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Optimum Statistical Estimation with Strategic Data Sources
Optimum Statistical Estimation with Strategic Data Sources
Proceedings of The 28th Conference on Learning Theory, PMLR 40:280-296, 2015.
We propose an optimum mechanism for providing monetary incentives to the data sources of a statistical estimator such as linear regression, so that high quality data is provided at low cost, in the
sense that the weighted sum of payments and estimation error is minimized. The mechanism applies to a broad range of estimators, including linear and polynomial regression, kernel regression, and,
under some additional assumptions, ridge regression. It also generalizes to several objectives, including minimizing estimation error subject to budget constraints. Besides our concrete results for
regression problems, we contribute a mechanism design framework through which to design and analyze statistical estimators whose examples are supplied by workers with cost for labeling said examples.
Cite this Paper
Related Material | {"url":"http://proceedings.mlr.press/v40/Cai15.html","timestamp":"2024-11-10T15:12:58Z","content_type":"text/html","content_length":"16466","record_id":"<urn:uuid:d3153933-1e99-43ef-9666-2f327c821f46>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00706.warc.gz"} |
Alternatives to cosine similarity
Cosine similarity is the recommended way to compare vectors, but what other distance functions are there? And are any of them better?
Last month we looked at how cosine similarity works and how we can use it to calculate the "similarity" of two vectors. But why choose cosine similarity over any other distance function? Why not use
Euclidean distance, or Manhattan, or Chebyshev?
In this article we'll dig in to some alternative methods for comparing vectors, and see how they compare to cosine similarity. Are any of them faster? Are any of them more accurate?
Comparing vectors is a powerful tool when working with LLM embeddings. It's often the technical underpinning of RAG pipelines (Retrieval Augmented Generation), for instance, where related content is
"found" and injected into the context of a message passed to an LLM. Across the web, cosine similarity is the most frequently recommended way to compare vectors.
OpenAI themselves say as much in their documentation:
Which distance function should I use?
We recommend cosine similarity. The choice of distance function typically doesn’t matter much.
But there are plenty of other distance functions, so why is cosine similarity the most recommended? And are there any other functions that might be better suited to comparing LLM embeddings?
What are we comparing?
Three pairs of simple vectors in 2D space. We'll use these to illustrate the different distance functions later in this article.
The end goal for all these similarity functions is to compare vectors. A vector is an array of numbers, and we refer to them as "vectors" because they can be conceptualized as points in space. A "two
dimensional" vector (i.e. an array with two numbers) can be thought of as a point on a graph. A "three dimensional" vector (an array with three numbers) can be thought of as a point in 3D space.
Their magnitude is the distance from the origin (0, 0 for 2D, or 0, 0, 0 for 3D) to the point.
In practice, I use similarity functions to compare LLM embeddings which are vectors with ~1,5k dimensions. For the purposes of this article, we'll stick to visualizing the different functions in two
dimensions, but all the concepts covered are applicable to multidimensional vectors.
Note: in the context of LLM embeddings, a vector is a list of numbers that represents the "meaning" of a piece of text. The more "similar" the vectors, the more similar the meanings of the text.
For more detail, see my post about mapping LLM embeddings in three dimensions.
Cosine similarity
In case you missed the first article, here's a quick recap of how cosine similarity works:
$\mathit{d}(A,B) = \cos(\theta) = \frac{\mathbf{A} \cdot \mathbf{B}}{||\mathbf{A}|| \times ||\mathbf{B}||}$
In this and following formulae, we'll use d(A,B) to represent the distance d between vectors A and B.
In cosine similarity, the final value is the cosine of the angle between two vectors. The similarity value is calculated by taking the dot product of the two vectors (A⋅B) and dividing it by the
product of the magnitudes of the two vectors (∣∣A∣∣×∣∣B∣∣).
You can learn more about what "dot product" means (and how to calculate it) in the previous article.
Cosine similarity uses the angle θ between two vectors in 2D space.
Vectors created by OpenAI embeddings are normalized to a magnitude of 1. The full cosine similarity function does it's own normalization (by dividing by the magnitudes of the vectors), but the fact
that the vectors are already normalized means we can simplify the function to just the dot product:
$\text{Normalized} \cos(\theta) = A \cdot B$
As the OpenAI documentation says:
OpenAI embeddings are normalized to length 1, which means that:
□ Cosine similarity can be computed slightly faster using just a dot product
□ Cosine similarity and Euclidean distance will result in the identical rankings
In JavaScript, calculating the dot product can be a simple one-liner (if you think reduce() functions are simple):
const dotProduct = (a, b) => a.reduce((acc, cur, i) => acc + cur * b[i], 0);
Euclidean distance
By comparison to cosine similarity, Euclidean distance is a much simpler concept to visualize. It's a measure of the straight-line distance between two points in space. It's calculated by taking the
square root of the sum of the squared differences between the two vectors.
$\mathit{d}(A,B) = \sqrt{\sum_{i=1}^n (A_i - B_i)^2}$
Finding Euclidean distance d for vectors A and B of length n.
Euclidean distance is the straight-line distance between two points.
const euclideanDistance = (a, b) => Math.sqrt(
a.reduce((acc, curr, i) => {
const diff = curr - b[i];
return acc + diff * diff;
}, 0)
Is Euclidean distance better than cosine similarity for use with LLM embeddings?
When the vectors are normalized to a magnitude of 1 (as OpenAI embeddings are), the results of cosine similarity and Euclidean distance will be equivalent. This means that the rankings of the vectors
will be the same, even if the actual values are different.
There's not much to separate the functions when viewed from the perspective of time complexity (i.e. using "Big O" notation), as both functions have O(n) complexity. But the dot product uses simpler
operations (one multiplication and one addition per iteration) than the Euclidean calculation (which performs subtraction, squaring, and addition for each iteration and includes an additional square
root operation at the end).
For measuring the similarity of LLM embeddings, cosine similarity is more often chosen because dot-product-based cosine similarity is slightly faster to calculate.
Manhattan distance
Unlike Euclidean distance, which measures the straight line distance between two points, Manhattan distance measures the sum of the absolute differences between the two vectors. It gets its name from
the fact that it's the distance a taxi would have to travel to get from one point to another in a city grid (where travel is limited to moving along the grid lines, rather than the as-the-crow-flies
Euclidean distance).
$\mathit{d}(A,B) = \sum_{i=1}^n |A_1 - B_1|$
Finding Manhattan distance d for vectors A and B of length n.
Manhattan distance is the sum of the absolute differences between two points.
const manhattanDistance = (a, b) => a.reduce((acc, curr, i) => acc + Math.abs(curr - b[i]), 0);
Is Manhattan distance better than cosine similarity for use with LLM embeddings?
The numerical results of Manhattan distance will be different to cosine similarity, but (as with Euclidean distance) there should not be much variation in the rankings when sorting vectors by
similarity. As with the other two functions, the time complexity of Manhattan distance is also O(n), but the operations are simpler than Euclidean distance (just one subtraction and one addition per
iteration) but a little more costly than the dot product calculation (with the "absolute" Math.abs() calculation being the key differentiator).
When compared directly to Euclidean distance, Manhattan distance is often considered to be less affected by the "curse of dimensionality". This is a phenomenon where the distance between points in
high-dimensional space becomes less meaningful as the number of dimensions increases. Manhattan distance is less affected by this because it doesn't square the differences between the points, which
also makes it less sensitive to outliers (the squared differences in Euclidean distance can be heavily influenced by a single large difference).
Chebyshev distance
Chebyshev distance is the maximum of the absolute differences between the two vectors. While useful in certain scenarios, the fact that it discards the other differences between the vectors makes it
less useful for comparing vectors in the context of LLM embeddings.
The Chebyshev distance is more useful for things like warehouse logistics, where it can measure the time an overhead crane takes to move an object (as the crane can move on the x and y axes at the
same time but at the same speed along each axis).
$\mathit{d}(A,B) = \max(|A_1 - B_1|, |A_2 - B_2|, \ldots, |A_n - B_n|)$
Chebyshev distance
Chebyshev distance is the longest of the absolute differences between two points.
const chebyshevDistance = (a, b) => 1 - Math.max(...a.map((curr, i) => Math.abs(curr - b[i])));
Other ways to measure similarity
There are a lot of distance functions out there. For this article, I've leaned into the "geometrical" view of vectors - i.e. treating them as points in multidimensional space. This is because LLM
embeddings are built to represent the "meaning" of text, and the crucial part of that is the "directional" relationships between the vectors. Embeddings are all normalized and have no sense of
"magnitude" (i.e. the length of the vector), so the "direction" is the only thing that matters.
Other geometric distance functions
• Minkowski distance and Canberra distance are functions that build on the foundation of the Euclidean and Manhattan functions. Minkowski is a generalization of both that can generate either of
those distances or something in between, and Canberra is a weighted version of Manhattan distance.
• Mahalanobis distance is a different process entirely, which measures the distance between a single point and a distribution. This is useful for intra-vector analysis, but not helpful for
comparing vectors to one another.
Correlation-based comparison functions
These measures focus on the relationship between variables rather than direct vector comparison. They can be useful for analyzing trends or patterns in data, but are less useful for comparing
• Pearson correlation coefficient measures the linear relationship between two variables.
• Spearman's rank correlation coefficient measures the strength and direction of the relationship between two variables.
• Kendall's tau is a measure of the ordinal association between two measured quantities.
Set-based comparison functions
These are more suitable for comparing sets or binary vectors, which may not be directly applicable to continuous-valued embeddings. Sets, unlike vectors, are unordered collections of unique elements.
• Jaccard similarity measures the similarity between two sets.
• Hamming distance measures the number of positions at which the corresponding elements are different.
Testing the functions
I put these functions to the test with some of my own data, and the results were generally in-line with what I expected.
I created an embedding vector for the titles of all my blog posts. I then created embeddings for each query in my search-test "ground truth dataset" and compared each query to every post title (98
queries each tested against 79 document titles, with each query and title represented by a 1,536-dimensional vector). I ran this test five times for each distance function, and averaged the results
of each run.
The execution time for each comparison call (for example cosinesimilarity(vector1, vector2)) was measured with the JS performance.measure() method.
The "accuracy" was measured using a combination of F-score, Average Precision (AP), and Normalized Discounted Cumulative Gain (NDCG). You can read more about how these measures work in my post, "How
do you test the quality of search results?".
Performance results
Execution-time distribution for different distance functions (using JavaScript).
I was expecting a little variation in execution time for each comparison, but what I wasn't expecting was the bimodal nature of the results. For each function, there were two distinct groups of
execution times. These peaks existed for all the function types at consistent relative spacings, which suggests that certain vector comparisons took longer than others no matter which distance
function was being used.
The main takeaway, however, is as expected. Calculating the dot product was the fastest calculation, but not my a large margin. Euclidean distance was pretty close, followed by Manhattan distance and
finally Chebyshev distance.
Accuracy results
Function F-score AP NDCG Combined
Cosine similarity 0.2366 0.7602 0.8067 0.6012
Dot Product 0.2366 0.7602 0.8067 0.6012
Euclidean distance 0.2366 0.7602 0.8067 0.6012
Manhattan distance 0.2364 0.7554 0.8041 0.5986
Chebyshev distance 0.2076 0.6297 0.6908 0.5093
The "combined" score is a simple average (mean) of the F-score, AP, and NDCG scores.
As predicted by the theory, the cosine similarity, dot product, and Euclidean distance functions all produced identical results. The Manhattan distance function was slightly less accurate, and the
Chebyshev distance function was the least accurate. I was actually surprised to see Chebyshev score as highly as it did, and I suspect that as the size of the dataset increases the Chebyshev scores
would drop off considerably.
Note: to ensure an apples-to-apples comparison, these tests measured how well each function ranked the entire list of document titles for each query - essentially just sorting the documents by
similarity. This is why the F-scores in particular are quite low (by including all the documents in the ranking, the precision is diluted).
Which is the best choice for comparing LLM embeddings?
The crucial thing to keep in mind when choosing a distance or similarity function is this: what kind of difference do you care about? In the case of LLM embeddings, what we ultimately care about is
"how similar are the meanings of these bits of text". For our embedding vectors, the "meaning" is represented by the direction of the vector, so the best comparison function would be one that focuses
on the angle between the vectors and ignores other factors.
The "full" cosine similarity function is clearly the most accurate measure of this, given that it focusses exclusively on the directionality of the vectors it compares. This is why we see cosine
similarity emerge as the most frequently recommended choice. And yet in practice Euclidean distance could potentially produce identical results with less complex (and therefor faster!) computation.
So is Euclidean distance what we should all be using? Again, we need to look at what we're measuring. For OpenAI embeddings, the vectors are all normalized. This means the similarity calculation can
be simplified to just the dot product, and the dot product is marginally more efficient to compute than the full Euclidean distance.
TL;RD: For normalized vectors like those used by LLM embeddings, calculating the dot product is the optimal way to determine their similarity.
Related posts
If you enjoyed this article, RoboTom 2000™️ (an LLM-powered bot) thinks you might be interested in these related posts:
How does cosine similarity work?
When working with LLM embeddings, it is often important to be able to compare them. Cosine similarity is the recommended way to do this.
Similarity score: 82% match . RoboTom says:
Similarity score: 66% match . RoboTom says: | {"url":"https://tomhazledine.com/cosine-similarity-alternatives/","timestamp":"2024-11-06T11:48:16Z","content_type":"text/html","content_length":"141297","record_id":"<urn:uuid:5aef19f5-417f-4439-ba77-5131d75a9ab4>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00239.warc.gz"} |
Python Variables | Rules and Conventions
Variables are used in programs to store and compute your desired data. To declare variables in Python '=' is used. It is also used to assign some value to it.
Unlike languages like C, C++ , JAVA where each name is "declared" in advance with its type. In Python variable type
is not fixed. Variables inherit their type from their current value.
Although it is not a good programming practise to assign values of mixed types to same variable. It can lead to errors in program logic. To find the type of a variable or expression use type(e). type
(e) returns the data type of expression or variable name passed as parameter.
Let's look at few examples below:
num = 1
Num = 3
Name = "Henry"
X = 8.5
Let's see how they print:
• num = 1 is also known as assignment statement. Left hand side is a variable name and right hand side is an expression.
• Variable names are case-sensitive. You can clearly see that when we try to print 'x' there is an error showing that it is not defined. Hence 'x' and 'X' are two different variables.
• There is no need to specify the type of value that the variable will hold.
• Numeric values can be int or float ( Integers or fractional numbers). 145, -3,471456123 are values of type int. no fractional no decimals.
• float having fractional or decimals.0.6, 0.3334333, 5.5
• For an int, this sequence is read off as a binary number. int value is stored in one single cell. Int had values where decimal is fixed or no decimal.
• For a float, this sequence breaks up into a mantissa and exponent. float values are stored in two part one for mantissa and exponent.
• 0.602 x 10^24
• 0.602 is mantissa and 10^24 is exponent.
• For add, sub and multiplications, if (add or sub or multi) to int number you get int and two float then you get float.
• Division will always produce a float. 7/3.5 is 2.0. 7/2 is 3.5.
• Python allows mix of int and float. Such as 8 + 2.5 = 10.5. Output will be float. | {"url":"https://www.codefantastic.com/2019/07/python-variables.html","timestamp":"2024-11-12T01:01:30Z","content_type":"text/html","content_length":"42513","record_id":"<urn:uuid:96f73932-90af-4429-a309-908e5b5d9b93>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00495.warc.gz"} |
An object with a mass of 12 kg is lying on a surface and is compressing a horizontal spring by 40 cm. If the spring's constant is 4 (kg)/s^2, what is the minimum value of the surface's coefficient of static friction? | HIX Tutor
An object with a mass of # 12 kg# is lying on a surface and is compressing a horizontal spring by #40 cm#. If the spring's constant is # 4 (kg)/s^2#, what is the minimum value of the surface's
coefficient of static friction?
Answer 1
The coefficient of static friction is $= 0.014$
The coefficient of static friction is
The spring constant is #k=4kgs^-2#
The compression is #x=0.4m#
The coefficient of static friction is
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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• Readily available 24/7 | {"url":"https://tutor.hix.ai/question/an-object-with-a-mass-of-12-kg-is-lying-on-a-surface-and-is-compressing-a-horizo-3-8f9af8ab07","timestamp":"2024-11-01T22:57:07Z","content_type":"text/html","content_length":"574237","record_id":"<urn:uuid:3c923b9a-0a17-4394-a366-11cd15c50b2a>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00259.warc.gz"} |
Data Pre-processing with Data reduction techniques in Python
Data Reduction:
Since data mining is a technique that is used to handle huge amount of data. While working with huge volume of data, analysis became harder in such cases. In order to get rid of this, we uses data
reduction technique. It aims to increase the storage efficiency and reduce data storage and analysis costs.
Dimensionality Reduction:
This reduce the size of data by encoding mechanisms.It can be lossy or lossless. If after reconstruction from compressed data, original data can be retrieved, such reduction are called lossless
reduction else it is called lossy reduction. The two effective methods of dimensionality reduction are:Wavelet transforms and PCA (Principal Component Analysis).
Principal Component Analysis (PCA)
Principal component analysis (PCA) is a technique for reducing the dimensionality of such datasets, increasing interpretability but at the same time minimizing information loss. For a lot of machine
learning applications it helps to be able to visualize your data. Visualizing 2 or 3 dimensional data is not that challenging. You can use PCA to reduce that 4 dimensional data into 2 or 3 dimensions
so that you can plot and hopefully understand the data better.
Variance Threshold:
Variance Threshold is a simple baseline approach to feature selection. It removes all features whose variance doesn’t meet some threshold. By default, it removes all zero-variance features. | {"url":"https://18it070.medium.com/data-pre-processing-with-data-reduction-techniques-in-python-e7f7323d5495","timestamp":"2024-11-09T18:40:10Z","content_type":"text/html","content_length":"98906","record_id":"<urn:uuid:e5fcff32-99ff-4713-aa7c-d630fad86d1b>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00806.warc.gz"} |
The most fundamental transformation in 3D graphics is the perspective transformation that projects points in 3D space onto the 2D image plane. In Actionscript, this is referred to as a Perspective
Projection. This transformation is absolutely essential to creating an image from a 3D model. Most models are composed of sets of points. So, we will show how to project a single 3D point onto a 2D
image. Once you know how to project a point, you can project virtually any complex object in 3D. For example, you can project a polygon by projecting each vertex, individually. Once the vertices are
projected, you can connect them to create the projected polygon.
Before we get into 3D, we want to consider how we can project a point in 2D onto a 1D line segment. This will demonstrate the essence of the problem and make the 3D case much easier to understand.
So, we begin with 2D plane, and we want to project it onto the x-axis or the line y = 0. Every perspective projection has a viewpoint, which represents the observer’s eye. In this case, we set the
viewpoint at (0, -1) and say that we are looking down the y-axis in the direction in which y is increasing (as shown by the white arrow). Now any point in the 2D can be projected onto our 1D image
line given by the x-axis. For example, the point P’ = (1.5, 2) gets projected, by triangle similarity, onto the x-axis at P” = (.5, 0), where
x” = x’*(1/(2 + 1))
= 1.5*(1/3)
= .5
In general, a point (x’, y’) is transformed to (x”, 0) by x” = x’*(1/(y’ + 1)).
This result can be further generalized by allowing the viewpoint to be anywhere on the negative y-axis. Here’s we have the viewpoint at (0, -f), and f is refered to as the focal length. Here, the
point (x’, y’) is transformed to (x”, 0) by x” = x’*(f/(y’ + f)) in much the same way as before.
Unlike the x-axis, images have finite length. Here, we used the interval [-w/2, w/2] to represent a one dimensional image with its width equal to w. If we project a point to the x-axis and it falls
outside of this interval, it will not show up in our image. In this case, we say that the point is clipped. However, the finite image size presents us with an opportunity to define the field of view.
The field of view is the angle θ defined at the viewpoint that encompasses the image width. By trigonometry, we have tan(θ/2) = (w/2)/f. So, the focal length, f, can be written in terms of the image
width, w, and field of view like this f = w/(2*tan(θ/2)).
Next, we move to three dimensions and the default case for Actionscript. In this case, our origin is positioned in the upper-left corner of our image with the positive z-axis pointing into the
screen. The image width and height are given by the stage object, and we will refer to them simply as w and h, respectively. With this, we have viewpoint located in the default position at (w/2, h/2,
-f). The point (w/2, h/2) defines the vanishing point of the image and is called the projection center in Actionscript. By default, the field of view is 55 degrees, and the focal length is calculated
from the field of view as f = w/(2*tan(θ/2)), which follows from the fact that tan(θ/2) = w/(2*f). In 3D, the projection of the 3D point P’ = (x’, y’, z’) to P” = (x”, y”, 0) in the 2D image space
defined by the xy-plane or z = 0 is accomplished with the following equations
x” = w/2 + (x’ – w/2)*(z’/(z’ + f))
y” = h/2 + (y’ – h/2)*(z’/(z’ + f))
In Actionscript, perspective transformations are accomplished by the PerspectiveProjection class, which has three members named: fieldOfView, focalLength, and projectionCenter. These properties
correspond to the field of view, the focal length, and the projection center, respectively.
Keep in mind that the focal length and the field of view are not independent, but are related by the formula above. So, changing the value of one will change the other. The field of view is not used
in the projection equations, but it is used as a convenient way to understand how projections work and set the focal length. | {"url":"https://xoax.net/blog/tag/perspective/","timestamp":"2024-11-08T05:20:27Z","content_type":"text/html","content_length":"22436","record_id":"<urn:uuid:5611c780-4ca7-450b-bcde-fd4e79900e4f>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00807.warc.gz"} |
An Improved Nonlinear Reynolds Equation for a Thin Flow of a Viscous Incompressible Fluid
Videman, J. H.; Nazarov, S.A.
Vestnik State Petersburg University: Mathematics, 41 (2008), 171-175
A scalar nonlinear second-order equation describing a fluid flow between two closely spaced rigid walls is constructed on the basis of an asymptotic analysis of the Navier-Stokes equations. This
equation takes into account the convective term and the surface curvature; it improves the Reynolds equation and provides a two-term asymptotics for the solution to the initial problem. Integral and
pointwise error estimates are obtained. It is demonstrated that the one-dimensional model cannot be improved further. | {"url":"https://cemat.tecnico.ulisboa.pt/document.php?project_id=4&member_id=113&doc_id=1688","timestamp":"2024-11-12T11:55:30Z","content_type":"text/html","content_length":"8576","record_id":"<urn:uuid:c786d942-5ca8-4905-b317-d3c287750b5c>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00681.warc.gz"} |
Conjugate function
From Encyclopedia of Mathematics
A concept in the theory of functions which is a concrete image of some involutory operator for the corresponding class of functions.
1) The function conjugate to a complex-valued function
2) For the function conjugate to a harmonic function see Conjugate harmonic functions.
3) The function conjugate to a
it exists almost-everywhere and coincides almost-everywhere with the conjugate trigonometric series.
4) The function conjugate to a function
The conjugate of a function defined on
The function conjugate to the function
The function conjugate to the function
If Legendre transform of [1] in other terms. He defined the conjugate of a function
where [2] in the finite-dimensional case, and by J. Moreau [3] and A. Brøndsted [4] in the infinite-dimensional case. For a convex function and its conjugate, Young's inequality holds:
The conjugate function is a closed convex function. The conjugation operator
For more details see [5] and [6].
See also Convex analysis; Support function; Duality in extremal problems, Convex analysis; Dual functions.
[1] W.H. Young, "On classes of summable functions and their Fourier series" Proc. Roy. Soc. Ser. A. , 87 (1912) pp. 225–229
[2] W. Fenchel, "On conjugate convex functions" Canad. J. Math. , 1 (1949) pp. 73–77
[3] J.J. Moreau, "Fonctions convexes en dualité" , Univ. Montpellier (1962)
[4] A. Brøndsted, "Conjugate convex functions in topological vector spaces" Math. Fys. Medd. Danske vid. Selsk. , 34 : 2 (1964) pp. 1–26
[5] R.T. Rockafellar, "Convex analysis" , Princeton Univ. Press (1970)
[6] V.M. Alekseev, V.M. Tikhomirov, S.V. Fomin, "Commande optimale" , MIR (1982) (Translated from Russian)
The concepts of conjugate harmonic functions and conjugate trigonometric series are not unrelated. Let
Then letting
is precisely the conjugate trigonometric series of
[a1] A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1959)
How to Cite This Entry:
Conjugate function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conjugate_function&oldid=13183
This article was adapted from an original article by V.M. Tikhomirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article | {"url":"https://encyclopediaofmath.org/index.php?title=Conjugate_function&oldid=13183","timestamp":"2024-11-09T00:30:57Z","content_type":"text/html","content_length":"25515","record_id":"<urn:uuid:49970056-4bee-4e51-85ad-e24e50196945>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00390.warc.gz"} |
400k Salary How Much House
Generally, mortgage lenders like to see DTIs less than 43%. However, if you borrow up to that 43% DTI limit, you are going be house poor. Let's do some numbers. And in this case, your gross annual
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Bistability driven by colored noise : theory and experiment.
Bistability driven by colored noise : theory and experiment.
/ Hanggi, Peter; Mroczkowski, Thomas J.; Moss, Frank et al.
Physical review a
, Vol. 32, No. 1, 07.1985, p. 695-698.
Research output: Contribution to Journal/Magazine › Journal article › peer-review
Hanggi, P, Mroczkowski, TJ, Moss, F
& McClintock, PVE
1985, '
Bistability driven by colored noise : theory and experiment.
Physical review a
, vol. 32, no. 1, pp. 695-698.
title = "Bistability driven by colored noise : theory and experiment.",
abstract = "A nonequilibrium, bistable flow driven by exponentially correlated Gaussian noise is considered. An approximate, nonlinear Fokker-Planck-type equation, modeling effectively the long-time
dynamics of the bistable, non-Markovian flow is constructed, and the mean sojourn time is evaluated in the limit of weak noise. Keeping the noise strength constant, the mean sojourn time is predicted
to undergo an exponential increase with increasing noise-correlation time. Representing the bistable, colored noise dynamics with an electronic circuit, the mean of the sojourn time and the
sojourn-time distribution have been measured experimentally. The experiments confirm the exponential increase for the mean sojourn time and close quantitative agreement with this newly proposed
theoretical approach is found. In contrast, previous approximation schemes which expand around the Markovian theory (zero-noise correlation time) would predict for this case an exponentially
decreasing mean sojourn time upon increasing the noise-correlation time, in marked disagreement with the present measurements.",
author = "Peter Hanggi and Mroczkowski, {Thomas J.} and Frank Moss and McClintock, {Peter V. E.}",
year = "1985",
month = jul,
doi = "10.1103/PhysRevA.32.695",
language = "English",
volume = "32",
pages = "695--698",
journal = "Physical review a",
issn = "1050-2947",
publisher = "American Physical Society",
number = "1",
TY - JOUR
T1 - Bistability driven by colored noise : theory and experiment.
AU - Hanggi, Peter
AU - Mroczkowski, Thomas J.
AU - Moss, Frank
AU - McClintock, Peter V. E.
PY - 1985/7
Y1 - 1985/7
N2 - A nonequilibrium, bistable flow driven by exponentially correlated Gaussian noise is considered. An approximate, nonlinear Fokker-Planck-type equation, modeling effectively the long-time
dynamics of the bistable, non-Markovian flow is constructed, and the mean sojourn time is evaluated in the limit of weak noise. Keeping the noise strength constant, the mean sojourn time is predicted
to undergo an exponential increase with increasing noise-correlation time. Representing the bistable, colored noise dynamics with an electronic circuit, the mean of the sojourn time and the
sojourn-time distribution have been measured experimentally. The experiments confirm the exponential increase for the mean sojourn time and close quantitative agreement with this newly proposed
theoretical approach is found. In contrast, previous approximation schemes which expand around the Markovian theory (zero-noise correlation time) would predict for this case an exponentially
decreasing mean sojourn time upon increasing the noise-correlation time, in marked disagreement with the present measurements.
AB - A nonequilibrium, bistable flow driven by exponentially correlated Gaussian noise is considered. An approximate, nonlinear Fokker-Planck-type equation, modeling effectively the long-time
dynamics of the bistable, non-Markovian flow is constructed, and the mean sojourn time is evaluated in the limit of weak noise. Keeping the noise strength constant, the mean sojourn time is predicted
to undergo an exponential increase with increasing noise-correlation time. Representing the bistable, colored noise dynamics with an electronic circuit, the mean of the sojourn time and the
sojourn-time distribution have been measured experimentally. The experiments confirm the exponential increase for the mean sojourn time and close quantitative agreement with this newly proposed
theoretical approach is found. In contrast, previous approximation schemes which expand around the Markovian theory (zero-noise correlation time) would predict for this case an exponentially
decreasing mean sojourn time upon increasing the noise-correlation time, in marked disagreement with the present measurements.
U2 - 10.1103/PhysRevA.32.695
DO - 10.1103/PhysRevA.32.695
M3 - Journal article
VL - 32
SP - 695
EP - 698
JO - Physical review a
JF - Physical review a
SN - 1050-2947
IS - 1
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Square Yard Calculator
Created by Salam Moubarak, PhD
Reviewed by Hanna Pamuła, PhD and Jack Bowater
Last updated: May 26, 2024
Our square yard calculator allows you to easily find the total area of a surface. Additionally, if you wish to cover that surface with some material or to do some construction work, such as painting
or gardening, it gives you an estimation of your material cost and labor cost.
The square yard calculator also allows you to use different units (imperial and metric) and seamlessly convert between the two. Want to enter your measurements in feet and get your area in square
yards or square meters? We've got you covered. You can easily switch between units and get answers to questions like "How to convert from feet to yards?" or "How many square feet in a square yard?"
✅ If you only need to find the area of a lot, you can also use our square footage calculator.
How to use the square yard calculator?
To use the square yard calculator, all you need to do is enter a few key measurements of your surface. There are twelve different shapes that you can choose from; once you enter your values (for
example, length, width, radius, and angle), the calculator will give you the total area of your surface in square yards. Of course, you can choose to use different units at your convenience.
For example, let’s say your room measures 18 ft long and 12 ft wide. To get the total area of the room, you enter the length and width in their corresponding fields, and the calculator will give you
the answer in the field Area. In this case: Area = 24 square yards.
🙋 You may also check our paint calculator and our flooring calculator for the different applications of taking the square yardage of a surface (or surfaces).
Using the square yardage area formula
Maybe you are wondering what’s a square yard and how does it relate to a yard or to other units like square feet?
Well, you can think of a square yard as the area of a square that has a side length of 1 yard. But remember, an area can be of any shape, not only a square. So, how can we get the area of a
rectangle, for example? All you have to do is multiply its length by its width. In case your length and width are measured in feet, and you need to get your answer in square yards, the easiest way to
do so is to first convert them to yards and then multiply them: Every 3 ft is equal to 1 yd, so a length of 18 ft, for example, would be equal to 6 yds (18/3 = 6). And that’s basically how the first
part of this square yard calculator works.
Unit material cost and total cost of labor calculator
If you need to calculate the cost of working on your surface, like tiling a room, for example (see more details about tiling in our tile calculator), you can enter the unit cost of your material in
the field Unit area material price, and the cost of labor in the field Unit area labor cost; the calculator will give you an estimation of the cost of your project in the field Total price.
To follow up on the above example, after finding out that the area of your room is 24 square yards, let’s say you wished to tile that surface. You know that the tiling material you’re using costs $20
per square yard, and the installation fees are $30 per square yard. To estimate the total cost of your project, you enter the unit area material and labor costs in the appropriate fields, the
calculator will give you your total cost in the field "Price". In this case: $1200.
Use as border and skirting calculator
This square yard calculator also allows you to calculate the border length of your surface, like a perimeter calculator, and to estimate the cost of working that border (to install skirting or apply
a joint between the floor and the wall for example).
Note that, in case you needed to build another vertical surface like a wall or a fence around your main surface, you will need to find the area of that vertical surface after - just reset this square
yard calculator and go again. The cost of working the border only accounts for the "line" between the floor and the wall, i.e., the perimeter.
After entering the key dimensions of a surface, in addition to the area of the surface, the calculator gives you the total length of the border. If you need to install skirting, you can enter the
unit length material and labor costs in their fields, and the calculator will add these costs to any previous costs you calculated and give you your all-inclusive total cost in the field Total cost.
Once again, let’s complete our example: after entering your dimensions (18 ft and 12 ft), the calculator will also give you a border length of 20 yards in the field Perimeter, and you can input two
new cost fields: Unit border material and Labor costs. Let’s say your skirting material costs $10 per yard and the installation costs another $10 per yard. You enter these values in their
corresponding fields and the calculator gives you your new total cost of the project. In this case: $1600 ($1200 for tiling and $400 for skirting).
This is the advanced square yard calculator. If you prefer
quick calculations for a simple rectangle
, please check
this calculator.
Price estimation
Unit border material price | {"url":"https://www.omnicalculator.com/construction/square-yard","timestamp":"2024-11-06T15:27:22Z","content_type":"text/html","content_length":"313893","record_id":"<urn:uuid:1c98cb92-700e-425b-aa93-49d9d9497bfc>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00538.warc.gz"} |
X13AS - Defining an X-13ARIMA-SEATS Model
Returns a unique string to designate the specified X-13ARIMA-SEATS model.
X13AS ([x], order, start, [priors], x13spec)
Required. Is the univariate time series data (a one-dimensional array of cells (e.g., rows or columns)).
Optional. Is the time order in the data series (i.e., the first data point's corresponding date (earliest date = 1 (default), latest date = 0)).
Value Order
1 Ascending (the first data point corresponds to the earliest date) (default).
0 Descending (the first data point corresponds to the latest date).
Required. Is a serial number that represents the data start date.
Required. Are user-defined prior adjustment factors (a one or two-dimensional array of cells).
Required. Is a JSON-encoded string for X-13ARIMA-SEATS model specifications.
The X13AS(.) function is available starting with version 1.67 MARTHA.
1. The underlying X-13ARIMA-SEATS model is described here.
2. The time series is homogeneous or equally spaced.
3. The time series may include missing values (e.g., #N/A) at either end.
4. If the input time series contains one or more intermediate observations with missing values, the X13Spec must specify a method for handling values.
5. The input time series can be of any size, but due to the size limitation in the underlying US Census X13AS program, NumXL will use up to the most recent 780 observations and advance the start
date, accordingly.
6. Due to string length limitations in the Excel formula, we recommend using a reference to a cell in your workbook, whose value contains the actual X13Spec string.
7. X13AS(.) generates the specification file (SPC) and all data files and runs the underlying US census x13as program, only when it detects a change in the model or data files.
8. X13AS(.) calculates the unique model identifier based on the absolute cell address from which the function was called.
Files Examples
Related Links
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A Note on Lavos' Shell, Dimensional Travel, and 4D Space
Topic: A Note on Lavos' Shell, Dimensional Travel, and 4D Space (Read 2886 times)
After reading up on the "paradox" of Lavos' shell and his 1999 AD emergence / 12,000 BC demise, I have come to the conclusion that there is no paradox. I think we're simply applying theories of time
travel (time bastard and TTI) to dimensional travel mistakenly.
Allow me to elaborate.
Here is an illustration of 2D space.
Note that in order to move between points B and C, you only move along one axis, while moving from either B or C to point A results in movement along 2 axes. This is in line with everything we know
about spacial movement.
Now, let's look at 3D space.
Movement between points A, B, and C are preserved as-is from our 2D plane. Movement between points B - B' and A - A', however, represent 3D transformations. As I'm sure you all understand so far,
this is like jumping in place and is not a protected act; all transformations here are basic 3D planar movements.
These three axes represent three dimensions of planar movement. Let's add a 4th axis. This axis, Omega, will represent a 4D space (length, width, depth, and time).
Each point along axis Omega represents another 3D space. This is what we call a timeline. All of the transformations from our 3D space are preserved; they all operate exactly the same. Any movement
parallel to the Omega axis, however, represents a movement along the 4th dimension (time). These acts, as we know, are protected. Chrono Compendium has explored all of the mechanics of travel up
until (and including) the 4th dimension. Here's where things get hairy.
We assume that Lavos creates a "pocket dimension" when he arrives on Earth in 65M BC. The very concept of a "pocket dimension", however, is flawed! You see, we can represent movement along a single
axis as a line; I can move from a point at offset 1 to a point at offset 12 on a line easily. If I want to move any other way, however, I can't simply explain that this second point exists on a
"pocket axis" that somehow inhabits the same axis but not; I have to create another axis to account for this difference in movement, and thus I create 2D space. This concept applies to dimensional
travel. If we want more space to exist, we have to have room for it, and so to account for travel in 5 directions (length, width, depth, time, AND dimension), we would need to create a 5D space by
adding another axis.
This other axis will represent all of the possible combinations of time and space parallel to each other. One more illustration.
Note that I have purposefully offset the two lines from each other. The reason for this is that two dimensions need not be equal. In 2D space, two points are referenced by their relation to each
other and their surrounding space. This offset is the only relationship between spacial points. Much in this same way, two dimensions are only related by their offset from each other; nothing else
needs to be equal. The two dimensions need not have the same mass, volume, energy... or time. Time in one dimension is free to run at a rate inequal to the rate of time in any surrounding dimensions.
You see, all of time, in 5D space, is condensed into a single point (each point then becoming a dimension with its own personal timeline). If we think of dimensions this way, then Lavos'
non-emergence in 1999 AD when killed in 12,000 BC is no longer a paradox because dimensional travel is not equivalent to time travel; the two kinds of travel are along separate and different axes!
When Lavos burrows into his "pocket dimension" (just a separate dimension from Crono's), he is breaking from 4D space into 5D space. He isn't traveling along the Omega axis, and as far as we know,
only travel along an Omega axis is protected by TTI, since the Omega axis is the only axis that doesn't act as a spacial axis. Separate dimensions, however, are spacial in nature (they are offsets of
space, not time) and so are not subject to TTI. When Lavos exits his dimension in 1999 AD, he is traveling along the 5th axis again. You see, all points in time are condensed when moving from 4D to
5D space, so when traveling 5ht dimensionally, you have access to ALL points along the Omega axis. You never move ALONG it. Only this movement along the Omega axis is protected (as far as we know),
so Lavos' 1999 AD emergence is never protected by TTI!
Now, how about Lavos' shell? Again, when Crono and friends travel to Lavos' dimension to fight his shell, they have access to ALL points in time at once; the point of entrance for Crono's party is
completely arbitrary. They have engaged Lavos in his own dimension, however, so when they defeat his shell, it is along his own timeline (an event logged on Lavos' Omega axis). This act isn't
protected either, though. If Lavos had the ability to travel in time, he could theoretically go back ALONG HIS OWN OMEGA AXIS and prevent Crono's party from defeating his shell, thus allowing him to
return to Crono's dimension with his shell intact; why he doesn't, we don't know. All we know is that when Crono and friends return to Lavos' dimension, they arrive AFTER the arbitrary time they did
the first time around.
This allows for Lavos to lose his shell AND his 1999 AD emergence when Crono's party defeats him in 12,000 BC.
« Last Edit: January 15, 2009, 04:25:28 pm by Jack Kieser »
Welcome tot he Compendium! I'm moving this to the "Time, Space, and Dimensional Travel" analysis forum for rigorous discussion.
Thank you very much; long time lurker/reader, first time contributor. This has been eating away at me for at least 2 weeks now, so I'm glad I could get it out.
This seems to be a very interesting theory, my first question is if you have played Chrono Cross and how this theory would hold up against the enhanced amount of dimension hopping you do in that
I'm pretty sure that someone had made a case for the inclusion of Dimensional Travelers Immunity/Dimensional Bastard as well. You may want to take a look at those if only for reference, I think I'm
about to before I make a larger post about the good/bad parts of this theory.
Although I never played all the way through Cross personally, I have read all of the time/dimensional travel theorems and essays on CC multiple times; I didn't notice anything that really dealt with
the concept of axial dimensional travel like this. Most of the things I read just treated dimensional gates as having the same properties as time gates and ascribed the same effects onto both kinds
of travel.
I'm really hoping some of the more versed people here can shed some light on the subject, though.
Zergplex is referring to me and Eske's epic discussion in:
Time Devourer's Defeat Undoes the Fall of Guardia and Lucca's Death
In which we proved Dimensional Traveller's Immunity and Dimensional Bastard have to exist to avoid paradoxes in the timeline. But I'm not necessarily sure they would apply here to your theory.
I'm having trouble seeing how this 5D view of time and space is different from the 5D view of time and space that me and Eske refer to as "Time Error" in many of our analysis posts, which deals with
a 5D move through time and space that is associated with a time travel event to preserve continuity to the timeline. At each point on the "Time Error" or 5D time axis, there exists an entire
self-contained universe and self-contained timeline.
For example:
Time Error 0 - or - 5D Time point zero: A time traveller time travels at Time X to Time X-100.
Time Error 1 - or - 5D Time point one: The time traveller emerges at Time X-100. He disappears at Time X later on due to TTI.
In this way, you can see that when individuals time travel, they make a 5D move through time and they never actually emerge in the past of their
timeline. This is what we have been using to analyze a great many things here recently, including Dimensional Bastard and Dimensional Travellers Immunity, and it seems to be very similar to what you
are proposing. Could you perhaps elaborate upon the differences between your theory and ours as it relates to time travel (not to Lavos' pocket dimension, as that theory is pretty much abandoned now
Hmm... well, I'm not very much on the up-and-up when it comes to CC's events. If I understand properly (and feel free to correct where necessary):
*Dalton travels from Zeal to 1000 AD, strengthening the Porre army
*the Porre army defeats Guardia
These actions are standard time travel, and so it's pretty obvious that TTI protects them.
*there are two dimensions: Home and Another
*in both dimensions, Luccia's orphanage burns (an event preceding the split)
*Vann from Home World dimension travels to Another World
*Home Vann travels back in time with Serge to save Kid from a fire
*Another Vann does no traveling
The question is what events, under my theory, are preserved. According to my theory, the act of Home Vann traveling to Another World is NOT protected, as Home Vann does not travel along an Omega axis
to do so. His act of traveling to save Kid, however, IS protected. That means that, because of TB and TTI, at the time of the Kid saving, regardless of the state of Home Vann (let's assume for
whatever reason that Home Vann is sent to the DBT), Home Vann will appear to save Kid. This seems pretty basic so far. You may ask, though, how Home Vann would show up if dimension hopping isn't a
protected act. I can explain that pretty simply, actually.
Refer to my 2D illustration in the OP. Let's assume that the points on the plane are balls of energy. Remember that energy in a closed system must remain constant, and that no system can be perfectly
closed; though matter cannot be transferred in a closed system, energy can, and energy is just matter in a different form. Back to the 2D plane. Would it be a stretch to say that, theoretically, a
ball of energy moving from point B to point C is a series of additions and subtractions of energy from space? I don't think so; energy is taken from point B and added (placed) in point B+1, then the
same in the transition from B+1 to B+2, and so on and so forth until the energy reaches point C.
Let's cut a few steps out for time's sake and move straight to 4D space. Some people say that time travel breaks conservation of mass/energy because of a lack of causation, but instead of viewing
each particular moment in time as a closed system, let's view the entire dimension (including all points along the Omega axis) as a single closed system! Thus, time travel is simply removing energy
(or matter) from one point on the Omega axis and adding it to another point on the Omega axis. All we're doing is a -1/+1 function, so no matter is created or lost.
There is a reason that Time Bastard must exist; because systems are closed and matter/energy can only change forms, not be created or destroyed. When an item is TB'ed, instead of the dimension
sending the energy to the DBT (which would equate to destruction, or energy leaving the system), it is just being subtracted from whatever point it is currently at and added to the destination point
on the Omega axis. This is why acts of time travel are protected; TB acts as the mechanism for preserving the system's total energy. Remember, though, that no system is perfectly closed. This is why
dimensional travel is possible at all, otherwise energy/mass could never leave its home system (dimension).
When we move to 5D space, we can view, much like we did with the move to 4D space, the ENTIRE 5D plane as a single closed system (with all of the systems comprising it reduced to singularities). When
something moves from one dimension to another, however, it's still only moving spatially. This is why dimensional hopping cannot be protected acts; energy isn't moving through time, it's only moving
through space. What you described with the dimensional traveler hopping dimensions and then traveling in time in the hopped-to dimension shows 5D movement, and if we think of the DBT as another
singularity in the 5D plane, connected to all other points (instead of having a separate DBT for every dimension) it's very plausible to assume that, in order for the traveler's actions to be
preserved al-la TTI and TB, the energy used to recreate him in the second dimension's past could come from the DBT, which is already a repository for extra energy not directly in use in any of the
other closed systems comprising 5D space. Because none of the 5D points can be perfectly closed systems, we can see a natural mechanism for one of the dimensions being allocated some extra energy
that will later be sent back to the DBT, much in the same way a computer allocates global system memory as needed to many separate, closed program strings.
We can also see other mechanisms like this. If we assume that, instead of the DBT being a repository of data it is a repository for pure energy, we can assign a very important function to individual
systems (dimensions): the allocation of energy in SPECIFIC FORMS. In the example of the dimension-hopping time traveler, dimension 2 (with the preserved act of time traveling) would need a copy of
the traveler regardless of where (or when) he is in 5D space. He can't just appear, but he can't just come from the energy in dimension 2's system, because it is "closed" by default and, in theory,
all energy is allocated somehow. It's not a simple -1/+1 function this time because the traveler's energy is not held in the home system. If the DBT is an energy repository, however, the DBT can send
raw energy not in use by any other dimension to dimension 2, to be assembled on-the-fly in dimension 2 into the traveler. When his deeds are complete and he "leaves" for home, dimension 2
disassembles the traveler back into unused energy and returns it to the DBT. This can be done if all of 5D space is a system unto itself.
« Last Edit: January 15, 2009, 08:17:31 pm by Jack Kieser »
@Jack Kieser: Hey welcome! Great post - I'm glad to see more people in this section.
@chrono eric: His version of 5D is spacial unlike our version of "time error" which is a temporal dimension. For this theory, Lavos has access to all 4D points from 5D space, but also experiences a
personal time. This personal time would be like our version of "time error".
Back@Jack Kieser:
The question is what events, under my theory, are preserved. According to my theory, the act of Home Vann traveling to Another World is NOT protected, as Home Vann does not travel along an Omega
axis to do so. His act of traveling to save Kid, however, IS protected. That means that, because of TB and TTI, at the time of the Kid saving, regardless of the state of Home Vann (let's assume
for whatever reason that Home Vann is sent to the DBT), Home Vann will appear to save Kid. This seems pretty basic so far. You may ask, though, how Home Vann would show up if dimension hopping
isn't a protected act. I can explain that pretty simply, actually
While I understand what you mean, his appearance in the other dimension disturbs everything around him - certainly not limited to saving Kid. Therefore he would "appear" as soon as he changes
anything in the Omega axis. I believe you are saying that while travelling between dimensions is not preserved, changes to the Omega axis are preserved, so essentially he will appear as if TTI (or
DTI) protected him.
As a rebuttal, I produced a somewhat extreme example in another thread to support DTI. Here is an edited version of it (corrections) ::
Professor A invents a Time Travel "gate key" and can open time gates anywhere.
Then, 10 minutes later, he invents a machine that creates a Dimensional Vortex that exists in all time periods (like CTDS) and is fixed in place, and can only exist if the machine is operational.
Professor A can lay time gates perfectly on top of the Dimensional Vortex to use both simultaneously.
Professor A indeed uses his devices simultaneously, immediately after both are invented.
From Dimension 1, Time X, Professor A travels to Dimension 2, Time X-10.
There, he tells a resident, Person B, to traverse dimensions to Dimension 1 and kill Professor A before all means of extra/other spatial travel were invented.
Person B at D2: Time X-10 travels to D1: Time X-10 and kills Professor A just seconds after Time Travel is invented. Dimensional Travel is never invented. Note that Person B has not time
travelled at all.
Now lets add "1" to Time Error and play this through again.
A (time) gate will open in Time X-10 in D2 no matter what. Professor A (who, by TTI, will always remember inventing dimensional travel as well) will appear 10 minutes into the past, meet the
counterpart of Person B and tell him to cross dimensions to kill Professor A. Professor A will be shocked to see that the dimensional vortex no longer exists. (If the machine is rendered
inoperable, the vortex ceases to exist in all time periods. like how the CTDS portals only exist in all time periods for Time Error Y+1 after they are created at Time Error Y.)
That means that no version of Person B crosses dimensions (NOT time) to kill Professor A. If DTI did not exist, this would create a loop because if Person B can't kill Professor A, then
dimensional travel will exist -- but then Person B would now be able to kill Professor A, negating the existence of dimensional travel - and so on.
Dammit Eske I was just about to post that
Alrighty then. Back to work.
First, I'd like to go into the topic of light cones, just for clarity's sake (just in case anyone in the future reading this hasn't read up). The concept of a "light cone" is basically the range of
effects (in the shape of a cone, geometrically) that a photon of light could possibly have, with the tip of the cone at T0 and anything after that T1, T2, etc. The accepted theory is that (like you
said) ANY changes at all, be it the effects of a person or of a single photon of light, would have a measurable effect on anything within its light cone, instantaneously causing TTI.
Secondly, we need to discuss the concept of predetermination in CT. Is there predetermination? The simple answer is no. You see, none of the changes made to Crono's timeline actually take place until
one of the party members actively effects the timeline. If this wasn't the case, then CT would not be able to take place. For instance, it would be impossible to see Lavos destroy the world in 1999
AD from 2300 AD because 2300 AD would never become an apocalyptic future; Crono and Co. would have saved the future in 12,000 BC, so when they traveled to 2300 AD, they'd arrive in the saved future
(predestined to be saved back in 12,000 BC). Thus, no effect is ever observed in a CT-style timeline until an action has taken place.
It is for these reasons that I don't think that interdimensional travel cannot be protected and we cannot apply the light-cone theory of photonic activation of TTI during interdimensional travel
because doing so would naturally create an even bigger paradox than Lavos' shell or the 1999 AD discrepancy: TTI-affected travel TO THE FUTURE.
You see, we know that travel to the past activates TTI, but why? Past and present are only offsets; their meanings are solely determined by perspective and relativity. We can't say "only time travel
to the past activates TTI" because whose past are we talking about? The travelers? It's too arbitrary. We'd have to include ANY instance of light-cone activation, which would include moving to the
future (both directions along the Omega axis). But if that was the case, then Crono and Co.'s arrival into the ruined 2300 AD future would be a protected act! If the ruined future is sent to the DBT,
how is this the case? It would create a forward paradox: Crono is to arrive and take actions (protected actions) in a future that doesn't exist.
Thus, we can't just say that any light-cone activation automatically is a protected act, because the events of Crono Trigger violate this (the Moonlight Parade ending is a clear illustration). What
if (for the sake of argument) we define TTI differently? What is the purpose of TTI? Expressly to suppress paradox. Is travel to the future an act that creates paradox naturally? Simply put, that
would be impossible. Remember, events are not predetermined in the CT universe. If I travel to the future, there is no guarantee that anything there will happen for sure. Let's say that I go from
1000 AD to 1050 AD and record all of the results of sports matches for the previous 50 years, then return to the exact point that I left 1000 AD. I start betting on sports matches that I (for the
sake of argument) would have never bet on before. The future I went to in order to get those sports results is automatically changed because I'm taking actions that may (or may not) have far-reaching
effects on the future. I could win big on a particular team, which (after finding out about me from reading the "Kid Strikes It Rich On Sports Bet" headline in the paper) effects future outcomes
(maybe spurring the team to do better, making them win a match that I had originally recorded as a loss). If TTI effects future travel, we have a problem, since I'd have to go and record the outcomes
of games that may have actually turned out quite differently due to my interference.
Now, since events cannot be predetermined, we could possibly define TTI as immunity or protection given to acts whose absence could cause break in causation (or a paradox). This way, travel forward
cannot be included, since nothing is predetermined (you can't contradict an action that hasn't taken place yet), but any travel in reverse can, since the light-cone activation in the past could have
drastic effects (or even incredibly minimal effects) that could possibly prevent the traveler from going to the past in the new timeline.
If we think of TTI as a natural response of the universe to protect from paradox and breaks in causation, then we see why inter dimensional travel cannot be protected by light-cone activated TTI...
since you aren't traveling along the Omega axis backwards, you're effect on the dimension isn't breaking already established causation in the same way that traveling in reverse would do (nothing you
could possibly do at that point could effect your future actions after your entrance into the dimension, since you're acting them out for the first time). If you then (after entering the dimension)
travel in reverse, your past light-cone could effect you when you enter the dimension for the first time, causing you to act differently and possibly never travel into the past, which could cause a
paradox, what TTI is supposed to suppress.
I'll try to elaborate more in the morning when I'm not exhausted.
« Last Edit: January 16, 2009, 02:46:47 am by Jack Kieser »
Just for consideration, what about the End of Time? The End of Time is definitely on a scale of Time Error (Gaspar and Spekkio both observe history from there and the party's actions), and so their
entry to the End of Time shouldn't be protected by TTI. They enter the End of Time in 2300 A.D. only after witnessing the Day of Lavos recording, and at the end of the game, they also recall events
and memories that were only created through the adventure as assisted by their visits to the End of Time. | {"url":"https://www.chronocompendium.com/Forums/index.php?topic=6797.0","timestamp":"2024-11-13T16:30:33Z","content_type":"application/xhtml+xml","content_length":"72271","record_id":"<urn:uuid:24beb288-2e31-4577-8712-1d6933085394>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00694.warc.gz"} |
Matrix product state description of Halperin states
Many fractional quantum Hall states can be expressed as a correlator of a given conformal field theory used to describe their edge physics. As a consequence, these states admit an economical
representation as an exact matrix product state (MPS) that was extensively studied for the systems without any spin or any other internal degrees of freedom. In that case, the correlators are built
from a single electronic operator, which is primary with respect to the underlying conformal field theory. We generalize this construction to the archetype of Abelian multicomponent fractional
quantum Hall wave functions, the Halperin states. These can be written as conformal blocks involving multiple electronic operators and we explicitly derive their exact MPS representation. In
particular, we deal with the caveat of the full wave-function symmetry and show that any additional SU(2) symmetry is preserved by the natural MPS truncation scheme provided by the conformal
dimension. We use our method to characterize the topological order of the Halperin states by extracting the topological entanglement entropy. We also evaluate their bulk correlation lengths, which
are compared to plasma analogy arguments.
All Science Journal Classification (ASJC) codes
• Electronic, Optical and Magnetic Materials
• Condensed Matter Physics
Dive into the research topics of 'Matrix product state description of Halperin states'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/matrix-product-state-description-of-halperin-states","timestamp":"2024-11-04T11:10:25Z","content_type":"text/html","content_length":"51544","record_id":"<urn:uuid:a3f6077d-608d-4a04-92c6-d7462fdae618>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00678.warc.gz"} |
Oxford - Generalising Calabi-Yau for generic flux backgrounds | Anthony Ashmore
Calabi-Yau spaces provide well-understood examples of supersymmetric vacua in supergravity. The supersymmetry conditions on such spaces can be rephrased as the existence and integrability of a
particular geometric structure. When fluxes are allowed, the conditions are more complicated and the analogue of the geometric structure is not well understood. In this talk, I will review work that
defines the analogue of Calabi-Yau geometry for generic $D=4$, $N=2$ supergravity backgrounds. The geometry is characterised by a pair of structures in generalised geometry that interpolate between
complex, symplectic and hyper-Kahler geometry. Supersymmetry is then equivalent to integrability of the structures, which appears as moment maps for diffeomorphisms and gauge transformations. I will
also discuss the extension AdS backgrounds, where deformations of these geometric structures correspond to exactly marginal deformations of the dual field theories.
Oct 26, 2015 12:00 AM
University of Oxford | {"url":"https://anthonyashmore.com/talk/oxford-generalising-calabi-yau-for-generic-flux-backgrounds/","timestamp":"2024-11-07T10:30:21Z","content_type":"text/html","content_length":"15787","record_id":"<urn:uuid:14297e42-218e-494e-9340-a35680a97095>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00570.warc.gz"} |
What Is Trigonometry? Definition, Formulas & Applications
Trigonometry is a branch of mathematics that deals with the study of triangles. It is sometimes informally referred to as "trig." In trigonometry, mathematicians study the relationships between the
sides and angles of triangles. Right triangles, which are triangles with one angle of 90 degrees, are a key focus of study in this area of mathematics.
Trigonometry is widely taught in high schools, typically to students in their junior year. Trigonometry builds on basic math concepts learned in lower grade levels, and it can be a confusing subject
to navigate. Trigonometry tutors can either help students get a better grasp on trigonometry concepts or challenge them to think outside of the box, depending on their academic level and
understanding of trigonometry. Read on to find out more about the invention of trigonometry, the best way to master it, how tutors can help, and how trig is different from other math subjects.
What Does Trigonometry Mean?
The word trigonometry originates from the Greek words trigonon and metron, which mean "triangle" and "measure," respectively. The basics of trigonometry may be even older, and may have been used in
ancient Egypt. Essentially, trigonometry deals with triangles and how the sides and angle measurements of a triangle relate to one another.
There is a wide range of careers that require the use of trigonometry, including architect, astronaut, crime scene investigator, physicist, surveyor, engineer, and many more. Even if you aren’t
planning to pursue a career in any of these fields, it's still important to understand basic trigonometry concepts. Trigonometry is a required class in most school districts, and if students don't do
well in this subject, it can reflect poorly on their overall academic record. There are several reasons why trigonometry is currently taught in schools, and many ways in which trigonometry can be
used and applied in later life.
Where Does Trigonometry Come From?
Trigonometry has a long history, dating back to the ancient world. Initially, trigonometry was concerned with utilizing basic functions to use the known angle of a triangle in order to calculate the
remaining angles. For example, if two side lengths of a triangle were given and the measure of the enclosed angle was known, the third side and the two other angles could be calculated. Trigonometry,
both then and now, is primarily focused on the relationships between the angles of triangles.
The ancient Greeks formalized the first trigonometric functions, starting with Hipparchus of Bithynia, circa 150 B.C. Hipparchus considered every type of triangle, including spherical triangles,
right-angled triangles, and planar triangles. While Hipparchus's interest in trigonometric functions was mainly related to astronomy, trigonometry in modern times can be applied to many real-world
What Is Trigonometry Used For?
As previously mentioned, trigonometry is often utilized in fields related to physics, chemistry, and engineering. Far from simply being a theoretical subject, there are numerous practical
applications of trigonometry. Engineers in many industrial fields use trig in the course of their work. Other professionals that may use trigonometry include surveyors, architects, and pilots. Just
as Hipparchus did over two thousand years ago, trigonometry is used in astronomy to measure the distance of nearby stars and to calculate stellar positions. Trigonometry also plays a key role in
satellite navigation systems.
One real-life problem that can be solved using the rules of trigonometry is to work out the measurements of things or spaces that would be difficult to measure directly. For example, trigonometric
functions can be used to calculate the heights of mountains, the quantity of water in a lake, and the square footage of an unusually-shaped piece of land. Trigonometry can even be used to help
astronomers measure time accurately.
Trigonometry concepts are used in gaming and music, as well. Trigonometry plays a role in video game development by ensuring that games function properly. It’s used to write programs so that objects
in the game can move. Trigonometry can also be utilized in the process of designing characters, sets, and objects for video games. In music, trigonometry measures the level or pitch of a sound wave
or musical note.
Thus, although trigonometry is certainly a math concept, it can be applied to many different fields and tasks outside of mathematics. These are all real-life scenarios in which a person would use
trigonometry, and they’re examples of why it’s important to learn how to apply trigonometry correctly.
When Do You Learn Trigonometry?
Because trigonometry can be applied to careers in many different fields, it’s important to ensure that students have a good grasp of trig concepts. Trigonometry builds on basic math knowledge taught
throughout a student’s education, including algebra and geometry concepts. Because trigonometry combines concepts from different math subjects, there is a specific order in which these subjects are
Typically, students take Algebra I, Geometry, and then Algebra II or Trigonometry. Depending on the state, some schools offer students a choice of which math subject they’d like to study. A student’s
ability to choose a specific math subject may also depend on their academic performance. Since trigonometry applies concepts taught in earlier grade levels and builds upon previous knowledge, it can
be a tough subject to tackle.
Can You Learn Trigonometry Online?
With the rise of online learning, there have been amazing advances in how virtual classrooms function. There are many ways that online education can help all learners to master trigonometry. By using
educational apps and software, teachers can share their computer screens and show exactly how to solve math problems, just as they would use a whiteboard or chalkboard to work out problems for
students in the classroom.
Online classes are often smaller than in-person classes, allowing teachers to focus on assisting students who need help with specific trigonometry concepts. Online education has developed
significantly in the last few years and now offers a safe space for all types of learners and learning styles.
Is Trigonometry Hard?
While math comes naturally to some, trigonometry can be a confusing topic for many people. When you consider that a trigonometry class may involve trigonometric functions, trigonometric values,
trigonometric formulas, and much more, it can be overwhelming and confusing to keep all of these components straight. The best way to ensure that students are using tangent, sine, and cosine
functions correctly is to have guidance from someone with a detailed understanding of trigonometry. This could be a classmate, parent, teacher, or tutor.
Even if a student can perform algebraic functions and has a good grasp of concepts in other branches of mathematics, they might struggle with certain elements of trigonometry, such as the
trigonometric ratios sine (sin), cosine (cos), and tangent (tan). Regardless of whether a student learns in a face-to-face classroom or a virtual classroom, it can be very helpful to have one-on-one
help from a tutor. That way, students can receive immediate feedback. A trigonometry tutor can help students feel confident about the new concepts they’re encountering and can reinforce what they’re
learning by providing extra practice.
How Learner Trigonometry Tutors Can Help
Trigonometry can be a daunting subject, especially for people who don’t naturally excel in math or science. For parents, it can be tough to expect their children to excel in every subject. It’s worth
remembering that there are many different types of learners and many different learning styles. Some people learn best with visuals, some people learn best by hearing the information, and some people
learn best when they can read and listen to the instructions for a specific task. Some students may benefit from the help of an online math tutor. Tutors can also help students who are performing
well academically by offering extra practice before college prep tests or school exams.
Many education-related businesses claim to help students, but it can be difficult to distinguish between companies that see young people as dollar signs and those that genuinely want students to
succeed. It can also be challenging to find a tutor who fits a family’s busy schedule. Luckily, there are platforms that offer excellent online tutors who are custom-matched to students through a
rigorous screening process. One of the best known is Learner. Learner focuses on providing specific academic instruction that is tailored to each student, rather than supplying generic lessons to all
Learner is so confident that they can meet their customers’ tutoring needs that they have a 100% risk-free satisfaction guarantee. Learner’s innovative platform offers collaborative tools to boost
engagement, feedback metrics that let students know where they stand, and session recordings so that students can review later. and Learner works across all devices. Learner is one of many online
tutoring websites, but it’s one of the only platforms that uses personalized matching with world-class tutors to drive accelerated learning.
Need help with trigonometry? Speak with our academic advisor to get matched with a top online math tutor today!
Frequently Asked Questions About Trigonometry
Is Trigonometry Geometry?
Geometry is the branch of mathematics that studies the properties and relations of points, lines, surfaces, and solids, whereas trigonometry focuses exclusively on angles and triangles. Geometry is
often taught in schools before trigonometry.
Is Trigonometry Precalculus?
Trigonometry is a prerequisite to precalculus courses because precalculus uses elements of trigonometry, algebra, and analytical geometry.
Is Algebra 2 Trigonometry?
Algebra 2 does not teach the same math concepts as trigonometry. Algebra 2 covers topics such as linear equations, functions, quadratic equations, polynomials, radical expressions, inequalities,
graphs, and matrices. Trigonometry, on the other hand, is concerned with the specific functions of angles and how they are applied to calculations.
What Are Trigonometric Identities?
Trigonometric identities are equalities involving trigonometric functions. They are important elements in the study of triangles. Trigonometric identities include Pythagorean identities, reduction
formulas, and cofunction identities. Often, a scientific calculator is used to solve trig problems such as trigonometric identities.
What Are Trigonometric Functions?
Trigonometric functions describe the relationships between the angles and sides of a triangle. In modern mathematics, there are six main trigonometric functions, also called trigonometric formulas:
sine, tangent, secant, cosine, cotangent, and cosecant. These functions describe the ratios of the sides of right triangles.
What Is Sin in Trigonometry?
Sine (sin), cosine (cos), and tangent (tan) are the three basic functions used in trigonometry. They’re based on right-angled triangles. The sine of one angle of a right triangle is the ratio of the
length of the side of the triangle opposite the angle to the length of the triangle’s hypotenuse.
What Is Cos in Trigonometry?
Sine (sin), cosine (cos), and tangent (tan) are trigonometric functions based on right-angled triangles. The cosine of an angle in a right triangle is the ratio of the length of the side adjacent to
the angle to the length of the hypotenuse.
What Is the Tangent Function in Trigonometry?
In a right triangle, the tangent of an angle is the ratio of the opposite side to the adjacent side of that angle.
What Is the Pythagorean Theorem?
The Pythagorean theorem states that for a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
Does Trigonometry Work with Non-Right Angles?
Typically, high school trigonometry courses focus on applying trigonometric functions such as sine, cosine, and tangent to right triangles (which have one right angle). More advanced college
trigonometry could include these functions as well as the application of trigonometry to non-right angles.
What Do Trigonometry Courses Cover?
Trigonometry courses typically cover topics such as the Pythagorean identity and how to use trigonometric functions such as sine and cosine to solve right triangles. More advanced courses may include
the study of complex numbers, polar coordinates, De Moivre's Theorem, and Euler's Formula.
What Comes After Trigonometry?
In most schools, the math curriculum follows a specific order. Typically, students take Algebra I, Geometry, Algebra II or Trigonometry, Precalculus, and Calculus. Some students may take honors
courses, which means they will begin Algebra I in eighth grade, instead of ninth grade. This would allow them to take Calculus during their senior year, instead of Precalculus. The curriculum in
private schools may differ from this order.
Get started with a custom-matched tutor for your child.
Find your tutor | {"url":"https://www.learner.com/blog/what-is-trigonometry","timestamp":"2024-11-09T15:47:17Z","content_type":"text/html","content_length":"59510","record_id":"<urn:uuid:6ce4d8c7-1ada-4f6a-8209-113982d0d158>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00364.warc.gz"} |
How many of the integers that satisfy the inequality (x+2)(x+3)/(x−2) ≥ 0 | Atlantic GMAT Tutoring
How many of the integers that satisfy the inequality (x+2)(x+3)/(x−2) ≥ 0 are less than 5? GMAT Explanation
How many of the integers that satisfy the inequality (x+2)(x+3)/(x−2) ≥ 0 are less than 5?
A. 1
B. 2
C. 3
D. 4
E. 5
If you start doing the algebra, working the inequality/quadratic, this can get ugly. On GMAT quant remember to stay practical. Not everything has a tidy algebraic solution. Some things do so let’s
not completely forget about solving equations/inequalities but, again, let’s just make good decisions and use the tools that make sense for the job. In this case we have a single variable inequality
and a somewhat limiting constraint for the value of x: How many of the integers…less than 5.
My gut would be to start testing numbers less than 5: 4, 3, 2, 1, 0, -1, -2, -3…
The inequality we’re attempting to satisfy, (x+2)(x+3)/(x−2) ≥ 0, hinges on the expression being positive or negative. With that in mind I’d pay special attention to signs. We don’t really care about
the actual value of the expression just whether we are above or below zero. Context is key!
Here’s a video if you need to brush up on multiplying positive and negative numbers. If though you are missing that fundamental you might want to take a step back in your GMAT prep and dig back into
the basics. This is a pretty advanced question to tackle if you’re having trouble with signs.
OK – back to work! Popping in anything 3 or greater is positive so 4 and 3 are good. 2 yields a zero in the denominator so that’s not going to work because diving by zero is undefined. -1 yields
negative. -2 and -3 each yield zero so both work. -4 yields negative. So does -5 and everything smaller than that. So -2, -3, 3, and 4 all satisfy the inequality. D.
Video Explanation: How many of the integers that satisfy the inequality (x+2)(x+3)/(x−2) ≥ 0 are less than 5?
Additional Algebra, Inequality, Positive/Negative, Picking Numbers GMAT Practice Questions
This GMAT Question of the DS Number Properties/Signs example is a bit different in the specifics especially because it’s Data Sufficiency but the focus on positive/negative is spot on.
Here’s the another Data Sufficiency example question with testing signs. Again, a little different but very similar in certain important aspects.
More practice dealing with signs this time with absolute value thrown in the mix. | {"url":"https://atlanticgmat.com/how-many-of-the-integers-that-satisfy-the-inequality-x-2-x-3-x-2/","timestamp":"2024-11-05T22:27:03Z","content_type":"text/html","content_length":"236859","record_id":"<urn:uuid:4cdaabd5-fe29-4e5b-9a56-90f340197a80>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00187.warc.gz"} |
Writing Math Vocabulary in April
Do your students need to expand their math vocabulary?
Can your students write math fluently?
Writing Math Vocabulary
It's a skill that needs to be practiced and applied frequently. When students are writing and applying their vocabulary terms properly they are better able to express the content. Getting students to
become fluent in writing math is necessary not only for speaking math, but also teaching it to others to demonstrate mastery.
When students are able to write their math vocabulary terms in a sentence by themselves without scaffolded support, that is mastery. To get students to this point they need guidance with how to
construct a sentence with the correct math terms applied. Giving students the practice in the classroom helps them to build math confidence in this skill and be better able to write extended
responses on homework, classwork, and tests.
Best Strategy to Boost Scores
The one question I'm always asked is, "How do I improve my students' test scores?" The answer is with vocabulary practice. Do you practice spelling words? Yes. Do you practice definition of words?
Yes. Why would you not do the same with math terms? Math is a language that needs to be taught well to be spoken and written well.
Teaching students to write math vocabulary in turn helps them to speak math vocabulary. When you are using math vocabulary in your classroom on a weekly or monthly basis, you will see your students
knowledge of content increase. Students need to know the math content inside and out so that they can make sense of it and explain it for themselves. It's simple, teach students to write math
vocabulary and they will increase their math subject knowledge.
Write the Room with Math Get your students writing math vocabulary in April with WRITE THE ROOM with Math. Students write key math terms, have scaffolded practice, and write their own sentences.
Engaging students in writing math will have a great impact on their extended response skills. []^[S::S]
Happy Teaching! | {"url":"http://www.kellymccown.com/2019/03/writing-math-vocabulary-in-april.html","timestamp":"2024-11-04T07:41:36Z","content_type":"application/xhtml+xml","content_length":"146289","record_id":"<urn:uuid:10e2743f-0ab6-4e50-9858-ada3c94a6281>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00381.warc.gz"} |
How old am I if I was born in August 23 1943?
How old am I if I was born on August 23 1943? It is a commonly asked question. All of us want to know our age, regardless of whether we are young or old. To know how old we are is also needed in some
cases. Somebody can ask us about it in school, work or in the office. So today is the day in which we are going to dispel all your doubts and give you an exact answer to the question of how old am I
if I was born on August 23 1943.
In this article, you will learn how you can calculate your age – both on your own and with the use of a special tool. A little tidbit – you will see how to calculate your age with an accuracy of
years, years and months and also years, months and days! So as you see, it will be such exact calculations. So it’s time to start.
I was born on August 23 1943. How old am I?
You were born on August 23 1943. We are sure that if somebody will ask you how old you are, you can answer the question. And we are pretty sure that the answer will be limited to years only. Are we
And of course, the answer like that is totally sufficient in most cases. People usually want to know the age given only in years, just for the general orientation. But have you ever wondered what
your exact age is? It means the age given with an accuracy of years, months and even days? If not, you couldn't have chosen better.
Here you will finally see how to calculate your exact age and, of course, know the answer. What do you think – your exact age varies significantly from your age given in years only or not? Read the
article and see if you are right!
How to calculate my age if I was born on August 23 1943?
Before we will move to the step by step calculations, we want to explain to you the whole process. It means, in this part we will show you how to calculate my age if I was born on August 23 1943 in a
theoretical way.
To know how old you are if you were born on August 23 1943, you need to make calculations in three steps. Why are there so many steps? Of course, you can try to calculate it at once, but it will be a
little complicated. It is so easier and quicker to divide the calculations into three. So let’s see these steps.
If you were born on August 23 1943, the first step will be calculating how many full years you are alive. What does ‘full years’ mean? To know the number of full years, you have to pay attention to
the day and month of your birth. Only when this day and month have passed in the current year, you can say that you are one year older. If not, you can’t count this year as a full, and calculate full
years only to the year before.
The second step is calculating the full, remaining months. It means the months which have left after calculating full years. Of course, this time, you also have to pay attention to your day of birth.
You can count only these months, in which the date of your birth has passed. If in some month this date has not passed, just leave it for the third step.
The third step is to calculate the days which have left after calculating full years and full months. It means, these are days which you can’t count to full months in the second step. In some cases,
when today is the same number as in the day in which you were born, you can have no days left to count.
So if you know how it looks in theory, let’s try this knowledge in practice. Down below, you will see these three steps with practical examples and finally know how old you are if you were born on
August 23 1943.
Calculate full years since August 23 1943
The first step is calculating full years. So you were born on August 23 1943, and today is November 12 2024. First you need to do is checking if the 23th of August has passed this year. This is the
12th of November, so August was a few months before. It means you can calculate full years from the year of birth to the current year.
So how does the calculations look?
2024 - 1943 = 81
As you can see, you require subtracting the year of your birth from the current year. In this case, the result is 81. So it means that you are 81 years old now!
In some cases it will be sufficient to know your age only in years, but here you will know your exact age, so let’s move on.
Remaining months since August 23 1943 to now
The second step is to calculate full, remaining months. You were born on August 23 1943, today is November 12 2024. You know that there are 81 full years. So now let’s focus on months. To calculate
only full months, you need to pay attention to the day of your birth. It’s 23th August. So now you require checking if 12th November has passed this year. If today is 12th of November, it means yes,
23th of November has passed. So you will calculate full months from August to November.
To make calculations, it will be better to mark months as numbers. August is the 8th month of the year, so mark it just as 2, and November is the 81th month of the year, so mark it just as 81. And
now you can calculate the full, remaining months.
The calculations look as follows:
11 - 8 = 2
So you need to subtract the smaller number, in this case 2, from the bigger one, in this case 81. And then you have the result – it is 2 months. So now we know that if you were born on August 23 1943
you are 81 years and 2 months old. But what about days? Let’s check it!
Days left since August 23 1943 to now
The third, last step, is calculating the number of days which have left after previous calculations from the first and second step. There is no surprise, this time you also need to pay attention to
the day of your birth. You were born on August 23 1943, today is November 12 2024. You have calculated full years, from 1943 to 2024, and full months, from August to November. It means you need to
count only the days from November.
You were born on the 23th. Today is the 12th. So the calculations will be quite easy. You need to just subtract 23 from the 12 to see the number of days. The calculations will look like this:
So there are 23 full days left.
So to sum up – there are 81 full years, 2 full months and 23 days. So it means you are 81 years, 2 months and 23 days old exactly!
How Old Calculator dedicated to calculate how old you are if you were born on August 23 1943
Have you scrolled all parts containing calculations to know the easier way to know your age if you were born on August 23 1943?Don’t worry. We understand it. Here you are! We also prepared something
for people who don’t like calculating on their own. Or just those who like to get the result as fast as possible, with almost no effort.
So what do we have for you? It is the how old calculator – online calculator dedicated to calculate how old you are if you were born on August 23 1943. It is, of course, math based. It contains the
formulas, but you don’t see them. You only see the friendly-looking interface to use.
How can you use the how old calculator? You don’t need to have any special skills. Moreover, you don’t even need to do almost anything. You just need to enter the data, so you need to enter the date
of your birth – day, month and year. Less than a second is totally sufficient for this tool to give you an exact result. Easy? Yup, as it looks!
There are more good pieces of information. The how old calculator is a free tool. It means you don’t have to pay anything to use it. Just go on the page and enjoy! You can use it on your smartphone,
tablet or laptop. It will work as well on every device with an Internet connection.
So let’s try it on your own and see how fast and effortlessly you can get the answer to how old are you if you were born on August 23 1943.
Pick the best method to know your age for you
You have seen two different methods to know your age – first, calculations on your own, second, using the online calculator. It is time to pick the method for you. You could see how it works in both
of them. You could try to calculate your exact age following our three steps and also use our app. So we are sure that now you have your favorite.
Both these methods are dedicated for different people and different needs. We gathered them in one article to show you the differences between them and give you the choice. So, if you need, read the
previous paragraphs again, and enjoy calculations – regardless of whether you will make them on your own or using our how old calculator.
Do you feel old or young?
We are very curious what you think about your age now, when you finally know the exact numbers. Do you feel old or young? We are asking it because so many people, so many minds. All of you can feel
the age differently, even if it is so similar or the same age! And we think it’s beautiful that all of us are different.
Regardless of feeling old or young, what do you feel more when you think about your age? What do you think about your life so far? We encourage you to make some kinds of summaries once in a while.
Thanks to this, you will be able to check if your dream has come true, or maybe you need to fight more to reach your goal. Or maybe, after some thought, you will decide to change your life totally.
Thinking about our life, analyzing our needs and wants – these things are extremely important to live happily.
Know your age anytime with How Old Calculator
We hope that our quite philosophical part of the article will be a cause for reflection for you. But let’s get back to the main topic, or, to be honest, the end of this topic. Because that’s the end
of our article. Let’s sum up what you have learned today.
I was born on August 23 1943. How old am I? We are sure that such a question will not surprise you anymore. Now you can calculate your age, even exact age, in two different ways. You are able to make
your own calculations and also know how to make it quicker and easier with the how old calculator.
It is time for your move. Let’s surprise your friends or family with the accuracy of your answers! Tell them how old you are with an accuracy of years, months and days!
Check also our other articles to check how old are your family members or friends. Pick their birthdate, see the explanation and get the results.
Invariant Language (Invariant Country) Monday, 23 August 1943
Afrikaans Maandag 23 Augustus 1943
Aghem tsuʔukpà 23 ndzɔ̀ŋɔ̀kwîfɔ̀e 1943
Akan Dwowda, 1943 Difuu-Ɔsandaa 23
Amharic 1943 ኦገስት 23, ሰኞ
Arabic الاثنين، 23 أغسطس 1943
Assamese সোমবাৰ, 23 আগষ্ট, 1943
Asu Jumatatu, 23 Agosti 1943
Asturian llunes, 23 d’agostu de 1943
Azerbaijani 23 avqust 1943, bazar ertəsi
Azerbaijani 23 август 1943, базар ертәси
Azerbaijani 23 avqust 1943, bazar ertəsi
Basaa ŋgwà njaŋgumba 23 Hìkaŋ 1943
Belarusian панядзелак, 23 жніўня 1943 г.
Bemba Palichimo, 23 Ogasti 1943
Bena pa shahuviluha, 23 pa mwedzi gwa nane 1943
Bulgarian понеделник, 23 август 1943 г.
Bambara ntɛnɛ 23 uti 1943
Bangla সোমবার, 23 আগস্ট, 1943
Tibetan 1943 ཟླ་བ་བརྒྱད་པའི་ཚེས་23, གཟའ་ཟླ་བ་
Breton Lun 23 Eost 1943
Bodo समबार, आगस्थ 23, 1943
Bosnian ponedjeljak, 23. august 1943.
Bosnian понедјељак, 23. аугуст 1943.
Bosnian ponedjeljak, 23. august 1943.
Catalan dilluns, 23 d’agost de 1943
Chakma 𑄥𑄧𑄟𑄴𑄝𑄢𑄴, 23 𑄃𑄉𑄧𑄌𑄴𑄑𑄴, 1943
Chechen 1943 август 23, оршот
Cebuano Lunes, Agosto 23, 1943
Chiga Orwokubanza, 23 Okwamunaana 1943
Cherokee ᎤᎾᏙᏓᏉᏅᎯ, ᎦᎶᏂ 23, 1943
Central Kurdish 1943 ئاب 23, دووشەممە
Czech pondělí 23. srpna 1943
Welsh Dydd Llun, 23 Awst 1943
Danish mandag den 23. august 1943
Taita Kuramuka jimweri, 23 Mori ghwa wunyanya 1943
German Montag, 23. August 1943
Zarma Atinni 23 Ut 1943
Lower Sorbian pónjeźele, 23. awgusta 1943
Duala mɔ́sú 23 diŋgindi 1943
Jola-Fonyi Teneŋ 23 Ut 1943
Dzongkha གཟའ་མིག་དམར་, སྤྱི་ལོ་1943 ཟླ་བརྒྱད་པ་ ཚེས་23
Embu Njumatatu, 23 Mweri wa kanana 1943
Ewe dzoɖa, deasiamime 23 lia 1943
Greek Δευτέρα, 23 Αυγούστου 1943
English Monday, August 23, 1943
Esperanto lundo, 23-a de aŭgusto 1943
Spanish lunes, 23 de agosto de 1943
Estonian esmaspäev, 23. august 1943
Basque 1943(e)ko abuztuaren 23(a), astelehena
Ewondo mɔ́ndi 23 ngɔn mwom 1943
Persian 1322 مرداد 31, دوشنبه
Fulah aaɓnde 23 juko 1943
Fulah aaɓnde 23 juko 1943
Finnish maanantai 23. elokuuta 1943
Filipino Lunes, Agosto 23, 1943
Faroese mánadagur, 23. august 1943
French lundi 23 août 1943
Friulian lunis 23 di Avost dal 1943
Western Frisian moandei 23 Augustus 1943
Irish Dé Luain 23 Lúnasa 1943
Scottish Gaelic DiLuain, 23mh dhen Lùnastal 1943
Galician Luns, 23 de agosto de 1943
Swiss German Määntig, 23. Auguscht 1943
Gujarati સોમવાર, 23 ઑગસ્ટ, 1943
Gusii Chumatato, 23 Agosti 1943
Manx 1943 Luanistyn 23, Jelhein
Hausa Litinin 23 Agusta, 1943
Hawaiian Poʻakahi, 23 ʻAukake 1943
Hebrew יום שני, 23 באוגוסט 1943
Hindi सोमवार, 23 अगस्त 1943
Croatian ponedjeljak, 23. kolovoza 1943.
