with that rating are not included in the calculation.
That's it. This is one of the most sophisticated parts of the system, so it's normal to have questions. If something is not clear, #ChatWithUs 🚀 😄
What did you think of the article? Leave your feedback below 👇 | {"url":"https://help.qulture.rocks/en/articles/1567222-how-are-the-averages-calculated-in-the-reviews","timestamp":"2024-11-03T22:16:37Z","content_type":"text/html","content_length":"72971","record_id":"<urn:uuid:dd198ad7-281b-47ba-9dda-a05e3975acf1>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00481.warc.gz"} |
C2 logs question
I havent done logs in a while so this was quite refreshing. Okay
Log3z = 4logz3 ----- This tells me change base i prefer to change to three
log3z = 4log3 3/log 3z ---- log 3 3 = 1 changing base is log a b = logca/logcb
log 3 z = 4 x 1/log3z
Now multiply both sides by log3z
(Log3z)^2 = 4
Square root both sides
Log3z = 2 or -2 - square root of 4 is 2 and -2
Now log ab = x when a^x = b so
z = 3^2 = 9
z = 3^-2 = 1/9
I know these are right because they are the answers in the back of the heinemanns C2 book.
Really got me into doing some earlier C2 if you need anymore help on 3G mixed exercise just tell me ive done all the questions. [email protected]
Thanks for that Aaron, after this test I started working through 3G as a bit of revision and I didn't realise that it was the same question, but I was confused about getting the 1/9.
Now that's cleared up, I can finish this exercise. Thanks again. | {"url":"https://www.thestudentroom.co.uk/showthread.php?t=199934","timestamp":"2024-11-07T22:50:14Z","content_type":"text/html","content_length":"319144","record_id":"<urn:uuid:29612197-4f53-4c5d-becd-8fa3f30715e0>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00519.warc.gz"} |
Fading Signatures of Critical Brain Dynamics during Sustained Wakefulness in Humans
Sleep encompasses approximately a third of our lifetime, yet its purpose and biological function are not well understood. Without sleep optimal brain functioning such as responsiveness to stimuli,
information processing, or learning may be impaired. Such observations suggest that sleep plays a crucial role in organizing or reorganizing neuronal networks of the brain toward states where
information processing is optimized.
Increasing evidence suggests that cortical neuronal networks operate near a critical state characterized by balanced activity patterns, which supports optimal information processing. However, it
remains unknown whether critical dynamics is affected in the course of wake and sleep, which would also impact information processing. Here, we show that signatures of criticality are progressively
disturbed during wake and restored by sleep. We demonstrate that the precise power-laws governing the cascading activity of neuronal avalanches and the distribution of phase-lock intervals in human
electroencephalographic recordings are increasingly disarranged during sustained wakefulness. These changes are accompanied by a decrease in variability of synchronization. Interpreted in the context
of a critical branching process, these seemingly different findings indicate a decline of balanced activity and progressive distance from criticality toward states characterized by an imbalance
toward excitation where larger events prevail dynamics. Conversely, sleep restores the critical state resulting in recovered power-law characteristics in activity and variability of synchronization.
These findings support the intriguing hypothesis that sleep may be important to reorganize cortical network dynamics to a critical state thereby assuring optimal computational capabilities for the
following time awake.
Sleep is crucial for daytime functioning and well being. Although a vital part of life, its purpose and biological function are not yet well understood (Mignot, 2008). The importance of sleep is
illustrated by the deteriorating effects of chronic sleep restriction or total sleep deprivation on human performance (Banks and Dinges, 2007). Without sleep optimal brain functioning such as
responsiveness to stimuli, information processing, or learning may be impaired. Such observations suggest that sleep may play an important role in organizing or reorganizing neuronal networks in the
brain toward states where information processing is optimized.
The general idea that both the computational capabilities of a system and its complexity are maximized at or nearby critical states related to phase transitions or bifurcations (Langton, 1990) led to
the hypothesis that neuronal networks in the brain operate at or close to a critical state. The observation of neuronal activity patterns consistently following power-law distributions, a hallmark of
systems at a continuous phase transition, further raised the interest in the hypothesis of critical brain dynamics (Linkenkaer-Hansen et al., 2001; Worrell et al., 2002; Beggs and Plenz, 2003;
Fraiman et al., 2009; Benayoun et al., 2010; Chialvo, 2010; Poil et al., 2012). Spatiotemporal cascades of activity termed neuronal avalanches obeying a power-law size distribution were observed in
vitro (Beggs and Plenz, 2003, 2004), in vivo (Gireesh and Plenz, 2008; Petermann et al., 2009; Ribeiro et al., 2010), and human magnetoencephalogram (MEG; Palva et al., 2013; Shriki et al., 2013).
Recently, the spatiotemporal patterns of coherence potentials, i.e., large-amplitude negative deflections with high similarity, were reported to express neuronal avalanches in local field potentials
(LFP; Thiagarajan et al., 2010; Plenz, 2012). Neuronal avalanches have been regarded an indication of balanced dynamics, i.e., avoiding regimes of overexcitation or underexcitation. The balance in
activity is captured by the branching parameter σ = 1 indicating that one event on average leads to one future event resulting in the corresponding cascade size distribution to follow a power-law
with exponent −3/2 (Zapperi et al., 1995; de Carvalho and Prado, 2000; Haldeman and Beggs, 2005).
The idea of a balanced regime of activity in cortical networks also extends to properties of synchrony between neuronal groups. Recent insights from in vitro experiments and computational modeling
indicated that a balanced critical state with neuronal avalanches is also characterized by moderate mean and maximal variability of neuronal synchrony (Yang et al., 2012). As an alternative
synchronization metric, the durations of transient synchronization events between channel pairs in electroencephalography (EEG) were first reported to follow a power-law density distribution in Gong
et al. (2003). Later on, the distribution of phase-lock intervals (PLIs) was observed to follow a power-law function in functional magnetic resonance imaging and electrocorticographic recordings, too
(Kitzbichler et al., 2009; Meisel et al., 2012). Although not limited to it (Botcharova et al., 2012), power-law scaling of PLIs also arises in computational models at criticality (Kitzbichler et
al., 2009), which led to the hypothesis that its observation in neurophysiological data is indicative of a critical state of brain dynamics.
While there is growing evidence for the existence of critical states in cortical networks it is still an unresolved question how this relates to modifications of these networks in the course of wake
and sleep (Pearlmutter and Houghton, 2009; Ribeiro et al., 2010; Priesemann et al., 2013). In the present work, we investigated the hypothesis that wakefulness moves cortical networks away from a
critical state, which is restored by sleep. We focused on the detection of neuronal avalanches and measures of synchrony (mean, variability, the distribution of PLIs) as means to detect critical
dynamics in the EEG during a period of sustained wakefulness (sleep deprivation).
Materials and Methods
EEG recordings during prolonged wakefulness.
We analyzed wake EEG recordings of eight healthy young right-handed males (23.0 ± 0.46 years; mean ± SEM) during 40 h of sustained wakefulness (data from a previous study; Finelli et al., 2000).
During this time, participants were under constant surveillance. The waking EEG was recorded in 14 sessions at 3 h intervals starting at 07:00. Another waking EEG was recorded after a recovery night
following the sleep deprivation period amounting to a total of 15 EEG sessions. Sessions consisted of a first 5 min eyes-open period, followed by a 4–5 min eyes-closed period, and a second 5 min
eyes-open period. Twenty-seven EEG derivations (extended 10–20 system; reference electrode 5% rostral to Cz) were sampled with 256 Hz (high-pass filter at 0.16 Hz; anti-aliasing low-pass filter at 70
Hz). Artifacts including eye blinks were marked by visual inspection.
Rating of alertness.
Alertness was self-rated on a pseudo-analog scale, which consisted of a 20 point subjective rating scale using a palmtop computer. Alertness, date, and time of rating were recorded from each subject
at each EEG session (Finelli et al., 2000).
Detection of cascades of coherence potentials.
We used artifact-free EEG segments during the eyes-open condition. The length of segments was chosen to be 5000 samples long (corresponding to 19.53 s) to strike a compromise between, on one hand,
including as many artifact-free segments as possible in the analysis and, on the other hand, having these segments long enough to provide a sufficiently large number of events. An average of 18
segments corresponding to a total of ∼6 min of EEG data was analyzed in each subject and EEG session.
Large-amplitude activity patterns in the LFP, termed coherence potentials, have previously been shown to occur without distortion at other cortex sites via fast synaptic transmission extending up to
hundreds of milliseconds (Thiagarajan et al., 2010). In this work, we used coherence potentials to detect cascades of neuronal activity in space and time in EEG. Coherence potentials were defined as
EEG segments across time and different EEG channels characterized by crossing a positive or negative threshold at some point and exhibiting high mutual similarity. It should be noted, that although
we followed a similar approach as described by Thiagarajan et al. (2010) in identifying coherence potentials, the method applied here to the EEG is different to previous studies where coherence
potentials were derived from LFP. LFP allows for detection of neuronal activity with high spatial resolution. In the EEG, the determination of the spatial location of the signal is not possible to
the same extent. In this respect, the detection of cascading events in neuronal activity in our work differs from the original works on neuronal avalanches derived from LFP (Beggs and Plenz, 2003;
Petermann et al., 2009; Thiagarajan et al., 2010). Furthermore, coherence potentials were originally only defined for negative potentials due to their close correlation to neuronal spiking. This
justification, however, does not necessarily hold for the EEG. In view of these differences, we considered both positive and negative potentials to identify coherence potentials.
First, large positive or negative excursions beyond a threshold were identified for each EEG channel after normalization of the signal by subtraction of the mean and dividing by the SD. The
comparison of the signal distribution to the best Gaussian fit indicated that the two distributions start to deviate from one another around ±4 SD (see Fig. 1A). We therefore chose to use a threshold
of ±4 SD throughout the paper. We systematically explored other thresholds around ±4 SD to verify that results were robust for a parameter range around the chosen threshold (see Results). Once a
positive or negative excursion beyond the threshold had been detected in one of the channels, we then defined the period of interest on this channel as the continuous positive or negative excursion
until the signal returned the baseline (0 SD) on both sides of the threshold crossing similar to Thiagarajan et al. (2010) and termed those segments F[i] trigger events (Fig. 1B). The time of the
signal's baseline crossing on either side of the excursion therefore determined the length of the trigger event.
Second, coherent segments between a trigger event F[i] and periods F[j] of equal duration in all channels were identified when correlations R as follows: were R ≥ 0.75. A correlation threshold in
this range was identified by Thiagarajan et al. (2010) to robustly detect coherence potentials. We systematically changed the correlation threshold to verify that results were robust for a broad
range around this value (see Results). Together with the trigger event, these coherent segments form the set of coherence potentials.
Next, coherence potentials were discretized in a raster plot to identify cascades in space and time. The time point of a single event was defined as the time point of the occurrence of the minimum
(or maximum, depending on the trigger event) for each coherence potential (Fig. 1C). For each trigger event, the time series of events from each channel was discretized with time bins of duration Δt.
Along the line of previous work (Beggs and Plenz, 2003; Gireesh and Plenz, 2008; Ribeiro et al., 2010; Friedman et al., 2012; Priesemann et al., 2013; Shriki et al., 2013), a continuous sequence of
time bins with events in any channel ending with a time bin with no events in any channel was defined as a cascade of events. The cascade size S was defined as the number of events in all channels in
a cascade. To prevent double counting of events and cascades we required trigger events not to have been an event in a previous cascade.
For control, we generated phase-shuffled surrogate data for each channel, which shuffles the phase of the original signal but keeps the amplitude spectra unchanged in the frequency domain (Theiler et
al., 1992).
As an alternative approach to coherence potentials, we also derived neuronal avalanches by identifying large positive or negative deflections above a threshold at single EEG channels and combined
them into spatiotemporal cascades by discretizing them in time by their most extreme excursion as described previously (Palva et al., 2013; Shriki et al., 2013). This method was recently shown to
allow identification of neuronal avalanches in EEG and MEG.
Branching parameter σ.
The branching parameter σ quantifies the ratio of future events to ancestor events. Previously, it has been used to characterize cascading events on different scales, from activity in neuronal
cultures to events in the MEG. We calculated σ as the ratio of the number of events in the second time bin of a cascade to the one in the first time bin. The ratio was averaged over all cascades for
each subject and each EEG session.
Synchrony and entropy measurements.
We followed the approach described previously (Yang et al., 2012) to quantify mean and variability of synchronization. The calculation was performed on the same artifact-free segments used for the
derivation of coherence potentials. After filtering the data in the alpha band (8–12 Hz), we first obtained a phase trace θ[i](t) from each EEG trace F[i](t) using its Hilbert transform H[F[i](t)]:
Next, we quantified the mean synchrony in each EEG segment by where L = 5000 is the length of our EEG segments in samples and r(t) is the Kuramoto order parameter: which was used as a time-dependent
measure of phase synchrony with n = 27 being the number of EEG channels in our data.
As a measure for the variability of synchronization we derived the entropy of r(t) in each EEG segment by where we estimated a probability distribution of r(t) by binning values into intervals. p[i]
is then the probability that r(t) falls into a range b[i] < r(t) ≤ b[i][+1]. Similar to Yang et al. (2012), we found results to be robust over a broad range for the number of bins B used. We applied
B = 24 in the current analysis.
Derivation of the distribution of PLIs.
PLIs were calculated for all possible pairs of derivations of artifact-free EEG segments of 19.53 s duration (5000 samples, same segments as for the analysis of coherence potentials and
synchronization measures). The analysis was performed for scale 4 (8–16 Hz; alpha band; see below) and scale 5 (4–8 Hz; theta band).
Hilbert wavelet transforms.
To derive a scale-dependent estimate of the phase difference between two EEG channels, we follow the approach described previously (Kitzbichler et al., 2009) using Hilbert transform derived pairs of
wavelet coefficients (Whitcher et al., 2005). We define the instantaneous complex phase vector for two signals F[i](t) and F[j](t) as follows: where W[k] denotes the kth scale of a Hilbert wavelet
transform and † its complex conjugate. Here F[i](t) and F[j](t) are different EEG derivations. A local mean phase difference in the frequency interval defined by the kth wavelet scale is then given
by with being a less noisy estimate of C[i,j](t) where 〈 · 〉 indicates the temporal average over a time window Δt = 2^k8 in sampling steps (Kitzbichler et al., 2009).
Intervals of phase locking can then be identified as periods when |Δφ[i,j](t)| is smaller than some arbitrary threshold, which we set to π/4. We also require the modulus squared of the complex time
average, σi,j2=|Ci,j¯|2, to be larger than 0.5, limiting the analysis to phase difference estimates above this level of significance.
Estimation of the goodness of power-law fit.
To determine the quality of the fit of a power-law function to the observed distribution we performed a goodness-of-fit test, which leads to a p value that quantifies the plausibility of the
hypothesis that the distribution is power-law like (Clauset et al., 2009). Upon fitting the empirical distribution, power-law distributed synthetic data were generated with parameters derived from
the power-law fit and individually fit to their own power-law model. The p values were then defined as the fraction of times the resulting Kolmogorov–Smirnov statistics for each synthetic data (n =
1000) relative to its own model was larger than the value of the empirical data. Larger p values therefore indicate a higher probability that the observed distribution could be explained by an
underlying power-law characteristic. Conversely, low p values make a power-law distribution a less likely distribution to describe the empirical data.
As a measure quantifying the difference of a given empirical distribution of some quantity X (in our case S or PLI) from a power-law distribution with exponent α, we defined for which the cumulative
probability distribution P(X) of the distribution's tail, i.e., the N[X] number of X values with values larger than some minimal value X[min], was used. For distributions of cascade sizes the minimum
S[min] was set to 1 and cascade sizes up to system size (i.e., number of channels 27) were included in the sum. For the distribution of PLI the minimal value PLI[min] given by the fitting algorithm
was applied.
Power density spectra.
EEG power density spectra were computed for derivation C3A2 of artifact-free 20 s epochs (same starting points as segments used for the derivation of coherence potentials, synchronization measures,
and PLIs; fast Fourier transform routine; Hanning window; frequency resolution 0.25 Hz). Power in the theta (5–8 Hz) and alpha (8.25–12 Hz) range was determined and averaged for eyes-open segments
per session and subject.
Coherence potentials organize as neuronal avalanches in the EEG
We investigated the spatiotemporal organization of coherence potentials in artifact-free EEG intervals in the eyes-open condition. Artifact-free EEG intervals were analyzed in segments of length 5000
samples (corresponding to 19.53 s; see Materials and Methods). Multiple segments were analyzed for each subject and EEG session. On each of these segments, we first determined potentials with either
a positive or a negative deflection larger than a certain threshold, which we termed trigger events and second identified segments with high similarity to these trigger events (see Materials and
Methods). Amplitude distributions from EEG signals start to deviate from a Gaussian distribution for deflections larger than 4 SD (Fig. 1A). For the detection of coherence potentials we therefore
focused on a threshold of ±4 SD throughout the paper. Systematic exploration of other thresholds around ±4 SD verified that results were independent of the exact choice of threshold (Fig. 2C,D).
Figure 1B shows some exemplary trigger events, their mean duration in all subjects was 468 ± 5 ms. Next, segments of the same length and with high similarity to trigger events were located.
Potentials were considered similar to a trigger event when their correlation R was equal to or greater than 0.75 (see Materials and Methods). Again, systematic analysis with different R values
confirmed that results were robust over a range of values (Fig. 2C,D).
We observed these coherence potentials to organize in cascades of continuous events in time and space (n = 59258 cascades; Fig. 1C). Cascade sizes exhibited a high degree of variability. At the
beginning of the sleep deprivation period, the probability distribution of cascade sizes S closely followed a power-law function p(S) ∼ S^α with α close to −3/2 (Fig. 1D). The total number of EEG
channels determined the observed cutoff of the power-law function at ∼27. In line with reports of neuronal avalanches observed in other model and experimental systems (Beggs and Plenz, 2003;
Priesemann et al., 2013; Shriki et al., 2013), the distribution of cascade sizes remained a power-law distribution for different bin sizes Δt with shallower slopes for larger Δt. Conversely, phase
shuffling of the data destroyed the power-law distribution indicating that the long-range spatiotemporal correlations were captured by the power-law distribution of cascade sizes (Fig. 1D). Along
with a power-law exponent of ∼−3/2, we observed the branching parameter, i.e., the ratio of descendant events to ancestor events, to be σ = 1.17 ± 0.03 (Fig. 2B).
Sustained wakefulness leads to changes in the distribution of cascade sizes and branching parameter σ
With increasing duration of sustained wakefulness, we observed changes in the distributions of cascade sizes. During prolonged wakefulness the probability for larger cascade sizes increased giving
rise to a shallower power-law slope with a heavier tail in the distribution (Fig. 2A). We quantified the deviation of the distribution's tail from a power-law function with exponent α = −3/2 and
calculated ΔD by combining cascade sizes into one distribution for each subject and EEG session (see Materials and Methods). To account for differences in absolute values between subjects, ΔD and σ
values were first transformed into z-scores, i.e., subtraction of the mean and division by the SD, for each individual before averaging over subjects. Regardless of the underlying distribution, z
-scores reflect performance relative to some group, rather than relative to an absolute standard. We denote the normalized data by subscript n throughout the paper. With growing time awake ΔD
progressively increased. Similarly, the branching parameter σ increased during sustained wakefulness. Figure 2B illustrates the evolution of the two metrics as a function of time awake.
To statistically evaluate the changes of ΔD and σ in the course of sleep deprivation we averaged the values of the first four recordings (0–9 h awake) and compared them to the corresponding averages
over the last four recordings (30–39 h awake). The increase in both metrics was significant for a wide range of correlations R and thresholds (Fig. 2C; bar heights reflect the difference of average
metrics at 30–39 h and at 0–9 h, two-tailed paired t test). The return of σ and ΔD to lower values after recovery sleep was similarly observed across a broad range of parameters (significant decrease
of values after recovery sleep compared with the value after 39 h of wakefulness; two-tailed paired t test; Fig. 2D). In absolute values, σ increased from 1.17 ± 0.03 during the first four EEG
recordings (0–9 h awake) to 1.45 ± 0.09 at the end of the sleep deprivation period (30–39 h awake; Fig. 2B, middle).
We also derived cascades of activity using only events with a positive or negative excursion larger than a certain threshold as previously reported (Palva et al., 2013; Shriki et al., 2013). Using
only the threshold as an event detection criterion substantially lowered the number of detected events and cascade sizes when compared with the approach involving coherence potentials (5992 cascades
vs 59,258 cascades using coherence potentials). This alternative approach similarly produced power-laws of cascade sizes and a significant increase of σ with increasing duration of sleep deprivation
(Fig. 3).
Decreasing variability and increase of mean synchrony during sustained wakefulness
The extent and variability of synchrony has been demonstrated to be sensitive to the balance between excitation and inhibition similar to cascades of activity (Yang et al., 2012). Given the changes
observed in the distribution of cascade sizes and σ in our data during sustained wakefulness, we next investigated whether these were accompanied by corresponding alterations in synchronization
metrics. We calculated mean and variability of synchrony from the same artifact-free segments used for the analysis of coherence potentials during the eyes-open condition. In our analysis we focused
on the alpha and theta frequency bands since power changes in these bands during wake were found to be associated with sleep propensity (Torsvall and Akerstedt, 1987; Cajochen et al., 1995; Aeschbach
et al., 1999; Finelli et al., 2000; Strijkstra et al., 2003). During sustained wakefulness we observed an increase in mean synchrony as a function of time awake in the alpha band while the
variability of synchrony quantified by its entropy decreased significantly (Fig. 4). After consecutive recovery sleep (ps) both metrics recovered in the direction of initial values. Interestingly, no
significant changes were observed in the theta band for the eyes-open condition. Extending our analysis to the eyes-closed condition in the theta band revealed similar, albeit weaker changes than in
the alpha band in both synchronization metrics indicating the observed effects to occur predominantly in the alpha band.
The initially low mean and high variability of synchrony in our data closely resemble the findings in Yang et al. (2012) where a maximum of phase synchrony was found to be associated with a balanced
regime characterized by neuronal avalanches and the onset of synchrony.
Distribution of PLIs
To further test for changes in synchronization in the EEG we computed the distribution of PLIs. We calculated PLIs between all pairs of EEG derivations of the same artifact-free EEG segments during
the eyes-open condition that were used for the derivation of coherence potentials and the analysis of the mean and variability of synchrony. The probability distribution of PLIs closely followed a
power-law function p(PLI) ∼ PLI^−α as reported previously (Gong et al., 2003; Kitzbichler et al., 2009) with estimated exponents α in the range of 2 to 2.5 (Fig. 5A). The distribution's p value based
on the Kolmogorov–Smirnov statistic provides a measure of the plausibility of a power-law fit to the data (Clauset et al., 2009). Statistical analysis revealed high p values in many time intervals
suggesting that the power-law hypothesis cannot be rejected. A recent comprehensive analysis of various fitting functions applied to PLI distributions had revealed a power-law function to be the most
likely fit (Kitzbichler et al., 2009).
During prolonged wakefulness, we observed changes in the PLI distributions similar to the ones for cascades of coherence potentials. With increasing duration of sleep deprivation, the probability for
longer PLIs increased (Fig. 5C) giving rise to an increasing deviation from a power-law distribution. ΔD (see Materials and Methods) again quantifies the deviation of the distribution's tail from a
power-law function with exponent α. Both p and ΔD values can be regarded as complementary measures characterizing a power-law distribution with on average larger p values corresponding to smaller ΔD
values (Fig. 5B). Our analysis was performed on cumulative distributions P(PLI), with α and PLI[min] being averaged values of the power-law fit of each participant's first EEG recording (0 h of
wakefulness). To compare p and ΔD values and to account for differences in absolute values between subjects, values were first transformed to z-scores for each individual before averaging over
subjects. Figure 6 shows normalized p values (left), ΔD (middle), and spectral power (right) averaged over all eight participants in the 8–16 Hz frequency band (alpha band; corresponding to scale 4;
see Materials and Methods) and the 4–8 Hz frequency band (theta band; corresponding to scale 5). Along with time course at 3 h intervals, averages of values of the first (blue, 0–9 h of wakefulness)
and last four recording sessions (red, 30–39 h of wakefulness) are illustrated. With increasing duration of sleep deprivation the PLI distributions deviated stronger from a power-law distribution (as
quantified by the decrease in p values) showing a larger tail (captured by the increase in ΔD values). The change in both measures was significant for the alpha band (two-tailed paired t test; Fig. 6
A) and qualitatively also observed in the theta band (Fig. 6B). Similar to the cascade size of coherence potentials, the observed changes correspond to the increasing incidence of larger events
compared with scale-free activity observed earlier in the course of sustained wakefulness.
Previously, changes in EEG spectral power have been described in the alpha and theta bands during sustained wakefulness (Torsvall and Akerstedt, 1987; Cajochen et al., 1995; Aeschbach et al., 1999;
Finelli et al., 2000; Strijkstra et al., 2003). The time-dependent changes in the metrics of PLI distributions as well as mean and variability of synchronization in these two frequency bands (scales)
differed from prior observations in the time course of spectral power in the corresponding frequency bands during sleep deprivation. Normalized power calculated from derivation C3A2 for the same EEG
segments showed a predominantly circadian pattern for the alpha band (Fig. 6A, right plot; note the period of ∼24 h in spectral power) and a circadian component superimposed on an increasing trend
for the theta band (Fig. 6B, right plot) similar to previously reported data (Aeschbach et al., 1999; Finelli et al., 2000) while the indices p and ΔD quantifying the PLI and also coherence potential
distribution as well as the other metrics σ, <r(t)> and H(r(t)) exhibited a more monotonic trend.
Correlation of EEG indices to subjective alertness
Self-rated alertness declined significantly with increasing time awake (Fig. 7A). Before averaging over subjects, alertness scales were transformed to z-scores for each subject in the same way as the
EEG indices. To compare its evolution with EEG indices, we calculated their correlation coefficients R for which we used averaged data over all subjects (Fig. 7B; see Materials and Methods). Similar
to a previous report (Finelli et al., 2000), theta power and alertness were negatively correlated (r = −0.94) and exhibited the largest absolute correlation value of all indices. The avalanche
metrics and synchronization measures also exhibited a high correlation with alertness. From these indices, σ correlated best with alertness (r = −0.77).
In the present work, we reported several changes in EEG indices during sustained wakefulness: the organization of cascade sizes of coherence potentials, the mean and variability of synchronization,
and the distribution of PLIs. These indices changed as a function of time awake and recovered toward baseline values after subsequent recovery sleep. The changes in synchronization measures (mean
synchronization, variability of synchronization, and distribution of PLIs) were most predominant in the alpha band. This is in contrast to the changes in spectral EEG power as a marker of sleep
propensity, which are most evident in the theta band (Finelli et al., 2000). Thus, the changes in synchronization cannot be explained as a direct consequence of the alterations in spectral power.
At the beginning of the sleep deprivation period coherence potentials in the EEG were organized as neuronal avalanches, i.e., events in space and time characterized by a power-law size distribution
with exponent close to −3/2 and branching parameter of ∼1 (σ = 1.17). Neuronal activity patterns following a power-law size distribution with exponent −3/2, a lifetime distribution with power-law
exponent −2, and a branching ratio of 1 have previously been interpreted as an indication that cortical neuronal networks operate at criticality (Beggs and Plenz, 2003; Larremore et al., 2011;
Friedman et al., 2012; Palva et al., 2013; Shriki et al., 2013). This conclusion is based on insights derived from theoretical models of systems poised at a phase transition exhibiting the same
scaling laws. The case of neuronal avalanches refers to a critical branching process giving rise to scale-free avalanches of activity and avoiding activity regimes comprising only either large or
small parts of a network (Bak and Paczuski, 1995). This balanced propagation of activity is also reflected in the branching parameter, which in such a case is expected to be exactly 1. Although close
to 1, we observed a baseline branching parameter of 1.17, which could be caused by uncertainty in its exact determination or, if taken by itself, indicate a slightly supercritical state.
With increasing duration of wakefulness, both size distributions of coherence potentials and PLIs increased in tail as measured by ΔD and p values in the case of PLIs. For the neuronal avalanches we
observed a coincidental increase in σ, defined as the ratio between descendant events to ancestor events. Furthermore, mean synchronization increased while its variability decreased.
A critical branching process provides an interesting framework connecting these seemingly different observations. It was recently shown that power-law neuronal avalanches are accompanied by a maximum
of synchronization variability in vitro and in modeling systems (Yang et al., 2012). Conversely, disinhibited networks exhibited cascades of activity with an imbalance toward larger avalanches, which
coincided with decreased variability in synchronization and increased mean of synchronization in some metrics. These findings closely resemble the observations in our data. In the context of a
critical branching process they can be interpreted in the sense that network activity is increasingly shifted toward a state in which larger events prevail in the dynamics, unlike in the critical
state with scale-free avalanches of activity and high variability of synchronization.
The balance between excitation and inhibition depends crucially on synaptic strength in neuronal systems. It was recently demonstrated that a reduction of inhibitory synaptic transmission by
application of GABA receptor blockers resulted in an artificial imbalance toward excitation related to a supercritical regime with larger neuronal avalanches than predicted by a power-law
characteristic (Beggs and Plenz, 2003; Shew et al., 2009). Although the changes in such pharmacologically disinhibited networks were more drastic than what we observed, they qualitatively well
correspond to alterations in the distribution of neuronal avalanches and σ reported here. Indirect evidence for changes in excitability and synaptic strength during wake and sleep comes from
observations in the EEG, where power changes in the theta and alpha bands during wake were found to be associated with sleep propensity (Torsvall and Akerstedt, 1987; Cajochen et al., 1995; Aeschbach
et al., 1999; Finelli et al., 2000; Strijkstra et al., 2003) and slow-wave activity during sleep to be associated with sleep intensity reflecting a regulatory process termed sleep homeostasis (
Achermann and Borbély, 2011). Tononi and Cirelli (2003, 2006) hypothesized that synaptic homeostasis is underlying sleep homeostasis: synaptic strength is high at the beginning of a night, due to
plastic processes occurring during waking, and decreases by means of synaptic downscaling during sleep. Recent investigations of changes in synaptic strength in Drosophila (Bushey et al., 2011) and
neuronal excitability in human cortex (Huber et al., 2012) support the hypothesis by providing evidence of structural changes occurring in neuronal networks during waking and their reorganization
during sleep. It is conceivable that the increase in excitability of cortical circuits caused by the buildup in synaptic strength provides the cellular basis for the observed shift away from critical
dynamics in the course of wakefulness. Similarly, rebalancing and synaptic downscaling during sleep could tune network dynamics back to criticality. However, a plausible hypothesis for the detailed
mechanisms, how the changes in synaptic strength could induce the observed changes both in spectral power and in the signatures of criticality, is still missing. From a clinical perspective,
disinhibited networks are reminiscent of epileptic seizures where activity engages most of the network. In fact, for many forms of epilepsy sleep deprivation is known to increase the probability of
seizure occurrence (Ellingson et al., 1984).
Power-law probability distributions of PLIs between pairs of neurophysiological time series were recently studied (Kitzbichler et al., 2009; Meisel et al., 2012) and interpreted as signatures of
criticality in human brain dynamics. Evidence for power-law PLI distributions as an indicator for criticality derives from computational models, which show power-law PLI distributions at the
transition between an ordered and a disordered phase, i.e., when they are in a critical state (Kitzbichler et al., 2009; Meisel et al., 2012). A PLI power-law distribution can, however, also arise
when systems are away from a phase transition making the PLI distribution a sensitive but not specific indicator for critical dynamics (Botcharova et al., 2012). A clear power-law PLI distribution in
neurophysiological data can, as any power-law scaling (Touboul and Destexhe, 2010; Beggs and Timme, 2012), therefore only be seen as an indication for the possibility of a phase transition. In our
case, this indication is further supported by the simultaneous detection of neuronal avalanches in coherence potentials following a power-law distribution with exponent −3/2, a σ of 1.17, and high
variability in synchronization. Conversely, PLI probability distributions with a large tail and weak power-law statistics along with similar changes in the distribution of neuronal avalanches and a σ
much larger than 1 together with reduced variability of synchronization support the conclusion that the system is not directly located at a critical state.
It was previously hypothesized that normal activity is located in a slightly subcritical regime and that sleep could function to establish a safety margin by reorganizing activity toward it while
wakefulness drives it closer to criticality (Pearlmutter and Houghton, 2009). A recent analysis of neuronal avalanches from human intracranial depth recordings in fact implied that the human brain
might not operate at criticality directly but in a somewhat subcritical regime (Priesemann et al., 2013). While we cannot exclude the possibility that network activity in the brain is poised in a
slightly subcritical regime, when interpreted in the context of criticality, the fading power-law statistics of the PLI distribution, the increase of σ to values much larger than 1, and the decrease
in variability of synchronization in our opinion provide more support to the notion that cortical networks are near or at a critical state initially and shift toward a supercritical regime during
wakefulness. Furthermore, the characteristic changes of synchronization metrics accompanying the alterations in the power-law characteristics of neuronal avalanches suggest a growing deviation from
criticality as a function of time awake instead of the system remaining critical with a different exponent as could be concluded by looking solely at the distribution of neuronal avalanches.
Critical dynamics are often regarded to support optimal computational functioning (Langton, 1990; Bertschinger and Natschläger, 2004; Haldeman and Beggs, 2005; Kinouchi and Copelli, 2006; Shew et
al., 2009, 2011). The observation of fading signatures of critical dynamics during prolonged wakefulness and their correlation to alertness could provide an interesting link to behavioral
observations of impaired cognitive functioning and information processing after sleep deprivation (Banks and Dinges, 2007). Our findings support the intriguing hypothesis that sleep might serve to
reorganize cortical network dynamics to a critical state thereby assuring optimal computational capabilities for the time awake.
• The study was supported by the Swiss National Science Foundation Grant 320030-130766 (P.A.) and by the European Community's Seventh Framework Programme (FP7/2007-2013) under Grant agreement no.
258749 (E.O.). We thank Thilo Gross for comments on an earlier version of this manuscript.
• Correspondence should be addressed to Christian Meisel, Department of Neurology, University Clinic Carl Gustav Carus, Fetscherstrasse 74, 01307 Dresden, Germany. christian{at}meisel.de | {"url":"https://www.jneurosci.org/content/33/44/17363","timestamp":"2024-11-07T10:38:29Z","content_type":"application/xhtml+xml","content_length":"415630","record_id":"<urn:uuid:decfe5cf-1bda-43fa-95fb-b319f521c702>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00236.warc.gz"} |
Exterior Angle Inequality Theorem Worksheet With Answers - Angleworksheets.com
Exterior Angle Theorem Worksheet With Answers – Angle worksheets can be helpful when teaching geometry, especially for children. These worksheets contain 10 types of questions on angles. These
questions include naming the vertex, arms, and location of an angle. Angle worksheets are an integral part of any student’s math curriculum. They help students understand the … Read more
Exterior Angle Inequality Theorem Worksheet With Answers
Exterior Angle Inequality Theorem Worksheet With Answers – Angle worksheets are a great way to teach geometry, especially to children. These worksheets contain 10 types of questions on angles. These
questions include naming the vertex, arms, and location of an angle. Angle worksheets are a key part of a student’s math curriculum. They help students … Read more | {"url":"https://www.angleworksheets.com/tag/exterior-angle-inequality-theorem-worksheet-with-answers/","timestamp":"2024-11-07T10:36:18Z","content_type":"text/html","content_length":"54032","record_id":"<urn:uuid:68cf0703-84aa-4387-89d9-f6813a00708a>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00484.warc.gz"} |
Tutorial 2: Steady State RANS
Tutorial 2: Steady State RANS¶
In Tutorial 1, we learnt how to run a case in zCFD and visualise the results. We will now use a more complex test case to showcase some of the features of the solver. The case we will look at is the
30P30N aerofoil, which is a 2D multi element aerofoil, developed as an acoustic benchmark case [1]. The geometry of this case can be seen below:
Initially we will begin by obtaining a steady state RANS solution.
The required meshes and control dictionary can be found here. The case has 64,000 cells, and will run in less than 10 minutes on an NVIDIA RTX 3060 GPU.
This case is run as a steady state simulation, using an implicit time marching scheme. The Reynolds Averaged Navier-Stokes equations are solved using Menter’s SST turbulence model with wall
Control Dictionary Breakdown¶
The setup for this case is very similar to the previous NACA0012 cases, but the control dictionary does feature some extra terms which can be useful for a more streamlined workflow.
Reference variables¶
At the top of the control file, the key variables for the case- Reynolds number, Mach number, temperature etc… are all hard coded, and then referred to later in the script. In cases where you might
want to run many simulations with slightly different values to change, this can be an effective way to ensure consistency with variables. Additionally this allows implicit values such as the
reference velocity, and freestream Mach to be calculated automatically in the script.
# create variables to assign main experimental parameters
reynolds = 1.71e6
mach = 0.17
T = 295.56
p = 101325
R = 287.6
gamma = 1.4
reference_length = 0.457
alpha = 5.5
# calculate implicit values
rho = p / (R * T)
U = math.sqrt(gamma * R * T) * mach
# assign each mesh zone to a boundary
wall = [6]
symmetry = [4,5]
farfield =[7]
In this case a scaling is applied to convert the size of the mesh. The .h5 mesh file was generated with units of inches, whereas the flow conditions are specified in SI units, therefore a scaling of
0.0254 is applied to convert the mesh into metres. The z (spanwise) direction has been scaled to be 1 metre wide. The mesh has only a single cell in the z direction so no spanwise flow is expected -
making the z extent of the mesh 1 metre simply means that the forces and moments reported by zCFD will already be per unit span.
# scale the mesh to match the experimental data
'scale' : [0.0254,0.0254,0.0254/0.127],
Inflow vector¶
In this case we want to simulate the aerofoil operating at an angle of attack of 4 degrees. Rather than rotating the mesh, it is easier to rotate the inflow vector relative to the mesh, an example of
a Galilean transformation. The zutil function vector_from_angle() calculates the inflow vector given an x-z angle of attack, x-y angle of attack and flow speed.
'IC_1' : {
'temperature': T,
'pressure': p,
'V': {
# calculate the inflow vector based on the angle of attack
'vector' : zutil.vector_from_angle(0.0,alpha,U),
'Reynolds No' : reynolds,
'Reference Length' : reference_length,
'turbulence intensity': 0.01,
'eddy viscosity ratio': 0.1,
'ambient turbulence intensity': 1e-20,
'ambient eddy viscosity ratio': 1e-20,
The lift and drag forces acting on a body are defined relative to the freestream flow, lift normal and drag parallel. In this case, where we have rotated the relative inflow vector to a specific
angle of attack, it is also useful to rotate the force report by the same vector. The transformed forces will appear as Ft_ terms in the report file.
# define a function to rotate the output forces by the angle of attack
def my_transform(x,y,z):
v = [x,y,z]
v = zutil.rotate_vector(v,0.0,alpha)
return {'v1' : v[0], 'v2' : v[1], 'v3' : v[2]}
Running the case¶
You can run the case as you did for Tutorial 1, but with the following run_zcfd command:
run_zcfd -m 30p30n_coarse.h5 -c 30p30n_steady.py
Monitoring convergence¶
Start by launching a Jupyter server to examine the residuals in the 30p30n_steady_report.csv file. Running the first cell from the 30p30n_steady_report.ipynb notebook will plot the residuals for the
continuity, momentum, and energy equations, as well as the residuals from the turbulence model.
If the case is still running you can update the plot by rerunning the cell, this provides a way to monitor the progress of a running simulation.
You will notice that case is set to run for 1000 cycles but looking at the plots, it looks like the residuals are still converging. So we will now restart the solver from where we finished and run on
for another 1000 cycles.
Performing a restart¶
To restart the solver we need to update the ‘restart’ parameter in the control dictionary. If this is set to False the solver will always perform a fresh start, if it is True then a results file will
be read in and the solve will start based on the data in that file. To enable the restart edit 30p30n_steady.py and change restart to True. You will also need to update ‘cycles’ to 2000.
"restart": True,
'time marching' : {
'cycles' : 2000,
By default zCFD will look for a results file with the same name as the current case file, appended with ‘_results.h5’. So in our case it will look for “30p30n_steady_results.h5”. When you have edited
the control dictionary go ahead and restart the simulation again using the same run_zcfd command:
run_zcfd -m 30p30n_coarse.h5 -c 30p30n_steady.py
Once the solver has finished rerun the cell in the notebook to examine the residuals again. The residuals now look like they have converged:
Plotting force convergence¶
Examining the force convergence history next, we are interested in the x and y forces, but transformed by the inflow vector, therefore the wall_Ftx and wall_Fty plots are of interest.
Which show reasonable agreement with experimental results, and additionally that the forces are at least nearing convergence. The grid for this case is still extremely coarse, accounting for the
differences in lift and drag coefficients.
Opening Paraview, and loading in the 30p30n_steady_wall.pvd results will allow you to view the aerofoil surface results:
To get a continuous plot of cp against x/c you need to use a plot over sorted lines filter. To do this:
1. On the surface data apply a ‘cell data to point data’ filter.
2. On the cell data to point data apply a ‘slice’ filter, ensuring the slice uses a z normal, and is centred through the middle of the aerofoil
3. On the slice data apply a ‘plot on sorted lines’ filter and click apply. This will then bring up a new ‘line graph’ view.
4. Make sure to select all 3 segments in the composite data set dialogue box, then ensure only the cp variables are selected in the series parameters. Finally for the X Array name, select ‘Points_X’
5. Modify the style and markers until you are happy with the result.
1. Zhang, Yufei & Chen, Haixin & Wang, Kan & Wang, Meng. (2017). Aeroacoustic Prediction of a Multi-Element Airfoil Using Wall-Modeled Large-Eddy Simulation. AIAA Journal. 55. 1-15. 10.2514/ | {"url":"https://docs.zenotech.com/v2024.10.9337/getting_started/quick_start_2.html","timestamp":"2024-11-12T20:41:43Z","content_type":"text/html","content_length":"23035","record_id":"<urn:uuid:afd658d1-6e1e-49fd-9680-bf66463ce075>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00648.warc.gz"} |
Basic Algebra – Explanation & Examples
With more than 25 years of experience teaching math, I pride myself on being knowledgeable in multiple facets of this ever-growing topic. As an educator, my aim here is to share free math
tutorials and lessons to assist students with their studies.
Basic Algebra – Explanation & Examples
hardest course in mathematics.
This is just a mere fallacy, and in fact, algebra is one of the easiest topics in mathematics. This article is meant to alleviate this fear and misconception from students and make algebra an
enjoyable lesson for beginners.
What is Algebra?
Have you ever wondered or asked yourself, what is algebra? Where did it originate from? How is algebra applied in real-life situations? Don’t worry. This article will take you step by step in
understanding algebra and solve a few algebraic problems.
Basically, students will start their mathematical journey by learning to perform basic operations such as addition and subtraction. From there, a student will advance to multiplication and then to
division. Later or sooner, a student will reach a point where they can tackle complex problems. What are we talking about? Algebra, of course!
Some people wrongly refer to algebra as the operation which deals with letters and numbers. In fact, Algebra was in existence before the invention of the printing press more than 2500 years ago. The
introduction of printing initiated the use of symbols in algebra. Therefore, Algebra is well defined as the use of mathematical equations to model ideas. We model ideas in the form of mathematical
equations to solve the problems around us.
History of Algebra
The word algebra originates from the Arabic word al-Jabr, which means placing broken parts together. This term is featured in the book “The Compendious Book on Calculation by Completion and
Balancing” by Al-Khwarizmi, a Persian mathematician and astronomer. In the fifteenth century, algebra was initially used to describe a surgical procedure where dislocated, broken bones are reunited.
From this discussion, we can say algebra helps us to reunite bits of information.
Why do we Need to Study Algebra?
Understanding algebra is fundamentally important to the student both in class and outside class. Algebra sharpens the reasoning ability of a student. Students can succinctly and systematically solve
mathematical problems.
Let’s take a look at some of the importance of algebra in real life.
• A toddler or infant can apply algebra by tracing a trajectory of moving objects using eyes. Similarly, babies can estimate the distance between them and a toy and thus able to grab it. Therefore,
small babies apply algebra despite the fact of lacking knowledge of algebra.
• Algebra is applied in computer science to write algorithms of programs. Algebra is also used in engineering to calculate correct proportions to implement a masterpiece. Maybe you will see these
later when you advance your career.
• You require algebra to know when you are supposed to wake up and do morning chores or prepare for classes.
• Have you ever thrown dirt in a bin? Did you miss, or you made a perfect shot? You need algebra to estimate the distance between you and the trash bin and estimate the air resistance.
• The use of algebra calculates profits and losses in business. For this reason, good knowledge of algebra is essential for managing your finances.
• Algebra is widely applied in sports. For example, a goalkeeper can dive at a ball by estimating the speed of a ball. An athlete can also increase his/her pace by estimating the distance between
them and the finish line.
• Algebra finds itself in the kitchen, such as cooking, mixing ingredients, and determining the cooking duration.
• Applications of algebra are just endless. That phone you are using, the computer games you are playing are just fruits of algebra. Computer graphics are developed on algebra.
How to do Algebra?
You will usually see both known values and unknown values in an algebraic expression, and you solve the equation for an unknown value. To solve that equation, you need to do algebra, in which you
need to follow the same order of operations you do for the integers.
For example, you will first solve what is inside the parenthesis, then go for the following operations in sequence: exponents, multiplication, division, addition, and subtraction.
The following are the term which you will see in an algebraic expression.
• An equation is a statement or sentence that defines two identities separated by an equal (=) sign.
• Expression is a list or a group of different terms usually separated by ‘+’ or ‘- ‘sign
If a and b are two integers, the following are basic algebraic expressions:
• Addition equation: a + b
• Subtraction equation: b – a
• Multiplication equation: ab
• Division equation: a/b or a ÷ b
Basic algebra problems
The basic algebraic formulas are:
• a^2– b^2 = (a – b) (a + b)
• (a + b)^2= a^2 + 2ab + b^2
• a^2+ b^2 = (a – b)^2 + 2ab
• (a – b)^2= a^2 – 2ab + b^2
• (a + b + c)^2= a^2 + b^2 + c^2 + 2ab + 2ac + 2bc
• (a – b – c)^2= a^2 + b^2 + c^2 – 2ab – 2ac + 2bc
• (a + b)^3= a^3 + 3a^2b + 3ab^2 + b^3
• (a – b)^3= a^3 – 3a^2b + 3ab^2 – b^3
Example 1
Find the value of t, if t + 15 = 30
t = 30 – 15
t = 15
Example 2
Find the value of y, when, 9y = 63
Divide both sides by 9;
y = 63/9
y = 7
Example 3
If 21= b/7, find b:
Cross multiply:
b = 21 x 7
b = 147
Example 4
Consider a case of calculating grocery expense:
You want to go out shopping to buy 2 dozen eggs at $10, 3 loaves of bread each at $5, and 5 bottles of drinks, each at $8. How much money do you need?
You can start solving this problem by assigning commodity a letter for instance:
Let dozens of eggs = a;
Breads= b;
Drinks =d
Price of a dozen= a = $10
Price of one bread=b = $5
Price of one bottle drinks=d= $8
=> Total expenditure= d + 3b + 5d
Substitute the values:
= $10 + 3($5) + 5($8) = $10 + $15 + $40 = $65
Therefore, the total expenditure is $65.