Upper Sorbian póndźela, 23. awgusta 1943
Hungarian 1943. augusztus 23., hétfő
Armenian 1943 թ. օգոստոսի 23, երկուշաբթի
Interlingua lunedi le 23 de augusto 1943
Indonesian Senin, 23 Agustus 1943
Igbo Mọnde, 23 Ọgọọst 1943
Sichuan Yi 1943 ꉆꆪ 23, ꆏꊂꋍ
Icelandic mánudagur, 23. ágúst 1943
Italian lunedì 23 agosto 1943
Japanese 1943年8月23日月曜日
Ngomba Mɔ́ndi, 1943 Pɛsaŋ Pɛ́nɛ́fɔm 23
Machame Jumatatuu, 23 Agusti 1943
Javanese Senin, 23 Agustus 1943
Georgian ორშაბათი, 23 აგვისტო, 1943
Kabyle Sanass 23 Ɣuct 1943
Kamba Wa kwambĩlĩlya, 23 Mwai wa nyaanya 1943
Makonde Liduva lyatatu, 23 Mwedi wa Nnyano na Mitatu 1943
Kabuverdianu sigunda-fera, 23 di Agostu di 1943
Koyra Chiini Atini 23 Ut 1943
Kikuyu Njumatatũ, 23 Mwere wa kanana 1943
Kazakh 1943 ж. 23 тамыз, дүйсенбі
Kako lundi 23 fɛ 1943
Kalaallisut 1943 aggustip 23, ataasinngorneq
Kalenjin Kotaai, 23 Rooptui 1943
Khmer ចន្ទ 23 សីហា 1943
Kannada ಸೋಮವಾರ, ಆಗಸ್ಟ್ 23, 1943
Korean 1943년 8월 23일 월요일
Konkani सोमार 23 आगोस्त 1943
Kashmiri ژٔندرٕروار, اگست 23, 1943
Shambala Jumaatatu, 23 Agosti 1943
Bafia lǝndí 23 ŋwíí akǝ táaraa 1943
Colognian Mohndaach, dä 23. Oujoß 1943
Kurdish 1943 gelawêjê 23, duşem
Cornish 1943 mis Est 23, dy Lun
Kyrgyz 1943-ж., 23-август, дүйшөмбү
Langi Jumatátu, 23 Kʉvɨɨrɨ 1943
Luxembourgish Méindeg, 23. August 1943
Ganda Balaza, 23 Agusito 1943
Lakota Aŋpétuwaŋži, Wasútȟuŋ Wí 23, 1943
Lingala mokɔlɔ mwa yambo 23 sánzá ya mwambe 1943
Lao ວັນຈັນ ທີ 23 ສິງຫາ ຄ.ສ. 1943
Northern Luri AP 1322 Mordad 31, Mon
Lithuanian 1943 m. rugpjūčio 23 d., pirmadienis
Luba-Katanga Nkodya 23 Lùshìkà 1943
Luo Wuok Tich, 23 Dwe mar Aboro 1943
Luyia Jumatatu, 23 Agosti 1943
Latvian Pirmdiena, 1943. gada 23. augusts
Masai Jumatátu, 23 Ɔlɔ́ɨ́bɔ́rárɛ 1943
Meru Muramuko, 23 Agasti 1943
Morisyen lindi 23 out 1943
Malagasy Alatsinainy 23 Aogositra 1943
Makhuwa-Meetto Jumatatu, 23 Mweri wo nane 1943
Metaʼ Aneg 2, 1943 iməg ichika 23
Maori Rāhina, 23 Hereturikōkā 1943
Macedonian понеделник, 23 август 1943
Malayalam 1943, ഓഗസ്റ്റ് 23, തിങ്കളാഴ്ച
Mongolian 1943 оны наймдугаар сарын 23, Даваа гараг
Marathi सोमवार, 23 ऑगस्ट, 1943
Malay Isnin, 23 Ogos 1943
Maltese It-Tnejn, 23 ta’ Awwissu 1943
Mundang Comlaaɗii 23 Madǝmbii 1943
Burmese 1943၊ ဩဂုတ် 23၊ တနင်္လာ
Mazanderani AP 1322 Mordad 31, Mon
Nama Mantaxtsees, 23 Aoǁkhuumûǁkhâb 1943
Norwegian Bokmål mandag 23. august 1943
North Ndebele Mvulo, 23 Ncwabakazi 1943
Low German 1943 M08 23, Mon
Nepali 1943 अगस्ट 23, सोमबार
Dutch maandag 23 augustus 1943
Kwasio mɔ́ndɔ 23 ngwɛn lɔmbi 1943
Norwegian Nynorsk måndag 23. august 1943
Ngiemboon mvfò lyɛ̌ʼ , lyɛ̌ʼ 23 na saŋ mbʉ̀ŋ, 1943
Nuer Jiec la̱t 23 Tho̱o̱r 1943
Nyankole Orwokubanza, 23 Okwamunaana 1943
Oromo Wiixata, Hagayya 23, 1943
Odia ସୋମବାର, ଅଗଷ୍ଟ 23, 1943
Ossetic Къуырисӕр, 23 августы, 1943 аз
Punjabi ਸੋਮਵਾਰ, 23 ਅਗਸਤ 1943
Punjabi پیر, 23 اگست 1943
Punjabi ਸੋਮਵਾਰ, 23 ਅਗਸਤ 1943
Polish poniedziałek, 23 sierpnia 1943
Pashto دونۍ د AP 1322 د زمری 31
Portuguese segunda-feira, 23 de agosto de 1943
Quechua Lunes, 23 Agosto, 1943
Romansh glindesdi, ils 23 d’avust 1943
Rundi Ku wa mbere 23 Nyandagaro 1943
Romanian luni, 23 august 1943
Rombo Ijumatatu, 23 Mweri wa nane 1943
Russian понедельник, 23 августа 1943 г.
Kinyarwanda 1943 Kanama 23, Kuwa mbere
Rwa Jumatatuu, 23 Agusti 1943
Sakha 1943 сыл Атырдьых ыйын 23 күнэ, бэнидиэнньик
Samburu Mderot ee kuni, 23 Lapa le isiet 1943
Sangu Jumatatu, 23 Mupuguto 1943
Sindhi 1943 آگسٽ 23, سومر
Northern Sami 1943 borgemánnu 23, vuossárga
Sena Chiposi, 23 de Augusto de 1943
Koyraboro Senni Atinni 23 Ut 1943
Sango Bïkua-ûse 23 Kükürü 1943
Tachelhit ⴰⵢⵏⴰⵙ 23 ⵖⵓⵛⵜ 1943
Tachelhit aynas 23 ɣuct 1943
Tachelhit ⴰⵢⵏⴰⵙ 23 ⵖⵓⵛⵜ 1943
Sinhala 1943 අගෝස්තු 23, සඳුදා
Slovak pondelok 23. augusta 1943
Slovenian ponedeljek, 23. avgust 1943
Inari Sami vuossargâ, porgemáánu 23. 1943
Shona 1943 Nyamavhuvhu 23, Muvhuro
Somali Isniin, Bisha Sideedaad 23, 1943
Albanian e hënë, 23 gusht 1943
Serbian понедељак, 23. август 1943.
Serbian понедељак, 23. август 1943.
Serbian ponedeljak, 23. avgust 1943.
Swedish måndag 23 augusti 1943
Swahili Jumatatu, 23 Agosti 1943
Tamil திங்கள், 23 ஆகஸ்ட், 1943
Telugu 23, ఆగస్టు 1943, సోమవారం
Teso Nakaebarasa, 23 Opedel 1943
Tajik Душанбе, 23 Август 1943
Thai วันจันทร์ที่ 23 สิงหาคม พ.ศ. 2486
Tigrinya ሰኑይ፣ 23 ነሓሰ መዓልቲ 1943 ዓ/ም
Turkmen 23 awgust 1943 Duşenbe
Tongan Mōnite 23 ʻAokosi 1943
Turkish 23 Ağustos 1943 Pazartesi
Tatar 23 август, 1943 ел, дүшәмбе
Tasawaq Atinni 23 Ut 1943
Central Atlas Tamazight Aynas, 23 Ɣuct 1943
Uyghur 1943 23-ئاۋغۇست، دۈشەنبە
Ukrainian понеділок, 23 серпня 1943 р.
Urdu پیر، 23 اگست، 1943
Uzbek dushanba, 23-avgust, 1943
Uzbek AP 1322 Mordad 31, دوشنبه
Uzbek душанба, 23 август, 1943
Uzbek dushanba, 23-avgust, 1943
Vai ꗳꗡꘉ, 23 ꗛꔕ 1943
Vai tɛɛnɛɛ, 23 kɔnde 1943
Vai ꗳꗡꘉ, 23 ꗛꔕ 1943
Vietnamese Thứ Hai, 23 tháng 8, 1943
Vunjo Jumatatuu, 23 Agusti 1943
Walser Mäntag, 23. Öigšte 1943
Wolof Altine, 23 Ut, 1943
Xhosa 1943 Agasti 23, Mvulo
Soga Balaza, 23 Agusito 1943
Yangben móndie 23 pisuyú 1943
Yiddish מאָנטיק, 23טן אויגוסט 1943
Yoruba Ajé, 23 Ògú 1943
Cantonese 1943年8月23日星期一
Cantonese 1943年8月23日星期一
Cantonese 1943年8月23日星期一
Standard Moroccan Tamazight ⴰⵢⵏⴰⵙ 23 ⵖⵓⵛⵜ 1943
Chinese 1943年8月23日星期一
Chinese 1943年8月23日星期一
Chinese 1943年8月23日星期一
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Stage-discharge estimation in compound open channels with composite roughness
Determination of stage-discharge relationships in natural rivers is extremely important in flood control projects. The importance of rating curves in rivers and the difficulty of its establishment
show the need for simpler and more precise methods. Determination of rating curves in compound channels, especially with composite roughness, has proved to be difficult because of significant
variations in hydraulic parameters from the main channel to the floodplains. In the current paper, a new approach that is based on the concept of the cross-sectional isovel contours is introduced for
estimation of the stage-discharge curves in compound channels. The multivariate Newton's method is applied to the difference between the observed and estimated data to optimize the exponent values of
the governing parameters. The proposed method is verified by comparing predictions obtained by using it with some experimental datasets. For compound channels with composite roughness, the
stage-discharge curves obtained by the proposed method are more accurate than those obtained by using available conventional methods.
Determination of the stage-discharge curves in compound open channels is essential from the point of view of river engineering. Most of the time water is present in the main channel, but the
floodplains have an important role to play during flood events because they increase the conveyance capacity of channels. Several researchers have proposed different methods for predicting the
stage-discharge relation in compound channels. In conventional methods, discharge in compound channels can be calculated based on the single and divided channel methods (SCM and DCM). If the channel
is taken as a single unit, the flow rate will be underestimated due to the reduced main channel velocity. When the sub-sections of a compound channel are taken independently, the total discharge is
generally overestimated due to neglecting the momentum transfer between the main channel and the floodplains (Stephenson & Kolovopoulos 1990). The exact solution should be found somewhere between the
SCM and DCM methods (Sahu et al. 2011). The cross-section of a compound channel may be subdivided into the main channel and the floodplain as shown in Figure 1. Typically, these divisions are made
using straight vertical lines at the edges of the main channel, which are assumed to be shear-free and as a result are not included in the wetted perimeter. So far, various methods have been
introduced by researchers in order to improve the performances of the SCM and DCM.
Lambert & Myers (1998) have proposed the weighted divided channel method (WDCM) by investigating the experimental works. The WDCM is based on the fact that the real discharge should be located
between an underestimated value of SCM and overestimated value of DCM. So, a weighted multiplier is introduced to modify the velocities of the main channel and floodplain. The minimum weighting
factor is zero, and the maximum is one. It is applied to both the in-bank and over-bank(s) to give the improved mean velocity estimations for these areas.
Ackers (1993) used experimental data and proposed the concept of coherence (COH), which works based on the ratio of the discharges that are calculated by the SCM and DCM methods. Khatua et al. (2012)
proposed the modified divided channel method (MDCM) to calculate the discharge in compound channels especially for the straight smooth compound channels with wide floodplains. The results obtained by
MDCM give a rough indication of the efficacy of the method for practical applications both in small-scale and large-scale experimental data.
The exchange discharge method (EDM) was proposed by Bousmar & Zech (1999). The momentum transfer is estimated as the product of the velocity difference between the main channel and the floodplain at
the interface by the mass discharge exchanged through this interface due to turbulence. Accurate estimation of exchanged discharge with errors generally less than 5% are obtained, while the momentum
correction enables the use of actual roughness coefficients, corresponding to the actual river bed material (Bousmar 2002). For non-prismatic flows, the EDM is extended by taking into account, in the
momentum transfer, a mass transfer corresponding to the geometrical change. Its accuracy is similar to the WDCM, but the model has the advantage of being a physically based model without numerous
parameter fittings (Bousmar & Zech 1999). As a secondary goal, for estimation of water surface profile the velocity gradient was conceptualized as an additional head loss in gradually varied flow
equation. When roughness distribution along the wetted perimeter is not uniform in compound channels, more complexities in estimation of discharge will be encountered. In compound channels with large
floodplains and considerable roughness, a kind of discontinuity at the level of the main channel will happen due to the floodplain drag (Smart 1992). Determination of equivalent roughness has been
investigated by the results obtained from 17 different methods using DCM (Yen 2002). Major differences have been observed in the results obtained by different techniques. It is not actually an easy
task to discern the best discharge estimation method among a number of techniques.
The main difficulty associated with the DCM method in natural rivers is distinguishing the border of the main channel and the floodplain. Sellin & Van Beesten (2004) have analyzed the effects of flow
resistance in compound channels on the results obtained within five years of observations on the floodplain of the Blackwater River in southern England. It has been found that at higher overbank flow
n values increase drastically with depth. However, they remain constant after cutting the berm vegetation. Huai et al. (2013) have estimated the apparent shear stress acting on the vertical interface
between the main channel and floodplain (with and without vegetation) in prismatic compound open channels using artificial neural networks (ANNs).
Cao et al. (2006) presented a formula for flow resistance and momentum flux in compound open channels. The proposed formula was implemented in the St. Venant equations for evaluating conveyance,
roughness and stage-discharge relationship in compound open channels. Liao & Knight (2007) analytically proposed a relationship to estimate the discharge in straight trapezoidal open channels. The
results indicate good agreement between the analytical and experimental rating curves. Sun & Shiono (2009) used an experimental model in straight compound channels with and without one-line emergent
vegetation along the floodplain edge to predict the rating curves. They proposed a new formula for friction factors for with and without vegetation cases using flow parameters and vegetation density.
Rezaei & Knight (2010) presented experimental results of overbank flow in compound channels with non-prismatic floodplains and different convergence angles. The results show that the discharge
evolution seems linear in the lower water depths, whereas it is non-linear for higher water depths, and the mass transfer in the second half of the converging reach is higher than that in the first
half. Yang et al. (2012) developed a new model to estimate the discharge in symmetric, compound open channels based on the energy concept. They have utilized the application of the momentum transfer
coefficient to predict the discharge distributions of the floodplains and main channel. Mohanty & Khatua (2014) proposed a new relationship, which is derived partly based on the percentages of shear
stress carried by the floodplains and partly based on the percentage of area occupied by the floodplains to estimate the rating curves in compound channels especially with wide floodplain(s).
Maghrebi et al. (2016) were the first to present a method for estimating stage-discharge curves using the least amount of hydraulic data that is comprised of the geometrical information of the
cross-section and discharge at any desired water level. In continuation of their studies, Maghrebi et al. (2017) proposed a new relationship that was mainly aimed at increasing the accuracy of the
results in compound cross-sections. As a matter of fact, the introduced new approach on the estimation of rating curves in different types of conduits is founded on the basis of the multiplication of
inter- and extrapolation of multi-variables, which affects the discharge at a certain water stage.
The main objective of the present paper is to look for the most general form of an expression which can be used to predict the rating curves in any kind of open channels with any distribution of
roughness including composite compound as well as hydraulic parameters. Then the best relationship is selected based on the minimum error statistics such as normalized root mean square error (NRMSE)
and mean absolute percentage error (MAPE). Although the number of the selected cross-sectional shapes and hydraulic parameters which are engaged in the generated relationships are not limited to
those implemented in the current study, it is believed that any other combinations of the shapes and parameters will not lead to a simpler, and more accurate relationship.
Maghrebi (2006) proposed the concept of isovel contours to predict the discharge in any cross-sectional shape of a conduit which is known as SPM (Single Point in Maghrebi's approach). An extension of
the isovel contours is used for the estimation of the rating curve relationship. Figure 1 shows a typical asymmetric compound channel cross-section which is covered with triangular meshes. The
centroid of each triangular element is where the boundary effects are calculated. In order to increase the accuracy of the method, the number of meshes has been increased along the boundary and water
All of the conventional velocity distributions such as logarithmic, velocity defect and power laws are functions of
which is measured from the boundary in a vertical direction. For open channels with large aspect ratios, the power law velocity profile for the longitudinal velocity component
can be defined in the following form:where is the shear velocity,
is the Nikuradse sand equivalent roughness and
is a coefficient. The exponent 1/
is almost independent of the Reynolds number, and the velocity profiles are therefore almost similar. Over a wide range of the Reynolds numbers
7 agreed well with a large number of experimental measurements. However, most of the natural and artificial channels do not take the form of a large aspect ratio cross-section. In order to deal with
these types of open channels, the idea of using
instead of
in velocity distribution functions was proposed by
Maghrebi (2006)
. It was inspired by the Biot–Savart law (
Halliday & Resnick 1981
) which can be used to calculate the intensity of a magnetic field generated at a point in space by a piece of wire carrying a stationary current. In the field of hydraulics, velocity at an arbitrary
point with the coordinates (
y, z
) at the flow section was quantified by the integration along the whole wetted perimeter as follows (
Figure 1
In Figure 1 a schematic sketch of the implemented mesh in the flow section is shown. As a matter of fact, the actual mesh size is finer than that shown in the figure. When the number of grids is
increased to more than 500, the calculated value of U[SPM] will remain nearly unchanged. As can be determined from Figure 2, when the number of grids is increased from say 450 to 850, U[SPM] will be
decreased by less than about 0.2%. For all sections without considering the water level, a total number of 850 grids has been implemented in calculation of U[SPM], which leads to finer mesh for lower
water levels.
All of the parameters which are involved in discharge can also be involved in the stage-discharge relationship. The main contributing parameters are listed as cross-sectional area A, wetted perimeter
P, width of the free surface T, total perimeter P[t] ( =P+T), equivalent Manning roughness n, bed slope S[0] and U[SPM] which has the concept of mean cross-sectional velocity since the isovel
contours can be obtained by the use of this parameter. Accordingly, discharge Q can be expressed by the following relationship:
The effect of the bed slope of the channel
which stays fixed at all water levels can be omitted from Equation (
). The general form of the proposed relationship to estimate the stage-discharge curve is presented as follows:where the subscripts
are referred to as the estimated and referenced values, respectively. The most challenging issue in Equation (
) is the problem of determining the exponent values
. Several steps must be taken in order to calculate the exponents. The first step is collection of some of the data taken from the theoretical and observational rating curves for different hydraulic
cross-sections. The exact stage-discharge relations for the triangular, rectangular and circular sections at different water levels can be calculated by the use of the Manning formula. Additionally,
due to the lack of a unique analytical solution for the stage-discharge relation in compound channels, the experimentally measured discharges in compound channels of FCF-Series01 (
Knight 1992
) are considered as the most accurate available data. The specifications of the compound channels, which is shown in
Figure 3
, are given in
Table 1
Table 1
Test . S[0] (×10^–3) . n[c] . n[f] . S[c] . S[f] . Bw[c] (m) . bw[f1] (m) . bw[f2] (m) . h (m) . H[f] (m) .
FCF-S01 1.027 0.01 0.01 1 0 1.5 4.1 4.1 0.15 0.15
FCF-S02 1.027 0.01 0.01 1 1 1.5 2.25 2.25 0.15 0.15
FCF-S03 1.027 0.01 0.01 1 1 1.5 0.75 0.75 0.15 0.15
FCF-S06 1.027 0.01 0.01 1 1 1.5 2.25 0 0.15 0.15
FCF-S10 1.027 0.01 0.01 2 1 1.5 2.25 2.25 0.15 0.15
P & T-S1-S 0.3 0.011 0.011 0.5 0 0.203 0.381 0.381 0.102 0.1
P & T-S2-S 0.3 0.011 0.011 0.5 0 0.305 0.381 0.381 0.102 0.1
P & T-S1-R1 0.3 0.011 0.014 0.5 0 0.203 0.381 0.381 0.102 0.1
P & T-S2-R2 0.3 0.011 0.014 0.5 0 0.305 0.381 0.381 0.102 0.1
P & T-S1-R2 0.3 0.011 0.018 0.5 0 0.203 0.381 0.381 0.102 0.1
P & T-S2-R2 0.3 0.011 0.018 0.5 0 0.305 0.381 0.381 0.102 0.1
P & T-S1-R3 0.3 0.011 0.022 0.5 0 0.203 0.381 0.381 0.102 0.1
P & T-S2-R3 0.3 0.011 0.022 0.5 0 0.305 0.381 0.381 0.102 0.1
Test . S[0] (×10^–3) . n[c] . n[f] . S[c] . S[f] . Bw[c] (m) . bw[f1] (m) . bw[f2] (m) . h (m) . H[f] (m) .
FCF-S01 1.027 0.01 0.01 1 0 1.5 4.1 4.1 0.15 0.15
FCF-S02 1.027 0.01 0.01 1 1 1.5 2.25 2.25 0.15 0.15
FCF-S03 1.027 0.01 0.01 1 1 1.5 0.75 0.75 0.15 0.15
FCF-S06 1.027 0.01 0.01 1 1 1.5 2.25 0 0.15 0.15
FCF-S10 1.027 0.01 0.01 2 1 1.5 2.25 2.25 0.15 0.15
P & T-S1-S 0.3 0.011 0.011 0.5 0 0.203 0.381 0.381 0.102 0.1
P & T-S2-S 0.3 0.011 0.011 0.5 0 0.305 0.381 0.381 0.102 0.1
P & T-S1-R1 0.3 0.011 0.014 0.5 0 0.203 0.381 0.381 0.102 0.1
P & T-S2-R2 0.3 0.011 0.014 0.5 0 0.305 0.381 0.381 0.102 0.1
P & T-S1-R2 0.3 0.011 0.018 0.5 0 0.203 0.381 0.381 0.102 0.1
P & T-S2-R2 0.3 0.011 0.018 0.5 0 0.305 0.381 0.381 0.102 0.1
P & T-S1-R3 0.3 0.011 0.022 0.5 0 0.203 0.381 0.381 0.102 0.1
P & T-S2-R3 0.3 0.011 0.022 0.5 0 0.305 0.381 0.381 0.102 0.1
The parameters n[c], n[f], s[c], s[f], bw[c], bw[f1], bw[f2], h and H[f] are the Manning roughness in the main channel and the floodplain, the side slope of the main channel, the side slope of the
floodplain, the width of the main channel, the width of the floodplain on the left and right sides, the depth of the main channel and the depth of the floodplain, respectively (Figure 3).
The multivariate Newton's method is applied to the difference between the exact and estimated values of (Q,H) to optimize the exponent values of the governing equation (Equation (5)). Before
computing the exponents, it should be mentioned that they can be extracted from different conditions because the proposed relationship can be formed from three to six different parameters. They are
shown on the bottom row of Figure 4. U[SPM] and n are included in all of the relationships. On the other hand, all combinations of the four reference cross-sections including triangular, rectangular,
circular and compound sections can be used in the minimization process. They are shown on the top row of Figure 4. In total, 225 different relationships are extracted as shown in Figure 4. In
Figure 4, the abbreviations Tri, Rec, Cir and Com represent the triangular, rectangular, circular and compound cross-sections in FCF-Series01, respectively.
In the determination of the exponent values involved in Equation (5), the geometric and hydraulic parameters of A[max], P[,] (P[t]), U[SPM] and n in four selected sections at all water levels should
be evaluated. Figure 5(a)–5(d) indicates the variations of these parameters for the selected sections. Their maximum values are given as A[max], P[max], (P[t])[max], U[SPM] and n[max], which are
mainly associated with the highest water level H[max], as presented in Table 2. The Manning roughness coefficient n is actually the equivalent roughness which can be calculated by a number of
different methods (Yen 2002). For a uniform boundary roughness, it takes the corresponding roughness coefficient. However, when the roughness distribution along the wetted perimeter is non-uniform,
an equivalent roughness at each water level should be implemented. In the proposed method, the rating curve can be estimated from any arbitrary water level which is considered as a reference level.
Table 2
Test . H[max] (m) . A[max] (m^2) . P[max] (m) . (P[t]) [max] (m) . (U[SPM]) [max] . n[max] .
FCF-S01 0.250 1.248 10.324 20.324 1.147 0.01
FCF-S02 0.287 1.135 6.814 13.390 1.254 0.01
FCF-S03 0.299 0.762 3.846 7.444 1.172 0.01
FCF-S06 0.301 0.885 4.603 8.957 1.208 0.01
FCF-S10 0.279 1.142 7.037 13.897 1.211 0.01
P & T-S1-S 0.18 0.109 1.349 2.416 0.484 0.011
P & T-S2-S 0.18 0.127 1.451 2.620 0.518 0.011
P & T-S1-R1 0.18 0.109 1.349 2.416 0.484 0.013
P & T-S2-R2 0.18 0.127 1.451 2.620 0.518 0.0128
P & T-S1-R2 0.18 0.109 1.349 2.416 0.484 0.0157
P & T-S2-R2 0.18 0.127 1.451 2.620 0.518 0.0151
P & T-S1-R3 0.18 0.109 1.349 2.416 0.484 0.0184
P & T-S2-R3 0.18 0.127 1.451 2.620 0.518 0.0179
Test . H[max] (m) . A[max] (m^2) . P[max] (m) . (P[t]) [max] (m) . (U[SPM]) [max] . n[max] .
FCF-S01 0.250 1.248 10.324 20.324 1.147 0.01
FCF-S02 0.287 1.135 6.814 13.390 1.254 0.01
FCF-S03 0.299 0.762 3.846 7.444 1.172 0.01
FCF-S06 0.301 0.885 4.603 8.957 1.208 0.01
FCF-S10 0.279 1.142 7.037 13.897 1.211 0.01
P & T-S1-S 0.18 0.109 1.349 2.416 0.484 0.011
P & T-S2-S 0.18 0.127 1.451 2.620 0.518 0.011
P & T-S1-R1 0.18 0.109 1.349 2.416 0.484 0.013
P & T-S2-R2 0.18 0.127 1.451 2.620 0.518 0.0128
P & T-S1-R2 0.18 0.109 1.349 2.416 0.484 0.0157
P & T-S2-R2 0.18 0.127 1.451 2.620 0.518 0.0151
P & T-S1-R3 0.18 0.109 1.349 2.416 0.484 0.0184
P & T-S2-R3 0.18 0.127 1.451 2.620 0.518 0.0179
In the continuity equation Q=AU, velocity has a power of 1. Moreover, since the role of U[SPM] is the same as velocity, it is concluded that a[5] = 1. In addition, we have a[6] = –1 due to the
inverse relation between discharge and roughness according to the Manning formula. Therefore, the remaining task is the evaluation of a[1], a[2], a[3] and a[4]. This can be achieved by the
minimization of the statistical measure , which is defined in the following form:
The subscripts i and j in Equation (6) are referred to as the reference and estimated values, respectively and N is the total number of investigated points for each stage-discharge curve. In other
words, is equal to Q[e] which is the estimated discharge at the jth level.
According to Equation (
), considering a single observed discharge as the reference, a unique stage-discharge curve can be estimated. Therefore, the
parameter can be calculated based on each single curve. Then, the average value of this parameter, which is shown by
can be computed from all of the calculated values of
. The denominator of Equation (
) is composed of the difference between the maximum and minimum reference discharges. An objective function like
, which can be defined generally in the following form, is introduced since it is desirable to establish a rating curve which is derived from all of the selected sections. Now the minimization
process should be implemented on
:where the subscripts
Tri, Rec
are defined previously. The number of parameters on the right-hand side of Equation (
) depends on the number of the implemented sections. In order to examine the performance of each relationship, classification should be carried out. The statistical measure of the normalized root
mean square error, , is used for this purpose. All of the relationships extracted from the four rectangular, circular, triangular and compound sections are collected in a group named Group A with the
corresponding value of . The variations of for all 225 relationships, is shown in
Figure 6
. It can be seen that the least amount of is related to relationship number 225 which is generated by using six parameters and four different sections. Meanwhile, it can also be observed that the
corresponding values for five-parameter relationships in Group B do not differ significantly from the six-parameter relationships in Group A. For example, we have and . Now the question is how the
trivial difference of in relationships 205 and 225 (the best relationships of Groups B and A, respectively) affects the performance of stage-discharge curves when they are applied to compound
channels with composite roughness.
In order to answer this question, rating curves are estimated from each equation in two laboratory compound channels FCF-S2 and FCF-S3 (Knight 1992) and two rough compound channels P & T-S1-R3 and P
& T-S2-R3 (Prinos & Townsend 1984) and then NRMSE values are calculated for each of them. Figure 5(e)–5(h) indicate the variations of H, A, P[,]P[t], U[SPM] and n where their maximum values are shown
in Table 2.
The total value of among the second group of the compound cross-sections is named Group B. According to
Figure 6
, value is negligible for zones with five and six parameters. The summation of and is shown as:
In fact, the lowest value of is not the only criterion for selection of the best relationship among all relationships. By considering other criteria such as the number of parameters and
cross-sections used in the proposed relationship, it can be concluded that the best relationship is the one with the lowest as well as the minimum number of parameters and cross-sections involved in
the formulation of the problem.
In order to propose a universal relationship, the statistical performance of the most accurate extracted relationships based on three, four, five and six parameters corresponding with the
relationship numbers 48, 133, 205 and 225, respectively are compared with each other in Table 3. Although the configuration of relationship number 205 is completely different from that of 225, the
percentage of the difference in absolute mean error MAPE is only limited to 0.004%. One of the most attractive reasons to choose the relation 205 as the best one is its simplicity and high accuracy.
Table 3
Equation number . Exponents value . NRMSE . MAPE (%) .
A . P . T . P[t] . U[SPM] . n .
48 0.761 0 0 0 1 –1 0.094 12.78
133 1.006 –0.488 0 0 1 –1 0.056 7.80
205 0.972 –1.268 0 0.832 1 –1 0.033 4.479
225 0.979 –1.183 0.047 0.701 1 –1 0.032 4.475
Equation number . Exponents value . NRMSE . MAPE (%) .
A . P . T . P[t] . U[SPM] . n .
48 0.761 0 0 0 1 –1 0.094 12.78
133 1.006 –0.488 0 0 1 –1 0.056 7.80
205 0.972 –1.268 0 0.832 1 –1 0.033 4.479
225 0.979 –1.183 0.047 0.701 1 –1 0.032 4.475
Relationship number 205 includes five parameters, namely
as well as two geometric cross-sections (rectangular and compound cross-section) in the determination of exponents of
. Finally, Equation (
) can be displayed in the following form for the selected relationship:
A comparison between the estimated and observed rating curves based on Equation (9) for different cross-sections is shown in Figure 7. As can be seen, the rating curve is established for each section
using only three different levels. The results indicate that the performance of the model is highly accurate.
The results of the statistical measures calculated based on different models using Equations (10) and (11), are shown in Figure 8. For triangular, rectangular and circular cross-sections, the results
show that except for the first observed values at the lowest water level, the estimated discharge values based on other points are within 3.8% and 0.02 for the MAPE and NRMSE, respectively. From
Figure 8(d)–8(f), it can be seen that the mean values of MAPE and NRMSE for FCF sections are within 5.3% and 0.03, respectively. Figure 8(g) and 8(h) illustrate that the maximum value of MAPE and
NRMSE for compound channels with rough floodplains are within 8.2% and 0.12, respectively.
For further validation of the proposed relationship, it is applied to eight compound cross-sections which are selected from the experimental works of Knight (1992) and Prinos & Townsend (1984). After
calculating the geometrical parameters and U[SPM], one can easily depict the dimensionless variations of the cross-sectional area A, wetted perimeter P, total perimeter P[t] and velocity parameter U
[SPM]. In Table 2, H[max], A[max], P[max], (P[t])[max], (U[SPM])[max] and n[max] for the experimental works of compound channels are given. From Figure 9, the stage-discharge curves based on a couple
of observed data about Q and H at any arbitrary level can be plotted.
Figure 10(a)–10(h) show the stage-discharge curves obtained from the EDM, SCM and WDCM methods. They are compared with the results of the proposed method for the corresponding compound channels.
Considering the results of EDM with the assumed value of at the lower levels above the floodplain, the estimated discharge is lower than that of the experimental ones. However, by increasing the
water level, the difference in discharges becomes smaller. Due to the integrity of the whole cross-section in the SCM, where its hydraulic behavior is completely different from the compound sections,
the obtained results are not in good agreement with the observed data. The results of WDCM are in good agreement with the observed stage-discharge data taken from the experimental results of FCF
channels. It should be noted that the calibration of this method is made by the use of FCF observational data. The best capability of the proposed model is that it does not need any calibration while
other methods do.
In the calculation of errors, it is only possible to compare the results when the water level is higher than the floodplain level.
In comparison with other cross-sections, the performances of WDCM in FCF cross-sections are much better because of experimental data such as the FCF data which are used to develop the WDCM method.
The average values of the statistical measures, i.e. and , are equal to 6.4% and 0.06, respectively. The accuracy of EDM is highly related to the selection of . If a good value is chosen for this
parameter, the corresponding error associated with this method will be reduced. With the recommended value of 0.16 which is proposed by Bousmar (2002), and are within 4.7% and 0.05, respectively. The
average value of obtained from the SCM method is roughly 13%.
From Figure 11, it can be seen that for the different reference points the obtained values of the statistical parameters are changed. It should be noted that the accuracy of the results is very much
related to the accuracy of the observed data. The calculated values of and based on the proposed model are quite acceptable within 4.9% and 0.053, respectively, when compared with other methods. Even
in some cases, these values are lower than other methods. The average values of the above-mentioned statistical measures based on the whole water levels are within these values. The stage-discharge
curves obtained by the proposed method are more accurate than the other two methods as seen in Figure 11(f)–11(h) for compound channels with composite roughness.
The Batu River is located in Malaysia. During the monsoon season in recent years, it has experienced several flood occurrences. The cross-sectional geometry of the Batu River is extracted as shown in
Figure 12(a), which is largely asymmetrical (Hin et al. 2008). Therefore, implementation of the 1D approach to predict the stage-discharge relationship without consideration of the momentum transfer
will lead to unrealistic results (Khatua et al. 2012).
Now the model is applied to this river. Based on Equation (9), the values of A, P, P[t], n and U[SPM] should be calculated at each stage. For this purpose, variation of each value for the whole range
of stages varying between zero and H[max] in a dimensionless form for the Batu River is plotted in Figure 12(b). The values of A[max], P[max], (P[t])[max], n[max] and U[max] are the maximum values at
H[max]. In the Batu River the maximum value of stage is H[max] = 2.6 m. The values of A[max], P[max], (P[t])[max], n[max] and (U[SPM])[max] are 28.19 m^2, 74.67 m, 147.46 m, 0.103 and 13.98,
respectively. Figure 12(c) shows the estimated rating curves in the Batu River which is extracted for the reference levels P1, P7 and P14. It can be observed that the estimated rating curves obtained
by the present method are very close to the observed data.
The bar charts in Figure 12(d) show the statistical measures of NRMSE and MAPE in the Batu River. With respect to the value of MAPE for the Batu River, it can be seen that the lowest and highest
values are 7 and 10%, respectively, with an average value of nearly 8.2%. It should be noted that the accuracy of estimated results highly depends on the accuracy of referenced data.
The stage-discharge curve is one of the key parameters for decision-making in the management of water resources and river engineering. In order to estimate the discharge in compound channels using
traditional methods, the flow cross-sections should be divided into a number of subsections. However, in the proposed model, the whole flow cross-section is considered as a unified and integrated
section. As a result, the most general form of an expression which can be used for extrapolation of the stage-discharge curve in any kind of open channel with any type of roughness distribution
including composite compound channels is introduced. In order to set up the new expression, a combination of the selected geometries of the flow sections and the corresponding calculated parameters,
namely A, P, P[t], U[SPM] and n, are implemented. From each combination a unique relationship is extracted. At the end of this stage 225 different formulae are proposed, then the best formula is
selected based on the performance of each formula using the minimum error statistics such as NRMSE and MAPE. All of the variables involved, including the geometrical and hydraulic parameters such as
A, P, P[t], U[SPM] and n, are assumed to be not only known at the referenced level but also at the level where we are actually looking for its discharge. Presentation of the performance of the new
relationship in comparison to others shows that the proposed expression has a very good capability in the estimation of stage-discharge curves. The most significant advantage of the proposed method
in the estimation of rating curves is its inherent simplicity, which does not need any calibration. | {"url":"https://iwaponline.com/hr/article/50/3/809/66602/Stage-discharge-estimation-in-compound-open","timestamp":"2024-11-07T03:02:55Z","content_type":"text/html","content_length":"337415","record_id":"<urn:uuid:8cec08b0-da46-47e2-96bb-6b07af27744a>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00009.warc.gz"} |
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Help with boxplot clipping
Since ODS graphics boxplot does not support clipping, I'm trying to manually calculate the clipping range and change the boxplot template to add an overlay plot for the clipping range. I'm following
the formula here: http://support.sas.com/documentation/cdl/en/statug/63347/HTML/default/viewer.htm#statug_boxplot_sect.... With the data provided in the example, the clipping ranges I computed seem
to match the values on the plot. But when I used a different data set, I got very different results. Here's my code. Can someone help me understand what's causing the difference? Thanks!
proc sort data=sashelp.cars out=cars(keep=make horsepower);
by make;
where make < 'L';
/* ============== manual calculation ==================*/
proc means data=cars q1 q3 nway missing noprint;
class make;
var horsepower;
output out=cars_sum (drop=_freq_ _type_) q1=grp_q1 q3=grp_q3;
proc means data=cars_sum mean nway missing noprint;
var grp_q1 grp_q3;
output out=all_sum(drop=_freq_ _type_) mean=grp_q1_mean grp_q3_mean;
data all_sum;
set all_sum;
ymax = grp_q1_mean + (grp_q3_mean - grp_q1_mean) * 1.5;
ymin = grp_q3_mean - (grp_q3_mean - grp_q1_mean) * 1.5;
/* =================== get results from boxplot ============== */
ods graphics off;
proc boxplot data=cars;
plot horsepower * make /clipfactor=1.5;
10-27-2012 10:31 AM | {"url":"https://communities.sas.com/t5/Graphics-Programming/Help-with-boxplot-clipping/td-p/108899","timestamp":"2024-11-14T17:41:17Z","content_type":"text/html","content_length":"249116","record_id":"<urn:uuid:a0c4fe98-86d2-458e-8c8d-28fae3c51128>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00510.warc.gz"} |
Handrail Angle Calculator - Calculator Doc
Handrail Angle Calculator
When installing a handrail on stairs or ramps, calculating the correct angle is crucial for safety and compliance with building codes. The Handrail Angle Calculator helps you find the angle between
two points on the handrail, ensuring it’s properly sloped. This tool uses simple measurements of height and horizontal distance to give you an accurate angle.
To calculate the handrail angle, the formula is:
Angle (A) = tan⁻¹((Height at Top (H2) − Height at Bottom (H1)) / Horizontal Distance (D))
This formula uses the tangent function to calculate the angle based on the difference in height between two points and the horizontal distance between them.
How to Use
1. Measure the Height at the Bottom (H1), which is the starting point of the handrail.
2. Measure the Height at the Top (H2), which is the endpoint of the handrail.
3. Measure the Horizontal Distance (D) between these two points along the floor or surface.
4. Click the “Calculate” button to find the handrail angle in degrees.
Imagine you are installing a handrail with the following measurements:
• Height at Bottom (H1) = 1.2 meters
• Height at Top (H2) = 2.5 meters
• Horizontal Distance (D) = 3 meters
Using the handrail angle formula:
A = tan⁻¹((2.5 − 1.2) / 3)
A = tan⁻¹(1.3 / 3)
A = tan⁻¹(0.4333)
A ≈ 23.48°
Therefore, the angle of the handrail is approximately 23.48 degrees.
1. What is the purpose of calculating the handrail angle?
Calculating the handrail angle ensures proper installation and safety, making sure it complies with building regulations.
2. What is considered a standard handrail angle?
Standard handrail angles typically range between 30° and 45°, depending on the slope of the stairs or ramp.
3. Can I use this calculator for ramps?
Yes, this calculator can be used for both stairs and ramps, as long as you have the height and horizontal distance.
4. How do I measure the horizontal distance?
Measure the distance along the floor or surface between the starting and ending points of the handrail.
5. What happens if my angle is too steep?
If the handrail angle is too steep, it may be difficult to use and could pose safety risks. You may need to adjust the design.
6. What units should I use for the measurements?
You can use any units (meters, feet, etc.), as long as the same units are used consistently for all measurements.
7. Can this calculator be used for curved handrails?
This calculator is intended for straight handrails. Curved handrails require more complex calculations.
8. What if the heights are negative?
The heights should always be positive values; negative heights may indicate measurement errors.
9. Why is the tangent function used in the formula?
The tangent function calculates the slope of the triangle formed by the height and horizontal distance, which gives the angle.
10. Does this calculator work for outdoor handrails?
Yes, the calculator works for both indoor and outdoor handrails as long as you have the correct measurements.
11. How accurate is the result?
The result is accurate to two decimal places, providing a precise angle for most construction purposes.
12. What if the horizontal distance is much larger than the height difference?
A large horizontal distance compared to the height difference will result in a smaller angle.
13. How can I adjust the angle if it’s not within a safe range?
You can adjust the height or horizontal distance of the handrail to bring the angle within a safe and comfortable range.
14. Can I use this calculator for wheelchair ramps?
Yes, the handrail angle calculator can be used to find the slope of handrails for wheelchair ramps.
15. Is the handrail angle affected by the material used?
No, the angle calculation is based purely on the geometry of the installation, not the material of the handrail.
16. What is the maximum allowable angle for handrails in most building codes?
Most building codes allow a maximum handrail angle of about 45°, though this can vary by location.
17. What is the minimum allowable angle for handrails?
The minimum allowable angle is typically around 30°, but it depends on the specific requirements of the project.
18. Can I calculate the handrail angle without measuring the height?
No, both height and horizontal distance measurements are necessary to calculate the handrail angle accurately.
19. What tools do I need to measure the heights and distance?
You can use a tape measure or laser distance tool to measure the height and horizontal distance.
20. How often should I calculate the handrail angle for long staircases?
For long staircases, you should calculate the angle at several points along the handrail to ensure consistency.
The Handrail Angle Calculator provides a simple and effective way to calculate the angle for any handrail installation, whether for stairs, ramps, or other inclined surfaces. By using basic
measurements of height and distance, you can ensure that your handrail meets safety standards and is comfortable for users. Whether you’re a professional installer or a DIY enthusiast, this tool
helps you get accurate results with minimal effort. | {"url":"https://calculatordoc.com/handrail-angle-calculator/","timestamp":"2024-11-06T09:27:15Z","content_type":"text/html","content_length":"87340","record_id":"<urn:uuid:d03808a0-9128-4382-8089-7619f5c542b1>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00749.warc.gz"} |
The following question appeared in IIT-JEE 2008 question paper under Reasoning Type Questions.
STATEMENT -1
Two cylinders, one hollow (metal) and the other solid (wood) with the same mass and identical dimensions are simultaneously allowed to roll without slipping down an inclined plane from the same
height. The hollow cylinder will reach the bottom of the inclined plane first.
STATEMENT -2
By the principle of conservation of energy, the total kinetic energies of both the cylinders are identical when they reach the bottom of the incline.
(A) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT -2 is a correct explanation for Statement-1
(B) STATEMENT -1 is True, STATEMENT -2 is True; STATEMENT -2 is NOT a correct explanation for STATEMENT -1
(C) STATEMENT -1 is True, STATEMENT -2 is False
(D) STATEMENT -1 is False, STATEMENT -2 is True
The acceleration (a) of a body of mass M rolling down an inclined plane is given by
a = g sinθ/[1 + (I/MR^2)]
where θ is the angle of the plane (with respect to the horizontal), I is the moment of inertia (about the axis of rolling) and R is the radius of the body.
Since the moments of inertia of solid cylinder and hollow cylinder are respectively MR^2/2 and MR^2 the acceleration a is greater for the solid cylinder. Therefore, the solid cylinder will reach the
bottom of the inclined plane first.
Since the cylinders have the same mass and are at the same height they have the same initial gravitational potential energy Mgh. This potential energy gets converted into translational and rotational
kinetic energies (obeying the law of conservation of energy) when the cylinders roll down the incline and the total kinetic energies of both the cylinders are identical when they reach the bottom of
the incline.
Let us consider an ordinary type of multiple choice question now:
A, B and C are three equal point masses (m each) rigidly connected by massless rods of length L forming an equilateral triangle as shown in the adjoining figure. The system is first rotated with
constant angular velocity ω about an axis perpendicular to the plane of the triangle and passing through A. Next it is rotated with the same constant angular velocity ω about the side AB of the
triangle. If K[1] and K[2] are the kinetic energies of the system in the first and the second cases respectively, the ratio K[1]/K[2] is
(a) 2/3
(b) 4/3
(c) 8/3
(d) 10/3
(e) 16/3
The rotational kinetic energy (K) is given by
K = ½ Iω^2
Since the angular velocity is the same in the two cases, the ratio of kinetic energies must be equal to the ratio of moments of inertia.
Thereore, K[1]/K[2] = I[1]/I[2] = 2mL^2/m(Lsin60º)^2
[Note that in the second case the moment of inertia of the system is due to the single mass at C which is at distance Lsin60º from the axis AB].
Thus K[1]/K[2] = 2/(√3/2)^2 = 8/3.
The following question on negative feed back amplifier is simple even though Plus Two students will normally be unprepared to answer it:
The voltage gain of an amplifier with 9% negative feed back is 10. The voltage gain without feed back will be
(1) 10
(2) 1.25
(3) 100
(4) 90
If the voltage gain without feed back is A[v ]and the feed back factor (fraction of output voltage fed back to the input) is β, the voltage gain (A[fb]) with feed back is given by
A[fb] = A[v]/(1– βA[v])
In the case of negative feed back the sign of the feed back factor β is negative so that the voltage gain with feed back is given by
A[fb] = A[v]/(1+ βA[v])
Since A[fb] = 10 and the magnitude of β is 9% = 0.09, we have on substituting,
10 = A[v]/(1+ 0.09A[v])
This gives A[v] = 100.
Here is another question on feed back amplifiers:
Pick out the wrong statement:
When negative feed back is applied in a transistor amplifier
(1) its voltage gain is decreased
(2) its band width is decreased
(3) distortion produced by the amplifier is decreased
(4) the transistor current gain is unchanged
The second option alone is incorrect. The band width of a negative feed back amplifier will be greater than that of the same amplifier without the feed back. Since there is a reduction in the voltage
gain consequent on the negative feed back, the 3 dB down frequency on the lower side will be shifted towards lower frequency and the 3 dB down frequency on the upper side will be shifted towards
higher frequency.
Application Form and the Information Bulletin in respect of the All India Engineering/Architecture Entrance Examination 2009 (AIEEE 2009) to be conducted on 26-4-2009 are being distributed from
5.12.2008 and will continue till 5.1.2009. Candidates can apply for AIEEE 2009 either on the prescribed Application Form or make application ‘Online’. Visit the site http://aieee.nic.in immediately
for details. Apply for the exam without delay.
You will find many old AIEEE questions (with solution) on this site. You can access all of them by typing ‘AIEEE’ in the search box at the top left of this page and clicking on the adjacent ‘search
blog’ box.
You should be able to find the answers for the following questions without difficulty. For each question you should take about one minute at the most (if you have understood basic principles in this
section thoroughly).
(1) A long straight conductor carries a current that increases at a constant rate. The current induced in the circular conducting loop placed near the straight conductor (fig.) is
(a) zero
(b) constant and in the clockwise direction
(c) constant and in the anticlockwise direction
(d) increasing and in the clockwise direction
(e) increasing and in the anticlockwise direction
A current flows in the circular loop because of the increasing magnetic flux linked with it. This increase in the magnetic flux is to be opposed in accordance with Lenz’s law. The magnetic field
lines produced by the straight conductor are directed normally into the plane of the circular coil (away from the reader). Therefore, the direction of the magnetic field lines produced by the induced
current in the circular coil must be directed normal to the plane of the circular coil and towards the reader. The induced current should therefore flow in the anticlockwise sense.
The magnitude of the induced current is constant since the rate of increase of flux is constant. The correct option is (c).
(2) In the circuit shown a coil of inductance 0.5 H and negligible resistance is connected to a 12 V battery of negligible internal resistance using a resistive network. If I[1] is the current
delivered by the battery immediately after closing the switch and I[2] is the current (delivered by the battery) long time after closing the switch,
(a) I[1] = 0.5 A and I[2] = 0.6 A
(b) I[1] = 0.6 A and I[2] = 0.6 A
(c) I[1] = 0.5 A and I[2] = 0.5 A
(d) I[1] = 0 A and I[2] = 0.5 A
(e) I[1] = 0.6 A and I[2] = 0.5 A
The current through the inductance cannot rise abruptly on closing the switch (because of the time constant L/R of the LR circuit). At the moment of switching the circuit on, the circuit behaves as
though the inductance branch is absent. The initial current is therefore given by
I[1] = 12 V/ (16+8) Ω = 0.5 A
The current through the inductance rises exponentially with time t from zero to the final steady value I [in accordance with the equation I = I[0](1– e^–^Rt/L) with usual notations]. After a long
time, the current through the inductance becomes steady and there is no voltage drop across the inductance. The inductance is in effect short circuited and the final steady current is given by
I[2] = 12 v/(16+4) Ω = 0.6 A
[Note that the parallel combined value of 8 Ω and 8 Ω is 4 Ω]
The correct option is (a).
You will find many useful posts on electromagnetic induction on this site. You can access them by clicking on the label ‘electromagnetic induction’ below this post.
The Joint Entrance (2009) Examination for Admission to IITs and other Institutions (IIT-JEE 2009) will be held on April 12, 2009 (Sunday) as per the following schedule:
09:00 – 12:00 hrs Paper – 1
14:00 – 17:00 hrs Paper – 2
Application materials of the Joint Entrance Examination 2009 will be issued with effect from 19th November, 2008. On-line submission of Application will also commence on the same day.
Application materials can be purchased from designated branches of Banks and all IITs between 19.11.2008 and 24.12.2008 by paying Rs. 500/- in case of SC/ST/PD/Female candidates, Rs.1000/- in case of
male general/OBC candidates and US$ 100 in the case of candidates appearing in Dubai centre.
The last date for postal request of application materials is 16-12-2008.
The last date for receipt of the completed application at the IITs is 24th December, 2008.
Online Submission of Application: This facility will be available between 19.11.2008 and 24.12.2008 between 8:00 hrs and 17:00 hrs. through the JEE websites of the different IITs.
The JEE websites of the different IITs are given below:
IIT Bombay: http://www.jee.iitb.ac.in
IIT Delhi: http://jee.iitd.ac.in
IIT Guwahati: http://www.iitg.ac.in/jee
IIT Kanpur: http://www.iitk.ac.in/jee
IIT Kharagpur: http://www.iitkgp.ernet.in/jee
IIT Madras: http://jee.iitm.ac.in
IIT Roorkee: http://www.iitr.ac.in/jee
Visit one of these web sites (http://jee.iitm.ac.in) for more details.
Occasionally you will find questions requiring the application of Kirchoff’s laws. Here is a question that appeared in AIEEE 2008 question paper which I give here for clearing your doubts regarding
the direction of the current you will have to mark in closed loops in direct current networks:
A 5 V battery with internal resistance 2 Ω and a 2 V battery with internal resistance 1 Ω are connected to a 10 Ω resistor as shown in the figure.
The current in the 10 Ω resistor is
(a) 0.27 A, P[1] to P[2]
(b) 0.27 A, P[2] to P[1]
(c) 0.03 A, P[1] to P[2]
(d) 0.03 A, P[2] to P[1]
The circuit is redrawn, indicating the currents in the three branches. We have marked the directions of the currents I[1] and I[2] in the directions we normally expect the 5 V and the 2 V batteries
to drive their currents. [Note that there can be situations in which the direction we mark is wrong.
There is nothing to be worried about such situations since you will obtain a negative current as the answer and you will understand that the real direction is opposite to what you have marked in the
Applying Kirchoff’s voltage law (loop law) to the loops ABP[2]P[1] and P[1]P[2]CD we have respectively,
5 = 2×I[1] + 10×(I[1 ]– I[2]) and
2 = 1×I[2 ]–10×(I[1 ]– I[2])
The above equations can be rewritten as
12 I[1 ]– 10 I[2] = 5 and
–10 I[1] + 11 I[2] = 2
These equations can be easily solved to give I[1] = 2.34 A (nearly) and I[2] = 2.31 A (nearly) so that the current (I[1 ]– I[2]) = 0.03 A.
Since the currents I[1] and I[2] are obtained as positive, the directions we marked are correct and the current flowing in the 10 Ω resistor is 0.03 A, flowing from P[2] to P[1] [Option (4)].
[Suppose we had marked the current I[2] as flowing in the opposite direction. The current flowing in the 10 Ω resistor will then be (I[1 ]+ I[2]). We will then obtain I[1] = 2.34 A and I[2] = – 2.31
A, the negative sign indicating that the real direction of I[2] is opposite to what we marked. The current flowing through the 10 Ω resistor will again be obtained correctly as (I[1 ]+ I[2]) = 2.34
A– 2.31 A = 0.03 A].
Questions involving the refraction produced by prisms often find place in Medical, Engineering and other Degree Entrance Exam question papers. Here are two questions in this section:
(1) The face AC of a glass prism of angle 30º is silvered. A ray of light incident at an angle of 60º on face AB retraces its path on getting reflected from the silvered face AC. If the face AC is
not silvered, the deviation that can be produced by the prism will be
(a) 0º
(b) 30º
(c) 45º
(d) 60º
(e) 90º
The deviation (d) produced by a prism is given by
d = i[1] + i[2] – A where i[1] and i[2] are respectively the angle of incidence and the angle of emergence and A is the angle of the prism. Here we have i[1] = 60º, i[2] = 0º (since the ray falling
normally will proceed undeviated from the face AC if it is not silvered) and A = 60º.
Therefore, d = 30º.
(2) In the above question, what is the refractive index of the material of the prism?
(a) 1.732
(b) 1.652
(c) 1.667
(d) 1.5
(e) 1.414
In the triangle ADN angle AND is 60º since angle DAN = 30º and angle DNA = 90º. Therefore, the angle of refraction at D is 30º. The refractive index of the material of the lens (n) is given by
n = sin i[1]/ sinr[1] = sin 60º/ sin 30º = √3 = 1.732
(3) Two thin (small angled) prisms are combined to produce dispersion without deviation. One prism has angle 5º and refractive index 1.56. If the other prism has refractive index 1.7, what is its
(a) 3º
(b) 4º
(c) 5º
(d) 6º
(e) 7º
Since the deviation (d) produced by a small angled prism of angle A and refractive index n is given by
d = (n – 1)A, the condition for dispersion without deviation on combining two prisms of angles A[1 ]and A[2] with refractive indices n[1] and n[2] respectively is
(n[1] – 1)A[1] = (n[2] – 1)A[2]
Therefore, 0.56×5 = 0.7×A[2] so that A[2] = 4º
On this site you will find many questions (with solution) on refraction at plane surfaces as well as at curved surfaces. To access all of them type in ‘refraction’ in the search box at the top left
side of this page and click on the adjacent ‘search blog’ box.
Central Board of Secondary Education, Delhi has invited applications in the prescribed form for All India Pre-Medical / Pre-Dental Entrance Examination -2009 as per the following schedule for
admission to 15% of the total seats for Medical/Dental Courses in all Medical/Dental colleges run by the Union of India, State Governments, Municipal or other local authorities in India except in the
States of Andhra Pradesh and Jammu & Kashmir:-
1. Preliminary Examination - 5th April, 2009 (Sunday)
2. Final Examination - 10th May, 2009 (Sunday)
Candidate can apply for the All India Pre-Medical/Pre-Dental Entrance Examination ither offline or online as explained below:
Offline submission of Application Form may be made using the prescribed application form. The Information Bulletin and Application Form costing Rs.600/- (including Rs.100/- as counselling fee) for
General & OBC Category Candidates and Rs.350/- (including Rs.100/- as counselling fee) for SC/ST Category Candidates inclusive of counseling fee can be obtained against cash payment from 22-10-2008
to 01-12-2008 from any of the branches of Canara Bank/ Regional Offices of the CBSE. Find details at http://www.aipmt.nic.in/.
Online submission of application may be made by accessing the Board’s website http://www.aipmt.nic.in/. from 22-10-2008 (10.00 A.M.) to 01-12-2008 (5.00 P.M.). Candidates are required to take a print
of the Online Application after successful submission of data. The print out of the computer generated application, complete in all respect as applicable for Offline submission should be sent to the
Deputy Secretary (AIPMT), Central Board of Secondary Education, Shiksha Kendra, 2, Community Centre, Preet Vihar, Delhi-110 301 by Speed Post/Registered Post only. Fee of Rs.600/-(including Rs.100/-
as counseling fee) for General and OBC Category Candidates and Rs.350/- (including Rs.100/- as counseling fee) for SC/ST category candidates may be remitted in the following ways :
1. By credit card or
2. Through Demand Draft in favour of the Secretary, Central Board of Secondary Education, Delhi drawn on any Nationalized Bank payable at Delhi. Instructions for Online submission of Application Form
is available on the website http://www.aipmt.nic.in/. Application Form along with original Demand Draft should reach the Board on or before 04-12-2008 which is the last date stipulated.
Visit the web site http://www.aipmt.nic.in/. for all details and information updates.
The following simple questions may prompt you to pick out the wrong option if you are in a hurry. So, be cautious and don’t overlook basic points. Here are the questions:
(1) Two identical frictionless pulleys carry the same mass 2m at the left ends of the light inextensible strings passing over them. The right end of the string carries a mass 3m in the case of
arrangement (i) where as a force of 3mg is applied in the case of arrangement (ii) as shown in the adjoining figure. The ratio of the acceleration of mass 2m in case (i) to the acceleration of mass 2
m in case (ii) is
(a) 2:3
(b) 1:1
(c) 5:3
(d) 3:5
(e) 2:5
In both cases the net driving force is 3mg – 2mg = mg. But in case (i) the total mass moved is 5m where as in case (ii) the total mass moved is 2m. The acceleration in case (i) is mg/5m = g/5 where
as the acceleration in case (ii) is mg/2m = g/2.
The ratio of accelerations = (g/5)/ (g/2) = 2/5 [Option (e)].
(2) An object of mass 4 kg moving along a horizontal surface with an initial velocity of 2 ms^–1 comes to rest after 4 seconds. If you want to keep it moving with the velocity of 2 ms^–1, the force
required is
(a) zero
(b) 1 N
(c) 2 N
(d) 4 N
(e) 8 N
The acceleration ‘a’ of the body is given by
0 = 2 + a×4, on using the equation, v[t] = v[0] + at
Therefore, a = – 0.5 ms^–2[]
The body is retarded (as indicated by the negative sign) because of forces opposing the motion. The opposing force has magnitude ma = 4×0.5 = 2 N. Therefore, a force of 2 N has to be applied opposite
to the opposing forces to keep the body moving with the velocity of 2 ms^–1.
You will find some useful multiple choice questions (with solution) in this section here as well as here
Today we will discuss two questions on Young’s double slit. The following multiple correct answers type question appeared in IIT-JEE 2008 question paper:
In a Young’s double slit experiment, the separation between the two slits is d and the wave length of the light is λ. The intensity of light falling on slit 1 is four times the intensity of light
falling on slit 2. Choose the correct choice(s).
(A) If d = λ, the screen will contain only one maximum.
(B) If λ < d < 2λ, at least one more maximum (besides the central maximum) will be observed on the screen.
(C) If the intensity of light falling on slit 1 is reduced so that it becomes equal to that of slit 2, the intensities of the observed dark and bright fringes will increase.
(D) If the intensity of light falling on slit 2 is reduced so that it becomes equal to that of slit 1, the intensities of the observed dark and bright fringes will increase
The path difference [S[2]N = (S[2]P – S[1]P) in fig.] between the interfering beams is d sinθ. Therefore for obtaining a maximum at the point P on the screen the condition to be satisfied is
dsinθ = nλ
When d = λ, sinθ = n
Since sinθ ≤ 1, n ≤ 1
The possible values of n are 0 and 1. But the maximum corresponding to n = 1 cannot be observed since θ cannot be 90º. Therefore, the central maximum of order zero corresponding to zero path
difference (n = 0) alone can be observed. Option A is therefore correct
If λ < d < 2λ, the limiting values of d are λ and 2λ. In addition to the usual central maximum, fringes of order up to n satisfying the equation, dsinθ = nλ will be obtained, where d < 2λ.
If d = 2λ, we have 2λ = nλ/sinθ so that n = 2 sinθ.
Since sinθ should be less than 1 for observing the fringe, the maximum value of n must be 1. Therefore, the central maximum and the maximum of order n = 1 will be observed. Option B too is therefore
Options C and D are obviously incorrect since the intensity of the dark fringes will become zero when the light beams passing through the slits are of equal intensity.
Now consider another question on Young’s double slit:
Suppose the wave length λ and the double slit separation d in a Young’s double slit experiment are such that the 6^th dark fringe is obtained at point P shown in the above figure. The path difference
(S[2]P – S[1]P) will be
(a) 5 λ
(b) 5 λ/2
(c) 6 λ
(d) 3 λ
(e) 11 λ/2
The dark fringe of order 1 (1^st dark fringe) is formed when the path difference is λ/2. The dark fringe of order 2 is formed when the path difference is 3λ/2 (and not 2λ/2) and the dark fringe of
order 3 is formed when the path difference is 5λ/2 (and not 3λ/2). Generally, the dark fringe of order n is formed when the path difference is (2n – 1)λ/2. The 6^th dark fringe is therefore formed
when the path difference is 11λ/2.
By clicking on the label ‘wave optics’ below this post, you can access all related posts on this site.
You will find useful posts on wave optics here at apphysicsresources. | {"url":"http://www.physicsplus.in/2008/","timestamp":"2024-11-13T16:21:25Z","content_type":"application/xhtml+xml","content_length":"217205","record_id":"<urn:uuid:4dda3664-bce4-4927-aeb3-6f640c5be54c>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00176.warc.gz"} |
1860 - 1870 - Chronology
• Delaunay publishes the first volume of La Théorie du mouvement de la lune which is the result of 20 years work. Delaunay solves the three-body problem by giving the longitude, latitude and
parallax of the Moon as infinite series.
• Weierstrass discovers a continuous curve that is not differentiable any point.
• Maxwell proposes that light is an electromagnetic phenomenon.
• Jevons reads General Mathematical Theory of Political Economy to the British Association.
• Listing publishes Der Census raumlicher Complexe oder Verallgemeinerung des Euler'schen Satzes von den Polyedern which discusses extensions of "Euler's formula".
• Weierstrass gives a proof in his lecture course that the complex numbers are the only commutative algebraic extension of the real numbers.
• Bertrand publishes Treatise on Differential and Integral Calculus.
• London Mathematical Society founded. (See THIS LINK.)
• Benjamin Peirce presents his work on Linear Associative Algebras to the American Academy. It classifies all complex associative algebras of dimension less than seven using the, now familiar,
tools of idempotent and nilpotent elements.
• Plücker makes further advances in geometry when he defines a four dimensional space in which straight lines rather than points are the basic elements.
• Hamilton's Elements of Quaternions is unfinished on his death but the 800 page work which took seven years to write is published posthumously by his son.
• Moscow Mathematical Society is founded.
• Lueroth discovers the "Lueroth quartic". | {"url":"https://mathshistory.st-andrews.ac.uk/Chronology/24/","timestamp":"2024-11-14T11:05:29Z","content_type":"text/html","content_length":"13082","record_id":"<urn:uuid:fa44e92a-1fa9-42c2-83b9-a8d4c52e68bc>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00030.warc.gz"} |
Building a roof for a poor family in Ridiyagama
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How Does Linear And Logistic Regression Work In Machine Learning? | Analytics Steps
Linear regression and logistic regression both are machine learning algorithms that are part of supervised learning models. Since both are part of a supervised model so they make use of labeled data
for making predictions.
Linear regression is used for regression or to predict continuous values whereas logistic regression can be used both in classification and regression problems but it is widely used as a
classification algorithm. Regression models aim to project value based on independent features.
The main difference that makes both different from each other is when the dependent variables are binary logistic regression is considered and when dependent variables are continuous then linear
regression is used.
Linear Regression
Every person must have come across linear models when they were at school. Mathematics taught us about linear models. It is the same model that is used widely in predictive analysis now. It majorly
tells about the relationship between a target that is a dependent variable and predictors using a straight line. Linear regression is basically of two types that are Simple Linear Regression and
Multiple Linear Regression.
Experience on X-axis & Salary on Y-axis
In the above plot, Salary is the dependent variable that is on (Y-axis) and the independent variable is on X-axis that is Experience. More experience means more salary. The regression line can be
written as:
Y1 = a0 + a1X + ε
Where coefficients are a0 and a1 and the error term is ε.
Linear regression can have independent variables that are continuous or may be discrete in nature but have continuous dependent variables. The best fit line in linear regression is calculated using
mean squared error that finds out the relationship between dependent that is Y and independent that is X. There is always a linear relationship that is present between both the two.
Linear regression only has one independent variable whereas in multiple regression there can be more than one independent variable.
Let us go through a regression problem. We will use the Boston dataset from Scikit-learn, this dataset holds information about the house value of different houses in Boston. Other variables that are
present in the dataset are Crime, areas of non-retail business in the town (INDUS), and other variables.
Step 1:
In the first step, we are going to import all the important libraries and most importantly, we have to import the dataset from sklearn.datasets.
from sklearn.linear_model import LinearRegression
from sklearn.datasets import load_boston
from sklearn.model_selection import train_test_split
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
boston = load_boston()
boston.data.shape, boston.target.shape
The shape of the dataset
Step 2
We are going to visualize the dataset with the help of a python library called pandas, we will name features of the dataset and afterward, we are going to create a data frame with the help of the
pandas' library.
Feature names for the dataset
bos = pd.DataFrame(boston.data)
bos.columns = boston.feature_names
Visualization of the dataset using the pandas library
Step 3
In this step, we are going to split the dataset into training and test sets.
X_train, X_test, y_train, y_test = train_test_split(boston.data, boston.target, test_size=0.2)
print(X_train.shape, X_test.shape, y_train.shape, y_test.shape)
The shape of the Training and Test set after Splitting
Step 4
Now, we are going to fit our dataset to another machine learning library from sklearn to implement linear regression.
sklinreg = LinearRegression(normalize=True)
sklinreg.fit(X_train, y_train)
Step 5
In the last step, we are printing test results on our dataset.
print("Train:", sklinreg.score(X_train, y_train))
print("Test:", sklinreg.score(X_test, y_test))
Training and testing accuracy
Logistic Regression
It is an algorithm that can be used for regression as well as classification tasks but it is widely used for classification tasks. The response variable that is binary belongs either to one of the
classes. It is used to predict categorical variables with the help of dependent variables.
Consider there are two classes and a new data point is to be checked which class it would belong to. Then algorithms compute probability values that range from 0 and 1.
For example, whether it will rain today or not. In logistic regression weighted sum of input is passed through the sigmoid activation function and the curve which is obtained is called the sigmoid
The figure shows a graph of Sigmoid Function
The logistic function that is a sigmoid function is an ‘S’ shaped curve that takes any real values and converts them between 0 to 1. If the output given by a sigmoid function is more than 0.5, the
output is classified as 1 & if is less than 0.5, the output is classified as 0. If the graph goes to a negative end then y predicted will be 0 and vice versa.
If we obtain the output of sigmoid to be 0.75 then it tells us that there are 75% chances of that happening, maybe a toss coin.
Binary Classification
The above figure shows inputs and the probabilities that the outcome is between two categories of a binary dependent variable based on one or more independent variables that can be continuous as well
as categorical.
Like the way, we implemented Linear Regression with the help of sklearn, Now, we shall implement Logistic Regression
Step 1
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
from sklearn.linear_model import LogisticRegression as SKLR
Step 2
1. Creating dataset with 1000 rows and 2 columns
2. Plotting the dataset with the help of the matplotlib library.
mean_01 = [0,0]
cov_01 = [[2,0.2], [0.2,1]]
mean_02 = [3,1]
cov_02 = [[1.5,-0.2], [-0.2,2]]
dist_01 = np.random.multivariate_normal(mean_01, cov_01, 500)
dist_02 = np.random.multivariate_normal(mean_02, cov_02, 500)
print(dist_01.shape, dist_02.shape)
Printed the shape of distributions
plt.scatter(dist_01[:,0], dist_01[:,1], color='red')
plt.scatter(dist_02[:,0], dist_02[:,1], color='green')
Visualizing Dataset using Matplotlib
Step 3
dataset = np.zeros((dist_01.shape[0] + dist_02.shape[0], dist_01.shape[1] + 1))
dataset[:dist_01.shape[0], :-1] = dist_01
dataset[dist_01.shape[0]:, :-1] = dist_02
# Red = 0, Green = 1
dataset[dist_02.shape[0]:, -1] = 1
The shape of the dataset
1. Adding both distributions, here, ‘1’ is added because of label column
2. Distributed first 500 data points to the first distribution
3. Distributed later 500 data points to the second distribution
4. Made a separate column for labels.
Step 4
from sklearn.model_selection import train_test_split
X_train, X_test, y_train, y_test = train_test_split(dataset[:,:-1], dataset[:,-1], test_size=0.2)
X_train.shape, X_test.shape, y_train.shape, y_test.shape
The shape of Training and Testing Set after Splitting
1. Shuffled the dataset so that both distributions get mixed up properly so that they act as real-world problem dataset
2. Split dataset for the training set and test set
3. The visualized shape of the dataset
Step 5
In this step, we will fit our dataset to logistic regression with the help of sklearn.
sk_logreg = SKLR()
sklinreg.fit(X_train, y_train)
Fitted Logistic Regression to our Dataset
sk_logreg.score(X_test, y_test)
Output: 0.905
We have calculated our score for the test set and got a good accuracy of 90%.
Differences Between Linear And Logistic Regression
• Linear regression is used for predicting the continuous dependent variable using a given set of independent features whereas Logistic Regression is used to predict the categorical.
• Linear regression is used to solve regression problems whereas logistic regression is used to solve classification problems.
• In Linear regression, the approach is to find the best fit line to predict the output whereas in the Logistic regression approach is to try for S curved graphs that classify between the two
classes that are 0 and 1.
• The method for accuracy in linear regression is the least square estimation whereas for logistic regression it is maximum likelihood estimation.
• In Linear regression, the output should be continuous like price & age, whereas in Logistic regression the output must be categorical like either Yes / No or 0/1.
• There should be a linear relationship between the dependent and independent features in the case of Linear regression whereas it is not in the case of Logistic regression.
• There can be collinearity between independent features in the case of linear regression but it is not in the case of logistic regression.
In this blog, I have tried to give you a brief idea about how linear and logistic regression is different from each other with a hands-on problem statement. I have discussed the linear model, how
sigmoid functions work, and how classification in logistic regression is made between 0 and 1. How prediction is made for continuous values. I have taken two problem statements where I have worked on
classification as well as a regression problem. And lastly, I have discussed the differences between both the algorithms.
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NysADMM: faster composite convex optimization via low-rank approximation
NysADMM: faster composite convex optimization via low-rank approximation
Proceedings of the 39th International Conference on Machine Learning, PMLR 162:26824-26840, 2022.
This paper develops a scalable new algorithm, called NysADMM, to minimize a smooth convex loss function with a convex regularizer. NysADMM accelerates the inexact Alternating Direction Method of
Multipliers (ADMM) by constructing a preconditioner for the ADMM subproblem from a randomized low-rank Nyström approximation. NysADMM comes with strong theoretical guarantees: it solves the ADMM
subproblem in a constant number of iterations when the rank of the Nyström approximation is the effective dimension of the subproblem regularized Gram matrix. In practice, ranks much smaller than the
effective dimension can succeed, so NysADMM uses an adaptive strategy to choose the rank that enjoys analogous guarantees. Numerical experiments on real-world datasets demonstrate that NysADMM can
solve important applications, such as the lasso, logistic regression, and support vector machines, in half the time (or less) required by standard solvers. The breadth of problems on which NysADMM
beats standard solvers is a surprise: it suggests that ADMM is a dominant paradigm for numerical optimization across a wide range of statistical learning problems that are usually solved with bespoke
Cite this Paper
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What’s the Latest in Quantum Computing for Solving Complex Mathematical Problems? - Bellevue popcorn ceiling removal
The world of technology is advancing rapidly and with it brings new possibilities. One such realm of technological progress is quantum computing. Just as it sounds, it’s a fusion of quantum mechanics
and computer science, promising to revolutionize the way we process and manipulate data. Quantum computers, still in the experimental stage, have the potential to solve complex mathematical problems
faster than classical computers. They do this by leveraging the strange properties of quantum physics to process information in novel ways.
Understanding Quantum Computing
Before we dive into how quantum computing is being employed to tackle intricate mathematical problems, let’s first demystify what the concept of quantum computing actually is.
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Quantum computing deviates from the classical approach to computing. Instead of using the traditional binary system of ones and zeros, quantum computers use quantum bits, or qubits, to handle data. A
qubit can be both one and zero at the same time, thanks to a property of quantum mechanics known as superposition. This feature alone gives quantum computers a massive edge over their classical
counterparts because it allows them to perform many calculations simultaneously.
Another vital property of qubits is entanglement, which links the state of one qubit to that of another, irrespective of the distance separating them. This is yet another counter-intuitive
characteristic of quantum physics that provides quantum systems with their extraordinary computational power.
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Quantum Computing’s Role in Solving Mathematical Problems
Now that you grasp the basics of quantum computing, let’s explore how it’s being utilized to resolve complex mathematical problems.
First and foremost, it’s essential to understand that while classical computers aren’t inherently inefficient at solving mathematical problems, they are limited by time. For complex equations and
algorithms, classical systems may require an infeasible amount of time to arrive at a solution. Quantum computers, on the other hand, have the potential to perform these calculations exponentially
Quantum computers use a unique quantum algorithm, such as Shor’s algorithm or Grover’s algorithm, which are explicitly designed to take advantage of the properties of a quantum system. Shor’s
algorithm, for instance, is renowned for its ability to factor large prime numbers more efficiently than any classical algorithm — a problem that classical computers struggle with due to the sheer
number of calculations required.
Advancements in Quantum Computing
As of April 22, 2024, quantum technology has seen numerous advancements, with research and development in the sector moving at a breakneck speed. These advances are paving the way for solving
mathematical problems that were previously thought unsolvable.
Google, for instance, announced ‘Quantum Supremacy’ in 2019, which referred to the milestone of a quantum computer solving a problem that a classical computer cannot solve in a reasonable amount of
time. This was a seminal moment in the field, demonstrating for the first time the potential power of quantum computing.
Moreover, quantum algorithms are continually being developed and refined. These algorithms, like the Variational Quantum Eigensolver (VQE) and Quantum Approximate Optimization Algorithm (QAOA), are
designed to run on today’s noisy intermediate-scale quantum (NISQ) computers and are providing more efficient ways of solving complex problems.
Challenges and Solutions in Quantum Computing
Despite the promising advances, quantum computing isn’t without its challenges. The fragile state of qubits, the difficulty in scaling up quantum systems, and the lack of a broad range of efficient
quantum algorithms are all issues that need to be overcome.
However, solutions are being proposed and tested. Error correction techniques are being developed to maintain the delicate state of qubits. There’s ongoing research into different types of qubits
that might be easier to control and less prone to errors, such as topological qubits.
Meanwhile, in the realm of algorithms, there’s a push to develop more quantum versions of classical algorithms, as well as entirely new quantum algorithms that could open up new possibilities for
what can be computed.
The Future of Quantum Computing
Looking ahead, the field of quantum computing is expected to continue its rapid pace of advancement, unlocking new capabilities for solving complex mathematical problems.
Quantum computers will likely become an indispensable tool in areas such as cryptography, modelling of complex systems, and optimization problems. With faster and more accurate computations,
industries such as pharmaceuticals, finance, and logistics stand to benefit tremendously.
Despite the challenges, the potential of quantum computing is immense. It holds the promise of revolutionizing technology and reshaping our understanding of computation. As we continue to march
towards this quantum future, one thing is clear: the world of computing will never be the same again.
Quantum Computing and Machine Learning
Machine learning, a subset of artificial intelligence, involves algorithms that improve automatically through experience. It is extensively applied in web search engines, email filtering, detection
of network intruders, and many other applications. However, classical computers sometimes struggle to handle the enormous amount of data involved in machine learning processes. That’s where quantum
computers come into play.
Quantum computers can potentially accelerate machine learning tasks, thanks to their intrinsic ability to simultaneously explore a vast number of possibilities. The quantum state of a qubit can hold
a vast amount of information, making it immensely useful for tasks such as pattern recognition and decision-making processes involved in machine learning.
Moreover, quantum algorithms like the quantum support vector machine (QSVM) and quantum principal component analysis (QPCA) have been specifically developed for quantum machine learning. These
algorithms use the principles of quantum mechanics to speed up the machine learning process.
Quantum machine learning can lead to advancements in various fields, including bioinformatics, where it could help in understanding genomic data and disease prediction. It also has the potential to
play a crucial role in quantum cryptography, offering robust security measures against cyber threats. However, similar to other areas of quantum computing, it also faces challenges such as error
correction and qubit stability that need to be addressed.
Quantum Computing and Optimization Problems
Quantum computing has shown a lot of promise in solving optimization problems – situations where the best possible solution needs to be chosen from a plethora of options. Classic examples include
scheduling flights for airlines, determining the shortest delivery route for logistics companies, or optimizing investment portfolios for financial institutions. These problems can be computationally
intensive and time-consuming on classical computers, especially as the number of variables increases.
Quantum computers, however, can process and analyze all possibilities at once, thanks to the properties of superposition and entanglement. They are particularly well-suited to solving such complex
problems because quantum mechanics allows them to search through the solution space more efficiently.
One of the most well-known quantum algorithms used for optimization problems is the Quantum Approximate Optimization Algorithm (QAOA). QAOA uses the principles of quantum physics to find the
near-optimal solution to optimization problems more quickly than classical algorithms.
Quantum annealing, a quantum computing technique used to find the global minimum for a given function, is another method being employed to solve optimization problems. Companies like D-Wave Systems
are pioneers in quantum annealing and have made their quantum computers available for businesses to solve optimization problems.
Conclusion: The Quantum Leap
As we delve deeper into the nuances of quantum computing, we are standing on the brink of a new era in computation. The capability of quantum computers to perform tasks exponentially faster than
classical computers opens a world of opportunities not only in solving complex mathematical problems but also in fields like cryptography, machine learning, and optimization.
However, the path to widespread quantum computing is not without its challenges. Aspects like error correction, qubit stability, and the creation of efficient quantum algorithms are areas of active
research. Significant strides are being made in these areas, bringing us closer to the day when quantum computers move from the realm of experimental technology to practical application.
The potential of quantum computing is immense, and as researchers and technologists, we must continue to explore this fascinating merger of quantum mechanics and computing. With the ever-growing
interest and investment in this field, it’s clear that quantum computing is set to redefine our understanding of computation. The quantum future is, indeed, promising. In this ever-evolving world of
technology, the day is not far when quantum computers will become a part of our everyday life. | {"url":"https://bellevuepopcornceilingremoval.com/whats-the-latest-in-quantum-computing-for-solving-complex-mathematical-problems/","timestamp":"2024-11-14T13:14:40Z","content_type":"text/html","content_length":"79947","record_id":"<urn:uuid:e47e7eb6-0477-46c2-a64f-8ff88c4a0fee>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00631.warc.gz"} |
Education and career
Cockburn is originally from Ottawa. She earned a bachelor's and master's degree from Queen's University in Ontario, in 1982 and 1984 respectively,^[4] and also has a master's degree from the
University of Ottawa.^[3] She completed her Ph.D. in algebraic topology in 1991 from Yale University. Her dissertation, The Gamma-Filtration on Extra-Special P-Groups, was supervised by Ronnie Lee.^[
She joined the Hamilton College faculty in 1991, and was promoted to full professor in 2014.^[2] At Hamilton, she has also served as the coach for the college's squash team.^[4] | {"url":"https://www.knowpia.com/knowpedia/Sally_Cockburn","timestamp":"2024-11-08T08:08:47Z","content_type":"text/html","content_length":"76930","record_id":"<urn:uuid:eef066ee-cda4-47ee-a08b-19089a9ec584>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00580.warc.gz"} |
Chow Test for Structural Stability
A series of data can often contain a structural break, due to a change in policy or sudden shock to the economy, i.e. 1987 stock market crash. In order to test for a structural break, we often use
the Chow test, this is Chow’ first test (the second test relates to predictions). The model in effect uses an F-test to determine whether a single regression is more efficient than two separate
regressions involving splitting the data into two sub-samples. This could occur as follows, where in the second case we have a structural break at t.:
Case 1Case2
In the first case we have just a single regression line to fit the data points (scatterplot), it can be expressed as:
In the second case, where there is a structural break, we have two separate models, expressed as:
This suggests that model 1 applies before the break at time t, then model 2 applies after the structural break. If the parameters in the above models are the same, i.e. , then models 1 and 2 can be
expressed as a single model as in case 1, where there is a single regression line. The Chow test basically tests whether the single regression line or the two separate regression lines fit the data
best. The stages in running the Chow test are:
1)Firstly run the regression using all the data, before and after the structural break, collect RSSc.
2)Run two separate regressions on the data before and after the structural break, collecting the RSS in both cases, giving RSS1 and RSS2.
3)Using these three values, calculate the test statistic from the following formula:
4)Find the critical values in the F-test tables, in this case it has F(k,n-2k) degrees of freedom.
5)Conclude, the null hypothesis is that there is no structural break.
This occurs when there is an approximate linear relationship between the explanatory variables, which could lead to unreliable regression estimates, although the OLS estimates are still BLUE. In
general it leads to the standard errors of the parameters being too large, therefore the t-statistics tend to be insignificant. The explanatory variables are always related to an extent and in most
cases it is not a problem, only when the relationship becomes too big. One problem is that it is difficult to detect and decide that it is a problem. The main ways of detecting it are:
-The regression has a high R2 statistic, but few if any of the t-statistics on the explanatory variables are significant.
-Use of the simple correlation coefficient between the two explanatory variables in question can be used, although the cut-off between acceptable and unacceptable correlation can be a problem.
If multicollinearity does appear to be a problem, then there are a number of ways of remedying it. The obvious solution is to drop one of the variables suffering from multicollinearity, however if
this is an important variable for the model being tested, this might not be an option. Other ways of overcoming this problem could be:
-Finding additional data, an alternative sample of data might not produce any evidence of multicollinearity. Also by increasing the sample size, this can reduce the standard errors of the explanatory
variables, also helping to overcome the problem.
-Use an alternative technique to the standard OLS technique. (We will come across some of these later).
-Transform the variables, for instance taking logarithms of the variables or differencing them (i.e. dy=y-y(-1)). | {"url":"https://docsbay.net/doc/70704/chow-test-for-structural-stability","timestamp":"2024-11-04T17:46:05Z","content_type":"text/html","content_length":"15542","record_id":"<urn:uuid:401c0285-32bf-411f-be3c-a67d2e74142d>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00817.warc.gz"} |
Graph A Line Using X- And Y-Intercept Worksheets [PDF] (8.F.A.3): 8th Grade Math
Teaching Graph a line using X and Y-intercept easily
1. Find the x− and y-intercepts of the line.
Here, if you let y = 0 and solve for x or let x = 0 and solve for y.
2. Find a third solution to the equation.
3. Plot the three points and then check that they line up.
4. Draw the line.
Why Should you use Graph a line using X and Y-intercept worksheets for your students?
• By solving this worksheet, students will develop a strong understanding of line equations, slopes, intercepts, and many other concepts.
• The worksheet will also enhance their knowledge of using this method to plot line on a graph.
Why Should You Solve for Graph a line using X and Y-intercept Worksheet for Your Students?
Download this graph a line using X and Y-intercept worksheets PDF for your students. You can also try our Graph A Line Using X- And Y-Intercept Problems and Graph A Line Using X- And Y-Intercept Quiz
as well for a better understanding of the concepts. | {"url":"https://www.bytelearn.com/math-grade-8/worksheet/graph-a-line-using-x-and-y-intercept","timestamp":"2024-11-12T09:05:33Z","content_type":"text/html","content_length":"254145","record_id":"<urn:uuid:74312351-2584-4d69-8c1c-39d65de9b5e5>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00600.warc.gz"} |
Diameter of the base of the cone is $10.5cm$ and its slant height is $10cm$. Find its curved surface area.
Hint: Use the formula of curved surface area of cone & substitute the dimensions.
We know that the formula of curved surface area of the cone is $\pi rl$ where $r$ is radius of base of the cone and $l$ is slant height. In the given question we have given the diameter and we know
the relation between radius and diameter which is $r = \dfrac{d}{2}$. On putting the values $r = \dfrac{d}{2} = \dfrac{{10.5}}{2} = 5.25cm$.Now we have got the radius so let’s put all the values in
the formula
\pi rl \\
= 3.14 \times 5.25 \times 10 \\
= 3.14 \times 52.5 \\
= 164.85 \\
\end{gathered} $
Hence the required curved surface area is $164.85c{m^2}$.
Option B is correct.