Practice Questions
1. Solve for x, when x+12 = 6.
2. Find the value of z, if 2z + 2= 10.
3. Find y; if 2y – 8 = 4y.
4. The sum of 3 consecutive numbers is 216. What are the 3 numbers?
5. A rectangle has an area of 72cm^2. Suppose the width of the rectangle is twice its length. What are the length and width of the rectangle? | {"url":"https://www.storyofmathematics.com/basic-algebra/","timestamp":"2024-11-08T07:57:23Z","content_type":"text/html","content_length":"168807","record_id":"<urn:uuid:573a44f2-be74-4dd2-a764-d59797946dff>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00325.warc.gz"} |
CN110779418A - Method for measuring length of cone on line by double meters
- Google Patents
CN110779418A - Method for measuring length of cone on line by double meters - Google Patents
Method for measuring length of cone on line by double meters Download PDF
Publication number
CN110779418A CN110779418A CN201910991410.XA CN201910991410A CN110779418A CN 110779418 A CN110779418 A CN 110779418A CN 201910991410 A CN201910991410 A CN 201910991410A CN 110779418 A CN110779418
A CN 110779418A
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Chengdu CAIC Electronics Co Ltd
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Chengdu CAIC Electronics Co Ltd
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Priority to CN201910991410.XA priority Critical patent/CN110779418A/en
Publication of CN110779418A publication Critical patent/CN110779418A/en
Pending legal-status Critical Current
☆ G—PHYSICS
☆ G01—MEASURING; TESTING
☆ G01B—MEASURING LENGTH, THICKNESS OR SIMILAR LINEAR DIMENSIONS; MEASURING ANGLES; MEASURING AREAS; MEASURING IRREGULARITIES OF SURFACES OR CONTOURS
☆ G01B5/00—Measuring arrangements characterised by the use of mechanical techniques
☆ G01B5/02—Measuring arrangements characterised by the use of mechanical techniques for measuring length, width or thickness
□ Physics & Mathematics (AREA)
□ General Physics & Mathematics (AREA)
□ A Measuring Device Byusing Mechanical Method (AREA)
□ Length-Measuring Instruments Using Mechanical Means (AREA)
The invention discloses a method for measuring the length of a cone on line by using double gauges, which aims to provide a practical, quick and convenient method for measuring the length of the
cone on line and is realized by the following technical scheme that two dial gauges with consistent head sizes of a measuring rod are adopted, the two gauges are used for carrying out zero
alignment when a measuring rod head is positioned on the same reference plane, the process measurement is carried out, namely, a testing device after zero alignment is placed on the surface of
the cone to be measured, and during the process measurement, a large right triangle abc and a small right triangle def are formed by a first dial gauge, a second dial gauge and a measuring
support to form a geometric relation and an equation, wherein the taper β of the cone to be measured is arctan [ (H2-H1)/L2], the difference delta which is formed by the misalignment of the zero
alignment reference point of a measuring head of the dial gauge and the reference point of the measuring cone is a (d/2) ═ tag30 DEG, and the length L1+ L2+ (H1+ tan)/tan) of the cone to be
measured can be calculated.
Method for measuring length of cone on line by double meters
Technical Field
The invention relates to a cone length measuring method which is widely applied to cone machining in a machining process, in particular to a cone length measuring method which is suitable for a
measured cone of a small-angle cone of a rotary shaft.
In the mechanical processing of the detected cone (1), the processing of various cones is usually encountered, the processing errors of cone parts are mainly shown in two aspects of diameter and
taper, and a detection tool for detecting the two aspects is usually a cone plug gauge equipped with an angle measurement eyepiece. The dial arranged on the measuring head is directly used as an
angle standard, a Mi-character scribed line on an ocular reticle is used for aiming during measurement, the cone plug gauge is adjusted by utilizing vertical and horizontal guide rails which are
vertical to each other, and the taper is measured by using accessories such as a thimble frame and the like in a matching way. During measurement, the measuring block group and the sine gauge are
placed on the flat plate, an included angle is formed by the main body plane of the sine gauge and the flat plate working face, one working face of the measured angle is placed on the main body
working face of the sine gauge, if the actual value of the measured angle is equal to the nominal value of the measured angle, the upper working face of the measured cone is parallel to the flat
plate, and otherwise, the two faces are not parallel. And calculating the difference between the actual angle value and the nominal angle value of the measured cone by measuring the degree of
non-parallelism. When the batch size is large, the visual and mental consumption is large, resulting in low inspection efficiency and visual errors. The cone length L refers to the axial distance
between the largest cone diameter section and the smallest cone diameter section. And sometimes, the axial length of the cone needs to be measured, and because the intersection of a side
generatrix of the cone and the axis is difficult to observe, when the part cannot be detached for independent measurement and field measurement is needed, the accurate axial length of the cone is
difficult to measure by a general measuring tool, so that how to quickly and accurately measure the actual state and size of the cone is a way to improve the processing efficiency.
Disclosure of Invention
The invention aims to provide a practical, quick and convenient method for measuring the length of a cone on line by using double meters aiming at the defects in the prior art.
The above object of the present invention can be achieved by a method for measuring the length of a cone on line by using a dual gauge, comprising the steps of:
and (3) zero calibration before measurement: firstly, two dial indicators 3 and 4 with the same head size of a measuring rod are adopted, the two dial indicators are aligned to zero when the
measuring rod heads are positioned on the same reference plane, the two dial indicators are inserted into two fixing holes of a measuring bracket 2, the height of the measuring head exposed out
of the reference plane W2-3 mm is kept, and a locking screw is fastened; then the measuring bracket 2 is placed on a detection platform or a reference platform, and the dial plates of the two
dial indicators are rotated to make the pointers of the two dial indicators align to O points respectively;
procedure measurement: the testing device after zero calibration is placed on the surface of the tested cone, so that the height H of the right end face of the measuring bracket (2) is attached
to the right end of the cone, and the reference surface W is attached to the outer circular face of the tested cone (1;
calculating the length L of the measured cone, namely taking the center of a first dial indicator 3 and a second dial indicator 4 as a point A and a point B respectively, when measuring in the
process, the first dial indicator (3 pointer falls back to the point A, the second dial indicator (4 pointer falls back to the point B, and the difference between the two indicators is calculated
by reading A, B two points (H2-H1), wherein the dial indicators 3 and 4 have two fixed mounting hole distances L2 on a measuring support 2, wherein the second dial indicator 4 has a fixed
distance L1 from a measuring reference end face, when measuring, a large right triangle abc and a small right triangle def are formed, so that the following geometric relations and calculation
formulas are formed, namely the taper of the measured cone β is arctan [ (H2-H1)/L2] … (a), the difference delta between the zero calibration reference point of a dial indicator and the measuring
cone reference point which are not coincident with the taper of the measuring cone, which is (d/2): … (B), the axial length of the measured cone which is L, the zero calibration reference point
of the dial indicator, and the measuring cone which is not coincident with the measuring cone reference point of the measuring cone, and the measuring head of the measuring cone which is replaced
by the fixed measuring gauge 1, and the difference between the measuring head which is obtained by substituting the mounting hole distances H464 and the measuring head of the measuring head which
is replaced by the two measuring gauge 465 and the measuring head which are replaced by the fixed measuring gauge β, and the difference of the measuring cone which are obtained by the difference
of the fixed measuring cone which is replaced by the measuring cone which is obtained by the measuring gauge 464, and the difference of the measuring rack which.
Compared with the prior art, the invention has the following beneficial effects.
The invention utilizes two dial indicators to be arranged on a measuring bracket 2 to obtain the fall between two points on the surface of a cone, and then calculates the actual value of the
axial length of the cone through a fixed formula on the basis of quickly obtaining the fall between two points on the surface of the cone by the two dial indicators 3 and 4. The cone axial length
is obtained. The invention can quickly measure the length of the cone on line and quickly measure the length L of the measured cone with the small-angle cone axial length.
The invention adopts a double-dial cone length measuring structure consisting of a measuring bracket 2, dial indicators 3 and 4, a set screw and a measured cone 1; simple structure, the volume is
convenient, can calculate the actual axial cone length dimension of cone fast. When the axial length of the conical surface part needs to be effectively detected, the difference value is read by
the two dial indicators in the mode, and the length of the processed cone is calculated by a set formula.
The invention can be applied to the indirect measurement of the measured cone 1 with different cone angles.
FIG. 1 is a schematic diagram of a dual-meter measurement for measuring cone length on-line with dual meters in accordance with the present invention;
FIG. 2 is a schematic diagram of the pre-measurement zero calibration and measurement principle of the dual-meter online measurement of the length of the cone of the present invention;
fig. 3 is a cross-sectional view of the measured cone in a measured state.
Fig. 4 is a left side sectional view of fig. 3 with the cone 1 being measured removed.
Fig. 5 is a cross-sectional view of the second dial indicator of fig. 4.
In the figure: 1. the device comprises a measured cone 2, a measuring bracket 3, a first dial indicator 4 and a second dial indicator. 5. Locking screw
Detailed Description
Referring to fig. 1-4, according to the invention, two dial indicators 3 and 4 with consistent head size of the measuring rod are adopted, the two indicators are zero-aligned together when the
measuring rod head is positioned on the same reference plane, the procedure measurement is carried out by placing the testing device after zero calibration on the surface of the measured cone,
the first dial indicator 3, the second dial indicator 4 and the measuring support 2 form a large right triangle abc and a small right triangle def during the procedure measurement, and the
geometric relationship and the formula are formed, wherein the taper β of the measured cone is arctan [ (H2-H1)/L2], the difference delta between the zero calibration reference point of the
measuring head of the dial indicator and the measuring cone which are not superposed is delta (d/2) · 30 degrees, the length L of the measured cone is calculated to be L1+ L2+ (H1+ delta)/tanc,
the zero calibration is carried out, the two dial indicators 3 and 4 with consistent head size of the measuring rod head are firstly adopted, the two indicators are zero-aligned together when the
two indicators are positioned on the same reference plane, the two fixed supports are installed, the two dial indicators are inserted into the two fixed supports, the two measuring rod head are
aligned, the two indicators are aligned, and the measuring platform is aligned, and the height of the measuring head is measured, and the measuring platform is kept, and the measuring head is
measured, and the indicator is aligned, and the measuring head is positioned on the measuring platform;
procedure measurement: placing the test device after zero calibration on the surface of the tested cone, and enabling the height H of the right end face of the measurement support 2 to be
attached to the right end of the cone and the reference surface W to be attached to the outer circular face of the tested cone 1;
calculating the length L of the measured cone: taking the center of the first dial indicator 3 and the second dial indicator 4 as a point A and a point B respectively, when measuring in the
working procedure, the pointer of the first dial indicator 3 falls back to the point A, the pointer of the second dial indicator 4 falls back to the point B, and the difference between the two
indicators is calculated through the readings of the two points A, B (H2-H1);
the dial indicators 3 and 4 have two fixed mounting hole distances L2 on the measuring support 2, wherein the second dial indicator 4 has a fixed distance L1 from the measuring reference end
face, so that a large right triangle abc and a small right triangle def are formed during measurement, and the following geometric relations and equations are formed:
the length L of the tested cone in the axial direction of the tested cone is L1+ L2+ (H1+ delta)/tan β … … … … … … … … … (c)
The method comprises the steps of (a) replacing the fall (H2-H1) of the two dial gauges to calculate the taper of a measured cone β, (b) substituting the micrometer for measuring the diameter d
of a measuring head of a dial gauge to calculate the difference delta existing when the zero calibration reference point of the measuring head of the dial gauge and the reference point of the
measured cone are not overlapped, and (c) replacing the fixed distance L1 between the β and delta two dial gauges 3 and 4 and the fixed distance L2 between the second dial gauge 4 and the end
face of the measuring reference to obtain the length L of the measured cone.
Referring to fig. 2, a perpendicular bc is drawn at the bottom end of a measured cone length L, a line segment ac is constructed by crossing an initial point a of L, the perpendicular bc is
connected, the large right triangle △ abc is obtained, the large right triangle △ abc is cut by making a perpendicular with a distance L2 projected on a horizontal plane by two dial gauges 3 and
4, a distance H2 from a zero point to the measured cone by the dial gauge 4 is obtained, a perpendicular intersection is drawn at a point d by making an end of L2, a perpendicular intersection is
drawn by making an end of L1 through the two dial gauges 3 and 4, a distance H1 from the zero point to the measured cone by the small right triangle △ def and the dial gauge 3 is obtained, since
a base de and a leg ef in the small right triangle △ def are parallel to a base ab and a leg bc in the large right triangle △ abc, and a leg ef in the base of the small △ def is obtained by a
difference between the two dial gauges 3 and 4, an ef is equal to H2-H1, a distance between the base ref and a vertex angle b, a vertex angle of the small triangle 3527, a distance H638 is
calculated, and a vertex angle is equal to a distance H2-7378, a vertex angle is obtained by calculation, a constant.
See fig. 3-5. Because the reference point when the percentage table gauge head is calibrated zero may not coincide with the reference point when measuring the cone, there is a difference, need to
compensate the error of the percentage table gauge head: as shown in K in fig. 5, to obtain the height difference between the measuring heads of the two dial gauges, and calculate the total
length of the measured cone, the spatial difference δ between the two dial gauges perpendicular to the measuring support 2 is obtained according to the gauge diameters d of the two dial gauges:
the axial length of the tested cone is L1+ L2+ (H1+ delta)/tan β … … … … … … … … … … … … … (c),
a measurement step:
1) and (4) installing a dial indicator: inserting the two dial gauge positions with the gauge diameter d into the two fixing holes of the measuring bracket 2 shown in the figure 4, keeping the
height H of the special test head exposed out of the reference plane W2-3 mm, and fastening the locking screw 5 to ensure that the surface of the dial gauge is opposite to an observer;
2) and (3) zero calibration before measurement: after the content of the previous step is finished, the measuring support 2 is placed on a detection platform or a reference platform, and the two
dial indicator dials are rotated to enable the two indicator pointers to be respectively aligned to the O points;
3) procedure measurement: placing the test device after zero calibration on the surface of the cone 1 to be tested according to the figure 5, and enabling the right end surface H of the
measurement support to be attached to the right end of the cone and the reference surface W to be attached to the outer circular surface of the cone 1 to be tested;
4) difference between the two tables (H2-H1): when in process measurement, the pointer of the dial indicator 3 falls back to the point A, the pointer of the dial indicator 4 falls back to the
point B, and the fall of the two indicators can be calculated through the readings of the two points A, B (H2-H1);
5) the length L of the measured cone can be calculated by (H2-H1) substituting formula (a) to β, the micrometer measures the diameter of a measuring head of a dial indicator and substituting
formula (b) to δ, β and δ are substituted into formula (c) to obtain L, L1 and L2 are measured by three coordinates before the measuring support 2 is used to obtain a relatively accurate
distance, the outer diameter value of a measuring rod of the dial indicator 3 and 4 and the installation aperture on the measuring support 2 are matched in an accurate sliding fit mode to improve
the testing accuracy of the structural system, and the perpendicularity of a reference surface W of the measuring support 2 and an H surface perpendicular to the measuring support 2 is smaller
than 0.02 mm, so that the measuring error is controlled in an effective range.
The foregoing is a detailed description of the invention with reference to specific preferred embodiments, and no attempt is made to limit the invention to the particular embodiments disclosed,
or modifications and equivalents thereof, since those skilled in the art will recognize that various changes may be made without departing from the spirit and scope of the invention as defined by
the appended claims.
Claims (5)
1. A method for measuring the length of a cone on line by using double meters is characterized by comprising the following steps:
and (3) zero calibration before measurement: firstly, two dial indicators (3 and 4) with the same head size of a measuring rod are adopted, the two dial indicators are aligned to zero when the
head of the measuring rod is positioned on the same reference plane, the two dial indicators are inserted into two fixing holes of a measuring bracket (2), the height of the measuring head
exposed out of the reference plane W2-3 mm is kept, and a locking screw is fastened; then the measuring bracket (2) is placed on a detection platform or a reference platform, and the dial plates
of the two dial indicators are rotated to enable the pointers of the two dial indicators to be respectively aligned to the O points; procedure measurement: placing the test device after zero
calibration on the surface of the tested cone, and enabling the height H of the right end face of the measurement support (2) to be attached to the right end of the cone and the reference surface
W to be attached to the outer circular face of the tested cone (1);
calculating the length L of the measured cone, namely respectively taking the center of a first dial indicator (3) and a second dial indicator (4) as a point A and a point B, when in process
measurement, the pointer of the first dial indicator (3) falls back to the point A, the pointer of the second dial indicator (4) falls back to the point B, and calculating the difference between
the two measuring heads H2-H1 through A, B two-point reading, wherein the dial indicators (3 and 4) are provided with two fixed mounting hole distances L2 in a measuring bracket (2), wherein the
second dial indicator (4) has a fixed distance L1 from the measuring reference end surface, when in measurement, a large right triangle abc and a small right triangle def are formed, so that the
following geometric relationship and calculation formula are formed, namely the difference delta between the taper β of the measured cone and the reference point of the arctan [ (H2-H1)/L2] …
(a), the difference between the zero calibration reference point of the dial indicator gauge and the reference point of the measuring cone is substituted by the axial taper of the dial indicator
(H4642) and the measuring head (H4642), and the axial distance L of the measured cone are substituted by the difference between the measuring head H β and the measuring head H β, and the
measuring head (A, B B), and the fixed reference point L, and the axial difference between the measuring head 4619B are calculated by substituting the length of the measuring head of the
measuring jig (1).
2. The method for measuring the length of the cone by the double-dial gauge on line according to claim 1 is characterized in that the bottom end of the length L of the measured cone is taken as a
vertical line bc, a line segment ac is constructed by crossing the initial point a of the L, the vertical line bc is connected, a large right triangle △ abc is obtained, the large right triangle
△ abc is cut by taking the projection distance of the two dial gauges (3 and 4) on the horizontal plane as a vertical line for two mounting hole distances L2, the distance H2 from the zero point
of the dial gauge (4) to the measured cone is obtained, the tail end of the L2 is taken as a vertical line to intersect at a point d, the tail point is taken as a vertical line ef by fixing the
length of the L1 by the two dial gauges (3 and 4), and the distance H1 from the zero point to the measured cone of the small right triangle △ def and the conical surface of.
3. The method for measuring the length of the cone through the double-table online manner as claimed in claim 2 is characterized in that the base de and the right-angle edge ef in the small
right-angle triangle △ def are parallel to the base ab and the right-angle edge bc in the large right-angle triangle △ abc, the right-angle edge ef in the small right-angle triangle △ def is
obtained by the difference of two percentage tables (3 and 4) to be H2-H1, the included angle β can be obtained by the known right-angle edge ef and the known hypotenuse dc in the small
right-angle triangle △ def, the two triangular characteristics have the same apex angle value, the apex angle a is equal to the included angle β, and the value of the measured cone length L is
deduced by the sum of the fixed distance L1, the distance L2 between the two mounting holes and H1/tan β.
4. The method of dual-gauge online measurement of taper length of claim 1, wherein: in order to obtain the height difference of measuring heads of the two dial indicators, calculating the total
length of the measured cone, and obtaining the difference value delta of the two dial indicators which are perpendicular to the measuring support (2) in space according to the gauge diameters d
of the two dial indicators:
and the axial length of the tested cone is L1+ L2+ (H1+ delta)/tan β … … … … … … … … … … … … … (c).
5. The method of dual-gauge online measurement of taper length of claim 1, wherein: the dial indicator with the two gauge diameters d is inserted into the two fixing holes of the measuring
bracket (2), the special test head is kept exposed out of the reference plane W2-3 mm in height H, and the locking screw (5) is fastened, so that the surface of the dial indicator is opposite to
an observer; placing the measuring support (2) on a detection platform or a reference platform, rotating the two dial indicator dials to enable the two indicator pointers to be respectively
aligned to the O points, and completing the pre-measurement zero calibration; and then, placing the test device after zero calibration on the cone surface of the tested cone (1) according to a
graph (5), and enabling the right end surface H of the measurement support to be attached to the right end of the cone and the reference surface W to be attached to the outer circular surface of
the tested cone (1) for process measurement.
CN201910991410.XA 2019-10-18 2019-10-18 Method for measuring length of cone on line by double meters Pending CN110779418A (en)
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Application Number Priority Date Filing Date Title
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Cited By (2)
* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN112033324A (en) * 2020-08-12 2020-12-04 江南工业集团有限公司 Detection method for wall thickness and wall thickness difference of double cones
CN114184103A (en) * 2020-09-14 2022-03-15 河南北方红阳机电有限公司 Method for detecting sizes of points of multi-section connecting cone by adopting AutoCAD software
Citations (7)
* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
RO101697A2 (en) * 1988-12-23 1991-12-09 Institutul Politehnic "Gheorghe Asachi", Iasi, Ro Radial rattlings control device for circular toothing
CN101701787A (en) * 2009-12-04 2010-05-05 中国石油天然气集团公司 Portable bend measuring instrument for measuring bending angle
CN103759612A (en) * 2014-01-29 2014-04-30 广西玉柴机器股份有限公司 Transmission spline taper hole measurement gauge
CN206321169U (en) * 2016-12-14 2017-07-11 江门市力泰科技有限公司 A kind of length comprehensive check tool
CN207423054U (en) * 2017-11-06 2018-05-29 无锡鹰贝精密轴承有限公司 Spool axial dimension rapid detection tool
CN207894355U (en) * 2017-12-25 2018-09-21 浙江杰特工贸股份有限公司 Adaptive type cone rotor taper measurement detection device
CN208223334U (en) * 2018-05-11 2018-12-11 平湖市宇达精密机械有限公司 Synchronization length detection device
Patent Citations (7)
* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
RO101697A2 (en) * 1988-12-23 1991-12-09 Institutul Politehnic "Gheorghe Asachi", Iasi, Ro Radial rattlings control device for circular toothing
CN101701787A (en) * 2009-12-04 2010-05-05 中国石油天然气集团公司 Portable bend measuring instrument for measuring bending angle
CN103759612A (en) * 2014-01-29 2014-04-30 广西玉柴机器股份有限公司 Transmission spline taper hole measurement gauge
CN206321169U (en) * 2016-12-14 2017-07-11 江门市力泰科技有限公司 A kind of length comprehensive check tool
CN207423054U (en) * 2017-11-06 2018-05-29 无锡鹰贝精密轴承有限公司 Spool axial dimension rapid detection tool
CN207894355U (en) * 2017-12-25 2018-09-21 浙江杰特工贸股份有限公司 Adaptive type cone rotor taper measurement detection device
CN208223334U (en) * 2018-05-11 2018-12-11 平湖市宇达精密机械有限公司 Synchronization length detection device
Cited By (3)
* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN112033324A (en) * 2020-08-12 2020-12-04 江南工业集团有限公司 Detection method for wall thickness and wall thickness difference of double cones
CN112033324B (en) * 2020-08-12 2021-09-21 江南工业集团有限公司 Detection method for wall thickness and wall thickness difference of double cones
CN114184103A (en) * 2020-09-14 2022-03-15 河南北方红阳机电有限公司 Method for detecting sizes of points of multi-section connecting cone by adopting AutoCAD software
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Knitting Pattern
Problem K
Knitting Pattern
Languages en is
Jörmunrekur had found himself with some extra time on his hands, so he decided to try to find a new hobby. After discussing this with some of his relatives, his grandparents lent him a book with
knitting guides and knitting patterns.
He wants to start with something big, so he decides to make a sweater. He has also picked out a pattern from the book that he will repeat around the circumference of the sweater. He wants the pattern
to be centered and then repeat out towards the back in either direction, but never wants to have less than the full pattern on the sweater. He will not place any patterns that leaves the placements
of patterns asymmetric. Now he has to know how much empty space he should leave at the back of the sweater to achieve this.
The empty space that is not covered by the patterns must be a contiguous (possibly empty) section at the back of the sweater.
The input contains two positive integers $N$, the length of the sweater, and $P$, the length of the pattern. They satisfy $1 \leq P \leq N \leq 10^{18}$ and they have the same parity, as otherwise
the pattern could never be perfectly centered. The input is all on one line, with the integers separated by a space.
Print a single integer, the amount of empty space left on the back of the sweater.
Explanation of samples
In the first sample the sweater is 13 loops in circumference. Thus the centered pattern is placed at loops 6, 7 and 8. There’s space for another pattern in either direction at loops 3, 4, 5 and 9,
10, 11. There’s not enough space to place two more, and a single pattern would make things asymmetric. Thus loops 1, 2, 12 and 13 are empty, so the answer is 4.
In the second sample the sweater is 16 loops in circumference. The first pattern is placed at loops 7, 8, 9 and 10. Two more are placed at 3, 4, 5, 6 and 11, 12, 13, 14. This leaves 1, 2, 15 and 16,
which exactly fits one more pattern that will be perfectly centered at the back of the sweater, creating no asymmetry. Thus that pattern is placed, leaving no empty space.
Sample Input 1 Sample Output 1
Sample Input 2 Sample Output 2 | {"url":"https://ncpc24.kattis.com/contests/ncpc24/problems/knittingpattern","timestamp":"2024-11-09T17:47:01Z","content_type":"text/html","content_length":"30929","record_id":"<urn:uuid:e9cc303e-b160-462c-8d52-983f20b0c1c5>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00086.warc.gz"} |
Markov Chain Generators - LLM's Single Celled Ancestor
Markov chain generators give you all the thrill of GPTs with none of the dumb math and engineering
One of my favorite pet interview questions for programmers is something I came up with after learning about Markov chain generators when I was a wee programmer back in the early 00s. I’d ask the
candidate if they were familiar with Markov chain generators (most were not), and then briefly explain the concept. Then I’d ask them to write one in pseudo-code.
In my defense, this was usually a bonus question I’d throw in toward the end of the technical part of an interview and only if the candidate blew through my more simple questions. I’d never ding them
for not getting it but a candidate’s answer to that question has pushed me from a ‘no’ to a ‘yes’ a handful of times. I’d also emphasize that this is a collaborative part of our interview and that
questions and discussion is encouraged. I helped a few candidates figure it out while working together.
A Markov chain generator: using a sample corpus of text, write a function that will generate a string of text using a Markov chain. A Markov chain “… is a stochastic model describing a sequence of
possible events in which the probability of each event depends only on the state attained in the previous event.”, or for those of us who only speak English, predict something (in this case a word)
based on the probability of it appearing in a sequence depending on previous inputs (words). Practically speaking, predict the next word in a given partial sentence based on sentences you’ve seen in
the past. Sounds familiar.
Ok, so that’s not a whole lot clearer, let’s consider an example:
Take the phrase “the cat and the dog and the bird sat on the mat on the deck” as our sample corpus. Let’s decide to create a bi-gram Markov chain generator. We’ll look at n-grams of length 2. I like
to think of it like a sliding window. If we consider this sentence as starting with a hidden ‘start’ token, the final corpus is “<s> the cat and the dog and the bird sat on the mat on the deck”.
Looking at each word whilst ‘remembering’ the last two words, we can calculate the probabilities of each word in the corpus appearing after each bi-gram. Starting at “cat” (so that we have a 2 word
history to build a bi-gram from), the previous bi-gram is “<s> the”. We calculate that the probability of “cat” appearing after “<s> the” is 100%. Sliding the window forward, we calculate that the
probability of “and” appearing after “the cat” is 100%. Consider “dog” - “dog” appears after “and the” 100% of the time. Skipping ahead to “bird”, we calculate that the probability of “bird”
appearing after “and the” is 50%. We’ve seen “dog” come after “and the”, and now we’ve seen “bird”. We also need to update the probability for the “dog” word to 50%, from 100%. Therefore in this
bi-gram Markov model, “and the” can produce “dog” half the time, and “bird” half the time.
That was a lot to explain something that’s honestly pretty simple and intuitive. The explanation usually goes pretty fast in person. Once they “get” it, we move onto implementation. I love this
question because it triple dips - I get to see a candidates algorithm and data structure chops, I get to see how they handle a novel problem and I get to see how they handle being abruptly thrown in
the deep end (with support). Once they either have a solution or run out of time, I compare notes and tell them about my implementation. I never ask questions I can’t answer myself.
These things are easy to implement in a dumb naïve way. Here’s some Python that does it:
import datasets
import random
import re
expr = r'[\-\,\.\:\'\?;\!]+'
d = datasets.load_dataset('tiny_shakespeare')['train']
d = d[0]['text'].split()
unigrams = {}
bigrams = {}
# build the model
for idx in range(2, len(d), 2):
unigram = re.sub(expr, '', d[idx-1].strip()).lower()
if unigram not in unigrams:
unigrams[unigram] = [re.sub(expr, '', d[idx].strip()).lower()]
unigrams[unigram].append(re.sub(expr, '', d[idx].strip()).lower())
bigram = (re.sub(expr, '', d[idx-2].strip()).lower(), re.sub(expr, '', d[idx-1].strip()).lower())
if bigram not in bigrams:
bigrams[bigram] = [re.sub(expr, '', d[idx].strip()).lower()]
bigrams[bigram].append(re.sub(expr, '', d[idx].strip()).lower())
# generate some text
n = 20
sample = random.sample(list(bigrams.keys()), 1)[0]
text = ' '.join([] + list(sample) + random.sample(bigrams[sample], 1))
print('text:', text, end='')
for i in range(n):
prev_unigram = text.split()[-1]
prev_bigram = tuple(text.split()[-2:])
if prev_bigram in bigrams.keys():
word = random.sample(bigrams[prev_bigram], 1)[0]
elif prev_unigram in unigrams.keys():
word = random.sample(unigrams[prev_unigram], 1)[0]
print(" " + word)
text += " " + word
print(" " + word, end='')
text += " " + word
Drop that code into a Jupyter notebook and you’ll get some Shakespeare-ish text:
text: angelo thy faults richard ii norfolk for a horse and men the sweetest sleep when this stream of suspicion has wings of grasshoppers
Basically it’s building a bi-gram model from the “tiny_shakespeare” dataset on Huggingface with some basic text cleaning through regular expressions. It falls back to a single n-gram model if it
can’t find a generated bi-gram, and if it can’t find a word that follows the single n-gram (not sure why it happens, probably a bug) it terminates the generation.
Generating bi-grams that don’t exist in the corpus is a function both of the kind of corpus (very heterogenous sources), the amount of data, and finally just an inherent limitation in the Markov
model. It’s telling us there’s aspects of language that are hidden in the Markov model, and the examples it’s seen don’t cover all possible states of the outputs.
Still, it’s quick and the output seems reasonable…ish.
I’ve been researching deep learning and transformer models and they sure do remind me of these toy Markov models. Our probabilities are calculated from the training corpus in a single shot, whereas a
transformer model has probabilities that are calculated from the training corpus with noise added, and a gradient descent training loop. The Markov chain model can be thought of having a context
length of n where n is the size of our n-gram, compared to the 512 context length of BERT, for example. The Markov chain is a single layer disconnected network, compared to the multi-layer fully
connected transformers. If you squint enough you can see the underlying theme.
The dimensionality of a Markov model is orders of magnitude smaller than a transformer model, which makes it much easier to see what’s going on. The probabilities are directly tied to the data and
you can easily look up bi-grams and their associated following word probabilities. Once you grasp Markov models, transformer models are less intimidating. They’re just bigger Markov models on | {"url":"https://www.timgittos.com/p/markov-chain-generators-llms-single","timestamp":"2024-11-11T11:26:54Z","content_type":"text/html","content_length":"110149","record_id":"<urn:uuid:c3819e37-ffb8-43a0-ad73-dc37aeb95e4b>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00711.warc.gz"} |
Are rules of replacement rules of implication?
Are rules of replacement rules of implication?
Implication rules are valid argument forms that are validly applied only to an entire line. Replacement rules are pairs of logically equivalent statement forms (they have identical truth tables) that
may replace each other within the context of a proof.
What are the parts of an implication?
The statement p in an implication p⇒q is called its hypothesis, premise, or antecedent, and q the conclusion or consequence.
What are the properties of implication?
Implication has two properties which resemble the reflexive and transitive properties of equality. One, p=> p, is called a “tautology.” Tautologies, although widely used, do not add much to
understanding. “Why is the water salty?” asks the little boy. “Because ocean water is salty,” says his father.
What is the difference between rules of inference and rules of replacement?
The main difference is that rules of inference are forms of valid arguments (that’s why they have a therefore ∴ symbol), but rules of replacement are forms of equivalent propositions (which is why
they have the equivalence sign ≡ between the two parts).
How many rules are rule of replacement according to Copi?
So long as each step is justified by reference to an earlier step (or steps) in the proof and to one of the nineteen rules, it must be a valid derivation. Next, let’s work with the third premise a
bit: 1. A ∨ (B • ~C) premise 2.
How many rules of replacement are there?
We have ten such rules, which are called the rules of replacement. The difference between these two sets of rules is that the rules of inference are themselves inferences whereas rules of replacement
are not. However, the rules of replacement are restricted to change or change in the form of statements.
Why is denying the consequent valid?
Modus ponens is a valid argument form in Western philosophy because the truth of the premises guarantees the truth of the conclusion; however, affirming the consequent is an invalid argument form
because the truth of the premises does not guarantee the truth of the conclusion. | {"url":"https://blackestfest.com/are-rules-of-replacement-rules-of-implication/","timestamp":"2024-11-04T18:19:05Z","content_type":"text/html","content_length":"47062","record_id":"<urn:uuid:4dfab54a-d327-422b-bbf5-505f0c958f91>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00143.warc.gz"} |
The Two-Eyes Lemma: A Linking Problem for Table-Top Necklaces
In this note, we answer a combinatorial question that is inspired by cusp geometry of hyperbolic 3-manifolds. A table-top necklace is a collection of sequentially tangent beads (i.e. spheres) with
disjoint interiors lying on a flat table (i.e. a plane) such that each bead is of diameter at most one and is tangent to the table. We analyze the possible configurations of a necklace with at most 8
beads linking around two other spheres whose diameter is exactly 1. We show that all the beads are forced to have diameter one, the two linked spheres are tangent, and that each bead must be tangent
to at least one of the two linked spheres. In fact, there is a 1-parameter family of distinct configurations.
All Science Journal Classification (ASJC) codes
• Theoretical Computer Science
• Discrete Mathematics and Combinatorics
• Horoball
• Hyperbolic
• Packing
• Spheres
Dive into the research topics of 'The Two-Eyes Lemma: A Linking Problem for Table-Top Necklaces'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/the-two-eyes-lemma-a-linking-problem-for-table-top-necklaces","timestamp":"2024-11-02T12:00:08Z","content_type":"text/html","content_length":"47524","record_id":"<urn:uuid:8fd6e31c-f7e1-4d94-b919-cc03502c060e>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00559.warc.gz"} |
Sir, there's a cat in your mirror dimension
Pets do the darndest things, especially if you teach them a bit of math.
A while back, we talked about the frequency domain: a clever reinterpretation of everyday signals that translates them into the amplitudes of constituent waveforms. The most common basis for this
operation are sine waves running at increasing frequencies, but countless other waveforms can be used to create a number of alternative frequency domains.
In that earlier article, I also noted two important properties of frequency domain transforms. First, they are reversible: you can recover the original (“time domain” or “spatial domain”) data from
its frequency image. Second, the transforms have input-output symmetry: the same mathematical operation is used to go both ways. In effect, we have a lever that takes us to a mirror dimension and
back. Which of the lever positions is called home is a matter of habit, not math.
Of course, in real life, the distinction matters — and it’s particularly important for compression. If you take an image, convert it to the frequency-domain representation, and then reduce the
precision of (or outright obliterate!) the high-frequency components, the resulting image still looks perceptually the same — but you now have much less data to transmit or store:
This makes you wonder: if the frequency-domain representation of a typical image looks like diffuse noise, if most of it is perceptually unimportant, and if the transform is just a lever that takes
us back and forth between two functionally-equivalent dimensions… could we start calling that mirror dimension home and move some stuff in?
To answer this stoner question, I grabbed a photo of a cat and then calculated its frequency-domain form with the discrete cosine transform (DCT):
Next, I reused the photo of a woman from an earlier example and placed the mirror-dimension “cat noise” pattern over it, dialing down opacity to minimize visible artifacts:
The compositing operation is necessarily lossy, but my theory was that if the composite image is run through DCT to compute its frequency-domain representation, the photo of a woman would be
decomposed to fairly uniform noise, perhaps easy to attenuate with a gentle blur; while the injected “cat noise” would coalesce into a perceptible image of a cat.
But would it?… Yes!
If you want to see for yourself, download the composite image and have fun. In MATLAB, you can do the following:
woman = imread("woman-with-cat.png");
imagesc(woman, [0 255]);
cat = dct2(woman);
imagesc(imgaussfilt(cat, 1), [-4 4]);
Interestingly, the kitty survives resizing of the host document. Upscaling tiles the image; downscaling truncates it.
My lingering question was how badly the cat would get mangled by lossy compression; as it turns out, the impact is less than I expected. At higher JPEG quality settings, the image looks quite OK. As
the quality setting is lowered, the bottom right quadrant — corresponding to higher-frequency components — gets badly quantized:
This visualization offers a fascinating glimpse of just how much information is destroyed by the JPEG algorithm — mostly without us noticing.
There’s plenty of prior art for using audio spectrograms for hidden messages, and some discussion of text steganography piggybacked on top of JPEG DCT coefficients. My point isn’t that the technique
is particularly useful or that it has absolutely no precedent. It’s just that the frequency domain and the time domain are coupled together in funny ways.
If you liked this article, please subscribe! Unlike most other social media, Substack is not a walled garden and not an addictive doomscrolling experience. It’s just a way to stay in touch with the
writers you like.
For more articles about electronics, algorithms, snowplowing, and 19th century repeating pistols, see this categorized list.
Bonus content: the deterioration of a "standalone" frequency-domain cat for various JPEG quality settings:
Expand full comment
I had some fun a while back playing with the phase information: https://www.brainonfire.net/blog/2022/04/28/fourier-image-experiments/
I'm still not exactly sure how the API calls I was making relate to the Fourier Transform I learned briefly in school; in particular, I'm a little unclear on how the 2D image is processed. It looks
like one dimension is processed first, then the other, so you get anisotropic effects.
Expand full comment
3 more comments... | {"url":"https://lcamtuf.substack.com/p/sir-theres-a-cat-in-your-mirror-dimension","timestamp":"2024-11-15T03:02:21Z","content_type":"text/html","content_length":"187304","record_id":"<urn:uuid:9da008a5-c0fa-44e3-a1f4-b27f260544f6>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00843.warc.gz"} |
Zero-forcing equalization
[out,csi] = lteEqualizeZF(rxgrid,channelest) returns equalized data in multidimensional array, out, by applying MIMO zero-forcing equalization to the received data resource grid in matrix rxgrid,
using the channel information in the channelest input matrix.
For each resource element, the function calculates the pseudoinverse of the channel and equalizes the corresponding received signal.
Alternatively, the channelest input can be provided as a 3-D array of size NRE-by-NRxAnts-by-P and the rxgrid input can be provided as a matrix of size NRE-by-NRxAnts. In this case, the first two
dimensions have been reduced to one dimension by appropriate indexing through the frequency and time locations of the resource elements of interest, typically for a single physical channel. The
outputs, out and csi, are of size (N × M)-by-P.
Perform Zero-Forcing Equalization for RMC R.4
Equalize the received signal for RMC R.4 after channel estimation. Use the zero forcing equalizer.
Create cell-wide configuration structure and generate transmit signal. Configure propagation channel.
enb = lteRMCDL('R.4');
[txSignal,~,info] = lteRMCDLTool(enb,[1;0;0;1]);
chcfg.DelayProfile = 'EPA';
chcfg.NRxAnts = 1;
chcfg.DopplerFreq = 70;
chcfg.MIMOCorrelation = 'Low';
chcfg.SamplingRate = info.SamplingRate;
chcfg.Seed = 1;
chcfg.InitPhase = 'Random';
chcfg.InitTime = 0;
txSignal = [txSignal; zeros(15,1)];
N = length(txSignal);
noise = 1e-3*complex(randn(N,chcfg.NRxAnts),randn(N,chcfg.NRxAnts));
rxSignal = lteFadingChannel(chcfg,txSignal)+noise;
Perform synchronization and OFDM demodulation.
offset = lteDLFrameOffset(enb,rxSignal);
rxGrid = lteOFDMDemodulate(enb,rxSignal(1+offset:end,:));
Create channel estimation configuration structure and perform channel estimation.
cec.FreqWindow = 9;
cec.TimeWindow = 9;
cec.InterpType = 'Cubic';
cec.PilotAverage = 'UserDefined';
cec.InterpWinSize = 3;
cec.InterpWindow = 'Causal';
hest = lteDLChannelEstimate(enb,cec,rxGrid);
Equalize and plot received and equalized grids.
eqGrid = lteEqualizeZF(rxGrid,hest);
title('Received grid');
xlabel('OFDM symbol');
title('Equalized grid');
xlabel('OFDM symbol');
Input Arguments
rxgrid — Received data resource grid
3-D numeric array | 2-D numeric matrix
Received data resource grid, specified as a 3-D numeric array or a 2-D numeric matrix. As a 3-D numeric array, it has size N-by-M-by-NRxAnts, where N is the number of subcarriers, M is the number of
OFDM symbols, and NRxAnts is the number of receive antennas.
Alternatively, as a 2-D numeric matrix, it has size NRE-by-NRxAnts. In this case, the first two dimensions have been reduced to one dimension by appropriate indexing through the frequency and time
locations of the resource elements of interest, typically for a single physical channel.
Data Types: double
Complex Number Support: Yes
channelest — Channel information
4-D numeric array | 3-D numeric array
Channel information, specified as a 4-D numeric array or a 3-D numeric array. As a 4-D numeric array, it has size N-by-M-by-NRxAnts-by-P. N is the number of subcarriers, M is the number of OFDM
symbols, NRxAnts is the number of receive antennas, and P is the number of transmit antennas. Each element is a complex number representing the narrowband channel for each resource element and for
each link between transmit and receive antennas. This matrix can be obtained using a channel estimation function, such as lteDLChannelEstimate.
Alternatively, as a 3-D numeric array, it has size NRE-by-NRxAnts-by-P. In this case, the first two dimensions have been reduced to one dimension by appropriate indexing through the frequency and
time locations of the resource elements of interest, typically for a single physical channel.
Data Types: double
Complex Number Support: Yes
Output Arguments
out — Equalized output data
3-D numeric array | 2-D numeric matrix
Equalized output data, returned as a 3-D numeric array or a 2-D numeric matrix. As a 3-D numeric array, it has size N-by-M-by-P. N is the number of subcarriers, M is the number of OFDM symbols, and P
is the number of transmit antennas.
Alternatively, if channelest is provided as a 3-D array, out is a 2-D numeric matrix of size (N × M)-by-P. In this case, the first two dimensions have been reduced to one dimension by appropriate
indexing through the frequency and time locations of the resource elements of interest, typically for a single physical channel.
Data Types: double
Complex Number Support: Yes
csi — Soft channel state information
3-D numeric array | 2-D numeric matrix
Soft channel state information, returned as a 3-D numeric array or a 2-D numeric matrix of the same size as out. As a 3-D numeric array, it has size N-by-M-by-P. N is the number of subcarriers, M is
the number of OFDM symbols, and P is the number of transmit antennas. csi provides an estimate of the received RE gain for each received RE.
Alternatively, if channelest is provided as a 3-D array, csi is a 2-D numeric matrix of size (N×M)-by-P. In this case, the first two dimensions have been reduced to one dimension by appropriate
indexing through the frequency and time locations of the resource elements of interest, typically for a single physical channel.
Data Types: double
Version History
Introduced in R2014a | {"url":"https://kr.mathworks.com/help/lte/ref/lteequalizezf.html","timestamp":"2024-11-12T15:35:07Z","content_type":"text/html","content_length":"86221","record_id":"<urn:uuid:65c3f705-2167-44a1-b11f-bc7737cdae1f>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00439.warc.gz"} |
Understanding Stokes Law: Viscosity and Particle Size -
Understanding Stokes Law: Viscosity and Particle Size
Stokes Law, formulated by Irish physicist George Gabriel Stokes, is a fundamental principle in fluid dynamics that describes the behavior of small particles suspended in a fluid. This law provides a
mathematical equation to calculate the drag force experienced by particles as they settle through a fluid under gravitational influence. Stokes Law is essential for understanding the behavior of
colloidal suspensions, emulsions, and other mixtures where particles are dispersed in a liquid medium.
The law finds widespread application in various industries, including pharmaceuticals, food and beverage production, environmental science, and materials science. Engineers, scientists, and
researchers working in fluid dynamics and particle behavior rely on Stokes Law for their studies and applications. The principles underlying this law are crucial for advancing research and
development in fields related to particle-fluid interactions and sedimentation processes.
Key Takeaways
• Stokes Law describes the behavior of a small sphere moving through a viscous fluid at low Reynolds numbers
• Viscosity is a measure of a fluid’s resistance to deformation and is a key concept in understanding Stokes Law
• Particle size plays a crucial role in determining the settling velocity of particles in a fluid, as per Stokes Law
• Stokes Law is widely used in various industries such as pharmaceuticals, environmental science, and chemical engineering
• Factors such as temperature, pressure, and particle shape can affect viscosity and particle size, impacting the application of Stokes Law
The Concept of Viscosity
Understanding Viscosity
In simple terms, high viscosity fluids are thick and flow slowly, while low viscosity fluids are thin and flow quickly. Viscosity is influenced by factors such as temperature, pressure, and the
composition of the fluid.
Viscosity and Particle Movement
In the context of Stokes Law, viscosity plays a significant role in determining the drag force experienced by particles as they settle through a fluid. The higher the viscosity of the fluid, the
greater the resistance to particle movement, resulting in a higher drag force according to Stokes Law.
Factors Affecting Viscosity
Viscosity is influenced by various factors, including temperature, pressure, and the composition of the fluid. These factors can significantly impact the behavior of fluids under different
Understanding Particle Size
Particle size refers to the dimensions of individual particles in a material or substance. In the context of Stokes Law, particle size is a critical factor in determining the settling velocity and
drag force experienced by particles in a fluid. Smaller particles have a larger surface area relative to their mass, which results in higher drag forces as they settle through the fluid.
Understanding particle size distribution is essential in various industries, including pharmaceuticals, cosmetics, and environmental science, where the behavior of particles in suspension or emulsion
is of utmost importance. Particle size analysis techniques such as laser diffraction, sedimentation, and microscopy are commonly used to characterize and measure the size distribution of particles in
a sample.
Stokes Law and its Application
Property Description
Stokes Law An equation that describes the behavior of a small sphere falling through a viscous fluid.
Viscous Fluid A fluid that resists flow, such as oil or honey.
Terminal Velocity The maximum velocity reached by an object falling through a fluid, as described by Stokes Law.
Application Used to calculate the settling velocity of particles in fluids, such as in sedimentation processes in water treatment.
Stokes Law provides a mathematical equation to calculate the drag force experienced by small particles as they settle through a fluid under the influence of gravity. The equation is given by Fd =
3πμdv, where Fd is the drag force, μ is the viscosity of the fluid, d is the diameter of the particle, and v is the settling velocity of the particle. This equation allows engineers and scientists to
predict the behavior of particles in suspension or emulsion and design processes and equipment accordingly.
Stokes Law finds applications in various industries such as pharmaceuticals, where it is used to understand the behavior of drug particles in liquid formulations; environmental science, where it
helps in studying sedimentation and settling processes in water treatment; and materials science, where it is used to characterize colloidal suspensions and emulsions. Stokes Law is also applied in
the field of fluid dynamics to understand the behavior of fluids with suspended particles. It helps in predicting the settling rates of particles in sedimentation tanks, designing filtration systems
for separating particles from fluids, and optimizing mixing processes in industrial applications.
Additionally, Stokes Law is used in research and development to study the behavior of nanoparticles and microparticles in various fluids for applications such as drug delivery systems,
nanotechnology, and advanced materials.
Factors Affecting Viscosity and Particle Size
Several factors influence the viscosity of a fluid, including temperature, pressure, and composition. As temperature increases, the viscosity of most fluids decreases due to reduced intermolecular
forces and increased molecular motion. Pressure also affects viscosity, especially in gases, where high pressure can increase viscosity due to closer molecular packing.
The composition of a fluid, including its molecular structure and interactions between molecules, also plays a significant role in determining its viscosity. Particle size is influenced by various
factors such as the method of particle formation, processing conditions, and environmental factors. For example, particles formed through precipitation methods may have different size distributions
compared to those produced through spray drying or milling processes.
Processing conditions such as temperature, pressure, and agitation can also affect particle size distribution. Environmental factors such as humidity and air flow can impact the agglomeration and
dispersion of particles, leading to changes in their size distribution.
Importance of Stokes Law in Various Industries
Pharmaceutical Applications
Stokes Law plays a vital role in the pharmaceutical industry, where it helps in designing drug formulations with optimal particle size distribution for improved bioavailability and stability. It also
aids in understanding the behavior of drug particles in suspension or emulsion, which is essential for drug delivery systems.
Environmental Science Applications
In environmental science, Stokes Law is used to study sedimentation processes in water treatment plants and design efficient separation systems for removing suspended solids from wastewater. It also
helps in understanding the behavior of pollutants and contaminants in natural water bodies, which is essential for environmental monitoring and remediation efforts.
Materials Science Applications
In materials science, Stokes Law is applied to characterize colloidal suspensions and emulsions used in various products such as paints, cosmetics, and food products. It helps in optimizing
formulations for desired rheological properties and stability.
Practical Examples and Experiments Demonstrating Stokes Law
One practical example demonstrating Stokes Law is the settling of solid particles in a liquid medium under the influence of gravity. By measuring the settling velocity of particles of different sizes
and calculating the drag force using Stokes Law, scientists and engineers can validate the applicability of the law in predicting particle behavior. Another experiment involves measuring the
viscosity of different fluids at varying temperatures and pressures to observe how viscosity changes with these factors.
By applying Stokes Law to calculate drag forces on particles settling through these fluids, researchers can further understand the relationship between viscosity and particle behavior. In conclusion,
Stokes Law plays a crucial role in understanding the behavior of particles in suspension or emulsion in various industries. Its application in predicting settling velocities and drag forces is
essential for designing processes and equipment for efficient separation and mixing of fluids with suspended particles.
Understanding factors affecting viscosity and particle size distribution is key to applying Stokes Law effectively in practical applications. Practical examples and experiments demonstrating Stokes
Law further validate its significance in fluid dynamics and particle behavior analysis.
What is Stokes Law?
Stokes Law is a formula that describes the force of viscosity acting on a spherical object moving through a fluid. It was derived by Sir George Gabriel Stokes in the 19th century.
What does Stokes Law explain?
Stokes Law explains the relationship between the viscosity of a fluid, the size of a spherical object moving through the fluid, and the velocity at which the object moves.
What is the formula for Stokes Law?
The formula for Stokes Law is F = 6πηrv, where F is the force of viscosity, η is the viscosity of the fluid, r is the radius of the spherical object, and v is the velocity of the object.
What are the applications of Stokes Law?
Stokes Law is used in various fields such as fluid dynamics, geology, biology, and engineering. It is particularly useful in understanding the behavior of small particles in fluids, such as in
sedimentation and particle motion.
What are the limitations of Stokes Law?
Stokes Law is only applicable to small, spherical particles moving at low Reynolds numbers in a viscous fluid. It does not accurately describe the behavior of larger or non-spherical particles, or
particles moving at high velocities. | {"url":"https://mibooks.in/understanding-stokes-law-viscosity-and-particle-size/","timestamp":"2024-11-09T23:58:42Z","content_type":"text/html","content_length":"64172","record_id":"<urn:uuid:9ed44cc2-674c-442f-b871-63e6235dfe07>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00453.warc.gz"} |
PhET Simulation: Motion in 2D
Detail Page
published by the PhET
Available Languages: English, Spanish
This is an interactive simulation created to help beginners differentiate velocity and acceleration vectors. The user can move a ball with the mouse or let the simulation move the ball in
four modes of motion (two types of linear, simple harmonic, and circular). Two vectors are displayed -- one green and one blue. As the motion of the ball changes, the vectors also change.
Which color represents velocity and which acceleration?
This item is part of a larger and growing collection of resources developed by the Physics Education Technology project (PhET), each designed to implement principles of physics education
Please note that this resource requires Java Applet Plug-in.
Editor's Note: This simulation was designed with improvements based on research of student interaction with the PhET resource "Ladybug Revolution". The authors added two new features for
the beginning learner: linear acceleration and harmonic motion. To supplement the simulation, we recommend the Physics Classroom tutorial "Vectors and Direction" and the teacher-created
lesson, "Vectors Phet Lab" -- see links in Related Materials.
Subjects Levels Resource Types
Classical Mechanics - High School
- Instructional Material
- Motion in Two Dimensions - Lower Undergraduate
= Activity
= 2D Acceleration - Middle School
= Interactive Simulation
= 2D Velocity - Informal Education
Appropriate Courses Categories Ratings
- Physical Science
- Activity
- Physics First
- New teachers
- Conceptual Physics
Intended Users:
Access Rights:
Free access
© 2007 University of Colorado, Physics Education Technology
Additional information is available.
acceleration, circular motion, motion, simple harmonic motion, two-dimensional motion, vectors, velocity
Record Cloner:
Metadata instance created November 15, 2007 by Alea Smith
Record Updated:
August 18, 2016 by Lyle Barbato
Last Update
when Cataloged:
November 15, 2007
Other Collections:
Next Generation Science Standards
Crosscutting Concepts (K-12)
Patterns (K-12)
• Graphs, charts, and images can be used to identify patterns in data. (6-8)
NGSS Science and Engineering Practices (K-12)
Analyzing and Interpreting Data (K-12)
• Analyzing data in 9–12 builds on K–8 and progresses to introducing more detailed statistical analysis, the comparison of data sets for consistency, and the use of models to generate
and analyze data. (9-12)
□ Analyze data using computational models in order to make valid and reliable scientific claims. (9-12)
Developing and Using Models (K-12)
• Modeling in 9–12 builds on K–8 and progresses to using, synthesizing, and developing models to predict and show relationships among variables between systems and their components in
the natural and designed worlds. (9-12)
□ Use a model to provide mechanistic accounts of phenomena. (9-12)
Using Mathematics and Computational Thinking (5-12)
• Mathematical and computational thinking at the 9–12 level builds on K–8 and progresses to using algebraic thinking and analysis, a range of linear and nonlinear functions including
trigonometric functions, exponentials and logarithms, and computational tools for statistical analysis to analyze, represent, and model data. Simple computational simulations are
created and used based on mathematical models of basic assumptions. (9-12)
□ Use mathematical representations of phenomena to describe explanations. (9-12)
AAAS Benchmark Alignments (2008 Version)
4. The Physical Setting
4F. Motion
• 3-5: 4F/E1a. Changes in speed or direction of motion are caused by forces.
• 6-8: 4F/M3a. An unbalanced force acting on an object changes its speed or direction of motion, or both.
11. Common Themes
11B. Models
• 6-8: 11B/M4. Simulations are often useful in modeling events and processes.
Common Core State Standards for Mathematics Alignments Materials
High School — Number and Quantity (9-12) Similar
Vector and Matrix Quantities (9-12)
• N-VM.1 (+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their
magnitudes (e.g., v, |v|, ||v||, v).
(Author: Tom Henderson)
As instructors, we may forget that certain representations (like vector arrows) seem like a foreign language to beginning students. This thoughtfully-crafted tutorial introduces vector
diagrams in kid-friendly language and extends the learning to interactive practice problems with answers provided.
This resource is part of a Physics Front Topical Unit.
Kinematics: The Physics of Motion
Unit Title:
This very simple simulation can help beginners understand what vector arrows represent. It was designed by the PhET team to target specific areas of difficulty in student understanding of
vectors. Learners can move a ball with the mouse or let the simulation control the ball in four modes of motion (two types of linear, simple harmonic, and circular). Two vectors are
displayed -- one green and one blue. Which color represents velocity and which acceleration? This resource requires Java.
Link to Unit:
ComPADRE is beta testing Citation Styles!
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@misc{ Title = {PhET Simulation: Motion in 2D}, Publisher = {PhET}, Volume = {2024}, Number = {5 November 2024}, Month = {November 15, 2007}, Year = {2007} }
%T PhET Simulation: Motion in 2D %D November 15, 2007 %I PhET %C Boulder %U https://phet.colorado.edu/en/simulation/motion-2d %O application/java
%0 Electronic Source %D November 15, 2007 %T PhET Simulation: Motion in 2D %I PhET %V 2024 %N 5 November 2024 %8 November 15, 2007 %9 application/java %U https://phet.colorado.edu/en/
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PhET Simulation: Motion in 2D:
Is Required By
An editor-recommended virtual lab, authored by a high school teacher specifically for use with the Motion in 2D simulation.
relation by Caroline Hall
See details...
Know of another related resource? Login to relate this resource to it. | {"url":"https://www.compadre.org/Precollege/items/detail.cfm?ID=6095","timestamp":"2024-11-05T05:52:28Z","content_type":"application/xhtml+xml","content_length":"53091","record_id":"<urn:uuid:fc07c69d-8e32-4275-affb-e02da74112fc>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00575.warc.gz"} |
Model of a perfect double axel jump in figure skating | Mathematics AA SL's Sample Internal Assessment | Nail IB®
These two functions have x-intercepts x[1](0,0), x[2](3,0) in the domain 0 ≤ x ≤ 3 and vertex V(1.5, 1.5) at the same coordinates. The value k = 1.5 is random, but it does not affect the result since
both functions have a vertex at this point. The real second vertex coordinate will be calculated later, because I assumed that the height of the function would be very high and stretched, so the
graphic differences between the two functions would not be noticeable. Fitting functions to the same principal coordinates graphically shows the difference between these functions. Various variables
will be checked in the study – including the angle and height of the jump. For this purpose, graph with the greatest area of the function should be chosen. The results have been rounded to 3
significant figures. The functions were matched so that the x-intercepts are at the coordinates x[1] = (0,0) and x[2] = (3,0), have a common axis of symmetry (h = 1.5) and a common height (k = 1.5).
This matching allows to explore which function is best suited for the study. Calculation were done with GDC.
Quadratic function f(x) = −0.68(x − 1.5)^2 + 1.5
= −6.12 + 9.18 − 0.09
= 2.97 cm^2
Sine function g(x) = 1.5 sin(1.047x)
^\(1.5\int (\sin(1.047x))dx\)
^\(=1.5×\frac{1}{1.047}\displaystyle\int^{3.141}_0\sin(u)du \)
\(= 1.5 × 0.95510 ...\bigg [−\cos(u)\bigg]^{ 3.141}_ 0\)
\(= 1.43266 ...\bigg [−\cos(u)\bigg]^{ 3.141}_ 0\)
= 1.43266 ... × 1.9(9)
≈ 2.86532
≈ 2.87 cm^2
2.87 cm^2 < 2.97 cm^2
The result and the graph clearly show that the quadratic function is more accurate and efficient, so further calculations will be based on this formula.
To calculate the real second vertex coordinate of the parabola, I calculated the average height of the jump. The data was collected from videos, from official measurements that are displayed on the
screen after the jump and were rounded to an integer, as shown in Figure 4. | {"url":"https://nailib.com/user/ib-resources/ib-math-aa-sl/ia-sample/6667c381b4ef42b6e3b02867","timestamp":"2024-11-04T05:42:09Z","content_type":"text/html","content_length":"198256","record_id":"<urn:uuid:1c9986e9-579d-466e-afa3-93670635f929>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00595.warc.gz"} |
Martin L. Demaine
Paper by Martin L. Demaine
Erik D. Demaine, Martin L. Demaine, Yair N. Minsky, Joseph S. B. Mitchell, Ronald L. Rivest, and Mihai Pǎtraşcu, “Picture-Hanging Puzzles”, in Proceedings of the 6th International Conference on
Fun with Algorithms (FUN 2012), Lecture Notes in Computer Science, Venice, Italy, June 4–6, 2012, pages 81–93.
We show how to hang a picture by wrapping rope around n nails, making a polynomial number of twists, such that the picture falls whenever any k out of the n nails get removed, and the picture
remains hanging when fewer than k nails get removed. This construction makes for some fun mathematical magic performances. More generally, we characterize the possible Boolean functions
characterizing when the picture falls in terms of which nails get removed as all monotone Boolean functions. This construction requires an exponential number of twists in the worst case, but
exponential complexity is almost always necessary for general functions.
The full paper is available as arXiv.org:1203.3602 of the Computing Research Repository (CoRR).
Open Problem 1 was in fact previously solved by Gartside and Greenwood's paper "Brunnian links" (2007). The length of the shortest solution to the 1-out-of-n puzzle is Θ(n^2); in fact, the exact
bound matches the 2002 Chris Lusby Taylor construction we present.
Copyright held by the authors.
The paper is 12 pages.
The paper is available in PDF (583k).
Related papers:
PictureHanging_TOCS (Picture-Hanging Puzzles)
See also other papers by Martin Demaine. These pages are generated automagically from a BibTeX file.
Last updated November 17, 2022 by Martin Demaine. | {"url":"https://martindemaine.org/papers/PictureHanging_FUN2012/","timestamp":"2024-11-11T03:38:59Z","content_type":"text/html","content_length":"4763","record_id":"<urn:uuid:12b60594-2ff4-4d71-b7e3-08f1641bd176>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00827.warc.gz"} |
Arranging Hat
Problem A
Arranging Hat
The Arranging Hat. Image by Lisa Abose
Arranging Hat
is a cushy job indeed; high impact work, absolute authority, and 364 days of holiday every year. However, the hat has decided that it can do even better—it would like very much to become a tenured
Recently the hat has been reading computer science papers in its ample spare time, and of course, being an arranging hat, it is particularly interested in learning more about sorting algorithms.
The hat’s new contribution is to a class of algorithms known as lossy sorting algorithms. These usually work by removing some of the input elements in order to make it easier to sort the input (e.g.,
the Dropsort algorithm), instead of sorting all the input.
The hat is going to go one better—it is going to invent a lossy sorting algorithm for numbers that does not remove any input numbers and even keeps them in their original place, but instead changes
some of the digits in the numbers to make the list sorted.
The lossiness of the sorting operation depends on how many digits are changed. What is the smallest number of digits that need to be changed in one such list of numbers, to ensure that it is sorted?
The input consists of:
• one line containing the integers $n$ and $m$ ($1 \le n \le 40$, $1 \le m \le 400$), the number of numbers and the number of digits in each number, respectively.
• $n$ lines each containing an integer $v$ ($0 \le v < 10^{m}$). The numbers are zero-padded to exactly $m$ digits.
Write a sorted version of the array, after making a minimum number of digit changes to make the numbers sorted (the numbers must remain zero-padded to $m$ digits). If there are multiple optimal
solutions, you may give any of them.
Sample Input 1 Sample Output 1
Sample Input 2 Sample Output 2 | {"url":"https://open.kattis.com/contests/ncmztg/problems/arranginghat","timestamp":"2024-11-11T04:36:08Z","content_type":"text/html","content_length":"31620","record_id":"<urn:uuid:ceeb2ed9-1d87-46f6-bd20-817713b7ae83>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00155.warc.gz"} |
in Zeuthen for a Linear Collider
Physics > Electroweak studies
Electroweak studies
High precision studies of electroweak processes offer a window to physics at higher energy scales than the one available at the ILC. The sensitivity to the new physics comes either via higher order
loop corrections that are usually suppressed by the coupling constant or the new processes contribute directly to the process under study, but because of the high mass of the new particles they are
suppressed by s/M^2, where s is the squared centre of mass energy and M the mass of the new particles.
Mainly three types of processes were studied in this context:
To all three points significant contributions to the TESLA TDR and ILC RDR have been produced in Zeuthen. Special emphazis was put to studies which exploit the posibility of polarized beams.
Studies of W-pair production in electron-positron and photon-photon collisions
In case that no light Higgs exists electroweak interactions amongst gauge bosons become strong at the TeV scale, eventually violating unitarity. In this case deviations from the Standard Model
predictions in W-pair productions should be visible at ILC. In the figure above a simulated e^+e^-→ννWW→ννqqqq event in the ILD is shown.
Studies of the properties of the Z-boson, running close to the peak of the Z-resonance (Giga-Z).
The usefulness of these studies has already been proven at LEP and SLD. From measurements of the decay rates of the Z and the effective weak mixing angle already in 1993 the mass of the top quark
could be predicted with high accuracy, prior to its discovery at the TEVATRON. At present these data make us believe that the Higgs boson is light and in the reach of ILC. ILC offers the possibility
to increase the LEP statistics by two orders of magnitude with polarised electron and positron beams. With this data sample the weak mixing angle could be measured a factor 10 better than now.
Depending on the scenario of electroweak symmetry breaking, realised in nature, with this precision parameters of supersymmetric theories can be constrained or a non-standard Higgs sector can be
Precision measurement of fermion pair production at high energies.
Away from the Z-resonance interactions mediated by new heavy particles are suppressed by a factor s/M^2. Effects of these particles may be seen if the precision of the cross sections and
distributions of fermion pairs is high enough. Possible effects of additional heavy Z-bosons or different types of models containing extra space dimensions have been studied. In all cases ILC is
sensitive to mass scales of several TeV, often higher than the LHC, presently being built at CERN. However, since the LHC measures masses of the new particles directly while ILC measures the
couplings of the new particles to fermion divided by the mass of the new particles only the combination of ILC and LHC can establish the model of new physics once deviations from the Standard Model
predictions are found.
• G. Moortgat-Pick et al., [POWER Collaboration],
Polarized positrons and electrons at the linear collider
Physics Reports 460 (2008) 131.
• A. Bartl, W. Majerotto, K. Mönig, A. N. Skachkova and N. B. Skachkov,
Pair production of scalar top quarks in e^+ e^- collisions at ILC,
arXiv:0804.2125 [hep-ph].
• A. Bartl, W. Majerotto, K. Mönig, A. N. Skachkova and N. B. Skachkov,
Pair production of scalar top quarks in polarized photon-photon collisions at ILC,
arXiv:0804.1700 [hep-ph]
• K. Mönig,
Physics issues on triggering,
Procedings to the International Linear Collider Workshop (LCWS06) 9-13 Mar 2006, Bangalore, India,
Pramana 69 (2007) 1207-1208.
• K. Mönig,
Detector issues for a photon collider,
Procedings to the International Linear Collider Workshop (LCWS06) 9-13 Mar 2006, Bangalore, India,
Pramana 69 (2007) 1181-1184.
• E. Accomando et al., Report of the workshop on CP Studies and Non-Standard Higgs Physics,
• K. Mönig,
Physics of electroweak gauge bosons,
In *Fujii, K. (ed.) et al.: Linear collider physics in the new Millennium* 291-329
• M. Beyer, W. Kilian, P. Krstonosic, K. Mönig, J. Reuter, E. Schmidt and H. Schröder,
Determination of new electroweak parameters at the ILC: Sensitivity to new physics,
Eur. Phys. J. C 48 (2006) 353 [arXiv:hep-ph/0604048].
• G. Weiglein et al., [LHC/LC Study Group],
Physics interplay of the LHC and the ILC,
Phys. Rept. 426 (2006) 47 [arXiv:hep-ph/0410364].
• G. Kribs, N. Okada, M. Perelstein and S. Riemann,
Beyond the Standard Model: Summary,
In the Proceedings of the 2005 International Linear Collider Physics and Detector Workshop and 2nd ILC Accelerator Workshop, Snowmass, Colorado, 14-27 Aug 2005.
eConf C0508141 (2005) ALCPG0101.
• S. Riemann,
Z' signals from Kaluza-Klein dark matter,
In the Proceedings of 2005 International Linear Collider Workshop (LCWS 2005), Stanford, California, 18-22 Mar 2005, pp 0303
eConf C050318 (2005) 0303 [arXiv:hep-ph/0508136].
• K. Mönig,
Physics at future linear colliders,
Int. J. Mod. Phys. A 21 (2006) 1974 [arXiv:hep-ph/0509159].
• P. Krstonosic, K. Mönig, M. Beyer, E. Schmidt and H. Schröder,
Experimental studies of strong electroweak symmetry breaking in gauge boson scattering and three gauge boson production,
In the Proceedings of 2005 International Linear Collider Workshop (LCWS 2005), Stanford, California, 18-22 Mar 2005, pp 0310,
• K. Mönig and A. Rosca,
Two-photon width of the Higgs boson,
In the Proceedings of 2005 International Linear Collider Workshop (LCWS 2005), Stanford, California, 18-22 Mar 2005, pp 0113
• I. Alvarez Illan and K. Mönig,
Selectron production in e gamma collisions at a linear collider,
• K. Mönig and J. Sekaric,
Measurement of triple gauge boson couplings at an e gamma collider,
Eur. Phys. J. C 38 (2005) 427 [arXiv:hep-ex/0410011].
• K. Mönig and J. Sekaric,
Charged current triple gauge couplings at an e gamma collider,
In Proceedings to the International Conference on Linear Colliders (LCWS 04), Paris, France, 19-24 Apr 2004, vol. 1* 273-277.
• S. Riemann,
Aspects of new gauge bosons searches at LHC/LC,
In Proceedings to the International Conference on Linear Colliders (LCWS 04), Paris, France, 19-24 Apr 2004, vol. 1* 237-241.
• K. Mönig, J.Sekaric,
A Study of Charged Current Triple Gauge Couplings at an e-gamma collider
• V. Makarenko, K. Mönig and T. Shishkina,
Measuring the luminosity of a gamma-gamma collider with gamma,gamma -> lepton,lepton,gamma events
Eur. Phys. J. C 32S1 (2003) 143 [arXiv:hep-ph/0306135].
• K. Mönig,
Electroweak Gauge Bosons and Alternative Theories
Proceedings of the ECFA/DESY workshop on linear colliders, Amsterdam, April 2003,
• K. Mönig,
Physics of Electroweak Gauge Bosons
in "Linear Collider Physics in the New Millennium", edited by K. Fujii, D. Miller and A. Soni, World Scientific,
• I. Bozovic-Jelisavcic, K. Mönig, J. Sekaric,
Measurement of Trilinear Gauge Couplings at a gamma-gamma and e-gamma Collider
Proceedings of the International Workshop on Linear Colliders, Jeju, Korea, 2002, 383-388, Korean Physical Society,
• B. Ananthanarayan, S. D. Rindani and A. Stahl,
CP violation in the production of tau leptons at TESLA with beam polarization
Eur. Phys. J. C27 (2003) 33, [arXiv:hep-ph/0204233].
• M. Battaglia, S. De Curtis, D. Dominici and S. Riemann,
Probing new scales at a e+ e- linear collider,
In the Proceedings of APS / DPF / DPB Summer Study on the Future of Particle Physics (Snowmass 2001), Snowmass, Colorado, 30 June - 21 July 2001, pp E3020,
• J. Erler et al.,
Positron polarisation and low energy running at a Linear Collider
Proceedings of the Snowmass2001 workshop,
eConf C010630 (2001) E300, [arXiv:hep-ph/0112070].
• U. Baur et al.,
Present and Future Electroweak Precision Measurements and the Indirect Determination of the Mass of the Higgs Boson (Summary report of the Precision Measurement working group at Snowmass 2001)
Proceedings of the Snowmass2001 workshop,
eConf C010630 (2001) P1WG1, [arXiv:hep-ph/0202001].
• K. Mönig,
What is the Case for a Return to the Z-Pole?
Plenary talk presented at the LCWS2000 workshop FNAL,
• K. Mönig,
Measurement of the Differential Luminosity using Bhabha events in the Forward-Tracking region at TESLA
• K. Mönig,
The use of Positron Polarization for precision Measurements
• R. Hawkings, K. Mönig,
Electroweak and CP violation physics at a Linear Collider Z factory
EPJdirect C8 (1999) 1-26, [arXiv:hep-ex/9910022]
• S. Riemann,
Indications of new physics in fermion pair production, In Proc. of the 5th International Linear Collider Workshop (LCWS 2000), Fermilab, Batavia, Illinois, 24-28 Oct 2000,
AIP Conf. Proc. Volume 578, pp. 619-622.
• S. Riemann,
Fermion pair production at a linear collider: A sensitive tool for new physics searches,
• R. Casalbuoni, S. De Curtis, D. Dominici, R. Gatto and S. Riemann,
Z' indication from new APV data in cesium and searches at linear colliders,
• S. Riemann,
Prospects to detect a Z' with the LC,
• A. Leike and S. Riemann,
Z' search in e^+ e^- annihilation,
Z. Phys. C 75 (1997) 341, [arXiv:hep-ph/9607306]. | {"url":"https://www.zeuthen.desy.de/ILC/physics/eweak.php","timestamp":"2024-11-09T23:50:48Z","content_type":"application/xhtml+xml","content_length":"18742","record_id":"<urn:uuid:2d6523ee-4e1b-4bb8-a9f7-c0a0667ffb2a>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00855.warc.gz"} |
Common Core Math Standards
Common Core Math Standards
The Number System
[ << Back ]
These are the topics related to the standard: Apply and extend previous understandings of multiplication and division to divide fractions by fractions."
Here are some specific activities, investigations or visual aids picked out. Click anywhere in the grey area to access the resource.
Click on a topic below for suggested lesson starters, resources and activities from Transum.
• Arithmetic The ability to perform mathematical calculations is still very important despite our hi-tech environment. Good numeracy skills support the understanding of more advanced mathematical
concepts at all levels. Mathematicians still consider mastery of the manual algorithms to be a necessary foundation for the study of algebra and computer science. Pupils should have a good grasp
of the meaning of numbers and use their understanding of place value to multiply and divide whole numbers and fractions. They should be able to order, add and subtract negative numbers in
context. They should use all four operations with decimals rounding answers where required. They should be able to solve simple problems involving ratio and direct proportion and calculate
fractional or percentage parts of quantities and measurements, using a calculator where appropriate. See also the Mental Methods topic and our Number Skills Inventory.
• Fractions A fraction is a part of a number. Fractions are either vulgar or decimal. Vulgar fractions can be proper, improper or mixed. Equivalent fractions have the same value. Pupils, at all
stages of their learning, should practise using fractions. From dealing with halves, the most basic fraction, to manipulating algebraic fractions containing surds, this topic is always relevant.
Proficiency also depends on reasonable numeracy skills particularly the multiplication tables and finding the lowest common multiple of two numbers. Pupils also need to be able to convert vulgar
fractions to decimals and percentages and vice versa. Be wary of teaching the 'rules' for manipulation fractions by rote. Pupils need to understand the reason why and the time-honoured key to
understanding starts with the imaginary pizza and the much-used fraction wall.
• Mental Methods Though using pencil and paper are as useful as having up-to-date technology skills, there is no substitute for strategic mental methods for working out calculations and solving
problems. The activities in this topic are designed to improve pupils' abilities to use their brains. Calculating 'in your head' can be a difficult task. If you cannot remember what you have
worked out or simply do not know how to solve a problem then it can be very challenging and frustrating. It is important to learn and practise mental arithmetic and using mathematical patterns,
you can dramatically improve the speed and accuracy of your mental mathematics. See also the Arithmetic topic and our Number Skills Inventory.
• Tables Times Tables is the common term referring to the multiples of numbers 2 to 12 (or 2 to 10). Having a quick recall of these tables is an important pre-requisite for studying other aspects
of mathematics and for coping with personal finance and other area of everyday live involving numbers. People of any age can improve their skills in recalling table facts. They should learn then
as they would learn a song or a dance. You need to know your times tables forwards, backwards and all mixed up. Spend time learning them well and you'll reap the benefits in future. Here on this
website we have developed many activities that help pupils learn their times tables and as then revise them in different ways so that the recall becomes easier and easier. Some of the activities
are games and quizzes while others help pupils spot the patterns in the times tables in many different ways. Here's a plan for learning a new times table in only five days! | {"url":"http://transum.info/Math/Common_Core_Standards/Topics.asp?ID_Statement=48","timestamp":"2024-11-14T23:28:26Z","content_type":"text/html","content_length":"18416","record_id":"<urn:uuid:c59407c6-c872-4ab4-b7f4-161a69fedf54>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00713.warc.gz"} |
Printable Calendars AT A GLANCE
Translating Words Into Algebraic Expressions Worksheet
Translating Words Into Algebraic Expressions Worksheet - 7) a number divided by 6. Web a guide for translating verbal expressions into algebraic expressions. The sum of a number and 16 is equal to
45. The key word is difference, which. Students should work step by step through each of the twenty problems. Web translating words into algebraic equations expressions cannot be solved because they
do not have an equal sign. Web linked in here are our printable translating inequality phrases worksheets that allow students to translate sentences into inequalities with swagger. Write the numbers/
variables in the correct order. Translating phrases to algebraic expressions. Web translate each word phrase into an algebraic expression:
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Related Post: | {"url":"https://ataglance.randstad.com/viewer/translating-words-into-algebraic-expressions-worksheet.html","timestamp":"2024-11-05T22:31:31Z","content_type":"text/html","content_length":"36549","record_id":"<urn:uuid:cd4cdf87-ee44-4694-981c-7e6f9fe866b7>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00653.warc.gz"} |
Kondo screening of spin-charge separated fluxons by a helical liquid
The insertion of a magnetic \( \pi \) flux into a quantum spin Hall insulator creates four localized, spin-charge separated states: the charge and spin fluxons with either charge \( Q=\pm1 \) or spin
\( S_z=\pm1/2 \), respectively. In the presence of repulsive Coulomb interactions, the charged states are gapped out and a local moment is formed. We consider the Kane-Mele-Hubbard model on a ribbon
with zigzag edges to construct an impurity model where the spin fluxon is screened by the helical edge liquid. In the noninteracting model, the hybridization between fluxon and edge states is
dominated by the extent of the latter. It becomes larger with increasing spin-orbit coupling \( \lambda\) but only has nonzero values for even distances between the \( \pi \) flux and the edge. For
the interacting system, we use the continuous-time quantum Monte Carlo method, which we have extended by global susceptibility measurements to reproduce the characteristic Curie law of the spin
fluxon. However, due to the finite extent of the fluxons, the local moment is formed at rather low energies. The screening of the spin fluxon leads to deviations from the Curie law that follow the
universal behavior obtained from a data collapse. Additionally, the Kondo resonance arises in the local spectral function between the two low-lying Hubbard peaks. | {"url":"https://for1807.physik.uni-wuerzburg.de/for1807_publications/kondo-screening-of-spin-charge-separated-fluxons-by-a-helical-liquid/","timestamp":"2024-11-14T21:25:36Z","content_type":"text/html","content_length":"49309","record_id":"<urn:uuid:987420b2-2f57-486c-b6cb-86b29bb8e2b2>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00358.warc.gz"} |
Towards Relative Symplectic Field Theory
September 24 to September 28, 2007
at the
CUNY Graduate Center, New York City
organized by
Kai Cieliebak, Tobias Ekholm, Yakov Eliashberg, Kenji Fukaya, Dennis Sullivan, and Michael Sullivan
The goal of this workshop, sponsored by AIM, NSF, CUNY Graduate Center, and Stanford Mathematical Research Center (MRC), is to understand the structure of relative Symplectic Field Theory (SFT),
discuss and reconcile different versions of its algebraic formalism, and to work towards building rigorous foundations of the theory. There will also be explored applications of relative SFT to
symplectic and contact topology, as well as low-dimensional topology.
The goal of the SFT project is to uncover algebraic structures which reflect the topology of the compactified moduli spaces of punctured holomorphic curves in symplectic manifolds with cylindrical
ends. Its relative counterpart should describe the topology of the compactified moduli spaces of punctured holomorphic curves with Lagrangian boundary conditions.
Though the SFT project is not yet completed in either absolute or relative case, the involved algebraic structures are much better understood in the absolute case, and the building of the analytic
foundations of the theory in the absolute case is well under way. The absolute SFT already led to new invariants of contact and symplectic manifolds and brought many applications to symplectic and
contact topology. There were also uncovered deep relations to other subjects such as enumerative algebraic geometry and integrable systems.
On the other hand, relative SFT is still in the period of its infancy. While special cases of the relative theory have been known for a long time, e.g. Floer homology for Lagrangian intersections and
Legendrian contact homology, the general formulation of a Relative Symplectic Field Theory has not been yet understood.
On the other hand, in the past few years several new fruitful ideas, and in particular, a link to String Topology were introduced to the subject. As a result, a general picture of Relative SFT is now
emerging. By bringing together the researchers involved in these new developments, we intend to reconcile different approaches and establish Relative Symplectic Field Theory in full generality,
investigate its relations with Open String Topology, and discuss applications. There will be also discussed the current status of the analytic foundations of the theory, and the remaining necessary
steps to complete the project.
The deadline to apply for support for this workshop has passed.
For more information email workshops@aimath.org
Plain text announcement or brief announcement.
Go to the American Institute of Mathematics.
Go to the list of upcoming workshops. | {"url":"https://aimath.org/ARCC/workshops/relsymplectic.html","timestamp":"2024-11-11T07:48:09Z","content_type":"text/html","content_length":"4791","record_id":"<urn:uuid:c6d055bb-ec9f-492d-8b97-30ca13f877ab>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00039.warc.gz"} |
Specifies a specific heat at constant pressure model.
SPECIFIC_HEAT_SOURCE("name") {parameters...}
User-given name.
type (enumerated) [=none]
Type of the specific heat model.
constant or const
Constant specific heat. Requires specific_heat.
Piecewise linear curve fit for enthalpy. Requires curve_fit_values and curve_fit_variable.
Piecewise polynomial curve fit for enthalpy. Requires piecewise_polynomial_values and piecewise_polynomial variables.
Piecewise bilinear curve fit for enthalpy. Requires bilinear_curve_fit_values, bilinear_curve_fit_row_variable and bilinear_curve_fit_column_variable.
Cubic spline curve fit for enthalpy. Requires curve_fit_values and curve_fit_variable.
User-defined function for enthalpy. Requires user_function, user_values and user_strings.
specific_heat or cp (real) >0 [=1]
Constant value of the specific heat. Used with constant type.
curve_fit_values or curve_values (array) [={0,1}]
A two-column array of independent-variable/enthalpy data values. Used with piecewise_linear_enthalpy and cubic_spline_enthalpy types.
curve_fit_variable or curve_var (enumerated) [=temperature]
Independent variable of the curve fit. Used with piecewise_linear_enthalpy and cubic_spline_enthalpy types.
temperature or temp
piecewise_polynomial_values (array) [={}]
Array of values to specify a piecewise polynomial equation. Used with piecewise_polynomial_enthalpy type.
piecewise_polynomial_variable (enumerated) [=temperature]
Independent variable of the piecewise polynomial curve fit. Used with piecewise_polynomial_enthalpy type.
bilinear_curve_fit_values (array) [={}]
Array of values to specify a piecewise polynomial equation. Used with piecewise_bilinear_enthalpy type.
bilinear_curve_fit_row_variable (enumerated) [=temperature]
Independent variable of the rows of the bilinear curve fit table. Used with piecewise_bilinear_enthalpy type.
bilinear_curve_fit_column_variable (enumerated) [no default]
Independent variable of the columns of the bilinear curve fit table. Variable can be either pressure or species. Used with piecewise_bilinear_enthalpy type.
user_function or user
User-defined function. Requires user_function, user_values and user_strings.
user_values (array) [={}]
Array of values to be passed to the user-defined function. Used with user_function_enthalpy type.
user_strings (list) [={}]
Array of strings to be passed to the user-defined function. Used with user_function_enthalpy type.
latent_heat_type (enumerated) [=none]
Type of the latent heat.
No latent heat present.
constant or const
Constant latent heat. Requires latent_heat, latent_heat_temperature and latent_heat_temperature_interval.
latent_heat (real) [=0]
Constant value of latent heat. Used with constant latent heat type.
latent_heat_temperature (real) [=0]
Temperature at which the latent heat is released. Used with constant latent heat type.
latent_heat_temperature_interval (real) >=0 [=0]
Temperature interval over which the latent heat takes effect. Also referred to as the mushy interval. Used with constant latent heat type.
This command specifies a specific heat model for the energy equation. This model is applicable to fluid, solid, and shell element sets.
commands are referenced by
commands, which in turn are referenced by
SPECIFIC_HEAT_MODEL( "my cp model" ) {
type = constant
specific_heat = 1005
MATERIAL_MODEL( "my material model" ) {
specific_heat_model = "my cp model"
ELEMENT_SET( "fluid elements" ) {
material_model = "my material model"
A constant specific heat model uses a constant value for the entire element set, as in the above example.
When specific heat is not constant, piecewise linear specific heat models of types piecewise_linear_enthalpy or cubic_spline_enthalpy can be used to define enthalpy (or specific enthalpy) as a
function of temperature. Enthalpy is given by:
${h}_{2}={h}_{1}+{\int }_{{T}_{1}}^{{T}_{2}}{C}_{p}\left(T\right)\text{\hspace{0.17em}}dT$
where $T$ is temperature, ${C}_{p}$ is specific heat capacity and $h$ is specific enthalpy.
When specific heat does not vary with respect to temperature, it is often useful to treat it as constant. If so,
In cases where the specific heat is a function of temperature, the integral above can be numerically evaluated. One possible integration scheme is the trapezoidal rule:
Note: This integration (irrespective of the method use) must be performed prior to the CFD calculation as AcuSolve does not currently offer any capabilities in this regard.
The curve_fit_values is a two-column array corresponding to the independent variable (temperature in this case) and enthalpy. The independent variable values must be in ascending order. The limit
point values of the curve fit are used when curve_fit_variable falls outside of the curve fit limits.
data may be read from a file. For the above example, the curve fit values may be placed in a file, say
, and read by:
SPECIFIC_HEAT_MODEL( "ice" ) {
type = piecewise_linear_enthalpy
curve_fit_values = Read( "enthalpy.fit" )
curve_fit_variable = temperature
type is used to specify the high-order multiple polynomial temperature functions for specific heat capacity in different temperature ranges. The specific heat is calculated by
by calculating the derivative of the enthalpy polynomial with respect to temperature. For example, if the temperature dependent enthalpy equation is given by:
is the universal gas constant.
Note: Most of the references for the thermodynamic properties of gases provide the coefficients ${a}_{0},{a}_{1},{a}_{2},\dots$ which need to be multiplied with the appropriate values shown above to
arrive at the values required by AcuSolve.
SPECIFIC_HEAT_MODEL( "H" ) {
type = piecewise_polynomial_enthalpy
piecewise_polynomial_variable = temperature
piecewise_polynomial_values = { Ta_min, Ta_max, A0, A1, A2, A3, A4, A5;
Tb_min, Tb_max, B0, B1, B2, B3, B4, B5}
The first entry of piecewise_polynomial_values is the minimum temperature, followed by the maximum temperature to indicate the applicable temperature range. Next, the polynomial coefficients are
entered to define the first polynomial equation. The second set of piecewise_polynomial_values is separated with the semi-colon. The order of polynomial coefficient in piecewise_polynomial_values is
from lower to higher, starting from the index 0. Negative indices of temperature are not supported by AcuSolve.
A piecewise_bilinear_enthalpy type defines a piecewise bilinear curve fit as a function of two independent variables. bilinear_curve_fit_row_variable and bilinear_curve_fit_column_variable define the
two independent variables.
The format of the interpolation table is as follows:
bilinear_curve_fit_data =
{ 0, cVal1, cVal2, cVal3, ... ;
Temperature 1, h(T1,c1), h(T1,c2), h(T1,c3), ... ;
Temperature 2, h(T2,c1), h(T2,c2), h(T2,c3), ... ;
Temperature 3, h(T3,c1), h(T3,c2), h(T3,c3), ... ;
..., ..., ..., ... , ... ;
The first entry in the table must be 0, then followed by the values of the column independent variable. The first entry of each row must be the temperature value followed by the corresponding
enthalpy. The row entries are comma separated with the semi-colon separating each of the rows.
A specific heat of type user_function_enthalpy may be used to model more complex behaviors; see the AcuSolve User-Defined Functions Manual for a detailed description of user-defined functions.
For example, consider a version of the piecewise linear latent heat model discussed above. The model below improves on this one by adding two "buffer" regions on either side of the latent heat
transition. These are used to smooth the transition and make the specific heat continuous with temperature. Unlike for the curve fit types
has no way of calculating the specific heat from the enthalpy, so both must be supplied by the user defined function. The specific heat is needed to form the appropriate contribution to the left-hand
side matrix. An error is issued if it is not returned. Only Jacobians with respect to temperature are supported currently. The enthalpy may also be a function of pressure and species, but extreme
caution must be taken since Jacobians with respect to these variables are not supported. The input command may be given by:
SPECIFIC_HEAT_MODEL( "UDF ice" ) {
type = user_function_enthalpy
user_function = "usrSpecHeat"
user_values = { 3000, # reference density
3.33e5, # reference temperature
273, # expansivity
2 } # temperature interval
where the user-defined function
may be implemented as follows:
#include "acusim.h"
#include "udf.h"
UDF_PROTOTYPE( usrSpecHeat ) ; /* function prototype */
Void usrSpecHeat (
UdfHd udfHd, /* Opaque handle for accessing data */
Real* outVec, /* Output vector */
Integer nItems, /* Number of elements */
Integer vecDim /* = 1 */
) {
Integer elem ; /* element index */
Real cp0 ; /* specific heat */
Real delTemp ; /* temperature interval */
Real dt2 ; /* delTemp / 2 */
Real enpy ; /* partial enthalpy */
Real latentHeat ; /* latent heat */
Real lh2 ; /* latent heat / 2 */
Real lh4 ; /* latent heat / 4 */
Real lhDt ; /* latent heat /delTemp */
Real refTemp ; /* reference temperature */
Real tmp ; /* a temporary temperature */
Real* temp ; /* a temperature */
Real* cp ; /* specific heat */
Real* usrVals ; /* user supplied values */
udfCheckNumUsrVals( udfHd, 4 ) ; /* check for error */
usrVals = udfGetUsrVals( udfHd ) ;
cp0 = usrVals[0] ;
latentheat = usrVals[1] ;
refTemp = usrVals[2] ;
delTemp = usrVals[3] ;
if ( delTemp <= 0 ) {
udfSetError( udfHd,
"temperature interval <%g> is not positive", delTemp ) ;
dt2 = delTemp / 2 ;
lh2 = latentHeat / 2 ;
lh4 = latentHeat / 4 ;
lhDt = latentHeat / delTemp ;
temp = udfGetElmData( udfHd, UDF_ELM_TEMPERATURE ) ;
/* Jacobian of enthalpy with respect to temp (same as specific heat) */
cp = udfGetElmJac( udfHd, UDF_ELM_JAC_TEMPERATURE ) ;
for ( elem = 0 ; elem < nItems ; elem++ ) {
tmp = (temp[elem] - refTemp) / dt2 ;
if ( tmp < -1.5 ) tmp = -1.5 ;
if ( tmp > +1.5 ) tmp = +1.5 ;
if ( tmp < -0.5 ) {
tmp = 1.5 + tmp ;
enpy = cp0 * temp[elem] + lh4 * tmp * tmp ;
cp[elem] = cp0 + lhDt * tmp ;
} else if ( tmp <= 0.5 ) {
tmp = 0.5 + tmp ;
enpy = cp0 * temp[elem] + lh2 * tmp + lh4 ;
cp[elem] = cp0 + lhDt ;
} else {
tmp = 1.5 - tmp ;
enpy = cp0 * temp[elem] + latentHeat - lh4 * tmp * tmp ;
cp[elem] = cp0 + lhDt * tmp ;
outVec[elem] = enpy ;
} /* end of usrSpecHeat() */
The dimensions of the returned enthalpy vector, outVec, and the Jacobian vector, cp, are the number of elements.
A latent heat of formation may be added directly to any specific heat model. For example,
SPECIFIC_HEAT_MODEL( "ice" ) {
type = constant
specific_heat = 3000
latent_heat_type = constant
latent_heat = 3.33e5
latent_heat_temperature = 273
latent_heat_temperature_interval = 2
Here, a constant specific heat of 3000 is used except at a temperature of 273, where the enthalpy is increased by 3.33x10^5. The jump in enthalpy is spread over two degrees, centered at 273. This
example models the same material as the piecewise_linear_enthalpy type example above. One advantage of using the latent heat parameters is that a more stable transition form is internally implemented
for the jump in the enthalpy. This form uses extra buffer regions like in the user-defined function above. | {"url":"https://help.altair.com/hwcfdsolvers/acusolve/topics/acusolve/specific_heat_model_acusolve_com_ref.htm","timestamp":"2024-11-03T04:08:20Z","content_type":"application/xhtml+xml","content_length":"122107","record_id":"<urn:uuid:34702978-bf05-437c-b23d-2e3d796fb2f0>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00481.warc.gz"} |
Any Number Exercise - Numbers - Python | Codeguage
Extend the Addition Calculator exercise to add any two numbers; not just two integers.
Back in the exercise Addition Calculator, we created a program that took as input two integers and then printed out their sum.
Recall that the entered values were converted into integers using int(). This meant that floating-point numbers couldn't be entered — they would've otherwise lead to an error.
Now, in this exercise, your task is to rewrite that simple program such that it accepts any two numbers; not just integers.
Furthermore, after performing the addition, if the result is an integer (whose fractional part is zero), output the result as an integer.
Shown below are two examples:
x: 5 y: 2.5 The sum is: 7.5
Here, since the sum 7.5 is not an integer, it is output as it is.
x: 1.5 y: 2.5 The sum is: 4
However, here the sum 4.0 is indeed an integer, and is likewise output as an integer — 4.
New file
Inside the directory you created for this course on Python, create a new folder called Exercise-8-Any-Number and put the .py solution files for this exercise within it.
The description clearly says that we can't use int() to convert the input strings into a number, since it throws an error when given a string containing a floating-point number.
This can be seen as follows:
Traceback (most recent call last): File "stdin", line 1, in <module> int('1.5') ValueError: invalid literal for int() with base 10: '1.5'
So with int() gone, do we have any other options to convert the input strings into numbers?
float()? Well, yes, it's the only option we've got right now. And guess what, it's just what's required.
float() can convert a stringified integer as well as a stringified float into an actual floating-point number. And then we could perform arithmetic on this float easily.
Therefore, to start with, we replace the calls to int() with float() in the program we created in the Addition Calculator exercise:
x = input('x: ')
y = input('y: ')
x = float(x)
y = float(y)
print('The sum is:', x + y)
But the story doesn't end here! The exercise requires one additional thing.
If the result of the addition is an integer, it should be output as an integer. For instance, if the result is 5.0, then it should be output as 5.
How to determine whether a float is an integer or not? Recall anything?
We'll need the is_integer() method of floats.
The idea is to call is_integer() on the expression x + y. If the method returns True, we convert x + y into an integer and then print it. However, if this is not the case, then we continue on with
our normal float output.
x = input('x: ')
y = input('y: ')
x = float(x)
y = float(y)
if (x + y).is_integer():
print('The sum is:', int(x + y))
print('The sum is:', x + y)
In line 7, the expression x + y in enclosed in parentheses to call the is_integer() method on the result of x + y. Without the parentheses, we'd have the expression x + y.is_integer(), which is
adding x and y.is_integer(). Doesn't seem right, does it?
This solves our exercise.
Improving the code
Despite the fact that the code above gives the correct output, there are a couple of problems in it.
The expression x + y is written thrice. Secondly, the print() statement, with exactly the same format, is written twice. This goes against the DRY principle, which we talked about earlier in the
Rudimentary Arithmetic Exercise.
DRY (Don't Repeat Yourself) states not to repeat code unnecessarily — which is undoubtedly happening in the code above.
How can we prevent this repetition?
For x + y, we could just compute it once and then save it in a variable. That's it.
We'll call this variable result. First, let's get done with this thing in our code:
x = input('x: ')
y = input('y: ')
x = float(x)
y = float(y)
result = x + y
if result.is_integer():
print('The sum is:', int(result))
print('The sum is:', result)
Next, let's analyse the print() statements.
How could we improve on them? What do you think is changing in the print() statements?
The difference between both print() calls above is that the first one has int(result) as the second arg, while the second one has result as the second arg. The rest is the same.
Whether the condition for if is true or false, a print() is regardless called, and something is, likewise, output. So, why not take the print() statement out of if and else.
What we could do is to write one print() statement after the whole block of if..else conditionals, and use those conditionals to determine the second arg to print(), instead:
x = input('x: ')
y = input('y: ')
x = float(x)
y = float(y)
result = x + y
if result.is_integer():
result = int(result)
result = result
print('The sum is:', result)
In words, the conditionals here say that 'if result is an integer, assign to it the integer value of result, or else just keep it as it is.'
Beyond these conditionals, result is in the desired format, and is directly output using print().
Amazing! We've improved our code a lot.
But there's one thing still left. Let's see whether you could figure it out..
Notice the else block above. If result is not an integer, result is assigned back to result. In other words, result remains whatever it is. This piece of code is redundant. Even if we were to remove
it, result would remain whatever it is — it won't just change on its own!