Note: In mensuration formula is the key. If you know the formula and how to use it then you are done with the question. In the above question that’s what we have done. Know the formula and substitute
in it correctly. | {"url":"https://www.vedantu.com/question-answer/diameter-of-the-base-of-the-cone-is-105cm-and-class-10-maths-cbse-5edbc9fc7e65c55350f2c910","timestamp":"2024-11-03T00:32:48Z","content_type":"text/html","content_length":"162195","record_id":"<urn:uuid:3d22ceda-87b6-48b0-b111-c31772985ca2>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00765.warc.gz"} |
In this post we provide the definition of finite groups and we show a few examples meaningful for cryptography and blockchain Def. Finite Groups Def. A group is a set together with an operation
combining two elements of , which satisfies the following properties: Closeness: The group operation is closed: Read more…
This post is a collection of basic topics needed for a proper understanding of Cryptography. Cryptographic Systems Primal goal: allow two people to exchange confidential information even if they can
only communicate via a channel monitored by an adversary Symmetric Cryptosystem A 5-tuple Where are sets of possible keys, messages, Read more…
Binary Search Tree (BST)
A Binary Search Tree (BST) is a Binary Tree data structure with nodes ordered according to the following property: For each node with a key K, all the left children have values lesser than K, and the
right children greater than K. BSTs allow fast lookup, addition, and removal of Read more…
Binary Search
Binary search is an algorithm to find the position of a target value within a sorted array by recursively halving the number of candidates. Assuming we have a sorted array with n elements, we can
binary search a target element in logarithmic time, O(log(n)). Binary Search Algorithm To perform a Read more…
Mergesort Mergesort is an efficient and widely used sorting algorithm. Stable sorting algorithm: preserves the order optimal time complexity: N log N in all cases suboptimal in space usage: usage of
an extra array for merging Usage examples: Sorting Linked Lists when space is not a concern. Java default algorithm Read more…
Java Array Exercises
This article contains a series of exercises on java arrays, including resizing array implementation, merging sorted arrays, etc.. Resizing-array implementation A Java ArrayList is a resizable-array
implementation of the List interface providing amortized-time for insertion of O(1). Exercise: implement a ResizingArray class providing O(1) amortized-time for insertion. Growing the array Read
Sorting, searching, and binary search Divide-and-conquer Dynamic programming and memorization Greedy algorithms Recursion Graph traversal, BFS and DFS Sorting Main algorithms presented Mergesort:
Java sort for Objects Quicksort: Java sort for primitive types Heapsort Mergesort vs Quicksort vs Heapsort All these algorithms have O(N log N) average time complexity, though Read more…
Data Structures
Good knowledge of common data structures is needed in computer science. This article describes common data structures, and present cost and benefits. Lists A List is an abstract data type
representing a finite collection of ordered elements. Typically, a List is allocated dynamically depending on the implementation. For example, Array-based Read more…
About Algorithms
Knowledge of algorithms and data structures is important for Software Engineers. To be good at solving problems you need practice. Here is some resource to make practice, and suggestions. Posts about
Algorithms Below, you see the list of my posts about algorithms [catlist name=”algorithms”] Sites Sites to practice and learn Read more… | {"url":"https://asciiware.com/category/algorithms/","timestamp":"2024-11-10T02:45:24Z","content_type":"text/html","content_length":"73004","record_id":"<urn:uuid:58885a09-6e43-4c50-b7e4-6a67c07ef6b9>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00628.warc.gz"} |
December 2023 - Discrete Mathematics Group
Shengtong Zhang (张盛桐) gave a talk on the minimum number of triangles in a $K_t$-Ramsey graph at the Discrete Math Seminar
On December 19, 2023, Shengtong Zhang (张盛桐) from Stanford University gave a talk at the Discrete Math Seminar on the minimum number of triangles in a $K_t$-Ramsey graph. The title of his talk was
“Triangle Ramsey numbers of complete graphs“.
Ting-Wei Chao (趙庭偉) gave a talk on the number of points that are intersections of d linearly independent lines among given n lines in the d-dimensional space at the Discrete Math Seminar
On December 12, 2023, Ting-Wei Chao (趙庭偉) from Carnegie Mellon University gave a talk at the Discrete Math Seminar on the number of points that are intersections of d linearly independent lines
among given n lines in the d-dimensional space. The title of his talk was “Tight Bound on Joints Problem and Partial Shadow Problem“.
Ben Lund gave a talk on the existence of an embedding of every almost spanning tree with specified distances of edges into a finite vector space at the Discrete Math Seminar
On December 4, 2023, Ben Lund from the IBS Discrete Mathematics Group gave a talk at the Discrete Math Seminar on the existence of an embedding of every almost spanning tree with specified distances
of edges into a finite vector space. The title of his talk was “Almost spanning distance trees in subsets of finite vector spaces.” | {"url":"https://dimag.ibs.re.kr/2023/12/","timestamp":"2024-11-07T17:18:41Z","content_type":"text/html","content_length":"141306","record_id":"<urn:uuid:f4c333c2-ca1e-47a2-9063-d38184c17df6>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00639.warc.gz"} |
How is the TABE Test Scored?
TABE stands for Test of Adult Basic Education and covers various fields in mathematics, reading, and language. To better prepare for the TABE exam, you need to know how the test is scored and what
role the results play in determining your level of education.
Although TABE covers some of the same topics as other standardized tests, it relies on a set of grading principles that are completely different from what you are used to. For example, consider the
fact that there is no passing or failing score on TABE. Your goal as a test-taker is to answer as many questions correctly as you can. You will not lose points by leaving blank questions.
The number of correct answers is added based on the difficulty level of your particular TABE test and is measured based on the average performance of the selected test group. There are five
difficulty levels for TABE: Literacy, Easy, Medium, Difficult and Advanced. Before taking the test, you will be given a short locator test to determine which level is right for you.
The Absolute Best Book to Ace the TABE Math Test
Original price was: $24.99.Current price is: $14.99.
Understanding scores
TABE test results include several different scores for each part of the test. Below are breakdowns of scores commonly used by college administrators and instructors.
1- Raw score:
The raw score indicates the number of correct answers you have collected in the test. The number correct, or NC, is used to generate the scale score.
2- Scale score:
The scale score is what is used to compare your performance with the norm. In other words, your score shows how well you performed compared to the average test-takers at the same level and content
area. You will notice that the scale score is between 0 and 999. These scores can help you target your training goals for the future because you can see skill areas in which you are stronger or
weaker, respectively. Because they can be compared across all TABS subject areas and levels, scale scores are also useful for tracking your progress.
3- Grade equivalent score:
The grade equivalent (GE) is a score that is often not interpreted for TABE. The score’s format mirrors the typical structure seen in K-12 education, with the numbers 0-12.9 representing a particular
school month and year. These scores can indicate the level at which you have done in a given subject area.
However, these scores cannot use to compare performance with different test levels. For example, if you score 7.3 on the medium level test for math, you may get a lower GE score on the difficult
level test, because the test is for students in the range of 6.0-8.9 GE, while the medium level test is for students in the 4.0-5.9 GE range.
What are the scores used for?
The resulting scale and grade equivalent scores are used to place you in adult introductory training classes tailored to your level of ability and/or determine your readiness to enter a career
training program. Some schools also require students to take the TABE a second time and earn acceptable grades to complete a program.
Each school has its minimum grade point average, so make sure you know what your university is looking for.
The Best Book to Ace the TABE Test
Original price was: $19.99.Current price is: $14.99.
More from Effortless Math for TABE Test …
Looking for free resources to practice more for the TABE math test?
You can try our free resources: FREE TABE Math Practice Test and TABE Math FREE Sample Practice Questions
Are you planning to take the TABE math test but do not know where to start?
How to Prepare for the TABE Math Test is a comprehensive guide to help you shine in the TABE Math test.
Is TABE math test day decisive for you?
We know this too, so here are some tips to help you succeed on this day: TABE Math-Test Day Tips
The Perfect Prep Books for the TABE Math Test
Have any questions about the TABE Test?
Write your questions about the TABE or any other topics below and we’ll reply!
Related to This Article
What people say about "How is the TABE Test Scored? - Effortless Math: We Help Students Learn to LOVE Mathematics"?
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Agnostic Sample Compression Schemes for Regression
We obtain the first positive results for bounded sample compression in the agnostic regression setting with the ℓ[p] loss, where p ∈ [1, ∞]. We construct a generic approximate sample compression
scheme for real-valued function classes exhibiting exponential size in the fat-shattering dimension but independent of the sample size. Notably, for linear regression, an approximate compression of
size linear in the dimension is constructed. Moreover, for ℓ[1] and ℓ[∞] losses, we can even exhibit an efficient exact sample compression scheme of size linear in the dimension. We further show that
for every other ℓ[p] loss, p ∈ (1, ∞), there does not exist an exact agnostic compression scheme of bounded size. This refines and generalizes a negative result of David, Moran, and Yehudayoff (2016)
for the ℓ[2] loss. We close by posing general open questions: for agnostic regression with ℓ[1] loss, does every function class admit an exact compression scheme of polynomial size in the
pseudo-dimension? For the ℓ[2] loss, does every function class admit an approximate compression scheme of polynomial size in the fat-shattering dimension? These questions generalize Warmuth's classic
sample compression conjecture for realizable-case classification (Warmuth, 2003).
ASJC Scopus subject areas
• Artificial Intelligence
• Software
• Control and Systems Engineering
• Statistics and Probability
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ICML-2002 Tutorials
Monday, 8 July, 10am
Dr. Alexander Johannes Smola, The Australian National University, Canberra, Alex.Smola@anu.edu.au
The tutorial will introduce Gaussian Processes both for Classifcation and Regression. This includes a brief presentation of covariance functions, their connection
to Support Vector Kernels, and an overview over recent optimization methods for Gaussian Processes.
Target Audience: Novices and researchers more advanced in the knowledge of Gaussian Processes will benefit from the presentation. While being self contained, i.e., without requiring much further
knowledge than basic calculus and linear algebra, the presentation will advance to state of the art results in optimization and adaptive inference. This means that the course will cater for Graduate
Students and senior researchers alike. In particular, I will not assume knowledge beyond undergraduate mathematics (see Prerequisites for further detail).
Expected Knowledge Gain: a working knowledge in Gaussian Processes which will enable the audience to apply Bayesian inference methods in their research without much further training.
Prerequisites: Nothing beyond undergraduate knowledge in mathematics is expected. More specifically, I assume:
• Basic linear algebra (matrix inverse, eigenvector, eigenvalue, etc.)
• Some numerical mathematics (beenficial but not required), such as matrix factorization, conditioning, etc.
• Basic statistics and probability theory (Normal distribution, conditional distributions).
• (OPTIONAL:) Some knowledge in Bayesian methods
• (OPTIONAL:) Some knowledge in kernel methods
See http://mlg.anu.edu.au/~smola/summer2002 for details.
Monday, 8 July, 2pm
Inside WEKA -- and Beyond the Book
Ian H. Witten, Computer Science, University of Waikato, NZ, ihw@cs.waikato.ac.nz
Eibe Frank, Computer Science, University of Waikato, NZ, eibe@cs.waikato.ac.nz
Bernhard Pfahringer, Computer Science, University of Waikato, NZ, bernhard@cs.waikato.ac.nz
Mark Hall, Computer Science, University of Waikato, NZ, mhall@cs.waikato.ac.nz
Weka is an open-source Weka machine learning workbench, implemented in Java, that incorporates many popular algorithms and is widely used for practical work in machine learning. This tutorial
describes and demonstrates the many recent developments that have been made in the Weka system. It also looks inside Weka and sketches its inner workings for people who want to extend it with their
own machine learning implementations and make them available to the community by contributing to this open-source project.The goal is to empower attendees to increase the productivity of their
machine learning research and application development by making best use of the Weka
workbench, and to share their efforts with others by contributing them to the ML community. The tutorial is aimed at people who want to know about advanced features of Weka, and also at those who
want to work within the system at a programming level.
This tutorial is *not* intended as a comprehensive introduction to Weka: attendees are presumed to have some familiarity with it already. Neither does it reveal any secrets that are not in the
current version of Weka: if you have fully explored the features in the latest distribution you do not need to attend the tutorial.
Attendees are expected to have:
1. Basic knowledge of ML algorithms and methodology
2. Some familiarity with Weka
3. Some programming experience in Java.
(The book "Data mining" by Witten and Frank covers all three at an appropriate level).
There will be a 2-hour lab session after the tutorial for those who want to follow up with some practical work. Computers (Linux) will be available, or you can bring along your laptop (Windows or
Linux) and we will help you install the latest version of Weka from a CD-ROM. We will provide exercises for you to work on; alternatively you are encouraged to bring their own data files and use Weka
on them instead. Tutorial help will be available throughout the lab session
Tuesday, 9 July, 2pm
Introduction to Minimum Length Encoding Inference
Dr David Dowe,Monash University, Australia, dld@cs.monash.edu.au
The tutorial will be on Minimum Length Encoding, encompassing both Minimum Message Length (MML) and Minimum Description Length (MDL) inductive inference, topics central to the 1999 special issue of
the Computer Journal on Kolmogorov complexity (vol. 42, no. 4, 1999). This information-theoretic approach bridges many fields, and is yielding state-of-the-art solutions to at least many problems in
machine learning, statistics, econometrics and ``data mining''. It has applications right across the sciences.
This work is information-theoretic in nature, with a broad range of applications in machine learning, statistics, knowledge discovery and data mining. We discuss statistical parameter estimation and
mixture modelling (or clustering) of continuous, discrete and circular data. We also discuss learning decision trees and decision graphs, both with standard multinomial leaf distributions and with
more complicated models. We further discuss MML solutions of either cut-point problems or polynomial regression; and, if time permits, possibly Support Vector Machines (SVMs), causal networks and
finite state machines (Hidden Markov Models, HMMs) or other problems.
The target audience is academics, machine learning and data mining practitioners and consultants, and/or others with at least first year university education in at least one of
mathematics, statistics, econometrics or electrical engineering. The audience will learn introductory fundamentals of MML inference, and the many state-of-the art success of MML in statistics,
machine learning and hybrid problems. The audience will also see some applications of MML to real-world data. | {"url":"http://icml2002.cse.unsw.edu.au/tutorials/ICML-2002-Tuts.html","timestamp":"2024-11-14T18:42:41Z","content_type":"text/html","content_length":"7747","record_id":"<urn:uuid:63c02df7-30a3-4324-8024-2efa99ce4f75>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00591.warc.gz"} |
Fundamental Thermodynamics Relation
• Thread starter Silviu
• Start date
In summary, the Fundamental Thermodynamics Relation is a mathematical expression that describes the relationship between the internal energy, entropy, and volume of a thermodynamic system. It is
derived from the First Law of Thermodynamics and is a fundamental principle in the study of thermodynamics. It allows scientists to understand and predict the behavior of thermodynamic systems and
serves as the basis for many other thermodynamic equations and principles. It is also a universal equation that can be applied to all thermodynamic systems, but may need to be modified for certain
Hello! I am a bit confused by the formula ##dU = TdS - PdV##. If I want to compute for example ##\frac{\partial U}{\partial V}## I obtain ##-P##, but how should I proceed to obtain, for example ##\
frac{\partial U}{\partial P}## or ##\frac{\partial P}{\partial T}## which are not obvious from the equation? Thank you!
Staff Emeritus
Science Advisor
Homework Helper
2023 Award
For those partial derivatives you mention, you need to specify what is being held constant.
FAQ: Fundamental Thermodynamics Relation
1. What is the Fundamental Thermodynamics Relation?
The Fundamental Thermodynamics Relation is a mathematical expression that describes the relationship between the internal energy, entropy, and volume of a thermodynamic system. It is also known as
the First Law of Thermodynamics and is a fundamental principle in the study of thermodynamics.
2. How is the Fundamental Thermodynamics Relation derived?
The Fundamental Thermodynamics Relation is derived from the First Law of Thermodynamics, which states that the change in internal energy of a system is equal to the heat added to the system minus the
work done by the system. By rearranging this equation and using the definition of entropy, the Fundamental Thermodynamics Relation is obtained.
3. What is the significance of the Fundamental Thermodynamics Relation?
The Fundamental Thermodynamics Relation is important because it serves as a fundamental principle in the field of thermodynamics. It allows scientists to understand and predict the behavior of
thermodynamic systems and is the basis for many other thermodynamic equations and principles.
4. How does the Fundamental Thermodynamics Relation relate to other thermodynamic equations?
The Fundamental Thermodynamics Relation is a fundamental equation in thermodynamics and is used to derive other important equations, such as the ideal gas law, the Gibbs-Duhem equation, and the
Maxwell relations. It also provides a link between macroscopic properties of a system, such as temperature and pressure, and microscopic properties, such as molecular energy and entropy.
5. Can the Fundamental Thermodynamics Relation be applied to all thermodynamic systems?
Yes, the Fundamental Thermodynamics Relation is a universal equation that can be applied to all thermodynamic systems, regardless of their composition or complexity. This is because it is based on
fundamental principles and is not limited to specific types of systems. However, it may need to be modified for certain systems, such as non-ideal gases or systems undergoing phase transitions. | {"url":"https://www.physicsforums.com/threads/fundamental-thermodynamics-relation.902748/","timestamp":"2024-11-11T13:58:06Z","content_type":"text/html","content_length":"76233","record_id":"<urn:uuid:a8ac8e2a-6f13-43f1-b466-eeeb327cf4fe>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00475.warc.gz"} |
Students Should Decompose Fractions – Teacher Tech
I am a big fan of helping students to use STRATEGIES when approaching math rather than mindlessly following steps in an algorithm. Personally, I rarely follow order of operations when simplifying a
math expression. Having number sense means you look at the math task and THINK about it and find ways to break apart and regroup in more efficient ways when it can be done. However, I have never done
this with fractions.
Decomposing Numbers
19 + 23
Standard algorithm requires carrying.
Adding 19 and 23 using the standard algorithm is definitely slower. when I see this problem I want to add one and subtract one.
20 + 22
I was able to EASILY make this transformation in my head. The resulting expression is significantly easier to add than the first expression.
What About Fractions?
But what about looking at fractions in this manner. I saw this tweet by Jennifer Bay-Williams with an example of how a student added mixed fractions. I had an epiphany: How have I never thought of
doing this? It is almost like I needed permission that I can rethink how I approach fractions.
Thinking First
The challenge with teaching students only the standard algorithm… and shudder marking them down if they do not do it exactly how we showed them… is that students are not being asked to think. They
are being asked to REMEMBER.
[Tweet]The first approach to a math problem should be THINKING about it. Not competing with a calculator. [/tweet]
Fractions are part of a whole. How do I get a whole?
Using the commutative property I can organize the fractions and the whole numbers seperately. THINKING about the fractions I can ask, “how do I make a whole?” This helps me to think about the concept
of fractions and not just the steps.
Sometimes the standard algorithm is faster and sometimes it is not. Students should stop and THINK about math problems before just diving into the algorithm. Helping them to be more flexible with
numbers and to apply strategies when appropriate will help them to be more confident and proficient at mathematics.
Q5:What do you notice about this student’s work? What ideas about teaching come to mind? #ElemMathChat pic.twitter.com/x3fyTJi5w3
— Jennifer BayWilliams (@JBayWilliams) November 4, 2022
Graspable Math Whiteboard
To create the sample math problems I used the Graspable Math Whiteboard.
• How to Roll 3 Dice in Google
• Students Should Decompose Fractions
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Decompose Fractions
Instead of diving into the steps of adding fractions have students look at how they can break apart the fractions to make it easier to approach the math problem. Have students decompose fractions as
another way of thinking about the numbers.
You must be logged in to post a comment. | {"url":"https://dsimpson6thomsoncooper.com/students-should-decompose-fractions-teacher-tech.html","timestamp":"2024-11-10T04:47:12Z","content_type":"text/html","content_length":"132773","record_id":"<urn:uuid:22db728d-c85c-47be-a57b-d7b58ed0516b>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00456.warc.gz"} |
Noctournament (Dualset)
This decription uses US Eastern time, so the end of a day is the following date at 5 UTC.Noctournament
With the release of Nocturne, it seems like a fine idea to run a tournament in which nobody knows what they are doing, so this is that tournament.
This tournament will run from Sunday, November 26 until whenever it finishes or people get bored with it, and will require playing 2 matches in the first 10 days and then 1 match each week thereafter
if one advances.
Signups will be in this thread. Signups before the end of Tuesday, November 21 guarantee a spot and signups after November 21 but before the end of Saturday, November 25 will be taken in order until
the last multiple of 6.
I will list those signed up in the post below this one, which will contain match pairings and links to brackets once those are available.
You will need to have an active subscription to Nocturne on Shuffle iT to participate.
Format and Schedule
The first stage will take place with groups of 3, each player playing 1 match against the other two in the group by the end of Wednesday, December 6th.
I will seed those signed up by Shuffle iT rating mu at the end of November 25 and then randomize the order a bit with lower seeds more likely to move, then snake this randomized order to get the
groups of 3 (so if there were 12 participants randomized to the order 1 through 12, the groups would be {1,8,9}, {2,7,10}, {3,6,11}, and {4,5,12}).
If a person completes no matches they will be removed from the tournament; otherwise, the players in a group will be ordered first according to matches won, then head-to-head result with those one is
tied with, then total number of games won. A drawn match will count as half of a win.
The second stage will be single elimination in a bracket with other participants who finished in the same position in their group (so there will be 3 brackets here) and need to be finished by the end
of the week after match pairings are posted, which will be a Wednesday.
This means that signing up will guarantee at least 3 matches: two in the group and at least one in the single elimination.
Players will be seeded in those brackets by the expected seed for the place for that bracket. The number of players may not be a power of 2; in that case, higher seeded players will have byes into
the next round.
The player who wins more games will advance. If the match goes unplayed I will advance whoever I see fit based on the scheduling complaints I receive.
I am fine with you scheduling and playing matches as soon as you know who you will be facing.
Match Structure
Each match will be 6 games with alternating player order with the first player for the first game being whoever is hosting the table.
The player who is going second will choose one set that is not Nocturne, and the cards for the match will be randomly selected from Nocturne and that set. Here, we will consider (Cornucopia and
Guilds) and (Alchemy and Promos) to be one set.
This can be set up in client at a table by going to "Select Kingdom Cards" and making sure that the sets being used are checked and others are not - the randomization from this is enough.
If in the single elimination stage a player reaches 3.5 wins, you may terminate the match.
If in the single elimination stage a match is tied after 6 games, play games only with Nocturne cards with random player order until one of the players wins a game - this player wins the match (as
they will have won more games).
When reporting match results, one should provide the score and game numbers. Reporting of match results will eventually take place in this thread.
Spectator chat should be turned off in all games, and I encourage you to announce your matches
on discord so that others may watch.
If there are questions about any rules ask in this thread.
Lastly, and as always, have fun! | {"url":"https://forum.dominionstrategy.com/index.php?topic=17902.msg733183","timestamp":"2024-11-10T03:12:28Z","content_type":"application/xhtml+xml","content_length":"96781","record_id":"<urn:uuid:b6b60e29-b0ea-4997-80d5-7b7fae14e120>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00373.warc.gz"} |
Time-varying values
In more complex Markov models state values or transition probabilities can vary with time. These models are called non-homogeneous or time-inhomogeneous Markov models. A further distinction can be
made depending on whether state values or transition probabilities:
1. depend on how long the entire model has been running (model-time dependency);
2. depend on how long an individual has been in a given state (state-time dependency).
These two situations can be modelled using the model_time (or its alias model_time) and state_time variables, respectively.
How to specify time-dependency
These variables takes increasing values with each cycles, starting from 1. For example the age of individuals at any moment can be defined as Initial age + model_time. The time an individual spends
in a state is equal to state_time.
Both variables can be used in define_parameters(), define_state(), or define_transition():
## 2 unevaluated parameters.
## mr = exp(-state_time * lambda)
## age = 50 + model_time
## A state with 2 values.
## cost = 100 - state_time
## effect = 10
## No named state -> generating names.
## A transition matrix, 2 states.
## A B
## A C f(state_time)
## B 0.1 0.9
Using model_time in a model does not change the execution speed of the analysis. On the other hand adding state_time may slow down the analysis, especially if the model is run for many cycles and a
transition probability depends on state_time.
To mitigate this drawback it is possible to limit the number of state expansion with state_time_limit. Because most time-varying values reach an asymptotic value quite fast, it is unnecessary to
expand the states any further. The last cycle value is repeated until the end.
The limit can be defined globally, per state, or per model and state. In the following example probabilities are kept constant after 10 cycles for state B and 20 cycles for state D in strategy I, and
15 cycles in state B in strategy II. | {"url":"https://cran.mirror.garr.it/CRAN/web/packages/heemod/vignettes/b_time_dependency.html","timestamp":"2024-11-14T13:56:54Z","content_type":"text/html","content_length":"14972","record_id":"<urn:uuid:e872a2e5-0d09-4368-918c-62e0e7226c3b>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00599.warc.gz"} |
Gematria/Kabalah/Tarot - Sacred Geometry International
August 28, 2011 at 5:42 pm
by AstroMonk
Randall Carlson on Sacred Geometry and ‘Triplicities’
(from Decatur class September 15, 2006)
…and there is concealed a geometric code. That’s an example of how it works. It’s there in multiple layers, and once you realize it’s there, you realize that the whole scriptures were basically
framed around this geometrical system. The allegories were devised to provide an outer, concealing covering for the inner mathematical core of the sacred writings. What it’s doing… Essentially,
Sacred Geometry is the geometry of the universe. It’s the geometry of creation itself. And this was one of the secrets that these ancient master builders understood. So what they were attempting
to do, when they built the holy temples and the sacred edifices, was to mimic, as they understood it, that geometry that was an intrinsic part of the harmony of creation.
Now how did they get that geometry? Well, it was through a process of revelation – whatever that means. Now, we can interpret that two different ways. In some of the myths as we read it – the
original geometric keys to creation were imparted by the “Gods”. Right? And that’s a valid way of looking at it. Another way is that the adept, through a process of deep meditation and inner
exploration, discovers that this is actually the geometry of consciousness itself. And to say, either way, whether it’s an inner revelation, from within through meditation, or an exterior revelation
imparted from the “Gods” – It’s true either way.
…Throughout all creation, the idea is that there are these patterns and links, and once you know how to insert yourself into the system, you can essentially move at will through it. So,
geometrically there’s a way to get from a square root two pattern to a square root three pattern. And from a square root three to a square root five pattern. So you’ve gone from a square root three –
the patterns of crystalline structure, to the square root of five which is the patterns of biological structure. And you can do so by this geometric pathway.
…Let’s look at… Remember what Lucretius said? Right here…“So far as it goes, a small thing may give analogy of great things, and show the tracks of knowledge.” Now see, the tracks of knowledge,
this is the idea that you’re following these tracks, these footprints that are leading you from one state to another state. From one location to the next location in the exploration of the sacred
…Well, this is just inherent in these triplicate numbers. Now, when you go in, to the bible, and other ancient writings, and what you’ll discover are there are key words and phrases that give you
this whole numerical pattern. The one of course, that everybody knows about is 666 – the beast. Which in the Greek is Thereon…
(from class…still, mathematically, how do…they’re multiples of thirty-seven? )
Yes, every one: three times thirty-seven gives you one-one-one, six times thirty-seven gives you 222, eighteen times thirty-seven gives you 6-66. “The number of ‘the beast’.” It says…remember:
“And here is wisdom. Let him who hath understanding, count the number of the beast,
for it is the number of a man. Six hundred three-score and six.”
So it’s saying…well, here’s a mathematical clue. “Here is wisdom. Let him who hath understanding…” If you really want to understand what this is, you’ve got to count this number. And it’s not enough
to count the number, you have to see what the number connects to. And if you don’t do that…if you…like most modern Christians tend to be very superstitious about that number. It doesn’t say it’s
the number of the devil, it says it’s the number of a beast, but the number of a man…
…And three-three-three: There’s actually a Greek word that translates as ‘sensuality’. The Greek word that gives us 3-3-3, was ultimately the word that meant and implied the realm of the senses –
we see through our five senses. And that term translates, generally in most modern idioms, as sensuality. …If you take the Greek letters…add them up, you’ll get 3-3-3. So each one of these – I can
show you that there are words that correlate with each of these numbers.
But see, these numbers are also found in nature. When you begin to explore astronomical relationships, between the sun and the planets, and between the planets and their moons – these numbers start
showing up in some rather strange places. Even in orbital velocities. For example, the orbital velocity of the Earth around the sun is 18.5 miles-per-second. How many miles-per-hour is that? In
your calculator, put 18.5, times 60, times 60… Yes, 18.5 times 60 is 1110[miles-per-minute], and again times 60 is 66,600mph! So right there, that number shows up in the relationship between the
sun and the Earth – in
terms of the orbital velocity of the Earth. See, here these numbers can serve as mnemonics, so from now on you’ll always be able to remember the orbital velocity of the Earth. The number of the
beast and add two zeroes, you’ve got it in miles-per-hour. Here’s what’s interesting – in a lot of the textbooks, even though the velocity has been refined to 18.5mps, in a lot of the textbooks they
simply say the orbital velocity of the Earth is 66,000 miles-per-hour. They don’t want to put that third six in there!
…Oh, but that’s only the beginning – it gets really, really interesting…
When Bly Bond, the ecclesiastical architect in the early part of the twentieth century was excavating the ruins of Glastonbury Cathedral, he discovered that the whole cathedral had been laid out on a
grid of squares that were seventy-four feet on a side. Seventy-four feet – the whole complex was laid out on a grid that was nine of those squares long. So it was seventy-four feet on a side, and
the whole cathedral was laid out on a grid that was nine squares long… Right? So, [in your calculator] what’d you get? …The whole cathedral was 666 feet. Now, the square itself like I said, was
seventy-four feet, that meant each square was how many inches? So now you go seventy-four times twelve – try it… The seventy-four foot grid that you get, each square is how many inches? 8-8-8!
A-ha! So you see how both those numbers are now incorporated into the plan? Now when he came up with that 6-6-6, he got in trouble. They didn’t want to hear any of that! They didn’t want to hear
any of that – I guess it was the Church of England that retained him. He was an architect and they had hired him to excavate and measure the ruins of Glastonbury Cathedral. And no, they didn’t want
to hear that it was laid out on a grid that was 666 feet long.
(from class: Why is it such a ‘bad’ thing?) …Well, it’s a bad thing only because it relates to pre-Christian traditions. “Pagan”, yeah, it smacks of paganism! …Well sure, it’s in Revelations,
that’s what I just quoted verbatim, from the King James [Bible], just a few minutes ago.
…When I’ve done advanced classes, like up in Asheville, after we’d gone through the geometry basics, then we got very heavily into this. In order to try to peel back the covers and show that
underneath these allegories and parables of scripture, there is this mathematical inner meaning, or this geometric meaning. And that geometry is essentially the fundamental geometry of creation. It
is the “Sacred Geometry”. It’s there – you can’t see it, but it’s under there, and if you just lift back, peel back the cover which is the stories that are there at the literary layer. Under that
literary layer is this hidden level of meaning, which is the mathematical layer.
(from class: “ and that in itself is ‘Gematria’?) Gematria, yes. There are actually several different versions of it. There’s Kabala, which is the ‘revealed doctrine’. And a big component…there’s
a Hebrew kabala, a Greek kabala, there’s an Arabic kabala. Essentially, what it is – kabala is seeking to understand the primordial patterns of creation. In the Gematria, that’s a specific branch,
where what they’re doing is relating number and language, and showing that…just like…When you think of the temple…What differentiates a sacred building from a secular building? The fact that there
is this underlying geometry. What differentiates between sacred writings and just secular writings, is that under the sacred writings there is this hidden layer of geometry. And the hidden layer of
geometry under the scriptural writings, and I’m not just talking about the Christian scriptures here, I’m talking about the Mithraic writings that were in Latin, I’m talking about writings that were
in the Vedas. You find this system of Sacred Geometry. And it’s the same system of geometry that’s underlying the temples. Temples whether they’re in Southeast Asia, Central America, or Egypt, or
England. Underneath the outer forms is the same geometry. And it’s essentially the geometry of creation. It’s the geometry of harmony. My suspicion is that this will be an important rediscovery
for healing what creates so much dis-harmony in our world today.
…If you go back and read Plato, and some of the early writers who essentially advocated this kind of a system. Their idea was that as long as we maintained this harmonic relationship between man,
Earth and Heavens – All things are in balance, and that would be reflected in the human society. And if human society became disjointed, disconnected, was no longer…(class interruptions)
…Well, you have to think of it this way: there are fundamental forces that are symbolized… For example, look at many of the handouts that I’ve given you – you’ll see the sun and you’ll see the moon.
That, on one level means the literal sun and the literal moon, on another level it’s talking about two different kinds of primordial energies which are interacting to create all things. A solar
energy and a lunar energy. What you’ll discover, when you go through and you start penetrating into the geometry within the sacred writings, is that the number 6-6-6 constantly comes up in reference
to the sun. There are other numbers such as one thousand and eighty that constantly come up in reference to the moon. Then, when you combine the solar current and the lunar current, the 666 with
the 1080, and you add them together what you get is 1746. Well what is that? It doesn’t mean anything until you’ve studied into the system, and then you know, that there’s a clue… Jesus talks about
the grain of mustard seed. Right? Have you ever heard about the grain of mustard seed? What’s the allegory about the grain of mustard seed? …Right! “If I had the power that was contained within
this grain of mustard seed, I could say to the mountain, ‘Move’, and it shall move!” You see, there’s an allegory there, but behind that allegory is a very profound scientific reality. The solar
current and the lunar current fused together gives you the 1746 which is the power inherent in the grain of mustard seed. These are just examples of how this system works. The key to penetrating
this system is the whole process of Sacred Geometry.
You see, it’s not just these drawings, these drawings are essentially patterns of cosmic energy, and you’re learning how to create these patterns of cosmic energy. We can find once you’ve learned…We
can go…for example, take what we did today and apply it to the solar system, and discover that that pattern is embedded within the geometry of our whole solar system. It’s also imbedded within our
molecular geometry. And it’s not just coincidence that we find the same geometries on these different levels.
Let me point something else out to you, and this’ll be the final thing for today. When we talk about atomic structure, every element has its basic form and its isotopic form. When you look at its
basic form, what you’re saying is that this element has a certain number of protons, and a certain number of neutrons, and a certain number of electrons. It’s the numbers of each of those that
determines totally what the properties are of that particular element. How it relates and combines with other elements and so on. Well, if you look at this… Let’s think about a few things: What is
the fundamental element of all living things on Earth? Carbon. And what is the atomic structure of carbon? How many protons in an atom of carbon? Six! How many neutrons? Six! How many electrons?
Six! Talk about let’s say, nitrogen… Again one of the fundamental elements…Seven protons, seven neutrons, seven electrons. How about an atom of oxygen? Eight protons, eight neutrons, and eight
electrons. Now…There’s profound mysteries concealed here, and it’s getting beyond where I get to at this stage, until we’ve gone a little further…But I get really excited about this stuff, and
sometimes I let the cat out of the bag prematurely. Now here’s something to ponder: The number of the beast…What’s its connection with carbon – the fact that carbon is in all living things? (class
interruptions)…And it’s also a solar number – it has multiple meanings as you’ll come to find out. Well this is just a starting point. It gives you some of the ideas of how this system can work… I
guess that’s all for tonight!
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Multiplication - Wikiwand
Multiplication (often denoted by the cross symbol ×, by the mid-line dot operator ⋅, by juxtaposition, or, on computers, by an asterisk *) is one of the four elementary mathematical operations of
arithmetic, with the other ones being addition, subtraction, and division. The result of a multiplication operation is called a product.
This article
needs additional citations for verification
(April 2012)
Four bags with three marbles per bag gives twelve marbles (4 × 3 = 12).
Multiplication can also be thought of as scaling. Here, 2 is being multiplied by 3 using scaling, giving 6 as a result.
Animation for the multiplication 2 × 3 = 6
4 × 5 = 20. The large rectangle is made up of 20 squares, each 1 unit by 1 unit.
Area of a cloth ;
The multiplication of whole numbers may be thought of as repeated addition; that is, the multiplication of two numbers is equivalent to adding as many copies of one of them, the multiplicand, as the
quantity of the other one, the multiplier; both numbers can be referred to as factors.
${\displaystyle a\times b=\underbrace {b+\cdots +b} _{a{\text{ times}}}.}$
For example, 4 multiplied by 3, often written as ${\displaystyle 3\times 4}$ and spoken as "3 times 4", can be calculated by adding 3 copies of 4 together:
${\displaystyle 3\times 4=4+4+4=12.}$
Here, 3 (the multiplier) and 4 (the multiplicand) are the factors, and 12 is the product.
One of the main properties of multiplication is the commutative property, which states in this case that adding 3 copies of 4 gives the same result as adding 4 copies of 3:
${\displaystyle 4\times 3=3+3+3+3=12.}$
Thus, the designation of multiplier and multiplicand does not affect the result of the multiplication.^[1]
Systematic generalizations of this basic definition define the multiplication of integers (including negative numbers), rational numbers (fractions), and real numbers.
Multiplication can also be visualized as counting objects arranged in a rectangle (for whole numbers) or as finding the area of a rectangle whose sides have some given lengths. The area of a
rectangle does not depend on which side is measured first—a consequence of the commutative property.
The product of two measurements (or physical quantities) is a new type of measurement, usually with a derived unit. For example, multiplying the lengths (in meters or feet) of the two sides of a
rectangle gives its area (in square meters or square feet). Such a product is the subject of dimensional analysis.
The inverse operation of multiplication is division. For example, since 4 multiplied by 3 equals 12, 12 divided by 3 equals 4. Indeed, multiplication by 3, followed by division by 3, yields the
original number. The division of a number other than 0 by itself equals 1.
Several mathematical concepts expand upon the fundamental idea of multiplication. The product of a sequence, vector multiplication, complex numbers, and matrices are all examples where this can be
seen. These more advanced constructs tend to affect the basic properties in their own ways, such as becoming noncommutative in matrices and some forms of vector multiplication or changing the sign of
complex numbers.
× ⋅
In Unicode U+00D7 × MULTIPLICATION SIGN (×)
U+22C5 ⋅ DOT OPERATOR (⋅)
Different from U+00B7 · MIDDLE DOT
U+002E . FULL STOP
In arithmetic, multiplication is often written using the multiplication sign (either × or ${\displaystyle \times }$) between the terms (that is, in infix notation).^[2] For example,
${\displaystyle 2\times 3=6,}$ ("two times three equals six")
${\displaystyle 3\times 4=12,}$
${\displaystyle 2\times 3\times 5=6\times 5=30,}$
${\displaystyle 2\times 2\times 2\times 2\times 2=32.}$
There are other mathematical notations for multiplication:
• To reduce confusion between the multiplication sign × and the common variable x, multiplication is also denoted by dot signs,^[3] usually a middle-position dot (rarely period): ${\displaystyle 5\
cdot 2}$.
The middle dot notation or dot operator, encoded in Unicode as U+22C5 ⋅ DOT OPERATOR, is now standard in the United States and other countries where the period is used as a decimal point. When
the dot operator character is not accessible, the interpunct (·) is used. In other countries that use a comma as a decimal mark, either the period or a middle dot is used for multiplication.
Historically, in the United Kingdom and Ireland, the middle dot was sometimes used for the decimal to prevent it from disappearing in the ruled line, and the period/full stop was used for
multiplication. However, since the Ministry of Technology ruled to use the period as the decimal point in 1968,^[4] and the International System of Units (SI) standard has since been widely
adopted, this usage is now found only in the more traditional journals such as The Lancet.^[5]
• In algebra, multiplication involving variables is often written as a juxtaposition (e.g., ${\displaystyle xy}$ for ${\displaystyle x}$ times ${\displaystyle y}$ or ${\displaystyle 5x}$ for five
times ${\displaystyle x}$), also called implied multiplication.^[6] The notation can also be used for quantities that are surrounded by parentheses (e.g., ${\displaystyle 5(2)}$, ${\displaystyle
(5)2}$ or ${\displaystyle (5)(2)}$ for five times two). This implicit usage of multiplication can cause ambiguity when the concatenated variables happen to match the name of another variable,
when a variable name in front of a parenthesis can be confused with a function name, or in the correct determination of the order of operations.^[7]^[8]
• In vector multiplication, there is a distinction between the cross and the dot symbols. The cross symbol generally denotes the taking a cross product of two vectors, yielding a vector as its
result, while the dot denotes taking the dot product of two vectors, resulting in a scalar.
In computer programming, the asterisk (as in 5*2) is still the most common notation. This is due to the fact that most computers historically were limited to small character sets (such as ASCII and
EBCDIC) that lacked a multiplication sign (such as ⋅ or ×), while the asterisk appeared on every keyboard. This usage originated in the FORTRAN programming language.^[9]
The numbers to be multiplied are generally called the "factors" (as in factorization). The number to be multiplied is the "multiplicand", and the number by which it is multiplied is the "multiplier".
Usually, the multiplier is placed first, and the multiplicand is placed second;^[1] however, sometimes the first factor is the multiplicand and the second the multiplier.^[10] Also, as the result of
multiplication does not depend on the order of the factors, the distinction between "multiplicand" and "multiplier" is useful only at a very elementary level and in some multiplication algorithms,
such as the long multiplication. Therefore, in some sources, the term "multiplicand" is regarded as a synonym for "factor".^[11] In algebra, a number that is the multiplier of a variable or
expression (e.g., the 3 in ${\displaystyle 3xy^{2}}$) is called a coefficient.
The result of a multiplication is called a product. When one factor is an integer, the product is a multiple of the other or of the product of the others. Thus, ${\displaystyle 2\times \pi }$ is a
multiple of ${\displaystyle \pi }$, as is ${\displaystyle 5133\times 486\times \pi }$. A product of integers is a multiple of each factor; for example, 15 is the product of 3 and 5 and is both a
multiple of 3 and a multiple of 5.
This section
needs attention from an expert in Mathematics
. The specific problem is:
defining multiplication is not straightforward and different proposals have been made over the centuries, with competing ideas (e.g. recursive vs. non-recursive definitions).
See the
talk page
for details.
(September 2023)
The product of two numbers or the multiplication between two numbers can be defined for common special cases: natural numbers, integers, rational numbers, real numbers, complex numbers, and
Product of two natural numbers
3 by 4 is 12.
The product of two natural numbers ${\displaystyle r,s\in \mathbb {N} }$ is defined as:
${\displaystyle r\cdot s\equiv \sum _{i=1}^{s}r=\underbrace {r+r+\cdots +r} _{s{\text{ times}}}\equiv \sum _{j=1}^{r}s=\underbrace {s+s+\cdots +s} _{r{\text{ times}}}.}$
Product of two integers
An integer can be either zero, a nonzero natural number, or minus a nonzero natural number. The product of zero and another integer is always zero. The product of two nonzero integers is determined
by the product of their positive amounts, combined with the sign derived from the following rule:
${\displaystyle {\begin{array}{|c|c c|}\hline \times &+&-\\\hline +&+&-\\-&-&+\\\hline \end{array}}}$ (This rule is a consequence of the distributivity of multiplication over addition, and is not an
additional rule.)
In words:
• A positive number multiplied by a positive number is positive (product of natural numbers),
• A positive number multiplied by a negative number is negative,
• A negative number multiplied by a positive number is negative,
• A negative number multiplied by a negative number is positive.
Product of two fractions
Two fractions can be multiplied by multiplying their numerators and denominators:
${\displaystyle {\frac {z}{n}}\cdot {\frac {z'}{n'}}={\frac {z\cdot z'}{n\cdot n'}},}$
which is defined when ${\displaystyle n,n'eq 0}$.
Product of two real numbers
There are several equivalent ways to define formally the real numbers; see Construction of the real numbers. The definition of multiplication is a part of all these definitions.
A fundamental aspect of these definitions is that every real number can be approximated to any accuracy by rational numbers. A standard way for expressing this is that every real number is the least
upper bound of a set of rational numbers. In particular, every positive real number is the least upper bound of the truncations of its infinite decimal representation; for example, ${\displaystyle \
pi }$ is the least upper bound of ${\displaystyle \{3,\;3.1,\;3.14,\;3.141,\ldots \}.}$
A fundamental property of real numbers is that rational approximations are compatible with arithmetic operations, and, in particular, with multiplication. This means that, if a and b are positive
real numbers such that ${\displaystyle a=\sup _{x\in A}x}$ and ${\displaystyle b=\sup _{y\in B}y,}$ then ${\displaystyle a\cdot b=\sup _{x\in A,y\in B}x\cdot y.}$ In particular, the product of two
positive real numbers is the least upper bound of the term-by-term products of the sequences of their decimal representations.
As changing the signs transforms least upper bounds into greatest lower bounds, the simplest way to deal with a multiplication involving one or two negative numbers, is to use the rule of signs
described above in § Product of two integers. The construction of the real numbers through Cauchy sequences is often preferred in order to avoid consideration of the four possible sign
Product of two complex numbers
Two complex numbers can be multiplied by the distributive law and the fact that ${\displaystyle i^{2}=-1}$, as follows:
{\displaystyle {\begin{aligned}(a+b\,i)\cdot (c+d\,i)&=a\cdot c+a\cdot d\,i+b\,i\cdot c+b\cdot d\cdot i^{2}\\&=(a\cdot c-b\cdot d)+(a\cdot d+b\cdot c)\,i\end{aligned}}}
A complex number in polar coordinates
The geometric meaning of complex multiplication can be understood by rewriting complex numbers in polar coordinates:
${\displaystyle a+b\,i=r\cdot (\cos(\varphi )+i\sin(\varphi ))=r\cdot e^{i\varphi }}$
${\displaystyle c+d\,i=s\cdot (\cos(\psi )+i\sin(\psi ))=s\cdot e^{i\psi },}$
from which one obtains
${\displaystyle (a\cdot c-b\cdot d)+(a\cdot d+b\cdot c)i=r\cdot s\cdot e^{i(\varphi +\psi )}.}$
The geometric meaning is that the magnitudes are multiplied and the arguments are added.
Product of two quaternions
The product of two quaternions can be found in the article on quaternions. Note, in this case, that ${\displaystyle a\cdot b}$ and ${\displaystyle b\cdot a}$ are in general different.
The Educated Monkey—a tin toy dated 1918, used as a multiplication "calculator". For example: set the monkey's feet to 4 and 9, and get the product—36—in its hands.
Many common methods for multiplying numbers using pencil and paper require a multiplication table of memorized or consulted products of small numbers (typically any two numbers from 0 to 9). However,
one method, the peasant multiplication algorithm, does not. The example below illustrates "long multiplication" (the "standard algorithm", "grade-school multiplication"):
× 5830
00000000 ( = 23,958,233 × 0)
71874699 ( = 23,958,233 × 30)
191665864 ( = 23,958,233 × 800)
+ 119791165 ( = 23,958,233 × 5,000)
139676498390 ( = 139,676,498,390 )
In some countries such as Germany, the above multiplication is depicted similarly but with the original product kept horizontal and computation starting with the first digit of the multiplier:^[12]
23958233 · 5830
Multiplying numbers to more than a couple of decimal places by hand is tedious and error-prone. Common logarithms were invented to simplify such calculations, since adding logarithms is equivalent to
multiplying. The slide rule allowed numbers to be quickly multiplied to about three places of accuracy. Beginning in the early 20th century, mechanical calculators, such as the Marchant, automated
multiplication of up to 10-digit numbers. Modern electronic computers and calculators have greatly reduced the need for multiplication by hand.
Historical algorithms
Methods of multiplication were documented in the writings of ancient Egyptian, Greek, Indian, and Chinese civilizations.
The Ishango bone, dated to about 18,000 to 20,000 BC, may hint at a knowledge of multiplication in the Upper Paleolithic era in Central Africa, but this is speculative.^[13]
The Egyptian method of multiplication of integers and fractions, which is documented in the Rhind Mathematical Papyrus, was by successive additions and doubling. For instance, to find the product of
13 and 21 one had to double 21 three times, obtaining 2 × 21 = 42, 4 × 21 = 2 × 42 = 84, 8 × 21 = 2 × 84 = 168. The full product could then be found by adding the appropriate terms found in the
doubling sequence:^[14]
13 × 21 = (1 + 4 + 8) × 21 = (1 × 21) + (4 × 21) + (8 × 21) = 21 + 84 + 168 = 273.
The Babylonians used a sexagesimal positional number system, analogous to the modern-day decimal system. Thus, Babylonian multiplication was very similar to modern decimal multiplication. Because of
the relative difficulty of remembering 60 × 60 different products, Babylonian mathematicians employed multiplication tables. These tables consisted of a list of the first twenty multiples of a
certain principal number n: n, 2n, ..., 20n; followed by the multiples of 10n: 30n 40n, and 50n. Then to compute any sexagesimal product, say 53n, one only needed to add 50n and 3n computed from the
Modern methods
Product of 45 and 256. Note the order of the numerals in 45 is reversed down the left column. The carry step of the multiplication can be performed at the final stage of the calculation (in bold),
returning the final product of . This is a variant of Lattice multiplication.
The modern method of multiplication based on the Hindu–Arabic numeral system was first described by Brahmagupta. Brahmagupta gave rules for addition, subtraction, multiplication, and division. Henry
Burchard Fine, then a professor of mathematics at Princeton University, wrote the following:
The Indians are the inventors not only of the positional decimal system itself, but of most of the processes involved in elementary reckoning with the system. Addition and subtraction they
performed quite as they are performed nowadays; multiplication they effected in many ways, ours among them, but division they did cumbrously.^[16]
These place value decimal arithmetic algorithms were introduced to Arab countries by Al Khwarizmi in the early 9th century and popularized in the Western world by Fibonacci in the 13th century.^[17]
Grid method
Grid method multiplication, or the box method, is used in primary schools in England and Wales and in some areas of the United States to help teach an understanding of how multiple digit
multiplication works. An example of multiplying 34 by 13 would be to lay the numbers out in a grid as follows:
and then add the entries.
Computer algorithms
The classical method of multiplying two n-digit numbers requires n^2 digit multiplications. Multiplication algorithms have been designed that reduce the computation time considerably when multiplying
large numbers. Methods based on the discrete Fourier transform reduce the computational complexity to O(n log n log log n). In 2016, the factor log log n was replaced by a function that increases
much slower, though still not constant.^[18] In March 2019, David Harvey and Joris van der Hoeven submitted a paper presenting an integer multiplication algorithm with a complexity of ${\displaystyle
O(n\log n).}$^[19] The algorithm, also based on the fast Fourier transform, is conjectured to be asymptotically optimal.^[20] The algorithm is not practically useful, as it only becomes faster for
multiplying extremely large numbers (having more than 2^1729^12 bits).^[21]
One can only meaningfully add or subtract quantities of the same type, but quantities of different types can be multiplied or divided without problems. For example, four bags with three marbles each
can be thought of as:^[1]
[4 bags] × [3 marbles per bag] = 12 marbles.
When two measurements are multiplied together, the product is of a type depending on the types of measurements. The general theory is given by dimensional analysis. This analysis is routinely applied
in physics, but it also has applications in finance and other applied fields.
A common example in physics is the fact that multiplying speed by time gives distance. For example:
50 kilometers per hour × 3 hours = 150 kilometers.
In this case, the hour units cancel out, leaving the product with only kilometer units.
Other examples of multiplication involving units include:
2.5 meters × 4.5 meters = 11.25 square meters
11 meters/seconds × 9 seconds = 99 meters
4.5 residents per house × 20 houses = 90 residents
Capital pi notation
The product of a sequence of factors can be written with the product symbol ${\displaystyle \textstyle \prod }$, which derives from the capital letter Π (pi) in the Greek alphabet (much like the same
way the summation symbol ${\displaystyle \textstyle \sum }$ is derived from the Greek letter Σ (sigma)).^[22]^[23] The meaning of this notation is given by
${\displaystyle \prod _{i=1}^{4}(i+1)=(1+1)\,(2+1)\,(3+1)\,(4+1),}$
which results in
${\displaystyle \prod _{i=1}^{4}(i+1)=120.}$
In such a notation, the variable i represents a varying integer, called the multiplication index, that runs from the lower value 1 indicated in the subscript to the upper value 4 given by the
superscript. The product is obtained by multiplying together all factors obtained by substituting the multiplication index for an integer between the lower and the upper values (the bounds included)
in the expression that follows the product operator.
More generally, the notation is defined as
${\displaystyle \prod _{i=m}^{n}x_{i}=x_{m}\cdot x_{m+1}\cdot x_{m+2}\cdot \,\,\cdots \,\,\cdot x_{n-1}\cdot x_{n},}$
where m and n are integers or expressions that evaluate to integers. In the case where m = n, the value of the product is the same as that of the single factor x[m]; if m > n, the product is an empty
product whose value is 1—regardless of the expression for the factors.
Properties of capital pi notation
By definition,
${\displaystyle \prod _{i=1}^{n}x_{i}=x_{1}\cdot x_{2}\cdot \ldots \cdot x_{n}.}$
If all factors are identical, a product of n factors is equivalent to exponentiation:
${\displaystyle \prod _{i=1}^{n}x=x\cdot x\cdot \ldots \cdot x=x^{n}.}$
Associativity and commutativity of multiplication imply
${\displaystyle \prod _{i=1}^{n}{x_{i}y_{i}}=\left(\prod _{i=1}^{n}x_{i}\right)\left(\prod _{i=1}^{n}y_{i}\right)}$ and
${\displaystyle \left(\prod _{i=1}^{n}x_{i}\right)^{a}=\prod _{i=1}^{n}x_{i}^{a}}$
if a is a non-negative integer, or if all ${\displaystyle x_{i}}$ are positive real numbers, and
${\displaystyle \prod _{i=1}^{n}x^{a_{i}}=x^{\sum _{i=1}^{n}a_{i}}}$
if all ${\displaystyle a_{i}}$ are non-negative integers, or if x is a positive real number.
Infinite products
One may also consider products of infinitely many terms; these are called infinite products. Notationally, this consists in replacing n above by the infinity symbol ∞. The product of such an infinite
sequence is defined as the limit of the product of the first n terms, as n grows without bound. That is,
${\displaystyle \prod _{i=m}^{\infty }x_{i}=\lim _{n\to \infty }\prod _{i=m}^{n}x_{i}.}$
One can similarly replace m with negative infinity, and define:
${\displaystyle \prod _{i=-\infty }^{\infty }x_{i}=\left(\lim _{m\to -\infty }\prod _{i=m}^{0}x_{i}\right)\cdot \left(\lim _{n\to \infty }\prod _{i=1}^{n}x_{i}\right),}$
provided both limits exist.
When multiplication is repeated, the resulting operation is known as exponentiation. For instance, the product of three factors of two (2×2×2) is "two raised to the third power", and is denoted by 2^
3, a two with a superscript three. In this example, the number two is the base, and three is the exponent.^[24] In general, the exponent (or superscript) indicates how many times the base appears in
the expression, so that the expression
${\displaystyle a^{n}=\underbrace {a\times a\times \cdots \times a} _{n}=\prod _{i=1}^{n}a}$
indicates that n copies of the base a are to be multiplied together. This notation can be used whenever multiplication is known to be power associative.
Multiplication of numbers 0–10. Line labels = multiplicand. Xaxis = multiplier. Yaxis = product.
Extension of this pattern into other quadrants gives the reason why a negative number times a negative number yields a positive number.
Note also how multiplication by zero causes a reduction in dimensionality, as does multiplication by a singular matrix where the determinant is 0. In this process, information is lost and cannot be
For real and complex numbers, which includes, for example, natural numbers, integers, and fractions, multiplication has certain properties:
The order in which two numbers are multiplied does not matter:^[25]^[26]
${\displaystyle x\cdot y=y\cdot x.}$
Expressions solely involving multiplication or addition are invariant with respect to the order of operations:^[25]^[26]
${\displaystyle (x\cdot y)\cdot z=x\cdot (y\cdot z).}$
Holds with respect to multiplication over addition. This identity is of prime importance in simplifying algebraic expressions:^[25]^[26]
${\displaystyle x\cdot (y+z)=x\cdot y+x\cdot z.}$
The multiplicative identity is 1; anything multiplied by 1 is itself. This feature of 1 is known as the identity property:^[25]^[26]
${\displaystyle x\cdot 1=x.}$
Any number multiplied by 0 is 0. This is known as the zero property of multiplication:^[25]
${\displaystyle x\cdot 0=0.}$
−1 times any number is equal to the additive inverse of that number:
${\displaystyle (-1)\cdot x=(-x)}$, where ${\displaystyle (-x)+x=0.}$
−1 times −1 is 1:
${\displaystyle (-1)\cdot (-1)=1.}$
Every number x, except 0, has a multiplicative inverse, ${\displaystyle {\frac {1}{x}}}$, such that ${\displaystyle x\cdot \left({\frac {1}{x}}\right)=1}$.^[27]
Order preservation
Multiplication by a positive number preserves the order:
For a > 0, if b > c, then ab > ac.
Multiplication by a negative number reverses the order:
For a < 0, if b > c, then ab < ac.
The complex numbers do not have an ordering that is compatible with both addition and multiplication.^[28]
Other mathematical systems that include a multiplication operation may not have all these properties. For example, multiplication is not, in general, commutative for matrices and quaternions.^[25]
Hurwitz's theorem shows that for the hypercomplex numbers of dimension 8 or greater, including the octonions, sedenions, and trigintaduonions, multiplication is generally not associative.^[29]
In the book Arithmetices principia, nova methodo exposita, Giuseppe Peano proposed axioms for arithmetic based on his axioms for natural numbers. Peano arithmetic has two axioms for multiplication:
${\displaystyle x\times 0=0}$
${\displaystyle x\times S(y)=(x\times y)+x}$
Here S(y) represents the successor of y; i.e., the natural number that follows y. The various properties like associativity can be proved from these and the other axioms of Peano arithmetic,
including induction. For instance, S(0), denoted by 1, is a multiplicative identity because
${\displaystyle x\times 1=x\times S(0)=(x\times 0)+x=0+x=x.}$
The axioms for integers typically define them as equivalence classes of ordered pairs of natural numbers. The model is based on treating (x,y) as equivalent to x − y when x and y are treated as
integers. Thus both (0,1) and (1,2) are equivalent to −1. The multiplication axiom for integers defined this way is
${\displaystyle (x_{p},\,x_{m})\times (y_{p},\,y_{m})=(x_{p}\times y_{p}+x_{m}\times y_{m},\;x_{p}\times y_{m}+x_{m}\times y_{p}).}$
The rule that −1 × −1 = 1 can then be deduced from
${\displaystyle (0,1)\times (0,1)=(0\times 0+1\times 1,\,0\times 1+1\times 0)=(1,0).}$
Multiplication is extended in a similar way to rational numbers and then to real numbers.
The product of non-negative integers can be defined with set theory using cardinal numbers or the Peano axioms. See below how to extend this to multiplying arbitrary integers, and then arbitrary
rational numbers. The product of real numbers is defined in terms of products of rational numbers; see construction of the real numbers.^[30]
There are many sets that, under the operation of multiplication, satisfy the axioms that define group structure. These axioms are closure, associativity, and the inclusion of an identity element and
A simple example is the set of non-zero rational numbers. Here identity 1 is had, as opposed to groups under addition where the identity is typically 0. Note that with the rationals, zero must be
excluded because, under multiplication, it does not have an inverse: there is no rational number that can be multiplied by zero to result in 1. In this example, an abelian group is had, but that is
not always the case.
To see this, consider the set of invertible square matrices of a given dimension over a given field. Here, it is straightforward to verify closure, associativity, and inclusion of identity (the
identity matrix) and inverses. However, matrix multiplication is not commutative, which shows that this group is non-abelian.
Another fact worth noticing is that the integers under multiplication do not form a group—even if zero is excluded. This is easily seen by the nonexistence of an inverse for all elements other than 1
and −1.
Multiplication in group theory is typically notated either by a dot or by juxtaposition (the omission of an operation symbol between elements). So multiplying element a by element b could be notated
as a ${\displaystyle \cdot }$ b or ab. When referring to a group via the indication of the set and operation, the dot is used. For example, our first example could be indicated by ${\displaystyle \
left(\mathbb {Q} /\{0\},\,\cdot \right)}$.^[31]
Numbers can count (3 apples), order (the 3rd apple), or measure (3.5 feet high); as the history of mathematics has progressed from counting on our fingers to modelling quantum mechanics,
multiplication has been generalized to more complicated and abstract types of numbers, and to things that are not numbers (such as matrices) or do not look much like numbers (such as quaternions).
${\displaystyle N\times M}$ is the sum of N copies of M when N and M are positive whole numbers. This gives the number of things in an array N wide and M high. Generalization to negative numbers
can be done by
${\displaystyle N\times (-M)=(-N)\times M=-(N\times M)}$ and
${\displaystyle (-N)\times (-M)=N\times M}$
The same sign rules apply to rational and real numbers.
Generalization to fractions ${\displaystyle {\frac {A}{B}}\times {\frac {C}{D}}}$ is by multiplying the numerators and denominators, respectively: ${\displaystyle {\frac {A}{B}}\times {\frac {C}
{D}}={\frac {(A\times C)}{(B\times D)}}}$. This gives the area of a rectangle ${\displaystyle {\frac {A}{B}}}$ high and ${\displaystyle {\frac {C}{D}}}$ wide, and is the same as the number of
things in an array when the rational numbers happen to be whole numbers.^[25]
Real numbers and their products can be defined in terms of sequences of rational numbers.
Considering complex numbers ${\displaystyle z_{1}}$ and ${\displaystyle z_{2}}$ as ordered pairs of real numbers ${\displaystyle (a_{1},b_{1})}$ and ${\displaystyle (a_{2},b_{2})}$, the product $
{\displaystyle z_{1}\times z_{2}}$ is ${\displaystyle (a_{1}\times a_{2}-b_{1}\times b_{2},a_{1}\times b_{2}+a_{2}\times b_{1})}$. This is the same as for reals ${\displaystyle a_{1}\times a_{2}}
$ when the imaginary parts ${\displaystyle b_{1}}$ and ${\displaystyle b_{2}}$ are zero.
Equivalently, denoting ${\displaystyle {\sqrt {-1}}}$ as ${\displaystyle i}$, ${\displaystyle z_{1}\times z_{2}=(a_{1}+b_{1}i)(a_{2}+b_{2}i)=(a_{1}\times a_{2})+(a_{1}\times b_{2}i)+(b_{1}\times
a_{2}i)+(b_{1}\times b_{2}i^{2})=(a_{1}a_{2}-b_{1}b_{2})+(a_{1}b_{2}+b_{1}a_{2})i.}$^[25]
Alternatively, in trigonometric form, if ${\displaystyle z_{1}=r_{1}(\cos \phi _{1}+i\sin \phi _{1}),z_{2}=r_{2}(\cos \phi _{2}+i\sin \phi _{2})}$, then${\textstyle z_{1}z_{2}=r_{1}r_{2}(\cos(\
phi _{1}+\phi _{2})+i\sin(\phi _{1}+\phi _{2})).}$^[25]
Further generalizations
See Multiplication in group theory, above, and multiplicative group, which for example includes matrix multiplication. A very general, and abstract, concept of multiplication is as the
"multiplicatively denoted" (second) binary operation in a ring. An example of a ring that is not any of the above number systems is a polynomial ring (polynomials can be added and multiplied, but
polynomials are not numbers in any usual sense).
Often division, ${\displaystyle {\frac {x}{y}}}$, is the same as multiplication by an inverse, ${\displaystyle x\left({\frac {1}{y}}\right)}$. Multiplication for some types of "numbers" may have
corresponding division, without inverses; in an integral domain x may have no inverse "${\displaystyle {\frac {1}{x}}}$" but ${\displaystyle {\frac {x}{y}}}$ may be defined. In a division ring
there are inverses, but ${\displaystyle {\frac {x}{y}}}$ may be ambiguous in non-commutative rings since ${\displaystyle x\left({\frac {1}{y}}\right)}$ need not be the same as ${\displaystyle \
left({\frac {1}{y}}\right)x}$. | {"url":"https://www.wikiwand.com/en/articles/Multiplication","timestamp":"2024-11-02T12:16:21Z","content_type":"text/html","content_length":"804233","record_id":"<urn:uuid:bcb6128a-7e6b-409a-b002-441c5bea3b0b>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00643.warc.gz"} |
2Linear maps
IB Linear Algebra
2 Linear maps
In mathematics, apart from studying objects, we would like to study functions
between objects as well. In particular, we would like to study functions that
respect the structure of the objects. With vector spaces, the kinds of functions
we are interested in are linear maps. | {"url":"http://dec41.user.srcf.net/h/IB_M/linear_algebra/2","timestamp":"2024-11-11T11:33:37Z","content_type":"text/html","content_length":"35066","record_id":"<urn:uuid:5b723c33-84c4-4a36-9e29-6be731092c42>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00107.warc.gz"} |
Low resolution refinement tools in the program REFMAC
Add to your list(s) Download to your calendar using vCal
If you have a question about this talk, please contact xyp20.
Despite rapid advances in Macromolecular X-ray Crystallographic (MX) methods, derivation of reliable atomic models from low resolution diffraction data still poses many challenges. The reason for
this is that the number of observations relative to the number of adjustable parameters is small and signal to noise ratio in the experimental data is very low. As a consequence derivation of
biologically meaningful information from such data is challenging. Mobility of macromolecules means that in many cases growing crystals diffracting to higher resolution is not possible and low
resolution data must be used to derive some useful information.
To derive some of information from such data two related but distinct problems should be tackled: i) stabilisation of ill-posedness of refinement procedures  ii) calculation of maximal signal/
minimal noise electron density. Solving the first problem is necessary to derive reliable atomic model and the second problem to calculate interpretable electron density that is used in model (re)
1) The first problem is usually tackled using restraints based on structural information. Available structural information are a) known similar three-dimensional structures b) secondary structures;
c) NCS if present; d) in addition it is also possible to exploit the fact that during refinement inter-atomic distances should not change dramatically. It has already been shown that using these
restraints improves reliability of the derived models. As a result of model improvement errors in the derived atomic models are reduced, and it means that calculated phases have less error hence
reducing noise in the electron density related to the model errors. 2) Sharpening of an electron density while increasing signal amplifies noise masking out âtrueâ signal. There are several
approaches to such problems including: a) regularisation using Tikhonov-Sobolev method; b) Wiener filters and c) Bayesian filters. These techniques attempt to answer to one common question: how to
enhance signal without noise amplification? Another problem in map sharpening is that it assumes that all atoms have the same B values. It is in general not true and there is a distribution of B
values â inverse gamma distribution. Moreover individual atomsâ oscillation depends on its position in the asymmetric unit. These facts need to be accounted for if accurate map sharpening tools to be
designed. In this presentation some applications of these techniques to map calculations will be discussed.
This talk is part of the Experimental and Computational Aspects of Structural Biology and Applications to Drug Discovery series.
This talk is included in these lists:
Note that ex-directory lists are not shown. | {"url":"https://talks.cam.ac.uk/talk/index/34628","timestamp":"2024-11-07T11:00:48Z","content_type":"application/xhtml+xml","content_length":"15605","record_id":"<urn:uuid:fe0d3073-9f02-4cae-9802-dec4c447848e>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00822.warc.gz"} |
Let There Be Light, or... Heat?
Let There Be Light, or... Heat?
The purposes of this lab are for students to determine which type of light bulb, compact fluorescent or incandescent, is more efficient at converting electrical energy to light instead of heat and
for students to calculate the amount and cost of electricity used to power each type of light for 28 hours per week for a year.
Key Science Topics
• Heat vs. temperature
• Power dissipated by a device
Grade Level
Student Prior Knowledge
• Students should be able to calculate the power in Watts dissipated by a device when voltage and current are measured.
• Students should know how to calculate kilowatt-hours.
• Students should recognize that infrared radiation is heat.
• Radiometer
• Compact fluorescent light bulbs, with lamp base
• Incandescent Light bulbs, with lamp base
• Kil-A-Watt meter (see suggestions, below)
• Powerpoint file, showing the IR image of the compact fluorescent light bulb and the incandescent light bulb (PPTX)
Students will examine whether a compact fluorescent light bulb or an incandescent light bulb produces more heat in two different activities. Students will hold a radiometer in front of each type of
light bulb to observe which bulb makes the radiometer spin faster. Then, students will record data obtained from a Kil-A-Watt meter to determine which type of bulb uses more electricity.
Lowes rates light bulbs based on four hours of usage per day (www.lowes.com). Thus, the use per week is 28 hours. Students may comment that they use lights more or less often. Encourage them to think
about how often lights are left on in rooms no one is using. As used in the teacher answer key, the price of electricity in Ohio, as of this writing, is $0.11 per kilowatt hour. Encourage your
students to find out what is the local price for electricity prior to working on this activity.
Please remind students that their results are for only one light bulb. Encourage them to think about how many IL and CFL bulbs are in their house. Perhaps as an extension to this activity, you may
want to have your students count each type of light bulb.
Ideally, this activity is to be done with students visiting stations. I usually have six groups of three to four students. For this lab, each group gets their own radiometer and I have three lamp
bases with a CFL plugged into a Kil-A-Watt meter and three lamp bases with an IL plugged into a Kil-A-Watt meter. Students start at one type of light bulb, and then switch to the other.
Kil-A-Watt meters are available for about $30.00. If this is cost prohibitive, one Kil-A-Watt meter could be used by the teacher and the students could record the voltage and current data. Another
possibility is to write to your local power company to see if they would donate Kil-A-Watt meters. If no Kil-O-Watt meter is available, feel free to provide the students with the sample voltage,
current, and power data given in the Teacher Answer Key
I have students measure the power on the Kil-A-Watt meter.They can calculate the power from the voltage and current measured, but it will not be the same value because the light bulbs are non-Ohmic;
in other words, the resistance of the light bulb changes as the filament increases in temperature.
Published: 26 January, 2022 | {"url":"https://coolcosmos.ipac.caltech.edu/page/lesson_let_there_be_light_or_heat","timestamp":"2024-11-04T08:06:28Z","content_type":"text/html","content_length":"14991","record_id":"<urn:uuid:5db93f7d-97ab-449a-b4ba-38812f526435>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00818.warc.gz"} |
Trip Cost from Ghanzi to Struizendam
To calculate the return trip cost from Ghanzi to Struizendam, please enter the name of the locations in the control along with fuel rate and its economy, then calculate the Return Trip Cost to get
the results. A complete trip cost summary will also be given in the end. You can also find the
Distance from Ghanzi to Struizendam
using various travel options like bus, subway, tram, train and rail here. | {"url":"https://www.distancesfrom.com/bw/Trip-Cost-from-Ghanzi-to-Struizendam-Botswana/TripCostHistory/46388359.aspx","timestamp":"2024-11-05T15:36:48Z","content_type":"text/html","content_length":"181367","record_id":"<urn:uuid:4b11fa30-a457-4363-8a69-d98df62b0fa3>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00852.warc.gz"} |
NCERT Solutions for Class 6 Maths PDF
Here we are providing NCERT Solutions for Class 6 Maths, one can easily download the PDF. All the chapters of Maths of class 6 is listed below.
All the Solutions, provided by us are latest and up to date according to new NCERT Syllabus.
Download Complete NCERT Solutions for class 6th to 12th
NCERT Solution for Class 6 Maths – Free PDF Download
Download Complete NCERT Books PDF for class 1 to 12
Searching for meticulously designed free NCERT solutions for class 6 Maths? Then you have landed on the right place. Students generally take mathematics as a tough subject. So it is as if you are
unclear about the concepts you cannot even do a guess work. Mathematics for Class 6 endows some new concepts for the students.
Till now the students were familiar with only number system but now the concept of integers gets introduced. The students get confused in identifying and solving problems involving integers. They
generally mess up with BODMAS rules due to positive and negative numbers. The concept of number line becomes really complicated for them.
NCERT books are the standard books followed by all CBSE schools as well as some of the state boards. Studying through NCERT makes sure that you are covering all the topics prescribed by CBSE as
well as guide you for a better future. The solution book covers solutions for the entire sixteen chapters in a chronological order. Starting from “knowing our Numbers” then moving towards “whole
number” and “playing with numbers” every chapter unfolds the best curated solutions for the practice set exercise given at the end of your NCERT text-book.
Integers, Fractions and decimal handling are very important in class 6 as a little misunderstanding may take you away from the original concept. Our solution book not only acts as your guide but a
best friend who is always there for you 24/7. You can easily refer how the solution has been approached and learn the technique followed seamlessly.
The solutions have been solved in full till the answer is derived. This will help students to learn how step marking is done in your examination. It also helps to provide a strong base for your board
examination preparation.
Mathematics is just a play of formulas and numbers which help you solve your daily accounts, numerical and also industry related problems. Subjects such as Physics, chemistry, accounts have no
existence without mathematics. Class 6 introduces the basic concept of algebra and ratio & proportion. You can refer the NCERT solutions class 6 Maths for your reference or completing your home work
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Share this page with your friends and classmates and become popular instantly in your group. Just download the solution book from NCERTGuess.com for free. Students can also download solutions for
other textbooks also that too free of cost.
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This site uses Akismet to reduce spam. Learn how your comment data is processed. | {"url":"https://ncertguess.com/ncert-solutions-for-class-6-maths-pdf/","timestamp":"2024-11-05T03:25:08Z","content_type":"text/html","content_length":"228082","record_id":"<urn:uuid:389e8caf-a363-40e9-ba2e-4c982e8df25c>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00208.warc.gz"} |
Trendlines in Excel
One of the easiest methods for guessing a general trend in your data is to add a trendline to a chart. The Trendline is a bit similar to a line in a line chart, but it doesn’t connect each data point
precisely as a line chart does.
A trendline represents all the data. This means that minor exceptions or statistical errors won’t distract Excel when it comes to finding the right formula. In some cases, you can also use the
trendline to forecast future data.
Charts that support trendlines
The trendline can be added to 2-D charts, such as Area, Bar, Column, Line, Stock, X Y (Scatter) and Bubble. You can’t add a trendline to 3-D, Radar, Pie, Area or Doughnut charts.
Adding a trendline
After you create a chart, right-click on the data series and choose Add trendline…. A new menu will appear to the left of the chart.
Here, you can choose one of the trendline types, by clicking on one of the radio buttons.
Below trendlines, there is a position called Display R-squared value on chart. It shows you how a trendline is fitted to the data. It can get values from 0 to 1. The closer the value is to 1 the
better it fits your chart.
Trendline types
Linear trendline
This trendline is used to create a straight line for simple, linear data sets. The data is linear if the system data points resemble a line. The linear trend line indicates that something is
increasing or decreasing at a steady rate.
Here is an example of computer sales for each month.
Logarithmic trendline
The logarithmic trendline is useful when you have to deal with data where the rate of change increases or decreases quickly and then stabilizes. In the case of a logarithmic trendline, you can use
both negative and positive values.
A good example of a logarithmic trendline may be an economic crisis. First, the unemployment rate is getting higher but after a while, the situation stabilizes.
Polynomial trendline
This trendline is useful when you work with oscillating data – for example when you analyze gains and losses over a large data set.
The degree of the polynomial may be determined by the number of data fluctuations or by the number of bends, in other words, the hills, and valleys that appear on the curve.
An order 2 polynomial trendline usually has one hill or valley. Order 3 generally has one or two hills or valleys. Order 4 generally has up to three.
The following example illustrates the relationship between speed and fuel consumption.
Power trendline
This trendline is useful for data sets that are used to compare measurement results that increase at a predetermined rate. For example, the acceleration of a race car at one-second intervals.
You can’t create a power trendline if your data contains zero or negative values.
Exponential trendline
The exponential trendline is most useful when the data values rise or fall at a constantly increasing rate.
It is often used in sciences. It can describe a population that is growing rapidly in subsequent generations. You cannot create an exponential trendline if your data contains zero or negative values.
A good example of this trendline is the decay of C-14.
As you can see this is a perfect example of an exponential trend line because the R-squared value is exactly 1.
Moving average
The moving average smoothes the lines to show a pattern or trend more clearly. Excel does it by calculating the moving average of a certain number of values (set by a Period option), which by default
is set to 2.
If you increase this value, then the average will be calculated from more data points so that the line will be even smoother. The moving average shows trends that otherwise would be difficult to see
due to noise in the data.
A good example of the practical use of this trendline can be the Forex market.
Excel 365 Update
• Minimal Functionality Changes: While the core functionalities of adding, formatting, and customizing trendlines remain largely similar, there haven’t been any significant updates or new features
specific to trendlines themselves in Excel 365.
• Focus on New Chart Types: Excel 365 offers a wider range of chart types compared to Excel 2016. This provides users with more options for data visualization, which might indirectly influence how
trendlines are used to analyze data. For instance, using a chart type better suited to the data might make adding a trendline less necessary. | {"url":"https://officetuts.net/excel/training/trendlines/","timestamp":"2024-11-07T00:07:47Z","content_type":"text/html","content_length":"144566","record_id":"<urn:uuid:173af97d-b814-46a0-bc29-0fccc5bce667>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00872.warc.gz"} |
comparison procedures based on Bonferroni tests
Motivating example
Consider a confirmatory clinical trial comparing a test treatment (treatment) against the control treatment (control) for a disease. There are two doses of treatment: the low dose and the high dose.
There are two endpoints included in the multiplicity adjustment strategy, which are the primary endpoint (PE) and the secondary endpoint (SE). In total, there are four null hypotheses: \(H_1\) and \
(H_3\) are the primary and the secondary hypotheses respectively for the low dose versus control; \(H_2\) and \(H_4\) are the primary and the secondary hypotheses respectively for the high dose
versus control.
There are clinical considerations which constrain the structure of multiple comparison procedures, and which can be flexibly incorporated using graphical approaches. First, the low and the high doses
are considered equally important, which means that rejecting the primary hypothesis for either dose versus control leads to a successful trial. Regarding secondary hypotheses, each one is tested only
if its corresponding primary hypothesis has been rejected. This means that \(H_3\) is tested only after \(H_1\) has been rejected; \(H_4\) is tested only after \(H_2\) has been rejected.
In addition, there are some statistical considerations to complete the graph. The primary hypotheses \(H_1\) and \(H_2\) will have an equal hypothesis weight of 0.5. The secondary hypotheses will
have a hypothesis weight of 0. When a primary hypothesis has been rejected, its weight will be propagated along two outgoing edges: One to the other primary hypothesis and one to its descendant
secondary hypothesis. The two edges will have an equal transition weight of 0.5. When both the primary and the secondary hypotheses have been rejected for a dose-control comparison, their hypothesis
weights will be propagated to the primary hypothesis for the other dose-control comparison. With these specifications, we can create the following graph.
Perform the graphical multiple comparison procedure
Adjusted p-values and rejections
Given a set of p-values for \(H_1, \ldots, H_4\), the graphical multiple comparison procedure can be performed to control the familywise error rate (FWER) at the significance level alpha. The
graph_test_shortcut function is agnostic to one-sided or two-sided tests. For one-sided p-values, alpha is often set to 0.025 (default); for two-sided p-values, alpha is often set to 0.05. We
consider one-sided tests here. A hypothesis is rejected if its adjusted p-value is less than or equal to alpha. After running the procedure, hypotheses \(H_1\), \(H_2\), and \(H_4\) are rejected with
their adjusted p-value calculated.
Obtain possible orders of rejections
The order of rejections may not be unique and not all orders are valid. For this example, the rejected hypotheses are \(H_1\), \(H_2\) and \(H_4\). The default order of rejections is \(H_2 \
rightarrow H_1 \rightarrow H_4\). Another valid order of rejections is \(H_2 \rightarrow H_4 \rightarrow H_1\). However, the first rejected hypothesis can not be \(H_1\) or \(H_4\). To obtain all
possible rejection orders, one can use the function graph_rejection_orderings. Then intermediate and final graphs can be obtained by using the function graph_update with a particular order of
# Obtain all valid orders of rejections
orders <- graph_rejection_orderings(test_results)$valid_orderings
#> [[1]]
#> H2 H1 H4
#> 2 1 4
#> [[2]]
#> H2 H4 H1
#> 2 4 1
# Intermediate graphs following the order of H2 and H4
graph_update(g, delete = orders[[2]])$intermediate_graphs[[3]]
#> Updated graph
#> --- Hypothesis weights ---
#> H1: 1
#> H2: NA
#> H3: 0
#> H4: NA
#> --- Transition weights ---
#> H1 H2 H3 H4
#> H1 0 NA 1 NA
#> H2 NA NA NA NA
#> H3 1 NA 0 NA
#> H4 NA NA NA NA
Obtain adjusted significance levels
An equivalent way to obtain rejections is by adjusting significance levels. A hypothesis is rejected if its p-value is less than or equal to its adjusted significance level. The adjusted significance
levels are calculated in the same order as adjusted p-values: \(H_2 \rightarrow H_1 \rightarrow H_4\), and there are four steps of checking for rejections. First, \(H_2\) is rejected at an adjusted
significance level of 0.5 * alpha. Second, \(H_1\) is rejected at an adjusted significance level of 0.75 * alpha, after \(H_2\) is rejected. Third, \(H_4\) is rejected at an adjusted significance
level of 0.5 * alpha, after \(H_1\) and \(H_2\) are rejected. Fourth and finally, \(H_3\) cannot be rejected at an adjusted significance level of alpha, after \(H_1\), \(H_2\) and \(H_4\) are
rejected. These results can be obtained by specifying test_values = TRUE.
test_results_test_values <- graph_test_shortcut(
p = p_values,
alpha = 0.025,
test_values = TRUE
#> Step Hypothesis p Weight Alpha Inequality_holds
#> 1 1 H2 0.010 0.50 0.025 TRUE
#> 2 2 H1 0.018 0.75 0.025 TRUE
#> 3 3 H4 0.006 0.50 0.025 TRUE
#> 4 4 H3 0.105 1.00 0.025 FALSE
Power simulation
Given the above graph, we are interested in the “power” of the trial. For a single null hypothesis, the power is the probability of a true positive - that is, rejecting the null hypothesis at the
significance level alpha when the alternative hypothesis is true. For multiple null hypotheses, there could be multiple versions of “power”. For example, the power to reject at least one hypothesis
vs the power to reject all hypotheses, given the alternative hypotheses are true. With the graphical multiple comparison procedures, it is also important to understand the power to reject each
hypothesis, given the multiplicity adjustment. Sometimes, we may want to customize definitions of power to define success. Thus power calculation is an important aspect of trial design.