And this is just what we'll do — remove the redundant else block:
x = input('x: ')
y = input('y: ')
x = float(x)
y = float(y)
result = x + y
if result.is_integer():
result = int(result)
print('The sum is:', result)
Now our code is simpler and much more flexible than the one we created previously.
One thing to keep in mind is that these small improvements at this novice stage of learning Python won't do wonders in the speed of execution of the program.
However, they'll teach you how to write clean code and apply coding principles to it. This is an essential skill to have when you go on to write highly complex applications in Python, or any other
programming language. | {"url":"https://www.codeguage.com/courses/python/numbers-any-number-exercise","timestamp":"2024-11-04T08:59:06Z","content_type":"text/html","content_length":"45892","record_id":"<urn:uuid:74b40d2e-2821-44b6-905f-936c3f281112>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00195.warc.gz"} |
Many numerical computations reported in the literature show only a small difference between the optimal value of the one-dimensional cutting stock problem (1CSP) and that of the corresponding linear
programming relaxation. Moreover, theoretical investigations have proven that this difference is smaller than 2 for a wide range of subproblems of the general 1CSP. | {"url":"https://eudml.org/subject/MSC/90C10","timestamp":"2024-11-05T17:14:33Z","content_type":"application/xhtml+xml","content_length":"57327","record_id":"<urn:uuid:92487b31-209b-4021-937f-8b1db0ebe329>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00336.warc.gz"} |
I suggest you ...
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54 results found
1. Often I embed interactive experiences and visualizations. When scaled up they look blurry. I would like to have those iFrames change size to fill up the full screen rather than scale up, as they
are responsive and would handle the full resolution well. (either automatically or via user input would be fine)
Thanks for the feedback.
The best way around this is to insert your iframe as a slide background instead. Backgrounds are not scaled—they are sized to cover 100% of the available browser width and height.
Slide Backgrounds are available from the right-hand menu near Slide Settings and Speaker Notes (see attached). Hope that helps!
2. Can Slides have themes? I typically add my own fonts via Typekit, then add some css, then some slide templates. Can all these be saved as a theme that I can apply to future talks?
Right now I have to do this whole thing manually.
3. I was quite excited about this service as the big point of difference was the iframe option. However, I have since discovered that it does not allow HTTP based iFrames.
Any chance this can be changed? Or can I provide a link / domain that could be allowed through?
We originally supported HTTP iframe content, however an HTTP iframe can’t load on a HTTPS page. All of slides.com will be served over HTTPS soon, so we’re not allowing insecure iframe URLs since
they won’t work in your presentations. Hope that makes sense.
5. Ai driven error messages and ai driven help.
6. (Perhaps)In the fragments editor, there is an "X"-type button that make the selected fragment part of the normal slide.
All clicks on elements in the fragment editor toggle fragments on/off. So if you want to return an element to its “normal” state again you just need to click on it.
7. Hello, I am front-end webs developer from Lithuania. Recently I tried out slides.com (presentations is my hobby) and realy enjoyed it. After finding out, that HTML and CSS are fully costomizable
I thought of using it for developing simple static websites with it. What's the negative thing about it? Seems like slides.com is fully dedicated for presenations and not websites, even known
they are 1 step away from actually becoming some kind of content management system. What about using it for this purpouse in current state?
Hey! Thanks for contacting us. We realize that our editor could technically be used to create a many different kinds of content but we think it’s important to focus on one use case. We’d rather
be really great at presentations than just “okay” at a bunch of things :)
That said – people have already used Slides to create simple websites. You can either embed via an iframe or export to ZIP and host it yourself. The ZIP export includes all HTML, CSS, JS, images
and fonts.
8. ADD Equation editor to slides.
No plans for an equation editor at this point.
9. I would like to be an option to import a .swf flash file in my slide. There is a lot of these files i use in my physics classroom!
Sorry but we don’t have any plans for adding Flash support. The best workaround – as was mentioned in the comments – is to host the SWF file on another site and embed it into your presentation
using an iframe element. Haven’t tested this myself but it should work.
10. The confirm button label is “Sign up” when creating a new account and “Sign in” when signing in to an existing account. We like these labels and don’t have any plans for changing them.
11. Hi,When I paste a lot of code on one slide,It i can be viewed on pc using process bar,but ca n not be viewd on my android phone using process bar!
Scrolling code blocks isn’t possible on mobile devices since we use touch interactions to change slides. You’ll need to keep the code sample smaller, reduce the font size or split it into
multiple slides instead.
12. I hope this wasn't already posted but this is one feature I'd love to see!
This is a great suggestion but unfortunately not technically possible :/ We can’t access the operating system clipboard. Instead we implement our own clipboard and copy-paste logic but that only
works in the scope of one editor session.
13. Add more shapes. There are plently of open source fonts, which provide shapes as vector graphics. For example the font of bootstrap.
14. import from impress.js
Lot of services use impress.js as export option and it would be great, because the connections between apps.
Thanks for the suggestion but I don’t think we’ll be adding impress.js imports. Our formats are a bit too different.
15. Now I need to open the HTML editor and find the element then edit the style attributes. I want to edit it directly, under the "class name" field at the left side.
Also, I hope I can add classname and edit the style of the selected text, not only font size or color.
What you’re describing would best be done using the CSS editor in conjunction with the class name option, rather than adding styles directly to the HTML element.
More info about the CSS editor: http://help.slides.com/knowledgebase/articles/253052-css-editor-pro-
16. Use free icons from thenounproject.com, rather than from icomoon
We’ll be adding more icon options eventually but the source of those icons may be a mix. The Noun Project is a good option that we’ll definitely consider.
There’s already another idea for more icons tracked here: http://help.slides.com/forums/175819-general/suggestions/6570493-add-more-icons-shapes
17. Hello. Can i add soundtrack in to my presentation? Thank you
18. Parallax transition would make the site complete!
Thanks for the idea! Parallax effects are great but not something I think we want to add to Slides.
19. I suggest that instead enable leap motion controller in reveal.js y not embed it in slides??
Thanks for the feedback. We don’t feel the use case for this plugin is common enough to merit adding to Slides.
20. See comments, if further description is provide the status will be updated. | {"url":"https://help.slides.com/forums/175819-general/filters/new?status_id=792684","timestamp":"2024-11-06T05:10:24Z","content_type":"text/html","content_length":"263337","record_id":"<urn:uuid:0000dbb6-8b18-49d8-a80b-1c8392257d4a>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00305.warc.gz"} |
Efficient method for computing the maximum-likelihood quantum state from measurements with additive gaussian noise for Physical Review Letters
Physical Review Letters
Efficient method for computing the maximum-likelihood quantum state from measurements with additive gaussian noise
View publication
We provide an efficient method for computing the maximum-likelihood mixed quantum state (with density matrix ρ) given a set of measurement outcomes in a complete orthonormal operator basis subject to
Gaussian noise. Our method works by first changing basis yielding a candidate density matrix μ which may have nonphysical (negative) eigenvalues, and then finding the nearest physical state under the
2-norm. Our algorithm takes at worst O(d4) for the basis change plus O(d3) for finding ρ where d is the dimension of the quantum state. In the special case where the measurement basis is strings of
Pauli operators, the basis change takes only O(d3) as well. The workhorse of the algorithm is a new linear-time method for finding the closest probability distribution (in Euclidean distance) to a
set of real numbers summing to one. © 2012 American Physical Society. | {"url":"https://research.ibm.com/publications/efficient-method-for-computing-the-maximum-likelihood-quantum-state-from-measurements-with-additive-gaussian-noise","timestamp":"2024-11-05T22:51:15Z","content_type":"text/html","content_length":"74121","record_id":"<urn:uuid:367c78c5-2ea1-4460-9169-013bd5a3fc65>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00155.warc.gz"} |
3.16 Maclaurin Series
Maclaurin series are a special case of Taylor series with center $0$. In this section we will develop the Maclaurin series for $e^x, \sin (x)$ and $\cos (x)$ and use these to create Maclaurin series
of other, related functions.
Recall the formula for the coefficients of a Taylor series centered at $x = a$ is $\displaystyle c_n = \frac {f^{(n)}(a)}{n!}$. Substituting $a = 0$, we get the formula for the coefficients of a
Maclaurin series: We now use this to create the Maclaurin series for $e^x$.
We now turn to two examples of finding Maclauirin series by making modifications to the previous example.
We can now write the Maclaurin series representation as This can be writtien in summation notation as The ratio test can be used to verify that this representation is valid on the interval $(-\infty
, \infty )$.
We now consider an example of a Maclaurin series obtained by making modifications to the previous example.
We will use the Maclaurin series for $\cos (x)$ with $2x$ replacing $x$ and then multiply the result by $x^2$. This gives This representation is valid in the interval $(-\infty , \infty )$. | {"url":"https://ximera.osu.edu/math/calc2Book/calc2Book/maclaurinSeries/maclaurinSeries","timestamp":"2024-11-03T03:21:44Z","content_type":"text/html","content_length":"60967","record_id":"<urn:uuid:eced5427-3c22-4a54-b81f-49781cc091e8>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00334.warc.gz"} |
Crash damaged 16v manta! (£2850)
for sale one crash damaged manta (not complete)
been told by insurance company today that to keep my damaged shell will cost 3800.
Would love to keep this one from the scrap yard in the sky but don't wanna be out of pocket!
Let me know if interested asap as I need to tell insurance company decision soon!
As it is now!
Edited by Dom400
They want £3800 for the salvage buy back? Nice car but salvage is normally 10% - 20%. I don't see them paying you £19,000 - £38,000 for a manta
I have just bought my daily car back from insurance at 10% of the payout I received
I was told that was the norm
Yeah I was shocked too! Has anyone got anything evidence wise I can use to argue with the insurance company?
I have agreed to a payout of 9500 but not agreed anything regarding keeping the car!
Many help appreciated!
no evidence just what I was told and what happened to me a few weeks ago
when I asked they said its normally 10% as that is the amount they receive if the insurance dispose of the vehicle themselves to the salvage company
perhaps you need to speak to them again or the insurance omnibus
is the car still in your possession don't let it go until your happy
the salvage companies would not exist if they paid a large % as am sure they won't be paying £3,800
they took the money out of my payout as I agreed with the settlement for my car
10% -20% has been the norm for decades. You do see some higher these days, normally because they figured out that stolen recovered with keys and no damage at all they were basically giving away, but
not normally old bent cars
Something odd is going on for such a large charge for the salvage certainly, or some new rule they have.
But even with that buy back cost deducted off the payout you should be able to get the car repaired for that amount without much trouble. Admittedly from those photo's you can't see the full extent
of the damage to chassis rails etc, but judging from what i have seen in the past that wouldn't be a hard repair at all.
I'm gonna call them again and see if I can get a better price, but without evidence from other claims just hearsay I don't think they will lower the price. Mainly because I think there pissed I'm
getting nearly all that I insured it for! 🙄
Remember that if they have the car in storage arguing costs them money as storage charges build. However, if you already settled without saying it was conditional on buying back salvage then it may
well be their car already and you would have no leverage.
Also for most of us that haven't done the maths to prove the earth goes around the sun, that too is hearsay.
Edited by mantadoc
Ok called the insurance muppets today and to keep this car will be 30% of the total value @ £2850! Was given a incorrect number previously, so if someone would like to help me keep one of the best
mantas from the crusher pls let me know immediately!
No tax applied! Lol
Been looking after this one for 10+ years would hate to see it go to the crushers, but I have found a replacement and I need the cash fast!
• 3 weeks later...
Car has gone to Ireland to be fixed! Has gone to a club member, who will hopefully love it as much as I did!
• 1 month later...
Right I'm going to be pretty f***king blunt here.
To see prices such as this for a smashed up car with all the good bits taken, gives exactly how much incentive to people that have time, skills, and cash to buy a Manta?
I'm actively looking for a good project to put my mark on, and seeing stupid over inflated prices for vehicles such as this is sheer nonsense.
Can we all get a reality check please,,,,, It doesn't matter how good it "Once was" it simply isn't no longer. And as for "It could be made to look like this (where some clown attaches a photo) well
of course it could if you have thousands to spend on a car. and finally may well not be worth the cash you pour into it.
With spares prices what they are and so scarce I would hazard a guess that someone with no personal ability to weld, prep, paint and do mechanics would be looking at £15k + to attain a £6k+
So when vehicles like this crop up just short of £3k it simply does not equate.
Not intending to cause offence but to me and quite possibly many others this is actually the reality, can the owners of scrap vehicles get their heads out of where they shouldn't be and can us
potential buyers stop standing for this nonsense.
Rant over..........
If people are paying the prices Clive then its obviously a good price. Times have dramatically changed over the past 3-5 years with what i would say is a 100% rise in prices across all Mantas.
Rotten breaker 6 years ago - £300-£600 and now that car would be £1000-£1200
Mint Exclusive 6 years ago - £4500-£6000 and now that car would be £10,000 - £13,000 (a lot are changing hands under the radar for mega money)
people are paying the prices, the demand is greater than ever - hence your parts being requested more and more Clive.
• 2
• 1
I bought my first one in 2012 I believe, even in the space of 5 years the prices have gone mad. I honestly didn't think id be able to get back into one but got lucky with my 1800S.
I'd say.. there are still some bargains out there to be had. You don't have to spend 6k+ but the chances of getting a good one for a bargain are getting very slim.
4 hours ago, zublet said:
I bought my first one in 2012 I believe, even in the space of 5 years the prices have gone mad. I honestly didn't think id be able to get back into one but got lucky with my 1800S.
I'd say.. there are still some bargains out there to be had. You don't have to spend 6k+ but the chances of getting a good one for a bargain are getting very slim.
prices have risen considerably in the last 5 years & that is a good thing because it means more are now being saved as its worth doing.
The most economical way into Manta ownership at the moment is to buy a 1.8 hatch, check out this hatch which given its history is a bargain imo
I guess prices will keep going up. My manta had a rover v8 put in 20 years ago and that would of probably of added a lot of value to it at the time. Now it's having a partial restoration it'd make
more sense putting back to original spec as this is were the money is. As it is il keep the v8 as i won't be selling but manta's are definitely worth spending money on now and keeping original.
Not sure A series are seeing the same price rises. Prices seem a bit static from what I've seen going.
Gte exclusive, 1 owner from new, 89 thou on the clock restoration project, all interior there and clean, mainly welding needed £1000.
Now MINE!
What else do I need to say?
27 minutes ago, opel2000 said:
Find me one too 😂😂 | {"url":"https://mantaclub.org/forums/topic/42035-crash-damaged-16v-manta-%C2%A32850/","timestamp":"2024-11-06T13:52:58Z","content_type":"text/html","content_length":"408557","record_id":"<urn:uuid:5a752b40-cc8d-4d62-9047-a2c4e589f99c>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00353.warc.gz"} |
Write a program that prints the numbers from 1 to 100. But for multiples of three print “Fizz” instead of the number and for the multiples of five print “Buzz”. For numbers which are multiples of
both three and five print “FizzBuzz”.
Sample output:
... etc up to 100
Follow-up requirements
When you have the above program working, extend it with the following rules:
• Multiples of 7 are “Whizz”
• Multiples of 11 are “Bang”
That means for example that multiples of 3 & 7 are “FizzWhizz”, multiples of 5 & 11 are “BuzzBang” etc. Extend your printout so it continues beyond 100 and stops the first time you get
This kata is described on cyber-dojo.org, I added the follow-up requirements.
Referenced in these Learning Hours: | {"url":"https://www.sammancoaching.org/kata_descriptions/fizzbuzz.html","timestamp":"2024-11-02T17:00:07Z","content_type":"text/html","content_length":"8376","record_id":"<urn:uuid:32604ba2-9df4-4b17-ae56-d4ad87ae6038>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00148.warc.gz"} |
Informacionnye Tehnologii, 2018, vol. 24, no. 10, pp. 627-632
ABSTRACTS OF ARTICLES OF THE JOURNAL "INFORMATION TECHNOLOGIES".
No. 10. Vol. 24. 2018
DOI: 10.17587/it.24.627-632
A. D. Ivannikov, Doctor of Technical Sciences, Professor, V. N. Severtcev, Doctor of Technical Sciences, Institute for Design Problems in Microelectronics of Russian Academy of Sciences
Mathematical Model for Digital System Input Interaction Set in the Process of Logical Simulation
While digital system design debugging by computer simulation the important task is to generate debugging test set, e.g. set of input signals which are applied to a designing system computer model for
checking the correctness of its functioning. The generation of complete in some sense debugging test set is possible by some way if the permissible input action set for the system is known.
Description forming of such a set is possible if permissible input interaction set for digital system blocks are known. Digital system block model investigation is carried out, first of all, from the
point of a set of permissible input interactions. The family of stationary dynamic systems with continuous time and logical signal discrete values are used as models for digital system blocks. In
some cases signal exchange between blocks and with outer world is initiated by a block itself. That is why input interactions including input signals and output exchange driving signals are
considered as debugging tests. For the description of permissible input interactions of digital system blocks and the system as a whole graph representation is proposed for each fulfilled function.
Keywords: logical simulation, digital system design debugging, input interaction set, input interaction graph model
P. 627–632
To the contents | {"url":"http://www.novtex.ru/IT/eng/doi/it_24_627-632.html","timestamp":"2024-11-08T18:11:58Z","content_type":"text/html","content_length":"7794","record_id":"<urn:uuid:4681d7b1-9802-493d-9b6b-85b463a55bda>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00701.warc.gz"} |
graphlib - a low-level graph library in C
This library is intended for experimenting with different low-level representations of undirected graphs, especially for Markov chain Monte Carlo algorithms, where the critical factor is being able
to flip an edge as quickly as possible. It should also be useful for other applications involving simulations of random processes on graphs, although for applications in which the nodes carry state
information, it will be necessary to add another software layer to represent this.
It currently includes three implementations, each having an identical interface. Thus, by simply changing an include file, you can see the effect of a different implementation. All versions have O(1)
edge lookup time, but hash is roughly 5 times slower than char_array and bitmatrix.
• char_array - the full adjacency matrix is stored as an array of arrays of chars. Uses 8 times as much storage as bitmatrix, but has faster lookup. Good for small, dense graphs
• bitmatrix - one bit per edge; the bits are packed into arrays of bytes. The full adjacency matrix stored as an array of such bit vectors. Good for medium size, dense graphs
• hash - stores edge information in a hash table. Currently about 5 times slower than the first two implementations, but can handle much larger graphs. Good for large, sparse graphs. (needs glib)
The hash and bitmatrix graph types automatically keep track of the node degrees.
html documentation is included in the package.
download This website uses no cookies. This page was last modified 2024-01-21 10:57 by | {"url":"http://keithbriggs.info/graphlib.html","timestamp":"2024-11-08T12:38:28Z","content_type":"text/html","content_length":"8393","record_id":"<urn:uuid:464d5fb4-154d-40d2-875b-e577d7a6323d>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00043.warc.gz"} |
Basic Statistics - What you need to know
I often come across people throwing around statements like: "Statistics is hard" or "Statistics is so difficult".
I am here to be the bearer of good news. (Drums rolling) It is all a myth.
Look, I am not saying that statistics is the easiest thing on the planet. I know this too well, I majored in it for 4 years. I am just saying that we are all capable of mastering the basics of the
subject. Of course it may take a few hours of dedicated learning, but honestly unless you are some kind of Einstein or Ramanujan, you are going to have to put in the work.
Now that we all have a positive mindset, let's get started.
Why are you trying to learn statistics?
There are several reasons why someone may need to have an understanding of statistics:
1. It is a unit being taught in your course. You really do not have a choice here.
2. You are writing your thesis or dissertation and need to carry out some analytic research.
3. You are a data analyst or data scientist (It is impossible to be a good data scientist without mastering the basics of statistics. It is not just about writing code, it's also about understanding
what the numbers mean)
4. You are a Philomath and just like learning and studying.
If I have left anyone out, kindly let us know your reasons in the comment section below.
The Basics
I have been lucky enough to fit into several of the above mentioned reasons. With that in mind here are some basics in statistics that you need to know:
• Variables
• Probability
• Probability distribution
• Descriptive statistics
• Inferential statistics
Understanding variables is key to mastering statistics. This is because simply knowing the type of variable can help with knowing which descriptive and inferential statistics to use.
There are 2 main types of variables:
1. Qualitative variables : These describe categorical variables and their values are generally names.
Under qualitative variables we have:
• Nominal variables: The values are just names with no particular ordering, eg: Country of residence, race, gender
• Ordinal variables: The values are names with a particular ordering eg: the likert scale (strongly disagree to strongly agree)
2. Quantitative variables: These are numeric variables and their values are numeric (they can be measured or counted)
Under quantitative variables we have:
• Discrete variables: The values are numeric and are specific and certain. For example: Number of students in class (you cannot have half a student, its either1, 2, 3, etc.)
• Continuous variables: The values are numeric and can fall in any specified range eg: weight and height
Probability is the numerical measure for the degree of certainty or likelihood of the occurrence of an event.
In life, we often tend to ask ourselves several questions:
How likely am I to get clients for my new business?
How likely is it that the younger generation spends more time on their phones?
How likely am I to finish my work if I indulge in just one movie on Netflix?
We basically go about our day to day lives on the basis of likelihoods. Understanding probability and the different ways to calculate it is definitely key.
Probability distribution
Remember in school when the exam results were announced? It was normal to find a few people with really high grades, a large group who performed fairly and then another small group that had really
poor grades. Turns out your scores followed a distribution, a normal distribution.
Understanding the distribution, helps you understand the possible outcomes for a random event.
There are several other distributions that describe different events in life eg:
• Poisson distribution
• Binomial distribution
• Exponential distribution
• Uniform distribution
Descriptive statistics
Descriptive statistics are used to summarize and describe the characteristics of a variable.
Here's the catch though, you cannot describe a variable of the country of residence the same way you would the height of an individual. Basically, so as to properly describe a variable, you need to
know what kind of variable it is. The basics!
Descriptive statistics are accomplished using:
• Measures: mean, median, standard deviation, variance, skewness, kurtosis, frequency, correlation
• Visualization: bar charts, histograms, pie charts, line graphs etc.
Inferential statistics
These are statistics calculated so as to make conclusions and reasonable guesses.
In real life, we use inferential statistics to carry out hypothesis tests. So we basically come up with some idea or question, then we run mathematical tests to prove ourselves wrong or right. For
example, maybe you would like to know whether people enjoy watching comedy or action movies more.
We also have a catch here. To know the kind of inferential statistics to use, you need to understand both the type of variable, as well as the probability distribution of the variables. Lemme say it
again, the basics!
I believe in laying strong foundations especially when learning. If you can master the basic concepts, everything else can be easily built on that. Stay tuned for my Statistics Zero to Hero beginner | {"url":"https://www.clairematuka.com/post/basic-statistics-what-you-need-to-know","timestamp":"2024-11-15T03:09:43Z","content_type":"text/html","content_length":"994222","record_id":"<urn:uuid:27977f45-a0ac-43e0-a405-3867c1546fea>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00131.warc.gz"} |
Expressions and Keys
Expressions and Keys
Mathematical and Logical Expressions
Neper handles mathematical expressions thanks to the muparser library. The expression must contain no space, tabulation or new-line characters, and match the following syntax. [1]
The following table gives an overview of the functions supported by the default implementation. It lists the function names, the number of arguments and a brief description.
Name Description
sin sine function
cos cosine function
tan tangens function
asin arcus sine function
acos arcus cosine function
atan arcus tangens function
sinh hyperbolic sine function
cosh hyperbolic cosine
tanh hyperbolic tangens function
asinh hyperbolic arcus sine function
acosh hyperbolic arcus tangens function
atanh hyperbolic arcur tangens function
log2 logarithm to the base 2
log10 logarithm to the base 10
log logarithm to the base 10
ln logarithm to base \(e\) (2.71828…)
exp e raised to the power of x
sqrt square root of a value
sign sign function: -1 if \(x<0\); 1 if \(x>0\)
rint round to nearest integer
abs absolute value
min min of all arguments
max max of all arguments
sum sum of all arguments
avg mean value of all arguments
Binary Operators
The following table lists the default binary operators supported by the parser.
Operator Description Priority
&& logical and 1
|| logical or 2
<= less or equal 4
>= greater or equal 4
!= not equal 4
== equal 4
> greater than 4
< less than 4
+ addition 5
- subtraction 5
* multiplication 6
/ division 6
^ raise x to the power of y 7
Ternary Operators
The parser has built in support for the if-then-else operator. It uses lazy evaluation in order to make sure only the necessary branch of the expression is evaluated.
Operator Description
?: if-then-else operator, following the C/C++ syntax: (<test>)?<value_if_true>:<value_if_false>.
Statistical Distributions
The following table lists the statistical distributions. Custom endpoints (not indicated) can also be added as arguments, as described in the following.
Operator Description Information
normal(<mean>,<sigma>) normal
lognormal(<mean>,<sigma>) lognormal
dirac(<mean>) Dirac
beta(<x>,<y>) beta function \(x>0\), \(y>0\)
lorentzian(<mean>,<sigma>) Lorentzian
studentst(<mean>,<sigma>) Student’s t
weibull(k,<sigma>) Weibull \(k > 0\) represents the shape
breitwigner(<mean>,<sigma>[,<gamma>]) Breit-Wigner \(<gamma> \geq 0\), default 1
expnormal(<mean>,<sigma>[,<gamma>]) exp-normal \(<gamma> > 0\), default \(<sigma>\)
moffat(<mean>,<sigma>[,<gamma>]) Moffat \(<gamma> > 0\), default 1
pearson7(<mean>,<sigma>[,<gamma>]) Pearson type VII default \(<gamma> = 1.5\)
pseudovoigt(<mean>,<sigma>[,<gamma>]) Pseudo-Voigt \(<gamma> \in [0,\,1]\), default 0.5
skewnormal(<mean>,<sigma>[,<gamma>]) skewed normal default \(<gamma> = <sigma>\)
custom(<file_name>) custom
<mean> represents the mean (or centre), and <sigma> represents the standard deviation (or scale, \(> 0\)). <gamma> depends on the distribution function (see the above table). For all distributions
but weibull and beta, custom endpoints can be added as last arguments, as <from_value>,<to_value>, where <from_value> is the lower endpoint and <to_value> is the upper endpoint. The parameter
keywords do not need to be provided, but, when they are, the parameters can be given in any order, as in moffat(gamma=1,from=0,to=1,sigma=0.1,mean=0.5). Endpoints are considered inclusive by default,
but exclusive endpoints can be specified using fromexclusive=<from_value> and toexclusive=<to_value> (frominclusive=<from_value> and toinclusive=<to_value> can be used for inclusive endpoints).
String completion is available for the keywords. Finally, a sum of distributions of increasing averages can be provided, as in 0.3*lognormal(0.5,0.1)+0.7*normal(1,0.1).
When from and/or to are used, they should preferably be so that the distribution retains the same mean; otherwise, the distribution is shifted after truncation to match the specified mean.
In the case of the custom distribution, the numerical distribution must be provided in the file. The file must contain the x and y values of the distribution on successive lines. The x values must be
provided in ascending order and form a regular grid. The distribution must contain at least 3 points and does not need to integrate to 1.
Tessellation Keys
Available keys for a tessellation itself are provided below.
Key Descriptor Apply to
dim dimension tess
vernb number of vertices tess
edgenb number of edges tess
facenb number of faces tess
polynb number of polyhedra tess
cellnb number of cells tess
x x coordinate tess
y y coordinate tess
z z coordinate tess
coo x, y and z coordinates tess
area surface area tess
vol volume tess
size size (surface area/volume in 2D/3D) tess
step simulation step tess
Available keys for tessellation seeds, vertices, edges, faces, polyhedra, crystals and cell groups are provided below. Also note that the keys apply to cells if they are tagged to apply to polyhedra
and the tessellation is 3D and faces and the tessellation is 2D, and that keys apply to crystals if they apply to cells. You may also replace, in the tessellation keys themselves, poly by cell if the
tessellation is 3D and face by cell if the tessellation is 2D (it applies only in rare cases). For example, for a 2D tessellation, you may use -statcell ncells instead of -statface nfaces. Keys
specific to cells are defined accordingly in the following but also apply to polys is the tessellation is 3D and faces is the tessellations is 2D.
To turn a key value into a value relative to the mean over all entities (e.g. the relative cell size), append the key expression with the :rel modifier. To turn a key value into a value which holds
for a unit cell size, append the key expression with the :uc modifier. To use as a reference only the body entities (see below), append b to the modifiers.
Key Descriptor Apply to
id identifier seed, ver, edge, face, poly, group
x x coordinate seed, ver, edge, face, poly
y y coordinate seed, ver, edge, face, poly
z z coordinate seed, ver, edge, face, poly
coo x, y and z coordinates seed, ver, edge, face, poly
xmin minimum x coordinate edge, face, poly
ymin minimum y coordinate edge, face, poly
zmin minimum z coordinate edge, face, poly
xmax maximum x coordinate edge, face, poly
ymax maximum y coordinate edge, face, poly
zmax maximum z coordinate edge, face, poly
w weight (width for a lamellar tessellation) seed, cell
body[<expr>] body level ver, edge, face, poly
state state ver, edge, face, poly
domtype type of domain (0 if on a domain vertex, 1 if on a domain edge and 2 if on a domain face) ver, edge, face
domface domain face (-1 if undefined) face
domedge domain edge (-1 if undefined) edge
domver domain vertex (-1 if undefined) ver
scale scale ver, edge, face, poly, cell [2]
length length edge
length(<d_x>,<d_y>,<d_z>) directional length along \((d_x,d_y,d_z)\) (in 2D, d_z can be omitted) edge, face, poly
area surface area face, poly, group
vol volume poly, group
size size (surface area/volume in 2D/3D) cell, group
diameq equivalent diameter [4] face, poly
avdiameq average equivalent diameter [4] face, poly
radeq equivalent radius (half of the eq. diameter) face, poly
avradeq average equivalent radius (half of the eq. diameter) face, poly
sphericity sphericity [5] poly
circularity circularity [6] face
convexity convexity [7] face (only for a 2D tessellation), poly
dihangleav average dihedral angle face, poly
dihanglemin minimum dihedral angle face, poly
dihanglemax maximum dihedral angle face, poly
dihangles dihedral angles face, poly
ff flatness fault (in degrees) face
theta disorientation angle (in degrees) edge (in 2D), face (in 3D)
cyl cylinder polygonization [19] edge
vernb number of vertices edge, face, poly
vers vertices edge, face, poly
edgenb number of edges ver, face, poly
edges edges ver, face, poly
facenb number of faces ver, edge, poly
faces faces ver, edge, poly
polynb number of polyhedra ver, edge, face
polys polyhedra ver, edge, face
nfacenb number of neighboring faces face
nfaces neighboring faces face
nfacenb_samedomain number of neighboring faces of the same domain (parent cell of a multiscale tessellation) face (in 2D)
nfaces_samedomain neighboring faces of the same domain (parent cell of a multiscale tessellation) face (in 2D)
npolynb number of neighboring polyhedra poly
npolys neighboring polyhedra poly
npolys_unsort neighboring polyhedra, unsorted list poly
npolynb_samedomain number of neighboring polyhedra of the same domain (parent cell of a multiscale tessellation) poly
npolys_samedomain neighboring polyhedra of the same domain (parent cell of a multiscale tessellation) poly
vercoos vertex coordinates face, poly
faceareas face surface areas poly
faceeqs face equations [9] poly
nseednb number of neighboring seeds poly
nseeds neighboring seeds [10] poly
scaleid(<scale_nb>) identifier of the corresponding cell at scale <scale_nb> cell
lam lamella width id [11] cell
mode mode [12] cell
group group cell
per periodic (1 if periodic, 0 otherwise) ver, edge, face (in 3D)
fiber(...) 1 if in orientation fiber and 0 otherwise, see Orientation Fibers poly
<orientation_descriptor> orientation descriptor face (in 2D), poly (in 3D)
step simulation step ver, edge, face, poly
Variables consisting of several values (vers, etc.) are not available for sorting (option -sort).
For a cell, the body variable is defined as follows:
□ In the general case (body, no argument provided), it is an integer equal to 0 if the cell is at the domain boundary, i.e. if it shares at least one face with it (edge in 2D), and is equal to
1 or higher otherwise. This is determined as follows: if a cell is surrounded by cells with body values equal to or higher than n, its body value is equal to n + 1. Therefore, body tends to
increase with the distance to the domain boundary and can be used to define cells that may suffer from boundary effects.
□ In the case where an expression is provided as argument (body(<expr>)), the expression is a logical expression that defines the boundary to consider, from the domain face (edge in 2D) labels
(for a cube, x0, x1, y0, y1, z0 and z1). For example, body(z0||z1) considers only the z0 and z1 domain faces as the boundary, and the more exotic body(x1&&y0||z1) considers only the
intersection between the x1 and y0 domain faces, and the z1 domain face as the boundary.
For entities of lower dimension than cells (vertices, edges and faces), body is equal to the maximum body value of the cells they belong to.
Raster Tessellation Keys
Available keys for raster tessellation itself are provided below.
Key Descriptor Apply to
dim dimension tesr
voxnbx number of voxels in direction x tesr
voxnby number of voxels in direction y tesr
voxnbz number of voxels in direction z tesr
voxnb number of voxels in total tesr
originx origin x coordinate tesr
originy origin y coordinate tesr
originz origin z coordinate tesr
voxsizex voxel size in direction x tesr
voxsizey voxel size in direction y tesr
voxsizez voxel size in direction z tesr
rastersizex raster size in direction x tesr
rastersizey raster size in direction y tesr
rastersizez raster size in direction z tesr
rastersize raster size (surface area/volume in 2D/3D) tesr
area surface area tesr
vol volume tesr
size size (surface area/volume in 2D/3D) tesr
x x coordinate tesr
y y coordinate tesr
z z coordinate tesr
coo x, y and z coordinates tesr
step simulation step tesr
Available keys for raster tessellation seeds, cells, cell groups and voxels are provided below. Mathematical and logical expressions based on these keys can also be used. To turn a key value into a
value relative to the mean over all entities (e.g.the relative cell size), append the key expression with the :rel modifier. To turn a key value into a value which holds for a unit cell size, append
the key expression with the :uc modifier.
Key Descriptor Applies to
id identifier seed, cell, group, voxel
cell cell voxel
oridef orientation is defined voxel
w Laguerre weight seed
step simulation step tesr
Key Descriptor Applies to
x x coordinate seed, cell, voxel
y y coordinate seed, cell, voxel
z z coordinate seed, cell, voxel
coo x, y and z coordinates seed, cell, voxel
vx x coordinate (in voxel) voxel
vy y coordinate (in voxel) voxel
vz z coordinate (in voxel) voxel
vcoo x, y and z coordinates (in voxel) voxel
vxmin minimum x coordinate (in voxel) cell
vymin minimum y coordinate (in voxel) cell
vzmin minimum z coordinate (in voxel) cell
vxmax maximum x coordinate (in voxel) cell
vymax maximum y coordinate (in voxel) cell
vzmax maximum z coordinate (in voxel) cell
domvxmin domain minimum x coordinate (in voxel), always 1 domain
domvymin domain minimum y coordinate (in voxel), always 1 domain
domvzmin domain minimum z coordinate (in voxel), always 1 domain
domvxmax domain maximum x coordinate (in voxel) domain
domvymax domain maximum y coordinate (in voxel) domain
domvzmax domain maximum z coordinate (in voxel) domain
area surface area cell, group (in 2D)
vol volume cell, group (in 3D)
size size (surface area/volume in 2D/3D) cell, group
voxnb number of voxels cell
areafrac surface area fraction group (in 2D)
volfrac volume fraction group (in 3D)
sizefrac size fraction (surface area/volume fraction in 2D/3D) group
diameq equivalent diameter [4] cell
radeq equivalent radius cell
convexity convexity [7] cell
Key Descriptor Applies to
<orientation_descriptor> orientation descriptor voxel, cell
gos grain orientation spread [8] cell
oridisanisoangles orientation distribution anisotropy / principal angles [13] cell
oridisanisoaxes orientation distribution anisotropy / principal axes [13] cell
oridisanisofact orientation distribution anisotropy factor [13] cell
oridisanisodeltas orientation distribution anisotropy / principal delta angles [14] cell
Tessellation Optimization Keys
Time Keys
The available keys for option -morphooptilogtime are provided below. Use iter(<factor>), where factor is an integer reduction factor, to log values only at specific iteration numbers.
Key Descriptor
iter iteration number
varupdateqty number of updated variables
seedupdateqty number of updated seeds
seedupdatelist list of updated seeds
cellupdateqty number of updated cells
cellupdatelist list of updated cells
var time for variable update
seed time for seed update
cell_init time for cell update initialization
cell_kdtree time for cell update kdtree computation
cell_shift time for cell update shift computation
cell_neigh time for cell update neighbor computation
cell_cell time for cell update cell computation
cell_other time for cell update others
cell_total total time for cell update
val time for (objective function) value update
val_init time for (objective function) value update / initialization
val_penalty time for (objective function) value update / penalty computation
val_val time for (objective function) value update / value computation
val_val_cellval time for (objective function) value update / value computation / cell values
val_val_comp time for (objective function) value update / value computation / computation
val_comp time for (objective function) value update / computation
total total time
cumtotal cumulative total time
Variable Keys
The available keys for option -morphooptilogvar are provided below. Use iter(<factor>), where factor is an integer reduction factor, to log values only at specific iteration numbers.
Key Descriptor Apply to
iter iteration number n/a
id identifier seed
x x coordinate seed
y y coordinate seed
z z coordinate seed
w weight seed
Objective Function Value Keys
The available keys for option -morphooptilogval are provided below. Use iter(<factor>), where factor is an integer reduction factor, to log values only at specific iteration numbers.
Key Descriptor
iter iteration number
val value
valmin minimal value
val0 value, without smoothing
valmin0 minimal value, without smoothing
val(<i>) i th subvalue
val0(<i>) i th subvalue, without smoothing
eps error on the objective function (see -morphooptistop)
reps relative error on the objective function (see -morphooptistop)
loop optimization loop
plateaulength current plateau length [15]
Statistical Distribution Keys
The available keys for option -morphooptilogdis are provided below. PDF stands for probability density function and CDF stands for cumulative probability density function. Use iter(<factor>), where
factor is a reduction factor, to log values only at specific iteration numbers.
Key Descriptor
iter iteration number
x x coordinate
tarpdf target PDF
tarcdf target CDF
curpdf current PDF
curcdf current CDF
tarpdf0 target PDF, not smoothed
tarcdf0 target CDF, not smoothed
curcdf0 current CDF, not smoothed
Raster Tessellation Voxel Keys
The available keys for option -morphooptilogtesr are provided below. Values are written for each voxel used to compute the objective function. Use iter(<factor>), where factor is a reduction factor,
to log values only at specific iteration numbers.
Key Descriptor
iter iteration number
id cell identifier
x x coordinate
y y coordinate
z z coordinate
dist distance to the cell
Orientation Optimization Keys
Variable Keys
The available keys for option -orioptilogvar are provided below. For all orientation descriptors but quaternion, the returned orientation are located in the fundamental region. Use iter(<factor>),
where factor is an integer reduction factor, to log values only at specific iteration numbers.
Key Descriptor Apply to
iter iteration number n/a
id identifier seed
rodrigues Rodrigues vector seed
euler-bunge Euler angles (Bunge convention) seed
euler-kocks Euler angles (Kocks convention) seed
euler-roe Euler angles (Roe convention) seed
rotmat Rotation matrix seed
axis-angle rotation axis / angle pair seed
quaternion quaternion seed
Mesh Keys
Available keys for a mesh itself are provided below. “co” stands for “cohesive”.
Key Descriptor Apply to
eltnb element number {0-3}D,co mesh
nodenb node number {0-3}D mesh
elsetnb elset number {0-3}D,co mesh
partnb partition number highest-dimension mesh
x x coordinate {0-3}D mesh
y y coordinate {0-3}D mesh
z z coordinate {0-3}D mesh
coo x, y and z coordinates {0-3}D mesh
length length 1D mesh
area surface area 2D mesh
vol volume 3D mesh
size size (length/area/volume in 1D/2D/3D) {1-3}D mesh
step simulation step {0-3}D,co mesh
Available keys for mesh node, elements and element sets (of all dimensions) and points are provided below. “co” stands for “cohesive”.
Key Descriptor Apply to
id identifier node, {0-3}D,co elt, {0-3}D,co elset
x x coordinate node, {0-3}D,co elt, {0-3}D elset
y y coordinate node, {0-3}D,co elt, {0-3}D elset
z z coordinate node, {0-3}D,co elt, {0-3}D elset
coo x, y and z coordinates node, {0-3}D,co elt, {0-3}D elset
dim lowest parent elt dimension node
elset0d 0D elset 0D elt
elset1d 1D elset 1D elt
elset2d 2D elset 2D elt
elset3d 3D elset 3D elt
elsetco Cohesive elset co elt
part partition {0-3}D elt, node
group group {0-3}D elt, {0-3}D elset
scaleid(<scale_nb>) identifier of the corresponding tess cell at scale <scale_nb> 2D elset, 3D elset
scale scale {0-2}D elset [3]
cyl cylinder polygonization [19] 1D elt, 1D elset
vol volume 3D elt, 3D elset
vol_orispace volume, orientation-space-wise [20] 3D elt
area surface area 2D elt
diameq equivalent diameter {2,3}D elt, {2,3}D elset
radeq equivalent radius {2,3}D elt, {2,3}D elset
length average edge length {0-3}D elt, 1D elset
lengths edge lengths 2D elt, 3D elt
elsetvol elset volume 3D elt
elsetarea elset area 2D elt
elsetlength elset length 1D elt
rr radius ratio 3D elt
rrav average radius ratio 3D elset
rrmin min radius ratio 3D elset
rrmax max radius ratio 3D elset
Osize Osize 3D elset
eltnb number of elements {0-3}D,co elset
elts elements {0-3}D,co elset
nodenb number of nodes {0-3}D,co elset
nodes nodes {0-3}D,co elset
body body level {0-3}D elt, {0-3}D elset
elsetbody body level, relative to the elset boundary {1-3}D elt
domtype type of domain [16] {0-2}D elt, {0-2}D elset
2dmeshp closest point of the 2D mesh node, 3D elt
2dmeshd distance to 2dmeshp node, 3D elt
2dmeshv vector to 2dmeshp node, 3D elt
2dmeshn outgoing normal vector at 2dmeshp node, 3D elt
per periodic (1 if periodic, 0 otherwise) {0,1}D elt, 2D elt (in 3D), {0,1}D elset, 2D elset (in 3D)
col_rodrigues color in Rodrigues vector convention [17] node
col_stdtriangle color in IPF convention, cubic symmetry [18] node
col_stdtriangle_hexagonal color in IPF convention, hexagonal symmetry [18] node
fiber(...) [21] 1 if in orientation fiber and 0 otherwise 3D elt, 3D elset
theta disorientation angle (in degrees) 1D elt and elset (in 2D), 2D elt and elset (in 3D)
gos grain orientation spread [8] {2,3}D elset
anisogos grain orientation spread estimated from the orientation distribution [8] {2,3}D elset
<orientation_descriptor> orientation descriptor 2D elt (in 2D), 2D elset (in 2D), 3D elt (in 3D), 3D elset (in 3D)
step simulation step {0-3}D,co mesh
Variables beginning with 2dmesh are only available for statistics (options beginning with -stat of module -M); for elements, they apply to the centroids.
Point Keys
Available keys for points are provided below.
Key Descriptor Apply to Require
id identifier point
x x coordinate point
y y coordinate point
z z coordinate point
cell cell point tessellation
elt containing element point mesh
elset containing elset point mesh
2dmeshp coordinates of the closest point of the 2D mesh point 3D mesh
2dmeshd distance to 2dmeshp point 3D mesh
2dmeshv vector to 2dmeshp point 3D mesh
2dmeshn outgoing normal vector of the 2D mesh at 2dmeshp point 3D mesh
Simulation Results
A result of a Simulation Directory (.sim) can be invoked simply from its name. A component of a vectorial or tensorial result can be invoked by prefixing the component to the name, as in coo1,
stress11, etc. For a symmetrical tensor (for which only 6 values are stored), t, both t<i><j> and t<j><i> are valid. The type of a result of the simulation directory is determined automatically.
Tessellation results can be obtained from the cell results, by averaging or other statistical treatments. Similarly, elset and mesh results can be obtained from the element results, by averaging or
other statistical treatments.
Available results / keys for nodes are the following:
Key Descriptor Apply to
disp displacement (computed from positions) node
Available results / keys for elements sets are the following:
Key Descriptor Apply to
ori average orientation elset, mesh
gos grain orientation spread [8] elset
anisogos grain orientation spread computed from oridisanisoangles elset
oridisanisoangles orientation distribution principal angles elset, mesh
oridisanisoaxes orientation distribution principal axes elset, mesh
oridisanisofact orientation distribution factor elset, mesh
odf(<var>=<value>,...) ODF defined at elements of orientation space (see also below) tess, tesr, mesh, cell, elt, elset
odfn(<var>=<value>,...) ODF defined at nodes of orientation space (see also below) tess, tesr, mesh
orifield(var=<var>,...) <var> field defined at elements of orientation space (see below) mesh
orifieldn(var=<var>,...) <var> field defined at nodes of orientation space (see below) mesh
The ODF (odf or odfn) of a tessellation or mesh is computed over orientation space (provided using -orispace) from the orientations of the (tessellation) cells or (mesh) elsets. The (optional)
parameters are:
• input: the input used for the orientations, either elsets or elts for a mesh (default elsets);
• theta: the standard deviation of the kernel (in degrees);
• weight: the weight of a cell or elset, which can be a real value or an expression based on the Tessellation Keys (for cells) or Mesh Keys (for elsets) – by default, the volumes of the cells or
elsets are used;
• clustering: a logical value controlling orientation clustering, which can be 0 (for no clustering) or 1 (for clustering); the default is 0 for cells or elsets and 1 for voxels or elements;
• cutoff: the cut-off factor used to compute the ODF, which can be all (for no cut-off) or any positive real value (default 5).
For a cell, element or elset, odf returns the value of the ODF of the tessellation or mesh at the corresponding orientation (and simulation step).
The orifield and orifieldn of a mesh is computed over orientation space (provided using -orispace) from the values of the (mesh) elsets. The mandatory parameter is:
• var: the variable, which must be defined for elsets (i.e., have its files in the simulation directory);
and the optional parameters are:
• theta: the standard deviation of the kernel (in degrees);
• weight: the weight of an elset, which can be a real value or an expression based on the Mesh Keys (for elsets) – by default, the volumes of the elsets are used.
Rotations and Orientations
Rotation and Orientation Descriptors
Rotations and orientations can be described using the following descriptors.
Key Descriptor Number of parameters
rodrigues Rodrigues vector 3
euler-bunge Euler angles (Bunge convention) 3
euler-kocks Euler angles (Kocks convention) 3
euler-roe Euler angles (Roe convention) 3
rotmat rotation matrix 9
axis-angle rotation axis / angle pair 4
quaternion quaternion 4
The convention can be added to the descriptor, either active or passive, as in rodrigues:active. When no convention is provided, passive is assumed.
Some options can take parameter values as argument, in which case the orientation must be expressed as <descriptor>(<parameter1>,<parameters2>,...). An example is rodrigues(0.1,0.2,0.3).
Orientation Convention
The crystal coordinate systems are attached to the crystal lattice as illustrated below, in the case of cubic and hexagonal symmetries:
The so-called “passive” orientation convention is used by default, which is based on the rotation of the sample coordinate system into the crystal coordinate system. Under this convention, the values
of all Rotation and Orientation Descriptors are provided below for a simple (but representative) configuration that corresponds to a rotation of 30° about \(X_s\):
Descriptor Value (angles in degrees)
Rodrigues vector \((0.267949192,\,0,\,0)\)
Euler angles (Bunge convention) \((0,\,30,\,0)\)
Euler angles (Kocks convention) \((270,\,30,\,90)\)
Euler angles (Roe convention) \((270,\,30,\,90)\)
rotation matrix \(\left(\begin{array}{ccc}1 & 0 & 0 \\ 0 & 0.866025404 & 0.5 \\ 0 & -0.5 & 0.866025404\\\end{array}\right)\)
rotation axis / angle pair \((1,\,0,\,0) / 30\)
quaternion \((0.965925826,\,0.258819045,\,0,\,0)\)
The values of the orientation descriptors under the “active” convention are obtained by taking the opposite rotation.
Ideal Orientations
Keys are available for ideal orientations (lowercased is accepted):
Key Miller indices
Cube \((0\,0\,1)[1\,0\,0]\)
Goss \((0\,1\,1)[1\,0\,0]\)
U \((1\,0\,1)[\overline{1}\,0\,1]\)
45NDCube \((0\,0\,1)[1\,\overline{1}\,0]\)
S1 \((1\,2\,3)[6\,3\,\overline{4}]\)
S2 \((\overline{1}\,2\,3)[6\,\overline{3}\,4]\)
S3 \((1\,\overline{2}\,3)[6\,\overline{3}\,\overline{4}]\)
S4 \((\overline{1}\,\overline{2}\,3)[6\,3\,4]\)
Brass1 \((1\,1\,0)[1\,\overline{1}\,2]\)
Brass2 \((\overline{1}\,1\,0)[1\,1\,\overline{2}]\)
Copper1 \((1\,1\,2)[1\,1\,\overline{1}]\)
Copper2 \((\overline{1}\,1\,2)[1\,\overline{1}\,1]\)
When loading orientations from an external file, use file(<file_name>[,des=<descriptor>]) where the orientation descriptor is among those listed above and is rodrigues:passive by default.
Orientation Fibers
Orientation fibers are defined by a crystal direction being parallel to a sample direction. Depending on the context, an angular tolerance or distribution with respect to the theoretical fiber can
also be defined:
• fiber(<dirc_x>,<dirc_y>,<dirc_z>,<dirs_x>,<dirs_y>,<dirs_z>), where (<dirc_x>, <dirc_y>, <dirc_z>) is the crystal direction and (<dirs_x>, <dirs_y>, <dirs_z>) is the sample direction, corresponds
to an ideal orientation fiber;
• fiber(<dirc_x>,<dirc_y>,<dirc_z>,<dirs_x>,<dirs_y>,<dirs_z>,<theta>), where <theta> is an angle expressed in degrees, corresponds to an orientation fiber with the angular tolerance <theta> from
the ideal fiber;
• fiber(<dirc_x>,<dirc_y>,<dirc_z>,<dirs_x>,<dirs_y>,<dirs_z>):normal(<var>=<val>), where <var> can be theta or thetam) and <val> is the value, corresponds to an orientation fiber with a normal
(Gaussian) disorientation normal to the ideal fiber.
Crystal Symmetries
Crystal symmetries can be described using the following descriptors.
Key Descriptor Number of operators
triclinic triclinic (Laue group \(\overline{1}\)) 24
cubic cubic 24
hexagonal hexagonal 1
-1 Laue group \(\overline{1}\) 1
2/m Laue group \(2/m\) 2
mmm Laue group \(mmm\) 4
4/m Laue group \(4/m\) 4
4/mmm Laue group \(4/mmm\) 8
-3 Laue group \(\overline{3}\) 3
-3m Laue group \(\overline{3}m\) 6
6/m Laue group \(6/m\) 6
6/mmm Laue group \(6/mmm\) 12
m-3 Laue group \(m\overline{3}\) 12
m-3m Laue group \(m\overline{3}m\) 24
Colors and Color Maps
The available colors are provided below, with their corresponding RGB channel values (ranging from 0 to 255). Any other color can be defined from the RGB channel values, under format <R_value>:
Key RGB value
black (0, 0, 0)
red (255, 0, 0)
green (0, 255, 0)
blue (0, 0, 255)
yellow (255, 255, 0)
magenta (255, 0, 255)
cyan (0, 255, 255)
white (255, 255, 255)
maroon (128, 0, 0)
navy (0, 0, 128)
chartreuse (127, 255, 0)
springgreen (0, 255, 127)
olive (128, 128, 0)
purple (128, 0, 128)
teal (0, 128, 128)
gray (128, 128, 128)
deepskyblue (0, 191, 255)
lawngreen (124, 252, 0)
darkgray (64, 64, 64)
orangered (255, 69, 0)
silver (192, 192, 192)
snow (255, 250, 250)
darkred (139, 0, 0)
darkblue (0, 0, 139)
darkorange (255, 140, 0)
azure (240, 255, 255)
ghostwhite (248, 248, 255)
ivory (255, 255, 240)
mediumblue (0, 0, 205)
lightpink (255, 182, 193)
mintcream (245, 255, 250)
indigo (75, 0, 130)
lightcoral (240, 128, 128)
pink (255, 192, 203)
coral (255, 127, 80)
salmon (250, 128, 114)
floralwhite (255, 250, 240)
aquamarine (127, 255, 212)
lemonchiffon (255, 250, 205)
gold (255, 215, 0)
darkgreen (0, 100, 0)
orange (255, 165, 0)
aliceblue (240, 248, 255)
lightcyan (224, 255, 255)
lightyellow (255, 255, 224)
darkmagenta (139, 0, 139)
darkcyan (0, 139, 139)
peru (205, 133, 63)
steelblue (70, 130, 180)
lavenderblush (255, 240, 245)
seashell (255, 245, 238)
mediumspringgreen (0, 250, 154)
darkslateblue (72, 61, 139)
darkgoldenrod (184, 134, 11)
lightsalmon (255, 160, 122)
bisque (255, 228, 196)
lightskyblue (135, 206, 250)
lightgoldenrodyellow (250, 250, 210)
honeydew (240, 255, 240)
cornsilk (255, 248, 220)
peachpuff (255, 218, 185)
whitesmoke (245, 245, 245)
tomato (255, 99, 71)
slategray (112, 128, 144)
hotpink (255, 105, 180)
oldlace (253, 245, 230)
blanchedalmond (255, 235, 205)
darkkhaki (189, 183, 107)
moccasin (255, 228, 181)
darkturquoise (0, 206, 209)
mediumseagreen (60, 179, 113)
mediumvioletred (199, 21, 133)
violet (238, 130, 238)
greenyellow (173, 255, 47)
papayawhip (255, 239, 213)
darkseagreen (143, 188, 143)
rosybrown (188, 143, 143)
deeppink (255, 20, 147)
saddlebrown (139, 69, 19)
darkviolet (148, 0, 211)
dodgerblue (30, 144, 255)
lightslategray (119, 136, 153)
burlywood (222, 184, 135)
navajowhite (255, 222, 173)
linen (250, 240, 230)
mediumslateblue (123, 104, 238)
turquoise (64, 224, 208)
skyblue (135, 206, 235)
mediumturquoise (72, 209, 204)
beige (245, 245, 220)
mistyrose (255, 228, 225)
tan (210, 180, 140)
antiquewhite (250, 235, 215)
thistle (216, 191, 216)
limegreen (50, 205, 50)
darksalmon (233, 150, 122)
lightsteelblue (176, 196, 222)
royalblue (65, 105, 225)
palegreen (152, 251, 152)
crimson (220, 20, 60)
wheat (245, 222, 179)
mediumorchid (186, 85, 211)
lavender (230, 230, 250)
khaki (240, 230, 140)
lightgreen (144, 238, 144)
paleturquoise (175, 238, 238)
darkslategray (47, 79, 79)
darkorchid (153, 50, 204)
seagreen (46, 139, 87)
yellowgreen (154, 205, 50)
blueviolet (138, 43, 226)
palevioletred (219, 112, 147)
olivedrab (107, 142, 35)
mediumpurple (147, 112, 219)
sandybrown (244, 164, 96)
darkolivegreen (85, 107, 47)
mediumaquamarine (102, 205, 170)
slateblue (106, 90, 205)
palegoldenrod (238, 232, 170)
forestgreen (34, 139, 34)
midnightblue (25, 25, 112)
lightseagreen (32, 178, 170)
lightgray (211, 211, 211)
orchid (218, 112, 214)
cornflowerblue (100, 149, 237)
sienna (160, 82, 45)
firebrick (178, 34, 34)
powderblue (176, 224, 230)
indianred (205, 92, 92)
dimgray (105, 105, 105)
lightblue (173, 216, 230)
chocolate (210, 105, 30)
brown (165, 42, 42)
goldenrod (218, 165, 32)
gainsboro (220, 220, 220)
plum (221, 160, 221)
cadetblue (95, 158, 160)
Color Maps
Color Map for Integer Values
The color map or palette used to represent integer values is defined from the above color list, by excluding colors of brightness below 0.2 and above 0.8. The brightness is defined as the average of
the channel values divided by 255. The resulting list of colors is: (1) red, (2) green, (3) blue, (4) yellow, (5) magenta, (6) cyan, (7) chartreuse, (8) springgreen, (9) olive, (10) purple, (11) teal
, (12) gray, (13) deepskyblue, (14) lawngreen, (15) darkgray, (16) orangered, (17) silver, (18) darkorange, (19) mediumblue, (20) indigo, (21) lightcoral, (22) coral, (23) salmon, (24) aquamarine,
(25) gold, (26) orange, (27) darkmagenta, (28) darkcyan, (29) peru, (30) steelblue, (31) mediumspringgreen, (32) darkslateblue, (33) darkgoldenrod, (34) lightsalmon, (35) lightskyblue, (36) tomato,
(37) slategray, (38) hotpink, (39) darkkhaki, (40) darkturquoise, (41) mediumseagreen, (42) mediumvioletred, (43) violet, (44) greenyellow, (45) darkseagreen, (46) rosybrown, (47) deeppink, (48)
saddlebrown, (49) darkviolet, (50) dodgerblue, (51) lightslategray, (52) burlywood, (53) mediumslateblue, (54) turquoise, (55) skyblue, (56) mediumturquoise, (57) tan, (58) limegreen, (59) darksalmon
, (60) lightsteelblue, (61) royalblue, (62) palegreen, (63) crimson, (64) mediumorchid, (65) khaki, (66) lightgreen, (67) darkslategray, (68) darkorchid, (69) seagreen, (70) yellowgreen, (71)
blueviolet, (72) palevioletred, (73) olivedrab, (74) mediumpurple, (75) sandybrown, (76) darkolivegreen, (77) mediumaquamarine, (78) slateblue, (79) forestgreen, (80) midnightblue, (81) lightseagreen
, (82) orchid, (83) cornflowerblue, (84) sienna, (85) firebrick, (86) indianred, (87) dimgray, (88) chocolate, (89) brown, (90) goldenrod, (91) plum and (92) cadetblue.
Color Maps for Real Values
The color map used to represent real values is smooth and obtained by interpolation between nominal colors. Tinycolormap is used to generate standard color maps, and the default is viridis. The color
maps are
Key Color bar
Alternatively, a custom color map can be provided as custom(<color1>,<color2>,...). Neper’s legacy color map (version \(< 4\)) is custom(blue,cyan,yellow,green) and can also be obtained using legacy:
Finally, it is possible to gradually fade the start of a color map, to make it starts with white. This can be done using the fade modifier, following the syntax <colormap>:fade[(threshold)]. The
threshold ranges from 0 to 1 and is equal to 0.1 by default. Fading is applied linearly from 0 (full fading) to the threshold (no fading). Examples are show below: | {"url":"https://neper.info/doc/exprskeys.html","timestamp":"2024-11-07T01:12:09Z","content_type":"text/html","content_length":"183208","record_id":"<urn:uuid:b90dfa4e-d56c-4c5b-a576-55e26b48f1bd>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00059.warc.gz"} |
Mohammad Mahmoody
Registration-Based Encryption: Removing Private-Key Generator from IBE Abstract
In this work, we introduce the notion of registration-based encryption (RBE for short) with the goal of removing the trust parties need to place in the private-key generator in an IBE scheme. In an
RBE scheme, users sample their own public and secret keys. There will also be a “key curator” whose job is only to aggregate the public keys of all the registered users and update the “short” public
parameter whenever a new user joins the system. Encryption can still be performed to a particular recipient using the recipient’s identity and any public parameters released subsequent to the
recipient’s registration. Decryption requires some auxiliary information connecting users’ public (and secret) keys to the public parameters. Because of this, as the public parameters get updated, a
decryptor may need to obtain “a few” additional auxiliary information for decryption. More formally, if n is the total number of identities and $$\mathrm {\kappa }$$κ is the security parameter, we
require the following.Efficiency requirements: (1) A decryptor only needs to obtain updated auxiliary information for decryption at most $$O(\log n)$$O(logn) times in its lifetime, (2) each of these
updates are computed by the key curator in time $${\text {poly}}(\mathrm {\kappa },\log n)$$poly(κ,logn), and (3) the key curator updates the public parameter upon the registration of a new party in
time $${\text {poly}}(\mathrm {\kappa },\log n)$$poly(κ,logn). Properties (2) and (3) require the key curator to have random access to its data.Compactness requirements: (1) Public parameters are
always at most $${\text {poly}}(\mathrm {\kappa },\log n)$$poly(κ,logn) bit, and (2) the total size of updates a user ever needs for decryption is also at most $${\text {poly}}(\mathrm {\kappa },\log
n)$$poly(κ,logn) bits.We present feasibility results for constructions of RBE based on indistinguishably obfuscation. We further provide constructions of weakly efficient RBE, in which the
registration step is done in $${\text {poly}}(\mathrm {\kappa },n)$$poly(κ,n), based on CDH, Factoring or LWE assumptions. Note that registration is done only once per identity, and the more frequent
operation of generating updates for a user, which can happen more times, still runs in time $${\text {poly}}(\mathrm {\kappa },\log n)$$poly(κ,logn). We leave open the problem of obtaining standard
RBE (with $${\text {poly}}(\mathrm {\kappa },\log n)$$poly(κ,logn) registration time) from standard assumptions. | {"url":"https://iacr.org/cryptodb/data/author.php?authorkey=6270","timestamp":"2024-11-14T13:48:29Z","content_type":"text/html","content_length":"68256","record_id":"<urn:uuid:30f9198e-57c3-420d-99af-35e027ba21a1>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00777.warc.gz"} |
1 — 16:20 — Solving a stochastic multi-horizon facility location problem with capacity expansion
We study the problem of locating production facilities and selecting production technology for hydrogen in Norway. Hydrogen demand is expected to increase over the next years but is also the main
source of uncertainty. Further, hydrogen production costs highly depend on electricity prices, which are also uncertain.
We present a stochastic multi-stage multi-horizon formulation, which enables the modelling of strategic as well as operational uncertainty. The strategic uncertainty is related to increasing but
uncertain future demand, while the operational uncertainty is related to uncertain future electricity costs directly affecting the operational costs. We consider two production technologies, which
differ in production flexibility and costs, and allow for their combination at one location. Therefore, capacity expansion is modelled as the opening of a new facility. The objective is to minimize
the expected sum of investment, production and distribution costs while satisfying customer demand.
We use Lagrangian relaxation to solve our problem. The lower bound is calculated by solving the linear relaxation of the Lagrangian subproblem. We implement a heuristic based on a restricted MIP
approach to obtain a feasible solution. The results show that our solution method can find high-quality solutions for large problem instances with up to 150 operational scenarios.
2 — 16:50 — A slope scaling heuristic for the multiperiod strategic planning of carbon capture and sequestration value chains
Around the world, various decarbonization strategies are being evaluated to support countries meeting their net-zero emissions objectives. One of these decarbonization strategies is carbon capture
and sequestration (CCS). In CCS, CO2 is captured at emitter sites (e.g. industrial plants) and transported to geological sequestration sites (e.g. saline aquifers, depleted oil fields), where it is
injected underground for long-term storage (10,000+ years). Recent studies indicate that without CCS, countries may not be able to meet their net-zero carbon emissions objectives.
The deployment of CCS involves billions of dollars of investment and requires planning ahead for decades. For this reason, in this presentation, we focus on the multiperiod strategic planning of a
pipeline-based CCS value chain. This problem combines characteristics of two classical problems in Operations Research: facility location and network design. Facility location decisions are related
to the activation and operation of CO2 capture units at emitter sites, and the opening and operation of geological reservoirs. Network design characteristics consist of the activation and operation
of the pipeline network.
To account for multiple potential sources of uncertainty, this problem may have to be solved thousands of times. Therefore, reaching high-quality solutions in a few minutes is important.
Computational experiments show that commercial solvers struggle to attain these requirements. Adding further problem attributes such as multiple CO2 transportation modes may exacerbate this issue. To
address this computational challenge, a novel slope scaling heuristic was previously introduced by Homsi, Ayotte-Sauvé, and Jena. This heuristic is based on existing work on single-period CCS
problems and network design problems. It approximates the cost of design variables, has long-term memory search strategies, generates upper bounds with dynamic programming, and has a final improving
phase where a restricted model is solved iteratively.
In this presentation, we provide updated results for the slope scaling heuristic along with additional performance insights. These new results show that the heuristic generates better solutions than
CPLEX for most (58\%) experiments (average relative improvement of 10\%), at a fraction (10\%) of the CPU time. For the same fraction of CPU time, results also show the satisfactory performance of
the heuristic when it underperforms CPLEX: the vast majority of SS solutions (95\%) have a relative cost that is at most 1\% larger than CPLEX.
3 — 17:20 — Cumulative Customer Demand in Facility Location and Network Design
Dynamic Facility Location and Network Design are a popular class of combinatorial optimization problems that provide the infrastructure necessary to satisfy customer demand. They have been applied to
a large variety of planning contexts, where customer demand may have different interpretations, such as consumer goods, manufacturing components, transportation services, or medical relief items.
Existing literature commonly assumes that customer demand quantities are defined independently for each time-period. In many planning contexts, however, unmet demand carries over to future time
periods. Demand that is unmet for some time periods may therefore affect decisions of subsequent time periods.
In this talk, we discuss the implications of cumulative customer demand in two representative planning contexts for multi-period network design and facility location: humanitarian supply chains,
where unmet demand for relief aid may result in a spread of disease, and therefore increases future demand; and temporary facility location, such as pop-up stores, where customer demand for necessary
items gradually builds up until it is satisfied by a nearby facility. We discuss two principal types of cumulative demand propagation and their respective modeling in mathematical programming
formulations. We show how a failure of modeling such behavior may result in severe economic underperformance, how the corresponding models can be modeled and solved efficiently, and how different
problem characteristics impact the computational complexity. | {"url":"https://ismp2024.gerad.ca/schedule/MC/202","timestamp":"2024-11-03T15:32:13Z","content_type":"text/html","content_length":"21514","record_id":"<urn:uuid:abe672a1-5adb-485a-b1fd-1d9cbf59caf5>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00778.warc.gz"} |
For years, the logic course in the first year of our Bachelor in Computer Science was a serious hurdle for many students. Most of the students computer science perceive the formal and abstract
mechanisms of logic as difficult and awkward to deal with. We have seen a lot of procrastination with regard to studying this subject. As a result, the basic principles were often insufficiently
known and quickly the students could no longer follow the explanations given during the lectures and exercise sessions. This in turn led to reduced motivation or even dropouts.
Since educational games are commended as an enjoyable and effective way for learning, we decided in 2015 to develop an educational game for the course. We decided to first focus on practicing the
truth tables of proposition logic, as a good knowledge of the truth tables is essential for understanding more complex topics introduced later on in the course. This resulted in the development of
the TrueBiters game.
A first version of the game was developed by a master student, Eman El Sayed, in the context of her master's thesis during the academic year 2015-2016. The master thesis describing this first version
of the game, its development and evaluation can be found here. This first, but limited evaluation was promising. Next, the game has been improved on various aspects, such as graphics, supported
platforms and game modes, and was embedded into the course. The Education Innovation Fund (OVP) of the VUB supported these developments.
TrueBiters was inspired by a card game called “bOOleO” on Boolean logic. We adapted the game to proposition logic and digitized it. Since most of our students have a smartphone and playing games on
smartphones is popular among youngsters, we decided to develop a game for which they could use their smartphone with typical gesture-based interactions.
The goal of the game is not to teach the truth tables but to practice their use. In this way, the game is complementary to the lectures, but can be used as a replacement for some of the exercise
The evaluations carried out in the autumn semester of 2017 with the university students, show that the game is a good complement for the traditional face-to-face exercise session, that students who
played the game have better results, but also that making the game available without obligation to use it, was not the best approach.
Here we provide a short description of the last version of the game. Related articles and links to manuals and the software are given below.
The principles of TrueBiters
The game allows for practicing the basic logical operators of propositional logic: AND, OR, IMPLY, EQUIVALENT, and NOT.
In principle it is a two-players competitive game, but the game can also be played alone.
At the start, the game generates six random binary values (1 or 0, called bits). As is common in logic, the value 1 represents TRUE and the value 0 represents FALSE. These bits are placed at the top
of a triangular board composed of tiles (see figure 1), and the goal for a player is to reduce the sequence of bits into a single bit, which should corresponds to the rightmost bit of the sequence.
Figure 1: The triangular board
Reducing the bits is done by filling the triangular board step by step, each time applying a logic operator on two bits; in this way each time two bits are reduced into one bit.
Each binary logical operator is represented by a fictive creature that can eat two bits and spits out one bit (see Figure 2 for some examples). Each creature type comes in two versions: one that
spits out the 1-value and one that spits out the 0-value. This is because each logical operator can result in TRUE or FALSE depending on the input values. The player should use the correct version of
the creatures when reducing the bits otherwise the reduction is invalid and the turn is over. This will force the players to be very familiar with the truth tables of the logical operators because
each mistake will result in a turn lost.
Figure 2: Some example of the creatures that eat bits
When two players play the game, both players has a similar goal, i.e. reducing a sequence of bits into a single bit but the players’ sequences are inverted, i.e. each bit is the inverse of the
corresponding bit given to the other player. Each player has his own triangular board to reduce his bits. The two players play in turn. The first player that achieves his goal is the winner.
The types of creatures, as well as the number of creatures that a player has at his disposal depend on the difficulty level of the game. In the difficulty levels "medium" and "hard", a timer is used,
i.e. the player has to make a reducing in a given time. At most 6 creatures are visible at a turn. There is no guarantee that the player can finish the game with the creatures received
In the two-players mode, each player can use his own device or they can share one device. Figure 3 shows a screenshot of the screen for the two players version. When they are sharing a single device,
the players can see each other’s creatures, which is not the case when each is using an own device. In that case the devices communicate by Bluetooth.
Figure 3: Two players sharing the same device
The game is available as iOS app, Android app and also a web-version is available.
Software and Manuals
A Dutch manual is available here
Here is a Dutch instruction clip:
An English manual and instruction clips will become available later.
The game is available for free. Download the Android version or the iOS version from the Apple App Store (name: Truebiters) or play the Web version .
Google Play Store:
Apple App Store:
Related Publications | {"url":"https://wise.vub.ac.be/index.php/project/truebiters","timestamp":"2024-11-11T20:29:25Z","content_type":"text/html","content_length":"38653","record_id":"<urn:uuid:b4ff9492-8ceb-403a-bc09-3ad3aaf7756e>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00237.warc.gz"} |
GSEB Class 7 Maths Notes Chapter 3 Data Handling
This GSEB Class 7 Maths Notes Chapter 3 Data Handling covers all the important topics and concepts as mentioned in the chapter.
Data Handling Class 7 GSEB Notes
Today data handling is one of the most important tasks in any organisation. In hospitals data needs to be maintained. In schools cumulative records of students are kept for further reference.
Data handling includes collection, interpretation or presentation of data by using various methods.
1. Data: A data is a collection of facts in the form of numerical figures that is used to provide some information.
2. Observation: Each numerical figure (entry) in a data is called an observation (or variate)
3. Frequency: The number of times a particular observation occurs in the data is called its frequency.
4. Tally Marks: Tally marks are useful in counting observations. We write tally marks in bunches of five like
5. Arrayed data: A data arranged in ascending or descending order is called arrayed data.
6. Frequency Table: A table showing the frequency of various observations is called a frequency distribution table or frequency table.
7. Arithmetic Mean: Arithmetic mean usually termed a mean, is one of the measures of central tendency as it gives the average value of the given data.
The average or Arithmetic Mean (A.M.) or simply mean is defined as follows:
Mean = \(\frac{\text { Sum of all observations }}{\text { Number of observations }}\)
8. Range: The difference between the highest observation and the lowest observation is called the range of the data. [Range = Highest observation – Lowest observation].
• Mode: The observation which occurs maximum number of times in the given data is called mode (or modal value) of the data.
• Mode of Larger Data: Putting the same observations together and counting them is not easy if the number of observations is large. In such cases we tabulate the data. Tabulation can begin by
putting tally marks and finding the frequency.
• Median: Median is that value which lies in the middle of the data. When observations are arranged in ascending or descending order with half of the observations above it and the other half below
Bar Graphs
A bar graph is a representation of numbers using bars of uniform width. In bar graphs, the height (or length) of a bar represents the frequency of the corresponding observation. All bars must be of
equal width and there should be equal gap between the adjoining bars.
Choosing a scale
In a bar graph where numbers in units are to be shown, the graph represents one unit length for our observation. If it has to show number in tens or hundreds one length can represent 10 or 100
Double Bar Graphs
A graph that displays two sets of data using two bars drawn besides each other is called a double bar graph. This graph helps us to compare two collections of data at a glance.
Chance is the occurrence of events. It is the possibility of something happening.
Look at the statements given below and try to understand these terms.
• The sun rises from the west.
• An ant growing to 3 m height.
• India winning the next cricket match.
If we look at the statements given above we would say that the sun rises from the West is impossible, an ant growing to 3 m is also not possible. Where as India can win the match or lose it. Both are
When a die is thrown probability of getting either of 1, 2, 3, 4, 5 or 6 is equal. For a die, there are 6 equally likely possible outcomes. We say that each of 1, 2, 3, 4, 5, 6 has one – sixth \(\
frac{1}{6}\) probability.
The probability of an event E, written as P (E), is defined as
P(E) = \(\frac{\text { Number of favourable outcomes }}{\text { Total number of outcomes }}\)
Number of favourable outcomes Total number of outcomes We shall study more about this in next classes.
Event that have many possibilities can have probability between 0 and 1. Those which have no chance of happening have probability 0 and those that are bound to happen have a probability 1.
• An experiment is a situation that involves a chance of the occurrence of a particular event.
• An outcome is the result of an experiment.
• Sample space is the set of all possible outcomes in an experiment.
• An event is a specific outcome of an experiment.
Leave a Comment | {"url":"https://gsebsolutions.in/gseb-class-7-maths-notes-chapter-3/","timestamp":"2024-11-07T04:29:04Z","content_type":"text/html","content_length":"238547","record_id":"<urn:uuid:3dec009c-ce1f-4978-b2c5-b2285d85828d>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00148.warc.gz"} |
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CPM.2020.22
URN: urn:nbn:de:0030-drops-121471
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/12147/
Lafond, Manuel ; Zhu, Binhai ; Zou, Peng
Genomic Problems Involving Copy Number Profiles: Complexity and Algorithms
Recently, due to the genomic sequence analysis in several types of cancer, genomic data based on copy number profiles (CNP for short) are getting more and more popular. A CNP is a vector where each
component is a non-negative integer representing the number of copies of a specific segment of interest. The motivation is that in the late stage of certain types of cancer, the genomes are
progressing rapidly by segmental duplications and deletions, and hence obtaining the exact sequences becomes difficult. Instead, the number of copies of important segments can be predicted from
expression analysis and carries important biological information. Therefore, significant research has recently been devoted to the analysis of genomic data represented as CNP’s.
In this paper, we present two streams of results. The first is the negative results on two open problems regarding the computational complexity of the Minimum Copy Number Generation (MCNG) problem
posed by Qingge et al. in 2018. The Minimum Copy Number Generation (MCNG) is defined as follows: given a string S in which each character represents a gene or segment, and a CNP C, compute a string T
from S, with the minimum number of segmental duplications and deletions, such that cnp(T)=C. It was shown by Qingge et al. that the problem is NP-hard if the duplications are tandem and they left the
open question of whether the problem remains NP-hard if arbitrary duplications and/or deletions are used. We answer this question affirmatively in this paper; in fact, we prove that it is NP-hard to
even obtain a constant factor approximation. This is achieved through a general-purpose lemma on set-cover reductions that require an exact cover in one direction, but not the other, which might be
of independent interest. We also prove that the corresponding parameterized version is W[1]-hard, answering another open question by Qingge et al.
The other result is positive and is based on a new (and more general) problem regarding CNP’s. The Copy Number Profile Conforming (CNPC) problem is formally defined as follows: given two CNP’s C₁ and
C₂, compute two strings S₁ and S₂ with cnp(S₁)=C₁ and cnp(S₂)=C₂ such that the distance between S₁ and S₂, d(S₁,S₂), is minimized. Here, d(S₁,S₂) is a very general term, which means it could be any
genome rearrangement distance (like reversal, transposition, and tandem duplication, etc). We make the first step by showing that if d(S₁,S₂) is measured by the breakpoint distance then the problem
is polynomially solvable. We expect that this will trigger some related research along the line in the near future.
BibTeX - Entry
author = {Manuel Lafond and Binhai Zhu and Peng Zou},
title = {{Genomic Problems Involving Copy Number Profiles: Complexity and Algorithms}},
booktitle = {31st Annual Symposium on Combinatorial Pattern Matching (CPM 2020)},
pages = {22:1--22:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-149-8},
ISSN = {1868-8969},
year = {2020},
volume = {161},
editor = {Inge Li G{\o}rtz and Oren Weimann},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/12147},
URN = {urn:nbn:de:0030-drops-121471},
doi = {10.4230/LIPIcs.CPM.2020.22},
annote = {Keywords: Computational genomics, cancer genomics, copy number profiles, NP-hardness, approximation algorithms, FPT algorithms}
Keywords: Computational genomics, cancer genomics, copy number profiles, NP-hardness, approximation algorithms, FPT algorithms
Collection: 31st Annual Symposium on Combinatorial Pattern Matching (CPM 2020)
Issue Date: 2020
Date of publication: 09.06.2020
DROPS-Home | Fulltext Search | Imprint | Privacy | {"url":"http://dagstuhl.sunsite.rwth-aachen.de/opus/frontdoor.php?source_opus=12147","timestamp":"2024-11-06T22:13:59Z","content_type":"text/html","content_length":"8333","record_id":"<urn:uuid:3ceeaaf9-ed76-4849-a56a-d1ae7f88863e>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00039.warc.gz"} |
American Mathematical Society
Critical points, critical values, and a determinant identity for complex polynomials
HTML articles powered by AMS MathViewer
by Michael Dougherty and Jon McCammond
Proc. Amer. Math. Soc. 148 (2020), 5277-5289
DOI: https://doi.org/10.1090/proc/15215
Published electronically: September 18, 2020
PDF | Request permission
Given any $n$-tuple of complex numbers, one can easily define a canonical polynomial of degree $n+1$ that has the entries of this $n$-tuple as its critical points. In 2002, Beardon, Carne, and Ng
studied a map $\theta \colon \mathbb {C}^n\to \mathbb {C}^n$ which outputs the critical values of the canonical polynomial constructed from the input, and they proved that this map is onto. Along the
way, they showed that $\theta$ is a local homeomorphism whenever the entries of the input are distinct and nonzero, and, implicitly, they produced a polynomial expression for the Jacobian determinant
of $\theta$. In this article we extend and generalize both the local homeomorphism result and the elegant determinant identity to analogous situations where the critical points occur with
multiplicities. This involves stratifying $\mathbb {C}^n$ according to which coordinates are equal and generalizing $\theta$ to a similar map $\mathbb {C}^\ell \to \mathbb {C}^\ell$ where $\ell$ is
the number of distinct critical points. The more complicated determinant identity that we establish is closely connected to the multinomial identity known as Dyson’s conjecture. Similar Articles
• Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 30C10, 30C15, 05A10, 57N80
• Retrieve articles in all journals with MSC (2010): 30C10, 30C15, 05A10, 57N80
Bibliographic Information
• Michael Dougherty
• Affiliation: Department of Mathematics and Statistics, Swarthmore College, Swarthmore, Pennsylvania 19081
• MR Author ID: 938590
• Email: mdoughe1@swarthmore.edu
• Jon McCammond
• Affiliation: Department of Mathematics, University of California Santa Barbara, Santa Barbara, California 93106
• MR Author ID: 311045
• Email: jon.mccammond@math.ucsb.edu
• Received by editor(s): September 5, 2019
• Received by editor(s) in revised form: June 2, 2020
• Published electronically: September 18, 2020
• Communicated by: Harold P. Boas
• © Copyright 2020 American Mathematical Society
• Journal: Proc. Amer. Math. Soc. 148 (2020), 5277-5289
• MSC (2010): Primary 30C10, 30C15; Secondary 05A10, 57N80
• DOI: https://doi.org/10.1090/proc/15215
• MathSciNet review: 4163840 | {"url":"https://www.ams.org/journals/proc/2020-148-12/S0002-9939-2020-15215-X/?active=current","timestamp":"2024-11-10T15:33:21Z","content_type":"text/html","content_length":"63409","record_id":"<urn:uuid:24aba84d-9d4e-4ced-8d68-58e5e9f6859d>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00333.warc.gz"} |
Introducing Opensource ERC-4337 Gas Estimation Package | Biconomy
Introducing Opensource ERC-4337 Gas Estimation Package
Estimating userOp gas limits for Entry Point v0.6 is one of the most complicated parts of Account Abstraction (ERC4337) infra. All AA infra providers have faced issues of inaccurate gas estimations.
If gas estimations are not accurate, it does more harm to UX and makes the flow and experience even more complicated.
To tackle these issues faced by the AA builder community, we have come up with a way to estimate each gas limit all wrapped inside an easy-to-use and lightweight SDK that is open source and for the
Why accurate gas estimation is important
If gas estimations are not done the right way, the following issues can come up:
For the userOps involving Token Paymasters, the amount of ERC20 tokens to be paid can get pretty off very quickly and may result in the user having the balance but still not being able to pay because
the fees shown are higher than what will get consumed
- If the gas limit for the call data execution is low, the execution will fail on the chain without the user coming to know and an unclear message will pop up on every explorer.
- Estimating various validation styles is hard and sometimes inaccurate as while estimating the dummy values will not run all the steps and not capture the actual gas usage meaning one needs to keep
adding buffers over the estimated gas values
- The Entry Point’s formula of max gas required gets pretty high in cases of paymasters even though it might not even require such high gas values and funds in the paymaster
- Calculating the roll-up cost and adjusting in preVerficaitonGas so that bundlers don’t lose money and users don’t overpay has been a challenge on various L2s where the math to do it is unclear.
Benefits of using the SDK:
• Bundlers don’t need to reinvent the wheel for gas estimations. They can focus on other challenging aspects of running a bundler like transaction management, memory management, scalability, or
• Apps don’t need to depend on the eth_estimateUserOperationGas always and in simplified cases can use the package and decrease a lot of latency. This is specifically useful for dapps looking for
very very low latency transactions
• The approach used in the SDK uses what node clients use to tackle the difficulties faced in estimating gas. It is the standard way to handle the edge cases involved in it. These cases cannot be
covered by calling the simulation methods on the Entry Point contract.
Gas values involved in a UserOp
First of all, let's look at the various gas values involved in a userOp:
1. callGasLimit: The gas required to execute the call data part of the userOp which is the call from the Entry Point to the Smart Account
2. verificationGasLimit: The gas required to run all validation checks and deploy the wallet if the case be
3. preVerificationGas: This gas value is the only value that is not a limit but a direct number that accounts for all the gas that cannot be measured on the chain. This typically involves the base
gas cost and for roll-ups, it has to take in the roll-up fee.
Problem with estimating with the simulateHandleOp method
Entry Point provides a simulateHandleOp method that simulates the validation and call data execution phases.
Issues with callGasLimit:
The problem with using simulateHandleOp to calculate callGasLimit is that on-chain it won’t calculate correctly because the current way is to capture the paid field from simulateHandleOp revert data
and divide it by the max fee values to get the gas used in the simulateHandleOp gas metering and then further subtract the preOpGas from it. This should work ideally but is not accurate for two major
• This above logic includes postOp gas and there is no way to separate it from the main callData execution gas in EP v6
• Another is the 63/64 EVM rule. Since EIP-150, the use of the CALL opcode (and all its variants) cannot consume more than 63/64 of the remaining gas. As a transaction’s call stack gets deeper,
more gas must be reserved upfront to meet the gas requirements of higher call frames.
• One needs to send a nonzero maxFeePerGas value to capture the value in paid which forces a smart account to have funds even though a paymaster might be involved in future steps of execution. This
gets solved for networks supporting state overrides but remains an issue on networks where eth_call does not support state overrides
Due to the complications of callGasLimit the general approach is to move the callGasLimit calculation outside of the entry point. This can be achieved by using this extra call execution inside
This allows us to call any contract with some data after the validation step is done. This is super helpful as we can now avoid call data being executed in the entry point flow by setting
callGasLimit as 0 and forcing the execution to happen in a different logical flow.
Issues with verificationGasLimit:
For verificationGasLimit using simulateHandleOp should ideally work but we have improved it by using on-chain Binary Search. Alchemy introduced a way to calculate callGasLimit with Binary Search on
the chain, we have taken inspiration from it and adopted it for the calculation of verificationGasLimit and modified the contract a bit for callGasLimit.
A thing to note is that verificationGasLimit can never be fully captured. It involves calling validation modules that have signature checks. While estimating one sends dummy signatures which will
revert in simulations hence the full gas will never be estimated.
How do we do gas estimation?
A lot of our code and inspiration comes from Alchemy’s and Pimlico’s gas estimation style.
We use two special contracts for estimating callGasLimit and verificationGasLimit which run a binary search algorithm to find the optimal gas limit. Alchemy’s Call Gas Estimation Proxy inspires these
contracts with modifications to handle edge cases that we observed with some Smart Account implementations.
Both sets of contracts extend the entry point contract and are never deployed but replace the entry point code using state overrides. This ensures that no edge case is broken where a particular
entity (SA, paymaster, etc) might enforce that calls should be made from the Entry Point only.
We call the estimateVerificationGas method which first calls the entry point methods with a gas limit of 30M (max block gas) to check if the user operation is even valid. Once that is successful we
start the binary search till the execution runs out of gas and we have a gas value that had the successful binary search which we return as the verificationGasLimit. We also override the callGasLimit
to 0 as we don’t need to run call data execution
The algorithm is the same as what happens in estimateVerificationGas.
In this also we override the callGasLimit to 0 to make sure that the call data execution does not run inside the executeUserOp method of simulateHandleOp and is fully executed inside the
estimateCallGas method.
preVerificationGas is tricky to calculate but essentially one needs to have a way to calculate how much total cost it would be for the bundler to send a userOp. This should include the unaccounted L2
cost plus the roll-up fee to the L1.
For L2s as we have to calculate the roll-up fee, the exact logic for each is described network side below:
• We create the handleOpsData that the L2 has to post on L1:
For Arbitrum:
For Scroll:
For Mantle:
This piece is authored by Yash Chaudhary. Follow him on twitter. | {"url":"https://www.biconomy.io/post/introducing-our-open-source-gas-estimation-package-for-erc-4337-infra","timestamp":"2024-11-12T22:32:22Z","content_type":"text/html","content_length":"103344","record_id":"<urn:uuid:bfd82a6e-968e-4010-b4ff-f72c07db6db6>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00604.warc.gz"} |
925 kilometers per square second to decameters per square second
4,925 Kilometers per square second = 492,500 Decameters per square second
Acceleration Converter - Kilometers per square second to decameters per square second - 4,925 decameters per square second to kilometers per square second
This conversion of 4,925 kilometers per square second to decameters per square second has been calculated by multiplying 4,925 kilometers per square second by 100 and the result is 492,500 decameters
per square second. | {"url":"https://unitconverter.io/kilometers-per-square-second/decameters-per-square-second/4925","timestamp":"2024-11-14T14:48:18Z","content_type":"text/html","content_length":"27217","record_id":"<urn:uuid:f0d8008f-5c15-4d06-a8c8-5459d9e161dc>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00537.warc.gz"} |
signum function pronunciation
Information about the function, including its domain, range, and key data relating to graphing, differentiation, and integration, is presented in the article. Overview/Introduction-Functions-Graph of
a Function-Classification of Functions-One-Valued and Many-Valued Functions-The Square Root-The Absolute Value Symbol-The Signum Function-Definition of a Limit-Theorems on Limits-Right-Hand and
Left-Hand Limits-Continuity-Missing Point Discontinuities-Finite Jumps-Infinite Discontinuities 3. signum function pronunciation - How to properly say signum function. The graph of a signum function
is as shown in the figure given above. Unit Impulse Function. Definition, domain, range, solution of Signum function. Therefore, its Fourier transform is (22) F q = 1 2 δ q + q 0 + δ q − q 0 − i 2 π
q − q 0 + i 2 π q + q 0. For all real we have . Alternatively, you could simply let the user define the matrix X and use it as an input for the function. Let’s Workout: Example 1: Find the greatest
integer function for following (a) ⌊-261 ⌋ (b) ⌊ 3.501 ⌋ (c) ⌊-1.898⌋ Solution: According to the greatest integer function definition A function cannot be continuous and discontinuous at the same
point. Similarly, . sign(x) Description. For a simple, outgoing source, (21) e i 2 π q 0 x = cos 2 π q 0 x + i sin 2 π q 0 x sgn x, where q 0 = 1 / λ. Suppose is a positive integer. Note that the
procedure signum/f is passed only the argument(s) to the function f. The value of the environment variable is set within signum, and can be checked by signum/f, but it is not passed as a parameter.
No, it is not continuous every where . Unit Step Function. The precise definition of the limit is discussed in the wiki Epsilon-Delta Definition of a Limit. Study Physics, Chemistry and Mathematics
at askIITians website and be a winner. Meaning of signum function. Hypernyms . Formal Definition of a Function Limit: The limit of f (x) f(x) f (x) as x x x approaches x 0 x_0 x 0 is L L L, i.e.
where denotes the magnitude (absolute value) of . Note that if the routine signum/f is not able to determine a value for signum(f()) then it can return FAIL. signum (plural signums or signa) A sign,
mark, or symbol. Listen to the audio pronunciation of Signum Fidei on pronouncekiwi How To Pronounce Signum Fidei: Signum Fidei pronunciation + Definition Sign in to disable ALL ads. The cosine
transform of an odd function can be evaluated as a convolution with the Fourier transform of a signum function sgn(x). Yes the function is discontinuous which is right as per your argument. Thus, at
x = 0, it is left undefined. »Returns the signum function of the argument«.. Apart from that, I believe both majority viewpoints about the right approach to define such a function are in a way
correct, and the "controversy" about it is actually a non-argument once you take into account two important caveats: Signum manipuli - Wikipedia "He carried a signum for a cohort or century." Syntax.
function; Translations . Watch Queue Queue Pronunciations of proper names are generally given as pronounced as in the original language. The signum function is the real valued function defined for
real as follows. It's not uncommon to need to know the arithmetic sign of a value, typically represented as -1 for values < 0, +1 for values > 0, or 0. In mathematics, the sign function or signum
function (from signum, Latin for "sign") is an odd mathematical function that extracts the sign of a real number. Permalink. Synonyms (bell): signum bell (math): signum function, sign function View a
complete list of particular functions on this wiki Definition. Some … I cannot find the American English (San Francisco) pronunciation of »signum« in a dictionary. Newbie; Posts: 17; Karma: 0 ; sgn /
sign / signum function suggestions. A statement function should appear before any executable statement in the program, but after any type declaration statements. Is the »s« voiced or voiceless?
Anthony, looking good. A statement function in Fortran is like a single line function definition in Basic. Definition. In applications such as the Laplace transform this definition is adequate, since
the value of a function at a single point does not change the analysis. Signifer - Wikipedia. It is written in the form y … The function’s value stays constant within an interval. Example Problems.
The signum function, denoted , is defined as follows: Note: In the definition given here, we define the value to The signum function is the real valued function defined for real as follows. Of course
it is continuous at every x € R except at x = 0 . For a complex argument it is defined by. Unit step function is denoted by u(t). For a complex argument it is defined by. A sinc pulse passes through
zero at all positive and negative integers (i.e., t = ± 1, ± 2, …), but at time t = 0, it reaches its maximum of 1.This is a very desirable property in a pulse, as it helps to avoid intersymbol
interference, a major cause of degradation in digital transmission systems. Information and translations of signum function in the most comprehensive dictionary definitions resource on the web. Jul
19, 2019, 05:35 pm. Sign function (signum function) collapse all in page. Graphing functions by plotting points Definition A relation f from a set A to a set B is said to be a function if every input
of a set A has only one output in a set B. See for example . In other words, the signum function project a non-zero complex number to the unit circle . All the engineering examinations including IIT
JEE and AIEEE study material is available online free of cost at askIITians.com. for the Diacritically Challenged, This guide includes most mathematicians and mathematical terms that are encountered
in high school and college. Any real number can be expressed as the product of its absolute value and its sign function: From equation (1) it follows that whenever x is not equal to 0 we have.
Solution for 1. (mathematics) A function that extracts the sign of a real number x, yielding -1 if x is negative, +1 if x is positive, or 0 if x is zero. I think the question wanted to convey this..
A medieval tower bell used particularly for ringing the 8 canonical hours. signum function (plural signum functions) (mathematics) The function that determines the sign of a real number, yielding -1
if negative, +1 if positive, or otherwise zero. In other words, the signum function project a non-zero complex number to the unit circle . These are useful in defining functions that can be expressed
with a single formula. That is why, it is not continuous everywhere over R . Signum Function . The signum function of a real number x … Community ♦ 1 1 1 silver badge. Mathematics Pronunciation
Guide. Listen to the audio pronunciation in several English accents. Similarly, . German: Signumfunktion f; Hungarian: előjelfüggvény; Russian: зна́ковая фу́нкция f (znákovaja fúnkcija) Turkish: işaret
fonksiyonu; Usage notes . Use the e-8 definition of the limit to show that x3 – 2x + 4 lim X-ix2 + 4x – 3 3 !3! of the . Signum function is an integer valued function defined over R . Use the
sequential criteria for limits, to show that the… It has a jumped discontinuity which means if the function is assigned some value at the point of discontinuity it cannot be made continuous. In
mathematical expressions the sign function is often represented as sgn. »U« as in »soon« or »number« or just a schwa? Post by Stefan Ram »Returns the signum function of the argument«.. A
Megametamathematical Guide. Sign function - Wikipedia "The signum manipuli (Latin for 'standard' of the maniple, Latin: manipulus) was a standard for both the centuriae and the legion." »I« as in
»it« or as in »seal«? answered Sep 16 '18 at 14:26. The format is simple - just type f(x,y,z,…) = formula . is called the signum function. where denotes the magnitude (absolute value) of . A sinc
function is an even function with unity area. Here, we should point out that the signum function is often defined simply as 1 for x > 0 and -1 for x < 0. Definition. function. This video is
unavailable. Serge Stroobandt Serge Stroobandt. 1 Definition; 2 Properties; 3 Complex signum; 4 Generalized signum function; 5 See also; 6 Notes; Definition. Stefan Ram 2014-05-14 22:55:02 UTC . Area
under unit step function is unity. davidhbrown. Contents. The signum function is defined as f(x) = |x|/x; x≠0 = 0; x = 0 . Definition of signum function in the Definitions.net dictionary. Y = sign(x)
returns an array Y the same size as x, where each element of Y is: 1 if the corresponding element of x is greater than 0. For instance, the value of function f(x) is equal to -5 in the interval [-5,
-4). Note that zero (0) is neither positive nor negative. For all real we have . Signum definition is - something that marks or identifies or represents : sign, signature. Put the code into a
function and use inputdlg to allow inputting the numbers more easily. If then also . SIGN(x) returns a number that indicates the sign x: -1 if x is negative; 0 if x equals 0; or 1 if x is positive.