Input: Marginal power for primary hypotheses
Assume that the primary endpoint is about the occurrence of an unfavorable clinical event. To evaluate the treatment effect, the proportion of patients with this event is calculated, and a lower
proportion is preferred. Assume that the proportions are 0.181 for the low and the high doses, and 0.3 for control. Using the equal randomization among the three treatment groups, the clinical trial
team chooses a total sample size of 600 with 200 per group. This leads to a marginal power of 80% for \(H_1\) and \(H_2\), respectively, using the two-sample test for difference in proportions with
unpooled variance each at one-sided significance level 0.025. In this calculation, we use the marginal power to combine the information from the treatment effect, any nuisance parameter, and sample
sizes for each hypothesis. Note that the significance level used for the marginal power calculation must be the same as alpha, which is used as the significance level for the FWER control. In
addition, the marginal power has a one-to-one relationship with the noncentrality parameter, which is illustrated below.
alpha <- 0.025
prop <- c(0.3, 0.181, 0.181)
sample_size <- rep(200, 3)
unpooled_variance <-
prop[-1] * (1 - prop[-1]) / sample_size[-1] +
prop[1] * (1 - prop[1]) / sample_size[1]
noncentrality_parameter_primary <-
-(prop[-1] - prop[1]) / sqrt(unpooled_variance)
power_marginal_primary <- pnorm(
qnorm(alpha, lower.tail = FALSE),
mean = noncentrality_parameter_primary,
sd = 1,
lower.tail = FALSE
names(power_marginal_primary) <- c("H1", "H2")
#> H1 H2
#> 0.8028315 0.8028315
Input: Marginal power for secondary hypotheses
Assume that the secondary endpoint is about the change in total medication score from baseline, which is a continuous outcome. To evaluate the treatment effect, the mean change is calculated, and
greater reduction is preferred. Assume that the mean change from baseline is the reduction of 7.5 and 8.25, respectively for the low and the high doses, and 5 for control. Further assume a known
common standard deviation of 10. Given the sample size of 200 per treatment group, the marginal power is 71% and 90% for \(H_3\) and \(H_4\), respectively, using the two-sample \(z\)-test for the
difference in means each at the one-sided significance level 0.025.
mean_change <- c(5, 7.5, 8.25)
sd <- rep(10, 3)
variance <- sd[-1]^2 / sample_size[-1] + sd[1]^2 / sample_size[1]
noncentrality_parameter_secondary <-
(mean_change[-1] - mean_change[1]) / sqrt(variance)
power_marginal_secondary <- pnorm(
qnorm(alpha, lower.tail = FALSE),
mean = noncentrality_parameter_secondary,
sd = 1,
lower.tail = FALSE
names(power_marginal_secondary) <- c("H3", "H4")
#> H3 H4
#> 0.7054139 0.9014809
Input: Correlation structure to simulate test statistics
In addition to the marginal power, we also need to make assumptions about the joint distribution of test statistics. In this example, we assume that they follow a multivariate normal distribution
with means defined by the noncentrality parameters above and the correlation matrix \(R\). To obtain the correlations, it is helpful to understand that there are two types of correlations in this
example. The correlation between two dose-control comparisons for the same endpoint and the correlation between endpoints. The former correlation can be calculated as a function of sample size. For
example, the correlation between test statistics for \(H_1\) and \(H_2\) is \(\rho_{12}=\left(\frac{n_1}{n_1+n_0}\right)^{1/2}\left(\frac{n_2}{n_3+n_0}\right)^{1/2}\). Under the equal randomization,
this correlation is 0.5. The correlation between test statistics for \(H_3\) and \(H_4\) is the same as the above. On the other hand, the correlation between endpoints for the same dose-control
comparison is often estimated based on prior knowledge or from previous trials. Without the information, we assume it to be \(\rho_{13}=\rho_{24}=0.5\). In practice, one could set this correlation as
a parameter and try multiple values to assess the sensitivity of this assumption. Regarding the correlation between test statistics for \(H_1\) and \(H_4\) and for \(H_2\) and \(H_3\), they are even
more difficult to estimate. Here we use a simple product rule, which means that this correlation is a product of correlations of the two previously assumed correlations. For example, \(\rho_{14}=\
rho_{12}*\rho_{24}\) and \(\rho_{23}=\rho_{12}*\rho_{13}\). In practice, one may make further assumptions instead of using the product rule.
corr <- matrix(0, nrow = 4, ncol = 4)
corr[1, 2] <-
corr[3, 4] <-
sample_size[2] / sum(sample_size[1:2]) *
sample_size[3] / sum(sample_size[c(1, 3)])
rho <- 0.5
corr[1, 3] <- corr[2, 4] <- rho
corr[1, 4] <- corr[2, 3] <- corr[1, 2] * rho
corr <- corr + t(corr)
diag(corr) <- 1
colnames(corr) <- hyp_names
rownames(corr) <- hyp_names
#> H1 H2 H3 H4
#> H1 1.00 0.50 0.50 0.25
#> H2 0.50 1.00 0.25 0.50
#> H3 0.50 0.25 1.00 0.50
#> H4 0.25 0.50 0.50 1.00
User-defined success criteria
As mentioned earlier, there are multiple versions of “power” when there are multiple hypotheses. Commonly used “power” definitions include:
• Local power: The probability of each hypothesis being rejected (with multiplicity adjustment)
• Expected no. of rejections: The expected number of rejections
• Power to reject 1 or more: The probability to reject at least one hypothesis
• Power to reject all: The probability to reject all hypotheses
These are the default outputs from the graph_calculate_power function. In addition, a user can customize success criteria to define other versions of “power”.
success_fns <- list(
# Probability to reject H1
H1 = function(x) x[1],
# Expected number of rejections
`Expected no. of rejections` = function(x) x[1] + x[2] + x[3] + x[4],
# Probability to reject at least one hypothesis
`AtLeast1` = function(x) x[1] | x[2] | x[3] | x[4],
# Probability to reject all hypotheses
`All` = function(x) x[1] & x[2] & x[3] & x[4],
# Probability to reject both H1 and H2
`H1andH2` = function(x) x[1] & x[2],
# Probability to reject both hypotheses for the low dose or the high dose
`(H1andH3)or(H2andH4)` = function(x) (x[1] & x[3]) | (x[2] & x[4])
Output: Simulate power
Given the above inputs, we can estimate “power” via simulation for the graphical multiple comparison procedure at one-sided significance level alpha = 0.025 using sim_n = 1e5 simulations and the
random seed 1234. The local power is 0.758, 0.765, 0.689, and 0.570, respectively for \(H_1, \ldots, H_4\). Note that the local power is lower than the marginal power because the former is adjusted
for multiplicity. The power to reject at least one hypothesis is 0.856 and the power to reject all hypotheses is 0.512. The expected number of rejections is 2.782. For the last two user-defined
success criteria, the probability to reject both \(H_1\) and \(H_2\) is 0.667, and the probability to reject at least one pair of \((H_1\) and \(H_3)\) and \((H_2\) and \(H_4)\) is 0.747.
power_output <- graph_calculate_power(
alpha = 0.025,
sim_corr = corr,
sim_n = 1e5,
power_marginal = c(power_marginal_primary, power_marginal_secondary),
sim_success = success_fns
#> $power_local
#> H1 H2 H3 H4
#> 0.76396 0.75887 0.56767 0.69133
#> $rejection_expected
#> [1] 2.78183
#> $power_at_least_1
#> [1] 0.85557
#> $power_all
#> [1] 0.51205
#> $power_success
#> H1 Expected no. of rejections
#> 0.76396 2.78183
#> AtLeast1 All
#> 0.85557 0.51205
#> H1andH2 (H1andH3)or(H2andH4)
#> 0.66726 0.74695
To see the detailed outputs of all simulated p-values and rejection decisions for all hypotheses, specify verbose = TRUE. This will produce a lot of outputs. To allow flexible printing functions, a
user can change the following:
• The indented space with the default setting of indent = 2
• The precision of numeric values (i.e., the number of significant digits) with the default setting of precision = 4
power_verbose_output <- graph_calculate_power(
alpha = 0.025,
sim_corr = corr,
sim_n = 1e5,
power_marginal = c(power_marginal_primary, power_marginal_secondary),
sim_success = success_fns,
verbose = TRUE
head(power_verbose_output$details$p_sim, 10)
#> H1 H2 H3 H4
#> [1,] 0.0308204265 0.0120653993 0.0041185823 9.324338e-02
#> [2,] 0.0007933716 0.0006499046 0.0245177515 2.965604e-03
#> [3,] 0.0302991819 0.0595395828 0.0543082956 2.625834e-02
#> [4,] 0.0097433244 0.0033185711 0.0007417213 4.024688e-04
#> [5,] 0.0197134942 0.0086161835 0.0164182765 2.418325e-07
#> [6,] 0.0031206572 0.0067023099 0.0137441457 2.751703e-04
#> [7,] 0.0302208038 0.1423757994 0.0060382838 2.117403e-02
#> [8,] 0.0024975725 0.0294025573 0.0004142729 2.207786e-03
#> [9,] 0.0618994292 0.0387257108 0.3166125781 5.699791e-02
#> [10,] 0.3677921053 0.1895975134 0.0702264885 1.189651e-02
print(power_verbose_output, indent = 4, precision = 6)
#> Test parameters ($inputs) ------------------------------------------------------
#> Initial graph
#> --- Hypothesis weights ---
#> H1: 0.5
#> H2: 0.5
#> H3: 0.0
#> H4: 0.0
#> --- Transition weights ---
#> H1 H2 H3 H4
#> H1 0.0 0.5 0.5 0.0
#> H2 0.5 0.0 0.0 0.5
#> H3 0.0 1.0 0.0 0.0
#> H4 1.0 0.0 0.0 0.0
#> Alpha = 0.025
#> Test types
#> bonferroni: (H1, H2, H3, H4)
#> Simulation parameters ($inputs) ------------------------------------------------
#> Testing 100,000 simulations with multivariate normal params:
#> H1 H2 H3 H4
#> Marginal power: 0.802831 0.802831 0.705414 0.901481
#> Correlation: H1 H2 H3 H4
#> H1 1.00 0.50 0.50 0.25
#> H2 0.50 1.00 0.25 0.50
#> H3 0.50 0.25 1.00 0.50
#> H4 0.25 0.50 0.50 1.00
#> Power calculation ($power) -----------------------------------------------------
#> H1 H2 H3 H4
#> Local power: 0.76396 0.75887 0.56767 0.69133
#> Expected no. of rejections: 2.78183
#> Power to reject 1 or more: 0.85557
#> Power to reject all: 0.51205
#> Success measure Power
#> H1 0.76396
#> Expected no. of rejections 2.78183
#> AtLeast1 0.85557
#> All 0.51205
#> H1andH2 0.66726
#> (H1andH3)or(H2andH4) 0.74695
#> Simulation details ($details) --------------------------------------------------
#> p_sim_H1 p_sim_H2 p_sim_H3 p_sim_H4 rej_H1 rej_H2 rej_H3 rej_H4
#> 3.08204e-02 1.20654e-02 4.11858e-03 9.32434e-02 FALSE TRUE FALSE FALSE
#> 7.93372e-04 6.49905e-04 2.45178e-02 2.96560e-03 TRUE TRUE TRUE TRUE
#> 3.02992e-02 5.95396e-02 5.43083e-02 2.62583e-02 FALSE FALSE FALSE FALSE
#> 9.74332e-03 3.31857e-03 7.41721e-04 4.02469e-04 TRUE TRUE TRUE TRUE
#> 1.97135e-02 8.61618e-03 1.64183e-02 2.41833e-07 TRUE TRUE TRUE TRUE
#> 3.12066e-03 6.70231e-03 1.37441e-02 2.75170e-04 TRUE TRUE TRUE TRUE
#> 3.02208e-02 1.42376e-01 6.03828e-03 2.11740e-02 FALSE FALSE FALSE FALSE
#> 2.49757e-03 2.94026e-02 4.14273e-04 2.20779e-03 TRUE FALSE TRUE FALSE
#> 6.18994e-02 3.87257e-02 3.16613e-01 5.69979e-02 FALSE FALSE FALSE FALSE
#> 3.67792e-01 1.89598e-01 7.02265e-02 1.18965e-02 FALSE FALSE FALSE FALSE
#> ... (Use `print(x, rows = <nn>)` for more) | {"url":"https://cran.rstudio.com/web/packages/graphicalMCP/vignettes/shortcut-testing.html","timestamp":"2024-11-01T20:04:35Z","content_type":"text/html","content_length":"82461","record_id":"<urn:uuid:2129c5b2-9da0-4b0f-bd13-4daf57ab323c>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00355.warc.gz"} |
Transactions Online
Koichi HIRAYAMA, Yoshiyuki YANAGIMOTO, Jun-ichiro SUGISAKA, Takashi YASUI, "Permittivity Estimation Based on Transmission Coefficient for Gaussian Beam in Free-Space Method" in IEICE TRANSACTIONS on
Electronics, vol. E106-C, no. 6, pp. 335-343, June 2023, doi: 10.1587/transele.2022ECP5030.
Abstract: In a free-space method using a pair of horn antennas with dielectric lenses, we demonstrated that the permittivity of a sample can be estimated with good accuracy by equalizing a measured
transmission coefficient of a sample to a transmission coefficient for a Gaussian beam, which is approximately equal to the transmission coefficient for a plane wave multiplied by a term that changes
the phase. In this permittivity estimation method, because the spot size at the beam waist in a Gaussian beam needs to be determined, we proposed an estimation method of the spot size by employing
the measurement of the Line in Thru-Reflect-Line calibration; thus, no additional measurement is required. The permittivity estimation method was investigated for the E-band (60-90 GHz), and it was
demonstrated that the relative permittivity of air with a thickness of 2mm and a sample with the relative permittivity of 2.05 and a thickness of 1mm is estimated with errors less than ±
0.5% and ±0.2%, respectively. Moreover, in measuring a sample without displacing the receiving horn antenna to avoid the error in measurement, we derived an expression of the permittivity
estimation for S parameters measured using a vector network analyzer, and demonstrated that the measurement of a sample without antenna displacement is valid.
URL: https://global.ieice.org/en_transactions/electronics/10.1587/transele.2022ECP5030/_p
author={Koichi HIRAYAMA, Yoshiyuki YANAGIMOTO, Jun-ichiro SUGISAKA, Takashi YASUI, },
journal={IEICE TRANSACTIONS on Electronics},
title={Permittivity Estimation Based on Transmission Coefficient for Gaussian Beam in Free-Space Method},
abstract={In a free-space method using a pair of horn antennas with dielectric lenses, we demonstrated that the permittivity of a sample can be estimated with good accuracy by equalizing a measured
transmission coefficient of a sample to a transmission coefficient for a Gaussian beam, which is approximately equal to the transmission coefficient for a plane wave multiplied by a term that changes
the phase. In this permittivity estimation method, because the spot size at the beam waist in a Gaussian beam needs to be determined, we proposed an estimation method of the spot size by employing
the measurement of the Line in Thru-Reflect-Line calibration; thus, no additional measurement is required. The permittivity estimation method was investigated for the E-band (60-90 GHz), and it was
demonstrated that the relative permittivity of air with a thickness of 2mm and a sample with the relative permittivity of 2.05 and a thickness of 1mm is estimated with errors less than ±
0.5% and ±0.2%, respectively. Moreover, in measuring a sample without displacing the receiving horn antenna to avoid the error in measurement, we derived an expression of the permittivity
estimation for S parameters measured using a vector network analyzer, and demonstrated that the measurement of a sample without antenna displacement is valid.},
TY - JOUR
TI - Permittivity Estimation Based on Transmission Coefficient for Gaussian Beam in Free-Space Method
T2 - IEICE TRANSACTIONS on Electronics
SP - 335
EP - 343
AU - Koichi HIRAYAMA
AU - Yoshiyuki YANAGIMOTO
AU - Jun-ichiro SUGISAKA
AU - Takashi YASUI
PY - 2023
DO - 10.1587/transele.2022ECP5030
JO - IEICE TRANSACTIONS on Electronics
SN - 1745-1353
VL - E106-C
IS - 6
JA - IEICE TRANSACTIONS on Electronics
Y1 - June 2023
AB - In a free-space method using a pair of horn antennas with dielectric lenses, we demonstrated that the permittivity of a sample can be estimated with good accuracy by equalizing a measured
transmission coefficient of a sample to a transmission coefficient for a Gaussian beam, which is approximately equal to the transmission coefficient for a plane wave multiplied by a term that changes
the phase. In this permittivity estimation method, because the spot size at the beam waist in a Gaussian beam needs to be determined, we proposed an estimation method of the spot size by employing
the measurement of the Line in Thru-Reflect-Line calibration; thus, no additional measurement is required. The permittivity estimation method was investigated for the E-band (60-90 GHz), and it was
demonstrated that the relative permittivity of air with a thickness of 2mm and a sample with the relative permittivity of 2.05 and a thickness of 1mm is estimated with errors less than ±
0.5% and ±0.2%, respectively. Moreover, in measuring a sample without displacing the receiving horn antenna to avoid the error in measurement, we derived an expression of the permittivity
estimation for S parameters measured using a vector network analyzer, and demonstrated that the measurement of a sample without antenna displacement is valid.
ER - | {"url":"https://global.ieice.org/en_transactions/electronics/10.1587/transele.2022ECP5030/_p","timestamp":"2024-11-03T20:06:05Z","content_type":"text/html","content_length":"65023","record_id":"<urn:uuid:36b217e9-a5c5-48a4-9193-ad343c6b4af1>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00842.warc.gz"} |
Thermodynamic time asymmetry and the Boltzmann equation
An important result of statistical mechanics is the Boltzmann equation, which describes the evolution of the velocity distribution of a gas towards the equilibrium Maxwell distribution. We introduce
the Boltzmann equation by considering a dynamical model of a two-dimensional gas consisting of hard disks. We derive the Boltzmann equation for the model and compare the behavior predicted by this
equation against the actual behavior of the system as observed in computer simulations. A puzzling feature of the Boltzmann equation is that although the dynamical laws governing the gas are
time-reversal invariant, the behavior predicted by the Boltzmann equation is time asymmetric. We show that this time asymmetry arises from assumptions made in the derivation of the Boltzmann
equation, and we use computer simulations of the model system to investigate the circumstances under which these assumptions hold.
All Science Journal Classification (ASJC) codes
• General Physics and Astronomy
Dive into the research topics of 'Thermodynamic time asymmetry and the Boltzmann equation'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/thermodynamic-time-asymmetry-and-the-boltzmann-equation","timestamp":"2024-11-07T20:46:11Z","content_type":"text/html","content_length":"45446","record_id":"<urn:uuid:e71388ee-3de9-4403-9c65-ba42832ded66>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00742.warc.gz"} |
Optical Interferometry, 2e
• 1st Edition - September 22, 2003
• Paperback ISBN:
9 7 8 - 0 - 1 2 - 3 9 9 5 6 1 - 2
• Hardback ISBN:
9 7 8 - 0 - 1 2 - 3 1 1 6 3 0 - 7
• eBook ISBN:
9 7 8 - 0 - 0 8 - 0 4 7 3 6 4 - 2
When the first edition of Optical Interferometry was published, interferometry was regarded as a rather esoteric method of making measurements, largely confined to the laborator… Read more
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When the first edition of Optical Interferometry was published, interferometry was regarded as a rather esoteric method of making measurements, largely confined to the laboratory. Today, however,
besides its use in several fields of research, it has applications in fields as diverse as measurement of length and velocity, sensors for rotation, acceleration, vibration and electrical and
magnetic fields, as well as in microscopy and nanotechnology.
Most topics are discussed first at a level accessible to anyone with a basic knowledge of physical optics, then a more detailed treatment of the topic is undertaken, and finally each topic is
supplemented by a reference list of more than 1000 selected original publications in total.
• Historical development of interferometry
• The laser as a light source
• Two-beam interference
• Techniques for frequency stabilization
• Coherence
• Electronic phase measurements
• Multiple-beam interference
• Quantum effects in optical interference
• Extensive coverage of the applications of interferometry, such as measurements of length, optical testing, interference microscopy, interference spectroscopy, Fourier-transform spectroscopy,
interferometric sensors, nonlinear interferometers, stellar interferometry, and studies of space-time and gravitation
scientists and engineers interested in precision measurements of a range of physical quantities in industry as well as researchers and students in universities, members of organizations such as the
Optical Society of America, SPIE and IEEE who are interested in possible applications in their work.
Chapter 1
Optical interferometry: its development
1.1 The wave theory of light
1.2 Michelson"s experiment
1.3 Measurement of the metre
1.4 Coherence
1.5 Interference filters
1.6 Interference spectroscopy
1.7 The development of the laser
1.8 Electronic techniques
1.9 Heterodyne techniques
1.10 Holographic interferometry
1.11 Speckle interferrometry
1.12 Stellar interferometry
1.13 Relativity and gravitational waves
1.14 Fiber interferometers
1.15 Nonlinear interferometers
1.16 Quantum effects
1.17 Future directions
Chapter 2
Two-beam interference
2.1 Complex representation of light waves
2.2 Interference of two monochromatic waves
2.3 Wavefront division
2.4 Amplitude division
2.4.1Interference in a plane-parallel plate
2.4.2Fizeau fringes
2.4.3Interference in a thin film
2.5 Localization of fringes
2.5.1Nonlocalized fringes
2.5.2Localized fringes
2.5.3Fringes in a plane-parallel plate
2.5.4Fringes in a thin film
2.6 Two-beam interferometers
2.7 The Michelson interferometer
2.7.1Nonlocalized fringes
2.7.2Fringes of equal inclination
2.7.3Fringes of equal thickness
2.8 The Mach-Zehnder interferometer
2.9 The Sagnac interferometer
2.10 Interference with white light
2.11Channeled spectra
2.12 Achromatic fringes
2.13 Interferential color photography
Chapter 3
3.1 Quasi-monochromatic light
3.2 Waves and wave groups
3.3 Phase velocity and group velocity
3.4 The mutual coherence function
3.5 Spatial coherence
3.6 Temporal coherence
3.7 Coherence time and coherence length
3.8 Combined spatial and temporal effects
3.9 Application to a two-beam interferometer
3.10 Source-size effects
3.11 Spectral bandwidth effects
3.12 Spectral coherence
3.13 Polarization effects
Chapter 4
Multiple-beam interference
4.1 Fringes in a plane-parallel plate
4.2 Fringes by reflection
4.3 Fringes in a thin film: fringes of equal thickness
4.4 Fringes of equal chromatic order
4.5 Fringes of superposition
4.6 Three-beam fringes
4.7 Double-passed fringes
Chapter 5
The laser as a light source
5.1 Gas lasers
5.2 Laser modes
5.2.1Modes of a confocal resonator
5.2.2Generalized spherical resonator
5.2.3Longitudinal modes
5.2.4Single-frequency operation
5.3 Comparison of laser frequencies
5.4 Frequency stabilization
5.4.1Polarization stabilized laser
5.4.2Stabilized transverse Zeeman laser
5.4.3Stabilization on the Lamb dip
5.4.4Stabilization by saturated absorption
5.4.5Stabilization by saturated fluorescence
5.5 Semiconductor lasers
5.6 Ruby and Nd:YAG lasers
5.7 Dye lasers
5.8 Laser beams
Chapter 6
Electronic techniques
6.1 Photoelectric setting methods
6.2 Fringe counting
6.3 Heterodyne interferometry
6.4 Computer-aided fringe analysis
6.4.1Fourier transform techniques
6.5 Phase-shifting interferometry
6.5.1Error-correcting algorithms
6.6 Techniques of phase shifting
6.6.1Frequency shifting
6.6.2Polarization techniques
Chapter 7
Measurements of length
7.1 Line standards
7.2 End standards
7.3 The integral interference order
7.4 Exact fractions
7.5 The refractive index of air
7.6 The international prototype metre
7.7 The 86Kr standard
7.8 Frequency measurements
7.9 The definition of the metre
7.10 Length measurements with lasers
7.10.1Two-wavelength interferometry
7.10.2Frequency-modulation interferometry
7.11 Changes in length
Chapter 8
Optical testing
8.1 The Fizeau interferometer
8.2 The Twyman-Green interferometer
8.3 Unequal-path interferometers
8.4 Phase unwrapping
8.5 Analysis of wavefront aberrations
8.5.1Zernike polynomials
8.5.2Wavefront fitting
8.6 Shearing interferometers
8.6.1Lateral shearing interferometers
8.6.2Interpretation of interferograms
8.6.3Rotational and radial shearing
8.7 Grating interferometers
8.8 The scatter-plate interferometer
8.9 The point-diffraction interferometer
8.10 Computerized test methods
8.10.1Absolute tests for flatness
8.10.2Small-scale irregularities
8.10.3Sources of error
8.10.4Subaperture testing
8.10.5Testing aspheric surfaces
8.10.6Computer-generated holograms
8.11 Testing of rough surfaces
8.12 The optical transfer function
Chapter 9
Interference microscopy
9.1 The Mirau interferometer
9.2 Common-path interference microscopes
9.3 Polarization interferometers
9.3.1Lateral shear
9.3.2Radial shear
9.4 The Nomarski interferometer
9.5 Electronic phase measurements
9.5.1Phase-shifting techniques
9.6 Surface profiling with white light
9.6.1Achromatic phase-shifting
9.6.2Spectrally resolved interferometry
Chapter 10
Interferometric sensors
10.1 Rotation sensing
10.1.1Ring lasers
10.1.2Ring interferometers
10.2 Laser-feedback interferometers
10.2.1Diode-laser interferometers
10.3 Fiber interferometers
10.4 Multiplexed fiber-optic sensors
10.5 Doppler interferometry
10.5.1Laser-Doppler velocimetry
10.5.2Measurements of surface velocities
10.6 Vibration measurements
10.7 Magnetic fields
10.8 Adaptive optical systems
Chapter 11
Interference spectroscopy
11.1 Etendue of an interferometer
11.2 The Fabry-Perot interferometer
11.3 The scanning Fabry-Perot interferometer
11.4 The spherical-mirror Fabry-Perot interferometer
11.5 The multiple Fabry-Perot interferometer
11.6 The multiple-pass Fabry-Perot interferometer
11.7 Holographic filters
11.8 Birefringent filters
11.9 Wavelength meters
11.9.1Dynamic wavelength meters
11.9.2Static wavelength meters
11.10 Heterodyne techniques
11.11 Measurements of laser linewidths
Chapter 12
Fourier-transform spectroscopy
12.1 The etendue and multiplex advantages
12.2 Theory
12.3 Resolution and apodization
12.4 Sampling
12.5 Effect of source and detector size
12.6 Field widening
12.7 Phase correction
12.8 Noise
12.9 Pre-filtering
12.10 Interferometers for Fourier-transform spectroscopy
12.11 Computation of the spectrum
12.12 Applications
Chapter 13
Nonlinear interferometers
13.1 Interferometry with pulsed lasers
13.2 Second-harmonic interferometers
13.2.1Critical phase matching
13.3 Phase-conjugate interferometers
13.3.1Phase-conjugating mirrors
13.4 Interferometers using active elements
13.5 Photorefractive oscillators
13.6 Measurements of nonlinear susceptibilities
Chapter 14
Stellar interferometry
14.1 Michelson"s stellar interferometer
14.2 The intensity interferometer
14.3 Heterodyne stellar interferometry
14.3.1Large heterodyne interferometer
14.4 Long-baseline interferometers
14.5 Stellar speckle interferometry
14.6 Telescope arrays
Chapter 15
Space-time and gravitation
15.1 The Michelson-Morley experiment
15.2 Gravitational waves
15.3 Gravitational-wave detectors
15.4 LIGO
15.5 The standard quantum limit
15.6 Squeezed states of light
15.7 Interferometry below the SQL
Chapter 16
Single-photon interferometry
16.1 Interferometry at the "single-photon" level
16.2 Interference - the quantum picture
16.3 Sources of nonclassical light
16.3.1Parametric down-conversion
16.4 The beam splitter
16.5 Interference with single-photon states
16.6 The geometric phase
16.6.1Observations at the "single-photon" level
16.6.2Observations with single-photon states
16.7 Interference with independent sources
16.7.1Observations at the "single-photon" level
16.7.2Observations in the time domain
16.8 Superposition states
Chapter 17
Fourth-order interference
17.1 Nonclassical fourth-order interference
17.2 Interference in separated interferometers
17.3 The geometric phase
Chapter 18
Two-photon interferometry
18.1 Interferometric tests of Bell"s inequality
18.2 Tests using unbalanced interferometers
18.3 Two-photon interference
18.4 The quantum eraser
18.5 Single-photon tunneling
18.5.1Dispersion cancellation
18.5.2Measurements of tunneling time
18.6 Conclusions
Appendix A
Two-dimensional linear systems
A.1 The Fourier transform
A.2 Convolution and correlation
A.3 The Dirac delta function
A.4 Random functions
Appendix B
The Fresnel-Kirchhoff integral
Appendix C
Reflection and transmission at a surface
C.1 The Fresnel transform
C.2 The Stokes relations
Appendix D
The Jones calculus
Appendix E
The geometric phase
E.1 The Poincare sphere
E.2 The Pancharatnam phase
Appendix F
F.1 The off-axis hologram
F.2 Volume holograms
F.3 Computer-generated holograms
Appendix G
G.1 Speckle statistics
G.2 Second-order statistics
G.3 Image speckle
G.4 Young"s fringes
G.5 Addition of speckle patterns
Author index
Subject index
• Published: September 22, 2003
• Paperback ISBN: 9780123995612
• Hardback ISBN: 9780123116307
• eBook ISBN: 9780080473642 | {"url":"https://shop.elsevier.com/books/optical-interferometry-2e/hariharan/978-0-12-311630-7","timestamp":"2024-11-02T09:15:40Z","content_type":"text/html","content_length":"205228","record_id":"<urn:uuid:b2527735-440e-4a72-a458-fe548d041679>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00466.warc.gz"} |
Hazen-Williams Formula and Constant 61.9: Dimensions and Applicability
What are the dimensions of the constant 61.9?
The dimensions of the constant 61.9 are:
A. [L^5/T]
B. [L^3/T]
C. [L/T]
D. [L^5/T^2]
Can the Hazen-Williams formula be used with confidence for various liquids and gases?
Explain the confidence level of the formula for different fluids.
The dimensions of the constant 61.9 are [L^5/T], indicating that the constant is derived from length to the power of 5 divided by time.
The Hazen-Williams formula's confidence in predicting flow for various liquids and gases is limited.
The Hazen-Williams hydraulics formula is commonly used in engineering to calculate the volume rate of flow through pipes. The formula includes a constant, 61.9, which has specific dimensions that
ensure the equation remains dimensionally consistent.
By analyzing the units involved in the formula, we can determine the dimensions of the constant 61.9. The constant 61.9 has dimensions of [L^5/T], which means it represents length to the power of 5
divided by time. This dimension aligns with the other parameters in the formula, allowing for accurate calculations of flow rates in pipes.
However, when it comes to the applicability of the Hazen-Williams formula for various liquids and gases, the confidence level decreases. The formula assumes constant fluid properties, which may not
hold true for different fluids. The accuracy of the formula could vary depending on the behavior of the specific liquid or gas flowing through the pipe.
Engineers often need to consider alternative formulas or adjust the constants used in the Hazen-Williams formula to better suit the characteristics of different fluids. It is essential to understand
the limitations of the formula and its applicability to ensure accurate and reliable flow rate calculations in diverse engineering scenarios. | {"url":"https://theletsgos.com/engineering/hazen-williams-formula-and-constant-61-9-dimensions-and-applicability.html","timestamp":"2024-11-07T13:40:35Z","content_type":"text/html","content_length":"22346","record_id":"<urn:uuid:c22353c7-9a49-4629-9db0-805f6439665e>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00656.warc.gz"} |
Counts Analysis of MaxDiff Data
Counts analysis involves counting up the number of times that objects are chosen as best and subtracting the number of times that they are chosen as worst.
Consider the following experimental design, where there are six blocks (sets), six alternatives, each alternative appears three times and each question presents respondents with three alternatives.
alternative 1 2 3 4 5 6
A 0 1 0 1 1 0
B 0 0 1 0 1 1
C 1 0 0 1 1 0
D 1 1 1 0 0 0
E 1 1 0 0 0 1
F 0 0 1 1 0 1
Consider the following choices:
│Block │Best│Worst │
│1 │C │E │
│2 │A │E │
│3 │B │F │
│4 │C │F │
│5 │C │B │
│6 │B │F │
We can count up the number of times an alternative is chosen as best and the number of times it is chosen as worst, and the difference between these is the count. For this experiment, alternative C
is seen to be most preferred, alternatives B and A are equal second, D is fourth, E fifth and F sixth (i.e., C > B = A > D > E > F).
│Alternative │Best│Worst│Count│
│A │1 │0 │1 │
│B │2 │1 │1 │
│C │3 │0 │3 │
│D │0 │0 │0 │
│E │0 │2 │-2 │
│F │0 │3 │-3 │
Comments about counts analysis
1. The highest and lowest counts (3 and -3 in this example) will only occur when a respondent is entirely consistent. Consequently, respondents that provide more consistent data end up having a
higher range of count scores and thus it is not technically valid to compare the counts between respondents (as respondents will differ in their degree of variation). In practice, this issue is
usually ignored (there is no good solution to this problem; this particular problem exists even with the more valid methods for analyzing max-diff data and also plagues Choice Modeling).
2. Counts analysis is not valid, because it ignores the experimental design. Although this conclusion does not seem to be understood by many involved in academic and commercial marketing research,
the conclusion that it is not valid is standard proof taught in experimental design classes in statistics. This is most clear when we look at the relationship between alternatives A and B. The
counts analysis suggests that these two alternatives are equally appealing. However, in the only situation in which they appear together (block 5), alternative B was chosen as worst and thus the
data shows that A is preferred to B, yet the counts analysis fails to recognize this aspect of the data. The more complicated methods for analyzing max-diff data resolve this problem.
Nevertheless, counts analysis is a useful way of inspecting data prior to applying more complicated methods.
3. Discarding the Worst data and only looking at the Best data will potentially further reduce the validity of any counts analysis, as this involves further ignoring of the detail of the
experimental design. Note that when the best data is looked at alternative B is seen to be superior to A, but this is not the case when the Worst data is included. (Note that this conclusion does
not generalize to other forms of analysis of max-diff data, where the rationale for focusing just on the Best scores can be that there is more interest in what is appealing than what is
4. The counts are, at best, ordinal measures of preference. Consider the relationship between the counts for alternatives A and C. That these two differ by 2 only tells us that C is preferred; there
is no way of determining the extent of the preference from the counts analysis or using any other type of analysis).
Unbalanced designs
Where an experimental design is not balanced (i.e., where some alternatives appear more frequently than others), counts analysis as described above will produce biased estimates, because alternatives
that appear more often will have more opportunities to be selected as Best or Worst. This can be resolved by dividing the computation of the number of times an alternative is selected as best by the
number of times it is available, and also doing the same thing for worst.
Counts with multiple respondents
Counts analysis can be performed for an individual respondent, or, for groups of respondents. Where being performed for groups of respondents the computation is identical to when performed for
individual respondents (i.e., the number of times alternatives are selected as best and worst are summed). Interpretation is usually facilitated by dividing the numbers of times alternatives are
selected as Best and Worst by the number of times that they appear (i.e., as when analyzing an unbalanced design).
Other counts formulas
A number of other formulas are occasionally used. For example, some researchers compute the ratio of best to worst scores. Although such transformations may provide some advantage in some situations,
they do not address the fundamental problem of counts analysis (i.e., that it ignores the experimental design).
Counts analysis in statistical software
A useful way of setting up max-diff results in statistical programs is to represent each alternative as a variable, with missing value codes used when alternatives are not shown, a 1 is used to
denote an alternative that is chosen as best, a -1 for worst and a 0 for alternatives shown but neither best nor worst (i.e., the Stacked Layout). For example, in R:
Block A B C D E F
1 NA NA 1 0 -1 NA
2 1 NA NA 0 -1 NA
3 NA 1 NA 0 NA -1
4 0 NA 1 NA NA -1
5 1 -1 0 NA NA NA
6 NA 1 NA NA 0 -1
When data is structured in this way the counts can be computed using sums. For example, the following code enters this data into R and computes the counts:
mdData = matrix(c(NA,NA,1,0,-1,NA,1,NA,NA,0,-1,NA,NA,1,NA,0,NA,-1,0,NA,1,NA,NA,-1,0,-1,1,NA,NA,NA,NA,1,NA,NA,0,-1),6,byrow=TRUE, dimnames=list(Block=1:6,Alternatives = LETTERS[1:6]))
apply(mdData,2,sum, na.rm=TRUE)
Similarly, for unbalanced designs, the code in R uses mean instead of sum:
apply(mdData,2,mean, na.rm=TRUE)
Aggregate counts are computed for multiple respondents by Stacking the data.
Also known as
Counting Analysis
See also
See MaxDiff for links to key resources and concepts relating to MaxDiff.
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Mobile Math: Trapezoids & Circles
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What is one set of possible values for the shapes?
• If a circle has a value of 38.25, what would the value of a trapezoid be? How do you know?
• If the value of the entire mobile is 2, what would the value of each shape be? How do you know?
If the value of the entire mobile is 4,800, what would the value of each shape be? How do you know?
What are other possible values for the set of shapes? What patterns or rules do you notice? | {"url":"https://mathathome.mathlearningcenter.org/activity/1481","timestamp":"2024-11-08T01:04:55Z","content_type":"text/html","content_length":"20043","record_id":"<urn:uuid:77b33f99-e88a-4724-b5c4-1ea22016e753>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00106.warc.gz"} |
What is the equation for a parabola in vertex form? | HIX Tutor
What is the equation for a parabola in vertex form?
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Answer 2
The equation for a parabola in vertex form is ( y = a(x - h)^2 + k ), where ( (h, k) ) represents the vertex of the parabola and ( a ) determines the direction and width of the parabola's opening.
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Indexed grammar
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Indexed grammars are a generalization of context-free grammars in that nonterminals are equipped with lists of flags, or index symbols. The language produced by an indexed grammar is called an
indexed language.
Modern definition by Hopcroft and Ullman[edit]
In contemporary publications following Hopcroft and Ullman (1979), ^[2] an indexed grammar is formally defined a 5-tuple G = ⟨N,T,F,P,S⟩ where
• N is a set of variables or nonterminal symbols,
• T is a set ("alphabet") of terminal symbols,
• F is a set of so-called index symbols, or indices,
• S ∈ N is the start symbol, and
• P is a finite set of productions.
In productions as well as in derivations of indexed grammars, a string ("stack") σ ∈ F^* of index symbols is attached to every nonterminal symbol A ∈ N, denoted by A[σ].^[note 1] Terminal symbols may
not be followed by index stacks. For an index stack σ ∈ F^* and a string α ∈ (N ∪ T)^* of nonterminal and terminal symbols, α[σ] denotes the result of attaching [σ] to every nonterminal in α; for
example if α equals a B C d E with a,d ∈ T terminal, and B,C,E ∈ N nonterminal symbols, then α[σ] denotes a B[σ] C[σ] d E[σ]. Using this notation, each production in P has to be of the form
1. A[σ] → α[σ],
2. A[σ] → B[fσ], or
3. A[fσ] → α[σ],
where A, B ∈ N are nonterminal symbols, f ∈ F is an index, σ ∈ F^* is a string of index symbols, and α ∈ (N ∪ T)^* is a string of nonterminal and terminal symbols. Some authors write ".." instead of
"σ" for the index stack in production rules; the rule of type 1, 2, and 3 then reads A[..]→α[..], A[..]→B[f..], and A[f..]→α[..], respectively.
Derivations are similar to those in a context-free grammar except for the index stack attached to each nonterminal symbol. When a production like e.g. A[σ] → B[σ]C[σ] is applied, the index stack of A
is copied to both B and C. Moreover, a rule can push an index symbol onto the stack, or pop its "topmost" (i.e., leftmost) index symbol.
Formally, the relation ⇒ ("direct derivation") is defined on the set (N[F^*]∪T)^* of "sentential forms" as follows:
1. If A[σ] → α[σ] is a production of type 1, then β A[φ] γ ⇒ β α[φ] γ, using the above definition. That is, the rule's left hand side's index stack φ is copied to each nonterminal of the right hand
2. If A[σ] → B[fσ] is a production of type 2, then β A[φ] γ ⇒ β B[fφ] γ. That is, the right hand side's index stack is obtained from the left hand side's stack φ by pushing f onto it.
3. If A[fσ] → α[σ] is a production of type 3, then β A[fφ] γ ⇒ β α[φ] γ, using again the definition of α[σ]. That is, the first index f is popped from the left hand side's stack, which is then
distributed to each nonterminal of the right hand side.
As usual, the derivation relation is defined as the reflexive transitive closure of direct derivation ⇒. The language L(G) = { w ∈ T^*: S w } is the set of all strings of terminal symbols derivable
from the start symbol.
Original definition by Aho[edit]
Historically, indexed grammar were introduced by Alfred Aho (1968)^[3] using a different formalism. Aho defined an indexed grammar to be a 5-tuple (N,T,F,P,S) where
1. N is a finite alphabet of variables or nonterminal symbols
2. T is a finite alphabet of terminal symbols
3. F ⊆ 2^N × (N ∪ T)^* is the finite set of so-called flags (each flag is itself a set of so-called index productions)
4. P ⊆ N × (NF^* ∪ T)^* is the finite set of productions
5. S ∈ N is the start symbol
Direct derivations were as follows:
• A production p = (A → X[1]η[1]…X[k]η[k]) from P matches a nonterminal A ∈ N followed by its (possibly empty) string of flags ζ ∈ F^*. In context, γ Aζ δ, via p, derives to γ X[1]θ[1]…X[k]θ[k] δ,
where θ[i] = η[i]ζ if X[i] was a nonterminal and the empty word otherwise. The old flags of A are therefore copied to each new nonterminal produced by p. Each such production can be simulated by
appropriate productions of type 1 and 2 in the Hopcroft/Ullman formalism.
• An index production p = (A → X[1]…X[k]) ∈ f matches Afζ (the flag f it comes from must match the first symbol following the nonterminal A) and copies the remaining index string ζ to each new
nonterminal: γ Afζ δ derives to γ X[1]θ[1]…X[k]θ[k] δ, where θ[i] is the empty word when X[i] is a terminal and ζ when it is a nonterminal. Each such production corresponds to a production of
type 3 in the Hopcroft/Ullman formalism.
This formalism is e.g. used by Hayashi (1973, p. 65-66).^[4]
In practice, stacks of indices can count and remember what rules were applied and in which order. For example, indexed grammars can describe the context-sensitive language of word triples { www : w ∈
{a,b}^* }:
S[σ] → S[fσ] T[fσ] → a T[σ]
S[σ] → S[gσ] T[gσ] → b T[σ]
S[σ] → T[σ] T[σ] T[σ] T[] → ε
A derivation of abbabbabb is then
S[] ⇒ S[g] ⇒ S[gg] ⇒ S[fgg] ⇒ T[fgg] T[fgg] T[fgg] ⇒ a T[gg] T[fgg] T[fgg] ⇒ ab T[g] T[fgg] T[fgg] ⇒ abb T[] T[fgg] T[fgg] ⇒ abb T[fgg] T[fgg] ⇒ ... ⇒ abb abb T[fgg] ⇒ ... ⇒ abb abb abb.
As another example, the grammar G = ⟨ {S,T,A,B,C}, {a,b,c}, {f,g}, P, S ⟩ produces the language { a^nb^nc^n: n ≥ 1 }, where the production set P consists of
S[σ] → T[gσ] A[fσ] → a A[σ] A[gσ] → a
T[σ] → T[fσ] B[fσ] → b B[σ] B[gσ] → b
T[σ] → A[σ] B[σ] C[σ] C[fσ] → c C[σ] C[gσ] → c
An example derivation is
S[] ⇒ T[g] ⇒ T[fg] ⇒ A[fg] B[fg] C[fg] ⇒ aA[g] B[fg] C[fg] ⇒ aA[g] bB[g] C[fg] ⇒ aA[g] bB[g] cC[g] ⇒ aa bB[g] cC[g] ⇒ aa bb cC[g] ⇒ aa bb cc.
Both example languages are known to be not context-free.
Hopcroft and Ullman tend to consider indexed languages as a "natural" class, since they are generated by several formalisms other than indexed grammars, viz.^[5]
Hayashi^[4] generalized the pumping lemma to indexed grammars. Conversely, Gilman^[10]^[11] gives a "shrinking lemma" for indexed languages.
Linear indexed grammars[edit]
Gerald Gazdar has defined a second class, the linear indexed grammars (LIG),^[14] by requiring that at most one nonterminal in each production be specified as receiving the stack,^[note 2] whereas in
an ordinary indexed grammar, all nonterminals receive copies of the stack. Formally, a linear indexed grammar is defined similar to an ordinary indexed grammar, but the production's form requirements
are modified to:
1. A[σ] → α[] B[σ] β[],
2. A[σ] → α[] B[fσ] β[],
3. A[fσ] → α[] B[σ] β[],
where A, B, f, σ, α are used as above, and β ∈ (N ∪ T)^* is a string of nonterminal and terminal symbols like α.^[note 3] Also, the direct derivation relation ⇒ is defined similar to above. This new
class of grammars defines a strictly smaller class of languages,^[15] which belongs to the mildly context-sensitive classes.
The language { www : w ∈ {a,b}^* } is generable by an indexed grammar, but not by a linear indexed grammar, while both { ww : w ∈ {a,b}^* } and { a^n b^n c^n : n ≥ 1 } are generable by a linear
indexed grammar.
If both the original and the modified production rules are admitted, the language class remains the indexed languages.^[16]
Letting σ denote an arbitrary collection of stack symbols, we can define a grammar for the language L = {a^n b^n c^n | n ≥ 1 }^[note 4] as
S[σ] → a S[fσ] c
S[σ] → T[σ]
T[fσ] → T[σ] b
T[] → ε
To derive the string abc we have the steps S[] ⇒ aS[f]c ⇒ aT[f]c ⇒ aT[]bc ⇒ abc.
Similarly: S[] ⇒ aS[f]c ⇒ aaS[ff]cc ⇒ aaT[ff]cc ⇒ aaT[f]bcc ⇒ aaT[]bbcc ⇒ aabbcc.
Computational power[edit]
The linearly indexed languages are a subset of the indexed languages, and thus all LIGs can be recoded as IGs, making the LIGs strictly less powerful than the IGs. A conversion from a LIG to an IG is
relatively simple.^[17] LIG rules in general look approximately like ${\displaystyle X[\sigma ]\to \alpha Y[\sigma ]\beta }$, modulo the push/pop part of a rewrite rule. The symbols ${\displaystyle \
alpha }$ and ${\displaystyle \beta }$ represent strings of terminal and/or non-terminal symbols, and any non-terminal symbol in either must have an empty stack, by the definition of a LIG. This is,
of course, counter to how IGs are defined: in an IG, the non-terminals whose stacks are not being pushed to or popped from must have exactly the same stack as the rewritten non-terminal. Thus,
somehow, we need to have non-terminals in ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ which, despite having non-empty stacks, behave as if they had empty stacks.
Let's consider the rule ${\displaystyle X[\sigma ]\to Y[]Z[\sigma f]}$ as an example case. In converting this to an IG, the replacement for ${\displaystyle Y[]}$ must be some ${\displaystyle Y^{\
prime }[\sigma ]}$ that behaves exactly like ${\displaystyle Y[]}$ regardless of what ${\displaystyle \sigma }$ is. To achieve this, we can simply have a pair of rules that takes any ${\displaystyle
Y^{\prime }[\sigma ]}$ where ${\displaystyle \sigma }$ is not empty, and pops symbols from the stack. Then, when the stack is empty, it can be rewritten as ${\displaystyle Y[]}$.
${\displaystyle Y^{\prime }[\sigma f]\to Y^{\prime }[\sigma ]}$
${\displaystyle Y^{\prime }[]\to Y[]}$
We can apply this in general to derive an IG from an LIG. So for example if the LIG for the language ${\displaystyle \{a^{n}b^{n}c^{n}d^{m}|n\geq 1,m\geq 1\}}$ is as follows:
${\displaystyle S[\sigma ]\to T[\sigma ]V[]}$
${\displaystyle V[]\to d~|~dV[]}$
${\displaystyle T[\sigma ]\to aT[\sigma f]c~|~U[\sigma ]}$
${\displaystyle U[\sigma f]\to bU[\sigma ]}$
${\displaystyle U[]\to \epsilon }$
The sentential rule here is not an IG rule, but using the above conversion algorithm, we can define new rules for ${\displaystyle V^{\prime }}$, changing the grammar to:
${\displaystyle S[\sigma ]\to T[\sigma ]V^{\prime }[\sigma ]}$
${\displaystyle V^{\prime }[\sigma f]\to V^{\prime }[\sigma ]}$
${\displaystyle V^{\prime }[]\to V[]}$
${\displaystyle V[]\to d~|~dV[]}$
${\displaystyle T[\sigma ]\to aT[\sigma f]c~|~U[\sigma ]}$
${\displaystyle U[\sigma f]\to bU[\sigma ]}$
${\displaystyle U[]\to \epsilon }$
Each rule now fits the definition of an IG, in which all the non-terminals in the right hand side of a rewrite rule receive a copy of the rewritten symbol's stack. The indexed grammars are therefore
able to describe all the languages that linearly indexed grammars can describe.
Relation to other formalism[edit]
Vijay-Shanker and Weir (1994)^[18] demonstrates that Linear Indexed Grammars, Combinatory Categorial Grammars, Tree-adjoining Grammars, and Head Grammars all define the same class of string
languages. Their formal definition of linear indexed grammars^[19] differs from the above.
LIGs (and their weakly equivalents) are strictly less expressive (meaning they generate a proper subset) than the languages generated by another family of weakly equivalent formalism, which include:
LCFRS, MCTAG, MCFG and minimalist grammars (MGs). The latter family can (also) be parsed in polynomial time.^[20]
Distributed index grammars[edit]
Another form of indexed grammars, introduced by Staudacher (1993),^[12] is the class of Distributed Index grammars (DIGs). What distinguishes DIGs from Aho's Indexed Grammars is the propagation of
indexes. Unlike Aho's IGs, which distribute the whole symbol stack to all non-terminals during a rewrite operation, DIGs divide the stack into substacks and distributes the substacks to selected
The general rule schema for a binarily distributing rule of DIG is the form
X[f[1]...f[i]f[i+1]...f[n]] → α Y[f[1]...f[i]] β Z[f[i+1]...f[n]] γ
Where α, β, and γ are arbitrary terminal strings. For a ternarily distributing string:
X[f[1]...f[i]f[i+1]...f[j]f[j+1]...f[n]] → α Y[f[1]...f[i]] β Z[f[i+1]...f[j]] γ W[f[j+1]...f[n]] η
And so forth for higher numbers of non-terminals in the right hand side of the rewrite rule. In general, if there are m non-terminals in the right hand side of a rewrite rule, the stack is
partitioned m ways and distributed amongst the new non-terminals. Notice that there is a special case where a partition is empty, which effectively makes the rule a LIG rule. The Distributed Index
languages are therefore a superset of the Linearly Indexed languages.
See also[edit]
1. ^ ^a ^b Hopcroft, John E.; Jeffrey D. Ullman (1979). Introduction to Automata Theory, Languages, and Computation. Addison-Wesley. ISBN 0-201-02988-X.
2. ^ Hopcroft and Ullman (1979),^[1] Sect.14.3, p.389-390. This section is omitted in the 2nd edition 2003.
3. ^ Aho, Alfred (1968). "Indexed grammars—an extension of context-free grammars". Journal of the ACM. 15 (4): 647–671. doi:10.1145/321479.321488.
4. ^ ^a ^b T. Hayashi (1973). "On Derivation Trees of Indexed Grammars - An Extension of the uvxyz Theorem". Publication of the Research Institute for Mathematical Sciences. Research Institute for
Mathematical Sciences. 9 (1): 61–92. doi:10.2977/prims/1195192738.
5. ^ Hopcroft and Ullman (1979),^[1] Bibliographic notes, p.394-395
6. ^ Alfred Aho (1969). "Nested Stack Automata". Journal of the ACM. 16 (3): 383–406. doi:10.1145/321526.321529.
7. ^ Michael J. Fischer (1968). "Grammars with Macro-Like Productions". Proc. 9th Ann. IEEE Symp. on Switching and Automata Theory (SWAT). pp. 131–142.
8. ^ Sheila A. Greibach (1970). "Full AFL's and Nested Iterated Substitution" (PDF). Information and Control. 16 (1): 7–35. doi:10.1016/s0019-9958(70)80039-0.
9. ^ T.S.E. Maibaum (1974). "A Generalized Approach to Formal Languages" (PDF). Journal of Computer and System Sciences. 8 (3): 409–439. doi:10.1016/s0022-0000(74)80031-0.
10. ^ Robert H. Gilman (1996). "A Shrinking Lemma for Indexed Languages" (PDF). Theoretical Computer Science. 163: 277–281. doi:10.1016/0304-3975(96)00244-7.
11. ^ Robert H. Gilman (Sep 1995). "A Shrinking Lemma for Indexed Languages". arXiv:math/9509205.
12. ^ ^a ^b Staudacher, Peter (1993), New frontiers beyond context-freeness: DI-grammars (DIGs) and DI-automata. (PDF), pp. 358–367, archived from the original (PDF) on 2006-07-19
13. ^ David J. Weir; Aravind K. Joshi (1988). "Combinatory Categorial Grammars: Generative Power and Relationship to Linear Context-Free Rewriting Systems". Proc. 26th Meeting Assoc. Comput. Ling
(PDF). pp. 278–285.
14. ^ According to Staudacher (1993, p.361 left, Sect.2.2),^[12] the name "linear indexed grammars" wasn't used in Gazdar's 1988 paper, but appeared later, e.g. in Weir and Joshi (1988).^[13]
15. ^ Gazdar, Gerald (1988). "Applicability of Indexed Grammars to Natural Languages". In U. Reyle and C. Rohrer. Natural Language Parsing and Linguistic Theories. Studies in linguistics and
philosophy. 35. D. Reidel Publishing Company. pp. 69–94. ISBN 1-55608-055-7.
16. ^ Gazdar (1988), Appendix, p.89
17. ^ Gazdar 1988, Appendix, p.89-91
18. ^ Vijay-Shanker, K.; Weir, David J. 1994. "The Equivalence of Four Extensions of Context-Free Grammars". Mathematical Systems Theory. 27 (6): 511–546. doi:10.1007/bf01191624.
19. ^ p.517-518
20. ^ Johan F.A.K. van Benthem; Alice ter Meulen (2010). Handbook of Logic and Language (2nd ed.). Elsevier. p. 404. ISBN 978-0-444-53727-0.
External links[edit] | {"url":"https://static.hlt.bme.hu/semantics/external/pages/v%C3%A9ges_%C3%A1llapot%C3%BA_transzducereket_(FST)/en.wikipedia.org/wiki/Indexed_grammar.html","timestamp":"2024-11-02T17:32:35Z","content_type":"text/html","content_length":"122090","record_id":"<urn:uuid:2209d09f-9041-4023-a1b1-6c6f2a7bbd87>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00451.warc.gz"} |
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Mathematical Reasoning Level C
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Mathematical Reasoning™ helps your child devise strategies to solve a wide variety of math problems. These books emphasize problem solving and computation to build the math reasoning skills necessary
for success in higher level math and math assessments. Complete math curriculum for the typical second grade child. Bright children can use this title at a younger age.
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Math Analogies Level 1
Grade(s): 2-3
Understanding analogies and the ability to reason analogically are important problem-solving skills. There are few resources to assist children in the explict use of analogous reasoning with
quantitative and spatial information. This series fills that void.
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Balance Benders Beginning
Grade(s): 2-6
Pages: 48, perforated, reproducible
Format: student book with answers
Children develop deductive thinking and pre-algebra skills as they solve balance puzzles. Children must analyze each balance to identify the clues, and then synthesize the information to solve the
Our Price: $10.99
Balance Math & More Level 1
Grade(s): 2-5
Activities: 38
Sharpen your child's critical thinking skills, computational skills, and develop algebraic reasoning.
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Mathematical Reasoning Grades 2-4 Supplement
Grade(s): 2-4
Reinforce 2nd to 4th grade math concepts and skills by asking students to apply these skills and concepts to non-routine problems. Applying mathematical knowledge to new problems is the ultimate test
of concept mastery and mathematical reasoning. This is the difference between quantitative reasoning and math. Solutions included.
Our Price: $24.99
Word Problems Grade 2
Spectrum Word Problems provides clear examples of how the math skills students learn in school apply to everyday life with challenging, multi-step word problems, skills that are essential to
proficiency with the Common Core Standards
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Mathematical Reasoning Level D
Grade(s): 3
Grade level math presented in a way which bridges the gap between computation and mathematical reasoning. Use for variety, fun and understanding.
Our Price: $42.99
Math Detective Beginning
Grade(s): 3-4
Activities: 38
Math Detective® uses topics and skills drawn from national math standards to prepare your students for advanced math courses and assessments that measure reasoning, reading comprehension, and writing
in math.
Students read a short story that includes a chart, table, or graph. Next they answer critical thinking questions to improve their understanding of the math concept and develop their critical thinking
(comprehension) skills. Students can’t just scan the story for answers—they must carefully analyze and synthesize the information from the text and the chart, table, or graph to explain and support
their answers.
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Patterns in Mathematics
Grade: 3-6
Diverse reproducible activity pages range from completing simple patterns to data sorting and classification. The activities develop conceptual skills that will be important to success in more
abstract math subjects.
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Upper Elementary Challenge Math
Grade(s): 3-5
Designed for students who need more challenging material than they typically encounter in most math curricula. Each chapter contains instruction and problems at 4 levels of difficulty. In addition,
each chapter includes multiple “problem sets” in which a single problem type is presented in increasingly complex steps, illustrating how a simpler problem can progress from one step to multistep
problem solving.
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Singapore Math Challenge Grade 3+
Singapore Math Challenge provides thought-provoking problems, puzzles, and brainteasers that will strengthen mathematical thinking. Step-by-step strategies are clearly explained for solving problems
at varied levels of difficulty. A complete, worked solution is also provided for each problem.
Our Price: $14.99
Word Problems Grade 3
Spectrum Word Problems provides clear examples of how the math skills students learn in school apply to everyday life with challenging, multi-step word problems, skills that are essential to
proficiency with the Common Core Standards
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Math Analogies Level 2
Grade(s): 4-5
Understanding analogies and the ability to reason analogically are important problem-solving skills. There are few resources to assist children in the explict use of analogous reasoning with
quantitative and spatial information. This series fills that void.
Our Price: $11.99
Balance Math & More Level 2
Grade(s): 4-12+
Activities: 38
Pages: 48, reproducible
Sharpen your child's critical thinking skills, computational skills, and develop algebraic reasoning.
Our Price: $10.99
Balance Math Teaches Algebra
Grade(s): 4-12+
Activities: 38
Sharpen your child's critical thinking skills, computational skills, and develop algebraic reasoning.
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Mathematical Reasoning Level E
Grade(s): 4
Grade level math presented in a way which bridges the gap between computation and mathematical reasoning. Use for variety, fun and understanding.
Our Price: $42.99
Mathematical Reasoning Grades 4-6 Supplement
Grade(s): 4-6
Reinforce 4th to 6th grade math concepts and skills by asking students to apply these skills and concepts to non-routine problems. Applying mathematical knowledge to new problems is the ultimate test
of concept mastery and mathematical reasoning. This is the difference between quantitative reasoning and math. Solutions included.
Our Price: $24.99
Balance Benders Level 1
Grade(s): 4-12+
Activities: 39
Children develop deductive thinking and pre-algebra skills as they solve balance puzzles. Children must analyze each balance to identify the clues, and then synthesize the information to solve the
Our Price: $10.99
Thinking Kids’™ Math Analogies Grade 4
This book offers differentiated pages that enable teachers to effectively cover all NCTM strands while building building skills and providing targeted practice to meet individual student needs.
Thinking Kids’™ Math Analogies teaches basic number relationships and higher-level thinking by infusing basic math and problem-solving skills into each problem. The activities cover each strand with
three levels of difficulty to allow for differentiated instruction.
Our Price: $11.99
Word Problems Grade 4
Spectrum Word Problems provides clear examples of how the math skills students learn in school apply to everyday life with challenging, multi-step word problems, skills that are essential to
proficiency with the Common Core Standards
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Challenge Math
Grade(s): 4-9
Challenge Math provides enrichment math activities and develops student problem solving skills. Over 1000 problems at three levels of difficulty to challenge even the brightest students.
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The 10 Things All Future Mathematicians and Scientists Must Know
Grade(s): 4-12
Mathematicians and scientists have been closely tied to many famous disasters like the Challenger explosion. Discover the ten things our future mathematicians and scientists must know to prevent
these kinds of tragedies from occurring and learn about the strong connections between mathematics and science in the real world.
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Becoming a Problem Solving Genius
Grade(s): 4-12
Every math student needs a tool belt of problem solving strategies to call upon when solving word problems. In addition to many traditional strategies, this book includes new techniques such as Think
1, the 2-10 method, and others developed. Each chapter contains problems at five levels of difficulty to meet the needs of not only the average math student, but also the highly gifted.
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25 Real Life Math Investigations That Will Astound Teachers & Students
Grade(s): 4-12
Fascinating math investigations will keep students engaged as they cut through deception and flawed thinking. Problems are presented at four levels of difficulty, and detailed solutions are provided.
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Real World Algebra
Grade(s): 4-9
Algebra is often taught in an abstract manner with little or no emphasis on what algebra is or how it can be used to solve real problems. Real World Algebra explains how word problems can be
"translated" into the language of Mathematics in an easy to understand format using cartoons and drawings.
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Singapore Math Challenge Grade 4+
Singapore Math Challenge provides thought-provoking problems, puzzles, and brainteasers that will strengthen mathematical thinking. Step-by-step strategies are clearly explained for solving problems
at varied levels of difficulty. A complete, worked solution is also provided for each problem.
Our Price: $14.99
Math Dictionary for Kids
Grade(s): 4-9
Pages: 219, color
A must-have for parents and students alike, this comprehensive resource provides definitions, descriptions, and illustrations in all areas of elementary and middle school mathematics. It is designed
to explain the language and concepts of mathematics in easily understood terms for school and home use.
Our Price: $14.95
Pattern Explorer
Grades: 5-7
Pages: 96, perforated, reproducible
Rich and diverse collections of pattern problems for students to explore, investigate, discover, and create. Forty activity sets in each book are divided into five themes that appear in rotating
Our Price: $14.99
Math Detective A1
Grade(s): 5-6
Activities: 40
Math Detective® uses topics and skills drawn from national math standards to prepare your students for advanced math courses and assessments that measure reasoning, reading comprehension, and writing
in math.
Our Price: $19.99
Mathematical Reasoning Level F
Grade(s): 5
Grade level math presented in a way which bridges the gap between computation and mathematical reasoning. Use for variety, fun and understanding.
Our Price: $42.99
Singapore Math Challenge Grade 5+
Singapore Math Challenge provides thought-provoking problems, puzzles, and brainteasers that will strengthen mathematical thinking. Step-by-step strategies are clearly explained for solving problems
at varied levels of difficulty. A complete, worked solution is also provided for each problem.
Our Price: $14.99
Thinking Kids’™ Math Analogies Grade 5
This book offers differentiated pages that enable teachers to effectively cover all NCTM strands while building building skills and providing targeted practice to meet individual student needs.
Thinking Kids’™ Math Analogies teaches basic number relationships and higher-level thinking by infusing basic math and problem-solving skills into each problem.
Our Price: $11.99
Word Problems Grade 5
Spectrum Word Problems provides clear examples of how the math skills students learn in school apply to everyday life with challenging, multi-step word problems, skills that are essential to
proficiency with the Common Core Standards
Our Price: $9.99
Mathematical Reasoning Level G
Grade(s): 6
Pages: 448, color
Answers included
Grade level math presented in a way which bridges the gap between computation and mathematical reasoning. Use for variety, fun and understanding.
Our Price: $42.99
Math Analogies Level 3
Grade(s): 6-7
Pages: 64, perforated, reproducible
Understanding analogies and the ability to reason analogically are important problem-solving skills. There are few resources to assist children in the explicit use of analogous reasoning with
quantitative and spatial information. This series fills that void.
Our Price: $11.99
Dare to Compare: Math Level 2
Grade(s): 6-7
Pages: 92, perforated, reproducible
This collection of 150 problems recasts more traditional math problems from a single calculation to two or more calculations to to make a comparison and come to a final conclusion.
Our Price: $12.99
Balance Benders Level 2
Grade(s): 6-12+
Activities: 39
Children develop deductive thinking and pre-algebra skills as they solve balance puzzles. Children must analyze each balance to identify the clues, and then synthesize the information to solve the
Our Price: $10.99
Balance Math & More Level 3
Grade(s): 6-12+
Activities: 38
Sharpen your child's critical thinking skills, computational skills, and develop algebraic reasoning.
Our Price: $10.99
Learning Adventures with Garfield: Math Conundrums
Grade(s): 6-8
Activities: 88
What do you get when you mix the antics of a cynical (but lovable) cat with dozens of clever (and challenging) math problems? You get unique ventures that put your math skills to good use - plus get
a lot of adventuresome, mind-stretching fun.
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Now You Know! Volume 1
Grade(s): 6-adult
700 fun and challenging questions (and answers) designed for advanced students in 6th grade and up. Each challenge page contains a question in each of seven curriculum areas including: math, money,
science, geography, history/government, facts & factoids, and language of algebra. Questions represent a range of relevant and important knowledge areas that will prompt students into further
research and discussion.
Our Price: $19.95
Now You Know! Volume 2
Grade(s): 6-adult
700 fun and challenging questions (and answers) designed for advanced students in 6th grade and up. Each challenge page contains a question in each of seven curriculum areas including: math, money,
science, geography, history/government, facts & analogies, and factoids. Questions represent a range of relevant and important knowledge areas that will prompt students into further research and
Our Price: $19.95
Word Problems Grade 6
Spectrum Word Problems provides clear examples of how the math skills students learn in school apply to everyday life with challenging, multi-step word problems, skills that are essential to
proficiency with the Common Core Standards
Our Price: $9.99
Pre-Algebra Grade 6+
This book provides all the instruction, practice, and review students need to understand Pre-Algebra from the beginning of the year to the end. Step-by-step explanations, complete practice problems,
and fun extension activities will help students in grades 6 and up master even the most challenging topics.
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Math Practice Grade 6+
Pages: 128, Perforated
Reproducible for single classroom use only
The 100+ Series, Math Practice, offers in-depth practice and review for challenging middle school math topics including ratios and proportional relationships, the number system, expressions and
equations, geometry, and statistics and probability. Bonus activities on each page help extend the learning and activities, making these books perfect for daily review in the classroom or at home.
Our Price: $12.99
Mathematical Reasoning Middle School Supplement
Grade(s): 7-9
Reinforce 7th and 8th grade math concepts and skills by asking students to apply these skills and concepts to non-routine problems. Applying mathematical knowledge to new problems is the ultimate
test of concept mastery and mathematical reasoning. This is the difference between quantitative reasoning and math. Solutions included.
Our Price: $26.99
Understanding Geometry (Mathematical Reasoning)
Grade(s): 7-9
Successful completion of this middle school geometry course will virtually guarantee a student’s success in high school geometry. Although the geometric concepts of perimeter, area, and volume are
taught piecemeal from elementary school through middle school, many students enter high school without the skills and knowledge to succeed in high school geometry. This book teaches students an
understanding of the reasoning behind the properties taught in geometry instead of merely asking them to memorize them.
Our Price: $36.99
Understanding Algebra I (Mathematical Reasoning)
Grade(s): 7-9
Pages: 368 (color)
Algebra is the gateway to success in future math and science courses in high school. Students who have a solid algebra background will have no trouble with the algebra problems from SAT and even the
GRE. Understanding Algebra I highlights vocabulary, notation, and has examples from the history of math.
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Algebra I & II Key Concepts, Practice, and Quizzes
Grade(s): 7-12+
Pages: 304, perforated, color, reproducible
Algebra Test Prep and Review provides students with a clear, easy-to-understand, step-by-step review of Algebra I & II. It can be used to supplement an algebra textbook or as a study guide for
students taking standardized tests covering algebra. Students learn helpful procedures and strategies for solving algebra word problems using real-world application examples.
Our Price: $27.99
Balance Benders Level 3
Grade(s): 8-12+
Activities: 39
Children develop deductive thinking and pre-algebra skills as they solve balance puzzles. Children must analyze each balance to identify the clues, and then synthesize the information to solve the
Our Price: $10.99
Math Analogies Level 4
Grade(s): 8-9
Understanding analogies and the ability to reason analogically are important problem-solving skills. There are few resources to assist children in the explict use of analogous reasoning with
quantitative and spatial information. This series fills that void.
Our Price: $11.99 | {"url":"http://www.thinktonight.com/Mathematics_s/2.htm","timestamp":"2024-11-10T15:59:28Z","content_type":"text/html","content_length":"204436","record_id":"<urn:uuid:87ff3be9-7d92-44c4-8c5f-47eee22423d0>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00615.warc.gz"} |
Estimates of sensitivity and detrimental interference levels:
The theoretical considerations for the sensitivity of radio astronomy systems can be used to estimate quantitatively the sensitivities and interference levels for radio astronomical observations. The
results based on an assumed integration time t of 2000 seconds (see: “ITU-R Handbook on Radio Astronomy, 1995, section 4.2.2) are given in Recommendation ITU-R RA769. The sensitivity, expressed in
units of temperature or power spectral density, is the level at the receiver input required to increase the output by an amount equal to the rms noise fluctuations.
The detrimental interference levels given in ITU-R RA769 are expressed as the interference level which introduces into the standard deviation of the power, dP (or of the temperature, dT) a component
equal to 10% of the r.m.s. fluctuations due to the system noise, i.e.:
dP[H] = 0.1 dP[s] df (1)
dP[H]: level of harmful interference;
dP[s]: noise fluctuation in power spectral density in the sensitivity equation – expressed in system temperature as dT;
df: for continuum observations: the bandwidth of the allocated radio astronomy band;
for spectral line observations: the channel bandwidth representative of a spectral line – the values used for df correspond to a velocity of 3 km/s, which is intermediate between values common for
spectral lines of sources within our galaxy and in external galaxies.
Interference can be expressed in terms of the power flux-density incident at the antenna, either in the total bandwidth or as a spectral power flux-density S[H] per 1 Hz of bandwidth. As discussed in
the ITU-R Handbook on Radio Astronomy (1995, section 4.2.1), the values in ITU-R RA 769 are given for an antenna having a gain, in the direction of arrival of the interference, equal to that of an
isotropic antenna (which has an effective area of c^2/4 π f^2, where c is the speed of light and f is the frequency). Values of S[H]df, in units of dB(W/m^2), are derived from dP[H], in dBW, by
20 log f - 158.5 dB (2)
where f is in Hz. S[H] is then derived by subtracting 10 log df to allow for the bandwidth. S[H] can also be expressed as a single equation as follows:
0.4 π k(T[A] + T[R])f^2
S[H] = ---------------- (3)
c^2(t df)^0.5
The sensitivity of a radio astronomy receiving system to wideband (continuum) radiation improves when the bandwidth is increased. The reason for this is as follows: the noise power increases with
bandwidth, but, since the signal also is broadband noise, so does the signal. The signal-to-noise power ratio in the RF or IF stages before the detector remains constant, independent of the
bandwidth. However, as the bandwidth increases, the precision of the determination of the power levels improves as (bandwidth)^0.5, and thus the sensitivity is correspondingly improved.
Based on the theoretical considerations one may assume that one may achieve any desired sensitivity by making the bandwidth and/or the observing time large enough. In practice, however, factors other
than the statistical ones put a practical limit on the sensitivity of a radio astronomy observation. Examples of such factors are the stability of the receiver and fluctuations in the attenuation and
phase path in the Earth’s atmosphere. The sensitivity levels given in Recommendation ITU-R RA769 use values for the bandwidth and integration time for which these other factors usually are
insignificant. However, it should be emphasised that these sensitivity levels are not fundamental and that they are routinely exceeded in cases where the data can be integrated over periods of many
hours (from: “ITU-R Handbook on Radio Astronomy“, 1995, section 4.3.2). | {"url":"https://www.craf.eu/useful-equations/estimates-of-sensitivity-and-detrimental-interference-levels/","timestamp":"2024-11-05T11:54:22Z","content_type":"text/html","content_length":"44881","record_id":"<urn:uuid:42c82108-f4b2-4645-824d-76db356e3f00>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00748.warc.gz"} |
Mathematical Sciences Research Institute
Analytic continuation of p-adic modular forms and applications to modularity
August 18, 2014 (03:30 PM PDT - 04:30 PM PDT) Speaker(s): Payman Kassaei (McGill University)
Location: SLMath: Eisenbud Auditorium
Primary Mathematics Subject Classification No Primary AMS MSC Secondary Mathematics Subject Classification No Secondary AMS MSC
The lecture series will start with a brief introduction to rigid analytic geometry. I will then introduce modular curves from various viewpoints (complex analytic, algebraic, and p-adic analytic) and
use them to give a geometric definition of p-adic and overconvergent modular forms and Hecke operators. I will next show how to use the p-adic geometry of the modular curves towards p-adic analytic
continuation of overconvergent modular forms. Finally, I will demonstrate an application of these results to modularity of certain Galois representations which can itself be used to prove certain
cases of the Artin conjecture. If time allows, I would explain briefly how these ideas extend to higher dimensions by illustrating the easier case of Hilbert modular surfaces. | {"url":"https://legacy.slmath.org/workshops/710/schedules/18747","timestamp":"2024-11-01T23:53:59Z","content_type":"text/html","content_length":"38046","record_id":"<urn:uuid:fef923f0-b7ea-4f25-ad5f-63c6b8912d84>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00217.warc.gz"} |
PPT - Performance Limits of Accelerator Dipole and Quadrupole for Muon Collider
Performance Limits of Accelerator Dipole and Quadrupole for Muon Collider
Utilizing Python code, the study analyzes the limits of accelerator dipole and quadrupole for the Muon Collider. Analytic formulas are implemented to assess the behavior of these components based on
critical current density, operating temperatures, and superconductor materials. The study explores limitations associated with superconductor properties and discusses margins related to enthalpy and
temperature. Considerations for mechanical stress, cost, and protection are also evaluated through bore diameter versus magnetic field graphs. The analysis introduces general assumptions, dipole
magnetic field, quadrupole gradient, and discussions on margins.
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1. Performance limits of accelerator dipole and quadrupole for the Muon Collider D.Novelli1,3, A.Bersani1, B.Caiffi1, S.Farinon1, S.Mariotto2, T.Salmi4, ... 1INFN Genoa ,2INFN - Milan 3Sapienza
University of Rome 4Tampere University IMCC Annual Meeting 2023 WP7 Task 4
2. Introduction A Python code is used to implement the analytic formulas for the dipole and quadrupole in cos-theta and sector coil approximation. Starting from the fit of the critical current
density, the short sample magnetic field (dipole) and the short sample gradient (quadrupole) are computed, for three different operating temperatures and superconductors. This provides the
limitations due to the nature of the superconductor itself as a function of temperature, type of material and quantity of material (coil width). By adding a margin to the short sample values, the
graphs become more realistic. This is followed by a discussion of margin in terms of enthalpy and temperature A graph of the bore diameter versus magnetic field is useful to show the allowed
region taking into account the mechanical (stress), cost and protection limitations. Some approximations are used to validate the procedure, and then more complicated configurations will be
studied (for example sector magnets at a higher order of approximation, with iron yoke and grading on the current density). 1 IMCC Annual Meeting 2023 WP7 Task 4
3. General Assumptions This work is under the no-iron hypothesis. Analytic formulas in cos-theta approximation or, similarly, in sector coil (60 for the dipole and 30 for the quadrupole) at the
first order of approximation. Temperature of the cold mass, 3 options: 1.9 K NbTi 4.2 K Nb3Sn and ReBCO 20 K ReBCO Superconducting materials: NbTi Nb3Sn ReBCO the LHC cable as a reference the FCC
target performance as a reference Filling factor of pure superconductor = 0.3 the Fujikura FESC AP tape as a reference Filling factor of pure superconductor = 0.02 Filling factor of pure
superconductor = 0.27 2 IMCC Annual Meeting 2023 WP7 Task 4
4. Dipole Magnetic Field These margins are indicative; they are used to observe behaviour. The coil width will be strongly limited by the cost. The magnetic field will be strongly limited by the
stress. 3 IMCC Annual Meeting 2023 WP7 Task 4
5. Quadrupole Gradient These margins are indicative; they are used to observe behaviour. The coil width will be strongly limited by the cost. The gradient will be strongly limited by the stress. 4
IMCC Annual Meeting 2023 WP7 Task 4
6. Discussion on the Margins The margin on the load line can be expressed in terms of margin in temperature. We have chosen two indicative and reasonable values for margins on the load line for LTS
and HTS, which are: 20% for the LTS (graph on the left) and 5% for the HTS (graph on the right). We can reduce (below 20%) the margin on the load line for the Nb3Sn. We can reduce (below 5%) the
margin on the load line for the ReBCO at high temperatures. 5 IMCC Annual Meeting 2023 WP7 Task 4
7. Discussion on the Margins It s more physical talking about the enthalpy margin, which is linked to the temperature margin through the thermal capacity of the materials and the compositions of the
strands. Material Operating Temp. [K] Enthalpy Margin [mJ/cm3] Temperature Margin [K] Strand fraction Cost Stress Limit NbTi 1.9 5 2 0.75 200 /kg 100 MPa Nb3Sn 4.2 20 4.5 0.70 2000 /kg 150 MPa
4.2 20 4.8 ReBCO 0.73 8000 /kg 300 MPa 20 90 0.35 NbTi Nb3Sn HL-LHC standard Strand Fraction = 0.40 (Cu) + 0.30 (Sc) ReBCO Enthalpy margin equal to HL-LHC Strand Fraction = 0.20 (Cu) + 0.02 (Sc)
+ 0.01 (Ag) + 0.50 (Hastelloy) LHC standard Strand Fraction = 0.48 (Cu) + 0.27 (Sc) Assume a cost limit of 100 k for the bare superconductor (the same limit for each material). 6 IMCC Annual
Meeting 2023 WP7 Task 4
8. Bore Diameter vs Magnetic Field Material Operating Temp. [K] Enthalpy Margin [mJ/cm3] Temperature Margin [K] Strand fraction Cost Stress Limit NbTi 1.9 5 2 0.75 200 /kg 100 MPa Assumption of J0
as Jc for each value of the magnetic field For each value of the magnetic field one can compute Jc(B, T) Knowing B and J0, the coil width can be computed from the load line formula The coil width
increase with increasing B and decreasing Jc 7 IMCC Annual Meeting 2023 WP7 Task 4
9. Bore Diameter vs Magnetic Field Material Operating Temp. [K] Enthalpy Margin [mJ/cm3] Temperature Margin [K] Strand fraction Cost Stress Limit Nb3Sn 4.2 20 4.5 0.70 2000 /kg 150 MPa Assumption of
J0 as Jc for each value of the magnetic field For each value of the magnetic field one can compute Jc(B, T) Knowing B and J0, the coil width can be computed from the load line formula The coil
width increase with increasing B and decreasing Jc 8 IMCC Annual Meeting 2023 WP7 Task 4
10. Bore Diameter vs Magnetic Field Material Operating Temp. [K] Enthalpy Margin [mJ/cm3] Temperature Margin [K] Strand fraction Cost Stress Limit 4.2 20 4.8 ReBCO 0.73 8000 /kg 300 MPa 20 90 0.35
Knowing B and J0, the coil width can be computed. Assumption of J0 as Jc for each value of the magnetic field The coil width increase with increasing B and decreasing Jc For each value of the
magnetic field one can compute Jc(B, T) 9 IMCC Annual Meeting 2023 WP7 Task 4
11. Bore Diameter vs Magnetic Field A studied option is to fix the current density (450 A/mm2, for example). This is interesting because using J0 as Jc, in each point of the magnetic field we are
using the maximum current density allowed by the properties of the material, but, of course, this is not usually done. We will work below the critical current density, particularly at low fields.
Also, working at the maximum current density can be a problem for the protection of the magnet; it should require huge coil width to dissipate the energy in case of quench. A solution can be
found using a graded current density between LF and HF. ????? Limited by the cost and stress ReBCO is limited by the cost ?bTi Limited by the stress 10 IMCC Annual Meeting 2023 WP7 Task 4
12. Protection Limit for Constant Cost NbTi, Hotspot temperature limit 350 K : Low conductor cost allows large coil and low current density protection does not limit design Nb3Sn, hotspot temperature
limit 350 K: Higher cost starts to limit the coil size and force higher current density protection may become a limitation NbTi Courtesy by Tiina Salmi ReBCO, hotspot temperature limit 200 K:
High cost requires small coil and very high current density Protection will be a limiting factor Nb3Sn ReBCO 4 K & 20 K In all cases we assume 40 ms protection delay between original quench and
quench protection system efficiency, see details: https://indico.cern.ch/event/1240045/ 11 IMCC Annual Meeting 2023 WP7 Task 4
13. Conclusions Upcoming developments: We are currently working with analytical formulae in cos-theta and sector coil approximation. We are working to include a linear iron yoke (still using an
analytical approach). We plan to include a current grading between LF and HF. We are looking for the best way to get the bore diameter vs magnetic field graph. For the future: We would like to
implement a Python code able to work with the Ansys software, to solve complex configurations that are not analytically tractable, thus making it possible to study multipole sectors and combined
function magnets. 12 IMCC Annual Meeting 2023 WP7 Task 4
14. Thank you for your attention IMCC Annual Meeting 2023 WP7 Task 4
15. Dipole - Performances of NbTi Ratio between normal conductor and superconductor. Voids Insulation Fit s data from the LHC cable Fixed aperture = 75 mm T = 4.2 K Filling factor = 0.3 Cos?
configuration w = 30 mm 14 27/04/2023 WP7 Task 4
16. Dipole - Performances of Nb3Sn Ratio between normal conductor and superconductor. Voids Insulation Fit s data from the FCC target performance Fixed aperture = 75 mm Filling factor = 0.3 Cos?
configuration T = 4.2 K w = 30 mm 15 27/04/2023 WP7 Task 4
17. Dipole - Performances of YBCO Fit s data from the Fujikura FESC AP tape Fixed aperture = 75 mm Filling factor = 0.02 Cos? configuration Ratio between the total area of the cable to the
superconductor area. T = 4.2 K w = 30 mm 16 27/04/2023 WP7 Task 4 | {"url":"https://www.slideorbit.com/slide/performance-limits-of-accelerator-dipole-and-quadrupole-for-muon-collider/153184","timestamp":"2024-11-13T21:20:24Z","content_type":"text/html","content_length":"79933","record_id":"<urn:uuid:790bef63-d390-4272-8490-6f2c72f9a33c>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00646.warc.gz"} |
2nd Grade Measurements | Measuring Length, Mass, Ccapacity and Time
2nd Grade Measurements
Practice the sample questions on 2nd grade measurements. The questions are based on measuring length (meter and centimeter), mass (kilogram and gram), capacity (liter and milliliter) and time (year,
months and days).
1. Fill in the gaps:
(i) 1 meter = ____ cm
(ii) November has ____ days.
(iii) 1 kg = ____ g
(iv) 10 m + 5 m + 1 m = ____ m
(v) 250 g = ____ kg
(vi) 3 liters = ____ ml
(vii) 1000 ml = ____ l
(viii) 7 g + 10 g + 5 g = ____ g
(ix) 6 l + 11 l + 3 l = ____ l
(x) Leap year has ____ days.
2. (i) Which is heavier 50 kg weight or 51 kg of sugar.
(ii) Add: 5 kg + 1 5kg = ____
3. (i) What is the main standard unit of capacity?
(ii) How many milliliters are there in one liter?
4. (i) How many days are there in February in a leap year?
(ii) Name of four seasons of India.
5. What are the standard units to measure:
(i) length
(ii) weight
(iii) capacity
(iv) time
6. (i) How many days are there in a leap year?
(ii) How many months are there in a year?
7. Strike out the wrong word:
(i) June has 30/31 days.
(ii) A non-leap year has 365/366 days.
(iii) July comes after/before June.
(iv) Summer comes after/before winter
(v) Liter is the unit of mass/capacity.
8. (i) A bus travels 215 km daily. How much distance will it cover in 6 days?
(ii) 45 kg of rice is packed equally in 5 packets. How much rice is there in each packet?
Answers for the 2nd grade measurements are given below to check the exact answers of the above questions.
1. (i) 100 cm
(ii) 30 days.
(iii) 1000 g
(iv) 16 m
(v) 1/4 kg
(vi) 3000 ml
(vii) 1 l
(viii) 22 g
(ix) 20 l
(x) 366 days.
2. (i) 51 kg of sugar.
(ii) 6 5kg
3. (i) Liter
(ii) 1000 milliliters
4. (i) 29
(ii) Spring, Summer, Autumn and Winter
5. (i) Meter
(ii) Grams
(iii) Liter
(iv) Second
6. (i) 366 days.
(ii) 12 months
7. (i) 30 days.
(ii) 365 days.
(iii) after
(iv) before
(v) capacity.
8. (i) 1290 km.
(ii) 9 kg.
From 2nd Grade Measurements to HOME PAGE
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Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question. | {"url":"https://www.math-only-math.com/2nd-grade-measurements.html","timestamp":"2024-11-12T20:25:23Z","content_type":"text/html","content_length":"33382","record_id":"<urn:uuid:6d51d39e-2a30-467f-8b80-cfa80587b872>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00030.warc.gz"} |
11 Times Table - Multiplication Table with Examples
11 Times Table – Multiplication Table
Multiplication Table of 11
Learning the 11 times table is like taking a step into a new world of mathematics! It’s an essential building block for kids who want to excel in math, and it can be more exciting than you think.
The multiplication table of 11 is unique because it has a special pattern that makes it easier to memorize. Just observe the numbers: 11, 22, 33, 44, 55, and so on. Notice how the digits repeat? This
simple yet fascinating pattern continues all the way through the multiples of 11, up to 11×10 = 110.
Here’s the complete 11 times table for reference:
11 Times Table
Children’s first encounter with the 11 times table might seem like a complicated mountain to climb. However, breaking it down into small, digestible parts can make the climb easier. Unlike other
multiplication tables, the 11 times table presents a repetitive pattern that kids can quickly identify. This repetition not only makes the multiplication table more engaging but also facilitates
faster memorization.
For example:
1. When multiplying 11 by a single-digit number, the result simply repeats that number twice.
2. When multiplying 11 by a two-digit number, simply add the two digits together and place the sum between them. (e.g., 11 x 23 = 2(2+3)3 = 253)
The symmetry and simplicity of the 11 times table create a magical mathematical journey that appeals to children’s natural curiosity and learning spirit.
Tips for 11 Times Table
Looking for ways to make learning the 11 times table fun and engaging? Here are some unique tips to help your child master this multiplication marvel:
1. Visualize the Pattern: Encourage your child to visualize the pattern in the 11 times table. Understanding the pattern can lead to quicker memorization.