In general, there is no standard signum function in C/C++, and the lack of such a fundamental function tells you a lot about these languages. What does signum function mean? The second property
implies that for real non-zero we have . If then also . share | improve this answer | follow | edited Jun 20 at 9:12. The signum function of a real number x is defined as follows: Properties. Topic:
sgn / sign / signum function suggestions (Read 2558 times) previous topic - next topic. The second property implies that for real non-zero we have . This function definition executes fast and yields
guaranteed correct results for 0, 0.0, -0.0, -4 and 5 (see comments to other incorrect answers). for the . The signum vector function is a function whose behavior in each coordinate is as per the
signum function. Explicitly, it is defined as the function: "The signum function is the derivative of the absolute value function, up to (but not including) the indeterminacy at zero." It is defined
as u(t) = $\left\{\begin{matrix} 1 & t \geqslant 0\\ 0 & t. 0 \end{matrix}\right.$ It is used as best test signal. example. Impulse function is denoted by δ(t). Proper American English Pronunciation
of Words and Names. Represented as sgn discontinuous at the same point which is right as your! Even function with unity area: 0 ; sgn / sign / signum function project a non-zero complex to. / signum
function is an even function with unity area Definition of the to. Signum manipuli - Wikipedia `` He carried a signum function suggestions ( Read 2558 times ) previous topic next. Or symbol complex
number to the unit circle with unity area in Basic t. The sign function is the real valued function defined over R the American English ( San Francisco ) pronunciation »! Function ’ s value stays
constant within an interval cohort or century. for 1 online free of at. Into a function signum function pronunciation not find the American English ( San Francisco ) pronunciation »... Right as per
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century. even function with unity area study Physics, Chemistry and at! To show that x3 – 2x + 4 lim X-ix2 + 4x – 3 3! 3! 3 3... Karma: 0 ; x = 0 -4 ) the sign function signum! Format is simple -
just type f ( x ) is equal to -5 in the figure given.! Implies that for real non-zero we have a function and use it as an input for the function functions... Be continuous and discontinuous at the
same point that zero ( 0 ) is neither positive nor negative |. And use it as an input for the function: solution for 1 for ringing the 8 hours... Function defined for real as follows = formula x, y,
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to the unit circle is positive. A schwa = formula to properly say signum function is denoted by U ( t ) € except! Alternatively, you could simply let the user define the matrix x and use to! It « or
just a schwa edited Jun 20 at 9:12, at x 0. Is as per the signum function suggestions ( Read 2558 times ) topic. Cohort or century. Posts: 17 ; Karma: 0 ; =. ( t ) all the engineering examinations
including IIT JEE and AIEEE study material is online... The precise Definition of the limit to show that x3 – 2x + 4 lim X-ix2 + –! Follow | edited Jun 20 at 9:12 an input for the function as the. Is
discontinuous which is right as per your argument is defined as f ( x is. R except at x = 0 written in the most comprehensive dictionary definitions resource on the web an. Solution for 1 x and use
it as an input for the function ’ value. A sinc function is as shown in the wiki Epsilon-Delta Definition of real... Except at x = 0 x = 0 signum function pronunciation x = 0 signa ) sign. In
mathematical expressions the sign function ( signum function suggestions ( Read 2558 times ) previous topic - next.! A non-zero complex number to the unit circle real non-zero we have 1 Definition ;
2 Properties 3. Defined over R an integer valued function defined over R most mathematicians and mathematical that. Askiitians website and be a winner why, it is defined as follows format simple. Can
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dictionary definitions resource on the web [ -5, -4 ) function 5. List of particular functions on this wiki Definition appear before any executable statement the... X-Ix2 + 4x – 3 3! 3! 3! 3! 3! 3!
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that for real non-zero we have and... ) pronunciation of » signum « in a dictionary second property implies that for real follows. -4 ) follows: Properties, it is continuous at every x € R except at
x = 0 sgn... Signum vector function is denoted by U ( t ) non-zero we.., domain, range, solution of signum function suggestions of particular functions on this wiki Definition project... Cost at
askIITians.com soon « or just a schwa ’ s value stays constant within interval... A single formula times ) previous topic - next topic He carried a signum for a cohort or.... Vector function is as
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as an input for the function ’ s value stays constant within an interval vector function is denoted δ! The figure given above ) previous topic - next topic of signum function ; 5 also... Number x …
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valued function defined for real non-zero we have unit circle ) neither! This guide includes most mathematicians and mathematical terms that are encountered in high school and college on this
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= 0 per signum... Study Physics, Chemistry and Mathematics at askIITians website and be a winner Definition of the is. This wiki Definition defined over R Posts: 17 ; Karma: 0 ; =., -4 ) all the
engineering examinations including IIT JEE and AIEEE study material is available online free cost... Diacritically Challenged, this guide includes most mathematicians and mathematical terms that are
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or century. 0, it is continuous at every x € R at., looking good x … Anthony, looking good of cost at askIITians.com type f ( )... Line function Definition in Basic each coordinate is as per the
signum suggestions... Alternatively, you could simply let the user define the matrix x and it... X = 0 δ ( t ), y, z, … ) = formula project a complex. It « or just a schwa continuous at every x € R
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4x – 3 3! 3 3. … ) = |x|/x ; x≠0 = 0 ; sgn / sign / signum function a.! Single formula the unit circle topic - next topic program, but after any type declaration.! In » soon « or as in » it « or as
in » «. Denotes the magnitude ( absolute value ) of U ( t ) implies that for real non-zero we.... Most mathematicians and mathematical terms that are encountered in high school and college as follows
are useful in functions... Whose behavior in each coordinate is as shown in the program, but after type... Like a single line function Definition in Basic « in a dictionary the most comprehensive
dictionary definitions resource on web... Is continuous at every x € R except at x = 0 ; sgn / sign / function! School and college follow | edited Jun 20 at 9:12 function of a limit an even function
with area! Y … Definition an integer valued function defined over R right as per the signum function suggestions ( Read times! Of course it is continuous at every x € R except at x = 0 discontinuous
which right. Encountered in high school and college just a schwa several English accents « in a dictionary real valued function for... Inputdlg to allow inputting the numbers more easily, y, z, … ) =
;. Next topic figure given above ; 6 Notes ; Definition signum « in a dictionary limit to show x3. The real valued function defined for real as follows at 9:12 find American... Pronounced as in »
seal « type declaration statements sign / signum function ; See... Manipuli - Wikipedia `` He carried a signum function ( Read 2558 times ) previous topic - topic... Written in the form y …
Definition as in » soon « or as »...: 0 ; x = 0 as per your argument absolute value of! Follows: Properties the engineering examinations including IIT JEE and AIEEE study material is online.
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SDGB 7844 HW 1: Chocolate & Nobel Prizes solved
Submit two files through Blackboard: (a) .Rmd R Markdown file with answers and code
and (b) Word document of knitted R Markdown file. Your file should be named as follows:
“HW[X]-[Full Name]-[Class Time]” and include those details in the body of your file.
Complete your work individually and comment your code for full credit. For an example of
how to format your homework see the files posted with Lecture 1 on Blackboard. Show all
of your code in the knitted Word document.
Read the New England Journal of Medicine article, “Chocolate Consumption, Cognitive
Function, and Nobel Laureates” (Messerli, F.H., Vol. 367(16), 1562-1564; 2012) which is
posted with this assignment. We will be using a reconstruction of Messerli’s data. The
variables in the data set you will use are (file: “nobel chocolate.txt” on Blackboard) are
“country”, “nobel rate”, and “chocolate”.
The information gathered in the data set you will be using is from several different sources.
The number of Nobel prize winners is from Wikipedia and includes winners through November 2012, population information (used to compute the “nobel rate” variable) is from the
World Bank, and chocolate market size is from the Euromonitor International’s Passport
Goal: In this assignment, you will be replicating Messerli’s analysis.
1. According to Messerli, what is the variable “number of Nobel laureates per capita”
supposed to measure? Do you think it is a reasonable measure? Justify your answer.
2. Are countries without Nobel prize recipients included in Messerli’s study? If not, what
types of bias(es) would that introduce?
3. Are the number of Nobel laureates per capita and chocolate consumption per capita
measured on the same temporal scale? If not, how could this affect the analysis?
4. Create a table of summary statistics for the following variables: Nobel laureates per
capita, GDP per capita, and chocolate consumption. Include the statistics: minimum,
maximum, median, mean, and standard deviation. Remember to include the units of
measurement in your table.
5. Create histograms for the following variables: Nobel laureates per capita, GDP per
capita, and chocolate consumption. Describe the shape of the distributions.
6. Construct a scatterplot of Nobel laureates per capita vs. chocolate consumption. Label
Sweden on your plot (on the computer, not by hand). Compute the correlation between these two variables and add it to the scatterplot. How would you describe this
relationship? Is correlation an appropriate measure? Why or why not?
7. What is Messerli’s correlation value? (Use the correlation value that includes Sweden.)
Why is your correlation different?
8. Why does Messerli consider Sweden an outlier? How does he explain it?
9. Regress Nobel laureates per capita against chocolate consumption (include Sweden):
(a) What is the regression equation? (Include units of measurement.)
(b) Interpret the slope.
(c) Conduct a residual analysis to check the regression assumptions. Make all plots
within one figure. Can we conduct hypothesis tests for this regression model?
Justify your answer.
(d) Is the slope significant (conduct a hypothesis test and include your regression output
in your answer)? Test at the α = 0.05 level and remember to specify the hypotheses
you are testing.
(e) Add the regression line to your scatterplot.
10. Using your model, what is the number of Nobel laureates expected to be for Sweden?
What is the residual? (Remember to include units of measurement.)
11. Now we will see if the variable GDP per capita (i.e., “GDP cap”) is a better way to
predict Nobel laureates.
(a) In one figure construct a scatter plot of (i) Nobel laureates vs. GDP per capita and
(ii) log(Nobel laureates) vs. GDP per capita. Which plot is more linear? Label
Sweden on both plots. On the second plot, label the two countries which appear
on the bottom left corner.
(b) Is Sweden still an outlier? Justify your answer.
(c) Regress log(Nobel laureates) against GDP per capita. Provide the output and add
the regression line to your scatterplot. (In practice, we would do a residual analysis
here, but we will skip it to reduce the length of this assignment.)
(d) The log-y model is a multiplicative model: log(y) = β0 + β1 is y = e
. For
such a model, the slope is interpreted as follows: a unit increase in x changes y
by approximately (e
β1 − 1) × 100%. For your regression, model interpret the slope
(remember to include units of measurement).
12. Does increasing chocolate consumption cause an increase in the number of Nobel Laureates? Justify your answer. | {"url":"https://codeshive.com/questions-and-answers/sdgb-7844-hw-1-chocolate-nobel-prizes-solved/","timestamp":"2024-11-14T07:16:34Z","content_type":"text/html","content_length":"104217","record_id":"<urn:uuid:6bcd401b-6fc4-403e-9745-c1fa03aedefc>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00385.warc.gz"} |
TI-Basic Developer
Coding Pitfalls
When you are coding, there are several different pitfalls that you have to be aware of. A pitfall is simply code that is syntactically correct, but does not produce the correct or desired results.
This list is not meant to be exhaustive, but rather a general collection of things to look out for.
Arrow Keys
One of the simplest pitfalls that people make is forgetting to use the proper values for the arrow keys. This is especially prevalent with beginners, since they are still learning the ins and outs of
the calculator. For example, when the user is making movement, and wants to update the player position on the board, you will see something like this:
:A+(K=25)-(K=34→A // Y coordinate
:B+(K=24)-(K=26→B // X coordinate
While this code looks right, it actually has the arrow directions flipped around (25 should be swapped with 34, and vice versa for 24 and 26). This code will not generate any errors by the
calculator, since it is syntactically correct, but figuring out the logic problem can be challenging if you don't recognize the mistake.
Boolean Expressions
Another common pitfall that people make is messing up their Boolean expressions. A big part of this is simply people not taking the time to learn and understand how the calculator reads Boolean
expressions and deals with operator precedence.
A Boolean expression is based on Boolean Logic, the principle that something can be either true or false. A true value is represented as 1 or any nonzero value, while a false value is represented as
0. In addition to the four math operators (*,/,+,-), there are six conditional operators (=,≠,>,<,≥,≤) and four logic operators (and,or,xor,not).
The operator precedence that the calculator follows is math operators are executed first, then the conditional operators and finally the logic operators. Of course, if there are parentheses, the
calculator executes what's inside the parentheses first, which can include any one of the three different kinds of operators. Here is an example to illustrate:
:If B=A and CD:Disp "B=A and C*D≠0
:If 5X=2Y+(Y/X≠5:Output(2,2,"Hello
Memory Leaks
Another pitfall to avoid is causing memory leaks with branching out of loops and conditionals and overusing subprograms. This is especially important because memory leaks not only take up more and
more memory, but also slow the calculator down (depending on the size of the memory leak, it can be to a halt).
To prevent memory leaks from occurring, you should always make sure that any loops and conditionals (anything with an End command) are executed to their completion. The reason is that the calculator
keeps track of the End commands for loops and conditionals, so if one of them isn't completed, the calculator isn't able to remove it from its list.
While it is possible to correct a memory leak problem in your pre-existing code, the best time to make those changes is when you are actually planning a program. This is because a properly planned
program can be made to not only have no memory leaks, but also be as fast and small as possible. Of course, if you don't feel like rewriting your code again, a simple fix will suffice.
:If A=5:Then
:Disp "A=5
:Goto A
should be
:If A=5:Disp "A=5
:If A=5:Goto A
One of the most common pitfalls that people make is forgetting about program portability. With all of the Assembly libraries available, and there being several different TI-83 based calculators
available, it is easy to see how portability becomes an issue.
In addition to the Assembly libraries, there are also several new commands that TI has added to the TI-Basic language for the newer TI-84+/SE calculators. While you can use these commands in your
programs, they will crash your programs if somebody tries to execute the program on a TI-83/+/SE calculator.
Unfortunately, the only thing you can do if you want your program to be TI-83/+/SE compatible is to not use these libraries and commands. This means you won't be able to include that functionality in
your program, which can be a big selling point of your program.
If you don't care about your program working on the TI-83/+/SE calculators, then portability isn't an issue for you and you don't have to worry about it. However, you should still tell the user at
the beginning of the program that the program is designed to work on the TI-84+/SE, and in fact will crash if used on a TI-83/+/SE.
The same goes for using Archive/UnArchive if you care about portability to the TI-83 calculator. Additionally, while programs with lowercase letters will work on the TI-83, they can't be sent from a
TI-83+ or higher to a TI-83 via the link cable.
Math Errors
Because of the way the calculator is designed, it has limited precision when doing math. Any math calculations that involve extremely small or large numbers (ranging from -[E]99 to [E]99) will
produce rounding errors that don't return the right number. There are a couple different ways you can deal with this problem.
The round( command will round a floating-point number so that it has the specified number of decimal digits. While you can hypothetically set it so that it has 25 digits, the calculator only allows a
number to have up to 14 digits, which means the range is 0-14 digits.
Another way to deal with the problem is by multiplying the number by a smaller number. The calculator will automatically treat the number in a similar fashion to the smaller number, which allows it
to make the math expression work. Truthfully, neither of these methods is fool-proof, so you should just be aware of the problem.
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12 Trillion Slices of Pi
JF Ptak Science Books Quick Post
The mathematician Hermann Schubert wrote in his 1889 text on the uselessness of calculating pi past 500 digits--I haven't located a copy of the original 1889 publication though the story is often
repeated: I've seen in in Petr Beckmann's A History of Pi (1993 edition) on page 101 and also in Cliff Pickover's Keys to Infinity (John Wiley, 1995).
"Conceive a sphere constructed with the earth at its center, and imagine its surface to pass through Sirius, whis is 8.8 light years distant from the earth [that is, light, traveling at a velocity of
186,000 miles per second, takes 8.8 years to cover this distance]. Then imagine this enormous sphere to be so packed with microbes that in every cubic millimeter millions of millions of these
diminuitive animalcula are present. Now conceive these microbes to be unpacked and so distributed singly along a straight line that every two microbes are as far distant from each other as Sirius
from us, 8.8 light years. Conceive the long line thus fixed by all the microbes as the diameter of a circle, and imagine its circumference to be calculated by multiplying its diameter by to 100
decimal places. Then, in the case of a circle of this enormous magnitude even, the circumference so calculated would not vary from the real circumference by a millionth part of a millimeter."
"This example will suffice to show that the calculation of to 100 or 500 decimal places is wholly useless."
Long before Schbert pi was being calculated to quite a degree: it was computed to 9 places by Francoise Viete in 1579; 15 places by Adriaan van Roonan, 1593; 32 by Ludolph van Ceulen in 1596; 35 by
Willebrord Snell in 1621; 38 by Christoph Grienberger; 75 by Abraham Sharp in 1699; 100 by John Machin in 1706; 137 by Jurj Vega in 1794; and 152 by Legendre in 1794, which is nearly 100 years before
Schubert. William Rutherford came in with 248 in 1847, and then William Shanks with 527 places in 1874. D.F. Fergusson would break 1000 places in 1949, followed by F. Genuys (using the IBM 704)
breaking 10,000 i 1958. Daniel Shanks reached 100,000 in 1961, Jean Guillyud finding 1 million in 1967, and then many others, right up to the 12 trillion mark by Shigeru Kondo in 2013.
All of which leave Dr. Schubert without very much crust.
Schubert was unfortunate to miss Feynman's justification for knowing pi to 762 digits. The desire to recite up to the six consecutive 9s which occur beginning at 762 was driven purely for the joy of
the joke... "Nine nine nine nine nine nine and so forth." | {"url":"https://longstreet.typepad.com/thesciencebookstore/2014/10/12-trillion-slices-of-pi.html","timestamp":"2024-11-07T15:48:19Z","content_type":"application/xhtml+xml","content_length":"47557","record_id":"<urn:uuid:b99a70c0-3f31-4e18-aa6c-85e67a094872>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00708.warc.gz"} |
Using a Two-Way Frequency Table to Determine the Probability of Complement of Event
Question Video: Using a Two-Way Frequency Table to Determine the Probability of Complement of Event Mathematics • Third Year of Preparatory School
The table represents the data collected from 200 conference attendees of different nationalities. Find the probability that a randomly selected participant does not speak English.
Video Transcript
The table represents the data collected from 200 conference attendees of different nationalities. Find the probability that a randomly selected participant does not speak English.
The rows in our table tell us whether the participant was male or female. The columns tell us which language they speak, whether they speak Arabic, English, or French. We are told in the question
that there are a total of 200 attendees. If we let 𝐸 be the event that the conference attendee speaks English, we can calculate the probability of event 𝐸. This will be the number of attendees that
speak English out of the total number of attendees.
There are 35 men who speak English and 30 women, giving us a total of 65 people. The probability that a randomly selected participant speaks English is 65 out of 200 or sixty-five two hundredths. We
are interested in the probability that the participant does not speak English. This is known as the complement. We know that the probability of any complementary event, 𝐴 bar, occurring is equal to
one minus the probability of 𝐴. In this question, the probability of 𝐸 bar, the participant not speaking English, is equal to one minus 65 out of 200. This is equal to 135 out of 200.
We can simplify this fraction by dividing the numerator and denominator by five. 135 divided by five is 27 and 200 divided by five is equal to 40. The probability that a randomly selected participant
does not speak English is 27 out of 40 or twenty-seven fortieths. We could also write this answer as a decimal by firstly considering the fraction 135 out of 200. Dividing the denominator by two
gives us 100. If we divide the numerator by two, we get 67.5 as a half of 100 is 50 and a half of 35 is 17.5. Dividing 67.5 by 100 gives us 0.675. The probability that the randomly selected
participant does not speak English, written as a decimal, is 0.675. We could also write this as a percentage by multiplying 100, giving us 67.5 percent. | {"url":"https://www.nagwa.com/en/videos/738154604209/","timestamp":"2024-11-12T08:42:49Z","content_type":"text/html","content_length":"250163","record_id":"<urn:uuid:348cc9dd-7374-48d5-8bc5-32cdd4196cd8>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00736.warc.gz"} |
Could Someone Please Help Me With Finding The Coordinate Pairs Of The Three Circled Questions I Can Do The Graphing On My Own
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Could someone please help me with finding the coordinate pairs of the three circled questions. I can do the graphing on my own
Sagot :
When you see how easy this is, you'll smack yourself upside the head.
Take #60. Those are two lines.
-- One line is x=-5. That's a vertical line that crosses the x-axis where x=-5, and EVERY POINT on it is at x=-5 no matter what 'y' is at that point.
-- The other line is y=2. That's a horizontal line that crosses the y-axis where y=2, and EVERY POINT on it is at y=2 no matter what 'x' is at that point.
So you have the intersection of two lines. On one of them, 'x' is always -5. On the other one, 'y' is always 2 . Now what do you suppose the coordinates will be at the point where they cross ?
Could it possibly be anything different from (-5, 2) ? ?
In #62:
On the first line, 'y' is always -6. On the other line, 'x' is always 1.
They MUST intersect at (1, -6) .
In #64:
On one line, 'y' is always -1. On the other line, 'x' is always zero.
(The line where 'x' is always zero happens to be the y-axis.)
I'm SURE that by now you know where these two lines intersect.
You don't even have to graph any of these to know where they intersect !
You can just look at the problem and the coordinate pair jumps out at you.
Answer Link
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Trying to simulate electric field lines from collection of point charges
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2 Replies
3 Total Likes
Trying to simulate electric field lines from collection of point charges
So, I'm fairly new to Mathematica; sorry if this ends up being a very dumb question. I've been trying to simulate and plot electric field lines from point charges in Mathematica. The first resource I
found is Wolfram's Electric Field Lines Due To A Collection of Point Charges, but I've been having trouble figuring out how one would incorporate more source charges. What I think I'm not quite
understanding is the first lines of code defining the electric field - I don't understand how p and pp are being defined, or why it wouldn't work to simply add a third term along the lines of qi and
so on. If anyone can offer any insight, I'd be very grateful.
eeX = Compile[{{q, _Real, 1}, {pp, _Real, 2}, {p, _Real, 1}},
Sum[{-((q[[i]] (p[[1]] - pp[[i, 1]]))/
((p[[1]] - pp[[i, 1]])^2 + (p[[2]] - pp[[i, 2]])^2)^(3/2)),
-((q[[i]] (p[[2]] - pp[[i, 2]]))/
((p[[1]] - pp[[i, 1]])^2 + (p[[2]] - pp[[i, 2]])^2)^(3/2)),
{i, Length[pp]}]]
2 Replies
Hi Amy,
This is not their code, but may be a little easier to see into. It defines a function for the potential generated by a single charge, sums this for a list of charges, and then uses E = - grad V to
get the field.
(* set the directory so we export plots it *)
(* define a vector norm that does not use Abs *)
norm[a_] := Sqrt[a.a]
(* the potential at point (x,y) generated by charge q at (px,py) *)
ePot[{x_, y_}, {px_, py_, q_}] := q/norm[{x, y} - {px, py}]
(* the potential of a point charge at the origin *)
p1 = Plot3D[ePot[{x, y}, {0, 0, 1}], {x, -2, 2}, {y, -2, 2}]
(* a list of charges *)
charges = {{-1, 0, 1}, {1, 0, 1}, {0, 1, -1}};
(* total potential at (x,y) from all the charges in a list *)
totPot = Total[ePot[{x, y}, #] & /@ charges];
(* the total potential *)
p2 = Plot3D[totPot, {x, -2, 2}, {y, -2, 2}]
(* the field is minus the gradient of the potential *)
totField = -Grad[totPot, {x, y}];
p3 = StreamPlot[totField, {x, -2, 2}, {y, -2, 2}]
Hey Amy,
If you are given a point p in the plane and two sources pp = {source_1 , source_2}, the field at this point should be the vector sum of the fields induced by the two point sources. In your code, p is
coordinate of a point you are interested and p[[1]] gives the x-coordinate of the point and so on. pp[[1,1]] gives the x-coordinate of source_1 and so on.
You do not have "tjhe third item" item because we are working on the 2D plane.
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Kuta Software Infinite Algebra 1 Answers Pdf Finding Slope From A Graph - Graphworksheets.com
Finding Slope From A Graph Kuta Software Infinite Algebra 1 – 7th Grade Graph Worksheets are a great resource for students studying graphs in school. These worksheets are available in PDF format for
downloading and contain worksheets for each type of graph a student might encounter. They are an excellent way to introduce a student … Read more
Finding Slope From A Graph Kuta Software
Finding Slope From A Graph Kuta Software – 7th Grade Graph Worksheets are a great resource for students studying graphs in school. These worksheets are available in PDF format for downloading and
contain worksheets for each type of graph a student might encounter. These worksheets are a great way to introduce students to graphs and … Read more | {"url":"https://www.graphworksheets.com/tag/kuta-software-infinite-algebra-1-answers-pdf-finding-slope-from-a-graph/","timestamp":"2024-11-11T05:06:21Z","content_type":"text/html","content_length":"54065","record_id":"<urn:uuid:19081d97-24c8-4a0d-8ac7-2fbe5d1e530f>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00514.warc.gz"} |
Quiet Power: Be Aware of Default Values in Circuit Simulators
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Estimated reading time: 8 minutes
August 27, 2020
Quiet Power: Be Aware of Default Values in Circuit Simulators
Simulators are very convenient for getting quick answers without lengthy, expensive, and time-consuming measurements. Simulators range from simple spreadsheet-based illustration tools^[1] to very
sophisticated 3D field solvers^[2]. Somewhere in the middle, we have the generic circuit simulators—the most well-known among them being SPICE. Berkeley SPICE has been the grand-daddy of all SPICE
tools^[3], and these days, there are many professional SPICE variants available. These tools have been around for a long time, and we usually take the validity of their output for granted. While the
tools may be bug-free, no tool can give us perfect answers for just any arbitrary numerical input; sometimes, we can be surprised if we forget about the numerical limits and the limitations imposed
by internal default values.
As an example, I will show a few simulation results on a simple ladder-like power distribution network, all done with the free LTspice simulator ^[4] from Linear Technologies, now part of Analog
Figure 1: LTspice schematics of a simple PDN.
Figure 1 shows the schematic diagram of a simplified ladder model of a point-of-load power distribution network (PDN). The PDN is represented by four cascaded blocks. On the left is an ideal voltage
source with series resistance and inductance modeling the DC source. To its right is a PI model of the PCB with plane resistance and inductance, as well as bulk and ceramic capacitors.
The next block describes the package with its series resistance, inductance, and capacitance. The 10-µF capacitance value suggests that this is not only the static capacitance of the package planes,
but it also represents package capacitors. The last block on the right describes the die with a series RL term, a parallel capacitance, and a parallel load resistance, which is determined by the
nominal voltage and the average power consumption.
Outside of these blocks is a 1A AC current source injecting test current into the silicon node. Since all elements are linear and time-invariant models, the actual current value does not matter, but
the 1A value is convenient because the simulated V (load) output voltage directly gives us impedance without the need for further scaling.
Figure 2: Impedance magnitude and phase of the simple PDN shown in Figure 1.
Figure 2 shows the result. The heavier line is the impedance magnitude with its scale on the left, the phase is the thin line with its scale on the right. We see four resonance peaks and one sharp
dip on the plot. Peaks 1, 2, and 3 come from the anti-resonances of neighboring capacitor banks. For instance, the first peak is formed by Lsrc and Cbulk, and the LC parallel resonance of the 100-nH
and 10000-µF values produce the 5-kHz resonance peak. To find the second peak, which comes from the series inductance of the Cbulk capacitor and the capacitance of Cceramic, we need to know the
assumed inductance of Cbulk.
You will notice that there are no series resistance and inductance symbols in series to the capacitors, so does it mean the simulation assumes zero values for those parasitics? In this
regard, LTspice is unique among the SPICE circuit simulators. We can specify the usual simple parasitics without adding the corresponding schematic elements.
Figure 3: Screenshot explaining the capacitor equivalent circuit in LTspice.
The equivalent circuit, as defined in LT Wiki^[5], is shown in Figure 3. We can specify not only the equivalent series resistance and inductance but also two parallel loss elements and a body
capacitance. These parameters will be frequency-independent entries. But how do we enter these parameters if we don’t want to type up the SPICE deck manually?
Figure 4: Options to enter parasitic values for capacitors in LTspice.
LTspice makes it easy, offering multiple options. In Figure 4, the left portion shows what happens if we move the cursor over a capacitor in the schematic diagram and right-click. A window pops up
where we can manually enter various attributes. On the right, you see the window which pops up when you hold the control key while you right-click. The two windows offer somewhat different choices.
On the left—in addition to the equivalent series resistance, inductance, and body capacitance—we have only one parallel resistance entry. On the right, we can enter every parameter listed in Figure
3, including the initial condition, temperature, and the multiplier (m or x), which is a convenient way to simplify the schematics if we have m number of identical capacitors connected in parallel.
We can also hide parameters or make them visible on the schematic using the checkmark in the last column. For the schematics shown, I turned on the feature only for the capacitance value; otherwise,
the view would become very crowded. Notice that I show the actual parasitic values that were used to generate figure 2. Now, we see that the series inductance of the bulk capacitor is 10 nH, and this
creates the anti-resonance with the 100-µF ceramic capacitor. From these two values, we get a 150-kHz antiresonance frequency, and that is exactly where Peak 2 is. Peak 3 is at 150 MHz, and it
appears to be split by the sharp and deep Notch 4.
Table 1: Parasitic values of capacitors that were used to generate Figure 2.
Table 1 summarizes the capacitor-parasitic values for all four capacitors. We may wonder if the values in this table represent reality because ESR and ESL for the ceramic capacitor appear to be
unrealistically low. Yes, it would be unrealistic to expect these values from a single capacitor, but if we imagine that these values represent ten pieces of 10-µF ceramic capacitor with 5-mOhm ESR
and 1-nH ESL in each, then it looks reasonable.
If we move on to look at the resonance at Peak 3, we realize that it is formed by the 10-nF Cdie capacitance and the equivalent inductance of the entire network looking back from the silicon, which
is the well-known die-package resonance. By the time we properly add up all series and parallel inductances, it comes out around 160 pH. The antiresonance with the 10-nF Cdie value comes out close to
100 MHz, where the split antiresonance peak happens.
Figure 5: Impedance magnitude and phase of the simple PDN showed in Figure 1, but all parallel body capacitance is set to zero.
We still need to understand where the two extra resonances—Notch 4 and Peak 5—come from. To get the answer, we need to go back to Figure 4 and check what happens with the parameters that we did not
fill out. On the left, there are two parameters we left empty: parallel capacitance and parallel resistance. What happens if we explicitly set the body capacitance to zero? The result is shown in
Figure 5. Notch 4 and Peak 5 disappeared, but the rest remained practically unchanged.
Figure 6: Equivalent circuit of inductor parasitics and attribute list.
Now, the resonance pattern makes sense, but there is still something happening. Why do we have 5-mOhm impedance at low frequencies, when the circuit calls out only 1 mOhm and three times 0.1-mOhm
resistor values in the series path, altogether 1.3-mOhm series resistance? We need to look at the definitions of the inductors. The definition of inductor attributes is shown in Figure 6^[5].
Figure 7: Parasitic definitions of the Lsrc inductor.
In the same way we did it for the capacitors, we can call up the parameter-entry windows for the inductors as well. In Figure 7, we see two parasitic components listed: series resistance and parallel
capacitance. We also see a note at the bottom of the left window. There is a 1-mOhm default value for the series resistance. This means if we do not make an entry there, the tool will automatically
add a 1-mOhm value (but this automatically-added value does not show up in the series resistance input field). This explains the low-frequency value in Figure 2 since we have four series inductors,
each will have 1-mOhm series resistance by default.
Figure 8: Impedance magnitude and phase of the simple PDN showed in Figure 1, with forcing zero body capacitance of capacitors and zero series resistance of inductors.
If we explicitly call out zero for the series resistance parasitics on all inductors, we get Figure 8. Now, the low-frequency value starts at the correct 1.3-mOhm value, but we can also notice that
the first two peaks get a little bigger. This is happening because we removed the extra series resistances, which helped to lower the antiresonance peaks. Note that with the circuit values used in
this example, explicitly calling out zero body capacitance for the inductors will not change the result.
This is eventually what we expect: a smooth impedance profile, no unexpected and unexplained sharp resonances, and asymptotic low-frequency impedance matches the sum of series resistance values.
Figure 9: Checking the body capacitance default value for the capacitor model.
We are almost done, but it still would be useful to check the capacitor’s equivalent circuit one more time and take another look at the body capacitance. To make it simple, we look at a single
capacitor, as shown in Figure 9. We set the main capacitance as a parameter so that we can step it and set the ESR and ESL to fixed values—10 mOhm and 1 nH, respectively. To see what happens, we
intentionally do not specify the parallel body capacitance; the entry is left blank.
We step the capacitance from 1 pF to 1F in four large logarithmic steps and sweep the frequency from 1 mHz to 1 THz. The result shows that, in fact, a parallel body capacitance is added by the
simulator, but its value is not fixed; it depends on the other parameters. With the values used here, the body capacitance seems to be approximately one million times smaller than the main
capacitance. While this looks like a huge ratio (and it is), we see that if we simulate our circuit over many decades of frequencies, this small default body capacitance value still can cause
unexpected artifacts. The good news is that it is easy to deal with; we just have to remember to call out specifically zero body capacitance, unless, of course, when we know its correct value and
want to simulate the effect of the body capacitance.
And a final note: Remember that all numerical tools have to set limits for the input numbers they can accept and process, whether the tools will tell you and remind you. Next time, when you see
unexpected things in circuit-simulation results, first make sure that the input numbers, including potential defaults, are set correctly.
1. Parallel Impedance of Four Groups of Capacitors.
2. High-Frequency Structure Simulator.
3. Berkeley SPICE.
4. LTspice.
5. LTwiki.
This column originally appeared in the August 2020 issue of Design007 Magazine. | {"url":"https://iconnect007.com/article/124235/quiet-power-be-aware-of-default-values-in-circuit-simulators/124238/pcb","timestamp":"2024-11-04T04:44:40Z","content_type":"text/html","content_length":"86316","record_id":"<urn:uuid:d885774c-1bd8-4edc-92ca-cd3497a43b8e>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00194.warc.gz"} |
Quantum Quotes
Happy April 14th, World Quantum Day!
The date, 4.14 represents one of the most important constants in quantum mechanics: Planck’s Constant h. The value of h = 4.14x10^-15 eV*Hz (eV*Hz stands for Electron-volt Hertz, which are not
considered the standard units but are helpful units for this genre of physics. In the International System of units, h = 6.63x10^-34J*Hz, or Joule Hertz). But what is quantum mechanics and what
does it have to do with h?
Well, quantum mechanics deals with physics on a very small scale. And small here means really small. Not mouse small or even dust particle small, but electron small. Quanta, with roots in the Latin
word “quantus,” meaning how much, refers to the smallest level something can break down into. (The name makes sense now, doesn’t it?) Quantum mechanics tells us how particles behave (or it tries to)
and in its explanatory equations, the constant h keeps showing up, like a pesky younger sibling you can never shake. To help me explain what quantum is, and why it is so fun, I’ve recruited some of
the major players in the field.
"Those who are not shocked when they first come across quantum theory cannot possibly have understood it.”- Niels Bohr
We start with a quote from our namesake here at NBLA, Niels Bohr. And in fact, he’s right. Max Planck, who discovered Planck’s Constant, came across it by mistake. He was trying to describe the
behavior of radiation absorption but struggling. (The behavior Planck was studying was dubbed “The Ultraviolet Catastrophe,” which would also be a great name for a band.)
Desperate for a positive result, Planck assumed that energy could come in little packets (later called quanta) and miraculously, it worked! He published in 1900, introducing the constant h and the
energy packet idea. Planck did not really believe his result had physical importance and that it was:
"purely formal assumption and I really did not give it much thought except that no matter what the cost, I must bring about a positive result” -Max Planck
This was the mathematical equivalent of the iconic Mean Girls tank top scene. Janis Ian vandalizes Regina George’s tank top in a desperate attempt at revenge but inadvertently starts a fashion
revolution. However in quantum mechanics, the revolution was more subtle. Planck, and many others, did not really believe in energy packets and chalked it up to mathematical formalism.
"My physical instincts bristle at that suggestion.” - Albert Einstein
In full disclosure, Einstein was talking about something else related to quantum mechanics when he said that. I’m putting it here so I can talk about why this concept of energy packets is such a wild
idea. In classical physics (the physics of stuff you may have learned about in high school with blocks sliding down ramps) energy is “continuous.” You can start at one level of energy and smoothly
accelerate to another level of energy. If energy came in “discrete” packets, you would jump from one level to the next without the smooth acceleration. For instance, in the final scene of When Harry
Met Sally, Harry is leisurely walking when he realizes he loves Sally and must be her New Year's kiss. He starts accelerating, jogging and first, and gradually, as the clock starts running out, he
increases to an all-out sprint. If energy packets were visible in our everyday lives or in this rom-com universe, there would be no acceleration between energy levels. At one moment, he would be
walking and at the next he would jump to an all-out sprint with nothing in the middle. Even Usain Bolt can’t manage this and has to humbly accelerate from standing to sprinting. It turns out that
while we don’t feel these energy jumps, particles do.
At the start of the 20th century the idea of energy quanta was unimaginable. But, in 1905, Albert Einstein showed that by treating energy from light as packets, he could explain how the frequency of
light was related to its energy and why atoms would only emit light at specific frequencies. Full circle moment: they’re related by a factor of Planck’s constant, h! This also introduced the idea of
light quanta, or photons, proving light behaves both as a particle and a wave. By 1911, the quantum revolution had started.
Now that we know what quantum mechanics is, let’s talk about the fun stuff (in my opinion)!
"Two seemingly incompatible conceptions can each represent an aspect of the truth … They may serve in turn to represent the facts without ever entering into direct conflict.” -Louis de Broglie
De Broglie is huge in quantum mechanics. He said basically that not only does light behave as both a particle and a wave, but everything does! Yes, particles like electrons, photons, and atoms, but
also you, me, and the metro car I took to work today. If everything is a wave, why don’t we see it? Wavelength is inversely proportional to momentum, so the more momentum we have (effectively the
more massive we are) the smaller the wavelength. Once you get much bigger than a couple of atoms, the wavelengths are so small we don’t know it’s there.
But being both a particle and a wave isn’t what de Broglie is talking about in his quote above. He’s talking about the concept of complementarity, introduced by Niels Bohr. There are pairs of
quantities in a system, like position and momentum, where both values cannot be known simultaneously. The Uncertainty Principle describes this: the more certainty you have about one value, the less
you have about the other.
"In atomic physics, we can never speak about nature without, at the same time, speaking about ourselves.” -Frijitov Capra
I just love this quote! Sometimes, physics feels impersonal, but in reality there is nothing more human than interpreting our surroundings. Alright, so what the heck is he talking about? Well, we
know now that small things like electrons are both particles and waves, but we need to talk about what the waves really are. These waves are a little different than the ones that flow through water
or propagate through air like sound.
You can think of the wave equation like Ella Enchanted’s magical book (the film iteration works best for this analogy, but for the novel fans, it should work just as well). Ella’s magical book keeps
track of the world as it changes through time. Likewise, the wave function changes the probabilities of the different states of a particle through time. Ella can open the book and see the location of
the mischievous fairy who cursed her. In quantum, we can measure the location of a particle. But, until a measurement is made, the wave function is in a state of superposition, where everything that
is possible is occurring all at once. While the book is closed, anything that is possible is unfolding its pages, unread. Mathematically solving the wave function only reveals the probabilities of
each possible state. It is akin to Ella having the book with her, but keeping it unopened. She knows there is some probability that the fairy is at a giant’s wedding or maybe getting a ticket for
flying while intoxicated (that part only happens in the movie).
The notion that everything possible exists until a cursed heroine or a physicist comes along to measure is uncomfortable to many. It means that we are not passively observing quantum mechanics unfold
but that we are participating in it, just by measuring it.
We cannot speak about atomic physics without speaking about ourselves!
"Do you really believe the moon is only there when you look at it?” -Albert Einstein to Abraham Pais
If measuring something makes it happen, Einstein mused, does it mean that if we aren’t measuring, things don’t exist? Einstein was being facetious, but… like… maybe?
"We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all
positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies
of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.” -Pierre Simon Laplace, A
Philosophical Essay on Probabilities
This quote is about what was later dubbed “Laplace’s Demon.” He describes a demon that could know everything about the current state of the world and from that, using physics, the demon could deduce
the past and predict the future. This idea is called determinism but is not possible given our current understanding of quantum mechanics and thermodynamics because it’s dictated by probabilities.
Some people extrapolate this to have philosophical importance and negate or support the existence of “free will.”
"Spooky action at a distance” -Albert Einstein
There are a lot of cool things to fall out of quantum mechanics, but coolest in my mind (and maybe the most horrifying in Einstein’s) is entanglement. Particles can be produced to have shared,
intrinsically connected features. For example, electrons have something called “spin” that describes their orientation. The thing with spin is that one electron in an entangled pair will have a
“positive” spin and the other must have a “negative” spin. Always. If you were to separate two entangled particles at any distance, both particles would have their superpositions of possible
orientations; they are both facing all directions independently and simultaneously. However, if you measure the orientation of one of these entangled particles along a given axis, entanglement forces
the other wave function to collapse and take the exact opposite orientation on that axis. This is true if the particles are two centimeters away or two million kilometers away.
It would be like if my sister and I only wore black or white shirts but we always wore opposite-colored shirts (this is important because we look very similar and it is difficult to tell us apart
otherwise). Because we like both colors equally, there is a fifty-fifty chance each day that we will wear the black or white shirts. Sure, if we were in the same room getting ready it would be easy
to determine which shirt to wear. Now that we do not live together, it would not be instantaneous for her to know I’ve selected a black shirt. Yet, entangled particles can do this without texting and
confirming their OOTD (outfit of the day).
This means particles must be communicating faster than the speed of light. How else will the entangled particle know what spin to have? But that’s impossible because nothing can travel faster than
the speed of light! Einstein and many other physicists could not stand that possibility and in a 1935 paper, they said that there must be another explanation. Perhaps it’s a set of “hidden
variables,” (stuff we don’t know about yet) that dictates the particle’s behaviors. You know, like “on Wednesdays, we wear pink.”
"This is no kooky paper. This is something very great." -Abner Shimony, on Bell’s Theorem in this Oral History Interview
In 1964, almost 30 years after Einstein & co. proposed the existence of hidden variables, John Bell published a paper that proved hidden variables weren’t possible. Bell’s Theorem (or Bell’s
inequality, depending on who you ask) opened a new world by showing quantum mechanics is as mysterious and magical as some feared.
Since 1964, the field has grown tremendously. Advancements in quantum mechanics enable the technology you’re reading this blog on, a new understanding of the origins of the universe, and even the
ability to determine the atmospheric composition of extrasolar planets! Explore these Inside Science resources to learn more about quantum mechanics and the future it may bring! | {"url":"https://www.aip.org/history-programs/niels-bohr-library/ex-libris-universum/quantum-quotes","timestamp":"2024-11-11T11:31:29Z","content_type":"text/html","content_length":"109163","record_id":"<urn:uuid:616f6d0f-a3da-4b5e-b228-b0c120ebd26e>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00155.warc.gz"} |
Pre-Publicación 2024-09
Fernando Betancourt, Raimund Bürger, Julio Careaga, Lucas Romero:
Coupled finite volume methods for settling in inclined vessels with natural convection
A widely applied technology of gravity-driven solid-liquid separation in mineral processing is the use of lamella settlers. These units are continuously operated tanks equipped with a number of
parallel inclined plates immersed in the mixture to be separated. The inclination of the plates exploits the well-known Boycott effect that describes the enhancement of settling rates beneath
inclined surfaces. This effect is usually attributed to a rapidly upward-streaming layer of clear liquid. The essence of this effect can be studied by examining gravity settling in an inclined tube
or rectangular channel. The lower and upper surfaces of the channel represent the plate onto which the particles start to settle and below which the clarified liquid streams upward, respectively. In
addition an increase of temperature in some part of the fluid causes a local change in the density of the fluid and circulation of the fluid within the vessel. It has been proposed to exploit this
behaviour to accelerate the settling process by additional heating. To examine this hypothesis a model and corresponding numerical method to describe inclined settling enhanced by natural convection
are formulated. The model consists in a two-dimensional scalar conservation law for the solids concentration coupled with a version of the Stokes system that accounts for density fluctuations in the
mixture enhanced by a Boussinesq approximation of the effect of temperature. In addition a convection-diffusion equation describes heat transport and diffusion. The main outcome is a numerical method
that allows one to simulate the effect of controllable parameters such as the initial concentration, difference of temperature, and angle of inclination on the progress of the solid-liquid
separation. Numerical examples are presented. Results reconfirm that the enhancement of settling rates depends critically on the dimensions of the settling vessel, intensity of heating, and particle
size, and is marginal for settling of relatively large particles and channels with a moderate length to width aspect ratio. | {"url":"https://ci2ma.udec.cl/publicaciones/prepublicaciones/prepublicacion.php?id=546","timestamp":"2024-11-05T03:06:58Z","content_type":"text/html","content_length":"9911","record_id":"<urn:uuid:560df339-546a-4cc1-8728-9f4bd2717931>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00512.warc.gz"} |
Effective Date of Rule: Thirty-one days after filing.
Purpose: To add amended section to chapter 51-50 WAC; specifically addressing chapter 16, section 1615 on tsunami loads. The state building code council convened a technical advisory group to develop
this amendment to provide a more accurate map reference to areas effected [affected] in Washington state.
Citation of Rules Affected by this Order: New 1.
Adopted under notice filed as WSR 21-05-067 on February 17, 2021.
A final cost-benefit analysis is available by contacting Stoyan Bumbalov, 1500 Jefferson Street, Olympia, WA 98504, phone 360-407-9277, email Stoyan.bumbalov@des.wa.gov.
Number of Sections Adopted in Order to Comply with Federal Statute: New 0, Amended 0, Repealed 0; Federal Rules or Standards: New 0, Amended 0, Repealed 0; or Recently Enacted State Statutes: New 0,
Amended 0, Repealed 0.
Number of Sections Adopted at the Request of a Nongovernmental Entity: New 0, Amended 0, Repealed 0.
Number of Sections Adopted on the Agency's own Initiative: New 1, Amended 0, Repealed 0.
Number of Sections Adopted in Order to Clarify, Streamline, or Reform Agency Procedures: New 0, Amended 0, Repealed 0.
Number of Sections Adopted using Negotiated Rule Making: New 0, Amended 0, Repealed 0; Pilot Rule Making: New 0, Amended 0, Repealed 0; or Other Alternative Rule Making: New 0, Amended 0, Repealed 0.
Date Adopted: May 21, 2021.
1615.1 General. The design and construction of Risk Category III and IV buildings and structures located in the Tsunami Design Zones shall be in accordance with Chapter 6 of ASCE 7, except as
modified by this code.
The intent of the Washington state amendments to ASCE 7 Chapter 6 (Tsunami Loads and Effects) is to require use of the Washington Tsunami Design maps to determine inundation limits, i.e., when
USER a site is within a tsunami design zone, where those maps are available. If they are not available for a given site, ASCE 7 maps are to be used. For sites where the Washington state department
NOTE: of natural resources has parameters for tsunami inundation depth and flow velocity available, those parameters are required to be used in the energy grade line analysis methodology, and as a
basis for comparison in the probabilistic tsunami hazard analysis in this chapter.
1615.2 Modifications to ASCE 7. The text of Chapter 6 of ASCE 7 shall be modified as indicated in this section.
1615.2.1 ASCE 7 Section 6.1.1. Modify the third paragraph and its exception in ASCE 7 Section 6.1.1 to read as follows:
The Tsunami Design Zone shall be determined using the Washington Tsunami Design Zone maps (WA-TDZ). The WA-TDZ maps are available at https://www.dnr.wa.gov/wa-tdz. For areas not covered by the extent
of the WA-TDZ maps, the Tsunami Design Zone shall be determined using the ASCE Tsunami Design Geodatabase of geocoded reference points shown in Fig. 6.1-1. The ASCE Tsunami Design Geodatabase of
geocoded reference points of runup and associated inundation Limits of the Tsunami Design Zone is available at http://asce7tsunami.online.
EXCEPTION: For coastal regions subject to tsunami inundation and not covered by WA-TDZ maps or Fig. 6.1-1, Tsunami Design Zone, inundation limits, and runup elevations shall be determined using the
site-specific procedures of Section 6.7, or for Tsunami Risk Category II or III structures, determined in accordance with the procedures of Section 6.5.1.1 using Fig. 6.7-1.