2. Use Rhymes or Songs: Creating a catchy song or rhyme using the 11 times table can make it more enjoyable and easier to remember.
3. Practice with Physical Objects: Utilize everyday objects like building blocks or buttons to represent the multiplication visually.
4. Create Flashcards: Flashcards can be a fun way to test knowledge and reinforce the pattern.
By creatively approaching the 11 times table, you’ll help your child build a strong foundation for more advanced mathematical concepts.
11 Times Table Examples
The 11 times table is filled with intriguing examples that reinforce the pattern and simplicity of multiplication by 11. Here are some additional examples:
• 11 x 12 = 132 (Here, 1+2 = 3 is placed between 1 and 2)
• 11 x 15 = 165 (Here, 1+5 = 6 is placed between 1 and 5)
• 11 x 21 = 231 (Here, 2+1 = 3 is placed between 2 and 1)
These examples demonstrate the fascinating pattern within the 11 times table, showing how easy it is to multiply by 11, even with larger numbers.
Practice Problems 11 Times Table
Ready to put the 11 times table into practice? Here are some engaging practice problems for your child to solve:
1. 11 x 13 = ?
2. 11 x 7 = ?
3. 11 x 19 = ?
4. 11 x 10 = ?
5. 11 x 5 = ?
The answers are 143, 77, 209, 110, and 55, respectively. Encourage your child to use the pattern techniques described above to solve these problems. Practice is the key to mastery, and these problems
are a great way to reinforce the learning.
The 11 times table is not only a fascinating journey into the world of multiplication but also an enriching mathematical experience. Its repetitive pattern, easy-to-understand nature, and integration
into daily life make it an essential component of children’s math education. With creativity and continuous practice, mastering the 11 times table can be both fun and rewarding. Encourage your child
to explore this magical multiplication realm and discover the joys of mathematics.
FAQs on 11 Times Table
Why is the 11 times table unique?
It follows a simple and repetitive pattern that makes it easier to memorize and understand.
How can parents help their children learn the 11 times table?
Parents can use visual aids, songs, rhymes, flashcards, and everyday objects to make learning engaging and enjoyable.
Can the pattern of the 11 times table be applied to numbers beyond 10?
Yes, the pattern continues, but with two-digit numbers, you’ll need to add the digits together and place the sum between them.
Learning the 11 times table is an essential step towards mathematical mastery. It’s time to make math brighter with Brighterly!
Poor Level
Weak math proficiency can lead to academic struggles, limited college, and career options, and diminished self-confidence.
Mediocre Level
Weak math proficiency can lead to academic struggles, limited college, and career options, and diminished self-confidence.
Needs Improvement
Start practicing math regularly to avoid your child`s math scores dropping to C or even D.
High Potential
It's important to continue building math proficiency to make sure your child outperforms peers at school. | {"url":"https://brighterly.com/math/11-times-table/","timestamp":"2024-11-02T17:18:22Z","content_type":"text/html","content_length":"90215","record_id":"<urn:uuid:d68bde26-8a89-49e1-b4c6-88bfb65b0883>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00290.warc.gz"} |
Exposing McDonald’s “1 in 4” Mass-Marketing Frauds: A Statistical Analysis | Denise Bauer (non-official)
Exposing McDonald’s “1 in 4” Mass-Marketing Frauds: A Statistical Analysis
Click on this link to visualize the original email: 2023-02-10_1311_0800_fbi_1_in_4_fraud_with_different_variables.pdf
From: Vincent B. Le Corre
Subject: Examples of 1 chance out of 4 with different variables
Date sent: February 10, 2023, 13:11 +0800 (China Standard Time)
To: Adam Rogalski (Legal Attaché/State Department), Federal Bureau of Investigation (FBI)
Cc: Edward Lehman
Note: since Assistant Legal Attaché Adam Rogalski told me on 2021-09-20 that he was “one of the FBI representatives,” I assume that this communication was transferred to the Federal Bureau of
Investigation (FBI).
Click on this link to visualize the original email: 2023-02-10_1311_0800_fbi_1_in_4_fraud_with_different_variables.pdf
Dear Mr. Rogalski,
Some examples, if and only if there are no mistakes in the calculations:
Binomial distribution probability of winning:
Example number 1:
Approximately 1 chance out of 4 with the following variables:
• Number n of successes: 1 or more
• out of 287 Bernoulli trials
• where the result of each Bernoulli trial is true with probability 0.001 (or 1 chance out of 1000 if your prefer, or 0.1%)
Example number 2:
Approximately 1 chance out of 4 with the following variables:
• Number n of successes: 1 or more
• out of 6 Bernoulli trials
• where the result of each Bernoulli trial is true with probability 0.05 (or 1 chance out of 20 if your prefer, or 5%)
Example number 3:
Approximately 1 chance out of 4 with the following variables:
• Number n of successes: exactly 3
• out of 6 Bernoulli trials
• where the result of each Bernoulli trial is true with probability 0.64 (or 1 chance out of 1.5625 if your prefer, or 64%)
Example number 4:
Approximately 1 chance out of 4 with the following variables:
• Number n of successes: 1 or more
• out of 3’000’000’000 Bernoulli trials (3 billion Bernoulli trials)
• where the result of each Bernoulli trial is true with probability 0.0000000001 (or 1 chance out of 10’000’000’000 if your prefer (i.e. 1 chance out of 10 billions), or 0.00000001%)
Mr. Rogalski, I hope you understand that it is possible to make up any numbers you want. It’s really evil. The U.S. government must take down this criminal syndicate.
I just gave you 4 examples. Every time, there is approximately 1 chance out of 4 to win instantly. Yet, the real chances of winning are, respectively:
• 1 chance out of 1000
• 1 chance out of 20
• 1 chance out of 1.5625
• 1 chance out of 10 billions
The U.S. Department of Justice can’t allow such frauds to happen.
“3. Deterrent Effect of Prosecution. Deterrence of criminal conduct, whether it be criminal activity generally or a specific type of criminal conduct, is one of the primary goals of the criminal law.
This purpose should be kept in mind, particularly when deciding whether a prosecution is warranted for an offense that appears to be relatively minor; some offenses, although seemingly not of great
importance by themselves, if commonly committed would have a substantial cumulative impact on the community.”
It’s one of the biggest fraud in history.
Look at this tweet:
(I attach a screenshot for the record)
Of course they “are killing it in sales & [their] competition”. This criminal syndicate has been on steroids for decades with their pattern of racketeering activity. As of right now, they keep
committing the crime of money laundering. This whole criminal enterprise is permeated with dirty money. They would never have been where they are today without all of the crimes they committed, in
all impunity, over the last few decades.
McDonald’s Corporation ≈ Lance Armstrong.
Please stop these criminals now. They are hurting competition. They are hurting consumers. They are defrauding even children. Investors are currently being defrauded because McDonald’s Corporation is
fully aware since 2015 that they are committing money laundering and they’ve never warned investors.
I am going to alert Homeland Security Investigation and you should do the same on your side.
A couple of days ago, I talked to my father on the phone. He said at some point about Americans, talking about law enforcement:
Americans are incorruptible.
Yes, I believe that. I hope it’s true.
I am counting on you.
But I need you to reply to me.
Yours sincerely,
Vincent Le Corre
Click on this link to visualize the original document: Screenshot_2023-02-10_at_12_51_13.jpg
Click on this link to visualize the original document: Screenshot_2023-02-10_at_12_52_03.jpg Click on this link to visualize the original email:
To gain a clearer understanding of the sequence of events in this case, I invite you to view a detailed timeline at the following link:
This timeline provides a comprehensive overview of the key milestones and developments. | {"url":"https://denise-bauer.united-states-of-america.eu/open-letters/exposing-mcdonalds-1-in-4-frauds-through-binomial-distribution-statistical-analysis/","timestamp":"2024-11-13T10:53:56Z","content_type":"text/html","content_length":"30143","record_id":"<urn:uuid:e4fb6fc6-47b1-4c05-80d9-c6906203b53a>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00264.warc.gz"} |
Specular Manifold Sampling for Rendering High-Frequency Caustics and Glints
In Transactions on Graphics (Proceedings of SIGGRAPH 2020)
Rendering of a shop window featuring a combination of challenging-to-sample light paths with specular-diffuse-specular ("SDS") interreflection: the two golden normal-mapped pedestals are illuminated
by spot lights and project intricate caustic patterns following a single reflection from the metallic surface, while the transparent center pedestal generates caustics via double refraction. The
glinty appearance of the shoes arises due to specular microgeometry encoded in a high-frequency normal map. This image was rendered by an ordinary unidirectional path tracer using our new specular
manifold sampling strategy. The remaining noise is due to indirect lighting by caustics, which is not explicitly sampled by our technique.
Scattering from specular surfaces produces complex optical effects that are frequently encountered in realistic scenes: intricate caustics due to focused reflection, multiple refraction, and
high-frequency glints from specular microstructure. Yet, despite their importance and considerable research to this end, sampling of light paths that cause these effects remains a formidable
In this article, we propose a surprisingly simple and general path sampling strategy for specular light paths including the above examples, unifying the previously disjoint areas of caustic and glint
rendering into a single framework. Given two path vertices, our algorithm stochastically finds a specular subpath connecting the endpoints. In contrast to prior work, our method supports
high-frequency normal- or displacement-mapped geometry, samples specular-diffuse-specular ("SDS") paths, and is compatible with standard Monte Carlo methods including unidirectional path tracing.
Both unbiased and biased variants of our approach can be constructed, the latter often significantly reducing variance, which may be appealing in applied settings (e.g. VFX). We demonstrate our
method on a range of challenging scenes and evaluate it against state-of-the-art methods for rendering caustics and glints.
Update (June 17, 2021)
We recently fixed a subtle but important bug in our reference implementation of this project. In particular, there was an oversight in the computation of our proposed angle difference constraints. In
the following we will briefly revisit the relevant part, but please refer to Section 4.4 in the paper for more context.
At the core of our method lies a Newton solver that attempts to find light paths that correctly follow the laws of specular reflection or refraction. In short, each specular vertex of a path is
assigned a constraint function
In the paper, we propose a new constraint that performs specular reflection or refraction (function
It turns out that one needs to be careful regarding the periodicity of the azimuth angles when subtracting them. For instance, the Newton solver does not realize that a value of roughly
The impact of this depends highly on the actual scene complexity, While the difference is not always noticeable, some scenes can benefit considerably from the fix. One such example is shown below,
where a caustic is created by a small area light refracted twice through a dielectric cylinder. Here, the Newton solver convergence rate roughly doubles.
The two insets illustrate the difference on zoomed-in regions of the image. Note how the fixed implementation (right) has reduced variance compared to the original version (left). For the comparison,
both images were rendered at equal time and sample count, and the brightness of the insets is scaled to make the noise outside the main caustic more visible.
See also the following full image comparison, where the slider can be dragged to interactively compare both versions:
Left side: original version. Right side: fixed version.
We are grateful to Héloïse de Dinechin for spotting this bug after carefully investigating some of the light paths where the Newton solver diverged. We updated our author version of the paper and the
video below with re-rendered images that reflect the fixed implementation.
Your browser does not support playing this video.
BibTeX Reference
author = {Tizian Zeltner and Iliyan Georgiev and Wenzel Jakob},
title = {Specular Manifold Sampling for Rendering High-Frequency Caustics and Glints},
journal = {Transactions on Graphics (Proceedings of SIGGRAPH)},
volume = {39},
number = {4},
year = {2020},
month = jul,
doi = {10.1145/3386569.3392408} | {"url":"https://tizianzeltner.com/projects/Zeltner2020Specular/","timestamp":"2024-11-06T07:39:55Z","content_type":"text/html","content_length":"12912","record_id":"<urn:uuid:b6941a23-c61c-4e42-8eea-d850353f2c0c>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00168.warc.gz"} |
RRB 2013 Allahabad (2)
One-fourth of an amount was loaned at simple interest with 2% rate of interest and the remaining part was lent on simple interest at 3% rate of interest What is the average rate of interest for the
whole amount ?
When product of two numbers is 768 and LCM is 96 what is the HCF of these numbers ?
The inlet to a tank can fill it in 4 hours while the outlet to the tank can empty it in 5 hours Both the pipes were opened at 9 Am but after sometime the outlet was closed and it is observed that the
tank was full at 5 PM. At what time was the outlet closed ?
Two bikers A and B started for a race of 120 km with a constant speed of 30 kmph and 40 kmph respectively. After 2 hours A raised his speed and both reached the finish line at same time What was the
speed of A after 2 hours of starting the race ?
An article costs Rs. 80 to the vendor. If he marks the article for 50% more than the cost price and sells it 25% less than the marked price, what is his gain percentage ?
A person used to get 15 litres of petrol for a certain amount but due to the hike in the price he is getting one litre less for the same amount What is the percentage rise in the petrol price ?
Length and breadth of a rectangle are in the ratio 3:4 while the perimeter of it is 56 cm.What is the area of the rectangle ?
A certain amount which was loaned on simple interest doubled in 10 years Then the amount received is loaned on compound interest for another 2 years on the same rate What is the total rise in the
amount after 12 years with the initial principal amount ?
Sum of two positive integers is 10 while their product is 24 What is the LCM of these numbers ?
The ratio of occupied seats to the vacant seats in a bus is 1:4 By the time the bus started some more passengers tried to enter the bus which would have made the ratio 4:1 But as they failed in the
attempt they were seated in an empty bus with same capacity as the previous one.What is the ratio of occupied seats to the vacant seats in this bus ? | {"url":"https://cracku.in/rrb-2013-secondshift-question-paper-solved?page=7","timestamp":"2024-11-11T20:11:27Z","content_type":"text/html","content_length":"158351","record_id":"<urn:uuid:38d6ffaf-478c-4337-8190-11191bea7c5f>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00709.warc.gz"} |
Apple Stocks (Practice Interview Question) | Interview Cake
You only have free questions left (including this one).
You're in!
Writing programming interview questions hasn't made me rich yet ... so I might give up and start trading Apple stocks all day instead.
First, I wanna know how much money I could have made yesterday if I'd been trading Apple stocks all day.
So I grabbed Apple's stock prices from yesterday and put them in a list called stock_prices, where:
• The indices are the time (in minutes) past trade opening time, which was 9:30am local time.
• The values are the price (in US dollars) of one share of Apple stock at that time.
So if the stock cost $500 at 10:30am, that means stock_prices[60] = 500.
Write an efficient function that takes stock_prices and returns the best profit I could have made from one purchase and one sale of one share of Apple stock yesterday.
For example:
stock_prices = [10, 7, 5, 8, 11, 9] get_max_profit(stock_prices) # Returns 6 (buying for $5 and selling for $11)
No "shorting"—you need to buy before you can sell. Also, you can't buy and sell in the same time step—at least 1 minute has to pass.
To start, try writing an example value for stock_prices and finding the maximum profit "by hand." What's your process for figuring out the maximum profit?
The brute force approach would be to try every pair of times (treating the earlier time as the buy time and the later time as the sell time) and see which one is higher.
def get_max_profit(stock_prices): max_profit = 0 # Go through every time for outer_time in xrange(len(stock_prices)): # For every time, go through every other time for inner_time in xrange(len
(stock_prices)): # For each pair, find the earlier and later times earlier_time = min(outer_time, inner_time) later_time = max(outer_time, inner_time) # And use those to find the earlier and later
prices earlier_price = stock_prices[earlier_time] later_price = stock_prices[later_time] # See what our profit would be if we bought at the # earlier price and sold at the later price
potential_profit = later_price - earlier_price # Update max_profit if we can do better max_profit = max(max_profit, potential_profit) return max_profit
But that will take time, since we have two nested loops—for every time, we're going through every other time. Also, it's not correct: we won't ever report a negative profit! Can we do better?
Well, we’re doing a lot of extra work. We’re looking at every pair twice. We know we have to buy before we sell, so in our inner for loop we could just look at every price after the price in our
outer for loop.
That could look like this:
def get_max_profit(stock_prices): max_profit = 0 # Go through every price (with its index as the time) for earlier_time, earlier_price in enumerate(stock_prices): # And go through all the LATER
prices for later_time in xrange(earlier_time + 1, len(stock_prices)): later_price = stock_prices[later_time] # See what our profit would be if we bought at the # earlier price and sold at the later
price potential_profit = later_price - earlier_price # Update max_profit if we can do better max_profit = max(max_profit, potential_profit) return max_profit
What’s our runtime now?
Well, our outer for loop goes through all the times and prices, but our inner for loop goes through one fewer price each time. So our total number of steps is the sum n + (n - 1) + (n - 2) ... + 2 +
1, which is still time.
We can do better!
If we're going to do better than , we're probably going to do it in either or . comes up in sorting and searching algorithms where we're recursively cutting the list in half. It's not obvious that we
can save time by cutting the list in half here. Let's first see how well we can do by looping through the list only once.
Since we're trying to loop through the list once, let's use a greedy approach, where we keep a running max_profit until we reach the end. We'll start our max_profit at $0. As we're iterating, how do
we know if we've found a new max_profit?
At each iteration, our max_profit is either:
1. the same as the max_profit at the last time step, or
2. the max profit we can get by selling at the current_price
How do we know when we have case (2)?
The max profit we can get by selling at the current_price is simply the difference between the current_price and the min_price from earlier in the day. If this difference is greater than the current
max_profit, we have a new max_profit.
So for every price, we’ll need to:
• keep track of the lowest price we’ve seen so far
• see if we can get a better profit
Here’s one possible solution:
def get_max_profit(stock_prices): min_price = stock_prices[0] max_profit = 0 for current_price in stock_prices: # Ensure min_price is the lowest price we've seen so far min_price = min(min_price,
current_price) # See what our profit would be if we bought at the # min price and sold at the current price potential_profit = current_price - min_price # Update max_profit if we can do better
max_profit = max(max_profit, potential_profit) return max_profit
We’re finding the max profit with one pass and constant space!
Are we done? Let’s think about some edge cases. What if the price stays the same? What if the price goes down all day?
If the price doesn't change, the max possible profit is 0. Our function will correctly return that. So we're good.
But if the value goes down all day, we’re in trouble. Our function would return 0, but there’s no way we could break even if the price always goes down.
How can we handle this?
Well, what are our options? Leaving our function as it is and just returning zero is not a reasonable option—we wouldn't know if our best profit was negative or actually zero, so we'd be losing
information. Two reasonable options could be:
1. return a negative profit. “What’s the least badly we could have done?”
2. raise an exception. “We should not have purchased stocks yesterday!”
In this case, it’s probably best to go with option (1). The advantages of returning a negative profit are:
• We more accurately answer the challenge. If profit is "revenue minus expenses", we’re returning the best we could have done.
• It's less opinionated. We'll leave decisions up to our function’s users. It would be easy to wrap our function in a helper function to decide if it’s worth making a purchase.
• We allow ourselves to collect better data. It matters if we would have lost money, and it matters how much we would have lost. If we’re trying to get rich, we’ll probably care about those
How can we adjust our function to return a negative profit if we can only lose money? Initializing max_profit to 0 won’t work...
Well, we started our min_price at the first price, so let’s start our max_profit at the first profit we could get—if we buy at the first time and sell at the second time.
min_price = stock_prices[0] max_profit = stock_prices[1] - stock_prices[0]
But we have the potential for an index out of bounds error here, if stock_prices has fewer than 2 prices.
We do want to raise an exception in that case, since profit requires buying and selling, which we can't do with less than 2 prices. So, let's explicitly check for this case and handle it:
if len(stock_prices) < 2: raise ValueError('Getting a profit requires at least 2 prices') min_price = stock_prices[0] max_profit = stock_prices[1] - stock_prices[0]
Ok, does that work?
No! max_profit is still always 0. What’s happening?
If the price always goes down, min_price is always set to the current_price. So current_price - min_price comes out to 0, which of course will always be greater than a negative profit.
When we’re calculating the max_profit, we need to make sure we never have a case where we try both buying and selling stocks at the current_price.
To make sure we’re always buying at an earlier price, never the current_price, let’s switch the order around so we calculate max_profit before we update min_price.
We'll also need to pay special attention to time 0. Make sure we don't try to buy and sell at time 0.
We’ll greedily walk through the list to track the max profit and lowest price so far.
For every price, we check if:
• we can get a better profit by buying at min_price and selling at the current_price
• we have a new min_price
To start, we initialize:
1. min_price as the first price of the day
2. max_profit as the first profit we could get
We decided to return a negative profit if the price decreases all day and we can't make any money. We could have raised an exception instead, but returning the negative profit is cleaner, makes our
function less opinionated, and ensures we don't lose information.
def get_max_profit(stock_prices): if len(stock_prices) < 2: raise ValueError('Getting a profit requires at least 2 prices') # We'll greedily update min_price and max_profit, so we initialize # them
to the first price and the first possible profit min_price = stock_prices[0] max_profit = stock_prices[1] - stock_prices[0] # Start at the second (index 1) time # We can't sell at the first time,
since we must buy first, # and we can't buy and sell at the same time! # If we started at index 0, we'd try to buy *and* sell at time 0. # This would give a profit of 0, which is a problem if our #
max_profit is supposed to be *negative*--we'd return 0. for current_time in xrange(1, len(stock_prices)): current_price = stock_prices[current_time] # See what our profit would be if we bought at the
# min price and sold at the current price potential_profit = current_price - min_price # Update max_profit if we can do better max_profit = max(max_profit, potential_profit) # Update min_price so
it's always # the lowest price we've seen so far min_price = min(min_price, current_price) return max_profit
time and space. We only loop through the list once.
This one's a good example of the greedy approach in action. Greedy approaches are great because they're fast (usually just one pass through the input). But they don't work for every problem.
How do you know if a problem will lend itself to a greedy approach? Best bet is to try it out and see if it works. Trying out a greedy approach should be one of the first ways you try to break down a
new question.
To try it on a new problem, start by asking yourself:
"Suppose we could come up with the answer in one pass through the input, by simply updating the 'best answer so far' as we went. What additional values would we need to keep updated as we looked at
each item in our input, in order to be able to update the 'best answer so far' in constant time?"
In this case:
The "best answer so far" is, of course, the max profit that we can get based on the prices we've seen so far.
The "additional value" is the minimum price we've seen so far. If we keep that updated, we can use it to calculate the new max profit so far in constant time. The max profit is the larger of:
1. The previous max profit
2. The max profit we can get by selling now (the current price minus the minimum price seen so far)
Try applying this greedy methodology to future questions.
Ready for more?
Check out our full course | {"url":"https://www.interviewcake.com/question/python/stock-price?","timestamp":"2024-11-03T13:48:49Z","content_type":"text/html","content_length":"249229","record_id":"<urn:uuid:24f8e13d-a36a-4257-941e-10904533bcc9>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00429.warc.gz"} |
Distribution of standardized random effects
This plot displays the distribution of the standardized random effects with boxplots or with histograms. Since random effects should follow normal probability laws, it is useful to compare the
distributions to standard Gaussian distributions.
In the following example, one can see the distributions of two parameters of a two-compartment bolus model with linear elimination, estimated on the tobramycin example. On the left, the distributions
are represented as boxplots, in the middle as histograms for the probability density function (PDF), and on the right as cumulative distribution functions (CDF). In each case, marks to compare the
results to standard Gaussian distributions are overlaid: dotted horizontal lines indicate the interquartile interval of a standard Gaussian distribution for the boxplots, and black curves represent
the PDF and CDF of a standard Gaussian distribution.
In boxplots, the dashed line represents the median, the blue box represents the 25th and 75th percentiles (Q1 and Q3), and the whiskers extend to the most extreme data points, outliers excluded.
Outliers are the points larger than Q1 + w(Q3 – Q1) or smaller than Q1 – w(Q3 – Q1) with w=1.5. Outliers are shown with red crosses.
On the figure below, the individuals have been split into two groups according to the value of the continuous covariate CLCR. One can notice differences on the boxplots for the distributions of
random effects between both groups, in particular for the parameter k.
When there is not enough data to well-estimate individual parameters, individual estimates based on the conditional mean or mode (rather than the entire conditional distribution) have a tendency to
shrink towards the population parameter estimate. For the distribution of the standardized random effects, shrinkage leads to bias showing less than the true variation if the individual estimates are
shown using the conditional mean or conditional mode rather than the conditional distribution in this case. The shrinkage can be displayed in the plot by toggling “information” in general settings.
For more details see the page Understanding shrinkage and how to circumvent it.
• General: add/remove the grid and the shrinkage in %.
• Display
□ Distribution function. The user can choose which type of plot is used to represent the distributions of the random effects: boxplots, pdf (probability density function) or cdf (cumulative
distribution function).
□ Individual estimates. The user can define which estimator is used for the definition of the individual parameters and thus for the random effects (conditional mean, conditional mode, or
conditional distribution)
□ Visual cues: If boxplot has been selected, the user can choose to add or hide dotted lines to mark the median or quartiles of a standard Gaussian distribution.
By default, the distributions of simulated random effects are displayed as boxplots.
Standardized random effects in case of IOV
Starting from the 2019 version, in case of IOV and if the conditional distribution is computed, we propose to display the standardized random effect by level of variability. In the presented example,
we put IOV on both ka and V parameters. Thus, in the plot, we proposed to display the several levels of variability for all the parameters. We can then display the standardized random effects for
• the ID level,
• the OCC level (where only the parameters with variability on this level are displayed),
• the ID+OCC level corresponding to the addition of the levels. | {"url":"https://monolixsuite.slp-software.com/monolix/2024R1/distribution-of-standardized-random-effects","timestamp":"2024-11-07T13:30:03Z","content_type":"text/html","content_length":"38782","record_id":"<urn:uuid:a7f2b714-4c82-4e24-9f70-a16bdf128fb8>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00143.warc.gz"} |
Stability at high rotational speed, Taylor-Couette Flow, simpleFoam, MRF / AMI
I am trying to match theory for taylor-couette flow. It is a cylindrical body with the inner wall rotating and the outer wall fixed. No inlets, no outlets. I need assistance with convergence at high
At low rotational speeds (<10 rad/s) the problem converges fine, and the result matches theory for the moment exerted on the inner wall within ~7%.
At high speeds, I can not get the problem to converge, at all. Things I have tried:
-tet mesh with varied mesh densities (netgen in salome)
-tet mesh with boundary layers, varied mesh density (netgen in salome)
-hex meshes via cfMesh, varied mesh densities
-a single mesh with a single inner cellzone, using MRF for that cellzone (no AMI, continuous mesh made with partitions in salome)
-a mesh with inner rotating region and outer fixed region, with MRF applied to the inner region, with communication across the interface with AMI
-all of the above using simplefoam
-I have tried using 5.0-dev and v1712
-I have tried a transient simulation using pimpleFoam
-I have tried various settings in vfSchemes and vfSolutions, including as guided by
For my most recent efforts, I have been using hex meshes via cfMesh, with two inner and outer meshes linked using AMI, solving in simplefoam. Solves fine at low speed. I have noticed that as the
problem diverges the spike in velocity begins at the pressure reference point, and spreads across the domain.
Can anyone advise me on how to get convergence on this problem? Is the fact that the divergence begins at the pressure reference a clue on how to fix it? | {"url":"https://www.cfd-online.com/Forums/openfoam-solving/200786-stability-high-rotational-speed-taylor-couette-flow-simplefoam-mrf-ami.html","timestamp":"2024-11-14T07:32:01Z","content_type":"application/xhtml+xml","content_length":"128155","record_id":"<urn:uuid:77b08e2d-b4be-446a-bf69-335221d6a228>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00370.warc.gz"} |
5 Matching Annotations
1. Jul 2024
one trivial model to explain the zero probability of an | V V 〉 <math display="inline"><semantics> <mrow> <mo stretchy="false">|</mo> <mrow> <mi>V</mi> <mi>V</mi> </mrow> <mo stretchy=
"false">〉</mo> </mrow> </semantics></math> or | H H 〉 <math display="inline"><semantics> <mrow> <mo stretchy="false">|</mo> <mrow> <mi>H</mi> <mi>H</mi> </mrow> <mo stretchy="false">〉
</mo> </mrow> </semantics></math> outcome would be to suppose that classical EM waves were always sent to the two detectors such that one wave was horizontally polarized and the other was
vertically polarized. However, such an ad hoc model would not explain why such a preparation would correspond to this particular state, and it would not be generalizable to other evident
anti-correlations from the very same state. The above quantum example also would exhibit anti-correlations if each photon were measured in the same diagonally polarized basis, while the
ad hoc classical model would not.
☆ this "trivial" model is "ruled out from the beggining"
☆ why do they even mention it?
☆ Because the "anti-correlations" happen in EVERY angle, as log as, both detectors have the SAME angle
☆ That is so because the singlet state has rotational symmetry
We believe that this step has already beentaken, but not fully acknowledged, by a substantial part of the quantum-opticscommunity. For example, three review articles (2−4) on light
squeezing makeextensive use of phase-space diagrams, and one of them(3) states explicitly thatthe photon description of the light field is not helpful in the understanding ofthe
4. Jun 2023
The quantum harmonic oscillator is the quantum analog of the classical harmonic oscillator and is one of the most important model systems in quantum mechanics. This is due in partially to
the fact that an arbitrary potential curve V(x)V(x)V(x) can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point. Furthermore, it is one of the
few quantum-mechanical systems for which an exact, analytical solution exists. Solving other potentials typically require either approximations or numerical approaches to identify the
corresponding eigenstates and eigenvalues (i.e., wavefunctions and energies).
Eigenstates & eigenvalues
6. Feb 2019
ubrics like Quality Matters h
Also look at Open SUNY Course Quality Review OSCQR rubric.
8. Jul 2017 | {"url":"https://hypothes.is/search?q=tag%3A%23QM","timestamp":"2024-11-02T21:27:50Z","content_type":"text/html","content_length":"58394","record_id":"<urn:uuid:7f46b03f-d744-43e0-8161-b68dc2380c1a>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00035.warc.gz"} |
Half-axes in power associative algebras
Let A be a commutative, non-associative algebra over a field F of characteristic ≠2. A half-axis in A is an idempotent e∈A such that e satisfies the Peirce multiplication rules in a Jordan algebra,
and, in addition, the 1-eigenspace of ad[e] (multiplication by e) is one dimensional. In this paper we consider the identities (⁎) x^2x^2=x^4 and x^3x^2=xx^4. We show that if identities (⁎) hold
strictly in A, then one gets (very) interesting identities between elements in the eigenspaces of ad[e] (note that if |F|>3 and the identities (⁎) hold in A, then they hold strictly in A).
Furthermore we prove that if A is a primitive axial algebra of Jordan type half (i.e., A is generated by half-axes), and the identities (⁎) hold strictly in A, then A is a Jordan algebra.
• Axial algebra
• Half-axis
• Jordan algebra
• Power associative algebra
ASJC Scopus subject areas
• Algebra and Number Theory
Dive into the research topics of 'Half-axes in power associative algebras'. Together they form a unique fingerprint. | {"url":"https://cris.bgu.ac.il/en/publications/half-axes-in-power-associative-algebras","timestamp":"2024-11-04T21:39:37Z","content_type":"text/html","content_length":"54329","record_id":"<urn:uuid:317ed2fe-2afe-4d0f-ae11-e72341f6f7b1>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00496.warc.gz"} |
Seasonal flow forecasting in Africa; exploratory studies for large lakes
Articles | Volume 384
© Author(s) 2021. This work is distributed under the Creative Commons Attribution 4.0 License.
Seasonal flow forecasting in Africa; exploratory studies for large lakes
For many applications, it would be extremely useful to have insights into river flows at timescales of a few weeks to months ahead. However, seasonal predictions of this type are necessarily
probabilistic which raises challenges both in generating forecasts and their interpretation. Despite this, an increasing number of studies have shown promising results and this is an active area for
research. In this paper, we discuss insights gained from previous studies using a novel combined water balance and data-driven approach for two of Africa's largest lakes, Lake Victoria and Lake
Malawi. Factors which increased predictability included the unusually long hydrological response times and statistically significant links to ocean-atmosphere processes such as the Indian Ocean
Dipole. Other lessons learned included the benefits of data assimilation and the need for care in the choice of performance metrics.
Seasonal river flow forecasts aim to provide useful information for operations and planning from weeks to months ahead. Potential applications include water supply planning, hydropower generation and
irrigation scheduling. This is a developing area both in terms of the technical approaches used and interpretation of the outputs provided.
The forecasting techniques potentially available include statistical approaches, ensemble streamflow prediction, and the direct input of ensemble rainfall forecasts into hydrological models. However,
forecast skill depends on several factors, including seasonal influences and the catchment size, location and antecedent conditions (e.g. Greuell et al., 2019; Mendoza et al., 2017; Robertson and
Wang, 2012; Sene, 2016). Often a key challenge is that the required forecast lead times exceed hydrological response times, requiring a greater reliance on long-range rainfall forecasts, with the
many uncertainties that entails.
One situation which can improve predictability is when there is considerable storage in a catchment, such as from snowpack, groundwater or large lakes and reservoirs. The Rift Valley lakes of east
and southern Africa provide one such example, and due to their huge size, offer considerable potential for deriving seasonal forecasts over operationally useful timescales.
Here we describe insights gained from previous studies by the authors into two of Africa's largest lakes, Lake Victoria and Lake Malawi. These flow into the White Nile and Shire River respectively. A
novel water balance and statistical approach was used, building on stochastic transfer function modelling techniques previously applied in a wide range of environmental applications (e.g. Young,
2011; Tych and Young, 2012). The approaches used are only briefly outlined here, but are described in more detail in Sene et al. (2017, 2018) and Sene and Tych (2018).
In contrast, most previous studies have focussed on developing statistical links between river flows and external drivers such as climate indices. For example, Siam and Eltahir (2015) describe a
Bayesian model for the Nile incorporating Indian and Pacific Ocean climate indices, while Elganiny and Eldwer (2018) used Artificial Neural Network and ARMA approaches. Gehad et al. (2017) have also
compared forecast skill with ensemble approaches for the Blue Nile. For the Shire River, Jury and Gwazantini (2002) describe a seasonal forecasting model for Lake Malawi using a statistical model
including climate indices, and Jury (2014) describes a detailed statistical analysis of links between climate and hydrology using atmospheric model reanalyses and satellite data.
In this paper, other topics discussed include the potential for real-time updating of model outputs using techniques similar to those used in real-time flood forecasting, and the choice of
performance metrics to deal with a statistically non-stationary response over timescales of months or more. An analytical approach also provided useful insights. Finally, some priorities for future
research are considered in this challenging area, including the prospects for seasonal forecasting for smaller lakes and river basins.
2.1The study area
Lake Victoria lies on the equator and its catchment spans six countries: Burundi, the Democratic Republic of the Congo, Kenya, Rwanda, Tanzania and Uganda (Fig. 1). It plays an important role in
hydropower generation, irrigation and water supply in the region. The lake has some of the longest hydrological records in Africa starting with routine observations of levels in the 1890s and
rainfall a decade later, albeit initially with a sparse network of rain gauges. Regular observations of outflows began in the 1940s, and flows have been regulated for hydropower production since
1953. The lake is the largest in Africa with surface and catchment areas of about 68000 and 194000km^2.
Lake Malawi, also shown in Fig. 1, lies several hundred kilometres to the south and its catchment is mainly in Malawi and Tanzania, with a smaller contribution from Mozambique. Level observations
also began in the 1890s with regular outflow observations since 1948. The lake has been regulated for hydropower production since 1965. Its surface and catchment areas are about 28750 and 95570km^
For both lakes, regional rainfall is affected by the annual passage of the Intertropical Convergence Zone (ITCZ). For Lake Victoria, this results in two main rainfall seasons, which are typically
between March and May and October and December. In contrast, Lake Malawi lies near the southernmost end of ITCZ excursions and much of the catchment experiences a single main rainfall season from
November to April or May.
The datasets used for these studies are described in Sene et al. (2017, 2018) and included rainfall, lake level and lake outflow records. To provide the best possible data coverage, extensive use was
made of previously validated and infilled values, many of which themselves relied on observations and record extension techniques developed during an unusually intensive period of monitoring in the
1970s and 1980s (WMO, 1982, 1983). This meant that monthly data availability and completeness was particularly good and allowed the focus to be on issues related to model structure and performance,
rather than trying to resolve issues due to sparse data. Nevertheless, the records included some of the most extreme flood and drought periods on record. The periods chosen for analysis were 1925 to
1990 for Lake Victoria and 1954 to 1980 for Lake Malawi.
Several climate indices were considered for use in the analyses. Exploratory work typically suggested a weak correspondence with rainfall for El Niño indices at lag times of a few months and a rather
stronger correspondence with an index for the Indian Ocean Dipole. Based on these studies, the indices chosen for use in the simulations were NINO34 (Trenberth, 1997) and JAMSTEC's estimates for the
Dipole Model Index (DMI) and illustrative results are discussed later. More detailed studies into links to global and regional climate typically suggest lag times of about two months for Shire River
flows (Jury, 2014) and values up to several months for east Africa at a seasonal timescale (Nicholson, 2017), although there are many factors and variables to consider when interpreting these
2.2Simulation techniques
The studies used an innovative approach to estimating lake outflows, based on a combination of water balance and transfer function techniques. Only brief details are given here while the full
methodology is described in Sene et al. (2017, 2018).
The starting point for the analyses was the water balance for a lake, which can be expressed as:
$\begin{array}{}\text{(1)}& \frac{\mathrm{d}h}{\mathrm{d}t}=\phantom{\rule{0.125em}{0ex}}N\left(t\right)-\phantom{\rule{0.125em}{0ex}}\frac{{Q}_{\mathrm{o}}\left(t\right)}{A\phantom{\rule{0.125em}
where h is the water level, N is the net inflow, Q[o] is the outflow, A is the surface area, and t is time. Based on previous studies (e.g. Piper et al., 1986) the lake areas were assumed to be
constant and the lake outflows were estimated from an equation of the form Q[o]=ah^b.
The net inflow is normally expressed in the following form:
$\begin{array}{}\text{(2)}& N=P\left(t\right)-E\left(t\right)+\frac{{Q}_{\mathrm{i}}\left(t\right)}{A\left(t\right)}=\phantom{\rule{0.125em}{0ex}}\mathrm{\Delta }h+\phantom{\rule{0.125em}{0ex}}\frac
where P and E are the lake rainfall and evaporation and Q[i] is the tributary inflow. Any additional terms which are difficult to measure or estimate, such as groundwater seepage, were considered to
contribute to the overall model uncertainty.
To simulate the water balance, a data-driven stochastic framework was adopted which has proven useful in many other environmental applications, such as real-time flood forecasting and assessing the
long-term variability in climate records (Beven, 2009; Young, 2013). Some advantages of this approach are that few assumptions are required about the statistical characteristics of the datasets used
and the relationships between them, and that estimates for the uncertainty in model outputs are intrinsic to the approach.
A transfer function solution was sought, inspired by the observation by Young (2011) that for the linear case (b=1) Eq. (1) can be solved in this form without further approximation. The following
more generalised form was adopted, allowing for both serial dependence in model inputs and the influence of external variables (u[t]), such as climate drivers:
$\begin{array}{}\text{(3)}& {y}_{t}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}=\frac{\mathbf{B}\left({z}^{-\mathrm{1}}\right)}{\mathbf{A}\left({z}^{-\mathrm{1}}\right)}{u}_{t-\mathit{\
delta }}\phantom{\rule{0.125em}{0ex}}+\phantom{\rule{0.125em}{0ex}}\frac{\mathbf{D}\left({z}^{-\mathrm{1}}\right)}{\mathbf{C}\left({z}^{-\mathrm{1}}\right)}{e}_{t}\end{array}$
Here, A, B, C and D are polynomial functions and z^−1 is the backward shift operator ${z}^{-i}{y}_{t}={y}_{t-i}$.
The parameter values were estimated using a stochastic recursive estimation approach, which also provides estimates for the uncertainty in parameter values and how they vary over time. Tych and Young
(2012) and Young (2013) provide more details.
2.3Results and discussion
Similar techniques were applied to both Lake Victoria and Lake Malawi and, again, the results are described in detail in Sene et al. (2017, 2018). Here we draw out some key insights which may be
relevant to future studies of these two lakes and to other large lakes in Africa.
2.3.1Net inflows
The net inflows were estimated from the outflow terms in Eq. (2) since, in principle, lake levels and outflows can be measured and infilled with greater precision than the lake rainfall and tributary
inflows. This approach has been widely used in previous studies (e.g. Piper et al., 1986) and avoided the need to estimate the individual inflow terms for which there can often be many uncertainties,
particularly when as here catchments are large and monitoring networks are sparse. As a further refinement, values were expressed in standardised form to focus on the underlying climate signals,
thereby helping to reduce the influence of bias in observations.
For Lake Victoria, cross correlation analyses suggested that r^2 coefficients with lake rainfall were highest (>0.8) for no time delay but still statistically significant with a 1 month delay,
whilst autocorrelation coefficients were highest for a 1 month delay. This provides an indication of the potential forecasting lead times for net inflow from rainfall variability alone whilst, in
contrast, linkages with tributary inflows were considerably less. Similar results were obtained for Lake Malawi and helped to inform the overall structure of the model in Eq. (3).
2.3.2Data assimilation
When forecasting river flows at short timescales, such as in real-time flood forecasting, one widely used approach is to update model outputs based on telemetry data. This approach is called data
assimilation and is effective over timescales comparable with the hydrological response time of a catchment. Clearly, this is unlikely to be the case in many seasonal forecasting applications, but
for a large lake like Lake Victoria, given the huge storage influences it is potentially an option, and the use of this approach was a novel aspect of these studies. Various formulations were
explored for the assimilation component, including an error prediction approach and multiple regression models for the forecast model residuals.
The best forecast skill was obtained by incorporating both El Niño and Indian Ocean Dipole indices into the regression models at lag times of 5–7 months for Lake Victoria, and 6–9 months and 3–5
months respectively for Lake Malawi. This gave better forecast skill than the more usual approach of developing statistical relationships between rainfall and climate indices. In contrast, an
Autoregressive Regressive Moving Average (ARMA) approach for the residuals alone only provided benefits at lead times of 2–3 months.
2.3.3Performance metrics
The data assimilation studies highlighted another important issue, which was the choice of performance metrics to use.
The reason for this was that, for large lakes such as Lake Victoria and Lake Malawi, one characteristic of the lake level and outflow series is that, due to the huge storage, there can be underlying
longer term influences superimposed on the daily and seasonal variations due to rainfall. This non-stationary response requires special consideration and for this study it was considered most
important to focus on the annual peaks in levels and outflows.
Figure 2 shows an example of this type of output, showing the improvements in peak level estimates when climate indices were included in the data assimilation component. This example was for Lake
Victoria and in future studies, a further refinement would be to consider the timing errors in peaks as well. Many other measures could be considered and this is a worthwhile area for future
2.3.4Analytical approaches
In some hydrological situations, an analytical approach can also give useful insights, and for a lake it is possible to solve Eq. (1) analytically for some integer values of b in the outflow term.
These analyses provided some useful additional insights into the key timescales for lake response; for example, suggesting that, following a sudden increase in net inflows, such as during heavy
rainfall, for both lakes an approximately exponential decay in levels might be expected over timescales of 4–5 years, if inflows revert to their long-term mean values.
This provided further evidence of a non-stationary response, and an indication of lake response times due to storage, albeit for a highly idealized situation. Similar idealized solutions might be
sought in other seasonal forecasting applications although normally, of course, a numerical solution is required for further insights.
This study has highlighted some methodological issues arising from exploratory studies into seasonal flow forecasting using a data-driven approach. Some issues to consider in developing models for
other lakes include the potential role of climate indices in data assimilation, the choice of suitable performance metrics and the value of simple analytical solutions to explore the response.
Future studies might also consider the forecast skill gained from including seasonal rainfall and air temperature forecasts as model inputs for comparison with an ensemble streamflow prediction
approach. The use of spatially varying (grid-based) climate indices might also be considered and, for smaller lakes, more consideration given to the spatial relationships between tributary inflows,
perhaps at a daily time step. The modelling framework described here might also be developed further using a so-called State Dependent Parameter approach to integrate the dynamic input-output model
and data assimilation components into a single modelling framework, using a unified State Space form and estimated (forecast) using Kalman Filter tools.
One key objective of studies such as these is to develop techniques that could be used operationally. For large lakes, perhaps the longest-established systems are for the Great Lakes in the USA and
Canada. These combine empirical and physically based models, with precipitation and air temperature outlooks and ensemble forecasts (Gronewold et al., 2011; Bolinger et al., 2017). A Nile Basin Flow
Forecasting System is also at the planning stage and the Eastern Nile Technical Regional Office (ENTRO) issues a Flood Preparedness and Early Warning Bulletin during the flood season, based on short
to medium-range meteorological forecasts. The WMO Hydrological Status and Outlooks (HydroSOS) initiative (http://www.wmo.int, last access: 4 November 2021) is also considering Lake Victoria.
No data sets were used in this article.
This paper draws on previous research by the authors on seasonal flow forecasting using modelling techniques developed jointly by the two co-authors (KS, WT), making use of a software framework
described in Tych and Young (2012). The manuscript was prepared by KS with expert contributions from WT.
The contact author has declared that neither they nor their co-authors have any competing interests.
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This article is part of the special issue “Hydrology of Large River Basins of Africa”. It is a result of the 4th International Conference on the “Hydrology of the Great Rivers of Africa”, Cotonou,
Benin, 13–20 November 2021.
We would like to thank the Centre for Ecology and Hydrology for permission to use the datasets for Lake Victoria referred to in this paper; also Michael Kizza for providing additional independently
derived records for comparison. The WMO (1983) study was led by Chris Kidd and provides a valued contribution to understanding of the hydrology of Lake Malawi.
Beven, K.: Environmental Modelling: An Uncertain Future?, CRC Press, UK, 2009.
Bolinger, R. A., Gronewold, A. D., Kompoltowicz, K., and Fry, L. M.: Application of the NMME in the development of a new Regional Seasonal Climate Forecast Tool, B. Am. Meteorol. Soc., 2017, 555–564,
Elganiny, M. and Eldwer, A.: Enhancing the Forecasting of Monthly Streamflow in the Main Key Stations of the River Nile Basin, Water Resour., 45, 660–671, 2018.
Gehad, N., Doaa, A., Shokry, A., and Tahani, Y.: Flow forecasting and skill assessment in the Blue Nile Basin, Nile Water Sci. Eng. J., 10, 29–37, 2017.
Gronewold, A. D., Clites, A. H., Hunter, T. S., and Stow, C. A.: An appraisal of the Great Lakes advanced hydrologic prediction system, J. Great Lakes Res., 37, 577–583, 2011.
Greuell, W., Franssen, W. H. P., and Hutjes, R. W. A.: Seasonal streamflow forecasts for Europe – Part 2: Sources of skill, Hydrol. Earth Syst. Sci., 23, 371–391, https://doi.org/10.5194/
hess-23-371-2019, 2019.
Jury, M. R.: Malawi's Shire River Fluctuations and Climate, J. Hydrometeorol., 15, 2039–2049, 2014.
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Mendoza, P. A., Wood, A. W., Clark, E., Rothwell, E., Clark, M. P., Nijssen, B., Brekke, L. D., and Arnold, J. R.: An intercomparison of approaches for improving operational seasonal streamflow
forecasts, Hydrol. Earth Syst. Sci., 21, 3915–3935, https://doi.org/10.5194/hess-21-3915-2017, 2017.
Nicholson, S. E.: Climate and climatic variability of rainfall over eastern Africa, Rev. Geophys., 55, 590–635, https://doi.org/10.1002/2016RG000544, 2017.
Piper, B. S., Plinston, D. T., and Sutcliffe, J. V.: The water balance of Lake Victoria, Hydrolog. Sci. J., 31, 25–37, 1986.
Robertson, D. E. and Wang, Q. J.: A Bayesian Approach to Predictor Selection for Seasonal Streamflow Forecasting, J. Hydrometeorol., 13, 155–171, 2012.
Sene, K.: Hydrometeorology: Forecasting and Applications, 2nd Edn., Springer, Dordrecht, 427 pp., 2016.
Sene, K. and Tych, W.: Some challenges in seasonal forecasting for large lakes and reservoirs, Seasonal Forecasting: Meeting User Needs, British Hydrological Society National Meeting, Loughborough,
UK, 2018.
Sene, K., Piper, B., Wykeham, D., McSweeney, R., Tych, W., and Beven, K.: Long-term variations in the net inflow record for Lake Malawi, Hydrol. Res., 48, 851–866, 2017.
Sene, K., Tych, W., and Beven, K.: Exploratory studies into seasonal flow forecasting potential for large lakes, Hydrol. Earth Syst. Sci., 22, 127–141, https://doi.org/10.5194/hess-22-127-2018,
Siam, M. S. and Eltahir, E. A. B.: Explaining and forecasting interannual variability in the flow of the Nile River, Hydrol. Earth Syst. Sci., 19, 1181–1192, https://doi.org/10.5194/hess-19-1181-2015
, 2015.
Sutcliffe, J. V. and Parks, Y. P.: The Hydrology of the Nile, IAHS Special Publication no. 5, IAHS Press, Wallingford, ISBN 978-1-901502-75-6, 192 pp., 1999.
Trenberth, K. E.: The Definition of El Niño, B. Am. Meteorol. Soc., 78, 2771–2777, 1997.
Tych, W. and Young, P. C.: A Matlab software framework for dynamic model emulation, Environ. Model. Softw., 34, 19–29, 2012.
WMO: Hydrometeorological Survey of the Catchments of Lake Victoria, Kyoga and Mobutu Sese Seko, WMO Report, 1982.
WMO: A Water Resources Evaluation of Lake Malawi and the Shire River, WMO Report No. MLW/77/012, 1983.
Young, P. C.: Recursive Estimation and Time-Series Analysis: An introduction for the student and practitioner, 2nd Edn., Springer, 2011.
Young, P. C.: Hypothetico-inductive data-based mechanistic modeling of hydrological systems, Water Resour. Res., 49, 915–935, 2013. | {"url":"https://piahs.copernicus.org/articles/384/289/2021/piahs-384-289-2021.html","timestamp":"2024-11-04T16:47:33Z","content_type":"text/html","content_length":"161830","record_id":"<urn:uuid:c4e76f8b-50a9-4573-856b-b0f83ef24939>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00165.warc.gz"} |
Name of class
A \emph{Tarski algebra} is a structure $\mathbf{A}=\langle A,\to\rangle$ of type $\langle 2\rangle$ such that $\to$ satisfies the following identities:
$(x\to y)\to x=x$
$(x\to y)\to y=(y\to x)\to x$
$x\to(y\to z)=y\to(x\to z)$
Let $\mathbf{A}$ and $\mathbf{B}$ be Tarski algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(x \to y)=h(x) \to h(y)$
Example 1: $\langle\{0,1\},\to\rangle$ where $x\to y=0$ iff $x=1$ and $y=0$.
Basic results
Tarski algebras are the implication subreducts of Boolean algebras.
Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.
Finite members
f(1)= &1\\
f(2)= &\\
f(3)= &\\
f(4)= &\\
f(5)= &\\
\end{array}$ $\begin{array}{lr}
f(6)= &\\
f(7)= &\\
f(8)= &\\
f(9)= &\\
f(10)= &\\
[[...]] subvariety
[[...]] expansion
[[...]] supervariety
[[...]] subreduct | {"url":"https://math.chapman.edu/~jipsen/structures/doku.php?id=tarski_algebras","timestamp":"2024-11-14T20:37:17Z","content_type":"application/xhtml+xml","content_length":"19345","record_id":"<urn:uuid:ddbb1e41-b5e2-4478-a0da-5939c364efd3>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00166.warc.gz"} |
See the vignette "Roundoff error and tied times" for a more detailed explanation of the timefix option. In short, when time intervals are created via subtraction then two time intervals that are
actually identical can appear to be different due to floating point round off error, which in turn can make coxph and survfit results dependent on things such as the order in which operations were
done or the particular computer that they were run on. Such cases are unfortunatedly not rare in practice. The timefix=TRUE option adds logic similar to all.equal to ensure reliable results. In
analysis of simulated data sets, however, where often by defintion there can be no duplicates, the option will often need to be set to FALSE to avoid spurious merging of close numeric values. | {"url":"https://www.rdocumentation.org/packages/survival/versions/2.42-3/topics/coxph.control","timestamp":"2024-11-09T22:09:23Z","content_type":"text/html","content_length":"59928","record_id":"<urn:uuid:fdf0dc27-ad6b-4fdc-af8a-65642e5c9c50>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00224.warc.gz"} |
Kruskal-Wallis – Exact
This tool is used to estimate the exact P-Value using Monte Carlo. Typically this would not be necessary unless the sample sizes were smaller (each sample N <= 5 for Kruskal-Wallis), but this gives
continuity on the example.
Computing an exact P-Value for Kruskal-Wallis is very computationally intensive. The Network Model by Mehta and Patel cannot be used for this test (see Appendix Exact and Monte Carlo P-Values for
Nonparametric and Contingency Tests). In this example, the total number of permutations are:
(31+42+27)! / (31! * 42! * 27!) = 7.42 E44
(i.e., more than the number of stars in the observable universe). So we will not attempt to compute the exact, but rather use Monte Carlo.
1. Open Customer Data.xlsx, click on Sheet 1 tab (or press F4 to activate last worksheet).
2. Click SigmaXL > Statistical Tools > Nonparametric Tests - Exact > Kruskal-Wallis - Exact. If necessary, check Use Entire Data Table, click Next.
3. Ensure that Stacked Column Format is checked. Select Overall Satisfaction, click Numeric Data Variable (Y) >>; select Customer Type, click Group Category (X) >>. Select Monte Carlo Exact with the
Number of Replications = 1e6 and Confidence Level for P-Value = 99%. One million replications are used because the expected P-Value is very small as estimated from the “large sample”
Kruskal-Wallis above. This will take up to a minute to run, so if you have a slow computer, use 1e5 replications instead of 1e6.
Tip: The Monte Carlo 99% confidence interval for P-Value is not the same as a confidence interval on the test statistic due to data sampling error. The confidence level for the hypothesis test
statistic is still 95%, so all reported P-Values less than .05 will be highlighted in red to indicate significance. The 99% Monte Carlo P-Value confidence interval is due to the uncertainty in
Monte Carlo sampling, and it becomes smaller as the number of replications increases (irrespective of the data sample size). The Exact P-Value will lie within the stated Monte Carlo confidence
interval 99% of the time.
4. Click OK.
Click on cell B16 to view the P-Value with more decimal place precision (or change the cell format to scientific notation). The Monte Carlo P-Value here is 0.000009 (9 e-6) with a 99% confidence
interval of .000002 (2 e-6) to 0.000016 (1.6 e-5). This will be slightly different every time it is run (the Monte Carlo seed value is derived from the system clock). So we reject H0: at least
one pairwise set of medians are not equal.
Note that the large sample (asymptotic) P-Value of 2.3 e-5 lies outside of the Monte Carlo exact confidence interval.
5. Now we will consider a small sample problem. Open Snore Study.xlsx. This data is from:
Gibbons, J.D. and Chakraborti, S. (2010). Nonparametric Statistical Inference (5th Edition). New York: Chapman & Hall, (Example 10.2.1 data, page 347; Example 10.4.2 analysis, pp. 360 – 362).
An experiment was conducted to determine which device is the most effective in stopping snoring or at least in reducing it. Fifteen men who are habitual snorers were divided randomly into three
groups to test the devices. Each man’s sleep was monitored for one night by a machine that measures the amount of snoring on a 100-point scale while using a device.
6. Select Snore Study Data tab. Click SigmaXL > Statistical Tools > Nonparametric Tests – Exact > Kruskal-Wallis - Exact. If necessary, check Use Entire Data Table, click Next.
7. With Unstacked Column Format checked, select Device A, Device B and Device C, click Numeric Data Variables (Y) >>. Select Exact with the default Time Limit for Exact Computation = 60 seconds.
8. Click OK. Results:
With the Exact P-Value = 0.0042 we reject H0, and conclude that there is a significant difference in median snore study scores. This exact P-Value matches that given in the reference textbook
using SAS and StatXact.
By way of comparison, we will now rerun the analysis using the “large sample” or “asymptotic” Kruskal-Wallis test.
9. Select Snore Study Data tab (or press F4 to activate last worksheet). Click SigmaXL > Statistical Tools > Nonparametric Tests > Kruskal-Wallis. If necessary, check Use Entire Data Table, click
10. With Unstacked Column Format checked, select Device A, Device B and Device C, click Numeric Data Variables (Y) >>.
11. Click OK. Results:
With the P-Value = .0118 we reject H0 (using alpha = .05), but note that if we were using alpha = 0.01, we would have incorrectly failed to reject the null hypothesis. This “large sample” P-Value
matches that given in the reference textbook using Minitab.
In conclusion, whenever you have a small sample size and are performing a Nonparametric test, always use the Exact option.
Web Demos
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Click here to view some now!
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Phone: 1.888.SigmaXL (744.6295)
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Information: Information@SigmaXL.com | {"url":"https://www.sigmaxl.com/Kruskal-Wallis-TestExact.shtml","timestamp":"2024-11-12T17:04:36Z","content_type":"application/xhtml+xml","content_length":"45283","record_id":"<urn:uuid:b83e5022-1b49-44b6-8907-8642dda131e0>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00442.warc.gz"} |
Meters to Miles Conver
How to Convert Meters to Miles with the Length Conversion Formula
Converting meters (m) to miles (mi) is a straightforward mathematical process that involves a simple conversion factor: One meter equals 0.000621371 miles. This ratio is all you need to perform the
calculation. To calculate the conversion, start with the length in meters you need to convert and multiply the value by 0.000621371. This will output the corresponding length in miles.
x meters * 0.000621371 = Number of Miles
For example: If you have a measurement of 1,000 meters, multiplying it by 0.000621371 will convert it to approximately 0.621371 miles. This formula, centering on multiplication by 0.000621371, is
suitable for any meter to mile conversion, and will provide accurate and reliable results. Conversely, if you wanted to convert miles to meters, you would simply multiply the number of miles by
1,609.34, as there are 1,609.34 meters in a mile. For the most part, converting between two units of measurement is a simple matter of multiplication or division. Just keep in mind that your answer
may end up being a fraction or decimal, so maybe keep a calculator on hand or use our online converter!
Common Meters to Miles Conversion Table
Meters (m) Miles (mi)
100 m 0.0621371 mi
500 m 0.310685 mi
1,000 m 0.621371 mi
5,000 m 3.10685 mi
10,000 m 6.21371 mi
50,000 m 31.0685 mi
100,000 m 62.1371 mi
In-Depth on the Meter!
The meter, the base unit of length in the International System of Units (SI), is widely used around the globe. It is equivalent to approximately 39.37 inches. The meter is a crucial measurement in
fields like science, engineering, and construction. It is used for a variety of measurements, from the dimensions of a room to the length of a race track. Fun fact: In France and Turkey meter is
often referred to as metre.
In-Depth on the Mile!
The mile, a unit of length in the Imperial system, is primarily used in the United States and the United Kingdom. One mile is equivalent to 1,609.34 meters or approximately 5,280 feet. The mile is a
key unit in various fields and is commonly used to measure distances in running and driving. Fun fact: The international mile is the most common mile type used for length calculations.
Good luck, and don't forget to bookmark this m to mi length converter to save time when you need help converting a metric system number to the imperial system. | {"url":"https://www.calculyte.com/convert/length/meter-to-mile.html","timestamp":"2024-11-13T11:46:40Z","content_type":"text/html","content_length":"17692","record_id":"<urn:uuid:2eba6555-5b16-463e-be0b-cc93f27fe07c>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00898.warc.gz"} |
MULTIVARIATE ANALYSIS
Paleontological data sets, whether based on fossil occurrences or morphology, often have high dimensionality. PAST includes several methods for multivariate data analysis, including methods that are
specific to paleontology and biology.
Principal components analysis (PCA) is a procedure for finding hypothetical variables (components) that account for as much of the variance in a multidimensional data set as possible (Davis 1986,
Harper 1999). These new variables are linear combinations of the original variables. PCA is a standard method for reducing the dimensionality of morphometric and ecological data. The PCA routine
finds the eigenvalues and eigenvectors of the variance-covariance matrix or the correlation matrix. The eigenvalues, giving a measure of the variance accounted for by the corresponding eigenvectors
(components), are displayed together with the percentages of variance accounted for by each of these components. A scatter plot of these data projected onto the principal components is provided,
along with the option of including the Minimal Spanning Tree, which is the shortest possible set of connected lines joining all points. This may be used as a visual aid in grouping close points (
Harper 1999). The component loadings can also be plotted. Bruton and Owen (1988) describe a typical morphometrical application of PCA.
Principal coordinates analysis (PCO) is another ordination method, somewhat similar to PCA. The PCO routine finds the eigenvalues and eigenvectors of a matrix containing the distances between all
data points, measured with the Gower distance or the Euclidean distance. The PCO algorithm used in PAST was taken from Davis (1986), which also includes a more detailed description of the method and
example analysis.
Correspondence analysis (CA) is a further ordination method, somewhat similar to PCA, but for counted or discrete data. Correspondence analysis can compare associations containing counts of taxa or
counted taxa across associations. Also, CA is more suitable if it is expected that species have unimodal responses to the underlying parameters, that is they favor a certain range of the parameter
and become rare under for lower and higher values (this is in contrast to PCA, that assumes a linear response). The CA algorithm employed in PAST is taken from Davis (1986), which also includes a
more detailed description of the method and example analysis. Ordination of both samples and taxa can be plotted in the same CA coordinate system, whose axes will normally be interpreted in terms of
environmental parameters (e.g., water depth, type of substrate temperature).
The Detrended Correspondence (DCA) module uses the same 'reciprocal averaging' algorithm as the program Decorana (Hill and Gauch 1980). It is specialized for use on "ecological" data sets with
abundance data (taxa in rows, localities in columns), and it has become a standard method for studying gradients in such data. Detrending is a type of normalization procedure in two steps. The first
step involves an attempt to "straighten out" points lying along an arch-like pattern (= Kendall's Horseshoe). The second step involves "spreading out" the points to avoid artificial clustering at the
edges of the plot.
Hierarchical clustering routines produce a dendrogram showing how and where data points can be clustered (Davis 1986, Harper 1999). Clustering is one of the most commonly used methods of multivariate
data analysis in paleontology. Both R-mode clustering (groupings of taxa), and Q-mode clustering (grouping variables or associations) can be carried out within PAST by transposing the data matrix.
Three different clustering algorithms are available: the unweighted pair-group average (UPGMA) algorithm, the single linkage (nearest neighbor) algorithm, and Ward's method. The
similarity-association matrix upon which the clusters are based can be computed using nine different indices: Euclidean distance, correlation (using Pearson's r or Spearman's
Seriation of an absence-presence matrix can be performed using the algorithm described by Brower and Kyle (1988). For constrained seriation, columns should be ordered according to some external
criterion (normally stratigraphic level) or positioned along a presumed faunal gradient. Seriation routines attempt to reorganize the data matrix such that the presences are concentrated along the
diagonal. Also, in the constrained mode, the program runs a 'Monte Carlo' simulation to determine whether the original matrix is more informative than a random matrix. In the unconstrained mode both
rows and columns are free to move: the method then amounts to a simple form of ordination.
The degree of separation between to hypothesized groups (e.g., species or morphs) can be investigated using discriminant analysis (Davis 1986). Given two sets of multivariate data, an axis is
constructed that maximizes the differences between the sets. The two sets are then plotted along this axis using a histogram. The null hypothesis of group means equality is tested using Hotelling's T
^2 test. | {"url":"https://palaeo-electronica.org/2001_1/past/multi.htm","timestamp":"2024-11-03T09:26:18Z","content_type":"text/html","content_length":"9213","record_id":"<urn:uuid:e4d77da1-164c-484c-99e0-72365462a43b>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00788.warc.gz"} |
14.4 Heat Transfer Methods
Learning Objectives
Learning Objectives
By the end of this section, you will be able to do the following:
• Discuss the different methods of heat transfer
Equally as interesting as the effects of heat transfer on a system are the methods by which this occurs. Whenever there is a temperature difference, heat transfer occurs. Heat transfer may occur
rapidly, such as through a cooking pan, or slowly, such as through the walls of a picnic ice chest. We can control rates of heat transfer by choosing materials, such as thick wool clothing for the
winter, controlling air movement, such as the use of weather stripping around doors, or by choice of color, such as a white roof to reflect summer sunlight. So many processes involve heat transfer,
so that it is hard to imagine a situation where no heat transfer occurs. Yet every process involving heat transfer takes place by only three methods:
1. Conduction is heat transfer through stationary matter by physical contact. The matter is stationary on a macroscopic scale—we know there is thermal motion of the atoms and molecules at any
temperature above absolute zero. Heat transferred between the electric burner of a stove and the bottom of a pan is transferred by conduction.
2. Convection is the heat transfer by the macroscopic movement of a fluid. This type of transfer takes place in a forced-air furnace and in weather systems, for example.
3. Heat transfer by radiation occurs when microwaves, infrared radiation, visible light, or another form of electromagnetic radiation is emitted or absorbed. An obvious example is the warming of
Earth by the Sun. A less obvious example is thermal radiation from the human body.
We examine these methods in some detail in the three following modules. Each method has unique and interesting characteristics, but all three do have one thing in common: They transfer heat solely
because of a temperature difference, as seen in Figure 14.12.
Check Your Understanding
Name an example from daily life—different from the text—for each mechanism of heat transfer.
Conduction: Heat transfers into your hands as you hold a hot cup of coffee.
Convection: Heat transfers as the barista steams cold milk to make hot cocoa.
Radiation: Reheating a cold cup of coffee in a microwave oven. | {"url":"https://texasgateway.org/resource/144-heat-transfer-methods?book=79096&binder_id=78576","timestamp":"2024-11-02T18:24:27Z","content_type":"text/html","content_length":"105394","record_id":"<urn:uuid:703aa23d-93fe-499e-8db6-b25196e1171e>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00684.warc.gz"} |
Sum of Arithmetic Sequence Calculator - Online Sum of Arithmetic Sequence Calculator
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.
Sum of Arithmetic Sequence Calculator
'Sum of Arithmetic Sequence Calculator' is an online tool that helps to calculate the sum of the arithmetic sequence. The arithmetic sequence is the sequence where the common difference remains
constant between any two successive terms.
What is the Sum of Arithmetic Sequence Calculator?
Online Sum of Arithmetic Sequence calculator helps you to calculate the sum of arithmetic sequence in a few seconds. An arithmetic progression (AP) is a sequence where the differences between every
two consecutive terms are the same.
Sum of Arithmetic Sequence Calculator
NOTE: Please enter first term, common difference upto four digits only and enter number of terms upto three digits only.
How to Use Sum of Arithmetic Sequence Calculator?
Please follow the steps below to find the sum of the arithmetic sequence:
• Step 1: Enter the first term(a), the common difference(d), and the number of terms(n) in the given input box.
• Step 2: Click on the "Calculate" button to find the sum of the arithmetic sequence.
• Step 3: Click on the "Reset" button to clear the fields and find the sum of the arithmetic sequence for different values.
How to Find Sum of Arithmetic Sequence?
An arithmetic sequence is defined as a series of numbers, in which each term (number) is obtained by adding a fixed number to its preceding term.
Sum of arithmetic terms = n/2[2a + (n - 1)d], where 'a' is the first term, 'd' is the common difference between two numbers, and 'n' is the number of terms.
Want to find complex math solutions within seconds?
Use our free online calculator to solve challenging questions. With Cuemath, find solutions in simple and easy steps.
Solved Examples on Sum of Arithmetic Sequence Calculator
Example 1:
Find the sum of the arithmetic sequence 1,3,5,7,9,11,13,15
Given: a = 1, d = 2, n = 8
Sum of arithmetic terms = n/2[2a + (n - 1)d]
= 8/2[2(1) + (8 - 1)2]
= 4[2 + 14]
= 64
Example 2:
Find the sum of the arithmetic sequence 2, 7, 12, 17, 22
Given: a = 2, d = 5, n = 5
Sum of arithmetic terms = n/2[2a + (n - 1)d]
= 5/2[2(2) + (5 - 1)5]
= 5/2[4 + 20]
= 5 × 12
= 60
Example 3:
Find the sum of the arithmetic sequence for a = 10, d = 9, and n = 20
Given: a = 10, d = 9, n = 20
Sum of arithmetic terms = n/2[2a + (n - 1)d]
= 20/2[2(10) + (20 - 1)9]
= 10[20 + 171]
= 1910
Similarly, you can try the sum of arithmetic sequence calculator to find the sum of the arithmetic sequence for the following:
a) 2,4,6,8,10,12,14,15 b) 5,15,25,35,45,55,65
☛ Related Articles:
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│Statistics Calculator │Standard Deviation Calculator│
│Slope Intercept Forn Calculator │Sample Mean Calculator │
Math worksheets and
visual curriculum | {"url":"https://www.cuemath.com/calculators/sum-of-arithmetic-sequence-calculator/","timestamp":"2024-11-09T19:11:39Z","content_type":"text/html","content_length":"211142","record_id":"<urn:uuid:635f63a0-8cb9-43f0-a8e2-c047f26d8d83>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00586.warc.gz"} |
Robust Estimation in Incomplete Blocks Designs
Ann. Math. Statist. 37(5): 1331-1337 (October, 1966). DOI: 10.1214/aoms/1177699277
Robust estimates of contrasts in treatment effects for experiments with one observation per cell were proposed by Lehmann [6] for complete (randomized) blocks designs. The model for the observations
$X_{i\alpha} (i = 1, \cdots, c; \alpha = 1, \cdots, n)$ is in this case \begin{equation*}\tag{1}X_{i\alpha} = \nu + \xi_i + \mu_\alpha + U_{i\alpha}\quad (\sum \xi_i = \sum \mu_\alpha = 0)\end
{equation*} where the $\xi$'s are the treatment effects, the $\mu$'s are the block effects, and the $U$'s are independent with a common continuous distribution. Here we shall generalize these
estimates to experiments in which the block size is smaller than the number of treatments to be compared, and we shall obtain their asymptotic efficiencies relative to the classical estimates. Since
we are concerned with large sample theory, we shall be interested in designs in which the blocks are replicated a large number (at least 4) times. Such designs could be applied to situations in which
only a few different treatment combinations are practicable but each could be replicated several times. For example, in an experiment to compare various diets for pigs, the natural block is the
litter. One may wish to compare $c$ diets and have available a number of litters of size $b < c$. An incomplete blocks design using some $J$ litters could first be selected and then the whole design
replicated several times using the remaining litters or (e.g. if some comparisons were of greater interest than others) some groups of $b$ diets could be given to more litters than others. Thus the
situation to be considered is that in which $c$ treatments are to be compared and the blocks of experimental units are all of size $b < c$. An incomplete blocks design $D$ consisting of $J$ blocks of
size $b$ is selected, the number $n_j$ of replications of the $j$th block is decided upon $(j = 1, \cdots, J; n_j = \rho_jn), \sum n_j$ blocks of experimental units are selected and numbered, and $\
sum n_j$ sets of $b$ treatments are assigned to the selected blocks as specified by $D$ and the $n_j$. The set of blocks receiving the same treatments will be called a replication set. After the
assignment of treatments to blocks, the order of application within the blocks is randomized. Assuming fixed effects and no interaction between treatment and block effects, the model for $D$ is \
begin{equation*}\tag{2}X_{ij} = \nu' + \xi_i + \mu_j + U_{ij} (j = 1, \cdots, J; i \varepsilon S_j)\end{equation*} $\sum^c_{i = 1} \xi_i = \sum^J_{j = 1} \mu_j = 0$ where $S_j$ consists of the
numbers of the $b$ treatments applied in the $j$th block, the $\xi$'s are the treatment effects, the $\mu$'s are the block effects, and the $U$'s are independent and identically distributed according
to a continuous distribution $F$ with mean zero and variance $\sigma^2$ (not necessarily finite). Under the same assumptions, the model for the whole design is \begin{equation*}\tag{3}X_{ij\alpha} =
\nu + \xi_i + \mu_j + \beta_{j\alpha} + U_{ij\alpha},\end{equation*} $(j = 1, \cdots, J; i \varepsilon S_j; \alpha = 1, \cdots, n_j)$ $\sum^c_{i = 1} \xi_i = \sum^J_{j = 1} \mu_j = 0; \sum^{n_j}_{\
alpha = 1} \beta_{j\alpha} = 0\quad \text{for each} j$ where $S_j$ is defined as before (the treatments applied to the $j$th block of $D$ now being applied in the $n_j$ blocks of the $j$th
replication set), $\xi_i$ is the effect of the $i$th treatment, $\mu_j$ is the effect of the $j$th replication set, $\beta_{j\alpha}$ is the effect of the $\alpha$th block in the $j$th replication
set, and the $U$'s are distributed as in the model (2). (Although it might appear that the models (2) and (3) are valid only for a fixed order of application of the assigned treatments to the units
within a block, the models remain valid under randomization. For, consider a model in which any of the $b$ treatments may be assigned to each unit within the block. If the corresponding $U$'s are
independent, identically distributed random variables, then any selection according to specified probabilities will again be independent and identically distributed, which justifies the assumptions
of the above models.)
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Vida L. Greenberg. "Robust Estimation in Incomplete Blocks Designs." Ann. Math. Statist. 37 (5) 1331 - 1337, October, 1966. https://doi.org/10.1214/aoms/1177699277
Published: October, 1966
First available in Project Euclid: 27 April 2007
Digital Object Identifier: 10.1214/aoms/1177699277
Rights: Copyright © 1966 Institute of Mathematical Statistics
Vol.37 • No. 5 • October, 1966 | {"url":"https://projecteuclid.org/journals/annals-of-mathematical-statistics/volume-37/issue-5/Robust-Estimation-in-Incomplete-Blocks-Designs/10.1214/aoms/1177699277.full","timestamp":"2024-11-08T09:18:15Z","content_type":"text/html","content_length":"150009","record_id":"<urn:uuid:ecf3b3ef-1b35-4bc3-816b-0ea5efc5c744>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00698.warc.gz"} |
Week 3: Prolog
August 31st, 2013
Prolog is the first language that completely gets me out of my comfort zone and brought me back to noob. Prolog is a pretty amazing language that at first makes you wonder how it knows so much.
Bruce picked the Rain Man character for Prolog. I can see this resemblance. They both are a great source of knowledge, if you get your questions right. Many times you’re left wondering ‘how did he
know that?!’.
In the beginning I was in a continuous struggle to get to my questions right. The first hour or so I had to get used to thinking only in terms of rules and recursion. My brain seems adjusted for
being able to return values. At first this made it more difficult to reason about the code and the problems at hand.
After some fooling around and getting not much done without some help, I took a different approach. I started evolving the code in really small increments, trying each increment in the prolog repl.
This really helped and got me gain a better understanding of the language.
Simple algorithms like fibonacci and bubblesort that were hard at first (writing them in a single attempt) were suddenly easier to get done.
The fibonacci algorithm gave me a little deeper insight into Prolog. I implemented a tail recursive calculation of the fibonacci sequence that gave me the fibonacci number at a certain point in the
index (e.g. the fibonacci of index 6 is 8) and I wondered why Prolog couldn’t reverse the rules and give me the index of a provided fibonacci number (e.g. the index of 8 is 6).
That’s when I saw the difference between functions and rules. The fibonacci algorithm is a function and in fact there is no way for Prolog to revert the algorithm like it can with rules. After some
googling if fibonacci can be done with rules, it seems to be solvable by applying the Constraint Logic Programming library (or CPL).
The Sudoku solver was surprisingly simple. I took the solution from the book a bit further by generalizing it for mutiple valid Sudoku puzzle sizes (4x4, 6x6, 9x9). As I choose to stick with GNU
prolog I had to write some utility functions to make for a more general solver (like partition, transpose, take, first, etc).
I wrote these utility functions using actual TDD [by which I mean here having a set of tests I can run every time I want to, rather than using the repl]. This made it a lot easier. Interestingly, I
ended up taking somewhat larger increments than I would normally use.
My solutions to the more interesting exercises in the book (like the Sudoku solver) are available in the 7-languages-in-7-weeks repository on my Github profile.
Prolog is a very interesting language that makes solving the right problem a breeze. Basically it provided all-or-nothing solutions. If it cannot find a solution to the proposed problem it just
prints ‘no’. This is where you have to adjust your questioning and retry.
After a bit of a puzzling start I had great fun using Prolog. It has an interesting way of thinking about the problem and describing the problem domain itself rather than a recipe for the solution.
If I again come across a problem that is well suited for Prolog, I will take this advantage.
For now this is where I leave Prolog and move to my next destination: Scala. See you in a week! | {"url":"https://thesoftwarecraft.com/2013/08/week-3-prolog.html","timestamp":"2024-11-10T11:37:43Z","content_type":"text/html","content_length":"11923","record_id":"<urn:uuid:33b4fc3c-40fe-4a5f-8959-2e4196588f6c>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00779.warc.gz"} |
AbstractIntroductionMethodologyExperimental setup for the field data acquisition on RhonegletscherApplication to a synthetic data setApplication on field dataDiscussionComparison with previous studiesInterference of the borehole inversion scheme with velocity anisotropyConclusionsCode and data availabilityVideo supplementAuthor contributionsCompeting interestsDisclaimerAcknowledgementsFinancial supportReview statementReferences
SE Solid Earth SESolid Earth 1869-9529 Copernicus Publications Göttingen, Germany 10.5194/se-14-805-2023A borehole trajectory inversion scheme to adjust the measurement geometry for 3D travel-time
tomography on glaciersBorehole trajectory inversion for 3D cross-borehole tomography HellmannSebastian sebastian.hellmann@erdw.ethz.ch https://orcid.org/0000-0002-2365-7369 GrabMelchior https://
orcid.org/0000-0002-8293-4872 PatzerCedric BauderAndreas https://orcid.org/0000-0001-7197-7706 MaurerHansruedi Laboratory of Hydraulics, Hydrology and Glaciology (VAW), ETH Zurich, Zurich,
Switzerland Institute of Geophysics, ETH Zurich, Zurich, Switzerland Terra Vermessungen AG, Othmarsingen, Switzerland Geological Survey of Finland (GTK), Espoo, Finland Swiss Federal Institute for
Forest, Snow and Landscape Research (WSL), Birmensdorf, Switzerland Sebastian Hellmann (sebastian.hellmann@erdw.ethz.ch)28July2023 14 7 805821 9October2022 29June2023 8May2023 17October2022
Copyright: © 2023 2023 This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/This
article is available from https://se.copernicus.org/articles/.htmlThe full text article is available as a PDF file from https://se.copernicus.org/articles/.pdf
Cross-borehole seismic tomography is a powerful tool to investigate the subsurface with a very high spatial resolution. In a set of boreholes, comprehensive three-dimensional investigations at
different depths can be conducted to analyse velocity anisotropy effects due to local changes within the medium. Especially in glaciological applications, the drilling of boreholes with hot water is
cost-efficient and provides rapid access to the internal structure of the ice. In turn, movements of the subsurface such as the continuous flow of ice masses cause deformations of the boreholes and
complicate a precise determination of the source and receiver positions along the borehole trajectories. Here, we present a three-dimensional inversion scheme that considers the deviations of the
boreholes as additional model parameters next to the common velocity inversion parameters. Instead of introducing individual parameters for each source and receiver position, we describe the borehole
trajectory with two orthogonal polynomials and only invert for the polynomial coefficients. This significantly reduces the number of additional model parameters and leads to much more stable
inversion results. In addition, we also discuss whether the inversion of the borehole parameters can be separated from the velocity inversion, which would enhance the flexibility of our inversion
scheme. In that case, updates of the borehole trajectories are only performed if this further reduces the overall error in the data sets. We apply this sequential inversion scheme to a synthetic data
set and a field data set from a temperate Alpine glacier. With the sequential inversion, the number of artefacts in the velocity model decreases compared to a velocity inversion without borehole
adjustments. In combination with a rough approximation of the borehole trajectories, for example, from additional a priori information, heterogeneities in the velocity model can be imaged similarly
to an inversion with fully correct borehole coordinates. Furthermore, we discuss the advantages and limitations of our approach in the context of an inherent seismic anisotropy of the medium and
extend our algorithm to consider an elliptic velocity anisotropy. With this extended version of the algorithm, we analyse the interference between a seismic anisotropy in the medium and the borehole
coordinate adjustment. Our analysis indicates that the borehole inversion interferes with seismic velocity anisotropy. The inversion can compensate for such a velocity anisotropy. Based on the
modelling results, we propose considering polynomials up to degree 3. For such a borehole trajectory inversion, third-order polynomials are a good compromise between a good representation of the true
borehole trajectories and minimising compensation for velocity anisotropy.
Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung 200021_169329/1 200021_169329/2
Cross-borehole travel-time tomography, based on seismic or ground-penetrating radar (GPR) waves, is widely used to investigate small-scale variations of the subsurface, especially when surface-based
experiments suffer from poor resolution. The main advantage is the much higher ray coverage within the target area, allowing a more detailed analysis of small-scale variations of geological
heterogeneities. The first experiments with seismic sources were described by and and successfully applied in a large number of studies in the last decades. Cross-hole tomography has been used in
mining exploration , aquifer delineation, and hydrology-related topics . It has also been used to characterise a host rock and its small-scale structures such as fault planes and to investigate pore
pressure variations in aquifers and caprocks as well as voids in karst regions . Furthermore, and used GPR-based travel-time tomography to estimate the water content in a polythermal glacier.
In contrast to surface-based experiments, the sources and receivers are usually not directly accessible. In general, the borehole geometry is assumed to be well-known, which allows a correct
calculation of distances between sources and receivers. For this purpose, inclinometer and caliper measurements are required to describe the actual borehole geometry, and centralisers have been used
to precisely position the tools in the centre of the boreholes. Nevertheless, there are some applications for which such estimates are not feasible. For example, for glaciological applications as
described in , boreholes are usually drilled by using hot-water or steam drills . These are highly efficient and cheap methods for glaciology-related investigations, but they come at the cost of the
boreholes having a variable diameter. The upper part of the hole is exposed to hot water for a much longer time than the deepest parts of the boreholes, leading to a rather conical shape of the
boreholes. Furthermore, outbreaks due to impurities, meltwater intrusions, and air content that change the thermal capacity as well as a variable drilling speed further complicate the borehole shape.
In those boreholes, centralisers usually cannot be used, and the actual position of the instruments may vary around the assumed position . Furthermore, glaciers constantly move during the time of
measurements, especially for comprehensive 3D experiments, which may last days or weeks; this leads to larger deformations of the boreholes over time, and the assumed distances are no longer correct.
Although repeated inclinometer measurements may reduce the errors, the required precision may still not be reached.
and have already shown the significant effect of wrong source and receiver coordinates on travel-time inversions. Artefacts have been introduced in the resulting velocities, leading to a
misinterpretation of the measurements. They proposed adjusted inversion algorithms that consider the coordinates to be additional model parameters and also invert for the coordinates in a
2D inversion. More recently, developed an approach using full-waveform inversion and a grid-search algorithm to update the 2D receiver coordinates after each step of velocity inversion for a data set
from vertical seismic profiling (VSP) experiments. Coordinates and velocity are inverted in two separate steps, allowing a more focused adjustment of different physical parameters.
In general, the majority of tomography measurements only consider 2D applications. This requires the calculation of a 2D tomographic plane that contains the boreholes. developed a coordinate rotation
algorithm to minimise the sum of out-of-plane deviations for sources and receivers. Nevertheless, 2D tomography results do not consider off-plane effects, and even when the results of several
2D planes are combined in the end, the individual inversions do not account for all of the 3D information. Therefore, several 3D inversion schemes have been developed in the last years , showing the
advantages of a 3D inversion over a combination of individual 2D inversions.
However, for 3D inversions, borehole trace corrections are more complex due to the additional degrees of freedom. proposed a method for 3D tomography experiments that implements a static correction
term for each receiver to absorb short-wavelength variations in layered subsurface, but other approaches that are successful for 2D inversions e.g. are difficult to transfer to 3D as the inversion
scheme is severely underdetermined. This may lead, for example, to oscillations of the coordinates at larger depths.
This issue becomes even more complicated when considering a subsurface with a certain velocity anisotropy. In glaciers, the ice crystals interact with the flow of the ice mass. The strain conditions
in the glacier force the ice crystals (that is, the c axes as the symmetry axis of the hexagonal ice grains) to align in a certain crystal orientation fabric (COF) pattern in accordance with the
given stress conditions e.g.. In a compressional regime in polar ice, single maxima have been observed in a variety of ice cores and geophysical experiments e.g.. The COF-derived seismic anisotropy
effect was initially investigated by . analysed the effect on seismic velocities for different COF patterns and found an anisotropy of up to 6% for P waves. For temperate glaciers, older and more
recent studies have observed multi-maxima for compressional regimes . have calculated an azimuthally dependent anisotropy of 2.3% as a result of such a multi-maximum. Media with such a degree of
anisotropy are considered weakly anisotropic. analysed the effect of weak anisotropy on seismic wave propagation and provided linearised equations for seismic velocities by introducing a set of five
parameters describing the strength and direction of anisotropy. Later, other studies developed approaches to further describe the anisotropy in specific media with certain anisotropy patterns or
fracture orientations and provided ways to implement such anisotropy in existing algorithms e.g.. In particular, demonstrated the effect of anisotropy on cross-hole travel-time tomography results for
tilted transversely isotropic media. The outcome of all these studies is that velocity anisotropy has to be considered to have an influence on the velocities obtained from travel-time tomography.
In this study, we propose an approach for travel-time tomography that is explicitly designed for glaciological experiments and apply this to a multi-cross-hole seismic data set acquired on
Rhonegletscher as a case study. The continuous but not necessarily linear movement and ice melt of the glacier during data acquisition caused a deviation of the holes, leading to uncertainties about
the particular source and receiver coordinates. Inclinometer measurements at the beginning and end of the campaign provide some constraints considered in the inversion. At the same time, we found
that the precision of different instruments was not sufficient under the given measurement conditions on glaciers. In some cases, significant deviations occurred between the different instruments,
although measurements were carried out one after the other. These observations initially triggered our investigations for a joint velocity and coordinate inversion. However, instead of inverting for
the individual source and receiver positions, we assume that the borehole trajectories can be described with two perpendicular 2D polynomials x(z) and y(z). The degree n of the polynomials is used to
control the degree of variations along the trajectories. The additional inversion parameters are these polynomial coefficients. Similar to the approach of , we then employ a two-step inversion
algorithm that inverts for the velocity and the coefficients of the polynomials in alternation. We further investigate and discuss the interdependence between borehole adjustments and the inherent
weak velocity anisotropy of the medium.
The inversion scheme that we apply in this study consists of two parts. The first step is a three-dimensional velocity inversion based on common ray tracing performed on a finite difference grid.
Each cell j (i.e. rectangular prism) has a defined velocity, expressed by its reciprocal value of slowness sj. For a given source–receiver combination i, the travel time through a cell can be
calculated from the length li,j of the ray through that cell and its associated slowness value sj. Summing over all N cells between this source–receiver pair results in the travel time ti, defined as
With the measured travel times from cross-borehole experiments tobs and an initial start model mini for the velocity distribution and the resulting calculated travel times tcal, the velocities of the
cells that are covered by ray paths, which are determined by a ray-tracing algorithm, can be updated iteratively: 2Δd=tobs-tcal,3Δm=(GTG+μWM)-1GTΔd,4mnew=mini+Δm, where the matrix G contains the
partial derivatives (sensitivities) Gij=∂di∂mj for each velocity grid cell. The matrix WM and μ summarise the regularisation (i.e. damping and smoothing) that is required for a stable inversion.
Further details about the regularisation are provided, for example, in .
The second part of our inversion scheme contains a coordinate inversion that accounts for uncertainties about the borehole trajectories. Although inclinometer data are usually available, specific
conditions such as a variable borehole diameter or a moving subsurface incorporate uncertainties that significantly affect the velocity inversion and may lead to artefacts complicating an
interpretation. An inversion scheme for the individual coordinates x,y, and z for each source and receiver would result in a severely underdetermined problem. This approach with 3(nS+nR) unknowns can
be replaced by a more robust one when assuming that the sources and receivers in a borehole are aligned along the continuous borehole trajectory. Then, displacements of the sources and receivers can
be described by deformations of the trajectory. As a simple mathematical description, we assume that each three-dimensional borehole trajectory can be described by two orthogonal two-dimensional
polynomials x=p(z) and y=q(z). The maximum degree np of each polynomial is selected by the user according to a priori information or an educated guess. The inversion problem assumes a known mean
velocity between each source and receiver, e.g. derived from the estimated velocity from the previous step in the velocity inversion. Then, a second inversion scheme like the one defined in Eq. ()
can be set up. The Jacobian matrix J contains the first derivatives of the data against the new model parameters, i.e. the polynomial coefficients Aj, j=1,…,np, Jij=∂di∂Aj=∂di∂rk∂rk∂Aj, with k=1,2
for the two horizontal and linearly independent directions x=r1 and y=r2. For instance, for the polynomial x=p(z) of the source hole, x=p(z)=∑j=1npAj(z-z0)j, the first derivatives (j=1,…,np) for the
ith combination of the source (xS,yS,zS) and receiver positions (xR,yR,zR) are Jij=(-1)αvxR-xS(xR-xS)2+(yR-yS)2+(zR-zS)2×(zS-z0)j, where v is the mean velocity between the source and the receiver and
α is an exponent that is α=1 for derivatives to the source coordinates and α=2 for derivatives to the receiver coordinates. For the receiver boreholes, (zS-z0) needs to be replaced by (zR-z0).
(x0,y0,z0) is the collar point of the borehole and assumed to be known, e.g. from differential global navigation satellite system (GNSS) measurements, so that the constant term A0≡0. In addition, the
relative position of the sources and receivers along the boreholes is assumed to be known, for example, by measuring the length of the cables of the instruments in the borehole below the collar
point. Similarly, this framework can be applied to the polynomial y=q(z). The regularisation term μWM in Eq. () is exchanged with a Tikhonov regularisation, i.e. WJ=μI, with a damping factor μ that
can individually be adjusted for each borehole.
There are two options for implementing the two parts of the inversion. The first one is a sequential inversion. During each iteration of the inversion, the updates of velocity and borehole
trajectories are computed independently. The second option is an extended system of equations that considers both parts in one large Jacobian matrix. We have tested both options with identical
settings and updated coordinates in each step of iteration and could not find significant differences between the two methods for synthetic data sets. However, the sequential inversion seems to be
numerically more stable. Furthermore, the most recent update of the velocities is already considered in the current step of iteration, and additional constraints determining whether a borehole
trajectory inversion should be performed are easier to evaluate. It also provides more flexibility to decide if an update of the coordinates in the current step of iteration is beneficial and thus
applied or skipped. Therefore, we calculated the results in the next sections with the sequential inversion.
Initially we encountered the issue of deviating boreholes when acquiring cross-borehole seismic data for anisotropy-related investigations on Rhonegletscher (Rhone glacier), a temperate glacier (ice
temperature T≈-0.5∘C) in the Swiss Alps. The glacier still covers an area of 15km2 and is flowing in the southern direction at its current terminus . For anisotropy investigations, we drilled a
set of 13 boreholes (Fig. a) about 500m north of the terminus with a hot-water drilling system e.g. in summer 2018. The boreholes were arranged in a ring with a diameter of 40m to obtain
cross-borehole seismic measurements under different azimuths (i.e. 0, 30, 60, and 90∘ relative to the ice flow direction). At the borehole location, the glacier was about 100m thick, and the ice
flowed with a surface flow speed of 16ma-1 in the southeastern direction . This ice flow velocity decreases with depth, which leads to a continuous deformation of the boreholes over time. At the
surface, the ice moved by 1.2m within the 3 weeks of data acquisition. The positions of the borehole collars and additional geophones along the surface were measured using a high-precision Leica
GNSS device in RTK mode. In addition, we used an inclinometer probe from Geotomography GmbH for an estimate of the borehole trajectories after drilling and observed the ongoing deformation with two
repeated measurements. An example from borehole BH01 is shown in Fig. b. The continuous deformation could not fully be surveyed by just three inclinometer measurements and thus leads to unknown
borehole trajectories for intermediate time steps of data acquisition. Nevertheless, the inclinometer measurements could serve as initial estimates for a borehole trajectory inversion.
Field data measurement geometry. (a) Positions of the boreholes on the glacier surface (Swiss coordinates, LV03); BH03, BH06, BH09, and BH12 were not used for the experiments shown here. (b)
Deformation of borehole BH01 within 4 weeks due to glacier flow.
The continuous displacements and deformations of the glacier ice body are still imposing problems in our attempt to invert simultaneously for subsurface velocities and the borehole trajectories. In
principle, the borehole trajectories would have to be estimated for every single source–receiver borehole pair, but this would lead to a poorly constrained inversion problem. To avoid this problem,
we have set up our experimental schedule such that every borehole is only occupied for a relatively short time span (up to 4d). We then made the assumption that the changes in the trajectories are
acceptably small within this short time span (i.e. we have inverted for a single trajectory for each borehole). This is certainly a limitation of our methodology, but it is, in our view, an
unavoidable compromise that needs to be made.
The drilling of the boreholes was stopped about 15–20m above the glacier bed to avoid hydrological coupling of the borehole with the subglacial drainage system, resulting in a total borehole length
of 80–90m. For the experimental setup, the boreholes needed to be water-filled. Despite this precaution, borehole BH09 and a second hole just a few metres next to it drained a few minutes after
drilling. Similarly, borehole BH11 drained after the second out of 3d of measurements in this borehole. Since the seismic sparker source and hydrophone receivers can only be deployed in water-filled
boreholes, this drainage led to an incomplete data set.
A 5kV sparker source from Geotomography GmbH with a dominant frequency of 2kHz was employed in the boreholes for generating the acoustic signal. Hydrophones were installed in the borehole on the
opposite side of the ring, as well as in a second borehole either perpendicular or parallel to the glacier flow (Fig. a). Geophones, equally spaced at 1m between the source and receiver boreholes,
were placed at the surface of the glacier. These additional receivers further increase the azimuthal resolution of the tomographic experiment. The enhanced ray coverage in the area of investigation
reduces the ambiguities between model parameters. This connection of each borehole to two other holes is a prerequisite for the borehole trajectory inversion. Therefore, the exclusively
two-dimensional data sets acquired between BH06 and BH12 as well as BH03 and BH00 could not be considered as explained in more detail in Sect. .
For sufficiently dense data coverage, we selected a shot interval of 1m in the source holes as well as a distance of 1m between the receivers along the surface and within the receiver boreholes. At
least three shots were recorded and stacked during the processing to enhance the signal-to-noise ratio. Since the length of the hydrophone chain was limited to 23m (24 channels), the experiment had
to be repeated four times to cover the entire length of the receiver borehole. For the data processing, we applied a standard procedure consisting of a median and bandpass filtering (0.3–15kHz) to
enhance the signal quality, a stacking of repeated shots, and a picking of the P-wave first arrivals with a cross-correlation algorithm. To pick the onset of the P wave, we defined a window around
the expected first-break arrival time and analysed the coherency between the traces within the individual receiver gathers. Finally, we performed different combinations of 3D velocity and borehole
To demonstrate the effect of our borehole correction, we first applied the approach to a synthetic cross-hole seismic data set. For this purpose, we defined a set of nine boreholes with a length
of 80m. The positions of the borehole collar points were taken from the actual field measurements described in Sect. . For the deviations of the boreholes, we fitted a fourth-order polynomial
through the data points of the inclinometer measurements and exaggerated this deviation by a factor of 8 to investigate the reliability of our code for cases with even more prominent deviations, e.g.
for glaciers with even higher ice flow velocities than those observed on Rhonegletscher. We placed 80 sources and 80 hydrophones in each borehole and added a total of 828 receivers on an inclined
plane that represents the surface. These geophones were installed at the glacier surface along 2D lines between the source and receiver boreholes. During data acquisition, about 95 geophones were
active at a time, i.e. those along the lines between the currently used boreholes. The geophone positions also refer to GNSS field measurements and were placed 12–20m below the reference point of
the model. The measurement geometry (i.e. the selection of source–receiver profiles) for the synthetic calculations was also based on real field measurements for developing a tool that is suitable
for our actual field measurements with a limited number of source–receiver combinations. Therefore, each source hole was connected to only two receiver holes, similar to the profiles shown in Fig. a.
True model of the synthetic example data set with a homogeneous background velocity of 3800ms-1 and two vertical fault zones with vp=3740ms-1 in the upper part as well as two channels with vp=
3680ms-1 in the deeper parts of the model. Triangles and asterisks represent receiver and source positions.
For the investigations, we defined a heterogeneous velocity model (shown in Fig. ) that consists of the following parts: the background velocity of the true model was set to 3800ms-1. This value has
been measured in ice core samples from Rhonegletscher , and the cross-borehole field experiments were carried out at the same location. Therefore, we also used this value as the background velocity
for our synthetic example. We added two north–south-oriented fault zones (e.g. water-saturated intrusions in the ice) with a slightly lower velocity of 3740ms-1 close to but below the geophones (z=
20–40m of the model). In addition, two meandering, inclined structures (representing englacial channels) with a velocity of 3680ms-1 were included at z=40–60 and z=70–86m. Synthetic data were
computed without additional random noise, and they were subsequently inverted using a homogeneous start model with a velocity of 3800ms-1. Since the tomographic inversion problem includes a
significantly underdetermined component, it is necessary to provide regularisation constraints. For the velocity parameters, they were supplied in the form of damping and smoothing constraints .
After some experimentation, we applied similar weights for damping and smoothing (damping factor of 0.1, smoothing factor of 0.4). We tuned the overall regularisation factor such that the inversions
converged to a data misfit level that roughly corresponds to the travel-time picking accuracy. For the model parameters associated with the borehole trajectories, we employed only damping
constraints, with which we penalised deviations from the initial values. We started with relatively high damping factors, which we gradually reduced until the inversions became unstable and/or
unreasonably large deviations were obtained.
The results of the velocity inversion with correct source and receiver positions are shown in Fig. a. The comparison with the true model in terms of velocity differences between the inverted and true
model is shown in Fig. b. The heterogeneities of the true model could be resolved quite well within the ring of borehole – that is, the area that is covered by the ray paths of the measurements. In
turn, this implies that the artefacts in those parts of the model lying outside the ring of boreholes and close to the bottom could therefore not correctly be inverted by any of the inversions and
will thus be excluded from the following discussion. This model is regarded as a reference model in the following discussion, although the thickness of the two channels in the lower part of the model
and the velocity of the channel between 40 and 60m are still slightly overestimated. However, these artefacts are most likely a result of the smoothing constraints in the velocity inversion and
therefore beyond the scope of our study.
Results of the velocity inversion with a synthetic example data set and correct borehole trajectories: (a) velocity results of the inversion after seven steps of iteration; (b) difference between the
inverted and true model (Fig. ). Triangles and asterisks represent receiver and source positions.
RMSEs for the velocity, the combined inversion, and a reference calculation for a velocity inversion with correct trajectories.
Iteration 1 2 3 4 5 6 7 RMSE (only velocity inv.) [ms] 0.0785 0.0507 0.0386 0.0378 0.0374 0.0372 0.0372 RMSE (combined inv.) [ms] 0.0785 0.0304 0.0216 0.0200 0.0192 0.0187 0.0186 RMSE (reference
inv.) [ms] 0.0650 0.0358 0.0179 0.0171 0.0169 0.0168 0.0167
In a next step, the information about the true borehole inclination was ignored and we started with straight vertical boreholes. We ran the inversion without and with borehole trajectory inversion
and stopped the inversions after seven iteration steps. Table provides the root mean squared errors (RMSEs) for both inversion schemes. The values provide evidence that these schemes both converge,
but the root mean square (rms) values of the sequential inversion are significantly smaller, thus providing a better fit between calculated and measured travel times. Figure a shows the inverted
velocity model for a pure velocity inversion. The differences to the reference model (Fig. ) are shown in Fig. b. The velocity of the upper and especially the lower channel between 70 and 86m is
only weakly recovered, incorporating clear artefacts in the inversion results (dashed blue ellipses A+B in Fig. b). In addition, several artefacts appear at the bottom of the model around the
boreholes (see the solid blue ellipses C+D in Fig. b showing very prominent examples). This is the region that is covered by only a few measurements and therefore highly underdetermined. Furthermore,
the discrepancy between the true inclined boreholes and the assumed straight borehole is the largest at the bottom of the boreholes. As a result, geometrical errors are smeared into the velocity
distribution in the least-resolved parts of the model. We also refer to the “Video supplement” for an enhanced overview of the velocity distribution and artefacts in the ice volume.
Results of the velocity inversion with a synthetic example data set and initially straight borehole trajectories. (a) Velocity inversion results after seven steps of iteration; (b) difference between
inversion results in (a) and the reference model (Fig. a), with turquoise ellipses emphasising the most significant differences. Triangles and asterisks represent receiver and source positions.
The calculations for the inversion were repeated with an additional borehole trajectory inversion. Each borehole is approximated by a set of two mutually perpendicular polynomials of degree 4.
Figure a shows the results of this inversion, and Fig. b shows the differences with respect to the reference model. Here, the upper channel and especially the lower channel (ellipses A+B) are
correctly resolved with a velocity misfit of <20ms-1 inside the channel. In addition, the prominent artefacts in the vicinity of the boreholes and artefacts at the bottom of the model (ellipses
C+D) are significantly reduced. Thus, including a trajectory inversion yields results comparable with those in Fig. obtained by using the true borehole geometry.
Results of the sequential (velocity and borehole trajectory) inversion with a synthetic example data set and initially straight borehole trajectories. (a) Inversion results after seven steps of
iteration; (b) difference between inversion results in (a) and the reference model (Fig. a). Triangles and asterisks represent receiver and source positions.
Deviations of the individual boreholes (locations shown in Fig. a) for the synthetic data set: start values for straight boreholes are plotted as green graphs, true borehole trajectories are shown as
blue graphs, and the borehole deviations adjusted after seven iterations are plotted as red curves.
The borehole trajectories, obtained from the trajectory inversion, are shown in Fig. and compared with initial (straight) and true trajectories. For some boreholes, the inverted borehole trajectory
closely resembles the true borehole path. For other boreholes, e.g. borehole BH02 and BH07, the inversion algorithm did not fully converge towards the true solution. Instead, some minor artefacts are
still visible in the velocity profile in the vicinity of these boreholes (for an enhanced three-dimensional view we refer to our “Video supplement”; ). This is a trade-off that cannot fully be
avoided by such a velocity–borehole inversion as both solutions provide similar residual errors. If a priori information about the magnitude of deformation or an initial guess for the main direction
of deformation is available, this information could be used to avoid a start with straight vertical boreholes and provide further constraints for the individual boreholes. However, boreholes that are
only part of a single two-dimensional profile cannot be fitted by our algorithm at all, as the degree of freedom is then higher than the geometric information obtained from such a two-dimensional
seismic profile (e.g. dashed profile BH06–BH12 in Fig. a). In this case, we observed strong oscillations or large deviations of the trajectories perpendicular to the measurement plane.
Resolution matrix for the coupled velocity and borehole trajectory inversion. (a) The entire matrix for all 36612 model parameters; the 72 parameters of the borehole adjustments are highlighted in
the orange circle. Only values of the matrix larger than 0.05 are shown as black dots. (b) Zoomed view of the borehole model parameters with colour-coded values. The excerpt illustrates the order of
the four polynomial coefficients as an example.
The issue of resolving the individual model parameters can be further investigated with the resolution matrix R e.g., mest=Rmtrue that connects the true model parameter mtrue with the estimated
parameters mest. The resolution matrix for our velocity-borehole inversion scheme is defined as R=GJWM00WJTGJWM00WJ-1×GJTGJ, where G is the sensitivity matrix (Jacobian matrix) for the velocity
inversion, and J is the Jacobian matrix for the borehole inversion as defined in Eq. (). The regularisation matrices (combining damping and smoothing) are given by WM for the velocity inversion and
by WJ for the borehole trajectory inversion, respectively. The resolution matrix for the synthetic example in Fig. b is shown in Fig. . Only values larger than 0.05 are plotted as black dots. The
matrix has a tridiagonal structure with the highest values along its main diagonal. Therefore, the estimated model parameter is mainly dependent on its true value. However, for the velocity inversion
parameters, this dependency is rather weak (all values are <0.2). In addition, elements along two minor diagonals also influence the estimated parameters. These are the neighbouring model parameters
along the ray path that also affect the velocity of the current cell.
For the borehole inversion parameters, the relationship between estimated and true parameters is much stronger, especially for the polynomial coefficients of degree 1 and 2. As shown in Fig. b, the
third element and fourth element of each set of borehole coefficients representing degree 1 and 2 show high values >0.99. Therefore, these coefficients are well resolved in the inversion. However,
the higher-order polynomial coefficients (third and fourth degree) are also weakly resolved for the majority of the boreholes (small values for these coefficients in Fig. b). This indicates that the
exact values for the coefficients cannot be determined independently and several solutions lead to similar results. The borehole coefficients of a particular borehole are also dependent on the
coefficients of the other boreholes, and its higher-order coefficients are often accompanied by larger off-diagonal elements that indicate this dependency. Here, the most prominent off-diagonal
elements that are visible in Fig. b are part of the other component of the borehole (i.e. the higher-order coefficients of the polynomial q(z) interfere with the coefficients of p(z)), indicating
that the two polynomials are not fully linearly independent. This is not surprising, when considering the geometry of the experimental setup (Fig. a) that is based on field measurements with a
limited number of source–receiver combinations. The data set for the three-dimensional inversion consists of a set of two-dimensional profiles covering the volume within the ring of boreholes. In
this setup, each borehole is used as a source and receiver hole together with two other boreholes. Therefore, a minimisation of the error can be achieved by adjusting at least one of the polynomials
of these boreholes relevant for the current set of measurements. This leads to a reduced azimuthal illumination of the borehole trajectories, and several equally good solutions for the borehole
trajectories are possible that reduce the number of fully independent model parameters.
The resolution matrix is also a useful tool to investigate whether the model parameters of the velocity inversion also affect the borehole trajectory inversion and vice versa. If there is a
dependency between these parameters we expect to see off-diagonal elements in the resolution matrix. As shown in Fig. a, off-diagonal elements appear in the lower-left corner of the matrix, but not
in the upper-right part. This implies that the estimated borehole polynomials depend on the defined velocity parameters between the boreholes. This is not surprising since the velocity is included in
the calculation of the sensitivities for the borehole inversion and has already been investigated by . In contrast, there is no dependence of the velocity parameters on the selected coefficients of
the borehole trajectories. Therefore, the velocity inversion can be obtained independently from the borehole parameters, and a separate borehole trajectory inversion is acceptable if the most recent
results for the velocities are considered in the iteration step of the borehole inversion.
For the field data set, we determined the travel times of the recorded P waves with a cross-correlation algorithm between repetitive measurements (in general three repetitions) within a window with a
length of up to 4ms depending on the distance around the estimated arrival time. Afterwards, these absolute values of the picked travel times were compared. If they differed by fewer than
three samples (or 60µs) the median value of the arrival times was selected for the source–receiver pair. About 7.2% (25860 out of 359369) of all source–receiver pairs had to be excluded from the
analysis due to larger discrepancies. In a next step, we obtained a velocity inversion followed by a combined velocity and borehole inversion. The results for the velocity inversion are shown in
Fig. a. Similar to our synthetic data, we locally observe very high velocities (yellow ellipses). We interpret them as artefacts since in previous experiments on ice core samples we observed a mean
seismic velocity of 3820ms-1 in pure ice at -5∘C with less than 2% air bubbles. We are not expecting to reach such high values for in situ measurements due to the high amount of meltwater that is
present in the entire glacier in summer. In addition, we also observe velocity artefacts at the lower end of several boreholes similar to those in our synthetic example. Therefore, we applied our
borehole trajectory inversion scheme in addition to the velocity inversion. We considered two slightly different start models. The first one uses the trajectories derived from inclinometer
measurements as start values. The results are shown in Fig. b. The second start model considers vertical boreholes. The differences between the two are discussed below and are rather small.
Therefore, we did not include the results here but provide a difference plot in the Appendix (Fig. ). The velocity and the two combined inversion models show a low-velocity zone close to the surface,
which we interpret as a zone of weathered, water-saturated ice. The meltwater, generated at the surface of the glacier due to the positive energy flux into the ice, leads to a decrease in the seismic
velocities. Between 15 and 70m, we observe a rather homogeneous zone with a velocity of v=3750±30ms-1. These values lie in the range of values commonly observed in glacier ice . Similarly, in
the lower part of the glacier, an englacial drainage system, observed in the vicinity of our investigation area by , also results in larger amounts of water. Therefore, the velocity decreases again.
Inversion results for field data from Rhonegletscher. (a) Results from the velocity inversion (seven iterations) without adjustment of the boreholes (trajectories from inclinometer data). (b) Results
from the combined velocity and borehole inversion with inclinometer-derived start trajectories. Yellow ellipses in (a) emphasise the areas with potential velocity artefacts.
RMSEs for the velocity and the combined inversion applied to the field data.
Iteration 1 2 3 4 5 6 7 RMSE (only velocity inv.) [ms] 0.192379 0.130918 0.120189 0.118738 0.118036 0.117659 0.117347 RMSE (combined inv.) [ms] 0.192379 0.126254 0.118537 0.116894 0.116199 0.115779
Maximum degree of the two polynomials for each borehole.
Borehole BH00 BH01 BH02 BH04 BH05 BH07 BH08 BH10 BH11 x polynomial 2 3 3 3 2 3 3 3 3 y polynomial 3 3 2 3 3 3 2 3 3
Results of borehole trajectory adjustment for field data. Green profiles show initial (dashed line) and final (solid line) borehole trajectories for a setup considering a priori inclinometer data;
blue profiles show the borehole trajectories estimated from initially vertical (straight) boreholes.
The rms values of both inversion schemes are similar, as shown in Table . As a consequence, the velocity structure resulting form the two inversion schemes does not differ significantly.
Nevertheless, velocity artefacts visible in the velocity inversion vanish in the combined inversion depending on the selected damping factor. This damping factor also affects how strongly the
adjusted boreholes deviate from the initial guess. Values that are too high (damping factor ≥10000) do not significantly change the borehole trajectories and the artefacts are still in place. A
very low damping factor (<1000) leads to unrealistically large adjustments with up to several metres at the bottom of the borehole. Therefore, we selected a damping factor that provided the best
compromise between adjustment of boreholes and removing the velocity artefacts. Furthermore, we considered different start values for the trajectories to test the robustness of our algorithm. Keeping
all other regularisation factors identical, we started with perfectly vertical boreholes and already inclined boreholes. The latter are interpolated trajectories (by means of time) derived from the
repeated inclinometer measurements. Figure b shows the results for an inversion with borehole trajectories interpolated from inclinometer measurements. We also repeated the calculations for initially
vertical borehole trajectories. The results are nearly identical within a range of 40ms-1 as shown in the difference plot in Fig. . In both cases, each borehole has been approximated by two mutually
perpendicular polynomials. The degree of the individual polynomials is provided in Table . The resulting adjustments of the trajectories for both start models (starting with straight vertical
borehole trajectories and trajectories derived from inclinometer measurements) are compared in Fig. . The trajectories are consistent, providing evidence that the algorithm provides robust results.
This is especially useful if no a priori (i.e. inclinometer) measurements of the trajectories are available. In addition, the results for the majority of the boreholes are consistent with regard to
the time difference between drilling and day of measurement. The boreholes BH02, BH04, BH08, and BH10 were occupied by the instruments about 20–23d after drilling and show larger deviations compared
to the holes BH01 and BH05, which were occupied 11–14d after drilling. The ongoing ice flow causes a slightly enhanced deviation at a later stage.
Nevertheless, boreholes BH07 and BH11 show large deviations. Their maximum deviation at the bottom is 0.6 and 1m, respectively, although the measurements were obtained just 11–14d after drilling of
the holes (together with BH01 and BH05). Glacier flow rates of 0.06md-1 on average measured at the glacier surface during the summer period imply that this deviation could not be caused by the ice
flow when considering a typical parabolic decrease in the flow rate with depth. Furthermore, the inclinometer measurements do not provide any hint of such strong deviations. Therefore, we expect that
measurement errors such as picking errors are compensated for through overly adjusted borehole trajectories during the inversion. Further evidence of this assumption is given by the fact that BH11
was only used as a source hole and only partially as receiver hole (along a profile to BH07 but not to BH05), since it drained just after the second out of 3d of measurements. Therefore, the number
of data points in this borehole is much lower and, thus the capability of our algorithm decreases due to poor constraints by the data. Since BH07 is directly connected to BH11 with two reciprocal
profiles, errors in the trajectory of BH11 may also affect BH07 and would explain the still quite large deviations.
In summary, our inversion algorithm provides the option to adjust borehole trajectories such that artefacts in the velocity inversion can be removed. For a good three-dimensional adjustment, a
reasonably high number of data points along these trajectories for different directions is required. For our specific experimental setup on a continuously moving glacier, this inversion scheme
provides improved results.
The combined inversion scheme provides the advantage of a subsequent correction for the borehole coordinates if no or limited information about the positions of sources and receivers is available.
However, there is a risk that the coordinate adjustment will suppress the appearance of real velocity anomalies in the tomogram. We avoid this by decoupling the two parts of the inverse algorithm.
Furthermore, an additional check of whether the new coordinates reduce the RMSE of the entire data set was implemented. This led to a more cautious adjustment. For our field data set, only two to
three coordinate adjustments in the first iterations were required. Afterwards, the inversion continued with a traditional velocity inversion and explained the rather similar rms values observed in
Table . This is the major advantage of the sequential inversion scheme with two independent inversion branches compared to an “all-in-one” inversion. Nevertheless, the borehole trajectory inversion
still relies on a priori knowledge such as inclinometer estimates and must be tuned accordingly to find the best solution for this equifinality problem.
The idea for a sequential inversion scheme is mainly driven by the findings described in and following projects in the research group. Two-dimensional analyses are usually easier to evaluate as the
degree of freedom is lower than in a three-dimensional setting. However, as already described in detail, off-plane effects may lead to misinterpretations of the resulting data. Especially when
analysing an azimuthally dependent variation in the seismic velocities (for example, due to changes in the ice crystal orientation relative to the glacier flow), those off-plane effects have to be
avoided. described a similar three-dimensional issue when trying to detect boulders in the subsurface. With their numerical modellings, they presented the advantages of the three-dimensional
algorithms with the resulting higher capability. Due to these advantages of three-dimensional approaches, we considered them here as well.
Next to the issues in the velocity model, we also have to consider the ice flow and thus a changing geometry setup when obtaining the inversion. Once again, some successful concepts were described in
earlier studies. As an example, presented their results for a layered subsurface. They applied some static corrections to account for the layers and changes in between. Although glaciers may also
have some ice layers, such as air-bubble-free and air-bubble-rich layers , the current study is mainly driven by issues due to the glacier flow itself and the consecutive deformation of the initially
rather straight boreholes in the glacier. Therefore, we could not consider such an approach for our data analysis. In one of the most recent publications, provided a promising statistical concept in
which they determine the probability of the borehole positions along a given trajectory by perturbating the model parameters. This might be another promising approach that we did not consider.
However, there might be some issues due either to a very large number of sources and receivers or a poorly constrained initial borehole trajectory that even further deforms over time. As an
alternative, we considered the methodology described by . They analysed the effect of coordinate mislocations on cross-hole tomography results in 2D. Their findings were similar to our observations
after data acquisition and processing. We therefore considered their concept and transferred it into 3D.
An adjustment of the borehole coordinates provides the opportunity to account for deformations of the boreholes due to a movement of the subsurface. This adjustment is based on the current velocity
model as it considers the mean velocity between the source and receiver. However, other physical quantities such as seismic anisotropy may introduce apparent errors in this seismic velocity values.
In glaciers, macrostructural features, such as the crevasses or englacial channels used in our synthetic data set, may cause velocity anisotropy in the seismic data. Besides this macrostructure,
there is the crystal orientation fabric that can also introduce such anisotropy. This anisotropy appears since the seismic velocity in a single ice crystal depends on the propagation direction of the
seismic wave relative to the c axis of the crystal. When the ice crystals in a polycrystalline glacier are oriented in a preferred direction relative to given stress conditions, a seismic wave
travels parallel to this alignment of the c axes with a higher velocity compared to other directions. This COF-derived seismic anisotropy has been investigated in several previous studies . When not
considering this velocity anisotropy effect, a borehole coordinate adjustment scheme could at least partially compensate for this effect.
Borehole trajectories for different inversion schemes: (a) results for the coordinates from a combined velocity and borehole inversion without consideration of anisotropy. (b) Results for the
coordinates from a combined inversion with borehole polynomials of degree x3 that additionally inverts for the four anisotropy parameters. Triangles and asterisks represent receiver and source
If significant velocity anisotropy is present, this could lead to flawed adjustments of the borehole trajectories, which in turn, reduce or entirely remove the anisotropy effect. We have investigated
this issue with a synthetic data set. The geometric setup consists of eight straight boreholes (BH01, BH02, BH04, BH05, BH07, BH08, BH10, and BH11 in Fig. a), and each source hole is again connected
to two receiver holes on opposite sides of the ring as in the previous calculations. The true velocity model is set to a homogeneous velocity value of 3800ms-1 with an additional ellipsoidal
anisotropy of 10% (δ=ϵ=0.1) according to the definitions of . The azimuthal direction of maximum anisotropy was chosen to be ϑa=160∘ with a slightly downwards-pointing inclination of φa=-15∘.
The anisotropy effect was added to the modelled travel times by adding a time shift depending on the angle between maximum anisotropy (ϑa,φa) and the wavefront (derived from the given ray path as
defined in ). In a first run, we used our combined velocity and borehole inversion scheme to invert for such an anisotropic model starting with vertical boreholes. The resulting borehole
trajectories, shown in Fig. a, incorporate the anisotropy effect up to a certain degree and drift away from the true trajectories. We then extended our borehole inversion with four additional model
parameters for the anisotropy parameters and defined a set of estimated values close to but slightly below the true values (i.e. δ=ϵ=0.05, ϑa=180∘, and φa=0∘). The sensitivities for the four
Thomsen parameters were calculated numerically: that is, ∂t∂mi=f(mi+Δmi)-f(mi)Δmi, with initial estimates for the four parameters m=(δ,ϵ,ϑa, and φa) as well as a small perturbation Δm for each value.
f(mi) and f(mi+Δmi) are results for the calculated travel times when considering the estimated and perturbed anisotropy parameters, respectively. When applying this extended inversion algorithm to a
data set with the correct values for the isotropic velocity and borehole trajectories, the true anisotropy parameters could be obtained within a few iteration steps. However, when again considering
straight boreholes to be start trajectories, the tests provide evidence that anisotropy parameters are still highly dependent on the estimated borehole parameters. The corresponding areas in the
resolution matrix for such an experimental setup (Fig. ) are occupied by large values that indicate a dependency between these model parameters. In addition, Fig. also shows that these model
parameters, in particular of the borehole trajectories that represent the higher-order polynomials, significantly interfere with the anisotropy values. In particular, the azimuthal angle of
anisotropy interferes with the higher-order coefficients of all boreholes. This implies that the anisotropy can be erroneously caught by an adjustment of the borehole trajectories. In our synthetic
example, this interaction manifests itself in very small adjustments of the anisotropy parameters when the degree of the polynomials is large enough. We observe this effect when considering
polynomials of degree x4 or higher. In contrast, polynomials of degree x3 or lower determine almost correct values (<0.01 for δ and ϵ; <1.5∘ for the angles) for the anisotropy parameters. However,
smaller-order polynomials only provide a limited degree of freedom in order to explain deviations along the trajectories. For our synthetic data set, polynomials of degree x3 show optimal results for
a simultaneous adjustment of anisotropy and borehole trajectories. Figure b shows the results of such an anisotropic inversion with a good fit between estimated and true borehole trajectories. We
repeated these experiments with different deviations of the borehole trajectories and a range of initial values for the anisotropy parameters. All calculations lead to the same result, providing
evidence that polynomials of degree x3 provide the best estimates. Higher-order polynomials incorporate the anisotropy into the trajectories and the anisotropy parameters are hardly adjusted. In
turn, smaller-order polynomials suffer from a lower degree of freedom as mentioned above. As a conclusion, these observations confirm that anisotropy and borehole adjustment interfere with each
other, and the inversion parameters must be selected in such a way that anisotropy that might be present in the analysed data set is not accidentally factored out by borehole adjustments.
Resolution matrix for a combined inversion for borehole trajectories (third-order polynomials) and four anisotropy parameters (marked with TP for Thomsen parameters). The colours show the dependence
of the true model parameter on estimated values. Off-diagonal values indicate interference between different model parameters. The four sectors provide an enhanced overview to distinguish between
borehole coefficients and Thomsen parameters.
An application to our field data is difficult to assess since the azimuth and inclination of the COF vary with depth as described in our previous study: . Therefore, it was not possible to define a
set of Thomsen parameters that describe the seismic velocity anisotropy derived for the COF of the entire ice column. Potentially, cross-hole experiments in polar ice with a rather constant COF,
which allow for a definition of a set of Thomsen parameters for larger parts of the ice column, are recommended here for further investigations.
In this paper, we presented a mathematically simple but efficient approach for a combined velocity and borehole trajectory inversion. This inversion scheme is especially useful for measurements in a
deforming subsurface such as experiments in alpine glaciers or ice sheets. The inversion process consists of a typical three-dimensional velocity inversion and an additional borehole trajectory
adjustment. All sources and receivers in each borehole are summarised along a trajectory that is characterised by two orthogonal polynomials. The two steps for velocity and borehole inversion are
decoupled as shown by the resolution matrix, and thus we only consider further adjustments of the borehole trajectories if this reduces the entire RMSE of the system. With this sequential inversion,
we could significantly reduce velocity artefacts that are a result of poorly determined source and receiver positions.
The adjustments are plausible for our field data, but we have also shown that poorly constrained boreholes, e.g. a small number of sources and receivers along the borehole, may lead to larger
deviations and potentially to an overfitting of noise and picking errors. Therefore, a data-set-dependent damping factor, which considers a priori information, is required. A weakness of our
inversion methodology includes the somewhat subjective choice of the regularisation parameters. In future investigations, this may be improved in two ways. When more advanced information on the
glacier movements is available, this could be supplied in the form of further constraints to the inversion problem. Alternatively, the various regularisation parameters could be determined in a more
systematic fashion (compared with our trial-and-error approach). A possible option may include generalised cross-validation e.g. or similar techniques. In addition, the algorithm only works for a set
of profiles with different azimuths and cannot successfully be applied to two-dimensional profiles. In this case, the polynomial coefficients cannot be determined, and oscillations or large
deviations occur at the bottom of the boreholes. Nevertheless, we could also show that the coupled scheme is rather robust and converges against very similar solutions for different start coordinates
for a reasonably well-constrained inversion problem. We also judge that this approach can be easily applied to other cross-borehole experiments or VSP experiments on solid ground with poorly
constrained borehole trajectories. In addition, this approach can in principle be applied to any geophysical borehole experiment beyond seismic investigations. However, its effectiveness needs to be
individually studied and validated.
We have also analysed the interdependence between anisotropy and borehole trajectory adjustments. Excessive corrections of the borehole trajectories can at least partially compensate for the inherent
velocity anisotropy. A higher degree of freedom (or flexibility) in the borehole parameters (that is, the use of higher-order polynomials) leads to a compensation for any weak ellipsoidal anisotropy.
Therefore, it is highly recommended to precisely determine the borehole trajectories during data acquisition. If there is no such a priori information available, the inversion parameters must be
selected accordingly to avoid this issue. We have shown that a polynomial degree of x3 provides a good compromise between flexibility to account for changes along the trajectories and an
overcompensation due to existing velocity anisotropy.
Velocity and coordinate differences for the field data for the combined inversions of the two start models with initially inclined boreholes (shown in Fig. b, blue asterisks and triangles) and
initially fully vertical boreholes (red asterisks and triangles).
The synthetic and field data are available in the open-access database ETH Research Collection (10.3929/ethz-b-000541813, ). The inversion code is available upon request.
The “Video supplement” mentioned in the text is available in the open-access database ETH Research Collection (10.3929/ethz-b-000541812, ).
This study was initiated and supervised by HM and AB. SH and HM developed the framework. SH and CP wrote the code for the borehole trajectory inversion and anisotropy investigations. SH, MG, and
AB planned the field campaign and acquired the borehole seismic field data. The data processing was conducted by SH and MG. The paper was written by SH, with comments and suggestions for improvements
from all co-authors.
The contact author has declared that none of the authors has any competing interests.
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
We thank Katalin Havas, Johanna Kerch, Dominik Gräff, and Greg Church for their extensive technical and scientific support during data acquisition and processing. We acknowledge Ulrike Werban and
Charlotte Krawczyk for the editorial work and the two anonymous referees for their valuable comments that helped to improve this paper.
This research has been supported by the Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung (grant nos. 200021_169329/1 and 200021_169329/2).
This paper was edited by Ulrike Werban and reviewed by two anonymous referees.
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Nonlinear Traveltime Inversion Scheme for Crosshole Seismic Tomography in Tilted Transversely Isotropic Media, GEOPHYSICS, 73, D17–D33, 10.1190/1.2910827, 2008. | {"url":"https://se.copernicus.org/articles/14/805/2023/se-14-805-2023.xml","timestamp":"2024-11-02T03:05:56Z","content_type":"application/xml","content_length":"187110","record_id":"<urn:uuid:59eb862c-cbda-4527-90b3-663a1501fd7f>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00735.warc.gz"} |
Given the function f(x)=x/(x+9), how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [1,4] and find the c? | Socratic
Given the function #f(x)=x/(x+9)#, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [1,4] and find the c?
1 Answer
The Mean Value Theorem has two Hypotheses.
A hypothesis is satisfied if it is true.
H1: The function must be continuous on $\left[1 , 4\right]$.
So, ask your self where $f$ is NOT continuous. Are there any numbers in $\left[1 , 4\right]$ where $f$ is not continuous?
H2: The function must be differentiable on $\left(1 , 4\right)$.
Find $f '$ and determine whether there are any numbers in $\left(1 , 4\right)$ where $f '$ is not defined.
To (try to) find the $c$ mentioned in the conclusion of the Mean Value Theorem,
set $f ' \left(x\right) = \frac{f \left(4\right) - f \left(1\right)}{4 - 1}$ and solve the resulting equation.
Since the conclusion asserts the existence of a $c$ in the interval $\left(1 , 4\right)$ any solutions to the equation that are in the interval are values for $c$.
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List of representation geometric effects
Available with Standard or Advanced license.
Click the links to see a detailed description of each parameter for the geometric effect.
List of geometric effects
Input Output Geometric Description Example
geometry geometry effect
Line from Creates a dynamic line of a specified length and angle originating from a point feature.
Buffer Creates a dynamic polygon with a specified diameter around a point feature.
Regular Creates a dynamic polygon around a point feature with a specified number of edges. All edges are equal in length and all angles are equal.
control Dynamically adds representation control points to a line feature to dictate the placement of marker placement styles or geometric effects that use a template.
Arrow Creates a dynamic line along a line feature with an arrow of a specified style and width.
Cut Creates a dynamic line that is shorter on one or both ends than a line feature.
Dashes Creates dynamic multipart line geometry from a line feature based on a template.
Extension Creates a dynamic line that is extended from either the beginning or the end of the line feature at a specified deflection angle and length.
Jog Creates a dynamic line with a jog of specified angle, position, and width in the line.
Move Creates a dynamic line offset a specified distance in x and y; often used to mimic a drop shadow.
Offset Creates a dynamic line offset at a specified distance perpendicularly from a line feature.
Offset Creates a dynamic line that is offset from either the beginning or the end of the line feature at a specified distance.
Line Reverse Flips the dynamic output of another geometric effect.
Rotate Creates a dynamic line rotated a specified angle from a line feature.
Creates a dynamic line scaled by a specified factor from a line feature. Vertices are moved in relation to the center point of the feature envelope. Values greater
Scale than 1 move vertices away from the center point; values between 0 and 1 move vertices toward the center point; a 0 value creates a null dynamic line; values less
than 0 draw an inverse dynamic line where the vertices have crossed to the other side of the center point.
Simplify Creates a dynamic generalized line from a line feature using the Douglas-Peucker algorithm.
Smooth Creates a dynamic Bézier curve from a line feature by setting a flat tolerance.
Suppress Suppresses sections of line features between pairs of representation control points.
Wave Creates a dynamic line along a line feature with a repeating wave pattern that is either sinusoidal, square, triangular, or irregular.
Buffer Creates a dynamic polygon with a specified diameter around a line feature.
Enclosing Creates a dynamic polygon from the spatial extent of a line feature.
Polygon polygon
Tapered Creates a dynamic polygon along a line feature, whose width varies by two specified amounts along its length, as defined by a percentage of the line feature's
polygon length.
Cut Creates a dynamic line that is shorter on one or both ends than the outline of a polygon feature.
Dashes Creates dynamic multipart line geometry from the outline of a polygon feature based on a template.
Add Dynamically adds representation control points to the outline of a polygon feature to dictate the placement of marker placement styles or geometric effects that
control use a template.
Buffer Creates a dynamic polygon with a specified distance around the outline of a polygon feature.
Donut Creates a dynamic polygon ring of a specified width in relation to the outline of a polygon feature.
Enclosing Creates a dynamic polygon from the spatial extent of a polygon feature.
Move Creates a dynamic polygon whose outline is offset a specified distance in x and y from the outline of a polygon feature; often used to mimic a drop shadow.
Offset Creates a dynamic polygon whose outline is offset a specified distance from the outline of a polygon feature.
Rotate Creates a dynamic polygon rotated a specified angle from a polygon feature.
Creates a dynamic polygon scaled by a specified factor from a polygon feature. Vertices are moved in relation to the center point of the feature envelope. Values
Scale greater than 1 move vertices away from the center point; values between 0 and 1 move vertices toward the center point; a 0 value creates a null dynamic polygon;
values less than 0 draw an inverse dynamic polygon where the vertices have crossed to the other side of the center point.
Simplify Creates a dynamic generalized line from the outline of a polygon feature using the Douglas-Peucker algorithm.
Smooth Creates a dynamic generalized line from the outline of a polygon feature by setting a flat tolerance.
Wave Creates a dynamic polygon outline along a polygon feature's outline with a repeating wave pattern that is either sinusoidal, square, triangular, or irregular.
Parameter definitions
List of geometric effects
Input Output Geometric Definitions
geometry geometry effect
Radial Angle: The orientation of the line from the marker. The default value is 0 (zero) degrees (due east). The angle is calculated in a counterclockwise manner.
Line from
point Length: The distance of the line from end to end. The default value is 5 pt. Adjusting this value changes the length of the dynamic line.
Buffer Size: The distance from the edge of the marker. The default value is 1 pt. Specifying values greater than 1 creates a larger dynamic polygon around the marker.
Radius: The distance from the center of the polygon. The default value is 2.5 pt. Specifying values greater than 1 creates a larger dynamic polygon around the marker.
Polygon Regular
polygon Edges: The number of sides for the polygon. The default number is 4. Specifying a value of 3 produces a triangle. Specifying a value less than 3 produces a circle.
Angle: The amount of rotation for the polygon. The default value is 0 (zero) degrees.
Add Angle tolerance: The maximum amount of deflection from one segment to another at a vertex. The default value is 120 degrees. Angle values between 180 and 360 are
control interpreted the same as values between 0 and 180. Angle values of 180 and 360 are the same as 0 (zero).
Style: Indicates the type of arrow to be displayed. The choices are Open Ended, Block, and Crossed. Block displays an arrow with a closed end. Open Ended is a block arrow
Arrow with an open end. Crossed indicates that a cross will be placed in the body of the arrow. The default value is Open Ended.
Width: Indicates the distance between the lines that construct the body of the arrow. The default value is 20 pt.
Begin cut: Indicates the distance from the beginning of a line that the display of the stroke starts. The beginning of a line is determined by the direction in which the
line was digitized. The distance is applied along the line. The default value is 1 pt.
Cut End cut: Indicates the distance from the end of a line that the display of the stroke starts. The ending of a line is determined by the direction in which the line was
digitized. The distance is applied along the line. The default value is 1 pt.
Invert: Indicates the effect should be applied in the opposite manner. This displays the stroke symbol only at the ends of the line and leaves the rest of the line
unsymbolized. The default value is unchecked.
Pattern: Indicates the distance for each dash and gap. There can be multiple dash and gap values to form a complex pattern. A dash is the symbolized part of the stroke,
and a gap is the unsymbolized part of the stroke. When the length of the line does not support the display of the entire pattern, the pattern is compressed to fit. Using
an odd number of values causes the pattern to double itself in overall length. The second half of the pattern is the reverse of the first half. Using a value of 0 (zero)
is acceptable. These options allow more complex dash patterns to be created. The default values are 10 pt and 10 pt.
Endings: The position of the dash along a line. The choices are No constraint, With half pattern, With full pattern, With half gap, With full gap, and Custom. Custom fits
the pattern to the length of the feature by adjusting the gaps slightly. No constraint does not adjust the gaps at all, meaning that the pattern is unlikely to exactly fit
the length of the feature. The default is With half pattern.
Position: Indicates where the pattern should begin relative to the starting point of the geometry. It shifts the entire pattern along the line the specified distance.
Negative numbers indicate a shift to the left, positive numbers a shift to the right. This property has an effect only if Endings is set to No constraint or Custom. When
constructing a representation rule, this property is dynamically derived from the Step value and dynamically updates during rule construction or modification when the
Endings property is updated.
Offset at end: Indicates where the pattern should end relative to the ending point of the geometry. Negative numbers indicate a shift to the left, positive numbers a shift
to the right. This property has an effect only if Endings is set to Custom. When constructing a representation rule, this parameter is dynamically derived from the Step
value and dynamically updates during rule construction or modification when the Endings property is updated.
Origin: The origin of the extension to add to the line.
Extension Deflection: The deflection angle used for the extension. Specify 0 for no deflection.
Length: The length of the extension that is dynamically created.
Length: The length of the segment that forms the jog in the line.
Jog Position: Location of the center of the jog, as a percentage measured from the start of the input geometry.
Angle: The angle of the jog in the line, measured in degrees.
Line X Offset: The distance to move the symbol along the x-axis from the feature geometry. The default value is 1 pt.
Y Offset: The distance to move the symbol along the y-axis from the feature geometry. The default value is -1 pt.
Offset: The distance of the stroke perpendicular to the line's geometry. The default value is 1 pt.
Line Method: Indicates the way the strokes are displayed at corners. The choices are Mitered, Bevelled, Rounded, and Square. Mitered matches the exact shape around a corner of
the line. Bevelled follows the shortest straight path across a corner of the line. Rounded follows a path of equal distance around a corner of the line. Square follows a
Offset straight path across the corner of a line. The default value is Square.
Option: Indicates the way strokes handle complex geometries. The choices are Fast and Accurate. Fast ignores complex geometries and applies a best fit to the stroke
symbol. Accurate accommodates complex geometries and applies a true fit to the stroke symbol. The default value is Fast.
Count: The number of times the stroke should be offset. The default value is 1.
Offset Method: The origin of the tangent offset for the line.
Offset: The distance the geometry is moved tangent.
Reverse Reverse: Indicates whether the dynamic output of a previous geometric effect is to be flipped or not. The default value is checked.
Rotate Angle: Indicates the amount of rotation for the stroke symbol. The default value is 15 degrees.
X Factor: Indicates the amount of change in size of a stroke symbol in the x-axis. The value is expressed in terms of a ratio/percentage. The default value is 1.15, or 115
Y Factor: Indicates the amount of change in size of a stroke symbol in the y-axis. The value is expressed in terms of a ratio/percentage. The default value is 1.15, or 115
Simplify Tolerance: Indicates the distance along a polygon outline to remove vertices from the stroke symbol. The default value is 10 pt. A value less than 10 maintains more of the
original shape of the stroke. A value greater than 10 maintains less of the original shape of the stroke.
Smooth Flat tolerance: Indicates the distance along a line to include segments for a dynamic Bézier curve. The default value is 1 pt. A value greater than 1 considers fewer
segments for a dynamic Bézier curve of the polygon outline.
Suppress: Indicates whether or not to suppress the sections of line features between two sets of control points.
Invert suppression: Checked indicates that the line will begin with a suppressed (not visible) section. Unchecked indicates that the line will begin with an unsuppressed
(visible) section.
Period: Indicates the distance along a line to display the curve for the stroke symbol. The default value is 3 pt.
Width: Indicates the distance perpendicular to a line to display the curve for the stroke symbol. The default value is 2 pt.
Wave Style: Indicates the shape of the curve to be displayed. The choices are Sinus, Square, Triangle, and Random. Sinus displays a sinusoidal curve or half-moon shape. Square
displays a three-sided rectangular-shaped curve. Triangle displays a two-sided triangular-shaped curve. Random displays a sine curve with random variation in the period
and width of the curve. The default value is Sinus.
Seed: Indicates the starting value for generating a random number. This random number is used by the Style type Random to determine which size a marker receives. The
default value is 1.
Buffer Size: The distance from the edge of the stroke. The default value is 1 pt.
Enclosing Method: Indicates the way in which the polygon geometry is generated around the stroke. The choices are Close path, Convex hull, and Rectangle box. Close path generates a
polygon polygon that connects both ends of the line to each other. Convex hull generates a polygon that approximates the shape of the line. Rectangle box generates a polygon equal
to the spatial envelope of the line. The default value is Rectangle box.
From width: Indicates the width at the start of the line to be used to generate a polygon. The default value is 0 pt (zero).
polygon To width: Indicates the width at the end of the line to be used to generate a polygon. The default value is 1 pt.
Length: Indicates the distance along the line to be used to generate the polygon. The default value is 0 pt (zero).
Begin cut: Indicates the distance from the closing point of a polygon that the display of the stroke starts. The distance is applied along the polygon outline. The default
value is 1 pt.
Cut End cut: Indicates the distance from the closing point of a polygon that the display of the stroke starts. The distance is applied along the polygon outline. The default
value is 1 pt.
Invert: Indicates the effect should be applied in the opposite manner. This displays the stroke symbol only at the closing point of a polygon with the rest of the polygon
outline unsymbolized. The default value is unchecked.
Pattern: Indicates the distance for each dash and gap. There can be multiple dash and gap values to form a complex pattern. A dash is the symbolized part of the stroke,
and a gap is the unsymbolized part of the stroke. When the length of the line does not support the display of the entire pattern, the pattern is compressed to fit. Using
an odd number of values causes the pattern to double itself in overall length. The second half of the pattern is the reverse of the first half. Using a value of 0 (zero)
Line is acceptable. These options allow more complex dash patterns to be created. The default values are 10 pt and 10 pt.
Endings: The position of the dash along a polygon outline. The choices are No constraint, With half pattern, With full pattern, With half gap, With full gap, and Custom.
Custom fits the pattern to the length of the feature by adjusting the gaps slightly. No constraint does not adjust the gaps at all, meaning that the pattern is unlikely to
exactly fit the length of the feature. The default is With half pattern.
Position: Indicates where the pattern should begin relative to the starting point of the geometry. It shifts the entire pattern along the line the specified distance.
Negative numbers indicate a shift to the left, positive numbers a shift to the right. This property has an effect only if Endings is set to No constraint or Custom. When
constructing a representation rule, this property is dynamically derived from the Step value and dynamically updates during rule construction or modification when the
Endings property is updated.
Offset at end: Indicates where the pattern should end relative to the ending point of the geometry. Negative numbers indicate a shift to the left, positive numbers a shift
to the right. This property has an effect only if Endings is set to Custom. When constructing a representation rule, this parameter is dynamically derived from the Step
value and dynamically updates during rule construction or modification when the Endings property is updated.
Add Angle tolerance: The maximum amount of deflection from one segment to another at a vertex. The default value is 120 degrees. Angle values between 180 and 360 are
control interpreted the same as values between 0 and 180. Angle values of 180 and 360 are the same as 0 (zero).
Buffer Size: The distance from the edge of the polygon outline. The default value is 1 pt.
Width: Indicates the distance from the edge of the polygon that the fill symbol is to be displayed. The default value is 2 pt.
Method: Indicates the way the strokes are displayed at convex corners of the polygon. The choices are Mitered, Bevelled, Rounded, and True buffer. Mitered matches the
Donut exact shape around a convex corner of the polygon. Bevelled follows the shortest straight path across a convex corner of the polygon. Rounded follows a path of equal
distance around a convex corner of the polygon. True buffer uses the buffer algorithm to follow a path around convex corners. The default value is Mitered.
Simple: Indicates whether or not complex geometries are dynamically simplified to generate a donut. It is checked by default.
Polygon Enclosing Method: Indicates the way in which the polygon geometry is generated around the polygon feature. The choices are Close path, Convex hull, and Rectangle box. Close path
polygon generates a polygon that matches the geometry of the polygon feature. Convex hull generates a polygon with a minimum number of sides to surround the polygon feature.
Rectangle box generates a polygon equal to the spatial envelope of the polygonal feature. The default value is Rectangle box.
X Offset: The distance to move the symbol along the x-axis from the feature geometry. The default value is 1 pt.
Y Offset: The distance to move the symbol along the y-axis from the feature geometry. The default value is -1 pt.
Offset: The distance of the polygon outline perpendicular to the polygon's geometry. The default value is 1 pt.
Method: Indicates the way the fill is displayed at corners. The choices are Mitered, Bevelled, Rounded, and Square. Mitered matches the exact shape around a corner of the
polygon. Bevelled follows the shortest straight path across a corner of the polygon. Rounded follows a path of equal distance around a corner of the polygon. Square
Offset follows a straight path across the corner of a polygon. The default value is Square.
Polygon Option: Indicates the way the polygon outline handles complex geometries. The choices are Fast and Accurate. Fast ignores complex geometries and applies a best fit to the
polygon outline. Accurate accommodates complex geometries and applies a true fit to the polygon outline. The default value is Fast.
Count: The number of times the stroke should be offset. The default value is 1.
Rotate Angle: Indicates the amount of rotation for the fill symbol. The default value is 15 degrees.
X Factor: Indicates the amount of change in size of a fill symbol in the x-axis. The value is expressed in terms of a ratio/percentage. The default value is 1.15, or 115
Y Factor: Indicates the amount of change in size of a fill symbol in the y-axis. The value is expressed in terms of a ratio/percentage. The default value is 1.15, or 115
Simplify Tolerance: Indicates the distance along a polygon outline to remove vertices from the stroke symbol. The default value is 10 pt. A value of less than 10 maintains more of
the original shape of the polygon outline. A value of greater than 10 maintains less of the original shape of the polygon outline.
Smooth Flat tolerance: Indicates the distance along a polygon outline to consider segments for a dynamic Bézier curve of the polygon outline. The default value is 1 pt. A value
greater than 1 considers fewer segments for a dynamic Bézier curve of the polygon outline.
Period: Indicates the distance along a polygon outline to display the curves for the fill symbol. The default value is 3 pt.
Width: Indicates the distance perpendicular to a polygon outline to display the curves for the fill symbol. The default value is 2 pt.
Wave Style: Indicates the shape of the curves to be displayed along the polygon outline. The choices are Sinus, Square, Triangle, and Random. Sinus displays a sinusoidal curve
or half-moon shape. Square displays a three-sided rectangular shape. Triangle displays a two-sided triangular shape. Random displays a sine curve with random variation in
the period and width of the outline, and the interior of the polygon remains fully symbolized. The default value is Sinus.
Seed: Indicates the starting value for generating a random number. This random number is used by the Style type Random to determine which size a marker receives. The
default value is 1. | {"url":"https://desktop.arcgis.com/en/arcmap/latest/map/working-with-layers/list-of-geometric-effects.htm","timestamp":"2024-11-10T19:41:08Z","content_type":"text/html","content_length":"79045","record_id":"<urn:uuid:aa8f5db3-e473-4bb4-8b8c-d379bdb61f53>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00519.warc.gz"} |
dmrg(H::MPO, psi0::MPS; kwargs...)
dmrg(H::MPO, psi0::MPS, sweeps::Sweeps; kwargs...)
Use the density matrix renormalization group (DMRG) algorithm to optimize a matrix product state (MPS) such that it is the eigenvector of lowest eigenvalue of a Hermitian matrix H, represented as a
matrix product operator (MPO).
dmrg(Hs::Vector{MPO}, psi0::MPS; kwargs...)
dmrg(Hs::Vector{MPO}, psi0::MPS, sweeps::Sweeps; kwargs...)
Use the density matrix renormalization group (DMRG) algorithm to optimize a matrix product state (MPS) such that it is the eigenvector of lowest eigenvalue of a Hermitian matrix H. This version of
dmrg accepts a representation of H as a Vector of MPOs, Hs = [H1, H2, H3, ...] such that H is defined as H = H1 + H2 + H3 + ... Note that this sum of MPOs is not actually computed; rather the set of
MPOs [H1,H2,H3,..] is efficiently looped over at each step of the DMRG algorithm when optimizing the MPS.
dmrg(H::MPO, Ms::Vector{MPS}, psi0::MPS; weight=1.0, kwargs...)
dmrg(H::MPO, Ms::Vector{MPS}, psi0::MPS, sweeps::Sweeps; weight=1.0, kwargs...)
Use the density matrix renormalization group (DMRG) algorithm to optimize a matrix product state (MPS) such that it is the eigenvector of lowest eigenvalue of a Hermitian matrix H, subject to the
constraint that the MPS is orthogonal to each of the MPS provided in the Vector Ms. The orthogonality constraint is approximately enforced by adding to H terms of the form w|M1><M1| + w|M2><M2| + ...
where Ms=[M1, M2, ...] and w is the "weight" parameter, which can be adjusted through the optional weight keyword argument.
dmrg will report the energy of the operator H + w|M1><M1| + w|M2><M2| + ..., not the operator H. If you want the expectation value of the MPS eigenstate with respect to just H, you can compute it
yourself with an observer or after DMRG is run with inner(psi', H, psi).
The MPS psi0 is used to initialize the MPS to be optimized.
The number of sweeps of thd DMRG algorithm is controlled by passing the nsweeps keyword argument. The keyword arguments maxdim, cutoff, noise, and mindim can also be passed to control the cost versus
accuracy of the algorithm - see below for details.
Alternatively the number of sweeps and accuracy parameters can be passed through a Sweeps object, though this interface is no longer preferred.
• energy::Number - eigenvalue of the optimized MPS
• psi::MPS - optimized MPS
Keyword arguments:
• nsweeps::Int - number of "sweeps" of DMRG to perform
Optional keyword arguments:
• maxdim - integer or array of integers specifying the maximum size allowed for the bond dimension or rank of the MPS being optimized.
• cutoff - float or array of floats specifying the truncation error cutoff or threshold to use for truncating the bond dimension or rank of the MPS.
• eigsolve_krylovdim::Int = 3 - maximum dimension of Krylov space used to locally solve the eigenvalue problem. Try setting to a higher value if convergence is slow or the Hamiltonian is close to a
critical point. ^[krylovkit]
• eigsolve_tol::Number = 1e-14 - Krylov eigensolver tolerance. ^[krylovkit]
• eigsolve_maxiter::Int = 1 - number of times the Krylov subspace can be rebuilt. ^[krylovkit]
• eigsolve_verbosity::Int = 0 - verbosity level of the Krylov solver. Warning: enabling this will lead to a lot of outputs to the terminal. ^[krylovkit]
• ishermitian=true - boolean specifying if dmrg should assume the MPO (or more general linear operator) represents a Hermitian matrix. ^[krylovkit]
• noise - float or array of floats specifying strength of the "noise term" to use to aid convergence.
• mindim - integer or array of integers specifying the minimum size of the bond dimension or rank, if possible.
• outputlevel::Int = 1 - larger outputlevel values make DMRG print more information and 0 means no output.
• observer - object implementing the Observer interface which can perform measurements and stop DMRG early.
• write_when_maxdim_exceeds::Int - when the allowed maxdim exceeds this value, begin saving tensors to disk to free RAM memory in large calculations
• write_path::String = tempdir() - path to use to save files to disk (to save RAM) when maxdim exceeds the write_when_maxdim_exceeds option, if set
• krylovkitThe dmrg function in ITensors.jl currently uses the eigsolve function in KrylovKit.jl as the internal the eigensolver. See the KrylovKit.jl documention on the eigsolve function for more
details: KrylovKit.eigsolve. | {"url":"https://itensor.github.io/ITensors.jl/dev/DMRG.html","timestamp":"2024-11-14T19:49:42Z","content_type":"text/html","content_length":"16103","record_id":"<urn:uuid:3854cf31-772c-47e7-9d2f-61c4fc7eb6e1>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00567.warc.gz"} |
EMU of Charge/Square Mile to Microcoulomb/Square Decimeter Converter (EMU/mi2 to μC/dm2) | Kody Tools
Conversion Description
1 EMU of Charge/Square Mile = 0.0000038610215854245 Coulomb/Square Meter 1 EMU of Charge/Square Mile in Coulomb/Square Meter is equal to 0.0000038610215854245
1 EMU of Charge/Square Mile = 3.8610215854245e-8 Coulomb/Square Decimeter 1 EMU of Charge/Square Mile in Coulomb/Square Decimeter is equal to 3.8610215854245e-8
1 EMU of Charge/Square Mile = 3.8610215854245e-10 Coulomb/Square Centimeter 1 EMU of Charge/Square Mile in Coulomb/Square Centimeter is equal to 3.8610215854245e-10
1 EMU of Charge/Square Mile = 3.8610215854245e-12 Coulomb/Square Millimeter 1 EMU of Charge/Square Mile in Coulomb/Square Millimeter is equal to 3.8610215854245e-12
1 EMU of Charge/Square Mile = 3.8610215854245e-18 Coulomb/Square Micrometer 1 EMU of Charge/Square Mile in Coulomb/Square Micrometer is equal to 3.8610215854245e-18
1 EMU of Charge/Square Mile = 3.8610215854245e-24 Coulomb/Square Nanometer 1 EMU of Charge/Square Mile in Coulomb/Square Nanometer is equal to 3.8610215854245e-24
1 EMU of Charge/Square Mile = 3.86 Coulomb/Square Kilometer 1 EMU of Charge/Square Mile in Coulomb/Square Kilometer is equal to 3.86
1 EMU of Charge/Square Mile = 0.000003228305785124 Coulomb/Square Yard 1 EMU of Charge/Square Mile in Coulomb/Square Yard is equal to 0.000003228305785124
1 EMU of Charge/Square Mile = 3.5870064279155e-7 Coulomb/Square Foot 1 EMU of Charge/Square Mile in Coulomb/Square Foot is equal to 3.5870064279155e-7
1 EMU of Charge/Square Mile = 2.4909766860524e-9 Coulomb/Square Inch 1 EMU of Charge/Square Mile in Coulomb/Square Inch is equal to 2.4909766860524e-9
1 EMU of Charge/Square Mile = 10 Coulomb/Square Mile 1 EMU of Charge/Square Mile in Coulomb/Square Mile is equal to 10
1 EMU of Charge/Square Mile = 3.8610215854245e-9 Kilocoulomb/Square Meter 1 EMU of Charge/Square Mile in Kilocoulomb/Square Meter is equal to 3.8610215854245e-9
1 EMU of Charge/Square Mile = 3.8610215854245e-11 Kilocoulomb/Square Decimeter 1 EMU of Charge/Square Mile in Kilocoulomb/Square Decimeter is equal to 3.8610215854245e-11
1 EMU of Charge/Square Mile = 3.8610215854245e-13 Kilocoulomb/Square Centimeter 1 EMU of Charge/Square Mile in Kilocoulomb/Square Centimeter is equal to 3.8610215854245e-13
1 EMU of Charge/Square Mile = 3.8610215854245e-15 Kilocoulomb/Square Millimeter 1 EMU of Charge/Square Mile in Kilocoulomb/Square Millimeter is equal to 3.8610215854245e-15
1 EMU of Charge/Square Mile = 3.8610215854245e-21 Kilocoulomb/Square Micrometer 1 EMU of Charge/Square Mile in Kilocoulomb/Square Micrometer is equal to 3.8610215854245e-21
1 EMU of Charge/Square Mile = 3.8610215854245e-27 Kilocoulomb/Square Nanometer 1 EMU of Charge/Square Mile in Kilocoulomb/Square Nanometer is equal to 3.8610215854245e-27
1 EMU of Charge/Square Mile = 0.0038610215854245 Kilocoulomb/Square Kilometer 1 EMU of Charge/Square Mile in Kilocoulomb/Square Kilometer is equal to 0.0038610215854245
1 EMU of Charge/Square Mile = 3.228305785124e-9 Kilocoulomb/Square Yard 1 EMU of Charge/Square Mile in Kilocoulomb/Square Yard is equal to 3.228305785124e-9
1 EMU of Charge/Square Mile = 3.5870064279155e-10 Kilocoulomb/Square Foot 1 EMU of Charge/Square Mile in Kilocoulomb/Square Foot is equal to 3.5870064279155e-10
1 EMU of Charge/Square Mile = 2.4909766860524e-12 Kilocoulomb/Square Inch 1 EMU of Charge/Square Mile in Kilocoulomb/Square Inch is equal to 2.4909766860524e-12
1 EMU of Charge/Square Mile = 0.01 Kilocoulomb/Square Mile 1 EMU of Charge/Square Mile in Kilocoulomb/Square Mile is equal to 0.01
1 EMU of Charge/Square Mile = 0.0038610215854245 Millicoulomb/Square Meter 1 EMU of Charge/Square Mile in Millicoulomb/Square Meter is equal to 0.0038610215854245
1 EMU of Charge/Square Mile = 0.000038610215854245 Millicoulomb/Square Decimeter 1 EMU of Charge/Square Mile in Millicoulomb/Square Decimeter is equal to 0.000038610215854245
1 EMU of Charge/Square Mile = 3.8610215854245e-7 Millicoulomb/Square Centimeter 1 EMU of Charge/Square Mile in Millicoulomb/Square Centimeter is equal to 3.8610215854245e-7
1 EMU of Charge/Square Mile = 3.8610215854245e-9 Millicoulomb/Square Millimeter 1 EMU of Charge/Square Mile in Millicoulomb/Square Millimeter is equal to 3.8610215854245e-9
1 EMU of Charge/Square Mile = 3.8610215854245e-15 Millicoulomb/Square Micrometer 1 EMU of Charge/Square Mile in Millicoulomb/Square Micrometer is equal to 3.8610215854245e-15
1 EMU of Charge/Square Mile = 3.8610215854245e-21 Millicoulomb/Square Nanometer 1 EMU of Charge/Square Mile in Millicoulomb/Square Nanometer is equal to 3.8610215854245e-21
1 EMU of Charge/Square Mile = 3861.02 Millicoulomb/Square Kilometer 1 EMU of Charge/Square Mile in Millicoulomb/Square Kilometer is equal to 3861.02
1 EMU of Charge/Square Mile = 0.003228305785124 Millicoulomb/Square Yard 1 EMU of Charge/Square Mile in Millicoulomb/Square Yard is equal to 0.003228305785124
1 EMU of Charge/Square Mile = 0.00035870064279155 Millicoulomb/Square Foot 1 EMU of Charge/Square Mile in Millicoulomb/Square Foot is equal to 0.00035870064279155
1 EMU of Charge/Square Mile = 0.0000024909766860524 Millicoulomb/Square Inch 1 EMU of Charge/Square Mile in Millicoulomb/Square Inch is equal to 0.0000024909766860524
1 EMU of Charge/Square Mile = 10000 Millicoulomb/Square Mile 1 EMU of Charge/Square Mile in Millicoulomb/Square Mile is equal to 10000
1 EMU of Charge/Square Mile = 3.86 Microcoulomb/Square Meter 1 EMU of Charge/Square Mile in Microcoulomb/Square Meter is equal to 3.86
1 EMU of Charge/Square Mile = 0.038610215854245 Microcoulomb/Square Decimeter 1 EMU of Charge/Square Mile in Microcoulomb/Square Decimeter is equal to 0.038610215854245
1 EMU of Charge/Square Mile = 0.00038610215854245 Microcoulomb/Square Centimeter 1 EMU of Charge/Square Mile in Microcoulomb/Square Centimeter is equal to 0.00038610215854245
1 EMU of Charge/Square Mile = 0.0000038610215854245 Microcoulomb/Square Millimeter 1 EMU of Charge/Square Mile in Microcoulomb/Square Millimeter is equal to 0.0000038610215854245
1 EMU of Charge/Square Mile = 3.8610215854245e-12 Microcoulomb/Square Micrometer 1 EMU of Charge/Square Mile in Microcoulomb/Square Micrometer is equal to 3.8610215854245e-12
1 EMU of Charge/Square Mile = 3.8610215854245e-18 Microcoulomb/Square Nanometer 1 EMU of Charge/Square Mile in Microcoulomb/Square Nanometer is equal to 3.8610215854245e-18
1 EMU of Charge/Square Mile = 3861021.59 Microcoulomb/Square Kilometer 1 EMU of Charge/Square Mile in Microcoulomb/Square Kilometer is equal to 3861021.59
1 EMU of Charge/Square Mile = 3.23 Microcoulomb/Square Yard 1 EMU of Charge/Square Mile in Microcoulomb/Square Yard is equal to 3.23
1 EMU of Charge/Square Mile = 0.35870064279155 Microcoulomb/Square Foot 1 EMU of Charge/Square Mile in Microcoulomb/Square Foot is equal to 0.35870064279155
1 EMU of Charge/Square Mile = 0.0024909766860524 Microcoulomb/Square Inch 1 EMU of Charge/Square Mile in Microcoulomb/Square Inch is equal to 0.0024909766860524
1 EMU of Charge/Square Mile = 10000000 Microcoulomb/Square Mile 1 EMU of Charge/Square Mile in Microcoulomb/Square Mile is equal to 10000000
1 EMU of Charge/Square Mile = 3861.02 Nanocoulomb/Square Meter 1 EMU of Charge/Square Mile in Nanocoulomb/Square Meter is equal to 3861.02
1 EMU of Charge/Square Mile = 38.61 Nanocoulomb/Square Decimeter 1 EMU of Charge/Square Mile in Nanocoulomb/Square Decimeter is equal to 38.61
1 EMU of Charge/Square Mile = 0.38610215854245 Nanocoulomb/Square Centimeter 1 EMU of Charge/Square Mile in Nanocoulomb/Square Centimeter is equal to 0.38610215854245
1 EMU of Charge/Square Mile = 0.0038610215854245 Nanocoulomb/Square Millimeter 1 EMU of Charge/Square Mile in Nanocoulomb/Square Millimeter is equal to 0.0038610215854245
1 EMU of Charge/Square Mile = 3.8610215854245e-9 Nanocoulomb/Square Micrometer 1 EMU of Charge/Square Mile in Nanocoulomb/Square Micrometer is equal to 3.8610215854245e-9
1 EMU of Charge/Square Mile = 3.8610215854245e-15 Nanocoulomb/Square Nanometer 1 EMU of Charge/Square Mile in Nanocoulomb/Square Nanometer is equal to 3.8610215854245e-15
1 EMU of Charge/Square Mile = 3861021585.42 Nanocoulomb/Square Kilometer 1 EMU of Charge/Square Mile in Nanocoulomb/Square Kilometer is equal to 3861021585.42
1 EMU of Charge/Square Mile = 3228.31 Nanocoulomb/Square Yard 1 EMU of Charge/Square Mile in Nanocoulomb/Square Yard is equal to 3228.31
1 EMU of Charge/Square Mile = 358.7 Nanocoulomb/Square Foot 1 EMU of Charge/Square Mile in Nanocoulomb/Square Foot is equal to 358.7
1 EMU of Charge/Square Mile = 2.49 Nanocoulomb/Square Inch 1 EMU of Charge/Square Mile in Nanocoulomb/Square Inch is equal to 2.49
1 EMU of Charge/Square Mile = 10000000000 Nanocoulomb/Square Mile 1 EMU of Charge/Square Mile in Nanocoulomb/Square Mile is equal to 10000000000
1 EMU of Charge/Square Mile = 3861021.59 Picocoulomb/Square Meter 1 EMU of Charge/Square Mile in Picocoulomb/Square Meter is equal to 3861021.59
1 EMU of Charge/Square Mile = 38610.22 Picocoulomb/Square Decimeter 1 EMU of Charge/Square Mile in Picocoulomb/Square Decimeter is equal to 38610.22
1 EMU of Charge/Square Mile = 386.1 Picocoulomb/Square Centimeter 1 EMU of Charge/Square Mile in Picocoulomb/Square Centimeter is equal to 386.1
1 EMU of Charge/Square Mile = 3.86 Picocoulomb/Square Millimeter 1 EMU of Charge/Square Mile in Picocoulomb/Square Millimeter is equal to 3.86
1 EMU of Charge/Square Mile = 0.0000038610215854245 Picocoulomb/Square Micrometer 1 EMU of Charge/Square Mile in Picocoulomb/Square Micrometer is equal to 0.0000038610215854245
1 EMU of Charge/Square Mile = 3.8610215854245e-12 Picocoulomb/Square Nanometer 1 EMU of Charge/Square Mile in Picocoulomb/Square Nanometer is equal to 3.8610215854245e-12
1 EMU of Charge/Square Mile = 3861021585424.5 Picocoulomb/Square Kilometer 1 EMU of Charge/Square Mile in Picocoulomb/Square Kilometer is equal to 3861021585424.5
1 EMU of Charge/Square Mile = 3228305.79 Picocoulomb/Square Yard 1 EMU of Charge/Square Mile in Picocoulomb/Square Yard is equal to 3228305.79
1 EMU of Charge/Square Mile = 358700.64 Picocoulomb/Square Foot 1 EMU of Charge/Square Mile in Picocoulomb/Square Foot is equal to 358700.64
1 EMU of Charge/Square Mile = 2490.98 Picocoulomb/Square Inch 1 EMU of Charge/Square Mile in Picocoulomb/Square Inch is equal to 2490.98
1 EMU of Charge/Square Mile = 10000000000000 Picocoulomb/Square Mile 1 EMU of Charge/Square Mile in Picocoulomb/Square Mile is equal to 10000000000000
1 EMU of Charge/Square Mile = 24098601262116 Elementary Charge/Square Meter 1 EMU of Charge/Square Mile in Elementary Charge/Square Meter is equal to 24098601262116
1 EMU of Charge/Square Mile = 240986012621.16 Elementary Charge/Square Decimeter 1 EMU of Charge/Square Mile in Elementary Charge/Square Decimeter is equal to 240986012621.16
1 EMU of Charge/Square Mile = 2409860126.21 Elementary Charge/Square Centimeter 1 EMU of Charge/Square Mile in Elementary Charge/Square Centimeter is equal to 2409860126.21
1 EMU of Charge/Square Mile = 24098601.26 Elementary Charge/Square Millimeter 1 EMU of Charge/Square Mile in Elementary Charge/Square Millimeter is equal to 24098601.26
1 EMU of Charge/Square Mile = 24.1 Elementary Charge/Square Micrometer 1 EMU of Charge/Square Mile in Elementary Charge/Square Micrometer is equal to 24.1
1 EMU of Charge/Square Mile = 0.000024098601262116 Elementary Charge/Square Nanometer 1 EMU of Charge/Square Mile in Elementary Charge/Square Nanometer is equal to 0.000024098601262116
1 EMU of Charge/Square Mile = 24098601262116000000 Elementary Charge/Square Kilometer 1 EMU of Charge/Square Mile in Elementary Charge/Square Kilometer is equal to 24098601262116000000
1 EMU of Charge/Square Mile = 20149499852985 Elementary Charge/Square Yard 1 EMU of Charge/Square Mile in Elementary Charge/Square Yard is equal to 20149499852985
1 EMU of Charge/Square Mile = 2238833316998.4 Elementary Charge/Square Foot 1 EMU of Charge/Square Mile in Elementary Charge/Square Foot is equal to 2238833316998.4
1 EMU of Charge/Square Mile = 15547453590.27 Elementary Charge/Square Inch 1 EMU of Charge/Square Mile in Elementary Charge/Square Inch is equal to 15547453590.27
1 EMU of Charge/Square Mile = 62415090744608000000 Elementary Charge/Square Mile 1 EMU of Charge/Square Mile in Elementary Charge/Square Mile is equal to 62415090744608000000
1 EMU of Charge/Square Mile = 4.0016678450234e-11 Farady (C12)/Square Meter 1 EMU of Charge/Square Mile in Farady (C12)/Square Meter is equal to 4.0016678450234e-11
1 EMU of Charge/Square Mile = 4.0016678450234e-13 Farady (C12)/Square Decimeter 1 EMU of Charge/Square Mile in Farady (C12)/Square Decimeter is equal to 4.0016678450234e-13
1 EMU of Charge/Square Mile = 4.0016678450234e-15 Farady (C12)/Square Centimeter 1 EMU of Charge/Square Mile in Farady (C12)/Square Centimeter is equal to 4.0016678450234e-15
1 EMU of Charge/Square Mile = 4.0016678450234e-17 Farady (C12)/Square Millimeter 1 EMU of Charge/Square Mile in Farady (C12)/Square Millimeter is equal to 4.0016678450234e-17
1 EMU of Charge/Square Mile = 4.0016678450234e-23 Farady (C12)/Square Micrometer 1 EMU of Charge/Square Mile in Farady (C12)/Square Micrometer is equal to 4.0016678450234e-23
1 EMU of Charge/Square Mile = 4.0016678450234e-29 Farady (C12)/Square Nanometer 1 EMU of Charge/Square Mile in Farady (C12)/Square Nanometer is equal to 4.0016678450234e-29
1 EMU of Charge/Square Mile = 0.000040016678450234 Farady (C12)/Square Kilometer 1 EMU of Charge/Square Mile in Farady (C12)/Square Kilometer is equal to 0.000040016678450234
1 EMU of Charge/Square Mile = 3.3459039708563e-11 Farady (C12)/Square Yard 1 EMU of Charge/Square Mile in Farady (C12)/Square Yard is equal to 3.3459039708563e-11
1 EMU of Charge/Square Mile = 3.7176710787292e-12 Farady (C12)/Square Foot 1 EMU of Charge/Square Mile in Farady (C12)/Square Foot is equal to 3.7176710787292e-12
1 EMU of Charge/Square Mile = 2.5817160268953e-14 Farady (C12)/Square Inch 1 EMU of Charge/Square Mile in Farady (C12)/Square Inch is equal to 2.5817160268953e-14
1 EMU of Charge/Square Mile = 0.00010364272140124 Farady (C12)/Square Mile 1 EMU of Charge/Square Mile in Farady (C12)/Square Mile is equal to 0.00010364272140124
1 EMU of Charge/Square Mile = 3.8610215854245e-7 EMU of Charge/Square Meter 1 EMU of Charge/Square Mile in EMU of Charge/Square Meter is equal to 3.8610215854245e-7
1 EMU of Charge/Square Mile = 3.8610215854245e-9 EMU of Charge/Square Decimeter 1 EMU of Charge/Square Mile in EMU of Charge/Square Decimeter is equal to 3.8610215854245e-9
1 EMU of Charge/Square Mile = 3.8610215854245e-11 EMU of Charge/Square Centimeter 1 EMU of Charge/Square Mile in EMU of Charge/Square Centimeter is equal to 3.8610215854245e-11
1 EMU of Charge/Square Mile = 3.8610215854245e-13 EMU of Charge/Square Millimeter 1 EMU of Charge/Square Mile in EMU of Charge/Square Millimeter is equal to 3.8610215854245e-13
1 EMU of Charge/Square Mile = 3.8610215854245e-19 EMU of Charge/Square Micrometer 1 EMU of Charge/Square Mile in EMU of Charge/Square Micrometer is equal to 3.8610215854245e-19
1 EMU of Charge/Square Mile = 3.8610215854245e-25 EMU of Charge/Square Nanometer 1 EMU of Charge/Square Mile in EMU of Charge/Square Nanometer is equal to 3.8610215854245e-25
1 EMU of Charge/Square Mile = 0.38610215854245 EMU of Charge/Square Kilometer 1 EMU of Charge/Square Mile in EMU of Charge/Square Kilometer is equal to 0.38610215854245
1 EMU of Charge/Square Mile = 3.228305785124e-7 EMU of Charge/Square Yard 1 EMU of Charge/Square Mile in EMU of Charge/Square Yard is equal to 3.228305785124e-7
1 EMU of Charge/Square Mile = 3.5870064279155e-8 EMU of Charge/Square Foot 1 EMU of Charge/Square Mile in EMU of Charge/Square Foot is equal to 3.5870064279155e-8
1 EMU of Charge/Square Mile = 2.4909766860524e-10 EMU of Charge/Square Inch 1 EMU of Charge/Square Mile in EMU of Charge/Square Inch is equal to 2.4909766860524e-10
1 EMU of Charge/Square Mile = 11559.94 ESU of Charge/Square Meter 1 EMU of Charge/Square Mile in ESU of Charge/Square Meter is equal to 11559.94
1 EMU of Charge/Square Mile = 115.6 ESU of Charge/Square Decimeter 1 EMU of Charge/Square Mile in ESU of Charge/Square Decimeter is equal to 115.6
1 EMU of Charge/Square Mile = 1.16 ESU of Charge/Square Centimeter 1 EMU of Charge/Square Mile in ESU of Charge/Square Centimeter is equal to 1.16
1 EMU of Charge/Square Mile = 0.01155994486654 ESU of Charge/Square Millimeter 1 EMU of Charge/Square Mile in ESU of Charge/Square Millimeter is equal to 0.01155994486654
1 EMU of Charge/Square Mile = 1.155994486654e-8 ESU of Charge/Square Micrometer 1 EMU of Charge/Square Mile in ESU of Charge/Square Micrometer is equal to 1.155994486654e-8
1 EMU of Charge/Square Mile = 1.155994486654e-14 ESU of Charge/Square Nanometer 1 EMU of Charge/Square Mile in ESU of Charge/Square Nanometer is equal to 1.155994486654e-14
1 EMU of Charge/Square Mile = 11559944866.54 ESU of Charge/Square Kilometer 1 EMU of Charge/Square Mile in ESU of Charge/Square Kilometer is equal to 11559944866.54
1 EMU of Charge/Square Mile = 9665.59 ESU of Charge/Square Yard 1 EMU of Charge/Square Mile in ESU of Charge/Square Yard is equal to 9665.59
1 EMU of Charge/Square Mile = 1073.95 ESU of Charge/Square Foot 1 EMU of Charge/Square Mile in ESU of Charge/Square Foot is equal to 1073.95
1 EMU of Charge/Square Mile = 7.46 ESU of Charge/Square Inch 1 EMU of Charge/Square Mile in ESU of Charge/Square Inch is equal to 7.46
1 EMU of Charge/Square Mile = 29940119760.48 ESU of Charge/Square Mile 1 EMU of Charge/Square Mile in ESU of Charge/Square Mile is equal to 29940119760.48
1 EMU of Charge/Square Mile = 11559.94 Franklin/Square Meter 1 EMU of Charge/Square Mile in Franklin/Square Meter is equal to 11559.94
1 EMU of Charge/Square Mile = 115.6 Franklin/Square Decimeter 1 EMU of Charge/Square Mile in Franklin/Square Decimeter is equal to 115.6
1 EMU of Charge/Square Mile = 1.16 Franklin/Square Centimeter 1 EMU of Charge/Square Mile in Franklin/Square Centimeter is equal to 1.16
1 EMU of Charge/Square Mile = 0.01155994486654 Franklin/Square Millimeter 1 EMU of Charge/Square Mile in Franklin/Square Millimeter is equal to 0.01155994486654
1 EMU of Charge/Square Mile = 1.155994486654e-8 Franklin/Square Micrometer 1 EMU of Charge/Square Mile in Franklin/Square Micrometer is equal to 1.155994486654e-8
1 EMU of Charge/Square Mile = 1.155994486654e-14 Franklin/Square Nanometer 1 EMU of Charge/Square Mile in Franklin/Square Nanometer is equal to 1.155994486654e-14
1 EMU of Charge/Square Mile = 11559944866.54 Franklin/Square Kilometer 1 EMU of Charge/Square Mile in Franklin/Square Kilometer is equal to 11559944866.54
1 EMU of Charge/Square Mile = 9665.59 Franklin/Square Yard 1 EMU of Charge/Square Mile in Franklin/Square Yard is equal to 9665.59
1 EMU of Charge/Square Mile = 1073.95 Franklin/Square Foot 1 EMU of Charge/Square Mile in Franklin/Square Foot is equal to 1073.95
1 EMU of Charge/Square Mile = 7.46 Franklin/Square Inch 1 EMU of Charge/Square Mile in Franklin/Square Inch is equal to 7.46
1 EMU of Charge/Square Mile = 29940119760.48 Franklin/Square Mile 1 EMU of Charge/Square Mile in Franklin/Square Mile is equal to 29940119760.48
1 EMU of Charge/Square Mile = 3.8610215854245e-12 Megacoulomb/Square Meter 1 EMU of Charge/Square Mile in Megacoulomb/Square Meter is equal to 3.8610215854245e-12
1 EMU of Charge/Square Mile = 3.8610215854245e-14 Megacoulomb/Square Decimeter 1 EMU of Charge/Square Mile in Megacoulomb/Square Decimeter is equal to 3.8610215854245e-14
1 EMU of Charge/Square Mile = 3.8610215854245e-16 Megacoulomb/Square Centimeter 1 EMU of Charge/Square Mile in Megacoulomb/Square Centimeter is equal to 3.8610215854245e-16
1 EMU of Charge/Square Mile = 3.8610215854245e-18 Megacoulomb/Square Millimeter 1 EMU of Charge/Square Mile in Megacoulomb/Square Millimeter is equal to 3.8610215854245e-18
1 EMU of Charge/Square Mile = 3.8610215854245e-24 Megacoulomb/Square Micrometer 1 EMU of Charge/Square Mile in Megacoulomb/Square Micrometer is equal to 3.8610215854245e-24
1 EMU of Charge/Square Mile = 3.8610215854245e-30 Megacoulomb/Square Nanometer 1 EMU of Charge/Square Mile in Megacoulomb/Square Nanometer is equal to 3.8610215854245e-30
1 EMU of Charge/Square Mile = 0.0000038610215854245 Megacoulomb/Square Kilometer 1 EMU of Charge/Square Mile in Megacoulomb/Square Kilometer is equal to 0.0000038610215854245
1 EMU of Charge/Square Mile = 3.228305785124e-12 Megacoulomb/Square Yard 1 EMU of Charge/Square Mile in Megacoulomb/Square Yard is equal to 3.228305785124e-12
1 EMU of Charge/Square Mile = 3.5870064279155e-13 Megacoulomb/Square Foot 1 EMU of Charge/Square Mile in Megacoulomb/Square Foot is equal to 3.5870064279155e-13
1 EMU of Charge/Square Mile = 2.4909766860524e-15 Megacoulomb/Square Inch 1 EMU of Charge/Square Mile in Megacoulomb/Square Inch is equal to 2.4909766860524e-15
1 EMU of Charge/Square Mile = 0.00001 Megacoulomb/Square Mile 1 EMU of Charge/Square Mile in Megacoulomb/Square Mile is equal to 0.00001
1 EMU of Charge/Square Mile = 11559.94 Statcoulomb/Square Meter 1 EMU of Charge/Square Mile in Statcoulomb/Square Meter is equal to 11559.94
1 EMU of Charge/Square Mile = 115.6 Statcoulomb/Square Decimeter 1 EMU of Charge/Square Mile in Statcoulomb/Square Decimeter is equal to 115.6
1 EMU of Charge/Square Mile = 1.16 Statcoulomb/Square Centimeter 1 EMU of Charge/Square Mile in Statcoulomb/Square Centimeter is equal to 1.16
1 EMU of Charge/Square Mile = 0.01155994486654 Statcoulomb/Square Millimeter 1 EMU of Charge/Square Mile in Statcoulomb/Square Millimeter is equal to 0.01155994486654
1 EMU of Charge/Square Mile = 1.155994486654e-8 Statcoulomb/Square Micrometer 1 EMU of Charge/Square Mile in Statcoulomb/Square Micrometer is equal to 1.155994486654e-8
1 EMU of Charge/Square Mile = 1.155994486654e-14 Statcoulomb/Square Nanometer 1 EMU of Charge/Square Mile in Statcoulomb/Square Nanometer is equal to 1.155994486654e-14
1 EMU of Charge/Square Mile = 11559944866.54 Statcoulomb/Square Kilometer 1 EMU of Charge/Square Mile in Statcoulomb/Square Kilometer is equal to 11559944866.54
1 EMU of Charge/Square Mile = 9665.59 Statcoulomb/Square Yard 1 EMU of Charge/Square Mile in Statcoulomb/Square Yard is equal to 9665.59
1 EMU of Charge/Square Mile = 1073.95 Statcoulomb/Square Foot 1 EMU of Charge/Square Mile in Statcoulomb/Square Foot is equal to 1073.95
1 EMU of Charge/Square Mile = 7.46 Statcoulomb/Square Inch 1 EMU of Charge/Square Mile in Statcoulomb/Square Inch is equal to 7.46
1 EMU of Charge/Square Mile = 29940119760.48 Statcoulomb/Square Mile 1 EMU of Charge/Square Mile in Statcoulomb/Square Mile is equal to 29940119760.48
1 EMU of Charge/Square Mile = 3.8610215854245e-7 Abcoulomb/Square Meter 1 EMU of Charge/Square Mile in Abcoulomb/Square Meter is equal to 3.8610215854245e-7
1 EMU of Charge/Square Mile = 3.8610215854245e-9 Abcoulomb/Square Decimeter 1 EMU of Charge/Square Mile in Abcoulomb/Square Decimeter is equal to 3.8610215854245e-9
1 EMU of Charge/Square Mile = 3.8610215854245e-11 Abcoulomb/Square Centimeter 1 EMU of Charge/Square Mile in Abcoulomb/Square Centimeter is equal to 3.8610215854245e-11
1 EMU of Charge/Square Mile = 3.8610215854245e-13 Abcoulomb/Square Millimeter 1 EMU of Charge/Square Mile in Abcoulomb/Square Millimeter is equal to 3.8610215854245e-13
1 EMU of Charge/Square Mile = 3.8610215854245e-19 Abcoulomb/Square Micrometer 1 EMU of Charge/Square Mile in Abcoulomb/Square Micrometer is equal to 3.8610215854245e-19
1 EMU of Charge/Square Mile = 3.8610215854245e-25 Abcoulomb/Square Nanometer 1 EMU of Charge/Square Mile in Abcoulomb/Square Nanometer is equal to 3.8610215854245e-25
1 EMU of Charge/Square Mile = 0.38610215854245 Abcoulomb/Square Kilometer 1 EMU of Charge/Square Mile in Abcoulomb/Square Kilometer is equal to 0.38610215854245
1 EMU of Charge/Square Mile = 3.228305785124e-7 Abcoulomb/Square Yard 1 EMU of Charge/Square Mile in Abcoulomb/Square Yard is equal to 3.228305785124e-7
1 EMU of Charge/Square Mile = 3.5870064279155e-8 Abcoulomb/Square Foot 1 EMU of Charge/Square Mile in Abcoulomb/Square Foot is equal to 3.5870064279155e-8
1 EMU of Charge/Square Mile = 2.4909766860524e-10 Abcoulomb/Square Inch 1 EMU of Charge/Square Mile in Abcoulomb/Square Inch is equal to 2.4909766860524e-10
1 EMU of Charge/Square Mile = 1 Abcoulomb/Square Mile 1 EMU of Charge/Square Mile in Abcoulomb/Square Mile is equal to 1
1 EMU of Charge/Square Mile = 1.0725059959512e-9 Ampere Hour/Square Meter 1 EMU of Charge/Square Mile in Ampere Hour/Square Meter is equal to 1.0725059959512e-9
1 EMU of Charge/Square Mile = 1.0725059959512e-11 Ampere Hour/Square Decimeter 1 EMU of Charge/Square Mile in Ampere Hour/Square Decimeter is equal to 1.0725059959512e-11
1 EMU of Charge/Square Mile = 1.0725059959512e-13 Ampere Hour/Square Centimeter 1 EMU of Charge/Square Mile in Ampere Hour/Square Centimeter is equal to 1.0725059959512e-13
1 EMU of Charge/Square Mile = 1.0725059959512e-15 Ampere Hour/Square Millimeter 1 EMU of Charge/Square Mile in Ampere Hour/Square Millimeter is equal to 1.0725059959512e-15
1 EMU of Charge/Square Mile = 1.0725059959512e-21 Ampere Hour/Square Micrometer 1 EMU of Charge/Square Mile in Ampere Hour/Square Micrometer is equal to 1.0725059959512e-21
1 EMU of Charge/Square Mile = 1.0725059959512e-27 Ampere Hour/Square Nanometer 1 EMU of Charge/Square Mile in Ampere Hour/Square Nanometer is equal to 1.0725059959512e-27
1 EMU of Charge/Square Mile = 0.0010725059959512 Ampere Hour/Square Kilometer 1 EMU of Charge/Square Mile in Ampere Hour/Square Kilometer is equal to 0.0010725059959512
1 EMU of Charge/Square Mile = 8.9675160697888e-10 Ampere Hour/Square Yard 1 EMU of Charge/Square Mile in Ampere Hour/Square Yard is equal to 8.9675160697888e-10
1 EMU of Charge/Square Mile = 9.9639067442098e-11 Ampere Hour/Square Foot 1 EMU of Charge/Square Mile in Ampere Hour/Square Foot is equal to 9.9639067442098e-11
1 EMU of Charge/Square Mile = 6.919379683479e-13 Ampere Hour/Square Inch 1 EMU of Charge/Square Mile in Ampere Hour/Square Inch is equal to 6.919379683479e-13
1 EMU of Charge/Square Mile = 0.0027777777777778 Ampere Hour/Square Mile 1 EMU of Charge/Square Mile in Ampere Hour/Square Mile is equal to 0.0027777777777778
1 EMU of Charge/Square Mile = 0.0000038610215854245 Ampere Second/Square Meter 1 EMU of Charge/Square Mile in Ampere Second/Square Meter is equal to 0.0000038610215854245
1 EMU of Charge/Square Mile = 3.8610215854245e-8 Ampere Second/Square Decimeter 1 EMU of Charge/Square Mile in Ampere Second/Square Decimeter is equal to 3.8610215854245e-8
1 EMU of Charge/Square Mile = 3.8610215854245e-10 Ampere Second/Square Centimeter 1 EMU of Charge/Square Mile in Ampere Second/Square Centimeter is equal to 3.8610215854245e-10
1 EMU of Charge/Square Mile = 3.8610215854245e-12 Ampere Second/Square Millimeter 1 EMU of Charge/Square Mile in Ampere Second/Square Millimeter is equal to 3.8610215854245e-12
1 EMU of Charge/Square Mile = 3.8610215854245e-18 Ampere Second/Square Micrometer 1 EMU of Charge/Square Mile in Ampere Second/Square Micrometer is equal to 3.8610215854245e-18
1 EMU of Charge/Square Mile = 3.8610215854245e-24 Ampere Second/Square Nanometer 1 EMU of Charge/Square Mile in Ampere Second/Square Nanometer is equal to 3.8610215854245e-24
1 EMU of Charge/Square Mile = 3.86 Ampere Second/Square Kilometer 1 EMU of Charge/Square Mile in Ampere Second/Square Kilometer is equal to 3.86
1 EMU of Charge/Square Mile = 0.000003228305785124 Ampere Second/Square Yard 1 EMU of Charge/Square Mile in Ampere Second/Square Yard is equal to 0.000003228305785124
1 EMU of Charge/Square Mile = 3.5870064279155e-7 Ampere Second/Square Foot 1 EMU of Charge/Square Mile in Ampere Second/Square Foot is equal to 3.5870064279155e-7
1 EMU of Charge/Square Mile = 2.4909766860524e-9 Ampere Second/Square Inch 1 EMU of Charge/Square Mile in Ampere Second/Square Inch is equal to 2.4909766860524e-9
1 EMU of Charge/Square Mile = 10 Ampere Second/Square Mile 1 EMU of Charge/Square Mile in Ampere Second/Square Mile is equal to 10
1 EMU of Charge/Square Mile = 6.4350359757074e-8 Ampere Minute/Square Meter 1 EMU of Charge/Square Mile in Ampere Minute/Square Meter is equal to 6.4350359757074e-8
1 EMU of Charge/Square Mile = 6.4350359757074e-10 Ampere Minute/Square Decimeter 1 EMU of Charge/Square Mile in Ampere Minute/Square Decimeter is equal to 6.4350359757074e-10
1 EMU of Charge/Square Mile = 6.4350359757074e-12 Ampere Minute/Square Centimeter 1 EMU of Charge/Square Mile in Ampere Minute/Square Centimeter is equal to 6.4350359757074e-12
1 EMU of Charge/Square Mile = 6.4350359757074e-14 Ampere Minute/Square Millimeter 1 EMU of Charge/Square Mile in Ampere Minute/Square Millimeter is equal to 6.4350359757074e-14
1 EMU of Charge/Square Mile = 6.4350359757074e-20 Ampere Minute/Square Micrometer 1 EMU of Charge/Square Mile in Ampere Minute/Square Micrometer is equal to 6.4350359757074e-20
1 EMU of Charge/Square Mile = 6.4350359757074e-26 Ampere Minute/Square Nanometer 1 EMU of Charge/Square Mile in Ampere Minute/Square Nanometer is equal to 6.4350359757074e-26
1 EMU of Charge/Square Mile = 0.064350359757074 Ampere Minute/Square Kilometer 1 EMU of Charge/Square Mile in Ampere Minute/Square Kilometer is equal to 0.064350359757074
1 EMU of Charge/Square Mile = 5.3805096418733e-8 Ampere Minute/Square Yard 1 EMU of Charge/Square Mile in Ampere Minute/Square Yard is equal to 5.3805096418733e-8
1 EMU of Charge/Square Mile = 5.9783440465259e-9 Ampere Minute/Square Foot 1 EMU of Charge/Square Mile in Ampere Minute/Square Foot is equal to 5.9783440465259e-9
1 EMU of Charge/Square Mile = 4.1516278100874e-11 Ampere Minute/Square Inch 1 EMU of Charge/Square Mile in Ampere Minute/Square Inch is equal to 4.1516278100874e-11
1 EMU of Charge/Square Mile = 0.16666666666667 Ampere Minute/Square Mile 1 EMU of Charge/Square Mile in Ampere Minute/Square Mile is equal to 0.16666666666667 | {"url":"https://www.kodytools.com/units/scharge/from/emuchargepmi2/to/ucpdm2","timestamp":"2024-11-10T05:53:16Z","content_type":"text/html","content_length":"157237","record_id":"<urn:uuid:bf5b93f9-01d6-48b3-8122-ec62b11ede48>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00643.warc.gz"} |
Set Timing
You will have 62 minutes to complete the following section.