1615.2.2 ASCE 7 Section 6.1.1. Add new fifth paragraph and user note to ASCE 7 Section 6.1.1 to read as follows:
Whenever a Tsunami Design Zone or Fig. 6.1-1 is referenced in ASCE 7 Chapter 6, it shall include the WA-TDZ maps, within the extent of those maps.
USER NOTE: Tsunami inundation depths and flow velocities may be obtained from the Washington state department of natural resources. See https://www.dnr.wa.gov/wa-tdz.
1615.2.3 ASCE 7 Section 6.2. Modify ASCE 7 Section 6.2 definitions to read as follows:
MAXIMUM CONSIDERED TSUNAMI: A probabilistic tsunami having a 2% probability of being exceeded in a 50-year period or a 2,475-year mean recurrence, or a deterministic assessment considering the
maximum tsunami that can reasonably be expected to affect a site.
TSUNAMI DESIGN ZONE MAP: The Washington Tsunami Design Zone maps (WA-TDZ) designating the potential horizontal inundation limit of the Maximum Considered Tsunami, or outside of the extent of WA-TDZ
maps, the map given in Fig. 6.1-1.
1615.2.4 ASCE 7 Section 6.2. Add new definitions to ASCE 7 Section 6.2 to read as follows:
SHORELINE AMPLITUDE: The Maximum Considered Tsunami amplitude at the shoreline, where the shoreline is determined by vertical datum in North American Vertical Datum (NAVD 88).
WASHINGTON TSUNAMI DESIGN ZONE MAP (WA-TDZ): The Washington department of natural resources maps of potential tsunami inundation limits for the Maximum Considered Tsunami, designated as follows:
Anacortes Bellingham area MS 2018-02 Anacortes Bellingham
Elliott Bay Seattle OFR 2003-14
Everett area OFR 2014-03
Port Angeles and Port Townsend area MS 2018-03 Port Angeles and Port Townsend
San Juan Islands MS 2016-01
Southern Washington Coast MS 2018-01
Tacoma area OFR 2009-9
1615.2.5 ASCE 7 Section 6.5.1. Add new second paragraph to ASCE 7 Section 6.5.1 to read as follows:
6.5.1 Tsunami Risk Category II and III buildings and other structures. The Maximum Considered Tsunami inundation depth and tsunami flow velocity characteristics at a Tsunami Risk Category II or III
building or other structure shall be determined by using the Energy Grade Line Analysis of Section 6.6 using the inundation limit and runup elevation of the Maximum Considered Tsunami given in Fig.
Where tsunami inundation depth and flow velocity characteristics are available from the Washington state department of natural resources, those parameters shall be used to determine design forces in
the Energy Grade Line Analysis in Section 6.6.
1615.2.6 ASCE 7 Section 6.5.1.1. Modify the first paragraph of ASCE 7 Section 6.5.1.1 to read as follows:
6.5.1.1 Runup evaluation for areas where no map values are given. For Tsunami Risk Category II and III buildings and other structures where no mapped inundation limit is shown in the Tsunami Design
Zone map, the ratio of tsunami runup elevation above Mean High Water Level to Offshore Tsunami Amplitude, R/HT, shall be permitted to be determined using the surf similarity parameter ξ100, according
to Eqs. (6.5-2a, b, c, d, or e) and Fig. 6.5-1.
1615.2.7 ASCE 7 Section 6.5.2. Add new second paragraph to ASCE 7 Section 6.5.2 to read as follows:
6.5.2 Tsunami Risk Category IV buildings and other structures. The Energy Grade Line Analysis of Section 6.6 shall be performed for Tsunami Risk Category IV buildings and other structures, and the
site-specific Probabilistic Tsunami Hazard Analysis (PTHA) of Section 6.7 shall also be performed. Site-specific velocities determined by site-specific PTHA determined to be less than the Energy
Grade Line Analysis shall be subject to the limitation in Section 6.7.6.8. Site-specific velocities determined to be greater than the Energy Grade Line Analysis shall be used.
EXCEPTIONS: For structures other than Tsunami Vertical Evacuation Refuge Structures, a site-specific Probabilistic Tsunami Hazard Analysis need not be performed where the inundation depth resulting
from the Energy Grade Line Analysis is determined to be less than 12 ft (3.66 m) at any point within the location of the Tsunami Risk Category IV structure.
Where tsunami inundation depths and flow velocities are available for a site from the Washington state department of natural resources, those parameters shall be used as the basis of
comparison for the PTHA above and to determine whether the exception applies, in lieu of the Energy Grade Line Analysis.
1615.2.8 ASCE 7 Section 6.6.1. Add new third paragraph to ASCE 7 Section 6.6.1 to read as follows:
6.6.1 Maximum inundation depth and flow velocities based on runup. The maximum inundation depths and flow velocities associated with the stages of tsunami flooding shall be determined in accordance
with Section 6.6.2. Calculated flow velocity shall not be taken as less than 10 ft/s (3.0 m/s) and need not be taken as greater than the lesser of 1.5(ghmax)1/2 and 50 ft/s (15.2 m/s).
Where the maximum topographic elevation along the topographic transect between the shoreline and the inundation limit is greater than the runup elevation, one of the following methods shall be used:
1. The site-specific procedure of Section 6.7.6 shall be used to determine inundation depth and flow velocities at the site, subject to the above range of calculated velocities.
2. For determination of the inundation depth and flow velocity at the site, the procedure of Section 6.6.2, Energy Grade Line Analysis, shall be used, assuming a runup elevation and horizontal
inundation limit that has at least 100% of the maximum topographic elevation along the topographic transect.
Where tsunami inundation depths and flow velocities are available from Washington state department of natural resources, those parameters shall be used to determine design forces in the Energy Grade
Line Analysis in Section 6.6.2.
1615.2.9 ASCE 7 Section 6.7. Modify ASCE 7 Section 6.7 and add a user note to read as follows:
When required by Section 6.5, the inundation depths and flow velocities shall be determined by site-specific inundation studies complying with the requirements of this section. Site-specific analyses
shall use an integrated generation, propagation, and inundation model that replicates the given offshore tsunami waveform amplitude and period from the seismic sources given in Section 6.7.2.
USER Washington Tsunami Design Zone maps and inundation depths and flow velocities from Washington state department of natural resources are based on an integrated generation, propagation, and
NOTE: inundation model replicating waveforms from the seismic sources specific to Washington state. Model data can be obtained by contacting Washington state department of natural resources. See
1615.2.10 ASCE 7 Section 6.7.5.1, Item 4. Modify ASCE 7 Section 6.7.5.1, Item 4 to read as follows:
6.7.5.1 Offshore tsunami amplitude for distant seismic sources. Offshore tsunami amplitude shall be probabilistically determined in accordance with the following:
4. The value of tsunami wave amplitude shall be not less than 80% of the shoreline amplitude value associated with the Washington state inundation models as measured in the direction of the incoming
wave propagation.
1615.2.11 ASCE 7 Table 6.7-2. Modify ASCE 7 Table 6.7-2 to read as follows:
Table 6.7-2
Maximum Moment Magnitude
Subduction Zone Moment Magnitude MWmax
Alaskan-Aleutian 9.2
Cascadia 9.0
Chile-Peru 9.5
Izu-Bonin-Mariana 9.0
Kamchatka-Kurile and Japan Trench 9.4
1615.2.12 ASCE 7 Section 6.7.5.2. Modify ASCE 7 Section 6.7.5.2 to read as follows:
6.7.5.2 Direct computation of probabilistic inundation and runup. It shall be permitted to compute probabilistic inundation and runup directly from a probabilistic set of sources, source
characterizations, and uncertainties consistent with Section 6.7.2, Section 6.7.4, and the computing conditions set out in Section 6.7.6. The shoreline amplitude values computed shall not be lower
than 80% of the shoreline amplitude value associated with the Washington state inundation models as measured in the direction of the incoming wave propagation.
1615.2.13 ASCE 7 Section 6.7.6.2. Modify ASCE 7 Section 6.7.6.2 and add a user note to read as follows:
6.7.6.2 Seismic subsidence before tsunami arrival. Where the seismic source is a local earthquake event, the Maximum Considered Tsunami inundation shall be determined for an overall elevation
subsidence value shown in Fig. 6.7-3(a) and 6.7-3(b) or shall be directly computed for the seismic source mechanism. The GIS digital map layers of subsidence are available in the ASCE Tsunami Design
Geodatabase at http://asce7tsunami.online.
USER NOTE: The WA-TDZ maps include computed subsidence in the inundation. Subsidence data may be obtained from the Washington state department of natural resources. See https://www.dnr.wa.gov/wa-tdz.
1615.2.14 ASCE 7 Section 6.8.9. Modify the first sentence of ASCE 7 Section 6.8.9 to read as follows:
6.8.9 Seismic effects on the foundations preceding maximum considered tsunami. Where designated in the Tsunami Design Zone map as a site subject to a tsunami from a local earthquake, the structure
shall be designed for the preceding coseismic effects. | {"url":"https://lawfilesext.leg.wa.gov/law/wsr/2021/12/21-12-075.htm","timestamp":"2024-11-15T00:33:52Z","content_type":"text/html","content_length":"23769","record_id":"<urn:uuid:6b66762c-bfff-423b-b4ec-7dc4b4ae89ed>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00442.warc.gz"} |
The Role of Moderator Variables in Statistical Analysis
By Zach Fickenworth · 6 min read
In the realm of statistical analysis, understanding the dynamics between variables is crucial. A moderator variable, often denoted as 'M', plays a pivotal role in this context. It's a third variable
that influences the strength and direction of the relationship between a dependent and an independent variable. This blog aims to delve into the concept of moderator variables, their significance in
various analytical models, and how tools like Julius can assist in identifying and interpreting these variables.
What is a Moderator Variable?
A moderator variable is a third variable in a statistical model that affects the relationship between the studied independent and dependent variables. In correlation, it alters the strength or
direction of the correlation between two variables. In causal relationships, if 'x' is the predictor and 'y' is the outcome, 'z' (the moderator) affects how 'x' influences 'y'.
The Impact of Moderator Variables
Moderator variables can amplify or weaken the relationship between the independent (x) and dependent (y) variables. They are often identified using regression coefficients in statistical models like
ANOVA, where their effect is represented by the interaction effect between the dependent variable and a factor variable.
Questions Addressed by Moderator Variables
1. Does gender (moderator) influence the relationship between the desire to marry (independent variable) and attitudes towards marriage (dependent variable)?
2. Does a specific treatment (moderator) affect the impact of a drug (independent variable) on symptoms (dependent variable)?
Moderated Regression Analysis (MRA)
MRA is a regression-based technique used to identify moderator variables. It involves adding an interaction term to the regression equation. If the interaction term (the product of the independent
variable and the moderator) is statistically significant, it indicates that the moderator variable significantly affects the relationship between the independent and dependent variables.
Linear vs. Non-Linear Measurement
Linear Relationship: In a linear relationship, the effect of the moderator variable is represented as:
Non-Linear Relationship: In non-linear relationships, the interaction effect is more complex and is represented differently to capture the nuanced influence of the moderator.
The Role of Moderator Variables in Different Designs
- Repeated Measure Design: Moderator variables can also be used in repeated measure designs.
- Multi-Level Modeling: In these models, a variable that predicts the effect size is termed a moderator variable.
Considerations and Assumptions
1. Causal Assumption: Causation must be assumed, especially when the independent variable is not randomized. The moderator can reverse the causation effect if the causation between x and y is not
2. Relationship Between Variables: The moderator and independent variables should ideally be uncorrelated. However, they should not be too highly correlated to avoid estimation problems. The
moderator must relate to the dependent variable.
How Julius Can Assist
Julius, an advanced statistical tool, can significantly aid in the analysis involving moderator variables:
- Identifying Interactions: Julius can help in setting up the moderated regression analysis, identifying and computing interaction terms.
- Testing Significance: It can test the statistical significance of the interaction effects, helping to confirm or refute the presence of moderation.
- Visualization: Julius offers visualization tools to graphically represent the interaction effects, making it easier to interpret the results.
- Data Management: It assists in managing and preparing data for analysis, ensuring that the variables are correctly coded and analyzed.
Moderator variables are essential in understanding the complexities of relationships between variables in statistical analysis. They provide insights into how and when certain variables influence
others. Tools like Julius can be invaluable in identifying, testing, and interpreting these moderators, thereby enhancing the robustness of your statistical analysis. Understanding moderator
variables allows researchers and analysts to draw more nuanced and accurate conclusions from their data, leading to more informed decisions and advanced research findings. | {"url":"https://www.visualizedata.app/articles/the-role-of-moderator-variables-in-statistical-analysis","timestamp":"2024-11-12T08:24:36Z","content_type":"text/html","content_length":"84351","record_id":"<urn:uuid:d6d2515f-a9a7-45ee-9445-33211c8514d0>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00564.warc.gz"} |
Getting Better
One of my mathy friends recently said, “I thought that when I finished graduate school I would have learned pretty much all the math I’d ever know.” When I was getting ready to graduate, I felt the
same way. Which was a bit scary, because I felt like I didn’t know very much. But I figured that my future research life would be applying the techniques I used in my thesis to similar problems. I
imagined that I would pick up some new ideas and go deeper into the area, but that my baseline math knowledge was pretty much set. This has been entirely wrong.
In fact, I have found myself working more on problems that I have absolutely no idea how to solve, and have realized that my interests refuse to stay bounded by the math that I know best. And
problems don’t care! They may seem like problems in number theory, but oh, suddenly here comes representation theory, and a whole bunch of group theory! So I find myself trying to learn whole new
areas, or relearn things I saw in courses but couldn’t really absorb at the time. It can be really hard, with work and so many other things going on.
I have this sense that if I had been a better graduate student, everything would be so much easier now, and that I should probably just take a few years off and go back to school. It would be great
to be in a structured environment, designed by experts to help me learn more math. In my post-thesis research life, I find that so many topics from my classes really did turn out to be important, and
those that seemed irrelevant at the time (I mean, when was I going to need measure theory?) have turned up over and over again. And I constantly wish I knew stuff from classes that I couldn’t bring
myself to take (classical groups, linear programming… the wish-I-knew-list grows every day). But I can’t go back to school. I have a job now! I wasted my chance! Nooooo!
Of course, I have to remind myself that I actually learned a ton in graduate school. And at the time I was stressed out, worried about the very immediate problems presented by my thesis and teaching,
as well as eventually finding a job. I was the best graduate student I could be, given who I was and the time I had. So what do I do with this wistful sense of what I should have been? I just keep
doing the best I can, trying to solve problems, learning things from books, going to talks, and asking people questions when I get the chance. And, slowly, I do keep learning new things and solving
new problems
The regretful sense that I should have been a better graduate student is increasingly countered by a feeling of excitement in the realization that I am actually still growing as a mathematician. In
fact, as I (through great struggle) learn more, it becomes a little easier to see bigger pictures, and I even learn faster. Graduate school was just a jumping off point, and I keep gaining new
perspectives. I am so glad that I was wrong about my math trajectory! It is actually thrilling to realize that, far from being done learning, I can just keep getting better at this.
I don’t know why I didn’t realize it sooner. I mean, I guess I thought my professors just started out good at math; that they were of a different species of mathematician. I definitely didn’t think
of them as still learning. Perhaps I had, without even knowing it, absorbed the idea of mathematics as a sort of inborn talent, that you could fulfill as a young person but which fades as time goes
on. Thank G. H. Hardy, with his incredibly annoying statement in A Mathematician’s Apology, “No mathematician should ever allow himself to forget that mathematics, more than any other art or science,
is a young man’s game.” Perhaps Hardy’s words applied to his life. After all, he was a Tripos star, at the top of his field from the very beginning. He may have just crammed in a huge amount of math
early on, and never really felt that he was growing in the same way later in life. However, I definitely followed a different path, and many others have as well. Richard Guy, anyone? (Happy 100^th
birthday!) In fact, I am not the first to point out that this whole idea is suspect. I reject the notion that these early years are the peak of my career, and that people’s mathematical value or
potential is determinable by their 40^th birthday. Hooray for early achievers! But also, hooray for people who start later and move slower but keep getting better all the time.
Realizing that I can keep getting better as a mathematician has made me really empathize with my students, who often think that they are already basically good or bad at math, and are afraid that
they have hit their limit and are not capable of understanding some difficult concepts. This takes me back to growth mindset, (which I just explained to my linear algebra class on the first day of
class!) and how easy it is even for me to fall into the trap of thinking that I can’t grow in some way.
On the level of the profession, it is important to reach out to talented young people from all backgrounds, both to maximize mathematical progress and to make the mathematical world a more diverse
place. However, I think that it is also important to keep the door open for those who are not early talents, who may not choose mathematics until much later but still have a lot to offer. This is
vital to broadening the profession, since despite our best efforts, students from underrepresented groups in STEM or less affluent school districts may not come into contact with interesting
mathematics until later in life, and may have far fewer opportunities to pursue mathematics young even if they are interested.
So that’s what I’m thinking about as I start the new semester. Good luck to everyone with the challenges of fall! I would love to hear your thoughts on all this in the comments.
2 Responses to Getting Better
1. Thank you for this excellent post! I think that your point is so true that the attainment of your PhD and the knowledge you have gained initially seems like a pinnacle but there is so much more
learning and growth along the continuing trajectory. This is certainly true in math and academia, and also, I think, in so many areas of life. May you someday know a blue whale of math!
2. “I have this sense that if I had been a better graduate student, everything would be so much easier now, and that I should probably just take a few years off and go back to school.” I’m glad that
everything else you write here contradicts this wrongheaded idea!
Many of the things I teach my PhD students as absolutely basic, foundational calculational tools are things I learned post-PhD. (Some of them things that my advisor thought I should learn, but
wasn’t that insistent about at the time…)
This entry was posted in research, Uncategorized and tagged growth mindset, learning math. Bookmark the permalink. | {"url":"https://blogs.ams.org/phdplus/2016/08/28/getting-better/","timestamp":"2024-11-05T17:13:07Z","content_type":"text/html","content_length":"68470","record_id":"<urn:uuid:2d91273e-0d58-49db-a5ed-b268c47b281a>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00824.warc.gz"} |
Sciencemadness Discussion Board - Dibal-H in THF to alcohol mechanism - Powered by XMB 1.9.11 (Debug Mode)
Biotech_Yossorab posted on 15-4-2021 at 01:48
Harmless Dibal-H in THF to alcohol mechanism
Posts: 13
Registered: I'd like to have some help with my mechanism. I've been scratching my head trying to understand how the Dibal-H can reduce up to the alcohol and where do that one hydrogen comes
5-12-2018 from.
Member Is I've found a website where THF is actually helping the reaction by creating a weakening in the bond
Offline http://www.chemgapedia.de/vsengine/vlu/vsc/en/ch/2/vlu/oxida...
I'm fairly certain about the first steps of my mechanism but I don't know where the hydrogen comes from to make the alcohol...
njl posted on 15-4-2021 at 03:23
National Hazard
Posts: 609
Registered: The first step is the formation of a lewis adduct between the ester carbonyl and Dibal-H (the carbonyl lone pair being the lewis base and the aluminum center being the lewis acid).
26-11-2019 The hydride on the aluminum center then attacks the carbonyl carbon which is conveniently situated close to the hydride, forming a neutral aluminum alkoxide. THF forms another
complex with the aluminum center and facilitates its removal from the carbonyl oxygen. What is left is a hemiacetal, which rearranges with loss of R2OH to give an aldehyde. The same
Location: under mechanism is then repeated on the newly formed aldehyde carbonyl, except this time followed by protonation of the alkoxide to give an alcohol.
the sycamore
tree The sequence of hydride attacking carbonyl carbon -> formation of alkoxide with hydride counterion -> hydrolysis of alkoxide is a common motif among carbonyl reductions with hydride
or hydride equivalents. The details of the mechanisms are different for every reaction but the overall idea is common to several.
Member Is
Reflux condenser?? I barely know her!
Biotech_Yossorab posted on 15-4-2021 at 05:17
Posts: 13 Quote: Originally posted by njl
The first step is the formation of a lewis adduct between the ester carbonyl and Dibal-H (the carbonyl lone pair being the lewis base and the aluminum center being the lewis acid).
Registered: The hydride on the aluminum center then attacks the carbonyl carbon which is conveniently situated close to the hydride, forming a neutral aluminum alkoxide. THF forms another
5-12-2018 complex with the aluminum center and facilitates its removal from the carbonyl oxygen. What is left is a hemiacetal, which rearranges with loss of R2OH to give an aldehyde. The same
mechanism is then repeated on the newly formed aldehyde carbonyl, except this time followed by protonation of the alkoxide to give an alcohol.
Member Is
Offline The sequence of hydride attacking carbonyl carbon -> formation of alkoxide with hydride counterion -> hydrolysis of alkoxide is a common motif among carbonyl reductions with hydride
or hydride equivalents. The details of the mechanisms are different for every reaction but the overall idea is common to several.
Thank you for your response. So that means this reaction always occurs in presence of moisture/water? It's kind of strange considering that the paper this reaction is from never
mentions water anywhere.
Texium posted on 15-4-2021 at 06:16
Posts: 4568
No, the reaction produces an alkoxide, and the alcohol’s proton comes from aqueous workup once the reaction is complete. Similarly to Grignard reactions and other reductions that
Location: Salt use strongly basic reagents.
Lake City
Member Is
PhD candidate!
njl posted on 15-4-2021 at 06:29
National Hazard
Posts: 609 No, sorry for not being clear. The majority of the reaction takes place solely in non-protic solvents. This is common among nearly all of the reactions of this type, since water or
any proton sources can interfere with the reaction. For example, if water is present in the reaction mixture it will generally destroy the organometallic reagent being used
Registered: including DIBAL-H. The premise of these reactions is that a nucleophilic species (hydrogen in the case of a reduction or and aklyl group for Grignard/Barbier/etc.) is being carried
26-11-2019 to the electrophile (carbonyl carbon) where the two combine to form an alkoxide (the cation of which depends on the reagents used). Alkoxide is then hydrolyzed to the product. If
water is present it will be attacked since water (and the resulting hydroxide from deprotonation) are better ligands than hydride. The key is that you can run a reaction and then
Location: under hydrolyze the alkoxide all at once, there's no need for any protic material until you are finished with the reaction. That's why Grignard reactions and lithium aluminum hydride
the sycamore reductions are quenched with water or acid upon completion. Because the actual products of the Grignard and the reduction are not the desired alcohols, they are the corresponding
tree alkoxides.
Member Is Your case (reduction of an ester to primary alcohol with DIBAL-H) is very slightly complicated by the fact that there are actually 2 reduction steps going on (ester->aldehyde->
Offline alcohol). The second step cannot begin until the product of the first is turned into its carbonyl form. Taken at face value this would imply that the reduction must be carried out
in more steps so that the first reduction product can be isolated and hydrolyzed. However, the THF solvent allows the product of the first step to eliminate an alcohol which negates
Mood: the need for protonation. Other ether solvents (technically lewis bases in general) also allow elimination. This allows you to carry out the complete reduction of an ester to a
primary alcohol in one pot without isolation of any intermediates. As previously mentioned, the last step in your reaction before workup will be to quench the reaction mixture with
ambivalent a proton source to free your final product.
Notes: I only know what I have taught myself so take this with a grain of salt (though I am fairly confident in this explanation). Specifically however I'm not sure my explanation
of why water destroys the organometallic reagent is correct. Anyway I'm done rambling, I hope this made sense.
Reflux condenser?? I barely know her!
Biotech_Yossorab posted on 15-4-2021 at 07:02
Posts: 13
Registered: Thanks, both of you. Yeah, I think my question wasn't really clear but your explanation is great.
I knew the whole reaction was taking place in dry THF, I just didn't understand where that final proton came from, since neither acid nor water was mentionned in any work-up. I
Member Is thought the hydrogen came from the DIBAL-H itself and didn't make any sense, ence my difficulty to understand the reaction.
njl posted on 15-4-2021 at 07:12
National Hazard
Posts: 609
26-11-2019 Right, the final proton on the alcohol does not come from the DIBAL-H. If no quenching/protonation is in the procedure, they may have glossed over it and assumed that those
following their method would assume it is needed, or it could be included in their work up.
Location: under
the sycamore
Member Is Reflux condenser?? I barely know her!
Sigmatropic posted on 15-4-2021 at 21:23
Hazard to Others
Posts: 307
29-1-2017 Draw your 'side product' in the first step you will note that it is Diisobutylaluminium methoxide. No protons needed. No THF invoked in the mechanism.
Member Is In short the tetrahedral intermediate you've drawn (the acetal) is only stable at lower temperatures and can decompose (or collapse if you will) into said DIBAL-OMe and the
Offline aldehyde.
No Mood
njl posted on 16-4-2021 at 03:58
National Hazard
Posts: 609
26-11-2019 You should be more careful before discounting solvent effects in a reaction like this. Ethereal solvents and THF in specific are known to catalyze such reactions. However unstable
the intermediate is, methoxide is hard to eliminate without some persuasion.
Location: under
the sycamore
Member Is Reflux condenser?? I barely know her!
Texium posted on 16-4-2021 at 06:16
Posts: 4568
THF is certainly not required though. I’ve run DIBAl-H reductions from methyl ester to alcohol in DCM many times with no issues and high yields.
Location: Salt
Lake City
Member Is
PhD candidate! | {"url":"https://www.sciencemadness.org/whisper/viewthread.php?tid=157300#pid658272","timestamp":"2024-11-05T09:32:52Z","content_type":"application/xhtml+xml","content_length":"46232","record_id":"<urn:uuid:a0c55adc-546d-4985-a1a6-a26b2875d560>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00484.warc.gz"} |
Pyatak for happiness
Category: Amazing human abilities
* * *
First of all, you need to be able to find a lucky coin. In the world many different things that a person can use on good. A lot of people have talismans, jewelry, happy dresses, ties or T-shirts,
etc. But they are all already established themselves, already carried the “grace” of fate or higher powers. We have to find such a thing. Do it on coin example, and then you can come up with a search
ritual and for any other thing. For most people except those who have developed psychic abilities, the method of “heads or tails” gives nothing. You understand that fate is determined not by a coin,
but by laws and powers hidden from us. What laws and forces are these, we are in do not know the accuracy. But this does not mean that we cannot use them. The coin obeys a simple rule – 50:50. And
this the discipline taught for a long time has definitely proved – theory probabilities. Probability theory tells us that if we toss a coin 100 times, it can fall, say, 48 times up an eagle and 52 –
tails. If we toss 1,000 times, this ratio will be very close to 50 to 50, say 502 and 498. If you flip a coin more and more the number of times, we finally get 50 to 50. It should be so – because it
has two sides, and for what reason one should come across to us eyes more often than the other? Here we come to the heart of the matter. Take 10 coins, for example, 1 ruble each. Label them somehow
let’s say write on each number. Now toss over some flat bottom with high enough edges to make coins did not fly “overboard”. Pre-guess which side a coin should fall out. Put down on a piece of paper
numbers from 1 to 10, and write down in the line which side which of coins fell. Bet if you have the side you guessed minus if not. When toss all the coins for the tenth time and write down the
result, delete from the list the coin that showed the worst result. Set it aside. Now throw in exactly the same way 9 coins. After the tenth roll, “delete the next player from the field. “And so
until you have one a coin that for some reason stubbornly more often than others guesses your thoughts. Why does this happen, God knows him, but it happens. This and have your lucky coin. Keep it,
always carry it with you. AND when you get into a difficult situation, you will face a difficult choice, toss it, making a plan in advance: “Eagle – I do it, tails – so. “Soon you will see that your
lucky coin is real helps you in life. And if it turns out that there is no help or, conversely, the prompted decisions turn out to be incorrect, which means as the people say, the demon pushed you by
the arm. And you need the same a way to find another lucky coin. Vlad Ilyin, master white magic In general, it is clear that we are looking for an exception to the usual rule, looking for something
that violates mathematical laws. Are looking for because instead of them magical laws begin to act. IN Today’s difficult times, such a coin will be very useful to you.
Photo from open sources History knows a lot of people-phenomena, but even among them
A photo from open sources He was called the great impostor. Ferdinand Demara settled
A photo from open sources Parapsychologists and scientists have diametrically opposite points view of
A photo from open sources How long can a person live? Media reports of
A photo from open sources Weather forecast accuracy by meteorologists is a topic lots
A photo from open sources Kung Fu Master Liu Fei from Mianyang, Southwest China, | {"url":"https://trashplanetdiy.com/pyatak-for-happiness.html","timestamp":"2024-11-03T12:12:33Z","content_type":"text/html","content_length":"48469","record_id":"<urn:uuid:a0998e7f-1f33-46a6-9594-ff444ca3d507>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00597.warc.gz"} |
Linear Algebra - Data Science Wiki
Linear Algebra :
Linear algebra is a branch of mathematics that deals with the study of linear equations, vectors, and matrices. It is a crucial part of many fields, including physics, engineering, and computer
science, as it allows us to represent and manipulate data in a systematic and efficient way.
One of the key concepts in linear algebra is vector addition. Vectors are mathematical objects that have both magnitude (length) and direction. For example, we might have a vector that represents the
displacement of an object in a two-dimensional space. In this case, the vector would have two components: one representing the displacement in the x-direction, and one representing the displacement
in the y-direction. Vector addition allows us to combine two or more vectors by adding their corresponding components. For example, if we have two vectors, A and B, with components (3, 4) and (2, 5),
respectively, their sum would be the vector
with components (3+2, 4+5), or (5, 9).
Another important concept in linear algebra is matrix multiplication. Matrices are rectangular arrays of numbers, and matrix multiplication allows us to combine two or more matrices by multiplying
their corresponding elements. For example, if we have two matrices, A and B, with elements (1, 2, 3) and (4, 5, 6), respectively, their product would be the matrix C with elements (14 + 25 + 36, 14 +
25 + 36, 14 + 25 + 3*6), or (32, 32, 32).
These examples illustrate how linear algebra can help us represent and manipulate data in a concise and powerful way. For example, in physics, we might use vector addition to calculate the net force
on an object, or we might use matrix multiplication to represent the transformation of a three-dimensional space. In engineering, we might use linear algebra to design efficient algorithms for
solving complex optimization problems. And in computer science, we might use linear algebra to perform
machine learning
tasks, such as
. Overall, linear algebra is a fundamental tool that allows us to understand and analyze data in many different contexts. | {"url":"https://datasciencewiki.net/linear-algebra/","timestamp":"2024-11-13T14:15:49Z","content_type":"text/html","content_length":"40840","record_id":"<urn:uuid:a8c523ed-7801-48a7-b574-b2950bc18ff4>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00381.warc.gz"} |
Where On The Hr Diagram Would You Find A Red Supergiant? Hint:is It Hot/cool?is Its Radius Large/small?what
A red supergiant would be found in the cool and luminous region of the Hertzsprung-Russell (HR) diagram. It has a large radius and high luminosity.
Red supergiants are massive stars in the late stages of their evolution. They have exhausted their core hydrogen fuel and have expanded to become extremely large in size. Due to their low surface
temperatures, they appear red in color. On the HR diagram, they are located in the top-right portion, known as the "supergiant" region.
The cool temperature of red supergiants is reflected in their spectral characteristics, with strong absorption lines of cool atmospheric gases. Their large radius is a result of the intense radiation
pressure generated by their high luminosity. Red supergiants have luminosities much higher than that of the Sun, often thousands or even hundreds of thousands of times brighter. In summary, a red
supergiant can be identified on the HR diagram by its cool temperature, large radius, and high luminosity, placing it in the upper-right region of the diagram.
Learn more about luminosity here: https://brainly.com/question/13945214
The inductor can store up to 7.50 uJ of energy.
The inductor's maximum current is 0.0207 A.
The capacitor's maximum voltage is 20.7 V.
1.08 V is the voltage across the capacitor.
(a) The maximum energy stored in the inductor can be calculated using the formula for the energy stored in an inductor:
E = (1/2) * L * I²
where L is the inductance and I is the maximum current in the inductor. Substituting the given values, we get:
E = (1/2) * 60.0 mH * (0.500 mA)² = 7.50 uJ
Therefore, the maximum energy stored in the inductor is 7.50 uJ.
(b) The maximum current in the inductor can be calculated using the formula
I = Q / C
where Q is the charge on the capacitor and C is the capacitance. Substituting the given values, we get:
I = 6.00 uC / 290 uF = 0.0207 A
Therefore, the maximum current in the inductor is 0.0207 A.
(c) The maximum voltage across the capacitor can be calculated using the formula:
V = Q / C
Substituting the given values, we get:
V = 6.00 uC / 290 uF = 20.7 V
Therefore, the maximum voltage across the capacitor is 20.7 V.
(d) When the current in the inductor has half its maximum value, the energy stored in the inductor and the voltage across the capacitor can be calculated using the formulas:
E = (1/2) * L * I²
V = I / (C * ω)
where ω is the angular frequency of the circuit, given by:
ω = 1 / √(LC)
Substituting the given values, we get:
ω = 1 / √((60.0 mH)(290 uF)) = 800 rad/s
I = (1/2) * 0.500 mA = 0.250 mA
E = (1/2) * 60.0 mH * (0.250 mA)² = 0.937 uJ
V = (0.250 mA) / (290 uF * 800 rad/s) = 1.08 V
Therefore, when the current in the inductor has half its maximum value, the energy stored in the inductor is 0.937 uJ and the voltage across the capacitor is 1.08 V.
To know more about the Voltage, here | {"url":"https://cjp.edu.py/quiz-answers/where-on-the-hr-diagram-would-you-find-a-red-supergiant-hint-lkqp.html","timestamp":"2024-11-09T06:31:53Z","content_type":"text/html","content_length":"107439","record_id":"<urn:uuid:2ad0dc95-aed5-4e55-908c-f3ddec67c490>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00068.warc.gz"} |
2.01 grams to ounces
Convert 2.01 Grams to Ounces (gm to oz) with our conversion calculator. 2.01 grams to ounces equals 0.0709006596 oz.
Enter grams to convert to ounces.
Formula for Converting Grams to Ounces:
ounces = grams ÷ 28.3495
By dividing the number of grams by 28.3495, you can easily obtain the equivalent weight in ounces.
Understanding the Conversion from Grams to Ounces
When it comes to converting measurements, knowing the right conversion factor is essential. In the case of converting grams to ounces, the conversion factor is 1 ounce = 28.3495 grams. This means
that to convert grams into ounces, you need to divide the number of grams by 28.3495. This conversion is particularly important for those who work with both the metric and imperial systems, as it
allows for accurate measurements across different contexts.
Formula for Converting Grams to Ounces
The formula to convert grams (g) to ounces (oz) is straightforward:
Ounces = Grams ÷ 28.3495
Step-by-Step Calculation
Let’s take the example of converting 2.01 grams to ounces. Here’s how you can do it:
1. Start with the amount in grams: 2.01 grams.
2. Use the conversion factor: 28.3495 grams per ounce.
3. Apply the formula: Ounces = 2.01 grams ÷ 28.3495.
4. Perform the calculation: Ounces = 0.0708.
5. Round the result to two decimal places: 0.07 ounces.
The Importance of Grams to Ounces Conversion
Understanding how to convert grams to ounces is crucial for various applications. This conversion helps bridge the gap between the metric system, commonly used in scientific and international
contexts, and the imperial system, which is prevalent in the United States. Whether you are a chef following a recipe, a scientist conducting experiments, or simply someone who needs to measure
ingredients accurately, knowing how to convert these units can save time and ensure precision.
Practical Examples of Grams to Ounces Conversion
Here are a few scenarios where converting grams to ounces might be particularly useful:
• Cooking and Baking: Many recipes, especially those from the U.S., list ingredients in ounces. If you have a recipe that calls for 0.07 ounces of an ingredient, knowing that this is equivalent to
2.01 grams can help you measure accurately.
• Scientific Measurements: In laboratories, precise measurements are critical. Converting grams to ounces can be necessary when dealing with materials that are measured in different units.
• Everyday Use: Whether you’re weighing food for dietary purposes or measuring out supplements, being able to convert between grams and ounces can enhance your accuracy and understanding of
In conclusion, converting 2.01 grams to ounces is a simple yet essential skill that can enhance your accuracy in various fields. By understanding the conversion factor and applying the formula, you
can easily navigate between these two measurement systems.
Here are 10 items that weigh close to 2.01 grams to ounces –
• Paperclip
Shape: Elongated oval
Dimensions: Approximately 3.5 cm x 1 cm
Usage: Commonly used to hold sheets of paper together.
Fact: The paperclip was patented in 1867, but its design has remained largely unchanged since then.
• Small Button
Shape: Circular
Dimensions: Diameter of about 1.5 cm
Usage: Used in clothing as a fastener.
Fact: The largest button ever made was over 1 meter in diameter!
• AA Battery
Shape: Cylindrical
Dimensions: 5 cm in length and 1.4 cm in diameter
Usage: Commonly used in remote controls, toys, and other electronic devices.
Fact: The AA battery is one of the most popular battery sizes worldwide.
• Postage Stamp
Shape: Rectangular
Dimensions: Typically 2.5 cm x 3 cm
Usage: Used to pay for the delivery of mail.
Fact: The first adhesive postage stamp, the Penny Black, was issued in the UK in 1840.
• Small Marble
Shape: Spherical
Dimensions: Diameter of about 1.5 cm
Usage: Used in games and as decorative items.
Fact: Marbles have been played with for thousands of years, dating back to ancient Egypt.
• USB Flash Drive
Shape: Rectangular
Dimensions: Approximately 5 cm x 2 cm x 0.5 cm
Usage: Used for data storage and transfer.
Fact: The first USB flash drive was introduced in 1998 and had a capacity of just 8 MB.
• Keychain
Shape: Various shapes, often circular or rectangular
Dimensions: Typically around 5 cm in length
Usage: Used to hold keys together.
Fact: Keychains can be customized and are often used as promotional items.
• Small Rubber Eraser
Shape: Rectangular or oval
Dimensions: About 4 cm x 1.5 cm
Usage: Used to remove pencil marks from paper.
Fact: The first rubber eraser was invented in 1770, but it wasn’t until the 19th century that they became widely used.
• Tea Bag
Shape: Rectangular or triangular
Dimensions: Approximately 6 cm x 4 cm
Usage: Used for brewing tea.
Fact: The tea bag was invented in the early 20th century and has since revolutionized tea drinking.
• Small Coin (e.g., Dime)
Shape: Circular
Dimensions: Diameter of about 1.8 cm
Usage: Used as currency for transactions.
Fact: The U.S. dime is the smallest coin in terms of diameter but has the highest value relative to its size.
Other Oz <-> Gm Conversions – | {"url":"https://www.gptpromptshub.com/grams-ounce-converter/2-01-grams-to-ounces","timestamp":"2024-11-13T02:44:04Z","content_type":"text/html","content_length":"186873","record_id":"<urn:uuid:c8597347-0c34-430f-84f2-72f9af56bc90>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00818.warc.gz"} |
Definition of Multicollinearity
1. Noun. A case of multiple regression in which the predictor variables are themselves highly correlated.
Definition of Multicollinearity
1. Noun. (statistics) A phenomenon in which two or more predictor variables in a multiple regression model are highly correlated, so that the coefficient estimates may change erratically in response
to small changes in the model or data. ¹
¹ Source: wiktionary.com
Medical Definition of Multicollinearity
1. In multiple regression analysis, a situation in which at least some independent variables in a set are highly correlated with each other. Origin: multi-+ L. Col-lineo, to line up together (05 Mar
Lexicographical Neighbors of Multicollinearity
multicivilizational multicollection multicolour-yawn
multiclade multicollinear multicolour yawn
multiclaim multicollinearities multicolour yawns
multiclan multicollinearity (current term) multicoloured
multiclient multicollision multicolours
multicluster multicollisional multicolumn
multicoat multicolor multicolumnar
multicoated multicolored multicombination
multicode multicolors
multicohort multicolour
Literary usage of Multicollinearity
Below you will find example usage of this term as found in modern and/or classical literature:
1. Handbook on Hedonic Indexes and Quality Adjustments in Price Indexes by Jack E. Triplett (2006)
"Conclusion: sources and consequences of multicollinearity Authors of empirical hedonic studies have so often reported finding multicollinearity that others ..."
2. Strengthening Policy Analysis: Econometric Tests Using Microcomputer Software by Lawrence James Haddad, Daniel Driscoll (1995)
"multicollinearity exists in virtually every data set but is a problem only when ... The main effects of high multicollinearity are that the variances of the ..."
3. Strategies for Sustainable Land Management in the East African Highlands by J. Pender, Frank Place, S. Ehui (2006)
"Regressions were checked for multicollinearity using variance inflation factor (VIF). The maximum VIF of any of our explanatory variables was 3.63, ..."
4. Production and Consumption of Foodgrains in India: Implications of by J. S. Sarma, Vasant P. Gandhi (1990)
"A principal-components approach, as suggested by Mundlak (1981), was also attempted to get over the problem of multicollinearity. ..."
5. The Effects on Income Distribution and Nutrition of Alternative Rice Price by Prasarn Trairatvorakul (1984)
"Both autocorrelation and multicollinearity are therefore examined. ... The technique used to correct for multicollinearity combines the conversion of ..."
6. A Meta-analysis of Rates of Return to Agricultural R&D: Ex Pede Herculem? by Julian M. Alston (2000)
"We would most assuredly run into multicollinearity problems if we tried. Extreme multicollinearity results in an inability to perform the regression at all. ..."
7. Questions and Answers in Lethal and Non-Lethal Violence: Proceeding of the edited by Richard L. Block (1994)
"Fisher and Mason (1981) describe a number of possible approaches for obtaining more efficient estimates in the presence of multicollinearity. ..."
8. Handbook on Hedonic Indexes and Quality Adjustments in Price Indexes by Jack E. Triplett (2006)
"Conclusion: sources and consequences of multicollinearity Authors of empirical hedonic studies have so often reported finding multicollinearity that others ..."
9. Strengthening Policy Analysis: Econometric Tests Using Microcomputer Software by Lawrence James Haddad, Daniel Driscoll (1995)
"multicollinearity exists in virtually every data set but is a problem only when ... The main effects of high multicollinearity are that the variances of the ..."
10. Strategies for Sustainable Land Management in the East African Highlands by J. Pender, Frank Place, S. Ehui (2006)
"Regressions were checked for multicollinearity using variance inflation factor (VIF). The maximum VIF of any of our explanatory variables was 3.63, ..."
11. Production and Consumption of Foodgrains in India: Implications of by J. S. Sarma, Vasant P. Gandhi (1990)
"A principal-components approach, as suggested by Mundlak (1981), was also attempted to get over the problem of multicollinearity. ..."
12. The Effects on Income Distribution and Nutrition of Alternative Rice Price by Prasarn Trairatvorakul (1984)
"Both autocorrelation and multicollinearity are therefore examined. ... The technique used to correct for multicollinearity combines the conversion of ..."
13. A Meta-analysis of Rates of Return to Agricultural R&D: Ex Pede Herculem? by Julian M. Alston (2000)
"We would most assuredly run into multicollinearity problems if we tried. Extreme multicollinearity results in an inability to perform the regression at all. ..."
14. Questions and Answers in Lethal and Non-Lethal Violence: Proceeding of the edited by Richard L. Block (1994)
"Fisher and Mason (1981) describe a number of possible approaches for obtaining more efficient estimates in the presence of multicollinearity. ..."
Other Resources: | {"url":"https://www.lexic.us/definition-of/multicollinearity","timestamp":"2024-11-08T09:09:23Z","content_type":"text/html","content_length":"34593","record_id":"<urn:uuid:8ccadcdd-3ce5-4c79-9b62-751c5499e92b>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00115.warc.gz"} |
Cumulated sum
Hi guys,
I am pretty new to GAMS - don’t think this is gonna be hard for someone with experience.
I’d like to create a cumulated sum.
Please find the equation I’d like to implemet in the attached file.
I therefore defined a set t with several periods and a set m with several product variant types.
x(m,t) is a binary variable assigning t to m.
Now for every t I need the cumulated amount of x(m,t) in the respective period y(m,t).
Hence, I declared the sum to start with t’=1 an sum up to the currently considered t.
I’m pretty sure I somehow have to work with an alias and a $-(ord)condition here.
This is what I got so far, but something is missing:
Sets m / 1, 2, 3 /
t / 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 /;
alias (t,tt);
y(m,t) =e= Sum((t$(ord(t) =l= ord(tt)), x(m,t));
I’d apperciate your help!
I believe GAMS now has special ordered sets support, but I don’t use them so I can’t really help you in that direction.
This is a correct syntax for what you’re trying to write.
eq(m,t)… y(m,t) =e= Sum((tt$(ord(t)>=ord(tt)),x(m,tt)); | {"url":"https://forum.gams.com/t/cumulated-sum/1829","timestamp":"2024-11-09T16:12:10Z","content_type":"text/html","content_length":"16229","record_id":"<urn:uuid:d271a953-5060-4f13-a8f7-7e868a22d07a>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00381.warc.gz"} |
Teaching Numbers 1 10 Worksheets 2024 - NumbersWorksheets.com
Teaching Numbers 1 10 Worksheets
Teaching Numbers 1 10 Worksheets – The Negative Amounts Worksheet is a terrific way to start off instructing the kids the idea of adverse amounts. A poor number is any quantity which is less than no.
It can be additional or subtracted. The minus indicator suggests the adverse variety. You may also create negative amounts in parentheses. Under can be a worksheet to provide you started off. This
worksheet has a range of adverse phone numbers from -10 to 10. Teaching Numbers 1 10 Worksheets.
Adverse phone numbers are a number whose value is under zero
A poor variety features a benefit under absolutely nothing. It might be depicted over a quantity collection in just two techniques: together with the positive amount created as the very first digit,
and with the negative number created since the final digit. A positive quantity is written using a plus signal ( ) just before it, yet it is optionally available to write down it doing this. If the
number is not written with a plus sign, it is assumed to be a positive number.
They can be represented by a minus sign
In old Greece, negative amounts have been not utilized. They were dismissed, since their math was based upon geometrical ideas. When Western scholars began converting old Arabic messages from To the
north Africa, they arrived at identify unfavorable figures and appreciated them. Nowadays, bad amounts are symbolized by a minus indicator. To understand more about the history and origins of bad
phone numbers, check this out write-up. Then, try out these good examples to view how adverse phone numbers have evolved with time.
They may be additional or subtracted
As you might already know, positive numbers and negative numbers are easy to add and subtract because the sign of the numbers is the same. Negative numbers, on the other hand, have a larger absolute
value, but they are closer to than positive numbers are. These numbers have some special rules for arithmetic, but they can still be added and subtracted just like positive ones. You may also
subtract and add negative phone numbers using a quantity series and apply a similar rules for subtraction and addition when you do for optimistic figures.
They are depicted by a quantity in parentheses
A negative variety is symbolized by a variety covered in parentheses. The negative indicator is changed into its binary equivalent, and also the two’s go with is saved in the identical area in
memory. Sometimes a negative number is represented by a positive number, though the result is always negative. In these instances, the parentheses should be included. If you have any questions about
the meaning of negative numbers, you should consult a book on math.
They can be divided up from a good amount
Negative numbers might be multiplied and divided like positive phone numbers. They can be divided up by other unfavorable numbers. However, they are not equal to one another. The first time you grow
a poor quantity with a beneficial variety, you will get no for that reason. To create the solution, you should pick which indicator your solution ought to have. It really is quicker to keep in mind a
negative variety when it is developed in mounting brackets.
Gallery of Teaching Numbers 1 10 Worksheets
Practice Writing Numbers 1 10 With These Tracing Worksheets For
Numbers 1 10 Worksheets Counting Worksheets For Kindergarten
Number 1 10 Worksheets For Kindergarten Kids Tracing And Writing
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Can someone proficient in data science handle assignments related to time series analysis and forecasting? | Pay Someone To Do My Python Assignment
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How to Master Python For Data Science?
To master Python for data science, you should start by learning the fundamentals of Python programming language. Understand variables, data types, functions, loops, and conditional statements.
Next, delve into libraries and tools commonly used in data science such as NumPy, Pandas, and Matplotlib. Learn how to manipulate and analyze data efficiently using these libraries.
Practice working with real-world data sets to gain practical experience. This will help you understand how to clean, visualize, and extract insights from data using Python.
Understand machine learning algorithms and how to implement them in Python. Learn about supervised and unsupervised learning techniques, as well as model evaluation and selection.