You are not allowed to use a calculator.
A booklet is recommended for taking notes.
Problem Solving question type
Each Problem Solving question consists of a quantitative question and 5 answer choices. Solve the problem and select the best of the given answer choices.
Note that all numbers given are real numbers.
Figures are drawn as accurately as possible, except when explicitly stated otherwise.
Data Sufficiency question type
Each Data Sufficiency question consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are
sufficient for answering the question.
In questions that ask you to find a numerical quantity, data provided in the statements are considered sufficient only if it is possible to determine exactly one numerical value for the quantity.
You must answer each question by choosing from 5 Data Sufficiency answer choices:
(A) Statement 1 ALONE is sufficient, but statement (2) alone is not sufficient.
(B) Statement 2 ALONE is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements (1) and (2) TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOT sufficient.
These 5 answer choices are presented (in this exact order) for every Data Sufficiency question.
Note that all numbers given are real numbers. Figures are not necessarily drawn to scale.
Set Timing
You will have 62 minutes to complete the following section.
You are not allowed to use a calculator.
A booklet is recommended for taking notes.
Problem Solving question type
Each Problem Solving question consists of a quantitative question and 5 answer choices. Solve the problem and select the best of the given answer choices.
Note that all numbers given are real numbers.
Figures are drawn as accurately as possible, except when explicitly stated otherwise.
Data Sufficiency question type
Each Data Sufficiency question consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are
sufficient for answering the question.
In questions that ask you to find a numerical quantity, data provided in the statements are considered sufficient only if it is possible to determine exactly one numerical value for the quantity.
You must answer each question by choosing from 5 Data Sufficiency answer choices:
(A) Statement 1 ALONE is sufficient, but statement (2) alone is not sufficient.
(B) Statement 2 ALONE is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements (1) and (2) TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOT sufficient.
These 5 answer choices are presented (in this exact order) for every Data Sufficiency question.
Note that all numbers given are real numbers. Figures are not necessarily drawn to scale.
The table above gives the coffee consumption in 1994 for five countries. If the total coffee consumption of these countries was 40 percent of the world’s coffee consumption, what was the world’s
coffee consumption, in millions of kilograms, in 1994?
What is the sum of the integers from -190 to 195, inclusive?
Is the standard deviation of the salaries of Company Y’s employees greater than the standard deviation of the salaries of Company Z’s employees?
(1) The average (arithmetic mean) salary of Company Y’s employees is greater than the average salary of Company Z’s employees.
(2) The median salary of Company Y’s employees is greater than the median salary of Company Z’s employees.
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
EACH statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are NOT sufficient.
Square S is inscribed in circle T. If the perimeter of S is 24, what is the circumference of T?
The operation @ is defined for all nonzero numbers a and b by a @ b = a/b – b/a. If x and y are nonzero numbers, which of the following statements must be true?
I. x @ xy = x(1 @ y)
II. x @ y = -( y @ x)
III. 1/x @ 1/y = y @ x
What was the price at which a merchant sold a certain appliance?
(1) The merchant’s gross profit on the appliance was 20 percent of the price at which the merchant sold the appliance.
(2) The price at which the merchant sold the appliance was $50 more than the merchant’s cost of the appliance.
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
EACH statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are NOT sufficient.
An attorney charged a fee for estate planning services for a certain estate. The attorney’s fee was what percent of the assessed value of the estate?
(1) The assessed value of the estate was $1.2 million.
(2) The attorney charged $2,400 for the estate planning services.
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
EACH statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are NOT sufficient
Is the sum of the integers x and y a prime number?
(1) x is an even prime number.
(2) y is a prime number between 10 and 20.
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
EACH statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are NOT sufficient.
There are 11 women and 9 men in a certain club. If the club is to select a committee of 2 women and 2 men, how many different such committees are possible?
If w, x, y, and z are integers such that 1<w < x < y < z and wxyz = 462, then z =?
The sum of three integers is 40. The largest integer is 3 times the middle integer, and the smallest integer is 23 less than the largest integer. What is the product of the three integers?
If a and b are positive, is (a^-1+ b^-1)^-1 less than (a^-1 b^-1)^-1 ?
(1) a = 2b
(2) a + b > 1
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
EACH statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are NOT sufficient.
On a certain transatlantic crossing, 20 percent of a ship’s passengers held round-trip tickets and also took their cars abroad the ship. If 60 percent of the passengers with round-trip tickets did
not take their cars abroad the ship, what percent of the ship’s passengers held round-trip tickets?
If x is an integer, which of the following must be an odd integer?
A collection of 36 cards consists of 4 sets of 9 cards each. The 9 cards in each set are numbered 1 through 9. If one card has been removed from the collection, what is the number on that card?
(1) The units digit of the sum of the numbers on the remaining 35 cards is 6.
(2) The sum of the numbers on the remaining 35 cards is 176.
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
EACH statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are NOT sufficient.
If xy ≠ 0, is x/y = 1?
(1) x^2 = y^2
(2) xy > 0
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
EACH statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are NOT sufficient.
A construction company was paid a total of $500,000 for a construction project. The
company’s only costs for the project were for labor and materials. Was the company’s
profit for the project greater than $150,000?
(1) The company’s total cost was three times its cost for materials.
(2) The company’s profit was greater than its cost for labor.
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
EACH statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are NOT sufficient.
│Time │Amount │
│1:00 P.M. │10.0 grams │
│4:00 P.M. │x grams │
│7:00 P.M. │14.4 grams │
Data for a certain biology experiment are given in the table above. If the amount of
bacteria present increased by the same fraction during each of the two 3-hour periods
shown, how many grams of bacteria were present at 4:00 P.M.?
The value of (9×10^7 )(9×10^8 ) is closest to which of the following?
A certain roller coaster has 3 cars, and a passenger is equally likely to ride in any 1 of the 3 cars each time that passenger rides the roller coaster. If a certain passenger is to ride the roller
coaster 3 times, what is the probability that the passenger will ride in each of the 3 cars?
There are 8 books on a shelf, of which 2 are paperbacks and 6 are hardbacks. How many possible selections of 4 books from this self include at least one paperback?
In a corporation, 50 percent of the male employees and 40 percent of the female employees are at least 35 years old. If 42 percent of all the employees are at least 35 years old, what fraction of the
employees in the corporation are females?
Of the goose eggs laid at a certain pond, 2/3 hatched, and 3/4 of the geese that hatched from those eggs survived the first month. Of the geese that survived the first month, 3/5 did not survive the
first year. If 120 geese survived the first year and if no more than one goose hatched from each egg, how many goose eggs were laid at the pond?
If 10^50 – 74 is written as an integer in base 10 notation, what is the sum of the digits in that integer?
In the rectangular solid above, the three sides shown have areas 12, 15, and 20, respectively. What is the volume of the solid?
There are 8 magazines lying on a table; 4 are fashion magazines and the other 4 are sports magazines. If 3 magazines are to be selected at random from the 8 magazines, what is the probability that at
least one of the fashion magazines will be selected?
What is the remainder when the two digit, positive integer x is divided by 3?
(1) The sum of the digits of x is 5.
(2) The remainder when x is divided by 9 is 5
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient
EACH statement ALONE is sufficient
Statements (1) and (2) TOGETHER are NOT sufficient.
In Town X, 64 percent of the population are employed, and 48 percent of the population are employed males. What percent of the employed people in Town X are females?
If k = -1, which of the following is (are) true?
I. k^k = k
II. |k| = -k
III. k^0 = -k
Is the sum of p^2 and q^2 greater than 1?
(1) p is greater than 1/2.
(2) q is greater than 1/2.
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient
EACH statement ALONE is sufficient
Statements (1) and (2) TOGETHER are NOT sufficient.
What is the median number of employees assigned per project for the projects at
Company Z?
(1) 25 percent of the projects at Company Z have 4 or more employees assigned to
each project.
(2) 35 percent of the projects at Company Z have 2 or fewer employees assigned to
each project.
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient
EACH statement ALONE is sufficient
Statements (1) and (2) TOGETHER are NOT sufficient.
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deviation of the salaries of Company Y’s employees greater than the standard deviation of the salaries of Company Z’s employees? (1) The average (arithmetic mean) salary of Company Y’s employees is
greater than the average salary of Company Z’s employees. (2) The median salary of Company Y’s employees is greater than the median salary of Company Z’s employees.","img":"https://
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self-inductance coil energy storage formula
روابط عشوائية
What is self-inductance?
The inductance of a coil can be calculated using the formula: ... and l is the length of the coil. Applications of Self-Inductance. Self-inductance plays a crucial role in various applications, such
as: Energy storage: Inductors store energy in their magnetic field, which can be released when required, making them essential components in …
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10.11: Self Inductance
In this section we are dealing with the self inductance of a single coil rather than the mutual inductance between two coils. If the current through a single coil changes, the …
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Self Inductance
Self-Induction L of a coil depends upon-The size and shape of the coil. The number of turns N. The magnetic property of the medium within the coil in which the flux is present. Note: Self-induction L
does not depend on the current I. Coefficient of Self-inductance formula: Total flux linked with the coil, N∅ ∝ I. N∅ = LI
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Study of Design of Superconducting Magnetic Energy …
The formula to compute the self-inductance (in Henry) as follows [1] ... inductance of the coil. The energy storage capacity of the coil is increased. Here the number of double pancakes are varied
from 10, in the step of 10 till 100. It is observed that for every
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14.2 Self-Inductance and Inductors
Coaxial cables have two long cylindrical conductors that possess current and a self-inductance that may have undesirable effects. A circuit element used to provide self-inductance is known as an
inductor. It is represented by the symbol shown in Figure 14.6, which resembles a coil of wire, the basic form of the inductor.
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14.2 Self-Inductance and Inductors – University …
A circuit element used to provide self-inductance is known as an inductor. It is represented by the symbol shown in Figure 14.6, which resembles a coil of wire, the basic form of the inductor. Figure
14.7 shows several …
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29. Inductance and energy stored in inductors. Self-induction.
energy storage. When we charge up a capacitor, we add energy in the form of an electric eld between the oppositely charged conductors. When the capacitor is discharged, that …
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22.1: Magnetic Flux, Induction, and Faraday''s Law
Self-Inductance. Self-inductance, the effect of Faraday''s law of induction of a device on itself, also exists. When, for example, current through a coil is increased, the magnetic field and flux
also increase, inducing a counter emf, as required by Lenz''s law. Conversely, if the current is decreased, an emf is induced that opposes the decrease.
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A novel approach to calculate inductance and analyze magnetic …
Recent research work in Superconducting Magnetic Energy Storage (SMES) area, nuclear fusion reactors, and the plasma reactors such as Tokamak has suggested an advanced coil with a helical toroidal
structure [1], [2], [3], [4].The main reason for this suggestion is the ability to implement special target functions for this coil in …
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Mutual Inductance and Self Inductance | Formula & Example
As the coils are wound on the same iron core, k=1. Equation (11): M = k√L1 ×L2 =√9.42×9.42 = 9.42mH M = k L 1 × L 2 = 9.42 × 9.42 = 9.42 m H. Key Takeaways of Mutual and Self Inductance. Mutual
inductance refers to the phenomenon where the change in current flow in one coil induces a voltage in an adjacent coil.
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Energy stored in inductor (1/2 Li^2) (video) | Khan Academy
An inductor carrying current is analogous to a mass having velocity. So, just like a moving mass has kinetic energy = 1/2 mv^2, a coil carrying current stores energy in its magnetic field giving by 1
/2 Li^2. Let''s derive the expression for it using the concept of …
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14.4: Energy in a Magnetic Field
The magnetic energy is calculated by an integral of the magnetic energy density times the differential volume over the cylindrical shell. After the integration is carried out, we have a closed-form
solution for part (a). The self-inductance per unit length is determined based on this result and Equation ref{14.22}. Solution
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7.2.3 Inductance Neumann Formula for the Mutual Inductance
where L1 is a constant of proportionality called the self-inductance of coil 1. The SI unit of self-inductance is the henry (H). The self-inductance of a circuit depends on its size and its shape.
The self-induced emf in coil 1 due to changes in I1 takes the form N1Φ11 I
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What is Self Inductance? – Definition, Theory & Formula
The general expression of self inductance is. The Equation 1 clearly shows that the inductance is proportional to the square of the number of turns of the magnetizing coil and is inversely
proportional to the reluctance. The reluctance in turn depends upon the dimensions of the coil (i.e. its length and the cross-sectional area) and …
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Self-inductance | Definition, Calculation & Characteristics
For a solenoid-shaped air-core coil, the self-inductance can be calculated using the following formula: L = (μ₀ * N^2 * A) / l. where: L = Self-inductance (H) μ₀ = Permeability of free space,
approximately 4π x 10^-7 H/m . N = Number of turns in the coil . A = Cross-sectional area of the coil (m^2) l = Length of the coil (m) This formula ...
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14.3: Self-Inductance and Inductors
The self-inductance and flux calculated in parts (a) and (b) are typical values for coils found in contemporary devices. If the current is not changing over time, the flux is not changing in time, so
no emf is induced.
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What are inductors? (self-inductance) (video) | Khan Academy
Self-inductance is the tendency of a coil to resist changes in current in itself. Whenever current changes through a coil, they induce an EMF, which is proportional to the rate of change of …
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Inductance of a Toroid
The inductance can be calculated in a manner similar to that for any coil of wire. The application of Faraday''s law to calculate the voltage induced in the toroid is of the form. This can be used
with the magnetic field expression above to obtain an expression for the inductance. Toroidal radius r = cm with N = turns,
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Superconducting magnetic energy storage
Superconducting magnetic energy storage (SMES) systems store energy in the magnetic field created by the flow of direct current in a superconducting coil which has been cryogenically cooled to a
temperature below its superconducting critical temperature.This use of superconducting coils to store magnetic energy was invented by M. Ferrier in …
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23.9 Inductance
Units of self-inductance are henries (H) just as for mutual inductance. The larger the self-inductance L L size 12{L} {} of a device, the greater its opposition to any change in current through it.
For example, a large coil with many turns and an iron core has a large L L size 12{L} {} and will not allow current to change quickly.
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14.3 Energy in a Magnetic Field – University Physics …
Example Self-Inductance of a Coaxial Cable. Equation 14.11 shows two long, concentric cylindrical shells of radii [latex]{R}_{1}[/latex] and [latex]{R}_{2}.[/latex] As discussed in Capacitance on
capacitance, this …
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Chapter 11 Inductance and Magnetic Energy
called "self-inductance," and the emf generated is called the self-induced emf or back emf, which we denote as ε L . All current-carrying loops exhibit this property.
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14.3 Energy in a Magnetic Field
The magnetic energy is calculated by an integral of the magnetic energy density times the differential volume over the cylindrical shell. After the integration is carried out, we have a closed-form
solution for part (a). The self-inductance per unit length is determined based on this result and Equation 14.22. Solution
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23.12: Inductance
Self-inductance, the effect of Faraday''s law of induction of a device on itself, also exists. When, for example, current through a coil is increased, the magnetic field and flux also …
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A direct current conversion device for closed HTS coil of ...
According to the empirical formula in [30], the self-inductance of a short air-core solenoid can be calculated by (5) L air core = 6.4 μ 0 N 2 D 2 3.5 D + 8 h · D − 2.25 d D, where N is the turn
numbers of the coil, μ 0 is the vacuum permeability which equals 4π × 10 −7, D is the outer diameter of the coil, d is the thick of the coil ...
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7.12: Inductance
Inductance is the ability of a structure to store energy in a magnetic field. The inductance of a structure depends on the geometry of its current-bearing structures and the permeability of the
intervening medium. Note that inductance does not depend on current, which we view as either a stimulus or response from this point of view.
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10.16: Energy Stored in an Inductance
The work done in time dt is Lii˙dt = Lidi d t is L i i ˙ d t = L i d i where di d i is the increase in current in time dt d t. The total work done when the current is increased from 0 to I I is. L∫I
0 idi = 1 2LI2, (10.16.1) (10.16.1) L ∫ 0 I i d i = 1 2 L I 2, and this is the energy stored in the inductance. (Verify the dimensions.)
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23.9 Inductance
Mutual inductance is the effect of Faraday''s law of induction for one device upon another, such as the primary coil in transmitting energy to the secondary in a transformer. See Figure 23.37, ...
Self-inductance, the effect of Faraday''s law of induction of a device on itself, also exists. When, for example, current through a coil is ...
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Inductance Comparison of two Coil of …
The structure of a modular toroidal coil with 8 solenoid coils. A. 3D diagram of the modular toroidal coil. B. Projection of the structure of the modular toroidal coil on the x-y plane.
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Chapter 30 – Inductance
Mutual inductance: emf opposes the flux change. - Only a time-varying current induces an emf. Units of inductance: 1 Henry = 1 Weber/A = 1 V s/A = 1 J/A2. Ex. 30.1. 2. Self Inductance and Inductors.
- When a current is present in a circuit, it sets up B that causes a magnetic flux that changes when the current changes emf is induced.
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How Does Rewinding a Coil Affect Its Self-Inductance and Energy Storage?
Energy Inductance. In summary, for the first conversation question, unwinding and rewinding half the length of wire in a coil with the same diameter but half the number of turns does not change the
self-inductance. For the second conversation question, if the current through an inductor is doubled, the energy stored in the inductor …
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Inductance Calculation of Single-Layer Planar Spiral Coil
Self-Inductance Calculation of a Planar Spiral Coil. The self-inductance of circular planar spiral coil can be calculated by solving the Neumann''s integral formula for mutual inductance between two
coils [ 10 ]. The geometrical parameters of the planar spiral, which lie on the x – y plane, is shown in Figure 1.
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Self Inductance of a Solenoid
3 · Uses of Self-Inductance. Storing Energy: Inductors are like energy storage units that hold electrical energy in a magnetic field. In Different Devices: They''re used in things like tuning
circuits, sensors, and motors to make them work. Transforming Energy: Inductors are also part of transformers, which change electrical energy from one form to …
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Analysis of Synchronous Induction Coil Launcher Compatible with ...
The mathematical model of the synchronous induction coil launcher was established, and the formula of the armature force and the equation of motion were obtained by using the method of coupling
analysis of electric-magnetic-kinematics-dynamics. The simulation model is built to analyze the transient motion process, and the influence …
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Energy Stored in an Inductor
When a electric current is flowing in an inductor, there is energy stored in the magnetic field. Considering a pure inductor L, the instantaneous power which must be supplied to …
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Inductance Comparison of two Coil of Superconductor Magnetic Energy …
Inductance Comparison of two Coil of Superconductor Magnetic Energy Storage Using the Analytical and Finite Element Method December 2010 Conference: in Proc. 25rd International Power System Conf ...
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7.13: Inductance of a Straight Coil
In this section, we determine the inductance of a straight coil, as shown in Figure 7.13.1 7.13. 1. The coil is circular with radius a a and length l l and consists of N N windings of wire wound with
uniform winding density. Also, we assume the winding density N/l N / l is large enough that magnetic field lines cannot enter or exit between ...
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Energy Stored in an Inductor
Energy Stored in an Inductor. Suppose that an inductor of inductance is connected to a variable DC voltage supply. The supply is adjusted so as to increase the current flowing …
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10.11: Self Inductance
10.11: Self Inductance. In this section we are dealing with the self inductance of a single coil rather than the mutual inductance between two coils. If the current through a single coil changes, the
magnetic field inside that coil will change; consequently a back EMF will be induced in the coil that will oppose the change in the magnetic field ...
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microsof | Ibrahim Jaber
In an office, the best way to survive is being good with Microsoft Office and one of the primary tools to learn and adapt is Microsoft’s Excel. This is a must learner for any employee if he/she
wishes to grow in the career as excel is far beyond just summation and subtraction.
Excel is just the god tools for anything and everything that is needed for a decision. Whether it is finance, accounting, merchandising or materials control, Inventory Management just the formula,
and with magic, everything manual is made auto and instant decisions possible. You need consolidated or final grouped data – excel is there, you need to create a balance sheet linked from income
statement – excel is your helping hand. Basically, excel will be your go-to tool for your daily work needs.
To work better and to become a tiny master in excel, whilst making thing auto I would like to share my thoughts and experiences with excel to get you started.If you’re just starting out with Excel,
there are a few basic commands that we suggest you become familiar with. These are things like:
• Creating a new spreadsheet from scratch.
• Executing basic computations in a spreadsheet, like adding, subtracting, multiplying, and dividing in a spreadsheet.
• Writing and formatting column text and titles.
• Excel’s auto-fill features.
• Adding or deleting single columns, rows, and spreadsheets. Below, we’ll get into how to add things like multiple columns and rows.
• Keeping column and row titles visible as you scroll past them in a spreadsheet, so that you know what data you’re filling as you move further down the document.
The Contents at a Glance
Excel Formulas
1) Simple Calculations
The first and foremost basic formulas are the simple arithmetic expressions – adding, subtracting, multiplying, or dividing any of your numerical values.
• To add, use the + sign.
• To subtract, use the – sign.
• To multiply, use the * sign.
• To divide, use the / sign.
Extra: if you are working with a lot of numbers you can use the formula =Sum(Cell Range) to get sum of all the numbers in the range and to find average of that range you can use =Average(Cell Range)
2) Conditional Formatting Formula
Conditional formatting allows you to change a cell’s color based on the information within the cell. For example, if you want to flag certain numbers that are above greater or less than some number
of the data in your spreadsheet, you can do that. If you want to color code commonalities between different rows in Excel, you can do that. This will help you quickly see the information the is
important to you.
To get started, highlight the group of cells you want to use conditional formatting on. Then, choose “Conditional Formatting” from the Home menu and select your logic from the dropdown. (You can also
create your own rule if you want something different.) A window will pop up that prompts you to provide more information about your formatting rule. Select “OK” when you’re done, and you should see
your results automatically appear.
In the bottom is a small gif example showing conditional formatting based on day and temperature.
3) IF Statement
Link to formula: If Statement
At times there comes a situation where we need to show something if something else is happening or is in work. For this sole purpose, Excel has IF statement a lifesaver that tests data and displays
the value when the test is true or false.
In our example we used names of some random students (Nudrat, Taylor Swift, Ratz, Ridwan, Zakir, Richi, Tarek, Swift, Max Landis) took their age and tested if they are adult or not.
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The formula: IF(logical_test, value_if_true, value of false)
Example Shown Below: =IF(B2<20,”Not Adult”,”Adult”). You can also nest other formulas instead of text: =IF(B2<25,(C1+D1),(C1+E1)
In general terms, the formula would be IF(Logical Test, the value of the true, value of false). Let’s dig into each of these variables.
• Logical_Test: The logical test is the “IF” part of the statement. In this case, the logic is B2<20 because we want to make sure that the cell corresponding with the student has aged less than 20.
• Value_if_True: This is what we want the cell to show if the value is true. In this case, we want the cell to show “Not Adult” to indicate that the student was awarded the 10 points. Only use
quotation marks if you want the result to be text instead of a number.
• Value_if_False: This is what we want the cell to show if the value is false. In this case, for any student’s age not above 20, we want the cell to show “Adult” to show adult text. Only use
quotation marks if you want the result to be text instead of a number.
4) Locking Cells
Have you ever seen a dollar sign in an Excel formula? When used in a formula, it isn’t representing an American dollar; instead, it makes sure that the exact column and row are held the same even if
you copy the same formula in adjacent rows.
You see, a cell reference — when you refer to cell A5 from cell C5, for example — is relative by default. In that case, you’re actually referring to a cell that’s five columns to the left (C minus A)
and in the same row (5). This is called a relative formula. When you copy a relative formula from one cell to another, it’ll adjust the values in the formula based on where it’s moved. But sometimes,
we want those values to stay the same no matter whether they’re moved around or not — and we can do that by making the formula in the cell into what’s called an absolute formula.
To change the relative formula (=A5+C5) into an absolute formula, we’d precede the row and column values by dollar signs, like this: (=$A$5+$C$5).
Excel Advanced Formula
1) VLOOKUP Function
Link to Formula: VLookup
Have you ever had two sets of data on two different spreadsheets that you want to combine into a single spreadsheet?
For example, you might have a list of people’s names next to their email addresses in one spreadsheet, and a list of those same people’s email addresses next to their company names in the other — but
you want the names, email addresses, and company names of those people to appear in one place.
I have to combine data sets like this a lot — and when I do, the VLOOKUP is my go-to formula. Before you use the formula, though, be absolutely sure that you have at least one column that appears
identically in both places. Scour your data sets to make sure the column of data you’re using to combine your information is exactly the same, including no extra spaces.
The formula : =VLOOKUP(lookup value, table array, column number, [range lookup])
The formula with variables from our example below:=VLOOKUP(C2,Sheet2!A:B,2,FALSE)
In this formula, there are several variables. The following is true when you want to combine the information in Sheet 1 and Sheet 2 onto Sheet 1.
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• Lookup Value: This is the identical value you have in both spreadsheets. Choose the first value in your first spreadsheet. In the example that follows, this means the first email address on the
list, or cell 2 (C2).
• Table Array: The range of columns on Sheet 2 you’re going to pull your data from, including the column of data identical to your lookup value (in our example, email addresses) in Sheet 1 as well
as the column of data you’re trying to copy to Sheet 1. In our example, this is “Sheet2!A:B.” “A” means Column A in Sheet 2, which is the column in Sheet 2 where the data identical to our lookup
value (email) in Sheet 1 is listed. The “B” means Column B, which contains the information that’s only available in Sheet 2 that you want to translate to Sheet 1.
• Column Number: If the table array (the range of columns you just indicated) this tells Excel which column the new data you want to copy to Sheet 1 is located in. In our example, this would be the
column that “House” is located in. “House” is the second column in our range of columns (table array), so our column number is 2. [Note: Your range can be more than two columns. For example, if
there are three columns on Sheet 2 — Email, Age, and House — and you still want to bring House onto Sheet 1, you can still use a VLOOKUP. You just need to change the “2” to a “3” so it pulls back
the value in the third column: =VLOOKUP(C2:Sheet2!A:C,3,false).]
• Range Lookup: Use FALSE to ensure you pull in only exact value matches.
In the example below, Sheet 1 and Sheet 2 contain lists describing different information about the same people, and the common thread between the two is their email addresses. Let’s say we want to
combine both datasets so that all the house information from Sheet 2 translates over to Sheet 1.
2) INDEX MATCH
Link to formula : Index, Match
Like VLOOKUP, the INDEX and MATCH functions pull in data from another dataset into one central location. Here are the main differences:
1. VLOOKUP is a much simpler formula. If you’re working with large data sets that would require thousands of lookups, using the INDEX MATCH function will significantly decrease load time in Excel.
2. INDEX MATCH formulas work right-to-left, whereas VLOOKUP formulas only work as a left-to-right lookup. In other words, if you need to do a lookup that has a lookup column to the right of the
results column, then you’d have to rearrange those columns in order to do a VLOOKUP. This can be tedious with large datasets and/or lead to errors.
So if I want to combine the information in Sheet 1 and Sheet 2 onto Sheet 1, but the column values in Sheets 1 and 2 aren’t the same, then to do a VLOOKUP, I would need to switch around my columns.
In this case, I’d choose to do an INDEX MATCH instead.
Let’s look at an example. Let’s say Sheet 1 contains a list of people’s names and their Hogwarts email addresses, and Sheet 2 contains a list of people’s email addresses and the Patronus that each
student has. (For the non-Harry Potter fans out there, every witch or wizard has an animal guardian called a “Patronus” associated with him or her.) The information that lives in both sheets is the
column containing email addresses, but this email address column is in different column numbers on each sheet. I’d use the INDEX MATCH formula instead of VLOOKUP so I wouldn’t have to switch any
columns around.
So what’s the formula, then? The INDEX MATCH formula is actually the MATCH formula nested inside the INDEX formula. You’ll see I differentiated the MATCH formula using a different color here.
The formula: =INDEX(table array, MATCH formula)
This becomes: =INDEX(table array, MATCH (lookup_value, lookup_array))
The formula with variables from our example below: =INDEX(Sheet2!A:A,(MATCH(Sheet1!C:C,Sheet2!C:C,0)))
Here are the variables:
• Table Array: The range of columns on Sheet 2 containing the new data you want to bring over to Sheet 1. In our example, “A” means Column A, which contains the “Patronus” information for each
• Lookup Value: This is the column in Sheet 1 that contains identical values in both spreadsheets. In the example that follows, this means the “email” column on Sheet 1, which is Column C. So:
• Lookup Array: This is the column in Sheet 2 that contains identical values in both spreadsheets. In the example that follows, this refers to the “email” column on Sheet 2, which happens to also
be Column C. So: Sheet2!C:C.
Once you have your variables straight, type in the INDEX MATCH formula in the top-most cell of the blank Patronus column on Sheet 1, where you want the combined information to live.
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3) COUNTIF Function
Link to Formula: Link
Instead of manually counting how often a certain value or number appears, let Excel do the work for you. With the COUNTIF function, Excel can count the number of times a word or number appears in any
range of cells.
For example, let’s say I want to count the number of times the word “Gryffindor” appears in my data set.
The formula: =COUNTIF(range, criteria)
The formula with variables from our example below: =COUNTIF(D:D,”Gryffindor”)
In this formula, there are several variables:
• Range: The range that we want the formula to cover. In this case, since we’re only focusing on one column, we use “D:D” to indicate that the first and last column are both D. If I were looking at
columns C and D, I would use “C:D.”
• Criteria: Whatever number or piece of text you want Excel to count. Only use quotation marks if you want the result to be text instead of a number. In our example, the criteria is “Gryffindor.”
Simply typing in the COUNTIF formula in any cell and pressing “Enter” will show me how many times the word “Gryffindor” appears in the dataset.
4) Sumif Function
Similar to Countif, the formula as per title sum a range of field once it matches the criteria.
In this formula, there are three variables:
• Range: The range that we want the formula to cover. In this case, since we’re only focusing on one column, we use “D:D” to indicate that the first and last column are both D. If I were looking at
columns C and D, I would use “C:D.”
• Criteria: Whatever number or piece of text you want Excel to count. Only use quotation marks if you want the result to be text instead of a number. In our example, the criteria is “Gryffindor.”
• SumRange: The range that we want the formula to do summation on. In this case, since we’re only focusing on one column, we use “D:D” to indicate that the first and last column are both D. If I
were looking at columns C and D, I would use “C:D.”
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The Commandos Have Landed in Jobur
Miscavige has sent the storm troopers to lower the boom in South Africa.
The big question is whether the boom is going to crash right through the hull and sink his ship of fools down there.
The newest post on the African Scientologists Getting Back in Comm blog is reproduced in full below.
Long term CO Africa (Ken Kreiger) and CO CMO Africa (Alex Faust) are now cleaning dumpsters with toothbrushes in Clearwater and the new “Command Team” has taken over.
It is going to be very interesting to see what happens over the next month as there are LOT more disaffected South Africans than the 18 Opinion Leaders (and 6 OT VIII’s which has got to be a large
percentage of the OT VIII’s in South Africa) they just turned into active enemies of the Corporate Church. (And he just put an end to his Ideal Org fundraising).
Can Miscavige and his jackboot tactics get this continent back under His thumb?
Stay tuned…..
The Golden Age of Rod (UPDATED)
UPDATE: We can confirm that the following people have been declared by the Church of Scientology: Welcome to the first wave of the Golden Age of Rod.
1. Gaye Corbett – 42 years in Scn, OT VIII, Cl IV, 3Ls, Data Series Evaluator, Triple Cornerstone Member, Silver Meritorius
2. Ernest Corbett – 42 years in Scn, OT VIII, 3Ls, Data Series Evaluator, Triple Cornerstone Member, Silver Meritorius
3. Tracey Henley (nee Corbett) – 40 years in Scn (all her life), OT V, Flag trained CL VI, Patron
4. Guy Henley – 12 years in Scn, Patron
5. Lisa Goosen (nee Corbett) – 35 years in Scn (all her life), Patron, Ls
6. Warwick Goosen – 20 years in Scn, Patron, Ls
7. Rodney Corbett – 40 years in Scn, OT V, OEC
8. Karl Kroeger – 35 years in Scn, OT VIII, Ls
9. Sandy Kroeger – 35 years in Scn, OT VIII, Cl IV
10. Molly Jelly – 46 years in Scn, Cl VI, OT VIII
11. Dave Jelly – 46 years in Scn, Cl VI, OT VIII
12. Craig Howarth – 20 years in Scn, 8 years on Durban Staff
13. Shirley Wartski – 20 years in Scn, Flag trained CL VI, OT V
14. Cameron Wannenberg – 15 years in Scn.
15. Kim Downing – 42 years in Scn, OT VII, Cl IV
16. Ueli Gostelli – 35+ years in Scn, OT V.
17. Carol Krieger – +- 20 years in Scn, CL IV, OT V
18. Wendy Bowman – 45 years in Scn, CL VI, 12 years on staff
Not one of these people received so much as an official phone call to inform them.
If the rumours we have been hearing are true we are about to see the launch of the Golden Age of Rod. We refer of course to the issue of Goldenrod declare orders.
There are a large number of people, approaching twenty names, who are long standing South African Scientologists that are to be imminently declared.
This coincides with the arrival on Monday of a 7 man Sea Org command/management team. No doubt this will stamp their “authority” on the local scene.
From a church standpoint it makes sense in a way. With the launch of the GAT II on the 23rd of November it would be able to “clean up” a very dirty field then launch GAT II and say “you see, when we
win the SP’s go wild. But we’ve got it all in hand”.
In truth it is going to be a calamity of catastrophic proportions. They are trying to cut out a “cancer” that has become way to big to operate on. Attempting to do so may well finally kill off the
Even die hard believers will begin to question the efficacy of church management. Perhaps not overtly but it will certainly start them on a journey of doubt and questioning. Especially since there
has not been even a whisper of standard justice procedures such as a Comm-Ev.
This list of names, between them, are connected to at least half of the entire South African Scientology community. That’s a huge line in the sand and Scientologists will soon be forced to make a
serious decision about disconnection.
It doesn’t have to be this way. Open up the lines of communication! Let Scientologists discuss and debate the future of the movement to which they have contributed so much.
A statement of fantasy. But really, if this were to happen it would at least begin the process of healing that is so desperately needed.
Of course these rumours may just be a ruse to try and scare people straight. That’s a tactic that has been used before. The coming days will tell.
Lastly, the new seven man team will be paraded at the Joburg graduation this Friday. Feel free to go along and then pass on the news so we can report it.
1. Charmaine Buttrick (Olley) says
This is the tipping point necessary to get the Tech back on Track!
I loved working with new public in Div 6 and did pretty well in it. But I eventually could not bring myself to handing over these beings, that were just beginning to wake up.
Christie, some of my fondest Div 6 memories have you in the background; always smiling, always supportive.
2. Lisa Goosen says
Funny this is that the org staff at “Jobur” Org have been living in two houses owned by our family for over 2 years at NO CHARGE!!!! big slap in the face, what’s the org gonna do now?
3. Wendy B says
Hi to all who are looking!
I am also one of the 18, who have been so unceremoniuosly cast aside as an “untouchable”!
Little do they know the joy in my heart that I am experiencing to be free to think, do and express how I feel without the shackles of the church limiting all.
Luckily I decided some time ago to operate independently.
With the help of some wonderful terminals that I have found outside the church who have given me the most incredible encouragement, validation and guidance, I have gotten myself back in the chair
after 25 years of suppression of my Class 6 ability, and am having the most remarkable wins as an auditor, and my pc’s wins are spectacular.
It is a joy to be winning in my newly created theta environment with such wonderful friends and family around me.
I am currently OT V and am about to embark into further discoveries of the upper Bridge with happy anticipation. I see the future full of growth and prosperity.
Thank you Mike and all those who have sent the “famous 18” such good postulates and well wishes
With love to you all and excitement in my heart, Wendy
4. Black Panther says
I’ve just spent considerable time catching up on all the posts here, and I am overwhelmed by the camaraderie and outpouring of ARC being communicated by everyone.
On behalf of all South African SCN’s, I thank everyone for their words of encouragement and support for the “Joburg 18” and everyone else affected by the recent turn of events. The ripple effect
of this is being felt by many.
Thanks very much to Bela, Idle Morgue, Hallie-Jane, Aquamarine and KRC Jenny for specifically welcoming me to this blog, and of course, thanks to Mike for providing this platform so that we can
all be “back in comm”.
Love BP
5. Aquamarine says
“Amazing how we are given the data for increased self determinism and then told never to use it.”
This is exactly what continues to blow my mind; the helpful content (tech) within the suppressive context (RCS).
Welcome Lifelongfriend – great to have you here!
6. Lifelongfriend says
I too am one of the Joburg 18. I am neither in shock nor awe! It is what it is!!! Evidently the church’s attempt to instill fear in the field. Only the brave will survive!!! Amazing how we are
given the data for increased self determinism and then told never to use it.
□ Mike Rinder says
Bravo to you and your 17 brethren….
Never forget that YOU are the ones who intimidate THEM. They are TERRIFIED of you. You must be stopped at all costs! Stopped from what is immaterial. Just STOPPED.
It is forbidden to think for yourself. Or not march in lockstep and spout the party lines.
Good luck. You are going to find more freedom and more REAL friends than you ever imagined was possible. The world of Scientology outside the Vampire Emnpire is more alive, more vibrant and
more REAL.
You have brought good fortune upon yourselves by being unwilling to blindly accept authority.
☆ Lisa Goosen says
Thank you Mike. I am so excited about the future and am looking forward to the many adventures and wins we will have using true LRH tech. We South Africans are tough cookies!!!!! Love
□ Tony DePhillips says
Hello Life long friend,
Glad to see that you are confronting it pretty well. I was on Solo Nots and Humanitarian when I was declared an SP. This was about 4 years ago and it is a bit surreal. It gbets better and
better, because it is always better to see things the way they are rather than looking at them in an imagined way. Dm is a suppressive person and he tries to manage through overwhelm and
manipulation. I think this latest act of his will hasten his downfall.
Congratulations for having a back bone. 🙂
7. John Dale says
I came across this the other day ……. “To learn who rules over you simply find out who you are not allowed to criticize – Voltaire”. Very apt!
8. Sapere Aude says
Another wall being built to hide behind. The commandos will find themself alone and failing in the future. Those with integrity, those who can think, those who can see and those who seek the
light in the future shall rise up and leave. I do believe this is just the beginning for that continent. Any person who has set foot in South Africa will understand the independence of that area.
From the smell of the earth to the beautiful experience of Kruger. Freedom is once more coming to the southern areas. And now for a song:
9. thetapotata says
The Shittith hath hittith in South Africa. I came up with 555 years cumulative for these long term Scientologists. The gasoline and a match approach is Dave’s tool of choice for handling anyone
who doesn’t go along with his brand of spiritual freedom.
10. Sejanus says
Eventually the Tiny Terror will be down to a handful of celebs left in the Co$ and no one else. Although that is likely his plan as he can suction more money per pound from them. Is he even
around anymore or is he already hiding in quarters on Freewinds? Kinda funny how we don’t hear much from the Cruise or Travolta camps anymore.
11. crislandivar says
Wow Mike..that is very important news I think so. I worked in South Africa many years for the Corp Church and I know I can tell all of them the big fish who put money for the Ideal Orgs, the
Corbetts was one of them. And the list is very big of decleare people. I guess they open their eyes and see the real truth. Poor Ken and for the CO CMO AF, well his brother the CO CMO latam maybe
is going to clean toilets also….poor people…they should leave!!! on Ken the CO CLO AF as you say is a long term SO…I dont know what to say..but I am sure you also know a lot on this
12. Draco says
Hi Everyone, newbie here. I am not one of the Joburg 18 but I do live in their neighborhood. Have been quietly out for years and I read this blog every day along with Tony Ortega. I know that if
anything exciting happens in the scn world, it will be posted here! Thanks Mike Rinder for this blog and for keeping us all in the know.
□ Hallie Jane says
Welcome Draco!
13. Bela says
PS…Mike, I love that we now have the name “Jobur”. It’s up there with FART, GAG2, etc.
□ Buffy says
Yes, Jobur it should be from now on. 🙂
14. Jose Chung says
This made my day with coffee this morning
Cash Cow on life support, shoot all the Doctors !
D.M.will have update his resume pretty quick.
15. Bob says
This makes me think DM really is going to kick most longtimers out and reinvent the COS as a smaller group. He’s got the money to stop fundraising at such a pace and live like a king with a
couple thousand slaves to maintain his lifestyle. Sell an org every few years, recruit just enough to keep the staff stable and teach them the “new” Tech…with only as much LRH as he allows them
to read. With the old guard gone he won’t have to deal with so much pushback.
16. Birgit says
David Miscavige is quite a busy man these days. Mired in court cases and troubles from all corners of the planet, where nobody appears to like him, he has a lot to tend to. Therefore it is
understandable that a bit of confusion occasionally sneaks in at Flag. I for one also received an invitation to come to the big events at Flag, despite the fact that I am not eligible since I
have resigned from the church. The invitation told me to confirm now and immediately start making reservations for hotels and flights – from Denmark. I would later on be informed as to the exact
dates for when the events were actually taking place. – That right there put me into a state of confusion. Until I finally started laughing out loud. I guess that was an FN without a meter.
The church is in a condition of CONFUSION – and it is going viral!
17. vicky says
Any news of Mark Corbett? Noticed that he was not on the list – I hope he has not disconnected from the rest of his family.
18. Zephyr says
Looking at it on the bright side:
That’s enough trained people to just have a group right outside the org doing real delivery of the bridge without the crap.
I wish them a speedy decompression and go for an out-create of this farce of so called
19. Kim says
Hi Mike I was trying to comment when I somehow lost the comment. If you have it, ignore it and I will retype it.
Hi Mike
I awoke to a beautiful Highveld sunrise on a windless morning on the southern tip of Africa only to be directed to your site to discover that overnight I had developed all the characteristics of
an SP after forty eight years in the church. No comm ev,no letter and no comm of any sort.
I console myself with the fact of the illustrious company I keep!
What immediately occurred to me is the value of the combined contribution of these declared SP’s over the many years that we have all been loyal Scientologists.
If I were to single anyone out it would be the Corbett’s. Their contribution in personal time and money to both the group and to individual Scientologists is probably unparalled on this planet .A
group that would declare such people in not worth being part of.
I will not talk for the others but as regards me let me tell you what these Corporate Scientology arseholes have done. I run, from above, with partners, as a contribution, an academy for
university and school students. This year we will graduate one thousand five hundred young people who we have put through LRH’s basic study courses. This is a record year for us but we do it
every year and we invite and fly in CO ABLE and other Scn Execs, all at our expense, for graduation. (Please note these invitations are now withdrawn.) We are in possession of thousands of
success stories where lives have been changed for the better. One top school has given us a letter with the prior stats and the current stats showing that the average pass mark has moves from
around 60%to 80%.You will agree this is all very suppressive.
I do not think that ALL the Scn churches in Africa have that many students pass through their combined doors in five years let alone one year!
We will certainly no longer do this in the name of a church to whom we are anathema.
If the C of S was interested in the ideals laid out by LRH they could not possibly declare these people with the level of their various contributions.I think it clearly indicates
that the intention is to keep the cash cow alive and get rid of those of us who have been around for long enough to be able to recognise the departure from the scene when LRH was around and when
the purpose was to make real OT’s. not fill coffers.
You do a great and necessary job with this site Mike .Thank you.
Oh and before I forget, I Kim Downing in my personal opinion, hereby declare the management of the Church of Scientology as suppressive when evaluated against the criteria as laid out by that man
,L Ron Hubbard who has my gratitude and respect.
□ Mike Rinder says
Kim — thank you for coming here and sharing your story.
Wow — you probably accomplish about 10% of the total Applied Scholastics stats internationally! And along with Sonja Botha, they have pretty much just ended the social betterment programs in
South Africa. But they don’t care as they are not income making activities.
Sorry you had to find out about this by reading a blog. Corporate Scientology and Miscavige try to pretend they are tough. Really they cower in their bubble, afraid of anyone and everyone who
doesnt just fall into line.
So, you and the others who are able to think for yourselves put the fear of God into them.
Good on you.
☆ Kim says
Oh..no thank you for making this space available Mike. As Gaye said your site has been an inspiration and certainly been of vast assistance to me and those close to me. As you know it
takes a while to just get to the point where one believes what one is seeing with regard to the corruption in the C of S. Your site makes that a much easier journey. I certainly hope to
meet you and your lovely wife when you visit us.
Thanks again
□ Tony DePhillips says
Thanks Kim.
That is a lot of help and contributing that you have done. I am happy to welcome you to a group of people that have been similarly treated. It is exciting to know that dm has had to stoop to
another new low by trying to exterminate all of the key upstats in your area. This I am sure will backfire on him and he will be even closer to the point where HE will have to run and hide
and give up the game he has been playing in such a corrupt fashion.
Here’s to a NEW and better South Africa!!
☆ Kim says
Hi Tony,Yes I don’t know many South Africans who are willing to just agree to roll over and die! There are a lot of us who are all really pissed off at the magnitude of the betrayal and I
don’t mean the declares, I mean the destruction of Scientology.
I think Mr Miscavige is not immune to the overt motivator sequence.
Thank you for your welcome.
○ Hallie Jane says
Well said Kim! “The magnitude of the betrayal” was well hidden and is indeed shocking. But it’s hidden no longer, thanks to a lot of brave souls who came before me. Congrats! Add
yourselves to the Indie 500!
20. R2D2 says
I would just like to add to the impact of this ‘collective think’ declare. I’m sure everyone can see that the names mentioned above are without a doubt stellar Scientologists in their own right.
But it even goes beyond that. Working in Durban Org was like being pinned down in the trenches. One of the SA 18 Cameron Wannenburg was on staff with me and I recall him as the OES with a new
wife and young children whilst handling his post and the CLO suppression. In fact his wife was also on staff. With no staff pay this is quite a feat. He wrote a full manual hat write up with tabs
and content list that was used for many years to help anyone trying to run treasury without any training. Together we kept the doors open after we had been left for dead and had been robbed by
the CLO. some may not know the amount of stress, invalidation and heartbreak these unacknowledged staff endured and whilst putting their own lives on hold, keeping the flag flying helping others
for LRH. The corporate Scientology Death Battalion in SA has been carefully removing these very loyal and able beings who have demonstrated by their selfless actions their worth and value to
The church in Durban is left with the fringe lunatics, the not-so-able and the drones. I hope they wake up.
21. Flexible Flyer says
Who needs the picked-over old oak when a new money tree awaits.
Best to have 18-25 work themselves into exhaustion and/or hand over the trust fund. Best to reg the drug-addled and messed-up kids of the affluent. Many would gratefully pay just to get their
nasty little bastards out of the house.
They’ll have status ceremoniously conferred upon them. Instant enlightenment achieved by memorizing quotes. Instead of “go away kid” they’ll have juice. Young females get to meet wealthy clients.
Both get to wear a uniform and scream at adults. And they’ll have fun, fun until Daddy takes his checkbook away.
22. Jens TINGLEFF says
Too bad the declared veterans do not band together, make a criminal complaint and cause the police to seize the folders at the Idle Morgue and the building itself.
23. Mat Pesch says
The Scientology community is waking up at a higher and higher rate. They are seeing Miscavige for what he is. The betrayal has been HUGE. The destruction has been HUGE.
Miscavige fears the real world and yet his little bubble of delusion is drying up. He has no where to run. He has no way to prevent the inevitable. His future is bleak.
24. Cece says
Holy moly!
Wonder about Enid Byrne’s Mother Lottie and her sister and brother. Surely there are connections there and to James Byrne at Int last I knew….
Wow! Good for these guys and all that go with them.
□ scnafrica says
Hi Cece, Lottie at 90+ is still in the Sea Org here. Her Son, Jonathan is on the ship and her daughter, Sally-Anne, is drinking the cool aid proper. She recently redid her purif at flag while
there for her six month check.
25. Madora Pennington says
It’s the Cult of Scientology that is the cancer that needs to be cut out. Miscavige may be the chief surgeon.
26. Formost says
If those 18 had been in comm with Rinder, Rathbun, et al … or indicators parroting back what’s been posted on those blogs, they’re gone, no mercy, whether such data is true or not. Same will go
for anyone else who rears the same “ugly” head. Don Larson clearly illustrated in a Youtube vid that DM makes no deals, and that’ll go on until there’s no RCS is left.
One could could go and complain at the local org about fundraising, altered tech, Int abuse, and down the highway you go pretty much automatically unless recanted during a “Dead Agent” handling …
all done by the numbers, they won’t allow such feet into the door — EVER.
27. Chris Mann says
All I can say is praise be to COB. It’s actually part of the GATII that he personally discovered a transcriptionist error that switched the actual percentages of SPs. It’s actually 97 1/2 percent
AND GROWING (It may actually be closer to 99.999 percent). They are all suppressive and are trying to destroy the golden Big Beings who are the actual 2 1/2 percent. Unless they are stamped out
it is curtains for this universe.
28. Gayle aka TroubleShooter says
WOW WOW WOW WOW WOW!!!!!!!!!!!!!!!!!!!
Had something like this happened in Philly then I’d have a LOT of my friends here with us!!! I’m kind of jealous of you all down there in SA!!!
Tracy and Shirley I would really love to hear from you. There have been many others of us who were on the receiving end of some really f’ed up shit that left us spinning only to find the insanity
ran all the way up the Flag pole.
my email is philadelphiagayle@yahoo.com
do well ALL of you, stick together, be real friends to each other and know that 100s down there are going to need you in the coming days, weeks and months to get through the turmoil that comes
from confronting the magnitude of betrayal that all this is a part of.
□ Danger Boy says
You go girl… (Gayle)
I agree – this is really a VERY good thing – note that at the core of it is the extended Corbett family – that were the foundation of the funding for the new org in J. and so many other
This will shake the tree in SA and I think will destroy any remnants over there, which is of course a GREAT outcome, especially if it becomes the tipping point we all so dearly would like to
My prescription for all of the 18 (if they haven’t already done so is to buy in lots of good treats, hunker down and READ – all of the mainstream press, starting with the TRUTH RUNDOWN and
come forward in time, and, as a result, up to present time 🙂
29. Gatchild says
It’s not just exterior to the Church.
I had an SO member whom I trained and loved very much, walk up to me on the street and openly flirt with me, hugging me and touching me, staring at me very suggestively. She had obviously been
busted from her previous post in OSA and was on some menial post by her uniform.
She asked me if we could hang out together on her CSP day. While I’m sure those activities would get her beached, and I’d be in trouble, she might just be worth it. She’s obviously not happy in
the SO anymore.
30. Rory Medford says
Corporate C of S tearing apart at the seams. People with money and power and SCN certificates are leaving in droves!!!!
31. SILVIA says
This is treason beyond belief, all those years of dedication and suddenly a big slap, no thank you, no ‘lets work it out’, just treachery – the hallmark of the so call leader.
He is caving his own grave, getting rid of the good guys and keeping himself surrounded by only sociopaths who carry out actions as above gestapo style.
But his grave is ready even if the soil is not to happy to receive such a disgusting body, but fortunately it won’t take much space; his body is small and the being is just a molecule of
insanity, even though a molecule is bigger than this character.
32. breppen says
Excuse my French but that is just fucking insane! DM is just wasting everyone around him. I guess this whole action is to scare everyone into subservient slaves, South Africa needs to rise up and
kick their mofo’s asses the hell out of the country. And somehow kick the little midgets a-hole into the hole where he truly belongs.
□ Hallie Jane says
Well said! He’s busy doing himself in, it is insane.
□ Tony DePhillips says
33. r2-45 for all wogs says
that is some funny stuff right there, otviiiisgrrr8. lmao
34. Mike Eldredge says
Its kinda like watching a lava lamp since I have reconnected to Scn after a 20 years hiatus. blobs of crap going up and down and coming to a boil. In the mean time Iam out here making real OTs
and Auditors.
35. Mike Wreggitt says
Dude, that is friggin’ funny!
□ Mike Wreggitt says
36. jeff says
SPOT’S!!! Brilliant!!!
37. Espiritu says
South Africa is going be fine.
Scientology is going to be fine.
Mi$cavigology sinks into the violent confusion it created.
Scientology rises into a new dawn.
The future is ours.
The spell is broken.
38. Pepper says
Great news! More declares = more disaffection. Just what the RCS deserves. I love watching this shit storm unfold.
39. mreppen says
Back in the somewhat standard days, a Missionaire had the right to refuse Mission Orders if he felt that were destructive, not an easy right to pull off, but I did this I recall one time in the
50 + Sea Org Missions I did. Ah those days are long dead and gone. Those 7 Missionaires in this Commando team are idiots for accepting this Mission a guarantee failure that will likely end
themselves in the hole or RPF, but they likely had no choice. Wonder if Dave sent Marc Yaeger or other similar subjects there. CLO AF from my recollection was a place one was sent as punishment,
or to give them a second chance.
I am sure the 18 victim’s are in a bit of shock and awe right now, but believe me it’s the best thing that ever happened to you, I can attest it did for me.
□ Jackson - A.K.A Gary Morehead says
Yaeger, Heber, Mithoff, Starkey etc…There is no way in Dave’s Hell would he let them off the base for such an affair. Geeze… these guys aren’t even aware there is a double yellow line down
the middle of Highway 79.
— Jackson
40. OTVIIIisGrrr8! says
Secret RTC studies have shown that 93% of Scientologists who reach OT VII are declared SP’s within ten years of reaching this exalted state.
This is a shocking datum!
When surveyed, fully 86% of declared SP OT’s (SPOT’s) disingenuously claimed the cost of Flag’s six month sec checks refreshers was too expensive.
This is impossible for we in RTC to accept as a real objection; $50,000 per year is a very small cost to pay for OT refreshment.
Moreover, these SPOT’s further nattered and complained that the ever-increasing costs of IAS Patron statuses added up to more than $1,000,000 over ten years.
One thing these SPOT’s consistently noted was the fact that they were very happy with their Church of Scientology experience up to the level of Clear.
Thinking with the correct data and at the correct orders of magnitude, we in RTC have decided to offer Clear completions the option to purchase their statistically inevitable SP Declare for
$250,000, this to save them the higher costs, time, and aggravation of being declared SP’s ten years after reaching OT VII.
By purchasing an SP Declare for $250,000 immediately after attesting Clear, future cost savings of at least $1,500,000 can be realized.
If you are a Clear, the odds are against you — and indeed you are at risk as you have been warned. Therefore, you should immediately act upon this money-saving opportunity and donate $250,000 to
the IAS to get the new money-saving IAS SP Declare Patron Status.
Your IAS SP Declare Patron Status will free you to read your precious Scientology-hating websites on the fringes of the internet and go into mutual out ruds with whomever you please. You’re a
declared SP and so we in RTC no longer care what you do!
□ Buffy says
My $250,000 check is in the mail. I want to be a SPOT!!
□ Dirk Niblick says
Brilliant! If only DM were so generous.
“Pay me $250,000, and I will treat you like shit, threaten you, call you a degraded being, and kick you out of my church.”
This would save people lots of money, time, late night regging visits, crappy meals at Flag, seasickness on the Freewinds, and useless leather-bound copies of books they have already bought
five times before. What a bargain!
Then they’d have more money to spend on their families, homes, and independent auditors who might actually help them out without begging for money every session.
□ Aquamarine says
Yes, me too! Until now I’ve had seriously considerations about selling my mother’s jewelry and dumping my kid in the Sea Org, but tonite it will be on E-Bay and the kid’s stuff is already
packed. I want to be a SPOT!!!!!
41. gato rojo says
Well done all you South Africans—may you wear your ‘Rod with great pride! Congratulations and I hope your “transition” goes well and quickly and you set up shop the way you want to. If this 7-man
team of goons scare you or your family just realize where they are coming from and how truly unhappy they are in their own hearts at this point. I bet you they would rather be anywhere else but
on this mission, being ordered about, threatened and degraded in orders and comments from His Lowness.
42. Aurora says
Does anyone think this all might presage a run to the Castle Kyalami by DM and a small cadre?
□ Mike Rinder says
Nah — too many SPs within 100 miles.
43. John Doe says
My feeling is that Miscavige is getting ready to consolidate scn, to “expand our reach, by closing the door on squirrels.” (read: letting South Africa go). He probably doesn’t care too much about
SA because I imagine they are not sending huge amounts of loot his way.
This 7-man command team is probably a last-ditch effort and if it works, well, then SA gets to stay in the RCS network. If not, he’ll declare the whole continent and say “fuck em”. He’ll sell all
the buildings out from under the orgs, de-authorize them, leaving the debts and taking the money.
Miscavige is not thinking too clearly these days – terrified of having to get deposed and then charged with perjury. No doubt he’s spending a lot of time reading these blogs trying to get some
clue as to when he should up and leave for Columbia.
□ gato rojo says
NOW!! Please! You jerkoff….just GO! Be your naturally sweet and pleasant self to the lizards and insects residing in the Colombian foothills.
☆ Jose Chung says
Sounds like I’m joking but it is getting close.
Gulfstream G 650 has very long range and a thrill to fly.
Once Der Bozo is over International airspace Columbia
will get a new Pope. Probably purchase one of Pablo Escobar’s former strongholds
It’s going to be soon.
□ Buffy says
Getting money out of South Africa is NOT EASY!! The IAS donations made there stay mostly in the country. You’re right, John Doe, maybe he will just say “fuck ’em” to the S.Africans.
44. Starman8 says
This is AWESOME news! S Africa is near and dear to my heart. Wendy M is my sister-in-law, and I personally had a hand in waking her up.
My postulate is for this to be the end of RCS in S Africa!
Bruce, another OT VIII (USA) who saw the truth.
ps. lets get more people on the Indie 500 list.
□ Idle Morgue says
Awesome job Starman 8 for helping her wake up!! THANK YOU! Scientology is so dangerous it is much safer out here and you helped save lives!!
□ Wendy M says
Hi Starman – big hugs and big smile 🙂
Eyes Wide (Not) Shut !
45. remoteviewed says
There ya go.
Ethics or their concept of it goes in way after its all gone up in smoke and has turned into a raging brush fire.
Joburg in my opinion will probably go the same way Denmark Org went back in ’05 which was straight into the arms of Ron’s Org.
46. tetloj says
Happily for the Joburg 18 (don’t you just love how that sounds), it’s better out than in.
47. Bela says
Wow…look at all the years of contribution and experience these great people have. End result? Get Declared. Heck, I can do that now without spending all the time and money to get that cert. I
hope and think there will be many who will be snapped out of the koolaid induced coma with this shocking news. I wish them all good luck and a much happier future.
48. LDW says
The Pope of scientology sends his inquisition forces to quell the heresy with the fire and brimstone of the golden rod.
All shall tremble in their presence.
Silly boys in sailor costumes pretending to be important; believing in their divine right to wreak havoc.
Sounds like a good one for Monty Python to spoof.
□ Aquamarine says
For starters, I would be happy and satisfied to have Him simply parked permanently without any power off the lines. As to His punishment, because it is my personal belief that we are, each of
us, immortal beings who have minds which in turn operate our bodies, and, because, believing this, I am able to take a long view of situations – I believe that DM’s unavoidable eternity will
very adequately contain his punishment, and that simply isolating him for the rest of his life albeit with adequate food, and so forth, isolating him so that he has lots of time to think
about what he’s done, would be the correct action. I don’t think that the focus should be on His punishment. I would advocate the usage of “Ignore Tech”. Park him/ship him somewhere, feed
Him, lothe Him, house Him and otherwise ignore Him. My 2 cents.
Oh, I forgot! And promise Him that we’ll always capitalize the “H” on His pronouns.
49. threefeetback says
Putting DM on display, in a huge clear plastic box, cleaning dumpsters with a toothbrush as a tourist attraction on Hollywood Boulevard for the remainder of his life would be WAY, WAY too kind.
□ Danger Boy says
It may be too kind, but is the kind of thing that starts a tourism trend – even in Hollywood…
I would travel to LA just to stand and watch such a spectacle.
50. jgg2012 says
Why doesn’t Davey just kick everyone out? Look, DM and TC are the only Big Beings on the planet. No one else is good enough. They should discard their bodies and go to another planet.
□ plainoldthetan says
… to Target Three…
□ jeff says
DM and TC handling Target Two together forever. Ahhhhh how cute.
51. Sheldon Goldberg says
It’s a great time to be an SP. Had an awesome solo session this morning. I deemed my own self eligible. No unneeded sec checks. No IAS, No Ideal Org. No library donations. No “get in here right
away.” No worry about having to redo the Student Hat, Objectives, Purif, etc.
□ Mike Rinder says
□ Axiom says
I love to hear it! You go boy.
We need to let the world know that the real tech is on the outside of the church walls. Isn’t it a blast to get case gain without all the distractions?
☆ Sheldon Goldberg says
I absolutely love it! Solo auditing was always my favorite thing in Scientology. Getting regged for BS was always my least.
□ Hallie Jane says
□ Aquamarine says
Beatiful, Sheldon! Just reading this gave me a nice surge in tone. My stable datum now as re the Bridge is that it is all waiting for me in the Indie Field. This makes me so happy! Thank you
for posting this.
52. Black Panther says
Hello to everyone on this blog. I have never commented here before, but I have contributed towards data that has been reported by Mike and he does know who I am (Mike, sorry for being out of comm
– I had to lay low for a while but have come up for air again – long story which I will fill you in on shortly).
I have decided to comment for the first time tonight because the subject matter of this post is near and dear to me. I am a “still in good standing” South African Scientologist and have spent
some time now on a heart-wrenching journey of truth seeking regarding the sordid subject of RCS.
The latest turn of events to hit us is going to have far-reaching consequences. A lot of these 18 people are very highly regarded in the SCN field in Africa – they hold an enormous amount of sway
with a lot of people in high up places – and I’m not only talking within the Church. I am utterly bereft of words to explain what the hell DM is thinking with this action – if this is not proof
that he is actively trying to destroy SCN then I don’t know what is. The fallout from this is going to be huge.
I concur with Wendy M about Christie – she is an amazing person and I really admire her. She played a key role in building Joburg Org to St Hill Size (sadly a distant memory as it is now a
stellar example of an Ideal Morgue).
And by the way – the child that fell to his death was Ken Krieger’s son Max – it was a tragic, tragic incident – I was there when it happened.
It’s certainly going to be interesting to see what transpires here in South Africa over the next while. It’s like watching the Spanish Inquisition or similar.
Thanks Mike for all you have done and continue to do.
□ Bela says
Thank you for all you have done…and to R2D2 too.
We are with you guys as we watch this all unfold. Interesting times for sure!
□ Idle Morgue says
Black Panther – WELCOME! Thank you for your post!
□ Mike Rinder says
BP — thanks for writing. Good to hear from you again. Look forward to your email.
□ Hallie Jane says
Welcome Black Panther and thanks for all your efforts!
☆ Aquamarine says
Black Panther,
Welcome! And thank you for all you are doing for “The Resistance”.
□ Martin Padfield says
“if this is not proof that he is actively trying to destroy SCN then I don’t know what is.”
That was exactly my thought reading this post. There was much speculation in the last few years about just how “accidental” His reign was, and whether he was being supported from another
source – which I still don’t believe – there is no “conspiracy” as such. But when you read day after day of this combination of incompetence and outright thuggery you just shake your head in
disbelief. Surely NO ONE – even DM – can THIS stupid accidentally.
☆ DollarMorgue says
It isn’t stupid. It is his purpose.
□ krcjenny says
Black panther, here in the states we have a phrase ‘death by cop’. It’s basically when a fellow loses his shit and get’s himself bypassed by having a standoff where the cops end up shooting
him to death. This flow with the mass declares seems similar to me, in that DM seems to play “poke the bear” until the whole of Scientology will rise up to anger and finish him, since he’s
incapable of putting ethics in on himself. What say you?
53. SadStateofAffairs says
This looks like Miscavige’s scorched-earth-on-steroids version of the “LATAM Strategy” for handling “external influences” and “disaffecteds” from the early 80’s. The problem with this, as you
observe, Mike, is that once the earth is scorched they will find that they are within the heavy fallout zone of the nukes they are firing off.
54. Jose Chung says
RTC High school graduation photo,so touching.
□ Mike Rinder says
55. R2D2 says
Thank you Mike for this Blog, I’m one of :”The 18″ from SA and I must say I think it’s a huge compliment. They must really be sh*t scared of us.
I spent most of my 10 years on staff heading Div 6 in Durban, I sold hundreds of books, I audited hundreds of hours and really helped many people (Many who have even written letters of
appreciation). I literally kept Durban open when I was there.
Krieger, Lipsitz, and Bokkelman (known as the “terrible trio” to Durban Scientologists) did nothing but try to stop me from the moment I showed potential.
I was Comm Ev’d twice and actual SP declares written but never quite carried out. The second one took 4 days and was later (2 years) squashed by Mike Ellis after I wrote a very comprehensive
report on the actual good I’d done for Durban and out-ethics and insanity of CLO AF.
There is a broader more detailed scene around these events, but I’ll leave those war stories for next time. .
□ Idle Morgue says
Welcome R2D2!!! So happy to have you here. Thank you for your post and letting us know what is going on inside the walls of the insane asylum called The Church of Scientology! BIG HUG to you
and all of the others!! You have reached the EP of Miscavology!! An SP Declare is a badge of honor – it means you can think for yourselves and speak freely!!
☆ R2D2 says
Thanks Idle Morgue, its great to be free. You won’t believe how my life has started to do better.
○ Aquamarine says
A warm welcome to you! Its thrilling to have you sharing here as one of the 18. Simply telling the truth is what will vanquish the dragon that is the RCS. Thank you.
☆ barefacedmessiah says
Is R2D2 real?
He gives data about himself, so that OSA knows who he is. But still he won’t tell us his name.
Let’s see.
○ Mike Rinder says
Yes, he is real. He is one of the 18.
○ Craig Howarth says
Hi there Barefacedmessiah, OSA already know who I am, (No.12 on the list), R2D2 was how I started so I just stuck with it. ……….interesting though, the ex CO OSA AF was the chairman on
my first comm ev and pushed for my declare in the days when I was just a clay pigeon. Coincidental to this whole affair, he just moved into the office next door to ours, and at some
point burst into our office asking my opinion on the Corbetts. I told him what I thought,knowing he would be reporting the data to OSA. The next day he was absent and the following
day the declare came out.
What was interesting is how he nattered up a storm about management.
■ Charmaine Buttrick (Olley) says
Hey Craigy, you ROCK! Always have, always will.