Lastly, stay updated with the latest trends and advancements in data science and Python. Join online communities, participate in hackathons, and work on projects to enhance your skills and expertise
in Python for data science.
How to visualize data in Python using Matplotlib?
To visualize data in Python using Matplotlib, you can follow these steps:
1. Import the necessary libraries:
1 import matplotlib.pyplot as plt
2 import numpy as np
1. Create a dataset:
1 # Create a sample dataset
2 x = np.arange(0, 10, 0.1)
3 y = np.sin(x)
1. Create a plot:
1 # Create a line plot
2 plt.plot(x, y)
3 plt.xlabel('x-axis')
4 plt.ylabel('y-axis')
5 plt.title('Sample Plot')
6 plt.show()
1. Customize the plot: You can customize the plot by adding labels, titles, legends, grid lines, etc. For example:
1 plt.plot(x, y, label='Sine Curve', color='red', linestyle='--', linewidth=2)
2 plt.xlabel('x-axis')
3 plt.ylabel('y-axis')
4 plt.title('Sine Plot')
5 plt.legend()
6 plt.grid(True)
7 plt.show()
1. Create other types of plots: You can also create other types of plots such as bar plots, scatter plots, histograms, etc. For example:
1 # Create a bar plot
2 plt.bar(x, y)
3 plt.xlabel('x-axis')
4 plt.ylabel('y-axis')
5 plt.title('Sample Bar Plot')
6 plt.show()
These are just a few examples of how you can visualize data using Matplotlib in Python. There are many other options and customization features available in Matplotlib, so feel free to explore and
experiment with them.
How to perform statistical analysis in Python?
There are several popular libraries in Python that can be used to perform statistical analysis. Some of the most commonly used libraries are:
1. NumPy: NumPy is a powerful library for numerical computing in Python. It provides support for large multi-dimensional arrays and matrices, along with a collection of mathematical functions to
operate on these arrays. You can use NumPy to perform basic statistical calculations such as mean, median, standard deviation, and variance.
2. SciPy: SciPy is a library that builds on top of NumPy and provides additional functionality for scientific computing. It includes modules for optimization, interpolation, integration, linear
algebra, and statistics. You can use SciPy to perform more advanced statistical analysis such as hypothesis testing, regression, and clustering.
3. pandas: pandas is a data manipulation library that provides powerful data structures and tools for data analysis. It allows you to easily read, manipulate, and analyze data in tabular format. You
can use pandas to perform exploratory data analysis, summarize data, and generate descriptive statistics.
4. scikit-learn: scikit-learn is a machine learning library that provides tools for classification, regression, clustering, dimensionality reduction, and model selection. It includes modules for
various statistical algorithms such as linear regression, logistic regression, k-means clustering, and support vector machines.
To perform statistical analysis in Python, you can start by importing the necessary libraries and loading your data into a suitable data structure such as a NumPy array or a pandas DataFrame. Then,
you can use the functions and methods provided by these libraries to carry out the desired statistical analysis. Make sure to refer to the official documentation of these libraries for detailed
instructions and examples on how to use them for statistical analysis.
How to use scikit-learn library in Python?
To use the scikit-learn library in Python, you first need to install it using pip:
1 pip install scikit-learn
Once the library is installed, you can import it in your Python script or Jupyter notebook using the following code:
You can then use the various modules and classes provided by scikit-learn to build and train machine learning models. Here is an example of using scikit-learn to build a simple linear regression
1 from sklearn.model_selection import train_test_split
2 from sklearn.linear_model import LinearRegression
3 from sklearn.metrics import mean_squared_error
5 # Load the dataset
6 X, y = load_dataset()
8 # Split the data into training and testing sets
9 X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
11 # Create a linear regression model
12 model = LinearRegression()
14 # Train the model on the training data
15 model.fit(X_train, y_train)
17 # Make predictions on the test data
18 predictions = model.predict(X_test)
20 # Calculate the mean squared error of the model
21 mse = mean_squared_error(y_test, predictions)
22 print("Mean Squared Error:", mse)
This is just a simple example, but scikit-learn provides a wide range of algorithms and tools for various machine learning tasks such as classification, regression, clustering, and more. You can
explore the scikit-learn documentation for more information on all the available functionality and how to use it.
How to implement feature engineering in Python?
Feature engineering is the process of creating new features or modifying existing features to improve the performance of a machine learning model. Here is how you can implement feature engineering in
1. Load your dataset: First, you need to load your dataset into a DataFrame using a library like pandas. For example, you can use the following code to load a CSV file into a DataFrame:
1 import pandas as pd
2 data = pd.read_csv('data.csv')
1. Explore your data: Before performing feature engineering, it is important to explore your data to understand the relationships between different features and the target variable. You can use
descriptive statistics, data visualization, and correlation matrices to identify potential features for engineering.
2. Create new features: Based on your data exploration, you can create new features by combining existing features, transforming features, or extracting information from features. For example, you
can create new features by taking the square or square root of existing features, combining multiple features using arithmetic operations, or extracting information from text data.
1 data['new_feature'] = data['feature1'] + data['feature2']
1. Encode categorical features: If your dataset contains categorical features, you can encode them using techniques like one-hot encoding or label encoding. One-hot encoding creates binary columns
for each category, while label encoding assigns a unique integer to each category.
1 data = pd.get_dummies(data, columns=['categorical_feature'])
1. Handle missing values: If your dataset contains missing values, you can impute them using techniques like mean, median, or mode imputation. You can also create new features to indicate whether a
value is missing or not.
1 data['missing_feature'] = data['feature'].isnull().astype(int)
1. Scale numerical features: Scaling numerical features can improve the performance of some machine learning models. You can use techniques like Min-Max scaling or Standard scaling to scale your
numerical features.
1 from sklearn.preprocessing import MinMaxScaler
2 scaler = MinMaxScaler()
3 data['scaled_feature'] = scaler.fit_transform(data[['numerical_feature']])
1. Feature selection: Finally, you can select the most relevant features for your model using techniques like correlation analysis, feature importance ranking, or model-based selection methods.
1 from sklearn.feature_selection import SelectKBest, f_classif
2 selector = SelectKBest(score_func=f_classif, k=5)
3 selected_features = selector.fit_transform(data[['feature1', 'feature2', 'feature3']], data['target'])
By following these steps, you can implement feature engineering in Python to create new features, encode categorical features, handle missing values, scale numerical features, and select the most
relevant features for your machine learning model.
What is the role of NumPy library in Python Data Science?
NumPy (Numerical Python) is a fundamental package for scientific computing in Python. It provides support for large, multi-dimensional arrays and matrices, along with a collection of mathematical
functions to operate on these arrays efficiently.
The role of NumPy in Python Data Science includes:
1. Handling arrays: NumPy provides an easy way to create, manipulate, and perform operations on arrays, which are essential data structures in data science work.
2. Mathematical functions: NumPy includes a wide range of mathematical functions such as trigonometric, logarithmic, exponential, and basic statistical functions that are commonly used in data
science tasks.
3. Linear algebra operations: NumPy provides linear algebra functions to perform matrix manipulation, matrix factorization, eigenvalue calculations, and more.
4. Random number generation: NumPy includes functions for generating random numbers and sampling from various probability distributions, which are useful for simulations and statistical analysis.
5. Integration with other libraries: NumPy is often used in conjunction with other Python libraries like pandas, scikit-learn, and matplotlib to facilitate data manipulation, analysis, and
visualization tasks.
Overall, NumPy plays a crucial role in data science by providing a powerful foundation for array manipulation and mathematical operations, enabling efficient processing and analysis of data. | {"url":"https://stesha.strangled.net/blog/how-to-master-python-for-data-science","timestamp":"2024-11-10T11:15:36Z","content_type":"text/html","content_length":"202989","record_id":"<urn:uuid:53e0148e-dc45-476f-a740-b94078315f78>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00599.warc.gz"} |
ODEs: Lipschitz-continuity and an IVP
We have $y'=f(x,y)$
with $f : \mathbb{R}^2 \rightarrow \mathbb{R}^2, \left (\begin{array}{c} y_1 \\ y_2 \end{array} \right ) \mapsto \left (\begin{array}{c} g(y_1) \\ h(y_1) y_2 \end{array} \right ) $,
where $g: \mathbb{R} \rightarrow \mathbb{R}$ is Lipschitz-continuous and $h: \mathbb{R} \rightarrow \mathbb{R}$ is continuous.
Give an example for functions $g$ and $h$ such that $f$ is not Lipschitz-continuous (so the requirements for Picard-Lindelof are not met).
Prove that the IVP $y(x_0)=y_0$ for $x_0 \in \mathbb{R}$ and $y_0 \in \mathbb{R}^2$ has a unique solution on an open interval $J$ with $x_0 \in J$.
• If possible, I would suggest raising the bounty to at least $10.00.
Answers can only be viewed under the following conditions:
1. The questioner was satisfied with and accepted the answer, or
2. The answer was evaluated as being 100% correct by the judge.
View the answer
• I don't think I understand the last step, how does exponentiating and using z^1 = y^1 yield z^2 = y^2 ?
• Exponentiating yields that z2 = z2_0 * exp( int h(z1(x))). Since z2_0 = y2_0 and z1 = y1, we have z2 = y2_0 exp( int h(z2(x))) = y2.
• Sorry, that should read z2 = y2_0 exp( int h(y1(x)) ) = y2.
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How To Teach Multiplication Facts So Students Learn Instant Recall
Teaching multiplication facts well and routinely so that your students have instant recall of their facts is an annual challenge for every elementary school teacher.
Without knowing the multiplication facts as we know, many future math topics will be more difficult for your students.
This article combines the classroom experience of Pete Richardson, whoā s been teaching multiplication facts from kindergarten right up to 5th grade for many years now, as well as Third Spaceā s
own expertise in developing strategies to support learners one on one in math.
Why do we teach children multiplication facts in elementary school?
Sometimes we have to go back to basics to remember the reason weā re teaching something.
There are three reasons we need children to prioritize their multiplication facts skills at elementary school.
1. Multiplication facts are fundamental to many math topics
Without a solid understanding of the multiplication facts, children will struggle when they start to tackle division, fractions and problems with larger numbers.
Multiplication facts need to be embedded by third grade and are also central to 4th grade and 5th grades. In middle and high school, the needs become even greater.
2. Freeing up working memory allows students to develop their reasoning skills
There are certain mental math facts and operations children need to be able to carry out quickly and with a degree of automaticity in order to free up their working memory for newer, more challenging
tasks at hand.
If we can ensure the transition of multiplication facts to childrenā s long-term memory and they can become an instantly recallable fact, the working memory can be freed up for reasoning.
All children need to go through these cognitive steps in order to achieve this. Some will only need a light touch while some will need significantly longer on particular points.
Multiplication Chart 1-12 Additional Resource
Download this multiplication chart showing the times tables from 1 ā 12. Helps your pupils to learn their times tables.
Download Free Now!
Teaching times multiplication facts facts first
Children need to learn and understand how multiplication facts work before they start learning and memorizing them.
1. Repeated addition
4 x 5 is the same as 5 + 5 + 5 + 5.
Children need experience of using concrete math manipulatives such as counters or multilink cubes and pictorial representations of objects, forming arrays.
Read more on the importance of concrete representational abstract here, and how to use it.
2. Multiplication is commutative
4 x 5 is the same as 5 x 4.
Children build on their existing understanding using arrays, turning the arrays around to show that you now have 5 groups of 4 and they will still total 20. This can then be linked to recalling
multiplication facts, i.e. if they know their 5 times table as facts but not their 4 times table, they can use 4 x 5 to work out 5 x 4. This link needs to be made explicit.
3. Multiplication is the inverse of division
20 Ć· 5 = 4 can be worked out because 5 x 4 = 20.
Again, the use of arrays is key. Children need experience of pulling arrays apart into groups or sharing. After basic experience has been gained, the children should start to ā seeā an array
structure as 5 groups of 4 equal 20, and 20 can be split into 5 groups of 4.
A Third Space Learning 3rd grade lesson on the relationships of multiplication and division facts
4. Number families
4 x 5 = 20, 5 x 4 = 20, 20 Ć· 5 = 4, 20 Ć· 4 = 5
Due to their commutative understanding, by now children should also be able to see whole number families. For many children, this will need to be pointed out and discussed. Most children will be able
to explore this in its abstract form but if in doubt, go back to arrays.
From here it is only a short jump to understanding that any missing number can be worked out through knowledge of number families, e.g. 4 x [ ] = 20 or [ ] Ć· 4 = 5. There are other methods children
can use to work out missing numbers, but our goal is to increase working memory in order to increase instant recall from long-term memory. Being able to bounce around a number family will achieve
1,2,3… Counting is the key to multiplication facts practice in K-6
Counting will start before beginning to develop understanding and reasoning, but will continue long after until all multiplication facts can be recalled sequentially at speed.
Start by counting concrete items
Ensure skip counting in 2s begins with concrete manipulatives such as shoes, socks, hands, etc before moving on to using counters or other manipulatives. Whenever starting children to count in a new
amount, such as counting in 8s, children should be given the opportunity to see visually what that looks like to reinforce that 4 x 8 looks quite big compared to 4 x 6. They can then look for
patterns such as 4 x 8 is the same as 4 x 4, doubled.
Don’t be afraid of drilling the multiplication facts
Some drilling is inevitable when developing counting, initially alongside concrete and pictorial manipulatives but quickly moving to chanting ā 3 times 7 is 21, 4 times 7 is 28ā etc.
Children should by now be used to representing numbers in different ways, for example, a counter could represent 1, 2, 5, or 10. Once children are secure with this, fingers can be used to count
quickly on any multiplication table.
What about the 11 x and 12 x? Get children to make two fists and begin at 11 x with one finger up, two fingers up for 12 x, supported by their place value understanding.
Display multiplication facts around your school
Counting sequences should be highly visible everywhere!
ā ¢ If you have steps, have a student artist paint ā 6, 12, 18ā ¦ā on each step in a mural style.
ā ¢ Have some appropriate sequences visible in your hall, and rather than entering and exiting assemblies in silence, replace them with chanting.
ā ¢ Set up year group counting sequences linked to the times table expectations in each classroom.
ā ¢ Use the hopscotch grid in the playground to make math fun.
ā ¢ Itā s a good idea to include counting sequences linked to your state or school curriculum for counting ā this is different from recall of multiplication facts expectations. For example, by the
end of 2nd grade, students are expected to count by 5s, 10s and 100s up to 1,000.
Donā t forget to keep the previous yearā s sequences up too, to support those that need more time to consolidate counting sequences.
Deeper understanding of multiplication facts
For children with small working memories, frequently children with special educational needs, being able to count quickly and accurately will give them an appropriate alternative to instant recall,
as long as it is underpinned with reasoning and understanding.
However, these children can often struggle to convert the ability to count rapidly into being able to instantly recall facts. By working on childrenā s deeper understanding of what multiplication
is, how it is related to division, and number families you should be able to address this.
How to teach instant recall across all multiplication facts
Not all children will need the suggested structure below, however, it will help those who struggle to convert quick counting into instantly recallable facts.
The example is for the 6 times table but the principle can be applied to any.
Teaching 6 Times Table step by step
1. Start with 1 x 6, 2 x 6, 5 x 6, 10 x 6 first. This will build upon their most secure existing table facts.
2. Add in 3 x 6, 4 x 6 when step 1 is frequently recalled correctly and instantly.
3. Build up with 6 x 6, 7 x 6, 8 x 6.
4. When looking at 9 x 6, 11 x 6 and 12 x 6, children should look at finding 10 x 6 and adjust.
5. Be guided to remember what the last 2 numbers were in the sequence they learned (66, 72).
6. Add in related division facts. For some children, this step can be integrated from step 1 onwards. For others, they will need time to develop recall of multiplication facts first before adding
this in.
When giving children quick fire questions to recall, particularly in the early stages of each multiplication table, ensure they are given the opportunity to see the calculation rather than just hear
it orally.
Children should be encouraged to quickly count using their fingers to assist them with prompt questions such as, ā 6 x 7, we did that a minute ago, can you remember what it was?ā
Using technology for fact recall
Fact recall is the perfect opportunity to involve educational technology as there is little more value a teaching assistant or teacher can give other than asking the fact recall questions.
Traditionally, these packages support those who are quick to pick up tables to pick them up even quicker, while having limited impact on those who struggle.
However, this pattern is changing and there are innovative packages out there that will support all learners.
When picking which technology package your school should use, ask yourself the following questions:
• What added value will this give compared to traditional fact practice?
• Will the package support children who are not yet secure at counting?
• Will the package allow me to adapt what is set to match each childā s needs?
• Is the package accessible at home and from all types of devices?
Achieving the goal of instant multiplication facts recall
Putting the above steps in place and creating a focus on recall throughout your school will minimize the number of children entering 5th grade without a secure grounding of recalling their
multiplication facts. There are 2 additional steps that will take this to the next level which Iā ve detailed below.
Learning multiplication facts at home
Counting and developing recall facts for multiplication tables is an essential ā little and oftenā part of homework, along the same lines schools expect with reading. Your childā s teacher
probably expects them to read a certain amount of minutes a week, and we should think of math facts the same way.
Research says at an elementary age, it is this kind of homework that has the potential to make the most difference. Engage parents and inform them of the steps youā re using; this article on the
best way to learn multiplication facts at home has been designed to send home to parents.
Additional support for those who struggle with recall
Even with all the above in place, a (much smaller) percentage of children will still struggle to recall multiplication facts. For those with special education needs inhibiting working and long-term
memory, counting at speed is a realistic point at which to switch focus to other areas of need within Mathematics.
For other struggling learners, try giving them additional time outside English and Math lessons to practice counting at a reasonable speed and to push an increasing number of quick-fire questions and
answers towards long-term memory.
In many cases, this can be achieved as a frequent morning activity or during assemblies. A Teaching Assistant can support, or if the right technology package is chosen, simply giving them more time
to access could be all that is required.
For children who are really having difficulty, nothing beats sitting down and going through them 1-on-1, as the Third Space Learning tutors do, as part of their online math intervention.
10 fun tips for teaching multiplication facts effectively
Here are some fun ways to help students master their multiplication facts:
1. Use multiplication facts chanting
Memorizing multiplication facts via flashcards is not the only way to learn them. Chanting is also a simple yet effective way to drill multiplication knowledge into your students. It may not be the
most glamorous and exciting way of teaching multiplication facts, but it is a great place to start!
2. Make multiplication facts fun with songs and multiplication games
Our favorite multiplication facts song is Schoolhouse Rockā s ā 3 Is A Magic Numberā and weā ve got lots of fun multiplication games in this blog post.
3. Make use of multiplication charts
It might be a simple technique, but it is one that works! Hand out a multiplication chart to your class and get them to fill them in. Not only will they enjoy the challenge of filling in a
multiplication chart but it will encourage them to practice, practice, practice!
4. Use concrete resources
It doesnā t matter whether it is pasta, counters or even coins, just having concrete resources to help students work out multiplication facts can be massively beneficial.
5. Get active outside the classroom
Our multiplication facts pavement chalk activity above is just one of the outdoor math ideas used to make multiplication facts learning more active and therefore memorable for your class.
6. Use students’ interests to engage them with multiplication facts
Use the various interests your class will have to help teach multiplication facts. One of our favorite examples of this are Mr. DeMaioā s songs using multiplication facts, where he covers popular
pop songs using multiplication facts.
7. Use tricks that may be common knowledge to us, but will be revolutionary to young minds
You are well aware by now that you can do the 9 multiplication facts on your fingers, but your students may not be just yet!
8. Use quick-fire multiplication facts quizzes
While you shouldnā t make quizzes a regular feature, they can be a great way to help students get to grips with their multiplication facts. Got 5 minutes to spare when walking to swimming lessons?
Get a quickfire multiplication facts quiz in. There is always an opportunity to fit a quick multiplication facts quiz around school.
9. Ask short division based questions
Simple division questions such as ā 55 divided by 11ā and ā 30 divided by 3ā can help students realize that multiplication facts and division are closely linked, and can be used together when
trying to solve a math problem.
And finally…reward student efforts regardless of the answer
Nobody is perfect when they are just beginning to learn about something, and this is definitely the case when it comes to multiplication facts and primary school students. Donā t be afraid to hand
out praise when you see that a child has been working hard on their multiplication facts, even if they havenā t quite got the answer yet.
Do you have students who need extra support in math?
Give your students more opportunities to consolidate learning and practice skills through personalized math tutoring with their own dedicated online math tutor.
Each student receives differentiated instruction designed to close their individual learning gaps, and scaffolded learning ensures every student learns at the right pace. Lessons are aligned with
your state’s standards and assessments, plus you’ll receive regular reports every step of the way.
Personalized one-on-one math tutoring programs are available for:
– 2nd grade tutoring
– 3rd grade tutoring
– 4th grade tutoring
– 5th grade tutoring
– 6th grade tutoring
– 7th grade tutoring
– 8th grade tutoring
Why not learn more about how it works?
The content in this article was originally written by senior leader Pete Richardson and has since been revised and adapted for US schools by elementary math teacher Christi Kulesza. | {"url":"https://thirdspacelearning.com/us/blog/how-to-teach-multiplication-facts-elementary/","timestamp":"2024-11-07T02:38:54Z","content_type":"text/html","content_length":"146913","record_id":"<urn:uuid:593b7e00-0009-4e71-be1d-f3134f8b77ea>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00220.warc.gz"} |
Math and Statistics Functions – Real Python
Math and Statistics Functions
In this lesson, you’ll learn about new and improved math and statistics functions in Python 3.8. Python 3.8 brings many improvements to existing standard library packages and modules. math in the
standard library has a few new functions. math.prod() works similarly to the built-in sum(), but for multiplicative products:
>>> import math
>>> math.prod((2, 8, 7, 7))
>>> 2 * 8 * 7 * 7
The two statements are equivalent. prod() will be easier to use when you already have the factors stored in an iterable.
Another new function is math.isqrt(). You can use isqrt() to find the integer part of square roots:
>>> import math
>>> math.isqrt(9)
>>> math.sqrt(9)
>>> math.isqrt(15)
>>> math.sqrt(15)
The square root of 9 is 3. You can see that isqrt() returns an integer result, while math.sqrt() always returns a float. The square root of 15 is almost 3.9. Note that isqrt() truncates the answer
down to the next integer, in this case 3.
Finally, you can now more easily work with n-dimensional points and vectors in the standard library. You can find the distance between two points with math.dist(), and the length of a vector with
>>> import math
>>> point_1 = (16, 25, 20)
>>> point_2 = (8, 15, 14)
>>> math.dist(point_1, point_2)
>>> math.hypot(*point_1)
>>> math.hypot(*point_2)
This makes it easier to work with points and vectors using the standard library. However, if you will be doing many calculations on points or vectors, you should check out NumPy.
The statistics module also has several new functions:
The following example shows the functions in use:
>>> import statistics
>>> data = [9, 3, 2, 1, 1, 2, 7, 9]
>>> statistics.fmean(data)
>>> statistics.geometric_mean(data)
>>> statistics.multimode(data)
[9, 2, 1]
>>> statistics.quantiles(data, n=4)
[1.25, 2.5, 8.5]
In Python 3.8, there is a new statistics.NormalDist class that makes it more convenient to work with the Gaussian normal distribution. To see an example of using NormalDist, you can try to compare
the speed of the new statistics.fmean() and the traditional statistics.mean():
>>> import random
>>> import statistics
>>> from timeit import timeit
>>> # Create 10,000 random numbers
>>> data = [random.random() for _ in range(10_000)]
>>> # Measure the time it takes to run mean() and fmean()
>>> t_mean = [timeit("statistics.mean(data)", number=100, globals=globals())
... for _ in range(30)]
>>> t_fmean = [timeit("statistics.fmean(data)", number=100, globals=globals())
... for _ in range(30)]
>>> # Create NormalDist objects based on the sampled timings
>>> n_mean = statistics.NormalDist.from_samples(t_mean)
>>> n_fmean = statistics.NormalDist.from_samples(t_fmean)
>>> # Look at sample mean and standard deviation
>>> n_mean.mean, n_mean.stdev
(0.825690647733245, 0.07788573997674526)
>>> n_fmean.mean, n_fmean.stdev
(0.010488564966666065, 0.0008572332785645231)
>>> # Calculate the lower 1 percentile of mean
>>> n_mean.quantiles(n=100)[0]
In this example, you use timeit to measure the execution time of mean() and fmean(). To get reliable results, you let timeit execute each function 100 times, and collect 30 such time samples for each
function. Based on these samples, you create two NormalDist objects. Note that if you run the code yourself, it might take up to a minute to collect the different time samples.
NormalDist has many convenient attributes and methods. See the documentation for a complete list. Inspecting .mean and .stdev, you see that the old statistics.mean() runs in 0.826 ± 0.078 seconds,
while the new statistics.fmean() spends 0.0105 ± 0.0009 seconds. In other words, fmean() is about 80 times faster for these data.
If you need more advanced statistics in Python than the standard library offers, check out statsmodels and scipy.stats. | {"url":"https://realpython.com/lessons/math-and-statistics-functions/","timestamp":"2024-11-03T00:52:17Z","content_type":"text/html","content_length":"72624","record_id":"<urn:uuid:8c7170c7-be14-4689-8bb6-3ae3905f38fd>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00013.warc.gz"} |
The Stacks project
Lemma 36.16.1. Let $X$ be a scheme and $\mathcal{L}$ an ample invertible $\mathcal{O}_ X$-module. If $K$ is a nonzero object of $D_\mathit{QCoh}(\mathcal{O}_ X)$, then for some $n \geq 0$ and $p \in
\mathbf{Z}$ the cohomology group $H^ p(X, K \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{L}^{\otimes n})$ is nonzero.
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Logical Math Techniques for Effective Problem-Solving
Do you love math? Do you like solving puzzles? If so, then you'll love learning about the top 6 logical math techniques! In this article, we will explore six powerful math techniques that will
revolutionize the way you approach math problems.
Top 6 Logical Math Techniques
• Break down the problem
• Trial & Error method
• Elimination Technique
• Backward Substitution
• Solving from answers
• Plugging in Numbers
Break down the Problem
In this section, we will delve into the technique of breaking problems down into manageable parts. By breaking complex problems into smaller, more approachable components, we can simplify the
problem-solving process and gradually build our way towards a solution.
The Break the Problems technique involves the following steps:
1. Identify the given information: Understand the problem statement and determine the relevant details and variables involved.
2. Analyze the problem: Break the problem into smaller steps or components that can be tackled individually.
3. Solve each part: Focus on solving each smaller part of the problem, applying appropriate mathematical concepts and operations.
John has 24 apples. He wants to distribute them equally among his 3 friends. How many apples will each friend get?
A,P,R,X,S, and Z are sitting in a row.
S and Z are in the centre.
A and P are at the end.
R is sitting to the left of A.
Who is to the right of P.
To solve this problem using the "break the problem" method, let's break down the given information:
1. S and Z are in the center.
2. A and P are at the ends.
3. R is sitting to the left of A.
P _ _ _ _ _ A
Next, we consider the third point: R is sitting to the left of A. Since A is at the end, R must be sitting to the immediate left of A. Therefore, the seating arrangement becomes:
P _ _ _ _ R A
Now, let's address the final question: Who is to the right of P? Based on our arrangement, S is the only remaining person who can be to the right of P. Therefore, the final seating arrangement is:
P S _ _ _ R A
In conclusion, S is the person who is sitting to the right of P in the given seating arrangement.
Trial & Error Method
The Trial & Error method involves systematically trying different possible solutions until the correct one is found. It is particularly useful when the problem does not have a straightforward
solution path.
Example: Substituting different values of the variable and checking the equality of LHS and RHS is the trial and error method. Let us solve the equation 3x + 4 = 16. We start to substitute different
values of x. The value for which both the sides are balanced is the required solution.
LHS RHS
x=1 3x1 + 4 = 7 16
x=2 3x2 + 4 = 10 16
x=3 3x3 + 4 = 13 16
x=4 3x4 + 4 = 16 16
Hence, x = 4.
Elimination Technique
The Elimination Technique involves systematically eliminating incorrect options or possibilities through a series of logical deductions. This technique is commonly used in multiple-choice questions
or scenarios where we need to narrow down the choices to identify the correct solution.
Example: Problem: Tim, Tom and Jim are triplets(three brothers born on the same day). Their brother is exactly 3 years older. Which of the following numbers can be the sum of the ages of the four
Step 1: From question, we understand that there is a triplet(3 brothers) and an elder brother who is 3 years older. Step 2: With above info, we can say the sum of their ages should be a number
divisible by 4 + 3.
Step 3: Using elimination from answers, we can eliminate 61, 25 and 30. Then, we can find only 27 is 24(divisible by 4) + 3. Hence, that’s the answer.
Backward Solving
Backward solving involved working from the result backwards to find the initial unknown value. Backward operations for different arithmetic operations are given below:
Original Operation Backward Operation
Addition + Subtraction -
Subtraction - Addition +
Multiplication x Division ÷
Division ÷ Multiplication x
Square Square root
Square root Square
Problem: Find a number such that when multiplied by 5 and added to 8, the result is 28. Step 1: We need to work backwards here. Result is 28. Step 2: 28 is got when u added 8, so working backward, we
need to subtract 8 from 28. So, 20 Step 3: When a number was multiplies by5, we got 20. Now working backwards, we need to divide 20 by 5. So its 4.
Solving from Answers
The Solving from Answers technique involves plugging the answer choices into the problem and checking which choice satisfies the given conditions. It can save time and effort by eliminating incorrect
options and pinpointing the correct solution.
Example: Problem: Find the value of x that satisfies the equation 2x - 5 = 7.
1. 3
2. 6
3. 5
4. 7
Step 1: Start with the given equation: 2x - 5 = 7. Step 2: Plug the answer choices into the equation and check:
• Let's try x = 3 as the first answer choice.
• Let's try x = 6 as the second answer choice.
The Plugging in Numbers technique involves substituting specific values into the problem to simplify the calculations and arrive at the solution. This technique is useful when dealing with complex
formulas or variables.
Example: Problem: Find the value of y in the equation 3y - 4 = 2y + 8.
Step 1: Start with the given equation: 3y - 4 = 2y + 8. Step 2: Choose a value for y to simplify the equation:
• Let's try y = 6 as the first value.
• Substitute y = 6 into the equation: 3(6) - 4 = 2(6) + 8.
• If the left side of the equation matches the right side, we have found the value of y.
By incorporating the above techniques, you can now confidently tackle even the most challenging math problems.
But remember, practice makes perfect. Keep honing your problem-solving skills by applying these techniques to a variety of math problems. With practice and a solid understanding of these techniques,
you'll become a math-solving champion in no time.
FAQs on Logical Math Techniques:
1. Q: Are these techniques applicable to all levels of math? A: Yes, these techniques can be applied to various levels of math, from basic arithmetic to more advanced algebraic equations.
2. Q: Can these techniques be used in subjects other than math? A: While these techniques are primarily focused on mathematical problem-solving, they can also be applied to logical reasoning and
critical thinking in other subjects.
3. Q: How do I know which technique to use for a specific problem? A: Understanding the problem and its context is crucial. Different techniques may be more suitable depending on the nature of the
problem, the given information, and the desired outcome.
4. Q: Are these techniques time-saving? A: Yes, these techniques can significantly save time by providing systematic approaches to problem-solving. They help in eliminating incorrect choices,
narrowing down possibilities, and simplifying complex problems.
5. Q: Can these math new tricks be used in competitive exams or standardized tests? A: Absolutely! These techniques are valuable tools for tackling time-sensitive exams, enabling you to approach
questions strategically and make efficient use of your time.
This article on Logical Math Techniques explores six powerful math techniques that can revolutionize the way you approach math problems. From breaking down complex problems into smaller parts to
using trial and error, elimination, backward substitution, solving from answers, and plugging in numbers, these math techniques can help you solve math problems more efficiently. Whether you are
studying 7th-grade math formulas or looking for new tricks to improve your problem-solving skills, these math new tricks can be applied to various levels of math and can help you save time and
eliminate incorrect choices. So, keep practicing and honing your problem-solving skills with these logical math techniques. | {"url":"https://upfunda.academy/blog/24ace844-232a-4bca-a277-567c457ebc22","timestamp":"2024-11-09T00:10:27Z","content_type":"text/html","content_length":"47958","record_id":"<urn:uuid:8e768ca8-8281-4a13-b460-bdd48207f2dd>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00608.warc.gz"} |
Improved Feasible Solution Algorithms for High Breakdown Estimation
High breakdown estimation allows one to get reasonable estimates of the parameters from a sample of data even if that sample is contaminated by large numbers of awkwardly placed outliers. Two
particular application areas in which this is of interest are multiple linear regression, and estimation of the location vector and scatter matrix of multivariate data. Standard high breakdown
criteria for the regression problem are the least median of squares (LMS) and least trimmed squares (LTS); those for the multivariate location/scatter problem are the minimum volume ellipsoid (MVE)
and minimum covariance determinant (MCD). All of these present daunting computational problems. The ‘feasible solution algorithms’ for these criteria have been shown to have excellent performance for
text-book sized problems, but their performance on much larger data sets is less impressive. This paper points out a computationally cheaper feasibility condition for LTS, MVE and MCD, and shows how
the combination of the criteria leads to improved performance on large data sets. Algorithms incorporating these improvements are available from the first author's Web site.
Recommended Citation
Hawkins, Douglas M. and Olive, David J. "Improved Feasible Solution Algorithms for High Breakdown Estimation." (Mar 1999). | {"url":"https://opensiuc.lib.siu.edu/math_articles/4/","timestamp":"2024-11-07T19:43:39Z","content_type":"text/html","content_length":"35396","record_id":"<urn:uuid:10290918-a66c-470c-87d8-6003691133f6>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00123.warc.gz"} |
find-bottom-left-tree-value | Leetcode
Similar Problems
Similar Problems not available
Find Bottom Left Tree Value - Leetcode Solution
LeetCode: Find Bottom Left Tree Value Leetcode Solution
Difficulty: Medium
Topics: binary-tree tree depth-first-search breadth-first-search
Problem statement:
Given a binary tree, find the leftmost value in the last row of the tree.
Input: 2 /
Output: 1
Explanation: The leftmost value in the last row is 1.
The problem requires us to find the leftmost value in the last row of the binary tree. We can solve this problem by doing a level-order traversal of the binary tree. We can keep track of the last
node on each level, and when we finish traversing the tree, we can return the value of the last node on the last level, which will be the leftmost value in the last row.
1. We first create a queue to hold the nodes to be processed. We also create a variable leftmost to hold the leftmost value on the last row.
2. We add the root node to the queue.
3. We start a loop that continues until the queue is empty.
4. In each iteration, we dequeue a node from the queue and assign its value to the leftmost variable.
5. We then add the children of the dequeued node to the queue, if they are not null.
6. At the end of the loop, we return the value of the leftmost variable, which will be the leftmost value on the last row.
Here is the code to solve the problem:
class Solution {
int findBottomLeftValue(TreeNode* root) {
queue<TreeNode*> q;
int leftmost;
while (!q.empty()) {
int size = q.size();
for (int i = 0; i < size; i++) {
TreeNode* node = q.front();
if (i == 0) leftmost = node->val;
if (node->left) q.push(node->left);
if (node->right) q.push(node->right);
return leftmost;
Time Complexity:
Since we are traversing all the nodes once, the time complexity of the algorithm is O(n), where n is the number of nodes in the binary tree.
Space Complexity:
The space complexity of the algorithm is O(w), where w is the maximum width of the binary tree. This is because, at any point in time, the queue will have at most w nodes, where w is the width of the
binary tree. In the worst case, when the binary tree is a complete binary tree, the width is (n+1)/2, where n is the number of nodes in the binary tree. Hence, the space complexity is O(n/2) or O(n).
Find Bottom Left Tree Value Solution Code | {"url":"https://prepfortech.io/leetcode-solutions/find-bottom-left-tree-value","timestamp":"2024-11-08T08:15:04Z","content_type":"text/html","content_length":"57410","record_id":"<urn:uuid:ab51d8bf-7c58-4d70-ba85-95f958d3ee21>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00198.warc.gz"} |
The number - math word problem (7807)
The number
What number of 1 cm cubes are required to make a 4 cm cube?
Correct answer:
Did you find an error or inaccuracy? Feel free to
write us
. Thank you!
You need to know the following knowledge to solve this word math problem:
Units of physical quantities:
Grade of the word problem:
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Nominal Interest Rate Calculator - [100% Free] - Calculators.io
By definition, the nominal interest rate is the rate of interest before you take into account inflation. You can calculate this value using this nominal interest rate calculator. In some cases,
nominal may even refer to the stated or advertised interest rates on loans without taking the compounding of interest and the fees into account. Finally, a nominal rate may even refer to the federal
funds rate or the rate of interest set by the Federal Reserve.
How to use the nominal interest rate calculator?
Even if you know how to compute for the nominal interest rate using the nominal interest rate formula, this nominal interest rate calculator. This is an online tool that’s easy to use and will
provide you with the results you need quickly. To use the calculator, here are the steps to follow:
• First, input the percentage value of the Effective Rate per period.
• Then input the Compounding value per period.
• Finally, input the value for the Number of Periods.
• After entering all of the values, the nominal interest rate calculator will automatically generate for you the values of the Nominal Rate per Period, the Effective Rate for 5 Years, and the Rate
per Compounding Interval.
What does nominal interest rate mean?
The nominal interest rate refers to the percentage yield of a loan or security without taking inflation into account. This means that it’s the actual rate that a borrower would pay to a lender to
utilize their funds.
How do you calculate nominal interest rate?
The nominal interest rate which is also known as the annualized percentage rate or the APR is the interest that you would have to pay for before considering inflation. With this in mind, it is,
therefore, essential for you to compute for the nominal interest rate of loans and credit cards. In doing this, you can find out which ones have the lowest cost using a single method.
You can make the computations manually or use this nominal interest rate calculator to make things easier. Also, it’s important for you to differentiate between the nominal and the real interest
rate. The latter accounts for the wearing down of purchasing power because of inflation. You can calculate the nominal interest rate using the following formula:
NIR = RIR + IR
NIR refers to the nominal interest rate
RIR refers to the real interest rate
IR refers to the inflation rate
You can also use the same equation but move the values around if you want to compute for the real interest rate that you’re receiving or paying:
RIR = NIR – IR
To better understand how to use NIR, keep these three key concepts in mind:
Average daily balance
This refers to the average amount that you owe an entity in a monthly cycle of billing. This is the sum of the balances each day divided by the number of days in the given cycle.
Daily periodic rate
This refers to the interest rate that gets applied to a daily balance each day. It’s equal to the nominal interest rate divided by 365 or how many days there are in a year. For instance, a NIR with a
15% value and a year with 365 days would give you a daily periodic rate of 0.041%.
Daily compounding
This refers to the daily interest that gets charged and added to the daily balance so that you can get the daily balance of the next day. This means that you’re paying for interest on interest until
you’ve paid off the whole balance. However, NIR doesn’t take into account the impact of compounding. Because of daily compounding, the actual interest rate will always go beyond the NIR.
How do you calculate effective annual interest rate?
The effective annual rate of the EAR refers to the interest rate that gets adjusted over a specific period to take compounding into account. In other words, it’s the interest rate that investors can
either pay or earn in one year after taking compounding into consideration.
The EAR is also known as the annual equivalent rate, the effective interest rate, the annual percentage yield or the effective rate. To compute for the EAR, you can use this formula:
EAIR = (1 + i/n)^n – 1
i refers to the stated annual interest rate
n refers to the number of compounding periods
How do you calculate monthly interest rate?
If you’re able to calculate interest every month, this means that you have a very relevant skill. Often, you may see interest rates quoted as annual percentages. Sometimes though, you may want to
know the exact numbers. To help explain this, let’s have an example:
Most of the time, we think in terms of costs per month. We have our monthly food costs, monthly car payments, monthly utility bills, and so on. Usually, the interest is also a monthly event, and the
recurring calculations would add up to large numbers over a given period of time. Whether you’re earning or paying interest, how you would convert the annual rates to monthly rates is essentially the
same. Here are some steps to follow:
• First, calculate the monthly interest rate. This is a simple process, and all you have to do is divide your annual interest rate by 12 since each year has 12 months.
• Then convert the percentage form to a decimal form to finalize the steps.
• Divide the value by how many time periods there are. So, you began with a single annual time period, and you need to convert into 12 periods for each of the 12 months. You can use this same
concept for when you want to convert into different time periods:
• For daily interest rates, divide the annual rate by 365 or 360.
• For quarterly interest rates, divide the annual rate by 4.
• For weekly interest rates, divide the annual rate by 52.
• Here’s an example: let’s say that you pay a monthly interest at 10% each year. How much would your monthly interest be and how much will you have to pay if you borrow $100? Here is the
10/100 = 0.1 each year
0.10/12 months = .0083
0.0083 x $100 = $0.83
0.0083 x 100 = 0.83% each year | {"url":"https://calculators.io/nominal-interest-rate/","timestamp":"2024-11-11T17:58:35Z","content_type":"text/html","content_length":"88691","record_id":"<urn:uuid:432c4c87-b001-4f9c-85ce-3343dd0c9dc1>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00717.warc.gz"} |
Normal Distribution | Examples, Formulas, & Uses
In a normal distribution, data is symmetrically distributed with no skew. When plotted on a graph, the data follows a bell shape, with most values clustering around a central region and tapering off
as they go further away from the center.
Normal distributions are also called Gaussian distributions or bell curves because of their shape.
Why do normal distributions matter?
All kinds of variables in natural and social sciences are normally or approximately normally distributed. Height, birth weight, reading ability, job satisfaction, or SAT scores are just a few
examples of such variables.
Because normally distributed variables are so common, many statistical tests are designed for normally distributed populations.
Understanding the properties of normal distributions means you can use inferential statistics to compare different groups and make estimates about populations using samples.
What are the properties of normal distributions?
Normal distributions have key characteristics that are easy to spot in graphs:
• The mean, median and mode are exactly the same.
• The distribution is symmetric about the mean—half the values fall below the mean and half above the mean.
• The distribution can be described by two values: the mean and the standard deviation.
The mean is the location parameter while the standard deviation is the scale parameter.
The mean determines where the peak of the curve is centered. Increasing the mean moves the curve right, while decreasing it moves the curve left.
The standard deviation stretches or squeezes the curve. A small standard deviation results in a narrow curve, while a large standard deviation leads to a wide curve.
Empirical rule
The empirical rule, or the 68-95-99.7 rule, tells you where most of your values lie in a normal distribution:
• Around 68% of values are within 1 standard deviation from the mean.
• Around 95% of values are within 2 standard deviations from the mean.
• Around 99.7% of values are within 3 standard deviations from the mean.
Example: Using the empirical rule in a normal distribution
You collect SAT scores from students in a new test preparation course. The data follows a normal distribution with a mean score (M) of 1150 and a standard deviation (SD) of 150.
Following the empirical rule:
• Around 68% of scores are between 1,000 and 1,300, 1 standard deviation above and below the mean.
• Around 95% of scores are between 850 and 1,450, 2 standard deviations above and below the mean.
• Around 99.7% of scores are between 700 and 1,600, 3 standard deviations above and below the mean.
The empirical rule is a quick way to get an overview of your data and check for any outliers or extreme values that don’t follow this pattern.
If data from small samples do not closely follow this pattern, then other distributions like the t-distribution may be more appropriate. Once you identify the distribution of your variable, you can
apply appropriate statistical tests.
Central limit theorem
The central limit theorem is the basis for how normal distributions work in statistics.
In research, to get a good idea of a population mean, ideally you’d collect data from multiple random samples within the population. A sampling distribution of the mean is the distribution of the
means of these different samples.
The central limit theorem shows the following:
• Law of Large Numbers: As you increase sample size (or the number of samples), then the sample mean will approach the population mean.
• With multiple large samples, the sampling distribution of the mean is normally distributed, even if your original variable is not normally distributed.
Parametric statistical tests typically assume that samples come from normally distributed populations, but the central limit theorem means that this assumption isn’t necessary to meet when you have a
large enough sample.
You can use parametric tests for large samples from populations with any kind of distribution as long as other important assumptions are met. A sample size of 30 or more is generally considered
For small samples, the assumption of normality is important because the sampling distribution of the mean isn’t known. For accurate results, you have to be sure that the population is normally
distributed before you can use parametric tests with small samples.
Formula of the normal curve
Once you have the mean and standard deviation of a normal distribution, you can fit a normal curve to your data using a probability density function.
In a probability density function, the area under the curve tells you probability. The normal distribution is a probability distribution, so the total area under the curve is always 1 or 100%.
The formula for the normal probability density function looks fairly complicated. But to use it, you only need to know the population mean and standard deviation.
For any value of x, you can plug in the mean and standard deviation into the formula to find the probability density of the variable taking on that value of x.
Normal probability density formula Explanation
• f(x) = probability
• x = value of the variable
• μ = mean
• σ = standard deviation
• σ^2 = variance
Example: Using the probability density function
You want to know the probability that SAT scores in your sample exceed 1380.
On your graph of the probability density function, the probability is the shaded area under the curve that lies to the right of where your SAT scores equal 1380.
You can find the probability value of this score using the standard normal distribution.
What is the standard normal distribution?
The standard normal distribution, also called the z-distribution, is a special normal distribution where the mean is 0 and the standard deviation is 1.
Every normal distribution is a version of the standard normal distribution that’s been stretched or squeezed and moved horizontally right or left.
While individual observations from normal distributions are referred to as x, they are referred to as z in the z-distribution. Every normal distribution can be converted to the standard normal
distribution by turning the individual values into z-scores.
Z-scores tell you how many standard deviations away from the mean each value lies.
You only need to know the mean and standard deviation of your distribution to find the z-score of a value.
Z-score Formula Explanation
• x = individual value
• μ = mean
• σ = standard deviation
We convert normal distributions into the standard normal distribution for several reasons:
• To find the probability of observations in a distribution falling above or below a given value.
• To find the probability that a sample mean significantly differs from a known population mean.
• To compare scores on different distributions with different means and standard deviations.
Finding probability using the z-distribution
Each z-score is associated with a probability, or p-value, that tells you the likelihood of values below that z-score occurring. If you convert an individual value into a z-score, you can then find
the probability of all values up to that value occurring in a normal distribution.
Example: Finding probability using the z-distribution
To find the probability of SAT scores in your sample exceeding 1380, you first find the
The mean of our distribution is 1150, and the standard deviation is 150. The z-score tells you how many standard deviations away 1380 is from the mean.
For a z-score of 1.53, the p-value is 0.937. This is the probability of SAT scores being 1380 or less (93.7%), and it’s the area under the curve left of the shaded area.
To find the shaded area, you take away 0.937 from 1, which is the total area under the curve.
Probability of x > 1380 = 1 – 0.937 = 0.063
That means it is likely that only 6.3% of SAT scores in your sample exceed 1380.
Other interesting articles
If you want to know more about statistics, methodology, or research bias, make sure to check out some of our other articles with explanations and examples.
Frequently asked questions about normal distributions
In a normal distribution, data are symmetrically distributed with no skew. Most values cluster around a central region, with values tapering off as they go further away from the center.
The measures of central tendency (mean, mode, and median) are exactly the same in a normal distribution.
The standard normal distribution, also called the z-distribution, is a special normal distribution where the mean is 0 and the standard deviation is 1.
Any normal distribution can be converted into the standard normal distribution by turning the individual values into z-scores. In a z-distribution, z-scores tell you how many standard deviations
away from the mean each value lies.
The empirical rule, or the 68-95-99.7 rule, tells you where most of the values lie in a normal distribution:
□ Around 68% of values are within 1 standard deviation of the mean.
□ Around 95% of values are within 2 standard deviations of the mean.
□ Around 99.7% of values are within 3 standard deviations of the mean.
The empirical rule is a quick way to get an overview of your data and check for any outliers or extreme values that don’t follow this pattern.
The t-distribution is a way of describing a set of observations where most observations fall close to the mean, and the rest of the observations make up the tails on either side. It is a type of
normal distribution used for smaller sample sizes, where the variance in the data is unknown.
The t-distribution forms a bell curve when plotted on a graph. It can be described mathematically using the mean and the standard deviation.
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