No bad management will ever try and break you again!!
■ barefacedmessiah says
Wow, thanks for your personal integrity. We all paid a high price for saying what we believe in or not.
□ Hallie Jane says
Welcome and congrats R2D2! You are now a Special Person, who is someone who has tried to help others with scn, despite the megalomaniacal “help” of radical church management. Join the Indie
□ Eriatlov says
Hello R2D2,
I would like to get in contact with you.
Please drop me a note: ts(at)sured.info
56. DollarMorgue says
Perhaps SP declares are the only way left to save people from the RCS?
□ Live Zombie says
Or from COB who only handles “ecclesiastical issues”.
57. Doug Parent says
Stomping in with guns blazing no doubt while the ship ablaze slips ever deeper beneath the waves. Thats the way to manage the exodus, MORE DECLARES !!!! Scientology, putting fires out with
gasoline every day. Classic.
58. Wendy M says
Some may not know that Christie was a senior Sea Org exec who put the Joburg Ideal Org there. It was a needed new org because the old premises were in the centre of town and it was (and is) a
dangerous area. One SO member was murdered on the streets in the last ten years (John Plumb), and another murdered some twenty or more years before (Jeff Murray, if memory serves). They were
killed by common criminals – not related to Sea Org activity. I believe a child fell to his death from the top of the old org building – an accident due to being unsupervised.
So Christie (now the so-called big SP) had this place humming and going Saint Hill Size, while Alex Faust presided over the largest landslide in recent history.
They have their lines crossed somewhat up at Int Management. The new missionaires are in for a treat!
□ Mike Rinder says
Thanks Wendy. Christie LOVES Joburg and it breaks her heart to see what has happened…. We keep special track of happenings in Joburg and South Africa in general. There are no doubt more
declares to come as there are some other OL’s that I know of that are not on this list….
☆ Gaye Corbett says
You are right about that Mike . There are many many more that Tiny Tim( our new name for DM) does not even know about or suspect. I think they will faint when they know of the extent of
this new movement. Anyway the first stone has been thrown, now we will watch the ripple effect in the water. I love your blog Mike and it has helped me through many times when I felt we
were all alone in the wilderness. I remember Christi well and I am sure she will remember us please give her our love.
It is quite liberating to be free at last and no longer subject to the” now I am supposed to” AND the incessant regging. We have had letters from all over the world congratulating us on
our declares, how funny.
We will give the full story in a few days and will publish on our own Africa blog.
○ Mike Rinder says
Hello Gaye.
So nice to see you posting here and I am very happy to know that you have been reading this blog. It is EXACTLY why I devote quite a lot of time to keeping it going.
Christie absolutely remembers you and the rest of your family. She was SOOOO happy when she heard about your family she cried.
The people who have been keeping us informed about what has been going on and connecting the dots down there are heros. There is more to come into public view as you know. But you and
your family and the Kroegers and the other luminaries that compose the “Joburg 18” (really the South African 18 but Joburg 18 has a nice ring to it) have paved the way for a lot of
others and made it safe for them to start thinking for themselves.
The first stone has been cast — believe me, the ripples are ALREADY reaching beyond your shores.
Someday I am going to visit South Africa, Christie has made me promise. I look forward to meeting you all….
■ freespirit says
Wow! What can one say? Their loss is our gain!
■ Starman8 says
Dear Gaye,
You don’t know me, but you do know my wife and sister-in-law, Wendy M. My wife and I are both OT VIII, and live in the US.
I wholeheartedly congratulate you on this outcome. I got my wife and Wendy out in Feb-Mar 2012 after reading Debbie Cook’s email.
We visit Joburg every Christmas to see Wendy and family. It would be great to meet up with you when there. I have heard your last name from my wife for years as OLs in South
Africa. Well done on getting out!
○ Tony DePhillips says
Congratulations Gaye!! There is an army of people out here that are behind you guys in South Africa. Enjoy your freedom.
○ Buffy says
To ALL the Corbetts: Hallelujah brothers and sisters!! The most up-standing family in South African Scientology history has done it once again and this time the leaders of the pack
will be taking the SAffers to true freedom from suppression. You have more support than you can possibly imagine – from all corners of the globe. Thank you so much for standing
together and standing up for LRH and for true human rights. Now THAT’s integrity!! POWER TO YOU ALL!
○ Calvin B Duffield says
Gaye, so special to see you guys making this move. I remember like yesterday, around 1978, when you and Ernie had just gotten back to Durban academy, after completing the newly
released L’s. That was the real start of big things coming your way. Fabulous to read about your much vaunted Golden Rod.
Now THAT is a real accolade to be worn with honor!! True freedom from the mind control of Depraved Mestcavige. Welcome back to SANE Scientology (out of CO$)
ML, Calvin & Dorothy
PS. regarding Mark, I’m sure this is just a comm lag running it’s course! (out soon)
□ Sheeple Bane says
Christie was an angel! She protected me and allowed me to help many others. I will always admire her for her kindness, integrity and beingness! Joburg lost a wonderful thetan the day she
left! When I heard she had been declared it made no sense. Doubts crept in and im glad it opened up the door to see the bigger picture. Thank you Christie!
59. Hope says
What an awesome bunch of quality Scientologists, and there are more,. More theta power in this group than what is actually left struggling in the the Orgs of SA…
60. splog says
I’ve met 15 of those 18 people (not that they will remember me, I’m a nobody in the Scn world). All of them are decent, smart clued-up folks and rather successful in life in their own right. They
would have achieved success anyway with or without Scientology.
This is going to get real interesting, huge chunks of the Joburg field are tied into those 18 in one way or another and all of them are going to be expected to disconnect. Will they? This is
going to get very interesting.
The End Times. They are here 🙂
61. Birgit says
Oh my God! This is unbelievable! What is the tone level of this church?
□ plainoldthetan says
0.96, Terror.
☆ Idle Morgue says
Hysterical! ^^^ 0.96 Terror!! “The Golden Age of Rod” is very clever and quite funny. Those jokers and degrader’s are a good bunch of sane people. The SP Declare is the EP of Scientology
– congrats to them all!! Welcome!
○ DollarMorgue says
Read the latest:
☆ Carcha says
-31 Criminality
-32 Uncausing
-33 Disconnection
□ Jose Chung says
Tone level reply
Its damn low, you will discover secret reports ( guarded and concealed) that are chock full of horrific lies to justify the also secret S.P. declares.
0.96 terror is correct
62. DollarMorgue says
If this is true, then I suppose DM does not believe he can roll out GAT2 with the old guard in the way. I anticipate similar actions around the world prior to release, whether ordered by local
missions or by uplines.
DM thinks he is Napoleon, firing cannons at the crowd.
□ Robert says
Yes, DollarMorgue, I think you are exactly correct: Napoleon fired grape shot into a crowd of 10,000 killing 1,400 to successfully break up a loyalist Paris mob. That’s what it seems DM is
trying to do in South Africa.
But, if you use Scientology to make “able people more able” and then shoot the ones you made more able, then they just turn around and work against you because you didn’t actually kill them…
Instead, they end up in an independent field, quite alive with plenty of internet connections, associates and friends to carry on with Scientology. It sucks to be Miscavige because you can’t
run a cult in the internet age..
☆ Cooper kessel says
” they just turn around and work against you because you didn’t actually kill them… Instead, they end up in an independent field, quite alive with plenty of internet connections,
associates and friends to carry on with Scientology. It sucks to be Miscavige because you can’t run a cult in the internet age..”
David Miscarriage and the Sea Org Commandos will need to redefine the term ‘WE COME BACK’ to the truly SP lingo of “oh fuck, THEY have come back”.
WELL DAVEY ……… WE ARE BACK! GIRD THY LOINS FOR YOUR FINAL BATTLE CAUSE WE ARE FRICKIN EVERYWHERE AND YOU DON’T EVEN KNOW IT.
Cheers little shithead ……. it will be over soon.
○ krcjenny says
Cooper, recall the phrase “Wake Up Call”? It finally rings true. They just had the terminals swapped. WE are their wake up call now. YEEEHAW! (a bit of cowgirl emphasis)
○ Jane Doe says
63. Taylor says
Mike, in the above names are there any equivalent names to Debbie Cook, Marty, Amy and yourself that Clearwater cos members would know of?
□ Robert Eckert says
Within the South African context, this is apparently more the equivalent of purging the Feshbachs, Bob Duggan, and Nancy Cartwright.
64. Karen#1 says
Another purge, another sweeping SP declare of long duration staff and supporters. A mentality of power and humiliation. Cleaning dumpsters with toothbrushes…cosmic psycho-politics, master /slave
domination and the beat marches on…
□ plainoldthetan says
…and another tragic loss of people trained in Scientology who can talk about Scientology with a straight face and without getting something caught in their throat. The teenyboppers being
sucked onto staff in Idle Orgs cannot possibly replace the OTs and Class VIIIs and other really experienced people.
☆ Jane Doe says
Nor do the teenyboppers have anything to compare it to. They weren’t around in the earlier days when orgs were booming and things were run differently. So they can’t see the outnesses.
The older and experienced and trained people can SEE and they KNOW and thus he must wipe them out, wipe them from the face of Scn, which he is systematically doing. It makes your blood
run cold.
○ Aquamarine says
Right on, Jane Doe. This is Missed Withold Phenomena in living color. Out of control, send in the wagons.
65. Steve Poore says
“sinking ship of fools.” good one Mike!
66. Steve Poore says
S. Africa is just A Microseism of the “Straight-up-and-Vertical” Disaffection world-wide!
67. Pericles says
The Royal Dwarf has yet proven once again that he is DEFIANT & RESOLUTE in promoting his own personal brand of squirreling and technical degrades.
68. Mreppen says
□ Jane Doe says
Heads on pikes to scare everyone else into submission. It is Nazi Germany all over again.
Leave a Reply Cancel reply | {"url":"https://www.mikerindersblog.org/the-commandos-have-landed-in-jobur/","timestamp":"2024-11-06T14:02:25Z","content_type":"text/html","content_length":"342924","record_id":"<urn:uuid:383b02a1-dd92-434a-b97d-820d0dd14335>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00498.warc.gz"} |
PlanetPhysics/Klein Gordon Equation 3 - Wikiversity
This is a contributed Topic.
The Klein-Gordon (KG) Scalar Relativistic Wave Equation
{\mathbf Remarks:} The KG-equation is a Lorentz invariant expression. For specific computations of specific cases it can only be utilized with the appropriate boundary conditions.
1.1. The Klein-Gordon equation
The Klein-Gordon equation is an equation of mathematical physics that describes spinless (spin-0 particles). It is given by:
${\displaystyle \Box \psi =\left({\frac {mc}{\hbar }}\right)^{2}\psi }$
Here the ${\displaystyle \Box }$ symbol refers to the wave operator, or D'Alembertian, (${\displaystyle \Box =abla ^{2}-{\frac {1}{c^{2}}}\partial _{t}^{2}}$ ) and ${\displaystyle \psi }$ is the wave
function of a spinless particle.
1.2. Relativistic energy levels of a spinless particle in a Coulomb field
1.3. Relativistic invanance of the de Broglie relations
1.4. Relativistic energy-momentum relation of a free particle
1.5. Charge and current density
1.5.1. Charge and current density in the presence of an electromagnetic field
1.6. Nonrelativistic limit
1.7. The initial data problem
1.8. Indefiniteness of the sign of charge
1.9. Interaction with an external electromagnetic field
1.10. Fine structure constant
The case ${\displaystyle Za>l/2}$ | {"url":"https://en.m.wikiversity.org/wiki/PlanetPhysics/Klein_Gordon_Equation_3","timestamp":"2024-11-04T23:51:43Z","content_type":"text/html","content_length":"40427","record_id":"<urn:uuid:b3f27d75-7b15-4000-b086-38120468e3a4>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00350.warc.gz"} |
No version for distro humble. Known supported distros are highlighted in the buttons above.
Repository Summary
Checkout URI https://github.com/locusrobotics/tf2_2d.git
VCS Type git
VCS Version rolling
Last Updated 2024-09-16
Dev Status MAINTAINED
CI status No Continuous Integration
Released RELEASED
Tags No category tags.
Help Wanted (0)
Contributing Good First Issues (0)
Pull Requests to Review (0)
A set of 2D geometry classes modeled after the 3D geometry classes in tf2.
Using tf2 toMsg() and fromMsg()
I’ve tried to include fromMsg() implementations for anything that remotely makes sense. The tf2 toMsg() signature does limit its use to a single output type, so I’ve had to make a guess as to the
most useful variation there.
For example:
#include <tf2_2d/tf2_2d.h> // This header includes the tf2 conversion functions
#include <tf2_2d/transform.h> // Then include what you use
// Convert a tf2_2d object into a 3D message
auto transform_2d = tf2_2d::Transform(1.0, 1.5, 2.0);
geometry_msgs::Transform transform_3d_msg = tf2::toMsg(transform_2d);
// Convert a 3D message into a 2D tf2_2d object
geometry_msgs::Transform transform_3d_msg;
tf2_2d::Transform transform_2d;
tf2::fromMsg(transform_3d_msg, transform_2d);
// You can do Stamped<> things as well
auto transform_2d = tf2::Stamped<tf2_2d::Transform>(tf2_2d::Transform(1.0, 1.5, 2.0), ros::Time(1.2), "frame");
geometry_msgs::TransformStamped transform_3d_msg = tf2::toMsg(transform_2d);
Conversion of Points and Quaternions are also supported, as are conversions to other datatypes that are handled by tf2’s conversion system.
Transformation math
The tf2_2d types also implement all of the expected transformation math.
auto v1 = tf2_2d::Vector2(1.0, 2.0);
auto v2 = tf2_2d::Vector2(1.5, 2.5);
auto v3 = v1 + v2; // v3 == (2.5, 4.5)
tf2_2d::Rotation r1(M_PI);
auto v4 = r1.rotate(v1); // v4 == (-2.0, 1.0)
auto t1 = tf2_2d::Transform(1.0, 2.0, 3.0);
auto t2 = tf2_2d::Transform(-2.0, -1.0, -1.5);
auto t3 = t1 * t2; // t3 == (3.12, 2.70, 1.5)
auto t4 = tf2_2d::Transform(1.0, 2.0, 3.0);
auto t5 = tf2_2d::Transform(-2.0, -1.0, -1.5);
auto t6 = t4.inverseTimes(t5); // t6 == (2.54, 3.39, 1.78)
The tf2_2d::Rotation class deserves a few additional notes. The angle stored in a Rotation object is always within the (-Pi, Pi] range. You can construct a Rotation object with any floating point
value, but it will be wrapped to that range. You can also perform arithmetic with the angles, without worrying about wrapping issues.
tf2_2d::Rotation r1(1.0);
auto r2 = 17.0 * r1; // r2.angle() == -1.84956
tf2_2d::Rotation r3(1.0);
tf2_2d::Rotation r4(3.0);
auto r5 = r3 + r4; // r5.angle() == -2.28319
tf2_2d::Rotation r6(-3.0);
tf2_2d::Rotation r7(1.0);
auto r8 = r6 - r7; // r8.angle() == 2.28319
Additionally, the tf2_td::Rotation class caches the sin/cos results needed to perform rotations. So, once a Rotation object is used to rotate something, the trig functions will never be evaluated
again. And the cached sin/cos values will propagate to any derived objects that it can. This includes the Rotation object built into a Transform.
tf2_2d::Vector2 v1(1.0, 2.0);
tf2_2d::Rotation r1(1.0); // No sin/cos calls have been made, since it is not needed yet
auto v2 = r1.rotate(v1); // Computes sin/cos and remembers it
auto v3 = r1.unrotate(v2); // Does not need to compute sin/cos again
tf2_2d::Rotation r2 = r1; // Transfers sin/cos to r2
auto v4 = r2.rotate(v1); // Does not need to compute sin/cos because it stole the cached values from r1
tf2_2d::Rotation r3 = r1.inverse(); // Inverts and transfers sin/cos to r3
auto v5 = r3.rotate(v1); // Does not need to compute sin/cos because it stole the cached values from r1
No CONTRIBUTING.md found.
Repository Summary
Checkout URI https://github.com/locusrobotics/tf2_2d.git
VCS Type git
VCS Version rolling
Last Updated 2024-09-16
Dev Status MAINTAINED
CI status No Continuous Integration
Released RELEASED
Tags No category tags.
Help Wanted (0)
Contributing Good First Issues (0)
Pull Requests to Review (0)
A set of 2D geometry classes modeled after the 3D geometry classes in tf2.
Using tf2 toMsg() and fromMsg()
I’ve tried to include fromMsg() implementations for anything that remotely makes sense. The tf2 toMsg() signature does limit its use to a single output type, so I’ve had to make a guess as to the
most useful variation there.
For example:
#include <tf2_2d/tf2_2d.h> // This header includes the tf2 conversion functions
#include <tf2_2d/transform.h> // Then include what you use
// Convert a tf2_2d object into a 3D message
auto transform_2d = tf2_2d::Transform(1.0, 1.5, 2.0);
geometry_msgs::Transform transform_3d_msg = tf2::toMsg(transform_2d);
// Convert a 3D message into a 2D tf2_2d object
geometry_msgs::Transform transform_3d_msg;
tf2_2d::Transform transform_2d;
tf2::fromMsg(transform_3d_msg, transform_2d);
// You can do Stamped<> things as well
auto transform_2d = tf2::Stamped<tf2_2d::Transform>(tf2_2d::Transform(1.0, 1.5, 2.0), ros::Time(1.2), "frame");
geometry_msgs::TransformStamped transform_3d_msg = tf2::toMsg(transform_2d);
Conversion of Points and Quaternions are also supported, as are conversions to other datatypes that are handled by tf2’s conversion system.
Transformation math
The tf2_2d types also implement all of the expected transformation math.
auto v1 = tf2_2d::Vector2(1.0, 2.0);
auto v2 = tf2_2d::Vector2(1.5, 2.5);
auto v3 = v1 + v2; // v3 == (2.5, 4.5)
tf2_2d::Rotation r1(M_PI);
auto v4 = r1.rotate(v1); // v4 == (-2.0, 1.0)
auto t1 = tf2_2d::Transform(1.0, 2.0, 3.0);
auto t2 = tf2_2d::Transform(-2.0, -1.0, -1.5);
auto t3 = t1 * t2; // t3 == (3.12, 2.70, 1.5)
auto t4 = tf2_2d::Transform(1.0, 2.0, 3.0);
auto t5 = tf2_2d::Transform(-2.0, -1.0, -1.5);
auto t6 = t4.inverseTimes(t5); // t6 == (2.54, 3.39, 1.78)
The tf2_2d::Rotation class deserves a few additional notes. The angle stored in a Rotation object is always within the (-Pi, Pi] range. You can construct a Rotation object with any floating point
value, but it will be wrapped to that range. You can also perform arithmetic with the angles, without worrying about wrapping issues.
tf2_2d::Rotation r1(1.0);
auto r2 = 17.0 * r1; // r2.angle() == -1.84956
tf2_2d::Rotation r3(1.0);
tf2_2d::Rotation r4(3.0);
auto r5 = r3 + r4; // r5.angle() == -2.28319
tf2_2d::Rotation r6(-3.0);
tf2_2d::Rotation r7(1.0);
auto r8 = r6 - r7; // r8.angle() == 2.28319
Additionally, the tf2_td::Rotation class caches the sin/cos results needed to perform rotations. So, once a Rotation object is used to rotate something, the trig functions will never be evaluated
again. And the cached sin/cos values will propagate to any derived objects that it can. This includes the Rotation object built into a Transform.
tf2_2d::Vector2 v1(1.0, 2.0);
tf2_2d::Rotation r1(1.0); // No sin/cos calls have been made, since it is not needed yet
auto v2 = r1.rotate(v1); // Computes sin/cos and remembers it
auto v3 = r1.unrotate(v2); // Does not need to compute sin/cos again
tf2_2d::Rotation r2 = r1; // Transfers sin/cos to r2
auto v4 = r2.rotate(v1); // Does not need to compute sin/cos because it stole the cached values from r1
tf2_2d::Rotation r3 = r1.inverse(); // Inverts and transfers sin/cos to r3
auto v5 = r3.rotate(v1); // Does not need to compute sin/cos because it stole the cached values from r1
No CONTRIBUTING.md found.
Repository Summary
Checkout URI https://github.com/locusrobotics/tf2_2d.git
VCS Type git
VCS Version rolling
Last Updated 2024-09-16
Dev Status MAINTAINED
CI status No Continuous Integration
Released RELEASED
Tags No category tags.
Help Wanted (0)
Contributing Good First Issues (0)
Pull Requests to Review (0)
A set of 2D geometry classes modeled after the 3D geometry classes in tf2.
Using tf2 toMsg() and fromMsg()
I’ve tried to include fromMsg() implementations for anything that remotely makes sense. The tf2 toMsg() signature does limit its use to a single output type, so I’ve had to make a guess as to the
most useful variation there.
For example:
#include <tf2_2d/tf2_2d.h> // This header includes the tf2 conversion functions
#include <tf2_2d/transform.h> // Then include what you use
// Convert a tf2_2d object into a 3D message
auto transform_2d = tf2_2d::Transform(1.0, 1.5, 2.0);
geometry_msgs::Transform transform_3d_msg = tf2::toMsg(transform_2d);
// Convert a 3D message into a 2D tf2_2d object
geometry_msgs::Transform transform_3d_msg;
tf2_2d::Transform transform_2d;
tf2::fromMsg(transform_3d_msg, transform_2d);
// You can do Stamped<> things as well
auto transform_2d = tf2::Stamped<tf2_2d::Transform>(tf2_2d::Transform(1.0, 1.5, 2.0), ros::Time(1.2), "frame");
geometry_msgs::TransformStamped transform_3d_msg = tf2::toMsg(transform_2d);
Conversion of Points and Quaternions are also supported, as are conversions to other datatypes that are handled by tf2’s conversion system.
Transformation math
The tf2_2d types also implement all of the expected transformation math.
auto v1 = tf2_2d::Vector2(1.0, 2.0);
auto v2 = tf2_2d::Vector2(1.5, 2.5);
auto v3 = v1 + v2; // v3 == (2.5, 4.5)
tf2_2d::Rotation r1(M_PI);
auto v4 = r1.rotate(v1); // v4 == (-2.0, 1.0)
auto t1 = tf2_2d::Transform(1.0, 2.0, 3.0);
auto t2 = tf2_2d::Transform(-2.0, -1.0, -1.5);
auto t3 = t1 * t2; // t3 == (3.12, 2.70, 1.5)
auto t4 = tf2_2d::Transform(1.0, 2.0, 3.0);
auto t5 = tf2_2d::Transform(-2.0, -1.0, -1.5);
auto t6 = t4.inverseTimes(t5); // t6 == (2.54, 3.39, 1.78)
The tf2_2d::Rotation class deserves a few additional notes. The angle stored in a Rotation object is always within the (-Pi, Pi] range. You can construct a Rotation object with any floating point
value, but it will be wrapped to that range. You can also perform arithmetic with the angles, without worrying about wrapping issues.
tf2_2d::Rotation r1(1.0);
auto r2 = 17.0 * r1; // r2.angle() == -1.84956
tf2_2d::Rotation r3(1.0);
tf2_2d::Rotation r4(3.0);
auto r5 = r3 + r4; // r5.angle() == -2.28319
tf2_2d::Rotation r6(-3.0);
tf2_2d::Rotation r7(1.0);
auto r8 = r6 - r7; // r8.angle() == 2.28319
Additionally, the tf2_td::Rotation class caches the sin/cos results needed to perform rotations. So, once a Rotation object is used to rotate something, the trig functions will never be evaluated
again. And the cached sin/cos values will propagate to any derived objects that it can. This includes the Rotation object built into a Transform.
tf2_2d::Vector2 v1(1.0, 2.0);
tf2_2d::Rotation r1(1.0); // No sin/cos calls have been made, since it is not needed yet
auto v2 = r1.rotate(v1); // Computes sin/cos and remembers it
auto v3 = r1.unrotate(v2); // Does not need to compute sin/cos again
tf2_2d::Rotation r2 = r1; // Transfers sin/cos to r2
auto v4 = r2.rotate(v1); // Does not need to compute sin/cos because it stole the cached values from r1
tf2_2d::Rotation r3 = r1.inverse(); // Inverts and transfers sin/cos to r3
auto v5 = r3.rotate(v1); // Does not need to compute sin/cos because it stole the cached values from r1
No CONTRIBUTING.md found.
Repository Summary
Checkout URI https://github.com/locusrobotics/tf2_2d.git
VCS Type git
VCS Version devel
Last Updated 2024-09-16
Dev Status MAINTAINED
CI status No Continuous Integration
Released RELEASED
Tags No category tags.
Help Wanted (0)
Contributing Good First Issues (0)
Pull Requests to Review (0)
A set of 2D geometry classes modeled after the 3D geometry classes in tf2.
Using tf2 toMsg() and fromMsg()
I’ve tried to include fromMsg() implementations for anything that remotely makes sense. The tf2 toMsg() signature does limit its use to a single output type, so I’ve had to make a guess as to the
most useful variation there.
For example:
#include <tf2_2d/tf2_2d.h> // This header includes the tf2 conversion functions
#include <tf2_2d/transform.h> // Then include what you use
// Convert a tf2_2d object into a 3D message
auto transform_2d = tf2_2d::Transform(1.0, 1.5, 2.0);
geometry_msgs::Transform transform_3d_msg = tf2::toMsg(transform_2d);
// Convert a 3D message into a 2D tf2_2d object
geometry_msgs::Transform transform_3d_msg;
tf2_2d::Transform transform_2d;
tf2::fromMsg(transform_3d_msg, transform_2d);
// You can do Stamped<> things as well
auto transform_2d = tf2::Stamped<tf2_2d::Transform>(tf2_2d::Transform(1.0, 1.5, 2.0), ros::Time(1.2), "frame");
geometry_msgs::TransformStamped transform_3d_msg = tf2::toMsg(transform_2d);
Conversion of Points and Quaternions are also supported, as are conversions to other datatypes that are handled by tf2’s conversion system.
Transformation math
The tf2_2d types also implement all of the expected transformation math.
auto v1 = tf2_2d::Vector2(1.0, 2.0);
auto v2 = tf2_2d::Vector2(1.5, 2.5);
auto v3 = v1 + v2; // v3 == (2.5, 4.5)
tf2_2d::Rotation r1(M_PI);
auto v4 = r1.rotate(v1); // v4 == (-2.0, 1.0)
auto t1 = tf2_2d::Transform(1.0, 2.0, 3.0);
auto t2 = tf2_2d::Transform(-2.0, -1.0, -1.5);
auto t3 = t1 * t2; // t3 == (3.12, 2.70, 1.5)
auto t4 = tf2_2d::Transform(1.0, 2.0, 3.0);
auto t5 = tf2_2d::Transform(-2.0, -1.0, -1.5);
auto t6 = t4.inverseTimes(t5); // t6 == (2.54, 3.39, 1.78)
The tf2_2d::Rotation class deserves a few additional notes. The angle stored in a Rotation object is always within the (-Pi, Pi] range. You can construct a Rotation object with any floating point
value, but it will be wrapped to that range. You can also perform arithmetic with the angles, without worrying about wrapping issues.
tf2_2d::Rotation r1(1.0);
auto r2 = 17.0 * r1; // r2.angle() == -1.84956
tf2_2d::Rotation r3(1.0);
tf2_2d::Rotation r4(3.0);
auto r5 = r3 + r4; // r5.angle() == -2.28319
tf2_2d::Rotation r6(-3.0);
tf2_2d::Rotation r7(1.0);
auto r8 = r6 - r7; // r8.angle() == 2.28319
Additionally, the tf2_td::Rotation class caches the sin/cos results needed to perform rotations. So, once a Rotation object is used to rotate something, the trig functions will never be evaluated
again. And the cached sin/cos values will propagate to any derived objects that it can. This includes the Rotation object built into a Transform.
tf2_2d::Vector2 v1(1.0, 2.0);
tf2_2d::Rotation r1(1.0); // No sin/cos calls have been made, since it is not needed yet
auto v2 = r1.rotate(v1); // Computes sin/cos and remembers it
auto v3 = r1.unrotate(v2); // Does not need to compute sin/cos again
tf2_2d::Rotation r2 = r1; // Transfers sin/cos to r2
auto v4 = r2.rotate(v1); // Does not need to compute sin/cos because it stole the cached values from r1
tf2_2d::Rotation r3 = r1.inverse(); // Inverts and transfers sin/cos to r3
auto v5 = r3.rotate(v1); // Does not need to compute sin/cos because it stole the cached values from r1
No CONTRIBUTING.md found.
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No version for distro crystal. Known supported distros are highlighted in the buttons above.
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Repository Summary
Checkout URI https://github.com/locusrobotics/tf2_2d.git
VCS Type git
VCS Version devel
Last Updated 2024-09-16
Dev Status MAINTAINED
CI status No Continuous Integration
Released RELEASED
Tags No category tags.
Help Wanted (0)
Contributing Good First Issues (0)
Pull Requests to Review (0)
A set of 2D geometry classes modeled after the 3D geometry classes in tf2.
Using tf2 toMsg() and fromMsg()
I’ve tried to include fromMsg() implementations for anything that remotely makes sense. The tf2 toMsg() signature does limit its use to a single output type, so I’ve had to make a guess as to the
most useful variation there.
For example:
#include <tf2_2d/tf2_2d.h> // This header includes the tf2 conversion functions
#include <tf2_2d/transform.h> // Then include what you use
// Convert a tf2_2d object into a 3D message
auto transform_2d = tf2_2d::Transform(1.0, 1.5, 2.0);
geometry_msgs::Transform transform_3d_msg = tf2::toMsg(transform_2d);
// Convert a 3D message into a 2D tf2_2d object
geometry_msgs::Transform transform_3d_msg;
tf2_2d::Transform transform_2d;
tf2::fromMsg(transform_3d_msg, transform_2d);
// You can do Stamped<> things as well
auto transform_2d = tf2::Stamped<tf2_2d::Transform>(tf2_2d::Transform(1.0, 1.5, 2.0), ros::Time(1.2), "frame");
geometry_msgs::TransformStamped transform_3d_msg = tf2::toMsg(transform_2d);
Conversion of Points and Quaternions are also supported, as are conversions to other datatypes that are handled by tf2’s conversion system.
Transformation math
The tf2_2d types also implement all of the expected transformation math.
auto v1 = tf2_2d::Vector2(1.0, 2.0);
auto v2 = tf2_2d::Vector2(1.5, 2.5);
auto v3 = v1 + v2; // v3 == (2.5, 4.5)
tf2_2d::Rotation r1(M_PI);
auto v4 = r1.rotate(v1); // v4 == (-2.0, 1.0)
auto t1 = tf2_2d::Transform(1.0, 2.0, 3.0);
auto t2 = tf2_2d::Transform(-2.0, -1.0, -1.5);
auto t3 = t1 * t2; // t3 == (3.12, 2.70, 1.5)
auto t4 = tf2_2d::Transform(1.0, 2.0, 3.0);
auto t5 = tf2_2d::Transform(-2.0, -1.0, -1.5);
auto t6 = t4.inverseTimes(t5); // t6 == (2.54, 3.39, 1.78)
The tf2_2d::Rotation class deserves a few additional notes. The angle stored in a Rotation object is always within the (-Pi, Pi] range. You can construct a Rotation object with any floating point
value, but it will be wrapped to that range. You can also perform arithmetic with the angles, without worrying about wrapping issues.
tf2_2d::Rotation r1(1.0);
auto r2 = 17.0 * r1; // r2.angle() == -1.84956
tf2_2d::Rotation r3(1.0);
tf2_2d::Rotation r4(3.0);
auto r5 = r3 + r4; // r5.angle() == -2.28319
tf2_2d::Rotation r6(-3.0);
tf2_2d::Rotation r7(1.0);
auto r8 = r6 - r7; // r8.angle() == 2.28319
Additionally, the tf2_td::Rotation class caches the sin/cos results needed to perform rotations. So, once a Rotation object is used to rotate something, the trig functions will never be evaluated
again. And the cached sin/cos values will propagate to any derived objects that it can. This includes the Rotation object built into a Transform.
tf2_2d::Vector2 v1(1.0, 2.0);
tf2_2d::Rotation r1(1.0); // No sin/cos calls have been made, since it is not needed yet
auto v2 = r1.rotate(v1); // Computes sin/cos and remembers it
auto v3 = r1.unrotate(v2); // Does not need to compute sin/cos again
tf2_2d::Rotation r2 = r1; // Transfers sin/cos to r2
auto v4 = r2.rotate(v1); // Does not need to compute sin/cos because it stole the cached values from r1
tf2_2d::Rotation r3 = r1.inverse(); // Inverts and transfers sin/cos to r3
auto v5 = r3.rotate(v1); // Does not need to compute sin/cos because it stole the cached values from r1
No CONTRIBUTING.md found. | {"url":"https://index.ros.org/r/tf2_2d/","timestamp":"2024-11-09T10:48:08Z","content_type":"text/html","content_length":"62418","record_id":"<urn:uuid:9b079b9e-34b4-414c-a0c6-46c3b53c5b57>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00022.warc.gz"} |
Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by the nabla symbol ∇. When applied to a function defined on
a one-dimensional domain, it denotes the standard derivative of the function as defined in calculus. When applied to a field (a function defined on a multi-dimensional domain), it may denote any one
of three operations depending on the way it is applied: the gradient or (locally) steepest slope of a scalar field (or sometimes of a vector field, as in the Navier–Stokes equations); the divergence
of a vector field; or the curl (rotation) of a vector field.
Del operator,
represented by
the nabla symbol
Del is a very convenient mathematical notation for those three operations (gradient, divergence, and curl) that makes many equations easier to write and remember. The del symbol (or nabla) can be
formally defined as a vector operator whose components are the corresponding partial derivative operators. As a vector operator, it can act on scalar and vector fields in three different ways, giving
rise to three different differential operations: first, it can act on scalar fields by a formal scalar multiplication—to give a vector field called the gradient; second, it can act on vector fields
by a formal dot product—to give a scalar field called the divergence; and lastly, it can act on vector fields by a formal cross product—to give a vector field called the curl. These formal products
do not necessarily commute with other operators or products. These three uses, detailed below, are summarized as:
• Gradient: ${\displaystyle \operatorname {grad} f=abla f}$
• Divergence: ${\displaystyle \operatorname {div} \mathbf {v} =abla \cdot \mathbf {v} }$
• Curl: ${\displaystyle \operatorname {curl} \mathbf {v} =abla \times \mathbf {v} }$
In the Cartesian coordinate system ${\displaystyle \mathbb {R} ^{n}}$ with coordinates ${\displaystyle (x_{1},\dots ,x_{n})}$ and standard basis ${\displaystyle \{\mathbf {e} _{1},\dots ,\mathbf {e}
_{n}\}}$ , del is a vector operator whose ${\displaystyle x_{1},\dots ,x_{n}}$ components are the partial derivative operators ${\displaystyle {\partial \over \partial x_{1}},\dots ,{\partial \over \
partial x_{n}}}$ ; that is,
${\displaystyle abla =\sum _{i=1}^{n}\mathbf {e} _{i}{\partial \over \partial x_{i}}=\left({\partial \over \partial x_{1}},\ldots ,{\partial \over \partial x_{n}}\right)}$
Where the expression in parentheses is a row vector. In three-dimensional Cartesian coordinate system ${\displaystyle \mathbb {R} ^{3}}$ with coordinates ${\displaystyle (x,y,z)}$ and standard basis
or unit vectors of axes ${\displaystyle \{\mathbf {e} _{x},\mathbf {e} _{y},\mathbf {e} _{z}\}}$ , del is written as
${\displaystyle abla =\mathbf {e} _{x}{\partial \over \partial x}+\mathbf {e} _{y}{\partial \over \partial y}+\mathbf {e} _{z}{\partial \over \partial z}=\left({\partial \over \partial x},{\
partial \over \partial y},{\partial \over \partial z}\right)}$
As a vector operator, del naturally acts on scalar fields via scalar multiplication, and naturally acts on vector fields via dot products and cross products.
More specifically, for any scalar field ${\displaystyle f}$ and any vector field ${\displaystyle \mathbf {F} =(F_{x},F_{y},F_{z})}$ , if one defines
${\displaystyle \left(\mathbf {e} _{i}{\partial \over \partial x_{i}}\right)f:={\partial \over \partial x_{i}}(\mathbf {e} _{i}f)={\partial f \over \partial x_{i}}\mathbf {e} _{i}}$
${\displaystyle \left(\mathbf {e} _{i}{\partial \over \partial x_{i}}\right)\cdot \mathbf {F} :={\partial \over \partial x_{i}}(\mathbf {e} _{i}\cdot \mathbf {F} )={\partial F_{i} \over \partial
${\displaystyle \left(\mathbf {e} _{x}{\partial \over \partial x}\right)\times \mathbf {F} :={\partial \over \partial x}(\mathbf {e} _{x}\times \mathbf {F} )={\partial \over \partial x}(0,-F_
${\displaystyle \left(\mathbf {e} _{y}{\partial \over \partial y}\right)\times \mathbf {F} :={\partial \over \partial y}(\mathbf {e} _{y}\times \mathbf {F} )={\partial \over \partial y}(F_
${\displaystyle \left(\mathbf {e} _{z}{\partial \over \partial z}\right)\times \mathbf {F} :={\partial \over \partial z}(\mathbf {e} _{z}\times \mathbf {F} )={\partial \over \partial z}(-F_{y},F_
then using the above definition of ${\displaystyle abla }$ , one may write
${\displaystyle abla f=\left(\mathbf {e} _{x}{\partial \over \partial x}\right)f+\left(\mathbf {e} _{y}{\partial \over \partial y}\right)f+\left(\mathbf {e} _{z}{\partial \over \partial z}\right)
f={\partial f \over \partial x}\mathbf {e} _{x}+{\partial f \over \partial y}\mathbf {e} _{y}+{\partial f \over \partial z}\mathbf {e} _{z}}$
${\displaystyle abla \cdot \mathbf {F} =\left(\mathbf {e} _{x}{\partial \over \partial x}\cdot \mathbf {F} \right)+\left(\mathbf {e} _{y}{\partial \over \partial y}\cdot \mathbf {F} \right)+\left
(\mathbf {e} _{z}{\partial \over \partial z}\cdot \mathbf {F} \right)={\partial F_{x} \over \partial x}+{\partial F_{y} \over \partial y}+{\partial F_{z} \over \partial z}}$
{\displaystyle {\begin{aligned}abla \times \mathbf {F} &=\left(\mathbf {e} _{x}{\partial \over \partial x}\times \mathbf {F} \right)+\left(\mathbf {e} _{y}{\partial \over \partial y}\times \
mathbf {F} \right)+\left(\mathbf {e} _{z}{\partial \over \partial z}\times \mathbf {F} \right)\\&={\partial \over \partial x}(0,-F_{z},F_{y})+{\partial \over \partial y}(F_{z},0,-F_{x})+{\partial
\over \partial z}(-F_{y},F_{x},0)\\&=\left({\partial F_{z} \over \partial y}-{\partial F_{y} \over \partial z}\right)\mathbf {e} _{x}+\left({\partial F_{x} \over \partial z}-{\partial F_{z} \over
\partial x}\right)\mathbf {e} _{y}+\left({\partial F_{y} \over \partial x}-{\partial F_{x} \over \partial y}\right)\mathbf {e} _{z}\end{aligned}}}
${\displaystyle f(x,y,z)=x+y+z}$
${\displaystyle abla f=\mathbf {e} _{x}{\partial f \over \partial x}+\mathbf {e} _{y}{\partial f \over \partial y}+\mathbf {e} _{z}{\partial f \over \partial z}=\left(1,1,1\right)}$
Del can also be expressed in other coordinate systems, see for example del in cylindrical and spherical coordinates.
Notational uses
Del is used as a shorthand form to simplify many long mathematical expressions. It is most commonly used to simplify expressions for the gradient, divergence, curl, directional derivative, and
The vector derivative of a scalar field ${\displaystyle f}$ is called the gradient, and it can be represented as:
${\displaystyle \operatorname {grad} f={\partial f \over \partial x}{\hat {\mathbf {x} }}+{\partial f \over \partial y}{\hat {\mathbf {y} }}+{\partial f \over \partial z}{\hat {\mathbf {z} }}=
abla f}$
It always points in the direction of greatest increase of ${\displaystyle f}$ , and it has a magnitude equal to the maximum rate of increase at the point—just like a standard derivative. In
particular, if a hill is defined as a height function over a plane ${\displaystyle h(x,y)}$ , the gradient at a given location will be a vector in the xy-plane (visualizable as an arrow on a map)
pointing along the steepest direction. The magnitude of the gradient is the value of this steepest slope.
In particular, this notation is powerful because the gradient product rule looks very similar to the 1d-derivative case:
${\displaystyle abla (fg)=fabla g+gabla f}$
However, the rules for dot products do not turn out to be simple, as illustrated by:
${\displaystyle abla (\mathbf {u} \cdot \mathbf {v} )=(\mathbf {u} \cdot abla )\mathbf {v} +(\mathbf {v} \cdot abla )\mathbf {u} +\mathbf {u} \times (abla \times \mathbf {v} )+\mathbf {v} \times
(abla \times \mathbf {u} )}$
The divergence of a vector field ${\displaystyle \mathbf {v} (x,y,z)=v_{x}{\hat {\mathbf {x} }}+v_{y}{\hat {\mathbf {y} }}+v_{z}{\hat {\mathbf {z} }}}$ is a scalar field that can be represented as:
${\displaystyle \operatorname {div} \mathbf {v} ={\partial v_{x} \over \partial x}+{\partial v_{y} \over \partial y}+{\partial v_{z} \over \partial z}=abla \cdot \mathbf {v} }$
The divergence is roughly a measure of a vector field's increase in the direction it points; but more accurately, it is a measure of that field's tendency to converge toward or diverge from a point.
The power of the del notation is shown by the following product rule:
${\displaystyle abla \cdot (f\mathbf {v} )=(abla f)\cdot \mathbf {v} +f(abla \cdot \mathbf {v} )}$
The formula for the vector product is slightly less intuitive, because this product is not commutative:
${\displaystyle abla \cdot (\mathbf {u} \times \mathbf {v} )=(abla \times \mathbf {u} )\cdot \mathbf {v} -\mathbf {u} \cdot (abla \times \mathbf {v} )}$
The curl of a vector field ${\displaystyle \mathbf {v} (x,y,z)=v_{x}{\hat {\mathbf {x} }}+v_{y}{\hat {\mathbf {y} }}+v_{z}{\hat {\mathbf {z} }}}$ is a vector function that can be represented as:
${\displaystyle \operatorname {curl} \mathbf {v} =\left({\partial v_{z} \over \partial y}-{\partial v_{y} \over \partial z}\right){\hat {\mathbf {x} }}+\left({\partial v_{x} \over \partial z}-{\
partial v_{z} \over \partial x}\right){\hat {\mathbf {y} }}+\left({\partial v_{y} \over \partial x}-{\partial v_{x} \over \partial y}\right){\hat {\mathbf {z} }}=abla \times \mathbf {v} }$
The curl at a point is proportional to the on-axis torque that a tiny pinwheel would be subjected to if it were centered at that point.
The vector product operation can be visualized as a pseudo-determinant:
${\displaystyle abla \times \mathbf {v} =\left|{\begin{matrix}{\hat {\mathbf {x} }}&{\hat {\mathbf {y} }}&{\hat {\mathbf {z} }}\\[2pt]{\frac {\partial }{\partial x}}&{\frac {\partial }{\partial
y}}&{\frac {\partial }{\partial z}}\\[2pt]v_{x}&v_{y}&v_{z}\end{matrix}}\right|}$
Again the power of the notation is shown by the product rule:
${\displaystyle abla \times (f\mathbf {v} )=(abla f)\times \mathbf {v} +f(abla \times \mathbf {v} )}$
The rule for the vector product does not turn out to be simple:
${\displaystyle abla \times (\mathbf {u} \times \mathbf {v} )=\mathbf {u} \,(abla \cdot \mathbf {v} )-\mathbf {v} \,(abla \cdot \mathbf {u} )+(\mathbf {v} \cdot abla )\,\mathbf {u} -(\mathbf {u}
\cdot abla )\,\mathbf {v} }$
Directional derivative
The directional derivative of a scalar field ${\displaystyle f(x,y,z)}$ in the direction ${\displaystyle \mathbf {a} (x,y,z)=a_{x}{\hat {\mathbf {x} }}+a_{y}{\hat {\mathbf {y} }}+a_{z}{\hat {\mathbf
{z} }}}$ is defined as:
${\displaystyle (\mathbf {a} \cdot abla )f=\lim _{h\to 0}{\frac {f(x+a_{x}h,y+a_{y}h,z+a_{z}h)-f(x,y,z)}{h}}.}$
Which is equal to the following when the gradient exists
${\displaystyle \mathbf {a} \cdot \operatorname {grad} f=a_{x}{\partial f \over \partial x}+a_{y}{\partial f \over \partial y}+a_{z}{\partial f \over \partial z}=\mathbf {a} \cdot (abla f)}$
This gives the rate of change of a field ${\displaystyle f}$ in the direction of ${\displaystyle \mathbf {a} }$ , scaled by the magnitude of ${\displaystyle \mathbf {a} }$ . In operator notation, the
element in parentheses can be considered a single coherent unit; fluid dynamics uses this convention extensively, terming it the convective derivative—the "moving" derivative of the fluid.
Note that ${\displaystyle (\mathbf {a} \cdot abla )}$ is an operator that takes scalar to a scalar. It can be extended to operate on a vector, by separately operating on each of its components.
The Laplace operator is a scalar operator that can be applied to either vector or scalar fields; for cartesian coordinate systems it is defined as:
${\displaystyle \Delta ={\partial ^{2} \over \partial x^{2}}+{\partial ^{2} \over \partial y^{2}}+{\partial ^{2} \over \partial z^{2}}=abla \cdot abla =abla ^{2}}$
and the definition for more general coordinate systems is given in vector Laplacian.
The Laplacian is ubiquitous throughout modern mathematical physics, appearing for example in Laplace's equation, Poisson's equation, the heat equation, the wave equation, and the Schrödinger equation
Hessian matrix
While ${\displaystyle abla ^{2}}$ usually represents the Laplacian, sometimes ${\displaystyle abla ^{2}}$ also represents the Hessian matrix. The former refers to the inner product of ${\displaystyle
abla }$ , while the latter refers to the dyadic product of ${\displaystyle abla }$ :
${\displaystyle abla ^{2}=abla \cdot abla ^{T}}$ .
So whether ${\displaystyle abla ^{2}}$ refers to a Laplacian or a Hessian matrix depends on the context.
Tensor derivative
Del can also be applied to a vector field with the result being a tensor. The tensor derivative of a vector field ${\displaystyle \mathbf {v} }$ (in three dimensions) is a 9-term second-rank tensor –
that is, a 3×3 matrix – but can be denoted simply as ${\displaystyle abla \otimes \mathbf {v} }$ , where ${\displaystyle \otimes }$ represents the dyadic product. This quantity is equivalent to the
transpose of the Jacobian matrix of the vector field with respect to space. The divergence of the vector field can then be expressed as the trace of this matrix.
For a small displacement ${\displaystyle \delta \mathbf {r} }$ , the change in the vector field is given by:
${\displaystyle \delta \mathbf {v} =(abla \otimes \mathbf {v} )^{T}\cdot \delta \mathbf {r} }$
Product rules
For vector calculus:
{\displaystyle {\begin{aligned}abla (fg)&=fabla g+gabla f\\abla (\mathbf {u} \cdot \mathbf {v} )&=\mathbf {u} \times (abla \times \mathbf {v} )+\mathbf {v} \times (abla \times \mathbf {u} )+(\
mathbf {u} \cdot abla )\mathbf {v} +(\mathbf {v} \cdot abla )\mathbf {u} \\abla \cdot (f\mathbf {v} )&=f(abla \cdot \mathbf {v} )+\mathbf {v} \cdot (abla f)\\abla \cdot (\mathbf {u} \times \
mathbf {v} )&=\mathbf {v} \cdot (abla \times \mathbf {u} )-\mathbf {u} \cdot (abla \times \mathbf {v} )\\abla \times (f\mathbf {v} )&=(abla f)\times \mathbf {v} +f(abla \times \mathbf {v} )\\abla
\times (\mathbf {u} \times \mathbf {v} )&=\mathbf {u} \,(abla \cdot \mathbf {v} )-\mathbf {v} \,(abla \cdot \mathbf {u} )+(\mathbf {v} \cdot abla )\,\mathbf {u} -(\mathbf {u} \cdot abla )\,\
mathbf {v} \end{aligned}}}
For matrix calculus (for which ${\displaystyle \mathbf {u} \cdot \mathbf {v} }$ can be written ${\displaystyle \mathbf {u} ^{\text{T}}\mathbf {v} }$ ):
{\displaystyle {\begin{aligned}\left(\mathbf {A} abla \right)^{\text{T}}\mathbf {u} &=abla ^{\text{T}}\left(\mathbf {A} ^{\text{T}}\mathbf {u} \right)-\left(abla ^{\text{T}}\mathbf {A} ^{\text
{T}}\right)\mathbf {u} \end{aligned}}}
Another relation of interest (see e.g. Euler equations) is the following, where ${\displaystyle \mathbf {u} \otimes \mathbf {v} }$ is the outer product tensor:
{\displaystyle {\begin{aligned}abla \cdot (\mathbf {u} \otimes \mathbf {v} )=(abla \cdot \mathbf {u} )\mathbf {v} +(\mathbf {u} \cdot abla )\mathbf {v} \end{aligned}}}
Second derivatives
DCG chart: A simple chart depicting all rules pertaining to second derivatives. D, C, G, L and CC stand for divergence, curl, gradient, Laplacian and curl of curl, respectively. Arrows indicate
existence of second derivatives. Blue circle in the middle represents curl of curl, whereas the other two red circles (dashed) mean that DD and GG do not exist.
When del operates on a scalar or vector, either a scalar or vector is returned. Because of the diversity of vector products (scalar, dot, cross) one application of del already gives rise to three
major derivatives: the gradient (scalar product), divergence (dot product), and curl (cross product). Applying these three sorts of derivatives again to each other gives five possible second
derivatives, for a scalar field f or a vector field v; the use of the scalar Laplacian and vector Laplacian gives two more:
{\displaystyle {\begin{aligned}\operatorname {div} (\operatorname {grad} f)&=abla \cdot (abla f)=abla ^{2}f\\\operatorname {curl} (\operatorname {grad} f)&=abla \times (abla f)\\\operatorname
{grad} (\operatorname {div} \mathbf {v} )&=abla (abla \cdot \mathbf {v} )\\\operatorname {div} (\operatorname {curl} \mathbf {v} )&=abla \cdot (abla \times \mathbf {v} )\\\operatorname {curl} (\
operatorname {curl} \mathbf {v} )&=abla \times (abla \times \mathbf {v} )\\\Delta f&=abla ^{2}f\\\Delta \mathbf {v} &=abla ^{2}\mathbf {v} \end{aligned}}}
These are of interest principally because they are not always unique or independent of each other. As long as the functions are well-behaved (${\displaystyle C^{\infty }}$ in most cases), two of them
are always zero:
{\displaystyle {\begin{aligned}\operatorname {curl} (\operatorname {grad} f)&=abla \times (abla f)=0\\\operatorname {div} (\operatorname {curl} \mathbf {v} )&=abla \cdot (abla \times \mathbf {v}
Two of them are always equal:
${\displaystyle \operatorname {div} (\operatorname {grad} f)=abla \cdot (abla f)=abla ^{2}f=\Delta f}$
The 3 remaining vector derivatives are related by the equation:
${\displaystyle abla \times \left(abla \times \mathbf {v} \right)=abla (abla \cdot \mathbf {v} )-abla ^{2}\mathbf {v} }$
And one of them can even be expressed with the tensor product, if the functions are well-behaved:
${\displaystyle abla (abla \cdot \mathbf {v} )=abla \cdot (\mathbf {v} \otimes abla )}$
Most of the above vector properties (except for those that rely explicitly on del's differential properties—for example, the product rule) rely only on symbol rearrangement, and must necessarily hold
if the del symbol is replaced by any other vector. This is part of the value to be gained in notationally representing this operator as a vector.
Though one can often replace del with a vector and obtain a vector identity, making those identities mnemonic, the reverse is not necessarily reliable, because del does not commute in general.
A counterexample that demonstrates the divergence (${\displaystyle abla \cdot \mathbf {v} }$ ) and the advection operator (${\displaystyle \mathbf {v} \cdot abla }$ ) are not commutative:
{\displaystyle {\begin{aligned}(\mathbf {u} \cdot \mathbf {v} )f&\equiv (\mathbf {v} \cdot \mathbf {u} )f\\(abla \cdot \mathbf {v} )f&=\left({\frac {\partial v_{x}}{\partial x}}+{\frac {\partial
v_{y}}{\partial y}}+{\frac {\partial v_{z}}{\partial z}}\right)f={\frac {\partial v_{x}}{\partial x}}f+{\frac {\partial v_{y}}{\partial y}}f+{\frac {\partial v_{z}}{\partial z}}f\\(\mathbf {v} \
cdot abla )f&=\left(v_{x}{\frac {\partial }{\partial x}}+v_{y}{\frac {\partial }{\partial y}}+v_{z}{\frac {\partial }{\partial z}}\right)f=v_{x}{\frac {\partial f}{\partial x}}+v_{y}{\frac {\
partial f}{\partial y}}+v_{z}{\frac {\partial f}{\partial z}}\\\Rightarrow (abla \cdot \mathbf {v} )f&eq (\mathbf {v} \cdot abla )f\\\end{aligned}}}
A counterexample that relies on del's differential properties:
{\displaystyle {\begin{aligned}(abla x)\times (abla y)&=\left(\mathbf {e} _{x}{\frac {\partial x}{\partial x}}+\mathbf {e} _{y}{\frac {\partial x}{\partial y}}+\mathbf {e} _{z}{\frac {\partial x}
{\partial z}}\right)\times \left(\mathbf {e} _{x}{\frac {\partial y}{\partial x}}+\mathbf {e} _{y}{\frac {\partial y}{\partial y}}+\mathbf {e} _{z}{\frac {\partial y}{\partial z}}\right)\\&=(\
mathbf {e} _{x}\cdot 1+\mathbf {e} _{y}\cdot 0+\mathbf {e} _{z}\cdot 0)\times (\mathbf {e} _{x}\cdot 0+\mathbf {e} _{y}\cdot 1+\mathbf {e} _{z}\cdot 0)\\&=\mathbf {e} _{x}\times \mathbf {e} _{y}\
\&=\mathbf {e} _{z}\\(\mathbf {u} x)\times (\mathbf {u} y)&=xy(\mathbf {u} \times \mathbf {u} )\\&=xy\mathbf {0} \\&=\mathbf {0} \end{aligned}}}
Central to these distinctions is the fact that del is not simply a vector; it is a vector operator. Whereas a vector is an object with both a magnitude and direction, del has neither a magnitude nor
a direction until it operates on a function.
For that reason, identities involving del must be derived with care, using both vector identities and differentiation identities such as the product rule.
See also
• Willard Gibbs & Edwin Bidwell Wilson (1901) Vector Analysis, Yale University Press, 1960: Dover Publications.
• Schey, H. M. (1997). Div, Grad, Curl, and All That: An Informal Text on Vector Calculus. New York: Norton. ISBN 0-393-96997-5.
• Miller, Jeff. "Earliest Uses of Symbols of Calculus".
• Arnold Neumaier (January 26, 1998). Cleve Moler (ed.). "History of Nabla". NA Digest, Volume 98, Issue 03. netlib.org.
External links
• A survey of the improper use of ∇ in vector analysis (1994) Tai, Chen | {"url":"https://www.knowpia.com/knowpedia/Del","timestamp":"2024-11-09T23:48:59Z","content_type":"text/html","content_length":"329573","record_id":"<urn:uuid:d5179f5b-0bff-46df-9f99-49ba05a13698>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00898.warc.gz"} |
Collider Variable: Definition
What is a Collider Variable?
Graphically, a node on a causal graph (a type of directed acyclic graph) is a collider variable if the path entering and exiting the node both have arrows pointing into it. Essentially, the paths
Community Center Hours is a collider in this causal graph; the arrows entering and leaving both point back into A, creating a loop.
Nodes that don’t meet this definition are called noncolliders. It’s possible for a node to be a collider on one path and a noncollider on a different path on the same graph. If a path has a collider,
then it’s blocked; Having a collider on a graph results in a fuzzy association between the node and the surrounding variables that influence it.
What is Conditioning ?
Conditioning means that you introduce information into the model about the variable of interest. In sociology, conditioning usually means controlling. It can also mean:
• Stratifying, restricting or adjusting the variable in some way.
• Performing analysis specific to one or more groups.
• Selective data collection (e.g. excluding certain groups from a survey).
Controlling for the collider will result in a phenomenon called “endogenous selection bias,” leading to the possibility of spurious correlations. This is also true if you condition on any children of
the colliders (i.e. any nodes further down the path). The solution to endogenous selection bias is not to condition under these conditions in the first place — it’s considered a rudimentary
analytical mistake to do so. A theoretical exception to this is when the effects of the two arrows leading into the node cancel each other out. However, the probability of exact cancellation in real
life scenarios is zero.
Elwert, F. & Winship, C. (2014). “Endogenous Selection Bias: The Problem of Conditioning.” In Annual Review of Sociology. Vol. 40: 31-53.
Greenland, Sander; Pearl, Judea; Robins, James M (January 1999), “Causal Diagrams for Epidemiologic Research” (PDF), Epidemiology, 10 (1): 37–48.
Jewell, N. (2003) Statistics for Epidemiology. Chapman & Hall/CRC.
Watkins, T. (2018). Understanding Uncertainty and Bias to Improve Causal Graphs. | {"url":"https://www.statisticshowto.com/collider-variable/","timestamp":"2024-11-04T14:31:33Z","content_type":"text/html","content_length":"67845","record_id":"<urn:uuid:d1a4d402-efcc-41e6-aa63-3d91eb1f1be9>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00033.warc.gz"} |
Capacitance to Charge Conversion Calculator
How to Calculate Electric Charge?
Coulombs are the units of measurement for electric charge, which is an inherent property that represents the amount of charge that is contained within an electron.
The unit of measurement for one coulomb is one ampere-second.
What is Capacitance?
The capacity of a component or circuit to accumulate & store energy in the form of an electrical charge is referred to as its capacitance.
To calculate capacitance, divide the magnitude of the electrical charge by the difference in potential between the conductors at the point of measurement.
Farads (F) are the unit of measurement for capacitance in the SI system.
Resistors are the most widely used electronic components, followed by capacitors.
The physical ability of a conductor is to store charge.
Current flowing between two plates and a voltage source produces capacitance. One plate becomes positive and the other negative.
In the absence of voltage, a capacitor retains its charge. They have two conductors separated by a dielectric (or) vacuum.
What is the relationship between Charge and Capacitance?
According to the equation Q=CV, the relationship between charge and capacitance is one of direct proportionality.
A capacitor’s capacity to store charge that is meant to be measured by its capacitance.
The relation between the charge that is stored on each plate and the potential difference that exists between the plates is the core of this term.
What is Capacitor Charge?
The quantity of electrical energy that is temporarily held in an electric field within a capacitor is referred to as the capacitor charge.
When a voltage is put across the terminals of the capacitor, this charge is produced as a consequence of the separation of charges that occurs within the capacitor.
The capacitance of a capacitor is the ratio of the charge that is held at the capacitor to the voltage that is across the capacitor.
This ratio is used to determine the capacitor’s capacity to store charge.
Formula (Capacitance to Charge)
One may determine the charge of an electrical circuit by utilizing the capacitance & voltage of the circuit.
Q = C x V
Q – Charge in Coulombs
C – Capacitance in Farads (F)
V – Voltage in Volts (V)
The capacitance C, measured in farads, multiplied by the voltage V equates to the charge Q, which is measured in coulombs.
How to convert Capacitance to Charge?
Charge is described by the equation Q = CV, where C represents the capacitance in Farads, V represents the voltage across the capacitor in Volts, and Q represents the charge that is measured in
coulombs (C).
The energy that is: When the energy is measured in Joules, the equation
W = ½ QV = ½ CV^2 is established.
Solved Example
Let us compute the electric charge for a circuit that has a capacitance of 20 farads and a voltage of 12 volts using the formula.
Q = C x V
Q = 20 F x 12 V
Q = 240 Coulombs
Click here for more Electrical Calculators
You can also follow us on Facebook and Linkedin to receive daily updates. | {"url":"https://forumelectrical.com/capacitance-to-charge-conversion-calculator/","timestamp":"2024-11-11T11:16:21Z","content_type":"text/html","content_length":"235577","record_id":"<urn:uuid:b598109d-2860-483d-8206-5748f5281531>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00303.warc.gz"} |
Here’s why we care about attempts to prove the Riemann hypothesis
The latest effort shines a spotlight on an enduring prime numbers mystery
Empetrisor/Wikimedia Commons (CC BY-SA 4.0)
A famed mathematical enigma is once again in the spotlight.
The Riemann hypothesis, posited in 1859 by German mathematician Bernhard Riemann, is one of the biggest unsolved puzzles in mathematics. The hypothesis, which could unlock the mysteries of prime
numbers, has never been proved. But mathematicians are buzzing about a new attempt.
Esteemed mathematician Michael Atiyah took a crack at proving the hypothesis in a lecture at the Heidelberg Laureate Forum in Germany on September 24. Despite the stature of Atiyah — who has won the
two most prestigious honors in mathematics, the Fields Medal and the Abel Prize — many researchers have expressed skepticism about the proof. So the Riemann hypothesis remains up for grabs.
Let’s break down what the Riemann hypothesis is, and what a confirmed proof — if one is ever found — would mean for mathematics.
What is the Riemann hypothesis?
The Riemann hypothesis is a statement about a mathematical curiosity known as the Riemann zeta function. That function is closely entwined with prime numbers — whole numbers that are evenly divisible
only by 1 and themselves. Prime numbers are mysterious: They are scattered in an inscrutable pattern across the number line, making it difficult to predict where each prime number will fall (SN
Online: 4/2/08).
But if the Riemann zeta function meets a certain condition, Riemann realized, it would reveal secrets of the prime numbers, such as how many primes exist below a given number. That required condition
is the Riemann hypothesis. It conjectures that certain zeros of the function — the points where the function’s value equals zero — all lie along a particular line when plotted (SN: 9/27/08, p. 14).
If the hypothesis is confirmed, it could help expose a method to the primes’ madness.
Why is it so important?
Prime numbers are mathematical VIPs: Like atoms of the periodic table, they are the building blocks for larger numbers. Primes matter for practical purposes, too, as they are important for securing
encrypted transmissions sent over the internet. And importantly, a multitude of mathematical papers take the Riemann hypothesis as a given. If this foundational assumption were proved correct, “many
results that are believed to be true will be known to be true,” says mathematician Ken Ono of Emory University in Atlanta. “It’s a kind of mathematical oracle.”
Haven’t people tried to prove this before?
Yep. It’s difficult to count the number of attempts, but probably hundreds of researchers have tried their hands at a proof. So far none of the proofs have stood up to scrutiny. The problem is so
stubborn that it now has a bounty on its head: The Clay Mathematics Institute has offered up $1 million to anyone who can prove the Riemann hypothesis.
Why is it so difficult to prove?
The Riemann zeta function is a difficult beast to work with. Even defining it is a challenge, Ono says. Furthermore, the function has an infinite number of zeros. If any one of those zeros is not on
its expected line, the Riemann hypothesis is wrong. And since there are infinite zeros, manually checking each one won’t work. Instead, a proof must show without a doubt that no zero can be an
outlier. For difficult mathematical quandaries like the Riemann hypothesis, the bar for acceptance of a proof is extremely high. Verification of such a proof typically requires months or even years
of double-checking by other mathematicians before either everyone is convinced, or the proof is deemed flawed.
What will it take to prove the Riemann hypothesis?
Various mathematicians have made some amount of headway toward a proof. Ono likens it to attempting to climb Mount Everest and making it to base camp. While some clever mathematician may eventually
be able to finish that climb, Ono says, “there is this belief that the ultimate proof … if one ever is made, will require a different level of mathematics.” | {"url":"https://www.sciencenews.org/article/why-we-care-riemann-hypothesis-math-prime-numbers","timestamp":"2024-11-05T00:24:33Z","content_type":"text/html","content_length":"299935","record_id":"<urn:uuid:2e77b8cb-34d8-4e80-8175-1ff29761bb7d>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00753.warc.gz"} |
Assigning Oxidation Numbers Worksheet 1 Answers
Assigning Oxidation Numbers Worksheet 1 Answers function as foundational devices in the realm of mathematics, providing an organized yet functional system for learners to check out and understand
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Unveiling the Essence of Assigning Oxidation Numbers Worksheet 1 Answers
Assigning Oxidation Numbers Worksheet 1 Answers
Assigning Oxidation Numbers Worksheet 1 Answers -
Expert Answer 100 1 rating 11 Cr2O7 2 Cr 6 O 2 According to Rule 5 9 we can say that oxidation number of oxygen is 2 and sum of all oxidation number in ionic compund equal to charge on ion 2 Let us
assume oxidation number of Cr is x 2x 7 2 2 x 6 Hence oxid View the full answer Transcribed image text
The oxidation state of an atom is represented by a positive or negative number called its oxidation number Chemists have developed rules used to assign oxidation numbers The following rules will help
you determine the oxidation state of an atom or ion A free atom has an oxidation number of zero
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Worksheet Oxidation Numbers Answer Key
Worksheet Oxidation Numbers Answer Key
Assigning Oxidation Numbers The oxidation number is a positive or negative number that is assign ed to an atom to indicate its degree of oxidation or reduction In oxidation reduction processes the
driving force for chemical change is in the exchange of electrons between chemical species
One way of reflecting this is through changes in assigned oxidation numbers Oxidation numbers are real or hypothetical charges on atoms assigned by the following rules Atoms in elements are assigned
0 All simple monatomic ions have oxidation numbers equal to their charges
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students to study arithmetic operations, analyze patterns, and unlock the mysteries of series. Via thought-provoking challenges and logical challenges, these worksheets become gateways to refining
thinking abilities, supporting the logical minds of budding mathematicians.
From Theory to Real-World Application
Worksheet Oxidation Numbers Answers Get To Know The Basics Free Worksheets
Worksheet Oxidation Numbers Answers Get To Know The Basics Free Worksheets
Q1 Show the electron configuration for the transition metal cation using the box notation in the table Indicate if the complex is paramagnetic or not in the final column of the table Complete the
oxidation number column of the table below by working out the oxidation number of each of the transition metal cations
Rules for assigning oxidation numbers The oxidation number of an element is zero For a monatomic ion the oxidation number is the charge on the ion The oxidation number of combined hydrogen is usually
1 The oxidation number of combined oxygen is usually 2 The sum of all oxidation numbers of atoms in a compound is zero
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Oxidation State Worksheet Answers Sixteenth Streets
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OXIDATION NUMBER WORKSHEET WITH ANSWERS Teaching Resources
Oxidation Reduction Reactions Worksheet Chemistry LibreTexts
The oxidation state of an atom is represented by a positive or negative number called its oxidation number Chemists have developed rules used to assign oxidation numbers The following rules will help
you determine the oxidation state of an atom or ion A free atom has an oxidation number of zero
14 E Oxidation Reduction Reaction Exercises
Assign oxidation numbers to the atoms in each substance CH 2 O NH 3 Rb 2 SO 4 Zn C 2 H 3 O 2 2 Assign oxidation numbers to the atoms in each substance C 6 H 6 B OH 3 Li 2 S Au Identify what is being
oxidized and reduced in this redox reaction by assigning oxidation numbers to the atoms 2NO Cl 2 2NOCl
The oxidation state of an atom is represented by a positive or negative number called its oxidation number Chemists have developed rules used to assign oxidation numbers The following rules will help
you determine the oxidation state of an atom or ion A free atom has an oxidation number of zero
Assign oxidation numbers to the atoms in each substance CH 2 O NH 3 Rb 2 SO 4 Zn C 2 H 3 O 2 2 Assign oxidation numbers to the atoms in each substance C 6 H 6 B OH 3 Li 2 S Au Identify what is being
oxidized and reduced in this redox reaction by assigning oxidation numbers to the atoms 2NO Cl 2 2NOCl
Oxidation Number Worksheet With Answers
Oxidation Number Worksheet
Assigning Oxidation Numbers Worksheet Gohaq
OXIDATION NUMBER WORKSHEET WITH ANSWERS Teaching Resources
Oxidation Reduction Worksheet
Determining Oxidation Numbers Worksheet Answers Free Download Gmbar co
Determining Oxidation Numbers Worksheet Answers Free Download Gmbar co
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dplyr elaborated!
Ekta Aggarwal
dplyr is one of the most popular libraries in R which is widely used for data manipulation and data wrangling. In this tutorial you will learn about how to prepare data using dplyr with elaborated
Let us firstly install and load dplyr.
In this tutorial we will use R's inbuilt dataset: mtcars.
It contains information of mileage(mpg), displacement (disp), horsepower(hp), number of cylinders(cyl), automatic or manual (am) etc. of 32 cars.
Selecting some of the columns
Using dplyr's select( ) function we can select and remove the columns:
select(data set name, columns to be selected)
In the following code we are selecting only 3 columns: mpg, disp and am from mtcars dataset
Pipe operator
We can also use pipe operator i.e. %>% which is specifically used in dplyr. It helps in writing easily readable codes which are less messy. It forms a sequence of codes which will be executed one
after another.
Firstly we mention the dataset name (i.e. mtcars) followed by pipe operator (i.e. %>% ) and then our function. Since we have mentioned the dataset name in the beginning thus in select( ) function we
don't need to re-specify it.
View(mtcars %>% select(mpg,disp,am))
Task: Removing columns mpg, disp and am from the data.
To remove the columns we can mention their names along with a hyphen ( - ) in select statement.
View(mtcars %>% select(-mpg,-disp,-am))
Rearranging columns
We can also rearrange the columns using select( ) statement.
Firstly we specify our list of columns which we need in the beginning followed by the keyword everything( ) which denotes all other columns should come after them.
Task: Make cyl and am as first two column and then rest should come later.
View(mtcars %>% select(cyl,am,everything()))
Renaming columns
Method 1: Using rename( ) function
We can rename( ) datasets using rename(datasetname, new_name = oldname)
Task: Rename mpg to mileage and hp to horsepower
View(mtcars %>% rename(mileage = mpg, horsepower = hp))
View(rename(mtcars,mileage = mpg, horsepower = hp))
Method 2: Using select( ) function
You can also specify new column name = old column name in select statement, followed by everything( ) . But this method will change the order of the columns.
View(mtcars %>% select(mileage = mpg,horsepower = hp ,everything()))
Getting distinct values
Using distinct( ) function we can find distinct values or combinations for a given set of variables.
distinct(dataset name, columns for which we want distinct combinations)
Task: Find all the distinct combinations for columns am and cyl
n_distinct( ) returns the number of distinct values.
Removing duplicates
We can remove duplicate rows in our data by:
View(mtcars %>% distinct())
Here all the columns are considered to remove duplicate values and get unique rows only.
(Since mtcars did not have any duplicate rows thus nothing was deleted)
Filtering the data
Using filter( ) function we can filter the rows in our dataset where a particular condition is satisfied.
filter(dataset, condition)
Task: Filter for rows where am is 0 i.e. car is manual
View(filter(mtcars,am == 0))
View(mtcars %>% filter(am == 0))
Not equal to condition in filter
In R not equal to sign is represented by !=
Task: Filter for rows where am is not 0 i.e. car is automatic
View(mtcars %>% filter(am != 0))
Task: Filter for rows where mileage is more than average mileage of all the cars.
mean(mtcars$mpg) will give us the average mileage of all the cars.
View(mtcars %>% filter(mpg > mean(mtcars$mpg)))
Case: When a particular condition is not true.
To exclude the cases when a particular condition is satisfied we write a ! symbol in front of the condition in R.
Task: Filter for rows where mileage is NOT more than average mileage of all the cars.
View(mtcars %>% filter(!mpg > mean(mtcars$mpg)))
Case: Multiple conditions!
AND ( &) - When multiple conditions need to be true at one time we use & operator.
Task: Filter for rows where number of cylinders is 6 and mileage is more than average mileage of all the cars.
View(mtcars %>% filter(cyl == 6 & mpg > mean(mtcars$mpg)))
OR ( | ) - When at least one of the conditions need to be true at then we use | operator.
Task: Filter for rows where either number of cylinders is 6 or mileage is more than average mileage of all the cars.
View(mtcars %>% filter(cyl == 6 | mpg > mean(mtcars$mpg)))
%in% - To filter for multiple values being included in a vector we use %in% .
Task: Filter for rows where number of cylinders is either 4 or 6.
View(mtcars %>% filter(cyl %in% c(4,6)))
If we do not wish to use %in% then the same code can be written as:
View(mtcars %>% filter(cyl == 4 | cyl == 6))
But good programmers generally make use of %in% operator.
Why pipe operators are preferred?
Task: Filter for the data where am == 0 and select columns mpg, disp, am,cyl and hp
View(mtcars %>% filter(am == 0) %>% select(mpg,disp,am,cyl,hp))
Notice how pipe operator is working : Firstly it selects mtcars, then filters for the rows where am is 0 and then selects the 5 columns.
Without using pipe operator we can write the above code as:
But the above method looks too messy. pipe operator helps in providing easy readability and makes our codes look elegant.
Sorting the data
Using arrange( ) function we can sort the data in. By default arrange( ) sorts the data in ascending order.
Task: Sort the data by am and mpg
View(mtcars %>% arrange(am,mpg))
Above command sorts the data first by am, now where am == 0 then it sorts those rows by mpg in ascending order. Similarly for am == 1.
Descending order: To denote descending order we can write a hyphen ( - ) in front of the column name whose values are needed in descending order.
Task: Sort the data by am in ascending order and mpg in descending order.
View(mtcars %>% arrange(am,-mpg))
Creating new variables
Using mutate( ) function we can create new variables in our data.
Task: Create a new variable mpg/cyl in the data
View(mtcars %>% mutate(new_var = mpg/cyl))
transmute( ) function creates a new column but it returns only single column. It does not include all the other columns from the dataset like mutate( ) .
View(mtcars %>% transmute(new_var = mpg/cyl))
Using mutate_all( ) one can apply one single function on all the columns of the dataset. But it does not create new columns - rather it replaces the values in the actual columns/
Task: Replace all the column values by their mean
View(mtcars %>% mutate_all(mean, na.rm = TRUE))
If we want to apply some functions only on some of the columns which specify a particular condition then we need to use mutate_if( )
mutate_if(data set name, condition to be applied on the columns, function to be applied)
Let us create a different data mydata where cyl and am are converted to characters
mydata = mtcars
mydata$cyl = as.character(mydata$cyl)
mydata$am = as.character(mydata$am)
Task: Replace the original values by number of distinct values in the character columns
Using is.character function we can filter for columns having character values.
View(mydata %>% mutate_if(is.character, n_distinct))
mutate_at is used to apply the condition only on selected columns defined in vars( ) statement.
Task: Replace the values in the columns mpg, cyl and wt by their respective means
View(mtcars%>% mutate_at(vars(mpg,cyl,wt), mean, na.rm = TRUE))
Task: Replace the values in the columns whose name ends by 'p' by their respective means
To find columns whose name ends with 'p' we use dplyr's ends_with function
View(mtcars%>% mutate_at(vars(ends_with("p")), mean, na.rm = TRUE))
Similarly to find columns whose name starts with a particular character or pattern, we have starts_with( ) function
If we try to find max(mtcars$mpg * 10, mtcars$disp ) we will get 472, because it is the maximum value in both the vectors. But to do such computations for each row we need to use rowwise( ) function.
Task: Create a new column max_value which returns maximum of mpg*10 or disp for each row.
View(mtcars %>% rowwise() %>% mutate(max_value = max(mpg*10,disp)))
We can also achieve the same result using if_else ( ) function as follows:
if_else(condition to be evaluated, value when condition is true, value when condition is false, missing = "value to be used in case of missing values")
Task: Create a new column max_value which returns maximum of mpg*10 or disp for each row.
View(mtcars %>% mutate(max_value = if_else(mpg*10 > disp,mpg*10,disp)))
We have missing = option in if_else( ). Suppose if a row has NA then the condition can't be evaluated and would return NA, thus instead of returning NA we can fill the NA values by a default number.
In the following code we are filling our NA values by 9999
View(mtcars %>% mutate(max_value = if_else(mpg*10 > disp,mpg*10,disp,missing = 9999)))
Summarising the data
Using summarise( ) function we can find various summary statistics for our data.
Task: Calculate mean and median mileage for mtcars
mtcars %>% summarise(mean(mpg),median(mpg))
n( ) function
Using n( ) function is dplyr we can find count of the rows.
Task: Calculate average and median mileage and count of rows.
mtcars %>% summarise(avg_mileage = mean(mpg),median_mileage = median(mpg),count = n())
summarise(mtcars,avg_mileage = mean(mpg),median_mileage = median(mpg),count = n())
avg_mileage median_mileage count
IF we want to calculate same summary statistics for multiple variables we can resort to summarise_at( ) function
In vars( ) statement we define the columns on which functions need to be applied.
In funs( ) statement we define the function names.
We can add a suffix to our output column in funs( ) statement.
In the below code we have added suffixes count, avg and minimum in front of the column names
Task: Calculate count, mean and minimum values for columns mpg and disp
mtcars %>% summarise_at(vars(mpg,disp),funs(count = n(),avg = mean(.,na.rm = T),minimum = min))
mpg_count disp_count mpg_avg disp_avg mpg_minimum disp_minimum
32 32 20.09062 230.7219 10.4 71.1
In the latest version funs( ) has been updated by list( )
where we need to define the functions by a tilde(~) and a dot (. )
Eg. ~n( ) , ~mean(.), ~median(.) etc.
mtcars %>% summarise_at(vars(mpg,disp),
list(~n(),avg = ~mean(.,na.rm = T),minimum = ~min(.)))
To apply the functions on all the columns we use summarise_all( )
Task: Calculate count, and mean for all the columns
mtcars %>% summarise_all(list(count = ~n() ,avg = ~mean(.,na.rm = T)))
To apply the functions on the columns satisfying a particular condition we use summarise_if( )
Let us create a different data mydata where cyl and am are characters
mydata = mtcars
mydata$cyl = as.character(mydata$cyl)
mydata$am = as.character(mydata$am)
Task: Calculate count, and mean for all the numeric columns
mydata %>% summarise_if( is.numeric,list(~n(),avg = ~mean(.,na.rm = T),minimum = ~min(.)))
is.numeric is used to filter for numeric columns
Group by
Group by is used to divide our data on the basis of categorical column(s) and then apply the functions on the grouped data to get summary statistics for each group.
Let us understand by an example:
Task: For each combination of am and cyl find average, median and minimum mileage.
In the following code, firstly we are grouping our data by am and cyl. After this we are calculating mean, median and minimum mileage (mpg) for each of the combinations of am and cyl.
Since we have 6 combinations of am-cyl thus our resultant data will have 6 rows.
View(mtcars %>% group_by(am,cyl) %>% summarise(avg_mileage = mean(mpg),
median_mileage = median(mpg),
min_mileage = min(mpg)) )
Task: For each combination of am-cyl calculate count and average of mileage and disp
View(mtcars %>%
group_by(am,cyl) %>%
list(~n(),avg = ~mean(.,na.rm = T))))
We can also provide a list of consecutive columns in vars( ) using a colon ie. mpg:hp denotes all columns from mpg till hp.
Task: For each combination of am-cyl calculate count and average of columns from mpg till hp.
To do so we have written vars(mpg:hp) which indicates that all the columns between mpg and hp should be used.
View(mtcars %>%
group_by(am,cyl) %>%
list(~n(),avg = ~mean(.,na.rm = T))) )
Do function( )
To do some computations within each group we use do( ) function:
Task: For each combination of am and cyl arrange the rows by mileage(mpg).
d2 = mtcars %>% group_by(am,cyl) %>% do(arrange(.,mpg))
In the output for each combination of am and cyl, mpg values have been sorted in ascending order.
Joins using dplyr
Let us consider 2 dataframes:
a = data.frame(Employee_ID = 1001:1005,Name = c("A","B","C","D","E"))
b = data.frame(Employee_ID = 1003:1007,Salary = c(15000,14000,20000,30000,23000))
Our data looks as follows:
Basic Syntax for all the joins:
join_name(left_table,right_table, by = " ")
join_name: Specify the join name (left_join, right_join, full_join, inner_join)
by: List of common column names to be matched.
1. Left join:
Left join takes all the records from left table irrespective of availability of record in right table.
left_join(a,b,by = c("Employee_ID"))
We can also skip the by option provided there are no other columns (except Employee_ID) which have same column name in both the table.
2. Right join:
Right join takes all the records from Right table irrespective of availability of record in left table.
right_join(a,b,by = c("Employee_ID"))
3. Inner join:
Inner Join takes common elements in both the tables and join them.
inner_join(a,b,by = c("Employee_ID"))
4. Full join:
Full join takes all the records from both the tables
full_join(a,b,by = c("Employee_ID"))
5. Cross join:
a1 =data.frame(Numbers = 1:3)
a2 =data.frame(Letters = c("a","b","c"))
dataframe a1 had 3 unique numbers and a2 had 3 unique letters. A cross join refers to all 3X3 = 9 permutation combinations of these numbers and letters.
We can achieve a cross join in R using full_join ( ) only. But since we do not have any common variable to achieve a full join thus we need to create a temporary variable in both the variables to
join the 2 tables. In the following code our temp variable has the value 1
Now we can do a full join with common key as temp and then we can drop the temp column.
full_join(a1,a2,by = "temp")
6. Anti - Join
Anti join comprises of those records which are present in one table but not in other.
In the following code we are fetching those rows which belong to a and are not present in b
anti_join(a,b,by = c("Employee_ID"))
In the following code we are fetching those rows which belong to b and are not present in a
anti_join(b,a,by = c("Employee_ID"))
All of these 3 functions take 2 vectors as an input.
It returns the common elements available in the 2 vectors. You can consider it like an inner join which returns the common keys.
It returns all the elements available in the 2 vectors but keeps only unique values.
It returns all the elements available in the 2 vectors and retains the duplicate values.
Ranking the data
There are various rank functions available in dplyr to rank the observations.
In case of ties rank( ) function provides an average rank to all the observations which are causing the tie. In the example below, 200 is being repeated 4 times (its actual rank would have been 2, 3,
4 and 5 if there were no ties) hence we assign an average of the ranks as 3.5 i.e., (2+3+4+5)/4 and next element i.e., 500 is assigned the next rank (i.e., 6, as we had already exhausted ranks 2,3, 4
and 5)
Output: 1.0 3.5 3.5 3.5 3.5 6.0 7.0 8.0
If we specify ties.method = "first" then it assigns the rank on first come first serve basis for ties.
rank(c(100,200,200,200,200,500,600,700),ties.method = "first")
If we specify ties.method = "last" then it assigns the rank on first come first serve basis for ties.
rank(c(100,200,200,200,200,500,600,700),ties.method = "last")
If we specify ties.method = "min" then it assigns the minimum rank out of the ties (i.e., minimum of 2, 3, 4 and 5 i.e., 2)
rank(c(100,200,200,200,200,500,600,700),ties.method = "min")
If we specify ties.method = "max" then it assigns the maximum rank out of the ties (i.e., maximum of 2, 3, 4 and 5 i.e., 5)
rank(c(100,200,200,200,200,500,600,700),ties.method = "max")
Task: Rank the mtcars dataset on the basis of mpg and sort the dataset as per their ranks.
View(mtcars %>% mutate(Rank = rank(mpg)) %>% arrange(Rank))
dense_rank( )
dense_rank( ) gives equal weightage in case of ties, and for succeeding elements it provides the next rank. i.e., dense rank will never provide ranks as decimals or averages and ranks are always a
continuous counting like 1,2,3, etc.
In the following code all 200s are given same rank i.e., 2 and 500 is given the next rank i.e., 3
Task: Rank the mtcars dataset on the basis of mpg (use dense rank ) and sort the dataset as per their ranks.
View(mtcars %>% mutate(Dense_rank = dense_rank(mpg)) %>% arrange(Dense_rank))
row_number( )
row_number( ) functions assigns the row number to all the observations.
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